# LongBench v2 / 66f40e44821e116aacb30b45

task_id: db6a69f6-3090-52c4-aead-16bcbc8c2838
task_key: train--66f40e44821e116aacb30b45
task_revision_id: 3

{"choice_A":"mtype = 1，phase = 23, iparm[6] = 0, iparm[23] = 10","choice_B":"mtype = 4，phase = 33, iparm[7] = 0, iparm[8] = 0","choice_C":"mtype = -2，phase = 23, iparm[7] = 0, iparm[20] = 2","choice_D":"mtype = -4，phase = 33, iparm[7] = 0, iparm[9] = 0","context":"Contents\nChapter 1: Developer Reference for Intel® oneAPI Math Kernel\nLibrary - C\nGetting Help and Support ......................................................................... 17\nWhat's New ............................................................................................ 18\nNotational Conventions ............................................................................ 18\nOverview................................................................................................ 19\nPerformance Enhancements.............................................................. 24\nParallelism ..................................................................................... 24\nC Datatypes Specific to Intel MKL ...................................................... 25\nOpenMP* Offload..................................................................................... 26\nOpenMP* Offload for Intel® oneAPI Math Kernel Library ........................ 26\nBLAS and Sparse BLAS Routines................................................................ 33\nBLAS Routines ................................................................................ 33\nNaming Conventions for BLAS Routines...................................... 33\nC Interface Conventions for BLAS Routines................................. 35\nMatrix Storage Schemes for BLAS Routines ................................ 36\nBLAS Level 1 Routines and Functions......................................... 37\nBLAS Level 2 Routines ............................................................. 54\nBLAS Level 3 Routines ............................................................. 97\nSparse BLAS Level 1 Routines......................................................... 119\nVector Arguments ................................................................. 119\nNaming Conventions for Sparse BLAS Routines ......................... 120\nRoutines and Data Types........................................................ 120\nBLAS Level 1 Routines That Can Work With Sparse Vectors......... 120\ncblas_?axpyi ........................................................................ 121\ncblas_?doti .......................................................................... 122\ncblas_?dotci ......................................................................... 122\ncblas_?dotui......................................................................... 123\ncblas_?gthr .......................................................................... 124\ncblas_?gthrz......................................................................... 125\ncblas_?roti ........................................................................... 125\ncblas_?sctr........................................................................... 126\nSparse BLAS Level 2 and Level 3 Routines ........................................ 127\nNaming Conventions in Sparse BLAS Level 2 and Level 3............ 127\nSparse Matrix Storage Formats for Sparse BLAS Routines........... 128\nRoutines and Supported Operations......................................... 129\nInterface Consideration.......................................................... 130\nSparse BLAS Level 2 and Level 3 Routines................................ 134\nSparse QR Routines....................................................................... 241\nmkl_sparse_set_qr_hint ........................................................ 241\nmkl_sparse_?_qr .................................................................. 242\nmkl_sparse_qr_reorder.......................................................... 244\nmkl_sparse_?_qr_factorize..................................................... 245\nmkl_sparse_?_qr_solve ......................................................... 246\nmkl_sparse_?_qr_qmult ........................................................ 248\nmkl_sparse_?_qr_rsolve ........................................................ 250\nCompact BLAS and LAPACK Functions .............................................. 251\nmkl_?gemm_compact............................................................ 255\nDeveloper Reference for Intel® oneAPI Math Kernel Library for C\n2\n\n\nmkl_?trsm_compact.............................................................. 258\nmkl_?potrf_compact.............................................................. 260\nmkl_?getrfnp_compact .......................................................... 261\nmkl_?geqrf_compact ............................................................. 262\nmkl_?getrinp_compact .......................................................... 264\nNumerical Limitations for Compact BLAS and Compact LAPACK\nRoutines .......................................................................... 265\nmkl_?get_size_compact......................................................... 266\nmkl_get_format_compact ...................................................... 266\nmkl_?gepack_compact .......................................................... 267\nmkl_?geunpack_compact ....................................................... 269\nInspector-executor Sparse BLAS Routines......................................... 270\nNaming Conventions in Inspector-Executor Sparse BLAS Routines 270\nSparse Matrix Storage Formats for Inspector-executor Sparse\nBLAS Routines .................................................................. 272\nSupported Inspector-executor Sparse BLAS Operations.............. 272\nTwo-stage Algorithm in Inspector-Executor Sparse BLAS Routines 273\nMatrix Manipulation Routines .................................................. 274\nInspector-Executor Sparse BLAS Analysis Routines.................... 294\nInspector-Executor Sparse BLAS Execution Routines.................. 311\nBLAS-like Extensions ..................................................................... 355\ncblas_?axpy_batch................................................................ 357\ncblas_?axpy_batch_strided .................................................... 358\ncblas_?axpby ....................................................................... 359\ncblas_?gemmt...................................................................... 360\ncblas_?gemm3m................................................................... 363\ncblas_?gemm_batch.............................................................. 367\ncblas_?gemm_batch_strided .................................................. 370\ncblas_?gemm3m_batch_strided .............................................. 373\ncblas_?gemm3m_batch ......................................................... 377\ncblas_?trsm_batch ................................................................ 380\ncblas_?trsm_batch_strided..................................................... 383\nmkl_?imatcopy ..................................................................... 385\nmkl_?imatcopy_batch............................................................ 387\nmkl_?imatcopy_batch_strided ................................................ 388\nmkl_?omatadd_batch_strided................................................. 390\nmkl_?omatcopy .................................................................... 392\nmkl_?omatcopy_batch........................................................... 394\nmkl_?omatcopy_batch_strided ............................................... 396\nmkl_?omatcopy2 .................................................................. 398\nmkl_?omatadd ..................................................................... 400\ncblas_?gemm_pack_get_size, cblas_gemm_*_pack_get_size ..... 402\ncblas_?gemm_pack............................................................... 404\ncblas_gemm_*_pack............................................................. 407\ncblas_?gemm_compute ......................................................... 412\ncblas_gemm_*_compute ....................................................... 415\ncblas_gemm_bf16bf16f32_compute ........................................ 421\ncblas_gemm_bf16bf16f32 ...................................................... 425\ncblas_gemm_f16f16f32_compute............................................ 427\ncblas_gemm_f16f16f32 ......................................................... 431\ncblas_?gemm_free................................................................ 434\ncblas_gemm_* ..................................................................... 435\ncblas_?gemv_batch_strided ................................................... 439\ncblas_?gemv_batch............................................................... 440\ncblas_?dgmm_batch_strided .................................................. 442\nContents\n3\n\n\ncblas_?dgmm_batch.............................................................. 444\nmkl_jit_create_?gemm .......................................................... 446\nmkl_jit_get_?gemm_ptr ........................................................ 448\nmkl_jit_destroy .................................................................... 451\nLAPACK Routines................................................................................... 452\nChoosing a LAPACK Routine............................................................ 452\nC Interface Conventions for LAPACK Routines.................................... 452\nMatrix Layout for LAPACK Routines .................................................. 454\nMatrix Storage Schemes for LAPACK Routines ................................... 456\nMathematical Notation for LAPACK Routines...................................... 464\nError Analysis ............................................................................... 465\nLAPACK Linear Equation Routines .................................................... 465\nLAPACK Linear Equation Computational Routines....................... 466\nLAPACK Linear Equation Driver Routines .................................. 676\nLAPACK Least Squares and Eigenvalue Problem Routines.................... 780\nLAPACK Least Squares and Eigenvalue Problem Computational\nRoutines .......................................................................... 781\nLAPACK Least Squares and Eigenvalue Problem Driver Routines .1000\nLAPACK Auxiliary Routines.............................................................1174\n?lacgv ................................................................................1174\n?lacrm................................................................................1175\n?syconv ..............................................................................1176\n?syr ...................................................................................1177\ni?max1 ...............................................................................1179\n?sum1................................................................................1179\n?gelq2................................................................................1180\n?geqr2 ...............................................................................1181\n?geqrt2 ..............................................................................1183\n?geqrt3 ..............................................................................1185\n?getf2 ................................................................................1187\n?lacn2 ................................................................................1188\n?lacpy ................................................................................1190\n?lakf2.................................................................................1191\n?lange ................................................................................1192\n?lansy ................................................................................1193\n?lanhe................................................................................1194\n?lantr .................................................................................1195\nLAPACKE_set_nancheck........................................................1197\nLAPACKE_get_nancheck........................................................1197\n?lapmr ...............................................................................1197\n?lapmt................................................................................1199\n?lapy2 ................................................................................1200\n?lapy3 ................................................................................1200\n?laran.................................................................................1201\n?larfb .................................................................................1201\n?larfg .................................................................................1204\n?larft..................................................................................1206\n?larfx .................................................................................1208\n?large ................................................................................1209\n?larnd ................................................................................1210\n?larnv.................................................................................1211\n?laror .................................................................................1212\n?larot .................................................................................1214\n?lartgp ...............................................................................1217\n?lartgs................................................................................1218\nDeveloper Reference for Intel® oneAPI Math Kernel Library for C\n4\n\n\n?lascl .................................................................................1219\n?lasd0 ................................................................................1220\n?lasd1 ................................................................................1221\n?lasd2 ................................................................................1224\n?lasd3 ................................................................................1226\n?lasd4 ................................................................................1228\n?lasd5 ................................................................................1229\n?lasd6 ................................................................................1230\n?lasd7 ................................................................................1233\n?lasd8 ................................................................................1236\n?lasd9 ................................................................................1238\n?lasda ................................................................................1239\n?lasdq ................................................................................1242\n?lasdt.................................................................................1244\n?laset.................................................................................1244\n?lasrt .................................................................................1246\n?laswp................................................................................1247\n?latm1................................................................................1248\n?latm2................................................................................1250\n?latm3................................................................................1252\n?latm5................................................................................1256\n?latm6................................................................................1259\n?latme................................................................................1261\n?latmr ................................................................................1265\n?lauum...............................................................................1271\n?syswapr ............................................................................1272\n?heswapr............................................................................1273\n?sfrk ..................................................................................1275\n?hfrk ..................................................................................1276\n?tfsm .................................................................................1278\n?tfttp .................................................................................1280\n?tfttr ..................................................................................1281\n?tpqrt2 ...............................................................................1283\n?tprfb.................................................................................1285\n?tpttf .................................................................................1288\n?tpttr .................................................................................1289\n?trttf ..................................................................................1291\n?trttp .................................................................................1292\n?lacp2 ................................................................................1293\n?larcm................................................................................1294\nmkl_?tppack .......................................................................1295\nmkl_?tpunpack ....................................................................1297\nLAPACK Utility Functions and Routines ............................................1299\nilaver .................................................................................1300\nilaenv.................................................................................1300\n?lamch ...............................................................................1303\nLAPACK Test Functions and Routines...............................................1304\n?lagge ................................................................................1304\n?laghe ................................................................................1305\n?lagsy ................................................................................1306\n?latms................................................................................1307\nAdditional LAPACK Routines (Included for Compatibility with Netlib\nLAPACK) .................................................................................1311\nScaLAPACK Routines.............................................................................1315\nOverview of ScaLAPACK Routines ...................................................1315\nContents\n5\n\n\nScaLAPACK Array Descriptors.........................................................1316\nNaming Conventions for ScaLAPACK Routines ..................................1318\nScaLAPACK Computational Routines................................................1319\nSystems of Linear Equations: ScaLAPACK Computational Routines1319\nMatrix Factorization: ScaLAPACK Computational Routines ..........1320\nSolving Systems of Linear Equations: ScaLAPACK Computational\nRoutines .........................................................................1334\nEstimating the Condition Number: ScaLAPACK Computational\nRoutines .........................................................................1350\nRefining the Solution and Estimating Its Error: ScaLAPACK\nComputational Routines ....................................................1358\nMatrix Inversion: ScaLAPACK Computational Routines...............1368\nMatrix Equilibration: ScaLAPACK Computational Routines ..........1372\nOrthogonal Factorizations: ScaLAPACK Computational Routines..1376\nSymmetric Eigenvalue Problems: ScaLAPACK Computational\nRoutines .........................................................................1445\nNonsymmetric Eigenvalue Problems: ScaLAPACK Computational\nRoutines .........................................................................1478\nSingular Value Decomposition: ScaLAPACK Driver Routines........1492\nGeneralized Symmetric-Definite Eigenvalue Problems:\nScaLAPACK Computational Routines ...................................1504\nScaLAPACK Driver Routines ...........................................................1508\np?geevx .............................................................................1508\np?gesv ...............................................................................1512\np?gesvx..............................................................................1513\np?gbsv ...............................................................................1518\np?dbsv ...............................................................................1521\np?dtsv................................................................................1523\np?posv ...............................................................................1525\np?posvx..............................................................................1527\np?pbsv ...............................................................................1532\np?ptsv................................................................................1534\np?gels ................................................................................1536\np?syev ...............................................................................1539\np?syevd..............................................................................1542\np?syevr ..............................................................................1544\np?syevx..............................................................................1548\np?heev ...............................................................................1554\np?heevd .............................................................................1557\np?heevr..............................................................................1559\np?heevx .............................................................................1564\np?gesvd..............................................................................1571\np?sygvx..............................................................................1575\np?hegvx .............................................................................1582\nScaLAPACK Auxiliary Routines........................................................1589\np?lacgv...............................................................................1594\np?max1 ..............................................................................1595\npilaver................................................................................1596\npmpcol ...............................................................................1597\npmpim2..............................................................................1598\n?combamax1.......................................................................1599\np?sum1 ..............................................................................1599\np?dbtrsv .............................................................................1600\np?dttrsv..............................................................................1603\np?gebal ..............................................................................1605\nDeveloper Reference for Intel® oneAPI Math Kernel Library for C\n6\n\n\np?gebd2 .............................................................................1608\np?gehd2 .............................................................................1611\np?gelq2 ..............................................................................1613\np?geql2 ..............................................................................1615\np?geqr2..............................................................................1617\np?gerq2..............................................................................1619\np?getf2...............................................................................1621\np?labrd...............................................................................1623\np?lacon ..............................................................................1626\np?laconsb ...........................................................................1628\np?lacp2 ..............................................................................1629\np?lacp3 ..............................................................................1630\np?lacpy...............................................................................1632\np?laevswp...........................................................................1633\np?lahrd...............................................................................1635\np?laiect ..............................................................................1637\np?lamve .............................................................................1638\np?lange ..............................................................................1639\np?lanhs ..............................................................................1641\np?lansy, p?lanhe..................................................................1643\np?lantr ...............................................................................1645\np?lapiv ...............................................................................1646\np?lapv2 ..............................................................................1649\np?laqge ..............................................................................1651\np?laqr0...............................................................................1652\np?laqr1...............................................................................1655\np?laqr2...............................................................................1658\np?laqr3...............................................................................1660\np?laqr5...............................................................................1663\np?laqsy...............................................................................1665\np?lared1d ...........................................................................1667\np?lared2d ...........................................................................1668\np?larf .................................................................................1669\np?larfb ...............................................................................1672\np?larfc................................................................................1675\np?larfg ...............................................................................1677\np?larft ................................................................................1679\np?larz.................................................................................1681\np?larzb ...............................................................................1684\np?larzc ...............................................................................1688\np?larzt................................................................................1690\np?lascl................................................................................1693\np?lase2 ..............................................................................1695\np?laset ...............................................................................1696\np?lasmsub ..........................................................................1698\np?lasrt................................................................................1699\np?lassq...............................................................................1701\np?laswp ..............................................................................1702\np?latra ...............................................................................1704\np?latrd ...............................................................................1705\np?latrs................................................................................1708\np?latrz................................................................................1710\np?lauu2 ..............................................................................1712\np?lauum .............................................................................1714\np?lawil................................................................................1715\nContents\n7\n\n\np?org2l/p?ung2l...................................................................1716\np?org2r/p?ung2r..................................................................1718\np?orgl2/p?ungl2...................................................................1720\np?orgr2/p?ungr2..................................................................1722\np?orm2l/p?unm2l.................................................................1724\np?orm2r/p?unm2r................................................................1727\np?orml2/p?unml2.................................................................1731\np?ormr2/p?unmr2................................................................1734\np?pbtrsv .............................................................................1737\np?pttrsv..............................................................................1741\np?potf2...............................................................................1743\np?rot..................................................................................1745\np?rscl .................................................................................1747\np?sygs2/p?hegs2 .................................................................1748\np?sytd2/p?hetd2..................................................................1750\np?trord...............................................................................1753\np?trsen...............................................................................1757\np?trti2................................................................................1761\n?lahqr2...............................................................................1762\n?lamsh ...............................................................................1764\n?lapst.................................................................................1765\n?laqr6 ................................................................................1766\n?lar1va...............................................................................1769\n?laref .................................................................................1770\n?larrb2 ...............................................................................1773\n?larrd2 ...............................................................................1775\n?larre2 ...............................................................................1778\n?larre2a..............................................................................1781\n?larrf2 ................................................................................1785\n?larrv2 ...............................................................................1786\n?lasorte ..............................................................................1791\n?lasrt2................................................................................1792\n?stegr2...............................................................................1793\n?stegr2a .............................................................................1796\n?stegr2b .............................................................................1799\n?stein2 ...............................................................................1802\n?dbtf2 ................................................................................1804\n?dbtrf.................................................................................1805\n?dttrf .................................................................................1806\n?dttrsv ...............................................................................1807\n?pttrsv ...............................................................................1809\n?steqr2...............................................................................1810\n?trmvt................................................................................1812\npilaenv ...............................................................................1814\npilaenvx .............................................................................1815\npjlaenv...............................................................................1817\nAdditional ScaLAPACK Routines..............................................1818\nScaLAPACK Utility Functions and Routines .......................................1820\np?labad ..............................................................................1821\np?lachkieee.........................................................................1822\np?lamch .............................................................................1822\np?lasnbt .............................................................................1823\ndescinit ..............................................................................1824\nnumroc ..............................................................................1825\nScaLAPACK Redistribution/Copy Routines ........................................1826\nDeveloper Reference for Intel® oneAPI Math Kernel Library for C\n8\n\n\np?gemr2d ...........................................................................1826\np?trmr2d ............................................................................1828\nSparse Solver Routines .........................................................................1830\noneMKL PARDISO - Parallel Direct Sparse Solver Interface.................1830\npardiso...............................................................................1837\npardisoinit ..........................................................................1844\npardiso_64..........................................................................1845\nmkl_pardiso_pivot ...............................................................1846\npardiso_getdiag...................................................................1847\npardiso_export ....................................................................1848\npardiso_handle_store ...........................................................1850\npardiso_handle_restore ........................................................1851\npardiso_handle_delete..........................................................1851\npardiso_handle_store_64......................................................1852\npardiso_handle_restore_64 ...................................................1853\npardiso_handle_delete_64 ....................................................1854\noneMKL PARDISO Parameters in Tabular Form .........................1854\npardiso iparm Parameter.......................................................1859\nPARDISO_DATA_TYPE...........................................................1873\nParallel Direct Sparse Solver for Clusters Interface............................1873\ncluster_sparse_solver...........................................................1875\ncluster_sparse_solver_64......................................................1880\ncluster_sparse_solver_get_csr_size........................................1880\ncluster_sparse_solver_set_csr_ptrs ........................................1882\ncluster_sparse_solver_set_ptr ...............................................1884\ncluster_sparse_solver_export ................................................1886\ncluster_sparse_solver iparm Parameter...................................1887\nDirect Sparse Solver (DSS) Interface Routines .................................1896\nDSS Interface Description .....................................................1898\nDSS Implementation Details..................................................1899\nDSS Routines ......................................................................1900\nIterative Sparse Solvers based on Reverse Communication Interface\n(RCI ISS)................................................................................1911\nCG Interface Description .......................................................1912\nFGMRES Interface Description ...............................................1917\nRCI ISS Routines .................................................................1923\nRCI ISS Implementation Details.............................................1936\nPreconditioners based on Incomplete LU Factorization Technique ........1937\nILU0 and ILUT Preconditioners Interface Description.................1937\ndcsrilu0 ..............................................................................1938\ndcsrilut...............................................................................1941\nSparse Matrix Checker Routines .....................................................1944\nsparse_matrix_checker.........................................................1944\nsparse_matrix_checker_init...................................................1946\nExtended Eigensolver Routines...............................................................1947\nThe FEAST Algorithm ....................................................................1947\nExtended Eigensolver Functionality.................................................1949\nParallelism in Extended Eigensolver Routines ...........................1950\nAchieving Performance With Extended Eigensolver Routines.......1950\nExtended Eigensolver Interfaces for Eigenvalues within Interval .........1951\nExtended Eigensolver Naming Conventions..............................1951\nfeastinit..............................................................................1952\nExtended Eigensolver Input Parameters ..................................1952\nExtended Eigensolver Output Details ......................................1954\nExtended Eigensolver RCI Routines ........................................1955\nContents\n9\n\n\nExtended Eigensolver Predefined Interfaces.............................1960\nExtended Eigensolver Interfaces for Extremal Eigenvalues/Singular\nValues ....................................................................................1973\nExtended Eigensolver Interfaces to find largest/smallest\neigenvalues.....................................................................1973\nExtended Eigensolver Interfaces to find largest/smallest singular\nvalues ............................................................................1978\nmkl_sparse_ee_init ..............................................................1980\nExtended Eigensolver Input Parameters for Extremal Eigenvalue\nProblem..........................................................................1980\nVector Mathematical Functions ...............................................................1982\nVM Data Types, Accuracy Modes, and Performance Tips.....................1983\nVM Naming Conventions ...............................................................1983\nVM Function Interfaces .........................................................1984\nVector Indexing Methods...............................................................1986\nVM Error Diagnostics ....................................................................1987\nVM Mathematical Functions ...........................................................1988\nSpecial Value Notations.........................................................1990\nArithmetic Functions.............................................................1991\nPower and Root Functions .....................................................2008\nExponential and Logarithmic Functions ...................................2026\nTrigonometric Functions........................................................2041\nHyperbolic Functions ............................................................2071\nSpecial Functions .................................................................2084\nRounding Functions..............................................................2100\nVM Pack/Unpack Functions ............................................................2111\nv?Pack ...............................................................................2111\nv?Unpack............................................................................2112\nVM Service Functions....................................................................2113\nvmlSetMode ........................................................................2114\nvmlGetMode........................................................................2116\nMKLFreeTls .........................................................................2116\nvmlSetErrStatus ..................................................................2117\nvmlGetErrStatus ..................................................................2118\nvmlClearErrStatus................................................................2118\nvmlSetErrorCallBack.............................................................2119\nvmlGetErrorCallBack ............................................................2121\nvmlClearErrorCallBack ..........................................................2121\nMiscellaneous VM Functions ...........................................................2121\nv?CopySign.........................................................................2121\nv?NextAfter.........................................................................2122\nv?Fdim ...............................................................................2124\nv?Fmax ..............................................................................2125\nv?Fmin ...............................................................................2126\nv?MaxMag...........................................................................2127\nv?MinMag ...........................................................................2129\nStatistical Functions..............................................................................2130\nRandom Number Generators..........................................................2131\nRandom Number Generators Conventions ...............................2131\nBasic Generators..................................................................2137\nError Reporting....................................................................2140\nVS RNG Usage ModelIntel® oneMKL RNG Usage Model...............2142\nService Routines ..................................................................2143\nDistribution Generators.........................................................2164\nAdvanced Service Routines....................................................2209\nDeveloper Reference for Intel® oneAPI Math Kernel Library for C\n10\n\n\nConvolution and Correlation...........................................................2214\nConvolution and Correlation Naming Conventions.....................2215\nConvolution and Correlation Data Types ..................................2216\nConvolution and Correlation Parameters..................................2216\nConvolution and Correlation Task Status and Error Reporting .....2218\nConvolution and Correlation Task Constructors.........................2219\nConvolution and Correlation Task Editors.................................2226\nTask Execution Routines........................................................2231\nConvolution and Correlation Task Destructors ..........................2238\nConvolution and Correlation Task Copiers ................................2239\nConvolution and Correlation Usage Examples...........................2240\nConvolution and Correlation Mathematical Notation and\nDefinitions ......................................................................2244\nConvolution and Correlation Data Allocation.............................2245\nSummary Statistics ......................................................................2247\nSummary Statistics Naming Conventions ................................2248\nSummary Statistics Data Types..............................................2248\nSummary Statistics Parameters .............................................2249\nSummary Statistics Task Status and Error Reporting.................2249\nSummary Statistics Task Constructors ....................................2253\nSummary Statistics Task Editors ............................................2255\nSummary Statistics Task Computation Routines .......................2282\nSummary Statistics Task Destructor .......................................2287\nSummary Statistics Usage Examples ......................................2287\nSummary Statistics Mathematical Notation and Definitions ........2289\nFourier Transform Functions...................................................................2293\nFFT Functions ..............................................................................2294\nFFT Interface.......................................................................2295\nComputing an FFT................................................................2295\nConfiguration Settings ..........................................................2296\nFFT Descriptor Manipulation Functions ....................................2311\nFFT Descriptor Configuration Functions ...................................2315\nFFT Computation Functions ...................................................2317\nStatus Checking Functions ....................................................2324\nCluster FFT Functions ...................................................................2326\nComputing Cluster FFT .........................................................2327\nDistributing Data Among Processes ........................................2328\nCluster FFT Interface............................................................2329\nCluster FFT Descriptor Manipulation Functions..........................2330\nCluster FFT Computation Functions.........................................2332\nCluster FFT Descriptor Configuration Functions.........................2335\nError Codes.........................................................................2339\nPBLAS Routines....................................................................................2339\nPBLAS Routines Overview..............................................................2340\nPBLAS Routine Naming Conventions ...............................................2341\nPBLAS Level 1 Routines.................................................................2342\np?amax ..............................................................................2343\np?asum ..............................................................................2344\np?axpy ...............................................................................2345\np?copy ...............................................................................2346\np?dot .................................................................................2347\np?dotc................................................................................2349\np?dotu................................................................................2350\np?nrm2 ..............................................................................2351\np?scal ................................................................................2352\nContents\n11\n\n\np?swap...............................................................................2353\nPBLAS Level 2 Routines.................................................................2354\np?gemv ..............................................................................2355\np?agemv ............................................................................2357\np?ger .................................................................................2360\np?gerc................................................................................2361\np?geru ...............................................................................2363\np?hemv ..............................................................................2365\np?ahemv ............................................................................2366\np?her .................................................................................2368\np?her2 ...............................................................................2370\np?symv ..............................................................................2372\np?asymv.............................................................................2374\np?syr .................................................................................2375\np?syr2................................................................................2377\np?trmv ...............................................................................2379\np?atrmv..............................................................................2381\np?trsv ................................................................................2383\nPBLAS Level 3 Routines.................................................................2385\np?geadd .............................................................................2386\np?tradd ..............................................................................2387\np?gemm .............................................................................2389\np?hemm .............................................................................2391\np?herk................................................................................2393\np?her2k..............................................................................2395\np?symm .............................................................................2397\np?syrk................................................................................2399\np?syr2k ..............................................................................2401\np?tran ................................................................................2404\np?tranu ..............................................................................2405\np?tranc...............................................................................2406\np?trmm ..............................................................................2407\np?trsm ...............................................................................2410\nPartial Differential Equations Support ......................................................2412\nTrigonometric Transform Routines...................................................2412\nTrigonometric Transforms Implemented ..................................2413\nSequence of Invoking TT Routines..........................................2414\nTrigonometric Transform Interface Description .........................2415\nTT Routines.........................................................................2416\nCommon Parameters of the Trigonometric Transforms...............2423\nTrigonometric Transform Implementation Details......................2426\nFast Poisson Solver Routines .........................................................2427\nPoisson Solver Implementation ..............................................2427\nSequence of Invoking Poisson Solver Routines .........................2433\nFast Poisson Solver Interface Description ................................2435\nRoutines for the Cartesian Solver ...........................................2436\nRoutines for the Spherical Solver ...........................................2445\nCommon Parameters for the Poisson Solver.............................2452\nPoisson Solver Implementation Details....................................2461\nNonlinear Optimization Problem Solvers ..................................................2462\nNonlinear Solver Organization and Implementation...........................2462\nNonlinear Solver Routine Naming Conventions .................................2464\nNonlinear Least Squares Problem without Constraints .......................2464\n?trnlsp_init .........................................................................2465\n?trnlsp_check ......................................................................2467\nDeveloper Reference for Intel® oneAPI Math Kernel Library for C\n12\n\n\n?trnlsp_solve.......................................................................2468\n?trnlsp_get .........................................................................2470\n?trnlsp_delete .....................................................................2471\nNonlinear Least Squares Problem with Linear (Bound) Constraints ......2472\n?trnlspbc_init ......................................................................2472\n?trnlspbc_check...................................................................2474\n?trnlspbc_solve....................................................................2476\n?trnlspbc_get ......................................................................2477\n?trnlspbc_delete ..................................................................2479\nJacobian Matrix Calculation Routines...............................................2479\n?jacobi_init .........................................................................2480\n?jacobi_solve ......................................................................2481\n?jacobi_delete .....................................................................2482\n?jacobi ...............................................................................2482\n?jacobix..............................................................................2483\nSupport Functions ................................................................................2485\nVersion Information......................................................................2488\nmkl_get_version ..................................................................2488\nmkl_get_version_string ........................................................2490\nThreading Control ........................................................................2490\nmkl_set_num_threads..........................................................2491\nmkl_domain_set_num_threads ..............................................2492\nmkl_set_num_threads_local..................................................2493\nmkl_set_dynamic.................................................................2495\nmkl_get_max_threads..........................................................2496\nmkl_domain_get_max_threads..............................................2497\nmkl_get_dynamic ................................................................2498\nmkl_set_num_stripes ...........................................................2499\nmkl_get_num_stripes...........................................................2499\nError Handling .............................................................................2500\nError Handling for Linear Algebra Routines ..............................2500\nHandling Fatal Errors ............................................................2503\nCharacter Equality Testing .............................................................2504\nlsame.................................................................................2504\nlsamen ...............................................................................2505\nTiming........................................................................................2505\nsecond/dsecnd ....................................................................2505\nmkl_get_cpu_clocks .............................................................2506\nmkl_get_cpu_frequency........................................................2507\nmkl_get_max_cpu_frequency ................................................2507\nmkl_get_clocks_frequency ....................................................2508\nMemory Management ...................................................................2508\nmkl_free_buffers .................................................................2508\nmkl_thread_free_buffers.......................................................2509\nmkl_disable_fast_mm ..........................................................2510\nmkl_mem_stat ....................................................................2510\nmkl_peak_mem_usage.........................................................2511\nmkl_malloc .........................................................................2512\nmkl_calloc ..........................................................................2513\nmkl_realloc .........................................................................2514\nmkl_free.............................................................................2514\nmkl_set_memory_limit .........................................................2515\nUsage Examples for the Memory Functions..............................2516\nSingle Dynamic Library Control ......................................................2517\nmkl_set_interface_layer........................................................2517\nContents\n13\n\n\nmkl_set_threading_layer ......................................................2518\nmkl_set_xerbla....................................................................2519\nmkl_set_progress ................................................................2520\nmkl_set_pardiso_pivot..........................................................2521\nConditional Numerical Reproducibility Control...................................2521\nmkl_cbwr_set......................................................................2522\nmkl_cbwr_get .....................................................................2523\nmkl_cbwr_get_auto_branch ..................................................2524\nNamed Constants for CNR Control..........................................2525\nReproducibility Conditions .....................................................2526\nUsage Examples for CNR Support Functions.............................2527\nMiscellaneous ..............................................................................2528\nmkl_progress ......................................................................2528\nmkl_enable_instructions .......................................................2529\nmkl_set_env_mode..............................................................2532\nmkl_verbose .......................................................................2532\nmkl_verbose_output_file.......................................................2533\nmkl_set_mpi .......................................................................2534\nmkl_finalize ........................................................................2535\nBLACS Routines ...................................................................................2536\nMatrix Shapes..............................................................................2537\nRepeatability and Coherence..........................................................2538\nBLACS Combine Operations ...........................................................2541\n?gamx2d ............................................................................2542\n?gamn2d ............................................................................2543\n?gsum2d ............................................................................2545\nBLACS Point To Point Communication ..............................................2546\n?gesd2d .............................................................................2548\n?trsd2d...............................................................................2549\n?gerv2d..............................................................................2549\n?trrv2d...............................................................................2550\nBLACS Broadcast Routines.............................................................2550\n?gebs2d .............................................................................2552\n?trbs2d...............................................................................2552\n?gebr2d..............................................................................2553\n?trbr2d...............................................................................2554\nBLACS Support Routines ...............................................................2555\nInitialization Routines ...........................................................2555\nDestruction Routines ............................................................2561\nInformational Routines .........................................................2563\nMiscellaneous Routines .........................................................2565\nBLACS Routines Usage Examples....................................................2566\nData Fitting Functions ...........................................................................2566\nData Fitting Function Naming Conventions.......................................2566\nData Fitting Function Data Types ....................................................2567\nMathematical Conventions for Data Fitting Functions.........................2567\nData Fitting Usage Model...............................................................2570\nData Fitting Usage Examples .........................................................2570\nData Fitting Function Task Status and Error Reporting .......................2576\nData Fitting Task Creation and Initialization Routines ........................2578\ndf?NewTask1D ......................................................................2578\nTask Configuration Routines...........................................................2580\ndf?EditPPSpline1D ..............................................................2581\ndf?EditPtr .........................................................................2588\nDeveloper Reference for Intel® oneAPI Math Kernel Library for C\n14\n\n\ndfiEditVal .........................................................................2589\ndf?EditIdxPtr.....................................................................2591\ndf?QueryPtr........................................................................2593\ndfiQueryVal........................................................................2593\ndf?QueryIdxPtr ...................................................................2594\nData Fitting Computational Routines ...............................................2595\ndf?Construct1D ...................................................................2596\ndf?Interpolate1D/df?InterpolateEx1D ..................................2597\ndf?Integrate1D/df?IntegrateEx1D ........................................2605\ndf?SearchCells1D/df?SearchCellsEx1D ..................................2609\ndf?InterpCallBack ..............................................................2611\ndf?IntegrCallBack ..............................................................2612\ndf?SearchCellsCallBack ......................................................2614\nData Fitting Task Destructors .........................................................2615\ndfDeleteTask ......................................................................2615\nAppendix A: Linear Solvers Basics ..........................................................2616\nSparse Linear Systems..................................................................2616\nMatrix Fundamentals ............................................................2617\nDirect Method......................................................................2618\nSparse Matrix Storage Formats ......................................................2624\nDSS Symmetric Matrix Storage..............................................2625\nDSS Nonsymmetric Matrix Storage.........................................2626\nDSS Structurally Symmetric Matrix Storage.............................2626\nDSS Distributed Symmetric Matrix Storage..............................2627\nSparse BLAS CSR Matrix Storage Format.................................2628\nSparse BLAS CSC Matrix Storage Format.................................2630\nSparse BLAS Coordinate Matrix Storage Format .......................2631\nSparse BLAS Diagonal Matrix Storage Format ..........................2632\nSparse BLAS Skyline Matrix Storage Format ............................2633\nSparse BLAS BSR Matrix Storage Format.................................2634\nAppendix B: Routine and Function Arguments ..........................................2636\nVector Arguments in BLAS.............................................................2636\nVector Arguments in Vector Math ...................................................2637\nMatrix Arguments.........................................................................2638\nAppendix C: FFTW Interface to Intel® oneAPI Math Kernel Library (oneMKL) .2643\nNotational Conventions .................................................................2643\nFFTW2 Interface to Intel® oneAPI Math Kernel Library (oneMKL) .........2643\nWrappers Reference .............................................................2644\nLimitations of the FFTW2 Interface to Intel® oneAPI Math Kernel\nLibrary (oneMKL) .............................................................2646\nInstalling FFTW2 Interface Wrappers ......................................2647\nMPI FFTW2 Wrappers ...........................................................2648\nFFTW3 Interface to Intel® oneAPI Math Kernel Library (oneMKL) .........2651\nUsing FFTW3 Wrappers.........................................................2651\nBuilding Your Own Wrapper Library.........................................2653\nBuilding an Application With FFTW3 Interface Wrappers ............2653\nRunning FFTW3 Interface Wrapper Examples ...........................2654\nMPI FFTW3 Wrappers ...........................................................2654\nAppendix D: Code Examples ..................................................................2655\nBLAS Code Examples....................................................................2655\nFourier Transform Functions Code Examples ....................................2661\nFFT Code Examples ..............................................................2661\nExamples for Cluster FFT Functions ........................................2667\nAuxiliary Data Transformations ..............................................2669\nContents\n15\n\n\nAppendix F: oneMKL Functionality...........................................................2670\nBLAS Functionality ....................................................................2670\nTransposition Functionality .......................................................2671\nLAPACK Functionality ................................................................2671\nDFT Functionality ......................................................................2672\nSparse BLAS Functionality.........................................................2673\nSparse Solvers Functionality .....................................................2678\nRandom Number Generators Functionality ................................2678\nVector Math Functionality .........................................................2680\nData Fitting Functionality ..........................................................2680\nSummary Statistics Functionality ..............................................2681\nBibliography ........................................................................................2682\nGlossary..............................................................................................2687\nNotices and Disclaimers.........................................................................2692\nDeveloper Reference for Intel® oneAPI Math Kernel Library for C\n16\n\n\nDeveloper Reference for Intel®\noneAPI Math Kernel Library - C\n1\nFor detailed information on setting up and using Intel® oneAPI Math Kernel Library (oneMKL), refer to the \nDeveloper Guide for Linux and the Developer Guide for Windows.\nFor more documentation on this and other products, visit the oneAPI Documentation Library.\nIntel® Math Kernel Library is now Intel® oneAPI Math Kernel Library (oneMKL).\nDocumentation for versions of Intel® Math Kernel Library older than 2023.0 is available for download only.\nSee Downloadable Documentation.\nThis publication describes the C interface.\nBasic Linear Algebra\nSubprograms (BLAS)\nThe BLAS routines provide vector, matrix-vector, and matrix-matrix operations.\nSparse BLAS\nThe Sparse BLAS routines provide basic operations on sparse vectors and\nmatrices.\nSparse QR\nThe Sparse QR Routines provide a multifrontal sparse QR factorization method\nfor solving a sparse system of linear equations.\nLAPACK\nThe LAPACK routines solve systems of linear equations, least square problems,\neigenvalue and singular value problems, and Sylvester's equations.\nStatistical Functions\nThe Statistical Functions provides a set of routines implementing commonly used\npseudorandom random number generators (RNG) with continuous distribution.\nDirect and Iterative\nSparse Solvers\nAmong several options for solving sparse linear systems of equations, oneMKL\noffers a direct sparse solver based on PARDISO*, which is referred to here as \nIntel MKL PARDISO.\nVector Mathematics\nFunctions\nThe Vector Mathematics (VM) functions compute core mathematical functions on\nvector arguments.\nVector Statistics Functions The Vector Statistics (VS) functions generate vectors of pseudorandom numbers\nwith different types of statistical distributions and perform convolution and\ncorrelation computations.\nFourier Transform\nFunctions\nThe Fourier Transform Functions offer several options for computing Fast Fourier\nTransforms (FFTs).\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nGetting Help and Support\nIntel provides a support web site that contains a rich repository of self help information, including getting\nstarted tips, known product issues, product errata, license information, user forums, and more. Visit the\nIntel® oneAPI Math Kernel Library (oneMKL) support website athttp://www.intel.com/software/products/\nsupport/.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n17\n\n\nWhat's New\nThis Developer Reference documents Intel® oneAPI Math Kernel Library (oneMKL) release for the C interface.\nIntel® Math Kernel Library is now Intel® oneAPI Math Kernel Library (oneMKL). Documentation for older\nversions of Intel® Math Kernel Library is available for download only. For a list of available documentation\ndownloads by product version, see these pages:\n•\nDownload Documentation for Intel® Parallel Studio XE\n•\nDownload Documentation for Intel® System Studio\nThe manual has been updated to reflect enhancements to the product, besides improvements and error\ncorrections.\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nNotational Conventions\nThis manual uses the following terms to refer to operating systems:\nWindows* OS\nThis term refers to information that is valid on all supported Windows* operating\nsystems.\nLinux* OS\nThis term refers to information that is valid on all supported Linux* operating\nsystems.\nmacOS*\nThis term refers to information that is valid on Intel®-based systems running the\nmacOS* operating system.\nThis manual uses the following notational conventions:\n•\nRoutine name shorthand (for example, ?ungqr instead of cungqr/zungqr).\n•\nFont conventions used for distinction between the text and the code.\nRoutine Name Shorthand\nFor shorthand, names that contain a question mark \"?\" represent groups of routines with similar\nfunctionality. Each group typically consists of routines used with four basic data types: single-precision real,\ndouble-precision real, single-precision complex, and double-precision complex. The question mark is used to\nindicate any or all possible varieties of a function; for example:\n?swap\nRefers to all four data types of the vector-vector ?swap routine:\nsswap, dswap, cswap, and zswap.\nFont Conventions\nThe following font conventions are used:\nlowercase courier\nCode examples:\na[k+i][j] = matrix[i][j];\ndata types; for example, const float*\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n18\n\n\nlowercase courier mixed with\nUpperCase courier\nFunction names; for example, vmlSetMode\nlowercase courier italic\nVariables in arguments and parameters description. For example, incx.\n*\nUsed as a multiplication symbol in code examples and equations and\nwhere required by the programming language syntax.\nOverview\nIntel® oneAPI Math Kernel Library (oneMKL) is optimized for performance on Intel processors. oneMKL also\nruns on non-Intel x86-compatible processors.\nNOTE\noneMKL provides limited input validation to minimize the performance overheads. It is your\nresponsibility when using oneMKL to ensure that input data has the required format and does not\ncontain invalid characters. These can cause unexpected behavior of the library. Examples of the inputs\nthat may result in unexpected behavior:\n•\nNot-a-number (NaN) and other special floating point values\n•\nLarge inputs may lead to accumulator overflow\nAs the oneMKL API accepts raw pointers, it is your application's responsibility to validate the buffer\nsizes before passing them to the library. The library requires subroutine and function parameters to be\nvalid before being passed. While some oneMKL routines do limited checking of parameter errors, your\napplication should check for NULL pointers, for example.\nThe Intel® oneAPI Math Kernel Library includes Fortran routines and functions optimized for Intel® processor-\nbased computers running operating systems that support multiprocessing. In addition to the Fortran\ninterface, Intel® oneAPI Math Kernel Library (oneMKL) includes a C-language interface for the Discrete\nFourier transform functions, as well as for the Vector Mathematics, Vector Statistics, and many other\nfunctions. For hardware and software requirements to use Intel® oneAPI Math Kernel Library (oneMKL),\nseeIntel® oneAPI Math Kernel Library (oneMKL) Release Notes.\nNOTE\nFunction calls at runtime for Intel® oneAPI Math Kernel Library (oneMKL) libraries on the Microsoft\nWindows* operating system can utilize the functionLoadLibrary() and related loading functions in\nstatic, dynamic, and single-dynamic library linking models. These functions attempt to access the\nloader lock which when used within or at the same time as another DllMainfunction call, can lead to a\ndeadlock. If possible, avoid making your calls to Intel® oneAPI Math Kernel Library (oneMKL) in\naDllMain function or at the same time as other calls to DllMain even on separate threads. Refer to\nthe Microsoft documentation about DllMain and Dynamic-Link Library Best Practices for more details.\nBLAS Routines\nThe BLAS routines and functions are divided into the following groups according to the operations they\nperform:\n•\nBLAS Level 1 Routines perform operations of both addition and reduction on vectors of data. Typical\noperations include scaling and dot products.\n•\nBLAS Level 2 Routines perform matrix-vector operations, such as matrix-vector multiplication, rank-1 and\nrank-2 matrix updates, and solution of triangular systems.\n•\nBLAS Level 3 Routines perform matrix-matrix operations, such as matrix-matrix multiplication, rank-k\nupdate, and solution of triangular systems.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n19\n\n\nStarting from release 8.0, Intel® oneAPI Math Kernel Library (oneMKL) also supports the Fortran 95 interface\nto the BLAS routines.\nStarting from release 10.1, a number of BLAS-like Extensions are added to enable the user to perform\ncertain data manipulation, including matrix in-place and out-of-place transposition operations combined with\nsimple matrix arithmetic operations.\nSparse BLAS Routines\nThe Sparse BLAS Level 1 Routines and Functions and Sparse BLAS Level 2 and Level 3 Routinesroutines and\nfunctions operate on sparse vectors and matrices. These routines perform vector operations similar to the\nBLAS Level 1, 2, and 3 routines. The Sparse BLAS routines take advantage of vector and matrix sparsity:\nthey allow you to store only non-zero elements of vectors and matrices. Intel® oneAPI Math Kernel Library\n(oneMKL) also supports Fortran 95 interface to Sparse BLAS routines.\nSparse QR\nSparse QRin Intel® oneAPI Math Kernel Library (oneMKL) is a set of routines used to solve sparse matrices\nwith real coefficients and general structure. All Sparse QR routines can be divided into three steps:\nreordering, factorization, and solving. Currently, only CSR format is supported for the input matrix, and\nSparse QR operates on the matrix handle used in all SpBLAS IE routines. (For details on how to create a\nmatrix handle, refer tomkl-sparse-create-csr.)\nLAPACK Routines\nThe Intel® oneAPI Math Kernel Library fully supports the LAPACK 3.7 set of computational, driver, auxiliary\nand utility routines.\nThe original versions of LAPACK from which that part of Intel® oneAPI Math Kernel Library (oneMKL) was\nderived can be obtained fromhttp://www.netlib.org/lapack/index.html. The authors of LAPACK are E.\nAnderson, Z. Bai, C. Bischof, S. Blackford, J. Demmel, J. Dongarra, J. Du Croz, A. Greenbaum, S.\nHammarling, A. McKenney, and D. Sorensen.\nThe LAPACK routines can be divided into the following groups according to the operations they perform:\n•\nRoutines for solving systems of linear equations, factoring and inverting matrices, and estimating\ncondition numbers (see LAPACK Routines: Linear Equations).\n•\nRoutines for solving least squares problems, eigenvalue and singular value problems, and Sylvester's\nequations (see LAPACK Routines: Least Squares and Eigenvalue Problems).\nStarting from release 8.0, Intel® oneAPI Math Kernel Library (oneMKL) also supports the Fortran 95 interface\nto LAPACK computational and driver routines. This interface provides an opportunity for simplified calls of\nLAPACK routines with fewer required arguments.\nSparse Solver Routines\nDirect sparse solver routines in Intel® oneAPI Math Kernel Library (oneMKL) (seeSparse Solver Routines )\nsolve symmetric and symmetrically-structured sparse matrices with real or complex coefficients. For\nsymmetric matrices, these Intel® oneAPI Math Kernel Library (oneMKL) subroutines can solve both positive-\ndefinite and indefinite systems. Intel® oneAPI Math Kernel Library (oneMKL) includes a solver based on the\nPARDISO* sparse solver, referred to as Intel® oneAPI Math Kernel Library (oneMKL) PARDISO, as well as an\nalternative set of user callable direct sparse solver routines.\nIf you use the Intel® oneAPI Math Kernel Library (oneMKL) PARDISO sparse solver, please cite:\nO.Schenk and K.Gartner. Solving unsymmetric sparse systems of linear equations with PARDISO. J. of Future\nGeneration Computer Systems, 20(3):475-487, 2004.\nIntel® oneAPI Math Kernel Library (oneMKL) provides also an iterative sparse solver (seeSparse Solver\nRoutines) that uses Sparse BLAS level 2 and 3 routines and works with different sparse data formats.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n20\n\n\nExtended Eigensolver Routines\nTheExtended Eigensolver RCI Routines is a set of high-performance numerical routines for solving standard\n(Ax = λx) and generalized (Ax = λBx) eigenvalue problems, where A and B are symmetric or Hermitian. It\nyields all the eigenvalues and eigenvectors within a given search interval. It is based on the Feast algorithm,\nan innovative fast and stable numerical algorithm presented in [Polizzi09], which deviates fundamentally\nfrom the traditional Krylov subspace iteration based techniques (Arnoldi and Lanczos algorithms [Bai00]) or\nother Davidson-Jacobi techniques [Sleijpen96]. The Feast algorithm is inspired by the density-matrix\nrepresentation and contour integration technique in quantum mechanics.\nIt is free from orthogonalization procedures. Its main computational tasks consist of solving very few inner\nindependent linear systems with multiple right-hand sides and one reduced eigenvalue problem orders of\nmagnitude smaller than the original one. The Feast algorithm combines simplicity and efficiency and offers\nmany important capabilities for achieving high performance, robustness, accuracy, and scalability on parallel\narchitectures. This algorithm is expected to significantly augment numerical performance in large-scale\nmodern applications.\nSome of the characteristics of the Feast algorithm [Polizzi09] are:\n•\nConverges quickly in 2-3 iterations with very high accuracy\n•\nNaturally captures all eigenvalue multiplicities\n•\nNo explicit orthogonalization procedure\n•\nCan reuse the basis of pre-computed subspace as suitable initial guess for performing outer-refinement\niterations\nThis capability can also be used for solving a series of eigenvalue problems that are close one another.\n•\nThe number of internal iterations is independent of the size of the system and the number of eigenpairs in\nthe search interval\n•\nThe inner linear systems can be solved either iteratively (even with modest relative residual error) or\ndirectly\nVM Functions\nThe Vector Mathematics functions (see Vector Mathematical Functions) include a set of highly optimized\nimplementations of certain computationally expensive core mathematical functions (power, trigonometric,\nexponential, hyperbolic, etc.) that operate on vectors of real and complex numbers.\nApplication programs that might significantly improve performance with VM include nonlinear programming\nsoftware, integrals computation, and many others. VM provides interfaces both for Fortran and C languages.\nStatistical Functions\nVector Statistics (VS) contains three sets of functions (see Statistical Functions) providing:\n•\nPseudorandom, quasi-random, and non-deterministic random number generator subroutines\nimplementing basic continuous and discrete distributions. To provide best performance, the VS\nsubroutines use calls to highly optimized Basic Random Number Generators (BRNGs) and a set of vector\nmathematical functions.\n•\nA wide variety of convolution and correlation operations.\n•\nInitial statistical analysis of raw single and double precision multi-dimensional datasets.\nFourier Transform Functions\nThe Intel® oneAPI Math Kernel Library (oneMKL) multidimensional Fast Fourier Transform (FFT) functions with\nmixed radix support (see Fourier Transform Functions) provide uniformity of discrete Fourier transform\ncomputation and combine functionality with ease of use. Both Fortran and C interface specifications are\ngiven. There is also a cluster version of FFT functions, which runs on distributed-memory architectures and is\nprovided only for Intel® 64 architectures.\nThe FFT functions provide fast computation via the FFT algorithms for arbitrary lengths. See the Intel®\noneAPI Math Kernel Library (oneMKL) Developer Guide for the specific radices supported.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n21\n\n\nPartial Differential Equations Support\nIntel® oneAPI Math Kernel Library (oneMKL) provides tools for solving Partial Differential Equations (PDE)\n(seePartial Differential Equations Support). These tools are Trigonometric Transform interface routines and\nPoisson Solver.\nThe Trigonometric Transform routines may be helpful to users who implement their own solvers similar to the\nIntel® oneAPI Math Kernel Library (oneMKL) Poisson Solver. The users can improve performance of their\nsolvers by using fast sine, cosine, and staggered cosine transforms implemented in the Trigonometric\nTransform interface.\nThe Poisson Solver is designed for fast solving of simple Helmholtz, Poisson, and Laplace problems. The\nTrigonometric Transform interface, which underlies the solver, is based on the Intel® oneAPI Math Kernel\nLibrary (oneMKL) FFT interface (refer toFourier Transform Functions), optimized for Intel® processors.\nSupport Functions\nThe Intel® oneAPI Math Kernel Library (oneMKL) support functions (seeSupport Functions) are used to\nsupport the operation of the Intel® oneAPI Math Kernel Library (oneMKL) software and provide basic\ninformation on the library and library operation, such as the current library version, timing, setting and\nmeasuring of CPU frequency, error handling, and memory allocation.\nStarting from release 10.0, the Intel® oneAPI Math Kernel Library (oneMKL) support functions provide\nadditional threading control.\nStarting from release 10.1, Intel® oneAPI Math Kernel Library (oneMKL) selectively supports aProgress\nRoutine feature to track progress of a lengthy computation and/or interrupt the computation using a callback\nfunction mechanism. The user application can define a function called mkl_progressthat is regularly called\nfrom the Intel® oneAPI Math Kernel Library (oneMKL) routine supporting the progress routine feature.\nSeeProgress Routine in Support Functions for reference. Refer to a specific LAPACK or DSS/PARDISO function\ndescription to see whether the function supports this feature or not.\noneMKL Initialization on CPU\nWhen a user first invokes any oneMKL functions, there is an initialization cost to keep in mind. Here are some\ndetails about running oneMKL C/Fortran functions:\nWhen we run an application with oneMKL C/Fortran functions on CPU, we spend time on some service\nroutines. Here's what is happening inside the library when we call oneMKL C/Fortran functions:\n•\nThe first step is setting xerbla. It's a oneMKL routine that acts as an error handler for BLAS, LAPACK, VS,\nand VM domains if an input parameter has an invalid value. See xerbla for more information.\n•\nThe next step is to check which oneMKL verbose mode was chosen. oneMKL verbose mode is needed to\nprofile oneMKL usage in the application. You can read more about oneMKL Verbose mode in the\ndocumentation here:\nLinux\nUsing oneMKL Verbose Mode\nWindows\nUsing oneMKL Verbose Mode\nThe oneMKL Verbose feature is enabled only for certain domains such as BLAS (and BLAS-like\nextensions), LAPACK, selected functionality in ScaLAPACK and FFT, and (in the DPC++ API only) RNG.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n22\n\n\n•\nThe next item in the list is the oneMKL dispatcher. oneMKL dispatcher checks the hardware used for\nrunning the application and the available instruction set. Based on the results from dispatcher, different\nfunction implementations (optimized for different hardware and instruction-sets) will be called. More\ndetails can be found in the oneMKL documentation here:\nLinux\nInstruction Set–Specific Dispatching\nWindows\nInstruction Set–Specific Dispatching\n•\nDuring the function run (or even before), you may need to allocate the memory. oneMKL has a memory\nmanager that provides a list of support functions, the ability to redefine memory functions, and internal\nfast memory allocations with memory reuse. See the following for more information:\nMemory Management\nRedefining Memory Functions (Linux)\nRedefining Memory Functions (Windows)\n \n•\nIf you're in the threading mode, oneMKL will also call its own threading manager where it will check for\ndifferent environment variables and set the number of threads. You can read more about this in oneMKL\ndocumentation here:\nLinux\nImproving Performance with Threading\nWindows\nImproving Performance with Threading\nAs an example, BLAS dgemm was run on the 4th Gen Intel® Xeon® Scalable Processors system. Sizes of\nmatrices A and B were 10000x10000. Running the dgemm function in sequential mode took 32.5 seconds\n(32500 milliseconds), from which:\n•\nSetting oneMKL xerbla took 0.001 millisecond.\n•\nSetting/checking oneMKL verbose mode took 0.009 milliseconds.\n•\nChecking for MKL_CBWR settings and detecting CPU using MKL dispatcher took 0.004 milliseconds.\n•\nAdditional internal memory allocations in dgemm took 0.009 milliseconds followed by 0.002 milliseconds of\ndeallocation.\nAs you can see in the example, before the dgemm function runs there are several mkl_malloc calls to\nallocate memory for the A, B, and C matrices. Overall memory allocation took around 0.084 milliseconds.\nAfter the dgemm function completes, there are several mkl_free calls to free the A, B, and C matrix memory.\nThis took around 5.159 milliseconds.\nIf you run dgemm with intel omp threading, you'll spend 24 milliseconds in the oneMKL threading manager.\nIf you run dgemm with tbb threading, you'll spend around 5 milliseconds in oneMKL threading manager.\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n23\n\n\nProduct and Performance Information\nNotice revision #20201201\nPerformance Enhancements\nThe Intel® oneAPI Math Kernel Library has been optimized by exploiting both processor and system features\nand capabilities. Special care has been given to those routines that most profit from cache-management\ntechniques. These especially include matrix-matrix operation routines such asdgemm().\nIn addition, code optimization techniques have been applied to minimize dependencies of scheduling integer\nand floating-point units on the results within the processor.\nThe major optimization techniques used throughout the library include:\n•\nLoop unrolling to minimize loop management costs\n•\nBlocking of data to improve data reuse opportunities\n•\nCopying to reduce chances of data eviction from cache\n•\nData prefetching to help hide memory latency\n•\nMultiple simultaneous operations (for example, dot products in dgemm) to eliminate stalls due to\narithmetic unit pipelines\n•\nUse of hardware features such as the SIMD arithmetic units, where appropriate\nThese are techniques from which the arithmetic code benefits the most.\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nParallelism\nIntel® oneAPI Math Kernel Library (oneMKL) offers performance gains through parallelism provided by the\nsymmetric multiprocessing performance (SMP) feature. You can obtain improvements from SMP in the\nfollowing ways:\n•\nOne way is based on user-managed threads in the program and further distribution of the operations over\nthe threads based on data decomposition, domain decomposition, control decomposition, or some other\nparallelizing technique. Each thread can use any of the Intel® oneAPI Math Kernel Library (oneMKL)\nfunctions (except for the deprecated?lacon LAPACK routine) because the library has been designed to be\nthread-safe.\n•\nAnother method is to use the FFT and BLAS level 3 routines. They have been parallelized and require no\nalterations of your application to gain the performance enhancements of multiprocessing. Performance\nusing multiple processors on the level 3 BLAS shows excellent scaling. Since the threads are called and\nmanaged within the library, the application does not need to be recompiled thread-safe.\n•\nYet another method is to use tuned LAPACK routines. Currently these include the single- and double\nprecision flavors of routines for QR factorization of general matrices, triangular factorization of general\nand symmetric positive-definite matrices, solving systems of equations with such matrices, as well as\nsolving symmetric eigenvalue problems.\nFor instructions on setting the number of available processors for the BLAS level 3 and LAPACK routines, see\nIntel® oneAPI Math Kernel Library (oneMKL) Developer Guide.\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n24\n\n\nProduct and Performance Information\nNotice revision #20201201\nC Datatypes Specific to Intel MKL\nThe mkl_types.hfile defines datatypes specific to Intel® oneAPI Math Kernel Library (oneMKL).\nC/C++ Type\nFortran Type\nLP32\nEquivalent\n(Size in\nBytes)\nLP64 Equivalent\n(Size in Bytes)\nILP64 Equivalent\n(Size in Bytes)\nMKL_INT\n(MKL integer)\nINTEGER\n(default\nINTEGER)\nC/C++:\nint\nFortran:\nINTEGER*4\n(4 bytes)\nC/C++: int\nFortran: INTEGER*4\n(4 bytes)\nC/C++: long long\n(or define MKL_ILP64\nmacros\nFortran: INTEGER*8\n(8 bytes)\nMKL_UINT\n(MKL unsigned\ninteger)\nN/A\nC/C++:\nunsigned\nint\n(4 bytes)\nC/C++: unsigned\nint\n(4 bytes)\nC/C++: unsigned\nlong long\n(8 bytes)\nMKL_LONG\n(MKL long integer)\nN/A\nC/C++:\nlong\n(4 bytes)\nC/C++: long\n(Windows: 4 bytes)\n(Linux, Mac: 8\nbytes)\nC/C++: long\n(8 bytes)\nMKL_Complex8\n(Like C99 complex\nfloat)\nCOMPLEX*8\n(8 bytes)\n(8 bytes)\n(8 bytes)\nMKL_Complex16\n(Like C99 complex\ndouble)\nCOMPLEX*16\n(16 bytes)\n(16 bytes)\n(16 bytes)\nYou can redefine datatypes specific to Intel® oneAPI Math Kernel Library (oneMKL). One reason to do this is if\nyou have your own types which are binary-compatible with Intel® oneAPI Math Kernel Library (oneMKL)\ndatatypes, with the same representation or memory layout. To redefine a datatype, use one of these\nmethods:\n•\nInsert the #define statement redefining the datatype before the mkl.h header file #include statement.\nFor example,\n#define MKL_INT size_t\n#include \"mkl.h\"\n•\nUse the compiler -D option to redefine the datatype. For example,\n...-DMKL_INT=size_t...\nNOTE\nAs the user, if you redefine Intel® oneAPI Math Kernel Library (oneMKL) datatypes you are responsible\nfor making sure that your definition is compatible with that of Intel® oneAPI Math Kernel Library\n(oneMKL). If not, it might cause unpredictable results or crash the application.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n25\n\n\nOpenMP* Offload\nThis section describes how to perform OpenMP offload computations using Intel® oneAPI Math Kernel Library.\nOpenMP* Offload for Intel® oneAPI Math Kernel Library\nYou can use Intel® oneAPI Math Kernel Library (oneMKL) and OpenMP* offload to run standard oneMKL\ncomputations on Intel GPUs. You can find the list of oneMKL features that support OpenMP offload in the\nmkl_omp_offload.h header file, which includes:\n•\nAll Level 1, 2, and 3 BLAS functions through the CBLAS and BLAS interfaces, supporting both synchronous\nand asynchronous execution\n•\nBLAS-like extensions: cblas_?axpby, cblas_?axpy_batch{_strided},\ncblas_?copy_batch{_strided}, cblas_?gemv_batch{_strided}, cblas_?dgmm_batch{_strided},\ncblas_hgemm, cblas_gemm_bf16bf16f32, cblas_gemm_s8u8s32, cblas_?gemm_batch{_strided},\ncblas_?trsm_batch{_strided}, and cblas_?gemmt functionality through the CBLAS and BLAS\ninterfaces as well as mkl_?omatcopy_batch_strided, mkl_?imatcopy_batch_strided, and\nmkl_?omatadd_batch_strided, supporting both synchronous and asynchronous execution\n•\nLAPACK, including LAPACK-like extensions\n•\nAll computations on the Intel GPU (supports both synchronous and asynchronous execution):\n•\n?getrf_batch\n•\n?getrf_batch_strided\n•\n?getrfnp_batch\n•\n?getrfnp_batch_strided\n•\n?getri\n•\n?getri_oop_batch\n•\n?getri_oop_batch_strided\n•\n?getrs\n•\n?getrs_batch_strided\n•\n?getrsnp_batch_strided\n•\n?gels_batch_strided\n•\n?potrf\n•\n?potri\n•\n?potrs\n•\n?trtri\n•\n?trtrs\n•\nHybrid; some computations on the Intel GPU (supports synchronous execution):\n•\n?geqrf\n•\n?getrf (all computations on the CPU for n <= 256)\n•\nmkl_?getrfnp (all computations on the CPU for n <= 512)\n•\n?ormqr, ?unmqr\n•\ndsyevd\n•\ndsygvd\n•\nInterface support only; all computations on the CPU (supports synchronous execution):\n•\n?gebrd\n•\n?gesvd\n•\n?gesvda_batch_strided\n•\n?orgqr, ?ungqr\n•\n?steqr\n•\n?syev, ?heev\n•\nssyevd, ?heevd\n•\n?syevx, ?heevx\n•\nssygvd, ?hegvd\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n26\n\n\n•\n?sygvx, ?hegvx\n•\n?sytrd, ?hetrd\n•\nVector Statistics\n•\nRandom number generators\nNOTE\nAll distributions are supported. See https://www.intel.com/content/www/us/en/docs/onemkl/\ndeveloper-reference-c/current/distribution-generators.html\nBasic random number generators:\n•\nVSL_BRNG_MCG31\n•\nVSL_BRNG_MCG59\n•\nVSL_BRNG_PHILOX4X32X10\n•\nVSL_BRNG_MRG32K3A\n•\nVSL_BRNG_MT19937\n•\nVSL_BRNG_MT2203\n•\nVSL_BRNG_SOBOL\nImportant Check the oneMKL DPC++ developer reference for the BRNG data type used in the\ndistributions in case the offload device doesn't have sycl::aspect::fp64 support.\n•\nSummary statistics\nSupports the vsl?SSCompute routine for the following estimates:\n•\nVSL_SS_MEAN\n•\nVSL_SS_SUM\n•\nVSL_SS_2R_MOM\n•\nVSL_SS_2R_SUM\n•\nVSL_SS_3R_MOM\n•\nVSL_SS_3R_SUM\n•\nVSL_SS_4R_MOM\n•\nVSL_SS_4R_SUM\n•\nVSL_SS_2C_MOM\n•\nVSL_SS_2C_SUM\n•\nVSL_SS_3C_MOM\n•\nVSL_SS_3C_SUM\n•\nVSL_SS_4C_MOM\n•\nVSL_SS_4C_SUM\n•\nVSL_SS_KURTOSIS\n•\nVSL_SS_SKEWNESS\n•\nVSL_SS_MIN\n•\nVSL_SS_MAX\n•\nVSL_SS_VARIATION\nSupported methods:\n•\nVSL_SS_METHOD_FAST\n•\nVSL_SS_METHOD_FAST_USER_MEAN\n•\nFFTs through both DFTI and FFTW3 interfaces in one, two, and three dimensions.\n•\nFor COMPLEX_STORAGE, only the DFTI_COMPLEX_COMPLEX format is currently supported on CPU and\nGPU devices.\n•\nBoth synchronous and asynchronous computations are supported.\n•\nFor R2C/C2R transforms on the GPU, only\nDFTI_CONJUGATE_EVEN_STORAGE=DFTI_COMPLEX_COMPLEX is supported (implying\nDFTI_PACKED_FORMAT=DFTI_CCE_FORMAT).\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n27\n\n\n•\nNOTEINCONSISTENT_CONFIGURATION errors at compute time indicate an invalid descriptor or invalid\ndata pointer. Double check your data mapping if you encounter such errors.\n•\nArbitrary strides and batch distances are not supported for multi-dimensional R2C transforms offloaded\nto the GPU. Considering the last dimension of the data, every element must be separated from its two\nnearest peers (along another dimension and/or in another batch) by a constant distance. For example,\nto compute a batched, two-dimensional R2C FFT of size [N2, N1] with input strides [0, S2, 1]\n(row-major layout with unit elementary stride and no offset), INPUT_DISTANCE must be equal to\nN2*S2 so that every element is separated from its nearest last-dimension counterpart(s) by a distance\nS2 (in this example), even across batches.\n•\nDue to the variadic implementation of DftiComputeForward and DftiComputeBackward, out-of-place\ncompute calls using the DFTI API with the OpenMP 5.1 dispatch construct differ from common dispatch\nconstruct usage by requiring a \"need_device_ptr\" clause. The oneMKL examples provided on\ninstallation demonstrate this usage.\n•\nTransforms on GPU devices may overwrite FFT-irrelevant, padding entries in the output data.\n•\nSparse BLAS\n•\nmkl_sparse_{s, d}_create_csr\n•\nmkl_sparse_{s, d}_export_csr\n•\nmkl_sparse_destroy\n•\nmkl_sparse_order\n•\nCurrently supports only CSR matrix format.\n•\nmkl_sparse_set_mv_hint\n•\nCurrently supports only SPARSE_OPERATION_NON_TRANSPOSE with CSR matrix format for general\nMV (SPARSE_MATRIX_TYPE_GENERAL) and triangular MV (SPARSE_MATRIX_TYPE_TRIANGULAR with\nfill modes SPARSE_FILL_MODE_LOWER/SPARSE_FILL_MODE_UPPER).\n•\nmkl_sparse_set_sv_hint\n•\nmkl_sparse_optimize\n•\nSupports optimization for mkl_sparse_{s, d}_mv functionality based on supported hints added\nthrough mkl_sparse_set_mv_hint offload.\n•\nSupports optimization for mkl_sparse_{s, d}_trsv functionality based on supported hints added\nthrough mkl_sparse_set_sv_hint offload.\n•\nBoth synchronous and asynchronous executions are supported.\nNOTE Note that although you can run the mkl_sparse_optimize offload function asynchronously,\nyou are responsible for the data dependency between the optimization routine and the execution\nroutines.\n•\nmkl_sparse_{s, d}_mv:\n•\nCurrently supports only SPARSE_OPERATION_NON_TRANSPOSE with the following combinations of\nmatrix types:\n•\nSPARSE_MATRIX_TYPE_GENERAL\n•\nSPARSE_MATRIX_TYPE_TRIANGULAR with fill modes SPARSE_FILL_MODE_LOWER/\nSPARSE_FILL_MODE_UPPER and diagonal types SPARSE_DIAG_UNIT/SPARSE_DIAG_NON_UNIT\n•\nSPARSE_MATRIX_TYPE_SYMMETRIC fill modes SPARSE_FILL_MODE_LOWER/\nSPARSE_FILL_MODE_UPPER and diagonal type SPARSE_DIAG_NON_UNIT (currently,\nSPARSE_DIAG_UNIT is not supported)\n•\nBoth synchronous and asynchronous computations are supported.\n•\nmkl_sparse_{s, d}_mm:\n•\nCurrently supported only with SPARSE_MATRIX_TYPE_GENERAL and\nSPARSE_OPERATION_NON_TRANSPOSE.\n•\nBoth SPARSE_LAYOUT_ROW_MAJOR and SPARSE_LAYOUT_COLUMN_MAJOR are supported.\n•\nBoth synchronous and asynchronous computations are supported.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n28\n\n\n•\nmkl_sparse_{s, d}_trsv\n•\nCurrently only supported with SPARSE_MATRIX_TYPE_TRIANGULAR,\nSPARSE_OPERATION_NON_TRANSPOSE, and alpha == 1.0\n•\nBoth synchronous and asynchronous computations are supported\n•\nmkl_sparse_sp2m\n•\nCurrently supported only with SPARSE_MATRIX_TYPE_GENERAL.\n•\nBoth synchronous and asynchronous computations are supported with Level Zero backend, and\ncurrently only synchronous computations are supported with OpenCL backend.\n•\nNote that you can run the mkl_sparse_sp2m offload function asynchronously, but you are\nresponsible for the data dependency between the first stage and the second stage of\nmkl_sparse_sp2m.\n•\nmkl_sparse_sp2m internally creates arrays for the sparse C matrix output. As they may be\nexpected to be used subsequently on both host and device, they are created internally using USM\nshared memory. The arrays are managed by the library and will be cleaned up when the\ncorresponding C matrix handle is destroyed; however, direct access to the arrays is provided by the\nmkl_sparse_{s,d}_export_csr() OpenMP offload function. Users are recommended to make a\ncopy to their own arrays if they want to have such data beyond the scope of the C matrix handle.\nThe choice of USM shared memory for C arrays is made for functional support of the OpenMP\nOffload paradigm and has a performance impact over choosing USM device memory, which would\nbe more performant but not functional in all subsequent use cases.\n•\nThe created C matrix in the provided handle is not guaranteed to be sorted, so the\nmkl_sparse_order() OpenMP offload API is provided for user convenience if that property is\nneeded.\n•\nThe input matrix handle A is not required to be sorted on input, but the input matrix handle B is\nrequired to be sorted on input.\n•\nIn Sparse BLAS, the usage model consists of the creation stage, the inspection stage, the execution\nstage, and the destruction stage. For Sparse BLAS with C OpenMP Offload, all stages can be\nasynchronously executed, provided any data dependencies are already respected.\nThe OpenMP offload feature from Intel® oneAPI Math Kernel Library (oneMKL) enables you to run oneMKL\ncomputations on Intel GPUs through the standard oneMKL APIs within an omp target variant dispatch\nsection. For example, the standard CBLAS API for single precision real data type matrix multiply is:\nvoid cblas_sgemm(const CBLAS_LAYOUT Layout, const CBLAS_TRANSPOSE TransA,\n    const CBLAS_TRANSPOSE TransB, const MKL_INT M, const MKL_INT N,\n    const MKL_INT K, const float alpha, const float *A, const MKL_INT lda,\n    const float *B, const MKL_INT ldb, const float beta, float *C,\n    const MKL_INT ldc);\nIf the oneMKL function (for example, cblas_sgemm) is called outside of an omp target variant dispatch\nsection or if offload is disabled, then the CPU implementation is dispatched. If the same function is called\nwithin an omp target variant dispatch section and offload is possible then the GPU implementation is\ndispatched. By default the execution of the oneMKL function within a dispatch variant construct is\nsynchronous. OpenMP offload computations may be done asynchronously by adding the nowait clause to the\ntarget variant dispatch construct. This ensures that the host thread encountering the task region\ngenerated by this target construct will not be blocked by the oneMKL call. Rather, the host thread is returned\nto the caller for further use. To finish the asynchronous (nowait) computations and ensure memory and\nexecution model consistency (for example, that the results of a computation will be ready in memory to\nmap), the last such nowait computation is followed by the stand-alone construct #pragma omp taskwait.\nFrom the OpenMP Application Programming Interface version 5.0 specification: \"The taskwait region binds to\nthe current task region [i.e., in this case, the last nowait computation]. The current task region is suspended\nat an implicit task scheduling point associated with the construct. The current task region remains suspended\nuntil all child tasks that it generated before the taskwait region complete execution [currently, depend clause\nis not supported].\"\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n29\n\n\nExample\nExamples for using the OpenMP offload for oneMKL are located in the Intel® oneAPI Math Kernel Library\n(oneMKL) installation directory, under:\nexamples/c_offload\nThe following code snippet shows how to use OpenMP offload for single-call oneMKL features such as most\ndense linear algebra functionality.\n#include <omp.h>\n#include \"mkl.h\"\n#include \"mkl_omp_offload.h\" // MKL header file for OpenMP offload\nint dnum = 0; \nint main() {\n    float *a, *b, *c, alpha = 1.0, beta = 1.0;\n    MKL_INT m = 150, n = 200, k = 128, lda = m, ldb = k, ldc = m;\n    MKL_INT sizea = lda * k, sizeb = ldb * n, sizec = ldc * n;\n    // allocate matrices and check pointers\n    a = (float *)mkl_malloc(sizea * sizeof(float), 64);\n    ...\n    // initialize matrices\n#pragma omp target map(c[0:sizec])\n    {\n        for (i = 0; i < sizec; i++) {\n            c[i] = 42;\n        }\n        ...\n    }\n    // run gemm on host, use standard MKL interface\n    cblas_sgemm(CblasColMajor, CblasNoTrans, CblasNoTrans, m, n, k, alpha, a, \nlda, b, ldb, beta, c, ldc);\n    // map the a, b, and c matrices on the device memory\n#pragma omp target data map(to:a[0:sizea],b[0:sizeb]) map(tofrom:c[0:sizec]) \ndevice(dnum)\n    {\n        // run gemm on gpu, use standard MKL interface within a variant dispatch \nconstruct\n        // if offload is not possible, default to cpu \n        // use the use_device_ptr clause to specify that a, b and c are device \nmemory\n#pragma omp target variant dispatch device(dnum) use_device_ptr(a, b, c)\n        {\n            cblas_sgemm(CblasColMajor, CblasNoTrans, CblasNoTrans, m, n, k, \nalpha, a, lda, b, ldb, beta, c, ldc);\n        }\n    }\n    // Free matrices\n    mkl_free(a);\n        …\n}\nSome of the oneMKL functionality requires to call a set of functions to perform the corresponding\ncomputation. This is the case, for example, for the Discrete Fourier Transform which for a typical computation\ninvolves calling the functions.\nDFTI_EXTERN MKL_LONG DftiCreateDescriptor(DFTI_DESCRIPTOR_HANDLE*,\n                              enum DFTI_CONFIG_VALUE,\n                              enum DFTI_CONFIG_VALUE,\n                              MKL_LONG, ...);\nDFTI_EXTERN MKL_LONG DftiCommitDescriptor(DFTI_DESCRIPTOR_HANDLE);\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n30\n\n\nDFTI_EXTERN MKL_LONG DftiComputeForward(DFTI_DESCRIPTOR_HANDLE, void*, ...);\nDFTI_EXTERN MKL_LONG DftiComputeBackward(DFTI_DESCRIPTOR_HANDLE, void*, ...);\nDFTI_EXTERN MKL_LONG DftiFreeDescriptor(DFTI_DESCRIPTOR_HANDLE*);\nIn that case, only a subset of the calls must be wrapped in an omp target variant dispatch construct as\nshown in the following code snippet for DFTI.\n#include <omp.h>\n#include \"mkl.h\"\n#include \"mkl_omp_offload.h\"\nint main(void)\n{\n    const int devNum = 0;\n    const MKL_LONG N = 64;      // Size of 1D transform\n    MKL_LONG status    = 0;\n    MKL_LONG statusGPU = 0;\n    DFTI_DESCRIPTOR_HANDLE descHandle    = NULL;\n    DFTI_DESCRIPTOR_HANDLE descHandleGPU = NULL;\n    MKL_Complex8 *x    = NULL;\n    MKL_Complex8 *xGPU = NULL;\n    printf(\"Create DFTI descriptor\\n\");\n    status = DftiCreateDescriptor(&descHandle, DFTI_SINGLE, DFTI_COMPLEX, 1, N);\n    printf(\"Create GPU DFTI descriptor\\n\");\n    statusGPU = DftiCreateDescriptor(&descHandleGPU, DFTI_SINGLE, DFTI_COMPLEX, \n1, N);\n    printf(\"Commit DFTI descriptor\\n\");\n    status = DftiCommitDescriptor(descHandle);\n    printf(\"Commit GPU DFTI descriptor\\n\");\n#pragma omp target variant dispatch device(devNum)\n    {\n        statusGPU = DftiCommitDescriptor(descHandleGPU);\n    }\n    printf(\"Allocate memory for input array\\n\");\n    x = (MKL_Complex8 *)mkl_malloc(N*sizeof(MKL_Complex8), 64);\n    printf(\"Allocate memory for GPU input array\\n\");\n    xGPU = (MKL_Complex8 *)mkl_malloc(N*sizeof(MKL_Complex8), 64);\n    printf(\"Initialize input for forward FFT\\n\");\n    // init x and xGPU ...\n    printf(\"Compute forward FFT in-place\\n\");\n    status = DftiComputeForward(descHandle, x);\n    printf(\"Compute GPU forward FFT in-place\\n\");\n#pragma omp target data map(tofrom:xGPU[0:N]) device(devNum)\n    {\n#pragma omp target variant dispatch use_device_ptr(xGPU) device(devNum)\n        {\n            statusGPU = DftiComputeForward(descHandleGPU, xGPU);\n        }\n    }\n    // use results now in x and xGPU ...\n cleanup:\n    DftiFreeDescriptor(&descHandle);\n    DftiFreeDescriptor(&descHandleGPU);\n    mkl_free(x);\n    mkl_free(xGPU);\n}\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n31\n\n\nFor asynchronous execution of multi-call oneMKL computation, the nowait clause needs to be used only on\nthe call to the function performing the actual computation (for example,\nDftiCompute{Forward,Backward}). For instance, the following snippet shows how the DFTI example above\ncould be changed to have two, back-to-back, asynchronous (nowait) computations dispatched, with a\ntaskwait at the end of the second to ensure the completion of both computations before their results are\naccessed:\nprintf(\"Compute Intel GPU forward FFT 1 in-place\\n\");\n#pragma omp target data map(tofrom:x1GPU[0:N1], x2GPU[0:N2]) device(devNum)\n    {\n#pragma omp target variant dispatch use_device_ptr(x1GPU) device(devNum) nowait\n        {\n            status1GPU = DftiComputeForward(descHandle1GPU, x1GPU);\n        }\n    printf(\"Compute Intel GPU forward FFT 2 in-place\\n\");\n#pragma omp target variant dispatch use_device_ptr(x2GPU) device(devNum) nowait\n        {\n            status2GPU = DftiComputeForward(descHandle2GPU, x2GPU);\n        }\n#pragma omp taskwait\n    }\n    if (status1GPU != DFTI_NO_ERROR) goto failed;\n    if (status2GPU != DFTI_NO_ERROR) goto failed;\nFor sparse BLAS computations, the workflow ‘create a CSR matrix handle’ → ‘compute’ → ‘destroy the CSR\nmatrix handle’ must be done so that the offloaded data arrays are alive through the full workflow. For\ninstance, if you are using a target data map, then the workflow must be contained in a single target data\nregion. On the other hand, if the arrays were allocated directly using omp_target_alloc() or the Intel\nExtensions omp_target_alloc_host/omp_target_alloc_device/omp_target_alloc_shared, then the\nworkflow must be contained at least in a subset of the scope where those arrays are usable; that is, before\nthe corresponding calls to omp_target_free. The following snippet shows how the Sparse BLAS OpenMP\nOffload example for mkl_sparse_s_mv() could be run using a target data map region, where N is the\nnumber of rows, M is the number of columns, and NNZ is the number of non-zero entries of the sparse\nmatrix csrA_gpu, x is the input vector, and the output is stored in the z array:\n#pragma omp target data map(to:ia[0:N+1],ja[0:NNZ],a[0:NNZ],x[0:M]) map(tofrom:z[0:N]) \ndevice(devNum)\n{                                                                            \n#pragma omp target variant dispatch device(devNum) use_device_ptr(ia, ja, a)\n        {                                                                   \n            status_gpu1 = mkl_sparse_s_create_csr(&csrA_gpu, SPARSE_INDEX_BASE_ZERO, N, M, ia, \nia + 1, ja, a);                            \n        }                                                                            \n#pragma omp target variant dispatch device(devNum) use_device_ptr(x, z)\n        {                                                              \n            status_gpu2 = mkl_sparse_s_mv(SPARSE_OPERATION_NON_TRANSPOSE, alpha, csrA_gpu, \ndescrA, x, beta, z);                                            \n        }                                                                            \n#pragma omp target variant dispatch device(devNum)\n        {                                         \n            status_gpu3 = mkl_sparse_destroy(csrA_gpu);\n        }                                              \n    }\nFor asynchronous execution of multi-call oneMKL Sparse BLAS computation, the nowait clause can be added\nto the call of the function performing the actual computation (for example, calls to the\nmkl_sparse_{s,d}_mv() function).\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n32\n\n\nAs an example, the following snippet shows how the Sparse BLAS example above could be changed to have\ntwo asynchronous (nowait) computations using the same matrix handle, csrA_gpu, but unrelated vector\ndata so there is no read/write dependency between them. Add a taskwait at the end of the second\nexecution to ensure the completion of both computations before the mkl_sparse_destroy() function is\ncalled:\n#pragma omp target data map(to:ia[0:N+1],ja[0:NNZ],a[0:NNZ],x[0:M],w[0:M]) \nmap(tofrom:y[0:N],z[0:N]) device(devNum)\n    {\n#pragma omp target variant dispatch device(devNum) use_device_ptr(ia, ja, a)\n        {\n            status_gpu1 = mkl_sparse_s_create_csr(&csrA_gpu, SPARSE_INDEX_BASE_ZERO, N, M, ia, \nia + 1, ja, a);\n        }\n#pragma omp target variant dispatch device(devNum) use_device_ptr(x, z) nowait\n        {\n            status_gpu2 = mkl_sparse_s_mv(SPARSE_OPERATION_NON_TRANSPOSE, alpha, csrA_gpu, \ndescrA, x, beta, z);\n        }\n#pragma omp target variant dispatch device(devNum) use_device_ptr(w, y) nowait\n        {\n            status_gpu3 = mkl_sparse_s_mv(SPARSE_OPERATION_NON_TRANSPOSE, alpha, csrA_gpu, \ndescrA, w, beta, y);\n        }\n#pragma omp taskwait\n       \n#pragma omp target variant dispatch device(devNum)\n        {\n            status_gpu4 = mkl_sparse_destroy(csrA_gpu);\n        }\n    }\nBLAS and Sparse BLAS Routines\nIntel® oneAPI Math Kernel Library (oneMKL)implements the BLAS and Sparse BLAS routines, and BLAS-like\nextensions. The routine descriptions are arranged in several sections:\n•\nBLAS Level 1 Routines (vector-vector operations)\n•\nBLAS Level 2 Routines (matrix-vector operations)\n•\nBLAS Level 3 Routines (matrix-matrix operations)\n•\nSparse BLAS Level 1 Routines (vector-vector operations).\n•\nSparse BLAS Level 2 and Level 3 Routines (matrix-vector and matrix-matrix operations)\n•\nBLAS-like Extensions\nThe question mark in the group name corresponds to different character codes indicating the data type (s, d,\nc, and z or their combination); see Routine Naming Conventions.\nWhen BLAS or Sparse BLAS routines encounter an error, they call the error reporting routine xerbla.\nBLAS Routines\nNaming Conventions for BLAS Routines\nBLAS routine names have the following structure:\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n33\n\n\n<character> <name> <mod> [_64]\nThe <character> field indicates the data type:\ns\nreal, single precision\nc\ncomplex, single precision\nd\nreal, double precision\nz\ncomplex, double precision\nSome routines and functions can have combined character codes, such as sc or dz.\nFor example, the function scasum uses a complex input array and returns a real value.\nThe <name> field, in BLAS level 1, indicates the operation type. For example, the BLAS level 1\nroutines ?dot, ?rot, ?swap compute a vector dot product, vector rotation, and vector swap, respectively.\nIn BLAS level 2 and 3, <name> reflects the matrix argument type:\nge\ngeneral matrix\ngb\ngeneral band matrix\nsy\nsymmetric matrix\nsp\nsymmetric matrix (packed storage)\nsb\nsymmetric band matrix\nhe\nHermitian matrix\nhp\nHermitian matrix (packed storage)\nhb\nHermitian band matrix\ntr\ntriangular matrix\ntp\ntriangular matrix (packed storage)\ntb\ntriangular band matrix.\nThe <mod> field, if present, provides additional details of the operation. BLAS level 1 names can have the\nfollowing characters in the <mod> field:\nc\nconjugated vector\nu\nunconjugated vector\ng\nGivens rotation construction\nm\nmodified Givens rotation\nmg\nmodified Givens rotation construction\nBLAS level 2 names can have the following characters in the <mod> field:\nmv\nmatrix-vector product\nsv\nsolving a system of linear equations with a single unknown vector\nr\nrank-1 update of a matrix\nr2\nrank-2 update of a matrix.\nBLAS level 3 names can have the following characters in the <mod> field:\nmm\nmatrix-matrix product\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n34\n\n\nsm\nsolving a system of linear equations with multiple unknown vectors\nrk\nrank-k update of a matrix\nr2k\nrank-2k update of a matrix.\nOn 64-bit platforms, routines with the _64 suffix support large data arrays in the LP64 interface library and\nenable you to mix integer types in one application. For example, when an application is linked with the LP64\ninterface library, SGEMM indexes arrays with the 32-bit integer type, while SGEMM_64 indexes arrays with the\n64-bit integer type. For more interface library details, see \"Using the ILP64 Interface vs. LP64 Interface\" in\nthe developer guide.\nThe examples below illustrate how to interpret BLAS routine names:\nddot\n<d> <dot>: real and double precision, vector-vector dot product\ncdotc\n<c> <dot> <c>: complex and single precision, vector-vector dot product,\nconjugated\ncdotu\n<c> <dot> <u>: complex and single precision, vector-vector dot product,\nunconjugated\nscasum\n<sc> <asum>: real and single-precision output, complex and single-precision\ninput, sum of magnitudes of vector elements\nsgemv\n<s> <ge> <mv>: real and single precision, general matrix, matrix-vector product\nztrmm\n<z> <tr> <mm> _64: complex and double precision, triangular matrix, matrix-\nmatrix product, 64-bit integer type\nSparse BLAS level 1 naming conventions are similar to those of BLAS level 1. For more information, see \nNaming Conventions.\nC Interface Conventions for BLAS Routines\nCBLAS, the C interface to the Basic Linear Algebra Subprograms (BLAS), provides a C language interface to\nBLAS routines for Intel® oneAPI Math Kernel Library (oneMKL). While you can call the Fortran implementation\nof BLAS, for coding in C the CBLAS interface has some advantages such as allowing you to specify column-\nmajor or row-major ordering with thelayout parameter.\nFor more information about calling Fortran routines from C in general, and specifically about calling BLAS and\nCBLAS routines, see \" Mixed-language Programming with the Intel® oneAPI Math Kernel Library\" in theIntel®\noneAPI Math Kernel Library Developer Guide.\nNOTE\nThis reference contains syntax in C for both the CBLAS interface and the Fortran BLAS routines.\nIn CBLAS, the Fortran routine names are prefixed with cblas_ (for example, dasum becomes cblas_dasum).\nNames of all CBLAS functions are in lowercase letters. Like BLAS routines, Intel® oneAPI Math Kernel Library\nprovides CBLAS routines with the _64 suffix (for example, cblas_dasum_64) to support large data arrays in\nthe LP64 interface library on 64-bit platforms. For more interface library details, see \"Using the ILP64\nInterface vs. LP64 Interface\" in the developer guide.\nComplex functions ?dotc and ?dotu become CBLAS subroutines (void functions); they return the complex\nresult via a void pointer, added as the last parameter. CBLAS names of these functions are suffixed with\n_sub. For example, the BLAS function cdotc corresponds to cblas_cdotc_sub.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n35\n\n\nWARNING\nUsers of the CBLAS interface should be aware that the CBLAS are just a C interface to the BLAS, which\nis based on the FORTRAN standard and subject to the FORTRAN standard restrictions. In particular, the\noutput parameters should not be referenced through more than one argument.\nNOTE\nThis interface is not implemented in the Sparse BLAS Level 2 and Level 3 routines.\nThe arguments of CBLAS functions comply with the following rules:\n•\nInput arguments are declared with the const modifier.\n•\nNon-complex scalar input arguments are passed by value.\n•\nComplex scalar input arguments are passed as void pointers.\n•\nArray arguments are passed by address.\n•\nBLAS character arguments are replaced by the appropriate enumerated type.\n•\nLevel 2 and Level 3 routines acquire an additional parameter of type CBLAS_LAYOUT as their first\nargument. This parameter specifies whether two-dimensional arrays are row-major (CblasRowMajor) or\ncolumn-major (CblasColMajor).\nEnumerated Types\nThe CBLAS interface uses the following enumerated types:\nenum CBLAS_LAYOUT {\n   CblasRowMajor=101,  /* row-major arrays */\n   CblasColMajor=102};   /* column-major arrays */\nenum CBLAS_TRANSPOSE {\n   CblasNoTrans=111,     /* trans='N' */\n   CblasTrans=112,       /* trans='T' */\n   CblasConjTrans=113};  /* trans='C' */\nenum CBLAS_UPLO {\n   CblasUpper=121,        /* uplo ='U' */\n   CblasLower=122};       /* uplo ='L' */\nenum CBLAS_DIAG {\n   CblasNonUnit=131,      /* diag ='N' */\n   CblasUnit=132};        /* diag ='U' */\nenum CBLAS_SIDE {\n   CblasLeft=141,         /* side ='L' */\n   CblasRight=142};       /* side ='R' */\nMatrix Storage Schemes for BLAS Routines\nMatrix arguments of BLAS and CBLAS routines can use the following storage schemes:\n•\nFull storage: a matrix A is stored in a two-dimensional array a, with the matrix element Aij stored in the\narray element a[i + j*lda] for column-major layout and a[j + i*lda] for row-major layout, where\nlda is the leading dimension for the array.\n•\nPacked storage scheme allows you to store symmetric, Hermitian, or triangular matrices more compactly.\nFor column-major layout, the upper or lower triangle of the matrix is packed by columns in a one\ndimensional array. For row-major layout, the upper or lower triangle of the matrix is packed by rows in a\none dimensional array.\n•\nBand storage: a band matrix is stored compactly in a two-dimensional array. For column-major layout,\ncolumns of the matrix are stored in the corresponding columns of the array, and diagonals of the matrix\nare stored in a specific row of the array. For row-major layout, rows of the matrix are stored in the\ncorresponding rows of the array, and diagonals of the matrix are stored in a specific column of the array.\nFor more information on matrix storage schemes, see Matrix Arguments in the Appendix “Routine and\nFunction Arguments”.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n36\n\n\nRow-Major and Column-Major Layout\nThe BLAS routines follow the Fortran convention of storing two-dimensional arrays using column-major\nlayout. When calling BLAS routines from C, remember that they require arrays to be in column-major format,\nnot the row-major format that is the convention for C. Unless otherwise specified, the psuedo-code examples\nfor the BLAS routines illustrate matrices stored using column-major layout.\nThe CBLAS interface allows you to specify either column-major or row-major layout for BLAS Level 2 and\nLevel 3 routines, by setting the layout parameter to CblasColMajor or CblasRowMajor.\nBLAS Level 1 Routines and Functions\nBLAS Level 1 includes routines and functions, which perform vector-vector operations. The following table\nlists the BLAS Level 1 routine and function groups and the data types associated with them.\nBLAS Level 1 Routine and Function Groups and Their Data Types\nRoutine or\nFunction Group\nData Types\nDescription\ncblas_?asum\ns, d, sc, dz\nSum of vector magnitudes (functions)\ncblas_?axpy\ns, d, c, z\nScalar-vector product (routines)\ncblas_?copy\ns, d, c, z\nCopy vector (routines)\ncblas_?dot\ns, d\nDot product (functions)\ncblas_?sdot\nsd, d\nDot product with double precision (functions)\ncblas_?dotc\nc, z\nDot product conjugated (functions)\ncblas_?dotu\nc, z\nDot product unconjugated (functions)\ncblas_?nrm2\ns, d, sc, dz\nVector 2-norm (Euclidean norm) (functions)\ncblas_?rot\ns, d, c, z, cs, zd\nPlane rotation of points (routines)\ncblas_?rotg\ns, d, c, z\nGenerate Givens rotation of points (routines)\ncblas_?rotm\ns, d\nModified Givens plane rotation of points (routines)\ncblas_?rotmg\ns, d\nGenerate modified Givens plane rotation of points\n(routines)\ncblas_?scal\ns, d, c, z, cs, zd\nVector-scalar product (routines)\ncblas_?swap\ns, d, c, z\nVector-vector swap (routines)\ncblas_i?amax\ns, d, c, z\nIndex of the maximum absolute value element of a vector\n(functions)\ncblas_i?amin\ns, d, c, z\nIndex of the minimum absolute value element of a vector\n(functions)\ncblas_?cabs1\ns, d\nAuxiliary functions, compute the absolute value of a\ncomplex number of single or double precision\ncblas_?asum\nComputes the sum of magnitudes of the vector\nelements.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n37\n\n\nSyntax\nfloat cblas_sasum (const MKL_INT n, const float *x, const MKL_INT incx);\nfloat cblas_scasum (const MKL_INT n, const void *x, const MKL_INT incx);\ndouble cblas_dasum (const MKL_INT n, const double *x, const MKL_INT incx);\ndouble cblas_dzasum (const MKL_INT n, const void *x, const MKL_INT incx);\nInclude Files\n•\nmkl.h\nDescription\nThe ?asum routine computes the sum of the magnitudes of elements of a real vector, or the sum of\nmagnitudes of the real and imaginary parts of elements of a complex vector:\nres = |Rex1| + |Imx1| + |Rex2| + Imx2|+ ... + |Rexn| + |Imxn|,\nresult = ∑\ni = 1\nn\nRe Xi\n+\nIm Xi\nwhere x is a vector with n elements.\nInput Parameters\nn\nSpecifies the number of elements in vector x.\nx\nArray, size at least (1 + (n-1)*abs(incx)).\nincx\nSpecifies the increment for indexing vector x.\nOutput Parameters\nres\nContains the sum of magnitudes of real and imaginary parts of all elements\nof the vector.\nReturn Values\nContains the sum of magnitudes of real and imaginary parts of all elements of the vector.\ncblas_?axpy\nComputes a vector-scalar product and adds the result\nto a vector.\nSyntax\nvoid cblas_saxpy (const MKL_INT n, const float a, const float *x, const MKL_INT incx,\nfloat *y, const MKL_INT incy);\nvoid cblas_daxpy (const MKL_INT n, const double a, const double *x, const MKL_INT incx,\ndouble *y, const MKL_INT incy);\nvoid cblas_caxpy (const MKL_INT n, const void *a, const void *x, const MKL_INT incx,\nvoid *y, const MKL_INT incy);\nvoid cblas_zaxpy (const MKL_INT n, const void *a, const void *x, const MKL_INT incx,\nvoid *y, const MKL_INT incy);\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n38\n\n\nInclude Files\n•\nmkl.h\nDescription\nThe ?axpy routines perform a vector-vector operation defined as\ny := a*x + y\nwhere:\na is a scalar\nx and y are vectors each with a number of elements that equals n.\nInput Parameters\nn\nSpecifies the number of elements in vectors x and y.\na\nSpecifies the scalar a.\nx\nArray, size at least (1 + (n-1)*abs(incx)).\nincx\nSpecifies the increment for the elements of x.\ny\nArray, size at least (1 + (n-1)*abs(incy)).\nincy\nSpecifies the increment for the elements of y.\nOutput Parameters\ny\nContains the updated vector y.\ncblas_?copy\nCopies a vector to another vector.\nSyntax\nvoid cblas_scopy (const MKL_INT n, const float *x, const MKL_INT incx, float *y, const\nMKL_INT incy);\nvoid cblas_dcopy (const MKL_INT n, const double *x, const MKL_INT incx, double *y,\nconst MKL_INT incy);\nvoid cblas_ccopy (const MKL_INT n, const void *x, const MKL_INT incx, void *y, const\nMKL_INT incy);\nvoid cblas_zcopy (const MKL_INT n, const void *x, const MKL_INT incx, void *y, const\nMKL_INT incy);\nInclude Files\n•\nmkl.h\nDescription\nThe ?copy routines perform a vector-vector operation defined as\ny = x,\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n39\n\n\nwhere x and y are vectors.\nInput Parameters\nn\nSpecifies the number of elements in vectors x and y.\nx\nArray, size at least (1 + (n-1)*abs(incx)).\nincx\nSpecifies the increment for the elements of x.\ny\nArray, size at least (1 + (n-1)*abs(incy)).\nincy\nSpecifies the increment for the elements of y.\nOutput Parameters\ny\nContains a copy of the vector x if n is positive. Otherwise, parameters are\nunaltered.\ncblas_?copy_batch\nComputes a group of vector copies.\nSyntax\nvoid cblas_scopy_batch (const MKL_INT *n_array, const float **x_array, const MKL_INT\n*incx_array, float **y_array, const MKL_INT *incy_array, const MKL_INT group_count,\nconst MKL_INT *group_size_array);\nvoid cblas_dcopy_batch (const MKL_INT *n_array, const double **x_array, const MKL_INT\n*incx_array, double **y_array, const MKL_INT *incy_array, const MKL_INT group_count,\nconst MKL_INT *group_size_array);\nvoid cblas_ccopy_batch (const MKL_INT *n_array, const void **x_array, const MKL_INT\n*incx_array, void **y_array, const MKL_INT *incy_array, const MKL_INT group_count,\nconst MKL_INT *group_size_array);\nvoid cblas_zcopy_batch (const MKL_INT *n_array, const void **x_array, const MKL_INT\n*incx_array, void **y_array, const MKL_INT *incy_array, const MKL_INT group_count,\nconst MKL_INT *group_size_array);\nDescription\nThe cblas_?copy_batch routines perform a series of vector copies. They are similar to their cblas_?copy\nroutine counterparts, but the cblas_?copy_batch routines perform vector operations with a group of\nvectors. Each groups contains vectors with the same parameters (size and increment), while those\nparameters may vary between groups.\nThe operation is defined as follows:\nidx = 0\nfor i = 0 … group_count – 1\n    n, incx, incy and group_size at position i in n_array, alpha_array, incx_array, incy_array \nand group_size_array\n    for j = 0 … group_size – 1\n        x and y are vectors of size n at position idx in x_array and y_array\n        y := x\n        idx := idx + 1\n    end for\nend for\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n40\n\n\nThe number of entries in x_array and y_array is total_batch_count, which is the sum of all the\ngroup_size entries.\nInput Parameters\nn_array\nArray of size group_count. For the group i, n_i = n_array[i] is the\nnumber of elements in the vectors x and y.\nx_array\nArray of size total_batch_count of pointers used to store x vectors.\nThe array allocated for the x vectors of the group i must be of size\nat least (1 + (n_i - 1)*abs(incx_i)).\nincx_array\nArray of size group_count. For the group i, incx_i = incx_array[i]\nis the increment (or stride) between two consecutive elements of\nthe vector x.\ny_array\nArray of size total_batch_count of pointers used to store the output\nvectors y. The array allocated for the y vectors of the group i must\nbe of size at least (1 + (n_i - 1)*abs(incy_i)).\nincy_array\nArray of size group_count. For the group i, incy_i = incy_array[i]\nis the increment (or stride) between two consecutive elements of\nthe vector y.\ngroup_count\nNumber of groups. Must be at least 0.\ngroup_size_array\nArray of size group_count. The element group_size_array[i] is the\nnumber of vectors in the group i. Each element in\ngroup_size_array must be at least 0.\nOutput Parameters\ny_array\nArray of pointers holding the total_batch_count copied vectors y.\ncblas_?copy_batch_strided\nComputes a group of vector copies.\nSyntax\nvoid cblas_scopy_batch_strided (const MKL_INT n, const float *x, const MKL_INT incx,\nconst MKL_INT stridex, float *y, const MKL_INT incy, const MKL_INT stridey, const\nMKL_INT batch_size);\nvoid cblas_dcopy_batch_strided (const MKL_INT n, const double *x, const MKL_INT incx,\nconst MKL_INT stridex, double *y, const MKL_INT incy, const MKL_INT stridey, const\nMKL_INT batch_size);\nvoid cblas_ccopy_batch_strided (const MKL_INT n, const void *x, const MKL_INT incx,\nconst MKL_INT stridex, void *y, const MKL_INT incy, const MKL_INT stridey, const\nMKL_INT batch_size);\nvoid cblas_zcopy_batch_strided (const MKL_INT n, const void *x, const MKL_INT incx,\nconst MKL_INT stridex, void *y, const MKL_INT incy, const MKL_INT stridey, const\nMKL_INT batch_size);\nDescription\nThe cblas_?copy_batch_strided routines perform a series of vector copies. They are similar to their\ncblas_?copy routine counterparts, but the cblas_?copy_batch_strided routines perform vector\noperations with a group of vectors.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n41\n\n\nAll vectors x and y have the same parameters (size, increments) and are stored at constant distance\nstridex (respectively, stridey) from each other. The operation is defined as follows:\nfor i = 0 … batch_size – 1\n    X and Y are vectors at offset i * stridex and i * stridey in x and y\n    Y = X\nend for\nInput Parameters\nn\nNumber of elements in vectors x and y. Must be at least 0.\nx\nArray containing the input vectors. Must be of size at least (1 +\n(n-1)*abs(incx)) + (batch_size – 1) * stridex.\nincx\nIncrement between two consecutive elements of a single vector in x.\nstridex\nStride between two consecutive vectors in x. Must be at least (1 +\n(n-1)*abs(incx)).\ny\nArray holding the output vectors. Must be of size at least (1 +\n(n-1)*abs(incy)) + (batch_size – 1) * stridey.\nincy\nIncrement between two consecutive elements of a single vector in y.\nstridey\nStride between two consecutive y vectors. Must be at least (1 +\n(n-1)*abs(incy)).\nbatch_size\nNumber of copy computations to perform; also the number of x and\ny vectors. Must be at least 0.\nOutput Parameters\ny\nArray holding the batch_size copied vectors y.\ncblas_?dot\nComputes a vector-vector dot product.\nSyntax\nfloat cblas_sdot (const MKL_INT n, const float *x, const MKL_INT incx, const float *y,\nconst MKL_INT incy);\ndouble cblas_ddot (const MKL_INT n, const double *x, const MKL_INT incx, const double\n*y, const MKL_INT incy);\nInclude Files\n•\nmkl.h\nDescription\nThe ?dot routines perform a vector-vector reduction operation defined as\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n42\n\n\nwhere xi and yi are elements of vectors x and y.\nInput Parameters\nn\nSpecifies the number of elements in vectors x and y.\nx\nArray, size at least (1+(n-1)*abs(incx)).\nincx\nSpecifies the increment for the elements of x.\ny\nArray, size at least (1+(n-1)*abs(incy)).\nincy\nSpecifies the increment for the elements of y.\nReturn Values\nThe result of the dot product of x and y, if n is positive. Otherwise, returns 0.\ncblas_?sdot\nComputes a vector-vector dot product with double\nprecision.\nSyntax\nfloat cblas_sdsdot (const MKL_INT n, const float sb, const float *sx, const MKL_INT\nincx, const float *sy, const MKL_INT incy);\ndouble cblas_dsdot (const MKL_INT n, const float *sx, const MKL_INT incx, const float\n*sy, const MKL_INT incy);\nInclude Files\n•\nmkl.h\nDescription\nThe ?sdot routines compute the inner product of two vectors with double precision. Both routines use double\nprecision accumulation of the intermediate results, but the sdsdot routine outputs the final result in single\nprecision, whereas the dsdot routine outputs the double precision result. The function sdsdot also adds\nscalar value sb to the inner product.\nInput Parameters\nn\nSpecifies the number of elements in the input vectors sx and sy.\nsb\nSingle precision scalar to be added to inner product (for the function\nsdsdot only).\nsx, sy\nArrays, size at least (1+(n -1)*abs(incx)) and (1+(n-1)*abs(incy)),\nrespectively. Contain the input single precision vectors.\nincx\nSpecifies the increment for the elements of sx.\nincy\nSpecifies the increment for the elements of sy.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n43\n\n\nOutput Parameters\nres\nContains the result of the dot product of sx and sy (with sb added for\nsdsdot), if n is positive. Otherwise, res contains sb for sdsdot and 0 for\ndsdot.\nReturn Values\nThe result of the dot product of sx and sy (with sb added for sdsdot), if n is positive. Otherwise, returns sb\nfor sdsdot and 0 for dsdot.\ncblas_?dotc\nComputes a dot product of a conjugated vector with\nanother vector.\nSyntax\nvoid cblas_cdotc_sub (const MKL_INT n, const void *x, const MKL_INT incx, const void\n*y, const MKL_INT incy, void *dotc);\nvoid cblas_zdotc_sub (const MKL_INT n, const void *x, const MKL_INT incx, const void\n*y, const MKL_INT incy, void *dotc);\nInclude Files\n•\nmkl.h\nDescription\nThe ?dotc routines perform a vector-vector operation defined as:\nwhere xi and yi are elements of vectors x and y.\nInput Parameters\nn\nSpecifies the number of elements in vectors x and y.\nx\nArray, size at least (1 + (n -1)*abs(incx)).\nincx\nSpecifies the increment for the elements of x.\ny\nArray, size at least (1 + (n -1)*abs(incy)).\nincy\nSpecifies the increment for the elements of y.\nOutput Parameters\ndotc\nContains the result of the dot product of the conjugated x and unconjugated\ny, if n is positive. Otherwise, it contains 0.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n44\n\n\ncblas_?dotu\nComputes a complex vector-vector dot product.\nSyntax\nvoid cblas_cdotu_sub (const MKL_INT n, const void *x, const MKL_INT incx, const void\n*y, const MKL_INT incy, void *dotu);\nvoid cblas_zdotu_sub (const MKL_INT n, const void *x, const MKL_INT incx, const void\n*y, const MKL_INT incy, void *dotu);\nInclude Files\n•\nmkl.h\nDescription\nThe ?dotu routines perform a vector-vector reduction operation defined as\nwhere xi and yi are elements of complex vectors x and y.\nNOTE The _sub suffix on cblas_cdotu_sub and cblas_zdotu_sub is to emphasize that these\nare subroutines rather than functions (the return value is stored into the dotu pointer).\nInput Parameters\nn\nSpecifies the number of elements in vectors x and y.\nx\nArray, size at least (1 + (n -1)*abs(incx)).\nincx\nSpecifies the increment for the elements of x.\ny\nArray, size at least (1 + (n -1)*abs(incy)).\nincy\nSpecifies the increment for the elements of y.\nOutput Parameters\ndotu\nContains the result of the dot product of x and y, if n is positive. Otherwise,\nit contains 0.\ncblas_?nrm2\nComputes the Euclidean norm of a vector.\nSyntax\nfloat cblas_snrm2 (const MKL_INT n, const float *x, const MKL_INT incx);\ndouble cblas_dnrm2 (const MKL_INT n, const double *x, const MKL_INT incx);\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n45\n\n\nfloat cblas_scnrm2 (const MKL_INT n, const void *x, const MKL_INT incx);\ndouble cblas_dznrm2 (const MKL_INT n, const void *x, const MKL_INT incx);\nInclude Files\n•\nmkl.h\nDescription\nThe ?nrm2 routines perform a vector reduction operation defined as\nres = ||x||,\nwhere:\nx is a vector,\nres is a value containing the Euclidean norm of the elements of x.\nInput Parameters\nn\nSpecifies the number of elements in vector x.\nx\nArray, size at least (1 + (n -1)*abs (incx)).\nincx\nSpecifies the increment for the elements of x.\nReturn Values\nThe Euclidean norm of the vector x.\ncblas_?rot\nPerforms rotation of points in the plane.\nSyntax\nvoid cblas_srot (const MKL_INT n, float *x, const MKL_INT incx, float *y, const MKL_INT incy, \nconst float c, const float s);\nvoid cblas_drot (const MKL_INT n, double *x, const MKL_INT incx, double *y, const MKL_INT incy, \nconst double c, const double s);\nvoid cblas_crot (const MKL_INT n, void *x, const MKL_INT incx, void *y, const MKL_INT incy, \nconst float c, const void* s);\nvoid cblas_zrot (const MKL_INT n, void *x, const MKL_INT incx, void *y, const MKL_INT incy, \nconst double c, const void* s); \nvoid cblas_csrot (const MKL_INT n, void *x, const MKL_INT incx, void *y, const MKL_INT incy, \nconst float c, const float s);\nvoid cblas_zdrot (const MKL_INT n, void *x, const MKL_INT incx, void *y, const MKL_INT incy, \nconst double c, const double s);\nDescription\nGiven two complex vectors x and y, each vector element of these vectors is replaced as follows:\nxi = c*xi + s*yi\nyi = c*yi - s*xi\nIf s is a complex type, each vector element is replaced as follows:\nxi = c*xi + s*yi\nyi = c*yi - conj(s)*xi\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n46\n\n\nInput Parameters\nn\nSpecifies the number of elements in vectors x and y.\nx\nArray, size at least (1 + (n-1)*abs(incx)).\nincx\nSpecifies the increment for the elements of x.\ny\nArray, size at least (1 + (n -1)*abs(incy)).\nincy\nSpecifies the increment for the elements of y.\nc\nA scalar.\ns\nA scalar.\nOutput Parameters\nx\nEach element is replaced by c*x + s*y.\ny\nEach element is replaced by c*y - s*x, or by c*y-conj(s)*x if s is a\ncomplex type.\ncblas_?rotg\nComputes the parameters for a Givens rotation.\nSyntax\nvoid cblas_srotg (float *a, float *b, float *c, float *s);\nvoid cblas_drotg (double *a, double *b, double *c, double *s);\nvoid cblas_crotg (void *a, const void *b, float *c, void *s);\nvoid cblas_zrotg (void *a, const void *b, double *c, void *s);\nInclude Files\n•\nmkl.h\nDescription\nGiven the Cartesian coordinates (a, b) of a point, these routines return the parameters c, s, r, and z\nassociated with the Givens rotation. The parameters c and s define a unitary matrix such that:\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n47\n\n\nThe parameter z is defined such that if |a| > |b|, z is s; otherwise if c is not 0 z is 1/c; otherwise z is 1.\nInput Parameters\na\nProvides the x-coordinate of the point p.\nb\nProvides the y-coordinate of the point p.\nOutput Parameters\na\nContains the parameter r associated with the Givens rotation.\nb\nContains the parameter z associated with the Givens rotation.\nc\nContains the parameter c associated with the Givens rotation.\ns\nContains the parameter s associated with the Givens rotation.\ncblas_?rotm\nPerforms modified Givens rotation of points in the\nplane.\nSyntax\nvoid cblas_srotm (const MKL_INT n, float *x, const MKL_INT incx, float *y, const\nMKL_INT incy, const float *param);\nvoid cblas_drotm (const MKL_INT n, double *x, const MKL_INT incx, double *y, const\nMKL_INT incy, const double *param);\nInclude Files\n•\nmkl.h\nDescription\nGiven two vectors x and y, each vector element of these vectors is replaced as follows:\nxi\nyi\n= H\nxi\nyi\nfor i=1 to n, where H is a modified Givens transformation matrix whose values are stored in the param[1]\nthrough param[4] array. See discussion on the param argument.\nInput Parameters\nn\nSpecifies the number of elements in vectors x and y.\nx\nArray, size at least (1 + (n -1)*abs(incx)).\nincx\nSpecifies the increment for the elements of x.\ny\nArray, size at least (1 + (n -1)*abs(incy)).\nincy\nSpecifies the increment for the elements of y.\nparam\nArray, size 5.\nThe elements of the param array are:\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n48\n\n\nparam[0] contains a switch, flag. param[1-4] contain h11, h21, h12, and\nh22, respectively, the components of the array H.\nDepending on the values of flag, the components of H are set as follows:\nflag = -1.0: H =\nh11 h12\nh21 h22\nflag = 0.0: H =\n1.0 h12\nh21 1.0\nflag = 1.0: H =\nh11\n1.0\n−1.0 h22\nflag = -2.0: H = 1.0 0.0\n0.0 1.0\nIn the last three cases, the matrix entries of 1.0, -1.0, and 0.0 are assumed\nbased on the value of flag and are not required to be set in the param\nvector.\nOutput Parameters\nx\nEach element x[i] is replaced by h11*x[i] + h12*y[i].\ny\nEach element y[i] is replaced by h21*x[i] + h22*y[i].\ncblas_?rotmg\nComputes the parameters for a modified Givens\nrotation.\nSyntax\nvoid cblas_srotmg (float *d1, float *d2, float *x1, const float y1, float *param);\nvoid cblas_drotmg (double *d1, double *d2, double *x1, const double y1, double *param);\nInclude Files\n•\nmkl.h\nDescription\nGiven Cartesian coordinates (x1, y1) of an input vector, these routines compute the components of a\nmodified Givens transformation matrix H that zeros the y-component of the resulting vector:\nx1\n0\n= H x1 d1\ny1 d2\nInput Parameters\nd1\nProvides the scaling factor for the x-coordinate of the input vector.\nd2\nProvides the scaling factor for the y-coordinate of the input vector.\nx1\nProvides the x-coordinate of the input vector.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n49\n\n\ny1\nProvides the y-coordinate of the input vector.\nOutput Parameters\nd1\nProvides the first diagonal element of the updated matrix.\nd2\nProvides the second diagonal element of the updated matrix.\nx1\nProvides the x-coordinate of the rotated vector before scaling.\nparam\nArray, size 5.\nThe elements of the param array are:\nparam[0] contains a switch, flag. the other array elements param[1-4]\ncontain the components of the array H: h11, h21, h12, and h22, respectively.\nDepending on the values of flag, the components of H are set as follows:\nflag = -1.0: H =\nh11 h12\nh21 h22\nflag = 0.0: H =\n1.0 h12\nh21 1.0\nflag = 1.0: H =\nh11\n1.0\n−1.0 h22\nflag = -2.0: H = 1.0 0.0\n0.0 1.0\nIn the last three cases, the matrix entries of 1.0, -1.0, and 0.0 are assumed\nbased on the value of flag and are not required to be set in the param\nvector.\ncblas_?scal\nComputes the product of a vector by a scalar.\nSyntax\nvoid cblas_sscal (const MKL_INT n, const float a, float *x, const MKL_INT incx);\nvoid cblas_dscal (const MKL_INT n, const double a, double *x, const MKL_INT incx);\nvoid cblas_cscal (const MKL_INT n, const void *a, void *x, const MKL_INT incx);\nvoid cblas_zscal (const MKL_INT n, const void *a, void *x, const MKL_INT incx);\nvoid cblas_csscal (const MKL_INT n, const float a, void *x, const MKL_INT incx);\nvoid cblas_zdscal (const MKL_INT n, const double a, void *x, const MKL_INT incx);\nInclude Files\n•\nmkl.h\nDescription\nThe ?scal routines perform a vector operation defined as\nx = a*x\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n50\n\n\nwhere:\na is a scalar, x is an n-element vector.\nInput Parameters\nn\nSpecifies the number of elements in vector x.\na\nSpecifies the scalar a.\nx\nArray, size at least (1 + (n -1)*abs(incx)).\nincx\nSpecifies the increment for the elements of x.\nOutput Parameters\nx\nUpdated vector x.\ncblas_?swap\nSwaps a vector with another vector.\nSyntax\nvoid cblas_sswap (const MKL_INT n, float *x, const MKL_INT incx, float *y, const\nMKL_INT incy);\nvoid cblas_dswap (const MKL_INT n, double *x, const MKL_INT incx, double *y, const\nMKL_INT incy);\nvoid cblas_cswap (const MKL_INT n, void *x, const MKL_INT incx, void *y, const MKL_INT\nincy);\nvoid cblas_zswap (const MKL_INT n, void *x, const MKL_INT incx, void *y, const MKL_INT\nincy);\nInclude Files\n•\nmkl.h\nDescription\nGiven two vectors x and y, the ?swap routines return vectors y and x swapped, each replacing the other.\nInput Parameters\nn\nSpecifies the number of elements in vectors x and y.\nx\nArray, size at least (1 + (n-1)*abs(incx)).\nincx\nSpecifies the increment for the elements of x.\ny\nArray, size at least (1 + (n-1)*abs(incy)).\nincy\nSpecifies the increment for the elements of y.\nOutput Parameters\nx\nContains the resultant vector x, that is, the input vector y.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n51\n\n\ny\nContains the resultant vector y, that is, the input vector x.\ncblas_i?amax\nFinds the index of the element with maximum\nabsolute value.\nSyntax\nCBLAS_INDEX cblas_isamax (const MKL_INT n, const float *x, const MKL_INT incx);\nCBLAS_INDEX cblas_idamax (const MKL_INT n, const double *x, const MKL_INT incx);\nCBLAS_INDEX cblas_icamax (const MKL_INT n, const void *x, const MKL_INT incx);\nCBLAS_INDEX cblas_izamax (const MKL_INT n, const void *x, const MKL_INT incx);\nInclude Files\n•\nmkl.h\nDescription\nGiven a vector x, the i?amax functions return the position of the vector element x[i] that has the largest\nabsolute value for real flavors, or the largest sum |Re(x[i])|+|Im(x[i])| for complex flavors.\nIf either n or incx are not positive, the routine returns 0.\nIf more than one vector element is found with the same largest absolute value, the index of the first one\nencountered is returned.\nIf the vector contains NaN values, then the routine returns the index of the first NaN.\nInput Parameters\nn\nSpecifies the number of elements in vector x.\nx\nArray, size at least (1+(n-1)*abs(incx)).\nincx\nSpecifies the increment for the elements of x.\nReturn Values\nReturns the position of vector element that has the largest absolute value such that x[index-1] has the\nlargest absolute value. The index returned is zero-based.\ncblas_i?amin\nFinds the index of the element with the smallest\nabsolute value.\nSyntax\nCBLAS_INDEX cblas_isamin (const MKL_INT n, const float *x, const MKL_INT incx);\nCBLAS_INDEX cblas_idamin (const MKL_INT n, const double *x, const MKL_INT incx);\nCBLAS_INDEX cblas_icamin (const MKL_INT n, const void *x, const MKL_INT incx);\nCBLAS_INDEX cblas_izamin (const MKL_INT n, const void *x, const MKL_INT incx);\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n52\n\n\nInclude Files\n•\nmkl.h\nDescription\nGiven a vector x, the i?amin functions return the position of the vector element x[i] that has the smallest\nabsolute value for real flavors, or the smallest sum |Re(x[i])|+|Im(x[i])| for complex flavors.\nIf either n or incx are not positive, the routine returns 0.\nIf more than one vector element is found with the same smallest absolute value, the index of the first one\nencountered is returned.\nIf the vector contains NaN values, then the routine returns the index of the first NaN.\nInput Parameters\nn\nOn entry, n specifies the number of elements in vector x.\nx\nArray, size at least (1+(n-1)*abs(incx)).\nincx\nSpecifies the increment for the elements of x.\nReturn Values\nIndicates the position of vector element with the smallest absolute value such that x[index-1] has the\nsmallest absolute value. The index returned is zero-based.\ncblas_?cabs1\nComputes absolute value of complex number.\nSyntax\nfloat cblas_scabs1 (const void *z);\ndouble cblas_dcabs1 (const void *z);\nInclude Files\n•\nmkl.h\nDescription\nThe ?cabs1 is an auxiliary routine for a few BLAS Level 1 routines. This routine performs an operation\ndefined as\nres=|Re(z)|+|Im(z)|,\nwhere z is a scalar, and res is a value containing the absolute value of a complex number z.\nInput Parameters\nz\nScalar.\nReturn Values\nThe absolute value of a complex number z.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n53\n\n\nBLAS Level 2 Routines\nThis section describes BLAS Level 2 routines, which perform matrix-vector operations. The following table\nlists the BLAS Level 2 routine groups and the data types associated with them.\nBLAS Level 2 Routine Groups and Their Data Types\nRoutine Groups\nData Types\nDescription\ncblas_?gbmv\ns, d, c, z\nMatrix-vector product using a general band matrix\ncblas?_gemv\ns, d, c, z\nMatrix-vector product using a general matrix\ncblas_?ger\ns, d\nRank-1 update of a general matrix\ncblas_?gerc\nc, z\nRank-1 update of a conjugated general matrix\ncblas_?geru\nc, z\nRank-1 update of a general matrix, unconjugated\ncblas_?hbmv\nc, z\nMatrix-vector product using a Hermitian band matrix\ncblas_?hemv\nc, z\nMatrix-vector product using a Hermitian matrix\ncblas_?her\nc, z\nRank-1 update of a Hermitian matrix\ncblas_?her2\nc, z\nRank-2 update of a Hermitian matrix\ncblas_?hpmv\nc, z\nMatrix-vector product using a Hermitian packed matrix\ncblas_?hpr\nc, z\nRank-1 update of a Hermitian packed matrix\ncblas_?hpr2\nc, z\nRank-2 update of a Hermitian packed matrix\ncblas_?sbmv\ns, d\nMatrix-vector product using symmetric band matrix\ncblas_?spmv\ns, d\nMatrix-vector product using a symmetric packed matrix\ncblas_?spr\ns, d\nRank-1 update of a symmetric packed matrix\ncblas_?spr2\ns, d\nRank-2 update of a symmetric packed matrix\ncblas_?symv\ns, d\nMatrix-vector product using a symmetric matrix\ncblas_?syr\ns, d\nRank-1 update of a symmetric matrix\ncblas_?syr2\ns, d\nRank-2 update of a symmetric matrix\ncblas_?tbmv\ns, d, c, z\nMatrix-vector product using a triangular band matrix\ncblas_?tbsv\ns, d, c, z\nSolution of a linear system of equations with a triangular\nband matrix\ncblas_?tpmv\ns, d, c, z\nMatrix-vector product using a triangular packed matrix\ncblas_?tpsv\ns, d, c, z\nSolution of a linear system of equations with a triangular\npacked matrix\ncblas_?trmv\ns, d, c, z\nMatrix-vector product using a triangular matrix\ncblas_?trsv\ns, d, c, z\nSolution of a linear system of equations with a triangular\nmatrix\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n54\n\n\ncblas_?gbmv\nComputes a matrix-vector product with a general\nband matrix.\nSyntax\nvoid cblas_sgbmv (const CBLAS_LAYOUT Layout, const CBLAS_TRANSPOSE trans, const MKL_INT\nm, const MKL_INT n, const MKL_INT kl, const MKL_INT ku, const float alpha, const float\n*a, const MKL_INT lda, const float *x, const MKL_INT incx, const float beta, float *y,\nconst MKL_INT incy);\nvoid cblas_dgbmv (const CBLAS_LAYOUT Layout, const CBLAS_TRANSPOSE trans, const MKL_INT\nm, const MKL_INT n, const MKL_INT kl, const MKL_INT ku, const double alpha, const\ndouble *a, const MKL_INT lda, const double *x, const MKL_INT incx, const double beta,\ndouble *y, const MKL_INT incy);\nvoid cblas_cgbmv (const CBLAS_LAYOUT Layout, const CBLAS_TRANSPOSE trans, const MKL_INT\nm, const MKL_INT n, const MKL_INT kl, const MKL_INT ku, const void *alpha, const void\n*a, const MKL_INT lda, const void *x, const MKL_INT incx, const void *beta, void *y,\nconst MKL_INT incy);\nvoid cblas_zgbmv (const CBLAS_LAYOUT Layout, const CBLAS_TRANSPOSE trans, const MKL_INT\nm, const MKL_INT n, const MKL_INT kl, const MKL_INT ku, const void *alpha, const void\n*a, const MKL_INT lda, const void *x, const MKL_INT incx, const void *beta, void *y,\nconst MKL_INT incy);\nInclude Files\n•\nmkl.h\nDescription\nThe ?gbmv routines perform a matrix-vector operation defined as\ny := alpha*A*x + beta*y,\nor\ny := alpha*A'*x + beta*y,\nor\ny := alpha *conjg(A')*x + beta*y,\nwhere:\nalpha and beta are scalars,\nx and y are vectors,\nA is an m-by-n band matrix, with kl sub-diagonals and ku super-diagonals.\nInput Parameters\nLayout\nSpecifies whether two-dimensional array storage is row-major\n(CblasRowMajor) or column-major (CblasColMajor).\ntrans\nSpecifies the operation:\nIf trans=CblasNoTrans, then y := alpha*A*x + beta*y\nIf trans=CblasTrans, then y := alpha*A'*x + beta*y\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n55\n\n\nIf trans=CblasConjTrans, then y := alpha *conjg(A')*x + beta*y\nm\nSpecifies the number of rows of the matrix A.\nThe value of m must be at least zero.\nn\nSpecifies the number of columns of the matrix A.\nThe value of n must be at least zero.\nkl\nSpecifies the number of sub-diagonals of the matrix A.\nThe value of kl must satisfy 0≤kl.\nku\nSpecifies the number of super-diagonals of the matrix A.\nThe value of ku must satisfy 0≤ku.\nalpha\nSpecifies the scalar alpha.\na\nArray, size lda*n.\nLayout = CblasColMajor: Before entry, the leading (kl + ku + 1) by n\npart of the array a must contain the matrix of coefficients. This matrix must\nbe supplied column-by-column, with the leading diagonal of the matrix in\nrow (ku) of the array, the first super-diagonal starting at position 1 in row\n(ku - 1), the first sub-diagonal starting at position 0 in row (ku + 1),\nand so on. Elements in the array a that do not correspond to elements in\nthe band matrix (such as the top left ku by ku triangle) are not referenced.\nThe following program segment transfers a band matrix from conventional\nfull matrix storage (matrix, with leading dimension ldm) to band storage (a,\nwith leading dimension lda):\nfor (j = 0; j < n; j++) {\n    k = ku - j;\n    for (i = max(0, j-ku); i < min(m, j+kl+1); i++) {\n        a[(k+i) + j*lda] = matrix[i + j*ldm];\n    }\n}\nLayout = CblasRowMajor: Before entry, the leading (kl + ku + 1) by m\npart of the array a must contain the matrix of coefficients. This matrix must\nbe supplied row-by-row, with the leading diagonal of the matrix in column\n(kl) of the array, the first super-diagonal starting at position 0 in column\n(kl + 1), the first sub-diagonal starting at position 1 in row (kl - 1),\nand so on. Elements in the array a that do not correspond to elements in\nthe band matrix (such as the top left kl by kl triangle) are not referenced.\nThe following program segment transfers a band matrix from row-major full\nmatrix storage (matrix, with leading dimension ldm) to band storage (a,\nwith leading dimension lda):\nfor (i = 0; i < m; i++) {\n    k = kl - i;\n    for (j = max(0, i-kl); j < min(n, i+ku+1); j++) {\n        a[(k+j) + i*lda] = matrix[j + i*ldm];\n    }\n}\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n56\n\n\nlda\nSpecifies the leading dimension of a as declared in the calling\n(sub)program. The value of lda must be at least (kl + ku + 1).\nx\nArray, size at least (1 + (n - 1)*abs(incx)) when\ntrans=CblasNoTrans, and at least (1 + (m - 1)*abs(incx))\notherwise. Before entry, the array x must contain the vector x.\nincx\nSpecifies the increment for the elements of x. incx must not be zero.\nbeta\nSpecifies the scalar beta. When beta is equal to zero, then y need not be\nset on input.\ny\nArray, size at least (1 +(m - 1)*abs(incy)) when\ntrans=CblasNoTrans and at least (1 +(n - 1)*abs(incy)) otherwise.\nBefore entry, the incremented array y must contain the vector y.\nincy\nSpecifies the increment for the elements of y.\nThe value of incy must not be zero.\nOutput Parameters\ny\nBuffer holding the updated vector y.\ncblas_?gemv\nComputes a matrix-vector product using a general\nmatrix.\nSyntax\nvoid cblas_sgemv (const CBLAS_LAYOUT Layout, const CBLAS_TRANSPOSE trans, const MKL_INT\nm, const MKL_INT n, const float alpha, const float *a, const MKL_INT lda, const float\n*x, const MKL_INT incx, const float beta, float *y, const MKL_INT incy);\nvoid cblas_dgemv (const CBLAS_LAYOUT Layout, const CBLAS_TRANSPOSE trans, const MKL_INT\nm, const MKL_INT n, const double alpha, const double *a, const MKL_INT lda, const\ndouble *x, const MKL_INT incx, const double beta, double *y, const MKL_INT incy);\nvoid cblas_cgemv (const CBLAS_LAYOUT Layout, const CBLAS_TRANSPOSE trans, const MKL_INT\nm, const MKL_INT n, const void *alpha, const void *a, const MKL_INT lda, const void *x,\nconst MKL_INT incx, const void *beta, void *y, const MKL_INT incy);\nvoid cblas_zgemv (const CBLAS_LAYOUT Layout, const CBLAS_TRANSPOSE trans, const MKL_INT\nm, const MKL_INT n, const void *alpha, const void *a, const MKL_INT lda, const void *x,\nconst MKL_INT incx, const void *beta, void *y, const MKL_INT incy);\nInclude Files\n•\nmkl.h\nDescription\nThe ?gemv routines perform a matrix-vector operation defined as:\ny := alpha*A*x + beta*y,\nor\ny := alpha*A'*x + beta*y,\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n57\n\n\nor\ny := alpha*conjg(A')*x + beta*y,\nwhere:\nalpha and beta are scalars,\nx and y are vectors,\nA is an m-by-n matrix.\nInput Parameters\nLayout\nSpecifies whether two-dimensional array storage is row-major\n(CblasRowMajor) or column-major (CblasColMajor).\ntrans\nSpecifies the operation:\nif trans=CblasNoTrans, then y := alpha*A*x + beta*y;\nif trans=CblasTrans, then y := alpha*A'*x + beta*y;\nif trans=CblasConjTrans, then y := alpha *conjg(A')*x + beta*y.\nm\nSpecifies the number of rows of the matrix A. The value of m must be at\nleast zero.\nn\nSpecifies the number of columns of the matrix A. The value of n must be at\nleast zero.\nalpha\nSpecifies the scalar alpha.\na\nArray, size lda*k.\nFor Layout = CblasColMajor, k is n. Before entry, the leading m-by-n part\nof the array a must contain the matrix A.\nFor Layout = CblasRowMajor, k is m. Before entry, the leading n-by-m part\nof the array a must contain the matrix A.\nlda\nSpecifies the leading dimension of a as declared in the calling\n(sub)program.\nFor Layout = CblasColMajor, the value of lda must be at least max(1,\nm).\nFor Layout = CblasRowMajor, the value of lda must be at least max(1,\nn).\nx\nArray, size at least (1+(n-1)*abs(incx)) when trans=CblasNoTrans\nand at least (1+(m - 1)*abs(incx)) otherwise. Before entry, the\nincremented array x must contain the vector x.\nincx\nSpecifies the increment for the elements of x.\nThe value of incx must not be zero.\nbeta\nSpecifies the scalar beta. When beta is set to zero, then y need not be set\non input.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n58\n\n\ny\nArray, size at least (1 +(m - 1)*abs(incy)) when\ntrans=CblasNoTrans and at least (1 +(n - 1)*abs(incy)) otherwise.\nBefore entry with non-zero beta, the incremented array y must contain the\nvector y.\nincy\nSpecifies the increment for the elements of y.\nThe value of incy must not be zero.\nOutput Parameters\ny\nUpdated vector y.\ncblas_?ger\nPerforms a rank-1 update of a general matrix.\nSyntax\nvoid cblas_sger (const CBLAS_LAYOUT Layout, const MKL_INT m, const MKL_INT n, const\nfloat alpha, const float *x, const MKL_INT incx, const float *y, const MKL_INT incy,\nfloat *a, const MKL_INT lda);\nvoid cblas_dger (const CBLAS_LAYOUT Layout, const MKL_INT m, const MKL_INT n, const\ndouble alpha, const double *x, const MKL_INT incx, const double *y, const MKL_INT incy,\ndouble *a, const MKL_INT lda);\nInclude Files\n•\nmkl.h\nDescription\nThe ?ger routines perform a matrix-vector operation defined as\nA := alpha*x*y'+ A,\nwhere:\nalpha is a scalar,\nx is an m-element vector,\ny is an n-element vector,\nA is an m-by-n general matrix.\nInput Parameters\nLayout\nSpecifies whether two-dimensional array storage is row-major\n(CblasRowMajor) or column-major (CblasColMajor).\nm\nSpecifies the number of rows of the matrix A.\nThe value of m must be at least zero.\nn\nSpecifies the number of columns of the matrix A.\nThe value of n must be at least zero.\nalpha\nSpecifies the scalar alpha.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n59\n\n\nx\nArray, size at least (1 + (m - 1)*abs(incx)). Before entry, the\nincremented array x must contain the m-element vector x.\nincx\nSpecifies the increment for the elements of x.\nThe value of incx must not be zero.\ny\nArray, size at least (1 + (n - 1)*abs(incy)). Before entry, the\nincremented array y must contain the n-element vector y.\nincy\nSpecifies the increment for the elements of y.\nThe value of incy must not be zero.\na\nArray, size lda*k.\nFor Layout = CblasColMajor, k is n. Before entry, the leading m-by-n part\nof the array a must contain the matrix A.\nFor Layout = CblasRowMajor, k is m. Before entry, the leading n-by-m part\nof the array a must contain the matrix A.\nlda\nSpecifies the leading dimension of a as declared in the calling\n(sub)program.\nFor Layout = CblasColMajor, the value of lda must be at least max(1,\nm).\nFor Layout = CblasRowMajor, the value of lda must be at least max(1,\nn).\nOutput Parameters\na\nOverwritten by the updated matrix.\ncblas_?gerc\nPerforms a rank-1 update (conjugated) of a general\nmatrix.\nSyntax\nvoid cblas_cgerc (const CBLAS_LAYOUT Layout, const MKL_INT m, const MKL_INT n, const\nvoid *alpha, const void *x, const MKL_INT incx, const void *y, const MKL_INT incy, void\n*a, const MKL_INT lda);\nvoid cblas_zgerc (const CBLAS_LAYOUT Layout, const MKL_INT m, const MKL_INT n, const\nvoid *alpha, const void *x, const MKL_INT incx, const void *y, const MKL_INT incy, void\n*a, const MKL_INT lda);\nInclude Files\n•\nmkl.h\nDescription\nThe ?gerc routines perform a matrix-vector operation defined as\nA := alpha*x*conjg(y') + A,\nwhere:\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n60\n\n\nalpha is a scalar,\nx is an m-element vector,\ny is an n-element vector,\nA is an m-by-n matrix.\nInput Parameters\nLayout\nSpecifies whether two-dimensional array storage is row-major\n(CblasRowMajor) or column-major (CblasColMajor).\nm\nSpecifies the number of rows of the matrix A.\nThe value of m must be at least zero.\nn\nSpecifies the number of columns of the matrix A.\nThe value of n must be at least zero.\nalpha\nSpecifies the scalar alpha.\nx\nArray, size at least (1 + (m - 1)*abs(incx)). Before entry, the\nincremented array x must contain the m-element vector x.\nincx\nSpecifies the increment for the elements of x.\nThe value of incx must not be zero.\ny\nArray, size at least (1 + (n - 1)*abs(incy)). Before entry, the\nincremented array y must contain the n-element vector y.\nincy\nSpecifies the increment for the elements of y.\nThe value of incy must not be zero.\na\nArray, size lda*k.\nFor Layout = CblasColMajor, k is n. Before entry, the leading m-by-n part\nof the array a must contain the matrix A.\nFor Layout = CblasRowMajor, k is m. Before entry, the leading n-by-m part\nof the array a must contain the matrix A.\nlda\nSpecifies the leading dimension of a as declared in the calling\n(sub)program.\nFor Layout = CblasColMajor, the value of lda must be at least max(1,\nm).\nFor Layout = CblasRowMajor, the value of lda must be at least max(1,\nn).\nOutput Parameters\na\nOverwritten by the updated matrix.\ncblas_?geru\nPerforms a rank-1 update (unconjugated) of a general\nmatrix.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n61\n\n\nSyntax\nvoid cblas_cgeru (const CBLAS_LAYOUT Layout, const MKL_INT m, const MKL_INT n, const\nvoid *alpha, const void *x, const MKL_INT incx, const void *y, const MKL_INT incy, void\n*a, const MKL_INT lda);\nvoid cblas_zgeru (const CBLAS_LAYOUT Layout, const MKL_INT m, const MKL_INT n, const\nvoid *alpha, const void *x, const MKL_INT incx, const void *y, const MKL_INT incy, void\n*a, const MKL_INT lda);\nInclude Files\n•\nmkl.h\nDescription\nThe ?geru routines perform a matrix-vector operation defined as\nA := alpha*x*y ' + A,\nwhere:\nalpha is a scalar,\nx is an m-element vector,\ny is an n-element vector,\nA is an m-by-n matrix.\nInput Parameters\nLayout\nSpecifies whether two-dimensional array storage is row-major\n(CblasRowMajor) or column-major (CblasColMajor).\nm\nSpecifies the number of rows of the matrix A.\nThe value of m must be at least zero.\nn\nSpecifies the number of columns of the matrix A.\nThe value of n must be at least zero.\nalpha\nSpecifies the scalar alpha.\nx\nArray, size at least (1 + (m - 1)*abs(incx)). Before entry, the\nincremented array x must contain the m-element vector x.\nincx\nSpecifies the increment for the elements of x.\nThe value of incx must not be zero.\ny\nArray, size at least (1 + (n - 1)*abs(incy)). Before entry, the\nincremented array y must contain the n-element vector y.\nincy\nSpecifies the increment for the elements of y.\nThe value of incy must not be zero.\na\nArray, size lda*k.\nFor Layout = CblasColMajor, k is n. Before entry, the leading m-by-n part\nof the array a must contain the matrix A.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n62\n\n\nFor Layout = CblasRowMajor, k is m. Before entry, the leading n-by-m part\nof the array a must contain the matrix A.\nlda\nSpecifies the leading dimension of a as declared in the calling\n(sub)program.\nFor Layout = CblasColMajor, the value of lda must be at least max(1,\nm).\nFor Layout = CblasRowMajor, the value of lda must be at least max(1,\nn).\nOutput Parameters\na\nOverwritten by the updated matrix.\ncblas_?hbmv\nComputes a matrix-vector product using a Hermitian\nband matrix.\nSyntax\nvoid cblas_chbmv (const CBLAS_LAYOUT Layout, const CBLAS_UPLO uplo, const MKL_INT n,\nconst MKL_INT k, const void *alpha, const void *a, const MKL_INT lda, const void *x,\nconst MKL_INT incx, const void *beta, void *y, const MKL_INT incy);\nvoid cblas_zhbmv (const CBLAS_LAYOUT Layout, const CBLAS_UPLO uplo, const MKL_INT n,\nconst MKL_INT k, const void *alpha, const void *a, const MKL_INT lda, const void *x,\nconst MKL_INT incx, const void *beta, void *y, const MKL_INT incy);\nInclude Files\n•\nmkl.h\nDescription\nThe ?hbmv routines perform a matrix-vector operation defined as y := alpha*A*x + beta*y,\nwhere:\nalpha and beta are scalars,\nx and y are n-element vectors,\nA is an n-by-n Hermitian band matrix, with k super-diagonals.\nInput Parameters\nLayout\nSpecifies whether two-dimensional array storage is row-major\n(CblasRowMajor) or column-major (CblasColMajor).\nuplo\nSpecifies whether the upper or lower triangular part of the Hermitian band\nmatrix A is used:\nIf uplo = CblasUpper, then the upper triangular part of the matrix A is\nused.\nIf uplo = CblasLower, then the low triangular part of the matrix A is used.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n63\n\n\nn\nSpecifies the order of the matrix A. The value of n must be at least zero.\nk\nFor uplo = CblasUpper: Specifies the number of super-diagonals of the\nmatrix A.\nFor uplo = CblasLower: Specifies the number of sub-diagonals of the\nmatrix A.\nThe value of k must satisfy 0≤k.\nalpha\nSpecifies the scalar alpha.\na\nArray, size lda*n.\nLayout = CblasColMajor:\nBefore entry with uplo = CblasUpper, the leading (k + 1) by n part of\nthe array a must contain the upper triangular band part of the Hermitian\nmatrix. The matrix must be supplied column-by-column, with the leading\ndiagonal of the matrix in row k of the array, the first super-diagonal starting\nat position 1 in row (k - 1), and so on. The top left k by k triangle of the\narray a is not referenced.\nThe following program segment transfers the upper triangular part of a\nHermitian band matrix from conventional full matrix storage (matrix, with\nleading dimension ldm) to band storage (a, with leading dimension lda):\nfor (j = 0; j < n; j++) {\n    m = k - j;\n    for (i = max( 0, j - k); i <= j; i++) {\n        a[(m+i) + j*lda] = matrix[i + j*ldm];\n    }\n}\nBefore entry with uplo = CblasLower, the leading (k + 1) by n part of\nthe array a must contain the lower triangular band part of the Hermitian\nmatrix, supplied column-by-column, with the leading diagonal of the matrix\nin row 0 of the array, the first sub-diagonal starting at position 0 in row 1,\nand so on. The bottom right k by k triangle of the array a is not referenced.\nThe following program segment transfers the lower triangular part of a\nHermitian band matrix from conventional full matrix storage (matrix, with\nleading dimension ldm) to band storage (a, with leading dimension lda):\nfor (j = 0; j < n; j++) {\n    m = -j;\n    for (i = j; i < min(n, j + k + 1); i++) {\n        a[(m+i) + j*lda] = matrix[i + j*ldm];\n    }\n}\nLayout = CblasRowMajor:\nBefore entry with uplo = CblasUpper, the leading (k + 1)-by-n part of\narray a must contain the upper triangular band part of the Hermitian\nmatrix. The matrix must be supplied row-by-row, with the leading diagonal\nof the matrix in column 0 of the array, the first super-diagonal starting at\nposition 0 in column 1, and so on. The bottom right k-by-k triangle of array\na is not referenced.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n64\n\n\nThe following program segment transfers the upper triangular part of a\nHermitian band matrix from row-major full matrix storage (matrix with\nleading dimension ldm) to row-major band storage (a, with leading\ndimension lda):\nfor (i = 0; i < n; i++) {\n    m = -i;\n    for (j = i; j < MIN(n, i+k+1); j++) {\n        a[(m+j) + i*lda] = matrix[j + i*ldm];\n    }\n}\nBefore entry with uplo = CblasLower, the leading (k + 1)-by-n part of\narray a must contain the lower triangular band part of the Hermitian matrix,\nsupplied row-by-row, with the leading diagonal of the matrix in column k of\nthe array, the first sub-diagonal starting at position 1 in column k-1, and so\non. The top left k-by-k triangle of array a is not referenced.\nThe following program segment transfers the lower triangular part of a\nHermitian row-major band matrix from row-major full matrix storage\n(matrix, with leading dimension ldm) to row-major band storage (a, with\nleading dimension lda):\nfor (i = 0; i < n; i++) {\n    m = k - i;\n    for (j = max(0, i-k); j <= i; j++) {\n         a[(m+j) + i*lda] = matrix[j + i*ldm];\n    }\n}\nThe imaginary parts of the diagonal elements need not be set and are\nassumed to be zero.\nlda\nSpecifies the leading dimension of a as declared in the calling\n(sub)program. The value of lda must be at least (k + 1).\nx\nArray, size at least (1 + (n - 1)*abs(incx)). Before entry, the\nincremented array x must contain the vector x.\nincx\nSpecifies the increment for the elements of x.\nThe value of incx must not be zero.\nbeta\nSpecifies the scalar beta.\ny\nArray, size at least (1 + (n - 1)*abs(incy)). Before entry, the\nincremented array y must contain the vector y.\nincy\nSpecifies the increment for the elements of y.\nThe value of incy must not be zero.\nOutput Parameters\ny\nOverwritten by the updated vector y.\ncblas_?hemv\nComputes a matrix-vector product using a Hermitian\nmatrix.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n65\n\n\nSyntax\nvoid cblas_chemv (const CBLAS_LAYOUT Layout, const CBLAS_UPLO uplo, const MKL_INT n,\nconst void *alpha, const void *a, const MKL_INT lda, const void *x, const MKL_INT incx,\nconst void *beta, void *y, const MKL_INT incy);\nvoid cblas_zhemv (const CBLAS_LAYOUT Layout, const CBLAS_UPLO uplo, const MKL_INT n,\nconst void *alpha, const void *a, const MKL_INT lda, const void *x, const MKL_INT incx,\nconst void *beta, void *y, const MKL_INT incy);\nInclude Files\n•\nmkl.h\nDescription\nThe ?hemv routines perform a matrix-vector operation defined as\ny := alpha*A*x + beta*y,\nwhere:\nalpha and beta are scalars,\nx and y are n-element vectors,\nA is an n-by-n Hermitian matrix.\nInput Parameters\nLayout\nSpecifies whether two-dimensional array storage is row-major\n(CblasRowMajor) or column-major (CblasColMajor).\nuplo\nSpecifies whether the upper or lower triangular part of the array a is used.\nIf uplo = CblasUpper, then the upper triangular of the array a is used.\nIf uplo = CblasLower, then the low triangular of the array a is used.\nn\nSpecifies the order of the matrix A. The value of n must be at least zero.\nalpha\nSpecifies the scalar alpha.\na\nArray, size lda*n.\nBefore entry with uplo = CblasUpper, the leading n-by-n upper triangular\npart of the array a must contain the upper triangular part of the Hermitian\nmatrix and the strictly lower triangular part of a is not referenced. Before\nentry with uplo = CblasLower, the leading n-by-n lower triangular part of\nthe array a must contain the lower triangular part of the Hermitian matrix\nand the strictly upper triangular part of a is not referenced.\nThe imaginary parts of the diagonal elements need not be set and are\nassumed to be zero.\nlda\nSpecifies the leading dimension of a as declared in the calling\n(sub)program. The value of lda must be at least max(1, n).\nx\nArray, size at least (1 + (n - 1)*abs(incx)). Before entry, the\nincremented array x must contain the n-element vector x.\nincx\nSpecifies the increment for the elements of x.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n66\n\n\nThe value of incx must not be zero.\nbeta\nSpecifies the scalar beta. When beta is supplied as zero then y need not be\nset on input.\ny\nArray, size at least (1 + (n - 1)*abs(incy)). Before entry, the\nincremented array y must contain the n-element vector y.\nincy\nSpecifies the increment for the elements of y.\nThe value of incy must not be zero.\nOutput Parameters\ny\nOverwritten by the updated vector y.\ncblas_?her\nPerforms a rank-1 update of a Hermitian matrix.\nSyntax\nvoid cblas_cher (const CBLAS_LAYOUT Layout, const CBLAS_UPLO uplo, const MKL_INT n,\nconst float alpha, const void *x, const MKL_INT incx, void *a, const MKL_INT lda);\nvoid cblas_zher (const CBLAS_LAYOUT Layout, const CBLAS_UPLO uplo, const MKL_INT n,\nconst double alpha, const void *x, const MKL_INT incx, void *a, const MKL_INT lda);\nInclude Files\n•\nmkl.h\nDescription\nThe ?her routines perform a matrix-vector operation defined as\nA := alpha*x*conjg(x') + A,\nwhere:\nalpha is a real scalar,\nx is an n-element vector,\nA is an n-by-n Hermitian matrix.\nInput Parameters\nLayout\nSpecifies whether two-dimensional array storage is row-major\n(CblasRowMajor) or column-major (CblasColMajor).\nuplo\nSpecifies whether the upper or lower triangular part of the array a is used.\nIf uplo = CblasUpper, then the upper triangular of the array a is used.\nIf uplo = CblasLower, then the low triangular of the array a is used.\nn\nSpecifies the order of the matrix A. The value of n must be at least zero.\nalpha\nSpecifies the scalar alpha.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n67\n\n\nx\nArray, dimension at least (1 + (n - 1)*abs(incx)). Before entry, the\nincremented array x must contain the n-element vector x.\nincx\nSpecifies the increment for the elements of x.\nThe value of incx must not be zero.\na\nArray, size lda*n.\nBefore entry with uplo = CblasUpper, the leading n-by-n upper triangular\npart of the array a must contain the upper triangular part of the Hermitian\nmatrix and the strictly lower triangular part of a is not referenced.\nBefore entry with uplo = CblasLower, the leading n-by-n lower triangular\npart of the array a must contain the lower triangular part of the Hermitian\nmatrix and the strictly upper triangular part of a is not referenced.\nThe imaginary parts of the diagonal elements need not be set and are\nassumed to be zero.\nlda\nSpecifies the leading dimension of a as declared in the calling\n(sub)program. The value of lda must be at least max(1, n).\nOutput Parameters\na\nWith uplo = CblasUpper, the upper triangular part of the array a is\noverwritten by the upper triangular part of the updated matrix.\nWith uplo = CblasLower, the lower triangular part of the array a is\noverwritten by the lower triangular part of the updated matrix.\nIf alpha is zero, matrix A is unchanged; otherwise, the imaginary parts of\nthe diagonal elements are set to zero.\ncblas_?her2\nPerforms a rank-2 update of a Hermitian matrix.\nSyntax\nvoid cblas_cher2 (const CBLAS_LAYOUT Layout, const CBLAS_UPLO uplo, const MKL_INT n,\nconst void *alpha, const void *x, const MKL_INT incx, const void *y, const MKL_INT\nincy, void *a, const MKL_INT lda);\nvoid cblas_zher2 (const CBLAS_LAYOUT Layout, const CBLAS_UPLO uplo, const MKL_INT n,\nconst void *alpha, const void *x, const MKL_INT incx, const void *y, const MKL_INT\nincy, void *a, const MKL_INT lda);\nInclude Files\n•\nmkl.h\nDescription\nThe ?her2 routines perform a matrix-vector operation defined as\nA := alpha *x*conjg(y') + conjg(alpha)*y *conjg(x') + A,\nwhere:\nalpha is scalar,\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n68\n\n\nx and y are n-element vectors,\nA is an n-by-n Hermitian matrix.\nInput Parameters\nLayout\nSpecifies whether two-dimensional array storage is row-major\n(CblasRowMajor) or column-major (CblasColMajor).\nuplo\nSpecifies whether the upper or lower triangular part of the array a is used.\nIf uplo = CblasUpper, then the upper triangular of the array a is used.\nIf uplo = CblasLower, then the low triangular of the array a is used.\nn\nSpecifies the order of the matrix A. The value of n must be at least zero.\nalpha\nSpecifies the scalar alpha.\nx\nArray, size at least (1 + (n - 1)*abs(incx)). Before entry, the\nincremented array x must contain the n-element vector x.\nincx\nSpecifies the increment for the elements of x.\nThe value of incx must not be zero.\ny\nArray, size at least (1 + (n - 1)*abs(incy)). Before entry, the\nincremented array y must contain the n-element vector y.\nincy\nSpecifies the increment for the elements of y.\nThe value of incy must not be zero.\na\nArray, size lda*n.\nBefore entry with uplo = CblasUpper, the leading n-by-n upper triangular\npart of the array a must contain the upper triangular part of the Hermitian\nmatrix and the strictly lower triangular part of a is not referenced.\nBefore entry with uplo = CblasLower, the leading n-by-n lower triangular\npart of the array a must contain the lower triangular part of the Hermitian\nmatrix and the strictly upper triangular part of a is not referenced.\nThe imaginary parts of the diagonal elements need not be set and are\nassumed to be zero.\nlda\nSpecifies the leading dimension of a as declared in the calling\n(sub)program. The value of lda must be at least max(1, n).\nOutput Parameters\na\nWith uplo = CblasUpper, the upper triangular part of the array a is\noverwritten by the upper triangular part of the updated matrix.\nWith uplo = CblasLower, the lower triangular part of the array a is\noverwritten by the lower triangular part of the updated matrix.\nIf alpha is zero, matrix A is unchanged; otherwise, the imaginary parts of\nthe diagonal elements are set to zero.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n69\n\n\ncblas_?hpmv\nComputes a matrix-vector product using a Hermitian\npacked matrix.\nSyntax\nvoid cblas_chpmv (const CBLAS_LAYOUT Layout, const CBLAS_UPLO uplo, const MKL_INT n,\nconst void *alpha, const void *ap, const void *x, const MKL_INT incx, const void *beta,\nvoid *y, const MKL_INT incy);\nvoid cblas_zhpmv (const CBLAS_LAYOUT Layout, const CBLAS_UPLO uplo, const MKL_INT n,\nconst void *alpha, const void *ap, const void *x, const MKL_INT incx, const void *beta,\nvoid *y, const MKL_INT incy);\nInclude Files\n•\nmkl.h\nDescription\nThe ?hpmv routines perform a matrix-vector operation defined as\ny := alpha*A*x + beta*y,\nwhere:\nalpha and beta are scalars,\nx and y are n-element vectors,\nA is an n-by-n Hermitian matrix, supplied in packed form.\nInput Parameters\nLayout\nSpecifies whether two-dimensional array storage is row-major\n(CblasRowMajor) or column-major (CblasColMajor).\nuplo\nSpecifies whether the upper or lower triangular part of the matrix A is\nsupplied in the packed array ap.\nIf uplo = CblasUpper, then the upper triangular part of the matrix A is\nsupplied in the packed array ap .\nIf uplo = CblasLower, then the low triangular part of the matrix A is\nsupplied in the packed array ap .\nn\nSpecifies the order of the matrix A. The value of n must be at least zero.\nalpha\nSpecifies the scalar alpha.\nap\nArray, size at least ((n*(n + 1))/2).\nFor Layout = CblasColMajor:\nBefore entry with uplo = CblasUpper, the array ap must contain the upper\ntriangular part of the Hermitian matrix packed sequentially, column-by-\ncolumn, so that ap[0] contains A1, 1, ap[1] and ap[2] contain A1, 2 and\nA2, 2 respectively, and so on. Before entry with uplo = CblasLower, the\narray ap must contain the lower triangular part of the Hermitian matrix\npacked sequentially, column-by-column, so that ap[0] contains A1, 1,\nap[1] and ap[2] contain A2, 1 and A3, 1 respectively, and so on.\nFor Layout = CblasRowMajor:\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n70\n\n\nBefore entry with uplo = CblasUpper, the array ap must contain the upper\ntriangular part of the Hermitian matrix packed sequentially, row-by-row,\nap[0] contains A1, 1, ap[1] and ap[2] contain A1, 2 and A1, 3 respectively,\nand so on. Before entry with uplo = CblasLower, the array ap must\ncontain the lower triangular part of the Hermitian matrix packed\nsequentially, row-by-row, so that ap[0] contains A1, 1, ap[1] and ap[2]\ncontain A2, 1 and A2, 2 respectively, and so on.\nThe imaginary parts of the diagonal elements need not be set and are\nassumed to be zero.\nx\nArray, size at least (1 +(n - 1)*abs(incx)). Before entry, the\nincremented array x must contain the n-element vector x.\nincx\nSpecifies the increment for the elements of x.\nThe value of incx must not be zero.\nbeta\nSpecifies the scalar beta.\nWhen beta is equal to zero then y need not be set on input.\ny\nArray, size at least (1 + (n - 1)*abs(incy)). Before entry, the\nincremented array y must contain the n-element vector y.\nincy\nSpecifies the increment for the elements of y.\nThe value of incy must not be zero.\nOutput Parameters\ny\nOverwritten by the updated vector y.\ncblas_?hpr\nPerforms a rank-1 update of a Hermitian packed\nmatrix.\nSyntax\nvoid cblas_chpr (const CBLAS_LAYOUT Layout, const CBLAS_UPLO uplo, const MKL_INT n,\nconst float alpha, const void *x, const MKL_INT incx, void *ap);\nvoid cblas_zhpr (const CBLAS_LAYOUT Layout, const CBLAS_UPLO uplo, const MKL_INT n,\nconst double alpha, const void *x, const MKL_INT incx, void *ap);\nInclude Files\n•\nmkl.h\nDescription\nThe ?hpr routines perform a matrix-vector operation defined as\nA := alpha*x*conjg(x') + A,\nwhere:\nalpha is a real scalar,\nx is an n-element vector,\nA is an n-by-n Hermitian matrix, supplied in packed form.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n71\n\n\nInput Parameters\nLayout\nSpecifies whether two-dimensional array storage is row-major\n(CblasRowMajor) or column-major (CblasColMajor).\nuplo\nSpecifies whether the upper or lower triangular part of the matrix A is\nsupplied in the packed array ap.\nIf uplo = CblasUpper, the upper triangular part of the matrix A is supplied\nin the packed array ap .\nIf uplo = CblasLower, the low triangular part of the matrix A is supplied in\nthe packed array ap .\nn\nSpecifies the order of the matrix A. The value of n must be at least zero.\nalpha\nSpecifies the scalar alpha.\nx\nArray, size at least (1 + (n - 1)*abs(incx)). Before entry, the\nincremented array x must contain the n-element vector x.\nincx\nSpecifies the increment for the elements of x. incx must not be zero.\nap\nArray, size at least ((n*(n + 1))/2).\nFor Layout = CblasColMajor:\nBefore entry with uplo = CblasUpper, the array ap must contain the upper\ntriangular part of the Hermitian matrix packed sequentially, column-by-\ncolumn, so that ap[0] contains A1, 1, ap[1] and ap[2] contain A1, 2 and\nA2, 2 respectively, and so on.\nBefore entry with uplo = CblasLower, the array ap must contain the lower\ntriangular part of the Hermitian matrix packed sequentially, column-by-\ncolumn, so that ap[0] contains A1, 1, ap[1] and ap[2] contain A2, 1 and\nA3, 1 respectively, and so on.\nFor Layout = CblasRowMajor:\nBefore entry with uplo = CblasUpper, the array ap must contain the upper\ntriangular part of the Hermitian matrix packed sequentially, row-by-row,\nap[0] contains A1, 1, ap[1] and ap[2] contain A1, 2 and A1, 3 respectively,\nand so on.\nBefore entry with uplo = CblasLower, the array ap must contain the lower\ntriangular part of the Hermitian matrix packed sequentially, row-by-row, so\nthat ap[0] contains A1, 1, ap[1] and ap[2] contain A2, 1 and A2, 2\nrespectively, and so on.\nThe imaginary parts of the diagonal elements need not be set and are\nassumed to be zero.\nOutput Parameters\nap\nWith uplo = CblasUpper, overwritten by the upper triangular part of the\nupdated matrix.\nWith uplo = CblasLower, overwritten by the lower triangular part of the\nupdated matrix.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n72\n\n\nIf alpha is zero, matrix A is unchanged; otherwise, the imaginary parts of\nthe diagonal elements are set to zero.\ncblas_?hpr2\nPerforms a rank-2 update of a Hermitian packed\nmatrix.\nSyntax\nvoid cblas_chpr2 (const CBLAS_LAYOUT Layout, const CBLAS_UPLO uplo, const MKL_INT n,\nconst void *alpha, const void *x, const MKL_INT incx, const void *y, const MKL_INT\nincy, void *ap);\nvoid cblas_zhpr2 (const CBLAS_LAYOUT Layout, const CBLAS_UPLO uplo, const MKL_INT n,\nconst void *alpha, const void *x, const MKL_INT incx, const void *y, const MKL_INT\nincy, void *ap);\nInclude Files\n•\nmkl.h\nDescription\nThe ?hpr2 routines perform a matrix-vector operation defined as\nA := alpha*x*conjg(y') + conjg(alpha)*y*conjg(x') + A,\nwhere:\nalpha is a scalar,\nx and y are n-element vectors,\nA is an n-by-n Hermitian matrix, supplied in packed form.\nInput Parameters\nLayout\nSpecifies whether two-dimensional array storage is row-major\n(CblasRowMajor) or column-major (CblasColMajor).\nuplo\nSpecifies whether the upper or lower triangular part of the matrix A is\nsupplied in the packed array ap.\nIf uplo = CblasUpper, then the upper triangular part of the matrix A is\nsupplied in the packed array ap .\nIf uplo = CblasLower, then the low triangular part of the matrix A is\nsupplied in the packed array ap .\nn\nSpecifies the order of the matrix A. The value of n must be at least zero.\nalpha\nSpecifies the scalar alpha.\nx\nArray, dimension at least (1 +(n - 1)*abs(incx)). Before entry, the\nincremented array x must contain the n-element vector x.\nincx\nSpecifies the increment for the elements of x.\nThe value of incx must not be zero.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n73\n\n\ny\nArray, size at least (1 +(n - 1)*abs(incy)). Before entry, the\nincremented array y must contain the n-element vector y.\nincy\nSpecifies the increment for the elements of y.\nThe value of incy must not be zero.\nap\nArray, size at least ((n*(n + 1))/2).\nFor Layout = CblasColMajor:\nBefore entry with uplo = CblasUpper, the array ap must contain the upper\ntriangular part of the Hermitian matrix packed sequentially, column-by-\ncolumn, so that ap[0] contains A1, 1, ap[1] and ap[2] contain A1, 2 and\nA2, 2 respectively, and so on.\nBefore entry with uplo = CblasLower, the array ap must contain the lower\ntriangular part of the Hermitian matrix packed sequentially, column-by-\ncolumn, so that ap[0] contains A1, 1, ap[1] and ap[2] contain A2, 1 and\nA3, 1 respectively, and so on.\nFor Layout = CblasRowMajor:\nBefore entry with uplo = CblasUpper, the array ap must contain the upper\ntriangular part of the Hermitian matrix packed sequentially, row-by-row,\nap[0] contains A1, 1, ap[1] and ap[2] contain A1, 2 and A1, 3 respectively,\nand so on.\nBefore entry with uplo = CblasLower, the array ap must contain the lower\ntriangular part of the Hermitian matrix packed sequentially, row-by-row, so\nthat ap[0] contains A1, 1, ap[1] and ap[2] contain A2, 1 and A2, 2\nrespectively, and so on.\nThe imaginary parts of the diagonal elements need not be set and are\nassumed to be zero.\nOutput Parameters\nap\nWith uplo = CblasUpper, overwritten by the upper triangular part of the\nupdated matrix.\nWith uplo = CblasLower, overwritten by the lower triangular part of the\nupdated matrix.\nIf alpha is zero, matrix A is unchanged; otherwise, the imaginary parts of\nthe diagonal elements need are set to zero.\ncblas_?sbmv\nComputes a matrix-vector product with a symmetric\nband matrix.\nSyntax\nvoid cblas_ssbmv (const CBLAS_LAYOUT Layout, const CBLAS_UPLO uplo, const MKL_INT n,\nconst MKL_INT k, const float alpha, const float *a, const MKL_INT lda, const float *x,\nconst MKL_INT incx, const float beta, float *y, const MKL_INT incy);\nvoid cblas_dsbmv (const CBLAS_LAYOUT Layout, const CBLAS_UPLO uplo, const MKL_INT n,\nconst MKL_INT k, const double alpha, const double *a, const MKL_INT lda, const double\n*x, const MKL_INT incx, const double beta, double *y, const MKL_INT incy);\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n74\n\n\nInclude Files\n•\nmkl.h\nDescription\nThe ?sbmv routines perform a matrix-vector operation defined as\ny := alpha*A*x + beta*y,\nwhere:\nalpha and beta are scalars,\nx and y are n-element vectors,\nA is an n-by-n symmetric band matrix, with k super-diagonals.\nInput Parameters\nLayout\nSpecifies whether two-dimensional array storage is row-major\n(CblasRowMajor) or column-major (CblasColMajor).\nuplo\nSpecifies whether the upper or lower triangular part of the band matrix A is\nused:\nif uplo = CblasUpper - upper triangular part;\nif uplo = CblasLower - low triangular part.\nn\nSpecifies the order of the matrix A. The value of n must be at least zero.\nk\nSpecifies the number of super-diagonals of the matrix A.\nThe value of k must satisfy 0≤k.\nalpha\nSpecifies the scalar alpha.\na\nArray, size lda*n. Before entry with uplo = CblasUpper, the leading (k +\n1) by n part of the array a must contain the upper triangular band part of\nthe symmetric matrix, supplied column-by-column, with the leading\ndiagonal of the matrix in row k of the array, the first super-diagonal starting\nat position 1 in row (k - 1), and so on. The top left k by k triangle of the\narray a is not referenced.\nThe following program segment transfers the upper triangular part of a\nsymmetric band matrix from conventional full matrix storage (matrix, with\nleading dimension ldm) to band storage (a, with leading dimension lda):\nfor (j = 0; j < n; j++) {\n    m = k - j;\n    for (i = max( 0, j - k); i <= j; i++) {\n        a[(m+i) + j*lda] = matrix[i + j*ldm];\n    }\n}\nBefore entry with uplo = CblasLower, the leading (k + 1) by n part of\nthe array a must contain the lower triangular band part of the symmetric\nmatrix, supplied column-by-column, with the leading diagonal of the matrix\nin row 0 of the array, the first sub-diagonal starting at position 0 in row 1,\nand so on. The bottom right k by k triangle of the array a is not referenced.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n75\n\n\nThe following program segment transfers the lower triangular part of a\nsymmetric band matrix from conventional full matrix storage (matrix, with\nleading dimension ldm) to band storage (a, with leading dimension lda):\nfor (j = 0; j < n; j++) {\n    m = -j;\n    for (i = j; i < min(n, j + k + 1); i++) {\n        a[(m+i) + j*lda] = matrix[i + j*ldm];\n    }\n}\nLayout = CblasRowMajor:\nBefore entry with uplo = CblasUpper, the leading (k + 1)-by-n part of\narray a must contain the upper triangular band part of the symmetric\nmatrix. The matrix must be supplied row-by-row, with the leading diagonal\nof the matrix in column 0 of the array, the first super-diagonal starting at\nposition 0 in column 1, and so on. The bottom right k-by-k triangle of array\na is not referenced.\nThe following program segment transfers the upper triangular part of a\nsymmetric band matrix from row-major full matrix storage (matrix with\nleading dimension ldm) to row-major band storage (a, with leading\ndimension lda):\nfor (i = 0; i < n; i++) {\n    m = -i;\n    for (j = i; j < MIN(n, i+k+1); j++) {\n        a[(m+j) + i*lda] = matrix[j + i*ldm];\n    }\n}\nBefore entry with uplo = CblasLower, the leading (k + 1)-by-n part of\narray a must contain the lower triangular band part of the symmetric\nmatrix, supplied row-by-row, with the leading diagonal of the matrix in\ncolumn k of the array, the first sub-diagonal starting at position 1 in column\nk-1, and so on. The top left k-by-k triangle of array a is not referenced.\nThe following program segment transfers the lower triangular part of a\nsymmetric row-major band matrix from row-major full matrix storage\n(matrix, with leading dimension ldm) to row-major band storage (a, with\nleading dimension lda):\nfor (i = 0; i < n; i++) {\n    m = k - i;\n    for (j = max(0, i-k); j <= i; j++) {\n         a[(m+j) + i*lda] = matrix[j + i*ldm];\n    }\n}\nlda\nSpecifies the leading dimension of a as declared in the calling\n(sub)program. The value of lda must be at least (k + 1).\nx\nArray, size at least (1 + (n - 1)*abs(incx)). Before entry, the\nincremented array x must contain the vector x.\nincx\nSpecifies the increment for the elements of x.\nThe value of incx must not be zero.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n76\n\n\nbeta\nSpecifies the scalar beta.\ny\nArray, size at least (1 + (n - 1)*abs(incy)). Before entry, the\nincremented array y must contain the vector y.\nincy\nSpecifies the increment for the elements of y.\nThe value of incy must not be zero.\nOutput Parameters\ny\nOverwritten by the updated vector y.\ncblas_?spmv\nComputes a matrix-vector product with a symmetric\npacked matrix.\nSyntax\nvoid cblas_sspmv (const CBLAS_LAYOUT Layout, const CBLAS_UPLO uplo, const MKL_INT n,\nconst float alpha, const float *ap, const float *x, const MKL_INT incx, const float\nbeta, float *y, const MKL_INT incy);\nvoid cblas_dspmv (const CBLAS_LAYOUT Layout, const CBLAS_UPLO uplo, const MKL_INT n,\nconst double alpha, const double *ap, const double *x, const MKL_INT incx, const double\nbeta, double *y, const MKL_INT incy);\nInclude Files\n•\nmkl.h\nDescription\nThe ?spmv routines perform a matrix-vector operation defined as\ny := alpha*A*x + beta*y,\nwhere:\nalpha and beta are scalars,\nx and y are n-element vectors,\nA is an n-by-n symmetric matrix, supplied in packed form.\nInput Parameters\nLayout\nSpecifies whether two-dimensional array storage is row-major\n(CblasRowMajor) or column-major (CblasColMajor).\nuplo\nSpecifies whether the upper or lower triangular part of the matrix A is\nsupplied in the packed array ap.\nIf uplo = CblasUpper, then the upper triangular part of the matrix A is\nsupplied in the packed array ap .\nIf uplo = CblasLower, then the low triangular part of the matrix A is\nsupplied in the packed array ap .\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n77\n\n\nn\nSpecifies the order of the matrix A. The value of n must be at least zero.\nalpha\nSpecifies the scalar alpha.\nap\nArray, size at least ((n*(n + 1))/2).\nFor Layout = CblasColMajor:\nBefore entry with uplo = CblasUpper, the array ap must contain the upper\ntriangular part of the symmetric matrix packed sequentially, column-by-\ncolumn, so that ap[0] contains A1, 1, ap[1] and ap[2] contain A1, 2 and\nA2, 2 respectively, and so on. Before entry with uplo = CblasLower, the\narray ap must contain the lower triangular part of the symmetric matrix\npacked sequentially, column-by-column, so that ap[0] contains A1, 1,\nap[1] and ap[2] contain A2, 1 and A3, 1 respectively, and so on.\nFor Layout = CblasRowMajor:\nBefore entry with uplo = CblasUpper, the array ap must contain the upper\ntriangular part of the symmetric matrix packed sequentially, row-by-row,\nap[0] contains A1, 1, ap[1] and ap[2] contain A1, 2 and A1, 3 respectively,\nand so on. Before entry with uplo = CblasLower, the array ap must\ncontain the lower triangular part of the symmetric matrix packed\nsequentially, row-by-row, so that ap[0] contains A1, 1, ap[1] and ap[2]\ncontain A2, 1 and A2, 2 respectively, and so on.\nx\nArray, size at least (1 + (n - 1)*abs(incx)). Before entry, the\nincremented array x must contain the n-element vector x.\nincx\nSpecifies the increment for the elements of x.\nThe value of incx must not be zero.\nbeta\nSpecifies the scalar beta.\nWhen beta is supplied as zero, then y need not be set on input.\ny\nArray, size at least (1 + (n - 1)*abs(incy)). Before entry, the\nincremented array y must contain the n-element vector y.\nincy\nSpecifies the increment for the elements of y.\nThe value of incy must not be zero.\nOutput Parameters\ny\nOverwritten by the updated vector y.\ncblas_?spr\nPerforms a rank-1 update of a symmetric packed\nmatrix.\nSyntax\nvoid cblas_sspr (const CBLAS_LAYOUT Layout, const CBLAS_UPLO uplo, const MKL_INT n,\nconst float alpha, const float *x, const MKL_INT incx, float *ap);\nvoid cblas_dspr (const CBLAS_LAYOUT Layout, const CBLAS_UPLO uplo, const MKL_INT n,\nconst double alpha, const double *x, const MKL_INT incx, double *ap);\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n78\n\n\nInclude Files\n•\nmkl.h\nDescription\nThe ?spr routines perform a matrix-vector operation defined as\na:= alpha*x*x'+ A,\nwhere:\nalpha is a real scalar,\nx is an n-element vector,\nA is an n-by-n symmetric matrix, supplied in packed form.\nInput Parameters\nLayout\nSpecifies whether two-dimensional array storage is row-major\n(CblasRowMajor) or column-major (CblasColMajor).\nuplo\nSpecifies whether the upper or lower triangular part of the matrix A is\nsupplied in the packed array ap.\nIf uplo = CblasUpper, then the upper triangular part of the matrix A is\nsupplied in the packed array ap .\nIf uplo = CblasLower, then the low triangular part of the matrix A is\nsupplied in the packed array ap .\nn\nSpecifies the order of the matrix A. The value of n must be at least zero.\nalpha\nSpecifies the scalar alpha.\nx\nArray, size at least (1 + (n - 1)*abs(incx)). Before entry, the\nincremented array x must contain the n-element vector x.\nincx\nSpecifies the increment for the elements of x.\nThe value of incx must not be zero.\nap\nFor Layout = CblasColMajor:\nBefore entry with uplo = CblasUpper, the array ap must contain the upper\ntriangular part of the symmetric matrix packed sequentially, column-by-\ncolumn, so that ap[0] contains A1, 1, ap[1] and ap[2] contain A1, 2 and\nA2, 2 respectively, and so on.\nBefore entry with uplo = CblasLower, the array ap must contain the lower\ntriangular part of the symmetric matrix packed sequentially, column-by-\ncolumn, so that ap[0] contains A1, 1, ap[1] and ap[2] contain A2, 1 and\nA3, 1 respectively, and so on.\nFor Layout = CblasRowMajor:\nBefore entry with uplo = CblasUpper, the array ap must contain the upper\ntriangular part of the symmetric matrix packed sequentially, row-by-row,\nap[0] contains A1, 1, ap[1] and ap[2] contain A1, 2 and A1, 3 respectively,\nand so on.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n79\n\n\nBefore entry with uplo = CblasLower, the array ap must contain the lower\ntriangular part of the symmetric matrix packed sequentially, row-by-row, so\nthat ap[0] contains A1, 1, ap[1] and ap[2] contain A2, 1 and A2, 2\nrespectively, and so on.\nOutput Parameters\nap\nWith uplo = CblasUpper, overwritten by the upper triangular part of the\nupdated matrix.\nWith uplo = CblasLower, overwritten by the lower triangular part of the\nupdated matrix.\ncblas_?spr2\nComputes a rank-2 update of a symmetric packed\nmatrix.\nSyntax\nvoid cblas_sspr2 (const CBLAS_LAYOUT Layout, const CBLAS_UPLO uplo, const MKL_INT n,\nconst float alpha, const float *x, const MKL_INT incx, const float *y, const MKL_INT\nincy, float *ap);\nvoid cblas_dspr2 (const CBLAS_LAYOUT Layout, const CBLAS_UPLO uplo, const MKL_INT n,\nconst double alpha, const double *x, const MKL_INT incx, const double *y, const MKL_INT\nincy, double *ap);\nInclude Files\n•\nmkl.h\nDescription\nThe ?spr2 routines perform a matrix-vector operation defined as\nA:= alpha*x*y'+ alpha*y*x' + A,\nwhere:\nalpha is a scalar,\nx and y are n-element vectors,\nA is an n-by-n symmetric matrix, supplied in packed form.\nInput Parameters\nLayout\nSpecifies whether two-dimensional array storage is row-major\n(CblasRowMajor) or column-major (CblasColMajor).\nuplo\nSpecifies whether the upper or lower triangular part of the matrix A is\nsupplied in the packed array ap.\nIf uplo = CblasUpper, then the upper triangular part of the matrix A is\nsupplied in the packed array ap .\nIf uplo = CblasLower, then the low triangular part of the matrix A is\nsupplied in the packed array ap .\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n80\n\n\nn\nSpecifies the order of the matrix A. The value of n must be at least zero.\nalpha\nSpecifies the scalar alpha.\nx\nArray, size at least (1 + (n - 1)*abs(incx)). Before entry, the\nincremented array x must contain the n-element vector x.\nincx\nSpecifies the increment for the elements of x.\nThe value of incx must not be zero.\ny\nArray, size at least (1 + (n - 1)*abs(incy)). Before entry, the\nincremented array y must contain the n-element vector y.\nincy\nSpecifies the increment for the elements of y. The value of incy must not be\nzero.\nap\nFor Layout = CblasColMajor:\nBefore entry with uplo = CblasUpper, the array ap must contain the upper\ntriangular part of the symmetric matrix packed sequentially, column-by-\ncolumn, so that ap[0] contains A1, 1, ap[1] and ap[2] contain A1, 2 and\nA2, 2 respectively, and so on.\nBefore entry with uplo = CblasLower, the array ap must contain the lower\ntriangular part of the symmetric matrix packed sequentially, column-by-\ncolumn, so that ap[0] contains A1, 1, ap[1] and ap[2] contain A2, 1 and\nA3, 1 respectively, and so on.\nFor Layout = CblasRowMajor:\nBefore entry with uplo = CblasUpper, the array ap must contain the upper\ntriangular part of the symmetric matrix packed sequentially, row-by-row,\nap[0] contains A1, 1, ap[1] and ap[2] contain A1, 2 and A1, 3 respectively,\nand so on.\nBefore entry with uplo = CblasLower, the array ap must contain the lower\ntriangular part of the symmetric matrix packed sequentially, row-by-row, so\nthat ap[0] contains A1, 1, ap[1] and ap[2] contain A2, 1 and A2, 2\nrespectively, and so on.\nOutput Parameters\nap\nWith uplo = CblasUpper, overwritten by the upper triangular part of the\nupdated matrix.\nWith uplo = CblasLower, overwritten by the lower triangular part of the\nupdated matrix.\ncblas_?symv\nComputes a matrix-vector product for a symmetric\nmatrix.\nSyntax\nvoid cblas_ssymv (const CBLAS_LAYOUT Layout, const CBLAS_UPLO uplo, const MKL_INT n,\nconst float alpha, const float *a, const MKL_INT lda, const float *x, const MKL_INT\nincx, const float beta, float *y, const MKL_INT incy);\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n81\n\n\nvoid cblas_dsymv (const CBLAS_LAYOUT Layout, const CBLAS_UPLO uplo, const MKL_INT n,\nconst double alpha, const double *a, const MKL_INT lda, const double *x, const MKL_INT\nincx, const double beta, double *y, const MKL_INT incy);\nInclude Files\n•\nmkl.h\nDescription\nThe ?symv routines perform a matrix-vector operation defined as\ny := alpha*A*x + beta*y,\nwhere:\nalpha and beta are scalars,\nx and y are n-element vectors,\nA is an n-by-n symmetric matrix.\nInput Parameters\nLayout\nSpecifies whether two-dimensional array storage is row-major\n(CblasRowMajor) or column-major (CblasColMajor).\nuplo\nSpecifies whether the upper or lower triangular part of the array a is used.\nIf uplo = CblasUpper, then the upper triangular part of the array a is\nused.\nIf uplo = CblasLower, then the low triangular part of the array a is used.\nn\nSpecifies the order of the matrix A. The value of n must be at least zero.\nalpha\nSpecifies the scalar alpha.\na\nArray, size lda*n.\nBefore entry with uplo = CblasUpper, the leading n-by-n upper triangular\npart of the array a must contain the upper triangular part of the symmetric\nmatrix A and the strictly lower triangular part of a is not referenced. Before\nentry with uplo = CblasLower, the leading n-by-n lower triangular part of\nthe array a must contain the lower triangular part of the symmetric matrix\nA and the strictly upper triangular part of a is not referenced.\nlda\nSpecifies the leading dimension of a as declared in the calling\n(sub)program. The value of lda must be at least max(1, n).\nx\nArray, size at least (1 + (n - 1)*abs(incx)). Before entry, the\nincremented array x must contain the n-element vector x.\nincx\nSpecifies the increment for the elements of x.\nThe value of incx must not be zero.\nbeta\nSpecifies the scalar beta.\nWhen beta is supplied as zero, then y need not be set on input.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n82\n\n\ny\nArray, size at least (1 + (n - 1)*abs(incy)). Before entry, the\nincremented array y must contain the n-element vector y.\nincy\nSpecifies the increment for the elements of y.\nThe value of incy must not be zero.\nOutput Parameters\ny\nOverwritten by the updated vector y.\ncblas_?syr\nPerforms a rank-1 update of a symmetric matrix.\nSyntax\nvoid cblas_ssyr (const CBLAS_LAYOUT Layout, const CBLAS_UPLO uplo, const MKL_INT n,\nconst float alpha, const float *x, const MKL_INT incx, float *a, const MKL_INT lda);\nvoid cblas_dsyr (const CBLAS_LAYOUT Layout, const CBLAS_UPLO uplo, const MKL_INT n,\nconst double alpha, const double *x, const MKL_INT incx, double *a, const MKL_INT lda);\nInclude Files\n•\nmkl.h\nDescription\nThe ?syr routines perform a matrix-vector operation defined as\nA := alpha*x*x' + A ,\nwhere:\nalpha is a real scalar,\nx is an n-element vector,\nA is an n-by-n symmetric matrix.\nInput Parameters\nLayout\nSpecifies whether two-dimensional array storage is row-major\n(CblasRowMajor) or column-major (CblasColMajor).\nuplo\nSpecifies whether the upper or lower triangular part of the array a is used.\nIf uplo = CblasUpper, then the upper triangular part of the array a is\nused.\nIf uplo = CblasLower, then the low triangular part of the array a is used.\nn\nSpecifies the order of the matrix A. The value of n must be at least zero.\nalpha\nSpecifies the scalar alpha.\nx\nArray, size at least (1 + (n-1)*abs(incx)). Before entry, the\nincremented array x must contain the n-element vector x.\nincx\nSpecifies the increment for the elements of x.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n83\n\n\nThe value of incx must not be zero.\na\nArray, size lda*n.\nBefore entry with uplo = CblasUpper, the leading n-by-n upper triangular\npart of the array a must contain the upper triangular part of the symmetric\nmatrix A and the strictly lower triangular part of a is not referenced.\nBefore entry with uplo = CblasLower, the leading n-by-n lower triangular\npart of the array a must contain the lower triangular part of the symmetric\nmatrix A and the strictly upper triangular part of a is not referenced.\nlda\nSpecifies the leading dimension of a as declared in the calling\n(sub)program. The value of lda must be at least max(1, n).\nOutput Parameters\na\nWith uplo = CblasUpper, the upper triangular part of the array a is\noverwritten by the upper triangular part of the updated matrix.\nWith uplo = CblasLower, the lower triangular part of the array a is\noverwritten by the lower triangular part of the updated matrix.\ncblas_?syr2\nPerforms a rank-2 update of a symmetric matrix.\nSyntax\nvoid cblas_ssyr2 (const CBLAS_LAYOUT Layout, const CBLAS_UPLO uplo, const MKL_INT n,\nconst float alpha, const float *x, const MKL_INT incx, const float *y, const MKL_INT\nincy, float *a, const MKL_INT lda);\nvoid cblas_dsyr2 (const CBLAS_LAYOUT Layout, const CBLAS_UPLO uplo, const MKL_INT n,\nconst double alpha, const double *x, const MKL_INT incx, const double *y, const MKL_INT\nincy, double *a, const MKL_INT lda);\nInclude Files\n•\nmkl.h\nDescription\nThe ?syr2 routines perform a matrix-vector operation defined as\nA := alpha*x*y'+ alpha*y*x' + A,\nwhere:\nalpha is scalar,\nx and y are n-element vectors,\nA is an n-by-n symmetric matrix.\nInput Parameters\nLayout\nSpecifies whether two-dimensional array storage is row-major\n(CblasRowMajor) or column-major (CblasColMajor).\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n84\n\n\nuplo\nSpecifies whether the upper or lower triangular part of the array a is used.\nIf uplo = CblasUpper, then the upper triangular part of the array a is\nused.\nIf uplo = CblasLower, then the low triangular part of the array a is used.\nn\nSpecifies the order of the matrix A. The value of n must be at least zero.\nalpha\nSpecifies the scalar alpha.\nx\nArray, size at least (1 + (n - 1)*abs(incx)). Before entry, the\nincremented array x must contain the n-element vector x.\nincx\nSpecifies the increment for the elements of x.\nThe value of incx must not be zero.\ny\nArray, size at least (1 + (n - 1)*abs(incy)). Before entry, the\nincremented array y must contain the n-element vector y.\nincy\nSpecifies the increment for the elements of y. The value of incy must not be\nzero.\na\nArray, size lda*n.\nBefore entry with uplo = CblasUpper, the leading n-by-n upper triangular\npart of the array a must contain the upper triangular part of the symmetric\nmatrix and the strictly lower triangular part of a is not referenced.\nBefore entry with uplo = CblasLower, the leading n-by-n lower triangular\npart of the array a must contain the lower triangular part of the symmetric\nmatrix and the strictly upper triangular part of a is not referenced.\nlda\nSpecifies the leading dimension of a as declared in the calling\n(sub)program. The value of lda must be at least max(1, n).\nOutput Parameters\na\nWith uplo = CblasUpper, the upper triangular part of the array a is\noverwritten by the upper triangular part of the updated matrix.\nWith uplo = CblasLower, the lower triangular part of the array a is\noverwritten by the lower triangular part of the updated matrix.\ncblas_?tbmv\nComputes a matrix-vector product using a triangular\nband matrix.\nSyntax\nvoid cblas_stbmv (const CBLAS_LAYOUT Layout, const CBLAS_UPLO uplo, const\nCBLAS_TRANSPOSE trans, const CBLAS_DIAG diag, const MKL_INT n, const MKL_INT k, const\nfloat *a, const MKL_INT lda, float *x, const MKL_INT incx);\nvoid cblas_dtbmv (const CBLAS_LAYOUT Layout, const CBLAS_UPLO uplo, const\nCBLAS_TRANSPOSE trans, const CBLAS_DIAG diag, const MKL_INT n, const MKL_INT k, const\ndouble *a, const MKL_INT lda, double *x, const MKL_INT incx);\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n85\n\n\nvoid cblas_ctbmv (const CBLAS_LAYOUT Layout, const CBLAS_UPLO uplo, const\nCBLAS_TRANSPOSE trans, const CBLAS_DIAG diag, const MKL_INT n, const MKL_INT k, const\nvoid *a, const MKL_INT lda, void *x, const MKL_INT incx);\nvoid cblas_ztbmv (const CBLAS_LAYOUT Layout, const CBLAS_UPLO uplo, const\nCBLAS_TRANSPOSE trans, const CBLAS_DIAG diag, const MKL_INT n, const MKL_INT k, const\nvoid *a, const MKL_INT lda, void *x, const MKL_INT incx);\nInclude Files\n•\nmkl.h\nDescription\nThe ?tbmv routines perform one of the matrix-vector operations defined as\nx := A*x, or x := A'*x, or x := conjg(A')*x,\nwhere:\nx is an n-element vector,\nA is an n-by-n unit, or non-unit, upper or lower triangular band matrix, with (k +1) diagonals.\nInput Parameters\nLayout\nSpecifies whether two-dimensional array storage is row-major\n(CblasRowMajor) or column-major (CblasColMajor).\nuplo\nSpecifies whether the matrix A is an upper or lower triangular matrix:\nuplo = CblasUpper\nif uplo = CblasLower, then the matrix is low triangular.\ntrans\nSpecifies the operation:\nif trans=CblasNoTrans, then x := A*x;\nif trans=CblasTrans, then x := A'*x;\nif trans=CblasConjTrans, then x := conjg(A')*x.\ndiag\nSpecifies whether the matrix A is unit triangular:\nif diag = CblasUnit then the matrix is unit triangular;\nif diag = CblasNonUnit , then the matrix is not unit triangular.\nn\nSpecifies the order of the matrix A. The value of n must be at least zero.\nk\nOn entry with uplo = CblasUpper specifies the number of super-diagonals\nof the matrix A. On entry with uplo = CblasLower, k specifies the number\nof sub-diagonals of the matrix a.\nThe value of k must satisfy 0≤k.\na\nArray, size lda*n.\nLayout = CblasColMajor:\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n86\n\n\nBefore entry with uplo = CblasUpper, the leading (k + 1) by n part of\nthe array a must contain the upper triangular band part of the matrix of\ncoefficients, supplied column-by-column, with the leading diagonal of the\nmatrix in row k of the array, the first super-diagonal starting at position 1 in\nrow (k - 1), and so on. The top left k by k triangle of the array a is not\nreferenced. The following program segment transfers an upper triangular\nband matrix from conventional full matrix storage (matrix, with leading\ndimension ldm) to band storage (a, with leading dimension lda):\nfor (j = 0; j < n; j++) {\n    m = k - j;\n    for (i = max( 0, j - k); i <= j; i++) {\n        a[(m+i) + j*lda] = matrix[i + j*ldm];\n    }\n}\nBefore entry with uplo = CblasLower, the leading (k + 1) by n part of\nthe array a must contain the lower triangular band part of the matrix of\ncoefficients, supplied column-by-column, with the leading diagonal of the\nmatrix in row 0 of the array, the first sub-diagonal starting at position 0 in\nrow 1, and so on. The bottom right k by k triangle of the array a is not\nreferenced. The following program segment transfers a lower triangular\nband matrix from conventional full matrix storage (matrix, with leading\ndimension ldm) to band storage (a, with leading dimension lda):\nfor (j = 0; j < n; j++) {\n    m = -j;\n    for (i = j; i < min(n, j + k + 1); i++) {\n        a[(m+i) + j*lda] = matrix[i + j*ldm];\n    }\n}\nNote that when diag = CblasUnit , the elements of the array a\ncorresponding to the diagonal elements of the matrix are not referenced,\nbut are assumed to be unity.\nLayout = CblasRowMajor:\nBefore entry with uplo = CblasUpper, the leading (k + 1)-by-n part of\narray a must contain the upper triangular band part of the matrix of\ncoefficients. The matrix must be supplied row-by-row, with the leading\ndiagonal of the matrix in column 0 of the array, the first super-diagonal\nstarting at position 0 in column 1, and so on. The bottom right k-by-k\ntriangle of array a is not referenced.\nThe following program segment transfers the upper triangular part of a\nHermitian band matrix from row-major full matrix storage (matrix with\nleading dimension ldm) to row-major band storage (a, with leading\ndimension lda):\nfor (i = 0; i < n; i++) {\n    m = -i;\n    for (j = i; j < MIN(n, i+k+1); j++) {\n        a[(m+j) + i*lda] = matrix[j + i*ldm];\n    }\n}\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n87\n\n\nBefore entry with uplo = CblasLower, the leading (k + 1)-by-n part of\narray a must contain the lower triangular band part of the matrix of\ncoefficients, supplied row-by-row, with the leading diagonal of the matrix in\ncolumn k of the array, the first sub-diagonal starting at position 1 in column\nk-1, and so on. The top left k-by-k triangle of array a is not referenced.\nThe following program segment transfers the lower triangular part of a\nHermitian row-major band matrix from row-major full matrix storage\n(matrix, with leading dimension ldm) to row-major band storage (a, with\nleading dimension lda):\nfor (i = 0; i < n; i++) {\n    m = k - i;\n    for (j = max(0, i-k); j <= i; j++) {\n         a[(m+j) + i*lda] = matrix[j + i*ldm];\n    }\n}\nlda\nSpecifies the leading dimension of a as declared in the calling\n(sub)program. The value of lda must be at least (k + 1).\nx\nArray, size at least (1 + (n - 1)*abs(incx)). Before entry, the\nincremented array x must contain the n-element vector x.\nincx\nSpecifies the increment for the elements of x.\nThe value of incx must not be zero.\nOutput Parameters\nx\nOverwritten with the transformed vector x.\ncblas_?tbsv\nSolves a system of linear equations whose coefficients\nare in a triangular band matrix.\nSyntax\nvoid cblas_stbsv (const CBLAS_LAYOUT Layout, const CBLAS_UPLO uplo, const\nCBLAS_TRANSPOSE trans, const CBLAS_DIAG diag, const MKL_INT n, const MKL_INT k, const\nfloat *a, const MKL_INT lda, float *x, const MKL_INT incx);\nvoid cblas_dtbsv (const CBLAS_LAYOUT Layout, const CBLAS_UPLO uplo, const\nCBLAS_TRANSPOSE trans, const CBLAS_DIAG diag, const MKL_INT n, const MKL_INT k, const\ndouble *a, const MKL_INT lda, double *x, const MKL_INT incx);\nvoid cblas_ctbsv (const CBLAS_LAYOUT Layout, const CBLAS_UPLO uplo, const\nCBLAS_TRANSPOSE trans, const CBLAS_DIAG diag, const MKL_INT n, const MKL_INT k, const\nvoid *a, const MKL_INT lda, void *x, const MKL_INT incx);\nvoid cblas_ztbsv (const CBLAS_LAYOUT Layout, const CBLAS_UPLO uplo, const\nCBLAS_TRANSPOSE trans, const CBLAS_DIAG diag, const MKL_INT n, const MKL_INT k, const\nvoid *a, const MKL_INT lda, void *x, const MKL_INT incx);\nInclude Files\n•\nmkl.h\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n88\n\n\nDescription\nThe ?tbsv routines solve one of the following systems of equations:\nA*x = b, or A'*x = b, or conjg(A')*x = b,\nwhere:\nb and x are n-element vectors,\nA is an n-by-n unit, or non-unit, upper or lower triangular band matrix, with (k + 1) diagonals.\nThe routine does not test for singularity or near-singularity.\nSuch tests must be performed before calling this routine.\nInput Parameters\nLayout\nSpecifies whether two-dimensional array storage is row-major\n(CblasRowMajor) or column-major (CblasColMajor).\nuplo\nSpecifies whether the matrix A is an upper or lower triangular matrix:\nif uplo = CblasUpper the matrix is upper triangular;\nif uplo = CblasLower, the matrix is low triangular.\ntrans\nSpecifies the system of equations:\nif trans=CblasNoTrans, then A*x = b;\nif trans=CblasTrans, then A'*x = b;\nif trans=CblasConjTrans, then conjg(A')*x = b.\ndiag\nSpecifies whether the matrix A is unit triangular:\nif diag = CblasUnit then the matrix is unit triangular;\nif diag = CblasNonUnit, then the matrix is not unit triangular.\nn\nSpecifies the order of the matrix A. The value of n must be at least zero.\nk\nOn entry with uplo = CblasUpper, k specifies the number of super-\ndiagonals of the matrix A. On entry with uplo = CblasLower, k specifies\nthe number of sub-diagonals of the matrix A.\nThe value of k must satisfy 0≤k.\na\nArray, size lda*n.\nLayout = CblasColMajor:\nBefore entry with uplo = CblasUpper, the leading (k + 1) by n part of\nthe array a must contain the upper triangular band part of the matrix of\ncoefficients, supplied column-by-column, with the leading diagonal of the\nmatrix in row k of the array, the first super-diagonal starting at position 1 in\nrow (k - 1), and so on. The top left k by k triangle of the array a is not\nreferenced.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n89\n\n\nThe following program segment transfers an upper triangular band matrix\nfrom conventional full matrix storage (matrix, with leading dimension ldm)\nto band storage (a, with leading dimension lda):\nfor (j = 0; j < n; j++) {\n    m = k - j;\n    for (i = max( 0, j - k); i <= j; i++) {\n        a[(m+i) + j*lda] = matrix[i + j*ldm];\n    }\n}\nBefore entry with uplo = CblasLower, the leading (k + 1) by n part of\nthe array a must contain the lower triangular band part of the matrix of\ncoefficients, supplied column-by-column, with the leading diagonal of the\nmatrix in row 0 of the array, the first sub-diagonal starting at position 0 in\nrow 1, and so on. The bottom right k by k triangle of the array a is not\nreferenced.\nThe following program segment transfers a lower triangular band matrix\nfrom conventional full matrix storage (matrix, with leading dimension ldm)\nto band storage (a, with leading dimension lda):\nfor (j = 0; j < n; j++) {\n    m = -j;\n    for (i = j; i < min(n, j + k + 1); i++) {\n        a[(m+i) + j*lda] = matrix[i + j*ldm];\n    }\n}\nWhen diag = CblasUnit, the elements of the array a corresponding to the\ndiagonal elements of the matrix are not referenced, but are assumed to be\nunity.\nLayout = CblasRowMajor:\nBefore entry with uplo = CblasUpper, the leading (k + 1)-by-n part of\narray a must contain the upper triangular band part of the matrix of\ncoefficients. The matrix must be supplied row-by-row, with the leading\ndiagonal of the matrix in column 0 of the array, the first super-diagonal\nstarting at position 0 in column 1, and so on. The bottom right k-by-k\ntriangle of array a is not referenced.\nThe following program segment transfers the upper triangular part of a\nHermitian band matrix from row-major full matrix storage (matrix with\nleading dimension ldm) to row-major band storage (a, with leading\ndimension lda):\nfor (i = 0; i < n; i++) {\n    m = -i;\n    for (j = i; j < MIN(n, i+k+1); j++) {\n        a[(m+j) + i*lda] = matrix[j + i*ldm];\n    }\n}\nBefore entry with uplo = CblasLower, the leading (k + 1)-by-n part of\narray a must contain the lower triangular band part of the matrix of\ncoefficients, supplied row-by-row, with the leading diagonal of the matrix in\ncolumn k of the array, the first sub-diagonal starting at position 1 in column\nk-1, and so on. The top left k-by-k triangle of array a is not referenced.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n90\n\n\nThe following program segment transfers the lower triangular part of a\nHermitian row-major band matrix from row-major full matrix storage\n(matrix, with leading dimension ldm) to row-major band storage (a, with\nleading dimension lda):\nfor (i = 0; i < n; i++) {\n    m = k - i;\n    for (j = max(0, i-k); j <= i; j++) {\n         a[(m+j) + i*lda] = matrix[j + i*ldm];\n    }\n}\nlda\nSpecifies the leading dimension of a as declared in the calling\n(sub)program. The value of lda must be at least (k + 1).\nx\nArray, size at least (1 + (n - 1)*abs(incx)). Before entry, the\nincremented array x must contain the n-element right-hand side vector b.\nincx\nSpecifies the increment for the elements of x.\nThe value of incx must not be zero.\nOutput Parameters\nx\nOverwritten with the solution vector x.\ncblas_?tpmv\nComputes a matrix-vector product using a triangular\npacked matrix.\nSyntax\nvoid cblas_stpmv (const CBLAS_LAYOUT Layout, const CBLAS_UPLO uplo, const\nCBLAS_TRANSPOSE trans, const CBLAS_DIAG diag, const MKL_INT n, const float *ap, float\n*x, const MKL_INT incx);\nvoid cblas_dtpmv (const CBLAS_LAYOUT Layout, const CBLAS_UPLO uplo, const\nCBLAS_TRANSPOSE trans, const CBLAS_DIAG diag, const MKL_INT n, const double *ap, double\n*x, const MKL_INT incx);\nvoid cblas_ctpmv (const CBLAS_LAYOUT Layout, const CBLAS_UPLO uplo, const\nCBLAS_TRANSPOSE trans, const CBLAS_DIAG diag, const MKL_INT n, const void *ap, void *x,\nconst MKL_INT incx);\nvoid cblas_ztpmv (const CBLAS_LAYOUT Layout, const CBLAS_UPLO uplo, const\nCBLAS_TRANSPOSE trans, const CBLAS_DIAG diag, const MKL_INT n, const void *ap, void *x,\nconst MKL_INT incx);\nInclude Files\n•\nmkl.h\nDescription\nThe ?tpmv routines perform one of the matrix-vector operations defined as\nx := A*x, or x := A'*x, or x := conjg(A')*x,\nwhere:\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n91\n\n\nx is an n-element vector,\nA is an n-by-n unit, or non-unit, upper or lower triangular matrix, supplied in packed form.\nInput Parameters\nLayout\nSpecifies whether two-dimensional array storage is row-major\n(CblasRowMajor) or column-major (CblasColMajor).\nuplo\nSpecifies whether the matrix A is upper or lower triangular:\nuplo = CblasUpper\nif uplo = CblasLower, then the matrix is low triangular.\ntrans\nSpecifies the operation:\nif trans=CblasNoTrans, then x := A*x;\nif trans=CblasTrans, then x := A'*x;\nif trans=CblasConjTrans, then x := conjg(A')*x.\ndiag\nSpecifies whether the matrix A is unit triangular:\nif diag = CblasUnit then the matrix is unit triangular;\nif diag = CblasNonUnit, then the matrix is not unit triangular.\nn\nSpecifies the order of the matrix A. The value of n must be at least zero.\nap\nArray, size at least ((n*(n + 1))/2).\nFor Layout = CblasColMajor:\nBefore entry with uplo = CblasUpper, the array ap must contain the upper\ntriangular matrix packed sequentially, column-by-column, so that\nrespectively, and so on. Before entry with uplo = CblasLowerap[0]\ncontains A1, 1, ap[1] and ap[2] contain A1, 2 and A2, 2, the array ap must\ncontain the lower triangular matrix packed sequentially, column-by-column,\nso thatap[0] contains A1, 1, ap[1] and ap[2] contain A2, 1 and A3, 1\nrespectively, and so on. When diag = CblasUnit, the diagonal elements of\na are not referenced, but are assumed to be unity.\nFor Layout = CblasRowMajor:\nBefore entry with uplo = CblasUpper, the array ap must contain the upper\ntriangular matrix packed sequentially, row-by-row, ap[0] contains A1, 1,\nap[1] and ap[2] contain A1, 2 and A1, 3 respectively, and so on.\nBefore entry with uplo = CblasLower, the array ap must contain the lower\ntriangular matrix packed sequentially, row-by-row, so that ap[0] contains\nA1, 1, ap[1] and ap[2] contain A2, 1 and A2, 2 respectively, and so on.\nx\nArray, size at least (1 + (n - 1)*abs(incx)). Before entry, the\nincremented array x must contain the n-element vector x.\nincx\nSpecifies the increment for the elements of x.\nThe value of incx must not be zero.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n92\n\n\nOutput Parameters\nx\nOverwritten with the transformed vector x.\ncblas_?tpsv\nSolves a system of linear equations whose coefficients\nare in a triangular packed matrix.\nSyntax\nvoid cblas_stpsv (const CBLAS_LAYOUT Layout, const CBLAS_UPLO uplo, const\nCBLAS_TRANSPOSE trans, const CBLAS_DIAG diag, const MKL_INT n, const float *ap, float\n*x, const MKL_INT incx);\nvoid cblas_dtpsv (const CBLAS_LAYOUT Layout, const CBLAS_UPLO uplo, const\nCBLAS_TRANSPOSE trans, const CBLAS_DIAG diag, const MKL_INT n, const double *ap, double\n*x, const MKL_INT incx);\nvoid cblas_ctpsv (const CBLAS_LAYOUT Layout, const CBLAS_UPLO uplo, const\nCBLAS_TRANSPOSE trans, const CBLAS_DIAG diag, const MKL_INT n, const void *ap, void *x,\nconst MKL_INT incx);\nvoid cblas_ztpsv (const CBLAS_LAYOUT Layout, const CBLAS_UPLO uplo, const\nCBLAS_TRANSPOSE trans, const CBLAS_DIAG diag, const MKL_INT n, const void *ap, void *x,\nconst MKL_INT incx);\nInclude Files\n•\nmkl.h\nDescription\nThe ?tpsv routines solve one of the following systems of equations\nA*x = b, or A'*x = b, or conjg(A')*x = b,\nwhere:\nb and x are n-element vectors,\nA is an n-by-n unit, or non-unit, upper or lower triangular matrix, supplied in packed form.\nThis routine does not test for singularity or near-singularity.\nSuch tests must be performed before calling this routine.\nInput Parameters\nLayout\nSpecifies whether two-dimensional array storage is row-major\n(CblasRowMajor) or column-major (CblasColMajor).\nuplo\nSpecifies whether the matrix A is upper or lower triangular:\nuplo = CblasUpper\nif uplo = CblasLower, then the matrix is low triangular.\ntrans\nSpecifies the system of equations:\nif trans=CblasNoTrans, then A*x = b;\nif trans=CblasTrans, then A'*x = b;\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n93\n\n\nif trans=CblasConjTrans, then conjg(A')*x = b.\ndiag\nSpecifies whether the matrix A is unit triangular:\nif diag = CblasUnit then the matrix is unit triangular;\nif diag = CblasNonUnit , then the matrix is not unit triangular.\nn\nSpecifies the order of the matrix A. The value of n must be at least zero.\nap\nArray, size at least ((n*(n + 1))/2).\nFor Layout = CblasColMajor:\nBefore entry with uplo = CblasUpper, the array ap must contain the upper\ntriangular part of the triangular matrix packed sequentially, column-by-\ncolumn, so that ap[0] contains A1, 1, ap[1] and ap[2] contain A1, 2 and\nA2, 2 respectively, and so on.\nBefore entry with uplo = CblasLower, the array ap must contain the lower\ntriangular part of the triangular matrix packed sequentially, column-by-\ncolumn, so that ap[0] contains A1, 1, ap[1] and ap[2] contain A2, 1 and\nA3, 1 respectively, and so on.\nFor Layout = CblasRowMajor:\nBefore entry with uplo = CblasUpper, the array ap must contain the upper\ntriangular part of the triangular matrix packed sequentially, row-by-row,\nap[0] contains A1, 1, ap[1] and ap[2] contain A1, 2 and A1, 3 respectively,\nand so on. Before entry with uplo = CblasLower, the array ap must\ncontain the lower triangular part of the triangular matrix packed\nsequentially, row-by-row, so that ap[0] contains A1, 1, ap[1] and ap[2]\ncontain A2, 1 and A2, 2 respectively, and so on.\nWhen diag = CblasUnit, the diagonal elements of a are not referenced,\nbut are assumed to be unity.\nx\nArray, size at least (1 + (n - 1)*abs(incx)). Before entry, the\nincremented array x must contain the n-element right-hand side vector b.\nincx\nSpecifies the increment for the elements of x.\nThe value of incx must not be zero.\nOutput Parameters\nx\nOverwritten with the solution vector x.\ncblas_?trmv\nComputes a matrix-vector product using a triangular\nmatrix.\nSyntax\nvoid cblas_strmv (const CBLAS_LAYOUT Layout, const CBLAS_UPLO uplo, const\nCBLAS_TRANSPOSE trans, const CBLAS_DIAG diag, const MKL_INT n, const float *a, const\nMKL_INT lda, float *x, const MKL_INT incx);\nvoid cblas_dtrmv (const CBLAS_LAYOUT Layout, const CBLAS_UPLO uplo, const\nCBLAS_TRANSPOSE trans, const CBLAS_DIAG diag, const MKL_INT n, const double *a, const\nMKL_INT lda, double *x, const MKL_INT incx);\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n94\n\n\nvoid cblas_ctrmv (const CBLAS_LAYOUT Layout, const CBLAS_UPLO uplo, const\nCBLAS_TRANSPOSE trans, const CBLAS_DIAG diag, const MKL_INT n, const void *a, const\nMKL_INT lda, void *x, const MKL_INT incx);\nvoid cblas_ztrmv (const CBLAS_LAYOUT Layout, const CBLAS_UPLO uplo, const\nCBLAS_TRANSPOSE trans, const CBLAS_DIAG diag, const MKL_INT n, const void *a, const\nMKL_INT lda, void *x, const MKL_INT incx);\nInclude Files\n•\nmkl.h\nDescription\nThe ?trmv routines perform one of the following matrix-vector operations defined as\nx := A*x, or x := A'*x, or x := conjg(A')*x,\nwhere:\nx is an n-element vector,\nA is an n-by-n unit, or non-unit, upper or lower triangular matrix.\nInput Parameters\nLayout\nSpecifies whether two-dimensional array storage is row-major\n(CblasRowMajor) or column-major (CblasColMajor).\nuplo\nSpecifies whether the matrix A is upper or lower triangular:\nuplo = CblasUpper\nif uplo = CblasLower, then the matrix is low triangular.\ntrans\nSpecifies the operation:\nif trans=CblasNoTrans, then x := A*x;\nif trans=CblasTrans, then x := A'*x;\nif trans=CblasConjTrans, then x := conjg(A')*x.\ndiag\nSpecifies whether the matrix A is unit triangular:\nif diag = CblasUnit then the matrix is unit triangular;\nif diag = CblasNonUnit , then the matrix is not unit triangular.\nn\nSpecifies the order of the matrix A. The value of n must be at least zero.\na\nArray, size lda*n. Before entry with uplo = CblasUpper, the leading n-by-\nn upper triangular part of the array a must contain the upper triangular\nmatrix and the strictly lower triangular part of a is not referenced. Before\nentry with uplo = CblasLower, the leading n-by-n lower triangular part of\nthe array a must contain the lower triangular matrix and the strictly upper\ntriangular part of a is not referenced.\nWhen diag = CblasUnit, the diagonal elements of a are not referenced\neither, but are assumed to be unity.\nlda\nSpecifies the leading dimension of a as declared in the calling\n(sub)program. The value of lda must be at least max(1, n).\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n95\n\n\nx\nArray, size at least (1 + (n - 1)*abs(incx)). Before entry, the\nincremented array x must contain the n-element vector x.\nincx\nSpecifies the increment for the elements of x.\nThe value of incx must not be zero.\nOutput Parameters\nx\nOverwritten with the transformed vector x.\ncblas_?trsv\nSolves a system of linear equations whose coefficients\nare in a triangular matrix.\nSyntax\nvoid cblas_strsv (const CBLAS_LAYOUT Layout, const CBLAS_UPLO uplo, const\nCBLAS_TRANSPOSE trans, const CBLAS_DIAG diag, const MKL_INT n, const float *a, const\nMKL_INT lda, float *x, const MKL_INT incx);\nvoid cblas_dtrsv (const CBLAS_LAYOUT Layout, const CBLAS_UPLO uplo, const\nCBLAS_TRANSPOSE trans, const CBLAS_DIAG diag, const MKL_INT n, const double *a, const\nMKL_INT lda, double *x, const MKL_INT incx);\nvoid cblas_ctrsv (const CBLAS_LAYOUT Layout, const CBLAS_UPLO uplo, const\nCBLAS_TRANSPOSE trans, const CBLAS_DIAG diag, const MKL_INT n, const void *a, const\nMKL_INT lda, void *x, const MKL_INT incx);\nvoid cblas_ztrsv (const CBLAS_LAYOUT Layout, const CBLAS_UPLO uplo, const\nCBLAS_TRANSPOSE trans, const CBLAS_DIAG diag, const MKL_INT n, const void *a, const\nMKL_INT lda, void *x, const MKL_INT incx);\nInclude Files\n•\nmkl.h\nDescription\nThe ?trsv routines solve one of the systems of equations:\nA*x = b, or A'*x = b, or conjg(A')*x = b,\nwhere:\nb and x are n-element vectors,\nA is an n-by-n unit, or non-unit, upper or lower triangular matrix.\nThe routine does not test for singularity or near-singularity.\nSuch tests must be performed before calling this routine.\nInput Parameters\nLayout\nSpecifies whether two-dimensional array storage is row-major\n(CblasRowMajor) or column-major (CblasColMajor).\nuplo\nSpecifies whether the matrix A is upper or lower triangular:\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n96\n\n\nuplo = CblasUpper\nif uplo = CblasLower, then the matrix is low triangular.\ntrans\nSpecifies the systems of equations:\nif trans=CblasNoTrans, then A*x = b;\nif trans=CblasTrans, then A'*x = b;\nif trans=CblasConjTrans, then oconjg(A')*x = b.\ndiag\nSpecifies whether the matrix A is unit triangular:\nif diag = CblasUnit then the matrix is unit triangular;\nif diag = CblasNonUnit, then the matrix is not unit triangular.\nn\nSpecifies the order of the matrix A. The value of n must be at least zero.\na\nArray, size lda*n . Before entry with uplo = CblasUpper, the leading n-\nby-n upper triangular part of the array a must contain the upper triangular\nmatrix and the strictly lower triangular part of a is not referenced. Before\nentry with uplo = CblasLower, the leading n-by-n lower triangular part of\nthe array a must contain the lower triangular matrix and the strictly upper\ntriangular part of a is not referenced.\nWhen diag = CblasUnit, the diagonal elements of a are not referenced\neither, but are assumed to be unity.\nlda\nSpecifies the leading dimension of a as declared in the calling\n(sub)program. The value of lda must be at least max(1, n).\nx\nArray, size at least (1 + (n - 1)*abs(incx)). Before entry, the\nincremented array x must contain the n-element right-hand side vector b.\nincx\nSpecifies the increment for the elements of x.\nThe value of incx must not be zero.\nOutput Parameters\nx\nOverwritten with the solution vector x.\nBLAS Level 3 Routines\nBLAS Level 3 routines perform matrix-matrix operations. The following table lists the BLAS Level 3 routine\ngroups and the data types associated with them.\nBLAS Level 3 Routine Groups and Their Data Types\nRoutine Group\nData Types\nDescription\ncblas_?gemm\ns, d, c, z\nComputes a matrix-matrix product with general matrices.\ncblas_?hemm\nc, z\nComputes a matrix-matrix product where one input matrix\nis Hermitian.\ncblas_?herk\nc, z\nPerforms a Hermitian rank-k update.\ncblas_?her2k\nc, z\nPerforms a Hermitian rank-2k update.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n97\n\n\nRoutine Group\nData Types\nDescription\ncblas_?symm\ns, d, c, z\nComputes a matrix-matrix product where one input matrix\nis symmetric.\ncblas_?syrk\ns, d, c, z\nPerforms a symmetric rank-k update.\ncblas_?syr2k\ns, d, c, z\nPerforms a symmetric rank-2k update.\ncblas_?trmm\ns, d, c, z\nComputes a matrix-matrix product where one input matrix\nis triangular.\ncblas_?trsm\ns, d, c, z\nSolves a triangular matrix equation.\nSymmetric Multiprocessing Version of Intel® MKL\nMany applications spend considerable time executing BLAS routines. This time can be scaled by the number\nof processors available on the system through using the symmetric multiprocessing (SMP) feature built into\nthe Intel® oneMKL. The performance enhancements based on the parallel use of the processors are available\nwithout any programming effort on your part.\nTo enhance performance, the library uses the following methods:\n•\nThe BLAS functions are blocked where possible to restructure the code in a way that increases the\nlocalization of data reference, enhances cache memory use, and reduces the dependency on the memory\nbus.\n•\nThe code is distributed across the processors to maximize parallelism.\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\ncblas_?gemm\nComputes a matrix-matrix product with general\nmatrices.\nSyntax\nvoid cblas_hgemm (const CBLAS_LAYOUT Layout, const CBLAS_TRANSPOSE transa, const\nCBLAS_TRANSPOSE transb, const MKL_INT m, const MKL_INT n, const MKL_INT k, const\nMKL_F16 alpha, const MKL_F16 *a, const MKL_INT lda, const MKL_F16 *b, const MKL_INT\nldb, const MKL_F16 beta, MKL_F16 *c, const MKL_INT ldc);\nvoid cblas_sgemm (const CBLAS_LAYOUT Layout, const CBLAS_TRANSPOSE transa, const\nCBLAS_TRANSPOSE transb, const MKL_INT m, const MKL_INT n, const MKL_INT k, const float\nalpha, const float *a, const MKL_INT lda, const float *b, const MKL_INT ldb, const\nfloat beta, float *c, const MKL_INT ldc);\nvoid cblas_dgemm (const CBLAS_LAYOUT Layout, const CBLAS_TRANSPOSE transa, const\nCBLAS_TRANSPOSE transb, const MKL_INT m, const MKL_INT n, const MKL_INT k, const double\nalpha, const double *a, const MKL_INT lda, const double *b, const MKL_INT ldb, const\ndouble beta, double *c, const MKL_INT ldc);\nvoid cblas_cgemm (const CBLAS_LAYOUT Layout, const CBLAS_TRANSPOSE transa, const\nCBLAS_TRANSPOSE transb, const MKL_INT m, const MKL_INT n, const MKL_INT k, const void\n*alpha, const void *a, const MKL_INT lda, const void *b, const MKL_INT ldb, const void\n*beta, void *c, const MKL_INT ldc);\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n98\n\n\nvoid cblas_zgemm (const CBLAS_LAYOUT Layout, const CBLAS_TRANSPOSE transa, const\nCBLAS_TRANSPOSE transb, const MKL_INT m, const MKL_INT n, const MKL_INT k, const void\n*alpha, const void *a, const MKL_INT lda, const void *b, const MKL_INT ldb, const void\n*beta, void *c, const MKL_INT ldc);\nInclude Files\n•\nmkl.h\nDescription\nThe ?gemm routines compute a scalar-matrix-matrix product and add the result to a scalar-matrix product,\nwith general matrices. The operation is defined as\nC := alpha*op(A)*op(B) + beta*C\nwhere:\nop(X) is one of op(X) = X, or op(X) = XT, or op(X) = XH,\nalpha and beta are scalars,\nA, B and C are matrices:\nop(A) is an m-by-k matrix,\nop(B) is a k-by-n matrix,\nC is an m-by-n matrix.\nSee also:\n•\n?gemm3m, BLAS-like extension routines, that use matrix multiplication for similar matrix-matrix operations\nInput Parameters\nLayout\nSpecifies whether two-dimensional array storage is row-major\n(CblasRowMajor) or column-major (CblasColMajor).\ntransa\nSpecifies the form of op(A) used in the matrix multiplication:\n•\nif transa=CblasNoTrans, then op(A) = A;\n•\nif transa=CblasTrans, then op(A) = AT;\n•\nif transa=CblasConjTrans, then op(A) = AH.\ntransb\nSpecifies the form of op(B) used in the matrix multiplication:\n•\nif transb=CblasNoTrans, then op(B) = B;\n•\nif transb=CblasTrans, then op(B) = BT;\n•\nif transb=CblasConjTrans, then op(B) = BH.\nm\nSpecifies the number of rows of the matrix op(A) and of the matrix C. The\nvalue of m must be at least zero.\nn\nSpecifies the number of columns of the matrix op(B) and the number of\ncolumns of the matrix C. The value of n must be at least zero.\nk\nSpecifies the number of columns of the matrix op(A) and the number of\nrows of the matrix op(B). The value of k must be at least zero.\nalpha\nSpecifies the scalar alpha.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n99\n\n\na\ntransa=CblasNoTrans\ntransa=CblasTrans or\ntransa=CblasConjTrans\nLayout =\nCblasColMajor\nArray, size lda*k.\nBefore entry, the leading\nm-by-k part of the array a\nmust contain the matrix\nA.\nArray, size lda*m.\nBefore entry, the leading k-\nby-m part of the array a\nmust contain the matrix A.\nLayout =\nCblasRowMajor\nArray, size lda* m.\nBefore entry, the leading\nk-by-m part of the array a\nmust contain the matrix\nA.\nArray, size lda*k.\nBefore entry, the leading m-\nby-k part of the array a\nmust contain the matrix A.\nlda\nSpecifies the leading dimension of a as declared in the calling\n(sub)program.\ntransa=CblasNoTrans\ntransa=CblasTrans or\ntransa=CblasConjTrans\nLayout =\nCblasColMajor\nlda must be at least\nmax(1, m).\nlda must be at least\nmax(1, k)\nLayout =\nCblasRowMajor\nlda must be at least\nmax(1, k)\nlda must be at least\nmax(1, m).\nb\ntransb=CblasNoTrans\ntransb=CblasTrans or\ntransb=CblasConjTrans\nLayout =\nCblasColMajor\nArray, size ldb by n.\nBefore entry, the leading\nk-by-n part of the array b\nmust contain the matrix\nB.\nArray, size ldb by k. Before\nentry the leading n-by-k\npart of the array b must\ncontain the matrix B.\nLayout =\nCblasRowMajor\nArray, size ldb by k.\nBefore entry the leading\nn-by-k part of the array b\nmust contain the matrix\nB.\nArray, size ldb by n. Before\nentry, the leading k-by-n\npart of the array b must\ncontain the matrix B.\nldb\nSpecifies the leading dimension of b as declared in the calling\n(sub)program.\nWhen transb=CblasNoTrans , then ldb must be at least max(1, k),\notherwise ldb must be at least max(1, n).\ntransb=CblasNoTrans\ntransb=CblasTrans or\ntransb=CblasConjTrans\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n100\n\n\nLayout =\nCblasColMajor\nldb must be at least\nmax(1, k).\nldb must be at least\nmax(1, n).\nLayout =\nCblasRowMajor\nldb must be at least\nmax(1, n).\nldb must be at least\nmax(1, k).\nbeta\nSpecifies the scalar beta. When beta is equal to zero, then c need not be\nset on input.\nc\nLayout =\nCblasColMajor\nArray, size ldc by n. Before entry, the leading m-\nby-n part of the array c must contain the matrix C,\nexcept when beta is equal to zero, in which case c\nneed not be set on entry.\nLayout =\nCblasRowMajor\nArray, size ldc by m. Before entry, the leading n-\nby-m part of the array c must contain the matrix C,\nexcept when beta is equal to zero, in which case c\nneed not be set on entry.\nldc\nSpecifies the leading dimension of c as declared in the calling\n(sub)program.\nLayout = CblasColMajor\nldc must be at least max(1, m).\nLayout = CblasRowMajor\nldc must be at least max(1, n).\nOutput Parameters\nc\nOverwritten by the m-by-n matrix (alpha*op(A)*op(B) + beta*C).\nExample\nFor examples of routine usage, see these code examples in the Intel® oneAPI Math Kernel Library (oneMKL)\ninstallation directory:\n•\ncblas_hgemm: examples\\cblas\\source\\cblas_hgemmx.c\n•\ncblas_sgemm: examples\\cblas\\source\\cblas_sgemmx.c\n•\ncblas_dgemm: examples\\cblas\\source\\cblas_dgemmx.c\n•\ncblas_cgemm: examples\\cblas\\source\\cblas_cgemmx.c\n•\ncblas_zgemm: examples\\cblas\\source\\cblas_zgemmx.c\ncblas_?hemm\nComputes a matrix-matrix product where one input\nmatrix is Hermitian.\nSyntax\nvoid cblas_chemm (const CBLAS_LAYOUT Layout, const CBLAS_SIDE side, const CBLAS_UPLO\nuplo, const MKL_INT m, const MKL_INT n, const void *alpha, const void *a, const MKL_INT\nlda, const void *b, const MKL_INT ldb, const void *beta, void *c, const MKL_INT ldc);\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n101\n\n\nvoid cblas_zhemm (const CBLAS_LAYOUT Layout, const CBLAS_SIDE side, const CBLAS_UPLO\nuplo, const MKL_INT m, const MKL_INT n, const void *alpha, const void *a, const MKL_INT\nlda, const void *b, const MKL_INT ldb, const void *beta, void *c, const MKL_INT ldc);\nInclude Files\n•\nmkl.h\nDescription\nThe ?hemm routines compute a scalar-matrix-matrix product using a Hermitian matrix A and a general matrix\nB and add the result to a scalar-matrix product using a general matrix C. The operation is defined as\nC := alpha*A*B + beta*C\nor\nC := alpha*B*A + beta*C\nwhere:\nalpha and beta are scalars,\nA is a Hermitian matrix,\nB and C are m-by-n matrices.\nInput Parameters\nLayout\nSpecifies whether two-dimensional array storage is row-major\n(CblasRowMajor) or column-major (CblasColMajor).\nside\nSpecifies whether the Hermitian matrix A appears on the left or right in the\noperation as follows:\nif side = CblasLeft, then C := alpha*A*B + beta*C;\nif side = CblasRight, then C := alpha*B*A + beta*C.\nuplo\nSpecifies whether the upper or lower triangular part of the Hermitian matrix\nA is used:\nIf uplo = CblasUpper, then the upper triangular part of the Hermitian\nmatrix A is used.\nIf uplo = CblasLower, then the low triangular part of the Hermitian matrix\nA is used.\nm\nSpecifies the number of rows of the matrix C.\nThe value of m must be at least zero.\nn\nSpecifies the number of columns of the matrix C.\nThe value of n must be at least zero.\nalpha\nSpecifies the scalar alpha.\na\nArray, size lda* ka, where ka is m when side = CblasLeft and is n\notherwise. Before entry with side = CblasLeft, the m-by-m part of the\narray a must contain the Hermitian matrix, such that when uplo =\nCblasUpper, the leading m-by-m upper triangular part of the array a must\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n102\n\n\ncontain the upper triangular part of the Hermitian matrix and the strictly\nlower triangular part of a is not referenced, and when uplo = CblasLower,\nthe leading m-by-m lower triangular part of the array a must contain the\nlower triangular part of the Hermitian matrix, and the strictly upper\ntriangular part of a is not referenced.\nBefore entry with side = CblasRight, the n-by-n part of the array a must\ncontain the Hermitian matrix, such that when uplo = CblasUpper, the\nleading n-by-n upper triangular part of the array a must contain the upper\ntriangular part of the Hermitian matrix and the strictly lower triangular part\nof a is not referenced, and when uplo = CblasLower, the leading n-by-n\nlower triangular part of the array a must contain the lower triangular part of\nthe Hermitian matrix, and the strictly upper triangular part of a is not\nreferenced. The imaginary parts of the diagonal elements need not be set,\nthey are assumed to be zero.\nlda\nSpecifies the leading dimension of a as declared in the calling (sub)\nprogram. When side = CblasLeft then lda must be at least max(1, m),\notherwise lda must be at least max(1,n).\nb\nFor Layout = CblasColMajor: array, size ldb*n. The leading m-by-n part\nof the array b must contain the matrix B.\nFor Layout = CblasRowMajor: array, size ldb*m. The leading n-by-m part\nof the array b must contain the matrix B\nldb\nSpecifies the leading dimension of b as declared in the calling\n(sub)program.When Layout = CblasColMajor, ldb must be at least\nmax(1, m); otherwise, ldb must be at least max(1, n) .\nbeta\nSpecifies the scalar beta.\nWhen beta is supplied as zero, then c need not be set on input.\nc\nFor Layout = CblasColMajor: array, size ldc*n. Before entry, the leading\nm-by-n part of the array c must contain the matrix C, except when beta is\nzero, in which case c need not be set on entry.\nFor Layout = CblasRowMajor: array, size ldc*m. Before entry, the leading\nn-by-m part of the array c must contain the matrix C, except when beta is\nzero, in which case c need not be set on entry.\nldc\nSpecifies the leading dimension of c as declared in the calling\n(sub)program. When Layout = CblasColMajor, ldc must be at least\nmax(1, m); otherwise, ldc must be at least max(1, n) .\nOutput Parameters\nc\nOverwritten by the m-by-n updated matrix.\ncblas_?herk\nPerforms a Hermitian rank-k update.\nSyntax\nvoid cblas_cherk (const CBLAS_LAYOUT Layout, const CBLAS_UPLO uplo, const\nCBLAS_TRANSPOSE trans, const MKL_INT n, const MKL_INT k, const float alpha, const void\n*a, const MKL_INT lda, const float beta, void *c, const MKL_INT ldc);\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n103\n\n\nvoid cblas_zherk (const CBLAS_LAYOUT Layout, const CBLAS_UPLO uplo, const\nCBLAS_TRANSPOSE trans, const MKL_INT n, const MKL_INT k, const double alpha, const void\n*a, const MKL_INT lda, const double beta, void *c, const MKL_INT ldc);\nInclude Files\n•\nmkl.h\nDescription\nThe ?herk routines perform a rank-k matrix-matrix operation using a general matrix A and a Hermitian\nmatrix C. The operation is defined as:\nC := alpha*A*AH + beta*C,\nor\nC := alpha*AH*A + beta*C,\nwhere:\nalpha and beta are real scalars,\nC is an n-by-n Hermitian matrix,\nA is an n-by-k matrix in the first case and a k-by-n matrix in the second case.\nInput Parameters\nLayout\nSpecifies whether two-dimensional array storage is row-major\n(CblasRowMajor) or column-major (CblasColMajor).\nuplo\nSpecifies whether the upper or lower triangular part of the array c is used.\nIf uplo = CblasUpper, then the upper triangular part of the array c is\nused.\nIf uplo = CblasLower, then the low triangular part of the array c is used.\ntrans\nSpecifies the operation:\nif trans=CblasNoTrans, then C := alpha*A*AH + beta*C;\nif trans=CblasConjTrans, then C := alpha*AH*A + beta*C.\nn\nSpecifies the order of the matrix C. The value of n must be at least zero.\nk\nWithtrans=CblasNoTrans, k specifies the number of columns of the\nmatrix A, and with trans=CblasConjTrans, k specifies the number of\nrows of the matrix A.\nThe value of k must be at least zero.\nalpha\nSpecifies the scalar alpha.\na\ntrans=CblasNoTrans\ntrans=CblasConjTrans\nLayout =\nCblasColMajor\nArray, size lda*k.\nArray, size lda*n.\nBefore entry, the leading k-\nby-n part of the array a\nmust contain the matrix A.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n104\n\n\nBefore entry, the leading\nn-by-k part of the array\na must contain the\nmatrix A.\nLayout =\nCblasRowMajor\nArray, size lda*n.\nBefore entry, the leading\nk-by-n part of the array\na must contain the\nmatrix A.\nArray, size lda*k.\nBefore entry, the leading n-\nby-k part of the array a\nmust contain the matrix A.\nlda\ntrans=CblasNoTrans\ntrans=CblasConjTrans\nLayout =\nCblasColMajor\nlda must be at least\nmax(1, n).\nlda must be at least\nmax(1, k)\nLayout =\nCblasRowMajor\nlda must be at least\nmax(1, k)\nlda must be at least\nmax(1, n).\nbeta\nSpecifies the scalar beta.\nc\nArray, size ldc by n.\nBefore entry with uplo = CblasUpper, the leading n-by-n upper triangular\npart of the array c must contain the upper triangular part of the Hermitian\nmatrix and the strictly lower triangular part of c is not referenced.\nBefore entry with uplo = CblasLower, the leading n-by-n lower triangular\npart of the array c must contain the lower triangular part of the Hermitian\nmatrix and the strictly upper triangular part of c is not referenced.\nThe imaginary parts of the diagonal elements need not be set, they are\nassumed to be zero.\nldc\nSpecifies the leading dimension of c as declared in the calling\n(sub)program. The value of ldc must be at least max(1, n).\nOutput Parameters\nc\nWith uplo = CblasUpper, the upper triangular part of the array c is\noverwritten by the upper triangular part of the updated matrix.\nWith uplo = CblasLower, the lower triangular part of the array c is\noverwritten by the lower triangular part of the updated matrix.\nThe imaginary parts of the diagonal elements are set to zero.\ncblas_?her2k\nPerforms a Hermitian rank-2k update.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n105\n\n\nSyntax\nvoid cblas_cher2k (const CBLAS_LAYOUT Layout, const CBLAS_UPLO uplo, const\nCBLAS_TRANSPOSE trans, const MKL_INT n, const MKL_INT k, const void *alpha, const void\n*a, const MKL_INT lda, const void *b, const MKL_INT ldb, const float beta, void *c,\nconst MKL_INT ldc);\nvoid cblas_zher2k (const CBLAS_LAYOUT Layout, const CBLAS_UPLO uplo, const\nCBLAS_TRANSPOSE trans, const MKL_INT n, const MKL_INT k, const void *alpha, const void\n*a, const MKL_INT lda, const void *b, const MKL_INT ldb, const double beta, void *c,\nconst MKL_INT ldc);\nInclude Files\n•\nmkl.h\nDescription\nThe ?her2k routines perform a rank-2k matrix-matrix operation using general matrices A and B and a\nHermitian matrix C. The operation is defined as\nC := alpha*A*BH + conjg(alpha)B*AH + beta*C\nor\nC := alpha*AH*B + conjg(alpha)*BH*A + beta*C\nwhere:\nalpha is a scalar and beta is a real scalar.\nC is an n-by-n Hermitian matrix.\nA and B are n-by-k matrices in the first case and k-by-n matrices in the second case.\nInput Parameters\nLayout\nSpecifies whether two-dimensional array storage is row-major\n(CblasRowMajor) or column-major (CblasColMajor).\nuplo\nSpecifies whether the upper or lower triangular part of the array c is used.\nIf uplo = CblasUpper, then the upper triangular of the array c is used.\nIf uplo = CblasLower, then the low triangular of the array c is used.\ntrans\nSpecifies the operation:\niftrans=CblasNoTrans, then C:=alpha*A*BH + alpha*B*AH + beta*C;\nif trans=CblasConjTrans, then C:=alpha*AH*B + alpha*BH*A +\nbeta*C.\nn\nSpecifies the order of the matrix C. The value of n must be at least zero.\nk\nWith trans=CblasNoTrans specifies the number of columns of the matrix\nA, and with trans=CblasConjTrans, k specifies the number of rows of the\nmatrix A.\nThe value of k must be at least equal to zero.\nalpha\nSpecifies the scalar alpha.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n106\n\n\na\ntrans=CblasNoTrans\ntrans=CblasConjTrans\nLayout =\nCblasColMajor\nArray, size lda*k.\nBefore entry, the leading\nn-by-k part of the array\na must contain the\nmatrix A.\nArray, size lda*n.\nBefore entry, the leading k-\nby-n part of the array a\nmust contain the matrix A.\nLayout =\nCblasRowMajor\nArray, size lda*n.\nBefore entry, the leading\nk-by-n part of the array\na must contain the\nmatrix A.\nArray, size lda*k.\nBefore entry, the leading n-\nby-k part of the array a\nmust contain the matrix A.\nlda\nSpecifies the leading dimension of a as declared in the calling\n(sub)program.\ntrans=CblasNoTrans\ntrans=CblasConjTrans\nLayout =\nCblasColMajor\nlda must be at least\nmax(1, n).\nlda must be at least\nmax(1, k)\nLayout =\nCblasRowMajor\nlda must be at least\nmax(1, k)\nlda must be at least\nmax(1, n).\nbeta\nSpecifies the scalar beta.\nb\ntrans=CblasNoTrans\ntrans=CblasConjTrans\nLayout =\nCblasColMajor\nArray, size ldb*k.\nBefore entry, the leading\nn-by-k part of the array\nb must contain the\nmatrix B.\nArray, size ldb*n.\nBefore entry, the leading k-\nby-n part of the array b\nmust contain the matrix B.\nLayout =\nCblasRowMajor\nArray, size lda*n.\nBefore entry, the leading\nk-by-n part of the array\nb must contain the\nmatrix B.\nArray, size lda*k.\nBefore entry, the leading n-\nby-k part of the array b\nmust contain the matrix B.\nldb\nSpecifies the leading dimension of a as declared in the calling\n(sub)program.\ntrans=CblasNoTrans\ntrans=CblasConjTrans\nLayout =\nCblasColMajor\nldb must be at least\nmax(1, n).\nldb must be at least\nmax(1, k)\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n107\n\n\nLayout =\nCblasRowMajor\nldb must be at least\nmax(1, k)\nldb must be at least\nmax(1, n).\nc\nArray, size ldc by n.\nBefore entry withuplo = CblasUpper, the leading n-by-n upper triangular\npart of the array c must contain the upper triangular part of the Hermitian\nmatrix and the strictly lower triangular part of c is not referenced.\nBefore entry with uplo = CblasLower, the leading n-by-n lower triangular\npart of the array c must contain the lower triangular part of the Hermitian\nmatrix and the strictly upper triangular part of c is not referenced.\nThe imaginary parts of the diagonal elements need not be set, they are\nassumed to be zero.\nldc\nSpecifies the leading dimension of c as declared in the calling\n(sub)program. The value of ldc must be at least max(1, n).\nOutput Parameters\nc\nWith uplo = CblasUpper, the upper triangular part of the array c is\noverwritten by the upper triangular part of the updated matrix.\nWith uplo = CblasLower, the lower triangular part of the array c is\noverwritten by the lower triangular part of the updated matrix.\nThe imaginary parts of the diagonal elements are set to zero.\ncblas_?symm\nComputes a matrix-matrix product where one input\nmatrix is symmetric.\nSyntax\nvoid cblas_ssymm (const CBLAS_LAYOUT Layout, const CBLAS_SIDE side, const CBLAS_UPLO\nuplo, const MKL_INT m, const MKL_INT n, const float alpha, const float *a, const\nMKL_INT lda, const float *b, const MKL_INT ldb, const float beta, float *c, const\nMKL_INT ldc);\nvoid cblas_dsymm (const CBLAS_LAYOUT Layout, const CBLAS_SIDE side, const CBLAS_UPLO\nuplo, const MKL_INT m, const MKL_INT n, const double alpha, const double *a, const\nMKL_INT lda, const double *b, const MKL_INT ldb, const double beta, double *c, const\nMKL_INT ldc);\nvoid cblas_csymm (const CBLAS_LAYOUT Layout, const CBLAS_SIDE side, const CBLAS_UPLO\nuplo, const MKL_INT m, const MKL_INT n, const void *alpha, const void *a, const MKL_INT\nlda, const void *b, const MKL_INT ldb, const void *beta, void *c, const MKL_INT ldc);\nvoid cblas_zsymm (const CBLAS_LAYOUT Layout, const CBLAS_SIDE side, const CBLAS_UPLO\nuplo, const MKL_INT m, const MKL_INT n, const void *alpha, const void *a, const MKL_INT\nlda, const void *b, const MKL_INT ldb, const void *beta, void *c, const MKL_INT ldc);\nInclude Files\n•\nmkl.h\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n108\n\n\nDescription\nThe ?symm routines compute a scalar-matrix-matrix product with one symmetric matrix and add the result to\na scalar-matrix product . The operation is defined as\nC := alpha*A*B + beta*C,\nor\nC := alpha*B*A + beta*C,\nwhere:\nalpha and beta are scalars,\nA is a symmetric matrix,\nB and C are m-by-n matrices.\nInput Parameters\nLayout\nSpecifies whether two-dimensional array storage is row-major\n(CblasRowMajor) or column-major (CblasColMajor).\nside\nSpecifies whether the symmetric matrix A appears on the left or right in the\noperation:\nif side = CblasLeft, then C := alpha*A*B + beta*C;\nif side = CblasRight, then C := alpha*B*A + beta*C.\nuplo\nSpecifies whether the upper or lower triangular part of the symmetric\nmatrix A is used:\nif uplo = CblasUpper, then the upper triangular part is used;\nif uplo = CblasLower, then the lower triangular part is used.\nm\nSpecifies the number of rows of the matrix C.\nThe value of m must be at least zero.\nn\nSpecifies the number of columns of the matrix C.\nThe value of n must be at least zero.\nalpha\nSpecifies the scalar alpha.\na\nArray, size lda* ka , where ka is m when side = CblasLeft and is n\notherwise.\nBefore entry with side = CblasLeft, the m-by-m part of the array a must\ncontain the symmetric matrix, such that when uplo = CblasUpper, the\nleading m-by-m upper triangular part of the array a must contain the upper\ntriangular part of the symmetric matrix and the strictly lower triangular part\nof a is not referenced, and when uplo = CblasLeft, the leading m-by-m\nlower triangular part of the array a must contain the lower triangular part of\nthe symmetric matrix and the strictly upper triangular part of a is not\nreferenced.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n109\n\n\nBefore entry with side = CblasRight, the n-by-n part of the array a must\ncontain the symmetric matrix, such that when uplo = CblasUppere array a\nmust contain the upper triangular part of the symmetric matrix and the\nstrictly lower triangular part of a is not referenced, and when uplo =\nCblasLeft, the leading n-by-n lower triangular part of the array a must\ncontain the lower triangular part of the symmetric matrix and the strictly\nupper triangular part of a is not referenced.\nlda\nSpecifies the leading dimension of a as declared in the calling\n(sub)program. When side = CblasLeft then lda must be at least max(1,\nm), otherwise lda must be at least max(1, n).\nb\nFor Layout = CblasColMajor: array, size ldb*n. The leading m-by-n part\nof the array b must contain the matrix B.\nFor Layout = CblasRowMajor: array, size ldb*m. The leading n-by-m part\nof the array b must contain the matrix B\nldb\nSpecifies the leading dimension of b as declared in the calling\n(sub)program. When Layout = CblasColMajor, ldb must be at least\nmax(1, m); otherwise, ldb must be at least max(1, n).\nbeta\nSpecifies the scalar beta.\nWhen beta is set to zero, then c need not be set on input.\nc\nFor Layout = CblasColMajor: array, size ldc*n. Before entry, the leading\nm-by-n part of the array c must contain the matrix C, except when beta is\nzero, in which case c need not be set on entry.\nFor Layout = CblasRowMajor: array, size ldc*m. Before entry, the leading\nn-by-m part of the array c must contain the matrix C, except when beta is\nzero, in which case c need not be set on entry.\nldc\nSpecifies the leading dimension of c as declared in the calling\n(sub)program. When Layout = CblasColMajor, ldc must be at least\nmax(1, m); otherwise, ldc must be at least max(1, n).\nOutput Parameters\nc\nOverwritten by the m-by-n updated matrix.\ncblas_?syrk\nPerforms a symmetric rank-k update.\nSyntax\nvoid cblas_ssyrk (const CBLAS_LAYOUT Layout, const CBLAS_UPLO uplo, const\nCBLAS_TRANSPOSE trans, const MKL_INT n, const MKL_INT k, const float alpha, const float\n*a, const MKL_INT lda, const float beta, float *c, const MKL_INT ldc);\nvoid cblas_dsyrk (const CBLAS_LAYOUT Layout, const CBLAS_UPLO uplo, const\nCBLAS_TRANSPOSE trans, const MKL_INT n, const MKL_INT k, const double alpha, const\ndouble *a, const MKL_INT lda, const double beta, double *c, const MKL_INT ldc);\nvoid cblas_csyrk (const CBLAS_LAYOUT Layout, const CBLAS_UPLO uplo, const\nCBLAS_TRANSPOSE trans, const MKL_INT n, const MKL_INT k, const void *alpha, const void\n*a, const MKL_INT lda, const void *beta, void *c, const MKL_INT ldc);\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n110\n\n\nvoid cblas_zsyrk (const CBLAS_LAYOUT Layout, const CBLAS_UPLO uplo, const\nCBLAS_TRANSPOSE trans, const MKL_INT n, const MKL_INT k, const void *alpha, const void\n*a, const MKL_INT lda, const void *beta, void *c, const MKL_INT ldc);\nInclude Files\n•\nmkl.h\nDescription\nThe ?syrk routines perform a rank-k matrix-matrix operation for a symmetric matrix C using a general\nmatrix A . The operation is defined as:\nC := alpha*A*A' + beta*C,\nor\nC := alpha*A'*A + beta*C,\nwhere:\nalpha and beta are scalars,\nC is an n-by-n symmetric matrix,\nA is an n-by-k matrix in the first case and a k-by-n matrix in the second case.\nInput Parameters\nLayout\nSpecifies whether two-dimensional array storage is row-major\n(CblasRowMajor) or column-major (CblasColMajor).\nuplo\nSpecifies whether the upper or lower triangular part of the array c is used.\nIf uplo = CblasUpper, then the upper triangular part of the array c is\nused.\nIf uplo = CblasLower, then the low triangular part of the array c is used.\ntrans\nSpecifies the operation:\nif trans=CblasNoTrans, then C := alpha*A*A' + beta*C;\nif trans=CblasTrans, then C := alpha*A'*A + beta*C;\nif trans=CblasConjTrans, then C := alpha*A'*A + beta*C.\nn\nSpecifies the order of the matrix C. The value of n must be at least zero.\nk\nOn entry with trans=CblasNoTrans, k specifies the number of columns of\nthe matrix a, and on entry with trans=CblasTrans or\ntrans=CblasConjTrans , k specifies the number of rows of the matrix a.\nThe value of k must be at least zero.\nalpha\nSpecifies the scalar alpha.\na\nArray, size lda* ka, where ka is k when trans=CblasNoTrans, and is n\notherwise. Before entry with trans=CblasNoTrans, the leading n-by-k\npart of the array a must contain the matrix A, otherwise the leading k-by-n\npart of the array a must contain the matrix A.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n111\n\n\ntrans=CblasNoTrans\ntrans=CblasConjTrans\nLayout =\nCblasColMajor\nArray, size lda*k.\nBefore entry, the leading\nn-by-k part of the array\na must contain the\nmatrix A.\nArray, size lda*n.\nBefore entry, the leading k-\nby-n part of the array a\nmust contain the matrix A.\nLayout =\nCblasRowMajor\nArray, size lda*n.\nBefore entry, the leading\nk-by-n part of the array\na must contain the\nmatrix A.\nArray, size lda*k.\nBefore entry, the leading n-\nby-k part of the array a\nmust contain the matrix A.\nlda\ntrans=CblasNoTrans\ntrans=CblasConjTrans\nLayout =\nCblasColMajor\nlda must be at least\nmax(1, n).\nlda must be at least\nmax(1, k)\nLayout =\nCblasRowMajor\nlda must be at least\nmax(1, k)\nlda must be at least\nmax(1, n).\nbeta\nSpecifies the scalar beta.\nc\nArray, size ldc* n. Before entry with uplo = CblasUpper, the leading n-\nby-n upper triangular part of the array c must contain the upper triangular\npart of the symmetric matrix and the strictly lower triangular part of c is not\nreferenced.\nBefore entry with uplo = CblasLower, the leading n-by-n lower triangular\npart of the array c must contain the lower triangular part of the symmetric\nmatrix and the strictly upper triangular part of c is not referenced.\nldc\nSpecifies the leading dimension of c as declared in the calling\n(sub)program. The value of ldc must be at least max(1, n).\nOutput Parameters\nc\nWith uplo = CblasUpper, the upper triangular part of the array c is\noverwritten by the upper triangular part of the updated matrix.\nWith uplo = CblasLower, the lower triangular part of the array c is\noverwritten by the lower triangular part of the updated matrix.\ncblas_?syr2k\nPerforms a symmetric rank-2k update.\nSyntax\nvoid cblas_ssyr2k (const CBLAS_LAYOUT Layout, const CBLAS_UPLO uplo, const\nCBLAS_TRANSPOSE trans, const MKL_INT n, const MKL_INT k, const float alpha, const float\n*a, const MKL_INT lda, const float *b, const MKL_INT ldb, const float beta, float *c,\nconst MKL_INT ldc);\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n112\n\n\nvoid cblas_dsyr2k (const CBLAS_LAYOUT Layout, const CBLAS_UPLO uplo, const\nCBLAS_TRANSPOSE trans, const MKL_INT n, const MKL_INT k, const double alpha, const\ndouble *a, const MKL_INT lda, const double *b, const MKL_INT ldb, const double beta,\ndouble *c, const MKL_INT ldc);\nvoid cblas_csyr2k (const CBLAS_LAYOUT Layout, const CBLAS_UPLO uplo, const\nCBLAS_TRANSPOSE trans, const MKL_INT n, const MKL_INT k, const void *alpha, const void\n*a, const MKL_INT lda, const void *b, const MKL_INT ldb, const void *beta, void *c,\nconst MKL_INT ldc);\nvoid cblas_zsyr2k (const CBLAS_LAYOUT Layout, const CBLAS_UPLO uplo, const\nCBLAS_TRANSPOSE trans, const MKL_INT n, const MKL_INT k, const void *alpha, const void\n*a, const MKL_INT lda, const void *b, const MKL_INT ldb, const void *beta, void *c,\nconst MKL_INT ldc);\nInclude Files\n•\nmkl.h\nDescription\nThe ?syr2k routines perform a rank-2k matrix-matrix operation for a symmetric matrix C using general\nmatrices A and BThe operation is defined as:\nC := alpha*A*B' + alpha*B*A' + beta*C,\nor\nC := alpha*A'*B + alpha*B'*A + beta*C,\nwhere:\nalpha and beta are scalars,\nC is an n-by-n symmetric matrix,\nA and B are n-by-k matrices in the first case, and k-by-n matrices in the second case.\nInput Parameters\nLayout\nSpecifies whether two-dimensional array storage is row-major\n(CblasRowMajor) or column-major (CblasColMajor).\nuplo\nSpecifies whether the upper or lower triangular part of the array c is used.\nIf uplo = CblasUpper, then the upper triangular part of the array c is\nused.\nIf uplo = CblasLower, then the low triangular part of the array c is used.\ntrans\nSpecifies the operation:\nif trans=CblasNoTrans, then C := alpha*A*B'+alpha*B*A'+beta*C;\nif trans=CblasTrans, then C := alpha*A'*B +alpha*B'*A +beta*C;\nif trans=CblasConjTrans, then C := alpha*A'*B +alpha*B'*A\n+beta*C.\nn\nSpecifies the order of the matrix C.The value of n must be at least zero.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n113\n\n\nk\nOn entry with trans=CblasNoTrans, k specifies the number of columns of\nthe matrices A and B, and on entry with trans=CblasTrans or\ntrans=CblasConjTrans, k specifies the number of rows of the matrices A\nand B. The value of k must be at least zero.\nalpha\nSpecifies the scalar alpha.\na\ntrans=CblasNoTrans\ntrans=CblasConjTrans\nLayout =\nCblasColMajor\nArray, size lda*k.\nBefore entry, the leading\nn-by-k part of the array\na must contain the\nmatrix A.\nArray, size lda*n.\nBefore entry, the leading k-\nby-n part of the array a\nmust contain the matrix A.\nLayout =\nCblasRowMajor\nArray, size lda*n.\nBefore entry, the leading\nk-by-n part of the array\na must contain the\nmatrix A.\nArray, size lda*k.\nBefore entry, the leading n-\nby-k part of the array a\nmust contain the matrix A.\nlda\nSpecifies the leading dimension of a as declared in the calling\n(sub)program.\ntrans=CblasNoTrans\ntrans=CblasConjTrans\nLayout =\nCblasColMajor\nlda must be at least\nmax(1, n).\nlda must be at least\nmax(1, k)\nLayout =\nCblasRowMajor\nlda must be at least\nmax(1, k)\nlda must be at least\nmax(1, n).\nb\ntrans=CblasNoTrans\ntrans=CblasConjTrans\nLayout =\nCblasColMajor\nArray, size ldb*k.\nBefore entry, the leading\nn-by-k part of the array\nb must contain the\nmatrix B.\nArray, size ldb*n.\nBefore entry, the leading k-\nby-n part of the array b\nmust contain the matrix B.\nLayout =\nCblasRowMajor\nArray, size lda*n.\nBefore entry, the leading\nk-by-n part of the array\nb must contain the\nmatrix B.\nArray, size lda*k.\nBefore entry, the leading n-\nby-k part of the array b\nmust contain the matrix B.\nldb\nSpecifies the leading dimension of a as declared in the calling\n(sub)program.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n114\n\n\ntrans=CblasNoTrans\ntrans=CblasConjTrans\nLayout =\nCblasColMajor\nldb must be at least\nmax(1, n).\nldb must be at least\nmax(1, k)\nLayout =\nCblasRowMajor\nldb must be at least\nmax(1, k)\nldb must be at least\nmax(1, n).\nbeta\nSpecifies the scalar beta.\nc\nArray, size ldc* n. Before entry with uplo = CblasUpper, the leading n-\nby-n upper triangular part of the array c must contain the upper triangular\npart of the symmetric matrix and the strictly lower triangular part of c is not\nreferenced.\nBefore entry with uplo = CblasLower, the leading n-by-n lower triangular\npart of the array c must contain the lower triangular part of the symmetric\nmatrix and the strictly upper triangular part of c is not referenced.\nldc\nSpecifies the leading dimension of c as declared in the calling\n(sub)program. The value of ldc must be at least max(1, n).\nOutput Parameters\nc\nWith uplo = CblasUpper, the upper triangular part of the array c is\noverwritten by the upper triangular part of the updated matrix.\nWith uplo = CblasLower, the lower triangular part of the array c is\noverwritten by the lower triangular part of the updated matrix.\ncblas_?trmm\nComputes a matrix-matrix product where one input\nmatrix is triangular.\nSyntax\nvoid cblas_strmm (const CBLAS_LAYOUT Layout, const CBLAS_SIDE side, const CBLAS_UPLO\nuplo, const CBLAS_TRANSPOSE transa, const CBLAS_DIAG diag, const MKL_INT m, const\nMKL_INT n, const float alpha, const float *a, const MKL_INT lda, float *b, const\nMKL_INT ldb);\nvoid cblas_dtrmm (const CBLAS_LAYOUT Layout, const CBLAS_SIDE side, const CBLAS_UPLO\nuplo, const CBLAS_TRANSPOSE transa, const CBLAS_DIAG diag, const MKL_INT m, const\nMKL_INT n, const double alpha, const double *a, const MKL_INT lda, double *b, const\nMKL_INT ldb);\nvoid cblas_ctrmm (const CBLAS_LAYOUT Layout, const CBLAS_SIDE side, const CBLAS_UPLO\nuplo, const CBLAS_TRANSPOSE transa, const CBLAS_DIAG diag, const MKL_INT m, const\nMKL_INT n, const void *alpha, const void *a, const MKL_INT lda, void *b, const MKL_INT\nldb);\nvoid cblas_ztrmm (const CBLAS_LAYOUT Layout, const CBLAS_SIDE side, const CBLAS_UPLO\nuplo, const CBLAS_TRANSPOSE transa, const CBLAS_DIAG diag, const MKL_INT m, const\nMKL_INT n, const void *alpha, const void *a, const MKL_INT lda, void *b, const MKL_INT\nldb);\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n115\n\n\nInclude Files\n•\nmkl.h\nDescription\nThe ?trmm routines compute a scalar-matrix-matrix product with one triangular matrix . The operation is\ndefined as\nB := alpha*op(A)*B\nor\nB := alpha*B*op(A)\nwhere:\nalpha is a scalar,\nB is an m-by-n matrix,\nA is a unit, or non-unit, upper or lower triangular matrix\nop(A) is one of op(A) = A, or op(A) = A', or op(A) = conjg(A').\nInput Parameters\nLayout\nSpecifies whether two-dimensional array storage is row-major\n(CblasRowMajor) or column-major (CblasColMajor).\nside\nSpecifies whether op(A) appears on the left or right of B in the operation:\nif side = CblasLeft, then B := alpha*op(A)*B;\nif side = CblasRight, then B := alpha*B*op(A).\nuplo\nSpecifies whether the matrix A is upper or lower triangular.\nuplo = CblasUpper\nif uplo = CblasLower, then the matrix is low triangular.\ntransa\nSpecifies the form of op(A) used in the matrix multiplication:\nif transa=CblasNoTrans, then op(A) = A;\nif transa=CblasTrans, then op(A) = A';\nif transa=CblasConjTrans, then op(A) = conjg(A').\ndiag\nSpecifies whether the matrix A is unit triangular:\nif diag = CblasUnit then the matrix is unit triangular;\nif diag = CblasNonUnit , then the matrix is not unit triangular.\nm\nSpecifies the number of rows of B. The value of m must be at least zero.\nn\nSpecifies the number of columns of B. The value of n must be at least zero.\nalpha\nSpecifies the scalar alpha.\nWhen alpha is zero, then a is not referenced and b need not be set before\nentry.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n116\n\n\na\nArray, size lda by k, where k is m when side = CblasLeft and is n when\nside = CblasRight. Before entry with uplo = CblasUpper, the leading k\nby k upper triangular part of the array a must contain the upper triangular\nmatrix and the strictly lower triangular part of a is not referenced.\nBefore entry with uplo = CblasLower, the leading k by k lower triangular\npart of the array a must contain the lower triangular matrix and the strictly\nupper triangular part of a is not referenced.\nWhen diag = CblasUnit, the diagonal elements of a are not referenced\neither, but are assumed to be unity.\nlda\nSpecifies the leading dimension of a as declared in the calling\n(sub)program. Whenside = CblasLeft, then lda must be at least max(1,\nm), when side = CblasRight, then lda must be at least max(1, n).\nb\nFor Layout = CblasColMajor: array, size ldb*n. Before entry, the leading\nm-by-n part of the array b must contain the matrix B.\nFor Layout = CblasRowMajor: array, size ldb*m. Before entry, the leading\nn-by-m part of the array b must contain the matrix B.\nldb\nSpecifies the leading dimension of b as declared in the calling\n(sub)program. When Layout = CblasColMajor, ldb must be at least\nmax(1, m); otherwise, ldb must be at least max(1, n).\nOutput Parameters\nb\nOverwritten by the transformed matrix.\ncblas_?trsm\nSolves a triangular matrix equation.\nSyntax\nvoid cblas_strsm (const CBLAS_LAYOUT Layout, const CBLAS_SIDE side, const CBLAS_UPLO\nuplo, const CBLAS_TRANSPOSE transa, const CBLAS_DIAG diag, const MKL_INT m, const\nMKL_INT n, const float alpha, const float *a, const MKL_INT lda, float *b, const\nMKL_INT ldb);\nvoid cblas_dtrsm (const CBLAS_LAYOUT Layout, const CBLAS_SIDE side, const CBLAS_UPLO\nuplo, const CBLAS_TRANSPOSE transa, const CBLAS_DIAG diag, const MKL_INT m, const\nMKL_INT n, const double alpha, const double *a, const MKL_INT lda, double *b, const\nMKL_INT ldb);\nvoid cblas_ctrsm (const CBLAS_LAYOUT Layout, const CBLAS_SIDE side, const CBLAS_UPLO\nuplo, const CBLAS_TRANSPOSE transa, const CBLAS_DIAG diag, const MKL_INT m, const\nMKL_INT n, const void *alpha, const void *a, const MKL_INT lda, void *b, const MKL_INT\nldb);\nvoid cblas_ztrsm (const CBLAS_LAYOUT Layout, const CBLAS_SIDE side, const CBLAS_UPLO\nuplo, const CBLAS_TRANSPOSE transa, const CBLAS_DIAG diag, const MKL_INT m, const\nMKL_INT n, const void *alpha, const void *a, const MKL_INT lda, void *b, const MKL_INT\nldb);\nInclude Files\n•\nmkl.h\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n117\n\n\nDescription\nThe ?trsm routines solve one of the following matrix equations:\nop(A)*X = alpha*B,\nor\nX*op(A) = alpha*B,\nwhere:\nalpha is a scalar,\nX and B are m-by-n matrices,\nA is a unit, or non-unit, upper or lower triangular matrix, and\nop(A) is one of op(A) = A, or op(A) = A', or op(A) = conjg(A').\nThe matrix B is overwritten by the solution matrix X.\nInput Parameters\nLayout\nSpecifies whether two-dimensional array storage is row-major\n(CblasRowMajor) or column-major (CblasColMajor).\nside\nSpecifies whether op(A) appears on the left or right of X in the equation:\nif side = CblasLeft, then op(A)*X = alpha*B;\nif side = CblasRight, then X*op(A) = alpha*B.\nuplo\nSpecifies whether the matrix A is upper or lower triangular.\nuplo = CblasUpper\nif uplo = CblasLower, then the matrix is low triangular.\ntransa\nSpecifies the form of op(A) used in the matrix multiplication:\nif transa=CblasNoTrans, then op(A) = A;\nif transa=CblasTrans;\nif transa=CblasConjTrans, then op(A) = conjg(A').\ndiag\nSpecifies whether the matrix A is unit triangular:\nif diag = CblasUnit then the matrix is unit triangular;\nif diag = CblasNonUnit , then the matrix is not unit triangular.\nm\nSpecifies the number of rows of B. The value of m must be at least zero.\nn\nSpecifies the number of columns of B. The value of n must be at least zero.\nalpha\nSpecifies the scalar alpha.\nWhen alpha is zero, then a is not referenced and b need not be set before\nentry.\na\nArray, size lda* k , where k is m when side = CblasLeft and is n when\nside = CblasRight. Before entry with uplo = CblasUpper, the leading k\nby k upper triangular part of the array a must contain the upper triangular\nmatrix and the strictly lower triangular part of a is not referenced.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n118\n\n\nBefore entry with uplo = CblasLower lower triangular part of the array a\nmust contain the lower triangular matrix and the strictly upper triangular\npart of a is not referenced.\nWhen diag = CblasUnit, the diagonal elements of a are not referenced\neither, but are assumed to be unity.\nlda\nSpecifies the leading dimension of a as declared in the calling\n(sub)program. When side = CblasLeft, then lda must be at least max(1,\nm), when side = CblasRight, then lda must be at least max(1, n).\nb\nFor Layout = CblasColMajor: array, size ldb*n. Before entry, the leading\nm-by-n part of the array b must contain the matrix B.\nFor Layout = CblasRowMajor: array, size ldb*m. Before entry, the leading\nn-by-m part of the array b must contain the matrix B.\nldb\nSpecifies the leading dimension of b as declared in the calling\n(sub)program. When Layout = CblasColMajor, ldb must be at least\nmax(1, m); otherwise, ldb must be at least max(1, n).\nOutput Parameters\nb\nOverwritten by the solution matrix X.\nSparse BLAS Level 1 Routines\nThis section describes Sparse BLAS Level 1, an extension of BLAS Level 1 included in the Intel® oneAPI Math\nKernel Library beginning with the Intel® oneAPI Math Kernel Library (oneMKL) release 2.1. Sparse BLAS Level\n1 is a group of routines and functions that perform a number of common vector operations on sparse vectors\nstored in compressed form.\nSparse vectors are those in which the majority of elements are zeros. Sparse BLAS routines and functions\nare specially implemented to take advantage of vector sparsity. This allows you to achieve large savings in\ncomputer time and memory. If nz is the number of non-zero vector elements, the computer time taken by\nSparse BLAS operations will be O(nz).\nVector Arguments\nCompressed sparse vectors. Let a be a vector stored in an array, and assume that the only non-zero\nelements of a are the following:\na[k1], a[k2], a[k3] . . . a[knz], \nwhere nz is the total number of non-zero elements in a.\nIn Sparse BLAS, this vector can be represented in compressed form by two arrays, x (values) and indx\n(indices). Each array has nz elements:\nx[0]=a[k1], x[1]=a[k2], . . . x[nz-1]= a[knz], \nindx[0]=k1, indx[1]=k2, . . . indx[nz-1]= knz. \nThus, a sparse vector is fully determined by the triple (nz, x, indx). If you pass a negative or zero value of nz\nto Sparse BLAS, the subroutines do not modify any arrays or variables.\nFull-storage vectors. Sparse BLAS routines can also use a vector argument fully stored in a single array (a\nfull-storage vector). If y is a full-storage vector, its elements must be stored contiguously: the first element\nin y[0], the second in y[1], and so on. This corresponds to an increment incy = 1 in BLAS Level 1. No\nincrement value for full-storage vectors is passed as an argument to Sparse BLAS routines or functions.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n119\n\n\nNaming Conventions for Sparse BLAS Routines\nSimilar to BLAS, the names of Sparse BLAS subprograms have prefixes that determine the data type\ninvolved: s and d for single- and double-precision real; c and z for single- and double-precision complex\nrespectively.\nIf a Sparse BLAS routine is an extension of a \"dense\" one, the subprogram name is formed by appending the\nsuffix i (standing for indexed) to the name of the corresponding \"dense\" subprogram. For example, the\nSparse BLAS routine saxpyi corresponds to the BLAS routine saxpy, and the Sparse BLAS function cdotci\ncorresponds to the BLAS function cdotc.\nRoutines and Data Types\nRoutines and data types supported in the Intel® oneAPI Math Kernel Library (oneMKL) implementation of\nSparse BLAS are listed inTable “Sparse BLAS Routines and Their Data Types”.\nSparse BLAS Routines and Their Data Types\nRoutine/\nFunction\nData Types\nDescription\ncblas_?axpyi\ns, d, c, z\nScalar-vector product plus vector (routines)\ncblas_?doti\ns, d\nDot product (functions)\ncblas_?dotci\nc, z\nComplex dot product conjugated (functions)\ncblas_?dotui\nc, z\nComplex dot product unconjugated (functions)\ncblas_?gthr\ns, d, c, z\nGathering a full-storage sparse vector into compressed\nform nz, x, indx (routines)\ncblas_?gthrz\ns, d, c, z\nGathering a full-storage sparse vector into compressed\nform and assigning zeros to gathered elements in the full-\nstorage vector (routines)\ncblas_?roti\ns, d\nGivens rotation (routines)\ncblas_?sctr\ns, d, c, z\nScattering a vector from compressed form to full-storage\nform (routines)\nBLAS Level 1 Routines That Can Work With Sparse Vectors\nThe following BLAS Level 1 routines will give correct results when you pass to them a compressed-form array\nx(with the increment incx=1):\ncblas_?asum\nsum of absolute values of vector elements\ncblas_?copy\ncopying a vector\ncblas_?nrm2\nEuclidean norm of a vector\ncblas_?scal\nscaling a vector\ncblas_i?amax\nindex of the element with the largest absolute value for real flavors, or the\nlargest sum |Re(x[i])|+|Im(x[i])| for complex flavors.\ncblas_i?amin\nindex of the element with the smallest absolute value for real flavors, or the\nsmallest sum |Re(x[i])|+|Im(x[i])| for complex flavors.\nThe result i returned by i?amax and i?amin should be interpreted as index in the compressed-form array, so\nthat the largest (smallest) value is x[i-1]; the corresponding index in full-storage array is indx[i-1].\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n120\n\n\nYou can also call cblas_?rotg to compute the parameters of Givens rotation and then pass these\nparameters to the Sparse BLAS routines cblas_?roti.\ncblas_?axpyi\nAdds a scalar multiple of compressed sparse vector to\na full-storage vector.\nSyntax\nvoid cblas_saxpyi (const MKL_INT nz, const float a, const float *x, const MKL_INT\n*indx, float *y);\nvoid cblas_daxpyi (const MKL_INT nz, const double a, const double *x, const MKL_INT\n*indx, double *y);\nvoid cblas_caxpyi (const MKL_INT nz, const void *a, const void *x, const MKL_INT *indx,\nvoid *y);\nvoid cblas_zaxpyi (const MKL_INT nz, const void *a, const void *x, const MKL_INT *indx,\nvoid *y);\nInclude Files\n•\nmkl.h\nDescription\nThe ?axpyi routines perform a vector-vector operation defined as\ny := a*x + y\nwhere:\na is a scalar,\nx is a sparse vector stored in compressed form,\ny is a vector in full storage form.\nThe ?axpyi routines reference or modify only the elements of y whose indices are listed in the array indx.\nThe values in indx must be distinct.\nInput Parameters\nnz\nThe number of elements in x and indx.\na\nSpecifies the scalar a.\nx\nArray, size at least nz.\nindx\nSpecifies the indices for the elements of x.\nArray, size at least nz.\ny\nArray, size at least max(indx[i]).\nOutput Parameters\ny\nContains the updated vector y.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n121\n\n\ncblas_?doti\nComputes the dot product of a compressed sparse real\nvector by a full-storage real vector.\nSyntax\nfloat cblas_sdoti (const MKL_INT nz, const float *x, const MKL_INT *indx, const float\n*y);\ndouble cblas_ddoti (const MKL_INT nz, const double *x, const MKL_INT *indx, const\ndouble *y);\nInclude Files\n•\nmkl.h\nDescription\nThe ?doti routines return the dot product of x and y defined as\nres = x[0]*y[indx[0]] + x[1]*y[indx[1]] +...+ x[nz-1]*y[indx[nz-1]]\nwhere the triple (nz, x, indx) defines a sparse real vector stored in compressed form, and y is a real vector in\nfull storage form. The functions reference only the elements of y whose indices are listed in the array indx.\nThe values in indx must be distinct.\nInput Parameters\nnz\nThe number of elements in x and indx .\nx\nArray, size at least nz.\nindx\nSpecifies the indices for the elements of x.\nArray, size at least nz.\ny\nArray, size at least max(indx[i]).\nOutput Parameters\nres\nContains the dot product of x and y, if nz is positive. Otherwise, res\ncontains 0.\ncblas_?dotci\nComputes the conjugated dot product of a\ncompressed sparse complex vector with a full-storage\ncomplex vector.\nSyntax\nvoid cblas_cdotci_sub (const MKL_INT nz, const void *x, const MKL_INT *indx, const void\n*y, void *dotui);\nvoid cblas_zdotci_sub (const MKL_INT nz, const void *x, const MKL_INT *indx, const void\n*y, void *dotui);\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n122\n\n\nInclude Files\n•\nmkl.h\nDescription\nThe ?dotci routines return the dot product of x and y defined as\nconjg(x[0])*y[indx[0]] + ... + conjg(x[nz-1])*y[indx[nz-1]]\nwhere the triple (nz, x, indx) defines a sparse complex vector stored in compressed form, and y is a real\nvector in full storage form. The functions reference only the elements of y whose indices are listed in the\narray indx. The values in indx must be distinct.\nInput Parameters\nnz\nThe number of elements in x and indx .\nx\nArray, size at least nz.\nindx\nSpecifies the indices for the elements of x.\nArray, size at least nz.\ny\nArray, size at least max(indx[i]).\nOutput Parameters\ndotui\nContains the conjugated dot product of x and y, if nz is positive. Otherwise,\nit contains 0.\ncblas_?dotui\nComputes the dot product of a compressed sparse\ncomplex vector by a full-storage complex vector.\nSyntax\nvoid cblas_cdotui_sub (const MKL_INT nz, const void *x, const MKL_INT *indx, const void\n*y, void *dotui);\nvoid cblas_zdotui_sub (const MKL_INT nz, const void *x, const MKL_INT *indx, const void\n*y, void *dotui);\nInclude Files\n•\nmkl.h\nDescription\nThe ?dotui routines return the dot product of x and y defined as\nres = x[0]*y[indx[0]] + x[1]*y(indx[1]) +...+ x[nz - 1]*y[indx[nz - 1]]\nwhere the triple (nz, x, indx) defines a sparse complex vector stored in compressed form, and y is a real\nvector in full storage form. The functions reference only the elements of y whose indices are listed in the\narray indx. The values in indx must be distinct.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n123\n\n\nInput Parameters\nnz\nThe number of elements in x and indx.\nx\nArray, size at least nz.\nindx\nSpecifies the indices for the elements of x.\nArray, size at least nz.\ny\nArray, size at least max(indx[i]).\nOutput Parameters\ndotui\nContains the dot product of x and y, if nz is positive. Otherwise, res\ncontains 0.\ncblas_?gthr\nGathers a full-storage sparse vector's elements into\ncompressed form.\nSyntax\nvoid cblas_sgthr (const MKL_INT nz, const float *y, float *x, const MKL_INT *indx);\nvoid cblas_dgthr (const MKL_INT nz, const double *y, double *x, const MKL_INT *indx);\nvoid cblas_cgthr (const MKL_INT nz, const void *y, void *x, const MKL_INT *indx);\nvoid cblas_zgthr (const MKL_INT nz, const void *y, void *x, const MKL_INT *indx);\nInclude Files\n•\nmkl.h\nDescription\nThe ?gthr routines gather the specified elements of a full-storage sparse vector y into compressed form(nz,\nx, indx). The routines reference only the elements of y whose indices are listed in the array indx:\nx[i] = y]indx[i]], for i=0,1,... ,nz-1.\nInput Parameters\nnz\nThe number of elements of y to be gathered.\nindx\nSpecifies indices of elements to be gathered.\nArray, size at least nz.\ny\nArray, size at least max(indx[i]).\nOutput Parameters\nx\nArray, size at least nz.\nContains the vector converted to the compressed form.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n124\n\n\ncblas_?gthrz\nGathers a sparse vector's elements into compressed\nform, replacing them by zeros.\nSyntax\nvoid cblas_sgthrz (const MKL_INT nz, float *y, float *x, const MKL_INT *indx);\nvoid cblas_dgthrz (const MKL_INT nz, double *y, double *x, const MKL_INT *indx);\nvoid cblas_cgthrz (const MKL_INT nz, void *y, void *x, const MKL_INT *indx);\nvoid cblas_zgthrz (const MKL_INT nz, void *y, void *x, const MKL_INT *indx);\nInclude Files\n•\nmkl.h\nDescription\nThe ?gthrz routines gather the elements with indices specified by the array indx from a full-storage vector y\ninto compressed form (nz, x, indx) and overwrite the gathered elements of y by zeros. Other elements of y\nare not referenced or modified (see also ?gthr).\nInput Parameters\nnz\nThe number of elements of y to be gathered.\nindx\nSpecifies indices of elements to be gathered.\nArray, size at least nz.\ny\nArray, size at least max(indx[i]).\nOutput Parameters\nx\nArray, size at least nz.\nContains the vector converted to the compressed form.\ny\nThe updated vector y.\ncblas_?roti\nApplies Givens rotation to sparse vectors one of which\nis in compressed form.\nSyntax\nvoid cblas_sroti (const MKL_INT nz, float *x, const MKL_INT *indx, float *y, const\nfloat c, const float s);\nvoid cblas_droti (const MKL_INT nz, double *x, const MKL_INT *indx, double *y, const\ndouble c, const double s);\nInclude Files\n•\nmkl.h\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n125\n\n\nDescription\nThe ?roti routines apply the Givens rotation to elements of two real vectors, x (in compressed form nz, x,\nindx) and y (in full storage form):\nx[i] = c*x[i] + s*y[indx[i]]\ny[indx[i]] = c*y[indx[i]]- s*x[i]\nThe routines reference only the elements of y whose indices are listed in the array indx. The values in indx\nmust be distinct.\nInput Parameters\nnz\nThe number of elements in x and indx.\nx\nArray, size at least nz.\nindx\nSpecifies the indices for the elements of x.\nArray, size at least nz.\ny\nArray, size at least max(indx[i]).\nc\nA scalar.\ns\nA scalar.\nOutput Parameters\nx and y\nThe updated arrays.\ncblas_?sctr\nConverts compressed sparse vectors into full storage\nform.\nSyntax\nvoid cblas_ssctr (const MKL_INT nz, const float *x, const MKL_INT *indx, float *y);\nvoid cblas_dsctr (const MKL_INT nz, const double *x, const MKL_INT *indx, double *y);\nvoid cblas_csctr (const MKL_INT nz, const void *x, const MKL_INT *indx, void *y);\nvoid cblas_zsctr (const MKL_INT nz, const void *x, const MKL_INT *indx, void *y);\nInclude Files\n•\nmkl.h\nDescription\nThe ?sctr routines scatter the elements of the compressed sparse vector (nz, x, indx) to a full-storage\nvector y. The routines modify only the elements of y whose indices are listed in the array indx:\ny[indx[i]] = x[i], for i=0,1,... ,nz-1.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n126\n\n\nInput Parameters\nnz\nThe number of elements of x to be scattered.\nindx\nSpecifies indices of elements to be scattered.\nArray, size at least nz.\nx\nArray, size at least nz.\nContains the vector to be converted to full-storage form.\nOutput Parameters\ny\nArray, size at least max(indx[i]).\nContains the vector y with updated elements.\nSparse BLAS Level 2 and Level 3 Routines\nNOTE The Intel® oneAPI Math Kernel Library (oneMKL) Sparse BLAS Level 2 and Level 3 routines are\ndeprecated. Use the corresponding routine from the Intel® oneAPI Math Kernel Library (oneMKL)\nInspector-executor Sparse BLAS interface as indicated in the description for each routine.\nThis section describes Sparse BLAS Level 2 and Level 3 routines included in the Intel® oneAPI Math Kernel\nLibrary (oneMKL) . Sparse BLAS Level 2 is a group of routines and functions that perform operations between\na sparse matrix and dense vectors. Sparse BLAS Level 3 is a group of routines and functions that perform\noperations between a sparse matrix and dense matrices.\nThe terms and concepts required to understand the use of the Intel® oneAPI Math Kernel Library (oneMKL)\nSparse BLAS Level 2 and Level 3 routines are discussed in theLinear Solvers Basics appendix.\nThe Sparse BLAS routines can be useful to implement iterative methods for solving large sparse systems of\nequations or eigenvalue problems. For example, these routines can be considered as building blocks for \nIterative Sparse Solvers based on Reverse Communication Interface (RCI ISS).\nIntel® oneAPI Math Kernel Library (oneMKL) provides Sparse BLAS Level 2 and Level 3 routines with typical\n(or conventional) interface similar to the interface used in the NIST* Sparse BLAS library [Rem05].\nSome software packages and libraries (the PARDISO* Solverused in Intel® oneAPI Math Kernel Library\n(oneMKL),Sparskit 2 [Saad94], the Compaq* Extended Math Library (CXML)[CXML01]) use different (early)\nvariation of the compressed sparse row (CSR) format and support only Level 2 operations with simplified\ninterfaces. Intel® oneAPI Math Kernel Library (oneMKL) provides an additional set of Sparse BLAS Level 2\nroutines with similar simplified interfaces. Each of these routines operates only on a matrix of the fixed type.\nThe routines described in this section support both one-based indexing and zero-based indexing of the input\ndata (see details in the section One-based and Zero-based Indexing).\nNaming Conventions in Sparse BLAS Level 2 and Level 3\nEach Sparse BLAS Level 2 and Level 3 routine has a six- or eight-character base name preceded by the prefix\nmkl_ or mkl_cspblas_ .\nThe routines with typical (conventional) interface have six-character base names in accordance with the\ntemplate:\nmkl_<character > <data> <operation>( )\nThe routines with simplified interfaces have eight-character base names in accordance with the templates:\nmkl_<character > <data> <mtype> <operation>( )\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n127\n\n\nfor routines with one-based indexing; and\nmkl_cspblas_<character> <data><mtype><operation>( )\nfor routines with zero-based indexing.\nThe <character> field indicates the data type:\ns\nreal, single precision\nc\ncomplex, single precision\nd\nreal, double precision\nz\ncomplex, double precision\nThe <data> field indicates the sparse matrix storage format (see section Sparse Matrix Storage Formats):\ncoo\ncoordinate format\ncsr\ncompressed sparse row format and its variations\ncsc\ncompressed sparse column format and its variations\ndia\ndiagonal format\nsky\nskyline storage format\nbsr\nblock sparse row format and its variations\nThe <operation> field indicates the type of operation:\nmv\nmatrix-vector product (Level 2)\nmm\nmatrix-matrix product (Level 3)\nsv\nsolving a single triangular system (Level 2)\nsm\nsolving triangular systems with multiple right-hand sides (Level 3)\nThe field <mtype> indicates the matrix type:\nge\nsparse representation of a general matrix\nsy\nsparse representation of the upper or lower triangle of a symmetric matrix\ntr\nsparse representation of a triangular matrix\nSparse Matrix Storage Formats for Sparse BLAS Routines\nThe current version of Intel® oneAPI Math Kernel Library (oneMKL) Sparse BLAS Level 2 and Level 3 routines\nsupport the following point entry [Duff86] storage formats for sparse matrices:\n•\ncompressed sparse row format (CSR) and its variations;\n•\ncompressed sparse column format (CSC);\n•\ncoordinate format;\n•\ndiagonal format;\n•\nskyline storage format;\nand one block entry storage format:\n•\nblock sparse row format (BSR) and its variations.\nFor more information see \"Sparse Matrix Storage Formats\" in the Appendix\"Linear Solvers Basics\".\nIntel® oneAPI Math Kernel Library (oneMKL) provides auxiliary routines -matrix converters - that convert\nsparse matrix from one storage format to another.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n128\n\n\nRoutines and Supported Operations\nThis section describes operations supported by the Intel® oneAPI Math Kernel Library (oneMKL) Sparse BLAS\nLevel 2 and Level 3 routines. The following notations are used here:\n \nA is a sparse matrix;\n \nB and C are dense matrices;\n \nD is a diagonal scaling matrix;\n \nx and y are dense vectors;\n \nalpha and beta are scalars;\nop(A) is one of the possible operations:\n \nop(A) = A;\n \nop(A) = AT - transpose of A;\n \nop(A) = AH - conjugated transpose of A.\ninv(op(A)) denotes the inverse of op(A).\nThe Intel® oneAPI Math Kernel Library (oneMKL) Sparse BLAS Level 2 and Level 3 routines support the\nfollowing operations:\n•\ncomputing the vector product between a sparse matrix and a dense vector:\ny := alpha*op(A)*x + beta*y\n•\nsolving a single triangular system:\ny := alpha*inv(op(A))*x\n•\ncomputing a product between sparse matrix and dense matrix:\nC := alpha*op(A)*B + beta*C\n•\nsolving a sparse triangular system with multiple right-hand sides:\nC := alpha*inv(op(A))*B\nIntel® oneAPI Math Kernel Library (oneMKL) provides an additional set of the Sparse BLAS Level 2 routines\nwithsimplified interfaces. Each of these routines operates on a matrix of the fixed type. The following\noperations are supported:\n•\ncomputing the vector product between a sparse matrix and a dense vector (for general and symmetric\nmatrices):\ny := op(A)*x\n•\nsolving a single triangular system (for triangular matrices):\ny := inv(op(A))*x\nMatrix type is indicated by the field <mtype> in the routine name (see section Naming Conventions in Sparse\nBLAS Level 2 and Level 3).\nNOTE\nThe routines with simplified interfaces support only four sparse matrix storage formats, specifically:\n \nCSR format in the 3-array variation accepted in the direct sparse solvers and in the CXML;\n \ndiagonal format accepted in the CXML;\n \ncoordinate format;\n \nBSR format in the 3-array variation.\nNote that routines with both typical (conventional) and simplified interfaces use the same computational\nkernels that work with certain internal data structures.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n129\n\n\nThe Intel® oneAPI Math Kernel Library (oneMKL) Sparse BLAS Level 2 and Level 3 routines do not support in-\nplace operations.\nComplete list of all routines is given in the “Sparse BLAS Level 2 and Level 3 Routines”.\nInterface Consideration\nOne-Based and Zero-Based Indexing\nThe Intel® oneAPI Math Kernel Library (oneMKL) Sparse BLAS Level 2 and Level 3 routines support one-based\nand zero-based indexing of data arrays.\nRoutines with typical interfaces support zero-based indexing for the following sparse data storage formats:\nCSR, CSC, BSR, and COO. Routines with simplified interfaces support zero based indexing for the following\nsparse data storage formats: CSR, BSR, and COO. See the complete list of Sparse BLAS Level 2 and Level 3\nRoutines.\nThe one-based indexing uses the convention of starting array indices at 1. The zero-based indexing uses the\nconvention of starting array indices at 0. For example, indices of the 5-element array x can be presented in\ncase of one-based indexing as follows:\nElement index: 1 2 3 4 5\nElement value: 1.0 5.0 7.0 8.0 9.0\nand in case of zero-based indexing as follows:\nElement index: 0 1 2 3 4\nElement value: 1.0 5.0 7.0 8.0 9.0\nThe detailed descriptions of the one-based and zero-based variants of the sparse data storage formats are\ngiven in the \"Sparse Matrix Storage Formats\" in the Appendix \"Linear Solvers Basics\".\nMost parameters of the routines are identical for both one-based and zero-based indexing, but some of them\nhave certain differences. The following table lists all these differences.\nParameter\nOne-based Indexing\nZero-based Indexing\nval\nArray containing non-zero elements of the\nmatrix A, its length is . pntre[m] -\npntrb[1]\nArray containing non-zero elements of\nthe matrix A, its length is . pntre[m—1]\n- pntrb[0]\npntrb\nArray of length m. This array contains row\nindices, such that pntrb[i] -\npntrb[1]+1 is the first index of row i in\nthe arrays val and indx\nArray of length m. This array contains row\nindices, such that pntrb[i] - pntrb[0]\nis the first index of row i in the arrays\nval and indx.\npntre\nArray of length m. This array contains row\nindices, such that pntre[I] - pntrb[1]\nis the last index of row i in the arrays\nval and indx.\nArray of length m. This array contains row\nindices, such that pntre[i] -\npntrb[0]-1 is the last index of row i in\nthe arrays val and indx.\nia\nArray of length m + 1, containing indices\nof elements in the array a, such that\nia[i] is the index in the array a of the\nfirst non-zero element from the row i.\nThe value of the last element ia[m + 1]\nis equal to the number of non-zeros plus\none.\nArray of length m+1, containing indices of\nelements in the array a, such that ia[i]\nis the index in the array a of the first\nnon-zero element from the row i. The\nvalue of the last element ia[m] is equal\nto the number of non-zeros.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n130\n\n\nParameter\nOne-based Indexing\nZero-based Indexing\nldb\nSpecifies the leading dimension of b as\ndeclared in the calling (sub)program.\nSpecifies the second dimension of b as\ndeclared in the calling (sub)program.\nldc\nSpecifies the leading dimension of c as\ndeclared in the calling (sub)program.\nSpecifies the second dimension of c as\ndeclared in the calling (sub)program.\nDifferences Between Intel MKL and NIST* Interfaces\nThe Intel® oneAPI Math Kernel Library (oneMKL) Sparse BLAS Level 3 routines have the following\nconventional interfaces:\nmkl_xyyymm(transa, m, n, k, alpha, matdescra, arg(A), b, ldb, beta, c, ldc), for matrix-\nmatrix product;\nmkl_xyyysm(transa, m, n, alpha, matdescra, arg(A), b, ldb, c, ldc), for triangular solvers\nwith multiple right-hand sides.\nHere x denotes data type, and yyy - sparse matrix data structure (storage format).\nThe analogous NIST* Sparse BLAS (NSB) library routines have the following interfaces:\nxyyymm(transa, m, n, k, alpha, descra, arg(A), b, ldb, beta, c, ldc, work, lwork), for\nmatrix-matrix product;\nxyyysm(transa, m, n, unitd, dv, alpha, descra, arg(A), b, ldb, beta, c, ldc, work,\nlwork), for triangular solvers with multiple right-hand sides.\nSome similar arguments are used in both libraries. The argument transa indicates what operation is\nperformed and is slightly different in the NSB library (see Table \"Parameter transa\"). The arguments m and k\nare the number of rows and column in the matrix A, respectively, n is the number of columns in the matrix C.\nThe arguments alpha and beta are scalar alpha and beta respectively (betais not used in the Intel® oneAPI\nMath Kernel Library (oneMKL) triangular solvers.) The argumentsb and c are rectangular arrays with the\nleading dimension ldb and ldc, respectively. arg(A) denotes the list of arguments that describe the sparse\nrepresentation of A.\nParameter transa\n \nMKL interface\nNSB interface\nOperation\ndata type\nchar *\nINTEGER\n \nvalue\nN or n\n0\nop(A) = A\n \nT or t\n1\nop(A) = AT\n \nC or c\n2\nop(A) = AT or op(A) =\nAH\nParameter matdescra\nThe parameter matdescra describes the relevant characteristic of the matrix A. This manual describes\nmatdescraas an array of six elements in line with the NIST* implementation. However, only the first four\nelements of the array are used in the current versions of the Intel® oneAPI Math Kernel Library (oneMKL)\nSparse BLAS routines. Elementsmatdescra[4] and matdescra[5] are reserved for future use. Note that\nwhether matdescrais described in your application as an array of length 6 or 4 is of no importance because\nthe array is declared as a pointer in the Intel® oneAPI Math Kernel Library (oneMKL) routines. To learn more\nabout declaration of thematdescraarray, see the Sparse BLAS examples located in the Intel® oneAPI Math\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n131\n\n\nKernel Library (oneMKL) installation directory:examples/spblasc/ for C. The table below lists elements of\nthe parameter matdescra, their Fortran values, and their meanings. The parameter matdescra corresponds\nto the argument descra from NSB library.\nPossible Values of the Parameter matdescra [descra - 1]\n \nMKL interface\nNSB\ninterface\nMatrix characteristics\none-based\nindexing\nzero-based\nindexing\ndata type\nchar *\nchar *\nint *\n \n1st element\nmatdescra[1]\nmatdescra[0]\ndescra[0]\nmatrix structure\nvalue\nG\nG\n0\ngeneral\n \nS\nS\n1\nsymmetric (A = AT)\n \nH\nH\n2\nHermitian (A = (AH))\n \nT\nT\n3\ntriangular\n \nA\nA\n4\nskew(anti)-symmetric (A = -AT)\n \nD\nD\n5\ndiagonal\n2nd element\nmatdescra[2]\nmatdescra[1]\ndescra[1]\nupper/lower triangular indicator\nvalue\nL\nL\n1\nlower\n \nU\nU\n2\nupper\n3rd element\nmatdescra[3]\nmatdescra[2]\ndescra[2]\nmain diagonal type\nvalue\nN\nN\n0\nnon-unit\n \nU\nU\n1\nunit\n4th element\nmatdescra[4]\nmatdescra[3]\ndescra[3]\ntype of indexing\nvalue\nF\n1\none-based indexing\nC\n0\nzero-based indexing\nIn some cases possible element values of the parameter matdescra depend on the values of other elements.\nThe Table \"Possible Combinations of Element Values of the Parameter matdescra\" lists all possible\ncombinations of element values for both multiplication routines and triangular solvers.\nPossible Combinations of Element Values of the Parameter matdescra\nRoutines\nmatdescra[0]\nmatdescra[1]\nmatdescra[2]\nmatdescra[3]\nMultiplication\nRoutines\nG\nignored\nignored\nF (default) or C\nS or H\nL (default)\nN (default)\nF (default) or C\nS or H\nL (default)\nU\nF (default) or C\nS or H\nU\nN (default)\nF (default) or C\nS or H\nU\nU\nF (default) or C\nA\nL (default)\nignored\nF (default) or C\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n132\n\n\nRoutines\nmatdescra[0]\nmatdescra[1]\nmatdescra[2]\nmatdescra[3]\nA\nU\nignored\nF (default) or C\nMultiplication\nRoutines and\nTriangular Solvers\nT\nL\nU\nF (default) or C\nT\nL\nN\nF (default) or C\nT\nU\nU\nF (default) or C\nT\nU\nN\nF (default) or C\nD\nignored\nN (default)\nF (default) or C\nD\nignored\nU\nF (default) or C\nFor a matrix in the skyline format with the main diagonal declared to be a unit, diagonal elements must be\nstored in the sparse representation even if they are zero. In all other formats, diagonal elements can be\nstored (if needed) in the sparse representation if they are not zero.\nOperations with Partial Matrices\nOne of the distinctive feature of the Intel® oneAPI Math Kernel Library (oneMKL) Sparse BLAS routines is a\npossibility to perform operations only on partial matrices composed of certain parts (triangles and the main\ndiagonal) of the input sparse matrix. It can be done by setting properly first three elements of the\nparametermatdescra.\nAn arbitrary sparse matrix A can be decomposed as\nA = L + D + U\nwhere L is the strict lower triangle of A, U is the strict upper triangle of A, D is the main diagonal.\nTable \"Output Matrices for Multiplication Routines\" shows correspondence between the output matrices and\nvalues of the parameter matdescra for the sparse matrix A for multiplication routines.\nOutput Matrices for Multiplication Routines\nmatdescra[0]\nmatdescra[1]\nmatdescra[2]\nOutput Matrix\nG\nignored\nignored\nalpha*op(A)*x + beta*y\nalpha*op(A)*B + beta*C\nS or H\nL\nN\nalpha*op(L+D+L')*x + beta*y\nalpha*op(L+D+L')*B + beta*C\nS or H\nL\nU\nalpha*op(L+I+L')*x + beta*y\nalpha*op(L+I+L')*B + beta*C\nS or H\nU\nN\nalpha*op(U'+D+U)*x + beta*y\nalpha*op(U'+D+U)*B + beta*C\nS or H\nU\nU\nalpha*op(U'+I+U)*x + beta*y\nalpha*op(U'+I+U)*B + beta*C\nT\nL\nU\nalpha*op(L+I)*x + beta*y\nalpha*op(L+I)*B + beta*C\nT\nL\nN\nalpha*op(L+D)*x + beta*y\nalpha*op(L+D)*B + beta*C\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n133\n\n\nmatdescra[0]\nmatdescra[1]\nmatdescra[2]\nOutput Matrix\nT\nU\nU\nalpha*op(U+I)*x + beta*y\nalpha*op(U+I)*B + beta*C\nT\nU\nN\nalpha*op(U+D)*x + beta*y\nalpha*op(U+D)*B + beta*C\nA\nL\nignored\nalpha*op(L-L')*x + beta*y\nalpha*op(L-L')*B + beta*C\nA\nU\nignored\nalpha*op(U-U')*x + beta*y\nalpha*op(U-U')*B + beta*C\nD\nignored\nN\nalpha*D*x + beta*y\nalpha*D*B + beta*C\nD\nignored\nU\nalpha*x + beta*y\nalpha*B + beta*C\nTable \"Output Matrices for Triangular Solvers\" shows correspondence between the output matrices and values\nof the parameter matdescra for the sparse matrix A for triangular solvers.\nOutput Matrices for Triangular Solvers\nmatdescra[0]\nmatdescra[1]\nmatdescra[2]\nOutput Matrix\nT\nL\nN\nalpha*inv(op(L))*x\nalpha*inv(op(L))*B\nT\nL\nU\nalpha*inv(op(L))*x\nalpha*inv(op(L))*B\nT\nU\nN\nalpha*inv(op(U))*x\nalpha*inv(op(U))*B\nT\nU\nU\nalpha*inv(op(U))*x\nalpha*inv(op(U))*B\nD\nignored\nN\nalpha*inv(D)*x\nalpha*inv(D)*B\nD\nignored\nU\nalpha*x\nalpha*B\nSparse BLAS Level 2 and Level 3 Routines.\nNOTE The Intel® oneAPI Math Kernel Library (oneMKL) Sparse BLAS Level 2 and Level 3 routines are\ndeprecated. Use the corresponding routine from the Intel® oneAPI Math Kernel Library (oneMKL)\nInspector-executor Sparse BLAS interface as indicated in the description for each routine.\nTable “Sparse BLAS Level 2 and Level 3 Routines” lists the sparse BLAS Level 2 and Level 3 routines\ndescribed in more detail later in this section.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n134\n\n\nSparse BLAS Level 2 and Level 3 Routines\nRoutine/Function\nDescription\nSimplified interface, one-based indexing\nmkl_?csrgemv\nComputes matrix - vector product of a sparse general matrix\nin the CSR format (3-array variation)\nmkl_?bsrgemv\nComputes matrix - vector product of a sparse general matrix\nin the BSR format (3-array variation).\nmkl_?coogemv\nComputes matrix - vector product of a sparse general matrix\nin the coordinate format.\nmkl_?diagemv\nComputes matrix - vector product of a sparse general matrix\nin the diagonal format.\nmkl_?csrsymv\nComputes matrix - vector product of a sparse symmetrical\nmatrix in the CSR format (3-array variation)\nmkl_?bsrsymv\nComputes matrix - vector product of a sparse symmetrical\nmatrix in the BSR format (3-array variation).\nmkl_?coosymv\nComputes matrix - vector product of a sparse symmetrical\nmatrix in the coordinate format.\nmkl_?diasymv\nComputes matrix - vector product of a sparse symmetrical\nmatrix in the diagonal format.\nmkl_?csrtrsv\nTriangular solvers with simplified interface for a sparse matrix\nin the CSR format (3-array variation).\nmkl_?bsrtrsv\nTriangular solver with simplified interface for a sparse matrix\nin the BSR format (3-array variation).\nmkl_?cootrsv\nTriangular solvers with simplified interface for a sparse matrix\nin the coordinate format.\nmkl_?diatrsv\nTriangular solvers with simplified interface for a sparse matrix\nin the diagonal format.\nSimplified interface, zero-based indexing\nmkl_cspblas_?csrgemv\nComputes matrix - vector product of a sparse general matrix\nin the CSR format (3-array variation) with zero-based\nindexing.\nmkl_cspblas_?bsrgemv\nComputes matrix - vector product of a sparse general matrix\nin the BSR format (3-array variation)with zero-based indexing.\nmkl_cspblas_?coogemv\nComputes matrix - vector product of a sparse general matrix\nin the coordinate format with zero-based indexing.\nmkl_cspblas_?csrsymv\nComputes matrix - vector product of a sparse symmetrical\nmatrix in the CSR format (3-array variation) with zero-based\nindexing\nmkl_cspblas_?bsrsymv\nComputes matrix - vector product of a sparse symmetrical\nmatrix in the BSR format (3-array variation) with zero-based\nindexing.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n135\n\n\nRoutine/Function\nDescription\nmkl_cspblas_?coosymv\nComputes matrix - vector product of a sparse symmetrical\nmatrix in the coordinate format with zero-based indexing.\nmkl_cspblas_?csrtrsv\nTriangular solvers with simplified interface for a sparse matrix\nin the CSR format (3-array variation) with zero-based\nindexing.\nmkl_cspblas_?bsrtrsv\nTriangular solver with simplified interface for a sparse matrix\nin the BSR format (3-array variation) with zero-based\nindexing.\nmkl_cspblas_?cootrsv\nTriangular solver with simplified interface for a sparse matrix\nin the coordinate format with zero-based indexing.\nTypical (conventional) interface, one-based and zero-based indexing\nmkl_?csrmv\nComputes matrix - vector product of a sparse matrix in the\nCSR format.\nmkl_?bsrmv\nComputes matrix - vector product of a sparse matrix in the\nBSR format.\nmkl_?cscmv\nComputes matrix - vector product for a sparse matrix in the\nCSC format.\nmkl_?coomv\nComputes matrix - vector product for a sparse matrix in the\ncoordinate format.\nmkl_?csrsv\nSolves a system of linear equations for a sparse matrix in the\nCSR format.\nmkl_?bsrsv\nSolves a system of linear equations for a sparse matrix in the\nBSR format.\nmkl_?cscsv\nSolves a system of linear equations for a sparse matrix in the\nCSC format.\nmkl_?coosv\nSolves a system of linear equations for a sparse matrix in the\ncoordinate format.\nmkl_?csrmm\nComputes matrix - matrix product of a sparse matrix in the\nCSR format\nmkl_?bsrmm\nComputes matrix - matrix product of a sparse matrix in the\nBSR format.\nmkl_?cscmm\nComputes matrix - matrix product of a sparse matrix in the\nCSC format\nmkl_?coomm\nComputes matrix - matrix product of a sparse matrix in the\ncoordinate format.\nmkl_?csrsm\nSolves a system of linear matrix equations for a sparse matrix\nin the CSR format.\nmkl_?bsrsm\nSolves a system of linear matrix equations for a sparse matrix\nin the BSR format.\nmkl_?cscsm\nSolves a system of linear matrix equations for a sparse matrix\nin the CSC format.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n136\n\n\nRoutine/Function\nDescription\nmkl_?coosm\nSolves a system of linear matrix equations for a sparse matrix\nin the coordinate format.\nTypical (conventional) interface, one-based indexing\nmkl_?diamv\nComputes matrix - vector product of a sparse matrix in the\ndiagonal format.\nmkl_?skymv\nComputes matrix - vector product for a sparse matrix in the\nskyline storage format.\nmkl_?diasv\nSolves a system of linear equations for a sparse matrix in the\ndiagonal format.\nmkl_?skysv\nSolves a system of linear equations for a sparse matrix in the\nskyline format.\nmkl_?diamm\nComputes matrix - matrix product of a sparse matrix in the\ndiagonal format.\nmkl_?skymm\nComputes matrix - matrix product of a sparse matrix in the\nskyline storage format.\nmkl_?diasm\nSolves a system of linear matrix equations for a sparse matrix\nin the diagonal format.\nmkl_?skysm\nSolves a system of linear matrix equations for a sparse matrix\nin the skyline storage format.\nAuxiliary routines\nMatrix converters\nmkl_?dnscsr\nConverts a sparse matrix in uncompressed representation to\nCSR format (3-array variation) and vice versa.\nmkl_?csrcoo\nConverts a sparse matrix in CSR format (3-array variation) to\ncoordinate format and vice versa.\nmkl_?csrbsr\nConverts a sparse matrix in CSR format to BSR format (3-\narray variations) and vice versa.\nmkl_?csrcsc\nConverts a sparse matrix in CSR format to CSC format and\nvice versa (3-array variations).\nmkl_?csrdia\nConverts a sparse matrix in CSR format (3-array variation) to\ndiagonal format and vice versa.\nmkl_?csrsky\nConverts a sparse matrix in CSR format (3-array variation) to\nsky line format and vice versa.\nOperations on sparse matrices\nmkl_?csradd\nComputes the sum of two sparse matrices stored in the CSR\nformat (3-array variation) with one-based indexing.\nmkl_?csrmultcsr\nComputes the product of two sparse matrices stored in the\nCSR format (3-array variation) with one-based indexing.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n137\n\n\nRoutine/Function\nDescription\nmkl_?csrmultd\nComputes product of two sparse matrices stored in the CSR\nformat (3-array variation) with one-based indexing. The result\nis stored in the dense matrix.\nmkl_?csrgemv\nComputes matrix - vector product of a sparse general\nmatrix stored in the CSR format (3-array variation)\nwith one-based indexing (deprecated).\nSyntax\nvoid mkl_scsrgemv (const char *transa , const MKL_INT *m , const float *a , const\nMKL_INT *ia , const MKL_INT *ja , const float *x , float *y );\nvoid mkl_dcsrgemv (const char *transa , const MKL_INT *m , const double *a , const\nMKL_INT *ia , const MKL_INT *ja , const double *x , double *y );\nvoid mkl_ccsrgemv (const char *transa , const MKL_INT *m , const MKL_Complex8 *a ,\nconst MKL_INT *ia , const MKL_INT *ja , const MKL_Complex8 *x , MKL_Complex8 *y );\nvoid mkl_zcsrgemv (const char *transa , const MKL_INT *m , const MKL_Complex16 *a ,\nconst MKL_INT *ia , const MKL_INT *ja , const MKL_Complex16 *x , MKL_Complex16 *y );\nInclude Files\n•\nmkl.h\nDescription\nThis routine is deprecated. Use mkl_sparse_?_mvfrom the Intel® oneAPI Math Kernel Library (oneMKL)\nInspector-executor Sparse BLAS interface instead.\nThe mkl_?csrgemv routine performs a matrix-vector operation defined as\ny := A*x\nor\ny := AT*x,\nwhere:\nx and y are vectors,\nA is an m-by-m sparse square matrix in the CSR format (3-array variation), AT is the transpose of A.\nNOTE\nThis routine supports only one-based indexing of the input arrays.\nInput Parameters\ntransa\nSpecifies the operation.\nIf transa = 'N' or 'n', then as y := A*x\nIf transa = 'T' or 't' or 'C' or 'c', then y := AT*x,\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n138\n\n\nm\nNumber of rows of the matrix A.\na\nArray containing non-zero elements of the matrix A. Its length is equal to\nthe number of non-zero elements in the matrix A. Refer to values array\ndescription in Sparse Matrix Storage Formats for more details.\nia\nArray of length m + 1, containing indices of elements in the array a, such\nthat ia[i] - ia[0] is the index in the array a of the first non-zero\nelement from the row i. The value of the last element ia[m] - ia[0] is\nequal to the number of non-zeros. Refer to rowIndex array description in \nSparse Matrix Storage Formats for more details.\nja\nArray containing the column indices plus one for each non-zero element of\nthe matrix A.\nIts length is equal to the length of the array a. Refer to columns array\ndescription in Sparse Matrix Storage Formats for more details.\nx\nArray, size is m.\nOn entry, the array x must contain the vector x.\nOutput Parameters\ny\nArray, size at least m.\nOn exit, the array y must contain the vector y.\nmkl_?bsrgemv\nComputes matrix - vector product of a sparse general\nmatrix stored in the BSR format (3-array variation)\nwith one-based indexing (deprecated).\nSyntax\nvoid mkl_sbsrgemv (const char *transa , const MKL_INT *m , const MKL_INT *lb , const\nfloat *a , const MKL_INT *ia , const MKL_INT *ja , const float *x , float *y );\nvoid mkl_dbsrgemv (const char *transa , const MKL_INT *m , const MKL_INT *lb , const\ndouble *a , const MKL_INT *ia , const MKL_INT *ja , const double *x , double *y );\nvoid mkl_cbsrgemv (const char *transa , const MKL_INT *m , const MKL_INT *lb , const\nMKL_Complex8 *a , const MKL_INT *ia , const MKL_INT *ja , const MKL_Complex8 *x ,\nMKL_Complex8 *y );\nvoid mkl_zbsrgemv (const char *transa , const MKL_INT *m , const MKL_INT *lb , const\nMKL_Complex16 *a , const MKL_INT *ia , const MKL_INT *ja , const MKL_Complex16 *x ,\nMKL_Complex16 *y );\nInclude Files\n•\nmkl.h\nDescription\nThis routine is deprecated. Use mkl_sparse_?_mvfrom the Intel® oneAPI Math Kernel Library (oneMKL)\nInspector-executor Sparse BLAS interface instead.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n139\n\n\nThe mkl_?bsrgemv routine performs a matrix-vector operation defined as\ny := A*x\nor\ny := AT*x,\nwhere:\nx and y are vectors,\nA is an m-by-m block sparse square matrix in the BSR format (3-array variation), AT is the transpose of A.\nNOTE\nThis routine supports only one-based indexing of the input arrays.\nInput Parameters\ntransa\nSpecifies the operation.\nIf transa = 'N' or 'n', then the matrix-vector product is computed as\ny := A*x\nIf transa = 'T' or 't' or 'C' or 'c', then the matrix-vector product is\ncomputed as y := AT*x,\nm\nNumber of block rows of the matrix A.\nlb\nSize of the block in the matrix A.\na\nArray containing elements of non-zero blocks of the matrix A. Its length is\nequal to the number of non-zero blocks in the matrix A multiplied by lb*lb.\nRefer to values array description in BSR Format for more details.\nia\nArray of length (m + 1), containing indices of block in the array a, such\nthat ia[i] - ia[0] is the index in the array a of the first non-zero\nelement from the row i. The value of the last element ia[m] - ia[0] is\nequal to the number of non-zero blocks. Refer to rowIndex array\ndescription in BSR Format for more details.\nja\nArray containing the column indices plus one for each non-zero block in the\nmatrix A.\nIts length is equal to the number of non-zero blocks of the matrix A. Refer\nto columns array description in BSR Format for more details.\nx\nArray, size (m*lb).\nOn entry, the array x must contain the vector x.\nOutput Parameters\ny\nArray, size at least (m*lb).\nOn exit, the array y must contain the vector y.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n140\n\n\nmkl_?coogemv\nComputes matrix-vector product of a sparse general\nmatrix stored in the coordinate format with one-based\nindexing (deprecated).\nSyntax\nvoid mkl_scoogemv (const char *transa , const MKL_INT *m , const float *val , const\nMKL_INT *rowind , const MKL_INT *colind , const MKL_INT *nnz , const float *x , float\n*y );\nvoid mkl_dcoogemv (const char *transa , const MKL_INT *m , const double *val , const\nMKL_INT *rowind , const MKL_INT *colind , const MKL_INT *nnz , const double *x , double\n*y );\nvoid mkl_ccoogemv (const char *transa , const MKL_INT *m , const MKL_Complex8 *val ,\nconst MKL_INT *rowind , const MKL_INT *colind , const MKL_INT *nnz , const MKL_Complex8\n*x , MKL_Complex8 *y );\nvoid mkl_zcoogemv (const char *transa , const MKL_INT *m , const MKL_Complex16 *val ,\nconst MKL_INT *rowind , const MKL_INT *colind , const MKL_INT *nnz , const\nMKL_Complex16 *x , MKL_Complex16 *y );\nInclude Files\n•\nmkl.h\nDescription\nThis routine is deprecated. Use mkl_sparse_?_mvfrom the Intel® oneAPI Math Kernel Library (oneMKL)\nInspector-executor Sparse BLAS interface instead.\nThe mkl_?coogemv routine performs a matrix-vector operation defined as\ny := A*x\nor\ny := AT*x,\nwhere:\nx and y are vectors,\nA is an m-by-m sparse square matrix in the coordinate format, AT is the transpose of A.\nNOTE\nThis routine supports only one-based indexing of the input arrays.\nInput Parameters\ntransa\nSpecifies the operation.\nIf transa = 'N' or 'n', then the matrix-vector product is computed as\ny := A*x\nIf transa = 'T' or 't' or 'C' or 'c', then the matrix-vector product is\ncomputed as y := AT*x,\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n141\n\n\nm\nNumber of rows of the matrix A.\nval\nArray of length nnz, contains non-zero elements of the matrix A in the\narbitrary order.\nRefer to values array description in Coordinate Format for more details.\nrowind\nArray of length nnz, contains the row indices plus one for each non-zero\nelement of the matrix A.\nRefer to rows array description in Coordinate Format for more details.\ncolind\nArray of length nnz, contains the column indices plus one for each non-zero\nelement of the matrix A. Refer to columns array description in Coordinate\nFormat for more details.\nnnz\nSpecifies the number of non-zero element of the matrix A.\nRefer to nnz description in Coordinate Format for more details.\nx\nArray, size is m.\nOne entry, the array x must contain the vector x.\nOutput Parameters\ny\nArray, size at least m.\nOn exit, the array y must contain the vector y.\nmkl_?diagemv\nComputes matrix - vector product of a sparse general\nmatrix stored in the diagonal format with one-based\nindexing (deprecated).\nSyntax\nvoid mkl_sdiagemv (const char *transa , const MKL_INT *m , const float *val , const\nMKL_INT *lval , const MKL_INT *idiag , const MKL_INT *ndiag , const float *x , float\n*y );\nvoid mkl_ddiagemv (const char *transa , const MKL_INT *m , const double *val , const\nMKL_INT *lval , const MKL_INT *idiag , const MKL_INT *ndiag , const double *x , double\n*y );\nvoid mkl_cdiagemv (const char *transa , const MKL_INT *m , const MKL_Complex8 *val ,\nconst MKL_INT *lval , const MKL_INT *idiag , const MKL_INT *ndiag , const MKL_Complex8\n*x , MKL_Complex8 *y );\nvoid mkl_zdiagemv (const char *transa , const MKL_INT *m , const MKL_Complex16 *val ,\nconst MKL_INT *lval , const MKL_INT *idiag , const MKL_INT *ndiag , const MKL_Complex16\n*x , MKL_Complex16 *y );\nInclude Files\n•\nmkl.h\nDescription\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n142\n\n\nThis routine is deprecated, but no replacement is available yet in the Inspector-Executor Sparse BLAS API\ninterfaces. You can continue using this routine until a replacement is provided and this can be fully removed.\nThe mkl_?diagemv routine performs a matrix-vector operation defined as\ny := A*x\nor\ny := AT*x,\nwhere:\nx and y are vectors,\nA is an m-by-m sparse square matrix in the diagonal storage format, AT is the transpose of A.\nNOTE\nThis routine supports only one-based indexing of the input arrays.\nInput Parameters\ntransa\nSpecifies the operation.\nIf transa = 'N' or 'n', then y := A*x\nIf transa = 'T' or 't' or 'C' or 'c', then y := AT*x,\nm\nNumber of rows of the matrix A.\nval\nTwo-dimensional array of size lval*ndiag, contains non-zero diagonals of\nthe matrix A. Refer to values array description in Diagonal Storage Scheme\nfor more details.\nlval\nLeading dimension of vallval≥m. Refer to lval description in Diagonal\nStorage Scheme for more details.\nidiag\nArray of length ndiag, contains the distances between main diagonal and\neach non-zero diagonals in the matrix A.\nRefer to distance array description in Diagonal Storage Scheme for more\ndetails.\nndiag\nSpecifies the number of non-zero diagonals of the matrix A.\nx\nArray, size is m.\nOn entry, the array x must contain the vector x.\nOutput Parameters\ny\nArray, size at least m.\nOn exit, the array y must contain the vector y.\nmkl_?csrsymv\nComputes matrix - vector product of a sparse\nsymmetrical matrix stored in the CSR format (3-array\nvariation) with one-based indexing (deprecated).\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n143\n\n\nSyntax\nvoid mkl_scsrsymv (const char *uplo , const MKL_INT *m , const float *a , const MKL_INT\n*ia , const MKL_INT *ja , const float *x , float *y );\nvoid mkl_dcsrsymv (const char *uplo , const MKL_INT *m , const double *a , const\nMKL_INT *ia , const MKL_INT *ja , const double *x , double *y );\nvoid mkl_ccsrsymv (const char *uplo , const MKL_INT *m , const MKL_Complex8 *a , const\nMKL_INT *ia , const MKL_INT *ja , const MKL_Complex8 *x , MKL_Complex8 *y );\nvoid mkl_zcsrsymv (const char *uplo , const MKL_INT *m , const MKL_Complex16 *a , const\nMKL_INT *ia , const MKL_INT *ja , const MKL_Complex16 *x , MKL_Complex16 *y );\nInclude Files\n•\nmkl.h\nDescription\nThis routine is deprecated. Use mkl_sparse_?_mvfrom the Intel® oneAPI Math Kernel Library (oneMKL)\nInspector-executor Sparse BLAS interface instead.\nThe mkl_?csrsymv routine performs a matrix-vector operation defined as\ny := A*x\nwhere:\nx and y are vectors,\nA is an upper or lower triangle of the symmetrical sparse matrix in the CSR format (3-array variation).\nNOTE\nThis routine supports only one-based indexing of the input arrays.\nInput Parameters\n \nuplo\nSpecifies whether the upper or low triangle of the matrix A is used.\nIf uplo = 'U' or 'u', then the upper triangle of the matrix A is used.\nIf uplo = 'L' or 'l', then the low triangle of the matrix A is used.\nm\nNumber of rows of the matrix A.\na\nArray containing non-zero elements of the matrix A. Its length is equal to\nthe number of non-zero elements in the matrix A. Refer to values array\ndescription in Sparse Matrix Storage Formats for more details.\nia\nArray of length m + 1, containing indices of elements in the array a, such\nthat ia[i] - ia[0] is the index in the array a of the first non-zero\nelement from the row i. The value of the last element ia[m] - ia[0] is\nequal to the number of non-zeros. Refer to rowIndex array description in \nSparse Matrix Storage Formats for more details.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n144\n\n\nja\nArray containing the column indices plus one for each non-zero element of\nthe matrix A.\nIts length is equal to the length of the array a. Refer to columns array\ndescription in Sparse Matrix Storage Formats for more details.\nx\nArray, size is m.\nOn entry, the array x must contain the vector x.\nOutput Parameters\ny\nArray, size at least m.\nOn exit, the array y must contain the vector y.\nmkl_?bsrsymv\nComputes matrix-vector product of a sparse\nsymmetrical matrix stored in the BSR format (3-array\nvariation) with one-based indexing (deprecated).\nSyntax\nvoid mkl_sbsrsymv (const char *uplo , const MKL_INT *m , const MKL_INT *lb , const\nfloat *a , const MKL_INT *ia , const MKL_INT *ja , const float *x , float *y );\nvoid mkl_dbsrsymv (const char *uplo , const MKL_INT *m , const MKL_INT *lb , const\ndouble *a , const MKL_INT *ia , const MKL_INT *ja , const double *x , double *y );\nvoid mkl_cbsrsymv (const char *uplo , const MKL_INT *m , const MKL_INT *lb , const\nMKL_Complex8 *a , const MKL_INT *ia , const MKL_INT *ja , const MKL_Complex8 *x ,\nMKL_Complex8 *y );\nvoid mkl_zbsrsymv (const char *uplo , const MKL_INT *m , const MKL_INT *lb , const\nMKL_Complex16 *a , const MKL_INT *ia , const MKL_INT *ja , const MKL_Complex16 *x ,\nMKL_Complex16 *y );\nInclude Files\n•\nmkl.h\nDescription\nThis routine is deprecated. Use mkl_sparse_?_mvfrom the Intel® oneAPI Math Kernel Library (oneMKL)\nInspector-executor Sparse BLAS interface instead.\nThe mkl_?bsrsymv routine performs a matrix-vector operation defined as\ny := A*x\nwhere:\nx and y are vectors,\nA is an upper or lower triangle of the symmetrical sparse matrix in the BSR format (3-array variation).\nNOTE\nThis routine supports only one-based indexing of the input arrays.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n145\n\n\nInput Parameters\n \nuplo\nSpecifies whether the upper or low triangle of the matrix A is considered.\nIf uplo = 'U' or 'u', then the upper triangle of the matrix A is used.\nIf uplo = 'L' or 'l', then the low triangle of the matrix A is used.\nm\nNumber of block rows of the matrix A.\nlb\nSize of the block in the matrix A.\na\nArray containing elements of non-zero blocks of the matrix A. Its length is\nequal to the number of non-zero blocks in the matrix A multiplied by lb*lb.\nRefer to values array description in BSR Format for more details.\nia\nArray of length (m + 1), containing indices of block in the array a, such\nthat ia[i] - ia[0] is the index in the array a of the first non-zero\nelement from the row i. The value of the last element ia[m] - ia[0] is\nequal to the number of non-zero blocks. Refer to rowIndex array\ndescription in BSR Format for more details.\nja\nArray containing the column indices plus one for each non-zero block in the\nmatrix A.\nIts length is equal to the number of non-zero blocks of the matrix A. Refer\nto columns array description in BSR Format for more details.\nx\nArray, size (m*lb).\nOn entry, the array x must contain the vector x.\nOutput Parameters\ny\nArray, size at least (m*lb).\nOn exit, the array y must contain the vector y.\nmkl_?coosymv\nComputes matrix - vector product of a sparse\nsymmetrical matrix stored in the coordinate format\nwith one-based indexing (deprecated).\nSyntax\nvoid mkl_scoosymv (const char *uplo , const MKL_INT *m , const float *val , const\nMKL_INT *rowind , const MKL_INT *colind , const MKL_INT *nnz , const float *x , float\n*y );\nvoid mkl_dcoosymv (const char *uplo , const MKL_INT *m , const double *val , const\nMKL_INT *rowind , const MKL_INT *colind , const MKL_INT *nnz , const double *x , double\n*y );\nvoid mkl_ccoosymv (const char *uplo , const MKL_INT *m , const MKL_Complex8 *val ,\nconst MKL_INT *rowind , const MKL_INT *colind , const MKL_INT *nnz , const MKL_Complex8\n*x , MKL_Complex8 *y );\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n146\n\n\nvoid mkl_zcoosymv (const char *uplo , const MKL_INT *m , const MKL_Complex16 *val ,\nconst MKL_INT *rowind , const MKL_INT *colind , const MKL_INT *nnz , const\nMKL_Complex16 *x , MKL_Complex16 *y );\nInclude Files\n•\nmkl.h\nDescription\nThis routine is deprecated. Use mkl_sparse_?_mvfrom the Intel® oneAPI Math Kernel Library (oneMKL)\nInspector-executor Sparse BLAS interface instead.\nThe mkl_?coosymv routine performs a matrix-vector operation defined as\ny := A*x\nwhere:\nx and y are vectors,\nA is an upper or lower triangle of the symmetrical sparse matrix in the coordinate format.\nNOTE\nThis routine supports only one-based indexing of the input arrays.\nInput Parameters\nuplo\nSpecifies whether the upper or low triangle of the matrix A is used.\nIf uplo = 'U' or 'u', then the upper triangle of the matrix A is used.\nIf uplo = 'L' or 'l', then the low triangle of the matrix A is used.\nm\nNumber of rows of the matrix A.\nval\nArray of length nnz, contains non-zero elements of the matrix A in the\narbitrary order.\nRefer to values array description in Coordinate Format for more details.\nrowind\nArray of length nnz, contains the row indices plus one for each non-zero\nelement of the matrix A.\nRefer to rows array description in Coordinate Format for more details.\ncolind\nArray of length nnz, contains the column indices plus one for each non-zero\nelement of the matrix A. Refer to columns array description in Coordinate\nFormat for more details.\nnnz\nSpecifies the number of non-zero element of the matrix A.\nRefer to nnz description in Coordinate Format for more details.\nx\nArray, size is m.\nOn entry, the array x must contain the vector x.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n147\n\n\nOutput Parameters\ny\nArray, size at least m.\nOn exit, the array y must contain the vector y.\nmkl_?diasymv\nComputes matrix - vector product of a sparse\nsymmetrical matrix stored in the diagonal format with\none-based indexing (deprecated).\nSyntax\nvoid mkl_sdiasymv (const char *uplo , const MKL_INT *m , const float *val , const\nMKL_INT *lval , const MKL_INT *idiag , const MKL_INT *ndiag , const float *x , float\n*y );\nvoid mkl_ddiasymv (const char *uplo , const MKL_INT *m , const double *val , const\nMKL_INT *lval , const MKL_INT *idiag , const MKL_INT *ndiag , const double *x , double\n*y );\nvoid mkl_cdiasymv (const char *uplo , const MKL_INT *m , const MKL_Complex8 *val ,\nconst MKL_INT *lval , const MKL_INT *idiag , const MKL_INT *ndiag , const MKL_Complex8\n*x , MKL_Complex8 *y );\nvoid mkl_zdiasymv (const char *uplo , const MKL_INT *m , const MKL_Complex16 *val ,\nconst MKL_INT *lval , const MKL_INT *idiag , const MKL_INT *ndiag , const MKL_Complex16\n*x , MKL_Complex16 *y );\nInclude Files\n•\nmkl.h\nDescription\nThis routine is deprecated, but no replacement is available yet in the Inspector-Executor Sparse BLAS API\ninterfaces. You can continue using this routine until a replacement is provided and this can be fully removed.\nThe mkl_?diasymv routine performs a matrix-vector operation defined as\ny := A*x\nwhere:\nx and y are vectors,\nA is an upper or lower triangle of the symmetrical sparse matrix.\nNOTE\nThis routine supports only one-based indexing of the input arrays.\nInput Parameters\nuplo\nSpecifies whether the upper or low triangle of the matrix A is used.\nIf uplo = 'U' or 'u', then the upper triangle of the matrix A is used.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n148\n\n\nIf uplo = 'L' or 'l', then the low triangle of the matrix A is used.\nm\nNumber of rows of the matrix A.\nval\nTwo-dimensional array of size lval by ndiag, contains non-zero diagonals\nof the matrix A. Refer to values array description in Diagonal Storage\nScheme for more details.\nlval\nLeading dimension of val, lval≥m. Refer to lval description in Diagonal\nStorage Scheme for more details.\nidiag\nArray of length ndiag, contains the distances between main diagonal and\neach non-zero diagonals in the matrix A.\nRefer to distance array description in Diagonal Storage Scheme for more\ndetails.\nndiag\nSpecifies the number of non-zero diagonals of the matrix A.\nx\nArray, size is m.\nOn entry, the array x must contain the vector x.\nOutput Parameters\ny\nArray, size at least m.\nOn exit, the array y must contain the vector y.\nmkl_?csrtrsv\nTriangular solvers with simplified interface for a sparse\nmatrix in the CSR format (3-array variation) with one-\nbased indexing (deprecated).\nSyntax\nvoid mkl_scsrtrsv (const char *uplo , const char *transa , const char *diag , const\nMKL_INT *m , const float *a , const MKL_INT *ia , const MKL_INT *ja , const float *x ,\nfloat *y );\nvoid mkl_dcsrtrsv (const char *uplo , const char *transa , const char *diag , const\nMKL_INT *m , const double *a , const MKL_INT *ia , const MKL_INT *ja , const double\n*x , double *y );\nvoid mkl_ccsrtrsv (const char *uplo , const char *transa , const char *diag , const\nMKL_INT *m , const MKL_Complex8 *a , const MKL_INT *ia , const MKL_INT *ja , const\nMKL_Complex8 *x , MKL_Complex8 *y );\nvoid mkl_zcsrtrsv (const char *uplo , const char *transa , const char *diag , const\nMKL_INT *m , const MKL_Complex16 *a , const MKL_INT *ia , const MKL_INT *ja , const\nMKL_Complex16 *x , MKL_Complex16 *y );\nInclude Files\n•\nmkl.h\nDescription\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n149\n\n\nThis routine is deprecated. Use mkl_sparse_?_trsvfrom the Intel® oneAPI Math Kernel Library (oneMKL)\nInspector-executor Sparse BLAS interface instead.\nThe mkl_?csrtrsv routine solves a system of linear equations with matrix-vector operations for a sparse\nmatrix stored in the CSR format (3 array variation):\nA*y = x\nor\nAT*y = x,\nwhere:\nx and y are vectors,\nA is a sparse upper or lower triangular matrix with unit or non-unit main diagonal, AT is the transpose of A.\nNOTE\nThis routine supports only one-based indexing of the input arrays.\nInput Parameters\nuplo\nSpecifies whether the upper or low triangle of the matrix A is used.\nIf uplo = 'U' or 'u', then the upper triangle of the matrix A is used.\nIf uplo = 'L' or 'l', then the low triangle of the matrix A is used.\ntransa\nSpecifies the system of linear equations.\nIf transa = 'N' or 'n', then A*y = x\nIf transa = 'T' or 't' or 'C' or 'c', then AT*y = x,\ndiag\nSpecifies whether A is unit triangular.\nIf diag = 'U' or 'u', then A is a unit triangular.\nIf diag = 'N' or 'n', then A is not unit triangular.\nm\nNumber of rows of the matrix A.\na\nArray containing non-zero elements of the matrix A. Its length is equal to\nthe number of non-zero elements in the matrix A. Refer to values array\ndescription in Sparse Matrix Storage Formats for more details.\nNOTE\nThe non-zero elements of the given row of the matrix must be\nstored in the same order as they appear in the row (from left to\nright).\nNo diagonal element can be omitted from a sparse storage if the solver\nis called with the non-unit indicator.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n150\n\n\nia\nArray of length m + 1, containing indices of elements in the array a, such\nthat ia[i] - ia[0] is the index in the array a of the first non-zero\nelement from the row i. The value of the last element ia[m] - ia[0] is\nequal to the number of non-zeros. Refer to rowIndex array description in \nSparse Matrix Storage Formats for more details.\nja\nArray containing the column indices plus one for each non-zero element of\nthe matrix A.\nIts length is equal to the length of the array a. Refer to columns array\ndescription in Sparse Matrix Storage Formats for more details.\nNOTE\nColumn indices must be sorted in increasing order for each row.\nx\nArray, size is m.\nOn entry, the array x must contain the vector x.\nOutput Parameters\ny\nArray, size at least m.\nContains the vector y.\nmkl_?bsrtrsv\nTriangular solver with simplified interface for a sparse\nmatrix stored in the BSR format (3-array variation)\nwith one-based indexing (deprecated).\nSyntax\nvoid mkl_sbsrtrsv (const char *uplo , const char *transa , const char *diag , const\nMKL_INT *m , const MKL_INT *lb , const float *a , const MKL_INT *ia , const MKL_INT\n*ja , const float *x , float *y );\nvoid mkl_dbsrtrsv (const char *uplo , const char *transa , const char *diag , const\nMKL_INT *m , const MKL_INT *lb , const double *a , const MKL_INT *ia , const MKL_INT\n*ja , const double *x , double *y );\nvoid mkl_cbsrtrsv (const char *uplo , const char *transa , const char *diag , const\nMKL_INT *m , const MKL_INT *lb , const MKL_Complex8 *a , const MKL_INT *ia , const\nMKL_INT *ja , const MKL_Complex8 *x , MKL_Complex8 *y );\nvoid mkl_zbsrtrsv (const char *uplo , const char *transa , const char *diag , const\nMKL_INT *m , const MKL_INT *lb , const MKL_Complex16 *a , const MKL_INT *ia , const\nMKL_INT *ja , const MKL_Complex16 *x , MKL_Complex16 *y );\nInclude Files\n•\nmkl.h\nDescription\nThis routine is deprecated. Use mkl_sparse_?_trsvfrom the Intel® oneAPI Math Kernel Library (oneMKL)\nInspector-executor Sparse BLAS interface instead.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n151\n\n\nThe mkl_?bsrtrsv routine solves a system of linear equations with matrix-vector operations for a sparse\nmatrix stored in the BSR format (3-array variation) :\ny := A*x\nor\ny := AT*x,\nwhere:\nx and y are vectors,\nA is a sparse upper or lower triangular matrix with unit or non-unit main diagonal, AT is the transpose of A.\nNOTE\nThis routine supports only one-based indexing of the input arrays.\nInput Parameters\nuplo\nSpecifies the upper or low triangle of the matrix A is used.\nIf uplo = 'U' or 'u', then the upper triangle of the matrix A is used.\nIf uplo = 'L' or 'l', then the low triangle of the matrix A is used.\ntransa\nSpecifies the operation.\nIf transa = 'N' or 'n', then the matrix-vector product is computed as\ny := A*x\nIf transa = 'T' or 't' or 'C' or 'c', then the matrix-vector product is\ncomputed as y := AT*x.\ndiag\nSpecifies whether A is a unit triangular matrix.\nIf diag = 'U' or 'u', then A is a unit triangular.\nIf diag = 'N' or 'n', then A is not a unit triangular.\nm\nNumber of block rows of the matrix A.\nlb\nSize of the block in the matrix A.\na\nArray containing elements of non-zero blocks of the matrix A. Its length is\nequal to the number of non-zero blocks in the matrix A multiplied by lb*lb.\nRefer to values array description in BSR Format for more details.\nNOTE\nThe non-zero elements of the given row of the matrix must be\nstored in the same order as they appear in the row (from left to\nright).\nNo diagonal element can be omitted from a sparse storage if the solver\nis called with the non-unit indicator.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n152\n\n\nia\nArray of length (m + 1), containing indices of block in the array a, such\nthat ia[I] - ia[0] is the index in the array a of the first non-zero\nelement from the row I. The value of the last element ia[m] - ia[0] is\nequal to the number of non-zero blocks. Refer to rowIndex array\ndescription in BSR Format for more details.\nja\nArray containing the column indices plus one for each non-zero block in the\nmatrix A.\nIts length is equal to the number of non-zero blocks of the matrix A. Refer\nto columns array description in BSR Format for more details.\nx\nArray, size (m*lb).\nOn entry, the array x must contain the vector x.\nOutput Parameters\ny\nArray, size at least (m*lb).\nOn exit, the array y must contain the vector y.\nmkl_?cootrsv\nTriangular solvers with simplified interface for a sparse\nmatrix in the coordinate format with one-based\nindexing (deprecated).\nSyntax\nvoid mkl_scootrsv (const char *uplo , const char *transa , const char *diag , const\nMKL_INT *m , const float *val , const MKL_INT *rowind , const MKL_INT *colind , const\nMKL_INT *nnz , const float *x , float *y );\nvoid mkl_dcootrsv (const char *uplo , const char *transa , const char *diag , const\nMKL_INT *m , const double *val , const MKL_INT *rowind , const MKL_INT *colind , const\nMKL_INT *nnz , const double *x , double *y );\nvoid mkl_ccootrsv (const char *uplo , const char *transa , const char *diag , const\nMKL_INT *m , const MKL_Complex8 *val , const MKL_INT *rowind , const MKL_INT *colind ,\nconst MKL_INT *nnz , const MKL_Complex8 *x , MKL_Complex8 *y );\nvoid mkl_zcootrsv (const char *uplo , const char *transa , const char *diag , const\nMKL_INT *m , const MKL_Complex16 *val , const MKL_INT *rowind , const MKL_INT *colind ,\nconst MKL_INT *nnz , const MKL_Complex16 *x , MKL_Complex16 *y );\nInclude Files\n•\nmkl.h\nDescription\nThis routine is deprecated. Use mkl_sparse_?_trsvfrom the Intel® oneAPI Math Kernel Library (oneMKL)\nInspector-executor Sparse BLAS interface instead.\nThe mkl_?cootrsv routine solves a system of linear equations with matrix-vector operations for a sparse\nmatrix stored in the coordinate format:\nA*y = x\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n153\n\n\nor\nAT*y = x,\nwhere:\nx and y are vectors,\nA is a sparse upper or lower triangular matrix with unit or non-unit main diagonal, AT is the transpose of A.\nNOTE\nThis routine supports only one-based indexing of the input arrays.\nInput Parameters\nuplo\nSpecifies whether the upper or low triangle of the matrix A is considered.\nIf uplo = 'U' or 'u', then the upper triangle of the matrix A is used.\nIf uplo = 'L' or 'l', then the low triangle of the matrix A is used.\ntransa\nSpecifies the system of linear equations.\nIf transa = 'N' or 'n', then A*y = x\nIf transa = 'T' or 't' or 'C' or 'c', then AT*y = x,\ndiag\nSpecifies whether A is unit triangular.\nIf diag = 'U' or 'u', then A is unit triangular.\nIf diag = 'N' or 'n', then A is not unit triangular.\nm\nNumber of rows of the matrix A.\nval\nArray of length nnz, contains non-zero elements of the matrix A in the\narbitrary order.\nRefer to values array description in Coordinate Format for more details.\nrowind\nArray of length nnz, contains the row indices plus one for each non-zero\nelement of the matrix A.\nRefer to rows array description in Coordinate Format for more details.\ncolind\nArray of length nnz, contains the column indices plus one for each non-zero\nelement of the matrix A. Refer to columns array description in Coordinate\nFormat for more details.\nnnz\nSpecifies the number of non-zero element of the matrix A.\nRefer to nnz description in Coordinate Format for more details.\nx\nArray, size is m.\nOn entry, the array x must contain the vector x.\nOutput Parameters\ny\nArray, size at least m.\nContains the vector y.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n154\n\n\nmkl_?diatrsv\nTriangular solvers with simplified interface for a sparse\nmatrix in the diagonal format with one-based indexing\n(deprecated).\nSyntax\nvoid mkl_sdiatrsv (const char *uplo , const char *transa , const char *diag , const\nMKL_INT *m , const float *val , const MKL_INT *lval , const MKL_INT *idiag , const\nMKL_INT *ndiag , const float *x , float *y );\nvoid mkl_ddiatrsv (const char *uplo , const char *transa , const char *diag , const\nMKL_INT *m , const double *val , const MKL_INT *lval , const MKL_INT *idiag , const\nMKL_INT *ndiag , const double *x , double *y );\nvoid mkl_cdiatrsv (const char *uplo , const char *transa , const char *diag , const\nMKL_INT *m , const MKL_Complex8 *val , const MKL_INT *lval , const MKL_INT *idiag ,\nconst MKL_INT *ndiag , const MKL_Complex8 *x , MKL_Complex8 *y );\nvoid mkl_zdiatrsv (const char *uplo , const char *transa , const char *diag , const\nMKL_INT *m , const MKL_Complex16 *val , const MKL_INT *lval , const MKL_INT *idiag ,\nconst MKL_INT *ndiag , const MKL_Complex16 *x , MKL_Complex16 *y );\nInclude Files\n•\nmkl.h\nDescription\nThis routine is deprecated, but no replacement is available yet in the Inspector-Executor Sparse BLAS API\ninterfaces. You can continue using this routine until a replacement is provided and this can be fully removed.\nThe mkl_?diatrsv routine solves a system of linear equations with matrix-vector operations for a sparse\nmatrix stored in the diagonal format:\nA*y = x\nor\nAT*y = x,\nwhere:\nx and y are vectors,\nA is a sparse upper or lower triangular matrix with unit or non-unit main diagonal, AT is the transpose of A.\nNOTE\nThis routine supports only one-based indexing of the input arrays.\nInput Parameters\nuplo\nSpecifies whether the upper or low triangle of the matrix A is used.\nIf uplo = 'U' or 'u', then the upper triangle of the matrix A is used.\nIf uplo = 'L' or 'l', then the low triangle of the matrix A is used.\ntransa\nSpecifies the system of linear equations.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n155\n\n\nIf transa = 'N' or 'n', then A*y = x\nIf transa = 'T' or 't' or 'C' or 'c', then AT*y = x,\ndiag\nSpecifies whether A is unit triangular.\nIf diag = 'U' or 'u', then A is unit triangular.\nIf diag = 'N' or 'n', then A is not unit triangular.\nm\nNumber of rows of the matrix A.\nval\nTwo-dimensional array of size lval by ndiag, contains non-zero diagonals\nof the matrix A. Refer to values array description in Diagonal Storage\nScheme for more details.\nlval\nLeading dimension of val, lval≥m. Refer to lval description in Diagonal\nStorage Scheme for more details.\nidiag\nArray of length ndiag, contains the distances between main diagonal and\neach non-zero diagonals in the matrix A.\nNOTE\nAll elements of this array must be sorted in increasing order.\nRefer to distance array description in Diagonal Storage Scheme for more\ndetails.\nndiag\nSpecifies the number of non-zero diagonals of the matrix A.\nx\nArray, size is m.\nOn entry, the array x must contain the vector x.\nOutput Parameters\ny\nArray, size at least m.\nContains the vector y.\nmkl_cspblas_?csrgemv\nComputes matrix - vector product of a sparse general\nmatrix stored in the CSR format (3-array variation)\nwith zero-based indexing (deprecated).\nSyntax\nvoid mkl_cspblas_scsrgemv (const char *transa , const MKL_INT *m , const float *a ,\nconst MKL_INT *ia , const MKL_INT *ja , const float *x , float *y );\nvoid mkl_cspblas_dcsrgemv (const char *transa , const MKL_INT *m , const double *a ,\nconst MKL_INT *ia , const MKL_INT *ja , const double *x , double *y );\nvoid mkl_cspblas_ccsrgemv (const char *transa , const MKL_INT *m , const MKL_Complex8\n*a , const MKL_INT *ia , const MKL_INT *ja , const MKL_Complex8 *x , MKL_Complex8 *y );\nvoid mkl_cspblas_zcsrgemv (const char *transa , const MKL_INT *m , const MKL_Complex16\n*a , const MKL_INT *ia , const MKL_INT *ja , const MKL_Complex16 *x , MKL_Complex16\n*y );\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n156\n\n\nInclude Files\n•\nmkl.h\nDescription\nThis routine is deprecated. Use mkl_sparse_?_mvfrom the Intel® oneAPI Math Kernel Library (oneMKL)\nInspector-executor Sparse BLAS interface instead.\nThe mkl_cspblas_?csrgemv routine performs a matrix-vector operation defined as\ny := A*x\nor\ny := AT*x,\nwhere:\nx and y are vectors,\nA is an m-by-m sparse square matrix in the CSR format (3-array variation) with zero-based indexing, AT is\nthe transpose of A.\nNOTE\nThis routine supports only zero-based indexing of the input arrays.\nInput Parameters\ntransa\nSpecifies the operation.\nIf transa = 'N' or 'n', then the matrix-vector product is computed as\ny := A*x\nIf transa = 'T' or 't' or 'C' or 'c', then the matrix-vector product is\ncomputed as y := AT*x,\nm\nNumber of rows of the matrix A.\na\nArray containing non-zero elements of the matrix A. Its length is equal to\nthe number of non-zero elements in the matrix A. Refer to values array\ndescription in Sparse Matrix Storage Formats for more details.\nia\nArray of length m + 1, containing indices of elements in the array a, such\nthat ia[I] is the index in the array a of the first non-zero element from the\nrow I. The value of the last element ia[m] is equal to the number of non-\nzeros. Refer to rowIndex array description in Sparse Matrix Storage\nFormats for more details.\nja\nArray containing the column indices for each non-zero element of the\nmatrix A.\nIts length is equal to the length of the array a. Refer to columns array\ndescription in Sparse Matrix Storage Formats for more details.\nx\nArray, size is m.\nOne entry, the array x must contain the vector x.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n157\n\n\nOutput Parameters\ny\nArray, size at least m.\nOn exit, the array y must contain the vector y.\nmkl_cspblas_?bsrgemv\nComputes matrix - vector product of a sparse general\nmatrix stored in the BSR format (3-array variation)\nwith zero-based indexing (deprecated).\nSyntax\nvoid mkl_cspblas_sbsrgemv (const char *transa , const MKL_INT *m , const MKL_INT *lb ,\nconst float *a , const MKL_INT *ia , const MKL_INT *ja , const float *x , float *y );\nvoid mkl_cspblas_dbsrgemv (const char *transa , const MKL_INT *m , const MKL_INT *lb ,\nconst double *a , const MKL_INT *ia , const MKL_INT *ja , const double *x , double\n*y );\nvoid mkl_cspblas_cbsrgemv (const char *transa , const MKL_INT *m , const MKL_INT *lb ,\nconst MKL_Complex8 *a , const MKL_INT *ia , const MKL_INT *ja , const MKL_Complex8 *x ,\nMKL_Complex8 *y );\nvoid mkl_cspblas_zbsrgemv (const char *transa , const MKL_INT *m , const MKL_INT *lb ,\nconst MKL_Complex16 *a , const MKL_INT *ia , const MKL_INT *ja , const MKL_Complex16\n*x , MKL_Complex16 *y );\nInclude Files\n•\nmkl.h\nDescription\nThis routine is deprecated. Use mkl_sparse_?_mvfrom the Intel® oneAPI Math Kernel Library (oneMKL)\nInspector-executor Sparse BLAS interface instead.\nThe mkl_cspblas_?bsrgemv routine performs a matrix-vector operation defined as\ny := A*x\nor\ny := AT*x,\nwhere:\nx and y are vectors,\nA is an m-by-m block sparse square matrix in the BSR format (3-array variation) with zero-based indexing,\nAT is the transpose of A.\nNOTE\nThis routine supports only zero-based indexing of the input arrays.\nInput Parameters\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n158\n\n\ntransa\nSpecifies the operation.\nIf transa = 'N' or 'n', then the matrix-vector product is computed as\ny := A*x\nIf transa = 'T' or 't' or 'C' or 'c', then the matrix-vector product is\ncomputed as y := AT*x,\nm\nNumber of block rows of the matrix A.\nlb\nSize of the block in the matrix A.\na\nArray containing elements of non-zero blocks of the matrix A. Its length is\nequal to the number of non-zero blocks in the matrix A multiplied by lb*lb.\nRefer to values array description in BSR Format for more details.\nia\nArray of length (m + 1), containing indices of block in the array a, such\nthat ia[i] is the index in the array a of the first non-zero element from the\nrow i. The value of the last element ia[m] is equal to the number of non-\nzero blocks. Refer to rowIndex array description in BSR Format for more\ndetails.\nja\nArray containing the column indices for each non-zero block in the matrix A.\nIts length is equal to the number of non-zero blocks of the matrix A. Refer\nto columns array description in BSR Format for more details.\nx\nArray, size (m*lb).\nOn entry, the array x must contain the vector x.\nOutput Parameters\ny\nArray, size at least (m*lb).\nOn exit, the array y must contain the vector y.\nmkl_cspblas_?coogemv\nComputes matrix - vector product of a sparse general\nmatrix stored in the coordinate format with zero-\nbased indexing (deprecated).\nSyntax\nvoid mkl_cspblas_scoogemv (const char *transa , const MKL_INT *m , const float *val ,\nconst MKL_INT *rowind , const MKL_INT *colind , const MKL_INT *nnz , const float *x ,\nfloat *y );\nvoid mkl_cspblas_dcoogemv (const char *transa , const MKL_INT *m , const double *val ,\nconst MKL_INT *rowind , const MKL_INT *colind , const MKL_INT *nnz , const double *x ,\ndouble *y );\nvoid mkl_cspblas_ccoogemv (const char *transa , const MKL_INT *m , const MKL_Complex8\n*val , const MKL_INT *rowind , const MKL_INT *colind , const MKL_INT *nnz , const\nMKL_Complex8 *x , MKL_Complex8 *y );\nvoid mkl_cspblas_zcoogemv (const char *transa , const MKL_INT *m , const MKL_Complex16\n*val , const MKL_INT *rowind , const MKL_INT *colind , const MKL_INT *nnz , const\nMKL_Complex16 *x , MKL_Complex16 *y );\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n159\n\n\nInclude Files\n•\nmkl.h\nDescription\nThis routine is deprecated. Use mkl_sparse_?_mvfrom the Intel® oneAPI Math Kernel Library (oneMKL)\nInspector-executor Sparse BLAS interface instead.\nThe mkl_cspblas_dcoogemv routine performs a matrix-vector operation defined as\ny := A*x\nor\ny := AT*x,\nwhere:\nx and y are vectors,\nA is an m-by-m sparse square matrix in the coordinate format with zero-based indexing, AT is the transpose\nof A.\nNOTE\nThis routine supports only zero-based indexing of the input arrays.\nInput Parameters\ntransa\nSpecifies the operation.\nIf transa = 'N' or 'n', then the matrix-vector product is computed as\ny := A*x\nIf transa = 'T' or 't' or 'C' or 'c', then the matrix-vector product is\ncomputed as y := AT*x.\nm\nNumber of rows of the matrix A.\nval\nArray of length nnz, contains non-zero elements of the matrix A in the\narbitrary order.\nRefer to values array description in Coordinate Format for more details.\nrowind\nArray of length nnz, contains the row indices for each non-zero element of\nthe matrix A.\nRefer to rows array description in Coordinate Format for more details.\ncolind\nArray of length nnz, contains the column indices for each non-zero element\nof the matrix A. Refer to columns array description in Coordinate Format\nfor more details.\nnnz\nSpecifies the number of non-zero element of the matrix A.\nRefer to nnz description in Coordinate Format for more details.\nx\nArray, size is m.\nOn entry, the array x must contain the vector x.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n160\n\n\nOutput Parameters\ny\nArray, size at least m.\nOn exit, the array y must contain the vector y.\nmkl_cspblas_?csrsymv\nComputes matrix-vector product of a sparse\nsymmetrical matrix stored in the CSR format (3-array\nvariation) with zero-based indexing (deprecated).\nSyntax\nvoid mkl_cspblas_scsrsymv (const char *uplo , const MKL_INT *m , const float *a , const\nMKL_INT *ia , const MKL_INT *ja , const float *x , float *y );\nvoid mkl_cspblas_dcsrsymv (const char *uplo , const MKL_INT *m , const double *a ,\nconst MKL_INT *ia , const MKL_INT *ja , const double *x , double *y );\nvoid mkl_cspblas_ccsrsymv (const char *uplo , const MKL_INT *m , const MKL_Complex8\n*a , const MKL_INT *ia , const MKL_INT *ja , const MKL_Complex8 *x , MKL_Complex8 *y );\nvoid mkl_cspblas_zcsrsymv (const char *uplo , const MKL_INT *m , const MKL_Complex16\n*a , const MKL_INT *ia , const MKL_INT *ja , const MKL_Complex16 *x , MKL_Complex16\n*y );\nInclude Files\n•\nmkl.h\nDescription\nThis routine is deprecated. Use mkl_sparse_?_mvfrom the Intel® oneAPI Math Kernel Library (oneMKL)\nInspector-executor Sparse BLAS interface instead.\nThe mkl_cspblas_?csrsymv routine performs a matrix-vector operation defined as\ny := A*x\nwhere:\nx and y are vectors,\nA is an upper or lower triangle of the symmetrical sparse matrix in the CSR format (3-array variation) with\nzero-based indexing.\nNOTE\nThis routine supports only zero-based indexing of the input arrays.\nInput Parameters\n \nuplo\nSpecifies whether the upper or low triangle of the matrix A is used.\nIf uplo = 'U' or 'u', then the upper triangle of the matrix A is used.\nIf uplo = 'L' or 'l', then the low triangle of the matrix A is used.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n161\n\n\nm\nNumber of rows of the matrix A.\na\nArray containing non-zero elements of the matrix A. Its length is equal to\nthe number of non-zero elements in the matrix A. Refer to values array\ndescription in Sparse Matrix Storage Formats for more details.\nia\nArray of length m + 1, containing indices of elements in the array a, such\nthat ia[i] is the index in the array a of the first non-zero element from the\nrow i. The value of the last element ia[m] is equal to the number of non-\nzeros. Refer to rowIndex array description in Sparse Matrix Storage\nFormats for more details.\nja\nArray containing the column indices for each non-zero element of the\nmatrix A.\nIts length is equal to the length of the array a. Refer to columns array\ndescription in Sparse Matrix Storage Formats for more details.\nx\nArray, size is m.\nOn entry, the array x must contain the vector x.\nOutput Parameters\ny\nArray, size at least m.\nOn exit, the array y must contain the vector y.\nmkl_cspblas_?bsrsymv\nComputes matrix-vector product of a sparse\nsymmetrical matrix stored in the BSR format (3-arrays\nvariation) with zero-based indexing (deprecated).\nSyntax\nvoid mkl_cspblas_sbsrsymv (const char *uplo , const MKL_INT *m , const MKL_INT *lb ,\nconst float *a , const MKL_INT *ia , const MKL_INT *ja , const float *x , float *y );\nvoid mkl_cspblas_dbsrsymv (const char *uplo , const MKL_INT *m , const MKL_INT *lb ,\nconst double *a , const MKL_INT *ia , const MKL_INT *ja , const double *x , double\n*y );\nvoid mkl_cspblas_cbsrsymv (const char *uplo , const MKL_INT *m , const MKL_INT *lb ,\nconst MKL_Complex8 *a , const MKL_INT *ia , const MKL_INT *ja , const MKL_Complex8 *x ,\nMKL_Complex8 *y );\nvoid mkl_cspblas_zbsrsymv (const char *uplo , const MKL_INT *m , const MKL_INT *lb ,\nconst MKL_Complex16 *a , const MKL_INT *ia , const MKL_INT *ja , const MKL_Complex16\n*x , MKL_Complex16 *y );\nInclude Files\n•\nmkl.h\nDescription\nThis routine is deprecated. Use mkl_sparse_?_mvfrom the Intel® oneAPI Math Kernel Library (oneMKL)\nInspector-executor Sparse BLAS interface instead.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n162\n\n\nThe mkl_cspblas_?bsrsymv routine performs a matrix-vector operation defined as\ny := A*x\nwhere:\nx and y are vectors,\nA is an upper or lower triangle of the symmetrical sparse matrix in the BSR format (3-array variation) with\nzero-based indexing.\nNOTE\nThis routine supports only zero-based indexing of the input arrays.\nInput Parameters\n \nuplo\nSpecifies whether the upper or low triangle of the matrix A is used.\nIf uplo = 'U' or 'u', then the upper triangle of the matrix A is used.\nIf uplo = 'L' or 'l', then the low triangle of the matrix A is used.\nm\nNumber of block rows of the matrix A.\nlb\nSize of the block in the matrix A.\na\nArray containing elements of non-zero blocks of the matrix A. Its length is\nequal to the number of non-zero blocks in the matrix A multiplied by lb*lb.\nRefer to values array description in BSR Format for more details.\nia\nArray of length (m + 1), containing indices of block in the array a, such\nthat ia[i] is the index in the array a of the first non-zero element from the\nrow i. The value of the last element ia[m] is equal to the number of non-\nzero blocks. Refer to rowIndex array description in BSR Format for more\ndetails.\nja\nArray containing the column indices for each non-zero block in the matrix A.\nIts length is equal to the number of non-zero blocks of the matrix A. Refer\nto columns array description in BSR Format for more details.\nx\nArray, size (m*lb).\nOn entry, the array x must contain the vector x.\nOutput Parameters\ny\nArray, size at least (m*lb).\nOn exit, the array y must contain the vector y.\nmkl_cspblas_?coosymv\nComputes matrix - vector product of a sparse\nsymmetrical matrix stored in the coordinate format\nwith zero-based indexing (deprecated).\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n163\n\n\nSyntax\nvoid mkl_cspblas_scoosymv (const char *uplo , const MKL_INT *m , const float *val ,\nconst MKL_INT *rowind , const MKL_INT *colind , const MKL_INT *nnz , const float *x ,\nfloat *y );\nvoid mkl_cspblas_dcoosymv (const char *uplo , const MKL_INT *m , const double *val ,\nconst MKL_INT *rowind , const MKL_INT *colind , const MKL_INT *nnz , const double *x ,\ndouble *y );\nvoid mkl_cspblas_ccoosymv (const char *uplo , const MKL_INT *m , const MKL_Complex8\n*val , const MKL_INT *rowind , const MKL_INT *colind , const MKL_INT *nnz , const\nMKL_Complex8 *x , MKL_Complex8 *y );\nvoid mkl_cspblas_zcoosymv (const char *uplo , const MKL_INT *m , const MKL_Complex16\n*val , const MKL_INT *rowind , const MKL_INT *colind , const MKL_INT *nnz , const\nMKL_Complex16 *x , MKL_Complex16 *y );\nInclude Files\n•\nmkl.h\nDescription\nThis routine is deprecated. Use mkl_sparse_?_mvfrom the Intel® oneAPI Math Kernel Library (oneMKL)\nInspector-executor Sparse BLAS interface instead.\nThe mkl_cspblas_?coosymv routine performs a matrix-vector operation defined as\ny := A*x\nwhere:\nx and y are vectors,\nA is an upper or lower triangle of the symmetrical sparse matrix in the coordinate format with zero-based\nindexing.\nNOTE\nThis routine supports only zero-based indexing of the input arrays.\nInput Parameters\nuplo\nSpecifies whether the upper or low triangle of the matrix A is used.\nIf uplo = 'U' or 'u', then the upper triangle of the matrix A is used.\nIf uplo = 'L' or 'l', then the low triangle of the matrix A is used.\nm\nNumber of rows of the matrix A.\nval\nArray of length nnz, contains non-zero elements of the matrix A in the\narbitrary order.\nRefer to values array description in Coordinate Format for more details.\nrowind\nArray of length nnz, contains the row indices for each non-zero element of\nthe matrix A.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n164\n\n\nRefer to rows array description in Coordinate Format for more details.\ncolind\nArray of length nnz, contains the column indices for each non-zero element\nof the matrix A. Refer to columns array description in Coordinate Format\nfor more details.\nnnz\nSpecifies the number of non-zero element of the matrix A.\nRefer to nnz description in Coordinate Format for more details.\nx\nArray, size is m.\nOn entry, the array x must contain the vector x.\nOutput Parameters\ny\nArray, size at least m.\nOn exit, the array y must contain the vector y.\nmkl_cspblas_?csrtrsv\nTriangular solvers with simplified interface for a sparse\nmatrix in the CSR format (3-array variation) with\nzero-based indexing (deprecated).\nSyntax\nvoid mkl_cspblas_scsrtrsv (const char *uplo , const char *transa , const char *diag ,\nconst MKL_INT *m , const float *a , const MKL_INT *ia , const MKL_INT *ja , const float\n*x , float *y );\nvoid mkl_cspblas_dcsrtrsv (const char *uplo , const char *transa , const char *diag ,\nconst MKL_INT *m , const double *a , const MKL_INT *ia , const MKL_INT *ja , const\ndouble *x , double *y );\nvoid mkl_cspblas_ccsrtrsv (const char *uplo , const char *transa , const char *diag ,\nconst MKL_INT *m , const MKL_Complex8 *a , const MKL_INT *ia , const MKL_INT *ja ,\nconst MKL_Complex8 *x , MKL_Complex8 *y );\nvoid mkl_cspblas_zcsrtrsv (const char *uplo , const char *transa , const char *diag ,\nconst MKL_INT *m , const MKL_Complex16 *a , const MKL_INT *ia , const MKL_INT *ja ,\nconst MKL_Complex16 *x , MKL_Complex16 *y );\nInclude Files\n•\nmkl.h\nDescription\nThis routine is deprecated. Use mkl_sparse_?_trsvfrom the Intel® oneAPI Math Kernel Library (oneMKL)\nInspector-executor Sparse BLAS interface instead.\nThe mkl_cspblas_?csrtrsv routine solves a system of linear equations with matrix-vector operations for a\nsparse matrix stored in the CSR format (3-array variation) with zero-based indexing:\nA*y = x\nor\nAT*y = x,\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n165\n\n\nwhere:\nx and y are vectors,\nA is a sparse upper or lower triangular matrix with unit or non-unit main diagonal, AT is the transpose of A.\nNOTE\nThis routine supports only zero-based indexing of the input arrays.\nInput Parameters\nuplo\nSpecifies whether the upper or low triangle of the matrix A is used.\nIf uplo = 'U' or 'u', then the upper triangle of the matrix A is used.\nIf uplo = 'L' or 'l', then the low triangle of the matrix A is used.\ntransa\nSpecifies the system of linear equations.\nIf transa = 'N' or 'n', then A*y = x\nIf transa = 'T' or 't' or 'C' or 'c', then AT*y = x,\ndiag\nSpecifies whether matrix A is unit triangular.\nIf diag = 'U' or 'u', then A is unit triangular.\nIf diag = 'N' or 'n', then A is not unit triangular.\nm\nNumber of rows of the matrix A.\na\nArray containing non-zero elements of the matrix A. Its length is equal to\nthe number of non-zero elements in the matrix A. Refer to values array\ndescription in Sparse Matrix Storage Formats for more details.\nNOTE\nThe non-zero elements of the given row of the matrix must be\nstored in the same order as they appear in the row (from left to\nright).\nNo diagonal element can be omitted from a sparse storage if the solver\nis called with the non-unit indicator.\nia\nArray of length m+1, containing indices of elements in the array a, such that\nia[i] is the index in the array a of the first non-zero element from the row\ni. The value of the last element ia[m] is equal to the number of non-zeros.\nRefer to rowIndex array description in Sparse Matrix Storage Formats for\nmore details.\nja\nArray containing the column indices for each non-zero element of the\nmatrix A.\nIts length is equal to the length of the array a. Refer to columns array\ndescription in Sparse Matrix Storage Formats for more details.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n166\n\n\nNOTE\nColumn indices must be sorted in increasing order for each row.\nx\nArray, size is m.\nOn entry, the array x must contain the vector x.\nOutput Parameters\ny\nArray, size at least m.\nContains the vector y.\nmkl_cspblas_?bsrtrsv\nTriangular solver with simplified interface for a sparse\nmatrix stored in the BSR format (3-array variation)\nwith zero-based indexing (deprecated).\nSyntax\nvoid mkl_cspblas_sbsrtrsv (const char *uplo , const char *transa , const char *diag ,\nconst MKL_INT *m , const MKL_INT *lb , const float *a , const MKL_INT *ia , const\nMKL_INT *ja , const float *x , float *y );\nvoid mkl_cspblas_dbsrtrsv (const char *uplo , const char *transa , const char *diag ,\nconst MKL_INT *m , const MKL_INT *lb , const double *a , const MKL_INT *ia , const\nMKL_INT *ja , const double *x , double *y );\nvoid mkl_cspblas_cbsrtrsv (const char *uplo , const char *transa , const char *diag ,\nconst MKL_INT *m , const MKL_INT *lb , const MKL_Complex8 *a , const MKL_INT *ia ,\nconst MKL_INT *ja , const MKL_Complex8 *x , MKL_Complex8 *y );\nvoid mkl_cspblas_zbsrtrsv (const char *uplo , const char *transa , const char *diag ,\nconst MKL_INT *m , const MKL_INT *lb , const MKL_Complex16 *a , const MKL_INT *ia ,\nconst MKL_INT *ja , const MKL_Complex16 *x , MKL_Complex16 *y );\nInclude Files\n•\nmkl.h\nDescription\nThis routine is deprecated. Use mkl_sparse_?_trsvfrom the Intel® oneAPI Math Kernel Library (oneMKL)\nInspector-executor Sparse BLAS interface instead.\nThe mkl_cspblas_?bsrtrsv routine solves a system of linear equations with matrix-vector operations for a\nsparse matrix stored in the BSR format (3-array variation) with zero-based indexing:\ny := A*x\nor\ny := AT*x,\nwhere:\nx and y are vectors,\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n167\n\n\nA is a sparse upper or lower triangular matrix with unit or non-unit main diagonal, AT is the transpose of A.\nNOTE\nThis routine supports only zero-based indexing of the input arrays.\nInput Parameters\nuplo\nSpecifies the upper or low triangle of the matrix A is used.\nIf uplo = 'U' or 'u', then the upper triangle of the matrix A is used.\nIf uplo = 'L' or 'l', then the low triangle of the matrix A is used.\ntransa\nSpecifies the operation.\nIf transa = 'N' or 'n', then the matrix-vector product is computed as\ny := A*x\nIf transa = 'T' or 't' or 'C' or 'c', then the matrix-vector product is\ncomputed as y := AT*x.\ndiag\nSpecifies whether matrix A is unit triangular or not.\nIf diag = 'U' or 'u', A is unit triangular.\nIf diag = 'N' or 'n', A is not unit triangular.\nm\nNumber of block rows of the matrix A.\nlb\nSize of the block in the matrix A.\na\nArray containing elements of non-zero blocks of the matrix A. Its length is\nequal to the number of non-zero blocks in the matrix A multiplied by lb*lb.\nRefer to values array description in BSR Format for more details.\nNOTE\nThe non-zero elements of the given row of the matrix must be\nstored in the same order as they appear in the row (from left to\nright).\nNo diagonal element can be omitted from a sparse storage if the solver\nis called with the non-unit indicator.\nia\nArray of length (m + 1), containing indices of block in the array a, such\nthat ia[I] is the index in the array a of the first non-zero element from the\nrow I. The value of the last element ia[m] is equal to the number of non-\nzero blocks. Refer to rowIndex array description in BSR Format for more\ndetails.\nja\nArray containing the column indices for each non-zero block in the matrix A.\nIts length is equal to the number of non-zero blocks of the matrix A. Refer\nto columns array description in BSR Format for more details.\nx\nArray, size (m*lb).\nOn entry, the array x must contain the vector x.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n168\n\n\nOutput Parameters\ny\nArray, size at least (m*lb).\nOn exit, the array y must contain the vector y.\nmkl_cspblas_?cootrsv\nTriangular solvers with simplified interface for a sparse\nmatrix in the coordinate format with zero-based\nindexing (deprecated).\nSyntax\nvoid mkl_cspblas_scootrsv (const char *uplo , const char *transa , const char *diag ,\nconst MKL_INT *m , const float *val , const MKL_INT *rowind , const MKL_INT *colind ,\nconst MKL_INT *nnz , const float *x , float *y );\nvoid mkl_cspblas_dcootrsv (const char *uplo , const char *transa , const char *diag ,\nconst MKL_INT *m , const double *val , const MKL_INT *rowind , const MKL_INT *colind ,\nconst MKL_INT *nnz , const double *x , double *y );\nvoid mkl_cspblas_ccootrsv (const char *uplo , const char *transa , const char *diag ,\nconst MKL_INT *m , const MKL_Complex8 *val , const MKL_INT *rowind , const MKL_INT\n*colind , const MKL_INT *nnz , const MKL_Complex8 *x , MKL_Complex8 *y );\nvoid mkl_cspblas_zcootrsv (const char *uplo , const char *transa , const char *diag ,\nconst MKL_INT *m , const MKL_Complex16 *val , const MKL_INT *rowind , const MKL_INT\n*colind , const MKL_INT *nnz , const MKL_Complex16 *x , MKL_Complex16 *y );\nInclude Files\n•\nmkl.h\nDescription\nThis routine is deprecated. Use mkl_sparse_?_trsvfrom the Intel® oneAPI Math Kernel Library (oneMKL)\nInspector-executor Sparse BLAS interface instead.\nThe mkl_cspblas_?cootrsv routine solves a system of linear equations with matrix-vector operations for a\nsparse matrix stored in the coordinate format with zero-based indexing:\nA*y = x\nor\nAT*y = x,\nwhere:\nx and y are vectors,\nA is a sparse upper or lower triangular matrix with unit or non-unit main diagonal, AT is the transpose of A.\nNOTE\nThis routine supports only zero-based indexing of the input arrays.\nInput Parameters\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n169\n\n\nuplo\nSpecifies whether the upper or low triangle of the matrix A is considered.\nIf uplo = 'U' or 'u', then the upper triangle of the matrix A is used.\nIf uplo = 'L' or 'l', then the low triangle of the matrix A is used.\ntransa\nSpecifies the system of linear equations.\nIf transa = 'N' or 'n', then A*y = x\nIf transa = 'T' or 't' or 'C' or 'c', then AT*y = x,\ndiag\nSpecifies whether A is unit triangular.\nIf diag = 'U' or 'u', then A is unit triangular.\nIf diag = 'N' or 'n', then A is not unit triangular.\nm\nNumber of rows of the matrix A.\nval\nArray of length nnz, contains non-zero elements of the matrix A in the\narbitrary order.\nRefer to values array description in Coordinate Format for more details.\nrowind\nArray of length nnz, contains the row indices for each non-zero element of\nthe matrix A.\nRefer to rows array description in Coordinate Format for more details.\ncolind\nArray of length nnz, contains the column indices for each non-zero element\nof the matrix A. Refer to columns array description in Coordinate Format\nfor more details.\nnnz\nSpecifies the number of non-zero element of the matrix A.\nRefer to nnz description in Coordinate Format for more details.\nx\nArray, size is m.\nOn entry, the array x must contain the vector x.\nOutput Parameters\ny\nArray, size at least m.\nContains the vector y.\nmkl_?csrmv\nComputes matrix - vector product of a sparse matrix\nstored in the CSR format (deprecated).\nSyntax\nvoid mkl_scsrmv (const char *transa , const MKL_INT *m , const MKL_INT *k , const float\n*alpha , const char *matdescra , const float *val , const MKL_INT *indx , const MKL_INT\n*pntrb , const MKL_INT *pntre , const float *x , const float *beta , float *y );\nvoid mkl_dcsrmv (const char *transa , const MKL_INT *m , const MKL_INT *k , const\ndouble *alpha , const char *matdescra , const double *val , const MKL_INT *indx , const\nMKL_INT *pntrb , const MKL_INT *pntre , const double *x , const double *beta , double\n*y );\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n170\n\n\nvoid mkl_ccsrmv (const char *transa , const MKL_INT *m , const MKL_INT *k , const\nMKL_Complex8 *alpha , const char *matdescra , const MKL_Complex8 *val , const MKL_INT\n*indx , const MKL_INT *pntrb , const MKL_INT *pntre , const MKL_Complex8 *x , const\nMKL_Complex8 *beta , MKL_Complex8 *y );\nvoid mkl_zcsrmv (const char *transa , const MKL_INT *m , const MKL_INT *k , const\nMKL_Complex16 *alpha , const char *matdescra , const MKL_Complex16 *val , const MKL_INT\n*indx , const MKL_INT *pntrb , const MKL_INT *pntre , const MKL_Complex16 *x , const\nMKL_Complex16 *beta , MKL_Complex16 *y );\nInclude Files\n•\nmkl.h\nDescription\nThis routine is deprecated. Use mkl_sparse_?_mvfrom the Intel® oneAPI Math Kernel Library (oneMKL)\nInspector-executor Sparse BLAS interface instead.\nThe mkl_?csrmv routine performs a matrix-vector operation defined as\ny := alpha*A*x + beta*y\nor\ny := alpha*AT*x + beta*y,\nwhere:\nalpha and beta are scalars,\nx and y are vectors,\nA is an m-by-k sparse matrix in the CSR format, AT is the transpose of A.\nNOTE\nThis routine supports a CSR format both with one-based indexing and zero-based indexing.\nInput Parameters\ntransa\nSpecifies the operation.\nIf transa = 'N' or 'n', then y := alpha*A*x + beta*y\nIf transa = 'T' or 't' or 'C' or 'c', then y := alpha*AT*x + beta*y,\nm\nNumber of rows of the matrix A.\nk\nNumber of columns of the matrix A.\nalpha\nSpecifies the scalar alpha.\nmatdescra\nArray of six elements, specifies properties of the matrix used for operation.\nOnly first four array elements are used, their possible values are given in \nTable “Possible Values of the Parameter matdescra (descra)”. Possible\ncombinations of element values of this parameter are given in Table\n“Possible Combinations of Element Values of the Parameter matdescra”.\nval\nArray containing non-zero elements of the matrix A.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n171\n\n\nIts length is pntre[m-1] - pntrb[0].\nRefer to values array description in CSR Format for more details.\nindx\nFor one-based indexing, array containing the column indices plus one for\neach non-zero element of the matrix A. For zero-based indexing, array\ncontaining the column indices for each non-zero element of the matrix A.\nIts length is equal to length of the val array.\nRefer to columns array description in CSR Format for more details.\npntrb\nArray of length m.\nThis array contains row indices, such that pntrb[i] - pntrb[0] is the\nfirst index of row i in the arrays val and indx.\nRefer to pointerb array description in CSR Format for more details.\npntre\nArray of length m.\nThis array contains row indices, such that pntre[i] - pntrb[0]-1 is the\nlast index of row i in the arrays val and indx.\nRefer to pointerE array description in CSR Format for more details.\nx\nArray, size at least k if transa = 'N' or 'n' and at least m otherwise. On\nentry, the array x must contain the vector x.\nbeta\nSpecifies the scalar beta.\ny\nArray, size at least m if transa = 'N' or 'n' and at least k otherwise. On\nentry, the array y must contain the vector y.\nOutput Parameters\ny\nOverwritten by the updated vector y.\nmkl_?bsrmv\nComputes matrix - vector product of a sparse matrix\nstored in the BSR format (deprecated).\nSyntax\nvoid mkl_sbsrmv (const char *transa , const MKL_INT *m , const MKL_INT *k , const\nMKL_INT *lb , const float *alpha , const char *matdescra , const float *val , const\nMKL_INT *indx , const MKL_INT *pntrb , const MKL_INT *pntre , const float *x , const\nfloat *beta , float *y );\nvoid mkl_dbsrmv (const char *transa , const MKL_INT *m , const MKL_INT *k , const\nMKL_INT *lb , const double *alpha , const char *matdescra , const double *val , const\nMKL_INT *indx , const MKL_INT *pntrb , const MKL_INT *pntre , const double *x , const\ndouble *beta , double *y );\nvoid mkl_cbsrmv (const char *transa , const MKL_INT *m , const MKL_INT *k , const\nMKL_INT *lb , const MKL_Complex8 *alpha , const char *matdescra , const MKL_Complex8\n*val , const MKL_INT *indx , const MKL_INT *pntrb , const MKL_INT *pntre , const\nMKL_Complex8 *x , const MKL_Complex8 *beta , MKL_Complex8 *y );\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n172\n\n\nvoid mkl_zbsrmv (const char *transa , const MKL_INT *m , const MKL_INT *k , const\nMKL_INT *lb , const MKL_Complex16 *alpha , const char *matdescra , const MKL_Complex16\n*val , const MKL_INT *indx , const MKL_INT *pntrb , const MKL_INT *pntre , const\nMKL_Complex16 *x , const MKL_Complex16 *beta , MKL_Complex16 *y );\nInclude Files\n•\nmkl.h\nDescription\nThis routine is deprecated. Use mkl_sparse_?_mvfrom the Intel® oneAPI Math Kernel Library (oneMKL)\nInspector-executor Sparse BLAS interface instead.\nThe mkl_?bsrmv routine performs a matrix-vector operation defined as\ny := alpha*A*x + beta*y\nor\ny := alpha*AT*x + beta*y,\nwhere:\nalpha and beta are scalars,\nx and y are vectors,\nA is an m-by-k block sparse matrix in the BSR format, AT is the transpose of A.\nNOTE\nThis routine supports a BSR format both with one-based indexing and zero-based indexing.\nInput Parameters\ntransa\nSpecifies the operation.\nIf transa = 'N' or 'n', then the matrix-vector product is computed as\ny := alpha*A*x + beta*y\nIf transa = 'T' or 't' or 'C' or 'c', then the matrix-vector product is\ncomputed as y := alpha*AT*x + beta*y,\nm\nNumber of block rows of the matrix A.\nk\nNumber of block columns of the matrix A.\nlb\nSize of the block in the matrix A.\nalpha\nSpecifies the scalar alpha.\nmatdescra\nArray of six elements, specifies properties of the matrix used for operation.\nOnly first four array elements are used, their possible values are given in \nTable “Possible Values of the Parameter matdescra (descra)”. Possible\ncombinations of element values of this parameter are given in Table\n“Possible Combinations of Element Values of the Parameter matdescra”.\nval\nArray containing elements of non-zero blocks of the matrix A. Its length is\nequal to the number of non-zero blocks in the matrix A multiplied by lb*lb.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n173\n\n\nRefer to values array description in BSR Format for more details.\nindx\nFor one-based indexing, array containing the column indices plus one for\neach non-zero block of the matrix A. For zero-based indexing, array\ncontaining the column indices for each non-zero block of the matrix A.\nIts length is equal to the number of non-zero blocks in the matrix A.\nRefer to columns array description in BSR Format for more details.\npntrb\nArray of length m.\nThis array contains row indices, such that pntrb[i] - pntrb[0] is the\nfirst index of block row i in the array indx\nRefer to pointerB array description in BSR Format for more details.\npntre\nArray of length m.\nFor zero-based indexing this array contains row indices, such that\npntre[i] - pntrb[0] - 1 is the last index of block row i in the array\nindx.\nRefer to pointerE array description in BSR Format for more details.\nx\nArray, size at least (k*lb) if transa = 'N' or 'n', and at least (m*lb)\notherwise. On entry, the array x must contain the vector x.\nbeta\nSpecifies the scalar beta.\ny\nArray, size at least (m*lb) if transa = 'N' or 'n', and at least (k*lb)\notherwise. On entry, the array y must contain the vector y.\nOutput Parameters\ny\nOverwritten by the updated vector y.\nmkl_?cscmv\nComputes matrix-vector product for a sparse matrix in\nthe CSC format (deprecated).\nSyntax\nvoid mkl_scscmv (const char *transa , const MKL_INT *m , const MKL_INT *k , const float\n*alpha , const char *matdescra , const float *val , const MKL_INT *indx , const MKL_INT\n*pntrb , const MKL_INT *pntre , const float *x , const float *beta , float *y );\nvoid mkl_dcscmv (const char *transa , const MKL_INT *m , const MKL_INT *k , const\ndouble *alpha , const char *matdescra , const double *val , const MKL_INT *indx , const\nMKL_INT *pntrb , const MKL_INT *pntre , const double *x , const double *beta , double\n*y );\nvoid mkl_ccscmv (const char *transa , const MKL_INT *m , const MKL_INT *k , const\nMKL_Complex8 *alpha , const char *matdescra , const MKL_Complex8 *val , const MKL_INT\n*indx , const MKL_INT *pntrb , const MKL_INT *pntre , const MKL_Complex8 *x , const\nMKL_Complex8 *beta , MKL_Complex8 *y );\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n174\n\n\nvoid mkl_zcscmv (const char *transa , const MKL_INT *m , const MKL_INT *k , const\nMKL_Complex16 *alpha , const char *matdescra , const MKL_Complex16 *val , const MKL_INT\n*indx , const MKL_INT *pntrb , const MKL_INT *pntre , const MKL_Complex16 *x , const\nMKL_Complex16 *beta , MKL_Complex16 *y );\nInclude Files\n•\nmkl.h\nDescription\nThis routine is deprecated. Use mkl_sparse_?_mvfrom the Intel® oneAPI Math Kernel Library (oneMKL)\nInspector-executor Sparse BLAS interface instead.\nThe mkl_?cscmv routine performs a matrix-vector operation defined as\ny := alpha*A*x + beta*y\nor\ny := alpha*AT*x + beta*y,\nwhere:\nalpha and beta are scalars,\nx and y are vectors,\nA is an m-by-k sparse matrix in compressed sparse column (CSC) format, AT is the transpose of A.\nNOTE\nThis routine supports CSC format both with one-based indexing and zero-based indexing.\nInput Parameters\ntransa\nSpecifies the operation.\nIf transa = 'N' or 'n', then y := alpha*A*x + beta*y\nIf transa = 'T' or 't' or 'C' or 'c', then y := alpha*AT*x + beta*y,\nm\nNumber of rows of the matrix A.\nk\nNumber of columns of the matrix A.\nalpha\nSpecifies the scalar alpha.\nmatdescra\nArray of six elements, specifies properties of the matrix used for operation.\nOnly first four array elements are used, their possible values are given in \nTable “Possible Values of the Parameter matdescra (descra)”. Possible\ncombinations of element values of this parameter are given in Table\n“Possible Combinations of Element Values of the Parameter matdescra”.\nval\nArray containing non-zero elements of the matrix A.\nIts length is pntre[k-1] - pntrb[0].\nRefer to values array description in CSC Format for more details.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n175\n\n\nindx\nFor one-based indexing, array containing the column indices plus one for\neach non-zero element of the matrix A. For zero-based indexing, array\ncontaining the column indices for each non-zero element of the matrix A.\nIts length is equal to length of the val array.\nRefer to rows array description in CSC Format for more details.\npntrb\nArray of length k.\nThis array contains column indices, such that pntrb[i] - pntrb[0] + 1\nis the first index of column i in the arrays val and indx.\nRefer to pointerb array description in CSC Format for more details.\npntre\nArray of length k.\nFor one-based indexing this array contains column indices, such that\npntre[i] - pntrb[1] is the last index of column i in the arrays val and\nindx.\nFor zero-based indexing this array contains column indices, such that\npntre[i] - pntrb[1] - 1 is the last index of column i in the arrays val\nand indx.\nRefer to pointerE array description in CSC Format for more details.\nx\nArray, size at least k if transa = 'N' or 'n' and at least m otherwise. On\nentry, the array x must contain the vector x.\nbeta\nSpecifies the scalar beta.\ny\nArray, size at least m if transa = 'N' or 'n' and at least k otherwise. On\nentry, the array y must contain the vector y.\nOutput Parameters\ny\nOverwritten by the updated vector y.\nmkl_?coomv\nComputes matrix - vector product for a sparse matrix\nin the coordinate format (deprecated).\nSyntax\nvoid mkl_scoomv (const char *transa , const MKL_INT *m , const MKL_INT *k , const float\n*alpha , const char *matdescra , const float *val , const MKL_INT *rowind , const\nMKL_INT *colind , const MKL_INT *nnz , const float *x , const float *beta , float *y );\nvoid mkl_dcoomv (const char *transa , const MKL_INT *m , const MKL_INT *k , const\ndouble *alpha , const char *matdescra , const double *val , const MKL_INT *rowind ,\nconst MKL_INT *colind , const MKL_INT *nnz , const double *x , const double *beta ,\ndouble *y );\nvoid mkl_ccoomv (const char *transa , const MKL_INT *m , const MKL_INT *k , const\nMKL_Complex8 *alpha , const char *matdescra , const MKL_Complex8 *val , const MKL_INT\n*rowind , const MKL_INT *colind , const MKL_INT *nnz , const MKL_Complex8 *x , const\nMKL_Complex8 *beta , MKL_Complex8 *y );\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n176\n\n\nvoid mkl_zcoomv (const char *transa , const MKL_INT *m , const MKL_INT *k , const\nMKL_Complex16 *alpha , const char *matdescra , const MKL_Complex16 *val , const MKL_INT\n*rowind , const MKL_INT *colind , const MKL_INT *nnz , const MKL_Complex16 *x , const\nMKL_Complex16 *beta , MKL_Complex16 *y );\nInclude Files\n•\nmkl.h\nDescription\nThis routine is deprecated. Use mkl_sparse_?_mvfrom the Intel® oneAPI Math Kernel Library (oneMKL)\nInspector-executor Sparse BLAS interface instead.\nThe mkl_?coomv routine performs a matrix-vector operation defined as\ny := alpha*A*x + beta*y\nor\ny := alpha*AT*x + beta*y,\nwhere:\nalpha and beta are scalars,\nx and y are vectors,\nA is an m-by-k sparse matrix in compressed coordinate format, AT is the transpose of A.\nNOTE\nThis routine supports a coordinate format both with one-based indexing and zero-based indexing.\nInput Parameters\ntransa\nSpecifies the operation.\nIf transa = 'N' or 'n', then y := alpha*A*x + beta*y\nIf transa = 'T' or 't' or 'C' or 'c', then y := alpha*AT*x + beta*y,\nm\nNumber of rows of the matrix A.\nk\nNumber of columns of the matrix A.\nalpha\nSpecifies the scalar alpha.\nmatdescra\nArray of six elements, specifies properties of the matrix used for operation.\nOnly first four array elements are used, their possible values are given in \nTable “Possible Values of the Parameter matdescra (descra)”. Possible\ncombinations of element values of this parameter are given in Table\n“Possible Combinations of Element Values of the Parameter matdescra”.\nval\nArray of length nnz, contains non-zero elements of the matrix A in the\narbitrary order.\nRefer to values array description in Coordinate Format for more details.\nrowind\nArray of length nnz.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n177\n\n\nFor one-based indexing, contains the row indices plus one for each non-zero\nelement of the matrix A.\nFor zero-based indexing, contains the row indices for each non-zero\nelement of the matrix A.\nRefer to rows array description in Coordinate Format for more details.\ncolind\nArray of length nnz.\nFor one-based indexing, contains the column indices plus one for each non-\nzero element of the matrix A.\nFor zero-based indexing, contains the column indices for each non-zero\nelement of the matrix A.\nRefer to columns array description in Coordinate Format for more details.\nnnz\nSpecifies the number of non-zero element of the matrix A.\nRefer to nnz description in Coordinate Format for more details.\nx\nArray, size at least k if transa = 'N' or 'n' and at least m otherwise. On\nentry, the array x must contain the vector x.\nbeta\nSpecifies the scalar beta.\ny\nArray, size at least m if transa = 'N' or 'n' and at least k otherwise. On\nentry, the array y must contain the vector y.\nOutput Parameters\ny\nOverwritten by the updated vector y.\nmkl_?csrsv\nSolves a system of linear equations for a sparse\nmatrix in the CSR format (deprecated).\nSyntax\nvoid mkl_scsrsv (const char *transa , const MKL_INT *m , const float *alpha , const\nchar *matdescra , const float *val , const MKL_INT *indx , const MKL_INT *pntrb , const\nMKL_INT *pntre , const float *x , float *y );\nvoid mkl_dcsrsv (const char *transa , const MKL_INT *m , const double *alpha , const\nchar *matdescra , const double *val , const MKL_INT *indx , const MKL_INT *pntrb ,\nconst MKL_INT *pntre , const double *x , double *y );\nvoid mkl_ccsrsv (const char *transa , const MKL_INT *m , const MKL_Complex8 *alpha ,\nconst char *matdescra , const MKL_Complex8 *val , const MKL_INT *indx , const MKL_INT\n*pntrb , const MKL_INT *pntre , const MKL_Complex8 *x , MKL_Complex8 *y );\nvoid mkl_zcsrsv (const char *transa , const MKL_INT *m , const MKL_Complex16 *alpha ,\nconst char *matdescra , const MKL_Complex16 *val , const MKL_INT *indx , const MKL_INT\n*pntrb , const MKL_INT *pntre , const MKL_Complex16 *x , MKL_Complex16 *y );\nInclude Files\n•\nmkl.h\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n178\n\n\nDescription\nThis routine is deprecated. Use mkl_sparse_?_trsvfrom the Intel® oneAPI Math Kernel Library (oneMKL)\nInspector-executor Sparse BLAS interface instead.\nThe mkl_?csrsv routine solves a system of linear equations with matrix-vector operations for a sparse\nmatrix in the CSR format:\ny := alpha*inv(A)*x\nor\ny := alpha*inv(AT)*x,\nwhere:\nalpha is scalar, x and y are vectors, A is a sparse upper or lower triangular matrix with unit or non-unit main\ndiagonal, AT is the transpose of A.\nNOTE\nThis routine supports a CSR format both with one-based indexing and zero-based indexing.\nInput Parameters\ntransa\nSpecifies the system of linear equations.\nIf transa = 'N' or 'n', then y := alpha*inv(A)*x\nIf transa = 'T' or 't' or 'C' or 'c', then y := alpha*inv(AT)*x,\nm\nNumber of columns of the matrix A.\nalpha\nSpecifies the scalar alpha.\nmatdescra\nArray of six elements, specifies properties of the matrix used for operation.\nOnly first four array elements are used, their possible values are given in \nTable “Possible Values of the Parameter matdescra (descra)”. Possible\ncombinations of element values of this parameter are given in Table\n“Possible Combinations of Element Values of the Parameter matdescra”.\nval\nArray containing non-zero elements of the matrix A.\nIts length is pntre[m - 1] - pntrb[0].\nRefer to values array description in CSR Format for more details.\nNOTE\nThe non-zero elements of the given row of the matrix must be\nstored in the same order as they appear in the row (from left to\nright).\nNo diagonal element can be omitted from a sparse storage if the solver\nis called with the non-unit indicator.\nindx\nFor one-based indexing, array containing the column indices plus one for\neach non-zero element of the matrix A. For zero-based indexing, array\ncontaining the column indices for each non-zero element of the matrix A.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n179\n\n\nIts length is equal to length of the val array.\nRefer to columns array description in CSR Format for more details.\nNOTE\nColumn indices must be sorted in increasing order for each row.\npntrb\nArray of length m.\nThis array contains row indices, such that pntrb[i] - pntrb[0] is the\nfirst index of row i in the arrays val and indx.\nRefer to pointerb array description in CSR Format for more details.\npntre\nArray of length m.\nThis array contains row indices, such that pntre[i] - pntrb[0] - 1 is\nthe last index of row i in the arrays val and indx.\nRefer to pointerE array description in CSR Format for more details.\nx\nArray, size at least m.\nOn entry, the array x must contain the vector x. The elements are accessed\nwith unit increment.\ny\nArray, size at least m.\nOn entry, the array y must contain the vector y. The elements are accessed\nwith unit increment.\nOutput Parameters\ny\nContains solution vector x.\nmkl_?bsrsv\nSolves a system of linear equations for a sparse\nmatrix in the BSR format (deprecated).\nSyntax\nvoid mkl_sbsrsv (const char *transa , const MKL_INT *m , const MKL_INT *lb , const\nfloat *alpha , const char *matdescra , const float *val , const MKL_INT *indx , const\nMKL_INT *pntrb , const MKL_INT *pntre , const float *x , float *y );\nvoid mkl_dbsrsv (const char *transa , const MKL_INT *m , const MKL_INT *lb , const\ndouble *alpha , const char *matdescra , const double *val , const MKL_INT *indx , const\nMKL_INT *pntrb , const MKL_INT *pntre , const double *x , double *y );\nvoid mkl_cbsrsv (const char *transa , const MKL_INT *m , const MKL_INT *lb , const\nMKL_Complex8 *alpha , const char *matdescra , const MKL_Complex8 *val , const MKL_INT\n*indx , const MKL_INT *pntrb , const MKL_INT *pntre , const MKL_Complex8 *x ,\nMKL_Complex8 *y );\nvoid mkl_zbsrsv (const char *transa , const MKL_INT *m , const MKL_INT *lb , const\nMKL_Complex16 *alpha , const char *matdescra , const MKL_Complex16 *val , const MKL_INT\n*indx , const MKL_INT *pntrb , const MKL_INT *pntre , const MKL_Complex16 *x ,\nMKL_Complex16 *y );\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n180\n\n\nInclude Files\n•\nmkl.h\nDescription\nThis routine is deprecated. Use mkl_sparse_?_trsvfrom the Intel® oneAPI Math Kernel Library (oneMKL)\nInspector-executor Sparse BLAS interface instead.\nThe mkl_?bsrsv routine solves a system of linear equations with matrix-vector operations for a sparse\nmatrix in the BSR format:\ny := alpha*inv(A)*x\nor\ny := alpha*inv(AT)* x,\nwhere:\nalpha is scalar, x and y are vectors, A is a sparse upper or lower triangular matrix with unit or non-unit main\ndiagonal, AT is the transpose of A.\nNOTE\nThis routine supports a BSR format both with one-based indexing and zero-based indexing.\nInput Parameters\ntransa\nSpecifies the operation.\nIf transa = 'N' or 'n', then y := alpha*inv(A)*x\nIf transa = 'T' or 't' or 'C' or 'c', then y := alpha*inv(AT)* x,\nm\nNumber of block columns of the matrix A.\nlb\nSize of the block in the matrix A.\nalpha\nSpecifies the scalar alpha.\nmatdescra\nArray of six elements, specifies properties of the matrix used for operation.\nOnly first four array elements are used, their possible values are given in \nTable “Possible Values of the Parameter matdescra (descra)”. Possible\ncombinations of element values of this parameter are given in Table\n“Possible Combinations of Element Values of the Parameter matdescra”.\nval\nArray containing elements of non-zero blocks of the matrix A. Its length is\nequal to the number of non-zero blocks in the matrix A multiplied by lb*lb.\nRefer to the values array description in BSR Format for more details.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n181\n\n\nNOTE\nThe non-zero elements of the given row of the matrix must be\nstored in the same order as they appear in the row (from left to\nright).\nNo diagonal element can be omitted from a sparse storage if the solver\nis called with the non-unit indicator.\nindx\nFor one-based indexing, array containing the column indices plus one for\neach non-zero element of the matrix A. For zero-based indexing, array\ncontaining the column indices for each non-zero element of the matrix A.\nIts length is equal to the number of non-zero blocks in the matrix A.\nRefer to the columns array description in BSR Format for more details.\npntrb\nArray of length m.\nThis array contains row indices, such that pntrb[i] - pntrb[0] is the\nfirst index of block row i in the array indx\nRefer to pointerB array description in BSR Format for more details.\npntre\nArray of length m.\nFor one-based indexing this array contains row indices, such that pntre[i]\n- pntrb[1] is the last index of block row i in the array indx.\nFor zero-based indexing this array contains row indices, such that\npntre[i] - pntrb[0] - 1 is the last index of block row i in the array\nindx.\nRefer to pointerE array description in BSR Format for more details.\nx\nArray, size at least (m*lb).\nOn entry, the array x must contain the vector x. The elements are accessed\nwith unit increment.\ny\nArray, size at least (m*lb).\nOn entry, the array y must contain the vector y. The elements are accessed\nwith unit increment.\nOutput Parameters\ny\nContains solution vector x.\nmkl_?cscsv\nSolves a system of linear equations for a sparse\nmatrix in the CSC format (deprecated).\nSyntax\nvoid mkl_scscsv (const char *transa , const MKL_INT *m , const float *alpha , const\nchar *matdescra , const float *val , const MKL_INT *indx , const MKL_INT *pntrb , const\nMKL_INT *pntre , const float *x , float *y );\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n182\n\n\nvoid mkl_dcscsv (const char *transa , const MKL_INT *m , const double *alpha , const\nchar *matdescra , const double *val , const MKL_INT *indx , const MKL_INT *pntrb ,\nconst MKL_INT *pntre , const double *x , double *y );\nvoid mkl_ccscsv (const char *transa , const MKL_INT *m , const MKL_Complex8 *alpha ,\nconst char *matdescra , const MKL_Complex8 *val , const MKL_INT *indx , const MKL_INT\n*pntrb , const MKL_INT *pntre , const MKL_Complex8 *x , MKL_Complex8 *y );\nvoid mkl_zcscsv (const char *transa , const MKL_INT *m , const MKL_Complex16 *alpha ,\nconst char *matdescra , const MKL_Complex16 *val , const MKL_INT *indx , const MKL_INT\n*pntrb , const MKL_INT *pntre , const MKL_Complex16 *x , MKL_Complex16 *y );\nInclude Files\n•\nmkl.h\nDescription\nThis routine is deprecated. Use mkl_sparse_?_trsvfrom the Intel® oneAPI Math Kernel Library (oneMKL)\nInspector-executor Sparse BLAS interface instead.\nThe mkl_?cscsv routine solves a system of linear equations with matrix-vector operations for a sparse\nmatrix in the CSC format:\ny := alpha*inv(A)*x\nor\ny := alpha*inv(AT)* x,\nwhere:\nalpha is scalar, x and y are vectors, A is a sparse upper or lower triangular matrix with unit or non-unit main\ndiagonal, AT is the transpose of A.\nNOTE\nThis routine supports a CSC format both with one-based indexing and zero-based indexing.\nInput Parameters\ntransa\nSpecifies the operation.\nIf transa = 'N' or 'n', then y := alpha*inv(A)*x\nIf transa= 'T' or 't' or 'C' or 'c', then y := alpha*inv(AT)* x,\nm\nNumber of columns of the matrix A.\nalpha\nSpecifies the scalar alpha.\nmatdescra\nArray of six elements, specifies properties of the matrix used for operation.\nOnly first four array elements are used, their possible values are given in \nTable “Possible Values of the Parameter matdescra (descra)”. Possible\ncombinations of element values of this parameter are given in Table\n“Possible Combinations of Element Values of the Parameter matdescra”.\nval\nArray containing non-zero elements of the matrix A.\nIts length is pntre[m-1] - pntrb[0].\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n183\n\n\nRefer to values array description in CSC Format for more details.\nNOTE\nThe non-zero elements of the given row of the matrix must be\nstored in the same order as they appear in the row (from left to\nright).\nNo diagonal element can be omitted from a sparse storage if the solver\nis called with the non-unit indicator.\nindx\nFor one-based indexing, array containing the row indices plus one for each\nnon-zero element of the matrix A.\nFor zero-based indexing, array containing the row indices for each non-zero\nelement of the matrix A.\nIts length is equal to length of the val array.\nRefer to columns array description in CSC Format for more details.\nNOTE\nRow indices must be sorted in increasing order for each column.\npntrb\nArray of length m.\nThis array contains column indices, such that pntrb[i] - pntrb[0] is the\nfirst index of column i in the arrays val and indx.\nRefer to pointerb array description in CSC Format for more details.\npntre\nArray of length m.\nThis array contains column indices, such that pntre[i] - pntrb[0] - 1\nis the last index of column i in the arrays val and indx.\nRefer to pointerE array description in CSC Format for more details.\nx\nArray, size at least m.\nOn entry, the array x must contain the vector x. The elements are accessed\nwith unit increment.\ny\nArray, size at least m.\nOn entry, the array y must contain the vector y. The elements are accessed\nwith unit increment.\nOutput Parameters\ny\nContains the solution vector x.\nmkl_?coosv\nSolves a system of linear equations for a sparse\nmatrix in the coordinate format (deprecated).\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n184\n\n\nSyntax\nvoid mkl_scoosv (const char *transa , const MKL_INT *m , const float *alpha , const\nchar *matdescra , const float *val , const MKL_INT *rowind , const MKL_INT *colind ,\nconst MKL_INT *nnz , const float *x , float *y );\nvoid mkl_dcoosv (const char *transa , const MKL_INT *m , const double *alpha , const\nchar *matdescra , const double *val , const MKL_INT *rowind , const MKL_INT *colind ,\nconst MKL_INT *nnz , const double *x , double *y );\nvoid mkl_ccoosv (const char *transa , const MKL_INT *m , const MKL_Complex8 *alpha ,\nconst char *matdescra , const MKL_Complex8 *val , const MKL_INT *rowind , const MKL_INT\n*colind , const MKL_INT *nnz , const MKL_Complex8 *x , MKL_Complex8 *y );\nvoid mkl_zcoosv (const char *transa , const MKL_INT *m , const MKL_Complex16 *alpha ,\nconst char *matdescra , const MKL_Complex16 *val , const MKL_INT *rowind , const\nMKL_INT *colind , const MKL_INT *nnz , const MKL_Complex16 *x , MKL_Complex16 *y );\nInclude Files\n•\nmkl.h\nDescription\nThis routine is deprecated. Use mkl_sparse_?_trsvfrom the Intel® oneAPI Math Kernel Library (oneMKL)\nInspector-executor Sparse BLAS interface instead.\nThe mkl_?coosv routine solves a system of linear equations with matrix-vector operations for a sparse\nmatrix in the coordinate format:\ny := alpha*inv(A)*x\nor\ny := alpha*inv(AT)*x,\nwhere:\nalpha is scalar, x and y are vectors, A is a sparse upper or lower triangular matrix with unit or non-unit main\ndiagonal, AT is the transpose of A.\nNOTE\nThis routine supports a coordinate format both with one-based indexing and zero-based indexing.\nInput Parameters\ntransa\nSpecifies the system of linear equations.\nIf transa = 'N' or 'n', then y := alpha*inv(A)*x\nIf transa = 'T' or 't' or 'C' or 'c', then y := alpha*inv(AT)* x,\nm\nNumber of rows of the matrix A.\nalpha\nSpecifies the scalar alpha.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n185\n\n\nmatdescra\nArray of six elements, specifies properties of the matrix used for operation.\nOnly first four array elements are used, their possible values are given in \nTable “Possible Values of the Parameter matdescra (descra)”. Possible\ncombinations of element values of this parameter are given in Table\n“Possible Combinations of Element Values of the Parameter matdescra”.\nval\nArray of length nnz, contains non-zero elements of the matrix A in the\narbitrary order.\nRefer to values array description in Coordinate Format for more details.\nrowind\nArray of length nnz.\nFor one-based indexing, contains the row indices plus one for each non-zero\nelement of the matrix A.\nFor zero-based indexing, contains the row indices for each non-zero\nelement of the matrix A.\nRefer to rows array description in Coordinate Format for more details.\ncolind\nArray of length nnz.\nFor one-based indexing, contains the column indices plus one for each non-\nzero element of the matrix A.\nFor zero-based indexing, contains the column indices for each non-zero\nelement of the matrix A.\nRefer to columns array description in Coordinate Format for more details.\nnnz\nSpecifies the number of non-zero element of the matrix A.\nRefer to nnz description in Coordinate Format for more details.\nx\nArray, size at least m.\nOn entry, the array x must contain the vector x. The elements are accessed\nwith unit increment.\ny\nArray, size at least m.\nOn entry, the array y must contain the vector y. The elements are accessed\nwith unit increment.\nOutput Parameters\ny\nContains solution vector x.\nmkl_?csrmm\nComputes matrix - matrix product of a sparse matrix\nstored in the CSR format (deprecated).\nSyntax\nvoid mkl_scsrmm (const char *transa , const MKL_INT *m , const MKL_INT *n , const\nMKL_INT *k , const float *alpha , const char *matdescra , const float *val , const\nMKL_INT *indx , const MKL_INT *pntrb , const MKL_INT *pntre , const float *b , const\nMKL_INT *ldb , const float *beta , float *c , const MKL_INT *ldc );\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n186\n\n\nvoid mkl_dcsrmm (const char *transa , const MKL_INT *m , const MKL_INT *n , const\nMKL_INT *k , const double *alpha , const char *matdescra , const double *val , const\nMKL_INT *indx , const MKL_INT *pntrb , const MKL_INT *pntre , const double *b , const\nMKL_INT *ldb , const double *beta , double *c , const MKL_INT *ldc );\nvoid mkl_ccsrmm (const char *transa , const MKL_INT *m , const MKL_INT *n , const\nMKL_INT *k , const MKL_Complex8 *alpha , const char *matdescra , const MKL_Complex8\n*val , const MKL_INT *indx , const MKL_INT *pntrb , const MKL_INT *pntre , const\nMKL_Complex8 *b , const MKL_INT *ldb , const MKL_Complex8 *beta , MKL_Complex8 *c ,\nconst MKL_INT *ldc );\nvoid mkl_zcsrmm (const char *transa , const MKL_INT *m , const MKL_INT *n , const\nMKL_INT *k , const MKL_Complex16 *alpha , const char *matdescra , const MKL_Complex16\n*val , const MKL_INT *indx , const MKL_INT *pntrb , const MKL_INT *pntre , const\nMKL_Complex16 *b , const MKL_INT *ldb , const MKL_Complex16 *beta , MKL_Complex16 *c ,\nconst MKL_INT *ldc );\nInclude Files\n•\nmkl.h\nDescription\nThis routine is deprecated. Use Use mkl_sparse_?_mmfrom the Intel® oneAPI Math Kernel Library (oneMKL)\nInspector-executor Sparse BLAS interface instead.\nThe mkl_?csrmm routine performs a matrix-matrix operation defined as\nC := alpha*A*B + beta*C\nor\nC := alpha*AT*B + beta*C\nor\nC := alpha*AH*B + beta*C,\nwhere:\nalpha and beta are scalars,\nB and C are dense matrices, A is an m-by-k sparse matrix in compressed sparse row (CSR) format, AT is the\ntranspose of A, and AH is the conjugate transpose of A.\nNOTE\nThis routine supports a CSR format both with one-based indexing and zero-based indexing.\nInput Parameters\ntransa\nSpecifies the operation.\nIf transa = 'N' or 'n', then C := alpha*A*B + beta*C,\nIf transa = 'T' or 't', then C := alpha*AT*B + beta*C,\nIf transa = 'C' or 'c', then C := alpha*AH*B + beta*C.\nm\nNumber of rows of the matrix A.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n187\n\n\nn\nNumber of columns of the matrix C.\nk\nNumber of columns of the matrix A.\nalpha\nSpecifies the scalar alpha.\nmatdescra\nArray of six elements, specifies properties of the matrix used for operation.\nOnly first four array elements are used, their possible values are given in \nTable \"Possible Values of the Parameter matdescra (descra)\". Possible\ncombinations of element values of this parameter are given in Table\n\"Possible Combinations of Element Values of the Parameter matdescra\".\nval\nArray containing non-zero elements of the matrix A.\nFor zero-based indexing its length is pntre[m—1] - pntrb[0].\nRefer to values array description in CSR Format for more details.\nindx\nFor one-based indexing, array containing the column indices plus one for\neach non-zero element of the matrix A.\nFor zero-based indexing, array containing the column indices for each non-\nzero element of the matrix A.\nIts length is equal to length of the val array.\nRefer to columns array description in CSR Format for more details.\npntrb\nArray of length m.\nThis array contains row indices, such that pntrb[I] - pntrb[0] is the\nfirst index of row I in the arrays val and indx.\nRefer to pointerb array description in CSR Format for more details.\npntre\nArray of length m.\nThis array contains row indices, such that pntre[I] - pntrb[0] - 1 is\nthe last index of row I in the arrays val and indx.\nRefer to pointerE array description in CSR Format for more details.\nb\nArray, size ldb by at least n for non-transposed matrix A and at least m for\ntransposed for one-based indexing, and (at least k for non-transposed\nmatrix A and at least m for transposed, ldb) for zero-based indexing.\nOn entry with transa='N' or 'n', the leading k-by-n part of the array b\nmust contain the matrix B, otherwise the leading m-by-n part of the array b\nmust contain the matrix B.\nldb\nSpecifies the leading dimension of b for one-based indexing, and the second\ndimension of b for zero-based indexing, as declared in the calling\n(sub)program.\nbeta\nSpecifies the scalar beta.\nc\nArray, size ldc by n for one-based indexing, and (m, ldc) for zero-based\nindexing.\nOn entry, the leading m-by-n part of the array c must contain the matrix C,\notherwise the leading k-by-n part of the array c must contain the matrix C.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n188\n\n\nldc\nSpecifies the leading dimension of c for one-based indexing, and the second\ndimension of c for zero-based indexing, as declared in the calling\n(sub)program.\nOutput Parameters\nc\nOverwritten by the matrix (alpha*A*B + beta* C), (alpha*AT*B +\nbeta*C), or (alpha*AH*B + beta*C).\nmkl_?bsrmm\nComputes matrix - matrix product of a sparse matrix\nstored in the BSR format (deprecated).\nSyntax\nvoid mkl_sbsrmm (const char *transa , const MKL_INT *m , const MKL_INT *n , const\nMKL_INT *k , const MKL_INT *lb , const float *alpha , const char *matdescra , const\nfloat *val , const MKL_INT *indx , const MKL_INT *pntrb , const MKL_INT *pntre , const\nfloat *b , const MKL_INT *ldb , const float *beta , float *c , const MKL_INT *ldc );\nvoid mkl_dbsrmm (const char *transa , const MKL_INT *m , const MKL_INT *n , const\nMKL_INT *k , const MKL_INT *lb , const double *alpha , const char *matdescra , const\ndouble *val , const MKL_INT *indx , const MKL_INT *pntrb , const MKL_INT *pntre , const\ndouble *b , const MKL_INT *ldb , const double *beta , double *c , const MKL_INT *ldc );\nvoid mkl_cbsrmm (const char *transa , const MKL_INT *m , const MKL_INT *n , const\nMKL_INT *k , const MKL_INT *lb , const MKL_Complex8 *alpha , const char *matdescra ,\nconst MKL_Complex8 *val , const MKL_INT *indx , const MKL_INT *pntrb , const MKL_INT\n*pntre , const MKL_Complex8 *b , const MKL_INT *ldb , const MKL_Complex8 *beta ,\nMKL_Complex8 *c , const MKL_INT *ldc );\nvoid mkl_zbsrmm (const char *transa , const MKL_INT *m , const MKL_INT *n , const\nMKL_INT *k , const MKL_INT *lb , const MKL_Complex16 *alpha , const char *matdescra ,\nconst MKL_Complex16 *val , const MKL_INT *indx , const MKL_INT *pntrb , const MKL_INT\n*pntre , const MKL_Complex16 *b , const MKL_INT *ldb , const MKL_Complex16 *beta ,\nMKL_Complex16 *c , const MKL_INT *ldc );\nInclude Files\n•\nmkl.h\nDescription\nThis routine is deprecated. Use Use mkl_sparse_?_mmfrom the Intel® oneAPI Math Kernel Library (oneMKL)\nInspector-executor Sparse BLAS interface instead.\nThe mkl_?bsrmm routine performs a matrix-matrix operation defined as\nC := alpha*A*B + beta*C\nor\nC := alpha*AT*B + beta*C\nor\nC := alpha*AH*B + beta*C,\nwhere:\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n189\n\n\nalpha and beta are scalars,\nB and C are dense matrices, A is an m-by-k sparse matrix in block sparse row (BSR) format, AT is the\ntranspose of A, and AH is the conjugate transpose of A.\nNOTE\nThis routine supports a BSR format both with one-based indexing and zero-based indexing.\nInput Parameters\ntransa\nSpecifies the operation.\nIf transa = 'N' or 'n', then the matrix-matrix product is computed as\nC := alpha*A*B + beta*C\nIf transa = 'T' or 't', then the matrix-vector product is computed as\nC := alpha*AT*B + beta*C\nIf transa = 'C' or 'c', then the matrix-vector product is computed as\nC := alpha*AH*B + beta*C,\nm\nNumber of block rows of the matrix A.\nn\nNumber of columns of the matrix C.\nk\nNumber of block columns of the matrix A.\nlb\nSize of the block in the matrix A.\nalpha\nSpecifies the scalar alpha.\nmatdescra\nArray of six elements, specifies properties of the matrix used for operation.\nOnly first four array elements are used, their possible values are given in \nTable “Possible Values of the Parameter matdescra (descra)”. Possible\ncombinations of element values of this parameter are given in Table\n“Possible Combinations of Element Values of the Parameter matdescra”.\nval\nArray containing elements of non-zero blocks of the matrix A. Its length is\nequal to the number of non-zero blocks in the matrix A multiplied by lb*lb.\nRefer to the values array description in BSR Format for more details.\nindx\nFor one-based indexing, array containing the column indices plus one for\neach non-zero block in the matrix A.\nFor zero-based indexing, array containing the column indices for each non-\nzero block in the matrix A.\nIts length is equal to the number of non-zero blocks in the matrix A. Refer\nto the columns array description in BSR Format for more details.\npntrb\nArray of length m.\nThis array contains row indices, such that pntrb[I] - pntrb[0] is the\nfirst index of block row I in the array indx.\nRefer to pointerB array description in BSR Format for more details.\npntre\nArray of length m.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n190\n\n\nThis array contains row indices, such that pntre[I] - pntrb[0] - 1 is\nthe last index of block row I in the array indx.\nRefer to pointerE array description in BSR Format for more details.\nb\nArray, size ldb by at least n for non-transposed matrix A and at least m for\ntransposed for one-based indexing, and (at least k for non-transposed\nmatrix A and at least m for transposed, ldb) for zero-based indexing.\nOn entry with transa='N' or 'n', the leading n-by-k block part of the\narray b must contain the matrix B, otherwise the leading m-by-n block part\nof the array b must contain the matrix B.\nldb\nSpecifies the leading dimension (in blocks) of b as declared in the calling\n(sub)program.\nbeta\nSpecifies the scalar beta.\nc\nArray, size ldc* n for one-based indexing, size k* ldc for zero-based\nindexing.\nOn entry, the leading m-by-n block part of the array c must contain the\nmatrix C, otherwise the leading n-by-k block part of the array c must\ncontain the matrix C.\nldc\nSpecifies the leading dimension (in blocks) of c as declared in the calling\n(sub)program.\nOutput Parameters\nc\nOverwritten by the matrix (alpha*A*B + beta*C) or (alpha*AT*B +\nbeta*C) or (alpha*AH*B + beta*C).\nmkl_?cscmm\nComputes matrix-matrix product of a sparse matrix\nstored in the CSC format (deprecated).\nSyntax\nvoid mkl_scscmm (const char *transa , const MKL_INT *m , const MKL_INT *n , const\nMKL_INT *k , const float *alpha , const char *matdescra , const float *val , const\nMKL_INT *indx , const MKL_INT *pntrb , const MKL_INT *pntre , const float *b , const\nMKL_INT *ldb , const float *beta , float *c , const MKL_INT *ldc );\nvoid mkl_dcscmm (const char *transa , const MKL_INT *m , const MKL_INT *n , const\nMKL_INT *k , const double *alpha , const char *matdescra , const double *val , const\nMKL_INT *indx , const MKL_INT *pntrb , const MKL_INT *pntre , const double *b , const\nMKL_INT *ldb , const double *beta , double *c , const MKL_INT *ldc );\nvoid mkl_ccscmm (const char *transa , const MKL_INT *m , const MKL_INT *n , const\nMKL_INT *k , const MKL_Complex8 *alpha , const char *matdescra , const MKL_Complex8\n*val , const MKL_INT *indx , const MKL_INT *pntrb , const MKL_INT *pntre , const\nMKL_Complex8 *b , const MKL_INT *ldb , const MKL_Complex8 *beta , MKL_Complex8 *c ,\nconst MKL_INT *ldc );\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n191\n\n\nvoid mkl_zcscmm (const char *transa , const MKL_INT *m , const MKL_INT *n , const\nMKL_INT *k , const MKL_Complex16 *alpha , const char *matdescra , const MKL_Complex16\n*val , const MKL_INT *indx , const MKL_INT *pntrb , const MKL_INT *pntre , const\nMKL_Complex16 *b , const MKL_INT *ldb , const MKL_Complex16 *beta , MKL_Complex16 *c ,\nconst MKL_INT *ldc );\nInclude Files\n•\nmkl.h\nDescription\nThis routine is deprecated. Use Use mkl_sparse_?_mmfrom the Intel® oneAPI Math Kernel Library (oneMKL)\nInspector-executor Sparse BLAS interface instead.\nThe mkl_?cscmm routine performs a matrix-matrix operation defined as\nC := alpha*A*B + beta*C\nor\nC := alpha*AT*B + beta*C,\nor\nC := alpha*AH*B + beta*C,\nwhere:\nalpha and beta are scalars,\nB and C are dense matrices, A is an m-by-k sparse matrix in compressed sparse column (CSC) format, AT is\nthe transpose of A, and AH is the conjugate transpose of A.\nNOTE\nThis routine supports CSC format both with one-based indexing and zero-based indexing.\nInput Parameters\ntransa\nSpecifies the operation.\nIf transa = 'N' or 'n', then C := alpha*A* B + beta*C\nIf transa = 'T' or 't', then C := alpha*AT*B + beta*C,\nIf transa ='C' or 'c', then C := alpha*AH*B + beta*C\nm\nNumber of rows of the matrix A.\nn\nNumber of columns of the matrix C.\nk\nNumber of columns of the matrix A.\nalpha\nSpecifies the scalar alpha.\nmatdescra\nArray of six elements, specifies properties of the matrix used for operation.\nOnly first four array elements are used, their possible values are given in \nTable “Possible Values of the Parameter matdescra (descra)”. Possible\ncombinations of element values of this parameter are given in Table\n“Possible Combinations of Element Values of the Parameter matdescra”.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n192\n\n\nval\nArray containing non-zero elements of the matrix A.\nIts length is pntrb[k-1] - pntrb[0].\nRefer to values array description in CSC Format for more details.\nindx\nFor one-based indexing, array containing the row indices plus one for each\nnon-zero element of the matrix A.\nFor zero-based indexing, array containing the column indices for each non-\nzero element of the matrix A.\nIts length is equal to length of the val array.\nRefer to rows array description in CSC Format for more details.\npntrb\nArray of length k.\nThis array contains column indices, such that pntrb[i] - pntrb[0] is the\nfirst index of column i in the arrays val and indx.\nRefer to pointerb array description in CSC Format for more details.\npntre\nArray of length k.\nThis array contains column indices, such that pntre[i] - pntrb[0] - 1\nis the last index of column i in the arrays val and indx.\nRefer to pointerE array description in CSC Format for more details.\nb\nArray, size ldb by at least n for non-transposed matrix A and at least m for\ntransposed for one-based indexing, and (at least k for non-transposed\nmatrix A and at least m for transposed, ldb) for zero-based indexing.\nOn entry with transa = 'N' or 'n', the leading k-by-n part of the array b\nmust contain the matrix B, otherwise the leading m-by-n part of the array b\nmust contain the matrix B.\nldb\nSpecifies the leading dimension of b for one-based indexing, and the second\ndimension of b for zero-based indexing, as declared in the calling\n(sub)program.\nbeta\nSpecifies the scalar beta.\nc\nArray, size ldc by n for one-based indexing, and (m, ldc) for zero-based\nindexing.\nOn entry, the leading m-by-n part of the array c must contain the matrix C,\notherwise the leading k-by-n part of the array c must contain the matrix C.\nldc\nSpecifies the leading dimension of c for one-based indexing, and the second\ndimension of c for zero-based indexing, as declared in the calling\n(sub)program.\nOutput Parameters\nc\nOverwritten by the matrix (alpha*A*B + beta* C) or (alpha*AT*B +\nbeta*C) or (alpha*AH*B + beta*C).\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n193\n\n\nmkl_?coomm\nComputes matrix-matrix product of a sparse matrix\nstored in the coordinate format (deprecated).\nSyntax\nvoid mkl_scoomm (const char *transa , const MKL_INT *m , const MKL_INT *n , const\nMKL_INT *k , const float *alpha , const char *matdescra , const float *val , const\nMKL_INT *rowind , const MKL_INT *colind , const MKL_INT *nnz , const float *b , const\nMKL_INT *ldb , const float *beta , float *c , const MKL_INT *ldc );\nvoid mkl_dcoomm (const char *transa , const MKL_INT *m , const MKL_INT *n , const\nMKL_INT *k , const double *alpha , const char *matdescra , const double *val , const\nMKL_INT *rowind , const MKL_INT *colind , const MKL_INT *nnz , const double *b , const\nMKL_INT *ldb , const double *beta , double *c , const MKL_INT *ldc );\nvoid mkl_ccoomm (const char *transa , const MKL_INT *m , const MKL_INT *n , const\nMKL_INT *k , const MKL_Complex8 *alpha , const char *matdescra , const MKL_Complex8\n*val , const MKL_INT *rowind , const MKL_INT *colind , const MKL_INT *nnz , const\nMKL_Complex8 *b , const MKL_INT *ldb , const MKL_Complex8 *beta , MKL_Complex8 *c ,\nconst MKL_INT *ldc );\nvoid mkl_zcoomm (const char *transa , const MKL_INT *m , const MKL_INT *n , const\nMKL_INT *k , const MKL_Complex16 *alpha , const char *matdescra , const MKL_Complex16\n*val , const MKL_INT *rowind , const MKL_INT *colind , const MKL_INT *nnz , const\nMKL_Complex16 *b , const MKL_INT *ldb , const MKL_Complex16 *beta , MKL_Complex16 *c ,\nconst MKL_INT *ldc );\nInclude Files\n•\nmkl.h\nDescription\nThis routine is deprecated. Use Use mkl_sparse_?_mmfrom the Intel® oneAPI Math Kernel Library (oneMKL)\nInspector-executor Sparse BLAS interface instead.\nThe mkl_?coomm routine performs a matrix-matrix operation defined as\nC := alpha*A*B + beta*C\nor\nC := alpha*AT*B + beta*C,\nor\nC := alpha*AH*B + beta*C,\nwhere:\nalpha and beta are scalars,\nB and C are dense matrices, A is an m-by-k sparse matrix in the coordinate format, AT is the transpose of A,\nand AH is the conjugate transpose of A.\nNOTE\nThis routine supports a coordinate format both with one-based indexing and zero-based indexing.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n194\n\n\nInput Parameters\ntransa\nSpecifies the operation.\nIf transa = 'N' or 'n', then C := alpha*A*B + beta*C\nIf transa = 'T' or 't', then C := alpha*AT*B + beta*C,\nIf transa = 'C' or 'c', then C := alpha*AH*B + beta*C.\nm\nNumber of rows of the matrix A.\nn\nNumber of columns of the matrix C.\nk\nNumber of columns of the matrix A.\nalpha\nSpecifies the scalar alpha.\nmatdescra\nArray of six elements, specifies properties of the matrix used for operation.\nOnly first four array elements are used, their possible values are given in \nTable “Possible Values of the Parameter matdescra (descra)”. Possible\ncombinations of element values of this parameter are given in Table\n“Possible Combinations of Element Values of the Parameter matdescra”.\nval\nArray of length nnz, contains non-zero elements of the matrix A in the\narbitrary order.\nRefer to values array description in Coordinate Format for more details.\nrowind\nArray of length nnz.\nFor one-based indexing, contains the row indices plus one for each non-zero\nelement of the matrix A.\nFor zero-based indexing, contains the row indices for each non-zero\nelement of the matrix A.\nRefer to rows array description in Coordinate Format for more details.\ncolind\nArray of length nnz.\nFor one-based indexing, contains the column indices plus one for each non-\nzero element of the matrix A.\nFor zero-based indexing, contains the column indices for each non-zero\nelement of the matrix A.\nRefer to columns array description in Coordinate Format for more details.\nnnz\nSpecifies the number of non-zero element of the matrix A.\nRefer to nnz description in Coordinate Format for more details.\nb\nArray, size ldb by at least n for non-transposed matrix A and at least m for\ntransposed for one-based indexing, and (at least k for non-transposed\nmatrix A and at least m for transposed, ldb) for zero-based indexing.\nOn entry with transa = 'N' or 'n', the leading k-by-n part of the array b\nmust contain the matrix B, otherwise the leading m-by-n part of the array b\nmust contain the matrix B.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n195\n\n\nldb\nSpecifies the leading dimension of b for one-based indexing, and the second\ndimension of b for zero-based indexing, as declared in the calling\n(sub)program.\nbeta\nSpecifies the scalar beta.\nc\nArray, size ldc by n for one-based indexing, and (m, ldc) for zero-based\nindexing.\nOn entry, the leading m-by-n part of the array c must contain the matrix C,\notherwise the leading k-by-n part of the array c must contain the matrix C.\nldc\nSpecifies the leading dimension of c for one-based indexing, and the second\ndimension of c for zero-based indexing, as declared in the calling\n(sub)program.\nOutput Parameters\nc\nOverwritten by the matrix (alpha*A*B + beta*C), (alpha*AT*B +\nbeta*C), or (alpha*AH*B + beta*C).\nmkl_?csrsm\nSolves a system of linear matrix equations for a\nsparse matrix in the CSR format (deprecated).\nSyntax\nvoid mkl_scsrsm (const char *transa , const MKL_INT *m , const MKL_INT *n , const float\n*alpha , const char *matdescra , const float *val , const MKL_INT *indx , const MKL_INT\n*pntrb , const MKL_INT *pntre , const float *b , const MKL_INT *ldb , float *c , const\nMKL_INT *ldc );\nvoid mkl_dcsrsm (const char *transa , const MKL_INT *m , const MKL_INT *n , const\ndouble *alpha , const char *matdescra , const double *val , const MKL_INT *indx , const\nMKL_INT *pntrb , const MKL_INT *pntre , const double *b , const MKL_INT *ldb , double\n*c , const MKL_INT *ldc );\nvoid mkl_ccsrsm (const char *transa , const MKL_INT *m , const MKL_INT *n , const\nMKL_Complex8 *alpha , const char *matdescra , const MKL_Complex8 *val , const MKL_INT\n*indx , const MKL_INT *pntrb , const MKL_INT *pntre , const MKL_Complex8 *b , const\nMKL_INT *ldb , MKL_Complex8 *c , const MKL_INT *ldc );\nvoid mkl_zcsrsm (const char *transa , const MKL_INT *m , const MKL_INT *n , const\nMKL_Complex16 *alpha , const char *matdescra , const MKL_Complex16 *val , const MKL_INT\n*indx , const MKL_INT *pntrb , const MKL_INT *pntre , const MKL_Complex16 *b , const\nMKL_INT *ldb , MKL_Complex16 *c , const MKL_INT *ldc );\nInclude Files\n•\nmkl.h\nDescription\nThis routine is deprecated. Use mkl_sparse_?_trsmfrom the Intel® oneAPI Math Kernel Library (oneMKL)\nInspector-executor Sparse BLAS interface instead.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n196\n\n\nThe mkl_?csrsm routine solves a system of linear equations with matrix-matrix operations for a sparse\nmatrix in the CSR format:\nC := alpha*inv(A)*B\nor\nC := alpha*inv(AT)*B,\nwhere:\nalpha is scalar, B and C are dense matrices, A is a sparse upper or lower triangular matrix with unit or non-\nunit main diagonal, AT is the transpose of A.\nNOTE\nThis routine supports a CSR format both with one-based indexing and zero-based indexing.\nInput Parameters\ntransa\nSpecifies the system of linear equations.\nIf transa = 'N' or 'n', then C := alpha*inv(A)*B\nIf transa = 'T' or 't' or 'C' or 'c', then C := alpha*inv(AT)*B,\nm\nNumber of columns of the matrix A.\nn\nNumber of columns of the matrix C.\nalpha\nSpecifies the scalar alpha.\nmatdescra\nArray of six elements, specifies properties of the matrix used for operation.\nOnly first four array elements are used, their possible values are given in \nTable \"Possible Values of the Parameter matdescra (descra)\". Possible\ncombinations of element values of this parameter are given in Table\n\"Possible Combinations of Element Values of the Parameter matdescra\".\nval\nArray containing non-zero elements of the matrix A.\nFor zero-based indexing its length is pntre[m—1] - pntrb[0].\nRefer to values array description in CSR Format for more details.\nNOTE\nThe non-zero elements of the given row of the matrix must be\nstored in the same order as they appear in the row (from left to\nright).\nNo diagonal element can be omitted from a sparse storage if the solver\nis called with the non-unit indicator.\nindx\nFor one-based indexing, array containing the column indices plus one for\neach non-zero element of the matrix A.\nFor zero-based indexing, array containing the column indices for each non-\nzero element of the matrix A.\nIts length is equal to length of the val array.\nRefer to columns array description in CSR Format for more details.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n197\n\n\nNOTE\nColumn indices must be sorted in increasing order for each row.\npntrb\nArray of length m.\nThis array contains row indices, such that pntrb[i] - pntrb[0] is the\nfirst index of row i in the arrays val and indx.\nRefer to pointerb array description in CSR Format for more details.\npntre\nArray of length m.\nFor zero-based indexing this array contains row indices, such that\npntre[i] - pntrb[0] - 1 is the last index of row i in the arrays val and\nindx.\nRefer to pointerE array description in CSR Format for more details.\nb\nArray, size ldb* n for one-based indexing, and (m, ldb) for zero-based\nindexing.\nOn entry the leading m-by-n part of the array b must contain the matrix B.\nldb\nSpecifies the leading dimension of b for one-based indexing, and the second\ndimension of b for zero-based indexing, as declared in the calling\n(sub)program.\nldc\nSpecifies the leading dimension of c for one-based indexing, and the second\ndimension of c for zero-based indexing, as declared in the calling\n(sub)program.\nOutput Parameters\nc\nArray, size ldc by n for one-based indexing, and (m, ldc) for zero-based\nindexing.\nThe leading m-by-n part of the array c contains the output matrix C.\nmkl_?cscsm\nSolves a system of linear matrix equations for a\nsparse matrix in the CSC format (deprecated).\nSyntax\nvoid mkl_scscsm (const char *transa , const MKL_INT *m , const MKL_INT *n , const float\n*alpha , const char *matdescra , const float *val , const MKL_INT *indx , const MKL_INT\n*pntrb , const MKL_INT *pntre , const float *b , const MKL_INT *ldb , float *c , const\nMKL_INT *ldc );\nvoid mkl_dcscsm (const char *transa , const MKL_INT *m , const MKL_INT *n , const\ndouble *alpha , const char *matdescra , const double *val , const MKL_INT *indx , const\nMKL_INT *pntrb , const MKL_INT *pntre , const double *b , const MKL_INT *ldb , double\n*c , const MKL_INT *ldc );\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n198\n\n\nvoid mkl_ccscsm (const char *transa , const MKL_INT *m , const MKL_INT *n , const\nMKL_Complex8 *alpha , const char *matdescra , const MKL_Complex8 *val , const MKL_INT\n*indx , const MKL_INT *pntrb , const MKL_INT *pntre , const MKL_Complex8 *b , const\nMKL_INT *ldb , MKL_Complex8 *c , const MKL_INT *ldc );\nvoid mkl_zcscsm (const char *transa , const MKL_INT *m , const MKL_INT *n , const\nMKL_Complex16 *alpha , const char *matdescra , const MKL_Complex16 *val , const MKL_INT\n*indx , const MKL_INT *pntrb , const MKL_INT *pntre , const MKL_Complex16 *b , const\nMKL_INT *ldb , MKL_Complex16 *c , const MKL_INT *ldc );\nInclude Files\n•\nmkl.h\nDescription\nThis routine is deprecated. Use mkl_sparse_?_trsmfrom the Intel® oneAPI Math Kernel Library (oneMKL)\nInspector-executor Sparse BLAS interface instead.\nThe mkl_?cscsm routine solves a system of linear equations with matrix-matrix operations for a sparse\nmatrix in the CSC format:\nC := alpha*inv(A)*B\nor\nC := alpha*inv(AT)*B,\nwhere:\nalpha is scalar, B and C are dense matrices, A is a sparse upper or lower triangular matrix with unit or non-\nunit main diagonal, AT is the transpose of A.\nNOTE\nThis routine supports a CSC format both with one-based indexing and zero-based indexing.\nInput Parameters\ntransa\nSpecifies the system of equations.\nIf transa = 'N' or 'n', then C := alpha*inv(A)*B\nIf transa = 'T' or 't' or 'C' or 'c', then C := alpha*inv(AT)*B,\nm\nNumber of columns of the matrix A.\nn\nNumber of columns of the matrix C.\nalpha\nSpecifies the scalar alpha.\nmatdescra\nArray of six elements, specifies properties of the matrix used for operation.\nOnly first four array elements are used, their possible values are given in \nTable “Possible Values of the Parameter matdescra (descra)”. Possible\ncombinations of element values of this parameter are given in Table\n“Possible Combinations of Element Values of the Parameter matdescra”.\nval\nArray containing non-zero elements of the matrix A.\nFor zero-based indexing its length is pntre[m] - pntrb[0].\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n199\n\n\nRefer to values array description in CSC Format for more details.\nNOTE\nThe non-zero elements of the given row of the matrix must be\nstored in the same order as they appear in the row (from left to\nright).\nNo diagonal element can be omitted from a sparse storage if the solver\nis called with the non-unit indicator.\nindx\nFor one-based indexing, array containing the row indices plus one for each\nnon-zero element of the matrix A. For zero-based indexing, array containing\nthe row indices for each non-zero element of the matrix A.\nRefer to rows array description in CSC Format for more details.\nNOTE\nRow indices must be sorted in increasing order for each column.\npntrb\nArray of length m.\nThis array contains column indices, such that pntrb[I] - pntrb[0] is the\nfirst index of column I in the arrays val and indx.\nRefer to pointerb array description in CSC Format for more details.\npntre\nArray of length m.\nThis array contains column indices, such that pntre[I] - pntrb[1]-1 is\nthe last index of column I in the arrays val and indx.\nRefer to pointerE array description in CSC Format for more details.\nb\nArray, size ldb by n for one-based indexing, and (m, ldb) for zero-based\nindexing.\nOn entry the leading m-by-n part of the array b must contain the matrix B.\nldb\nSpecifies the leading dimension of b for one-based indexing, and the second\ndimension of b for zero-based indexing, as declared in the calling\n(sub)program.\nldc\nSpecifies the leading dimension of c for one-based indexing, and the second\ndimension of c for zero-based indexing, as declared in the calling\n(sub)program.\nOutput Parameters\nc\nArray, size ldc by n for one-based indexing, and (m, ldc) for zero-based\nindexing.\nThe leading m-by-n part of the array c contains the output matrix C.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n200\n\n\nmkl_?coosm\nSolves a system of linear matrix equations for a\nsparse matrix in the coordinate format (deprecated).\nSyntax\nvoid mkl_scoosm (const char *transa , const MKL_INT *m , const MKL_INT *n , const float\n*alpha , const char *matdescra , const float *val , const MKL_INT *rowind , const\nMKL_INT *colind , const MKL_INT *nnz , const float *b , const MKL_INT *ldb , float *c ,\nconst MKL_INT *ldc );\nvoid mkl_dcoosm (const char *transa , const MKL_INT *m , const MKL_INT *n , const\ndouble *alpha , const char *matdescra , const double *val , const MKL_INT *rowind ,\nconst MKL_INT *colind , const MKL_INT *nnz , const double *b , const MKL_INT *ldb ,\ndouble *c , const MKL_INT *ldc );\nvoid mkl_ccoosm (const char *transa , const MKL_INT *m , const MKL_INT *n , const\nMKL_Complex8 *alpha , const char *matdescra , const MKL_Complex8 *val , const MKL_INT\n*rowind , const MKL_INT *colind , const MKL_INT *nnz , const MKL_Complex8 *b , const\nMKL_INT *ldb , MKL_Complex8 *c , const MKL_INT *ldc );\nvoid mkl_zcoosm (const char *transa , const MKL_INT *m , const MKL_INT *n , const\nMKL_Complex16 *alpha , const char *matdescra , const MKL_Complex16 *val , const MKL_INT\n*rowind , const MKL_INT *colind , const MKL_INT *nnz , const MKL_Complex16 *b , const\nMKL_INT *ldb , MKL_Complex16 *c , const MKL_INT *ldc );\nInclude Files\n•\nmkl.h\nDescription\nThis routine is deprecated. Use mkl_sparse_?_trsmfrom the Intel® oneAPI Math Kernel Library (oneMKL)\nInspector-executor Sparse BLAS interface instead.\nThe mkl_?coosm routine solves a system of linear equations with matrix-matrix operations for a sparse\nmatrix in the coordinate format:\nC := alpha*inv(A)*B\nor\nC := alpha*inv(AT)*B,\nwhere:\nalpha is scalar, B and C are dense matrices, A is a sparse upper or lower triangular matrix with unit or non-\nunit main diagonal, AT is the transpose of A.\nNOTE\nThis routine supports a coordinate format both with one-based indexing and zero-based indexing.\nInput Parameters\ntransa\nSpecifies the system of linear equations.\nIf transa = 'N' or 'n', then the matrix-matrix product is computed as\nC := alpha*inv(A)*B\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n201\n\n\nIf transa = 'T' or 't' or 'C' or 'c', then the matrix-vector product is\ncomputed as C := alpha*inv(AT)*B,\nm\nNumber of rows of the matrix A.\nn\nNumber of columns of the matrix C.\nalpha\nSpecifies the scalar alpha.\nmatdescra\nArray of six elements, specifies properties of the matrix used for operation.\nOnly first four array elements are used, their possible values are given in \nTable “Possible Values of the Parameter matdescra (descra)”. Possible\ncombinations of element values of this parameter are given in Table\n“Possible Combinations of Element Values of the Parameter matdescra”.\nval\nArray of length nnz, contains non-zero elements of the matrix A in the\narbitrary order.\nRefer to values array description in Coordinate Format for more details.\nrowind\nArray of length nnz.\nFor one-based indexing, contains the row indices plus one for each non-zero\nelement of the matrix A.\nFor zero-based indexing, contains the row indices for each non-zero\nelement of the matrix A.\nRefer to rows array description in Coordinate Format for more details.\ncolind\nArray of length nnz.\nFor one-based indexing, contains the column indices plus one for each non-\nzero element of the matrix A\nFor zero-based indexing, contains the row indices for each non-zero\nelement of the matrix A\nRefer to columns array description in Coordinate Format for more details.\nnnz\nSpecifies the number of non-zero element of the matrix A.\nRefer to nnz description in Coordinate Format for more details.\nb\nArray, size ldb by n for one-based indexing, and (m, ldb) for zero-based\nindexing.\nBefore entry the leading m-by-n part of the array b must contain the matrix\nB.\nldb\nSpecifies the leading dimension of b for one-based indexing, and the second\ndimension of b for zero-based indexing, as declared in the calling\n(sub)program.\nldc\nSpecifies the leading dimension of c for one-based indexing, and the second\ndimension of c for zero-based indexing, as declared in the calling\n(sub)program.\nOutput Parameters\nc\nArray, size ldc by n for one-based indexing, and (m, ldc) for zero-based\nindexing.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n202\n\n\nThe leading m-by-n part of the array c contains the output matrix C.\nmkl_?bsrsm\nSolves a system of linear matrix equations for a\nsparse matrix in the BSR format (deprecated).\nSyntax\nvoid mkl_sbsrsm (const char *transa , const MKL_INT *m , const MKL_INT *n , const\nMKL_INT *lb , const float *alpha , const char *matdescra , const float *val , const\nMKL_INT *indx , const MKL_INT *pntrb , const MKL_INT *pntre , const float *b , const\nMKL_INT *ldb , float *c , const MKL_INT *ldc );\nvoid mkl_dbsrsm (const char *transa , const MKL_INT *m , const MKL_INT *n , const\nMKL_INT *lb , const double *alpha , const char *matdescra , const double *val , const\nMKL_INT *indx , const MKL_INT *pntrb , const MKL_INT *pntre , const double *b , const\nMKL_INT *ldb , double *c , const MKL_INT *ldc );\nvoid mkl_cbsrsm (const char *transa , const MKL_INT *m , const MKL_INT *n , const\nMKL_INT *lb , const MKL_Complex8 *alpha , const char *matdescra , const MKL_Complex8\n*val , const MKL_INT *indx , const MKL_INT *pntrb , const MKL_INT *pntre , const\nMKL_Complex8 *b , const MKL_INT *ldb , MKL_Complex8 *c , const MKL_INT *ldc );\nvoid mkl_zbsrsm (const char *transa , const MKL_INT *m , const MKL_INT *n , const\nMKL_INT *lb , const MKL_Complex16 *alpha , const char *matdescra , const MKL_Complex16\n*val , const MKL_INT *indx , const MKL_INT *pntrb , const MKL_INT *pntre , const\nMKL_Complex16 *b , const MKL_INT *ldb , MKL_Complex16 *c , const MKL_INT *ldc );\nInclude Files\n•\nmkl.h\nDescription\nThis routine is deprecated. Use mkl_sparse_?_trsmfrom the Intel® oneAPI Math Kernel Library (oneMKL)\nInspector-executor Sparse BLAS interface instead.\nThe mkl_?bsrsm routine solves a system of linear equations with matrix-matrix operations for a sparse\nmatrix in the BSR format:\nC := alpha*inv(A)*B\nor\nC := alpha*inv(AT)*B,\nwhere:\nalpha is scalar, B and C are dense matrices, A is a sparse upper or lower triangular matrix with unit or non-\nunit main diagonal, AT is the transpose of A.\nNOTE\nThis routine supports a BSR format both with one-based indexing and zero-based indexing.\nInput Parameters\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n203\n\n\ntransa\nSpecifies the operation.\nIf transa = 'N' or 'n', then the matrix-matrix product is computed as\nC := alpha*inv(A)*B.\nIf transa = 'T' or 't' or 'C' or 'c', then the matrix-vector product is\ncomputed as C := alpha*inv(AT)*B.\nm\nNumber of block columns of the matrix A.\nn\nNumber of columns of the matrix C.\nlb\nSize of the block in the matrix A.\nalpha\nSpecifies the scalar alpha.\nmatdescra\nArray of six elements, specifies properties of the matrix used for operation.\nOnly first four array elements are used, their possible values are given in \nTable “Possible Values of the Parameter matdescra (descra)”. Possible\ncombinations of element values of this parameter are given in Table\n“Possible Combinations of Element Values of the Parameter matdescra”.\nval\nArray containing elements of non-zero blocks of the matrix A. Its length is\nequal to the number of non-zero blocks in the matrix A multiplied by lb*lb.\nRefer to the values array description in BSR Format for more details.\nNOTE\nThe non-zero elements of the given row of the matrix must be\nstored in the same order as they appear in the row (from left to\nright).\nNo diagonal element can be omitted from a sparse storage if the solver\nis called with the non-unit indicator.\nindx\nFor one-based indexing, array containing the column indices plus one for\neach non-zero element of the matrix A. For zero-based indexing, array\ncontaining the column indices for each non-zero element of the matrix A.\nIts length is equal to the number of non-zero blocks in the matrix A.\nRefer to the columns array description in BSR Format for more details.\npntrb\nArray of length m.\nThis array contains row indices, such that pntrb[i] - pntrb[0] is the\nfirst index of block row i in the array indx.\nRefer to pointerB array description in BSR Format for more details.\npntre\nArray of length m.\nThis array contains row indices, such that pntre[i] - pntrb[0] - 1 is\nthe last index of block row i in the arrays val and indx.\nRefer to pointerE array description in BSR Format for more details.\nb\nArray, size ldb* n for one-based indexing, size m* ldb for zero-based\nindexing.\nOn entry the leading m-by-n part of the array b must contain the matrix B.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n204\n\n\nldb\nSpecifies the leading dimension (in blocks) of b as declared in the calling\n(sub)program.\nldc\nSpecifies the leading dimension (in blocks) of c as declared in the calling\n(sub)program.\nOutput Parameters\nc\nArray, size ldc* n for one-based indexing, size m* ldc for zero-based\nindexing.\nThe leading m-by-n part of the array c contains the output matrix C.\nmkl_?diamv\nComputes matrix - vector product for a sparse matrix\nin the diagonal format with one-based indexing\n(deprecated).\nSyntax\nvoid mkl_sdiamv (const char *transa , const MKL_INT *m , const MKL_INT *k , const float\n*alpha , const char *matdescra , const float *val , const MKL_INT *lval , const MKL_INT\n*idiag , const MKL_INT *ndiag , const float *x , const float *beta , float *y );\nvoid mkl_ddiamv (const char *transa , const MKL_INT *m , const MKL_INT *k , const\ndouble *alpha , const char *matdescra , const double *val , const MKL_INT *lval , const\nMKL_INT *idiag , const MKL_INT *ndiag , const double *x , const double *beta , double\n*y );\nvoid mkl_cdiamv (const char *transa , const MKL_INT *m , const MKL_INT *k , const\nMKL_Complex8 *alpha , const char *matdescra , const MKL_Complex8 *val , const MKL_INT\n*lval , const MKL_INT *idiag , const MKL_INT *ndiag , const MKL_Complex8 *x , const\nMKL_Complex8 *beta , MKL_Complex8 *y );\nvoid mkl_zdiamv (const char *transa , const MKL_INT *m , const MKL_INT *k , const\nMKL_Complex16 *alpha , const char *matdescra , const MKL_Complex16 *val , const MKL_INT\n*lval , const MKL_INT *idiag , const MKL_INT *ndiag , const MKL_Complex16 *x , const\nMKL_Complex16 *beta , MKL_Complex16 *y );\nInclude Files\n•\nmkl.h\nDescription\nThis routine is deprecated, but no replacement is available yet in the Inspector-Executor Sparse BLAS API\ninterfaces. You can continue using this routine until a replacement is provided and this can be fully removed.\nThe mkl_?diamv routine performs a matrix-vector operation defined as\ny := alpha*A*x + beta*y\nor\ny := alpha*AT*x + beta*y,\nwhere:\nalpha and beta are scalars,\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n205\n\n\nx and y are vectors,\nA is an m-by-k sparse matrix stored in the diagonal format, AT is the transpose of A.\nNOTE\nThis routine supports only one-based indexing of the input arrays.\nInput Parameters\ntransa\nSpecifies the operation.\nIf transa = 'N' or 'n', then y := alpha*A*x + beta*y,\nIf transa = 'T' or 't' or 'C' or 'c', then y := alpha*AT*x + beta*y.\nm\nNumber of rows of the matrix A.\nk\nNumber of columns of the matrix A.\nalpha\nSpecifies the scalar alpha.\nmatdescra\nArray of six elements, specifies properties of the matrix used for operation.\nOnly first four array elements are used, their possible values are given in \nTable “Possible Values of the Parameter matdescra (descra)”. Possible\ncombinations of element values of this parameter are given in Table\n“Possible Combinations of Element Values of the Parameter matdescra”.\nval\nTwo-dimensional array of size lval by ndiag, contains non-zero diagonals\nof the matrix A. Refer to values array description in Diagonal Storage\nScheme for more details.\nlval\nLeading dimension of val, lval≥m. Refer to lval description in Diagonal\nStorage Scheme for more details.\nidiag\nArray of length ndiag, contains the distances between main diagonal and\neach non-zero diagonals in the matrix A.\nRefer to distance array description in Diagonal Storage Scheme for more\ndetails.\nndiag\nSpecifies the number of non-zero diagonals of the matrix A.\nx\nArray, size at least k if transa = 'N' or 'n', and at least m otherwise. On\nentry, the array x must contain the vector x.\nbeta\nSpecifies the scalar beta.\ny\nArray, size at least m if transa = 'N' or 'n', and at least k otherwise. On\nentry, the array y must contain the vector y.\nOutput Parameters\ny\nOverwritten by the updated vector y.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n206\n\n\nmkl_?skymv\nComputes matrix - vector product for a sparse matrix\nin the skyline storage format with one-based indexing\n(deprecated).\nSyntax\nvoid mkl_sskymv (const char *transa , const MKL_INT *m , const MKL_INT *k , const float\n*alpha , const char *matdescra , const float *val , const MKL_INT *pntr , const float\n*x , const float *beta , float *y );\nvoid mkl_dskymv (const char *transa , const MKL_INT *m , const MKL_INT *k , const\ndouble *alpha , const char *matdescra , const double *val , const MKL_INT *pntr , const\ndouble *x , const double *beta , double *y );\nvoid mkl_cskymv (const char *transa , const MKL_INT *m , const MKL_INT *k , const\nMKL_Complex8 *alpha , const char *matdescra , const MKL_Complex8 *val , const MKL_INT\n*pntr , const MKL_Complex8 *x , const MKL_Complex8 *beta , MKL_Complex8 *y );\nvoid mkl_zskymv (const char *transa , const MKL_INT *m , const MKL_INT *k , const\nMKL_Complex16 *alpha , const char *matdescra , const MKL_Complex16 *val , const MKL_INT\n*pntr , const MKL_Complex16 *x , const MKL_Complex16 *beta , MKL_Complex16 *y );\nInclude Files\n•\nmkl.h\nDescription\nThis routine is deprecated, but no replacement is available yet in the Inspector-Executor Sparse BLAS API\ninterfaces. You can continue using this routine until a replacement is provided and this can be fully removed.\nThe mkl_?skymv routine performs a matrix-vector operation defined as\ny := alpha*A*x + beta*y\nor\ny := alpha*AT*x + beta*y,\nwhere:\nalpha and beta are scalars,\nx and y are vectors,\nA is an m-by-k sparse matrix stored using the skyline storage scheme, AT is the transpose of A.\nNOTE\nThis routine supports only one-based indexing of the input arrays.\nInput Parameters\ntransa\nSpecifies the operation.\nIf transa = 'N' or 'n', then y := alpha*A*x + beta*y\nIf transa = 'T' or 't' or 'C' or 'c', then y := alpha*AT*x + beta*y,\nm\nNumber of rows of the matrix A.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n207\n\n\nk\nNumber of columns of the matrix A.\nalpha\nSpecifies the scalar alpha.\nmatdescra\nArray of six elements, specifies properties of the matrix used for operation.\nOnly first four array elements are used, their possible values are given in \nTable “Possible Values of the Parameter matdescra (descra)”. Possible\ncombinations of element values of this parameter are given in Table\n“Possible Combinations of Element Values of the Parameter matdescra”.\nNOTE\nGeneral matrices (matdescra[0]='G') is not supported.\nval\nArray containing the set of elements of the matrix A in the skyline profile\nform.\nIf matdescrsa[1]= 'L', then val contains elements from the low triangle\nof the matrix A.\nIf matdescrsa[1]= 'U', then val contains elements from the upper\ntriangle of the matrix A.\nRefer to values array description in Skyline Storage Scheme for more\ndetails.\npntr\nArray of length (m + 1) for lower triangle, and (k + 1) for upper triangle.\nIt contains the indices specifying in the val the positions of the first\nelement in each row (column) of the matrix A. Refer to pointers array\ndescription in Skyline Storage Scheme for more details.\nx\nArray, size at least k if transa = 'N' or 'n' and at least m otherwise. On\nentry, the array x must contain the vector x.\nbeta\nSpecifies the scalar beta.\ny\nArray, size at least m if transa = 'N' or 'n' and at least k otherwise. On\nentry, the array y must contain the vector y.\nOutput Parameters\ny\nOverwritten by the updated vector y.\nmkl_?diasv\nSolves a system of linear equations for a sparse\nmatrix in the diagonal format with one-based indexing\n(deprecated).\nSyntax\nvoid mkl_sdiasv (const char *transa , const MKL_INT *m , const float *alpha , const\nchar *matdescra , const float *val , const MKL_INT *lval , const MKL_INT *idiag , const\nMKL_INT *ndiag , const float *x , float *y );\nvoid mkl_ddiasv (const char *transa , const MKL_INT *m , const double *alpha , const\nchar *matdescra , const double *val , const MKL_INT *lval , const MKL_INT *idiag ,\nconst MKL_INT *ndiag , const double *x , double *y );\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n208\n\n\nvoid mkl_cdiasv (const char *transa , const MKL_INT *m , const MKL_Complex8 *alpha ,\nconst char *matdescra , const MKL_Complex8 *val , const MKL_INT *lval , const MKL_INT\n*idiag , const MKL_INT *ndiag , const MKL_Complex8 *x , MKL_Complex8 *y );\nvoid mkl_zdiasv (const char *transa , const MKL_INT *m , const MKL_Complex16 *alpha ,\nconst char *matdescra , const MKL_Complex16 *val , const MKL_INT *lval , const MKL_INT\n*idiag , const MKL_INT *ndiag , const MKL_Complex16 *x , MKL_Complex16 *y );\nInclude Files\n•\nmkl.h\nDescription\nThis routine is deprecated, but no replacement is available yet in the Inspector-Executor Sparse BLAS API\ninterfaces. You can continue using this routine until a replacement is provided and this can be fully removed.\nThe mkl_?diasv routine solves a system of linear equations with matrix-vector operations for a sparse\nmatrix stored in the diagonal format:\ny := alpha*inv(A)*x\nor\ny := alpha*inv(AT)* x,\nwhere:\nalpha is scalar, x and y are vectors, A is a sparse upper or lower triangular matrix with unit or non-unit main\ndiagonal, AT is the transpose of A.\nNOTE\nThis routine supports only one-based indexing of the input arrays.\nInput Parameters\ntransa\nSpecifies the system of linear equations.\nIf transa = 'N' or 'n', then y := alpha*inv(A)*x\nIf transa = 'T' or 't' or 'C' or 'c', then y := alpha*inv(AT)*x,\nm\nNumber of rows of the matrix A.\nalpha\nSpecifies the scalar alpha.\nmatdescra\nArray of six elements, specifies properties of the matrix used for operation.\nOnly first four array elements are used, their possible values are given in \nTable “Possible Values of the Parameter matdescra (descra)”. Possible\ncombinations of element values of this parameter are given in Table\n“Possible Combinations of Element Values of the Parameter matdescra”.\nval\nTwo-dimensional array of size lval by ndiag, contains non-zero diagonals\nof the matrix A. Refer to values array description in Diagonal Storage\nScheme for more details.\nlval\nLeading dimension of val, lval≥m. Refer to lval description in Diagonal\nStorage Scheme for more details.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n209\n\n\nidiag\nArray of length ndiag, contains the distances between main diagonal and\neach non-zero diagonals in the matrix A.\nNOTE\nAll elements of this array must be sorted in increasing order.\nRefer to distance array description in Diagonal Storage Scheme for more\ndetails.\nndiag\nSpecifies the number of non-zero diagonals of the matrix A.\nx\nArray, size at least m.\nOn entry, the array x must contain the vector x. The elements are accessed\nwith unit increment.\ny\nArray, size at least m.\nOn entry, the array y must contain the vector y. The elements are accessed\nwith unit increment.\nOutput Parameters\ny\nContains solution vector x.\nmkl_?skysv\nSolves a system of linear equations for a sparse\nmatrix in the skyline format with one-based indexing\n(deprecated).\nSyntax\nvoid mkl_sskysv (const char *transa , const MKL_INT *m , const float *alpha , const\nchar *matdescra , const float *val , const MKL_INT *pntr , const float *x , float *y );\nvoid mkl_dskysv (const char *transa , const MKL_INT *m , const double *alpha , const\nchar *matdescra , const double *val , const MKL_INT *pntr , const double *x , double\n*y );\nvoid mkl_cskysv (const char *transa , const MKL_INT *m , const MKL_Complex8 *alpha ,\nconst char *matdescra , const MKL_Complex8 *val , const MKL_INT *pntr , const\nMKL_Complex8 *x , MKL_Complex8 *y );\nvoid mkl_zskysv (const char *transa , const MKL_INT *m , const MKL_Complex16 *alpha ,\nconst char *matdescra , const MKL_Complex16 *val , const MKL_INT *pntr , const\nMKL_Complex16 *x , MKL_Complex16 *y );\nInclude Files\n•\nmkl.h\nDescription\nThis routine is deprecated, but no replacement is available yet in the Inspector-Executor Sparse BLAS API\ninterfaces. You can continue using this routine until a replacement is provided and this can be fully removed.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n210\n\n\nThe mkl_?skysv routine solves a system of linear equations with matrix-vector operations for a sparse\nmatrix in the skyline storage format:\ny := alpha*inv(A)*x\nor\ny := alpha*inv(AT)*x,\nwhere:\nalpha is scalar, x and y are vectors, A is a sparse upper or lower triangular matrix with unit or non-unit main\ndiagonal, AT is the transpose of A.\nNOTE\nThis routine supports only one-based indexing of the input arrays.\nInput Parameters\ntransa\nSpecifies the system of linear equations.\nIf transa = 'N' or 'n', then y := alpha*inv(A)*x\nIf transa = 'T' or 't' or 'C' or 'c', then y := alpha*inv(AT)* x,\nm\nNumber of rows of the matrix A.\nalpha\nSpecifies the scalar alpha.\nmatdescra\nArray of six elements, specifies properties of the matrix used for operation.\nOnly first four array elements are used, their possible values are given in \nTable “Possible Values of the Parameter matdescra (descra)”. Possible\ncombinations of element values of this parameter are given in Table\n“Possible Combinations of Element Values of the Parameter matdescra”.\nNOTE\nGeneral matrices (matdescra[0]='G') is not supported.\nval\nArray containing the set of elements of the matrix A in the skyline profile\nform.\nIf matdescra[2]= 'L', then val contains elements from the low triangle\nof the matrix A.\nIf matdescsa[2]= 'U', then val contains elements from the upper\ntriangle of the matrix A.\nRefer to values array description in Skyline Storage Scheme for more\ndetails.\npntr\nArray of length (m + 1) for lower triangle, and (k + 1) for upper triangle.\nIt contains the indices specifying in the val the positions of the first\nelement in each row (column) of the matrix A. Refer to pointers array\ndescription in Skyline Storage Scheme for more details.\nx\nArray, size at least m.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n211\n\n\nOn entry, the array x must contain the vector x. The elements are accessed\nwith unit increment.\ny\nArray, size at least m.\nOn entry, the array y must contain the vector y. The elements are accessed\nwith unit increment.\nOutput Parameters\ny\nContains solution vector x.\nmkl_?diamm\nComputes matrix-matrix product of a sparse matrix\nstored in the diagonal format with one-based indexing\n(deprecated).\nSyntax\nvoid mkl_sdiamm (const char *transa , const MKL_INT *m , const MKL_INT *n , const\nMKL_INT *k , const float *alpha , const char *matdescra , const float *val , const\nMKL_INT *lval , const MKL_INT *idiag , const MKL_INT *ndiag , const float *b , const\nMKL_INT *ldb , const float *beta , float *c , const MKL_INT *ldc );\nvoid mkl_ddiamm (const char *transa , const MKL_INT *m , const MKL_INT *n , const\nMKL_INT *k , const double *alpha , const char *matdescra , const double *val , const\nMKL_INT *lval , const MKL_INT *idiag , const MKL_INT *ndiag , const double *b , const\nMKL_INT *ldb , const double *beta , double *c , const MKL_INT *ldc );\nvoid mkl_cdiamm (const char *transa , const MKL_INT *m , const MKL_INT *n , const\nMKL_INT *k , const MKL_Complex8 *alpha , const char *matdescra , const MKL_Complex8\n*val , const MKL_INT *lval , const MKL_INT *idiag , const MKL_INT *ndiag , const\nMKL_Complex8 *b , const MKL_INT *ldb , const MKL_Complex8 *beta , MKL_Complex8 *c ,\nconst MKL_INT *ldc );\nvoid mkl_zdiamm (const char *transa , const MKL_INT *m , const MKL_INT *n , const\nMKL_INT *k , const MKL_Complex16 *alpha , const char *matdescra , const MKL_Complex16\n*val , const MKL_INT *lval , const MKL_INT *idiag , const MKL_INT *ndiag , const\nMKL_Complex16 *b , const MKL_INT *ldb , const MKL_Complex16 *beta , MKL_Complex16 *c ,\nconst MKL_INT *ldc );\nInclude Files\n•\nmkl.h\nDescription\nThis routine is deprecated, but no replacement is available yet in the Inspector-Executor Sparse BLAS API\ninterfaces. You can continue using this routine until a replacement is provided and this can be fully removed.\nThe mkl_?diamm routine performs a matrix-matrix operation defined as\nC := alpha*A*B + beta*C\nor\nC := alpha*AT*B + beta*C,\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n212\n\n\nor\nC := alpha*AH*B + beta*C,\nwhere:\nalpha and beta are scalars,\nB and C are dense matrices, A is an m-by-k sparse matrix in the diagonal format, AT is the transpose of A,\nand AH is the conjugate transpose of A.\nNOTE\nThis routine supports only one-based indexing of the input arrays.\nInput Parameters\ntransa\nSpecifies the operation.\nIf transa = 'N' or 'n', then C := alpha*A*B + beta*C,\nIf transa = 'T' or 't', then C := alpha*AT*B + beta*C,\nIf transa = 'C' or 'c', then C := alpha*AH*B + beta*C.\nm\nNumber of rows of the matrix A.\nn\nNumber of columns of the matrix C.\nk\nNumber of columns of the matrix A.\nalpha\nSpecifies the scalar alpha.\nmatdescra\nArray of six elements, specifies properties of the matrix used for operation.\nOnly first four array elements are used, their possible values are given in \nTable “Possible Values of the Parameter matdescra (descra)”. Possible\ncombinations of element values of this parameter are given in Table\n“Possible Combinations of Element Values of the Parameter matdescra”.\nval\nTwo-dimensional array of size lval by ndiag, contains non-zero diagonals\nof the matrix A. Refer to values array description in Diagonal Storage\nScheme for more details.\nlval\nLeading dimension of val, lval≥m. Refer to lval description in Diagonal\nStorage Scheme for more details.\nidiag\nArray of length ndiag, contains the distances between main diagonal and\neach non-zero diagonals in the matrix A.\nRefer to distance array description in Diagonal Storage Scheme for more\ndetails.\nndiag\nSpecifies the number of non-zero diagonals of the matrix A.\nb\nArray, size ldb* n.\nOn entry with transa = 'N' or 'n', the leading k-by-n part of the array b\nmust contain the matrix B, otherwise the leading m-by-n part of the array b\nmust contain the matrix B.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n213\n\n\nldb\nSpecifies the leading dimension of b as declared in the calling\n(sub)program.\nbeta\nSpecifies the scalar beta.\nc\nArray, size ldc by n.\nOn entry, the leading m-by-n part of the array c must contain the matrix C,\notherwise the leading k-by-n part of the array c must contain the matrix C.\nldc\nSpecifies the leading dimension of c as declared in the calling\n(sub)program.\nOutput Parameters\nc\nOverwritten by the matrix (alpha*A*B + beta*C), (alpha*AT*B +\nbeta*C), or (alpha*AH*B + beta*C).\nmkl_?skymm\nComputes matrix-matrix product of a sparse matrix\nstored using the skyline storage scheme with one-\nbased indexing (deprecated).\nSyntax\nvoid mkl_sskymm (const char *transa , const MKL_INT *m , const MKL_INT *n , const\nMKL_INT *k , const float *alpha , const char *matdescra , const float *val , const\nMKL_INT *pntr , const float *b , const MKL_INT *ldb , const float *beta , float *c ,\nconst MKL_INT *ldc );\nvoid mkl_dskymm (const char *transa , const MKL_INT *m , const MKL_INT *n , const\nMKL_INT *k , const double *alpha , const char *matdescra , const double *val , const\nMKL_INT *pntr , const double *b , const MKL_INT *ldb , const double *beta , double *c ,\nconst MKL_INT *ldc );\nvoid mkl_cskymm (const char *transa , const MKL_INT *m , const MKL_INT *n , const\nMKL_INT *k , const MKL_Complex8 *alpha , const char *matdescra , const MKL_Complex8\n*val , const MKL_INT *pntr , const MKL_Complex8 *b , const MKL_INT *ldb , const\nMKL_Complex8 *beta , MKL_Complex8 *c , const MKL_INT *ldc );\nvoid mkl_zskymm (const char *transa , const MKL_INT *m , const MKL_INT *n , const\nMKL_INT *k , const MKL_Complex16 *alpha , const char *matdescra , const MKL_Complex16\n*val , const MKL_INT *pntr , const MKL_Complex16 *b , const MKL_INT *ldb , const\nMKL_Complex16 *beta , MKL_Complex16 *c , const MKL_INT *ldc );\nInclude Files\n•\nmkl.h\nDescription\nThis routine is deprecated, but no replacement is available yet in the Inspector-Executor Sparse BLAS API\ninterfaces. You can continue using this routine until a replacement is provided and this can be fully removed.\nThe mkl_?skymm routine performs a matrix-matrix operation defined as\nC := alpha*A*B + beta*C\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n214\n\n\nor\nC := alpha*AT*B + beta*C,\nor\nC := alpha*AH*B + beta*C,\nwhere:\nalpha and beta are scalars,\nB and C are dense matrices, A is an m-by-k sparse matrix in the skyline storage format, AT is the transpose\nof A, and AH is the conjugate transpose of A.\nNOTE\nThis routine supports only one-based indexing of the input arrays.\nInput Parameters\ntransa\nSpecifies the operation.\nIf transa = 'N' or 'n', then C := alpha*A*B + beta*C,\nIf transa = 'T' or 't', then C := alpha*AT*B + beta*C,\nIf transa = 'C' or 'c', then C := alpha*AH*B + beta*C.\nm\nNumber of rows of the matrix A.\nn\nNumber of columns of the matrix C.\nk\nNumber of columns of the matrix A.\nalpha\nSpecifies the scalar alpha.\nmatdescra\nArray of six elements, specifies properties of the matrix used for operation.\nOnly first four array elements are used, their possible values are given in \nTable “Possible Values of the Parameter matdescra (descra)”. Possible\ncombinations of element values of this parameter are given in Table\n“Possible Combinations of Element Values of the Parameter matdescra”.\nNOTE\nGeneral matrices (matdescra [0]='G') is not supported.\nval\nArray containing the set of elements of the matrix A in the skyline profile\nform.\nIf matdescrsa[2]= 'L', then val contains elements from the low triangle\nof the matrix A.\nIf matdescrsa[2]= 'U', then val contains elements from the upper\ntriangle of the matrix A.\nRefer to values array description in Skyline Storage Scheme for more\ndetails.\npntr\nArray of length (m + 1) for lower triangle, and (k + 1) for upper triangle.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n215\n\n\nIt contains the indices specifying the positions of the first element of the\nmatrix A in each row (for the lower triangle) or column (for upper triangle)\nin the val array such that val[pntr[i] - 1] is the first element in row or\ncolumn i + 1. Refer to pointers array description in Skyline Storage\nScheme for more details.\nb\nArray, size ldb* n.\nOn entry with transa = 'N' or 'n', the leading k-by-n part of the array b\nmust contain the matrix B, otherwise the leading m-by-n part of the array b\nmust contain the matrix B.\nldb\nSpecifies the leading dimension of b as declared in the calling\n(sub)program.\nbeta\nSpecifies the scalar beta.\nc\nArray, size ldc by n.\nOn entry, the leading m-by-n part of the array c must contain the matrix C,\notherwise the leading k-by-n part of the array c must contain the matrix C.\nldc\nSpecifies the leading dimension of c as declared in the calling\n(sub)program.\nOutput Parameters\nc\nOverwritten by the matrix (alpha*A*B + beta*C), (alpha*AT*B +\nbeta*C), or (alpha*AH*B + beta*C).\nmkl_?diasm\nSolves a system of linear matrix equations for a\nsparse matrix in the diagonal format with one-based\nindexing (deprecated).\nSyntax\nvoid mkl_sdiasm (const char *transa , const MKL_INT *m , const MKL_INT *n , const float\n*alpha , const char *matdescra , const float *val , const MKL_INT *lval , const MKL_INT\n*idiag , const MKL_INT *ndiag , const float *b , const MKL_INT *ldb , float *c , const\nMKL_INT *ldc );\nvoid mkl_ddiasm (const char *transa , const MKL_INT *m , const MKL_INT *n , const\ndouble *alpha , const char *matdescra , const double *val , const MKL_INT *lval , const\nMKL_INT *idiag , const MKL_INT *ndiag , const double *b , const MKL_INT *ldb , double\n*c , const MKL_INT *ldc );\nvoid mkl_cdiasm (const char *transa , const MKL_INT *m , const MKL_INT *n , const\nMKL_Complex8 *alpha , const char *matdescra , const MKL_Complex8 *val , const MKL_INT\n*lval , const MKL_INT *idiag , const MKL_INT *ndiag , const MKL_Complex8 *b , const\nMKL_INT *ldb , MKL_Complex8 *c , const MKL_INT *ldc );\nvoid mkl_zdiasm (const char *transa , const MKL_INT *m , const MKL_INT *n , const\nMKL_Complex16 *alpha , const char *matdescra , const MKL_Complex16 *val , const MKL_INT\n*lval , const MKL_INT *idiag , const MKL_INT *ndiag , const MKL_Complex16 *b , const\nMKL_INT *ldb , MKL_Complex16 *c , const MKL_INT *ldc );\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n216\n\n\nInclude Files\n•\nmkl.h\nDescription\nThis routine is deprecated, but no replacement is available yet in the Inspector-Executor Sparse BLAS API\ninterfaces. You can continue using this routine until a replacement is provided and this can be fully removed.\nThe mkl_?diasm routine solves a system of linear equations with matrix-matrix operations for a sparse\nmatrix in the diagonal format:\nC := alpha*inv(A)*B\nor\nC := alpha*inv(AT)*B,\nwhere:\nalpha is scalar, B and C are dense matrices, A is a sparse upper or lower triangular matrix with unit or non-\nunit main diagonal, AT is the transpose of A.\nNOTE\nThis routine supports only one-based indexing of the input arrays.\nInput Parameters\ntransa\nSpecifies the system of linear equations.\nIf transa = 'N' or 'n', then C := alpha*inv(A)*B,\nIf transa = 'T' or 't' or 'C' or 'c', then C := alpha*inv(AT)*B.\nm\nNumber of rows of the matrix A.\nn\nNumber of columns of the matrix C.\nalpha\nSpecifies the scalar alpha.\nmatdescra\nArray of six elements, specifies properties of the matrix used for operation.\nOnly first four array elements are used, their possible values are given in \nTable “Possible Values of the Parameter matdescra (descra)”. Possible\ncombinations of element values of this parameter are given in Table\n“Possible Combinations of Element Values of the Parameter matdescra”.\nval\nTwo-dimensional array of size lval by ndiag, contains non-zero diagonals\nof the matrix A. Refer to values array description in Diagonal Storage\nScheme for more details.\nlval\nLeading dimension of val, lval≥m. Refer to lval description in Diagonal\nStorage Scheme for more details.\nidiag\nArray of length ndiag, contains the distances between main diagonal and\neach non-zero diagonals in the matrix A.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n217\n\n\nNOTE\nAll elements of this array must be sorted in increasing order.\nRefer to distance array description in Diagonal Storage Scheme for more\ndetails.\nndiag\nSpecifies the number of non-zero diagonals of the matrix A.\nb\nArray, size ldb* n.\nOn entry the leading m-by-n part of the array b must contain the matrix B.\nldb\nSpecifies the leading dimension of b as declared in the calling\n(sub)program.\nldc\nSpecifies the leading dimension of c as declared in the calling\n(sub)program.\nOutput Parameters\nc\nArray, size ldc by n.\nThe leading m-by-n part of the array c contains the matrix C.\nmkl_?skysm\nSolves a system of linear matrix equations for a\nsparse matrix stored using the skyline storage scheme\nwith one-based indexing (deprecated).\nSyntax\nvoid mkl_sskysm (const char *transa , const MKL_INT *m , const MKL_INT *n , const float\n*alpha , const char *matdescra , const float *val , const MKL_INT *pntr , const float\n*b , const MKL_INT *ldb , float *c , const MKL_INT *ldc );\nvoid mkl_dskysm (const char *transa , const MKL_INT *m , const MKL_INT *n , const\ndouble *alpha , const char *matdescra , const double *val , const MKL_INT *pntr , const\ndouble *b , const MKL_INT *ldb , double *c , const MKL_INT *ldc );\nvoid mkl_cskysm (const char *transa , const MKL_INT *m , const MKL_INT *n , const\nMKL_Complex8 *alpha , const char *matdescra , const MKL_Complex8 *val , const MKL_INT\n*pntr , const MKL_Complex8 *b , const MKL_INT *ldb , MKL_Complex8 *c , const MKL_INT\n*ldc );\nvoid mkl_zskysm (const char *transa , const MKL_INT *m , const MKL_INT *n , const\nMKL_Complex16 *alpha , const char *matdescra , const MKL_Complex16 *val , const MKL_INT\n*pntr , const MKL_Complex16 *b , const MKL_INT *ldb , MKL_Complex16 *c , const MKL_INT\n*ldc );\nInclude Files\n•\nmkl.h\nDescription\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n218\n\n\nThis routine is deprecated, but no replacement is available yet in the Inspector-Executor Sparse BLAS API\ninterfaces. You can continue using this routine until a replacement is provided and this can be fully removed.\nThe mkl_?skysm routine solves a system of linear equations with matrix-matrix operations for a sparse\nmatrix in the skyline storage format:\nC := alpha*inv(A)*B\nor\nC := alpha*inv(AT)*B,\nwhere:\nalpha is scalar, B and C are dense matrices, A is a sparse upper or lower triangular matrix with unit or non-\nunit main diagonal, AT is the transpose of A.\nNOTE\nThis routine supports only one-based indexing of the input arrays.\nInput Parameters\ntransa\nSpecifies the system of linear equations.\nIf transa = 'N' or 'n', then C := alpha*inv(A)*B,\nIf transa = 'T' or 't' or 'C' or 'c', then C := alpha*inv(AT)*B,\nm\nNumber of rows of the matrix A.\nn\nNumber of columns of the matrix C.\nalpha\nSpecifies the scalar alpha.\nmatdescra\nArray of six elements, specifies properties of the matrix used for operation.\nOnly first four array elements are used, their possible values are given in \nTable “Possible Values of the Parameter matdescra (descra)”. Possible\ncombinations of element values of this parameter are given in Table\n“Possible Combinations of Element Values of the Parameter matdescra”.\nNOTE\nGeneral matrices (matdescra[0]='G') is not supported.\nval\nArray containing the set of elements of the matrix A in the skyline profile\nform.\nIf matdescrsa[2]= 'L', then val contains elements from the low triangle\nof the matrix A.\nIf matdescrsa[2]= 'U', then val contains elements from the upper\ntriangle of the matrix A.\nRefer to values array description in Skyline Storage Scheme for more\ndetails.\npntr\nArray of length (m + 1) for lower triangle, and (n + 1) for upper triangle.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n219\n\n\nIt contains the indices specifying the positions of the first element of the\nmatrix A in each row (for the lower triangle) or column (for upper triangle)\nin the val array such that val[pntr[i] - 1] is the first element in row or\ncolumn i + 1. Refer to pointers array description in Skyline Storage\nScheme for more details.\nb\nArray, size ldb* n.\nOn entry the leading m-by-n part of the array b must contain the matrix B.\nldb\nSpecifies the leading dimension of b as declared in the calling\n(sub)program.\nldc\nSpecifies the leading dimension of c as declared in the calling\n(sub)program.\nOutput Parameters\nc\nArray, size ldc by n.\nThe leading m-by-n part of the array c contains the matrix C.\nmkl_?dnscsr\nConvert a sparse matrix in uncompressed\nrepresentation to the CSR format and vice versa\n(deprecated).\nSyntax\nvoid mkl_ddnscsr (const MKL_INT *job , const MKL_INT *m , const MKL_INT *n , double\n*adns , const MKL_INT *lda , double *acsr , MKL_INT *ja , MKL_INT *ia , MKL_INT *info );\nvoid mkl_sdnscsr (const MKL_INT *job , const MKL_INT *m , const MKL_INT *n , float\n*adns , const MKL_INT *lda , float *acsr , MKL_INT *ja , MKL_INT *ia , MKL_INT *info );\nvoid mkl_cdnscsr (const MKL_INT *job , const MKL_INT *m , const MKL_INT *n ,\nMKL_Complex8 *adns , const MKL_INT *lda , MKL_Complex8 *acsr , MKL_INT *ja , MKL_INT\n*ia , MKL_INT *info );\nvoid mkl_zdnscsr (const MKL_INT *job , const MKL_INT *m , const MKL_INT *n ,\nMKL_Complex16 *adns , const MKL_INT *lda , MKL_Complex16 *acsr , MKL_INT *ja , MKL_INT\n*ia , MKL_INT *info );\nInclude Files\n•\nmkl.h\nDescription\nThis routine is deprecated, but no replacement is available yet in the Inspector-Executor Sparse BLAS API\ninterfaces. Either write your own (see the examples/c/sparse_blas/source/sparse_converters.c\nexample for hints) or continue using this routine until a replacement is provided and this can be fully\nremoved.\nThis routine converts a sparse matrix A between formats: stored as a rectangular array (dense\nrepresentation) and stored using compressed sparse row (CSR) format (3-array variation).\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n220\n\n\nInput Parameters\njob\nArray, contains the following conversion parameters:\n•\njob[0]: Conversion type.\n•\nIf job[0]=0, the rectangular matrix A is converted to the CSR\nformat;\n•\nif job[0]=1, the rectangular matrix A is restored from the CSR\nformat.\n•\njob[1]: index base for the rectangular matrix A.\n•\nIf job[1]=0, zero-based indexing for the rectangular matrix A is\nused;\n•\nif job[1]=1, one-based indexing for the rectangular matrix A is used.\n•\njob[2]: Index base for the matrix in CSR format.\n•\nIf job[2]=0, zero-based indexing for the matrix in CSR format is\nused;\n•\nif job[2]=1, one-based indexing for the matrix in CSR format is\nused.\n•\njob[3]: Portion of matrix.\n•\nIf job[3]=0, adns is a lower triangular part of matrix A;\n•\nIf job[3]=1, adns is an upper triangular part of matrix A;\n•\nIf job[3]=2, adns is a whole matrix A.\n•\njob[4]=nzmax: maximum number of the non-zero elements allowed if\njob[0]=0.\n•\njob[5]: job indicator for conversion to CSR format.\n•\nIf job[5]=0, only array ia is generated for the output storage.\n•\nIf job[5]>0, arrays acsr, ia, ja are generated for the output\nstorage.\nm\nNumber of rows of the matrix A.\nn\nNumber of columns of the matrix A.\nadns\n(input/output)\nIf the conversion type is from uncompressed to CSR, on input adns\ncontains an uncompressed (dense) representation of matrix A.\nlda\nSpecifies the leading dimension of adns as declared in the calling\n(sub)program.\nFor zero-based indexing of A, lda must be at least max(1, n).\nFor one-based indexing of A, lda must be at least max(1, m).\nacsr\n(input/output)\nIf conversion type is from CSR to uncompressed, on input acsr contains\nthe non-zero elements of the matrix A. Its length is equal to the number of\nnon-zero elements in the matrix A. Refer to values array description in \nSparse Matrix Storage Formats for more details.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n221\n\n\nja\n(input/output). If conversion type is from CSR to uncompressed, on input\nfor zero-based indexing of A ja contains the column indices plus one for\neach non-zero element of the matrix A. For one-based indexing of A ja\ncontains the column indices for each non-zero element of the matrix A.\nIts length is equal to the length of the array acsr. Refer to columns array\ndescription in Sparse Matrix Storage Formats for more details.\nia\n(input/output). Array of length m + 1.\nIf conversion type is from CSR to uncompressed, on input for zero-based\nindexing of A ia contains indices of elements in the array acsr, such that\nia[i] - 1 is the index in the array acsr of the first non-zero element from\nthe row i. For one-based indexing of A ia contains indices of elements in\nthe array acsr, such that ia[i] is the index in the array acsr of the first\nnon-zero element from the row i.\nThe value ofia[m] - ia[0] is equal to the number of non-zeros. Refer to\nrowIndex array description in Sparse Matrix Storage Formats for more\ndetails.\nOutput Parameters\nadns\nIf conversion type is from CSR to uncompressed, on output adns contains\nthe uncompressed (dense) representation of matrix A.\nacsr, ja, ia\nIf conversion type is from uncompressed to CSR, on output acsr, ja, and\nia contain the compressed sparse row (CSR) format (3-array variation) of\nmatrix A (see Sparse Matrix Storage Formats for a description of the\nstorage format).\ninfo\nInteger info indicator only for restoring the matrix A from the CSR format.\nIf info=0, the execution is successful.\nIf info=i, the routine is interrupted processing the i-th row because there\nis no space in the arrays acsr and ja according to the value nzmax.\nmkl_?csrcoo\nConverts a sparse matrix in the CSR format to the\ncoordinate format and vice versa (deprecated).\nSyntax\nvoid mkl_scsrcoo (const MKL_INT *job , const MKL_INT *n , float *acsr , MKL_INT *ja ,\nMKL_INT *ia , MKL_INT *nnz , float *acoo , MKL_INT *rowind , MKL_INT *colind , MKL_INT\n*info );\nvoid mkl_dcsrcoo (const MKL_INT *job , const MKL_INT *n , double *acsr , MKL_INT *ja ,\nMKL_INT *ia , MKL_INT *nnz , double *acoo , MKL_INT *rowind , MKL_INT *colind , MKL_INT\n*info );\nvoid mkl_ccsrcoo (const MKL_INT *job , const MKL_INT *n , MKL_Complex8 *acsr , MKL_INT\n*ja , MKL_INT *ia , MKL_INT *nnz , MKL_Complex8 *acoo , MKL_INT *rowind , MKL_INT\n*colind , MKL_INT *info );\nvoid mkl_zcsrcoo (const MKL_INT *job , const MKL_INT *n , MKL_Complex16 *acsr , MKL_INT\n*ja , MKL_INT *ia , MKL_INT *nnz , MKL_Complex16 *acoo , MKL_INT *rowind , MKL_INT\n*colind , MKL_INT *info );\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n222\n\n\nInclude Files\n•\nmkl.h\nDescription\nThis routine is deprecated. Use the matrix manipulation routinesfrom the Intel® oneAPI Math Kernel Library\n(oneMKL) Inspector-executor Sparse BLAS interface instead.\nThis routine converts a sparse matrix A stored in the compressed sparse row (CSR) format (3-array\nvariation) to coordinate format and vice versa.\nInput Parameters\njob\nArray, contains the following conversion parameters:\njob[0]\nIf job[0]=0, the matrix in the CSR format is converted to the coordinate\nformat;\nif job[0]=1, the matrix in the coordinate format is converted to the CSR\nformat.\nif job[0]=2, the matrix in the coordinate format is converted to the CSR\nformat, and the column indices in CSR representation are sorted in the\nincreasing order within each row.\njob[1]\nIf job[1]=0, zero-based indexing for the matrix in CSR format is used;\nif job[1]=1, one-based indexing for the matrix in CSR format is used.\njob[2]\nIf job[2]=0, zero-based indexing for the matrix in coordinate format is\nused;\nif job[2]=1, one-based indexing for the matrix in coordinate format is\nused.\njob[4]\njob[4]=nzmax - maximum number of the non-zero elements allowed if\njob[0]=0.\njob[5] - job indicator.\nFor conversion to the coordinate format:\nIf job[5]=1, only array rowind is filled in for the output storage.\nIf job[5]=2, arrays rowind, colind are filled in for the output storage.\nIf job[5]=3, all arrays rowind, colind, acoo are filled in for the output\nstorage.\nFor conversion to the CSR format:\nIf job[5]=0, all arrays acsr, ja, ia are filled in for the output storage.\nIf job[5]=1, only array ia is filled in for the output storage.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n223\n\n\nIf job[5]=2, then it is assumed that the routine already has been called\nwith the job[5]=1, and the user allocated the required space for storing\nthe output arrays acsr and ja.\nn\nDimension of the matrix A.\nnnz\nSpecifies the number of non-zero elements of the matrix A for job[0]≠0.\nRefer to nnz description in Coordinate Format for more details.\nacsr\n(input/output)\nArray containing non-zero elements of the matrix A. Its length is equal to\nthe number of non-zero elements in the matrix A. Refer to values array\ndescription in Sparse Matrix Storage Formats for more details.\nja\n(input/output). For job[1] = 1 (one-based indexing for the matrix in CSR\nformat), array containing the column indices plus one for each non-zero\nelement of the matrix A.\nFor job[1] = 0 (zero-based indexing for the matrix in CSR format), array\ncontaining the column indices for each non-zero element of the matrix A.\nIts length is equal to the length of the array acsr. Refer to columns array\ndescription in Sparse Matrix Storage Formats for more details.\nia\n(input/output). Array of length n + 1, containing indices of elements in the\narray acsr, such that ia[i] - ia[0] is the index in the array acsr of the\nfirst non-zero element from the row i. The value of the last element ia[n]\n- ia[0] is equal to the number of non-zeros plus one. Refer to rowIndex\narray description in Sparse Matrix Storage Formats for more details.\nacoo\n(input/output)\nArray containing non-zero elements of the matrix A. Its length is equal to\nthe number of non-zero elements in the matrix A. Refer to values array\ndescription in Sparse Matrix Storage Formats for more details.\nrowind\n(input/output). Array of length nnz, contains the row indices for each non-\nzero element of the matrix A.\nRefer to rows array description in Coordinate Format for more details.\ncolind\n(input/output). Array of length nnz, contains the column indices for each\nnon-zero element of the matrix A. Refer to columns array description in \nCoordinate Format for more details.\nOutput Parameters\nnnz\nReturns the number of converted elements of the matrix A for job[0]=0.\ninfo\nInteger info indicator only for converting the matrix A from the CSR format.\nIf info=0, the execution is successful.\nIf info=1, the routine is interrupted because there is no space in the arrays\nacoo, rowind, colind according to the value nzmax.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n224\n\n\nmkl_?csrbsr\nConverts a square sparse matrix in the CSR format to\nthe BSR format and vice versa (deprecated).\nSyntax\nvoid mkl_scsrbsr (const MKL_INT *job , const MKL_INT *m , const MKL_INT *mblk , const\nMKL_INT *ldabsr , float *acsr , MKL_INT *ja , MKL_INT *ia , float *absr , MKL_INT *jab ,\nMKL_INT *iab , MKL_INT *info );\nvoid mkl_dcsrbsr (const MKL_INT *job , const MKL_INT *m , const MKL_INT *mblk , const\nMKL_INT *ldabsr , double *acsr , MKL_INT *ja , MKL_INT *ia , double *absr , MKL_INT\n*jab , MKL_INT *iab , MKL_INT *info );\nvoid mkl_ccsrbsr (const MKL_INT *job , const MKL_INT *m , const MKL_INT *mblk , const\nMKL_INT *ldabsr , MKL_Complex8 *acsr , MKL_INT *ja , MKL_INT *ia , MKL_Complex8 *absr ,\nMKL_INT *jab , MKL_INT *iab , MKL_INT *info );\nvoid mkl_zcsrbsr (const MKL_INT *job , const MKL_INT *m , const MKL_INT *mblk , const\nMKL_INT *ldabsr , MKL_Complex16 *acsr , MKL_INT *ja , MKL_INT *ia , MKL_Complex16\n*absr , MKL_INT *jab , MKL_INT *iab , MKL_INT *info );\nInclude Files\n•\nmkl.h\nDescription\nThis routine is deprecated. Use the matrix manipulation routinesfrom the Intel® oneAPI Math Kernel Library\n(oneMKL) Inspector-executor Sparse BLAS interface instead.\nThis routine converts a square sparse matrix A stored in the compressed sparse row (CSR) format (3-array\nvariation) to the block sparse row (BSR) format and vice versa.\nInput Parameters\njob\nArray, contains the following conversion parameters:\njob[0]\nIf job[0]=0, the matrix in the CSR format is converted to the BSR format;\nif job[0]=1, the matrix in the BSR format is converted to the CSR format.\njob[1]\nIf job[1]=0, zero-based indexing for the matrix in CSR format is used;\nif job[1]=1, one-based indexing for the matrix in CSR format is used.\njob[2]\nIf job[2]=0, zero-based indexing for the matrix in the BSR format is used;\nif job[2]=1, one-based indexing for the matrix in the BSR format is used.\njob[3] is only used for conversion to CSR format. By default, the converter\nsaves the blocks without checking whether an element is zero or not. If\njob[3]=1, then the converter only saves non-zero elements in blocks.\njob[5] - job indicator.\nFor conversion to the BSR format:\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n225\n\n\nIf job[5]=0, only arrays jab, iab are generated for the output storage.\nIf job[5]>0, all output arrays absr, jab, and iab are filled in for the\noutput storage.\nIf job[5]=-1, iab[m] - iab[0] returns the number of non-zero blocks.\nFor conversion to the CSR format:\nIf job[5]=0, only arrays ja, ia are generated for the output storage.\nm\nActual row dimension of the matrix A for convert to the BSR format; block\nrow dimension of the matrix A for convert to the CSR format.\nmblk\nSize of the block in the matrix A.\nldabsr\nLeading dimension of the array absr as declared in the calling program.\nldabsr must be greater than or equal to mblk*mblk.\nacsr\n(input/output)\nArray containing non-zero elements of the matrix A. Its length is equal to\nthe number of non-zero elements in the matrix A. Refer to values array\ndescription in Sparse Matrix Storage Formats for more details.\nja\n(input/output). Array containing the column indices for each non-zero\nelement of the matrix A.\nIts length is equal to the length of the array acsr. Refer to columns array\ndescription in Sparse Matrix Storage Formats for more details.\nia\n(input/output). Array of length m + 1, containing indices of elements in the\narray acsr, such that ia[I]] - iab[0] is the index in the array acsr of\nthe first non-zero element from the row I. The value of ia[m]] - iab[0]\nis equal to the number of non-zeros. Refer to rowIndex array description in \nSparse Matrix Storage Formats for more details.\nabsr\n(input/output)\nArray containing elements of non-zero blocks of the matrix A. Its length is\nequal to the number of non-zero blocks in the matrix A multiplied by\nmblk*mblk. Refer to values array description in BSR Format for more\ndetails.\njab\n(input/output). Array containing the column indices for each non-zero block\nof the matrix A.\nIts length is equal to the number of non-zero blocks of the matrix A. Refer\nto columns array description in BSR Format for more details.\niab\n(input/output). Array of length (m + 1), containing indices of blocks in the\narray absr, such that iab[i] - iab[0] is the index in the array absr of\nthe first non-zero element from the i-th row . The value of iab[m] is equal\nto the number of non-zero blocks. Refer to rowIndex array description in \nBSR Format for more details.\nOutput Parameters\ninfo\nInteger info indicator only for converting the matrix A from the CSR format.\nIf info=0, the execution is successful.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n226\n\n\nIf info=1, it means that mblk is equal to 0.\nIf info=2, it means that ldabsr is less than mblk*mblk and there is no\nspace for all blocks.\nmkl_?csrcsc\nConverts a square sparse matrix in the CSR format to\nthe CSC format and vice versa (deprecated).\nSyntax\nvoid mkl_dcsrcsc (const MKL_INT *job , const MKL_INT *n , double *acsr , MKL_INT *ja ,\nMKL_INT *ia , double *acsc , MKL_INT *ja1 , MKL_INT *ia1 , MKL_INT *info );\nvoid mkl_scsrcsc (const MKL_INT *job , const MKL_INT *n , float *acsr , MKL_INT *ja ,\nMKL_INT *ia , float *acsc , MKL_INT *ja1 , MKL_INT *ia1 , MKL_INT *info );\nvoid mkl_ccsrcsc (const MKL_INT *job , const MKL_INT *n , MKL_Complex8 *acsr , MKL_INT\n*ja , MKL_INT *ia , MKL_Complex8 *acsc , MKL_INT *ja1 , MKL_INT *ia1 , MKL_INT *info );\nvoid mkl_zcsrcsc (const MKL_INT *job , const MKL_INT *n , MKL_Complex16 *acsr , MKL_INT\n*ja , MKL_INT *ia , MKL_Complex16 *acsc , MKL_INT *ja1 , MKL_INT *ia1 , MKL_INT *info );\nInclude Files\n•\nmkl.h\nDescription\nThis routine is deprecated. Use the matrix manipulation routinesfrom the Intel® oneAPI Math Kernel Library\n(oneMKL) Inspector-executor Sparse BLAS interface instead.\nThis routine converts a square sparse matrix A stored in the compressed sparse row (CSR) format (3-array\nvariation) to the compressed sparse column (CSC) format and vice versa.\nInput Parameters\njob\nArray, contains the following conversion parameters:\njob[0]\nIf job[0]=0, the matrix in the CSR format is converted to the CSC format;\nif job[0]=1, the matrix in the CSC format is converted to the CSR format.\njob[1]\nIf job[1]=0, zero-based indexing for the matrix in CSR format is used;\nif job[1]=1, one-based indexing for the matrix in CSR format is used.\njob[2]\nIf job[2]=0, zero-based indexing for the matrix in the CSC format is used;\nif job[2]=1, one-based indexing for the matrix in the CSC format is used.\njob[5] - job indicator.\nFor conversion to the CSC format:\nIf job[5]=0, only arrays ja1, ia1 are filled in for the output storage.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n227\n\n\nIf job[5]≠0, all output arrays acsc, ja1, and ia1 are filled in for the\noutput storage.\nFor conversion to the CSR format:\nIf job[5]=0, only arrays ja, ia are filled in for the output storage.\nIf job[5]≠0, all output arrays acsr, ja, and ia are filled in for the output\nstorage.\nm\nDimension of the square matrix A.\nacsr\n(input/output)\nArray containing non-zero elements of the square matrix A. Its length is\nequal to the number of non-zero elements in the matrix A. Refer to values\narray description in Sparse Matrix Storage Formats for more details.\nja\n(input/output). Array containing the column indices for each non-zero\nelement of the matrix A.\nIts length is equal to the length of the array acsr. Refer to columns array\ndescription in Sparse Matrix Storage Formats for more details.\nia\n(input/output). Array of length m + 1, containing indices of elements in the\narray acsr, such that ia[i] - ia[0] is the index in the array acsr of the\nfirst non-zero element from the row i. The value of ia[m] - ia[0] is equal\nto the number of non-zeros. Refer to rowIndex array description in Sparse\nMatrix Storage Formats for more details.\nacsc\n(input/output)\nArray containing non-zero elements of the square matrix A. Its length is\nequal to the number of non-zero elements in the matrix A. Refer to values\narray description in Sparse Matrix Storage Formats for more details.\nja1\n(input/output). Array containing the row indices for each non-zero element\nof the matrix A.\nIts length is equal to the length of the array acsc. Refer to columns array\ndescription in Sparse Matrix Storage Formats for more details.\nia1\n(input/output). Array of length m + 1, containing indices of elements in the\narray acsc, such that ia1[i] - ia1[0] is the index in the array acsc of\nthe first non-zero element from the column i. The value of ia1[m] -\nia1[0] is equal to the number of non-zeros. Refer to rowIndex array\ndescription in Sparse Matrix Storage Formats for more details.\nOutput Parameters\ninfo\nThis parameter is not used now.\nmkl_?csrdia\nConverts a sparse matrix in the CSR format to the\ndiagonal format and vice versa (deprecated).\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n228\n\n\nSyntax\nvoid mkl_dcsrdia (const MKL_INT *job , const MKL_INT *n , double *acsr , MKL_INT *ja ,\nMKL_INT *ia , double *adia , const MKL_INT *ndiag , MKL_INT *distance , MKL_INT\n*idiag , double *acsr_rem , MKL_INT *ja_rem , MKL_INT *ia_rem , MKL_INT *info );\nvoid mkl_scsrdia (const MKL_INT *job , const MKL_INT *n , float *acsr , MKL_INT *ja ,\nMKL_INT *ia , float *adia , const MKL_INT *ndiag , MKL_INT *distance , MKL_INT *idiag ,\nfloat *acsr_rem , MKL_INT *ja_rem , MKL_INT *ia_rem , MKL_INT *info );\nvoid mkl_ccsrdia (const MKL_INT *job , const MKL_INT *n , MKL_Complex8 *acsr , MKL_INT\n*ja , MKL_INT *ia , MKL_Complex8 *adia , const MKL_INT *ndiag , MKL_INT *distance ,\nMKL_INT *idiag , MKL_Complex8 *acsr_rem , MKL_INT *ja_rem , MKL_INT *ia_rem , MKL_INT\n*info );\nvoid mkl_zcsrdia (const MKL_INT *job , const MKL_INT *n , MKL_Complex16 *acsr , MKL_INT\n*ja , MKL_INT *ia , MKL_Complex16 *adia , const MKL_INT *ndiag , MKL_INT *distance ,\nMKL_INT *idiag , MKL_Complex16 *acsr_rem , MKL_INT *ja_rem , MKL_INT *ia_rem , MKL_INT\n*info );\nInclude Files\n•\nmkl.h\nDescription\nThis routine is deprecated, but no replacement is available yet in the Inspector-Executor Sparse BLAS API\ninterfaces. Either write your own (see the examples/c/sparse_blas/source/sparse_converters.c\nexample for hints) or continue using this routine until a replacement is provided and this can be fully\nremoved.\nThis routine converts a sparse matrix A stored in the compressed sparse row (CSR) format (3-array\nvariation) to the diagonal format and vice versa.\nInput Parameters\njob\nArray, contains the following conversion parameters:\njob[0]\nIf job[0]=0, the matrix in the CSR format is converted to the diagonal\nformat;\nif job[0]=1, the matrix in the diagonal format is converted to the CSR\nformat.\njob[1]\nIf job[1]=0, zero-based indexing for the matrix in CSR format is used;\nif job[1]=1, one-based indexing for the matrix in CSR format is used.\njob[2]\nIf job[2]=0, zero-based indexing for the matrix in the diagonal format is\nused;\nif job[2]=1, one-based indexing for the matrix in the diagonal format is\nused.\njob[5] - job indicator.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n229\n\n\nFor conversion to the diagonal format:\nIf job[5]=0, diagonals are not selected internally, and acsr_rem, ja_rem,\nia_rem are not filled in for the output storage.\nIf job[5]=1, diagonals are not selected internally, and acsr_rem, ja_rem,\nia_rem are filled in for the output storage.\nIf job[5]=10, diagonals are selected internally, and acsr_rem, ja_rem,\nia_rem are not filled in for the output storage.\nIf job[5]=11, diagonals are selected internally, and csr_rem, ja_rem,\nia_rem are filled in for the output storage.\nFor conversion to the CSR format:\nIf job[5]=0, each entry in the array adia is checked whether it is zero.\nZero entries are not included in the array acsr.\nIf job[5]≠0, each entry in the array adia is not checked whether it is zero.\nm\nDimension of the matrix A.\nacsr\n(input/output)\nArray containing non-zero elements of the matrix A. Its length is equal to\nthe number of non-zero elements in the matrix A. Refer to values array\ndescription in Sparse Matrix Storage Formats for more details.\nja\n(input/output). Array containing the column indices for each non-zero\nelement of the matrix A.\nIts length is equal to the length of the array acsr. Refer to columns array\ndescription in Sparse Matrix Storage Formats for more details.\nia\n(input/output). Array of length m + 1, containing indices of elements in the\narray acsr, such that ia[i] - ia[0] is the index in the array acsr of the\nfirst non-zero element from the row i. The value of ia[m] - ia[0] is equal\nto the number of non-zeros. Refer to rowIndex array description in Sparse\nMatrix Storage Formats for more details.\nadia\n(input/output)\nArray of size (ndiag*idiag) containing diagonals of the matrix A.\nThe key point of the storage is that each element in the array adia retains\nthe row number of the original matrix. To achieve this diagonals in the\nlower triangular part of the matrix are padded from the top, and those in\nthe upper triangular part are padded from the bottom.\nndiag\nSpecifies the leading dimension of the array adia as declared in the calling\n(sub)program, must be at least max(1, m).\ndistance\nArray of length idiag, containing the distances between the main diagonal\nand each non-zero diagonal to be extracted. The distance is positive if the\ndiagonal is above the main diagonal, and negative if the diagonal is below\nthe main diagonal. The main diagonal has a distance equal to zero.\nidiag\nNumber of diagonals to be extracted. For conversion to diagonal format on\nreturn this parameter may be modified.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n230\n\n\nacsr_rem, ja_rem, ia_rem\nRemainder of the matrix in the CSR format if it is needed for conversion to\nthe diagonal format.\nOutput Parameters\ninfo\nThis parameter is not used now.\nmkl_?csrsky\nConverts a sparse matrix in CSR format to the skyline\nformat and vice versa (deprecated).\nSyntax\nvoid mkl_dcsrsky (const MKL_INT *job , const MKL_INT *m , double *acsr , MKL_INT *ja ,\nMKL_INT *ia , double *asky , MKL_INT *pointers , MKL_INT *info );\nvoid mkl_scsrsky (const MKL_INT *job , const MKL_INT *m , float *acsr , MKL_INT *ja ,\nMKL_INT *ia , float *asky , MKL_INT *pointers , MKL_INT *info );\nvoid mkl_ccsrsky (const MKL_INT *job , const MKL_INT *m , MKL_Complex8 *acsr , MKL_INT\n*ja , MKL_INT *ia , MKL_Complex8 *asky , MKL_INT *pointers , MKL_INT *info );\nvoid mkl_zcsrsky (const MKL_INT *job , const MKL_INT *m , MKL_Complex16 *acsr , MKL_INT\n*ja , MKL_INT *ia , MKL_Complex16 *asky , MKL_INT *pointers , MKL_INT *info );\nInclude Files\n•\nmkl.h\nDescription\nThis routine is deprecated, but no replacement is available yet in the Inspector-Executor Sparse BLAS API\ninterfaces. Either write your own (see the examples/c/sparse_blas/source/sparse_converters.c\nexample for hints) or continue using this routine until a replacement is provided and this can be fully\nremoved.\nThis routine converts a sparse matrix A stored in the compressed sparse row (CSR) format (3-array\nvariation) to the skyline format and vice versa.\nInput Parameters\njob\nArray, contains the following conversion parameters:\njob[0]\nIf job[0]=0, the matrix in the CSR format is converted to the skyline\nformat;\nif job[0]=1, the matrix in the skyline format is converted to the CSR\nformat.\njob[1]\nIf job[1]=0, zero-based indexing for the matrix in CSR format is used;\nif job[1]=1, one-based indexing for the matrix in CSR format is used.\njob[2]\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n231\n\n\nIf job[2]=0, zero-based indexing for the matrix in the skyline format is\nused;\nif job[2]=1, one-based indexing for the matrix in the skyline format is\nused.\njob[3]\nFor conversion to the skyline format:\nIf job[3]=0, the upper part of the matrix A in the CSR format is converted.\nIf job[3]=1, the lower part of the matrix A in the CSR format is converted.\nFor conversion to the CSR format:\nIf job[3]=0, the matrix is converted to the upper part of the matrix A in\nthe CSR format.\nIf job[3]=1, the matrix is converted to the lower part of the matrix A in\nthe CSR format.\njob[4]\njob[4]=nzmax - maximum number of the non-zero elements of the matrix\nA if job[0]=0.\njob[5] - job indicator.\nOnly for conversion to the skyline format:\nIf job[5]=0, only arrays pointers is filled in for the output storage.\nIf job[5]=1, all output arrays asky and pointers are filled in for the\noutput storage.\nm\nDimension of the matrix A.\nacsr\n(input/output)\nArray containing non-zero elements of the matrix A. Its length is equal to\nthe number of non-zero elements in the matrix A. Refer to values array\ndescription in Sparse Matrix Storage Formats for more details.\nja\n(input/output). Array containing the column indices for each non-zero\nelement of the matrix A.\nIts length is equal to the length of the array acsr. Refer to columns array\ndescription in Sparse Matrix Storage Formats for more details.\nia\n(input/output). Array of length m + 1, containing indices of elements in the\narray acsr, such that ia[i] - ia[0] is the index in the array acsr of the\nfirst non-zero element from the row i. The value of ia[m] - ia[0] is equal\nto the number of non-zeros. Refer to rowIndex array description in Sparse\nMatrix Storage Formats for more details.\nasky\n(input/output)\nArray, for a lower triangular part of A it contains the set of elements from\neach row starting from the first none-zero element to and including the\ndiagonal element. For an upper triangular matrix it contains the set of\nelements from each column of the matrix starting with the first non-zero\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n232\n\n\nelement down to and including the diagonal element. Encountered zero\nelements are included in the sets. Refer to values array description in \nSkyline Storage Format for more details.\npointers\n(input/output).\nArray with dimension (m+1), where m is number of rows for lower triangle\n(columns for upper triangle), pointers[i-1] - pointers[0] gives the\nindex of element in the array asky that is first non-zero element in row\n(column)i . The value of pointers[m] is set to nnz + pointers[0],\nwhere nnz is the number of elements in the array asky. Refer to pointers\narray description in Skyline Storage Format for more details\nOutput Parameters\ninfo\nInteger info indicator only for converting the matrix A from the CSR format.\nIf info=0, the execution is successful.\nIf info=1, the routine is interrupted because there is no space in the array\nasky according to the value nzmax.\nmkl_?csradd\nComputes the sum of two matrices stored in the CSR\nformat (3-array variation) with one-based indexing\n(deprecated).\nSyntax\nvoid mkl_dcsradd (const char *trans , const MKL_INT *request , const MKL_INT *sort ,\nconst MKL_INT *m , const MKL_INT *n , double *a , MKL_INT *ja , MKL_INT *ia , const\ndouble *beta , double *b , MKL_INT *jb , MKL_INT *ib , double *c , MKL_INT *jc , MKL_INT\n*ic , const MKL_INT *nzmax , MKL_INT *info );\nvoid mkl_scsradd (const char *trans , const MKL_INT *request , const MKL_INT *sort ,\nconst MKL_INT *m , const MKL_INT *n , float *a , MKL_INT *ja , MKL_INT *ia , const\nfloat *beta , float *b , MKL_INT *jb , MKL_INT *ib , float *c , MKL_INT *jc , MKL_INT\n*ic , const MKL_INT *nzmax , MKL_INT *info );\nvoid mkl_ccsradd (const char *trans , const MKL_INT *request , const MKL_INT *sort ,\nconst MKL_INT *m , const MKL_INT *n , MKL_Complex8 *a , MKL_INT *ja , MKL_INT *ia ,\nconst MKL_Complex8 *beta , MKL_Complex8 *b , MKL_INT *jb , MKL_INT *ib , MKL_Complex8\n*c , MKL_INT *jc , MKL_INT *ic , const MKL_INT *nzmax , MKL_INT *info );\nvoid mkl_zcsradd (const char *trans , const MKL_INT *request , const MKL_INT *sort ,\nconst MKL_INT *m , const MKL_INT *n , MKL_Complex16 *a , MKL_INT *ja , MKL_INT *ia ,\nconst MKL_Complex16 *beta , MKL_Complex16 *b , MKL_INT *jb , MKL_INT *ib ,\nMKL_Complex16 *c , MKL_INT *jc , MKL_INT *ic , const MKL_INT *nzmax , MKL_INT *info );\nInclude Files\n•\nmkl.h\nDescription\nThis routine is deprecated. Use mkl_sparse_?_addfrom the Intel® oneAPI Math Kernel Library (oneMKL)\nInspector-executor Sparse BLAS interface instead.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n233\n\n\nThe mkl_?csradd routine performs a matrix-matrix operation defined as\nC := A+beta*op(B)\nwhere:\nA, B, C are the sparse matrices in the CSR format (3-array variation).\nop(B) is one of op(B) = B, or op(B) = BT, or op(B) = BH\nbeta is a scalar.\nThe routine works correctly if and only if the column indices in sparse matrix representations of matrices A\nand B are arranged in the increasing order for each row. If not, use the parameter sort (see below) to\nreorder column indices and the corresponding elements of the input matrices.\nNOTE\nThis routine supports only one-based indexing of the input arrays.\nInput Parameters\ntrans\nSpecifies the operation.\nIf trans = 'N' or 'n', then C := A+beta*B\nIf trans = 'T' or 't', then C := A+beta*BT\nIf trans = 'C' or 'c', then C := A+beta*BH.\nrequest\nIf request=0, the routine performs addition. The memory for the output\narrays ic, jc, c must be allocated beforehand.\nIf request=1, the routine only computes the values of the array ic of\nlength m + 1. The memory for the ic array must be allocated beforehand.\nOn exit the value ic[m] - 1 is the actual number of the elements in the\narrays c and jc.\nIf request=2, after the routine is called previously with the parameter\nrequest=1 and after the output arrays jc and c are allocated in the calling\nprogram with length at least ic[m] - 1, the routine performs addition.\nsort\nSpecifies the type of reordering. If this parameter is not set (default), the\nroutine does not perform reordering.\nIf sort=1, the routine arranges the column indices ja for each row in the\nincreasing order and reorders the corresponding values of the matrix A in\nthe array a.\nIf sort=2, the routine arranges the column indices jb for each row in the\nincreasing order and reorders the corresponding values of the matrix B in\nthe array b.\nIf sort=3, the routine performs reordering for both input matrices A and B.\nm\nNumber of rows of the matrix A.\nn\nNumber of columns of the matrix A.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n234\n\n\na\nArray containing non-zero elements of the matrix A. Its length is equal to\nthe number of non-zero elements in the matrix A. Refer to values array\ndescription in Sparse Matrix Storage Formats for more details.\nja\nArray containing the column indices plus one for each non-zero element of\nthe matrix A. For each row the column indices must be arranged in the\nincreasing order.\nThe length of this array is equal to the length of the array a. Refer to\ncolumns array description in Sparse Matrix Storage Formats for more\ndetails.\nia\nArray of length m + 1, containing indices of elements in the array a, such\nthat ia[i] - ia[0] is the index in the array a of the first non-zero\nelement from the row i. The value of the last element ia[m] is equal to the\nnumber of non-zero elements of the matrix A plus one. Refer to rowIndex\narray description in Sparse Matrix Storage Formats for more details.\nbeta\nSpecifies the scalar beta.\nb\nArray containing non-zero elements of the matrix B. Its length is equal to\nthe number of non-zero elements in the matrix B. Refer to values array\ndescription in Sparse Matrix Storage Formats for more details.\njb\nArray containing the column indices plus one for each non-zero element of\nthe matrix B. For each row the column indices must be arranged in the\nincreasing order.\nThe length of this array is equal to the length of the array b. Refer to\ncolumns array description in Sparse Matrix Storage Formats for more\ndetails.\nib\nArray of length m + 1 when trans = 'N' or 'n', or n + 1 otherwise.\nThis array contains indices of elements in the array b, such that ib[i] -\nib[0] is the index in the array b of the first non-zero element from the row\ni. The value of the last element ib[m] or ib[n] is equal to the number of\nnon-zero elements of the matrix B plus one. Refer to rowIndex array\ndescription in Sparse Matrix Storage Formats for more details.\nnzmax\nThe length of the arrays c and jc.\nThis parameter is used only if request=0. The routine stops calculation if\nthe number of elements in the result matrix C exceeds the specified value\nof nzmax.\nOutput Parameters\nc\nArray containing non-zero elements of the result matrix C. Its length is\nequal to the number of non-zero elements in the matrix C. Refer to values\narray description in Sparse Matrix Storage Formats for more details.\njc\nArray containing the column indices plus one for each non-zero element of\nthe matrix C.\nThe length of this array is equal to the length of the array c. Refer to\ncolumns array description in Sparse Matrix Storage Formats for more\ndetails.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n235\n\n\nic\nArray of length m + 1, containing indices of elements in the array c, such\nthat ic[i] - ic[0] is the index in the array c of the first non-zero\nelement from the row i. The value of the last element ic[m] is equal to the\nnumber of non-zero elements of the matrix C plus one. Refer to rowIndex\narray description in Sparse Matrix Storage Formats for more details.\ninfo\nIf info=0, the execution is successful.\nIf info=I>0, the routine stops calculation in the I-th row of the matrix C\nbecause number of elements in C exceeds nzmax.\nIf info=-1, the routine calculates only the size of the arrays c and jc and\nreturns this value plus 1 as the last element of the array ic.\nmkl_?csrmultcsr\nComputes product of two sparse matrices stored in\nthe CSR format (3-array variation) with one-based\nindexing (deprecated).\nSyntax\nvoid mkl_dcsrmultcsr (const char *trans , const MKL_INT *request , const MKL_INT\n*sort , const MKL_INT *m , const MKL_INT *n , const MKL_INT *k , double *a , MKL_INT\n*ja , MKL_INT *ia , double *b , MKL_INT *jb , MKL_INT *ib , double *c , MKL_INT *jc ,\nMKL_INT *ic , const MKL_INT *nzmax , MKL_INT *info );\nvoid mkl_scsrmultcsr (const char *trans , const MKL_INT *request , const MKL_INT\n*sort , const MKL_INT *m , const MKL_INT *n , const MKL_INT *k , float *a , MKL_INT\n*ja , MKL_INT *ia , float *b , MKL_INT *jb , MKL_INT *ib , float *c , MKL_INT *jc ,\nMKL_INT *ic , const MKL_INT *nzmax , MKL_INT *info );\nvoid mkl_ccsrmultcsr (const char *trans , const MKL_INT *request , const MKL_INT\n*sort , const MKL_INT *m , const MKL_INT *n , const MKL_INT *k , MKL_Complex8 *a ,\nMKL_INT *ja , MKL_INT *ia , MKL_Complex8 *b , MKL_INT *jb , MKL_INT *ib , MKL_Complex8\n*c , MKL_INT *jc , MKL_INT *ic , const MKL_INT *nzmax , MKL_INT *info );\nvoid mkl_zcsrmultcsr (const char *trans , const MKL_INT *request , const MKL_INT\n*sort , const MKL_INT *m , const MKL_INT *n , const MKL_INT *k , MKL_Complex16 *a ,\nMKL_INT *ja , MKL_INT *ia , MKL_Complex16 *b , MKL_INT *jb , MKL_INT *ib ,\nMKL_Complex16 *c , MKL_INT *jc , MKL_INT *ic , const MKL_INT *nzmax , MKL_INT *info );\nInclude Files\n•\nmkl.h\nDescription\nThis routine is deprecated. Use mkl_sparse_spmmfrom the Intel® oneAPI Math Kernel Library (oneMKL)\nInspector-executor Sparse BLAS interface instead.\nThe mkl_?csrmultcsr routine performs a matrix-matrix operation defined as\nC := op(A)*B\nwhere:\nA, B, C are the sparse matrices in the CSR format (3-array variation);\nop(A) is one of op(A) = A, or op(A) =AT, or op(A) = AH .\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n236\n\n\nYou can use the parameter sort to perform or not perform reordering of non-zero entries in input and output\nsparse matrices. The purpose of reordering is to rearrange non-zero entries in compressed sparse row matrix\nso that column indices in compressed sparse representation are sorted in the increasing order for each row.\nThe following table shows correspondence between the value of the parameter sort and the type of\nreordering performed by this routine for each sparse matrix involved:\nValue of the parameter\nsort\nReordering of A (arrays\na, ja, ia)\nReordering of B (arrays\nb, ja, ib)\nReordering of C (arrays\nc, jc, ic)\n1\nyes\nno\nyes\n2\nno\nyes\nyes\n3\nyes\nyes\nyes\n4\nyes\nno\nno\n5\nno\nyes\nno\n6\nyes\nyes\nno\n7\nno\nno\nno\narbitrary value not equal to\n1, 2,..., 7\nno\nno\nyes\nNOTE\nThis routine supports only one-based indexing of the input arrays.\nInput Parameters\ntrans\nSpecifies the operation.\nIf trans = 'N' or 'n', then C := A*B\nIf trans = 'T' or 't' or 'C' or 'c', then C := AT*B.\nrequest\nIf request=0, the routine performs multiplication, the memory for the\noutput arrays ic, jc, c must be allocated beforehand.\nIf request=1, the routine computes only values of the array ic of length m\n+ 1, the memory for this array must be allocated beforehand. On exit the\nvalue ic[m] - 1 is the actual number of the elements in the arrays c and\njc.\nIf request=2, the routine has been called previously with the parameter\nrequest=1, the output arrays jc and c are allocated in the calling program\nand they are of the length ic[m] - 1 at least.\nsort\nSpecifies whether the routine performs reordering of non-zeros entries in\ninput and/or output sparse matrices (see table above).\nm\nNumber of rows of the matrix A.\nn\nNumber of columns of the matrix A.\nk\nNumber of columns of the matrix B.\na\nArray containing non-zero elements of the matrix A. Its length is equal to\nthe number of non-zero elements in the matrix A. Refer to values array\ndescription in Sparse Matrix Storage Formats for more details.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n237\n\n\nja\nArray containing the column indices plus one for each non-zero element of\nthe matrix A. For each row the column indices must be arranged in the\nincreasing order.\nThe length of this array is equal to the length of the array a. Refer to\ncolumns array description in Sparse Matrix Storage Formats for more\ndetails.\nia\nArray of length m + 1.\nThis array contains indices of elements in the array a, such that ia[i] -\nia[0] is the index in the array a of the first non-zero element from the row\ni. The value of the last element ia[m] is equal to the number of non-zero\nelements of the matrix A plus one. Refer to rowIndex array description in \nSparse Matrix Storage Formats for more details.\nb\nArray containing non-zero elements of the matrix B. Its length is equal to\nthe number of non-zero elements in the matrix B. Refer to values array\ndescription in Sparse Matrix Storage Formats for more details.\njb\nArray containing the column indices plus one for each non-zero element of\nthe matrix B. For each row the column indices must be arranged in the\nincreasing order.\nThe length of this array is equal to the length of the array b. Refer to\ncolumns array description in Sparse Matrix Storage Formats for more\ndetails.\nib\nArray of length n + 1 when trans = 'N' or 'n', or m + 1 otherwise.\nThis array contains indices of elements in the array b, such that ib[i] -\nib[0] is the index in the array b of the first non-zero element from the row\ni. The value of the last element ib[n] or ib[m] is equal to the number of\nnon-zero elements of the matrix B plus one. Refer to rowIndex array\ndescription in Sparse Matrix Storage Formats for more details.\nnzmax\nThe length of the arrays c and jc.\nThis parameter is used only if request=0. The routine stops calculation if\nthe number of elements in the result matrix C exceeds the specified value\nof nzmax.\nOutput Parameters\nc\nArray containing non-zero elements of the result matrix C. Its length is\nequal to the number of non-zero elements in the matrix C. Refer to values\narray description in Sparse Matrix Storage Formats for more details.\njc\nArray containing the column indices plus one for each non-zero element of\nthe matrix C.\nThe length of this array is equal to the length of the array c. Refer to\ncolumns array description in Sparse Matrix Storage Formats for more\ndetails.\nic\nArray of length m + 1 when trans = 'N' or 'n', or n + 1 otherwise.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n238\n\n\nThis array contains indices of elements in the array c, such that ic[i] -\nic[0] is the index in the array c of the first non-zero element from the row\ni. The value of the last element ic[m] or ic[n] is equal to the number of\nnon-zero elements of the matrix C plus one. Refer to rowIndex array\ndescription in Sparse Matrix Storage Formats for more details.\ninfo\nIf info=0, the execution is successful.\nIf info=I>0, the routine stops calculation in the I-th row of the matrix C\nbecause number of elements in C exceeds nzmax.\nIf info=-1, the routine calculates only the size of the arrays c and jc and\nreturns this value plus 1 as the last element of the array ic.\nmkl_?csrmultd\nComputes product of two sparse matrices stored in\nthe CSR format (3-array variation) with one-based\nindexing. The result is stored in the dense matrix\n(deprecated).\nSyntax\nvoid mkl_dcsrmultd (const char *trans , const MKL_INT *m , const MKL_INT *n , const\nMKL_INT *k , double *a , MKL_INT *ja , MKL_INT *ia , double *b , MKL_INT *jb , MKL_INT\n*ib , double *c , MKL_INT *ldc );\nvoid mkl_scsrmultd (const char *trans , const MKL_INT *m , const MKL_INT *n , const\nMKL_INT *k , float *a , MKL_INT *ja , MKL_INT *ia , float *b , MKL_INT *jb , MKL_INT\n*ib , float *c , MKL_INT *ldc );\nvoid mkl_ccsrmultd (const char *trans , const MKL_INT *m , const MKL_INT *n , const\nMKL_INT *k , MKL_Complex8 *a , MKL_INT *ja , MKL_INT *ia , MKL_Complex8 *b , MKL_INT\n*jb , MKL_INT *ib , MKL_Complex8 *c , MKL_INT *ldc );\nvoid mkl_zcsrmultd (const char *trans , const MKL_INT *m , const MKL_INT *n , const\nMKL_INT *k , MKL_Complex16 *a , MKL_INT *ja , MKL_INT *ia , MKL_Complex16 *b , MKL_INT\n*jb , MKL_INT *ib , MKL_Complex16 *c , MKL_INT *ldc );\nInclude Files\n•\nmkl.h\nDescription\nThis routine is deprecated. Use mkl_sparse_?_spmmdfrom the Intel® oneAPI Math Kernel Library (oneMKL)\nInspector-executor Sparse BLAS interface instead.\nThe mkl_?csrmultd routine performs a matrix-matrix operation defined as\nC := op(A)*B\nwhere:\nA, B are the sparse matrices in the CSR format (3-array variation), C is dense matrix;\nop(A) is one of op(A) = A, or op(A) =AT, or op(A) = AH .\nThe routine works correctly if and only if the column indices in sparse matrix representations of matrices A\nand B are arranged in the increasing order for each row.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n239\n\n\nNOTE\nThis routine supports only one-based indexing of the input arrays.\nInput Parameters\ntrans\nSpecifies the operation.\nIf trans = 'N' or 'n', then C := A*B\nIf trans = 'T' or 't' or 'C' or 'c', then C := AT*B.\nm\nNumber of rows of the matrix A.\nn\nNumber of columns of the matrix A.\nk\nNumber of columns of the matrix B.\na\nArray containing non-zero elements of the matrix A. Its length is equal to\nthe number of non-zero elements in the matrix A. Refer to values array\ndescription in Sparse Matrix Storage Formats for more details.\nja\nArray containing the column indices plus one for each non-zero element of\nthe matrix A. For each row the column indices must be arranged in the\nincreasing order.\nThe length of this array is equal to the length of the array a. Refer to\ncolumns array description in Sparse Matrix Storage Formats for more\ndetails.\nia\nArray of length m + 1 when trans = 'N' or 'n', or n + 1 otherwise.\nThis array contains indices of elements in the array a, such that ia[i] -\nia[0] is the index in the array a of the first non-zero element from the row\ni. The value of the last element ia[m] or ia[n] is equal to the number of\nnon-zero elements of the matrix A plus one. Refer to rowIndex array\ndescription in Sparse Matrix Storage Formats for more details.\nb\nArray containing non-zero elements of the matrix B. Its length is equal to\nthe number of non-zero elements in the matrix B. Refer to values array\ndescription in Sparse Matrix Storage Formats for more details.\njb\nArray containing the column indices plus one for each non-zero element of\nthe matrix B. For each row the column indices must be arranged in the\nincreasing order.\nThe length of this array is equal to the length of the array b. Refer to\ncolumns array description in Sparse Matrix Storage Formats for more\ndetails.\nib\nArray of length m + 1.\nThis array contains indices of elements in the array b, such that ib[i] -\nib[0] is the index in the array b of the first non-zero element from the row\ni. The value of the last element ib[m] is equal to the number of non-zero\nelements of the matrix B plus one. Refer to rowIndex array description in \nSparse Matrix Storage Formats for more details.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n240\n\n\nOutput Parameters\nc\nArray containing non-zero elements of the result matrix C.\nldc\nSpecifies the leading dimension of the dense matrix C as declared in the\ncalling (sub)program. Must be at least max(m, 1) when trans = 'N' or\n'n', or max(1, n) otherwise.\nSparse QR Routines\nSparse QR routines and their data types\nRoutine or function group\nData types\nDescription\nmkl_sparse_set_qr_hint\n \nEnables a pivot strategy for an ill-conditioned matrix.\nmkl_sparse_?_qr\ns,d\nCalculates the solution of a sparse system of linear equations\nusing QR factorization.\nmkl_sparse_qr_reorder\n \nPerforms reordering and symbolic analysis of the matrix A.\nmkl_sparse_?_qr_factorize\ns,d\nPerforms numerical factorization of the matrix A.\nmkl_sparse_?_qr_solve\ns,d\nSolves the system A*x = b using QR factorization of the matrix\nA.\nmkl_sparse_?_qr_qmult\ns,d\nPerforms x := Q^(-1)*b.\nmkl_sparse_?_qr_rsolve\ns,d\nPerforms x := R^(-1)*b.\nNOTE The underdetermined systems of equations are not supported. The number of columns should\nbe less or equal to the number or rows.\nFor more information about the workflow of sparse QR functionality, refer to oneMKL Sparse QR solver.\nMultifrontal Sparse QR Factorization Method for Solving a Sparse System of Linear Equations.\nmkl_sparse_set_qr_hint\nDefine the pivot strategy for further calls of\nmkl_sparse_?_qr.\nSyntax\nsparse_status_t mkl_sparse_set_qr_hint (sparse_matrix_t A, sparse_qr_hint_t hint);\nInclude Files\n•\nmkl_sparse_qr.h\nDescription\nYou can use this routine to enable a pivot strategy in the case of an ill-conditioned matrix.\nInput Parameters\nA\nHandle containing a sparse matrix in an internal data structure.\nhint\nValue specifying whether to use pivoting.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n241\n\n\nNOTE The only value currently supported is\nSPARSE_QR_WITH_PIVOTS, which enables the use of a pivot\nstrategy for an ill-conditioned matrix.\nReturn Values\nSPARSE_STATUS_SUCCESS The operation was successful.\nSPARSE_STATUS_NOT_INI\nTIALIZED\nThe routine encountered an empty handle or matrix array.\nSPARSE_STATUS_ALLOC_F\nAILED\nInternal memory allocation failed.\nSPARSE_STATUS_INVALID\n_VALUE\nThe input parameters contain an invalid value.\nSPARSE_STATUS_EXECUTI\nON_FAILED\nExecution failed.\nSPARSE_STATUS_INTERNA\nL_ERROR\nAn error in algorithm implementation occurred.\nSPARSE_STATUS_NOT_SUP\nPORTED\nThe requested operation is not supported.\nmkl_sparse_?_qr\nComputes the QR decomposition for the matrix of a\nsparse linear system and calculates the solution.\nSyntax\nsparse_status_t mkl_sparse_d_qr ( sparse_operation_t operation, sparse_matrix_t A,\nstruct matrix_descr descr, sparse_layout_t layout, MKL_INT columns, double *x, MKL_INT\nldx, const double *b, MKL_INT ldb );\nsparse_status_t mkl_sparse_s_qr ( sparse_operation_t operation, sparse_matrix_t A,\nstruct matrix_descr descr, sparse_layout_t layout, MKL_INT columns, float *x, MKL_INT\nldx, const float *b, MKL_INT ldb );\nInclude Files\n•\nmkl_sparse_qr.h\nDescription\nThe mkl_sparse_?_qr routine computes the QR decomposition for the matrix of a sparse linear system A*x\n= b, so that A = Q*R where Q is the orthogonal matrix and R is upper triangular, and calculates the solution.\nNOTE\nCurrently, mkl_sparse_?_qr supports only square and overdetermined systems. For\nunderdetermined systems you can manually transpose the system matrix and use QR\ndecomposition for AT to get the minimum-norm solution for the original underdetermined\nsystem.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n242\n\n\nNOTE Currently, mkl_sparse_?_qr supports only CSR format for the input matrix, non-\ntranspose operation, and single right-hand side.\nInput Parameters\noperation\nSpecifies the operation to perform.\nNOTE Currently, the only suppored value is\nSPARSE_OPERATION_NON_TRANSPOSE (non-transpose case; that is, A*x\n= b is solved).\nA\nHandle containing a sparse matrix in an internal data structure.\ndescr\nStructure specifying sparse matrix properties. Only the parameters listed here\nare currently supported.\ntype\nSpecifies the type of sparse matrix.\nNOTE Currently, the only supported value is\nSPARSE_MATRIX_TYPE_GENERAL (the matrix is processed as-is).\nlayout\nDescribes the storage scheme for the dense matrix:\nSPARSE_LAYOUT_COLUMN_MAJOR\nStorage of elements uses column-major\nlayout.\nSPARSE_LAYOUT_ROW_MAJOR\nStorage of elements uses row-major layout.\nx\nArray with a size of at least rows*cols:\n \nlayout =\nSPARSE_LAYOUT_COLUMN_MAJOR\nlayout =\nSPARSE_LAYOUT_ROW_MAJOR\nrows\n(number of\nrows in x)\nldx\nNumber of columns in A\ncols\n(number of\ncolumns in\nx)\ncolumns\nldx\ncolumns\nNumber of columns in matrix b.\nldx\nSpecifies the leading dimension of matrix x.\nb\nArray with a size of at least rows*cols:\n \nlayout =\nSPARSE_LAYOUT_COLUMN_MAJOR\nlayout =\nSPARSE_LAYOUT_ROW_MAJOR\nrows\n(number of\nrows in b)\nldb\nNumber of columns in A\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n243\n\n\ncols\n(number of\ncolumns in\nb)\ncolumns\nldb\nldb\nSpecifies the leading dimension of matrix b.\nOutput Parameters\nx\nOverwritten by the updated matrix y.\nReturn Values\nSPARSE_STATUS_SUCCESS The operation was successful.\nSPARSE_STATUS_NOT_INI\nTIALIZED\nThe routine encountered an empty handle or matrix array.\nSPARSE_STATUS_ALLOC_F\nAILED\nInternal memory allocation failed.\nSPARSE_STATUS_INVALID\n_VALUE\nThe input parameters contain an invalid value.\nSPARSE_STATUS_EXECUTI\nON_FAILED\nExecution failed.\nSPARSE_STATUS_INTERNA\nL_ERROR\nAn error in algorithm implementation occurred.\nSPARSE_STATUS_NOT_SUP\nPORTED\nThe requested operation is not supported.\nmkl_sparse_qr_reorder\nReordering step of SPARSE QR solver.\nSyntax\nsparse_status_t mkl_sparse_qr_reorder (sparse_matrix_t A, struct matrix_descr descr);\nInclude Files\n•\nmkl_sparse_qr.h\nDescription\nThe mkl_sparse_qr_reorder routine performs ordering and symbolic analysis of matrix A.\nNOTE Currently, mkl_sparse_qr_reorder supports only general structure and CSR format\nfor the input matrix.\nInput Parameters\nA\nHandle containing a sparse matrix in an internal data structure.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n244\n\n\ndescr\nStructure specifying sparse matrix properties. Only the parameters listed here\nare currently supported.\nReturn Values\nSPARSE_STATUS_SUCCESS The operation was successful.\nSPARSE_STATUS_NOT_INI\nTIALIZED\nThe routine encountered an empty handle or matrix array.\nSPARSE_STATUS_ALLOC_F\nAILED\nInternal memory allocation failed.\nSPARSE_STATUS_INVALID\n_VALUE\nThe input parameters contain an invalid value.\nSPARSE_STATUS_EXECUTI\nON_FAILED\nExecution failed.\nSPARSE_STATUS_INTERNA\nL_ERROR\nAn error in algorithm implementation occurred.\nSPARSE_STATUS_NOT_SUP\nPORTED\nThe requested operation is not supported.\nmkl_sparse_?_qr_factorize\nFactorization step of the SPARSE QR solver.\nSyntax\nsparse_status_t mkl_sparse_d_qr_factorize (sparse_matrix_t A, double *alt_values);\nsparse_status_t mkl_sparse_s_qr_factorize (sparse_matrix_t A, float *alt_values);\nInclude Files\n•\nmkl_sparse_qr.h\nDescription\nThe mkl_sparse_?_qr_factorize routine performs numerical factorization of matrix A. Prior to calling this\nroutine, the mkl_sparse_?_qr_reorder routine must be called for the matrix handle A. For more\ninformation about the workflow of sparse QR functionality, refer to oneMKL Sparse QR solver. Multifrontal\nSparse QR Factorization Method for Solving a Sparse System of Linear Equations.\nNOTE Currently, mkl_sparse_?_qr_factorize supports only CSR format for the input matrix.\nInput Parameters\nA\nHandle containing a sparse matrix in an internal data structure.\nalt_values\nArray with alternative values. Must be the size of the non-zeroes in the initial\ninput matrix. When passed to the routine, these values will be used during the\nfactorization step instead of the values stored in handle A.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n245\n\n\nReturn Values\nSPARSE_STATUS_SUCCESS The operation was successful.\nSPARSE_STATUS_NOT_INI\nTIALIZED\nThe routine encountered an empty handle or matrix array.\nSPARSE_STATUS_ALLOC_F\nAILED\nInternal memory allocation failed.\nSPARSE_STATUS_INVALID\n_VALUE\nThe input parameters contain an invalid value.\nSPARSE_STATUS_EXECUTI\nON_FAILED\nExecution failed.\nSPARSE_STATUS_INTERNA\nL_ERROR\nAn error in algorithm implementation occurred.\nSPARSE_STATUS_NOT_SUP\nPORTED\nThe requested operation is not supported.\nmkl_sparse_?_qr_solve\nSolving step of the SPARSE QR solver.\nSyntax\nsparse_status_t mkl_sparse_d_qr_solve ( sparse_operation_t operation, sparse_matrix_t\nA, double *alt_values, sparse_layout_t layout, MKL_INT columns, double *x, MKL_INT ldx,\nconst double *b, MKL_INT ldb );\nsparse_status_t mkl_sparse_s_qr_solve ( sparse_operation_t operation, sparse_matrix_t\nA, float *alt_values, sparse_layout_t layout, MKL_INT columns, float *x, MKL_INT ldx,\nconst float *b, MKL_INT ldb );\nInclude Files\n•\nmkl_sparse_qr.h\nDescription\nThe mkl_sparse_?_qr_solve routine computes the solution of sparse systems of linear equations A*x =\nb. Prior to calling this routine, the mkl_sparse_?_qr_factorize routine must be called for the matrix\nhandle A. For more information about the workflow of sparse QR functionality, refer to oneMKL Sparse QR\nsolver. Multifrontal Sparse QR Factorization Method for Solving a Sparse System of Linear Equations.\nNOTE\nCurrently, mkl_sparse_?_qr_solve supports only CSR format for the input matrix, non-\ntranspose operation, and single right-hand side.\nAlternative values are not supported and must be set to NULL.\nInput Parameters\noperation\nSpecifies the operation to perform.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n246\n\n\nNOTE Currently, the only supported value is\nSPARSE_OPERATION_NON_TRANSPOSE (non-transpose case; that is, A*x\n= b is solved).\nA\nHandle containing a sparse matrix in an internal data structure.\nalt_values\nReserved for future use.\nlayout\nDescribes the storage scheme for the dense matrix:\nSPARSE_LAYOUT_COLUMN_MAJOR\nStorage of elements uses column-major\nlayout.\nSPARSE_LAYOUT_ROW_MAJOR\nStorage of elements uses row-major layout.\nx\nArray with a size of at least rows*cols:\n \nlayout =\nSPARSE_LAYOUT_COLUMN_MAJOR\nlayout =\nSPARSE_LAYOUT_ROW_MAJOR\nrows\n(number of\nrows in x)\nldx\nNumber of columns in A\ncols\n(number of\ncolumns in\nx)\ncolumns\nldx\ncolumns\nNumber of columns in matrix b.\nldx\nSpecifies the leading dimension of matrix x.\nb\nArray with a size of at least rows*cols:\n \nlayout =\nSPARSE_LAYOUT_COLUMN_MAJOR\nlayout =\nSPARSE_LAYOUT_ROW_MAJOR\nrows\n(number of\nrows in b)\nldb\nNumber of columns in A\ncols\n(number of\ncolumns in\nb)\ncolumns\nldb\nldb\nSpecifies the leading dimension of matrix b.\nOutput Parameters\nx\nContains the solution of system A*x = b.\nReturn Values\nSPARSE_STATUS_SUCCESS The operation was successful.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n247\n\n\nSPARSE_STATUS_NOT_INI\nTIALIZED\nThe routine encountered an empty handle or matrix array.\nSPARSE_STATUS_ALLOC_F\nAILED\nInternal memory allocation failed.\nSPARSE_STATUS_INVALID\n_VALUE\nThe input parameters contain an invalid value.\nSPARSE_STATUS_EXECUTI\nON_FAILED\nExecution failed.\nSPARSE_STATUS_INTERNA\nL_ERROR\nAn error in algorithm implementation occurred.\nSPARSE_STATUS_NOT_SUP\nPORTED\nThe requested operation is not supported.\nmkl_sparse_?_qr_qmult\nFirst stage of the solving step of the SPARSE QR\nsolver.\nSyntax\nsparse_status_t mkl_sparse_d_qr_qmult ( sparse_operation_t operation, sparse_matrix_t\nA, sparse_layout_t layout, MKL_INT columns, double *x, MKL_INT ldx, const double *b,\nMKL_INT ldb );\nsparse_status_t mkl_sparse_s_qr_qmult ( sparse_operation_t operation, sparse_matrix_t\nA, sparse_layout_t layout, MKL_INT columns, float *x, MKL_INT ldx, const float *b,\nMKL_INT ldb );\nInclude Files\n•\nmkl_sparse_qr.h\nDescription\nThe mkl_sparse_?_qr_qmult routine computes multiplication of inversed matrix Q and right-hand side\nmatrix b. This routine can be used to perform the solving step in two separate calls as an alternative to a\nsingle call of mkl_sparse_?_qr_solve.\nNOTE Currently, mkl_sparse_?_qr_qmult supports only CSR format for the input matrix,\nnon-transpose operation, and single right-hand side.\nInput Parameters\noperation\nSpecifies the operation to perform.\nNOTE Currently, the only supported value is\nSPARSE_OPERATION_NON_TRANSPOSE (non-transpose case; that is, A*x\n= b is solved).\nA\nHandle containing a sparse matrix in an internal data structure.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n248\n\n\nlayout\nDescribes the storage scheme for the dense matrix:\nSPARSE_LAYOUT_COLUMN_MAJOR\nStorage of elements uses column-major\nlayout.\nSPARSE_LAYOUT_ROW_MAJOR\nStorage of elements uses row-major layout.\nx\nArray with a size of at least rows*cols:\n \nlayout =\nSPARSE_LAYOUT_COLUMN_MAJOR\nlayout =\nSPARSE_LAYOUT_ROW_MAJOR\nrows\n(number of\nrows in x)\nldx\nNumber of columns in A\ncols\n(number of\ncolumns in\nx)\ncolumns\nldx\ncolumns\nNumber of columns in matrix b.\nldx\nSpecifies the leading dimension of matrix x.\nb\nArray with a size of at least rows*cols:\n \nlayout =\nSPARSE_LAYOUT_COLUMN_MAJOR\nlayout =\nSPARSE_LAYOUT_ROW_MAJOR\nrows\n(number of\nrows in b)\nldb\nNumber of columns in A\ncols\n(number of\ncolumns in\nb)\ncolumns\nldb\nldb\nSpecifies the leading dimension of matrix b.\nOutput Parameters\nx\nOverwritten by the updated matrix x = Q-1*b.\nReturn Values\nSPARSE_STATUS_SUCCESS The operation was successful.\nSPARSE_STATUS_NOT_INI\nTIALIZED\nThe routine encountered an empty handle or matrix array.\nSPARSE_STATUS_ALLOC_F\nAILED\nInternal memory allocation failed.\nSPARSE_STATUS_INVALID\n_VALUE\nThe input parameters contain an invalid value.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n249\n\n\nSPARSE_STATUS_EXECUTI\nON_FAILED\nExecution failed.\nSPARSE_STATUS_INTERNA\nL_ERROR\nAn error in algorithm implementation occurred.\nSPARSE_STATUS_NOT_SUP\nPORTED\nThe requested operation is not supported.\nmkl_sparse_?_qr_rsolve\nSecond stage of the solving step of the SPARSE QR\nsolver.\nSyntax\nsparse_status_t mkl_sparse_d_qr_rsolve ( sparse_operation_t operation, sparse_matrix_t\nA, sparse_layout_t layout, MKL_INT columns, double *x, MKL_INT ldx, const double *b,\nMKL_INT ldb );\nsparse_status_t mkl_sparse_s_qr_rsolve ( sparse_operation_t operation, sparse_matrix_t\nA, sparse_layout_t layout, MKL_INT columns, float *x, MKL_INT ldx, const float *b,\nMKL_INT ldb );\nInclude Files\n•\nmkl_sparse_qr.h\nDescription\nThe mkl_sparse_?_qr_rsolve routine computes the solution of A*x = b.\nNOTE Currently, mkl_sparse_?_qr_rsolve supports only CSR format for the input matrix,\nnon-transpose operation, and single right-hand side.\nInput Parameters\noperation\nSpecifies the operation to perform.\nNOTE Currently, the only supported value is\nSPARSE_OPERATION_NON_TRANSPOSE (non-transpose case; that is, A*x\n= b is solved).\nA\nHandle containing a sparse matrix in an internal data structure.\nlayout\nDescribes the storage scheme for the dense matrix:\nSPARSE_LAYOUT_COLUMN_MAJOR\nStorage of elements uses column-major\nlayout.\nSPARSE_LAYOUT_ROW_MAJOR\nStorage of elements uses row-major layout.\nx\nArray with a size of at least rows*cols:\n \nlayout =\nSPARSE_LAYOUT_COLUMN_MAJOR\nlayout =\nSPARSE_LAYOUT_ROW_MAJOR\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n250\n\n\nrows\n(number of\nrows in x)\nldx\nNumber of columns in A\ncols\n(number of\ncolumns in\nx)\ncolumns\nldx\ncolumns\nNumber of columns in matrix b.\nldx\nSpecifies the leading dimension of matrix x.\nb\nArray with a size of at least rows*cols:\n \nlayout =\nSPARSE_LAYOUT_COLUMN_MAJOR\nlayout =\nSPARSE_LAYOUT_ROW_MAJOR\nrows\n(number of\nrows in b)\nldb\nNumber of columns in A\ncols\n(number of\ncolumns in\nb)\ncolumns\nldb\nldb\nSpecifies the leading dimension of matrix b.\nOutput Parameters\nx\nContains the solution of the triangular system R*x = b.\nReturn Values\nSPARSE_STATUS_SUCCESS The operation was successful.\nSPARSE_STATUS_NOT_INI\nTIALIZED\nThe routine encountered an empty handle or matrix array.\nSPARSE_STATUS_ALLOC_F\nAILED\nInternal memory allocation failed.\nSPARSE_STATUS_INVALID\n_VALUE\nThe input parameters contain an invalid value.\nSPARSE_STATUS_EXECUTI\nON_FAILED\nExecution failed.\nSPARSE_STATUS_INTERNA\nL_ERROR\nAn error in algorithm implementation occurred.\nSPARSE_STATUS_NOT_SUP\nPORTED\nThe requested operation is not supported.\nCompact BLAS and LAPACK Functions\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n251\n\n\nOverview\nMany HPC applications rely on the application of BLAS and LAPACK operations on groups of very small\nmatrices. While existing batch Intel® oneAPI Math Kernel Library (oneMKL) BLAS routines already provide\nmeaningful speedup over OpenMP* loops around BLAS operations for these sizes, another customization\noffers potential speedup by allocating matrices in aSIMD-friendly format, thus allowing for cross-matrix\nvectorization in the BLAS and LAPACK routines of the Intel® oneAPI Math Kernel Library (oneMKL)\ncalledCompact BLAS and LAPACK.\nThe main idea behind these compact methods is to create true SIMD computations in which subgroups of\nmatrices are operated on with kernels that abstractly appear as scalar kernels, while registers are filled by\ncross-matrix vectorization.\nThese are the BLAS/LAPACK compact functions:\n•\nmkl_?gemm_compact\n•\nmkl_?trsm_compact\n•\nmkl_?potrf_compact\n•\nmkl_?getrfnp_compact\n•\nmkl_?geqrf_compact\n•\nmkl_?getrinp_compact\nThe compact API provides additional service functions to refactor data. Because this capability is not specific\nto any particular BLAS or LAPACK operation, this data manipulation can be executed once for an application's\ndata, allowing the entire program -- consisting of any number of BLAS and LAPACK operations for which\ncompact kernels have been written -- to be performed on the compact data without any refactoring. For\napplications working on data in compact format, the packing function need not be used.\nSee \"About the Compact Format\" below for more details.\nAlong with this new data format, the API consists of two components:\n•\nBLAS and LAPACK Compact Kernels: The first component of the API is a compact kernel that works on\nmatrices stored in compact format.\n•\nService Functions for the Compact Format: The second component of the API is a compact service\nfunction allowing for data to be factored into and out of compact format. These are:\n•\nmkl_?gepack_compact\n•\nmkl_?geunpack_compact\n•\nmkl_get_format_compact\n•\nmkl_?get_size_compact\nNote that there are some Numerical Limitations for the routines mentioned above.\nAbout the Compact Format\nIn compact format, for calculations involving real precision, matrices are organized in packs of size V, where\nV is the SIMD vector length of the underlying architecture. Each pack is a 3D-tensor with the matrix index\nincrementing the fastest. These packs are then loaded into registers and operated on using SIMD\ninstructions.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n252\n\n\nThe figure below demonstrates the packing of a set of four 3 x 3 real-precision matrices into compact format.\nThe pack length for this example is V = 2, resulting in 2 compact packs.\nInterleaved Data for Compact BLAS and LAPACK\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n253\n\n\nFor calculations involving complex precision, the real and imaginary parts of each matrix are packed\nseparately. In the figure below, the group of four 3 x 3 complex matrices is packed into compact format with\npack length V = 2. The first pack consists of the real parts of the first two matrices, and the second pack\nconsists of the imaginary parts of the first two matrices. Real and imaginary packs alternate in memory. This\nstorage format means that all compact arrays can be handled as a real type.\nCompact Format for Complex Precision\nThe particular specifications (size and number) of the compact packs for the architecture and problem-\nprecision definition are specified by an MKL_COMPACT_PACK enum type. For example: given a double-\nprecision problem involving a group of 128 matrices working on an architecture with a 256-bit SIMD vector\nlength, the optimal pack length is V = 4, and the number of packs is 32.\nThe initially-permitted values for the enum are:\n•\nMKL_COMPACT_SSE - pack length 2 for double precision, pack length 4 for single precision.\n•\nMKL_COMPACT_AVX - pack length 4 for double precision, pack length 8 for single precision.\n•\nMKL_COMPACT_AVX512 - pack length 8 for double precision, pack length 16 for single precision.\nFor calculations involving complex precision, the pack length is the same; however, half of the packs store\nthe real parts of matrices, and half store the imaginary parts. The means that it takes double the number of\npacks to store the same number of matrices.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n254\n\n\nThe above examples illustrate the case when the number of matrices is evenly-divisible by the pack length.\nWhen this is not the case, there will be partially-unfilled packs at the end of the memory segment, and the\ncompact-packing routine will pad these partially unfilled packs with identity matrices, so that compact\nroutines use only the completely-filled registers in their calculations. The next figure illustrates this padding\nfor a group of three 3 x 3 real-precision matrices with a pack length of 2.\nCompact Format with Padding\nBefore calling a BLAS or LAPACK compact function, the input data must be packed in compact format. After\nexecution, the output data should be unpacked from this compact format, unless another compact routine\nwill be called immediately following the first. Two service functions, mkl_?gepack_compact, and mkl_?\ngeunpack_compact, facilitate the process of storing matrices in compact format. It is recommended that the\nuser call the function mkl_get_format_compact before calling the mkl_?gepack_compactroutine to obtain the\noptimal format for performance. Advanced users can pack and unpack the matrices themselves and still use\nIntel® oneAPI Math Kernel Library (oneMKL) compact functions on the packed set.\nCompact routines can only be called for groups of matrices that have the same dimensions, leading\ndimension, and storage format. For example, the routine mkl_?getrfnp_compact, which calculates the LU\nfactorization of a group of m x n matrices without pivoting, can only be called for a group of matrices with\nthe same number of rows (m) and the same number of columns (n). All of the matrices must also be stored\nin arrays with the same leading dimension, and all must be stored in the same storage format (column-major\nor row-major).\nmkl_?gemm_compact\nComputes a matrix-matrix product of a set of compact\nformat general matrices.\nSyntax\nvoid mkl_sgemm_compact (MKL_LAYOUT layout, MKL_TRANSPOSE transa, MKL_TRANSPOSE transb,\nMKL_INT m, MKL_INT n, MKL_INT k, float alpha, const float *ap, MKL_INT ldap, const float\n*bp, MKL_INT ldbp, float beta, float *cp, MKL_INT ldcp, MKL_COMPACT_PACK format, MKL_INT\nnm);\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n255\n\n\nvoid mkl_dgemm_compact (MKL_LAYOUT layout, MKL_TRANSPOSE transa, MKL_TRANSPOSE transb,\nMKL_INT m, MKL_INT n, MKL_INT k, double alpha, const double *ap, MKL_INT ldap, const\ndouble *bp, MKL_INT ldbp, double beta, double *cp, MKL_INT ldcp, MKL_COMPACT_PACK\nformat, MKL_INT nm);\nvoid mkl_cgemm_compact (MKL_LAYOUT layout, MKL_TRANSPOSE transa, MKL_TRANSPOSE transb,\nMKL_INT m, MKL_INT n, MKL_INT k, mkl_compact_complex_float *alpha, const float *ap,\nMKL_INT ldap, const float *bp, MKL_INT ldbp, mkl_compact_complex_float *beta, float\n*cp, MKL_INT ldcp, MKL_COMPACT_PACK format, MKL_INT nm);\nvoid mkl_zgemm_compact (MKL_LAYOUT layout, MKL_TRANSPOSE transa, MKL_TRANSPOSE transb,\nMKL_INT m, MKL_INT n, MKL_INT k, mkl_compact_complex_double *alpha, const double *ap,\nMKL_INT ldap, const double *bp, MKL_INT ldbp, mkl_compact_complex_double *beta, double\n*cp, MKL_INT ldcp, MKL_COMPACT_PACK format, MKL_INT nm);\nDescription\nThe mkl_?gemm_compact routine computes a scalar-matrix-matrix product and adds the result to a scalar-\nmatrix product for a group of nm general matrices Ac that have been stored in compact format. The operation\nis defined for each matrix as:\nCc := alpha*op(Ac)*op(Bc) + beta*Cc\nWhere\n•\nop(Xc) is one of op(Xc) = Xc, or op(Xc) = XcT, or op(Xc) = XcH,\n•\nalpha and beta are scalars,\n•\nAc, Bc, and Cc are matrices that have been stored in compact format,\n•\nop(Ac) is an m-by-k matrix for each matrix in the group,\n•\nop(Bc) is a k-by-n matrix for each matrix in the group,\n•\nand Cc is an m-by-n matrix.\nInput Parameters\nlayout\nSpecifies whether two-dimensional array storage is row-major\n(MKL_ROW_MAJOR) or column-major (MKL_COL_MAJOR).\ntransa\nSpecifies the operation:\nIf transa=MKL_NOTRANS, then op(Ac):=Ac.\nIf transa=MKL_TRANS, then op(Ac):=AcT.\nIf transa=MKL_CONJTRANS, then op(Ac):=AcH.\ntransb\nSpecifies the operation:\nIf transb=MKL_NOTRANS, then op(Bc):=Bc.\nIf transb=MKL_TRANS, then op(Bc):=BcT.\nIf transb=MKL_CONJTRANS, then op(Bc):=BcH.\nm\nThe number of rows of the matrices op(Ac), m >= 0.\nn\nThe number of columns of matrices op(Bc) and Cc. n≥0.\nk\nThe number of columns of matrices op(Ac) and the number of rows of\nmatrices op(Bc). k≥0.\nalpha\nSpecifies the scalar alpha.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n256\n\n\nap\nPoints to the beginning of the array that stores the nmAc matrices. See \nCompact Format for more details.\ntransa=MKL_NOTRANS\ntransa=MKL_TRANS or\ntransa=MKL_CONJTRANS\nlayout =\nMKL_COL_MAJOR\nap has size ldap*k*nm.\nap has size ldap*m*nm.\nlayout =\nMKL_ROW_MAJOR\nap has size ldap*m*nm.\nap has size ldap*k*nm.\nldap\nSpecifies the leading dimension of Ac.\nbp\nPoints to the beginning of the array that stores the nmBc matrices. See \nCompact Format for more details.\ntransb=MKL_NOTRANS\ntransb=MKL_TRANS or\ntransb=MKL_CONJTRANS\nlayout =\nMKL_COL_MAJOR\nbp has size ldbp*n*nm.\nbp has size ldbp*k*nm.\nlayout =\nMKL_ROW_MAJOR\nbp has size ldbp*k*nm.\nbp has size ldbp*n*nm.\nldbp\nSpecifies the leading dimension of Bc.\nbeta\nSpecifies the scalar beta.\ncp\nBefore entry, cp points to the beginning of the array that stores the nmCc\nmatrices, except when beta is equal to zero, in which case cp need not be\nset on entry.\n \n \nlayout = MKL_COL_MAJOR\ncp has size ldap*n*nm.\nlayout = MKL_ROW_MAJOR\ncp has size ldap*m*nm.\nldcp\nSpecifies the leading dimension of Cc.\n \n \nlayout = MKL_COL_MAJOR\nldcp must be at least max (1,m).\nlayout = MKL_ROW_MAJOR\nldcp must be at least max (1,n).\nformat\nSpecifies the format of the compact matrices. See Compact Format or\nmkl_get_format_compact for details.\nnm\nTotal number of matrices stored in compact format in the group of matrices.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n257\n\n\nNOTE\nThe values of ldap, ldbp, and ldcp used in mkl_?gemm_compact must be consistent with the\nvalues used in mkl_?get_size_compact, mkl_?gepack_compact, and mkl_?geunpack_compact.\nOutput Parameters\ncp\nEach matrix Cc is overwritten by the m-by-n matrix (alpha*op(Ac)*op(Bc)\n+ beta*Cc).\nmkl_?trsm_compact\nSolves a triangular matrix equation for a set of\ngeneral, m x n matrices that have been stored in\nCompact format.\nSyntax\nmkl_strsm_compact (MKL_LAYOUT layout, MKL_SIDE side, MKL_UPLO uplo, MKL_TRANSPOSE\ntransa, MKL_DIAG diag, MKL_INT m, MKL_INT n, float alpha, const float *ap, MKL_INT\na_stride, float *bp, MKL_INT b_stide, MKL_COMPACT_PACK format, MKL_INT nm);\nmkl_dtrsm_compact (MKL_LAYOUT layout, MKL_SIDE side, MKL_UPLO uplo, MKL_TRANSPOSE\ntransa, MKL_DIAG diag, MKL_INT m, MKL_INT n, double alpha, const double*ap, MKL_INT\na_stride, double *bp, MKL_INT b_stride, MKL_COMPACT_PACK format, MKL_INT nm);\nmkl_ctrsm_compact (MKL_LAYOUT layout, MKL_SIDE side, MKL_UPLO uplo, MKL_TRANSPOSE\ntransa, MKL_DIAG diag, MKL_INT m, MKL_INT n, mkl_compact_complex_float *alpha, const\nfloat *ap, MKL_INT a_stride, float *bp, MKL_INT b_stride, MKL_COMPACT_PACK format,\nMKL_INT nm);\nmkl_ztrsm_compact (MKL_LAYOUT layout, MKL_SIDE side, MKL_UPLO uplo, MKL_TRANSPOSE\ntransa, MKL_DIAG diag, MKL_INT m, MKL_INT n, mkl_compact_complex_double *alpha, const\ndouble *ap, MKL_INT a_stride, double *bp, MKL_INT b_stride, MKL_COMPACT_PACK format,\nMKL_INT nm);\nDescription\nThe routine solves one of the following matrix equations for a group of nm matrices:\nop(Ac)*Xc = alpha*Bc,\nor\nXc*op(Ac) = alpha*Bc\nwhere:\nalpha is a scalar, Xc and Bc are m-by-n matrices that have been stored in compact format, and Ac is a m-by-\nm unit, or non-unit, upper or lower triangular matrix that has been stored in compact format.\nop(Ac) is one of op(Ac) = Ac, or op(Ac) = AcT, or op(Ac) = AcH,\nBc is overwritten by the solution matrix Xc.\nInput Parameters\nlayout\nSpecifies whether two-dimensional array storage is row-major\n(MKL_ROW_MAJOR) or column-major (MKL_COL_MAJOR).\nside\nSpecifies whether op(Ac) appears on the left or right of Xc in the equation:\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n258\n\n\nif side = MKL_LEFT, then op(Ac)*Xc = alpha*Bc, if side = MKL_RIGHT, then\nXc*op(Ac) = alpha*Bc\nuplo\nSpecifies whether matrix Ac is upper or lower triangular.\nIf uplo = MKL_UPPER, Ac is upper triangular.\nIf uplo = MKL_LOWER, Ac is lower triangular.\ntransa\nSpecifies the operation:\nIf transa=MKL_NOTRANS, then op(Ac) = Ac;\nIf transa=MKL_TRANS, then op(Ac) = AcT;\nIf transa=MKL_CONJTRANS, then op(Ac) = AcH ;\ndiag\nSpecifies whether the matrix Ac is unit triangular:\nIf diag=MKL_UNIT, then the matrix is unit triangular;\nif diag=MKL_NONUNIT, then the matrix is not unit triangular.\nm\nThe number of rows of Bc and the number of rows and columns of\nAc when side=MKL_LEFT; m >= 0.\nn\nThe number of columns of Bc and the number of rows and columns\nof Ac when side=MKL_RIGHT; n >= 0.\nalpha\nSpecifies the scalar alpha. When alpha is zero, then ap is not\nreferenced and bp need not be set before entry.\nap\nArray, size ldap*k*nm, where k is m when side= MKL_LEFTand n\nwhen side = MKL_RIGHT. ap points to the beginning of nm Ac matrices\nstored in compact format. When uplo = MKL_UPPER, Ac is assumed\nto be an upper triangular matrix and the lower triangular part of Ac\nis not referenced. With uplo = MKL_LOWER, Ac is assumed to be a\nlower triangular matrix and the upper triangular part of Ac is not\nreferenced. With diag = MKL_UNIT, the diagonal elements of Ac are\nnot referenced either, but are assumed to be unity.\nldap\nColumn stride (column-major layout) or row stride (row-major\nlayout) of Ac.\nWhen side=MKL_LEFT, ldap must be at least max (1,m).\nWhen side=MKL_RIGHT, ldap must be at least max (1,n).\nbp\nArray, size ldbp*n*nm when layout = MKL_COL_MAJOR; size\nldbp*m*nm when layout = MKL_ROW_MAJOR. Before entry, bp\npoints to the beginning of nm Bc matrices stored in compact format.\nldbp\nColumn stride (column-major layout) or row stride (row-major\nlayout) of Bc.\n \n \nlayout = MKL_COL_MAJOR\nldbp must be at least max (1,m).\nlayout = MKL_ROW_MAJOR\n*ldbp must be at least max (1,n).\nformat\nSpecifies the format of the compact matrices. See <Compact\nFormat> or mkl_get_format_compact for details.\nnm\nTotal number of matrices stored in Compact format; nm >= 0.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n259\n\n\nNOTE\nThe values of ldap and ldbp used in mkl_?trsm_compact must be consistent with the values\nused in mkl_?get_size_compact, mkl_?gepack_compact, and mkl_?geunpack_compact.\nOutput Parameters\nbp\nOn exit, Bc is overwritten by the solution matrix Xc. bp points to the\nbeginning of nm such Xc matrices.\nmkl_?potrf_compact\nComputes the Cholesky factorization of a set of\nsymmetric (Hermitian), positive-definite matrices,\nstored in Compact format (see Compact Format for\ndetails).\nSyntax\nvoid mkl_spotrf_compact (MKL_LAYOUT layout, MKL_UPLO uplo, MKL_INT n, float * ap,\nMKL_INT ldap, MKL_INT * info, MKL_COMPACT_PACK format, MKL_INT nm);\nvoid mkl_cpotrf_compact (MKL_LAYOUT layout, MKL_UPLO uplo, MKL_INT n, float * ap,\nMKL_INT ldap, MKL_INT * info, MKL_COMPACT_PACK format, MKL_INT nm);\nvoid mkl_dpotrf_compact (MKL_LAYOUT layout, MKL_UPLO uplo, MKL_INT n, double * ap,\nMKL_INT ldap, MKL_INT * info, MKL_COMPACT_PACK format, MKL_INT nm);\nvoid mkl_zpotrf_compact (MKL_LAYOUT layout, MKL_UPLO uplo, MKL_INT n, double * ap,\nMKL_INT ldap, MKL_INT * info, MKL_COMPACT_PACK format, MKL_INT nm);\nDescription\nThe routine forms the Cholesky factorization of a set of symmetric, positive definite (or, for complex data,\nHermitian, positive-definite), n x n matrices Ac, stored in Compact format, as:\n•\nAc = Uc T*Uc (for real data), Ac = Uc H*Uc (for complex data), if uplo = MKL_UPPER\n•\nAc = Lc*Lc T (for real data), Ac = Lc*Lc H (for complex data), if uplo = MKL_LOWER\nwhere Lc is a lower triangular matrix, and Uc is upper triangular. The factorization (output) data will also be\nstored in Compact format.\nBefore calling this routine, call mkl_?gepack_compact to store the matrices in the Compact format.\nNOTE\nCompact routines have some limitations; see Numerical Limitations.\nInput Parameters\nlayout\nSpecifies whether two-dimensional array storage is row-major\n(MKL_ROW_MAJOR) or column-major (MKL_COL_MAJOR).\nuplo\nMust be MKL_UPPER or MKL_LOWER\nIndicates whether the upper or lower triangular part of Ac has been stored\nand will be factored.\nIf uplo = MKL_UPPER, the upper triangular part of Ac is stored, and the\nstrictly lower triangular part of Ac is not referenced.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n260\n\n\nIf uplo = MKL_LOWER, the lower triangular part of Ac is stored, and the\nstrictly upper triangular part of Ac is not referenced.\nn\nThe order of Ac; n >= 0.\nldap\nColumn stride (column-major layout) or row stride (row-major\nlayout) of Ac.\nap\nPoints to the beginning of the nm Ac matrices. On entry, ap contains\neither the upper or the lower triangular part of Ac (see uplo).\nformat\nSpecifies the format of the compact matrices. See Compact Format or \nmkl_get_format_compact for details.\nnm\nTotal number of matrices stored in Compact format; nm >= 0.\nApplication Notes:\nBefore calling this routine,mkl_?gepack_compact must be called. After calling this routine,\nmkl_?geunpack_compact should be called, unless another compact routine will be called for the Compact\nformat matrices.\nThe total number of floating-point operations is approximately nm* (1/3) n 3 for real flavors and nm* (4/3) n\n3 for complex flavors.\nOutput Parameters\nap\nThe upper or lower triangular part of Ac, stored in Compact format\nin ap, is overwritten by its Cholesky factor Uc or Lc (as specified by\nuplo). ap now points to the beginning of this set of factors, stored in\nCompact format.\ninfo\nThe parameter is not currently used in this routine. It is reserved for\nthe future use.\nmkl_?getrfnp_compact\nThe routine computes the LU factorization, without\npivoting, of a set of general, m x n matrices that have\nbeen stored in Compact format (see Compact\nFormat).\nSyntax\nvoid mkl_sgetrfnp_compact (MKL_LAYOUT layout, MKL_INT m, MKL_INT n, float * ap, MKL_INT\nldap, MKL_INT * info, MKL_COMPACT_PACK format, MKL_INT nm);\nvoid mkl_dgetrfnp_compact (MKL_LAYOUT layout, MKL_INT m, MKL_INT n, double * ap,\nMKL_INT ldap, MKL_INT * info, MKL_COMPACT_PACK format, MKL_INT nm);\nvoid mkl_cgetrfnp_compact (MKL_LAYOUT layout, MKL_INT m, MKL_INT n, float * ap, MKL_INT\nldap, MKL_INT * info, MKL_COMPACT_PACK format, MKL_INT nm);\nvoid mkl_zgetrfnp_compact (MKL_LAYOUT layout, MKL_INT m, MKL_INT n, double * ap,\nMKL_INT ldap, MKL_INT * info, MKL_COMPACT_PACK format, MKL_INT nm);\nDescription\nThe mkl_?getrfnp_compact routine calculates the LU factorizations of a set of nm general (m x n) matrices\nA, stored in Compact format, as Ac = Lc*Uc. The factorization (output) data will also be stored in Compact\nformat.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n261\n\n\nNOTE\nCompact routines have some limitations; see Numerical Limitations.\nInput Parameters\nlayout\nSpecifies whether two-dimensional array storage is row-major\n(MKL_ROW_MAJOR) or column-major (MKL_COL_MAJOR).\nm\nThe number of rows of A; m ≥ 0.\nn\nThe number of columns of A; n ≥ 0.\nap\nPoints to the beginning of the the array which stores nm Ac\nmatrices.\nSee Compact Format for more details.\nldap\nLeading dimension of Ac.\nformat\nSpecifies the format of the compact matrices. See Compact Format\nor mkl_get_format_compact for details.\nnm\nTotal number of matrices stored in Compact format.\nApplication Notes:\nBefore calling this routine, mkl_?gepack_compact must be called. After calling this routine, mkl_?\ngeunpack_compact should be called, unless another compact routine will be subsequently called for the\nCompact format matrices.\nThe approximate number of floating-point operations for real flavors is:\nnm*(2/3)n3, if m = n,\nnm*(1/3)n2(3m-n), if m > n,\nnm*(1/3)m2(3n-m), if m < n.\nThe number of operations for complex flavors is four times greater. Directly after calling this routine, you can\ncall the following:\nmkl_?getrinp_compact, for computing the inverse of the nm input matrices in Compact format\nOutput Parameters\nap\nOn exit, Ac is overwritten by its factorization data. ap points to the\nbeginning of nm Lc and Uc factors of Ac. The unit diagonal elements\nof Lc are not stored.\ninfo\nThe parameter is not currently used in this routine. It is reserved for\nthe future use.\nmkl_?geqrf_compact\nComputes the QR factorization of a set of general m x\nn, matrices, stored in Compact format (see Compact\nFormat for details).\nSyntax\nvoid mkl_sgeqrf_compact (MKL_LAYOUT layout, MKL_INT m, MKL_INT n, float * ap, MKL_INT\nldap, float * taup, float * work, MKL_INT lwork, MKL_INT * info, MKL_COMPACT_PACK\nformat, MKL_INT nm);\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n262\n\n\nvoid mkl_cgeqrf_compact (MKL_LAYOUT layout, MKL_INT m, MKL_INT n, float * ap, MKL_INT\nldap, float * taup, float * work, MKL_INT lwork, MKL_INT * info, MKL_COMPACT_PACK\nformat, MKL_INT nm);\nvoid mkl_dgeqrf_compact (MKL_LAYOUT layout, MKL_INT m, MKL_INT n, double * ap, MKL_INT\nldap, double * taup, double * work, MKL_INT lwork, MKL_INT * info, MKL_COMPACT_PACK\nformat, MKL_INT nm);\nvoid mkl_zgeqrf_compact (MKL_LAYOUT layout, MKL_INT m, MKL_INT n, double * ap, MKL_INT\nldap, double * taup, double * work, MKL_INT lwork, MKL_INT * info, MKL_COMPACT_PACK\nformat, MKL_INT nm);\nDescription\nThe routine forms the QR factorization of a set of general, m x n matrices A, stored in Compact format. The\nroutine does not form the Q factors explicitly. Instead, Q is represented as a product of min(m,n) elementary\nreflectors. The factorization (output) data will also be stored in Compact format.\nNOTE\nCompact routines have some limitations; see Numerical Limitations.\nInput Parameters\nlayout\nSpecifies whether two-dimensional array storage is row-major\n(MKL_ROW_MAJOR) or column-major (MKL_COL_MAJOR).\nm\nThe number of rows of Ac; m≥ 0.\nn\nThe number of columns of Ac; n≥ 0.\nap\nPoints to the beginning of the nm Ac matrices. On entry, ap contains\neither the upper or the lower triangular part of Ac (see uplo).\nldap\nColumn stride (column-major layout) or row stride (row-major\nlayout) of Ac.\nwork\nPoints to the beginning of the workspace array.\nlwork\nThe size of the work array. If lwork = -1, a workspace query is\nassumed; the routine only calculates the optimal size of the work\narray and returns this value as the first entry of the work array.\nformat\nSpecifies the format of the compact matrices. See Compact Format\nor mkl_get_format_compact for details.\nnm\nTotal number of matrices stored in Compact format.\nApplication Notes:\nThe compact array that will store the elementary reflectors needs to be allocated before the routine is called\nand unpacked after. First, the routine mkl_?get_size_compact should be called, to determine the size of taup,\nand memory for taup should be allocated. After calling mkl_?geqrf_compact, taup stores the elementary\nreflectors in compact form, so should be unpacked using mkl_?geunpack_compact. See Compact Format for\nmore details, or reference the example below. (Note: the following example is meant to demonstrate the\ncalling sequence to allocate memory and unpack taup. All other parameters are assumed to be already set\nup before the sequence below is executed.)\nMKL_R_TYPE *tau_array[nm];\n// ...\ntau_buffer_size = mkl_?get_size_compact(min(m, n), 1, format, nm);\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n263\n\n\nMKL_R_TYPE *tau_compact = (MKL_R_TYPE *)mkl_malloc(tau_buffer_size, 128);\nmkl_?geqrf_compact(layout, m, n, a_compact, ldap, tau_compact, work, lwork, &info, format, nm);\n// Note that here MKL_COL_MAJOR is used because tau is a 1-d array\nmkl_?geunpack_compact(MKL_COL_MAJOR, min(m, n), 1, tau_array, min(m, n), tau_compact, min(m, n), \nformat, nm);\nOutput Parameters\nap\nOn exit, A c is overwritten by its factorization data. ap points to the\nbeginning of nm factorizations of A c , stored in Compact format.\nThe factorization data is stored as follows: The elements on and\nabove the diagonal contain the min( m , n )-by- n upper trapezoidal\nmatrix R c ( R c is upper triangular if m ≥ n ); the elements below\nthe diagonal, with tau , present the orthogonal matrix Q c as a\nproduct of min( m , n ) elementary reflectors (see Orthogonal\nFactorizations: LAPACK Computational Routines). See Compact\nFormat for more details.\ntaup\nPoints to the beginning of a set of the tauc arrays, each of which has size\nmin(m,n), stored in Compact format. tauc contains scalars that define\nelementary reflectors for Qc in its decomposition in a product of elementary\nreflectors. taup needs to be allocated by the user before calling this routine.\nSee the application notes (below the description) for more details.\nwork[0]\nOn exit contains the minimum value of lwork required for optimum\nperformance. Use this lwork for subsequent runs.\ninfo\nThe parameter is not currently used in this routine. It is reserved for\nthe future use.\nmkl_?getrinp_compact\nComputes the inverse of a set of LU-factorized general\nmatrices, without pivoting, stored in the compact\nformat (see Compact Format for details).\nSyntax\nvoid mkl_sgetrinp_compact (MKL_LAYOUT layout, MKL_INT n, float * ap, MKL_INT ldap,\nfloat * work, MKL_INT lwork, MKL_INT * info, MKL_COMPACT_PACK format, MKL_INT nm);\nvoid mkl_dgetrinp_compact (MKL_LAYOUT layout, MKL_INT n, double * ap, MKL_INT ldap,\ndouble * work, MKL_INT lwork, MKL_INT * info, MKL_COMPACT_PACK format, MKL_INT nm);\nvoid mkl_cgetrinp_compact (MKL_LAYOUT layout, MKL_INT n, float * ap, MKL_INT ldap,\nfloat * work, MKL_INT lwork, MKL_INT * info, MKL_COMPACT_PACK format, MKL_INT nm);\nvoid mkl_zgetrinp_compact (MKL_LAYOUT layout, MKL_INT n, double * ap, MKL_INT ldap,\ndouble * work, MKL_INT lwork, MKL_INT * info, MKL_COMPACT_PACK format, MKL_INT nm);\nDescription\nThis routine computes the inverse inv( Ac) of a set of general, n x n matrices Ac, that have been stored in\nCompact format. The factorization (output) data will also be stored in Compact format.\nNOTE\nCompact routines have some limitations; see Numerical Limitations.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n264\n\n\nInput Parameters\nlayout\nSpecifies whether two-dimensional array storage is row-major\n(MKL_ROW_MAJOR) or column-major (MKL_COL_MAJOR).\nn\nThe order of Ac; n >= 0.\nap\nPoints to the beginning of the nm Ac matrices. On entry, ap contains\nthe LU factorizations of Ac, stored in Compact format, as returned\nby mkl_?getrfnp_compact : Ac=Lc*Uc.\nSee Compact Format for more details.\nldap\nColumn stride (column-major layout) or row stride (row-major\nlayout) of Ac.\nwork\nPoints to the beginning of the work array.\nlwork\nThe size of the work array. If lwork = -1, a workspace query is\nassumed; the routine calculates only the optimal size of the work\narray and returns this value as the first entry of the work array.\nformat\nSpecifies the format of the compact matrices. See Compact Format\nor mkl_get_format_compact for details.\nnm\nTotal number of matrices stored in Compact format.\nApplication Notes:\nBefore calling this routine, mkl_?gepack_compact must be called. After calling this routine,\nmkl_?geunpack_compact should be called, unless another compact routine will be subsequently called on\nthe Compact format matrices.\nThe total number of floating-point operations is approximately nm* (4/3) n 3 for real flavors and nm* (16/3) n\n3 for complex flavors.\nOutput Parameters\nap\nOn exit, A c is overwritten by inv(Ac). ap points to the beginning of\nnm inv(Ac) matrices stored in Compact format.\nwork[0]\nOn exit, work[0] contains the minimum value of lwork required for\noptimum performance. Use this lwork for subsequent runs.\ninfo\nThe parameter is not currently used in this routine. It is reserved for\nthe future use.\nNumerical Limitations for Compact BLAS and Compact LAPACK Routines\nCompact routines are subject to a set of numerical limitations. They also skip most of the checks presented\nin regular BLAS and LAPACK routines in order to provide effective vectorization. The following limitations\napply to at least one compact routine.\nComplex division: BLAS and LAPACK compact routines rely on a naïve method for complex division that does\nnot protect the solution against overflow, underflow, or loss of precision.\nError checking : the LAPACK compact routines skip error checking for performance reasons ; therefore, the\nuser is responsible for passing correct parameters. There are no checks for incorrect matrices (such as\nsingular for LU, non-positive-definite for Cholesky) - it is always assumed that the algorithm for the input\nmatrix can be completed without error.\nNo pivoting: the generic LU factorization routine, ?getrf , calculates the factorization using partial pivoting.\nHowever, because pivoting includes comparisons which cannot be effectively vectorized, only non-pivoting\nversions of LU mkl_?getrfnp_compact and Inverse from LU (mkl_?getrinp_compact) are provided as\ncompact routines.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n265\n\n\nMatrices scaled near underflow/overflow: the LAPACK compact routines do not provide safe handling for\nvalues near underflow/overflow. This means that Compact routines may return incorrect results for such\nmatrices. This limitation is related to compact routine for QR: mkl_?geqrf_compact.\nIt is the responsibility of the user to ensure that the input matrices can be factorized, inverted, and/or solved\ngiven these numerical limitations.\nmkl_?get_size_compact\nReturns the buffer size, in bytes, needed to pack data\nin Compact format.\nSyntax\nMKL_INT mkl_sget_size_compact (MKL_INT ld, MKL_INT sd, MKL_COMPACT_PACK format, MKL_INT\nnm);\nMKL_INT mkl_dget_size_compact (MKL_INT ld, MKL_INT sd, MKL_COMPACT_PACK format, MKL_INT\nnm);\nMKL_INT mkl_cget_size_compact (MKL_INT ld, MKL_INT sd, MKL_COMPACT_PACK format, MKL_INT\nnm);\nMKL_INT mkl_zget_size_compact (MKL_INT ld, MKL_INT sd, MKL_COMPACT_PACK format, MKL_INT\nnm);\nDescription\nThe routine returns the buffer size, in bytes, required for mkl_?gepack_compact.\nInput Parameters\nld\nLeading dimension of the matrices in Compact format.\nsd\nSecond dimension of the matrices in Compact format.\nformat\nDescribes the compact packing format according to the\nMKL_COMPACT_PACK enum type.\nnm\nTotal number of matrices to be packed in Compact format.\nApplication Notes:\nBefore calling this routine, mkl_?get_format_compact can be called to determine the optimal format.\nAfter calling this routine and allocating the amount of memory indicated by size, the user can call\nmkl_?gepack_compact to pack the nm input matrices in Compact format.\nReturn Values\nThis function returns a value size.\nsize\nThe buffer size, in bytes, required by the packing function\nmkl_?gepack_compact.\nmkl_get_format_compact\nReturns the optimal compact packing format for the\narchitecture, needed for all compact routines.\nSyntax\nMKL_COMPACT_PACK mkl_get_format_compact ();\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n266\n\n\nDescription\nThe routine returns the optimal compact packing format, which is an MKL_COMPACT_PACK type, for the\ncurrent architecture. The optimal value of format is determined by the architecture's vector-register length.\nformat is a required parameter for any packing, unpacking, or BLAS/LAPACK compact routine. See Compact\nFormat for details.\nReturn Values\nThe function returns a value format.\nformat\nformat can be returned as any of the following three\nvalues. MKL_COMPACT_AVX512 is the optimal format\nvalue for:\n•\nIntel® Advanced Vector Extensions 512 (Intel®\nAVX-512)-enabled processors.\n•\nIntel® Advanced Vector Extensions 512 (Intel®\nAVX-512) for Intel® Many Integrated Core\nArchitecture (Intel® MIC Architecture)-enabled\nprocessors.\n•\nIntel® Advanced Vector Extensions 512 (Intel®\nAVX-512) for Intel® Many Integrated Core\nArchitecture (Intel® MIC Architecture) with\nsupport of AVX512_4FMAPS and\nAVX512_4VNNIW instruction groups processors.\nMKL_COMPACT_AVX is the optimal format value for:\n•\nIntel® Advanced Vector Extensions (Intel® AVX)-\nenabled processors.\n•\nIntel® Advanced Vector Extensions 2 (Intel®\nAVX2)-enabled processors.\nMKL_COMPACT_SSE is the optimal format value for all\nother processors.\nApplication Notes:\nAfter calling this routine, mkl_?get_size_compact can be called to calculate the buffer size needed for\nmkl_?gepack_compact.\nmkl_?gepack_compact\nPacks matrices from standard (row or column-major)\nformat to Compact format.\nSyntax\nmkl_sgepack_compact(MKL_LAYOUT layout, MKL_INT rows, MKL_INT columns, const float *\nconst *a, MKL_INT lda, float *ap, MKL_INT ldap, MKL_COMPACT_PACK format, MKL_INT nm);\nmkl_dgepack_compact(MKL_LAYOUT layout, MKL_INT rows, MKL_INT columns, const double *\nconst *a, MKL_INT lda, double *ap, MKL_INT ldap, MKL_COMPACT_PACK format, MKL_INT nm);\nmkl_cgepack_compact (MKL_LAYOUT layout, MKL_INT rows, MKL_INT columns, const\nmkl_compact_complex_float * const *a, MKL_INT lda, float *ap, MKL_INT ldap,\nMKL_COMPACT_PACK format, const MKL_INT nm);\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n267\n\n\nmkl_zgepack_compact (MKL_LAYOUT layout, MKL_INT rows, MKL_INT columns, const\nmkl_compact_complex_double * const *a, MKL_INT lda, double *ap, MKL_INT ldap,\nMKL_COMPACT_PACK format, MKL_INT nm);\nDescription\nThe routine packs nm matrices A from standard format (row or column-major, pointer to pointer) in a into\nCompact format, storing the new compact format matrices Ac in array ap.\nInput Parameters\nlayout\nSpecifies whether two-dimensional array storage is row-major\n(MKL_ROW_MAJOR) or column-major (MKL_COL_MAJOR).\nrows\nThe number of rows of A; rows >= 0.\ncolumns\nThe number of columns of A; columns >= 0.\na\nA standard format (row or column-major, pointer-to-pointer) array,\nstoring nm input A matrices.\nlda\nLeading dimension of A.\n \n \nlayout = MKL_COL_MAJOR\nlda must be at least max (1,rows).\nlayout = MKL_ROW_MAJOR\nlda must be at least max (1,columns).\nldap\nLeading dimension of Ac.\n \n \nlayout = MKL_COL_MAJOR\nldap must be at least max (1,rows).\nlayout = MKL_ROW_MAJOR\nldap must be at least max (1,columns).\nNOTE\nThe values of ldap used in mkl_?gepack_compact must be\nconsistent with the values used in mkl_?get_size_compact and\nmkl_?geunpack_compact.\nformat\nSpecifies the format of the compact matrices. See Compact Format\nor mkl_get_format_compact for details.\nnm\nTotal number of matrices that will be stored in Compact format.\nApplication Notes:\nDirectly after calling this routine, any BLAS or LAPACK compact routine can be called. Unpacking matrices\nfrom Compact format can be done by calling mkl_?geunpack_compact.\nOutput Parameters\nap\nArray storing the compact format input matrices Ac. ap must have\nsize at least size = mkl_?get_size_compact.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n268\n\n\nmkl_?geunpack_compact\nUnpacks matrices from Compact format to standard\n(row- or column-major, pointer-to-pointer) format.\nSyntax\nmkl_sgeunpack_compact (MKL_LAYOUT layout, MKL_INT rows, MKL_INT columns, float * const\n*a, MKL_INT lda, const float *ap, MKL_INT ldap, MKL_COMPACT_PACK format, MKL_INT nm);\nmkl_dgeunpack_compact (MKL_LAYOUT layout, MKL_INT rows, MKL_INT columns, double * const\n*a, MKL_INT lda, const double *ap, MKL_INT ldap, MKL_COMPACT_PACK format, MKL_INT nm);\nmkl_cgeunpack_compact (MKL_LAYOUT layout, MKL_INT rows, MKL_INT columns,\nmkl_compact_complex_float * const *a, MKL_INT lda, const float *ap, MKL_INT ldap,\nMKL_COMPACT_PACK format, MKL_INT nm);\nmkl_zgeunpack_compact (MKL_LAYOUT layout, MKL_INT rows, MKL_INT columns,\nmkl_compact_complex_double * const *a, MKL_INT lda, const double *ap, MKL_INT ldap,\nMKL_COMPACT_PACK format, MKL_INT nm);\nDescription\nThe routine unpacks nm Compact format matrices Ac from array ap into standard (row- or column-major,\npointer-to-pointer) format in array A.\nInput Parameters\nlayout\nSpecifies whether two-dimensional array storage is row-major\n(MKL_ROW_MAJOR) or column-major (MKL_COL_MAJOR).\nrows\nThe number of rows of A; rows >= 0.\ncolumns\nThe number of columns of A; columns >= 0.\nlda\nLeading dimension of A.\n \n \nlayout = MKL_COL_MAJOR\nlda must be at least max (1,rows).\nlayout = MKL_ROW_MAJOR\nlda must be at least max (1,columns).\nap\nArray storing the compact format of input matrices Ac. See Compact\nFormator mkl_get_format_compact for details.\n \n \nlayout = MKL_COL_MAJOR\nap has size ldap*columns*nm.\nlayout = MKL_ROW_MAJOR\nap has size ldap*rows*nm.\nldap\nLeading dimension of of Ac.\n \n \nlayout = MKL_COL_MAJOR\nldap must be at least max (1,rows).\nlayout = MKL_ROW_MAJOR\nldap must be at least max (1,columns).\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n269\n\n\nNOTE\nThe values of ldap used in mkl_?geunpack_compact must be\nconsistent with the values used in mkl_?get_size_compact and\nmkl_?gepack_compact.\nformat\nSpecifies the format of the compact matrices. See Compact Format\normkl_get_format_compact for details.\nnm\nTotal number of matrices that will be stored in Compact format.\nOutput Parameters\na\nA standard format (row- or column-major, pointer-to-pointer) array,\nstoring nm output A matrices.\n \n \nlayout = MKL_COL_MAJOR\na has size lda*columns*nm.\nlayout = MKL_ROW_MAJOR\na has size lda*rows*nm.\nInspector-executor Sparse BLAS Routines\nThe inspector-executor API for Sparse BLAS divides operations into two stages: analysis and execution.\nDuring the initial analysis stage, the API inspects the matrix sparsity pattern and applies matrix structure\nchanges. In the execution stage, subsequent routine calls reuse this information in order to improve\nperformance.\nThe inspector-executor API supports key Sparse BLAS operations for iterative sparse solvers:\n•\nSparse matrix-vector multiplication\n•\nSparse matrix-matrix multiplication with a sparse or dense result\n•\nSolution of triangular systems\n•\nSparse matrix addition\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nNaming Conventions in Inspector-Executor Sparse BLAS Routines\nThe Inspector-Executor Sparse BLAS API routine names use the following convention:\nmkl_sparse_[<character>_]<operation>[_<format>]\nThe <character> field indicates the data type:\ns\nreal, single precision\nc\ncomplex, single precision\nd\nreal, double precision\nz\ncomplex, double precision\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n270\n\n\nThe data type is included in the name only if the function accepts dense matrix or scalar floating point\nparameters.\nThe <operation> field indicates the type of operation:\ncreate\ncreate matrix handle\ncopy\ncreate a copy of matrix handle\nconvert\nconvert matrix between sparse formats\nexport\nexport matrix from internal representation to CSR or BSR format\ndestroy\nfrees memory allocated for matrix handle\nset_<op>_hint provide information about number of upcoming compute operations and\noperation type for optimization purposes, where <op> is mv, sv, mm, sm, dotmv,\nsymgs, or memory\noptimize\nanalyze the matrix using hints and store optimization information in matrix\nhandle\nmv\ncompute sparse matrix-vector product\nmm\ncompute sparse matrix by dense matrix product (batch mv)\nset_value\nchange a value in a matrix\nspmm/spmmd\ncompute sparse matrix by sparse matrix product and store the result as a\nsparse/dense matrix\ntrsv\nsolve a triangular system\ntrsm\nsolve a triangular system with multiple right-hand sides\nadd\ncompute sum of two sparse matrices\nsymgs\ncompute a symmetric Gauss-Zeidel preconditioner\nsymgs_mv\ncompute a symmetric Gauss-Zeidel preconditioner with a final matrix-vector\nmultiplication\nsorv\ncomputes forward, backward sweeps or symmetric successive over-relaxation\npreconditioner\nsypr\ncompute the symmetric or Hermitian product of sparse matrices and store the\nresult as a sparse matrix\nsyprd\ncompute the symmetric or Hermitian product of sparse and dense matrices and\nstore the result as a dense matrix\nsyrk\ncompute the product of sparse matrix with its transposed matrix and store the\nresult as a sparse matrix\nsyrkd\ncompute the product of sparse matrix with its transposed matrix and store the\nresult as a dense matrix\norder\nperform ordering of column indexes of the matrix in CSR format\ndotmv\ncompute a sparse matrix-vector product with dot product\nThe <format> field indicates the sparse matrix storage format:\ncoo\ncoordinate format\nbsr\nblock sparse row format plus variations. Fill out either rows_start and rows_end\n(for 4-arrays representation) or rowIndex array (for 3-array BSR/CSR).\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n271\n\n\ncsr\ncompressed sparse row format plus variations. Fill out either rows_start and\nrows_end (for 4-arrays representation) or rowIndex array (for 3-array BSR/\nCSR).\ncsc\ncompressed sparse column format plus variations. Fill out either cols_start\nand cols_end (for 4-arrays representation) or colIndex array (for 3 array\nCSC).\nThe format is included in the function name only if the function parameters include an explicit sparse matrix\nin one of the conventional sparse matrix formats.\nSparse Matrix Storage Formats for Inspector-executor Sparse BLAS Routines\nInspector-executor Sparse BLAS routines support four conventional sparse matrix storage formats:\n•\ncompressed sparse row format (CSR) plus variations\n•\ncompressed sparse column format (CSC) plus variations\n•\ncoordinate format (COO)\n•\nblock sparse row format (BSR) plus variations\nComputational routines operate on a matrix handle that stores a matrix in CSR or BSR formats. Other\nformats should be converted to CSR or BSR format before calling any computational routines. For more\ninformation see Sparse Matrix Storage Formats.\nSupported Inspector-executor Sparse BLAS Operations\nThe Inspector-executor Sparse BLAS API can perform several operations involving sparse matrices. These\nnotations are used in the description of the operations:\n•\nA, G, V are sparse matrices\n•\nB and C are dense matrices\n•\nx and y are dense vectors\n•\nalpha and beta are scalars\nop(A) represents a possible transposition of matrix A\n \nop(A) = A\n \nop(A) = AT - transpose of A\n \nop(A) = AH - conjugate transpose of A\nop(A)-1 denotes the inverse of op(A).\nThe Inspector-executor Sparse BLAS routines support the following operations:\n•\ncomputing the vector product between a sparse matrix and a dense vector:\ny := alpha*op(A)*x + beta*y\n•\nsolving a single triangular system:\ny := alpha*inv(op(A))*x\n•\ncomputing a product between a sparse matrix and a dense matrix:\nC := alpha*op(A)*B + beta*C\n•\ncomputing a product between sparse matrices with a sparse result:\nV := alpha*op(A)*op(G)\n•\ncomputing a product between sparse matrices with a dense result:\nC := alpha*op(A)*op(G)\n•\ncomputing a sum of sparse matrices with a sparse result:\nV := alpha*op(A) + G\n•\nsolving a sparse triangular system with multiple right-hand sides:\nC := alpha*inv(op(A))*B\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n272\n\n\nTwo-stage Algorithm in Inspector-Executor Sparse BLAS Routines\nYou can use a two-stage algorithm in Inspector-executor Sparse BLAS routines which produce a sparse\nmatrix. The applicable routines are:\n•\nmkl_sparse_sp2m (BSR/CSR/CSC formats)\n•\nmkl_sparse_sypr (CSR format)\nThe two-stage algorithm allows you to split computations into stages. The main purpose of the splitting is to\nprovide an estimate for the memory required for the output prior to allocating the largest part of the memory\n(for the indices and values of the non-zero elements). Additionally, the two-stage approach extends the\nfunctionality and allows more complex usage models.\nNOTE The multistage approach currently does not allow you to allocate memory for the output matrix\noutside oneMKL.\nIn the two-stage algorithm:\n1.\nThe first stage allocates data which is necessary for the memory estimation (arrays rows_start/\nrows_end or cols_start/cols_end depending on the format, (see Sparse Matrix Storage Formats) and\ncomputes the number of entries or the full structure of the matrix.\nNOTE The format of the output is decided internally but can be checked using the export functionality\nmkl_sparse_?_export_<format>.\n2.\nThe second stage allocates data and computes column or row indices (depending on the format) of\nnon-zero elements and/or values of the output matrix.\nSpecifying the stage for execution is supported through the sparse_request_t parameter in the API with\nthe following options:\nValues for sparse_request_t parameter\nValue\nDescription\nSPARSE_STAGE_NNZ_COUN\nT\nAllocates and computes only the rows_start/rows_end (CSR/BSR format) or\ncols_start/cols_end (CSC format) arrays for the output matrix. After this\nstage, by calling mkl_sparse_?_export_<format>, you can obtain the\nnumber of non-zeros in the output matrix and calculate the amount of\nmemory required for the output matrix.\nSPARSE_STAGE_FINALIZE_\nMULT_NO_VAL\nAllocates and computes row/column indices provided that rows_start/\nrows_end or cols_start/cols_end have already been computed in a prior call\nwith the request SPARSE_STAGE_NNZ_COUNT. The values of the output\nmatrix are not computed.\nSPARSE_STAGE_FINALIZE_\nMULT\nDepending on the state of the output matrix C on entry to the routine, this\nstage does one of the following:\n•\nAllocates and computes row/column indices and values of nonzero\nelements, if only rows_start/rows_end or cols_start/cols_end are present\n•\nallocates and computes values of nonzero elements, if rows_start/\nrows_end or cols_start/cols_end and row/column indices of non-zero\nelements are present\nSPARSE_STAGE_FULL_MULT\n_NO_VAL\nAllocates and computes the output matrix structure in a single step. The\nvalues of the output matrix are not computed.\nSPARSE_STAGE_FULL_MULT\nAllocates and computes the entire output matrix (structure and values) in a\nsingle step.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n273\n\n\nThe example below shows how you can use the two-stage approach for estimating the memory requirements\nfor the output matrix in CSR format:\nFirst stage (sparse_request_t = SPARSE_STAGE_NNZ_COUNT)\n1.\nThe routine mkl_sparse_sp2m is called with the request parameter SPARSE_STAGE_NNZ_COUNT.\n2.\nThe arrays rows_start and rows_end are exported using the mkl_sparse_x_export_csr routine.\n3.\nThese arrays are used to calculate the number of non-zeros (nnz) of the resulting output matrix.\nNote that by the end of the first stage, the arrays associated with column indices and values of the output\nmatrix have not been allocated or computed yet.\nsparse_matrix_t csrC = NULL;\nstatus = mkl_sparse_sp2m (opA, descrA, csrA, opB, descrB, csrB, SPARSE_STAGE_NNZ_COUNT, &csrC);\n/* optional calculation of nnz in the output matrix for getting a memory estimate */\nstatus = mkl_sparse_?_export_csr (csrC, &indexing, &nrows, &ncols, &rows_start, &rows_end, \n&col_indx, &values);\nMKL_INT nnz = rows_end[nrows-1] - rows_start[0];\nSecond stage (sparse_request_t = SPARSE_STAGE_FINALIZE_MULT)\nThis stage allocates and computes the remaining output arrays (associated with column indices and values of\noutput matrix entries) and completes the matrix-matrix multiplication.\nstatus = mkl_sparse_sp2m (opA, descrA, csrA, opB, descrB, csrB, SPARSE_STAGE_FINALIZE_MULT, \n&csrC);\nWhen the two-stage approach is not needed, you can perform both stages in a single call:\nSingle stage operation (sparse_request_t = SPARSE_STAGE_FULL_MULT)\nstatus = mkl_sparse_sp2m (opA, descrA, csrA, opB, descrB, csrB, SPARSE_STAGE_FULL_MULT, &csrC);\nMatrix Manipulation Routines\nThe Matrix Manipulation Routines table lists the matrix manipulation routines and the data types associated\nwith them.\nMatrix Manipulation Routines and Their Data Types\nRoutine or\nFunction Group\nData Types\nDescription\nmkl_sparse_?\n_create_csr\ns, d, c, z\nCreates a handle for a CSR-format matrix.\nmkl_sparse_?\n_create_csc\ns, d, c, z\nCreates a handle for a CSC format matrix.\nmkl_sparse_?\n_create_coo\ns, d, c, z\nCreates a handle for a matrix in COO format.\nmkl_sparse_?\n_create_bsr\ns, d, c, z\nCreates a handle for a matrix in BSR format.\nmkl_sparse_copy\nNA\nCreates a copy of a matrix handle.\nmkl_sparse_destro\ny\nNA\nFrees memory allocated for matrix handle.\nmkl_sparse_conve\nrt_csr\nNA\nConverts internal matrix representation to CSR format.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n274\n\n\nRoutine or\nFunction Group\nData Types\nDescription\nmkl_sparse_conve\nrt_bsr\nNA\nConverts internal matrix representation to BSR format or\nchanges BSR block size.\nmkl_sparse_?\n_export_csr\ns, d, c, z\nExports CSR matrix from internal representation.\nmkl_sparse_?\n_export_csc\ns, d, c, z\nExports CSC matrix from internal representation.\nmkl_sparse_?\n_export_bsr\ns, d, c, z\nExports BSR matrix from internal representation.\nmkl_sparse_?\n_set_value\ns, d, c, z\nChanges a single value of matrix in internal\nrepresentation.\nmkl_sparse_?\n_update_values\ns, d, c, z\nChanges all or selected matrix values in internal\nrepresentation.\nmkl_sparse_order\nNA\nPerforms ordering of column indexes of the matrix in CSR\nformat.\nmkl_sparse_?_create_csr\nCreates a handle for a CSR-format matrix.\nSyntax\nsparse_status_t mkl_sparse_s_create_csr (sparse_matrix_t *A, const sparse_index_base_t \nindexing, const MKL_INT rows, const MKL_INT cols, MKL_INT *rows_start, MKL_INT\n*rows_end, MKL_INT *col_indx, float *values);\nsparse_status_t mkl_sparse_d_create_csr (sparse_matrix_t *A, const sparse_index_base_t \nindexing, const MKL_INT rows, const MKL_INT cols, MKL_INT *rows_start, MKL_INT\n*rows_end, MKL_INT *col_indx, double *values);\nsparse_status_t mkl_sparse_c_create_csr (sparse_matrix_t *A, const sparse_index_base_t \nindexing, const MKL_INT rows, const MKL_INT cols, MKL_INT *rows_start, MKL_INT\n*rows_end, MKL_INT *col_indx, MKL_Complex8 *values);\nsparse_status_t mkl_sparse_z_create_csr (sparse_matrix_t *A, const sparse_index_base_t \nindexing, const MKL_INT rows, const MKL_INT cols, MKL_INT *rows_start, MKL_INT\n*rows_end, MKL_INT *col_indx, MKL_Complex16 *values);\nInclude Files\n•\nmkl_spblas.h\nDescription\nThe mkl_sparse_?_create_csr routine creates a handle for an m-by-k matrix A in CSR format.\nNOTE\nThe input arrays provided are left unchanged except for the call to mkl_sparse_order, which\nperforms ordering of column indexes of the matrix. To avoid any changes to the input data,\nuse mkl_sparse_copy.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n275\n\n\nInput Parameters\nindexing\nIndicates how input arrays are indexed.\nSPARSE_INDEX_BASE_ZER\nO\nZero-based (C-style) indexing: indices start at\n0.\nSPARSE_INDEX_BASE_ONE\nOne-based (Fortran-style) indexing: indices\nstart at 1.\nrows\nNumber of rows of matrix A.\ncols\nNumber of columns of matrix A.\nrows_start\nArray of length at least rows. This array contains row indices, such that\nrows_start[i] - indexing is the first index of row i in the arrays values\nand col_indx. The value of indexing is 0 for zero-based indexing and 1\nfor one-based indexing.\nRefer to pointerB array description in CSR Format for more details.\nrows_end\nArray of at least length rows. This array contains row indices, such that\nrows_end[i] - indexing - 1 is the last index of row i in the arrays\nvalues and col_indx. The value of indexing is 0 for zero-based indexing\nand 1 for one-based indexing.\nRefer to pointerE array description in CSR Format for more details.\ncol_indx\nFor one-based indexing, array containing the column indices plus one for\neach non-zero element of the matrix A. For zero-based indexing, array\ncontaining the column indices for each non-zero element of the matrix A.\nIts length is at least rows_end[rows - 1] - indexing.\nThe value of indexing is 0 for zero-based indexing and 1 for one-based\nindexing.\nvalues\nArray containing non-zero elements of the matrix A. Its length is equal to\nlength of the col_indx array.\nRefer to values array description in CSR Format for more details.\nOutput Parameters\nA\nHandle containing internal data for subsequent Inspector-executor Sparse\nBLAS operations.\nReturn Values\nThe function returns a value indicating whether the operation was successful or not, and why.\nSPARSE_STATUS_SUCCESS\nThe operation was successful.\nSPARSE_STATUS_NOT_INITIALIZED\nThe routine encountered an empty handle or matrix array.\nSPARSE_STATUS_ALLOC_FAILED\nInternal memory allocation failed.\nSPARSE_STATUS_INVALID_VALUE\nThe input parameters contain an invalid value.\nSPARSE_STATUS_EXECUTION_FAILED\nExecution failed.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n276\n\n\nSPARSE_STATUS_INTERNAL_ERROR\nAn error in algorithm implementation occurred.\nSPARSE_STATUS_NOT_SUPPORTED\nThe requested operation is not supported.\nmkl_sparse_?_create_csc\nCreates a handle for a CSC format matrix.\nSyntax\nsparse_status_t mkl_sparse_s_create_csc (sparse_matrix_t *A, const sparse_index_base_t \nindexing, const MKL_INT rows, const MKL_INT cols, MKL_INT *cols_start, MKL_INT\n*cols_end, MKL_INT *row_indx, float *values);\nsparse_status_t mkl_sparse_d_create_csc (sparse_matrix_t *A, const sparse_index_base_t \nindexing, const MKL_INT rows, const MKL_INT cols, MKL_INT *cols_start, MKL_INT\n*cols_end, MKL_INT *row_indx, double *values);\nsparse_status_t mkl_sparse_c_create_csc (sparse_matrix_t *A, const sparse_index_base_t \nindexing, const MKL_INT rows, const MKL_INT cols, MKL_INT *cols_start, MKL_INT\n*cols_end, MKL_INT *row_indx, MKL_Complex8 *values);\nsparse_status_t mkl_sparse_z_create_csc (sparse_matrix_t *A, const sparse_index_base_t \nindexing, const MKL_INT rows, const MKL_INT cols, MKL_INT *cols_start, MKL_INT\n*cols_end, MKL_INT *row_indx, MKL_Complex16 *values);\nInclude Files\n•\nmkl_spblas.h\nDescription\nThe mkl_sparse_?_create_csc routine creates a handle for an m-by-k matrix A in CSC format.\nNOTE\nThe input arrays provided are left unchanged except for the call to mkl_sparse_order, which\nperforms ordering of column indexes of the matrix. To avoid any changes to the input data,\nuse mkl_sparse_copy.\nInput Parameters\nindexing\nIndicates how input arrays are indexed.\nSPARSE_INDEX_BASE_ZER\nO\nZero-based (C-style) indexing: indices start at\n0.\nSPARSE_INDEX_BASE_ONE\nOne-based (Fortran-style) indexing: indices\nstart at 1.\nrows\nNumber of rows of the matrix A.\ncols\nNumber of columns of the matrix A.\ncols_start\nArray of length at least m. This array contains col indices, such that\ncols_start[i] - ind is the first index of col i in the arrays values and\nrow_indx. ind takes 0 for zero-based indexing and 1 for one-based\nindexing.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n277\n\n\nRefer to pointerB array description in CSC Format for more details.\ncols_end\nArray of at least length m. This array contains col indices, such that\ncols_end[i] - ind - 1 is the last index of col i in the arrays values and\nrow_indx. ind takes 0 for zero-based indexing and 1 for one-based\nindexing.\nRefer to pointerE array description in CSC Format for more details.\nrow_indx\nFor one-based indexing, array containing the row indices plus one for each\nnon-zero element of the matrix A. For zero-based indexing, array containing\nthe row indices for each non-zero element of the matrix A. Its length is at\nleast cols_end[cols - 1] - ind. ind takes 0 for zero-based indexing and\n1 for one-based indexing.\nvalues\nArray containing non-zero elements of the matrix A. Its length is equal to\nlength of the row_indx array.\nRefer to values array description in CSC Format for more details.\nOutput Parameters\nA\nHandle containing internal data.\nReturn Values\nThe function returns a value indicating whether the operation was successful or not, and why.\nSPARSE_STATUS_SUCCESS\nThe operation was successful.\nSPARSE_STATUS_NOT_INITIALIZED\nThe routine encountered an empty handle or matrix array.\nSPARSE_STATUS_ALLOC_FAILED\nInternal memory allocation failed.\nSPARSE_STATUS_INVALID_VALUE\nThe input parameters contain an invalid value.\nSPARSE_STATUS_EXECUTION_FAILED\nExecution failed.\nSPARSE_STATUS_INTERNAL_ERROR\nAn error in algorithm implementation occurred.\nSPARSE_STATUS_NOT_SUPPORTED\nThe requested operation is not supported.\nmkl_sparse_?_create_coo\nCreates a handle for a matrix in COO format.\nSyntax\nsparse_status_t mkl_sparse_s_create_coo (sparse_matrix_t *A, const sparse_index_base_t \nindexing, const MKL_INT rows, const MKL_INT cols, const MKL_INT nnz, MKL_INT *row_indx,\nMKL_INT * col_indx, float *values);\nsparse_status_t mkl_sparse_d_create_coo (sparse_matrix_t *A, const sparse_index_base_t \nindexing, const MKL_INT rows, const MKL_INT cols, const MKL_INT nnz, MKL_INT *row_indx,\nMKL_INT * col_indx, double *values);\nsparse_status_t mkl_sparse_c_create_coo (sparse_matrix_t *A, const sparse_index_base_t \nindexing, const MKL_INT rows, const MKL_INT cols, const MKL_INT nnz, MKL_INT *row_indx,\nMKL_INT * col_indx, MKL_Complex8 *values);\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n278\n\n\nsparse_status_t mkl_sparse_z_create_coo (sparse_matrix_t *A, const sparse_index_base_t \nindexing, const MKL_INT rows, const MKL_INT cols, const MKL_INT nnz, MKL_INT *row_indx,\nMKL_INT * col_indx, MKL_Complex16 *values);\nInclude Files\n•\nmkl_spblas.h\nDescription\nThe mkl_sparse_?_create_coo routine creates a handle for an m-by-k matrix A in COO format.\nNOTE\nThe input arrays provided are left unchanged except for the call to mkl_sparse_order, which\nperforms ordering of column indexes of the matrix. To avoid any changes to the input data,\nuse mkl_sparse_copy.\nInput Parameters\nindexing\nIndicates how input arrays are indexed.\nSPARSE_INDEX_BASE_ZER\nO\nZero-based (C-style) indexing: indices start at\n0.\nSPARSE_INDEX_BASE_ONE\nOne-based (Fortran-style) indexing: indices\nstart at 1.\nrows\nNumber of rows of matrix A.\ncols\nNumber of columns of matrix A.\nnnz\nSpecifies the number of non-zero elements of the matrix A.\nRefer to nnz description in Coordinate Format for more details.\nrow_indx\nArray of length nnz, containing the row indices for each non-zero element\nof matrix A.\nRefer to rows array description in Coordinate Format for more details.\ncol_indx\nArray of length nnz, containing the column indices for each non-zero\nelement of matrix A.\nRefer to columns array description in Coordinate Format for more details.\nvalues\nArray of length nnz, containing the non-zero elements of matrix A in\narbitrary order.\nRefer to values array description in Coordinate Format for more details.\nOutput Parameters\nA\nHandle containing internal data.\nReturn Values\nThe function returns a value indicating whether the operation was successful or not, and why.\nSPARSE_STATUS_SUCCESS\nThe operation was successful.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n279\n\n\nSPARSE_STATUS_NOT_INITIALIZED\nThe routine encountered an empty handle or matrix array.\nSPARSE_STATUS_ALLOC_FAILED\nInternal memory allocation failed.\nSPARSE_STATUS_INVALID_VALUE\nThe input parameters contain an invalid value.\nSPARSE_STATUS_EXECUTION_FAILED\nExecution failed.\nSPARSE_STATUS_INTERNAL_ERROR\nAn error in algorithm implementation occurred.\nSPARSE_STATUS_NOT_SUPPORTED\nThe requested operation is not supported.\nmkl_sparse_?_create_bsr\nCreates a handle for a matrix in BSR format.\nSyntax\nsparse_status_t mkl_sparse_s_create_bsr (sparse_matrix_t *A, const sparse_index_base_t \nindexing, const sparse_layout_t block_layout, const MKL_INT rows, const MKL_INT cols,\nconst MKL_INT block_size, MKL_INT *rows_start, MKL_INT *rows_end, MKL_INT *col_indx,\nfloat *values);\nsparse_status_t mkl_sparse_d_create_bsr (sparse_matrix_t *A, const sparse_index_base_t \nindexing, const sparse_layout_t block_layout, const MKL_INT rows, const MKL_INT cols,\nconst MKL_INT block_size, MKL_INT *rows_start, MKL_INT *rows_end, MKL_INT *col_indx,\ndouble *values);\nsparse_status_t mkl_sparse_c_create_bsr (sparse_matrix_t *A, const sparse_index_base_t \nindexing, const sparse_layout_t block_layout, const MKL_INT rows, const MKL_INT cols,\nconst MKL_INT block_size, MKL_INT *rows_start, MKL_INT *rows_end, MKL_INT *col_indx,\nMKL_Complex8 *values);\nsparse_status_t mkl_sparse_z_create_bsr (sparse_matrix_t *A, const sparse_index_base_t \nindexing, const sparse_layout_t block_layout, const MKL_INT rows, const MKL_INT cols,\nconst MKL_INT block_size, MKL_INT *rows_start, MKL_INT *rows_end, MKL_INT *col_indx,\nMKL_Complex16 *values);\nInclude Files\n•\nmkl_spblas.h\nDescription\nThe mkl_sparse_?_create_bsr routine creates a handle for an m-by-k matrix A in BSR format.\nNOTE\nThe input arrays provided are left unchanged except for the call to mkl_sparse_order, which\nperforms ordering of column indexes of the matrix. To avoid any changes to the input data,\nuse mkl_sparse_copy.\nInput Parameters\nindexing\nIndicates how input arrays are indexed.\nSPARSE_INDEX_BASE_ZER\nO\nZero-based (C-style) indexing: indices start at\n0.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n280\n\n\nSPARSE_INDEX_BASE_ONE\nOne-based (Fortran-style) indexing: indices\nstart at 1.\nblock_layout\nSpecifies layout of blocks:\nSPARSE_LAYOUT_ROW_MAJ\nOR\nStorage of elements of blocks uses row major\nlayout.\nSPARSE_LAYOUT_COLUMN_\nMAJOR\nStorage of elements of blocks uses column\nmajor layout.\nrows\nNumber of block rows of matrix A.\ncols\nNumber of block columns of matrix A.\nblock_size\nSize of blocks in matrix A.\nrows_start\nArray of length m. This array contains row indices, such that\nrows_start[i] - ind is the first index of block row i in the arrays values\nand col_indx. ind takes 0 for zero-based indexing and 1 for one-based\nindexing.\nRefer to pointerB array description in CSR Format for more details.\nrows_end\nArray of length m. This array contains row indices, such that rows_end[i]\n- ind- 1 is the last index of block row i in the arrays values and\ncol_indx. ind takes 0 for zero-based indexing and 1 for one-based\nindexing.\nRefer to pointerE array description in CSR Format for more details.\ncol_indx\nFor one-based indexing, array containing the column indices plus one for\neach non-zero block of the matrix A. For zero-based indexing, array\ncontaining the column indices for each non-zero block of the matrix A. Its\nlength is rows_end[rows - 1] - ind. ind takes 0 for zero-based indexing\nand 1 for one-based indexing.\nvalues\nArray containing non-zero elements of the matrix A. Its length is equal to\nlength of the col_indx array multiplied by block_size*block_size.\nRefer to the values array description in BSR Format for more details.\nOutput Parameters\nA\nHandle containing internal data.\nReturn Values\nThe function returns a value indicating whether the operation was successful or not, and why.\nSPARSE_STATUS_SUCCESS\nThe operation was successful.\nSPARSE_STATUS_NOT_INITIALIZED\nThe routine encountered an empty handle or matrix array.\nSPARSE_STATUS_ALLOC_FAILED\nInternal memory allocation failed.\nSPARSE_STATUS_INVALID_VALUE\nThe input parameters contain an invalid value.\nSPARSE_STATUS_EXECUTION_FAILED\nExecution failed.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n281\n\n\nSPARSE_STATUS_INTERNAL_ERROR\nAn error in algorithm implementation occurred.\nSPARSE_STATUS_NOT_SUPPORTED\nThe requested operation is not supported.\nmkl_sparse_copy\nCreates a copy of a matrix handle.\nSyntax\nsparse_status_t mkl_sparse_copy (const sparse_matrix_t source, const struct\nmatrix_descr descr, sparse_matrix_t *dest);\nInclude Files\n•\nmkl_spblas.h\nDescription\nThe mkl_sparse_copy routine creates a copy of a matrix handle.\nNOTE\nCurrently, the mkl_sparse_copy routine does not support the descriptor argument and\ncreates an exact (deep) copy of the input matrix.\nInput Parameters\nsource\nSpecifies handle containing internal data.\ndescr\nStructure specifying sparse matrix properties.\nsparse_matrix_type_t type - Specifies the type of a sparse matrix:\nSPARSE_MATRIX_TYPE_GE\nNERAL\nThe matrix is processed as is.\nSPARSE_MATRIX_TYPE_SY\nMMETRIC\nThe matrix is symmetric (only the requested\ntriangle is processed).\nSPARSE_MATRIX_TYPE_HE\nRMITIAN\nThe matrix is Hermitian (only the requested\ntriangle is processed).\nSPARSE_MATRIX_TYPE_TR\nIANGULAR\nThe matrix is triangular (only the requested\ntriangle is processed).\nSPARSE_MATRIX_TYPE_DI\nAGONAL\nThe matrix is diagonal (only diagonal elements\nare processed).\nSPARSE_MATRIX_TYPE_BL\nOCK_TRIANGULAR\nThe matrix is block-triangular (only requested\ntriangle is processed). Applies to BSR format\nonly.\nSPARSE_MATRIX_TYPE_BL\nOCK_DIAGONAL\nThe matrix is block-diagonal (only diagonal\nblocks are processed). Applies to BSR format\nonly.\nsparse_fill_mode_t mode - Specifies the triangular matrix part for\nsymmetric, Hermitian, triangular, and block-triangular matrices:\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n282\n\n\nSPARSE_FILL_MODE_LOWE\nR\nThe lower triangular matrix part is processed.\nSPARSE_FILL_MODE_UPPE\nR\nThe upper triangular matrix part is processed.\nsparse_diag_type_t diag - Specifies diagonal type for non-general\nmatrices:\nSPARSE_DIAG_NON_UNIT\nDiagonal elements might not be equal to one.\nSPARSE_DIAG_UNIT\nDiagonal elements are equal to one.\nOutput Parameters\ndest\nHandle containing internal data.\nReturn Values\nThe function returns a value indicating whether the operation was successful or not, and why.\nSPARSE_STATUS_SUCCESS\nThe operation was successful.\nSPARSE_STATUS_NOT_INITIALIZED\nThe routine encountered an empty handle or matrix array.\nSPARSE_STATUS_ALLOC_FAILED\nInternal memory allocation failed.\nSPARSE_STATUS_INVALID_VALUE\nThe input parameters contain an invalid value.\nSPARSE_STATUS_EXECUTION_FAILED\nExecution failed.\nSPARSE_STATUS_INTERNAL_ERROR\nAn error in algorithm implementation occurred.\nSPARSE_STATUS_NOT_SUPPORTED\nThe requested operation is not supported.\nmkl_sparse_destroy\nFrees memory allocated for matrix handle.\nSyntax\nsparse_status_t mkl_sparse_destroy (sparse_matrix_t A);\nInclude Files\n•\nmkl_spblas.h\nDescription\nThe mkl_sparse_destroy routine frees memory allocated for matrix handle.\nNOTE\nYou must free memory allocated for matrices after completing use of them. The mkl_sparse_destroy\nroutine provides a utility to do so.\nInput Parameters\nA\nHandle containing internal data.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n283\n\n\nReturn Values\nThe function returns a value indicating whether the operation was successful or not, and why.\nSPARSE_STATUS_SUCCESS\nThe operation was successful.\nSPARSE_STATUS_NOT_INITIALIZED\nThe routine encountered an empty handle or matrix array.\nSPARSE_STATUS_ALLOC_FAILED\nInternal memory allocation failed.\nSPARSE_STATUS_INVALID_VALUE\nThe input parameters contain an invalid value.\nSPARSE_STATUS_EXECUTION_FAILED\nExecution failed.\nSPARSE_STATUS_INTERNAL_ERROR\nAn error in algorithm implementation occurred.\nSPARSE_STATUS_NOT_SUPPORTED\nThe requested operation is not supported.\nmkl_sparse_convert_csr\nConverts internal matrix representation to CSR\nformat.\nSyntax\nsparse_status_t mkl_sparse_convert_csr (const sparse_matrix_t source, const\nsparse_operation_t operation, sparse_matrix_t *dest);\nInclude Files\n•\nmkl_spblas.h\nDescription\nThe mkl_sparse_convert_csr routine converts internal matrix representation to CSR format.\nWhen the source matrix is in COO format, the routine performs a sum reduction on duplicate elements.\nInput Parameters\nsource\nHandle containing internal data.\noperation\nSpecifies operation op() on input matrix.\nSPARSE_OPERATION_NON_\nTRANSPOSE\nNon-transpose, op(A) = A.\nSPARSE_OPERATION_TRAN\nSPOSE\nTranspose, op(A) = AT.\nSPARSE_OPERATION_CONJ\nUGATE_TRANSPOSE\nConjugate transpose, op(A) = AH.\nOutput Parameters\ndest\nHandle containing internal data.\nReturn Values\nThe function returns a value indicating whether the operation was successful or not, and why.\nSPARSE_STATUS_SUCCESS\nThe operation was successful.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n284\n\n\nSPARSE_STATUS_NOT_INITIALIZED\nThe routine encountered an empty handle or matrix array.\nSPARSE_STATUS_ALLOC_FAILED\nInternal memory allocation failed.\nSPARSE_STATUS_INVALID_VALUE\nThe input parameters contain an invalid value.\nSPARSE_STATUS_EXECUTION_FAILED\nExecution failed.\nSPARSE_STATUS_INTERNAL_ERROR\nAn error in algorithm implementation occurred.\nSPARSE_STATUS_NOT_SUPPORTED\nThe requested operation is not supported.\nmkl_sparse_convert_bsr\nConverts internal matrix representation to BSR format\nor changes BSR block size.\nSyntax\nsparse_status_t mkl_sparse_convert_bsr (const sparse_matrix_t source, const MKL_INT\nblock_size, const sparse_layout_t block_layout, const sparse_operation_t operation,\nsparse_matrix_t *dest);\nInclude Files\n•\nmkl_spblas.h\nDescription\nThemkl_sparse_convert_bsr routine converts internal matrix representation to BSR format or changes\nBSR block size.\nWhen the source matrix is in COO format, the routine performs a sum reduction on duplicate elements.\nInput Parameters\nsource\nHandle containing internal data.\nblock_size\nSize of the block in the output structure.\nblock_layout\nSpecifies layout of blocks:\nSPARSE_LAYOUT_ROW_MAJ\nOR\nStorage of elements of blocks uses row major\nlayout.\nSPARSE_LAYOUT_COLUMN_\nMAJOR\nStorage of elements of blocks uses column\nmajor layout.\noperation\nSpecifies operation op() on input matrix.\nSPARSE_OPERATION_NON_\nTRANSPOSE\nNon-transpose, op(A) = A.\nSPARSE_OPERATION_TRAN\nSPOSE\nTranspose, op(A) = AT.\nSPARSE_OPERATION_CONJ\nUGATE_TRANSPOSE\nConjugate transpose, op(A) = AH.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n285\n\n\nOutput Parameters\ndest\nHandle containing internal data.\nReturn Values\nThe function returns a value indicating whether the operation was successful or not, and why.\nSPARSE_STATUS_SUCCESS\nThe operation was successful.\nSPARSE_STATUS_NOT_INITIALIZED\nThe routine encountered an empty handle or matrix array.\nSPARSE_STATUS_ALLOC_FAILED\nInternal memory allocation failed.\nSPARSE_STATUS_INVALID_VALUE\nThe input parameters contain an invalid value.\nSPARSE_STATUS_EXECUTION_FAILED\nExecution failed.\nSPARSE_STATUS_INTERNAL_ERROR\nAn error in algorithm implementation occurred.\nSPARSE_STATUS_NOT_SUPPORTED\nThe requested operation is not supported.\nmkl_sparse_?_export_csr\nExports CSR matrix from internal representation.\nSyntax\nsparse_status_t mkl_sparse_s_export_csr (const sparse_matrix_t source,\nsparse_index_base_t *indexing, MKL_INT *rows, MKL_INT *cols, MKL_INT **rows_start,\nMKL_INT **rows_end, MKL_INT **col_indx, float **values);\nsparse_status_t mkl_sparse_d_export_csr (const sparse_matrix_t source,\nsparse_index_base_t *indexing, MKL_INT *rows, MKL_INT *cols, MKL_INT **rows_start,\nMKL_INT **rows_end, MKL_INT **col_indx, double **values);\nsparse_status_t mkl_sparse_c_export_csr (const sparse_matrix_t source,\nsparse_index_base_t *indexing, MKL_INT *rows, MKL_INT *cols, MKL_INT **rows_start,\nMKL_INT **rows_end, MKL_INT **col_indx, MKL_Complex8 **values);\nsparse_status_t mkl_sparse_z_export_csr (const sparse_matrix_t source,\nsparse_index_base_t *indexing, MKL_INT *rows, MKL_INT *cols, MKL_INT **rows_start,\nMKL_INT **rows_end, MKL_INT **col_indx, MKL_Complex16 **values);\nInclude Files\n•\nmkl_spblas.h\nDescription\nIf the matrix specified by the source handle is in CSR format, the mkl_sparse_?_export_csr routine\nexports an m-by-k matrix A in CSR format matrix from the internal representation. The routine returns\npointers to the internal representation and does not allocate additional memory.\nIf the matrix is not already in CSR format, the routine returns SPARSE_STATUS_INVALID_VALUE.\nInput Parameters\nsource\nHandle containing internal data.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n286\n\n\nOutput Parameters\nindexing\nIndicates how input arrays are indexed.\nSPARSE_INDEX_BASE_ZER\nO\nZero-based (C-style) indexing: indices start at\n0.\nSPARSE_INDEX_BASE_ONE\nOne-based (Fortran-style) indexing: indices\nstart at 1.\nrows\nNumber of rows of the matrix source.\ncols\nNumber of columns of the matrix source.\nrows_start\nPointer to array of length m. This array contains row indices, such that\nrows_start[i] - ind is the first index of row i in the arrays values and\ncol_indx. ind takes 0 for zero-based indexing and 1 for one-based\nindexing.\nRefer to pointerB array description in CSR Format for more details.\nrows_end\nPointer to array of length m. This array contains row indices, such that\nrows_end[i] - ind - 1 is the last index of row i in the arrays values and\ncol_indx. ind takes 0 for zero-based indexing and 1 for one-based\nindexing.\nRefer to pointerE array description in CSR Format for more details.\ncol_indx\nFor one-based indexing, pointer to array containing the column indices plus\none for each non-zero element of the matrix source. For zero-based\nindexing, pointer to array containing the column indices for each non-zero\nelement of the matrix source. Its length is rows_end[rows - 1] - ind.\nind takes 0 for zero-based indexing and 1 for one-based indexing.\nvalues\nPointer to array containing non-zero elements of the matrix A. Its length is\nequal to length of the col_indx array.\nRefer to values array description in CSR Format for more details.\nReturn Values\nThe function returns a value indicating whether the operation was successful or not, and why.\nSPARSE_STATUS_SUCCESS\nThe operation was successful.\nSPARSE_STATUS_NOT_INITIALIZED\nThe routine encountered an empty handle or matrix array.\nSPARSE_STATUS_ALLOC_FAILED\nInternal memory allocation failed.\nSPARSE_STATUS_INVALID_VALUE\nThe input parameters contain an invalid value.\nSPARSE_STATUS_EXECUTION_FAILED\nExecution failed.\nSPARSE_STATUS_INTERNAL_ERROR\nAn error in algorithm implementation occurred.\nSPARSE_STATUS_NOT_SUPPORTED\nThe requested operation is not supported.\nmkl_sparse_?_export_csc\nExports CSC matrix from internal representation.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n287\n\n\nSyntax\nsparse_status_t mkl_sparse_s_export_csc (const sparse_matrix_t source,\nsparse_index_base_t *indexing, MKL_INT *rows, MKL_INT *cols, MKL_INT **cols_start,\nMKL_INT **cols_end, MKL_INT **row_indx, float **values);\nsparse_status_t mkl_sparse_d_export_csc (const sparse_matrix_t source,\nsparse_index_base_t *indexing, MKL_INT *rows, MKL_INT *cols, MKL_INT **cols_start,\nMKL_INT **cols_end, MKL_INT **row_indx, double **values);\nsparse_status_t mkl_sparse_c_export_csc (const sparse_matrix_t source,\nsparse_index_base_t *indexing, MKL_INT *rows, MKL_INT *cols, MKL_INT **cols_start,\nMKL_INT **cols_end, MKL_INT **row_indx, MKL_Complex8 **values);\nsparse_status_t mkl_sparse_z_export_csc (const sparse_matrix_t source,\nsparse_index_base_t *indexing, MKL_INT *rows, MKL_INT *cols, MKL_INT **cols_start,\nMKL_INT **cols_end, MKL_INT **row_indx, MKL_Complex16 **values);\nInclude Files\n•\nmkl_spblas.h\nDescription\nIf the matrix specified by the source handle is in CSC format, the mkl_sparse_?_export_csc routine\nexports an m-by-k matrix A in CSC format matrix from the internal representation. The routine returns\npointers to the internal representation and does not allocate additional memory.\nIf the matrix is not already in CSC format, the routine returns SPARSE_STATUS_INVALID_VALUE.\nInput Parameters\nsource\nHandle containing internal data.\nOutput Parameters\nindexing\nIndicates how input arrays are indexed.\nSPARSE_INDEX_BASE_ZER\nO\nZero-based (C-style) indexing: indices start at\n0.\nSPARSE_INDEX_BASE_ONE\nOne-based (Fortran-style) indexing: indices\nstart at 1.\nrows\nNumber of rows of the matrix source.\ncols\nNumber of columns of the matrix source.\ncols_start\nArray of length m. This array contains column indices, such that\ncols_start[i] - cols_start[0] is the first index of column i in the\narrays values and row_indx.\nRefer to pointerb array description in csc Format for more details.\ncols_end\nPointer to array of length m. This array contains row indices, such that\ncols_end[i] - cols_start[0] - 1 is the last index of column i in the\narrays values and row_indx.\nRefer to pointerE array description in csc Format for more details.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n288\n\n\nrow_indx\nFor one-based indexing, pointer to array containing the row indices plus one\nfor each non-zero element of the matrix source. For zero-based indexing,\npointer to array containing the row indices for each non-zero element of the\nmatrix source. Its length is cols_end[cols - 1] - cols_start[0].\nvalues\nPointer to array containing non-zero elements of the matrix A. Its length is\nequal to length of the row_indx array.\nRefer to values array description in csc Format for more details.\nReturn Values\nThe function returns a value indicating whether the operation was successful or not, and why.\nSPARSE_STATUS_SUCCESS\nThe operation was successful.\nSPARSE_STATUS_NOT_INITIALIZED\nThe routine encountered an empty handle or matrix array.\nSPARSE_STATUS_ALLOC_FAILED\nInternal memory allocation failed.\nSPARSE_STATUS_INVALID_VALUE\nThe input parameters contain an invalid value.\nSPARSE_STATUS_EXECUTION_FAILED\nExecution failed.\nSPARSE_STATUS_INTERNAL_ERROR\nAn error in algorithm implementation occurred.\nSPARSE_STATUS_NOT_SUPPORTED\nThe requested operation is not supported.\nmkl_sparse_?_export_bsr\nExports BSR matrix from internal representation.\nSyntax\nsparse_status_t mkl_sparse_s_export_bsr (const sparse_matrix_t source,\nsparse_index_base_t *indexing, sparse_layout_t *block_layout, MKL_INT *rows, MKL_INT\n*cols, MKL_INT *block_size, MKL_INT **rows_start, MKL_INT **rows_end, MKL_INT\n**col_indx, float **values);\nsparse_status_t mkl_sparse_d_export_bsr (const sparse_matrix_t source,\nsparse_index_base_t *indexing, sparse_layout_t *block_layout, MKL_INT *rows, MKL_INT\n*cols, MKL_INT *block_size, MKL_INT **rows_start, MKL_INT **rows_end, MKL_INT\n**col_indx, double **values);\nsparse_status_t mkl_sparse_c_export_bsr (const sparse_matrix_t source,\nsparse_index_base_t *indexing, sparse_layout_t *block_layout, MKL_INT *rows, MKL_INT\n*cols, MKL_INT *block_size, MKL_INT **rows_start, MKL_INT **rows_end, MKL_INT\n**col_indx, MKL_Complex8 **values);\nsparse_status_t mkl_sparse_z_export_bsr (const sparse_matrix_t source,\nsparse_index_base_t *indexing, sparse_layout_t *block_layout, MKL_INT *rows, MKL_INT\n*cols, MKL_INT *block_size, MKL_INT **rows_start, MKL_INT **rows_end, MKL_INT\n**col_indx, MKL_Complex16 **values);\nInclude Files\n•\nmkl_spblas.h\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n289\n\n\nDescription\nIf the matrix specified by the source handle is in BSR format, the mkl_sparse_?_export_bsr routine\nexports an (block_size * rows)-by-(block_size * cols) matrix A in BSR format from the internal\nrepresentation. The routine returns pointers to the internal representation and does not allocate additional\nmemory.\nIf the matrix is not already in BSR format, the routine returns SPARSE_STATUS_INVALID_VALUE.\nInput Parameters\nsource\nHandle containing internal data.\nOutput Parameters\nindexing\nIndicates how input arrays are indexed.\nSPARSE_INDEX_BASE_ZER\nO\nZero-based (C-style) indexing: indices start at\n0.\nSPARSE_INDEX_BASE_ONE\nOne-based (Fortran-style) indexing: indices\nstart at 1.\nblock_layout\nSpecifies layout of blocks:\nSPARSE_LAYOUT_ROW_MAJ\nOR\nStorage of elements of blocks uses row major\nlayout.\nSPARSE_LAYOUT_COLUMN_\nMAJOR\nStorage of elements of blocks uses column\nmajor layout.\nrows\nNumber of block rows of the matrix source.\ncols\nNumber of block columns of matrix source.\nblock_size\nSize of the square block in matrix source.\nrows_start\nPointer to array of length rows. This array contains row indices, such that\nrows_start[i] - ind is the first index of block row i in the arrays values\nand col_indx. ind takes 0 for zero-based indexing and 1 for one-based\nindexing.\nRefer to pointerB array description in BSR Format for more details.\nrows_end\nPointer to array of length rows. This array contains row indices, such that\nrows_end[i] - ind - 1 is the last index of block row i in the arrays values\nand col_indx. ind takes 0 for zero-based indexing and 1 for one-based\nindexing.\nRefer to pointerE array description in BSR Format for more details.\ncol_indx\nFor one-based indexing, pointer to array containing the column indices plus\none for each non-zero blocks of the matrix source. For zero-based indexing,\npointer to array containing the column indices for each non-zero blocks of\nthe matrix source. Its length is rows_end[rows - 1] - ind[0]. ind takes\n0 for zero-based indexing and 1 for one-based indexing.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n290\n\n\nvalues\nPointer to array containing non-zero elements of matrix source. Its length is\nequal to length of the col_indx array multiplied by\nblock_size*block_size.\nRefer to the values array description in BSR Format for more details.\nReturn Values\nThe function returns a value indicating whether the operation was successful or not, and why.\nSPARSE_STATUS_SUCCESS\nThe operation was successful.\nSPARSE_STATUS_NOT_INITIALIZED\nThe routine encountered an empty handle or matrix array.\nSPARSE_STATUS_ALLOC_FAILED\nInternal memory allocation failed.\nSPARSE_STATUS_INVALID_VALUE\nThe input parameters contain an invalid value.\nSPARSE_STATUS_EXECUTION_FAILED\nExecution failed.\nSPARSE_STATUS_INTERNAL_ERROR\nAn error in algorithm implementation occurred.\nSPARSE_STATUS_NOT_SUPPORTED\nThe requested operation is not supported.\nmkl_sparse_?_set_value\nChanges a single value of matrix in internal\nrepresentation.\nSyntax\nsparse_status_t mkl_sparse_s_set_value (const sparse_matrix_t A, const MKL_INT row,\nconst MKL_INT col, const float value);\nsparse_status_t mkl_sparse_d_set_value (const sparse_matrix_t A, const MKL_INT row,\nconst MKL_INT col, const double value);\nsparse_status_t mkl_sparse_c_set_value (const sparse_matrix_t A, const MKL_INT row,\nconst MKL_INT col, const MKL_Complex8 value);\nsparse_status_t mkl_sparse_z_set_value (const sparse_matrix_t A, const MKL_INT row,\nconst MKL_INT col, const MKL_Complex16 value);\nInclude Files\n•\nmkl_spblas.h\nDescription\nUse the mkl_sparse_?_set_value routine to change a single value of a matrix in the internal Inspector-\nexecutor Sparse BLAS format. The value should already be presented in a matrix structure.\nInput Parameters\nA\nSpecifies handle containing internal data.\nrow\nIndicates row of matrix in which to set value.\ncol\nIndicates column of matrix in which to set value.\nvalue\nIndicates value\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n291\n\n\nOutput Parameters\nA\nHandle containing modified internal data.\nReturn Values\nThe function returns a value indicating whether the operation was successful or not, and why.\nSPARSE_STATUS_SUCCESS\nThe operation was successful.\nSPARSE_STATUS_NOT_INITIALIZED\nThe routine encountered an empty handle or matrix array.\nSPARSE_STATUS_INVALID_VALUE\nThe input parameters contain an invalid value.\nSPARSE_STATUS_INTERNAL_ERROR\nAn error in algorithm implementation occurred.\nmkl_sparse_?_update_values\nChanges all or selected matrix values in internal\nrepresentation.\nSyntax\nNOTE\nThis routine is supported for sparse matrices in BSR format only.\nsparse_status_t mkl_sparse_s_update_values (sparse_matrix_t A, MKL_INT nvalues, MKL_INT\n*indx, MKL_INT *indy, float *values);\nsparse_status_t mkl_sparse_d_update_values (sparse_matrix_t A, MKL_INT nvalues, MKL_INT\n*indx, MKL_INT *indy, double *values);\nsparse_status_t mkl_sparse_c_update_values (sparse_matrix_t A, MKL_INT nvalues, MKL_INT\n*indx, MKL_INT *indy, MKL_Complex8 *values);\nsparse_status_t mkl_sparse_z_update_values (sparse_matrix_t A, MKL_INT nvalues, MKL_INT\n*indx, MKL_INT *indy, MKL_Complex16 *values);\nInclude Files\n•\nmkl_spblas.h\nDescription\nUse the mkl_sparse_?_update_values routine to change all or selected values of a matrix in the internal\nInspector-Executor Sparse BLAS format.\nThe values to be updated should already be present in the matrix structure.\n•\nTo change selected values, you must provide an array values (with new values) and also the\ncorresponding row and column indices for each value via indx and indy arrays as well as the overall\nnumber of changed elements nvalues.\nSo that, for example, to change A(0, 0) to 1 and A(0, 1) to 2, pass the following input parameters:\nnvalues = 2, indx = {0, 0}, indy = {0, 1} and values = {1, 2}.\n•\nTo change all the values in the matrix, provide the values array and explicitly set nvalues to 0 or the\nactual number of non zero elements. There is no need to supply indx and indy arrays.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n292\n\n\nInput Parameters\nA\nSpecifies handle containing internal data.\nnvalues\nTotal number of elements changed.\nindx\nRow indices for the new values.\nNOTE\nCurrently, only updating the full matrix is supported. Set indx\nand indy as NULL.\nindy\nColumn indices for the new values.\nNOTE\nCurrently, only updating the full matrix is supported. Set indx\nand indy as NULL.\nvalues\nNew values.\nOutput Parameters\nA\nHandle containing modified internal data.\nReturn Values\nThe function returns a value indicating whether the operation was successful or not, and why.\nSPARSE_STATUS_SUCCESS\nThe operation was successful.\nSPARSE_STATUS_NOT_INITIALIZED\nThe routine encountered an empty handle or matrix array.\nSPARSE_STATUS_ALLOC_FAILED\nInternal memory allocation failed.\nSPARSE_STATUS_INVALID_VALUE\nThe input parameters contain an invalid value.\nSPARSE_STATUS_EXECUTION_FAILED\nExecution failed.\nSPARSE_STATUS_INTERNAL_ERROR\nAn error in algorithm implementation occurred.\nSPARSE_STATUS_NOT_SUPPORTED\nThe requested operation is not supported.\nmkl_sparse_order\nPerforms ordering of column indexes of the matrix in\nCSR format\nSyntax\nsparse_status_t mkl_sparse_order (const sparse_matrix_t csrA);\nInclude Files\n•\nmkl_spblas.h\nDescription\nUse the mkl_sparse_order routine to perform ordering of column indexes of the matrix in CSR format.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n293\n\n\nInput Parameters\ncsrA\nCSR data\nOutput Parameters\ncsrA\nHandle containing modified internal data.\nReturn Values\nThe function returns a value indicating whether the operation was successful or not, and why.\nSPARSE_STATUS_SUCCESS\nThe operation was successful.\nSPARSE_STATUS_NOT_INITIALIZED\nThe routine encountered an empty handle or matrix array.\nSPARSE_STATUS_INVALID_VALUE\nThe input parameters contain an invalid value.\nSPARSE_STATUS_INTERNAL_ERROR\nAn error in algorithm implementation occurred.\nInspector-Executor Sparse BLAS Analysis Routines\nAnalysis Routines and Their Data Types\nRoutine or Function\nGroup\nDescription\nmkl_sparse_set_lu_smoot\nher_hint\nProvides and estimate of the number and type of upcoming calls to LU\nsmoother functionality.\nmkl_sparse_set_mv_hint\nProvides estimate of number and type of upcoming matrix-vector operations.\nmkl_sparse_set_sv_hint\nProvides estimate of number and type of upcoming triangular system solver\noperations.\nmkl_sparse_set_mm_hint\nProvides estimate of number and type of upcoming matrix-matrix\nmultiplication operations.\nmkl_sparse_set_sm_hint\nProvides estimate of number and type of upcoming triangular matrix solve\nwith multiple right hand sides operations.\nmkl_sparse_set_dotmv_h\nint\nSets estimate of the number and type of upcoming matrix-vector operations.\nmkl_sparse_set_symgs_h\nint\nSets estimate of number and type of upcoming mkl_sparse_?_symgs\noperations.\nmkl_sparse_set_sorv_hin\nt\nSets estimate of number and type of upcoming mkl_sparse_?_symgs\noperations.\nmkl_sparse_set_memory\n_hint\nProvides memory requirements for performance optimization purposes.\nmkl_sparse_optimize\nAnalyzes matrix structure and performs optimizations using the hints\nprovided in the handle.\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n294\n\n\nmkl_sparse_set_lu_smoother_hint\nProvides an estimate of the number and type of\nupcoming calls to LU smoother functionality.\nSyntax\nsparse_status_t mkl_sparse_set_lu_smoother_hint (sparse_matrix_t A, const\nsparse_operation_t operation, struct matrix_descr descr, MKL_INT expected_calls);\nInclude Files\n•\nmkl_spblas.h\nDescription\nThe mkl_sparse_set_lu_smoother_hint function provides subsequent Inspector-Executor Sparse BLAS\ncalls an estimate of the number of upcoming calls to the lu_smoother routine that ultimately may influence\nthe optimizations applied and specifies whether or not to perform an operation on the matrix.\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nInput Parameters\noperation\nSpecifies the operation op() on input matrix.\nSPARSE_OPERATION_NON_\nTRANSPOSE\nNon-transpose, op(A)= A.\nSPARSE_OPERATION_TRAN\nSPOSE\nTranspose, op(A)= AT.\nSPARSE_OPERATION_CONJ\nUGATE_TRANSPOSE\nConjugate transpose, op(A)= AH.\ndescr\nStructure specifying sparse matrix properties.\nsparse_matrix_type_ttype - Specifies the type of a sparse matrix:\nSPARSE_MATRIX_TYPE_GE\nNERAL\nThe matrix is processed as is.\nSPARSE_MATRIX_TYPE_SY\nMMETRIC\nThe matrix is symmetric (only the requested\ntriangle is processed).\nSPARSE_MATRIX_TYPE_HE\nRMITIAN\nThe matrix is Hermitian (only the requested\ntriangle is processed).\nSPARSE_MATRIX_TYPE_TR\nIANGULAR\nThe matrix is triangular (only the requested\ntriangle is processed).\nSPARSE_MATRIX_TYPE_DI\nAGONAL\nThe matrix is diagonal (only diagonal elements\nare processed).\nSPARSE_MATRIX_TYPE_BL\nOCK_TRIANGULAR\nThe matrix is block-triangular (only the\nrequested triangle is processed). Applies to BSR\nformat only.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n295\n\n\nSPARSE_MATRIX_TYPE_BL\nOCK_DIAGONAL\nThe matrix is block-diagonal (only diagonal\nblocks are processed). Applies to BSR format\nonly.\nsparse_fill_mode_tmode - Specifies the triangular matrix part for\nsymmetric, Hermitian, triangular, and block-triangular matrices:\nSPARSE_FILL_MODE_LOWE\nR\nThe lower triangular matrix part is processed.\nSPARSE_FILL_MODE_UPPE\nR\nThe upper triangular matrix part is processed.\nsparse_diag_type_tdiag - Specifies the diagonal type for non-general\nmatrices:\nSPARSE_DIAG_NON_UNIT\nDiagonal elements might not be equal to one.\nSPARSE_DIAG_UNIT\nDiagonal elements are equal to one.\nexpected_calls\nNumber of expected calls to execution routine.\nA\nHandle containing internal data.\nReturn Values\nThe function returns a value indicating whether the operation was successful or not, and why.\nSPARSE_STATUS_SUCCESS\nThe operation was successful.\nSPARSE_STATUS_NOT_INITIALIZED\nThe routine encountered an empty handle or matrix array.\nSPARSE_STATUS_ALLOC_FAILED\nInternal memory allocation failed.\nSPARSE_STATUS_INVALID_VALUE\nThe input parameters contain an invalid value.\nSPARSE_STATUS_EXECUTION_FAILED\nExecution failed.\nSPARSE_STATUS_INTERNAL_ERROR\nAn error in algorithm implementation occurred.\nSPARSE_STATUS_NOT_SUPPORTED\nThe requested operation is not supported.\nmkl_sparse_set_mv_hint\nProvides estimate of number and type of upcoming\nmatrix-vector operations.\nSyntax\nsparse_status_t mkl_sparse_set_mv_hint (const sparse_matrix_t A, const\nsparse_operation_t operation, const struct matrix_descr descr, const MKL_INT\nexpected_calls);\nInclude Files\n•\nmkl_spblas.h\nDescription\nUse the mkl_sparse_set_mv_hint routine to provide the Inspector-executor Sparse BLAS API an estimate\nof the number of upcoming matrix-vector multiplication operations for performance optimization, and specify\nwhether or not to perform an operation on the matrix.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n296\n\n\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nInput Parameters\noperation\nSpecifies operation op() on input matrix.\nSPARSE_OPERATION_NON_\nTRANSPOSE\nNon-transpose, op(A) = A.\nSPARSE_OPERATION_TRAN\nSPOSE\nTranspose, op(A) = AT.\nSPARSE_OPERATION_CONJ\nUGATE_TRANSPOSE\nConjugate transpose, op(A) = AH.\ndescr\nStructure specifying sparse matrix properties.\nsparse_matrix_type_t type - Specifies the type of a sparse matrix:\nSPARSE_MATRIX_TYPE_GE\nNERAL\nThe matrix is processed as is.\nSPARSE_MATRIX_TYPE_SY\nMMETRIC\nThe matrix is symmetric (only the requested\ntriangle is processed).\nSPARSE_MATRIX_TYPE_HE\nRMITIAN\nThe matrix is Hermitian (only the requested\ntriangle is processed).\nSPARSE_MATRIX_TYPE_TR\nIANGULAR\nThe matrix is triangular (only the requested\ntriangle is processed).\nSPARSE_MATRIX_TYPE_DI\nAGONAL\nThe matrix is diagonal (only diagonal elements\nare processed).\nSPARSE_MATRIX_TYPE_BL\nOCK_TRIANGULAR\nThe matrix is block-triangular (only requested\ntriangle is processed). Applies to BSR format\nonly.\nSPARSE_MATRIX_TYPE_BL\nOCK_DIAGONAL\nThe matrix is block-diagonal (only diagonal\nblocks are processed). Applies to BSR format\nonly.\nsparse_fill_mode_t mode - Specifies the triangular matrix part for\nsymmetric, Hermitian, triangular, and block-triangular matrices:\nSPARSE_FILL_MODE_LOWE\nR\nThe lower triangular matrix part is processed.\nSPARSE_FILL_MODE_UPPE\nR\nThe upper triangular matrix part is processed.\nsparse_diag_type_t diag - Specifies diagonal type for non-general\nmatrices:\nSPARSE_DIAG_NON_UNIT\nDiagonal elements might not be equal to one.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n297\n\n\nSPARSE_DIAG_UNIT\nDiagonal elements are equal to one.\nexpected_calls\nNumber of expected calls to execution routine.\nOutput Parameters\nA\nHandle containing internal data.\nReturn Values\nThe function returns a value indicating whether the operation was successful or not, and why.\nSPARSE_STATUS_SUCCESS\nThe operation was successful.\nSPARSE_STATUS_NOT_INITIALIZED\nThe routine encountered an empty handle or matrix array.\nSPARSE_STATUS_ALLOC_FAILED\nInternal memory allocation failed.\nSPARSE_STATUS_INVALID_VALUE\nThe input parameters contain an invalid value.\nSPARSE_STATUS_EXECUTION_FAILED\nExecution failed.\nSPARSE_STATUS_INTERNAL_ERROR\nAn error in algorithm implementation occurred.\nSPARSE_STATUS_NOT_SUPPORTED\nThe requested operation is not supported.\nmkl_sparse_set_sv_hint\nProvides estimate of number and type of upcoming\ntriangular system solver operations.\nSyntax\nsparse_status_t mkl_sparse_set_sv_hint (const sparse_matrix_t A, const\nsparse_operation_t operation, const struct matrix_descr descr, const MKL_INT\nexpected_calls);\nInclude Files\n•\nmkl_spblas.h\nDescription\nThe mkl_sparse_set_sv_hint routine provides an estimate of the number of upcoming triangular system\nsolver operations and type of these operations for performance optimization.\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nInput Parameters\noperation\nSpecifies operation op() on input matrix.\nSPARSE_OPERATION_NON_\nTRANSPOSE\nNon-transpose, op(A) = A.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n298\n\n\nSPARSE_OPERATION_TRAN\nSPOSE\nTranspose, op(A) = AT.\nSPARSE_OPERATION_CONJ\nUGATE_TRANSPOSE\nConjugate transpose, op(A) = AH.\ndescr\nStructure specifying sparse matrix properties.\nsparse_matrix_type_t type - Specifies the type of a sparse matrix:\nSPARSE_MATRIX_TYPE_GE\nNERAL\nThe matrix is processed as is.\nSPARSE_MATRIX_TYPE_SY\nMMETRIC\nThe matrix is symmetric (only the requested\ntriangle is processed).\nSPARSE_MATRIX_TYPE_HE\nRMITIAN\nThe matrix is Hermitian (only the requested\ntriangle is processed).\nSPARSE_MATRIX_TYPE_TR\nIANGULAR\nThe matrix is triangular (only the requested\ntriangle is processed).\nSPARSE_MATRIX_TYPE_DI\nAGONAL\nThe matrix is diagonal (only diagonal elements\nare processed).\nSPARSE_MATRIX_TYPE_BL\nOCK_TRIANGULAR\nThe matrix is block-triangular (only requested\ntriangle is processed). Applies to BSR format\nonly.\nSPARSE_MATRIX_TYPE_BL\nOCK_DIAGONAL\nThe matrix is block-diagonal (only diagonal\nblocks are processed). Applies to BSR format\nonly.\nsparse_fill_mode_t mode - Specifies the triangular matrix part for\nsymmetric, Hermitian, triangular, and block-triangular matrices:\nSPARSE_FILL_MODE_LOWE\nR\nThe lower triangular matrix part is processed.\nSPARSE_FILL_MODE_UPPE\nR\nThe upper triangular matrix part is processed.\nsparse_diag_type_t diag - Specifies diagonal type for non-general\nmatrices:\nSPARSE_DIAG_NON_UNIT\nDiagonal elements might not be equal to one.\nSPARSE_DIAG_UNIT\nDiagonal elements are equal to one.\nexpected_calls\nNumber of expected calls to execution routine.\nOutput Parameters\nA\nHandle containing internal data.\nReturn Values\nThe function returns a value indicating whether the operation was successful or not, and why.\nSPARSE_STATUS_SUCCESS\nThe operation was successful.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n299\n\n\nSPARSE_STATUS_NOT_INITIALIZED\nThe routine encountered an empty handle or matrix array.\nSPARSE_STATUS_ALLOC_FAILED\nInternal memory allocation failed.\nSPARSE_STATUS_INVALID_VALUE\nThe input parameters contain an invalid value.\nSPARSE_STATUS_EXECUTION_FAILED\nExecution failed.\nSPARSE_STATUS_INTERNAL_ERROR\nAn error in algorithm implementation occurred.\nSPARSE_STATUS_NOT_SUPPORTED\nThe requested operation is not supported.\nmkl_sparse_set_mm_hint\nProvides estimate of number and type of upcoming\nmatrix-matrix multiplication operations.\nSyntax\nsparse_status_t mkl_sparse_set_mm_hint (const sparse_matrix_t A, const\nsparse_operation_t operation, const struct matrix_descr descr, const sparse_layout_t\nlayout, const MKL_INT dense_matrix_size, const MKL_INT expected_calls);\nInclude Files\n•\nmkl_spblas.h\nDescription\nThe mkl_sparse_set_mm_hint routine provides an estimate of the number of upcoming matrix-matrix\nmultiplication operations and type of these operations for performance optimization purposes.\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nInput Parameters\noperation\nSpecifies operation op() on input matrix.\nSPARSE_OPERATION_NON_\nTRANSPOSE\nNon-transpose, op(A) = A.\nSPARSE_OPERATION_TRAN\nSPOSE\nTranspose, op(A) = AT.\nSPARSE_OPERATION_CONJ\nUGATE_TRANSPOSE\nConjugate transpose, op(A) = AH.\ndescr\nStructure specifying sparse matrix properties.\nsparse_matrix_type_t type - Specifies the type of a sparse matrix:\nSPARSE_MATRIX_TYPE_GE\nNERAL\nThe matrix is processed as is.\nSPARSE_MATRIX_TYPE_SY\nMMETRIC\nThe matrix is symmetric (only the requested\ntriangle is processed).\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n300\n\n\nSPARSE_MATRIX_TYPE_HE\nRMITIAN\nThe matrix is Hermitian (only the requested\ntriangle is processed).\nSPARSE_MATRIX_TYPE_TR\nIANGULAR\nThe matrix is triangular (only the requested\ntriangle is processed).\nSPARSE_MATRIX_TYPE_DI\nAGONAL\nThe matrix is diagonal (only diagonal elements\nare processed).\nSPARSE_MATRIX_TYPE_BL\nOCK_TRIANGULAR\nThe matrix is block-triangular (only requested\ntriangle is processed). Applies to BSR format\nonly.\nSPARSE_MATRIX_TYPE_BL\nOCK_DIAGONAL\nThe matrix is block-diagonal (only diagonal\nblocks are processed). Applies to BSR format\nonly.\nsparse_fill_mode_t mode - Specifies the triangular matrix part for\nsymmetric, Hermitian, triangular, and block-triangular matrices:\nSPARSE_FILL_MODE_LOWE\nR\nThe lower triangular matrix part is processed.\nSPARSE_FILL_MODE_UPPE\nR\nThe upper triangular matrix part is processed.\nsparse_diag_type_t diag - Specifies diagonal type for non-general\nmatrices:\nSPARSE_DIAG_NON_UNIT\nDiagonal elements might not be equal to one.\nSPARSE_DIAG_UNIT\nDiagonal elements are equal to one.\nlayout\nSpecifies layout of elements:\nSPARSE_LAYOUT_COLUMN_\nMAJOR\nStorage of elements uses column major layout.\nSPARSE_LAYOUT_ROW_MAJ\nOR\nStorage of elements uses row major layout.\ndense_matrix_size\nNumber of columns in dense matrix.\nexpected_calls\nNumber of expected calls to execution routine.\nOutput Parameters\nA\nHandle containing internal data.\nReturn Values\nThe function returns a value indicating whether the operation was successful or not, and why.\nSPARSE_STATUS_SUCCESS\nThe operation was successful.\nSPARSE_STATUS_NOT_INITIALIZED\nThe routine encountered an empty handle or matrix array.\nSPARSE_STATUS_ALLOC_FAILED\nInternal memory allocation failed.\nSPARSE_STATUS_INVALID_VALUE\nThe input parameters contain an invalid value.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n301\n\n\nSPARSE_STATUS_EXECUTION_FAILED\nExecution failed.\nSPARSE_STATUS_INTERNAL_ERROR\nAn error in algorithm implementation occurred.\nSPARSE_STATUS_NOT_SUPPORTED\nThe requested operation is not supported.\nmkl_sparse_set_sm_hint\nProvides estimate of number and type of upcoming\ntriangular matrix solve with multiple right hand sides\noperations.\nSyntax\nsparse_status_t mkl_sparse_set_sm_hint (const sparse_matrix_t A, const\nsparse_operation_t operation, const struct matrix_descr descr, const sparse_layout_t\nlayout, const MKL_INT dense_matrix_size, const MKL_INT expected_calls);\nInclude Files\n•\nmkl_spblas.h\nDescription\nThe mkl_sparse_set_sm_hint routine provides an estimate of the number of upcoming triangular matrix\nsolve with multiple right hand sides operations and type of these operations for performance optimization\npurposes.\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nInput Parameters\noperation\nSpecifies operation op() on input matrix.\nSPARSE_OPERATION_NON_\nTRANSPOSE\nNon-transpose, op(A) = A.\nSPARSE_OPERATION_TRAN\nSPOSE\nTranspose, op(A) = AT.\nSPARSE_OPERATION_CONJ\nUGATE_TRANSPOSE\nConjugate transpose, op(A) = AH.\ndescr\nStructure specifying sparse matrix properties.\nsparse_matrix_type_t type - Specifies the type of a sparse matrix:\nSPARSE_MATRIX_TYPE_GE\nNERAL\nThe matrix is processed as is.\nSPARSE_MATRIX_TYPE_SY\nMMETRIC\nThe matrix is symmetric (only the requested\ntriangle is processed).\nSPARSE_MATRIX_TYPE_HE\nRMITIAN\nThe matrix is Hermitian (only the requested\ntriangle is processed).\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n302\n\n\nSPARSE_MATRIX_TYPE_TR\nIANGULAR\nThe matrix is triangular (only the requested\ntriangle is processed).\nSPARSE_MATRIX_TYPE_DI\nAGONAL\nThe matrix is diagonal (only diagonal elements\nare processed).\nSPARSE_MATRIX_TYPE_BL\nOCK_TRIANGULAR\nThe matrix is block-triangular (only requested\ntriangle is processed). Applies to BSR format\nonly.\nSPARSE_MATRIX_TYPE_BL\nOCK_DIAGONAL\nThe matrix is block-diagonal (only diagonal\nblocks are processed). Applies to BSR format\nonly.\nsparse_fill_mode_t mode - Specifies the triangular matrix part for\nsymmetric, Hermitian, triangular, and block-triangular matrices:\nSPARSE_FILL_MODE_LOWE\nR\nThe lower triangular matrix part is processed.\nSPARSE_FILL_MODE_UPPE\nR\nThe upper triangular matrix part is processed.\nsparse_diag_type_t diag - Specifies diagonal type for non-general\nmatrices:\nSPARSE_DIAG_NON_UNIT\nDiagonal elements might not be equal to one.\nSPARSE_DIAG_UNIT\nDiagonal elements are equal to one.\nlayout\nSpecifies layout of elements:\nSPARSE_LAYOUT_COLUMN_\nMAJOR\nStorage of elements uses column major layout.\nSPARSE_LAYOUT_ROW_MAJ\nOR\nStorage of elements uses row major layout.\ndense_matrix_size\nNumber of right-hand-side.\nexpected_calls\nNumber of expected calls to execution routine.\nOutput Parameters\nA\nHandle containing internal data.\nReturn Values\nThe function returns a value indicating whether the operation was successful or not, and why.\nSPARSE_STATUS_SUCCESS\nThe operation was successful.\nSPARSE_STATUS_NOT_INITIALIZED\nThe routine encountered an empty handle or matrix array.\nSPARSE_STATUS_ALLOC_FAILED\nInternal memory allocation failed.\nSPARSE_STATUS_INVALID_VALUE\nThe input parameters contain an invalid value.\nSPARSE_STATUS_EXECUTION_FAILED\nExecution failed.\nSPARSE_STATUS_INTERNAL_ERROR\nAn error in algorithm implementation occurred.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n303\n\n\nSPARSE_STATUS_NOT_SUPPORTED\nThe requested operation is not supported.\nmkl_sparse_set_dotmv_hint\nSets estimate of the number and type of upcoming\nmatrix-vector operations.\nSyntax\nsparse_status_t mkl_sparse_set_dotmv_hint (const sparse_matrix_t A, const\nsparse_operation_t operation, const struct matrix_descr descr, const MKL_INT\nexpected_calls);\nInclude Files\n•\nmkl_spblas.h\nDescription\nUse the mkl_sparse_set_dotmv_hint routine to provide the Inspector-executor Sparse BLAS API an\nestimate of the number of upcoming matrix-vector multiplication operations for performance optimization,\nand specify whether or not to perform an operation on the matrix.\nInput Parameters\noperation\nSpecifies the operation performed on matrix A.\nIf operation = SPARSE_OPERATION_NON_TRANSPOSE, op(A) = A.\nIf operation = SPARSE_OPERATION_TRANSPOSE, op(A) = AT.\nIf operation = SPARSE_OPERATION_CONJUGATE_TRANSPOSE, op(A) = AH.\ndescr\nStructure specifying sparse matrix properties.\nsparse_matrix_type_t type - Specifies the type of a sparse matrix:\nSPARSE_MATRIX_TYPE_GE\nNERAL\nThe matrix is processed as is.\nSPARSE_MATRIX_TYPE_SY\nMMETRIC\nThe matrix is symmetric (only the requested\ntriangle is processed).\nSPARSE_MATRIX_TYPE_HE\nRMITIAN\nThe matrix is Hermitian (only the requested\ntriangle is processed).\nSPARSE_MATRIX_TYPE_TR\nIANGULAR\nThe matrix is triangular (only the requested\ntriangle is processed).\nSPARSE_MATRIX_TYPE_DI\nAGONAL\nThe matrix is diagonal (only diagonal elements\nare processed).\nSPARSE_MATRIX_TYPE_BL\nOCK_TRIANGULAR\nThe matrix is block-triangular (only requested\ntriangle is processed). Applies to BSR format\nonly.\nSPARSE_MATRIX_TYPE_BL\nOCK_DIAGONAL\nThe matrix is block-diagonal (only diagonal\nblocks are processed). Applies to BSR format\nonly.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n304\n\n\nsparse_fill_mode_t mode - Specifies the triangular matrix part for\nsymmetric, Hermitian, triangular, and block-triangular matrices:\nSPARSE_FILL_MODE_LOWE\nR\nThe lower triangular matrix part is processed.\nSPARSE_FILL_MODE_UPPE\nR\nThe upper triangular matrix part is processed.\nsparse_diag_type_t diag - Specifies diagonal type for non-general\nmatrices:\nSPARSE_DIAG_NON_UNIT\nDiagonal elements might not be equal to one.\nSPARSE_DIAG_UNIT\nDiagonal elements are equal to one.\nexpected_calls\nExpected number of calls to the execution routine.\nOutput Parameters\nA\nHandle containing internal data.\nReturn Values\nThe function returns a value indicating whether the operation was successful or not, and why.\nSPARSE_STATUS_SUCCESS\nThe operation was successful.\nSPARSE_STATUS_NOT_INITIALIZED\nThe routine encountered an empty handle or matrix array.\nSPARSE_STATUS_ALLOC_FAILED\nInternal memory allocation failed.\nSPARSE_STATUS_INVALID_VALUE\nThe input parameters contain an invalid value.\nSPARSE_STATUS_EXECUTION_FAILED\nExecution failed.\nSPARSE_STATUS_INTERNAL_ERROR\nAn error in algorithm implementation occurred.\nSPARSE_STATUS_NOT_SUPPORTED\nThe requested operation is not supported.\nmkl_sparse_set_symgs_hint\nSyntax\nSets estimate of number and type of upcoming mkl_sparse_?_symgs operations.\nsparse_status_t mkl_sparse_set_symgs_hint (const sparse_matrix_t A, const\nsparse_operation_t operation, const struct matrix_descr descr, const MKL_INT\nexpected_calls);\nInclude Files\n•\nmkl_spblas.h\nDescription\nUse the mkl_sparse_set_symgs_hint routine to provide the Inspector-executor Sparse BLAS API an\nestimate of the number of upcoming symmetric Gauss-Zeidel preconditioner operations for performance\noptimization, and specify whether or not to perform an operation on the matrix.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n305\n\n\nInput Parameters\noperation\nSpecifies the operation performed on matrix A.\nIf operation = SPARSE_OPERATION_NON_TRANSPOSE, op(A) = A.\nIf operation = SPARSE_OPERATION_TRANSPOSE, op(A) = AT.\nIf operation = SPARSE_OPERATION_CONJUGATE_TRANSPOSE, op(A) = AH.\ndescr\nStructure specifying sparse matrix properties.\nsparse_matrix_type_t type - Specifies the type of a sparse matrix:\nSPARSE_MATRIX_TYPE_GE\nNERAL\nThe matrix is processed as is.\nSPARSE_MATRIX_TYPE_SY\nMMETRIC\nThe matrix is symmetric (only the requested\ntriangle is processed).\nSPARSE_MATRIX_TYPE_HE\nRMITIAN\nThe matrix is Hermitian (only the requested\ntriangle is processed).\nSPARSE_MATRIX_TYPE_TR\nIANGULAR\nThe matrix is triangular (only the requested\ntriangle is processed).\nSPARSE_MATRIX_TYPE_DI\nAGONAL\nThe matrix is diagonal (only diagonal elements\nare processed).\nSPARSE_MATRIX_TYPE_BL\nOCK_TRIANGULAR\nThe matrix is block-triangular (only requested\ntriangle is processed). Applies to BSR format\nonly.\nSPARSE_MATRIX_TYPE_BL\nOCK_DIAGONAL\nThe matrix is block-diagonal (only diagonal\nblocks are processed). Applies to BSR format\nonly.\nsparse_fill_mode_t mode - Specifies the triangular matrix part for\nsymmetric, Hermitian, triangular, and block-triangular matrices:\nSPARSE_FILL_MODE_LOWE\nR\nThe lower triangular matrix part is processed.\nSPARSE_FILL_MODE_UPPE\nR\nThe upper triangular matrix part is processed.\nsparse_diag_type_t diag - Specifies diagonal type for non-general\nmatrices:\nSPARSE_DIAG_NON_UNIT\nDiagonal elements might not be equal to one.\nSPARSE_DIAG_UNIT\nDiagonal elements are equal to one.\ndiag\nSpecifies diagonal type for non-general matrices\nmode\nSpecifies the triangular matrix part for symmetric, Hermitian, triangular,\nand block-triangular matrices.\ntype\nSpecifies the type of a sparse matrix.\nexpected_calls\nEstimate of the number to the execution routine.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n306\n\n\nOutput Parameters\nA\nHandle containing internal data.\nReturn Values\nThe function returns a value indicating whether the operation was successful or not, and why.\nSPARSE_STATUS_SUCCESS\nThe operation was successful.\nSPARSE_STATUS_NOT_INITIALIZED\nThe routine encountered an empty handle or matrix array.\nSPARSE_STATUS_ALLOC_FAILED\nInternal memory allocation failed.\nSPARSE_STATUS_INVALID_VALUE\nThe input parameters contain an invalid value.\nSPARSE_STATUS_EXECUTION_FAILED\nExecution failed.\nSPARSE_STATUS_INTERNAL_ERROR\nAn error in algorithm implementation occurred.\nSPARSE_STATUS_NOT_SUPPORTED\nThe requested operation is not supported.\nmkl_sparse_set_sorv_hint\nSets an estimate of the number and type of upcoming\nmkl_sparse_?_sorv operations.\nSyntax\nsparse_status_t  mkl_sparse_set_sorv_hint(\n    const sparse_sor_type_t type,\n    const sparse_matrix_t A,\n    const struct matrix_descr descr,\n    const MKL_INT expected_calls\n);\n      \nInclude Files\n•\nmkl_spblas.h\nDescription\nUse the mkl_sparse_set_sorv_hint routine to provide the Inspector-Executor Sparse BLAS API an\nestimate of the number of upcoming forward/backward sweeps or symmetric SOR preconditioner operations\nfor performance optimization.\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nInput Parameters\ntype\nSpecifies the operation performed by the SORV preconditioner.\nSPARSE_SOR_FORWARD\nPerforms forward sweep as defined by:\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n307\n\n\nSPARSE_SOR_BACKWARD\nPerforms backward sweep as defined by:\nSPARSE_SOR_SYMMETRIC\nPreconditioner matrix could be expressed as:\ndescr\nStructure specifying sparse matrix properties.\nsparse_matrix_type_t\ntype\nSpecifies the type of a sparse matrix:\n•\nSPARSE_MATRIX_TYPE_GENERAL\nThe matrix is processed as-is.\n•\nSPARSE_MATRIX_TYPE_SYMMETRIC\nThe matrix is symmetric (only the requested\ntriangle is processed).\n•\nSPARSE_MATRIX_TYPE_HERMITIAN\nThe matrix is Hermitian (only the requested\ntriangle is processed).\n•\nSPARSE_MATRIX_TYPE_TRIANGULAR\nThe matrix is triangular (only the requested\ntriangle is processed).\n•\nSPARSE_MATRIX_TYPE_DIAGONAL\nThe matrix is diagonal (only diagonal\nelements are processed).\n•\nSPARSE_MATRIX_TYPE_BLOCK_TRIANGULAR\nThe matrix is block-triangular (only\nrequested triangle is processed). Applies to\nBSR format only.\n•\nSPARSE_MATRIX_TYPE_BLOCK_DIAGONAL\nThe matrix is block-diagonal (only diagonal\nblocks are processed). Applies to BSR format\nonly.\nsparse_fill_mode_t\nmode\nSpecifies the triangular matrix part for\nsymmetric, Hermitian, triangular, and block-\ntriangular matrices:\n•\nSPARSE_FILL_MODE_LOWER\nThe lower triangular matrix part is processed.\n•\nSPARSE_FILL_MODE_UPPER\nThe upper triangular matrix part is\nprocessed.\nsparse_diag_type_t\ndiag\nSpecifies diagonal type for non-general\nmatrices:\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n308\n\n\n•\nSPARSE_DIAG_NON_UNIT\nDiagonal elements might not be equal to\none.\n•\nSPARSE_DIAG_UNIT\nDiagonal elements are equal to one.\nA\nHandle containing internal data.\nexpected_calls\nEstimate of the number of calls to the execution routine.\nOutput Parameters\nA\nHandle containing internal data.\nReturn Values\nThe function returns a value indicating whether the operation was successful or not, and why.\nSPARSE_STATUS_SUCCESS\nThe operation was successful.\nSPARSE_STATUS_NOT_INITIALIZED\nThe routine encountered an empty handle or matrix array.\nSPARSE_STATUS_ALLOC_FAILED\nInternal memory allocation failed.\nSPARSE_STATUS_INVALID_VALUE\nThe input parameters contain an invalid value.\nSPARSE_STATUS_EXECUTION_FAILED\nExecution failed.\nSPARSE_STATUS_INTERNAL_ERROR\nAn error in algorithm implementation occurred.\nSPARSE_STATUS_NOT_SUPPORTED\nThe requested operation is not supported.\nmkl_sparse_set_memory_hint\nProvides memory requirements for performance\noptimization purposes.\nSyntax\nsparse_status_t mkl_sparse_set_memory_hint (const sparse_matrix_t A, const\nsparse_memory_usage_t policy);\nInclude Files\n•\nmkl_spblas.h\nDescription\nThe mkl_sparse_set_memory_hint routine allocates additional memory for further performance\noptimization purposes.\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n309\n\n\nInput Parameters\npolicy\nSpecify memory utilization policy for optimization routine using these types:\nSPARSE_MEMORY_NONE\nRoutine can allocate memory only for auxiliary\nstructures (such as for workload balancing); the\namount of memory is proportional to vector\nsize.\nSPARSE_MEMORY_AGGRESS\nIVE\nDefault.\nRoutine can allocate memory up to the size of\nmatrix A for converting into the appropriate\nsparse format.\nOutput Parameters\nA\nHandle containing internal data.\nReturn Values\nThe function returns a value indicating whether the operation was successful or not, and why.\nSPARSE_STATUS_SUCCESS\nThe operation was successful.\nSPARSE_STATUS_NOT_INITIALIZED\nThe routine encountered an empty handle or matrix array.\nSPARSE_STATUS_ALLOC_FAILED\nInternal memory allocation failed.\nSPARSE_STATUS_INVALID_VALUE\nThe input parameters contain an invalid value.\nSPARSE_STATUS_EXECUTION_FAILED\nExecution failed.\nSPARSE_STATUS_INTERNAL_ERROR\nAn error in algorithm implementation occurred.\nSPARSE_STATUS_NOT_SUPPORTED\nThe requested operation is not supported.\nmkl_sparse_optimize\nAnalyzes matrix structure and performs optimizations\nusing the hints provided in the handle.\nSyntax\nsparse_status_t mkl_sparse_optimize (sparse_matrix_t A);\nInclude Files\n•\nmkl_spblas.h\nDescription\nThe mkl_sparse_optimize routine analyzes matrix structure and performs optimizations using the hints\nprovided in the handle. Generally, specifying a higher number of expected operations allows for more\naggressive and time consuming optimizations.\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n310\n\n\nProduct and Performance Information\nNotice revision #20201201\nInput Parameters\nA\nHandle containing internal data.\nReturn Values\nThe function returns a value indicating whether the operation was successful or not, and why.\nSPARSE_STATUS_SUCCESS\nThe operation was successful.\nSPARSE_STATUS_NOT_INITIALIZED\nThe routine encountered an empty handle or matrix array.\nSPARSE_STATUS_ALLOC_FAILED\nInternal memory allocation failed.\nSPARSE_STATUS_INVALID_VALUE\nThe input parameters contain an invalid value.\nSPARSE_STATUS_EXECUTION_FAILED\nExecution failed.\nSPARSE_STATUS_INTERNAL_ERROR\nAn error in algorithm implementation occurred.\nSPARSE_STATUS_NOT_SUPPORTED\nThe requested operation is not supported.\nInspector-Executor Sparse BLAS Execution Routines\nExecution Routines and Their Data Types\nRoutine or\nFunction Group\nData Types\nDescription\nmkl_sparse_?\n_lu_smoother\ns, d, c, z\nComputes an action of a preconditioner which corresponds\nto the approximate matrix decomposition A ≈ (L+D)*E*(U\n+D) for the system Ax = b\nmkl_sparse_?_mv\ns, d, c, z\nComputes a sparse matrix-vector product.\nmkl_sparse_?_\ntrsv\ns, d, c, z\nSolves a system of linear equations for a square sparse\nmatrix.\nmkl_sparse_?_mm\ns, d, c, z\nComputes the product of a sparse matrix and a dense\nmatrix and stores the result as a dense matrix.\nmkl_sparse_?\n_trsm\ns, d, c, z\nSolves a system of linear equations with multiple right-\nhand sides for a square sparse matrix.\nmkl_sparse_?_add\ns, d, c, z\nComputes the sum of two sparse matrices. The result is\nstored in a newly allocated sparse matrix.\nmkl_sparse_spmm\ns, d, c, z\nComputes the product of two sparse matrices and stores\nthe result in a newly allocated sparse matrix.\nmkl_sparse_?\n_spmmd\ns, d, c, z\nComputes the product of two sparse matrices and stores\nthe result as a dense matrix.\nmkl_sparse_sp2m\ns, d, c, z\nComputes the product of two sparse matrices (support\noperations on both matrices) and stores the result in a\nnewly allocated sparse matrix.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n311\n\n\nRoutine or\nFunction Group\nData Types\nDescription\nmkl_sparse_?\n_sp2md\ns, d, c, z\nComputes the product of two sparse matrices (support\noperations on both matrices) and stores the result as a\ndense matrix.\nmkl_sparse_sypr\ns, d, c, z\nComputes the symmetric product of three sparse matrices\nand stores the result in a newly allocated sparse matrix.\nmkl_sparse_?\n_syprd\ns, d, c, z\nComputes the symmetric triple product of a sparse matrix\nand a dense matrix and stores the result as a dense\nmatrix.\nmkl_sparse_?\n_symgs\ns, d, c, z\nComputes an action of a symmetric Gauss-Seidel\npreconditioner.\nmkl_sparse_?\n_symgs_mv\ns, d, c, z\nComputes an action of a symmetric Gauss-Seidel\npreconditioner followed by a matrix-vector multiplication\nat the end.\nmkl_sparse_?\n_syrkd\ns, d, c, z\nComputes the product of sparse matrix with its transpose\n(or conjugate transpose) and stores the result as a dense\nmatrix.\nmkl_sparse_syrk\ns, d, c, z\nComputes the product of a sparse matrix with its\ntranspose (or conjugate transpose) and stores the result\nin a newly allocated sparse matrix.\nmkl_sparse_?\n_dotmv\ns, d, c, z\nComputes a sparse matrix-vector product followed by a\ndot product.\nmkl_sparse_?_lu_smoother\nComputes an action of a preconditioner which\ncorresponds to the approximate matrix decomposition\nA ≈\nL + D\n× E × U + D  for the system Ax = b (see\ndescription below).\nSyntax\nsparse_status_t mkl_sparse_s_lu_smoother (const sparse_operation_t op, const\nsparse_matrix_t A, const struct matrix descr descr, const float *diag, const float\n*approx_diag_inverse, float *x, const float *b);\nsparse_status_t mkl_sparse_d_lu_smoother (const sparse_operation_t op, const\nsparse_matrix_t A, const struct matrix descr descr, const double *diag, const double\n*approx_diag_inverse, double *x, const double *b);\nsparse_status_t mkl_sparse_c_lu_smoother (const sparse_operation_t op, const\nsparse_matrix_t A, const struct matrix descr descr, const MKL_COMPLEX8 *diag, const\nMKL_COMPLEX8 *approx_diag_inverse, MKL_COMPLEX8 *x, const MKL_COMPLEX8 *b);\nsparse_status_t mkl_sparse_z_lu_smoother (const sparse_operation_t op, const\nsparse_matrix_t A, const struct matrix descr descr, const MKL_COMPLEX16 *diag, const\nMKL_COMPLEX16 *approx_diag_inverse, MKL_COMPLEX16 *x, const MKL_COMPLEX16 *b);\nInclude Files\n•\nmkl_spblas.h\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n312\n\n\nDescription\nThis routine computes an update for an iterative solution x of the system Ax=b by means of applying one\niteration of an approximate preconditioner which is based on the following approximation:\nA\nL + D * E * U + D , where E is an approximate inverse of the diagonal (using exact inverse will result in\nGauss-Seidel preconditioner), L and U are lower/upper triangular parts of A, D is the diagonal (block diagonal\nin case of BSR format) of A.\nThe mkl_sparse_?_lu_smoother routine performs these operations:\nr = b - A*x    /* 1. Computes the residual */\n(L + D)*E*(U + D)*dx = r    /* 2. Finds the update dx by solving the system */\ny = x + dx    /* 3. Performs an update */\nThis is also equal to the Symmetric Gauss-Seidel operation in the case of a CSR format and 1x1 diagonal\nblocks:\n(L + D)*x^1 = b - U*x  /* Lower solve for intermediate x^1 */\n(U + D)*x = b - L*x^1  /* Upper solve */\nNOTE\nThis routine is supported only for non-transpose operation, real data types, and CSR/BSR\nsparse formats. In a BSR format, both diagonal values and approximate diagonal inverse\narrays should be passed explicitly. For CSR format, diagonal values should be passed\nexplicitly.\nInput Parameters\noperation\nSpecifies the operation performed on matrix A.\nSPARSE_OPERATION_NON_\nTRANSPOSE, op(A) := A\nNOTE\nTranspose and conjugate transpose\n(SPARSE_OPERATION_TRANSPOSE and\nSPARSE_OPERATION_CONJUGATE_TRANSPOSE)\nare not supported.\nNon-transpose, op(A)= A.\nA\nHandle which contains the sparse matrix A.\ndescr\nStructure specifying sparse matrix properties.\nsparse_matrix_type_ttype - Specifies the type of a sparse matrix:\nSPARSE_MATRIX_TYPE_GE\nNERAL\nThe matrix is processed as is.\nSPARSE_MATRIX_TYPE_SY\nMMETRIC\nThe matrix is symmetric (only the requested\ntriangle is processed).\nSPARSE_MATRIX_TYPE_HE\nRMITIAN\nThe matrix is Hermitian (only the requested\ntriangle is processed).\nSPARSE_MATRIX_TYPE_TR\nIANGULAR\nThe matrix is triangular (only the requested\ntriangle is processed).\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n313\n\n\nSPARSE_MATRIX_TYPE_DI\nAGONAL\nThe matrix is diagonal (only diagonal elements\nare processed).\nSPARSE_MATRIX_TYPE_BL\nOCK_TRIANGULAR\nThe matrix is block-triangular (only the\nrequested triangle is processed). Applies to BSR\nformat only.\nSPARSE_MATRIX_TYPE_BL\nOCK_DIAGONAL\nThe matrix is block-diagonal (only diagonal\nblocks are processed). Applies to BSR format\nonly.\nsparse_fill_mode_tmode - Specifies the triangular matrix part for\nsymmetric, Hermitian, triangular, and block-triangular matrices:\nSPARSE_FILL_MODE_LOWE\nR\nThe lower triangular matrix part is processed.\nSPARSE_FILL_MODE_UPPE\nR\nThe upper triangular matrix part is processed.\nsparse_diag_type_tdiag - Specifies the diagonal type for non-general\nmatrices:\nSPARSE_DIAG_NON_UNIT\nDiagonal elements might not be equal to one.\nSPARSE_DIAG_UNIT\nDiagonal elements are equal to one.\nNOTE\nOnly SPARSE_MATRIX_TYPE_GENERAL is supported.\ndiag\nArray of size at least m, where m is the number of rows (or nrows *\nblock_size * block_size in case of BSR format) of matrix A.\nThe array diag must contain the diagonal values of matrix A.\napprox_diag_inverse\nArray of size at least m, where m is the number of rows (or the number of\nrows * block_size * block_size in case of BSR format) of matrix A.\nThe array approx_diag_inverse will be used as E, approximate inverse of\nthe diagonal of the matrix A.\nx\nArray of size at least k, where k is the number of columns (or columns *\nblock_size in case of BSR format) of matrix A.\nOn entry, the array x must contain the input vector.\nb\nArray of size at least m, where m is the number of rows ( or rows *\nblock_size in case of BSR format ) of matrix A. The array b must contain\nthe values of the right-hand side of the system.\nOutput Parameters\nx\nOverwritten by the computed vector y.\nReturn Values\nThe function returns a value indicating whether the operation was successful or not, and why.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n314\n\n\nSPARSE_STATUS_SUCCESS\nThe operation was successful.\nSPARSE_STATUS_NOT_INITIALIZED\nThe routine encountered an empty handle or matrix array.\nSPARSE_STATUS_ALLOC_FAILED\nInternal memory allocation failed.\nSPARSE_STATUS_INVALID_VALUE\nThe input parameters contain an invalid value.\nSPARSE_STATUS_EXECUTION_FAILED\nExecution failed.\nSPARSE_STATUS_INTERNAL_ERROR\nAn error in algorithm implementation occurred.\nSPARSE_STATUS_NOT_SUPPORTED\nThe requested operation is not supported.\nmkl_sparse_?_mv\nComputes a sparse matrix- vector product.\nSyntax\nsparse_status_t mkl_sparse_s_mv (const sparse_operation_t operation, const float alpha, \nconst sparse_matrix_t A, const struct matrix_descr descr, const float *x, const float\nbeta, float *y);\nsparse_status_t mkl_sparse_d_mv (const sparse_operation_t operation, const double\nalpha, const sparse_matrix_t A, const struct matrix_descr descr, const double *x, const\ndouble beta, double *y);\nsparse_status_t mkl_sparse_c_mv (const sparse_operation_t operation, const MKL_Complex8 \nalpha, const sparse_matrix_t A, const struct matrix_descr descr, const MKL_Complex8 *x,\nconst MKL_Complex8 beta, MKL_Complex8 *y);\nsparse_status_t mkl_sparse_z_mv (const sparse_operation_t operation, const\nMKL_Complex16 alpha, const sparse_matrix_t A, const struct matrix_descr descr, const \nMKL_Complex16 *x, const MKL_Complex16 beta, MKL_Complex16 *y);\nInclude Files\n•\nmkl_spblas.h\nDescription\nThe mkl_sparse_?_mv routine computes a sparse matrix-dense vector product defined as\ny := alpha*op(A)*x + beta*y\nwhere:\nalpha and beta are scalars, x and y are vectors, and A is a sparse matrix handle of a matrix with m rows and\nk columns, and op is a matrix modifier for matrix A.\nInput Parameters\noperation\nSpecifies operation op() on input matrix.\nSPARSE_OPERATION_NON_\nTRANSPOSE\nNon-transpose, op(A) = A.\nSPARSE_OPERATION_TRAN\nSPOSE\nTranspose, op(A) = AT.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n315\n\n\nSPARSE_OPERATION_CONJ\nUGATE_TRANSPOSE\nConjugate transpose, op(A) = AH.\nalpha\nSpecifies the scalar alpha.\nA\nHandle which contains the input matrix A.\ndescr\nStructure specifying sparse matrix properties.\nsparse_matrix_type_t type - Specifies the type of a sparse matrix:\nSPARSE_MATRIX_TYPE_GE\nNERAL\nThe matrix is processed as is.\nSPARSE_MATRIX_TYPE_SY\nMMETRIC\nThe matrix is symmetric (only the requested\ntriangle is processed).\nSPARSE_MATRIX_TYPE_HE\nRMITIAN\nThe matrix is Hermitian (only the requested\ntriangle is processed).\nSPARSE_MATRIX_TYPE_TR\nIANGULAR\nThe matrix is triangular (only the requested\ntriangle is processed).\nSPARSE_MATRIX_TYPE_DI\nAGONAL\nThe matrix is diagonal (only diagonal elements\nare processed).\nSPARSE_MATRIX_TYPE_BL\nOCK_TRIANGULAR\nThe matrix is block-triangular (only requested\ntriangle is processed). Applies to BSR format\nonly.\nSPARSE_MATRIX_TYPE_BL\nOCK_DIAGONAL\nThe matrix is block-diagonal (only diagonal\nblocks are processed). Applies to BSR format\nonly.\nsparse_fill_mode_t mode - Specifies the triangular matrix part for\nsymmetric, Hermitian, triangular, and block-triangular matrices:\nSPARSE_FILL_MODE_LOWE\nR\nThe lower triangular matrix part is processed.\nSPARSE_FILL_MODE_UPPE\nR\nThe upper triangular matrix part is processed.\nsparse_diag_type_t diag - Specifies diagonal type for non-general\nmatrices:\nSPARSE_DIAG_NON_UNIT\nDiagonal elements might not be equal to one.\nSPARSE_DIAG_UNIT\nDiagonal elements are equal to one.\nx\nArray of size equal to the number of columns, k of A if operation =\nSPARSE_OPERATION_NON_TRANSPOSE and at least the number of rows, m,\nof A otherwise. On entry, the array must contain the vector x.\nbeta\nSpecifies the scalar beta.\ny\nArray with size at least m if\noperation=SPARSE_OPERATION_NON_TRANSPOSE and at least k otherwise.\nOn entry, the array y must contain the vector y. Array of size equal to the\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n316\n\n\nnumber of rows, m of A if operation =\nSPARSE_OPERATION_NON_TRANSPOSE and at least the number of columns,\nk, of A otherwise. On entry, the array y must contain the vector y.\nOutput Parameters\ny\nOverwritten by the updated vector y.\nReturn Values\nThe function returns a value indicating whether the operation was successful or not, and why.\nSPARSE_STATUS_SUCCESS\nThe operation was successful.\nSPARSE_STATUS_NOT_INITIALIZED\nThe routine encountered an empty handle or matrix array.\nSPARSE_STATUS_ALLOC_FAILED\nInternal memory allocation failed.\nSPARSE_STATUS_INVALID_VALUE\nThe input parameters contain an invalid value.\nSPARSE_STATUS_EXECUTION_FAILED\nExecution failed.\nSPARSE_STATUS_INTERNAL_ERROR\nAn error in algorithm implementation occurred.\nSPARSE_STATUS_NOT_SUPPORTED\nThe requested operation is not supported.\nmkl_sparse_?_trsv\nSolves a system of linear equations for a triangular\nsparse matrix.\nSyntax\nsparse_status_t mkl_sparse_s_trsv (const sparse_operation_t operation, const float\nalpha, const sparse_matrix_t A, const struct matrix_descr descr, const float *x, float\n*y);\nsparse_status_t mkl_sparse_d_trsv (const sparse_operation_t operation, const double\nalpha, const sparse_matrix_t A, const struct matrix_descr descr, const double *x,\ndouble *y);\nsparse_status_t mkl_sparse_c_trsv (const sparse_operation_t operation, const\nMKL_Complex8 alpha, const sparse_matrix_t A, const struct matrix_descr descr, const\nMKL_Complex8 *x, MKL_Complex8 *y);\nsparse_status_t mkl_sparse_z_trsv (const sparse_operation_t operation, const\nMKL_Complex16 alpha, const sparse_matrix_t A, const struct matrix_descr descr, const\nMKL_Complex16 *x, MKL_Complex16 *y);\nInclude Files\n•\nmkl_spblas.h\nDescription\nThe mkl_sparse_?_trsv routine solves a system of linear equations for a matrix:\nop(A)*y = alpha * x\nwhere A is a triangular sparse matrix , op is a matrix modifier for matrix A, alpha is a scalar, and x and y are\nvectors .\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n317\n\n\nNOTE\nFor sparse matrices in the BSR format, the supported combinations of\n(indexing,block_layout) are:\n•\n(SPARSE_INDEX_BASE_ZERO, SPARSE_LAYOUT_ROW_MAJOR)\n•\n(SPARSE_INDEX_BASE_ONE, SPARSE_LAYOUT_COLUMN_MAJOR)\nInput Parameters\noperation\nSpecifies operation op() on input matrix.\nSPARSE_OPERATION_NON_\nTRANSPOSE\nNon-transpose, op(A) = A.\nSPARSE_OPERATION_TRAN\nSPOSE\nTranspose, op(A) = AT.\nSPARSE_OPERATION_CONJ\nUGATE_TRANSPOSE\nConjugate transpose, op(A) = AH.\nalpha\nSpecifies the scalar alpha.\nA\nHandle which contains the input matrix A.\ndescr\nStructure specifying sparse matrix properties.\nsparse_matrix_type_t type - Specifies the type of a sparse matrix:\nSPARSE_MATRIX_TYPE_GE\nNERAL\nThe matrix is processed as is.\nSPARSE_MATRIX_TYPE_SY\nMMETRIC\nThe matrix is symmetric (only the requested\ntriangle is processed).\nSPARSE_MATRIX_TYPE_HE\nRMITIAN\nThe matrix is Hermitian (only the requested\ntriangle is processed).\nSPARSE_MATRIX_TYPE_TR\nIANGULAR\nThe matrix is triangular (only the requested\ntriangle is processed).\nSPARSE_MATRIX_TYPE_DI\nAGONAL\nThe matrix is diagonal (only diagonal elements\nare processed).\nSPARSE_MATRIX_TYPE_BL\nOCK_TRIANGULAR\nThe matrix is block-triangular (only requested\ntriangle is processed). Applies to BSR format\nonly.\nSPARSE_MATRIX_TYPE_BL\nOCK_DIAGONAL\nThe matrix is block-diagonal (only diagonal\nblocks are processed). Applies to BSR format\nonly.\nsparse_fill_mode_t mode - Specifies the triangular matrix part for\nsymmetric, Hermitian, triangular, and block-triangular matrices:\nSPARSE_FILL_MODE_LOWE\nR\nThe lower triangular matrix part is processed.\nSPARSE_FILL_MODE_UPPE\nR\nThe upper triangular matrix part is processed.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n318\n\n\nsparse_diag_type_t diag - Specifies diagonal type for non-general\nmatrices:\nSPARSE_DIAG_NON_UNIT\nDiagonal elements might not be equal to one.\nSPARSE_DIAG_UNIT\nDiagonal elements are equal to one.\nx\nArray of size at least m, where m is the number of rows of matrix A. On\nentry, the array must contain the vector x.\nOutput Parameters\ny\nArray of size at least m containing the solution to the system of linear\nequations.\nReturn Values\nThe function returns a value indicating whether the operation was successful or not, and why.\nSPARSE_STATUS_SUCCESS\nThe operation was successful.\nSPARSE_STATUS_NOT_INITIALIZED\nThe routine encountered an empty handle or matrix array.\nSPARSE_STATUS_ALLOC_FAILED\nInternal memory allocation failed.\nSPARSE_STATUS_INVALID_VALUE\nThe input parameters contain an invalid value.\nSPARSE_STATUS_EXECUTION_FAILED\nExecution failed.\nSPARSE_STATUS_INTERNAL_ERROR\nAn error in algorithm implementation occurred.\nSPARSE_STATUS_NOT_SUPPORTED\nThe requested operation is not supported.\nmkl_sparse_?_mm\nComputes the product of a sparse matrix and a dense\nmatrix and stores the result as a dense matrix.\nSyntax\nsparse_status_t mkl_sparse_s_mm (const sparse_operation_t operation, const float alpha,\nconst sparse_matrix_t A, const struct matrix_descr descr, const sparse_layout_t layout,\nconst float *B, const MKL_INT columns, const MKL_INT ldb, const float beta, float *C,\nconst MKL_INT ldc);\nsparse_status_t mkl_sparse_d_mm (const sparse_operation_t operation, const double \nalpha, const sparse_matrix_t A, const struct matrix_descr descr, const sparse_layout_t\nlayout, const double *B, const MKL_INT columns, const MKL_INT ldb, const double beta,\ndouble *C, const MKL_INT ldc);\nsparse_status_t mkl_sparse_c_mm (const sparse_operation_t operation, const MKL_Complex8 \nalpha, const sparse_matrix_t A, const struct matrix_descr descr, const sparse_layout_t\nlayout, const MKL_Complex8 *B, const MKL_INT columns, const MKL_INT ldb, const\nMKL_Complex8 beta, MKL_Complex8 *C, const MKL_INT ldc);\nsparse_status_t mkl_sparse_z_mm (const sparse_operation_t operation, const\nMKL_Complex16 alpha, const sparse_matrix_t A, const struct matrix_descr descr, const\nsparse_layout_t layout, const MKL_Complex16 *B, const MKL_INT columns, const MKL_INT\nldb, const MKL_Complex16 beta, MKL_Complex16 *C, const MKL_INT ldc);\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n319\n\n\nInclude Files\n•\nmkl_spblas.h\nDescription\nThe mkl_sparse_?_mm routine performs a matrix-matrix operation:\nC := alpha*op(A)*B + beta*C\nwhere alpha and beta are scalars, A is a sparse matrix, op is a matrix modifier for matrix A, and B and C are\ndense matrices.\nThe mkl_sparse_?_mm and mkl_sparse_?_trsm routines support these configurations:\nColumn-major dense matrix:\nlayout =\nSPARSE_LAYOUT_COLUMN_MAJOR\nRow-major dense matrix: layout\n= SPARSE_LAYOUT_ROW_MAJOR\n0-based sparse matrix:\nSPARSE_INDEX_BASE_ZERO\nCSR\nBSR: general non-transposed\nmatrix multiplication only\nAll formats\n1-based sparse matrix:\nSPARSE_INDEX_BASE_ONE\nAll formats\nCSR\nBSR: general non-transposed\nmatrix multiplication only\nNOTE\nFor sparse matrices in the BSR format, the supported combinations of\n(indexing,block_layout) are:\n•\n(SPARSE_INDEX_BASE_ZERO, SPARSE_LAYOUT_ROW_MAJOR )\n•\n(SPARSE_INDEX_BASE_ONE, SPARSE_LAYOUT_COLUMN_MAJOR )\nInput Parameters\noperation\nSpecifies operation op() on input matrix.\nSPARSE_OPERATION_NON_\nTRANSPOSE\nNon-transpose, op(A) = A.\nSPARSE_OPERATION_TRAN\nSPOSE\nTranspose, op(A) = AT.\nSPARSE_OPERATION_CONJ\nUGATE_TRANSPOSE\nConjugate transpose, op(A) = AH.\nalpha\nSpecifies the scalar alpha.\nA\nHandle which contains the sparse matrix A.\ndescr\nStructure specifying sparse matrix properties.\nsparse_matrix_type_t type - Specifies the type of a sparse matrix:\nSPARSE_MATRIX_TYPE_GE\nNERAL\nThe matrix is processed as is.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n320\n\n\nSPARSE_MATRIX_TYPE_SY\nMMETRIC\nThe matrix is symmetric (only the requested\ntriangle is processed).\nSPARSE_MATRIX_TYPE_HE\nRMITIAN\nThe matrix is Hermitian (only the requested\ntriangle is processed).\nSPARSE_MATRIX_TYPE_TR\nIANGULAR\nThe matrix is triangular (only the requested\ntriangle is processed).\nSPARSE_MATRIX_TYPE_DI\nAGONAL\nThe matrix is diagonal (only diagonal elements\nare processed).\nSPARSE_MATRIX_TYPE_BL\nOCK_TRIANGULAR\nThe matrix is block-triangular (only requested\ntriangle is processed). Applies to BSR format\nonly.\nSPARSE_MATRIX_TYPE_BL\nOCK_DIAGONAL\nThe matrix is block-diagonal (only diagonal\nblocks are processed). Applies to BSR format\nonly.\nsparse_fill_mode_t mode - Specifies the triangular matrix part for\nsymmetric, Hermitian, triangular, and block-triangular matrices:\nSPARSE_FILL_MODE_LOWE\nR\nThe lower triangular matrix part is processed.\nSPARSE_FILL_MODE_UPPE\nR\nThe upper triangular matrix part is processed.\nsparse_diag_type_t diag - Specifies diagonal type for non-general\nmatrices:\nSPARSE_DIAG_NON_UNIT\nDiagonal elements might not be equal to one.\nSPARSE_DIAG_UNIT\nDiagonal elements are equal to one.\nlayout\nDescribes the storage scheme for the dense matrix:\nSPARSE_LAYOUT_COLUMN_\nMAJOR\nStorage of elements uses column major layout.\nSPARSE_LAYOUT_ROW_MAJ\nOR\nStorage of elements uses row major layout.\nB\nArray of size at least rows*cols.\nlayout =\nSPARSE_LAYOUT_COLU\nMN_MAJOR\nlayout =\nSPARSE_LAYOUT_ROW_MA\nJOR\nrows (number of\nrows in B)\nldb\nIf op(A) = A, number\nof columns in A\nIf op(A) = AT, number\nof rows in A\ncols (number of\ncolumns in B)\ncolumns\nldb\ncolumns\nNumber of columns of matrix C.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n321\n\n\nldb\nSpecifies the leading dimension of matrix B.\nbeta\nSpecifies the scalar beta\nC\nArray of size at least rows*cols, where\nlayout =\nSPARSE_LAYOUT_COLU\nMN_MAJOR\nlayout =\nSPARSE_LAYOUT_ROW_MA\nJOR\nrows (number of\nrows in C)\nldc\nIf op(A) = A, number\nof rows in A\nIf op(A) = AT, number\nof columns in A\ncols (number of\ncolumns in C)\ncolumns\nldc\nldc\nSpecifies the leading dimension of matrix C.\nOutput Parameters\nC\nOverwritten by the updated matrix C.\nReturn Values\nThe function returns a value indicating whether the operation was successful or not, and why.\nSPARSE_STATUS_SUCCESS\nThe operation was successful.\nSPARSE_STATUS_NOT_INITIALIZED\nThe routine encountered an empty handle or matrix array.\nSPARSE_STATUS_ALLOC_FAILED\nInternal memory allocation failed.\nSPARSE_STATUS_INVALID_VALUE\nThe input parameters contain an invalid value.\nSPARSE_STATUS_EXECUTION_FAILED\nExecution failed.\nSPARSE_STATUS_INTERNAL_ERROR\nAn error in algorithm implementation occurred.\nSPARSE_STATUS_NOT_SUPPORTED\nThe requested operation is not supported.\nmkl_sparse_?_trsm\nSolves a system of linear equations with multiple right\nhand sides for a triangular sparse matrix.\nSyntax\nsparse_status_t mkl_sparse_s_trsm (const sparse_operation_t operation, const float\nalpha, const sparse_matrix_t A, const struct matrix_descr descr, const sparse_layout_t\nlayout, const float *x, const MKL_INT columns, const MKL_INT ldx, float *y, const\nMKL_INT ldy);\nsparse_status_t mkl_sparse_d_trsm (const sparse_operation_t operation, const double \nalpha, const sparse_matrix_t A, const struct matrix_descr descr, const sparse_layout_t\nlayout, const double *x, const MKL_INT columns, const MKL_INT ldx, double *y, const\nMKL_INT ldy);\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n322\n\n\nsparse_status_t mkl_sparse_c_trsm (const sparse_operation_t operation, const\nMKL_Complex8 alpha, const sparse_matrix_t A, const struct matrix_descr descr, const\nsparse_layout_t layout, const MKL_Complex8 *x, const MKL_INT columns, const MKL_INT\nldx, MKL_Complex8 *y, const MKL_INT ldy);\nsparse_status_t mkl_sparse_z_trsm (const sparse_operation_t operation, const\nMKL_Complex16 alpha, const sparse_matrix_t A, const struct matrix_descr descr, const\nsparse_layout_t layout, const MKL_Complex16 *x, const MKL_INT columns, const MKL_INT\nldx, MKL_Complex16 *y, const MKL_INT ldy);\nInclude Files\n•\nmkl_spblas.h\nDescription\nThe mkl_sparse_?_trsm routine solves a system of linear equations with multiple right hand sides for a\ntriangular sparse matrix:\nY := alpha*inv(op(A))*X\nwhere:\nalpha is a scalar, X and Y are dense matrices, A is a sparse matrix, and op is a matrix modifier for matrix A.\nThe mkl_sparse_?_mm and mkl_sparse_?_trsm routines support these configurations:\nColumn-major dense matrix:\nlayout =\nSPARSE_LAYOUT_COLUMN_MAJOR\nRow-major dense matrix: layout\n= SPARSE_LAYOUT_ROW_MAJOR\n0-based sparse matrix:\nSPARSE_INDEX_BASE_ZERO\nCSR\nBSR: general non-transposed\nmatrix multiplication only\nAll formats\n1-based sparse matrix:\nSPARSE_INDEX_BASE_ONE\nAll formats\nCSR\nBSR: general non-transposed\nmatrix multiplication only\nNOTE\nFor sparse matrices in the BSR format, the supported combinations of\n(indexing,block_layout) are:\n•\n(SPARSE_INDEX_BASE_ZERO, SPARSE_LAYOUT_ROW_MAJOR )\n•\n(SPARSE_INDEX_BASE_ONE, SPARSE_LAYOUT_COLUMN_MAJOR )\nInput Parameters\noperation\nSpecifies operation op() on input matrix.\nSPARSE_OPERATION_NON_\nTRANSPOSE\nNon-transpose, op(A) = A.\nSPARSE_OPERATION_TRAN\nSPOSE\nTranspose, op(A) = AT.\nSPARSE_OPERATION_CONJ\nUGATE_TRANSPOSE\nConjugate transpose, op(A) = AH.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n323\n\n\nalpha\nSpecifies the scalar alpha.\nA\nHandle which contains the sparse matrix A.\ndescr\nStructure specifying sparse matrix properties.\nsparse_matrix_type_t type - Specifies the type of a sparse matrix:\nSPARSE_MATRIX_TYPE_GE\nNERAL\nThe matrix is processed as is.\nSPARSE_MATRIX_TYPE_SY\nMMETRIC\nThe matrix is symmetric (only the requested\ntriangle is processed).\nSPARSE_MATRIX_TYPE_HE\nRMITIAN\nThe matrix is Hermitian (only the requested\ntriangle is processed).\nSPARSE_MATRIX_TYPE_TR\nIANGULAR\nThe matrix is triangular (only the requested\ntriangle is processed).\nSPARSE_MATRIX_TYPE_DI\nAGONAL\nThe matrix is diagonal (only diagonal elements\nare processed).\nSPARSE_MATRIX_TYPE_BL\nOCK_TRIANGULAR\nThe matrix is block-triangular (only requested\ntriangle is processed). Applies to BSR format\nonly.\nSPARSE_MATRIX_TYPE_BL\nOCK_DIAGONAL\nThe matrix is block-diagonal (only diagonal\nblocks are processed). Applies to BSR format\nonly.\nsparse_fill_mode_t mode - Specifies the triangular matrix part for\nsymmetric, Hermitian, triangular, and block-triangular matrices:\nSPARSE_FILL_MODE_LOWE\nR\nThe lower triangular matrix part is processed.\nSPARSE_FILL_MODE_UPPE\nR\nThe upper triangular matrix part is processed.\nsparse_diag_type_t diag - Specifies diagonal type for non-general\nmatrices:\nSPARSE_DIAG_NON_UNIT\nDiagonal elements might not be equal to one.\nSPARSE_DIAG_UNIT\nDiagonal elements are equal to one.\nlayout\nDescribes the storage scheme for the dense matrix:\nSPARSE_LAYOUT_COLUMN_\nMAJOR\nStorage of elements uses column major layout.\nSPARSE_LAYOUT_ROW_MAJ\nOR\nStorage of elements uses row major layout.\nx\nArray of size at least rows*cols.\nlayout =\nSPARSE_LAYOUT_COLU\nMN_MAJOR\nlayout =\nSPARSE_LAYOUT_ROW_MA\nJOR\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n324\n\n\nrows (number of\nrows in x)\nldx\nnumber of rows in A\ncols (number of\ncolumns in x)\ncolumns\nldx\nOn entry, the array x must contain the matrix X.\ncolumns\nNumber of columns in matrix Y.\nldx\nSpecifies the leading dimension of matrix X.\ny\nArray of size at least rows*cols, where\nlayout =\nSPARSE_LAYOUT_COLU\nMN_MAJOR\nlayout =\nSPARSE_LAYOUT_ROW_MA\nJOR\nrows (number of\nrows in y)\nldy\nnumber of rows in A\ncols (number of\ncolumns in y)\ncolumns\nldy\nOutput Parameters\ny\nOverwritten by the updated matrix Y.\nReturn Values\nThe function returns a value indicating whether the operation was successful or not, and why.\nSPARSE_STATUS_SUCCESS\nThe operation was successful.\nSPARSE_STATUS_NOT_INITIALIZED\nThe routine encountered an empty handle or matrix array.\nSPARSE_STATUS_ALLOC_FAILED\nInternal memory allocation failed.\nSPARSE_STATUS_INVALID_VALUE\nThe input parameters contain an invalid value.\nSPARSE_STATUS_EXECUTION_FAILED\nExecution failed.\nSPARSE_STATUS_INTERNAL_ERROR\nAn error in algorithm implementation occurred.\nSPARSE_STATUS_NOT_SUPPORTED\nThe requested operation is not supported.\nmkl_sparse_?_add\nComputes the sum of two sparse matrices. The result\nis stored in a newly allocated sparse matrix.\nSyntax\nsparse_status_t mkl_sparse_s_add (const sparse_operation_t operation, const\nsparse_matrix_t A, const float alpha, const sparse_matrix_t B, sparse_matrix_t *C);\nsparse_status_t mkl_sparse_d_add (const sparse_operation_t operation, const\nsparse_matrix_t A, const double alpha, const sparse_matrix_t B, sparse_matrix_t *C);\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n325\n\n\nsparse_status_t mkl_sparse_c_add (const sparse_operation_t operation, const\nsparse_matrix_t A, const MKL_Complex8 alpha, const sparse_matrix_t B, sparse_matrix_t \n*C);\nsparse_status_t mkl_sparse_z_add (const sparse_operation_t operation, const\nsparse_matrix_t A, const MKL_Complex16 alpha, const sparse_matrix_t B, sparse_matrix_t \n*C);\nInclude Files\n•\nmkl_spblas.h\nDescription\nThe mkl_sparse_?_add routine performs a matrix-matrix operation:\nC := alpha*op(A) + B\nwhere alpha is a scalar, op is a matrix modifier, and A, B, and C are sparse matrices.\nNOTE\nThis routine is only supported for sparse matrices in CSR and BSR formats. It is not\nsupported for COO or CSC formats.\nInput Parameters\nA\nHandle which contains the sparse matrix A.\nalpha\nSpecifies the scalar alpha.\noperation\nSpecifies operation op() on input matrix.\nSPARSE_OPERATION_NON_\nTRANSPOSE\nNon-transpose, op(A) = A.\nSPARSE_OPERATION_TRAN\nSPOSE\nTranspose, op(A) = AT.\nSPARSE_OPERATION_CONJ\nUGATE_TRANSPOSE\nConjugate transpose, op(A) = AH.\nB\nHandle which contains the sparse matrix B.\nOutput Parameters\nC\nHandle which contains the resulting sparse matrix.\nReturn Values\nThe function returns a value indicating whether the operation was successful or not, and why.\nSPARSE_STATUS_SUCCESS\nThe operation was successful.\nSPARSE_STATUS_NOT_INITIALIZED\nThe routine encountered an empty handle or matrix array.\nSPARSE_STATUS_ALLOC_FAILED\nInternal memory allocation failed.\nSPARSE_STATUS_INVALID_VALUE\nThe input parameters contain an invalid value.\nSPARSE_STATUS_EXECUTION_FAILED\nExecution failed.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n326\n\n\nSPARSE_STATUS_INTERNAL_ERROR\nAn error in algorithm implementation occurred.\nSPARSE_STATUS_NOT_SUPPORTED\nThe requested operation is not supported.\nmkl_sparse_spmm\nComputes the product of two sparse matrices. The\nresult is stored in a newly allocated sparse matrix.\nSyntax\nsparse_status_t mkl_sparse_spmm (const sparse_operation_t operation, const\nsparse_matrix_t A, const sparse_matrix_t B, sparse_matrix_t *C);\nInclude Files\n•\nmkl_spblas.h\nDescription\nThe mkl_sparse_spmm routine performs a matrix-matrix operation:\nC := op(A) *B\nwhere A, B, and C are sparse matrices and op is a matrix modifier for matrix A.\nNotes\n•\nThis routine is supported only for sparse matrices in CSC, CSR, and BSR formats. It is not\nsupported for sparse matrices in COO format.\n•\nThe column indices of the output matrix (if in CSR format) can appear unsorted due to the\nalgorithm chosen internally. To ensure sorted column indices (if that is important), call \nmkl_sparse_order().\nInput Parameters\noperation\nSpecifies operation op() on input matrix.\nSPARSE_OPERATION_NON_\nTRANSPOSE\nNon-transpose, op(A) = A.\nSPARSE_OPERATION_TRAN\nSPOSE\nTranspose, op(A) = AT.\nSPARSE_OPERATION_CONJ\nUGATE_TRANSPOSE\nConjugate transpose, op(A) = AH.\nA\nHandle which contains the sparse matrix A.\nB\nHandle which contains the sparse matrix B.\nOutput Parameters\nC\nHandle which contains the resulting sparse matrix.\nReturn Values\nThe function returns a value indicating whether the operation was successful or not, and why.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n327\n\n\nSPARSE_STATUS_SUCCESS\nThe operation was successful.\nSPARSE_STATUS_NOT_INITIALIZED\nThe routine encountered an empty handle or matrix array.\nSPARSE_STATUS_ALLOC_FAILED\nInternal memory allocation failed.\nSPARSE_STATUS_INVALID_VALUE\nThe input parameters contain an invalid value.\nSPARSE_STATUS_EXECUTION_FAILED\nExecution failed.\nSPARSE_STATUS_INTERNAL_ERROR\nAn error in algorithm implementation occurred.\nSPARSE_STATUS_NOT_SUPPORTED\nThe requested operation is not supported.\nmkl_sparse_?_spmmd\nComputes the product of two sparse matrices and\nstores the result as a dense matrix.\nSyntax\nsparse_status_t mkl_sparse_s_spmmd (const sparse_operation_t operation, const\nsparse_matrix_t A, const sparse_matrix_t B, const sparse_layout_t layout, float *C,\nconst MKL_INT ldc);\nsparse_status_t mkl_sparse_d_spmmd (const sparse_operation_t operation, const\nsparse_matrix_t A, const sparse_matrix_t B, const sparse_layout_t layout, double *C,\nconst MKL_INT ldc);\nsparse_status_t mkl_sparse_c_spmmd (const sparse_operation_t operation, const\nsparse_matrix_t A, const sparse_matrix_t B, const sparse_layout_t layout, MKL_Complex8\n*C, const MKL_INT ldc);\nsparse_status_t mkl_sparse_z_spmmd (const sparse_operation_t operation, const\nsparse_matrix_t A, const sparse_matrix_t B, const sparse_layout_t layout, MKL_Complex16\n*C, const MKL_INT ldc);\nInclude Files\n•\nmkl_spblas.h\nDescription\nThe mkl_sparse_?_spmmd routine performs a matrix-matrix operation:\nC := op(A)*B\nwhere A and B are sparse matrices, op is a matrix modifier for matrix A, and C is a dense matrix.\nNOTE\nThis routine is not supported for sparse matrices in the COO format. For sparse matrices in\nBSR format, these combinations of (indexing, block_layout) are supported:\n•\n(SPARSE_INDEX_BASE_ZERO, SPARSE_LAYOUT_ROW_MAJOR)\n•\n(SPARSE_INDEX_BASE_ONE, SPARSE_LAYOUT_COLUMN_MAJOR)\nInput Parameters\noperation\nSpecifies operation op() on input matrix.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n328\n\n\nSPARSE_OPERATION_NON_\nTRANSPOSE\nNon-transpose, op(A) = A.\nSPARSE_OPERATION_TRAN\nSPOSE\nTranspose, op(A) = AT.\nSPARSE_OPERATION_CONJ\nUGATE_TRANSPOSE\nConjugate transpose, op(A) = AH.\nA\nHandle which contains the sparse matrix A.\nB\nHandle which contains the sparse matrix B.\nlayout\nDescribes the storage scheme for the dense matrix:\nSPARSE_LAYOUT_COLUMN_\nMAJOR\nStorage of elements uses column major layout.\nSPARSE_LAYOUT_ROW_MAJ\nOR\nStorage of elements uses row major layout.\nldC\nLeading dimension of matrix C.\nOutput Parameters\nC\nResulting dense matrix.\nReturn Values\nThe function returns a value indicating whether the operation was successful or not, and why.\nSPARSE_STATUS_SUCCESS\nThe operation was successful.\nSPARSE_STATUS_NOT_INITIALIZED\nThe routine encountered an empty handle or matrix array.\nSPARSE_STATUS_ALLOC_FAILED\nInternal memory allocation failed.\nSPARSE_STATUS_INVALID_VALUE\nThe input parameters contain an invalid value.\nSPARSE_STATUS_EXECUTION_FAILED\nExecution failed.\nSPARSE_STATUS_INTERNAL_ERROR\nAn error in algorithm implementation occurred.\nSPARSE_STATUS_NOT_SUPPORTED\nThe requested operation is not supported.\nmkl_sparse_sp2m\nComputes the product of two sparse matrices. The\nresult is stored in a newly allocated sparse matrix.\nSyntax\nsparse_status_t mkl_sparse_sp2m (const sparse_operation_t transA, const struct\nmatrix_descr descrA, const sparse_matrix_t A, const sparse_operation_t transB, const\nstruct matrix_descr descrB, const sparse_matrix_t B, const sparse_request_t request,\nsparse_matrix_t *C);\nInclude Files\n•\nmkl_spblas.h\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n329\n\n\nDescription\nThe mkl_sparse_sp2m routine performs a matrix-matrix operation:\nC := opA(A) *opB(B)\nwhere A,B, and C are sparse matrices, opA and opB are matrix modifiers for matrices A and B, respectively.\nNOTE\nThe column indices of the output matrix (if in CSR format) can appear unsorted due to the\nalgorithm chosen internally. To ensure sorted column indices (if that is important), call \nmkl_sparse_order().\nInput Parameters\nopA\nSpecifies operation on input matrix.\nSPARSE_OPERATION_NON_TRANSPOSE\nNon-transpose, op(A)=A\nSPARSE_OPERATION_TRANSPOSE\nTranspose, op(A)=AT\nSPARSE_OPERATION_CONJUGATE_TRANSP\nOSE\nConjugate transpose,\nop(A)=AH\nopB\nSpecifies operation on input matrix.\nSPARSE_OPERATION_NON_TRANSPOSE\nNon-transpose, op(B)=B\nSPARSE_OPERATION_TRANSPOSE\nTranspose, op(B)=BT\nSPARSE_OPERATION_CONJUGATE_TRANSP\nOSE\nConjugate transpose,\nop(B)=BH\ndescrA\nStructure that specifies sparse matrix properties.\nNOTE Currently, only SPARSE_MATRIX_TYPE_GENERAL is\nsupported.\nsparse_matrix_type_ttype specifies the type of sparse matrix.\nSPARSE_MATRIX_TYPE_GENERAL\nThe matrix is processed as is.\nSPARSE_MATRIX_TYPE_SYMMETRIC\nThe matrix is symmetric (only the\nrequested triangle is processed).\nSPARSE_MATRIX_TYPE_HERMITIAN\nThe matrix is Hermitian (only the\nrequested triangle is processed).\nSPARSE_MATRIX_TYPE_TRIANGULA\nR\nThe matrix is triangular (only the\nrequested triangle is processed).\nSPARSE_MATRIX_TYPE_DIAGONAL\nThe matrix is diagonal (only\ndiagonal elements are processed).\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n330\n\n\nSPARSE_MATRIX_TYPE_BLOCK_TRI\nANGULAR\nThe matrix is block-triangular (only\nthe requested triangle is\nprocessed). This applies to BSR\nformat only.\nSPARSE_MATRIX_TYPE_BLOCK_DIA\nGONAL\nThe matrix is block-diagonal (only\nthe requested triangle is\nprocessed). This applies to BSR\nformat only.\nsparse_fill_mode_tmode specifies the triangular matrix portion for\nsymmetric, Hermitian, triangular, and block-triangular matrices.\nSPARSE_FILL_MODE_LOWER\nThe lower triangular matrix is\nprocessed.\nSPARSE_FILL_MODE_UPPER\nThe upper triangular matrix is\nprocessed.\nsparse_diag_type_tdiag specifies the type of diagonal for non-general\nmatrices.\nSPARSE_DIAG_NON_UNIT\nDiagonal elements must not be\nequal to 1.\nSPARSE_DIAG_UNIT\nDiagonal elements are equal to 1.\ndescrB\nStructure that specifies sparse matrix properties.\nNOTE Currently, only SPARSE_MATRIX_TYPE_GENERAL is\nsupported.\nsparse_matrix_type_ttype specifies the type of sparse matrix.\nSPARSE_MATRIX_TYPE_GENERAL\nThe matrix is processed as is.\nSPARSE_MATRIX_TYPE_SYMMETRIC\nThe matrix is symmetric (only the\nrequested triangle is processed).\nSPARSE_MATRIX_TYPE_HERMITIAN\nThe matrix is Hermitian (only the\nrequested triangle is processed).\nSPARSE_MATRIX_TYPE_TRIANGULA\nR\nThe matrix is triangular (only the\nrequested triangle is processed).\nSPARSE_MATRIX_TYPE_DIAGONAL\nThe matrix is diagonal (only\ndiagonal elements are processed).\nSPARSE_MATRIX_TYPE_BLOCK_TRI\nANGULAR\nThe matrix is block-triangular (only\nthe requested triangle is\nprocessed). This applies to BSR\nformat only.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n331\n\n\nSPARSE_MATRIX_TYPE_BLOCK_DIA\nGONAL\nThe matrix is block-diagonal (only\nthe requested triangle is\nprocessed). This applies to BSR\nformat only.\nsparse_fill_mode_tmode specifies the triangular matrix portion for\nsymmetric, Hermitian, triangular, and block-triangular matrices.\nSPARSE_FILL_MODE_LOWER\nThe lower triangular matrix is\nprocessed.\nSPARSE_FILL_MODE_UPPER\nThe upper triangular matrix is\nprocessed.\nsparse_diag_type_tdiag specifies the type of diagonal for non-general\nmatrices.\nSPARSE_DIAG_NON_UNIT\nDiagonal elements must not be\nequal to 1.\nSPARSE_DIAG_UNIT\nDiagonal elements are equal to 1.\nA\nHandle which contains the sparse matrix A.\nB\nHandle which contains the sparse matrix B.\nrequest\nSpecifies whether the full computations are performed at once or using the\ntwo-stage algorithm. See Two-stage Algorithm for Inspector-executor\nSparse BLAS Routines.\nSPARSE_STAGE_NNZ_COUNT\nOnly rowIndex (BSR/CSR format) or\ncolIndex (CSC format) array of the\nmatrix is computed internally. The\ncomputation can be extracted to\nmeasure the memory required for full\noperation.\nSPARSE_STAGE_FINALIZE_MULT_NO_\nVAL\nFinalize computations of the matrix\nstructure (values will not be\ncomputed). Use only after the call with\nSPARSE_STAGE_NNZ_COUNT\nparameter.\nSPARSE_STAGE_FINALIZE_MULT\nFinalize computation. Can also be used\nwhen the matrix structure remains\nunchanged and only values of the\nresulting matrix C need to be\nrecomputed.\nSPARSE_STAGE_FULL_MULT_NO_VAL\nPerform computations of the matrix\nstructure.\nSPARSE_STAGE_FULL_MULT\nPerform the entire computation in a\nsingle step.\nOutput Parameters\nC\nHandle which contains the resulting sparse matrix.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n332\n\n\nReturn Values\nThe function returns a value indicating whether the operation was successful, or the reason why it failed.\nSPARSE_STATUS_SUCCESS\nThe operation was successful.\nSPARSE_STATUS_NOT_INITIALIZED\nThe routine encountered an empty handle or matrix array.\nSPARSE_STATUS_ALLOC_FAILED\nThe internal memory allocation failed.\nSPARSE_STATUS_INVALID_VALUE\nThe input parameters contain an invalid value.\nSPARSE_STATUS_EXECUTION_FAILE\nD\nThe execution failed.\nSPARSE_STATUS_INTERNAL_ERROR\nAn error occurred in the implementation of the algorithm.\nSPARSE_STATUS_NOT_SUPPORTED\nThe requested operation is not supported.\nmkl_sparse_?_sp2md\nComputes the product of two sparse matrices (support\noperations on both matrices) and stores the result as\na dense matrix.\nSyntax\nsparse_status_t mkl_sparse_s_sp2md ( const sparse_operation_t transA, const struct\nmatrix_descr descrA, const sparse_matrix_t A, const sparse_operation_t transB, const\nstruct matrix_descr descrB, const sparse_matrix_t B, const float alpha, const float\nbeta, float *C, const sparse_layout_t layout, const MKL_INT ldc );\nsparse_status_t mkl_sparse_d_sp2md ( const sparse_operation_t transA, const struct\nmatrix_descr descrA, const sparse_matrix_t A, const sparse_operation_t transB, const\nstruct matrix_descr descrB, const sparse_matrix_t B, const double alpha, const double\nbeta, double *C, const sparse_layout_t layout, const MKL_INT ldc );\nsparse_status_t mkl_sparse_c_sp2md ( const sparse_operation_t transA, const struct\nmatrix_descr descrA, const sparse_matrix_t A, const sparse_operation_t transB, const\nstruct matrix_descr descrB, const sparse_matrix_t B, const MKL_Complex8 alpha, const\nMKL_Complex8 beta, MKL_Complex8 *C, const sparse_layout_t layout, const MKL_INT ldc );\nsparse_status_t mkl_sparse_z_sp2md ( const sparse_operation_t transA, const struct\nmatrix_descr descrA, const sparse_matrix_t A, const sparse_operation_t transB, const\nstruct matrix_descr descrB, const sparse_matrix_t B, const MKL_Complex16 alpha, const\nMKL_Complex16 beta, MKL_Complex16 *C, const sparse_layout_t layout, const MKL_INT\nldc );\nInclude Files\n•\nmkl_spblas.h\nDescription\nThe mkl_sparse_?_sp2md routine performs a matrix-matrix operation:\nC = alpha * opA(A) *opB(B) + beta*C\nwhere A and B are sparse matrices, opA is a matrix modifier for matrix A, opB is a matrix modifier for matrix\nB, and C is a dense matrix, alpha and beta are scalars.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n333\n\n\nNOTE\nThis routine is not supported for sparse matrices in the COO format. For sparse matrices in\nBSR format, these combinations of (indexing, block_layout) are supported:\n•\n(SPARSE_INDEX_BASE_ZERO, SPARSE_LAYOUT_ROW_MAJOR)\n•\n(SPARSE_INDEX_BASE_ONE, SPARSE_LAYOUT_COLUMN_MAJOR)\nInput Parameters\ntransA\nSpecifies operation op() on the input matrix.\nSPARSE_OPERATION_NON_TRANSPOSE\nNon-transpose, op(A)=A\nSPARSE_OPERATION_TRANSPOSE\nTranspose, op(A)=AT\nSPARSE_OPERATION_CONJUGATE_TRANSP\nOSE\nConjugate transpose,\nop(A)=AH\ndescrA\nStructure that specifies the sparse matrix properties.\nNOTE Currently, only SPARSE_MATRIX_TYPE_GENERAL is\nsupported.\nsparse_matrix_type_ttype specifies the type of sparse matrix.\nSPARSE_MATRIX_TYPE_GENERAL\nThe matrix is processed as is.\nSPARSE_MATRIX_TYPE_SYMMETRIC\nThe matrix is symmetric (only the\nrequested triangle is processed).\nSPARSE_MATRIX_TYPE_HERMITIAN\nThe matrix is Hermitian (only the\nrequested triangle is processed).\nSPARSE_MATRIX_TYPE_TRIANGULA\nR\nThe matrix is triangular (only the\nrequested triangle is processed).\nSPARSE_MATRIX_TYPE_DIAGONAL\nThe matrix is diagonal (only\ndiagonal elements are processed).\nSPARSE_MATRIX_TYPE_BLOCK_TRI\nANGULAR\nThe matrix is block-triangular (only\nthe requested triangle is\nprocessed). This applies to BSR\nformat only.\nSPARSE_MATRIX_TYPE_BLOCK_DIA\nGONAL\nThe matrix is block-diagonal (only\nthe requested triangle is\nprocessed). This applies to BSR\nformat only.\nsparse_fill_mode_tmode specifies the triangular matrix portion for\nsymmetric, Hermitian, triangular, and block-triangular matrices.\nSPARSE_FILL_MODE_LOWER\nThe lower triangular matrix is\nprocessed.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n334\n\n\nSPARSE_FILL_MODE_UPPER\nThe upper triangular matrix is\nprocessed.\nsparse_diag_type_tdiag specifies the type of diagonal for non-general\nmatrices.\nSPARSE_DIAG_NON_UNIT\nDiagonal elements must not be\nequal to 1.\nSPARSE_DIAG_UNIT\nDiagonal elements are equal to 1.\nA\nHandle which contains the sparse matrix A.\ntransB\nSpecifies operation opB() on the input matrix.\nSPARSE_OPERATION_NON_TRANSPO\nSE\nNon-transpose, opB(B)=B.\nSPARSE_OPERATION_TRANSPOSE\nTranspose, opB(B)=BT .\nSPARSE_OPERATION_CONJUGATE_T\nRANSPOSE\nConjugate transpose, opB(B)=BH .\ndescrB\nStructure that specifies the sparse matrix properties.\nNOTE\nCurrently, only SPARSE_MATRIX_TYPE_GENERAL is supported.\nsparse_matrix_type_ttype specifies the type of sparse matrix.\nSPARSE_MATRIX_TYPE_GENERAL\nThe matrix is processed as is.\nSPARSE_MATRIX_TYPE_SYMMETRIC\nThe matrix is symmetric (only the\nrequested triangle is processed).\nSPARSE_MATRIX_TYPE_HERMITIAN\nThe matrix is Hermitian (only the\nrequested triangle is processed).\nSPARSE_MATRIX_TYPE_TRIANGULA\nR\nThe matrix is triangular (only the\nrequested triangle is processed).\nSPARSE_MATRIX_TYPE_DIAGONAL\nThe matrix is diagonal (only\ndiagonal elements are processed).\nSPARSE_MATRIX_TYPE_BLOCK_TRI\nANGULAR\nThe matrix is block-triangular (only\nthe requested triangle is\nprocessed). This applies to BSR\nformat only.\nSPARSE_MATRIX_TYPE_BLOCK_DIA\nGONAL\nThe matrix is block-diagonal (only\nthe requested triangle is\nprocessed). This applies to BSR\nformat only.\nsparse_fill_mode_tmode specifies the triangular matrix portion for\nsymmetric, Hermitian, triangular, and block-triangular matrices.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n335\n\n\nSPARSE_FILL_MODE_LOWER\nThe lower triangular matrix is\nprocessed.\nSPARSE_FILL_MODE_UPPER\nThe upper triangular matrix is\nprocessed.\nsparse_diag_type_tdiag specifies the type of diagonal for non-general\nmatrices.\nSPARSE_DIAG_NON_UNIT\nDiagonal elements must not be\nequal to 1.\nSPARSE_DIAG_UNIT\nDiagonal elements are equal to 1.\nB\nHandle which contains the sparse matrix B.\nalpha\nSpecifies the scalar alpha.\nbeta\nSpecifies the scalar beta.\nlayout\nDescribes the storage scheme for the dense matrix:\nSPARSE_LAYOUT_COLUMN_MAJOR\nStorage of elements uses column\nmajor layout.\nSPARSE_LAYOUT_ROW_MAJOR\nStorage of elements uses row\nmajor layout.\nldc\nLeading dimension of matrix C.\nOutput Parameters\nC\nThe resulting dense matrix.\nReturn Values\nThe function returns a value indicating whether the operation was successful, or the reason why it failed.\nSPARSE_STATUS_SUCCESS\nThe operation was successful.\nSPARSE_STATUS_NOT_INITIALIZED\nThe routine encountered an empty handle or matrix array.\nSPARSE_STATUS_ALLOC_FAILED\nThe internal memory allocation failed.\nSPARSE_STATUS_INVALID_VALUE\nThe input parameters contain an invalid value.\nSPARSE_STATUS_EXECUTION_FAILE\nD\nThe execution failed.\nSPARSE_STATUS_INTERNAL_ERROR\nAn error occurred in the implementation of the algorithm.\nSPARSE_STATUS_NOT_SUPPORTED\nThe requested operation is not supported.\nmkl_sparse_sypr\nComputes the symmetric product of three sparse\nmatrices and stores the result in a newly allocated\nsparse matrix.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n336\n\n\nSyntax\nsparse_status_t mkl_sparse_sypr (const sparse_operation_t operation , const\nsparse_matrix_t A, const sparse_matrix_t B, const struct matrix_descr B,\nsparse_matrix_t *C, const sparse_request_t request);\nInclude Files\n•\nmkl_spblas.h\nDescription\nThe mkl_sparse_sypr routine performs a multiplication of three sparse matrices that results in a symmetric\nor Hermitian matrix, C.\nC:=A*B*opA(A)\nor\nC:=opA(A)*B*A\ndepending on the matrix modifier operation.\nHere, A, B, and C are sparse matrices, where A has a general structure while B and C are symmetric (for real\ndata types) or Hermitian (for complex data types) matrices. opA is the transpose (real data types) or\nconjugate transpose (complex data types) operator.\nNOTE\nThis routine is not supported for sparse matrices in COO or CSC formats. This routine\nsupports only CSR and BSR formats. In addition, it supports only the sorted CSR and sorted\nBSR formats for the input matrix. If the data is unsorted, call the mkl_sparse_order routine\nbefore either mkl_sparse_sypr or mkl_sparse_?_syprd.\nInput Parameters\noperation\nSpecifies operation on the input sparse matrices.\nSPARSE_OPERATION_NON_TRANSPOSE\nNon-transpose case.\nC:=A*B*(AT) for real\nprecision\nC:=A*B*(AH) for\ncomplex precision.\nSPARSE_OPERATION_TRANSPOSE\nTranspose case. This is\nnot supported for\ncomplex matrices.\nC:=(AT)*B*A\nSPARSE_OPERATION_CONJUGATE_TRAN\nSPOSE\nConjugate transpose\ncase. This is not\nsupported for real\nmatrices.\nC:=(AH)*B*A\nA\nHandle which contains the sparse matrix A.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n337\n\n\nB\nHandle which contains the sparse matrix B.\ndescrB\nStructure specifying properties of the sparse matrix.\nsparse_matrix_type_t type specifies the type of a sparse matrix\nSPARSE_MATRIX_TYPE_SYMMETRIC\nThe matrix is symmetric (only\nthe specified triangle is\nprocessed).\nSPARSE_MATRIX_TYPE_HERMITIAN\nThe matrix is Hermitian (only\nthe specified triangle is\nprocessed).\nsparse_fill_mode_t mode specifies the triangular matrix part.\nSPARSE_FILL_MODE_LOWER\nThe lower triangular matrix part\nis processed.\nSPARSE_FILL_MODE_UPPER\nThe upper triangular matrix part\nis processed.\nsparse_diag_type_t diag specifies the type of diagonal.\nSPARSE_DIAG_NON_UNIT\nDiagonal elements cannot be\nequal to one.\nNOTE\nThis routine also supports C=AAT,H with these parameters:\ndescrB.type=SPARSE_MATRIX_TYPE_DIAGONAL\ndescrB.diag=SPARSE_DIAG_UNIT\nIn this case, you do not need to allocate structure B. Use the\nroutine as a 2-stage version of mkl_sparse_syrk.\nrequest\nUse this routine to specify if the computations should be performed in\na single step or using the two-stage algorithm. See Two-stage\nAlgorithm for Inspector-executor Sparse BLAS Routines for more\ninformation.\nSPARSE_STAGE_NNZ_COUNT\nOnly rowIndex (BSR/CSR format)\nor colIndex (CSC format) array\nof the matrix is computed internally.\nThe computation can be extracted\nto measure the memory required\nfor full operation.\nSPARSE_STAGE_FINALIZE_MULT_N\nO_VAL\nFinalize computations of the matrix\nstructure (values will not be\ncomputed). Use only after the call\nwith SPARSE_STAGE_NNZ_COUNT\nparameter.\nSPARSE_STAGE_FINALIZE_MULT\nFinalize computation. Can be used\nafter the call with the\nSPARSE_STAGE_NNZ_COUNT or\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n338\n\n\nSPARSE_STAGE_FINALIZE_MULT_N\nO_VAL. Can also be used when the\nmatrix structure remains unchanged\nand only values of the resulting\nmatrix C need to be recomputed.\nSPARSE_STAGE_FULL_MULT_NO_V\nAL\nPerform computations of the matrix\nstructure.\nSPARSE_STAGE_FULL_MULT\nPerform the entire computation in a\nsingle step.\nOutput Parameters\nC\nHandle which contains the resulting sparse matrix. Only the upper-\ntriangular part of the matrix is computed.\nReturn Values\nThe function returns a value indicating whether the operation was successful, or the reason why it failed.\nSPARSE_STATUS_SUCCESS\nThe operation was successful.\nSPARSE_STATUS_NOT_INITIALIZED\nThe routine encountered an empty handle or matrix array.\nSPARSE_STATUS_ALLOC_FAILED\nInternal memory allocation failed.\nSPARSE_STATUS_INVALID_VALUE\nThe input parameters contain an invalid value.\nSPARSE_STATUS_EXECUTION_FAILED\nExecution failed.\nSPARSE_STATUS_INTERNAL_ERROR\nAn error in algorithm implementation occurred.\nSPARSE_STATUS_NOT_SUPPORTED\nThe requested operation is not supported.\nmkl_sparse_?_syprd\nComputes the symmetric triple product of a sparse\nmatrix and a dense matrix and stores the result as a\ndense matrix.\nSyntax\nsparse_status_t mkl_sparse_s_syprd (const sparse_operation_t op, const sparse_matrix_t\nA, const float *B, const sparse_layout_t layoutB, const MKL_INT ldb, const float alpha,\nconst float beta, float *C, const sparse_layout_t layoutC, const MKL_INT ldc);\nsparse_status_t mkl_sparse_d_syprd (const sparse_operation_t op, const sparse_matrix_t\nA, const double *B, const sparse_layout_t layoutB, const MKL_INT ldb, const double\nalpha, const double beta, double *C, const sparse_layout_t layoutC, const MKL_INT ldc);\nsparse_status_t mkl_sparse_c_syprd (const sparse_operation_t op, const sparse_matrix_t\nA, const MKL_Complex8 *B, const sparse_layout_t layoutB, const MKL_INT ldb, const\nMKL_Complex8 alpha, const MKL_Complex8 beta, MKL_Complex8 *C, const sparse_layout_t\nlayoutC, const MKL_INT ldc);\nsparse_status_t mkl_sparse_z_syprd (const sparse_operation_t op, const sparse_matrix_t\nA, const MKL_Complex16 *B, const sparse_layout_t layoutB, const MKL_INT ldb, const\nMKL_Complex16 alpha, const MKL_Complex16 beta, MKL_Complex16 *C, const sparse_layout_t\nlayoutC, const MKL_INT ldc);\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n339\n\n\nInclude Files\n•\nmkl_spblas.h\nDescription\nThe mkl_sparse_?_syprd routine performs a multiplication of three sparse matrices that results in a\nsymmetric or Hermitian matrix, C.\nC:=alpha*A*B*op(A) + beta*C\nor\nC:=alpha*op(A)*B*A + beta*C\ndepending on the matrix modifier operation. Here A is a sparse matrix, B and C are dense and symmetric\n(or Hermitian) matrices.\nop is the transpose (real precision) or conjugate transpose (complex precision) operator.\nNOTE\nThis routine is not supported for sparse matrices in COO or CSC formats. It supports only\nCSR and BSR formats. In addition, this routine supports only the sorted CSR and sorted BSR\nformats for the input matrix. If the data is unsorted, call the mkl_sparse_order routine\nbefore either mkl_sparse_sypr or mkl_sparse_?_syprd.\nInput Parameters\noperation\nSpecifies operation on the input sparse matrix.\nSPARSE_OPERATION_NON_TRANSPOSE\nNon-transpose case.\nC:=alpha*A*B*(AT)\n+beta*C for real\nprecision.\nC:=alpha*A*B*(AH)\n+beta*C for complex\nprecision.\nSPARSE_OPERATION_TRANSPOSE\nTranspose case. This is\nnot supported for\ncomplex matrices.\nC:=alpha*(AT)*B*A\n+beta*C\nSPARSE_OPERATION_CONJUGATE_TRAN\nSPOSE\nConjugate transpose\ncase. This is not\nsupported for real\nmatrices.\nC:=alpha*(AH)*B*A\n+beta*C\nA\nHandle which contains the sparse matrix A.\nB\nInput dense matrix. Only the upper triangular part of the matrix is\nused for computation.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n340\n\n\ndenselayoutB\nStructure that describes the storage scheme for the dense matrix.\nSPARSE_LAYOUT_COLUMN_MAJOR\nStore elements in a column-\nmajor layout.\nSPARSE_LAYOUT_ROW_MAJOR\nStore elements in a row-major\nlayout.\nldb\nLeading dimension of matrix B.\nalpha\nScalar parameter.\nbeta\nScalar parameter.\nNOTE\nSince the upper triangular part of matrix C is the only\nportion that is processed, set real values of alpha and beta\nin the complex case to obtain the Hermitian matrix.\ndenselayoutC\nStructure that describes the storage scheme for the dense matrix.\nSPARSE_LAYOUT_COLUMN_MAJOR\nStore elements in a column-\nmajor layout.\nSPARSE_LAYOUT_ROW_MAJOR\nStore elements in a row-major\nlayout.\nldc\nLeading dimension of matrix C.\nOutput Parameters\nC\nHandle which contains the resulting dense matrix. Only the upper-\ntriangular part of the matrix is computed.\nReturn Values\nThe function returns a value indicating whether the operation was successful, or the reason why it failed.\nSPARSE_STATUS_SUCCESS\nThe operation was successful.\nSPARSE_STATUS_NOT_INITIALIZED\nThe routine encountered an empty handle or matrix array.\nSPARSE_STATUS_ALLOC_FAILED\nThe internal memory allocation failed.\nSPARSE_STATUS_INVALID_VALUE\nThe input parameters contain an invalid value.\nSPARSE_STATUS_EXECUTION_FAILE\nD\nThe execution failed.\nSPARSE_STATUS_INTERNAL_ERROR\nAn error occurred in the implementation of the algorithm.\nSPARSE_STATUS_NOT_SUPPORTED\nThe requested operation is not supported.\nmkl_sparse_?_symgs\nComputes a symmetric Gauss-Seidel preconditioner.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n341\n\n\nSyntax\nsparse_status_t mkl_sparse_s_symgs (const sparse_operation_t operation, const\nsparse_matrix_t A, const struct matrix_descr descr, const float alpha, const float *b,\nfloat *x);\nsparse_status_t mkl_sparse_d_symgs (const sparse_operation_t operation, const\nsparse_matrix_t A, const struct matrix_descr descr, const double alpha, const double\n*b, double *x);\nsparse_status_t mkl_sparse_c_symgs (const sparse_operation_t operation, const\nsparse_matrix_t A, const struct matrix_descr descr, const MKL_Complex8 alpha, const\nMKL_Complex8 *b, MKL_Complex8 *x);\nsparse_status_t mkl_sparse_z_symgs (const sparse_operation_t operation, const\nsparse_matrix_t A, const struct matrix_descr descr, const MKL_Complex16 alpha, const\nMKL_Complex16 *b, MKL_Complex16 *x);\nInclude Files\n•\nmkl_spblas.h\nDescription\nThe mkl_sparse_?_symgs routine performs this operation:\nx0 := x*alpha;\n(L + D)*x1 = b - U*x0;\n(U + D)*x = b - L*x1;\nwhere A = L + D + U.\nNOTE\nThis routine is not supported for sparse matrices in BSR, COO, or CSC formats. It supports\nonly the CSR format. Additionally, only symmetric matrices are supported, so the desc.type\nmust be SPARSE_MATRIX_TYPE_SYMMETRIC.\nInput Parameters\noperation\nSpecifies the operation performed on matrix A.\nSPARSE_OPERATION_NON_TRANSPOSE, op(A) := A.\nNOTE\nTranspose (SPARSE_OPERATION_TRANSPOSE) and conjugate\ntranspose (SPARSE_OPERATION_CONJUGATE_TRANSPOSE) are not\nsupported.\nA\nHandle which contains the sparse matrix A.\nalpha\nSpecifies the scalar alpha.\ndescr\nStructure specifying sparse matrix properties.\nsparse_matrix_type_t type - Specifies the type of a sparse matrix:\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n342\n\n\nSPARSE_MATRIX_TYPE_GE\nNERAL\nThe matrix is processed as is.\nSPARSE_MATRIX_TYPE_SY\nMMETRIC\nThe matrix is symmetric (only the requested\ntriangle is processed).\nSPARSE_MATRIX_TYPE_HE\nRMITIAN\nThe matrix is Hermitian (only the requested\ntriangle is processed).\nSPARSE_MATRIX_TYPE_TR\nIANGULAR\nThe matrix is triangular (only the requested\ntriangle is processed).\nSPARSE_MATRIX_TYPE_DI\nAGONAL\nThe matrix is diagonal (only diagonal elements\nare processed).\nSPARSE_MATRIX_TYPE_BL\nOCK_TRIANGULAR\nThe matrix is block-triangular (only requested\ntriangle is processed). Applies to BSR format\nonly.\nSPARSE_MATRIX_TYPE_BL\nOCK_DIAGONAL\nThe matrix is block-diagonal (only diagonal\nblocks are processed). Applies to BSR format\nonly.\nsparse_fill_mode_t mode - Specifies the triangular matrix part for\nsymmetric, Hermitian, triangular, and block-triangular matrices:\nSPARSE_FILL_MODE_LOWE\nR\nThe lower triangular matrix part is processed.\nSPARSE_FILL_MODE_UPPE\nR\nThe upper triangular matrix part is processed.\nsparse_diag_type_t diag - Specifies diagonal type for non-general\nmatrices:\nSPARSE_DIAG_NON_UNIT\nDiagonal elements might not be equal to one.\nSPARSE_DIAG_UNIT\nDiagonal elements are equal to one.\nx\nArray of size at least m, where m is the number of rows of matrix A.\nOn entry, the array x must contain the vector x.\nb\nArray of size at least m, where m is the number of rows of matrix A.\nOn entry, the array b must contain the vector b.\nOutput Parameters\nx\nOverwritten by the computed vector x.\nReturn Values\nThe function returns a value indicating whether the operation was successful or not, and why.\nSPARSE_STATUS_SUCCESS\nThe operation was successful.\nSPARSE_STATUS_NOT_INITIALIZED\nThe routine encountered an empty handle or matrix array.\nSPARSE_STATUS_ALLOC_FAILED\nInternal memory allocation failed.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n343\n\n\nSPARSE_STATUS_INVALID_VALUE\nThe input parameters contain an invalid value.\nSPARSE_STATUS_EXECUTION_FAILED\nExecution failed.\nSPARSE_STATUS_INTERNAL_ERROR\nAn error in algorithm implementation occurred.\nSPARSE_STATUS_NOT_SUPPORTED\nThe requested operation is not supported.\nmkl_sparse_?_symgs_mv\nComputes a symmetric Gauss-Seidel preconditioner\nfollowed by a matrix-vector multiplication.\nSyntax\nsparse_status_t mkl_sparse_s_symgs_mv (const sparse_operation_t operation, const\nsparse_matrix_t A, const struct matrix_descr descr, const float alpha, const float *b,\nfloat *x, float *y);\nsparse_status_t mkl_sparse_d_symgs_mv (const sparse_operation_t operation, const\nsparse_matrix_t A, const struct matrix_descr descr, const double alpha, const double\n*b, double *x, double *y);\nsparse_status_t mkl_sparse_c_symgs_mv (const sparse_operation_t operation, const\nsparse_matrix_t A, const struct matrix_descr descr, const MKL_Complex8 alpha, const\nMKL_Complex8 *b, MKL_Complex8 *x, MKL_Complex8 *y);\nsparse_status_t mkl_sparse_z_symgs_mv (const sparse_operation_t operation, const\nsparse_matrix_t A, const struct matrix_descr descr, const MKL_Complex16 alpha, const\nMKL_Complex16 *b, MKL_Complex16 *x, MKL_Complex16 *y);\nInclude Files\n•\nmkl_spblas.h\nDescription\nThe mkl_sparse_?_symgs_mv routine performs this operation:\nx0 := x*alpha;\n(L + D)*x1 = b - U*x0;\n(U + D)*x = b - L*x1;\ny := A*x\nwhere A = L + D + U\nNOTE\nThis routine is not supported for sparse matrices in BSR, COO, or CSC formats. It supports\nonly the CSR format. Additionally, only symmetric matrices are supported, so the desc.type\nmust be SPARSE_MATRIX_TYPE_SYMMETRIC.\nInput Parameters\noperation\nSpecifies the operation performed on input matrix.\nSPARSE_OPERATION_NON_TRANSPOSE, op(A) = A.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n344\n\n\nNOTE\nTranspose (SPARSE_OPERATION_TRANSPOSE) and conjugate\ntranspose (SPARSE_OPERATION_CONJUGATE_TRANSPOSE) are not\nsupported.\nA\nHandle which contains the sparse matrix A.\nalpha\nSpecifies the scalar alpha.\ndescr\nStructure specifying sparse matrix properties.\nsparse_matrix_type_t type - Specifies the type of a sparse matrix:\nSPARSE_MATRIX_TYPE_GE\nNERAL\nThe matrix is processed as is.\nSPARSE_MATRIX_TYPE_SY\nMMETRIC\nThe matrix is symmetric (only the requested\ntriangle is processed).\nSPARSE_MATRIX_TYPE_HE\nRMITIAN\nThe matrix is Hermitian (only the requested\ntriangle is processed).\nSPARSE_MATRIX_TYPE_TR\nIANGULAR\nThe matrix is triangular (only the requested\ntriangle is processed).\nSPARSE_MATRIX_TYPE_DI\nAGONAL\nThe matrix is diagonal (only diagonal elements\nare processed).\nSPARSE_MATRIX_TYPE_BL\nOCK_TRIANGULAR\nThe matrix is block-triangular (only requested\ntriangle is processed). Applies to BSR format\nonly.\nSPARSE_MATRIX_TYPE_BL\nOCK_DIAGONAL\nThe matrix is block-diagonal (only diagonal\nblocks are processed). Applies to BSR format\nonly.\nsparse_fill_mode_t mode - Specifies the triangular matrix part for\nsymmetric, Hermitian, triangular, and block-triangular matrices:\nSPARSE_FILL_MODE_LOWE\nR\nThe lower triangular matrix part is processed.\nSPARSE_FILL_MODE_UPPE\nR\nThe upper triangular matrix part is processed.\nsparse_diag_type_t diag - Specifies diagonal type for non-general\nmatrices:\nSPARSE_DIAG_NON_UNIT\nDiagonal elements might not be equal to one.\nSPARSE_DIAG_UNIT\nDiagonal elements are equal to one.\nx\nArray of size at least m, where m is the number of rows of matrix A.\nOn entry, the array x must contain the vector x.\nb\nArray of size at least m, where m is the number of rows of matrix A.\nOn entry, the array b must contain the vector b.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n345\n\n\nOutput Parameters\nx\nOverwritten by the computed vector x.\ny\nArray of size at least m, where m is the number of rows of matrix A.\nOverwritten by the computed vector y.\nReturn Values\nThe function returns a value indicating whether the operation was successful or not, and why.\nSPARSE_STATUS_SUCCESS\nThe operation was successful.\nSPARSE_STATUS_NOT_INITIALIZED\nThe routine encountered an empty handle or matrix array.\nSPARSE_STATUS_ALLOC_FAILED\nInternal memory allocation failed.\nSPARSE_STATUS_INVALID_VALUE\nThe input parameters contain an invalid value.\nSPARSE_STATUS_EXECUTION_FAILED\nExecution failed.\nSPARSE_STATUS_INTERNAL_ERROR\nAn error in algorithm implementation occurred.\nSPARSE_STATUS_NOT_SUPPORTED\nThe requested operation is not supported.\nmkl_sparse_syrk\nComputes the product of sparse matrix with its\ntranspose (or conjugate transpose) and stores the\nresult in a newly allocated sparse matrix.\nSyntax\nsparse_status_t mkl_sparse_syrk (const sparse_operation_t operation, const\nsparse_matrix_t A, sparse_matrix_t *C);\nInclude Files\n•\nmkl_spblas.h\nDescription\nThe mkl_sparse_syrk routine performs a sparse matrix-matrix operation which results in a sparse matrix C\nthat is either Symmetric (real) or Hermitian (complex):\nC := A*op(A)\nwhere op(*) is the transpose for real matrices and conjugate transpose for complex matrices OR\nC := op(A)*A\ndepending on the matrix modifier op which can be the transpose for real matrices or conjugate transpose for\ncomplex matrices.\nHere, A and C are sparse matrices.\nNOTE This routine is not supported for sparse matrices in COO or CSC formats. It supports\nonly CSR and BSR formats. Additionally, this routine supports only the sorted CSR and\nsorted BSR formats for the input matrix. If data is unsorted, call the mkl_sparse_order\nroutine before either mkl_sparse_syrk or mkl_sparse_?_syrkd.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n346\n\n\nInput Parameters\noperation\nSpecifies the operation op() on input matrix .\nSPARSE_OPERATION_NON_TRANSPOSE, Non-transpose,C := A*op(A) where\nop(*) is the transpose for real matrices and conjugate transpose for\ncomplex matrices\nSPARSE_OPERATION_TRANSPOSE, Transpose,C := (AT)*Afor real matrix A\nSPARSE_OPERATION_CONJUGATE_TRANSPOSE, Conjugate transpose, C :=\n(AH)*A for complex matrix A.\nA\nHandle which contains the sparse matrix A.\nOutput Parameters\nC\nHandle which contains the resulting sparse matrix. Only the upper-\ntriangular part of the matrix is computed.\nReturn Values\nThe function returns a value indicating whether the operation was successful or not, and why.\nSPARSE_STATUS_SUCCESS\nThe operation was successful.\nSPARSE_STATUS_NOT_INITIALIZED\nThe routine encountered an empty handle or matrix array.\nSPARSE_STATUS_ALLOC_FAILED\nInternal memory allocation failed.\nSPARSE_STATUS_INVALID_VALUE\nThe input parameters contain an invalid value.\nSPARSE_STATUS_EXECUTION_FAILED\nExecution failed.\nSPARSE_STATUS_INTERNAL_ERROR\nAn error in algorithm implementation occurred.\nSPARSE_STATUS_NOT_SUPPORTED\nThe requested operation is not supported.\nmkl_sparse_?_syrkd\nComputes the product of sparse matrix with its\ntranspose (or conjugate transpose) and stores the\nresult as a dense matrix.\nSyntax\nsparse_status_t mkl_sparse_s_syrkd (sparse_operation_t operation, const sparse_matrix_t\nA, float alpha, float beta, float *C, sparse_layout_t layout, MKL_INT ldc);\nsparse_status_t mkl_sparse_d_syrkd (sparse_operation_t operation, const sparse_matrix_t\nA, double alpha, double beta, double *C, sparse_layout_t layout, MKL_INT ldc);\nsparse_status_t mkl_sparse_c_syrkd (sparse_operation_t operation, const sparse_matrix_t\nA, const MKL_Complex8 alpha, MKL_Complex8 beta, MKL_Complex8 *C, sparse_layout_t\nlayout, MKL_INT ldc);\nsparse_status_t mkl_sparse_z_syrkd (sparse_operation_t operation, const sparse_matrix_t\nA, MKL_Complex16 alpha, MKL_Complex16 beta, MKL_Complex16 *C, sparse_layout_t layout,\nconst MKL_INT ldc);\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n347\n\n\nInclude Files\n•\nmkl_spblas.h\nDescription\nThe mkl_sparse_?_syrkd routine performs a sparse matrix-matrix operation which results in a dense\nmatrix C that is either symmetric (real case) or Hermitian (complex case):\nC := beta*C + alpha*A*op(A)\nor\nC := beta*C + alpha*op(A)*A\ndepending on the matrix modifier op which can be the transpose for real matrices or conjugate transpose for\ncomplex matrices. Here, A is a sparse matrix and C is a dense matrix.\nNOTE This routine is not supported for sparse matrices in COO or CSC formats. It supports\nonly CSR and BSR formats. Additionally, this routine supports only the sorted CSR and\nsorted BSR formats for the input matrix. If data is unsorted, call the mkl_sparse_order\nroutine before either mkl_sparse_syrk or mkl_sparse_?_syrkd.\nInput Parameters\noperation\nSpecifies the operation op() performed on the input matrix.\nSPARSE_OPERATION_NON_TRANSPOSE, Non-transpose, C := beta*C +\nalpha*A*op(A) where op(*) is the transpose (real matrices) or conjugate\ntranspose (complex matrices).\nSPARSE_OPERATION_TRANSPOSE, Transpose,C := beta*C + alpha*AT*A\nfor real matrix A.\nSPARSE_OPERATION_CONJUGATE_TRANSPOSE Conjugate transpose,C :=\nbeta*C + alpha*AH*A for complex matrix A.\nA\nHandle which contains the sparse matrix A.\nalpha\nScalar parameter alpha.\nbeta\nScalar parameter beta.\nlayout\nDescribes the storage scheme for the dense matrix.\nlayout = SPARSE_LAYOUT_COLUMN_MAJOR\nStorage of elements uses\ncolumn-major layout.\nlayout = SPARSE_LAYOUT_ROW_MAJOR\nStorage of elements uses\nrow-major layout.\nldc\nLeading dimension of matrix C.\nNOTE\nOnly the upper triangular part of matrix C is processed. Therefore, you must set real values\nof alpha and beta for complex matrices in order to obtain a Hermitian matrix.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n348\n\n\nOutput Parameters\nC\nResulting dense matrix. Only the upper triangular part of the matrix is\ncomputed.\nReturn Values\nThe function returns a value indicating whether the operation was successful or not, and why.\nSPARSE_STATUS_SUCCESS\nThe operation was successful.\nSPARSE_STATUS_NOT_INITIALIZED\nThe routine encountered an empty handle or matrix array.\nSPARSE_STATUS_ALLOC_FAILED\nInternal memory allocation failed.\nSPARSE_STATUS_INVALID_VALUE\nThe input parameters contain an invalid value.\nSPARSE_STATUS_EXECUTION_FAILED\nExecution failed.\nSPARSE_STATUS_INTERNAL_ERROR\nAn error in algorithm implementation occurred.\nSPARSE_STATUS_NOT_SUPPORTED\nThe requested operation is not supported.\nmkl_sparse_?_dotmv\nComputes a sparse matrix-vector product followed by\na dot product.\nSyntax\nsparse_status_t mkl_sparse_s_dotmv (const sparse_operation_t operation, const float\nalpha, const sparse_matrix_t A, const struct matrix_descr descr, const float *x, const\nfloat beta, float *y, float *d);\nsparse_status_t mkl_sparse_d_dotmv (const sparse_operation_t operation, const double\nalpha, const sparse_matrix_t A, const struct matrix_descr descr, const double *x, const\ndouble beta, double *y, double *d);\nsparse_status_t mkl_sparse_c_dotmv (const sparse_operation_t operation, const\nMKL_Complex8 alpha, const sparse_matrix_t A, const struct matrix_descr descr, const\nMKL_Complex8 *x, const MKL_Complex8 beta, MKL_Complex8 *y, MKL_Complex8 *d);\nsparse_status_t mkl_sparse_z_dotmv (const sparse_operation_t operation, const\nMKL_Complex16 alpha, const sparse_matrix_t A, const struct matrix_descr descr, const\nMKL_Complex16 *x, const MKL_Complex16 beta, MKL_Complex16 *y, MKL_Complex16 *d);\nInclude Files\n•\nmkl_spblas.h\nDescription\nThe mkl_sparse_?_dotmv routine computes a sparse matrix-vector product and dot product:\ny := alpha*op(A)*x + beta*yd := ∑ixi*yi (real case)\nd := ∑iconj(xi)*yi (complex case)\nwhere\n•\nalpha and beta are scalars.\n•\nx and y are vectors.\n•\nA is an m-by-k matrix.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n349\n\n\n•\nconj represents complex conjugation.\n•\nop(A) is a matrix modifier.\nAvailable options for op(A) are A, AT, or AH.\nNOTE\nFor sparse matrices in the BSR format, the supported combinations of\n(indexing,block_layout) are:\n•\n(SPARSE_INDEX_BASE_ZERO, SPARSE_LAYOUT_ROW_MAJOR )\n•\n(SPARSE_INDEX_BASE_ONE, SPARSE_LAYOUT_COLUMN_MAJOR )\nInput Parameters\noperation\nSpecifies the operation performed on matrix A.\nIf operation = SPARSE_OPERATION_NON_TRANSPOSE, op(A) = A.\nIf operation = SPARSE_OPERATION_TRANSPOSE, op(A) = AT.\nIf operation = SPARSE_OPERATION_CONJUGATE_TRANSPOSE, op(A) = AH.\nalpha\nSpecifies the scalar alpha.\nA\nHandle which contains the sparse matrix A.\ndescr\nStructure specifying sparse matrix properties.\nsparse_matrix_type_t type - Specifies the type of a sparse matrix:\nSPARSE_MATRIX_TYPE_GE\nNERAL\nThe matrix is processed as is.\nSPARSE_MATRIX_TYPE_SY\nMMETRIC\nThe matrix is symmetric (only the requested\ntriangle is processed).\nSPARSE_MATRIX_TYPE_HE\nRMITIAN\nThe matrix is Hermitian (only the requested\ntriangle is processed).\nSPARSE_MATRIX_TYPE_TR\nIANGULAR\nThe matrix is triangular (only the requested\ntriangle is processed).\nSPARSE_MATRIX_TYPE_DI\nAGONAL\nThe matrix is diagonal (only diagonal elements\nare processed).\nSPARSE_MATRIX_TYPE_BL\nOCK_TRIANGULAR\nThe matrix is block-triangular (only requested\ntriangle is processed). Applies to BSR format\nonly.\nSPARSE_MATRIX_TYPE_BL\nOCK_DIAGONAL\nThe matrix is block-diagonal (only diagonal\nblocks are processed). Applies to BSR format\nonly.\nsparse_fill_mode_t mode - Specifies the triangular matrix part for\nsymmetric, Hermitian, triangular, and block-triangular matrices:\nSPARSE_FILL_MODE_LOWE\nR\nThe lower triangular matrix part is processed.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n350\n\n\nSPARSE_FILL_MODE_UPPE\nR\nThe upper triangular matrix part is processed.\nsparse_diag_type_t diag - Specifies diagonal type for non-general\nmatrices:\nSPARSE_DIAG_NON_UNIT\nDiagonal elements might not be equal to one.\nSPARSE_DIAG_UNIT\nDiagonal elements are equal to one.\nx\nIf operation = SPARSE_OPERATION_NON_TRANSPOSE, array of size at least\nk, where k is the number of columns of matrix A.\nOtherwise, array of size at least m, where m is the number of rows of\nmatrix A.\nOn entry, the array x must contain the vector x.\nbeta\nSpecifies the scalar beta.\ny\nIf operation = SPARSE_OPERATION_NON_TRANSPOSE, array of size at least\nm, where k is the number of rows of matrix A.\nOtherwise, array of size at least k, where k is the number of columns of\nmatrix A.\nOn entry, the array y must contain the vector y.\nOutput Parameters\ny\nOverwritten by the updated vector y.\nd\nOverwritten by the dot product of x and y.\nReturn Values\nThe function returns a value indicating whether the operation was successful or not, and why.\nSPARSE_STATUS_SUCCESS\nThe operation was successful.\nSPARSE_STATUS_NOT_INITIALIZED\nThe routine encountered an empty handle or matrix array.\nSPARSE_STATUS_ALLOC_FAILED\nInternal memory allocation failed.\nSPARSE_STATUS_INVALID_VALUE\nThe input parameters contain an invalid value.\nSPARSE_STATUS_EXECUTION_FAILED\nExecution failed.\nSPARSE_STATUS_INTERNAL_ERROR\nAn error in algorithm implementation occurred.\nSPARSE_STATUS_NOT_SUPPORTED\nThe requested operation is not supported.\nmkl_sparse_?_sorv\nComputes forward, backward sweeps or a symmetric\nsuccessive over-relaxation preconditioner operation.\nSyntax\nsparse_status_t mkl_sparse_s_sorv(\n    const sparse_sor_type_t type,\n    const struct matrix_descr descrA,\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n351\n\n\n    const sparse_matrix_t A,\n    float omega,\n    float alpha,\n    float* x,\n    float* b\n);\n      \nsparse_status_t mkl_sparse_d_sorv(\n    const sparse_sor_type_t type,\n    const struct matrix_descr descrA,\n    const sparse_matrix_t A,\n    double omega,\n    double alpha,\n    double* x,\n    double* b\n);\n      \nInclude Files\n•\nmkl_spblas.h\nDescription\nThe mkl_sparse_?_sorv routine performs one of the following operations:\nSPARSE_SOR_FORWARD:\nSPARSE_SOR_BACKWARD:\nSPARSE_SOR_SYMMETRIC: Performs application of a\npreconditioner.\nwhere A = L + D + U and x^0 is an input vector x scaled by input parameter alpha vector and x^1 is an\noutput stored in vector x.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n352\n\n\nNOTE\nCurrently this routine only supports the following configuration:\n•\nCSR format of the input matrix\n•\nSPARSE_SOR_FORWARD operation\n•\nGeneral matrix (descr.type is SPARSE_MATRIX_TYPE_GENERAL) or symmetric matrix with full\nportrait and unit diagonal (descr.type is SPARSE_MATRIX_TYPE_SYMMETRIC, descr.mode is\nSPARSE_FILL_MODE_FULL, and descr.diag is SPARSE_DIAG_UNIT)\nNOTE\nCurrently, this routine is optimized only for sequential threading execution mode.\nWarning It is currently not allowed to place a sorv call in a parallel section (e.g., under\n#pragma omp parallel), because it is not thread-safe in this scenario. This limitation will be\naddressed in one of the upcoming releases.\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nInput Parameters\ntype\nSpecifies the operation performed by the SORV preconditioner.\nSPARSE_SOR_FORWARD\nPerforms forward sweep as defined by:\nSPARSE_SOR_BACKWARD\nPerforms backward sweep as defined by:\nSPARSE_SOR_SYMMETRIC\nPreconditioner matrix could be expressed as:\ndescr\nStructure specifying sparse matrix properties.\nsparse_matrix_type_t\ntype\nSpecifies the type of a sparse matrix:\n•\nSPARSE_MATRIX_TYPE_GENERAL\nThe matrix is processed as-is.\n•\nSPARSE_MATRIX_TYPE_SYMMETRIC\nThe matrix is symmetric (only the requested\ntriangle is processed).\n•\nSPARSE_MATRIX_TYPE_HERMITIAN\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n353\n\n\nThe matrix is Hermitian (only the requested\ntriangle is processed).\n•\nSPARSE_MATRIX_TYPE_TRIANGULAR\nThe matrix is triangular (only the requested\ntriangle is processed).\n•\nSPARSE_MATRIX_TYPE_DIAGONAL\nThe matrix is diagonal (only diagonal\nelements are processed).\n•\nSPARSE_MATRIX_TYPE_BLOCK_TRIANGULAR\nThe matrix is block-triangular (only\nrequested triangle is processed). Applies to\nBSR format only.\n•\nSPARSE_MATRIX_TYPE_BLOCK_DIAGONAL\nThe matrix is block-diagonal (only diagonal\nblocks are processed). Applies to BSR format\nonly.\nsparse_fill_mode_t\nmode\nSpecifies the triangular matrix part for\nsymmetric, Hermitian, triangular, and block-\ntriangular matrices:\n•\nSPARSE_FILL_MODE_LOWER\nThe lower triangular matrix part is processed.\n•\nSPARSE_FILL_MODE_UPPER\nThe upper triangular matrix part is\nprocessed.\nsparse_diag_type_t\ndiag\nSpecifies diagonal type for non-general\nmatrices:\n•\nSPARSE_DIAG_NON_UNIT\nDiagonal elements might not be equal to\none.\n•\nSPARSE_DIAG_UNIT\nDiagonal elements are equal to one.\nA\nHandle containing internal data.\nomega\nRelaxation factor.\nalpha\nParameter that could be used to normalize or set to zero the vector x that\nholds the initial guess.\nx\nInitial guess on input.\nb\nRight-hand side.\nOutput Parameters\nx\nSolution vector on output.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n354\n\n\nReturn Values\nThe function returns a value indicating whether the operation was successful or not, and why.\nSPARSE_STATUS_SUCCESS\nThe operation was successful.\nSPARSE_STATUS_NOT_INITIALIZED\nThe routine encountered an empty handle or matrix array.\nSPARSE_STATUS_ALLOC_FAILED\nInternal memory allocation failed.\nSPARSE_STATUS_INVALID_VALUE\nThe input parameters contain an invalid value.\nSPARSE_STATUS_EXECUTION_FAILED\nExecution failed.\nSPARSE_STATUS_INTERNAL_ERROR\nAn error in algorithm implementation occurred.\nSPARSE_STATUS_NOT_SUPPORTED\nThe requested operation is not supported.\nBLAS-like Extensions\nIntel® oneAPI Math Kernel Library provides C and Fortran routines to extend the functionality of the BLAS\nroutines. These include routines to compute vector products, matrix-vector products, and matrix-matrix\nproducts.\nIntel® oneAPI Math Kernel Library also provides routines to perform certain data manipulation, including\nmatrix in-place and out-of-place transposition operations combined with simple matrix arithmetic operations.\nTransposition operations are Copy As Is, Conjugate transpose, Transpose, and Conjugate. Each routine adds\nthe possibility of scaling during the transposition operation by giving some alpha and/or beta parameters.\nEach routine supports both row-major orderings and column-major orderings.\nTable “BLAS-like Extensions” lists these routines.\nThe <?> symbol in the routine short names is a precision prefix that indicates the data type:\ns\nfloat\nd\ndouble\nc\nMKL_Complex8\nz\nMKL_Complex16\nBLAS-like Extensions\nRoutine\nData Types\nDescription\ncblas_?axpby\ns, d, c, z\nScales two vectors, adds them to one another and stores\nresult in the vector (routines).\ncblas_?axpy_batch\ncblas_?axpy_batch_strided\ns, d, c, z\nComputes groups of vector-scalar products added to a\nvector.\ncblas_?dgmm_batch_strided\ncblas_?dgmm_batch\ns, d, c, z\nComputes groups of diagonal matrix-general matrix\nproduct\ncblas_?gemm_batch\ncblas_?gemm_batch_strided\ns, d, c, z\nComputes scalar-matrix-matrix products and adds the\nresults to scalar matrix products for groups of general\nmatrices.\ncblas_gemm_bf16bf16f32\nbfloat16\nComputes a matrix-matrix product with general matrices\nof bfloat16 data type.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n355\n\n\nRoutine\nData Types\nDescription\ncblas_gemm_bf16bf16f32_compute\nbfloat16\nComputes a matrix-matrix product with general matrices\nof bfloat16 data type where one or both input matrices\nare stored in a packed data structure, and adds the result\nto a scalar-matrix product.\ncblas_gemm_f16f16f32 half precision\nComputes a matrix-matrix product with general matrices\nof half precision data type.\ncblas_gemm_f16f16f32_compute\nhalf precision\nComputes a matrix-matrix product with general matrices\nof half precision data type where one or both input\nmatrices are stored in a packed data structure, and adds\nthe result to a scalar-matrix product.\ncblas_gemm_*\nInteger\nComputes a matrix-matrix product with general integer\nmatrices.\ncblas_?gemm_compute\nh, s, d\nComputes a matrix-matrix product with general matrices\nwhere one or both input matrices are stored in a packed\ndata structure and adds the result to a scalar-matrix\nproduct.\ncblas_gemm_*_compute Integer\nComputes a matrix-matrix product with general integer\nmatrices where one or both input matrices are stored in a\npacked data structure and adds the result to a scalar-\nmatrix product.\ncblas_?gemm_pack\nh, s, d\nPerforms scaling and packing of the matrix into the\npreviously allocated buffer.\ncblas_gemm_*_pack\nInteger, bfloat16\nPack the matrix into the buffer allocated previously.\ncblas_?gemm_pack_get_size\nh, s, d\nReturns the number of bytes required to store the packed\nmatrix.\ncblas_gemm_*_pack_get_size\nInteger, bfloat16\nReturns the number of bytes required to store the packed\nmatrix.\ncblas_?gemm3m\nc, z\nComputes a scalar-matrix-matrix product using matrix\nmultiplications and adds the result to a scalar-matrix\nproduct.\ncblas_?gemm3m_batch\ncblas_?gemm3m_batch_strided\nc, z\nComputes a scalar-matrix-matrix product using matrix\nmultiplications and adds the result to a scalar-matrix\nproduct.\ncblas_?gemmt\ns, d, c, z\nComputes a matrix-matrix product with general matrices\nbut updates only the upper or lower triangular part of the\nresult matrix.\ncblas_?gemv_batch_strided\ncblas_?gemv_batch\ns, d, c, z\nComputes groups of matrix-vector product using general\nmatrices.\ncblas_?trsm_batch\n?cblas_?trsm_batch_strided\ns, d, c, z\nSolves a triangular matrix equation for a group of matrices.\nmkl_?imatcopy\ns, d, c, z\nPerforms scaling and in-place transposition/copying of\nmatrices.\nmkl_?imatcopy_batch_strided\nmkl_?imatcopy_batch\ns, d, c, z\nComputes groups of in-place matrix copy/transposition\nwith scaling using general matrices.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n356\n\n\nRoutine\nData Types\nDescription\nmkl_?omatadd\ns, d, c, z\nPerforms scaling and sum of two matrices including their\nout-of-place transposition/copying.\nmkl_?omatcopy\ns, d, c, z\nPerforms scaling and out-of-place transposition/copying of\nmatrices.\nmkl_?\nomatcopy_batch_stride\nd\nmkl_?omatcopy_batch\ns, d, c, z\nComputes groups of out of place matrix copy/transposition\nwith scaling using general matrices.\nmkl_?omatcopy2\ns, d, c, z\nPerforms two-strided scaling and out-of-place\ntransposition/copying of matrices.\nmkl_jit_create_?gemm s, d, c, z\nCreates a handle on a jitter and generates a GEMM kernel\nthat computes a scalar-matrix-matrix product and adds\nthe result to a scalar-matrix product, with general\nmatrices.\nmkl_jit_destroy\n \nDeletes the previously created jitter and the generated\nGEMM kernel.\nmkl_jit_get_?gemm_ptrs, d, c, z\nReturns the GEMM kernel previously generated.\ncblas_?axpy_batch\nComputes a group of vector-scalar products added to\na vector.\nSyntax\nvoid cblas_saxpy_batch (const MKL_INT *n_array, const float *alpha_array, const float\n**x_array, const MKL_INT *incx_array, float **y_array, const MKL_INT *incy_array, const\nMKL_INT group_count, const MKL_INT *group_size_array);\nvoid cblas_daxpy_batch (const MKL_INT *n_array, const double *alpha_array, const double\n**x_array, const MKL_INT *incx_array, double **y_array, const MKL_INT *incy_array,\nconst MKL_INT group_count, const MKL_INT *group_size_array);\nvoid cblas_caxpy_batch (const MKL_INT *n_array, const void *alpha_array, const void\n**x_array, const MKL_INT *incx_array, void **y_array, const MKL_INT *incy_array, const\nMKL_INT group_count, const MKL_INT *group_size_array);\nvoid cblas_zaxpy_batch (const MKL_INT *n_array, const void *alpha_array, const void\n**x_array, const MKL_INT *incx_array, void **y_array, const MKL_INT *incy_array, const\nMKL_INT group_count, const MKL_INT *group_size_array);\nDescription\nThe cblas_?axpy_batch routines perform a series of scalar-vector product added to a vector. They are\nsimilar to the cblas_?axpy routine counterparts, but the cblas_?axpy_batch routines perform vector\noperations with a group of vectors. The groups contain vectors with the same parameters.\nThe operation is defined as\nidx = 0\nfor i = 0 … group_count – 1\n    n, alpha, incx, incy and group_size at position i in n_array, alpha_array, incx_array, \nincy_array and group_size_array\n    for j = 0 … group_size – 1\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n357\n\n\n        x and y are vectors of size n at position idx in x_array and y_array\n        y := alpha * x + y\n        idx := idx + 1\n    end for\nend for\nThe number of entries in x_array, and y_array is total_batch_count = the sum of all of the group_size\nentries.\nInput Parameters\nn_array\nArray of size group_count. For the group i, ni = n_array[i] is the\nnumber of elements in vectors x and y.\nalpha_array\nArray of size group_count. For the group i, alphai = alpha_array[i] is\nthe scalar alpha.\nx_array\nArray of size total_batch_count of pointers used to store x vectors. The\narray allocated for the x vectors of the group i must be of size at least (1 +\n(ni – 1)*abs(incxi)).\nincx_array\nArray of size group_count. For the group i, incxi = incx_array[i] is the\nstride of vector x.\ny_array\nArray of size total_batch_count of pointers used to store y vectors. The\narray allocated for the y vectors of the group i must be of size at least (1 +\n(ni – 1)*abs(incyi)).\nincy_array\nArray of size group_count. For the group i, incyi = incy_array[i] is the\nstride of vector y.\ngroup_count\nNumber of groups. Must be at least 0.\ngroup_size_array\nArray of size group_count. The element group_size_array[i] is the\nnumber of vector in the group i. Each element in group_size_array must be\nat least 0.\nOutput Parameters\ny_array\nArray of pointers holding the total_batch_count updated vector y.\ncblas_?axpy_batch_strided\nComputes a group of vector-scalar products added to\na vector.\nSyntax\nvoid cblas_saxpy_batch_strided (const MKL_INT n, const float alpha, const float *x,\nconst MKL_INT incx, const MKL_INT stridex, float *y, const MKL_INT incy, const MKL_INT\nstridey, const MKL_INT batch_size);\nvoid cblas_daxpy_batch_strided (const MKL_INT n, const double alpha, const double *x,\nconst MKL_INT incx, const MKL_INT stridex, double *y, const MKL_INT incy, const MKL_INT\nstridey, const MKL_INT batch_size);\nvoid cblas_caxpy_batch_strided (const MKL_INT n, const void alpha, const void *x, const\nMKL_INT incx, const MKL_INT stridex, void *y, const MKL_INT incy, const MKL_INT\nstridey, const MKL_INT batch_size);\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n358\n\n\nvoid cblas_zaxpy_batch_strided (const MKL_INT n, const void alpha, const void *x, const\nMKL_INT incx, const MKL_INT stridex, void *y, const MKL_INT incy, const MKL_INT\nstridey, const MKL_INT batch_size);\nInclude Files\n•\nmkl.h\nDescription\nThe cblas_?axpy_batch_strided routines perform a series of scalar-vector product added to a vector.\nThey are similar to the cblas_?axpy routine counterparts, but the cblas_?axpy_batch_strided routines\nperform vector operations with a group of vectors.\nAll vector x (respectively, y) have the same parameters (size, increments) and are stored at constant stridex\n(respectively, stridey) from each other. The operation is defined as\nFor i = 0 … batch_size – 1\n    X and Y are vectors at offset i * stridex and i * stridey in x and y\n    Y = alpha * X + Y\nend for\nInput Parameters\nn\nNumber of elements in vectors x and y.\nalpha\nSpecifies the scalar alpha.\nx\nArray of size at least stridex*batch_size holding the x vectors.\nincx\nSpecifies the increment for the elements of x.\nstridex\nStride between two consecutive x vectors; must be at least zero.\ny\nArray of size at least stridey*batch_size holding the y vectors.\nincy\nSpecifies the increment for the elements of y.\nstridey\nStride between two consecutive y vectors; must be at least (1 +\n(n-1)*abs(incy)).\nbatch_size\nNumber of axpy computations to perform and x and y vectors. Must be at\nleast 0.\nOutput Parameters\ny\nArray holding the batch_size updated vector y.\ncblas_?axpby\nScales two vectors, adds them to one another and\nstores result in the vector.\nSyntax\nvoid cblas_saxpby (const MKL_INT n, const float a, const float *x, const MKL_INT incx,\nconst float b, float *y, const MKL_INT incy);\nvoid cblas_daxpby (const MKL_INT n, const double a, const double *x, const MKL_INT\nincx, const double b, double *y, const MKL_INT incy);\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n359\n\n\nvoid cblas_caxpby (const MKL_INT n, const void *a, const void *x, const MKL_INT incx,\nconst void *b, void *y, const MKL_INT incy);\nvoid cblas_zaxpby (const MKL_INT n, const void *a, const void *x, const MKL_INT incx,\nconst void *b, void *y, const MKL_INT incy);\nInclude Files\n•\nmkl.h\nDescription\nThe ?axpby routines perform a vector-vector operation defined as\ny := a*x + b*y\nwhere:\na and b are scalars\nx and y are vectors each with n elements.\nInput Parameters\nn\nSpecifies the number of elements in vectors x and y.\na\nSpecifies the scalar a.\nx\nArray, size at least (1 + (n-1)*abs(incx)).\nincx\nSpecifies the increment for the elements of x.\nb\nSpecifies the scalar b.\ny\nArray, size at least (1 + (n-1)*abs(incy)).\nincy\nSpecifies the increment for the elements of y.\nOutput Parameters\ny\nContains the updated vector y.\nExample\nFor examples of routine usage, see these code examples in the Intel® oneAPI Math Kernel Library (oneMKL)\ninstallation directory:\n•\ncblas_saxpby: examples\\cblas\\source\\cblas_saxpbyx.c\n•\ncblas_daxpby: examples\\cblas\\source\\cblas_daxpbyx.c\n•\ncblas_caxpby: examples\\cblas\\source\\cblas_caxpbyx.c\n•\ncblas_zaxpby: examples\\cblas\\source\\cblas_zaxpbyx.c\ncblas_?gemmt\nComputes a matrix-matrix product with general\nmatrices but updates only the upper or lower\ntriangular part of the result matrix.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n360\n\n\nSyntax\nvoid cblas_sgemmt (const CBLAS_LAYOUT Layout, const CBLAS_UPLO uplo, const\nCBLAS_TRANSPOSE transa, const CBLAS_TRANSPOSE transb, const MKL_INT n, const MKL_INT k,\nconst float alpha, const float *a, const MKL_INT lda, const float *b, const MKL_INT\nldb, const float beta, float *c, const MKL_INT ldc);\nvoid cblas_dgemmt (const CBLAS_LAYOUT Layout, const CBLAS_UPLO uplo, const\nCBLAS_TRANSPOSE transa, const CBLAS_TRANSPOSE transb, const MKL_INT n, const MKL_INT k,\nconst double alpha, const double *a, const MKL_INT lda, const double *b, const MKL_INT\nldb, const double beta, double *c, const MKL_INT ldc);\nvoid cblas_cgemmt (const CBLAS_LAYOUT Layout, const CBLAS_UPLO uplo, const\nCBLAS_TRANSPOSE transa, const CBLAS_TRANSPOSE transb, const MKL_INT n, const MKL_INT k,\nconst void *alpha, const void *a, const MKL_INT lda, const void *b, const MKL_INT ldb,\nconst void *beta, void *c, const MKL_INT ldc);\nvoid cblas_zgemmt (const CBLAS_LAYOUT Layout, const CBLAS_UPLO uplo, const\nCBLAS_TRANSPOSE transa, const CBLAS_TRANSPOSE transb, const MKL_INT n, const MKL_INT k,\nconst void *alpha, const void *a, const MKL_INT lda, const void *b, const MKL_INT ldb,\nconst void *beta, void *c, const MKL_INT ldc);\nInclude Files\n•\nmkl.h\nDescription\nThe ?gemmt routines compute a scalar-matrix-matrix product with general matrices and add the result to the\nupper or lower part of a scalar-matrix product. These routines are similar to the ?gemm routines, but they\nonly access and update a triangular part of the square result matrix (see Application Notes below).\nThe operation is defined as\nC := alpha*op(A)*op(B) + beta*C,\nwhere:\nop(X) is one of op(X) = X, or op(X) = XT, or op(X) = XH,\nalpha and beta are scalars,\nA, B and C are matrices:\nop(A) is an n-by-k matrix,\nop(B) is a k-by-n matrix,\nC is an n-by-n upper or lower triangular matrix.\nInput Parameters\nLayout\nSpecifies whether two-dimensional array storage is row-major\n(CblasRowMajor) or column-major (CblasColMajor).\nuplo\nSpecifies whether the upper or lower triangular part of the array c is used.\nIf uplo = 'CblasUpper', then the upper triangular part of the array c is\nused. If uplo = 'CblasLower', then the lower triangular part of the array\nc is used.\ntransa\nSpecifies the form of op(A) used in the matrix multiplication:\nif transa = 'CblasNoTrans', then op(A) = A;\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n361\n\n\nif transa = 'CblasTrans', then op(A) = AT;\nif transa = 'CblasConjTrans', then op(A) = AH.\ntransb\nSpecifies the form of op(B) used in the matrix multiplication:\nif transb = 'CblasNoTrans', then op(B) = B;\nif transb = 'CblasTrans', then op(B) = BT;\nif transb = 'CblasConjTrans', then op(B) = BH.\nn\nSpecifies the order of the matrix C. The value of n must be at least zero.\nk\nSpecifies the number of columns of the matrix op(A) and the number of\nrows of the matrix op(B). The value of k must be at least zero.\nalpha\nSpecifies the scalar alpha.\na\ntransa='CblasNoTr\nans'\ntransa='CblasTran\ns' or\n'CblasConjTrans'\nLayout='CblasColMaj\nor'\nArray, size lda * k.\nBefore entry, the leading\nn-by-k part of the array\na must contain the\nmatrix A.\nArray, size lda * n.\nBefore entry, the leading\nk-by-n part of the array\na must contain the\nmatrix A.\nLayout='CblasRowMaj\nor'\nArray, size lda * n.\nBefore entry, the leading\nk-by-n part of the array\na must contain the\nmatrix A.\nArray, size lda * k.\nBefore entry, the leading\nn-by-k part of the array\na must contain the\nmatrix A.\nlda\nSpecifies the leading dimension of a as declared in the calling\n(sub)program.\ntransa='CblasNoTr\nans'\ntransa='CblasTran\ns' or\n'CblasConjTrans'\nLayout='CblasColMaj\nor'\nlda must be at least\nmax(1, n).\nlda must be at least\nmax(1, k).\nLayout='CblasRowMaj\nor'\nlda must be at least\nmax(1, k).\nlda must be at least\nmax(1, n).\nb\ntransb='CblasNoTr\nans'\ntransb='CblasTran\ns' or\n'CblasConjTrans'\nLayout='CblasColMaj\nor'\nArray, size ldb * n.\nBefore entry, the leading\nk-by-n part of the array\nb must contain the\nmatrix B.\nArray, size ldb * k.\nBefore entry, the leading\nn-by-k part of the array\nb must contain the\nmatrix B.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n362\n\n\nLayout='CblasRowMaj\nor'\nArray, size ldb * k.\nBefore entry, the leading\nn-by-k part of the array\nb must contain the\nmatrix B.\nArray, size ldb * n.\nBefore entry, the leading\nk-by-n part of the array\nb must contain the\nmatrix B.\nldb\nSpecifies the leading dimension of b as declared in the calling\n(sub)program.\ntransb='CblasNoTr\nans'\ntransb='CblasTran\ns' or\n'CblasConjTrans'\nLayout='CblasColMaj\nor'\nldb must be at least\nmax(1, k).\nldb must be at least\nmax(1, n).\nLayout='CblasRowMaj\nor'\nldb must be at least\nmax(1, n).\nldb must be at least\nmax(1, k).\nbeta\nSpecifies the scalar beta. When beta is equal to zero, then c need not be\nset on input.\nc\nArray, size ldc by n.\nWhen beta is equal to zero, c need not be set on input.\nuplo = 'CblasUpper'\nuplo = 'CblasLower'\nThe leading n-by-n upper triangular\npart of the array c must contain the\nupper triangular part of the matrix C\nand the strictly lower triangular part of\nc is not referenced.\nThe leading n-by-n lower triangular\npart of the array c must contain the\nlower triangular part of the matrix C\nand the strictly upper triangular part of\nc is not referenced.\nldc\nSpecifies the leading dimension of c as declared in the calling\n(sub)program. The value of ldc must be at least max(1, n).\nOutput Parameters\nc\nWhen uplo = 'CblasUpper', the upper triangular part of the array c\nis overwritten by the upper triangular part of the updated matrix.\nWhen uplo = 'CblasLower', the lower triangular part of the array c\nis overwritten by the lower triangular part of the updated matrix.\nApplication Notes\nThese routines only access and update the upper or lower triangular part of the result matrix. This can be\nuseful when the result is known to be symmetric; for example, when computing a product of the form C :=\nalpha*B*S*BT + beta*C , where S and C are symmetric matrices and B is a general matrix. In this case,\nfirst compute A := B*S (which can be done using the corresponding ?symm routine), then compute C :=\nalpha*A*BT + beta*C using the ?gemmt routine.\ncblas_?gemm3m\nComputes a scalar-matrix-matrix product using matrix\nmultiplications and adds the result to a scalar-matrix\nproduct.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n363\n\n\nSyntax\nvoid cblas_cgemm3m (const CBLAS_LAYOUT Layout, const CBLAS_TRANSPOSE transa, const\nCBLAS_TRANSPOSE transb, const MKL_INT m, const MKL_INT n, const MKL_INT k, const void\n*alpha, const void *a, const MKL_INT lda, const void *b, const MKL_INT ldb, const void\n*beta, void *c, const MKL_INT ldc);\nvoid cblas_zgemm3m (const CBLAS_LAYOUT Layout, const CBLAS_TRANSPOSE transa, const\nCBLAS_TRANSPOSE transb, const MKL_INT m, const MKL_INT n, const MKL_INT k, const void\n*alpha, const void *a, const MKL_INT lda, const void *b, const MKL_INT ldb, const void\n*beta, void *c, const MKL_INT ldc);\nInclude Files\n•\nmkl.h\nDescription\nThe ?gemm3m routines perform a matrix-matrix operation with general complex matrices. These routines are\nsimilar to the ?gemm routines, but they use fewer matrix multiplication operations (see Application Notes\nbelow).\nThe operation is defined as\nC := alpha*op(A)*op(B) + beta*C,\nwhere:\nop(x) is one of op(x) = x, or op(x) = x', or op(x) = conjg(x'),\nalpha and beta are scalars,\nA, B and C are matrices:\nop(A) is an m-by-k matrix,\nop(B) is a k-by-n matrix,\nC is an m-by-n matrix.\nInput Parameters\nLayout\nSpecifies whether two-dimensional array storage is row-major\n(CblasRowMajor) or column-major (CblasColMajor).\ntransa\nSpecifies the form of op(A) used in the matrix multiplication:\nif transa=CblasNoTrans, then op(A) = A;\nif transa=CblasTrans, then op(A) = A';\nif transa=CblasConjTrans, then op(A) = conjg(A').\ntransb\nSpecifies the form of op(B) used in the matrix multiplication:\nif transb=CblasNoTrans, then op(B) = B;\nif transb=CblasTrans, then op(B) = B';\nif transb=CblasConjTrans, then op(B) = conjg(B').\nm\nSpecifies the number of rows of the matrix op(A) and of the matrix C. The\nvalue of m must be at least zero.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n364\n\n\nn\nSpecifies the number of columns of the matrix op(B) and the number of\ncolumns of the matrix C.\nThe value of n must be at least zero.\nk\nSpecifies the number of columns of the matrix op(A) and the number of\nrows of the matrix op(B).\nThe value of k must be at least zero.\nalpha\nSpecifies the scalar alpha.\na\ntransa=CblasNoTrans\ntransa=CblasTrans or\ntransa=CblasConjTrans\nLayout =\nCblasColMajor\nArray, size lda*k.\nBefore entry, the leading\nm-by-k part of the array a\nmust contain the matrix\nA.\nArray, size lda*m.\nBefore entry, the leading k-\nby-m part of the array a\nmust contain the matrix A.\nLayout =\nCblasRowMajor\nArray, size lda* m.\nBefore entry, the leading\nk-by-m part of the array a\nmust contain the matrix\nA.\nArray, size lda*k.\nBefore entry, the leading m-\nby-k part of the array a\nmust contain the matrix A.\nlda\nSpecifies the leading dimension of a as declared in the calling\n(sub)program.\ntransa=CblasNoTrans\ntransa=CblasTrans or\ntransa=CblasConjTrans\nLayout =\nCblasColMajor\nlda must be at least\nmax(1, m).\nlda must be at least\nmax(1, k)\nLayout =\nCblasRowMajor\nlda must be at least\nmax(1, k)\nlda must be at least\nmax(1, m).\nb\ntransb=CblasNoTrans\ntransb=CblasTrans or\ntransb=CblasConjTrans\nLayout =\nCblasColMajor\nArray, size ldb by n.\nBefore entry, the leading\nk-by-n part of the array b\nmust contain the matrix\nB.\nArray, size ldb by k. Before\nentry the leading n-by-k\npart of the array b must\ncontain the matrix B.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n365\n\n\nLayout =\nCblasRowMajor\nArray, size ldb by k.\nBefore entry the leading\nn-by-k part of the array b\nmust contain the matrix\nB.\nArray, size ldb by n. Before\nentry, the leading k-by-n\npart of the array b must\ncontain the matrix B.\nldb\nSpecifies the leading dimension of b as declared in the calling\n(sub)program.\ntransb=CblasNoTrans\ntransb=CblasTrans or\ntransb=CblasConjTrans\nLayout =\nCblasColMajor\nldb must be at least\nmax(1, k).\nldb must be at least\nmax(1, n).\nLayout =\nCblasRowMajor\nldb must be at least\nmax(1, n).\nldb must be at least\nmax(1, k).\nbeta\nSpecifies the scalar beta.\nWhen beta is equal to zero, then c need not be set on input.\nc\nLayout =\nCblasColMajor\nArray, size ldc by n. Before entry, the leading m-\nby-n part of the array c must contain the matrix C,\nexcept when beta is equal to zero, in which case c\nneed not be set on entry.\nLayout =\nCblasRowMajor\nArray, size ldc by m. Before entry, the leading n-\nby-m part of the array c must contain the matrix C,\nexcept when beta is equal to zero, in which case c\nneed not be set on entry.\nldc\nSpecifies the leading dimension of c as declared in the calling\n(sub)program.\nLayout = CblasColMajor\nldc must be at least max(1, m).\nLayout = CblasRowMajor\nldc must be at least max(1, n).\nOutput Parameters\nc\nOverwritten by the m-by-n matrix (alpha*op(A)*op(B) + beta*C).\nApplication Notes\nThese routines perform a complex matrix multiplication by forming the real and imaginary parts of the input\nmatrices. This uses three real matrix multiplications and five real matrix additions instead of the conventional\nfour real matrix multiplications and two real matrix additions. The use of three real matrix multiplications\nreduces the time spent in matrix operations by 25%, resulting in significant savings in compute time for\nlarge matrices.\nIf the errors in the floating point calculations satisfy the following conditions:\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n366\n\n\nfl(x op y)=(x op y)(1+δ),|δ|≤u, op=×,/, fl(x±y)=x(1+α)±y(1+β), |α|,|β|≤u\nthen for an n-by-n matrix Ĉ=fl(C1+iC2)= fl((A1+iA2)(B1+iB2))=Ĉ1+iĈ2, the following bounds are\nsatisfied:\n║Ĉ1-C1║≤ 2(n+1)u║A║∞║B║∞+O(u2),\n║Ĉ2-C2║≤ 4(n+4)u║A║∞║B║∞+O(u2),\nwhere ║A║∞=max(║A1║∞,║A2║∞), and ║B║∞=max(║B1║∞,║B2║∞).\nThus the corresponding matrix multiplications are stable.\ncblas_?gemm_batch\nComputes scalar-matrix-matrix products and adds the\nresults to scalar matrix products for groups of general\nmatrices.\nSyntax\nvoid cblas_sgemm_batch (const CBLAS_LAYOUT Layout, const CBLAS_TRANSPOSE* transa_array,\nconst CBLAS_TRANSPOSE* transb_array, const MKL_INT* m_array, const MKL_INT* n_array,\nconst MKL_INT* k_array, const float* alpha_array, const float **a_array, const MKL_INT*\nlda_array, const float **b_array, const MKL_INT* ldb_array, const float* beta_array,\nfloat **c_array, const MKL_INT* ldc_array, const MKL_INT group_count, const MKL_INT*\ngroup_size);\nvoid cblas_dgemm_batch (const CBLAS_LAYOUT Layout, const CBLAS_TRANSPOSE* transa_array,\nconst CBLAS_TRANSPOSE* transb_array, const MKL_INT* m_array, const MKL_INT* n_array,\nconst MKL_INT* k_array, const double* alpha_array, const double **a_array, const\nMKL_INT* lda_array, const double **b_array, const MKL_INT* ldb_array, const double*\nbeta_array, double **c_array, const MKL_INT* ldc_array, const MKL_INT group_count,\nconst MKL_INT* group_size);\nvoid cblas_cgemm_batch (const CBLAS_LAYOUT Layout, const CBLAS_TRANSPOSE* transa_array,\nconst CBLAS_TRANSPOSE* transb_array, const MKL_INT* m_array, const MKL_INT* n_array,\nconst MKL_INT* k_array, const void *alpha_array, const void **a_array, const MKL_INT*\nlda_array, const void **b_array, const MKL_INT* ldb_array, const void *beta_array, void\n**c_array, const MKL_INT* ldc_array, const MKL_INT group_count, const MKL_INT*\ngroup_size);\nvoid cblas_zgemm_batch (const CBLAS_LAYOUT Layout, const CBLAS_TRANSPOSE* transa_array,\nconst CBLAS_TRANSPOSE* transb_array, const MKL_INT* m_array, const MKL_INT* n_array,\nconst MKL_INT* k_array, const void *alpha_array, const void **a_array, const MKL_INT*\nlda_array, const void **b_array, const MKL_INT* ldb_array, const void *beta_array, void\n**c_array, const MKL_INT* ldc_array, const MKL_INT group_count, const MKL_INT*\ngroup_size);\nInclude Files\n•\nmkl.h\nDescription\nThe ?gemm_batch routines perform a series of matrix-matrix operations with general matrices. They are\nsimilar to the ?gemm routine counterparts, but the ?gemm_batch routines perform matrix-matrix operations\nwith groups of matrices, processing a number of groups at once. The groups contain matrices with the same\nparameters.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n367\n\n\nThe operation is defined as\nidx = 0\nfor i = 0..group_count - 1\n     alpha and beta in alpha_array[i] and beta_array[i]\n     for j = 0..group_size[i] - 1 \n          A, B, and C matrix in a_array[idx], b_array[idx], and c_array[idx]\n          C := alpha*op(A)*op(B) + beta*C,\n          idx = idx + 1\n     end for\n end for\nwhere:\nop(X) is one of op(X) = X, or op(X) = XT, or op(X) = XH,\nalpha and beta are scalar elements of alpha_array and beta_array,\nA, B and C are matrices such that for m, n, and k which are elements of m_array, n_array, and k_array:\nop(A) is an m-by-k matrix,\nop(B) is a k-by-n matrix,\nC is an m-by-n matrix.\nA, B, and C represent matrices stored at addresses pointed to by a_array, b_array, and c_array,\nrespectively. The number of entries in a_array, b_array, and c_array is total_batch_count = the sum of all\nof the group_size entries.\nSee also gemm for a detailed description of multiplication for general matrices and ?gemm3m_batch, BLAS-\nlike extension routines for similar matrix-matrix operations.\nNOTE\nError checking is not performed for oneMKL Windows* single dynamic libraries for\nthe?gemm_batch routines.\nInput Parameters\nLayout\nSpecifies whether two-dimensional array storage is row-major\n(CblasRowMajor) or column-major (CblasColMajor).\ntransa_array\nArray of size group_count. For the group i, transai = transa_array[i]\nspecifies the form of op(A) used in the matrix multiplication:\nif transai = CblasNoTrans, then op(A) = A;\nif transai = CblasTrans, then op(A) = AT;\nif transai = CblasConjTrans, then op(A) = AH.\ntransb_array\nArray of size group_count. For the group i, transbi = transb_array[i]\nspecifies the form of op(Bi) used in the matrix multiplication:\nif transbi = CblasNoTrans, then op(B) = B;\nif transbi = CblasTrans, then op(B) = BT;\nif transbi = CblasConjTrans, then op(B) = BH.\nm_array\nArray of size group_count. For the group i, mi = m_array[i] specifies the\nnumber of rows of the matrix op(A) and of the matrix C.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n368\n\n\nThe value of each element of m_array must be at least zero.\nn_array\nArray of size group_count. For the group i, ni = n_array[i] specifies the\nnumber of columns of the matrix op(B) and the number of columns of the\nmatrix C.\nThe value of each element of n_array must be at least zero.\nk_array\nArray of size group_count. For the group i, ki = k_array[i] specifies the\nnumber of columns of the matrix op(A) and the number of rows of the\nmatrix op(B).\nThe value of each element of k_array must be at least zero.\nalpha_array\nArray of size group_count. For the group i, alpha_array[i] specifies the\nscalar alphai.\na_array\nArray, size total_batch_count, of pointers to arrays used to store A\nmatrices.\nlda_array\nArray of size group_count. For the group i, ldai = lda_array[i]\nspecifies the leading dimension of the array storing matrix A as declared in\nthe calling (sub)program.\ntransai=CblasNoTrans\ntransai=CblasTrans or\ntransai=CblasConjTrans\nLayout =\nCblasColMajor\nldai must be at least\nmax(1, mi).\nldai must be at least\nmax(1, ki)\nLayout =\nCblasRowMajor\nldai must be at least\nmax(1, ki)\nldai must be at least\nmax(1, mi).\nb_array\nArray, size total_batch_count, of pointers to arrays used to store B\nmatrices.\nldb_array\nArray of size group_count. For the group i, ldbi = ldb_array[i]\nspecifies the leading dimension of the array storing matrix B as declared in\nthe calling (sub)program.\ntransbi=CblasNoTrans\ntransbi=CblasTrans or\ntransbi=CblasConjTrans\nLayout =\nCblasColMajor\nldbi must be at least\nmax(1, ki).\nldbi must be at least\nmax(1, ni).\nLayout =\nCblasRowMajor\nldbi must be at least\nmax(1, ni).\nldbi must be at least\nmax(1, ki).\nbeta_array\nArray of size group_count. For the group i, beta_array[i] specifies the\nscalar betai.\nWhen betai is equal to zero, then C matrices in group i need not be set on\ninput.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n369\n\n\nc_array\nArray, size total_batch_count, of pointers to arrays used to store C\nmatrices.\nldc_array\nArray of size group_count. For the group i, ldci = ldc_array[i]\nspecifies the leading dimension of all arrays storing matrix C in group i as\ndeclared in the calling (sub)program.\nWhen Layout = CblasColMajorldci must be at least max(1, mi).\nWhen Layout = CblasRowMajorldci must be at least max(1, ni).\ngroup_count\nSpecifies the number of groups. Must be at least 0.\ngroup_size\nArray of size group_count. The element group_size[i] specifies the\nnumber of matrices in group i. Each element in group_size must be at\nleast 0.\nOutput Parameters\nc_array\nOutput buffer, overwritten by total_batch_count matrix multiply operations\nof the form alpha*op(A)*op(B) + beta*C.\ncblas_?gemm_batch_strided\nComputes groups of matrix-matrix product with\ngeneral matrices.\nSyntax\nvoid cblas_sgemm_batch_strided (const CBLAS_LAYOUT layout, const CBLAS_TRANSPOSE\ntransa, const CBLAS_TRANSPOSE transb, const MKL_INT m, const MKL_INT n, const MKL_INT\nk, const float alpha, const float *a, const MKL_INT lda, const MKL_INT stridea, const\nfloat *b, const MKL_INT ldb, const MKL_INT strideb, const float beta, float *c, const\nMKL_INT ldc, const MKL_INT stridec, const MKL_INT batch_size);\nvoid cblas_dgemm_batch_strided (const CBLAS_LAYOUT layout, const CBLAS_TRANSPOSE\ntransa, const CBLAS_TRANSPOSE transb, const MKL_INT m, const MKL_INT n, const MKL_INT\nk, const double alpha, const double *a, const MKL_INT lda, const MKL_INT stridea, const\ndouble *b, const MKL_INT ldb, const MKL_INT strideb, const double beta, double *c,\nconst MKL_INT ldc, const MKL_INT stridec, const MKL_INT batch_size);\nvoid cblas_cgemm_batch_strided (const CBLAS_LAYOUT layout, const CBLAS_TRANSPOSE\ntransa, const CBLAS_TRANSPOSE transb, const MKL_INT m, const MKL_INT n, const MKL_INT\nk, const void *alpha, const void *a, const MKL_INT lda, const MKL_INT stridea, const\nvoid *b, const MKL_INT ldb, const MKL_INT strideb, const void *beta, void *c, const\nMKL_INT ldc, const MKL_INT stridec, const MKL_INT batch_size);\nvoid cblas_zgemm_batch_strided (const CBLAS_LAYOUT layout, const CBLAS_TRANSPOSE\ntransa, const CBLAS_TRANSPOSE transb, const MKL_INT m, const MKL_INT n, const MKL_INT\nk, const void *alpha, const void *a, const MKL_INT lda, const MKL_INT stridea, const\nvoid *b, const MKL_INT ldb, const MKL_INT strideb, const void *beta, void *c, const\nMKL_INT ldc, const MKL_INT stridec, const MKL_INT batch_size);\nInclude Files\n•\nmkl.h\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n370\n\n\nDescription\nThe cblas_?gemm_batch_strided routines perform a series of matrix-matrix operations with general\nmatrices. They are similar to the cblas_?gemm routine counterparts, but the cblas_?gemm_batch_strided\nroutines perform matrix-matrix operations with groups of matrices. The groups contain matrices with the\nsame parameters.\nAll matrix a (respectively, b or c) have the same parameters (size, leading dimension, transpose operation,\nalpha, beta scaling) and are stored at constant stridea (respectively, strideb or stridec) from each other. The\noperation is defined as\nFor i = 0 … batch_size – 1\n    Ai, Bi and Ci are matrices at offset i * stridea, i * strideb and i * stridec in a, b and c\n    Ci = alpha * Ai * Bi +  beta * Ci\nend for\nInput Parameters\nlayout\nSpecifies whether two-dimensional array storage is row-major\n(CblasRowMajor) or column-major (CblasColMajor).\ntransa\nSpecifies op(A) the transposition operation applied to the matrices A.\nif transa = CblasNoTrans, then op(A) = A;\nif transa = CblasTrans, then op(A) = AT;\nif transa = CblasConjTrans, then op(A) = AH.\ntransb\nSpecifies op(B) the transposition operation applied to the matrices B.\nif transb = CblasNoTrans, then op(B) = B;\nif transb = CblasTrans, then op(B) = BT;\nif transb = CblasConjTrans, then op(B) = BH.\nm\nNumber of rows of the op(A) and C matrices. Must be at least 0.\nn\nNumber of columns of the op(B) and C matrices. Must be at least 0.\nk\nNumber of columns of the op(A) matrix and number of rows of the op(B)\nmatrix. Must be at least 0.\nalpha\nSpecifies the scalar alpha.\na\nArray of size at least stridea*batch_size holding the a matrices.\ntransa=CblasNoTrans\ntransa=CblasTrans or\nCblasConjTrans\nlayout =\nCblasColMajor\nBefore entry, the leading\nm-by-k part of the array a\n+ i * stridea must contain\nthe matrix Ai.\nBefore entry, the leading\nk-by-m part of the array\na + i * stridea must\ncontain the matrix Ai.\nlayout =\nCblasRowMajor\nBefore entry, the leading k-\nby-m part of the array a + i\n* stridea must contain the\nmatrix Ai.\nBefore entry, the leading\nm-by-k part of the array\na + i * stridea must\ncontain the matrix Ai.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n371\n\n\nlda\nSpecifies the leading dimension of the a matrices.\ntransa=CblasNoTrans\ntransa=CblasTrans\nor CblasConjTrans\nlayout =\nCblasColMajor\nlda must be at least\nmax(1,m)\nlda must be at least\nmax(1,k).\nlayout =\nCblasRowMajor\nlda must be at least\nmax(1,k).\nlda must be at least\nmax(1,m)\nstridea\nStride between two consecutive a matrices.\ntransa=CblasNoTrans\ntransa=CblasTrans or\nCblasConjTrans\nlayout =\nCblasColMajor\nMust be at least lda*k\nMust be at least lda*m\nlayout =\nCblasRowMajor\nMust be at least lda*m\nMust be at least lda*k\nb\nArray of size at least strideb*batch_size holding the b matrices.\ntransb=CblasNoTrans\ntransb=CblasTrans or\nCblasConjTrans\nlayout =\nCblasColMajor\nBefore entry, the leading k-\nby-n part of the array b + i\n* strideb must contain the\nmatrix Bi.\nBefore entry, the leading\nn-by-k part of the array b\n+ i * strideb must\ncontain the matrix Bi.\nlayout =\nCblasRowMajor\nBefore entry, the leading n-\nby-k part of the array b + i\n* strideb must contain the\nmatrix Bi.\nBefore entry, the leading\nk-by-n part of the array b\n+ i * strideb must\ncontain the matrix Bi.\nldb\nSpecifies the leading dimension of the b matrices.\ntransab=CblasNoTrans\ntransb=CblasTrans\nor CblasConjTrans\nlayout =\nCblasColMajor\nldb must be at least\nmax(1,k)\nldb must be at least\nmax(1,n).\nlayout =\nCblasRowMajor\nldb must be at least\nmax(1,n).\nldb must be at least\nmax(1,k)\nstrideb\nStride between two consecutive b matrices.\ntransa=CblasNoTrans\ntransa=CblasTrans or\nCblasConjTrans\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n372\n\n\nlayout =\nCblasColMajor\nMust be at least ldb*n\nMust be at least ldb*k\nlayout =\nCblasRowMajor\nMust be at least ldb*k\nMust be at least ldb*n\nbeta\nSpecifies the scalar beta.\nc\nArray of size at least stridec*batch_size holding the c matrices.\nIf layout=CblasColMajor, before entry, the leading m-by-n part of the array\nc + i * stridec must contain the matrix Ci.\nIf layout=CblasRowMajor, before entry, the leading n-by-m part of the array\nc + i * stridec must contain the matrix Ci.\nldc\nSpecifies the leading dimension of the c matrices.\nMust be at least max(1,m) if layout=CblasColMajor or max(1,n) if\nlayout=CblasRowMajor.\nstridec\nSpecifies the stride between two consecutive c matrices.\nMust be at least ldc*nif layout=CblasColMajor or ldc*m if\nlayout=CblasRowMajor.\nbatch_size\nNumber of gemm computations to perform and a, b and c matrices. Must be\nat least 0.\nOutput Parameters\nc\nArray holding the batch_size updated c matrices.\ncblas_?gemm3m_batch_strided\nComputes groups of matrix-matrix product with\ngeneral matrices.\nSyntax\nvoid cblas_cgemm3m_batch_strided (const CBLAS_LAYOUT layout, const CBLAS_TRANSPOSE\ntransa, const CBLAS_TRANSPOSE transb, const MKL_INT m, const MKL_INT n, const MKL_INT\nk, const void *alpha, const void *a, const MKL_INT lda, const MKL_INT stridea, const\nvoid *b, const MKL_INT ldb, const MKL_INT strideb, const void *beta, void *c, const\nMKL_INT ldc, const MKL_INT stridec, const MKL_INT batch_size);\nvoid cblas_zgemm3m_batch_strided (const CBLAS_LAYOUT layout, const CBLAS_TRANSPOSE\ntransa, const CBLAS_TRANSPOSE transb, const MKL_INT m, const MKL_INT n, const MKL_INT\nk, const void *alpha, const void *a, const MKL_INT lda, const MKL_INT stridea, const\nvoid *b, const MKL_INT ldb, const MKL_INT strideb, const void *beta, void *c, const\nMKL_INT ldc, const MKL_INT stridec, const MKL_INT batch_size);\nInclude Files\n•\nmkl.h\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n373\n\n\nDescription\nThe cblas_?gemm3m_batch_strided routines perform a series of matrix-matrix operations with general\nmatrices. They are similar to the cblas_?gemm routine counterparts, but the\ncblas_?gemm3m_batch_strided routines perform matrix-matrix operations with groups of matrices. The\ngroups contain matrices with the same parameters.\nAll matrix a (respectively, b or c) have the same parameters (size, leading dimension, transpose operation,\nalpha, beta scaling) and are stored at constant stridea (respectively, strideb or stridec) from each other. The\noperation is defined as\nFor i = 0 … batch_size – 1\n    Ai, Bi and Ci are matrices at offset i * stridea, i * strideb and i * stridec in a, b and c\n    Ci = alpha * Ai * Bi +  beta * Ci\nend for\nThe cblas_?gemm3m_batch_strided routines use fewer matrix multiplications than the cblas_?gemm\nroutines, as described in the Application Notes below.\nInput Parameters\nlayout\nSpecifies whether two-dimensional array storage is row-major\n(CblasRowMajor) or column-major (CblasColMajor).\ntransa\nSpecifies op(A) the transposition operation applied to the matrices A.\nif transa = CblasNoTrans, then op(A) = A;\nif transa = CblasTrans, then op(A) = AT;\nif transa = CblasConjTrans, then op(A) = AH.\ntransb\nSpecifies op(B) the transposition operation applied to the matrices B.\nif transb = CblasNoTrans, then op(B) = B;\nif transb = CblasTrans, then op(B) = BT;\nif transb = CblasConjTrans, then op(B) = BH.\nm\nNumber of rows of the op(A) and C matrices. Must be at least 0.\nn\nNumber of columns of the op(B) and C matrices. Must be at least 0.\nk\nNumber of columns of the op(A) matrix and number of rows of the op(B)\nmatrix. Must be at least 0.\nalpha\nSpecifies the scalar alpha.\na\nArray of size at least stridea*batch_size holding the a matrices.\ntransa=CblasNoTrans\ntransa=CblasTrans or\nCblasConjTrans\nlayout =\nCblasColMajor\nBefore entry, the leading\nm-by-k part of the array a\n+ i * stridea must contain\nthe matrix Ai.\nBefore entry, the leading\nk-by-m part of the array\na + i * stridea must\ncontain the matrix Ai.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n374\n\n\nlayout =\nCblasRowMajor\nBefore entry, the leading k-\nby-m part of the array a + i\n* stridea must contain the\nmatrix Ai.\nBefore entry, the leading\nm-by-k part of the array\na + i * stridea must\ncontain the matrix Ai.\nlda\nSpecifies the leading dimension of the a matrices.\ntransa=CblasNoTrans\ntransa=CblasTrans\nor CblasConjTrans\nlayout =\nCblasColMajor\nlda must be at least\nmax(1,m).\nlda must be at least\nmax(1,k).\nlayout =\nCblasRowMajor\nlda must be at least\nmax(1,k).\nlda must be at least\nmax(1,m).\nstridea\nStride between two consecutive a matrices.\ntransa=CblasNoTrans\ntransa=CblasTrans or\nCblasConjTrans\nlayout =\nCblasColMajor\nMust be at least lda*k.\nMust be at least lda*m.\nlayout =\nCblasRowMajor\nMust be at least lda*m.\nMust be at least lda*k.\nb\nArray of size at least strideb*batch_size holding the b matrices.\ntransb=CblasNoTrans\ntransb=CblasTrans or\nCblasConjTrans\nlayout =\nCblasColMajor\nBefore entry, the leading k-\nby-n part of the array b + i\n* strideb must contain the\nmatrix Bi.\nBefore entry, the leading\nn-by-k part of the array b\n+ i * strideb must\ncontain the matrix Bi.\nlayout =\nCblasRowMajor\nBefore entry, the leading n-\nby-k part of the array b + i\n* strideb must contain the\nmatrix Bi.\nBefore entry, the leading\nk-by-n part of the array b\n+ i * strideb must\ncontain the matrix Bi.\nldb\nSpecifies the leading dimension of the b matrices.\ntransab=CblasNoTrans\ntransb=CblasTrans\nor CblasConjTrans\nlayout =\nCblasColMajor\nldb must be at least\nmax(1,k).\nldb must be at least\nmax(1,n).\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n375\n\n\nlayout =\nCblasRowMajor\nldb must be at least\nmax(1,n).\nldb must be at least\nmax(1,k).\nstrideb\nStride between two consecutive b matrices.\ntransa=CblasNoTrans\ntransa=CblasTrans or\nCblasConjTrans\nlayout =\nCblasColMajor\nMust be at least ldb*n.\nMust be at least ldb*k.\nlayout =\nCblasRowMajor\nMust be at least ldb*k.\nMust be at least ldb*n.\nbeta\nSpecifies the scalar beta.\nc\nArray of size at least stridec*batch_size holding the c matrices.\nIf layout=CblasColMajor, before entry, the leading m-by-n part of the array\nc + i * stridec must contain the matrix Ci.\nIf layout=CblasRowMajor, before entry, the leading n-by-m part of the array\nc + i * stridec must contain the matrix Ci.\nldc\nSpecifies the leading dimension of the c matrices.\nMust be at least max(1,m) if layout=CblasColMajor or max(1,n) if\nlayout=CblasRowMajor.\nstridec\nSpecifies the stride between two consecutive c matrices.\nMust be at least ldc*nif layout=CblasColMajor or ldc*m if\nlayout=CblasRowMajor.\nbatch_size\nNumber of gemm computations to perform and a, b and c matrices. Must be\nat least 0.\nOutput Parameters\nc\nArray holding the batch_size updated c matrices.\nApplication Notes\nThese routines perform a complex matrix multiplication by forming the real and imaginary parts of the input\nmatrices. This uses three real matrix multiplications and five real matrix additions instead of the conventional\nfour real matrix multiplications and two real matrix additions. The use of three real matrix multiplications\nreduces the time spent in matrix operations by 25%, resulting in significant savings in compute time for\nlarge matrices.\nIf the errors in the floating point calculations satisfy the following conditions:\nfl(x op y)=(x op y)(1+δ),|δ|≤u, op=×,/, fl(x±y)=x(1+α)±y(1+β), |α|,|β|≤u\nthen for an n-by-n matrix Ĉ=fl(C1+iC2)=fl((A1+iA2)(B1+iB2))=Ĉ1+iĈ2, the following bounds are\nsatisfied:\n║Ĉ1-C1║≤ 2(n+1)u║A║∞║B║∞+O(u2),\n║Ĉ2-C2║≤ 4(n+4)u║A║∞║B║∞+O(u2),\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n376\n\n\nwhere ║A║∞=max(║A1║∞,║A2║∞), and ║B║∞=max(║B1║∞,║B2║∞).\nThus the corresponding matrix multiplications are stable.\ncblas_?gemm3m_batch\nComputes scalar-matrix-matrix products and adds the\nresults to scalar matrix products for groups of general\nmatrices.\nSyntax\nvoid cblas_cgemm3m_batch (const CBLAS_LAYOUT Layout, const CBLAS_TRANSPOSE*\ntransa_array, const CBLAS_TRANSPOSE* transb_array, const MKL_INT* m_array, const\nMKL_INT* n_array, const MKL_INT* k_array, const void *alpha_array, const void\n**a_array, const MKL_INT* lda_array, const void **b_array, const MKL_INT* ldb_array,\nconst void *beta_array, void **c_array, const MKL_INT* ldc_array, const MKL_INT\ngroup_count, const MKL_INT* group_size);\nvoid cblas_zgemm3m_batch (const CBLAS_LAYOUT Layout, const CBLAS_TRANSPOSE*\ntransa_array, const CBLAS_TRANSPOSE* transb_array, const MKL_INT* m_array, const\nMKL_INT* n_array, const MKL_INT* k_array, const void *alpha_array, const void\n**a_array, const MKL_INT* lda_array, const void **b_array, const MKL_INT* ldb_array,\nconst void *beta_array, void **c_array, const MKL_INT* ldc_array, const MKL_INT\ngroup_count, const MKL_INT* group_size);\nInclude Files\n•\nmkl.h\nDescription\nThe ?gemm3m_batch routines perform a series of matrix-matrix operations with general matrices. They are\nsimilar to the ?gemm3m routine counterparts, but the ?gemm3m_batch routines perform matrix-matrix\noperations with groups of matrices, processing a number of groups at once. The groups contain matrices with\nthe same parameters. The ?gemm3m_batch routines use fewer matrix multiplications than the ?gemm_batch\nroutines, as described in the Application Notes.\nThe operation is defined as\nidx = 0\nfor i = 0..group_count - 1\n     alpha and beta in alpha_array[i] and beta_array[i]\n     for j = 0..group_size[i] - 1 \n          A, B, and C matrix in a_array[idx], b_array[idx], and c_array[idx]\n          C := alpha*op(A)*op(B) + beta*C,\n          idx = idx + 1\n     end for\n end for\nwhere:\nop(X) is one of op(X) = X, or op(X) = XT, or op(X) = XH,\nalpha and beta are scalar elements of alpha_array and beta_array,\nA, B and C are matrices such that for m, n, and k which are elements of m_array, n_array, and k_array:\nop(A) is an m-by-k matrix,\nop(B) is a k-by-n matrix,\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n377\n\n\nC is an m-by-n matrix.\nA, B, and C represent matrices stored at addresses pointed to by a_array, b_array, and c_array,\nrespectively. The number of entries in a_array, b_array, and c_array is total_batch_count = the sum of all\nthe group_size entries.\nSee also gemm for a detailed description of multiplication for general matrices and gemm_batch, BLAS-like\nextension routines for similar matrix-matrix operations.\nNOTE\nError checking is not performed for Intel® oneAPI Math Kernel Library (oneMKL) Windows*\nsingle dynamic libraries for the?gemm3m_batch routines.\nInput Parameters\nLayout\nSpecifies whether two-dimensional array storage is row-major\n(CblasRowMajor) or column-major (CblasColMajor).\ntransa_array\nArray of size group_count. For the group i, transai = transa_array[i]\nspecifies the form of op(A) used in the matrix multiplication:\nif transai = CblasNoTrans, then op(A) = A;\nif transai = CblasTrans, then op(A) = AT;\nif transai = CblasConjTrans, then op(A) = AH.\ntransb_array\nArray of size group_count. For the group i, transbi = transb_array[i]\nspecifies the form of op(Bi) used in the matrix multiplication:\nif transbi = CblasNoTrans, then op(B) = B;\nif transbi = CblasTrans, then op(B) = BT;\nif transbi = CblasConjTrans, then op(B) = BH.\nm_array\nArray of size group_count. For the group i, mi = m_array[i] specifies the\nnumber of rows of the matrix op(A) and of the matrix C.\nThe value of each element of m_array must be at least zero.\nn_array\nArray of size group_count. For the group i, ni = n_array[i] specifies the\nnumber of columns of the matrix op(B) and the number of columns of the\nmatrix C.\nThe value of each element of n_array must be at least zero.\nk_array\nArray of size group_count. For the group i, ki = k_array[i] specifies the\nnumber of columns of the matrix op(A) and the number of rows of the\nmatrix op(B).\nThe value of each element of k_array must be at least zero.\nalpha_array\nArray of size group_count. For the group i, alpha_array[i] specifies the\nscalar alphai.\na_array\nArray, size total_batch_count, of pointers to arrays used to store A\nmatrices.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n378\n\n\nlda_array\nArray of size group_count. For the group i, ldai = lda_array[i]\nspecifies the leading dimension of the array storing matrix A as declared in\nthe calling (sub)program.\ntransai=CblasNoTrans\ntransai=CblasTrans or\ntransai=CblasConjTrans\nLayout =\nCblasColMajor\nldai must be at least\nmax(1, mi).\nldai must be at least\nmax(1, ki)\nLayout =\nCblasRowMajor\nldai must be at least\nmax(1, ki)\nldai must be at least\nmax(1, mi).\nb_array\nArray, size total_batch_count, of pointers to arrays used to store B\nmatrices.\nldb_array\nArray of size group_count. For the group i, ldbi = ldb_array[i]\nspecifies the leading dimension of the array storing matrix B as declared in\nthe calling (sub)program.\ntransbi=CblasNoTrans\ntransbi=CblasTrans or\ntransbi=CblasConjTrans\nLayout =\nCblasColMajor\nldbi must be at least\nmax(1, ki).\nldbi must be at least\nmax(1, ni).\nLayout =\nCblasRowMajor\nldbi must be at least\nmax(1, ni).\nldbi must be at least\nmax(1, ki).\nbeta_array\nFor the group i, beta_array[i] specifies the scalar betai.\nWhen betai is equal to zero, then C matrices in group i need not be set on\ninput.\nc_array\nArray, size total_batch_count, of pointers to arrays used to store C\nmatrices.\nldc_array\nArray of size group_count. For the group i, ldci = ldc_array[i]\nspecifies the leading dimension of all arrays storing matrix C in group i as\ndeclared in the calling (sub)program.\nWhen Layout = CblasColMajorldci must be at least max(1, mi).\nWhen Layout = CblasRowMajorldci must be at least max(1, ni).\ngroup_count\nSpecifies the number of groups. Must be at least 0.\ngroup_size\nArray of size group_count. The element group_size[i] specifies the\nnumber of matrices in group i. Each element in group_size must be at\nleast 0.\nOutput Parameters\nc_array\nOverwritten by the mi-by-ni matrix (alphai*op(A)*op(B) + betai*C) for\ngroup i.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n379\n\n\nApplication Notes\nThese routines perform a complex matrix multiplication by forming the real and imaginary parts of the input\nmatrices. This uses three real matrix multiplications and five real matrix additions instead of the conventional\nfour real matrix multiplications and two real matrix additions. The use of three real matrix multiplications\nreduces the time spent in matrix operations by 25%, resulting in significant savings in compute time for\nlarge matrices.\nIf the errors in the floating point calculations satisfy the following conditions:\nfl(x op y)=(x op y)(1+δ),|δ|≤u, op=×,/, fl(x±y)=x(1+α)±y(1+β), |α|,|β|≤u\nthen for an n-by-n matrix Ĉ=fl(C1+iC2)= fl((A1+iA2)(B1+iB2))=Ĉ1+iĈ2, the following bounds are\nsatisfied:\n║Ĉ1-C1║≤ 2(n+1)u║A║∞║B║∞+O(u2),\n║Ĉ2-C2║≤ 4(n+4)u║A║∞║B║∞+O(u2),\nwhere ║A║∞=max(║A1║∞,║A2║∞), and ║B║∞=max(║B1║∞,║B2║∞).\nThus the corresponding matrix multiplications are stable.\ncblas_?trsm_batch\nSolves a triangular matrix equation for a group of\nmatrices.\nSyntax\nvoid cblas_strsm_batch (const CBLAS_LAYOUT Layout, const CBLAS_SIDE *Side_Array, const\nCBLAS_UPLO *Uplo_Array, const CBLAS_TRANSPOSE *TransA_Array, const CBLAS_DIAG\n*Diag_Array, const MKL_INT *M_Array, const MKL_INT *N_Array, const float *alpha_Array,\nconst float * *A_Array, const MKL_INT *lda_Array, float * *B_Array, const MKL_INT\n*ldb_Array, const MKL_INT group_count, const MKL_INT *group_size );\nvoid cblas_dtrsm_batch (const CBLAS_LAYOUT Layout, const CBLAS_SIDE *Side_Array, const\nCBLAS_UPLO *Uplo_Array, const CBLAS_TRANSPOSE *Transa_Array, const CBLAS_DIAG\n*Diag_Array, const MKL_INT *M_Array, const MKL_INT *N_Array, const double *alpha_Array,\nconst double * *A_Array, const MKL_INT *lda_Array, double * *B_Array, const MKL_INT\n*ldb_Array, const MKL_INT group_count, const MKL_INT *group_size );\nvoid cblas_ctrsm_batch (const CBLAS_LAYOUT Layout, const CBLAS_SIDE *Side_Array, const\nCBLAS_UPLO *Uplo_Array, const CBLAS_TRANSPOSE *Transa_Array, const CBLAS_DIAG\n*Diag_Array, const MKL_INT *M_Array, const MKL_INT *N_Array, const void *alpha_Array,\nconst void * *A_Array, const MKL_INT *lda_Array, void * *B_Array, const MKL_INT\n*ldb_Array, const MKL_INT group_count, const MKL_INT *group_size );\nvoid cblas_ztrsm_batch (const CBLAS_LAYOUT Layout, const CBLAS_SIDE *Side_Array, const\nCBLAS_UPLO *Uplo_Array, const CBLAS_TRANSPOSE *Transa_Array, const CBLAS_DIAG\n*Diag_Array, const MKL_INT *M_Array, const MKL_INT *N_Array, const void *alpha_Array,\nconst void * *A_Array, const MKL_INT *lda_Array, void * *B_Array, const MKL_INT\n*ldb_Array, const MKL_INT group_count, const MKL_INT *group_size );\nInclude Files\n•\nmkl.h\nDescription\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n380\n\n\nThe ?trsm_batch routines solve a series of matrix equations. They are similar to the ?trsm routines except\nthat they operate on groups of matrices which have the same parameters. The ?trsm_batch routines\nprocess a number of groups at once.\nidx = 0\nfor i = 0..group_count - 1\n    alpha in alpha_array[i]\n    for j = 0..group_size[i] - 1\n        A and B matrix in a_array[idx] and b_array[idx]\n        Solve op(A)*X = alpha*B\n          or\n        Solve X*op(A) = alpha*B\n        idx = idx + 1\n    end for\nend for                                                                                        \nwhere:\nalpha is a scalar element of alpha_array,\nX and B are m-by-n matrices for m and n which are elements of m_array and n_array, respectively,\nA is a unit, or non-unit, upper or lower triangular matrix,\nand op(A) is one of op(A) = A, or op(A) = AT, or op(A) = conjg(AT).\nA and B represent matrices stored at addresses pointed to by a_array and b_array, respectively. There are\ntotal_batch_count entries in each of a_array and b_array, where total_batch_count is the sum of all the\ngroup_size entries.\nInput Parameters\nLayout\nSpecifies whether two-dimensional array storage is row-major\n(CblasRowMajor) or column-major (CblasColMajor).\nside_array\nArray of size group_count. For group i, 0 ≤i≤group_count - 1, sidei =\nside_array[i] specifies whether op(A) appears on the left or right of X in\nthe equation:\nif sidei = CblasLeft, then op(A)*X = alpha*B;\nif sidei = CblasRight, then X*op(A) = alpha*B.\nuplo_array\nArray of size group_count. For group i, 0 ≤i≤group_count - 1, uploi =\nuplo_array[i] specifies whether the matrix A is upper or lower triangular:\nuploi = CblasUpper\nif uploi = CblasLower, then the matrix is low triangular.\ntransa_array\nArray of size group_count. For group i, 0 ≤i≤group_count - 1, transai =\ntransa_array[i] specifies the form of op(A) used in the matrix\nmultiplication:\nif transai=CblasNoTrans, then op(A) = A;\nif transai=CblasTrans;\nif transai=CblasConjTrans, then op(A) = conjg(A').\ndiag_array\nArray of size group_count. For group i, 0 ≤i≤group_count - 1, diagi =\ndiag_array[i] specifies whether the matrix A is unit triangular:\nif diagi = CblasUnit then the matrix is unit triangular;\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n381\n\n\nif diagi = CblasNonUnit , then the matrix is not unit triangular.\nm_array\nArray of size group_count. For group i, 0 ≤i≤group_count - 1, mi =\nm_array[i] specifies the number of rows of B. The value of mi must be at\nleast zero.\nn_array\nArray of size group_count. For group i, 0 ≤i≤group_count - 1, ni =\nn_array[i] specifies the number of columns of B. The value of ni must be\nat least zero.\nalpha_array\nArray of size group_count. For group i, 0 ≤i≤group_count - 1,\nalpha_array[i] specifies the scalar alphai.\na_array\nArray, size total_batch_count, of pointers to arrays used to store A\nmatrices.\nFor group i, 0 ≤i≤group_count - 1, k is mi when sidei = CblasLeft and is ni\nwhen sidei = CblasRight and a is any of the group_size[i] arrays\nstarting with a_array[group_size[0] + group_size[1] + ... +\ngroup_size(i - 1)]:\nBefore entry with uploi = CblasUpper, the leading k by k upper triangular\npart of the array a must contain the upper triangular matrix and the strictly\nlower triangular part of a is not referenced.\nBefore entry with uploi = CblasLower lower triangular part of the array a\nmust contain the lower triangular matrix and the strictly upper triangular\npart of a is not referenced.\nWhen diagi = CblasUnit, the diagonal elements of a are not referenced\neither, but are assumed to be unity.\nlda_array\nArray of size group_count. For group i, 0 ≤i≤group_count - 1, ldai =\nlda_array[i] specifies the leading dimension of a as declared in the\ncalling (sub)program. When sidei = CblasLeft, then ldai must be at least\nmax(1, mi), when sidei = CblasRight, then ldai must be at least max(1,\nni).\nb_array\nArray, size total_batch_count, of pointers to arrays used to store B\nmatrices.\nFor group i, 0 ≤i≤group_count - 1, b is any of the group_size[i] arrays\nstarting with b_array[group_size[0] + group_size[1] + ... +\ngroup_size(i - 1)]:\nFor Layout = CblasColMajor: before entry, the leading mi-by-ni part of\nthe array b must contain the matrix B.\nFor Layout = CblasRowMajor: before entry, the leading ni-by-mi part of\nthe array b must contain the matrix B.\nldb_array\nArray of size group_count. Specifies the leading dimension of b as declared\nin the calling (sub)program. When Layout = CblasColMajor, ldb must be\nat least max(1, m); otherwise, ldb must be at least max(1, n).\nArray of size group_count. For group i, 0 ≤i≤group_count - 1, ldbi =\nldb_array[i] specifies the leading dimension of b as declared in the\ncalling (sub)program. When Layout = CblasColMajor, ldbi must be at\nleast max(1, mi); otherwise, ldbi must be at least max(1, ni).\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n382\n\n\ngroup_count\nSpecifies the number of groups. Must be at least 0.\ngroup_size\nArray of size group_count. The element group_size[i] specifies the\nnumber of matrices in group i. Each element in group_size must be at\nleast 0.\nOutput Parameters\nb_array\nOverwritten by the solution matrix X.\ncblas_?trsm_batch_strided\nSolves groups of triangular matrix equations.\nSyntax\nvoid cblas_strsm_batch_strided(const CBLAS_LAYOUT layout, const CBLAS_SIDE side, const\nCBLAS_UPLO uplo, const CBLAS_TRANSPOSE transa, const CBLAS_DIAG diag, const MKL_INT m,\nconst MKL_INT n, const float alpha, const float *a, const MKL_INT lda, const MKL_INT\nstridea, float *b, const MKL_INT ldb, const MKL_INT strideb, MKL_INT batch_size);\nvoid cblas_dtrsm_batch_strided(const CBLAS_LAYOUT layout, const CBLAS_SIDE side, const\nCBLAS_UPLO uplo, const CBLAS_TRANSPOSE transa, const CBLAS_DIAG diag, const MKL_INT m,\nconst MKL_INT n, const double alpha, const double *a, const MKL_INT lda, const MKL_INT\nstridea, double *b, const MKL_INT ldb, const MKL_INT strideb, const MKL_INT\nbatch_size);\nvoid cblas_ctrsm_batch_strided(const CBLAS_LAYOUT layout, const CBLAS_SIDE side, const\nCBLAS_UPLO uplo, const CBLAS_TRANSPOSE transa, const CBLAS_DIAG diag, const MKL_INT m,\nconst MKL_INT n, const void *alpha, const void *a, const MKL_INT lda, const MKL_INT\nstridea, void *b, const MKL_INT ldb, const MKL_INT strideb, const MKL_INT batch_size);\nvoid zblas_ctrsm_batch_strided(const CBLAS_LAYOUT layout, const CBLAS_SIDE side, const\nCBLAS_UPLO uplo, const CBLAS_TRANSPOSE transa, const CBLAS_DIAG diag, const MKL_INT m,\nconst MKL_INT n, const void *alpha, const void *a, const MKL_INT lda, const MKL_INT\nstridea, void *b, const MKL_INT ldb, const MKL_INT strideb, const MKL_INT batch_size);\nInclude Files\n•\nmkl.h\nDescription\nThe cblas_?trsm_batch_strided routines solve a series of triangular matrix equations. They are similar to\nthe cblas_?trsm routine counterparts, but the cblas_?trsm_batch_strided routines solve triangular\nmatrix equations with groups of matrices. All matrix a have the same parameters (size, leading dimension,\nside, uplo, diag, transpose operation) and are stored at constant stridea from each other. Similarly, all matrix\nb have the same parameters (size, leading dimension, alpha scaling) and are stored at constant strideb from\neach other.\nThe operation is defined as\nInput Parameters\nLayout\nSpecifies whether two-dimensional array storage is row-major\n(CblasRowMajor) or column-major (CblasColMajor).\nside\nSpecifies whether op(A) appears on the left or right of X in the equation.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n383\n\n\nif side = CblasLeft, then op(A)*X = alpha*B;\nif side = CblasRight, then X*op(A) = alpha*B.\nuplo\nSpecifies whether the matrices A are upper or lower triangular.\nif uplo = CblasUpper, then A are upper triangular;\nif uplo = CblasLower, then A are lower triangular.\ntransa\nSpecifies op(A) the transposition operation applied to the matrices A.\nif transa = CblasNoTrans, then op(A) = A;\nif transa = CblasTrans, then op(A) = AT;\nif transa = CblasConjTrans, then op(A) = AH;\ndiag\nSpecifies whether the matrices A are unit triangular.\nif diag = CblasUnit, then A are unit triangular;\nif diag = CblasLower, then A are non-unit triangular.\nm\nNumber of rows of B matrices. Must be at least 0\nn\nNumber of columns of B matrices. Must be at least 0\nalpha\nSpecifies the scalar alpha.\na\nArray of size at least stridea*batch_size holding the A matrices. Each A\nmatrix is stored at constant stridea from each other.\nEach A matrix has size lda* k, where k is m when side = CblasLeft and is\nn when side = CblasRight.\nBefore entry with uplo = CblasUpper, the leading k-by-k upper triangular\npart of the array A must contain the upper triangular matrix and the strictly\nlower triangular part of A is not referenced.\nBefore entry with uplo = CblasLower lower triangular part of the array A\nmust contain the lower triangular matrix and the strictly upper triangular\npart of A is not referenced.\nWhen diag = CblasUnit, the diagonal elements of A are not referenced\neither, but are assumed to be unity.\nlda\nSpecifies the leading dimension of the A matrices. When side = CblasLeft,\nthen lda must be at least max(1, m), when side = side = CblasRight, then\nlda must be at least max(1, n).\nstridea\nStride between two consecutive A matrices.\nWhen side = CblasLeft, then stridea must be at least lda*m.\nWhen side = side = CblasRight, then stridea must be at least lda*n.\nb\nArray of size at least strideb*batch_size holding the B matrices. Each B\nmatrix is stored at constant strideb from each other.\nWhen layout= CblasColMajor, each B matrix has size ldb* n. Before entry,\nthe leading m-by-n part of the array B must contain the matrix B.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n384\n\n\nWhen layout= CblasRowMajor, each B matrix has size ldb* m. Before entry,\nthe leading n-by-m part of the array B must contain the matrix B.\nldb\nSpecifies the leading dimension of the B matrices.\nWhen layout= CblasColMajor, strideb must be at least max(1,m).\nOtherwise, strideb must be at least max(1,n).\nstrideb\nStride between two consecutive B matrices.\nWhen layout= CblasColMajor, strideb must be at least ldb*n. Otherwise,\nstrideb must be at least ldb*m.\nbatch_size\nNumber of trsm computations to perform. Must be at least 0.\nOutput Parameters\nb\nOverwritten by the solution batch_size X matrices.\nmkl_?imatcopy\nPerforms scaling and in-place transposition/copying of\nmatrices.\nSyntax\nvoid mkl_simatcopy (const char ordering, const char trans, size_t rows, size_t cols,\nconst float alpha, float * AB, size_t lda, size_t ldb);\nvoid mkl_dimatcopy (const char ordering, const char trans, size_t rows, size_t cols,\nconst double alpha, double * AB, size_t lda, size_t ldb);\nvoid mkl_cimatcopy (const char ordering, const char trans, size_t rows, size_t cols,\nconst MKL_Complex8 alpha, MKL_Complex8 * AB, size_t lda, size_t ldb);\nvoid mkl_zimatcopy (const char ordering, const char trans, size_t rows, size_t cols,\nconst MKL_Complex16 alpha, MKL_Complex16 * AB, size_t lda, size_t ldb);\nInclude Files\n•\nmkl.h\nDescription\nThe mkl_?imatcopy routine performs scaling and in-place transposition/copying of matrices. A transposition\noperation can be a normal matrix copy, a transposition, a conjugate transposition, or just a conjugation. The\noperation is defined as follows:\nAB := alpha*op(AB).\nNOTE\nDifferent arrays must not overlap.\nInput Parameters\nordering\nOrdering of the matrix storage.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n385\n\n\nIf ordering = 'R' or 'r', the ordering is row-major.\nIf ordering = 'C' or 'c', the ordering is column-major.\ntrans\nParameter that specifies the operation type.\nIf trans = 'N' or 'n', op(AB)=AB and the matrix AB is assumed\nunchanged on input.\nIf trans = 'T' or 't', it is assumed that AB should be transposed.\nIf trans = 'C' or 'c', it is assumed that AB should be conjugate\ntransposed.\nIf trans = 'R' or 'r', it is assumed that AB should be only conjugated.\nIf the data is real, then trans = 'R' is the same as trans = 'N', and\ntrans = 'C' is the same as trans = 'T'.\nrows\nThe number of rows in matrix AB before the transpose operation.\ncols\nThe number of columns in matrix AB before the transpose operation.\nab\nArray.\nalpha\nThis parameter scales the input matrix by alpha.\nlda\nDistance between the first elements in adjacent columns (in the case of the\ncolumn-major order) or rows (in the case of the row-major order) in the\nsource matrix; measured in the number of elements.\nThis parameter must be at least rows if ordering = 'C' or 'c', and\nmax(1,cols) otherwise.\nldb\nDistance between the first elements in adjacent columns (in the case of the\ncolumn-major order) or rows (in the case of the row-major order) in the\ndestination matrix; measured in the number of elements.\nTo determine the minimum value of ldb on output, consider the following\nguideline:\nIf ordering = 'C' or 'c', then\n•\nIf trans = 'T' or 't' or 'C' or 'c', this parameter must be at least\nmax(1,cols)\n•\nIf trans = 'N' or 'n' or 'R' or 'r', this parameter must be at least\nmax(1,rows)\nIf ordering = 'R' or 'r', then\n•\nIf trans = 'T' or 't' or 'C' or 'c', this parameter must be at least\nmax(1,rows)\n•\nIf trans = 'N' or 'n' or 'R' or 'r', this parameter must be at least\nmax(1,cols)\nOutput Parameters\nab\nArray.\nContains the matrix AB.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n386\n\n\nApplication Notes\nFor threading to be active in mkl_?imatcopy, the pointer AB must be aligned on the 64-byte boundary. This\nrequirement can be met by allocating AB with mkl_malloc.\nInterfaces\nmkl_?imatcopy_batch\nComputes a group of in-place scaled matrix copy or\ntransposition operations on general matrices.\nSyntax\nvoid mkl_simatcopy_batch (char layout, const char * trans_array, const size_t *\nrows_array, const size_t * cols_array, const float * alpha_array, float ** ab_array,\nconst size_t * lda_array, const size_t * ldb_array, size_t group_count, const size_t *\ngroup_size);\nvoid mkl_dimatcopy_batch (char layout, const char * trans_array, const size_t *\nrows_array, const size_t * cols_array, const double * alpha_array, double ** ab_array,\nconst size_t * lda_array, const size_t * ldb_array, size_t group_count, const size_t *\ngroup_size);\nvoid mkl_cimatcopy_batch (char layout, const char * trans_array, const size_t *\nrows_array, const size_t * cols_array, const MKL_Complex8 * alpha_array, MKL_Complex8 **\nab_array, const size_t * lda_array, const size_t * ldb_array, size_t group_count, const\nsize_t * group_size);\nvoid mkl_zimatcopy_batch (char layout, const char * trans_array, const size_t *\nrows_array, const size_t * cols_array, const MKL_Complex16 * alpha_array, MKL_Complex16\n** ab_array, const size_t * lda_array, const size_t * ldb_array, size_t group_count,\nconst size_t * group_size);\nDescription\nThe mkl_?imatcopy_batch routine performs a series of in-place scaled matrix copies or transpositions. They\nare similar to the mkl_?imatcopy routine counterparts, but the mkl_?imatcopy_batch routine performs\nmatrix operations with groups of matrices. Each group has the same parameters (matrix size, leading\ndimension, and scaling parameter), but a single call to mkl_?imatcopy_batch operates on multiple groups,\nand each group can have different parameters, unlike the related mkl_?imatcopy_batch_strided routines.\nThe operation is defined as\nidx = 0\nfor i = 0..group_count - 1\n     m in rows_array[i], n in cols_array[i], and alpha in alpha_array[i]\n     for j = 0..group_size[i] - 1 \n          AB matrices in AB_array[idx]\n          AB := alpha*op(AB)\n          idx = idx + 1\n     end for\nend for\nWhere op(X) is one of op(X)=X, op(X)=X', op(X)=conjg(X'), or op(X)=conjg(X). On entry, AB is a m-\nby-n matrix such that m and n are elements of rows_array and cols_array.\nAB represents a matrix stored at addresses pointed to by AB_array. The number of entries in AB_array is\ntotal_batch_count = the sum of all of the group_size entries.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n387\n\n\nInput Parameters\nlayout\nSpecifies whether two-dimensional array storage is row-major (R) or\ncolumn-major (C).\ntrans_array\nArray of size group_count. For the group i, trans = trans_array[i]\nspecifies the form of op(AB), the transposition operation applied to the AB\nmatrix:\nIf trans = 'N' or 'n', op(AB)=AB.\nIf trans = 'T' or 't', op(AB)=AB'\nIf trans = 'C' or 'c', op(AB)=conjg(AB')\nIf trans = 'R' or 'r', op(AB)=conjg(AB)\nrows_array\nArray of size group_count. Specifies the number of rows of the input\nmatrix AB. The value of each element must be at least zero.\ncols_array\nArray of size group_count. Specifies the number of columns of the input\nmatrix AB. The value of each element must be at least zero.\nalpha_array\nArray of size group_count. Specifies the scalar alpha.\nAB_array\nArray of size total_batch_count, holding pointers to arrays used to store AB\nmatrices.\nlda_array\nArray of size group_count. The leading dimension of the matrix input AB.\nIt must be positive and at least m if column major layout is used or at least\nn if row major layout is used.\nldb_array\nArray of size group_count. The leading dimension of the matrix input AB.\nIt must be positive and at least\nm if column major layout is used and op(AB) = AB or conjg(AB)\nn if row major layout is used and op(AB) = AB' or conjg(AB')\nn otherwise\ngroup_count\nSpecifies the number of groups. Must be at least 0\ngroup_size\nArray of size group_count. The element group_size[i] specifies the\nnumber of matrices in group i. Each element in group_size must be at\nleast 0.\nOutput Parameters\nAB_array\nOutput array of size total_batch_count, holding pointers to arrays used to\nstore the updated AB matrices.\nmkl_?imatcopy_batch_strided\nComputes a group of in-place scaled matrix copy or\ntransposition using general matrices.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n388\n\n\nSyntax\nvoid mkl_simatcopy_batch_strided (const char layout, const char trans, size_t row,\nsize_t col, const float alpha, float * ab, size_t lda, size_t ldb, size_t stride, size_t\nbatch_size);\nvoid mkl_dimatcopy_batch_strided (const char layout, const char trans, size_t row,\nsize_t col, const double alpha, double * ab, size_t lda, size_t ldb, size_t stride,\nsize_t batch_size);\nvoid mkl_cimatcopy_batch_strided (const char layout, const char trans, size_t row,\nsize_t col, MKL_complex8 alpha, MKL_complex8 * ab, size_t lda, size_t ldb, size_t\nstride, size_t batch_size);\nvoid mkl_zimatcopy_batch_strided (const char layout, const char trans, size_t row,\nsize_t col, MKL_complex16 alpha, MKL_complex16 * ab, size_t lda, size_t ldb, size_t\nstride, size_t batch_size);\nDescription\nThe mkl_?imatcopy_batch_strided routine performs a series of scaled matrix copy or transposition. They\nare similar to the mkl_?imatcopy routine counterparts, but the mkl_?imatcopy_batch_strided routine\nperforms matrix operations with a group of matrices.\nAll matrices ab have the same parameters (size, transposition operation…) and are stored at constant stride\nfrom each other. The operation is defined as\nfor i = 0 … batch_size – 1\n    AB is a matrix at offset i * stride in ab\n    AB = alpha * op(AB)\nend for\nInput Parameters\nlayout\nSpecifies whether two-dimensional array storage is row-major\n(CblasRowMajor) or column-major (CblasColMajor)\ntrans\nSpecifies op(AB), the transposition operation applied to the AB matrices.\nIf trans = 'N' or 'n', op(AB)=AB.\nIf trans = 'T' or 't', op(AB)=AB'\nIf trans = 'C' or 'c', op(AB)=conjg(AB')\nIf trans = 'R' or 'r', op(AB)=conjg(AB)\nrow\nSpecifies the number of rows of the matrices AB. The value of row must be\nat least zero.\ncol\nSpecifies the number of columns of the matrices AB. The value of col must\nbe at least zero.\nalpha\nSpecifies the scalar alpha.\nab\nArray holding all the input matrix AB. Must be of size at least batch_size\n* stride.\nlda\nThe leading dimension of the matrix input AB. It must be positive and at\nleast row if column major layout is used or at least col if row major layout\nis used.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n389\n\n\nldb\nThe leading dimension of the matrix input AB. It must be positive and at\nleast\nrow if column major layout is used and op(AB) = AB or conjg(AB)\nrow if row major layout is used and op(AB) = AB' or conjg(AB')\ncol otherwise\nstride\nStride between two consecutive AB matrices, must be at least\nmax(ldb,lda)*max(ka, kb) where\n•\nka is row if column major layout is used or col if row major layout is\nused\n•\nkb is col if column major layout is used and op(AB) = AB or\nconjg(AB) or row major layout is used and op(AB) = AB' or\nconjg(AB'); kb is row otherwise.\nbatch_size\nNumber of imatcopy computations to perform and AB matrices. Must be at\nleast 0.\nOutput Parameters\nab\nArray holding the batch_size updated matrices AB.\nmkl_?omatadd_batch_strided\nComputes a group of out-of-place scaled matrix\nadditions using general matrices.\nSyntax\nvoid mkl_somatadd_batch_strided(char ordering, char transa, char transb, size_t rows,\nsize_t cols, float alpha, const float * A, size_t lda, size_t stridea, float beta,\nconst float * B, size_t ldb, size_t strideb, float * C, size_t ldc, size_t stridec,\nsize_t batch_size);\nvoid mkl_domatadd_batch_strided(char ordering, char transa, char transb, size_t rows,\nsize_t cols, double alpha, const double * A, size_t lda, size_t stridea, double beta,\nconst double * B, size_t ldb, size_t strideb, double * C, size_t ldc, size_t stridec,\nsize_t batch_size);\nvoid mkl_comatadd_batch_strided(char ordering, char transa, char transb, size_t rows,\nsize_t cols, MKL_Complex8 alpha, const MKL_Complex8 * A, size_t lda, size_t stridea,\nMKL_Complex8 beta, const MKL_Complex8 * B, size_t ldb, size_t strideb, MKL_Complex8 *\nC, size_t ldc, size_t stridec, size_t batch_size);\nvoid mkl_zomatadd_batch_strided(char ordering, char transa, char transb, size_t rows,\nsize_t cols, MKL_Complex16 alpha, const MKL_Complex16 * A, size_t lda, size_t stridea,\nMKL_Complex16 beta, const MKL_Complex16 * B, size_t ldb, size_t strideb, MKL_Complex16\n* C, size_t ldc, size_t stridec, size_t batch_size);\nDescription\nThe mkl_omatadd_batch_strided routines perform a series of scaled matrix additions. They are similar to\nthe mkl_omatadd routines, but the mkl_omatadd_batch_strided routines perform matrix operations with a\ngroup of matrices.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n390\n\n\nThe matrices A, B, and C are stored at a constant stride from each other in memory, given by the parameters\nstridea, strideb, and stridec. The operation is defined as:\nfor i = 0 … batch_size – 1\n    A is a matrix at offset i * stridea in the array a\n    B is a matrix at offset i * strideb in the array b\n    C is a matrix at offset i * stridec in the array c\n    C = alpha * op(A) + beta * op(B)\nend for\nwhere:\n•\nop(X) is one of op(X) = X, op(X) = X', op(X) = conjg(X) or op(X) = conjg(X').\n•\nalpha and beta are scalars.\n•\nA, B, and C are matrices.\nThe input arrays a and b contain all the input matrices, and the single output array c contains all the output\nmatrices. The locations of the individual matrices within the array are given by stride lengths, while the\nnumber of matrices is given by the batch_size parameter.\nIn general, the a, b, and c arrays must not overlap in memory, with the exception of the following in-place\noperations:\n•\na and c can point to the same memory if transa is non-transpose and all the A matrices within a have\nthe same parameters as all the respective C matrices within c.\n•\nb and c can point to the same memory if transb is non-transpose and all the B matrices within b have\nthe same parameters as all the respective C matrices within c.\nInput Parameters\nlayout\nSpecifies whether two-dimensional array storage is row-major\n(CblasRowMajor) or column-major (CblasColMajor).\ntransa\nSpecifies op(A), the transposition operation applied to the matrices\nA. 'N' or 'n' indicates no operation, 'T' or 't' is transposition, 'R' or 'r'\nis complex conjugation wtihout tranpsosition, and 'C' or 'c' is\nconjugate transposition.\ntransb\nSpecifies op(B), the transposition operation applied to the matrices\nB.\nrows\nNumber of rows for the result matrix C. Must be at least zero.\ncols\nNumber of columns for the result matrix C. Must be at least zero.\nalpha\nScaling factor for the matrices A.\na\nArray holding the input matrices A. Must have size at least\nstride_a*batch_size.\nlda\nLeading dimension of the A matrices. If matrices are stored using\ncolumn major layout, lda must be at least rows if A is not\ntransposed or cols if A is transposed. If matrices are stored using\nrow major layout, lda must be at least cols if A is not transposed or\nat least rows if A is transposed. Must be positive.\nstride_a\nStride between the different A matrices. If matrices are stored using\ncolumn major layout, stride_a must be at least lda*rows if A is not\ntransposed or at least lda*cols if A is transposed. If matrices are\nstored using row major layout, stride_a must be at least lda*rows\nif B is not transposed or at least lda*cols if A is transposed.\nbeta\nScaling factor for the matrices B.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n391\n\n\nb\nArray holding the input matrices B. Must have size at least\nstride_b*batch_size.\nldb\nLeading dimension of the B matrices. If matrices are stored using\ncolumn major layout, ldb must be at least rows if B is not\ntransposed or cols if B is transposed. If matrices are stored using\nrow major layout, ldb must be at least cols if B is not transposed or\nat least rows if B is transposed. Must be positive.\nstride_b\nStride between the different B matrices. If matrices are stored using\ncolumn major layout, stride_b must be at least ldb*cols if B is not\ntransposed or at least ldb*rows if B is transposed. If matrices are\nstored using row major layout, stride_b must be at least ldb*rows\nif B is not transposed or at least ldb*cols if B is transposed.\nc\nOutput array, overwritten by batch_size matrix addition operations\nof the form alpha*op(A) + beta*op(B). Must have size at least\nstride_c*batch_size.\nldc\nLeading dimension of the A matrices. If matrices are stored using\ncolumn major layout, lda must be at least rows. If matrices are\nstored using row major layout, lda must be at least cols. Must be\npositive.\nstride_c\nStride between the different C matrices. If matrices are stored using\ncolumn major layout, stride_c must be at least ldc*cols. If\nmatrices are stored using row major layout, stride_c must be at\nleast ldc*rows.\nbatch_size\nSpecifies the number of input and output matrices to add.\nOutput Parameters\nc\nArray holding the updated matrices C.\nmkl_?omatcopy\nPerforms scaling and out-place transposition/copying\nof matrices.\nSyntax\nvoid mkl_somatcopy (char ordering, char trans, size_t rows, size_t cols, const float\nalpha, const float * A, size_t lda, float * B, size_t ldb);\nvoid mkl_domatcopy (char ordering, char trans, size_t rows, size_t cols, const double\nalpha, const double * A, size_t lda, double * B, size_t ldb);\nvoid mkl_comatcopy (char ordering, char trans, size_t rows, size_t cols, const\nMKL_Complex8 alpha, const MKL_Complex8 * A, size_t lda, MKL_Complex8 * B, size_t ldb);\nvoid mkl_zomatcopy (char ordering, char trans, size_t rows, size_t cols, const\nMKL_Complex16 alpha, const MKL_Complex16 * A, size_t lda, MKL_Complex16 * B, size_t\nldb);\nInclude Files\n•\nmkl.h\nDescription\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n392\n\n\nThe mkl_?omatcopy routine performs scaling and out-of-place transposition/copying of matrices. A\ntransposition operation can be a normal matrix copy, a transposition, a conjugate transposition, or just a\nconjugation. The operation is defined as follows:\nB := alpha*op(A)\nNOTE\nDifferent arrays must not overlap.\nInput Parameters\nordering\nOrdering of the matrix storage.\nIf ordering = 'R' or 'r', the ordering is row-major.\nIf ordering = 'C' or 'c', the ordering is column-major.\ntrans\nParameter that specifies the operation type.\nIf trans = 'N' or 'n', op(A)=A and the matrix A is assumed unchanged\non input.\nIf trans = 'T' or 't', it is assumed that A should be transposed.\nIf trans = 'C' or 'c', it is assumed that A should be conjugate\ntransposed.\nIf trans = 'R' or 'r', it is assumed that A should be only conjugated.\nIf the data is real, then trans = 'R' is the same as trans = 'N', and\ntrans = 'C' is the same as trans = 'T'.\nrows\nThe number of rows in matrix A (the input matrix).\ncols\nThe number of columns in matrix A (the input matrix).\nalpha\nThis parameter scales the input matrix by alpha.\na\nInput array.\nIf ordering = 'R' or 'r', the size of a is lda*rows.\nIf ordering = 'C' or 'c', the size of a is lda*cols.\nlda\nIf ordering = 'R' or 'r', lda represents the number of elements in array\na between adjacent rows of matrix A; lda must be at least equal to the\nnumber of columns of matrix A.\nIf ordering = 'C' or 'c', lda represents the number of elements in array\na between adjacent columns of matrix A; lda must be at least equal to the\nnumber of row in matrix A.\nb\nOutput array.\nIf ordering = 'R' or 'r';\n•\nIf trans = 'T' or 't' or 'C' or 'c', the size of b is ldb * cols.\n•\nIf trans = 'N' or 'n' or 'R' or 'r', the size of b is ldb * rows.\nIf ordering = 'C' or 'c';\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n393\n\n\n•\nIf trans = 'T' or 't' or 'C' or 'c', the size of b is ldb * rows.\n•\nIf trans = 'N' or 'n' or 'R' or 'r', the size of b is ldb * cols.\nldb\nIf ordering = 'R' or 'r', ldb represents the number of elements in array\nb between adjacent rows of matrix B.\n•\nIf trans = 'T' or 't' or 'C' or 'c', ldb must be at least equal to\nrows.\n•\nIf trans = 'N' or 'n' or 'R' or 'r', ldb must be at least equal to\ncols.\nIf ordering = 'C' or 'c', ldb represents the number of elements in array\nb between adjacent columns of matrix B.\n•\nIf trans = 'T' or 't' or 'C' or 'c', ldb must be at least equal to\ncols.\n•\nIf trans = 'N' or 'n' or 'R' or 'r', ldb must be at least equal to\nrows.\nOutput Parameters\nb\nOutput array.\nContains the destination matrix.\nInterfaces\nmkl_?omatcopy_batch\nComputes a group of out of place scaled matrix copy\nor transposition operations on general matrices.\nSyntax\nvoid mkl_somatcopy_batch (char layout, const char * trans_array, const size_t *\nrows_array, const size_t * cols_array, const float * alpha_array, float ** A_array,\nconst size_t * lda_array, float ** B_array, const size_t * ldb_array, size_t\ngroup_count, const size_t * group_size);\nvoid mkl_domatcopy_batch (char layout, const char * trans_array, const size_t *\nrows_array, const size_t * cols_array, const double * alpha_array, float ** A_array,\nconst size_t * lda_array, double ** B_array, const size_t * ldb_array, size_t\ngroup_count, const size_t * group_size);\nvoid mkl_comatcopy_batch (char layout, const char * trans_array, const size_t *\nrows_array, const size_t * cols_array, const MKL_Complex8 * alpha_array, MKL_Complex8 **\nA_array, const size_t * lda_array, MKL_Complex8 ** B_array, const size_t * ldb_array,\nsize_t group_count, const size_t * group_size);\nvoid mkl_zomatcopy_batch (char layout, const char * trans_array, const size_t *\nrows_array, const size_t * cols_array, const MKL_Complex16 * alpha_array, MKL_Complex16\n** A_array, const size_t * lda_array, MKL_Complex16 ** B_array, const size_t *\nldb_array, size_t group_count, const size_t * group_size);\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n394\n\n\nDescription\nThe mkl_?omatcopy_batch routine performs a series of out-of-place scaled matrix copies or transpositions.\nThey are similar to the mkl_?omatcopy routine counterparts, but the mkl_?omatcopy_batch routine\nperforms matrix operations with groups of matrices. Each group has the same parameters (matrix size,\nleading dimension, and scaling parameter), but a single call to mkl_?omatcopy_batch operates on multiple\ngroups, and each group can have different parameters, unlike the related mkl_?omatcopy_batch_strided\nroutines.\nThe operation is defined as\nidx = 0\nfor i = 0..group_count - 1\n     m in rows_array[i], n in cols_array[i], and alpha in alpha_array[i]\n     for j = 0..group_size[i] - 1 \n          A and B matrices in a_array[idx] and b_array[idx], respectively\n          B := alpha*op(A)\n          idx = idx + 1\n     end for\nend for\nWhere op(X) is one of op(X)=X, op(X)=X', op(X)=conjg(X'), or op(X)=conjg(X). A is a m-by-n matrix\nsuch that m and n are elements of rows_array and cols_array.\nA and B represent matrices stored at addresses pointed to by A_array and B_array. The number of entries in\nA_array and B_array is total_batch_count = the sum of all of the group_size entries.\nInput Parameters\nlayout\nSpecifies whether two-dimensional array storage is row-major (R) or\ncolumn-major (C).\ntrans_array\nArray of size group_count. For the group i, trans = trans_array[i]\nspecifies the form of op(A), the transposition operation applied to the A\nmatrix:\nIf trans = 'N' or 'n', op(A)=A.\nIf trans = 'T' or 't', op(A)=A'\nIf trans = 'C' or 'c', op(A)=conjg(A')\nIf trans = 'R' or 'r', op(A)=conjg(A)\nrows_array\nArray of size group_count. Specifies the number of rows of the matrix A.\nThe value of each element must be at least zero.\ncols_array\nArray of size group_count. Specifies the number of columns of the matrix\nA. The value of each element must be at least zero.\nalpha_array\nArray of size group_count. Specifies the scalar alpha.\nA_array\nArray of size total_batch_count, holding pointers to arrays used to store A\ninput matrices.\nlda_array\nArray of size group_count. The leading dimension of the input matrix A. It\nmust be positive and at least m if column major layout is used or at least n\nif row major layout is used.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n395\n\n\nldb_array\nArray of size group_count. The leading dimension of the output matrix B.\nIt must be positive and at least\nm if column major layout is used and op(A) = A or conjg(A)\nn if row major layout is used and op(A) = A' or conjg(A')\nn otherwise\ngroup_count\nSpecifies the number of groups. Must be at least 0\ngroup_size\nArray of size group_count. The element group_size[i] specifies the\nnumber of matrices in group i. Each element in group_size must be at\nleast 0.\nOutput Parameters\nB_array\nOutput array of size total_batch_count, holding pointers to arrays used to\nstore the B output matrices, the contents of which are overwritten by the\noperation of the form alpha*op(A).\nmkl_?omatcopy_batch_strided\nComputes a group of out of place scaled matrix copy\nor transposition using general matrices.\nSyntax\nvoid mkl_somatcopy_batch_strided (const char layout, const char trans, size_t row,\nsize_t col, const float alpha, const float * a, size_t lda, size_t stridea, float * b,\nsize_t ldb, size_t strideb, size_t batch_size);\nvoid mkl_domatcopy_batch_strided (const char layout, const char trans, size_t row,\nsize_t col, const double alpha, const double * a, size_t lda, size_t stridea, double *\nb, size_t ldb, size_t strideb, size_t batch_size);\nvoid mkl_comatcopy_batch_strided (const char layout, const char trans, size_t row,\nsize_t col, const MKL_complex8 alpha, const MKL_complex8 * a, size_t lda, size_t\nstridea, MKL_complex8 * b, size_t ldb, size_t strideb, size_t batch_size);\nvoid mkl_zomatcopy_batch_strided (const char layout, const char trans, size_t row,\nsize_t col, const MKL_complex16 alpha, const MKL_complex16 * a, size_t lda, size_t\nstridea, MKL_complex16 * b, size_t ldb, size_t strideb, size_t batch_size);\nDescription\nThe mkl_?omatcopy_batch_strided routine performs a series of out-of-place scaled matrix copy or\ntransposition. They are similar to the mkl_?omatcopy routine counterparts, but the\nmkl_?omatcopy_batch_strided routine performs matrix operations with group of matrices.\nAll matrices a and b have the same parameters (size, transposition operation…) and are stored at constant\nstride from each other respectively given by stridea and strideb. The operation is defined as\nfor i = 0 … batch_size – 1\n    A and B are matrices at offset i * stridea in a and I * strideb in b\n    B = alpha * op(A)\nend for\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n396\n\n\nInput Parameters\nlayout\nSpecifies whether two-dimensional array storage is row-major\n(CblasRowMajor) or column-major (CblasColMajor).\ntrans\nSpecifies op(A), the transposition operation applied to the AB matrices.\nIf trans = 'N' or 'n', op(A)=A.\nIf trans = 'T' or 't', op(A)=A'\nIf trans = 'C' or 'c', op(A)=conig(A')\nIf trans = 'R' or 'r', op(A)=conig(A)\nrow\nSpecifies the number of rows of the matrices A and B. The value of row\nmust be at least zero.\ncol\nSpecifies the number of columns of the matrices A and B. The value of col\nmust be at least zero.\nalpha\nSpecifies the scalar alpha.\na\nArray holding all the input matrices A. Must be of size at least lda * k +\nstridea * (batch_size - 1) * stridea where k is col if column\nmajor is used and row otherwise.\nlda\nThe leading dimension of the matrix input A. It must be positive and at\nleast row if column major layout is used or at least col if row major layout\nis used.\nstridea\nStride between two consecutive A matrices, must be at least 0.\nb\nArray holding all the output matrices B. Must be of size at least batch_size\n* strideb. The b array must be independent from the a array.\nldb\nThe leading dimension of the output matrix B. It must be positive and at\nleast:\n•\nrow if column major layout is used and op(A) = A or conjg(A)\n•\nrow if row major layout is used and op(A) = A' or conjg(A')\n•\ncol otherwise\nstrideb\nStride between two consecutive B matrices. It must be positive and at\nleast:\n•\nldb* col if column major layout is used and op(A) = A or conjg(A)\n•\nldb* col if row major layout is used and op(A) = A' or conjg(A')\n•\nldb*row otherwise\nbatch_size\nOutput Parameters\nb\nArray holding the batch_size updated matrices B.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n397\n\n\nmkl_?omatcopy2\nPerforms two-strided scaling and out-of-place\ntransposition/copying of matrices.\nSyntax\nvoid mkl_somatcopy2 (char ordering, char trans, size_t rows, size_t cols, const float\nalpha, const float * A, size_t lda, size_t stridea, float * B, size_t ldb, size_t\nstrideb);\nvoid mkl_domatcopy2 (char ordering, char trans, size_t rows, size_t cols, const double\nalpha, const double * A, size_t lda, size_t stridea, double * B, size_t ldb, size_t\nstrideb);\nvoid mkl_comatcopy2 (char ordering, char trans, size_t rows, size_t cols, const\nMKL_Complex8 alpha, const MKL_Complex8 * A, size_t lda, size_t stridea, MKL_Complex8 *\nB, size_t ldb, size_t strideb);\nvoid mkl_zomatcopy2 (char ordering, char trans, size_t rows, size_t cols, const\nMKL_Complex16 alpha, const MKL_Complex16 * A, size_t lda, size_t stridea, MKL_Complex16\n* B, size_t ldb, size_t strideb);\nInclude Files\n•\nmkl.h\nDescription\nThe mkl_?omatcopy2 routine performs two-strided scaling and out-of-place transposition/copying of\nmatrices. A transposition operation can be a normal matrix copy, a transposition, a conjugate transposition,\nor just a conjugation. The operation is defined as follows:\nB := alpha*op(A)\nNormally, matrices in the BLAS or LAPACK are specified by a single stride index. For instance, in the column-\nmajor order, A(2,1) is stored in memory one element away from A(1,1), but A(1,2) is a leading dimension\naway. The leading dimension in this case is at least the number of rows of the source matrix. If a matrix has\ntwo strides, then both A(2,1) and A(1,2) may be an arbitrary distance from A(1,1).\nNOTE\nDifferent arrays must not overlap.\nInput Parameters\nordering\nOrdering of the matrix storage.\nIf ordering = 'R' or 'r', the ordering is row-major.\nIf ordering = 'C' or 'c', the ordering is column-major.\ntrans\nParameter that specifies the operation type.\nIf trans = 'N' or 'n', op(A)=A and the matrix A is assumed unchanged\non input.\nIf trans = 'T' or 't', it is assumed that A should be transposed.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n398\n\n\nIf trans = 'C' or 'c', it is assumed that A should be conjugate\ntransposed.\nIf trans = 'R' or 'r', it is assumed that A should be only conjugated.\nIf the data is real, then trans = 'R' is the same as trans = 'N', and\ntrans = 'C' is the same as trans = 'T'.\nrows\nnumber of rows for the input matrix A. Must be at least zero.\ncols\nNumber of columns for the input matrix A. Must be at least zero.\nalpha\nScaling factor for the matrix transposition or copy.\na\nArray holding the input matrix A. Must have size at least lda * n for column\nmajor ordering and at least lda * m for row major ordering.\nlda\nLeading dimension of the matrix A. If matrices are stored using column\nmajor layout, lda is the number of elements in the array between adjacent\ncolumns of the matrix and must be at least stridea * (m-1) + 1. If\nusing row major layout, lda is the number of elements between adjacent\nrows of the matrix and must be at least stridea * (n-1) + 1.\nstridea\nThe second stride of the matrix A. For column major layout, stridea is the\nnumber of elements in the array between adjacent rows of the matrix. For\nrow major layout stridea is the number of elements between adjacent\ncolumns of the matrix. In both cases stridea must be at least 1.\nb\nArray holding the output matrix B.\n \ntrans =\ntranspose::nontrans\ntrans =\ntranspose::trans, or\ntrans =\ntranspose::conjtrans\nColumn major\nB is m x n matrix. Size of\narray b must be at least\nldb * n.\nB is n x m matrix. Size\nof array b must be at\nleast ldb * m.\nRow major\nB is m x n matrix. Size of\narray b must be at least\nldb * m.\nB is n x m matrix. Size\nof array b must be at\nleast ldb * n.\nldb\nThe leading dimension of the matrix B. Must be positive.\n \ntrans =\ntranspose::nontrans\ntrans =\ntranspose::trans, or\ntrans =\ntranspose::conjtrans\nColumn major\nldb must be at least\nstrideb * (m-1) +\n1.\nldb must be at least\nstrideb * (n-1) +\n1.\nRow major\nldb must be at least\nstrideb * (n-1) +\n1.\nldb must be at least\nstrideb * (m-1) +\n1.\nstrideb\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n399\n\n\nThe second stride of the matrix B. For column major layout, strideb is the\nnumber of elements in the array between adjacent rows of the matrix. For\nrow major layout, strideb is the number of elements between adjacent\ncolumns of the matrix. In both cases strideb must be at least 1.\nOutput Parameters\nb\nArray, size at least m.\nContains the destination matrix.\nInterfaces\nmkl_?omatadd\nScales and sums two matrices including in addition to\nperforming out-of-place transposition operations.\nSyntax\nvoid mkl_somatadd (char ordering, char transa, char transb, size_t m, size_t n, const\nfloat alpha, const float * A, size_t lda, const float beta, const float * B, size_t ldb,\nfloat * C, size_t ldc);\nvoid mkl_domatadd (char ordering, char transa, char transb, size_t m, size_t n, const\ndouble alpha, const double * A, size_t lda, const double beta, const double * B, size_t\nldb, double * C, size_t ldc);\nvoid mkl_comatadd (char ordering, char transa, char transb, size_t m, size_t n, const\nMKL_Complex8 alpha, const MKL_Complex8 * A, size_t lda, const MKL_Complex8 beta, const\nMKL_Complex8 * B, size_t ldb, MKL_Complex8 * C, size_t ldc);\nvoid mkl_zomatadd (char ordering, char transa, char transb, size_t m, size_t n, const\nMKL_Complex16 alpha, const MKL_Complex16 * A, size_t lda, const MKL_Complex16 beta,\nconst MKL_Complex16 * B, size_t ldb, MKL_Complex16 * C, size_t ldc);\nInclude Files\n•\nmkl.h\nDescription\nThe mkl_?omatadd routine scales and adds two matrices, as well as performing out-of-place transposition\noperations. A transposition operation can be no operation, a transposition, a conjugate transposition, or a\nconjugation (without transposition). The following out-of-place memory movement is done:\nC := alpha*op(A) + beta*op(B)\nwhere the op(A) and op(B) operations are transpose, conjugate-transpose, conjugate (no transpose), or no\ntranspose, depending on the values of transa and transb. If no transposition of the source matrices is\nrequired, m is the number of rows and n is the number of columns in the source matrices A and B. In this\ncase, the output matrix C is m-by-n.\nIn general, a, b, and c must not overlap in memory, with the exception of the following in-place operations:\n•\na and c can point to the same memory if transa is non-transpose and lda = ldc.\n•\nb and c can point to the same memory if transb is non-transpose and ldb = ldc.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n400\n\n\nInput Parameters\nordering\nOrdering of the matrix storage.\nIf ordering = 'R' or 'r', the ordering is row-major.\nIf ordering = 'C' or 'c', the ordering is column-major.\ntransa\nParameter that specifies the operation type on matrix A.\nIf transa = 'N' or 'n', op(A)=A and the matrix A is assumed unchanged\non input.\nIf transa = 'T' or 't', it is assumed that A should be transposed.\nIf transa = 'C' or 'c', it is assumed that A should be conjugate\ntransposed.\nIf transa = 'R' or 'r', it is assumed that A should be conjugated (and not\ntransposed).\nIf the data is real, then transa = 'R' is the same as transa = 'N', and\ntransa = 'C' is the same as transa = 'T'.\ntransb\nParameter that specifies the operation type on matrix B.\nIf transb = 'N' or 'n', op(B)=B and the matrix B is assumed unchanged\non input.\nIf transb = 'T' or 't', it is assumed that B should be transposed.\nIf transb = 'C' or 'c', it is assumed that B should be conjugate\ntransposed.\nIf transb = 'R' or 'r', it is assumed that B should be conjugated (and not\ntransposed).\nIf the data is real, then transb = 'R' is the same as transb = 'N', and\ntransb = 'C' is the same as transb = 'T'.\nm\nThe number of matrix rows in op(A), op(B), and C.\nn\nThe number of matrix columns in op(A), op(B), and C.\nalpha\nThis parameter scales the input matrix by alpha.\na\nArray.\nlda\nDistance between the first elements in adjacent columns (in the case of the\ncolumn-major order) or rows (in the case of the row-major order) in the\nsource matrix A; measured in the number of elements.\nFor ordering = 'C' or 'c': when transa = 'N', 'n', 'R', or 'r', lda\nmust be at least max(1,m); otherwise lda must be max(1,n).\nFor ordering = 'R' or 'r': when transa = 'N', 'n', 'R', or 'r', lda\nmust be at least max(1,n); otherwise lda must be max(1,m).\nbeta\nThis parameter scales the input matrix by beta.\nb\nArray.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n401\n\n\nldb\nDistance between the first elements in adjacent columns (in the case of the\ncolumn-major order) or rows (in the case of the row-major order) in the\nsource matrix B; measured in the number of elements.\nFor ordering = 'C' or 'c': when transa = 'N', 'n', 'R', or 'r', ldb\nmust be at least max(1,m); otherwise ldb must be max(1,n).\nFor ordering = 'R' or 'r': when transa = 'N', 'n', 'R', or 'r', ldb\nmust be at least max(1,n); otherwise ldb must be max(1,m).\nldc\nDistance between the first elements in adjacent columns (in the case of the\ncolumn-major order) or rows (in the case of the row-major order) in the\ndestination matrix C; measured in the number of elements.\nIf ordering = 'C' or 'c', then ldc must be at least max(1, m),\notherwise ldc must be at least max(1, n).\nOutput Parameters\nc\nArray.\nInterfaces\ncblas_?gemm_pack_get_size, cblas_gemm_*_pack_get_size\nReturns the number of bytes required to store the\npacked matrix.\nSyntax\nsize_t cblas_hgemm_pack_get_size (const CBLAS_IDENTIFIER identifier, const MKL_INT m,\nconst MKL_INT n, const MKL_INT k)\nsize_t cblas_sgemm_pack_get_size (const CBLAS_IDENTIFIER identifier, const MKL_INT m,\nconst MKL_INT n, const MKL_INT k)\nsize_t cblas_dgemm_pack_get_size (const CBLAS_IDENTIFIER identifier, const MKL_INT m,\nconst MKL_INT n, const MKL_INT k)\nsize_t cblas_gemm_s8u8s32_pack_get_size (const CBLAS_IDENTIFIER identifier, const\nMKL_INT m, const MKL_INT n, const MKL_INT k)\nsize_t cblas_gemm_s16s16s32_pack_get_size (const CBLAS_IDENTIFIER identifier, const\nMKL_INT m, const MKL_INT n, const MKL_INT k)\nsize_t cblas_gemm_bf16bf16f32_pack_get_size (const CBLAS_IDENTIFIER identifier, const\nMKL_INT m, const MKL_INT n, const MKL_INT k)\nsize_t cblas_gemm_f16f16f32_pack_get_size (const CBLAS_IDENTIFIER identifier, const\nMKL_INT m, const MKL_INT n, const MKL_INT k)\nInclude Files\n•\nmkl.h\nDescription\nThe cblas_?gemm_pack_get_size and cblas_gemm_*_pack_get_size routines belong to a set of related\nroutines that enable the use of an internal packed storage. Call the cblas_?gemm_pack_get_size and\ncblas_gemm_*_pack_get_size routines first to query the size of storage required for a packed matrix\nstructure to be used in subsequent calls. Ultimately, the packed matrix structure is used to compute\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n402\n\n\nC := alpha*op(A)*op(B) + beta*C for bfloat16, half, single and double precision or\nC := alpha*(op(A)+ A_offset)*(op(B)+ B_offset) + beta*C + C_offset for integer type.\nwhere:\nop(X) is one of the operations op(X) = X or op(X) = XT\nalpha and beta are scalars,\nA , A_offset,B, B_offset,C, and C_offset are matrices\nop(A) is an m-by-k matrix,\nop(B) is a k-by-n matrix,\nC is an m-by-n matrix.\nA_offset is an m-by-k matrix.\nB_offset is an k-by-n matrix.\nC_offset is an m-by-n matrix.\nInput Parameters\nParameter\nType\nDescription\nidentifier\nCBLAS_IDENTIFIER\nSpecifies which matrix is to be packed:\nIf identifier = CblasAMatrix, the size\nreturned is the size required to store matrix A\nin an internal format.\nIf identifier = CblasBMatrix, the size\nreturned is the size required to store matrix B\nin an internal format.\nm\nMKL_INT\nSpecifies the number of rows of matrix op(A)\nand of the matrix C. The value of m must be\nat least zero.\nn\nMKL_INT\nSpecifies the number of columns of matrix\nop(B) and the number of columns of matrix\nC. The value of n must be at least zero.\nk\nMKL_INT\nSpecifies the number of columns of matrix\nop(A) and the number of rows of matrix\nop(B). The value of k must be at least zero.\nReturn Values\nParameter\nType\nDescription\nsize\nsize_t\nReturns the size (in bytes) required to store\nthe matrix when packed into the internal\nformat of Intel® oneAPI Math Kernel Library\n(oneMKL).\nExample\nSee the following examples in the MKL installation directory to understand the use of these routines:\ncblas_hgemm_pack_get_size: examples\\cblas\\source\\cblas_hgemm_computex.c\ncblas_sgemm_pack_get_size: examples\\cblas\\source\\cblas_sgemm_computex.c\ncblas_dgemm_pack_get_size: examples\\cblas\\source\\cblas_dgemm_computex.c\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n403\n\n\ncblas_gemm_s8u8s32_pack_get_size: examples\\cblas\\source\\cblas_gemm_s8u8s32_computex.c\ncblas_gemm_s16u16s32_pack_get_size: examples\\cblas\\source\\cblas_gemm_s16s16s32_computex.c\ncblas_gemm_bf16bf16f32_pack_get_size: examples\\cblas\\source\\cblas_gemm_bf16bf16f32_computex.c\ncblas_gemm_f16f16f32_pack_get_size: examples\\cblas\\source\\cblas_gemm_f16f16f32_computex.c\nSee Also\ncblas_?gemm_pack and cblas_gemm_*_pack\n to pack the matrix into a buffer allocated previously.\ncblas_?gemm_compute and cblas_gemm_*_compute\n to compute a matrix-matrix product with general matrices (where one or both input matrices are stored in\na packed data structure) and add the result to a scalar-matrix product.\ncblas_?gemm_pack\nPerforms scaling and packing of the matrix into the\npreviously allocated buffer.\nSyntax\nvoid cblas_hgemm_pack (const CBLAS_LAYOUT Layout, const CBLAS_IDENTIFIER identifier,\nconst CBLAS_TRANSPOSE trans, const MKL_INT m, const MKL_INT n, const MKL_INT k, const\nMKL_F16 alpha, const MKL_F16 *src, const MKL_INT ld, MKL_F16 *dest);\nvoid cblas_sgemm_pack (const CBLAS_LAYOUT Layout, const CBLAS_IDENTIFIER identifier,\nconst CBLAS_TRANSPOSE trans, const MKL_INT m, const MKL_INT n, const MKL_INT k, const\nfloat alpha, const float *src, const MKL_INT ld, float *dest);\nvoid cblas_dgemm_pack (const CBLAS_LAYOUT Layout, const CBLAS_IDENTIFIER identifier,\nconst CBLAS_TRANSPOSE trans, const MKL_INT m, const MKL_INT n, const MKL_INT k, const\ndouble alpha, const double *src, const MKL_INT ld, double *dest);\nInclude Files\n•\nmkl.h\nDescription\nThe cblas_?gemm_pack routine is one of a set of related routines that enable use of an internal packed\nstorage. Call cblas_?gemm_pack after you allocate a buffer whose size is given by\ncblas_?gemm_pack_getsize. The cblas_?gemm_pack routine scales the identified matrix by alpha and\npacks it into the buffer allocated previously.\nNOTE\nDo not copy the packed matrix to a different address because the internal implementation\ndepends on the alignment of internally-stored metadata.\nThe cblas_?gemm_pack routine performs this operation:\ndest := alpha*op(src) as part of the computation C := alpha*op(A)*op(B) + beta*C\nwhere:\nop(X) is one of the operations op(X) = X, op(X) = XT, or op(X) = XH,\nalpha and beta are scalars,\nsrc is a matrix,\nA , B, and C are matrices\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n404\n\n\nop(src) is an m-by-k matrix if identifier = CblasAMatrix,\nop(src) is a k-by-n matrix if identifier = CblasBMatrix,\ndest is an internal packed storage buffer.\nNOTE\nYou must use the same value of the Layout parameter for the entire sequence of related\ncblas_?gemm_pack and cblas_?gemm_compute calls.\nFor best performance, use the same number of threads for packing and for computing.\nIf packing for both A and B matrices, you must use the same number of threads for packing A as for\npacking B.\nInput Parameters\nLayout\nSpecifies whether two-dimensional array storage is row-major\n(CblasRowMajor) or column-major (CblasColMajor).\nidentifier\nSpecifies which matrix is to be packed:\nIf identifier = CblasAMatrix, the routine allocates storage to pack\nmatrix A.\nIf identifier = CblasBMatrix, the routine allocates storage to pack\nmatrix B.\ntrans\nSpecifies the form of op(src) used in the packing:\nIf trans = CblasNoTrans  op(src) = src.\nIf trans = CblasTrans  op(src) = srcT.\nIf trans = CblasConjTrans  op(src) = srcH.\nm\nSpecifies the number of rows of the matrix op(A) and of the matrix C. The\nvalue of m must be at least zero.\nn\nSpecifies the number of columns of the matrix op(B) and the number of\ncolumns of the matrix C. The value of n must be at least zero.\nk\nSpecifies the number of columns of the matrix op(A) and the number of\nrows of the matrix op(B). The value of k must be at least zero.\nalpha\nSpecifies the scalar alpha.\nsrc\nArray:\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n405\n\n\nidentifier =\nCblasAMatrix\nidentifier = CblasBMatrix\ntrans =\nCblasNoT\nrans\ntrans =\nCblasTra\nns or\ntrans =\nCblasCon\njTrans\ntrans =\nCblasNoTrans\ntrans =\nCblasTrans\nor trans =\nCblasConjTra\nns\nLayout =\nCblasCol\nMajor\nSize\nld*k.\nBefore\nentry, the\nleading m-\nby-k part\nof the\narray src\nmust\ncontain\nthe matrix\nA.\nSize\nld*m.\nBefore\nentry, the\nleading k-\nby-m part\nof the\narray src\nmust\ncontain\nthe matrix\nA.\nSize ld*n.\nBefore entry,\nthe leading k-\nby-n part of\nthe array src\nmust contain\nthe matrix B.\nSize ld*k.\nBefore entry,\nthe leading n-\nby-k part of\nthe array src\nmust contain\nthe matrix B.\nLayout =\nCblasRow\nMajor\nSize\nld*m.\nBefore\nentry, the\nleading k-\nby-m part\nof the\narray src\nmust\ncontain\nthe matrix\nA.\nSize\nld*k.\nBefore\nentry, the\nleading m-\nby-k part\nof the\narray src\nmust\ncontain\nthe matrix\nA.\nSize ld*k.\nBefore entry,\nthe leading n-\nby-k part of\nthe array src\nmust contain\nthe matrix B.\nSize ld*n.\nBefore entry,\nthe leading k-\nby-n part of\nthe array src\nmust contain\nthe matrix B.\nld\nSpecifies the leading dimension of src as declared in the calling\n(sub)program.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n406\n\n\nidentifier =\nCblasAMatrix\nidentifier = CblasBMatrix\ntrans =\nCblasNoT\nrans\ntrans =\nCblasTra\nns or\ntrans =\nCblasCon\njTrans\ntrans =\nCblasNoTrans\ntrans =\nCblasTrans\nor trans =\nCblasConjTra\nns\nLayout =\nCblasCol\nMajor\nld must\nbe at\nleast\nmax(1,\nm).\nld must\nbe at\nleast\nmax(1,\nk).\nld must be at\nleast max(1,\nk).\nld must be at\nleast max(1,\nn).\nLayout =\nCblasRow\nMajor\nld must\nbe at\nleast\nmax(1,\nk).\nld must\nbe at\nleast\nmax(1,\nm).\nld must be at\nleast max(1,\nn).\nld must be at\nleast max(1,\nk).\ndest\nScaled and packed internal storage buffer.\nOutput Parameters\ndest\nOverwritten by the matrix alpha*op(src).\nSee Also\ncblas_?gemm_pack_get_size Returns the number of bytes required to store the packed matrix.\ncblas_?gemm_compute Computes a matrix-matrix product with general matrices where one or both\ninput matrices are stored in a packed data structure and adds the result to a scalar-matrix\nproduct.\ncblas_?gemm\n for a detailed description of general matrix multiplication.\ncblas_gemm_*_pack\nPack the matrix into the buffer allocated previously.\nSyntax\nvoid cblas_gemm_s8u8s32_pack (const CBLAS_LAYOUT Layout, const CBLAS_IDENTIFIER\nidentifier, const CBLAS_TRANSPOSE trans, const MKL_INT m, const MKL_INT n, const\nMKL_INT k, const void *src, const MKL_INT ld, void *dest);\nvoid cblas_gemm_s16s16s32_pack (const CBLAS_LAYOUT Layout, const CBLAS_IDENTIFIER\nidentifier, const CBLAS_TRANSPOSE trans, const MKL_INT m, const MKL_INT n, const\nMKL_INT k, const MKL_INT16 *src, const MKL_INT ld, MKL_INT16 *dest);\nvoid cblas_gemm_bf16bf16f32_pack (const CBLAS_LAYOUT Layout, const CBLAS_IDENTIFIER\nidentifier, const CBLAS_TRANSPOSE trans, const MKL_INT m, const MKL_INT n, const\nMKL_INT k, const MKL_BF16 *src, const MKL_INT ld, MKL_BF16 *dest);\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n407\n\n\nvoid cblas_gemm_f16f16f32_pack (const CBLAS_LAYOUT Layout, const CBLAS_IDENTIFIER\nidentifier, const CBLAS_TRANSPOSE trans, const MKL_INT m, const MKL_INT n, const\nMKL_INT k, const MKL_F16 *src, const MKL_INT ld, MKL_F16 *dest);\nInclude Files\n•\nmkl.h\nDescription\nThe cblas_gemm_*_pack routine is one of a set of related routines that enable the use of an internal packed\nstorage. Call cblas_gemm_*_pack after you allocate a buffer whose size is given by\ncblas_gemm_*_pack_get_size. The cblas_gemm_*_pack routine packs the identified matrix into the\nbuffer allocated previously.\nThe cblas_gemm_*_pack routine performs this operation:\ndest := op(src) as part of the computation C := alpha*(op(A) + A_offset)*(op(B) + B_offset) +\nbeta*C + C_offset for integer types.\nC := alpha*op(A) * op(B) + beta*C for bfloat16 type.\nwhere:\nop(X) is one of the operations op(X) = X or op(X) = XT\nalpha and beta are scalars,\nsrc is a matrix,\nA , A_offset,B, B_offset,c,and C_offset are matrices\nop(src) is an m-by-k matrix if identifier = CblasAMatrix,\nop(src) is a k-by-n matrix if identifier =CblasBMatrix ,\ndest is the buffer previously allocated to store the matrix packed into an internal format\nA_offset is an m-by-k matrix.\nB_offset is an k-by-n matrix.\nC_offset is an m-by-n matrix.\nNOTE\nYou must use the same value of the Layout parameter for the entire sequence of related\ncblas_gemm_*_pack and cblas_gemm_*_compute calls.\nFor best performance, use the same number of threads for packing and for computing.\nIf packing for both A and B matrices, you must use the same number of threads for packing A as for\npacking B.\nInput Parameters\nLayout\nCBLAS_LAYOUT\nSpecifies whether two-dimensional array storage is row-major\n(CblasRowMajor) or column-major(CblasColMajor).\nidentifier\nCBLAS_IDENTIFIER\nSpecifies which matrix is to be packed:\nIf identifier = CblasAMatrix, the A matrix is packed.\nIf identifier = CblasBMatrix, the B matrix is packed.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n408\n\n\ntrans\nCBLAS_TRANSPOSE\nSpecifies the form of op(src) used in the packing:\nIf trans = CblasNoTrans  op(src) = src.\nIf trans = CblasTrans  op(src) = srcT.\nm\nMKL_INT\nSpecifies the number of rows of matrix op(A) and of the matrix C. The value\nof m must be at least zero.\nn\nMKL_INT\nSpecifies the number of columns of matrix op(B) and the number of\ncolumns of matrix C. The value of n must be at least zero.\nk\nMKL_INT\nSpecifies the number of columns of matrix op(A) and the number of rows of\nmatrix op(B). The value of k must be at least zero.\nsrc\nMKL_BF16* for cblas_gemm_bf16bf16f32_pack, MKL_F16* for\ncblas_gemm_f16f16f32_pack, void* for cblas_gemm_s8u8s32_pack\nand MKL_INT16* for cblas_gemm_s16s16s32_pack\nidentifier =\nCblasAMatrix\nidentifier = CblasBMatrix\ntrans =\nCblasNoT\nrans\ntrans =\nCblasTra\nns\ntrans =\nCblasNoTrans\ntrans =\nCblasTrans\nLayout =\nCblasCol\nMajor\nSize\nld*k.\nBefore\nentry, the\nleading m-\nby-k part\nof the\narray src\nmust\ncontain\nthe matrix\nA.\nFor\ncblas_ge\nmm_s8u8s\n32_pack\nthe\nelement\nin src\narray\nmust be\nan 8-bit\nsigned\ninteger.\nSize\nld*m.\nBefore\nentry, the\nleading k-\nby-m part\nof the\narray src\nmust\ncontain\nthe matrix\nA.\nFor\ncblas_ge\nmm_s8u8s\n32_pack\nthe\nelement\nin src\narray\nmust be\nan 8-bit\nsigned\ninteger.\nSize ld*n.\nBefore entry,\nthe leading k-\nby-n part of\nthe array src\nmust contain\nthe matrix B.\nFor\ncblas_gemm_\ns8u8s32_pac\nk the element\nin src array\nmust be an 8-\nbit unsigned\ninteger.\nSize ld*k.\nBefore entry,\nthe leading n-\nby-k part of\nthe array src\nmust contain\nthe matrix B.\nFor\ncblas_gemm_\ns8u8s32_pac\nk the element\nin src array\nmust be an 8-\nbit unsigned\ninteger.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n409\n\n\nidentifier =\nCblasAMatrix\nidentifier = CblasBMatrix\ntrans =\nCblasNoT\nrans\ntrans =\nCblasTra\nns\ntrans =\nCblasNoTrans\ntrans =\nCblasTrans\nLayout =\nCblasRow\nMajor\nSize\nld*m.\nBefore\nentry, the\nleading k-\nby-m part\nof the\narray src\nmust\ncontain\nthe matrix\nA.\nFor\ncblas_ge\nmm_s8u8s\n32_pack\nthe\nelement\nin src\narray\nmust be\nan 8-bit\nunsigned\ninteger.\nSize\nld*k.\nBefore\nentry, the\nleading m-\nby-k part\nof the\narray src\nmust\ncontain\nthe matrix\nA.\nFor\ncblas_ge\nmm_s8u8s\n32_pack\nthe\nelement\nin src\narray\nmust be\nan 8-bit\nunsigned\ninteger.\nSize ld*k.\nBefore entry,\nthe leading n-\nby-k part of\nthe array src\nmust contain\nthe matrix B.\nFor\ncblas_gemm_\ns8u8s32_pac\nk the element\nin src array\nmust be an 8-\nbit signed\ninteger.\nSize ld*n.\nBefore entry,\nthe leading k-\nby-n part of\nthe array src\nmust contain\nthe matrix B.\nFor\ncblas_gemm_\ns8u8s32_pac\nk the element\nin src array\nmust be an 8-\nbit signed\ninteger.\nld\nMKL_INTSpecifies the leading dimension of src as declared in the calling\n(sub)program.\nidentifier =\nCblasAMatrix\nidentifier = CblasBMatrix\ntrans =\nCblasNoT\nrans\ntrans =\nCblasTra\nns\ntrans =\nCblasNoTrans\ntrans =\nCblasTrans\nLayout =\nCblasCol\nMajor\nld must\nbe at\nleast\nmax(1,\nm).\nld must\nbe at\nleast\nmax(1,\nk).\nld must be at\nleast max(1,\nk).\nld must be at\nleast max(1,\nn).\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n410\n\n\nidentifier =\nCblasAMatrix\nidentifier = CblasBMatrix\ntrans =\nCblasNoT\nrans\ntrans =\nCblasTra\nns\ntrans =\nCblasNoTrans\ntrans =\nCblasTrans\nLayout =\nCblasRow\nMajor\nld must\nbe at\nleast\nmax(1,\nk).\nld must\nbe at\nleast\nmax(1,\nm).\nld must be at\nleast max(1,\nn).\nld must be at\nleast max(1,\nk).\ndest\nMKL_BF16* for cblas_gemm_bf16bf16f32_pack, MKL_F16* for\ncblas_gemm_f16f16f32_pack, void* for cblas_gemm_s8u8s32_pack or\nMKL_INT16* for cblas_gemm_s16s16s32_pack\nBuffer for the packed matrix.\nOutput Parameters\ndest\nMKL_BF16* for cblas_gemm_bf16bf16f32_pack, MKL_F16* for\ncblas_gemm_f16f16f32_pack, void* for\ncblas_gemm_s8u8s32_pack or MKL_INT16* for\ncblas_gemm_s16s16s32_pack\nOverwritten by the matrix op(src)stored in a format internal to Intel®\noneAPI Math Kernel Library (oneMKL).\nExample\nSee the following examples in the MKL installation directory to understand the use of these routines:\ncblas_gemm_s8u8s32_pack: examples\\cblas\\source\\cblas_gemm_s8u8s32_computex.c\ncblas_gemm_s16s16s32_pack: examples\\cblas\\source\\cblas_gemm_s16s16s32_computex.c\ncblas_gemm_bf16bf16f32_pack: examples\\cblas\\source\\cblas_gemm_bf16bf16f32_computex.c\ncblas_gemm_f16f16f32_pack: examples\\cblas\\source\\cblas_gemm_f16f16f32_computex.c\nApplication Notes\nWhen using cblas_gemm_s8u8s32_pack with row-major layout , the data types of A and B must be\nswapped. That is, you must provide an 8-bit unsigned integer array for matrix A and an 8-bit signed integer\narray for matrix B .\nSee Also\ncblas_gemm_*_pack_get_size\n to return the number of bytes needed to store the packed matrix.\ncblas_gemm_*_compute\n to compute a matrix-matrix product with general integer matrices (where one or both input matrices are\nstored in a packed data structure) and add the result to a scalar-matrix product.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n411\n\n\ncblas_?gemm_compute\nComputes a matrix-matrix product with general\nmatrices where one or both input matrices are stored\nin a packed data structure and adds the result to a\nscalar-matrix product.\nSyntax\nvoid cblas_hgemm_compute (const CBLAS_LAYOUT Layout, const MKL_INT transa, const\nMKL_INT transb, const MKL_INT m, const MKL_INT n, const MKL_INT k, const MKL_F16 *a,\nconst MKL_INT lda, const MKL_F16 *b, const MKL_INT ldb, const MKL_F16 beta, MKL_F16 *c,\nconst MKL_INT ldc);\nvoid cblas_sgemm_compute (const CBLAS_LAYOUT Layout, const MKL_INT transa, const\nMKL_INT transb, const MKL_INT m, const MKL_INT n, const MKL_INT k, const float *a,\nconst MKL_INT lda, const float *b, const MKL_INT ldb, const float beta, float *c, const\nMKL_INT ldc);\nvoid cblas_dgemm_compute (const CBLAS_LAYOUT Layout, const MKL_INT transa, const\nMKL_INT transb, const MKL_INT m, const MKL_INT n, const MKL_INT k, const double *a,\nconst MKL_INT lda, const double *b, const MKL_INT ldb, const double beta, double *c,\nconst MKL_INT ldc);\nInclude Files\n•\nmkl.h\nDescription\nThe cblas_?gemm_compute routine is one of a set of related routines that enable use of an internal packed\nstorage. After calling cblas_?gemm_pack call cblas_?gemm_compute to compute\nC := op(A)*op(B) + beta*C,\nwhere:\nop(X) is one of the operations op(X) = X, op(X) = XT, or op(X) = XH,\nbeta is a scalar,\nA , B, and C are matrices:\nop(A) is an m-by-k matrix,\nop(B) is a k-by-n matrix,\nC is an m-by-n matrix.\nNOTE\nYou must use the same value of the Layout parameter for the entire sequence of related\ncblas_?gemm_pack and cblas_?gemm_compute calls.\nFor best performance, use the same number of threads for packing and for computing.\nIf packing for both A and B matrices, you must use the same number of threads for packing A as for\npacking B.\nInput Parameters\nLayout\nSpecifies whether two-dimensional array storage is row-major\n(CblasRowMajor) or column-major (CblasColMajor).\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n412\n\n\ntransa\nSpecifies the form of op(A) used in the matrix multiplication, one of the\nCBLAS_TRANSPOSE or CBLAS_STORAGE enumerated types:\nIf transa = CblasNoTrans  op(A) = A.\nIf transa = CblasTrans  op(A) = AT.\nIf transa = CblasConjTrans  op(A) = AH.\nIf transa = CblasPacked the matrix in array a is packed and lda is\nignored.\ntransb\nSpecifies the form of op(B) used in the matrix multiplication, one of the\nCBLAS_TRANSPOSE or CBLAS_STORAGE enumerated types:\nIf transb = CblasNoTrans  op(B) = B.\nIf transb = CblasTrans op(B) = BT.\nIf transb = CblasConjTrans op(B) = BH.\nIf transb = CblasPacked the matrix in array b is packed and ldb is\nignored.\nm\nSpecifies the number of rows of the matrix op(A) and of the matrix C. The\nvalue of m must be at least zero.\nn\nSpecifies the number of columns of the matrix op(B) and the number of\ncolumns of the matrix C. The value of n must be at least zero.\nk\nSpecifies the number of columns of the matrix op(A) and the number of\nrows of the matrix op(B). The value of k must be at least zero.\na\nArray:\ntransa =\nCblasNoTrans\ntransa =\nCblasTrans or\ntransa =\nCblasConjTrans\ntransa =\nCblasPacked\nLayout =\nCblasColMajor\nSize lda*k.\nBefore entry, the\nleading m-by-k\npart of the array\na must contain\nthe matrix A.\nSize lda*m.\nBefore entry, the\nleading k-by-m part\nof the array a must\ncontain the matrix\nA.\nStored in\ninternal\npacked\nformat.\nLayout =\nCblasRowMajor\nSize lda*m.\nBefore entry, the\nleading k-by-m\npart of the array\na must contain\nthe matrix A.\nSize lda*k.\nBefore entry, the\nleading m-by-k part\nof the array a must\ncontain the matrix\nA.\nStored in\ninternal\npacked\nformat.\nlda\nSpecifies the leading dimension of a as declared in the calling\n(sub)program.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n413\n\n\ntransa =\nCblasNoTrans\ntransa =\nCblasTrans or\ntransa =\nCblasConjTrans\ntransa =\nCblasPacked\nLayout =\nCblasColMajor\nlda must be at\nleast max(1,\nm).\nlda must be at\nleast max(1, k).\nlda is ignored.\nLayout =\nCblasRowMajor\nlda must be at\nleast max(1,\nk).\nlda must be at\nleast max(1, m).\nlda is ignored.\nb\nArray:\ntransb =\nCblasNoTrans\ntransb =\nCblasTrans or\ntransb =\nCblasConjTrans\ntransb =\nCblasPacked\nLayout =\nCblasColMajor\nSize ldb*n.\nBefore entry, the\nleading k-by-n\npart of the array\nb must contain\nthe matrix B.\nSize ldb*k.\nBefore entry, the\nleading n-by-k part\nof the array b must\ncontain the matrix\nB.\nStored in\ninternal\npacked\nformat.\nLayout =\nCblasRowMajor\nSize ldb*k.\nBefore entry, the\nleading n-by-k\npart of the array\nb must contain\nthe matrix B.\nSize ldb*n.\nBefore entry, the\nleading k-by-n part\nof the array b must\ncontain the matrix\nB.\nStored in\ninternal\npacked\nformat.\nldb\nSpecifies the leading dimension of b as declared in the calling\n(sub)program.\ntransb =\nCblasNoTrans\ntransb =\nCblasTransor\ntransb =\nCblasConjTrans\ntransb =\nCblasPacked\nLayout =\nCblasColMajor\nldb must be at\nleast max(1,\nk).\nldb must be at\nleast max(1, n).\nldb is ignored.\nLayout =\nCblasRowMajor\nldb must be at\nleast max(1,\nn).\nldb must be at\nleast max(1, k).\nldb is ignored.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n414\n\n\nbeta\nSpecifies the scalar beta. When beta is equal to zero, then c need not be\nset on input.\nc\nArray:\nLayout =\nCblasColMajor\nSize ldc*n.\nBefore entry, the leading m-by-n part of the array c\nmust contain the matrix C, except when beta is\nequal to zero, in which case c need not be set on\nentry.\nLayout =\nCblasRowMajor\nSize ldc*m.\nBefore entry, the leading n-by-m part of the array c\nmust contain the matrix C, except when beta is\nequal to zero, in which case c need not be set on\nentry.\nldc\nSpecifies the leading dimension of c as declared in the calling\n(sub)program.\nLayout = CblasColMajor\nldc must be at least max(1, m).\nLayout = CblasRowMajor\nldc must be at least max(1, n).\nOutput Parameters\nc\nOverwritten by the m-by-n matrix op(A)*op(B) + beta*C.\nSee Also\ncblas_?gemm_pack_get_size Returns the number of bytes required to store the packed matrix.\ncblas_?gemm_pack Performs scaling and packing of the matrix into the previously allocated buffer.\ncblas_?gemm\n for a detailed description of general matrix multiplication.\ncblas_gemm_*_compute\nComputes a matrix-matrix product with general\ninteger matrices (where one or both input matrices\nare stored in a packed data structure) and adds the\nresult to a scalar-matrix product.\nSyntax\nvoid cblas_gemm_s8u8s32_compute(const CBLAS_LAYOUT Layout, const MKL_INT transa, const\nMKL_INT transb, const CBLAS_OFFSET offsetc, const MKL_INT m, const MKL_INT n, const\nMKL_INT k, const float alpha, const void *a, const MKL_INT lda, const MKL_INT8 oa,\nconst void *b, const MKL_INT ldb, const MKL_INT8 ob, const float beta, MKL_INT32 *c,\nconst MKL_INT ldc, const MKL_INT32 *oc);\nvoid cblas_gemm_s16s16s32_compute(const CBLAS_LAYOUT Layout, const MKL_INT transa,\nconst MKL_INT transb, const CBLAS_OFFSET offsetc, const MKL_INT m, const MKL_INT n,\nconst MKL_INT k, const float alpha, const MKL_INT16 *a, const MKL_INT lda, const\nMKL_INT16 oa, const MKL_INT16 *b, const MKL_INT ldb, const MKL_INT16 ob, const float\nbeta, MKL_INT32 *c, const MKL_INT ldc, const MKL_INT32 *oc);\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n415\n\n\nInclude Files\n•\nmkl.h\nDescription\nThe cblas_gemm_*_compute routine is one of a set of related routines that enable use of an internal packed\nstorage. After calling cblas_gemm_*_pack call cblas_gemm_*_compute to compute\nC := alpha*(op(A) + A_offset)*(op(B) + B_offset) + beta*C + C_offset,\nwhere:\nop(X) is either op(X) = X or op(X) = XT\nalpha and betaare scalars\nA , B, and C are matrices:\nop(A) is an m-by-k matrix,\nop(B) is a k-by-n matrix,\nC is an m-by-n matrix.\nA_offset is an m-by-k matrix with every element equal to the value oa.\nB_offset is an k-by-n matrix with every element equal to the value ob.\nC_offset is an m-by-n matrix defined by the oc array as described in the description of the offsetc\nparameter.\nNOTE\nYou must use the same value of the Layout parameter for the entire sequence of related\ncblas_?gemm_pack and cblas_?gemm_compute calls.\nFor best performance, use the same number of threads for packing and for computing.\nIf you are packing for both A and B matrices, you must use the same number of threads for packing A\nas for packing B.\nInput Parameters\nLayout\nCBLAS_LAYOUT\nSpecifies whether two-dimensional array storage is row-major (CblasRowMajor) or column-\nmajor(CblasColMajor).\ntransa\nMKL_INTSpecifies the form of op(A) used in the packing:\nIf transa = CblasNoTrans  op(A) = A.\nIf transa = CblasTrans  op(A) = AT.\nIf transa = CblasPacked the matrix in array ais packed into a format internal to Intel® oneAPI\nMath Kernel Library (oneMKL) andlda is ignored.\ntransb\nMKL_INT Specifies the form of op(B) used in the packing:\nIf transb = CblasNoTrans  op(B) = B.\nIf transb = CblasTrans op(B) = BT.\nIf transb = CblasPacked the matrix in array bis packed into a format internal to Intel® oneAPI\nMath Kernel Library (oneMKL) andldb is ignored.\noffsetc\nCBLAS_OFFSET Specifies the form of C_offset used in the matrix multiplication.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n416\n\n\nIf offsetc=CblasFixOffset  :oc has a single element and every element of C_offset is equal to\nthis element.\nIf offsetc=CblasColOffset :oc has a size of m and every element of C_offset is equal to oc.\nIf offsetc=CblasRowOffset :oc has a size of n and every element of C_offset is equal to oc.\nm\nMKL_INTSpecifies the number of rows of the matrix op(A) and of the matrix C. The value of m\nmust be at least zero.\nn\nMKL_INTSpecifies the number of columns of the matrix op(B) and the number of columns of the\nmatrix C. The value of n must be at least zero.\nk\nMKL_INTSpecifies the number of columns of the matrix op(A) and the number of rows of the\nmatrix op(B). The value of k must be at least zero.\nalpha\nfloatSpecifies the scalar alpha.\na\nvoid* for gemm_s8u8s32_compute\nMKL_INT16* for gemm_s16s16s32_compute\nLayout = CblasColMajor\ntransa = CblasNoTrans\nArray, size lda*k.\nBefore entry, the leading m-by-k part of the\narray a must contain the matrix A.\nFor cblas_gemm_s8u8s32_compute, the\nelement in the a array must be an 8-bit\nsigned integer.\ntransa = CblasTrans\nArray, size lda*m.\nBefore entry, the leading k-by-m part of the\narray a must contain the matrix A.\nFor cblas_gemm_s8u8s32_compute, the\nelement in the a array must be an 8-bit\nsigned integer.\ntransa = CblasPacked\nArray of size returned by\ncblas_gemm_*_pack_get_size and\ninitialized using cblas_gemm_*_pack\nLayout = CblasRowMajor\ntransa = CblasNoTrans\nArray, size lda*m.\nBefore entry, the leading k-by-m part of the\narray a must contain the matrix A.\nFor cblas_gemm_s8u8s32_compute, the\nelement in the a array must be an 8-bit\nunsigned integer.\ntransa = CblasTrans\nArray, size lda*k.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n417\n\n\nLayout = CblasRowMajor\nBefore entry, the leading m-by-k part of the\narray a must contain the matrix A.\nFor cblas_gemm_s8u8s32_compute, the\nelement in the a array must be an 8-bit\nunsigned integer.\ntransa = CblasPacked\nArray size returned by\ncblas_gemm_*_pack_get_size and\ninitialized using cblas_gemm_*_pack\nlda\nMKL_INTSpecifies the leading dimension of a as declared in the calling (sub)program.\ntransa = CblasNoTrans\ntransa = CblasTrans\nLayout =\nCblasColMajor\nlda must be at least max(1, m).\nlda must be at least max(1, k).\nLayout =\nCblasRowMajor\nlda must be at least max(1, k).\nlda must be at least max(1, m).\noa\nMKL_INT8 for cblas_gemm_s8u8s32_compute\nMKL_INT16 for cblas_gemm_s16s16s32_compute\nSpecifies the scalar offset value for the matrix A.\nb\nvoid* for gemm_s8u8s32_compute\nMKL_INT16* for gemm_s16s16s32_compute\nLayout = CblasColMajor\ntransa = CblasNoTrans\nArray, size ldb*n.\nBefore entry, the leading k-by-n part of the\narray b must contain the matrix B.\nFor cblas_gemm_s8u8s32_compute, the\nelement in the b array must be an 8-bit\nunsigned integer.\ntransa = CblasTrans\nArray, size ldb*k.\nBefore entry, the leading n-by-k part of the\narray b must contain the matrix B.\nFor cblas_gemm_s8u8s32_compute, the\nelement in the b array must be an 8-bit\nunsigned integer.\ntransa = CblasPacked\nArray of size returned by\ncblas_gemm_*_pack_get_size and\ninitialized using cblas_gemm_*_pack\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n418\n\n\nLayout = CblasRowMajor\ntransa = CblasNoTrans\nArray, sizeldb*k.\nBefore entry, the leading n-by-k part of the\narray b must contain the matrix B.\nFor cblas_gemm_s8u8s32_compute, the\nelement in the b array must be an 8-bit\nsigned integer.\ntransa = CblasTrans\nArray, size ldb*n.\nBefore entry, the leading k-by-n part of the\narray b must contain the matrix B.\nFor cblas_gemm_s8u8s32_compute, the\nelement in the b array must be an 8-bit\nsigned integer.\ntransa = CblasPacked\nArray of size returned by\ncblas_gemm_*_pack_get_size and\ninitialized using cblas_gemm_*_pack\nldb\nMKL_INT Specifies the leading dimension of b as declared in the calling (sub)program.\ntransb = CblasNoTrans\ntransb = CblasTrans\nLayout =\nCblasColMajor\nldb must be at least max(1, k).\nldb must be at least max(1, n).\nLayout =\nCblasRowMajor\nldb must be at least max(1, n).\nldb must be at least max(1, k).\nob\nMKL_INT8 for cblas_gemm_s8u8s32_compute\nMKL_INT16 for cblas_gemm_s16s16s32_compute\nSpecifies the scalar offset value for the matrix B.\nbeta\nfloat\nSpecifies the scalar beta.\nc\nMKL_INT32*\nArray:\nLayout =\nCblasColMajor\nArray, size ldc*n.\nBefore entry, the leading m-by-n part of the array c must contain the\nmatrix C, except when beta is equal to zero, in which case c need not\nbe set on entry.\nLayout =\nCblasRowMajor\nArray, size ldc*m.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n419\n\n\nBefore entry, the leading n-by-m part of the array c must contain the\nmatrix C, except when beta is equal to zero, in which case c need not\nbe set on entry.\nldc\nMKL_INT Specifies the leading dimension of c as declared in the calling (sub)program.\nLayout = CblasColMajor\nldc must be at least max(1, m)\nLayout = CblasRowMajor\nldc must be at least max(1, n)\noc\nMKL_INT32*\nArray, size len. Specifies the scalar offset value for the matrix C.\nIf offsetc = CblasFixOffset , len must be at least 1.\nIf offsetc = CblasColOffset , len must be at least max(1, m).\nIf offsetc = CblasRowOffset , len must be at least max(1, n).\nOutput Parameters\nc\nMKL_INT32*\nOverwritten by the matrix alpha*(op(A) + A_offset)*(op(B) +\nB_offset) + beta*C + C_offset.\nExample\nSee the following examples in the MKL installation directory to understand the use of these routines:\ncblas_gemm_s8u8s32_compute: examples\\cblas\\source\\cblas_gemm_s8u8s32_computex.c\ncblas_gemm_s16s16s32_compute: examples\\cblas\\source\\cblas_gemm_s16s16s32_computex.c\nApplication Notes\nYou can expand the matrix-matrix product in this manner:\n(op(A) + A_offset)*(op(B) + B_offset) = op(A)*op(B) + op(A)*B_offset + A_offset*op(B) +\nA_offset*B_offset\nAfter computing these four multiplication terms separately, they are summed from left to right. The results\nfrom the matrix-matrix product and the C matrix are scaled with alpha and beta floating-point values\nrespectively using double-precision arithmetic. Before storing the results to the output c array, the floating-\npoint values are rounded to the nearest integers.\nIn the event of overflow or underflow, the results depend on the architecture. The results are either\nunsaturated (wrapped) or saturated to maximum or minimum representable integer values for the data type\nof the output matrix.\nWhen using cblas_gemm_s8u8s32_compute with row-major layout , the data types of A and B must be\nswapped. That is, you must provide an 8-bit unsigned integer array for matrix A and an 8-bit signed integer\narray for matrix B .\nSee Also\ncblas_gemm_*_pack_get_size\n to return the number of bytes needed to store the packed matrix.\ncblas_gemm_*_pack\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n420\n\n\n to pack the matrix into the buffer allocated previously.\ncblas_gemm_bf16bf16f32_compute\nComputes a matrix-matrix product with general\nbfloat16 matrices (where one or both input matrices\nare stored in a packed data structure) and adds the\nresult to a scalar-matrix product.\nSyntax\nC:\nvoid cblas_gemm_bf16bf16f32_compute (const CBLAS_LAYOUT Layout, const MKL_INT transa,\nconst MKL_INT transb, const MKL_INT m, const MKL_INT n, const MKL_INT k, const float\nalpha, const MKL_BF16 *a, const MKL_INT lda, const MKL_BF16 *b, const MKL_INT ldb,\nconst float beta, float *c, const MKL_INT ldc);\nInclude Files\n•\nmkl.h\nDescription\nThe cblas_gemm_bf16bf16f32_compute routine is one of a set of related routines that enable use of an\ninternal packed storage. After calling cblas_gemm_bf16bf16f32_pack call\ncblas_gemm_bf16bf16f32_compute to compute\nC := alpha* op(A)*op(B) + beta*C,\nwhere:\nop(X) is either op(X) = X or op(X) = XT,\nalpha and beta are scalars,\nA , B, and C are matrices:\nop(A) is an m-by-k matrix,\nop(B) is a k-by-n matrix,\nC is an m-by-n matrix.\nNOTE\nYou must use the same value of the Layout parameter for the entire sequence of related\ncblas_gemm_bf16bf16f32_pack and cblas_gemm_bf16bf16f32_compute calls.\nFor best performance, use the same number of threads for packing and for computing.\nIf packing for both A and B matrices, you must use the same number of threads for packing A as for\npacking B.\nInput Parameters\nLayout\nCBLAS_LAYOUT\nSpecifies whether two-dimensional array storage is row-major\n(CblasRowMajor) or column-major (CblasColMajor).\ntransa\nMKL_INT\nSpecifies the form of op(A) used in the packing:\nIf transa = CblasNoTrans  op(A) = A.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n421\n\n\nIf transa = CblasTrans  op(A) = AT.\nIf transa = CblasPacked the matrix in array a is packed into a\nformat internal to Intel® oneAPI Math Kernel Library (oneMKL) and\nlda is ignored.\ntransb\nMKL_INT\nSpecifies the form of op(B) used in the packing:\nIf transb = CblasNoTrans  op(B) = B.\nIf transb = CblasTrans op(B) = BT.\nIf transb = CblasPacked the matrix in array b is packed into a\nformat internal to Intel® oneAPI Math Kernel Library (oneMKL) and\nldb is ignored.\nm\nMKL_INT\nSpecifies the number of rows of the matrix op(A) and of the matrix C.\nThe value of m must be at least zero.\nn\nMKL_INT\nSpecifies the number of columns of the matrix op(B) and the number\nof columns of the matrix C. The value of n must be at least zero.\nk\nMKL_INT\nSpecifies the number of columns of the matrix op(A) and the number\nof rows of the matrix op(B). The value of k must be at least zero.\nalpha\nfloat\nSpecifies the scalar alpha.\na\nMKL_BF16*\ntransa =\nCblasNoTrans\ntransa =\nCblasTrans\ntransa = CblasPacked\nLayout =\nCblasColMajor\nArray, size\nlda*k.\nBefore entry,\nthe leading m-\nby-k part of\nthe array a\nmust contain\nthe matrix A.\nArray, size\nlda*m.\nBefore\nentry, the\nleading k-\nby-m part of\nthe array a\nmust\ncontain the\nmatrix A.\nArray of size returned by\ncblas_gemm_bf16bf16f32_pac\nand initialized using\ncblas_gemm_bf16bf16f32_pac\nLayout =\nCblasRowMajor\nArray, size\nlda*m.\nBefore entry,\nthe leading k-\nby-m part of\nArray, size\nlda*k.\nBefore\nentry, the\nleading m-\nArray size returned by\ncblas_gemm_bf16bf16f32_pac\nand initialized using\ncblas_gemm_bf16bf16f32_pac\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n422\n\n\nthe array a\nmust contain\nthe matrix A.\nby-k part of\nthe array a\nmust\ncontain the\nmatrix A.\nlda\nMKL_INT\nSpecifies the leading dimension of a as declared in the calling\n(sub)program.\ntransa =\nCblasNoTrans\ntransa =\nCblasTrans\nLayout =\nCblasColMajor\nlda must be at least\nmax(1, m).\nlda must be at least\nmax(1, k).\nLayout =\nCblasRowMajor\nlda must be at least\nmax(1, k).\nlda must be at least\nmax(1, m).\nb\nMKL_BF16*\ntransa =\nCblasNoTrans\ntransa =\nCblasTrans\ntransa = CblasPacked\nLayout =\nCblasColMajor\nArray, size\nldb*n.\nBefore entry,\nthe leading k-\nby-n part of\nthe array b\nmust contain\nthe matrix B.\nArray, size\nldb*k.\nBefore\nentry, the\nleading n-\nby-k part of\nthe array b\nmust\ncontain the\nmatrix B.\nArray of size returned by\ncblas_gemm_bf16bf16f32_pac\nand initialized using\ncblas_gemm_bf16bf16f32_pac\nLayout =\nCblasRowMajor\nArray, size\nldb*k.\nBefore entry,\nthe leading n-\nby-k part of\nthe array b\nmust contain\nthe matrix B.\nArray, size\nldb*n.\nBefore\nentry, the\nleading k-\nby-n part of\nthe array b\nmust\ncontain the\nmatrix B.\nArray size returned by\ncblas_gemm_bf16bf16f32_pac\nand initialized using\ncblas_gemm_bf16bf16f32_pac\nldb\nMKL_INT\nSpecifies the leading dimension of b as declared in the calling\n(sub)program.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n423\n\n\ntransb =\nCblasNoTrans\ntransb =\nCblasTrans\nLayout =\nCblasColMajor\nldb must be at least\nmax(1, k).\nldb must be at least\nmax(1, n).\nLayout =\nCblasRowMajor\nldb must be at least\nmax(1, n).\nldb must be at least\nmax(1, k).\nbeta\nfloat\nSpecifies the scalar beta.\nc\nfloat*\nLayout =\nCblasColMajor\nArray, size ldc*n.\nBefore entry, the leading m-by-n part of the\narray c must contain the matrix C, except\nwhen beta is equal to zero, in which case c\nneed not be set on entry.\nLayout =\nCblasRowMajor\nArray, size ldc*m.\nBefore entry, the leading n-by-m part of the\narray c must contain the matrix C, except\nwhen beta is equal to zero, in which case c\nneed not be set on entry.\nldc\nMKL_INT\nSpecifies the leading dimension of c as declared in the calling\n(sub)program.\nLayout = CblasColMajor\nldc must be at least max(1, m).\nLayout = CblasRowMajor\nldc must be at least max(1, n).\nOutput Parameters\nc\nfloat*\nOverwritten by the matrix alpha * op(A)*op(B) + beta*C.\nExample\nSee the following examples in the Intel® oneAPI Math Kernel Library (oneMKL) installation directory to\nunderstand the use of these routines:\ncblas_gemm_bf16bf16f32_compute:\n examples\\cblas\\source\\cblas_gemm_bf16bf16f32_computex.c\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n424\n\n\nApplication Notes\nOn architectures without native bfloat16 hardware instructions, matrix A and B are upconverted to single\nprecision and SGEMM is called to compute matrix multiplication operation.\ncblas_gemm_bf16bf16f32\nComputes a matrix-matrix product with general\nbfloat16 matrices.\nSyntax\nvoid cblas_gemm_bf16bf16f32 (const CBLAS_LAYOUT Layout, const CBLAS_TRANSPOSE transa,\nconst CBLAS_TRANSPOSE transb, const MKL_INT m, const MKL_INT n, const MKL_INT k, const\nfloat alpha, const MKL_BF16 *a, const MKL_INT lda, const MKL_BF16 *b, const MKL_INT\nldb, const float beta, float *c, const MKL_INT ldc);\nInclude Files\n•\nmkl.h\nDescription\nThe cblas_gemm_bf16bf16f32 routines compute a scalar-matrix-matrix product and adds the result to a\nscalar-matrix product. The operation is defined as:\nC := alpha*op(A) *op(B) + beta*C\nwhere :\nop(X) is one of op(X) = X or op(X) = XT,\nalpha and beta are scalars,\nA, B, and C are matrices\nop(A) is m-by-k matrix,\nop(B) is k-by-n matrix,\nC is an m-by-n matrix.\nInput Parameters\nLayout\nSpecifies whether two-dimensional array storage is row-major\n(CblasRowMajor) or column-major (CblasColMajor).\ntransa\nSpecifies the form of op(A) used in the matrix multiplication:\nif transa=CblasNoTrans, then op(A) = A;\nif transa=CblasTrans, then op(A) = AT.\ntransb\nSpecifies the form of op(B) used in the matrix multiplication:\nif transb=CblasNoTrans, then op(B) = B;\nif transb=CblasTrans, then op(B) = BT.\nm\nSpecifies the number of rows of the matrix op(A) and of the matrix C.\nThe value of m must be at least zero.\nn\nSpecifies the number of columns of the matrix op(B) and the number\nof columns of the matrix C. The value of n must be at least zero.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n425\n\n\nk\nSpecifies the number of columns of the matrix op(A) and the number\nof rows of the matrix op(B). The value of k must be at least zero.\nalpha\nSpecifies the scalar alpha.\na\ntransa=CblasNoTrans\ntransa=CblasTrans\nLayout =\nCblasColMajor\nArray, size lda*k\nBefore entry, the leading\nm-by-k part of the array\na must contain the\nmatrix A.\nArray, size lda*m\nBefore entry, the\nleading k-by-m part of\nthe array a must\ncontain the matrix A.\nLayout =\nCblasRowMajor\nArray, size lda* m\nBefore entry, the leading\nk-by-m part of the array\na must contain the\nmatrix.\nArray, size lda*k\nBefore entry, the\nleading m-by-k part of\nthe array a must\ncontain the matrix.\nlda\nSpecifies the leading dimension of a as declared in the calling\n(sub)program.\ntransa=CblasNoTrans\ntransa=CblasTrans\nLayout =\nCblasColMajor\nlda must be at least\nmax(1, m).\nlda must be at least\nmax(1, k).\nLayout =\nCblasRowMajor\nlda must be at least\nmax(1, k).\nlda must be at least\nmax(1, m).\nb\ntransb=CblasNoTrans\ntransb=CblasTrans\nLayout =\nCblasColMajor\nArray, size ldb by n\nBefore entry, the leading\nk-by-n part of the array\nb must contain the\nmatrix B.\nArray, size ldb by k\nBefore entry the\nleading n-by-k part of\nthe array b must\ncontain the matrix B.\nLayout =\nCblasRowMajor\nArray, size ldb by k\nBefore entry the leading\nn-by-k part of the array\nb must contain the\nmatrix B.\nArray, size ldb by n\nBefore entry, the\nleading k-by-n part of\nthe array b must\ncontain the matrix B.\nldb\nSpecifies the leading dimension of b as declared in the calling\n(sub)program.\ntransb=CblasNoTrans\ntransb=CblasTrans\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n426\n\n\nLayout =\nCblasColMajor\nldb must be at least\nmax(1, k).\nldb must be at least\nmax(1, n).\nLayout =\nCblasRowMajor\nldb must be at least\nmax(1, n).\nldb must be at least\nmax(1, k).\nbeta\nSpecifies the scalar beta. When beta is equal to zero, then c need not\nbe set on input.\nc\nLayout =\nCblasColMajor\nArray, size ldc by n. Before entry, the leading\nm-by-n part of the array c must contain the\nmatrix C, except when beta is equal to zero,\nin which case c need not be set on entry.\nLayout =\nCblasRowMajor\nArray, size ldc by m. Before entry, the leading\nn-by-m part of the array c must contain the\nmatrix C, except when beta is equal to zero,\nin which case c need not be set on entry.\nldc\nSpecifies the leading dimension of c as declared in the calling\n(sub)program.\nLayout = CblasColMajor\nldc must be at least max(1, m).\nLayout = CblasRowMajor\nldc must be at least max(1, n).\nOutput Parameters\nc\nOverwritten by alpha* op(A) * op(B) + beta*C.\nExample\nFor examples of routine usage, see these code examples in the Intel® oneAPI Math Kernel Library (oneMKL)\ninstallation directory:\n•\ncblas_gemm_bf16bf16f32: examples\\cblas\\source\\cblas_gemm_bf16bf16f32x.c\nApplication Notes\nOn architectures without native bfloat16 hardware instructions, matrix A and B are upconverted to single\nprecision and SGEMM is called to compute matrix multiplication operation.\ncblas_gemm_f16f16f32_compute\nComputes a matrix-matrix product with general\nmatrices of half-precision data type (where one or\nboth input matrices are stored in a packed data\nstructure) and adds the result to a scalar-matrix\nproduct.\nSyntax\nC:\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n427\n\n\nvoid cblas_gemm_f16f16f32_compute (const CBLAS_LAYOUT Layout, const MKL_INT transa,\nconst MKL_INT transb, const MKL_INT m, const MKL_INT n, const MKL_INT k, const float\nalpha, const MKL_F16 *a, const MKL_INT lda, const MKL_F16 *b, const MKL_INT ldb, const\nfloat beta, float *c, const MKL_INT ldc);\nInclude Files\n•\nmkl.h\nDescription\nThe cblas_gemm_f16f16f32_compute routine is one of a set of related routines that enable use of an\ninternal packed storage. After calling cblas_gemm_f16f16f32_pack call cblas_gemm_f16f16f32_compute\nto compute\nC := alpha* op(A)*op(B) + beta*C,\nwhere:\nop(X) is either op(X) = X or op(X) = XT,\nalpha and beta are scalars,\nA , B, and C are matrices:\nop(A) is an m-by-k matrix,\nop(B) is a k-by-n matrix,\nC is an m-by-n matrix.\nNOTE\nYou must use the same value of the Layout parameter for the entire sequence of related\ncblas_gemm_f16f16f32_pack and cblas_gemm_f16f16f32_compute calls.\nFor best performance, use the same number of threads for packing and for computing.\nIf packing for both A and B matrices, you must use the same number of threads for packing A as for\npacking B.\nInput Parameters\nLayout\nCBLAS_LAYOUT\nSpecifies whether two-dimensional array storage is row-major\n(CblasRowMajor) or column-major (CblasColMajor).\ntransa\nMKL_INT\nSpecifies the form of op(A) used in the packing:\nIf transa = CblasNoTrans  op(A) = A.\nIf transa = CblasTrans  op(A) = AT.\nIf transa = CblasPacked the matrix in array a is packed into a\nformat internal to Intel® oneAPI Math Kernel Library (oneMKL) and\nlda is ignored.\ntransb\nMKL_INT\nSpecifies the form of op(B) used in the packing:\nIf transb = CblasNoTrans  op(B) = B.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n428\n\n\nIf transb = CblasTrans op(B) = BT.\nIf transb = CblasPacked the matrix in array b is packed into a\nformat internal to Intel® oneAPI Math Kernel Library (oneMKL) and\nldb is ignored.\nm\nMKL_INT\nSpecifies the number of rows of the matrix op(A) and of the matrix C.\nThe value of m must be at least zero.\nn\nMKL_INT\nSpecifies the number of columns of the matrix op(B) and the number\nof columns of the matrix C. The value of n must be at least zero.\nk\nMKL_INT\nSpecifies the number of columns of the matrix op(A) and the number\nof rows of the matrix op(B). The value of k must be at least zero.\nalpha\nfloat\nSpecifies the scalar alpha.\na\nMKL_F16*\ntransa =\nCblasNoTrans\ntransa =\nCblasTrans\ntransa = CblasPacked\nLayout =\nCblasColMajor\nArray, size\nlda*k.\nBefore entry,\nthe leading m-\nby-k part of\nthe array a\nmust contain\nthe matrix A.\nArray, size\nlda*m.\nBefore\nentry, the\nleading k-\nby-m part of\nthe array a\nmust\ncontain the\nmatrix A.\nArray of size returned by\ncblas_gemm_f16f16f32_pack_\nand initialized using\ncblas_gemm_f16f16f32_pack.\nLayout =\nCblasRowMajor\nArray, size\nlda*m.\nBefore entry,\nthe leading k-\nby-m part of\nthe array a\nmust contain\nthe matrix A.\nArray, size\nlda*k.\nBefore\nentry, the\nleading m-\nby-k part of\nthe array a\nmust\ncontain the\nmatrix A.\nArray size returned by\ncblas_gemm_f16f16f32_pack_\nand initialized using\ncblas_gemm_f16f16f32_pack.\nlda\nMKL_INT\nSpecifies the leading dimension of a as declared in the calling\n(sub)program.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n429\n\n\ntransa =\nCblasNoTrans\ntransa =\nCblasTrans\nLayout =\nCblasColMajor\nlda must be at least\nmax(1, m).\nlda must be at least\nmax(1, k).\nLayout =\nCblasRowMajor\nlda must be at least\nmax(1, k).\nlda must be at least\nmax(1, m).\nb\nMKL_F16*\ntransa =\nCblasNoTrans\ntransa =\nCblasTrans\ntransa = CblasPacked\nLayout =\nCblasColMajor\nArray, size\nldb*n.\nBefore entry,\nthe leading k-\nby-n part of\nthe array b\nmust contain\nthe matrix B.\nArray, size\nldb*k.\nBefore\nentry, the\nleading n-\nby-k part of\nthe array b\nmust\ncontain the\nmatrix B.\nArray of size returned by\ncblas_gemm_f16f16f32_pack_\nand initialized using\ncblas_gemm_f16f16f32_pack.\nLayout =\nCblasRowMajor\nArray, size\nldb*k.\nBefore entry,\nthe leading n-\nby-k part of\nthe array b\nmust contain\nthe matrix B.\nArray, size\nldb*n.\nBefore\nentry, the\nleading k-\nby-n part of\nthe array b\nmust\ncontain the\nmatrix B.\nArray size returned by\ncblas_gemm_f16f16f32_pack_\nand initialized using\ncblas_gemm_f16f16f32_pack.\nldb\nMKL_INT\nSpecifies the leading dimension of b as declared in the calling\n(sub)program.\ntransb =\nCblasNoTrans\ntransb =\nCblasTrans\nLayout =\nCblasColMajor\nldb must be at least\nmax(1, k).\nldb must be at least\nmax(1, n).\nLayout =\nCblasRowMajor\nldb must be at least\nmax(1, n).\nldb must be at least\nmax(1, k).\nbeta\nfloat\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n430\n\n\nSpecifies the scalar beta.\nc\nfloat*\nLayout =\nCblasColMajor\nArray, size ldc*n.\nBefore entry, the leading m-by-n part of the\narray c must contain the matrix C, except\nwhen beta is equal to zero, in which case c\nneed not be set on entry.\nLayout =\nCblasRowMajor\nArray, size ldc*m.\nBefore entry, the leading n-by-m part of the\narray c must contain the matrix C, except\nwhen beta is equal to zero, in which case c\nneed not be set on entry.\nldc\nMKL_INT\nSpecifies the leading dimension of c as declared in the calling\n(sub)program.\nLayout = CblasColMajor\nldc must be at least max(1, m).\nLayout = CblasRowMajor\nldc must be at least max(1, n).\nOutput Parameters\nc\nfloat*\nOverwritten by the matrix alpha * op(A)*op(B) + beta*C.\nExample\nSee the following examples in the Intel® oneAPI Math Kernel Library (oneMKL) installation directory to\nunderstand the use of these routines:\ncblas_gemm_f16f16f32_compute:\n examples\\cblas\\source\\cblas_gemm_f16f16f32_computex.c\nApplication Notes\nOn architectures without native half precision hardware instructions, matrix A and B are upconverted to\nsingle precision and SGEMM is called to compute matrix multiplication operation.\ncblas_gemm_f16f16f32\nComputes a matrix-matrix product with general\nmatrices of half precision data type.\nSyntax\nvoid cblas_gemm_f16f16f32 (const CBLAS_LAYOUT Layout, const CBLAS_TRANSPOSE transa,\nconst CBLAS_TRANSPOSE transb, const MKL_INT m, const MKL_INT n, const MKL_INT k, const\nfloat alpha, const MKL_F16 *a, const MKL_INT lda, const MKL_F16 *b, const MKL_INT ldb,\nconst float beta, float *c, const MKL_INT ldc);\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n431\n\n\nInclude Files\n•\nmkl.h\nDescription\nThe cblas_gemm_f16f16f32 routines compute a scalar-matrix-matrix product and adds the result to a\nscalar-matrix product. The operation is defined as:\nC := alpha*op(A) *op(B) + beta*C\nwhere :\nop(X) is one of op(X) = X or op(X) = XT,\nalpha and beta are scalars,\nA, B, and C are matrices\nop(A) is m-by-k matrix,\nop(B) is k-by-n matrix,\nC is an m-by-n matrix.\nInput Parameters\nLayout\nSpecifies whether two-dimensional array storage is row-major\n(CblasRowMajor) or column-major (CblasColMajor).\ntransa\nSpecifies the form of op(A) used in the matrix multiplication:\nif transa=CblasNoTrans, then op(A) = A;\nif transa=CblasTrans, then op(A) = AT.\ntransb\nSpecifies the form of op(B) used in the matrix multiplication:\nif transb=CblasNoTrans, then op(B) = B;\nif transb=CblasTrans, then op(B) = BT.\nm\nSpecifies the number of rows of the matrix op(A) and of the matrix C.\nThe value of m must be at least zero.\nn\nSpecifies the number of columns of the matrix op(B) and the number\nof columns of the matrix C. The value of n must be at least zero.\nk\nSpecifies the number of columns of the matrix op(A) and the number\nof rows of the matrix op(B). The value of k must be at least zero.\nalpha\nSpecifies the scalar alpha.\na\ntransa=CblasNoTrans\ntransa=CblasTrans\nLayout =\nCblasColMajor\nArray, size lda*k\nBefore entry, the leading\nm-by-k part of the array\na must contain the\nmatrix A.\nArray, size lda*m\nBefore entry, the\nleading k-by-m part of\nthe array a must\ncontain the matrix A.\nLayout =\nCblasRowMajor\nArray, size lda* m\nArray, size lda*k\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n432\n\n\nBefore entry, the leading\nk-by-m part of the array\na must contain the\nmatrix.\nBefore entry, the\nleading m-by-k part of\nthe array a must\ncontain the matrix.\nlda\nSpecifies the leading dimension of a as declared in the calling\n(sub)program.\ntransa=CblasNoTrans\ntransa=CblasTrans\nLayout =\nCblasColMajor\nlda must be at least\nmax(1, m).\nlda must be at least\nmax(1, k).\nLayout =\nCblasRowMajor\nlda must be at least\nmax(1, k).\nlda must be at least\nmax(1, m).\nb\ntransb=CblasNoTrans\ntransb=CblasTrans\nLayout =\nCblasColMajor\nArray, size ldb by n\nBefore entry, the leading\nk-by-n part of the array\nb must contain the\nmatrix B.\nArray, size ldb by k\nBefore entry the\nleading n-by-k part of\nthe array b must\ncontain the matrix B.\nLayout =\nCblasRowMajor\nArray, size ldb by k\nBefore entry the leading\nn-by-k part of the array\nb must contain the\nmatrix B.\nArray, size ldb by n\nBefore entry, the\nleading k-by-n part of\nthe array b must\ncontain the matrix B.\nldb\nSpecifies the leading dimension of b as declared in the calling\n(sub)program.\ntransb=CblasNoTrans\ntransb=CblasTrans\nLayout =\nCblasColMajor\nldb must be at least\nmax(1, k).\nldb must be at least\nmax(1, n).\nLayout =\nCblasRowMajor\nldb must be at least\nmax(1, n).\nldb must be at least\nmax(1, k).\nbeta\nSpecifies the scalar beta. When beta is equal to zero, then c need not\nbe set on input.\nc\nLayout =\nCblasColMajor\nArray, size ldc by n. Before entry, the leading\nm-by-n part of the array c must contain the\nmatrix C, except when beta is equal to zero,\nin which case c need not be set on entry.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n433\n\n\nLayout =\nCblasRowMajor\nArray, size ldc by m. Before entry, the leading\nn-by-m part of the array c must contain the\nmatrix C, except when beta is equal to zero,\nin which case c need not be set on entry.\nldc\nSpecifies the leading dimension of c as declared in the calling\n(sub)program.\nLayout = CblasColMajor\nldc must be at least max(1, m).\nLayout = CblasRowMajor\nldc must be at least max(1, n).\nOutput Parameters\nc\nOverwritten by alpha* op(A) * op(B) + beta*C.\nExample\nFor examples of routine usage, see these code examples in the Intel® oneAPI Math Kernel Library (oneMKL)\ninstallation directory:\n•\ncblas_gemm_f16f16f32: examples\\cblas\\source\\cblas_gemm_f16f16f32x.c\nApplication Notes\nOn architectures without native half precision hardware instructions, matrix A and B are upconverted to\nsingle precision and SGEMM is called to compute matrix multiplication operation.\ncblas_?gemm_free\nFrees the storage previously allocated for the packed\nmatrix (deprecated).\nSyntax\nvoid cblas_sgemm_free (float *dest);\nvoid cblas_dgemm_free (double *dest);\nInclude Files\n•\nmkl.h\nDescription\nThe cblas_?gemm_free routine is one of a set of related routines that enable use of an internal packed\nstorage. Call the cblas_?gemm_free routine last to release storage for the packed matrix structure allocated\nwith cblas_?gemm_alloc (deprecated).\nInput Parameters\ndest\nPreviously allocated storage.\nOutput Parameters\ndest\nThe freed buffer.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n434\n\n\nSee Also\ncblas_?gemm_pack Performs scaling and packing of the matrix into the previously allocated buffer.\ncblas_?gemm_compute Computes a matrix-matrix product with general matrices where one or both\ninput matrices are stored in a packed data structure and adds the result to a scalar-matrix\nproduct.\ncblas_?gemm\n for a detailed description of general matrix multiplication.\ncblas_gemm_*\nComputes a matrix-matrix product with general\ninteger matrices.\nSyntax\nvoid cblas_gemm_s8u8s32 (const CBLAS_LAYOUT Layout, const CBLAS_TRANSPOSE transa, const\nCBLAS_TRANSPOSE transb, const CBLAS_OFFSET offsetc, const MKL_INT m, const MKL_INT n,\nconst MKL_INT k, const float alpha, const void *a, const MKL_INT lda, const MKL_INT8\noa, const void *b, const MKL_INT ldb, const MKL_INT8 ob, const float beta, MKL_INT32 *c,\nconst MKL_INT ldc, const MKL_INT32 *oc);\nvoid cblas_gemm_s16s16s32 (const CBLAS_LAYOUT Layout, const CBLAS_TRANSPOSE transa,\nconst CBLAS_TRANSPOSE transb, const CBLAS_OFFSET offsetc, const MKL_INT m, const\nMKL_INT n, const MKL_INT k, const float alpha, const MKL_INT16 *a, const MKL_INT lda,\nconst MKL_INT16 oa, const MKL_INT16 *b, const MKL_INT ldb, const MKL_INT16 ob, const\nfloat beta, MKL_INT32 *c, const MKL_INT ldc, const MKL_INT32 *oc);\nInclude Files\n•\nmkl.h\nDescription\nThe cblas_gemm_* routines compute a scalar-matrix-matrix product and adds the result to a scalar-matrix\nproduct. To get the final result, a vector is added to each row or column of the output matrix. The operation\nis defined as:\nC := alpha*(op(A) + A_offset)*(op(B) + B_offset) + beta*C + C_offset\nwhere :\nop(X) is either op(X) = X or op(X) = XT,\nA_offset is an m-by-k matrix with every element equal to the value oa,\nB_offset is a k-by-n matrix with every element equal to the value ob,\nC_offset is an m-by-n matrix defined by the oc array as described in the description of the offsetc\nparameter,\nalpha and beta are scalars,\nA is a matrix such that op(A) is m-by-k,\nB is a matrix such that op(B) is k-by-n,\nand C is an m-by-n matrix.\nInput Parameters\nLayout\nSpecifies whether two-dimensional array storage is row-major\n(CblasRowMajor) or column-major (CblasColMajor).\ntransa\nSpecifies the form of op(A) used in the matrix multiplication:\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n435\n\n\nif transa=CblasNoTrans, then op(A) = A;\nif transa=CblasTrans, then op(A) = AT.\ntransb\nSpecifies the form of op(B) used in the matrix multiplication:\nif transb=CblasNoTrans, then op(B) = B;\nif transb=CblasTrans, then op(B) = BT.\noffsetc\nSpecifies the form of C_offset used in the matrix multiplication.\noffsetc = CblasFixOffset: oc has a single element and every\nelement of C_offset is equal to this element.\noffsetc = CblasColOffset: oc has a size of m and every column of\nC_offset is equal to oc.\noffsetc = CblasRowOffset: oc has a size of n and every row of\nC_offset is equal to oc.\nm\nSpecifies the number of rows of the matrix op(A) and of the matrix C. The\nvalue of m must be at least zero.\nn\nSpecifies the number of columns of the matrix op(B) and the number of\ncolumns of the matrix C. The value of n must be at least zero.\nk\nSpecifies the number of columns of the matrix op(A) and the number of\nrows of the matrix op(B). The value of k must be at least zero.\nalpha\n. Specifies the scalar alpha.\na\ntransa=CblasNoTrans\ntransa=CblasTrans\nLayout =\nCblasColMajor\nArray, size lda*k\nBefore entry, the leading\nm-by-k part of the array a\nmust contain the matrix A\nof 8-bit signed integers for\ncblas_gemm_s8u8s32 or\n16-bit signed integers for\ncblas_gemm_s16s16s32.\nArray, size lda*m\nBefore entry, the leading\nk-by-m part of the array a\nmust contain the matrix A\nof 8-bit signed integers for\ncblas_gemm_s8u8s32 or\n16-bit signed integers for\ncblas_gemm_s16s16s32.\nLayout =\nCblasRowMajor\nArray, size lda* m\nBefore entry, the leading\nk-by-m part of the array a\nmust contain the matrix A\nof 8-bit unsigned integers\nfor cblas_gemm_s8u8s32\nor 16-bit signed integers\nfor\ncblas_gemm_s16s16s32.\nArray, size lda*k\nBefore entry, the leading\nm-by-k part of the array a\nmust contain the matrix A\nof 8-bit unsigned integers\nfor cblas_gemm_s8u8s32\nor 16-bit signed integers\nfor\ncblas_gemm_s16s16s32.\nlda\nSpecifies the leading dimension of a as declared in the calling\n(sub)program.\ntransa=CblasNoTrans\ntransa=CblasTrans\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n436\n\n\nLayout =\nCblasColMajor\nlda must be at least\nmax(1, m).\nlda must be at least\nmax(1, k).\nLayout =\nCblasRowMajor\nlda must be at least\nmax(1, k).\nlda must be at least\nmax(1, m).\noa\nSpecifies the scalar offset value for matrix A.\nb\ntransb=CblasNoTrans\ntransb=CblasTrans\nLayout =\nCblasColMajor\nArray, size ldb by n\nBefore entry, the leading\nk-by-n part of the array b\nmust contain the matrix B\nof 8-bit unsigned integers\nfor cblas_gemm_s8u8s32\nor 16-bit signed integers\nfor\ncblas_gemm_s16s16s32.\nArray, size ldb by k\nBefore entry the leading n-\nby-k part of the array b\nmust contain the matrix B\nof 8-bit unsigned integers\nfor cblas_gemm_s8u8s32\nor 16-bit signed integers\nfor\ncblas_gemm_s16s16s32.\nLayout =\nCblasRowMajor\nArray, size ldb by k\nBefore entry the leading n-\nby-k part of the array b\nmust contain the matrix B\nof 8-bit signed integers for\ncblas_gemm_s8u8s32 or\n16-bit signed integers for\ncblas_gemm_s16s16s32.\nArray, size ldb by n\nBefore entry, the leading\nk-by-n part of the array b\nmust contain the matrix B\nof 8-bit signed integers for\ncblas_gemm_s8u8s32 or\n16-bit signed integers for\ncblas_gemm_s16s16s32.\nldb\nSpecifies the leading dimension of b as declared in the calling\n(sub)program.\ntransb=CblasNoTrans\ntransb=CblasTrans\nLayout =\nCblasColMajor\nldb must be at least\nmax(1, k).\nldb must be at least\nmax(1, n).\nLayout =\nCblasRowMajor\nldb must be at least\nmax(1, n).\nldb must be at least\nmax(1, k).\nob\nSpecifies the scalar offset value for matrix B.\nbeta\nSpecifies the scalar beta. When beta is equal to zero, then c need not be\nset on input.\nc\nLayout =\nCblasColMajor\nArray, size ldc by n. Before entry, the leading m-\nby-n part of the array c must contain the matrix C,\nexcept when beta is equal to zero, in which case c\nneed not be set on entry.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n437\n\n\nLayout =\nCblasRowMajor\nArray, size ldc by m. Before entry, the leading n-\nby-m part of the array c must contain the matrix C,\nexcept when beta is equal to zero, in which case c\nneed not be set on entry.\nldc\nSpecifies the leading dimension of c as declared in the calling\n(sub)program.\nLayout = CblasColMajor\nldc must be at least max(1, m).\nLayout = CblasRowMajor\nldc must be at least max(1, n).\noc\nArray, size len. Specifies the offset values for matrix C.\nIf offsetc = CblasFixOffset: len must be at least 1.\nIf offsetc = CblasColOffset: len must be at least max(1, m).\nIf offsetc = CblasRowOffset: oc must be at least max(1, n).\nOutput Parameters\nc\nOverwritten by alpha*(op(A) + A_offset)*(op(B) + B_offset)\n+ beta*C+ C_offset.\nExample\nFor examples of routine usage, see the code in in the following links and in the Intel® oneAPI Math Kernel\nLibrary (oneMKL) installation directory:\n•\ncblas_gemm_s8u8s32: examples\\cblas\\source\\cblas_gemm_s8u8s32x.c\n•\ncblas_gemm_s16s16s32: examples\\cblas\\source\\cblas_gemm_s16s16s32x.c\nApplication Notes\nThe matrix-matrix product can be expanded:\n(op(A) + A_offset)*(op(B) + B_offset)\n= op(A)*op(B) + op(A)*B_offset + A_offset*op(B) + A_offset*B_offset\nAfter computing these four multiplication terms separately, they are summed from left to right. The results\nfrom the matrix-matrix product and the C matrix are scaled with alpha and beta floating-point values\nrespectively using double-precision arithmetic. Before storing the results to the output c array, the floating-\npoint values are rounded to the nearest integers. In the event of overflow or underflow, the results depend\non the architecture . The results are either unsaturated (wrapped) or saturated to maximum or minimum\nrepresentable integer values for the data type of the output matrix.\nWhen using cblas_gemm_s8u8s32 with row-major layout, the data types of A and B must be swapped. That\nis, you must provide an 8-bit unsigned integer array for matrix A and an 8-bit signed integer array for matrix\nB.\nIntermediate integer computations in cblas_gemm_s8u8s32 on 64-bit Intel® Advanced Vector Extensions 2\n(Intel® AVX2) and Intel® Advanced Vector Extensions 512 (Intel® AVX-512) architectures without Vector\nNeural Network Instructions (VNNI) extensions can saturate. This is because only 16-bits are available for\nthe accumulation of intermediate results. You can avoid integer saturation by maintaining all integer\nelements of A or B matrices under 8 bits.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n438\n\n\ncblas_?gemv_batch_strided\nComputes groups of matrix-vector product with\ngeneral matrices.\nSyntax\nvoid cblas_sgemv_batch_strided (const CBLAS_LAYOUT layout, const CBLAS_TRANSPOSE trans,\nconst MKL_INT m, const MKL_INT n, const float alpha, const float *a, const MKL_INT lda,\nconst MKL_INT stridea, const float *x, const MKL_INT incx, const MKL_INT stridex, const\nfloat beta, float *y, const MKL_INT incy, const MKL_INT stridey, const MKL_INT\nbatch_size);\nvoid cblas_dgemv_batch_strided (const CBLAS_LAYOUT layout, const CBLAS_TRANSPOSE trans,\nconst MKL_INT m, const MKL_INT n, const double alpha, const double *a, const MKL_INT\nlda, const MKL_INT stridea, const double *x, const MKL_INT incx, const MKL_INT stridex,\nconst double beta, double *y, const MKL_INT incy, const MKL_INT stridey, const MKL_INT\nbatch_size);\nvoid cblas_cgemv_batch_strided (const CBLAS_LAYOUT layout, const CBLAS_TRANSPOSE trans,\nconst MKL_INT m, const MKL_INT n, const void alpha, const void *a, const MKL_INT lda,\nconst MKL_INT stridea, const void *x, const MKL_INT incx, const MKL_INT stridex, const\nvoid beta, void *y, const MKL_INT incy, const MKL_INT stridey, const MKL_INT\nbatch_size);\nvoid cblas_zgemv_batch_strided (const CBLAS_LAYOUT layout, const CBLAS_TRANSPOSE trans,\nconst MKL_INT m, const MKL_INT n, const void alpha, const void *a, const MKL_INT lda,\nconst MKL_INT stridea, const void *x, const MKL_INT incx, const MKL_INT stridex, const\nvoid beta, void *y, const MKL_INT incy, const MKL_INT stridey, const MKL_INT\nbatch_size);\nInclude Files\n•\nmkl.h\nDescription\nThe cblas_?gemv_batch_strided routines perform a series of matrix-vector product added to a scaled\nvector. They are similar to the cblas_?gemv routine counterparts, but the cblas_?gemv_batch_strided\nroutines perform matrix-vector operations with groups of matrices and vectors.\nAll matrices a and vectors x and y have the same parameters (size, increments) and are stored at constant\nstridea, stridex, and stridey from each other. The operation is defined as\nfor i = 0 … batch_size – 1\n    A is a matrix at offset i * stridea in a\n    X and Y are vectors at offset i * stridex and i * stridey in x and y\n    Y = alpha * op(A) * X + beta * Y\nend for\nInput Parameters\nlayout\nSpecifies whether two-dimensional array storage is row-major\n(CblasRowMajor) or column-major (CblasColMajor).\ntrans\nSpecifies op(A) the transposition operation applied to the A matrices.\nif trans = CblasNoTrans, then op(A) = A;\nif trans = CblasTrans, then op(A) = A';\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n439\n\n\nif trans = CblasConjTrans, then op(A) = conjg(A').\nm\nNumber of rows of the matrices A. The value of m must be at least 0.\nn\nNumber of columns of the matrices A. The value of n must be at least 0.\nalpha\nSpecifies the scalar alpha.\na\nArray holding all the input matrix A. Must be of size at least lda*k + stridea\n* (batch_size -1) where k is n if column major layout is used or m if row\nmajor layout is used.\nlda\nSpecifies the leading dimension of the matrixA. It must be positive and at\nleast mif column major layout is used or at least n if row major layout is\nused.\nstridea\nStride between two consecutive A matrices. Must be at least 0.\nx\nArray holding all the input vector x. Must be of size at least (1 +\n(len-1)*abs(incx)) + stridex * (batch_size - 1) where len is n if the A\nmatrix is not transposed or m otherwise.\nincx\nStride between two consecutive elements of the x vectors. Must not be\nzero.\nstridex\nStride between two consecutive x vectors, must be at least 0.\nbeta\nSpecifies the scalar beta.\ny\nArray holding all the input vectors y. Must be of size at least batch_size *\nstridey.\nincy\nStride between two consecutive elements of the y vectors. Must not be\nzero.\nstridey\nStride between two consecutive y vectors, must be at least (1 +\n(len-1)*abs(incy)) where len is m if the matrix A is non transpose or n\notherwise.\nbatch_size\nNumber of gemv computations to perform and a matrices, x and y vectors.\nMust be at least 0.\nOutput Parameters\ny\nArray holding the batch_size updated vector y.\ncblas_?gemv_batch\nComputes groups of matrix-vector product with\ngeneral matrices.\nSyntax\nvoid cblas_sgemv_batch (const CBLAS_LAYOUT layout, const CBLAS_TRANSPOSE *trans_array,\nconst MKL_INT *m_array, const MKL_INT *n_array, const float *alpha_array, const float\n**a_array, const MKL_INT *lda_array, const float **x_array, const MKL_INT *incx_array,\nconst float *beta_array, float **y_array, const MKL_INT *incy_array, const MKL_INT\ngroup_count, const MKL_INT *group_size);\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n440\n\n\nvoid cblas_dgemv_batch (const CBLAS_LAYOUT layout, const CBLAS_TRANSPOSE *trans_array,\nconst MKL_INT *m_array, const MKL_INT *n_array, const double *alpha_array, const double\n**a_array, const MKL_INT *lda_array, const double **x_array, const MKL_INT *incx_array,\nconst double *beta_array, double **y_array, const MKL_INT *incy_array, const MKL_INT\ngroup_count, const MKL_INT *group_size);\nvoid cblas_cgemv_batch (const CBLAS_LAYOUT layout, const CBLAS_TRANSPOSE *trans_array,\nconst MKL_INT *m_array, const MKL_INT *n_array, const void *alpha_array, const void\n**a_array, const MKL_INT *lda_array, const void **x_array, const MKL_INT *incx_array,\nconst void *beta_array, void **y_array, const MKL_INT *incy_array, const MKL_INT\ngroup_count, const MKL_INT *group_size);\nvoid cblas_zgemv_batch (const CBLAS_LAYOUT layout, const CBLAS_TRANSPOSE *trans_array,\nconst MKL_INT *m_array, const MKL_INT *n_array, const void *alpha_array, const void\n**a_array, const MKL_INT *lda_array, const void **x_array, const MKL_INT *incx_array,\nconst void *beta_array, void **y_array, const MKL_INT *incy_array, const MKL_INT\ngroup_count, const MKL_INT *group_size);\nInclude Files\n•\nmkl.h\nDescription\nThe cblas_?gemv_batch routines perform a series of matrix-vector product added to a scaled vector. They\nare similar to the cblas_?gemv routine counterparts, but the cblas_?gemv_batch routines perform matrix-\nvector operations with groups of matrices and vectors.\nEach group contains matrices and vectors with the same parameters (size, increments). The operation is\ndefined as:\nidx = 0\nFor i = 0 … group_count – 1\n    trans, m, n, alpha, lda, incx, beta, incy and group_size at position i in trans_array, \nm_array, n_array, alpha_array, lda_array, incx_array, beta_array, incy_array and group_size_array\n    for j = 0 … group_size – 1\n        a is a matrix of size mxn at position idx in a_array\n        x and y are vectors of size m or n depending on trans, at position idx in x_array and \ny_array\n        y := alpha * op(a) * x + beta * y\n        idx := idx + 1\n    end for\nend for\nThe number of entries in a_array, x_array, and y_array is total_batch_count = the sum of all of the\ngroup_size entries.\nInput Parameters\nlayout\nSpecifies whether two-dimensional array storage is row-major\n(CblasRowMajor) or column-major (CblasColMajor).\ntrans_array\nArray of size group_count. For the group i, transi = trans_array[i] specifies\nthe transposition operation applied to A.\nif trans = CblasNoTrans, then op(A) = A;\nif trans = CblasTrans, then op(A) = A';\nif trans = CblasConjTrans, then op(A) = conjg(A').\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n441\n\n\nm_array\nArray of size group_count. For the group i, mi = m_array[i] is the number\nof rows of the matrix A.\nn_array\nArray of size group_count. For the group i, ni = n_array[i] is the number of\ncolumns in the matrix A.\nalpha_array\nArray of size group_count. For the group i, alphai = alpha_array[i] is the\nscalar alpha.\na_array\nArray of size total_batch_count of pointers used to store A matrices. The\narray allocated for the A matrices of the group i must be of size at least ldai\n* ni if column major layout is used or at least ldai * mi is row major layout\nis used.\nlda_array\nArray of size group_count. For the group i, ldai = lda_array[i] is the leading\ndimension of the matrix A. It must be positive and at least miif column\nmajor layout is used or at least ni if row major layout is used..\nx_array\nArray of size total_batch_count of pointers used to store x vectors. The\narray allocated for the x vectors of the group i must be of size at least (1 +\nleni – 1)*abs(incxi)) where leni is ni if the A matrix is not transposed or mi\notherwise.\nincx_array\nArray of size group_count. For the group i, incxi = incx_array[i] is the stride\nof vector x. Must not be zero.\nbeta_array\nArray of size group_count. For the group i, betai = beta_array[i] is the\nscalar beta.\ny_array\nArray of size total_batch_count of pointers used to store y vectors. The\narray allocated for the y vectors of the group i must be of size at least (1 +\nleni – 1)*abs(incyi)) where leni is mi if the A matrix is not transposed or ni\notherwise.\nincy_array\nArray of size group_count. For the group i, incyi = incy_array[i] is the stride\nof vector y. Must not be zero.\ngroup_count\nNumber of groups. Must be at least 0.\ngroup_size\nArray of size group_count. The element group_count[i] is the number of\noperations in the group i. Each element in group_count must be at least 0.\nOutput Parameters\ny_array\nArray of pointers holding the total_batch_count updated vector y.\ncblas_?dgmm_batch_strided\nComputes groups of matrix-vector product using\ngeneral matrices.\nSyntax\nvoid cblas_sdgmm_batch_strided (const CBLAS_LAYOUT layout, const CBLAS_SIDE left_right,\nconst MKL_INT m, const MKL_INT n, const float *a, const MKL_INT lda, const MKL_INT\nstridea, const float *x, const MKL_INT incx, const MKL_INT stridex, const float *c,\nconst MKL_INT ldc, const MKL_INT stridec, const MKL_INT batch_size);\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n442\n\n\nvoid cblas_ddgmm_batch_strided (const CBLAS_LAYOUT layout, const CBLAS_SIDE left_right,\nconst MKL_INT m, const MKL_INT n, const double *a, const MKL_INT lda, const MKL_INT\nstridea, const double *x, const MKL_INT incx, const MKL_INT stridex, const double *c,\nconst MKL_INT ldc, const MKL_INT stridec, const MKL_INT batch_size);\nvoid cblas_cdgmm_batch_strided (const CBLAS_LAYOUT layout, const CBLAS_SIDE left_right,\nconst MKL_INT m, const MKL_INT n, const void *a, const MKL_INT lda, const MKL_INT\nstridea, const void *x, const MKL_INT incx, const MKL_INT stridex, const void *c, const\nMKL_INT ldc, const MKL_INT stridec, const MKL_INT batch_size);\nvoid cblas_zdgmm_batch_strided (const CBLAS_LAYOUT layout, const CBLAS_SIDE left_right,\nconst MKL_INT m, const MKL_INT n, const void *a, const MKL_INT lda, const MKL_INT\nstridea, const void *x, const MKL_INT incx, const MKL_INT stridex, const void *c, const\nMKL_INT ldc, const MKL_INT stridec, const MKL_INT batch_size);\nInclude Files\n•\nmkl.h\nDescription\nThe cblas_?dgmm_batch_strided routines perform a series of diagonal matrix-matrix product. The\ndiagonal matrices are stored as dense vectors and the operations are performed with group of matrices and\nvectors.\nAll matrices a and c and vector x have the same parameters (size, increments) and are stored at constant\nstride, respectively, given by stridea, stridec, and stridex from each other. The operation is defined as\nfor i = 0 … batch_size – 1\n    A and C are matrices at offset i * stridea in a and i * stridec in c\n    X is a vector at offset i * stridex in x\n    C = diag(X) * A or C = A * diag(X)\nend for\nInput Parameters\nlayout\nSpecifies whether two-dimensional array storage is row-major\n(CblasRowMajor) or column-major (CblasColMajor).\nleft_right\nSpecifies the position of the diagonal matrix in the matrix product\nif left_right = CblasLeft, then C = diag(X) * A;\nif left_right = CblasRight, then C = A * diag(X).\nm\nNumber of rows of the matrices A and C. The value of m must be at least 0.\nn\nNumber of columns of the matrices A and C. The value of n must be at least\n0.\na\nArray holding all the input matrix A. Must be of size at least lda*k + stridea\n* (batch_size -1) where k is n if column major layout is used or m if row\nmajor layout is used.\nlda\nSpecifies the leading dimension of the matrixA. It must be positive and at\nleast mif column major layout is used or at least n if row major layout is\nused.\nstridea\nStride between two consecutive A matrices, must be at least 0.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n443\n\n\nx\nArray holding all the input vector x. Must be of size at least (1 + (len\n-1)*abs(incx)) + stridex * (batch_size - 1) where len is n if the diagonal\nmatrix is on the right of the product or m otherwise.\nincx\nStride between two consecutive elements of the x vectors.\nstridex\nStride between two consecutive x vectors, must be at least 0.\nc\nArray holding all the input matrix C. Must be of size at least batch_size *\nstridec.\nldc\nSpecifies the leading dimension of the matrix C. It must be positive and at\nleast mif column major layout is used or at least n if row major layout is\nused.\nstridec\nStride between two consecutive A matrices, must be at least ldc * nif\ncolumn major layout is used or ldc * m if row major layout is used.\nbatch_size\nNumber of dgmm computations to perform and a c matrices and x vectors.\nMust be at least 0.\nOutput Parameters\nc\nArray holding the batch_size updated matrices c.\ncblas_?dgmm_batch\nComputes groups of matrix-vector product using\ngeneral matrices.\nSyntax\nvoid cblas_sdgmm_batch (const CBLAS_LAYOUT layout, const CBLAS_SIDE *left_right_array,\nconst MKL_INT *m_array, const MKL_INT *n_array, const float **a_array, const MKL_INT\n*lda_array, const float **x_array, const MKL_INT *incx_array, float **c_array, const\nMKL_INT *ldc_array, const MKL_INT group_count, const MKL_INT *group_size);\nvoid cblas_ddgmm_batch (const CBLAS_LAYOUT layout, const CBLAS_SIDE *left_right_array,\nconst MKL_INT *m_array, const MKL_INT *n_array, const double **a_array, const MKL_INT\n*lda_array, const double **x_array, const MKL_INT *incx_array, double **c_array, const\nMKL_INT *ldc_array, const MKL_INT group_count, const MKL_INT *group_size);\nvoid cblas_cdgmm_batch (const CBLAS_LAYOUT layout, const CBLAS_SIDE *left_right_array,\nconst MKL_INT *m_array, const MKL_INT *n_array, const void **a_array, const MKL_INT\n*lda_array, const void **x_array, const MKL_INT *incx_array, void **c_array, const\nMKL_INT *ldc_array, const MKL_INT group_count, const MKL_INT *group_size);\nvoid cblas_zdgmm_batch (const CBLAS_LAYOUT layout, const CBLAS_SIDE *left_right_array,\nconst MKL_INT *m_array, const MKL_INT *n_array, const void **a_array, const MKL_INT\n*lda_array, const void **x_array, const MKL_INT *incx_array, void **c_array, const\nMKL_INT *ldc_array, const MKL_INT group_count, const MKL_INT *group_size);\nInclude Files\n•\nmkl.h\nDescription\nThe cblas_?dgmm_batch routines perform a series of diagonal matrix-matrix product. The diagonal matrices\nare stored as dense vectors and the operations are performed with group of matrices and vectors. .\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n444\n\n\nEach group contains matrices and vectors with the same parameters (size, increments). The operation is\ndefined as:\nidx = 0\nFor i = 0 … group_count – 1\n    left_right, m, n, lda, incx, ldc and group_size at position i in left_right_array, m_array, \nn_array, lda_array, incx_array, ldc_array and group_size_array\n    for j = 0 … group_size – 1\n        a and c are matrices of size mxn at position idx in a_array and c_array\n        x is a vector of size m or n depending on left_right, at position idx in x_array\n        if (left_right == oneapi::mkl::side::left) c := diag(x) * a\n        else c := a * diag(x)\n        idx := idx + 1\n    end for\nend for\nThe number of entries in a_array, x_array, and c_array is total_batch_count = the sum of all of the\ngroup_size entries.\nInput Parameters\nlayout\nSpecifies whether two-dimensional array storage is row-major\n(CblasRowMajor) or column-major (CblasColMajor).\nleft_right_array\nArray of size group_count. For the group i, left_righti = left_right_array[i]\nspecifies the position of the diagonal matrix in the matrix product.\nif left_righti = CblasLeft, then C = diag(X) * A.\nif left_righti = CblasRight, then C = A * diag(X).\nm_array\nArray of size group_count. For the group i, mi = m_array[i] is the number\nof rows of the matrix A and C.\nn_array\nArray of size group_count. For the group i, ni = n_array[i] is the number of\ncolumns in the matrix A and C.\na_array\nArray of size total_batch_count of pointers used to store A matrices. The\narray allocated for the A matrices of the group i must be of size at least ldai\n* niif column major layout is used or at least ldai * mi is row major layout is\nused.\nlda_array\nArray of size group_count. For the group i, ldai = lda_array[i] is the leading\ndimension of the matrix A. It must be positive and at least miif column\nmajor layout is used or at least ni if row major layout is used..\nx_array\nArray of size total_batch_count of pointers used to store x vectors. The\narray allocated for the x vectors of the group i must be of size at least (1 +\nleni – 1)*abs(incxi)) where leni is ni if the diagonal matrix is on the right of\nthe product or mi otherwise.\nincx_array\nArray of size group_count. For the group i, incxi = incx_array[i] is the stride\nof vector x.\nc_array\nArray of size total_batch_count of pointers used to store C matrices. The\narray allocated for the C matrices of the group i must be of size at least ldci\n* ni, if column major layout is used or at least ldci * mi if row major layout\nis used.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n445\n\n\nldc_array\nArray of size group_count. For the group i, ldci = ldc_array[i] is the leading\ndimension of the matrix C. It must be positive and at least miif column\nmajor layout is used or at least ni if row major layout is used..\ngroup_count\nNumber of groups. Must be at least 0.\ngroup_size\nArray of size group_count. The element group_count[i] is the number of\noperations in the group i. Each element in group_size must be at least 0.\nOutput Parameters\nc_array\nArray of pointers holding the total_batch_count updated matrix C.\nmkl_jit_create_?gemm\nCreate a GEMM kernel that computes a scalar-matrix-\nmatrix product and adds the result to a scalar-matrix\nproduct.\nSyntax\nmkl_jit_status_t mkl_jit_create_sgemm(void** jitter, const MKL_LAYOUT layout, const\nMKL_TRANPOSE transa, const MKL_TRANSPOSE transb, const MKL_INT m, const MKL_INT n,\nconst MKL_INT k, const float alpha, const MKL_INT lda, const MKL_INT ldb, const float\nbeta, const MKL_INT ldc);\nmkl_jit_status_t mkl_jit_create_dgemm(void** jitter, const MKL_LAYOUT layout, const\nMKL_TRANPOSE transa, const MKL_TRANSPOSE transb, const MKL_INT m, const MKL_INT n,\nconst MKL_INT k, const double alpha, const MKL_INT lda, const MKL_INT ldb, const double\nbeta, const MKL_INT ldc);\nmkl_jit_status_t mkl_jit_create_cgemm(void** jitter, const MKL_LAYOUT layout, const\nMKL_TRANPOSE transa, const MKL_TRANSPOSE transb, const MKL_INT m, const MKL_INT n,\nconst MKL_INT k, const void* alpha, const MKL_INT lda, const MKL_INT ldb, const void*\nbeta, const MKL_INT ldc);\nmkl_jit_status_t mkl_jit_create_zgemm(void** jitter, const MKL_LAYOUT layout, const\nMKL_TRANPOSE transa, const MKL_TRANSPOSE transb, const MKL_INT m, const MKL_INT n,\nconst MKL_INT k, const void* alpha, const MKL_INT lda, const MKL_INT ldb, const void*\nbeta, const MKL_INT ldc);\nInclude Files\n•\nmkl.h\nDescription\nThe mkl_jit_create_?gemm functions belong to a set of related routines that enable use of just-in-time\ncode generation.\nThe mkl_jit_create_?gemm functions create a handle to a just-in-time code generator (a jitter) and\ngenerate a GEMM kernel that computes a scalar-matrix-matrix product and adds the result to a scalar-matrix\nproduct, with general matrices. The operation of the generated GEMM kernel is defined as follows:\nC := alpha*op(A)*op(B) + beta*C\nWhere:\n•\nop(X) is either op(X) = X or op(X) = XT or op(X) = XH\n•\nalpha and beta are scalars\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n446\n\n\n•\nA, B, and C are matrices\n•\nop(A) is an m-by-k matrix\n•\nop(B) is a k-by-n matrix\n•\nC is an m-by-n matrix\nNOTE\nGenerating a new kernel with mkl_jit_create_?gemm involves moderate runtime overhead.\nTo benefit from JIT code generation, use this feature when you need to call the generated\nkernel many times (for example, several hundred calls).\nInput Parameters\nlayout\nSpecifies whether two-dimensional array storage is row-major\n(MKL_ROW_MAJOR) or column-major (MKL_COL_MAJOR).\ntransa\nSpecifies the form of op(A) used in the generated matrix multiplication:\n•\nif transa = MKL_NOTRANS, then op(A) = A\n•\nif transa = MKL_TRANS, then op(A) = AT\n•\nif transa = MKL_CONJTRANS, then op(A) = AH\ntransb\nSpecifies the form of op(B) used in the generated matrix multiplication:\n•\nif transb = MKL_NOTRANS, then op(B) = B\n•\nif transb = MKL_TRANS, then op(B) = BT\n•\nif transb = MKL_CONJTRANS, then op(B) = BH\nm\nSpecifies the number of rows of the matrix op(A) and of the matrix C. The\nvalue of m must be at least zero.\nn\nSpecifies the number of columns of the matrix op(B) and of the matrix C.\nThe value of n must be at least zero.\nk\nSpecifies the number of columns of the matrix op(A) and the number of\nrows of the matrix op(B). The value of k must be at least zero.\nalpha\nSpecifies the scalar alpha.\nNOTE\nalpha is passed by pointer for mkl_jit_create_cgemm and\nmkl_jit_create_zgemm.\nlda\nSpecifies the leading dimension of a.\ntransa=MKL_NOTRAN\nS\ntransa=MKL_TRANS\nor\ntransa=MKL_CONJTR\nANS\nlayout=MKL_ROW_MA\nJOR\nlda must be at least\nmax(1,k)\nlda must be at least\nmax(1,m)\nlayout=MKL_COL_MA\nJOR\nlda must be at least\nmax(1,m)\nlda must be at least\nmax(1,k)\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n447\n\n\nldb\nSpecifies the leading dimension of b:\ntransb=MKL_NOTRAN\nS\ntransb=MKL_TRANS\nor\ntransb=MKL_CONJTR\nANS\nlayout=MKL_ROW_MA\nJOR\nldb must be at least\nmax(1,n)\nldb must be at least\nmax(1,k)\nlayout=MKL_COL_MA\nJOR\nldb must be at least\nmax(1,k)\nldb must be at least\nmax(1,n)\nbeta\nSpecifies the scalar beta.\nNOTE\nbeta is passed by pointer for mkl_jit_create_cgemm and\nmkl_jit_create_zgemm.\nldc\nSpecifies the leading dimension of c.\nlayout=MKL_ROW_MAJOR\nldc must be at least max(1,n)\nlayout=MKL_COL_MAJOR\nldc must be at least max(1,m)\nOutput Parameters\njitter\nPointer to a handle to the newly created code generator.\nReturn Values\nstatus\nReturns one of the following:\n•\nMKL_JIT_ERROR if the handle cannot be created (no memory)\n—or—\n•\nMKL_JIT_SUCCESS if the jitter has been created and the GEMM kernel was\nsuccessfully created\n—or—\n•\nMKL_NO_JIT if the jitter has been created, but a JIT GEMM kernel was not\ncreated because JIT is not beneficial for the given input parameters. The\nfunction pointer returned by mkl_jit_get_?gemm_ptr will call standard\n(non-JIT) GEMM.\nmkl_jit_get_?gemm_ptr\nReturn the GEMM kernel associated with a jitter\npreviously created with mkl_jit_create_?gemm.\nSyntax\nsgemm_jit_kernel_t mkl_jit_get_sgemm_ptr(const void* jitter);\ndgemm_jit_kernel_t mkl_jit_get_dgemm_ptr(const void* jitter);\ncgemm_jit_kernel_t mkl_jit_get_cgemm_ptr(const void* jitter);\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n448\n\n\nzgemm_jit_kernel_t mkl_jit_get_zgemm_ptr(const void* jitter);\nInclude Files\n•\nmkl.h\nDescription\nThe mkl_jit_get_?gemm_ptr functions belong to a set of related routines that enable use of just-in-time\ncode generation.\nThe mkl_jit_get_?gemm_ptr functions take as input a jitter previously created with\nmkl_jit_create_?gemm, and return the GEMM kernel associated with that jitter. The returned GEMM kernel\ncomputes a scalar-matrix-matrix product and adds the result to a scalar-matrix product, with general\nmatrices. The operation is defined as follows:\nC := alpha*op(A)*op(B) + beta*C\nWhere:\n•\nop(X) is one of op(X) = X or op(X) = XT or op(X) = XH\n•\nalpha and beta are scalars\n•\nA, B, and C are matrices\n•\nop(A) is an m-by-k matrix\n•\nop(B) is a k-by-n matrix\n•\nC is an m-by-n matrix\nNOTE\nGenerating a new kernel with mkl_jit_create_?gemm involves moderate runtime overhead.\nTo benefit from JIT code generation, use this feature when you need to call the generated\nkernel many times (for example, several hundred calls).\nInput Parameter\njitter\nHandle to the code generator.\nReturn Values\nfunc\n•\nsgemm_jit_kernel_t – A function pointer type expecting four\ninputs of type void*, float*, float*, and float*\ntypedef void (*sgemm_jit_kernel_t)\n(void*,float*,float*,float*);\n•\ndgemm_jit_kernel_t – A function pointer type expecting four\ninputs of type void*, double*, double*, and double*\ntypedef void(*dgemm_jit_kernel_t)\n(void*,double*,double*,double*);\n•\ncgemm_jit_kernel_t – A function pointer type expecting four\ninputs of type void*, MKL_Complex8*, MKL_Complex8*, and\nMKL_Complex8*\ntypedef void(*cgemm_jit_kernel_t)\n(void*,MKL_Complex8*,MKL_Complex8*,MKL_Complex8*);\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n449\n\n\n•\nzgemm_jit_kernel_t – A function pointer type expecting four\ninputs of type void*, MKL_Complex16*, MKL_Complex16*, and\nMKL_Complex16*\ntypedef void(*zgemm_jit_kernel_t)\n(void*,MKL_Complex16*,MKL_Complex16*,MKL_Complex16*);\nIf the jitter input is not NULL, returns a function pointer to a GEMM\nkernel. The GEMM kernel is called with four parameters: the jitter and\nthe three matrices a, b, and c. Otherwise, returns NULL.\nIf layout, transa, transb, m, n, k, lda, ldb, and ldc are the parameters used during the creation of the\ninput jitter, then:\na\n \nlayout =\nMKL_COL_MAJOR\nlayout =\nMKL_ROW_MAJOR\ntransa =\nMKL_NOTRANS\nArray of size lda*k\nBefore calling the\nreturned function\npointer, the leading m-\nby-k part of the array\na must contain the\nmatrix A.\nArray of size lda*m\nBefore calling the\nreturned function\npointer, the leading k-\nby-m part of the array\na must contain the\nmatrix A.\ntransa =\nMKL_TRANS or\ntransa =\nMKL_CONJTRANS\nArray of size lda*m\nBefore calling the\nreturned function\npointer, the leading k-\nby-m part of the array\na must contain the\nmatrix A.\nArray of size lda*k\nBefore calling the\nreturned function\npointer, the leading m-\nby-k part of the array\na must contain the\nmatrix A.\nb\n \nlayout =\nMKL_COL_MAJOR\nlayout =\nMKL_ROW_MAJOR\ntransb =\nMKL_NOTRANS\nArray of size ldb*n\nBefore calling the\nreturned function\npointer, the leading k-\nby-n part of the array\nb must contain the\nmatrix B.\nArray of size ldb*k\nBefore calling the\nreturned function\npointer, the leading n-\nby-k part of the array\nb must contain the\nmatrix B.\ntransb =\nMKL_TRANS or\ntransb =\nMKL_CONJTRANS\nArray of size ldb*k\nBefore calling the\nreturned function\npointer, the leading n-\nby-k part of the array\nb must contain the\nmatrix B.\nArray of size ldb*n\nBefore calling the\nreturned function\npointer, the leading k-\nby-n part of the array\nb must contain the\nmatrix B.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n450\n\n\nc\nlayout = MKL_COL_MAJOR\nlayout = MKL_ROW_MAJOR\nArray of size ldc*n\nBefore calling the returned\nfunction pointer, the leading m-\nby-n part of the array c must\ncontain the matrix C.\nArray of size ldc*m\nBefore calling the returned\nfunction pointer, the leading n-\nby-m part of the array c must\ncontain the matrix C.\nmkl_jit_destroy\nDelete the jitter previously created with\nmkl_jit_create_?gemm as well as the GEMM kernel\nthat it contains.\nSyntax\nmkl_jit_status_t mkl_jit_destroy (void* jitter);\nInclude Files\n•\nmkl.h\nDescription\nThe mkl_jit_destroy function belongs to a set of related routines that enable use of just-in-time code\ngeneration.\nThe mkl_jit_destroy function takes as input a jitter previously created with mkl_jit_create_?gemm and\ndeletes the jitter as well as the GEMM kernel that it contains.\nNOTE\nGenerating a new kernel with mkl_jit_create_?gemm involves moderate runtime overhead.\nTo benefit from JIT code generation, use this feature when you need to call the generated\nkernel many times (for example, several hundred calls).\nInput Parameter\njitter\nJitter handle\nReturn Values\nstatus\nReturns one of the following:\n•\nMKL_JIT_ERROR if the pointer is not NULL and is not a handle on a jitter—that\nis, if it was not created with mkl_jit_create_?gemm\n—or—\n•\nMKL_JIT_SUCCESS if the jitter has been successfully destroyed\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n451\n\n\nLAPACK Routines\nIntel® oneAPI Math Kernel Library (oneMKL)implements routines from the LAPACK package that are used for\nsolving systems of linear equations, linear least squares problems, eigenvalue and singular value problems,\nand performing a number of related computational tasks. The library includes LAPACK routines for both real\nand complex data. Routines are supported for systems of equations with the following types of matrices:\n•\nGeneral\n•\nBanded\n•\nSymmetric or Hermitian positive-definite (full, packed, and rectangular full packed (RFP) storage)\n•\nSymmetric or Hermitian positive-definite banded\n•\nSymmetric or Hermitian indefinite (both full and packed storage)\n•\nSymmetric or Hermitian indefinite banded\n•\nTriangular (full, packed, and RFP storage)\n•\nTriangular banded\n•\nTridiagonal\n•\nDiagonally dominant tridiagonal.\nNOTE\nDifferent arrays used as parameters to Intel® MKL LAPACK routines must not overlap.\nWarning\nLAPACK routines assume that input matrices do not contain IEEE 754 special values such as INF or\nNaN values. Using these special values may cause LAPACK to return unexpected results or become\nunstable.\nChoosing a LAPACK Routine\nIntel® oneAPI Math Kernel Library (oneMKL) offers many LAPACK routines, and many perform similar\noperations. The Intel® oneAPI Math Kernel Library LAPACK Function Finding Advisor helps you understand the\ndifferences between routines so that you can choose the appropriate routine for your task:\nIntel® oneAPI Math Kernel Library LAPACK Function Finding Advisor:https://www.intel.com/\ncontent/www/us/en/developer/tools/oneapi/onemkl-function-finding-advisor.html.\nC Interface Conventions for LAPACK Routines\nThe C interfaces are implemented for most of the Intel® oneAPI Math Kernel Library (oneMKL) LAPACK driver\nand computational routines.\nNaN Checking in LAPACKE\nNaN checking can affect the performance of an application. By default, it is ON.\nSee the Support Functions section for details on the methods and options to turn NaN check off or back on\nwith LAPACKE:.\nFunction Prototypes\nIntel® oneAPI Math Kernel Library (oneMKL) supports four distinct floating-point precisions. Each\ncorresponding prototype looks similar, usually differing only in the data type. C interface LAPACK function\nnames follow the form<?><name>[_64], where <?> is:\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n452\n\n\n•\nLAPACKE_s for float\n•\nLAPACKE_d for double\n•\nLAPACKE_c for lapack_complex_float\n•\nLAPACKE_z for lapack_complex_double\nOn 64-bit platforms, Intel® oneAPI Math Kernel Library (oneMKL) provides LAPACK C interfaces with the _64\nsuffix to support large data arrays in the LP64 interface library. For more interface library details, see \"Using\nthe ILP64 Interface vs. LP64 Interface\" in the developer guide.\nA specific example follows. To solve a system of linear equations with a packed Cholesky-factored Hermitian\npositive-definite matrix with complex precision, use the following:\nlapack_int LAPACKE_cpptrs(int matrix_layout, char uplo, lapack_int n, lapack_int nrhs,\nconst lapack_complex_float* ap, lapack_complex_float* b, lapack_int ldb);\nFor matrices whose dimensions are greater than 231-1, you can use either LAPACKE_cpptrs in the ILP64\ninterface library or LAPACKE_cpptrs_64 in the LP64 interface library.\nWorkspace Arrays\nIn contrast to the Fortran interface, the LAPACK C interface omits workspace parameters because workspace\nis allocated during runtime and released upon completion of the function operation.\nIf you prefer to allocate workspace arrays yourself, the LAPACK C interface provides alternate interfaces with\nwork parameters. The name of the alternate interface is the same as the LAPACK C interface with _work\nappended. For example, the syntax for the singular value decomposition of a real bidiagonal matrix is:\nFortran:\ncall sbdsdc ( uplo, compq, n, d, e, u, ldu, vt, ldvt, q, iq,\nwork, iwork, info )\nC LAPACK interface:\nlapack_int LAPACKE_sbdsdc ( int matrix_layout, char uplo, char\ncompq, lapack_int n, float* d, float* e, float* u, lapack_int\nldu, float* vt, lapack_int ldvt, float* q, lapack_int* iq );\nAlternate C LAPACK\ninterface with work\nparameters:\nlapack_int LAPACKE_sbdsdc_work( int matrix_layout, char uplo,\nchar compq, lapack_int n, float* d, float* e, float* u,\nlapack_int ldu, float* vt, lapack_int ldvt, float* q, lapack_int*\niq, float* work, lapack_int* iwork );\nSee the install_dir/include/mkl_lapacke.h file for the full list of alternative C LAPACK interfaces.\nThe Intel® oneAPI Math Kernel Library (oneMKL) Fortran-specific documentation contains details about\nworkspace arrays.\nMapping Fortran Data Types against C Data Types\nFortran Data Types vs. C Data Types\nFORTRAN\nC\nINTEGER\nlapack_int\nLOGICAL\nlapack_logical\nREAL\nfloat\nDOUBLE PRECISION\ndouble\nCOMPLEX\nlapack_complex_float\nCOMPLEX*16/DOUBLE COMPLEX\nlapack_complex_double\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n453\n\n\nFORTRAN\nC\nCHARACTER\nchar\nC Type Definitions\nYou can find type definitions specific to Intel® oneAPI Math Kernel Library (oneMKL) such asMKL_INT,\nMKL_Complex8, and MKL_Complex16 in install_dir/mkl_types.h.\nC types\n#ifndef lapack_int\n#define lapack_int MKL_INT\n#endif\n#ifndef lapack_logical\n#define lapack_logical lapack_int\n#endif\nComplex Type Definitions\nComplex type for single precision:\n#ifndef lapack_complex_float\n#define lapack_complex_float   MKL_Complex8\n#endif\nComplex type for double precision:\n#ifndef lapack_complex_double\n#define lapack_complex_double   MKL_Complex16\n#endif\nMatrix Layout Definitions\n#define LAPACK_ROW_MAJOR  101\n#define LAPACK_COL_MAJOR  102\nSee Matrix Layout for LAPACK Routines above for an explanation of row-major\norder and column-major order storage.\nError Code Definitions\n#define LAPACK_WORK_MEMORY_ERROR       -1010  /* Failed to allocate \nmemory  \n                                            for a working array */\n#define LAPACK_TRANSPOSE_MEMORY_ERROR  -1011  /* Failed to allocate \nmemory \n                                            for transposed matrix */\nMatrix Layout for LAPACK Routines\nThere are two general methods of storing a two dimensional matrix in linear (one dimensional) memory:\ncolumn-wise (column major order) or row-wise (row major order). Consider an M-by-N matrix A:\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n454\n\n\nColumn Major Layout\nIn column major layout the first index, i, of matrix elements ai,j changes faster than the second index when\naccessing sequential memory locations. In other words, for 1 ≤i < M, if the element ai,j is stored in a specific\nlocation in memory, the element ai+1,j is stored in the next location, and, for 1 ≤j < N, the element aM,j is\nstored in the location previous to element a1,j+1. So the matrix elements are located in memory according to\nthis sequence:\n{a1,1a2,1 ... aM,1a1,2a2,2 ... aM,2 ... ... a1,Na2,N ... aM,N}\nRow Major Layout\nIn row major layout the second index, j, of matrix elements ai,j changes faster than the first index when\naccessing sequential memory locations. In other words, for 1 ≤j < N, if the element ai,j is stored in a specific\nlocation in memory, the element ai,j+1 is stored in the next location, and, for 1 ≤i < M, the element ai,N is\nstored in the location previous to element ai+1,1. So the matrix elements are located in memory according to\nthis sequence:\n{a1,1a1,2 ... a1,Na2,1a2,2 ... a2,N ... ... aN,1aN,2 ... aM,N}\nLeading Dimension Parameter\nA leading dimension parameter allows use of LAPACK routines on a submatrix of a larger matrix. For\nexample, the submatrix B can be extracted from the original matrix A defined previously:\nB is formed from rows with indices i0 + 1 to i0 + K and columns j0 + 1 to j0 + L of matrix A. To specify matrix\nB, LAPACK routines require four parameters:\n•\nthe number of rows K;\n•\nthe number of columns L;\n•\na pointer to the start of the array containing elements of B;\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n455\n\n\n•\nthe leading dimension of the array containing elements of B.\nThe leading dimension depends on the layout of the matrix:\n•\nColumn major layout\nLeading dimension ldb=M, the number of rows of matrix A.\nStarting address: offset by i0 + j0*ldb from a1,1.\n•\nRow major layout\nLeading dimension ldb=N, the number of columns of matrix A.\nStarting address: offset by i0*ldb + j0 from a1,1.\nMatrix Storage Schemes for LAPACK Routines\nLAPACK routines use the following matrix storage schemes:\n•\nFull Storage\n•\nPacked Storage\n•\nBand Storage\n•\nRectangular Full Packed (RFP) Storage\nFull Storage\nConsider an m-by-n matrix A :\nA =\na1, 1 a1, 2 a1, 3 ⋯a1, n\na2, 1 a2, 2 a2, 3 ⋯a2, n\na3, 1 a3, 2 a3, 3 ⋯a3, n\n⋮\n⋮\n⋮\n⋱\n⋮\nam, 1 am, 2 am, 3 ⋯am, n\nIt is stored in a one-dimensional array a of length at least lda*n for column major layout or m*lda for row\nmajor layout. Element ai,j is stored as array element a[k] where the mapping of k(i, j) is defined as\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n456\n\n\n•\ncolumn major layout: k(i, j) = i - 1 + (j - 1)*lda\n•\nrow major layout: k(i, j) = (i - 1)*lda + j - 1\nNOTE\nAlthough LAPACK accepts parameter values of zero for matrix size, in general the size of the array\nused to store an m-by-n matrix A with leading dimension lda should be greater than or equal to\nmax(1, n*lda) for column major layout and max (1, m*lda) for row major layout.\nNOTE\nEven though the array used to store a matrix is one-dimensional, for simplicity the documentation\nsometimes refers parts of the array such as rows, columns, upper and lower triangular part, and\ndiagonals. These refer to the parts of the matrix stored within the array. For example, the lower\ntriangle of array a is defined as the subset of elements a[k(i,j)] with i≥j.\nPacked Storage\nThe packed storage format compactly stores matrix elements when only one part of the matrix, the upper or\nlower triangle, is necessary to determine all of the elements of the matrix. This is the case when the matrix\nis upper triangular, lower triangular, symmetric, or Hermitian. For an n-by-n matrix of one of these types, a\nlinear array ap of length n*(n + 1)/2 is adequate. Two parameters define the storage scheme:\nmatrix_layout, which specifies column major (with the value LAPACK_COL_MAJOR) or row major (with the\nvalue LAPACK_ROW_MAJOR) matrix layout, and uplo, which specifies that the upper triangle (with the value\n'U') or the lower triangle (with the value 'L') is stored.\nElement ai,j is stored as array element a[k] where the mapping of k(i, j) is defined as\nmatrix_layout = LAPACK_COL_MAJOR\nmatrix_layout = LAPACK_ROW_MAJOR\nuplo = 'U'\n1 ≤i≤j≤n\nk(i, j) = i - 1 + j*(j - 1)/2\nk(i, j) = j - 1 + (i - 1)*(2*n - i)/2\nuplo = 'L'\n1 ≤j≤i≤n\nk(i, j) = i - 1 + (j - 1)*(2*n - j)/2\nk(i, j) = j - 1 + i*(i - 1)/2\nNOTE\nAlthough LAPACK accepts parameter values of zero for matrix size, in general the size of the array\nshould be greater than or equal to max(1, nx*(n + 1)/2).\nBand Storage\nWhen the non-zero elements of a matrix are confined to diagonal bands, it is possible to store the elements\nmore efficiently using band storage. For example, consider an m-by-n band matrix A with kl subdiagonals\nand ku superdiagonals:\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n457\n\n\nA =\na1, 1\na1, 2\n⋯\na1, ku + 1\n⋮\n⋮\n⋱\n⋱\n⋱\nakl + 1, 1 akl + 1, 2 ⋯akl + 1, ku + 1\n⋱\nakl + 1, kl + ku + 1\nakl + 2, 2 ⋱\n⋱\n⋱\n⋱\n⋱\n⋱\n⋱\n⋱\n⋱\n⋱⋱\nakl + j, j\nakl + 1, j + 1\n⋯\n⋯⋯akl + j, kl + ku + j\n⋱\n⋱\n⋱⋱\n⋱\n⋱\nThis matrix can be stored compactly in a one dimensional array ab. There are two operations involved in\nstoring the matrix: packing the band matrix into matrix AB, and converting the packed matrix to a one-\ndimensional array.\n•\nPacking the Band Matrix: How the band matrix is packed depends on the matrix layout.\n•\nColumn major layout: matrix A is packed in an ldab-by-n matrix AB column-wise so that the diagonals\nof A become rows of array AB.\nAB =\na1, ku + 1\na1, ku + 2\na1, ku + 3\n⋯\n⋰\n⋮\n⋮\n⋮\n⋯\na1, 3\n⋯\naku −1, ku + 1\naku, ku + 2\naku + 1, ku + 3\n⋯\na1, 2\na2, 3\n⋯\naku, ku + 1\naku + 1, ku + 2\naku + 2, ku + 3\n⋯\na1, 1\na2, 2\na3, 3\n⋯\naku + 1, ku + 1\naku + 2, ku + 2\naku + 3, ku + 3\n⋯\na2, 1\na3, 2\na4, 3\n⋯\naku + 2, ku + 1\naku + 3, ku + 2\naku + 4, ku + 3\n⋯\n⋮\n⋮\n⋮\n⋱\n⋮\n⋮\n⋮\n⋯\nakl + 1, 1 akl + 2, 2 akl + 3, 3 ⋯aku + kl. + 1, ku + 1 aku + kl. + 2, ku + 2 aku + kl. + 3, ku + 3 ⋯\nThe number of rows of ABldab≥kl + ku + 1, and the number of columns of AB is n.\n•\nRow major layout: matrix A is packed in an m-by-ldab matrix AB row-wise so that the diagonals of A\nbecome columns of AB.\nAB =\na1, 1\na1, 2\n⋯\na1, ku + 1\na2, 1\na2, 2\na2, 3\n⋯\na2, ku + 2\na3, 1 a3, 2\na3, 3\na3, 4\n⋯\na3, ku + 3\n⋰\n⋮\n⋮\n⋮\n⋮\n⋮\n⋮\nakl + 1, 1 akl + 1, 2\n⋯\n⋯\nakl + 1, kl + 1 akl + 1, kl + 2 ⋯akl + 1, ku + kl + 1\nakl + 2, 2 akl + 2, 3\n⋯\n⋯\nakl + 2, kl + 2 akl + 2, kl + 3 ⋯akl + 2, ku + kl + 2\nakl + 3, 3 akl + 3, 4\n⋯\n⋯\nakl + 3, kl + 3 akl + 3, kl + 4 ⋯akl + 3, ku + kl + 3\n⋮\n⋮\n⋮\n⋮\n⋮\n⋮\n⋮\n⋮\nThe number of columns of ABldab≥kl + ku + 1, and the number of rows of AB is m.\nNOTE\nFor both column major and row major layout, elements of the upper left triangle of AB are not used.\nDepending on the relationship of the dimensions m, n, kl, and ku, the lower right triangle might not be\nused.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n458\n\n\n•\nConverting the Packed Matrix to a One-Dimensional Array: The packed matrix AB is stored in a linear\narray ab as described in Full Storage . The size of ab should be greater than or equal to the total number\nof elements of matrix AB: ldab*n for column major layout or ldab*m for row major layout. The leading\ndimension of ab, ldab, must be greater than or equal to kl + ku + 1 (and some routines require it to be\neven larger).\nElement ai,j is stored as array element a[k(i, j)] where the mapping of k(i, j) is defined as\n•\ncolumn major layout: k(i, j) = i + ku - j + (j - 1)*ldab; 1 ≤j≤n, max(1, j - ku) ≤i≤ min(m, j + kl)\n•\nrow major layout: k(i,j) = j-i+kl+(i-1)(kl+ku+1), 1 ≤ i ≤ m, max(1, i - kl) ≤ j ≤ min(n, i + ku)\nNOTE\nAlthough LAPACK accepts parameter values of zero for matrix size, in general the size of the array\nshould be greater than or equal to max(1, n*ldab) for column major layout and max (1, m*ldab) for\nrow major layout.\nRectangular Full Packed Storage\nA combination of full and packed storage, rectangular full packed storage can be used to store the upper or\nlower triangle of a matrix which is upper triangular, lower triangular, symmetric, or Hermitian. It offers the\nstorage savings of packed storage plus the efficiency of using full storage Level 3 BLAS and LAPACK routines.\nThree parameters define the storage scheme: matrix_layout, which specifies column major (with the value\nLAPACK_COL_MAJOR) or row major (with the value LAPACK_ROW_MAJOR) matrix layout; uplo, which specifies\nthat the upper triangle (with the value 'U') or the lower triangle (with the value 'L') is stored;and transr,\nwhich specifies normal (with the value 'N'), transpose (with the value 'T'), or conjugate transpose (with the\nvalue 'C') operation on the matrix.\nConsider an N-by-N matrix A:\nA =\na0, 0\na0, 1\na0, 2\n⋯\na0, N −1\na1, 0\na1, 1\na1, 2\n⋯\na1, N −1\na2, 0\na2, 1\na2, 2\n⋯\na2, N −1\n⋮\n⋮\n⋮\n⋱\n⋮\naN −1, 0 aN −1, 1 aN −1, 2 ⋯aN −1, N −1\nThe upper or lower triangle of A can be stored in the array ap of length N*(N + 1)/2.\nAdditionally, define k as the integer part of N/2, such that N=2*k if N is even, and N=2*k + 1 if N is odd.\nStoring the matrix involves packing the matrix into a rectangular matrix, and then storing the matrix in a\none-dimensional array. The size of rectangular matrix AP required for the N-by-N matrix A is N + 1 by N/2 for\neven N, and N by (N + 1)/2 for odd N.\nThese examples illustrate the rectangular full packed storage method.\n•\nUpper triangular - uplo = 'U'\nConsider a matrix A with N = 6:\nA =\na0, 0 a0, 1 a0, 2 a0, 3 a0, 4 a0, 5\na1, 0 a1, 1 a1, 2 a1, 3 a1, 4 a1, 5\na2, 0 a2, 1 a2, 2 a2, 3 a2, 4 a2, 5\na3, 0 a3, 1 a3, 2 a3, 3 a3, 4 a3, 5\na4, 0 a4, 1 a4, 2 a4, 3 a4, 4 a4, 5\na5, 0 a5, 1 a5, 2 a5, 3 a5, 4 a5, 5\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n459\n\n\n•\nNot transposed - transr = 'N'\nThe elements of the upper triangle of A can be packed in a matrix with the dimensions (N + 1)-by-\n(N/2) = 7 by 3:\nAP =\na0, 3 a0, 4 a0, 5\na1, 3 a1, 4 a1, 5\na2, 3 a2, 4 a2, 5\na3, 3 a3, 4 a3, 5\na0, 0 a4, 4 a4, 5\na0, 1 a1, 1 a5, 5\na0, 2 a1, 2 a2, 2\n•\nTransposed or conjugate transposed - transr = 'T' or transr = 'C'\nThe elements of the upper triangle of A can be packed in a matrix with the dimensions (N/2) by (N +\n1) = 3 by 7:\nAP =\na0, 3 a1, 3 a2, 3 a3, 3 a0, 0 a0, 1 a0, 2\na0, 4 a1, 4 a2, 4 a3, 4 a4, 4 a1, 1 a1, 2\na0, 5 a1, 5 a2, 5 a3, 5 a4, 5 a5, 5 a2, 2\nConsider a matrix A with N = 5:\nA =\na0, 0 a0, 1 a0, 2 a0, 3 a0, 4\na1, 0 a1, 1 a1, 2 a1, 3 a1, 4\na2, 0 a2, 1 a2, 2 a2, 3 a2, 4\na3, 0 a3, 1 a3, 2 a3, 3 a3, 4\na4, 0 a4, 1 a4, 2 a4, 3 a4, 4\n•\nNot transposed - transr = 'N'\nThe elements of the upper triangle of A can be packed in a matrix with the dimensions (N)-by-((N\n+1)/2) = 5 by 3:\nAP =\na0, 2 a0, 3 a0, 4\na1, 2 a1, 3 a1, 4\na2, 2 a2, 3 a2, 4\na0, 0 a3, 3 a3, 4\na0, 1 a1, 1 a4, 4\n•\nTransposed or conjugate transposed - transr = 'T' or transr = 'C'\nThe elements of the upper triangle of A can be packed in a matrix with the dimensions ((N+1)/2) by\n(N ) = 5 by 3:\nAP =\na0, 2 a1, 2 a2, 3 a0, 0 a0, 1\na0, 3 a1, 3 a2, 3 a3, 3 a1, 1\na0, 4 a1, 4 a2, 4 a3, 4 a4, 4\n•\nLower triangular - uplo = 'L'\nConsider a matrix A with N = 6:\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n460\n\n\nA =\na0, 0 a0, 1 a0, 2 a0, 3 a0, 4 a0, 5\na1, 0 a1, 1 a1, 2 a1, 3 a1, 4 a1, 5\na2, 0 a2, 1 a2, 2 a2, 3 a2, 4 a2, 5\na3, 0 a3, 1 a3, 2 a3, 3 a3, 4 a3, 5\na4, 0 a4, 1 a4, 2 a4, 3 a4, 4 a4, 5\na5, 0 a5, 1 a5, 2 a5, 3 a5, 4 a5, 5\n•\nNot transposed - transr = 'N'\nThe elements of the lower triangle of A can be packed in a matrix with the dimensions (N + 1)-by-\n(N/2) = 7 by 3:\nAP =\na3, 3 a4, 3 a5, 3\na0, 0 a4, 4 a5, 4\na1, 0 a1, 1 a5, 5\na2, 0 a2, 1 a2, 2\na3, 0 a3, 1 a3, 2\na3, 0 a4, 1 a4, 2\na5, 0 a5, 1 a5, 2\n•\nTransposed or conjugate transposed - transr = 'T' or transr = 'C'\nThe elements of the lower triangle of A can be packed in a matrix with the dimensions (N/2) by (N +\n1) = 3 by 7:\nAP =\na3, 3 a0, 0 a1, 0 a2, 0 a3, 0 a4, 0 a5, 0\na4, 3 a4, 4 a1, 1 a2, 1 a3, 1 a4, 1 a5, 1\na5, 3 a5, 4 a5, 5 a2, 2 a3, 2 a4, 2 a5, 2\nConsider a matrix A with N = 5:\nA =\na0, 0 a0, 1 a0, 2 a0, 3 a0, 4\na1, 0 a1, 1 a1, 2 a1, 3 a1, 4\na2, 0 a2, 1 a2, 2 a2, 3 a2, 4\na3, 0 a3, 1 a3, 2 a3, 3 a3, 4\na4, 0 a4, 1 a4, 2 a4, 3 a4, 4\n•\nNot transposed - transr = 'N'\nThe elements of the lower triangle of A can be packed in a matrix with the dimensions (N)-by-((N\n+1)/2) = 5 by 3:\nAP =\na0, 0 a3, 3 a4, 3\na1, 0 a1, 1 a4, 4\na2, 0 a2, 1 a2, 2\na3, 0 a3, 1 a3, 2\na4, 0 a4, 1 a4, 2\n•\nTransposed or conjugate transposed - transr = 'T' or transr = 'C'\nThe elements of the lower triangle of A can be packed in a matrix with the dimensions ((N+1)/2) by\n(N ) = 5 by 3:\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n461\n\n\nAP =\na0, 0 a1, 0 a2, 0 a3, 0 a4, 0\na3, 3 a1, 1 a2, 1 a3, 1 a4, 1\na4, 3 a4, 4 a2, 2 a3, 2 a4, 2\nThe packed matrix AP can be stored using column major layout or row major layout.\nNOTE\nThe matrix_layout and transr parameters can specify the same storage scheme: for example, the\nstorage scheme for matrix_layout = LAPACK_COL_MAJOR and transr = 'N' is the same as that for\nmatrix_layout = LAPACK_ROW_MAJOR and transr = 'T'.\nElement ai,j is stored as array element ap[l] where the mapping of l(i, j) is defined in the following tables.\n•\nColumn major layout: matrix_layout = LAPACK_COL_MAJOR a\ntrans\nr\nuplo\nN\nl(i, j) =\ni\nj\n'N'\n'U'\n2*k\n(j - k)*(N + 1) + i\n0 ≤i < N\nmax(i, k) ≤j\n< N\ni*(N + 1) + j + k + 1\n0 ≤i < k\ni≤j < k\n2*k +\n1\n(j - k)*N + i\n0 ≤i < N\nmax(i, k) ≤j\n< N\ni*N + j + k + 1\n0 ≤i < k\ni≤j < k\n'L'\n2*k\nj*(N + 1) + i + 1\n0 ≤i < N\n0 ≤j≤ min(i,\nk)\n(i - k)*(N + 1) + j - k\nk≤i < N\nk≤j≤i\n2*k +\n1\nj*N + i\n0 ≤i < N\n0 ≤j≤ min(i,\nk)\n(i - k)*N + j - k - 1\nk + 1 ≤i < N\nk + 1 ≤j≤i\n'T' or\n'C'\n'U'\n2*k\ni*k + j - k\n0 ≤i < N\nmax(i, k) ≤j\n< N\n(j + k + 1)*k + i\n0 ≤i < k\ni≤j < k\n2*k +\n1\ni*(k + 1) + j - k\n0 ≤i < N\nmax(i, k) ≤j\n< N\n(j + k + 1)*(k + 1) + i\n0 ≤i < k\ni≤j < k\n'L'\n2*k\n(i + 1)*k + j\n0 ≤i < N\n0 ≤j≤ min(i,\nk)\n(j - k)*k + i - k\nk≤i < N\nk≤j≤i\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n462\n\n\ntrans\nr\nuplo\nN\nl(i, j) =\ni\nj\n2*k +\n1\ni*(k + 1) + j\n0 ≤i < N\n0 ≤j≤ min(i,\nk)\n(j - k - 1)*(k + 1) + i - k\nk + 1 ≤i < N\nk + 1 ≤j≤i\n•\nRow major layout: matrix_layout = LAPACK_ROW_MAJOR\ntrans\nr\nuplo\nN\nl(i, j) =\ni\nj\n'N'\n'U'\n2*k\ni*k + j - k\n0 ≤i < N\nmax(i, k) ≤j\n< N\n(k + j + 1)*k + i\n0 ≤i < k\ni≤j < k\n2*k +\n1\ni*(k + 1) + j - k\n0 ≤i < N\nmax(i, k) ≤j\n< N\n(k + j + 1)*(k + 1) + i\n0 ≤i < k\ni≤j < k\n'L'\n2*k\n(i + 1)*k + j\n0 ≤i < N\n0 ≤j≤ min(i,\nk)\n(j - k)*k + i - k\nk≤i < N\nk≤j≤i\n2*k +\n1\ni*(k + 1) + j\n0 ≤i < N\n0 ≤j≤ min(i,\nk)\n(j - k - 1)*(k + 1) + i - k\nk + 1 ≤i < N\nk + 1 ≤j≤i\n'T' or\n'C'\n'U'\n2*k\n(j - k)*(N + 1) + i\n0 ≤i < N\nmax(i, k) ≤j\n< N\ni*(N + 1) + k + j + 1\n0 ≤i < k\ni≤j < k\n2*k +\n1\n(j - k)*N + i\n0 ≤i < N\nmax(i, k) ≤j\n< N\ni*N + k + j + 1\n0 ≤i < k\ni≤j < k\n'L'\n2*k\nj*(N + 1) + i + 1\n0 ≤i < N\n0 ≤j≤ min(i,\nk)\n(i - k)*(N + 1) + j - k\nk≤i < N\nk≤j≤i\n2*k +\n1\nj*N + i\n0 ≤i < N\n0 ≤j≤ min(i,\nk)\n(i - k)*N + j - k - 1\nk + 1 ≤i < N\nk + 1 ≤j≤i\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n463\n\n\nNOTE\nAlthough LAPACK accepts parameter values of zero for matrix size, in general the size of the array\nshould be greater than or equal to max(1, N*(N + 1)/2).\nMathematical Notation for LAPACK Routines\nDescriptions of LAPACK routines use the following notation:\nAH\nFor an M-by-N matrix A, denotes the conjugate transposed N-by-M\nmatrix with elements:\nFor a real-valued matrix, AH = AT.\nx·y\nThe dot product of two vectors, defined as:\nAx = b\nA system of linear equations with an n-by-n matrix A = {aij}, a\nright-hand side vector b = {bi}, and an unknown vector x = {xi}.\nAX = B\nA set of systems with a common matrix A and multiple right-hand\nsides. The columns of B are individual right-hand sides, and the\ncolumns of X are the corresponding solutions.\n|x|\nthe vector with elements |xi| (absolute values of xi).\n|A|\nthe matrix with elements |aij| (absolute values of aij).\n||x||∞ = maxi|xi|\nThe infinity-norm of the vector x.\n||A||∞ = maxiΣj|aij|\nThe infinity-norm of the matrix A.\n||A||1 = maxjΣi|aij|\nThe one-norm of the matrix A. ||A||1 = ||AT||∞ = ||AH||∞\n||x||2\nThe 2-norm of the vector x: ||x||2 = (Σi|xi|2)1/2 = ||x||E (see\nthe definition for Euclidean norm in this topic).\n||A||2\nThe 2-norm (or spectral norm) of the matrix A.\n||A||E\nThe Euclidean norm of the matrix A: ||A||E2 = ΣiΣj|aij|2.\nκ(A) = ||A||·||A-1||\nThe condition number of the matrix A.\nλi\nEigenvalues of the matrix A (for the definition of eigenvalues, see \nEigenvalue Problems).\nσi\nSingular values of the matrix A. They are equal to square roots of the\neigenvalues of AHA. (For more information, see Singular Value\nDecomposition).\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n464\n\n\nError Analysis\nIn practice, most computations are performed with rounding errors. Besides, you often need to solve a\nsystem Ax = b, where the data (the elements of A and b) are not known exactly. Therefore, it is important\nto understand how the data errors and rounding errors can affect the solution x.\nData perturbations. If x is the exact solution of Ax = b, and x + δx is the exact solution of a perturbed\nproblem (A + δA)(x + δx) = (b + δb), then this estimate, given up to linear terms of perturbations,\nholds:\nwhere A + δA is nonsingular and\nIn other words, relative errors in A or b may be amplified in the solution vector x by a factor κ(A) = ||A||\n ||A-1|| called the condition number of A.\nRounding errors have the same effect as relative perturbations c(n)ε in the original data. Here ε is the\nmachine precision, defined as the smallest positive number x such that 1 + x > 1; and c(n) is a modest\nfunction of the matrix order n. The corresponding solution error is\n||δx||/||x||≤c(n)κ(A)ε. (The value of c(n) is seldom greater than 10n.)\nNOTE\nMachine precision depends on the data type used. For example, it is usually defined in the float.h\nfile as FLT_EPSILON the float datatype and DBL_EPSILON for the double datatype.\nThus, if your matrix A is ill-conditioned (that is, its condition number κ(A) is very large), then the error in\nthe solution x can also be large; you might even encounter a complete loss of precision. LAPACK provides\nroutines that allow you to estimate κ(A) (see Routines for Estimating the Condition Number) and also give\nyou a more precise estimate for the actual solution error (see Refining the Solution and Estimating Its Error).\nLAPACK Linear Equation Routines\nThis section describes routines for performing the following computations:\n–\nfactoring the matrix (except for triangular matrices)\n–\nequilibrating the matrix (except for RFP matrices)\n–\nsolving a system of linear equations\n–\nestimating the condition number of a matrix (except for RFP matrices)\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n465\n\n\n–\nrefining the solution of linear equations and computing its error bounds (except for RFP matrices)\n–\ninverting the matrix.\nTo solve a particular problem, you can call two or more computational routines or call a corresponding driver\nroutine that combines several tasks in one call. For example, to solve a system of linear equations with a\ngeneral matrix, call ?getrf (LU factorization) and then ?getrs (computing the solution). Then, call ?gerfs\nto refine the solution and get the error bounds. Alternatively, use the driver routine ?gesvx that performs all\nthese tasks in one call.\nLAPACK Linear Equation Computational Routines\nTable \"Computational Routines for Systems of Equations with Real Matrices\" lists the LAPACK computational\nroutines for factorizing, equilibrating, and inverting real matrices, estimating their condition numbers, solving\nsystems of equations with real matrices, refining the solution, and estimating its error. Table \"Computational\nRoutines for Systems of Equations with Complex Matrices\" lists similar routines for complex matrices.\nComputational Routines for Systems of Equations with Real Matrices\nMatrix type,\nstorage scheme\nFactorize\nmatrix\nEquilibrate\nmatrix\nSolve\nsystem\nCondition\nnumber\nEstimate\nerror\nInvert matrix\ngeneral\n?getrf\n?geequ,\n?geequb\n?getrs\n?gecon\n?gerfs,\n?gerfsx\n?getri\ngeneral band\n?gbtrf\n?gbequ,\n?gbequb\n?gbtrs\n?gbcon\n?gbrfs,\n?gbrfsx\n \ngeneral tridiagonal\n?gttrf\n \n?gttrs\n?gtcon\n?gtrfs\n \ndiagonally\ndominant\ntridiagonal\n?dttrfb\n \n?dttrsb\n \nsymmetric\npositive-definite\n?potrf\n?poequ,\n?poequb\n?potrs\n?pocon\n?porfs,\n?porfsx\n?potri\nsymmetric\npositive-definite,\npacked storage\n?pptrf\n?ppequ\n?pptrs\n?ppcon\n?pprfs\n?pptri\nsymmetric\npositive-definite,\nRFP storage\n?pftrf\n \n?pftrs\n \n \n?pftri\nsymmetric\npositive-definite,\nband\n?pbtrf\n?pbequ\n?pbtrs\n?pbcon\n?pbrfs\n \nsymmetric\npositive-definite,\ntridiagonal\n?pttrf\n \n?pttrs\n?ptcon\n?ptrfs\n \nsymmetric\nindefinite\n?sytrf\n?sytrf_rk\n?sytrf_aa\n?syequb\n?sytrs\n?sytrs2\n?sytrs3\n?sytrs_aa\n?sycon\n?sycon_3\n?syrfs,\n?syrfsx\n?sytri\n?sytri2\n?sytri2x\n?sytri_3\nsymmetric\nindefinite, packed\nstorage\n?sptrf\nmkl_?spffrt2, mkl_?spffrtx\n \n?sptrs\n?spcon\n?sprfs\n?sptri\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n466\n\n\nMatrix type,\nstorage scheme\nFactorize\nmatrix\nEquilibrate\nmatrix\nSolve\nsystem\nCondition\nnumber\nEstimate\nerror\nInvert matrix\ntriangular\n \n \n?trtrs\n?trcon\n?trrfs\n?trtri\ntriangular, packed\nstorage\n \n \n?tptrs\n?tpcon\n?tprfs\n?tptri\ntriangular, RFP\nstorage\n \n \n \n \n \n?tftri\ntriangular band\n \n \n?tbtrs\n?tbcon\n?tbrfs\n \nComputational Routines for Systems of Equations with Complex Matrices\nMatrix type,\nstorage scheme\nFactorize\nmatrix\nEquilibrate\nmatrix\nSolve\nsystem\nCondition\nnumber\nEstimate\nerror\nInvert matrix\ngeneral\n?getrf\n?geequ,\n?geequb\n?getrs\n?gecon\n?gerfs,\n?gerfsx\n?getri\ngeneral band\n?gbtrf\n?gbequ,\n?gbequb\n?gbtrs\n?gbcon\n?gbrfs,\n?gbrfsx\n \ngeneral tridiagonal\n?gttrf\n \n?gttrs\n?gtcon\n?gtrfs\n \nHermitian\npositive-definite\n?potrf\n?poequ,\n?poequb\n?potrs\n?pocon\n?porfs,\n?porfsx\n?potri\nHermitian\npositive-definite,\npacked storage\n?pptrf\n?ppequ\n?pptrs\n?ppcon\n?pprfs\n?pptri\nHermitian\npositive-definite,\nRFP storage\n?pftrf\n?pftrs\n?pftri\nHermitian\npositive-definite,\nband\n?pbtrf\n?pbequ\n?pbtrs\n?pbcon\n?pbrfs\n \nHermitian\npositive-definite,\ntridiagonal\n?pttrf\n \n?pttrs\n?ptcon\n?ptrfs\n \nHermitian\nindefinite\n?hetrf\n?hetrf_rk\n?hetrf_aa\n?heequb\n?hetrs\n?hetrs2\n?hetrs_3\n?hetrs_aa\n?hecon\n?hecon_3\n?herfs,\n?herfsx\n?hetri\n?hetri2\n?hetri2x\n?hetri_3\nsymmetric\nindefinite\n?sytrf\n?sytrf_rk\n?syequb\n?sytrs\n?sytrs2\n?sytrs3\n?sycon\n?sycon_3\n?syrfs,\n?syrfsx\n?sytri\n?sytri2\n?sytri2x\n?sytri_3\nHermitian\nindefinite, packed\nstorage\n?hptrf\n \n?hptrs\n?hpcon\n?hprfs\n?hptri\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n467\n\n\nMatrix type,\nstorage scheme\nFactorize\nmatrix\nEquilibrate\nmatrix\nSolve\nsystem\nCondition\nnumber\nEstimate\nerror\nInvert matrix\nsymmetric\nindefinite, packed\nstorage\n?sptrf\nmkl_?spffrt2, mkl_?spffrtx\n \n?sptrs\n?spcon\n?sprfs\n?sptri\ntriangular\n \n \n?trtrs\n?trcon\n?trrfs\n?trtri\ntriangular, packed\nstorage\n \n \n?tptrs\n?tpcon\n?tprfs\n?tptri\ntriangular, RFP\nstorage\n \n \n?tftri\ntriangular band\n \n \n?tbtrs\n?tbcon\n?tbrfs\n \nMatrix Factorization: LAPACK Computational Routines\nThis section describes the LAPACK routines for matrix factorization. The following factorizations are\nsupported:\n•\nLU factorization\n•\nCholesky factorization of real symmetric positive-definite matrices\n•\nCholesky factorization of real symmetric positive-definite matrices with pivoting\n•\nCholesky factorization of Hermitian positive-definite matrices\n•\nCholesky factorization of Hermitian positive-definite matrices with pivoting\n•\nBunch-Kaufman factorization of real and complex symmetric matrices\n•\nBunch-Kaufman factorization of Hermitian matrices.\nYou can compute:\n•\nthe LU factorization using full and band storage of matrices\n•\nthe Cholesky factorization using full, packed, RFP, and band storage\n•\nthe Bunch-Kaufman factorization using full and packed storage.\n?getrf\nComputes the LU factorization of a general m-by-n\nmatrix.\nSyntax\nlapack_int LAPACKE_sgetrf (int matrix_layout , lapack_int m , lapack_int n , float *\na , lapack_int lda , lapack_int * ipiv );\nlapack_int LAPACKE_dgetrf (int matrix_layout , lapack_int m , lapack_int n , double *\na , lapack_int lda , lapack_int * ipiv );\nlapack_int LAPACKE_cgetrf (int matrix_layout , lapack_int m , lapack_int n ,\nlapack_complex_float * a , lapack_int lda , lapack_int * ipiv );\nlapack_int LAPACKE_zgetrf (int matrix_layout , lapack_int m , lapack_int n ,\nlapack_complex_double * a , lapack_int lda , lapack_int * ipiv );\nInclude Files\n•\nmkl.h\nDescription\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n468\n\n\nThe routine computes the LU factorization of a general m-by-n matrix A as\nA = P*L*U,\nwhere P is a permutation matrix, L is lower triangular with unit diagonal elements (lower trapezoidal if m >\nn) and U is upper triangular (upper trapezoidal if m < n). The routine uses partial pivoting, with row\ninterchanges.\nNOTE\nThis routine supports the Progress Routine feature. See Progress Function for details.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\nm\nThe number of rows in the matrix A (m≥ 0).\nn\nThe number of columns in A; n≥ 0.\na\nArray, size at least max(1, lda*n) for column-major layout or max(1,\nlda*m) for row-major layout. Contains the matrix A.\nlda\nThe leading dimension of array a, which must be at least max(1, m)\nfor column-major layout or max(1, n) for row-major layout.\nOutput Parameters\na\nOverwritten by L and U. The unit diagonal elements of L are not\nstored.\nipiv\nArray, size at least max(1,min(m, n)). Contains the pivot indices; for\n1 ≤i≤ min(m, n), row i was interchanged with row ipiv(i).\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, parameter i had an illegal value.\nIf info = i, uii is 0. The factorization has been completed, but U is exactly singular. Division by 0 will\noccur if you use the factor U for solving a system of linear equations.\nApplication Notes\nThe computed L and U are the exact factors of a perturbed matrix A + E, where\n|E| ≤c(min(m,n))εP|L||U|\nc(n) is a modest linear function of n, and ε is the machine precision.\nThe approximate number of floating-point operations for real flavors is\n(2/3)n3\nIf m = n,\n(1/3)n2(3m-n)\nIf m>n,\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n469\n\n\n(1/3)m2(3n-m)\nIf m<n.\nThe number of operations for complex flavors is four times greater.\nAfter calling this routine with m = n, you can call the following:\n?getrs\nto solve A*X = B or ATX = B or AHX = B\n?gecon\nto estimate the condition number of A\n?getri\nto compute the inverse of A.\nSee Also\nmkl_progress\nMatrix Storage Schemes\nmkl_?getrfnp\nComputes the LU factorization of a general m-by-n\nmatrix without pivoting.\nSyntax\nlapack_int LAPACKE_mkl_sgetrfnp (int matrix_layout , lapack_int m , lapack_int n ,\nfloat * a , lapack_int lda );\nlapack_int LAPACKE_mkl_dgetrfnp (int matrix_layout , lapack_int m , lapack_int n ,\ndouble * a , lapack_int lda );\nlapack_int LAPACKE_mkl_cgetrfnp (int matrix_layout , lapack_int m , lapack_int n ,\nlapack_complex_float * a , lapack_int lda );\nlapack_int LAPACKE_mkl_zgetrfnp (int matrix_layout , lapack_int m , lapack_int n ,\nlapack_complex_double * a , lapack_int lda );\nInclude Files\n•\nmkl.h\nDescription\nThe routine computes the LU factorization of a general m-by-n matrix A as\nA = L*U,\nwhere L is lower triangular with unit-diagonal elements (lower trapezoidal if m > n) and U is upper triangular\n(upper trapezoidal if m < n). The routine does not use pivoting.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\nm\nThe number of rows in the matrix A (m≥ 0).\nn\nThe number of columns in A; n≥ 0.\na\nArray, size at least max(1, lda*n) for column-major layout or max(1,\nlda*m) for row-major layout. Contains the matrix A.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n470\n\n\nlda\nThe leading dimension of array a, which must be at least max(1, m)\nfor column-major layout or max(1, n) for row-major layout.\nOutput Parameters\na\nOverwritten by L and U. The unit diagonal elements of L are not\nstored.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, parameter i had an illegal value.\nIf info = i, uii is 0. The factorization has been completed, but U is exactly singular. Division by 0 will\noccur if you use the factor U for solving a system of linear equations.\nApplication Notes\nThe approximate number of floating-point operations for real flavors is\n(2/3)n3\nIf m = n,\n(1/3)n2(3m-n)\nIf m>n,\n(1/3)m2(3n-m)\nIf m<n.\nThe number of operations for complex flavors is four times greater.\nAfter calling this routine with m = n, you can call the following:\nmkl_?getrinp\nto compute the inverse of A\nSee Also\nmkl_progress\nMatrix Storage Schemes\nmkl_?getrfnpi\nPerforms LU factorization (complete or incomplete) of\na general matrix without pivoting.\nSyntax\nlapack_int LAPACKE_mkl_sgetrfnpi (int matrix_layout, lapack_int m, lapack_int n,\nlapack_int nfact, float* a, lapack_int lda);\nlapack_int LAPACKE_mkl_dgetrfnpi (int matrix_layout, lapack_int m, lapack_int n,\nlapack_int nfact, double* a, lapack_int lda);\nlapack_int LAPACKE_mkl_cgetrfnpi (int matrix_layout, lapack_int m, lapack_int n,\nlapack_int nfact, lapack_complex_float* a, lapack_int lda);\nlapack_int LAPACKE_mkl_zgetrfnpi (int matrix_layout, lapack_int m, lapack_int n,\nlapack_int nfact, lapack_complex_double* a, lapack_int lda);\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n471\n\n\nInclude Files\n•\nmkl.h\nDescription\nThe routine computes the LU factorization of a general m-by-n matrix A without using pivoting. It supports\nincomplete factorization. The factorization has the form:\nA = L*U,\nwhere L is lower triangular with unit diagonal elements (lower trapezoidal if m > n) and U is upper triangular\n(upper trapezoidal if m < n).\nIncomplete factorization has the form:\nwhere L is lower trapezoidal with unit diagonal elements, U is upper trapezoidal, and \nis the unfactored part of matrix A. See the application notes section for further details.\nNOTE\nUse ?getrf if it is possible that the matrix is not diagonal dominant.\nInput Parameters\nA <datatype> placeholder, if present, is used for the C interface data types in the C interface section above.\nSee C Interface Conventions for the C interface principal conventions and type definitions.\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nm\nThe number of rows in matrix A; m≥ 0.\nn\nThe number of columns in matrix A; n≥ 0.\nnfact\nThe number of rows and columns to factor; 0 ≤nfact≤ min(m, n). Note that\nif nfact < min(m, n), incomplete factorization is performed.\na\nArray of size at least lda*n for column major layout and at least lda*m for\nrow major layout. Contains the matrix A.\nlda\nThe leading dimension of array a. lda≥ max(1, m) for column major layout\nand lda≥ max(1, n) for row major layout.\nOutput Parameters\na\nOverwritten by L and U. The unit diagonal elements of L are not stored.\nWhen incomplete factorization is specified by setting nfact < min(m, n), a\nalso contains the unfactored submatrix \n. See the application notes section for further details.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n472\n\n\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nIf info = i, uii is 0. The requested factorization has been completed, but U is exactly singular. Division by 0\nwill occur if factorization is completed and factor U is used for solving a system of linear equations.\nApplication Notes\nThe computed L and U are the exact factors of a perturbed matrix A + E, with\n|E| ≤c(min(m, n))ε|L||U|\nwhere c(n) is a modest linear function of n, and ε is the machine precision.\nThe approximate number of floating-point operations for real flavors is\n(2/3)n3\nIf m = n = nfact\n(1/3)n2(3m-n)\nIf m>n = nfact\n(1/3)m2(3n-m)\nIf m = nfact<n\n(2/3)n3 - (n-nfact)3\nIf m = n,nfact< min(m, n)\n(1/3)(n2(3m-n) - (n-nfact)2(3m -\n2nfact - n))\nIf m>n > nfact\n(1/3)(m2(3n-m) - (m-nfact)2(3n -\n2nfact - m))\nIf nfact < m < n\nThe number of operations for complex flavors is four times greater.\nWhen incomplete factorization is specified, the first nfact rows and columns are factored, with the update of\nthe remaining rows and columns of A as follows:\nIf matrix A is represented as a block 2-by-2 matrix:\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n473\n\n\nwhere\n•\nA11 is a square matrix of order nfact,\n•\nA21 is an (m - nfact)-by-nfact matrix,\n•\nA12 is an nfact-by-(n - nfact) matrix, and\n•\nA22 is an (m - nfact)-by-(n - nfact) matrix.\nThe result is\nL1 is a lower triangular square matrix of order nfact with unit diagonal and U1 is an upper triangular square\nmatrix of order nfact. L1 and U1 result from LU factorization of matrix A11: A11 = L1U1.\nL2 is an (m - nfact)-by-nfact matrix and L2 = A21U1-1. U2 is an nfact-by-(n - nfact) matrix and U2 =\nL1-1A12.\nis an (m - nfact)-by-(n - nfact) matrix and \n= A22 - L2U2.\nOn exit, elements of the upper triangle U1 are stored in place of the upper triangle of block A11 in array a;\nelements of the lower triangle L1 are stored in the lower triangle of block A11 in array a (unit diagonal\nelements are not stored). Elements of L2 replace elements of A21; U2 replaces elements of A12 and \nreplaces elements of A22.\n?getrf2\nComputes LU factorization using partial pivoting with\nrow interchanges.\nSyntax\nlapack_int LAPACKE_sgetrf2 (int matrix_layout, lapack_int m, lapack_int n, float * a,\nlapack_int lda, lapack_int * ipiv);\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n474\n\n\nlapack_int LAPACKE_dgetrf2 (int matrix_layout, lapack_int m, lapack_int n, double * a,\nlapack_int lda, lapack_int * ipiv);\nlapack_int LAPACKE_cgetrf2 (int matrix_layout, lapack_int m, lapack_int n,\nlapack_complex_float * a, lapack_int lda, lapack_int * ipiv);\nlapack_int LAPACKE_zgetrf2 (int matrix_layout, lapack_int m, lapack_int n,\nlapack_complex_double * a, lapack_int lda, lapack_int * ipiv);\nInclude Files\n•\nmkl.h\nDescription\n?getrf2 computes an LU factorization of a general m-by-n matrix A using partial pivoting with row\ninterchanges.\nThe factorization has the form\nA = P * L * U\nwhere P is a permutation matrix, L is lower triangular with unit diagonal elements (lower trapezoidal if m >\nn), and U is upper triangular (upper trapezoidal if m < n).\nThis is the recursive version of the algorithm. It divides the matrix into four submatrices:\nA = A11 A12\nA21 A22\nwhere A11 is n1 by n1 and A22 is n2 by n2 with n1 = min(m, n), and n2 = n - n1.\nThe subroutine calls itself to factor A11\nA12 ,\ndo the swaps on A12\nA22 , solve A12, update A22, then it calls itself to factor A22 and do the swaps on A21.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nm\nThe number of rows of the matrix A. m >= 0.\nn\nThe number of columns of the matrix A. n >= 0.\na\nArray, size lda*n.\nOn entry, the m-by-n matrix to be factored.\nlda\nThe leading dimension of the array a. lda >= max(1,m).\nOutput Parameters\na\nOn exit, the factors L and U from the factorization A = P * L * U; the\nunit diagonal elements of L are not stored.\nipiv\nArray, size (min(m,n)).\nThe pivot indices; for 1 <= i <= min(m,n), row i of the matrix was\ninterchanged with row ipiv[i - 1].\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n475\n\n\nReturn Values\nThis function returns a value info.\n= 0: successful exit\n< 0: if info = -i, the i-th argument had an illegal value.\n> 0: if info = i, Ui, i is exactly zero. The factorization has been completed, but the factor U is exactly singular,\nand division by zero will occur if it is used to solve a system of equations.\n?gbtrf\nComputes the LU factorization of a general m-by-n\nband matrix.\nSyntax\nlapack_int LAPACKE_sgbtrf (int matrix_layout , lapack_int m , lapack_int n , lapack_int\nkl , lapack_int ku , float * ab , lapack_int ldab , lapack_int * ipiv );\nlapack_int LAPACKE_dgbtrf (int matrix_layout , lapack_int m , lapack_int n , lapack_int\nkl , lapack_int ku , double * ab , lapack_int ldab , lapack_int * ipiv );\nlapack_int LAPACKE_cgbtrf (int matrix_layout , lapack_int m , lapack_int n , lapack_int\nkl , lapack_int ku , lapack_complex_float * ab , lapack_int ldab , lapack_int * ipiv );\nlapack_int LAPACKE_zgbtrf (int matrix_layout , lapack_int m , lapack_int n , lapack_int\nkl , lapack_int ku , lapack_complex_double * ab , lapack_int ldab , lapack_int *\nipiv );\nInclude Files\n•\nmkl.h\nDescription\nThe routine forms the LU factorization of a general m-by-n band matrix A with kl non-zero subdiagonals and\nku non-zero superdiagonals, that is,\nA = P*L*U,\nwhere P is a permutation matrix; L is lower triangular with unit diagonal elements and at most kl non-zero\nelements in each column; U is an upper triangular band matrix with kl + ku superdiagonals. The routine uses\npartial pivoting, with row interchanges (which creates the additional kl superdiagonals in U).\nNOTE\nThis routine supports the Progress Routine feature. See Progress Function for details.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\nm\nThe number of rows in matrix A; m≥ 0.\nn\nThe number of columns in matrix A; n≥ 0.\nkl\nThe number of subdiagonals within the band of A; kl≥ 0.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n476\n\n\nku\nThe number of superdiagonals within the band of A; ku≥ 0.\nab\nArray, size at least max(1, ldab*n) for column-major layout or\nmax(1, ldab*m) for row-major layout.\nThe array ab contains the matrix A in band storage as described in \nBand Storage.\nldab\nThe leading dimension of the array ab. (ldab≥ 2*kl + ku + 1)\nOutput Parameters\nab\nOverwritten with elements of L and U. U is stored as an upper\ntriangular band matrix with kl + ku superdiagonals, and L is stored\nas a lower triangular band matrix with kl subdiagonals (diagonal unit\nvalues are not stored). Since the output array has more nonzero\nelements than the initial matrix A, there are limitations on the value of\nldab and the placement of elements of A in array ab.\nSee Application Notes below for further details.\nipiv\nArray, size at least max(1,min(m, n)). The pivot indices; for 1 ≤i≤\nmin(m, n) , row i was interchanged with row ipiv(i).\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info = -i, parameter i had an illegal value.\nIf info = i, uiiis 0. The factorization has been completed, but U is exactly singular. Division by 0 will occur\nif you use the factor U for solving a system of linear equations.\nApplication Notes\nThe computed L and U are the exact factors of a perturbed matrix A + E, where\n|E| ≤c(kl+ku+1) εP|L||U|\nc(k) is a modest linear function of k, and ε is the machine precision.\nThe total number of floating-point operations for real flavors varies between approximately 2n(ku+1)kl and\n2n(kl+ku+1)kl. The number of operations for complex flavors is four times greater. All these estimates\nassume that kl and ku are much less than min(m,n).\nAs described in Band Storage, storage of a band matrix can be considered in two steps: packing band matrix\nelements into a matrix AB, then storing the elements in a linear array ab using a full storage scheme. The\neffect of the ?gbtrf routine on matrix AB is illustrated by this example, for m = n = 6, kl = 2, ku = 1.\n•\nmatrix_layout = LAPACK_COL_MAJOR\nOn entry:\nOn exit:\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n477\n\n\n•\nmatrix_layout = LAPACK_ROW_MAJOR\nOn entry:\nOn exit:\nElements marked * are not used; elements marked + need not be set on entry, but are required by the\nroutine to store elements of U because of fill-in resulting from the row interchanges.\nAfter calling this routine with m = n, you can call the following routines:\ngbtrs\nto solve A*X = B or AT*X = B or AH*X = B\ngbcon\nto estimate the condition number of A.\nSee Also\nmkl_progress\nMatrix Storage Schemes\n?gttrf\nComputes the LU factorization of a tridiagonal matrix.\nSyntax\nlapack_int LAPACKE_sgttrf (lapack_int n , float * dl , float * d , float * du , float *\ndu2 , lapack_int * ipiv );\nlapack_int LAPACKE_dgttrf (lapack_int n , double * dl , double * d , double * du ,\ndouble * du2 , lapack_int * ipiv );\nlapack_int LAPACKE_cgttrf (lapack_int n , lapack_complex_float * dl ,\nlapack_complex_float * d , lapack_complex_float * du , lapack_complex_float * du2 ,\nlapack_int * ipiv );\nlapack_int LAPACKE_zgttrf (lapack_int n , lapack_complex_double * dl ,\nlapack_complex_double * d , lapack_complex_double * du , lapack_complex_double * du2 ,\nlapack_int * ipiv );\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n478\n\n\nInclude Files\n•\nmkl.h\nDescription\nThe routine computes the LU factorization of a real or complex tridiagonal matrix A using elimination with\npartial pivoting and row interchanges.\nThe factorization has the form\nA = L*U,\nwhere L is a product of permutation and unit lower bidiagonal matrices and U is upper triangular with\nnonzeroes in only the main diagonal and first two superdiagonals.\nInput Parameters\nn\nThe order of the matrix A; n≥ 0.\ndl, d, du\nArrays containing elements of A.\nThe array dl of dimension (n - 1) contains the subdiagonal elements\nof A.\nThe array d of dimension n contains the diagonal elements of A.\nThe array du of dimension (n - 1) contains the superdiagonal\nelements of A.\nOutput Parameters\ndl\nOverwritten by the (n-1) multipliers that define the matrix L from the\nLU factorization of A.\nd\nOverwritten by the n diagonal elements of the upper triangular matrix\nU from the LU factorization of A.\ndu\nOverwritten by the (n-1) elements of the first superdiagonal of U.\ndu2\nArray, dimension (n -2). On exit, du2 contains (n-2) elements of\nthe second superdiagonal of U.\nipiv\nArray, dimension (n). The pivot indices: for 1 ≤ i ≤ n, row i was\ninterchanged with row ipiv[i-1]. ipiv[i-1] is always i or i+1;\nipiv[i-1] = i indicates a row interchange was not required.\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info = -i, parameter i had an illegal value.\nIf info = i, uiiis 0. The factorization has been completed, but U is exactly singular. Division by zero will\noccur if you use the factor U for solving a system of linear equations.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n479\n\n\nApplication Notes\n?gbtrs\nto solve A*X = B or AT*X = B or AH*X = B\n?gbcon\nto estimate the condition number of A.\n?dttrfb\nComputes the factorization of a diagonally dominant\ntridiagonal matrix.\nSyntax\nvoid sdttrfb (const MKL_INT * n , float * dl , float * d , const float * du , MKL_INT *\ninfo );\nvoid ddttrfb (const MKL_INT * n , double * dl , double * d , const double * du ,\nMKL_INT * info );\nvoid cdttrfb (const MKL_INT * n , MKL_Complex8 * dl , MKL_Complex8 * d , const\nMKL_Complex8 * du , MKL_INT * info );\nvoid zdttrfb_ (const MKL_INT * n , MKL_Complex16 * dl , MKL_Complex16 * d , const\nMKL_Complex16 * du , MKL_INT * info );\nInclude Files\n•\nmkl.h\nDescription\nThe ?dttrfb routine computes the factorization of a real or complex tridiagonal matrix A with the BABE\n(Burning At Both Ends) algorithm without pivoting. The factorization has the form\nA = L1*U*L2\nwhere\n•\nL1 and L2 are unit lower bidiagonal with k and n - k - 1 subdiagonal elements, respectively, where k =\nn/2, and\n•\nU is an upper bidiagonal matrix with nonzeroes in only the main diagonal and first superdiagonal.\nInput Parameters\nn\nThe order of the matrix A; n≥ 0.\ndl, d, du\nArrays containing elements of A.\nThe array dl of dimension (n - 1) contains the subdiagonal\nelements of A.\nThe array d of dimension n contains the diagonal elements of A.\nThe array du of dimension (n - 1) contains the superdiagonal\nelements of A.\nOutput Parameters\ndl\nOverwritten by the (n -1) multipliers that define the matrix L from\nthe LU factorization of A.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n480\n\n\nd\nOverwritten by the n diagonal element reciprocals of the upper\ntriangular matrix U from the factorization of A.\ndu\nOverwritten by the (n-1) elements of the superdiagonal of U.\ninfo\nIf info = 0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nIf info = i, uii is 0. The factorization has been completed, but U is\nexactly singular. Division by zero will occur if you use the factor U for\nsolving a system of linear equations.\nApplication Notes\nA diagonally dominant tridiagonal system is defined such that |di| > |dli-1| + |dui| for any i:\n1 < i < n, and |d1| > |du1|, |dn| > |dln-1|\nThe underlying BABE algorithm is designed for diagonally dominant systems. Such systems are free from the\nnumerical stability issue unlike the canonical systems that use elimination with partial pivoting (see ?gttrf).\nThe diagonally dominant systems are much faster than the canonical systems.\nNOTE\n•\nThe current implementation of BABE has a potential accuracy issue on very small or large data\nclose to the underflow or overflow threshold respectively. Scale the matrix before applying the\nsolver in the case of such input data.\n•\nApplying the ?dttrfb factorization to non-diagonally dominant systems may lead to an accuracy\nloss, or false singularity detected due to no pivoting.\n?potrf\nComputes the Cholesky factorization of a symmetric\n(Hermitian) positive-definite matrix.\nSyntax\nlapack_int LAPACKE_spotrf (int matrix_layout , char uplo , lapack_int n , float * a ,\nlapack_int lda );\nlapack_int LAPACKE_dpotrf (int matrix_layout , char uplo , lapack_int n , double * a ,\nlapack_int lda );\nlapack_int LAPACKE_cpotrf (int matrix_layout , char uplo , lapack_int n ,\nlapack_complex_float * a , lapack_int lda );\nlapack_int LAPACKE_zpotrf (int matrix_layout , char uplo , lapack_int n ,\nlapack_complex_double * a , lapack_int lda );\nInclude Files\n•\nmkl.h\nDescription\nThe routine forms the Cholesky factorization of a symmetric positive-definite or, for complex data, Hermitian\npositive-definite matrix A:\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n481\n\n\nA = UT* U for real data, A = UH* U for complex data\nif uplo='U'\nA = L*LT for real data, A = L*LH for complex data\nif uplo='L'\nwhere L is a lower triangular matrix and U is upper triangular.\nNOTE\nThis routine supports the Progress Routine feature. See Progress Function for details.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nuplo\nMust be 'U' or 'L'.\nIndicates whether the upper or lower triangular part of A is stored and how\nA is factored:\nIf uplo = 'U', the array a stores the upper triangular part of the matrix A,\nand the strictly lower triangular part of the matrix is not referenced.\nIf uplo = 'L', the array a stores the lower triangular part of the matrix A,\nand the strictly upper triangular part of the matrix is not referenced.\nn\nSpecifies the order of the matrix A. The value of n must be at least zero.\na\nArray, size max(1, lda*n). The array a contains either the upper or the\nlower triangular part of the matrix A (see uplo).\nlda\nThe leading dimension of a. Must be at least max(1, n).\nOutput Parameters\na\nThe upper or lower triangular part of a is overwritten by the Cholesky factor\nU or L, as specified by uplo.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, parameter i had an illegal value.\nIf info = i, the leading minor of order i (and therefore the matrix A itself) is not positive-definite, and the\nfactorization could not be completed. This may indicate an error in forming the matrix A.\nApplication Notes\nIf uplo = 'U', the computed factor U is the exact factor of a perturbed matrix A + E, where\nc(n) is a modest linear function of n, and ε is the machine precision.\nA similar estimate holds for uplo = 'L'.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n482\n\n\nThe total number of floating-point operations is approximately (1/3)n3 for real flavors or (4/3)n3 for\ncomplex flavors.\nAfter calling this routine, you can call the following routines:\n?potrs\nto solve A*X = B\n?pocon\nto estimate the condition number of A\n?potri\nto compute the inverse of A.\nSee Also\nmkl_progress\nMatrix Storage Schemes\n?potrf2\nComputes Cholesky factorization using a recursive\nalgorithm.\nSyntax\nlapack_int LAPACKE_spotrf2 (int matrix_layout, char uplo, lapack_int n, float * a,\nlapack_int lda);\nlapack_int LAPACKE_dpotrf2 (int matrix_layout, char uplo, lapack_int n, double * a,\nlapack_int lda);\nlapack_int LAPACKE_cpotrf2 (int matrix_layout, char uplo, lapack_int n,\nlapack_complex_float * a, lapack_int lda);\nlapack_int LAPACKE_zpotrf2 (int matrix_layout, char uplo, lapack_int n,\nlapack_complex_double * a, lapack_int lda);\nInclude Files\n•\nmkl.h\nDescription\n?potrf2 computes the Cholesky factorization of a real or complex symmetric positive definite matrix A using\nthe recursive algorithm.\nThe factorization has the form\nfor real flavors:\nA = UT * U, if uplo = 'U', or\nA = L * LT, if uplo = 'L',\nfor complex flavors:\nA = UH * U, if uplo = 'U',\nor A = L * LH, if uplo = 'L',\nwhere U is an upper triangular matrix and L is lower triangular.\nThis is the recursive version of the algorithm. It divides the matrix into four submatrices:\nA = A11 A12\nA21 A22\nwhere A11 is n1 by n1 and A22 is n2 by n2, with n1 = n/2 and n2 = n-n1.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n483\n\n\nThe subroutine calls itself to factor A11. Update and scale A21 or A12, update A22 then call itself to factor\nA22.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nuplo\n= 'U': Upper triangle of A is stored;\n= 'L': Lower triangle of A is stored.\nn\nThe order of the matrix A.\nn≥ 0.\na\nArray, size (lda*n).\nOn entry, the symmetric matrix A.\nIf uplo = 'U', the leading n-by-n upper triangular part of a contains the\nupper triangular part of the matrix A, and the strictly lower triangular part\nof a is not referenced.\nIf uplo = 'L', the leading n-by-n lower triangular part of a contains the\nlower triangular part of the matrix A, and the strictly upper triangular part\nof a is not referenced.\nlda\nThe leading dimension of the array a.\nlda≥ max(1,n).\nOutput Parameters\na\nOn exit, if info = 0, the factor U or L from the Cholesky factorization.\nFor real flavors:\nA = UT*U or A = L*LT;\nFor complex flavors:\nA = UH*U or A = L*LH.\nReturn Values\nThis function returns a value info.\n= 0: successful exit\n< 0: if info = -i, the i-th argument had an illegal value\n> 0: if info = i, the leading minor of order i is not positive definite, and the factorization could not be\ncompleted.\n?pstrf\nComputes the Cholesky factorization with complete\npivoting of a real symmetric (complex Hermitian)\npositive semidefinite matrix.\nSyntax\nlapack_int LAPACKE_spstrf( int matrix_layout, char uplo, lapack_int n, float* a,\nlapack_int lda, lapack_int* piv, lapack_int* rank, float tol );\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n484\n\n\nlapack_int LAPACKE_dpstrf( int matrix_layout, char uplo, lapack_int n, double* a,\nlapack_int lda, lapack_int* piv, lapack_int* rank, double tol );\nlapack_int LAPACKE_cpstrf( int matrix_layout, char uplo, lapack_int n,\nlapack_complex_float* a, lapack_int lda, lapack_int* piv, lapack_int* rank, float tol );\nlapack_int LAPACKE_zpstrf( int matrix_layout, char uplo, lapack_int n,\nlapack_complex_double* a, lapack_int lda, lapack_int* piv, lapack_int* rank, double\ntol );\nInclude Files\n•\nmkl.h\nDescription\nThe routine computes the Cholesky factorization with complete pivoting of a real symmetric (complex\nHermitian) positive semidefinite matrix. The form of the factorization is:\n \nPT * A * P = UT * U, if uplo ='U' for real flavors,\n \nPT * A * P = UH * U, if uplo ='U' for complex flavors,\n \nPT * A * P = L * LT, if uplo ='L' for real flavors,\n \nPT * A * P = L * LH, if uplo ='L' for complex flavors,\nwhere P is a permutation matrix stored as vector piv, and U and L are upper and lower triangular matrices,\nrespectively.\nThis algorithm does not attempt to check that A is positive semidefinite. This version of the algorithm calls\nlevel 3 BLAS.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\nuplo\nMust be 'U' or 'L'.\nIndicates whether the upper or lower triangular part of A is stored:\nIf uplo = 'U', the array a stores the upper triangular part of the\nmatrix A, and the strictly lower triangular part of the matrix is not\nreferenced.\nIf uplo = 'L', the array a stores the lower triangular part of the\nmatrix A, and the strictly upper triangular part of the matrix is not\nreferenced.\nn\nThe order of matrix A; n≥ 0.\na\nArray a, size max(1,lda*n). The array a contains either the upper or\nthe lower triangular part of the matrix A (see uplo). .\ntol\nUser defined tolerance. If tol < 0, then n*ε*max(Ak,k), where ε is the\nmachine precision, will be used (see Error Analysis for the definition of\nmachine precision). The algorithm terminates at the (k-1)-st step, if\nthe pivot ≤tol.\nlda\nThe leading dimension of a; at least max(1, n).\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n485\n\n\nOutput Parameters\na\nIf info = 0, the factor U or L from the Cholesky factorization is as\ndescribed in Description.\npiv\nArray, size at least max(1, n). The array piv is such that the nonzero\nentries are Ppiv[k-1],k (1 ≤k≤n).\nrank\nThe rank of a given by the number of steps the algorithm completed.\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info = -k, the k-th argument had an illegal value.\nIf info > 0, the matrix A is either rank deficient with a computed rank as returned in rank, or is not\npositive semidefinite.\nSee Also\nMatrix Storage Schemes\n?pftrf\nComputes the Cholesky factorization of a symmetric\n(Hermitian) positive-definite matrix using the\nRectangular Full Packed (RFP) format .\nSyntax\nlapack_int LAPACKE_spftrf (int matrix_layout , char transr , char uplo , lapack_int n ,\nfloat * a );\nlapack_int LAPACKE_dpftrf (int matrix_layout , char transr , char uplo , lapack_int n ,\ndouble * a );\nlapack_int LAPACKE_cpftrf (int matrix_layout , char transr , char uplo , lapack_int n ,\nlapack_complex_float * a );\nlapack_int LAPACKE_zpftrf (int matrix_layout , char transr , char uplo , lapack_int n ,\nlapack_complex_double * a );\nInclude Files\n•\nmkl.h\nDescription\nThe routine forms the Cholesky factorization of a symmetric positive-definite or, for complex data, a\nHermitian positive-definite matrix A:\nA = UT*U for real data, A = UH*U for complex data\nif uplo='U'\nA = L*LT for real data, A = L*LH for complex data\nif uplo='L'\nwhere L is a lower triangular matrix and U is upper triangular.\nThe matrix A is in the Rectangular Full Packed (RFP) format. For the description of the RFP format, see Matrix\nStorage Schemes.\nThis is the block version of the algorithm, calling Level 3 BLAS.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n486\n\n\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\ntransr\nMust be 'N', 'T' (for real data) or 'C' (for complex data).\nIf transr = 'N', the Normal transr of RFP A is stored.\nIf transr = 'T', the Transpose transr of RFP A is stored.\nIf transr = 'C', the Conjugate-Transpose transr of RFP A is stored.\nuplo\nMust be 'U' or 'L'.\nIndicates whether the upper or lower triangular part of A is stored:\nIf uplo = 'U', the array a stores the upper triangular part of the\nmatrix A.\nIf uplo = 'L', the array a stores the lower triangular part of the\nmatrix A.\nn\nThe order of the matrix A; n≥ 0.\na\nArray, size (n*(n+1)/2). The array a contains the matrix A in the RFP\nformat.\nOutput Parameters\na\na is overwritten by the Cholesky factor U or L, as specified by uplo\nand trans.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, parameter i had an illegal value.\nIf info = i, the leading minor of order i (and therefore the matrix A itself) is not positive-definite, and the\nfactorization could not be completed. This may indicate an error in forming the matrix A.\nSee Also\nMatrix Storage Schemes\n?pptrf\nComputes the Cholesky factorization of a symmetric\n(Hermitian) positive-definite matrix using packed\nstorage.\nSyntax\nlapack_int LAPACKE_spptrf (int matrix_layout , char uplo , lapack_int n , float * ap );\nlapack_int LAPACKE_dpptrf (int matrix_layout , char uplo , lapack_int n , double *\nap );\nlapack_int LAPACKE_cpptrf (int matrix_layout , char uplo , lapack_int n ,\nlapack_complex_float * ap );\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n487\n\n\nlapack_int LAPACKE_zpptrf (int matrix_layout , char uplo , lapack_int n ,\nlapack_complex_double * ap );\nInclude Files\n•\nmkl.h\nDescription\nThe routine forms the Cholesky factorization of a symmetric positive-definite or, for complex data, Hermitian\npositive-definite packed matrix A:\nA = UT*U for real data, A = UH*U for complex data\nif uplo='U'\nA = L*LT for real data, A = L*LH for complex data\nif uplo='L'\nwhere L is a lower triangular matrix and U is upper triangular.\nNOTE\nThis routine supports the Progress Routine feature. See Progress Function for details.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\nuplo\nMust be 'U' or 'L'.\nIndicates whether the upper or lower triangular part of A is packed in\nthe array ap, and how A is factored:\nIf uplo = 'U', the array ap stores the upper triangular part of the\nmatrix A, and A is factored as UH*U.\nIf uplo = 'L', the array ap stores the lower triangular part of the\nmatrix A; A is factored as L*LH.\nn\nThe order of matrix A; n≥ 0.\nap\nArray, size at least max(1, n(n+1)/2). The array ap contains either the\nupper or the lower triangular part of the matrix A (as specified by\nuplo) in packed storage (see Matrix Storage Schemes).\nOutput Parameters\nap\nOverwritten by the Cholesky factor U or L, as specified by uplo.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, parameter i had an illegal value.\nIf info = i, the leading minor of order i (and therefore the matrix A itself) is not positive-definite, and the\nfactorization could not be completed. This may indicate an error in forming the matrix A.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n488\n\n\nApplication Notes\nIf uplo = 'U', the computed factor U is the exact factor of a perturbed matrix A + E, where\nc(n) is a modest linear function of n, and ε is the machine precision.\nA similar estimate holds for uplo = 'L'.\nThe total number of floating-point operations is approximately (1/3)n3 for real flavors and (4/3)n3 for\ncomplex flavors.\nAfter calling this routine, you can call the following routines:\n?pptrs\nto solve A*X = B\n?ppcon\nto estimate the condition number of A\n?pptri\nto compute the inverse of A.\nSee Also\nmkl_progress\nMatrix Storage Schemes\n?pbtrf\nComputes the Cholesky factorization of a symmetric\n(Hermitian) positive-definite band matrix.\nSyntax\nlapack_int LAPACKE_spbtrf (int matrix_layout , char uplo , lapack_int n , lapack_int\nkd , float * ab , lapack_int ldab );\nlapack_int LAPACKE_dpbtrf (int matrix_layout , char uplo , lapack_int n , lapack_int\nkd , double * ab , lapack_int ldab );\nlapack_int LAPACKE_cpbtrf (int matrix_layout , char uplo , lapack_int n , lapack_int\nkd , lapack_complex_float * ab , lapack_int ldab );\nlapack_int LAPACKE_zpbtrf (int matrix_layout , char uplo , lapack_int n , lapack_int\nkd , lapack_complex_double * ab , lapack_int ldab );\nInclude Files\n•\nmkl.h\nDescription\nThe routine forms the Cholesky factorization of a symmetric positive-definite or, for complex data, Hermitian\npositive-definite band matrix A:\nA = UT*U for real data, A = UH*U for complex data\nif uplo='U'\nA = L*LT for real data, A = L*LH for complex data\nif uplo='L'\nwhere L is a lower triangular matrix and U is upper triangular.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n489\n\n\nNOTE\nThis routine supports the Progress Routine feature. See Progress Function for details.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\nuplo\nMust be 'U' or 'L'.\nIndicates whether the upper or lower triangular part of A is stored in\nthe array ab, and how A is factored:\nIf uplo = 'U', the upper triangle of A is stored.\nIf uplo = 'L', the lower triangle of A is stored.\nn\nThe order of matrix A; n≥ 0.\nkd\nThe number of superdiagonals or subdiagonals in the matrix A; kd≥ 0.\nab\nArray, size max(1, ldab*n). The array ab contains either the upper or\nthe lower triangular part of the matrix A (as specified by uplo) in band\nstorage (see Matrix Storage Schemes).\nldab\nThe leading dimension of the array ab. (ldab≥kd + 1)\nOutput Parameters\nab\nThe upper or lower triangular part of A (in band storage) is\noverwritten by the Cholesky factor U or L, as specified by uplo.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, parameter i had an illegal value.\nIf info = i, the leading minor of order i (and therefore the matrix A itself) is not positive-definite, and the\nfactorization could not be completed. This may indicate an error in forming the matrix A.\nApplication Notes\nIf uplo = 'U', the computed factor U is the exact factor of a perturbed matrix A + E, where\nc(n) is a modest linear function of n, and ε is the machine precision.\nA similar estimate holds for uplo = 'L'.\nThe total number of floating-point operations for real flavors is approximately n(kd+1)2. The number of\noperations for complex flavors is 4 times greater. All these estimates assume that kd is much less than n.\nAfter calling this routine, you can call the following routines:\n?pbtrs\nto solve A*X = B\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n490\n\n\n?pbcon\nto estimate the condition number of A.\nSee Also\nmkl_progress\nMatrix Storage Schemes\n?pttrf\nComputes the factorization of a symmetric (Hermitian)\npositive-definite tridiagonal matrix.\nSyntax\nlapack_int LAPACKE_spttrf( lapack_int n, float* d, float* e );\nlapack_int LAPACKE_dpttrf( lapack_int n, double* d, double* e );\nlapack_int LAPACKE_cpttrf( lapack_int n, float* d, lapack_complex_float* e );\nlapack_int LAPACKE_zpttrf( lapack_int n, double* d, lapack_complex_double* e );\nInclude Files\n•\nmkl.h\nDescription\nThe routine forms the factorization of a symmetric positive-definite or, for complex data, Hermitian positive-\ndefinite tridiagonal matrix A:\nA = L*D*LT for real flavors, or\nA = L*D*LH for complex flavors,\nwhere D is diagonal and L is unit lower bidiagonal. The factorization may also be regarded as having the form\nA = UT*D*U for real flavors, or A = UH*D*U for complex flavors, where U is unit upper bidiagonal.\nInput Parameters\nn\nThe order of the matrix A; n≥ 0.\nd\nArray, dimension (n). Contains the diagonal elements of A.\ne\nArray, dimension (n -1). Contains the subdiagonal elements of A.\nOutput Parameters\nd\nOverwritten by the n diagonal elements of the diagonal matrix D from\nthe L*D*LT (for real flavors) or L*D*LH (for complex flavors)\nfactorization of A.\ne\nOverwritten by the (n - 1) sub-diagonal elements of the unit\nbidiagonal factor L or U from the factorization of A.\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n491\n\n\nIf info = -i, parameter i had an illegal value.\nIf info = i, the leading minor of order i (and therefore the matrix A itself) is not positive-definite; if i < n,\nthe factorization could not be completed, while if i = n, the factorization was completed, but d[n - 1] ≤\n0.\n?sytrf\nComputes the Bunch-Kaufman factorization of a\nsymmetric matrix.\nSyntax\nlapack_int LAPACKE_ssytrf (int matrix_layout , char uplo , lapack_int n , float * a ,\nlapack_int lda , lapack_int * ipiv );\nlapack_int LAPACKE_dsytrf (int matrix_layout , char uplo , lapack_int n , double * a ,\nlapack_int lda , lapack_int * ipiv );\nlapack_int LAPACKE_csytrf (int matrix_layout , char uplo , lapack_int n ,\nlapack_complex_float * a , lapack_int lda , lapack_int * ipiv );\nlapack_int LAPACKE_zsytrf (int matrix_layout , char uplo , lapack_int n ,\nlapack_complex_double * a , lapack_int lda , lapack_int * ipiv );\nInclude Files\n•\nmkl.h\nDescription\nThe routine computes the factorization of a real/complex symmetric matrix A using the Bunch-Kaufman\ndiagonal pivoting method. The form of the factorization is:\n \nif uplo='U', A = U*D*UT\n \nif uplo='L', A = L*D*LT\nwhere A is the input matrix, U and L are products of permutation and triangular matrices with unit diagonal\n(upper triangular for U and lower triangular for L), and D is a symmetric block-diagonal matrix with 1-by-1\nand 2-by-2 diagonal blocks. U and L have 2-by-2 unit diagonal blocks corresponding to the 2-by-2 blocks of\nD.\nNOTE This routine supports the Progress Routine feature. See Progress Routine for details.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\nuplo\nMust be 'U' or 'L'.\nIndicates whether the upper or lower triangular part of A is stored and\nhow A is factored:\nIf uplo = 'U', the array a stores the upper triangular part of the\nmatrix A, and A is factored as U*D*UT.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n492\n\n\nIf uplo = 'L', the array a stores the lower triangular part of the\nmatrix A, and A is factored as L*D*LT.\nn\nThe order of matrix A; n≥ 0.\na\nArray, size max(1, lda*n). The array a contains either the upper or\nthe lower triangular part of the matrix A (see uplo).\nlda\nThe leading dimension of a; at least max(1, n).\nOutput Parameters\na\nThe upper or lower triangular part of a is overwritten by details of the\nblock-diagonal matrix D and the multipliers used to obtain the factor U\n(or L).\nipiv\nArray, size at least max(1, n). Contains details of the interchanges\nand the block structure of D. If ipiv[i-1] = k >0, then dii is a 1-\nby-1 block, and the i-th row and column of A was interchanged with\nthe k-th row and column.\nIf uplo = 'U' and ipiv[i] =ipiv[i-1] = -m < 0, then D has a 2-by-2\nblock in rows/columns i and i+1, and i-th row and column of A was\ninterchanged with the m-th row and column.\nIf uplo = 'L' and ipiv[i] =ipiv[i-1] = -m < 0, then D has a 2-by-2\nblock in rows/columns i and i+1, and (i+1)-th row and column of A\nwas interchanged with the m-th row and column.\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info = -i, parameter i had an illegal value.\nIf info = i, Dii is 0. The factorization has been completed, but D is exactly singular. Division by 0 will occur\nif you use D for solving a system of linear equations.\nApplication Notes\nThe 2-by-2 unit diagonal blocks and the unit diagonal elements of U and L are not stored. The remaining\nelements of U and L are stored in the corresponding columns of the array a, but additional row interchanges\nare required to recover U or L explicitly (which is seldom necessary).\nIf ipiv[i-1] = i for all i =1...n, then all off-diagonal elements of U (L) are stored explicitly in the\ncorresponding elements of the array a.\nIf uplo = 'U', the computed factors U and D are the exact factors of a perturbed matrix A + E, where\n|E| ≤c(n)εP|U||D||UT|PT\nc(n) is a modest linear function of n, and ε is the machine precision. A similar estimate holds for the\ncomputed L and D when uplo = 'L'.\nThe total number of floating-point operations is approximately (1/3)n3 for real flavors or (4/3)n3 for\ncomplex flavors.\nAfter calling this routine, you can call the following routines:\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n493\n\n\n?sytrs\nto solve A*X = B\n?sycon\nto estimate the condition number of A\n?sytri\nto compute the inverse of A.\n \nIf uplo = 'U', then A = U*D*U', where\nU = P(n)*U(n)* ... *P(k)*U(k)*...,\n                \nthat is, U is a product of terms P(k)*U(k), where\n•\nk decreases from n to 1 in steps of 1 and 2.\n•\nD is a block diagonal matrix with 1-by-1 and 2-by-2 diagonal blocks D(k).\n•\nP(k) is a permutation matrix as defined by ipiv[k-1].\n•\nU(k) is a unit upper triangular matrix, such that if the diagonal block D(k) is of order s (s = 1 or 2), then\nIf s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k).\nIf s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k) and A(k,k), and v overwrites A(1:k-2,k\n-1:k).\n \nIf uplo = 'L', then A = L*D*L', where\nL = P(1)*L(1)* ... *P(k)*L(k)*...,\n                \nthat is, L is a product of terms P(k)*L(k), where\n•\nk increases from 1 to n in steps of 1 and 2.\n•\nD is a block diagonal matrix with 1-by-1 and 2-by-2 diagonal blocks D(k).\n•\nP(k) is a permutation matrix as defined by ipiv(k).\n•\nL(k) is a unit lower triangular matrix, such that if the diagonal block D(k) is of order s (s = 1 or 2), then\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n494\n\n\nIf s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k).\nIf s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k), and A(k+1,k+1), and v overwrites A(k\n+2:n,k:k+1).\nSee Also\nmkl_progress\nMatrix Storage Schemes\n?sytrf_aa\nComputes the factorization of a symmetric matrix\nusing Aasen's algorithm.\nlapack_int LAPACKE_ssytrf_aa (int matrix_layout, char uplo, lapack_int n, float * A,\nlapack_int lda, lapack_int * ipiv);\nlapack_int LAPACKE_dsytrf_aa (int matrix_layout, char uplo, lapack_int n, double * A,\nlapack_int lda, lapack_int * ipiv);\nlapack_int LAPACKE_csytrf_aa (int matrix_layout, char uplo, lapack_int n,\nlapack_complex_float * A, lapack_int lda, lapack_int * ipiv);\nlapack_int LAPACKE_zsytrf_aa (int matrix_layout, char uplo, lapack_int n,\nlapack_complex_double * A, lapack_int lda, lapack_int * ipiv);\nDescription\n?sytrf_aa computes the factorization of a symmetric matrix A using Aasen's algorithm. The form of the\nfactorization is A = U*T*UT or A = L*T*LT where U (or L) is a product of permutation and unit upper (lower)\ntriangular matrices, and T is a complex symmetric tridiagonal matrix.\nThis is the blocked version of the algorithm, calling Level 3 BLAS.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nuplo\n•\n= 'U': The upper triangle of A is stored.\n•\n= 'L': The lower triangle of A is stored.\nn\nThe order of the matrix A. n ≥ 0.\nA\nArray of size max(1, lda*n). The array A contains either the upper or the\nlower triangular part of the matrix A (see uplo).\nlda\nThe leading dimension of the array A.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n495\n\n\nOutput Parameters\nA\nOn exit, the tridiagonal matrix is stored in the diagonals and the\nsubdiagonals of A just below (or above) the diagonals, and L is stored below\n(or above) the subdiagonals, when uplo is 'L' (or 'U').\nipiv\nArray of size n. On exit, it contains the details of the interchanges; that is,\nthe row and column k of A were interchanged with the row and column\nipiv(k).\nReturn Values\nThis function returns a value info.\n= 0: Successful exit.\n< 0: If info = -i, the ith argument had an illegal value.\n> 0: If info = i, D(i,i) is exactly zero. The factorization has been completed, but the block diagonal matrix D\nis exactly singular, and division by zero will occur if it is used to solve a system of equations.\n?sytrf_rook\nComputes the bounded Bunch-Kaufman factorization\nof a symmetric matrix.\nSyntax\nlapack_int LAPACKE_ssytrf_rook (int matrix_layout, char uplo, lapack_int n, float * a,\nlapack_int lda, lapack_int * ipiv);\nlapack_int LAPACKE_dsytrf_rook (int matrix_layout, char uplo, lapack_int n, double * a,\nlapack_int lda, lapack_int * ipiv);\nlapack_int LAPACKE_csytrf_rook (int matrix_layout, char uplo, lapack_int n,\nlapack_complex_float * a, lapack_int lda, lapack_int * ipiv);\nlapack_int LAPACKE_zsytrf_rook (int matrix_layout, char uplo, lapack_int n,\nlapack_complex_double * a, lapack_int lda, lapack_int * ipiv);\nInclude Files\n•\nmkl.h\nDescription\nThe routine computes the factorization of a real/complex symmetric matrix A using the bounded Bunch-\nKaufman (\"rook\") diagonal pivoting method. The form of the factorization is:\n \nif uplo='U', A = U*D*UT\n \nif uplo='L', A = L*D*LT,\nwhere A is the input matrix, U and L are products of permutation and triangular matrices with unit diagonal\n(upper triangular for U and lower triangular for L), and D is a symmetric block-diagonal matrix with 1-by-1\nand 2-by-2 diagonal blocks. U and L have 2-by-2 unit diagonal blocks corresponding to the 2-by-2 blocks of\nD.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n496\n\n\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout for array b is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\nuplo\nMust be 'U' or 'L'.\nIndicates whether the upper or lower triangular part of A is stored and\nhow A is factored:\nIf uplo = 'U', the array a stores the upper triangular part of the\nmatrix A, and A is factored as U*D*UT.\nIf uplo = 'L', the array a stores the lower triangular part of the\nmatrix A, and A is factored as L*D*LT.\nn\nThe order of matrix A; n≥ 0.\na\nArray, size lda*n. The array a contains either the upper or the lower\ntriangular part of the matrix A (see uplo).\nlda\nThe leading dimension of a; at least max(1, n).\nOutput Parameters\na\nThe upper or lower triangular part of a is overwritten by details of the\nblock-diagonal matrix D and the multipliers used to obtain the factor U\n(or L).\nipiv\nIf ipiv(k) > 0, then rows and columns k and ipiv(k) were\ninterchanged and Dk, k is a 1-by-1 diagonal block.\nIf uplo = 'U' and ipiv(k) < 0 and ipiv(k - 1) < 0, then rows\nand columns k and -ipiv(k) were interchanged, rows and columns k -\n1 and -ipiv(k - 1) were interchanged, and Dk-1:k, k-1:k is a 2-by-2\ndiagonal block.\nIf uplo = 'L' and ipiv(k) < 0 and ipiv(k + 1) < 0, then rows\nand columns k and -ipiv(k) were interchanged, rows and columns k +\n1 and -ipiv(k + 1) were interchanged, and Dk:k+1, k:k+1 is a 2-by-2\ndiagonal block.\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nIf info = i, Dii is 0. The factorization has been completed, but D is exactly singular. Division by 0 will occur\nif you use D for solving a system of linear equations.\nApplication Notes\nThe total number of floating-point operations is approximately (1/3)n3 for real flavors or (4/3)n3 for\ncomplex flavors.\nAfter calling this routine, you can call the following routines:\n?sytrs_rook\nto solve A*X = B\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n497\n\n\n?sycon_rook (Fortran only)\nto estimate the condition number of A\n?sytri_rook (Fortran only)\nto compute the inverse of A.\n \nIf uplo = 'U', then A = U*D*U', where\nU = P(n)*U(n)* ... *P(k)*U(k)*...,\nthat is, U is a product of terms P(k)*U(k), where\n•\nk decreases from n to 1 in steps of 1 and 2.\n•\nD is a block diagonal matrix with 1-by-1 and 2-by-2 diagonal blocks D(k).\n•\nP(k) is a permutation matrix as defined by ipiv[k-1].\n•\nU(k) is a unit upper triangular matrix, such that if the diagonal block D(k) is of order s (s = 1 or 2), then\nIf s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k).\nIf s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k) and A(k,k), and v overwrites A(1:k-2,k\n-1:k).\n \nIf uplo = 'L', then A = L*D*L', where\nL = P(1)*L(1)* ... *P(k)*L(k)*...,\nthat is, L is a product of terms P(k)*L(k), where\n•\nk increases from 1 to n in steps of 1 and 2.\n•\nD is a block diagonal matrix with 1-by-1 and 2-by-2 diagonal blocks D(k).\n•\nP(k) is a permutation matrix as defined by ipiv(k).\n•\nL(k) is a unit lower triangular matrix, such that if the diagonal block D(k) is of order s (s = 1 or 2), then\nIf s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k).\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n498\n\n\nIf s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k), and A(k+1,k+1), and v overwrites A(k\n+2:n,k:k+1).\nSee Also\nMatrix Storage Schemes\n?sytrf_rk\nComputes the factorization of a real or complex\nsymmetric indefinite matrix using the bounded Bunch-\nKaufman (rook) diagonal pivoting method (BLAS3\nblocked algorithm).\nlapack_int LAPACKE_ssytrf_rk (int matrix_layout, char uplo, lapack_int n, float * A,\nlapack_int lda, float * e, lapack_int * ipiv);\nlapack_int LAPACKE_dsytrf_rk (int matrix_layout, char uplo, lapack_int n, double * A,\nlapack_int lda, double * e, lapack_int * ipiv);\nlapack_int LAPACKE_csytrf_rk (int matrix_layout, char uplo, lapack_int n,\nlapack_complex_float * A, lapack_int lda, lapack_complex_float * e, lapack_int * ipiv);\nlapack_int LAPACKE_zsytrf_rk (int matrix_layout, char uplo, lapack_int n,\nlapack_complex_double * A, lapack_int lda, lapack_complex_double * e, lapack_int *\nipiv);\nDescription\n?sytrf_rk computes the factorization of a real or complex symmetric matrix A using the bounded Bunch-\nKaufman (rook) diagonal pivoting method: A= P*U*D*(UT)*(PT) or A = P*L*D*(LT)*(PT), where U (or L) is\nunit upper (or lower) triangular matrix, UT (or LT) is the transpose of U (or L), P is a permutation matrix, PT\nis the transpose of P, and D is symmetric and block diagonal with 1-by-1 and 2-by-2 diagonal blocks.\nThis is the blocked version of the algorithm, calling Level-3 BLAS.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nuplo\nSpecifies whether the upper or lower triangular part of the symmetric\nmatrix A is stored:\n•\n= 'U': Upper triangular\n•\n= 'L': Lower triangular\nn\nThe order of the matrix A. n ≥ 0.\nA\nArray of size max(1, lda*n). On entry, the symmetric matrix A. If uplo =\n'U', the leading n-by-n upper triangular part of A contains the upper\ntriangular part of the matrix A, and the strictly lower triangular part of A is\nnot referenced. If uplo = 'L', the leading n-by-n lower triangular part of A\ncontains the lower triangular part of the matrix A, and the strictly upper\ntriangular part of A is not referenced.\nlda\nThe leading dimension of the array A.\nOutput Parameters\nA\nOn exit, contains:\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n499\n\n\n•\nOnly diagonal elements of the symmetric block diagonal matrix D on the\ndiagonal of A; that is, D(k,k) = A(k,k); (superdiagonal (or subdiagonal)\nelements of D are stored on exit in array e).\n•\nIf uplo = 'U', factor U in the superdiagonal part of A. If uplo = 'L',\nfactor L in the subdiagonal part of A.\ne\nArray of size n. On exit, contains the superdiagonal (or subdiagonal)\nelements of the symmetric block diagonal matrix D with 1-by-1 or 2-by-2\ndiagonal blocks. If uplo = 'U', e(i) = D(i-1,i), i=2:N, and e(1) is set to 0.\nIf uplo = 'L', e(i) = D(i+1,i), i=1:N-1, and e(n) is set to 0.\nNOTE For 1-by-1 diagonal block D(k), where 1 ≤ k ≤ n, the\nelement e[k-1] is set to 0 in both the uplo = 'U' and uplo =\n'L' cases.\nipiv\nArray of size n.ipiv describes the permutation matrix P in the factorization\nof matrix A as follows: The absolute value of ipiv(k) represents the index of\nthe row and column that were interchanged with the kth row and column.\nThe value of uplo describes the order in which the interchanges were\napplied. Also, the sign of ipiv represents the block structure of the\nsymmetric block diagonal matrix D with 1-by-1 or 2-by-2 diagonal blocks,\nwhich correspond to 1 or 2 interchanges at each factorization step. If uplo\n= 'U' (in factorization order, k decreases from n to 1):\n1.\nA single positive entry ipiv(k) > 0 means that D(k,k) is a 1-by-1\ndiagonal block. If ipiv(k) != k, rows and columns k and ipiv(k) were\ninterchanged in the matrix A(1:N,1:N). If ipiv(k) = k, no interchange\noccurred.\n2.\nA pair of consecutive negative entries ipiv(k) < 0 and ipiv(k-1). < 0\nmeans that D(k-1:k,k-1:k) is a 2-by-2 diagonal block. (Note that\nnegative entries in ipiv appear only in pairs.)\n•\nIf -ipiv(k) != k, rows and columns k and -ipiv(k) were\ninterchanged in the matrix A(1:N,1:N). If -ipiv(k) = k, no\ninterchange occurred.\n•\nIf -ipiv(k-1) != k-1, rows and columns k-1 and -ipiv(k-1) were\ninterchanged in the matrix A(1:N,1:N). If -ipiv(k-1) = k-1, no\ninterchange occurred.\n3.\nIn both cases 1 and 2, always ABS( ipiv(k) ) ≤ k.\nNOTE Any entry ipiv(k) is always nonzero on output.\nIf uplo = 'L' (in factorization order, k increases from 1 to n):\n1.\nA single positive entry ipiv(k) > 0 means that D(k,k) is a 1-by-1\ndiagonal block. If ipiv(k) != k, rows and columns k and ipiv(k) were\ninterchanged in the matrix A(1:N,1:N). If ipiv(k) = k, no interchange\noccurred.\n2.\nA pair of consecutive negative entries ipiv(k) < 0 and ipiv(k+1) < 0\nmeans that D(k:k+1,k:k+1) is a 2-by-2 diagonal block. (Note that\nnegative entries in ipiv appear only in pairs.)\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n500\n\n\n•\nIf -ipiv(k) != k, rows and columns k and -ipiv(k) were\ninterchanged in the matrix A(1:N,1:N). If -ipiv(k) = k, no\ninterchange occurred.\n•\nIf -ipiv(k+1) != k+1, rows and columns k-1 and -ipiv(k-1) were\ninterchanged in the matrix A(1:N,1:N). If -ipiv(k+1) = k+1, no\ninterchange occurred.\n3.\nIn both cases 1 and 2, always ABS( ipiv(k) ) ≥ k.\nNOTE Any entry ipiv(k) is always nonzero on output.\nReturn Values\nThis function returns a value info.\n= 0: Successful exit.\n< 0: If info = -k, the kth argument had an illegal value.\n> 0: If info = k, the matrix A is singular. If uplo = 'U', column k in the upper triangular part of A contains\nall zeros. If uplo = 'L', column k in the lower triangular part of A contains all zeros. Therefore, D(k,k) is\nexactly zero, and superdiagonal elements of column k of U (or subdiagonal elements of column k of L) are all\nzeros. The factorization has been completed, but the block diagonal matrix D is exactly singular, and division\nby zero will occur if it is used to solve a system of equations.\n?hetrf\nComputes the Bunch-Kaufman factorization of a\ncomplex Hermitian matrix.\nSyntax\nlapack_int LAPACKE_chetrf (int matrix_layout , char uplo , lapack_int n ,\nlapack_complex_float * a , lapack_int lda , lapack_int * ipiv );\nlapack_int LAPACKE_zhetrf (int matrix_layout , char uplo , lapack_int n ,\nlapack_complex_double * a , lapack_int lda , lapack_int * ipiv );\nInclude Files\n•\nmkl.h\nDescription\nThe routine computes the factorization of a complex Hermitian matrix A using the Bunch-Kaufman diagonal\npivoting method:\n \nif uplo='U', A = U*D*UH\n \nif uplo='L', A = L*D*LH,\nwhere A is the input matrix, U and L are products of permutation and triangular matrices with unit diagonal\n(upper triangular for U and lower triangular for L), and D is a Hermitian block-diagonal matrix with 1-by-1\nand 2-by-2 diagonal blocks. U and L have 2-by-2 unit diagonal blocks corresponding to the 2-by-2 blocks of\nD.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n501\n\n\nNOTE\nThis routine supports the Progress Routine feature. See Progress Routine for details.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\nuplo\nMust be 'U' or 'L'.\nIndicates whether the upper or lower triangular part of A is stored and\nhow A is factored:\nIf uplo = 'U', the array a stores the upper triangular part of the\nmatrix A, and A is factored as U*D*UH.\nIf uplo = 'L', the array a stores the lower triangular part of the\nmatrix A, and A is factored as L*D*LH.\nn\nThe order of matrix A; n≥ 0.\na\nArray, size max(1, lda*n).\nThe array a contains the upper or the lower triangular part of the\nmatrix A (see uplo).\nlda\nThe leading dimension of a; at least max(1, n).\nOutput Parameters\na\nThe upper or lower triangular part of a is overwritten by details of the\nblock-diagonal matrix D and the multipliers used to obtain the factor U\n(or L).\nipiv\nArray, size at least max(1, n). Contains details of the interchanges\nand the block structure of D. If ipiv[i-1] = k >0, then dii is a 1-\nby-1 block, and the i-th row and column of A was interchanged with\nthe k-th row and column.\nIf uplo = 'U' and ipiv[i] =ipiv[i-1] = -m < 0, then D has a 2-by-2\nblock in rows/columns i and i+1, and i-th row and column of A was\ninterchanged with the m-th row and column.\nIf uplo = 'L' and ipiv[i] =ipiv[i-1] = -m < 0, then D has a 2-by-2\nblock in rows/columns i and i+1, and (i+1)-th row and column of A\nwas interchanged with the m-th row and column.\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info = -i, parameter i had an illegal value.\nIf info = i, dii is 0. The factorization has been completed, but D is exactly singular. Division by 0 will occur\nif you use D for solving a system of linear equations.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n502\n\n\nApplication Notes\nThis routine is suitable for Hermitian matrices that are not known to be positive-definite. If A is in fact\npositive-definite, the routine does not perform interchanges, and no 2-by-2 diagonal blocks occur in D.\nThe 2-by-2 unit diagonal blocks and the unit diagonal elements of U and L are not stored. The remaining\nelements of U and L are stored in the corresponding columns of the array a, but additional row interchanges\nare required to recover U or L explicitly (which is seldom necessary).\nIfipiv[i-1] = i for all i =1...n, then all off-diagonal elements of U (L) are stored explicitly in the\ncorresponding elements of the array a.\nIf uplo = 'U', the computed factors U and D are the exact factors of a perturbed matrix A + E, where\n|E| ≤c(n)εP|U||D||UT|PT\nc(n) is a modest linear function of n, and ε is the machine precision.\nA similar estimate holds for the computed L and D when uplo = 'L'.\nThe total number of floating-point operations is approximately (4/3)n3.\nAfter calling this routine, you can call the following routines:\n?hetrs\nto solve A*X = B\n?hecon\nto estimate the condition number of A\n?hetri\nto compute the inverse of A.\nSee Also\nmkl_progress\nMatrix Storage Schemes\n?hetrf_aa\nComputes the factorization of a complex hermitian\nmatrix using Aasen's algorithm.\nLAPACK_DECL lapack_int LAPACKE_chetrf_aa (int matrix_layout, char uplo, lapack_int n,\nlapack_complex_float * a, lapack_int lda, lapack_int * ipiv );\nLAPACK_DECL lapack_int LAPACKE_zhetrf_aa (int matrix_layout, char uplo, lapack_int n,\nlapack_complex_double * a, lapack_int lda, lapack_int * ipiv );\nDescription\n?hetrf_aa computes the factorization of a complex Hermitian matrix A using Aasen's algorithm. The form of\nthe factorization is A = U * T * UH or a = L*T*LH where U (or L) is a product of permutation and unit upper\n(lower) triangular matrices, and T is a Hermitian tridiagonal matrix. This is the blocked version of the\nalgorithm, calling Level 3 BLAS.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nuplo\n= 'U': Upper triangle of A is stored; = 'L': Lower triangle of a is stored.\nn\nThe order of the matrix A. n≥ 0.\na\nArray of size lda*n. On entry, the Hermitian matrix A.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n503\n\n\nIf uplo = 'U', the leading n-by-n upper triangular part of a contains the\nupper triangular part of the matrix A, and the strictly lower triangular part\nof a is not referenced.\nIf uplo = 'L', the leading n-by-n lower triangular part of a contains the\nlower triangular part of the matrix A, and the strictly upper triangular part\nof a is not referenced.\nlda\nThe leading dimension of the array a. lda≥ max(1,n).\nlwork\nSee Syntax - Workspace. The length of work. lwork≥ 2*n. For optimum\nperformance lwork≥n*(1 + nb), where nb is the optimal block size. If\nlwork = -1, then a workspace query is assumed; the routine only\ncalculates the optimal size of the work array, returns this value as the first\nentry of the work array, and no error message related to lwork is issued by\nxerbla.\nOutput Parameters\na\nOn exit, the tridiagonal matrix is stored in the diagonals and the\nsubdiagonals of a just below (or above) the diagonals, and L is stored below\n(or above) the subdiagonals, when uplo is 'L' (or 'U').\nipiv\narray, dimension (n) On exit, it contains the details of the interchanges: the\nrow and column k of a were interchanged with the row and column\nipiv[k].\nwork\nSee Syntax - Workspace. Array of size (max(1, lwork)). On exit, if info =\n0, work[0] returns the optimal lwork.\nReturn Values\nThis function returns a value info.\nIf info = 0: successful exit < 0: if info = -i, the i-th argument had an illegal value,\nIf info > 0: if info = i, Di, i is exactly zero. The factorization has been completed, but the block diagonal\nmatrix D is exactly singular, and division by zero will occur if it is used to solve a system of equations.\nSyntax - Workspace\nUse this interface if you want to explicitly provide the workspace array.\nLAPACK_DECL lapack_int LAPACKE_chetrf_aa_work (int matrix_layout, char uplo, lapack_int\nn, lapack_complex_float * a, lapack_int lda, lapack_int * ipiv, lapack_complex_float *\nwork, lapack_int lwork );\nLAPACK_DECL lapack_int LAPACKE_zhetrf_aa_work (int matrix_layout, char uplo, lapack_int\nn, lapack_complex_double * a, lapack_int lda, lapack_int * ipiv, lapack_complex_double\n* work, lapack_int lwork );\n?hetrf_rook\nComputes the bounded Bunch-Kaufman factorization\nof a complex Hermitian matrix.\nSyntax\nlapack_int LAPACKE_chetrf_rook (int matrix_layout, char uplo, lapack_int n,\nlapack_complex_float * a, lapack_int lda, lapack_int * ipiv);\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n504\n\n\nlapack_int LAPACKE_zhetrf_rook (int matrix_layout, char uplo, lapack_int n,\nlapack_complex_double * a, lapack_int lda, lapack_int * ipiv);\nInclude Files\n•\nmkl.h\nDescription\nThe routine computes the factorization of a complex Hermitian matrix A using the bounded Bunch-Kaufman\ndiagonal pivoting method:\n \nif uplo='U', A = U*D*UH\n \nif uplo='L', A = L*D*LH,\nwhere A is the input matrix, U (or L ) is a product of permutation and unit upper ( or lower) triangular\nmatrices, and D is a Hermitian block-diagonal matrix with 1-by-1 and 2-by-2 diagonal blocks.\nThis is the blocked version of the algorithm, calling Level 3 BLAS.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout for array b is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\nuplo\nMust be 'U' or 'L'.\nIndicates whether the upper or lower triangular part of A is stored:\nIf uplo = 'U', the array a stores the upper triangular part of the\nmatrix A.\nIf uplo = 'L', the array a stores the lower triangular part of the\nmatrix A.\nn\nThe order of matrix A; n≥ 0.\na\nArray a, size (lda*n)\nThe array a contains the upper or the lower triangular part of the\nmatrix A (see uplo).\nIf uplo = 'U', the leading n-by-n upper triangular part of a contains\nthe upper triangular part of the matrix A, and the strictly lower\ntriangular part of a is not referenced. If uplo = 'L', the leading n-by-n\nlower triangular part of a contains the lower triangular part of the\nmatrix A, and the strictly upper triangular part of a is not referenced.\nlda\nThe leading dimension of a; at least max(1, n).\nOutput Parameters\na\nThe block diagonal matrix D and the multipliers used to obtain the\nfactor U or L (see Application Notes for further details).\nipiv\n•\nIf uplo = 'U':\nIf ipiv(k) > 0, then rows and columns k and ipiv(k) were\ninterchanged and Dk, k is a 1-by-1 diagonal block.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n505\n\n\nIf ipiv(k) < 0 and ipiv(k - 1) < 0, then rows and columns k and\n-ipiv(k) were interchanged and rows and columns k - 1 and -\nipiv(k - 1) were interchanged, Dk - 1:k,k - 1:k is a 2-by-2 diagonal\nblock.\n•\nIf uplo = 'L':\nIf ipiv(k) > 0, then rows and columns k and ipiv(k) were\ninterchanged and Dk,k is a 1-by-1 diagonal block.\nIf ipiv(k) < 0 and ipiv(k + 1) < 0, then rows and columns k and\n-ipiv(k) were interchanged and rows and columns k + 1 and -\nipiv(k + 1) were interchanged, Dk:k + 1,k:k + 1 is a 2-by-2 diagonal\nblock.\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nIf info = i, Dii is exactly 0. The factorization has been completed, but the block diagonal matrix D is\nexactly singular, and division by 0 will occur if you use D for solving a system of linear equations.\nApplication Notes\nIf uplo = 'U', thenA = U*D*UH, where\nU = P(n)*U(n)* ... *P(k)U(k)* ...,\ni.e., U is a product of terms P(k)*U(k), where k decreases from n to 1 in steps of 1 or 2, and D is a block\ndiagonal matrix with 1-by-1 and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as defined by\nipiv(k), and U(k) is a unit upper triangular matrix, such that if the diagonal block D(k) is of order s (s = 1\nor 2), then\nU k =\nk −s s n −k\nk −s\ns\nn −k\nI\nv\n0\n0\nI\n0\n0\n0\nI\nIf s = 1, D(k) overwrites A(k,k), and v overwrites A(1:k-1,k).\nIf s = 2, the upper triangle of D(k) overwrites A(k-1,k-1), A(k-1,k), and A(k,k), and v overwrites\nA(1:k-2,k-1:k).\nIf uplo = 'L', then A = L*D*LH, where\nL = P(1)*L(1)* ... *P(k)*L(k)* ...,\ni.e., L is a product of terms P(k)*L(k), where k increases from 1 to n in steps of 1 or 2, and D is a block\ndiagonal matrix with 1-by-1 and 2-by-2 diagonal blocks D(k). P(k) is a permutation matrix as defined by\nipiv(k), and L(k) is a unit lower triangular matrix, such that if the diagonal block D(k) is of order s (s = 1 or\n2), then\nL k =\nk −1 s n −k −s + 1\nk −1\ns\nn −k −s + 1\nI\n0\n0\n0\nI\n0\n0\nv\nI\nIf s = 1, D(k) overwrites A(k,k), and v overwrites A(k+1:n,k).\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n506\n\n\nIf s = 2, the lower triangle of D(k) overwrites A(k,k), A(k+1,k), and A(k+1,k+1), and v overwrites A(k\n+2:n,k:k+1).\nSee Also\nmkl_progress\nMatrix Storage Schemes\n?hetrf_rk\nComputes the factorization of a complex Hermitian\nindefinite matrix using the bounded Bunch-Kaufman\n(rook) diagonal pivoting method (BLAS3 blocked\nalgorithm).\nlapack_int LAPACKE_chetrf_rk (int matrix_layout, char uplo, lapack_int n,\nlapack_complex_float * A, lapack_int lda, lapack_complex_float * e, lapack_int * ipiv);\nlapack_int LAPACKE_zhetrf_rk (int matrix_layout, char uplo, lapack_int n,\nlapack_complex_double * A, lapack_int lda, lapack_complex_double * e, lapack_int *\nipiv);\nDescription\n?hetrf_rk computes the factorization of a complex Hermitian matrix A using the bounded Bunch-Kaufman\n(rook) diagonal pivoting method: A = P*U*D*(UH)*(PT) or A = P*L*D*(LH)*(PT), where U (or L) is unit upper\n(or lower) triangular matrix, UH (or LH) is the conjugate of U (or L), P is a permutation matrix, PT is the\ntranspose of P, and D is Hermitian and block diagonal with 1-by-1 and 2-by-2 diagonal blocks.\nThis is the blocked version of the algorithm, calling Level 3 BLAS.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nuplo\nSpecifies whether the upper or lower triangular part of the Hermitian matrix\nA is stored:\n•\n= 'U': Upper triangular.\n•\n= 'L': Lower triangular.\nn\nThe order of the matrix A. n ≥ 0.\nA\nArray of size max(1, lda*n). On entry, the Hermitian matrix A. If uplo =\n'U': The leading n-by-n upper triangular part of A contains the upper\ntriangular part of the matrix A, and the strictly lower triangular part of A is\nnot referenced. If uplo = 'L': The leading n-by-n lower triangular part of A\ncontains the lower triangular part of the matrix A, and the strictly upper\ntriangular part of A is not referenced.\nlda\nThe leading dimension of the array A.\nOutput Parameters\nA\nOn exit, contains:\n•\nOnly diagonal elements of the Hermitian block diagonal matrix D on the\ndiagonal of A; that is, D(k,k) = A(k,k). Superdiagonal (or subdiagonal)\nelements of D are stored on exit in array e.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n507\n\n\n—and—\n•\nIf uplo = 'U', factor U in the superdiagonal part of A. If uplo = 'L',\nfactor L in the subdiagonal part of A.\ne\nArray of size n. On exit, contains the superdiagonal (or subdiagonal)\nelements of the Hermitian block diagonal matrix D with 1-by-1 or 2-by-2\ndiagonal blocks. If uplo = 'U', e(i) = D(i-1,i), i=2:N, and e(1) is set to 0.\nIf uplo = 'L', e(i) = D(i+1,i), i=1:N-1, and e(n) is set to 0.\nNOTE For 1-by-1 diagonal block D(k), where 1 ≤ k ≤ n, the\nelement e[k-1] is set to 0 in both the uplo = 'U' and uplo =\n'L' cases.\nipiv\nArray of size n. ipiv describes the permutation matrix P in the factorization\nof matrix A as follows: The absolute value of ipiv[k-1] represents the\nindex of row and column that were interchanged with the kth row and\ncolumn. The value of uplo describes the order in which the interchanges\nwere applied. Also, the sign of ipiv represents the block structure of the\nHermitian block diagonal matrix D with 1-by-1 or 2-by-2 diagonal blocks\nthat correspond to 1 or 2 interchanges at each factorization step. If uplo =\n'U' (in factorization order, k decreases from n to 1):\n1.\nA single positive entry ipiv(k) > 0 means that D(k,k) is a 1-by-1\ndiagonal block. If ipiv(k) != k, rows and columns k and ipiv(k) were\ninterchanged in the matrix A(1:N,1:N). If ipiv(k) = k, no interchange\noccurred.\n2.\nA pair of consecutive negative entries ipiv(k) < 0 and ipiv(k-1) < 0\nmeans that D(k-1:k,k-1:k) is a 2-by-2 diagonal block. (Note that\nnegative entries in ipiv appear only in pairs.)\n•\nIf -ipiv(k) != k, rows and columns k and -ipiv(k) were\ninterchanged in the matrix A(1:N,1:N). If -ipiv(k) = k, no\ninterchange occurred.\n•\nIf -ipiv(k-1) != k-1, rows and columns k-1 and -ipiv(k-1) were\ninterchanged in the matrix A(1:N,1:N). If -ipiv(k-1) = k-1, no\ninterchange occurred.\n3.\nIn both cases 1 and 2, always ABS( ipiv(k) ) ≤ k.\nNOTE Any entry ipiv(k) is always nonzero on output.\nIf uplo = 'L' (in factorization order, k increases from 1 to n):\n1.\nA single positive entry ipiv(k) > 0 means that D(k,k) is a 1-by-1\ndiagonal block. If ipiv(k) != k, rows and columns k and ipiv(k) were\ninterchanged in the matrix A(1:N,1:N). If ipiv(k) = k, no interchange\noccurred.\n2.\nA pair of consecutive negative entries ipiv(k) < 0 and ipiv(k+1) < 0\nmeans that D(k:k+1,k:k+1) is a 2-by-2 diagonal block. (Note that\nnegative entries in ipiv appear only in pairs.)\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n508\n\n\n•\nIf -ipiv(k) != k, rows and columns k and -ipiv(k) were\ninterchanged in the matrix A(1:N,1:N). If -ipiv(k) = k, no\ninterchange occurred.\n•\nIf -ipiv(k+1) != k+1, rows and columns k-1 and -ipiv(k-1) were\ninterchanged in the matrix A(1:N,1:N). If -ipiv(k+1) = k+1, no\ninterchange occurred.\n3.\nIn both cases 1 and 2, always ABS( ipiv(k) ) ≥ k.\nNOTE Any entry ipiv(k) is always nonzero on output.\nReturn Values\nThis function returns a value info.\n= 0: Successful exit.\n< 0: If info = -k, the kth argument had an illegal value.\n> 0: If info = k, the matrix A is singular. If uplo = 'U', the column k in the upper triangular part of A\ncontains all zeros. If uplo = 'L', the column k in the lower triangular part of A contains all zeros. Therefore\nD(k,k) is exactly zero, and superdiagonal elements of column k of U (or subdiagonal elements of column k of\nL ) are all zeros. The factorization has been completed, but the block diagonal matrix D is exactly singular,\nand division by zero will occur if it is used to solve a system of equations.\n?sptrf\nComputes the Bunch-Kaufman factorization of a\nsymmetric matrix using packed storage.\nSyntax\nlapack_int LAPACKE_ssptrf (int matrix_layout , char uplo , lapack_int n , float * ap ,\nlapack_int * ipiv );\nlapack_int LAPACKE_dsptrf (int matrix_layout , char uplo , lapack_int n , double * ap ,\nlapack_int * ipiv );\nlapack_int LAPACKE_csptrf (int matrix_layout , char uplo , lapack_int n ,\nlapack_complex_float * ap , lapack_int * ipiv );\nlapack_int LAPACKE_zsptrf (int matrix_layout , char uplo , lapack_int n ,\nlapack_complex_double * ap , lapack_int * ipiv );\nInclude Files\n•\nmkl.h\nDescription\nThe routine computes the factorization of a real/complex symmetric matrix A stored in the packed format\nusing the Bunch-Kaufman diagonal pivoting method. The form of the factorization is:\n \nif uplo='U', A = U*D*UT\n \nif uplo='L', A = L*D*LT,\nwhere U and L are products of permutation and triangular matrices with unit diagonal (upper triangular for U\nand lower triangular for L), and D is a symmetric block-diagonal matrix with 1-by-1 and 2-by-2 diagonal\nblocks. U and L have 2-by-2 unit diagonal blocks corresponding to the 2-by-2 blocks of D.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n509\n\n\nNOTE\nThis routine supports the Progress Routine feature. See Progress Function for details.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\nuplo\nMust be 'U' or 'L'.\nIndicates whether the upper or lower triangular part of A is packed in\nthe array ap and how A is factored:\nIf uplo = 'U', the array ap stores the upper triangular part of the\nmatrix A, and A is factored as U*D*UT.\nIf uplo = 'L', the array ap stores the lower triangular part of the\nmatrix A, and A is factored as L*D*LT.\nn\nThe order of matrix A; n≥ 0.\nap\nArray, size at least max(1, n(n+1)/2). The array ap contains the upper\nor the lower triangular part of the matrix A (as specified by uplo) in\npacked storage (see Matrix Storage Schemes).\nOutput Parameters\nap\nThe upper or lower triangle of A (as specified by uplo) is overwritten\nby details of the block-diagonal matrix D and the multipliers used to\nobtain the factor U (or L).\nipiv\nArray, size at least max(1, n). Contains details of the interchanges\nand the block structure of D. If ipiv[i-1] = k >0, then dii is a 1-\nby-1 block, and the i-th row and column of A was interchanged with\nthe k-th row and column.\nIf uplo = 'U' and ipiv[i] =ipiv[i-1] = -m < 0, then D has a 2-by-2\nblock in rows/columns i and i+1, and i-th row and column of A was\ninterchanged with the m-th row and column.\nIf uplo = 'L' and ipiv[i] =ipiv[i-1] = -m < 0, then D has a 2-by-2\nblock in rows/columns i and i+1, and (i+1)-th row and column of A\nwas interchanged with the m-th row and column.\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info = -i, parameter i had an illegal value.\nIf info = i, dii is 0. The factorization has been completed, but D is exactly singular. Division by 0 will occur\nif you use D for solving a system of linear equations.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n510\n\n\nApplication Notes\nThe 2-by-2 unit diagonal blocks and the unit diagonal elements of U and L are not stored. The remaining\nelements of U and L overwrite elements of the corresponding columns of the array ap, but additional row\ninterchanges are required to recover U or L explicitly (which is seldom necessary).\nIf ipiv(i) = i for all i = 1...n, then all off-diagonal elements of U (L) are stored explicitly in packed form.\nIf uplo = 'U', the computed factors U and D are the exact factors of a perturbed matrix A + E, where\n|E| ≤c(n)εP|U||D||UT|PT\nc(n) is a modest linear function of n, and ε is the machine precision. A similar estimate holds for the\ncomputed L and D when uplo = 'L'.\nThe total number of floating-point operations is approximately (1/3)n3 for real flavors or (4/3)n3 for\ncomplex flavors.\nAfter calling this routine, you can call the following routines:\n?sptrs\nto solve A*X = B\n?spcon\nto estimate the condition number of A\n?sptri\nto compute the inverse of A.\nSee Also\nmkl_progress\nMatrix Storage Schemes\n?hptrf\nComputes the Bunch-Kaufman factorization of a\ncomplex Hermitian matrix using packed storage.\nSyntax\nlapack_int LAPACKE_chptrf (int matrix_layout , char uplo , lapack_int n ,\nlapack_complex_float * ap , lapack_int * ipiv );\nlapack_int LAPACKE_zhptrf (int matrix_layout , char uplo , lapack_int n ,\nlapack_complex_double * ap , lapack_int * ipiv );\nInclude Files\n•\nmkl.h\nDescription\nThe routine computes the factorization of a complex Hermitian packed matrix A using the Bunch-Kaufman\ndiagonal pivoting method:\n \nif uplo='U', A = U*D*UH\n \nif uplo='L', A = L*D*LH,\nwhere A is the input matrix, U and L are products of permutation and triangular matrices with unit diagonal\n(upper triangular for U and lower triangular for L), and D is a Hermitian block-diagonal matrix with 1-by-1\nand 2-by-2 diagonal blocks. U and L have 2-by-2 unit diagonal blocks corresponding to the 2-by-2 blocks of\nD.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n511\n\n\nNOTE\nThis routine supports the Progress Routine feature. See Progress Function for details.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\nuplo\nMust be 'U' or 'L'.\nIndicates whether the upper or lower triangular part of A is packed\nand how A is factored:\nIf uplo = 'U', the array ap stores the upper triangular part of the\nmatrix A, and A is factored as U*D*UH.\nIf uplo = 'L', the array ap stores the lower triangular part of the\nmatrix A, and A is factored as L*D*LH.\nn\nThe order of matrix A; n≥ 0.\nap\nArray, size at least max(1, n(n+1)/2). The array ap contains the upper\nor the lower triangular part of the matrix A (as specified by uplo) in\npacked storage (see Matrix Storage Schemes).\nOutput Parameters\nap\nThe upper or lower triangle of A (as specified by uplo) is overwritten\nby details of the block-diagonal matrix D and the multipliers used to\nobtain the factor U (or L).\nipiv\nArray, size at least max(1, n). Contains details of the interchanges\nand the block structure of D. If ipiv[i-1] = k >0, then dii is a 1-\nby-1 block, and the i-th row and column of A was interchanged with\nthe k-th row and column.\nIf uplo = 'U' and ipiv[i] =ipiv[i-1] = -m < 0, then D has a 2-by-2\nblock in rows/columns i and i+1, and i-th row and column of A was\ninterchanged with the m-th row and column.\nIf uplo = 'L' and ipiv[i] =ipiv[i-1] = -m < 0, then D has a 2-by-2\nblock in rows/columns i and i+1, and (i+1)-th row and column of A\nwas interchanged with the m-th row and column.\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info = -i, parameter i had an illegal value.\nIf info = i, dii is 0. The factorization has been completed, but D is exactly singular. Division by 0 will occur\nif you use D for solving a system of linear equations.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n512\n\n\nApplication Notes\nThe 2-by-2 unit diagonal blocks and the unit diagonal elements of U and L are not stored. The remaining\nelements of U and L are stored in the array ap, but additional row interchanges are required to recover U or L\nexplicitly (which is seldom necessary).\nIf ipiv[i-1] = i for all i = 1...n, then all off-diagonal elements of U (L) are stored explicitly in the\ncorresponding elements of the array a.\nIf uplo = 'U', the computed factors U and D are the exact factors of a perturbed matrix A + E, where\n|E| ≤c(n)εP|U||D||UT|PT\nc(n) is a modest linear function of n, and ε is the machine precision.\nA similar estimate holds for the computed L and D when uplo = 'L'.\nThe total number of floating-point operations is approximately (4/3)n3.\nAfter calling this routine, you can call the following routines:\n?hptrs\nto solve A*X = B\n?hpcon\nto estimate the condition number of A\n?hptri\nto compute the inverse of A.\nSee Also\nmkl_progress\nMatrix Storage Schemes\nmkl_?spffrt2, mkl_?spffrtx\nComputes the partial LDLT factorization of a\nsymmetric matrix using packed storage.\nSyntax\nvoid mkl_sspffrt2 (float *ap , const MKL_INT *n , const MKL_INT *ncolm , float *work ,\nfloat *work2 );\nvoid mkl_dspffrt2 (double *ap , const MKL_INT *n , const MKL_INT *ncolm , double\n*work , double *work2 );\nvoid mkl_cspffrt2 (MKL_Complex8 *ap , const MKL_INT *n , const MKL_INT *ncolm ,\nMKL_Complex8 *work , MKL_Complex8 *work2 );\nvoid mkl_zspffrt2 (MKL_Complex16 *ap , const MKL_INT *n , const MKL_INT *ncolm ,\nMKL_Complex16 *work , MKL_Complex16 *work2 );\nvoid mkl_sspffrtx (float *ap , const MKL_INT *n , const MKL_INT *ncolm , float *work ,\nfloat *work2 );\nvoid mkl_dspffrtx (double *ap , const MKL_INT *n , const MKL_INT *ncolm , double\n*work , double *work2 );\nvoid mkl_cspffrtx (MKL_Complex8 *ap , const MKL_INT *n , const MKL_INT *ncolm ,\nMKL_Complex8 *work , MKL_Complex8 *work2 );\nvoid mkl_zspffrtx (MKL_Complex16 *ap , const MKL_INT *n , const MKL_INT *ncolm ,\nMKL_Complex16 *work , MKL_Complex16 *work2 );\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n513\n\n\nInclude Files\n•\nmkl.h\nDescription\nThe routine computes the partial factorization A = LDLT , where L is a lower triangular matrix and D is a\ndiagonal matrix.\nCaution\nThe routine assumes that the matrix A is factorizable. The routine does not perform pivoting\nand does not handle diagonal elements which are zero, which cause the routine to produce\nincorrect results without any indication.\nConsider the matrix A = a bT\nb C\n, where a is the element in the first row and first column of A, b is a column\nvector of size n - 1 containing the elements from the second through n-th column of A, C is the lower-right\nsquare submatrix of A, and I is the identity matrix.\nThe mkl_?spffrt2 routine performs ncolm successive factorizations of the form\nA = a bT\nb C\n= a 0\nb I\na−1\n0\n0\nC −ba−1bT\na bT\n0 I\n.\nThe mkl_?spffrtx routine performs ncolm successive factorizations of the form\nA = a bT\nb C\n=\n1\n0\nba−1 I\na\n0\n0 C −ba−1bT\n1 ba−1 T\n0\nI\n.\nThe approximate number of floating point operations performed by real flavors of these routines is\n(1/6)*ncolm*(2*ncolm2 - 6*ncolm*n + 3*ncolm + 6*n2 - 6*n + 7).\nThe approximate number of floating point operations performed by complex flavors of these routines is\n(1/3)*ncolm*(4*ncolm2 - 12*ncolm*n + 9*ncolm + 12*n2 - 18*n + 8).\nInput Parameters\nap\nArray, size at least max(1, n(n+1)/2). The array ap contains the lower\ntriangular part of the matrix A in packed storage (see Matrix Storage\nSchemes for uplo = 'L').\nn\nThe order of matrix A; n≥ 0.\nncolm\nThe number of columns to factor, ncolm≤n.\nwork, work2\nWorkspace arrays, size of each at least n.\nOutput Parameters\nap\nOverwritten by the factor L. The first ncolm diagonal elements of the\ninput matrix A are replaced with the diagonal elements of D. The\nsubdiagonal elements of the first ncolm columns are replaced with the\ncorresponding elements of L. The rest of the input array is updated as\nindicated in the Description section.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n514\n\n\nNOTE\nSpecifying ncolm = n results in complete factorization A =\nLDLT.\nSee Also\nmkl_progress\nMatrix Storage Schemes\nSolving Systems of Linear Equations: LAPACK Computational Routines\nThis section describes the LAPACK routines for solving systems of linear equations. Before calling most of\nthese routines, you need to factorize the matrix of your system of equations (see Routines for Matrix\nFactorization). However, the factorization is not necessary if your system of equations has a triangular\nmatrix.\n?getrs\nSolves a system of linear equations with an LU-\nfactored square coefficient matrix, with multiple right-\nhand sides.\nSyntax\nlapack_int LAPACKE_sgetrs (int matrix_layout , char trans , lapack_int n , lapack_int\nnrhs , const float * a , lapack_int lda , const lapack_int * ipiv , float * b ,\nlapack_int ldb );\nlapack_int LAPACKE_dgetrs (int matrix_layout , char trans , lapack_int n , lapack_int\nnrhs , const double * a , lapack_int lda , const lapack_int * ipiv , double * b ,\nlapack_int ldb );\nlapack_int LAPACKE_cgetrs (int matrix_layout , char trans , lapack_int n , lapack_int\nnrhs , const lapack_complex_float * a , lapack_int lda , const lapack_int * ipiv ,\nlapack_complex_float * b , lapack_int ldb );\nlapack_int LAPACKE_zgetrs (int matrix_layout , char trans , lapack_int n , lapack_int\nnrhs , const lapack_complex_double * a , lapack_int lda , const lapack_int * ipiv ,\nlapack_complex_double * b , lapack_int ldb );\nInclude Files\n•\nmkl.h\nDescription\nThe routine solves for X the following systems of linear equations:\nA*X = B\nif trans='N',\nAT*X = B\nif trans='T',\nAH*X = B\nif trans='C' (for complex matrices only).\nBefore calling this routine, you must call ?getrf to compute the LU factorization of A.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n515\n\n\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\ntrans\nMust be 'N' or 'T' or 'C'.\nIndicates the form of the equations:\nIf trans = 'N', then A*X = B is solved for X.\nIf trans = 'T', then AT*X = B is solved for X.\nIf trans = 'C', then AH*X = B is solved for X.\nn\nThe order of A; the number of rows in B(n≥ 0).\nnrhs\nThe number of right-hand sides; nrhs≥ 0.\na\nArray of size max(1, lda*n).\nThe array a contains LU factorization of matrix A resulting from the\ncall of ?getrf.\nb\nArray of size max(1,ldb*nrhs) for column major layout, and\nmax(1,ldb*n) for row major layout.\nThe array b contains the matrix B whose columns are the right-hand\nsides for the systems of equations.\nlda\nThe leading dimension of a; lda≥ max(1, n).\nldb\nThe leading dimension of b; ldb≥ max(1, n) for column major layout\nand ldb≥nrhs for row major layout.\nipiv\nArray, size at least max(1, n). The ipiv array, as returned by ?getrf.\nOutput Parameters\nb\nOverwritten by the solution matrix X.\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info = -i, parameter i had an illegal value.\nApplication Notes\nFor each right-hand side b, the computed solution is the exact solution of a perturbed system of equations (A\n+ E)x = b, where\n|E| ≤c(n)εP|L||U|\nc(n) is a modest linear function of n, and ε is the machine precision.\nIf x0 is the true solution, the computed solution x satisfies this error bound:\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n516\n\n\nwhere cond(A,x)= || |A-1||A| |x| ||∞ / ||x||∞≤ ||A-1||∞ ||A||∞ = κ∞(A).\nNote that cond(A,x) can be much smaller than κ∞(A); the condition number of AT and AH might or might\nnot be equal to κ∞(A).\nThe approximate number of floating-point operations for one right-hand side vector b is 2n2 for real flavors\nand 8n2 for complex flavors.\nTo estimate the condition number κ∞(A), call ?gecon.\nTo refine the solution and estimate the error, call ?gerfs.\nSee Also\nMatrix Storage Schemes\n?gbtrs\nSolves a system of linear equations with an LU-\nfactored band coefficient matrix, with multiple right-\nhand sides.\nSyntax\nlapack_int LAPACKE_sgbtrs (int matrix_layout , char trans , lapack_int n , lapack_int\nkl , lapack_int ku , lapack_int nrhs , const float * ab , lapack_int ldab , const\nlapack_int * ipiv , float * b , lapack_int ldb );\nlapack_int LAPACKE_dgbtrs (int matrix_layout , char trans , lapack_int n , lapack_int\nkl , lapack_int ku , lapack_int nrhs , const double * ab , lapack_int ldab , const\nlapack_int * ipiv , double * b , lapack_int ldb );\nlapack_int LAPACKE_cgbtrs (int matrix_layout , char trans , lapack_int n , lapack_int\nkl , lapack_int ku , lapack_int nrhs , const lapack_complex_float * ab , lapack_int\nldab , const lapack_int * ipiv , lapack_complex_float * b , lapack_int ldb );\nlapack_int LAPACKE_zgbtrs (int matrix_layout , char trans , lapack_int n , lapack_int\nkl , lapack_int ku , lapack_int nrhs , const lapack_complex_double * ab , lapack_int\nldab , const lapack_int * ipiv , lapack_complex_double * b , lapack_int ldb );\nInclude Files\n•\nmkl.h\nDescription\nThe routine solves for X the following systems of linear equations:\nA*X = B\nif trans='N',\nAT*X = B\nif trans='T',\nAH*X = B\nif trans='C' (for complex matrices only).\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n517\n\n\nHere A is an LU-factored general band matrix of order n with kl non-zero subdiagonals and ku nonzero\nsuperdiagonals. Before calling this routine, call ?gbtrf to compute the LU factorization of A.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\ntrans\nMust be 'N' or 'T' or 'C'.\nn\nThe order of A; the number of rows in B; n≥ 0.\nkl\nThe number of subdiagonals within the band of A; kl≥ 0.\nku\nThe number of superdiagonals within the band of A; ku≥ 0.\nnrhs\nThe number of right-hand sides; nrhs≥ 0.\nab\nArray ab size max(1, ldab*n)\nThe array ab contains elements of the LU factors of the matrix A as\nreturned by gbtrf.\nb\nArray b size max(1, ldb*nrhs) for column major layout and max(1,\nldb*n) for row major layout.\nThe array b contains the matrix B whose columns are the right-hand\nsides for the systems of equations.\nldab\nThe leading dimension of the array ab; ldab≥ 2*kl + ku +1.\nldb\nThe leading dimension of b; ldb≥ max(1, n) for column major layout\nand ldb≥nrhs for row major layout.\nipiv\nArray, size at least max(1, n). The ipiv array, as returned by ?gbtrf.\nOutput Parameters\nb\nOverwritten by the solution matrix X.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, parameter i had an illegal value.\nApplication Notes\nFor each right-hand side b, the computed solution is the exact solution of a perturbed system of equations (A\n+ E)x = b, where\n|E| ≤c(kl + ku + 1)εP|L||U|\nc(k) is a modest linear function of k, and ε is the machine precision.\nIf x0 is the true solution, the computed solution x satisfies this error bound:\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n518\n\n\nwhere cond(A,x)= || |A-1||A| |x| ||∞ / ||x||∞≤ ||A-1||∞ ||A||∞ = κ∞(A).\nNote that cond(A,x) can be much smaller than κ∞(A); the condition number of AT and AH might or might\nnot be equal to κ∞(A).\nThe approximate number of floating-point operations for one right-hand side vector is 2n(ku + 2kl) for real\nflavors. The number of operations for complex flavors is 4 times greater. All these estimates assume that kl\nand ku are much less than min(m,n).\nTo estimate the condition number κ∞(A), call ?gbcon.\nTo refine the solution and estimate the error, call ?gbrfs.\nSee Also\nMatrix Storage Schemes\n?gttrs\nSolves a system of linear equations with a tridiagonal\ncoefficient matrix using the LU factorization computed\nby ?gttrf.\nSyntax\nlapack_int LAPACKE_sgttrs (int matrix_layout , char trans , lapack_int n , lapack_int\nnrhs , const float * dl , const float * d , const float * du , const float * du2 ,\nconst lapack_int * ipiv , float * b , lapack_int ldb );\nlapack_int LAPACKE_dgttrs (int matrix_layout , char trans , lapack_int n , lapack_int\nnrhs , const double * dl , const double * d , const double * du , const double * du2 ,\nconst lapack_int * ipiv , double * b , lapack_int ldb );\nlapack_int LAPACKE_cgttrs (int matrix_layout , char trans , lapack_int n , lapack_int\nnrhs , const lapack_complex_float * dl , const lapack_complex_float * d , const\nlapack_complex_float * du , const lapack_complex_float * du2 , const lapack_int *\nipiv , lapack_complex_float * b , lapack_int ldb );\nlapack_int LAPACKE_zgttrs (int matrix_layout , char trans , lapack_int n , lapack_int\nnrhs , const lapack_complex_double * dl , const lapack_complex_double * d , const\nlapack_complex_double * du , const lapack_complex_double * du2 , const lapack_int *\nipiv , lapack_complex_double * b , lapack_int ldb );\nInclude Files\n•\nmkl.h\nDescription\nThe routine solves for X the following systems of linear equations with multiple right hand sides:\nA*X = B\nif trans='N',\nAT*X = B\nif trans='T',\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n519\n\n\nAH*X = B\nif trans='C' (for complex matrices only).\nBefore calling this routine, you must call ?gttrf to compute the LU factorization of A.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout for array b is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\ntrans\nMust be 'N' or 'T' or 'C'.\nIndicates the form of the equations:\nIf trans = 'N', then A*X = B is solved for X.\nIf trans = 'T', then AT*X = B is solved for X.\nIf trans = 'C', then AH*X = B is solved for X.\nn\nThe order of A; n≥ 0.\nnrhs\nThe number of right-hand sides, that is, the number of columns in B;\nnrhs≥ 0.\ndl,d,du,du2\nArrays: dl(n -1), d(n), du(n -1), du2(n -2).\nThe array dl contains the (n - 1) multipliers that define the matrix L\nfrom the LU factorization of A.\nThe array d contains the n diagonal elements of the upper triangular\nmatrix U from the LU factorization of A.\nThe array du contains the (n - 1) elements of the first superdiagonal\nof U.\nThe array du2 contains the (n - 2) elements of the second\nsuperdiagonal of U.\nb\nArray of size max(1, ldb*nrhs) for column major layout and max(1,\nn*ldb) for row major layout. Contains the matrix B whose columns\nare the right-hand sides for the systems of equations.\nldb\nThe leading dimension of b; ldb≥ max(1, n) for column major\nlayout and ldb≥nrhs for row major layout.\nipiv\nArray, size (n). The ipiv array, as returned by ?gttrf.\nOutput Parameters\nb\nOverwritten by the solution matrix X.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, parameter i had an illegal value.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n520\n\n\nApplication Notes\nFor each right-hand side b, the computed solution is the exact solution of a perturbed system of equations (A\n+ E)x = b, where\n|E| ≤c(n)εP|L||U|\nc(n) is a modest linear function of n, and ε is the machine precision.\nIf x0 is the true solution, the computed solution x satisfies this error bound:\nwhere cond(A,x)= || |A-1||A| |x| ||∞ / ||x||∞≤ ||A-1||∞ ||A||∞ = κ∞(A).\nNote that cond(A,x) can be much smaller than κ∞(A); the condition number of AT and AH might or might\nnot be equal to κ∞(A).\nThe approximate number of floating-point operations for one right-hand side vector b is 7n (including n\ndivisions) for real flavors and 34n (including 2n divisions) for complex flavors.\nTo estimate the condition number κ∞(A), call ?gtcon.\nTo refine the solution and estimate the error, call ?gtrfs.\nSee Also\nMatrix Storage Schemes\n?dttrsb\nSolves a system of linear equations with a diagonally\ndominant tridiagonal coefficient matrix using the LU\nfactorization computed by ?dttrfb.\nSyntax\nvoid sdttrsb (const char * trans, const MKL_INT * n, const MKL_INT * nrhs, const float\n* dl, const float * d, const float * du, float * b, const MKL_INT * ldb, MKL_INT *\ninfo );\nvoid ddttrsb (const char * trans, const MKL_INT * n, const MKL_INT * nrhs, const double\n* dl, const double * d, const double * du, double * b, const MKL_INT * ldb, MKL_INT *\ninfo );\nvoid cdttrsb (const char * trans, const MKL_INT * n, const MKL_INT * nrhs, const\nMKL_Complex8 * dl, const MKL_Complex8 * d, const MKL_Complex8 * du, MKL_Complex8 * b,\nconst MKL_INT * ldb, MKL_INT * info );\nvoid zdttrsb (const char * trans, const MKL_INT * n, const MKL_INT * nrhs, const\nMKL_Complex16 * dl, const MKL_Complex16 * d, const MKL_Complex16 * du, MKL_Complex16 *\nb, const MKL_INT * ldb, MKL_INT * info );\nInclude Files\n•\nmkl.h\nDescription\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n521\n\n\nThe ?dttrsb routine solves the following systems of linear equations with multiple right hand sides for X:\nA*X = B\nif trans='N',\nAT*X = B\nif trans='T',\nAH*X = B\nif trans='C' (for complex matrices only).\nBefore calling this routine, call ?dttrfb to compute the factorization of A.\nInput Parameters\ntrans\nMust be 'N' or 'T' or 'C'.\nIndicates the form of the equations solved for X:\nIf trans = 'N', then A*X = B.\nIf trans = 'T', then AT*X = B.\nIf trans = 'C', then AH*X = B.\nn\nThe order of A; n≥ 0.\nnrhs\nThe number of right-hand sides, that is, the number of columns in B;\nnrhs≥ 0.\ndl, d, du\nArrays: dl(n -1), d(n), du(n -1).\nThe array dl contains the (n - 1) multipliers that define the\nmatrices L1, L2 from the factorization of A.\nThe array d contains the n diagonal elements of the upper triangular\nmatrix U from the factorization of A.\nThe array du contains the (n - 1) elements of the superdiagonal of U.\nb\nArray of size max(1, ldb*nrhs). Contains the matrix B whose\ncolumns are the right-hand sides for the systems of equations.\nldb\nThe leading dimension of b; ldb≥ max(1, n).\nOutput Parameters\nb\nOverwritten by the solution matrix X.\ninfo\nIf info = 0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\n?potrs\nSolves a system of linear equations with a Cholesky-\nfactored symmetric (Hermitian) positive-definite\ncoefficient matrix.\nSyntax\nlapack_int LAPACKE_spotrs (int matrix_layout , char uplo , lapack_int n , lapack_int\nnrhs , const float * a , lapack_int lda , float * b , lapack_int ldb );\nlapack_int LAPACKE_dpotrs (int matrix_layout , char uplo , lapack_int n , lapack_int\nnrhs , const double * a , lapack_int lda , double * b , lapack_int ldb );\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n522\n\n\nlapack_int LAPACKE_cpotrs (int matrix_layout , char uplo , lapack_int n , lapack_int\nnrhs , const lapack_complex_float * a , lapack_int lda , lapack_complex_float * b ,\nlapack_int ldb );\nlapack_int LAPACKE_zpotrs (int matrix_layout , char uplo , lapack_int n , lapack_int\nnrhs , const lapack_complex_double * a , lapack_int lda , lapack_complex_double * b ,\nlapack_int ldb );\nInclude Files\n•\nmkl.h\nDescription\nThe routine solves for X the system of linear equations A*X = B with a symmetric positive-definite or, for\ncomplex data, Hermitian positive-definite matrix A, given the Cholesky factorization of A:\nA = UT*U for real data, A = UH*U for complex data\nif uplo='U'\nA = L*LT for real data, A = L*LH for complex data\nif uplo='L'\nwhere L is a lower triangular matrix and U is upper triangular. The system is solved with multiple right-hand\nsides stored in the columns of the matrix B.\nBefore calling this routine, you must call ?potrf to compute the Cholesky factorization of A.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\nuplo\nMust be 'U' or 'L'.\nIndicates how the input matrix A has been factored:\nIf uplo = 'U', U is stored, whereA = UT*U for real data, A = UH*U\nfor complex data.\nIf uplo = 'L', L is stored, whereA = L*LT for real data, A = L*LH for\ncomplex data.\nn\nThe order of matrix A; n≥ 0.\nnrhs\nThe number of right-hand sides (nrhs≥ 0).\na\nArray A of size at least max(1, lda*n)\nThe array a contains the factor U or L (see uplo) as returned by \npotrf. .\nlda\nThe leading dimension of a. lda≥ max(1, n).\nb\nThe array b contains the matrix B whose columns are the right-hand\nsides for the systems of equations. The size of b must be at least\nmax(1, ldb*nrhs) for column major layout and max(1, ldb*n) for\nrow major layout.\nldb\nThe leading dimension of b. ldb≥ max(1, n) for column major layout\nand ldb≥nrhs for row major layout.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n523\n\n\nOutput Parameters\nb\nOverwritten by the solution matrix X.\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info = -i, parameter i had an illegal value.\nApplication Notes\nIf uplo = 'U', the computed solution for each right-hand side b is the exact solution of a perturbed system\nof equations (A + E)x = b, where\n|E| ≤c(n)ε |UH||U|\nc(n) is a modest linear function of n, and ε is the machine precision.\nA similar estimate holds for uplo = 'L'. If x0 is the true solution, the computed solution x satisfies this\nerror bound:\nwhere cond(A,x)= || |A-1||A| |x| ||∞ / ||x||∞≤ ||A-1||∞ ||A||∞ = κ∞(A).\nNote that cond(A,x) can be much smaller than κ∞ (A). The approximate number of floating-point operations\nfor one right-hand side vector b is 2n2 for real flavors and 8n2 for complex flavors.\nTo estimate the condition number κ∞(A), call ?pocon.\nTo refine the solution and estimate the error, call ?porfs.\nSee Also\nMatrix Storage Schemes\n?pftrs\nSolves a system of linear equations with a Cholesky-\nfactored symmetric (Hermitian) positive-definite\ncoefficient matrix using the Rectangular Full Packed\n(RFP) format.\nSyntax\nlapack_int LAPACKE_spftrs (int matrix_layout , char transr , char uplo , lapack_int n ,\nlapack_int nrhs , const float * a , float * b , lapack_int ldb );\nlapack_int LAPACKE_dpftrs (int matrix_layout , char transr , char uplo , lapack_int n ,\nlapack_int nrhs , const double * a , double * b , lapack_int ldb );\nlapack_int LAPACKE_cpftrs (int matrix_layout , char transr , char uplo , lapack_int n ,\nlapack_int nrhs , const lapack_complex_float * a , lapack_complex_float * b ,\nlapack_int ldb );\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n524\n\n\nlapack_int LAPACKE_zpftrs (int matrix_layout , char transr , char uplo , lapack_int n ,\nlapack_int nrhs , const lapack_complex_double * a , lapack_complex_double * b ,\nlapack_int ldb );\nInclude Files\n•\nmkl.h\nDescription\nThe routine solves a system of linear equations A*X = B with a symmetric positive-definite or, for complex\ndata, Hermitian positive-definite matrix A using the Cholesky factorization of A:\nA = UT*U for real data, A = UH*U for complex data\nif uplo='U'\nA = L*LT for real data, A = L*LH for complex data\nif uplo='L'\nBefore calling ?pftrs, you must call ?pftrf to compute the Cholesky factorization of A. L stands for a lower\ntriangular matrix and U for an upper triangular matrix.\nThe matrix A is in the Rectangular Full Packed (RFP) format. For the description of the RFP format, see Matrix\nStorage Schemes.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\ntransr\nMust be 'N', 'T' (for real data) or 'C' (for complex data).\nIf transr = 'N', the untransposed factor of Ais stored in RFP format.\nIf transr = 'T', the transposed factor of Ais stored in RFP format.\nIf transr = 'C', the conjugate-transposed factor of Ais stored in RFP\nformat.\nuplo\nMust be 'U' or 'L'.\nIndicates how the input matrix A has been factored:\nIf uplo = 'U', U is stored, where A = UT*U for real data, A = UH*U\nfor complex data.\nIf uplo = 'L', L is stored, where A = L*LT for real data, A = L*LH for\ncomplex data\nn\nThe order of the matrix A; n≥ 0.\nnrhs\nThe number of right-hand sides, that is, the number of columns of the\nmatrix B; nrhs≥ 0.\na\nArray a of size max(1,n*(n + 1)/2).\nThe array a contains, in the RFP format, the factor U or L obtained by\nfactorization of matrix A.\nb\nThe array b of size max(1, ldb*nrhs) for column major layout and\nmax(1,ldb*n) for row major layout contains the matrix B whose\ncolumns are the right-hand sides for the systems of equations.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n525\n\n\nldb\nThe leading dimension of b; ldb≥ max(1, n) for column major layout\nand ldb≥nrhs for row major layout.\nOutput Parameters\nb\nThe solution matrix X.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, parameter i had an illegal value.\nSee Also\nMatrix Storage Schemes\n?pptrs\nSolves a system of linear equations with a packed\nCholesky-factored symmetric (Hermitian) positive-\ndefinite coefficient matrix.\nSyntax\nlapack_int LAPACKE_spptrs (int matrix_layout , char uplo , lapack_int n , lapack_int\nnrhs , const float * ap , float * b , lapack_int ldb );\nlapack_int LAPACKE_dpptrs (int matrix_layout , char uplo , lapack_int n , lapack_int\nnrhs , const double * ap , double * b , lapack_int ldb );\nlapack_int LAPACKE_cpptrs (int matrix_layout , char uplo , lapack_int n , lapack_int\nnrhs , const lapack_complex_float * ap , lapack_complex_float * b , lapack_int ldb );\nlapack_int LAPACKE_zpptrs (int matrix_layout , char uplo , lapack_int n , lapack_int\nnrhs , const lapack_complex_double * ap , lapack_complex_double * b , lapack_int ldb );\nInclude Files\n•\nmkl.h\nDescription\nThe routine solves for X the system of linear equations A*X = B with a packed symmetric positive-definite or,\nfor complex data, Hermitian positive-definite matrix A, given the Cholesky factorization of A:\nA = UT*U for real data, A = UH*U for complex data\nif uplo='U'\nA = L*LT for real data, A = L*LH for complex data\nif uplo='L'\nwhere L is a lower triangular matrix and U is upper triangular. The system is solved with multiple right-hand\nsides stored in the columns of the matrix B.\nBefore calling this routine, you must call ?pptrf to compute the Cholesky factorization of A.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\nuplo\nMust be 'U' or 'L'.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n526\n\n\nIndicates how the input matrix A has been factored:\nIf uplo = 'U', U is stored, where A = UT*U for real data, A = UH*U\nfor complex data.\nIf uplo = 'L', L is stored, where A = L*LT for real data, A = L*LH for\ncomplex data\nn\nThe order of matrix A; n≥ 0.\nnrhs\nThe number of right-hand sides (nrhs≥ 0).\nap, b\nThe size of ap must be at least max(1,n(n+1)/2).\nThe array ap contains the factor U or L, as specified by uplo, in\npacked storage (see Matrix Storage Schemes).\nb\nThe array b of size max(1, ldb*nrhs) for column major layout and\nmax(1, ldb*n) for row major layout contains the matrix B whose\ncolumns are the right-hand sides for the systems of equations.\nldb\nThe leading dimension of b; ldb≥ max(1, n) for column major layout\nand ldb≥nrhs for row major layout.\nOutput Parameters\nb\nOverwritten by the solution matrix X.\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info = -i, parameter i had an illegal value.\nApplication Notes\nIf uplo = 'U', the computed solution for each right-hand side b is the exact solution of a perturbed system\nof equations (A + E)x = b, where\n|E| ≤c(n)ε |UH||U|\nc(n) is a modest linear function of n, and ε is the machine precision.\nA similar estimate holds for uplo = 'L'.\nIf x0 is the true solution, the computed solution x satisfies this error bound:\nwhere cond(A,x)= || |A-1||A| |x| ||∞ / ||x||∞≤ ||A-1||∞ ||A||∞ = κ∞(A).\nNote that cond(A,x) can be much smaller than κ∞(A).\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n527\n\n\nThe approximate number of floating-point operations for one right-hand side vector b is 2n2 for real flavors\nand 8n2 for complex flavors.\nTo estimate the condition number κ∞(A), call ?ppcon.\nTo refine the solution and estimate the error, call ?pprfs.\nSee Also\nMatrix Storage Schemes\n?pbtrs\nSolves a system of linear equations with a Cholesky-\nfactored symmetric (Hermitian) positive-definite band\ncoefficient matrix.\nSyntax\nlapack_int LAPACKE_spbtrs (int matrix_layout , char uplo , lapack_int n , lapack_int\nkd , lapack_int nrhs , const float * ab , lapack_int ldab , float * b , lapack_int\nldb );\nlapack_int LAPACKE_dpbtrs (int matrix_layout , char uplo , lapack_int n , lapack_int\nkd , lapack_int nrhs , const double * ab , lapack_int ldab , double * b , lapack_int\nldb );\nlapack_int LAPACKE_cpbtrs (int matrix_layout , char uplo , lapack_int n , lapack_int\nkd , lapack_int nrhs , const lapack_complex_float * ab , lapack_int ldab ,\nlapack_complex_float * b , lapack_int ldb );\nlapack_int LAPACKE_zpbtrs (int matrix_layout , char uplo , lapack_int n , lapack_int\nkd , lapack_int nrhs , const lapack_complex_double * ab , lapack_int ldab ,\nlapack_complex_double * b , lapack_int ldb );\nInclude Files\n•\nmkl.h\nDescription\nThe routine solves for real data a system of linear equations A*X = B with a symmetric positive-definite or,\nfor complex data, Hermitian positive-definite band matrix A, given the Cholesky factorization of A:\nA = UT*U for real data, A = UH*U for complex data\nif uplo='U'\nA = L*LT for real data, A = L*LH for complex data\nif uplo='L'\nwhere L is a lower triangular matrix and U is upper triangular. The system is solved with multiple right-hand\nsides stored in the columns of the matrix B.\nBefore calling this routine, you must call ?pbtrf to compute the Cholesky factorization of A in the band\nstorage form.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\nuplo\nMust be 'U' or 'L'.\nIndicates how the input matrix A has been factored:\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n528\n\n\nIf uplo = 'U', U is stored in ab, where A = UT*U for real matrices\nand A = UH*U for complex matrices.\nIf uplo = 'L', L is stored in ab, where A = L*LT for real matrices and\nA = L*LH for complex matrices.\nn\nThe order of matrix A; n≥ 0.\nkd\nThe number of superdiagonals or subdiagonals in the matrix A; kd≥ 0.\nnrhs\nThe number of right-hand sides; nrhs≥ 0.\nab\nArray ab is of size max (1, ldab*n).\nThe array ab contains the Cholesky factor, as returned by the\nfactorization routine, in band storage form.\nThe array b contains the matrix B whose columns are the right-hand\nsides for the systems of equations.\nb\nThe array b contains the matrix B whose columns are the right-hand\nsides for the systems of equations.\nThe size of b is at least max(1, ldb*nrhs) for column major layout\nand max(1, ldb*n) for row major layout.\nldab\nThe leading dimension of the array ab; ldab≥kd +1.\nldb\nThe leading dimension of b; ldb≥ max(1, n) for column major layout\nand ldb≥nrhs for row major layout.\nOutput Parameters\nb\nOverwritten by the solution matrix X.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, parameter i had an illegal value.\nApplication Notes\nFor each right-hand side b, the computed solution is the exact solution of a perturbed system of equations (A\n+ E)x = b, where\n|E| ≤c(kd + 1)εP|UH||U| or |E| ≤c(kd + 1)εP|LH||L|\nc(k) is a modest linear function of k, and ε is the machine precision.\nIf x0 is the true solution, the computed solution x satisfies this error bound:\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n529\n\n\nwhere cond(A,x)= || |A-1||A| |x| ||∞ / ||x||∞≤ ||A-1||∞ ||A||∞ = κ∞(A).\nNote that cond(A,x) can be much smaller than κ∞(A).\nThe approximate number of floating-point operations for one right-hand side vector is 4n*kd for real flavors\nand 16n*kd for complex flavors.\nTo estimate the condition number κ∞(A), call ?pbcon.\nTo refine the solution and estimate the error, call ?pbrfs.\nSee Also\nMatrix Storage Schemes\n?pttrs\nSolves a system of linear equations with a symmetric\n(Hermitian) positive-definite tridiagonal coefficient\nmatrix using the factorization computed by ?pttrf.\nSyntax\nlapack_int LAPACKE_spttrs( int matrix_layout, lapack_int n, lapack_int nrhs, const\nfloat* d, const float* e, float* b, lapack_int ldb );\nlapack_int LAPACKE_dpttrs( int matrix_layout, lapack_int n, lapack_int nrhs, const\ndouble* d, const double* e, double* b, lapack_int ldb );\nlapack_int LAPACKE_cpttrs( int matrix_layout, char uplo, lapack_int n, lapack_int nrhs,\nconst float* d, const lapack_complex_float* e, lapack_complex_float* b, lapack_int\nldb );\nlapack_int LAPACKE_zpttrs( int matrix_layout, char uplo, lapack_int n, lapack_int nrhs,\nconst double* d, const lapack_complex_double* e, lapack_complex_double* b, lapack_int\nldb );\nInclude Files\n•\nmkl.h\nDescription\nThe routine solves for X a system of linear equations A*X = B with a symmetric (Hermitian) positive-definite\ntridiagonal matrix A. Before calling this routine, call ?pttrf to compute the L*D*LT or UT*D*Ufor real data\nand the L*D*LH or UH*D*Ufactorization of A for complex data.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\nuplo\nUsed for cpttrs/zpttrs only. Must be 'U' or 'L'.\nSpecifies whether the superdiagonal or the subdiagonal of the\ntridiagonal matrix A is stored and how A is factored:\nIf uplo = 'U', the array e stores the conjugated values of the\nsuperdiagonal of U, and A is factored as UH*D*U.\nIf uplo = 'L', the array e stores the subdiagonal of L, and A is\nfactored as L*D*LH.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n530\n\n\nn\nThe order of A; n≥ 0.\nnrhs\nThe number of right-hand sides, that is, the number of columns of the\nmatrix B; nrhs≥ 0.\nd\nArray, dimension (n). Contains the diagonal elements of the diagonal\nmatrix D from the factorization computed by ?pttrf.\ne\nArray e is of size (n -1).\nThe array e contains the (n - 1) sub-diagonal elements of the unit\nbidiagonal factor L or the conjugated values of the superdiagonal of U\nfrom the factorization computed by ?pttrf (see uplo).\ne, b\nThe array b contains the matrix B whose columns are the right-hand\nsides for the systems of equations.\nThe size of b is at least max(1, ldb*nrhs) for column major layout\nand max(1, ldb*n) for row major layout.\nldb\nThe leading dimension of b; ldb≥ max(1, n) for column major layout\nand ldb≥nrhs for row major layout.\nOutput Parameters\nb\nOverwritten by the solution matrix X.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, parameter i had an illegal value.\nSee Also\nMatrix Storage Schemes\n?sytrs\nSolves a system of linear equations with a UDUT- or\nLDLT-factored symmetric coefficient matrix.\nSyntax\nlapack_int LAPACKE_ssytrs (int matrix_layout , char uplo , lapack_int n , lapack_int\nnrhs , const float * a , lapack_int lda , const lapack_int * ipiv , float * b ,\nlapack_int ldb );\nlapack_int LAPACKE_dsytrs (int matrix_layout , char uplo , lapack_int n , lapack_int\nnrhs , const double * a , lapack_int lda , const lapack_int * ipiv , double * b ,\nlapack_int ldb );\nlapack_int LAPACKE_csytrs (int matrix_layout , char uplo , lapack_int n , lapack_int\nnrhs , const lapack_complex_float * a , lapack_int lda , const lapack_int * ipiv ,\nlapack_complex_float * b , lapack_int ldb );\nlapack_int LAPACKE_zsytrs (int matrix_layout , char uplo , lapack_int n , lapack_int\nnrhs , const lapack_complex_double * a , lapack_int lda , const lapack_int * ipiv ,\nlapack_complex_double * b , lapack_int ldb );\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n531\n\n\nInclude Files\n•\nmkl.h\nDescription\nThe routine solves for X the system of linear equations A*X = B with a symmetric matrix A, given the Bunch-\nKaufman factorization of A:\nif uplo='U',\nA = U*D*UT\nif uplo='L',\nA = L*D*LT,\nwhere U and L are upper and lower triangular matrices with unit diagonal and D is a symmetric block-\ndiagonal matrix. The system is solved with multiple right-hand sides stored in the columns of the matrix B.\nYou must supply to this routine the factor U (or L) and the array ipiv returned by the factorization\nroutine ?sytrf.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\nuplo\nMust be 'U' or 'L'.\nIndicates how the input matrix A has been factored:\nIf uplo = 'U', the array a stores the upper triangular factor U of the\nfactorization A = U*D*UT.\nIf uplo = 'L', the array a stores the lower triangular factor L of the\nfactorization A = L*D*LT.\nn\nThe order of matrix A; n≥ 0.\nnrhs\nThe number of right-hand sides; nrhs≥ 0.\nipiv\nArray, size at least max(1, n). The ipiv array, as returned by ?sytrf.\na\nThe array aof size max(1, lda*n) contains the factor U or L (see\nuplo). .\nb\nThe array b contains the matrix B whose columns are the right-hand\nsides for the system of equations.\nThe size of b is at least max(1, ldb*nrhs) for column major layout\nand max(1, ldb*n) for row major layout.\nlda\nThe leading dimension of a; lda≥ max(1, n).\nldb\nThe leading dimension of b; ldb≥ max(1, n) for column major layout\nand ldb≥nrhs for row major layout.\nOutput Parameters\nb\nOverwritten by the solution matrix X.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n532\n\n\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, parameter i had an illegal value.\nApplication Notes\nFor each right-hand side b, the computed solution is the exact solution of a perturbed system of equations (A\n+ E)x = b, where\n|E| ≤c(n)εP|U||D||UT|PT or |E| ≤c(n)εP|L||D||UT|PT\nc(n) is a modest linear function of n, and ε is the machine precision.\nIf x0 is the true solution, the computed solution x satisfies this error bound:\nwhere cond(A,x)= || |A-1||A| |x| ||∞ / ||x||∞≤ ||A-1||∞ ||A||∞ = κ∞(A).\nNote that cond(A,x) can be much smaller than κ∞(A).\nThe total number of floating-point operations for one right-hand side vector is approximately 2n2 for real\nflavors or 8n2 for complex flavors.\nTo estimate the condition number κ∞(A), call ?sycon.\nTo refine the solution and estimate the error, call ?syrfs.\nSee Also\nMatrix Storage Schemes\n?sytrs_aa\nSolves a system of linear equations A * X = B with a\nsymmetric matrix.\nlapack_int LAPACKE_ssytrs_aa (int matrix_layout, char uplo, lapack_int n, lapack_int\nnrhs, const float * A, lapack_int lda, const lapack_int * ipiv, float * B, lapack_int\nldb);\nlapack_int LAPACKE_dsytrs_aa (int matrix_layout, char uplo, lapack_int n, lapack_int\nnrhs, const double * A, lapack_int lda, const lapack_int * ipiv, double * B, lapack_int\nldb);\nlapack_int LAPACKE_csytrs_aa (int matrix_layout, char uplo, lapack_int n, lapack_int\nnrhs, const lapack_complex_float * A, lapack_int lda, const lapack_int * ipiv,\nlapack_complex_float * B, lapack_int ldb);\nlapack_int LAPACKE_zsytrs_aa (int matrix_layout, char uplo, lapack_int n, lapack_int\nnrhs, const lapack_complex_double * A, lapack_int lda, const lapack_int * ipiv,\nlapack_complex_double * B, lapack_int ldb);\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n533\n\n\nDescription\n?sytrs_aa solves a system of linear equations A * X = B with a symmetric matrix A using the factorization A\n= U*T*UT or A = L*T*LT computed by ?sytrf_aa.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nuplo\nSpecifies whether the details of the factorization are stored as an upper or\nlower triangular matrix.\n•\n= 'U': Upper triangular; the form is A = U*T*UT.\n•\n= 'L': Lower triangular; the form is A = L*T*LT.\nn\nThe order of the matrix A. n ≥ 0.\nnrhs\nThe number of right-hand sides; that is, the number of columns of the\nmatrix B. nrhs ≥ 0.\nA\nArray of size max(1, lda*n). Details of factors computed by ?sytrf_aa.\nlda\nThe leading dimension of the array A.\nipiv\nArray of size n. Details of the interchanges as computed by ?sytrf_aa.\nB\nArray of size max(1, ldb*nrhs). On entry, the right-hand side matrix B.\nldb\nThe leading dimension of the array B. ldb ≥ max(1, n) for column-major\nlayout and ldb ≥ nrhs for row-major layout.\nOutput Parameters\nB\nOn exit, the solution matrix X.\nReturn Values\nThis function returns a value info.\n= 0: Successful exit.\n< 0: If info = -i, the ith argument had an illegal value.\n?sytrs_rook\nSolves a system of linear equations with a UDU- or\nLDL-factored symmetric coefficient matrix.\nSyntax\nlapack_int LAPACKE_ssytrs_rook (int matrix_layout, char uplo, lapack_int n, lapack_int\nnrhs, const float * a, lapack_int lda, const lapack_int * ipiv, float * b, lapack_int\nldb);\nlapack_int LAPACKE_dsytrs_rook (int matrix_layout, char uplo, lapack_int n, lapack_int\nnrhs, const double * a, lapack_int lda, const lapack_int * ipiv, double * b, lapack_int\nldb);\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n534\n\n\nlapack_int LAPACKE_csytrs_rook (int matrix_layout, char uplo, lapack_int n, lapack_int\nnrhs, const lapack_complex_float * a, lapack_int lda, const lapack_int * ipiv,\nlapack_complex_float * b, lapack_int ldb);\nlapack_int LAPACKE_zsytrs_rook (int matrix_layout, char uplo, lapack_int n, lapack_int\nnrhs, const lapack_complex_double * a, lapack_int lda, const lapack_int * ipiv,\nlapack_complex_double * b, lapack_int ldb);\nInclude Files\n•\nmkl.h\nDescription\nThe routine solves a system of linear equations A*X = B with a symmetric matrix A, using the factorization A\n= U*D*UT or A = L*D*LT computed by ?sytrf_rook.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout for array b is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\nuplo\nMust be 'U' or 'L'.\nIndicates how the input matrix A has been factored:\nIf uplo = 'U', the factorization is of the form A = U*D*UT.\nIf uplo = 'L', the factorization is of the form A = L*D*LT.\nn\nThe order of matrix A; n≥ 0.\nnrhs\nThe number of right-hand sides; nrhs≥ 0.\nipiv\nArray, size at least max(1, n). The ipiv array, as returned\nby ?sytrf_rook.\na, b\nArrays: a, size (lda*n), b size (ldb*nrhs).\nThe array a contains the block diagonal matrix D and the multipliers\nused to obtain U or L as computed by ?sytrf_rook (see uplo).\nThe array b contains the matrix B whose columns are the right-hand\nsides for the system of equations.\nlda\nThe leading dimension of a; lda≥ max(1, n).\nldb\nThe leading dimension of b; ldb≥ max(1, n) for column major layout\nand ldb≥nrhs) for row major layout.\nOutput Parameters\nb\nOverwritten by the solution matrix X.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n535\n\n\nIf info = -i, the i-th parameter had an illegal value.\nApplication Notes\nThe total number of floating-point operations for one right-hand side vector is approximately 2n2 for real\nflavors or 8n2 for complex flavors.\nSee Also\nMatrix Storage Schemes\n?hetrs\nSolves a system of linear equations with a UDUT- or\nLDLT-factored Hermitian coefficient matrix.\nSyntax\nlapack_int LAPACKE_chetrs (int matrix_layout , char uplo , lapack_int n , lapack_int\nnrhs , const lapack_complex_float * a , lapack_int lda , const lapack_int * ipiv ,\nlapack_complex_float * b , lapack_int ldb );\nlapack_int LAPACKE_zhetrs (int matrix_layout , char uplo , lapack_int n , lapack_int\nnrhs , const lapack_complex_double * a , lapack_int lda , const lapack_int * ipiv ,\nlapack_complex_double * b , lapack_int ldb );\nInclude Files\n•\nmkl.h\nDescription\nThe routine solves for X the system of linear equations A*X = B with a Hermitian matrix A, given the Bunch-\nKaufman factorization of A:\nif uplo='U',\nA = U*D*UH\nif uplo='L',\nA = L*D*LH,\nwhere U and L are upper and lower triangular matrices with unit diagonal and D is a symmetric block-\ndiagonal matrix. The system is solved with multiple right-hand sides stored in the columns of the matrix B.\nYou must supply to this routine the factor U (or L) and the array ipiv returned by the factorization\nroutine ?hetrf.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\nuplo\nMust be 'U' or 'L'.\nIndicates how the input matrix A has been factored:\nIf uplo = 'U', the array a stores the upper triangular factor U of the\nfactorization A = U*D*UH.\nIf uplo = 'L', the array a stores the lower triangular factor L of the\nfactorization A = L*D*LH.\nn\nThe order of matrix A; n≥ 0.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n536\n\n\nnrhs\nThe number of right-hand sides; nrhs≥ 0.\nipiv\nArray, size at least max(1, n).\nThe ipiv array, as returned by ?hetrf.\na\nThe array aof size max(1, lda*n) contains the factor U or L (see\nuplo).\nb\nThe array b contains the matrix B whose columns are the right-hand\nsides for the system of equations.\nThe size of b is at least max(1, ldb*nrhs) for column major layout\nand max(1, ldb*n) for row major layout.\nlda\nThe leading dimension of a; lda≥ max(1, n).\nldb\nThe leading dimension of b; ldb≥ max(1, n) for column major layout\nand ldb≥nrhs for row major layout.\nOutput Parameters\nb\nOverwritten by the solution matrix X.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, parameter i had an illegal value.\nApplication Notes\nFor each right-hand side b, the computed solution is the exact solution of a perturbed system of equations (A\n+ E)x = b, where\n|E| ≤c(n)εP|U||D||UH|PT or |E| ≤c(n)εP|L||D||LH|PT\nc(n) is a modest linear function of n, and ε is the machine precision.\nIf x0 is the true solution, the computed solution x satisfies this error bound:\nwhere cond(A,x)= || |A-1||A| |x| ||∞ / ||x||∞≤ ||A-1||∞ ||A||∞ = κ∞(A).\nNote that cond(A,x) can be much smaller than κ∞(A).\nThe total number of floating-point operations for one right-hand side vector is approximately 8n2.\nTo estimate the condition number κ∞(A), call ?hecon.\nTo refine the solution and estimate the error, call ?herfs.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n537\n\n\nSee Also\nMatrix Storage Schemes\n?hetrs_aa\nBSolves a system of linear equations A*X = with a\ncomplex Hermitian matrix.\nLAPACK_DECL lapack_int LAPACKE_chetrs_aa (int matrix_layout, char uplo, lapack_int n,\nlapack_int nrhs, const lapack_complex_float * a, lapack_int lda, const lapack_int *\nipiv, lapack_complex_float * b, lapack_int ldb );\nLAPACK_DECL lapack_int LAPACKE_zhetrs_aa (int matrix_layout, char uplo, lapack_int n,\nlapack_int nrhs, const lapack_complex_double * a, lapack_int lda, const lapack_int *\nipiv, lapack_complex_double * b, lapack_int ldb );\nDescription\n?hetrs_aa solves a system of linear equations A*X = X with a complex Hermitian matrix A using the\nfactorization A = U * T * UH or A = L * T * LH computed by ?hetrf_aa.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nuplo\nSpecifies whether the details of the factorization are stored as an upper or\nlower triangular matrix.\nIf uplo = 'U': Upper triangular of the form A = U * T * UH.\nIf uplo= 'L': Lower triangular of the form A = L * T * LH.\nn\nThe order of the matrix A. n≥ 0.\nnrhs\nThe number of right hand sides: the number of columns of the matrix b.\nnrhs≥ 0.\na\nArray of size lda*n. Details of factors computed by ?hetrf_aa.\nlda\nThe leading dimension of the array a. lda≥ max(1,n).\nipiv\nArray of size (n). Details of the interchanges as computed by ?hetrf_aa.\nb\nArray of size ldb*nrhs. On entry, the right hand side matrix B.\nldb\nThe leading dimension of the array b. ldb≥ max(1, n).\nwork\nSee Syntax - Workspace. Array of size (max(1, lwork)).\nlwork\nSee Syntax - Workspace. lwork≥ max(1, 3*n-2).\nOutput Parameters\nb\nOn exit, the solution matrix X.\nReturn Values\nThis function returns a value info.\nIf info = 0: successful exit.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n538\n\n\nIf info < 0: if info = -i, the i-th argument had an illegal value.\nSyntax - Workspace\nUse this interface if you want to explicitly provide the workspace array.\nLAPACK_DECL lapack_int LAPACKE_chetrs_aa_work (int matrix_layout, char uplo, lapack_int\nn, lapack_int nrhs, const lapack_complex_float * a, lapack_int lda, const lapack_int *\nipiv, lapack_complex_float * b, lapack_int ldb, lapack_complex_float * work, lapack_int\nlwork );\nLAPACK_DECL lapack_int LAPACKE_zhetrs_aa_work (int matrix_layout, char uplo, lapack_int\nn, lapack_int nrhs, const lapack_complex_double * a, lapack_int lda, const lapack_int *\nipiv, lapack_complex_double * b, lapack_int ldb, lapack_complex_double * work,\nlapack_int lwork );\n?hetrs_rook\nSolves a system of linear equations with a UDU- or\nLDL-factored Hermitian coefficient matrix.\nSyntax\nlapack_int LAPACKE_chetrs_rook (int matrix_layout, char uplo, lapack_int n, lapack_int\nnrhs, const lapack_complex_float * a, lapack_int lda, const lapack_int * ipiv,\nlapack_complex_float * b, lapack_int ldb);\nlapack_int LAPACKE_zhetrs_rook (int matrix_layout, char uplo, lapack_int n, lapack_int\nnrhs, const lapack_complex_double * a, lapack_int lda, const lapack_int * ipiv,\nlapack_complex_double * b, lapack_int ldb);\nInclude Files\n•\nmkl.h\nDescription\nThe routine solves for a system of linear equations A*X = B with a complex Hermitian matrix A using the\nfactorization A = U*D*UH or A = L*D*LH computed by ?hetrf_rook.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout for array b is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\nuplo\nMust be 'U' or 'L'.\nIndicates how the input matrix A has been factored:\nIf uplo = 'U', the factorization is of the form A = U*D*UH.\nIf uplo = 'L', the factorization is of the form A = L*D*LH.\nn\nThe order of matrix A; n≥ 0.\nnrhs\nThe number of right-hand sides; nrhs≥ 0.\nipiv\nArray, size at least max(1, n).\nThe ipiv array, as returned by ?hetrf_rook.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n539\n\n\na, b\nArrays: a (lda*n)), b(ldb*nrhs).\nThe array a contains the block diagonal matrix D and the multipliers\nused to obtain the factor U or L as computed by ?hetrf_rook (see\nuplo).\nThe array b contains the matrix B whose columns are the right-hand\nsides for the system of equations.\nlda\nThe leading dimension of a; lda≥ max(1, n).\nldb\nThe leading dimension of b; ldb≥ max(1, n) for column major layout\nand ldb≥nrhs) for row major layout.\nOutput Parameters\nb\nOverwritten by the solution matrix X.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\n?sytrs2\nSolves a system of linear equations with a UDU- or\nLDL-factored symmetric coefficient matrix.\nSyntax\nlapack_int LAPACKE_ssytrs2 (int matrix_layout , char uplo , lapack_int n , lapack_int\nnrhs , const float * a , lapack_int lda , const lapack_int * ipiv , float * b ,\nlapack_int ldb );\nlapack_int LAPACKE_dsytrs2 (int matrix_layout , char uplo , lapack_int n , lapack_int\nnrhs , const double * a , lapack_int lda , const lapack_int * ipiv , double * b ,\nlapack_int ldb );\nlapack_int LAPACKE_csytrs2 (int matrix_layout , char uplo , lapack_int n , lapack_int\nnrhs , const lapack_complex_float * a , lapack_int lda , const lapack_int * ipiv ,\nlapack_complex_float * b , lapack_int ldb );\nlapack_int LAPACKE_zsytrs2 (int matrix_layout , char uplo , lapack_int n , lapack_int\nnrhs , const lapack_complex_double * a , lapack_int lda , const lapack_int * ipiv ,\nlapack_complex_double * b , lapack_int ldb );\nInclude Files\n•\nmkl.h\nDescription\nThe routine solves a system of linear equations A*X = B with a symmetric matrix A using the factorization of\nA:\nif uplo='U',\nA = U*D*UT\nif uplo='L',\nA = L*D*LT\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n540\n\n\nwhere\n•\nU and L are upper and lower triangular matrices with unit diagonal\n•\nD is a symmetric block-diagonal matrix.\nThe factorization is computed by ?sytrf.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\nuplo\nMust be 'U' or 'L'.\nIndicates how the input matrix A has been factored:\nIf uplo = 'U', the array a stores the upper triangular factor U of the\nfactorization A = U*D*UT.\nIf uplo = 'L', the array a stores the lower triangular factor L of the\nfactorization A = L*D*LT.\nn\nThe order of matrix A; n≥ 0.\nnrhs\nThe number of right-hand sides; nrhs≥ 0.\na\nThe array aof size max(1, lda*n) contains the block diagonal matrix D\nand the multipliers used to obtain the factor U or L as computed\nby ?sytrf.\nb\nThe array b contains the right-hand side matrix B.\nThe size of b is at least max(1, ldb*nrhs) for column major layout\nand max(1, ldb*n) for row major layout.\nlda\nThe leading dimension of a; lda≥ max(1, n).\nldb\nThe leading dimension of b; ldb≥ max(1, n) for column major layout\nand ldb≥nrhs for row major layout.\nipiv\nArray of size n. The ipiv array contains details of the interchanges\nand the block structure of D as determined by ?sytrf.\nOutput Parameters\nb\nOverwritten by the solution matrix X.\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info = -i, parameter i had an illegal value.\nSee Also\n?sytrf\nMatrix Storage Schemes\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n541\n\n\n?hetrs2\nSolves a system of linear equations with a UDU- or\nLDL-factored Hermitian coefficient matrix.\nSyntax\nlapack_int LAPACKE_chetrs2 (int matrix_layout , char uplo , lapack_int n , lapack_int\nnrhs , const lapack_complex_float * a , lapack_int lda , const lapack_int * ipiv ,\nlapack_complex_float * b , lapack_int ldb );\nlapack_int LAPACKE_zhetrs2 (int matrix_layout , char uplo , lapack_int n , lapack_int\nnrhs , const lapack_complex_double * a , lapack_int lda , const lapack_int * ipiv ,\nlapack_complex_double * b , lapack_int ldb );\nInclude Files\n•\nmkl.h\nDescription\nThe routine solves a system of linear equations A*X = B with a complex Hermitian matrix A using the\nfactorization of A:\nif uplo='U',\nA = U*D*UH\nif uplo='L',\nA = L*D*LH\nwhere\n•\nU and L are upper and lower triangular matrices with unit diagonal\n•\nD is a Hermitian block-diagonal matrix.\nThe factorization is computed by ?hetrf.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\nuplo\nMust be 'U' or 'L'.\nIndicates how the input matrix A has been factored:\nIf uplo = 'U', the array a stores the upper triangular factor U of the\nfactorization A = U*D*UH.\nIf uplo = 'L', the array a stores the lower triangular factor L of the\nfactorization A = L*D*LH.\nn\nThe order of matrix A; n≥ 0.\nnrhs\nThe number of right-hand sides; nrhs≥ 0.\na\nThe array a of size max(1, lda*n) contains the block diagonal matrix\nD and the multipliers used to obtain the factor U or L as computed\nby ?hetrf.\nb\nThe array b of size max(1, ldb*nrhs) for column major layout and\nmax(1, ldb*n) for row major layout contains the right-hand side\nmatrix B.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n542\n\n\nlda\nThe leading dimension of a; lda≥ max(1, n).\nldb\nThe leading dimension of b; ldb≥ max(1, n) for column major layout\nand ldb≥nrhs for row major layout.\nipiv\nArray of size n. The ipiv array contains details of the interchanges\nand the block structure of D as determined by ?hetrf.\nOutput Parameters\nb\nOverwritten by the solution matrix X.\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info = -i, parameter i had an illegal value.\nSee Also\n?hetrf\nMatrix Storage Schemes\n?sytrs_3\nSolves a system of linear equations A * X = B with a\nreal or complex symmetric matrix.\nlapack_int LAPACKE_ssytrs_3 (int matrix_layout, char uplo, lapack_int n, lapack_int\nnrhs, const float * A, lapack_int lda, const float * e, const lapack_int * ipiv, float\n* B, lapack_int ldb);\nlapack_int LAPACKE_dsytrs_3 (int matrix_layout, char uplo, lapack_int n, lapack_int\nnrhs, const double * A, lapack_int lda, const double * e, const lapack_int * ipiv,\ndouble * B, lapack_int ldb);\nlapack_int LAPACKE_csytrs_3 (int matrix_layout, char uplo, lapack_int n, lapack_int\nnrhs, const lapack_complex_float * A, lapack_int lda, const lapack_complex_float * e,\nconst lapack_int * ipiv, lapack_complex_float * B, lapack_int ldb);\nlapack_int LAPACKE_zsytrs_3 (int matrix_layout, char uplo, lapack_int n, lapack_int\nnrhs, const lapack_complex_double * A, lapack_int lda, const lapack_complex_double * e,\nconst lapack_int * ipiv, lapack_complex_double * B, lapack_int ldb);\nDescription\n?sytrs_3 solves a system of linear equations A * X = B with a real or complex symmetric matrix A using the\nfactorization computed by ?sytrf_rk: A = P*U*D*(UT)*(PT) or A = P*L*D*(LT)*(PT), where U (or L) is unit\nupper (or lower) triangular matrix, UT (or LT) is the transpose of U (or L), P is a permutation matrix, PT is the\ntranspose of P, and D is a symmetric and block diagonal with 1-by-1 and 2-by-2 diagonal blocks.\nThis algorithm uses Level 3 BLAS.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nuplo\nSpecifies whether the details of the factorization are stored as an upper or\nlower triangular matrix:\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n543\n\n\n•\n= 'U': Upper triangular; the form is A= P*U*D*(UT)*(PT).\n•\n= 'L': Lower triangular; the form is A = P*L*D*(LT)*(PT).\nn\nThe order of the matrix A. n ≥ 0.\nnrhs\nThe number of right-hand sides; that is, the number of columns of the\nmatrix B. nrhs ≥ 0.\nA\nArray of size max(1, lda*n). Diagonal of the block diagonal matrix D and\nfactors U or L as computed by ?sytrf_rk:\n•\nOnly diagonal elements of the symmetric block diagonal matrix D on the\ndiagonal of A; that is, D(k,k) = A(k,k). Superdiagonal (or subdiagonal)\nelements of D should be provided on entry in array e.\n—and—\n•\nIf uplo = 'U', factor U in the superdiagonal part of A. If uplo = 'L',\nfactor L in the subdiagonal part of A.\nlda\nThe leading dimension of the array A.\ne\nArray of size n. On entry, contains the superdiagonal (or subdiagonal)\nelements of the symmetric block diagonal matrix D with 1-by-1 or 2-by-2\ndiagonal blocks. If uplo = 'U', e(i) = D(i-1,i),i=2:N, and e(1) is not\nreferenced. If uplo = 'L', e(i) = D(i+1,i), i=1:N-1, and e(n) is not\nreferenced.\nNOTE For 1-by-1 diagonal block D(k), where 1 ≤ k ≤ n, the\nelement e[k-1] is not referenced in both the uplo = 'U' and\nuplo = 'L' cases.\nipiv\nArray of size n. Details of the interchanges and the block structure of D as\ndetermined by ?sytrf_rk.\nB\nOn entry, the right-hand side matrix B.\nThe size of B is at least max(1, ldb*nrhs) for column-major layout and\nmax(1, ldb*n) for row-major layout.\nldb\nThe leading dimension of the array B. ldb ≥ max(1, n) for column-major\nlayout and ldb ≥ nrhs for row-major layout.\nOutput Parameters\nB\nOn exit, the solution matrix X.\nReturn Values\nThis function returns a value info.\n= 0: Successful exit.\n< 0: If info = -i, the ith argument had an illegal value.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n544\n\n\n?hetrs_3\nSolves a system of linear equations A * X = B with a\ncomplex Hermitian matrix using the factorization\ncomputed by ?hetrf_rk.\nlapack_int LAPACKE_chetrs_3 (int matrix_layout, char uplo, lapack_int n, lapack_int\nnrhs, const lapack_complex_float * A, lapack_int lda, const lapack_complex_float * e,\nconst lapack_int * ipiv, lapack_complex_float * B, lapack_int ldb);\nlapack_int LAPACKE_zhetrs_3 (int matrix_layout, char uplo, lapack_int n, lapack_int\nnrhs, const lapack_complex_double * A, lapack_int lda, const lapack_complex_double * e,\nconst lapack_int * ipiv, lapack_complex_double * B, lapack_int ldb);\nDescription\n?hetrs_3 solves a system of linear equations A * X = B with a complex Hermitian matrix A using the\nfactorization computed by ?hetrf_rk: A = P*U*D*(UH)*(PT) or A = P*L*D*(LH)*(PT), where U (or L) is unit\nupper (or lower) triangular matrix, UH (or LH) is the conjugate of U (or L), P is a permutation matrix, PT is the\ntranspose of P, and D is a Hermitian and block diagonal with 1-by-1 and 2-by-2 diagonal blocks.\nThis algorithm uses Level 3 BLAS.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nuplo\nSpecifies whether the details of the factorization are stored as an upper or\nlower triangular matrix:\n•\n= 'U': Upper triangular; form is A = P*U*D*(UH)*(PT).\n•\n= 'L': Lower triangular; form is A = P*L*D*(LH)*(PT).\nn\nThe order of the matrix A. n ≥ 0.\nnrhs\nThe number of right-hand sides; that is, the number of columns in the\nmatrix B. nrhs ≥ 0.\nA\nArray of size max(1, lda*n). Diagonal of the block diagonal matrix D and\nfactor U or L as computed by ?hetrf_rk:\n•\nOnly diagonal elements of the Hermitian block diagonal matrix D on the\ndiagonal of A; that is, D(k,k) = A(k,k). Superdiagonal (or subdiagonal)\nelements of D should be provided on entry in array e.\n•\nIf uplo = 'U', factor U in the superdiagonal part of A. If uplo = 'L',\nfactor L in the subdiagonal part of A.\nlda\nThe leading dimension of the array A.\ne\nArray of size n. On entry, contains the superdiagonal (or subdiagonal)\nelements of the Hermitian block diagonal matrix D with 1-by-1 or 2-by-2\ndiagonal blocks. If uplo = 'U', e(i) = D(i-1,i),i=2:N, and e(1) is not\nreferenced. If uplo = 'L', e(i) = D(i+1,i),i=1:N-1, and e(n) is not\nreferenced.\nNOTE For 1-by-1 diagonal block D(k), where 1 ≤ k ≤ n, the\nelement e[k-1] is not referenced in both the uplo = 'U' and\nuplo = 'L' cases.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n545\n\n\nipiv\nArray of size (n. Details of the interchanges and the block structure of D as\ndetermined by ?hetrf_rk.\nB\nOn entry, the right-hand side matrix B.\nThe size of B is at least max(1, ldb*nrhs) for column-major layout and\nmax(1, ldb*n) for row-major layout.\nldb\nThe leading dimension of the array B. ldb ≥ max(1, n) for column-major\nlayout and ldb ≥ nrhs for row-major layout.\nOutput Parameters\nB\nOn exit, the solution matrix X.\nReturn Values\nThis function returns a value info.\n= 0: Successful exit.\n< 0: If info = -i, the ith argument had an illegal value.\n?sptrs\nSolves a system of linear equations with a UDU- or\nLDL-factored symmetric coefficient matrix using\npacked storage.\nSyntax\nlapack_int LAPACKE_ssptrs (int matrix_layout , char uplo , lapack_int n , lapack_int\nnrhs , const float * ap , const lapack_int * ipiv , float * b , lapack_int ldb );\nlapack_int LAPACKE_dsptrs (int matrix_layout , char uplo , lapack_int n , lapack_int\nnrhs , const double * ap , const lapack_int * ipiv , double * b , lapack_int ldb );\nlapack_int LAPACKE_csptrs (int matrix_layout , char uplo , lapack_int n , lapack_int\nnrhs , const lapack_complex_float * ap , const lapack_int * ipiv , lapack_complex_float\n* b , lapack_int ldb );\nlapack_int LAPACKE_zsptrs (int matrix_layout , char uplo , lapack_int n , lapack_int\nnrhs , const lapack_complex_double * ap , const lapack_int * ipiv ,\nlapack_complex_double * b , lapack_int ldb );\nInclude Files\n•\nmkl.h\nDescription\nThe routine solves for X the system of linear equations A*X = B with a symmetric matrix A, given the Bunch-\nKaufman factorization of A:\nif uplo='U',\nA = U*D*UT\nif uplo='L',\nA = L*D*LT,\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n546\n\n\nwhere U and L are upper and lower packed triangular matrices with unit diagonal and D is a symmetric\nblock-diagonal matrix. The system is solved with multiple right-hand sides stored in the columns of the\nmatrix B. You must supply the factor U (or L) and the array ipiv returned by the factorization routine ?sptrf.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\nuplo\nMust be 'U' or 'L'.\nIndicates how the input matrix A has been factored:\nIf uplo = 'U', the array ap stores the packed factor U of the\nfactorization A = U*D*UT. If uplo = 'L', the array ap stores the\npacked factor L of the factorization A = L*D*LT.\nn\nThe order of matrix A; n≥ 0.\nnrhs\nThe number of right-hand sides; nrhs≥ 0.\nipiv\nArray, size at least max(1, n). The ipiv array, as returned by ?sptrf.\nap\nThe dimension of array ap must be at least max(1, n(n+1)/2). The\narray ap contains the factor U or L, as specified by uplo, in packed\nstorage (see Matrix Storage Schemes).\nb\nThe array b contains the matrix B whose columns are the right-hand\nsides for the system of equations. The size of b is max(1, ldb*nrhs)\nfor column major layout and max(1, ldb*n) for row major layout.\nldb\nThe leading dimension of b; ldb≥ max(1, n) for column major layout\nand ldb≥nrhs for row major layout.\nOutput Parameters\nb\nOverwritten by the solution matrix X.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, parameter i had an illegal value.\nApplication Notes\nFor each right-hand side b, the computed solution is the exact solution of a perturbed system of equations (A\n+ E)x = b, where\n|E| ≤c(n)εP|U||D||UT|PT or |E| ≤c(n)εP|L||D||LT|PT\nc(n) is a modest linear function of n, and ε is the machine precision.\nIf x0 is the true solution, the computed solution x satisfies this error bound:\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n547\n\n\nwhere cond(A,x)= || |A-1||A| |x| ||∞ / ||x||∞≤ ||A-1||∞ ||A||∞ = κ∞(A).\nNote that cond(A,x) can be much smaller than κ∞(A).\nThe total number of floating-point operations for one right-hand side vector is approximately 2n2 for real\nflavors or 8n2 for complex flavors.\nTo estimate the condition number κ∞(A), call ?spcon.\nTo refine the solution and estimate the error, call ?sprfs.\nSee Also\nMatrix Storage Schemes\n?hptrs\nSolves a system of linear equations with a UDU- or\nLDL-factored Hermitian coefficient matrix using\npacked storage.\nSyntax\nlapack_int LAPACKE_chptrs (int matrix_layout , char uplo , lapack_int n , lapack_int\nnrhs , const lapack_complex_float * ap , const lapack_int * ipiv , lapack_complex_float\n* b , lapack_int ldb );\nlapack_int LAPACKE_zhptrs (int matrix_layout , char uplo , lapack_int n , lapack_int\nnrhs , const lapack_complex_double * ap , const lapack_int * ipiv ,\nlapack_complex_double * b , lapack_int ldb );\nInclude Files\n•\nmkl.h\nDescription\nThe routine solves for X the system of linear equations A*X = B with a Hermitian matrix A, given the Bunch-\nKaufman factorization of A:\nif uplo='U',\nA = U*D*UH\nif uplo='L',\nA = L*D*LH,\nwhere U and L are upper and lower packed triangular matrices with unit diagonal and D is a symmetric\nblock-diagonal matrix. The system is solved with multiple right-hand sides stored in the columns of the\nmatrix B.\nYou must supply to this routine the arrays ap (containing U or L)and ipiv in the form returned by the\nfactorization routine ?hptrf.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n548\n\n\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\nuplo\nMust be 'U' or 'L'.\nIndicates how the input matrix A has been factored:\nIf uplo = 'U', the array ap stores the packed factor U of the\nfactorization A = U*D*UH. If uplo = 'L', the array ap stores the\npacked factor L of the factorization A = L*D*LH.\nn\nThe order of matrix A; n≥ 0.\nnrhs\nThe number of right-hand sides; nrhs≥ 0.\nipiv\nArray, size at least max(1, n). The ipiv array, as returned by ?hptrf.\nap\nThe dimension of array ap must be at least max(1,n(n+1)/2). The\narray ap contains the factor U or L, as specified by uplo, in packed\nstorage (see Matrix Storage Schemes).\nb\nThe array b contains the matrix B whose columns are the right-hand\nsides for the system of equations. The size of b is max(1, ldb*nrhs)\nfor column major layout and max(1, ldb*n) for row major layout.\nldb\nThe leading dimension of b; ldb≥ max(1, n) for column major layout\nand ldb≥nrhs for row major layout.\nOutput Parameters\nb\nOverwritten by the solution matrix X.\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info = -i, parameter i had an illegal value.\nApplication Notes\nFor each right-hand side b, the computed solution is the exact solution of a perturbed system of equations (A\n+ E)x = b, where\n|E| ≤c(n)εP|U||D||UH|PT or |E| ≤c(n)εP|L||D||LH|PT\nc(n) is a modest linear function of n, and ε is the machine precision.\nIf x0 is the true solution, the computed solution x satisfies this error bound:\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n549\n\n\nwhere cond(A,x)= || |A-1||A| |x| ||∞ / ||x||∞≤ ||A-1||∞ ||A||∞ = κ∞(A).\nNote that cond(A,x) can be much smaller than κ∞(A).\nThe total number of floating-point operations for one right-hand side vector is approximately 8n2 for complex\nflavors.\nTo estimate the condition number κ∞(A), call ?hpcon.\nTo refine the solution and estimate the error, call ?hprfs.\nSee Also\nMatrix Storage Schemes\n?trtrs\nSolves a system of linear equations with a triangular\ncoefficient matrix, with multiple right-hand sides.\nSyntax\nlapack_int LAPACKE_strtrs (int matrix_layout , char uplo , char trans , char diag ,\nlapack_int n , lapack_int nrhs , const float * a , lapack_int lda , float * b ,\nlapack_int ldb );\nlapack_int LAPACKE_dtrtrs (int matrix_layout , char uplo , char trans , char diag ,\nlapack_int n , lapack_int nrhs , const double * a , lapack_int lda , double * b ,\nlapack_int ldb );\nlapack_int LAPACKE_ctrtrs (int matrix_layout , char uplo , char trans , char diag ,\nlapack_int n , lapack_int nrhs , const lapack_complex_float * a , lapack_int lda ,\nlapack_complex_float * b , lapack_int ldb );\nlapack_int LAPACKE_ztrtrs (int matrix_layout , char uplo , char trans , char diag ,\nlapack_int n , lapack_int nrhs , const lapack_complex_double * a , lapack_int lda ,\nlapack_complex_double * b , lapack_int ldb );\nInclude Files\n•\nmkl.h\nDescription\nThe routine solves for X the following systems of linear equations with a triangular matrix A, with multiple\nright-hand sides stored in B:\nA*X = B\nif trans='N',\nAT*X = B\nif trans='T',\nAH*X = B\nif trans='C' (for complex matrices only).\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n550\n\n\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\nuplo\nMust be 'U' or 'L'.\nIndicates whether A is upper or lower triangular:\nIf uplo = 'U', then A is upper triangular.\nIf uplo = 'L', then A is lower triangular.\ntrans\nMust be 'N' or 'T' or 'C'.\nIf trans = 'N', then A*X = B is solved for X.\nIf trans = 'T', then AT*X = B is solved for X.\nIf trans = 'C', then AH*X = B is solved for X.\ndiag\nMust be 'N' or 'U'.\nIf diag = 'N', then A is not a unit triangular matrix.\nIf diag = 'U', then A is unit triangular: diagonal elements of A are\nassumed to be 1 and not referenced in the array a.\nn\nThe order of A; the number of rows in B; n≥ 0.\nnrhs\nThe number of right-hand sides; nrhs≥ 0.\na\nThe array a contains the matrix A.\nThe size of a is max(1, lda*n).\nb\nThe array b contains the matrix B whose columns are the right-hand\nsides for the systems of equations.\nThe size of b is max(1, ldb*nrhs) for column major layout and\nmax(1, ldb*n) for row major layout.\nlda\nThe leading dimension of a; lda≥ max(1, n).\nldb\nThe leading dimension of b; ldb≥ max(1, n) for column major layout\nand ldb≥nrhs for row major layout.\nOutput Parameters\nb\nOverwritten by the solution matrix X.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, parameter i had an illegal value.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n551\n\n\nApplication Notes\nFor each right-hand side b, the computed solution is the exact solution of a perturbed system of equations (A\n+ E)x = b, where\n|E| ≤c(n)ε |A|\nc(n) is a modest linear function of n, and ε is the machine precision. If x0 is the true solution, the computed\nsolution x satisfies this error bound:\nwhere cond(A,x)= || |A-1||A| |x| ||∞ / ||x||∞≤ ||A-1||∞ ||A||∞ = κ∞(A).\nNote that cond(A,x) can be much smaller than κ∞(A); the condition number of AT and AH might or might\nnot be equal to κ∞(A).\nThe approximate number of floating-point operations for one right-hand side vector b is n2 for real flavors\nand 4n2 for complex flavors.\nTo estimate the condition number κ∞(A), call ?trcon.\nTo estimate the error in the solution, call ?trrfs.\nSee Also\nMatrix Storage Schemes\n?tptrs\nSolves a system of linear equations with a packed\ntriangular coefficient matrix, with multiple right-hand\nsides.\nSyntax\nlapack_int LAPACKE_stptrs (int matrix_layout , char uplo , char trans , char diag ,\nlapack_int n , lapack_int nrhs , const float * ap , float * b , lapack_int ldb );\nlapack_int LAPACKE_dtptrs (int matrix_layout , char uplo , char trans , char diag ,\nlapack_int n , lapack_int nrhs , const double * ap , double * b , lapack_int ldb );\nlapack_int LAPACKE_ctptrs (int matrix_layout , char uplo , char trans , char diag ,\nlapack_int n , lapack_int nrhs , const lapack_complex_float * ap , lapack_complex_float\n* b , lapack_int ldb );\nlapack_int LAPACKE_ztptrs (int matrix_layout , char uplo , char trans , char diag ,\nlapack_int n , lapack_int nrhs , const lapack_complex_double * ap ,\nlapack_complex_double * b , lapack_int ldb );\nInclude Files\n•\nmkl.h\nDescription\nThe routine solves for X the following systems of linear equations with a packed triangular matrix A, with\nmultiple right-hand sides stored in B:\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n552\n\n\nA*X = B\nif trans='N',\nAT*X = B\nif trans='T',\nAH*X = B\nif trans='C' (for complex matrices only).\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\nuplo\nMust be 'U' or 'L'.\nIndicates whether A is upper or lower triangular:\nIf uplo = 'U', then A is upper triangular.\nIf uplo = 'L', then A is lower triangular.\ntrans\nMust be 'N' or 'T' or 'C'.\nIf trans = 'N', then A*X = B is solved for X.\nIf trans = 'T', then AT*X = B is solved for X.\nIf trans = 'C', then AH*X = B is solved for X.\ndiag\nMust be 'N' or 'U'.\nIf diag = 'N', then A is not a unit triangular matrix.\nIf diag = 'U', then A is unit triangular: diagonal elements are\nassumed to be 1 and not referenced in the array ap.\nn\nThe order of A; the number of rows in B; n≥ 0.\nnrhs\nThe number of right-hand sides; nrhs≥ 0.\nap\nThe dimension of arrayap must be at least max(1,n(n+1)/2). The\narray ap contains the matrix A in packed storage (see Matrix Storage\nSchemes).\nb\nThe array b contains the matrix B whose columns are the right-hand\nsides for the system of equations.\nThe size of b is max(1, ldb*nrhs) for column major layout and\nmax(1, ldb*n) for row major layout.\nldb\nThe leading dimension of b; ldb≥ max(1, n) for column major layout\nand ldb≥nrhs for row major layout.\nOutput Parameters\nb\nOverwritten by the solution matrix X.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, parameter i had an illegal value.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n553\n\n\nApplication Notes\nFor each right-hand side b, the computed solution is the exact solution of a perturbed system of equations (A\n+ E)x = b, where\n|E| ≤c(n)ε |A|\nc(n) is a modest linear function of n, and ε is the machine precision.\nIf x0 is the true solution, the computed solution x satisfies this error bound:\nwhere cond(A,x)= || |A-1||A| |x| ||∞ / ||x||∞≤ ||A-1||∞ ||A||∞ = κ∞(A).\nNote that cond(A,x) can be much smaller than κ∞(A); the condition number of AT and AH might or might\nnot be equal to κ∞(A).\nThe approximate number of floating-point operations for one right-hand side vector b is n2 for real flavors\nand 4n2 for complex flavors.\nTo estimate the condition number κ∞(A), call ?tpcon.\nTo estimate the error in the solution, call ?tprfs.\nSee Also\nMatrix Storage Schemes\n?tbtrs\nSolves a system of linear equations with a band\ntriangular coefficient matrix, with multiple right-hand\nsides.\nSyntax\nlapack_int LAPACKE_stbtrs (int matrix_layout , char uplo , char trans , char diag ,\nlapack_int n , lapack_int kd , lapack_int nrhs , const float * ab , lapack_int ldab ,\nfloat * b , lapack_int ldb );\nlapack_int LAPACKE_dtbtrs (int matrix_layout , char uplo , char trans , char diag ,\nlapack_int n , lapack_int kd , lapack_int nrhs , const double * ab , lapack_int ldab ,\ndouble * b , lapack_int ldb );\nlapack_int LAPACKE_ctbtrs (int matrix_layout , char uplo , char trans , char diag ,\nlapack_int n , lapack_int kd , lapack_int nrhs , const lapack_complex_float * ab ,\nlapack_int ldab , lapack_complex_float * b , lapack_int ldb );\nlapack_int LAPACKE_ztbtrs (int matrix_layout , char uplo , char trans , char diag ,\nlapack_int n , lapack_int kd , lapack_int nrhs , const lapack_complex_double * ab ,\nlapack_int ldab , lapack_complex_double * b , lapack_int ldb );\nInclude Files\n•\nmkl.h\nDescription\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n554\n\n\nThe routine solves for X the following systems of linear equations with a band triangular matrix A, with\nmultiple right-hand sides stored in B:\nA*X = B\nif trans='N',\nAT*X = B\nif trans='T',\nAH*X = B\nif trans='C' (for complex matrices only).\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\nuplo\nMust be 'U' or 'L'.\nIndicates whether A is upper or lower triangular:\nIf uplo = 'U', then A is upper triangular.\nIf uplo = 'L', then A is lower triangular.\ntrans\nMust be 'N' or 'T' or 'C'.\nIf trans = 'N', then A*X = B is solved for X.\nIf trans = 'T', then AT*X = B is solved for X.\nIf trans = 'C', then AH*X = B is solved for X.\ndiag\nMust be 'N' or 'U'.\nIf diag = 'N', then A is not a unit triangular matrix.\nIf diag = 'U', then A is unit triangular: diagonal elements are\nassumed to be 1 and not referenced in the array ab.\nn\nThe order of A; the number of rows in B; n≥ 0.\nkd\nThe number of superdiagonals or subdiagonals in the matrix A; kd≥ 0.\nnrhs\nThe number of right-hand sides; nrhs≥ 0.\nab\nThe array ab contains the matrix A in band storage form.\nThe size of ab must be max(1, ldab*n)\nb\nThe array b contains the matrix B whose columns are the right-hand\nsides for the systems of equations.\nThe size of b is max(1, ldb*nrhs) for column major layout and\nmax(1, ldb*n) for row major layout.\nldab\nThe leading dimension of ab; ldab≥kd + 1.\nldb\nThe leading dimension of b; ldb≥ max(1, n) for column major layout\nand ldb≥nrhs for row major layout.\nOutput Parameters\nb\nOverwritten by the solution matrix X.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n555\n\n\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, parameter i had an illegal value.\nApplication Notes\nFor each right-hand side b, the computed solution is the exact solution of a perturbed system of equations (A\n+ E)x = b, where\n|E|≤ c(n)ε|A|\nc(n) is a modest linear function of n, and ε is the machine precision. If x0 is the true solution, the computed\nsolution x satisfies this error bound:\nwhere cond(A,x)= || |A-1||A| |x| ||∞ / ||x||∞≤ ||A-1||∞ ||A||∞ = κ∞(A).\nNote that cond(A,x) can be much smaller than κ∞(A); the condition number of AT and AH might or might\nnot be equal to κ∞(A).\nThe approximate number of floating-point operations for one right-hand side vector b is 2n*kd for real\nflavors and 8n*kd for complex flavors.\nTo estimate the condition number κ∞(A), call ?tbcon.\nTo estimate the error in the solution, call ?tbrfs.\nSee Also\nMatrix Storage Schemes\nEstimating the Condition Number: LAPACK Computational Routines\nThis section describes the LAPACK routines for estimating the condition number of a matrix. The condition\nnumber is used for analyzing the errors in the solution of a system of linear equations (see Error Analysis).\nSince the condition number may be arbitrarily large when the matrix is nearly singular, the routines actually\ncompute the reciprocal condition number.\n?gecon\nEstimates the reciprocal of the condition number of a\ngeneral matrix in the 1-norm or the infinity-norm.\nSyntax\nlapack_int LAPACKE_sgecon( int matrix_layout, char norm, lapack_int n, const float* a,\nlapack_int lda, float anorm, float* rcond );\nlapack_int LAPACKE_dgecon( int matrix_layout, char norm, lapack_int n, const double* a,\nlapack_int lda, double anorm, double* rcond );\nlapack_int LAPACKE_cgecon( int matrix_layout, char norm, lapack_int n, const\nlapack_complex_float* a, lapack_int lda, float anorm, float* rcond );\nlapack_int LAPACKE_zgecon( int matrix_layout, char norm, lapack_int n, const\nlapack_complex_double* a, lapack_int lda, double anorm, double* rcond );\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n556\n\n\nInclude Files\n•\nmkl.h\nDescription\nThe routine estimates the reciprocal of the condition number of a general matrix A in the 1-norm or infinity-\nnorm:\nκ1(A) =||A||1||A-1||1 = κ∞(AT) = κ∞(AH)\nκ∞(A) =||A||∞||A-1||∞ = κ1(AT) = κ1(AH).\nAn estimate is obtained for ||A-1||, and the reciprocal of the condition number is computed as rcond =\n1 / (||A|| ||A-1||).\nBefore calling this routine:\n•\ncompute anorm (either ||A||1 = maxjΣi |aij| or ||A||∞ = maxiΣj |aij|)\n•\ncall ?getrf to compute the LU factorization of A.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\nnorm\nMust be '1' or 'O' or 'I'.\nIf norm = '1' or 'O', then the routine estimates the condition\nnumber of matrix A in 1-norm.\nIf norm = 'I', then the routine estimates the condition number of\nmatrix A in infinity-norm.\nn\nThe order of the matrix A; n≥ 0.\na\nThe array a contains the LU-factored matrix A, as returned\nby ?getrf.\nanorm\nThe norm of the original matrix A (see Description).\nlda\nThe leading dimension of a; lda≥ max(1, n).\nOutput Parameters\nrcond\nAn estimate of the reciprocal of the condition number. The routine sets\nrcond = 0 if the estimate underflows; in this case the matrix is\nsingular (to working precision). However, anytime rcond is small\ncompared to 1.0, for the working precision, the matrix may be poorly\nconditioned or even singular.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, parameter i had an illegal value.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n557\n\n\nApplication Notes\nThe computed rcond is never less than r (the reciprocal of the true condition number) and in practice is\nnearly always less than 10r. A call to this routine involves solving a number of systems of linear equations\nA*x = b or AH*x = b; the number is usually 4 or 5 and never more than 11. Each solution requires\napproximately 2*n2 floating-point operations for real flavors and 8*n2 for complex flavors.\nSee Also\nMatrix Storage Schemes\n?gbcon\nEstimates the reciprocal of the condition number of a\nband matrix in the 1-norm or the infinity-norm.\nSyntax\nlapack_int LAPACKE_sgbcon( int matrix_layout, char norm, lapack_int n, lapack_int kl,\nlapack_int ku, const float* ab, lapack_int ldab, const lapack_int* ipiv, float anorm,\nfloat* rcond );\nlapack_int LAPACKE_dgbcon( int matrix_layout, char norm, lapack_int n, lapack_int kl,\nlapack_int ku, const double* ab, lapack_int ldab, const lapack_int* ipiv, double anorm,\ndouble* rcond );\nlapack_int LAPACKE_cgbcon( int matrix_layout, char norm, lapack_int n, lapack_int kl,\nlapack_int ku, const lapack_complex_float* ab, lapack_int ldab, const lapack_int* ipiv,\nfloat anorm, float* rcond );\nlapack_int LAPACKE_zgbcon( int matrix_layout, char norm, lapack_int n, lapack_int kl,\nlapack_int ku, const lapack_complex_double* ab, lapack_int ldab, const lapack_int*\nipiv, double anorm, double* rcond );\nInclude Files\n•\nmkl.h\nDescription\nThe routine estimates the reciprocal of the condition number of a general band matrix A in the 1-norm or\ninfinity-norm:\nκ1(A) = ||A||1||A-1||1 = κ∞(AT) = κ∞(AH)\nκ∞(A) = ||A||∞||A-1||∞ = κ1(AT) = κ1(AH).\nAn estimate is obtained for ||A-1||, and the reciprocal of the condition number is computed as rcond =\n1 / (||A|| ||A-1||).\nBefore calling this routine:\n•\ncompute anorm (either ||A||1 = maxjΣi |aij| or ||A||∞ = maxiΣj |aij|)\n•\ncall ?gbtrf to compute the LU factorization of A.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\nnorm\nMust be '1' or 'O' or 'I'.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n558\n\n\nIf norm = '1' or 'O', then the routine estimates the condition\nnumber of matrix A in 1-norm.\nIf norm = 'I', then the routine estimates the condition number of\nmatrix A in infinity-norm.\nn\nThe order of the matrix A; n≥ 0.\nkl\nThe number of subdiagonals within the band of A; kl≥ 0.\nku\nThe number of superdiagonals within the band of A; ku≥ 0.\nldab\nThe leading dimension of the array ab. (ldab≥ 2*kl + ku +1).\nipiv\nArray, size at least max(1, n). The ipiv array, as returned by ?gbtrf.\nab\nThe array abof size max(1, ldab*n) contains the factored band matrix\nA, as returned by ?gbtrf.\nanorm\nThe norm of the original matrix A(see Description).\nOutput Parameters\nrcond\nAn estimate of the reciprocal of the condition number. The routine sets\nrcond =0 if the estimate underflows; in this case the matrix is singular\n(to working precision). However, anytime rcond is small compared to\n1.0, for the working precision, the matrix may be poorly conditioned\nor even singular.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, parameter i had an illegal value.\nApplication Notes\nThe computed rcond is never less than r (the reciprocal of the true condition number) and in practice is\nnearly always less than 10r. A call to this routine involves solving a number of systems of linear equations\nA*x = b or AH*x = b; the number is usually 4 or 5 and never more than 11. Each solution requires\napproximately 2n(ku + 2kl) floating-point operations for real flavors and 8n(ku + 2kl) for complex\nflavors.\nSee Also\nMatrix Storage Schemes\n?gtcon\nEstimates the reciprocal of the condition number of a\ntridiagonal matrix.\nSyntax\nlapack_int LAPACKE_sgtcon( char norm, lapack_int n, const float* dl, const float* d,\nconst float* du, const float* du2, const lapack_int* ipiv, float anorm, float* rcond );\nlapack_int LAPACKE_dgtcon( char norm, lapack_int n, const double* dl, const double* d,\nconst double* du, const double* du2, const lapack_int* ipiv, double anorm, double*\nrcond );\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n559\n\n\nlapack_int LAPACKE_cgtcon( char norm, lapack_int n, const lapack_complex_float* dl,\nconst lapack_complex_float* d, const lapack_complex_float* du, const\nlapack_complex_float* du2, const lapack_int* ipiv, float anorm, float* rcond );\nlapack_int LAPACKE_zgtcon( char norm, lapack_int n, const lapack_complex_double* dl,\nconst lapack_complex_double* d, const lapack_complex_double* du, const\nlapack_complex_double* du2, const lapack_int* ipiv, double anorm, double* rcond );\nInclude Files\n•\nmkl.h\nDescription\nThe routine estimates the reciprocal of the condition number of a real or complex tridiagonal matrix A in the\n1-norm or infinity-norm:\nκ1(A) = ||A||1||A-1||1\nκ∞(A) = ||A||∞||A-1||∞\nAn estimate is obtained for ||A-1||, and the reciprocal of the condition number is computed as rcond =\n1 / (||A|| ||A-1||).\nBefore calling this routine:\n•\ncompute anorm (either ||A||1 = maxjΣi |aij| or ||A||∞ = maxiΣj |aij|)\n•\ncall ?gttrf to compute the LU factorization of A.\nInput Parameters\nnorm\nMust be '1' or 'O' or 'I'.\nIf norm = '1' or 'O', then the routine estimates the condition\nnumber of matrix A in 1-norm.\nIf norm = 'I', then the routine estimates the condition number of\nmatrix A in infinity-norm.\nn\nThe order of the matrix A; n≥ 0.\ndl,d,du,du2\nArrays: dl(n -1), d(n), du(n -1), du2(n -2).\nThe array dl contains the (n - 1) multipliers that define the matrix L\nfrom the LU factorization of A as computed by ?gttrf.\nThe array d contains the n diagonal elements of the upper triangular\nmatrix U from the LU factorization of A.\nThe array du contains the (n - 1) elements of the first superdiagonal\nof U.\nThe array du2 contains the (n - 2) elements of the second\nsuperdiagonal of U.\nipiv\nArray, size (n). The array of pivot indices, as returned by ?gttrf.\nanorm\nThe norm of the original matrix A(see Description).\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n560\n\n\nOutput Parameters\nrcond\nAn estimate of the reciprocal of the condition number. The routine sets\nrcond=0 if the estimate underflows; in this case the matrix is singular\n(to working precision). However, anytime rcond is small compared to\n1.0, for the working precision, the matrix may be poorly conditioned\nor even singular.\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info = -i, parameter i had an illegal value.\nApplication Notes\nThe computed rcond is never less than r (the reciprocal of the true condition number) and in practice is\nnearly always less than 10r. A call to this routine involves solving a number of systems of linear equations\nA*x = b; the number is usually 4 or 5 and never more than 11. Each solution requires approximately 2n2\nfloating-point operations for real flavors and 8n2 for complex flavors.\n \n?pocon\nEstimates the reciprocal of the condition number of a\nsymmetric (Hermitian) positive-definite matrix.\nSyntax\nlapack_int LAPACKE_spocon( int matrix_layout, char uplo, lapack_int n, const float* a,\nlapack_int lda, float anorm, float* rcond );\nlapack_int LAPACKE_dpocon( int matrix_layout, char uplo, lapack_int n, const double* a,\nlapack_int lda, double anorm, double* rcond );\nlapack_int LAPACKE_cpocon( int matrix_layout, char uplo, lapack_int n, const\nlapack_complex_float* a, lapack_int lda, float anorm, float* rcond );\nlapack_int LAPACKE_zpocon( int matrix_layout, char uplo, lapack_int n, const\nlapack_complex_double* a, lapack_int lda, double anorm, double* rcond );\nInclude Files\n•\nmkl.h\nDescription\nThe routine estimates the reciprocal of the condition number of a symmetric (Hermitian) positive-definite\nmatrix A:\nκ1(A) = ||A||1 ||A-1||1 (since A is symmetric or Hermitian, κ∞(A) = κ1(A)).\nAn estimate is obtained for ||A-1||, and the reciprocal of the condition number is computed as rcond =\n1 / (||A|| ||A-1||).\nBefore calling this routine:\n•\ncompute anorm (either ||A||1 = maxjΣi |aij| or ||A||∞ = maxiΣj |aij|)\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n561\n\n\n•\ncall ?potrf to compute the Cholesky factorization of A.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\nuplo\nMust be 'U' or 'L'.\nIndicates how the input matrix A has been factored:\nIf uplo = 'U', A is factored as A = UT*U for real flavors or A = UH*U\nfor complex flavors, and U is stored.\nIf uplo = 'L', A is factored as A = L*LT for real flavors or A = L*LH\nfor complex flavors, and L is stored.\nn\nThe order of the matrix A; n≥ 0.\na\nThe array a of size max(1, lda*n) contains the factored matrix A, as\nreturned by ?potrf.\nlda\nThe leading dimension of a; lda≥ max(1, n).\nanorm\nThe norm of the original matrix A (see Description).\nOutput Parameters\nrcond\nAn estimate of the reciprocal of the condition number. The routine sets\nrcond =0 if the estimate underflows; in this case the matrix is singular\n(to working precision). However, anytime rcond is small compared to\n1.0, for the working precision, the matrix may be poorly conditioned\nor even singular.\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info = -i, parameter i had an illegal value.\nApplication Notes\nThe computed rcond is never less than r (the reciprocal of the true condition number) and in practice is\nnearly always less than 10r. A call to this routine involves solving a number of systems of linear equations\nA*x = b; the number is usually 4 or 5 and never more than 11. Each solution requires approximately 2n2\nfloating-point operations for real flavors and 8n2 for complex flavors.\nSee Also\nMatrix Storage Schemes\n?ppcon\nEstimates the reciprocal of the condition number of a\npacked symmetric (Hermitian) positive-definite\nmatrix.\nSyntax\nlapack_int LAPACKE_sppcon( int matrix_layout, char uplo, lapack_int n, const float* ap,\nfloat anorm, float* rcond );\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n562\n\n\nlapack_int LAPACKE_dppcon( int matrix_layout, char uplo, lapack_int n, const double*\nap, double anorm, double* rcond );\nlapack_int LAPACKE_cppcon( int matrix_layout, char uplo, lapack_int n, const\nlapack_complex_float* ap, float anorm, float* rcond );\nlapack_int LAPACKE_zppcon( int matrix_layout, char uplo, lapack_int n, const\nlapack_complex_double* ap, double anorm, double* rcond );\nInclude Files\n•\nmkl.h\nDescription\nThe routine estimates the reciprocal of the condition number of a packed symmetric (Hermitian) positive-\ndefinite matrix A:\nκ1(A) = ||A||1 ||A-1||1 (since A is symmetric or Hermitian, κ∞(A) = κ1(A)).\nAn estimate is obtained for ||A-1||, and the reciprocal of the condition number is computed as rcond =\n1 / (||A|| ||A-1||).\nBefore calling this routine:\n•\ncompute anorm (either ||A||1 = maxjΣi |aij| or ||A||∞ = maxiΣj |aij|)\n•\ncall ?pptrf to compute the Cholesky factorization of A.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\nuplo\nMust be 'U' or 'L'.\nIndicates how the input matrix A has been factored:\nIf uplo = 'U', A is factored as A = UT*U for real flavors or A = UH*U\nfor complex flavors, and U is stored.\nIf uplo = 'L', A is factored as A = L*LT for real flavors or A = L*LH\nfor complex flavors, and L is stored.\nn\nThe order of the matrix A; n≥ 0.\nap\nThe array ap contains the packed factored matrix A, as returned\nby ?pptrf. The dimension of ap must be at least max(1,n(n+1)/2).\nanorm\nThe norm of the original matrix A (see Description).\nOutput Parameters\nrcond\nAn estimate of the reciprocal of the condition number. The routine sets\nrcond =0 if the estimate underflows; in this case the matrix is singular\n(to working precision). However, anytime rcond is small compared to\n1.0, for the working precision, the matrix may be poorly conditioned\nor even singular.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n563\n\n\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info = -i, parameter i had an illegal value.\nApplication Notes\nThe computed rcond is never less than r (the reciprocal of the true condition number) and in practice is\nnearly always less than 10r. A call to this routine involves solving a number of systems of linear equations\nA*x = b; the number is usually 4 or 5 and never more than 11. Each solution requires approximately 2n2\nfloating-point operations for real flavors and 8n2 for complex flavors.\nSee Also\nMatrix Storage Schemes\n?pbcon\nEstimates the reciprocal of the condition number of a\nsymmetric (Hermitian) positive-definite band matrix.\nSyntax\nlapack_int LAPACKE_spbcon( int matrix_layout, char uplo, lapack_int n, lapack_int kd,\nconst float* ab, lapack_int ldab, float anorm, float* rcond );\nlapack_int LAPACKE_dpbcon( int matrix_layout, char uplo, lapack_int n, lapack_int kd,\nconst double* ab, lapack_int ldab, double anorm, double* rcond );\nlapack_int LAPACKE_cpbcon( int matrix_layout, char uplo, lapack_int n, lapack_int kd,\nconst lapack_complex_float* ab, lapack_int ldab, float anorm, float* rcond );\nlapack_int LAPACKE_zpbcon( int matrix_layout, char uplo, lapack_int n, lapack_int kd,\nconst lapack_complex_double* ab, lapack_int ldab, double anorm, double* rcond );\nInclude Files\n•\nmkl.h\nDescription\nThe routine estimates the reciprocal of the condition number of a symmetric (Hermitian) positive-definite\nband matrix A:\nκ1(A) = ||A||1 ||A-1||1 (since A is symmetric or Hermitian, κ∞(A) = κ1(A)).\nAn estimate is obtained for ||A-1||, and the reciprocal of the condition number is computed as rcond =\n1 / (||A|| ||A-1||).\nBefore calling this routine:\n•\ncompute anorm (either ||A||1 = maxjΣi |aij| or ||A||∞ = maxiΣj |aij|)\n•\ncall ?pbtrf to compute the Cholesky factorization of A.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n564\n\n\nuplo\nMust be 'U' or 'L'.\nIndicates how the input matrix A has been factored:\nIf uplo = 'U', A is factored as A = UT*U for real flavors or A = UH*U\nfor complex flavors, and U is stored.\nIf uplo = 'L', A is factored as A = L*LT for real flavors or A = L*LH\nfor complex flavors, and L is stored.\nn\nThe order of the matrix A; n≥ 0.\nkd\nThe number of superdiagonals or subdiagonals in the matrix A; kd≥ 0.\nldab\nThe leading dimension of the array ab. (ldab≥kd +1).\nab\nThe array ab of size max(1, ldab*n) contains the factored matrix A in\nband form, as returned by ?pbtrf.\nanorm\nThe norm of the original matrix A (see Description).\nOutput Parameters\nrcond\nAn estimate of the reciprocal of the condition number. The routine sets\nrcond =0 if the estimate underflows; in this case the matrix is singular\n(to working precision). However, anytime rcond is small compared to\n1.0, for the working precision, the matrix may be poorly conditioned\nor even singular.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, parameter i had an illegal value.\nApplication Notes\nThe computed rcond is never less than r (the reciprocal of the true condition number) and in practice is\nnearly always less than 10r. A call to this routine involves solving a number of systems of linear equations\nA*x = b; the number is usually 4 or 5 and never more than 11. Each solution requires approximately\n4*n(kd + 1) floating-point operations for real flavors and 16*n(kd + 1) for complex flavors.\nSee Also\nMatrix Storage Schemes\n?ptcon\nEstimates the reciprocal of the condition number of a\nsymmetric (Hermitian) positive-definite tridiagonal\nmatrix.\nSyntax\nlapack_int LAPACKE_sptcon( lapack_int n, const float* d, const float* e, float anorm,\nfloat* rcond );\nlapack_int LAPACKE_dptcon( lapack_int n, const double* d, const double* e, double\nanorm, double* rcond );\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n565\n\n\nlapack_int LAPACKE_cptcon( lapack_int n, const float* d, const lapack_complex_float* e,\nfloat anorm, float* rcond );\nlapack_int LAPACKE_zptcon( lapack_int n, const double* d, const lapack_complex_double*\ne, double anorm, double* rcond );\nInclude Files\n•\nmkl.h\nDescription\nThe routine computes the reciprocal of the condition number (in the 1-norm) of a real symmetric or complex\nHermitian positive-definite tridiagonal matrix using the factorization A = L*D*LT for real flavors and A =\nL*D*LH for complex flavors or A = UT*D*U for real flavors and A = UH*D*U for complex flavors computed\nby ?pttrf :\nκ1(A) = ||A||1 ||A-1||1 (since A is symmetric or Hermitian, κ∞(A) = κ1(A)).\nThe norm ||A-1|| is computed by a direct method, and the reciprocal of the condition number is computed\nas rcond = 1 / (||A|| ||A-1||).\nBefore calling this routine:\n•\ncompute anorm as ||A||1 = maxjΣi |aij|\n•\ncall ?pttrf to compute the factorization of A.\nInput Parameters\nn\nThe order of the matrix A; n≥ 0.\nd\nArrays, dimension (n).\nThe array d contains the n diagonal elements of the diagonal matrix D\nfrom the factorization of A, as computed by ?pttrf ;\ne\nArray, size (n -1).\nContains off-diagonal elements of the unit bidiagonal factor U or L\nfrom the factorization computed by ?pttrf .\nanorm\nThe 1- norm of the original matrix A (see Description).\nOutput Parameters\nrcond\nAn estimate of the reciprocal of the condition number. The routine sets\nrcond =0 if the estimate underflows; in this case the matrix is singular\n(to working precision). However, anytime rcond is small compared to\n1.0, for the working precision, the matrix may be poorly conditioned\nor even singular.\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info = -i, parameter i had an illegal value.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n566\n\n\nApplication Notes\nThe computed rcond is never less than r (the reciprocal of the true condition number) and in practice is\nnearly always less than 10r. A call to this routine involves solving a number of systems of linear equations\nA*x = b; the number is usually 4 or 5 and never more than 11. Each solution requires approximately\n4*n(kd + 1) floating-point operations for real flavors and 16*n(kd + 1) for complex flavors.\n \n?sycon\nEstimates the reciprocal of the condition number of a\nsymmetric matrix.\nSyntax\nlapack_int LAPACKE_ssycon( int matrix_layout, char uplo, lapack_int n, const float* a,\nlapack_int lda, const lapack_int* ipiv, float anorm, float* rcond );\nlapack_int LAPACKE_dsycon( int matrix_layout, char uplo, lapack_int n, const double* a,\nlapack_int lda, const lapack_int* ipiv, double anorm, double* rcond );\nlapack_int LAPACKE_csycon( int matrix_layout, char uplo, lapack_int n, const\nlapack_complex_float* a, lapack_int lda, const lapack_int* ipiv, float anorm, float*\nrcond );\nlapack_int LAPACKE_zsycon( int matrix_layout, char uplo, lapack_int n, const\nlapack_complex_double* a, lapack_int lda, const lapack_int* ipiv, double anorm, double*\nrcond );\nInclude Files\n•\nmkl.h\nDescription\nThe routine estimates the reciprocal of the condition number of a symmetric matrix A:\nκ1(A) = ||A||1 ||A-1||1 (since A is symmetric, κ∞(A) = κ1(A)).\nAn estimate is obtained for ||A-1||, and the reciprocal of the condition number is computed as rcond =\n1 / (||A|| ||A-1||).\nBefore calling this routine:\n•\ncompute anorm (either ||A||1 = maxjΣi |aij| or ||A||∞ = maxiΣj |aij|)\n•\ncall ?sytrf to compute the factorization of A.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\nuplo\nMust be 'U' or 'L'.\nIndicates how the input matrix A has been factored:\nIf uplo = 'U', the array a stores the upper triangular factor U of the\nfactorization A = U*D*UT.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n567\n\n\nIf uplo = 'L', the array a stores the lower triangular factor L of the\nfactorization A = L*D*LT.\nn\nThe order of matrix A; n≥ 0.\na\nThe array a of size max(1,lda*n) contains the factored matrix A, as\nreturned by ?sytrf.\nlda\nThe leading dimension of a; lda≥ max(1, n).\nipiv\nArray, size at least max(1, n).\nThe array ipiv, as returned by ?sytrf.\nanorm\nThe norm of the original matrix A (see Description).\nOutput Parameters\nrcond\nAn estimate of the reciprocal of the condition number. The routine sets\nrcond =0 if the estimate underflows; in this case the matrix is singular\n(to working precision). However, anytime rcond is small compared to\n1.0, for the working precision, the matrix may be poorly conditioned\nor even singular.\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info = -i, parameter i had an illegal value.\nApplication Notes\nThe computed rcond is never less than r (the reciprocal of the true condition number) and in practice is\nnearly always less than 10r. A call to this routine involves solving a number of systems of linear equations\nA*x = b; the number is usually 4 or 5 and never more than 11. Each solution requires approximately 2n2\nfloating-point operations for real flavors and 8n2 for complex flavors.\nSee Also\nMatrix Storage Schemes\n?sycon_3\nEstimates the reciprocal of the condition number (in\nthe 1-norm) of a real or complex symmetric matrix A\nusing the factorization computed by ?sytrf_rk.\nlapack_int LAPACKE_ssycon_3 (int matrix_layout, char uplo, lapack_int n, const float *\nA, lapack_int lda, const float * e, const lapack_int * ipiv, float anorm, float *\nrcond);\nlapack_int LAPACKE_dsycon_3 (int matrix_layout, char uplo, lapack_int n, const double *\nA, lapack_int lda, const double * e, const lapack_int * ipiv, double anorm, double *\nrcond);\nlapack_int LAPACKE_csycon_3 (int matrix_layout, char uplo, lapack_int n, const\nlapack_complex_float * A, lapack_int lda, const lapack_complex_float * e, const\nlapack_int * ipiv, float anorm, float * rcond);\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n568\n\n\nlapack_int LAPACKE_zsycon_3 (int matrix_layout, char uplo, lapack_int n, const\nlapack_complex_double * A, lapack_int lda, const lapack_complex_double * e, const\nlapack_int * ipiv, double anorm, double * rcond);\nDescription\n?sycon_3 estimates the reciprocal of the condition number (in the 1-norm) of a real or complex symmetric\nmatrix A using the factorization computed by ?sytrf_rk. A = P*U*D*(UT)*(PT) or A = P*L*D*(LT)*(PT),\nwhere U (or L) is unit upper (or lower) triangular matrix, UT (or LT) is the transpose of U (or L), P is a\npermutation matrix, PT is the transpose of P, and D is symmetric and block diagonal with 1-by-1 and 2-by-2\ndiagonal blocks.\nAn estimate is obtained for norm(inv(A)), and the reciprocal of the condition number is computed as rcond\n= 1 / (anorm * norm(inv(A))).\nThis routine uses BLAS3 solver ?sytrs_3.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nuplo\nSpecifies whether the details of the factorization are stored as an upper or\nlower triangular matrix:\n•\n= 'U': Upper triangular. The form is A = P*U*D*(UT)*(PT).\n•\n= 'L': Lower triangular. The form is A = P*L*D*(LT)*(PT).\nn\nThe order of the matrix A. n ≥ 0.\nA\nArray of size max(1, lda*n). Diagonal of the block diagonal matrix D and\nfactors U or L as computed by ?sytrf_rk:\n•\nOnly diagonal elements of the symmetric block diagonal matrix D on the\ndiagonal of A; that is, D(k,k) = A(k,k). Superdiagonal (or subdiagonal)\nelements of D should be provided on entry in array e).\n—and—\n•\nIf uplo = 'U', factor U in the superdiagonal part of A. If uplo = 'L',\nfactor L in the subdiagonal part of A.\nlda\nThe leading dimension of the array A.\ne\nArray of size n. On entry, contains the superdiagonal (or subdiagonal)\nelements of the symmetric block diagonal matrix D with 1-by-1 or 2-by-2\ndiagonal blocks. If uplo = 'U', e(i) = D(i-1,i), i=2:N, and e(1) is not\nreferenced. If uplo = 'L', e(i) = D(i+1,i), i=1:N-1, and e(n) is not\nreferenced.\nNOTE For 1-by-1 diagonal block D(k), where 1 ≤ k ≤ n, the\nelement e[k-1] is not referenced in both the uplo = 'U' and\nuplo = 'L' cases.\nipiv\nArray of size n. Details of the interchanges and the block structure of D as\ndetermined by ?sytrf_rk.\nanorm\nThe 1-norm of the original matrix A.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n569\n\n\nOutput Parameters\nrcond\nThe reciprocal of the condition number of the matrix A, computed as rcond\n= 1/(anorm * AINVNM), where AINVNM is an estimate of the 1-norm of\ninv(A) computed in this routine.\nReturn Values\nThis function returns a value info.\n= 0: Successful exit.\n< 0: If info = -i, the ith argument had an illegal value.\n?hecon\nEstimates the reciprocal of the condition number of a\nHermitian matrix.\nSyntax\nlapack_int LAPACKE_checon( int matrix_layout, char uplo, lapack_int n, const\nlapack_complex_float* a, lapack_int lda, const lapack_int* ipiv, float anorm, float*\nrcond );\nlapack_int LAPACKE_zhecon( int matrix_layout, char uplo, lapack_int n, const\nlapack_complex_double* a, lapack_int lda, const lapack_int* ipiv, double anorm, double*\nrcond );\nInclude Files\n•\nmkl.h\nDescription\nThe routine estimates the reciprocal of the condition number of a Hermitian matrix A:\nκ1(A) = ||A||1 ||A-1||1 (since A is Hermitian, κ∞(A) = κ1(A)).\nBefore calling this routine:\n•\ncompute anorm (either ||A||1 =maxjΣi |aij| or ||A||∞ =maxiΣj |aij|)\n•\ncall ?hetrf to compute the factorization of A.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\nuplo\nMust be 'U' or 'L'.\nIndicates how the input matrix A has been factored:\nIf uplo = 'U', the array a stores the upper triangular factor U of the\nfactorization A = U*D*UH.\nIf uplo = 'L', the array a stores the lower triangular factor L of the\nfactorization A = L*D*LH.\nn\nThe order of matrix A; n≥ 0.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n570\n\n\na\nThe array a of size max(1, lda*n) contains the factored matrix A, as\nreturned by ?hetrf.\nlda\nThe leading dimension of a; lda≥ max(1, n).\nipiv\nArray, size at least max(1, n).\nThe array ipiv, as returned by ?hetrf.\nanorm\nThe norm of the original matrix A (see Description).\nOutput Parameters\nrcond\nAn estimate of the reciprocal of the condition number. The routine sets\nrcond =0 if the estimate underflows; in this case the matrix is singular\n(to working precision). However, anytime rcond is small compared to\n1.0, for the working precision, the matrix may be poorly conditioned\nor even singular.\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info = -i, parameter i had an illegal value.\nApplication Notes\nThe computed rcond is never less than r (the reciprocal of the true condition number) and in practice is\nnearly always less than 10r. A call to this routine involves solving a number of systems of linear equations\nA*x = b; the number is usually 5 and never more than 11. Each solution requires approximately 8n2\nfloating-point operations.\nSee Also\nMatrix Storage Schemes\n?hecon_3\nEstimates the reciprocal of the condition number (in\nthe 1-norm) of a complex Hermitian matrix A.\nlapack_int LAPACKE_checon_3 (int matrix_layout, char uplo, lapack_int n, const\nlapack_complex_float * A, lapack_int lda, const lapack_complex_float * e, const\nlapack_int * ipiv, float anorm, float * rcond);\nlapack_int LAPACKE_zhecon_3 (int matrix_layout, char uplo, lapack_int n, const\nlapack_complex_double * A, lapack_int lda, const lapack_complex_double * e, const\nlapack_int * ipiv, double anorm, double * rcond);\nDescription\n?hecon_3 estimates the reciprocal of the condition number (in the 1-norm) of a complex Hermitian matrix A\nusing the factorization computed by ?hetrf_rk: A = P*U*D*(UH)*(PT) or A = P*L*D*(LH)*(PT), where U (or\nL) is unit upper (or lower) triangular matrix, UH (or LH) is the conjugate of U (or L), P is a permutation\nmatrix, PT is the transpose of P, and D is Hermitian and block diagonal with 1-by-1 and 2-by-2 diagonal\nblocks. An estimate is obtained for norm(inv(A)), and the reciprocal of the condition number is computed as\nrcond = 1 / (anorm * norm(inv(A))).\nThis routine uses BLAS3 solver ?hetrs_3.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n571\n\n\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nuplo\nSpecifies whether the details of the factorization are stored as an upper or\nlower triangular matrix: = 'U': Upper triangular, form is A =\nP*U*D*(UH)*(PT); = 'L': Lower triangular, form is A = P*L*D*(LH)*(PT).\nn\nThe order of the matrix A. n ≥ 0.\nA\nArray of size max(1, lda*n). Diagonal of the block diagonal matrix D and\nfactor U or L as computed by ?hetrf_rk:\n•\nOnly diagonal elements of the Hermitian block diagonal matrix D on the\ndiagonal of A—that is, D(k,k) = A(k,k). Superdiagonal (or subdiagonal)\nelements of D must be provided on entry in array e.\n—and—\n•\nIf uplo = 'U', factor U in the superdiagonal part of A. If uplo = 'L',\nfactor L in the subdiagonal part of A.\nlda\nThe leading dimension of the array A.\ne\nArray of size n. On entry, contains the superdiagonal (or subdiagonal)\nelements of the Hermitian block diagonal matrix D with 1-by-1 or 2-by-2\ndiagonal blocks. If uplo = 'U', e(i) = D(i-1, i),i=2:N, and e(1) is not\nreferenced. If uplo = 'L', e(i) = D(i+1,i), i=1:N-1, and e(n) is not\nreferenced.\nNOTE For 1-by-1 diagonal block D(k), where 1 ≤ k ≤ n, the\nelement e[k-1] is not referenced in both the uplo = 'U' and\nuplo = 'L' cases.\nipiv\nArray of size n. Details of the interchanges and the block structure of D as\ndetermined by ?hetrf_rk.\nanorm\nThe 1-norm of the original matrix A.\nOutput Parameters\nrcond\nThe reciprocal of the condition number of the matrix A, computed as rcond\n= 1/(anorm * AINVNM), where AINVNM is an estimate of the 1-norm of\ninv(A) computed in this routine.\nReturn Values\nThis function returns a value info.\n= 0: Successful exit.\n< 0: If info = -i, the ith argument had an illegal value.\n?spcon\nEstimates the reciprocal of the condition number of a\npacked symmetric matrix.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n572\n\n\nSyntax\nlapack_int LAPACKE_sspcon( int matrix_layout, char uplo, lapack_int n, const float* ap,\nconst lapack_int* ipiv, float anorm, float* rcond );\nlapack_int LAPACKE_dspcon( int matrix_layout, char uplo, lapack_int n, const double*\nap, const lapack_int* ipiv, double anorm, double* rcond );\nlapack_int LAPACKE_cspcon( int matrix_layout, char uplo, lapack_int n, const\nlapack_complex_float* ap, const lapack_int* ipiv, float anorm, float* rcond );\nlapack_int LAPACKE_zspcon( int matrix_layout, char uplo, lapack_int n, const\nlapack_complex_double* ap, const lapack_int* ipiv, double anorm, double* rcond );\nInclude Files\n•\nmkl.h\nDescription\nThe routine estimates the reciprocal of the condition number of a packed symmetric matrix A:\nκ1(A) = ||A||1 ||A-1||1 (since A is symmetric, κ∞(A) = κ1(A)).\nAn estimate is obtained for ||A-1||, and the reciprocal of the condition number is computed as rcond =\n1 / (||A|| ||A-1||).\nBefore calling this routine:\n•\ncompute anorm (either ||A||1 = maxjΣi |aij| or ||A||∞ = maxiΣj |aij|)\n•\ncall ?sptrf to compute the factorization of A.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\nuplo\nMust be 'U' or 'L'.\nIndicates how the input matrix A has been factored:\nIf uplo = 'U', the array ap stores the packed upper triangular factor\nU of the factorization A = U*D*UT.\nIf uplo = 'L', the array ap stores the packed lower triangular factor\nL of the factorization A = L*D*LT.\nn\nThe order of matrix A; n≥ 0.\nap\nThe array ap contains the packed factored matrix A, as returned\nby ?sptrf. The dimension of ap must be at least max(1,n(n+1)/2).\nipiv\nArray, size at least max(1, n).\nThe array ipiv, as returned by ?sptrf.\nanorm\nThe norm of the original matrix A (see Description).\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n573\n\n\nOutput Parameters\nrcond\nAn estimate of the reciprocal of the condition number. The routine sets\nrcond = 0 if the estimate underflows; in this case the matrix is\nsingular (to working precision). However, anytime rcond is small\ncompared to 1.0, for the working precision, the matrix may be poorly\nconditioned or even singular.\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info = -i, parameter i had an illegal value.\nApplication Notes\nThe computed rcond is never less than r (the reciprocal of the true condition number) and in practice is\nnearly always less than 10r. A call to this routine involves solving a number of systems of linear equations\nA*x = b; the number is usually 4 or 5 and never more than 11. Each solution requires approximately 2n2\nfloating-point operations for real flavors and 8n2 for complex flavors.\nSee Also\nMatrix Storage Schemes\n?hpcon\nEstimates the reciprocal of the condition number of a\npacked Hermitian matrix.\nSyntax\nlapack_int LAPACKE_chpcon( int matrix_layout, char uplo, lapack_int n, const\nlapack_complex_float* ap, const lapack_int* ipiv, float anorm, float* rcond );\nlapack_int LAPACKE_zhpcon( int matrix_layout, char uplo, lapack_int n, const\nlapack_complex_double* ap, const lapack_int* ipiv, double anorm, double* rcond );\nInclude Files\n•\nmkl.h\nDescription\nThe routine estimates the reciprocal of the condition number of a Hermitian matrix A:\nκ1(A) = ||A||1 ||A-1||1 (since A is Hermitian, κ∞(A) = k1(A)).\nAn estimate is obtained for ||A-1||, and the reciprocal of the condition number is computed as rcond =\n1 / (||A|| ||A-1||).\nBefore calling this routine:\n•\ncompute anorm (either ||A||1 =maxjΣi |aij| or ||A||∞ =maxiΣj |aij|)\n•\ncall ?hptrf to compute the factorization of A.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n574\n\n\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\nuplo\nMust be 'U' or 'L'.\nIndicates how the input matrix A has been factored:\nIf uplo = 'U', the array ap stores the packed upper triangular factor\nU of the factorization A = U*D*UT.\nIf uplo = 'L', the array ap stores the packed lower triangular factor\nL of the factorization A = L*D*LT.\nn\nThe order of matrix A; n≥ 0.\nap\nThe array ap contains the packed factored matrix A, as returned\nby ?hptrf. The dimension of ap must be at least max(1,n(n+1)/2).\nipiv\nArray, size at least max(1, n). The array ipiv, as returned by ?hptrf.\nanorm\nThe norm of the original matrix A (see Description).\nOutput Parameters\nrcond\nAn estimate of the reciprocal of the condition number. The routine sets\nrcond =0 if the estimate underflows; in this case the matrix is singular\n(to working precision). However, anytime rcond is small compared to\n1.0, for the working precision, the matrix may be poorly conditioned\nor even singular.\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info = -i, parameter i had an illegal value.\nApplication Notes\nThe computed rcond is never less than r (the reciprocal of the true condition number) and in practice is\nnearly always less than 10r. A call to this routine involves solving a number of systems of linear equations\nA*x = b; the number is usually 5 and never more than 11. Each solution requires approximately 8n2\nfloating-point operations.\nSee Also\nMatrix Storage Schemes\n?trcon\nEstimates the reciprocal of the condition number of a\ntriangular matrix.\nSyntax\nlapack_int LAPACKE_strcon( int matrix_layout, char norm, char uplo, char diag,\nlapack_int n, const float* a, lapack_int lda, float* rcond );\nlapack_int LAPACKE_dtrcon( int matrix_layout, char norm, char uplo, char diag,\nlapack_int n, const double* a, lapack_int lda, double* rcond );\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n575\n\n\nlapack_int LAPACKE_ctrcon( int matrix_layout, char norm, char uplo, char diag,\nlapack_int n, const lapack_complex_float* a, lapack_int lda, float* rcond );\nlapack_int LAPACKE_ztrcon( int matrix_layout, char norm, char uplo, char diag,\nlapack_int n, const lapack_complex_double* a, lapack_int lda, double* rcond );\nInclude Files\n•\nmkl.h\nDescription\nThe routine estimates the reciprocal of the condition number of a triangular matrix A in either the 1-norm or\ninfinity-norm:\nκ1(A) =||A||1 ||A-1||1 = κ∞(AT) = κ∞(AH)\nκ∞ (A) =||A||∞ ||A-1||∞ =k1 (AT) = κ1 (AH) .\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\nnorm\nMust be '1' or 'O' or 'I'.\nIf norm = '1' or 'O', then the routine estimates the condition\nnumber of matrix A in 1-norm.\nIf norm = 'I', then the routine estimates the condition number of\nmatrix A in infinity-norm.\nuplo\nMust be 'U' or 'L'.\nIndicates whether A is upper or lower triangular:\nIf uplo = 'U', the array a stores the upper triangle of A, other array\nelements are not referenced.\nIf uplo = 'L', the array a stores the lower triangle of A, other array\nelements are not referenced.\ndiag\nMust be 'N' or 'U'.\nIf diag = 'N', then A is not a unit triangular matrix.\nIf diag = 'U', then A is unit triangular: diagonal elements are\nassumed to be 1 and not referenced in the array a.\nn\nThe order of the matrix A; n≥ 0.\na\nThe array a of size max(1, lda*n) contains the matrix A.\nlda\nThe leading dimension of a; lda≥ max(1, n).\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n576\n\n\nOutput Parameters\nrcond\nAn estimate of the reciprocal of the condition number. The routine sets\nrcond =0 if the estimate underflows; in this case the matrix is singular\n(to working precision). However, anytime rcond is small compared to\n1.0, for the working precision, the matrix may be poorly conditioned\nor even singular.\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info = -i, parameter i had an illegal value.\nApplication Notes\nThe computed rcond is never less than r (the reciprocal of the true condition number) and in practice is\nnearly always less than 10r. A call to this routine involves solving a number of systems of linear equations\nA*x = b; the number is usually 4 or 5 and never more than 11. Each solution requires approximately n2\nfloating-point operations for real flavors and 4n2 operations for complex flavors.\nSee Also\nMatrix Storage Schemes\n?tpcon\nEstimates the reciprocal of the condition number of a\npacked triangular matrix.\nSyntax\nlapack_int LAPACKE_stpcon( int matrix_layout, char norm, char uplo, char diag,\nlapack_int n, const float* ap, float* rcond );\nlapack_int LAPACKE_dtpcon( int matrix_layout, char norm, char uplo, char diag,\nlapack_int n, const double* ap, double* rcond );\nlapack_int LAPACKE_ctpcon( int matrix_layout, char norm, char uplo, char diag,\nlapack_int n, const lapack_complex_float* ap, float* rcond );\nlapack_int LAPACKE_ztpcon( int matrix_layout, char norm, char uplo, char diag,\nlapack_int n, const lapack_complex_double* ap, double* rcond );\nInclude Files\n•\nmkl.h\nDescription\nThe routine estimates the reciprocal of the condition number of a packed triangular matrix A in either the 1-\nnorm or infinity-norm:\nκ1(A) =||A||1 ||A-1||1 = κ∞(AT) = κ∞(AH)\nκ∞(A) =||A||∞ ||A-1||∞ =κ1 (AT) = κ1(AH) .\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n577\n\n\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\nnorm\nMust be '1' or 'O' or 'I'.\nIf norm = '1' or 'O', then the routine estimates the condition\nnumber of matrix A in 1-norm.\nIf norm = 'I', then the routine estimates the condition number of\nmatrix A in infinity-norm.\nuplo\nMust be 'U' or 'L'. Indicates whether A is upper or lower triangular:\nIf uplo = 'U', the array ap stores the upper triangle of A in packed\nform.\nIf uplo = 'L', the array ap stores the lower triangle of A in packed\nform.\ndiag\nMust be 'N' or 'U'.\nIf diag = 'N', then A is not a unit triangular matrix.\nIf diag = 'U', then A is unit triangular: diagonal elements are\nassumed to be 1 and not referenced in the array ap.\nn\nThe order of the matrix A; n≥ 0.\nap\nThe array ap contains the packed matrix A. The dimension of ap must\nbe at least max(1,n(n+1)/2).\nOutput Parameters\nrcond\nAn estimate of the reciprocal of the condition number. The routine sets\nrcond =0 if the estimate underflows; in this case the matrix is singular\n(to working precision). However, anytime rcond is small compared to\n1.0, for the working precision, the matrix may be poorly conditioned\nor even singular.\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info = -i, parameter i had an illegal value.\nApplication Notes\nThe computed rcond is never less than r (the reciprocal of the true condition number) and in practice is\nnearly always less than 10r. A call to this routine involves solving a number of systems of linear equations\nA*x = b; the number is usually 4 or 5 and never more than 11. Each solution requires approximately n2\nfloating-point operations for real flavors and 4n2 operations for complex flavors.\nSee Also\nMatrix Storage Schemes\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n578\n\n\n?tbcon\nEstimates the reciprocal of the condition number of a\ntriangular band matrix.\nSyntax\nlapack_int LAPACKE_stbcon( int matrix_layout, char norm, char uplo, char diag,\nlapack_int n, lapack_int kd, const float* ab, lapack_int ldab, float* rcond );\nlapack_int LAPACKE_dtbcon( int matrix_layout, char norm, char uplo, char diag,\nlapack_int n, lapack_int kd, const double* ab, lapack_int ldab, double* rcond );\nlapack_int LAPACKE_ctbcon( int matrix_layout, char norm, char uplo, char diag,\nlapack_int n, lapack_int kd, const lapack_complex_float* ab, lapack_int ldab, float*\nrcond );\nlapack_int LAPACKE_ztbcon( int matrix_layout, char norm, char uplo, char diag,\nlapack_int n, lapack_int kd, const lapack_complex_double* ab, lapack_int ldab, double*\nrcond );\nInclude Files\n•\nmkl.h\nDescription\nThe routine estimates the reciprocal of the condition number of a triangular band matrix A in either the 1-\nnorm or infinity-norm:\nκ1(A) =||A||1 ||A-1||1 = κ∞(AT) = κ∞(AH)\nκ∞(A) =||A||∞ ||A-1||∞ =κ1 (AT) = κ1(AH) .\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\nnorm\nMust be '1' or 'O' or 'I'.\nIf norm = '1' or 'O', then the routine estimates the condition\nnumber of matrix A in 1-norm.\nIf norm = 'I', then the routine estimates the condition number of\nmatrix A in infinity-norm.\nuplo\nMust be 'U' or 'L'. Indicates whether A is upper or lower triangular:\nIf uplo = 'U', the array ap stores the upper triangle of A in packed\nform.\nIf uplo = 'L', the array ap stores the lower triangle of A in packed\nform.\ndiag\nMust be 'N' or 'U'.\nIf diag = 'N', then A is not a unit triangular matrix.\nIf diag = 'U', then A is unit triangular: diagonal elements are\nassumed to be 1 and not referenced in the array ab.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n579\n\n\nn\nThe order of the matrix A; n≥ 0.\nkd\nThe number of superdiagonals or subdiagonals in the matrix A; kd≥ 0.\nab\nThe array ab of size max(1, ldab*n) contains the band matrix A.\nldab\nThe leading dimension of the array ab. (ldab≥kd +1).\nOutput Parameters\nrcond\nAn estimate of the reciprocal of the condition number. The routine sets\nrcond =0 if the estimate underflows; in this case the matrix is singular\n(to working precision). However, anytime rcond is small compared to\n1.0, for the working precision, the matrix may be poorly conditioned\nor even singular.\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info = -i, parameter i had an illegal value.\nApplication Notes\nThe computed rcond is never less than r (the reciprocal of the true condition number) and in practice is\nnearly always less than 10r. A call to this routine involves solving a number of systems of linear equations\nA*x = b; the number is usually 4 or 5 and never more than 11. Each solution requires approximately\n2*n(kd + 1) floating-point operations for real flavors and 8*n(kd + 1) operations for complex flavors.\nSee Also\nMatrix Storage Schemes\nRefining the Solution and Estimating Its Error: LAPACK Computational Routines\nThis section describes the LAPACK routines for refining the computed solution of a system of linear equations\nand estimating the solution error. You can call these routines after factorizing the matrix of the system of\nequations and computing the solution (see Routines for Matrix Factorization and Routines for Solving\nSystems of Linear Equations).\n?gerfs\nRefines the solution of a system of linear equations\nwith a general coefficient matrix and estimates its\nerror.\nSyntax\nlapack_int LAPACKE_sgerfs( int matrix_layout, char trans, lapack_int n, lapack_int\nnrhs, const float* a, lapack_int lda, const float* af, lapack_int ldaf, const\nlapack_int* ipiv, const float* b, lapack_int ldb, float* x, lapack_int ldx, float* ferr,\nfloat* berr );\nlapack_int LAPACKE_dgerfs( int matrix_layout, char trans, lapack_int n, lapack_int\nnrhs, const double* a, lapack_int lda, const double* af, lapack_int ldaf, const\nlapack_int* ipiv, const double* b, lapack_int ldb, double* x, lapack_int ldx, double*\nferr, double* berr );\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n580\n\n\nlapack_int LAPACKE_cgerfs( int matrix_layout, char trans, lapack_int n, lapack_int\nnrhs, const lapack_complex_float* a, lapack_int lda, const lapack_complex_float* af,\nlapack_int ldaf, const lapack_int* ipiv, const lapack_complex_float* b, lapack_int ldb,\nlapack_complex_float* x, lapack_int ldx, float* ferr, float* berr );\nlapack_int LAPACKE_zgerfs( int matrix_layout, char trans, lapack_int n, lapack_int\nnrhs, const lapack_complex_double* a, lapack_int lda, const lapack_complex_double* af,\nlapack_int ldaf, const lapack_int* ipiv, const lapack_complex_double* b, lapack_int\nldb, lapack_complex_double* x, lapack_int ldx, double* ferr, double* berr );\nInclude Files\n•\nmkl.h\nDescription\nThe routine performs an iterative refinement of the solution to a system of linear equations A*X = B or AT*X\n= B or AH*X = B with a general matrix A, with multiple right-hand sides. For each computed solution vector\nx, the routine computes the component-wise backward errorβ. This error is the smallest relative\nperturbation in elements of A and b such that x is the exact solution of the perturbed system:\n|δaij| ≤β|aij|, |δbi| ≤β|bi| such that (A + δA)x = (b + δb).\nFinally, the routine estimates the component-wise forward error in the computed solution ||x - xe||∞/||\nx||∞ (here xe is the exact solution).\nBefore calling this routine:\n•\ncall the factorization routine ?getrf\n•\ncall the solver routine ?getrs.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\ntrans\nMust be 'N' or 'T' or 'C'.\nIndicates the form of the equations:\nIf trans = 'N', the system has the form A*X = B.\nIf trans = 'T', the system has the form AT*X = B.\nIf trans = 'C', the system has the form AH*X = B.\nn\nThe order of the matrix A; n≥ 0.\nnrhs\nThe number of right-hand sides; nrhs≥ 0.\na,af,b,x\nArrays:\na(size max(1, lda*n)) contains the original matrix A, as supplied\nto ?getrf.\naf(size max(1, ldaf*n)) contains the factored matrix A, as returned\nby ?getrf.\nbof size max(1, ldb*nrhs) for column major layout and max(1,\nldb*n) for row major layout contains the right-hand side matrix B.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n581\n\n\nxof size max(1, ldx*nrhs) for column major layout and max(1,\nldx*n) for row major layout contains the solution matrix X.\nlda\nThe leading dimension of a; lda≥ max(1, n).\nldaf\nThe leading dimension of af; ldaf≥ max(1, n).\nldb\nThe leading dimension of b; ldb≥ max(1, n) for column major layout\nand ldb≥nrhs for row major layout.\nldx\nThe leading dimension of x; ldx≥ max(1, n) for column major layout\nand ldx≥nrhs for row major layout.\nipiv\nArray, size at least max(1, n).\nThe ipiv array, as returned by ?getrf.\nOutput Parameters\nx\nThe refined solution matrix X.\nferr, berr\nArrays, size at least max(1, nrhs). Contain the component-wise\nforward and backward errors, respectively, for each solution vector.\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info = -i, parameter i had an illegal value.\nApplication Notes\nThe bounds returned in ferr are not rigorous, but in practice they almost always overestimate the actual\nerror.\nFor each right-hand side, computation of the backward error involves a minimum of 4n2 floating-point\noperations (for real flavors) or 16n2 operations (for complex flavors). In addition, each step of iterative\nrefinement involves 6n2 operations (for real flavors) or 24n2 operations (for complex flavors); the number of\niterations may range from 1 to 5. Estimating the forward error involves solving a number of systems of linear\nequations A*x = b with the same coefficient matrix A and different right hand sides b; the number is usually\n4 or 5 and never more than 11. Each solution requires approximately 2n2 floating-point operations for real\nflavors or 8n2 for complex flavors.\nSee Also\nMatrix Storage Schemes\n?gerfsx\nUses extra precise iterative refinement to improve the\nsolution to the system of linear equations with a\ngeneral coefficient matrix A and provides error bounds\nand backward error estimates.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n582\n\n\nSyntax\nlapack_int LAPACKE_sgerfsx( int matrix_layout, char trans, char equed, lapack_int n,\nlapack_int nrhs, const float* a, lapack_int lda, const float* af, lapack_int ldaf,\nconst lapack_int* ipiv, const float* r, const float* c, const float* b, lapack_int ldb,\nfloat* x, lapack_int ldx, float* rcond, float* berr, lapack_int n_err_bnds, float*\nerr_bnds_norm, float* err_bnds_comp, lapack_int nparams, float* params );\nlapack_int LAPACKE_dgerfsx( int matrix_layout, char trans, char equed, lapack_int n,\nlapack_int nrhs, const double* a, lapack_int lda, const double* af, lapack_int ldaf,\nconst lapack_int* ipiv, const double* r, const double* c, const double* b, lapack_int\nldb, double* x, lapack_int ldx, double* rcond, double* berr, lapack_int n_err_bnds,\ndouble* err_bnds_norm, double* err_bnds_comp, lapack_int nparams, double* params );\nlapack_int LAPACKE_cgerfsx( int matrix_layout, char trans, char equed, lapack_int n,\nlapack_int nrhs, const lapack_complex_float* a, lapack_int lda, const\nlapack_complex_float* af, lapack_int ldaf, const lapack_int* ipiv, const float* r,\nconst float* c, const lapack_complex_float* b, lapack_int ldb, lapack_complex_float* x,\nlapack_int ldx, float* rcond, float* berr, lapack_int n_err_bnds, float* err_bnds_norm,\nfloat* err_bnds_comp, lapack_int nparams, float* params );\nlapack_int LAPACKE_zgerfsx( int matrix_layout, char trans, char equed, lapack_int n,\nlapack_int nrhs, const lapack_complex_double* a, lapack_int lda, const\nlapack_complex_double* af, lapack_int ldaf, const lapack_int* ipiv, const double* r,\nconst double* c, const lapack_complex_double* b, lapack_int ldb, lapack_complex_double*\nx, lapack_int ldx, double* rcond, double* berr, lapack_int n_err_bnds, double*\nerr_bnds_norm, double* err_bnds_comp, lapack_int nparams, double* params );\nInclude Files\n•\nmkl.h\nDescription\nThe routine improves the computed solution to a system of linear equations and provides error bounds and\nbackward error estimates for the solution. In addition to a normwise error bound, the code provides a\nmaximum componentwise error bound, if possible. See comments for err_bnds_norm and err_bnds_comp\nfor details of the error bounds.\nThe original system of linear equations may have been equilibrated before calling this routine, as described\nby the parameters equed, r, and c below. In this case, the solution and error bounds returned are for the\noriginal unequilibrated system.\nInput Parameters\nmatrix_layout\nSpecifies whether two-dimensional array storage is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\ntrans\nMust be 'N', 'T', or 'C'.\nSpecifies the form of the system of equations:\nIf trans = 'N', the system has the form A*X = B (No transpose).\nIf trans = 'T', the system has the form AT*X = B (Transpose).\nIf trans = 'C', the system has the form AH*X = B (Conjugate\ntranspose for complex flavors, Transpose for real flavors).\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n583\n\n\nequed\nMust be 'N', 'R', 'C', or 'B'.\nSpecifies the form of equilibration that was done to A before calling\nthis routine.\nIf equed = 'N', no equilibration was done.\nIf equed = 'R', row equilibration was done, that is, A has been\npremultiplied by diag(r).\nIf equed = 'C', column equilibration was done, that is, A has been\npostmultiplied by diag(c).\nIf equed = 'B', both row and column equilibration was done, that is,\nA has been replaced by diag(r)*A*diag(c). The right-hand side B\nhas been changed accordingly.\nn\nThe number of linear equations; the order of the matrix A; n≥ 0.\nnrhs\nThe number of right-hand sides; the number of columns of the\nmatrices B and X; nrhs≥ 0.\na, af, b\nArrays: a (size max(1, lda*n)), af (size max(1, ldaf*n)), b (size\nmax(1, ldb*nrhs) for column major layout and max(1, ldb*n) for\nrow major layout).\nThe array a contains the original n-by-n matrix A.\nThe array af contains the factored form of the matrix A, that is, the\nfactors L and U from the factorization A = P*L*U as computed\nby ?getrf.\nThe array b contains the matrix B whose columns are the right-hand\nsides for the systems of equations.\nlda\nThe leading dimension of a; lda≥ max(1, n).\nldaf\nThe leading dimension of af; ldaf≥ max(1, n).\nipiv\nArray, size at least max(1, n). Contains the pivot indices as\ncomputed by ?getrf; for row 1 ≤i≤n, row i of the matrix was\ninterchanged with row ipiv(i).\nr, c\nArrays: r (size n), c (size n). The array r contains the row scale\nfactors for A, and the array c contains the column scale factors for A.\nequed = 'R' or 'B', A is multiplied on the left by diag(r); if equed =\n'N' or 'C', r is not accessed.\nIf equed = 'R' or 'B', each element of r must be positive.\nIf equed = 'C' or 'B', A is multiplied on the right by diag(c); if\nequed = 'N' or 'R', c is not accessed.\nIf equed = 'C' or 'B', each element of c must be positive.\nEach element of r or c should be a power of the radix to ensure a\nreliable solution and error estimates. Scaling by powers of the radix\ndoes not cause rounding errors unless the result underflows or\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n584\n\n\noverflows. Rounding errors during scaling lead to refining with a\nmatrix that is not equivalent to the input matrix, producing error\nestimates that may not be reliable.\nldb\nThe leading dimension of the array b; ldb≥ max(1, n) for column\nmajor layout and ldb≥nrhs for row major layout.\nx\nArray, of size max(1, ldx*nrhs) for column major layout and max(1,\nldx*n) for row major layout.\nThe solution matrix X as computed by ?getrs\nldx\nThe leading dimension of the output array x; ldx≥ max(1, n) for\ncolumn major layout and ldx≥nrhs for row major layout.\nn_err_bnds\nNumber of error bounds to return for each right hand side and each\ntype (normwise or componentwise). See err_bnds_norm and\nerr_bnds_comp descriptions in Output Arguments section below.\nnparams\nSpecifies the number of parameters set in params. If ≤ 0, the params\narray is never referenced and default values are used.\nparams\nArray, size nparams. Specifies algorithm parameters. If an entry is\nless than 0.0, that entry is filled with the default value used for that\nparameter. Only positions up to nparams are accessed; defaults are\nused for higher-numbered parameters. If defaults are acceptable, you\ncan pass nparams = 0, which prevents the source code from\naccessing the params argument.\nparams[0] : Whether to perform iterative refinement or not. Default:\n1.0\n=0.0\nNo refinement is performed and no error\nbounds are computed.\n=1.0\nUse the double-precision refinement\nalgorithm, possibly with doubled-single\ncomputations if the compilation environment\ndoes not support double precision.\n(Other values are reserved for future use.)\nparams[1] : Maximum number of residual computations allowed for\nrefinement.\nDefault\n10.0\nAggressive\nSet to 100.0 to permit convergence using\napproximate factorizations or factorizations\nother than LU. If the factorization uses a\ntechnique other than Gaussian elimination,\nthe guarantees in err_bnds_norm and\nerr_bnds_comp may no longer be\ntrustworthy.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n585\n\n\nparams[2] : Flag determining if the code will attempt to find a\nsolution with a small componentwise relative error in the double-\nprecision algorithm. Positive is true, 0.0 is false. Default: 1.0 (attempt\ncomponentwise convergence).\nOutput Parameters\nx\nThe improved solution matrix X.\nrcond\nReciprocal scaled condition number. An estimate of the reciprocal\nSkeel condition number of the matrix A after equilibration (if done). If\nrcond is less than the machine precision, in particular, if rcond = 0,\nthe matrix is singular to working precision. Note that the error may\nstill be small even if this number is very small and the matrix appears\nill-conditioned.\nberr\nArray, size at least max(1, nrhs). Contains the componentwise\nrelative backward error for each solution vector xj, that is, the\nsmallest relative change in any element of A or B that makes xj an\nexact solution.\nerr_bnds_norm\nArray of size nrhs*n_err_bnds. For each right-hand side, contains\ninformation about various error bounds and condition numbers\ncorresponding to the normwise relative error, which is defined as\nfollows:\nNormwise relative error in the i-th solution vector\nThe array is indexed by the type of error information as described\nbelow. There are currently up to three pieces of information returned.\nerr=1\n\"Trust/don't trust\" boolean. Trust the answer\nif the reciprocal condition number is less\nthan the threshold sqrt(n)*slamch(ε) for\nsingle precision flavors and\nsqrt(n)*dlamch(ε) for double precision\nflavors.\nerr=2\n\"Guaranteed\" error bound. The estimated\nforward error, almost certainly within a factor\nof 10 of the true error so long as the next\nentry is greater than the threshold\nsqrt(n)*slamch(ε) for single precision\nflavors and sqrt(n)*dlamch(ε) for double\nprecision flavors. This error bound should\nonly be trusted if the previous boolean is\ntrue.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n586\n\n\nerr=3\nReciprocal condition number. Estimated\nnormwise reciprocal condition number.\nCompared with the threshold\nsqrt(n)*slamch(ε) for single precision\nflavors and sqrt(n)*dlamch(ε) for double\nprecision flavors to determine if the error\nestimate is \"guaranteed\". These reciprocal\ncondition numbers for some appropriately\nscaled matrix Z are:\nLet z=s*a, where s scales each row by a\npower of the radix so all absolute row sums\nof z are approximately 1.\nThe information for right-hand side i, where 1 ≤i≤nrhs, and type of\nerror err is stored in:\n•\nColumn major layout: err_bnds_norm[(err - 1)*nrhs + i -\n1].\n•\nRow major layout: err_bnds_norm[err - 1 + (i -\n1)*n_err_bnds]\nerr_bnds_comp\nArray of size nrhs*n_err_bnds. For each right-hand side, contains\ninformation about various error bounds and condition numbers\ncorresponding to the componentwise relative error, which is defined as\nfollows:\nComponentwise relative error in the i-th solution vector:\nThe array is indexed by the type of error information as described\nbelow. There are currently up to three pieces of information returned\nfor each right-hand side. If componentwise accuracy is not requested\n(params[2] = 0.0), then err_bnds_comp is not accessed. If\nn_err_bnds < 3, then at most the first n_err_bnds columns of the\nerr_bnds_comp array are returned.\nerr=1\n\"Trust/don't trust\" boolean. Trust the answer\nif the reciprocal condition number is less\nthan the threshold sqrt(n)*slamch(ε) for\nsingle precision flavors and\nsqrt(n)*dlamch(ε) for double precision\nflavors.\nerr=2\n\"Guaranteed\" error bound. The estimated\nforward error, almost certainly within a factor\nof 10 of the true error so long as the next\nentry is greater than the threshold\nsqrt(n)*slamch(ε) for single precision\nflavors and sqrt(n)*dlamch(ε) for double\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n587\n\n\nprecision flavors. This error bound should\nonly be trusted if the previous boolean is\ntrue.\nerr=3\nReciprocal condition number. Estimated\ncomponentwise reciprocal condition number.\nCompared with the threshold\nsqrt(n)*slamch(ε) for single precision\nflavors and sqrt(n)*dlamch(ε) for double\nprecision flavors to determine if the error\nestimate is \"guaranteed\". These reciprocal\ncondition numbers for some appropriately\nscaled matrix Z are:\nLet z=s*(a*diag(x)), where x is the\nsolution for the current right-hand side and s\nscales each row of a*diag(x) by a power of\nthe radix so all absolute row sums of z are\napproximately 1.\nThe information for right-hand side i, where 1 ≤i≤nrhs, and type of\nerror err is stored in:\n•\nColumn major layout: err_bnds_comp[(err - 1)*nrhs + i -\n1].\n•\nRow major layout: err_bnds_comp[err - 1 + (i -\n1)*n_err_bnds]\nparams\nOutput parameter only if the input contains erroneous values, namely,\nin params[0], params[1], params[2]. In such a case, the\ncorresponding elements of params are filled with default values on\noutput.\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful. The solution to every right-hand side is guaranteed.\nIf info = -i, parameter i had an illegal value.\nIf 0 < info≤n: Uinfo,info is exactly zero. The factorization has been completed, but the factor U is exactly\nsingular, so the solution and error bounds could not be computed; rcond = 0 is returned.\nIf info = n+j: The solution corresponding to the j-th right-hand side is not guaranteed. The solutions\ncorresponding to other right-hand sides k with k > j may not be guaranteed as well, but only the first such\nright-hand side is reported. If a small componentwise error is not requested params[2] = 0.0, then the j-th\nright-hand side is the first with a normwise error bound that is not guaranteed (the smallest j such that for\ncolumn major layout err_bnds_norm[j - 1] = 0.0 or err_bnds_comp[j - 1] = 0.0; or for row major\nlayout err_bnds_norm[(j - 1)*n_err_bnds] = 0.0 or err_bnds_comp[(j - 1)*n_err_bnds] = 0.0).\nSee the definition of err_bnds_norm and err_bnds_comp for err = 1. To get information about all of the\nright-hand sides, check err_bnds_norm or err_bnds_comp.\nSee Also\nMatrix Storage Schemes\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n588\n\n\n?gbrfs\nRefines the solution of a system of linear equations\nwith a general band coefficient matrix and estimates\nits error.\nSyntax\nlapack_int LAPACKE_sgbrfs( int matrix_layout, char trans, lapack_int n, lapack_int kl,\nlapack_int ku, lapack_int nrhs, const float* ab, lapack_int ldab, const float* afb,\nlapack_int ldafb, const lapack_int* ipiv, const float* b, lapack_int ldb, float* x,\nlapack_int ldx, float* ferr, float* berr );\nlapack_int LAPACKE_dgbrfs( int matrix_layout, char trans, lapack_int n, lapack_int kl,\nlapack_int ku, lapack_int nrhs, const double* ab, lapack_int ldab, const double* afb,\nlapack_int ldafb, const lapack_int* ipiv, const double* b, lapack_int ldb, double* x,\nlapack_int ldx, double* ferr, double* berr );\nlapack_int LAPACKE_cgbrfs( int matrix_layout, char trans, lapack_int n, lapack_int kl,\nlapack_int ku, lapack_int nrhs, const lapack_complex_float* ab, lapack_int ldab, const\nlapack_complex_float* afb, lapack_int ldafb, const lapack_int* ipiv, const\nlapack_complex_float* b, lapack_int ldb, lapack_complex_float* x, lapack_int ldx,\nfloat* ferr, float* berr );\nlapack_int LAPACKE_zgbrfs( int matrix_layout, char trans, lapack_int n, lapack_int kl,\nlapack_int ku, lapack_int nrhs, const lapack_complex_double* ab, lapack_int ldab, const\nlapack_complex_double* afb, lapack_int ldafb, const lapack_int* ipiv, const\nlapack_complex_double* b, lapack_int ldb, lapack_complex_double* x, lapack_int ldx,\ndouble* ferr, double* berr );\nInclude Files\n•\nmkl.h\nDescription\nThe routine performs an iterative refinement of the solution to a system of linear equations A*X = B or AT*X\n= B or AH*X = B with a band matrix A, with multiple right-hand sides. For each computed solution vector x,\nthe routine computes the component-wise backward errorβ. This error is the smallest relative perturbation\nin elements of A and b such that x is the exact solution of the perturbed system:\n|δaij| ≤β|aij|, |δbi| ≤β|bi| such that (A + δA)x = (b + δb).\nFinally, the routine estimates the component-wise forward error in the computed solution ||x - xe||∞/||\nx||∞ (here xe is the exact solution).\nBefore calling this routine:\n•\ncall the factorization routine ?gbtrf\n•\ncall the solver routine ?gbtrs.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\ntrans\nMust be 'N' or 'T' or 'C'.\nIndicates the form of the equations:\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n589\n\n\nIf trans = 'N', the system has the form A*X = B.\nIf trans = 'T', the system has the form AT*X = B.\nIf trans = 'C', the system has the form AH*X = B.\nn\nThe order of the matrix A; n≥ 0.\nkl\nThe number of sub-diagonals within the band of A; kl≥ 0.\nku\nThe number of super-diagonals within the band of A; ku≥ 0.\nnrhs\nThe number of right-hand sides; nrhs≥ 0.\nab,afb,b,x\nArrays:\nab(size max(1, ldab*n)) contains the original band matrix A, as\nsupplied to ?gbtrf, but stored in rows from 1 to kl + ku + 1 for\ncolumn major layout, and columns from 1 to kl + ku + 1 for row\nmajor layout.\nafb(size max(1, ldafb*n)) contains the factored band matrix A, as\nreturned by ?gbtrf.\nbof size max(1, ldb*nrhs) for column major layout and max(1,\nldb*n) for row major layout contains the right-hand side matrix B.\nxof size max(1, ldx*nrhs) for column major layout and max(1,\nldx*n) for row major layout contains the solution matrix X.\nldab\nThe leading dimension of ab, ldab≥kl + ku + 1.\nldafb\nThe leading dimension of afb, ldafb≥ 2*kl + ku + 1.\nldb\nThe leading dimension of b; ldb≥ max(1, n) for column major layout\nand ldb≥nrhs for row major layout.\nldx\nThe leading dimension of x; ldx≥ max(1, n).\nipiv\nArray, size at least max(1, n). The ipiv array, as returned by ?gbtrf.\nOutput Parameters\nx\nThe refined solution matrix X.\nferr, berr\nArrays, size at least max(1, nrhs). Contain the component-wise\nforward and backward errors, respectively, for each solution vector.\nReturn Values\nThis function returns a value info.\nIf info =0, the execution is successful.\nIf info = -i, parameter i had an illegal value.\nApplication Notes\nThe bounds returned in ferr are not rigorous, but in practice they almost always overestimate the actual\nerror.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n590\n\n\nFor each right-hand side, computation of the backward error involves a minimum of 4n(kl + ku) floating-\npoint operations (for real flavors) or 16n(kl + ku) operations (for complex flavors). In addition, each step of\niterative refinement involves 2n(4kl + 3ku) operations (for real flavors) or 8n(4kl + 3ku) operations (for\ncomplex flavors); the number of iterations may range from 1 to 5. Estimating the forward error involves\nsolving a number of systems of linear equations A*x = b; the number is usually 4 or 5 and never more than\n11. Each solution requires approximately 2n2 floating-point operations for real flavors or 8n2 for complex\nflavors.\nSee Also\nMatrix Storage Schemes\n?gbrfsx\nUses extra precise iterative refinement to improve the\nsolution to the system of linear equations with a\nbanded coefficient matrix A and provides error bounds\nand backward error estimates.\nSyntax\nlapack_int LAPACKE_sgbrfsx( int matrix_layout, char trans, char equed, lapack_int n,\nlapack_int kl, lapack_int ku, lapack_int nrhs, const float* ab, lapack_int ldab, const\nfloat* afb, lapack_int ldafb, const lapack_int* ipiv, const float* r, const float* c,\nconst float* b, lapack_int ldb, float* x, lapack_int ldx, float* rcond, float* berr,\nlapack_int n_err_bnds, float* err_bnds_norm, float* err_bnds_comp, lapack_int nparams,\nfloat* params );\nlapack_int LAPACKE_dgbrfsx( int matrix_layout, char trans, char equed, lapack_int n,\nlapack_int kl, lapack_int ku, lapack_int nrhs, const double* ab, lapack_int ldab, const\ndouble* afb, lapack_int ldafb, const lapack_int* ipiv, const double* r, const double*\nc, const double* b, lapack_int ldb, double* x, lapack_int ldx, double* rcond, double*\nberr, lapack_int n_err_bnds, double* err_bnds_norm, double* err_bnds_comp, lapack_int\nnparams, double* params );\nlapack_int LAPACKE_cgbrfsx( int matrix_layout, char trans, char equed, lapack_int n,\nlapack_int kl, lapack_int ku, lapack_int nrhs, const lapack_complex_float* ab,\nlapack_int ldab, const lapack_complex_float* afb, lapack_int ldafb, const lapack_int*\nipiv, const float* r, const float* c, const lapack_complex_float* b, lapack_int ldb,\nlapack_complex_float* x, lapack_int ldx, float* rcond, float* berr, lapack_int\nn_err_bnds, float* err_bnds_norm, float* err_bnds_comp, lapack_int nparams, float*\nparams );\nlapack_int LAPACKE_zgbrfsx( int matrix_layout, char trans, char equed, lapack_int n,\nlapack_int kl, lapack_int ku, lapack_int nrhs, const lapack_complex_double* ab,\nlapack_int ldab, const lapack_complex_double* afb, lapack_int ldafb, const lapack_int*\nipiv, const double* r, const double* c, const lapack_complex_double* b, lapack_int ldb,\nlapack_complex_double* x, lapack_int ldx, double* rcond, double* berr, lapack_int\nn_err_bnds, double* err_bnds_norm, double* err_bnds_comp, lapack_int nparams, double*\nparams );\nInclude Files\n•\nmkl.h\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n591\n\n\nDescription\nThe routine improves the computed solution to a system of linear equations and provides error bounds and\nbackward error estimates for the solution. In addition to a normwise error bound, the code provides a\nmaximum componentwise error bound, if possible. See comments for err_bnds_norm and err_bnds_comp\nfor details of the error bounds.\nThe original system of linear equations may have been equilibrated before calling this routine, as described\nby the parameters equed, r, and c below. In this case, the solution and error bounds returned are for the\noriginal unequilibrated system.\nInput Parameters\nmatrix_layout\nSpecifies whether two-dimensional array storage is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\ntrans\nMust be 'N', 'T', or 'C'.\nSpecifies the form of the system of equations:\nIf trans = 'N', the system has the form A*X = B (No transpose).\nIf trans = 'T', the system has the form AT*X = B (Transpose).\nIf trans = 'C', the system has the form AH*X = B (Conjugate transpose\nfor complex flavors, Transpose for real flavors).\nequed\nMust be 'N', 'R', 'C', or 'B'.\nSpecifies the form of equilibration that was done to A before calling this\nroutine.\nIf equed = 'N', no equilibration was done.\nIf equed = 'R', row equilibration was done, that is, A has been\npremultiplied by diag(r).\nIf equed = 'C', column equilibration was done, that is, A has been\npostmultiplied by diag(c).\nIf equed = 'B', both row and column equilibration was done, that is, A has\nbeen replaced by diag(r)*A*diag(c). The right-hand side B has been\nchanged accordingly.\nn\nThe number of linear equations; the order of the matrix A; n≥ 0.\nkl\nThe number of subdiagonals within the band of A; kl≥ 0.\nku\nThe number of superdiagonals within the band of A; ku≥ 0.\nnrhs\nThe number of right-hand sides; the number of columns of the matrices B\nand X; nrhs≥ 0.\nab, afb, b\nThe array abof size max(1, ldab*n) contains the original matrix A in band\nstorage, in rows from 1 to kl+ku + 1 for column major layout, and in\ncolumns from 1 to kl+ku + 1 for row major layout.\nThe array afbof size max(1, ldafb*n) contains details of the LU\nfactorization of the banded matrix A as computed by ?gbtrf.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n592\n\n\nThe array bof size max(1, ldb*nrhs) for column major layout and max(1,\nldb*n) for row major layout contains the matrix B whose columns are the\nright-hand sides for the systems of equations.\nldab\nThe leading dimension of the array ab; ldab≥kl+ku+1.\nldafb\nThe leading dimension of the array afb; ldafb≥ 2*kl+ku+1.\nipiv\nArray, size at least max(1, n). Contains the pivot indices as computed\nby ?gbtrf; for row 1 ≤i≤n, row i of the matrix was interchanged with row\nipiv[i-1].\nr, c\nArrays: r(n), c(n). The array r contains the row scale factors for A, and\nthe array c contains the column scale factors for A.\nIf equed = 'R' or 'B', A is multiplied on the left by diag(r); if equed =\n'N' or 'C', r is not accessed.\nIf equed = 'R' or 'B', each element of r must be positive.\nIf equed = 'C' or 'B', A is multiplied on the right by diag(c); if equed =\n'N' or 'R', c is not accessed.\nIf equed = 'C' or 'B', each element of c must be positive.\nEach element of r or c should be a power of the radix to ensure a reliable\nsolution and error estimates. Scaling by powers of the radix does not cause\nrounding errors unless the result underflows or overflows. Rounding errors\nduring scaling lead to refining with a matrix that is not equivalent to the\ninput matrix, producing error estimates that may not be reliable.\nldb\nThe leading dimension of the array b; ldb≥ max(1, n) for column major\nlayout and ldb≥nrhs for row major layout.\nx\nArray, size max(1, ldx*nrhs) for column major layout and max(1, ldx*n)\nfor row major layout.\nThe solution matrix X as computed by sgbtrs/dgbtrs for real flavors or \ncgbtrs/zgbtrs for complex flavors.\nldx\nThe leading dimension of the output array x; ldx≥ max(1, n) for column\nmajor layout and ldx≥nrhs for row major layout.\nn_err_bnds\nNumber of error bounds to return for each right-hand side and each type\n(normwise or componentwise). See err_bnds_norm and err_bnds_comp\ndescriptions in Output Arguments section below.\nnparams\nSpecifies the number of parameters set in params. If ≤ 0, the params array\nis never referenced and default values are used.\nparams\nArray, size nparams. Specifies algorithm parameters. If an entry is less than\n0.0, that entry will be filled with the default value used for that parameter.\nOnly positions up to nparams are accessed; defaults are used for higher-\nnumbered parameters. If defaults are acceptable, you can pass nparams =\n0, which prevents the source code from accessing the params argument.\nparams[0] : Whether to perform iterative refinement or not. Default: 1.0\n(for single precision flavors), 1.0D+0 (for double precision flavors).\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n593\n\n\n=0.0\nNo refinement is performed and no error bounds\nare computed.\n=1.0\nUse the double-precision refinement algorithm,\npossibly with doubled-single computations if the\ncompilation environment does not support\ndouble precision.\n(Other values are reserved for future use.)\nparams[1] : Maximum number of residual computations allowed for\nrefinement.\nDefault\n10.0\nAggressive\nSet to 100.0 to permit convergence using\napproximate factorizations or factorizations\nother than LU. If the factorization uses a\ntechnique other than Gaussian elimination, the\nguarantees in err_bnds_norm and\nerr_bnds_comp may no longer be trustworthy.\nparams[2] : Flag determining if the code will attempt to find a solution\nwith a small componentwise relative error in the double-precision algorithm.\nPositive is true, 0.0 is false. Default: 1.0 (attempt componentwise\nconvergence).\nOutput Parameters\nx\nThe improved solution matrix X.\nrcond\nReciprocal scaled condition number. An estimate of the reciprocal Skeel\ncondition number of the matrix A after equilibration (if done). If rcond is\nless than the machine precision, in particular, if rcond = 0, the matrix is\nsingular to working precision. Note that the error may still be small even if\nthis number is very small and the matrix appears ill-conditioned.\nberr\nArray, size at least max(1, nrhs). Contains the componentwise relative\nbackward error for each solution vector xj, that is, the smallest relative\nchange in any element of A or B that makes xj an exact solution.\nerr_bnds_norm\nArray of size nrhs*n_err_bnds. For each right-hand side, contains\ninformation about various error bounds and condition numbers\ncorresponding to the normwise relative error, which is defined as follows:\nNormwise relative error in the i-th solution vector\nThe array is indexed by the type of error information as described below.\nThere are currently up to three pieces of information returned.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n594\n\n\nerr=1\n\"Trust/don't trust\" boolean. Trust the answer if\nthe reciprocal condition number is less than the\nthreshold sqrt(n)*slamch(ε) for single\nprecision flavors and sqrt(n)*dlamch(ε) for\ndouble precision flavors.\nerr=2\n\"Guaranteed\" error bound. The estimated\nforward error, almost certainly within a factor of\n10 of the true error so long as the next entry is\ngreater than the threshold sqrt(n)*slamch(ε)\nfor single precision flavors and\nsqrt(n)*dlamch(ε) for double precision\nflavors. This error bound should only be trusted\nif the previous boolean is true.\nerr=3\nReciprocal condition number. Estimated\nnormwise reciprocal condition number.\nCompared with the threshold\nsqrt(n)*slamch(ε) for single precision flavors\nand sqrt(n)*dlamch(ε) for double precision\nflavors to determine if the error estimate is\n\"guaranteed\". These reciprocal condition\nnumbers for some appropriately scaled matrix Z\nare\nLet z=s*a, where s scales each row by a power\nof the radix so all absolute row sums of z are\napproximately 1.\nThe information for right-hand side i, where 1 ≤i≤nrhs, and type of error\nerr is stored in:\n•\nColumn major layout: err_bnds_norm[(err - 1)*nrhs + i - 1].\n•\nRow major layout: err_bnds_norm[err - 1 + (i - 1)*n_err_bnds]\nerr_bnds_comp\nArray of size nrhs*n_err_bnds. For each right-hand side, contains\ninformation about various error bounds and condition numbers\ncorresponding to the componentwise relative error, which is defined as\nfollows:\nComponentwise relative error in the i-th solution vector:\nThe array is indexed by the type of error information as described below.\nThere are currently up to three pieces of information returned for each\nright-hand side. If componentwise accuracy is not requested (params[2] =\n0.0), then err_bnds_comp is not accessed.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n595\n\n\nerr=1\n\"Trust/don't trust\" boolean. Trust the answer if\nthe reciprocal condition number is less than the\nthreshold sqrt(n)*slamch(ε) for single\nprecision flavors and sqrt(n)*dlamch(ε) for\ndouble precision flavors.\nerr=2\n\"Guaranteed\" error bpound. The estimated\nforward error, almost certainly within a factor of\n10 of the true error so long as the next entry is\ngreater than the threshold sqrt(n)*slamch(ε)\nfor single precision flavors and\nsqrt(n)*dlamch(ε) for double precision\nflavors. This error bound should only be trusted\nif the previous boolean is true.\nerr=3\nReciprocal condition number. Estimated\ncomponentwise reciprocal condition number.\nCompared with the threshold\nsqrt(n)*slamch(ε) for single precision flavors\nand sqrt(n)*dlamch(ε) for double precision\nflavors to determine if the error estimate is\n\"guaranteed\". These reciprocal condition\nnumbers for some appropriately scaled matrix Z\nare\nLet z=s*(a*diag(x)), where x is the solution\nfor the current right-hand side and s scales each\nrow of a*diag(x) by a power of the radix so all\nabsolute row sums of z are approximately 1.\nThe information for right-hand side i, where 1 ≤i≤nrhs, and type of error\nerr is stored in:\n•\nColumn major layout: err_bnds_comp[(err - 1)*nrhs + i - 1].\n•\nRow major layout: err_bnds_comp[err - 1 + (i - 1)*n_err_bnds]\nparams\nOutput parameter only if the input contains erroneous values, namely, in\nparams[0], params[1], and params[2]. In such a case, the corresponding\nelements of params are filled with default values on output.\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful. The solution to every right-hand side is guaranteed.\nIf info = -i, parameter i had an illegal value.\nIf 0 < info≤n: Uinfo,info is exactly zero. The factorization has been completed, but the factor U is exactly\nsingular, so the solution and error bounds could not be computed; rcond = 0 is returned.\nIf info = n+j: The solution corresponding to the j-th right-hand side is not guaranteed. The solutions\ncorresponding to other right-hand sides k with k > j may not be guaranteed as well, but only the first such\nright-hand side is reported. If a small componentwise error is not requested params[2] = 0.0, then the j-th\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n596\n\n\nright-hand side is the first with a normwise error bound that is not guaranteed (the smallest j such that for\ncolumn major layout err_bnds_norm[j - 1] = 0.0 or err_bnds_comp[j - 1] = 0.0; or for row major\nlayout err_bnds_norm[(j - 1)*n_err_bnds] = 0.0 or err_bnds_comp[(j - 1)*n_err_bnds] = 0.0).\nSee the definition of err_bnds_norm and err_bnds_comp for err = 1. To get information about all of the\nright-hand sides, check err_bnds_norm or err_bnds_comp.\nSee Also\nMatrix Storage Schemes\n?gtrfs\nRefines the solution of a system of linear equations\nwith a tridiagonal coefficient matrix and estimates its\nerror.\nSyntax\nlapack_int LAPACKE_sgtrfs( int matrix_layout, char trans, lapack_int n, lapack_int\nnrhs, const float* dl, const float* d, const float* du, const float* dlf, const float*\ndf, const float* duf, const float* du2, const lapack_int* ipiv, const float* b,\nlapack_int ldb, float* x, lapack_int ldx, float* ferr, float* berr );\nlapack_int LAPACKE_dgtrfs( int matrix_layout, char trans, lapack_int n, lapack_int\nnrhs, const double* dl, const double* d, const double* du, const double* dlf, const\ndouble* df, const double* duf, const double* du2, const lapack_int* ipiv, const double*\nb, lapack_int ldb, double* x, lapack_int ldx, double* ferr, double* berr );\nlapack_int LAPACKE_cgtrfs( int matrix_layout, char trans, lapack_int n, lapack_int\nnrhs, const lapack_complex_float* dl, const lapack_complex_float* d, const\nlapack_complex_float* du, const lapack_complex_float* dlf, const lapack_complex_float*\ndf, const lapack_complex_float* duf, const lapack_complex_float* du2, const lapack_int*\nipiv, const lapack_complex_float* b, lapack_int ldb, lapack_complex_float* x,\nlapack_int ldx, float* ferr, float* berr );\nlapack_int LAPACKE_zgtrfs( int matrix_layout, char trans, lapack_int n, lapack_int\nnrhs, const lapack_complex_double* dl, const lapack_complex_double* d, const\nlapack_complex_double* du, const lapack_complex_double* dlf, const\nlapack_complex_double* df, const lapack_complex_double* duf, const\nlapack_complex_double* du2, const lapack_int* ipiv, const lapack_complex_double* b,\nlapack_int ldb, lapack_complex_double* x, lapack_int ldx, double* ferr, double* berr );\nInclude Files\n•\nmkl.h\nDescription\nThe routine performs an iterative refinement of the solution to a system of linear equations A*X = B or AT*X\n= B or AH*X = B with a tridiagonal matrix A, with multiple right-hand sides. For each computed solution\nvector x, the routine computes the component-wise backward errorβ. This error is the smallest relative\nperturbation in elements of A and b such that x is the exact solution of the perturbed system:\n|δaij|/|aij| ≤β|aij|, |δbi|/|bi| ≤β|bi| such that (A + δA)x = (b + δb).\nFinally, the routine estimates the component-wise forward error in the computed solution ||x - xe||∞/||\nx||∞ (here xe is the exact solution).\nBefore calling this routine:\n•\ncall the factorization routine ?gttrf\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n597\n\n\n•\ncall the solver routine ?gttrs.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\ntrans\nMust be 'N' or 'T' or 'C'.\nIndicates the form of the equations:\nIf trans = 'N', the system has the form A*X = B.\nIf trans = 'T', the system has the form AT*X = B.\nIf trans = 'C', the system has the form AH*X = B.\nn\nThe order of the matrix A; n≥ 0.\nnrhs\nThe number of right-hand sides, that is, the number of columns of the\nmatrix B; nrhs≥ 0.\ndl\nArray dl of size n -1 contains the subdiagonal elements of A.\nd\nArray d of size n contains the diagonal elements of A.\ndu\nArray du of size n -1 contains the superdiagonal elements of A.\ndlf\nArray dlf of size n -1 contains the (n - 1) multipliers that define the\nmatrix L from the LU factorization of A as computed by ?gttrf.\ndf\nArray df of size n contains the n diagonal elements of the upper\ntriangular matrix U from the LU factorization of A.\nduf\nArray duf of size n -1 contains the (n - 1) elements of the first\nsuperdiagonal of U.\ndu2\nArray du2 of size n -2 contains the (n - 2) elements of the second\nsuperdiagonal of U.\nb\nArray b (size max(1,ldb*nrhs) for column major layout and max(1,\nldb*n) for row major layout) contains the right-hand side matrix B.\nx\nArray x (size max(1,ldx*nrhs) for column major layout and max(1,\nldx*n) contains the solution matrix X, as computed by ?gttrs.\nldb\nThe leading dimension of b; ldb≥ max(1, n) for column major layout\nand ldb≥nrhs for row major layout.\nldx\nThe leading dimension of x; ldx≥ max(1, n) for column major layout\nand ldx≥nrhs for row major layout.\nipiv\nArray, size at least max(1, n). The ipiv array, as returned by ?gttrf.\nOutput Parameters\nx\nThe refined solution matrix X.\nferr, berr\nArrays, size at least max(1,nrhs). Contain the component-wise\nforward and backward errors, respectively, for each solution vector.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n598\n\n\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info = -i, parameter i had an illegal value.\nSee Also\nMatrix Storage Schemes\n?porfs\nRefines the solution of a system of linear equations\nwith a symmetric (Hermitian) positive-definite\ncoefficient matrix and estimates its error.\nSyntax\nlapack_int LAPACKE_sporfs( int matrix_layout, char uplo, lapack_int n, lapack_int nrhs,\nconst float* a, lapack_int lda, const float* af, lapack_int ldaf, const float* b,\nlapack_int ldb, float* x, lapack_int ldx, float* ferr, float* berr );\nlapack_int LAPACKE_dporfs( int matrix_layout, char uplo, lapack_int n, lapack_int nrhs,\nconst double* a, lapack_int lda, const double* af, lapack_int ldaf, const double* b,\nlapack_int ldb, double* x, lapack_int ldx, double* ferr, double* berr );\nlapack_int LAPACKE_cporfs( int matrix_layout, char uplo, lapack_int n, lapack_int nrhs,\nconst lapack_complex_float* a, lapack_int lda, const lapack_complex_float* af,\nlapack_int ldaf, const lapack_complex_float* b, lapack_int ldb, lapack_complex_float*\nx, lapack_int ldx, float* ferr, float* berr );\nlapack_int LAPACKE_zporfs( int matrix_layout, char uplo, lapack_int n, lapack_int nrhs,\nconst lapack_complex_double* a, lapack_int lda, const lapack_complex_double* af,\nlapack_int ldaf, const lapack_complex_double* b, lapack_int ldb, lapack_complex_double*\nx, lapack_int ldx, double* ferr, double* berr );\nInclude Files\n•\nmkl.h\nDescription\nThe routine performs an iterative refinement of the solution to a system of linear equations A*X = B with a\nsymmetric (Hermitian) positive definite matrix A, with multiple right-hand sides. For each computed solution\nvector x, the routine computes the component-wise backward errorβ. This error is the smallest relative\nperturbation in elements of A and b such that x is the exact solution of the perturbed system:\n|δaij| ≤β|aij|, |δbi| ≤β|bi| such that (A + δA)x = (b + δb).\nFinally, the routine estimates the component-wise forward error in the computed solution ||x - xe||∞/||\nx||∞ (here xe is the exact solution).\nBefore calling this routine:\n•\ncall the factorization routine ?potrf\n•\ncall the solver routine ?potrs.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n599\n\n\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\nuplo\nMust be 'U' or 'L'.\nIf uplo = 'U', the upper triangle of A is stored.\nIf uplo = 'L', the lower triangle of A is stored.\nn\nThe order of the matrix A; n≥ 0.\nnrhs\nThe number of right-hand sides; nrhs≥ 0.\na\nArray a (size max(1, lda*n)) contains the original matrix A, as\nsupplied to ?potrf.\naf\nArray af (size max(1, ldaf*n)) contains the factored matrix A, as\nreturned by ?potrf.\nb\nArray bof size max(1, ldb*nrhs) for column major layout and max(1,\nldb*n) for row major layout contains the right-hand side matrix B.\nThe second dimension of b must be at least max(1, nrhs).\nx\nArray x of size max(1, ldx*nrhs) for column major layout and max(1,\nldx*n) for row major layout contains the solution matrix X.\nlda\nThe leading dimension of a; lda≥ max(1, n).\nldaf\nThe leading dimension of af; ldaf≥ max(1, n).\nldb\nThe leading dimension of b; ldb≥ max(1, n) for column major layout\nand ldb≥nrhs for row major layout.\nldx\nThe leading dimension of x; ldx≥ max(1, n) for column major layout\nand ldx≥nrhs for row major layout.\nOutput Parameters\nx\nThe refined solution matrix X.\nferr, berr\nArrays, size at least max(1, nrhs). Contain the component-wise\nforward and backward errors, respectively, for each solution vector.\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info = -i, parameter i had an illegal value.\nApplication Notes\nThe bounds returned in ferr are not rigorous, but in practice they almost always overestimate the actual\nerror.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n600\n\n\nFor each right-hand side, computation of the backward error involves a minimum of 4n2 floating-point\noperations (for real flavors) or 16n2 operations (for complex flavors). In addition, each step of iterative\nrefinement involves 6n2 operations (for real flavors) or 24n2 operations (for complex flavors); the number of\niterations may range from 1 to 5. Estimating the forward error involves solving a number of systems of linear\nequations A*x = b; the number is usually 4 or 5 and never more than 11. Each solution requires\napproximately 2n2 floating-point operations for real flavors or 8n2 for complex flavors.\nSee Also\nMatrix Storage Schemes\n?porfsx\nUses extra precise iterative refinement to improve the\nsolution to the system of linear equations with a\nsymmetric/Hermitian positive-definite coefficient\nmatrix A and provides error bounds and backward\nerror estimates.\nSyntax\nlapack_int LAPACKE_sporfsx( int matrix_layout, char uplo, char equed, lapack_int n,\nlapack_int nrhs, const float* a, lapack_int lda, const float* af, lapack_int ldaf,\nconst float* s, const float* b, lapack_int ldb, float* x, lapack_int ldx, float* rcond,\nfloat* berr, lapack_int n_err_bnds, float* err_bnds_norm, float* err_bnds_comp,\nlapack_int nparams, float* params );\nlapack_int LAPACKE_dporfsx( int matrix_layout, char uplo, char equed, lapack_int n,\nlapack_int nrhs, const double* a, lapack_int lda, const double* af, lapack_int ldaf,\nconst double* s, const double* b, lapack_int ldb, double* x, lapack_int ldx, double*\nrcond, double* berr, lapack_int n_err_bnds, double* err_bnds_norm, double*\nerr_bnds_comp, lapack_int nparams, double* params );\nlapack_int LAPACKE_cporfsx( int matrix_layout, char uplo, char equed, lapack_int n,\nlapack_int nrhs, const lapack_complex_float* a, lapack_int lda, const\nlapack_complex_float* af, lapack_int ldaf, const float* s, const lapack_complex_float*\nb, lapack_int ldb, lapack_complex_float* x, lapack_int ldx, float* rcond, float* berr,\nlapack_int n_err_bnds, float* err_bnds_norm, float* err_bnds_comp, lapack_int nparams,\nfloat* params );\nlapack_int LAPACKE_zporfsx( int matrix_layout, char uplo, char equed, lapack_int n,\nlapack_int nrhs, const lapack_complex_double* a, lapack_int lda, const\nlapack_complex_double* af, lapack_int ldaf, const double* s, const\nlapack_complex_double* b, lapack_int ldb, lapack_complex_double* x, lapack_int ldx,\ndouble* rcond, double* berr, lapack_int n_err_bnds, double* err_bnds_norm, double*\nerr_bnds_comp, lapack_int nparams, double* params );\nInclude Files\n•\nmkl.h\nDescription\nThe routine improves the computed solution to a system of linear equations and provides error bounds and\nbackward error estimates for the solution. In addition to a normwise error bound, the code provides a\nmaximum componentwise error bound, if possible. See comments for err_bnds_norm and err_bnds_comp\nfor details of the error bounds.\nThe original system of linear equations may have been equilibrated before calling this routine, as described\nby the parameters equed and s below. In this case, the solution and error bounds returned are for the\noriginal unequilibrated system.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n601\n\n\nInput Parameters\nmatrix_layout\nSpecifies whether two-dimensional array storage is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\nuplo\nMust be 'U' or 'L'.\nIndicates whether the upper or lower triangular part of A is stored:\nIf uplo = 'U', the upper triangle of A is stored.\nIf uplo = 'L', the lower triangle of A is stored.\nequed\nMust be 'N' or 'Y'.\nSpecifies the form of equilibration that was done to A before calling this\nroutine.\nIf equed = 'N', no equilibration was done.\nIf equed = 'Y', both row and column equilibration was done, that is, A has\nbeen replaced by diag(s)*A*diag(s). The right-hand side B has been\nchanged accordingly.\nn\nThe number of linear equations; the order of the matrix A; n≥ 0.\nnrhs\nThe number of right-hand sides; the number of columns of the matrices B\nand X; nrhs≥ 0.\na\nThe array a (size max(1, lda*n)) contains the symmetric/Hermitian matrix\nA as specified by uplo. If uplo = 'U', the leading n-by-n upper triangular\npart of a contains the upper triangular part of the matrix A and the strictly\nlower triangular part of a is not referenced. If uplo = 'L', the leading n-\nby-n lower triangular part of a contains the lower triangular part of the\nmatrix A and the strictly upper triangular part of a is not referenced.\naf\nThe array af (size max(1, ldaf*n)) contains the triangular factor L or U\nfrom the Cholesky factorization A = UT*U or A = L*LT as computed by \nspotrf for real flavors or dpotrf for complex flavors.\nb\nThe array b (size max(1, ldb*nrhs for column major layout and max(1,\nldb*n) for row major layout) contains the matrix B whose columns are the\nright-hand sides for the systems of equations.\nlda\nThe leading dimension of a; lda≥ max(1, n).\nldaf\nThe leading dimension of af; ldaf≥ max(1, n).\ns\nArray of size n. The array s contains the scale factors for A.\nIf equed = 'N', s is not accessed.\nIf equed = 'Y', each element of s must be positive.\nEach element of s should be a power of the radix to ensure a reliable\nsolution and error estimates. Scaling by powers of the radix does not cause\nrounding errors unless the result underflows or overflows. Rounding errors\nduring scaling lead to refining with a matrix that is not equivalent to the\ninput matrix, producing error estimates that may not be reliable.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n602\n\n\nldb\nThe leading dimension of the array b; ldb≥ max(1, n) for column major\nlayout and ldb≥nrhs for row major layout.\nx\nArray, size max(1, ldx*nrhs) for column major layout and max(1, ldx*n)\nfor row major layout.\nThe solution matrix X as computed by ?potrs\nldx\nThe leading dimension of the output array x; ldx≥ max(1, n) for column\nmajor layout and ldx≥nrhs for row major layout.\nn_err_bnds\nNumber of error bounds to return for each right hand side and each type\n(normwise or componentwise). See err_bnds_norm and err_bnds_comp\ndescriptions in Output Arguments section below.\nnparams\nSpecifies the number of parameters set in params. If ≤ 0, the params array\nis never referenced and default values are used.\nparams\nArray, size nparams. Specifies algorithm parameters. If an entry is less than\n0.0, that entry will be filled with the default value used for that parameter.\nOnly positions up to nparams are accessed; defaults are used for higher-\nnumbered parameters. If defaults are acceptable, you can pass nparams =\n0, which prevents the source code from accessing the params argument.\nparams[0] : Whether to perform iterative refinement or not. Default: 1.0\n(for single precision flavors), 1.0D+0 (for double precision flavors).\n=0.0\nNo refinement is performed and no error bounds\nare computed.\n=1.0\nUse the double-precision refinement algorithm,\npossibly with doubled-single computations if the\ncompilation environment does not support\ndouble precision.\n(Other values are reserved for future use.)\nparams[1] : Maximum number of residual computations allowed for\nrefinement.\nDefault\n10.0\nAggressive\nSet to 100.0 to permit convergence using\napproximate factorizations or factorizations\nother than LU. If the factorization uses a\ntechnique other than Gaussian elimination, the\nguarantees in err_bnds_norm and\nerr_bnds_comp may no longer be trustworthy.\nparams[2] : Flag determining if the code will attempt to find a solution\nwith a small componentwise relative error in the double-precision algorithm.\nPositive is true, 0.0 is false. Default: 1.0 (attempt componentwise\nconvergence).\nOutput Parameters\nx\nThe improved solution matrix X.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n603\n\n\nrcond\nReciprocal scaled condition number. An estimate of the reciprocal Skeel\ncondition number of the matrix A after equilibration (if done). If rcond is\nless than the machine precision, in particular, if rcond = 0, the matrix is\nsingular to working precision. Note that the error may still be small even if\nthis number is very small and the matrix appears ill-conditioned.\nberr\nArray, size at least max(1, nrhs). Contains the componentwise relative\nbackward error for each solution vector xj, that is, the smallest relative\nchange in any element of A or B that makes xj an exact solution.\nerr_bnds_norm\nArray of size nrhs*n_err_bnds. For each right-hand side, contains\ninformation about various error bounds and condition numbers\ncorresponding to the normwise relative error, which is defined as follows:\nNormwise relative error in the i-th solution vector\nThe array is indexed by the type of error information as described below.\nThere are currently up to three pieces of information returned.\nerr=1\n\"Trust/don't trust\" boolean. Trust the answer if\nthe reciprocal condition number is less than the\nthreshold sqrt(n)*slamch(ε) for single\nprecision flavors and sqrt(n)*dlamch(ε) for\ndouble precision flavors.\nerr=2\n\"Guaranteed\" error bound. The estimated\nforward error, almost certainly within a factor of\n10 of the true error so long as the next entry is\ngreater than the threshold sqrt(n)*slamch(ε)\nfor single precision flavors and\nsqrt(n)*dlamch(ε) for double precision\nflavors. This error bound should only be trusted\nif the previous boolean is true.\nerr=3\nReciprocal condition number. Estimated\nnormwise reciprocal condition number.\nCompared with the threshold\nsqrt(n)*slamch(ε) for single precision flavors\nand sqrt(n)*dlamch(ε) for double precision\nflavors to determine if the error estimate is\n\"guaranteed\". These reciprocal condition\nnumbers for some appropriately scaled matrix Z\nare\nLet z=s*a, where s scales each row by a power\nof the radix so all absolute row sums of z are\napproximately 1.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n604\n\n\nThe information for right-hand side i, where 1 ≤i≤nrhs, and type of error\nerr is stored in:\n•\nColumn major layout: err_bnds_norm[(err - 1)*nrhs + i - 1].\n•\nRow major layout: err_bnds_norm[err - 1 + (i - 1)*n_err_bnds]\nerr_bnds_comp\nArray of size nrhs*n_err_bnds. For each right-hand side, contains\ninformation about various error bounds and condition numbers\ncorresponding to the componentwise relative error, which is defined as\nfollows:\nComponentwise relative error in the i-th solution vector:\nThe array is indexed by the type of error information as described below.\nThere are currently up to three pieces of information returned for each\nright-hand side. If componentwise accuracy is not requested (params[2] =\n0.0), then err_bnds_comp is not accessed.\nerr=1\n\"Trust/don't trust\" boolean. Trust the answer if\nthe reciprocal condition number is less than the\nthreshold sqrt(n)*slamch(ε) for single\nprecision flavors and sqrt(n)*dlamch(ε) for\ndouble precision flavors.\nerr=2\n\"Guaranteed\" error bpound. The estimated\nforward error, almost certainly within a factor of\n10 of the true error so long as the next entry is\ngreater than the threshold sqrt(n)*slamch(ε)\nfor single precision flavors and\nsqrt(n)*dlamch(ε) for double precision\nflavors. This error bound should only be trusted\nif the previous boolean is true.\nerr=3\nReciprocal condition number. Estimated\ncomponentwise reciprocal condition number.\nCompared with the threshold\nsqrt(n)*slamch(ε) for single precision flavors\nand sqrt(n)*dlamch(ε) for double precision\nflavors to determine if the error estimate is\n\"guaranteed\". These reciprocal condition\nnumbers for some appropriately scaled matrix Z\nare\nLet z=s*(a*diag(x)), where x is the solution\nfor the current right-hand side and s scales each\nrow of a*diag(x) by a power of the radix so all\nabsolute row sums of z are approximately 1.\nThe information for right-hand side i, where 1 ≤i≤nrhs, and type of error\nerr is stored in:\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n605\n\n\n•\nColumn major layout: err_bnds_comp[(err - 1)*nrhs + i - 1].\n•\nRow major layout: err_bnds_comp[err - 1 + (i - 1)*n_err_bnds]\nparams\nOutput parameter only if the input contains erroneous values, namely in\nparams[0], params[1], or params[2]. In such a case, the corresponding\nelements of params are filled with default values on output.\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful. The solution to every right-hand side is guaranteed.\nIf info = -i, parameter i had an illegal value.\nIf 0 < info≤n: Uinfo,info is exactly zero. The factorization has been completed, but the factor U is exactly\nsingular, so the solution and error bounds could not be computed; rcond = 0 is returned.\nIf info = n+j: The solution corresponding to the j-th right-hand side is not guaranteed. The solutions\ncorresponding to other right-hand sides k with k > j may not be guaranteed as well, but only the first such\nright-hand side is reported. If a small componentwise error is not requested params[2] = 0.0, then the j-th\nright-hand side is the first with a normwise error bound that is not guaranteed (the smallest j such that for\ncolumn major layout err_bnds_norm[j - 1] = 0.0 or err_bnds_comp[j - 1] = 0.0; or for row major\nlayout err_bnds_norm[(j - 1)*n_err_bnds] = 0.0 or err_bnds_comp[(j - 1)*n_err_bnds] = 0.0).\nSee the definition of err_bnds_norm and err_bnds_comp for err = 1. To get information about all of the\nright-hand sides, check err_bnds_norm or err_bnds_comp.\nSee Also\nMatrix Storage Schemes\n?pprfs\nRefines the solution of a system of linear equations\nwith a symmetric (Hermitian) positive-definite\ncoefficient matrix stored in a packed format and\nestimates its error.\nSyntax\nlapack_int LAPACKE_spprfs( int matrix_layout, char uplo, lapack_int n, lapack_int nrhs,\nconst float* ap, const float* afp, const float* b, lapack_int ldb, float* x, lapack_int\nldx, float* ferr, float* berr );\nlapack_int LAPACKE_dpprfs( int matrix_layout, char uplo, lapack_int n, lapack_int nrhs,\nconst double* ap, const double* afp, const double* b, lapack_int ldb, double* x,\nlapack_int ldx, double* ferr, double* berr );\nlapack_int LAPACKE_cpprfs( int matrix_layout, char uplo, lapack_int n, lapack_int nrhs,\nconst lapack_complex_float* ap, const lapack_complex_float* afp, const\nlapack_complex_float* b, lapack_int ldb, lapack_complex_float* x, lapack_int ldx,\nfloat* ferr, float* berr );\nlapack_int LAPACKE_zpprfs( int matrix_layout, char uplo, lapack_int n, lapack_int nrhs,\nconst lapack_complex_double* ap, const lapack_complex_double* afp, const\nlapack_complex_double* b, lapack_int ldb, lapack_complex_double* x, lapack_int ldx,\ndouble* ferr, double* berr );\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n606\n\n\nInclude Files\n•\nmkl.h\nDescription\nThe routine performs an iterative refinement of the solution to a system of linear equations A*X = B with a\nsymmetric (Hermitian) positive definite matrix A, with multiple right-hand sides. For each computed solution\nvector x, the routine computes the component-wise backward errorβ. This error is the smallest relative\nperturbation in elements of A and b such that x is the exact solution of the perturbed system:\n|δaij| ≤β|aij|, |δbi| ≤β|bi| such that (A + δA)x = (b + δb).\nFinally, the routine estimates the component-wise forward error in the computed solution\n||x - xe||∞/||x||∞\nwhere xe is the exact solution.\nBefore calling this routine:\n•\ncall the factorization routine ?pptrf\n•\ncall the solver routine ?pptrs.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\nuplo\nMust be 'U' or 'L'.\nIndicates how the input matrix A has been factored:\nIf uplo = 'U', the upper triangle of A is stored.\nIf uplo = 'L', the lower triangle of A is stored.\nn\nThe order of the matrix A; n≥ 0.\nnrhs\nThe number of right-hand sides; nrhs≥ 0.\nap\nap contains the original matrix A in a packed format, as supplied\nto ?pptrf. The dimension of ap must be at least max(1,n(n+1)/2).\nafp\nafp contains the factored matrix A in a packed format, as returned\nby ?pptrf. The dimension of afp must be at least max(1,n(n+1)/2).\nb\nArray b of size max(1, ldb*nrhs) for column major layout and\nmax(1, ldb*n) for row major layout contains the right-hand side\nmatrix B.\nx\nArray x of size max(1, ldx*nrhs) for column major layout and max(1,\nldx*n) for row major layout contains the solution matrix X.\nldb\nThe leading dimension of b; ldb≥ max(1, n) for column major layout\nand ldb≥nrhs for row major layout.\nldx\nThe leading dimension of x; ldx≥ max(1, n) for column major layout\nand ldx≥nrhs for row major layout.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n607\n\n\nOutput Parameters\nx\nThe refined solution matrix X.\nferr, berr\nArrays, size at least max(1, nrhs). Contain the component-wise\nforward and backward errors, respectively, for each solution vector.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, parameter i had an illegal value.\nApplication Notes\nThe bounds returned in ferr are not rigorous, but in practice they almost always overestimate the actual\nerror.\nFor each right-hand side, computation of the backward error involves a minimum of 4n2 floating-point\noperations (for real flavors) or 16n2 operations (for complex flavors). In addition, each step of iterative\nrefinement involves 6n2 operations (for real flavors) or 24n2 operations (for complex flavors); the number of\niterations may range from 1 to 5.\nEstimating the forward error involves solving a number of systems of linear equations A*x = b; the number\nof systems is usually 4 or 5 and never more than 11. Each solution requires approximately 2n2 floating-point\noperations for real flavors or 8n2 for complex flavors.\nSee Also\nMatrix Storage Schemes\n?pbrfs\nRefines the solution of a system of linear equations\nwith a band symmetric (Hermitian) positive-definite\ncoefficient matrix and estimates its error.\nSyntax\nlapack_int LAPACKE_spbrfs( int matrix_layout, char uplo, lapack_int n, lapack_int kd,\nlapack_int nrhs, const float* ab, lapack_int ldab, const float* afb, lapack_int ldafb,\nconst float* b, lapack_int ldb, float* x, lapack_int ldx, float* ferr, float* berr );\nlapack_int LAPACKE_dpbrfs( int matrix_layout, char uplo, lapack_int n, lapack_int kd,\nlapack_int nrhs, const double* ab, lapack_int ldab, const double* afb, lapack_int\nldafb, const double* b, lapack_int ldb, double* x, lapack_int ldx, double* ferr, double*\nberr );\nlapack_int LAPACKE_cpbrfs( int matrix_layout, char uplo, lapack_int n, lapack_int kd,\nlapack_int nrhs, const lapack_complex_float* ab, lapack_int ldab, const\nlapack_complex_float* afb, lapack_int ldafb, const lapack_complex_float* b, lapack_int\nldb, lapack_complex_float* x, lapack_int ldx, float* ferr, float* berr );\nlapack_int LAPACKE_zpbrfs( int matrix_layout, char uplo, lapack_int n, lapack_int kd,\nlapack_int nrhs, const lapack_complex_double* ab, lapack_int ldab, const\nlapack_complex_double* afb, lapack_int ldafb, const lapack_complex_double* b,\nlapack_int ldb, lapack_complex_double* x, lapack_int ldx, double* ferr, double* berr );\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n608\n\n\nInclude Files\n•\nmkl.h\nDescription\nThe routine performs an iterative refinement of the solution to a system of linear equations A*X = B with a\nsymmetric (Hermitian) positive definite band matrix A, with multiple right-hand sides. For each computed\nsolution vector x, the routine computes the component-wise backward errorβ. This error is the smallest\nrelative perturbation in elements of A and b such that x is the exact solution of the perturbed system:\n|δaij| ≤β|aij|, |δbi| ≤β|bi| such that (A + δA)x = (b + δb).\nFinally, the routine estimates the component-wise forward error in the computed solution ||x - xe||∞/||\nx||∞ (here xe is the exact solution).\nBefore calling this routine:\n•\ncall the factorization routine ?pbtrf\n•\ncall the solver routine ?pbtrs.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\nuplo\nMust be 'U' or 'L'.\nIndicates how the input matrix A has been factored:\nIf uplo = 'U', the upper triangle of A is stored.\nIf uplo = 'L', the lower triangle of A is stored.\nn\nThe order of the matrix A; n≥ 0.\nkd\nThe number of superdiagonals or subdiagonals in the matrix A; kd≥ 0.\nnrhs\nThe number of right-hand sides; nrhs≥ 0.\nab\nArray ab (size max(ldab*n)) contains the original band matrix A, as\nsupplied to ?pbtrf.\nafb\nArray afb (size max(ldafb*n)) contains the factored band matrix A,\nas returned by ?pbtrf.\nb\nArray b of size max(1, ldb*nrhs) for column major layout and\nmax(1, ldb*n) for row major layout contains the right-hand side\nmatrix B.\nx\nArray x of size max(1, ldx*nrhs) for column major layout and max(1,\nldx*n) for row major layout contains the solution matrix X.\nldab\nThe leading dimension of ab; ldab≥kd + 1.\nldafb\nThe leading dimension of afb; ldafb≥kd + 1.\nldb\nThe leading dimension of b; ldb≥ max(1, n) for column major layout\nand ldb≥nrhs for row major layout.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n609\n\n\nldx\nThe leading dimension of x; ldx≥ max(1, n) for column major layout\nand ldx≥nrhs for row major layout.\nOutput Parameters\nx\nThe refined solution matrix X.\nferr, berr\nArrays, size at least max(1, nrhs). Contain the component-wise\nforward and backward errors, respectively, for each solution vector.\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info = -i, parameter i had an illegal value.\nApplication Notes\nThe bounds returned in ferr are not rigorous, but in practice they almost always overestimate the actual\nerror.\nFor each right-hand side, computation of the backward error involves a minimum of 8n*kd floating-point\noperations (for real flavors) or 32n*kd operations (for complex flavors). In addition, each step of iterative\nrefinement involves 12n*kd operations (for real flavors) or 48n*kd operations (for complex flavors); the\nnumber of iterations may range from 1 to 5.\nEstimating the forward error involves solving a number of systems of linear equations A*x = b; the number\nis usually 4 or 5 and never more than 11. Each solution requires approximately 4n*kd floating-point\noperations for real flavors or 16n*kd for complex flavors.\nSee Also\nMatrix Storage Schemes\n?ptrfs\nRefines the solution of a system of linear equations\nwith a symmetric (Hermitian) positive-definite\ntridiagonal coefficient matrix and estimates its error.\nSyntax\nlapack_int LAPACKE_sptrfs( int matrix_layout, lapack_int n, lapack_int nrhs, const\nfloat* d, const float* e, const float* df, const float* ef, const float* b, lapack_int\nldb, float* x, lapack_int ldx, float* ferr, float* berr );\nlapack_int LAPACKE_dptrfs( int matrix_layout, lapack_int n, lapack_int nrhs, const\ndouble* d, const double* e, const double* df, const double* ef, const double* b,\nlapack_int ldb, double* x, lapack_int ldx, double* ferr, double* berr );\nlapack_int LAPACKE_cptrfs( int matrix_layout, char uplo, lapack_int n, lapack_int nrhs,\nconst float* d, const lapack_complex_float* e, const float* df, const\nlapack_complex_float* ef, const lapack_complex_float* b, lapack_int ldb,\nlapack_complex_float* x, lapack_int ldx, float* ferr, float* berr );\nlapack_int LAPACKE_zptrfs( int matrix_layout, char uplo, lapack_int n, lapack_int nrhs,\nconst double* d, const lapack_complex_double* e, const double* df, const\nlapack_complex_double* ef, const lapack_complex_double* b, lapack_int ldb,\nlapack_complex_double* x, lapack_int ldx, double* ferr, double* berr );\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n610\n\n\nInclude Files\n•\nmkl.h\nDescription\nThe routine performs an iterative refinement of the solution to a system of linear equations A*X = B with a\nsymmetric (Hermitian) positive definite tridiagonal matrix A, with multiple right-hand sides. For each\ncomputed solution vector x, the routine computes the component-wise backward errorβ. This error is the\nsmallest relative perturbation in elements of A and b such that x is the exact solution of the perturbed\nsystem:\n|δaij| ≤β|aij|, |δbi| ≤β|bi| such that (A + δA)x = (b + δb).\nFinally, the routine estimates the component-wise forward error in the computed solution ||x - xe||∞/||\nx||∞ (here xe is the exact solution).\nBefore calling this routine:\n•\ncall the factorization routine ?pttrf\n•\ncall the solver routine ?pttrs.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\nuplo\nUsed for complex flavors only. Must be 'U' or 'L'.\nSpecifies whether the superdiagonal or the subdiagonal of the\ntridiagonal matrix A is stored and how A is factored:\nIf uplo = 'U', the array e stores the superdiagonal of A, and A is\nfactored as UH*D*U.\nIf uplo = 'L', the array e stores the subdiagonal of A, and A is\nfactored as L*D*LH.\nn\nThe order of the matrix A; n≥ 0.\nnrhs\nThe number of right-hand sides; nrhs≥ 0.\nd\nThe array d (size n) contains the n diagonal elements of the\ntridiagonal matrix A.\ndf\nThe array df (size n) contains the n diagonal elements of the diagonal\nmatrix D from the factorization of A as computed by ?pttrf.\ne,ef,b,x\nThe array e (size n -1) contains the (n - 1) off-diagonal elements of\nthe tridiagonal matrix A (see uplo).\nThe array ef (size n -1) contains the (n - 1) off-diagonal elements of\nthe unit bidiagonal factor U or L from the factorization computed\nby ?pttrf (see uplo).\nThe array b of size max(1, ldb*nrhs) for column major layout and\nmax(1, ldb*n) for row major layout contains the matrix B whose\ncolumns are the right-hand sides for the systems of equations.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n611\n\n\nThe array x of size max(1, ldx*nrhs) for column major layout and\nmax(1, ldx*n) for row major layout contains the solution matrix X as\ncomputed by ?pttrs.\nldb\nThe leading dimension of b; ldb≥ max(1, n) for column major layout\nand ldb≥nrhs for row major layout.\nldx\nThe leading dimension of x; ldx≥ max(1, n) for column major layout\nand ldx≥nrhs for row major layout.\nOutput Parameters\nx\nThe refined solution matrix X.\nferr, berr\nArrays, size at least max(1, nrhs). Contain the component-wise\nforward and backward errors, respectively, for each solution vector.\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info = -i, parameter i had an illegal value.\nSee Also\nMatrix Storage Schemes\n?syrfs\nRefines the solution of a system of linear equations\nwith a symmetric coefficient matrix and estimates its\nerror.\nSyntax\nlapack_int LAPACKE_ssyrfs( int matrix_layout, char uplo, lapack_int n, lapack_int nrhs,\nconst float* a, lapack_int lda, const float* af, lapack_int ldaf, const lapack_int*\nipiv, const float* b, lapack_int ldb, float* x, lapack_int ldx, float* ferr, float*\nberr );\nlapack_int LAPACKE_dsyrfs( int matrix_layout, char uplo, lapack_int n, lapack_int nrhs,\nconst double* a, lapack_int lda, const double* af, lapack_int ldaf, const lapack_int*\nipiv, const double* b, lapack_int ldb, double* x, lapack_int ldx, double* ferr, double*\nberr );\nlapack_int LAPACKE_csyrfs( int matrix_layout, char uplo, lapack_int n, lapack_int nrhs,\nconst lapack_complex_float* a, lapack_int lda, const lapack_complex_float* af,\nlapack_int ldaf, const lapack_int* ipiv, const lapack_complex_float* b, lapack_int ldb,\nlapack_complex_float* x, lapack_int ldx, float* ferr, float* berr );\nlapack_int LAPACKE_zsyrfs( int matrix_layout, char uplo, lapack_int n, lapack_int nrhs,\nconst lapack_complex_double* a, lapack_int lda, const lapack_complex_double* af,\nlapack_int ldaf, const lapack_int* ipiv, const lapack_complex_double* b, lapack_int\nldb, lapack_complex_double* x, lapack_int ldx, double* ferr, double* berr );\nInclude Files\n•\nmkl.h\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n612\n\n\nDescription\nThe routine performs an iterative refinement of the solution to a system of linear equations A*X = B with a\nsymmetric full-storage matrix A, with multiple right-hand sides. For each computed solution vector x, the\nroutine computes the component-wise backward errorβ. This error is the smallest relative perturbation in\nelements of A and b such that x is the exact solution of the perturbed system:\n|δaij| ≤β|aij|, |δbi| ≤β|bi| such that (A + δA)x = (b + δb).\nFinally, the routine estimates the component-wise forward error in the computed solution ||x - xe||∞/||\nx||∞ (here xe is the exact solution).\nBefore calling this routine:\n•\ncall the factorization routine ?sytrf\n•\ncall the solver routine ?sytrs.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\nuplo\nMust be 'U' or 'L'.\nIf uplo = 'U', the upper triangle of A is stored.\nIf uplo = 'L', the lower triangle of A is stored.\nn\nThe order of the matrix A; n≥ 0.\nnrhs\nThe number of right-hand sides; nrhs≥ 0.\na\nArray a(size max(1, lda*n)) contains the original matrix A, as\nsupplied to ?sytrf.\naf\nArray af (size max(1, ldaf*n)) contains the factored matrix A, as\nreturned by ?sytrf.\nb\nArray b of size max(1, ldb*nrhs) for column major layout and\nmax(1, ldb*n) for row major layout contains the right-hand side\nmatrix B.\nx\nArray x of size max(1, ldx*nrhs) for column major layout and max(1,\nldx*n) for row major layout contains the solution matrix X.\nlda\nThe leading dimension of a; lda≥ max(1, n).\nldaf\nThe leading dimension of af; ldaf≥ max(1, n).\nldb\nThe leading dimension of b; ldb≥ max(1, n) for column major layout\nand ldb≥nrhs for row major layout.\nldx\nThe leading dimension of x; ldx≥ max(1, n) for column major layout\nand ldx≥nrhs for row major layout.\nipiv\nArray, size at least max(1, n). The ipiv array, as returned by ?sytrf.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n613\n\n\nOutput Parameters\nx\nThe refined solution matrix X.\nferr, berr\nArrays, size at least max(1, nrhs). Contain the component-wise\nforward and backward errors, respectively, for each solution vector.\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info = -i, parameter i had an illegal value.\nApplication Notes\nThe bounds returned in ferr are not rigorous, but in practice they almost always overestimate the actual\nerror.\nFor each right-hand side, computation of the backward error involves a minimum of 4n2 floating-point\noperations (for real flavors) or 16n2 operations (for complex flavors). In addition, each step of iterative\nrefinement involves 6n2 operations (for real flavors) or 24n2 operations (for complex flavors); the number of\niterations may range from 1 to 5. Estimating the forward error involves solving a number of systems of linear\nequations A*x = b; the number is usually 4 or 5 and never more than 11. Each solution requires\napproximately 2n2 floating-point operations for real flavors or 8n2 for complex flavors.\nSee Also\nMatrix Storage Schemes\n?syrfsx\nUses extra precise iterative refinement to improve the\nsolution to the system of linear equations with a\nsymmetric indefinite coefficient matrix A and provides\nerror bounds and backward error estimates.\nSyntax\nlapack_int LAPACKE_ssyrfsx( int matrix_layout, char uplo, char equed, lapack_int n,\nlapack_int nrhs, const float* a, lapack_int lda, const float* af, lapack_int ldaf,\nconst lapack_int* ipiv, const float* s, const float* b, lapack_int ldb, float* x,\nlapack_int ldx, float* rcond, float* berr, lapack_int n_err_bnds, float* err_bnds_norm,\nfloat* err_bnds_comp, lapack_int nparams, float* params );\nlapack_int LAPACKE_dsyrfsx( int matrix_layout, char uplo, char equed, lapack_int n,\nlapack_int nrhs, const double* a, lapack_int lda, const double* af, lapack_int ldaf,\nconst lapack_int* ipiv, const double* s, const double* b, lapack_int ldb, double* x,\nlapack_int ldx, double* rcond, double* berr, lapack_int n_err_bnds, double*\nerr_bnds_norm, double* err_bnds_comp, lapack_int nparams, double* params );\nlapack_int LAPACKE_csyrfsx( int matrix_layout, char uplo, char equed, lapack_int n,\nlapack_int nrhs, const lapack_complex_float* a, lapack_int lda, const\nlapack_complex_float* af, lapack_int ldaf, const lapack_int* ipiv, const float* s,\nconst lapack_complex_float* b, lapack_int ldb, lapack_complex_float* x, lapack_int ldx,\nfloat* rcond, float* berr, lapack_int n_err_bnds, float* err_bnds_norm, float*\nerr_bnds_comp, lapack_int nparams, float* params );\nlapack_int LAPACKE_zsyrfsx( int matrix_layout, char uplo, char equed, lapack_int n,\nlapack_int nrhs, const lapack_complex_double* a, lapack_int lda, const\nlapack_complex_double* af, lapack_int ldaf, const lapack_int* ipiv, const double* s,\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n614\n\n\nconst lapack_complex_double* b, lapack_int ldb, lapack_complex_double* x, lapack_int\nldx, double* rcond, double* berr, lapack_int n_err_bnds, double* err_bnds_norm, double*\nerr_bnds_comp, lapack_int nparams, double* params );\nInclude Files\n•\nmkl.h\nDescription\nThe routine improves the computed solution to a system of linear equations when the coefficient matrix is\nsymmetric indefinite, and provides error bounds and backward error estimates for the solution. In addition to\na normwise error bound, the code provides a maximum componentwise error bound, if possible. See\ncomments for err_bnds_norm and err_bnds_comp for details of the error bounds.\nThe original system of linear equations may have been equilibrated before calling this routine, as described\nby the parameters equed and s below. In this case, the solution and error bounds returned are for the\noriginal unequilibrated system.\nInput Parameters\nmatrix_layout\nSpecifies whether two-dimensional array storage is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\nuplo\nMust be 'U' or 'L'.\nIndicates whether the upper or lower triangular part of A is stored:\nIf uplo = 'U', the upper triangle of A is stored.\nIf uplo = 'L', the lower triangle of A is stored.\nequed\nMust be 'N' or 'Y'.\nSpecifies the form of equilibration that was done to A before calling this\nroutine.\nIf equed = 'N', no equilibration was done.\nIf equed = 'Y', both row and column equilibration was done, that is, A has\nbeen replaced by diag(s)*A*diag(s). The right-hand side B has been\nchanged accordingly.\nn\nThe number of linear equations; the order of the matrix A; n≥ 0.\nnrhs\nThe number of right-hand sides; the number of columns of the matrices B\nand X; nrhs≥ 0.\na, af, b\nThe array a (size max(1, lda*n)) contains the symmetric/Hermitian matrix\nA as specified by uplo. If uplo = 'U', the leading n-by-n upper triangular\npart of a contains the upper triangular part of the matrix A and the strictly\nlower triangular part of a is not referenced. If uplo = 'L', the leading n-\nby-n lower triangular part of a contains the lower triangular part of the\nmatrix A and the strictly upper triangular part of a is not referenced.\nThe array af (size max(1, ldaf*n)) contains the triangular factor L or U\nfrom the Cholesky factorization A = UT*U or A = L*LT as computed by \nssytrf for real flavors or dsytrf for complex flavors.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n615\n\n\nThe array b (size max(1, ldb*nrhs) for column major layout and max(1,\nldb*n) for row major layout) contains the matrix B whose columns are the\nright-hand sides for the systems of equations.\nlda\nThe leading dimension of a; lda≥ max(1, n).\nldaf\nThe leading dimension of af; ldaf≥ max(1, n).\nipiv\nArray, size at least max(1, n). Contains details of the interchanges and the\nblock structure of D as determined by ssytrf for real flavors or dsytrf for\ncomplex flavors.\ns\nArray, size (n). The array s contains the scale factors for A.\nIf equed = 'N', s is not accessed.\nIf equed = 'Y', each element of s must be positive.\nEach element of s should be a power of the radix to ensure a reliable\nsolution and error estimates. Scaling by powers of the radix does not cause\nrounding errors unless the result underflows or overflows. Rounding errors\nduring scaling lead to refining with a matrix that is not equivalent to the\ninput matrix, producing error estimates that may not be reliable.\nldb\nThe leading dimension of the array b; ldb≥ max(1, n) for column major\nlayout and ldb≥nrhs for row major layout.\nx\nArray, of size max(1, ldx*nrhs) for column major layout and max(1,\nldx*n) for row major layout.\nThe solution matrix X as computed by ?sytrs\nldx\nThe leading dimension of the output array x; ldx≥ max(1, n) for column\nmajor layout and ldx≥nrhs for row major layout.\nn_err_bnds\nNumber of error bounds to return for each right hand side and each type\n(normwise or componentwise). See err_bnds_norm and err_bnds_comp\ndescriptions in Output Arguments section below.\nnparams\nSpecifies the number of parameters set in params. If ≤ 0, the params array\nis never referenced and default values are used.\nparams\nArray, size nparams. Specifies algorithm parameters. If an entry is less than\n0.0, that entry will be filled with the default value used for that parameter.\nOnly positions up to nparams are accessed; defaults are used for higher-\nnumbered parameters. If defaults are acceptable, you can pass nparams =\n0, which prevents the source code from accessing the params argument.\nparams[0] : Whether to perform iterative refinement or not. Default: 1.0\n(for single precision flavors), 1.0D+0 (for double precision flavors).\n=0.0\nNo refinement is performed and no error bounds\nare computed.\n=1.0\nUse the double-precision refinement algorithm,\npossibly with doubled-single computations if the\ncompilation environment does not support\ndouble precision.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n616\n\n\n(Other values are reserved for future use.)\nparams[1] : Maximum number of residual computations allowed for\nrefinement.\nDefault\n10.0\nAggressive\nSet to 100.0 to permit convergence using\napproximate factorizations or factorizations\nother than LU. If the factorization uses a\ntechnique other than Gaussian elimination, the\nguarantees in err_bnds_norm and\nerr_bnds_comp may no longer be trustworthy.\nparams[2] : Flag determining if the code will attempt to find a solution\nwith a small componentwise relative error in the double-precision algorithm.\nPositive is true, 0.0 is false. Default: 1.0 (attempt componentwise\nconvergence).\nOutput Parameters\nx\nThe improved solution matrix X.\nrcond\nReciprocal scaled condition number. An estimate of the reciprocal Skeel\ncondition number of the matrix A after equilibration (if done). If rcond is\nless than the machine precision, in particular, if rcond = 0, the matrix is\nsingular to working precision. Note that the error may still be small even if\nthis number is very small and the matrix appears ill-conditioned.\nberr\nArray, size at least max(1, nrhs). Contains the componentwise relative\nbackward error for each solution vector xj, that is, the smallest relative\nchange in any element of A or B that makes xj an exact solution.\nerr_bnds_norm\nArray of size nrhs*n_err_bnds. For each right-hand side, contains\ninformation about various error bounds and condition numbers\ncorresponding to the normwise relative error, which is defined as follows:\nNormwise relative error in the i-th solution vector\nThe array is indexed by the type of error information as described below.\nThere are currently up to three pieces of information returned.\nerr=1\n\"Trust/don't trust\" boolean. Trust the answer if\nthe reciprocal condition number is less than the\nthreshold sqrt(n)*slamch(ε) for single\nprecision flavors and sqrt(n)*dlamch(ε) for\ndouble precision flavors.\nerr=2\n\"Guaranteed\" error bound. The estimated\nforward error, almost certainly within a factor of\n10 of the true error so long as the next entry is\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n617\n\n\ngreater than the threshold sqrt(n)*slamch(ε)\nfor single precision flavors and\nsqrt(n)*dlamch(ε) for double precision\nflavors. This error bound should only be trusted\nif the previous boolean is true.\nerr=3\nReciprocal condition number. Estimated\nnormwise reciprocal condition number.\nCompared with the threshold\nsqrt(n)*slamch(ε) for single precision flavors\nand sqrt(n)*dlamch(ε) for double precision\nflavors to determine if the error estimate is\n\"guaranteed\". These reciprocal condition\nnumbers for some appropriately scaled matrix Z\nare:\nThe information for right-hand side i, where 1 ≤i≤nrhs, and type of error\nerr is stored in:\n•\nColumn major layout: err_bnds_norm[(err - 1)*nrhs + i - 1].\n•\nRow major layout: err_bnds_norm[err - 1 + (i - 1)*n_err_bnds]\nerr_bnds_comp\nArray of size nrhs*n_err_bnds. For each right-hand side, contains\ninformation about various error bounds and condition numbers\ncorresponding to the componentwise relative error, which is defined as\nfollows:\nComponentwise relative error in the i-th solution vector:\nThe array is indexed by the type of error information as described below.\nThere are currently up to three pieces of information returned for each\nright-hand side. If componentwise accuracy is not requested (params[2] =\n0.0), then err_bnds_comp is not accessed.\nerr=1\n\"Trust/don't trust\" boolean. Trust the answer if\nthe reciprocal condition number is less than the\nthreshold sqrt(n)*slamch(ε) for single\nprecision flavors and sqrt(n)*dlamch(ε) for\ndouble precision flavors.\nerr=2\n\"Guaranteed\" error bpound. The estimated\nforward error, almost certainly within a factor of\n10 of the true error so long as the next entry is\ngreater than the threshold sqrt(n)*slamch(ε)\nfor single precision flavors and\nsqrt(n)*dlamch(ε) for double precision\nflavors. This error bound should only be trusted\nif the previous boolean is true.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n618\n\n\nerr=3\nReciprocal condition number. Estimated\ncomponentwise reciprocal condition number.\nCompared with the threshold\nsqrt(n)*slamch(ε) for single precision flavors\nand sqrt(n)*dlamch(ε) for double precision\nflavors to determine if the error estimate is\n\"guaranteed\". These reciprocal condition\nnumbers for some appropriately scaled matrix Z\nare:\nLet z=s*(a*diag(x)), where x is the solution\nfor the current right-hand side and s scales each\nrow of a*diag(x) by a power of the radix so all\nabsolute row sums of z are approximately 1.\nThe information for right-hand side i, where 1 ≤i≤nrhs, and type of error\nerr is stored in:\n•\nColumn major layout: err_bnds_comp[(err - 1)*nrhs + i - 1].\n•\nRow major layout: err_bnds_comp[err - 1 + (i - 1)*n_err_bnds]\nparams\nOutput parameter only if the input contains erroneous values, namely, in\nparams[0], params[1], params[2]. In such a case, the corresponding\nelements of params are filled with default values on output.\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful. The solution to every right-hand side is guaranteed.\nIf info = -i, parameter i had an illegal value.\nIf 0 < info≤n: Uinfo,info is exactly zero. The factorization has been completed, but the factor U is exactly\nsingular, so the solution and error bounds could not be computed; rcond = 0 is returned.\nIf info = n+j: The solution corresponding to the j-th right-hand side is not guaranteed. The solutions\ncorresponding to other right-hand sides k with k > j may not be guaranteed as well, but only the first such\nright-hand side is reported. If a small componentwise error is not requested params[2] = 0.0, then the j-th\nright-hand side is the first with a normwise error bound that is not guaranteed (the smallest j such that for\ncolumn major layout err_bnds_norm[j - 1] = 0.0 or err_bnds_comp[j - 1] = 0.0; or for row major\nlayout err_bnds_norm[(j - 1)*n_err_bnds] = 0.0 or err_bnds_comp[(j - 1)*n_err_bnds] = 0.0).\nSee the definition of err_bnds_norm and err_bnds_comp for err = 1. To get information about all of the\nright-hand sides, check err_bnds_norm or err_bnds_comp.\nSee Also\nMatrix Storage Schemes\n?herfs\nRefines the solution of a system of linear equations\nwith a complex Hermitian coefficient matrix and\nestimates its error.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n619\n\n\nSyntax\nlapack_int LAPACKE_cherfs( int matrix_layout, char uplo, lapack_int n, lapack_int nrhs,\nconst lapack_complex_float* a, lapack_int lda, const lapack_complex_float* af,\nlapack_int ldaf, const lapack_int* ipiv, const lapack_complex_float* b, lapack_int ldb,\nlapack_complex_float* x, lapack_int ldx, float* ferr, float* berr );\nlapack_int LAPACKE_zherfs( int matrix_layout, char uplo, lapack_int n, lapack_int nrhs,\nconst lapack_complex_double* a, lapack_int lda, const lapack_complex_double* af,\nlapack_int ldaf, const lapack_int* ipiv, const lapack_complex_double* b, lapack_int\nldb, lapack_complex_double* x, lapack_int ldx, double* ferr, double* berr );\nInclude Files\n•\nmkl.h\nDescription\nThe routine performs an iterative refinement of the solution to a system of linear equations A*X = B with a\ncomplex Hermitian full-storage matrix A, with multiple right-hand sides. For each computed solution vector x,\nthe routine computes the component-wise backward errorβ. This error is the smallest relative perturbation\nin elements of A and b such that x is the exact solution of the perturbed system:\n|δaij| ≤β|aij|, |δbi| ≤β|bi| such that (A + δA)x = (b + δb).\nFinally, the routine estimates the component-wise forward error in the computed solution ||x - xe||∞/||\nx||∞ (here xe is the exact solution).\nBefore calling this routine:\n•\ncall the factorization routine ?hetrf\n•\ncall the solver routine ?hetrs.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\nuplo\nMust be 'U' or 'L'.\nIf uplo = 'U', the upper triangle of A is stored.\nIf uplo = 'L', the lower triangle of A is stored.\nn\nThe order of the matrix A; n≥ 0.\nnrhs\nThe number of right-hand sides; nrhs≥ 0.\na,af,b,x\nArrays:\na(size max(1, lda*n)) contains the original matrix A, as supplied\nto ?hetrf.\naf(size max(1, ldaf*n)) contains the factored matrix A, as returned\nby ?hetrf.\nbof size max(1, ldb*nrhs) for column major layout and max(1,\nldb*n) for row major layout contains the right-hand side matrix B.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n620\n\n\nxof size max(1, ldx*nrhs) for column major layout and max(1,\nldx*n) for row major layout contains the solution matrix X.\nlda\nThe leading dimension of a; lda≥ max(1, n).\nldaf\nThe leading dimension of af; ldaf≥ max(1, n).\nldb\nThe leading dimension of b; ldb≥ max(1, n) for column major layout\nand ldb≥nrhs for row major layout.\nldx\nThe leading dimension of x; ldx≥ max(1, n) for column major layout\nand ldx≥nrhs for row major layout.\nipiv\nArray, size at least max(1, n). The ipiv array, as returned by ?hetrf.\nOutput Parameters\nx\nThe refined solution matrix X.\nferr, berr\nArrays, size at least max(1, nrhs). Contain the component-wise\nforward and backward errors, respectively, for each solution vector.\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info = -i, parameter i had an illegal value.\nApplication Notes\nThe bounds returned in ferr are not rigorous, but in practice they almost always overestimate the actual\nerror.\nFor each right-hand side, computation of the backward error involves a minimum of 16n2 operations. In\naddition, each step of iterative refinement involves 24n2 operations; the number of iterations may range\nfrom 1 to 5.\nEstimating the forward error involves solving a number of systems of linear equations A*x = b; the number\nis usually 4 or 5 and never more than 11. Each solution requires approximately 8n2 floating-point operations.\nThe real counterpart of this routine is ?ssyrfs/?dsyrfs\nSee Also\nMatrix Storage Schemes\n?herfsx\nUses extra precise iterative refinement to improve the\nsolution to the system of linear equations with a\nsymmetric indefinite coefficient matrix A and provides\nerror bounds and backward error estimates.\nSyntax\nlapack_int LAPACKE_cherfsx( int matrix_layout, char uplo, char equed, lapack_int n,\nlapack_int nrhs, const lapack_complex_float* a, lapack_int lda, const\nlapack_complex_float* af, lapack_int ldaf, const lapack_int* ipiv, const float* s,\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n621\n\n\nconst lapack_complex_float* b, lapack_int ldb, lapack_complex_float* x, lapack_int ldx,\nfloat* rcond, float* berr, lapack_int n_err_bnds, float* err_bnds_norm, float*\nerr_bnds_comp, lapack_int nparams, float* params );\nlapack_int LAPACKE_zherfsx( int matrix_layout, char uplo, char equed, lapack_int n,\nlapack_int nrhs, const lapack_complex_double* a, lapack_int lda, const\nlapack_complex_double* af, lapack_int ldaf, const lapack_int* ipiv, const double* s,\nconst lapack_complex_double* b, lapack_int ldb, lapack_complex_double* x, lapack_int\nldx, double* rcond, double* berr, lapack_int n_err_bnds, double* err_bnds_norm, double*\nerr_bnds_comp, lapack_int nparams, double* params );\nInclude Files\n•\nmkl.h\nDescription\nThe routine improves the computed solution to a system of linear equations when the coefficient matrix is\nHermitian indefinite, and provides error bounds and backward error estimates for the solution. In addition to\na normwise error bound, the code provides a maximum componentwise error bound, if possible. See\ncomments for err_bnds_norm and err_bnds_comp for details of the error bounds.\nThe original system of linear equations may have been equilibrated before calling this routine, as described\nby the parameters equed and s below. In this case, the solution and error bounds returned are for the\noriginal unequilibrated system.\nInput Parameters\nmatrix_layout\nSpecifies whether two-dimensional array storage is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\nuplo\nMust be 'U' or 'L'.\nIndicates whether the upper or lower triangular part of A is stored:\nIf uplo = 'U', the upper triangle of A is stored.\nIf uplo = 'L', the lower triangle of A is stored.\nequed\nMust be 'N' or 'Y'.\nSpecifies the form of equilibration that was done to A before calling this\nroutine.\nIf equed = 'N', no equilibration was done.\nIf equed = 'Y', both row and column equilibration was done, that is, A has\nbeen replaced by diag(s)*A*diag(s). The right-hand side B has been\nchanged accordingly.\nn\nThe number of linear equations; the order of the matrix A; n≥ 0.\nnrhs\nThe number of right-hand sides; the number of columns of the matrices B\nand X; nrhs≥ 0.\na, af, b\nThe array a of size max(1, lda*n) contains the Hermitian matrix A as\nspecified by uplo. If uplo = 'U', the leading n-by-n upper triangular part\nof a contains the upper triangular part of the matrix A and the strictly lower\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n622\n\n\ntriangular part of a is not referenced. If uplo = 'L', the leading n-by-n\nlower triangular part of a contains the lower triangular part of the matrix A\nand the strictly upper triangular part of a is not referenced.\nThe array af of size max(1, ldaf*n) contains the block diagonal matrix D\nand the multipliers used to obtain the factor U or L from the factorization A\n= U*D*UT or A = L*D*LT as computed by ssytrf for cherfsx or dsytrf\nfor zherfsx.\nThe array b of size max(1, ldb*nrhs) for row major layout and max(1,\nldb*n) for column major layout contains the matrix B whose columns are\nthe right-hand sides for the systems of equations.\nlda\nThe leading dimension of a; lda≥ max(1, n).\nldaf\nThe leading dimension of af; ldaf≥ max(1, n).\nipiv\nArray, size at least max(1, n). Contains details of the interchanges and the\nblock structure of D as determined by ssytrf for real flavors or dsytrf for\ncomplex flavors.\ns\nArray, size (n). The array s contains the scale factors for A.\nIf equed = 'N', s is not accessed.\nIf equed = 'Y', each element of s must be positive.\nEach element of s should be a power of the radix to ensure a reliable\nsolution and error estimates. Scaling by powers of the radix does not cause\nrounding errors unless the result underflows or overflows. Rounding errors\nduring scaling lead to refining with a matrix that is not equivalent to the\ninput matrix, producing error estimates that may not be reliable.\nldb\nThe leading dimension of the array b; ldb≥ max(1, n) for column major\nlayout and ldb≥nrhs for row major layout.\nx\nArray, size max(1, ldx*nrhs) for column major layout and max(1, ldx*n)\nfor row major layout.\nThe solution matrix X as computed by ?hetrs\nldx\nThe leading dimension of the output array x; ldx≥ max(1, n) for column\nmajor layout and ldx≥nrhs for row major layout.\nn_err_bnds\nNumber of error bounds to return for each right hand side and each type\n(normwise or componentwise). See err_bnds_norm and err_bnds_comp\ndescriptions in Output Arguments section below.\nnparams\nSpecifies the number of parameters set in params. If ≤ 0, the params array\nis never referenced and default values are used.\nparams\nArray, size nparams. Specifies algorithm parameters. If an entry is less than\n0.0, that entry will be filled with the default value used for that parameter.\nOnly positions up to nparams are accessed; defaults are used for higher-\nnumbered parameters. If defaults are acceptable, you can pass nparams =\n0, which prevents the source code from accessing the params argument.\nparams[0] : Whether to perform iterative refinement or not. Default: 1.0\n(for cherfsx), 1.0D+0 (for zherfsx).\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n623\n\n\n=0.0\nNo refinement is performed and no error bounds\nare computed.\n=1.0\nUse the double-precision refinement algorithm,\npossibly with doubled-single computations if the\ncompilation environment does not support\ndouble precision.\n(Other values are reserved for future use.)\nparams[1] : Maximum number of residual computations allowed for\nrefinement.\nDefault\n10\nAggressive\nSet to 100 to permit convergence using\napproximate factorizations or factorizations\nother than LU. If the factorization uses a\ntechnique other than Gaussian elimination, the\nguarantees in err_bnds_norm and\nerr_bnds_comp may no longer be trustworthy.\nparams[2] : Flag determining if the code will attempt to find a solution\nwith a small componentwise relative error in the double-precision algorithm.\nPositive is true, 0.0 is false. Default: 1.0 (attempt componentwise\nconvergence).\nOutput Parameters\nx\nThe improved solution matrix X.\nrcond\nReciprocal scaled condition number. An estimate of the reciprocal Skeel\ncondition number of the matrix A after equilibration (if done). If rcond is\nless than the machine precision, in particular, if rcond = 0, the matrix is\nsingular to working precision. Note that the error may still be small even if\nthis number is very small and the matrix appears ill-conditioned.\nberr\nArray, size at least max(1, nrhs). Contains the componentwise relative\nbackward error for each solution vector xj, that is, the smallest relative\nchange in any element of A or B that makes xj an exact solution.\nerr_bnds_norm\nArray of size nrhs*n_err_bnds. For each right-hand side, contains\ninformation about various error bounds and condition numbers\ncorresponding to the normwise relative error, which is defined as follows:\nNormwise relative error in the i-th solution vector\nThe array is indexed by the type of error information as described below.\nThere are currently up to three pieces of information returned.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n624\n\n\nerr=1\n\"Trust/don't trust\" boolean. Trust the answer if\nthe reciprocal condition number is less than the\nthreshold sqrt(n)*slamch(ε) for cherfsx and\nsqrt(n)*dlamch(ε) for zherfsx.\nerr=2\n\"Guaranteed\" error bound. The estimated\nforward error, almost certainly within a factor of\n10 of the true error so long as the next entry is\ngreater than the threshold sqrt(n)*slamch(ε)\nfor cherfsx and sqrt(n)*dlamch(ε) for\nzherfsx. This error bound should only be\ntrusted if the previous boolean is true.\nerr=3\nReciprocal condition number. Estimated\nnormwise reciprocal condition number.\nCompared with the threshold\nsqrt(n)*slamch(ε) for single precision flavors\nand sqrt(n)*dlamch(ε) for double precision\nflavors to determine if the error estimate is\n\"guaranteed\". These reciprocal condition\nnumbers for some appropriately scaled matrix Z\nare:\nLet z=s*a, where s scales each row by a power\nof the radix so all absolute row sums of z are\napproximately 1.\nThe information for right-hand side i, where 1 ≤i≤nrhs, and type of error\nerr is stored in:\n•\nColumn major layout: err_bnds_norm[(err - 1)*nrhs + i - 1].\n•\nRow major layout: err_bnds_norm[err - 1 + (i - 1)*n_err_bnds]\nerr_bnds_comp\nArray of size nrhs*n_err_bnds. For each right-hand side, contains\ninformation about various error bounds and condition numbers\ncorresponding to the componentwise relative error, which is defined as\nfollows:\nComponentwise relative error in the i-th solution vector:\nThe array is indexed by the type of error information as described below.\nThere are currently up to three pieces of information returned for each\nright-hand side. If componentwise accuracy is not requested (params[2] =\n0.0), then err_bnds_comp is not accessed.\nerr=1\n\"Trust/don't trust\" boolean. Trust the answer if\nthe reciprocal condition number is less than the\nthreshold sqrt(n)*slamch(ε) for cherfsx and\nsqrt(n)*dlamch(ε) for zherfsx.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n625\n\n\nerr=2\n\"Guaranteed\" error bpound. The estimated\nforward error, almost certainly within a factor of\n10 of the true error so long as the next entry is\ngreater than the threshold sqrt(n)*slamch(ε)\nfor cherfsx and sqrt(n)*dlamch(ε) for\nzherfsx. This error bound should only be\ntrusted if the previous boolean is true.\nerr=3\nReciprocal condition number. Estimated\ncomponentwise reciprocal condition number.\nCompared with the threshold\nsqrt(n)*slamch(ε) for single precision flavors\nand sqrt(n)*dlamch(ε) for double precision\nflavors to determine if the error estimate is\n\"guaranteed\". These reciprocal condition\nnumbers for some appropriately scaled matrix Z\nare:\nLet z=s*(a*diag(x)), where x is the solution\nfor the current right-hand side and s scales each\nrow of a*diag(x) by a power of the radix so all\nabsolute row sums of z are approximately 1.\nThe information for right-hand side i, where 1 ≤i≤nrhs, and type of error\nerr is stored in:\n•\nColumn major layout: err_bnds_comp[(err - 1)*nrhs + i - 1].\n•\nRow major layout: err_bnds_comp[err - 1 + (i - 1)*n_err_bnds]\nparams\nOutput parameter only if the input contains erroneous values, namely, in\nparams[0], params[1], params[2]. In such a case, the corresponding\nelements of params are filled with default values on output.\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful. The solution to every right-hand side is guaranteed.\nIf info = -i, parameter i had an illegal value.\nIf 0 < info≤n: Uinfo,info is exactly zero. The factorization has been completed, but the factor D is exactly\nsingular, so the solution and error bounds could not be computed; rcond = 0 is returned.\nIf info = n+j: The solution corresponding to the j-th right-hand side is not guaranteed. The solutions\ncorresponding to other right-hand sides k with k > j may not be guaranteed as well, but only the first such\nright-hand side is reported. If a small componentwise error is not requested params[2] = 0.0, then the j-th\nright-hand side is the first with a normwise error bound that is not guaranteed (the smallest j such that for\ncolumn major layout err_bnds_norm[j - 1] = 0.0 or err_bnds_comp[j - 1] = 0.0; or for row major\nlayout err_bnds_norm[(j - 1)*n_err_bnds] = 0.0 or err_bnds_comp[(j - 1)*n_err_bnds] = 0.0).\nSee the definition of err_bnds_norm and err_bnds_comp for err = 1. To get information about all of the\nright-hand sides, check err_bnds_norm or err_bnds_comp.\nSee Also\nMatrix Storage Schemes\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n626\n\n\n?sprfs\nRefines the solution of a system of linear equations\nwith a packed symmetric coefficient matrix and\nestimates the solution error.\nSyntax\nlapack_int LAPACKE_ssprfs( int matrix_layout, char uplo, lapack_int n, lapack_int nrhs,\nconst float* ap, const float* afp, const lapack_int* ipiv, const float* b, lapack_int\nldb, float* x, lapack_int ldx, float* ferr, float* berr );\nlapack_int LAPACKE_dsprfs( int matrix_layout, char uplo, lapack_int n, lapack_int nrhs,\nconst double* ap, const double* afp, const lapack_int* ipiv, const double* b,\nlapack_int ldb, double* x, lapack_int ldx, double* ferr, double* berr );\nlapack_int LAPACKE_csprfs( int matrix_layout, char uplo, lapack_int n, lapack_int nrhs,\nconst lapack_complex_float* ap, const lapack_complex_float* afp, const lapack_int*\nipiv, const lapack_complex_float* b, lapack_int ldb, lapack_complex_float* x,\nlapack_int ldx, float* ferr, float* berr );\nlapack_int LAPACKE_zsprfs( int matrix_layout, char uplo, lapack_int n, lapack_int nrhs,\nconst lapack_complex_double* ap, const lapack_complex_double* afp, const lapack_int*\nipiv, const lapack_complex_double* b, lapack_int ldb, lapack_complex_double* x,\nlapack_int ldx, double* ferr, double* berr );\nInclude Files\n•\nmkl.h\nDescription\nThe routine performs an iterative refinement of the solution to a system of linear equations A*X = B with a\npacked symmetric matrix A, with multiple right-hand sides. For each computed solution vector x, the routine\ncomputes the component-wise backward errorβ. This error is the smallest relative perturbation in elements\nof A and b such that x is the exact solution of the perturbed system:\n|δaij| ≤β|aij|, |δbi| ≤β|bi| such that (A + δA)x = (b + δb).\nFinally, the routine estimates the component-wise forward error in the computed solution ||x - xe||∞/||\nx||∞ (here xe is the exact solution).\nBefore calling this routine:\n•\ncall the factorization routine ?sptrf\n•\ncall the solver routine ?sptrs.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\nuplo\nMust be 'U' or 'L'.\nIf uplo = 'U', the upper triangle of A is stored.\nIf uplo = 'L', the lower triangle of A is stored.\nn\nThe order of the matrix A; n≥ 0.\nnrhs\nThe number of right-hand sides; nrhs≥ 0.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n627\n\n\nap,afp,b,x\nArrays:\nap of size max(1, n(n+1)/2) contains the original packed matrix A, as\nsupplied to ?sptrf.\nafp of size max(1, n(n+1)/2) contains the factored packed matrix A,\nas returned by ?sptrf.\nbof size max(1, ldb*nrhs) for column major layout and max(1,\nldb*n) for row major layout contains the right-hand side matrix B.\nxof size max(1, ldx*nrhs) for column major layout and max(1,\nldx*n) for row major layout contains the solution matrix X.\nldb\nThe leading dimension of b; ldb≥ max(1, n) for column major layout\nand ldb≥nrhs for row major layout.\nldx\nThe leading dimension of x; ldx≥ max(1, n) for column major layout\nand ldx≥ max(1,nrhs) for row major layout.\nipiv\nArray, size at least max(1, n). The ipiv array, as returned by ?sptrf.\nOutput Parameters\nx\nThe refined solution matrix X.\nferr, berr\nArrays, size at least max(1, nrhs). Contain the component-wise\nforward and backward errors, respectively, for each solution vector.\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info = -i, parameter i had an illegal value.\nApplication Notes\nThe bounds returned in ferr are not rigorous, but in practice they almost always overestimate the actual\nerror.\nFor each right-hand side, computation of the backward error involves a minimum of 4n2 floating-point\noperations (for real flavors) or 16n2 operations (for complex flavors). In addition, each step of iterative\nrefinement involves 6n2 operations (for real flavors) or 24n2 operations (for complex flavors); the number of\niterations may range from 1 to 5.\nEstimating the forward error involves solving a number of systems of linear equations A*x = b; the number\nof systems is usually 4 or 5 and never more than 11. Each solution requires approximately 2n2 floating-point\noperations for real flavors or 8n2 for complex flavors.\nSee Also\nMatrix Storage Schemes\n?hprfs\nRefines the solution of a system of linear equations\nwith a packed complex Hermitian coefficient matrix\nand estimates the solution error.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n628\n\n\nSyntax\nlapack_int LAPACKE_chprfs( int matrix_layout, char uplo, lapack_int n, lapack_int nrhs,\nconst lapack_complex_float* ap, const lapack_complex_float* afp, const lapack_int*\nipiv, const lapack_complex_float* b, lapack_int ldb, lapack_complex_float* x,\nlapack_int ldx, float* ferr, float* berr );\nlapack_int LAPACKE_zhprfs( int matrix_layout, char uplo, lapack_int n, lapack_int nrhs,\nconst lapack_complex_double* ap, const lapack_complex_double* afp, const lapack_int*\nipiv, const lapack_complex_double* b, lapack_int ldb, lapack_complex_double* x,\nlapack_int ldx, double* ferr, double* berr );\nInclude Files\n•\nmkl.h\nDescription\nThe routine performs an iterative refinement of the solution to a system of linear equations A*X = B with a\npacked complex Hermitian matrix A, with multiple right-hand sides. For each computed solution vector x, the\nroutine computes the component-wise backward errorβ. This error is the smallest relative perturbation in\nelements of A and b such that x is the exact solution of the perturbed system:\n|δaij| ≤β|aij|, |δbi| ≤β|bi| such that (A + δA)x = (b + δb).\nFinally, the routine estimates the component-wise forward error in the computed solution ||x - xe||∞/||\nx||∞ (here xe is the exact solution).\nBefore calling this routine:\n•\ncall the factorization routine ?hptrf\n•\ncall the solver routine ?hptrs.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\nuplo\nMust be 'U' or 'L'.\nIf uplo = 'U', the upper triangle of A is stored.\nIf uplo = 'L', the lower triangle of A is stored.\nn\nThe order of the matrix A; n≥ 0.\nnrhs\nThe number of right-hand sides; nrhs≥ 0.\nap,afp,b,x\nArrays:\napmax(1, n(n + 1)/2) contains the original packed matrix A, as\nsupplied to ?hptrf.\nafpmax(1, n(n + 1)/2) contains the factored packed matrix A, as\nreturned by ?hptrf.\nbof size max(1, ldb*nrhs) for column major layout and max(1,\nldb*n) for row major layout contains the right-hand side matrix B.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n629\n\n\nxof size max(1, ldx*nrhs) for column major layout and max(1,\nldx*n) for row major layout contains the solution matrix X.\nldb\nThe leading dimension of b; ldb≥ max(1, n) for column major layout\nand ldb≥nrhs for row major layout.\nldx\nThe leading dimension of x; ldx≥ max(1, n) for column major layout\nand ldx≥nrhs for row major layout.\nipiv\nArray, size at least max(1, n). The ipiv array, as returned by ?hptrf.\nOutput Parameters\nx\nThe refined solution matrix X.\nferr, berr\nArrays, size at least max(1,nrhs). Contain the component-wise\nforward and backward errors, respectively, for each solution vector.\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info = -i, parameter i had an illegal value.\nApplication Notes\nThe bounds returned in ferr are not rigorous, but in practice they almost always overestimate the actual\nerror.\nFor each right-hand side, computation of the backward error involves a minimum of 16n2 operations. In\naddition, each step of iterative refinement involves 24n2 operations; the number of iterations may range\nfrom 1 to 5.\nEstimating the forward error involves solving a number of systems of linear equations A*x = b; the number\nis usually 4 or 5 and never more than 11. Each solution requires approximately 8n2 floating-point operations.\nThe real counterpart of this routine is ?ssprfs/?dsprfs.\nSee Also\nMatrix Storage Schemes\n?trrfs\nEstimates the error in the solution of a system of\nlinear equations with a triangular coefficient matrix.\nSyntax\nlapack_int LAPACKE_strrfs( int matrix_layout, char uplo, char trans, char diag,\nlapack_int n, lapack_int nrhs, const float* a, lapack_int lda, const float* b,\nlapack_int ldb, const float* x, lapack_int ldx, float* ferr, float* berr );\nlapack_int LAPACKE_dtrrfs( int matrix_layout, char uplo, char trans, char diag,\nlapack_int n, lapack_int nrhs, const double* a, lapack_int lda, const double* b,\nlapack_int ldb, const double* x, lapack_int ldx, double* ferr, double* berr );\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n630\n\n\nlapack_int LAPACKE_ctrrfs( int matrix_layout, char uplo, char trans, char diag,\nlapack_int n, lapack_int nrhs, const lapack_complex_float* a, lapack_int lda, const\nlapack_complex_float* b, lapack_int ldb, const lapack_complex_float* x, lapack_int ldx,\nfloat* ferr, float* berr );\nlapack_int LAPACKE_ztrrfs( int matrix_layout, char uplo, char trans, char diag,\nlapack_int n, lapack_int nrhs, const lapack_complex_double* a, lapack_int lda, const\nlapack_complex_double* b, lapack_int ldb, const lapack_complex_double* x, lapack_int\nldx, double* ferr, double* berr );\nInclude Files\n•\nmkl.h\nDescription\nThe routine estimates the errors in the solution to a system of linear equations A*X = B or AT*X = B or\nAH*X = B with a triangular matrix A, with multiple right-hand sides. For each computed solution vector x, the\nroutine computes the component-wise backward errorβ. This error is the smallest relative perturbation in\nelements of A and b such that x is the exact solution of the perturbed system:\n|δaij| ≤β|aij|, |δbi| ≤β|bi| such that (A + δA)x = (b + δb).\nThe routine also estimates the component-wise forward error in the computed solution ||x - xe||∞/||\nx||∞ (here xe is the exact solution).\nBefore calling this routine, call the solver routine ?trtrs.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\nuplo\nMust be 'U' or 'L'.\nIndicates whether A is upper or lower triangular:\nIf uplo = 'U', then A is upper triangular.\nIf uplo = 'L', then A is lower triangular.\ntrans\nMust be 'N' or 'T' or 'C'.\nIndicates the form of the equations:\nIf trans = 'N', the system has the form A*X = B.\nIf trans = 'T', the system has the form AT*X = B.\nIf trans = 'C', the system has the form AH*X = B.\ndiag\nMust be 'N' or 'U'.\nIf diag = 'N', then A is not a unit triangular matrix.\nIf diag = 'U', then A is unit triangular: diagonal elements of A are\nassumed to be 1 and not referenced in the array a.\nn\nThe order of the matrix A; n≥ 0.\nnrhs\nThe number of right-hand sides; nrhs≥ 0.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n631\n\n\na, b, x\nArrays:\na(size max(1, lda*n)) contains the upper or lower triangular matrix\nA, as specified by uplo.\nbof size max(1, ldb*nrhs) for column major layout and max(1,\nldb*n) for row major layout contains the right-hand side matrix B.\nxof size max(1, ldx*nrhs) for column major layout and max(1,\nldx*n) for row major layout contains the solution matrix X.\nlda\nThe leading dimension of a; lda≥ max(1, n).\nldb\nThe leading dimension of b; ldb≥ max(1, n) for column major layout\nand ldb≥nrhs for row major layout.\nldx\nThe leading dimension of x; ldx≥ max(1, n) for column major layout\nand ldx≥nrhs for row major layout.\nOutput Parameters\nferr, berr\nArrays, size at least max(1, nrhs). Contain the component-wise\nforward and backward errors, respectively, for each solution vector.\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info = -i, parameter i had an illegal value.\nApplication Notes\nThe bounds returned in ferr are not rigorous, but in practice they almost always overestimate the actual\nerror.\nA call to this routine involves, for each right-hand side, solving a number of systems of linear equations A*x\n= b; the number of systems is usually 4 or 5 and never more than 11. Each solution requires approximately\nn2 floating-point operations for real flavors or 4n2 for complex flavors.\nSee Also\nMatrix Storage Schemes\n?tprfs\nEstimates the error in the solution of a system of\nlinear equations with a packed triangular coefficient\nmatrix.\nSyntax\nlapack_int LAPACKE_stprfs( int matrix_layout, char uplo, char trans, char diag,\nlapack_int n, lapack_int nrhs, const float* ap, const float* b, lapack_int ldb, const\nfloat* x, lapack_int ldx, float* ferr, float* berr );\nlapack_int LAPACKE_dtprfs( int matrix_layout, char uplo, char trans, char diag,\nlapack_int n, lapack_int nrhs, const double* ap, const double* b, lapack_int ldb, const\ndouble* x, lapack_int ldx, double* ferr, double* berr );\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n632\n\n\nlapack_int LAPACKE_ctprfs( int matrix_layout, char uplo, char trans, char diag,\nlapack_int n, lapack_int nrhs, const lapack_complex_float* ap, const\nlapack_complex_float* b, lapack_int ldb, const lapack_complex_float* x, lapack_int ldx,\nfloat* ferr, float* berr );\nlapack_int LAPACKE_ztprfs( int matrix_layout, char uplo, char trans, char diag,\nlapack_int n, lapack_int nrhs, const lapack_complex_double* ap, const\nlapack_complex_double* b, lapack_int ldb, const lapack_complex_double* x, lapack_int\nldx, double* ferr, double* berr );\nInclude Files\n•\nmkl.h\nDescription\nThe routine estimates the errors in the solution to a system of linear equations A*X = B or AT*X = B or\nAH*X = B with a packed triangular matrix A, with multiple right-hand sides. For each computed solution\nvector x, the routine computes the component-wise backward errorβ. This error is the smallest relative\nperturbation in elements of A and b such that x is the exact solution of the perturbed system:\n|δaij| ≤β|aij|, |δbi| ≤β|bi| such that (A + δA)x = (b + δb).\nThe routine also estimates the component-wise forward error in the computed solution ||x - xe||∞/||\nx||∞ (here xe is the exact solution).\nBefore calling this routine, call the solver routine ?tptrs.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\nuplo\nMust be 'U' or 'L'.\nIndicates whether A is upper or lower triangular:\nIf uplo = 'U', then A is upper triangular.\nIf uplo = 'L', then A is lower triangular.\ntrans\nMust be 'N' or 'T' or 'C'.\nIndicates the form of the equations:\nIf trans = 'N', the system has the form A*X = B.\nIf trans = 'T', the system has the form AT*X = B.\nIf trans = 'C', the system has the form AH*X = B.\ndiag\nMust be 'N' or 'U'.\nIf diag = 'N', A is not a unit triangular matrix.\nIf diag = 'U', A is unit triangular: diagonal elements of A are\nassumed to be 1 and not referenced in the array ap.\nn\nThe order of the matrix A; n≥ 0.\nnrhs\nThe number of right-hand sides; nrhs≥ 0.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n633\n\n\nap, b, x\nArrays:\napmax(1, n(n + 1)/2) contains the upper or lower triangular matrix A,\nas specified by uplo.\nbof size max(1, ldb*nrhs) for column major layout and max(1,\nldb*n) for row major layout contains the right-hand side matrix B.\nxof size max(1, ldx*nrhs) for column major layout and max(1,\nldx*n) for row major layout contains the solution matrix X.\nldb\nThe leading dimension of b; ldb≥ max(1, n) for column major layout\nand ldb≥nrhs for row major layout.\nldx\nThe leading dimension of x; ldb≥ max(1, n) for column major layout\nand ldb≥nrhs for row major layout.\nOutput Parameters\nferr, berr\nArrays, size at least max(1, nrhs). Contain the component-wise\nforward and backward errors, respectively, for each solution vector.\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info = -i, parameter i had an illegal value.\nApplication Notes\nThe bounds returned in ferr are not rigorous, but in practice they almost always overestimate the actual\nerror.\nA call to this routine involves, for each right-hand side, solving a number of systems of linear equations A*x\n= b; the number of systems is usually 4 or 5 and never more than 11. Each solution requires approximately\nn2 floating-point operations for real flavors or 4n2 for complex flavors.\nSee Also\nMatrix Storage Schemes\n?tbrfs\nEstimates the error in the solution of a system of\nlinear equations with a triangular band coefficient\nmatrix.\nSyntax\nlapack_int LAPACKE_stbrfs( int matrix_layout, char uplo, char trans, char diag,\nlapack_int n, lapack_int kd, lapack_int nrhs, const float* ab, lapack_int ldab, const\nfloat* b, lapack_int ldb, const float* x, lapack_int ldx, float* ferr, float* berr );\nlapack_int LAPACKE_dtbrfs( int matrix_layout, char uplo, char trans, char diag,\nlapack_int n, lapack_int kd, lapack_int nrhs, const double* ab, lapack_int ldab, const\ndouble* b, lapack_int ldb, const double* x, lapack_int ldx, double* ferr, double*\nberr );\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n634\n\n\nlapack_int LAPACKE_ctbrfs( int matrix_layout, char uplo, char trans, char diag,\nlapack_int n, lapack_int kd, lapack_int nrhs, const lapack_complex_float* ab,\nlapack_int ldab, const lapack_complex_float* b, lapack_int ldb, const\nlapack_complex_float* x, lapack_int ldx, float* ferr, float* berr );\nlapack_int LAPACKE_ztbrfs( int matrix_layout, char uplo, char trans, char diag,\nlapack_int n, lapack_int kd, lapack_int nrhs, const lapack_complex_double* ab,\nlapack_int ldab, const lapack_complex_double* b, lapack_int ldb, const\nlapack_complex_double* x, lapack_int ldx, double* ferr, double* berr );\nInclude Files\n•\nmkl.h\nDescription\nThe routine estimates the errors in the solution to a system of linear equations A*X = B or AT*X = B or\nAH*X = B with a triangular band matrix A, with multiple right-hand sides. For each computed solution vector\nx, the routine computes the component-wise backward errorβ. This error is the smallest relative\nperturbation in elements of A and b such that x is the exact solution of the perturbed system:\n|δaij| ≤β|aij|, |δbi| ≤β|bi| such that (A + δA)x = (b + δb).\nThe routine also estimates the component-wise forward error in the computed solution ||x - xe||∞/||\nx||∞ (here xe is the exact solution).\nBefore calling this routine, call the solver routine ?tbtrs.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\nuplo\nMust be 'U' or 'L'.\nIndicates whether A is upper or lower triangular:\nIf uplo = 'U', then A is upper triangular.\nIf uplo = 'L', then A is lower triangular.\ntrans\nMust be 'N' or 'T' or 'C'.\nIndicates the form of the equations:\nIf trans = 'N', the system has the form A*X = B.\nIf trans = 'T', the system has the form AT*X = B.\nIf trans = 'C', the system has the form AH*X = B.\ndiag\nMust be 'N' or 'U'.\nIf diag = 'N', A is not a unit triangular matrix.\nIf diag = 'U', A is unit triangular: diagonal elements of A are\nassumed to be 1 and not referenced in the array ab.\nn\nThe order of the matrix A; n≥ 0.\nkd\nThe number of super-diagonals or sub-diagonals in the matrix A; kd≥\n0.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n635\n\n\nnrhs\nThe number of right-hand sides; nrhs≥ 0.\nab, b, x\nArrays:\nab(size max(1, ldab*n)) contains the upper or lower triangular matrix\nA, as specified by uplo, in band storage format.\nbof size max(1, ldb*nrhs) for column major layout and max(1,\nldb*n) for row major layout contains the right-hand side matrix B.\nxof size max(1, ldx*nrhs) for column major layout and max(1,\nldx*n) for row major layout contains the solution matrix X.\nldab\nThe leading dimension of the array ab; ldab≥kd +1.\nldb\nThe leading dimension of b; ldb≥ max(1, n) for column major layout\nand ldb≥nrhs for row major layout.\nldx\nThe leading dimension of x; ldb≥ max(1, n) for column major layout\nand ldb≥nrhs for row major layout.\nOutput Parameters\nferr, berr\nArrays, size at least max(1, nrhs). Contain the component-wise\nforward and backward errors, respectively, for each solution vector.\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info = -i, parameter i had an illegal value.\nApplication Notes\nThe bounds returned in ferr are not rigorous, but in practice they almost always overestimate the actual\nerror.\nA call to this routine involves, for each right-hand side, solving a number of systems of linear equations A*x\n= b; the number of systems is usually 4 or 5 and never more than 11. Each solution requires approximately\n2n*kd floating-point operations for real flavors or 8n*kd operations for complex flavors.\nSee Also\nMatrix Storage Schemes\nMatrix Inversion: LAPACK Computational Routines\nIt is seldom necessary to compute an explicit inverse of a matrix. In particular, do not attempt to solve a\nsystem of equations Ax = b by first computing A-1 and then forming the matrix-vector product x = A-1b.\nCall a solver routine instead (see Routines for Solving Systems of Linear Equations); this is more efficient\nand more accurate.\nHowever, matrix inversion routines are provided for the rare occasions when an explicit inverse matrix is\nneeded.\n?getri\nComputes the inverse of an LU-factored general\nmatrix.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n636\n\n\nSyntax\nlapack_int LAPACKE_sgetri (int matrix_layout , lapack_int n , float * a , lapack_int\nlda , const lapack_int * ipiv );\nlapack_int LAPACKE_dgetri (int matrix_layout , lapack_int n , double * a , lapack_int\nlda , const lapack_int * ipiv );\nlapack_int LAPACKE_cgetri (int matrix_layout , lapack_int n , lapack_complex_float *\na , lapack_int lda , const lapack_int * ipiv );\nlapack_int LAPACKE_zgetri (int matrix_layout , lapack_int n , lapack_complex_double *\na , lapack_int lda , const lapack_int * ipiv );\nInclude Files\n•\nmkl.h\nDescription\nThe routine computes the inverse inv(A) of a general matrix A. Before calling this routine, call ?getrf to\nfactorize A.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\nn\nThe order of the matrix A; n≥ 0.\na\nArray a(size max(1, lda*n)) contains the factorization of the matrix\nA, as returned by ?getrf: A = P*L*U. The second dimension of a\nmust be at least max(1,n).\nlda\nThe leading dimension of a; lda≥ max(1, n).\nipiv\nArray, size at least max(1, n).\nThe ipiv array, as returned by ?getrf.\nOutput Parameters\na\nOverwritten by the n-by-n matrix inv(A).\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info = -i, parameter i had an illegal value.\nIf info = i, the i-th diagonal element of the factor U is zero, U is singular, and the inversion could not be\ncompleted.\nApplication Notes\nThe computed inverse X satisfies the following error bound:\n|XA - I| ≤c(n)ε|X|P|L||U|,\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n637\n\n\nwhere c(n) is a modest linear function of n; ε is the machine precision; I denotes the identity matrix; P, L,\nand U are the factors of the matrix factorization A = P*L*U.\nThe total number of floating-point operations is approximately (4/3)n3 for real flavors and (16/3)n3 for\ncomplex flavors.\nSee Also\nMatrix Storage Schemes\nmkl_?getrinp\nComputes the inverse of an LU-factored general\nmatrix without pivoting.\nSyntax\nlapack_int LAPACKE_mkl_sgetrinp (int matrix_layout , lapack_int n , float * a ,\nlapack_int lda );\nlapack_int LAPACKE_mkl_dgetrinp (int matrix_layout , lapack_int n , double * a ,\nlapack_int lda );\nlapack_int LAPACKE_mkl_cgetrinp (int matrix_layout , lapack_int n ,\nlapack_complex_float * a , lapack_int lda );\nlapack_int LAPACKE_mkl_zgetrinp (int matrix_layout , lapack_int n ,\nlapack_complex_double * a , lapack_int lda );\nInclude Files\n•\nmkl.h\nDescription\nThe routine computes the inverse inv(A) of a general matrix A. Before calling this routine, call \nmkl_?getrfnp to factorize A.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\nn\nThe order of the matrix A; n≥ 0.\na\nArray a(size max(1, lda*n)) contains the factorization of the matrix\nA, as returned by mkl_?getrfnp: A = L*U. The second dimension of\na must be at least max(1,n).\nlda\nThe leading dimension of a; lda≥ max(1, n).\nOutput Parameters\na\nOverwritten by the n-by-n matrix inv(A).\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info = -i, parameter i had an illegal value.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n638\n\n\nIf info = i, the i-th diagonal element of the factor U is zero, U is singular, and the inversion could not be\ncompleted.\nApplication Notes\nThe total number of floating-point operations is approximately (4/3)n3 for real flavors and (16/3)n3 for\ncomplex flavors.\nSee Also\nMatrix Storage Schemes\n?potri\nComputes the inverse of a symmetric (Hermitian)\npositive-definite matrix using the Cholesky\nfactorization.\nSyntax\nlapack_int LAPACKE_spotri (int matrix_layout , char uplo , lapack_int n , float * a ,\nlapack_int lda );\nlapack_int LAPACKE_dpotri (int matrix_layout , char uplo , lapack_int n , double * a ,\nlapack_int lda );\nlapack_int LAPACKE_cpotri (int matrix_layout , char uplo , lapack_int n ,\nlapack_complex_float * a , lapack_int lda );\nlapack_int LAPACKE_zpotri (int matrix_layout , char uplo , lapack_int n ,\nlapack_complex_double * a , lapack_int lda );\nInclude Files\n•\nmkl.h\nDescription\nThe routine computes the inverse inv(A) of a symmetric positive definite or, for complex flavors, Hermitian\npositive-definite matrix A. Before calling this routine, call ?potrf to factorize A.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\nuplo\nMust be 'U' or 'L'.\nIndicates how the input matrix A has been factored:\nIf uplo = 'U', the upper triangle of A is stored.\nIf uplo = 'L', the lower triangle of A is stored.\nn\nThe order of the matrix A; n≥ 0.\na\nArray a(size max(1, lda*n)). Contains the factorization of the matrix\nA, as returned by ?potrf.\nlda\nThe leading dimension of a. lda≥ max(1, n).\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n639\n\n\nOutput Parameters\na\nOverwritten by the upper or lower triangle of the inverse of A.\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info = -i, parameter i had an illegal value.\nIf info = i, the i-th diagonal element of the Cholesky factor (and therefore the factor itself) is zero, and the\ninversion could not be completed.\nApplication Notes\nThe computed inverse X satisfies the following error bounds:\n||XA - I||2≤c(n)εκ2(A), ||AX -  I||2≤c(n)εκ2(A),\nwhere c(n) is a modest linear function of n, and ε is the machine precision; I denotes the identity matrix.\nThe 2-norm ||A||2 of a matrix A is defined by ||A||2 = maxx·x=1(Ax·Ax)1/2, and the condition number\nκ2(A) is defined by κ2(A) = ||A||2 ||A-1||2.\nThe total number of floating-point operations is approximately (2/3)n3 for real flavors and (8/3)n3 for\ncomplex flavors.\nSee Also\nMatrix Storage Schemes\n?pftri\nComputes the inverse of a symmetric (Hermitian)\npositive-definite matrix in RFP format using the\nCholesky factorization.\nSyntax\nlapack_int LAPACKE_spftri (int matrix_layout , char transr , char uplo , lapack_int n ,\nfloat * a );\nlapack_int LAPACKE_dpftri (int matrix_layout , char transr , char uplo , lapack_int n ,\ndouble * a );\nlapack_int LAPACKE_cpftri (int matrix_layout , char transr , char uplo , lapack_int n ,\nlapack_complex_float * a );\nlapack_int LAPACKE_zpftri (int matrix_layout , char transr , char uplo , lapack_int n ,\nlapack_complex_double * a );\nInclude Files\n•\nmkl.h\nDescription\nThe routine computes the inverse inv(A) of a symmetric positive definite or, for complex data, Hermitian\npositive-definite matrix A using the Cholesky factorization:\nA = UT*U for real data, A = UH*U for complex data\nif uplo='U'\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n640\n\n\nA = L*LT for real data, A = L*LH for complex data\nif uplo='L'\nBefore calling this routine, call ?pftrf to factorize A.\nThe matrix A is in the Rectangular Full Packed (RFP) format. For the description of the RFP format, see Matrix\nStorage Schemes.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\ntransr\nMust be 'N', 'T' (for real data) or 'C' (for complex data).\nIf transr = 'N', the Normal transr of RFP U (if uplo = 'U') or L (if\nuplo = 'L') is stored.\nIf transr = 'T', the Transpose transr of RFP U (if uplo = 'U') or L\n(if uplo = 'L' is stored.\nIf transr = 'C', the Conjugate-Transpose transr of RFP U (if uplo\n= 'U') or L (if uplo = 'L' is stored.\nuplo\nMust be 'U' or 'L'.\nIndicates how the input matrix A has been factored:\nIf uplo = 'U', A = UT*U for real data or A = UH*U for complex data,\nand U is stored.\nIf uplo = 'L', A = L*LT for real data or A = L*LH for complex data,\nand L is stored.\nn\nThe order of the matrix A; n≥ 0.\na\nArray, size (n*(n+1)/2). The array a contains the factor U or L\nmatrix A in the RFP format.\nOutput Parameters\na\nThe symmetric/Hermitian inverse of the original matrix in the same\nstorage format.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, parameter i had an illegal value.\nIf info = i, the (i,i) element of the factor U or L is zero, and the inverse could not be computed.\nSee Also\nMatrix Storage Schemes\n?pptri\nComputes the inverse of a packed symmetric\n(Hermitian) positive-definite matrix using Cholesky\nfactorization.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n641\n\n\nSyntax\nlapack_int LAPACKE_spptri (int matrix_layout , char uplo , lapack_int n , float * ap );\nlapack_int LAPACKE_dpptri (int matrix_layout , char uplo , lapack_int n , double *\nap );\nlapack_int LAPACKE_cpptri (int matrix_layout , char uplo , lapack_int n ,\nlapack_complex_float * ap );\nlapack_int LAPACKE_zpptri (int matrix_layout , char uplo , lapack_int n ,\nlapack_complex_double * ap );\nInclude Files\n•\nmkl.h\nDescription\nThe routine computes the inverse inv(A) of a symmetric positive definite or, for complex flavors, Hermitian\npositive-definite matrix A in packed form. Before calling this routine, call ?pptrf to factorize A.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\nuplo\nMust be 'U' or 'L'.\nIndicates whether the upper or lower triangular factor is stored in ap:\nIf uplo = 'U', then the upper triangular factor is stored.\nIf uplo = 'L', then the lower triangular factor is stored.\nn\nThe order of the matrix A; n≥ 0.\nap\nArray, size at least max(1, n(n+1)/2).\nContains the factorization of the packed matrix A, as returned\nby ?pptrf.\nThe dimension ap must be at least max(1,n(n+1)/2).\nOutput Parameters\nap\nOverwritten by the packed n-by-n matrix inv(A).\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info = -i, parameter i had an illegal value.\nIf info = i, the i-th diagonal element of the Cholesky factor (and therefore the factor itself) is zero, and the\ninversion could not be completed.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n642\n\n\nApplication Notes\nThe computed inverse X satisfies the following error bounds:\n||XA - I||2≤c(n)εκ2(A), ||AX - I||2≤c(n)εκ2(A),\nwhere c(n) is a modest linear function of n, and ε is the machine precision; I denotes the identity matrix.\nThe 2-norm ||A||2 of a matrix A is defined by ||A||2 =maxx·x=1(Ax·Ax)1/2, and the condition number\nκ2(A) is defined by κ2(A) = ||A||2 ||A-1||2 .\nThe total number of floating-point operations is approximately (2/3)n3 for real flavors and (8/3)n3 for\ncomplex flavors.\nSee Also\nMatrix Storage Schemes\n?sytri\nComputes the inverse of a symmetric matrix using\nU*D*UT or L*D*LT Bunch-Kaufman factorization.\nSyntax\nlapack_int LAPACKE_ssytri (int matrix_layout , char uplo , lapack_int n , float * a ,\nlapack_int lda , const lapack_int * ipiv );\nlapack_int LAPACKE_dsytri (int matrix_layout , char uplo , lapack_int n , double * a ,\nlapack_int lda , const lapack_int * ipiv );\nlapack_int LAPACKE_csytri (int matrix_layout , char uplo , lapack_int n ,\nlapack_complex_float * a , lapack_int lda , const lapack_int * ipiv );\nlapack_int LAPACKE_zsytri (int matrix_layout , char uplo , lapack_int n ,\nlapack_complex_double * a , lapack_int lda , const lapack_int * ipiv );\nInclude Files\n•\nmkl.h\nDescription\nThe routine computes the inverse inv(A) of a symmetric matrix A. Before calling this routine, call ?sytrf to\nfactorize A.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\nuplo\nMust be 'U' or 'L'.\nIndicates how the input matrix A has been factored:\nIf uplo = 'U', the array a stores the Bunch-Kaufman factorization A\n= U*D*UT.\nIf uplo = 'L', the array a stores the Bunch-Kaufman factorization A\n= L*D*LT.\nn\nThe order of the matrix A; n≥ 0.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n643\n\n\na\na(size max(1, lda*n)) contains the factorization of the matrix A, as\nreturned by ?sytrf.\nlda\nThe leading dimension of a; lda≥ max(1, n).\nipiv\nArray, size at least max(1, n).\nThe ipiv array, as returned by ?sytrf.\nOutput Parameters\na\nOverwritten by the n-by-n matrix inv(A).\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info =-i, parameter i had an illegal value.\nIf info = i, the i-th diagonal element of D is zero, D is singular, and the inversion could not be completed.\nApplication Notes\nThe computed inverse X satisfies the following error bounds:\n|D*UT*PT*X*P*U - I| ≤c(n)ε(|D||UT|PT|X|P|U| + |D||D-1|)\nfor uplo = 'U', and\n|D*LT*PT*X*P*L - I| ≤c(n)ε(|D||LT|PT|X|P|L| + |D||D-1|)\nfor uplo = 'L'. Here c(n) is a modest linear function of n, and ε is the machine precision; I denotes the\nidentity matrix.\nThe total number of floating-point operations is approximately (2/3)n3 for real flavors and (8/3)n3 for\ncomplex flavors.\nSee Also\nMatrix Storage Schemes\n?hetri\nComputes the inverse of a complex Hermitian matrix\nusing U*D*UH or L*D*LH Bunch-Kaufman\nfactorization.\nSyntax\nlapack_int LAPACKE_chetri (int matrix_layout , char uplo , lapack_int n ,\nlapack_complex_float * a , lapack_int lda , const lapack_int * ipiv );\nlapack_int LAPACKE_zhetri (int matrix_layout , char uplo , lapack_int n ,\nlapack_complex_double * a , lapack_int lda , const lapack_int * ipiv );\nInclude Files\n•\nmkl.h\nDescription\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n644\n\n\nThe routine computes the inverse inv(A) of a complex Hermitian matrix A. Before calling this routine,\ncall ?hetrf to factorize A.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\nuplo\nMust be 'U' or 'L'.\nIndicates how the input matrix A has been factored:\nIf uplo = 'U', the array a stores the Bunch-Kaufman factorization A\n= U*D*UH.\nIf uplo = 'L', the array a stores the Bunch-Kaufman factorization A\n= L*D*LH.\nn\nThe order of the matrix A; n≥ 0.\na,\nArray a(size max(1, lda*n)) contains the factorization of the matrix\nA, as returned by ?hetrf. The second dimension of a must be at least\nmax(1,n).\nlda\nThe leading dimension of a; lda≥ max(1, n).\nipiv\nArray, size at least max(1, n). The ipiv array, as returned by ?hetrf.\nOutput Parameters\na\nOverwritten by the n-by-n matrix inv(A).\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info = -i, parameter i had an illegal value.\nIf info = i, the i-th diagonal element of D is zero, D is singular, and the inversion could not be completed.\nApplication Notes\nThe computed inverse X satisfies the following error bounds:\n|D*UH*PT*X*P*U - I| ≤c(n)ε(|D||UH|PT|X|P|U| + |D||D-1|)\nfor uplo = 'U', and\n|D*LH*PT*X*P*L - I| ≤c(n)ε(|D||LH|PT|X|P|L| + |D||D-1|)\nfor uplo = 'L'. Here c(n) is a modest linear function of n, and ε is the machine precision; I denotes the\nidentity matrix.\nThe total number of floating-point operations is approximately (8/3)n3 for complex flavors.\nThe real counterpart of this routine is ?sytri.\nSee Also\nMatrix Storage Schemes\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n645\n\n\n?sytri2\nComputes the inverse of a symmetric indefinite matrix\nthrough allocating memory and calling ?sytri2x.\nSyntax\nlapack_int LAPACKE_ssytri2 (int matrix_layout , char uplo , lapack_int n , float * a ,\nlapack_int lda , const lapack_int * ipiv );\nlapack_int LAPACKE_dsytri2 (int matrix_layout , char uplo , lapack_int n , double * a ,\nlapack_int lda , const lapack_int * ipiv );\nlapack_int LAPACKE_csytri2 (int matrix_layout , char uplo , lapack_int n ,\nlapack_complex_float * a , lapack_int lda , const lapack_int * ipiv );\nlapack_int LAPACKE_zsytri2 (int matrix_layout , char uplo , lapack_int n ,\nlapack_complex_double * a , lapack_int lda , const lapack_int * ipiv );\nInclude Files\n•\nmkl.h\nDescription\nThe routine computes the inverse inv(A) of a symmetric indefinite matrix A using the factorization A =\nU*D*UT or A = L*D*LT computed by ?sytrf.\nThe ?sytri2 routine allocates a temporary buffer before calling ?sytri2x that actually computes the\ninverse.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\nuplo\nMust be 'U' or 'L'.\nIndicates how the input matrix A has been factored:\nIf uplo = 'U', the array a stores the factorization A = U*D*UT.\nIf uplo = 'L', the array a stores the factorization A = L*D*LT.\nn\nThe order of the matrix A; n≥ 0.\na\nArray a(size max(1, lda*n)) contains the block diagonal matrix D and\nthe multipliers used to obtain the factor U or L as returned by ?sytrf.\nlda\nThe leading dimension of a; lda≥ max(1, n).\nipiv\nArray, size at least max(1, n).\nDetails of the interchanges and the block structure of D as returned\nby ?sytrf.\nOutput Parameters\na\nIf info = 0, the symmetric inverse of the original matrix.\nIf uplo = 'U', the upper triangular part of the inverse is formed and\nthe part of A below the diagonal is not referenced.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n646\n\n\nIf uplo = 'L', the lower triangular part of the inverse is formed and\nthe part of A above the diagonal is not referenced.\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info =-i, the i-th parameter had an illegal value.\nIf info = i, D(i,i) = 0; D is singular and its inversion could not be computed.\nSee Also\n?sytrf\n?sytri2x\nMatrix Storage Schemes\n?hetri2\nComputes the inverse of a Hermitian indefinite matrix\nthrough allocating memory and calling ?hetri2x.\nSyntax\nlapack_int LAPACKE_chetri2 (int matrix_layout , char uplo , lapack_int n ,\nlapack_complex_float * a , lapack_int lda , const lapack_int * ipiv );\nlapack_int LAPACKE_zhetri2 (int matrix_layout , char uplo , lapack_int n ,\nlapack_complex_double * a , lapack_int lda , const lapack_int * ipiv );\nInclude Files\n•\nmkl.h\nDescription\nThe routine computes the inverse inv(A) of a Hermitian indefinite matrix A using the factorization A =\nU*D*UH or A = L*D*LH computed by ?hetrf.\nThe ?hetri2 routine allocates a temporary buffer before calling ?hetri2x that actually computes the\ninverse.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\nuplo\nMust be 'U' or 'L'.\nIndicates how the input matrix A has been factored:\nIf uplo = 'U', the array a stores the factorization A = U*D*UH.\nIf uplo = 'L', the array a stores the factorization A = L*D*LH.\nn\nThe order of the matrix A; n≥ 0.\na\nArray a(size max(1, lda*n)) contains the block diagonal matrix D and\nthe multipliers used to obtain the factor U or L as returned by ?sytrf.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n647\n\n\nlda\nThe leading dimension of a; lda≥ max(1, n).\nipiv\nArray, size at least max(1, n).\nDetails of the interchanges and the block structure of D as returned\nby ?hetrf.\nOutput Parameters\na\nIf info = 0, the inverse of the original matrix.\nIf uplo = 'U', the upper triangular part of the inverse is formed and\nthe part of A below the diagonal is not referenced.\nIf uplo = 'L', the lower triangular part of the inverse is formed and\nthe part of A above the diagonal is not referenced.\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info =-i, parameter i had an illegal value.\nIf info = i, D(i,i) = 0; D is singular and its inversion could not be computed.\nSee Also\n?hetrf\n?hetri2x\nMatrix Storage Schemes\n?sytri2x\nComputes the inverse of a symmetric indefinite matrix\nafter ?sytri2allocates memory.\nSyntax\nlapack_int LAPACKE_ssytri2x (int matrix_layout , char uplo , lapack_int n , float * a ,\nlapack_int lda , const lapack_int * ipiv , lapack_int nb );\nlapack_int LAPACKE_dsytri2x (int matrix_layout , char uplo , lapack_int n , double *\na , lapack_int lda , const lapack_int * ipiv , lapack_int nb );\nlapack_int LAPACKE_csytri2x (int matrix_layout , char uplo , lapack_int n ,\nlapack_complex_float * a , lapack_int lda , const lapack_int * ipiv , lapack_int nb );\nlapack_int LAPACKE_zsytri2x (int matrix_layout , char uplo , lapack_int n ,\nlapack_complex_double * a , lapack_int lda , const lapack_int * ipiv , lapack_int nb );\nInclude Files\n•\nmkl.h\nDescription\nThe routine computes the inverse inv(A) of a symmetric indefinite matrix A using the factorization A =\nU*D*UT or A = L*D*LT computed by ?sytrf.\nThe ?sytri2x actually computes the inverse after the ?sytri2 routine allocates memory before\ncalling ?sytri2x.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n648\n\n\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\nuplo\nMust be 'U' or 'L'.\nIndicates how the input matrix A has been factored:\nIf uplo = 'U', the array a stores the factorization A = U*D*UT.\nIf uplo = 'L', the array a stores the factorization A = L*D*LT.\nn\nThe order of the matrix A; n≥ 0.\na\nArray a (size max(1, lda*n)) contains the nb (block size) diagonal\nmatrix D and the multipliers used to obtain the factor U or L as\nreturned by ?sytrf. The second dimension of a must be at least\nmax(1,n).\nlda\nThe leading dimension of a; lda≥ max(1, n).\nipiv\nArray, size at least max(1, n).\nDetails of the interchanges and the nb structure of D as returned\nby ?sytrf.\nnb\nBlock size.\nOutput Parameters\na\nIf info = 0, the symmetric inverse of the original matrix.\nIf info = 'U', the upper triangular part of the inverse is formed and\nthe part of A below the diagonal is not referenced.\nIf info = 'L', the lower triangular part of the inverse is formed and\nthe part of A above the diagonal is not referenced.\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info =-i, parameter i had an illegal value.\nIf info = i, Dii= 0; D is singular and its inversion could not be computed.\nSee Also\n?sytrf\n?sytri2\nMatrix Storage Schemes\n?hetri2x\nComputes the inverse of a Hermitian indefinite matrix\nafter ?hetri2allocates memory.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n649\n\n\nSyntax\nlapack_int LAPACKE_chetri2x (int matrix_layout , char uplo , lapack_int n ,\nlapack_complex_float * a , lapack_int lda , const lapack_int * ipiv , lapack_int nb );\nlapack_int LAPACKE_zhetri2x (int matrix_layout , char uplo , lapack_int n ,\nlapack_complex_double * a , lapack_int lda , const lapack_int * ipiv , lapack_int nb );\nInclude Files\n•\nmkl.h\nDescription\nThe routine computes the inverse inv(A) of a Hermitian indefinite matrix A using the factorization A =\nU*D*UH or A = L*D*LH computed by ?hetrf.\nThe ?hetri2x actually computes the inverse after the ?hetri2 routine allocates memory before\ncalling ?hetri2x.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\nuplo\nMust be 'U' or 'L'.\nIndicates how the input matrix A has been factored:\nIf uplo = 'U', the array a stores the factorization A = U*D*UH.\nIf uplo = 'L', the array a stores the factorization A = L*D*LH.\nn\nThe order of the matrix A; n≥ 0.\na\nArrays a(size max(1, lda*n)) contains the nb (block size) diagonal\nmatrix D and the multipliers used to obtain the factor U or L as\nreturned by ?hetrf.\nlda\nThe leading dimension of a; lda≥ max(1, n).\nipiv\nArray, size at least max(1, n).\nDetails of the interchanges and the nb structure of D as returned\nby ?hetrf.\nnb\nBlock size.\nOutput Parameters\na\nIf info = 0, the symmetric inverse of the original matrix.\nIf info = 'U', the upper triangular part of the inverse is formed and\nthe part of A below the diagonal is not referenced.\nIf info = 'L', the lower triangular part of the inverse is formed and\nthe part of A above the diagonal is not referenced.\nReturn Values\nThis function returns a value info.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n650\n\n\nIf info = 0, the execution is successful.\nIf info =-i, parameter i had an illegal value.\nIf info = i, Dii= 0; D is singular and its inversion could not be computed.\nSee Also\n?hetrf\n?hetri2\nMatrix Storage Schemes\n?sytri_3\nComputes the inverse of a real or complex symmetric\nmatrix.\nlapack_int LAPACKE_ssytri_3 (int matrix_layout, char uplo, lapack_int n, float * A,\nlapack_int lda, const float * e, const lapack_int * ipiv);\nlapack_int LAPACKE_dsytri_3 (int matrix_layout, char uplo, lapack_int n, double * A,\nlapack_int lda, const double * e, const lapack_int * ipiv);\nlapack_int LAPACKE_csytri_3 (int matrix_layout, char uplo, lapack_int n,\nlapack_complex_float * A, lapack_int lda, const lapack_complex_float * e, const\nlapack_int * ipiv);\nlapack_int LAPACKE_zsytri_3 (int matrix_layout, char uplo, lapack_int n,\nlapack_complex_double * A, lapack_int lda, const lapack_complex_double * e, const\nlapack_int * ipiv);\nDescription\n?sytri_3 computes the inverse of a real or complex symmetric matrix A using the factorization computed\nby ?sytrf_rk: A = P*U*D*(UT)*(PT) or A = P*L*D*(LT)*(PT), where U (or L) is a unit upper (or lower)\ntriangular matrix, UT (or LT) is the transpose of U (or L), P is a permutation matrix, PT is the transpose of P,\nand D is symmetric and block diagonal with 1-by-1 and 2-by-2 diagonal blocks.\n?sytri_3 sets the leading dimension of the workspace before calling ?sytri_3x, which actually computes\nthe inverse. This is the blocked version of the algorithm, calling Level-3 BLAS.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nuplo\nSpecifies whether the details of the factorization are stored as an upper or\nlower triangular matrix.\n•\n= 'U': The upper triangle of A is stored.\n•\n= 'L': The lower triangle of A is stored.\nn\nThe order of the matrix A. n ≥ 0.\nA\nArray of size max(1, lda*n). On entry, diagonal of the block diagonal\nmatrix D and factors U or L as computed by ?sytrf_rk:\n•\nOnly diagonal elements of the symmetric block diagonal matrix D on the\ndiagonal of A; that is, D(k,k) = A(k,k). Superdiagonal (or subdiagonal)\nelements of D should be provided on entry in array e.\n—and—\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n651\n\n\n•\nIf uplo = 'U', factor U in the superdiagonal part of A. If uplo = 'L',\nfactor L in the subdiagonal part of A.\nlda\nThe leading dimension of the array A.\ne\nArray of size n. On entry, contains the superdiagonal (or subdiagonal)\nelements of the symmetric block diagonal matrix D with 1-by-1 or 2-by-2\ndiagonal blocks. If uplo = 'U', e(i) = D(i-1,i), i=2:N, and e(1) is not\nreferenced. If uplo = 'L', e(i) = D(i+1,i), i=1:N-1, and e(n) is not\nreferenced.\nNOTE For 1-by-1 diagonal block D(k), where 1 ≤ k ≤ n, the\nelement e[k-1] is not referenced in both the uplo = 'U' and\nuplo = 'L' cases.\nipiv\nArray of size n. Details of the interchanges and the block structure of D as\ndetermined by ?sytrf_rk.\nOutput Parameters\nA\nOn exit, if info = 0, the symmetric inverse of the original matrix. If uplo =\n'U', the upper triangular part of the inverse is formed and the part of A\nbelow the diagonal is not referenced. If uplo = 'L', the lower triangular\npart of the inverse is formed and the part of A above the diagonal is not\nreferenced.\nReturn Values\nThis function returns a value info.\n= 0: Successful exit.\n< 0: If info = -i, the ith argument had an illegal value.\n> 0: If info = i, D(i,i) = 0; the matrix is singular and its inverse could not be computed.\n?hetri_3\nComputes the inverse of a complex Hermitian matrix\nusing the factorization computed by ?hetrf_rk.\nlapack_int LAPACKE_chetri_3 (int matrix_layout, char uplo, lapack_int n,\nlapack_complex_float * A, lapack_int lda, const lapack_complex_float * e, const\nlapack_int * ipiv);\nlapack_int LAPACKE_zhetri_3 (int matrix_layout, char uplo, lapack_int n,\nlapack_complex_double * A, lapack_int lda, const lapack_complex_double * e, const\nlapack_int * ipiv);\nDescription\n?hetri_3 computes the inverse of a complex Hermitian matrix A using the factorization computed\nby ?hetrf_rk: A = P*U*D*(UH)*(PT) or A = P*L*D*(LH)*(PT), where U (or L) is a unit upper (or lower)\ntriangular matrix, UH (or LH) is the conjugate of U (or L), P is a permutation matrix, PT is the transpose of P,\nand D is a Hermitian and block diagonal with 1-by-1 and 2-by-2 diagonal blocks.\n?hetri_3 sets the leading dimension of the workspace before calling ?hetri_3x, which actually computes\nthe inverse.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n652\n\n\nThis is the blocked version of the algorithm, calling Level-3 BLAS.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nuplo\nSpecifies whether the details of the factorization are stored as an upper or\nlower triangular matrix.\n•\n= 'U': The upper triangle of A is stored.\n•\n= 'L': The lower triangle of A is stored.\nn\nThe order of the matrix A. n ≥ 0.\nA\nArray of size max(1, lda*n). On entry, diagonal of the block diagonal\nmatrix D and factor U or L as computed by ?hetrf_rk:\n•\nOnly diagonal elements of the Hermitian block diagonal matrix D on the\ndiagonal of A; that is, D(k,k) = A(k,k). Superdiagonal (or subdiagonal)\nelements of D should be provided on entry in array e.\n•\nIf uplo = 'U', factor U in the superdiagonal part of A. If uplo = 'L',\nfactor L is the subdiagonal part of A.\nlda\nThe leading dimension of the array A.\ne\nArray of size n. On entry, contains the superdiagonal (or subdiagonal)\nelements of the Hermitian block diagonal matrix D with 1-by-1 or 2-by-2\ndiagonal blocks. If uplo = 'U', e(i) = D(i-1,i), i=2:N, and e(1) is not\nreferenced. If uplo = 'L', e(i) = D(i+1,i), i=1:N-1, and e(n) is not\nreferenced.\nNOTE For 1-by-1 diagonal block D(k), where 1 ≤ k ≤ n, the\nelement e[k-1] is not referenced in both the uplo = 'U' and\nuplo = 'L' cases.\nipiv\nArray of size n. Details of the interchanges and the block structure of D as\ndetermined by ?hetrf_rk.\nOutput Parameters\nA\nOn exit, if info = 0, the Hermitian inverse of the original matrix. If uplo =\n'U', the upper triangular part of the inverse is formed and the part of A\nbelow the diagonal is not referenced. If uplo = 'L', the lower triangular\npart of the inverse is formed and the part of A above the diagonal is not\nreferenced.\nReturn Values\nThis function returns a value info.\n= 0: Successful exit.\n< 0: If info = -i, the ith argument had an illegal value.\n> 0: If info = i, D(i,i) = 0; the matrix is singular and its inverse could not be computed.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n653\n\n\n?sptri\nComputes the inverse of a symmetric matrix using\nU*D*UT or L*D*LT Bunch-Kaufman factorization of\nmatrix in packed storage.\nSyntax\nlapack_int LAPACKE_ssptri (int matrix_layout , char uplo , lapack_int n , float * ap ,\nconst lapack_int * ipiv );\nlapack_int LAPACKE_dsptri (int matrix_layout , char uplo , lapack_int n , double * ap ,\nconst lapack_int * ipiv );\nlapack_int LAPACKE_csptri (int matrix_layout , char uplo , lapack_int n ,\nlapack_complex_float * ap , const lapack_int * ipiv );\nlapack_int LAPACKE_zsptri (int matrix_layout , char uplo , lapack_int n ,\nlapack_complex_double * ap , const lapack_int * ipiv );\nInclude Files\n•\nmkl.h\nDescription\nThe routine computes the inverse inv(A) of a packed symmetric matrix A. Before calling this routine,\ncall ?sptrf to factorize A.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\nuplo\nMust be 'U' or 'L'.\nIndicates how the input matrix A has been factored:\nIf uplo = 'U', the array ap stores the Bunch-Kaufman factorization A\n= U*D*UT.\nIf uplo = 'L', the array ap stores the Bunch-Kaufman factorization A\n= L*D*LT.\nn\nThe order of the matrix A; n≥ 0.\nap\nArrays ap (size max(1,n(n+1)/2)) contains the factorization of the\nmatrix A, as returned by ?sptrf.\nipiv\nArray, size at least max(1, n). The ipiv array, as returned by ?sptrf.\nOutput Parameters\nap\nOverwritten by the matrix inv(A) in packed form.\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n654\n\n\nIf info = -i, parameter i had an illegal value.\nIf info = i, the i-th diagonal element of D is zero, D is singular, and the inversion could not be completed.\nApplication Notes\nThe computed inverse X satisfies the following error bounds:\n|D*UT*PT*X*P*U - I| ≤c(n)ε(|D||UT|PT|X|P|U| + |D||D-1|)\nfor uplo = 'U', and\n|D*LT*PT*X*P*L - I| ≤c(n)ε(|D||LT|PT|X|P|L| + |D||D-1|)\nfor uplo = 'L'. Here c(n) is a modest linear function of n, and ε is the machine precision; I denotes the\nidentity matrix.\nThe total number of floating-point operations is approximately (2/3)n3 for real flavors and (8/3)n3 for\ncomplex flavors.\nSee Also\nMatrix Storage Schemes\n?hptri\nComputes the inverse of a complex Hermitian matrix\nusing U*D*UH or L*D*LH Bunch-Kaufman factorization\nof matrix in packed storage.\nSyntax\nlapack_int LAPACKE_chptri (int matrix_layout , char uplo , lapack_int n ,\nlapack_complex_float * ap , const lapack_int * ipiv );\nlapack_int LAPACKE_zhptri (int matrix_layout , char uplo , lapack_int n ,\nlapack_complex_double * ap , const lapack_int * ipiv );\nInclude Files\n•\nmkl.h\nDescription\nThe routine computes the inverse inv(A) of a complex Hermitian matrix A using packed storage. Before\ncalling this routine, call ?hptrf to factorize A.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\nuplo\nMust be 'U' or 'L'.\nIndicates how the input matrix A has been factored:\nIf uplo = 'U', the array ap stores the packed Bunch-Kaufman\nfactorization A = U*D*UH.\nIf uplo = 'L', the array ap stores the packed Bunch-Kaufman\nfactorization A = L*D*LH.\nn\nThe order of the matrix A; n≥ 0.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n655\n\n\nap\nArray ap (size max(1,n(n+1)/2)) contains the factorization of the\nmatrix A, as returned by ?hptrf.\nipiv\nArray, size at least max(1, n).\nThe ipiv array, as returned by ?hptrf.\nOutput Parameters\nap\nOverwritten by the matrix inv(A).\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info = -i, parameter i had an illegal value.\nIf info = i, the i-th diagonal element of D is zero, D is singular, and the inversion could not be completed.\nApplication Notes\nThe computed inverse X satisfies the following error bounds:\n|D*UH*PT*X*P*U - I| ≤c(n)ε(|D||UH|PT|X|P|U| + |D||D-1|)\nfor uplo = 'U', and\n|D*LH*PT*X*PL - I| ≤c(n)ε(|D||LH|PT|X|P|L| + |D||D-1|)\nfor uplo = 'L'. Here c(n) is a modest linear function of n, and ε is the machine precision; I denotes the\nidentity matrix.\nThe total number of floating-point operations is approximately (8/3)n3.\nThe real counterpart of this routine is ?sptri.\nSee Also\nMatrix Storage Schemes\n?trtri\nComputes the inverse of a triangular matrix.\nSyntax\nlapack_int LAPACKE_strtri (int matrix_layout , char uplo , char diag , lapack_int n ,\nfloat * a , lapack_int lda );\nlapack_int LAPACKE_dtrtri (int matrix_layout , char uplo , char diag , lapack_int n ,\ndouble * a , lapack_int lda );\nlapack_int LAPACKE_ctrtri (int matrix_layout , char uplo , char diag , lapack_int n ,\nlapack_complex_float * a , lapack_int lda );\nlapack_int LAPACKE_ztrtri (int matrix_layout , char uplo , char diag , lapack_int n ,\nlapack_complex_double * a , lapack_int lda );\nInclude Files\n•\nmkl.h\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n656\n\n\nDescription\nThe routine computes the inverse inv(A) of a triangular matrix A.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\nuplo\nMust be 'U' or 'L'.\nIndicates whether A is upper or lower triangular:\nIf uplo = 'U', then A is upper triangular.\nIf uplo = 'L', then A is lower triangular.\ndiag\nMust be 'N' or 'U'.\nIf diag = 'N', then A is not a unit triangular matrix.\nIf diag = 'U', A is unit triangular: diagonal elements of A are\nassumed to be 1 and not referenced in the array a.\nn\nThe order of the matrix A; n≥ 0.\na\nArray: . Contains the matrix A.\nlda\nThe first dimension of a; lda≥ max(1, n).\nOutput Parameters\na\nOverwritten by the matrix inv(A).\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info = -i, parameter i had an illegal value.\nIf info = i, the i-th diagonal element of A is zero, A is singular, and the inversion could not be completed.\nApplication Notes\nThe computed inverse X satisfies the following error bounds:\n|XA - I| ≤c(n)ε |X||A|\n|XA - I| ≤c(n)ε |A-1||A||X|,\nwhere c(n) is a modest linear function of n; ε is the machine precision; I denotes the identity matrix.\nThe total number of floating-point operations is approximately (1/3)n3 for real flavors and (4/3)n3 for\ncomplex flavors.\nSee Also\nMatrix Storage Schemes\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n657\n\n\n?tftri\nComputes the inverse of a triangular matrix stored in\nthe Rectangular Full Packed (RFP) format.\nSyntax\nlapack_int LAPACKE_stftri (int matrix_layout , char transr , char uplo , char diag ,\nlapack_int n , float * a );\nlapack_int LAPACKE_dtftri (int matrix_layout , char transr , char uplo , char diag ,\nlapack_int n , double * a );\nlapack_int LAPACKE_ctftri (int matrix_layout , char transr , char uplo , char diag ,\nlapack_int n , lapack_complex_float * a );\nlapack_int LAPACKE_ztftri (int matrix_layout , char transr , char uplo , char diag ,\nlapack_int n , lapack_complex_double * a );\nInclude Files\n•\nmkl.h\nDescription\nComputes the inverse of a triangular matrix A stored in the Rectangular Full Packed (RFP) format. For the\ndescription of the RFP format, see Matrix Storage Schemes.\nThis is the block version of the algorithm, calling Level 3 BLAS.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\ntransr\nMust be 'N', 'T' (for real data) or 'C' (for complex data).\nIf transr = 'N', the Normal transr of RFP A is stored.\nIf transr = 'T', the Transpose transr of RFP A is stored.\nIf transr = 'C', the Conjugate-Transpose transr of RFP A is stored.\nuplo\nMust be 'U' or 'L'.\nIndicates whether the upper or lower triangular part of RFP A is\nstored:\nIf uplo = 'U', the array a stores the upper triangular part of the\nmatrix A.\nIf uplo = 'L', the array a stores the lower triangular part of the\nmatrix A.\ndiag\nMust be 'N' or 'U'.\nIf diag = 'N', then A is not a unit triangular matrix.\nIf diag = 'U', A is unit triangular: diagonal elements of A are\nassumed to be 1 and not referenced in the array a.\nn\nThe order of the matrix A; n≥ 0.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n658\n\n\na\nArray, size max(1, n*(n + 1)/2). The array a contains the matrix A in\nthe RFP format.\nOutput Parameters\na\nThe (triangular) inverse of the original matrix in the same storage\nformat.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, parameter i had an illegal value.\nIf info = i, Ai, i is exactly zero. The triangular matrix is singular and its inverse cannot be computed.\nSee Also\nMatrix Storage Schemes\n?tptri\nComputes the inverse of a triangular matrix using\npacked storage.\nSyntax\nlapack_int LAPACKE_stptri (int matrix_layout , char uplo , char diag , lapack_int n ,\nfloat * ap );\nlapack_int LAPACKE_dtptri (int matrix_layout , char uplo , char diag , lapack_int n ,\ndouble * ap );\nlapack_int LAPACKE_ctptri (int matrix_layout , char uplo , char diag , lapack_int n ,\nlapack_complex_float * ap );\nlapack_int LAPACKE_ztptri (int matrix_layout , char uplo , char diag , lapack_int n ,\nlapack_complex_double * ap );\nInclude Files\n•\nmkl.h\nDescription\nThe routine computes the inverse inv(A) of a packed triangular matrix A.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\nuplo\nMust be 'U' or 'L'.\nIndicates whether A is upper or lower triangular:\nIf uplo = 'U', then A is upper triangular.\nIf uplo = 'L', then A is lower triangular.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n659\n\n\ndiag\nMust be 'N' or 'U'.\nIf diag = 'N', then A is not a unit triangular matrix.\nIf diag = 'U', A is unit triangular: diagonal elements of A are\nassumed to be 1 and not referenced in the array ap.\nn\nThe order of the matrix A; n≥ 0.\nap\nArray, size at least max(1,n(n+1)/2).\nContains the packed triangular matrix A.\nOutput Parameters\nap\nOverwritten by the packed n-by-n matrix inv(A) .\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info = -i, parameter i had an illegal value.\nIf info = i, the i-th diagonal element of A is zero, A is singular, and the inversion could not be completed.\nApplication Notes\nThe computed inverse X satisfies the following error bounds:\n|XA - I| ≤c(n)ε |X||A|\n|X - A-1| ≤c(n)ε |A-1||A||X|,\nwhere c(n) is a modest linear function of n; ε is the machine precision; I denotes the identity matrix.\nThe total number of floating-point operations is approximately (1/3)n3 for real flavors and (4/3)n3 for\ncomplex flavors.\nSee Also\nMatrix Storage Schemes\nMatrix Equilibration: LAPACK Computational Routines\nRoutines described in this section are used to compute scaling factors needed to equilibrate a matrix. Note\nthat these routines do not actually scale the matrices.\n?geequ\nComputes row and column scaling factors intended to\nequilibrate a general matrix and reduce its condition\nnumber.\nSyntax\nlapack_int LAPACKE_sgeequ( int matrix_layout, lapack_int m, lapack_int n, const float*\na, lapack_int lda, float* r, float* c, float* rowcnd, float* colcnd, float* amax );\nlapack_int LAPACKE_dgeequ( int matrix_layout, lapack_int m, lapack_int n, const double*\na, lapack_int lda, double* r, double* c, double* rowcnd, double* colcnd, double* amax );\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n660\n\n\nlapack_int LAPACKE_cgeequ( int matrix_layout, lapack_int m, lapack_int n, const\nlapack_complex_float* a, lapack_int lda, float* r, float* c, float* rowcnd, float*\ncolcnd, float* amax );\nlapack_int LAPACKE_zgeequ( int matrix_layout, lapack_int m, lapack_int n, const\nlapack_complex_double* a, lapack_int lda, double* r, double* c, double* rowcnd, double*\ncolcnd, double* amax );\nInclude Files\n•\nmkl.h\nDescription\nThe routine computes row and column scalings intended to equilibrate an m-by-n matrix A and reduce its\ncondition number. The output array r returns the row scale factors and the array c the column scale factors.\nThese factors are chosen to try to make the largest element in each row and column of the matrix B with\nelements bij=r[i-1]*aij*c[j-1] have absolute value 1.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\nm\nThe number of rows of the matrix A; m≥ 0.\nn\nThe number of columns of the matrix A; n≥ 0.\na\nArray: size max(1, lda*n) for column major layout and max(1,\nlda*m) for row major layout.\nContains the m-by-n matrix A whose equilibration factors are to be\ncomputed.\nlda\nThe leading dimension of a; lda≥ max(1, m).\nOutput Parameters\nr, c\nArrays: r (size m), c (size n).\nIf info = 0, or info>m, the array r contains the row scale factors of\nthe matrix A.\nIf info = 0, the array c contains the column scale factors of the\nmatrix A.\nrowcnd\nIf info = 0 or info>m, rowcnd contains the ratio of the smallest\nr[i] to the largest r[i].\ncolcnd\nIf info = 0, colcnd contains the ratio of the smallest c[i] to the\nlargest c[i].\namax\nAbsolute value of the largest element of the matrix A.\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n661\n\n\nIf info = -i, parameter i had an illegal value.\nIf info = i, i > 0, and\ni≤m, the i-th row of A is exactly zero;\ni>m, the (i-m)th column of A is exactly zero.\nApplication Notes\nAll the components of r and c are restricted to be between SMLNUM = smallest safe number and BIGNUM=\nlargest safe number. Use of these scaling factors is not guaranteed to reduce the condition number of A but\nworks well in practice.\nSMLNUM and BIGNUM are parameters representing machine precision. You can use the ?lamch routines to\ncompute them. For example, compute single precision values of SMLNUM and BIGNUM as follows:\nSMLNUM = slamch ('s')\nBIGNUM = 1 / SMLNUM\nIf rowcnd≥ 0.1 and amax is neither too large nor too small, it is not worth scaling by r.\nIf colcnd≥ 0.1, it is not worth scaling by c.\nIf amax is very close to SMLNUM or very close to BIGNUM, the matrix A should be scaled.\nSee Also\nError Analysis\nMatrix Storage Schemes\n?geequb\nComputes row and column scaling factors restricted to\na power of radix to equilibrate a general matrix and\nreduce its condition number.\nSyntax\nlapack_int LAPACKE_sgeequb( int matrix_layout, lapack_int m, lapack_int n, const float*\na, lapack_int lda, float* r, float* c, float* rowcnd, float* colcnd, float* amax );\nlapack_int LAPACKE_dgeequb( int matrix_layout, lapack_int m, lapack_int n, const\ndouble* a, lapack_int lda, double* r, double* c, double* rowcnd, double* colcnd, double*\namax );\nlapack_int LAPACKE_cgeequb( int matrix_layout, lapack_int m, lapack_int n, const\nlapack_complex_float* a, lapack_int lda, float* r, float* c, float* rowcnd, float*\ncolcnd, float* amax );\nlapack_int LAPACKE_zgeequb( int matrix_layout, lapack_int m, lapack_int n, const\nlapack_complex_double* a, lapack_int lda, double* r, double* c, double* rowcnd, double*\ncolcnd, double* amax );\nInclude Files\n•\nmkl.h\nDescription\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n662\n\n\nThe routine computes row and column scalings intended to equilibrate an m-by-n general matrix A and\nreduce its condition number. The output array r returns the row scale factors and the array c - the column\nscale factors. These factors are chosen to try to make the largest element in each row and column of the\nmatrix B with elements bi,j = r[i-1]*ai,j*c[j-1] have an absolute value of at most the radix.\nr[i-1] and c[j-1] are restricted to be a power of the radix between SMLNUM = smallest safe number and\nBIGNUM = largest safe number. Use of these scaling factors is not guaranteed to reduce the condition number\nof a but works well in practice.\nSMLNUM and BIGNUM are parameters representing machine precision. You can use the ?lamch routines to\ncompute them. For example, compute single precision values of SMLNUM and BIGNUM as follows:\nSMLNUM = slamch ('s')\nBIGNUM = 1 / SMLNUM\nThis routine differs from ?geequ by restricting the scaling factors to a power of the radix. Except for over-\nand underflow, scaling by these factors introduces no additional rounding errors. However, the scaled entries'\nmagnitudes are no longer equal to approximately 1 but lie between sqrt(radix) and 1/sqrt(radix).\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\nm\nThe number of rows of the matrix A; m≥ 0.\nn\nThe number of columns of the matrix A; n≥ 0.\na\nArray: size max(1, lda*n) for column major layout and max(1,\nlda*m) for row major layout.\nContains the m-by-n matrix A whose equilibration factors are to be\ncomputed.\nlda\nThe leading dimension of a; lda≥ max(1, m).\nOutput Parameters\nr, c\nArrays: r(m), c(n).\nIf info = 0, or info>m, the array r contains the row scale factors for\nthe matrix A.\nIf info = 0, the array c contains the column scale factors for the\nmatrix A.\nrowcnd\nIf info = 0 or info>m, rowcnd contains the ratio of the smallest\nr[i] to the largest r[i]. If rowcnd≥ 0.1, and amax is neither too\nlarge nor too small, it is not worth scaling by r.\ncolcnd\nIf info = 0, colcnd contains the ratio of the smallest c[i] to the\nlargest c[i]. If colcnd≥ 0.1, it is not worth scaling by c.\namax\nAbsolute value of the largest element of the matrix A. If amax is very\nclose to SMLNUM or very close to BIGNUM, the matrix should be scaled.\nReturn Values\nThis function returns a value info.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n663\n\n\nIf info = 0, the execution is successful.\nIf info = -i, parameter i had an illegal value.\nIf info = i, i > 0, and\ni≤m, the i-th row of A is exactly zero;\ni>m, the (i-m)-th column of A is exactly zero.\nSee Also\nError Analysis\nMatrix Storage Schemes\n?gbequ\nComputes row and column scaling factors intended to\nequilibrate a banded matrix and reduce its condition\nnumber.\nSyntax\nlapack_int LAPACKE_sgbequ( int matrix_layout, lapack_int m, lapack_int n, lapack_int\nkl, lapack_int ku, const float* ab, lapack_int ldab, float* r, float* c, float* rowcnd,\nfloat* colcnd, float* amax );\nlapack_int LAPACKE_dgbequ( int matrix_layout, lapack_int m, lapack_int n, lapack_int\nkl, lapack_int ku, const double* ab, lapack_int ldab, double* r, double* c, double*\nrowcnd, double* colcnd, double* amax );\nlapack_int LAPACKE_cgbequ( int matrix_layout, lapack_int m, lapack_int n, lapack_int\nkl, lapack_int ku, const lapack_complex_float* ab, lapack_int ldab, float* r, float* c,\nfloat* rowcnd, float* colcnd, float* amax );\nlapack_int LAPACKE_zgbequ( int matrix_layout, lapack_int m, lapack_int n, lapack_int\nkl, lapack_int ku, const lapack_complex_double* ab, lapack_int ldab, double* r, double*\nc, double* rowcnd, double* colcnd, double* amax );\nInclude Files\n•\nmkl.h\nDescription\nThe routine computes row and column scalings intended to equilibrate an m-by-n band matrix A and reduce\nits condition number. The output array r returns the row scale factors and the array c the column scale\nfactors. These factors are chosen to try to make the largest element in each row and column of the matrix B\nwith elements bij=r[i - 1]*aij*c[j - 1] have absolute value 1.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\nm\nThe number of rows of the matrix A; m≥ 0.\nn\nThe number of columns of the matrix A; n≥ 0.\nkl\nThe number of subdiagonals within the band of A; kl≥ 0.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n664\n\n\nku\nThe number of superdiagonals within the band of A; ku≥ 0.\nab\nArray, size max(1, ldab*n) for column major layout and max(1,\nldab*m) for row major layout. Contains the original band matrix A.\nldab\nThe leading dimension of ab; ldab≥kl+ku+1.\nOutput Parameters\nr, c\nArrays: r (size m), c (size n).\nIf info = 0, or info>m, the array r contains the row scale factors of\nthe matrix A.\nIf info = 0, the array c contains the column scale factors of the\nmatrix A.\nrowcnd\nIf info = 0 or info>m, rowcnd contains the ratio of the smallest\nr[i] to the largest r[i].\ncolcnd\nIf info = 0, colcnd contains the ratio of the smallest c[i] to the\nlargest c[i].\namax\nAbsolute value of the largest element of the matrix A.\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info = -i, parameter i had an illegal value.\nIf info = i and\ni≤m, the i-th row of A is exactly zero;\ni>m, the (i-m)th column of A is exactly zero.\nApplication Notes\nAll the components of r and c are restricted to be between SMLNUM = smallest safe number and BIGNUM=\nlargest safe number. Use of these scaling factors is not guaranteed to reduce the condition number of A but\nworks well in practice.\nSMLNUM and BIGNUM are parameters representing machine precision. You can use the ?lamch routines to\ncompute them. For example, compute single precision values of SMLNUM and BIGNUM as follows:\nSMLNUM = slamch ('s')\nBIGNUM = 1 / SMLNUM\nIf rowcnd≥ 0.1 and amax is neither too large nor too small, it is not worth scaling by r.\nIf colcnd≥ 0.1, it is not worth scaling by c.\nIf amax is very close to SMLNUM or very close to BIGNUM, the matrix A should be scaled.\nSee Also\nError Analysis\nMatrix Storage Schemes\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n665\n\n\n?gbequb\nComputes row and column scaling factors restricted to\na power of radix to equilibrate a banded matrix and\nreduce its condition number.\nSyntax\nlapack_int LAPACKE_sgbequb( int matrix_layout, lapack_int m, lapack_int n, lapack_int\nkl, lapack_int ku, const float* ab, lapack_int ldab, float* r, float* c, float* rowcnd,\nfloat* colcnd, float* amax );\nlapack_int LAPACKE_dgbequb( int matrix_layout, lapack_int m, lapack_int n, lapack_int\nkl, lapack_int ku, const double* ab, lapack_int ldab, double* r, double* c, double*\nrowcnd, double* colcnd, double* amax );\nlapack_int LAPACKE_cgbequb( int matrix_layout, lapack_int m, lapack_int n, lapack_int\nkl, lapack_int ku, const lapack_complex_float* ab, lapack_int ldab, float* r, float* c,\nfloat* rowcnd, float* colcnd, float* amax );\nlapack_int LAPACKE_zgbequb( int matrix_layout, lapack_int m, lapack_int n, lapack_int\nkl, lapack_int ku, const lapack_complex_double* ab, lapack_int ldab, double* r, double*\nc, double* rowcnd, double* colcnd, double* amax );\nInclude Files\n•\nmkl.h\nDescription\nThe routine computes row and column scalings intended to equilibrate an m-by-n banded matrix A and\nreduce its condition number. The output array r returns the row scale factors and the array c - the column\nscale factors. These factors are chosen to try to make the largest element in each row and column of the\nmatrix B with elements bi, j=r[i-1]*ai, j*c[j-1] have an absolute value of at most the radix.\nr[i] and c[j] are restricted to be a power of the radix between SMLNUM = smallest safe number and\nBIGNUM = largest safe number. Use of these scaling factors is not guaranteed to reduce the condition\nnumber of a but works well in practice.\nSMLNUM and BIGNUM are parameters representing machine precision. You can use the ?lamch routines to\ncompute them. For example, compute single precision values of SMLNUM and BIGNUM as follows:\nSMLNUM = slamch ('s')\nBIGNUM = 1 / SMLNUM\nThis routine differs from ?gbequ by restricting the scaling factors to a power of the radix. Except for over-\nand underflow, scaling by these factors introduces no additional rounding errors. However, the scaled entries'\nmagnitudes are no longer equal to approximately 1 but lie between sqrt(radix) and 1/sqrt(radix).\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\nm\nThe number of rows of the matrix A; m≥ 0.\nn\nThe number of columns of the matrix A; n≥ 0.\nkl\nThe number of subdiagonals within the band of A; kl≥ 0.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n666\n\n\nku\nThe number of superdiagonals within the band of A; ku≥ 0.\nab\nArray: size max(1, ldab*n) for column major layout and max(1,\nldab*m) for row major layout\nldab\nThe leading dimension of a; ldab≥ max(1, m).\nOutput Parameters\nr, c\nArrays: r (size m), c (size n).\nIf info = 0, or info>m, the array r contains the row scale factors for\nthe matrix A.\nIf info = 0, the array c contains the column scale factors for the\nmatrix A.\nrowcnd\nIf info = 0 or info>m, rowcnd contains the ratio of the smallest\nr(i) to the largest r(i). If rowcnd≥ 0.1, and amax is neither too\nlarge nor too small, it is not worth scaling by r.\ncolcnd\nIf info = 0, colcnd contains the ratio of the smallest c[i] to the\nlargest c[i]. If colcnd≥ 0.1, it is not worth scaling by c.\namax\nAbsolute value of the largest element of the matrix A. If amax is very\nclose to SMLNUM or BIGNUM, the matrix should be scaled.\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info = -i, parameter i had an illegal value.\nIf info = i, the i-th diagonal element of A is nonpositive.\ni≤m, the i-th row of A is exactly zero;\ni>m, the (i-m)-th column of A is exactly zero.\nSee Also\nError Analysis\nMatrix Storage Schemes\n?poequ\nComputes row and column scaling factors intended to\nequilibrate a symmetric (Hermitian) positive definite\nmatrix and reduce its condition number.\nSyntax\nlapack_int LAPACKE_spoequ( int matrix_layout, lapack_int n, const float* a, lapack_int\nlda, float* s, float* scond, float* amax );\nlapack_int LAPACKE_dpoequ( int matrix_layout, lapack_int n, const double* a, lapack_int\nlda, double* s, double* scond, double* amax );\nlapack_int LAPACKE_cpoequ( int matrix_layout, lapack_int n, const lapack_complex_float*\na, lapack_int lda, float* s, float* scond, float* amax );\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n667\n\n\nlapack_int LAPACKE_zpoequ( int matrix_layout, lapack_int n, const\nlapack_complex_double* a, lapack_int lda, double* s, double* scond, double* amax );\nInclude Files\n•\nmkl.h\nDescription\nThe routine computes row and column scalings intended to equilibrate a symmetric (Hermitian) positive-\ndefinite matrix A and reduce its condition number (with respect to the two-norm). The output array s returns\nscale factors such that contains\nThese factors are chosen so that the scaled matrix B with elements Bi,j=s[i-1]*Ai,j*s[j-1] has diagonal\nelements equal to 1.\nThis choice of s puts the condition number of B within a factor n of the smallest possible condition number\nover all possible diagonal scalings.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\nn\nThe order of the matrix A; n≥ 0.\na\nArray: size max(1, lda*n) .\nContains the n-by-n symmetric or Hermitian positive definite matrix A\nwhose scaling factors are to be computed. Only the diagonal elements\nof A are referenced.\nlda\nThe leading dimension of a; lda≥ max(1,n).\nOutput Parameters\ns\nArray, size n.\nIf info = 0, the array s contains the scale factors for A.\nscond\nIf info = 0, scond contains the ratio of the smallest s[i] to the\nlargest s[i].\namax\nAbsolute value of the largest element of the matrix A.\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info = -i, parameter i had an illegal value.\nIf info = i, the i-th diagonal element of A is nonpositive.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n668\n\n\nApplication Notes\nIf scond≥ 0.1 and amax is neither too large nor too small, it is not worth scaling by s.\nIf amax is very close to SMLNUM or very close to BIGNUM, the matrix A should be scaled.\nSee Also\nError Analysis\nMatrix Storage Schemes\n?poequb\nComputes row and column scaling factors intended to\nequilibrate a symmetric (Hermitian) positive definite\nmatrix and reduce its condition number.\nSyntax\nlapack_int LAPACKE_spoequb( int matrix_layout, lapack_int n, const float* a, lapack_int\nlda, float* s, float* scond, float* amax );\nlapack_int LAPACKE_dpoequb( int matrix_layout, lapack_int n, const double* a,\nlapack_int lda, double* s, double* scond, double* amax );\nlapack_int LAPACKE_cpoequb( int matrix_layout, lapack_int n, const\nlapack_complex_float* a, lapack_int lda, float* s, float* scond, float* amax );\nlapack_int LAPACKE_zpoequb( int matrix_layout, lapack_int n, const\nlapack_complex_double* a, lapack_int lda, double* s, double* scond, double* amax );\nInclude Files\n•\nmkl.h\nDescription\nThe routine computes row and column scalings intended to equilibrate a symmetric (Hermitian) positive-\ndefinite matrix A and reduce its condition number (with respect to the two-norm).\nThese factors are chosen so that the scaled matrix B with elements Bi,j=s[i-1]*Ai,j*s[j-1] has diagonal\nelements equal to 1. s[i - 1] is a power of two nearest to, but not exceeding 1/sqrt(Ai,i).\nThis choice of s puts the condition number of B within a factor n of the smallest possible condition number\nover all possible diagonal scalings.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\nn\nThe order of the matrix A; n≥ 0.\na\nArray: size max(1, lda*n) .\nContains the n-by-n symmetric or Hermitian positive definite matrix A\nwhose scaling factors are to be computed. Only the diagonal elements\nof A are referenced.\nlda\nThe leading dimension of a; lda≥ max(1, m).\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n669\n\n\nOutput Parameters\ns\nArray, size (n).\nIf info = 0, the array s contains the scale factors for A.\nscond\nIf info = 0, scond contains the ratio of the smallest s[i] to the\nlargest s[i]. If scond≥ 0.1, and amax is neither too large nor too\nsmall, it is not worth scaling by s.\namax\nAbsolute value of the largest element of the matrix A. If amax is very\nclose to SMLNUM or BIGNUM, the matrix should be scaled.\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info = -i, parameter i had an illegal value.\nIf info = i, the i-th diagonal element of A is nonpositive.\nSee Also\nError Analysis\nMatrix Storage Schemes\n?ppequ\nComputes row and column scaling factors intended to\nequilibrate a symmetric (Hermitian) positive definite\nmatrix in packed storage and reduce its condition\nnumber.\nSyntax\nlapack_int LAPACKE_sppequ( int matrix_layout, char uplo, lapack_int n, const float* ap,\nfloat* s, float* scond, float* amax );\nlapack_int LAPACKE_dppequ( int matrix_layout, char uplo, lapack_int n, const double*\nap, double* s, double* scond, double* amax );\nlapack_int LAPACKE_cppequ( int matrix_layout, char uplo, lapack_int n, const\nlapack_complex_float* ap, float* s, float* scond, float* amax );\nlapack_int LAPACKE_zppequ( int matrix_layout, char uplo, lapack_int n, const\nlapack_complex_double* ap, double* s, double* scond, double* amax );\nInclude Files\n•\nmkl.h\nDescription\nThe routine computes row and column scalings intended to equilibrate a symmetric (Hermitian) positive\ndefinite matrix A in packed storage and reduce its condition number (with respect to the two-norm). The\noutput array s returns scale factors such that contains\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n670\n\n\nThese factors are chosen so that the scaled matrix B with elements bij=s[i-1]*aij*s[j-1] has diagonal\nelements equal to 1.\nThis choice of s puts the condition number of B within a factor n of the smallest possible condition number\nover all possible diagonal scalings.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\nuplo\nMust be 'U' or 'L'.\nIndicates whether the upper or lower triangular part of A is packed in\nthe array ap:\nIf uplo = 'U', the array ap stores the upper triangular part of the\nmatrix A.\nIf uplo = 'L', the array ap stores the lower triangular part of the\nmatrix A.\nn\nThe order of matrix A; n≥ 0.\nap\nArray, size at least max(1,n(n+1)/2). The array ap contains the\nupper or the lower triangular part of the matrix A (as specified by\nuplo) in packed storage (see Matrix Storage Schemes).\nOutput Parameters\ns\nArray, size (n).\nIf info = 0, the array s contains the scale factors for A.\nscond\nIf info = 0, scond contains the ratio of the smallest s[i] to the\nlargest s[i].\namax\nAbsolute value of the largest element of the matrix A.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n671\n\n\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info = -i, parameter i had an illegal value.\nIf info = i, the i-th diagonal element of A is nonpositive.\nApplication Notes\nIf scond≥ 0.1 and amax is neither too large nor too small, it is not worth scaling by s.\nIf amax is very close to SMLNUM or very close to BIGNUM, the matrix A should be scaled.\nSee Also\nError Analysis\nMatrix Storage Schemes\n?pbequ\nComputes row and column scaling factors intended to\nequilibrate a symmetric (Hermitian) positive-definite\nband matrix and reduce its condition number.\nSyntax\nlapack_int LAPACKE_spbequ( int matrix_layout, char uplo, lapack_int n, lapack_int kd,\nconst float* ab, lapack_int ldab, float* s, float* scond, float* amax );\nlapack_int LAPACKE_dpbequ( int matrix_layout, char uplo, lapack_int n, lapack_int kd,\nconst double* ab, lapack_int ldab, double* s, double* scond, double* amax );\nlapack_int LAPACKE_cpbequ( int matrix_layout, char uplo, lapack_int n, lapack_int kd,\nconst lapack_complex_float* ab, lapack_int ldab, float* s, float* scond, float* amax );\nlapack_int LAPACKE_zpbequ( int matrix_layout, char uplo, lapack_int n, lapack_int kd,\nconst lapack_complex_double* ab, lapack_int ldab, double* s, double* scond, double*\namax );\nInclude Files\n•\nmkl.h\nDescription\nThe routine computes row and column scalings intended to equilibrate a symmetric (Hermitian) positive\ndefinite band matrix A and reduce its condition number (with respect to the two-norm). The output array s\nreturns scale factors such that contains\nThese factors are chosen so that the scaled matrix B with elements bij=s[i-1]*aij*s[j-1] has diagonal\nelements equal to 1. This choice of s puts the condition number of B within a factor n of the smallest possible\ncondition number over all possible diagonal scalings.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n672\n\n\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\nuplo\nMust be 'U' or 'L'.\nIndicates whether the upper or lower triangular part of A is stored in\nthe array ab:\nIf uplo = 'U', the array ab stores the upper triangular part of the\nmatrix A.\nIf uplo = 'L', the array ab stores the lower triangular part of the\nmatrix A.\nn\nThe order of matrix A; n≥ 0.\nkd\nThe number of superdiagonals or subdiagonals in the matrix A; kd≥ 0.\nab\nArray, size max(1, ldab*n) .\nThe array ap contains either the upper or the lower triangular part of\nthe matrix A (as specified by uplo) in band storage (see Matrix\nStorage Schemes).\nldab\nThe leading dimension of the array ab; ldab≥kd +1.\nOutput Parameters\ns\nArray, size (n).\nIf info = 0, the array s contains the scale factors for A.\nscond\nIf info = 0, scond contains the ratio of the smallest s[i] to the\nlargest s[i].\namax\nAbsolute value of the largest element of the matrix A.\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info = -i, parameter i had an illegal value.\nIf info = i, the i-th diagonal element of A is nonpositive.\nApplication Notes\nIf scond≥ 0.1 and amax is neither too large nor too small, it is not worth scaling by s.\nIf amax is very close to SMLNUM or very close to BIGNUM, the matrix A should be scaled.\nSee Also\nError Analysis\nMatrix Storage Schemes\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n673\n\n\n?syequb\nComputes row and column scaling factors intended to\nequilibrate a symmetric indefinite matrix and reduce\nits condition number.\nSyntax\nlapack_int LAPACKE_ssyequb( int matrix_layout, char uplo, lapack_int n, const float* a,\nlapack_int lda, float* s, float* scond, float* amax );\nlapack_int LAPACKE_dsyequb( int matrix_layout, char uplo, lapack_int n, const double*\na, lapack_int lda, double* s, double* scond, double* amax );\nlapack_int LAPACKE_csyequb( int matrix_layout, char uplo, lapack_int n, const\nlapack_complex_float* a, lapack_int lda, float* s, float* scond, float* amax );\nlapack_int LAPACKE_zsyequb( int matrix_layout, char uplo, lapack_int n, const\nlapack_complex_double* a, lapack_int lda, double* s, double* scond, double* amax );\nInclude Files\n•\nmkl.h\nDescription\nThe routine computes row and column scalings intended to equilibrate a symmetric indefinite matrix A and\nreduce its condition number (with respect to the two-norm).\nThe array s contains the scale factors, s[i-1] = 1/sqrt(A(i,i)). These factors are chosen so that the\nscaled matrix B with elements bi,j=s[i-1]*ai, j*s[j-1] has ones on the diagonal.\nThis choice of s puts the condition number of B within a factor n of the smallest possible condition number\nover all possible diagonal scalings.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\nuplo\nMust be 'U' or 'L'.\nIndicates whether the upper or lower triangular part of A is stored:\nIf uplo = 'U', the array a stores the upper triangular part of the\nmatrix A.\nIf uplo = 'L', the array a stores the lower triangular part of the\nmatrix A.\nn\nThe order of the matrix A; n≥ 0.\na\nArray a: max(1, lda*n) .\nContains the n-by-n symmetric indefinite matrix A whose scaling\nfactors are to be computed. Only the diagonal elements of A are\nreferenced.\nlda\nThe leading dimension of a; lda≥ max(1, m).\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n674\n\n\nOutput Parameters\ns\nArray, size (n).\nIf info = 0, the array s contains the scale factors for A.\nscond\nIf info = 0, scond contains the ratio of the smallest s[i] to the\nlargest s[i]. If scond≥ 0.1, and amax is neither too large nor too\nsmall, it is not worth scaling by s.\namax\nAbsolute value of the largest element of the matrix A. If amax is very\nclose to SMLNUM or BIGNUM, the matrix should be scaled.\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info = -i, parameter i had an illegal value.\nIf info = i, the i-th diagonal element of A is nonpositive.\nSee Also\nError Analysis\nMatrix Storage Schemes\n?heequb\nComputes row and column scaling factors intended to\nequilibrate a Hermitian indefinite matrix and reduce its\ncondition number.\nSyntax\nlapack_int LAPACKE_cheequb( int matrix_layout, char uplo, lapack_int n, const\nlapack_complex_float* a, lapack_int lda, float* s, float* scond, float* amax );\nlapack_int LAPACKE_zheequb( int matrix_layout, char uplo, lapack_int n, const\nlapack_complex_double* a, lapack_int lda, double* s, double* scond, double* amax );\nInclude Files\n•\nmkl.h\nDescription\nThe routine computes row and column scalings intended to equilibrate a Hermitian indefinite matrix A and\nreduce its condition number (with respect to the two-norm).\nThe array s contains the scale factors, s[i-1] = 1/sqrt(ai,i). These factors are chosen so that the scaled\nmatrix B with elements bi,j=s[i-1]*ai,j*s[j-1] has ones on the diagonal.\nThis choice of s puts the condition number of B within a factor n of the smallest possible condition number\nover all possible diagonal scalings.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n675\n\n\nuplo\nMust be 'U' or 'L'.\nIndicates whether the upper or lower triangular part of A is stored:\nIf uplo = 'U', the array a stores the upper triangular part of the\nmatrix A.\nIf uplo = 'L', the array a stores the lower triangular part of the\nmatrix A.\nn\nThe order of the matrix A; n≥ 0.\na\nArray a: size max(1, lda*n) .\nContains the n-by-n symmetric indefinite matrix A whose scaling\nfactors are to be computed. Only the diagonal elements of A are\nreferenced.\nlda\nThe leading dimension of a; lda≥ max(1, m).\nOutput Parameters\ns\nArray, size (n).\nIf info = 0, the array s contains the scale factors for A.\nscond\nIf info = 0, scond contains the ratio of the smallest s[i] to the\nlargest s[i]. If scond≥ 0.1, and amax is neither too large nor too\nsmall, it is not worth scaling by s.\namax\nAbsolute value of the largest element of the matrix A. If amax is very\nclose to SMLNUM or BIGNUM, the matrix should be scaled.\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info = -i, parameter i had an illegal value.\nIf info = i, the i-th diagonal element of A is nonpositive.\nSee Also\nError Analysis\nMatrix Storage Schemes\nLAPACK Linear Equation Driver Routines\nTable \"Driver Routines for Solving Systems of Linear Equations\" lists the LAPACK driver routines for solving\nsystems of linear equations with real or complex matrices.\nDriver Routines for Solving Systems of Linear Equations\nMatrix type, storage\nscheme\nSimple Driver\nExpert Driver\nExpert Driver using\nExtra-Precise\nInterative Refinement\ngeneral\n?gesv\n?gesvx\n?gesvxx\ngeneral band\n?gbsv\n?gbsvx\n?gbsvxx\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n676\n\n\nMatrix type, storage\nscheme\nSimple Driver\nExpert Driver\nExpert Driver using\nExtra-Precise\nInterative Refinement\ngeneral tridiagonal\n?gtsv\n?gtsvx\ndiagonally dominant\ntridiagonal\n?dtsvb\nsymmetric/Hermitian\npositive-definite\n?posv\n?posvx\n?posvxx\nsymmetric/Hermitian\npositive-definite,\nstorage\n?ppsv\n?ppsvx\nsymmetric/Hermitian\npositive-definite, band\n?pbsv\n?pbsvx\nsymmetric/Hermitian\npositive-definite,\ntridiagonal\n?ptsv\n?ptsvx\nsymmetric/Hermitian\nindefinite\n?sysv/?hesv\n?sysv_rook/?sysv_rk/\n?hesv_rk\n?sysv_aa/?hesv_aa\n?sysvx/?hesvx\n?sysvxx/?hesvxx\nsymmetric/Hermitian\nindefinite, packed\nstorage\n?spsv/?hpsv\n?spsvx/?hpsvx\ncomplex symmetric\n?sysv\n?sysv_rook\n?sysvx\ncomplex symmetric,\npacked storage\n?spsv\n?spsvx\nIn this table ? stands for s (single precision real), d (double precision real), c (single precision complex), or z\n(double precision complex). In the description of ?gesv and ?posv routines, the ? sign stands for combined\ncharacter codes ds and zc for the mixed precision subroutines.\n?gesv\nComputes the solution to the system of linear\nequations with a square coefficient matrix A and\nmultiple right-hand sides.\nSyntax\nlapack_int LAPACKE_sgesv (int matrix_layout , lapack_int n , lapack_int nrhs , float *\na , lapack_int lda , lapack_int * ipiv , float * b , lapack_int ldb );\nlapack_int LAPACKE_dgesv (int matrix_layout , lapack_int n , lapack_int nrhs , double *\na , lapack_int lda , lapack_int * ipiv , double * b , lapack_int ldb );\nlapack_int LAPACKE_cgesv (int matrix_layout , lapack_int n , lapack_int nrhs ,\nlapack_complex_float * a , lapack_int lda , lapack_int * ipiv , lapack_complex_float *\nb , lapack_int ldb );\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n677\n\n\nlapack_int LAPACKE_zgesv (int matrix_layout , lapack_int n , lapack_int nrhs ,\nlapack_complex_double * a , lapack_int lda , lapack_int * ipiv , lapack_complex_double\n* b , lapack_int ldb );\nlapack_int LAPACKE_dsgesv (int matrix_layout, lapack_int n, lapack_int nrhs, double *\na, lapack_int lda, lapack_int * ipiv, double * b, lapack_int ldb, double * x, lapack_int\nldx, lapack_int * iter);\nlapack_int LAPACKE_zcgesv (int matrix_layout, lapack_int n, lapack_int nrhs,\nlapack_complex_double * a, lapack_int lda, lapack_int * ipiv, lapack_complex_double *\nb, lapack_int ldb, lapack_complex_double * x, lapack_int ldx, lapack_int * iter);\nInclude Files\n•\nmkl.h\nDescription\nThe routine solves for X the system of linear equations A*X = B, where A is an n-by-n matrix, the columns\nof matrix B are individual right-hand sides, and the columns of X are the corresponding solutions.\nThe LU decomposition with partial pivoting and row interchanges is used to factor A as A = P*L*U, where P\nis a permutation matrix, L is unit lower triangular, and U is upper triangular. The factored form of A is then\nused to solve the system of equations A*X = B.\nThe dsgesv and zcgesv are mixed precision iterative refinement subroutines for exploiting fast single\nprecision hardware. They first attempt to factorize the matrix in single precision (dsgesv) or single complex\nprecision (zcgesv) and use this factorization within an iterative refinement procedure to produce a solution\nwith double precision (dsgesv) / double complex precision (zcgesv) normwise backward error quality (see\nbelow). If the approach fails, the method switches to a double precision or double complex precision\nfactorization respectively and computes the solution.\nThe iterative refinement is not going to be a winning strategy if the ratio single precision performance over\ndouble precision performance is too small. A reasonable strategy should take the number of right-hand sides\nand the size of the matrix into account. This might be done with a call to ilaenv in the future. At present,\niterative refinement is implemented.\nThe iterative refinement process is stopped if\niter > itermax\nor for all the right-hand sides:\nrnmr < sqrt(n)*xnrm*anrm*eps*bwdmax\nwhere\n•\niter is the number of the current iteration in the iterativerefinement process\n•\nrnmr is the infinity-norm of the residual\n•\nxnrm is the infinity-norm of the solution\n•\nanrm is the infinity-operator-norm of the matrix A\n•\neps is the machine epsilon returned by dlamch (‘Epsilon’).\nThe values itermax and bwdmax are fixed to 30 and 1.0d+00 respectively.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n678\n\n\nn\nThe number of linear equations, that is, the order of the matrix A; n≥\n0.\nnrhs\nThe number of right-hand sides, that is, the number of columns of the\nmatrix B; nrhs≥ 0.\na\nThe array a(size max(1, lda*n)) contains the n-by-n coefficient\nmatrix A.\nb\nThe array bof size max(1, ldb*nrhs) for column major layout and\nmax(1, ldb*n) for row major layout contains the n-by-nrhs matrix of\nright hand side matrix B.\nlda\nThe leading dimension of the array a; lda≥ max(1, n).\nldb\nThe leading dimension of the array b; ldb≥ max(1, n) for column\nmajor layout and ldb≥nrhs for row major layout.\nldx\nThe leading dimension of the array x; ldx≥ max(1, n) for column\nmajor layout and ldx≥nrhs for row major layout.\nOutput Parameters\na\nOverwritten by the factors L and U from the factorization of A =\nP*L*U; the unit diagonal elements of L are not stored.\nIf iterative refinement has been successfully used (info= 0 and\niter≥ 0), then A is unchanged.\nIf double precision factorization has been used (info= 0 and iter <\n0), then the array A contains the factors L and U from the\nfactorization A = P*L*U; the unit diagonal elements of L are not\nstored.\nb\nOverwritten by the solution matrix X for dgesv, sgesv,zgesv,zgesv.\nUnchanged for dsgesv and zcgesv.\nipiv\nArray, size at least max(1, n). The pivot indices that define the\npermutation matrix P; row i of the matrix was interchanged with row\nipiv[i-1]. Corresponds to the single precision factorization (if\ninfo= 0 and iter≥ 0) or the double precision factorization (if info=\n0 and iter < 0).\nx\nArray, size max(1, ldx*nrhs) for column major layout and max(1,\nldx*n) for row major layout. If info = 0, contains the n-by-nrhs\nsolution matrix X.\niter\nIf iter < 0: iterative refinement has failed, double precision\nfactorization has been performed\n•\nIf iter = -1: the routine fell back to full precision for\nimplementation- or machine-specific reason\n•\nIf iter = -2: narrowing the precision induced an overflow, the\nroutine fell back to full precision\n•\nIf iter = -3: failure of sgetrf for dsgesv, or cgetrf for zcgesv\n•\nIf iter = -31: stop the iterative refinement after the 30th\niteration.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n679\n\n\nIf iter > 0: iterative refinement has been successfully used. Returns\nthe number of iterations.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, parameter i had an illegal value.\nIf info = i, Ui, i (computed in double precision for mixed precision subroutines) is exactly zero. The\nfactorization has been completed, but the factor U is exactly singular, so the solution could not be computed.\nSee Also\ndlamch\nsgetrf\nMatrix Storage Schemes\n?gesvx\nComputes the solution to the system of linear\nequations with a square coefficient matrix A and\nmultiple right-hand sides, and provides error bounds\non the solution.\nSyntax\nlapack_int LAPACKE_sgesvx( int matrix_layout, char fact, char trans, lapack_int n,\nlapack_int nrhs, float* a, lapack_int lda, float* af, lapack_int ldaf, lapack_int* ipiv,\nchar* equed, float* r, float* c, float* b, lapack_int ldb, float* x, lapack_int ldx,\nfloat* rcond, float* ferr, float* berr, float* rpivot );\nlapack_int LAPACKE_dgesvx( int matrix_layout, char fact, char trans, lapack_int n,\nlapack_int nrhs, double* a, lapack_int lda, double* af, lapack_int ldaf, lapack_int*\nipiv, char* equed, double* r, double* c, double* b, lapack_int ldb, double* x,\nlapack_int ldx, double* rcond, double* ferr, double* berr, double* rpivot );\nlapack_int LAPACKE_cgesvx( int matrix_layout, char fact, char trans, lapack_int n,\nlapack_int nrhs, lapack_complex_float* a, lapack_int lda, lapack_complex_float* af,\nlapack_int ldaf, lapack_int* ipiv, char* equed, float* r, float* c,\nlapack_complex_float* b, lapack_int ldb, lapack_complex_float* x, lapack_int ldx,\nfloat* rcond, float* ferr, float* berr, float* rpivot );\nlapack_int LAPACKE_zgesvx( int matrix_layout, char fact, char trans, lapack_int n,\nlapack_int nrhs, lapack_complex_double* a, lapack_int lda, lapack_complex_double* af,\nlapack_int ldaf, lapack_int* ipiv, char* equed, double* r, double* c,\nlapack_complex_double* b, lapack_int ldb, lapack_complex_double* x, lapack_int ldx,\ndouble* rcond, double* ferr, double* berr, double* rpivot );\nInclude Files\n•\nmkl.h\nDescription\nThe routine uses the LU factorization to compute the solution to a real or complex system of linear equations\nA*X = B, where A is an n-by-n matrix, the columns of matrix B are individual right-hand sides, and the\ncolumns of X are the corresponding solutions.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n680\n\n\nError bounds on the solution and a condition estimate are also provided.\nThe routine ?gesvx performs the following steps:\n1.\nIf fact = 'E', real scaling factors r and c are computed to equilibrate the system:\ntrans = 'N': diag(r)*A*diag(c)*inv(diag(c))*X = diag(r)*B\ntrans = 'T': (diag(r)*A*diag(c))T*inv(diag(r))*X = diag(c)*B\ntrans = 'C': (diag(r)*A*diag(c))H*inv(diag(r))*X = diag(c)*B\nWhether or not the system will be equilibrated depends on the scaling of the matrix A, but if\nequilibration is used, A is overwritten by diag(r)*A*diag(c) and B by diag(r)*B (if trans='N') or\ndiag(c)*B (if trans = 'T' or 'C').\n2.\nIf fact = 'N' or 'E', the LU decomposition is used to factor the matrix A (after equilibration if fact\n= 'E') as A = P*L*U, where P is a permutation matrix, L is a unit lower triangular matrix, and U is\nupper triangular.\n3.\nIf some Ui,i= 0, so that U is exactly singular, then the routine returns with info = i. Otherwise, the\nfactored form of A is used to estimate the condition number of the matrix A. If the reciprocal of the\ncondition number is less than machine precision, info = n + 1 is returned as a warning, but the\nroutine still goes on to solve for X and compute error bounds as described below.\n4.\nThe system of equations is solved for X using the factored form of A.\n5.\nIterative refinement is applied to improve the computed solution matrix and calculate error bounds and\nbackward error estimates for it.\n6.\nIf equilibration was used, the matrix X is premultiplied by diag(c) (if trans = 'N') or diag(r) (if\ntrans = 'T' or 'C') so that it solves the original system before equilibration.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\nfact\nMust be 'F', 'N', or 'E'.\nSpecifies whether or not the factored form of the matrix A is supplied\non entry, and if not, whether the matrix A should be equilibrated\nbefore it is factored.\nIf fact = 'F': on entry, af and ipiv contain the factored form of A. If\nequed is not 'N', the matrix A has been equilibrated with scaling\nfactors given by r and c.\na, af, and ipiv are not modified.\nIf fact = 'N', the matrix A will be copied to af and factored.\nIf fact = 'E', the matrix A will be equilibrated if necessary, then\ncopied to af and factored.\ntrans\nMust be 'N', 'T', or 'C'.\nSpecifies the form of the system of equations:\nIf trans = 'N', the system has the form A*X = B (No transpose).\nIf trans = 'T', the system has the form AT*X = B (Transpose).\nIf trans = 'C', the system has the form AH*X = B (Transpose for\nreal flavors, conjugate transpose for complex flavors).\nn\nThe number of linear equations; the order of the matrix A; n≥ 0.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n681\n\n\nnrhs\nThe number of right hand sides; the number of columns of the\nmatrices B and X; nrhs≥ 0.\na\nThe array a(size max(1, lda*n)) contains the matrix A. If fact =\n'F' and equed is not 'N', then A must have been equilibrated by the\nscaling factors in r and/or c.\naf\nThe array afaf(size max(1, ldaf*n)) is an input argument if fact =\n'F'. It contains the factored form of the matrix A, that is, the factors\nL and U from the factorization A = P*L*U as computed by ?getrf. If\nequed is not 'N', then af is the factored form of the equilibrated\nmatrix A.\nb\nThe array bbof size max(1, ldb*nrhs) for column major layout and\nmax(1, ldb*n) for row major layout contains the matrix B whose\ncolumns are the right-hand sides for the systems of equations.\nlda\nThe leading dimension of a; lda≥ max(1, n).\nldaf\nThe leading dimension of af; ldaf≥ max(1, n).\nldb\nThe leading dimension of b; ldb≥ max(1, n) for column major layout\nand ldb≥nrhs for row major layout.\nipiv\nArray, size at least max(1, n). The array ipiv is an input argument if\nfact = 'F'. It contains the pivot indices from the factorization A =\nP*L*U as computed by ?getrf; row i of the matrix was interchanged\nwith row ipiv[i-1].\nequed\nMust be 'N', 'R', 'C', or 'B'.\nequed is an input argument if fact = 'F'. It specifies the form of\nequilibration that was done:\nIf equed = 'N', no equilibration was done (always true if fact =\n'N').\nIf equed = 'R', row equilibration was done, that is, A has been\npremultiplied by diag(r).\nIf equed = 'C', column equilibration was done, that is, A has been\npostmultiplied by diag(c).\nIf equed = 'B', both row and column equilibration was done, that is,\nA has been replaced by diag(r)*A*diag(c).\nr, c\nArrays: r (size n), c (size n). The array r contains the row scale\nfactors for A, and the array c contains the column scale factors for A.\nThese arrays are input arguments if fact = 'F' only; otherwise they\nare output arguments.\nIf equed = 'R' or 'B', A is multiplied on the left by diag(r); if equed\n= 'N' or 'C', r is not accessed.\nIf fact = 'F' and equed = 'R' or 'B', each element of r must be\npositive.\nIf equed = 'C' or 'B', A is multiplied on the right by diag(c); if\nequed = 'N' or 'R', c is not accessed.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n682\n\n\nIf fact = 'F' and equed = 'C' or 'B', each element of c must be\npositive.\nldx\nThe leading dimension of the output array x; ldx≥ max(1, n) for\ncolumn major layout and ldx≥nrhs for row major layout.\nOutput Parameters\nx\nArray, size max(1, ldx*nrhs) for column major layout and max(1,\nldx*n) for row major layout.\nIf info = 0 or info = n+1, the array x contains the solution matrix\nX to the original system of equations. Note that A and B are modified\non exit if equed≠'N', and the solution to the equilibrated system is:\ndiag(C)-1*X, if trans = 'N' and equed = 'C' or 'B';\ndiag(R)-1*X, if trans = 'T' or 'C' and equed = 'R' or 'B'. The\nsecond dimension of x must be at least max(1,nrhs).\na\nArray a is not modified on exit if fact = 'F' or 'N', or if fact =\n'E' and equed = 'N'. If equed≠'N', A is scaled on exit as follows:\nequed = 'R': A = diag(R)*A\nequed = 'C': A = A*diag(c)\nequed = 'B': A = diag(R)*A*diag(c).\naf\nIf fact = 'N' or 'E', then af is an output argument and on exit\nreturns the factors L and U from the factorization A = PLU of the\noriginal matrix A (if fact = 'N') or of the equilibrated matrix A (if\nfact = 'E'). See the description of a for the form of the equilibrated\nmatrix.\nb\nOverwritten by diag(r)*B if trans = 'N' and equed = 'R'or 'B';\noverwritten by diag(c)*B if trans = 'T' or 'C' and equed = 'C'\nor 'B';\nnot changed if equed = 'N'.\nr, c\nThese arrays are output arguments if fact≠'F'. See the description\nof r, c in Input Arguments section.\nrcond\nAn estimate of the reciprocal condition number of the matrix A after\nequilibration (if done). If rcond is less than the machine precision, in\nparticular, if rcond = 0, the matrix is singular to working precision.\nThis condition is indicated by a return code of info > 0.\nferr\nArray, size at least max(1, nrhs). Contains the estimated forward\nerror bound for each solution vector xj (the j-th column of the\nsolution matrix X). If xtrue is the true solution corresponding to xj,\nferr[j-1] is an estimated upper bound for the magnitude of the\nlargest element in (xj - xtrue) divided by the magnitude of the\nlargest element in xj. The estimate is as reliable as the estimate for\nrcond, and is almost always a slight overestimate of the true error.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n683\n\n\nberr\nArray, size at least max(1, nrhs). Contains the component-wise\nrelative backward error for each solution vector xj, that is, the\nsmallest relative change in any element of A or B that makes xj an\nexact solution.\nipiv\nIf fact = 'N'or 'E', then ipiv is an output argument and on exit\ncontains the pivot indices from the factorization A = P*L*U of the\noriginal matrix A (if fact = 'N') or of the equilibrated matrix A (if\nfact = 'E').\nequed\nIf fact≠'F', then equed is an output argument. It specifies the form\nof equilibration that was done (see the description of equed in Input\nArguments section).\nrpivot\nOn exit, rpivot contains the reciprocal pivot growth factor:\nIf rpivot is much less than 1, then the stability of the LU\nfactorization of the (equilibrated) matrix A could be poor. This also\nmeans that the solution x, condition estimator rcond, and forward\nerror bound ferr could be unreliable. If factorization fails with 0 <\ninfo≤n, then rpivot contains the reciprocal pivot growth factor for\nthe leading info columns of A.\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info = -i, parameter i had an illegal value.\nIf info = i, and i≤n, then U(i, i) is exactly zero. The factorization has been completed, but the factor U is\nexactly singular, so the solution and error bounds could not be computed; rcond = 0 is returned.\nIf info = n + 1, then U is nonsingular, but rcond is less than machine precision, meaning that the matrix is\nsingular to working precision. Nevertheless, the solution and error bounds are computed because there are a\nnumber of situations where the computed solution can be more accurate than the value of rcond would\nsuggest.\nSee Also\nMatrix Storage Schemes\n?gesvxx\nUses extra precise iterative refinement to compute the\nsolution to the system of linear equations with a\nsquare coefficient matrix A and multiple right-hand\nsides\nSyntax\nlapack_int LAPACKE_sgesvxx( int matrix_layout, char fact, char trans, lapack_int n,\nlapack_int nrhs, float* a, lapack_int lda, float* af, lapack_int ldaf, lapack_int* ipiv,\nchar* equed, float* r, float* c, float* b, lapack_int ldb, float* x, lapack_int ldx,\nfloat* rcond, float* rpvgrw, float* berr, lapack_int n_err_bnds, float* err_bnds_norm,\nfloat* err_bnds_comp, lapack_int nparams, const float* params );\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n684\n\n\nlapack_int LAPACKE_dgesvxx( int matrix_layout, char fact, char trans, lapack_int n,\nlapack_int nrhs, double* a, lapack_int lda, double* af, lapack_int ldaf, lapack_int*\nipiv, char* equed, double* r, double* c, double* b, lapack_int ldb, double* x,\nlapack_int ldx, double* rcond, double* rpvgrw, double* berr, lapack_int n_err_bnds,\ndouble* err_bnds_norm, double* err_bnds_comp, lapack_int nparams, const double*\nparams );\nlapack_int LAPACKE_cgesvxx( int matrix_layout, char fact, char trans, lapack_int n,\nlapack_int nrhs, lapack_complex_float* a, lapack_int lda, lapack_complex_float* af,\nlapack_int ldaf, lapack_int* ipiv, char* equed, float* r, float* c,\nlapack_complex_float* b, lapack_int ldb, lapack_complex_float* x, lapack_int ldx,\nfloat* rcond, float* rpvgrw, float* berr, lapack_int n_err_bnds, float* err_bnds_norm,\nfloat* err_bnds_comp, lapack_int nparams, const float* params );\nlapack_int LAPACKE_zgesvxx( int matrix_layout, char fact, char trans, lapack_int n,\nlapack_int nrhs, lapack_complex_double* a, lapack_int lda, lapack_complex_double* af,\nlapack_int ldaf, lapack_int* ipiv, char* equed, double* r, double* c,\nlapack_complex_double* b, lapack_int ldb, lapack_complex_double* x, lapack_int ldx,\ndouble* rcond, double* rpvgrw, double* berr, lapack_int n_err_bnds, double*\nerr_bnds_norm, double* err_bnds_comp, lapack_int nparams, const double* params );\nInclude Files\n•\nmkl.h\nDescription\nThe routine uses the LU factorization to compute the solution to a real or complex system of linear equations\nA*X = B, where A is an n-by-n matrix, the columns of the matrix B are individual right-hand sides, and the\ncolumns of X are the corresponding solutions.\nBoth normwise and maximum componentwise error bounds are also provided on request. The routine returns\na solution with a small guaranteed error (O(eps), where eps is the working machine precision) unless the\nmatrix is very ill-conditioned, in which case a warning is returned. Relevant condition numbers are also\ncalculated and returned.\nThe routine accepts user-provided factorizations and equilibration factors; see definitions of the fact and\nequed options. Solving with refinement and using a factorization from a previous call of the routine also\nproduces a solution with O(eps) errors or warnings but that may not be true for general user-provided\nfactorizations and equilibration factors if they differ from what the routine would itself produce.\nThe routine ?gesvxx performs the following steps:\n1.\nIf fact = 'E', scaling factors r and c are computed to equilibrate the system:\ntrans = 'N': diag(r)*A*diag(c)*inv(diag(c))*X = diag(r)*B\ntrans = 'T': (diag(r)*A*diag(c))T*inv(diag(r))*X = diag(c)*B\ntrans = 'C': (diag(r)*A*diag(c))H*inv(diag(r))*X = diag(c)*B\nWhether or not the system will be equilibrated depends on the scaling of the matrix A, but if\nequilibration is used, A is overwritten by diag(r)*A*diag(c) and B by diag(r)*B (if trans='N') or\ndiag(c)*B (if trans = 'T' or 'C').\n2.\nIf fact = 'N' or 'E', the LU decomposition is used to factor the matrix A (after equilibration if fact\n= 'E') as A = P*L*U, where P is a permutation matrix, L is a unit lower triangular matrix, and U is\nupper triangular.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n685\n\n\n3.\nIf some Ui,i= 0, so that U is exactly singular, then the routine returns with info = i. Otherwise, the\nfactored form of A is used to estimate the condition number of the matrix A (see the rcond parameter).\nIf the reciprocal of the condition number is less than machine precision, the routine still goes on to\nsolve for X and compute error bounds.\n4.\nThe system of equations is solved for X using the factored form of A.\n5.\nBy default, unless is set to zero, the routine applies iterative refinement to improve the computed\nsolution matrix and calculate error bounds. Refinement calculates the residual to at least twice the\nworking precision.\n6.\nIf equilibration was used, the matrix X is premultiplied by diag(c) (if trans = 'N') or diag(r) (if\ntrans = 'T' or 'C') so that it solves the original system before equilibration.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\nfact\nMust be 'F', 'N', or 'E'.\nSpecifies whether or not the factored form of the matrix A is supplied\non entry, and if not, whether the matrix A should be equilibrated\nbefore it is factored.\nIf fact = 'F', on entry, af and ipiv contain the factored form of A.\nIf equed is not 'N', the matrix A has been equilibrated with scaling\nfactors given by r and c. Parameters a, af, and ipiv are not\nmodified.\nIf fact = 'N', the matrix A will be copied to af and factored.\nIf fact = 'E', the matrix A will be equilibrated, if necessary, copied\nto af and factored.\ntrans\nMust be 'N', 'T', or 'C'.\nSpecifies the form of the system of equations:\nIf trans = 'N', the system has the form A*X = B (No transpose).\nIf trans = 'T', the system has the form AT*X = B (Transpose).\nIf trans = 'C', the system has the form AH*X = B (Conjugate\nTranspose = Transpose for real flavors, Conjugate Transpose for\ncomplex flavors).\nn\nThe number of linear equations; the order of the matrix A; n≥ 0.\nnrhs\nThe number of right hand sides; the number of columns of the\nmatrices B and X; nrhs≥ 0.\na, af, b\nArrays: a(size max(lda*n)), af(size max(ldaf*n)), b(size max(1,\nldb*nrhs) for column major layout and max(1, ldb*n) for row major\nlayout).\nThe array a contains the matrix A. If fact = 'F' and equed is not\n'N', then A must have been equilibrated by the scaling factors in r\nand/or c. .\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n686\n\n\nThe array af is an input argument if fact = 'F'. It contains the\nfactored form of the matrix A, that is, the factors L and U from the\nfactorization A = P*L*U as computed by ?getrf. If equed is not 'N',\nthen af is the factored form of the equilibrated matrix A.\nThe array b contains the matrix B whose columns are the right-hand\nsides for the systems of equations.\nlda\nThe leading dimension of a; lda≥ max(1, n).\nldaf\nThe leading dimension of af; ldaf≥ max(1, n).\nipiv\nArray, size at least max(1, n). The array ipiv is an input argument if\nfact = 'F'. It contains the pivot indices from the factorization A =\nP*L*U as computed by ?getrf; row i of the matrix was interchanged\nwith row ipiv[i-1].\nequed\nMust be 'N', 'R', 'C', or 'B'.\nequed is an input argument if fact = 'F'. It specifies the form of\nequilibration that was done:\nIf equed = 'N', no equilibration was done (always true if fact =\n'N').\nIf equed = 'R', row equilibration was done, that is, A has been\npremultiplied by diag(r).\nIf equed = 'C', column equilibration was done, that is, A has been\npostmultiplied by diag(c).\nIf equed = 'B', both row and column equilibration was done, that is,\nA has been replaced by diag(r)*A*diag(c).\nr, c\nArrays: r (size n), c (size n). The array r contains the row scale\nfactors for A, and the array c contains the column scale factors for A.\nThese arrays are input arguments if fact = 'F' only; otherwise they\nare output arguments.\nIf equed = 'R' or 'B', A is multiplied on the left by diag(r); if equed\n= 'N' or 'C', r is not accessed.\nIf fact = 'F' and equed = 'R'or 'B', each element of r must be\npositive.\nIf equed = 'C' or 'B', A is multiplied on the right by diag(c); if\nequed = 'N' or 'R', c is not accessed.\nIf fact = 'F' and equed = 'C' or 'B', each element of c must be\npositive.\nEach element of r or c should be a power of the radix to ensure a\nreliable solution and error estimates. Scaling by powers of the radix\ndoes not cause rounding errors unless the result underflows or\noverflows. Rounding errors during scaling lead to refining with a\nmatrix that is not equivalent to the input matrix, producing error\nestimates that may not be reliable.\nldb\nThe leading dimension of the array b; ldb≥ max(1, n) for column\nmajor layout and ldb≥nrhs for row major layout.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n687\n\n\nldx\nThe leading dimension of the output array x; ldx≥ max(1, n) for\ncolumn major layout and ldx≥nrhs for row major layout.\nn_err_bnds\nNumber of error bounds to return for each right hand side and each\ntype (normwise or componentwise). See err_bnds_norm and\nerr_bnds_comp descriptions in Output Arguments section below.\nnparams\nSpecifies the number of parameters set in params. If ≤ 0, the params\narray is never referenced and default values are used.\nparams\nArray, size max(1, nparams). Specifies algorithm parameters. If an\nentry is less than 0.0, that entry is filled with the default value used\nfor that parameter. Only positions up to nparams are accessed;\ndefaults are used for higher-numbered parameters. If defaults are\nacceptable, you can pass nparams = 0, which prevents the source\ncode from accessing the params argument.\nparams[0] : Whether to perform iterative refinement or not. Default:\n1.0\n=0.0\nNo refinement is performed and no error\nbounds are computed.\n=1.0\nUse the double-precision refinement\nalgorithm, possibly with doubled-single\ncomputations if the compilation environment\ndoes not support double precision.\n(Other values are reserved for future use.)\nparams[1] : Maximum number of residual computations allowed for\nrefinement.\nDefault\n10.0\nAggressive\nSet to 100.0 to permit convergence using\napproximate factorizations or factorizations\nother than LU. If the factorization uses a\ntechnique other than Gaussian elimination,\nthe guarantees in err_bnds_norm and\nerr_bnds_comp may no longer be\ntrustworthy.\nparams[2] : Flag determining if the code will attempt to find a\nsolution with a small componentwise relative error in the double-\nprecision algorithm. Positive is true, 0.0 is false. Default: 1.0 (attempt\ncomponentwise convergence).\nOutput Parameters\nx\nArray, size max(1, ldx*nrhs) for column major layout and max(1, ldx*n)\nfor row major layout.\nIf info = 0, the array x contains the solution n-by-nrhs matrix X to the\noriginal system of equations. Note that A and B are modified on exit if\nequed≠'N', and the solution to the equilibrated system is:\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n688\n\n\ninv(diag(c))*X, if trans = 'N' and equed = 'C' or 'B'; or\ninv(diag(r))*X, if trans = 'T' or 'C' and equed = 'R' or 'B'.\na\nArray a is not modified on exit if fact = 'F' or 'N', or if fact = 'E' and\nequed = 'N'.\nIf equed≠'N', A is scaled on exit as follows:\nequed = 'R': A = diag(r)*A\nequed = 'C': A = A*diag(c)\nequed = 'B': A = diag(r)*A*diag(c).\naf\nIf fact = 'N' or 'E', then af is an output argument and on exit returns\nthe factors L and U from the factorization A = PLU of the original matrix A\n(if fact = 'N') or of the equilibrated matrix A (if fact = 'E'). See the\ndescription of a for the form of the equilibrated matrix.\nb\nOverwritten by diag(r)*B if trans = 'N' and equed = 'R' or 'B';\noverwritten by trans = 'T' or 'C' and equed = 'C' or 'B';\nnot changed if equed = 'N'.\nr, c\nThese arrays are output arguments if fact≠'F'. Each element of these\narrays is a power of the radix. See the description of r, c in Input\nArguments section.\nrcond\nReciprocal scaled condition number. An estimate of the reciprocal Skeel\ncondition number of the matrix A after equilibration (if done). If rcond is\nless than the machine precision, in particular, if rcond = 0, the matrix is\nsingular to working precision. Note that the error may still be small even if\nthis number is very small and the matrix appears ill-conditioned.\nrpvgrw\nContains the reciprocal pivot growth factor:\nIf this is much less than 1, the stability of the LU factorization of the\n(equlibrated) matrix A could be poor. This also means that the solution X,\nestimated condition numbers, and error bounds could be unreliable. If\nfactorization fails with 0 < info≤n, this parameter contains the reciprocal\npivot growth factor for the leading info columns of A. In ?gesvx, this\nquantity is returned in rpivot.\nberr\nArray, size at least max(1, nrhs). Contains the componentwise relative\nbackward error for each solution vector xj, that is, the smallest relative\nchange in any element of A or B that makes xj an exact solution.\nerr_bnds_norm\nArray of size nrhs*n_err_bnds. For each right-hand side, contains\ninformation about various error bounds and condition numbers\ncorresponding to the normwise relative error, which is defined as follows:\nNormwise relative error in the i-th solution vector\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n689\n\n\nThe array is indexed by the type of error information as described below.\nThere are currently up to three pieces of information returned.\nerr=1\n\"Trust/don't trust\" boolean. Trust the answer if\nthe reciprocal condition number is less than the\nthreshold sqrt(n)*slamch(ε) for single\nprecision flavors and sqrt(n)*dlamch(ε) for\ndouble precision flavors.\nerr=2\n\"Guaranteed\" error bound. The estimated\nforward error, almost certainly within a factor of\n10 of the true error so long as the next entry is\ngreater than the threshold sqrt(n)*slamch(ε)\nfor single precision flavors and\nsqrt(n)*dlamch(ε) for double precision\nflavors. This error bound should only be trusted\nif the previous boolean is true.\nerr=3\nReciprocal condition number. Estimated\nnormwise reciprocal condition number.\nCompared with the threshold\nsqrt(n)*slamch(ε) for single precision flavors\nand sqrt(n)*dlamch(ε) for double precision\nflavors to determine if the error estimate is\n\"guaranteed\". These reciprocal condition\nnumbers for some appropriately scaled matrix Z\nare:\nLet z=s*a, where s scales each row by a power\nof the radix so all absolute row sums of z are\napproximately 1.\nThe information for right-hand side i, where 1 ≤i≤nrhs, and type of error\nerr is stored in err_bnds_norm[(err-1)*nrhs + i - 1].\nerr_bnds_comp\nArray of size nrhs*n_err_bnds. For each right-hand side, contains\ninformation about various error bounds and condition numbers\ncorresponding to the componentwise relative error, which is defined as\nfollows:\nComponentwise relative error in the i-th solution vector:\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n690\n\n\nThe array is indexed by the type of error information as described below.\nThere are currently up to three pieces of information returned for each\nright-hand side. If componentwise accuracy is not requested (params[2] =\n0.0), then err_bnds_comp is not accessed.\nerr=1\n\"Trust/don't trust\" boolean. Trust the answer if\nthe reciprocal condition number is less than the\nthreshold sqrt(n)*slamch(ε) for single\nprecision flavors and sqrt(n)*dlamch(ε) for\ndouble precision flavors.\nerr=2\n\"Guaranteed\" error bpound. The estimated\nforward error, almost certainly within a factor of\n10 of the true error so long as the next entry is\ngreater than the threshold sqrt(n)*slamch(ε)\nfor single precision flavors and\nsqrt(n)*dlamch(ε) for double precision\nflavors. This error bound should only be trusted\nif the previous boolean is true.\nerr=3\nReciprocal condition number. Estimated\ncomponentwise reciprocal condition number.\nCompared with the threshold\nsqrt(n)*slamch(ε) for single precision flavors\nand sqrt(n)*dlamch(ε) for double precision\nflavors to determine if the error estimate is\n\"guaranteed\". These reciprocal condition\nnumbers for some appropriately scaled matrix Z\nare:\nLet z=s*(a*diag(x)), where x is the solution\nfor the current right-hand side and s scales each\nrow of a*diag(x) by a power of the radix so all\nabsolute row sums of z are approximately 1.\nThe information for right-hand side i, where 1 ≤i≤nrhs, and type of error\nerr is stored in err_bnds_comp[(err-1)*nrhs + i - 1].\nipiv\nIf fact = 'N' or 'E', then ipiv is an output argument and on exit\ncontains the pivot indices from the factorization A = P*L*U of the original\nmatrix A (if fact = 'N') or of the equilibrated matrix A (if fact = 'E').\nequed\nIf fact≠'F', then equed is an output argument. It specifies the form of\nequilibration that was done (see the description of equed in Input\nArguments section).\nparams\nIf an entry is less than 0.0, that entry is filled with the default value used\nfor that parameter, otherwise the entry is not modified\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful. The solution to every right-hand side is guaranteed.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n691\n\n\nIf info = -i, parameter i had an illegal value.\nIf 0 < info≤n: Uinfo,info is exactly zero. The factorization has been completed, but the factor U is exactly\nsingular, so the solution and error bounds could not be computed; rcond = 0 is returned.\nIf info = n+j: The solution corresponding to the j-th right-hand side is not guaranteed. The solutions\ncorresponding to other right-hand sides k with k > j may not be guaranteed as well, but only the first such\nright-hand side is reported. If a small componentwise error is not requested params[2] = 0.0, then the j-th\nright-hand side is the first with a normwise error bound that is not guaranteed (the smallest j such that for\ncolumn major layout err_bnds_norm[j - 1] = 0.0 or err_bnds_comp[j - 1] = 0.0; or for row major\nlayout err_bnds_norm[(j - 1)*n_err_bnds] = 0.0 or err_bnds_comp[(j - 1)*n_err_bnds] = 0.0).\nSee the definition of err_bnds_norm and err_bnds_comp for err = 1. To get information about all of the\nright-hand sides, check err_bnds_norm or err_bnds_comp.\nSee Also\nMatrix Storage Schemes\n?gbsv\nComputes the solution to the system of linear\nequations with a band coefficient matrix A and\nmultiple right-hand sides.\nSyntax\nlapack_int LAPACKE_sgbsv (int matrix_layout , lapack_int n , lapack_int kl , lapack_int\nku , lapack_int nrhs , float * ab , lapack_int ldab , lapack_int * ipiv , float * b ,\nlapack_int ldb );\nlapack_int LAPACKE_dgbsv (int matrix_layout , lapack_int n , lapack_int kl , lapack_int\nku , lapack_int nrhs , double * ab , lapack_int ldab , lapack_int * ipiv , double * b ,\nlapack_int ldb );\nlapack_int LAPACKE_cgbsv (int matrix_layout , lapack_int n , lapack_int kl , lapack_int\nku , lapack_int nrhs , lapack_complex_float * ab , lapack_int ldab , lapack_int *\nipiv , lapack_complex_float * b , lapack_int ldb );\nlapack_int LAPACKE_zgbsv (int matrix_layout , lapack_int n , lapack_int kl , lapack_int\nku , lapack_int nrhs , lapack_complex_double * ab , lapack_int ldab , lapack_int *\nipiv , lapack_complex_double * b , lapack_int ldb );\nInclude Files\n•\nmkl.h\nDescription\nThe routine solves for X the real or complex system of linear equations A*X = B, where A is an n-by-n band\nmatrix with kl subdiagonals and ku superdiagonals, the columns of matrix B are individual right-hand sides,\nand the columns of X are the corresponding solutions.\nThe LU decomposition with partial pivoting and row interchanges is used to factor A as A = L*U, where L is a\nproduct of permutation and unit lower triangular matrices with kl subdiagonals, and U is upper triangular\nwith kl+ku superdiagonals. The factored form of A is then used to solve the system of equations A*X = B.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n692\n\n\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\nn\nThe order of A. The number of rows in B; n≥ 0.\nkl\nThe number of subdiagonals within the band of A; kl≥ 0.\nku\nThe number of superdiagonals within the band of A; ku≥ 0.\nnrhs\nThe number of right-hand sides. The number of columns in B; nrhs≥\n0.\nab, b\nArrays: ab(size max(1, ldab*n)), bof size max(1, ldb*nrhs) for\ncolumn major layout and max(1, ldb*n) for row major layout.\nThe array ab contains the matrix A in band storage (see Matrix\nStorage Schemes).\nThe array b contains the matrix B whose columns are the right-hand\nsides for the systems of equations.\nldab\nThe leading dimension of the array ab. (ldab≥ 2kl + ku +1)\nldb\nThe leading dimension of b; ldb≥ max(1, n) for column major layout\nand ldb≥nrhs for row major layout.\nOutput Parameters\nab\nOverwritten by L and U. U is stored as an upper triangular band\nmatrix with kl + ku superdiagonals and L is stored as a lower\ntriangular band matrix with kl subdiagonals. See Matrix Storage\nSchemes.\nb\nOverwritten by the solution matrix X.\nipiv\nArray, size at least max(1, n). The pivot indices: row i was\ninterchanged with row ipiv[i-1].\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info = -i, parameter i had an illegal value.\nIf info = i, Ui, i is exactly zero. The factorization has been completed, but the factor U is exactly singular,\nso the solution could not be computed.\nSee Also\nMatrix Storage Schemes\n?gbsvx\nComputes the solution to the real or complex system\nof linear equations with a band coefficient matrix A\nand multiple right-hand sides, and provides error\nbounds on the solution.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n693\n\n\nSyntax\nlapack_int LAPACKE_sgbsvx( int matrix_layout, char fact, char trans, lapack_int n,\nlapack_int kl, lapack_int ku, lapack_int nrhs, float* ab, lapack_int ldab, float* afb,\nlapack_int ldafb, lapack_int* ipiv, char* equed, float* r, float* c, float* b,\nlapack_int ldb, float* x, lapack_int ldx, float* rcond, float* ferr, float* berr, float*\nrpivot );\nlapack_int LAPACKE_dgbsvx( int matrix_layout, char fact, char trans, lapack_int n,\nlapack_int kl, lapack_int ku, lapack_int nrhs, double* ab, lapack_int ldab, double* afb,\nlapack_int ldafb, lapack_int* ipiv, char* equed, double* r, double* c, double* b,\nlapack_int ldb, double* x, lapack_int ldx, double* rcond, double* ferr, double* berr,\ndouble* rpivot );\nlapack_int LAPACKE_cgbsvx( int matrix_layout, char fact, char trans, lapack_int n,\nlapack_int kl, lapack_int ku, lapack_int nrhs, lapack_complex_float* ab, lapack_int\nldab, lapack_complex_float* afb, lapack_int ldafb, lapack_int* ipiv, char* equed,\nfloat* r, float* c, lapack_complex_float* b, lapack_int ldb, lapack_complex_float* x,\nlapack_int ldx, float* rcond, float* ferr, float* berr, float* rpivot );\nlapack_int LAPACKE_zgbsvx( int matrix_layout, char fact, char trans, lapack_int n,\nlapack_int kl, lapack_int ku, lapack_int nrhs, lapack_complex_double* ab, lapack_int\nldab, lapack_complex_double* afb, lapack_int ldafb, lapack_int* ipiv, char* equed,\ndouble* r, double* c, lapack_complex_double* b, lapack_int ldb, lapack_complex_double*\nx, lapack_int ldx, double* rcond, double* ferr, double* berr, double* rpivot );\nInclude Files\n•\nmkl.h\nDescription\nThe routine uses the LU factorization to compute the solution to a real or complex system of linear equations\nA*X = B, AT*X = B, or AH*X = B, where A is a band matrix of order n with kl subdiagonals and ku\nsuperdiagonals, the columns of matrix B are individual right-hand sides, and the columns of X are the\ncorresponding solutions.\nError bounds on the solution and a condition estimate are also provided.\nThe routine ?gbsvx performs the following steps:\n1.\nIf fact = 'E', real scaling factors r and c are computed to equilibrate the system:\ntrans = 'N': diag(r)*A*diag(c) *inv(diag(c))*X = diag(r)*B\ntrans = 'T': (diag(r)*A*diag(c))T *inv(diag(r))*X = diag(c)*B\ntrans = 'C': (diag(r)*A*diag(c))H *inv(diag(r))*X = diag(c)*B\nWhether the system will be equilibrated depends on the scaling of the matrix A, but if equilibration is\nused, A is overwritten by diag(r)*A*diag(c) and B by diag(r)*B (if trans='N') or diag(c)*B (if\ntrans = 'T'or 'C').\n2.\nIf fact = 'N'or 'E', the LU decomposition is used to factor the matrix A (after equilibration if fact =\n'E') as A = L*U, where L is a product of permutation and unit lower triangular matrices with kl\nsubdiagonals, and U is upper triangular with kl+ku superdiagonals.\n3.\nIf some Ui,i = 0, so that U is exactly singular, then the routine returns with info = i. Otherwise, the\nfactored form of A is used to estimate the condition number of the matrix A. If the reciprocal of the\ncondition number is less than machine precision, info = n + 1 is returned as a warning, but the\nroutine still goes on to solve for X and compute error bounds as described below.\n4.\nThe system of equations is solved for X using the factored form of A.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n694\n\n\n5.\nIterative refinement is applied to improve the computed solution matrix and calculate error bounds and\nbackward error estimates for it.\n6.\nIf equilibration was used, the matrix X is premultiplied by diag(c) (if trans = 'N') or diag(r) (if\ntrans = 'T' or 'C') so that it solves the original system before equilibration.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\nfact\nMust be 'F', 'N', or 'E'.\nSpecifies whether the factored form of the matrix A is supplied on\nentry, and if not, whether the matrix A should be equilibrated before it\nis factored.\nIf fact = 'F': on entry, afb and ipiv contain the factored form of A.\nIf equed is not 'N', the matrix A is equilibrated with scaling factors\ngiven by r and c.\nab, afb, and ipiv are not modified.\nIf fact = 'N', the matrix A will be copied to afb and factored.\nIf fact = 'E', the matrix A will be equilibrated if necessary, then\ncopied to afb and factored.\ntrans\nMust be 'N', 'T', or 'C'.\nSpecifies the form of the system of equations:\nIf trans = 'N', the system has the form A*X = B (No transpose).\nIf trans = 'T', the system has the form AT*X = B (Transpose).\nIf trans = 'C', the system has the form AH*X = B (Transpose for\nreal flavors, conjugate transpose for complex flavors).\nn\nThe number of linear equations, the order of the matrix A; n≥ 0.\nkl\nThe number of subdiagonals within the band of A; kl≥ 0.\nku\nThe number of superdiagonals within the band of A; ku≥ 0.\nnrhs\nThe number of right hand sides, the number of columns of the\nmatrices B and X; nrhs≥ 0.\nab, afb, b\nArrays: ab (max(ldab*n)), afb (max(ldafb*n)), b(max(1,\nldb*nrhs) for column major layout and max(1, ldb*n) for row major\nlayout).\nThe array ab contains the matrix A in band storage (see Matrix\nStorage Schemes). If fact = 'F' and equed is not 'N', then A must\nhave been equilibrated by the scaling factors in r and/or c.\nThe array afb is an input argument if fact = 'F'. It contains the\nfactored form of the matrix A, that is, the factors L and U from the\nfactorization A = P*L*U as computed by ?gbtrf. U is stored as an\nupper triangular band matrix with kl + ku superdiagonals.L is stored\nas lower triangular band matrix with kl subdiagonals. If equed is not\n'N', then afb is the factored form of the equilibrated matrix A.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n695\n\n\nThe array b contains the matrix B whose columns are the right-hand\nsides for the systems of equations.\nldab\nThe leading dimension of ab; ldab≥kl+ku+1.\nldafb\nThe leading dimension of afb; ldafb≥ 2*kl+ku+1.\nldb\nThe leading dimension of b; ldb≥ max(1, n) for column major layout\nand ldb≥nrhs for row major layout.\nipiv\nArray, size at least max(1, n). The array ipiv is an input argument if\nfact = 'F'. It contains the pivot indices from the factorization A =\nP*L*U as computed by ?gbtrf; row i of the matrix was interchanged\nwith row ipiv[i-1].\nequed\nMust be 'N', 'R', 'C', or 'B'.\nequed is an input argument if fact = 'F'. It specifies the form of\nequilibration that was done:\nIf equed = 'N', no equilibration was done (always true if fact =\n'N').\nIf equed = 'R', row equilibration was done, that is, A has been\npremultiplied by diag(r).\nIf equed = 'C', column equilibration was done, that is, A has been\npostmultiplied by diag(c).\nif equed = 'B', both row and column equilibration was done, that is,\nA has been replaced by diag(r)*A*diag(c).\nr, c\nArrays: r (size n), c (size n).\nThe array r contains the row scale factors for A, and the array c\ncontains the column scale factors for A. These arrays are input\narguments if fact = 'F' only; otherwise they are output arguments.\nIf equed = 'R'or 'B', A is multiplied on the left by diag(r); if\nequed = 'N' or 'C', r is not accessed.\nIf fact = 'F' and equed = 'R' or 'B', each element of r must be\npositive.\nIf equed = 'C'or 'B', A is multiplied on the right by diag(c); if\nequed = 'N'or 'R', c is not accessed.\nIf fact = 'F' and equed = 'C'or 'B', each element of c must be\npositive.\nldx\nThe leading dimension of the output array x; ldx≥ max(1, n) for\ncolumn major layout and ldx≥nrhs for row major layout.\nOutput Parameters\nx\nArray, size max(1, ldx*nrhs) for column major layout and max(1,\nldx*n) for row major layout.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n696\n\n\nIf info = 0 or info = n+1, the array x contains the solution matrix\nX to the original system of equations. Note that A and B are modified\non exit if equed≠'N', and the solution to the equilibrated system is:\ninv(diag(c))*X, if trans = 'N' and equed = 'C'or 'B';\ninv(diag(r))*X, if trans = 'T' or 'C' and equed = 'R' or 'B'.\nab\nArray ab is not modified on exit if fact = 'F' or 'N', or if fact =\n'E' and equed = 'N'.\nIf equed≠'N', A is scaled on exit as follows:\nequed = 'R': A = diag(r)*A\nequed = 'C': A = A*diag(c)\nequed = 'B': A = diag(r)*A*diag(c).\nafb\nIf fact = 'N' or 'E', then afb is an output argument and on exit\nreturns details of the LU factorization of the original matrix A (if fact\n= 'N') or of the equilibrated matrix A (if fact = 'E'). See the\ndescription of ab for the form of the equilibrated matrix.\nb\nOverwritten by diag(r)*b if trans = 'N' and equed = 'R' or 'B';\noverwritten by diag(c)*b if trans = 'T' or 'C' and equed = 'C'\nor 'B';\nnot changed if equed = 'N'.\nr, c\nThese arrays are output arguments if fact≠'F'. See the description\nof r, c in Input Arguments section.\nrcond\nAn estimate of the reciprocal condition number of the matrix A after\nequilibration (if done).\nIf rcond is less than the machine precision (in particular, if rcond =0),\nthe matrix is singular to working precision. This condition is indicated\nby a return code of info>0.\nferr\nArray, size at least max(1, nrhs). Contains the estimated forward\nerror bound for each solution vector xj (the j-th column of the solution\nmatrix X). If xtrue is the true solution corresponding to xj, ferr[j-1]\nis an estimated upper bound for the magnitude of the largest element\nin (xj - xtrue) divided by the magnitude of the largest element in xj.\nThe estimate is as reliable as the estimate for rcond, and is almost\nalways a slight overestimate of the true error.\nberr\nArray, size at least max(1, nrhs). Contains the component-wise\nrelative backward error for each solution vector xj, that is, the\nsmallest relative change in any element of A or B that makes xj an\nexact solution.\nipiv\nIf fact = 'N' or 'E', then ipiv is an output argument and on exit\ncontains the pivot indices from the factorization A = L*U of the\noriginal matrix A (if fact = 'N') or of the equilibrated matrix A (if\nfact = 'E').\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n697\n\n\nequed\nIf fact≠'F', then equed is an output argument. It specifies the form\nof equilibration that was done (see the description of equed in Input\nArguments section).\nrpivot\nOn exit, rpivot contains the reciprocal pivot growth factor:\nIf rpivot is much less than 1, then the stability of the LU\nfactorization of the (equilibrated) matrix A could be poor. This also\nmeans that the solution x, condition estimator rcond, and forward\nerror bound ferr could be unreliable. If factorization fails with 0 <\ninfo≤n, then rpivot contains the reciprocal pivot growth factor for\nthe leading info columns of A.\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info = -i, parameter i had an illegal value.\nIf info = i, and i≤n, then Ui, i is exactly zero. The factorization has been completed, but the factor U is\nexactly singular, so the solution and error bounds could not be computed; rcond = 0 is returned. If info =\ni, and i = n+1, then U is nonsingular, but rcond is less than machine precision, meaning that the matrix is\nsingular to working precision. Nevertheless, the solution and error bounds are computed because there are a\nnumber of situations where the computed solution can be more accurate than the value of rcond would\nsuggest.\nSee Also\nMatrix Storage Schemes\n?gbsvxx\nUses extra precise iterative refinement to compute the\nsolution to the system of linear equations with a\nbanded coefficient matrix A and multiple right-hand\nsides\nSyntax\nlapack_int LAPACKE_sgbsvxx( int matrix_layout, char fact, char trans, lapack_int n,\nlapack_int kl, lapack_int ku, lapack_int nrhs, float* ab, lapack_int ldab, float* afb,\nlapack_int ldafb, lapack_int* ipiv, char* equed, float* r, float* c, float* b,\nlapack_int ldb, float* x, lapack_int ldx, float* rcond, float* rpvgrw, float* berr,\nlapack_int n_err_bnds, float* err_bnds_norm, float* err_bnds_comp, lapack_int nparams,\nconst float* params );\nlapack_int LAPACKE_dgbsvxx( int matrix_layout, char fact, char trans, lapack_int n,\nlapack_int kl, lapack_int ku, lapack_int nrhs, double* ab, lapack_int ldab, double* afb,\nlapack_int ldafb, lapack_int* ipiv, char* equed, double* r, double* c, double* b,\nlapack_int ldb, double* x, lapack_int ldx, double* rcond, double* rpvgrw, double* berr,\nlapack_int n_err_bnds, double* err_bnds_norm, double* err_bnds_comp, lapack_int\nnparams, const double* params );\nlapack_int LAPACKE_cgbsvxx( int matrix_layout, char fact, char trans, lapack_int n,\nlapack_int kl, lapack_int ku, lapack_int nrhs, lapack_complex_float* ab, lapack_int\nldab, lapack_complex_float* afb, lapack_int ldafb, lapack_int* ipiv, char* equed,\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n698\n\n\nfloat* r, float* c, lapack_complex_float* b, lapack_int ldb, lapack_complex_float* x,\nlapack_int ldx, float* rcond, float* rpvgrw, float* berr, lapack_int n_err_bnds, float*\nerr_bnds_norm, float* err_bnds_comp, lapack_int nparams, const float* params );\nlapack_int LAPACKE_zgbsvxx( int matrix_layout, char fact, char trans, lapack_int n,\nlapack_int kl, lapack_int ku, lapack_int nrhs, lapack_complex_double* ab, lapack_int\nldab, lapack_complex_double* afb, lapack_int ldafb, lapack_int* ipiv, char* equed,\ndouble* r, double* c, lapack_complex_double* b, lapack_int ldb, lapack_complex_double*\nx, lapack_int ldx, double* rcond, double* rpvgrw, double* berr, lapack_int n_err_bnds,\ndouble* err_bnds_norm, double* err_bnds_comp, lapack_int nparams, const double*\nparams );\nInclude Files\n•\nmkl.h\nDescription\nThe routine uses the LU factorization to compute the solution to a real or complex system of linear equations\nA*X = B, AT*X = B, or AH*X = B, where A is an n-by-n banded matrix, the columns of the matrix B are\nindividual right-hand sides, and the columns of X are the corresponding solutions.\nBoth normwise and maximum componentwise error bounds are also provided on request. The routine returns\na solution with a small guaranteed error (O(eps), where eps is the working machine precision) unless the\nmatrix is very ill-conditioned, in which case a warning is returned. Relevant condition numbers are also\ncalculated and returned.\nThe routine accepts user-provided factorizations and equilibration factors; see definitions of the fact and\nequed options. Solving with refinement and using a factorization from a previous call of the routine also\nproduces a solution with O(eps) errors or warnings but that may not be true for general user-provided\nfactorizations and equilibration factors if they differ from what the routine would itself produce.\nThe routine ?gbsvxx performs the following steps:\n1.\nIf fact = 'E', scaling factors r and c are computed to equilibrate the system:\ntrans = 'N': diag(r)*A*diag(c)*inv(diag(c))*X = diag(r)*B\ntrans = 'T': (diag(r)*A*diag(c))T*inv(diag(r))*X = diag(c)*B\ntrans = 'C': (diag(r)*A*diag(c))H*inv(diag(r))*X = diag(c)*B\nWhether or not the system will be equilibrated depends on the scaling of the matrix A, but if\nequilibration is used, A is overwritten by diag(r)*A*diag(c) and B by diag(r)*B (if trans='N') or\ndiag(c)*B (if trans = 'T' or 'C').\n2.\nIf fact = 'N' or 'E', the LU decomposition is used to factor the matrix A (after equilibration if fact\n= 'E') as A = P*L*U, where P is a permutation matrix, L is a unit lower triangular matrix, and U is\nupper triangular.\n3.\nIf some Ui,i= 0, so that U is exactly singular, then the routine returns with info = i. Otherwise, the\nfactored form of A is used to estimate the condition number of the matrix A (see the rcond parameter).\nIf the reciprocal of the condition number is less than machine precision, the routine still goes on to\nsolve for X and compute error bounds.\n4.\nThe system of equations is solved for X using the factored form of A.\n5.\nBy default, unless params[0] is set to zero, the routine applies iterative refinement to improve the\ncomputed solution matrix and calculate error bounds. Refinement calculates the residual to at least\ntwice the working precision.\n6.\nIf equilibration was used, the matrix X is premultiplied by diag(c) (if trans = 'N') or diag(r) (if\ntrans = 'T' or 'C') so that it solves the original system before equilibration.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n699\n\n\nInput Parameters\nmatrix_layout\nSpecifies whether two-dimensional array storage is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\nfact\nMust be 'F', 'N', or 'E'.\nSpecifies whether or not the factored form of the matrix A is supplied\non entry, and if not, whether the matrix A should be equilibrated\nbefore it is factored.\nIf fact = 'F', on entry, afb and ipiv contain the factored form of\nA. If equed is not 'N', the matrix A has been equilibrated with scaling\nfactors given by r and c. Parameters ab, afb, and ipiv are not\nmodified.\nIf fact = 'N', the matrix A will be copied to afb and factored.\nIf fact = 'E', the matrix A will be equilibrated, if necessary, copied\nto afb and factored.\ntrans\nMust be 'N', 'T', or 'C'.\nSpecifies the form of the system of equations:\nIf trans = 'N', the system has the form A*X = B (No transpose).\nIf trans = 'T', the system has the form AT*X = B (Transpose).\nIf trans = 'C', the system has the form AH*X = B (Conjugate\nTranspose = Transpose for real flavors, Conjugate Transpose for\ncomplex flavors).\nn\nThe number of linear equations; the order of the matrix A; n≥ 0.\nkl\nThe number of subdiagonals within the band of A; kl≥ 0.\nku\nThe number of superdiagonals within the band of A; ku≥ 0.\nnrhs\nThe number of right-hand sides; the number of columns of the\nmatrices B and X; nrhs≥ 0.\nab, afb, b\nArrays: ab (max(ldab*n)), afb (max(ldafb*n)), b(max(1,\nldb*nrhs) for column major layout and max(1, ldb*n) for row major\nlayout).\nThe array ab contains the matrix A in band storage.\nIf fact = 'F' and equed is not 'N', then AB must have been\nequilibrated by the scaling factors in r and/or c.\nThe array afb is an input argument if fact = 'F'. It contains the\nfactored form of the banded matrix A, that is, the factors L and U from\nthe factorization A = P*L*U as computed by ?gbtrf. U is stored as\nan upper triangular banded matrix with kl + ku superdiagonals. L is\nstored as lower triangular band matrix with kl subdiagonals. If equed\nis not 'N', then afb is the factored form of the equilibrated matrix A.\nThe array b contains the matrix B whose columns are the right-hand\nsides for the systems of equations.\nldab\nThe leading dimension of the array ab; ldab≥kl+ku+1.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n700\n\n\nldafb\nThe leading dimension of the array afb; ldafb≥ 2*kl+ku+1.\nipiv\nArray, size at least max(1, n). The array ipiv is an input argument if\nfact = 'F'. It contains the pivot indices from the factorization A =\nP*L*U as computed by ?gbtrf; row i of the matrix was interchanged\nwith row ipiv[i-1].\nequed\nMust be 'N', 'R', 'C', or 'B'.\nequed is an input argument if fact = 'F'. It specifies the form of\nequilibration that was done:\nIf equed = 'N', no equilibration was done (always true if fact =\n'N').\nIf equed = 'R', row equilibration was done, that is, A has been\npremultiplied by diag(r).\nIf equed = 'C', column equilibration was done, that is, A has been\npostmultiplied by diag(c).\nIf equed = 'B', both row and column equilibration was done, that is,\nA has been replaced by diag(r)*A*diag(c).\nr, c\nArrays: r (size n), c (size n). The array r contains the row scale factors\nfor A, and the array c contains the column scale factors for A. These\narrays are input arguments if fact = 'F' only; otherwise they are\noutput arguments.\nIf equed = 'R' or 'B', A is multiplied on the left by diag(r); if equed\n= 'N'or 'C', r is not accessed.\nIf fact = 'F' and equed = 'R' or 'B', each element of r must be\npositive.\nIf equed = 'C' or 'B', A is multiplied on the right by diag(c); if\nequed = 'N' or 'R', c is not accessed.\nIf fact = 'F' and equed = 'C' or 'B', each element of c must be\npositive.\nEach element of r or c should be a power of the radix to ensure a\nreliable solution and error estimates. Scaling by powers of the radix\ndoes not cause rounding errors unless the result underflows or\noverflows. Rounding errors during scaling lead to refining with a\nmatrix that is not equivalent to the input matrix, producing error\nestimates that may not be reliable.\nldb\nThe leading dimension of the array b; ldb≥ max(1, n) for column\nmajor layout and ldb≥nrhs for row major layout.\nldx\nThe leading dimension of the output array x; ldx≥ max(1, n) for\ncolumn major layout and ldx≥nrhs for row major layout.\nn_err_bnds\nNumber of error bounds to return for each right hand side and each\ntype (normwise or componentwise). See err_bnds_norm and\nerr_bnds_comp descriptions in Output Arguments section below.\nnparams\nSpecifies the number of parameters set in params. If ≤ 0, the params\narray is never referenced and default values are used.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n701\n\n\nparams\nArray, size max(1, nparams). Specifies algorithm parameters. If an\nentry is less than 0.0, that entry is filled with the default value used\nfor that parameter. Only positions up to nparams are accessed;\ndefaults are used for higher-numbered parameters. If defaults are\nacceptable, you can pass nparams = 0, which prevents the source\ncode from accessing the params argument.\nparams[0] : Whether to perform iterative refinement or not. Default:\n1.0 (for single precision flavors), 1.0D+0 (for double precision\nflavors).\n=0.0\nNo refinement is performed and no error\nbounds are computed.\n=1.0\nUse the extra-precise refinement algorithm.\n(Other values are reserved for future use.)\nparams[1] : Maximum number of residual computations allowed for\nrefinement.\nDefault\n10.0\nAggressive\nSet to 100.0 to permit convergence using\napproximate factorizations or factorizations\nother than LU. If the factorization uses a\ntechnique other than Gaussian elimination,\nthe guarantees in err_bnds_norm and\nerr_bnds_comp may no longer be\ntrustworthy.\nparams[2] : Flag determining if the code will attempt to find a\nsolution with a small componentwise relative error in the double-\nprecision algorithm. Positive is true, 0.0 is false. Default: 1.0 (attempt\ncomponentwise convergence).\nOutput Parameters\nx\nArray, size max(1, ldx*nrhs) for column major layout and max(1, ldx*n)\nfor row major layout.\nIf info = 0, the array x contains the solution n-by-nrhs matrix X to the\noriginal system of equations. Note that A and B are modified on exit if\nequed≠'N', and the solution to the equilibrated system is:\ninv(diag(c))*X, if trans = 'N' and equed = 'C' or 'B'; or\ninv(diag(r))*X, if trans = 'T' or 'C' and equed = 'R' or 'B'.\nab\nArray ab is not modified on exit if fact = 'F' or 'N', or if fact = 'E'\nand equed = 'N'.\nIf equed≠'N', A is scaled on exit as follows:\nequed = 'R': A = diag(r)*A\nequed = 'C': A = A*diag(c)\nequed = 'B': A = diag(r)*A*diag(c).\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n702\n\n\nafb\nIf fact = 'N' or 'E', then afb is an output argument and on exit returns\nthe factors L and U from the factorization A = PLU of the original matrix A\n(if fact = 'N') or of the equilibrated matrix A (if fact = 'E').\nb\nOverwritten by diag(r)*B if trans = 'N' and equed = 'R' or 'B';\noverwritten by trans = 'T' or 'C' and equed = 'C' or 'B';\nnot changed if equed = 'N'.\nr, c\nThese arrays are output arguments if fact≠'F'. Each element of these\narrays is a power of the radix. See the description of r, c in Input\nArguments section.\nrcond\nReciprocal scaled condition number. An estimate of the reciprocal Skeel\ncondition number of the matrix A after equilibration (if done). If rcond is\nless than the machine precision, in particular, if rcond = 0, the matrix is\nsingular to working precision. Note that the error may still be small even if\nthis number is very small and the matrix appears ill-conditioned.\nrpvgrw\nContains the reciprocal pivot growth factor:\nIf this is much less than 1, the stability of the LU factorization of the\n(equlibrated) matrix A could be poor. This also means that the solution X,\nestimated condition numbers, and error bounds could be unreliable. If\nfactorization fails with 0 < info≤n, this parameter contains the reciprocal\npivot growth factor for the leading info columns of A. In ?gbsvx, this\nquantity is returned in rpivot.\nberr\nArray, size at least max(1, nrhs). Contains the componentwise relative\nbackward error for each solution vector xj, that is, the smallest relative\nchange in any element of A or B that makes xj an exact solution.\nerr_bnds_norm\nArray of size nrhs*n_err_bnds. For each right-hand side, contains\ninformation about various error bounds and condition numbers\ncorresponding to the normwise relative error, which is defined as follows:\nNormwise relative error in the i-th solution vector\nThe array is indexed by the type of error information as described below.\nThere are currently up to three pieces of information returned.\nerr=1\n\"Trust/don't trust\" boolean. Trust the answer if\nthe reciprocal condition number is less than the\nthreshold sqrt(n)*slamch(ε) for single\nprecision flavors and sqrt(n)*dlamch(ε) for\ndouble precision flavors.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n703\n\n\nerr=2\n\"Guaranteed\" error bound. The estimated\nforward error, almost certainly within a factor of\n10 of the true error so long as the next entry is\ngreater than the threshold sqrt(n)*slamch(ε)\nfor single precision flavors and\nsqrt(n)*dlamch(ε) for double precision\nflavors. This error bound should only be trusted\nif the previous boolean is true.\nerr=3\nReciprocal condition number. Estimated\nnormwise reciprocal condition number.\nCompared with the threshold\nsqrt(n)*slamch(ε) for single precision flavors\nand sqrt(n)*dlamch(ε) for double precision\nflavors to determine if the error estimate is\n\"guaranteed\". These reciprocal condition\nnumbers for some appropriately scaled matrix Z\nare:\nLet z=s*a, where s scales each row by a power\nof the radix so all absolute row sums of z are\napproximately 1.\nThe information for right-hand side i, where 1 ≤i≤nrhs, and type of error\nerr is stored in err_bnds_norm[(err-1)*nrhs + i - 1].\nerr_bnds_comp\nArray of size nrhs*n_err_bnds. For each right-hand side, contains\ninformation about various error bounds and condition numbers\ncorresponding to the componentwise relative error, which is defined as\nfollows:\nComponentwise relative error in the i-th solution vector:\nThe array is indexed by the type of error information as described below.\nThere are currently up to three pieces of information returned for each\nright-hand side. If componentwise accuracy is not requested (params[2] =\n0.0), then err_bnds_comp is not accessed.\nerr=1\n\"Trust/don't trust\" boolean. Trust the answer if\nthe reciprocal condition number is less than the\nthreshold sqrt(n)*slamch(ε) for single\nprecision flavors and sqrt(n)*dlamch(ε) for\ndouble precision flavors.\nerr=2\n\"Guaranteed\" error bpound. The estimated\nforward error, almost certainly within a factor of\n10 of the true error so long as the next entry is\ngreater than the threshold sqrt(n)*slamch(ε)\nfor single precision flavors and\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n704\n\n\nsqrt(n)*dlamch(ε) for double precision\nflavors. This error bound should only be trusted\nif the previous boolean is true.\nerr=3\nReciprocal condition number. Estimated\ncomponentwise reciprocal condition number.\nCompared with the threshold\nsqrt(n)*slamch(ε) for single precision flavors\nand sqrt(n)*dlamch(ε) for double precision\nflavors to determine if the error estimate is\n\"guaranteed\". These reciprocal condition\nnumbers for some appropriately scaled matrix Z\nare:\nLet z=s*(a*diag(x)), where x is the solution\nfor the current right-hand side and s scales each\nrow of a*diag(x) by a power of the radix so all\nabsolute row sums of z are approximately 1.\nThe information for right-hand side i, where 1 ≤i≤nrhs, and type of error\nerr is stored in err_bnds_comp[(err-1)*nrhs + i - 1].\nipiv\nIf fact = 'N' or 'E', then ipiv is an output argument and on exit\ncontains the pivot indices from the factorization A = P*L*U of the original\nmatrix A (if fact = 'N') or of the equilibrated matrix A (if fact = 'E').\nequed\nIf fact≠'F', then equed is an output argument. It specifies the form of\nequilibration that was done (see the description of equed in Input\nArguments section).\nparams\nIf an entry is less than 0.0, that entry is filled with the default value used\nfor that parameter, otherwise the entry is not modified.\ninfo\nIf info = 0, the execution is successful. The solution to every right-hand\nside is guaranteed.\nIf info = -i, the i-th parameter had an illegal value.\nIf 0 < info≤n: Uinfo,info is exactly zero. The factorization has been\ncompleted, but the factor U is exactly singular, so the solution and error\nbounds could not be computed; rcond = 0 is returned.\nIf info = n+j: The solution corresponding to the j-th right-hand side is not\nguaranteed. The solutions corresponding to other right-hand sides k with k\n> j may not be guaranteed as well, but only the first such right-hand side is\nreported. If a small componentwise error is not requested params[2] =\n0.0, then the j-th right-hand side is the first with a normwise error bound\nthat is not guaranteed (the smallest j such that err_bnds_norm[j - 1] =\n0.0 or err_bnds_comp[j - 1] = 0.0. See the definition of\nerr_bnds_norm and err_bnds_comp for err = 1. To get information about\nall of the right-hand sides, check err_bnds_norm or err_bnds_comp.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n705\n\n\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful. The solution to every right-hand side is guaranteed.\nIf info = -i, parameter i had an illegal value.\nIf 0 < info≤n: Uinfo,info is exactly zero. The factorization has been completed, but the factor U is exactly\nsingular, so the solution and error bounds could not be computed; rcond = 0 is returned.\nIf info = n+j: The solution corresponding to the j-th right-hand side is not guaranteed. The solutions\ncorresponding to other right-hand sides k with k > j may not be guaranteed as well, but only the first such\nright-hand side is reported. If a small componentwise error is not requested params[2] = 0.0, then the j-th\nright-hand side is the first with a normwise error bound that is not guaranteed (the smallest j such that for\ncolumn major layout err_bnds_norm[j - 1] = 0.0 or err_bnds_comp[j - 1] = 0.0; or for row major\nlayout err_bnds_norm[(j - 1)*n_err_bnds] = 0.0 or err_bnds_comp[(j - 1)*n_err_bnds] = 0.0).\nSee the definition of err_bnds_norm and err_bnds_comp for err = 1. To get information about all of the\nright-hand sides, check err_bnds_norm or err_bnds_comp.\nSee Also\nMatrix Storage Schemes\n?gtsv\nComputes the solution to the system of linear\nequations with a tridiagonal coefficient matrix A and\nmultiple right-hand sides.\nSyntax\nlapack_int LAPACKE_sgtsv (int matrix_layout , lapack_int n , lapack_int nrhs , float *\ndl , float * d , float * du , float * b , lapack_int ldb );\nlapack_int LAPACKE_dgtsv (int matrix_layout , lapack_int n , lapack_int nrhs , double *\ndl , double * d , double * du , double * b , lapack_int ldb );\nlapack_int LAPACKE_cgtsv (int matrix_layout , lapack_int n , lapack_int nrhs ,\nlapack_complex_float * dl , lapack_complex_float * d , lapack_complex_float * du ,\nlapack_complex_float * b , lapack_int ldb );\nlapack_int LAPACKE_zgtsv (int matrix_layout , lapack_int n , lapack_int nrhs ,\nlapack_complex_double * dl , lapack_complex_double * d , lapack_complex_double * du ,\nlapack_complex_double * b , lapack_int ldb );\nInclude Files\n•\nmkl.h\nDescription\nThe routine solves for X the system of linear equations A*X = B, where A is an n-by-n tridiagonal matrix, the\ncolumns of matrix B are individual right-hand sides, and the columns of X are the corresponding solutions.\nThe routine uses Gaussian elimination with partial pivoting.\nNote that the equation AT*X = B may be solved by interchanging the order of the arguments du and dl.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n706\n\n\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\nn\nThe order of A, the number of rows in B; n≥ 0.\nnrhs\nThe number of right-hand sides, the number of columns in B; nrhs≥\n0.\ndl\nThe array dl (size n - 1) contains the (n - 1) subdiagonal elements\nof A.\nd\nThe array d (size n) contains the diagonal elements of A.\ndu\nThe array du (size n - 1) contains the (n - 1) superdiagonal\nelements of A.\nb\nThe array b  of size max(1, ldb*nrhs) for column major layout and\nmax(1, ldb*n) for row major layout contains the matrix B whose\ncolumns are the right-hand sides for the systems of equations.\nldb\nThe leading dimension of b; ldb≥ max(1, n) for column major layout\nand ldb≥nrhs for row major layout.\nOutput Parameters\ndl\nOverwritten by the (n-2) elements of the second superdiagonal of the\nupper triangular matrix U from the LU factorization of A. These\nelements are stored in dl[0], ..., dl[n - 3].\nd\nOverwritten by the n diagonal elements of U.\ndu\nOverwritten by the (n-1) elements of the first superdiagonal of U.\nb\nOverwritten by the solution matrix X.\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info = -i, parameter i had an illegal value.\nIf info = i, Ui, i is exactly zero, and the solution has not been computed. The factorization has not been\ncompleted unless i = n.\nSee Also\nMatrix Storage Schemes\n?gtsvx\nComputes the solution to the real or complex system\nof linear equations with a tridiagonal coefficient matrix\nA and multiple right-hand sides, and provides error\nbounds on the solution.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n707\n\n\nSyntax\nlapack_int LAPACKE_sgtsvx( int matrix_layout, char fact, char trans, lapack_int n,\nlapack_int nrhs, const float* dl, const float* d, const float* du, float* dlf, float*\ndf, float* duf, float* du2, lapack_int* ipiv, const float* b, lapack_int ldb, float* x,\nlapack_int ldx, float* rcond, float* ferr, float* berr );\nlapack_int LAPACKE_dgtsvx( int matrix_layout, char fact, char trans, lapack_int n,\nlapack_int nrhs, const double* dl, const double* d, const double* du, double* dlf,\ndouble* df, double* duf, double* du2, lapack_int* ipiv, const double* b, lapack_int ldb,\ndouble* x, lapack_int ldx, double* rcond, double* ferr, double* berr );\nlapack_int LAPACKE_cgtsvx( int matrix_layout, char fact, char trans, lapack_int n,\nlapack_int nrhs, const lapack_complex_float* dl, const lapack_complex_float* d, const\nlapack_complex_float* du, lapack_complex_float* dlf, lapack_complex_float* df,\nlapack_complex_float* duf, lapack_complex_float* du2, lapack_int* ipiv, const\nlapack_complex_float* b, lapack_int ldb, lapack_complex_float* x, lapack_int ldx,\nfloat* rcond, float* ferr, float* berr );\nlapack_int LAPACKE_zgtsvx( int matrix_layout, char fact, char trans, lapack_int n,\nlapack_int nrhs, const lapack_complex_double* dl, const lapack_complex_double* d, const\nlapack_complex_double* du, lapack_complex_double* dlf, lapack_complex_double* df,\nlapack_complex_double* duf, lapack_complex_double* du2, lapack_int* ipiv, const\nlapack_complex_double* b, lapack_int ldb, lapack_complex_double* x, lapack_int ldx,\ndouble* rcond, double* ferr, double* berr );\nInclude Files\n•\nmkl.h\nDescription\nThe routine uses the LU factorization to compute the solution to a real or complex system of linear equations\nA*X = B, AT*X = B, or AH*X = B, where A is a tridiagonal matrix of order n, the columns of matrix B are\nindividual right-hand sides, and the columns of X are the corresponding solutions.\nError bounds on the solution and a condition estimate are also provided.\nThe routine ?gtsvx performs the following steps:\n1.\nIf fact = 'N', the LU decomposition is used to factor the matrix A as A = L*U, where L is a product\nof permutation and unit lower bidiagonal matrices and U is an upper triangular matrix with nonzeroes in\nonly the main diagonal and first two superdiagonals.\n2.\nIf some Ui,i= 0, so that U is exactly singular, then the routine returns with info = i. Otherwise, the\nfactored form of A is used to estimate the condition number of the matrix A. If the reciprocal of the\ncondition number is less than machine precision, info = n + 1 is returned as a warning, but the\nroutine still goes on to solve for X and compute error bounds as described below.\n3.\nThe system of equations is solved for X using the factored form of A.\n4.\nIterative refinement is applied to improve the computed solution matrix and calculate error bounds and\nbackward error estimates for it.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\nfact\nMust be 'F' or 'N'.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n708\n\n\nSpecifies whether or not the factored form of the matrix A has been\nsupplied on entry.\nIf fact = 'F': on entry, dlf, df, duf, du2, and ipiv contain the\nfactored form of A; arrays dl, d, du, dlf, df, duf, du2, and ipiv will not\nbe modified.\nIf fact = 'N', the matrix A will be copied to dlf, df, and duf and\nfactored.\ntrans\nMust be 'N', 'T', or 'C'.\nSpecifies the form of the system of equations:\nIf trans = 'N', the system has the form A*X = B (No transpose).\nIf trans = 'T', the system has the form AT*X = B (Transpose).\nIf trans = 'C', the system has the form AH*X = B (Conjugate\ntranspose).\nn\nThe number of linear equations, the order of the matrix A; n≥ 0.\nnrhs\nThe number of right hand sides, the number of columns of the\nmatrices B and X; nrhs≥ 0.\ndl,d,du,dlf,df, duf,du2,b\nArrays:\ndl, size (n -1), contains the subdiagonal elements of A.\nd, size (n), contains the diagonal elements of A.\ndu, size (n -1), contains the superdiagonal elements of A.\ndlf, size (n -1). If fact = 'F', then dlf is an input argument and on\nentry contains the (n -1) multipliers that define the matrix L from the\nLU factorization of A as computed by ?gttrf.\ndf, size (n). If fact = 'F', then df is an input argument and on\nentry contains the n diagonal elements of the upper triangular matrix\nU from the LU factorization of A.\nduf, size (n -1). If fact = 'F', then duf is an input argument and on\nentry contains the (n -1) elements of the first superdiagonal of U.\ndu2, size (n -2). If fact = 'F', then du2 is an input argument and\non entry contains the (n-2) elements of the second superdiagonal of\nU.\nb, size max(ldb*nrhs) for column major layout and max(ldb*n) for\nrow major layout, contains the right-hand side matrix B.\nldb\nThe leading dimension of b; ldb≥ max(1, n) for column major layout\nand ldb≥nrhs for row major layout.\nldx\nThe leading dimension of x; ldx≥ max(1, n) for column major layout\nand ldx≥nrhs for row major layout.\nipiv\nArray, size at least max(1, n). If fact = 'F', then ipiv is an input\nargument and on entry contains the pivot indices, as returned\nby ?gttrf.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n709\n\n\nOutput Parameters\nx\nArray, size max(1, ldx*nrhs) for column major layout and max(1,\nldx*n) for row major layout.\nIf info = 0 or info = n+1, the array x contains the solution matrix\nX.\ndlf\nIf fact = 'N', then dlf is an output argument and on exit contains\nthe (n-1) multipliers that define the matrix L from the LU\nfactorization of A.\ndf\nIf fact = 'N', then df is an output argument and on exit contains\nthe n diagonal elements of the upper triangular matrix U from the LU\nfactorization of A.\nduf\nIf fact = 'N', then duf is an output argument and on exit contains\nthe (n-1) elements of the first superdiagonal of U.\ndu2\nIf fact = 'N', then du2 is an output argument and on exit contains\nthe (n-2) elements of the second superdiagonal of U.\nipiv\nThe array ipiv is an output argument if fact = 'N'and, on exit,\ncontains the pivot indices from the factorization A = L*U ; row i of\nthe matrix was interchanged with row ipiv[i-1]. The value of ipiv[i-1]\nwill always be i or i+1; ipiv[i-1]=i indicates a row interchange was not\nrequired.\nrcond\nAn estimate of the reciprocal condition number of the matrix A. If\nrcond is less than the machine precision (in particular, if rcond =0),\nthe matrix is singular to working precision. This condition is indicated\nby a return code of info>0.\nferr\nArray, size at least max(1, nrhs). Contains the estimated forward\nerror bound for each solution vector xj (the j-th column of the solution\nmatrix X). If xtrue is the true solution corresponding to xj, ferr[j-1]\nis an estimated upper bound for the magnitude of the largest element\nin xj - xtrue divided by the magnitude of the largest element in xj. The\nestimate is as reliable as the estimate for rcond, and is almost always\na slight overestimate of the true error.\nberr\nArray, size at least max(1, nrhs). Contains the component-wise\nrelative backward error for each solution vector xj, that is, the\nsmallest relative change in any element of A or B that makes xj an\nexact solution.\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info = -i, parameter i had an illegal value.\nIf info = i, and i≤n, then Ui, i is exactly zero. The factorization has not been completed unless i = n, but\nthe factor U is exactly singular, so the solution and error bounds could not be computed; rcond = 0 is\nreturned. If info = i, and i = n + 1, then U is nonsingular, but rcond is less than machine precision,\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n710\n\n\nmeaning that the matrix is singular to working precision. Nevertheless, the solution and error bounds are\ncomputed because there are a number of situations where the computed solution can be more accurate than\nthe value of rcond would suggest.\nSee Also\nMatrix Storage Schemes\n?dtsvb\nComputes the solution to the system of linear\nequations with a diagonally dominant tridiagonal\ncoefficient matrix A and multiple right-hand sides.\nSyntax\nvoid sdtsvb (const MKL_INT * n, const MKL_INT * nrhs, float * dl, float * d, const\nfloat * du, float * b, const MKL_INT * ldb, MKL_INT * info );\nvoid ddtsvb (const MKL_INT * n, const MKL_INT * nrhs, double * dl, double * d, const\ndouble * du, double * b, const MKL_INT * ldb, MKL_INT * info );\nvoid cdtsvb (const MKL_INT * n, const MKL_INT * nrhs, MKL_Complex8 * dl, MKL_Complex8 *\nd, const MKL_Complex8 * du, MKL_Complex8 * b, const MKL_INT * ldb, MKL_INT * info );\nvoid zdtsvb (const MKL_INT * n, const MKL_INT * nrhs, MKL_Complex16 * dl, MKL_Complex16\n* d, const MKL_Complex16 * du, MKL_Complex16 * b, const MKL_INT * ldb, MKL_INT *\ninfo );\nInclude Files\n•\nmkl.h\nDescription\nThe ?dtsvb routine solves a system of linear equations A*X = B for X, where A is an n-by-n diagonally\ndominant tridiagonal matrix, the columns of matrix B are individual right-hand sides, and the columns of X\nare the corresponding solutions. The routine uses the BABE (Burning At Both Ends) algorithm.\nNote that the equation AT*X = B may be solved by interchanging the order of the arguments du and dl.\nInput Parameters\nn\nThe order of A, the number of rows in B; n≥ 0.\nnrhs\nThe number of right-hand sides, the number of columns in B; nrhs≥\n0.\ndl, d, du, b\nArrays: dl (size n - 1), d (size n), du (size n - 1), b(max(ldb*nrhs)\nfor column major layout and max(ldb*n) for row major layout).\nThe array dl contains the (n - 1) subdiagonal elements of A.\nThe array d contains the diagonal elements of A.\nThe array du contains the (n - 1) superdiagonal elements of A.\nThe array b contains the matrix B whose columns are the right-hand\nsides for the systems of equations.\nldb\nThe leading dimension of b; ldb≥ max(1, n).\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n711\n\n\nOutput Parameters\ndl\nOverwritten by the (n-1) elements of the subdiagonal of the lower\ntriangular matrices L1, L2 from the factorization of A (see dttrfb).\nd\nOverwritten by the n diagonal element reciprocals of U.\nb\nOverwritten by the solution matrix X.\ninfo\nIf info = 0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nIf info = i, uii is exactly zero, and the solution has not been\ncomputed. The factorization has not been completed unless i = n.\nApplication Notes\nA diagonally dominant tridiagonal system is defined such that |di| > |dli-1| + |dui| for any i:\n1 < i < n, and |d1| > |du1|, |dn| > |dln-1|\nThe underlying BABE algorithm is designed for diagonally dominant systems. Such systems have no\nnumerical stability issue unlike the canonical systems that use elimination with partial pivoting (see ?gtsv).\nThe diagonally dominant systems are much faster than the canonical systems.\nNOTE\n•\nThe current implementation of BABE has a potential accuracy issue on very small or large data\nclose to the underflow or overflow threshold respectively. Scale the matrix before applying the\nsolver in the case of such input data.\n•\nApplying the ?dtsvb factorization to non-diagonally dominant systems may lead to an accuracy\nloss, or false singularity detected due to no pivoting.\n?posv\nComputes the solution to the system of linear\nequations with a symmetric or Hermitian positive-\ndefinite coefficient matrix A and multiple right-hand\nsides.\nSyntax\nlapack_int LAPACKE_sposv (int matrix_layout, char uplo, lapack_int n, lapack_int nrhs,\nfloat * a, lapack_int lda, float * b, lapack_int ldb);\nlapack_int LAPACKE_dposv (int matrix_layout, char uplo, lapack_int n, lapack_int nrhs,\ndouble * a, lapack_int lda, double * b, lapack_int ldb);\nlapack_int LAPACKE_cposv (int matrix_layout, char uplo, lapack_int n, lapack_int nrhs,\nlapack_complex_float * a, lapack_int lda, lapack_complex_float * b, lapack_int ldb);\nlapack_int LAPACKE_zposv (int matrix_layout, char uplo, lapack_int n, lapack_int nrhs,\nlapack_complex_double * a, lapack_int lda, lapack_complex_double * b, lapack_int ldb);\nlapack_int LAPACKE_dsposv (int matrix_layout, char uplo, lapack_int n, lapack_int nrhs,\ndouble * a, lapack_int lda, double * b, lapack_int ldb, double * x, lapack_int ldx,\nlapack_int * iter);\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n712\n\n\nlapack_int LAPACKE_zcposv (int matrix_layout, char uplo, lapack_int n, lapack_int nrhs,\nlapack_complex_double * a, lapack_int lda, lapack_complex_double * b, lapack_int ldb,\nlapack_complex_double * x, lapack_int ldx, lapack_int * iter);\nInclude Files\n•\nmkl.h\nDescription\nThe routine solves for X the real or complex system of linear equations A*X = B, where A is an n-by-n\nsymmetric/Hermitian positive-definite matrix, the columns of matrix B are individual right-hand sides, and\nthe columns of X are the corresponding solutions.\nThe Cholesky decomposition is used to factor A as\nA = UT*U (real flavors) and A = UH*U (complex flavors), if uplo = 'U'\nor A = L*LT (real flavors) and A = L*LH (complex flavors), if uplo = 'L',\nwhere U is an upper triangular matrix and L is a lower triangular matrix. The factored form of A is then used\nto solve the system of equations A*X = B.\nThe dsposv and zcposv are mixed precision iterative refinement subroutines for exploiting fast single\nprecision hardware. They first attempt to factorize the matrix in single precision (dsposv) or single complex\nprecision (zcposv) and use this factorization within an iterative refinement procedure to produce a solution\nwith double precision (dsposv) / double complex precision (zcposv) normwise backward error quality (see\nbelow). If the approach fails, the method switches to a double precision or double complex precision\nfactorization respectively and computes the solution.\nThe iterative refinement is not going to be a winning strategy if the ratio single precision/complex\nperformance over double precision/double complex performance is too small. A reasonable strategy should\ntake the number of right-hand sides and the size of the matrix into account. This might be done with a call to\nilaenv in the future. At present, iterative refinement is implemented.\nThe iterative refinement process is stopped if\niter > itermax\nor for all the right-hand sides:\nrnmr < sqrt(n)*xnrm*anrm*eps*bwdmax,\nwhere\n•\niter is the number of the current iteration in the iterative refinement process\n•\nrnmr is the infinity-norm of the residual\n•\nxnrm is the infinity-norm of the solution\n•\nanrm is the infinity-operator-norm of the matrix A\n•\neps is the machine epsilon returned by dlamch (‘Epsilon’).\nThe values itermax and bwdmax are fixed to 30 and 1.0d+00 respectively.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\nuplo\nMust be 'U' or 'L'.\nIndicates whether the upper or lower triangular part of A is stored:\nIf uplo = 'U', the upper triangle of A is stored.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n713\n\n\nIf uplo = 'L', the lower triangle of A is stored.\nn\nThe order of matrix A; n≥ 0.\nnrhs\nThe number of right-hand sides, the number of columns in B; nrhs≥\n0.\na, b\nArrays: a(size max(1, lda)), b, size max(ldb*nrhs) for column major\nlayout and max(ldb*n) for row major layout,. The array a contains\nthe upper or the lower triangular part of the matrix A (see uplo).\nNote that in the case of zcposv the imaginary parts of the diagonal\nelements need not be set and are assumed to be zero.\nThe array b contains the matrix B whose columns are the right-hand\nsides for the systems of equations.\nlda\nThe leading dimension of a; lda≥ max(1, n).\nldb\nThe leading dimension of b; ldb≥ max(1, n) for column major layout\nand ldb≥nrhs for row major layout.\nldx\nThe leading dimension of the array x; ldx≥ max(1, n) for column\nmajor layout and ldx≥nrhs for row major layout.\nOutput Parameters\na\nIf info = 0, the upper or lower triangular part of a is overwritten by\nthe Cholesky factor U or L, as specified by uplo.\nIf iterative refinement has been successfully used (info= 0 and\niter≥ 0), then A is unchanged.\nIf double precision factorization has been used (info= 0 and iter <\n0), then the array A contains the factors L or U from the Cholesky\nfactorization.\nb\nOverwritten by the solution matrix X.\nx\nArray, size max(1, ldx*nrhs) for column major layout and max(1,\nldx*n) for row major layout. If info = 0, contains the n-by-nrhs\nsolution matrix X.\niter\nIf iter < 0: iterative refinement has failed, double precision\nfactorization has been performed\n•\nIf iter = -1: the routine fell back to full precision for\nimplementation- or machine-specific reason\n•\nIf iter = -2: narrowing the precision induced an overflow, the\nroutine fell back to full precision\n•\nIf iter = -3: failure of spotrf for dsposv, or cpotrf for zcposv\n•\nIf iter = -31: stop the iterative refinement after the 30th\niteration.\nIf iter > 0: iterative refinement has been successfully used. Returns\nthe number of iterations.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n714\n\n\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info = -i, parameter i had an illegal value.\nIf info = i, the leading minor of order i (and therefore the matrix A itself) is not positive definite, so the\nfactorization could not be completed, and the solution has not been computed.\nSee Also\nMatrix Storage Schemes\n?posvx\nUses the Cholesky factorization to compute the\nsolution to the system of linear equations with a\nsymmetric or Hermitian positive-definite coefficient\nmatrix A, and provides error bounds on the solution.\nSyntax\nlapack_int LAPACKE_sposvx( int matrix_layout, char fact, char uplo, lapack_int n,\nlapack_int nrhs, float* a, lapack_int lda, float* af, lapack_int ldaf, char* equed,\nfloat* s, float* b, lapack_int ldb, float* x, lapack_int ldx, float* rcond, float* ferr,\nfloat* berr );\nlapack_int LAPACKE_dposvx( int matrix_layout, char fact, char uplo, lapack_int n,\nlapack_int nrhs, double* a, lapack_int lda, double* af, lapack_int ldaf, char* equed,\ndouble* s, double* b, lapack_int ldb, double* x, lapack_int ldx, double* rcond, double*\nferr, double* berr );\nlapack_int LAPACKE_cposvx( int matrix_layout, char fact, char uplo, lapack_int n,\nlapack_int nrhs, lapack_complex_float* a, lapack_int lda, lapack_complex_float* af,\nlapack_int ldaf, char* equed, float* s, lapack_complex_float* b, lapack_int ldb,\nlapack_complex_float* x, lapack_int ldx, float* rcond, float* ferr, float* berr );\nlapack_int LAPACKE_zposvx( int matrix_layout, char fact, char uplo, lapack_int n,\nlapack_int nrhs, lapack_complex_double* a, lapack_int lda, lapack_complex_double* af,\nlapack_int ldaf, char* equed, double* s, lapack_complex_double* b, lapack_int ldb,\nlapack_complex_double* x, lapack_int ldx, double* rcond, double* ferr, double* berr );\nInclude Files\n•\nmkl.h\nDescription\nThe routine uses the Cholesky factorization A=UT*U (real flavors) / A=UH*U (complex flavors) or A=L*LT (real\nflavors) / A=L*LH (complex flavors) to compute the solution to a real or complex system of linear equations\nA*X = B, where A is a n-by-n real symmetric/Hermitian positive definite matrix, the columns of matrix B are\nindividual right-hand sides, and the columns of X are the corresponding solutions.\nError bounds on the solution and a condition estimate are also provided.\nThe routine ?posvx performs the following steps:\n1.\nIf fact = 'E', real scaling factors s are computed to equilibrate the system:\ndiag(s)*A*diag(s)*inv(diag(s))*X = diag(s)*B.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n715\n\n\nWhether or not the system will be equilibrated depends on the scaling of the matrix A, but if\nequilibration is used, A is overwritten by diag(s)*A*diag(s) and B by diag(s)*B.\n2.\nIf fact = 'N' or 'E', the Cholesky decomposition is used to factor the matrix A (after equilibration if\nfact = 'E') as\nA = UT*U (real), A = UH*U (complex), if uplo = 'U',\nor A = L*LT (real), A = L*LH (complex), if uplo = 'L',\nwhere U is an upper triangular matrix and L is a lower triangular matrix.\n3.\nIf the leading i-by-i principal minor is not positive-definite, then the routine returns with info = i.\nOtherwise, the factored form of A is used to estimate the condition number of the matrix A. If the\nreciprocal of the condition number is less than machine precision, info = n + 1 is returned as a\nwarning, but the routine still goes on to solve for X and compute error bounds as described below.\n4.\nThe system of equations is solved for X using the factored form of A.\n5.\nIterative refinement is applied to improve the computed solution matrix and calculate error bounds and\nbackward error estimates for it.\n6.\nIf equilibration was used, the matrix X is premultiplied by diag(s) so that it solves the original system\nbefore equilibration.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\nfact\nMust be 'F', 'N', or 'E'.\nSpecifies whether or not the factored form of the matrix A is supplied\non entry, and if not, whether the matrix A should be equilibrated\nbefore it is factored.\nIf fact = 'F': on entry, af contains the factored form of A. If equed\n= 'Y', the matrix A has been equilibrated with scaling factors given\nby s.\na and af will not be modified.\nIf fact = 'N', the matrix A will be copied to af and factored.\nIf fact = 'E', the matrix A will be equilibrated if necessary, then\ncopied to af and factored.\nuplo\nMust be 'U' or 'L'.\nIndicates whether the upper or lower triangular part of A is stored:\nIf uplo = 'U', the upper triangle of A is stored.\nIf uplo = 'L', the lower triangle of A is stored.\nn\nThe order of matrix A; n≥ 0.\nnrhs\nThe number of right-hand sides, the number of columns in B; nrhs≥\n0.\na, af, b\nArrays: a(size max(1, lda*n)), af(size max(1, ldaf*n)), b, size\nmax(ldb*nrhs) for column major layout and max(ldb*n) for row\nmajor layout, .\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n716\n\n\nThe array a contains the matrix A as specified by uplo. If fact = 'F'\nand equed = 'Y', then A must have been equilibrated by the scaling\nfactors in s, and a must contain the equilibrated matrix\ndiag(s)*A*diag(s).\nThe array af is an input argument if fact = 'F'. It contains the\ntriangular factor U or L from the Cholesky factorization of A in the\nsame storage format as A. If equed is not 'N', then af is the factored\nform of the equilibrated matrix diag(s)*A*diag(s).\nThe array b contains the matrix B whose columns are the right-hand\nsides for the systems of equations.\nlda\nThe leading dimension of a; lda≥ max(1, n).\nldaf\nThe leading dimension of af; ldaf≥ max(1, n).\nldb\nThe leading dimension of b; ldb≥ max(1, n) for column major layout\nand ldb≥nrhs for row major layout.\nequed\nMust be 'N' or 'Y'.\nequed is an input argument if fact = 'F'. It specifies the form of\nequilibration that was done:\nif equed = 'N', no equilibration was done (always true if fact =\n'N');\nif equed = 'Y', equilibration was done, that is, A has been replaced\nby diag(s)*A*diag(s).\ns\nArray, size (n). The array s contains the scale factors for A. This array\nis an input argument if fact = 'F' only; otherwise it is an output\nargument.\nIf equed = 'N', s is not accessed.\nIf fact = 'F' and equed = 'Y', each element of s must be positive.\nldx\nThe leading dimension of the output array x; ldx≥ max(1, n) for\ncolumn major layout and ldx≥nrhs for row major layout.\nOutput Parameters\nx\nArray, size max(1, ldx*nrhs) for column major layout and max(1,\nldx*n) for row major layout.\nIf info = 0 or info = n+1, the array x contains the solution matrix\nX to the original system of equations. Note that if equed = 'Y', A\nand B are modified on exit, and the solution to the equilibrated system\nis inv(diag(s))*X.\na\nArray a is not modified on exit if fact = 'F' or 'N', or if fact =\n'E' and equed = 'N'.\nIf fact = 'E' and equed = 'Y', A is overwritten by\ndiag(s)*A*diag(s).\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n717\n\n\naf\nIf fact = 'N' or 'E', then af is an output argument and on exit\nreturns the triangular factor U or L from the Cholesky factorization\nA=UT*U or A=L*LT (real routines), A=UH*U or A=L*LH (complex\nroutines) of the original matrix A (if fact = 'N'), or of the\nequilibrated matrix A (if fact = 'E'). See the description of a for the\nform of the equilibrated matrix.\nb\nOverwritten by diag(s)*B, if equed = 'Y'; not changed if equed =\n'N'.\ns\nThis array is an output argument if fact≠'F'. See the description of s\nin Input Arguments section.\nrcond\nAn estimate of the reciprocal condition number of the matrix A after\nequilibration (if done). If rcond is less than the machine precision (in\nparticular, if rcond =0), the matrix is singular to working precision.\nThis condition is indicated by a return code of info>0.\nferr\nArray, size at least max(1, nrhs). Contains the estimated forward\nerror bound for each solution vector xj (the j-th column of the solution\nmatrix X). If xtrue is the true solution corresponding to xj, ferr[j-1]\nis an estimated upper bound for the magnitude of the largest element\nin (xj) - xtrue) divided by the magnitude of the largest element in xj.\nThe estimate is as reliable as the estimate for rcond, and is almost\nalways a slight overestimate of the true error.\nberr\nArray, size at least max(1, nrhs). Contains the component-wise\nrelative backward error for each solution vector xj, that is, the\nsmallest relative change in any element of A or B that makes xj an\nexact solution.\nequed\nIf fact≠'F', then equed is an output argument. It specifies the form\nof equilibration that was done (see the description of equed in Input\nArguments section).\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info = -i, parameter i had an illegal value.\nIf info = i, and i≤n, the leading minor of order i (and therefore the matrix A itself) is not positive-definite,\nso the factorization could not be completed, and the solution and error bounds could not be computed; rcond\n=0 is returned.\nIf info = i, and i = n + 1, then U is nonsingular, but rcond is less than machine precision, meaning that the\nmatrix is singular to working precision. Nevertheless, the solution and error bounds are computed because\nthere are a number of situations where the computed solution can be more accurate than the value of rcond\nwould suggest.\nSee Also\nMatrix Storage Schemes\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n718\n\n\n?posvxx\nUses extra precise iterative refinement to compute the\nsolution to the system of linear equations with a\nsymmetric or Hermitian positive-definite coefficient\nmatrix A applying the Cholesky factorization.\nSyntax\nlapack_int LAPACKE_sposvxx( int matrix_layout, char fact, char uplo, lapack_int n,\nlapack_int nrhs, float* a, lapack_int lda, float* af, lapack_int ldaf, char* equed,\nfloat* s, float* b, lapack_int ldb, float* x, lapack_int ldx, float* rcond, float*\nrpvgrw, float* berr, lapack_int n_err_bnds, float* err_bnds_norm, float* err_bnds_comp,\nlapack_int nparams, const float* params );\nlapack_int LAPACKE_dposvxx( int matrix_layout, char fact, char uplo, lapack_int n,\nlapack_int nrhs, double* a, lapack_int lda, double* af, lapack_int ldaf, char* equed,\ndouble* s, double* b, lapack_int ldb, double* x, lapack_int ldx, double* rcond, double*\nrpvgrw, double* berr, lapack_int n_err_bnds, double* err_bnds_norm, double*\nerr_bnds_comp, lapack_int nparams, const double* params );\nlapack_int LAPACKE_cposvxx( int matrix_layout, char fact, char uplo, lapack_int n,\nlapack_int nrhs, lapack_complex_float* a, lapack_int lda, lapack_complex_float* af,\nlapack_int ldaf, char* equed, float* s, lapack_complex_float* b, lapack_int ldb,\nlapack_complex_float* x, lapack_int ldx, float* rcond, float* rpvgrw, float* berr,\nlapack_int n_err_bnds, float* err_bnds_norm, float* err_bnds_comp, lapack_int nparams,\nconst float* params );\nlapack_int LAPACKE_zposvxx( int matrix_layout, char fact, char uplo, lapack_int n,\nlapack_int nrhs, lapack_complex_double* a, lapack_int lda, lapack_complex_double* af,\nlapack_int ldaf, char* equed, double* s, lapack_complex_double* b, lapack_int ldb,\nlapack_complex_double* x, lapack_int ldx, double* rcond, double* rpvgrw, double* berr,\nlapack_int n_err_bnds, double* err_bnds_norm, double* err_bnds_comp, lapack_int\nnparams, const double* params );\nInclude Files\n•\nmkl.h\nDescription\nThe routine uses the Cholesky factorization A=UT*U (real flavors) / A=UH*U (complex flavors) or A=L*LT (real\nflavors) / A=L*LH (complex flavors) to compute the solution to a real or complex system of linear equations\nA*X = B, where A is an n-by-n real symmetric/Hermitian positive definite matrix, the columns of matrix B\nare individual right-hand sides, and the columns of X are the corresponding solutions.\nBoth normwise and maximum componentwise error bounds are also provided on request. The routine returns\na solution with a small guaranteed error (O(eps), where eps is the working machine precision) unless the\nmatrix is very ill-conditioned, in which case a warning is returned. Relevant condition numbers are also\ncalculated and returned.\nThe routine accepts user-provided factorizations and equilibration factors; see definitions of the fact and\nequed options. Solving with refinement and using a factorization from a previous call of the routine also\nproduces a solution with O(eps) errors or warnings but that may not be true for general user-provided\nfactorizations and equilibration factors if they differ from what the routine would itself produce.\nThe routine ?posvxx performs the following steps:\n1.\nIf fact = 'E', scaling factors are computed to equilibrate the system:\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n719\n\n\ndiag(s)*A*diag(s) *inv(diag(s))*X = diag(s)*B\nWhether or not the system will be equilibrated depends on the scaling of the matrix A, but if\nequilibration is used, A is overwritten by diag(s)*A*diag(s) and B by diag(s)*B.\n2.\nIf fact = 'N' or 'E', the Cholesky decomposition is used to factor the matrix A (after equilibration if\nfact = 'E') as\nA = UT*U (real), A = UH*U (complex), if uplo = 'U',\nor A = L*LT (real), A = L*LH (complex), if uplo = 'L',\nwhere U is an upper triangular matrix and L is a lower triangular matrix.\n3.\nIf the leading i-by-i principal minor is not positive-definite, the routine returns with info = i.\nOtherwise, the factored form of A is used to estimate the condition number of the matrix A (see the\nrcond parameter). If the reciprocal of the condition number is less than machine precision, the routine\nstill goes on to solve for X and compute error bounds.\n4.\nThe system of equations is solved for X using the factored form of A.\n5.\nBy default, unless params[0] is set to zero, the routine applies iterative refinement to get a small error\nand error bounds. Refinement calculates the residual to at least twice the working precision.\n6.\nIf equilibration was used, the matrix X is premultiplied by diag(s) so that it solves the original system\nbefore equilibration.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\nfact\nMust be 'F', 'N', or 'E'.\nSpecifies whether or not the factored form of the matrix A is supplied\non entry, and if not, whether the matrix A should be equilibrated\nbefore it is factored.\nIf fact = 'F', on entry, af contains the factored form of A. If equed\nis not 'N', the matrix A has been equilibrated with scaling factors\ngiven by s. Parameters a and af are not modified.\nIf fact = 'N', the matrix A will be copied to af and factored.\nIf fact = 'E', the matrix A will be equilibrated, if necessary, copied\nto af and factored.\nuplo\nMust be 'U' or 'L'.\nIndicates whether the upper or lower triangular part of A is stored:\nIf uplo = 'U', the upper triangle of A is stored.\nIf uplo = 'L', the lower triangle of A is stored.\nn\nThe number of linear equations; the order of the matrix A; n≥ 0.\nnrhs\nThe number of right-hand sides; the number of columns of the\nmatrices B and X; nrhs≥ 0.\na, af, b\nArrays: a(size max(lda*n)), af(size max(ldaf*n)), b)size max(1,\nldb*nrhs) for column major layout and max(1, ldb*n) for row major\nlayout).\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n720\n\n\nThe array a contains the matrix A as specified by uplo . If fact =\n'F' and equed = 'Y', then A must have been equilibrated by the\nscaling factors in s, and a must contain the equilibrated matrix\ndiag(s)*A*diag(s).\nThe array af is an input argument if fact = 'F'. It contains the\ntriangular factor U or L from the Cholesky factorization of A in the\nsame storage format as A. If equed is not 'N', then af is the factored\nform of the equilibrated matrix diag(s)*A*diag(s).\nThe array b contains the matrix B whose columns are the right-hand\nsides for the systems of equations.\nlda\nThe leading dimension of the array a; lda≥ max(1,n).\nldaf\nThe leading dimension of the array af; ldaf≥ max(1,n).\nequed\nMust be 'N' or 'Y'.\nequed is an input argument if fact = 'F'. It specifies the form of\nequilibration that was done:\nIf equed = 'N', no equilibration was done (always true if fact =\n'N').\nif equed = 'Y', both row and column equilibration was done, that is,\nA has been replaced by diag(s)*A*diag(s).\ns\nArray, size (n). The array s contains the scale factors for A. This array\nis an input argument if fact = 'F' only; otherwise it is an output\nargument.\nIf equed = 'N', s is not accessed.\nIf fact = 'F' and equed = 'Y', each element of s must be positive.\nEach element of s should be a power of the radix to ensure a reliable\nsolution and error estimates. Scaling by powers of the radix does not\ncause rounding errors unless the result underflows or overflows.\nRounding errors during scaling lead to refining with a matrix that is\nnot equivalent to the input matrix, producing error estimates that may\nnot be reliable.\nldb\nThe leading dimension of the array b; ldb≥ max(1, n) for column\nmajor layout and ldb≥nrhs for row major layout.\nldx\nThe leading dimension of the output array x; ldx≥ max(1, n) for\ncolumn major layout and ldx≥nrhs for row major layout.\nn_err_bnds\nNumber of error bounds to return for each right hand side and each\ntype (normwise or componentwise). See err_bnds_norm and\nerr_bnds_comp descriptions in the Output Arguments section below.\nnparams\nSpecifies the number of parameters set in params. If ≤ 0, the params\narray is never referenced and default values are used.\nparams\nArray, size max(1,nparams). Specifies algorithm parameters. If an\nentry is less than 0.0, that entry is filled with the default value used\nfor that parameter. Only positions up to nparams are accessed;\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n721\n\n\ndefaults are used for higher-numbered parameters. If defaults are\nacceptable, you can pass nparams = 0, which prevents the source\ncode from accessing the params argument.\nparams[0] : Whether to perform iterative refinement or not. Default:\n1.0 (for single precision flavors), 1.0D+0 (for double precision\nflavors).\n=0.0\nNo refinement is performed and no error\nbounds are computed.\n=1.0\nUse the extra-precise refinement algorithm.\n(Other values are reserved for future use.)\nparams[1] : Maximum number of residual computations allowed for\nrefinement.\nDefault\n10.0\nAggressive\nSet to 100.0 to permit convergence using\napproximate factorizations or factorizations\nother than LU. If the factorization uses a\ntechnique other than Gaussian elimination,\nthe guarantees in err_bnds_norm and\nerr_bnds_comp may no longer be\ntrustworthy.\nparams[2] : Flag determining if the code will attempt to find a\nsolution with a small componentwise relative error in the double-\nprecision algorithm. Positive is true, 0.0 is false. Default: 1.0 (attempt\ncomponentwise convergence).\nOutput Parameters\nx\nArray, size max(1, ldx*nrhs) for column major layout and max(1, ldx*n)\nfor row major layout.\nIf info = 0, the array x contains the solution n-by-nrhs matrix X to the\noriginal system of equations. Note that A and B are modified on exit if\nequed≠'N', and the solution to the equilibrated system is:\ninv(diag(s))*X.\na\nArray a is not modified on exit if fact = 'F' or 'N', or if fact = 'E' and\nequed = 'N'.\nIf fact = 'E' and equed = 'Y', A is overwritten by diag(s)*A*diag(s).\naf\nIf fact = 'N' or 'E', then af is an output argument and on exit returns\nthe triangular factor U or L from the Cholesky factorization A=UT*U or\nA=L*LT (real routines), A=UH*U or A=L*LH (complex routines) of the original\nmatrix A (if fact = 'N'), or of the equilibrated matrix A (if fact = 'E').\nSee the description of a for the form of the equilibrated matrix.\nb\nIf equed = 'N', B is not modified.\nIf equed = 'Y', B is overwritten by diag(s)*B.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n722\n\n\ns\nThis array is an output argument if fact≠'F'. Each element of this array is\na power of the radix. See the description of s in Input Arguments section.\nrcond\nReciprocal scaled condition number. An estimate of the reciprocal Skeel\ncondition number of the matrix A after equilibration (if done). If rcond is\nless than the machine precision, in particular, if rcond = 0, the matrix is\nsingular to working precision. Note that the error may still be small even if\nthis number is very small and the matrix appears ill-conditioned.\nrpvgrw\nContains the reciprocal pivot growth factor:\nIf this is much less than 1, the stability of the LU factorization of the\n(equlibrated) matrix A could be poor. This also means that the solution X,\nestimated condition numbers, and error bounds could be unreliable. If\nfactorization fails with 0 < info≤n, this parameter contains the reciprocal\npivot growth factor for the leading info columns of A.\nberr\nArray, size at least max(1, nrhs). Contains the componentwise relative\nbackward error for each solution vector xj, that is, the smallest relative\nchange in any element of A or B that makes xj an exact solution.\nerr_bnds_norm\nArray of size nrhs*n_err_bnds. For each right-hand side, contains\ninformation about various error bounds and condition numbers\ncorresponding to the normwise relative error, which is defined as follows:\nNormwise relative error in the i-th solution vector\nThe array is indexed by the type of error information as described below.\nThere are currently up to three pieces of information returned.\nerr=1\n\"Trust/don't trust\" boolean. Trust the answer if\nthe reciprocal condition number is less than the\nthreshold sqrt(n)*slamch(ε) for single\nprecision flavors and sqrt(n)*dlamch(ε) for\ndouble precision flavors.\nerr=2\n\"Guaranteed\" error bound. The estimated\nforward error, almost certainly within a factor of\n10 of the true error so long as the next entry is\ngreater than the threshold sqrt(n)*slamch(ε)\nfor single precision flavors and\nsqrt(n)*dlamch(ε) for double precision\nflavors. This error bound should only be trusted\nif the previous boolean is true.\nerr=3\nReciprocal condition number. Estimated\nnormwise reciprocal condition number.\nCompared with the threshold\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n723\n\n\nsqrt(n)*slamch(ε) for single precision flavors\nand sqrt(n)*dlamch(ε) for double precision\nflavors to determine if the error estimate is\n\"guaranteed\". These reciprocal condition\nnumbers for some appropriately scaled matrix Z\nare:\nLet z=s*a, where s scales each row by a power\nof the radix so all absolute row sums of z are\napproximately 1.\nThe information for right-hand side i, where 1 ≤i≤nrhs, and type of error\nerr is stored in err_bnds_norm[(err-1)*nrhs + i - 1].\nerr_bnds_comp\nArray of size nrhs*n_err_bnds. For each right-hand side, contains\ninformation about various error bounds and condition numbers\ncorresponding to the componentwise relative error, which is defined as\nfollows:\nComponentwise relative error in the i-th solution vector:\nThe array is indexed by the type of error information as described below.\nThere are currently up to three pieces of information returned for each\nright-hand side. If componentwise accuracy is not requested (params[2] =\n0.0), then err_bnds_comp is not accessed.\nerr=1\n\"Trust/don't trust\" boolean. Trust the answer if\nthe reciprocal condition number is less than the\nthreshold sqrt(n)*slamch(ε) for single\nprecision flavors and sqrt(n)*dlamch(ε) for\ndouble precision flavors.\nerr=2\n\"Guaranteed\" error bpound. The estimated\nforward error, almost certainly within a factor of\n10 of the true error so long as the next entry is\ngreater than the threshold sqrt(n)*slamch(ε)\nfor single precision flavors and\nsqrt(n)*dlamch(ε) for double precision\nflavors. This error bound should only be trusted\nif the previous boolean is true.\nerr=3\nReciprocal condition number. Estimated\ncomponentwise reciprocal condition number.\nCompared with the threshold\nsqrt(n)*slamch(ε) for single precision flavors\nand sqrt(n)*dlamch(ε) for double precision\nflavors to determine if the error estimate is\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n724\n\n\n\"guaranteed\". These reciprocal condition\nnumbers for some appropriately scaled matrix Z\nare:\nLet z=s*(a*diag(x)), where x is the solution\nfor the current right-hand side and s scales each\nrow of a*diag(x) by a power of the radix so all\nabsolute row sums of z are approximately 1.\nThe information for right-hand side i, where 1 ≤i≤nrhs, and type of error\nerr is stored in err_bnds_comp[(err-1)*nrhs + i - 1].\nequed\nIf fact≠'F', then equed is an output argument. It specifies the form of\nequilibration that was done (see the description of equed in Input\nArguments section).\nparams\nIf an entry is less than 0.0, that entry is filled with the default value used\nfor that parameter, otherwise the entry is not modified.\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful. The solution to every right-hand side is guaranteed.\nIf info = -i, parameter i had an illegal value.\nIf 0 < info≤n: Uinfo,info is exactly zero. The factorization has been completed, but the factor U is exactly\nsingular, so the solution and error bounds could not be computed; rcond = 0 is returned.\nIf info = n+j: The solution corresponding to the j-th right-hand side is not guaranteed. The solutions\ncorresponding to other right-hand sides k with k > j may not be guaranteed as well, but only the first such\nright-hand side is reported. If a small componentwise error is not requested params[2] = 0.0, then the j-th\nright-hand side is the first with a normwise error bound that is not guaranteed (the smallest j such that for\ncolumn major layout err_bnds_norm[j - 1] = 0.0 or err_bnds_comp[j - 1] = 0.0; or for row major\nlayout err_bnds_norm[(j - 1)*n_err_bnds] = 0.0 or err_bnds_comp[(j - 1)*n_err_bnds] = 0.0).\nSee the definition of err_bnds_norm and err_bnds_comp for err = 1. To get information about all of the\nright-hand sides, check err_bnds_norm or err_bnds_comp.\nSee Also\nMatrix Storage Schemes\n?ppsv\nComputes the solution to the system of linear\nequations with a symmetric (Hermitian) positive\ndefinite packed coefficient matrix A and multiple right-\nhand sides.\nSyntax\nlapack_int LAPACKE_sppsv (int matrix_layout , char uplo , lapack_int n , lapack_int\nnrhs , float * ap , float * b , lapack_int ldb );\nlapack_int LAPACKE_dppsv (int matrix_layout , char uplo , lapack_int n , lapack_int\nnrhs , double * ap , double * b , lapack_int ldb );\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n725\n\n\nlapack_int LAPACKE_cppsv (int matrix_layout , char uplo , lapack_int n , lapack_int\nnrhs , lapack_complex_float * ap , lapack_complex_float * b , lapack_int ldb );\nlapack_int LAPACKE_zppsv (int matrix_layout , char uplo , lapack_int n , lapack_int\nnrhs , lapack_complex_double * ap , lapack_complex_double * b , lapack_int ldb );\nInclude Files\n•\nmkl.h\nDescription\nThe routine solves for X the real or complex system of linear equations A*X = B, where A is an n-by-n real\nsymmetric/Hermitian positive-definite matrix stored in packed format, the columns of matrix B are individual\nright-hand sides, and the columns of X are the corresponding solutions.\nThe Cholesky decomposition is used to factor A as\nA = UT*U (real flavors) and A = UH*U (complex flavors), if uplo = 'U'\nor A = L*LT (real flavors) and A = L*LH (complex flavors), if uplo = 'L',\nwhere U is an upper triangular matrix and L is a lower triangular matrix. The factored form of A is then used\nto solve the system of equations A*X = B.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\nuplo\nMust be 'U' or 'L'.\nIndicates whether the upper or lower triangular part of A is stored:\nIf uplo = 'U', the upper triangle of A is stored.\nIf uplo = 'L', the lower triangle of A is stored.\nn\nThe order of matrix A; n≥ 0.\nnrhs\nThe number of right-hand sides, the number of columns in B; nrhs≥\n0.\nap, b\nArrays: ap (size max(1,n*(n+1)/2), b, size max(ldb*nrhs) for\ncolumn major layout and max(ldb*n) for row major layout,. The\narray ap contains the upper or the lower triangular part of the matrix\nA (as specified by uplo) in packed storage (see Matrix Storage\nSchemes). .\nThe array b contains the matrix B whose columns are the right-hand\nsides for the systems of equations.\nldb\nThe leading dimension of b; ldb≥ max(1, n) for column major layout\nand ldb≥nrhs for row major layout.\nOutput Parameters\nap\nIf info = 0, the upper or lower triangular part of A in packed storage is\noverwritten by the Cholesky factor U or L, as specified by uplo.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n726\n\n\nb\nOverwritten by the solution matrix X.\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info = -i, parameter i had an illegal value.\nIf info = i, the leading minor of order i (and therefore the matrix A itself) is not positive-definite, so the\nfactorization could not be completed, and the solution has not been computed.\nSee Also\nMatrix Storage Schemes\n?ppsvx\nUses the Cholesky factorization to compute the\nsolution to the system of linear equations with a\nsymmetric (Hermitian) positive definite packed\ncoefficient matrix A, and provides error bounds on the\nsolution.\nSyntax\nlapack_int LAPACKE_sppsvx( int matrix_layout, char fact, char uplo, lapack_int n,\nlapack_int nrhs, float* ap, float* afp, char* equed, float* s, float* b, lapack_int ldb,\nfloat* x, lapack_int ldx, float* rcond, float* ferr, float* berr );\nlapack_int LAPACKE_dppsvx( int matrix_layout, char fact, char uplo, lapack_int n,\nlapack_int nrhs, double* ap, double* afp, char* equed, double* s, double* b, lapack_int\nldb, double* x, lapack_int ldx, double* rcond, double* ferr, double* berr );\nlapack_int LAPACKE_cppsvx( int matrix_layout, char fact, char uplo, lapack_int n,\nlapack_int nrhs, lapack_complex_float* ap, lapack_complex_float* afp, char* equed,\nfloat* s, lapack_complex_float* b, lapack_int ldb, lapack_complex_float* x, lapack_int\nldx, float* rcond, float* ferr, float* berr );\nlapack_int LAPACKE_zppsvx( int matrix_layout, char fact, char uplo, lapack_int n,\nlapack_int nrhs, lapack_complex_double* ap, lapack_complex_double* afp, char* equed,\ndouble* s, lapack_complex_double* b, lapack_int ldb, lapack_complex_double* x,\nlapack_int ldx, double* rcond, double* ferr, double* berr );\nInclude Files\n•\nmkl.h\nDescription\nThe routine uses the Cholesky factorization A=UT*U (real flavors) / A=UH*U (complex flavors) or A=L*LT (real\nflavors) / A=L*LH (complex flavors) to compute the solution to a real or complex system of linear equations\nA*X = B, where A is a n-by-n symmetric or Hermitian positive-definite matrix stored in packed format, the\ncolumns of matrix B are individual right-hand sides, and the columns of X are the corresponding solutions.\nError bounds on the solution and a condition estimate are also provided.\nThe routine ?ppsvx performs the following steps:\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n727\n\n\n1.\nIf fact = 'E', real scaling factors s are computed to equilibrate the system:\ndiag(s)*A*diag(s)*inv(diag(s))*X = diag(s)*B.\nWhether or not the system will be equilibrated depends on the scaling of the matrix A, but if\nequilibration is used, A is overwritten by diag(s)*A*diag(s) and B by diag(s)*B.\n2.\nIf fact = 'N' or 'E', the Cholesky decomposition is used to factor the matrix A (after equilibration if\nfact = 'E') as\nA = UT*U (real), A = UH*U (complex), if uplo = 'U',\nor A = L*LT (real), A = L*LH (complex), if uplo = 'L',\nwhere U is an upper triangular matrix and L is a lower triangular matrix.\n3.\nIf the leading i-by-i principal minor is not positive-definite, then the routine returns with info = i.\nOtherwise, the factored form of A is used to estimate the condition number of the matrix A. If the\nreciprocal of the condition number is less than machine precision, info = n+1 is returned as a\nwarning, but the routine still goes on to solve for X and compute error bounds as described below.\n4.\nThe system of equations is solved for X using the factored form of A.\n5.\nIterative refinement is applied to improve the computed solution matrix and calculate error bounds and\nbackward error estimates for it.\n6.\nIf equilibration was used, the matrix X is premultiplied by diag(s) so that it solves the original system\nbefore equilibration.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\nfact\nMust be 'F', 'N', or 'E'.\nSpecifies whether or not the factored form of the matrix A is supplied\non entry, and if not, whether the matrix A should be equilibrated\nbefore it is factored.\nIf fact = 'F': on entry, afp contains the factored form of A. If\nequed = 'Y', the matrix A has been equilibrated with scaling factors\ngiven by s.\nap and afp will not be modified.\nIf fact = 'N', the matrix A will be copied to afp and factored.\nIf fact = 'E', the matrix A will be equilibrated if necessary, then\ncopied to afp and factored.\nuplo\nMust be 'U' or 'L'.\nIndicates whether the upper or lower triangular part of A is stored:\nIf uplo = 'U', the upper triangle of A is stored.\nIf uplo = 'L', the lower triangle of A is stored.\nn\nThe order of matrix A; n≥ 0.\nnrhs\nThe number of right-hand sides; the number of columns in B; nrhs≥\n0.\nap, afp, b\nArrays: (size max(1,n*(n+1)/2), afp (size max(1,n*(n+1)/2), bof size\nmax(1, ldb*nrhs) for column major layout and max(1, ldb*n) for\nrow major layout.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n728\n\n\nThe array ap contains the upper or lower triangle of the original\nsymmetric/Hermitian matrix A in packed storage (see Matrix Storage\nSchemes). In case when fact = 'F' and equed = 'Y', ap must\ncontain the equilibrated matrix diag(s)*A*diag(s).\nThe array afp is an input argument if fact = 'F' and contains the\ntriangular factor U or L from the Cholesky factorization of A in the\nsame storage format as A. If equed is not 'N', then afp is the\nfactored form of the equilibrated matrix A.\nThe array b contains the matrix B whose columns are the right-hand\nsides for the systems of equations.\nldb\nThe leading dimension of b; ldb≥ max(1, n) for column major layout\nand ldb≥nrhs for row major layout.\nequed\nMust be 'N' or 'Y'.\nequed is an input argument if fact = 'F'. It specifies the form of\nequilibration that was done:\nif equed = 'N', no equilibration was done (always true if fact =\n'N');\nif equed = 'Y', equilibration was done, that is, A has been replaced\nby diag(s)A*diag(s).\ns\nArray, size (n). The array s contains the scale factors for A. This array\nis an input argument if fact = 'F' only; otherwise it is an output\nargument.\nIf equed = 'N', s is not accessed.\nIf fact = 'F' and equed = 'Y', each element of s must be positive.\nldx\nThe leading dimension of the output array x; ldx≥ max(1, n) for\ncolumn major layout and ldx≥nrhs for row major layout.\nOutput Parameters\nx\nArray, size max(1, ldx*nrhs) for column major layout and max(1,\nldx*n) for row major layout.\nIf info = 0 or info = n+1, the array x contains the solution matrix\nX to the original system of equations. Note that if equed = 'Y', A\nand B are modified on exit, and the solution to the equilibrated system\nis inv(diag(s))*X.\nap\nArray ap is not modified on exit if fact = 'F' or 'N', or if fact =\n'E'and equed = 'N'.\nIf fact = 'E' and equed = 'Y', ap is overwritten by\ndiag(s)*A*diag(s).\nafp\nIf fact = 'N'or 'E', then afp is an output argument and on exit\nreturns the triangular factor U or L from the Cholesky factorization\nA=UT*U or A=L*LT (real routines), A=UH*U or A=L*LH (complex\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n729\n\n\nroutines) of the original matrix A (if fact = 'N'), or of the\nequilibrated matrix A (if fact = 'E'). See the description of ap for\nthe form of the equilibrated matrix.\nb\nOverwritten by diag(s)*B, if equed = 'Y'; not changed if equed =\n'N'.\ns\nThis array is an output argument if fact≠'F'. See the description of s\nin Input Arguments section.\nrcond\nAn estimate of the reciprocal condition number of the matrix A after\nequilibration (if done). If rcond is less than the machine precision (in\nparticular, if rcond = 0), the matrix is singular to working precision.\nThis condition is indicated by a return code of info > 0.\nferr\nArray, size at least max(1, nrhs). Contains the estimated forward\nerror bound for each solution vector xj(the j-th column of the solution\nmatrix X). If xtrue is the true solution corresponding to xj,\nferr[j-1] is an estimated upper bound for the magnitude of the\nlargest element in (xj - xtrue) divided by the magnitude of the\nlargest element in xj. The estimate is as reliable as the estimate for\nrcond, and is almost always a slight overestimate of the true error.\nberr\nArray, size at least max(1, nrhs). Contains the component-wise\nrelative backward error for each solution vector xj, that is, the\nsmallest relative change in any element of A or B that makes xj an\nexact solution.\nequed\nIf fact≠'F', then equed is an output argument. It specifies the form\nof equilibration that was done (see the description of equed in Input\nArguments section).\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, parameter i had an illegal value.\nIf info = i, and i≤n, the leading minor of order i (and therefore the matrix A itself) is not positive-definite,\nso the factorization could not be completed, and the solution and error bounds could not be computed; rcond\n= 0 is returned.\nIf info = i, and i = n + 1, then U is nonsingular, but rcond is less than machine precision, meaning that the\nmatrix is singular to working precision. Nevertheless, the solution and error bounds are computed because\nthere are a number of situations where the computed solution can be more accurate than the value of rcond\nwould suggest.\nSee Also\nMatrix Storage Schemes\n?pbsv\nComputes the solution to the system of linear\nequations with a symmetric or Hermitian positive-\ndefinite band coefficient matrix A and multiple right-\nhand sides.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n730\n\n\nSyntax\nlapack_int LAPACKE_spbsv (int matrix_layout , char uplo , lapack_int n , lapack_int\nkd , lapack_int nrhs , float * ab , lapack_int ldab , float * b , lapack_int ldb );\nlapack_int LAPACKE_dpbsv (int matrix_layout , char uplo , lapack_int n , lapack_int\nkd , lapack_int nrhs , double * ab , lapack_int ldab , double * b , lapack_int ldb );\nlapack_int LAPACKE_cpbsv (int matrix_layout , char uplo , lapack_int n , lapack_int\nkd , lapack_int nrhs , lapack_complex_float * ab , lapack_int ldab ,\nlapack_complex_float * b , lapack_int ldb );\nlapack_int LAPACKE_zpbsv (int matrix_layout , char uplo , lapack_int n , lapack_int\nkd , lapack_int nrhs , lapack_complex_double * ab , lapack_int ldab ,\nlapack_complex_double * b , lapack_int ldb );\nInclude Files\n•\nmkl.h\nDescription\nThe routine solves for X the real or complex system of linear equations A*X = B, where A is an n-by-n\nsymmetric/Hermitian positive definite band matrix, the columns of matrix B are individual right-hand sides,\nand the columns of X are the corresponding solutions.\nThe Cholesky decomposition is used to factor A as\nA = UT*U (real flavors) and A = UH*U (complex flavors), if uplo = 'U'\nor A = L*LT (real flavors) and A = L*LH (complex flavors), if uplo = 'L',\nwhere U is an upper triangular band matrix and L is a lower triangular band matrix, with the same number of\nsuperdiagonals or subdiagonals as A. The factored form of A is then used to solve the system of equations\nA*X = B.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\nuplo\nMust be 'U' or 'L'.\nIndicates whether the upper or lower triangular part of A is stored:\nIf uplo = 'U', the upper triangle of A is stored.\nIf uplo = 'L', the lower triangle of A is stored.\nn\nThe order of matrix A; n≥ 0.\nkd\nThe number of superdiagonals of the matrix A if uplo = 'U', or the\nnumber of subdiagonals if uplo = 'L';kd≥ 0.\nnrhs\nThe number of right-hand sides, the number of columns in B; nrhs≥\n0.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n731\n\n\nab, b\nArrays: ab(size max(1, ldab*n)), bof size max(1, ldb*nrhs) for\ncolumn major layout and max(1, ldb*n) for row major layout. The\narray ab contains the upper or the lower triangular part of the matrix\nA (as specified by uplo) in band storage (see Matrix Storage\nSchemes).\nThe array b contains the matrix B whose columns are the right-hand\nsides for the systems of equations.\nldab\nThe leading dimension of the array ab; ldab≥kd +1.\nldb\nThe leading dimension of b; ldb≥ max(1, n) for column major layout\nand ldb≥nrhs for row major layout.\nOutput Parameters\nab\nThe upper or lower triangular part of A (in band storage) is\noverwritten by the Cholesky factor U or L, as specified by uplo, in the\nsame storage format as A.\nb\nOverwritten by the solution matrix X.\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info = -i, parameter i had an illegal value.\nIf info = i, the leading minor of order i (and therefore the matrix A itself) is not positive-definite, so the\nfactorization could not be completed, and the solution has not been computed.\nSee Also\nMatrix Storage Schemes\n?pbsvx\nUses the Cholesky factorization to compute the\nsolution to the system of linear equations with a\nsymmetric (Hermitian) positive-definite band\ncoefficient matrix A, and provides error bounds on the\nsolution.\nSyntax\nlapack_int LAPACKE_spbsvx( int matrix_layout, char fact, char uplo, lapack_int n,\nlapack_int kd, lapack_int nrhs, float* ab, lapack_int ldab, float* afb, lapack_int\nldafb, char* equed, float* s, float* b, lapack_int ldb, float* x, lapack_int ldx, float*\nrcond, float* ferr, float* berr );\nlapack_int LAPACKE_dpbsvx( int matrix_layout, char fact, char uplo, lapack_int n,\nlapack_int kd, lapack_int nrhs, double* ab, lapack_int ldab, double* afb, lapack_int\nldafb, char* equed, double* s, double* b, lapack_int ldb, double* x, lapack_int ldx,\ndouble* rcond, double* ferr, double* berr );\nlapack_int LAPACKE_cpbsvx( int matrix_layout, char fact, char uplo, lapack_int n,\nlapack_int kd, lapack_int nrhs, lapack_complex_float* ab, lapack_int ldab,\nlapack_complex_float* afb, lapack_int ldafb, char* equed, float* s,\nlapack_complex_float* b, lapack_int ldb, lapack_complex_float* x, lapack_int ldx,\nfloat* rcond, float* ferr, float* berr );\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n732\n\n\nlapack_int LAPACKE_zpbsvx( int matrix_layout, char fact, char uplo, lapack_int n,\nlapack_int kd, lapack_int nrhs, lapack_complex_double* ab, lapack_int ldab,\nlapack_complex_double* afb, lapack_int ldafb, char* equed, double* s,\nlapack_complex_double* b, lapack_int ldb, lapack_complex_double* x, lapack_int ldx,\ndouble* rcond, double* ferr, double* berr );\nInclude Files\n•\nmkl.h\nDescription\nThe routine uses the Cholesky factorization A=UT*U (real flavors) / A=UH*U (complex flavors) or A=L*LT (real\nflavors) / A=L*LH (complex flavors) to compute the solution to a real or complex system of linear equations\nA*X = B, where A is a n-by-n symmetric or Hermitian positive definite band matrix, the columns of matrix B\nare individual right-hand sides, and the columns of X are the corresponding solutions.\nError bounds on the solution and a condition estimate are also provided.\nThe routine ?pbsvx performs the following steps:\n1.\nIf fact = 'E', real scaling factors s are computed to equilibrate the system:\ndiag(s)*A*diag(s)*inv(diag(s))*X = diag(s)*B.\nWhether or not the system will be equilibrated depends on the scaling of the matrix A, but if\nequilibration is used, A is overwritten by diag(s)*A*diag(s) and B by diag(s)*B.\n2.\nIf fact = 'N' or 'E', the Cholesky decomposition is used to factor the matrix A (after equilibration if\nfact = 'E') as\nA = UT*U (real), A = UH*U (complex), if uplo = 'U',\nor A = L*LT (real), A = L*LH (complex), if uplo = 'L',\nwhere U is an upper triangular band matrix and L is a lower triangular band matrix.\n3.\nIf the leading i-by-i principal minor is not positive definite, then the routine returns with info = i.\nOtherwise, the factored form of A is used to estimate the condition number of the matrix A. If the\nreciprocal of the condition number is less than machine precision, info = n+1 is returned as a\nwarning, but the routine still goes on to solve for X and compute error bounds as described below.\n4.\nThe system of equations is solved for X using the factored form of A.\n5.\nIterative refinement is applied to improve the computed solution matrix and calculate error bounds and\nbackward error estimates for it.\n6.\nIf equilibration was used, the matrix X is premultiplied by diag(s) so that it solves the original system\nbefore equilibration.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\nfact\nMust be 'F', 'N', or 'E'.\nSpecifies whether or not the factored form of the matrix A is supplied\non entry, and if not, whether the matrix A should be equilibrated\nbefore it is factored.\nIf fact = 'F': on entry, afb contains the factored form of A. If\nequed = 'Y', the matrix A has been equilibrated with scaling factors\ngiven by s.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n733\n\n\nab and afb will not be modified.\nIf fact = 'N', the matrix A will be copied to afb and factored.\nIf fact = 'E', the matrix A will be equilibrated if necessary, then\ncopied to afb and factored.\nuplo\nMust be 'U' or 'L'.\nIndicates whether the upper or lower triangular part of A is stored:\nIf uplo = 'U', the upper triangle of A is stored.\nIf uplo = 'L', the lower triangle of A is stored.\nn\nThe order of matrix A; n≥ 0.\nkd\nThe number of superdiagonals or subdiagonals in the matrix A; kd≥ 0.\nnrhs\nThe number of right-hand sides, the number of columns in B; nrhs≥\n0.\nab, afb, b\nArrays: ab(size max(1, ldab*n)), afb(size max(1, ldafb*n)), bof size\nmax(1, ldb*nrhs) for column major layout and max(1, ldb*n) for\nrow major layout.\nThe array ab contains the upper or lower triangle of the matrix A in\nband storage (see Matrix Storage Schemes).\nIf fact = 'F' and equed = 'Y', then ab must contain the\nequilibrated matrix diag(s)*A*diag(s).\nThe array afb is an input argument if fact = 'F'. It contains the\ntriangular factor U or L from the Cholesky factorization of the band\nmatrix A in the same storage format as A. If equed = 'Y', then afb\nis the factored form of the equilibrated matrix A.\nThe array b contains the matrix B whose columns are the right-hand\nsides for the systems of equations.\nldab\nThe leading dimension of ab; ldab≥kd+1.\nldafb\nThe leading dimension of afb; ldafb≥kd+1.\nldb\nThe leading dimension of b; ldb≥ max(1, n) for column major layout\nand ldb≥nrhs for row major layout.\nequed\nMust be 'N' or 'Y'.\nequed is an input argument if fact = 'F'. It specifies the form of\nequilibration that was done:\nif equed = 'N', no equilibration was done (always true if fact =\n'N')\nif equed = 'Y', equilibration was done, that is, A has been replaced\nby diag(s)*A*diag(s).\ns\nArray, size (n). The array s contains the scale factors for A. This array\nis an input argument if fact = 'F' only; otherwise it is an output\nargument.\nIf equed = 'N', s is not accessed.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n734\n\n\nIf fact = 'F' and equed = 'Y', each element of s must be positive.\nldx\nThe leading dimension of the output array x; ldx≥ max(1, n) for\ncolumn major layout and ldx≥nrhs for row major layout.\nOutput Parameters\nx\nArray, size max(1, ldx*nrhs) for column major layout and max(1,\nldx*n) for row major layout.\nIf info = 0 or info = n+1, the array x contains the solution matrix X\nto the original system of equations. Note that if equed = 'Y', A and\nB are modified on exit, and the solution to the equilibrated system is\ninv(diag(s))*X.\nab\nOn exit, if fact = 'E'and equed = 'Y', A is overwritten by\ndiag(s)*A*diag(s).\nafb\nIf fact = 'N'or 'E', then afb is an output argument and on exit\nreturns the triangular factor U or L from the Cholesky factorization\nA=UT*U or A=L*LT (real routines), A=UH*U or A=L*LH (complex\nroutines) of the original matrix A (if fact = 'N'), or of the\nequilibrated matrix A (if fact = 'E'). See the description of ab for\nthe form of the equilibrated matrix.\nb\nOverwritten by diag(s)*B, if equed = 'Y'; not changed if equed =\n'N'.\ns\nThis array is an output argument if fact≠'F'. See the description of s\nin Input Arguments section.\nrcond\nAn estimate of the reciprocal condition number of the matrix A after\nequilibration (if done). If rcond is less than the machine precision (in\nparticular, if rcond = 0), the matrix is singular to working precision.\nThis condition is indicated by a return code of info > 0.\nferr\nArray, size at least max(1, nrhs). Contains the estimated forward\nerror bound for each solution vector xj (the j-th column of the\nsolution matrix X). If xtrue is the true solution corresponding to xj,\nferr[j-1] is an estimated upper bound for the magnitude of the\nlargest element in (xj - xtrue) divided by the magnitude of the\nlargest element in xj. The estimate is as reliable as the estimate for\nrcond, and is almost always a slight overestimate of the true error.\nberr\nArray, size at least max(1, nrhs). Contains the component-wise\nrelative backward error for each solution vector xj, that is, the\nsmallest relative change in any element of A or B that makes xj an\nexact solution.\nequed\nIf fact≠'F', then equed is an output argument. It specifies the form\nof equilibration that was done (see the description of equed in Input\nArguments section).\nReturn Values\nThis function returns a value info.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n735\n\n\nIf info = 0, the execution is successful.\nIf info = -i, parameter i had an illegal value.\nIf info = i, and i≤n, the leading minor of order i (and therefore the matrix A itself) is not positive definite,\nso the factorization could not be completed, and the solution and error bounds could not be computed; rcond\n=0 is returned. If info = i, and i = n + 1, then U is nonsingular, but rcond is less than machine precision,\nmeaning that the matrix is singular to working precision. Nevertheless, the solution and error bounds are\ncomputed because there are a number of situations where the computed solution can be more accurate than\nthe value of rcond would suggest.\nSee Also\nMatrix Storage Schemes\n?ptsv\nComputes the solution to the system of linear\nequations with a symmetric or Hermitian positive\ndefinite tridiagonal coefficient matrix A and multiple\nright-hand sides.\nSyntax\nlapack_int LAPACKE_sptsv( int matrix_layout, lapack_int n, lapack_int nrhs, float* d,\nfloat* e, float* b, lapack_int ldb );\nlapack_int LAPACKE_dptsv( int matrix_layout, lapack_int n, lapack_int nrhs, double* d,\ndouble* e, double* b, lapack_int ldb );\nlapack_int LAPACKE_cptsv( int matrix_layout, lapack_int n, lapack_int nrhs, float* d,\nlapack_complex_float* e, lapack_complex_float* b, lapack_int ldb );\nlapack_int LAPACKE_zptsv( int matrix_layout, lapack_int n, lapack_int nrhs, double* d,\nlapack_complex_double* e, lapack_complex_double* b, lapack_int ldb );\nInclude Files\n•\nmkl.h\nDescription\nThe routine solves for X the real or complex system of linear equations A*X = B, where A is an n-by-n\nsymmetric/Hermitian positive-definite tridiagonal matrix, the columns of matrix B are individual right-hand\nsides, and the columns of X are the corresponding solutions.\nA is factored as A = L*D*LT (real flavors) or A = L*D*LH (complex flavors), and the factored form of A is\nthen used to solve the system of equations A*X = B.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\nn\nThe order of matrix A; n≥ 0.\nnrhs\nThe number of right-hand sides, the number of columns in B; nrhs≥\n0.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n736\n\n\nd\nArray, dimension at least max(1, n). Contains the diagonal elements\nof the tridiagonal matrix A.\ne, b\nArrays: e (size n - 1), bof size max(1, ldb*nrhs) for column major\nlayout and max(1, ldb*n) for row major layout. The array e contains\nthe (n - 1) subdiagonal elements of A.\nThe array b contains the matrix B whose columns are the right-hand\nsides for the systems of equations.\nldb\nThe leading dimension of b; ldb≥ max(1, n) for column major layout\nand ldb≥nrhs for row major layout.\nOutput Parameters\nd\nOverwritten by the n diagonal elements of the diagonal matrix D from\nthe L*D*LT (real)/ L*D*LH (complex) factorization of A.\ne\nOverwritten by the (n - 1) subdiagonal elements of the unit\nbidiagonal factor L from the factorization of A.\nb\nOverwritten by the solution matrix X.\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info = -i, parameter i had an illegal value.\nIf info = i, the leading minor of order i (and therefore the matrix A itself) is not positive-definite, and the\nsolution has not been computed. The factorization has not been completed unless i = n.\nSee Also\nMatrix Storage Schemes\n?ptsvx\nUses factorization to compute the solution to the\nsystem of linear equations with a symmetric\n(Hermitian) positive definite tridiagonal coefficient\nmatrix A, and provides error bounds on the solution.\nSyntax\nlapack_int LAPACKE_sptsvx( int matrix_layout, char fact, lapack_int n, lapack_int nrhs,\nconst float* d, const float* e, float* df, float* ef, const float* b, lapack_int ldb,\nfloat* x, lapack_int ldx, float* rcond, float* ferr, float* berr );\nlapack_int LAPACKE_dptsvx( int matrix_layout, char fact, lapack_int n, lapack_int nrhs,\nconst double* d, const double* e, double* df, double* ef, const double* b, lapack_int\nldb, double* x, lapack_int ldx, double* rcond, double* ferr, double* berr );\nlapack_int LAPACKE_cptsvx( int matrix_layout, char fact, lapack_int n, lapack_int nrhs,\nconst float* d, const lapack_complex_float* e, float* df, lapack_complex_float* ef,\nconst lapack_complex_float* b, lapack_int ldb, lapack_complex_float* x, lapack_int ldx,\nfloat* rcond, float* ferr, float* berr );\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n737\n\n\nlapack_int LAPACKE_zptsvx( int matrix_layout, char fact, lapack_int n, lapack_int nrhs,\nconst double* d, const lapack_complex_double* e, double* df, lapack_complex_double* ef,\nconst lapack_complex_double* b, lapack_int ldb, lapack_complex_double* x, lapack_int\nldx, double* rcond, double* ferr, double* berr );\nInclude Files\n•\nmkl.h\nDescription\nThe routine uses the Cholesky factorization A = L*D*LT (real)/A = L*D*LH (complex) to compute the\nsolution to a real or complex system of linear equations A*X = B, where A is a n-by-n symmetric or\nHermitian positive definite tridiagonal matrix, the columns of matrix B are individual right-hand sides, and\nthe columns of X are the corresponding solutions.\nError bounds on the solution and a condition estimate are also provided.\nThe routine ?ptsvx performs the following steps:\n1.\nIf fact = 'N', the matrix A is factored as A = L*D*LT (real flavors)/A = L*D*LH (complex flavors),\nwhere L is a unit lower bidiagonal matrix and D is diagonal. The factorization can also be regarded as\nhaving the form A = UT*D*U (real flavors)/A = UH*D*U (complex flavors).\n2.\nIf the leading i-by-i principal minor is not positive-definite, then the routine returns with info = i.\nOtherwise, the factored form of A is used to estimate the condition number of the matrix A. If the\nreciprocal of the condition number is less than machine precision, info = n+1 is returned as a\nwarning, but the routine still goes on to solve for X and compute error bounds as described below.\n3.\nThe system of equations is solved for X using the factored form of A.\n4.\nIterative refinement is applied to improve the computed solution matrix and calculate error bounds and\nbackward error estimates for it.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\nfact\nMust be 'F' or 'N'.\nSpecifies whether or not the factored form of the matrix A is supplied\non entry.\nIf fact = 'F': on entry, df and ef contain the factored form of A.\nArrays d, e, df, and ef will not be modified.\nIf fact = 'N', the matrix A will be copied to df and ef, and factored.\nn\nThe order of matrix A; n≥ 0.\nnrhs\nThe number of right-hand sides, the number of columns in B; nrhs≥\n0.\nd, df\nArrays: d (size n), df (size n).\nThe array d contains the n diagonal elements of the tridiagonal matrix\nA.\nThe array df is an input argument if fact = 'F' and on entry\ncontains the n diagonal elements of the diagonal matrix D from the\nL*D*LT (real)/ L*D*LH (complex) factorization of A.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n738\n\n\ne,ef,b\nArrays: e (size n -1), ef (size n -1), b, size max(ldb*nrhs) for column\nmajor layout and max(ldb*n) for row major layout. The array e\ncontains the (n - 1) subdiagonal elements of the tridiagonal matrix\nA.\nThe array ef is an input argument if fact = 'F' and on entry\ncontains the (n - 1) subdiagonal elements of the unit bidiagonal\nfactor L from the L*D*LT (real)/ L*D*LH (complex) factorization of A.\nThe array b contains the matrix B whose columns are the right-hand\nsides for the systems of equations.\nldb\nThe leading dimension of b; ldb≥ max(1, n) for column major layout\nand ldb≥nrhs for row major layout.\nldx\nThe leading dimension of x; ldx≥ max(1, n) for column major layout\nand ldx≥nrhs for row major layout.\nOutput Parameters\nx\nArray, size max(1, ldx*nrhs) for column major layout and max(1,\nldx*n) for row major layout.\nIf info = 0 or info = n+1, the array x contains the solution matrix\nX to the system of equations.\ndf, ef\nThese arrays are output arguments if fact = 'N'. See the\ndescription of df, ef in Input Arguments section.\nrcond\nAn estimate of the reciprocal condition number of the matrix A after\nequilibration (if done). If rcond is less than the machine precision (in\nparticular, if rcond = 0), the matrix is singular to working precision.\nThis condition is indicated by a return code of info > 0.\nferr\nArray, size at least max(1, nrhs). Contains the estimated forward\nerror bound for each solution vector xj (the j-th column of the\nsolution matrix X). If xtrue is the true solution corresponding to xj,\nferrj is an estimated upper bound for the magnitude of the largest\nelement in (xj - xtrue) divided by the magnitude of the largest\nelement in xj. The estimate is as reliable as the estimate for rcond,\nand is almost always a slight overestimate of the true error.\nberr\nArray, size at least max(1, nrhs). Contains the component-wise\nrelative backward error for each solution vector xj, that is, the\nsmallest relative change in any element of A or B that makes xj an\nexact solution.\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info = -i, parameter i had an illegal value.\nIf info = i, and i≤n, the leading minor of order i (and therefore the matrix A itself) is not positive-definite,\nso the factorization could not be completed, and the solution and error bounds could not be computed; rcond\n=0 is returned.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n739\n\n\nIf info = i, and i = n + 1, then U is nonsingular, but rcond is less than machine precision, meaning that the\nmatrix is singular to working precision. Nevertheless, the solution and error bounds are computed because\nthere are a number of situations where the computed solution can be more accurate than the value of rcond\nwould suggest.\nSee Also\nMatrix Storage Schemes\n?sysv\nComputes the solution to the system of linear\nequations with a real or complex symmetric coefficient\nmatrix A and multiple right-hand sides.\nSyntax\nlapack_int LAPACKE_ssysv (int matrix_layout , char uplo , lapack_int n , lapack_int\nnrhs , float * a , lapack_int lda , lapack_int * ipiv , float * b , lapack_int ldb );\nlapack_int LAPACKE_dsysv (int matrix_layout , char uplo , lapack_int n , lapack_int\nnrhs , double * a , lapack_int lda , lapack_int * ipiv , double * b , lapack_int ldb );\nlapack_int LAPACKE_csysv (int matrix_layout , char uplo , lapack_int n , lapack_int\nnrhs , lapack_complex_float * a , lapack_int lda , lapack_int * ipiv ,\nlapack_complex_float * b , lapack_int ldb );\nlapack_int LAPACKE_zsysv (int matrix_layout , char uplo , lapack_int n , lapack_int\nnrhs , lapack_complex_double * a , lapack_int lda , lapack_int * ipiv ,\nlapack_complex_double * b , lapack_int ldb );\nInclude Files\n•\nmkl.h\nDescription\nThe routine solves for X the real or complex system of linear equations A*X = B, where A is an n-by-n\nsymmetric matrix, the columns of matrix B are individual right-hand sides, and the columns of X are the\ncorresponding solutions.\nThe diagonal pivoting method is used to factor A as A = U*D*UT or A = L*D*LT, where U (or L) is a product\nof permutation and unit upper (lower) triangular matrices, and D is symmetric and block diagonal with 1-\nby-1 and 2-by-2 diagonal blocks.\nThe factored form of A is then used to solve the system of equations A*X = B.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\nuplo\nMust be 'U' or 'L'.\nIndicates whether the upper or lower triangular part of A is stored:\nIf uplo = 'U', the upper triangle of A is stored.\nIf uplo = 'L', the lower triangle of A is stored.\nn\nThe order of matrix A; n≥ 0.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n740\n\n\nnrhs\nThe number of right-hand sides; the number of columns in B; nrhs≥\n0.\na, b\nArrays: a(size max(1, lda*n)), bof size max(1, ldb*nrhs) for column\nmajor layout and max(1, ldb*n) for row major layout.\nThe array a contains the upper or the lower triangular part of the\nsymmetric matrix A (see uplo).\nThe array b contains the matrix B whose columns are the right-hand\nsides for the systems of equations.\nlda\nThe leading dimension of a; lda≥ max(1, n).\nldb\nThe leading dimension of b; ldb≥ max(1, n) for column major layout\nand ldb≥nrhs for row major layout.\nOutput Parameters\na\nIf info = 0, a is overwritten by the block-diagonal matrix D and the\nmultipliers used to obtain the factor U (or L) from the factorization of\nA as computed by ?sytrf.\nb\nIf info = 0, b is overwritten by the solution matrix X.\nipiv\nArray, size at least max(1, n). Contains details of the interchanges\nand the block structure of D, as determined by ?sytrf.\nIf ipiv[i-1] = k >0, then dii is a 1-by-1 diagonal block, and the i-\nth row and column of A was interchanged with the k-th row and\ncolumn.\nIf uplo = 'U' and ipiv[i] = ipiv[i-1] = -m < 0, then D has a 2-by-2\nblock in rows/columns i and i+1, and (i)-th row and column of A was\ninterchanged with the m-th row and column.\nIf uplo = 'L'and ipiv[i] = ipiv[i-1] = -m < 0, then D has a 2-by-2\nblock in rows/columns i and i+1, and (i+1)-th row and column of A\nwas interchanged with the m-th row and column.\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info = -i, parameter i had an illegal value.\nIf info = i, dii is 0. The factorization has been completed, but D is exactly singular, so the solution could\nnot be computed.\nSee Also\nMatrix Storage Schemes\n?sysv_aa\nComputes the solution to a system of linear equations\nA * X = B for symmetric matrices.\nlapack_int LAPACKE_ssysv_aa (int matrix_layout, char uplo, lapack_int n, lapack_int\nnrhs, float * A, lapack_int lda, lapack_int * ipiv, float * B, lapack_int ldb);\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n741\n\n\nlapack_int LAPACKE_dsysv_aa (int matrix_layout, char uplo, lapack_int n, lapack_int\nnrhs, double * A, lapack_int lda, lapack_int * ipiv, double * B, lapack_int ldb);\nlapack_int LAPACKE_csysv_aa (int matrix_layout, char uplo, lapack_int n, lapack_int\nnrhs, lapack_complex_float * A, lapack_int lda, lapack_int * ipiv, lapack_complex_float\n* B, lapack_int ldb);\nlapack_int LAPACKE_zsysv_aa (int matrix_layout, char uplo, lapack_int n, lapack_int\nnrhs, lapack_complex_double * A, lapack_int lda, lapack_int * ipiv,\nlapack_complex_double * B, lapack_int ldb);\nDescription\nThe ?sysv routine computes the solution to a complex system of linear equations A * X = B, where A is an\nn-by-n symmetric matrix and X and B are n-by-nrhs matrices.\nAasen's algorithm is used to factor A as A = U * T * UT, if uplo = 'U', or A = L * T * LT, if uplo = 'L',\nwhere U (or L) is a product of permutation and unit upper (lower) triangular matrices, and T is symmetric tri-\ndiagonal. The factored form of A is then used to solve the system of equations A * X= B.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nuplo\n•\n= 'U': The upper triangle of A is stored.\n•\n= 'L': The lower triangle of A is stored.\nn\nThe number of linear equations; that is, the order of the matrix A. n ≥ 0.\nnrhs\nThe number of right-hand sides; that is, the number of columns of the\nmatrix B. nrhs ≥ 0.\nA\nArray of size max(1, lda*n). On entry, the symmetric matrix A. If uplo =\n'U', the leading n-by-n upper triangular part of A contains the upper\ntriangular part of the matrix A, and the strictly lower triangular part of A is\nnot referenced. If uplo = 'L', the leading n-by-n lower triangular part of A\ncontains the lower triangular part of the matrix A, and the strictly upper\ntriangular part of A is not referenced.\nlda\nThe leading dimension of the array A.\nB\nArray of size max(1, ldb*nrhs) for column-major layout and max(1,\nldb*n) for row-major layout. On entry, the n-by-nrhs right-hand side\nmatrix B.\nldb\nThe leading dimension of the array B. ldb ≥ max(1, n) for column-major\nlayout and ldb ≥ nrhs for row-major layout.\nOutput Parameters\nA\nOn exit, if info = 0, the tridiagonal matrix T and the multipliers used to\nobtain the factor U or L from the factorization A = U*T*UT or A = L*T*LT as\ncomputed by ?sytrf.\nipiv\nArray of size n. On exit, it contains the details of the interchanges; that is,\nthe row and column k of A were interchanged with the row and column\nipiv(k).\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n742\n\n\nB\nOn exit, if info = 0, the n-by-nrhs solution matrix X.\nReturn Values\nThis function returns a value info.\n= 0: Successful exit.\n< 0: If info = -i, the ith argument had an illegal value.\n> 0: If info = i, D(i,i) is exactly zero. The factorization has been completed, but the block diagonal matrix D\nis exactly singular, so the solution could not be computed.\n?sysv_rook\nComputes the solution to the system of linear\nequations with a real or complex symmetric coefficient\nmatrix A and multiple right-hand sides.\nSyntax\nlapack_int LAPACKE_ssysv_rook (int matrix_layout , char uplo , lapack_int n ,\nlapack_int nrhs , float * a , lapack_int lda , lapack_int * ipiv , float * b ,\nlapack_int ldb );\nlapack_int LAPACKE_dsysv_rook (int matrix_layout , char uplo , lapack_int n ,\nlapack_int nrhs , double * a , lapack_int lda , lapack_int * ipiv , double * b ,\nlapack_int ldb );\nlapack_int LAPACKE_csysv_rook (int matrix_layout , char uplo , lapack_int n ,\nlapack_int nrhs , lapack_complex_float * a , lapack_int lda , lapack_int * ipiv ,\nlapack_complex_float * b , lapack_int ldb );\nlapack_int LAPACKE_zsysv_rook (int matrix_layout , char uplo , lapack_int n ,\nlapack_int nrhs , lapack_complex_double * a , lapack_int lda , lapack_int * ipiv ,\nlapack_complex_double * b , lapack_int ldb );\nInclude Files\n•\nmkl.h\nDescription\nThe routine solves for X the real or complex system of linear equations A*X = B, where A is an n-by-n\nsymmetric matrix, the columns of matrix B are individual right-hand sides, and the columns of X are the\ncorresponding solutions.\nThe diagonal pivoting method is used to factor A as A = U*D*UT or A = L*D*LT, where U (or L) is a product\nof permutation and unit upper (lower) triangular matrices, and D is symmetric and block diagonal with 1-\nby-1 and 2-by-2 diagonal blocks.\nThe ?sysv_rook routine is called to compute the factorization of a complex symmetric matrix A using the\nbounded Bunch-Kaufman (\"rook\") diagonal pivoting method.\nThe factored form of A is then used to solve the system of equations A*X = B.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n743\n\n\nuplo\nMust be 'U' or 'L'.\nIndicates whether the upper or lower triangular part of A is stored:\nIf uplo = 'U', the upper triangle of A is stored.\nIf uplo = 'L', the lower triangle of A is stored.\nn\nThe order of matrix A; n≥ 0.\nnrhs\nThe number of right-hand sides; the number of columns in B; nrhs≥\n0.\na, b\nArrays: a(size max(1, lda*n)), bof size max(1, ldb*nrhs) for column\nmajor layout and max(1, ldb*n) for row major layout.\nThe array a contains the upper or the lower triangular part of the\nsymmetric matrix A (see uplo). The second dimension of a must be at\nleast max(1, n).\nThe array b contains the matrix B whose columns are the right-hand\nsides for the systems of equations. The second dimension of b must\nbe at least max(1,nrhs).\nlda\nThe leading dimension of a; lda≥ max(1, n).\nldb\nThe leading dimension of b; ldb≥ max(1, n) for column major layout\nand ldb≥nrhs) for row major layout.\nOutput Parameters\na\nIf info = 0, a is overwritten by the block-diagonal matrix D and the\nmultipliers used to obtain the factor U (or L) from the factorization of\nA.\nb\nIf info = 0, b is overwritten by the solution matrix X.\nipiv\nArray, size at least max(1, n). Contains details of the interchanges\nand the block structure of D.\nIf ipiv[k - 1] > 0, then rows and columns k and ipiv[k - 1] were\ninterchanged and Dk, k is a 1-by-1 diagonal block.\nIf uplo = 'U' and ipiv[k - 1] < 0 and ipiv[k - 2] < 0, then\nrows and columns k and -ipiv[k - 1] were interchanged, rows and\ncolumns k - 1 and -ipiv[k - 2] were interchanged, and Dk-1:k, k-1:k is\na 2-by-2 diagonal block.\nIf uplo = 'L' and ipiv[k - 1] < 0 and ipiv[k] < 0, then rows\nand columns k and -ipiv[k - 1] were interchanged, rows and columns\nk + 1 and -ipiv[k ] were interchanged, and Dk:k+1, k:k+1 is a 2-by-2\ndiagonal block.\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n744\n\n\nIf info = i, dii is 0. The factorization has been completed, but D is exactly singular, so the solution could\nnot be computed.\nSee Also\nMatrix Storage Schemes\n?sysv_rk\nComputes the solution to system of linear equations A\n* X = B for SY matrices.\nlapack_int LAPACKE_ssysv_rk (int matrix_layout, char uplo, lapack_int n, lapack_int\nnrhs, float * A, lapack_int lda, float * e, lapack_int * ipiv, float * B, lapack_int\nldb);\nlapack_int LAPACKE_dsysv_rk (int matrix_layout, char uplo, lapack_int n, lapack_int\nnrhs, double * A, lapack_int lda, double * e, lapack_int * ipiv, double * B, lapack_int\nldb);\nlapack_int LAPACKE_csysv_rk (int matrix_layout, char uplo, lapack_int n, lapack_int\nnrhs, lapack_complex_float * A, lapack_int lda, lapack_complex_float * e, lapack_int *\nipiv, lapack_complex_float * B, lapack_int ldb);\nlapack_int LAPACKE_zsysv_rk (int matrix_layout, char uplo, lapack_int n, lapack_int\nnrhs, lapack_complex_double * A, lapack_int lda, lapack_complex_double * e, lapack_int\n* ipiv, lapack_complex_double * B, lapack_int ldb);\nDescription\n?sysv_rk computes the solution to a real or complex system of linear equations A * X = B, where A is an n-\nby-n symmetric matrix and X and B are n-by-nrhs matrices.\nThe bounded Bunch-Kaufman (rook) diagonal pivoting method is used to factor A as A= P*U*D*(UT)*(PT), if\nuplo = 'U', or A= P*L*D*(LT)*(PT), if uplo = 'L', where U (or L) is unit upper (or lower) triangular matrix,\nUT (or LT) is the transpose of U (or L), P is a permutation matrix, PT is the transpose of P, and D is symmetric\nand block diagonal with 1-by-1 and 2-by-2 diagonal blocks.\n?sytrf_rk is called to compute the factorization of a real or complex symmetric matrix. The factored form of\nA is then used to solve the system of equations A * X = B by calling BLAS3 routine ?sytrs_3.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nuplo\nSpecifies whether the upper or lower triangular part of the symmetric\nmatrix A is stored:\n•\n= 'U': The upper triangle of A is stored.\n•\n= 'L': The lower triangle of A is stored.\nn\nThe number of linear equations; that is, the order of the matrix A. n ≥ 0.\nnrhs\nThe number of right-hand sides; that is, the number of columns of the\nmatrix B. nrhs ≥ 0.\nA\nArray of size max(1, lda*n). On entry, the symmetric matrix A. If uplo =\n'U', the leading n-by-n upper triangular part of A contains the upper\ntriangular part of the matrix A, and the strictly lower triangular part of A is\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n745\n\n\nnot referenced. If uplo = 'L', the leading n-by-n lower triangular part of A\ncontains the lower triangular part of the matrix A, and the strictly upper\ntriangular part of A is not referenced.\nlda\nThe leading dimension of the array A.\nB\nArray of size max(1, ldb*nrhs). On entry, the n-by-nrhs right-hand side\nmatrix B.\nldb\nThe leading dimension of the array B. ldb ≥ max(1, n) for column-major\nlayout and ldb ≥ nrhs for row-major layout.\nOutput Parameters\nA\nOn exit, if info = 0, the diagonal of the block diagonal matrix D and factors\nU or L as computed by ?sytrf_rk:\n•\nOnly diagonal elements of the symmetric block diagonal matrix D on the\ndiagonal of A; that is, D(k,k) = A(k,k). Superdiagonal (or subdiagonal)\nelements of D are stored on exit in array e.\n•\nIf uplo = 'U', factor U in the superdiagonal part of A. If uplo = 'L',\nfactor L in the subdiagonal part of A. For more information, see the\ndescription of the ?sytrf_rk routine.\ne\nArray of size n. On exit, contains the output computed by the factorization\nroutine ?sytrf_rk; that is, the superdiagonal (or subdiagonal) elements of\nthe symmetric block diagonal matrix D with 1-by-1 or 2-by-2 diagonal\nblocks. If uplo = 'U', e(i) = D(i-1,i), i=1:N-1, and e(1) is set to 0. If uplo\n= 'L', e(i) = D(i+1,i), i=1:N-1, and e(n) is set to 0.\nNOTE For 1-by-1 diagonal block D(k), where 1 ≤ k ≤ n, the\nelement e(k) is set to 0 in both the uplo = 'U' and uplo = 'L'\ncases. For more information, see the description of\nthe?sytrf_rk routine.\nipiv\nArray of size n. Details of the interchanges and the block structure of D, as\ndetermined by ?sytrf_rk. For more information, see the description of\nthe ?sytrf_rk routine.\nB\nOn exit, if info = 0, the n-by-nrhs solution matrix X.\nReturn Values\nThis function returns a value info.\n= 0: Successful exit.\n< 0: If info = -k, the kth argument had an illegal value.\n> 0: If info = k, the matrix A is singular. If uplo = 'U', column k in the upper triangular part of A contains\nall zeros. If uplo = 'L', column k in the lower triangular part of A contains all zeros. Therefore D(k,k) is\nexactly zero, and superdiagonal elements of column k of U (or subdiagonal elements of column k of L) are all\nzeros. The factorization has been completed, but the block diagonal matrix D is exactly singular, and division\nby zero will occur if it is used to solve a system of equations.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n746\n\n\n?sysvx\nUses the diagonal pivoting factorization to compute\nthe solution to the system of linear equations with a\nreal or complex symmetric coefficient matrix A, and\nprovides error bounds on the solution.\nSyntax\nlapack_int LAPACKE_ssysvx( int matrix_layout, char fact, char uplo, lapack_int n,\nlapack_int nrhs, const float* a, lapack_int lda, float* af, lapack_int ldaf,\nlapack_int* ipiv, const float* b, lapack_int ldb, float* x, lapack_int ldx, float*\nrcond, float* ferr, float* berr );\nlapack_int LAPACKE_dsysvx( int matrix_layout, char fact, char uplo, lapack_int n,\nlapack_int nrhs, const double* a, lapack_int lda, double* af, lapack_int ldaf,\nlapack_int* ipiv, const double* b, lapack_int ldb, double* x, lapack_int ldx, double*\nrcond, double* ferr, double* berr );\nlapack_int LAPACKE_csysvx( int matrix_layout, char fact, char uplo, lapack_int n,\nlapack_int nrhs, const lapack_complex_float* a, lapack_int lda, lapack_complex_float*\naf, lapack_int ldaf, lapack_int* ipiv, const lapack_complex_float* b, lapack_int ldb,\nlapack_complex_float* x, lapack_int ldx, float* rcond, float* ferr, float* berr );\nlapack_int LAPACKE_zsysvx( int matrix_layout, char fact, char uplo, lapack_int n,\nlapack_int nrhs, const lapack_complex_double* a, lapack_int lda, lapack_complex_double*\naf, lapack_int ldaf, lapack_int* ipiv, const lapack_complex_double* b, lapack_int ldb,\nlapack_complex_double* x, lapack_int ldx, double* rcond, double* ferr, double* berr );\nInclude Files\n•\nmkl.h\nDescription\nThe routine uses the diagonal pivoting factorization to compute the solution to a real or complex system of\nlinear equations A*X = B, where A is a n-by-n symmetric matrix, the columns of matrix B are individual\nright-hand sides, and the columns of X are the corresponding solutions.\nError bounds on the solution and a condition estimate are also provided.\nThe routine ?sysvx performs the following steps:\n1.\nIf fact = 'N', the diagonal pivoting method is used to factor the matrix A. The form of the\nfactorization is A = U*D*UT or A = L*D*LT, where U (or L) is a product of permutation and unit upper\n(lower) triangular matrices, and D is symmetric and block diagonal with 1-by-1 and 2-by-2 diagonal\nblocks.\n2.\nIf some di,i= 0, so that D is exactly singular, then the routine returns with info = i. Otherwise, the\nfactored form of A is used to estimate the condition number of the matrix A. If the reciprocal of the\ncondition number is less than machine precision, info = n+1 is returned as a warning, but the routine\nstill goes on to solve for X and compute error bounds as described below.\n3.\nThe system of equations is solved for X using the factored form of A.\n4.\nIterative refinement is applied to improve the computed solution matrix and calculate error bounds and\nbackward error estimates for it.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n747\n\n\nfact\nMust be 'F' or 'N'.\nSpecifies whether or not the factored form of the matrix A has been\nsupplied on entry.\nIf fact = 'F': on entry, af and ipiv contain the factored form of A.\nArrays a, af, and ipiv will not be modified.\nIf fact = 'N', the matrix A will be copied to af and factored.\nuplo\nMust be 'U' or 'L'.\nIndicates whether the upper or lower triangular part of A is stored:\nIf uplo = 'U', the upper triangle of A is stored.\nIf uplo = 'L', the lower triangle of A is stored.\nn\nThe order of matrix A; n≥ 0.\nnrhs\nThe number of right-hand sides, the number of columns in B; nrhs≥\n0.\na, af, b\nArrays: a(size max(1, lda*n)), af(size max(1, ldaf*n)), bof size\nmax(1, ldb*nrhs) for column major layout and max(1, ldb*n) for\nrow major layout .\nThe array a contains the upper or the lower triangular part of the\nsymmetric matrix A (see uplo).\nThe array af is an input argument if fact = 'F'. It contains the block\ndiagonal matrix D and the multipliers used to obtain the factor U or L\nfrom the factorization A = U*D*UT orA = L*D*LT as computed\nby ?sytrf.\nThe array b contains the matrix B whose columns are the right-hand\nsides for the systems of equations.\nlda\nThe leading dimension of a; lda≥ max(1, n).\nldaf\nThe leading dimension of af; ldaf≥ max(1, n).\nldb\nThe leading dimension of b; ldb≥ max(1, n) for column major layout\nand ldb≥nrhs for row major layout.\nipiv\nArray, size at least max(1, n). The array ipiv is an input argument if\nfact = 'F'. It contains details of the interchanges and the block\nstructure of D, as determined by ?sytrf.\nIf ipiv[i-1] = k > 0, then dii is a 1-by-1 diagonal block, and the\ni-th row and column of A was interchanged with the k-th row and\ncolumn.\nIf uplo = 'U'and ipiv[i] = ipiv[i-1] = -m < 0, then D has a\n2-by-2 block in rows/columns i and i+1, and (i)-th row and column of\nA was interchanged with the m-th row and column.\nIf uplo = 'L'and ipiv[i] = ipiv[i-1] = -m < 0, then D has a\n2-by-2 block in rows/columns i and i+1, and (i+1)-th row and column\nof A was interchanged with the m-th row and column.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n748\n\n\nldx\nThe leading dimension of the output array x; ldx≥ max(1, n) for\ncolumn major layout and ldx≥nrhs for row major layout.\nOutput Parameters\nx\nArray, size max(1, ldx*nrhs) for column major layout and max(1,\nldx*n) for row major layout.\nIf info = 0 or info = n+1, the array x contains the solution matrix\nX to the system of equations.\naf, ipiv\nThese arrays are output arguments if fact = 'N'.\nSee the description of af, ipiv in Input Arguments section.\nrcond\nAn estimate of the reciprocal condition number of the matrix A. If\nrcond is less than the machine precision (in particular, if rcond = 0),\nthe matrix is singular to working precision. This condition is indicated\nby a return code of info > 0.\nferr\nArray, size at least max(1, nrhs). Contains the estimated forward\nerror bound for each solution vector xj (the j-th column of the solution\nmatrix X). If xtrue is the true solution corresponding to xj, ferr[j-1]\nis an estimated upper bound for the magnitude of the largest element\nin (xj - xtrue) divided by the magnitude of the largest element in xj.\nThe estimate is as reliable as the estimate for rcond, and is almost\nalways a slight overestimate of the true error.\nberr\nArray, size at least max(1, nrhs). Contains the component-wise\nrelative backward error for each solution vector xj, that is, the\nsmallest relative change in any element of A or B that makes xj an\nexact solution.\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info = -i, parameter i had an illegal value.\nIf info = i, and i≤n, then dii is exactly zero. The factorization has been completed, but the block diagonal\nmatrix D is exactly singular, so the solution and error bounds could not be computed; rcond = 0 is returned.\nIf info = i, and i = n + 1, then D is nonsingular, but rcond is less than machine precision, meaning that the\nmatrix is singular to working precision. Nevertheless, the solution and error bounds are computed because\nthere are a number of situations where the computed solution can be more accurate than the value of rcond\nwould suggest.\nSee Also\nMatrix Storage Schemes\n?sysvxx\nUses extra precise iterative refinement to compute the\nsolution to the system of linear equations with a\nsymmetric indefinite coefficient matrix A applying the\ndiagonal pivoting factorization.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n749\n\n\nSyntax\nlapack_int LAPACKE_ssysvxx( int matrix_layout, char fact, char uplo, lapack_int n,\nlapack_int nrhs, float* a, lapack_int lda, float* af, lapack_int ldaf, lapack_int* ipiv,\nchar* equed, float* s, float* b, lapack_int ldb, float* x, lapack_int ldx, float* rcond,\nfloat* rpvgrw, float* berr, lapack_int n_err_bnds, float* err_bnds_norm, float*\nerr_bnds_comp, lapack_int nparams, const float* params );\nlapack_int LAPACKE_dsysvxx( int matrix_layout, char fact, char uplo, lapack_int n,\nlapack_int nrhs, double* a, lapack_int lda, double* af, lapack_int ldaf, lapack_int*\nipiv, char* equed, double* s, double* b, lapack_int ldb, double* x, lapack_int ldx,\ndouble* rcond, double* rpvgrw, double* berr, lapack_int n_err_bnds, double*\nerr_bnds_norm, double* err_bnds_comp, lapack_int nparams, const double* params );\nlapack_int LAPACKE_csysvxx( int matrix_layout, char fact, char uplo, lapack_int n,\nlapack_int nrhs, lapack_complex_float* a, lapack_int lda, lapack_complex_float* af,\nlapack_int ldaf, lapack_int* ipiv, char* equed, float* s, lapack_complex_float* b,\nlapack_int ldb, lapack_complex_float* x, lapack_int ldx, float* rcond, float* rpvgrw,\nfloat* berr, lapack_int n_err_bnds, float* err_bnds_norm, float* err_bnds_comp,\nlapack_int nparams, const float* params );\nlapack_int LAPACKE_zsysvxx( int matrix_layout, char fact, char uplo, lapack_int n,\nlapack_int nrhs, lapack_complex_double* a, lapack_int lda, lapack_complex_double* af,\nlapack_int ldaf, lapack_int* ipiv, char* equed, double* s, lapack_complex_double* b,\nlapack_int ldb, lapack_complex_double* x, lapack_int ldx, double* rcond, double*\nrpvgrw, double* berr, lapack_int n_err_bnds, double* err_bnds_norm, double*\nerr_bnds_comp, lapack_int nparams, const double* params );\nInclude Files\n•\nmkl.h\nDescription\nThe routine uses the diagonal pivoting factorization to compute the solution to a real or complex system of\nlinear equations A*X = B, where A is an n-by-n real symmetric/Hermitian matrix, the columns of matrix B\nare individual right-hand sides, and the columns of X are the corresponding solutions.\nBoth normwise and maximum componentwise error bounds are also provided on request. The routine returns\na solution with a small guaranteed error (O(eps), where eps is the working machine precision) unless the\nmatrix is very ill-conditioned, in which case a warning is returned. Relevant condition numbers are also\ncalculated and returned.\nThe routine accepts user-provided factorizations and equilibration factors; see definitions of the fact and\nequed options. Solving with refinement and using a factorization from a previous call of the routine also\nproduces a solution with O(eps) errors or warnings but that may not be true for general user-provided\nfactorizations and equilibration factors if they differ from what the routine would itself produce.\nThe routine ?sysvxx performs the following steps:\n1.\nIf fact = 'E', scaling factors are computed to equilibrate the system:\ndiag(s)*A*diag(s) *inv(diag(s))*X = diag(s)*B\nWhether or not the system will be equilibrated depends on the scaling of the matrix A, but if\nequilibration is used, A is overwritten by diag(s)*A*diag(s) and B by diag(s)*B.\n2.\nIf fact = 'N' or 'E', the LU decomposition is used to factor the matrix A (after equilibration if fact\n= 'E') as\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n750\n\n\nA = U*D*UT, if uplo = 'U',\nor A = L*D*LT, if uplo = 'L',\nwhere U or L is a product of permutation and unit upper (lower) triangular matrices, and D is a\nsymmetric and block diagonal with 1-by-1 and 2-by-2 diagonal blocks.\n3.\nIf some D(i,i)=0, so that D is exactly singular, the routine returns with info = i. Otherwise, the\nfactored form of A is used to estimate the condition number of the matrix A (see the rcond parameter).\nIf the reciprocal of the condition number is less than machine precision, the routine still goes on to\nsolve for X and compute error bounds.\n4.\nThe system of equations is solved for X using the factored form of A.\n5.\nBy default, unless params[0] is set to zero, the routine applies iterative refinement to get a small error\nand error bounds. Refinement calculates the residual to at least twice the working precision.\n6.\nIf equilibration was used, the matrix X is premultiplied by diag(r) so that it solves the original system\nbefore equilibration.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\nfact\nMust be 'F', 'N', or 'E'.\nSpecifies whether or not the factored form of the matrix A is supplied\non entry, and if not, whether the matrix A should be equilibrated\nbefore it is factored.\nIf fact = 'F', on entry, af and ipiv contain the factored form of A.\nIf equed is not 'N', the matrix A has been equilibrated with scaling\nfactors given by s. Parameters a, af, and ipiv are not modified.\nIf fact = 'N', the matrix A will be copied to af and factored.\nIf fact = 'E', the matrix A will be equilibrated, if necessary, copied\nto af and factored.\nuplo\nMust be 'U' or 'L'.\nIndicates whether the upper or lower triangular part of A is stored:\nIf uplo = 'U', the upper triangle of A is stored.\nIf uplo = 'L', the lower triangle of A is stored.\nn\nThe number of linear equations; the order of the matrix A; n≥ 0.\nnrhs\nThe number of right-hand sides; the number of columns of the\nmatrices B and X; nrhs≥ 0.\na, af, b\nArrays: a(size max(1, lda*n)), af(size max(1, ldaf*n)), b, size\nmax(ldb*nrhs) for column major layout and max(ldb*n) for row\nmajor layout,.\nThe array a contains the symmetric matrix A as specified by uplo. If\nuplo = 'U', the leading n-by-n upper triangular part of a contains the\nupper triangular part of the matrix A and the strictly lower triangular\npart of a is not referenced. If uplo = 'L', the leading n-by-n lower\ntriangular part of a contains the lower triangular part of the matrix A\nand the strictly upper triangular part of a is not referenced.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n751\n\n\nThe array af is an input argument if fact = 'F'. It contains the\nblock diagonal matrix D and the multipliers used to obtain the factor U\nand L from the factorization A = U*D*UT or A = L*D*LT as computed\nby ?sytrf.\nThe array b contains the matrix B whose columns are the right-hand\nsides for the systems of equations.\nlda\nThe leading dimension of the array a; lda≥ max(1,n).\nldaf\nThe leading dimension of the array af; ldaf≥ max(1,n).\nipiv\nArray, size at least max(1, n). The array ipiv is an input argument if\nfact = 'F'. It contains details of the interchanges and the block\nstructure of D as determined by ?sytrf. If ipiv[k-1] > 0, rows and\ncolumns k and ipiv[k-1] were interchanged and D(k,k) is a 1-by-1\ndiagonal block.\nIf uplo = 'U' and ipiv[i] = ipiv[i - 1] = m < 0, D has a 2-\nby-2 diagonal block in rows and columns i and i + 1, and the i-th row\nand column of A were interchanged with the m-th row and column.\nIf uplo = 'L' and ipiv[i] = ipiv[i - 1] = m < 0, D has a 2-\nby-2 diagonal block in rows and columns i and i + 1, and the (i + 1)-st\nrow and column of A were interchanged with the m-th row and\ncolumn.\nequed\nMust be 'N' or 'Y'.\nequed is an input argument if fact = 'F'. It specifies the form of\nequilibration that was done:\nIf equed = 'N', no equilibration was done (always true if fact =\n'N').\nif equed = 'Y', both row and column equilibration was done, that is,\nA has been replaced by diag(s)*A*diag(s).\ns\nArray, size (n). The array s contains the scale factors for A. If equed\n= 'Y', A is multiplied on the left and right by diag(s).\nThis array is an input argument if fact = 'F' only; otherwise it is an\noutput argument.\nIf fact = 'F' and equed = 'Y', each element of s must be positive.\nEach element of s should be a power of the radix to ensure a reliable\nsolution and error estimates. Scaling by powers of the radix does not\ncause rounding errors unless the result underflows or overflows.\nRounding errors during scaling lead to refining with a matrix that is\nnot equivalent to the input matrix, producing error estimates that may\nnot be reliable.\nldb\nThe leading dimension of the array b; ldb≥ max(1, n) for column\nmajor layout and ldb≥nrhs for row major layout.\nldx\nThe leading dimension of the output array x; ldx≥ max(1, n) for\ncolumn major layout and ldx≥nrhs for row major layout.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n752\n\n\nn_err_bnds\nNumber of error bounds to return for each right hand side and each\ntype (normwise or componentwise). See err_bnds_norm and\nerr_bnds_comp descriptions in the Output Arguments section below.\nnparams\nSpecifies the number of parameters set in params. If ≤ 0, the params\narray is never referenced and default values are used.\nparams\nArray, size max(1,nparams). Specifies algorithm parameters. If an\nentry is less than 0.0, that entry is filled with the default value used\nfor that parameter. Only positions up to nparams are accessed;\ndefaults are used for higher-numbered parameters. If defaults are\nacceptable, you can pass nparams = 0, which prevents the source\ncode from accessing the params argument.\nparams[0] : Whether to perform iterative refinement or not. Default:\n1.0 (for single precision flavors), 1.0D+0 (for double precision\nflavors).\n=0.0\nNo refinement is performed and no error\nbounds are computed.\n=1.0\nUse the extra-precise refinement algorithm.\n(Other values are reserved for future use.)\nparams[1] : Maximum number of residual computations allowed for\nrefinement.\nDefault\n10.0\nAggressive\nSet to 100.0 to permit convergence using\napproximate factorizations or factorizations\nother than LU. If the factorization uses a\ntechnique other than Gaussian elimination,\nthe guarantees in err_bnds_norm and\nerr_bnds_comp may no longer be\ntrustworthy.\nparams[2] : Flag determining if the code will attempt to find a\nsolution with a small componentwise relative error in the double-\nprecision algorithm. Positive is true, 0.0 is false. Default: 1.0 (attempt\ncomponentwise convergence).\nOutput Parameters\nx\nArray, size max(1, ldx*nrhs) for column major layout and max(1, ldx*n)\nfor row major layout).\nIf info = 0, the array x contains the solution n-by-nrhs matrix X to the\noriginal system of equations. Note that A and B are modified on exit if\nequed≠'N', and the solution to the equilibrated system is:\ninv(diag(s))*X.\na\nIf fact = 'E' and equed = 'Y', overwritten by diag(s)*A*diag(s).\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n753\n\n\naf\nIf fact = 'N', af is an output argument and on exit returns the block\ndiagonal matrix D and the multipliers used to obtain the factor U or L from\nthe factorization A = U*D*UT or A = L*D*LT.\nb\nIf equed = 'N', B is not modified.\nIf equed = 'Y', B is overwritten by diag(s)*B.\ns\nThis array is an output argument if fact≠'F'. Each element of this array is\na power of the radix. See the description of s in Input Arguments section.\nrcond\nReciprocal scaled condition number. An estimate of the reciprocal Skeel\ncondition number of the matrix A after equilibration (if done). If rcond is\nless than the machine precision, in particular, if rcond = 0, the matrix is\nsingular to working precision. Note that the error may still be small even if\nthis number is very small and the matrix appears ill-conditioned.\nrpvgrw\nContains the reciprocal pivot growth factor:\nIf this is much less than 1, the stability of the LU factorization of the\n(equlibrated) matrix A could be poor. This also means that the solution X,\nestimated condition numbers, and error bounds could be unreliable. If\nfactorization fails with 0 < info≤n, this parameter contains the reciprocal\npivot growth factor for the leading info columns of A.\nberr\nArray, size at least max(1, nrhs). Contains the componentwise relative\nbackward error for each solution vector xj, that is, the smallest relative\nchange in any element of A or B that makes xj an exact solution.\nerr_bnds_norm\nArray of size nrhs*n_err_bnds. For each right-hand side, contains\ninformation about various error bounds and condition numbers\ncorresponding to the normwise relative error, which is defined as follows:\nNormwise relative error in the i-th solution vector\nThe array is indexed by the type of error information as described below. Up\nto three pieces of information are returned.\nerr=1\n\"Trust/don't trust\" boolean. Trust the answer if\nthe reciprocal condition number is less than the\nthreshold sqrt(n)*slamch(ε) for single\nprecision flavors and sqrt(n)*dlamch(ε) for\ndouble precision flavors.\nerr=2\n\"Guaranteed\" error bound. The estimated\nforward error, almost certainly within a factor of\n10 of the true error so long as the next entry is\ngreater than the threshold sqrt(n)*slamch(ε)\nfor single precision flavors and\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n754\n\n\nsqrt(n)*dlamch(ε) for double precision\nflavors. This error bound should only be trusted\nif the previous boolean is true.\nerr=3\nReciprocal condition number. Estimated\nnormwise reciprocal condition number.\nCompared with the threshold\nsqrt(n)*slamch(ε) for single precision flavors\nand sqrt(n)*dlamch(ε) for double precision\nflavors to determine if the error estimate is\n\"guaranteed\". These reciprocal condition\nnumbers for some appropriately scaled matrix Z\nare:\nLet z=s*a, where s scales each row by a power\nof the radix so all absolute row sums of z are\napproximately 1.\nThe information for right-hand side i, where 1 ≤i≤nrhs, and type of error\nerr is stored in err_bnds_norm[(err-1)*nrhs + i - 1].\nerr_bnds_comp\nArray of size nrhs*n_err_bnds. For each right-hand side, contains\ninformation about various error bounds and condition numbers\ncorresponding to the componentwise relative error, which is defined as\nfollows:\nComponentwise relative error in the i-th solution vector:\nThe array is indexed by the type of error information as described below.\nThere are currently up to three pieces of information returned for each\nright-hand side. If componentwise accuracy is not requested (params[2] =\n0.0), then err_bnds_comp is not accessed.\nerr=1\n\"Trust/don't trust\" boolean. Trust the answer if\nthe reciprocal condition number is less than the\nthreshold sqrt(n)*slamch(ε) for single\nprecision flavors and sqrt(n)*dlamch(ε) for\ndouble precision flavors.\nerr=2\n\"Guaranteed\" error bpound. The estimated\nforward error, almost certainly within a factor of\n10 of the true error so long as the next entry is\ngreater than the threshold sqrt(n)*slamch(ε)\nfor single precision flavors and\nsqrt(n)*dlamch(ε) for double precision\nflavors. This error bound should only be trusted\nif the previous boolean is true.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n755\n\n\nerr=3\nReciprocal condition number. Estimated\ncomponentwise reciprocal condition number.\nCompared with the threshold\nsqrt(n)*slamch(ε) for single precision flavors\nand sqrt(n)*dlamch(ε) for double precision\nflavors to determine if the error estimate is\n\"guaranteed\". These reciprocal condition\nnumbers for some appropriately scaled matrix Z\nare:\nLet z=s*(a*diag(x)), where x is the solution\nfor the current right-hand side and s scales each\nrow of a*diag(x) by a power of the radix so all\nabsolute row sums of z are approximately 1.\nThe information for right-hand side i, where 1 ≤i≤nrhs, and type of error\nerr is stored in err_bnds_comp[(err-1)*nrhs + i - 1].\nipiv\nIf fact = 'N', ipiv is an output argument and on exit contains details of\nthe interchanges and the block structure D, as determined by ssytrf for\nsingle precision flavors and dsytrf for double precision flavors.\nequed\nIf fact≠'F', then equed is an output argument. It specifies the form of\nequilibration that was done (see the description of equed in Input\nArguments section).\nparams\nIf an entry is less than 0.0, that entry is filled with the default value used\nfor that parameter, otherwise the entry is not modified.\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful. The solution to every right-hand side is guaranteed.\nIf info = -i, parameter i had an illegal value.\nIf 0 < info≤n: Uinfo,info is exactly zero. The factorization has been completed, but the factor U is exactly\nsingular, so the solution and error bounds could not be computed; rcond = 0 is returned.\nIf info = n+j: The solution corresponding to the j-th right-hand side is not guaranteed. The solutions\ncorresponding to other right-hand sides k with k > j may not be guaranteed as well, but only the first such\nright-hand side is reported. If a small componentwise error is not requested params[2] = 0.0, then the j-th\nright-hand side is the first with a normwise error bound that is not guaranteed (the smallest j such that for\ncolumn major layout err_bnds_norm[j - 1] = 0.0 or err_bnds_comp[j - 1] = 0.0; or for row major\nlayout err_bnds_norm[(j - 1)*n_err_bnds] = 0.0 or err_bnds_comp[(j - 1)*n_err_bnds] = 0.0).\nSee the definition of err_bnds_norm and err_bnds_comp for err = 1. To get information about all of the\nright-hand sides, check err_bnds_norm or err_bnds_comp.\nSee Also\nMatrix Storage Schemes\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n756\n\n\n?hesv\nComputes the solution to the system of linear\nequations with a Hermitian matrix A and multiple\nright-hand sides.\nSyntax\nlapack_int LAPACKE_chesv (int matrix_layout , char uplo , lapack_int n , lapack_int\nnrhs , lapack_complex_float * a , lapack_int lda , lapack_int * ipiv ,\nlapack_complex_float * b , lapack_int ldb );\nlapack_int LAPACKE_zhesv (int matrix_layout , char uplo , lapack_int n , lapack_int\nnrhs , lapack_complex_double * a , lapack_int lda , lapack_int * ipiv ,\nlapack_complex_double * b , lapack_int ldb );\nInclude Files\n•\nmkl.h\nDescription\nThe routine solves for X the complex system of linear equations A*X = B, where A is an n-by-n symmetric\nmatrix, the columns of matrix B are individual right-hand sides, and the columns of X are the corresponding\nsolutions.\nThe diagonal pivoting method is used to factor A as A = U*D*UH or A = L*D*LH, where U (or L) is a product\nof permutation and unit upper (lower) triangular matrices, and D is Hermitian and block diagonal with 1-by-1\nand 2-by-2 diagonal blocks.\nThe factored form of A is then used to solve the system of equations A*X = B.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\nuplo\nMust be 'U' or 'L'.\nIndicates whether the upper or lower triangular part of A is stored and\nhow A is factored:\nIf uplo = 'U', the array a stores the upper triangular part of the\nmatrix A, and A is factored as U*D*UH.\nIf uplo = 'L', the array a stores the lower triangular part of the\nmatrix A, and A is factored as L*D*LH.\nn\nThe order of matrix A; n≥ 0.\nnrhs\nThe number of right-hand sides, the number of columns in B; nrhs≥\n0.\na, b\nArrays: a(size max(1, lda*n)), bbof size max(1, ldb*nrhs) for\ncolumn major layout and max(1, ldb*n) for row major layout. The\narray a contains the upper or the lower triangular part of the\nHermitian matrix A (see uplo).\nThe array b contains the matrix B whose columns are the right-hand\nsides for the systems of equations.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n757\n\n\nlda\nThe leading dimension of a; lda≥ max(1, n).\nldb\nThe leading dimension of b; ldb≥ max(1, n) for column major layout\nand ldb≥nrhs for row major layout.\nOutput Parameters\na\nIf info = 0, a is overwritten by the block-diagonal matrix D and the\nmultipliers used to obtain the factor U (or L) from the factorization of\nA as computed by ?hetrf.\nb\nIf info = 0, b is overwritten by the solution matrix X.\nipiv\nArray, size at least max(1, n). Contains details of the interchanges\nand the block structure of D, as determined by ?hetrf.\nIf ipiv[i-1] = k > 0, then dii is a 1-by-1 diagonal block, and the\ni-th row and column of A was interchanged with the k-th row and\ncolumn.\nIf uplo = 'U'and ipiv[i] =ipiv[i-1] = -m < 0, then D has a 2-by-2\nblock in rows/columns i and i+1, and (i)-th row and column of A was\ninterchanged with the m-th row and column.\nIf uplo = 'L'and ipiv[i] =ipiv[i-1] = -m < 0, then D has a 2-by-2\nblock in rows/columns i and i+1, and (i+1)-th row and column of A\nwas interchanged with the m-th row and column.\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info = -i, parameter i had an illegal value.\nIf info = i, dii is 0. The factorization has been completed, but D is exactly singular, so the solution could\nnot be computed.\nSee Also\nMatrix Storage Schemes\n?hesv_aa\nComputes the solution to system of linear equations\nfor HE matrices.\nLAPACK_DECL lapack_int LAPACKE_chesv_aa (int matrix_layout, char uplo, lapack_int n,\nlapack_int nrhs, lapack_complex_float * a, lapack_int lda, lapack_int * ipiv,\nlapack_complex_float * b, lapack_int ldb );\nLAPACK_DECL lapack_int LAPACKE_chesv_aa_work (int matrix_layout, char uplo, lapack_int\nn, lapack_int nrhs, lapack_complex_float * a, lapack_int lda, lapack_int * ipiv,\nlapack_complex_float * b, lapack_int ldb, lapack_complex_float * work, lapack_int\nlwork );\nDescription\n?hesv_aa computes the solution to a complex system of linear equations A * X = B, where A is an n-by-n\nHermitian matrix and X and B are n-by-nrhs matrices. Aasen's algorithm is used to factor A as\nA = U * T * UH if uplo = 'U', or\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n758\n\n\nA = L * T * LH if uplo = 'L',\nwhere U (or L) is a product of permutation and unit upper (lower) triangular matrices, and T is Hermitian and\ntridiagonal. The factored form of A is then used to solve the system of equations A * X = B.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nuplo\nIf uplo = 'U': The upper triangle of A is stored.\nIf uplo = 'L': the lower triangle of A is stored.\nn\nThe number of linear equations or the order of the matrix A. n≥ 0.\nnrhs\nThe number of right hand sides or the number of columns of the matrix B.\nnrhs≥ 0.\na\nArray of size lda*n. On entry, the Hermitian matrix A.\nIf uplo = 'U', the leading n-by-n upper triangular part of a contains the\nupper triangular part of the matrix A, and the strictly lower triangular part\nof a is not referenced.\nIf uplo = 'L', the leading n-by-n lower triangular part of a contains the\nlower triangular part of the matrix A, and the strictly upper triangular part\nof a is not referenced.\nlda\nThe leading dimension of the array a. lda≥ max(1,n).\nb\nArray of size ldb*nrhs. On entry, the n-by-nrhs right hand side matrix B.\nldb\nThe leading dimension of the array b. ldb≥ max(1,n).\nlwork\nThe length of work. lwork≥ max(1, 2*n, 3*n-2), and for best performance\nlwork≥ max(1,n*nb), where nb is the optimal blocksize for ?hetrf.\nIf lwork < n, TRS is done with Level BLAS 2. If lwork≥n, TRS is done with\nLevel BLAS 3.\nIf lwork = -1, then a workspace query is assumed; the routine only\ncalculates the optimal size of the work array, returns this value as the first\nentry of the work array, and no error message related to lwork is issued by\nxerbla.\nOutput Parameters\na\nOn exit, if info = 0, the tridiagonal matrix T and the multipliers used to\nobtain the factor U or L from the factorization A = U*T*UH or A = L*T*LH as\ncomputed by ?hetrf_aa.\nipiv\nArray of size (n) On exit, it contains the details of the interchanges: row\nand column k of A were interchanged with the row and column ipiv[k].\nb\nOn exit, if info = 0, the n-by-nrhs solution matrix X.\nwork\nArray of size (max(1, lwork)). On exit, if info = 0, work[0] returns the\noptimal lwork.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n759\n\n\nReturn Values\nThis function returns a value info.\nIf info = 0: successful exit.\nIf info < 0: if info = -i, the i-th argument had an illegal value.\nIf info > 0: if info = i, Di, i is exactly zero. The factorization has been completed, but the block diagonal\nmatrix D is exactly singular, so the solution could not be computed.\n?hesv_rk\n?hesv_rk computes the solution to a system of linear\nequations A * X = B for Hermitian matrices.\nlapack_int LAPACKE_chesv_rk (int matrix_layout, char uplo, lapack_int n, lapack_int\nnrhs, lapack_complex_float * A, lapack_int lda, lapack_complex_float * e, lapack_int *\nipiv, lapack_complex_float * B, lapack_int ldb);\nlapack_int LAPACKE_zhesv_rk (int matrix_layout, char uplo, lapack_int n, lapack_int\nnrhs, lapack_complex_double * A, lapack_int lda, lapack_complex_double * e, lapack_int\n* ipiv, lapack_complex_double * B, lapack_int ldb);\nDescription\n?hesv_rk computes the solution to a complex system of linear equations A * X = B, where A is an n-by-n\nHermitian matrix and X and B are n-by-nrhs matrices.\nThe bounded Bunch-Kaufman (rook) diagonal pivoting method is used to factor A as A = P*U*D*(UH)*(PT), if\nuplo = 'U', or A = P*L*D*(LH)*(PT), if uplo = 'L', where U (or L) is unit upper (or lower) triangular\nmatrix, UH (or LH) is the conjugate of U (or L), P is a permutation matrix, PT is the transpose of P, and D is\nHermitian and block diagonal with 1-by-1 and 2-by-2 diagonal blocks.\n?hetrf_rk is called to compute the factorization of a complex Hermitian matrix. The factored form of A is\nthen used to solve the system of equations A * X = B by calling BLAS3 routine ?hetrs_3.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nuplo\nSpecifies whether the upper or lower triangular part of the Hermitian matrix\nA is stored:\n•\n= 'U': The upper triangle of A is stored.\n•\n= 'L': The lower triangle of A is stored.\nn\nThe number of linear equations; that is, the order of the matrix A. n ≥ 0.\nnrhs\nThe number of right-hand sides; that is, the number of columns of the\nmatrix B. nrhs ≥ 0.\nA\nArray of size max(1, lda*n). On entry, the Hermitian matrix A. If uplo =\n'U': the leading n-by-n upper triangular part of A contains the upper\ntriangular part of the matrix A, and the strictly lower triangular part of A is\nnot referenced. If uplo = 'L': the leading n-by-n lower triangular part of A\ncontains the lower triangular part of the matrix A, and the strictly upper\ntriangular part of A is not referenced.\nlda\nThe leading dimension of the array A.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n760\n\n\nB\nOn entry, the n-by-nrhs right-hand side matrix B.\nThe size of B is max(1, ldb*nrhs) for column-major layout and max(1,\nldb*n) for row-major layout.\nldb\nThe leading dimension of the array B. ldb ≥ max(1, n) for column-major\nlayout and ldb ≥ nrhs for row-major layout.\nOutput Parameters\nA\nOn exit, if info = 0, diagonal of the block diagonal matrix D and factors U\nor L as computed by ?hetrf_rk:\n•\nOnly diagonal elements of the Hermitian block diagonal matrix D on the\ndiagonal of A; that is, D(k,k) = A(k,k); (superdiagonal (or subdiagonal)\nelements of D are stored on exit in array e).\n—and—\n•\nIf uplo = 'U', factor U in the superdiagonal part of A. If uplo = 'L',\nfactor L in the subdiagonal part of A.\nFor more information, see the description of the ?hetrf_rk routine.\ne\nArray of size n. On exit, contains the output computed by the factorization\nroutine ?hetrf_rk; that is, the superdiagonal (or subdiagonal) elements of\nthe Hermitian block diagonal matrix D with 1-by-1 or 2-by-2 diagonal\nblocks:\n•\nIf uplo = 'U', e(i) = D(i-1,i), i=2:N, e(1) is set to 0.\n•\nIf uplo = 'L', e(i) = D(i+1,i), i=1:N-1, e(n) is set to 0.\nNOTE For a 1-by-1 diagonal block D(k), where 1 ≤ k ≤ n, the\nelement e(k) is set to 0 in both the uplo = 'U' and uplo = 'L'\ncases.\nFor more information, see the description of the ?hetrf_rk routine.\nipiv\nArray of size n. Details of the interchanges and the block structure of D, as\ndetermined by ?hetrf_rk.\nB\nOn exit, if info = 0, the n-by-nrhs solution matrix X.\nReturn Values\nThis function returns a value info.\n= 0: Successful exit.\n< 0: If info = -k, the kth argument had an illegal value.\n> 0: If info = k, the matrix A is singular. If uplo = 'U', column k in the upper triangular part of A contains\nall zeros. If uplo = 'L', column k in the lower triangular part of A contains all zeros. Therefore D(k,k) is\nexactly zero, and superdiagonal elements of column k of U (or subdiagonal elements of column k of L ) are\nall zeros. The factorization has been completed, but the block diagonal matrix D is exactly singular, and\ndivision by zero will occur if it is used to solve a system of equations.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n761\n\n\n?hesvx\nUses the diagonal pivoting factorization to compute\nthe solution to the complex system of linear equations\nwith a Hermitian coefficient matrix A, and provides\nerror bounds on the solution.\nSyntax\nlapack_int LAPACKE_chesvx( int matrix_layout, char fact, char uplo, lapack_int n,\nlapack_int nrhs, const lapack_complex_float* a, lapack_int lda, lapack_complex_float*\naf, lapack_int ldaf, lapack_int* ipiv, const lapack_complex_float* b, lapack_int ldb,\nlapack_complex_float* x, lapack_int ldx, float* rcond, float* ferr, float* berr );\nlapack_int LAPACKE_zhesvx( int matrix_layout, char fact, char uplo, lapack_int n,\nlapack_int nrhs, const lapack_complex_double* a, lapack_int lda, lapack_complex_double*\naf, lapack_int ldaf, lapack_int* ipiv, const lapack_complex_double* b, lapack_int ldb,\nlapack_complex_double* x, lapack_int ldx, double* rcond, double* ferr, double* berr );\nInclude Files\n•\nmkl.h\nDescription\nThe routine uses the diagonal pivoting factorization to compute the solution to a complex system of linear\nequations A*X = B, where A is an n-by-n Hermitian matrix, the columns of matrix B are individual right-hand\nsides, and the columns of X are the corresponding solutions.\nError bounds on the solution and a condition estimate are also provided.\nThe routine ?hesvx performs the following steps:\n1.\nIf fact = 'N', the diagonal pivoting method is used to factor the matrix A. The form of the\nfactorization is A = U*D*UH or A = L*D*LH, where U (or L) is a product of permutation and unit upper\n(lower) triangular matrices, and D is Hermitian and block diagonal with 1-by-1 and 2-by-2 diagonal\nblocks.\n2.\nIf some di,i= 0, so that D is exactly singular, then the routine returns with info = i. Otherwise, the\nfactored form of A is used to estimate the condition number of the matrix A. If the reciprocal of the\ncondition number is less than machine precision, info = n+1 is returned as a warning, but the routine\nstill goes on to solve for X and compute error bounds as described below.\n3.\nThe system of equations is solved for X using the factored form of A.\n4.\nIterative refinement is applied to improve the computed solution matrix and calculate error bounds and\nbackward error estimates for it.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\nfact\nMust be 'F' or 'N'.\nSpecifies whether or not the factored form of the matrix A has been\nsupplied on entry.\nIf fact = 'F': on entry, af and ipiv contain the factored form of A.\nArrays a, af, and ipiv are not modified.\nIf fact = 'N', the matrix A is copied to af and factored.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n762\n\n\nuplo\nMust be 'U' or 'L'.\nIndicates whether the upper or lower triangular part of A is stored and\nhow A is factored:\nIf uplo = 'U', the array a stores the upper triangular part of the\nHermitian matrix A, and A is factored as U*D*UH.\nIf uplo = 'L', the array a stores the lower triangular part of the\nHermitian matrix A; A is factored as L*D*LH.\nn\nThe order of matrix A; n≥ 0.\nnrhs\nThe number of right-hand sides, the number of columns in B; nrhs≥\n0.\na, af, b\nArrays: a(size max(1, lda*n)), af(size max(1, ldaf*n)), bof size\nmax(1, ldb*nrhs) for column major layout and max(1, ldb*n) for\nrow major layout.\nThe array a contains the upper or the lower triangular part of the\nHermitian matrix A (see uplo).\nThe array af is an input argument if fact = 'F'. It contains he block\ndiagonal matrix D and the multipliers used to obtain the factor U or L\nfrom the factorization A = U*D*UH or A = L*D*LH as computed\nby ?hetrf.\nThe array b contains the matrix B whose columns are the right-hand\nsides for the systems of equations.\nlda\nThe leading dimension of a; lda≥ max(1, n).\nldaf\nThe leading dimension of af; ldaf≥ max(1, n).\nldb\nThe leading dimension of b; ldb≥ max(1, n) for column major layout\nand ldb≥nrhs for row major layout.\nipiv\nArray, size at least max(1, n). The array ipiv is an input argument if\nfact = 'F'. It contains details of the interchanges and the block\nstructure of D, as determined by ?hetrf.\nIf ipiv[i-1] = k > 0, then dii is a 1-by-1 diagonal block, and the\ni-th row and column of A was interchanged with the k-th row and\ncolumn.\nIf uplo = 'U'and ipiv[i] =ipiv[i-1] = -m < 0, then D has a 2-by-2\nblock in rows/columns i and i+1, and (i)-th row and column of A was\ninterchanged with the m-th row and column.\nIf uplo = 'L'and ipiv[i] =ipiv[i-1] = -m < 0, then D has a 2-by-2\nblock in rows/columns i and i+1, and (i+1)-th row and column of A\nwas interchanged with the m-th row and column.\nldx\nThe leading dimension of the output array x; ldx≥ max(1, n) for\ncolumn major layout and ldx≥nrhs for row major layout.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n763\n\n\nOutput Parameters\nx\nArray, size max(1, ldx*nrhs) for column major layout and max(1,\nldx*n) for row major layout.\nIf info = 0 or info = n+1, the array x contains the solution matrix\nX to the system of equations.\naf, ipiv\nThese arrays are output arguments if fact = 'N'. See the\ndescription of af, ipiv in Input Arguments section.\nrcond\nAn estimate of the reciprocal condition number of the matrix A. If\nrcond is less than the machine precision (in particular, if rcond = 0),\nthe matrix is singular to working precision. This condition is indicated\nby a return code of info > 0.\nferr\nArray, size at least max(1, nrhs). Contains the estimated forward\nerror bound for each solution vector xj (the j-th column of the solution\nmatrix X). If xtrue is the true solution corresponding to xj, ferr[j-1]\nis an estimated upper bound for the magnitude of the largest element\nin (xj) - xtrue) divided by the magnitude of the largest element in xj.\nThe estimate is as reliable as the estimate for rcon, and is almost\nalways a slight overestimate of the true error.\nberr\nArray, size at least max(1, nrhs). Contains the component-wise\nrelative backward error for each solution vector xj, that is, the\nsmallest relative change in any element of A or B that makes xj an\nexact solution.\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info = -i, parameter i had an illegal value.\nIf info = i, and i≤n, then dii is exactly zero. The factorization has been completed, but the block diagonal\nmatrix D is exactly singular, so the solution and error bounds could not be computed; rcond = 0 is returned.\nIf info = i, and i = n + 1, then D is nonsingular, but rcond is less than machine precision, meaning that the\nmatrix is singular to working precision. Nevertheless, the solution and error bounds are computed because\nthere are a number of situations where the computed solution can be more accurate than the value of rcond\nwould suggest.\nSee Also\nMatrix Storage Schemes\n?hesvxx\nUses extra precise iterative refinement to compute the\nsolution to the system of linear equations with a\nHermitian indefinite coefficient matrix A applying the\ndiagonal pivoting factorization.\nSyntax\nlapack_int LAPACKE_chesvxx( int matrix_layout, char fact, char uplo, lapack_int n,\nlapack_int nrhs, lapack_complex_float* a, lapack_int lda, lapack_complex_float* af,\nlapack_int ldaf, lapack_int* ipiv, char* equed, float* s, lapack_complex_float* b,\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n764\n\n\nlapack_int ldb, lapack_complex_float* x, lapack_int ldx, float* rcond, float* rpvgrw,\nfloat* berr, lapack_int n_err_bnds, float* err_bnds_norm, float* err_bnds_comp,\nlapack_int nparams, const float* params );\nlapack_int LAPACKE_zhesvxx( int matrix_layout, char fact, char uplo, lapack_int n,\nlapack_int nrhs, lapack_complex_double* a, lapack_int lda, lapack_complex_double* af,\nlapack_int ldaf, lapack_int* ipiv, char* equed, double* s, lapack_complex_double* b,\nlapack_int ldb, lapack_complex_double* x, lapack_int ldx, double* rcond, double*\nrpvgrw, double* berr, lapack_int n_err_bnds, double* err_bnds_norm, double*\nerr_bnds_comp, lapack_int nparams, const double* params );\nInclude Files\n•\nmkl.h\nDescription\nThe routine uses the diagonal pivoting factorization to compute the solution to a complex/double complex\nsystem of linear equations A*X = B, where A is an n-by-n Hermitian matrix, the columns of matrix B are\nindividual right-hand sides, and the columns of X are the corresponding solutions.\nBoth normwise and maximum componentwise error bounds are also provided on request. The routine returns\na solution with a small guaranteed error (O(eps), where eps is the working machine precision) unless the\nmatrix is very ill-conditioned, in which case a warning is returned. Relevant condition numbers are also\ncalculated and returned.\nThe routine accepts user-provided factorizations and equilibration factors; see definitions of the fact and\nequed options. Solving with refinement and using a factorization from a previous call of the routine also\nproduces a solution with O(eps) errors or warnings but that may not be true for general user-provided\nfactorizations and equilibration factors if they differ from what the routine would itself produce.\nThe routine ?hesvxx performs the following steps:\n1.\nIf fact = 'E', scaling factors are computed to equilibrate the system:\ndiag(s)*A*diag(s) *inv(diag(s))*X = diag(s)*B\nWhether or not the system will be equilibrated depends on the scaling of the matrix A, but if\nequilibration is used, A is overwritten by diag(s)*A*diag(s) and B by diag(s)*B.\n2.\nIf fact = 'N' or 'E', the LU decomposition is used to factor the matrix A (after equilibration if fact\n= 'E') as\nA = U*D*UT, if uplo = 'U',\nor A = L*D*LT, if uplo = 'L',\nwhere U or L is a product of permutation and unit upper (lower) triangular matrices, and D is a\nsymmetric and block diagonal with 1-by-1 and 2-by-2 diagonal blocks.\n3.\nIf some D(i,i)=0, so that D is exactly singular, the routine returns with info = i. Otherwise, the\nfactored form of A is used to estimate the condition number of the matrix A (see the rcond parameter).\nIf the reciprocal of the condition number is less than machine precision, the routine still goes on to\nsolve for X and compute error bounds.\n4.\nThe system of equations is solved for X using the factored form of A.\n5.\nBy default, unless params[0] is set to zero, the routine applies iterative refinement to get a small error\nand error bounds. Refinement calculates the residual to at least twice the working precision.\n6.\nIf equilibration was used, the matrix X is premultiplied by diag(r) so that it solves the original system\nbefore equilibration.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n765\n\n\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\nfact\nMust be 'F', 'N', or 'E'.\nSpecifies whether or not the factored form of the matrix A is supplied\non entry, and if not, whether the matrix A should be equilibrated\nbefore it is factored.\nIf fact = 'F', on entry, af and ipiv contain the factored form of A.\nIf equed is not 'N', the matrix A has been equilibrated with scaling\nfactors given by s. Parameters a, af, and ipiv are not modified.\nIf fact = 'N', the matrix A will be copied to af and factored.\nIf fact = 'E', the matrix A will be equilibrated, if necessary, copied\nto af and factored.\nuplo\nMust be 'U' or 'L'.\nIndicates whether the upper or lower triangular part of A is stored:\nIf uplo = 'U', the upper triangle of A is stored.\nIf uplo = 'L', the lower triangle of A is stored.\nn\nThe number of linear equations; the order of the matrix A; n≥ 0.\nnrhs\nThe number of right-hand sides; the number of columns of the\nmatrices B and X; nrhs≥ 0.\na, af, b\nArrays: a(size max(lda*n)), af(size max(ldaf*n)), b, (size\nmax(ldb*nrhs) for column major layout and max(ldb*n) for row\nmajor layout),.\nThe array a contains the Hermitian matrix A as specified by uplo. If\nuplo = 'U', the leading n-by-n upper triangular part of a contains the\nupper triangular part of the matrix A and the strictly lower triangular\npart of a is not referenced. If uplo = 'L', the leading n-by-n lower\ntriangular part of a contains the lower triangular part of the matrix A\nand the strictly upper triangular part of a is not referenced.\nThe array af is an input argument if fact = 'F'. It contains the\nblock diagonal matrix D and the multipliers used to obtain the factor U\nand L from the factorization A = U*D*UT or A = L*D*LT as computed\nby ?hetrf.\nThe array b contains the matrix B whose columns are the right-hand\nsides for the systems of equations.\nlda\nThe leading dimension of the array a; lda≥ max(1,n).\nldaf\nThe leading dimension of the array af; ldaf≥ max(1,n).\nipiv\nArray, size at least max(1, n). The array ipiv is an input argument if\nfact = 'F'. It contains details of the interchanges and the block\nstructure of D as determined by ?sytrf.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n766\n\n\nIf ipiv[k-1] > 0, rows and columns k and ipiv[k-1] were\ninterchanged and Dk,k) is a 1-by-1 diagonal block.\nIf uplo = 'U' and ipiv[i] = ipiv[i - 1] = m < 0, D has a 2-\nby-2 diagonal block in rows and columns i and i + 1, and the i-th row\nand column of A were interchanged with the m-th row and column.\nIf uplo = 'L' and ipiv[i] = ipiv[i - 1] = m < 0, D has a 2-\nby-2 diagonal block in rows and columns i and i + 1, and the (i + 1)-st\nrow and column of A were interchanged with the m-th row and\ncolumn.\nequed\nMust be 'N' or 'Y'.\nequed is an input argument if fact = 'F'. It specifies the form of\nequilibration that was done:\nIf equed = 'N', no equilibration was done (always true if fact =\n'N').\nif equed = 'Y', both row and column equilibration was done, that is,\nA has been replaced by diag(s)*A*diag(s).\ns\nArray, size (n). The array s contains the scale factors for A. If equed\n= 'Y', A is multiplied on the left and right by diag(s).\nThis array is an input argument if fact = 'F' only; otherwise it is an\noutput argument.\nIf fact = 'F' and equed = 'Y', each element of s must be positive.\nEach element of s should be a power of the radix to ensure a reliable\nsolution and error estimates. Scaling by powers of the radix does not\ncause rounding errors unless the result underflows or overflows.\nRounding errors during scaling lead to refining with a matrix that is\nnot equivalent to the input matrix, producing error estimates that may\nnot be reliable.\nldb\nThe leading dimension of the array b; ldb≥ max(1, n) for column\nmajor layout and ldb≥nrhs for row major layout.\nldx\nThe leading dimension of the output array x; ldx≥ max(1, n) for\ncolumn major layout and ldx≥nrhs for row major layout.\nn_err_bnds\nNumber of error bounds to return for each right hand side and each\ntype (normwise or componentwise). See err_bnds_norm and\nerr_bnds_comp descriptions in the Output Arguments section below.\nnparams\nSpecifies the number of parameters set in params. If ≤ 0, the params\narray is never referenced and default values are used.\nparams\nArray, size max(1,nparams). Specifies algorithm parameters. If an\nentry is less than 0.0, that entry is filled with the default value used\nfor that parameter. Only positions up to nparams are accessed;\ndefaults are used for higher-numbered parameters. If defaults are\nacceptable, you can pass nparams = 0, which prevents the source\ncode from accessing the params argument.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n767\n\n\nparams[0] : Whether to perform iterative refinement or not. Default:\n1.0 (for single precision flavors), 1.0D+0 (for double precision\nflavors).\n=0.0\nNo refinement is performed and no error\nbounds are computed.\n=1.0\nUse the extra-precise refinement algorithm.\n(Other values are reserved for future use.)\nparams[1] : Maximum number of residual computations allowed for\nrefinement.\nDefault\n10\nAggressive\nSet to 100 to permit convergence using\napproximate factorizations or factorizations\nother than LU. If the factorization uses a\ntechnique other than Gaussian elimination,\nthe guarantees in err_bnds_norm and\nerr_bnds_comp may no longer be\ntrustworthy.\nparams[2] : Flag determining if the code will attempt to find a\nsolution with a small componentwise relative error in the double-\nprecision algorithm. Positive is true, 0.0 is false. Default: 1.0 (attempt\ncomponentwise convergence).\nOutput Parameters\nx\nArray, size max(1, ldx*nrhs) for column major layout and max(1, ldx*n)\nfor row major layout.\nIf info = 0, the array x contains the solution n-by-nrhs matrix X to the\noriginal system of equations. Note that A and B are modified on exit if\nequed≠'N', and the solution to the equilibrated system is:\ninv(diag(s))*X.\na\nIf fact = 'E' and equed = 'Y', overwritten by diag(s)*A*diag(s).\naf\nIf fact = 'N', af is an output argument and on exit returns the block\ndiagonal matrix D and the multipliers used to obtain the factor U or L from\nthe factorization A = U*D*UT or A = L*D*LT.\nb\nIf equed = 'N', B is not modified.\nIf equed = 'Y', B is overwritten by diag(s)*B.\ns\nThis array is an output argument if fact≠'F'. Each element of this array is\na power of the radix. See the description of s in Input Arguments section.\nrcond\nReciprocal scaled condition number. An estimate of the reciprocal Skeel\ncondition number of the matrix A after equilibration (if done). If rcond is\nless than the machine precision, in particular, if rcond = 0, the matrix is\nsingular to working precision. Note that the error may still be small even if\nthis number is very small and the matrix appears ill-conditioned.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n768\n\n\nrpvgrw\nContains the reciprocal pivot growth factor:\nIf this is much less than 1, the stability of the LU factorization of the\n(equlibrated) matrix A could be poor. This also means that the solution X,\nestimated condition numbers, and error bounds could be unreliable. If\nfactorization fails with 0 < info≤n, this parameter contains the reciprocal\npivot growth factor for the leading info columns of A.\nberr\nArray, size at least max(1, nrhs). Contains the component-wise relative\nbackward error for each solution vector xj, that is, the smallest relative\nchange in any element of A or B that makes xj an exact solution.\nerr_bnds_norm\nArray of size nrhs*n_err_bnds. For each right-hand side, contains\ninformation about various error bounds and condition numbers\ncorresponding to the normwise relative error, which is defined as follows:\nNormwise relative error in the i-th solution vector\nThe array is indexed by the type of error information as described below.\nThere are currently up to three pieces of information returned.\nerr=1\n\"Trust/don't trust\" boolean. Trust the answer if\nthe reciprocal condition number is less than the\nthreshold sqrt(n)*slamch(ε) for chesvxx and\nsqrt(n)*dlamch(ε) for zhesvxx.\nerr=2\n\"Guaranteed\" error bound. The estimated\nforward error, almost certainly within a factor of\n10 of the true error so long as the next entry is\ngreater than the threshold sqrt(n)*slamch(ε)\nfor chesvxx and sqrt(n)*dlamch(ε) for\nzhesvxx. This error bound should only be\ntrusted if the previous boolean is true.\nerr=3\nReciprocal condition number. Estimated\nnormwise reciprocal condition number.\nCompared with the threshold\nsqrt(n)*slamch(ε) for chesvxx and\nsqrt(n)*dlamch(ε)for zhesvxx to determine if\nthe error estimate is \"guaranteed\". These\nreciprocal condition numbers for some\nappropriately scaled matrix Z are:\nLet z=s*a, where s scales each row by a power\nof the radix so all absolute row sums of z are\napproximately 1.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n769\n\n\nThe information for right-hand side i, where 1 ≤i≤nrhs, and type of error\nerr is stored in err_bnds_norm[(err-1)*nrhs + i - 1].\nerr_bnds_comp\nArray of size nrhs*n_err_bnds. For each right-hand side, contains\ninformation about various error bounds and condition numbers\ncorresponding to the componentwise relative error, which is defined as\nfollows:\nComponentwise relative error in the i-th solution vector:\nThe array is indexed by the type of error information as described below.\nThere are currently up to three pieces of information returned for each\nright-hand side. If componentwise accuracy is not requested (params[2] =\n0.0), then err_bnds_comp is not accessed.\nerr=1\n\"Trust/don't trust\" boolean. Trust the answer if\nthe reciprocal condition number is less than the\nthreshold sqrt(n)*slamch(ε) for chesvxx and\nsqrt(n)*dlamch(ε) for zhesvxx.\nerr=2\n\"Guaranteed\" error bpound. The estimated\nforward error, almost certainly within a factor of\n10 of the true error so long as the next entry is\ngreater than the threshold sqrt(n)*slamch(ε)\nfor chesvxx and sqrt(n)*dlamch(ε) for\nzhesvxx. This error bound should only be\ntrusted if the previous boolean is true.\nerr=3\nReciprocal condition number. Estimated\ncomponentwise reciprocal condition number.\nCompared with the threshold\nsqrt(n)*slamch(ε) for chesvxx and\nsqrt(n)*dlamch(ε) for zhesvxx to determine\nif the error estimate is \"guaranteed\". These\nreciprocal condition numbers for some\nappropriately scaled matrix Z are:\nLet z=s*(a*diag(x)), where x is the solution\nfor the current right-hand side and s scales each\nrow of a*diag(x) by a power of the radix so all\nabsolute row sums of z are approximately 1.\nThe information for right-hand side i, where 1 ≤i≤nrhs, and type of error\nerr is stored in err_bnds_comp[(err-1)*nrhs + i - 1].\nipiv\nIf fact = 'N', ipiv is an output argument and on exit contains details of\nthe interchanges and the block structure D, as determined by ssytrf for\nsingle precision flavors and dsytrf for double precision flavors.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n770\n\n\nequed\nIf fact≠'F', then equed is an output argument. It specifies the form of\nequilibration that was done (see the description of equed in Input\nArguments section).\nparams\nIf an entry is less than 0.0, that entry is filled with the default value used\nfor that parameter, otherwise the entry is not modified.\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful. The solution to every right-hand side is guaranteed.\nIf info = -i, the i-th parameter had an illegal value.\nIf 0 < info≤n: Uinfo,info is exactly zero. The factorization has been completed, but the factor U is exactly\nsingular, so the solution and error bounds could not be computed; rcond = 0 is returned.\nIf info = n+j: The solution corresponding to the j-th right-hand side is not guaranteed. The solutions\ncorresponding to other right-hand sides k with k > j may not be guaranteed as well, but only the first such\nright-hand side is reported. If a small componentwise error is not requested params[2] = 0.0, then the j-th\nright-hand side is the first with a normwise error bound that is not guaranteed (the smallest j such that for\ncolumn major layout err_bnds_norm[j - 1] = 0.0 or err_bnds_comp[j - 1] = 0.0; or for row major\nlayout err_bnds_norm[(j - 1)*n_err_bnds] = 0.0 or err_bnds_comp[(j - 1)*n_err_bnds] = 0.0).\nSee the definition of err_bnds_norm and err_bnds_comp for err = 1. To get information about all of the\nright-hand sides, check err_bnds_norm or err_bnds_comp.\nSee Also\nMatrix Storage Schemes\n?spsv\nComputes the solution to the system of linear\nequations with a real or complex symmetric coefficient\nmatrix A stored in packed format, and multiple right-\nhand sides.\nSyntax\nlapack_int LAPACKE_sspsv (int matrix_layout , char uplo , lapack_int n , lapack_int\nnrhs , float * ap , lapack_int * ipiv , float * b , lapack_int ldb );\nlapack_int LAPACKE_dspsv (int matrix_layout , char uplo , lapack_int n , lapack_int\nnrhs , double * ap , lapack_int * ipiv , double * b , lapack_int ldb );\nlapack_int LAPACKE_cspsv (int matrix_layout , char uplo , lapack_int n , lapack_int\nnrhs , lapack_complex_float * ap , lapack_int * ipiv , lapack_complex_float * b ,\nlapack_int ldb );\nlapack_int LAPACKE_zspsv (int matrix_layout , char uplo , lapack_int n , lapack_int\nnrhs , lapack_complex_double * ap , lapack_int * ipiv , lapack_complex_double * b ,\nlapack_int ldb );\nInclude Files\n•\nmkl.h\nDescription\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n771\n\n\nThe routine solves for X the real or complex system of linear equations A*X = B, where A is an n-by-n\nsymmetric matrix stored in packed format, the columns of matrix B are individual right-hand sides, and the\ncolumns of X are the corresponding solutions.\nThe diagonal pivoting method is used to factor A as A = U*D*UT or A = L*D*LT, where U (or L) is a product\nof permutation and unit upper (lower) triangular matrices, and D is symmetric and block diagonal with 1-\nby-1 and 2-by-2 diagonal blocks.\nThe factored form of A is then used to solve the system of equations A*X = B.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\nuplo\nMust be 'U' or 'L'.\nIndicates whether the upper or lower triangular part of A is stored:\nIf uplo = 'U', the upper triangle of A is stored.\nIf uplo = 'L', the lower triangle of A is stored.\nn\nThe order of matrix A; n≥ 0.\nnrhs\nThe number of right-hand sides, the number of columns in B; nrhs≥\n0.\nap, b\nArrays: ap (size max(1,n*(n+1)/2), bof size max(1, ldb*nrhs) for\ncolumn major layout and max(1, ldb*n) for row major layout.\nThe array ap contains the factor U or L, as specified by uplo, in\npacked storage (see Matrix Storage Schemes).\nThe array b contains the matrix B whose columns are the right-hand\nsides for the systems of equations.\nldb\nThe leading dimension of b; ldb≥ max(1, n) for column major layout\nand ldb≥nrhs for row major layout.\nOutput Parameters\nap\nThe block-diagonal matrix D and the multipliers used to obtain the\nfactor U (or L) from the factorization of A as computed by ?sptrf,\nstored as a packed triangular matrix in the same storage format as A.\nb\nIf info = 0, b is overwritten by the solution matrix X.\nipiv\nArray, size at least max(1, n). Contains details of the interchanges\nand the block structure of D, as determined by ?sptrf.\nIf ipiv[i-1] = k > 0, then dii is a 1-by-1 block, and the i-th row\nand column of A was interchanged with the k-th row and column.\nIf uplo = 'U'and ipiv[i]=ipiv[i-1] = -m < 0, then D has a 2-by-2\nblock in rows/columns i and i+1, and i-th row and column of A was\ninterchanged with the m-th row and column.\nIf uplo = 'L'and ipiv[i-1] =ipiv[i] = -m < 0, then D has a 2-by-2\nblock in rows/columns i and i+1, and (i+1)-th row and column of A\nwas interchanged with the m-th row and column.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n772\n\n\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info = -i, parameter i had an illegal value.\nIf info = i, dii is 0. The factorization has been completed, but D is exactly singular, so the solution could\nnot be computed.\nSee Also\nMatrix Storage Schemes\n?spsvx\nUses the diagonal pivoting factorization to compute\nthe solution to the system of linear equations with a\nreal or complex symmetric coefficient matrix A stored\nin packed format, and provides error bounds on the\nsolution.\nSyntax\nlapack_int LAPACKE_sspsvx( int matrix_layout, char fact, char uplo, lapack_int n,\nlapack_int nrhs, const float* ap, float* afp, lapack_int* ipiv, const float* b,\nlapack_int ldb, float* x, lapack_int ldx, float* rcond, float* ferr, float* berr );\nlapack_int LAPACKE_dspsvx( int matrix_layout, char fact, char uplo, lapack_int n,\nlapack_int nrhs, const double* ap, double* afp, lapack_int* ipiv, const double* b,\nlapack_int ldb, double* x, lapack_int ldx, double* rcond, double* ferr, double* berr );\nlapack_int LAPACKE_cspsvx( int matrix_layout, char fact, char uplo, lapack_int n,\nlapack_int nrhs, const lapack_complex_float* ap, lapack_complex_float* afp, lapack_int*\nipiv, const lapack_complex_float* b, lapack_int ldb, lapack_complex_float* x,\nlapack_int ldx, float* rcond, float* ferr, float* berr );\nlapack_int LAPACKE_zspsvx( int matrix_layout, char fact, char uplo, lapack_int n,\nlapack_int nrhs, const lapack_complex_double* ap, lapack_complex_double* afp,\nlapack_int* ipiv, const lapack_complex_double* b, lapack_int ldb,\nlapack_complex_double* x, lapack_int ldx, double* rcond, double* ferr, double* berr );\nInclude Files\n•\nmkl.h\nDescription\nThe routine uses the diagonal pivoting factorization to compute the solution to a real or complex system of\nlinear equations A*X = B, where A is a n-by-n symmetric matrix stored in packed format, the columns of\nmatrix B are individual right-hand sides, and the columns of X are the corresponding solutions.\nError bounds on the solution and a condition estimate are also provided.\nThe routine ?spsvx performs the following steps:\n1.\nIf fact = 'N', the diagonal pivoting method is used to factor the matrix A. The form of the\nfactorization is A = U*D*UT orA = L*D*LT, where U (or L) is a product of permutation and unit upper\n(lower) triangular matrices, and D is symmetric and block diagonal with 1-by-1 and 2-by-2 diagonal\nblocks.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n773\n\n\n2.\nIf some di,i= 0, so that D is exactly singular, then the routine returns with info = i. Otherwise, the\nfactored form of A is used to estimate the condition number of the matrix A. If the reciprocal of the\ncondition number is less than machine precision, info = n+1 is returned as a warning, but the routine\nstill goes on to solve for X and compute error bounds as described below.\n3.\nThe system of equations is solved for X using the factored form of A.\n4.\nIterative refinement is applied to improve the computed solution matrix and calculate error bounds and\nbackward error estimates for it.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\nfact\nMust be 'F' or 'N'.\nSpecifies whether or not the factored form of the matrix A has been\nsupplied on entry.\nIf fact = 'F': on entry, afp and ipiv contain the factored form of A.\nArrays ap, afp, and ipiv are not modified.\nIf fact = 'N', the matrix A is copied to afp and factored.\nuplo\nMust be 'U' or 'L'.\nIndicates whether the upper or lower triangular part of A is stored and\nhow A is factored:\nIf uplo = 'U', the array ap stores the upper triangular part of the\nsymmetric matrix A, and A is factored as U*D*UT.\nIf uplo = 'L', the array ap stores the lower triangular part of the\nsymmetric matrix A; A is factored as L*D*LT.\nn\nThe order of matrix A; n≥ 0.\nnrhs\nThe number of right-hand sides, the number of columns in B; nrhs≥\n0.\nap, afp, b\nArrays: ap (size max(1,n*(n+1)/2), afp (size max(1,n*(n+1)/2), bof\nsize max(1, ldb*nrhs) for column major layout and max(1, ldb*n)\nfor row major layout.\nThe array ap contains the upper or lower triangle of the symmetric\nmatrix A in packed storage (see Matrix Storage Schemes).\nThe array afp is an input argument if fact = 'F'. It contains the\nblock diagonal matrix D and the multipliers used to obtain the factor U\nor L from the factorization A = U*D*UT or A = L*D*LT as computed\nby ?sptrf, in the same storage format as A.\nThe array b contains the matrix B whose columns are the right-hand\nsides for the systems of equations.\nldb\nThe leading dimension of b; ldb≥ max(1, n) for column major layout\nand ldb≥nrhs for row major layout.\nipiv\nArray, size at least max(1, n). The array ipiv is an input argument if\nfact = 'F'. It contains details of the interchanges and the block\nstructure of D, as determined by ?sptrf.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n774\n\n\nIf ipiv[i-1] = k > 0, then dii is a 1-by-1 block, and the i-th row\nand column of A was interchanged with the k-th row and column.\nIf uplo = 'U'and ipiv[i]=ipiv[i-1] = -m < 0, then D has a 2-by-2\nblock in rows/columns i and i+1, and i-th row and column of A was\ninterchanged with the m-th row and column.\nIf uplo = 'L'and ipiv[i-1] =ipiv[i] = -m < 0, then D has a 2-by-2\nblock in rows/columns i and i+1, and (i+1)-th row and column of A\nwas interchanged with the m-th row and column.\nldx\nThe leading dimension of the output array x; ldx≥ max(1, n) for\ncolumn major layout and ldx≥nrhs for row major layout.\nOutput Parameters\nx\nArray, size max(1, ldx*nrhs) for column major layout and max(1,\nldx*n) for row major layout.\nIf info = 0 or info = n+1, the array x contains the solution matrix\nX to the system of equations.\nafp, ipiv\nThese arrays are output arguments if fact = 'N'. See the\ndescription of afp, ipiv in Input Arguments section.\nrcond\nAn estimate of the reciprocal condition number of the matrix A. If\nrcond is less than the machine precision (in particular, if rcond = 0),\nthe matrix is singular to working precision. This condition is indicated\nby a return code of info > 0.\nferr, berr\nArrays, size at least max(1, nrhs). Contain the component-wise\nforward and relative backward errors, respectively, for each solution\nvector.\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info = -i, parameter i had an illegal value.\nIf info = i, and i≤n, then dii is exactly zero. The factorization has been completed, but the block diagonal\nmatrix D is exactly singular, so the solution and error bounds could not be computed; rcond = 0 is returned.\nIf info = i, and i = n + 1, then D is nonsingular, but rcond is less than machine precision, meaning that the\nmatrix is singular to working precision. Nevertheless, the solution and error bounds are computed because\nthere are a number of situations where the computed solution can be more accurate than the value of rcond\nwould suggest.\nSee Also\nMatrix Storage Schemes\n?hpsv\nComputes the solution to the system of linear\nequations with a Hermitian coefficient matrix A stored\nin packed format, and multiple right-hand sides.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n775\n\n\nSyntax\nlapack_int LAPACKE_chpsv (int matrix_layout , char uplo , lapack_int n , lapack_int\nnrhs , lapack_complex_float * ap , lapack_int * ipiv , lapack_complex_float * b ,\nlapack_int ldb );\nlapack_int LAPACKE_zhpsv (int matrix_layout , char uplo , lapack_int n , lapack_int\nnrhs , lapack_complex_double * ap , lapack_int * ipiv , lapack_complex_double * b ,\nlapack_int ldb );\nInclude Files\n•\nmkl.h\nDescription\nThe routine solves for X the system of linear equations A*X = B, where A is an n-by-n Hermitian matrix\nstored in packed format, the columns of matrix B are individual right-hand sides, and the columns of X are\nthe corresponding solutions.\nThe diagonal pivoting method is used to factor A as A = U*D*UH or A = L*D*LH, where U (or L) is a product\nof permutation and unit upper (lower) triangular matrices, and D is Hermitian and block diagonal with 1-by-1\nand 2-by-2 diagonal blocks.\nThe factored form of A is then used to solve the system of equations A*X = B.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\nuplo\nMust be 'U' or 'L'.\nIndicates whether the upper or lower triangular part of A is stored:\nIf uplo = 'U', the upper triangle of A is stored.\nIf uplo = 'L', the lower triangle of A is stored.\nn\nThe order of matrix A; n≥ 0.\nnrhs\nThe number of right-hand sides; the number of columns in B; nrhs≥\n0.\nap, b\nArrays: ap (size max(1,n*(n+1)/2), bof size max(1, ldb*nrhs) for\ncolumn major layout and max(1, ldb*n) for row major layout.\nThe array ap contains the factor U or L, as specified by uplo, in\npacked storage (see Matrix Storage Schemes).\nThe array b contains the matrix B whose columns are the right-hand\nsides for the systems of equations.\nldb\nThe leading dimension of b; ldb≥ max(1, n) for column major layout\nand ldb≥nrhs for row major layout.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n776\n\n\nOutput Parameters\nap\nThe block-diagonal matrix D and the multipliers used to obtain the\nfactor U (or L) from the factorization of A as computed by ?hptrf,\nstored as a packed triangular matrix in the same storage format as A.\nb\nIf info = 0, b is overwritten by the solution matrix X.\nipiv\nArray, size at least max(1, n). Contains details of the interchanges\nand the block structure of D, as determined by ?hptrf.\nIf ipiv[i-1] = k > 0, then dii is a 1-by-1 block, and the i-th row\nand column of A was interchanged with the k-th row and column.\nIf uplo = 'U'and ipiv[i]=ipiv[i-1] = -m < 0, then D has a 2-by-2\nblock in rows/columns i and i+1, and i-th row and column of A was\ninterchanged with the m-th row and column.\nIf uplo = 'L'and ipiv[i-1] =ipiv[i] = -m < 0, then D has a 2-by-2\nblock in rows/columns i and i+1, and (i+1)-th row and column of A\nwas interchanged with the m-th row and column.\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info = -i, parameter i had an illegal value.\nIf info = i, dii is 0. The factorization has been completed, but D is exactly singular, so the solution could\nnot be computed.\nSee Also\nMatrix Storage Schemes\n?hpsvx\nUses the diagonal pivoting factorization to compute\nthe solution to the system of linear equations with a\nHermitian coefficient matrix A stored in packed\nformat, and provides error bounds on the solution.\nSyntax\nlapack_int LAPACKE_chpsvx( int matrix_layout, char fact, char uplo, lapack_int n,\nlapack_int nrhs, const lapack_complex_float* ap, lapack_complex_float* afp, lapack_int*\nipiv, const lapack_complex_float* b, lapack_int ldb, lapack_complex_float* x,\nlapack_int ldx, float* rcond, float* ferr, float* berr );\nlapack_int LAPACKE_zhpsvx( int matrix_layout, char fact, char uplo, lapack_int n,\nlapack_int nrhs, const lapack_complex_double* ap, lapack_complex_double* afp,\nlapack_int* ipiv, const lapack_complex_double* b, lapack_int ldb,\nlapack_complex_double* x, lapack_int ldx, double* rcond, double* ferr, double* berr );\nInclude Files\n•\nmkl.h\nDescription\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n777\n\n\nThe routine uses the diagonal pivoting factorization to compute the solution to a complex system of linear\nequations A*X = B, where A is a n-by-n Hermitian matrix stored in packed format, the columns of matrix B\nare individual right-hand sides, and the columns of X are the corresponding solutions.\nError bounds on the solution and a condition estimate are also provided.\nThe routine ?hpsvx performs the following steps:\n1.\nIf fact = 'N', the diagonal pivoting method is used to factor the matrix A. The form of the\nfactorization is A = U*D*UH or A = L*D*LH, where U (or L) is a product of permutation and unit upper\n(lower) triangular matrices, and D is a Hermitian and block diagonal with 1-by-1 and 2-by-2 diagonal\nblocks.\n2.\nIf some di,i = 0, so that D is exactly singular, then the routine returns with info = i. Otherwise, the\nfactored form of A is used to estimate the condition number of the matrix A. If the reciprocal of the\ncondition number is less than machine precision, info = n+1 is returned as a warning, but the routine\nstill goes on to solve for X and compute error bounds as described below.\n3.\nThe system of equations is solved for X using the factored form of A.\n4.\nIterative refinement is applied to improve the computed solution matrix and calculate error bounds and\nbackward error estimates for it.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\nfact\nMust be 'F' or 'N'.\nSpecifies whether or not the factored form of the matrix A has been\nsupplied on entry.\nIf fact = 'F': on entry, afp and ipiv contain the factored form of A.\nArrays ap, afp, and ipiv are not modified.\nIf fact = 'N', the matrix A is copied to afp and factored.\nuplo\nMust be 'U' or 'L'.\nIndicates whether the upper or lower triangular part of A is stored and\nhow A is factored:\nIf uplo = 'U', the array ap stores the upper triangular part of the\nHermitian matrix A, and A is factored as U*D*UH.\nIf uplo = 'L', the array ap stores the lower triangular part of the\nHermitian matrix A, and A is factored as L*D*LH.\nn\nThe order of matrix A; n≥ 0.\nnrhs\nThe number of right-hand sides, the number of columns in B; nrhs≥\n0.\nap, afp, b\nArrays: ap (size max(1,n*(n+1)/2), afp (size max(1,n*(n+1)/2), bof\nsize max(1, ldb*nrhs) for column major layout and max(1, ldb*n)\nfor row major layout.\nThe array ap contains the upper or lower triangle of the Hermitian\nmatrix A in packed storage (see Matrix Storage Schemes).\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n778\n\n\nThe array afp is an input argument if fact = 'F'. It contains the\nblock diagonal matrix D and the multipliers used to obtain the factor U\nor L from the factorization A = U*D*UH or A = L*D*LH as computed\nby ?hptrf, in the same storage format as A.\nThe array b contains the matrix B whose columns are the right-hand\nsides for the systems of equations.\nldb\nThe leading dimension of b; ldb≥ max(1, n) for column major layout\nand ldb≥nrhs for row major layout.\nipiv\nArray, size at least max(1, n). The array ipiv is an input argument if\nfact = 'F'. It contains details of the interchanges and the block\nstructure of D, as determined by ?hptrf.\nIf ipiv[i-1] = k > 0, then dii is a 1-by-1 block, and the i-th row\nand column of A was interchanged with the k-th row and column.\nIf uplo = 'U'and ipiv[i]=ipiv[i-1] = -m < 0, then D has a 2-by-2\nblock in rows/columns i and i+1, and i-th row and column of A was\ninterchanged with the m-th row and column.\nIf uplo = 'L'and ipiv[i-1] =ipiv[i] = -m < 0, then D has a 2-by-2\nblock in rows/columns i and i+1, and (i+1)-th row and column of A\nwas interchanged with the m-th row and column.\nldx\nThe leading dimension of the output array x; ldx≥ max(1, n) for\ncolumn major layout and ldx≥nrhs for row major layout.\nOutput Parameters\nx\nArray, size max(1, ldx*nrhs) for column major layout and max(1,\nldx*n) for row major layout.\nIf info = 0 or info = n+1, the array x contains the solution matrix\nX to the system of equations.\nafp, ipiv\nThese arrays are output arguments if fact = 'N'. See the\ndescription of afp, ipiv in Input Arguments section.\nrcond\nAn estimate of the reciprocal condition number of the matrix A. If\nrcond is less than the machine precision (in particular, if rcond = 0),\nthe matrix is singular to working precision. This condition is indicated\nby a return code of info > 0.\nferr\nArray, size at least max(1, nrhs). Contains the estimated forward\nerror bound for each solution vector xj (the j-th column of the solution\nmatrix X). If xtrue is the true solution corresponding to xj, ferr[j-1]\nis an estimated upper bound for the magnitude of the largest element\nin (xj - xtrue) divided by the magnitude of the largest element in xj.\nThe estimate is as reliable as the estimate for rcond, and is almost\nalways a slight overestimate of the true error.\nberr\nArray, size at least max(1, nrhs). Contains the component-wise\nrelative backward error for each solution vector xj, that is, the\nsmallest relative change in any element of A or B that makes xj an\nexact solution.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n779\n\n\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info = -i, parameter i had an illegal value.\nIf info = i, and i≤n, then dii is exactly zero. The factorization has been completed, but the block diagonal\nmatrix D is exactly singular, so the solution and error bounds could not be computed; rcond = 0 is returned.\nIf info = i, and i = n + 1, then D is nonsingular, but rcond is less than machine precision, meaning that the\nmatrix is singular to working precision. Nevertheless, the solution and error bounds are computed because\nthere are a number of situations where the computed solution can be more accurate than the value of rcond\nwould suggest.\nSee Also\nMatrix Storage Schemes\nLAPACK Least Squares and Eigenvalue Problem Routines\nThis section includes descriptions of LAPACK computational routines and driver routines for solving linear\nleast squares problems, eigenvalue and singular value problems, and performing a number of related\ncomputational tasks. For a full reference on LAPACK routines and related information see [LUG].\nLeast Squares Problems. A typical least squares problem is as follows: given a matrix A and a vector b,\nfind the vector x that minimizes the sum of squares Σi((Ax)i - bi)2 or, equivalently, find the vector x that\nminimizes the 2-norm ||Ax - b||2.\nIn the most usual case, A is an m-by-n matrix with m ≥ n and rank(A) = n. This problem is also referred to\nas finding the least squares solution to an overdetermined system of linear equations (here we have more\nequations than unknowns). To solve this problem, you can use the QR factorization of the matrix A (see QR\nFactorization).\nIf m < n and rank(A) = m, there exist an infinite number of solutions x which exactly satisfy Ax = b, and\nthus minimize the norm ||Ax - b||2. In this case it is often useful to find the unique solution that\nminimizes ||x||2. This problem is referred to as finding the minimum-norm solution to an\nunderdetermined system of linear equations (here we have more unknowns than equations). To solve this\nproblem, you can use the LQ factorization of the matrix A (see LQ Factorization).\nIn the general case you may have a rank-deficient least squares problem, with rank(A)< min(m, n): find\nthe minimum-norm least squares solution that minimizes both ||x||2 and ||Ax - b||2. In this case (or\nwhen the rank of A is in doubt) you can use the QR factorization with pivoting or singular value\ndecomposition (see Singular Value Decomposition).\nEigenvalue Problems. The eigenvalue problems (from German eigen \"own\") are stated as follows: given a\nmatrix A, find the eigenvaluesλ and the corresponding eigenvectorsz that satisfy the equation\nAz = λz (right eigenvectors z)\nor the equation\nzHA = λzH (left eigenvectors z).\nIf A is a real symmetric or complex Hermitian matrix, the above two equations are equivalent, and the\nproblem is called a symmetric eigenvalue problem. Routines for solving this type of problems are described\nin the topic Symmetric Eigenvalue Problems.\nRoutines for solving eigenvalue problems with nonsymmetric or non-Hermitian matrices are described in the\ntopic Nonsymmetric Eigenvalue Problems.\nThe library also includes routines that handle generalized symmetric-definite eigenvalue problems: find\nthe eigenvalues λ and the corresponding eigenvectors x that satisfy one of the following equations:\nAz = λBz, ABz = λz, or BAz = λz,\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n780\n\n\nwhere A is symmetric or Hermitian, and B is symmetric positive-definite or Hermitian positive-definite.\nRoutines for reducing these problems to standard symmetric eigenvalue problems are described in the topic \nGeneralized Symmetric-Definite Eigenvalue Problems.\nTo solve a particular problem, you usually call several computational routines. Sometimes you need to\ncombine the routines of this chapter with other LAPACK routines described in \"LAPACK Routines: Linear\nEquations\" as well as with BLAS routines described in \"BLAS and Sparse BLAS Routines\".\nFor example, to solve a set of least squares problems minimizing ||Ax - b||2 for all columns b of a given\nmatrix B (where A and B are real matrices), you can call ?geqrf to form the factorization A = QR, then\ncall ?ormqr to compute C = QHB and finally call the BLAS routine ?trsm to solve for X the system of\nequations RX = C.\nAnother way is to call an appropriate driver routine that performs several tasks in one call. For example, to\nsolve the least squares problem the driver routine ?gels can be used.\nLAPACK Least Squares and Eigenvalue Problem Computational Routines\nIn the topics that follow, the descriptions of LAPACK computational routines are given. These routines\nperform distinct computational tasks that can be used for:\nOrthogonal Factorizations\nSingular Value Decomposition\nSymmetric Eigenvalue Problems\nGeneralized Symmetric-Definite Eigenvalue Problems\nNonsymmetric Eigenvalue Problems\nGeneralized Nonsymmetric Eigenvalue Problems\nGeneralized Singular Value Decomposition\nSee also the respective driver routines.\nOrthogonal Factorizations: LAPACK Computational Routines\nThis topic describes the LAPACK routines for the QR (RQ) and LQ (QL) factorization of matrices. Routines for\nthe RZ factorization as well as for generalized QR and RQ factorizations are also included.\nQR Factorization. Assume that A is an m-by-n matrix to be factored.\nIf m≥n, the QR factorization is given by\nwhere R is an n-by-n upper triangular matrix with real diagonal elements, and Q is an m-by-m orthogonal (or\nunitary) matrix.\nYou can use the QR factorization for solving the following least squares problem: minimize ||Ax - b||2\nwhere A is a full-rank m-by-n matrix (m≥n). After factoring the matrix, compute the solution x by solving Rx\n= (Q1)Tb.\nIf m < n, the QR factorization is given by\nA = QR = Q(R1R2)\nwhere R is trapezoidal, R1 is upper triangular and R2 is rectangular.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n781\n\n\nQ is represented as a product of min(m, n) elementary reflectors. Routines are provided to work with Q in\nthis representation.\nLQ Factorization LQ factorization of an m-by-n matrix A is as follows. If m≤n,\nwhere L is an m-by-m lower triangular matrix with real diagonal elements, and Q is an n-by-n orthogonal (or\nunitary) matrix.\nIf m > n, the LQ factorization is\nwhere L1 is an n-by-n lower triangular matrix, L2 is rectangular, and Q is an n-by-n orthogonal (or unitary)\nmatrix.\nYou can use the LQ factorization to find the minimum-norm solution of an underdetermined system of linear\nequations Ax = b where A is an m-by-n matrix of rank m (m < n). After factoring the matrix, compute the\nsolution vector x as follows: solve Ly = b for y, and then compute x = (Q1)Hy.\nTable \"Computational Routines for Orthogonal Factorization\" lists LAPACK routines that perform orthogonal\nfactorization of matrices.\nComputational Routines for Orthogonal Factorization\nMatrix type, factorization\nFactorize without\npivoting\nFactorize with\npivoting\nGenerate\nmatrix Q\nApply\nmatrix Q\ngeneral matrices, QR factorization\ngeqrf\ngeqrfp\ngeqpf\ngeqp3\norgqr \nungqr\normqr \nunmqr\ngeneral matrices, blocked QR\nfactorization\ngeqrt\n \n \ngemqrt\ngeneral matrices, RQ factorization\ngerqf\n \norgrq \nungrq\normrq \nunmrq\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n782\n\n\nMatrix type, factorization\nFactorize without\npivoting\nFactorize with\npivoting\nGenerate\nmatrix Q\nApply\nmatrix Q\ngeneral matrices, LQ factorization\ngelqf\n \norglq \nunglq\normlq \nunmlq\ngeneral matrices, QL factorization\ngeqlf\n \norgql \nungql\normql \nunmql\ntrapezoidal matrices, RZ\nfactorization\ntzrzf\n \n \normrz \nunmrz\npair of matrices, generalized QR\nfactorization\nggqrf\n \n \n \npair of matrices, generalized RQ\nfactorization\nggrqf\n \n \n \ntriangular-pentagonal matrices,\nblocked QR factorization\ntpqrt\n \n \ntpmqrt\n?geqrf\nComputes the QR factorization of a general m-by-n\nmatrix.\nSyntax\nlapack_int LAPACKE_sgeqrf (int matrix_layout, lapack_int m, lapack_int n, float* a,\nlapack_int lda, float* tau);\nlapack_int LAPACKE_dgeqrf (int matrix_layout, lapack_int m, lapack_int n, double* a,\nlapack_int lda, double* tau);\nlapack_int LAPACKE_cgeqrf (int matrix_layout, lapack_int m, lapack_int n,\nlapack_complex_float* a, lapack_int lda, lapack_complex_float* tau);\nlapack_int LAPACKE_zgeqrf (int matrix_layout, lapack_int m, lapack_int n,\nlapack_complex_double* a, lapack_int lda, lapack_complex_double* tau);\nInclude Files\n•\nmkl.h\nDescription\nThe routine forms the QR factorization of a general m-by-n matrix A (see Orthogonal Factorizations). No\npivoting is performed.\nThe routine does not form the matrix Q explicitly. Instead, Q is represented as a product of min(m, n)\nelementary reflectors. Routines are provided to work with Q in this representation.\nNOTE\nThis routine supports the Progress Routine feature. See Progress Function for details.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n783\n\n\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nm\nThe number of rows in the matrix A (m≥ 0).\nn\nThe number of columns in A (n≥ 0).\na\nArray a of size max(1, lda*n) for column major layout and max(1, lda*m)\nfor row major layout contains the matrix A.\nlda\nThe leading dimension of a; at least max(1, m) for column major layout and\nat least max(1, n) for row major layout.\nOutput Parameters\na\nOverwritten by the factorization data as follows:\nThe elements on and above the diagonal of the array contain the min(m,n)-\nby-n upper trapezoidal matrix R (R is upper triangular if m≥n); the elements\nbelow the diagonal, with the array tau, present the orthogonal matrix Q as\na product of min(m,n) elementary reflectors (see Orthogonal Factorizations).\ntau\nArray, size at least max (1, min(m, n)). Contains scalars that define\nelementary reflectors for the matrix Q in its decomposition in a product of\nelementary reflectors (see Orthogonal Factorizations).\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nApplication Notes\nThe computed factorization is the exact factorization of a matrix A + E, where\n||E||2 = O(ε)||A||2.\nThe approximate number of floating-point operations for real flavors is\n(4/3)n3\nif m = n,\n(2/3)n2(3m-n)\nif m > n,\n(2/3)m2(3n-m)\nif m < n.\nThe number of operations for complex flavors is 4 times greater.\nTo solve a set of least squares problems minimizing ||A*x - b||2 for all columns b of a given matrix B, you\ncan call the following:\n?geqrf (this routine)\nto factorize A = QR;\normqr\nto compute C = QT*B (for real matrices);\nunmqr\nto compute C = QH*B (for complex matrices);\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n784\n\n\ntrsm (a BLAS routine)\nto solve R*X = C.\n(The columns of the computed X are the least squares solution vectors x.)\nTo compute the elements of Q explicitly, call\norgqr\n(for real matrices)\nungqr\n(for complex matrices).\nSee Also\nmkl_progress\nMatrix Storage Schemes\n?geqrfp\nComputes the QR factorization of a general m-by-n\nmatrix with non-negative diagonal elements.\nSyntax\nlapack_int LAPACKE_sgeqrfp (int matrix_layout, lapack_int m, lapack_int n, float* a,\nlapack_int lda, float* tau);\nlapack_int LAPACKE_dgeqrfp (int matrix_layout, lapack_int m, lapack_int n, double* a,\nlapack_int lda, double* tau);\nlapack_int LAPACKE_cgeqrfp (int matrix_layout, lapack_int m, lapack_int n,\nlapack_complex_float* a, lapack_int lda, lapack_complex_float* tau);\nlapack_int LAPACKE_zgeqrfp (int matrix_layout, lapack_int m, lapack_int n,\nlapack_complex_double* a, lapack_int lda, lapack_complex_double* tau);\nInclude Files\n•\nmkl.h\nDescription\nThe routine forms the QR factorization of a general m-by-n matrix A (see Orthogonal Factorizations). No\npivoting is performed. The diagonal entries of R are real and nonnegative.\nThe routine does not form the matrix Q explicitly. Instead, Q is represented as a product of min(m, n)\nelementary reflectors. Routines are provided to work with Q in this representation.\nNOTE\nThis routine supports the Progress Routine feature. See Progress Function for details.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nm\nThe number of rows in the matrix A (m≥ 0).\nn\nThe number of columns in A (n≥ 0).\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n785\n\n\na\nArray, size max(1,lda*n) for column major layout and max(1,lda*m) for\nrow major layout, containing the matrix A.\nlda\nThe leading dimension of a; at least max(1, m) for column major layout and\nat least max(1, n) for row major layout.\nOutput Parameters\na\nOverwritten by the factorization data as follows:\nThe elements on and above the diagonal of the array contain the min(m,n)-\nby-n upper trapezoidal matrix R (R is upper triangular if m≥n); the elements\nbelow the diagonal, with the array tau, present the orthogonal matrix Q as\na product of min(m,n) elementary reflectors (see Orthogonal Factorizations).\nThe diagonal elements of the matrix R are real and non-negative.\ntau\nArray, size at least max (1, min(m, n)). Contains scalars that define\nelementary reflectors for the matrix Qin its decomposition in a product of\nelementary reflectors (see Orthogonal Factorizations).\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nApplication Notes\nThe computed factorization is the exact factorization of a matrix A + E, where\n||E||2 = O(ε)||A||2.\nThe approximate number of floating-point operations for real flavors is\n(4/3)n3\nif m = n,\n(2/3)n2(3m-n)\nif m > n,\n(2/3)m2(3n-m)\nif m < n.\nThe number of operations for complex flavors is 4 times greater.\nTo solve a set of least squares problems minimizing ||A*x - b||2 for all columns b of a given matrix B, you\ncan call the following:\n?geqrfp (this routine)\nto factorize A = QR;\normqr\nto compute C = QT*B (for real matrices);\nunmqr\nto compute C = QH*B (for complex matrices);\ntrsm (a BLAS routine)\nto solve R*X = C.\n(The columns of the computed X are the least squares solution vectors x.)\nTo compute the elements of Q explicitly, call\norgqr\n(for real matrices)\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n786\n\n\nungqr\n(for complex matrices).\nSee Also\nmkl_progress\nMatrix Storage Schemes\n?geqrt\nComputes a blocked QR factorization of a general real\nor complex matrix using the compact WY\nrepresentation of Q.\nSyntax\nlapack_int LAPACKE_sgeqrt (int matrix_layout, lapack_int m, lapack_int n, lapack_int\nnb, float* a, lapack_int lda, float* t, lapack_int ldt);\nlapack_int LAPACKE_dgeqrt (int matrix_layout, lapack_int m, lapack_int n, lapack_int\nnb, double* a, lapack_int lda, double* t, lapack_int ldt);\nlapack_int LAPACKE_cgeqrt (int matrix_layout, lapack_int m, lapack_int n, lapack_int\nnb, lapack_complex_float* a, lapack_int lda, lapack_complex_float* t, lapack_int ldt);\nlapack_int LAPACKE_zgeqrt (int matrix_layout, lapack_int m, lapack_int n, lapack_int\nnb, lapack_complex_double* a, lapack_int lda, lapack_complex_double* t, lapack_int\nldt);\nInclude Files\n•\nmkl.h\nDescription\nThe strictly lower triangular matrix V contains the elementary reflectors H(i) in the ith column below the\ndiagonal. For example, if m=5 and n=3, the matrix V is\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n787\n\n\nwhere vi represents one of the vectors that define H(i). The vectors are returned in the lower triangular part\nof array a.\nNOTE\nThe 1s along the diagonal of V are not stored in a.\nLet k = min(m,n). The number of blocks is b = ceiling(k/nb), where each block is of order nb except for\nthe last block, which is of order ib = k - (b-1)*nb. For each of the b blocks, a upper triangular block\nreflector factor is computed:t1, t2, ..., tb. The nb-by-nb (and ib-by-ib for the last block) ts are stored\nin the nb-by-n array t as\nt = (t1t2 ... tb).\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nm\nThe number of rows in the matrix A (m ≥ 0).\nn\nThe number of columns in A (n ≥ 0).\nnb\nThe block size to be used in the blocked QR (min(m, n) ≥ nb ≥ 1).\na\nArray a of size max(1, lda*n) for column major layout and max(1, lda*m)\nfor row major layout contains the m-by-n matrix A.\nlda\nThe leading dimension of a; at least max(1, m) for column major layout and\nmax(1, n) for row major layout.\nldt\nThe leading dimension of t; at least nb for column major layout and max(1,\nmin(m, n)) for row major layout.\nOutput Parameters\na\nOverwritten by the factorization data as follows:\nThe elements on and above the diagonal of the array contain the min(m,n)-\nby-n upper trapezoidal matrix R (R is upper triangular if m≥n); the elements\nbelow the diagonal, with the array t, present the orthogonal matrix Q as a\nproduct of min(m,n) elementary reflectors (see Orthogonal Factorizations).\nt\nArray, size max(1, ldt*min(m, n)) for column major layout and max(1,\nldt*nb) for row major layout.\nThe upper triangular block reflector's factors stored as a sequence of upper\ntriangular blocks.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info < 0 and info = -i, the i-th parameter had an illegal value.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n788\n\n\n?gemqrt\nMultiplies a general matrix by the orthogonal/unitary\nmatrix Q of the QR factorization formed by ?geqrt.\nSyntax\nlapack_int LAPACKE_sgemqrt (int matrix_layout, char side, char trans, lapack_int m,\nlapack_int n, lapack_int k, lapack_int nb, const float* v, lapack_int ldv, const float*\nt, lapack_int ldt, float* c, lapack_int ldc);\nlapack_int LAPACKE_dgemqrt (int matrix_layout, char side, char trans, lapack_int m,\nlapack_int n, lapack_int k, lapack_int nb, const double* v, lapack_int ldv, const\ndouble* t, lapack_int ldt, double* c, lapack_int ldc);\nlapack_int LAPACKE_cgemqrt (int matrix_layout, char side, char trans, lapack_int m,\nlapack_int n, lapack_int k, lapack_int nb, const lapack_complex_float* v, lapack_int\nldv, const lapack_complex_float* t, lapack_int ldt, lapack_complex_float* c, lapack_int\nldc);\nlapack_int LAPACKE_zgemqrt (int matrix_layout, char side, char trans, lapack_int m,\nlapack_int n, lapack_int k, lapack_int nb, const lapack_complex_double* v, lapack_int\nldv, const lapack_complex_double* t, lapack_int ldt, lapack_complex_double* c,\nlapack_int ldc);\nInclude Files\n•\nmkl.h\nDescription\nThe ?gemqrt routine overwrites the general real or complex m-by-n matrixC with\nside ='L'\nside ='R'\ntrans = 'N':\nQ*C\nC*Q\ntrans = 'T':\nQT*C\nC*QT\ntrans = 'C':\nQH*C\nC*QH\nwhere Q is a real orthogonal (complex unitary) matrix defined as the product of k elementary reflectors\nQ = H(1) H(2)... H(k) = I - V*T*VT for real flavors, and\nQ = H(1) H(2)... H(k) = I - V*T*VH for complex flavors,\ngenerated using the compact WY representation as returned by geqrt. Q is of order m if side = 'L' and of\norder n if side = 'R'.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nside\n='L': apply Q, QT, or QH from the left.\n='R': apply Q, QT, or QH from the right.\ntrans\n='N', no transpose, apply Q.\n='T', transpose, apply QT.\n='C', transpose, apply QH.\nm\nThe number of rows in the matrix C, (m ≥ 0).\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n789\n\n\nn\nThe number of columns in the matrix C, (n ≥ 0).\nk\nThe number of elementary reflectors whose product defines the matrix Q.\nConstraints:\nIf side = 'L', m ≥ k≥0\nIf side = 'R', n ≥ k≥0.\nnb\nThe block size used for the storage of t, k ≥ nb ≥ 1. This must be the same\nvalue of nb used to generate t in geqrt.\nv\nArray of size max(1, ldv*k) for column major layout, max(1, ldv*m) for\nrow major layout and side = 'L', and max(1, ldv*n) for row major layout\nand side = 'R'.\nThe ith column must contain the vector which defines the elementary\nreflector H(i), for i = 1,2,...,k, as returned by geqrt in the first k columns of\nits array argument a.\nldv\nThe leading dimension of the array v.\nif side = 'L', ldv must be at least max(1,m) for column major layout and\nmax(1, k) for row major layout;\nif side = 'R', ldv must be at least max(1,n) for column major layout and\nmax(1, k) for row major layout.\nt\nArray, size max(1, ldt*min(m, n)) for column major layout and max(1,\nldt*nb) for row major layout.\nThe upper triangular factors of the block reflectors as returned by geqrt.\nldt\nThe leading dimension of the array t. ldt must be at least nb for column\nmajor layout and max(1, k) for row major layout.\nc\nThe m-by-n matrix C.\nldc\nThe leadinng dimension of the array c. ldc must be at least max(1, m) for\ncolumn major layout and max(1, n) for row major layout.\nOutput Parameters\nc\nOverwritten by the product Q*C, C*Q, QT*C, C*QT, QH*C, or C*QH as\nspecified by side and trans.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\n?geqpf\nComputes the QR factorization of a general m-by-n\nmatrix with pivoting.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n790\n\n\nSyntax\nlapack_int LAPACKE_sgeqpf (int matrix_layout, lapack_int m, lapack_int n, float* a,\nlapack_int lda, lapack_int* jpvt, float* tau);\nlapack_int LAPACKE_dgeqpf (int matrix_layout, lapack_int m, lapack_int n, double* a,\nlapack_int lda, lapack_int* jpvt, double* tau);\nlapack_int LAPACKE_cgeqpf (int matrix_layout, lapack_int m, lapack_int n,\nlapack_complex_float* a, lapack_int lda, lapack_int* jpvt, lapack_complex_float* tau);\nlapack_int LAPACKE_zgeqpf (int matrix_layout, lapack_int m, lapack_int n,\nlapack_complex_double* a, lapack_int lda, lapack_int* jpvt, lapack_complex_double*\ntau);\nInclude Files\n•\nmkl.h\nDescription\nThe routine is deprecated and has been replaced by routine geqp3.\nThe routine ?geqpf forms the QR factorization of a general m-by-n matrix A with column pivoting: A*P =\nQ*R (see Orthogonal Factorizations). Here P denotes an n-by-n permutation matrix.\nThe routine does not form the matrix Q explicitly. Instead, Q is represented as a product of min(m, n)\nelementary reflectors. Routines are provided to work with Q in this representation.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nm\nThe number of rows in the matrix A (m≥ 0).\nn\nThe number of columns in A (n≥ 0).\na\nArray a of size max(1, lda*n) for column major layout and max(1, lda*m)\nfor row major layout contains the matrix A.\nlda\nThe leading dimension of a; at least max(1, m)for column major layout and\nmax(1, n) for row major layout.\njpvt\nArray, size at least max(1, n).\nOn entry, if jpvt[i - 1] > 0, the i-th column of A is moved to the\nbeginning of A*P before the computation, and fixed in place during the\ncomputation.\nIf jpvt[i - 1] = 0, the ith column of A is a free column (that is, it may\nbe interchanged during the computation with any other free column).\nOutput Parameters\na\nOverwritten by the factorization data as follows:\nThe elements on and above the diagonal of the array contain the min(m,n)-\nby-n upper trapezoidal matrix R (R is upper triangular if m≥n); the elements\nbelow the diagonal, with the array tau, present the orthogonal matrix Q as\na product of min(m,n) elementary reflectors (see Orthogonal Factorizations).\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n791\n\n\ntau\nArray, size at least max (1, min(m, n)). Contains additional information on\nthe matrix Q.\njpvt\nOverwritten by details of the permutation matrix P in the factorization A*P\n= Q*R. More precisely, the columns of A*P are the columns of A in the\nfollowing order:\njpvt[0], jpvt[1], ..., jpvt[n - 1].\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nApplication Notes\nThe computed factorization is the exact factorization of a matrix A + E, where\n||E||2 = O(ε)||A||2.\nThe approximate number of floating-point operations for real flavors is\n(4/3)n3\nif m = n,\n(2/3)n2(3m-n)\nif m > n,\n(2/3)m2(3n-m)\nif m < n.\nThe number of operations for complex flavors is 4 times greater.\nTo solve a set of least squares problems minimizing ||A*x - b||2 for all columns b of a given matrix B, you\ncan call the following:\n?geqpf (this routine)\nto factorize A*P = Q*R;\normqr\nto compute C = QT*B (for real matrices);\nunmqr\nto compute C = QH*B (for complex matrices);\ntrsm (a BLAS routine)\nto solve R*X = C.\n(The columns of the computed X are the permuted least squares solution vectors x; the output array jpvt\nspecifies the permutation order.)\nTo compute the elements of Q explicitly, call\norgqr\n(for real matrices)\nungqr\n(for complex matrices).\n?geqp3\nComputes the QR factorization of a general m-by-n\nmatrix with column pivoting using level 3 BLAS.\nSyntax\nlapack_int LAPACKE_sgeqp3 (int matrix_layout, lapack_int m, lapack_int n, float* a,\nlapack_int lda, lapack_int* jpvt, float* tau);\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n792\n\n\nlapack_int LAPACKE_dgeqp3 (int matrix_layout, lapack_int m, lapack_int n, double* a,\nlapack_int lda, lapack_int* jpvt, double* tau);\nlapack_int LAPACKE_cgeqp3 (int matrix_layout, lapack_int m, lapack_int n,\nlapack_complex_float* a, lapack_int lda, lapack_int* jpvt, lapack_complex_float* tau);\nlapack_int LAPACKE_zgeqp3 (int matrix_layout, lapack_int m, lapack_int n,\nlapack_complex_double* a, lapack_int lda, lapack_int* jpvt, lapack_complex_double*\ntau);\nInclude Files\n•\nmkl.h\nDescription\nThe routine forms the QR factorization of a general m-by-n matrix A with column pivoting: A*P = Q*R (see \nOrthogonal Factorizations) using Level 3 BLAS. Here P denotes an n-by-n permutation matrix. Use this\nroutine instead of geqpf for better performance.\nThe routine does not form the matrix Q explicitly. Instead, Q is represented as a product of min(m, n)\nelementary reflectors. Routines are provided to work with Q in this representation.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nm\nThe number of rows in the matrix A (m≥ 0).\nn\nThe number of columns in A (n≥ 0).\na\nArray a of size max(1, lda*n) for column major layout and max(1, lda*m)\nfor row major layout contains the matrix A.\nlda\nThe leading dimension of a; at least max(1, m)for column major layout and\nmax(1, n) for row major layout.\njpvt\nArray, size at least max(1, n).\nOn entry, if jpvt[i - 1]≠ 0, the i-th column of A is moved to the\nbeginning of AP before the computation, and fixed in place during the\ncomputation.\nIf jpvt[i - 1] = 0, the i-th column of A is a free column (that is, it may\nbe interchanged during the computation with any other free column).\nOutput Parameters\na\nOverwritten by the factorization data as follows:\nThe elements on and above the diagonal of the array contain the min(m,n)-\nby-n upper trapezoidal matrix R (R is upper triangular if m≥n); the elements\nbelow the diagonal, with the array tau, present the orthogonal matrix Q as\na product of min(m,n) elementary reflectors (see Orthogonal Factorizations).\ntau\nArray, size at least max (1, min(m, n)). Contains scalar factors of the\nelementary reflectors for the matrix Q.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n793\n\n\njpvt\nOverwritten by details of the permutation matrix P in the factorization A*P\n= Q*R. More precisely, the columns of AP are the columns of A in the\nfollowing order:\njpvt[0], jpvt[1], ..., jpvt[n - 1].\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nApplication Notes\nTo solve a set of least squares problems minimizing ||A*x - b||2 for all columns b of a given matrix B, you\ncan call the following:\n?geqp3 (this routine)\nto factorize A*P = Q*R;\normqr\nto compute C = QT*B (for real matrices);\nunmqr\nto compute C = QH*B (for complex matrices);\ntrsm (a BLAS routine)\nto solve R*X = C.\n(The columns of the computed X are the permuted least squares solution vectors x; the output array jpvt\nspecifies the permutation order.)\nTo compute the elements of Q explicitly, call\norgqr\n(for real matrices)\nungqr\n(for complex matrices).\n?orgqr\nGenerates the real orthogonal matrix Q of the QR\nfactorization formed by ?geqrf.\nSyntax\nlapack_int LAPACKE_sorgqr (int matrix_layout, lapack_int m, lapack_int n, lapack_int k,\nfloat* a, lapack_int lda, const float* tau);\nlapack_int LAPACKE_dorgqr (int matrix_layout, lapack_int m, lapack_int n, lapack_int k,\ndouble* a, lapack_int lda, const double* tau);\nInclude Files\n•\nmkl.h\nDescription\nThe routine generates the whole or part of m-by-m orthogonal matrix Q of the QR factorization formed by\nthe routine ?geqrf or geqpf. Use this routine after a call to sgeqrf/dgeqrf or sgeqpf/dgeqpf.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n794\n\n\nUsually Q is determined from the QR factorization of an m by p matrix A with m≥p. To compute the whole\nmatrix Q, use:\nLAPACKE_?orgqr(matrix_layout, m, m, p, a, lda, tau)\nTo compute the leading p columns of Q (which form an orthonormal basis in the space spanned by the\ncolumns of A):\nLAPACKE_?orgqr(matrix_layout, m, p, p, a, lda)\nTo compute the matrix Qk of the QR factorization of leading k columns of the matrix A:\nLAPACKE_?orgqr(matrix_layout, m, m, k, a, lda, tau)\nTo compute the leading k columns of Qk (which form an orthonormal basis in the space spanned by leading k\ncolumns of the matrix A):\nLAPACKE_?orgqr(matrix_layout, m, k, k, a, lda, tau)\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nm\nThe order of the orthogonal matrix Q (m≥ 0).\nn\nThe number of columns of Q to be computed\n(0 ≤n≤m).\nk\nThe number of elementary reflectors whose product defines the matrix Q (0\n≤k≤n).\na, tau\nArrays:\na and tau are the arrays returned by sgeqrf / dgeqrf or sgeqpf / dgeqpf.\nThe size of a is max(1, lda*n) for column major layout and max(1, lda*m)\nfor row major layout .\nThe size of tau must be at least max(1, k).\nlda\nThe leading dimension of a; at least max(1, m)for column major layout and\nmax(1, n) for row major layout.\nOutput Parameters\na\nOverwritten by n leading columns of the m-by-m orthogonal matrix Q.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nApplication Notes\nThe computed Q differs from an exactly orthogonal matrix by a matrix E such that\n||E||2 = O(ε)|*|A||2 where ε is the machine precision.\nThe total number of floating-point operations is approximately 4*m*n*k - 2*(m + n)*k2 + (4/3)*k3.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n795\n\n\nIf n = k, the number is approximately (2/3)*n2*(3m - n).\nThe complex counterpart of this routine is ungqr.\n?ormqr\nMultiplies a real matrix by the orthogonal matrix Q of\nthe QR factorization formed by ?geqrf or ?geqpf.\nSyntax\nlapack_int LAPACKE_sormqr (int matrix_layout, char side, char trans, lapack_int m,\nlapack_int n, lapack_int k, const float* a, lapack_int lda, const float* tau, float* c,\nlapack_int ldc);\nlapack_int LAPACKE_dormqr (int matrix_layout, char side, char trans, lapack_int m,\nlapack_int n, lapack_int k, const double* a, lapack_int lda, const double* tau, double*\nc, lapack_int ldc);\nInclude Files\n•\nmkl.h\nDescription\nThe routine multiplies a real matrix C by Q or QT, where Q is the orthogonal matrix Q of the QR factorization\nformed by the routine ?geqrf or ?geqpf.\nDepending on the parameters sideleft_right and trans, the routine can form one of the matrix products\nQ*C, QT*C, C*Q, or C*QT (overwriting the result on C).\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nside\nMust be either 'L' or 'R'.\nIf side='L', Q or QT is applied to C from the left.\nIf side='R', Q or QT is applied to C from the right.\ntrans\nMust be either 'N' or 'T'.\nIf trans='N', the routine multiplies C by Q.\nIf trans='T', the routine multiplies C by QT.\nm\nThe number of rows in the matrix C (m≥ 0).\nn\nThe number of columns in C (n≥ 0).\nk\nThe number of elementary reflectors whose product defines the matrix Q.\nConstraints:\n0 ≤k≤m if side='L';\n0 ≤k≤n if side='R'.\na, tau, c\nArrays:\na and tau are the arrays returned by sgeqrf / dgeqrf or sgeqpf / dgeqpf.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n796\n\n\nThe size of a is max(1, lda*k) for column major layout, max(1, lda*m) for\nrow major layout and side = 'L', and max(1, lda*n) for row major layout\nand side = 'R'.\nThe size of tau must be at least max(1, k).\nArray c of size max(1, ldc*n) for column major layout and max(1, ldc*m)\nfor row major layout contains the m-by-n matrix C.\nlda\nThe leading dimension of a. Constraints:\nif side = 'L', lda≥ max(1, m)for column major layout and max(1, k) for\nrow major layout ;\nif side = 'R', lda≥ max(1, n)for column major layout and max(1, k) for\nrow major layout.\nldc\nThe leading dimension of c. Constraint:\nldc≥ max(1, m)for column major layout and max(1, n) for row major layout.\nOutput Parameters\nc\nOverwritten by the product Q*C, QT*C, C*Q, or C*QT (as specified by side\nand trans).\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nApplication Notes\nThe complex counterpart of this routine is unmqr.\n?ungqr\nGenerates the complex unitary matrix Q of the QR\nfactorization formed by ?geqrf.\nSyntax\nlapack_int LAPACKE_cungqr (int matrix_layout, lapack_int m, lapack_int n, lapack_int k,\nlapack_complex_float* a, lapack_int lda, const lapack_complex_float* tau);\nlapack_int LAPACKE_zungqr (int matrix_layout, lapack_int m, lapack_int n, lapack_int k,\nlapack_complex_double* a, lapack_int lda, const lapack_complex_double* tau);\nInclude Files\n•\nmkl.h\nDescription\nThe routine generates the whole or part of m-by-m unitary matrix Q of the QR factorization formed by the\nroutines ?geqrf or geqpf. Use this routine after a call to cgeqrf/zgeqrf or cgeqpf/zgeqpf.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n797\n\n\nUsually Q is determined from the QR factorization of an m by p matrix A with m≥p. To compute the whole\nmatrix Q, use:\nLAPACKE_?ungqr(matrix_layout, m, m, p, a, lda, tau)\nTo compute the leading p columns of Q (which form an orthonormal basis in the space spanned by the\ncolumns of A):\nLAPACKE_?ungqr(matrix_layout, m, p, p, a, lda, tau)\nTo compute the matrix Qk of the QR factorization of the leading k columns of the matrix A:\nLAPACKE_?ungqr(matrix_layout, m, m, k, a, lda, tau)\nTo compute the leading k columns of Qk (which form an orthonormal basis in the space spanned by the\nleading k columns of the matrix A):\nLAPACKE_?ungqr(matrix_layout, m, k, k, a, lda, tau)\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nm\nThe order of the unitary matrix Q (m≥ 0).\nn\nThe number of columns of Q to be computed\n(0 ≤n≤m).\nk\nThe number of elementary reflectors whose product defines the matrix Q (0\n≤k≤n).\na, tau\nArrays: a and tau are the arrays returned by cgeqrf/zgeqrf or cgeqpf/\nzgeqpf.\nThe size of a is max(1, lda*n) for column major layout and max(1, lda*m)\nfor row major layout .\nThe size of tau must be at least max(1, k).\nlda\nThe leading dimension of a; at least max(1, m)for column major layout and\nmax(1, n) for row major layout.\nOutput Parameters\na\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nApplication Notes\nThe computed Q differs from an exactly unitary matrix by a matrix E such that ||E||2 = O(ε)*||A||2,\nwhere ε is the machine precision.\nThe total number of floating-point operations is approximately 16*m*n*k - 8*(m + n)*k2 + (16/3)*k3.\nIf n = k, the number is approximately (8/3)*n2*(3m - n).\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n798\n\n\nThe real counterpart of this routine is orgqr.\n?unmqr\nMultiplies a complex matrix by the unitary matrix Q of\nthe QR factorization formed by ?geqrf.\nSyntax\nlapack_int LAPACKE_cunmqr (int matrix_layout, char side, char trans, lapack_int m,\nlapack_int n, lapack_int k, const lapack_complex_float* a, lapack_int lda, const\nlapack_complex_float* tau, lapack_complex_float* c, lapack_int ldc);\nlapack_int LAPACKE_zunmqr (int matrix_layout, char side, char trans, lapack_int m,\nlapack_int n, lapack_int k, const lapack_complex_double* a, lapack_int lda, const\nlapack_complex_double* tau, lapack_complex_double* c, lapack_int ldc);\nInclude Files\n•\nmkl.h\nDescription\nThe routine multiplies a rectangular complex matrix C by Q or QH, where Q is the unitary matrix Q of the QR\nfactorization formed by the routines ?geqrf or geqpf.\nDepending on the parameters side and trans, the routine can form one of the matrix products Q*C, QH*C,\nC*Q, or C*QH (overwriting the result on C).\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nside\nMust be either 'L' or 'R'.\nIf side = 'L', Q or QH is applied to C from the left.\nIf side = 'R', Q or QH is applied to C from the right.\ntrans\nMust be either 'N' or 'C'.\nIf trans = 'N', the routine multiplies C by Q.\nIf trans = 'C', the routine multiplies C by QH.\nm\nThe number of rows in the matrix C (m≥ 0).\nn\nThe number of columns in C (n≥ 0).\nk\nThe number of elementary reflectors whose product defines the matrix Q.\nConstraints:\n0 ≤k≤m if side = 'L';\n0 ≤k≤n if side = 'R'.\na, c, tau\nArrays:\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n799\n\n\na size max(1, lda*k) for column major layout, max(1, lda*m) for row\nmajor layout when side ='L', and max(1, lda*n) for row major layout\nwhen side ='R' and tau are the arrays returned by cgeqrf / zgeqrf or\ncgeqpf / zgeqpf.\nThe size of tau must be at least max(1, k).\nc(size max(1, ldc*n) for column major layout and max(1, ldc*m for row\nmajor layout) contains the m-by-n matrix C.\nlda\nThe leading dimension of a. Constraints:\nlda≥ max(1, m) for column major layout and lda≥ max(1, k) for row\nmajor layout if side = 'L';\nlda≥ max(1, n) for column major layout and lda≥ max(1, k) for row\nmajor layout if side = 'R'.\nldc\nThe leading dimension of c. Constraint:\nldc≥ max(1, m) for column major layout and max(1, n) for row major\nlayout.\nOutput Parameters\nc\nOverwritten by the product Q*C, QH*C, C*Q, or C*QH (as specified by side\nand trans).\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nApplication Notes\nThe real counterpart of this routine is ormqr.\n?gelqf\nComputes the LQ factorization of a general m-by-n\nmatrix.\nSyntax\nlapack_int LAPACKE_sgelqf (int matrix_layout, lapack_int m, lapack_int n, float* a,\nlapack_int lda, float* tau);\nlapack_int LAPACKE_dgelqf (int matrix_layout, lapack_int m, lapack_int n, double* a,\nlapack_int lda, double* tau);\nlapack_int LAPACKE_cgelqf (int matrix_layout, lapack_int m, lapack_int n,\nlapack_complex_float* a, lapack_int lda, lapack_complex_float* tau);\nlapack_int LAPACKE_zgelqf (int matrix_layout, lapack_int m, lapack_int n,\nlapack_complex_double* a, lapack_int lda, lapack_complex_double* tau);\nInclude Files\n•\nmkl.h\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n800\n\n\nDescription\nThe routine forms the LQ factorization of a general m-by-n matrix A (see Orthogonal Factorizations). No\npivoting is performed.\nThe routine does not form the matrix Q explicitly. Instead, Q is represented as a product of min(m, n)\nelementary reflectors. Routines are provided to work with Q in this representation.\nNOTE\nThis routine supports the Progress Routine feature. See Progress Function for details.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nm\nThe number of rows in the matrix A (m≥ 0).\nn\nThe number of columns in A (n≥ 0).\na\nArray a of size max(1, lda*n) for column major layout and max(1, lda*m)\nfor row major layout contains the matrix A.\nlda\nThe leading dimension of a; at least max(1, m) for column major layout and\nmax(1, n) for row major layout.\nOutput Parameters\na\nOverwritten by the factorization data as follows:\nThe elements on and below the diagonal of the array contain the m-by-\nmin(m,n) lower trapezoidal matrix L (L is lower triangular if m≤n); the\nelements above the diagonal, with the array tau, represent the orthogonal\nmatrix Q as a product of elementary reflectors.\ntau\nArray, size at least max(1, min(m, n)).\nContains scalars that define elementary reflectors for the matrix Q (see \nOrthogonal Factorizations).\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nApplication Notes\nThe computed factorization is the exact factorization of a matrix A + E, where\n||E||2 = O(ε) ||A||2.\nThe approximate number of floating-point operations for real flavors is\n(4/3)n3\nif m = n,\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n801\n\n\n(2/3)n2(3m-n)\nif m > n,\n(2/3)m2(3n-m)\nif m < n.\nThe number of operations for complex flavors is 4 times greater.\nTo find the minimum-norm solution of an underdetermined least squares problem minimizing ||A*x - b||2\nfor all columns b of a given matrix B, you can call the following:\n?gelqf (this routine)\nto factorize A = L*Q;\ntrsm (a BLAS routine)\nto solve L*Y = B for Y;\normlq\nto compute X = (Q1)T*Y (for real matrices);\nunmlq\nto compute X = (Q1)H*Y (for complex matrices).\n(The columns of the computed X are the minimum-norm solution vectors x. Here A is an m-by-n matrix with\nm < n; Q1 denotes the first m columns of Q).\nTo compute the elements of Q explicitly, call\norglq\n(for real matrices)\nunglq\n(for complex matrices).\nSee Also\nmkl_progress\nMatrix Storage Schemes\n?orglq\nGenerates the real orthogonal matrix Q of the LQ\nfactorization formed by ?gelqf.\nSyntax\nlapack_int LAPACKE_sorglq (int matrix_layout, lapack_int m, lapack_int n, lapack_int k,\nfloat* a, lapack_int lda, const float* tau);\nlapack_int LAPACKE_dorglq (int matrix_layout, lapack_int m, lapack_int n, lapack_int k,\ndouble* a, lapack_int lda, const double* tau);\nInclude Files\n•\nmkl.h\nDescription\nThe routine generates the whole or part of n-by-n orthogonal matrix Q of the LQ factorization formed by the\nroutines gelqf. Use this routine after a call to sgelqf/dgelqf.\nUsually Q is determined from the LQ factorization of an p-by-n matrix A with n≥p. To compute the whole\nmatrix Q, use:\ninfo = LAPACKE_?orglq(matrix_layout, n, n, p, a, lda, tau)\nTo compute the leading p rows of Q, which form an orthonormal basis in the space spanned by the rows of A,\nuse:\ninfo = LAPACKE_?orglq(matrix_layout, p, n, p, a, lda, tau)\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n802\n\n\nTo compute the matrix Qk of the LQ factorization of the leading k rows of A, use:\ninfo = LAPACKE_?orglq(matrix_layout, n, n, k, a, lda, tau)\nTo compute the leading k rows of Qk, which form an orthonormal basis in the space spanned by the leading k\nrows of A, use:\ninfo = LAPACKE_?orgqr(matrix_layout, k, n, k, a, lda, tau)\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nm\nThe number of rows of Q to be computed\n(0 ≤m≤n).\nn\nThe order of the orthogonal matrix Q (n≥m).\nk\nThe number of elementary reflectors whose product defines the matrix Q (0\n≤k≤m).\na, tau\nArrays: a (size max(1, lda*n) for column major layout and max(1, lda*m)\nfor row major layout) and tau are the arrays returned by sgelqf/dgelqf.\nThe size of tau must be at least max(1, k).\nlda\nThe leading dimension of a; at least max(1, m)for column major layout and\nmax(1, n) for row major layout.\nOutput Parameters\na\nOverwritten by m leading rows of the n-by-n orthogonal matrix Q.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nApplication Notes\nThe computed Q differs from an exactly orthogonal matrix by a matrix E such that ||E||2 = O(ε)*||A||2,\nwhere ε is the machine precision.\nThe total number of floating-point operations is approximately 4*m*n*k - 2*(m + n)*k2 + (4/3)*k3.\nIf m = k, the number is approximately (2/3)*m2*(3n - m).\nThe complex counterpart of this routine is unglq.\n?ormlq\nMultiplies a real matrix by the orthogonal matrix Q of\nthe LQ factorization formed by ?gelqf.\nSyntax\nlapack_int LAPACKE_sormlq (int matrix_layout, char side, char trans, lapack_int m,\nlapack_int n, lapack_int k, const float* a, lapack_int lda, const float* tau, float* c,\nlapack_int ldc);\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n803\n\n\nlapack_int LAPACKE_dormlq (int matrix_layout, char side, char trans, lapack_int m,\nlapack_int n, lapack_int k, const double* a, lapack_int lda, const double* tau, double*\nc, lapack_int ldc);\nInclude Files\n•\nmkl.h\nDescription\nThe routine multiplies a real m-by-n matrix C by Q or QT, where Q is the orthogonal matrix Q of the LQ\nfactorization formed by the routine gelqf.\nDepending on the parameters side and trans, the routine can form one of the matrix products Q*C, QT*C,\nC*Q, or C*QT (overwriting the result on C).\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nside\nMust be either 'L' or 'R'.\nIf side = 'L', Q or QT is applied to C from the left.\nIf side = 'R', Q or QT is applied to C from the right.\ntrans\nMust be either 'N' or 'T'.\nIf trans = 'N', the routine multiplies C by Q.\nIf trans = 'T', the routine multiplies C by QT.\nm\nThe number of rows in the matrix C (m≥ 0).\nn\nThe number of columns in C (n≥ 0).\nk\nThe number of elementary reflectors whose product defines the matrix Q.\nConstraints:\n0 ≤k≤m if side = 'L';\n0 ≤k≤n if side = 'R'.\na, c, tau\nArrays:\na and tau are arrays returned by ?gelqf.\nThe size of a must be:\nFor side = 'L' and column major layout, max(1, lda*m).\nFor side = 'R' and column major layout, max(1, lda*n).\nFor row major layout regardless of side, max(1, lda*k).\nThe dimension of tau must be at least max(1, k).\nc(size max(1, ldc*n) for column major layout and max(1, ldc*m for row\nmajor layout) contains the m-by-n matrix C.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n804\n\n\nlda\nThe leading dimension of a. For column major layout, lda≥ max(1, k). For\nrow major layout, if side = 'L', lda≥ max(1, m), or, if side = 'R', lda≥\nmax(1, n).\nldc\nThe leading dimension of c; ldc≥ max(1, m) for column major layout and\nmax(1, n) for row major layout.\nOutput Parameters\nc\nOverwritten by the product Q*C, QT*C, C*Q, or C*QT (as specified by side\nand trans).\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nApplication Notes\nThe complex counterpart of this routine is unmlq.\n?unglq\nGenerates the complex unitary matrix Q of the LQ\nfactorization formed by ?gelqf.\nSyntax\nlapack_int LAPACKE_cunglq (int matrix_layout, lapack_int m, lapack_int n, lapack_int k,\nlapack_complex_float* a, lapack_int lda, const lapack_complex_float* tau);\nlapack_int LAPACKE_zunglq (int matrix_layout, lapack_int m, lapack_int n, lapack_int k,\nlapack_complex_double* a, lapack_int lda, const lapack_complex_double* tau);\nInclude Files\n•\nmkl.h\nDescription\nThe routine generates the whole or part of n-by-n unitary matrix Q of the LQ factorization formed by the\nroutines gelqf. Use this routine after a call to cgelqf/zgelqf.\nUsually Q is determined from the LQ factorization of an p-by-n matrix A with n < p. To compute the whole\nmatrix Q, use:\ninfo = LAPACKE_?unglq(matrix_layout, n, n, p, a, lda, tau)\nTo compute the leading p rows of Q, which form an orthonormal basis in the space spanned by the rows of A,\nuse:\ninfo = LAPACKE_?unglq(matrix_layout, p, n, p, a, lda, tau)\nTo compute the matrix Qk of the LQ factorization of the leading k rows of A, use:\ninfo = LAPACKE_?unglq(matrix_layout, n, n, k, a, lda, tau)\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n805\n\n\nTo compute the leading k rows of Qk, which form an orthonormal basis in the space spanned by the leading k\nrows of A, use:\ninfo = LAPACKE_?ungqr(matrix_layout, k, n, k, a, lda, tau)\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nm\nThe number of rows of Q to be computed (0 ≤m≤n).\nn\nThe order of the unitary matrix Q (n≥m).\nk\nThe number of elementary reflectors whose product defines the matrix Q (0\n≤k≤m).\na, tau\nArrays: a (size max(1, lda*n) for column major layout and max(1, lda*m)\nfor row major layout) and tau are the arrays returned by cgelqf/zgelqf.\nThe dimension of tau must be at least max(1, k).\nlda\nThe leading dimension of a; at least max(1, m)for column major layout and\nmax(1, n) for row major layout.\nOutput Parameters\na\nOverwritten by m leading rows of the n-by-n unitary matrix Q.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nApplication Notes\nThe computed Q differs from an exactly unitary matrix by a matrix E such that ||E||2 = O(ε)*||A||2,\nwhere ε is the machine precision.\nThe total number of floating-point operations is approximately 16*m*n*k - 8*(m + n)*k2 + (16/3)*k3.\nIf m = k, the number is approximately (8/3)*m2*(3n - m) .\nThe real counterpart of this routine is orglq.\n?unmlq\nMultiplies a complex matrix by the unitary matrix Q of\nthe LQ factorization formed by ?gelqf.\nSyntax\nlapack_int LAPACKE_cunmlq (int matrix_layout, char side, char trans, lapack_int m,\nlapack_int n, lapack_int k, const lapack_complex_float* a, lapack_int lda, const\nlapack_complex_float* tau, lapack_complex_float* c, lapack_int ldc);\nlapack_int LAPACKE_zunmlq (int matrix_layout, char side, char trans, lapack_int m,\nlapack_int n, lapack_int k, const lapack_complex_double* a, lapack_int lda, const\nlapack_complex_double* tau, lapack_complex_double* c, lapack_int ldc);\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n806\n\n\nInclude Files\n•\nmkl.h\nDescription\nThe routine multiplies a real m-by-n matrix C by Q or QH, where Q is the unitary matrix Q of the LQ\nfactorization formed by the routine gelqf.\nDepending on the parameters side and trans, the routine can form one of the matrix products Q*C, QH*C,\nC*Q, or C*QH (overwriting the result on C).\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nside\nMust be either 'L' or 'R'.\nIf side = 'L', Q or QH is applied to C from the left.\nIf side = 'R', Q or QH is applied to C from the right.\ntrans\nMust be either 'N' or 'C'.\nIf trans = 'N', the routine multiplies C by Q.\nIf trans = 'C', the routine multiplies C by QH.\nm\nThe number of rows in the matrix C (m≥ 0).\nn\nThe number of columns in C (n≥ 0).\nk\nThe number of elementary reflectors whose product defines the matrix Q.\nConstraints:\n0 ≤k≤m if side = 'L';\n0 ≤k≤n if side = 'R'.\na, c, tau\nArrays:\na and tau are arrays returned by ?gelqf.\nThe size of a must be:\nFor side = 'L' and column major layout, max(1, lda*m).\nFor side = 'R' and column major layout, max(1, lda*n).\nFor row major layout regardless of side, max(1, lda*k).\nThe size of tau must be at least max(1, k).\nc(size max(1, ldc*n) for column major layout and max(1, ldc*m for row\nmajor layout) contains the m-by-n matrix C.\nlda\nThe leading dimension of a. For column major layout, lda≥ max(1, k). For\nrow major layout, if side = 'L', lda≥ max(1, m), or, if side = 'R', lda≥\nmax(1, n).\nldc\nThe leading dimension of c; ldc≥ max(1, m) for column major layout and\nmax(1, n) for row major layout.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n807\n\n\nOutput Parameters\nc\nOverwritten by the product Q*C, QH*C, C*Q, or C*QH (as specified by side\nand trans).\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nApplication Notes\nThe real counterpart of this routine is ormlq.\n?geqlf\nComputes the QL factorization of a general m-by-n\nmatrix.\nSyntax\nlapack_int LAPACKE_sgelqf (int matrix_layout, lapack_int m, lapack_int n, float* a,\nlapack_int lda, float* tau);\nlapack_int LAPACKE_dgelqf (int matrix_layout, lapack_int m, lapack_int n, double* a,\nlapack_int lda, double* tau);\nlapack_int LAPACKE_cgelqf (int matrix_layout, lapack_int m, lapack_int n,\nlapack_complex_float* a, lapack_int lda, lapack_complex_float* tau);\nlapack_int LAPACKE_zgelqf (int matrix_layout, lapack_int m, lapack_int n,\nlapack_complex_double* a, lapack_int lda, lapack_complex_double* tau);\nInclude Files\n•\nmkl.h\nDescription\nThe routine forms the QL factorization of a general m-by-n matrix A (see Orthogonal Factorizations). No\npivoting is performed.\nThe routine does not form the matrix Q explicitly. Instead, Q is represented as a product of min(m, n)\nelementary reflectors. Routines are provided to work with Q in this representation.\nNOTE\nThis routine supports the Progress Routine feature. See Progress Function for details.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nm\nThe number of rows in the matrix A (m≥ 0).\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n808\n\n\nn\nThe number of columns in A (n≥ 0).\na\nArray a of size max(1, lda*n) for column major layout and max(1, lda*m)\nfor row major layout contains the matrix A.\nlda\nThe leading dimension of a; at least max(1, m)for column major layout and\nmax(1, n) for row major layout.\nOutput Parameters\na\nOverwritten on exit by the factorization data as follows:\nif m≥n, the lower triangle of the subarray a(m-n+1:m, 1:n) contains the n-\nby-n lower triangular matrix L; if m≤n, the elements on and below the (n-\nm)-th superdiagonal contain the m-by-n lower trapezoidal matrix L; in both\ncases, the remaining elements, with the array tau, represent the\northogonal/unitary matrix Q as a product of elementary reflectors.\ntau\nArray, size at least max(1, min(m, n)). Contains scalar factors of the\nelementary reflectors for the matrix Q (see Orthogonal Factorizations).\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nApplication Notes\nRelated routines include:\norgql\nto generate matrix Q (for real matrices);\nungql\nto generate matrix Q (for complex matrices);\normql\nto apply matrix Q (for real matrices);\nunmql\nto apply matrix Q (for complex matrices).\nSee Also\nmkl_progress\nMatrix Storage Schemes\n?orgql\nGenerates the real matrix Q of the QL factorization\nformed by ?geqlf.\nSyntax\nlapack_int LAPACKE_sorgql (int matrix_layout, lapack_int m, lapack_int n, lapack_int k,\nfloat* a, lapack_int lda, const float* tau);\nlapack_int LAPACKE_dorgql (int matrix_layout, lapack_int m, lapack_int n, lapack_int k,\ndouble* a, lapack_int lda, const double* tau);\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n809\n\n\nInclude Files\n•\nmkl.h\nDescription\nThe routine generates an m-by-n real matrix Q with orthonormal columns, which is defined as the last n\ncolumns of a product of k elementary reflectors H(i) of order m: Q = H(k) *...* H(2)*H(1) as returned\nby the routines geqlf. Use this routine after a call to sgeqlf/dgeqlf.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nm\nThe number of rows of the matrix Q (m≥ 0).\nn\nThe number of columns of the matrix Q (m≥ n≥ 0).\nk\nThe number of elementary reflectors whose product defines the matrix Q\n(n≥ k≥ 0).\na, tau\nArrays: a (size max(1, lda*n) for column major layout and max(1, lda*m)\nfor row major layout), tau.\nOn entry, the (n - k + i)th column of a must contain the vector which\ndefines the elementary reflector H(i), for i = 1,2,...,k, as returned by\nsgeqlf/dgeqlf in the last k columns of its array argument a; tau[i - 1]\nmust contain the scalar factor of the elementary reflector H(i), as returned\nby sgeqlf/dgeqlf;\nThe size of tau must be at least max(1, k).\nlda\nThe leading dimension of a; at least max(1, m)for column major layout and\nmax(1, n) for row major layout.\nOutput Parameters\na\nOverwritten by the last n columns of the m-by-m orthogonal matrix Q.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nApplication Notes\nThe complex counterpart of this routine is ungql.\n?ungql\nGenerates the complex matrix Q of the QL\nfactorization formed by ?geqlf.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n810\n\n\nSyntax\nlapack_int LAPACKE_cungql (int matrix_layout, lapack_int m, lapack_int n, lapack_int k,\nlapack_complex_float* a, lapack_int lda, const lapack_complex_float* tau);\nlapack_int LAPACKE_zungql (int matrix_layout, lapack_int m, lapack_int n, lapack_int k,\nlapack_complex_double* a, lapack_int lda, const lapack_complex_double* tau);\nInclude Files\n•\nmkl.h\nDescription\nThe routine generates an m-by-n complex matrix Q with orthonormal columns, which is defined as the last n\ncolumns of a product of k elementary reflectors H(i) of order m: Q = H(k) *...* H(2)*H(1) as returned\nby the routines geqlf/geqlf . Use this routine after a call to cgeqlf/zgeqlf.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nm\nThe number of rows of the matrix Q (m≥0).\nn\nThe number of columns of the matrix Q (m≥n≥0).\nk\nThe number of elementary reflectors whose product defines the matrix Q\n(n≥k≥0).\na, tau\nArrays: a (size max(1, lda*n) for column major layout and max(1, lda*m)\nfor row major layout), tau.\nOn entry, the (n - k + i)th column of a must contain the vector which\ndefines the elementary reflector H(i), for i = 1,2,...,k, as returned by\ncgeqlf/zgeqlf in the last k columns of its array argument a;\ntau[i - 1] must contain the scalar factor of the elementary reflector H(i), as\nreturned by cgeqlf/zgeqlf;\nThe size of tau must be at least max(1, k).\nlda\nThe leading dimension of a; at least max(1, m)for column major layout and\nmax(1, n) for row major layout.\nOutput Parameters\na\nOverwritten by the last n columns of the m-by-m unitary matrix Q.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nApplication Notes\nThe real counterpart of this routine is orgql.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n811\n\n\n?ormql\nMultiplies a real matrix by the orthogonal matrix Q of\nthe QL factorization formed by ?geqlf.\nSyntax\nlapack_int LAPACKE_sormql (int matrix_layout, char side, char trans, lapack_int m,\nlapack_int n, lapack_int k, const float* a, lapack_int lda, const float* tau, float* c,\nlapack_int ldc);\nlapack_int LAPACKE_dormql (int matrix_layout, char side, char trans, lapack_int m,\nlapack_int n, lapack_int k, const double* a, lapack_int lda, const double* tau, double*\nc, lapack_int ldc);\nInclude Files\n•\nmkl.h\nDescription\nThe routine multiplies a real m-by-n matrix C by Q or QT, where Q is the orthogonal matrix Q of the QL\nfactorization formed by the routine geqlf.\nDepending on the parameters side and trans, the routine ormql can form one of the matrix products Q*C,\nQT*C, C*Q, or C*QT (overwriting the result over C).\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nside\nMust be either 'L' or 'R'.\nIf side = 'L', Q or QT is applied to C from the left.\nIf side = 'R', Q or QT is applied to C from the right.\ntrans\nMust be either 'N' or 'T'.\nIf trans = 'N', the routine multiplies C by Q.\nIf trans = 'T', the routine multiplies C by QT.\nm\nThe number of rows in the matrix C (m≥ 0).\nn\nThe number of columns in C (n≥ 0).\nk\nThe number of elementary reflectors whose product defines the matrix Q.\nConstraints:\n0 ≤k≤m if side = 'L';\n0 ≤k≤n if side = 'R'.\na, tau, c\nArrays: a, tau, c.\nThe size of a must be:\nFor column major layout regardless of side, max(1, lda*k).\nFor side = 'L' and row major layout, max(1, lda*m).\nFor side = 'R' and row major layout, max(1, lda*n).\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n812\n\n\nOn entry, the ith column of a must contain the vector which defines the\nelementary reflector Hi, for i = 1,2,...,k, as returned by sgeqlf/dgeqlf in\nthe last k columns of its array argument a.\ntau[i - 1] must contain the scalar factor of the elementary reflector Hi, as\nreturned by sgeqlf/dgeqlf.\nThe size of tau must be at least max(1, k).\nc(size max(1, ldc*n) for column major layout and max(1, ldc*m) for row\nmajor layout) contains the m-by-n matrix C.\nlda\nThe leading dimension of a;\nif side = 'L', lda≥ max(1, m)for column major layout and max(1, k) for\nrow major layout ;\nif side = 'R', lda≥ max(1, n)for column major layout and max(1, k) for\nrow major layout.\nldc\nThe leading dimension of c; ldc≥ max(1, m)for column major layout and\nmax(1, n) for row major layout.\nOutput Parameters\nc\nOverwritten by the product Q*C, QT*C, C*Q, or C*QT (as specified by side\nand trans).\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nApplication Notes\nThe complex counterpart of this routine is unmql.\n?unmql\nMultiplies a complex matrix by the unitary matrix Q of\nthe QL factorization formed by ?geqlf.\nSyntax\nlapack_int LAPACKE_cunmql (int matrix_layout, char side, char trans, lapack_int m,\nlapack_int n, lapack_int k, const lapack_complex_float* a, lapack_int lda, const\nlapack_complex_float* tau, lapack_complex_float* c, lapack_int ldc);\nlapack_int LAPACKE_zunmql (int matrix_layout, char side, char trans, lapack_int m,\nlapack_int n, lapack_int k, const lapack_complex_double* a, lapack_int lda, const\nlapack_complex_double* tau, lapack_complex_double* c, lapack_int ldc);\nInclude Files\n•\nmkl.h\nDescription\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n813\n\n\nThe routine multiplies a complex m-by-n matrix C by Q or QH, where Q is the unitary matrix Q of the QL\nfactorization formed by the routine geqlf.\nDepending on the parameters side and trans, the routine unmql can form one of the matrix products Q*C,\nQH*C, C*Q, or C*QH (overwriting the result over C).\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nside\nMust be either 'L' or 'R'.\nIf side = 'L', Q or QH is applied to C from the left.\nIf side = 'R', Q or QH is applied to C from the right.\ntrans\nMust be either 'N' or 'C'.\nIf trans = 'N', the routine multiplies C by Q.\nIf trans = 'C', the routine multiplies C by QH.\nm\nThe number of rows in the matrix C (m≥ 0).\nn\nThe number of columns in C (n≥ 0).\nk\nThe number of elementary reflectors whose product defines the matrix Q.\nConstraints:\n0 ≤k≤m if side = 'L';\n0 ≤k≤n if side = 'R'.\na, tau, c\nArrays: a, tau, c.\nThe size of a must be:\nFor column major layout regardless of side, max(1, lda*k).\nFor side = 'L' and row major layout, max(1, lda*m).\nFor side = 'R' and row major layout, max(1, lda*n).\nOn entry, the i-th column of a must contain the vector which defines the\nelementary reflector H(i), for i = 1,2,...,k, as returned by cgeqlf/zgeqlf\nin the last k columns of its array argument a.\ntau[i - 1] must contain the scalar factor of the elementary reflector H(i),\nas returned by cgeqlf/zgeqlf.\nThe size of tau must be at least max(1, k).\nc(size max(1, ldc*n) for column major layout and max(1, ldc*m for row\nmajor layout) contains the m-by-n matrix C.\nlda\nThe leading dimension of a.\nIf side = 'L', lda≥ max(1, m)for column major layout and max(1, k) for\nrow major layout.\nIf side = 'R', lda≥ max(1, n)for column major layout and max(1, k) for\nrow major layout.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n814\n\n\nldc\nThe leading dimension of c; ldc≥ max(1, m)for column major layout and\nmax(1, n) for row major layout.\nOutput Parameters\nc\nOverwritten by the product Q*C, QH*C, C*Q, or C*QH (as specified by side\nand trans).\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nApplication Notes\nThe real counterpart of this routine is ormql.\n?gerqf\nComputes the RQ factorization of a general m-by-n\nmatrix.\nSyntax\nlapack_int LAPACKE_sgerqf (int matrix_layout, lapack_int m, lapack_int n, float* a,\nlapack_int lda, float* tau);\nlapack_int LAPACKE_dgerqf (int matrix_layout, lapack_int m, lapack_int n, double* a,\nlapack_int lda, double* tau);\nlapack_int LAPACKE_cgerqf (int matrix_layout, lapack_int m, lapack_int n,\nlapack_complex_float* a, lapack_int lda, lapack_complex_float* tau);\nlapack_int LAPACKE_zgerqf (int matrix_layout, lapack_int m, lapack_int n,\nlapack_complex_double* a, lapack_int lda, lapack_complex_double* tau);\nInclude Files\n•\nmkl.h\nDescription\nThe routine forms the RQ factorization of a general m-by-n matrix A(see Orthogonal Factorizations). No\npivoting is performed.\nThe routine does not form the matrix Q explicitly. Instead, Q is represented as a product of min(m, n)\nelementary reflectors. Routines are provided to work with Q in this representation.\nNOTE\nThis routine supports the Progress Routine feature. See Progress Function for details.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n815\n\n\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nm\nThe number of rows in the matrix A (m≥ 0).\nn\nThe number of columns in A (n≥ 0).\na\nArray a of size max(1, lda*n) for column major layout and max(1, lda*m)\nfor row major layout contains the m-by-n matrix A.\nlda\nThe leading dimension of a; at least max(1, m)for column major layout and\nmax(1, n) for row major layout.\nOutput Parameters\na\nOverwritten on exit by the factorization data as follows:\nif m≤n, the upper triangle of the subarray\na(1:m, n-m+1:n ) contains the m-by-m upper triangular matrix R;\nif m≥n, the elements on and above the (m-n)th subdiagonal contain the m-\nby-n upper trapezoidal matrix R;\nin both cases, the remaining elements, with the array tau, represent the\northogonal/unitary matrix Q as a product of min(m,n) elementary\nreflectors.\ntau\nArray, size at least max (1, min(m, n)). (See Orthogonal Factorizations.)\nContains scalar factors of the elementary reflectors for the matrix Q.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nApplication Notes\nRelated routines include:\norgrq\nto generate matrix Q (for real matrices);\nungrq\nto generate matrix Q (for complex matrices);\normrq\nto apply matrix Q (for real matrices);\nunmrq\nto apply matrix Q (for complex matrices).\nSee Also\nmkl_progress\nMatrix Storage Schemes\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n816\n\n\n?orgrq\nGenerates the real matrix Q of the RQ factorization\nformed by ?gerqf.\nSyntax\nlapack_int LAPACKE_sorgrq (int matrix_layout, lapack_int m, lapack_int n, lapack_int k,\nfloat* a, lapack_int lda, const float* tau);\nlapack_int LAPACKE_dorgrq (int matrix_layout, lapack_int m, lapack_int n, lapack_int k,\ndouble* a, lapack_int lda, const double* tau);\nInclude Files\n•\nmkl.h\nDescription\nThe routine generates an m-by-n real matrix with orthonormal rows, which is defined as the last m rows of a\nproduct of k elementary reflectors H(i) of order n: Q = H(1)* H(2)*...*H(k)as returned by the routines \ngerqf. Use this routine after a call to sgerqf/dgerqf.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nm\nThe number of rows of the matrix Q (m≥ 0).\nn\nThe number of columns of the matrix Q (n≥ m).\nk\nThe number of elementary reflectors whose product defines the matrix Q\n(m≥ k≥ 0).\na, tau\nArrays: a(size max(1, lda*n) for column major layout and max(1, lda*m)\nfor row major layout), tau.\nOn entry, the (m - k + i)-th row of a must contain the vector which defines\nthe elementary reflector H(i), for i = 1,2,...,k, as returned by sgerqf/\ndgerqf in the last k rows of its array argument a;\ntau[i - 1] must contain the scalar factor of the elementary reflector H(i),\nas returned by sgerqf/dgerqf;\nThe size of tau must be at least max(1, k).\nlda\nThe leading dimension of a; at least max(1, m)for column major layout and\nmax(1, n) for row major layout.\nOutput Parameters\na\nOverwritten by the last m rows of the n-by-n orthogonal matrix Q.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n817\n\n\nApplication Notes\nThe complex counterpart of this routine is ungrq.\n?ungrq\nGenerates the complex matrix Q of the RQ\nfactorization formed by ?gerqf.\nSyntax\nlapack_int LAPACKE_cungrq (int matrix_layout, lapack_int m, lapack_int n, lapack_int k,\nlapack_complex_float* a, lapack_int lda, const lapack_complex_float* tau);\nlapack_int LAPACKE_zungrq (int matrix_layout, lapack_int m, lapack_int n, lapack_int k,\nlapack_complex_double* a, lapack_int lda, const lapack_complex_double* tau);\nInclude Files\n•\nmkl.h\nDescription\nThe routine generates an m-by-n complex matrix with orthonormal rows, which is defined as the last m rows\nof a product of k elementary reflectors H(i) of order n: Q = H(1)H* H(2)H*...*H(k)H as returned by the\nroutines gerqf. Use this routine after a call to cgerqf/zgerqf.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nm\nThe number of rows of the matrix Q (m≥0).\nn\nThe number of columns of the matrix Q (n≥m ).\nk\nThe number of elementary reflectors whose product defines the matrix Q\n(m≥k≥0).\na, tau\nArrays: a(size max(1, lda*n) for column major layout and max(1, lda*m)\nfor row major layout), tau.\nOn entry, the (m - k + i)th row of a must contain the vector which defines\nthe elementary reflector H(i), for i = 1,2,...,k, as returned by cgerqf/\nzgerqf in the last k rows of its array argument a;\ntau[i - 1] must contain the scalar factor of the elementary reflector H(i), as\nreturned by cgerqf/zgerqf;\nThe size of tau must be at least max(1, k).\nlda\nThe leading dimension of a; at least max(1, m)for column major layout and\nmax(1, n) for row major layout.\nOutput Parameters\na\nOverwritten by the m last rows of the n-by-n unitary matrix Q.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n818\n\n\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nApplication Notes\nThe real counterpart of this routine is orgrq.\n?ormrq\nMultiplies a real matrix by the orthogonal matrix Q of\nthe RQ factorization formed by ?gerqf.\nSyntax\nlapack_int LAPACKE_sormrq (int matrix_layout, char side, char trans, lapack_int m,\nlapack_int n, lapack_int k, const float* a, lapack_int lda, const float* tau, float* c,\nlapack_int ldc);\nlapack_int LAPACKE_dormrq (int matrix_layout, char side, char trans, lapack_int m,\nlapack_int n, lapack_int k, const double* a, lapack_int lda, const double* tau, double*\nc, lapack_int ldc);\nInclude Files\n•\nmkl.h\nDescription\nThe routine multiplies a real m-by-n matrix C by Q or QT, where Q is the real orthogonal matrix defined as a\nproduct of k elementary reflectors Hi : Q = H1H2 ... Hk as returned by the RQ factorization routine gerqf.\nDepending on the parameters side and trans, the routine can form one of the matrix products Q*C, QT*C,\nC*Q, or C*QT (overwriting the result over C).\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nside\nMust be either 'L' or 'R'.\nIf side = 'L', Q or QT is applied to C from the left.\nIf side = 'R', Q or QT is applied to C from the right.\ntrans\nMust be either 'N' or 'T'.\nIf trans = 'N', the routine multiplies C by Q.\nIf trans = 'T', the routine multiplies C by QT.\nm\nThe number of rows in the matrix C (m≥ 0).\nn\nThe number of columns in C (n≥ 0).\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n819\n\n\nk\nThe number of elementary reflectors whose product defines the matrix Q.\nConstraints:\n0 ≤k≤m, if side = 'L';\n0 ≤k≤n, if side = 'R'.\na, tau, c\nArrays: a(size for side = 'L': max(1, lda*m) for column major layout and\nmax(1, lda*k) for row major layout; for side = 'R': max(1, lda*n) for\ncolumn major layout and max(1, lda*k) for row major layout), tau, c (size\nmax(1, ldc*n) for column major layout and max(1, ldc*m) for row major\nlayout).\nOn entry, the ith row of a must contain the vector which defines the\nelementary reflector Hi, for i = 1,2,...,k, as returned by sgerqf/dgerqf in\nthe last k rows of its array argument a.\ntau[i - 1] must contain the scalar factor of the elementary reflector Hi, as\nreturned by sgerqf/dgerqf.\nThe size of tau must be at least max(1, k).\nc contains the m-by-n matrix C.\nlda\nThe leading dimension of a; lda≥ max(1, k)for column major layout. For\nrow major layout, lda≥ max(1, m) if side = 'L', and lda≥ max(1, n) if\nside = 'R'.\nldc\nThe leading dimension of c; ldc≥ max(1, m)for column major layout and\nmax(1, n) for row major layout.\nOutput Parameters\nc\nOverwritten by the product Q*C, QT*C, C*Q, or C*QT (as specified by side\nand trans).\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nApplication Notes\nThe complex counterpart of this routine is unmrq.\n?unmrq\nMultiplies a complex matrix by the unitary matrix Q of\nthe RQ factorization formed by ?gerqf.\nSyntax\nlapack_int LAPACKE_cunmrq (int matrix_layout, char side, char trans, lapack_int m,\nlapack_int n, lapack_int k, const lapack_complex_float* a, lapack_int lda, const\nlapack_complex_float* tau, lapack_complex_float* c, lapack_int ldc);\nlapack_int LAPACKE_zunmrq (int matrix_layout, char side, char trans, lapack_int m,\nlapack_int n, lapack_int k, const lapack_complex_double* a, lapack_int lda, const\nlapack_complex_double* tau, lapack_complex_double* c, lapack_int ldc);\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n820\n\n\nInclude Files\n•\nmkl.h\nDescription\nThe routine multiplies a complex m-by-n matrix C by Q or QH, where Q is the complex unitary matrix defined\nas a product of k elementary reflectors H(i) of order n: Q = H(1)H* H(2)H*...*H(k)Has returned by the\nRQ factorization routine gerqf .\nDepending on the parameters side and trans, the routine can form one of the matrix products Q*C, QH*C,\nC*Q, or C*QH (overwriting the result over C).\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nside\nMust be either 'L' or 'R'.\nIf side = 'L', Q or QH is applied to C from the left.\nIf side = 'R', Q or QH is applied to C from the right.\ntrans\nMust be either 'N' or 'C'.\nIf trans = 'N', the routine multiplies C by Q.\nIf trans = 'C', the routine multiplies C by QH.\nm\nThe number of rows in the matrix C (m≥ 0).\nn\nThe number of columns in C (n≥ 0).\nk\nThe number of elementary reflectors whose product defines the matrix Q.\nConstraints:\n0 ≤k≤m, if side = 'L';\n0 ≤k≤n, if side = 'R'.\na, tau, c\nArrays: a(size for side = 'L': max(1, lda*m) for column major layout and\nmax(1, lda*k) for row major layout; for side = 'R': max(1, lda*n) for\ncolumn major layout and max(1, lda*k) for row major layout), tau, c (size\nmax(1, ldc*n) for column major layout and max(1, ldc*m) for row major\nlayout).\nOn entry, the ith row of a must contain the vector which defines the\nelementary reflector H(i), for i = 1,2,...,k, as returned by cgerqf/zgerqf in\nthe last k rows of its array argument a.\ntau[i - 1] must contain the scalar factor of the elementary reflector H(i), as\nreturned by cgerqf/zgerqf.\nThe size of tau must be at least max(1, k).\nc(size max(1, ldc*n) for column major layout and max(1, ldc*m for row\nmajor layout) contains the m-by-n matrix C.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n821\n\n\nlda\nThe leading dimension of a; lda≥ max(1, k)for column major layout. For row\nmajor layout, lda≥ max(1, m) if side = 'L', and lda≥ max(1, n) if side\n= 'R' .\nldc\nThe leading dimension of c; ldc≥ max(1, m)for column major layout and\nmax(1, n) for row major layout.\nOutput Parameters\nc\nOverwritten by the product Q*C, QH*C, C*Q, or C*QH (as specified by side\nand trans).\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nApplication Notes\nThe real counterpart of this routine is ormrq.\n?tzrzf\nReduces the upper trapezoidal matrix A to upper\ntriangular form.\nSyntax\nlapack_int LAPACKE_stzrzf (int matrix_layout, lapack_int m, lapack_int n, float* a,\nlapack_int lda, float* tau);\nlapack_int LAPACKE_dtzrzf (int matrix_layout, lapack_int m, lapack_int n, double* a,\nlapack_int lda, double* tau);\nlapack_int LAPACKE_ctzrzf (int matrix_layout, lapack_int m, lapack_int n,\nlapack_complex_float* a, lapack_int lda, lapack_complex_float* tau);\nlapack_int LAPACKE_ztzrzf (int matrix_layout, lapack_int m, lapack_int n,\nlapack_complex_double* a, lapack_int lda, lapack_complex_double* tau);\nInclude Files\n•\nmkl.h\nDescription\nThe routine reduces the m-by-n (m≤n) real/complex upper trapezoidal matrix A to upper triangular form by\nmeans of orthogonal/unitary transformations. The upper trapezoidal matrix A = [A1 A2] = [A1:m, 1:m, A1:m, m\n+1:n] is factored as\nA = [R0]*Z,\nwhere Z is an n-by-n orthogonal/unitary matrix, R is an m-by-m upper triangular matrix, and 0 is the m-by-\n(n-m) zero matrix.\nThe ?tzrzf routine replaces the deprecated ?tzrqf routine.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n822\n\n\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nm\nThe number of rows in the matrix A (m≥ 0).\nn\nThe number of columns in A (n≥m).\na\nArray a is of size max(1, lda*n) for column major layout and max(1,\nlda*m) for row major layout.\nThe leading m-by-n upper trapezoidal part of the array a contains the\nmatrix A to be factorized.\nlda\nThe leading dimension of a; at least max(1, m)for column major layout and\nmax(1, n) for row major layout.\nOutput Parameters\na\nOverwritten on exit by the factorization data as follows:\nthe leading m-by-m upper triangular part of a contains the upper triangular\nmatrix R, and elements m +1 to n of the first m rows of a, with the array\ntau, represent the orthogonal matrix Z as a product of m elementary\nreflectors.\ntau\nArray, size at least max (1, m). Contains scalar factors of the elementary\nreflectors for the matrix Z.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nApplication Notes\nThe factorization is obtained by Householder's method. The k-th transformation matrix, Z(k), which is used\nto introduce zeros into the (m - k + 1)-th row of A, is given in the form\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n823\n\n\nwhere for real flavors\nand for complex flavors\ntau is a scalar and z(k) is an l-element vector. tau and z(k) are chosen to annihilate the elements of the k-th\nrow of A2.\nThe scalar tau is returned in the k-th element of tau and the vector u(k) in the k-th row of A, such that the\nelements of z(k) are stored in the last m - n elements of the k-th row of array a.\nThe elements of R are returned in the upper triangular part of A.\nThe matrix Z is given by\nZ = Z(1)*Z(2)*...*Z(m).\nRelated routines include:\normrz\nto apply matrix Q (for real matrices)\nunmrz\nto apply matrix Q (for complex matrices).\n?ormrz\nMultiplies a real matrix by the orthogonal matrix\ndefined from the factorization formed by ?tzrzf.\nSyntax\nlapack_int LAPACKE_sormrz (int matrix_layout, char side, char trans, lapack_int m,\nlapack_int n, lapack_int k, lapack_int l, const float* a, lapack_int lda, const float*\ntau, float* c, lapack_int ldc);\nlapack_int LAPACKE_dormrz (int matrix_layout, char side, char trans, lapack_int m,\nlapack_int n, lapack_int k, lapack_int l, const double* a, lapack_int lda, const\ndouble* tau, double* c, lapack_int ldc);\nInclude Files\n•\nmkl.h\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n824\n\n\nDescription\nThe ?ormrz routine multiplies a real m-by-n matrix C by Q or QT, where Q is the real orthogonal matrix\ndefined as a product of k elementary reflectors H(i) of order n: Q = H(1)* H(2)*...*H(k) as returned by\nthe factorization routine tzrzf .\nDepending on the parameters side and trans, the routine can form one of the matrix products Q*C, QT*C,\nC*Q, or C*QT (overwriting the result over C).\nThe matrix Q is of order m if side = 'L' and of order n if side = 'R'.\nThe ?ormrz routine replaces the deprecated ?latzm routine.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nside\nMust be either 'L' or 'R'.\nIf side = 'L', Q or QT is applied to C from the left.\nIf side = 'R', Q or QT is applied to C from the right.\ntrans\nMust be either 'N' or 'T'.\nIf trans = 'N', the routine multiplies C by Q.\nIf trans = 'T', the routine multiplies C by QT.\nm\nThe number of rows in the matrix C (m≥ 0).\nn\nThe number of columns in C (n≥ 0).\nk\nThe number of elementary reflectors whose product defines the matrix Q.\nConstraints:\n0 ≤k≤m, if side = 'L';\n0 ≤k≤n, if side = 'R'.\nl\nThe number of columns of the matrix A containing the meaningful part of\nthe Householder reflectors. Constraints:\n0 ≤l≤m, if side = 'L';\n0 ≤l≤n, if side = 'R'.\na, tau, c\nArrays: a(size for side = 'L': max(1, lda*m) for column major layout and\nmax(1, lda*k) for row major layout; for side = 'R': max(1, lda*b) for\ncolumn major layout and max(1, lda*k) for row major layout), tau, c (size\nmax(1, ldc*n) for column major layout and max(1, ldc*m) for row major\nlayout).\nOn entry, the ith row of a must contain the vector which defines the\nelementary reflector H(i), for i = 1,2,...,k, as returned by stzrzf/dtzrzf\nin the last k rows of its array argument a.\ntau[i - 1] must contain the scalar factor of the elementary reflector H(i),\nas returned by stzrzf/dtzrzf.\nThe size of tau must be at least max(1, k).\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n825\n\n\nc contains the m-by-n matrix C.\nlda\nThe leading dimension of a; lda≥ max(1, k)for column major layout. For row\nmajor layout, lda≥ max(1, m) if side = 'L', and lda≥ max(1, n) if side\n= 'R' .\nldc\nThe leading dimension of c; ldc≥ max(1, m)for column major layout and\nmax(1, n) for row major layout.\nOutput Parameters\nc\nOverwritten by the product Q*C, QT*C, C*Q, or C*QT (as specified by side\nand trans).\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nApplication Notes\nThe complex counterpart of this routine is unmrz.\n?unmrz\nMultiplies a complex matrix by the unitary matrix\ndefined from the factorization formed by ?tzrzf.\nSyntax\nlapack_int LAPACKE_cunmrz (int matrix_layout, char side, char trans, lapack_int m,\nlapack_int n, lapack_int k, lapack_int l, const lapack_complex_float* a, lapack_int\nlda, const lapack_complex_float* tau, lapack_complex_float* c, lapack_int ldc);\nlapack_int LAPACKE_zunmrz (int matrix_layout, char side, char trans, lapack_int m,\nlapack_int n, lapack_int k, lapack_int l, const lapack_complex_double* a, lapack_int\nlda, const lapack_complex_double* tau, lapack_complex_double* c, lapack_int ldc);\nInclude Files\n•\nmkl.h\nDescription\nThe routine multiplies a complex m-by-n matrix C by Q or QH, where Q is the unitary matrix defined as a\nproduct of k elementary reflectors H(i):\nQ = H(1)H* H(2)H*...*H(k)H as returned by the factorization routine tzrzf.\nDepending on the parameters side and trans, the routine can form one of the matrix products Q*C, QH*C,\nC*Q, or C*QH (overwriting the result over C).\nThe matrix Q is of order m if side = 'L' and of order n if side = 'R'.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n826\n\n\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nside\nMust be either 'L' or 'R'.\nIf side = 'L', Q or QH is applied to C from the left.\nIf side = 'R', Q or QH is applied to C from the right.\ntrans\nMust be either 'N' or 'C'.\nIf trans = 'N', the routine multiplies C by Q.\nIf trans = 'C', the routine multiplies C by QH.\nm\nThe number of rows in the matrix C (m≥ 0).\nn\nThe number of columns in C (n≥ 0).\nk\nThe number of elementary reflectors whose product defines the matrix Q.\nConstraints:\n0 ≤k≤m, if side = 'L';\n0 ≤k≤n, if side = 'R'.\nl\nThe number of columns of the matrix A containing the meaningful part of\nthe Householder reflectors. Constraints:\n0 ≤l≤m, if side = 'L';\n0 ≤l≤n, if side = 'R'.\na, tau, c\nArrays: a(size for side = 'L': max(1, lda*m) for column major layout and\nmax(1, lda*k) for row major layout; for side = 'R': max(1, lda*b) for\ncolumn major layout and max(1, lda*k) for row major layout), tau, c (size\nmax(1, ldc*n) for column major layout and max(1, ldc*m) for row major\nlayout).\nOn entry, the ith row of a must contain the vector which defines the\nelementary reflector H(i), for i = 1,2,...,k, as returned by ctzrzf/ztzrzf\nin the last k rows of its array argument a.\ntau[i - 1] must contain the scalar factor of the elementary reflector H(i),\nas returned by ctzrzf/ztzrzf.\nThe size of tau must be at least max(1, k).\nc contains the m-by-n matrix C.\nlda\nThe leading dimension of a; lda≥ max(1, k)for column major layout. For\nrow major layout, lda≥ max(1, m) if side = 'L', and lda≥ max(1, n) if\nside = 'R'.\nldc\nThe leading dimension of c; ldc≥ max(1, m)for column major layout and\nmax(1, n) for row major layout.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n827\n\n\nOutput Parameters\nc\nOverwritten by the product Q*C, QH*C, C*Q, or C*QH (as specified by side\nand trans).\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nApplication Notes\nThe real counterpart of this routine is ormrz.\n?ggqrf\nComputes the generalized QR factorization of two\nmatrices.\nSyntax\nlapack_int LAPACKE_sggqrf (int matrix_layout, lapack_int n, lapack_int m, lapack_int p,\nfloat* a, lapack_int lda, float* taua, float* b, lapack_int ldb, float* taub);\nlapack_int LAPACKE_dggqrf (int matrix_layout, lapack_int n, lapack_int m, lapack_int p,\ndouble* a, lapack_int lda, double* taua, double* b, lapack_int ldb, double* taub);\nlapack_int LAPACKE_cggqrf (int matrix_layout, lapack_int n, lapack_int m, lapack_int p,\nlapack_complex_float* a, lapack_int lda, lapack_complex_float* taua,\nlapack_complex_float* b, lapack_int ldb, lapack_complex_float* taub);\nlapack_int LAPACKE_zggqrf (int matrix_layout, lapack_int n, lapack_int m, lapack_int p,\nlapack_complex_double* a, lapack_int lda, lapack_complex_double* taua,\nlapack_complex_double* b, lapack_int ldb, lapack_complex_double* taub);\nInclude Files\n•\nmkl.h\nDescription\nThe routine forms the generalized QR factorization of an n-by-m matrix A and an n-by-p matrix B as A =\nQ*R, B = Q*T*Z, where Q is an n-by-n orthogonal/unitary matrix, Z is a p-by-p orthogonal/unitary matrix,\nand R and T assume one of the forms:\nor\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n828\n\n\nwhere R11 is upper triangular, and\nwhere T12 or T21 is a p-by-p upper triangular matrix.\nIn particular, if B is square and nonsingular, the GQR factorization of A and B implicitly gives the QR\nfactorization of B-1A as:\nB-1*A = ZT*(T-1*R) (for real flavors) or B-1*A = ZH*(T-1*R) (for complex flavors).\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nn\nThe number of rows of the matrices A and B (n≥ 0).\nm\nThe number of columns in A (m≥ 0).\np\nThe number of columns in B (p≥ 0).\na, b\nArray a of size max(1, lda*m) for column major layout and max(1, lda*n)\nfor row major layout contains the matrix A.\nArray b of size max(1, ldb*p) for column major layout and max(1, ldb*n)\nfor row major layout contains the matrix B.\nlda\nThe leading dimension of a; at least max(1, n) for column major layout and\nat least max(1, m) for row major layout.\nldb\nThe leading dimension of b; at least max(1, n) for column major layout and\nat least max(1, p) for row major layout.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n829\n\n\nOutput Parameters\na, b\nOverwritten by the factorization data as follows:\non exit, the elements on and above the diagonal of the array a contain the\nmin(n,m)-by-m upper trapezoidal matrix R (R is upper triangular if n≥m);the\nelements below the diagonal, with the array taua, represent the orthogonal/\nunitary matrix Q as a product of min(n,m) elementary reflectors ;\nif n≤p, the upper triangle of the subarray b(1:n, p-n+1:p ) contains the n-\nby-n upper triangular matrix T;\nif n > p, the elements on and above the (n-p)th subdiagonal contain the n-\nby-p upper trapezoidal matrix T; the remaining elements, with the array\ntaub, represent the orthogonal/unitary matrix Z as a product of elementary\nreflectors.\ntaua, taub\nArrays, size at least max (1, min(n, m)) for taua and at least max (1,\nmin(n, p)) for taub. The array taua contains the scalar factors of the\nelementary reflectors which represent the orthogonal/unitary matrix Q.\nThe array taub contains the scalar factors of the elementary reflectors\nwhich represent the orthogonal/unitary matrix Z.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nApplication Notes\nThe matrix Q is represented as a product of elementary reflectors\nQ = H(1)H(2)...H(k), where k = min(n,m).\nEach H(i) has the form\nH(i) = I - τa*v*vT for real flavors, or\nH(i) = I - τa*v*vH for complex flavors,\nwhere τa is a real/complex scalar, and v is a real/complex vector with vj = 0 for 1 ≤j≤i - 1, vi = 1.\nOn exit, fori + 1 ≤j≤n, vj is stored in a[(j - 1) + (i - 1)*lda] for column major layout and in a[(j -\n1)*lda + (i - 1)] for row major layout and τa is stored in taua[i - 1]\nThe matrix Z is represented as a product of elementary reflectors\nZ = H(1)H(2)...H(k), where k = min(n,p).\nEach H(i) has the form\nH(i) = I - τb*v*vT for real flavors, or\nH(i) = I - τb*v*vH for complex flavors,\nwhere τb is a real/complex scalar, and v is a real/complex vector with vp - k + 1 = 1, vj = 0 for p - k + 1 ≤j≤p -\n1, .\nOn exit, for 1 ≤j≤p - k + i - 1, vj is stored in b[(n - k + i - 1) + (j - 1)*ldb] for column major layout\nand in b[(n - k + i - 1)*ldb + (j - 1)] for row major layout and τb is stored in taub[i - 1].\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n830\n\n\n?ggrqf\nComputes the generalized RQ factorization of two\nmatrices.\nSyntax\nlapack_int LAPACKE_sggrqf (int matrix_layout, lapack_int m, lapack_int p, lapack_int n,\nfloat* a, lapack_int lda, float* taua, float* b, lapack_int ldb, float* taub);\nlapack_int LAPACKE_dggrqf (int matrix_layout, lapack_int m, lapack_int p, lapack_int n,\ndouble* a, lapack_int lda, double* taua, double* b, lapack_int ldb, double* taub);\nlapack_int LAPACKE_cggrqf (int matrix_layout, lapack_int m, lapack_int p, lapack_int n,\nlapack_complex_float* a, lapack_int lda, lapack_complex_float* taua,\nlapack_complex_float* b, lapack_int ldb, lapack_complex_float* taub);\nlapack_int LAPACKE_zggrqf (int matrix_layout, lapack_int m, lapack_int p, lapack_int n,\nlapack_complex_double* a, lapack_int lda, lapack_complex_double* taua,\nlapack_complex_double* b, lapack_int ldb, lapack_complex_double* taub);\nInclude Files\n•\nmkl.h\nDescription\nThe routine forms the generalized RQ factorization of an m-by-n matrix A and an p-by-n matrix B as A =\nR*Q, B = Z*T*Q, where Q is an n-by-n orthogonal/unitary matrix, Z is a p-by-p orthogonal/unitary matrix,\nand R and T assume one of the forms:\nor\nwhere R11 or R21 is upper triangular, and\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n831\n\n\nor\nwhere T11 is upper triangular.\nIn particular, if B is square and nonsingular, the GRQ factorization of A and B implicitly gives the RQ\nfactorization of A*B-1 as:\nA*B-1 = (R*T-1)*ZT (for real flavors) or A*B-1 = (R*T-1)*ZH (for complex flavors).\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nm\nThe number of rows of the matrix A (m≥ 0).\np\nThe number of rows in B (p≥ 0).\nn\nThe number of columns of the matrices A and B (n≥ 0).\na, b\nArrays:\na(size max(1, lda*n) for column major layout and max(1, lda*m) for row\nmajor layout) contains the m-by-n matrix A.\nb(size max(1, ldb*n) for column major layout and max(1, ldb*p) for row\nmajor layout) contains the p-by-n matrix B.\nlda\nThe leading dimension of a; at least max(1, m)for column major layout and\nmax(1, n) for row major layout.\nldb\nThe leading dimension of b; at least max(1, p)for column major layout and\nmax(1, n) for row major layout.\nOutput Parameters\na, b\nOverwritten by the factorization data as follows:\non exit, if m≤n, element Ri j (1<=i≤j≤m) of upper triangular matrix R is\nstored in a[(i - 1) + (n - m + j - 1)*lda] for column major layout\nand in a[(i - 1)*lda + (n - m + j - 1)] for row major layout.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n832\n\n\nif m > n, the elements on and above the (m-n)th subdiagonal contain the\nm-by-n upper trapezoidal matrix R;\nthe remaining elements, with the array taua, represent the orthogonal/\nunitary matrix Q as a product of elementary reflectors.\nThe elements on and above the diagonal of the array b contain the\nmin(p,n)-by-n upper trapezoidal matrix T (T is upper triangular if p≥n); the\nelements below the diagonal, with the array taub, represent the orthogonal/\nunitary matrix Z as a product of elementary reflectors.\ntaua, taub\nArrays, size at least max (1, min(m, n)) for taua and at least max (1,\nmin(p, n)) for taub.\nThe array taua contains the scalar factors of the elementary reflectors\nwhich represent the orthogonal/unitary matrix Q.\nThe array taub contains the scalar factors of the elementary reflectors\nwhich represent the orthogonal/unitary matrix Z.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nApplication Notes\nThe matrix Q is represented as a product of elementary reflectors\nQ = H(1)H(2)...H(k), where k = min(m,n).\nEach H(i) has the form\nH(i) = I - taua*v*vT for real flavors, or\nH(i) = I - taua*v*vH for complex flavors,\nwhere taua is a real/complex scalar, and v is a real/complex vector with vn - k + i = 1, vn - k + i + 1:n = 0.\nOn exit, v1:n - k + i - 1 is stored in a(m-k+i,1:n-k+i-1) and taua is stored in taua[i - 1].\nThe matrix Z is represented as a product of elementary reflectors\nZ = H(1)H(2)...H(k), where k = min(p,n).\nEach H(i) has the form\nH(i) = I - taub*v*vT for real flavors, or\nH(i) = I - taub*v*vH for complex flavors,\nwhere taub is a real/complex scalar, and v is a real/complex vector with v1:i - 1 = 0, vi = 1.\nOn exit, vi + 1:p is stored in b(i+1:p, i) and taub is stored in taub[i - 1].\n?tpqrt\nComputes a blocked QR factorization of a real or\ncomplex \"triangular-pentagonal\" matrix, which is\ncomposed of a triangular block and a pentagonal\nblock, using the compact WY representation for Q.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n833\n\n\nSyntax\nlapack_int LAPACKE_stpqrt (int matrix_layout, lapack_int m, lapack_int n, lapack_int l,\nlapack_int nb, float* a, lapack_int lda, float* b, lapack_int ldb, float* t, lapack_int\nldt);\nlapack_int LAPACKE_dtpqrt (int matrix_layout, lapack_int m, lapack_int n, lapack_int l,\nlapack_int nb, double* a, lapack_int lda, double* b, lapack_int ldb, double* t,\nlapack_int ldt);\nlapack_int LAPACKE_ctpqrt (int matrix_layout, lapack_int m, lapack_int n, lapack_int l,\nlapack_int nb, lapack_complex_float* a, lapack_int lda, lapack_complex_float* b,\nlapack_int ldb, lapack_complex_float* t, lapack_int ldt);\nlapack_int LAPACKE_ztpqrt (int matrix_layout, lapack_int m, lapack_int n, lapack_int l,\nlapack_int nb, lapack_complex_double* a, lapack_int lda, lapack_complex_double* b,\nlapack_int ldb, lapack_complex_double* t, lapack_int ldt);\nInclude Files\n•\nmkl.h\nDescription\nThe input matrix C is an (n+m)-by-n matrix\nwhere A is an n-by-n upper triangular matrix, and B is an m-by-n pentagonal matrix consisting of an (m-l)-\nby-n rectangular matrix B1 on top of an l-by-n upper trapezoidal matrix B2:\nThe upper trapezoidal matrix B2 consists of the first l rows of an n-by-n upper triangular matrix, where 0 ≤\nl ≤ min(m,n). If l=0, B is an m-by-n rectangular matrix. If m=l=n, B is upper triangular. The elementary\nreflectors H(i) are stored in the ith column below the diagonal in the (n+m)-by-n input matrix C. The\nstructure of vectors defining the elementary reflectors is illustrated by:\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n834\n\n\nThe elements of the unit matrix I are not stored. Thus, V contains all of the necessary information, and is\nreturned in array b.\nNOTE\nNote that V has the same form as B:\nThe columns of V represent the vectors which define the H(i)s.\nThe number of blocks is k = ceiling(n/nb), where each block is of order nb except for the last block, which is\nof order ib = n - (k-1)*nb. For each of the k blocks, an upper triangular block reflector factor is computed:\nT1, T2, ..., Tk. The nb-by-nb (ib-by-ib for the last block) Tis are stored in the nb-by-n array t as\nt = [T1T2 ... Tk] .\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nm\nThe total number of rows in the matrix B (m ≥ 0).\nn\nThe number of columns in B and the order of the triangular matrix A (n ≥\n0).\nl\nThe number of rows of the upper trapezoidal part of B (min(m, n) ≥ l ≥ 0).\nnb\nThe block size to use in the blocked QR factorization (n ≥ nb ≥ 1).\na, b\nArrays: a size lda*n contains the n-by-n upper triangular matrix A.\nb size max(1, ldb*n) for column major layout and max(1, ldb*m) for row\nmajor layout, the pentagonal m-by-n matrix B. The first (m-l) rows contain\nthe rectangular B1 matrix, and the next l rows contain the upper\ntrapezoidal B2 matrix.\nlda\nThe leading dimension of a; at least max(1, n).\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n835\n\n\nldb\nThe leading dimension of b; at least max(1, m) for column major layout and\nat least max(1, n) for row major layout.\nldt\nThe leading dimension of t; at least nb for column major layout and at least\nmax(1, n) for row major layout.\nOutput Parameters\na\nThe elements on and above the diagonal of the array contain the upper\ntriangular matrix R.\nb\nThe pentagonal matrix V.\nt\nArray, size ldt*n for column major layout and ldt*nb for row major\nlayout.\nThe upper triangular block reflectors stored in compact form as a sequence\nof upper triangular blocks.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\n?tpmqrt\nApplies a real or complex orthogonal matrix obtained\nfrom a \"triangular-pentagonal\" complex block reflector\nto a general real or complex matrix, which consists of\ntwo blocks.\nSyntax\nlapack_int LAPACKE_stpmqrt (int matrix_layout, char side, char trans, lapack_int m,\nlapack_int n, lapack_int k, lapack_int l, lapack_int nb, const float* v, lapack_int ldv,\nconst float* t, lapack_int ldt, float* a, lapack_int lda, float* b, lapack_int ldb);\nlapack_int LAPACKE_dtpmqrt (int matrix_layout, char side, char trans, lapack_int m,\nlapack_int n, lapack_int k, lapack_int l, lapack_int nb, const double* v, lapack_int\nldv, const double* t, lapack_int ldt, double* a, lapack_int lda, double* b, lapack_int\nldb);\nlapack_int LAPACKE_ctpmqrt (int matrix_layout, char side, char trans, lapack_int m,\nlapack_int n, lapack_int k, lapack_int l, lapack_int nb, const lapack_complex_float* v,\nlapack_int ldv, const lapack_complex_float* t, lapack_int ldt, lapack_complex_float* a,\nlapack_int lda, lapack_complex_float* b, lapack_int ldb);\nlapack_int LAPACKE_ztpmqrt (int matrix_layout, char side, char trans, lapack_int m,\nlapack_int n, lapack_int k, lapack_int l, lapack_int nb, const lapack_complex_double*\nv, lapack_int ldv, const lapack_complex_double* t, lapack_int ldt,\nlapack_complex_double* a, lapack_int lda, lapack_complex_double* b, lapack_int ldb);\nInclude Files\n•\nmkl.h\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n836\n\n\nDescription\nThe columns of the pentagonal matrix V contain the elementary reflectors H(1), H(2), ..., H(k); V is\ncomposed of a rectangular block V1 and a trapezoidal block V2:\nThe size of the trapezoidal block V2 is determined by the parameter l, where 0 ≤ l ≤ k. V2 is upper\ntrapezoidal, consisting of the first l rows of a k-by-k upper triangular matrix.\nIf l=k, V2 is upper triangular;\nIf l=0, there is no trapezoidal block, so V = V1 is rectangular.\nIf side = 'L':\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n837\n\n\nwhere A is k-by-n, B is m-by-n and V is m-by-k.\nIf side = 'R':\nwhere A is m-by-k, B is m-by-n and V is n-by-k.\nThe real/complex orthogonal matrix Q is formed from V and T.\nIf trans='N' and side='L', c contains Q * C on exit.\nIf trans='T' and side='L', C contains QT * C on exit.\nIf trans='C' and side='L', C contains QH * C on exit.\nIf trans='N' and side='R', C contains C * Q on exit.\nIf trans='T' and side='R', C contains C * QT on exit.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n838\n\n\nIf trans='C' and side='R', C contains C * QH on exit.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nside\n='L': apply Q, QT, or QH from the left.\n='R': apply Q, QT, or QH from the right.\ntrans\n='N', no transpose, apply Q.\n='T', transpose, apply QT.\n='C', transpose, apply QH.\nm\nThe number of rows in the matrix B, (m ≥ 0).\nn\nThe number of columns in the matrix B, (n ≥ 0).\nk\nThe number of elementary reflectors whose product defines the matrix Q,\n(k ≥ 0).\nl\nThe order of the trapezoidal part of V (k ≥ l ≥ 0).\nnb\nThe block size used for the storage of t, k ≥ nb ≥ 1. This must be the same\nvalue of nb used to generate t in tpqrt.\nv\nSize ldv*k for column major layout; ldv*m for row major layout and side\n= 'L', ldv*n for row major layout and side = 'R'.\nThe ith column must contain the vector which defines the elementary\nreflector H(i), for i = 1,2,...,k, as returned by tpqrt in array argument b.\nldv\nThe leading dimension of the array v.\nIf side = 'L', ldv must be at least max(1,m) for column major layout and\nmax(1, k for row major layout;\nIf side = 'R', ldv must be at least max(1,n) for column major layout and\nmax(1, k for row major layout.\nt\nArray, size ldt*k for column major layout and ldt*nb for row major\nlayout.\nThe upper triangular factors of the block reflectors as returned by tpqrt\nldt\nThe leading dimension of the array t. ldt must be at least nb for column\nmajor layout and max(1, k for row major layout.\na\nIf side = 'L', size lda*n for column major layout and lda*k for row major\nlayout ..\nIf side = 'R', size lda*k for column major layout and lda*m for row major\nlayout ..\nThe k-by-n or m-by-k matrix A.\nlda\nThe leading dimension of the array a.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n839\n\n\nIf side = 'L', lda must be at least max(1,k) for column major layout and\nmax(1, n for row major layout.\nIf side = 'R', lda must be at least max(1,m) for column major layout and\nmax(1, k for row major layout.\nb\nSize ldb*n for column major layout and ldb*m for row major layout.\nThe m-by-n matrix B.\nldb\nThe leading dimension of the array b. ldb must be at least max(1,m) for\ncolumn major layout and max(1, n for row major layout.\nOutput Parameters\na\nOverwritten by the corresponding block of the product Q*C, C*Q, QT*C,\nC*QT, QH*C, or C*QH.\nb\nOverwritten by the corresponding block of the product Q*C, C*Q, QT*C,\nC*QT, QH*C, or C*QH.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nSingular Value Decomposition: LAPACK Computational Routines\nThis topic describes LAPACK routines for computing the singular value decomposition (SVD) of a general m-\nby-n matrix A:\nA = UΣVH.\nIn this decomposition, U and V are unitary (for complex A) or orthogonal (for real A); Σ is an m-by-n\ndiagonal matrix with real diagonal elements σi:\nσ1≥σ2≥ ... ≥σmin(m, n)≥ 0.\nThe diagonal elements σi are singular values of A. The first min(m, n) columns of the matrices U and V are,\nrespectively, left and right singular vectors of A. The singular values and singular vectors satisfy\nAvi = σiui and AHui = σivi\nwhere ui and vi are the i-th columns of U and V, respectively.\nTo find the SVD of a general matrix A, call the LAPACK routine ?gebrd or ?gbbrd for reducing A to a\nbidiagonal matrix B by a unitary (orthogonal) transformation: A = QBPH. Then call ?bdsqr, which forms the\nSVD of a bidiagonal matrix: B = U1ΣV1H.\nThus, the sought-for SVD of A is given by A = UΣVH =(QU1)Σ(V1HPH).\nTable \"Computational Routines for Singular Value Decomposition (SVD)\" lists LAPACK routines that perform\nsingular value decomposition of matrices.\nComputational Routines for Singular Value Decomposition (SVD)\nOperation\nReal matrices\nComplex matrices\nReduce A to a bidiagonal matrix B: A = QBPH\n(full storage)\n?gebrd\n?gebrd\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n840\n\n\nOperation\nReal matrices\nComplex matrices\nReduce A to a bidiagonal matrix B: A = QBPH\n(band storage)\n?gbbrd\n?gbbrd\nGenerate the orthogonal (unitary) matrix Q or\nP\n?orgbr\n?ungbr\nApply the orthogonal (unitary) matrix Q or P\n?ormbr\n?unmbr\nForm singular value decomposition of the\nbidiagonal matrix B: B = UΣVH\n?bdsqr  ?bdsdc\n?bdsqr\nYou can use the SVD to find a minimum-norm solution to a (possibly) rank-deficient least squares problem of\nminimizing ||Ax - b||2. The effective rank k of the matrix A can be determined as the number of singular\nvalues which exceed a suitable threshold. The minimum-norm solution is\nx = Vk(Σk)-1c\nwhere Σk is the leading k-by-k submatrix of Σ, the matrix Vk consists of the first k columns of V = PV1, and\nthe vector c consists of the first k elements of UHb = U1HQHb.\n?gebrd\nReduces a general matrix to bidiagonal form.\nSyntax\nlapack_int LAPACKE_sgebrd( int matrix_layout, lapack_int m, lapack_int n, float* a,\nlapack_int lda, float* d, float* e, float* tauq, float* taup );\nlapack_int LAPACKE_dgebrd( int matrix_layout, lapack_int m, lapack_int n, double* a,\nlapack_int lda, double* d, double* e, double* tauq, double* taup );\nlapack_int LAPACKE_cgebrd( int matrix_layout, lapack_int m, lapack_int n,\nlapack_complex_float* a, lapack_int lda, float* d, float* e, lapack_complex_float*\ntauq, lapack_complex_float* taup );\nlapack_int LAPACKE_zgebrd( int matrix_layout, lapack_int m, lapack_int n,\nlapack_complex_double* a, lapack_int lda, double* d, double* e, lapack_complex_double*\ntauq, lapack_complex_double* taup );\nInclude Files\n•\nmkl.h\nDescription\nThe routine reduces a general m-by-n matrix A to a bidiagonal matrix B by an orthogonal (unitary)\ntransformation.\nIf m≥n, the reduction is given by A = QBPH =\nB1\n0\nPH = Q1B1PH,\nwhere B1 is an n-by-n upper diagonal matrix, Q and P are orthogonal or, for a complex A, unitary matrices;\nQ1 consists of the first n columns of Q.\nIf m < n, the reduction is given by\nA = Q*B*PH = Q*(B10)*PH = Q1*B1*P1H,\nwhere B1 is an m-by-m lower diagonal matrix, Q and P are orthogonal or, for a complex A, unitary matrices;\nP1 consists of the first m columns of P.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n841\n\n\nThe routine does not form the matrices Q and P explicitly, but represents them as products of elementary\nreflectors. Routines are provided to work with the matrices Q and P in this representation:\nIf the matrix A is real,\n•\nto compute Q and P explicitly, call orgbr.\n•\nto multiply a general matrix by Q or P, call ormbr.\nIf the matrix A is complex,\n•\nto compute Q and P explicitly, call ungbr.\n•\nto multiply a general matrix by Q or P, call unmbr.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nm\nThe number of rows in the matrix A (m≥ 0).\nn\nThe number of columns in A (n≥ 0).\na\nArrays:\na(size max(1, lda*n) for column major layout and max(1, lda*m) for row\nmajor layout) contains the matrix A.\nlda\nThe leading dimension of a; at least max(1, m) for column major layout\nand at least max(1, n) for row major layout.\nOutput Parameters\na\nIf m≥n, the diagonal and first super-diagonal of a are overwritten by the\nupper bidiagonal matrix B. The elements below the diagonal, with the array\ntauq, represent the orthogonal matrix Q as a product of elementary\nreflectors, and the elements above the first superdiagonal, with the array\ntaup, represent the orthogonal matrix P as a product of elementary\nreflectors.\nIf m < n, the diagonal and first sub-diagonal of a are overwritten by the\nlower bidiagonal matrix B. The elements below the first subdiagonal, with\nthe array tauq, represent the orthogonal matrix Q as a product of\nelementary reflectors, and the elements above the diagonal, with the array\ntaup, represent the orthogonal matrix P as a product of elementary\nreflectors.\nd\nArray, size at least max(1, min(m, n)).\nContains the diagonal elements of B.\ne\nArray, size at least max(1, min(m, n) - 1). Contains the off-diagonal\nelements of B.\ntauq, taup\nArrays, size at least max (1, min(m, n)). The scalar factors of the\nelementary reflectors which represent the orthogonal or unitary matrices P\nand Q.\nReturn Values\nThis function returns a value info.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n842\n\n\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nApplication Notes\nThe computed matrices Q, B, and P satisfy QBPH = A + E, where ||E||2 = c(n)ε ||A||2, c(n) is a\nmodestly increasing function of n, and ε is the machine precision.\nThe approximate number of floating-point operations for real flavors is\n(4/3)*n2*(3*m - n) for m≥n,\n(4/3)*m2*(3*n - m) for m < n.\nThe number of operations for complex flavors is four times greater.\nIf n is much less than m, it can be more efficient to first form the QR factorization of A by calling geqrf and\nthen reduce the factor R to bidiagonal form. This requires approximately 2*n2*(m + n) floating-point\noperations.\nIf m is much less than n, it can be more efficient to first form the LQ factorization of A by calling gelqf and\nthen reduce the factor L to bidiagonal form. This requires approximately 2*m2*(m + n) floating-point\noperations.\n?gbbrd\nReduces a general band matrix to bidiagonal form.\nSyntax\nlapack_int LAPACKE_sgbbrd( int matrix_layout, char vect, lapack_int m, lapack_int n,\nlapack_int ncc, lapack_int kl, lapack_int ku, float* ab, lapack_int ldab, float* d,\nfloat* e, float* q, lapack_int ldq, float* pt, lapack_int ldpt, float* c, lapack_int\nldc );\nlapack_int LAPACKE_dgbbrd( int matrix_layout, char vect, lapack_int m, lapack_int n,\nlapack_int ncc, lapack_int kl, lapack_int ku, double* ab, lapack_int ldab, double* d,\ndouble* e, double* q, lapack_int ldq, double* pt, lapack_int ldpt, double* c, lapack_int\nldc );\nlapack_int LAPACKE_cgbbrd( int matrix_layout, char vect, lapack_int m, lapack_int n,\nlapack_int ncc, lapack_int kl, lapack_int ku, lapack_complex_float* ab, lapack_int\nldab, float* d, float* e, lapack_complex_float* q, lapack_int ldq,\nlapack_complex_float* pt, lapack_int ldpt, lapack_complex_float* c, lapack_int ldc );\nlapack_int LAPACKE_zgbbrd( int matrix_layout, char vect, lapack_int m, lapack_int n,\nlapack_int ncc, lapack_int kl, lapack_int ku, lapack_complex_double* ab, lapack_int\nldab, double* d, double* e, lapack_complex_double* q, lapack_int ldq,\nlapack_complex_double* pt, lapack_int ldpt, lapack_complex_double* c, lapack_int ldc );\nInclude Files\n•\nmkl.h\nDescription\nThe routine reduces an m-by-n band matrix A to upper bidiagonal matrix B: A = Q*B*PH. Here the matrices\nQ and P are orthogonal (for real A) or unitary (for complex A). They are determined as products of Givens\nrotation matrices, and may be formed explicitly by the routine if required. The routine can also update a\nmatrix C as follows: C = QH*C.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n843\n\n\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nvect\nMust be 'N' or 'Q' or 'P' or 'B'.\nIf vect = 'N', neither Q nor PH is generated.\nIf vect = 'Q', the routine generates the matrix Q.\nIf vect = 'P', the routine generates the matrix PH.\nIf vect = 'B', the routine generates both Q and PH.\nm\nThe number of rows in the matrix A (m≥ 0).\nn\nThe number of columns in A (n≥ 0).\nncc\nThe number of columns in C (ncc≥ 0).\nkl\nThe number of sub-diagonals within the band of A (kl≥ 0).\nku\nThe number of super-diagonals within the band of A (ku≥ 0).\nab, c\nArrays:\nab(size max(1, ldab*n) for column major layout and max(1, ldab*m) for\nrow major layout) contains the matrix A in band storage (see Matrix\nStorage Schemes).\nc(size max(1, ldc*ncc) for column major layout and max(1, ldc*m) for\nrow major layout) contains an m-by-ncc matrix C.\nIf ncc = 0, the array c is not referenced.\nldab\nThe leading dimension of the array ab (ldab≥kl + ku + 1).\nldq\nThe leading dimension of the output array q.\nldq≥ max(1, m) if vect = 'Q' or 'B', ldq≥ 1 otherwise.\nldpt\nThe leading dimension of the output array pt.\nldpt≥ max(1, n) if vect = 'P' or 'B', ldpt≥ 1 otherwise.\nldc\nThe leading dimension of the array c.\nldc≥ max(1, m) if ncc > 0; ldc≥ 1 if ncc = 0.\nOutput Parameters\nab\nOverwritten by values generated during the reduction.\nd\nArray, size at least max(1, min(m, n)). Contains the diagonal elements of\nthe matrix B.\ne\nArray, size at least max(1, min(m, n) - 1).\nContains the off-diagonal elements of B.\nq, pt\nArrays:\nqsize max(1, ldq*m) contains the output m-by-m matrix Q.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n844\n\n\npsize max(1, ldpt*n) contains the output n-by-n matrix PT.\nc\nOverwritten by the product QH*C.\nc is not referenced if ncc = 0.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nApplication Notes\nThe computed matrices Q, B, and P satisfy Q*B*PH = A + E, where ||E||2 = c(n)ε ||A||2, c(n) is a\nmodestly increasing function of n, and ε is the machine precision.\nIf m = n, the total number of floating-point operations for real flavors is approximately the sum of:\n6*n2*(kl + ku) if vect = 'N' and ncc = 0,\n3*n2*ncc*(kl + ku - 1)/(kl + ku) if C is updated, and\n3*n3*(kl + ku - 1)/(kl + ku) if either Q or PH is generated (double this if both).\nTo estimate the number of operations for complex flavors, use the same formulas with the coefficients 20\nand 10 (instead of 6 and 3).\n?orgbr\nGenerates the real orthogonal matrix Q or PT\ndetermined by ?gebrd.\nSyntax\nlapack_int LAPACKE_sorgbr (int matrix_layout, char vect, lapack_int m, lapack_int n,\nlapack_int k, float* a, lapack_int lda, const float* tau);\nlapack_int LAPACKE_dorgbr (int matrix_layout, char vect, lapack_int m, lapack_int n,\nlapack_int k, double* a, lapack_int lda, const double* tau);\nInclude Files\n•\nmkl.h\nDescription\nThe routine generates the whole or part of the orthogonal matrices Q and PT formed by the routines gebrd.\nUse this routine after a call to sgebrd/dgebrd. All valid combinations of arguments are described in Input\nparameters. In most cases you need the following:\nTo compute the whole m-by-m matrix Q:\nLAPACKE_?orgbr(matrix_layout, 'Q', m, m, n, a, lda, tau )\n(note that the array a must have at least m columns).\nTo form the n leading columns of Q if m > n:\nLAPACKE_?orgbr(matrix_layout, 'Q', m, n, n, a, lda, tau )\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n845\n\n\nTo compute the whole n-by-n matrix PT:\nLAPACKE_?orgbr(matrix_layout, 'P', n, n, m, a, lda, tau )\n(note that the array a must have at least n rows).\nTo form the m leading rows of PT if m < n:\nLAPACKE_?orgbr(matrix_layout, 'P', m, n, m, a, lda, tau )\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nvect\nMust be 'Q' or 'P'.\nIf vect = 'Q', the routine generates the matrix Q.\nIf vect = 'P', the routine generates the matrix PT.\nm, n\nThe number of rows (m) and columns (n) in the matrix Q or PT to be\nreturned (m≥ 0, n≥ 0).\nIf vect = 'Q', m ≥ n ≥ min(m, k).\nIf vect = 'P', n ≥ m ≥ min(n, k).\nk\nIf vect = 'Q', the number of columns in the original m-by-k matrix\nreduced by gebrd.\nIf vect = 'P', the number of rows in the original k-by-n matrix reduced\nby gebrd.\na\nArray, size at least lda*n for column major layout and lda*m for row major\nlayout. The vectors which define the elementary reflectors, as returned by \ngebrd.\nlda\nThe leading dimension of the array a. lda ≥ max(1, m) for column major\nlayout and at least max(1, n) for row major layout .\ntau\nArray, size min (m,k) if vect = 'Q', min (n,k) if vect = 'P'.\nScalar factor of the elementary reflector H(i) or G(i), which determines Q\nand PT as returned by gebrd in the array tauq or taup.\nOutput Parameters\na\nOverwritten by the orthogonal matrix Q or PT (or the leading rows or\ncolumns thereof) as specified by vect, m, and n.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nApplication Notes\nThe computed matrix Q differs from an exactly orthogonal matrix by a matrix E such that ||E||2 = O(ε).\nThe approximate numbers of floating-point operations for the cases listed in Description are as follows:\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n846\n\n\nTo form the whole of Q:\n(4/3)*n*(3m2 - 3m*n + n2) if m > n;\n(4/3)*m3 if m≤n.\nTo form the n leading columns of Q when m > n:\n(2/3)*n2*(3m - n) if m > n.\nTo form the whole of PT:\n(4/3)*n3 if m≥n;\n(4/3)*m*(3n2 - 3m*n + m2) if m < n.\nTo form the m leading columns of PT when m < n:\n(2/3)*n2*(3m - n) if m > n.\nThe complex counterpart of this routine is ungbr.\n?ormbr\nMultiplies an arbitrary real matrix by the real\northogonal matrix Q or PT determined by ?gebrd.\nSyntax\nlapack_int LAPACKE_sormbr (int matrix_layout, char vect, char side, char trans,\nlapack_int m, lapack_int n, lapack_int k, const float* a, lapack_int lda, const float*\ntau, float* c, lapack_int ldc);\nlapack_int LAPACKE_dormbr (int matrix_layout, char vect, char side, char trans,\nlapack_int m, lapack_int n, lapack_int k, const double* a, lapack_int lda, const\ndouble* tau, double* c, lapack_int ldc);\nInclude Files\n•\nmkl.h\nDescription\nGiven an arbitrary real matrix C, this routine forms one of the matrix products Q*C, QT*C, C*Q, C*QT, P*C,\nPT*C, C*P, C*PT, where Q and P are orthogonal matrices computed by a call to gebrd. The routine overwrites\nthe product on C.\nInput Parameters\nIn the descriptions below, r denotes the order of Q or PT:\nIf side = 'L', r = m; if side = 'R', r = n.\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nvect\nMust be 'Q' or 'P'.\nIf vect = 'Q', then Q or QT is applied to C.\nIf vect = 'P', then P or PT is applied to C.\nside\nMust be 'L' or 'R'.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n847\n\n\nIf side = 'L', multipliers are applied to C from the left.\nIf side = 'R', they are applied to C from the right.\ntrans\nMust be 'N' or 'T'.\nIf trans = 'N', then Q or P is applied to C.\nIf trans = 'T', then QT or PT is applied to C.\nm\nThe number of rows in C.\nn\nThe number of columns in C.\nk\nOne of the dimensions of A in ?gebrd:\nIf vect = 'Q', the number of columns in A;\nIf vect = 'P', the number of rows in A.\nConstraints: m≥ 0, n≥ 0, k≥ 0.\na, c\nArrays:\na is the array a as returned by ?gebrd.\nThe size of a depends on the value of the matrix_layout, vect, and side\nparameters:\nmatrix_layout\nvect\nside\nsize\ncolumn major\n'Q'\n-\nmax(1, lda*k)\ncolumn major\n'P'\n'L'\nmax(1, lda*m)\ncolumn major\n'P'\n'R'\nmax(1, lda*n)\nrow major\n'Q'\n'L'\nmax(1, lda*m)\nrow major\n'Q'\n'R'\nmax(1, lda*n)\nrow major\n'P'\n-\nmax(1, lda*k)\nc(size max(1, ldc*n) for column major layout and max(1, ldc*m) for row\nmajor layout) holds the matrix C.\nlda\nThe leading dimension of a. Constraints:\nlda≥ max(1, r) for column major layout and at least max(1, k) for row\nmajor layout if vect = 'Q';\nlda≥ max(1, min(r,k)) for column major layout and at least max(1, r) for\nrow major layout if vect = 'P'.\nldc\nThe leading dimension of c; ldc≥ max(1, m) for column major layout and\nldc≥ max(1, n) for row major layout .\ntau\nArray, size at least max (1, min(r, k)).\nFor vect = 'Q', the array tauq as returned by ?gebrd. For vect = 'P',\nthe array taup as returned by ?gebrd.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n848\n\n\nOutput Parameters\nc\nOverwritten by the product Q*C, QT*C, C*Q, C*Q,T, P*C, PT*C, C*P, or C*PT,\nas specified by vect, side, and trans.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nApplication Notes\nThe computed product differs from the exact product by a matrix E such that ||E||2 = O(ε)*||C||2.\nThe total number of floating-point operations is approximately\n2*n*k(2*m - k) if side = 'L' and m≥k;\n2*m*k(2*n - k) if side = 'R' and n≥k;\n2*m2*n if side = 'L' and m < k;\n2*n2*m if side = 'R' and n < k.\nThe complex counterpart of this routine is unmbr.\n?ungbr\nGenerates the complex unitary matrix Q or PH\ndetermined by ?gebrd.\nSyntax\nlapack_int LAPACKE_cungbr (int matrix_layout, char vect, lapack_int m, lapack_int n,\nlapack_int k, lapack_complex_float* a, lapack_int lda, const lapack_complex_float*\ntau);\nlapack_int LAPACKE_zungbr (int matrix_layout, char vect, lapack_int m, lapack_int n,\nlapack_int k, lapack_complex_double* a, lapack_int lda, const lapack_complex_double*\ntau);\nInclude Files\n•\nmkl.h\nDescription\nThe routine generates the whole or part of the unitary matrices Q and PH formed by the routines gebrd. Use\nthis routine after a call to cgebrd/zgebrd. All valid combinations of arguments are described in Input\nParameters; in most cases you need the following:\nTo compute the whole m-by-m matrix Q, use:\nLAPACKE_?ungbr(matrix_layout, 'Q', m, m, n, a, lda, tau)\n(note that the arraya must have at least m columns).\nTo form the n leading columns of Q if m > n, use:\nLAPACKE_?ungbr(matrix_layout, 'Q', m, n, n, a, lda, tau)\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n849\n\n\nTo compute the whole n-by-n matrix PH, use:\nLAPACKE_?ungbr(matrix_layout, 'P', n, n, m, a, lda, tau)\n(note that the array a must have at least n rows).\nTo form the m leading rows of PH if m < n, use:\nLAPACKE_?ungbr(matrix_layout, 'P', m, m, n, a, lda, tau)\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nvect\nMust be 'Q' or 'P'.\nIf vect = 'Q', the routine generates the matrix Q.\nIf vect = 'P', the routine generates the matrix PH.\nm\nThe number of required rows of Q or PH.\nn\nThe number of required columns of Q or PH.\nk\nOne of the dimensions of A in ?gebrd:\nIf vect = 'Q', the number of columns in A;\nIf vect = 'P', the number of rows in A.\nConstraints: m≥ 0, n≥ 0, k≥ 0.\nFor vect = 'Q': k≤n≤m if m > k, or m = n if m≤k.\nFor vect = 'P': k≤m≤n if n > k, or m = n if n≤k.\na\nArrays:\na, size at least lda*n for column major layout and lda*m for row major\nlayout, is the array a as returned by ?gebrd.\nlda\nThe leading dimension of a; at least max(1, m)for column major layout and\nmax(1, n) for row major layout.\ntau\nFor vect = 'Q', the array tauq as returned by ?gebrd. For vect = 'P',\nthe array taup as returned by ?gebrd.\nThe dimension of tau must be at least max(1, min(m, k)) for vect = 'Q',\nor max(1, min(m, k)) for vect = 'P'.\nOutput Parameters\na\nOverwritten by the orthogonal matrix Q or PT (or the leading rows or\ncolumns thereof) as specified by vect, m, and n.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n850\n\n\nApplication Notes\nThe computed matrix Q differs from an exactly orthogonal matrix by a matrix E such that ||E||2 = O(ε).\nThe approximate numbers of possible floating-point operations are listed below:\nTo compute the whole matrix Q:\n(16/3)n(3m2 - 3m*n + n2) if m > n;\n(16/3)m3 if m≤n.\nTo form the n leading columns of Q when m > n:\n(8/3)n2(3m - n2).\nTo compute the whole matrix PH:\n(16/3)n3 if m≥n;\n(16/3)m(3n2 - 3m*n + m2) if m < n.\nTo form the m leading columns of PH when m < n:\n(8/3)n2(3m - n2) if m > n.\nThe real counterpart of this routine is orgbr.\n?unmbr\nMultiplies an arbitrary complex matrix by the unitary\nmatrix Q or P determined by ?gebrd.\nSyntax\nlapack_int LAPACKE_cunmbr (int matrix_layout, char vect, char side, char trans,\nlapack_int m, lapack_int n, lapack_int k, const lapack_complex_float* a, lapack_int\nlda, const lapack_complex_float* tau, lapack_complex_float* c, lapack_int ldc);\nlapack_int LAPACKE_zunmbr (int matrix_layout, char vect, char side, char trans,\nlapack_int m, lapack_int n, lapack_int k, const lapack_complex_double* a, lapack_int\nlda, const lapack_complex_double* tau, lapack_complex_double* c, lapack_int ldc);\nInclude Files\n•\nmkl.h\nDescription\nGiven an arbitrary complex matrix C, this routine forms one of the matrix products Q*C, QH*C, C*Q, C*QH,\nP*C, PH*C, C*P, or C*PH, where Q and P are unitary matrices computed by a call to gebrd/gebrd. The routine\noverwrites the product on C.\nInput Parameters\nIn the descriptions below, r denotes the order of Q or PH:\nIf side = 'L', r = m; if side = 'R', r = n.\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nvect\nMust be 'Q' or 'P'.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n851\n\n\nIf vect = 'Q', then Q or QH is applied to C.\nIf vect = 'P', then P or PH is applied to C.\nside\nMust be 'L' or 'R'.\nIf side = 'L', multipliers are applied to C from the left.\nIf side = 'R', they are applied to C from the right.\ntrans\nMust be 'N' or 'C'.\nIf trans = 'N', then Q or P is applied to C.\nIf trans = 'C', then QH or PH is applied to C.\nm\nThe number of rows in C.\nn\nThe number of columns in C.\nk\nOne of the dimensions of A in ?gebrd:\nIf vect = 'Q', the number of columns in A;\nIf vect = 'P', the number of rows in A.\nConstraints: m≥ 0, n≥ 0, k≥ 0.\na, c\nArrays:\na is the array a as returned by ?gebrd.\nThe size of a depends on the value of the matrix_layout, vect, and side\nparameters:\nmatrix_layout\nvect\nside\nsize\ncolumn major\n'Q'\n-\nmax(1, lda*k)\ncolumn major\n'P'\n'L'\nmax(1, lda*m)\ncolumn major\n'P'\n'R'\nmax(1, lda*n)\nrow major\n'Q'\n'L'\nmax(1, lda*m)\nrow major\n'Q'\n'R'\nmax(1, lda*n)\nrow major\n'P'\n-\nmax(1, lda*k)\n \n \n \n \n \n \n \n \nc(size max(1, ldc*n) for column major layout and max(1, ldc*m for row\nmajor layout) holds the matrix C.\nlda\nThe leading dimension of a. Constraints:\nlda≥ max(1, r) for column major layout and at least max(1, k) for row\nmajor layout if vect = 'Q';\nlda≥ max(1, min(r,k)) for column major layout and at least max(1, r) for\nrow major layout if vect = 'P'.\nldc\nThe leading dimension of c; ldc≥ max(1, m).\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n852\n\n\ntau\nArray, size at least max (1, min(r, k)).\nFor vect = 'Q', the array tauq as returned by ?gebrd. For vect = 'P',\nthe array taup as returned by ?gebrd.\nOutput Parameters\nc\nOverwritten by the product Q*C, QH*C, C*Q, C*QH, P*C, PH*C, C*P, or\nC*PH, as specified by vect, side, and trans.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nApplication Notes\nThe computed product differs from the exact product by a matrix E such that ||E||2 = O(ε)*||C||2.\nThe total number of floating-point operations is approximately\n8*n*k(2*m - k) if side = 'L' and m≥k;\n8*m*k(2*n - k) if side = 'R' and n≥k;\n8*m2*n if side = 'L' and m < k;\n8*n2*m if side = 'R' and n < k.\nThe real counterpart of this routine is ormbr.\n?bdsqr\nComputes the singular value decomposition of a\ngeneral matrix that has been reduced to bidiagonal\nform.\nSyntax\nlapack_int LAPACKE_sbdsqr( int matrix_layout, char uplo, lapack_int n, lapack_int ncvt,\nlapack_int nru, lapack_int ncc, float* d, float* e, float* vt, lapack_int ldvt, float*\nu, lapack_int ldu, float* c, lapack_int ldc );\nlapack_int LAPACKE_dbdsqr( int matrix_layout, char uplo, lapack_int n, lapack_int ncvt,\nlapack_int nru, lapack_int ncc, double* d, double* e, double* vt, lapack_int ldvt,\ndouble* u, lapack_int ldu, double* c, lapack_int ldc );\nlapack_int LAPACKE_cbdsqr( int matrix_layout, char uplo, lapack_int n, lapack_int ncvt,\nlapack_int nru, lapack_int ncc, float* d, float* e, lapack_complex_float* vt,\nlapack_int ldvt, lapack_complex_float* u, lapack_int ldu, lapack_complex_float* c,\nlapack_int ldc );\nlapack_int LAPACKE_zbdsqr( int matrix_layout, char uplo, lapack_int n, lapack_int ncvt,\nlapack_int nru, lapack_int ncc, double* d, double* e, lapack_complex_double* vt,\nlapack_int ldvt, lapack_complex_double* u, lapack_int ldu, lapack_complex_double* c,\nlapack_int ldc );\nInclude Files\n•\nmkl.h\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n853\n\n\nDescription\nThe routine computes the singular values and, optionally, the right and/or left singular vectors from the \nSingular Value Decomposition (SVD) of a real n-by-n (upper or lower) bidiagonal matrix B using the implicit\nzero-shift QR algorithm. The SVD of B has the form B = Q*S*PH where S is the diagonal matrix of singular\nvalues, Q is an orthogonal matrix of left singular vectors, and P is an orthogonal matrix of right singular\nvectors. If left singular vectors are requested, this subroutine actually returns U *Q instead of Q, and, if right\nsingular vectors are requested, this subroutine returns PH *VT instead of PH, for given real/complex input\nmatrices U and VT. When U and VT are the orthogonal/unitary matrices that reduce a general matrix A to\nbidiagonal form: A = U*B*VT, as computed by ?gebrd, then\nA = (U*Q)*S*(PH*VT)\nis the SVD of A. Optionally, the subroutine may also compute QH *C for a given real/complex input matrix C.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nuplo\nMust be 'U' or 'L'.\nIf uplo = 'U', B is an upper bidiagonal matrix.\nIf uplo = 'L', B is a lower bidiagonal matrix.\nn\nThe order of the matrix B (n≥ 0).\nncvt\nThe number of columns of the matrix VT, that is, the number of right\nsingular vectors (ncvt≥ 0).\nSet ncvt = 0 if no right singular vectors are required.\nnru\nThe number of rows in U, that is, the number of left singular vectors (nru≥\n0).\nSet nru = 0 if no left singular vectors are required.\nncc\nThe number of columns in the matrix C used for computing the product\nQH*C (ncc≥ 0). Set ncc = 0 if no matrix C is supplied.\nd, e\nArrays:\nd contains the diagonal elements of B.\nThe size of d must be at least max(1, n).\ne contains the (n-1) off-diagonal elements of B.\nThe size of e must be at least max(1, n - 1).\nvt, u, c\nArrays:\nvt, size max(1, ldvt*ncvt) for column major layout and max(1, ldvt*n)\nfor row major layout, contains an n-by-ncvt matrix VT.\nvt is not referenced if ncvt = 0.\nu, size max(1, ldu*n) for column major layout and max(1, ldu*nru) for\nrow major layout, contains an nru by n matrix U.\nu is not referenced if nru = 0.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n854\n\n\nc, size max(1, ldc*ncc) for column major layout and max(1, ldc*n) for\nrow major layout, contains the n-by-ncc matrix C for computing the\nproduct QH*C.\nldvt\nThe leading dimension of vt. Constraints:\nldvt≥ max(1, n) if ncvt > 0 for column major layout and ldvt≥ max(1,\nncvt) for row major layout;\nldvt≥ 1 if ncvt = 0.\nldu\nThe leading dimension of u. Constraint:\nldu≥ max(1, nru) for column major layout and ldu≥ max(1, n) for row\nmajor layout .\nldc\nThe leading dimension of c. Constraints:\nldc≥ max(1, n) if ncc > 0 for column major layout and ldc≥ max(1, ncc)\nfor row major layout; ldc≥ 1 otherwise.\nOutput Parameters\nd\nOn exit, if info = 0, overwritten by the singular values in decreasing order\n(see info).\ne\nOn exit, if info = 0, e is destroyed. See also info below.\nc\nOverwritten by the product QH*C.\nvt\nOn exit, this array is overwritten by PH *VT. Not referenced if ncvt = 0.\nu\nOn exit, this array is overwritten by U *Q. Not referenced if nru = 0.\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nIf info > 0,\nIf ncvt = nru = ncc = 0,\n•\ninfo = 1, a split was marked by a positive value in e\n•\ninfo = 2, the current block of z not diagonalized after 100*n iterations (in the inner while loop)\n•\ninfo = 3, termination criterion of the outer while loop is not met (the program created more than n\nunreduced blocks).\nIn all other cases when ncvt, nru, or ncc > 0, the algorithm did not converge; d and e contain the elements\nof a bidiagonal matrix that is orthogonally similar to the input matrix B; if info = i, i elements of e have\nnot converged to zero.\nApplication Notes\nEach singular value and singular vector is computed to high relative accuracy. However, the reduction to\nbidiagonal form (prior to calling the routine) may decrease the relative accuracy in the small singular values\nof the original matrix if its singular values vary widely in magnitude.\nIf si is an exact singular value of B, and si is the corresponding computed value, then\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n855\n\n\n|si - σi| ≤p*(m,n)*ε*σi\nwhere p(m, n) is a modestly increasing function of m and n, and ε is the machine precision.\nIf only singular values are computed, they are computed more accurately than when some singular vectors\nare also computed (that is, the function p(m, n) is smaller).\nIf ui is the corresponding exact left singular vector of B, and wi is the corresponding computed left singular\nvector, then the angle θ(ui, wi) between them is bounded as follows:\nθ(ui, wi) ≤p(m,n)*ε / min i≠j(|σi - σj|/|σi + σj|).\nHere mini≠j(|σi - σj|/|σi + σj|) is the relative gap between σi and the other singular values. A similar\nerror bound holds for the right singular vectors.\nThe total number of real floating-point operations is roughly proportional to n2 if only the singular values are\ncomputed. About 6n2*nru additional operations (12n2*nru for complex flavors) are required to compute the\nleft singular vectors and about 6n2*ncvt operations (12n2*ncvt for complex flavors) to compute the right\nsingular vectors.\n?bdsdc\nComputes the singular value decomposition of a real\nbidiagonal matrix using a divide and conquer method.\nSyntax\nlapack_int LAPACKE_sbdsdc (int matrix_layout, char uplo, char compq, lapack_int n,\nfloat* d, float* e, float* u, lapack_int ldu, float* vt, lapack_int ldvt, float* q,\nlapack_int* iq);\nlapack_int LAPACKE_dbdsdc (int matrix_layout, char uplo, char compq, lapack_int n,\ndouble* d, double* e, double* u, lapack_int ldu, double* vt, lapack_int ldvt, double* q,\nlapack_int* iq);\nInclude Files\n•\nmkl.h\nDescription\nThe routine computes the Singular Value Decomposition (SVD) of a real n-by-n (upper or lower) bidiagonal\nmatrix B: B = U*Σ*VT, using a divide and conquer method, where Σ is a diagonal matrix with non-negative\ndiagonal elements (the singular values of B), and U and V are orthogonal matrices of left and right singular\nvectors, respectively. ?bdsdc can be used to compute all singular values, and optionally, singular vectors or\nsingular vectors in compact form.\nThis rotuine\nuses ?lasd0, ?lasd1, ?lasd2, ?lasd3, ?lasd4, ?lasd5, ?lasd6, ?lasd7, ?lasd8, ?lasd9, ?lasda,\n?lasdq, ?lasdt.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nuplo\nMust be 'U' or 'L'.\nIf uplo = 'U', B is an upper bidiagonal matrix.\nIf uplo = 'L', B is a lower bidiagonal matrix.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n856\n\n\ncompq\nMust be 'N', 'P', or 'I'.\nIf compq = 'N', compute singular values only.\nIf compq = 'P', compute singular values and compute singular vectors in\ncompact form.\nIf compq = 'I', compute singular values and singular vectors.\nn\nThe order of the matrix B (n ≥ 0).\nd, e\nArrays:\nd contains the n diagonal elements of the bidiagonal matrix B. The size of d\nmust be at least max(1, n).\ne contains the off-diagonal elements of the bidiagonal matrix B. The size of\ne must be at least max(1, n).\nldu\nThe leading dimension of the output array u; ldu≥ 1.\nIf singular vectors are desired, then ldu≥ max(1, n), regardless of the value\nof matrix_layout.\nldvt\nThe leading dimension of the output array vt; ldvt≥ 1.\nIf singular vectors are desired, then ldvt≥ max(1, n), regardless of the value\nof matrix_layout.\nOutput Parameters\nd\nIf info = 0, overwritten by the singular values of B.\ne\nOn exit, e is overwritten.\nu, vt, q\nArrays: u(size ldu*n), vt(size ldvt*n), q(size ≥n*(11 + 2*smlsiz +\n8*int(log2(n/(smlsiz+1)))) where smlsiz is returned by ilaenv and is\nequal to maximum size of the subproblems at the bottom of the\ncomputation tree )..\nIf compq = 'I', then on exit u contains the left singular vectors of the\nbidiagonal matrix B, unless info≠ 0 (seeinfo). For other values of compq, u\nis not referenced.\nif compq = 'I', then on exit vtT contains the right singular vectors of the\nbidiagonal matrix B, unless info≠ 0 (seeinfo). For other values of compq,\nvt is not referenced.\nIf compq = 'P', then on exit, if info = 0, q and iq contain the left and\nright singular vectors in a compact form. Specifically, q contains all the\nfloat (for sbdsdc) or double (for dbdsdc) data for singular vectors. For\nother values of compq, q is not referenced.\niq\nArray: iq(size ≥n*(3 + 3*int(log2(n/(smlsiz+1)))) where smlsiz is\nreturned by ilaenv and is equal to maximum size of the subproblems at\nthe bottom of the computation tree.).\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n857\n\n\nIf compq = 'P', then on exit, if info = 0, q and iq contain the left and\nright singular vectors in a compact form. Specifically, iq contains all the\nlapack_int data for singular vectors. For other values of compq, iq is not\nreferenced.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nIf info = i, the algorithm failed to compute a singular value. The update process of divide and conquer\nfailed.\nSymmetric Eigenvalue Problems: LAPACK Computational Routines\nSymmetric eigenvalue problems are posed as follows: given an n-by-n real symmetric or complex\nHermitian matrix A, find the eigenvalues λ and the corresponding eigenvectors z that satisfy the equation\nAz = λz (or, equivalently, zHA = λzH).\nIn such eigenvalue problems, all n eigenvalues are real not only for real symmetric but also for complex\nHermitian matrices A, and there exists an orthonormal system of n eigenvectors. If A is a symmetric or\nHermitian positive-definite matrix, all eigenvalues are positive.\nTo solve a symmetric eigenvalue problem with LAPACK, you usually need to reduce the matrix to tridiagonal\nform and then solve the eigenvalue problem with the tridiagonal matrix obtained. LAPACK includes routines\nfor reducing the matrix to a tridiagonal form by an orthogonal (or unitary) similarity transformation A =\nQTQH as well as for solving tridiagonal symmetric eigenvalue problems. These routines are listed in Table\n\"Computational Routines for Solving Symmetric Eigenvalue Problems\".\nThere are different routines for symmetric eigenvalue problems, depending on whether you need all\neigenvectors or only some of them or eigenvalues only, whether the matrix A is positive-definite or not, and\nso on.\nThese routines are based on three primary algorithms for computing eigenvalues and eigenvectors of\nsymmetric problems: the divide and conquer algorithm, the QR algorithm, and bisection followed by inverse\niteration. The divide and conquer algorithm is generally more efficient and is recommended for computing all\neigenvalues and eigenvectors. Furthermore, to solve an eigenvalue problem using the divide and conquer\nalgorithm, you need to call only one routine. In general, more than one routine has to be called if the QR\nalgorithm or bisection followed by inverse iteration is used.\nComputational Routines for Solving Symmetric Eigenvalue Problems\nOperation\nReal symmetric matrices\nComplex Hermitian\nmatrices\nReduce to tridiagonal form A = QTQH (full\nstorage)\nsytrd \nhetrd \nReduce to tridiagonal form A = QTQH\n(packed storage)\nsptrd\nhptrd\nReduce to tridiagonal form A = QTQH (band\nstorage).\nsbtrd\nhbtrd\nGenerate matrix Q (full storage)\norgtr\nungtr\nGenerate matrix Q (packed storage)\nopgtr\nupgtr\nApply matrix Q (full storage)\normtr\nunmtr\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n858\n\n\nOperation\nReal symmetric matrices\nComplex Hermitian\nmatrices\nApply matrix Q (packed storage)\nopmtr\nupmtr\nFind all eigenvalues of a tridiagonal matrix\nT\nsterf\n \nFind all eigenvalues and eigenvectors of a\ntridiagonal matrix T\nsteqr  stedc\nsteqr  stedc\nFind all eigenvalues and eigenvectors of a\ntridiagonal positive-definite matrix T.\npteqr\npteqr\nFind selected eigenvalues of a tridiagonal\nmatrix T\nstebz  stegr\nstegr\nFind selected eigenvectors of a tridiagonal\nmatrix T\nstein  stegr\nstein  stegr\nFind selected eigenvalues and eigenvectors\nof f a real symmetric tridiagonal matrix T\nstemr\nstemr\nCompute the reciprocal condition numbers\nfor the eigenvectors\ndisna\ndisna\n?sytrd\nReduces a real symmetric matrix to tridiagonal form.\nSyntax\nlapack_int LAPACKE_ssytrd (int matrix_layout, char uplo, lapack_int n, float* a,\nlapack_int lda, float* d, float* e, float* tau);\nlapack_int LAPACKE_dsytrd (int matrix_layout, char uplo, lapack_int n, double* a,\nlapack_int lda, double* d, double* e, double* tau);\nInclude Files\n•\nmkl.h\nDescription\nThe routine reduces a real symmetric matrix A to symmetric tridiagonal form T by an orthogonal similarity\ntransformation: A = Q*T*QT. The orthogonal matrix Q is not formed explicitly but is represented as a\nproduct of n-1 elementary reflectors. Routines are provided for working with Q in this representation (see\nApplication Notes below).\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nuplo\nMust be 'U' or 'L'.\nIf uplo = 'U', a stores the upper triangular part of A.\nIf uplo = 'L', a stores the lower triangular part of A.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n859\n\n\nn\nThe order of the matrix A (n≥ 0).\na\na (size max(1, lda*n)) is an array containing either upper or lower\ntriangular part of the matrix A, as specified by uplo. If uplo = 'U', the\nleading n-by-n upper triangular part of a contains the upper triangular part\nof the matrix A, and the strictly lower triangular part of A is not referenced.\nIf uplo = 'L', the leading n-by-n lower triangular part of a contains the\nlower triangular part of the matrix A, and the strictly upper triangular part\nof A is not referenced.\nlda\nThe leading dimension of a; at least max(1, n).\nOutput Parameters\na\nOn exit,\nif uplo = 'U', the diagonal and first superdiagonal of A are overwritten by\nthe corresponding elements of the tridiagonal matrix T, and the elements\nabove the first superdiagonal, with the array tau, represent the orthogonal\nmatrix Q as a product of elementary reflectors;\nif uplo = 'L', the diagonal and first subdiagonal of A are overwritten by\nthe corresponding elements of the tridiagonal matrix T, and the elements\nbelow the first subdiagonal, with the array tau, represent the orthogonal\nmatrix Q as a product of elementary reflectors.\nd, e, tau\nArrays:\nd contains the diagonal elements of the matrix T.\nThe size of d must be at least max(1, n).\ne contains the off-diagonal elements of T.\nThe size of e must be at least max(1, n-1).\ntau stores (n-1) scalars that define elementary reflectors in decomposition\nof the orthogonal matrix Q in a product of n-1 elementary reflectors. tau(n)\nis used as workspace.\nThe size of tau must be at least max(1, n).\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nApplication Notes\nThe computed matrix T is exactly similar to a matrix A+E, where ||E||2 = c(n)*ε*||A||2, c(n) is a\nmodestly increasing function of n, and ε is the machine precision.\nThe approximate number of floating-point operations is (4/3)n3.\nAfter calling this routine, you can call the following:\norgtr\nto form the computed matrix Q explicitly\normtr\nto multiply a real matrix by Q.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n860\n\n\nThe complex counterpart of this routine is ?hetrd.\n?orgtr\nGenerates the real orthogonal matrix Q determined\nby ?sytrd.\nSyntax\nlapack_int LAPACKE_sorgtr (int matrix_layout, char uplo, lapack_int n, float* a,\nlapack_int lda, const float* tau);\nlapack_int LAPACKE_dorgtr (int matrix_layout, char uplo, lapack_int n, double* a,\nlapack_int lda, const double* tau);\nInclude Files\n•\nmkl.h\nDescription\nThe routine explicitly generates the n-by-n orthogonal matrix Q formed by ?sytrd when reducing a real\nsymmetric matrix A to tridiagonal form: A = Q*T*QT. Use this routine after a call to ?sytrd.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nuplo\nMust be 'U' or 'L'.\nUse the same uplo as supplied to ?sytrd.\nn\nThe order of the matrix Q (n≥ 0).\na, tau\nArrays:\na (size max(1, lda*n)) is the array a as returned by ?sytrd.\ntau is the array tau as returned by ?sytrd.\nThe size of tau must be at least max(1, n-1).\nlda\nThe leading dimension of a; at least max(1, n).\nOutput Parameters\na\nOverwritten by the orthogonal matrix Q.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nApplication Notes\nThe computed matrix Q differs from an exactly orthogonal matrix by a matrix E such that ||E||2 = O(ε),\nwhere ε is the machine precision.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n861\n\n\nThe approximate number of floating-point operations is (4/3)n3.\nThe complex counterpart of this routine is ungtr.\n?ormtr\nMultiplies a real matrix by the real orthogonal matrix\nQ determined by ?sytrd.\nSyntax\nlapack_int LAPACKE_sormtr (int matrix_layout, char side, char uplo, char trans,\nlapack_int m, lapack_int n, const float* a, lapack_int lda, const float* tau, float* c,\nlapack_int ldc);\nlapack_int LAPACKE_dormtr (int matrix_layout, char side, char uplo, char trans,\nlapack_int m, lapack_int n, const double* a, lapack_int lda, const double* tau, double*\nc, lapack_int ldc);\nInclude Files\n•\nmkl.h\nDescription\nThe routine multiplies a real matrix C by Q or QT, where Q is the orthogonal matrix Q formed by sytrd when\nreducing a real symmetric matrix A to tridiagonal form: A = Q*T*QT. Use this routine after a call to ?sytrd.\nDepending on the parameters side and trans, the routine can form one of the matrix products Q*C, QT*C,\nC*Q, or C*QT (overwriting the result on C).\nInput Parameters\nIn the descriptions below, r denotes the order of Q:\nIf side = 'L', r = m; if side = 'R', r = n.\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nside\nMust be either 'L' or 'R'.\nIf side = 'L', Q or QT is applied to C from the left.\nIf side = 'R', Q or QT is applied to C from the right.\nuplo\nMust be 'U' or 'L'.\nUse the same uplo as supplied to ?sytrd.\ntrans\nMust be either 'N' or 'T'.\nIf trans = 'N', the routine multiplies C by Q.\nIf trans = 'T', the routine multiplies C by QT.\nm\nThe number of rows in the matrix C (m≥ 0).\nn\nThe number of columns in C (n≥ 0).\na, c, tau\na (size max(1, lda*r)) and tau are the arrays returned by ?sytrd.\nThe size of tau must be at least max(1, r-1).\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n862\n\n\nc(size max(1, ldc*n) for column major layout and max(1, ldc*m) for row\nmajor layout) contains the matrix C.\nlda\nThe leading dimension of a; lda≥ max(1, r).\nldc\nThe leading dimension of c; ldc≥ max(1, m) for column major layout and\nat least max(1, n) for row major layout .\nOutput Parameters\nc\nOverwritten by the product Q*C, QT*C, C*Q, or C*QT (as specified by side\nand trans).\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nApplication Notes\nThe computed product differs from the exact product by a matrix E such that ||E||2 = O(ε)*||C||2.\nThe total number of floating-point operations is approximately 2*m2*n, if side = 'L', or 2*n2*m, if side =\n'R'.\nThe complex counterpart of this routine is unmtr.\n?hetrd\nReduces a complex Hermitian matrix to tridiagonal\nform.\nSyntax\nlapack_int LAPACKE_chetrd( int matrix_layout, char uplo, lapack_int n,\nlapack_complex_float* a, lapack_int lda, float* d, float* e, lapack_complex_float*\ntau );\nlapack_int LAPACKE_zhetrd( int matrix_layout, char uplo, lapack_int n,\nlapack_complex_double* a, lapack_int lda, double* d, double* e, lapack_complex_double*\ntau );\nInclude Files\n•\nmkl.h\nDescription\nThe routine reduces a complex Hermitian matrix A to symmetric tridiagonal form T by a unitary similarity\ntransformation: A = Q*T*QH. The unitary matrix Q is not formed explicitly but is represented as a product of\nn-1 elementary reflectors. Routines are provided to work with Q in this representation. (They are described\nlater in this topic.)\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n863\n\n\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nuplo\nMust be 'U' or 'L'.\nIf uplo = 'U', a stores the upper triangular part of A.\nIf uplo = 'L', a stores the lower triangular part of A.\nn\nThe order of the matrix A (n≥ 0).\na\na (size max(1, lda*n)) is an array containing either upper or lower\ntriangular part of the matrix A, as specified by uplo. If uplo = 'U', the\nleading n-by-n upper triangular part of a contains the upper triangular part\nof the matrix A, and the strictly lower triangular part of A is not referenced.\nIf uplo = 'L', the leading n-by-n lower triangular part of a contains the\nlower triangular part of the matrix A, and the strictly upper triangular part\nof A is not referenced.\nlda\nThe leading dimension of a; at least max(1, n).\nOutput Parameters\na\nOn exit,\nif uplo = 'U', the diagonal and first superdiagonal of A are overwritten by\nthe corresponding elements of the tridiagonal matrix T, and the elements\nabove the first superdiagonal, with the array tau, represent the orthogonal\nmatrix Q as a product of elementary reflectors;\nif uplo = 'L', the diagonal and first subdiagonal of A are overwritten by\nthe corresponding elements of the tridiagonal matrix T, and the elements\nbelow the first subdiagonal, with the array tau, represent the orthogonal\nmatrix Q as a product of elementary reflectors.\nd, e\nArrays:\nd contains the diagonal elements of the matrix T.\nThe dimension of d must be at least max(1, n).\ne contains the off-diagonal elements of T.\nThe dimension of e must be at least max(1, n-1).\ntau\nArray, size at least max(1, n-1). Stores (n-1) scalars that define elementary\nreflectors in decomposition of the unitary matrix Q in a product of n-1\nelementary reflectors.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n864\n\n\nApplication Notes\nThe computed matrix T is exactly similar to a matrix A + E, where ||E||2 = c(n)*ε*||A||2, c(n) is a\nmodestly increasing function of n, and ε is the machine precision.\nThe approximate number of floating-point operations is (16/3)n3.\nAfter calling this routine, you can call the following:\nungtr\nto form the computed matrix Q explicitly\nunmtr\nto multiply a complex matrix by Q.\nThe real counterpart of this routine is ?sytrd.\n?ungtr\nGenerates the complex unitary matrix Q determined\nby ?hetrd.\nSyntax\nlapack_int LAPACKE_cungtr (int matrix_layout, char uplo, lapack_int n,\nlapack_complex_float* a, lapack_int lda, const lapack_complex_float* tau);\nlapack_int LAPACKE_zungtr (int matrix_layout, char uplo, lapack_int n,\nlapack_complex_double* a, lapack_int lda, const lapack_complex_double* tau);\nInclude Files\n•\nmkl.h\nDescription\nThe routine explicitly generates the n-by-n unitary matrix Q formed by ?hetrd when reducing a complex\nHermitian matrix A to tridiagonal form: A = Q*T*QH. Use this routine after a call to ?hetrd.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nuplo\nMust be 'U' or 'L'.\nUse the same uplo as supplied to ?hetrd.\nn\nThe order of the matrix Q (n≥ 0).\na, tau\nArrays:\na (size max(1, lda*n)) is the array a as returned by ?hetrd.\ntau is the array tau as returned by ?hetrd.\nThe dimension of tau must be at least max(1, n-1).\nlda\nThe leading dimension of a; at least max(1, n).\nOutput Parameters\na\nOverwritten by the unitary matrix Q.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n865\n\n\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nApplication Notes\nThe computed matrix Q differs from an exactly unitary matrix by a matrix E such that ||E||2 = O(ε), where\nε is the machine precision.\nThe approximate number of floating-point operations is (16/3)n3.\nThe real counterpart of this routine is orgtr.\n?unmtr\nMultiplies a complex matrix by the complex unitary\nmatrix Q determined by ?hetrd.\nSyntax\nlapack_int LAPACKE_cunmtr (int matrix_layout, char side, char uplo, char trans,\nlapack_int m, lapack_int n, const lapack_complex_float* a, lapack_int lda, const\nlapack_complex_float* tau, lapack_complex_float* c, lapack_int ldc);\nlapack_int LAPACKE_zunmtr (int matrix_layout, char side, char uplo, char trans,\nlapack_int m, lapack_int n, const lapack_complex_double* a, lapack_int lda, const\nlapack_complex_double* tau, lapack_complex_double* c, lapack_int ldc);\nInclude Files\n•\nmkl.h\nDescription\nThe routine multiplies a complex matrix C by Q or QH, where Q is the unitary matrix Q formed by ?hetrd\nwhen reducing a complex Hermitian matrix A to tridiagonal form: A = Q*T*QH. Use this routine after a call\nto ?hetrd.\nDepending on the parameters side and trans, the routine can form one of the matrix products Q*C, QH*C,\nC*Q, or C*QH (overwriting the result on C).\nInput Parameters\nIn the descriptions below, r denotes the order of Q:\nIf side = 'L', r = m; if side = 'R', r = n.\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nside\nMust be either 'L' or 'R'.\nIf side = 'L', Q or QH is applied to C from the left.\nIf side = 'R', Q or QH is applied to C from the right.\nuplo\nMust be 'U' or 'L'.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n866\n\n\nUse the same uplo as supplied to ?hetrd.\ntrans\nMust be either 'N' or 'T'.\nIf trans = 'N', the routine multiplies C by Q.\nIf trans = 'C', the routine multiplies C by QH.\nm\nThe number of rows in the matrix C (m≥ 0).\nn\nThe number of columns in C (n≥ 0).\na, c, tau\na (size max(1, lda*r)) and tau are the arrays returned by ?hetrd.\nThe dimension of tau must be at least max(1, r-1).\nc(size max(1, ldc*n) for column major layout and max(1, ldc*m) for row\nmajor layout) contains the matrix C.\nlda\nThe leading dimension of a; lda≥ max(1, r).\nldc\nThe leading dimension of c; ldc≥ max(1, n) for column major layout and\nldc≥ max(1, m) for row major layout .\nOutput Parameters\nc\nOverwritten by the product Q*C, QH*C, C*Q, or C*QH (as specified by side\nand trans).\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nApplication Notes\nThe computed product differs from the exact product by a matrix E such that ||E||2 = O(ε)*||C||2, where\nε is the machine precision.\nThe total number of floating-point operations is approximately 8*m2*n if side = 'L' or 8*n2*m if side =\n'R'.\nThe real counterpart of this routine is ormtr.\n?sptrd\nReduces a real symmetric matrix to tridiagonal form\nusing packed storage.\nSyntax\nlapack_int LAPACKE_ssptrd (int matrix_layout, char uplo, lapack_int n, float* ap,\nfloat* d, float* e, float* tau);\nlapack_int LAPACKE_dsptrd (int matrix_layout, char uplo, lapack_int n, double* ap,\ndouble* d, double* e, double* tau);\nInclude Files\n•\nmkl.h\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n867\n\n\nDescription\nThe routine reduces a packed real symmetric matrix A to symmetric tridiagonal form T by an orthogonal\nsimilarity transformation: A = Q*T*QT. The orthogonal matrix Q is not formed explicitly but is represented as\na product of n-1 elementary reflectors. Routines are provided for working with Q in this representation. See\nApplication Notes below for details.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nuplo\nMust be 'U' or 'L'.\nIf uplo = 'U', ap stores the packed upper triangle of A.\nIf uplo = 'L', ap stores the packed lower triangle of A.\nn\nThe order of the matrix A (n≥ 0).\nap\nArray, size at least max(1, n(n+1)/2). Contains either upper or lower\ntriangle of A (as specified by uplo) in the packed form described in Matrix\nStorage Schemes.\nOutput Parameters\nap\nOverwritten by the tridiagonal matrix T and details of the orthogonal matrix\nQ, as specified by uplo.\nd, e, tau\nArrays:\nd contains the diagonal elements of the matrix T.\nThe dimension of d must be at least max(1, n).\ne contains the off-diagonal elements of T.\nThe dimension of e must be at least max(1, n-1).\ntau Stores (n-1) scalars that define elementary reflectors in decomposition\nof the matrix Q in a product of n-1 reflectors.\nThe dimension of tau must be at least max(1, n-1).\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nApplication Notes\nThe matrix Q is represented as a product of n-1 elementary reflectors, as follows :\n•\nIf uplo = 'U', Q = H(n-1) ... H(2)H(1)\nEach H(i) has the form\nH(i) = I - tau*v*vT\nwhere tau is a real scalar and v is a real vector with v(i+1:n) = 0 and v(i) = 1.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n868\n\n\nOn exit, tau is stored in tau[i - 1], and v(1:i-1) is stored in AP, overwriting A(1:i-1, i+1).\n•\nIf uplo = 'L', Q = H(1)H(2) ... H(n-1)\nEach H(i) has the form\nH(i) = I - tau*v*vT\nwhere tau is a real scalar and v is a real vector with v(1:i) = 0 and v(i+1) = 1.\nOn exit, tau is stored in tau[i - 1], and v(i+2:n) is stored in AP, overwriting A(i+2:n, i).\nThe computed matrix T is exactly similar to a matrix A+E, where ||E||2 = c(n)*ε*||A||2, c(n) is a\nmodestly increasing function of n, and ε is the machine precision. The approximate number of floating-point\noperations is (4/3)n3.\nAfter calling this routine, you can call the following:\nopgtr\nto form the computed matrix Q explicitly\nopmtr\nto multiply a real matrix by Q.\nThe complex counterpart of this routine is hptrd.\n?opgtr\nGenerates the real orthogonal matrix Q determined\nby ?sptrd.\nSyntax\nlapack_int LAPACKE_sopgtr (int matrix_layout, char uplo, lapack_int n, const float* ap,\nconst float* tau, float* q, lapack_int ldq);\nlapack_int LAPACKE_dopgtr (int matrix_layout, char uplo, lapack_int n, const double*\nap, const double* tau, double* q, lapack_int ldq);\nInclude Files\n•\nmkl.h\nDescription\nThe routine explicitly generates the n-by-n orthogonal matrix Q formed by sptrd when reducing a packed real\nsymmetric matrix A to tridiagonal form: A = Q*T*QT. Use this routine after a call to ?sptrd.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nuplo\nMust be 'U' or 'L'. Use the same uplo as supplied to ?sptrd.\nn\nThe order of the matrix Q (n≥ 0).\nap, tau\nArrays ap and tau, as returned by ?sptrd.\nThe size of ap must be at least max(1, n(n+1)/2).\nThe size of tau must be at least max(1, n-1).\nldq\nThe leading dimension of the output array q; at least max(1, n).\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n869\n\n\nOutput Parameters\nq\nArray, size (size max(1, ldq*n)) .\nContains the computed matrix Q.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nApplication Notes\nThe computed matrix Q differs from an exactly orthogonal matrix by a matrix E such that ||E||2 = O(ε),\nwhere ε is the machine precision.\nThe approximate number of floating-point operations is (4/3)n3.\nThe complex counterpart of this routine is upgtr.\n?opmtr\nMultiplies a real matrix by the real orthogonal matrix\nQ determined by ?sptrd.\nSyntax\nlapack_int LAPACKE_sopmtr (int matrix_layout, char side, char uplo, char trans,\nlapack_int m, lapack_int n, const float* ap, const float* tau, float* c, lapack_int\nldc);\nlapack_int LAPACKE_dopmtr (int matrix_layout, char side, char uplo, char trans,\nlapack_int m, lapack_int n, const double* ap, const double* tau, double* c, lapack_int\nldc);\nInclude Files\n•\nmkl.h\nDescription\nThe routine multiplies a real matrix C by Q or QT, where Q is the orthogonal matrix Q formed by sptrd when\nreducing a packed real symmetric matrix A to tridiagonal form: A = Q*T*QT. Use this routine after a call\nto ?sptrd.\nDepending on the parameters side and trans, the routine can form one of the matrix products Q*C, QT*C,\nC*Q, or C*QT (overwriting the result on C).\nInput Parameters\nIn the descriptions below, r denotes the order of Q:\nIf side = 'L', r = m; if side = 'R', r = n.\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nside\nMust be either 'L' or 'R'.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n870\n\n\nIf side = 'L', Q or QT is applied to C from the left.\nIf side = 'R', Q or QT is applied to C from the right.\nuplo\nMust be 'U' or 'L'.\nUse the same uplo as supplied to ?sptrd.\ntrans\nMust be either 'N' or 'T'.\nIf trans = 'N', the routine multiplies C by Q.\nIf trans = 'T', the routine multiplies C by QT.\nm\nThe number of rows in the matrix C (m≥ 0).\nn\nThe number of columns in C (n≥ 0).\nap, tau, c\nap and tau are the arrays returned by ?sptrd.\nThe dimension of ap must be at least max(1, r(r+1)/2).\nThe dimension of tau must be at least max(1, r-1).\nc(size max(1, ldc*n) for column major layout and max(1, ldc*m) for row\nmajor layout) contains the matrix C.\nldc\nThe leading dimension of c; ldc≥ max(1, n) for column major layout and\nldc≥ max(1, m) for row major layout .\nOutput Parameters\nc\nOverwritten by the product Q*C, QT*C, C*Q, or C*QT (as specified by side\nand trans).\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nApplication Notes\nThe computed product differs from the exact product by a matrix E such that ||E||2 = O(ε) ||C||2, where\nε is the machine precision.\nThe total number of floating-point operations is approximately 2*m2*n if side = 'L', or 2*n2*m if side =\n'R'.\nThe complex counterpart of this routine is upmtr.\n?hptrd\nReduces a complex Hermitian matrix to tridiagonal\nform using packed storage.\nSyntax\nlapack_int LAPACKE_chptrd( int matrix_layout, char uplo, lapack_int n,\nlapack_complex_float* ap, float* d, float* e, lapack_complex_float* tau );\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n871\n\n\nlapack_int LAPACKE_zhptrd( int matrix_layout, char uplo, lapack_int n,\nlapack_complex_double* ap, double* d, double* e, lapack_complex_double* tau );\nInclude Files\n•\nmkl.h\nDescription\nThe routine reduces a packed complex Hermitian matrix A to symmetric tridiagonal form T by a unitary\nsimilarity transformation: A = Q*T*QH. The unitary matrix Q is not formed explicitly but is represented as a\nproduct of n-1 elementary reflectors. Routines are provided for working with Q in this representation (see\nApplication Notes below).\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nuplo\nMust be 'U' or 'L'.\nIf uplo = 'U', ap stores the packed upper triangle of A.\nIf uplo = 'L', ap stores the packed lower triangle of A.\nn\nThe order of the matrix A (n≥ 0).\nap\nArray, size at least max(1, n(n+1)/2). Contains either upper or lower\ntriangle of A (as specified by uplo) in the packed form described in \"Matrix\nStorage Schemes.\nOutput Parameters\nap\nOverwritten by the tridiagonal matrix T and details of the unitary matrix Q,\nas specified by uplo.\nd, e\nArrays:\nd contains the diagonal elements of the matrix T.\nThe size of d must be at least max(1, n).\ne contains the off-diagonal elements of T.\nThe size of e must be at least max(1, n-1).\ntau\nArray, size at least max(1, n-1). Stores (n-1) scalars that define elementary\nreflectors in decomposition of the unitary matrix Q in a product of\nreflectors.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n872\n\n\nApplication Notes\nThe computed matrix T is exactly similar to a matrix A + E, where ||E||2 = c(n)*ε*||A||2, c(n) is a\nmodestly increasing function of n, and ε is the machine precision.\nThe approximate number of floating-point operations is (16/3)n3.\nAfter calling this routine, you can call the following:\nupgtr\nto form the computed matrix Q explicitly\nupmtr\nto multiply a complex matrix by Q.\nThe real counterpart of this routine is sptrd.\n?upgtr\nGenerates the complex unitary matrix Q determined\nby ?hptrd.\nSyntax\nlapack_int LAPACKE_cupgtr (int matrix_layout, char uplo, lapack_int n, const\nlapack_complex_float* ap, const lapack_complex_float* tau, lapack_complex_float* q,\nlapack_int ldq);\nlapack_int LAPACKE_zupgtr (int matrix_layout, char uplo, lapack_int n, const\nlapack_complex_double* ap, const lapack_complex_double* tau, lapack_complex_double* q,\nlapack_int ldq);\nInclude Files\n•\nmkl.h\nDescription\nThe routine explicitly generates the n-by-n unitary matrix Q formed by hptrd when reducing a packed\ncomplex Hermitian matrix A to tridiagonal form: A = Q*T*QH. Use this routine after a call to ?hptrd.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nuplo\nMust be 'U' or 'L'. Use the same uplo as supplied to ?hptrd.\nn\nThe order of the matrix Q (n≥ 0).\nap, tau\nArrays ap and tau, as returned by ?hptrd.\nThe dimension of ap must be at least max(1, n(n+1)/2).\nThe dimension of tau must be at least max(1, n-1).\nldq\nThe leading dimension of the output array q;\nat least max(1, n).\nOutput Parameters\nq\nArray, size (size max(1, ldq*n)) .\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n873\n\n\nContains the computed matrix Q.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nApplication Notes\nThe computed matrix Q differs from an exactly orthogonal matrix by a matrix E such that ||E||2 = O(ε),\nwhere ε is the machine precision.\nThe approximate number of floating-point operations is (16/3)n3.\nThe real counterpart of this routine is opgtr.\n?upmtr\nMultiplies a complex matrix by the unitary matrix Q\ndetermined by ?hptrd.\nSyntax\nlapack_int LAPACKE_cupmtr (int matrix_layout, char side, char uplo, char trans,\nlapack_int m, lapack_int n, const lapack_complex_float* ap, const lapack_complex_float*\ntau, lapack_complex_float* c, lapack_int ldc);\nlapack_int LAPACKE_zupmtr (int matrix_layout, char side, char uplo, char trans,\nlapack_int m, lapack_int n, const lapack_complex_double* ap, const\nlapack_complex_double* tau, lapack_complex_double* c, lapack_int ldc);\nInclude Files\n•\nmkl.h\nDescription\nThe routine multiplies a complex matrix C by Q or QH, where Q is the unitary matrix formed by hptrd when\nreducing a packed complex Hermitian matrix A to tridiagonal form: A = Q*T*QH. Use this routine after a call\nto ?hptrd.\nDepending on the parameters side and trans, the routine can form one of the matrix products Q*C, QH*C,\nC*Q, or C*QH (overwriting the result on C).\nInput Parameters\nIn the descriptions below, r denotes the order of Q:\nIf side = 'L', r = m; if side = 'R', r = n.\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nside\nMust be either 'L' or 'R'.\nIf side = 'L', Q or QH is applied to C from the left.\nIf side = 'R', Q or QH is applied to C from the right.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n874\n\n\nuplo\nMust be 'U' or 'L'.\nUse the same uplo as supplied to ?hptrd.\ntrans\nMust be either 'N' or 'T'.\nIf trans = 'N', the routine multiplies C by Q.\nIf trans = 'T', the routine multiplies C by QH.\nm\nThe number of rows in the matrix C (m≥ 0).\nn\nThe number of columns in C (n≥ 0).\nap, tau, c, \nap and tau are the arrays returned by ?hptrd.\nThe size of ap must be at least max(1, r(r+1)/2).\nThe size of tau must be at least max(1, r-1).\nc(size max(1, ldc*n) for column major layout and max(1, ldc*m) for row\nmajor layout) contains the matrix C.\nldc\nThe leading dimension of c; ldc≥ max(1, m) for column major layout and\nldc≥ max(1, n) for row major layout .\nOutput Parameters\nc\nOverwritten by the product Q*C, QH*C, C*Q, or C*QH (as specified by side\nand trans).\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nApplication Notes\nThe computed product differs from the exact product by a matrix E such that ||E||2 = O(ε)*||C||2, where\nε is the machine precision.\nThe total number of floating-point operations is approximately 8*m2*n if side = 'L' or 8*n2*m if side =\n'R'.\nThe real counterpart of this routine is opmtr.\n?sbtrd\nReduces a real symmetric band matrix to tridiagonal\nform.\nSyntax\nlapack_int LAPACKE_ssbtrd (int matrix_layout, char vect, char uplo, lapack_int n,\nlapack_int kd, float* ab, lapack_int ldab, float* d, float* e, float* q, lapack_int\nldq);\nlapack_int LAPACKE_dsbtrd (int matrix_layout, char vect, char uplo, lapack_int n,\nlapack_int kd, double* ab, lapack_int ldab, double* d, double* e, double* q, lapack_int\nldq);\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n875\n\n\nInclude Files\n•\nmkl.h\nDescription\nThe routine reduces a real symmetric band matrix A to symmetric tridiagonal form T by an orthogonal\nsimilarity transformation: A = Q*T*QT. The orthogonal matrix Q is determined as a product of Givens\nrotations.\nIf required, the routine can also form the matrix Q explicitly.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nvect\nMust be 'V', 'N', or 'U'.\nIf vect = 'V', the routine returns the explicit matrix Q.\nIf vect = 'N', the routine does not return Q.\nIf vect = 'U', the routine updates matrix X by forming X*Q.\nuplo\nMust be 'U' or 'L'.\nIf uplo = 'U', ab stores the upper triangular part of A.\nIf uplo = 'L', ab stores the lower triangular part of A.\nn\nThe order of the matrix A (n≥0).\nkd\nThe number of super- or sub-diagonals in A\n(kd≥0).\nab, q\nab(size at least max(1, ldab*n) for column major layout and at least\nmax(1, ldab*(kd+ 1)) for row major layout) is an array containing either\nupper or lower triangular part of the matrix A (as specified by uplo) in band\nstorage format.\nq (size max(1, ldq*n)) is an array.\nIf vect = 'U', the q array must contain an n-by-n matrix X.\nIf vect = 'N' or 'V', the q parameter need not be set.\nldab\nThe leading dimension of ab; at least kd+1 for column major layout and n\nfor row major layout .\nldq\nThe leading dimension of q. Constraints:\nldq≥ max(1, n) if vect = 'V' or 'U';\nldq≥ 1 if vect = 'N'.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n876\n\n\nOutput Parameters\nab\nOn exit, the diagonal elements of the array ab are overwritten by the\ndiagonal elements of the tridiagonal matrix T. If kd > 0, the elements on\nthe first superdiagonal (if uplo = 'U') or the first subdiagonal (if uplo =\n'L') are ovewritten by the off-diagonal elements of T. The rest of ab is\noverwritten by values generated during the reduction.\nd, e, q\nArrays:\nd contains the diagonal elements of the matrix T.\nThe size of d must be at least max(1, n).\ne contains the off-diagonal elements of T.\nThe size of e must be at least max(1, n-1).\nq is not referenced if vect = 'N'.\nIf vect = 'V', q contains the n-by-n matrix Q.\nIf vect = 'U', q contains the product X* Q.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nApplication Notes\nThe computed matrix T is exactly similar to a matrix A+E, where ||E||2 = c(n)*ε*||A||2, c(n) is a\nmodestly increasing function of n, and ε is the machine precision. The computed matrix Q differs from an\nexactly orthogonal matrix by a matrix E such that ||E||2 = O(ε).\nThe total number of floating-point operations is approximately 6n2*kd if vect = 'N', with 3n3*(kd-1)/kd\nadditional operations if vect = 'V'.\nThe complex counterpart of this routine is hbtrd.\n?hbtrd\nReduces a complex Hermitian band matrix to\ntridiagonal form.\nSyntax\nlapack_int LAPACKE_chbtrd( int matrix_layout, char vect, char uplo, lapack_int n,\nlapack_int kd, lapack_complex_float* ab, lapack_int ldab, float* d, float* e,\nlapack_complex_float* q, lapack_int ldq );\nlapack_int LAPACKE_zhbtrd( int matrix_layout, char vect, char uplo, lapack_int n,\nlapack_int kd, lapack_complex_double* ab, lapack_int ldab, double* d, double* e,\nlapack_complex_double* q, lapack_int ldq );\nInclude Files\n•\nmkl.h\nDescription\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n877\n\n\nThe routine reduces a complex Hermitian band matrix A to symmetric tridiagonal form T by a unitary\nsimilarity transformation: A = Q*T*QH. The unitary matrix Q is determined as a product of Givens rotations.\nIf required, the routine can also form the matrix Q explicitly.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nvect\nMust be 'V', 'N', or 'U'.\nIf vect = 'V', the routine returns the explicit matrix Q.\nIf vect = 'N', the routine does not return Q.\nIf vect = 'U', the routine updates matrix X by forming Q*X.\nuplo\nMust be 'U' or 'L'.\nIf uplo = 'U', ab stores the upper triangular part of A.\nIf uplo = 'L', ab stores the lower triangular part of A.\nn\nThe order of the matrix A (n≥ 0).\nkd\nThe number of super- or sub-diagonals in A\n(kd≥ 0).\nab\nab(size at least max(1, ldab*n) for column major layout and at least\nmax(1, ldab*(kd+ 1)) for row major layout) is an array containing either\nupper or lower triangular part of the matrix A (as specified by uplo) in band\nstorage format.\nq\nq (size max(1, ldq*n)) is an array.\nIf vect = 'U', the q array must contain an n-by-n matrix X.\nIf vect = 'N' or 'V', the q parameter need not be set.'\nldab\nThe leading dimension of ab; at least kd+1 for column major layout and n\nfor row major layout.\nldq\nThe leading dimension of q. Constraints:\nldq≥ max(1, n) if vect = 'V' or 'U';\nldq≥ 1 if vect = 'N'.\nOutput Parameters\nab\nOn exit, the diagonal elements of the array ab are overwritten by the\ndiagonal elements of the tridiagonal matrix T. If kd > 0, the elements on\nthe first superdiagonal (if uplo = 'U') or the first subdiagonal (if uplo =\n'L') are ovewritten by the off-diagonal elements of T. The rest of ab is\noverwritten by values generated during the reduction.\nd, e\nArrays:\nd contains the diagonal elements of the matrix T.\nThe dimension of d must be at least max(1, n).\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n878\n\n\ne contains the off-diagonal elements of T.\nThe dimension of e must be at least max(1, n-1).\nq\nIf vect = 'N', q is not referenced.\nIf vect = 'V', q contains the n-by-n matrix Q.\nIf vect = 'U', q contains the product X* Q.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nApplication Notes\nThe computed matrix T is exactly similar to a matrix A + E, where ||E||2 = c(n)*ε*||A||2, c(n) is a\nmodestly increasing function of n, and ε is the machine precision. The computed matrix Q differs from an\nexactly unitary matrix by a matrix E such that ||E||2 = O(ε).\nThe total number of floating-point operations is approximately 20n2*kd if vect = 'N', with 10n3*(kd-1)/\nkd additional operations if vect = 'V'.\nThe real counterpart of this routine is sbtrd.\n?sterf\nComputes all eigenvalues of a real symmetric\ntridiagonal matrix using QR algorithm.\nSyntax\nlapack_int LAPACKE_ssterf (lapack_int n, float* d, float* e);\nlapack_int LAPACKE_dsterf (lapack_int n, double* d, double* e);\nInclude Files\n•\nmkl.h\nDescription\nThe routine computes all the eigenvalues of a real symmetric tridiagonal matrix T (which can be obtained by\nreducing a symmetric or Hermitian matrix to tridiagonal form). The routine uses a square-root-free variant of\nthe QR algorithm.\nIf you need not only the eigenvalues but also the eigenvectors, call steqr.\nInput Parameters\nn\nThe order of the matrix T (n≥ 0).\nd, e\nArrays:\nd contains the diagonal elements of T.\nThe dimension of d must be at least max(1, n).\ne contains the off-diagonal elements of T.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n879\n\n\nThe dimension of e must be at least max(1, n-1).\nOutput Parameters\nd\nThe n eigenvalues in ascending order, unless info > 0.\nSee also info.\ne\nOn exit, the array is overwritten; see info.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = i, the algorithm failed to find all the eigenvalues after 30n iterations:\ni off-diagonal elements have not converged to zero. On exit, d and e contain, respectively, the diagonal and\noff-diagonal elements of a tridiagonal matrix orthogonally similar to T.\nIf info = -i, the i-th parameter had an illegal value.\nApplication Notes\nThe computed eigenvalues and eigenvectors are exact for a matrix T+E such that ||E||2 = O(ε)*||T||2,\nwhere ε is the machine precision.\nIf λi is an exact eigenvalue, and mi is the corresponding computed value, then\n|μi - λi| ≤c(n)*ε*||T||2\nwhere c(n) is a modestly increasing function of n.\nThe total number of floating-point operations depends on how rapidly the algorithm converges. Typically, it is\nabout 14n2.\n?steqr\nComputes all eigenvalues and eigenvectors of a\nsymmetric or Hermitian matrix reduced to tridiagonal\nform (QR algorithm).\nSyntax\nlapack_int LAPACKE_ssteqr( int matrix_layout, char compz, lapack_int n, float* d,\nfloat* e, float* z, lapack_int ldz );\nlapack_int LAPACKE_dsteqr( int matrix_layout, char compz, lapack_int n, double* d,\ndouble* e, double* z, lapack_int ldz );\nlapack_int LAPACKE_csteqr( int matrix_layout, char compz, lapack_int n, float* d,\nfloat* e, lapack_complex_float* z, lapack_int ldz );\nlapack_int LAPACKE_zsteqr( int matrix_layout, char compz, lapack_int n, double* d,\ndouble* e, lapack_complex_double* z, lapack_int ldz );\nInclude Files\n•\nmkl.h\nDescription\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n880\n\n\nThe routine computes all the eigenvalues and (optionally) all the eigenvectors of a real symmetric tridiagonal\nmatrix T. In other words, the routine can compute the spectral factorization: T = Z*Λ*ZT. Here Λ is a\ndiagonal matrix whose diagonal elements are the eigenvalues λi; Z is an orthogonal matrix whose columns\nare eigenvectors. Thus,\nT*zi = λi*zi for i = 1, 2, ..., n.\nThe routine normalizes the eigenvectors so that ||zi||2 = 1.\nYou can also use the routine for computing the eigenvalues and eigenvectors of an arbitrary real symmetric\n(or complex Hermitian) matrix A reduced to tridiagonal form T: A = Q*T*QH. In this case, the spectral\nfactorization is as follows: A = Q*T*QH = (Q*Z)*Λ*(Q*Z)H. Before calling ?steqr, you must reduce A to\ntridiagonal form and generate the explicit matrix Q by calling the following routines:\n \nfor real matrices:\nfor complex matrices:\nfull storage\n?sytrd, ?orgtr\n?hetrd, ?ungtr\npacked storage\n?sptrd, ?opgtr\n?hptrd, ?upgtr\nband storage\n?sbtrd(vect='V')\n?hbtrd(vect='V')\nIf you need eigenvalues only, it's more efficient to call sterf. If T is positive-definite, pteqr can compute small\neigenvalues more accurately than ?steqr.\nTo solve the problem by a single call, use one of the divide and conquer routines stevd, syevd, spevd, or \nsbevd for real symmetric matrices or heevd, hpevd, or hbevd for complex Hermitian matrices.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\ncompz\nMust be 'N' or 'I' or 'V'.\nIf compz = 'N', the routine computes eigenvalues only.\nIf compz = 'I', the routine computes the eigenvalues and eigenvectors of\nthe tridiagonal matrix T.\nIf compz = 'V', the routine computes the eigenvalues and eigenvectors of\nthe original symmetric matrix. On entry, z must contain the orthogonal\nmatrix used to reduce the original matrix to tridiagonal form.\nn\nThe order of the matrix T (n≥ 0).\nd, e\nArrays:\nd contains the diagonal elements of T.\nThe size of d must be at least max(1, n).\ne contains the off-diagonal elements of T.\nThe size of e must be at least max(1, n-1).\nz\nArray, size max(1, ldz*n).\nIf compz = 'N' or 'I', z need not be set.\nIf vect = 'V', z must contain the orthogonal matrix used in the reduction\nto tridiagonal form.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n881\n\n\nldz\nThe leading dimension of z. Constraints:\nldz≥ 1 if compz = 'N';\nldz≥ max(1, n) if compz = 'V' or 'I'.\nOutput Parameters\nd\nThe n eigenvalues in ascending order, unless info > 0.\nSee also info.\ne\nOn exit, the array is overwritten; see info.\nz\nIf info = 0, contains the n-by-n matrix the columns of which are\northonormal eigenvectors (the i-th column corresponds to the i-th\neigenvalue).\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = i, the algorithm failed to find all the eigenvalues after 30n iterations: i off-diagonal elements have\nnot converged to zero. On exit, d and e contain, respectively, the diagonal and off-diagonal elements of a\ntridiagonal matrix orthogonally similar to T.\nIf info = -i, the i-th parameter had an illegal value.\nApplication Notes\nThe computed eigenvalues and eigenvectors are exact for a matrix T+E such that ||E||2 = O(ε)*||T||2,\nwhere ε is the machine precision.\nIf λi is an exact eigenvalue, and μi is the corresponding computed value, then\n|μi - λi| ≤c(n)*ε*||T||2\nwhere c(n) is a modestly increasing function of n.\nIf zi is the corresponding exact eigenvector, and wi is the corresponding computed vector, then the angle\nθ(zi, wi) between them is bounded as follows:\nθ(zi, wi) ≤c(n)*ε*||T||2 / mini≠j|λi - λj|.\nThe total number of floating-point operations depends on how rapidly the algorithm converges. Typically, it is\nabout\n24n2 if compz = 'N';\n7n3 (for complex flavors, 14n3) if compz = 'V' or 'I'.\n?stemr\nComputes selected eigenvalues and eigenvectors of a\nreal symmetric tridiagonal matrix.\nSyntax\nlapack_int LAPACKE_sstemr( int matrix_layout, char jobz, char range, lapack_int n,\nconst float* d, float* e, float vl, float vu, lapack_int il, lapack_int iu, lapack_int*\nm, float* w, float* z, lapack_int ldz, lapack_int nzc, lapack_int* isuppz,\nlapack_logical* tryrac );\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n882\n\n\nlapack_int LAPACKE_dstemr( int matrix_layout, char jobz, char range, lapack_int n,\nconst double* d, double* e, double vl, double vu, lapack_int il, lapack_int iu,\nlapack_int* m, double* w, double* z, lapack_int ldz, lapack_int nzc, lapack_int* isuppz,\nlapack_logical* tryrac );\nlapack_int LAPACKE_cstemr( int matrix_layout, char jobz, char range, lapack_int n,\nconst float* d, float* e, float vl, float vu, lapack_int il, lapack_int iu, lapack_int*\nm, float* w, lapack_complex_float* z, lapack_int ldz, lapack_int nzc, lapack_int*\nisuppz, lapack_logical* tryrac );\nlapack_int LAPACKE_zstemr( int matrix_layout, char jobz, char range, lapack_int n,\nconst double* d, double* e, double vl, double vu, lapack_int il, lapack_int iu,\nlapack_int* m, double* w, lapack_complex_double* z, lapack_int ldz, lapack_int nzc,\nlapack_int* isuppz, lapack_logical* tryrac );\nInclude Files\n•\nmkl.h\nDescription\nThe routine computes selected eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal\nmatrix T. Any such unreduced matrix has a well defined set of pairwise different real eigenvalues, the\ncorresponding real eigenvectors are pairwise orthogonal.\nThe spectrum may be computed either completely or partially by specifying either an interval (vl,vu] or a\nrange of indices il:iu for the desired eigenvalues.\nDepending on the number of desired eigenvalues, these are computed either by bisection or the dqds\nalgorithm. Numerically orthogonal eigenvectors are computed by the use of various suitable L*D*LT\nfactorizations near clusters of close eigenvalues (referred to as RRRs, Relatively Robust Representations). An\ninformal sketch of the algorithm follows.\nFor each unreduced block (submatrix) of T,\na.\nCompute T - sigma*I = L*D*LT, so that L and D define all the wanted eigenvalues to high relative\naccuracy. This means that small relative changes in the entries of L and D cause only small relative\nchanges in the eigenvalues and eigenvectors. The standard (unfactored) representation of the\ntridiagonal matrix T does not have this property in general.\nb.\nCompute the eigenvalues to suitable accuracy. If the eigenvectors are desired, the algorithm attains full\naccuracy of the computed eigenvalues only right before the corresponding vectors have to be\ncomputed, see steps c and d.\nc.\nFor each cluster of close eigenvalues, select a new shift close to the cluster, find a new factorization,\nand refine the shifted eigenvalues to suitable accuracy.\nd.\nFor each eigenvalue with a large enough relative separation compute the corresponding eigenvector by\nforming a rank revealing twisted factorization. Go back to step c for any clusters that remain.\nNormal execution of ?stemr may create NaNs and infinities and may abort due to a floating point exception\nin environments that do not handle NaNs and infinities in the IEEE standard default manner.\nFor more details, see: [Dhillon04], [Dhillon04-02], [Dhillon97]\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\njobz\nMust be 'N' or 'V'.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n883\n\n\nIf jobz = 'N', then only eigenvalues are computed.\nIf jobz = 'V', then eigenvalues and eigenvectors are computed.\nrange\nMust be 'A' or 'V' or 'I'.\nIf range = 'A', the routine computes all eigenvalues.\nIf range = 'V', the routine computes all eigenvalues in the half-open\ninterval: (vl, vu].\nIf range = 'I', the routine computes eigenvalues with indices il to iu.\nn\nThe order of the matrix T (n≥0).\nd\nArray, size n.\nContains n diagonal elements of the tridiagonal matrix T.\ne\nArray, size n.\nContains (n-1) off-diagonal elements of the tridiagonal matrix T in\nelements 0 to n-2 of e. e[n - 1] need not be set on input, but is used\ninternally as workspace.\nvl, vu\nIf range = 'V', the lower and upper bounds of the interval to be searched\nfor eigenvalues. Constraint: vl<vu.\nIf range = 'A' or 'I', vl and vu are not referenced.\nil, iu\nIf range = 'I', the indices in ascending order of the smallest and largest\neigenvalues to be returned.\nConstraint: 1≤il≤iu≤n, if n>0.\nIf range = 'A' or 'V', il and iu are not referenced.\nldz\nThe leading dimension of the output array z.\nif jobz = 'V', then ldz ≥ max(1, n) for column major layout and ldz≥\nmax(1, m) for row major layout ;\nldz ≥ 1 otherwise.\nnzc\nThe number of eigenvectors to be held in the array z.\nIf range = 'A', then nzc≥max(1, n);\nIf range = 'V', then nzc is greater than or equal to the number of\neigenvalues in the half-open interval: (vl, vu].\nIf range = 'I', then nzc≥iu-il+1.\nIf nzc = -1, then a workspace query is assumed; the routine calculates the\nnumber of columns of the array z that are needed to hold the eigenvectors.\nThis value is returned as the first entry of the array z, and no error\nmessage related to nzc is issued by the routine xerbla.\ntryrac\nIf tryrac is true, it indicates that the code should check whether the\ntridiagonal matrix defines its eigenvalues to high relative accuracy. If so,\nthe code uses relative-accuracy preserving algorithms that might be (a bit)\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n884\n\n\nslower depending on the matrix. If the matrix does not define its\neigenvalues to high relative accuracy, the code can uses possibly faster\nalgorithms.\nIf tryrac is not true, the code is not required to guarantee relatively\naccurate eigenvalues and can use the fastest possible techniques.\nOutput Parameters\nd\nOn exit, the array d is overwritten.\ne\nOn exit, the array e is overwritten.\nm\nThe total number of eigenvalues found, 0≤m≤n.\nIf range = 'A', then m=n, and if range = 'I', then m=iu-il+1.\nw\nArray, size n.\nThe first m elements contain the selected eigenvalues in ascending order.\nz\nArray z(size max(1, ldz*m) for column major layout and max(1, ldz*n) for\nrow major layout) .\nIf jobz = 'V', and info = 0, then the first m columns of z contain the\northonormal eigenvectors of the matrix T corresponding to the selected\neigenvalues, with the i-th column of z holding the eigenvector associated\nwith w(i).\nIf jobz = 'N', then z is not referenced.\nNote: the exact value of m is not known in advance and can be computed\nwith a workspace query by setting nzc=-1, see description of the\nparameter nzc.\nisuppz\nArray, size (2*max(1, m)).\nThe support of the eigenvectors in z, that is the indices indicating the\nnonzero elements in z. The i-th computed eigenvector is nonzero only in\nelements isuppz[2*i - 2] through isuppz[2*i - 1]. This is relevant in\nthe case when the matrix is split. isuppz is only accessed when jobz =\n'V' and n>0.\ntryrac\nOn exit, , set to true. tryrac is set to false if the matrix does not define its\neigenvalues to high relative accuracy.\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nIf info > 0, an internal error occurred.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n885\n\n\n?stedc\nComputes all eigenvalues and eigenvectors of a\nsymmetric tridiagonal matrix using the divide and\nconquer method.\nSyntax\nlapack_int LAPACKE_sstedc( int matrix_layout, char compz, lapack_int n, float* d,\nfloat* e, float* z, lapack_int ldz );\nlapack_int LAPACKE_dstedc( int matrix_layout, char compz, lapack_int n, double* d,\ndouble* e, double* z, lapack_int ldz );\nlapack_int LAPACKE_cstedc( int matrix_layout, char compz, lapack_int n, float* d,\nfloat* e, lapack_complex_float* z, lapack_int ldz );\nlapack_int LAPACKE_zstedc( int matrix_layout, char compz, lapack_int n, double* d,\ndouble* e, lapack_complex_double* z, lapack_int ldz );\nInclude Files\n•\nmkl.h\nDescription\nThe routine computes all the eigenvalues and (optionally) all the eigenvectors of a symmetric tridiagonal\nmatrix using the divide and conquer method. The eigenvectors of a full or band real symmetric or complex\nHermitian matrix can also be found if sytrd/hetrd or sptrd/hptrd or sbtrd/hbtrd has been used to reduce this\nmatrix to tridiagonal form.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\ncompz\nMust be 'N' or 'I' or 'V'.\nIf compz = 'N', the routine computes eigenvalues only.\nIf compz = 'I', the routine computes the eigenvalues and eigenvectors of\nthe tridiagonal matrix.\nIf compz = 'V', the routine computes the eigenvalues and eigenvectors of\noriginal symmetric/Hermitian matrix. On entry, the array z must contain the\northogonal/unitary matrix used to reduce the original matrix to tridiagonal\nform.\nn\nThe order of the symmetric tridiagonal matrix (n≥ 0).\nd, e\nArrays:\nd contains the diagonal elements of the tridiagonal matrix.\nThe dimension of d must be at least max(1, n).\ne contains the subdiagonal elements of the tridiagonal matrix.\nThe dimension of e must be at least max(1, n-1).\nz\nArray z is of size max(1, ldz*n).\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n886\n\n\nIf compz = 'V', then, on entry, z must contain the orthogonal/unitary\nmatrix used to reduce the original matrix to tridiagonal form.\nldz\nThe leading dimension of z. Constraints:\nldz≥ 1 if compz = 'N';\nldz≥ max(1, n) if compz = 'V' or 'I'.\nOutput Parameters\nd\nThe n eigenvalues in ascending order, unless info≠ 0.\nSee also info.\ne\nOn exit, the array is overwritten; see info.\nz\nIf info = 0, then if compz = 'V', z contains the orthonormal eigenvectors\nof the original symmetric/Hermitian matrix, and if compz = 'I', z contains\nthe orthonormal eigenvectors of the symmetric tridiagonal matrix. If compz\n= 'N', z is not referenced.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nIf info = i, the algorithm failed to compute an eigenvalue while working on the submatrix lying in rows and\ncolumns i/(n+1) through mod(i, n+1).\n?stegr\nComputes selected eigenvalues and eigenvectors of a\nreal symmetric tridiagonal matrix.\nSyntax\nlapack_int LAPACKE_sstegr( int matrix_layout, char jobz, char range, lapack_int n,\nfloat* d, float* e, float vl, float vu, lapack_int il, lapack_int iu, float abstol,\nlapack_int* m, float* w, float* z, lapack_int ldz, lapack_int* isuppz );\nlapack_int LAPACKE_dstegr( int matrix_layout, char jobz, char range, lapack_int n,\ndouble* d, double* e, double vl, double vu, lapack_int il, lapack_int iu, double abstol,\nlapack_int* m, double* w, double* z, lapack_int ldz, lapack_int* isuppz );\nlapack_int LAPACKE_cstegr( int matrix_layout, char jobz, char range, lapack_int n,\nfloat* d, float* e, float vl, float vu, lapack_int il, lapack_int iu, float abstol,\nlapack_int* m, float* w, lapack_complex_float* z, lapack_int ldz, lapack_int* isuppz );\nlapack_int LAPACKE_zstegr( int matrix_layout, char jobz, char range, lapack_int n,\ndouble* d, double* e, double vl, double vu, lapack_int il, lapack_int iu, double abstol,\nlapack_int* m, double* w, lapack_complex_double* z, lapack_int ldz, lapack_int*\nisuppz );\nInclude Files\n•\nmkl.h\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n887\n\n\nDescription\nThe routine computes selected eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal\nmatrix T.\nThe spectrum may be computed either completely or partially by specifying either an interval (vl,vu] or a\nrange of indices il:iu for the desired eigenvalues.\n?stegr is a compatibility wrapper around the improved stemr routine. See its description for further details.\nNote that the abstol parameter no longer provides any benefit and hence is no longer used.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\njobz\nMust be 'N' or 'V'.\nIf job = 'N', then only eigenvalues are computed.\nIf job = 'V', then eigenvalues and eigenvectors are computed.\nrange\nMust be 'A' or 'V' or 'I'.\nIf range = 'A', the routine computes all eigenvalues.\nIf range = 'V', the routine computes eigenvalues w[i] in the half-open\ninterval:\nvl<w[i]≤vu.\nIf range = 'I', the routine computes eigenvalues with indices il to iu.\nn\nThe order of the matrix T (n≥ 0).\nd, e\nArrays:\nd contains the diagonal elements of T.\nThe dimension of d must be at least max(1, n).\ne contains the subdiagonal elements of T in elements 1 to n-1; e(n) need\nnot be set on input, but it is used as a workspace.\nThe dimension of e must be at least max(1, n).\nvl, vu\nIf range = 'V', the lower and upper bounds of the interval to be searched\nfor eigenvalues.\nConstraint: vl< vu.\nIf range = 'A' or 'I', vl and vu are not referenced.\nil, iu\nIf range = 'I', the indices in ascending order of the smallest and largest\neigenvalues to be returned.\nConstraint: 1 ≤il≤iu≤n, if n > 0.\nIf range = 'A' or 'V', il and iu are not referenced.\nabstol\nUnused. Was the absolute error tolerance for the eigenvalues/eigenvectors\nin previous versions.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n888\n\n\nldz\nThe leading dimension of the output array z. Constraints:\nldz≥ 1 if jobz = 'N';\nldz≥ max(1, n) if jobz = 'V'.\nOutput Parameters\nd, e\nOn exit, d and e are overwritten.\nm\nThe total number of eigenvalues found,\n0 ≤m≤n.\nIf range = 'A', m = n, and if range = 'I', m = iu-il+1.\nw\nArray, size at least max(1, n).\nThe selected eigenvalues in ascending order, stored in w[0] to w[m - 1].\nz\nArray z(size max(1,ldz*m)).\nIf jobz = 'V', and if info = 0, the first m columns of z contain the\northonormal eigenvectors of the matrix T corresponding to the selected\neigenvalues, with the i-th column of z holding the eigenvector associated\nwith w[i - 1].\nIf jobz = 'N', then z is not referenced.\nNote: if range = 'V', the exact value of m is not known in advance and an\nupper bound must be used. Using n = m is always safe.\nisuppz\nArray, size at least (2*max(1, m)).\nThe support of the eigenvectors in z, that is the indices indicating the\nnonzero elements in z. The i-th computed eigenvector is nonzero only in\nelements isuppz[2*i - 2] through isuppz[2*i - 1]. This is relevant in\nthe case when the matrix is split. isuppz is only accessed when jobz =\n'V', and n > 0.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nIf info > 0, an internal error occurred.\n?pteqr\nComputes all eigenvalues and (optionally) all\neigenvectors of a real symmetric positive-definite\ntridiagonal matrix.\nSyntax\nlapack_int LAPACKE_spteqr( int matrix_layout, char compz, lapack_int n, float* d,\nfloat* e, float* z, lapack_int ldz );\nlapack_int LAPACKE_dpteqr( int matrix_layout, char compz, lapack_int n, double* d,\ndouble* e, double* z, lapack_int ldz );\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n889\n\n\nlapack_int LAPACKE_cpteqr( int matrix_layout, char compz, lapack_int n, float* d,\nfloat* e, lapack_complex_float* z, lapack_int ldz );\nlapack_int LAPACKE_zpteqr( int matrix_layout, char compz, lapack_int n, double* d,\ndouble* e, lapack_complex_double* z, lapack_int ldz );\nInclude Files\n•\nmkl.h\nDescription\nThe routine computes all the eigenvalues and (optionally) all the eigenvectors of a real symmetric positive-\ndefinite tridiagonal matrix T. In other words, the routine can compute the spectral factorization: T =\nZ*Λ*ZT.\nHere Λ is a diagonal matrix whose diagonal elements are the eigenvalues λi; Z is an orthogonal matrix whose\ncolumns are eigenvectors. Thus,\nT*zi = λi*zi for i = 1, 2, ..., n.\n(The routine normalizes the eigenvectors so that ||zi||2 = 1.)\nYou can also use the routine for computing the eigenvalues and eigenvectors of real symmetric (or complex\nHermitian) positive-definite matrices A reduced to tridiagonal form T: A = Q*T*QH. In this case, the spectral\nfactorization is as follows: A = Q*T*QH = (QZ)*Λ*(QZ)H. Before calling ?pteqr, you must reduce A to\ntridiagonal form and generate the explicit matrix Q by calling the following routines:\n \nfor real matrices:\nfor complex matrices:\nfull storage\n?sytrd, ?orgtr\n?hetrd, ?ungtr\npacked storage\n?sptrd, ?opgtr\n?hptrd, ?upgtr\nband storage\n?sbtrd(vect='V')\n?hbtrd(vect='V')\nThe routine first factorizes T as L*D*LH where L is a unit lower bidiagonal matrix, and D is a diagonal matrix.\nThen it forms the bidiagonal matrix B = L*D1/2 and calls ?bdsqr to compute the singular values of B, which\nare the square roots of the eigenvalues of T.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\ncompz\nMust be 'N' or 'I' or 'V'.\nIf compz = 'N', the routine computes eigenvalues only.\nIf compz = 'I', the routine computes the eigenvalues and eigenvectors of\nthe tridiagonal matrix T.\nIf compz = 'V', the routine computes the eigenvalues and eigenvectors of\nA (and the array z must contain the matrix Q on entry).\nn\nThe order of the matrix T (n≥ 0).\nd, e\nArrays:\nd contains the diagonal elements of T.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n890\n\n\nThe size of d must be at least max(1, n).\ne contains the off-diagonal elements of T.\nThe size of e must be at least max(1, n-1).\nz\nArray, size max(1, ldz*n)\nIf compz = 'N' or 'I', z need not be set.\nIf compz = 'V', z must contain the orthogonal matrix used in the\nreduction to tridiagonal form..\nldz\nThe leading dimension of z. Constraints:\nldz≥ 1 if compz = 'N';\nldz≥ max(1, n) if compz = 'V' or 'I'.\nOutput Parameters\nd\nThe n eigenvalues in descending order, unless info > 0.\nSee also info.\ne\nOn exit, the array is overwritten.\nz\nIf info = 0, contains an n-byn matrix the columns of which are\northonormal eigenvectors. (The i-th column corresponds to the i-th\neigenvalue.)\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = i, the leading minor of order i (and hence T itself) is not positive-definite.\nIf info = n + i, the algorithm for computing singular values failed to converge; i off-diagonal elements\nhave not converged to zero.\nIf info = -i, the i-th parameter had an illegal value.\nApplication Notes\nIf λi is an exact eigenvalue, and μi is the corresponding computed value, then\n|μi - λi| ≤c(n)*ε*K*λi\nwhere c(n) is a modestly increasing function of n, ε is the machine precision, and K = ||DTD||2 *||\n(DTD)-1||2, D is diagonal with dii = tii-1/2.\nIf zi is the corresponding exact eigenvector, and wi is the corresponding computed vector, then the angle θ(zi,\nwi) between them is bounded as follows:\nθ(ui, wi) ≤c(n)εK / mini≠j(|λi - λj|/|λi + λj|).\nHere mini≠j(|λi - λj|/|λi + λj|) is the relative gap between λi and the other eigenvalues.\nThe total number of floating-point operations depends on how rapidly the algorithm converges.\nTypically, it is about\n30n2 if compz = 'N';\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n891\n\n\n6n3 (for complex flavors, 12n3) if compz = 'V' or 'I'.\n?stebz\nComputes selected eigenvalues of a real symmetric\ntridiagonal matrix by bisection.\nSyntax\nlapack_int LAPACKE_sstebz (char range, char order, lapack_int n, float vl, float vu,\nlapack_int il, lapack_int iu, float abstol, const float* d, const float* e, lapack_int*\nm, lapack_int* nsplit, float* w, lapack_int* iblock, lapack_int* isplit);\nlapack_int LAPACKE_dstebz (char range, char order, lapack_int n, double vl, double vu,\nlapack_int il, lapack_int iu, double abstol, const double* d, const double* e,\nlapack_int* m, lapack_int* nsplit, double* w, lapack_int* iblock, lapack_int* isplit);\nInclude Files\n•\nmkl.h\nDescription\nThe routine computes some (or all) of the eigenvalues of a real symmetric tridiagonal matrix T by bisection.\nThe routine searches for zero or negligible off-diagonal elements to see if T splits into block-diagonal form T\n= diag(T1, T2, ...). Then it performs bisection on each of the blocks Ti and returns the block index of\neach computed eigenvalue, so that a subsequent call to stein can also take advantage of the block structure.\nInput Parameters\nrange\nMust be 'A' or 'V' or 'I'.\nIf range = 'A', the routine computes all eigenvalues.\nIf range = 'V', the routine computes eigenvalues w[i] in the half-open\ninterval: vl < w[i]≤vu.\nIf range = 'I', the routine computes eigenvalues with indices il to iu.\norder\nMust be 'B' or 'E'.\nIf order = 'B', the eigenvalues are to be ordered from smallest to largest\nwithin each split-off block.\nIf order = 'E', the eigenvalues for the entire matrix are to be ordered\nfrom smallest to largest.\nn\nThe order of the matrix T (n≥ 0).\nvl, vu\nIf range = 'V', the routine computes eigenvalues w[i] in the half-open\ninterval:\nvl < w[i]) ≤vu.\nIf range = 'A' or 'I', vl and vu are not referenced.\nil, iu\nConstraint: 1 ≤il≤iu≤n.\nIf range = 'I', the routine computes eigenvalues w[i] such that il≤i≤iu\n(assuming that the eigenvalues w[i] are in ascending order).\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n892\n\n\nIf range = 'A' or 'V', il and iu are not referenced.\nabstol\nThe absolute tolerance to which each eigenvalue is required. An eigenvalue\n(or cluster) is considered to have converged if it lies in an interval of width\nabstol.\nIf abstol≤ 0.0, then the tolerance is taken as eps*|T|, where eps is the\nmachine precision, and |T| is the 1-norm of the matrix T.\nd, e\nArrays:\nd contains the diagonal elements of T.\nThe size of d must be at least max(1, n).\ne contains the off-diagonal elements of T.\nThe size of e must be at least max(1, n-1).\nOutput Parameters\nm\nThe actual number of eigenvalues found.\nnsplit\nThe number of diagonal blocks detected in T.\nw\nArray, size at least max(1, n). The computed eigenvalues, stored in w[0] to\nw[m - 1].\niblock, isplit\nArrays, size at least max(1, n).\nA positive value iblock[i] is the block number of the eigenvalue stored in\nw[i] (see also info).\nThe leading nsplit elements of isplit contain points at which T splits into\nblocks Ti as follows: the block T1 contains rows/columns 1 to isplit[0];\nthe block T2 contains rows/columns isplit[0]+1 to isplit[1], and so\non.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = 1, for range = 'A' or 'V', the algorithm failed to compute some of the required eigenvalues to\nthe desired accuracy; iblock[i] < 0 indicates that the eigenvalue stored in w[i] failed to converge.\nIf info = 2, for range = 'I', the algorithm failed to compute some of the required eigenvalues. Try calling\nthe routine again with range = 'A'.\nIf info = 3:\nfor range = 'A' or 'V', same as info = 1;\nfor range = 'I', same as info = 2.\nIf info = 4, no eigenvalues have been computed. The floating-point arithmetic on the computer is not\nbehaving as expected.\nIf info = -i, the i-th parameter had an illegal value.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n893\n\n\nApplication Notes\nThe eigenvalues of T are computed to high relative accuracy which means that if they vary widely in\nmagnitude, then any small eigenvalues will be computed more accurately than, for example, with the\nstandard QR method. However, the reduction to tridiagonal form (prior to calling the routine) may exclude\nthe possibility of obtaining high relative accuracy in the small eigenvalues of the original matrix if its\neigenvalues vary widely in magnitude.\n?stein\nComputes the eigenvectors corresponding to specified\neigenvalues of a real symmetric tridiagonal matrix.\nSyntax\nlapack_int LAPACKE_sstein( int matrix_layout, lapack_int n, const float* d, const\nfloat* e, lapack_int m, const float* w, const lapack_int* iblock, const lapack_int*\nisplit, float* z, lapack_int ldz, lapack_int* ifailv );\nlapack_int LAPACKE_dstein( int matrix_layout, lapack_int n, const double* d, const\ndouble* e, lapack_int m, const double* w, const lapack_int* iblock, const lapack_int*\nisplit, double* z, lapack_int ldz, lapack_int* ifailv );\nlapack_int LAPACKE_cstein( int matrix_layout, lapack_int n, const float* d, const\nfloat* e, lapack_int m, const float* w, const lapack_int* iblock, const lapack_int*\nisplit, lapack_complex_float* z, lapack_int ldz, lapack_int* ifailv );\nlapack_int LAPACKE_zstein( int matrix_layout, lapack_int n, const double* d, const\ndouble* e, lapack_int m, const double* w, const lapack_int* iblock, const lapack_int*\nisplit, lapack_complex_double* z, lapack_int ldz, lapack_int* ifailv );\nInclude Files\n•\nmkl.h\nDescription\nThe routine computes the eigenvectors of a real symmetric tridiagonal matrix T corresponding to specified\neigenvalues, by inverse iteration. It is designed to be used in particular after the specified eigenvalues have\nbeen computed by ?stebz with order = 'B', but may also be used when the eigenvalues have been\ncomputed by other routines.\nIf you use this routine after ?stebz, it can take advantage of the block structure by performing inverse\niteration on each block Ti separately, which is more efficient than using the whole matrix T.\nIf T has been formed by reduction of a full symmetric or Hermitian matrix A to tridiagonal form, you can\ntransform eigenvectors of T to eigenvectors of A by calling ?ormtr or ?opmtr (for real flavors) or by\ncalling ?unmtr or ?upmtr (for complex flavors).\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nn\nThe order of the matrix T (n≥ 0).\nm\nThe number of eigenvectors to be returned.\nd, e, w\nArrays:\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n894\n\n\nd contains the diagonal elements of T.\nThe size of d must be at least max(1, n).\ne contains the sub-diagonal elements of T stored in elements 1 to n-1\nThe size of e must be at least max(1, n-1).\nw contains the eigenvalues of T, stored in w[0] to w[m - 1] (as returned\nby stebz). Eigenvalues of T1 must be supplied first, in non-decreasing\norder; then those of T2, again in non-decreasing order, and so on.\nConstraint:\nif iblock[i] = iblock[i+1], w[i] ≤w[i+1].\nThe size of w must be at least max(1, n).\niblock, isplit\nArrays, size at least max(1, n). The arrays iblock and isplit, as returned\nby ?stebz with order = 'B'.\nIf you did not call ?stebz with order = 'B', set all elements of iblock to\n1, and isplit[0] to n.)\nldz\nThe leading dimension of the output array z; ldz≥ max(1, n) for column\nmajor layout and ldz>=max(1,m) for row major layout.\nOutput Parameters\nz\nArray, size at least max(1,ldz*m) for column major layout and\nmax(1,ldz*n) for row major layout.\nIf info = 0, z contains an n-by-n matrix the columns of which are\northonormal eigenvectors. (The i-th column corresponds to the ith\neigenvalue.)\nifailv\nArray, size at least max(1, m).\nIf info = i > 0, the first i elements of ifailv contain the indices of any\neigenvectors that failed to converge.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = i, then i eigenvectors (as indicated by the parameter ifailv) each failed to converge in 5 iterations.\nThe current iterates are stored in the corresponding columns/rows of the array z.\nIf info = -i, the i-th parameter had an illegal value.\nApplication Notes\nEach computed eigenvector zi is an exact eigenvector of a matrix T+Ei, where ||Ei||2 = O(ε)*||T||2.\nHowever, a set of eigenvectors computed by this routine may not be orthogonal to so high a degree of\naccuracy as those computed by ?steqr.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n895\n\n\n?disna\nComputes the reciprocal condition numbers for the\neigenvectors of a symmetric/ Hermitian matrix or for\nthe left or right singular vectors of a general matrix.\nSyntax\nlapack_int LAPACKE_sdisna (char job, lapack_int m, lapack_int n, const float* d, float*\nsep);\nlapack_int LAPACKE_ddisna (char job, lapack_int m, lapack_int n, const double* d,\ndouble* sep);\nInclude Files\n•\nmkl.h\nDescription\nThe routine computes the reciprocal condition numbers for the eigenvectors of a real symmetric or complex\nHermitian matrix or for the left or right singular vectors of a general m-by-n matrix.\nThe reciprocal condition number is the 'gap' between the corresponding eigenvalue or singular value and the\nnearest other one.\nThe bound on the error, measured by angle in radians, in the i-th computed vector is given by\n?lamch('E')*(anorm/sep(i))\nwhere anorm = ||A||2 = max( |d(j)| ). sep(i) is not allowed to be smaller than slamch('E')*anorm in\norder to limit the size of the error bound.\n?disna may also be used to compute error bounds for eigenvectors of the generalized symmetric definite\neigenproblem.\nInput Parameters\njob\nMust be 'E','L', or 'R'. Specifies for which problem the reciprocal\ncondition numbers should be computed:\njob = 'E': for the eigenvectors of a symmetric/Hermitian matrix;\njob = 'L': for the left singular vectors of a general matrix;\njob = 'R': for the right singular vectors of a general matrix.\nm\nThe number of rows of the matrix (m≥ 0).\nn\nIf job = 'L', or 'R', the number of columns of the matrix (n≥ 0). Ignored\nif job = 'E'.\nd\nArray, dimension at least max(1,m) if job = 'E', and at least max(1,\nmin(m,n)) if job = 'L' or 'R'.\nThis array must contain the eigenvalues (if job = 'E') or singular values\n(if job = 'L' or 'R') of the matrix, in either increasing or decreasing\norder.\nIf singular values, they must be non-negative.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n896\n\n\nOutput Parameters\nsep\nArray, dimension at least max(1,m) if job = 'E', and at least max(1,\nmin(m,n)) if job = 'L' or 'R'. The reciprocal condition numbers of the\nvectors.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nGeneralized Symmetric-Definite Eigenvalue Problems: LAPACK Computational Routines\nGeneralized symmetric-definite eigenvalue problems are as follows: find the eigenvalues λ and the\ncorresponding eigenvectors z that satisfy one of these equations:\nAz = λBz, ABz = λz, or BAz = λz,\nwhere A is an n-by-n symmetric or Hermitian matrix, and B is an n-by-n symmetric positive-definite or\nHermitian positive-definite matrix.\nIn these problems, there exist n real eigenvectors corresponding to real eigenvalues (even for complex\nHermitian matrices A and B).\nRoutines described in this topic allow you to reduce the above generalized problems to standard symmetric\neigenvalue problem Cy = λy, which you can solve by calling LAPACK routines described earlier in this\nchapter (see Symmetric Eigenvalue Problems).\nDifferent routines allow the matrices to be stored either conventionally or in packed storage. Prior to\nreduction, the positive-definite matrix B must first be factorized using either potrf or pptrf.\nThe reduction routine for the banded matrices A and B uses a split Cholesky factorization for which a specific\nroutine pbstf is provided. This refinement halves the amount of work required to form matrix C.\nTable \"Computational Routines for Reducing Generalized Eigenproblems to Standard Problems\" lists LAPACK\nroutines that can be used to solve generalized symmetric-definite eigenvalue problems.\nComputational Routines for Reducing Generalized Eigenproblems to Standard Problems\nMatrix type\nReduce to standard\nproblems (full\nstorage)\nReduce to standard\nproblems (packed\nstorage)\nReduce to standard\nproblems (band\nmatrices)\nFactorize\nband\nmatrix\nreal\nsymmetric\nmatrices\nsygst\nspgst\nsbgst\npbstf\ncomplex\nHermitian\nmatrices\nhegst\nhpgst\nhbgst\npbstf\n?sygst\nReduces a real symmetric-definite generalized\neigenvalue problem to the standard form.\nSyntax\nlapack_int LAPACKE_ssygst (int matrix_layout, lapack_int itype, char uplo, lapack_int\nn, float* a, lapack_int lda, const float* b, lapack_int ldb);\nlapack_int LAPACKE_dsygst (int matrix_layout, lapack_int itype, char uplo, lapack_int\nn, double* a, lapack_int lda, const double* b, lapack_int ldb);\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n897\n\n\nInclude Files\n•\nmkl.h\nDescription\nThe routine reduces real symmetric-definite generalized eigenproblems\nA*z = λ*B*z, A*B*z = λ*z, or B*A*z = λ*z\nto the standard form C*y = λ*y. Here A is a real symmetric matrix, and B is a real symmetric positive-\ndefinite matrix. Before calling this routine, call ?potrf to compute the Cholesky factorization: B = UT*U or B\n= L*LT.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nitype\nMust be 1 or 2 or 3.\nIf itype = 1, the generalized eigenproblem is A*z = lambda*B*z\nfor uplo = 'U': C = inv(UT)*A*inv(U), z = inv(U)*y;\nfor uplo = 'L': C = inv(L)*A*inv(LT), z = inv(LT)*y.\nIf itype = 2, the generalized eigenproblem is A*B*z = lambda*z\nfor uplo = 'U': C = U*A*UT, z = inv(U)*y;\nfor uplo = 'L': C = LT*A*L, z = inv(LT)*y.\nIf itype = 3, the generalized eigenproblem is B*A*z = lambda*z\nfor uplo = 'U': C = U*A*UT, z = UT*y;\nfor uplo = 'L': C = LT*A*L, z = L*y.\nuplo\nMust be 'U' or 'L'.\nIf uplo = 'U', the array a stores the upper triangle of A; you must supply\nB in the factored form B = UT*U.\nIf uplo = 'L', the array a stores the lower triangle of A; you must supply\nB in the factored form B = L*LT.\nn\nThe order of the matrices A and B (n≥ 0).\na, b\nArrays:\na (size max(1, lda*n)) contains the upper or lower triangle of A.\nb (size max(1, ldb*n)) contains the Cholesky-factored matrix B:\nB = UT*U or B = L*LT (as returned by ?potrf).\nlda\nThe leading dimension of a; at least max(1, n).\nldb\nThe leading dimension of b; at least max(1, n).\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n898\n\n\nOutput Parameters\na\nThe upper or lower triangle of A is overwritten by the upper or lower\ntriangle of C, as specified by the arguments itype and uplo.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nApplication Notes\nForming the reduced matrix C is a stable procedure. However, it involves implicit multiplication by inv(B) (if\nitype = 1) or B (if itype = 2 or 3). When the routine is used as a step in the computation of eigenvalues\nand eigenvectors of the original problem, there may be a significant loss of accuracy if B is ill-conditioned\nwith respect to inversion.\nThe approximate number of floating-point operations is n3.\n?hegst\nReduces a complex Hermitian positive-definite\ngeneralized eigenvalue problem to the standard form.\nSyntax\nlapack_int LAPACKE_chegst (int matrix_layout, lapack_int itype, char uplo, lapack_int\nn, lapack_complex_float* a, lapack_int lda, const lapack_complex_float* b, lapack_int\nldb);\nlapack_int LAPACKE_zhegst (int matrix_layout, lapack_int itype, char uplo, lapack_int\nn, lapack_complex_double* a, lapack_int lda, const lapack_complex_double* b, lapack_int\nldb);\nInclude Files\n•\nmkl.h\nDescription\nThe routine reduces a complex Hermitian positive-definite generalized eigenvalue problem to standard form.\nitype\nProblem\nResult\n1\nA*x = λ*B*x\nA overwritten by inv(UH)*A*inv(U) or\ninv(L)*A*inv(LH)\n2\nA*B*x = λ*x\nA overwritten by U*A*UH or LH*A*L\n3\nB*A*x = λ*x\nBefore calling this routine, you must call ?potrf to compute the Cholesky factorization: B = UH*U or B =\nL*LH.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n899\n\n\nInput Parameters\nitype\nMust be 1 or 2 or 3.\nIf itype = 1, the generalized eigenproblem is A*z = lambda*B*z\nfor uplo = 'U': C = (UH)-1*A*U-1;\nfor uplo = 'L': C = L-1*A*(LH)-1.\nIf itype = 2, the generalized eigenproblem is A*B*z = lambda*z\nfor uplo = 'U': C = U*A*UH;\nfor uplo = 'L': C = LH*A*L.\nIf itype = 3, the generalized eigenproblem is B*A*z = lambda*z\nfor uplo = 'U': C = U*A*UH;\nfor uplo = 'L': C = LH*A*L.\nuplo\nMust be 'U' or 'L'.\nIf uplo = 'U', the array a stores the upper triangle of A; you must supply\nB in the factored form B = UH*U.\nIf uplo = 'L', the array a stores the lower triangle of A; you must supply\nB in the factored form B = L*LH.\nn\nThe order of the matrices A and B (n≥ 0).\na, b\nArrays:\na (size max(1, lda*n)) contains the upper or lower triangle of A.\nb (size max(1, ldb*n)) contains the Cholesky-factored matrix B:\nB = UH*U or B = L*LH (as returned by ?potrf).\nlda\nThe leading dimension of a; at least max(1, n).\nldb\nThe leading dimension of b; at least max(1, n).\nOutput Parameters\na\nThe upper or lower triangle of A is overwritten by the upper or lower\ntriangle of C, as specified by the arguments itype and uplo.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nApplication Notes\nForming the reduced matrix C is a stable procedure. However, it involves implicit multiplication by B-1 (if\nitype = 1) or B (if itype = 2 or 3). When the routine is used as a step in the computation of eigenvalues\nand eigenvectors of the original problem, there may be a significant loss of accuracy if B is ill-conditioned\nwith respect to inversion.\nThe approximate number of floating-point operations is n3.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n900\n\n\n?spgst\nReduces a real symmetric-definite generalized\neigenvalue problem to the standard form using packed\nstorage.\nSyntax\nlapack_int LAPACKE_sspgst (int matrix_layout, lapack_int itype, char uplo, lapack_int\nn, float* ap, const float* bp);\nlapack_int LAPACKE_dspgst (int matrix_layout, lapack_int itype, char uplo, lapack_int\nn, double* ap, const double* bp);\nInclude Files\n•\nmkl.h\nDescription\nThe routine reduces real symmetric-definite generalized eigenproblems\nA*x = λ*B*x, A*B*x = λ*x, or B*A*x = λ*x\nto the standard form C*y = λ*y, using packed matrix storage. Here A is a real symmetric matrix, and B is a\nreal symmetric positive-definite matrix. Before calling this routine, call ?pptrf to compute the Cholesky\nfactorization: B = UT*U or B = L*LT.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nitype\nMust be 1 or 2 or 3.\nIf itype = 1, the generalized eigenproblem is A*z = lambda*B*z\nfor uplo = 'U': C = inv(UT)*A*inv(U), z = inv(U)*y;\nfor uplo = 'L': C = inv(L)*A*inv(LT), z = inv(LT)*y.\nIf itype = 2, the generalized eigenproblem is A*B*z = lambda*z\nfor uplo = 'U': C = U*A*UT, z = inv(U)*y;\nfor uplo = 'L': C = LT*A*L, z = inv(LT)*y.\nIf itype = 3, the generalized eigenproblem is B*A*z = lambda*z\nfor uplo = 'U': C = U*A*UT, z = UT*y;\nfor uplo = 'L': C = LT*A*L, z = L*y.\nuplo\nMust be 'U' or 'L'.\nIf uplo = 'U', ap stores the packed upper triangle of A;\nyou must supply B in the factored form B = UT*U.\nIf uplo = 'L', ap stores the packed lower triangle of A;\nyou must supply B in the factored form B = L*LT.\nn\nThe order of the matrices A and B (n≥ 0).\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n901\n\n\nap, bp\nArrays:\nap contains the packed upper or lower triangle of A.\nThe dimension of ap must be at least max(1, n*(n+1)/2).\nbp contains the packed Cholesky factor of B (as returned by ?pptrf with\nthe same uplo value).\nThe dimension of bp must be at least max(1, n*(n+1)/2).\nOutput Parameters\nap\nThe upper or lower triangle of A is overwritten by the upper or lower\ntriangle of C, as specified by the arguments itype and uplo.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nApplication Notes\nForming the reduced matrix C is a stable procedure. However, it involves implicit multiplication by inv(B) (if\nitype = 1) or B (if itype = 2 or 3). When the routine is used as a step in the computation of eigenvalues\nand eigenvectors of the original problem, there may be a significant loss of accuracy if B is ill-conditioned\nwith respect to inversion.\nThe approximate number of floating-point operations is n3.\n?hpgst\nReduces a generalized eigenvalue problem with a\nHermitian matrix to a standard eigenvalue problem\nusing packed storage.\nSyntax\nlapack_int LAPACKE_chpgst (int matrix_layout, lapack_int itype, char uplo, lapack_int\nn, lapack_complex_float* ap, const lapack_complex_float* bp);\nlapack_int LAPACKE_zhpgst (int matrix_layout, lapack_int itype, char uplo, lapack_int\nn, lapack_complex_double* ap, const lapack_complex_double* bp);\nInclude Files\n•\nmkl.h\nDescription\nThe routine reduces generalized eigenproblems with Hermitian matrices\nA*z = λ*B*z, A*B*z = λ*z, or B*A*z = λ*z.\nto standard eigenproblems C*y = λ*y, using packed matrix storage. Here A is a complex Hermitian matrix,\nand B is a complex Hermitian positive-definite matrix. Before calling this routine, you must call ?pptrf to\ncompute the Cholesky factorization: B = UH*U or B = L*LH.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n902\n\n\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nitype\nMust be 1 or 2 or 3.\nIf itype = 1, the generalized eigenproblem is A*z = lambda*B*z\nfor uplo = 'U': C = inv(UH)*A*inv(U), z = inv(U)*y;\nfor uplo = 'L': C = inv(L)*A*inv(LH), z = inv(LH)*y.\nIf itype = 2, the generalized eigenproblem is A*B*z = lambda*z\nfor uplo = 'U': C = U*A*UH, z = inv(U)*y;\nfor uplo = 'L': C = LH*A*L, z = inv(LH)*y.\nIf itype = 3, the generalized eigenproblem is B*A*z = lambda*z\nfor uplo = 'U': C = U*A*UH, z = UH*y;\nfor uplo = 'L': C = LH*A*L, z = L*y.\nuplo\nMust be 'U' or 'L'.\nIf uplo = 'U', ap stores the packed upper triangle of A; you must supply\nB in the factored form B = UH*U.\nIf uplo = 'L', ap stores the packed lower triangle of A; you must supply B\nin the factored form B = L*LH.\nn\nThe order of the matrices A and B (n≥ 0).\nap, bp\nArrays:\nap contains the packed upper or lower triangle of A.\nThe dimension of a must be at least max(1, n*(n+1)/2).\nbp contains the packed Cholesky factor of B (as returned by ?pptrf with\nthe same uplo value).\nThe dimension of b must be at least max(1, n*(n+1)/2).\nOutput Parameters\nap\nThe upper or lower triangle of A is overwritten by the upper or lower\ntriangle of C, as specified by the arguments itype and uplo.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nApplication Notes\nForming the reduced matrix C is a stable procedure. However, it involves implicit multiplication by inv(B) (if\nitype = 1) or B (if itype = 2 or 3). When the routine is used as a step in the computation of eigenvalues\nand eigenvectors of the original problem, there may be a significant loss of accuracy if B is ill-conditioned\nwith respect to inversion.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n903\n\n\nThe approximate number of floating-point operations is n3.\n?sbgst\nReduces a real symmetric-definite generalized\neigenproblem for banded matrices to the standard\nform using the factorization performed by ?pbstf.\nSyntax\nlapack_int LAPACKE_ssbgst (int matrix_layout, char vect, char uplo, lapack_int n,\nlapack_int ka, lapack_int kb, float* ab, lapack_int ldab, const float* bb, lapack_int\nldbb, float* x, lapack_int ldx);\nlapack_int LAPACKE_dsbgst (int matrix_layout, char vect, char uplo, lapack_int n,\nlapack_int ka, lapack_int kb, double* ab, lapack_int ldab, const double* bb, lapack_int\nldbb, double* x, lapack_int ldx);\nInclude Files\n•\nmkl.h\nDescription\nTo reduce the real symmetric-definite generalized eigenproblem A*z = λ*B*z to the standard form C*y=λ*y,\nwhere A, B and C are banded, this routine must be preceded by a call to pbstf, which computes the split\nCholesky factorization of the positive-definite matrix B: B=ST*S. The split Cholesky factorization, compared\nwith the ordinary Cholesky factorization, allows the work to be approximately halved.\nThis routine overwrites A with C = XT*A*X, where X = inv(S)*Q and Q is an orthogonal matrix chosen\n(implicitly) to preserve the bandwidth of A. The routine also has an option to allow the accumulation of X,\nand then, if z is an eigenvector of C, X*z is an eigenvector of the original system.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nvect\nMust be 'N' or 'V'.\nIf vect = 'N', then matrix X is not returned;\nIf vect = 'V', then matrix X is returned.\nuplo\nMust be 'U' or 'L'.\nIf uplo = 'U', ab stores the upper triangular part of A.\nIf uplo = 'L', ab stores the lower triangular part of A.\nn\nThe order of the matrices A and B (n≥ 0).\nka\nThe number of super- or sub-diagonals in A\n(ka≥ 0).\nkb\nThe number of super- or sub-diagonals in B\n(ka≥kb≥ 0).\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n904\n\n\nab, bb\nab(size at least max(1, ldab*n) for column major layout and at least\nmax(1, ldab*(ka + 1)) for row major layout) is an array containing either\nupper or lower triangular part of the symmetric matrix A (as specified by\nuplo) in band storage format.\nbb(size at least max(1, ldbb*n) for column major layout and at least\nmax(1, ldbb*(kb + 1)) for row major layout) is an array containing the\nbanded split Cholesky factor of B as specified by uplo, n and kb and\nreturned by pbstf/pbstf.\nldab\nThe leading dimension of the array ab; must be at least ka+1 for column\nmajor layout and max(1, n) for row major layout.\nldbb\nThe leading dimension of the array bb; must be at least kb+1 for column\nmajor layout and max(1, n) for row major layout.\nldx\nThe leading dimension of the output array x. Constraints: if vect = 'N',\nthen ldx≥ 1;\nif vect = 'V', then ldx≥ max(1, n).\nOutput Parameters\nab\nOn exit, this array is overwritten by the upper or lower triangle of C as\nspecified by uplo.\nx\nArray.\nIf vect = 'V', then x (size at least max(1, ldx*n)) contains the n-by-n\nmatrix X = inv(S)*Q.\nIf vect = 'N', then x is not referenced.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nApplication Notes\nForming the reduced matrix C involves implicit multiplication by inv(B). When the routine is used as a step\nin the computation of eigenvalues and eigenvectors of the original problem, there may be a significant loss of\naccuracy if B is ill-conditioned with respect to inversion.\nIf ka and kb are much less than n then the total number of floating-point operations is approximately\n6n2*kb, when vect = 'N'. Additional (3/2)n3*(kb/ka) operations are required when vect = 'V'.\n?hbgst\nReduces a complex Hermitian positive-definite\ngeneralized eigenproblem for banded matrices to the\nstandard form using the factorization performed\nby ?pbstf.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n905\n\n\nSyntax\nlapack_int LAPACKE_chbgst (int matrix_layout, char vect, char uplo, lapack_int n,\nlapack_int ka, lapack_int kb, lapack_complex_float* ab, lapack_int ldab, const\nlapack_complex_float* bb, lapack_int ldbb, lapack_complex_float* x, lapack_int ldx);\nlapack_int LAPACKE_zhbgst (int matrix_layout, char vect, char uplo, lapack_int n,\nlapack_int ka, lapack_int kb, lapack_complex_double* ab, lapack_int ldab, const\nlapack_complex_double* bb, lapack_int ldbb, lapack_complex_double* x, lapack_int ldx);\nInclude Files\n•\nmkl.h\nDescription\nTo reduce the complex Hermitian positive-definite generalized eigenproblem A*z = λ*B*z to the standard\nform C*x = λ*y, where A, B and C are banded, this routine must be preceded by a call to pbstf/pbstf, which\ncomputes the split Cholesky factorization of the positive-definite matrix B: B = SH*S. The split Cholesky\nfactorization, compared with the ordinary Cholesky factorization, allows the work to be approximately halved.\nThis routine overwrites A with C = XH*A*X, where X = inv(S)*Q, and Q is a unitary matrix chosen\n(implicitly) to preserve the bandwidth of A. The routine also has an option to allow the accumulation of X,\nand then, if z is an eigenvector of C, X*z is an eigenvector of the original system.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nvect\nMust be 'N' or 'V'.\nIf vect = 'N', then matrix X is not returned;\nIf vect = 'V', then matrix X is returned.\nuplo\nMust be 'U' or 'L'.\nIf uplo = 'U', ab stores the upper triangular part of A.\nIf uplo = 'L', ab stores the lower triangular part of A.\nn\nThe order of the matrices A and B (n≥ 0).\nka\nThe number of super- or sub-diagonals in A\n(ka≥ 0).\nkb\nThe number of super- or sub-diagonals in B\n(ka≥kb≥ 0).\nab, bb\nab(size at least max(1, ldab*n) for column major layout and at least\nmax(1, ldab*(ka + 1)) for row major layout) is an array containing either\nupper or lower triangular part of the Hermitian matrix A (as specified by\nuplo) in band storage format.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n906\n\n\nbb(size at least max(1, ldbb*n) for column major layout and at least\nmax(1, ldbb*(kb + 1)) for row major layout) is an array containing the\nbanded split Cholesky factor of B as specified by uplo, n and kb and\nreturned by pbstf/pbstf.\nldab\nThe leading dimension of the array ab; must be at least ka+1 for column\nmajor layout and max(1, n) for row major layout.\nldbb\nThe leading dimension of the array bb; must be at least kb+1 for column\nmajor layout and max(1, n) for row major layout.\nldx\nThe leading dimension of the output array x. Constraints:\nif vect = 'N', then ldx≥ 1;\nif vect = 'V', then ldx≥ max(1, n).\nOutput Parameters\nab\nOn exit, this array is overwritten by the upper or lower triangle of C as\nspecified by uplo.\nx\nArray.\nIf vect = 'V', then x (size at least max(1, ldx*n)) contains the n-by-n\nmatrix X = inv(S)*Q.\nIf vect = 'N', then x is not referenced.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nApplication Notes\nForming the reduced matrix C involves implicit multiplication by inv(B). When the routine is used as a step\nin the computation of eigenvalues and eigenvectors of the original problem, there may be a significant loss of\naccuracy if B is ill-conditioned with respect to inversion. The total number of floating-point operations is\napproximately 20n2*kb, when vect = 'N'. Additional 5n3*(kb/ka) operations are required when vect =\n'V'. All these estimates assume that both ka and kb are much less than n.\n?pbstf\nComputes a split Cholesky factorization of a real\nsymmetric or complex Hermitian positive-definite\nbanded matrix used in ?sbgst/?hbgst .\nSyntax\nlapack_int LAPACKE_spbstf (int matrix_layout, char uplo, lapack_int n, lapack_int kb,\nfloat* bb, lapack_int ldbb);\nlapack_int LAPACKE_dpbstf (int matrix_layout, char uplo, lapack_int n, lapack_int kb,\ndouble* bb, lapack_int ldbb);\nlapack_int LAPACKE_cpbstf (int matrix_layout, char uplo, lapack_int n, lapack_int kb,\nlapack_complex_float* bb, lapack_int ldbb);\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n907\n\n\nlapack_int LAPACKE_zpbstf (int matrix_layout, char uplo, lapack_int n, lapack_int kb,\nlapack_complex_double* bb, lapack_int ldbb);\nInclude Files\n•\nmkl.h\nDescription\nThe routine computes a split Cholesky factorization of a real symmetric or complex Hermitian positive-\ndefinite band matrix B. It is to be used in conjunction with sbgst/hbgst.\nThe factorization has the form B = ST*S (or B = SH*S for complex flavors), where S is a band matrix of the\nsame bandwidth as B and the following structure: S is upper triangular in the first (n+kb)/2 rows and lower\ntriangular in the remaining rows.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nuplo\nMust be 'U' or 'L'.\nIf uplo = 'U', bb stores the upper triangular part of B.\nIf uplo = 'L', bb stores the lower triangular part of B.\nn\nThe order of the matrix B (n≥ 0).\nkb\nThe number of super- or sub-diagonals in B\n(kb≥ 0).\nbb\nbb(size at least max(1, ldbb*n) for column major layout and at least\nmax(1, ldbb*(kb + 1)) for row major layout) is an array containing either\nupper or lower triangular part of the matrix B (as specified by uplo) in band\nstorage format.\nldbb\nThe leading dimension of bb; must be at least kb+1for column major and at\nleast max(1, n) for row major.\nOutput Parameters\nbb\nOn exit, this array is overwritten by the elements of the split Cholesky\nfactor S.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = i, then the factorization could not be completed, because the updated element bii would be the\nsquare root of a negative number; hence the matrix B is not positive-definite.\nIf info = -i, the i-th parameter had an illegal value.\nApplication Notes\nThe computed factor S is the exact factor of a perturbed matrix B + E, where\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n908\n\n\nc(n) is a modest linear function of n, and ε is the machine precision.\nThe total number of floating-point operations for real flavors is approximately n(kb+1)2. The number of\noperations for complex flavors is 4 times greater. All these estimates assume that kb is much less than n.\nAfter calling this routine, you can call sbgst/hbgst to solve the generalized eigenproblem Az = λBz, where A\nand B are banded and B is positive-definite.\nNonsymmetric Eigenvalue Problems: LAPACK Computational Routines\nThis topic describes LAPACK routines for solving nonsymmetric eigenvalue problems, computing the Schur\nfactorization of general matrices, as well as performing a number of related computational tasks.\nA nonsymmetric eigenvalue problem is as follows: given a nonsymmetric (or non-Hermitian) matrix A, find\nthe eigenvaluesλ and the corresponding eigenvectorsz that satisfy the equation\nAz = λz (right eigenvectors z)\nor the equation\nzHA = λzH (left eigenvectors z).\nNonsymmetric eigenvalue problems have the following properties:\n•\nThe number of eigenvectors may be less than the matrix order (but is not less than the number of\ndistinct eigenvalues of A).\n•\nEigenvalues may be complex even for a real matrix A.\n•\nIf a real nonsymmetric matrix has a complex eigenvalue a+bi corresponding to an eigenvector z, then a-\nbi is also an eigenvalue. The eigenvalue a-bi corresponds to the eigenvector whose elements are\ncomplex conjugate to the elements of z.\nTo solve a nonsymmetric eigenvalue problem with LAPACK, you usually need to reduce the matrix to the\nupper Hessenberg form and then solve the eigenvalue problem with the Hessenberg matrix obtained. Table\n\"Computational Routines for Solving Nonsymmetric Eigenvalue Problems\" lists LAPACK routines to reduce the\nmatrix to the upper Hessenberg form by an orthogonal (or unitary) similarity transformation A = QHQH as\nwell as routines to solve eigenvalue problems with Hessenberg matrices, forming the Schur factorization of\nsuch matrices and computing the corresponding condition numbers.\nComputational Routines for Solving Nonsymmetric Eigenvalue Problems\nOperation performed\nRoutines for real matrices\nRoutines for complex matrices\nReduce to Hessenberg form\nA = QHQH\n?gehrd,\n?gehrd\nGenerate the matrix Q\n?orghr\n?unghr\nApply the matrix Q\n?ormhr\n?unmhr\nBalance matrix\n?gebal\n?gebal\nTransform eigenvectors of\nbalanced matrix to those of\nthe original matrix\n?gebak\n?gebak\nFind eigenvalues and Schur\nfactorization (QR algorithm)\n?hseqr\n?hseqr\nFind eigenvectors from\nHessenberg form (inverse\niteration)\n?hsein\n?hsein\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n909\n\n\nOperation performed\nRoutines for real matrices\nRoutines for complex matrices\nFind eigenvectors from\nSchur factorization\n?trevc\n?trevc\nEstimate sensitivities of\neigenvalues and\neigenvectors\n?trsna\n?trsna\nReorder Schur factorization\n?trexc\n?trexc\nReorder Schur factorization,\nfind the invariant subspace\nand estimate sensitivities\n?trsen\n?trsen\nSolves Sylvester's equation.\n?trsyl\n?trsyl\n?gehrd\nReduces a general matrix to upper Hessenberg form.\nSyntax\nlapack_int LAPACKE_sgehrd (int matrix_layout, lapack_int n, lapack_int ilo, lapack_int\nihi, float* a, lapack_int lda, float* tau);\nlapack_int LAPACKE_dgehrd (int matrix_layout, lapack_int n, lapack_int ilo, lapack_int\nihi, double* a, lapack_int lda, double* tau);\nlapack_int LAPACKE_cgehrd (int matrix_layout, lapack_int n, lapack_int ilo, lapack_int\nihi, lapack_complex_float* a, lapack_int lda, lapack_complex_float* tau);\nlapack_int LAPACKE_zgehrd (int matrix_layout, lapack_int n, lapack_int ilo, lapack_int\nihi, lapack_complex_double* a, lapack_int lda, lapack_complex_double* tau);\nInclude Files\n•\nmkl.h\nDescription\nThe routine reduces a general matrix A to upper Hessenberg form H by an orthogonal or unitary similarity\ntransformation A = Q*H*QH. Here H has real subdiagonal elements.\nThe routine does not form the matrix Q explicitly. Instead, Q is represented as a product of elementary\nreflectors. Routines are provided to work with Q in this representation.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nn\nThe order of the matrix A (n≥ 0).\nilo, ihi\nIf A is an output by ?gebal, then ilo and ihi must contain the values\nreturned by that routine. Otherwise ilo = 1 and ihi = n. (If n > 0, then\n1 ≤ilo≤ihi≤n; if n = 0, ilo = 1 and ihi = 0.)\na\nArrays:\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n910\n\n\na (size max(1, lda*n)) contains the matrix A.\nlda\nThe leading dimension of a; at least max(1, n).\nOutput Parameters\na\nThe elements on and above the subdiagonal contain the upper Hessenberg\nmatrix H. The subdiagonal elements of H are real. The elements below the\nsubdiagonal, with the array tau, represent the orthogonal matrix Q as a\nproduct of n elementary reflectors.\ntau\nArray, size at least max (1, n-1).\nContains scalars that define elementary reflectors for the matrix Q.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nApplication Notes\nThe computed Hessenberg matrix H is exactly similar to a nearby matrix A + E, where ||E||2 < c(n)ε||\nA||2, c(n) is a modestly increasing function of n, and ε is the machine precision.\nThe approximate number of floating-point operations for real flavors is (2/3)*(ihi - ilo)2(2ihi + 2ilo\n+ 3n); for complex flavors it is 4 times greater.\n?orghr\nGenerates the real orthogonal matrix Q determined\nby ?gehrd.\nSyntax\nlapack_int LAPACKE_sorghr (int matrix_layout, lapack_int n, lapack_int ilo, lapack_int\nihi, float* a, lapack_int lda, const float* tau);\nlapack_int LAPACKE_dorghr (int matrix_layout, lapack_int n, lapack_int ilo, lapack_int\nihi, double* a, lapack_int lda, const double* tau);\nInclude Files\n•\nmkl.h\nDescription\nThe routine explicitly generates the orthogonal matrix Q that has been determined by a preceding call to\nsgehrd/dgehrd. (The routine ?gehrd reduces a real general matrix A to upper Hessenberg form H by an\northogonal similarity transformation, A = Q*H*QT, and represents the matrix Q as a product of ihi-\niloelementary reflectors. Here ilo and ihi are values determined by sgebal/dgebal when balancing the\nmatrix; if the matrix has not been balanced, ilo = 1 and ihi = n.)\nThe matrix Q generated by ?orghr has the structure:\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n911\n\n\nwhere Q22 occupies rows and columns ilo to ihi.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nn\nThe order of the matrix Q (n≥ 0).\nilo, ihi\nThese must be the same parameters ilo and ihi, respectively, as supplied\nto ?gehrd. (If n > 0, then 1 ≤ilo≤ihi≤n; if n = 0, ilo = 1 and ihi =\n0.)\na, tau\nArrays: a (size max(1, lda*n)) contains details of the vectors which define\nthe elementary reflectors, as returned by ?gehrd.\ntau contains further details of the elementary reflectors, as returned\nby ?gehrd.\nThe dimension of tau must be at least max (1, n-1).\nlda\nThe leading dimension of a; at least max(1, n).\nOutput Parameters\na\nOverwritten by the n-by-n orthogonal matrix Q.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n912\n\n\nApplication Notes\nThe computed matrix Q differs from the exact result by a matrix E such that ||E||2 = O(ε), where ε is the\nmachine precision.\nThe approximate number of floating-point operations is (4/3)(ihi-ilo)3.\nThe complex counterpart of this routine is unghr.\n?ormhr\nMultiplies an arbitrary real matrix C by the real\northogonal matrix Q determined by ?gehrd.\nSyntax\nlapack_int LAPACKE_sormhr (int matrix_layout, char side, char trans, lapack_int m,\nlapack_int n, lapack_int ilo, lapack_int ihi, const float* a, lapack_int lda, const\nfloat* tau, float* c, lapack_int ldc);\nlapack_int LAPACKE_dormhr (int matrix_layout, char side, char trans, lapack_int m,\nlapack_int n, lapack_int ilo, lapack_int ihi, const double* a, lapack_int lda, const\ndouble* tau, double* c, lapack_int ldc);\nInclude Files\n•\nmkl.h\nDescription\nThe routine multiplies a matrix C by the orthogonal matrix Q that has been determined by a preceding call to\nsgehrd/dgehrd. (The routine ?gehrd reduces a real general matrix A to upper Hessenberg form H by an\northogonal similarity transformation, A = Q*H*QT, and represents the matrix Q as a product of ihi-\niloelementary reflectors. Here ilo and ihi are values determined by sgebal/dgebal when balancing the\nmatrix;if the matrix has not been balanced, ilo = 1 and ihi = n.)\nWith ?ormhr, you can form one of the matrix products Q*C, QT*C, C*Q, or C*QT, overwriting the result on C\n(which may be any real rectangular matrix).\nA common application of ?ormhr is to transform a matrix V of eigenvectors of H to the matrix QV of\neigenvectors of A.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nside\nMust be 'L' or 'R'.\nIf side= 'L', then the routine forms Q*C or QT*C.\nIf side= 'R', then the routine forms C*Q or C*QT.\ntrans\nMust be 'N' or 'T'.\nIf trans= 'N', then Q is applied to C.\nIf trans= 'T', then QT is applied to C.\nm\nThe number of rows in C (m≥ 0).\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n913\n\n\nn\nThe number of columns in C (n≥ 0).\nilo, ihi\nThese must be the same parameters ilo and ihi, respectively, as supplied\nto ?gehrd.\nIf m > 0 and side = 'L', then 1 ≤ilo≤ihi≤m.\nIf m = 0 and side = 'L', then ilo = 1 and ihi = 0.\nIf n > 0 and side = 'R', then 1 ≤ilo≤ihi≤n.\nIf n = 0 and side = 'R', then ilo = 1 and ihi = 0.\na, tau, c\nArrays:\na(size max(1,lda*n) for side='R' and size max(1,lda*m) for side='L')\ncontains details of the vectors which define the elementary reflectors, as\nreturned by ?gehrd.\ntau contains further details of the elementary reflectors, as returned\nby ?gehrd .\nThe dimension of tau must be at least max (1, m-1) if side = 'L' and at\nleast max (1, n-1) if side = 'R'.\nc(size max(1, ldc*n) for column major layout and max(1, ldc*m for row\nmajor layout) contains the m by n matrix C.\nlda\nThe leading dimension of a; at least max(1, m) if side = 'L' and at least\nmax (1, n) if side = 'R'.\nldc\nThe leading dimension of c; at least max(1, m) for column major layout and\nat least max(1, n) for row major layout .\nOutput Parameters\nc\nC is overwritten by product Q*C, QT*C, C*Q, or C*QT as specified by side and\ntrans.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nApplication Notes\nThe computed matrix Q differs from the exact result by a matrix E such that ||E||2 = O(ε)|*|C||2, where\nε is the machine precision.\nThe approximate number of floating-point operations is\n2n(ihi-ilo)2 if side = 'L';\n2m(ihi-ilo)2 if side = 'R'.\nThe complex counterpart of this routine is unmhr.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n914\n\n\n?unghr\nGenerates the complex unitary matrix Q determined\nby ?gehrd.\nSyntax\nlapack_int LAPACKE_cunghr (int matrix_layout, lapack_int n, lapack_int ilo, lapack_int\nihi, lapack_complex_float* a, lapack_int lda, const lapack_complex_float* tau);\nlapack_int LAPACKE_zunghr (int matrix_layout, lapack_int n, lapack_int ilo, lapack_int\nihi, lapack_complex_double* a, lapack_int lda, const lapack_complex_double* tau);\nInclude Files\n•\nmkl.h\nDescription\nThe routine is intended to be used following a call to cgehrd/zgehrd, which reduces a complex matrix A to\nupper Hessenberg form H by a unitary similarity transformation: A = Q*H*QH. ?gehrd represents the matrix\nQ as a product of ihi-iloelementary reflectors. Here ilo and ihi are values determined by cgebal/zgebal\nwhen balancing the matrix; if the matrix has not been balanced, ilo = 1 and ihi = n.\nUse the routine unghr to generate Q explicitly as a square matrix. The matrix Q has the structure:\nwhere Q22 occupies rows and columns ilo to ihi.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nn\nThe order of the matrix Q (n≥ 0).\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n915\n\n\nilo, ihi\nThese must be the same parameters ilo and ihi, respectively, as supplied\nto ?gehrd . (If n > 0, then 1 ≤ilo≤ihi≤n. If n = 0, then ilo = 1 and\nihi = 0.)\na, tau\nArrays:\na (size max(1, lda*n)) contains details of the vectors which define the\nelementary reflectors, as returned by ?gehrd.\ntau contains further details of the elementary reflectors, as returned\nby ?gehrd .\nThe dimension of tau must be at least max (1, n-1).\nlda\nThe leading dimension of a; at least max(1, n).\nOutput Parameters\na\nOverwritten by the n-by-n unitary matrix Q.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nApplication Notes\nThe computed matrix Q differs from the exact result by a matrix E such that ||E||2 = O(ε), where ε is the\nmachine precision.\nThe approximate number of real floating-point operations is (16/3)(ihi-ilo)3.\nThe real counterpart of this routine is orghr.\n?unmhr\nMultiplies an arbitrary complex matrix C by the\ncomplex unitary matrix Q determined by ?gehrd.\nSyntax\nlapack_int LAPACKE_cunmhr (int matrix_layout, char side, char trans, lapack_int m,\nlapack_int n, lapack_int ilo, lapack_int ihi, const lapack_complex_float* a, lapack_int\nlda, const lapack_complex_float* tau, lapack_complex_float* c, lapack_int ldc);\nlapack_int LAPACKE_zunmhr (int matrix_layout, char side, char trans, lapack_int m,\nlapack_int n, lapack_int ilo, lapack_int ihi, const lapack_complex_double* a,\nlapack_int lda, const lapack_complex_double* tau, lapack_complex_double* c, lapack_int\nldc);\nInclude Files\n•\nmkl.h\nDescription\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n916\n\n\nThe routine multiplies a matrix C by the unitary matrix Q that has been determined by a preceding call to\ncgehrd/zgehrd. (The routine ?gehrd reduces a real general matrix A to upper Hessenberg form H by an\northogonal similarity transformation, A = Q*H*QH, and represents the matrix Q as a product of ihi-ilo\nelementary reflectors. Here ilo and ihi are values determined by cgebal/zgebal when balancing the\nmatrix; if the matrix has not been balanced, ilo = 1 and ihi = n.)\nWith ?unmhr, you can form one of the matrix products Q*C, QH*C, C*Q, or C*QH, overwriting the result on C\n(which may be any complex rectangular matrix). A common application of this routine is to transform a\nmatrix V of eigenvectors of H to the matrix QV of eigenvectors of A.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nside\nMust be 'L' or 'R'.\nIf side = 'L', then the routine forms Q*C or QH*C.\nIf side = 'R', then the routine forms C*Q or C*QH.\ntrans\nMust be 'N' or 'C'.\nIf trans = 'N', then Q is applied to C.\nIf trans = 'T', then QH is applied to C.\nm\nThe number of rows in C (m≥ 0).\nn\nThe number of columns in C (n≥ 0).\nilo, ihi\nThese must be the same parameters ilo and ihi, respectively, as supplied\nto ?gehrd .\nIf m > 0 and side = 'L', then 1 ≤ilo≤ihi≤m.\nIf m = 0 and side = 'L', then ilo = 1 and ihi = 0.\nIf n > 0 and side = 'R', then 1 ≤ilo≤ihi≤n.\nIf n = 0 and side = 'R', then ilo =1 and ihi = 0.\na, tau, c\nArrays:\na(size max(1,lda*n) for side='R' and size max(1,lda*m) for side='L')\ncontains details of the vectors which define the elementary reflectors, as\nreturned by ?gehrd.\ntau contains further details of the elementary reflectors, as returned\nby ?gehrd.\nThe dimension of tau must be at least max (1, m-1)\nif side = 'L' and at least max (1, n-1) if side = 'R'.\nc(size max(1, ldc*n) for column major layout and max(1, ldc*m for row\nmajor layout) contains the m-by-n matrix C.\nlda\nThe leading dimension of a; at least max(1, m) if side = 'L' and at least\nmax (1, n) if side = 'R'.\nldc\nThe leading dimension of c; at least max(1, m) for column major layout and\nat least max(1, n) for row major layout.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n917\n\n\nOutput Parameters\nc\nC is overwritten by Q*C, or QH*C, or C*QH, or C*Q as specified by side and\ntrans.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nApplication Notes\nThe computed matrix Q differs from the exact result by a matrix E such that ||E||2 = O(ε)*||C||2, where\nε is the machine precision.\nThe approximate number of floating-point operations is\n8n(ihi-ilo)2 if side = 'L';\n8m(ihi-ilo)2 if side = 'R'.\nThe real counterpart of this routine is ormhr.\n?gebal\nBalances a general matrix to improve the accuracy of\ncomputed eigenvalues and eigenvectors.\nSyntax\nlapack_int LAPACKE_sgebal( int matrix_layout, char job, lapack_int n, float* a,\nlapack_int lda, lapack_int* ilo, lapack_int* ihi, float* scale );\nlapack_int LAPACKE_dgebal( int matrix_layout, char job, lapack_int n, double* a,\nlapack_int lda, lapack_int* ilo, lapack_int* ihi, double* scale );\nlapack_int LAPACKE_cgebal( int matrix_layout, char job, lapack_int n,\nlapack_complex_float* a, lapack_int lda, lapack_int* ilo, lapack_int* ihi, float*\nscale );\nlapack_int LAPACKE_zgebal( int matrix_layout, char job, lapack_int n,\nlapack_complex_double* a, lapack_int lda, lapack_int* ilo, lapack_int* ihi, double*\nscale );\nInclude Files\n•\nmkl.h\nDescription\nThe routine balances a matrix A by performing either or both of the following two similarity transformations:\n(1) The routine first attempts to permute A to block upper triangular form:\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n918\n\n\nwhere P is a permutation matrix, and A'11 and A'33 are upper triangular. The diagonal elements of A'11 and\nA'33 are eigenvalues of A. The rest of the eigenvalues of A are the eigenvalues of the central diagonal block\nA'22, in rows and columns ilo to ihi. Subsequent operations to compute the eigenvalues of A (or its Schur\nfactorization) need only be applied to these rows and columns; this can save a significant amount of work if\nilo > 1 and ihi < n.\nIf no suitable permutation exists (as is often the case), the routine sets ilo = 1 and ihi = n, and A'22 is\nthe whole of A.\n(2) The routine applies a diagonal similarity transformation to A', to make the rows and columns of A'22 as\nclose in norm as possible:\nThis scaling can reduce the norm of the matrix (that is, ||A''22|| < ||A'22||), and hence reduce the\neffect of rounding errors on the accuracy of computed eigenvalues and eigenvectors.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\njob\nMust be 'N' or 'P' or 'S' or 'B'.\nIf job = 'N', then A is neither permuted nor scaled (but ilo, ihi, and scale\nget their values).\nIf job = 'P', then A is permuted but not scaled.\nIf job = 'S', then A is scaled but not permuted.\nIf job = 'B', then A is both scaled and permuted.\nn\nThe order of the matrix A (n≥ 0).\na\nArray a (size max(1, lda*n)) contains the matrix A.\nlda\nThe leading dimension of a; at least max(1, n).\nOutput Parameters\na\nOverwritten by the balanced matrix (a is not referenced if job = 'N').\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n919\n\n\nilo, ihi\nThe values ilo and ihi such that on exit a(i,j) is zero if i > j and 1 ≤j <\nilo or ihi < j≤n.\nIf job = 'N' or 'S', then ilo = 1 and ihi = n.\nscale\nArray, size at least max(1, n).\nContains details of the permutations and scaling factors.\nMore precisely, if pj is the index of the row and column interchanged with\nrow and column j, and dj is the scaling factor used to balance row and\ncolumn j, then\nscale[j - 1] = pj for j = 1, 2,..., ilo-1, ihi+1,..., n;\nscale[j - 1] = dj for j = ilo, ilo + 1,..., ihi.\nThe order in which the interchanges are made is n to ihi+1, then 1 to ilo-1.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nApplication Notes\nThe errors are negligible, compared with those in subsequent computations.\nIf the matrix A is balanced by this routine, then any eigenvectors computed subsequently are eigenvectors of\nthe matrix A'' and hence you must call gebak to transform them back to eigenvectors of A.\nIf the Schur vectors of A are required, do not call this routine with job = 'S' or 'B', because then the\nbalancing transformation is not orthogonal (not unitary for complex flavors).\nIf you call this routine with job = 'P', then any Schur vectors computed subsequently are Schur vectors of\nthe matrix A'', and you need to call gebak (with side = 'R') to transform them back to Schur vectors of A.\nThe total number of floating-point operations is proportional to n2.\n?gebak\nTransforms eigenvectors of a balanced matrix to those\nof the original nonsymmetric matrix.\nSyntax\nlapack_int LAPACKE_sgebak( int matrix_layout, char job, char side, lapack_int n,\nlapack_int ilo, lapack_int ihi, const float* scale, lapack_int m, float* v, lapack_int\nldv );\nlapack_int LAPACKE_dgebak( int matrix_layout, char job, char side, lapack_int n,\nlapack_int ilo, lapack_int ihi, const double* scale, lapack_int m, double* v,\nlapack_int ldv );\nlapack_int LAPACKE_cgebak( int matrix_layout, char job, char side, lapack_int n,\nlapack_int ilo, lapack_int ihi, const float* scale, lapack_int m, lapack_complex_float*\nv, lapack_int ldv );\nlapack_int LAPACKE_zgebak( int matrix_layout, char job, char side, lapack_int n,\nlapack_int ilo, lapack_int ihi, const double* scale, lapack_int m,\nlapack_complex_double* v, lapack_int ldv );\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n920\n\n\nInclude Files\n•\nmkl.h\nDescription\nThe routine is intended to be used after a matrix A has been balanced by a call to ?gebal, and eigenvectors\nof the balanced matrix A''22 have subsequently been computed. For a description of balancing, see gebal. The\nbalanced matrix A'' is obtained as A''= D*P*A*PT*inv(D), where P is a permutation matrix and D is a\ndiagonal scaling matrix. This routine transforms the eigenvectors as follows:\nif x is a right eigenvector of A'', then PT*inv(D)*x is a right eigenvector of A; if y is a left eigenvector of A'',\nthen PT*D*y is a left eigenvector of A.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\njob\nMust be 'N' or 'P' or 'S' or 'B'. The same parameter job as supplied\nto ?gebal.\nside\nMust be 'L' or 'R'.\nIf side = 'L', then left eigenvectors are transformed.\nIf side = 'R', then right eigenvectors are transformed.\nn\nThe number of rows of the matrix of eigenvectors (n≥ 0).\nilo, ihi\nThe values ilo and ihi, as returned by ?gebal. (If n > 0, then 1\n≤ilo≤ihi≤n;\nif n = 0, then ilo = 1 and ihi = 0.)\nscale\nArray, size at least max(1, n).\nContains details of the permutations and/or the scaling factors used to\nbalance the original general matrix, as returned by ?gebal.\nm\nThe number of columns of the matrix of eigenvectors (m≥ 0).\nv\nArrays:\nv(size max(1, ldv*n) for column major layout and max(1, ldv*m) for row\nmajor layout) contains the matrix of left or right eigenvectors to be\ntransformed.\nldv\nThe leading dimension of v; at least max(1, n) for column major layout and\nat least max(1, m) for row major layout .\nOutput Parameters\nv\nOverwritten by the transformed eigenvectors.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n921\n\n\nIf info = -i, the i-th parameter had an illegal value.\nApplication Notes\nThe errors in this routine are negligible.\nThe approximate number of floating-point operations is approximately proportional to m*n.\n?hseqr\nComputes all eigenvalues and (optionally) the Schur\nfactorization of a matrix reduced to Hessenberg form.\nSyntax\nlapack_int LAPACKE_shseqr( int matrix_layout, char job, char compz, lapack_int n,\nlapack_int ilo, lapack_int ihi, float* h, lapack_int ldh, float* wr, float* wi, float*\nz, lapack_int ldz );\nlapack_int LAPACKE_dhseqr( int matrix_layout, char job, char compz, lapack_int n,\nlapack_int ilo, lapack_int ihi, double* h, lapack_int ldh, double* wr, double* wi,\ndouble* z, lapack_int ldz );\nlapack_int LAPACKE_chseqr( int matrix_layout, char job, char compz, lapack_int n,\nlapack_int ilo, lapack_int ihi, lapack_complex_float* h, lapack_int ldh,\nlapack_complex_float* w, lapack_complex_float* z, lapack_int ldz );\nlapack_int LAPACKE_zhseqr( int matrix_layout, char job, char compz, lapack_int n,\nlapack_int ilo, lapack_int ihi, lapack_complex_double* h, lapack_int ldh,\nlapack_complex_double* w, lapack_complex_double* z, lapack_int ldz );\nInclude Files\n•\nmkl.h\nDescription\nThe routine computes all the eigenvalues, and optionally the Schur factorization, of an upper Hessenberg\nmatrix H: H = Z*T*ZH, where T is an upper triangular (or, for real flavors, quasi-triangular) matrix (the\nSchur form of H), and Z is the unitary or orthogonal matrix whose columns are the Schur vectors zi.\nYou can also use this routine to compute the Schur factorization of a general matrix A which has been\nreduced to upper Hessenberg form H:\nA = Q*H*QH, where Q is unitary (orthogonal for real flavors);\nA = (QZ)*T*(QZ)H.\nIn this case, after reducing A to Hessenberg form by gehrd, call orghr to form Q explicitly and then pass Q\nto ?hseqr with compz = 'V'.\nYou can also call gebal to balance the original matrix before reducing it to Hessenberg form by ?hseqr, so\nthat the Hessenberg matrix H will have the structure:\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n922\n\n\nwhere H11 and H33 are upper triangular.\nIf so, only the central diagonal block H22 (in rows and columns ilo to ihi) needs to be further reduced to\nSchur form (the blocks H12 and H23 are also affected). Therefore the values of ilo and ihi can be supplied\nto ?hseqr directly. Also, after calling this routine you must call gebak to permute the Schur vectors of the\nbalanced matrix to those of the original matrix.\nIf ?gebal has not been called, however, then ilo must be set to 1 and ihi to n. Note that if the Schur\nfactorization of A is required, ?gebal must not be called with job = 'S' or 'B', because the balancing\ntransformation is not unitary (for real flavors, it is not orthogonal).\n?hseqr uses a multishift form of the upper Hessenberg QR algorithm. The Schur vectors are normalized so\nthat ||zi||2 = 1, but are determined only to within a complex factor of absolute value 1 (for the real\nflavors, to within a factor ±1).\nInput Parameters\njob\nMust be 'E' or 'S'.\nIf job = 'E', then eigenvalues only are required.\nIf job = 'S', then the Schur form T is required.\ncompz\nMust be 'N' or 'I' or 'V'.\nIf compz = 'N', then no Schur vectors are computed (and the array z is not\nreferenced).\nIf compz = 'I', then the Schur vectors of H are computed (and the array z\nis initialized by the routine).\nIf compz = 'V', then the Schur vectors of A are computed (and the array z\nmust contain the matrix Q on entry).\nn\nThe order of the matrix H (n≥ 0).\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n923\n\n\nilo, ihi\nIf A has been balanced by ?gebal, then ilo and ihi must contain the values\nreturned by ?gebal. Otherwise, ilo must be set to 1 and ihi to n.\nh, z\nArrays:\nh (size max(1, ldh*n)) ) The n-by-n upper Hessenberg matrix H.\nz (size max(1, ldz*n))\nIf compz = 'V', then z must contain the matrix Q from the reduction to\nHessenberg form.\nIf compz = 'I', then z need not be set.\nIf compz = 'N', then z is not referenced.\nldh\nThe leading dimension of h; at least max(1, n).\nldz\nThe leading dimension of z;\nIf compz = 'N', then ldz≥ 1.\nIf compz = 'V' or 'I', then ldz≥ max(1, n).\nOutput Parameters\nw\nArray, size at least max (1, n). Contains the computed eigenvalues, unless\ninfo>0. The eigenvalues are stored in the same order as on the diagonal of\nthe Schur form T (if computed).\nwr, wi\nArrays, size at least max (1, n) each.\nContain the real and imaginary parts, respectively, of the computed\neigenvalues, unless info > 0. Complex conjugate pairs of eigenvalues\nappear consecutively with the eigenvalue having positive imaginary part\nfirst. The eigenvalues are stored in the same order as on the diagonal of the\nSchur form T (if computed).\nh\nIf info = 0 and job = 'S', h contains the upper quasi-triangular matrix T\nfrom the Schur decomposition (the Schur form).\nIf info = 0 and job = 'E', the contents of h are unspecified on exit. (The\noutput value of h when info > 0 is given under the description of info\nbelow.)\nz\nIf compz = 'V' and info = 0, then z contains Q*Z.\nIf compz = 'I' and info = 0, then z contains the unitary or orthogonal\nmatrix Z of the Schur vectors of H.\nIf compz = 'N', then z is not referenced.\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nIf info = i, ?hseqr failed to compute all of the eigenvalues. Elements 1,2, ..., ilo-1 and i+1, i+2, ..., n of\nthe eigenvalue arrays (wr and wi for real flavors and w for complex flavors) contain the real and imaginary\nparts of those eigenvalues that have been successfully found.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n924\n\n\nIf info > 0, and job = 'E', then on exit, the remaining unconverged eigenvalues are the eigenvalues of\nthe upper Hessenberg matrix rows and columns ilo through info of the final output value of H.\nIf info > 0, and job = 'S', then on exit (initial value of H)*U = U*(final value of H), where U is a unitary\nmatrix. The final value of H is upper Hessenberg and triangular in rows and columns info+1 through ihi.\nIf info > 0, and compz = 'V', then on exit (final value of Z) = (initial value of Z)*U, where U is the\nunitary matrix (regardless of the value of job).\nIf info > 0, and compz = 'I', then on exit (final value of Z) = U, where U is the unitary matrix (regardless\nof the value of job).\nIf info > 0, and compz = 'N', then Z is not accessed.\nApplication Notes\nThe computed Schur factorization is the exact factorization of a nearby matrix H + E, where ||E||2 < O(ε)\n||H||2/si, and ε is the machine precision.\nIf λi is an exact eigenvalue, and μi is the corresponding computed value, then |λi - μi|≤c(n)*ε*||H||2/si,\nwhere c(n) is a modestly increasing function of n, and si is the reciprocal condition number of λi. The\ncondition numbers si may be computed by calling trsna.\nThe total number of floating-point operations depends on how rapidly the algorithm converges; typical\nnumbers are as follows.\nIf only eigenvalues are computed:\n7n3 for real flavors\n25n3 for complex flavors.\nIf the Schur form is computed:\n10n3 for real flavors\n35n3 for complex flavors.\nIf the full Schur factorization is\ncomputed:\n20n3 for real flavors\n70n3 for complex flavors.\n?hsein\nComputes selected eigenvectors of an upper\nHessenberg matrix that correspond to specified\neigenvalues.\nSyntax\nlapack_int LAPACKE_shsein( int matrix_layout, char side, char eigsrc, char initv,\nlapack_logical* select, lapack_int n, const float* h, lapack_int ldh, float* wr, const\nfloat* wi, float* vl, lapack_int ldvl, float* vr, lapack_int ldvr, lapack_int mm,\nlapack_int* m, lapack_int* ifaill, lapack_int* ifailr );\nlapack_int LAPACKE_dhsein( int matrix_layout, char side, char eigsrc, char initv,\nlapack_logical* select, lapack_int n, const double* h, lapack_int ldh, double* wr,\nconst double* wi, double* vl, lapack_int ldvl, double* vr, lapack_int ldvr, lapack_int\nmm, lapack_int* m, lapack_int* ifaill, lapack_int* ifailr );\nlapack_int LAPACKE_chsein( int matrix_layout, char side, char eigsrc, char initv, const\nlapack_logical* select, lapack_int n, const lapack_complex_float* h, lapack_int ldh,\nlapack_complex_float* w, lapack_complex_float* vl, lapack_int ldvl,\nlapack_complex_float* vr, lapack_int ldvr, lapack_int mm, lapack_int* m, lapack_int*\nifaill, lapack_int* ifailr );\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n925\n\n\nlapack_int LAPACKE_zhsein( int matrix_layout, char side, char eigsrc, char initv, const\nlapack_logical* select, lapack_int n, const lapack_complex_double* h, lapack_int ldh,\nlapack_complex_double* w, lapack_complex_double* vl, lapack_int ldvl,\nlapack_complex_double* vr, lapack_int ldvr, lapack_int mm, lapack_int* m, lapack_int*\nifaill, lapack_int* ifailr );\nInclude Files\n•\nmkl.h\nDescription\nThe routine computes left and/or right eigenvectors of an upper Hessenberg matrix H, corresponding to\nselected eigenvalues.\nThe right eigenvector x and the left eigenvector y, corresponding to an eigenvalue λ, are defined by: H*x =\nλ*x and yH*H = λ*yH (or HH*y = λ**y). Here λ* denotes the conjugate of λ.\nThe eigenvectors are computed by inverse iteration. They are scaled so that, for a real eigenvector x, max|\nxi| = 1, and for a complex eigenvector, max(|Rexi| + |Imxi|) = 1.\nIf H has been formed by reduction of a general matrix A to upper Hessenberg form, then eigenvectors of H\nmay be transformed to eigenvectors of A by ormhr or unmhr.\nInput Parameters\nside\nMust be 'R' or 'L' or 'B'.\nIf side = 'R', then only right eigenvectors are computed.\nIf side = 'L', then only left eigenvectors are computed.\nIf side = 'B', then all eigenvectors are computed.\neigsrc\nMust be 'Q' or 'N'.\nIf eigsrc = 'Q', then the eigenvalues of H were found using hseqr; thus if\nH has any zero sub-diagonal elements (and so is block triangular), then the\nj-th eigenvalue can be assumed to be an eigenvalue of the block containing\nthe j-th row/column. This property allows the routine to perform inverse\niteration on just one diagonal block. If eigsrc = 'N', then no such\nassumption is made and the routine performs inverse iteration using the\nwhole matrix.\ninitv\nMust be 'N' or 'U'.\nIf initv = 'N', then no initial estimates for the selected eigenvectors are\nsupplied.\nIf initv = 'U', then initial estimates for the selected eigenvectors are\nsupplied in vl and/or vr.\nselect\nArray, size at least max (1, n). Specifies which eigenvectors are to be\ncomputed.\nFor real flavors:\nTo obtain the real eigenvector corresponding to the real eigenvalue wr[j],\nset select[j] to 1\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n926\n\n\nTo select the complex eigenvector corresponding to the complex eigenvalue\n(wr[j - 1], wi[j - 1]) with complex conjugate (wr[j], wi[j]), set select[j - 1]\nand/or select[j] to 1; the eigenvector corresponding to the first eigenvalue\nin the pair is computed.\nFor complex flavors:\nTo select the eigenvector corresponding to the eigenvalue w[j], set select[j]\nto 1\nn\nThe order of the matrix H (n≥ 0).\nh, vl, vr\nArrays:\nh (size max(1, ldh*n)) The n-by-n upper Hessenberg matrix H. If an NAN\nvalue is detected in h, the routine returns with info = -6.\nvl(size max(1, ldvl*mm) for column major layout and max(1, ldvl*n) for\nrow major layout)\nIf initv = 'V' and side = 'L' or 'B', then vl must contain starting\nvectors for inverse iteration for the left eigenvectors. Each starting vector\nmust be stored in the same column or columns as will be used to store the\ncorresponding eigenvector.\nIf initv = 'N', then vl need not be set.\nThe array vl is not referenced if side = 'R'.\nvr(size max(1, ldvr*mm) for column major layout and max(1, ldvr*n) for\nrow major layout)\nIf initv = 'V' and side = 'R' or 'B', then vr must contain starting\nvectors for inverse iteration for the right eigenvectors. Each starting vector\nmust be stored in the same column or columns as will be used to store the\ncorresponding eigenvector.\nIf initv = 'N', then vr need not be set.\nThe array vr is not referenced if side = 'L'.\nldh\nThe leading dimension of h; at least max(1, n).\nw\nArray, size at least max (1, n).\nContains the eigenvalues of the matrix H.\nIf eigsrc = 'Q', the array must be exactly as returned by ?hseqr.\nwr, wi\nArrays, size at least max (1, n) each.\nContain the real and imaginary parts, respectively, of the eigenvalues of the\nmatrix H. Complex conjugate pairs of values must be stored in consecutive\nelements of the arrays. If eigsrc = 'Q', the arrays must be exactly as\nreturned by ?hseqr.\nldvl\nThe leading dimension of vl.\nIf side = 'L' or 'B', ldvl≥ max(1,n) for column major layout and ldvl≥\nmax(1, mm) for row major layout .\nIf side = 'R', ldvl≥ 1.\nldvr\nThe leading dimension of vr.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n927\n\n\nIf side = 'R' or 'B', ldvr≥ max(1,n) for column major layout and ldvr≥\nmax(1, mm) for row major layout .\nIf side = 'L', ldvr≥1.\nmm\nThe number of columns in vl and/or vr.\nMust be at least m, the actual number of columns required (see Output\nParameters below).\nFor real flavors, m is obtained by counting 1 for each selected real\neigenvector and 2 for each selected complex eigenvector (see select).\nFor complex flavors, m is the number of selected eigenvectors (see select).\nConstraint:\n0 ≤mm≤n.\nOutput Parameters\nselect\nOverwritten for real flavors only.\nIf a complex eigenvector was selected as specified above, then select[j - 1]\nis set to 1 and select[j] to 0\nw\nThe real parts of some elements of w may be modified, as close eigenvalues\nare perturbed slightly in searching for independent eigenvectors.\nwr\nSome elements of wr may be modified, as close eigenvalues are perturbed\nslightly in searching for independent eigenvectors.\nvl, vr\nIf side = 'L' or 'B', vl contains the computed left eigenvectors (as\nspecified by select).\nIf side = 'R' or 'B', vr contains the computed right eigenvectors (as\nspecified by select).\nThe eigenvectors treated column-wise form a rectangular n-by-mm matrix.\nFor real flavors: a real eigenvector corresponding to a real eigenvalue\noccupies one column of the matrix; a complex eigenvector corresponding to\na complex eigenvalue occupies two columns: the first column holds the real\npart of the eigenvector and the second column holds the imaginary part of\nthe eigenvector. The matrix is stored in a one-dimensional array as\ndescribed by matrix_layout (using either column major or row major\nlayout).\nm\nFor real flavors: the number of columns of vl and/or vr required to store the\nselected eigenvectors.\nFor complex flavors: the number of selected eigenvectors.\nifaill, ifailr\nArrays, size at least max(1, mm) each.\nifaill[i - 1] = 0 if the ith column of vl converged;\nifaill[i - 1] = j > 0 if the eigenvector stored in the i-th column of vl\n(corresponding to the jth eigenvalue) failed to converge.\nifailr[i - 1] = 0 if the ith column of vr converged;\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n928\n\n\nifailr[i - 1] = j > 0 if the eigenvector stored in the i-th column of vr\n(corresponding to the jth eigenvalue) failed to converge.\nFor real flavors: if the ith and (i+1)th columns of vl contain a selected\ncomplex eigenvector, then ifaill[i - 1] and ifaill[i] are set to the same value.\nA similar rule holds for vr and ifailr.\nThe array ifaill is not referenced if side = 'R'. The array ifailr is not\nreferenced if side = 'L'.\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nIf info > 0, then i eigenvectors (as indicated by the parameters ifaill and/or ifailr above) failed to converge.\nThe corresponding columns of vl and/or vr contain no useful information.\nApplication Notes\nEach computed right eigenvector x i is the exact eigenvector of a nearby matrix A + Ei, such that ||Ei|| <\nO(ε)||A||. Hence the residual is small:\n||Axi - λixi|| = O(ε)||A||.\nHowever, eigenvectors corresponding to close or coincident eigenvalues may not accurately span the relevant\nsubspaces.\nSimilar remarks apply to computed left eigenvectors.\n?trevc\nComputes selected eigenvectors of an upper (quasi-)\ntriangular matrix computed by ?hseqr.\nSyntax\nlapack_int LAPACKE_strevc( int matrix_layout, char side, char howmny, lapack_logical*\nselect, lapack_int n, const float* t, lapack_int ldt, float* vl, lapack_int ldvl, float*\nvr, lapack_int ldvr, lapack_int mm, lapack_int* m );\nlapack_int LAPACKE_dtrevc( int matrix_layout, char side, char howmny, lapack_logical*\nselect, lapack_int n, const double* t, lapack_int ldt, double* vl, lapack_int ldvl,\ndouble* vr, lapack_int ldvr, lapack_int mm, lapack_int* m );\nlapack_int LAPACKE_ctrevc( int matrix_layout, char side, char howmny, const\nlapack_logical* select, lapack_int n, lapack_complex_float* t, lapack_int ldt,\nlapack_complex_float* vl, lapack_int ldvl, lapack_complex_float* vr, lapack_int ldvr,\nlapack_int mm, lapack_int* m );\nlapack_int LAPACKE_ztrevc( int matrix_layout, char side, char howmny, const\nlapack_logical* select, lapack_int n, lapack_complex_double* t, lapack_int ldt,\nlapack_complex_double* vl, lapack_int ldvl, lapack_complex_double* vr, lapack_int ldvr,\nlapack_int mm, lapack_int* m );\nInclude Files\n•\nmkl.h\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n929\n\n\nDescription\nThe routine computes some or all of the right and/or left eigenvectors of an upper triangular matrix T (or, for\nreal flavors, an upper quasi-triangular matrix T). Matrices of this type are produced by the Schur\nfactorization of a general matrix: A = Q*T*QH, as computed by hseqr.\nThe right eigenvector x and the left eigenvector y of T corresponding to an eigenvalue w, are defined by:\nT*x = w*x, yH*T = w*yH, where yH denotes the conjugate transpose of y.\nThe eigenvalues are not input to this routine, but are read directly from the diagonal blocks of T.\nThis routine returns the matrices X and/or Y of right and left eigenvectors of T, or the products Q*X and/or\nQ*Y, where Q is an input matrix.\nIf Q is the orthogonal/unitary factor that reduces a matrix A to Schur form T, then Q*X and Q*Y are the\nmatrices of right and left eigenvectors of A.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nside\nMust be 'R' or 'L' or 'B'.\nIf side = 'R', then only right eigenvectors are computed.\nIf side = 'L', then only left eigenvectors are computed.\nIf side = 'B', then all eigenvectors are computed.\nhowmny\nMust be 'A' or 'B' or 'S'.\nIf howmny = 'A', then all eigenvectors (as specified by side) are\ncomputed.\nIf howmny = 'B', then all eigenvectors (as specified by side) are computed\nand backtransformed by the matrices supplied in vl and vr.\nIf howmny = 'S', then selected eigenvectors (as specified by side and\nselect) are computed.\nselect\nArray, size at least max (1, n).\nIf howmny = 'S', select specifies which eigenvectors are to be computed.\nIf howmny = 'A' or 'B', select is not referenced.\nFor real flavors:\nIf omega[j] is a real eigenvalue, the corresponding real eigenvector is\ncomputed if select[j] is 1.\nIf omega[j - 1] and omega[j] are the real and imaginary parts of a complex\neigenvalue, the corresponding complex eigenvector is computed if either\nselect[j - 1] or select[j] is 1, and on exit select[j - 1] is set to 1and select[j]\nis set to 0.\nFor complex flavors:\nThe eigenvector corresponding to the j-th eigenvalue is computed if select[j\n- 1] is 1.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n930\n\n\nn\nThe order of the matrix T (n≥ 0).\nt, vl, vr\nArrays:\nt (size max(1, ldt*n)) contains the n-by-n matrix T in Schur canonical\nform. For complex flavors ctrevc and ztrevc, contains the upper\ntriangular matrix T.\nvl(size max(1, ldvl*mm) for column major layout and max(1, ldvl*n) for\nrow major layout)\nIf howmny = 'B' and side = 'L' or 'B', then vl must contain an n-by-n\nmatrix Q (usually the matrix of Schur vectors returned by ?hseqr).\nIf howmny = 'A' or 'S', then vl need not be set.\nThe array vl is not referenced if side = 'R'.\nvr(size max(1, ldvr*mm) for column major layout and max(1, ldvr*n) for\nrow major layout)\nIf howmny = 'B' and side = 'R' or 'B', then vr must contain an n-by-n\nmatrix Q (usually the matrix of Schur vectors returned by ?hseqr). .\nIf howmny = 'A' or 'S', then vr need not be set.\nThe array vr is not referenced if side = 'L'.\nldt\nThe leading dimension of t; at least max(1, n).\nldvl\nThe leading dimension of vl.\nIf side = 'L' or 'B', ldvl≥n.\nIf side = 'R', ldvl≥ 1.\nldvr\nThe leading dimension of vr.\nIf side = 'R' or 'B', ldvr≥n.\nIf side = 'L', ldvr≥ 1.\nmm\nThe number of columns in the arrays vl and/or vr. Must be at least m (the\nprecise number of columns required).\nIf howmny = 'A' or 'B', mm = n.\nIf howmny = 'S': for real flavors, mm is obtained by counting 1 for each\nselected real eigenvector and 2 for each selected complex eigenvector;\nfor complex flavors, mm is the number of selected eigenvectors (see\nselect).\nConstraint: 0 ≤mm≤n.\nOutput Parameters\nselect\nIf a complex eigenvector of a real matrix was selected as specified above,\nthen select[j] is set to 1 and select[j + 1] to 0\nt\nctrevc/ztrevc modify the t array, which is restored on exit.\nvl, vr\nIf side = 'L' or 'B', vl contains the computed left eigenvectors (as\nspecified by howmny and select).\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n931\n\n\nIf side = 'R' or 'B', vr contains the computed right eigenvectors (as\nspecified by howmny and select).\nThe eigenvectors treated column-wise form a rectangular n-by-mm matrix.\nFor real flavors: a real eigenvector corresponding to a real eigenvalue\noccupies one column of the matrix; a complex eigenvector corresponding to\na complex eigenvalue occupies two columns: the first column holds the real\npart of the eigenvector and the second column holds the imaginary part of\nthe eigenvector. The matrix is stored in a one-dimensional array as\ndescribed by matrix_layout (using either column major or row major\nlayout).\nm\nFor complex flavors: the number of selected eigenvectors.\nIf howmny = 'A' or 'B', m is set to n.\nFor real flavors: the number of columns of vl and/or vr actually used to\nstore the selected eigenvectors.\nIf howmny = 'A' or 'B', m is set to n.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nApplication Notes\nIf xi is an exact right eigenvector and yi is the corresponding computed eigenvector, then the angle θ(yi,\nxi) between them is bounded as follows: θ(yi,xi)≤(c(n)ε||T||2)/sepi where sepi is the reciprocal\ncondition number of xi. The condition number sepi may be computed by calling ?trsna.\n?trevc3\nComputes selected eigenvectors of an upper (quasi-)\ntriangular matrix computed by ?hseqr using Level 3\nBLAS\nSyntax\ncall strevc3(side, howmny, select, n, t, ldt, vl, ldvl, vr, ldvr, mm, m, work, lwork,\ninfo)\ncall dtrevc3(side, howmny, select, n, t, ldt, vl, ldvl, vr, ldvr, mm, m, work, lwork,\ninfo)\ncall ctrevc3(side, howmny, select, n, t, ldt, vl, ldvl, vr, ldvr, mm, m, work, lwork,\nrwork, lrwork, info)\ncall ztrevc3(side, howmny, select, n, t, ldt, vl, ldvl, vr, ldvr, mm, m, work, lwork,\nrwork, lrwork, info)\nInclude Files\n•\nmkl.fi\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n932\n\n\nDescription\nThis routine computes some or all of the right and left eigenvectors of an upper triangular matrix T (or, for\nreal flavors, an upper quasi-triangular matrix T) using Level 3 BLAS. Matrices of this type are produced by\nthe Schur factorization of a general matrix: A =Q*T*QH, as computed by hseqr.\nThe right eigenvector x and the left eigenvector y of T corresponding to an eigenvalue w are defined by the\nfollowing:\nT*x = w*x, yH*T = w*yH\nwhere yH denotes the conjugate transpose of y.\nThe eigenvalues are not passed to this routine but are read directly from the diagonal blocks of T.\nThis routine returns one or both of the matrices X and Y of the right and left eigenvectors of T, or one or both\nof the products Q*X and Q*Y, where Q is an input matrix.\nIf Q is the orthogonal/unitary factor that reduces a matrix A to Schur form T, then Q*X and Q*Y are the\nmatrices of the right and left eigenvectors of A.\nInput Parameters\nside\nCHARACTER*1\nMust be 'R', 'L', or 'B'.\n•\nIf side = 'R', only right eigenvectors are computed.\n•\nIf side = 'L', only left eigenvectors are computed.\n•\nIf side = 'B', all eigenvectors are computed.\nhowmny\nCHARACTER*1\nMust be 'A', 'B', or 'S'.\n•\nIf howmny = 'A', all eigenvectors (as specified by side) are\ncomputed.\n•\nIf howmny = 'B', all eigenvectors (as specified by side) are\ncomputed and back-transformed by the matrices supplied in vl\nand vr.\n•\nIf howmny = 'S', selected eigenvectors (as specified by side and\nselect) are computed.\nselect\nArray with a size of at least max (1, n)\nIf howmny = 'S', select specifies which eigenvectors are to be\ncomputed. If howmny = 'A' or howmny = 'B', select is not\nreferenced.\nFor real flavors:\n•\nIf omega(j) is a real eigenvalue and select(j) is .TRUE., the\ncorresponding real eigenvector is computed.\n•\nIf omega(j) and omega(j + 1) are the real and imaginary parts\nof a complex eigenvalue and either select(j) or select(j + 1)\nis .TRUE., the corresponding complex eigenvector is computed,\nand on exit select(j) is set to .TRUE. and select(j + 1) is set\nto .FALSE..\nFor complex flavors:\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n933\n\n\n•\nIf select(j) is .TRUE., the eigenvector corresponding to the jth\neigenvalue is computed.\nn\nINTEGER\nThe order of the matrix T (n≥ 0).\nt, vl, vr, work\n•\nREAL for strevc3\n•\nDOUBLE PRECISION for dtrevc3\n•\nCOMPLEX for ctrevc3\n•\nDOUBLE COMPLEX for ztrevc3\nArrays:\n•\nt(ldt,*) contains the n-by-n matrix T in Schur canonical form.\nFor complex flavors ctrevc3 and ztrevc3, the array contains the\nupper triangular matrix T.\nThe second dimension of t must be at least max(1, n).\n•\nvl(ldvl,*)\nIf howmny = 'B' and side = 'L' or 'B', then vl must contain an\nn-by-n matrix Q (usually the matrix of Schur vectors returned\nby ?hseqr).\nIf howmny = 'A' or 'S', vl need not be set.\nThe second dimension of vl must be at least max(1, mm) if side\n= 'L' or 'B', and at least 1 if side = 'R'.\nThe array vl is not referenced if side = 'R'.\n•\nvr(ldvr,*)\nIf howmny = 'B' and side = 'R' or 'B', vr must contain an n-by-\nn matrix Q (usually the matrix of Schur vectors returned\nby ?hseqr).\nIf howmny = 'A' or 'S', vr need not be set.\nThe second dimension of vr must be at least max(1, mm) if side\n= 'R' or 'B', and at least 1 if side = 'L'.\nThe array vr is not referenced if side = 'L'.\n•\nwork(*) is a workspace array, and its dimension is max (1,\nlwork).\nlwork\nINTEGER\nThe size of the work array. Must be at least max(1, 3*n) for real\nflavors, and at least max(1, 2*n) for complex flavors.\nIf lwork = -1, a workspace query is assumed; the routine calculates\nonly the optimal size of the work array and returns this value as the\nfirst entry of the work array, and no error message related to lwork is\nissued by xerbla. For details, see \"Application Notes\" below.\nldt\nINTEGER\nThe leading dimension of t. It is at least max(1, n).\nldvl\nINTEGER\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n934\n\n\nThe leading dimension of vl.\n•\nIf side = 'L' or 'B', ldvl≥n.\n•\nIf side = 'R', ldvl≥ 1.\nldvr\nINTEGER\nThe leading dimension of vr.\n•\nIf side = 'R' or 'B', ldvr≥n.\n•\nIf side = 'L', ldvr≥ 1.\nmm\nINTEGER\nThe number of columns in one or both of the arrays vl and vr. Must\nbe at least m (the precise number of columns required).\n•\nIf howmny = 'A' or 'B', mm = n.\n•\nIf howmny = 'S': for real flavors, mm is obtained by counting 1 for\neach selected real eigenvector and 2 for each selected complex\neigenvector; for complex flavors, mm is the number of selected\neigenvectors (see select).\nConstraint: 0 ≤mm≤n.\nrwork\n•\nREAL for ctrevc3\n•\nDOUBLE PRECISION for ztrevc3\nThe workspace array is used in complex flavors only. Its dimensionis\nmax (1, lrwork).\nlrwork\nINTEGER\nThe size of the rwork array. It must be at least max(1, n).\nIf lrwork = -1, a workspace query is assumed; the routine calculates\nonly the optimal size of the work array and returns this value as the\nfirst entry of the rwork array, and no error message related to\nlrwork is issued by xerbla. For details, see \"Application Notes\" below.\nOutput Parameters\nselect\nIf a complex eigenvector of a real matrix was selected as specified\nabove, then select(j) is set to .TRUE. and select(j + 1) is set\nto .FALSE..\nt\nCOMPLEX for ctrevc3\nDOUBLE COMPLEX for ztrevc3\nctrevc3 or ztrevc3 modifies the t(ldt,*) array, which is restored\non exit.\nvl, vr\nIf side = 'L' or 'B', vl contains the computed left eigenvectors (as\nspecified by howmny and select).\nIf side = 'R' or 'B', vr contains the computed right eigenvectors\n(as specified by howmny and select).\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n935\n\n\nTreated column-wise, the eigenvectors form a rectangular n-by-mm\nmatrix.\nFor real flavors A real eigenvector corresponding to a real\neigenvalue occupies one column of the matrix; a complex\neigenvector corresponding to a complex eigenvalue\noccupies two columns. The first column holds the real part\nof the eigenvector, and the second column holds the\nimaginary part of the eigenvector. The matrix is stored in a\none-dimensional array as described by matrix_layout\n(using either column major or row major layout).\nm\nINTEGER\nFor complex flavors The number of selected\neigenvectors. If howmny = 'A' or 'B', m is set to n.\nFor real flavors The number of columns of one or both of\nvl and vr actually used to store the selected eigenvectors.\nIf howmny = 'A' or 'B', m is set to n.\nwork(1)\nOn exit, if info = 0, work(1) returns the required optimal size of\nlwork.\nrwork(1)\nOn exit, if info = 0, then rwork(1) returns the required optimal size\nof lrwork.\ninfo\nINTEGER\nIf info = 0, the execution is successful.\nIf info = -i, the ith parameter contained an illegal value.\nApplication Notes\nIf xi is an exact right eigenvector and yi is the corresponding computed eigenvector, the angle θ(yi, xi)\nbetween them is bounded as follows:\nθ(yi,xi)≤(c(n)ε||T||2)/sepi\nwhere sepi is the reciprocal condition number of xi. You can compute the condition number sepi by\ncalling ?trsna.\nSee Also\nMatrix Storage Schemes\n?trsna\nEstimates condition numbers for specified eigenvalues\nand right eigenvectors of an upper (quasi-) triangular\nmatrix.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n936\n\n\nSyntax\nlapack_int LAPACKE_strsna( int matrix_layout, char job, char howmny, const\nlapack_logical* select, lapack_int n, const float* t, lapack_int ldt, const float* vl,\nlapack_int ldvl, const float* vr, lapack_int ldvr, float* s, float* sep, lapack_int mm,\nlapack_int* m );\nlapack_int LAPACKE_dtrsna( int matrix_layout, char job, char howmny, const\nlapack_logical* select, lapack_int n, const double* t, lapack_int ldt, const double*\nvl, lapack_int ldvl, const double* vr, lapack_int ldvr, double* s, double* sep,\nlapack_int mm, lapack_int* m );\nlapack_int LAPACKE_ctrsna( int matrix_layout, char job, char howmny, const\nlapack_logical* select, lapack_int n, const lapack_complex_float* t, lapack_int ldt,\nconst lapack_complex_float* vl, lapack_int ldvl, const lapack_complex_float* vr,\nlapack_int ldvr, float* s, float* sep, lapack_int mm, lapack_int* m );\nlapack_int LAPACKE_ztrsna( int matrix_layout, char job, char howmny, const\nlapack_logical* select, lapack_int n, const lapack_complex_double* t, lapack_int ldt,\nconst lapack_complex_double* vl, lapack_int ldvl, const lapack_complex_double* vr,\nlapack_int ldvr, double* s, double* sep, lapack_int mm, lapack_int* m );\nInclude Files\n•\nmkl.h\nDescription\nThe routine estimates condition numbers for specified eigenvalues and/or right eigenvectors of an upper\ntriangular matrix T (or, for real flavors, upper quasi-triangular matrix T in canonical Schur form). These are\nthe same as the condition numbers of the eigenvalues and right eigenvectors of an original matrix A =\nZ*T*ZH (with unitary or, for real flavors, orthogonal Z), from which T may have been derived.\nThe routine computes the reciprocal of the condition number of an eigenvalue λi as si = |vT*u|/(||u||E||\nv||E) for real flavors and si = |vH*u|/(||u||E||v||E) for complex flavors,\nwhere:\n•\nu and v are the right and left eigenvectors of T, respectively, corresponding to λi.\n•\nvT/vH denote transpose/conjugate transpose of v, respectively.\nThis reciprocal condition number always lies between zero (ill-conditioned) and one (well-conditioned).\nAn approximate error estimate for a computed eigenvalue λi is then given by ε*||T||/si, where ε is the\nmachine precision.\nTo estimate the reciprocal of the condition number of the right eigenvector corresponding to λi, the routine\nfirst calls trexc to reorder the diagonal elements of matrix T so that λi is in the leading position:\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n937\n\n\nThe reciprocal condition number of the eigenvector is then estimated as sepi, the smallest singular value of\nthe matrix T22 - λi*I.\nAn approximate error estimate for a computed right eigenvector u corresponding to λi is then given by ε*||\nT||/sepi.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\njob\nMust be 'E' or 'V' or 'B'.\nIf job = 'E', then condition numbers for eigenvalues only are computed.\nIf job = 'V', then condition numbers for eigenvectors only are computed.\nIf job = 'B', then condition numbers for both eigenvalues and\neigenvectors are computed.\nhowmny\nMust be 'A' or 'S'.\nIf howmny = 'A', then the condition numbers for all eigenpairs are\ncomputed.\nIf howmny = 'S', then condition numbers for selected eigenpairs (as\nspecified by select) are computed.\nselect\nArray, size at least max (1, n) if howmny = 'S' and at least 1 otherwise.\nSpecifies the eigenpairs for which condition numbers are to be computed if\nhowmny= 'S'.\nFor real flavors:\nTo select condition numbers for the eigenpair corresponding to the real\neigenvalue λj, select[j] must be set 1;\nto select condition numbers for the eigenpair corresponding to a complex\nconjugate pair of eigenvalues λj and λj + 1), select[j - 1] and/or select[j]\nmust be set 1\nFor complex flavors\nTo select condition numbers for the eigenpair corresponding to the\neigenvalue λj, select[j] must be set 1select is not referenced if howmny =\n'A'.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n938\n\n\nn\nThe order of the matrix T (n≥ 0).\nt, vl, vr\nArrays:\nt (size max(1, ldt*n)) contains the n-by-n matrix T.\nvl(size max(1, ldvl*mm) for column major layout and max(1, ldvl*n) for\nrow major layout)\nIf job = 'E' or 'B', then vl must contain the left eigenvectors of T (or of\nany matrix Q*T*QH with Q unitary or orthogonal) corresponding to the\neigenpairs specified by howmny and select. The eigenvectors must be\nstored in consecutive columns of vl, as returned by trevc or hsein.\nThe array vl is not referenced if job = 'V'.\nvr(size max(1, ldvr*mm) for column major layout and max(1, ldvr*n) for\nrow major layout)\nIf job = 'E' or 'B', then vr must contain the right eigenvectors of T (or of\nany matrix Q*T*QH with Q unitary or orthogonal) corresponding to the\neigenpairs specified by howmny and select. The eigenvectors must be\nstored in consecutive columns of vr, as returned by trevc or hsein.\nThe array vr is not referenced if job = 'V'.\nldt\nThe leading dimension of t; at least max(1, n).\nldvl\nThe leading dimension of vl.\nIf job = 'E' or 'B', ldvl≥ max(1,n) for column major layout and ldvl≥\nmax(1, mm) for row major layout .\nIf job = 'V', ldvl≥ 1.\nldvr\nThe leading dimension of vr.\nIf job = 'E' or 'B', ldvr≥ max(1,n) for column major layout and ldvr≥\nmax(1, mm) for row major layout .\nIf job = 'R', ldvr≥ 1.\nmm\nThe number of elements in the arrays s and sep, and the number of\ncolumns in vl and vr (if used). Must be at least m (the precise number\nrequired).\nIf howmny = 'A', mm = n;\nif howmny = 'S', for real flavorsmm is obtained by counting 1 for each\nselected real eigenvalue and 2 for each selected complex conjugate pair of\neigenvalues.\nfor complex flavorsmm is the number of selected eigenpairs (see select).\nConstraint:\n0 ≤mm≤n.\nOutput Parameters\ns\nArray, size at least max(1, mm) if job = 'E' or 'B' and at least 1 if job =\n'V'.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n939\n\n\nContains the reciprocal condition numbers of the selected eigenvalues if job\n= 'E' or 'B', stored in consecutive elements of the array. Thus s[j - 1],\nsep[j - 1] and the j-th columns of vl and vr all correspond to the same\neigenpair (but not in general the j th eigenpair unless all eigenpairs have\nbeen selected).\nFor real flavors: for a complex conjugate pair of eigenvalues, two\nconsecutive elements of s are set to the same value. The array s is not\nreferenced if job = 'V'.\nsep\nArray, size at least max(1, mm) if job = 'V' or 'B' and at least 1 if job =\n'E'. Contains the estimated reciprocal condition numbers of the selected\nright eigenvectors if job = 'V' or 'B', stored in consecutive elements of\nthe array.\nFor real flavors: for a complex eigenvector, two consecutive elements of sep\nare set to the same value; if the eigenvalues cannot be reordered to\ncompute sep[j - 1], then sep[j - 1] is set to zero; this can only occur when\nthe true value would be very small anyway. The array sep is not referenced\nif job = 'E'.\nm\nFor complex flavors: the number of selected eigenpairs.\nIf howmny = 'A', m is set to n.\nFor real flavors: the number of elements of s and/or sep actually used to\nstore the estimated condition numbers.\nIf howmny = 'A', m is set to n.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nApplication Notes\nThe computed values sepi may overestimate the true value, but seldom by a factor of more than 3.\n?trexc\nReorders the Schur factorization of a general matrix.\nSyntax\nlapack_int LAPACKE_strexc( int matrix_layout, char compq, lapack_int n, float* t,\nlapack_int ldt, float* q, lapack_int ldq, lapack_int* ifst, lapack_int* ilst );\nlapack_int LAPACKE_dtrexc( int matrix_layout, char compq, lapack_int n, double* t,\nlapack_int ldt, double* q, lapack_int ldq, lapack_int* ifst, lapack_int* ilst );\nlapack_int LAPACKE_ctrexc( int matrix_layout, char compq, lapack_int n,\nlapack_complex_float* t, lapack_int ldt, lapack_complex_float* q, lapack_int ldq,\nlapack_int ifst, lapack_int ilst );\nlapack_int LAPACKE_ztrexc( int matrix_layout, char compq, lapack_int n,\nlapack_complex_double* t, lapack_int ldt, lapack_complex_double* q, lapack_int ldq,\nlapack_int ifst, lapack_int ilst );\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n940\n\n\nInclude Files\n•\nmkl.h\nDescription\nThe routine reorders the Schur factorization of a general matrix A = Q*T*QH, so that the diagonal element or\nblock of T with row index ifst is moved to row ilst.\nThe reordered Schur form S is computed by an unitary (or, for real flavors, orthogonal) similarity\ntransformation: S = ZH*T*Z. Optionally the updated matrix P of Schur vectors is computed as P = Q*Z,\ngiving A = P*S*PH.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\ncompq\nMust be 'V' or 'N'.\nIf compq = 'V', then the Schur vectors (Q) are updated.\nIf compq = 'N', then no Schur vectors are updated.\nn\nThe order of the matrix T (n≥ 0).\nt, q\nArrays:\nt (size max(1, ldt*n)) contains the n-by-n matrix T.\nq (size max(1, ldq*n))\nIf compq = 'V', then q must contain Q (Schur vectors).\nIf compq = 'N', then q is not referenced.\nldt\nThe leading dimension of t; at least max(1, n).\nldq\nThe leading dimension of q;\nIf compq = 'N', then ldq≥ 1.\nIf compq = 'V', then ldq≥ max(1, n).\nifst, ilst\n1 ≤ifst≤n; 1 ≤ilst≤n.\nMust specify the reordering of the diagonal elements (or blocks, which is\npossible for real flavors) of the matrix T. The element (or block) with row\nindex ifst is moved to row ilst by a sequence of exchanges between\nadjacent elements (or blocks).\nOutput Parameters\nt\nOverwritten by the updated matrix S.\nq\nIf compq = 'V', q contains the updated matrix of Schur vectors.\nifst, ilst\nOverwritten for real flavors only.\nIf ifst pointed to the second row of a 2 by 2 block on entry, it is changed to\npoint to the first row; ilst always points to the first row of the block in its\nfinal position (which may differ from its input value by ±1).\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n941\n\n\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nApplication Notes\nThe computed matrix S is exactly similar to a matrix T+E, where ||E||2 = O(ε)*||T||2, and ε is the\nmachine precision.\nNote that if a 2 by 2 diagonal block is involved in the re-ordering, its off-diagonal elements are in general\nchanged; the diagonal elements and the eigenvalues of the block are unchanged unless the block is\nsufficiently ill-conditioned, in which case they may be noticeably altered. It is possible for a 2 by 2 block to\nbreak into two 1 by 1 blocks, that is, for a pair of complex eigenvalues to become purely real.\nThe approximate number of floating-point operations is\nfor real flavors:\n6n(ifst-ilst) if compq = 'N';\n12n(ifst-ilst) if compq = 'V';\nfor complex flavors:\n20n(ifst-ilst) if compq = 'N';\n40n(ifst-ilst) if compq = 'V'.\n?trsen\nReorders the Schur factorization of a matrix and\n(optionally) computes the reciprocal condition\nnumbers for the selected cluster of eigenvalues and\nrespective invariant subspace.\nSyntax\nlapack_int LAPACKE_strsen( int matrix_layout, char job, char compq, const\nlapack_logical* select, lapack_int n, float* t, lapack_int ldt, float* q, lapack_int\nldq, float* wr, float* wi, lapack_int* m, float* s, float* sep );\nlapack_int LAPACKE_dtrsen( int matrix_layout, char job, char compq, const\nlapack_logical* select, lapack_int n, double* t, lapack_int ldt, double* q, lapack_int\nldq, double* wr, double* wi, lapack_int* m, double* s, double* sep );\nlapack_int LAPACKE_ctrsen( int matrix_layout, char job, char compq, const\nlapack_logical* select, lapack_int n, lapack_complex_float* t, lapack_int ldt,\nlapack_complex_float* q, lapack_int ldq, lapack_complex_float* w, lapack_int* m, float*\ns, float* sep );\nlapack_int LAPACKE_ztrsen( int matrix_layout, char job, char compq, const\nlapack_logical* select, lapack_int n, lapack_complex_double* t, lapack_int ldt,\nlapack_complex_double* q, lapack_int ldq, lapack_complex_double* w, lapack_int* m,\ndouble* s, double* sep );\nInclude Files\n•\nmkl.h\nDescription\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n942\n\n\nThe routine reorders the Schur factorization of a general matrix A = Q*T*QT (for real flavors) or A = Q*T*QH\n(for complex flavors) so that a selected cluster of eigenvalues appears in the leading diagonal elements (or,\nfor real flavors, diagonal blocks) of the Schur form. The reordered Schur form R is computed by a unitary\n(orthogonal) similarity transformation: R = ZH*T*Z. Optionally the updated matrix P of Schur vectors is\ncomputed as P = Q*Z, giving A = P*R*PH.\nLet\nwhere the selected eigenvalues are precisely the eigenvalues of the leading m-by-m submatrix T11. Let P be\ncorrespondingly partitioned as (Q1Q2) where Q1 consists of the first m columns of Q. Then A*Q1 = Q1*T11,\nand so the m columns of Q1 form an orthonormal basis for the invariant subspace corresponding to the\nselected cluster of eigenvalues.\nOptionally the routine also computes estimates of the reciprocal condition numbers of the average of the\ncluster of eigenvalues and of the invariant subspace.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\njob\nMust be 'N' or 'E' or 'V' or 'B'.\nIf job = 'N', then no condition numbers are required.\nIf job = 'E', then only the condition number for the cluster of eigenvalues\nis computed.\nIf job = 'V', then only the condition number for the invariant subspace is\ncomputed.\nIf job = 'B', then condition numbers for both the cluster and the invariant\nsubspace are computed.\ncompq\nMust be 'V' or 'N'.\nIf compq = 'V', then Q of the Schur vectors is updated.\nIf compq = 'N', then no Schur vectors are updated.\nselect\nArray, size at least max (1, n).\nSpecifies the eigenvalues in the selected cluster. To select an eigenvalue λj,\nselect[j] must be 1\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n943\n\n\nFor real flavors: to select a complex conjugate pair of eigenvalues λj and λj\n+1 (corresponding 2 by 2 diagonal block), select[j - 1] and/or select[j] must\nbe 1; the complex conjugate λjand λj + 1 must be either both included in the\ncluster or both excluded.\nn\nThe order of the matrix T (n≥ 0).\nt, q\nArrays:\nt (size max(1, ldt*n)) Theupper quasi-triangular n-by-n matrix T, in Schur\ncanonical form.\nq (size max(1, ldq*n))\nIf compq = 'V', then q must contain the matrix Q of Schur vectors.\nIf compq = 'N', then q is not referenced.\nldt\nThe leading dimension of t; at least max(1, n).\nldq\nThe leading dimension of q;\nIf compq = 'N', then ldq≥ 1.\nIf compq = 'V', then ldq≥ max(1, n).\nOutput Parameters\nt\nOverwritten by the reordered matrix R in Schur canonical form with the\nselected eigenvalues in the leading diagonal blocks.\nq\nIf compq = 'V', q contains the updated matrix of Schur vectors; the first\nm columns of the Q form an orthogonal basis for the specified invariant\nsubspace.\nw\nArray, size at least max(1, n). The recorded eigenvalues of R. The\neigenvalues are stored in the same order as on the diagonal of R.\nwr, wi\nArrays, size at least max(1, n). Contain the real and imaginary parts,\nrespectively, of the reordered eigenvalues of R. The eigenvalues are stored\nin the same order as on the diagonal of R. Note that if a complex\neigenvalue is sufficiently ill-conditioned, then its value may differ\nsignificantly from its value before reordering.\nm\nFor complex flavors: the dimension of the specified invariant subspaces,\nwhich is the same as the number of selected eigenvalues (see select).\nFor real flavors: the dimension of the specified invariant subspace. The\nvalue of m is obtained by counting 1 for each selected real eigenvalue and 2\nfor each selected complex conjugate pair of eigenvalues (see select).\nConstraint: 0 ≤m≤n.\ns\nIf job = 'E' or 'B', s is a lower bound on the reciprocal condition number\nof the average of the selected cluster of eigenvalues.\nIf m = 0 or n, then s = 1.\nFor real flavors: if info = 1, then s is set to zero.s is not referenced if job\n= 'N' or 'V'.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n944\n\n\nsep\nIf job = 'V' or 'B', sep is the estimated reciprocal condition number of\nthe specified invariant subspace.\nIf m = 0 or n, then sep = |T|.\nFor real flavors: if info = 1, then sep is set to zero.\nsep is not referenced if job = 'N' or 'E'.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nIf info = 1, the reordering of T failed because some eigenvalues are too close to separate (the problem is\nvery ill-conditioned); T may have been partially reordered, and wr and wi contain the eigenvalues in the\nsame order as in T; s and sep (if requested) are set to zero.\nApplication Notes\nThe computed matrix R is exactly similar to a matrix T+E, where ||E||2 = O(ε)*||T||2, and ε is the\nmachine precision. The computed s cannot underestimate the true reciprocal condition number by more than\na factor of (min(m, n-m))1/2; sep may differ from the true value by (m*n-m2)1/2. The angle between the\ncomputed invariant subspace and the true subspace is O(ε)*||A||2/sep. Note that if a 2-by-2 diagonal\nblock is involved in the re-ordering, its off-diagonal elements are in general changed; the diagonal elements\nand the eigenvalues of the block are unchanged unless the block is sufficiently ill-conditioned, in which case\nthey may be noticeably altered. It is possible for a 2-by-2 block to break into two 1-by-1 blocks, that is, for a\npair of complex eigenvalues to become purely real.\n?trsyl\nSolves Sylvester equation for real quasi-triangular or\ncomplex triangular matrices.\nSyntax\nlapack_int LAPACKE_strsyl( int matrix_layout, char trana, char tranb, lapack_int isgn,\nlapack_int m, lapack_int n, const float* a, lapack_int lda, const float* b, lapack_int\nldb, float* c, lapack_int ldc, float* scale );\nlapack_int LAPACKE_dtrsyl( int matrix_layout, char trana, char tranb, lapack_int isgn,\nlapack_int m, lapack_int n, const double* a, lapack_int lda, const double* b,\nlapack_int ldb, double* c, lapack_int ldc, double* scale );\nlapack_int LAPACKE_ctrsyl( int matrix_layout, char trana, char tranb, lapack_int isgn,\nlapack_int m, lapack_int n, const lapack_complex_float* a, lapack_int lda, const\nlapack_complex_float* b, lapack_int ldb, lapack_complex_float* c, lapack_int ldc,\nfloat* scale );\nlapack_int LAPACKE_ztrsyl( int matrix_layout, char trana, char tranb, lapack_int isgn,\nlapack_int m, lapack_int n, const lapack_complex_double* a, lapack_int lda, const\nlapack_complex_double* b, lapack_int ldb, lapack_complex_double* c, lapack_int ldc,\ndouble* scale );\nInclude Files\n•\nmkl.h\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n945\n\n\nDescription\nThe routine solves the Sylvester matrix equation op(A)*X±X*op(B) = α*C, where op(A) = A or AH, and the\nmatrices A and B are upper triangular (or, for real flavors, upper quasi-triangular in canonical Schur form); α≤\n1 is a scale factor determined by the routine to avoid overflow in X; A is m-by-m, B is n-by-n, and C and X\nare both m-by-n. The matrix X is obtained by a straightforward process of back substitution.\nThe equation has a unique solution if and only if αi±βi≠ 0, where {αi} and {βi} are the eigenvalues of A and\nB, respectively, and the sign (+ or -) is the same as that used in the equation to be solved.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\ntrana\nMust be 'N' or 'T' or 'C'.\nIf trana = 'N', then op(A) = A.\nIf trana = 'T', then op(A) = AT (real flavors only).\nIf trana = 'C' then op(A) = AH.\ntranb\nMust be 'N' or 'T' or 'C'.\nIf tranb = 'N', then op(B) = B.\nIf tranb = 'T', then op(B) = BT (real flavors only).\nIf tranb = 'C', then op(B) = BH.\nisgn\nIndicates the form of the Sylvester equation.\nIf isgn = +1, op(A)*X + X*op(B) = alpha*C.\nIf isgn = -1, op(A)*X - X*op(B) = alpha*C.\nm\nThe order of A, and the number of rows in X and C (m≥ 0).\nn\nThe order of B, and the number of columns in X and C (n≥ 0).\na, b, c\nArrays:\na (size max(1, lda*m)) contains the matrix A.\nb (size max(1, ldb*n)) contains the matrix B.\nc(size max(1, ldc*n) for column major layout and max(1, ldc*m for row\nmajor layout) contains the matrix C.\nlda\nThe leading dimension of a; at least max(1, m)for column major layout and\nmax(1, n) for row major layout.\nldb\nThe leading dimension of b; at least max(1, n).\nldc\nThe leading dimension of c; at least max(1, m) for column major layout and\nat least max(1, n) for row major layout .\nOutput Parameters\nc\nOverwritten by the solution matrix X.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n946\n\n\nscale\nThe value of the scale factor α.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nIf info = 1, A and B have common or close eigenvalues; perturbed values were used to solve the equation.\nApplication Notes\nLet X be the exact, Y the corresponding computed solution, and R the residual matrix: R = C - (AY±YB).\nThen the residual is always small:\n||R||F = O(ε)*(||A||F +||B||F)*||Y||F.\nHowever, Y is not necessarily the exact solution of a slightly perturbed equation; in other words, the solution\nis not backwards stable.\nFor the forward error, the following bound holds:\n||Y - X||F≤||R||F/sep(A,B)\nbut this may be a considerable overestimate. See [Golub96] for a definition of sep(A, B).\nThe approximate number of floating-point operations for real flavors is m*n*(m + n). For complex flavors it\nis 4 times greater.\nGeneralized Nonsymmetric Eigenvalue Problems: LAPACK Computational Routines\nThis topic describes LAPACK routines for solving generalized nonsymmetric eigenvalue problems, reordering\nthe generalized Schur factorization of a pair of matrices, as well as performing a number of related\ncomputational tasks.\nA generalized nonsymmetric eigenvalue problem is as follows: given a pair of nonsymmetric (or non-\nHermitian) n-by-n matrices A and B, find the generalized eigenvaluesλ and the corresponding generalized\neigenvectorsx and y that satisfy the equations\nAx = λBx (right generalized eigenvectors x)\nand\nyHA = λyHB (left generalized eigenvectors y).\nTable \"Computational Routines for Solving Generalized Nonsymmetric Eigenvalue Problems\" lists LAPACK\nroutines used to solve the generalized nonsymmetric eigenvalue problems and the generalized Sylvester\nequation.\nComputational Routines for Solving Generalized Nonsymmetric Eigenvalue Problems\nRoutine\nname\nOperation performed\ngghrd\nReduces a pair of matrices to generalized upper Hessenberg form using orthogonal/\nunitary transformations.\nggbal\nBalances a pair of general real or complex matrices.\nggbak\nForms the right or left eigenvectors of a generalized eigenvalue problem.\ngghd3\nReduces a pair of matrices to generalized upper Hessenberg form.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n947\n\n\nRoutine\nname\nOperation performed\nhgeqz\nImplements the QZ method for finding the generalized eigenvalues of the matrix pair\n(H,T).\ntgevc\nComputes some or all of the right and/or left generalized eigenvectors of a pair of upper\ntriangular matrices\ntgexc\nReorders the generalized Schur decomposition of a pair of matrices (A,B) so that one\ndiagonal block of (A,B) moves to another row index.\ntgsen\nReorders the generalized Schur decomposition of a pair of matrices (A,B) so that a\nselected cluster of eigenvalues appears in the leading diagonal blocks of (A,B).\ntgsyl\nSolves the generalized Sylvester equation.\ntgsyl\nEstimates reciprocal condition numbers for specified eigenvalues and/or eigenvectors of a\npair of matrices in generalized real Schur canonical form.\n?gghrd\nReduces a pair of matrices to generalized upper\nHessenberg form using orthogonal/unitary\ntransformations.\nSyntax\nlapack_int LAPACKE_sgghrd (int matrix_layout, char compq, char compz, lapack_int n,\nlapack_int ilo, lapack_int ihi, float* a, lapack_int lda, float* b, lapack_int ldb,\nfloat* q, lapack_int ldq, float* z, lapack_int ldz);\nlapack_int LAPACKE_dgghrd (int matrix_layout, char compq, char compz, lapack_int n,\nlapack_int ilo, lapack_int ihi, double* a, lapack_int lda, double* b, lapack_int ldb,\ndouble* q, lapack_int ldq, double* z, lapack_int ldz);\nlapack_int LAPACKE_cgghrd (int matrix_layout, char compq, char compz, lapack_int n,\nlapack_int ilo, lapack_int ihi, lapack_complex_float* a, lapack_int lda,\nlapack_complex_float* b, lapack_int ldb, lapack_complex_float* q, lapack_int ldq,\nlapack_complex_float* z, lapack_int ldz);\nlapack_int LAPACKE_zgghrd (int matrix_layout, char compq, char compz, lapack_int n,\nlapack_int ilo, lapack_int ihi, lapack_complex_double* a, lapack_int lda,\nlapack_complex_double* b, lapack_int ldb, lapack_complex_double* q, lapack_int ldq,\nlapack_complex_double* z, lapack_int ldz);\nInclude Files\n•\nmkl.h\nDescription\nThe routine reduces a pair of real/complex matrices (A,B) to generalized upper Hessenberg form using\northogonal/unitary transformations, where A is a general matrix and B is upper triangular. The form of the\ngeneralized eigenvalue problem is A*x = λ*B*x, and B is typically made upper triangular by computing its\nQR factorization and moving the orthogonal matrix Q to the left side of the equation.\nThis routine simultaneously reduces A to a Hessenberg matrix H:\nQH*A*Z = H\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n948\n\n\nand transforms B to another upper triangular matrix T:\nQH*B*Z = T\nin order to reduce the problem to its standard form H*y = λ*T*y, where y = ZH*x.\nThe orthogonal/unitary matrices Q and Z are determined as products of Givens rotations. They may either be\nformed explicitly, or they may be postmultiplied into input matrices Q1 and Z1, so that\nQ1*A*Z1H = (Q1*Q)*H*(Z1*Z)H\nQ1*B*Z1H = (Q1*Q)*T*(Z1*Z)H\nIf Q1 is the orthogonal/unitary matrix from the QR factorization of B in the original equation A*x = λ*B*x,\nthen the routine ?gghrd reduces the original problem to generalized Hessenberg form.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\ncompq\nMust be 'N', 'I', or 'V'.\nIf compq = 'N', matrix Q is not computed.\nIf compq = 'I', Q is initialized to the unit matrix, and the orthogonal/\nunitary matrix Q is returned;\nIf compq = 'V', Q must contain an orthogonal/unitary matrix Q1 on entry,\nand the product Q1*Q is returned.\ncompz\nMust be 'N', 'I', or 'V'.\nIf compz = 'N', matrix Z is not computed.\nIf compz = 'I', Z is initialized to the unit matrix, and the orthogonal/\nunitary matrix Z is returned;\nIf compz = 'V', Z must contain an orthogonal/unitary matrix Z1 on entry,\nand the product Z1*Z is returned.\nn\nThe order of the matrices A and B (n≥ 0).\nilo, ihi\nilo and ihi mark the rows and columns of A which are to be reduced. It is\nassumed that A is already upper triangular in rows and columns 1:ilo-1 and\nihi+1:n. Values of ilo and ihi are normally set by a previous call to ggbal;\notherwise they should be set to 1 and n respectively.\nConstraint:\nIf n > 0, then 1 ≤ilo≤ihi≤n;\nif n = 0, then ilo = 1 and ihi = 0.\na, b, q, z\nArrays:\na (size max(1, lda*n)) contains the n-by-n general matrix A.\nb (size max(1, ldb*n)) contains the n-by-n upper triangular matrix B.\nq (size max(1, ldq*n))\nIf compq = 'N', then q is not referenced.\nIf compq = 'V', then q must contain the orthogonal/unitary matrix Q1,\ntypically from the QR factorization of B.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n949\n\n\nz (size max(1, ldz*n))\nIf compz = 'N', then z is not referenced.\nIf compz = 'V', then z must contain the orthogonal/unitary matrix Z1.\nlda\nThe leading dimension of a; at least max(1, n).\nldb\nThe leading dimension of b; at least max(1, n).\nldq\nThe leading dimension of q;\nIf compq = 'N', then ldq≥ 1.\nIf compq = 'I'or 'V', then ldq≥ max(1, n).\nldz\nThe leading dimension of z;\nIf compz = 'N', then ldz≥ 1.\nIf compz = 'I'or 'V', then ldz≥ max(1, n).\nOutput Parameters\na\nOn exit, the upper triangle and the first subdiagonal of A are overwritten\nwith the upper Hessenberg matrix H, and the rest is set to zero.\nb\nOn exit, overwritten by the upper triangular matrix T = QH*B*Z. The\nelements below the diagonal are set to zero.\nq\nIf compq = 'I', then q contains the orthogonal/unitary matrix Q, ;\nIf compq = 'V', then q is overwritten by the product Q1*Q.\nz\nIf compz = 'I', then z contains the orthogonal/unitary matrix Z;\nIf compz = 'V', then z is overwritten by the product Z1*Z.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\n?ggbal\nBalances a pair of general real or complex matrices.\nSyntax\nlapack_int LAPACKE_sggbal( int matrix_layout, char job, lapack_int n, float* a,\nlapack_int lda, float* b, lapack_int ldb, lapack_int* ilo, lapack_int* ihi, float*\nlscale, float* rscale );\nlapack_int LAPACKE_dggbal( int matrix_layout, char job, lapack_int n, double* a,\nlapack_int lda, double* b, lapack_int ldb, lapack_int* ilo, lapack_int* ihi, double*\nlscale, double* rscale );\nlapack_int LAPACKE_cggbal( int matrix_layout, char job, lapack_int n,\nlapack_complex_float* a, lapack_int lda, lapack_complex_float* b, lapack_int ldb,\nlapack_int* ilo, lapack_int* ihi, float* lscale, float* rscale );\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n950\n\n\nlapack_int LAPACKE_zggbal( int matrix_layout, char job, lapack_int n,\nlapack_complex_double* a, lapack_int lda, lapack_complex_double* b, lapack_int ldb,\nlapack_int* ilo, lapack_int* ihi, double* lscale, double* rscale );\nInclude Files\n•\nmkl.h\nDescription\nThe routine balances a pair of general real/complex matrices (A,B). This involves, first, permuting A and B by\nsimilarity transformations to isolate eigenvalues in the first 1 to ilo-1 and last ihi+1 to n elements on the\ndiagonal;and second, applying a diagonal similarity transformation to rows and columns ilo to ihi to make the\nrows and columns as close in norm as possible. Both steps are optional. Balancing may reduce the 1-norm of\nthe matrices, and improve the accuracy of the computed eigenvalues and/or eigenvectors in the generalized\neigenvalue problem A*x = λ*B*x.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\njob\nSpecifies the operations to be performed on A and B. Must be 'N' or 'P' or\n'S' or 'B'.\nIf job = 'N ', then no operations are done; simply set ilo =1, ihi=n,\nlscale[i] =1.0 and rscale[i]=1.0 for\ni = 0,..., n - 1.\nIf job = 'P', then permute only.\nIf job = 'S', then scale only.\nIf job = 'B', then both permute and scale.\nn\nThe order of the matrices A and B (n≥ 0).\na, b\nArrays:\na (size max(1, lda*n)) contains the matrix A.\nb (size max(1, ldb*n)) contains the matrix B.\nIf job = 'N', a and b are not referenced.\nlda\nThe leading dimension of a; at least max(1, n).\nldb\nThe leading dimension of b; at least max(1, n).\nOutput Parameters\na, b\nOverwritten by the balanced matrices A and B, respectively.\nilo, ihi\nilo and ihi are set to integers such that on exit Ai, j = 0 and Bi, j = 0 if i>j\nand j=1,...,ilo-1 or i=ihi+1,..., n.\nIf job = 'N'or 'S', then ilo = 1 and ihi = n.\nlscale, rscale\nArrays, size at least max(1, n).\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n951\n\n\nlscale contains details of the permutations and scaling factors applied to the\nleft side of A and B.\nIf Pj is the index of the row interchanged with row j, and Dj is the scaling\nfactor applied to row j, then\nlscale[j - 1] = Pj, for j = 1,..., ilo-1\n= Dj, for j = ilo,...,ihi\n= Pj, for j = ihi+1,..., n.\nrscale contains details of the permutations and scaling factors applied to the\nright side of A and B.\nIf Pj is the index of the column interchanged with column j, and Dj is the\nscaling factor applied to column j, then\nrscale[j - 1] = Pj, for j = 1,..., ilo-1\n= Dj, for j = ilo,...,ihi\n= Pj, for j = ihi+1,..., n\nThe order in which the interchanges are made is n to ihi+1, then 1 to ilo-1.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\n?ggbak\nForms the right or left eigenvectors of a generalized\neigenvalue problem.\nSyntax\nlapack_int LAPACKE_sggbak( int matrix_layout, char job, char side, lapack_int n,\nlapack_int ilo, lapack_int ihi, const float* lscale, const float* rscale, lapack_int m,\nfloat* v, lapack_int ldv );\nlapack_int LAPACKE_dggbak( int matrix_layout, char job, char side, lapack_int n,\nlapack_int ilo, lapack_int ihi, const double* lscale, const double* rscale, lapack_int\nm, double* v, lapack_int ldv );\nlapack_int LAPACKE_cggbak( int matrix_layout, char job, char side, lapack_int n,\nlapack_int ilo, lapack_int ihi, const float* lscale, const float* rscale, lapack_int m,\nlapack_complex_float* v, lapack_int ldv );\nlapack_int LAPACKE_zggbak( int matrix_layout, char job, char side, lapack_int n,\nlapack_int ilo, lapack_int ihi, const double* lscale, const double* rscale, lapack_int\nm, lapack_complex_double* v, lapack_int ldv );\nInclude Files\n•\nmkl.h\nDescription\nThe routine forms the right or left eigenvectors of a real/complex generalized eigenvalue problem\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n952\n\n\nA*x = λ*B*x\nby backward transformation on the computed eigenvectors of the balanced pair of matrices output by ggbal.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\njob\nSpecifies the type of backward transformation required. Must be 'N', 'P',\n'S', or 'B'.\nIf job = 'N', then no operations are done; return.\nIf job = 'P', then do backward transformation for permutation only.\nIf job = 'S', then do backward transformation for scaling only.\nIf job = 'B', then do backward transformation for both permutation and\nscaling. This argument must be the same as the argument job supplied\nto ?ggbal.\nside\nMust be 'L' or 'R'.\nIf side = 'L', then v contains left eigenvectors.\nIf side = 'R', then v contains right eigenvectors.\nn\nThe number of rows of the matrix V (n≥ 0).\nilo, ihi\nThe integers ilo and ihi determined by ?gebal. Constraint:\nIf n > 0, then 1 ≤ilo≤ihi≤n;\nif n = 0, then ilo = 1 and ihi = 0.\nlscale, rscale\nArrays, size at least max(1, n).\nThe array lscale contains details of the permutations and/or scaling factors\napplied to the left side of A and B, as returned by ?ggbal.\nThe array rscale contains details of the permutations and/or scaling factors\napplied to the right side of A and B, as returned by ?ggbal.\nm\nThe number of columns of the matrix V\n(m≥ 0).\nv\nArray v(size max(1, ldv*m) for column major layout and max(1, ldv*n) for\nrow major layout) . Contains the matrix of right or left eigenvectors to be\ntransformed, as returned by tgevc.\nldv\nThe leading dimension of v; at least max(1, n) for column major layout and\nat least max(1, m) for row major layout .\nOutput Parameters\nv\nOverwritten by the transformed eigenvectors\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n953\n\n\nIf info = -i, the i-th parameter had an illegal value.\n?gghd3\nReduces a pair of matrices to generalized upper\nHessenberg form.\nSyntax\nlapack_int LAPACKE_sgghd3 (int matrix_layout, char compq, char compz, lapack_int n,\nlapack_int ilo, lapack_int ihi, float * a, lapack_int lda, float * b, lapack_int ldb,\nfloat * q, lapack_int ldq, float * z, lapack_int ldz);\nlapack_int LAPACKE_dgghd3 (int matrix_layout, char compq, char compz, lapack_int n,\nlapack_int ilo, lapack_int ihi, double * a, lapack_int lda, double * b, lapack_int ldb,\ndouble * q, lapack_int ldq, double * z, lapack_int ldz);\nlapack_int LAPACKE_cgghd3 (int matrix_layout, char compq, char compz, lapack_int n,\nlapack_int ilo, lapack_int ihi, lapack_complex_float * a, lapack_int lda,\nlapack_complex_float * b, lapack_int ldb, lapack_complex_float * q, lapack_int ldq,\nlapack_complex_float * z, lapack_int ldz);\nlapack_int LAPACKE_zgghd3 (int matrix_layout, char compq, char compz, lapack_int n,\nlapack_int ilo, lapack_int ihi, lapack_complex_double * a, lapack_int lda,\nlapack_complex_double * b, lapack_int ldb, lapack_complex_double * q, lapack_int ldq,\nlapack_complex_double * z, lapack_int ldz);\nInclude Files\n•\nmkl.h\nDescription\n?gghd3 reduces a pair of real or complex matrices (A, B) to generalized upper Hessenberg form using\northogonal/unitary transformations, where A is a general matrix and B is upper triangular. The form of the\ngeneralized eigenvalue problem is\nA*x = λ*B*x,\nand B is typically made upper triangular by computing its QR factorization and moving the orthogonal/unitary\nmatrix Q to the left side of the equation.\nThis subroutine simultaneously reduces A to a Hessenberg matrix H:\nQT*A*Z = H for real flavors\nor\nQT*A*Z = H for complex flavors\nand transforms B to another upper triangular matrix T:\nQT*B*Z = T for real flavors\nor\nQT*B*Z = T for complex flavors\nin order to reduce the problem to its standard form\nH*y = λ*T*y\nwhere y = ZT*x for real flavors\nor\ny = ZT*x for complex flavors.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n954\n\n\nThe orthogonal/unitary matrices Q and Z are determined as products of Givens rotations. They may either be\nformed explicitly, or they may be postmultiplied into input matrices Q1 and Z1, so that\nfor real flavors:\nQ1 * A * Z1T = (Q1*Q) * H * (Z1*Z)T\nQ1 * B * Z1T = (Q1*Q) * T * (Z1*Z)T\nfor complex flavors:\nQ1 * A * Z1H = (Q1*Q) * H * (Z1*Z)T\nQ1 * B * Z1T = (Q1*Q) * T * (Z1*Z)T\nIf Q1 is the orthogonal/unitary matrix from the QR factorization of B in the original equation A*x = λ*B*x,\nthen ?gghd3 reduces the original problem to generalized Hessenberg form.\nThis is a blocked variant of ?gghrd, using matrix-matrix multiplications for parts of the computation to\nenhance performance.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\ncompq\n= 'N': do not compute q;\n= 'I': q is initialized to the unit matrix, and the orthogonal/unitary matrix Q\nis returned;\n= 'V': q must contain an orthogonal/unitary matrix Q1 on entry, and the\nproduct Q1*q is returned.\ncompz\n= 'N': do not compute z;\n= 'I': z is initialized to the unit matrix, and the orthogonal/unitary matrix Z\nis returned;\n= 'V': z must contain an orthogonal/unitary matrix Z1 on entry, and the\nproduct Z1*z is returned.\nn\nThe order of the matrices A and B.\nn≥ 0.\nilo, ihi\nilo and ihi mark the rows and columns of a which are to be reduced. It is\nassumed that a is already upper triangular in rows and columns 1:ilo - 1\nand ihi + 1:n. ilo and ihi are normally set by a previous call to ?ggbal;\notherwise they should be set to 1 and n, respectively.\n1 ≤ilo≤ihi≤n, if n > 0; ilo=1 and ihi=0, if n=0.\na\nArray, size (lda*n).\nOn entry, the n-by-n general matrix to be reduced.\nlda\nThe leading dimension of the array a.\nlda≥ max(1,n).\nb\nArray, (ldb*n).\nOn entry, then-by-n upper triangular matrix B.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n955\n\n\nldb\nThe leading dimension of the array b.\nldb≥ max(1,n).\nq\nArray, size (ldq*n).\nOn entry, if compq = 'V', the orthogonal/unitary matrix Q1, typically from\nthe QR factorization of b.\nldq\nThe leading dimension of the array q.\nldq≥n if compq='V' or 'I'; ldq≥ 1 otherwise.\nz\nArray, size (ldz*n).\nOn entry, if compz = 'V', the orthogonal/unitary matrix Z1.\nNot referenced if compz='N'.\nldz\nThe leading dimension of the array z. ldz≥n if compz='V' or 'I'; ldz≥ 1\notherwise.\nOutput Parameters\na\nOn exit, the upper triangle and the first subdiagonal of a are\noverwritten with the upper Hessenberg matrix H, and the rest is set to\nzero.\nb\nOn exit, the upper triangular matrix T = QTBZ for real flavors or T =\nQHBZ for complex flavors. The elements below the diagonal are set to\nzero.\nq\nOn exit, if compq='I', the orthogonal/unitary matrix Q, and if compq =\n'V', the product Q1*Q.\nNot referenced if compq='N'.\nz\nOn exit, if compz='I', the orthogonal/unitary matrix Z, and if compz =\n'V', the product Z1*Z.\nNot referenced if compz='N'.\nReturn Values\nThis function returns a value info.\n= 0: successful exit.\n< 0: if info = -i, the i-th argument had an illegal value.\nApplication Notes\nThis routine reduces A to Hessenberg form and maintains B in using a blocked variant of Moler and Stewart's\noriginal algorithm, as described by Kagstrom, Kressner, Quintana-Orti, and Quintana-Orti (BIT 2008).\n?hgeqz\nImplements the QZ method for finding the generalized\neigenvalues of the matrix pair (H,T).\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n956\n\n\nSyntax\nlapack_int LAPACKE_shgeqz( int matrix_layout, char job, char compq, char compz,\nlapack_int n, lapack_int ilo, lapack_int ihi, float* h, lapack_int ldh, float* t,\nlapack_int ldt, float* alphar, float* alphai, float* beta, float* q, lapack_int ldq,\nfloat* z, lapack_int ldz );\nlapack_int LAPACKE_dhgeqz( int matrix_layout, char job, char compq, char compz,\nlapack_int n, lapack_int ilo, lapack_int ihi, double* h, lapack_int ldh, double* t,\nlapack_int ldt, double* alphar, double* alphai, double* beta, double* q, lapack_int ldq,\ndouble* z, lapack_int ldz );\nlapack_int LAPACKE_chgeqz( int matrix_layout, char job, char compq, char compz,\nlapack_int n, lapack_int ilo, lapack_int ihi, lapack_complex_float* h, lapack_int ldh,\nlapack_complex_float* t, lapack_int ldt, lapack_complex_float* alpha,\nlapack_complex_float* beta, lapack_complex_float* q, lapack_int ldq,\nlapack_complex_float* z, lapack_int ldz );\nlapack_int LAPACKE_zhgeqz( int matrix_layout, char job, char compq, char compz,\nlapack_int n, lapack_int ilo, lapack_int ihi, lapack_complex_double* h, lapack_int ldh,\nlapack_complex_double* t, lapack_int ldt, lapack_complex_double* alpha,\nlapack_complex_double* beta, lapack_complex_double* q, lapack_int ldq,\nlapack_complex_double* z, lapack_int ldz );\nInclude Files\n•\nmkl.h\nDescription\nThe routine computes the eigenvalues of a real/complex matrix pair (H,T), where H is an upper Hessenberg\nmatrix and T is upper triangular, using the double-shift version (for real flavors) or single-shift version (for\ncomplex flavors) of the QZ method. Matrix pairs of this type are produced by the reduction to generalized\nupper Hessenberg form of a real/complex matrix pair (A,B):\nA = Q1*H*Z1H, B = Q1*T*Z1H,\nas computed by ?gghrd.\nFor real flavors:\nIf job = 'S', then the Hessenberg-triangular pair (H,T) is reduced to generalized Schur form,\nH = Q*S*ZT, T = Q*P*ZT,\nwhere Q and Z are orthogonal matrices, P is an upper triangular matrix, and S is a quasi-triangular matrix\nwith 1-by-1 and 2-by-2 diagonal blocks. The 1-by-1 blocks correspond to real eigenvalues of the matrix pair\n(H,T) and the 2-by-2 blocks correspond to complex conjugate pairs of eigenvalues.\nAdditionally, the 2-by-2 upper triangular diagonal blocks of P corresponding to 2-by-2 blocks of S are reduced\nto positive diagonal form, that is, if Sj + 1, j is non-zero, then Pj + 1, j = Pj, j + 1 = 0, Pj, j > 0, and Pj +\n1, j + 1 > 0.\nFor complex flavors:\nIf job = 'S', then the Hessenberg-triangular pair (H,T) is reduced to generalized Schur form,\nH = Q* S*ZH, T = Q*P*ZH,\nwhere Q and Z are unitary matrices, and S and P are upper triangular.\nFor all function flavors:\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n957\n\n\nOptionally, the orthogonal/unitary matrix Q from the generalized Schur factorization may be post-multiplied\nby an input matrix Q1, and the orthogonal/unitary matrix Z may be post-multiplied by an input matrix Z1.\nIf Q1 and Z1 are the orthogonal/unitary matrices from ?gghrd that reduced the matrix pair (A,B) to\ngeneralized upper Hessenberg form, then the output matrices Q1Q and Z1Z are the orthogonal/unitary\nfactors from the generalized Schur factorization of (A,B):\nA = (Q1Q)*S *(Z1Z)H, B = (Q1Q)*P*(Z1Z)H.\nTo avoid overflow, eigenvalues of the matrix pair (H,T) (equivalently, of (A,B)) are computed as a pair of\nvalues (alpha,beta). For chgeqz/zhgeqz, alpha and beta are complex, and for shgeqz/dhgeqz, alpha is\ncomplex and beta real. If beta is nonzero, λ = alpha/beta is an eigenvalue of the generalized\nnonsymmetric eigenvalue problem (GNEP)\nA*x = λ*B*x\nand if alpha is nonzero, μ = beta/alpha is an eigenvalue of the alternate form of the GNEP\nμ*A*y = B*y .\nReal eigenvalues (for real flavors) or the values of alpha and beta for the i-th eigenvalue (for complex\nflavors) can be read directly from the generalized Schur form:\nalpha = Si, i, beta = Pi, i.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\njob\nSpecifies the operations to be performed. Must be 'E' or 'S'.\nIf job = 'E', then compute eigenvalues only;\nIf job = 'S', then compute eigenvalues and the Schur form.\ncompq\nMust be 'N', 'I', or 'V'.\nIf compq = 'N', left Schur vectors (q) are not computed;\nIf compq = 'I', q is initialized to the unit matrix and the matrix of left\nSchur vectors of (H,T) is returned;\nIf compq = 'V', q must contain an orthogonal/unitary matrix Q1 on entry\nand the product Q1*Q is returned.\ncompz\nMust be 'N', 'I', or 'V'.\nIf compz = 'N', right Schur vectors (z) are not computed;\nIf compz = 'I', z is initialized to the unit matrix and the matrix of right\nSchur vectors of (H,T) is returned;\nIf compz = 'V', z must contain an orthogonal/unitary matrix Z1 on entry\nand the product Z1*Z is returned.\nn\nThe order of the matrices H, T, Q, and Z\n(n≥ 0).\nilo, ihi\nilo and ihi mark the rows and columns of H which are in Hessenberg form.\nIt is assumed that H is already upper triangular in rows and columns 1:ilo-1\nand ihi+1:n.\nConstraint:\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n958\n\n\nIf n > 0, then 1 ≤ilo≤ihi≤n;\nif n = 0, then ilo = 1 and ihi = 0.\nh, t, q, z\nArrays:\nOn entry, h (size max(1, ldh*n)) contains the n-by-n upper Hessenberg\nmatrix H.\nOn entry, t (size max(1, ldt*n)) contains the n-by-n upper triangular\nmatrix T.\nq (size max(1, ldq*n)) :\nOn entry, if compq = 'V', this array contains the orthogonal/unitary matrix\nQ1 used in the reduction of (A,B) to generalized Hessenberg form.\nIf compq = 'N', then q is not referenced.\nz (size max(1, ldz*n)) :\nOn entry, if compz = 'V', this array contains the orthogonal/unitary matrix\nZ1 used in the reduction of (A,B) to generalized Hessenberg form.\nIf compz = 'N', then z is not referenced.\nldh\nThe leading dimension of h; at least max(1, n).\nldt\nThe leading dimension of t; at least max(1, n).\nldq\nThe leading dimension of q;\nIf compq = 'N', then ldq≥ 1.\nIf compq = 'I'or 'V', then ldq≥ max(1, n).\nldz\nThe leading dimension of z;\nIf compq = 'N', then ldz≥ 1.\nIf compq = 'I'or 'V', then ldz≥ max(1, n).\nOutput Parameters\nh\nFor real flavors:\nIf job = 'S', then on exit h contains the upper quasi-triangular matrix S\nfrom the generalized Schur factorization.\nIf job = 'E', then on exit the diagonal blocks of h match those of S, but\nthe rest of h is unspecified.\nFor complex flavors:\nIf job = 'S', then, on exit, h contains the upper triangular matrix S from\nthe generalized Schur factorization.\nIf job = 'E', then on exit the diagonal of h matches that of S, but the rest\nof h is unspecified.\nt\nIf job = 'S', then, on exit, t contains the upper triangular matrix P from\nthe generalized Schur factorization.\nFor real flavors:\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n959\n\n\n2-by-2 diagonal blocks of P corresponding to 2-by-2 blocks of S are reduced\nto positive diagonal form, that is, if h(j+1,j) is non-zero, then t(j\n+1,j)=t(j,j+1)=0 and t(j,j) and t(j+1,j+1) will be positive.\nIf job = 'E', then on exit the diagonal blocks of t match those of P, but\nthe rest of t is unspecified.\nFor complex flavors:\nif job = 'E', then on exit the diagonal of t matches that of P, but the rest\nof t is unspecified.\nalphar, alphai\nArrays, size at least max(1, n). The real and imaginary parts, respectively,\nof each scalar alpha defining an eigenvalue of GNEP.\nIf alphai[j - 1] is zero, then the j-th eigenvalue is real; if positive, then the\nj-th and (j+1)-th eigenvalues are a complex conjugate pair, with\nalphai[j] = -alphai[j - 1].\nalpha\nArray, size at least max(1, n).\nThe complex scalars alpha that define the eigenvalues of GNEP. alphai[i\n- 1] = Si, i in the generalized Schur factorization.\nbeta\nArray, size at least max(1, n).\nFor real flavors:\nThe scalars beta that define the eigenvalues of GNEP.\nTogether, the quantities alpha = (alphar[j - 1], alphai[j - 1]) and\nbeta = beta[j - 1] represent the j-th eigenvalue of the matrix pair\n(A,B), in one of the forms lambda = alpha/beta or mu = beta/alpha.\nSince either lambda or mu may overflow, they should not, in general, be\ncomputed.\nFor complex flavors:\nThe real non-negative scalars beta that define the eigenvalues of GNEP.\nbeta[i - 1] = Pi, i in the generalized Schur factorization. Together, the\nquantities alpha = alpha[j - 1] and beta = beta[j - 1] represent\nthe j-th eigenvalue of the matrix pair (A,B), in one of the forms lambda =\nalpha/beta or mu = beta/alpha. Since either lambda or mu may\noverflow, they should not, in general, be computed.\nq\nOn exit, if compq = 'I', q is overwritten by the orthogonal/unitary matrix\nof left Schur vectors of the pair (H,T), and if compq = 'V', q is overwritten\nby the orthogonal/unitary matrix of left Schur vectors of (A,B).\nz\nOn exit, if compz = 'I', z is overwritten by the orthogonal/unitary matrix\nof right Schur vectors of the pair (H,T), and if compz = 'V', z is\noverwritten by the orthogonal/unitary matrix of right Schur vectors of\n(A,B).\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n960\n\n\nIf info = 1,..., n, the QZ iteration did not converge.\n(H,T) is not in Schur form, but alphar[i - 1], alphai[i - 1] (for real flavors), alpha[i - 1] (for complex flavors),\nand beta[i - 1], i=info+1,..., n should be correct.\nIf info = n+1,...,2n, the shift calculation failed.\n(H,T) is not in Schur form, but alphar[i - 1], alphai[i - 1] (for real flavors), alpha[i - 1] (for complex flavors),\nand beta[i - 1], i =info-n+1,..., n should be correct.\n?tgevc\nComputes some or all of the right and/or left\ngeneralized eigenvectors of a pair of upper triangular\nmatrices.\nSyntax\nlapack_int LAPACKE_stgevc (int matrix_layout, char side, char howmny, const\nlapack_logical* select, lapack_int n, const float* s, lapack_int lds, const float* p,\nlapack_int ldp, float* vl, lapack_int ldvl, float* vr, lapack_int ldvr, lapack_int mm,\nlapack_int* m);\nlapack_int LAPACKE_dtgevc (int matrix_layout, char side, char howmny, const\nlapack_logical* select, lapack_int n, const double* s, lapack_int lds, const double* p,\nlapack_int ldp, double* vl, lapack_int ldvl, double* vr, lapack_int ldvr, lapack_int mm,\nlapack_int* m);\nlapack_int LAPACKE_ctgevc (int matrix_layout, char side, char howmny, const\nlapack_logical* select, lapack_int n, const lapack_complex_float* s, lapack_int lds,\nconst lapack_complex_float* p, lapack_int ldp, lapack_complex_float* vl, lapack_int\nldvl, lapack_complex_float* vr, lapack_int ldvr, lapack_int mm, lapack_int* m);\nlapack_int LAPACKE_ztgevc (int matrix_layout, char side, char howmny, const\nlapack_logical* select, lapack_int n, const lapack_complex_double* s, lapack_int lds,\nconst lapack_complex_double* p, lapack_int ldp, lapack_complex_double* vl, lapack_int\nldvl, lapack_complex_double* vr, lapack_int ldvr, lapack_int mm, lapack_int* m);\nInclude Files\n•\nmkl.h\nDescription\nThe routine computes some or all of the right and/or left eigenvectors of a pair of real/complex matrices\n(S,P), where S is quasi-triangular (for real flavors) or upper triangular (for complex flavors) and P is upper\ntriangular.\nMatrix pairs of this type are produced by the generalized Schur factorization of a real/complex matrix pair\n(A,B):\nA = Q*S*ZH, B = Q*P*ZH\nas computed by ?gghrd plus ?hgeqz.\nThe right eigenvector x and the left eigenvector y of (S,P) corresponding to an eigenvalue w are defined by:\nS*x = w*P*x, yH*S = w*yH*P\nThe eigenvalues are not input to this routine, but are computed directly from the diagonal blocks or diagonal\nelements of S and P.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n961\n\n\nThis routine returns the matrices X and/or Y of right and left eigenvectors of (S,P), or the products Z*X\nand/or Q*Y, where Z and Q are input matrices.\nIf Q and Z are the orthogonal/unitary factors from the generalized Schur factorization of a matrix pair (A,B),\nthen Z*X and Q*Y are the matrices of right and left eigenvectors of (A,B).\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nside\nMust be 'R', 'L', or 'B'.\nIf side = 'R', compute right eigenvectors only.\nIf side = 'L', compute left eigenvectors only.\nIf side = 'B', compute both right and left eigenvectors.\nhowmny\nMust be 'A', 'B', or 'S'.\nIf howmny = 'A', compute all right and/or left eigenvectors.\nIf howmny = 'B', compute all right and/or left eigenvectors,\nbacktransformed by the matrices in vr and/or vl.\nIf howmny = 'S', compute selected right and/or left eigenvectors, specified\nby the logical array select.\nselect\nArray, size at least max (1, n).\nIf howmny = 'S', select specifies the eigenvectors to be computed.\nIf howmny = 'A'or 'B', select is not referenced.\nFor real flavors:\nIf w[j] is a real eigenvalue, the corresponding real eigenvector is computed\nif select[j] is 1.\nIf w[j] and omega[j + 1] are the real and imaginary parts of a complex\neigenvalue, the corresponding complex eigenvector is computed if either\nselect[j] or select[j + 1] is 1, and on exit select[j] is set to 1and select[j +\n1] is set to 0.\nFor complex flavors:\nThe eigenvector corresponding to the j-th eigenvalue is computed if\nselect[j] is 1.\nn\nThe order of the matrices S and P (n≥ 0).\ns, p, vl, vr\nArrays:\ns (size max(1, lds*n)) contains the matrix S from a generalized Schur\nfactorization as computed by ?hgeqz. This matrix is upper quasi-triangular\nfor real flavors, and upper triangular for complex flavors.\np (size max(1, ldp*n)) contains the upper triangular matrix P from a\ngeneralized Schur factorization as computed by ?hgeqz.\nFor real flavors, 2-by-2 diagonal blocks of P corresponding to 2-by-2 blocks\nof S must be in positive diagonal form.\nFor complex flavors, P must have real diagonal elements.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n962\n\n\nIf side = 'L' or 'B' and howmny = 'B', vl(size max(1, ldvl*mm) for\ncolumn major layout and max(1, ldvl*n) for row major layout) must\ncontain an n-by-n matrix Q (usually the orthogonal/unitary matrix Q of left\nSchur vectors returned by ?hgeqz).\nIf side = 'R', vl is not referenced.\nIf side = 'R' or 'B' and howmny = 'B', vr(size max(1, ldvr*mm) for\ncolumn major layout and max(1, ldvr*n) for row major layout) must\ncontain an n-by-n matrix Z (usually the orthogonal/unitary matrix Z of right\nSchur vectors returned by ?hgeqz).\nIf side = 'L', vr is not referenced.\nlds\nThe leading dimension of s; at least max(1, n).\nldp\nThe leading dimension of p; at least max(1, n).\nldvl\nThe leading dimension of vl;\nIf side = 'L' or 'B', then ldvl≥n for column major layout and ldvl≥\nmax(1, mm) for row major layout.\nIf side = 'R', then ldvl≥ 1 .\nldvr\nThe leading dimension of vr;\nIf side = 'R' or 'B', then ldvr≥n for column major layout and ldvr≥\nmax(1, mm) for row major layout.\nIf side = 'L', then ldvr≥ 1.\nmm\nThe number of columns in the arrays vl and/or vr (mm≥m).\nOutput Parameters\nvl\nOn exit, if side = 'L' or 'B', vl contains:\nif howmny = 'A', the matrix Y of left eigenvectors of (S,P);\nif howmny = 'B', the matrix Q*Y;\nif howmny = 'S', the left eigenvectors of (S,P) specified by select, stored\nconsecutively in the columns of vl, in the same order as their eigenvalues.\nFor real flavors:\nA complex eigenvector corresponding to a complex eigenvalue is stored in\ntwo consecutive columns, the first holding the real part, and the second the\nimaginary part.\nvr\nOn exit, if side = 'R' or 'B', vr contains:\nif howmny = 'A', the matrix X of right eigenvectors of (S,P);\nif howmny = 'B', the matrix Z*X;\nif howmny = 'S', the right eigenvectors of (S,P) specified by select, stored\nconsecutively in the columns of vr, in the same order as their eigenvalues.\nFor real flavors:\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n963\n\n\nA complex eigenvector corresponding to a complex eigenvalue is stored in\ntwo consecutive columns, the first holding the real part, and the second the\nimaginary part.\nm\nThe number of columns in the arrays vl and/or vr actually used to store the\neigenvectors.\nIf howmny = 'A' or 'B', m is set to n.\nFor real flavors:\nEach selected real eigenvector occupies one column and each selected\ncomplex eigenvector occupies two columns.\nFor complex flavors:\nEach selected eigenvector occupies one column.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nFor real flavors:\nif info = i>0, the 2-by-2 block (i:i+1) does not have a complex eigenvalue.\n?tgexc\nReorders the generalized Schur decomposition of a\npair of matrices (A,B) so that one diagonal block of\n(A,B) moves to another row index.\nSyntax\nlapack_int LAPACKE_stgexc (int matrix_layout, lapack_logical wantq, lapack_logical\nwantz, lapack_int n, float* a, lapack_int lda, float* b, lapack_int ldb, float* q,\nlapack_int ldq, float* z, lapack_int ldz, lapack_int* ifst, lapack_int* ilst);\nlapack_int LAPACKE_dtgexc (int matrix_layout, lapack_logical wantq, lapack_logical\nwantz, lapack_int n, double* a, lapack_int lda, double* b, lapack_int ldb, double* q,\nlapack_int ldq, double* z, lapack_int ldz, lapack_int* ifst, lapack_int* ilst);\nlapack_int LAPACKE_ctgexc (int matrix_layout, lapack_logical wantq, lapack_logical\nwantz, lapack_int n, lapack_complex_float* a, lapack_int lda, lapack_complex_float* b,\nlapack_int ldb, lapack_complex_float* q, lapack_int ldq, lapack_complex_float* z,\nlapack_int ldz, lapack_int ifst, lapack_int ilst);\nlapack_int LAPACKE_ztgexc (int matrix_layout, lapack_logical wantq, lapack_logical\nwantz, lapack_int n, lapack_complex_double* a, lapack_int lda, lapack_complex_double*\nb, lapack_int ldb, lapack_complex_double* q, lapack_int ldq, lapack_complex_double* z,\nlapack_int ldz, lapack_int ifst, lapack_int ilst);\nInclude Files\n•\nmkl.h\nDescription\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n964\n\n\nThe routine reorders the generalized real-Schur/Schur decomposition of a real/complex matrix pair (A,B)\nusing an orthogonal/unitary equivalence transformation\n(A,B) = Q*(A,B)*ZH,\nso that the diagonal block of (A, B) with row index ifst is moved to row ilst. Matrix pair (A, B) must be in a\ngeneralized real-Schur/Schur canonical form (as returned by gges), that is, A is block upper triangular with\n1-by-1 and 2-by-2 diagonal blocks and B is upper triangular. Optionally, the matrices Q and Z of generalized\nSchur vectors are updated.\nQin*Ain*ZinT = Qout*Aout*ZoutT\nQin*Bin*ZinT = Qout*Bout*ZoutT.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nwantq, wantz\nIf wantq = 1, update the left transformation matrix Q;\nIf wantq = 0, do not update Q;\nIf wantz = 1, update the right transformation matrix Z;\nIf wantz = 0, do not update Z.\nn\nThe order of the matrices A and B (n≥ 0).\na, b, q, z\nArrays:\na (size max(1, lda*n)) contains the matrix A.\nb (size max(1, ldb*n)) contains the matrix B.\nq (size at least 1 if wantq = 0 and at least max(1, ldq*n) if wantq = 1)\nIf wantq = 0, then q is not referenced.\nIf wantq = 1, then q must contain the orthogonal/unitary matrix Q.\nz (size at least 1 if wantz = 0 and at least max(1, ldz*n) if wantz = 1)\nIf wantz = 0, then z is not referenced.\nIf wantz = 1, then z must contain the orthogonal/unitary matrix Z.\nlda\nThe leading dimension of a; at least max(1, n).\nldb\nThe leading dimension of b; at least max(1, n).\nldq\nThe leading dimension of q;\nIf wantq = 0, then ldq≥ 1.\nIf wantq = 1, then ldq≥ max(1, n).\nldz\nThe leading dimension of z;\nIf wantz = 0, then ldz≥ 1.\nIf wantz = 1, then ldz≥ max(1, n).\nifst, ilst\nSpecify the reordering of the diagonal blocks of (A, B). The block with row\nindex ifst is moved to row ilst, by a sequence of swapping between adjacent\nblocks. Constraint: 1 ≤ifst, ilst≤n.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n965\n\n\nOutput Parameters\na, b, q, z\nOverwritten by the updated matrices A,B, Q, and Z respectively.\nifst, ilst\nOverwritten for real flavors only.\nIf ifst pointed to the second row of a 2 by 2 block on entry, it is changed to\npoint to the first row; ilst always points to the first row of the block in its\nfinal position (which may differ from its input value by ±1).\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nIf info = 1, the transformed matrix pair (A, B) would be too far from generalized Schur form; the problem\nis ill-conditioned. (A, B) may have been partially reordered, and ilst points to the first row of the current\nposition of the block being moved.\n?tgsen\nReorders the generalized Schur decomposition of a\npair of matrices (A,B) so that a selected cluster of\neigenvalues appears in the leading diagonal blocks of\n(A,B).\nSyntax\nlapack_int LAPACKE_stgsen( int matrix_layout, lapack_int ijob, lapack_logical wantq,\nlapack_logical wantz, const lapack_logical* select, lapack_int n, float* a, lapack_int\nlda, float* b, lapack_int ldb, float* alphar, float* alphai, float* beta, float* q,\nlapack_int ldq, float* z, lapack_int ldz, lapack_int* m, float* pl, float* pr, float*\ndif );\nlapack_int LAPACKE_dtgsen( int matrix_layout, lapack_int ijob, lapack_logical wantq,\nlapack_logical wantz, const lapack_logical* select, lapack_int n, double* a, lapack_int\nlda, double* b, lapack_int ldb, double* alphar, double* alphai, double* beta, double* q,\nlapack_int ldq, double* z, lapack_int ldz, lapack_int* m, double* pl, double* pr,\ndouble* dif );\nlapack_int LAPACKE_ctgsen( int matrix_layout, lapack_int ijob, lapack_logical wantq,\nlapack_logical wantz, const lapack_logical* select, lapack_int n, lapack_complex_float*\na, lapack_int lda, lapack_complex_float* b, lapack_int ldb, lapack_complex_float*\nalpha, lapack_complex_float* beta, lapack_complex_float* q, lapack_int ldq,\nlapack_complex_float* z, lapack_int ldz, lapack_int* m, float* pl, float* pr, float*\ndif );\nlapack_int LAPACKE_ztgsen( int matrix_layout, lapack_int ijob, lapack_logical wantq,\nlapack_logical wantz, const lapack_logical* select, lapack_int n,\nlapack_complex_double* a, lapack_int lda, lapack_complex_double* b, lapack_int ldb,\nlapack_complex_double* alpha, lapack_complex_double* beta, lapack_complex_double* q,\nlapack_int ldq, lapack_complex_double* z, lapack_int ldz, lapack_int* m, double* pl,\ndouble* pr, double* dif );\nInclude Files\n•\nmkl.h\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n966\n\n\nDescription\nThe routine reorders the generalized real-Schur/Schur decomposition of a real/complex matrix pair (A, B) (in\nterms of an orthogonal/unitary equivalence transformation QT*(A,B)*Z for real flavors or QH*(A,B)*Z for\ncomplex flavors), so that a selected cluster of eigenvalues appears in the leading diagonal blocks of the pair\n(A, B). The leading columns of Q and Z form orthonormal/unitary bases of the corresponding left and right\neigenspaces (deflating subspaces).\n(A, B) must be in generalized real-Schur/Schur canonical form (as returned by gges), that is, A and B are\nboth upper triangular.\n?tgsen also computes the generalized eigenvalues\nωj = (alphar(j) + alphai(j)*i)/beta(j) (for real flavors)\nωj = alpha(j)/beta(j) (for complex flavors)\nof the reordered matrix pair (A, B).\nOptionally, the routine computes the estimates of reciprocal condition numbers for eigenvalues and\neigenspaces. These are Difu[(A11, B11), (A22, B22)] and Difl[(A11, B11), (A22, B22)], that is, the\nseparation(s) between the matrix pairs (A11, B11) and (A22, B22) that correspond to the selected cluster and\nthe eigenvalues outside the cluster, respectively, and norms of \"projections\" onto left and right eigenspaces\nwith respect to the selected cluster in the (1,1)-block.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nijob\nSpecifies whether condition numbers are required for the cluster of\neigenvalues (pl and pr) or the deflating subspaces Difu and Difl.\nIf ijob =0, only reorder with respect to select;\nIf ijob =1, reciprocal of norms of \"projections\" onto left and right\neigenspaces with respect to the selected cluster (pl and pr);\nIf ijob =2, compute upper bounds on Difu and Difl, using F-norm-based\nestimate (dif (1:2));\nIf ijob =3, compute estimate of Difu and Difl, using 1-norm-based\nestimate (dif (1:2)). This option is about 5 times as expensive as ijob =2;\nIf ijob =4,>compute pl, pr and dif (i.e., options 0, 1 and 2 above). This is\nan economic version to get it all;\nIf ijob =5, compute pl, pr and dif (i.e., options 0, 1 and 3 above).\nwantq, wantz\nIf wantq = 1, update the left transformation matrix Q;\nIf wantq = 0, do not update Q;\nIf wantz = 1, update the right transformation matrix Z;\nIf wantz = 0, do not update Z.\nselect\nArray, size at least max (1, n). Specifies the eigenvalues in the selected\ncluster.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n967\n\n\nTo select an eigenvalue ωj, select[j - 1] must be 1For real flavors: to select\na complex conjugate pair of eigenvalues ωj and ωj + 1 (corresponding 2 by 2\ndiagonal block), select[j - 1] and/or select[j] must be set to 1; the complex\nconjugate ωj and ωj + 1 must be either both included in the cluster or both\nexcluded.\nn\nThe order of the matrices A and B (n≥ 0).\na, b, q, z\nArrays:\na (size max(1, lda*n)) contains the matrix A.\nFor real flavors: A is upper quasi-triangular, with (A, B) in generalized real\nSchur canonical form.\nFor complex flavors: A is upper triangular, in generalized Schur canonical\nform.\nb (size max(1, ldb*n)) contains the matrix B.\nFor real flavors: B is upper triangular, with (A, B) in generalized real Schur\ncanonical form.\nFor complex flavors: B is upper triangular, in generalized Schur canonical\nform.\nq (size at least 1 if wantq = 0 and at least max(1, ldq*n) if wantq = 1)\nIf wantq = 1, then q is an n-by-n matrix;\nIf wantq = 0, then q is not referenced.\nz (size at least 1 if wantz = 0 and at least max(1, ldz*n) if wantz = 1)\nIf wantz = 1, then z is an n-by-n matrix;\nIf wantz = 0, then z is not referenced.\nlda\nThe leading dimension of a; at least max(1, n).\nldb\nThe leading dimension of b; at least max(1, n).\nldq\nThe leading dimension of q; ldq≥ 1.\nIf wantq = 1, then ldq≥ max(1, n).\nldz\nThe leading dimension of z; ldz≥ 1.\nIf wantz = 1, then ldz≥ max(1, n).\nOutput Parameters\na, b\nOverwritten by the reordered matrices A and B, respectively.\nalphar, alphai\nArrays, size at least max(1, n). Contain values that form generalized\neigenvalues in real flavors.\nSee beta.\nalpha\nArray, size at least max(1, n). Contain values that form generalized\neigenvalues in complex flavors.\nSee beta.\nbeta\nArray, size at least max(1, n).\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n968\n\n\nFor real flavors:\nOn exit, (alphar[j] + alphai[j]*i)/beta[j], j=0,..., n - 1, will be the\ngeneralized eigenvalues.\nalphar[j] + alphai[j]*i and beta[j], j=0,..., n - 1 are the diagonals\nof the complex Schur form (S,T) that would result if the 2-by-2 diagonal\nblocks of the real generalized Schur form of (A,B) were further reduced to\ntriangular form using complex unitary transformations.\nIf alphai[j - 1] is zero, then the j-th eigenvalue is real; if positive, then\nthe j-th and (j + 1)-st eigenvalues are a complex conjugate pair, with\nalphai[j] negative.\nFor complex flavors:\nThe diagonal elements of A and B, respectively, when the pair (A,B) has\nbeen reduced to generalized Schur form. alpha[i]/beta[i],i=0,..., n - 1\nare the generalized eigenvalues.\nq\nIf wantq = 1, then, on exit, Q has been postmultiplied by the left\northogonal transformation matrix which reorder (A, B). The leading m\ncolumns of Q form orthonormal bases for the specified pair of left\neigenspaces (deflating subspaces).\nz\nIf wantz = 1, then, on exit, Z has been postmultiplied by the left orthogonal\ntransformation matrix which reorder (A, B). The leading m columns of Z\nform orthonormal bases for the specified pair of left eigenspaces (deflating\nsubspaces).\nm\nThe dimension of the specified pair of left and right eigen-spaces (deflating\nsubspaces); 0 ≤m≤n.\npl, pr\nIf ijob = 1, 4, or 5, pl and pr are lower bounds on the reciprocal of the\nnorm of \"projections\" onto left and right eigenspaces with respect to the\nselected cluster.\n0 < pl, pr≤ 1. If m = 0 or m = n, pl = pr = 1.\nIf ijob = 0, 2 or 3, pl and pr are not referenced\ndif\nArray, size (2).\nIf ijob≥ 2, dif(1:2) store the estimates of Difu and Difl.\nIf ijob = 2 or 4, dif(1:2) are F-norm-based upper bounds on Difu and\nDifl.\nIf ijob = 3 or 5, dif(1:2) are 1-norm-based estimates of Difu and Difl.\nIf m = 0 or m = n, dif(1:2) = F-norm([A, B]).\nIf ijob = 0 or 1, dif is not referenced.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n969\n\n\nIf info = 1, Reordering of (A, B) failed because the transformed matrix pair (A, B) would be too far from\ngeneralized Schur form; the problem is very ill-conditioned. (A, B) may have been partially reordered.\nIf ijob > 0, 0 is returned in dif, pl and pr.\n?tgsyl\nSolves the generalized Sylvester equation.\nSyntax\nlapack_int LAPACKE_stgsyl( int matrix_layout, char trans, lapack_int ijob, lapack_int\nm, lapack_int n, const float* a, lapack_int lda, const float* b, lapack_int ldb, float*\nc, lapack_int ldc, const float* d, lapack_int ldd, const float* e, lapack_int lde,\nfloat* f, lapack_int ldf, float* scale, float* dif );\nlapack_int LAPACKE_dtgsyl( int matrix_layout, char trans, lapack_int ijob, lapack_int\nm, lapack_int n, const double* a, lapack_int lda, const double* b, lapack_int ldb,\ndouble* c, lapack_int ldc, const double* d, lapack_int ldd, const double* e, lapack_int\nlde, double* f, lapack_int ldf, double* scale, double* dif );\nlapack_int LAPACKE_ctgsyl( int matrix_layout, char trans, lapack_int ijob, lapack_int\nm, lapack_int n, const lapack_complex_float* a, lapack_int lda, const\nlapack_complex_float* b, lapack_int ldb, lapack_complex_float* c, lapack_int ldc, const\nlapack_complex_float* d, lapack_int ldd, const lapack_complex_float* e, lapack_int lde,\nlapack_complex_float* f, lapack_int ldf, float* scale, float* dif );\nlapack_int LAPACKE_ztgsyl( int matrix_layout, char trans, lapack_int ijob, lapack_int\nm, lapack_int n, const lapack_complex_double* a, lapack_int lda, const\nlapack_complex_double* b, lapack_int ldb, lapack_complex_double* c, lapack_int ldc,\nconst lapack_complex_double* d, lapack_int ldd, const lapack_complex_double* e,\nlapack_int lde, lapack_complex_double* f, lapack_int ldf, double* scale, double* dif );\nInclude Files\n•\nmkl.h\nDescription\nThe routine solves the generalized Sylvester equation:\nA*R-L*B = scale*C\nD*R-L*E = scale*F\nwhere R and L are unknown m-by-n matrices, (A, D), (B, E) and (C, F) are given matrix pairs of size m-by-\nm, n-by-n and m-by-n, respectively, with real/complex entries. (A, D) and (B, E) must be in generalized real-\nSchur/Schur canonical form, that is, A, B are upper quasi-triangular/triangular and D, E are upper triangular.\nThe solution (R, L) overwrites (C, F). The factor scale, 0≤scale≤1, is an output scaling factor chosen to avoid\noverflow.\nIn matrix notation the above equation is equivalent to the following: solve Z*x = scale*b, where Z is\ndefined as\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n970\n\n\nHere Ik is the identity matrix of size k and XT is the transpose/conjugate-transpose of X. kron(X, Y) is the\nKronecker product between the matrices X and Y.\nIf trans = 'T' (for real flavors), or trans = 'C' (for complex flavors), the routine ?tgsyl solves the\ntransposed/conjugate-transposed system ZT*y = scale*b, which is equivalent to solve for R and L in\nAT*R+DT*L = scale*C\nR*BT+L*ET = scale*(-F)\nThis case (trans = 'T' for stgsyl/dtgsyl or trans = 'C' for ctgsyl/ztgsyl) is used to compute an\none-norm-based estimate of Dif[(A, D), (B, E)], the separation between the matrix pairs (A,D) and\n(B,E).\nIf ijob ≥ 1, ?tgsyl computes a Frobenius norm-based estimate of Dif[(A, D), (B,E)]. That is, the\nreciprocal of a lower bound on the reciprocal of the smallest singular value of Z. This is a level 3 BLAS\nalgorithm.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\ntrans\nMust be 'N', 'T', or 'C'.\nIf trans = 'N', solve the generalized Sylvester equation.\nIf trans = 'T', solve the 'transposed' system (for real flavors only).\nIf trans = 'C', solve the ' conjugate transposed' system (for complex\nflavors only).\nijob\nSpecifies what kind of functionality to be performed:\nIf ijob =0, solve the generalized Sylvester equation only;\nIf ijob =1, perform the functionality of ijob =0 and ijob =3;\nIf ijob =2, perform the functionality of ijob =0 and ijob =4;\nIf ijob =3, only an estimate of Dif[(A, D), (B, E)] is computed (look ahead\nstrategy is used);\nIf ijob =4, only an estimate of Dif[(A, D), (B,E)] is computed (?gecon on\nsub-systems is used). If trans = 'T' or 'C', ijob is not referenced.\nm\nThe order of the matrices A and D, and the row dimension of the matrices\nC, F, R and L.\nn\nThe order of the matrices B and E, and the column dimension of the\nmatrices C, F, R and L.\na, b, c, d, e, f\nArrays:\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n971\n\n\na (size max(1, lda*m)) contains the upper quasi-triangular (for real flavors)\nor upper triangular (for complex flavors) matrix A.\nb (size max(1, ldb*n)) contains the upper quasi-triangular (for real flavors)\nor upper triangular (for complex flavors) matrix B.\nc(size max(1, ldc*n) for column major layout and max(1, ldc*m) for row\nmajor layout) contains the right-hand-side of the first matrix equation in\nthe generalized Sylvester equation (as defined by trans)\nd (size max(1, ldd*m)) contains the upper triangular matrix D.\ne (size max(1, lde*n)) contains the upper triangular matrix E.\nf(size max(1, ldf*n) for column major layout and max(1, ldf*m) for row\nmajor layout) contains the right-hand-side of the second matrix equation in\nthe generalized Sylvester equation (as defined by trans)\nlda\nThe leading dimension of a; at least max(1, m).\nldb\nThe leading dimension of b; at least max(1, n).\nldc\nThe leading dimension of c; at least max(1, m) for column major layout and\nat least max(1, n) for row major layout .\nldd\nThe leading dimension of d; at least max(1, m).\nlde\nThe leading dimension of e; at least max(1, n).\nldf\nThe leading dimension of f; at least max(1, m) for column major layout and\nat least max(1, n) for row major layout .\nOutput Parameters\nc\nIf ijob=0, 1, or 2, overwritten by the solution R.\nIf ijob=3 or 4 and trans = 'N', c holds R, the solution achieved during the\ncomputation of the Dif-estimate.\nf\nIf ijob=0, 1, or 2, overwritten by the solution L.\nIf ijob=3 or 4 and trans = 'N', f holds L, the solution achieved during\nthe computation of the Dif-estimate.\ndif\nOn exit, dif is the reciprocal of a lower bound of the reciprocal of the Dif-\nfunction, that is, dif is an upper bound of Dif[(A, D), (B, E)] =\nsigma_min(Z), where Z as defined in the description.\nIf ijob = 0, or trans = 'T' (for real flavors), or trans = 'C' (for\ncomplex flavors), dif is not touched.\nscale\nOn exit, scale is the scaling factor in the generalized Sylvester equation.\nIf 0 < scale < 1, c and f hold the solutions R and L, respectively, to a\nslightly perturbed system but the input matrices A, B, D and E have not\nbeen changed.\nIf scale = 0, c and f hold the solutions R and L, respectively, to the\nhomogeneous system with C = F = 0. Normally, scale = 1.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n972\n\n\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nIf info > 0, (A, D) and (B, E) have common or close eigenvalues.\n?tgsna\nEstimates reciprocal condition numbers for specified\neigenvalues and/or eigenvectors of a pair of matrices\nin generalized real Schur canonical form.\nSyntax\nlapack_int LAPACKE_stgsna( int matrix_layout, char job, char howmny, const\nlapack_logical* select, lapack_int n, const float* a, lapack_int lda, const float* b,\nlapack_int ldb, const float* vl, lapack_int ldvl, const float* vr, lapack_int ldvr,\nfloat* s, float* dif, lapack_int mm, lapack_int* m );\nlapack_int LAPACKE_dtgsna( int matrix_layout, char job, char howmny, const\nlapack_logical* select, lapack_int n, const double* a, lapack_int lda, const double* b,\nlapack_int ldb, const double* vl, lapack_int ldvl, const double* vr, lapack_int ldvr,\ndouble* s, double* dif, lapack_int mm, lapack_int* m );\nlapack_int LAPACKE_ctgsna( int matrix_layout, char job, char howmny, const\nlapack_logical* select, lapack_int n, const lapack_complex_float* a, lapack_int lda,\nconst lapack_complex_float* b, lapack_int ldb, const lapack_complex_float* vl,\nlapack_int ldvl, const lapack_complex_float* vr, lapack_int ldvr, float* s, float* dif,\nlapack_int mm, lapack_int* m );\nlapack_int LAPACKE_ztgsna( int matrix_layout, char job, char howmny, const\nlapack_logical* select, lapack_int n, const lapack_complex_double* a, lapack_int lda,\nconst lapack_complex_double* b, lapack_int ldb, const lapack_complex_double* vl,\nlapack_int ldvl, const lapack_complex_double* vr, lapack_int ldvr, double* s, double*\ndif, lapack_int mm, lapack_int* m );\nInclude Files\n•\nmkl.h\nDescription\nThe real flavors stgsna/dtgsna of this routine estimate reciprocal condition numbers for specified\neigenvalues and/or eigenvectors of a matrix pair (A, B) in generalized real Schur canonical form (or of any\nmatrix pair (Q*A*ZT, Q*B*ZT) with orthogonal matrices Q and Z.\n(A, B) must be in generalized real Schur form (as returned by gges/gges), that is, A is block upper triangular\nwith 1-by-1 and 2-by-2 diagonal blocks. B is upper triangular.\nThe complex flavors ctgsna/ztgsna estimate reciprocal condition numbers for specified eigenvalues and/or\neigenvectors of a matrix pair (A, B). (A, B) must be in generalized Schur canonical form, that is, A and B are\nboth upper triangular.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n973\n\n\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\njob\nSpecifies whether condition numbers are required for eigenvalues or\neigenvectors. Must be 'E' or 'V' or 'B'.\nIf job = 'E', for eigenvalues only (compute s ).\nIf job = 'V', for eigenvectors only (compute dif ).\nIf job = 'B', for both eigenvalues and eigenvectors (compute both s and\ndif).\nhowmny\nMust be 'A' or 'S'.\nIf howmny = 'A', compute condition numbers for all eigenpairs.\nIf howmny = 'S', compute condition numbers for selected eigenpairs\nspecified by the logical array select.\nselect\nArray, size at least max (1, n).\nIf howmny = 'S', select specifies the eigenpairs for which condition\nnumbers are required.\nIf howmny = 'A', select is not referenced.\nFor real flavors:\nTo select condition numbers for the eigenpair corresponding to a real\neigenvalue ωj, select[j - 1] must be set to 1; to select condition numbers\ncorresponding to a complex conjugate pair of eigenvalues ωj and ωj + 1,\neither select[j - 1] or select[j] must be set to 1.\nFor complex flavors:\nTo select condition numbers for the corresponding j-th eigenvalue and/or\neigenvector, select[j - 1] must be set to 1.\nn\nThe order of the square matrix pair (A, B)\n(n≥ 0).\na, b, vl, vr\nArrays:\na (size max(1, lda*n)) contains the upper quasi-triangular (for real flavors)\nor upper triangular (for complex flavors) matrix A in the pair (A, B).\nb (size max(1, ldb*n)) contains the upper triangular matrix B in the pair\n(A, B).\nIf job = 'E' or 'B', vl(size max(1, ldvl*m) for column major layout and\nmax(1, ldvl*n) for row major layout) must contain left eigenvectors of (A,\nB), corresponding to the eigenpairs specified by howmny and select. The\neigenvectors must be stored in consecutive columns of vl, as returned\nby ?tgevc.\nIf job = 'V', vl is not referenced.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n974\n\n\nIf job = 'E' or 'B', vr(size max(1, ldvr*m) for column major layout and\nmax(1, ldvr*n) for row major layout) must contain right eigenvectors of\n(A, B), corresponding to the eigenpairs specified by howmny and select.\nThe eigenvectors must be stored in consecutive columns of vr, as returned\nby ?tgevc.\nIf job = 'V', vr is not referenced.\nlda\nThe leading dimension of a; at least max(1, n).\nldb\nThe leading dimension of b; at least max(1, n).\nldvl\nThe leading dimension of vl; ldvl≥ 1.\nIf job = 'E' or 'B', then ldvl≥ max(1, n) for column major layout and\nldvl≥ max(1, m) for row major layout .\nldvr\nThe leading dimension of vr; ldvr≥ 1.\nIf job = 'E' or 'B', then ldvr≥ max(1, n) for column major layout and\nldvr≥ max(1, m) for row major layout.\nmm\nThe number of elements in the arrays s and dif (mm≥m).\nOutput Parameters\ns\nArray, size mm.\nIf job = 'E' or 'B', contains the reciprocal condition numbers of the\nselected eigenvalues, stored in consecutive elements of the array.\nIf job = 'V', s is not referenced.\nFor real flavors:\nFor a complex conjugate pair of eigenvalues two consecutive elements of s\nare set to the same value. Thus, s[j - 1], dif[j - 1], and the j-th columns of\nvl and vr all correspond to the same eigenpair (but not in general the j-th\neigenpair, unless all eigenpairs are selected).\ndif\nArray, size mm.\nIf job = 'V' or 'B', contains the estimated reciprocal condition numbers\nof the selected eigenvectors, stored in consecutive elements of the array.\nIf the eigenvalues cannot be reordered to compute dif[j], dif[j] is set to 0;\nthis can only occur when the true value would be very small anyway.\nIf job = 'E', dif is not referenced.\nFor real flavors:\nFor a complex eigenvector, two consecutive elements of dif are set to the\nsame value.\nFor complex flavors:\nFor each eigenvalue/vector specified by select, dif stores a Frobenius norm-\nbased estimate of Difl.\nm\nThe number of elements in the arrays s and dif used to store the specified\ncondition numbers; for each selected eigenvalue one element is used.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n975\n\n\nIf howmny = 'A', m is set to n.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nGeneralized Singular Value Decomposition: LAPACK Computational Routines\nThis topic describes LAPACK computational routines used for finding the generalized singular value\ndecomposition (GSVD) of two matrices A and B as\nUHAQ = D1*(0 R),\nVHBQ = D2*(0 R),\nwhere U, V, and Q are orthogonal/unitary matrices, R is a nonsingular upper triangular matrix, and D1, D2\nare “diagonal” matrices of the structure detailed in the routines description section.\nTable “Computational Routines for Generalized Singular Value Decomposition” lists LAPACK routines that\nperform generalized singular value decomposition of matrices.\nComputational Routines for Generalized Singular Value Decomposition\nRoutine name\nOperation performed\nggsvp\nComputes the preprocessing decomposition for the generalized SVD\nggsvp3\nPerforms preprocessing for a generalized SVD.\nggsvd3\nComputes generalized SVD.\ntgsja\nComputes the generalized SVD of two upper triangular or trapezoidal\nmatrices\nYou can use routines listed in the above table as well as the driver routine ggsvd to find the GSVD of a pair of\ngeneral rectangular matrices.\n?ggsvp\nComputes the preprocessing decomposition for the\ngeneralized SVD (deprecated).\nSyntax\nlapack_int LAPACKE_sggsvp( int matrix_layout, char jobu, char jobv, char jobq,\nlapack_int m, lapack_int p, lapack_int n, float* a, lapack_int lda, float* b, lapack_int\nldb, float tola, float tolb, lapack_int* k, lapack_int* l, float* u, lapack_int ldu,\nfloat* v, lapack_int ldv, float* q, lapack_int ldq );\nlapack_int LAPACKE_dggsvp( int matrix_layout, char jobu, char jobv, char jobq,\nlapack_int m, lapack_int p, lapack_int n, double* a, lapack_int lda, double* b,\nlapack_int ldb, double tola, double tolb, lapack_int* k, lapack_int* l, double* u,\nlapack_int ldu, double* v, lapack_int ldv, double* q, lapack_int ldq );\nlapack_int LAPACKE_cggsvp( int matrix_layout, char jobu, char jobv, char jobq,\nlapack_int m, lapack_int p, lapack_int n, lapack_complex_float* a, lapack_int lda,\nlapack_complex_float* b, lapack_int ldb, float tola, float tolb, lapack_int* k,\nlapack_int* l, lapack_complex_float* u, lapack_int ldu, lapack_complex_float* v,\nlapack_int ldv, lapack_complex_float* q, lapack_int ldq );\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n976\n\n\nlapack_int LAPACKE_zggsvp( int matrix_layout, char jobu, char jobv, char jobq,\nlapack_int m, lapack_int p, lapack_int n, lapack_complex_double* a, lapack_int lda,\nlapack_complex_double* b, lapack_int ldb, double tola, double tolb, lapack_int* k,\nlapack_int* l, lapack_complex_double* u, lapack_int ldu, lapack_complex_double* v,\nlapack_int ldv, lapack_complex_double* q, lapack_int ldq );\nInclude Files\n•\nmkl.h\nDescription\nThis routine is deprecated; use ggsvp3.\nThe routine computes orthogonal matrices U, V and Q such that\nwhere the k-by-k matrix A12 and l-by-l matrix B13 are nonsingular upper triangular; A23 is l-by-l upper\ntriangular if m-k-l≥0, otherwise A23 is (m-k)-by-l upper trapezoidal. The sum k+l is equal to the effective\nnumerical rank of the (m+p)-by-n matrix (AH,BH)H.\nThis decomposition is the preprocessing step for computing the Generalized Singular Value Decomposition\n(GSVD), see subroutine ?tgsja.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\njobu\nMust be 'U' or 'N'.\nIf jobu = 'U', orthogonal/unitary matrix U is computed.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n977\n\n\nIf jobu = 'N', U is not computed.\njobv\nMust be 'V' or 'N'.\nIf jobv = 'V', orthogonal/unitary matrix V is computed.\nIf jobv = 'N', V is not computed.\njobq\nMust be 'Q' or 'N'.\nIf jobq = 'Q', orthogonal/unitary matrix Q is computed.\nIf jobq = 'N', Q is not computed.\nm\nThe number of rows of the matrix A (m≥ 0).\np\nThe number of rows of the matrix B (p≥ 0).\nn\nThe number of columns of the matrices A and B (n≥ 0).\na, b\nArrays:\na(size at least max(1, lda*n) for column major layout and max(1, lda*m)\nfor row major layout) contains the m-by-n matrix A.\nb(size at least max(1, ldb*n) for column major layout and max(1, ldb*p)\nfor row major layout) contains the p-by-n matrix B.\nlda\nThe leading dimension of a; at least max(1, m)for column major layout and\nmax(1, n) for row major layout.\nldb\nThe leading dimension of b; at least max(1, p)for column major layout and\nmax(1, n) for row major layout.\ntola, tolb\ntola and tolb are the thresholds to determine the effective numerical rank of\nmatrix B and a subblock of A. Generally, they are set to\ntola = max(m, n)*||A||*MACHEPS,\ntolb = max(p, n)*||B||*MACHEPS.\nThe size of tola and tolb may affect the size of backward errors of the\ndecomposition.\nldu\nThe leading dimension of the output array u . ldu≥ max(1, m) if jobu =\n'U'; ldu≥ 1 otherwise.\nldv\nThe leading dimension of the output array v . ldv≥ max(1, p) if jobv =\n'V'; ldv≥ 1 otherwise.\nldq\nThe leading dimension of the output array q . ldq≥ max(1, n) if jobq =\n'Q'; ldq≥ 1 otherwise.\nOutput Parameters\na\nOverwritten by the triangular (or trapezoidal) matrix described in the\nDescription section.\nb\nOverwritten by the triangular matrix described in the Description section.\nk, l\nOn exit, k and l specify the dimension of subblocks. The sum k + l is equal\nto effective numerical rank of (AH, BH)H.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n978\n\n\nu, v, q\nArrays:\nIf jobu = 'U', u (size max(1, ldu*m)) contains the orthogonal/unitary\nmatrix U.\nIf jobu = 'N', u is not referenced.\nIf jobv = 'V', v (size max(1, ldv*p)) contains the orthogonal/unitary\nmatrix V.\nIf jobv = 'N', v is not referenced.\nIf jobq = 'Q', q (size max(1, ldq*n)) contains the orthogonal/unitary\nmatrix Q.\nIf jobq = 'N', q is not referenced.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\n?ggsvp3\nPerforms preprocessing for a generalized SVD.\nSyntax\nlapack_int LAPACKE_sggsvp3 (int matrix_layout, char jobu, char jobv, char jobq,\nlapack_int m, lapack_int p, lapack_int n, float * a, lapack_int lda, float * b,\nlapack_int ldb, float tola, float tolb, lapack_int * k, lapack_int * l, float * u,\nlapack_int ldu, float * v, lapack_int ldv, float * q, lapack_int ldq);\nlapack_int LAPACKE_dggsvp3 (int matrix_layout, char jobu, char jobv, char jobq,\nlapack_int m, lapack_int p, lapack_int n, double * a, lapack_int lda, double * b,\nlapack_int ldb, double tola, double tolb, lapack_int * k, lapack_int * l, double * u,\nlapack_int ldu, double * v, lapack_int ldv, double * q, lapack_int ldq);\nlapack_int LAPACKE_cggsvp3 (int matrix_layout, char jobu, char jobv, char jobq,\nlapack_int m, lapack_int p, lapack_int n, lapack_complex_float * a, lapack_int lda,\nlapack_complex_float * b, lapack_int ldb, float tola, float tolb, lapack_int * k,\nlapack_int * l, lapack_complex_float * u, lapack_int ldu, lapack_complex_float * v,\nlapack_int ldv, lapack_complex_float * q, lapack_int ldq);\nlapack_int LAPACKE_zggsvp3 (int matrix_layout, char jobu, char jobv, char jobq,\nlapack_int m, lapack_int p, lapack_int n, lapack_complex_double * a, lapack_int lda,\nlapack_complex_double * b, lapack_int ldb, double tola, double tolb, lapack_int * k,\nlapack_int * l, lapack_complex_double * u, lapack_int ldu, lapack_complex_double * v,\nlapack_int ldv, lapack_complex_double * q, lapack_int ldq);\nInclude Files\n•\nmkl_lapack.h\nInclude Files\n•\nmkl.h\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n979\n\n\nDescription\n?ggsvp3 computes orthogonal or unitary matrices U, V, and Q such that\nfor real flavors:\nUTAQ =\nn −k −l k l\nk\nl\nm −k −l\n0\nA12\nA13\n0\n0\nA23\n0\n0\n0\n if m - k - l≥ 0;\nUTAQ =\nn −k −l k l\nk\nm −k\n0\nA12\nA13\n0\n0\nA23\n if m - k - l< 0;\nVTBQ =\nn −k −l k l\nl\np −l\n0 0 B13\n0 0\n0\nfor complex flavors:\nUHAQ =\nn −k −l k l\nk\nl\nm −k −l\n0 A12 A13\n0\n0\nA23\n0\n0\n0\n if m - k - l≥ 0;\nUHAQ =\nn −k −l k l\nk\nm −k\n0 A12 A13\n0\n0\nA23\n if m - k-l< 0;\nVHBQ =\nn −k −l k l\nl\np −l\n0 0 B13\n0 0\n0\nwhere the k-by-k matrix A12 and l-by-l matrix B13 are nonsingular upper triangular; A23 is l-by-l upper\ntriangular if m-k-l≥ 0, otherwise A23 is (m-k-by-l upper trapezoidal. k + l = the effective numerical rank of\nthe (m + p)-by-n matrix (AT,BT)T for real flavors or (AH,BH)H for complex flavors.\nThis decomposition is the preprocessing step for computing the Generalized Singular Value Decomposition\n(GSVD), see ?ggsvd3.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\njobu\n= 'U': Orthogonal/unitary matrix U is computed;\n= 'N': U is not computed.\njobv\n= 'V': Orthogonal/unitary matrix V is computed;\n= 'N': V is not computed.\njobq\n= 'Q': Orthogonal/unitary matrix Q is computed;\n= 'N': Q is not computed.\nm\nThe number of rows of the matrix A.\nm≥ 0.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n980\n\n\np\nThe number of rows of the matrix B.\np≥ 0.\nn\nThe number of columns of the matrices A and B.\nn≥ 0.\na\nArray, size (lda*n).\nOn entry, the m-by-n matrix A.\nlda\nThe leading dimension of the array a.\nlda≥ max(1,m).\nb\nArray, size (ldb*n).\nOn entry, the p-by-n matrix B.\nldb\nThe leading dimension of the array b.\nldb≥ max(1,p).\ntola, tolb\ntola and tolb are the thresholds to determine the effective numerical rank\nof matrix B and a subblock of A. Generally, they are set to\ntola = max(m,n)*norm(a)*MACHEPS,\ntolb = max(p,n)*norm(b)*MACHEPS.\nThe size of tola and tolb may affect the size of backward errors of the\ndecomposition.\nldu\nThe leading dimension of the array u.\nldu≥ max(1,m) if jobu = 'U'; ldu≥ 1 otherwise.\nldv\nThe leading dimension of the array v.\nldv≥ max(1,p) if jobv = 'V'; ldv≥ 1 otherwise.\nldq\nThe leading dimension of the array q.\nldq≥ max(1,n) if jobq = 'Q'; ldq≥ 1 otherwise.\nOutput Parameters\na\nOn exit, a contains the triangular (or trapezoidal) matrix described in\nthe Description section.\nb\nOn exit, b contains the triangular matrix described in the Description\nsection.\nk, l\nOn exit, k and l specify the dimension of the subblocks described in\nDescription section.\nk + l = effective numerical rank of (AT,BT)T for real flavors or\n(AH,BH)H for complex flavors.\nu\nArray, size (ldu*m).\nIf jobu = 'U', u contains the orthogonal/unitary matrix U.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n981\n\n\nIf jobu = 'N', u is not referenced.\nv\nArray, size (ldv*p).\nIf jobv = 'V', v contains the orthogonal/unitary matrix V.\nIf jobv = 'N', v is not referenced.\nq\nArray, size (ldq*n).\nIf jobq = 'Q', q contains the orthogonal/unitary matrix Q.\nIf jobq = 'N', q is not referenced.\nReturn Values\nThis function returns a value info.\n= 0: successful exit.\n< 0: if info = -i, the i-th argument had an illegal value.\nApplication Notes\nThe subroutine uses LAPACK subroutine ?geqp3 for the QR factorization with column pivoting to detect the\neffective numerical rank of the A matrix. It may be replaced by a better rank determination strategy.\n?ggsvp3 replaces the deprecated subroutine ?ggsvp.\n?ggsvd3\nComputes generalized SVD.\nSyntax\nlapack_int LAPACKE_sggsvd3 (int matrix_layout, char jobu, char jobv, char jobq,\nlapack_int m, lapack_int n, lapack_int p, lapack_int * k, lapack_int * l, float * a,\nlapack_int lda, float * b, lapack_int ldb, float * alpha, float * beta, float * u,\nlapack_int ldu, float * v, lapack_int ldv, float * q, lapack_int ldq, lapack_int *\niwork);\nlapack_int LAPACKE_dggsvd3 (int matrix_layout, char jobu, char jobv, char jobq,\nlapack_int m, lapack_int n, lapack_int p, lapack_int * k, lapack_int * l, double * a,\nlapack_int lda, double * b, lapack_int ldb, double * alpha, double * beta, double * u,\nlapack_int ldu, double * v, lapack_int ldv, double * q, lapack_int ldq, lapack_int *\niwork);\nlapack_int LAPACKE_cggsvd3 (int matrix_layout, char jobu, char jobv, char jobq,\nlapack_int m, lapack_int n, lapack_int p, lapack_int * k, lapack_int * l,\nlapack_complex_float * a, lapack_int lda, lapack_complex_float * b, lapack_int ldb,\nfloat * alpha, float * beta, lapack_complex_float * u, lapack_int ldu,\nlapack_complex_float * v, lapack_int ldv, lapack_complex_float * q, lapack_int ldq,\nlapack_int * iwork);\nlapack_int LAPACKE_zggsvd3 (int matrix_layout, char jobu, char jobv, char jobq,\nlapack_int m, lapack_int n, lapack_int p, lapack_int * k, lapack_int * l,\nlapack_complex_double * a, lapack_int lda, lapack_complex_double * b, lapack_int ldb,\ndouble * alpha, double * beta, lapack_complex_double * u, lapack_int ldu,\nlapack_complex_double * v, lapack_int ldv, lapack_complex_double * q, lapack_int ldq,\nlapack_int * iwork);\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n982\n\n\nInclude Files\n•\nmkl.h\nDescription\n?ggsvd3 computes the generalized singular value decomposition (GSVD) of an m-by-n real or complex matrix\nA and p-by-n real or complex matrix B:\nUT*A*Q = D1*( 0 R ), VT*B*Q = D2*( 0 R ) for real flavors\nor\nUH*A*Q = D1*( 0 R ), VH*B*Q = D2*( 0 R ) for complex flavors\nwhere U, V and Q are orthogonal/unitary matrices.\nLet k+l = the effective numerical rank of the matrix (ATBT)T for real flavors or the matrix (AH,BH)H for\ncomplex flavors, then R is a (k + l)-by-(k + l) nonsingular upper triangular matrix, D1 and D2 are m-by-(k +\nl) and p-by-(k + l) \"diagonal\" matrices and of the following structures, respectively:\nIf m-k-l≥ 0,\nD1 =\nk l\nk\nl\nm −k −l\nI 0\n0 C\n0 0\nD2 =\nk l\nl\np −l\n0 S\n0 0\n0 R =\nn −k −l k l\nk\nl\n0\nR11 R12\n0\n0\nR22\nwhere\nC = diag( alpha(k+1), ... , alpha(k+l) ),\nS = diag( beta(k+1), ... , beta(k+l) ),\nC2 + S2 = I.\nIf m - k - l < 0,\nD1 =\nk m −k k + l −m\nk\nm −k\nI\n0\n0\n0\nC\n0\nD2 =\nk m −k k + l −m\nm −k\nk + l −m\np −l\n0\nS\n0\n0\n0\nI\n0\n0\n0\n0 R =\nn −k −l k m −k k + l −m\nk\nm −k\nk + l −m\n0\nR11\nR12\nR13\n0\n0\nR22\nR23\n0\n0\n0\nR33\nwhere\nC = diag(alpha(k + 1), ... , alpha(m)),\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n983\n\n\nS = diag(beta(k + 1), ... , beta(m)),\nC2 + S2 = I.\nThe routine computes C, S, R, and optionally the orthogonal/unitary transformation matrices U, V and Q.\nIn particular, if B is an n-by-n nonsingular matrix, then the GSVD of A and B implicitly gives the SVD of\nA*inv(B):\nA*inv(B) = U*(D1*inv(D2))*VT for real flavors\nor\nA*inv(B) = U*(D1*inv(D2))*VH for complex flavors.\nIf (AT,BT)T for real flavors or (AH,BH)H for complex flavors has orthonormal columns, then the GSVD of A and\nB is also equal to the CS decomposition of A and B. Furthermore, the GSVD can be used to derive the\nsolution of the eigenvalue problem:\nAT*AX = λ* BT*BX for real flavors\nor\nAH*AX = λ* BH*BX for complex flavors\nIn some literature, the GSVD of A and B is presented in the form\nUT*A*X = ( 0 D1 ), VT*B*X = ( 0 D2 ) for real (A, B)\nor\nUH*A*X = ( 0 D1 ), VH*B*X = ( 0 D2 ) for complex (A, B)\nwhere U and V are orthogonal and X is nonsingular, D1 and D2 are \"diagonal''. The former GSVD form can be\nconverted to the latter form by taking the nonsingular matrix X as\nX = Q * I\n0\n0 inv R\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\njobu\n= 'U': Orthogonal/unitary matrix U is computed;\n= 'N': U is not computed.\njobv\n= 'V': Orthogonal/unitary matrix V is computed;\n= 'N': V is not computed.\njobq\n= 'Q': Orthogonal/unitary matrix Q is computed;\n= 'N': Q is not computed.\nm\nThe number of rows of the matrix A.\nm≥ 0.\nn\nThe number of columns of the matrices A and B.\nn≥ 0.\np\nThe number of rows of the matrix B.\np≥ 0.\na\nArray, size (lda*n).\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n984\n\n\nOn entry, the m-by-n matrix A.\nlda\nThe leading dimension of the array a.\nlda≥ max(1,m).\nb\nArray, size (ldb*n).\nOn entry, the p-by-n matrix B.\nldb\nThe leading dimension of the array b.\nldb≥ max(1,p).\nldu\nThe leading dimension of the array u.\nldu≥ max(1,m) if jobu = 'U'; ldu≥ 1 otherwise.\nldv\nThe leading dimension of the array v.\nldv≥ max(1,p) if jobv = 'V'; ldv≥ 1 otherwise.\nldq\nThe leading dimension of the array q.\nldq≥ max(1,n) if jobq = 'Q'; ldq≥ 1 otherwise.\niwork\nArray, size (n).\nOutput Parameters\nk, l\nOn exit, k and l specify the dimension of the subblocks described in\nthe Description section.\nk + l = effective numerical rank of (AT,BT)T for real flavors or\n(AH,BH)H for complex flavors.\na\nOn exit, a contains the triangular matrix R, or part of R.\nIf m-k-l≥ 0, R is stored in the elements of array a corresponding to A1:\nk + l,n - k - l + 1:n.\nIf m - k - l < 0, R11 R12 R13\n0\nR22 R23  is stored in the elements of array a\ncorresponding to A(1:m, n - k - l + 1:n, and R33 is stored in bthe\nelements of array a corresponding to Am - k + 1:l,n + m - k - l + 1:n on\nexit.\nb\nOn exit, b contains part of the triangular matrix R if m - k - l < 0.\nSee Description for details.\nalpha\nArray, size (n)\nbeta\nArray, size (n)\nOn exit, alpha and beta contain the generalized singular value pairs\nof a and b;\nalpha[0: k - 1] = 1,\nbeta[0: k - 1] = 0,\nand if m - k - l≥ 0,\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n985\n\n\nalpha[k:k + l - 1] = C,\nbeta[k:k + l - 1] = S,\nor if m - k - l < 0,\nalpha[k:m - 1] = C, alpha[m: k + l - 1] = 0\nbeta[k: m - 1] =S, beta[m: k + l - 1] = 1\nand\nalpha[k + l: n - 1] = 0\nbeta[k + l : n - 1] = 0\nu\nArray, size (ldu*m).\nIf jobu = 'U', u contains the m-by-m orthogonal/unitary matrix U.\nIf jobu = 'N', u is not referenced.\nv\nArray, size (ldv*p).\nIf jobv = 'V', v contains the p-by-p orthogonal/unitary matrix V.\nIf jobv = 'N', v is not referenced.\nq\nArray, size (ldq*n).\nIf jobq = 'Q', q contains the n-by-n orthogonal/unitary matrix Q.\nIf jobq = 'N', q is not referenced.\niwork\nOn exit, iwork stores the sorting information. More precisely, the\nfollowing loop uses iwork to sort alpha:\nfor (i = k; i<min(m,k + l); i++) {\n    swap (alpha[i], alpha[iwork[i] - 1]);\n}\nsuch that alpha[0] ≥alpha[1] ≥ ... ≥alpha[n - 1].\nReturn Values\nThis function returns a value info.\n= 0: successful exit.\n< 0: if info = -i, the i-th argument had an illegal value.\n> 0: if info = 1, the Jacobi-type procedure failed to converge.\nFor further details, see subroutine ?tgsja.\nApplication Notes\n?ggsvd3 replaces the deprecated subroutine ?ggsvd.\n?tgsja\nComputes the generalized SVD of two upper triangular\nor trapezoidal matrices.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n986\n\n\nSyntax\nlapack_int LAPACKE_stgsja( int matrix_layout, char jobu, char jobv, char jobq,\nlapack_int m, lapack_int p, lapack_int n, lapack_int k, lapack_int l, float* a,\nlapack_int lda, float* b, lapack_int ldb, float tola, float tolb, float* alpha, float*\nbeta, float* u, lapack_int ldu, float* v, lapack_int ldv, float* q, lapack_int ldq,\nlapack_int* ncycle );\nlapack_int LAPACKE_dtgsja( int matrix_layout, char jobu, char jobv, char jobq,\nlapack_int m, lapack_int p, lapack_int n, lapack_int k, lapack_int l, double* a,\nlapack_int lda, double* b, lapack_int ldb, double tola, double tolb, double* alpha,\ndouble* beta, double* u, lapack_int ldu, double* v, lapack_int ldv, double* q,\nlapack_int ldq, lapack_int* ncycle );\nlapack_int LAPACKE_ctgsja( int matrix_layout, char jobu, char jobv, char jobq,\nlapack_int m, lapack_int p, lapack_int n, lapack_int k, lapack_int l,\nlapack_complex_float* a, lapack_int lda, lapack_complex_float* b, lapack_int ldb, float\ntola, float tolb, float* alpha, float* beta, lapack_complex_float* u, lapack_int ldu,\nlapack_complex_float* v, lapack_int ldv, lapack_complex_float* q, lapack_int ldq,\nlapack_int* ncycle );\nlapack_int LAPACKE_ztgsja( int matrix_layout, char jobu, char jobv, char jobq,\nlapack_int m, lapack_int p, lapack_int n, lapack_int k, lapack_int l,\nlapack_complex_double* a, lapack_int lda, lapack_complex_double* b, lapack_int ldb,\ndouble tola, double tolb, double* alpha, double* beta, lapack_complex_double* u,\nlapack_int ldu, lapack_complex_double* v, lapack_int ldv, lapack_complex_double* q,\nlapack_int ldq, lapack_int* ncycle );\nInclude Files\n•\nmkl.h\nDescription\nThe routine computes the generalized singular value decomposition (GSVD) of two real/complex upper\ntriangular (or trapezoidal) matrices A and B. On entry, it is assumed that matrices A and B have the following\nforms, which may be obtained by the preprocessing subroutine ggsvp from a general m-by-n matrix A and p-\nby-n matrix B:\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n987\n\n\nwhere the k-by-k matrix A12 and l-by-l matrix B13 are nonsingular upper triangular; A23 is l-by-l upper\ntriangular if m-k-l≥0, otherwise A23 is (m-k)-by-l upper trapezoidal.\nOn exit,\nUH*A*Q = D1*(0 R), VH*B*Q = D2*(0 R),\nwhere U, V and Q are orthogonal/unitary matrices, R is a nonsingular upper triangular matrix, and D1 and D2\nare \"diagonal\" matrices, which are of the following structures:\nIf m-k-l≥0,\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n988\n\n\nwhere\nC = diag(alpha[k],...,alpha[k+l-1])\nS = diag(beta[k],...,beta[k+l-1])\nC2 + S2 = I\nR is stored in a(1:k+l, n-k-l+1:n ) on exit.\nIf m-k-l < 0,\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n989\n\n\nwhere\nC = diag(alpha[k],...,alpha[m-1]),\nS = diag(beta[k],...,beta[m-1]),\nC2 + S2 = I\nOn exit, \nis stored in a(1:m, n-k-l+1:n ) and R33 is stored\nin b(m-k+1:l, n+m-k-l+1:n ).\nThe computation of the orthogonal/unitary transformation matrices U, V or Q is optional. These matrices may\neither be formed explicitly, or they may be postmultiplied into input matrices U1, V1, or Q1.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\njobu\nMust be 'U', 'I', or 'N'.\nIf jobu = 'U', u must contain an orthogonal/unitary matrix U1 on entry.\nIf jobu = 'I', u is initialized to the unit matrix.\nIf jobu = 'N', u is not computed.\njobv\nMust be 'V', 'I', or 'N'.\nIf jobv = 'V', v must contain an orthogonal/unitary matrix V1 on entry.\nIf jobv = 'I', v is initialized to the unit matrix.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n990\n\n\nIf jobv = 'N', v is not computed.\njobq\nMust be 'Q', 'I', or 'N'.\nIf jobq = 'Q', q must contain an orthogonal/unitary matrix Q1 on entry.\nIf jobq = 'I', q is initialized to the unit matrix.\nIf jobq = 'N', q is not computed.\nm\nThe number of rows of the matrix A (m≥ 0).\np\nThe number of rows of the matrix B (p≥ 0).\nn\nThe number of columns of the matrices A and B (n≥ 0).\nk, l\nSpecify the subblocks in the input matrices A and B, whose GSVD is\ncomputed.\na, b, u, v, q\nArrays:\na(size at least max(1, lda*n) for column major layout and max(1, lda*m)\nfor row major layout) contains the m-by-n matrix A.\nb(size at least max(1, ldb*n) for column major layout and max(1, ldb*p)\nfor row major layout) contains the p-by-n matrix B.\nIf jobu = 'U', u (size max(1, ldu*m)) must contain a matrix U1 (usually\nthe orthogonal/unitary matrix returned by ?ggsvp).\nIf jobv = 'V', v (size at least max(1, ldv*p)) must contain a matrix V1\n(usually the orthogonal/unitary matrix returned by ?ggsvp).\nIf jobq = 'Q', q (size at least max(1, ldq*n)) must contain a matrix Q1\n(usually the orthogonal/unitary matrix returned by ?ggsvp).\nlda\nThe leading dimension of a; at least max(1, m)for column major layout and\nmax(1, n) for row major layout.\nldb\nThe leading dimension of b; at least max(1, p) for column major layout and\nmax(1, n) for row major layout.\nldu\nThe leading dimension of the array u .\nldu≥ max(1, m) if jobu = 'U'; ldu≥ 1 otherwise.\nldv\nThe leading dimension of the array v .\nldv≥ max(1, p) if jobv = 'V'; ldv≥ 1 otherwise.\nldq\nThe leading dimension of the array q .\nldq≥ max(1, n) if jobq = 'Q'; ldq≥ 1 otherwise.\ntola, tolb\ntola and tolb are the convergence criteria for the Jacobi-Kogbetliantz\niteration procedure. Generally, they are the same as used in ?ggsvp:\ntola = max(m, n)*|A|*MACHEPS,\ntolb = max(p, n)*|B|*MACHEPS.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n991\n\n\nOutput Parameters\na\nOn exit, a(n-k+1:n, 1:min(k+l, m)) contains the triangular matrix R or part\nof R.\nb\nOn exit, if necessary, b(m-k+1: l, n+m-k-l+1: n)) contains a part of R.\nalpha, beta\nArrays, size at least max(1, n). Contain the generalized singular value pairs\nof A and B:\nalpha(1:k) = 1,\nbeta(1:k) = 0,\nand if m-k-l≥ 0,\nalpha(k+1:k+l) = diag(C),\nbeta(k+1:k+l) = diag(S),\nor if m-k-l < 0,\nalpha(k+1:m)= diag(C), alpha(m+1:k+l)=0\nbeta(k+1:m) = diag(S),\nbeta(m+1:k+l) = 1.\nFurthermore, if k+l < n,\nalpha(k+l+1:n)= 0 and\nbeta(k+l+1:n) = 0.\nu\nIf jobu = 'I', u contains the orthogonal/unitary matrix U.\nIf jobu = 'U', u contains the product U1*U.\nIf jobu = 'N', u is not referenced.\nv\nIf jobv = 'I', v contains the orthogonal/unitary matrix U.\nIf jobv = 'V', v contains the product V1*V.\nIf jobv = 'N', v is not referenced.\nq\nIf jobq = 'I', q contains the orthogonal/unitary matrix U.\nIf jobq = 'Q', q contains the product Q1*Q.\nIf jobq = 'N', q is not referenced.\nncycle\nThe number of cycles required for convergence.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nIf info = 1, the procedure does not converge after MAXIT cycles.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n992\n\n\nCosine-Sine Decomposition: LAPACK Computational Routines\nThis topic describes LAPACK computational routines for computing the cosine-sine decomposition (CS\ndecomposition) of a partitioned unitary/orthogonal matrix. The algorithm computes a complete 2-by-2 CS\ndecomposition, which requires simultaneous diagonalization of all the four blocks of a unitary/orthogonal\nmatrix partitioned into a 2-by-2 block structure.\nThe computation has the following phases:\n1.\nThe matrix is reduced to a bidiagonal block form.\n2.\nThe blocks are simultaneously diagonalized using techniques from the bidiagonal SVD algorithms.\nTable \"Computational Routines for Cosine-Sine Decomposition (CSD)\" lists LAPACK routines that perform CS\ndecomposition of matrices.\nComputational Routines for Cosine-Sine Decomposition (CSD)\nOperation\nReal matrices\nComplex matrices\nCompute the CS decomposition of an\northogonal/unitary matrix in bidiagonal-block\nform\nbbcsd/bbcsd\nbbcsd/bbcsd\nSimultaneously bidiagonalize the blocks of a\npartitioned orthogonal matrix\norbdb unbdb\nSimultaneously bidiagonalize the blocks of a\npartitioned unitary matrix\norbdb unbdb\nSee Also\nCS Driver Routine \n?bbcsd\nComputes the CS decomposition of an orthogonal/\nunitary matrix in bidiagonal-block form.\nSyntax\nlapack_int LAPACKE_sbbcsd( int matrix_layout, char jobu1, char jobu2, char jobv1t, char\njobv2t, char trans, lapack_int m, lapack_int p, lapack_int q, float* theta, float* phi,\nfloat* u1, lapack_int ldu1, float* u2, lapack_int ldu2, float* v1t, lapack_int ldv1t,\nfloat* v2t, lapack_int ldv2t, float* b11d, float* b11e, float* b12d, float* b12e, float*\nb21d, float* b21e, float* b22d, float* b22e );\nlapack_int LAPACKE_dbbcsd( int matrix_layout, char jobu1, char jobu2, char jobv1t, char\njobv2t, char trans, lapack_int m, lapack_int p, lapack_int q, double* theta, double*\nphi, double* u1, lapack_int ldu1, double* u2, lapack_int ldu2, double* v1t, lapack_int\nldv1t, double* v2t, lapack_int ldv2t, double* b11d, double* b11e, double* b12d, double*\nb12e, double* b21d, double* b21e, double* b22d, double* b22e );\nlapack_int LAPACKE_cbbcsd( int matrix_layout, char jobu1, char jobu2, char jobv1t, char\njobv2t, char trans, lapack_int m, lapack_int p, lapack_int q, float* theta, float* phi,\nlapack_complex_float* u1, lapack_int ldu1, lapack_complex_float* u2, lapack_int ldu2,\nlapack_complex_float* v1t, lapack_int ldv1t, lapack_complex_float* v2t, lapack_int\nldv2t, float* b11d, float* b11e, float* b12d, float* b12e, float* b21d, float* b21e,\nfloat* b22d, float* b22e );\nlapack_int LAPACKE_zbbcsd( int matrix_layout, char jobu1, char jobu2, char jobv1t, char\njobv2t, char trans, lapack_int m, lapack_int p, lapack_int q, double* theta, double*\nphi, lapack_complex_double* u1, lapack_int ldu1, lapack_complex_double* u2, lapack_int\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n993\n\n\nldu2, lapack_complex_double* v1t, lapack_int ldv1t, lapack_complex_double* v2t,\nlapack_int ldv2t, double* b11d, double* b11e, double* b12d, double* b12e, double* b21d,\ndouble* b21e, double* b22d, double* b22e );\nInclude Files\n•\nmkl.h\nDescription\nmkl_lapack.fiThe routine ?bbcsd computes the CS decomposition of an orthogonal or unitary matrix in\nbidiagonal-block form:\nor\nrespectively.\nx is m-by-m with the top-left block p-by-q. Note that q must not be larger than p, m-p, or m-q. If q is not\nthe smallest index, x must be transposed and/or permuted in constant time using the trans option.\nSee ?orcsd/?uncsd for details.\nThe bidiagonal matrices b11, b12, b21, and b22 are represented implicitly by angles theta(1:q) and\nphi(1:q-1).\nThe orthogonal/unitary matrices u1, u2, v1t, and v2t are input/output. The input matrices are pre- or post-\nmultiplied by the appropriate singular vector matrices.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\njobu1\nIf equals Y, then u1 is updated. Otherwise, u1 is not updated.\njobu2\nIf equals Y, then u2 is updated. Otherwise, u2 is not updated.\njobv1t\nIf equals Y, then v1t is updated. Otherwise, v1t is not updated.\njobv2t\nIf equals Y, then v2t is updated. Otherwise, v2t is not updated.\ntrans\n= 'T':\nx, u1, u2, v1t, v2t are stored in row-major order.\notherwise\nx, u1, u2, v1t, v2t are stored in column-major\norder.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n994\n\n\nm\nThe number of rows and columns of the orthogonal/unitary matrix X in\nbidiagonal-block form.\np\nThe number of rows in the top-left block of x. 0 ≤p≤m.\n≤\nq\nThe number of columns in the top-left block of x. 0 q≤ min(p,m-p,m-q).\ntheta\nArray, size q.\nOn entry, the angles theta[0], ..., theta[q - 1] that, along with\nphi[0], ..., phi[q - 2], define the matrix in bidiagonal-block form as\nreturned by orbdb/unbdb.\nphi\nArray, size q-1.\nThe angles phi[0], ..., phi[q - 2] that, along with theta[0], ...,\ntheta[q - 1], define the matrix in bidiagonal-block form as returned by \norbdb/unbdb.\nu1\nArray, size at least max(1, ldu1*p).\nOn entry, a p-by-p matrix.\nldu1\nThe leading dimension of the array u1, ldu1≤ max(1, p).\nu2\nArray, size max(1, ldu2*(m-p)).\nOn entry, an (m-p)-by-(m-p) matrix.\nldu2\nThe leading dimension of the array u2, ldu2≤ max(1, m-p).\nv1t\nArray, size max(1, ldv1t*q).\nOn entry, a q-by-q matrix.\nldv1t\nThe leading dimension of the array v1t, ldv1t≤ max(1, q).\nv2t\nArray, size.\nOn entry, an (m-q)-by-(m-q) matrix.\nldv2t\nThe leading dimension of the array v2t, ldv2t≤ max(1, m-q).\nOutput Parameters\ntheta\nOn exit, the angles whose cosines and sines define the diagonal blocks in\nthe CS decomposition.\nu1\nOn exit, u1 is postmultiplied by the left singular vector matrix common to\n[ b11 ; 0 ] and [ b12 0 0 ; 0 -I 0 ].\nu2\nOn exit, u2 is postmultiplied by the left singular vector matrix common to\n[ b21 ; 0 ] and [ b22 0 0 ; 0 0 I ].\nv1t\nArray, size q.\nOn exit, v1t is premultiplied by the transpose of the right singular vector\nmatrix common to [ b11 ; 0 ] and [ b21 ; 0 ].\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n995\n\n\nv2t\nOn exit, v2t is premultiplied by the transpose of the right singular vector\nmatrix common to [ b12 0 0 ; 0 -I 0 ] and [ b22 0 0 ; 0 0 I ].\nb11d\nArray, size q.\nWhen ?bbcsd converges, b11d contains the cosines of theta[0], ...,\ntheta[q - 1]. If ?bbcsd fails to converge, b11d contains the diagonal of\nthe partially reduced top left block.\nb11e\nArray, size q-1.\nWhen ?bbcsd converges, b11e contains zeros. If ?bbcsd fails to converge,\nb11e contains the superdiagonal of the partially reduced top left block.\nb12d\nArray, size q.\nWhen ?bbcsd converges, b12d contains the negative sines of\ntheta[0], ..., theta[q - 1]. If ?bbcsd fails to converge, b12d contains\nthe diagonal of the partially reduced top right block.\nb12e\nArray, size q-1.\nWhen ?bbcsd converges, b12e contains zeros. If ?bbcsd fails to converge,\nb11e contains the superdiagonal of the partially reduced top right block.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nIf info > 0 and if ?bbcsd did not converge, info specifies the number of nonzero entries in phi, and b11d,\nb11e, etc. contain the partially reduced matrix.\nSee Also\n?orcsd/?uncsd\nxerbla\n?orbdb/?unbdb\nSimultaneously bidiagonalizes the blocks of a\npartitioned orthogonal/unitary matrix.\nSyntax\nlapack_int LAPACKE_sorbdb( int matrix_layout, char trans, char signs, lapack_int m,\nlapack_int p, lapack_int q, float* x11, lapack_int ldx11, float* x12, lapack_int ldx12,\nfloat* x21, lapack_int ldx21, float* x22, lapack_int ldx22, float* theta, float* phi,\nfloat* taup1, float* taup2, float* tauq1, float* tauq2 );\nlapack_int LAPACKE_dorbdb( int matrix_layout, char trans, char signs, lapack_int m,\nlapack_int p, lapack_int q, double* x11, lapack_int ldx11, double* x12, lapack_int\nldx12, double* x21, lapack_int ldx21, double* x22, lapack_int ldx22, double* theta,\ndouble* phi, double* taup1, double* taup2, double* tauq1, double* tauq );\nlapack_int LAPACKE_cunbdb( int matrix_layout, char trans, char signs, lapack_int m,\nlapack_int p, lapack_int q, lapack_complex_float* x11, lapack_int ldx11,\nlapack_complex_float* x12, lapack_int ldx12, lapack_complex_float* x21, lapack_int\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n996\n\n\nldx21, lapack_complex_float* x22, lapack_int ldx22, float* theta, float* phi,\nlapack_complex_float* taup1, lapack_complex_float* taup2, lapack_complex_float* tauq1,\nlapack_complex_float* tauq2 );\nlapack_int LAPACKE_zunbdb( int matrix_layout, char trans, char signs, lapack_int m,\nlapack_int p, lapack_int q, lapack_complex_double* x11, lapack_int ldx11,\nlapack_complex_double* x12, lapack_int ldx12, lapack_complex_double* x21, lapack_int\nldx21, lapack_complex_double* x22, lapack_int ldx22, double* theta, double* phi,\nlapack_complex_double* taup1, lapack_complex_double* taup2, lapack_complex_double*\ntauq1, lapack_complex_double* tauq2 );\nInclude Files\n•\nmkl.h\nDescription\nThe routines ?orbdb/?unbdb simultaneously bidiagonalizes the blocks of an m-by-m partitioned orthogonal\nmatrix X:\nor unitary matrix:\nx11 is p-by-q. q must not be larger than p, m-p, or m-q. Otherwise, x must be transposed and/or permuted\nin constant time using the trans and signs options.\nThe orthogonal/unitary matrices p1, p2, q1, and q2 are p-by-p, (m-p)-by-(m-p), q-by-q, (m-q)-by-(m-q),\nrespectively. They are represented implicitly by Housholder vectors.\nThe bidiagonal matrices b11, b12, b21, and b22 are q-by-q bidiagonal matrices represented implicitly by angles\ntheta[0], ..., theta[q - 1] and phi[0], ..., phi[q - 2]. b11 and b12 are upper bidiagonal, while b21 and\nb22 are lower bidiagonal. Every entry in each bidiagonal band is a product of a sine or cosine of theta with a\nsine or cosine of phi. See [Sutton09] for details.\np1, p2, q1, and q2 are represented as products of elementary reflectors. .\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\ntrans\n= 'T':\nx, u1, u2, v1t, v2t are stored in row-major order.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n997\n\n\notherwise\nx, u1, u2, v1t, v2t are stored in column-major\norder.\nsigns\n= 'O':\nThe lower-left block is made nonpositive (the\n\"other\" convention).\notherwise\nThe upper-right block is made nonpositive (the\n\"default\" convention).\nm\nThe number of rows and columns of the matrix X.\np\nThe number of rows in x11 and x12. 0 ≤p≤m.\nq\nThe number of columns in x11 and x21. 0 ≤q≤ min(p,m-p,m-q).\nx11\nArray, size (size max(1, ldx11*q) for column major layout and max(1,\nldx11*p) for row major layout) .\nOn entry, the top-left block of the orthogonal/unitary matrix to be reduced.\nldx11\nThe leading dimension of the array X11. If trans = 'T', ldx11≥p for column\nmajor layout and ldx11≥q for row major layout. Otherwise, ldx11≥q.\nx12\nArray, size (size max(1, ldx12*(m-q)) for column major layout and max(1,\nldx12*p) for row major layout).\nOn entry, the top-right block of the orthogonal/unitary matrix to be\nreduced.\nldx12\nThe leading dimension of the array X12. If trans = 'N', ldx12≥p for column\nmajor layout and ldx12≥m - q for row major layout. . Otherwise,\nldx12≥m-q.\nx21\nArray, size (size max(1, ldx21*q) for column major layout and max(1,\nldx21*(m-p)) for row major layout).\nOn entry, the bottom-left block of the orthogonal/unitary matrix to be\nreduced.\nldx21\nThe leading dimension of the array X21. If trans = 'N', ldx21≥m-p for\ncolumn major layout and ldx12≥q for row major layout. . Otherwise,\nldx21≥q.\nx22\nArray, size ((size max(1, ldx22*(m-q)) for column major layout and max(1,\nldx22*(m - p)) for row major layout).\nOn entry, the bottom-right block of the orthogonal/unitary matrix to be\nreduced.\nldx22\nThe leading dimension of the array X21. If trans = 'N', ldx22≥m-p for\ncolumn major layout and ldx22≥m - q for row major layout. . Otherwise,\nldx22≥m-q.\nOutput Parameters\nx11\nOn exit, the form depends on trans:\nIf trans='N',\nthe columns of the lower triangle of x11 specify\nreflectors for p1, the rows of the upper triangle of\nx11(1:q - 1, q:q - 1) specify reflectors for q1\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n998\n\n\notherwise\ntrans='T',\nthe rows of the upper triangle of x11 specify reflectors\nfor p1, the columns of the lower triangle of x11(1:q -\n1, q:q - 1) specify reflectors for q1\nx12\nOn exit, the form depends on trans:\nIf trans='N',\nthe columns of the upper triangle of x12 specify the first\np reflectors for q2\notherwise\ntrans='T',\nthe columns of the lower triangle of x12 specify the first\np reflectors for q2\nx21\nOn exit, the form depends on trans:\nIf trans='N',\nthe columns of the lower triangle of x21 specify the\nreflectors for p2\notherwise\ntrans='T',\nthe columns of the upper triangle of x21 specify the\nreflectors for p2\nx22\nOn exit, the form depends on trans:\nIf trans='N',\nthe rows of the upper triangle of x22(q+1:m-p,p+1:m-\nq) specify the last m-p-q reflectors for q2\notherwise\ntrans='T',\nthe columns of the lower triangle of x22(p+1:m-q,q\n+1:m-p) specify the last m-p-q reflectors for p2\ntheta\nArray, size q. The entries of bidiagonal blocks b11, b12, b21, and b22 can be\ncomputed from the angles theta and phi. See the Description section for\ndetails.\nphi\nArray, size q-1. The entries of bidiagonal blocks b11, b12, b21, and b22 can\nbe computed from the angles theta and phi. See the Description section\nfor details.\ntaup1\nArray, size p.\nScalar factors of the elementary reflectors that define p1.\ntaup2\nArray, size m-p.\nScalar factors of the elementary reflectors that define p2.\ntauq1\nArray, size q.\nScalar factors of the elementary reflectors that define q1.\ntauq2\nArray, size m-q.\nScalar factors of the elementary reflectors that define q2.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nSee Also\n?orcsd/?uncsd\n?orgqr\n?ungqr\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n999\n\n\n?orglq\n?unglq\nxerbla\nLAPACK Least Squares and Eigenvalue Problem Driver Routines\nEach of the LAPACK driver routines solves a complete problem. To arrive at the solution, driver routines\ntypically call a sequence of appropriate computational routines.\nDriver routines are described in the following topics :\nLinear Least Squares (LLS) Problems\nGeneralized LLS Problems\nSymmetric Eigenproblems\nNonsymmetric Eigenproblems\nSingular Value Decomposition\nCosine-Sine Decomposition\nGeneralized Symmetric Definite Eigenproblems\nGeneralized Nonsymmetric Eigenproblems\nLinear Least Squares (LLS) Problems: LAPACK Driver Routines\nThis topic describes LAPACK driver routines used for solving linear least squares problems. Table \"Driver\nRoutines for Solving LLS Problems\" lists all such routines.\nDriver Routines for Solving LLS Problems\nRoutine Name\nOperation performed\ngels\nUses QR or LQ factorization to solve a overdetermined or underdetermined linear\nsystem with full rank matrix.\ngelsy\nComputes the minimum-norm solution to a linear least squares problem using a\ncomplete orthogonal factorization of A.\ngelss\nComputes the minimum-norm solution to a linear least squares problem using the\nsingular value decomposition of A.\ngelsd\nComputes the minimum-norm solution to a linear least squares problem using the\nsingular value decomposition of A and a divide and conquer method.\n?gels\nUses QR or LQ factorization to solve a overdetermined\nor underdetermined linear system with full rank\nmatrix.\nSyntax\nlapack_int LAPACKE_sgels (int matrix_layout, char trans, lapack_int m, lapack_int n,\nlapack_int nrhs, float* a, lapack_int lda, float* b, lapack_int ldb);\nlapack_int LAPACKE_dgels (int matrix_layout, char trans, lapack_int m, lapack_int n,\nlapack_int nrhs, double* a, lapack_int lda, double* b, lapack_int ldb);\nlapack_int LAPACKE_cgels (int matrix_layout, char trans, lapack_int m, lapack_int n,\nlapack_int nrhs, lapack_complex_float* a, lapack_int lda, lapack_complex_float* b,\nlapack_int ldb);\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1000\n\n\nlapack_int LAPACKE_zgels (int matrix_layout, char trans, lapack_int m, lapack_int n,\nlapack_int nrhs, lapack_complex_double* a, lapack_int lda, lapack_complex_double* b,\nlapack_int ldb);\nInclude Files\n•\nmkl.h\nDescription\nThe routine solves overdetermined or underdetermined real/ complex linear systems involving an m-by-n\nmatrix A, or its transpose/ conjugate-transpose, using a QR or LQ factorization of A. It is assumed that A has\nfull rank.\nThe following options are provided:\n1. If trans = 'N' and m≥n: find the least squares solution of an overdetermined system, that is, solve the\nleast squares problem\nminimize ||b - A*x||2\n2. If trans = 'N' and m < n: find the minimum norm solution of an underdetermined system A*X = B.\n3. If trans = 'T' or 'C' and m≥n: find the minimum norm solution of an undetermined system AH*X = B.\n4. If trans = 'T' or 'C' and m < n: find the least squares solution of an overdetermined system, that is,\nsolve the least squares problem\nminimize ||b - AH*x||2\nSeveral right hand side vectors b and solution vectors x can be handled in a single call; they are formed by\nthe columns of the right hand side matrix B and the solution matrix X (when coefficient matrix is A, B is m-\nby-nrhs and X is n-by-nrhs; if the coefficient matrix is AT or AH, B isn-by-nrhs and X is m-by-nrhs.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\ntrans\nMust be 'N', 'T', or 'C'.\nIf trans = 'N', the linear system involves matrix A;\nIf trans = 'T', the linear system involves the transposed matrix AT (for\nreal flavors only);\nIf trans = 'C', the linear system involves the conjugate-transposed\nmatrix AH (for complex flavors only).\nm\nThe number of rows of the matrix A (m≥ 0).\nn\nThe number of columns of the matrix A\n(n≥ 0).\nnrhs\nThe number of right-hand sides; the number of columns in B (nrhs≥ 0).\na, b\nArrays:\na(size max(1, lda*n) for column major layout and max(1, lda*m) for row\nmajor layout) contains the m-by-n matrix A.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1001\n\n\nb(size max(1, ldb*nrhs) for column major layout and max(1, ldb*max(m,\nn)) for row major layout) contains the matrix B of right hand side vectors.\nlda\nThe leading dimension of a; at least max(1, m) for column major layout and\nat least max(1, n) for row major layout.\nldb\nThe leading dimension of b; must be at least max(1, m, n) for column\nmajor layout if trans='N' and at least max(1, n) if trans='T' and at\nleast max(1, nrhs) for row major layout regardless of the value of trans.\nOutput Parameters\na\nOn exit, overwritten by the factorization data as follows:\nif m≥n, array a contains the details of the QR factorization of the matrix A as\nreturned by ?geqrf;\nif m < n, array a contains the details of the LQ factorization of the matrix A\nas returned by ?gelqf.\nb\nIf info = 0, b overwritten by the solution vectors, stored columnwise:\nif trans = 'N' and m≥n, rows 1 to n of b contain the least squares solution\nvectors; the residual sum of squares for the solution in each column is\ngiven by the sum of squares of modulus of elements n+1 to m in that\ncolumn;\nif trans = 'N' and m < n, rows 1 to n of b contain the minimum norm\nsolution vectors;\nif trans = 'T' or 'C' and m≥n, rows 1 to m of b contain the minimum\nnorm solution vectors;\nif trans = 'T' or 'C' and m < n, rows 1 to m of b contain the least\nsquares solution vectors; the residual sum of squares for the solution in\neach column is given by the sum of squares of modulus of elements m+1 to\nn in that column.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nIf info = i, the i-th diagonal element of the triangular factor of A is zero, so that A does not have full rank;\nthe least squares solution could not be computed.\n?gelsy\nComputes the minimum-norm solution to a linear least\nsquares problem using a complete orthogonal\nfactorization of A.\nSyntax\nlapack_int LAPACKE_sgelsy( int matrix_layout, lapack_int m, lapack_int n, lapack_int\nnrhs, float* a, lapack_int lda, float* b, lapack_int ldb, lapack_int* jpvt, float rcond,\nlapack_int* rank );\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1002\n\n\nlapack_int LAPACKE_dgelsy( int matrix_layout, lapack_int m, lapack_int n, lapack_int\nnrhs, double* a, lapack_int lda, double* b, lapack_int ldb, lapack_int* jpvt, double\nrcond, lapack_int* rank );\nlapack_int LAPACKE_cgelsy( int matrix_layout, lapack_int m, lapack_int n, lapack_int\nnrhs, lapack_complex_float* a, lapack_int lda, lapack_complex_float* b, lapack_int ldb,\nlapack_int* jpvt, float rcond, lapack_int* rank );\nlapack_int LAPACKE_zgelsy( int matrix_layout, lapack_int m, lapack_int n, lapack_int\nnrhs, lapack_complex_double* a, lapack_int lda, lapack_complex_double* b, lapack_int\nldb, lapack_int* jpvt, double rcond, lapack_int* rank );\nInclude Files\n•\nmkl.h\nDescription\nThe ?gelsy routine computes the minimum-norm solution to a real/complex linear least squares problem:\nminimize ||b - A*x||2\nusing a complete orthogonal factorization of A. A is an m-by-n matrix which may be rank-deficient. Several\nright hand side vectors b and solution vectors x can be handled in a single call; they are stored as the\ncolumns of the m-by-nrhs right hand side matrix B and the n-by-nrhs solution matrix X.\nThe routine first computes a QR factorization with column pivoting:\nwith R11 defined as the largest leading submatrix whose estimated condition number is less than 1/rcond.\nThe order of R11, rank, is the effective rank of A. Then, R22 is considered to be negligible, and R12 is\nannihilated by orthogonal/unitary transformations from the right, arriving at the complete orthogonal\nfactorization:\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1003\n\n\nThe minimum-norm solution is then\nfor real flavors and\nfor complex flavors,\nwhere Q1 consists of the first rank columns of Q.\nThe ?gelsy routine is identical to the original deprecated ?gelsx routine except for the following\ndifferences:\n•\nThe call to the subroutine ?geqpf has been substituted by the call to the subroutine ?geqp3, which is a\nBLAS-3 version of the QR factorization with column pivoting.\n•\nThe matrix B (the right hand side) is updated with BLAS-3.\n•\nThe permutation of the matrix B (the right hand side) is faster and more simple.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nm\nThe number of rows of the matrix A (m≥ 0).\nn\nThe number of columns of the matrix A\n(n≥ 0).\nnrhs\nThe number of right-hand sides; the number of columns in B (nrhs≥ 0).\na, b\nArrays:\na(size max(1, lda*n) for column major layout and max(1, lda*m) for row\nmajor layout) contains the m-by-n matrix A.\nb(size max(1, ldb*nrhs) for column major layout and max(1, ldb*max(m,\nn)) for row major layout) contains the m-by-nrhs right hand side matrix B.\nlda\nThe leading dimension of a; at least max(1, m)for column major layout and\nmax(1, n) for row major layout.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1004\n\n\nldb\nThe leading dimension of b; must be at least max(1, m, n) for column\nmajor layout and at least max(1, nrhs) for row major layout.\njpvt\nArray, size at least max(1, n).\nOn entry, if jpvt[i - 1]≠ 0, the i-th column of A is permuted to the front of\nAP, otherwise the i-th column of A is a free column.\nrcond\nrcond is used to determine the effective rank of A, which is defined as the\norder of the largest leading triangular submatrix R11 in the QR factorization\nwith pivoting of A, whose estimated condition number < 1/rcond.\nOutput Parameters\na\nOn exit, overwritten by the details of the complete orthogonal factorization\nof A.\nb\nOverwritten by the n-by-nrhs solution matrix X.\njpvt\nOn exit, if jpvt[i - 1]= k, then the i-th column of AP was the k-th column of\nA.\nrank\nThe effective rank of A, that is, the order of the submatrix R11. This is the\nsame as the order of the submatrix T11 in the complete orthogonal\nfactorization of A.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\n?gelss\nComputes the minimum-norm solution to a linear least\nsquares problem using the singular value\ndecomposition of A.\nSyntax\nlapack_int LAPACKE_sgelss( int matrix_layout, lapack_int m, lapack_int n, lapack_int\nnrhs, float* a, lapack_int lda, float* b, lapack_int ldb, float* s, float rcond,\nlapack_int* rank );\nlapack_int LAPACKE_dgelss( int matrix_layout, lapack_int m, lapack_int n, lapack_int\nnrhs, double* a, lapack_int lda, double* b, lapack_int ldb, double* s, double rcond,\nlapack_int* rank );\nlapack_int LAPACKE_cgelss( int matrix_layout, lapack_int m, lapack_int n, lapack_int\nnrhs, lapack_complex_float* a, lapack_int lda, lapack_complex_float* b, lapack_int ldb,\nfloat* s, float rcond, lapack_int* rank );\nlapack_int LAPACKE_zgelss( int matrix_layout, lapack_int m, lapack_int n, lapack_int\nnrhs, lapack_complex_double* a, lapack_int lda, lapack_complex_double* b, lapack_int\nldb, double* s, double rcond, lapack_int* rank );\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1005\n\n\nInclude Files\n•\nmkl.h\nDescription\nThe routine computes the minimum norm solution to a real linear least squares problem:\nminimize ||b - A*x||2\nusing the singular value decomposition (SVD) of A. A is an m-by-n matrix which may be rank-deficient.\nSeveral right hand side vectors b and solution vectors x can be handled in a single call; they are stored as\nthe columns of the m-by-nrhs right hand side matrix B and the n-by-nrhs solution matrix X. The effective\nrank of A is determined by treating as zero those singular values which are less than rcond times the largest\nsingular value.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nm\nThe number of rows of the matrix A (m≥ 0).\nn\nThe number of columns of the matrix A\n(n≥ 0).\nnrhs\nThe number of right-hand sides; the number of columns in B\n(nrhs≥ 0).\na, b\nArrays:\na(size max(1, lda*n) for column major layout and max(1, lda*m) for row\nmajor layout) contains the m-by-n matrix A.\nb(size max(1, ldb*nrhs) for column major layout and max(1, ldb*max(m,\nn)) for row major layout) contains the m-by-nrhs right hand side matrix B.\nlda\nThe leading dimension of a; at least max(1, m)for column major layout and\nmax(1, n) for row major layout.\nldb\nThe leading dimension of b; must be at least max(1, m, n) for column\nmajor layout and at least max(1, nrhs) for row major layout.\nrcond\nrcond is used to determine the effective rank of A. Singular values s(i)\n≤rcond *s(1) are treated as zero.\nIf rcond <0, machine precision is used instead.\nOutput Parameters\na\nOn exit, the first min(m, n) rows of a are overwritten with the matrix of\nright singular vectors of A, stored row-wise.\nb\nOverwritten by the n-by-nrhs solution matrix X.\nIf m≥n and rank = n, the residual sum-of-squares for the solution in the i-\nth column is given by the sum of squares of modulus of elements n+1:m in\nthat column.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1006\n\n\ns\nArray, size at least max(1, min(m, n)). The singular values of A in\ndecreasing order. The condition number of A in the 2-norm is\nk2(A) = s(1)/ s(min(m, n)) .\nrank\nThe effective rank of A, that is, the number of singular values which are\ngreater than rcond *s(1).\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nIf info = i, then the algorithm for computing the SVD failed to converge; i indicates the number of off-\ndiagonal elements of an intermediate bidiagonal form which did not converge to zero.\n?gelsd\nComputes the minimum-norm solution to a linear least\nsquares problem using the singular value\ndecomposition of A and a divide and conquer method.\nSyntax\nlapack_int LAPACKE_sgelsd( int matrix_layout, lapack_int m, lapack_int n, lapack_int\nnrhs, float* a, lapack_int lda, float* b, lapack_int ldb, float* s, float rcond,\nlapack_int* rank );\nlapack_int LAPACKE_dgelsd( int matrix_layout, lapack_int m, lapack_int n, lapack_int\nnrhs, double* a, lapack_int lda, double* b, lapack_int ldb, double* s, double rcond,\nlapack_int* rank );\nlapack_int LAPACKE_cgelsd( int matrix_layout, lapack_int m, lapack_int n, lapack_int\nnrhs, lapack_complex_float* a, lapack_int lda, lapack_complex_float* b, lapack_int ldb,\nfloat* s, float rcond, lapack_int* rank );\nlapack_int LAPACKE_zgelsd( int matrix_layout, lapack_int m, lapack_int n, lapack_int\nnrhs, lapack_complex_double* a, lapack_int lda, lapack_complex_double* b, lapack_int\nldb, double* s, double rcond, lapack_int* rank );\nInclude Files\n•\nmkl.h\nDescription\nThe routine computes the minimum-norm solution to a real linear least squares problem:\nminimize ||b - A*x||2\nusing the singular value decomposition (SVD) of A. A is an m-by-n matrix which may be rank-deficient.\nSeveral right hand side vectors b and solution vectors x can be handled in a single call; they are stored as\nthe columns of the m-by-nrhs right hand side matrix B and the n-by-nrhs solution matrix X.\nThe problem is solved in three steps:\n1.\nReduce the coefficient matrix A to bidiagonal form with Householder transformations, reducing the\noriginal problem into a \"bidiagonal least squares problem\" (BLS).\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1007\n\n\n2.\nSolve the BLS using a divide and conquer approach.\n3.\nApply back all the Householder transformations to solve the original least squares problem.\nThe effective rank of A is determined by treating as zero those singular values which are less than rcond\ntimes the largest singular value.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nm\nThe number of rows of the matrix A (m≥ 0).\nn\nThe number of columns of the matrix A\n(n≥ 0).\nnrhs\nThe number of right-hand sides; the number of columns in B (nrhs≥ 0).\na, b\nArrays:\na(size max(1, lda*n) for column major layout and max(1, lda*m) for row\nmajor layout) contains the m-by-n matrix A.\nb(size max(1, ldb*nrhs) for column major layout and max(1, ldb*max(m,\nn)) for row major layout) contains the m-by-nrhs right hand side matrix B.\nlda\nThe leading dimension of a; at least max(1, m)for column major layout and\nmax(1, n) for row major layout.\nldb\nThe leading dimension of b; must be at least max(1, m, n) for column\nmajor layout and at least max(1, nrhs) for row major layout.\nrcond\nrcond is used to determine the effective rank of A. Singular values s(i)\n≤rcond *s(1) are treated as zero. If rcond≤ 0, machine precision is used\ninstead.\nOutput Parameters\na\nOn exit, A has been overwritten.\nb\nOverwritten by the n-by-nrhs solution matrix X.\nIf m≥n and rank = n, the residual sum-of-squares for the solution in the i-\nth column is given by the sum of squares of modulus of elements n+1:m in\nthat column.\ns\nArray, size at least max(1, min(m, n)). The singular values of A in\ndecreasing order. The condition number of A in the 2-norm is\nk2(A) = s(1)/ s(min(m, n)).\nrank\nThe effective rank of A, that is, the number of singular values which are\ngreater than rcond *s(1).\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1008\n\n\nIf info = i, then the algorithm for computing the SVD failed to converge; i indicates the number of off-\ndiagonal elements of an intermediate bidiagonal form that did not converge to zero.\nGeneralized Linear Least Squares (LLS) Problems: LAPACK Driver Routines\nThis topic describes LAPACK driver routines used for solving generalized linear least squares problems. Table\n\"Driver Routines for Solving Generalized LLS Problems\" lists all such routines.\nDriver Routines for Solving Generalized LLS Problems\nRoutine Name\nOperation performed\ngglse\nSolves the linear equality-constrained least squares problem using a generalized RQ\nfactorization.\nggglm\nSolves a general Gauss-Markov linear model problem using a generalized QR\nfactorization.\n?gglse\nSolves the linear equality-constrained least squares\nproblem using a generalized RQ factorization.\nSyntax\nlapack_int LAPACKE_sgglse (int matrix_layout, lapack_int m, lapack_int n, lapack_int p,\nfloat* a, lapack_int lda, float* b, lapack_int ldb, float* c, float* d, float* x);\nlapack_int LAPACKE_dgglse (int matrix_layout, lapack_int m, lapack_int n, lapack_int p,\ndouble* a, lapack_int lda, double* b, lapack_int ldb, double* c, double* d, double* x);\nlapack_int LAPACKE_cgglse (int matrix_layout, lapack_int m, lapack_int n, lapack_int p,\nlapack_complex_float* a, lapack_int lda, lapack_complex_float* b, lapack_int ldb,\nlapack_complex_float* c, lapack_complex_float* d, lapack_complex_float* x);\nlapack_int LAPACKE_zgglse (int matrix_layout, lapack_int m, lapack_int n, lapack_int p,\nlapack_complex_double* a, lapack_int lda, lapack_complex_double* b, lapack_int ldb,\nlapack_complex_double* c, lapack_complex_double* d, lapack_complex_double* x);\nInclude Files\n•\nmkl.h\nDescription\nThe routine solves the linear equality-constrained least squares (LSE) problem:\nminimize ||c - A*x||2 subject to B*x = d\nwhere A is an m-by-n matrix, B is a p-by-n matrix, c is a given m-vector, andd is a given p-vector. It is\nassumed that p≤n≤m+p, and\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1009\n\n\nThese conditions ensure that the LSE problem has a unique solution, which is obtained using a generalized\nRQ factorization of the matrices (B, A) given by\nB=(0 R)*Q, A=Z*T*Q\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nm\nThe number of rows of the matrix A (m≥ 0).\nn\nThe number of columns of the matrices A and B (n≥ 0).\np\nThe number of rows of the matrix B\n(0 ≤p≤n≤m+p).\na, b, c, d\nArrays:\na(size max(1, lda*n) for column major layout and max(1, lda*m) for row\nmajor layout) contains the m-by-n matrix A.\nb(size max(1, ldb*n) for column major layout and max(1, ldb*p) for row\nmajor layout) contains the p-by-nmatrix B.\nc size at least max(1, m), contains the right hand side vector for the least\nsquares part of the LSE problem.\nd, size at least max(1, p), contains the right hand side vector for the\nconstrained equation.\nlda\nThe leading dimension of a; at least max(1, m)for column major layout and\nmax(1, n) for row major layout.\nldb\nThe leading dimension of b; at least max(1, p)for column major layout and\nmax(1, n) for row major layout.\nOutput Parameters\na\nThe elements on and above the diagonal contain the min(m, n)-by-n upper\ntrapezoidal matrix T as returned by ?ggrqf.\nx\nThe solution of the LSE problem.\nb\nOn exit, the upper right triangle contains the p-by-p upper triangular matrix\nR as returned by ?ggrqf.\nd\nOn exit, d is destroyed.\nc\nOn exit, the residual sum-of-squares for the solution is given by the sum of\nsquares of elements n-p+1 to m of vector c.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1010\n\n\nIf info = 1, the upper triangular factor R associated with B in the generalized RQ factorization of the pair\n(B, A) is singular, so that rank(B) < p; the least squares solution could not be computed.\nIf info = 2, the (n-p)-by-(n-p) part of the upper trapezoidal factor T associated with A in the generalized\nRQ factorization of the pair (B, A) is singular, so that\n; the least squares solution could not be computed.\n?ggglm\nSolves a general Gauss-Markov linear model problem\nusing a generalized QR factorization.\nSyntax\nlapack_int LAPACKE_sggglm (int matrix_layout, lapack_int n, lapack_int m, lapack_int p,\nfloat* a, lapack_int lda, float* b, lapack_int ldb, float* d, float* x, float* y);\nlapack_int LAPACKE_dggglm (int matrix_layout, lapack_int n, lapack_int m, lapack_int p,\ndouble* a, lapack_int lda, double* b, lapack_int ldb, double* d, double* x, double* y);\nlapack_int LAPACKE_cggglm (int matrix_layout, lapack_int n, lapack_int m, lapack_int p,\nlapack_complex_float* a, lapack_int lda, lapack_complex_float* b, lapack_int ldb,\nlapack_complex_float* d, lapack_complex_float* x, lapack_complex_float* y);\nlapack_int LAPACKE_zggglm (int matrix_layout, lapack_int n, lapack_int m, lapack_int p,\nlapack_complex_double* a, lapack_int lda, lapack_complex_double* b, lapack_int ldb,\nlapack_complex_double* d, lapack_complex_double* x, lapack_complex_double* y);\nInclude Files\n•\nmkl.h\nDescription\nThe routine solves a general Gauss-Markov linear model (GLM) problem:\nminimizex ||y||2 subject to d = A*x + B*y\nwhere A is an n-by-m matrix, B is an n-by-p matrix, and d is a given n-vector. It is assumed that m≤n≤m+p,\nand rank(A) = m and rank(AB) = n.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1011\n\n\nUnder these assumptions, the constrained equation is always consistent, and there is a unique solution x and\na minimal 2-norm solution y, which is obtained using a generalized QR factorization of the matrices (A, B )\ngiven by\nIn particular, if matrix B is square nonsingular, then the problem GLM is equivalent to the following weighted\nlinear least squares problem\nminimizex ||B-1(d-A*x)||2.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nn\nThe number of rows of the matrices A and B (n≥ 0).\nm\nThe number of columns in A (m≥ 0).\np\nThe number of columns in B (p≥n - m).\na, b, d\nArrays:\na(size max(1, lda*m) for column major layout and max(1, lda*n) for row\nmajor layout) contains the n-by-m matrix A.\nb(size max(1, ldb*p) for column major layout and max(1, ldb*n) for row\nmajor layout) contains the n-by-p matrix B.\nd, size at least max(1, n), contains the left hand side of the GLM equation.\nlda\nThe leading dimension of a; at least max(1, n)for column major layout and\nmax(1, m) for row major layout.\nldb\nThe leading dimension of b; at least max(1, n)for column major layout and\nmax(1, p) for row major layout.\nOutput Parameters\nx, y\nArrays x, y. size at least max(1, m) for x and at least max(1, p) for y.\nOn exit, x and y are the solutions of the GLM problem.\na\nOn exit, the upper triangular part of the array a contains the m-by-m upper\ntriangular matrix R.\nb\nOn exit, if n ≤ p, the upper right triangle contains the n-by-n upper\ntriangular matrix T as returned by ?ggrqf; if n > p, the elements on and\nabove the (n-p)-th subdiagonal contain the n-by-p upper trapezoidal\nmatrix T.\nd\nOn exit, d is destroyed\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1012\n\n\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nIf info = 1, the upper triangular factor R associated with A in the generalized QR factorization of the pair\n(A, B) is singular, so that rank(A) < m; the least squares solution could not be computed.\nIf info = 2, the bottom (n-m)-by-(n-m) part of the upper trapezoidal factor T associated with B in the\ngeneralized QR factorization of the pair (A, B) is singular, so that rank(AB) < n; the least squares solution\ncould not be computed.\nSymmetric Eigenvalue Problems: LAPACK Driver Routines\nThis topic describes LAPACK driver routines used for solving symmetric eigenvalue problems. See also \ncomputational routines that can be called to solve these problems. Table \"Driver Routines for Solving\nSymmetric Eigenproblems\" lists all such driver routines.\nDriver Routines for Solving Symmetric Eigenproblems\nRoutine Name\nOperation performed\nsyev/heev\nComputes all eigenvalues and, optionally, eigenvectors of a real symmetric /\nHermitian matrix.\nsyevd/heevd\nComputes all eigenvalues and (optionally) all eigenvectors of a real symmetric /\nHermitian matrix using divide and conquer algorithm.\nsyevx/heevx\nComputes selected eigenvalues and, optionally, eigenvectors of a symmetric /\nHermitian matrix.\nsyevr/heevr\nComputes selected eigenvalues and, optionally, eigenvectors of a real symmetric /\nHermitian matrix using the Relatively Robust Representations.\nspev/hpev\nComputes all eigenvalues and, optionally, eigenvectors of a real symmetric /\nHermitian matrix in packed storage.\nspevd/hpevd\nUses divide and conquer algorithm to compute all eigenvalues and (optionally) all\neigenvectors of a real symmetric / Hermitian matrix held in packed storage.\nspevx/hpevx\nComputes selected eigenvalues and, optionally, eigenvectors of a real symmetric /\nHermitian matrix in packed storage.\nsbev /hbev\nComputes all eigenvalues and, optionally, eigenvectors of a real symmetric /\nHermitian band matrix.\nsbevd/hbevd\nComputes all eigenvalues and (optionally) all eigenvectors of a real symmetric /\nHermitian band matrix using divide and conquer algorithm.\nsbevx/hbevx\nComputes selected eigenvalues and, optionally, eigenvectors of a real symmetric /\nHermitian band matrix.\nstev\nComputes all eigenvalues and, optionally, eigenvectors of a real symmetric\ntridiagonal matrix.\nstevd\nComputes all eigenvalues and (optionally) all eigenvectors of a real symmetric\ntridiagonal matrix using divide and conquer algorithm.\nstevx\nComputes selected eigenvalues and eigenvectors of a real symmetric tridiagonal\nmatrix.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1013\n\n\nRoutine Name\nOperation performed\nstevr\nComputes selected eigenvalues and, optionally, eigenvectors of a real symmetric\ntridiagonal matrix using the Relatively Robust Representations.\n?syev\nComputes all eigenvalues and, optionally,\neigenvectors of a real symmetric matrix.\nSyntax\nlapack_int LAPACKE_ssyev (int matrix_layout, char jobz, char uplo, lapack_int n, float*\na, lapack_int lda, float* w);\nlapack_int LAPACKE_dsyev (int matrix_layout, char jobz, char uplo, lapack_int n,\ndouble* a, lapack_int lda, double* w);\nInclude Files\n•\nmkl.h\nDescription\nThe routine computes all eigenvalues and, optionally, eigenvectors of a real symmetric matrix A.\nNote that for most cases of real symmetric eigenvalue problems the default choice should be syevr function\nas its underlying algorithm is faster and uses less workspace.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\njobz\nMust be 'N' or 'V'.\nIf jobz = 'N', then only eigenvalues are computed.\nIf jobz = 'V', then eigenvalues and eigenvectors are computed.\nuplo\nMust be 'U' or 'L'.\nIf uplo = 'U', a stores the upper triangular part of A.\nIf uplo = 'L', a stores the lower triangular part of A.\nn\nThe order of the matrix A (n≥ 0).\na\na (size max(1, lda*n)) is an array containing either upper or lower\ntriangular part of the symmetric matrix A, as specified by uplo.\nlda\nThe leading dimension of the array a.\nMust be at least max(1, n).\nOutput Parameters\na\nOn exit, if jobz = 'V', then if info = 0, array a contains the orthonormal\neigenvectors of the matrix A.\nIf jobz = 'N', then on exit the lower triangle\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1014\n\n\n(if uplo = 'L') or the upper triangle (if uplo = 'U') of A, including the\ndiagonal, is overwritten.\nw\nArray, size at least max(1, n).\nIf info = 0, contains the eigenvalues of the matrix A in ascending order.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nIf info = i, then the algorithm failed to converge; i indicates the number of elements of an intermediate\ntridiagonal form which did not converge to zero.\n?heev\nComputes all eigenvalues and, optionally,\neigenvectors of a Hermitian matrix.\nSyntax\nlapack_int LAPACKE_cheev( int matrix_layout, char jobz, char uplo, lapack_int n,\nlapack_complex_float* a, lapack_int lda, float* w );\nlapack_int LAPACKE_zheev( int matrix_layout, char jobz, char uplo, lapack_int n,\nlapack_complex_double* a, lapack_int lda, double* w );\nInclude Files\n•\nmkl.h\nDescription\nThe routine computes all eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix A.\nNote that for most cases of complex Hermitian eigenvalue problems the default choice should be heevr\nfunction as its underlying algorithm is faster and uses less workspace.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\njobz\nMust be 'N' or 'V'.\nIf jobz = 'N', then only eigenvalues are computed.\nIf jobz = 'V', then eigenvalues and eigenvectors are computed.\nuplo\nMust be 'U' or 'L'.\nIf uplo = 'U', a stores the upper triangular part of A.\nIf uplo = 'L', a stores the lower triangular part of A.\nn\nThe order of the matrix A (n≥ 0).\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1015\n\n\na\na (size max(1, lda*n)) is an array containing either upper or lower\ntriangular part of the Hermitian matrix A, as specified by uplo.\nlda\nThe leading dimension of the array a. Must be at least max(1, n).\nOutput Parameters\na\nOn exit, if jobz = 'V', then if info = 0, array a contains the orthonormal\neigenvectors of the matrix A.\nIf jobz = 'N', then on exit the lower triangle\n(if uplo = 'L') or the upper triangle (if uplo = 'U') of A, including the\ndiagonal, is overwritten.\nw\nArray, size at least max(1, n).\nIf info = 0, contains the eigenvalues of the matrix A in ascending order.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nIf info = i, then the algorithm failed to converge; i indicates the number of elements of an intermediate\ntridiagonal form which did not converge to zero.\n?syevd\nComputes all eigenvalues and, optionally, all\neigenvectors of a real symmetric matrix using divide\nand conquer algorithm.\nSyntax\nlapack_int LAPACKE_ssyevd (int matrix_layout, char jobz, char uplo, lapack_int n,\nfloat* a, lapack_int lda, float* w);\nlapack_int LAPACKE_dsyevd (int matrix_layout, char jobz, char uplo, lapack_int n,\ndouble* a, lapack_int lda, double* w);\nInclude Files\n•\nmkl.h\nDescription\nThe routine computes all the eigenvalues, and optionally all the eigenvectors, of a real symmetric matrix A.\nIn other words, it can compute the spectral factorization of A as: A = Z*λ*ZT.\nHere Λ is a diagonal matrix whose diagonal elements are the eigenvalues λi, and Z is the orthogonal matrix\nwhose columns are the eigenvectors zi. Thus,\nA*zi = λi*zi for i = 1, 2, ..., n.\nIf the eigenvectors are requested, then this routine uses a divide and conquer algorithm to compute\neigenvalues and eigenvectors. However, if only eigenvalues are required, then it uses the Pal-Walker-Kahan\nvariant of the QL or QR algorithm.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1016\n\n\nNote that for most cases of real symmetric eigenvalue problems the default choice should be syevr function\nas its underlying algorithm is faster and uses less workspace. ?syevd requires more workspace but is faster\nin some cases, especially for large matrices.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\njobz\nMust be 'N' or 'V'.\nIf jobz = 'N', then only eigenvalues are computed.\nIf jobz = 'V', then eigenvalues and eigenvectors are computed.\nuplo\nMust be 'U' or 'L'.\nIf uplo = 'U', a stores the upper triangular part of A.\nIf uplo = 'L', a stores the lower triangular part of A.\nn\nThe order of the matrix A (n≥ 0).\na\nArray, size (lda, *).\na (size max(1, lda*n)) is an array containing either upper or lower\ntriangular part of the symmetric matrix A, as specified by uplo.\nlda\nThe leading dimension of the array a.\nMust be at least max(1, n).\nOutput Parameters\nw\nArray, size at least max(1, n).\nIf info = 0, contains the eigenvalues of the matrix A in ascending order.\nSee also info.\na\nIf jobz = 'V', then on exit this array is overwritten by the orthogonal\nmatrix Z which contains the eigenvectors of A.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = i, and jobz = 'N', then the algorithm failed to converge; i indicates the number of off-diagonal\nelements of an intermediate tridiagonal form which did not converge to zero.\nIf info = i, and jobz = 'V', then the algorithm failed to compute an eigenvalue while working on the\nsubmatrix lying in rows and columns info/(n+1) through mod(info,n+1).\nIf info = -i, the i-th parameter had an illegal value.\nApplication Notes\nThe computed eigenvalues and eigenvectors are exact for a matrix A+E such that ||E||2 = O(ε)*||A||2,\nwhere ε is the machine precision.\nThe complex analogue of this routine is heevd\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1017\n\n\n?heevd\nComputes all eigenvalues and, optionally, all\neigenvectors of a complex Hermitian matrix using\ndivide and conquer algorithm.\nSyntax\nlapack_int LAPACKE_cheevd( int matrix_layout, char jobz, char uplo, lapack_int n,\nlapack_complex_float* a, lapack_int lda, float* w );\nlapack_int LAPACKE_zheevd( int matrix_layout, char jobz, char uplo, lapack_int n,\nlapack_complex_double* a, lapack_int lda, double* w );\nInclude Files\n•\nmkl.h\nDescription\nThe routine computes all the eigenvalues, and optionally all the eigenvectors, of a complex Hermitian matrix\nA. In other words, it can compute the spectral factorization of A as: A = Z*Λ*ZH.\nHere Λ is a real diagonal matrix whose diagonal elements are the eigenvalues λi, and Z is the (complex)\nunitary matrix whose columns are the eigenvectors zi. Thus,\nA*zi = λi*zi for i = 1, 2, ..., n.\nIf the eigenvectors are requested, then this routine uses a divide and conquer algorithm to compute\neigenvalues and eigenvectors. However, if only eigenvalues are required, then it uses the Pal-Walker-Kahan\nvariant of the QL or QR algorithm.\nNote that for most cases of complex Hermetian eigenvalue problems the default choice should be heevr\nfunction as its underlying algorithm is faster and uses less workspace. ?heevd requires more workspace but\nis faster in some cases, especially for large matrices.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\njobz\nMust be 'N' or 'V'.\nIf jobz = 'N', then only eigenvalues are computed.\nIf jobz = 'V', then eigenvalues and eigenvectors are computed.\nuplo\nMust be 'U' or 'L'.\nIf uplo = 'U', a stores the upper triangular part of A.\nIf uplo = 'L', a stores the lower triangular part of A.\nn\nThe order of the matrix A (n≥ 0).\na\na (size max(1, lda*n)) is an array containing either upper or lower\ntriangular part of the Hermitian matrix A, as specified by uplo.\nlda\nThe leading dimension of the array a. Must be at least max(1, n).\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1018\n\n\nOutput Parameters\nw\nArray, size at least max(1, n).\nIf info = 0, contains the eigenvalues of the matrix A in ascending order.\nSee also info.\na\nIf jobz = 'V', then on exit this array is overwritten by the unitary matrix\nZ which contains the eigenvectors of A.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = i, and jobz = 'N', then the algorithm failed to converge; i off-diagonal elements of an\nintermediate tridiagonal form did not converge to zero;\nif info = i, and jobz = 'V', then the algorithm failed to compute an eigenvalue while working on the\nsubmatrix lying in rows and columns info/(n+1) through mod(info, n+1).\nIf info = -i, the i-th parameter had an illegal value.\nApplication Notes\nThe computed eigenvalues and eigenvectors are exact for a matrix A + E such that ||E||2 = O(ε)*||A||2,\nwhere ε is the machine precision.\nThe real analogue of this routine is syevd. See also hpevd for matrices held in packed storage, and hbevd for\nbanded matrices.\n?syevx\nComputes selected eigenvalues and, optionally,\neigenvectors of a symmetric matrix.\nSyntax\nlapack_int LAPACKE_ssyevx (int matrix_layout, char jobz, char range, char uplo,\nlapack_int n, float* a, lapack_int lda, float vl, float vu, lapack_int il, lapack_int\niu, float abstol, lapack_int* m, float* w, float* z, lapack_int ldz, lapack_int* ifail);\nlapack_int LAPACKE_dsyevx (int matrix_layout, char jobz, char range, char uplo,\nlapack_int n, double* a, lapack_int lda, double vl, double vu, lapack_int il, lapack_int\niu, double abstol, lapack_int* m, double* w, double* z, lapack_int ldz, lapack_int*\nifail);\nInclude Files\n•\nmkl.h\nDescription\nThe routine computes selected eigenvalues and, optionally, eigenvectors of a real symmetric matrix A.\nEigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for\nthe desired eigenvalues.\nNote that for most cases of real symmetric eigenvalue problems the default choice should be syevr function\nas its underlying algorithm is faster and uses less workspace. ?syevx is faster for a few selected\neigenvalues.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1019\n\n\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\njobz\nMust be 'N' or 'V'.\nIf jobz = 'N', then only eigenvalues are computed.\nIf jobz = 'V', then eigenvalues and eigenvectors are computed.\nrange\nMust be 'A', 'V', or 'I'.\nIf range = 'A', all eigenvalues will be found.\nIf range = 'V', all eigenvalues in the half-open interval (vl, vu] will be\nfound.\nIf range = 'I', the eigenvalues with indices il through iu will be found.\nuplo\nMust be 'U' or 'L'.\nIf uplo = 'U', a stores the upper triangular part of A.\nIf uplo = 'L', a stores the lower triangular part of A.\nn\nThe order of the matrix A (n≥ 0).\na\na (size max(1, lda*n)) is an array containing either upper or lower\ntriangular part of the symmetric matrix A, as specified by uplo.\nlda\nThe leading dimension of the array a. Must be at least max(1, n) .\nvl, vu\nIf range = 'V', the lower and upper bounds of the interval to be searched\nfor eigenvalues; vl≤vu. Not referenced if range = 'A'or 'I'.\nil, iu\nIf range = 'I', the indices of the smallest and largest eigenvalues to be\nreturned.\nConstraints: 1 ≤il≤iu≤n, if n > 0;\nil = 1 and iu = 0, if n = 0.\nNot referenced if range = 'A'or 'V'.\nabstol\nThe absolute error tolerance for the eigenvalues. See Application Notes for\nmore information.\nldz\nThe leading dimension of the output array z; ldz≥ 1.\nIf jobz = 'V', then ldz≥ max(1, n) for column major layout and lda≥\nmax(1, m) for row major layout .\nOutput Parameters\na\nOn exit, the lower triangle (if uplo = 'L') or the upper triangle (if uplo =\n'U') of A, including the diagonal, is overwritten.\nm\nThe total number of eigenvalues found;\n0 ≤m≤n.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1020\n\n\nIf range = 'A', m = n, and if range = 'I', m = iu-il+1.\nw\nArray, size at least max(1, n). The first m elements contain the selected\neigenvalues of the matrix A in ascending order.\nz\nArray z(size max(1, ldz*m) for column major layout and max(1, ldz*n) for\nrow major layout) contains eigenvectors.\nIf jobz = 'V', then if info = 0, the first m columns of z contain the\northonormal eigenvectors of the matrix A corresponding to the selected\neigenvalues, with the i-th column of z holding the eigenvector associated\nwith w(i).\nIf an eigenvector fails to converge, then that column of z contains the latest\napproximation to the eigenvector, and the index of the eigenvector is\nreturned in ifail.\nIf jobz = 'N', then z is not referenced.\nNote: you must ensure that at least max(1,m) columns are supplied in the\narray z; if range = 'V', the exact value of m is not known in advance and\nan upper bound must be used.\nifail\nArray, size at least max(1, n).\nIf jobz = 'V', then if info = 0, the first m elements of ifail are zero; if\ninfo > 0, then ifail contains the indices of the eigenvectors that failed to\nconverge.\nIf jobz = 'V', then ifail is not referenced.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nIf info = i, then i eigenvectors failed to converge; their indices are stored in the array ifail.\nApplication Notes\nAn approximate eigenvalue is accepted as converged when it is determined to lie in an interval [a,b] of width\nless than or equal to abstol+ε*max(|a|,|b|), where ε is the machine precision.\nIf abstol is less than or equal to zero, then ε*||T|} is used as tolerance, where ||T|| is the 1-norm of the\ntridiagonal matrix obtained by reducing A to tridiagonal form. Eigenvalues are computed most accurately\nwhen abstol is set to twice the underflow threshold 2*?lamch('S'), not zero.\nIf this routine returns with info > 0, indicating that some eigenvectors did not converge, try setting abstol\nto 2*?lamch('S').\n?heevx\nComputes selected eigenvalues and, optionally,\neigenvectors of a Hermitian matrix.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1021\n\n\nSyntax\nlapack_int LAPACKE_cheevx( int matrix_layout, char jobz, char range, char uplo,\nlapack_int n, lapack_complex_float* a, lapack_int lda, float vl, float vu, lapack_int\nil, lapack_int iu, float abstol, lapack_int* m, float* w, lapack_complex_float* z,\nlapack_int ldz, lapack_int* ifail );\nlapack_int LAPACKE_zheevx( int matrix_layout, char jobz, char range, char uplo,\nlapack_int n, lapack_complex_double* a, lapack_int lda, double vl, double vu,\nlapack_int il, lapack_int iu, double abstol, lapack_int* m, double* w,\nlapack_complex_double* z, lapack_int ldz, lapack_int* ifail );\nInclude Files\n•\nmkl.h\nDescription\nThe routine computes selected eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix A.\nEigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for\nthe desired eigenvalues.\nNote that for most cases of complex Hermetian eigenvalue problems the default choice should be heevr\nfunction as its underlying algorithm is faster and uses less workspace. ?heevx is faster for a few selected\neigenvalues.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\njobz\nMust be 'N' or 'V'.\nIf jobz = 'N', then only eigenvalues are computed.\nIf jobz = 'V', then eigenvalues and eigenvectors are computed.\nrange\nMust be 'A', 'V', or 'I'.\nIf range = 'A', all eigenvalues will be found.\nIf range = 'V', all eigenvalues in the half-open interval (vl, vu] will be\nfound.\nIf range = 'I', the eigenvalues with indices il through iu will be found.\nuplo\nMust be 'U' or 'L'.\nIf uplo = 'U', a stores the upper triangular part of A.\nIf uplo = 'L', a stores the lower triangular part of A.\nn\nThe order of the matrix A (n ≥ 0).\na\na (size max(1, lda*n)) is an array containing either upper or lower\ntriangular part of the Hermitian matrix A, as specified by uplo.\nlda\nThe leading dimension of the array a. Must be at least max(1, n).\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1022\n\n\nvl, vu\nIf range = 'V', the lower and upper bounds of the interval to be searched\nfor eigenvalues; vl≤vu. Not referenced if range = 'A'or 'I'.\nil, iu\nIf range = 'I', the indices of the smallest and largest eigenvalues to be\nreturned. Constraints:\n1 ≤il≤iu≤n, if n > 0;il = 1 and iu = 0, if n = 0. Not referenced if range =\n'A'or 'V'.\nabstol\nldz\nThe leading dimension of the output array z; ldz≥ 1.\nIf jobz = 'V', then ldz≥max(1, n) for column major layout and lda≥\nmax(1, m) for row major layout.\nOutput Parameters\na\nOn exit, the lower triangle (if uplo = 'L') or the upper triangle (if uplo =\n'U') of A, including the diagonal, is overwritten.\nm\nThe total number of eigenvalues found; 0 ≤m≤n.\nIf range = 'A', m = n, and if range = 'I', m = iu-il+1.\nw\nArray, size max(1, n). The first m elements contain the selected eigenvalues\nof the matrix A in ascending order.\nz\nArray z(size max(1, ldz*m) for column major layout and max(1, ldz*n) for\nrow major layout) contains eigenvectors.\nIf jobz = 'V', then if info = 0, the first m columns of z contain the\northonormal eigenvectors of the matrix A corresponding to the selected\neigenvalues, with the i-th column of z holding the eigenvector associated\nwith w(i).\nIf an eigenvector fails to converge, then that column of z contains the latest\napproximation to the eigenvector, and the index of the eigenvector is\nreturned in ifail.\nIf jobz = 'N', then z is not referenced.\nifail\nArray, size at least max(1, n).\nIf jobz = 'V', then if info = 0, the first m elements of ifail are zero; if\ninfo > 0, then ifail contains the indices of the eigenvectors that failed to\nconverge.\nIf jobz = 'V', then ifail is not referenced.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nIf info = i, then i eigenvectors failed to converge; their indices are stored in the array ifail.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1023\n\n\nApplication Notes\nAn approximate eigenvalue is accepted as converged when it is determined to lie in an interval [a,b] of width\nless than or equal to abstol+ε*max(|a|,|b|), where ε is the machine precision.\nIf abstol is less than or equal to zero, then ε*||T|| will be used in its place, where ||T|| is the 1-norm of\nthe tridiagonal matrix obtained by reducing A to tridiagonal form. Eigenvalues will be computed most\naccurately when abstol is set to twice the underflow threshold 2*?lamch('S'), not zero.\nIf this routine returns with info > 0, indicating that some eigenvectors did not converge, try setting abstol\nto 2*?lamch('S').\n?syevr\nComputes selected eigenvalues and, optionally,\neigenvectors of a real symmetric matrix using the\nRelatively Robust Representations.\nSyntax\nlapack_int LAPACKE_ssyevr (int matrix_layout, char jobz, char range, char uplo,\nlapack_int n, float* a, lapack_int lda, float vl, float vu, lapack_int il, lapack_int\niu, float abstol, lapack_int* m, float* w, float* z, lapack_int ldz, lapack_int*\nisuppz);\nlapack_int LAPACKE_dsyevr (int matrix_layout, char jobz, char range, char uplo,\nlapack_int n, double* a, lapack_int lda, double vl, double vu, lapack_int il, lapack_int\niu, double abstol, lapack_int* m, double* w, double* z, lapack_int ldz, lapack_int*\nisuppz);\nInclude Files\n•\nmkl.h\nDescription\nThe routine computes selected eigenvalues and, optionally, eigenvectors of a real symmetric matrix A.\nEigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for\nthe desired eigenvalues.\nThe routine first reduces the matrix A to tridiagonal form T. Then, whenever possible, ?syevr calls stemr to\ncompute the eigenspectrum using Relatively Robust Representations. stemr computes eigenvalues by the\ndqds algorithm, while orthogonal eigenvectors are computed from various \"good\" L*D*LT representations\n(also known as Relatively Robust Representations). Gram-Schmidt orthogonalization is avoided as far as\npossible. More specifically, the various steps of the algorithm are as follows. For the each unreduced block of\nT:\na.\nCompute T - σ*I = L*D*LT, so that L and D define all the wanted eigenvalues to high relative\naccuracy. This means that small relative changes in the entries of D and L cause only small relative\nchanges in the eigenvalues and eigenvectors. The standard (unfactored) representation of the\ntridiagonal matrix T does not have this property in general.\nb.\nCompute the eigenvalues to suitable accuracy. If the eigenvectors are desired, the algorithm attains full\naccuracy of the computed eigenvalues only right before the corresponding vectors have to be\ncomputed, see Steps c) and d).\nc.\nFor each cluster of close eigenvalues, select a new shift close to the cluster, find a new factorization,\nand refine the shifted eigenvalues to suitable accuracy.\nd.\nFor each eigenvalue with a large enough relative separation, compute the corresponding eigenvector by\nforming a rank revealing twisted factorization. Go back to Step c) for any clusters that remain.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1024\n\n\nThe desired accuracy of the output can be specified by the input parameter abstol.\nThe routine ?syevr calls stemr when the full spectrum is requested on machines that conform to the\nIEEE-754 floating point standard. ?syevr calls stebz and stein on non-IEEE machines and when partial\nspectrum requests are made.\nNormal execution of ?dsyevr may create NaNs and infinities and may abort due to a floating point exception\nin environments that do not handle NaNs and infinities in the IEEE standard default manner.\nNote that ?syevr is preferable for most cases of real symmetric eigenvalue problems as its underlying\nalgorithm is fast and uses less workspace.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\njobz\nMust be 'N' or 'V'.\nIf jobz = 'N', then only eigenvalues are computed.\nIf jobz = 'V', then eigenvalues and eigenvectors are computed.\nrange\nMust be 'A' or 'V' or 'I'.\nIf range = 'A', the routine computes all eigenvalues.\nIf range = 'V', the routine computes eigenvalues w[i] in the half-open\ninterval:\nvl < w[i]≤vu.\nIf range = 'I', the routine computes eigenvalues with indices il to iu.\nFor range = 'V'or 'I' and iu-il < n-1, sstebz/dstebz and sstein/\ndstein are called.\nuplo\nMust be 'U' or 'L'.\nIf uplo = 'U', a stores the upper triangular part of A.\nIf uplo = 'L', a stores the lower triangular part of A.\nn\nThe order of the matrix A (n≥ 0).\na\na (size max(1, lda*n)) is an array containing either upper or lower\ntriangular part of the symmetric matrix A, as specified by uplo.\nlda\nThe leading dimension of the array a. Must be at least max(1, n).\nvl, vu\nIf range = 'V', the lower and upper bounds of the interval to be searched\nfor eigenvalues.\nConstraint: vl< vu.\nIf range = 'A' or 'I', vl and vu are not referenced.\nil, iu\nIf range = 'I', the indices in ascending order of the smallest and largest\neigenvalues to be returned.\nConstraint:\n1 ≤il≤iu≤n, if n > 0;\nil=1 and iu=0, if n = 0.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1025\n\n\nIf range = 'A' or 'V', il and iu are not referenced.\nabstol\nIf jobz = 'V', the eigenvalues and eigenvectors output have residual\nnorms bounded by abstol, and the dot products between different\neigenvectors are bounded by abstol.\nIf abstol < n *eps*||T||, then n *eps*||T|| is used instead, where\neps is the machine precision, and ||T|| is the 1-norm of the matrix T. The\neigenvalues are computed to an accuracy of eps*||T|| irrespective of\nabstol.\nIf high relative accuracy is important, set abstol to ?lamch('S').\nldz\nThe leading dimension of the output array z.\nConstraints:\nldz≥ 1 if jobz = 'N' and\nldz≥ max(1, n) for column major layout and ldz≥ max(1, m) for row major\nlayout if jobz = 'V'.\nOutput Parameters\na\nOn exit, the lower triangle (if uplo = 'L') or the upper triangle (if uplo =\n'U') of A, including the diagonal, is overwritten.\nm\nThe total number of eigenvalues found, 0 ≤m≤n.\nIf range = 'A', m = n, if range = 'I', m = iu-il+1, and if range =\n'V' the exact value of m is not known in advance.\nw, z\nArrays:\nw, size at least max(1, n), contains the selected eigenvalues in ascending\norder, stored in w[0] to w[m - 1];\nz(size max(1, ldz*m) for column major layout and max(1, ldz*n) for row\nmajor layout) .\nIf jobz = 'V', then if info = 0, the first m columns of z contain the\northonormal eigenvectors of the matrix A corresponding to the selected\neigenvalues, with the i-th column of z holding the eigenvector associated\nwith w[i - 1].\nIf jobz = 'N', then z is not referenced.\nisuppz\nArray, size at least 2 *max(1, m).\nThe support of the eigenvectors in z, i.e., the indices indicating the nonzero\nelements in z. The i-th eigenvector is nonzero only in elements isuppz[2i\n- 2] through isuppz[2i - 1]. Referenced only if eigenvectors are needed\n(jobz = 'V') and all eigenvalues are needed, that is, range = 'A' or\nrange = 'I' and il = 1 and iu = n.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1026\n\n\nIf info = -i, the i-th parameter had an illegal value.\nIf info = i, an internal error has occurred.\nApplication Notes\n?heevr\nComputes selected eigenvalues and, optionally,\neigenvectors of a Hermitian matrix using the\nRelatively Robust Representations.\nSyntax\nlapack_int LAPACKE_cheevr( int matrix_layout, char jobz, char range, char uplo,\nlapack_int n, lapack_complex_float* a, lapack_int lda, float vl, float vu, lapack_int\nil, lapack_int iu, float abstol, lapack_int* m, float* w, lapack_complex_float* z,\nlapack_int ldz, lapack_int* isuppz );\nlapack_int LAPACKE_zheevr( int matrix_layout, char jobz, char range, char uplo,\nlapack_int n, lapack_complex_double* a, lapack_int lda, double vl, double vu,\nlapack_int il, lapack_int iu, double abstol, lapack_int* m, double* w,\nlapack_complex_double* z, lapack_int ldz, lapack_int* isuppz );\nInclude Files\n•\nmkl.h\nDescription\nThe routine computes selected eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix A.\nEigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for\nthe desired eigenvalues.\nThe routine first reduces the matrix A to tridiagonal form T with a call to hetrd. Then, whenever\npossible, ?heevr calls stegr to compute the eigenspectrum using Relatively Robust Representations. ?stegr\ncomputes eigenvalues by the dqds algorithm, while orthogonal eigenvectors are computed from various\n\"good\" L*D*LT representations (also known as Relatively Robust Representations). Gram-Schmidt\northogonalization is avoided as far as possible. More specifically, the various steps of the algorithm are as\nfollows. For each unreduced block (submatrix) of T:\na.\nCompute T - σ*I = L*D*LT, so that L and D define all the wanted eigenvalues to high relative\naccuracy. This means that small relative changes in the entries of D and L cause only small relative\nchanges in the eigenvalues and eigenvectors. The standard (unfactored) representation of the\ntridiagonal matrix T does not have this property in general.\nb.\nCompute the eigenvalues to suitable accuracy. If the eigenvectors are desired, the algorithm attains full\naccuracy of the computed eigenvalues only right before the corresponding vectors have to be\ncomputed, see Steps c) and d).\nc.\nFor each cluster of close eigenvalues, select a new shift close to the cluster, find a new factorization,\nand refine the shifted eigenvalues to suitable accuracy.\nd.\nFor each eigenvalue with a large enough relative separation, compute the corresponding eigenvector by\nforming a rank revealing twisted factorization. Go back to Step c) for any clusters that remain.\nThe desired accuracy of the output can be specified by the input parameter abstol.\nThe routine ?heevr calls stemr when the full spectrum is requested on machines which conform to the\nIEEE-754 floating point standard, or stebz and stein on non-IEEE machines and when partial spectrum\nrequests are made.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1027\n\n\nNote that the routine ?heevr is preferable for most cases of complex Hermitian eigenvalue problems as its\nunderlying algorithm is fast and uses less workspace.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\njobz\nMust be 'N' or 'V'.\nIf job = 'N', then only eigenvalues are computed.\nIf job = 'V', then eigenvalues and eigenvectors are computed.\nrange\nMust be 'A' or 'V' or 'I'.\nIf range = 'A', the routine computes all eigenvalues.\nIf range = 'V', the routine computes eigenvalues lambda(i) in the half-\nopen interval: vl< lambda(i)≤vu.\nIf range = 'I', the routine computes eigenvalues with indices il to iu.\nFor range = 'V'or 'I', sstebz/dstebz and cstein/zstein are called.\nuplo\nMust be 'U' or 'L'.\nIf uplo = 'U', a stores the upper triangular part of A.\nIf uplo = 'L', a stores the lower triangular part of A.\nn\nThe order of the matrix A (n≥ 0).\na\na (size max(1, lda*n)) is an array containing either upper or lower\ntriangular part of the Hermitian matrix A, as specified by uplo.\nlda\nThe leading dimension of the array a.\nMust be at least max(1, n).\nvl, vu\nIf range = 'V', the lower and upper bounds of the interval to be searched\nfor eigenvalues.\nConstraint: vl< vu.\nIf range = 'A' or 'I', vl and vu are not referenced.\nil, iu\nIf range = 'I', the indices in ascending order of the smallest and largest\neigenvalues to be returned.\nConstraint: 1 ≤il≤iu≤n, if n > 0; il=1 and iu=0 if n = 0.\nIf range = 'A' or 'V', il and iu are not referenced.\nabstol\nThe absolute error tolerance to which each eigenvalue/eigenvector is\nrequired.\nIf jobz = 'V', the eigenvalues and eigenvectors output have residual\nnorms bounded by abstol, and the dot products between different\neigenvectors are bounded by abstol.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1028\n\n\nIf abstol < n *eps*||T||, then n *eps*||T|| is used instead, where\neps is the machine precision, and ||T|| is the 1-norm of the matrix T. The\neigenvalues are computed to an accuracy of eps*||T|| irrespective of\nabstol.\nIf high relative accuracy is important, set abstol to ?lamch('S').\nldz\nThe leading dimension of the output array z. Constraints:\nldz≥ 1 if jobz = 'N';\nldz≥ max(1, n) for column major layout and ldz≥ max(1, m) for row major\nlayout if jobz = 'V'.\nOutput Parameters\na\nOn exit, the lower triangle (if uplo = 'L') or the upper triangle (if uplo =\n'U') of A, including the diagonal, is overwritten.\nm\nThe total number of eigenvalues found,\n0 ≤m≤n.\nIf range = 'A', m = n, if range = 'I', m = iu-il+1, and if range =\n'V' the exact value of m is not known in advance.\nw\nArray, size at least max(1, n), contains the selected eigenvalues in\nascending order, stored in w[0] to w[m - 1].\nz\nArray z(size max(1, ldz*m) for column major layout and max(1, ldz*n) for\nrow major layout) .\nIf jobz = 'V', then if info = 0, the first m columns of z contain the\northonormal eigenvectors of the matrix A corresponding to the selected\neigenvalues, with the i-th column of z holding the eigenvector associated\nwith w[i - 1].\nIf jobz = 'N', then z is not referenced.\nisuppz\nArray, size at least 2 *max(1, m).\nThe support of the eigenvectors in z, i.e., the indices indicating the nonzero\nelements in z. The i-th eigenvector is nonzero only in elements isuppz[2i\n- 2] through isuppz[2i - 1]. Referenced only if eigenvectors are needed\n(jobz = 'V') and all eigenvalues are needed, that is, range = 'A' or\nrange = 'I' and il = 1 and iu = n.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nIf info = i, an internal error has occurred.\nApplication Notes\nNormal execution of ?stemr may create NaNs and infinities and hence may abort due to a floating point\nexception in environments which do not handle NaNs and infinities in the IEEE standard default manner.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1029\n\n\nFor more details, see ?stemr and these references:\n•\nInderjit S. Dhillon and Beresford N. Parlett: \"Multiple representations to compute orthogonal eigenvectors\nof symmetric tridiagonal matrices,\" Linear Algebra and its Applications, 387(1), pp. 1-28, August 2004.\n•\nInderjit Dhillon and Beresford Parlett: \"Orthogonal Eigenvectors and Relative Gaps,\" SIAM Journal on\nMatrix Analysis and Applications, Vol. 25, 2004. Also LAPACK Working Note 154.\n•\nInderjit Dhillon: \"A new O(n^2) algorithm for the symmetric tridiagonal eigenvalue/eigenvector problem\",\nComputer Science Division Technical Report No. UCB/CSD-97-971, UC Berkeley, May 1997.\n?spev\nComputes all eigenvalues and, optionally,\neigenvectors of a real symmetric matrix in packed\nstorage.\nSyntax\nlapack_int LAPACKE_sspev (int matrix_layout, char jobz, char uplo, lapack_int n, float*\nap, float* w, float* z, lapack_int ldz);\nlapack_int LAPACKE_dspev (int matrix_layout, char jobz, char uplo, lapack_int n,\ndouble* ap, double* w, double* z, lapack_int ldz);\nInclude Files\n•\nmkl.h\nDescription\nThe routine computes all the eigenvalues and, optionally, eigenvectors of a real symmetric matrix A in\npacked storage.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\njobz\nMust be 'N' or 'V'.\nIf job = 'N', then only eigenvalues are computed.\nIf job = 'V', then eigenvalues and eigenvectors are computed.\nuplo\nMust be 'U' or 'L'.\nIf uplo = 'U', ap stores the packed upper triangular part of A.\nIf uplo = 'L', ap stores the packed lower triangular part of A.\nn\nThe order of the matrix A (n≥ 0).\nap\nArray ap contains the packed upper or lower triangle of symmetric matrix A,\nas specified by uplo.\nThe size of ap must be at least max(1, n*(n+1)/2).\nldz\nThe leading dimension of the output array z. Constraints:\nif jobz = 'N', then ldz≥ 1;\nif jobz = 'V', then ldz≥ max(1, n).\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1030\n\n\nOutput Parameters\nw, z\nArrays:\nw, size at least max(1, n).\nIf info = 0, w contains the eigenvalues of the matrix A in ascending order.\nz (size max(1, ldz*n)).\nIf jobz = 'V', then if info = 0, z contains the orthonormal eigenvectors\nof the matrix A, with the i-th column of z holding the eigenvector associated\nwith w[i - 1].\nIf jobz = 'N', then z is not referenced.\nap\nOn exit, this array is overwritten by the values generated during the\nreduction to tridiagonal form. The elements of the diagonal and the off-\ndiagonal of the tridiagonal matrix overwrite the corresponding elements of\nA.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nIf info = i, then the algorithm failed to converge; i indicates the number of elements of an intermediate\ntridiagonal form which did not converge to zero.\n?hpev\nComputes all eigenvalues and, optionally,\neigenvectors of a Hermitian matrix in packed storage.\nSyntax\nlapack_int LAPACKE_chpev( int matrix_layout, char jobz, char uplo, lapack_int n,\nlapack_complex_float* ap, float* w, lapack_complex_float* z, lapack_int ldz );\nlapack_int LAPACKE_zhpev( int matrix_layout, char jobz, char uplo, lapack_int n,\nlapack_complex_double* ap, double* w, lapack_complex_double* z, lapack_int ldz );\nInclude Files\n•\nmkl.h\nDescription\nThe routine computes all the eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix A in\npacked storage.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\njobz\nMust be 'N' or 'V'.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1031\n\n\nIf job = 'N', then only eigenvalues are computed.\nIf job = 'V', then eigenvalues and eigenvectors are computed.\nuplo\nMust be 'U' or 'L'.\nIf uplo = 'U', ap stores the packed upper triangular part of A.\nIf uplo = 'L', ap stores the packed lower triangular part of A.\nn\nThe order of the matrix A (n≥ 0).\nap\nArray ap contains the packed upper or lower triangle of Hermitian matrix A,\nas specified by uplo.\nThe size of ap must be at least max(1, n*(n+1)/2).\nldz\nThe leading dimension of the output array z.\nConstraints:\nif jobz = 'N', then ldz≥ 1;\nif jobz = 'V', then ldz≥ max(1, n) .\nOutput Parameters\nw\nArray, size at least max(1, n).\nIf info = 0, w contains the eigenvalues of the matrix A in ascending order.\nz\nArray z (size at least max(1, ldz*n)).\nIf jobz = 'V', then if info = 0, z contains the orthonormal eigenvectors\nof the matrix A, with the i-th column of z holding the eigenvector associated\nwith w[i - 1].\nIf jobz = 'N', then z is not referenced.\nap\nOn exit, this array is overwritten by the values generated during the\nreduction to tridiagonal form. The elements of the diagonal and the off-\ndiagonal of the tridiagonal matrix overwrite the corresponding elements of\nA.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nIf info = i, then the algorithm failed to converge; i indicates the number of elements of an intermediate\ntridiagonal form which did not converge to zero.\n?spevd\nUses divide and conquer algorithm to compute all\neigenvalues and (optionally) all eigenvectors of a real\nsymmetric matrix held in packed storage.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1032\n\n\nSyntax\nlapack_int LAPACKE_sspevd (int matrix_layout, char jobz, char uplo, lapack_int n,\nfloat* ap, float* w, float* z, lapack_int ldz);\nlapack_int LAPACKE_dspevd (int matrix_layout, char jobz, char uplo, lapack_int n,\ndouble* ap, double* w, double* z, lapack_int ldz);\nInclude Files\n•\nmkl.h\nDescription\nThe routine computes all the eigenvalues, and optionally all the eigenvectors, of a real symmetric matrix A\n(held in packed storage). In other words, it can compute the spectral factorization of A as:\nA = Z*Λ*ZT.\nHere Λ is a diagonal matrix whose diagonal elements are the eigenvalues λi, and Z is the orthogonal matrix\nwhose columns are the eigenvectors zi. Thus,\nA*zi = λi*zi for i = 1, 2, ..., n.\nIf the eigenvectors are requested, then this routine uses a divide and conquer algorithm to compute\neigenvalues and eigenvectors. However, if only eigenvalues are required, then it uses the Pal-Walker-Kahan\nvariant of the QL or QR algorithm.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\njobz\nMust be 'N' or 'V'.\nIf jobz = 'N', then only eigenvalues are computed.\nIf jobz = 'V', then eigenvalues and eigenvectors are computed.\nuplo\nMust be 'U' or 'L'.\nIf uplo = 'U', ap stores the packed upper triangular part of A.\nIf uplo = 'L', ap stores the packed lower triangular part of A.\nn\nThe order of the matrix A (n≥ 0).\nap\nap contains the packed upper or lower triangle of symmetric matrix A, as\nspecified by uplo.\nThe dimension of ap must be max(1, n*(n+1)/2)\nldz\nThe leading dimension of the output array z.\nConstraints:\nif jobz = 'N', then ldz≥ 1;\nif jobz = 'V', then ldz≥ max(1, n).\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1033\n\n\nOutput Parameters\nw, z\nArrays:\nw, size at least max(1, n).\nIf info = 0, contains the eigenvalues of the matrix A in ascending order.\nSee also info.\nz (size max(1, ldz*n)).\nIf jobz = 'V', then this array is overwritten by the orthogonal matrix Z\nwhich contains the eigenvectors of A. If jobz = 'N', then z is not\nreferenced.\nap\nOn exit, this array is overwritten by the values generated during the\nreduction to tridiagonal form. The elements of the diagonal and the off-\ndiagonal of the tridiagonal matrix overwrite the corresponding elements of\nA.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = i, then the algorithm failed to converge; i indicates the number of elements of an intermediate\ntridiagonal form which did not converge to zero.\nIf info = -i, the i-th parameter had an illegal value.\nApplication Notes\nThe computed eigenvalues and eigenvectors are exact for a matrix A+E such that ||E||2 = O(ε)*||A||2,\nwhere ε is the machine precision.\nThe complex analogue of this routine is hpevd.\nSee also syevd for matrices held in full storage, and sbevd for banded matrices.\n?hpevd\nUses divide and conquer algorithm to compute all\neigenvalues and, optionally, all eigenvectors of a\ncomplex Hermitian matrix held in packed storage.\nSyntax\nlapack_int LAPACKE_chpevd( int matrix_layout, char jobz, char uplo, lapack_int n,\nlapack_complex_float* ap, float* w, lapack_complex_float* z, lapack_int ldz );\nlapack_int LAPACKE_zhpevd( int matrix_layout, char jobz, char uplo, lapack_int n,\nlapack_complex_double* ap, double* w, lapack_complex_double* z, lapack_int ldz );\nInclude Files\n•\nmkl.h\nDescription\nThe routine computes all the eigenvalues, and optionally all the eigenvectors, of a complex Hermitian matrix\nA (held in packed storage). In other words, it can compute the spectral factorization of A as: A = Z*Λ*ZH.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1034\n\n\nHere Λ is a real diagonal matrix whose diagonal elements are the eigenvalues λi, and Z is the (complex)\nunitary matrix whose columns are the eigenvectors zi. Thus,\nA*zi = λi*zi for i = 1, 2, ..., n.\nIf the eigenvectors are requested, then this routine uses a divide and conquer algorithm to compute\neigenvalues and eigenvectors. However, if only eigenvalues are required, then it uses the Pal-Walker-Kahan\nvariant of the QL or QR algorithm.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\njobz\nMust be 'N' or 'V'.\nIf jobz = 'N', then only eigenvalues are computed.\nIf jobz = 'V', then eigenvalues and eigenvectors are computed.\nuplo\nMust be 'U' or 'L'.\nIf uplo = 'U', ap stores the packed upper triangular part of A.\nIf uplo = 'L', ap stores the packed lower triangular part of A.\nn\nThe order of the matrix A (n≥ 0).\nap\nap contains the packed upper or lower triangle of Hermitian matrix A, as\nspecified by uplo.\nThe dimension of ap must be at least max(1, n*(n+1)/2).\nldz\nThe leading dimension of the output array z.\nConstraints:\nif jobz = 'N', then ldz≥ 1;\nif jobz = 'V', then ldz≥ max(1, n).\nOutput Parameters\nw\nArray, size at least max(1, n).\nIf info = 0, contains the eigenvalues of the matrix A in ascending order.\nSee also info.\nz\nArray, size 1 if jobz = 'N' and max(1, ldz*n) if jobz = 'V'.\nIf jobz = 'V', then this array is overwritten by the unitary matrix Z which\ncontains the eigenvectors of A.\nIf jobz = 'N', then z is not referenced.\nap\nOn exit, this array is overwritten by the values generated during the\nreduction to tridiagonal form. The elements of the diagonal and the off-\ndiagonal of the tridiagonal matrix overwrite the corresponding elements of\nA.\nReturn Values\nThis function returns a value info.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1035\n\n\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nIf info = i, then the algorithm failed to converge; i indicates the number of elements of an intermediate\ntridiagonal form which did not converge to zero.\nApplication Notes\nThe computed eigenvalues and eigenvectors are exact for a matrix A + E such that ||E||2 = O(ε)*||A||2,\nwhere ε is the machine precision.\nThe real analogue of this routine is spevd.\nSee also heevd for matrices held in full storage, and hbevd for banded matrices.\n?spevx\nComputes selected eigenvalues and, optionally,\neigenvectors of a real symmetric matrix in packed\nstorage.\nSyntax\nlapack_int LAPACKE_sspevx (int matrix_layout, char jobz, char range, char uplo,\nlapack_int n, float* ap, float vl, float vu, lapack_int il, lapack_int iu, float abstol,\nlapack_int* m, float* w, float* z, lapack_int ldz, lapack_int* ifail);\nlapack_int LAPACKE_dspevx (int matrix_layout, char jobz, char range, char uplo,\nlapack_int n, double* ap, double vl, double vu, lapack_int il, lapack_int iu, double\nabstol, lapack_int* m, double* w, double* z, lapack_int ldz, lapack_int* ifail);\nInclude Files\n•\nmkl.h\nDescription\nThe routine computes selected eigenvalues and, optionally, eigenvectors of a real symmetric matrix A in\npacked storage. Eigenvalues and eigenvectors can be selected by specifying either a range of values or a\nrange of indices for the desired eigenvalues.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\njobz\nMust be 'N' or 'V'.\nIf job = 'N', then only eigenvalues are computed.\nIf job = 'V', then eigenvalues and eigenvectors are computed.\nrange\nMust be 'A' or 'V' or 'I'.\nIf range = 'A', the routine computes all eigenvalues.\nIf range = 'V', the routine computes eigenvalues w[i] in the half-open\ninterval: vl< w[i]≤vu.\nIf range = 'I', the routine computes eigenvalues with indices il to iu.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1036\n\n\nuplo\nMust be 'U' or 'L'.\nIf uplo = 'U', ap stores the packed upper triangular part of A.\nIf uplo = 'L', ap stores the packed lower triangular part of A.\nn\nThe order of the matrix A (n≥ 0).\nap\nArray ap contains the packed upper or lower triangle of the symmetric\nmatrix A, as specified by uplo.\nThe size of ap must be at least max(1, n*(n+1)/2).\nvl, vu\nIf range = 'V', the lower and upper bounds of the interval to be searched\nfor eigenvalues.\nConstraint: vl< vu.\nIf range = 'A' or 'I', vl and vu are not referenced.\nil, iu\nIf range = 'I', the indices in ascending order of the smallest and largest\neigenvalues to be returned.\nConstraint: 1 ≤il≤iu≤n, if n > 0; il=1 and iu=0\nif n = 0.\nIf range = 'A' or 'V', il and iu are not referenced.\nabstol\nThe absolute error tolerance to which each eigenvalue is required. See\nApplication notes for details on error tolerance.\nldz\nThe leading dimension of the output array z.\nConstraints:\nif jobz = 'N', then ldz≥ 1;\nif jobz = 'V', then ldz≥ max(1, n) for column major layout and ldz≥\nmax(1, m) for row major layout.\nOutput Parameters\nap\nOn exit, this array is overwritten by the values generated during the\nreduction to tridiagonal form. The elements of the diagonal and the off-\ndiagonal of the tridiagonal matrix overwrite the corresponding elements of\nA.\nm\nThe total number of eigenvalues found,\n0 ≤m≤n. If range = 'A', m = n, if range = 'I', m = iu-il+1, and if\nrange = 'V' the exact value of m is not known in advance..\nw, z\nArrays:\nw, size at least max(1, n).\nIf info = 0, contains the selected eigenvalues of the matrix A in ascending\norder.\nz(size max(1, ldz*m) for column major layout and max(1, ldz*n) for row\nmajor layout).\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1037\n\n\nIf jobz = 'V', then if info = 0, the first m columns of z contain the\northonormal eigenvectors of the matrix A corresponding to the selected\neigenvalues, with the i-th column of z holding the eigenvector associated\nwith w[i - 1].\nIf an eigenvector fails to converge, then that column of z contains the latest\napproximation to the eigenvector, and the index of the eigenvector is\nreturned in ifail.\nIf jobz = 'N', then z is not referenced.\nifail\nArray, size at least max(1, n).\nIf jobz = 'V', then if info = 0, the first m elements of ifail are zero; if\ninfo > 0, the ifail contains the indices the eigenvectors that failed to\nconverge.\nIf jobz = 'N', then ifail is not referenced.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nIf info = i, then i eigenvectors failed to converge; their indices are stored in the array ifail.\nApplication Notes\nAn approximate eigenvalue is accepted as converged when it is determined to lie in an interval [a,b] of width\nless than or equal to abstol+ε*max(|a|,|b|), where ε is the machine precision.\nIf abstol is less than or equal to zero, then ε*||T||1 will be used in its place, where T is the tridiagonal\nmatrix obtained by reducing A to tridiagonal form. Eigenvalues will be computed most accurately when abstol\nis set to twice the underflow threshold 2*?lamch('S'), not zero.\nIf this routine returns with info > 0, indicating that some eigenvectors did not converge, try setting abstol\nto 2*?lamch('S').\n?hpevx\nComputes selected eigenvalues and, optionally,\neigenvectors of a Hermitian matrix in packed storage.\nSyntax\nlapack_int LAPACKE_chpevx( int matrix_layout, char jobz, char range, char uplo,\nlapack_int n, lapack_complex_float* ap, float vl, float vu, lapack_int il, lapack_int\niu, float abstol, lapack_int* m, float* w, lapack_complex_float* z, lapack_int ldz,\nlapack_int* ifail );\nlapack_int LAPACKE_zhpevx( int matrix_layout, char jobz, char range, char uplo,\nlapack_int n, lapack_complex_double* ap, double vl, double vu, lapack_int il,\nlapack_int iu, double abstol, lapack_int* m, double* w, lapack_complex_double* z,\nlapack_int ldz, lapack_int* ifail );\nInclude Files\n•\nmkl.h\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1038\n\n\nDescription\nThe routine computes selected eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix A in\npacked storage. Eigenvalues and eigenvectors can be selected by specifying either a range of values or a\nrange of indices for the desired eigenvalues.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\njobz\nMust be 'N' or 'V'.\nIf job = 'N', then only eigenvalues are computed.\nIf job = 'V', then eigenvalues and eigenvectors are computed.\nrange\nMust be 'A' or 'V' or 'I'.\nIf range = 'A', the routine computes all eigenvalues.\nIf range = 'V', the routine computes eigenvalues w[i] in the half-open\ninterval: vl< w[i]≤vu.\nIf range = 'I', the routine computes eigenvalues with indices il to iu.\nuplo\nMust be 'U' or 'L'.\nIf uplo = 'U', ap stores the packed upper triangular part of A.\nIf uplo = 'L', ap stores the packed lower triangular part of A.\nn\nThe order of the matrix A (n≥ 0).\nap\nArray ap contains the packed upper or lower triangle of the Hermitian\nmatrix A, as specified by uplo.\nThe size of ap must be at least max(1, n*(n+1)/2).\nvl, vu\nIf range = 'V', the lower and upper bounds of the interval to be searched\nfor eigenvalues.\nConstraint: vl< vu.\nIf range = 'A' or 'I', vl and vu are not referenced.\nil, iu\nIf range = 'I', the indices in ascending order of the smallest and largest\neigenvalues to be returned.\nConstraint: 1 ≤il≤iu≤n, if n > 0; il=1 and iu=0 if n = 0.\nIf range = 'A' or 'V', il and iu are not referenced.\nabstol\nThe absolute error tolerance to which each eigenvalue is required. See\nApplication notes for details on error tolerance.\nldz\nThe leading dimension of the output array z.\nConstraints:\nif jobz = 'N', then ldz≥ 1;\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1039\n\n\nif jobz = 'V', then ldz≥ max(1, n) for column major layout and ldz≥\nmax(1, m) for row major layout.\nOutput Parameters\nap\nOn exit, this array is overwritten by the values generated during the\nreduction to tridiagonal form. The elements of the diagonal and the off-\ndiagonal of the tridiagonal matrix overwrite the corresponding elements of\nA.\nm\nThe total number of eigenvalues found, 0 ≤m≤n.\n0 ≤m≤n. If range = 'A', m = n, if range = 'I', m = iu-il+1, and if\nrange = 'V' the exact value of m is not known in advance..\nw\nArray, size at least max(1, n).\nIf info = 0, contains the selected eigenvalues of the matrix A in ascending\norder.\nz\nArray z(size max(1, ldz*m) for column major layout and max(1, ldz*n) for\nrow major layout).\nIf jobz = 'V', then if info = 0, the first m columns of z contain the\northonormal eigenvectors of the matrix A corresponding to the selected\neigenvalues, with the i-th column of z holding the eigenvector associated\nwith w(i).\nIf an eigenvector fails to converge, then that column of z contains the latest\napproximation to the eigenvector, and the index of the eigenvector is\nreturned in ifail.\nIf jobz = 'N', then z is not referenced.\nifail\nArray, size at least max(1, n).\nIf jobz = 'V', then if info = 0, the first m elements of ifail are zero; if\ninfo > 0, the ifail contains the indices the eigenvectors that failed to\nconverge.\nIf jobz = 'N', then ifail is not referenced.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nIf info = i, then i eigenvectors failed to converge; their indices are stored in the array ifail.\nApplication Notes\nAn approximate eigenvalue is accepted as converged when it is determined to lie in an interval [a,b] of width\nless than or equal to abstol+ε*max(|a|,|b|), where ε is the machine precision.\nIf abstol is less than or equal to zero, then ε*||T||1 will be used in its place, where T is the tridiagonal\nmatrix obtained by reducing A to tridiagonal form. Eigenvalues will be computed most accurately when abstol\nis set to twice the underflow threshold 2*?lamch('S'), not zero.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1040\n\n\nIf this routine returns with info > 0, indicating that some eigenvectors did not converge, try setting abstol\nto 2*?lamch('S').\n?sbev\nComputes all eigenvalues and, optionally,\neigenvectors of a real symmetric band matrix.\nSyntax\nlapack_int LAPACKE_ssbev (int matrix_layout, char jobz, char uplo, lapack_int n,\nlapack_int kd, float* ab, lapack_int ldab, float* w, float* z, lapack_int ldz);\nlapack_int LAPACKE_dsbev (int matrix_layout, char jobz, char uplo, lapack_int n,\nlapack_int kd, double* ab, lapack_int ldab, double* w, double* z, lapack_int ldz);\nInclude Files\n•\nmkl.h\nDescription\nThe routine computes all eigenvalues and, optionally, eigenvectors of a real symmetric band matrix A.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\njobz\nMust be 'N' or 'V'.\nIf jobz = 'N', then only eigenvalues are computed.\nIf jobz = 'V', then eigenvalues and eigenvectors are computed.\nuplo\nMust be 'U' or 'L'.\nIf uplo = 'U', ab stores the upper triangular part of A.\nIf uplo = 'L', ab stores the lower triangular part of A.\nn\nThe order of the matrix A (n≥ 0).\nkd\nThe number of super- or sub-diagonals in A\n(kd≥ 0).\nab\nab (size at least max(1, ldab*n) for column major layout and at least\nmax(1, ldab*(kd + 1)) for row major layout) is an array containing either\nupper or lower triangular part of the symmetric matrix A (as specified by\nuplo) in band storage format.\nldab\nThe leading dimension of ab; must be at least kd +1 for column major\nlayout and n for row major layout.\nldz\nThe leading dimension of the output array z.\nConstraints:\nif jobz = 'N', then ldz≥ 1;\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1041\n\n\nif jobz = 'V', then ldz≥ max(1, n) .\nOutput Parameters\nw, z\nArrays:\nw, size at least max(1, n).\nIf info = 0, contains the eigenvalues of the matrix A in ascending order.\nz(size max(1, ldz*n).\nIf jobz = 'V', then if info = 0, z contains the orthonormal eigenvectors\nof the matrix A, with the i-th column of z holding the eigenvector associated\nwith w[i - 1].\nIf jobz = 'N', then z is not referenced.\nab\nOn exit, this array is overwritten by the values generated during the\nreduction to tridiagonal form (see the description of ?sbtrd).\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\n?hbev\nComputes all eigenvalues and, optionally,\neigenvectors of a Hermitian band matrix.\nSyntax\nlapack_int LAPACKE_chbev( int matrix_layout, char jobz, char uplo, lapack_int n,\nlapack_int kd, lapack_complex_float* ab, lapack_int ldab, float* w,\nlapack_complex_float* z, lapack_int ldz );\nlapack_int LAPACKE_zhbev( int matrix_layout, char jobz, char uplo, lapack_int n,\nlapack_int kd, lapack_complex_double* ab, lapack_int ldab, double* w,\nlapack_complex_double* z, lapack_int ldz );\nInclude Files\n•\nmkl.h\nDescription\nThe routine computes all eigenvalues and, optionally, eigenvectors of a complex Hermitian band matrix A.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\njobz\nMust be 'N' or 'V'.\nIf jobz = 'N', then only eigenvalues are computed.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1042\n\n\nIf jobz = 'V', then eigenvalues and eigenvectors are computed.\nuplo\nMust be 'U' or 'L'.\nIf uplo = 'U', ab stores the upper triangular part of A.\nIf uplo = 'L', ab stores the lower triangular part of A.\nn\nThe order of the matrix A (n≥ 0).\nkd\nThe number of super- or sub-diagonals in A\n(kd≥ 0).\nab\nab (size at least max(1, ldab*n) for column major layout and at least\nmax(1, ldab*(kd + 1)) for row major layout) is an array containing either\nupper or lower triangular part of the Hermitian matrix A (as specified by\nuplo) in band storage format.\nldab\nThe leading dimension of ab; must be at least kd +1 for column major\nlayout and n for row major layout.\nldz\nThe leading dimension of the output array z.\nConstraints:\nif jobz = 'N', then ldz≥ 1;\nif jobz = 'V', then ldz≥ max(1, n) .\nOutput Parameters\nw\nArray, size at least max(1, n).\nIf info = 0, contains the eigenvalues in ascending order.\nz\nArray z(size max(1, ldz*n).\nIf jobz = 'V', then if info = 0, z contains the orthonormal eigenvectors\nof the matrix A, with the i-th column of z holding the eigenvector associated\nwith w[i - 1].\nIf jobz = 'N', then z is not referenced.\nab\nOn exit, this array is overwritten by the values generated during the\nreduction to tridiagonal form(see the description of hbtrd).\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nIf info = i, then the algorithm failed to converge;\ni indicates the number of elements of an intermediate tridiagonal form which did not converge to zero.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1043\n\n\n?sbevd\nComputes all eigenvalues and, optionally, all\neigenvectors of a real symmetric band matrix using\ndivide and conquer algorithm.\nSyntax\nlapack_int LAPACKE_ssbevd (int matrix_layout, char jobz, char uplo, lapack_int n,\nlapack_int kd, float* ab, lapack_int ldab, float* w, float* z, lapack_int ldz);\nlapack_int LAPACKE_dsbevd (int matrix_layout, char jobz, char uplo, lapack_int n,\nlapack_int kd, double* ab, lapack_int ldab, double* w, double* z, lapack_int ldz);\nInclude Files\n•\nmkl.h\nDescription\nThe routine computes all the eigenvalues, and optionally all the eigenvectors, of a real symmetric band\nmatrix A. In other words, it can compute the spectral factorization of A as:\nA = Z*Λ*ZT\nHere Λ is a diagonal matrix whose diagonal elements are the eigenvalues λi, and Z is the orthogonal matrix\nwhose columns are the eigenvectors zi. Thus,\nA*zi = λi*zi for i = 1, 2, ..., n.\nIf the eigenvectors are requested, then this routine uses a divide and conquer algorithm to compute\neigenvalues and eigenvectors. However, if only eigenvalues are required, then it uses the Pal-Walker-Kahan\nvariant of the QL or QR algorithm.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\njobz\nMust be 'N' or 'V'.\nIf jobz = 'N', then only eigenvalues are computed.\nIf jobz = 'V', then eigenvalues and eigenvectors are computed.\nuplo\nMust be 'U' or 'L'.\nIf uplo = 'U', ab stores the upper triangular part of A.\nIf uplo = 'L', ab stores the lower triangular part of A.\nn\nThe order of the matrix A (n≥ 0).\nkd\nThe number of super- or sub-diagonals in A\n(kd≥ 0).\nab\nab (size at least max(1, ldab*n) for column major layout and at least\nmax(1, ldab*(kd + 1)) for row major layout) is an array containing either\nupper or lower triangular part of the symmetric matrix A (as specified by\nuplo) in band storage format.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1044\n\n\nldab\nThe leading dimension of ab; must be at least kd+1 for column major\nlayout and n for row major layout.\nldz\nThe leading dimension of the output array z.\nConstraints:\nif jobz = 'N', then ldz≥ 1;\nif jobz = 'V', then ldz≥ max(1, n) .\nOutput Parameters\nw, z\nArrays:\nw, size at least max(1, n).\nIf info = 0, contains the eigenvalues of the matrix A in ascending order.\nSee also info.\nz(size max(1, ldz*n if job = 'V' and at least 1 if job = 'N').\nIf job = 'V', then this array is overwritten by the orthogonal matrix Z\nwhich contains the eigenvectors of A. The i-th column of Z contains the\neigenvector which corresponds to the eigenvalue w[i - 1].\nIf job = 'N', then z is not referenced.\nab\nOn exit, this array is overwritten by the values generated during the\nreduction to tridiagonal form.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = i, then the algorithm failed to converge; i indicates the number of elements of an intermediate\ntridiagonal form which did not converge to zero.\nIf info = -i, the i-th parameter had an illegal value.\nApplication Notes\nThe computed eigenvalues and eigenvectors are exact for a matrix A+E such that ||E||2=O(ε)*||A||2,\nwhere ε is the machine precision.\nThe complex analogue of this routine is hbevd.\nSee also syevd for matrices held in full storage, and spevd for matrices held in packed storage.\n?hbevd\nComputes all eigenvalues and, optionally, all\neigenvectors of a complex Hermitian band matrix\nusing divide and conquer algorithm.\nSyntax\nlapack_int LAPACKE_chbevd( int matrix_layout, char jobz, char uplo, lapack_int n,\nlapack_int kd, lapack_complex_float* ab, lapack_int ldab, float* w,\nlapack_complex_float* z, lapack_int ldz );\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1045\n\n\nlapack_int LAPACKE_zhbevd( int matrix_layout, char jobz, char uplo, lapack_int n,\nlapack_int kd, lapack_complex_double* ab, lapack_int ldab, double* w,\nlapack_complex_double* z, lapack_int ldz );\nInclude Files\n•\nmkl.h\nDescription\nThe routine computes all the eigenvalues, and optionally all the eigenvectors, of a complex Hermitian band\nmatrix A. In other words, it can compute the spectral factorization of A as: A = Z*Λ*ZH.\nHere Λ is a real diagonal matrix whose diagonal elements are the eigenvalues λi, and Z is the (complex)\nunitary matrix whose columns are the eigenvectors zi. Thus,\nA*zi = λi*zi for i = 1, 2, ..., n.\nIf the eigenvectors are requested, then this routine uses a divide and conquer algorithm to compute\neigenvalues and eigenvectors. However, if only eigenvalues are required, then it uses the Pal-Walker-Kahan\nvariant of the QL or QR algorithm.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\njobz\nMust be 'N' or 'V'.\nIf jobz = 'N', then only eigenvalues are computed.\nIf jobz = 'V', then eigenvalues and eigenvectors are computed.\nuplo\nMust be 'U' or 'L'.\nIf uplo = 'U', ab stores the upper triangular part of A.\nIf uplo = 'L', ab stores the lower triangular part of A.\nn\nThe order of the matrix A (n≥ 0).\nkd\nThe number of super- or sub-diagonals in A\n(kd≥ 0).\nab\nab (size at least max(1, ldab*n) for column major layout and at least\nmax(1, ldab*(kd + 1)) for row major layout) is an array containing either\nupper or lower triangular part of the Hermitian matrix A (as specified by\nuplo) in band storage format.\nldab\nThe leading dimension of ab; must be at least kd+1 for column major\nlayout and n for row major layout.\nldz\nThe leading dimension of the output array z.\nConstraints:\nif jobz = 'N', then ldz≥ 1;\nif jobz = 'V', then ldz≥ max(1, n) .\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1046\n\n\nOutput Parameters\nw\nArray, size at least max(1, n).\nIf info = 0, contains the eigenvalues of the matrix A in ascending order.\nSee also info.\nz\nArray, size max(1, ldz*n if job = 'V' and at least 1 if job = 'N'.\nIf jobz = 'V', then this array is overwritten by the unitary matrix Z which\ncontains the eigenvectors of A. The i-th column of Z contains the\neigenvector which corresponds to the eigenvalue w[i - 1].\nIf jobz = 'N', then z is not referenced.\nab\nOn exit, this array is overwritten by the values generated during the\nreduction to tridiagonal form.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nApplication Notes\nThe computed eigenvalues and eigenvectors are exact for a matrix A + E such that ||E||2 = O(ε)||A||2,\nwhere ε is the machine precision.\nThe real analogue of this routine is sbevd.\nSee also heevd for matrices held in full storage, and hpevd for matrices held in packed storage.\n?sbevx\nComputes selected eigenvalues and, optionally,\neigenvectors of a real symmetric band matrix.\nSyntax\nlapack_int LAPACKE_ssbevx (int matrix_layout, char jobz, char range, char uplo,\nlapack_int n, lapack_int kd, float* ab, lapack_int ldab, float* q, lapack_int ldq, float\nvl, float vu, lapack_int il, lapack_int iu, float abstol, lapack_int* m, float* w,\nfloat* z, lapack_int ldz, lapack_int* ifail);\nlapack_int LAPACKE_dsbevx (int matrix_layout, char jobz, char range, char uplo,\nlapack_int n, lapack_int kd, double* ab, lapack_int ldab, double* q, lapack_int ldq,\ndouble vl, double vu, lapack_int il, lapack_int iu, double abstol, lapack_int* m,\ndouble* w, double* z, lapack_int ldz, lapack_int* ifail);\nInclude Files\n•\nmkl.h\nDescription\nThe routine computes selected eigenvalues and, optionally, eigenvectors of a real symmetric band matrix A.\nEigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for\nthe desired eigenvalues.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1047\n\n\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\njobz\nMust be 'N' or 'V'.\nIf jobz = 'N', then only eigenvalues are computed.\nIf jobz = 'V', then eigenvalues and eigenvectors are computed.\nrange\nMust be 'A' or 'V' or 'I'.\nIf range = 'A', the routine computes all eigenvalues.\nIf range = 'V', the routine computes eigenvalues w[i] in the half-open\ninterval: vl<w[i]≤vu.\nIf range = 'I', the routine computes eigenvalues with indices in range il\nto iu.\nuplo\nMust be 'U' or 'L'.\nIf uplo = 'U', ab stores the upper triangular part of A.\nIf uplo = 'L', ab stores the lower triangular part of A.\nn\nThe order of the matrix A (n≥ 0).\nkd\nThe number of super- or sub-diagonals in A\n(kd≥ 0).\nab\nArrays:\nArray ab (size at least max(1, ldab*n) for column major layout and at least\nmax(1, ldab*(kd + 1)) for row major layout) contains either upper or\nlower triangular part of the symmetric matrix A (as specified by uplo) in\nband storage format.\nldab\nThe leading dimension of ab; must be at least kd +1 for column major\nlayout and n for row major layout.\nvl, vu\nIf range = 'V', the lower and upper bounds of the interval to be searched\nfor eigenvalues.\nConstraint: vl< vu.\nIf range = 'A' or 'I', vl and vu are not referenced.\nil, iu\nIf range = 'I', the indices in ascending order of the smallest and largest\neigenvalues to be returned.\nConstraint: 1 ≤il≤iu≤n, if n > 0; il=1 and iu=0\nif n = 0.\nIf range = 'A' or 'V', il and iu are not referenced.\nabstol\nThe absolute error tolerance to which each eigenvalue is required. See\nApplication notes for details on error tolerance.\nldq, ldz\nThe leading dimensions of the output arrays q and z, respectively.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1048\n\n\nConstraints:\nldq≥ 1, ldz≥ 1;\nIf jobz = 'V', then ldq≥ max(1, n) and ldz≥ max(1, n) for column\nmajor layout and ldz≥ max(1, m) for row major layout .\nOutput Parameters\nq\nArray, size max(1, ldz*n).\nIf jobz = 'V', the n-by-n orthogonal matrix is used in the reduction to\ntridiagonal form.\nIf jobz = 'N', the array q is not referenced.\nm\nThe total number of eigenvalues found, 0 ≤m≤n.\nIf range = 'A', m = n, if range = 'I', m = iu-il+1, and if range =\n'V', the exact value of m is not known in advance.\nw, z\nArrays:\nw, size at least max(1, n). The first m elements of w contain the selected\neigenvalues of the matrix A in ascending order.\nz(size at least max(1, ldz*m) for column major layout and max(1, ldz*n)\nfor row major layout).\nIf jobz = 'V', then if info = 0, the first m columns of z contain the\northonormal eigenvectors of the matrix A corresponding to the selected\neigenvalues, with the i-th column of z holding the eigenvector associated\nwith w[i - 1].\nIf an eigenvector fails to converge, then that column of z contains the latest\napproximation to the eigenvector, and the index of the eigenvector is\nreturned in ifail.\nIf jobz = 'N', then z is not referenced.\nab\nOn exit, this array is overwritten by the values generated during the\nreduction to tridiagonal form.\nifail\nArray, size at least max(1, n).\nIf jobz = 'V', then if info = 0, the first m elements of ifail are zero; if\ninfo > 0, the ifail contains the indices the eigenvectors that failed to\nconverge.\nIf jobz = 'N', then ifail is not referenced.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nIf info = i, then i eigenvectors failed to converge; their indices are stored in the array ifail.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1049\n\n\nApplication Notes\nAn approximate eigenvalue is accepted as converged when it is determined to lie in an interval [a,b] of width\nless than or equal to abstol+ε*max(|a|,|b|), where ε is the machine precision.\nIf abstol is less than or equal to zero, then ε*||T||1 is used as tolerance, where T is the tridiagonal matrix\nobtained by reducing A to tridiagonal form. Eigenvalues will be computed most accurately when abstol is set\nto twice the underflow threshold 2*?lamch('S'), not zero.\nIf this routine returns with info > 0, indicating that some eigenvectors did not converge, try setting abstol\nto 2*?lamch('S').\n?hbevx\nComputes selected eigenvalues and, optionally,\neigenvectors of a Hermitian band matrix.\nSyntax\nlapack_int LAPACKE_chbevx( int matrix_layout, char jobz, char range, char uplo,\nlapack_int n, lapack_int kd, lapack_complex_float* ab, lapack_int ldab,\nlapack_complex_float* q, lapack_int ldq, float vl, float vu, lapack_int il, lapack_int\niu, float abstol, lapack_int* m, float* w, lapack_complex_float* z, lapack_int ldz,\nlapack_int* ifail );\nlapack_int LAPACKE_zhbevx( int matrix_layout, char jobz, char range, char uplo,\nlapack_int n, lapack_int kd, lapack_complex_double* ab, lapack_int ldab,\nlapack_complex_double* q, lapack_int ldq, double vl, double vu, lapack_int il,\nlapack_int iu, double abstol, lapack_int* m, double* w, lapack_complex_double* z,\nlapack_int ldz, lapack_int* ifail );\nInclude Files\n•\nmkl.h\nDescription\nThe routine computes selected eigenvalues and, optionally, eigenvectors of a complex Hermitian band matrix\nA. Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices\nfor the desired eigenvalues.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\njobz\nMust be 'N' or 'V'.\nIf job = 'N', then only eigenvalues are computed.\nIf job = 'V', then eigenvalues and eigenvectors are computed.\nrange\nMust be 'A' or 'V' or 'I'.\nIf range = 'A', the routine computes all eigenvalues.\nIf range = 'V', the routine computes eigenvalues w[i] in the half-open\ninterval: vl< w[i]≤vu.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1050\n\n\nIf range = 'I', the routine computes eigenvalues with indices il to iu.\nuplo\nMust be 'U' or 'L'.\nIf uplo = 'U', ab stores the upper triangular part of A.\nIf uplo = 'L', ab stores the lower triangular part of A.\nn\nThe order of the matrix A (n≥ 0).\nkd\nThe number of super- or sub-diagonals in A\n(kd≥ 0).\nab\nab (size at least max(1, ldab*n) for column major layout and at least\nmax(1, ldab*(kd + 1)) for row major layout) is an array containing either\nupper or lower triangular part of the Hermitian matrix A (as specified by\nuplo) in band storage format.\nldab\nThe leading dimension of ab; must be at least kd +1 for column major\nlayout and n for row major layout.\nvl, vu\nIf range = 'V', the lower and upper bounds of the interval to be searched\nfor eigenvalues.\nConstraint: vl< vu.\nIf range = 'A' or 'I', vl and vu are not referenced.\nil, iu\nIf range = 'I', the indices in ascending order of the smallest and largest\neigenvalues to be returned.\nConstraint: 1 ≤il≤iu≤n, if n > 0; il=1 and iu=0 if n = 0.\nIf range = 'A' or 'V', il and iu are not referenced.\nabstol\nThe absolute error tolerance to which each eigenvalue is required. See\nApplication notes for details on error tolerance.\nldq, ldz\nThe leading dimensions of the output arrays q and z, respectively.\nConstraints:\nldq≥ 1, ldz≥ 1;\nIf jobz = 'V', then ldq≥ max(1, n) and ldz≥ max(1, n) for column major\nlayout and ldz≥ max(1, m) for row major layout.\nOutput Parameters\nq\nArray, size max(1, ldz*n).\nIf jobz = 'V', the n-by-n unitary matrix is used in the reduction to\ntridiagonal form.\nIf jobz = 'N', the array q is not referenced.\nm\nThe total number of eigenvalues found,\n0 ≤m≤n.\nIf range = 'A', m = n, if range = 'I', m = iu-il+1, and if range =\n'V', the exact value of m is not known in advance..\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1051\n\n\nw\nArray, size at least max(1, n). The first m elements contain the selected\neigenvalues of the matrix A in ascending order.\nz\nArray z(size at least max(1, ldz*m) for column major layout and max(1,\nldz*n) for row major layout).\nIf jobz = 'V', then if info = 0, the first m columns of z contain the\northonormal eigenvectors of the matrix A corresponding to the selected\neigenvalues, with the i-th column of z holding the eigenvector associated\nwith w[i - 1].\nIf an eigenvector fails to converge, then that column of z contains the latest\napproximation to the eigenvector, and the index of the eigenvector is\nreturned in ifail.\nIf jobz = 'N', then z is not referenced.\nab\nOn exit, this array is overwritten by the values generated during the\nreduction to tridiagonal form.\nIf uplo = 'U', the first superdiagonal and the diagonal of the tridiagonal\nmatrix T are returned in rows kd and kd+1 of ab, and if uplo = 'L', the\ndiagonal and first subdiagonal of T are returned in the first two rows of ab.\nifail\nArray, size at least max(1, n).\nIf jobz = 'V', then if info = 0, the first m elements of ifail are zero; if\ninfo > 0, the ifail contains the indices of the eigenvectors that failed to\nconverge.\nIf jobz = 'N', then ifail is not referenced.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nIf info = i, then i eigenvectors failed to converge; their indices are stored in the array ifail.\nApplication Notes\nAn approximate eigenvalue is accepted as converged when it is determined to lie in an interval [a,b] of width\nless than or equal to abstol + ε * max( |a|,|b| ), where ε is the machine precision.\nIf abstol is less than or equal to zero, then ε*||T||1 will be used in its place, where T is the tridiagonal\nmatrix obtained by reducing A to tridiagonal form. Eigenvalues will be computed most accurately when abstol\nis set to twice the underflow threshold 2*?lamch('S'), not zero.\nIf this routine returns with info > 0, indicating that some eigenvectors did not converge, try setting abstol\nto 2*?lamch('S').\n?stev\nComputes all eigenvalues and, optionally,\neigenvectors of a real symmetric tridiagonal matrix.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1052\n\n\nSyntax\nlapack_int LAPACKE_sstev (int matrix_layout, char jobz, lapack_int n, float* d, float*\ne, float* z, lapack_int ldz);\nlapack_int LAPACKE_dstev (int matrix_layout, char jobz, lapack_int n, double* d,\ndouble* e, double* z, lapack_int ldz);\nInclude Files\n•\nmkl.h\nDescription\nThe routine computes all eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix A.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\njobz\nMust be 'N' or 'V'.\nIf jobz = 'N', then only eigenvalues are computed.\nIf jobz = 'V', then eigenvalues and eigenvectors are computed.\nn\nThe order of the matrix A (n≥ 0).\nd, e\nArrays:\nArray d contains the n diagonal elements of the tridiagonal matrix A.\nThe size of d must be at least max(1, n).\nArray e contains the n-1 subdiagonal elements of the tridiagonal matrix A.\nThe size of e must be at least max(1, n). The n-th element of this array is\nused as workspace.\nldz\nThe leading dimension of the output array z; ldz≥ 1. If jobz = 'V' then\nldz≥ max(1, n).\nOutput Parameters\nd\nOn exit, if info = 0, contains the eigenvalues of the matrix A in ascending\norder.\nz\nArray, size (size max(1, ldz*n)).\nIf jobz = 'V', then if info = 0, z contains the orthonormal eigenvectors\nof the matrix A, with the i-th column of z holding the eigenvector associated\nwith the eigenvalue returned in d[i - 1].\nIf job = 'N', then z is not referenced.\ne\nOn exit, this array is overwritten with intermediate results.\nReturn Values\nThis function returns a value info.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1053\n\n\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nIf info = i, then the algorithm failed to converge;\ni elements of e did not converge to zero.\n?stevd\nComputes all eigenvalues and, optionally, all\neigenvectors of a real symmetric tridiagonal matrix\nusing divide and conquer algorithm.\nSyntax\nlapack_int LAPACKE_sstevd (int matrix_layout, char jobz, lapack_int n, float* d, float*\ne, float* z, lapack_int ldz);\nlapack_int LAPACKE_dstevd (int matrix_layout, char jobz, lapack_int n, double* d,\ndouble* e, double* z, lapack_int ldz);\nInclude Files\n•\nmkl.h\nDescription\nThe routine computes all the eigenvalues, and optionally all the eigenvectors, of a real symmetric tridiagonal\nmatrix T. In other words, the routine can compute the spectral factorization of T as: T = Z*Λ*ZT.\nHere Λ is a diagonal matrix whose diagonal elements are the eigenvalues λi, and Z is the orthogonal matrix\nwhose columns are the eigenvectors zi. Thus,\nT*zi = λi*zi for i = 1, 2, ..., n.\nIf the eigenvectors are requested, then this routine uses a divide and conquer algorithm to compute\neigenvalues and eigenvectors. However, if only eigenvalues are required, then it uses the Pal-Walker-Kahan\nvariant of the QL or QR algorithm.\nThere is no complex analogue of this routine.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\njobz\nMust be 'N' or 'V'.\nIf jobz = 'N', then only eigenvalues are computed.\nIf jobz = 'V', then eigenvalues and eigenvectors are computed.\nn\nThe order of the matrix T (n ≥ 0).\nd, e\nArrays:\nd contains the n diagonal elements of the tridiagonal matrix T.\nThe dimension of d must be at least max(1, n).\ne contains the n-1 off-diagonal elements of T.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1054\n\n\nThe dimension of e must be at least max(1, n). The n-th element of this\narray is used as workspace.\nldz\nThe leading dimension of the output array z. Constraints:\nldz≥ 1 if job = 'N';\nldz≥ max(1, n) if job = 'V'.\nOutput Parameters\nd\nOn exit, if info = 0, contains the eigenvalues of the matrix T in ascending\norder.\nSee also info.\nz\nArray, size max(1, ldz*n) if jobz = 'V' and 1 if jobz = 'N' .\nIf jobz = 'V', then this array is overwritten by the orthogonal matrix Z\nwhich contains the eigenvectors of T.\nIf jobz = 'N', then z is not referenced.\ne\nOn exit, this array is overwritten with intermediate results.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = i, then the algorithm failed to converge; i indicates the number of elements of an intermediate\ntridiagonal form which did not converge to zero.\nIf info = -i, the i-th parameter had an illegal value.\nApplication Notes\nThe computed eigenvalues and eigenvectors are exact for a matrix T+E such that ||E||2 = O(ε)*||T||2,\nwhere ε is the machine precision.\nIf λi is an exact eigenvalue, and μi is the corresponding computed value, then\n|μi - λi| ≤ c(n)*ε*||T||2\nwhere c(n) is a modestly increasing function of n.\nIf zi is the corresponding exact eigenvector, and wi is the corresponding computed vector, then the angle\nθ(zi, wi) between them is bounded as follows:\nθ(zi, wi) ≤ c(n)*ε*||T||2 / min i≠j|λi - λj|.\nThus the accuracy of a computed eigenvector depends on the gap between its eigenvalue and all the other\neigenvalues.\n?stevx\nComputes selected eigenvalues and eigenvectors of a\nreal symmetric tridiagonal matrix.\nSyntax\nlapack_int LAPACKE_sstevx (int matrix_layout, char jobz, char range, lapack_int n,\nfloat* d, float* e, float vl, float vu, lapack_int il, lapack_int iu, float abstol,\nlapack_int* m, float* w, float* z, lapack_int ldz, lapack_int* ifail);\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1055\n\n\nlapack_int LAPACKE_dstevx (int matrix_layout, char jobz, char range, lapack_int n,\ndouble* d, double* e, double vl, double vu, lapack_int il, lapack_int iu, double abstol,\nlapack_int* m, double* w, double* z, lapack_int ldz, lapack_int* ifail);\nInclude Files\n•\nmkl.h\nDescription\nThe routine computes selected eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal\nmatrix A. Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of\nindices for the desired eigenvalues.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\njobz\nMust be 'N' or 'V'.\nIf job = 'N', then only eigenvalues are computed.\nIf job = 'V', then eigenvalues and eigenvectors are computed.\nrange\nMust be 'A' or 'V' or 'I'.\nIf range = 'A', the routine computes all eigenvalues.\nIf range = 'V', the routine computes eigenvalues w[i] in the half-open\ninterval: vl<w[i]≤vu.\nIf range = 'I', the routine computes eigenvalues with indices il to iu.\nn\nThe order of the matrix A (n≥ 0).\nd, e\nArrays:\nd contains the n diagonal elements of the tridiagonal matrix A.\nThe dimension of d must be at least max(1, n).\ne contains the n-1 subdiagonal elements of A.\nThe dimension of e must be at least max(1, n-1). The n-th element of this\narray is used as workspace.\nvl, vu\nIf range = 'V', the lower and upper bounds of the interval to be searched\nfor eigenvalues.\nConstraint: vl< vu.\nIf range = 'A' or 'I', vl and vu are not referenced.\nil, iu\nIf range = 'I', the indices in ascending order of the smallest and largest\neigenvalues to be returned.\nConstraint: 1 ≤il≤iu≤n, if n > 0; il=1 and iu=0 if n = 0.\nIf range = 'A' or 'V', il and iu are not referenced.\nabstol\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1056\n\n\nldz\nThe leading dimensions of the output array z; ldz≥ 1. If jobz = 'V', then\nldz≥ max(1, n) for column major layout and ldz≥ max(1, m) for row major\nlayout.\nOutput Parameters\nm\nThe total number of eigenvalues found,\n0 ≤m≤n.\nIf range = 'A', m = n, if range = 'I', m = iu-il+1, and if range =\n'V' the exact value of m is unknown.\nw, z\nArrays:\nw, size at least max(1, n).\nThe first m elements of w contain the selected eigenvalues of the matrix A\nin ascending order.\nz(size at least max(1, ldz*m) for column major layout and max(1, ldz*n)\nfor row major layout) .\nIf jobz = 'V', then if info = 0, the first m columns of z contain the\northonormal eigenvectors of the matrix A corresponding to the selected\neigenvalues, with the i-th column of z holding the eigenvector associated\nwith w[i - 1].\nIf an eigenvector fails to converge, then that column of z contains the latest\napproximation to the eigenvector, and the index of the eigenvector is\nreturned in ifail.\nIf jobz = 'N', then z is not referenced.\nd, e\nOn exit, these arrays may be multiplied by a constant factor chosen to\navoid overflow or underflow in computing the eigenvalues.\nifail\nArray, size at least max(1, n).\nIf jobz = 'V', then if info = 0, the first m elements of ifail are zero; if\ninfo > 0, the ifail contains the indices of the eigenvectors that failed to\nconverge.\nIf jobz = 'N', then ifail is not referenced.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nIf info = i, then i eigenvectors failed to converge; their indices are stored in the array ifail.\nApplication Notes\nAn approximate eigenvalue is accepted as converged when it is determined to lie in an interval [a,b] of width\nless than or equal to abstol+ε*max(|a|,|b|), where ε is the machine precision.\nIf abstol is less than or equal to zero, then ε*|A|1 is used instead. Eigenvalues are computed most accurately\nwhen abstol is set to twice the underflow threshold 2*?lamch('S'), not zero.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1057\n\n\nIf this routine returns with info > 0, indicating that some eigenvectors did not converge, set abstol to\n2*?lamch('S').\n?stevr\nComputes selected eigenvalues and, optionally,\neigenvectors of a real symmetric tridiagonal matrix\nusing the Relatively Robust Representations.\nSyntax\nlapack_int LAPACKE_sstevr (int matrix_layout, char jobz, char range, lapack_int n,\nfloat* d, float* e, float vl, float vu, lapack_int il, lapack_int iu, float abstol,\nlapack_int* m, float* w, float* z, lapack_int ldz, lapack_int* isuppz);\nlapack_int LAPACKE_dstevr (int matrix_layout, char jobz, char range, lapack_int n,\ndouble* d, double* e, double vl, double vu, lapack_int il, lapack_int iu, double abstol,\nlapack_int* m, double* w, double* z, lapack_int ldz, lapack_int* isuppz);\nInclude Files\n•\nmkl.h\nDescription\nThe routine computes selected eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal\nmatrix T. Eigenvalues and eigenvectors can be selected by specifying either a range of values or a range of\nindices for the desired eigenvalues.\nWhenever possible, the routine calls stemr to compute the eigenspectrum using Relatively Robust\nRepresentations. stegr computes eigenvalues by the dqds algorithm, while orthogonal eigenvectors are\ncomputed from various \"good\" L*D*LT representations (also known as Relatively Robust Representations).\nGram-Schmidt orthogonalization is avoided as far as possible. More specifically, the various steps of the\nalgorithm are as follows. For the i-th unreduced block of T:\na.\nCompute T - σi = Li*Di*LiT, such that Li*Di*LiT is a relatively robust representation.\nb.\nCompute the eigenvalues, λj, of Li*Di*LiT to high relative accuracy by the dqds algorithm.\nc.\nIf there is a cluster of close eigenvalues, \"choose\" σi close to the cluster, and go to Step (a).\nd.\nGiven the approximate eigenvalue λj of Li*Di*LiT, compute the corresponding eigenvector by forming a\nrank-revealing twisted factorization.\nThe desired accuracy of the output can be specified by the input parameter abstol.\nThe routine ?stevr calls stemr when the full spectrum is requested on machines which conform to the\nIEEE-754 floating point standard. ?stevr calls stebz and stein on non-IEEE machines and when partial\nspectrum requests are made.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\njobz\nMust be 'N' or 'V'.\nIf jobz = 'N', then only eigenvalues are computed.\nIf jobz = 'V', then eigenvalues and eigenvectors are computed.\nrange\nMust be 'A' or 'V' or 'I'.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1058\n\n\nIf range = 'A', the routine computes all eigenvalues.\nIf range = 'V', the routine computes eigenvalues w[i]in the half-open\ninterval:\nvl<w[i]≤vu.\nIf range = 'I', the routine computes eigenvalues with indices il to iu.\nFor range = 'V'or 'I' and iu-il < n-1, sstebz/dstebz and sstein/\ndstein are called.\nn\nThe order of the matrix T (n≥ 0).\nd, e\nArrays:\nd contains the n diagonal elements of the tridiagonal matrix T.\nThe dimension of d must be at least max(1, n).\necontains the n-1 subdiagonal elements of A.\nThe dimension of e must be at least max(1, n-1). The n-th element of this\narray is used as workspace.\nvl, vu\nIf range = 'V', the lower and upper bounds of the interval to be searched\nfor eigenvalues.\nConstraint: vl< vu.\nIf range = 'A' or 'I', vl and vu are not referenced.\nil, iu\nIf range = 'I', the indices in ascending order of the smallest and largest\neigenvalues to be returned.\nConstraint: 1 ≤il≤iu≤n, if n > 0; il=1 and iu=0 if n = 0.\nIf range = 'A' or 'V', il and iu are not referenced.\nabstol\nThe absolute error tolerance to which each eigenvalue/eigenvector is\nrequired.\nIf jobz = 'V', the eigenvalues and eigenvectors output have residual\nnorms bounded by abstol, and the dot products between different\neigenvectors are bounded by abstol. If abstol < n *eps*||T||, then n\n*eps*||T|| will be used in its place, where eps is the machine precision,\nand ||T|| is the 1-norm of the matrix T. The eigenvalues are computed to\nan accuracy of eps*||T|| irrespective of abstol.\nIf high relative accuracy is important, set abstol to ?lamch('S').\nldz\nThe leading dimension of the output array z.\nConstraints:\nldz≥ 1 if jobz = 'N';\nldz≥ max(1, n) for column major layout and ldz≥ max(1, m) for row major\nlayout if jobz = 'V'.\nOutput Parameters\nm\nThe total number of eigenvalues found,\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1059\n\n\n0 ≤m≤n. If range = 'A', m = n, if range = 'I', m = iu-il+1, and if\nrange = 'V' the exact value of m is unknown..\nw, z\nArrays:\nw, size at least max(1, n).\nThe first m elements of w contain the selected eigenvalues of the matrix T\nin ascending order.\nz(size at least max(1, ldz*m) for column major layout and max(1, ldz*n)\nfor row major layout).\nIf jobz = 'V', then if info = 0, the first m columns of z contain the\northonormal eigenvectors of the matrix T corresponding to the selected\neigenvalues, with the i-th column of z holding the eigenvector associated\nwith w[i - 1].\nIf jobz = 'N', then z is not referenced.\nd, e\nOn exit, these arrays may be multiplied by a constant factor chosen to\navoid overflow or underflow in computing the eigenvalues.\nisuppz\nArray, size at least 2 *max(1, m).\nThe support of the eigenvectors in z, i.e., the indices indicating the nonzero\nelements in z. The i-th eigenvector is nonzero only in elements isuppz[2i\n- 2] through isuppz[2i - 1].\nImplemented only for range = 'A' or 'I' and iu-il = n-1.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nIf info = i, an internal error has occurred.\nApplication Notes\nNormal execution of the routine ?stegr may create NaNs and infinities and hence may abort due to a floating\npoint exception in environments which do not handle NaNs and infinities in the IEEE standard default manner.\nNonsymmetric Eigenvalue Problems: LAPACK Driver Routines\nThis topic describes LAPACK driver routines used for solving nonsymmetric eigenproblems. See also \ncomputational routines that can be called to solve these problems.\nTable \"Driver Routines for Solving Nonsymmetric Eigenproblems\" lists all such driver routines.\nDriver Routines for Solving Nonsymmetric Eigenproblems\nRoutine Name\nOperation performed\ngees\nComputes the eigenvalues and Schur factorization of a general matrix, and orders\nthe factorization so that selected eigenvalues are at the top left of the Schur form.\ngeesx\nComputes the eigenvalues and Schur factorization of a general matrix, orders the\nfactorization and computes reciprocal condition numbers.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1060\n\n\nRoutine Name\nOperation performed\ngeev\nComputes the eigenvalues and left and right eigenvectors of a general matrix.\ngeevx\nComputes the eigenvalues and left and right eigenvectors of a general matrix, with\npreliminary matrix balancing, and computes reciprocal condition numbers for the\neigenvalues and right eigenvectors.\n?gees\nComputes the eigenvalues and Schur factorization of a\ngeneral matrix, and orders the factorization so that\nselected eigenvalues are at the top left of the Schur\nform.\nSyntax\nlapack_int LAPACKE_sgees( int matrix_layout, char jobvs, char sort, LAPACK_S_SELECT2\nselect, lapack_int n, float* a, lapack_int lda, lapack_int* sdim, float* wr, float* wi,\nfloat* vs, lapack_int ldvs );\nlapack_int LAPACKE_dgees( int matrix_layout, char jobvs, char sort, LAPACK_D_SELECT2\nselect, lapack_int n, double* a, lapack_int lda, lapack_int* sdim, double* wr, double*\nwi, double* vs, lapack_int ldvs );\nlapack_int LAPACKE_cgees( int matrix_layout, char jobvs, char sort, LAPACK_C_SELECT1\nselect, lapack_int n, lapack_complex_float* a, lapack_int lda, lapack_int* sdim,\nlapack_complex_float* w, lapack_complex_float* vs, lapack_int ldvs );\nlapack_int LAPACKE_zgees( int matrix_layout, char jobvs, char sort, LAPACK_Z_SELECT1\nselect, lapack_int n, lapack_complex_double* a, lapack_int lda, lapack_int* sdim,\nlapack_complex_double* w, lapack_complex_double* vs, lapack_int ldvs );\nInclude Files\n•\nmkl.h\nDescription\nThe routine computes for an n-by-n real/complex nonsymmetric matrix A, the eigenvalues, the real Schur\nform T, and, optionally, the matrix of Schur vectors Z. This gives the Schur factorization A = Z*T*ZH.\nOptionally, it also orders the eigenvalues on the diagonal of the real-Schur/Schur form so that selected\neigenvalues are at the top left. The leading columns of Z then form an orthonormal basis for the invariant\nsubspace corresponding to the selected eigenvalues.\nA real matrix is in real-Schur form if it is upper quasi-triangular with 1-by-1 and 2-by-2 blocks. 2-by-2 blocks\nwill be standardized in the form\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1061\n\n\nwhere b*c < 0. The eigenvalues of such a block are \nA complex matrix is in Schur form if it is upper triangular.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\njobvs\nMust be 'N' or 'V'.\nIf jobvs = 'N', then Schur vectors are not computed.\nIf jobvs = 'V', then Schur vectors are computed.\nsort\nMust be 'N' or 'S'. Specifies whether or not to order the eigenvalues on\nthe diagonal of the Schur form.\nIf sort = 'N', then eigenvalues are not ordered.\nIf sort = 'S', eigenvalues are ordered (see select).\nselect\nIf sort = 'S', select is used to select eigenvalues to sort to the top left of\nthe Schur form.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1062\n\n\nIf sort = 'N', select is not referenced.\nFor real flavors:\nAn eigenvalue wr[j]+sqrt(-1)*wi[j] is selected if select(wr[j], wi[j]) is\ntrue; that is, if either one of a complex conjugate pair of eigenvalues is\nselected, then both complex eigenvalues are selected.\nFor complex flavors:\nAn eigenvalue w[j] is selected if select(w[j]) is true.\nNote that a selected complex eigenvalue may no longer satisfy select(wr[j],\nwi[j])= 1 after ordering, since ordering may change the value of complex\neigenvalues (especially if the eigenvalue is ill-conditioned); in this case info\nmay be set to n+2 (see info below).\nn\nThe order of the matrix A (n≥ 0).\na\nArrays:\na (size at least max(1, lda*n)) is an array containing the n-by-n matrix A.\nlda\nThe leading dimension of the array a. Must be at least max(1, n).\nldvs\nThe leading dimension of the output array vs. Constraints:\nldvs≥ 1;\nldvs≥ max(1, n) if jobvs = 'V'.\nOutput Parameters\na\nOn exit, this array is overwritten by the real-Schur/Schur form T.\nsdim\nIf sort = 'N', sdim= 0.\nIf sort = 'S', sdim is equal to the number of eigenvalues (after sorting)\nfor which select is true.\nNote that for real flavors complex conjugate pairs for which select is true for\neither eigenvalue count as 2.\nwr, wi\nArrays, size at least max (1, n) each. Contain the real and imaginary parts,\nrespectively, of the computed eigenvalues, in the same order that they\nappear on the diagonal of the output real-Schur form T. Complex conjugate\npairs of eigenvalues appear consecutively with the eigenvalue having\npositive imaginary part first.\nw\nArray, size at least max(1, n). Contains the computed eigenvalues. The\neigenvalues are stored in the same order as they appear on the diagonal of\nthe output Schur form T.\nvs\nArray vs (size at least max(1, ldvs*n)) .\nIf jobvs = 'V', vs contains the orthogonal/unitary matrix Z of Schur\nvectors.\nIf jobvs = 'N', vs is not referenced.\nReturn Values\nThis function returns a value info.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1063\n\n\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nIf info = i, and\ni≤n:\nthe QR algorithm failed to compute all the eigenvalues; elements 1:ilo-1 and i+1:n of wr and wi (for real\nflavors) or w (for complex flavors) contain those eigenvalues which have converged; if jobvs = 'V', vs\ncontains the matrix which reduces A to its partially converged Schur form;\ni = n+1:\nthe eigenvalues could not be reordered because some eigenvalues were too close to separate (the problem is\nvery ill-conditioned);\ni = n+2:\nafter reordering, round-off changed values of some complex eigenvalues so that leading eigenvalues in the\nSchur form no longer satisfy select = 1. This could also be caused by underflow due to scaling.\n?geesx\nComputes the eigenvalues and Schur factorization of a\ngeneral matrix, orders the factorization and computes\nreciprocal condition numbers.\nSyntax\nlapack_int LAPACKE_sgeesx( int matrix_layout, char jobvs, char sort, LAPACK_S_SELECT2\nselect, char sense, lapack_int n, float* a, lapack_int lda, lapack_int* sdim, float* wr,\nfloat* wi, float* vs, lapack_int ldvs, float* rconde, float* rcondv );\nlapack_int LAPACKE_dgeesx( int matrix_layout, char jobvs, char sort, LAPACK_D_SELECT2\nselect, char sense, lapack_int n, double* a, lapack_int lda, lapack_int* sdim, double*\nwr, double* wi, double* vs, lapack_int ldvs, double* rconde, double* rcondv );\nlapack_int LAPACKE_cgeesx( int matrix_layout, char jobvs, char sort, LAPACK_C_SELECT1\nselect, char sense, lapack_int n, lapack_complex_float* a, lapack_int lda, lapack_int*\nsdim, lapack_complex_float* w, lapack_complex_float* vs, lapack_int ldvs, float*\nrconde, float* rcondv );\nlapack_int LAPACKE_zgeesx( int matrix_layout, char jobvs, char sort, LAPACK_Z_SELECT1\nselect, char sense, lapack_int n, lapack_complex_double* a, lapack_int lda, lapack_int*\nsdim, lapack_complex_double* w, lapack_complex_double* vs, lapack_int ldvs, double*\nrconde, double* rcondv );\nInclude Files\n•\nmkl.h\nDescription\nThe routine computes for an n-by-n real/complex nonsymmetric matrix A, the eigenvalues, the real-Schur/\nSchur form T, and, optionally, the matrix of Schur vectors Z. This gives the Schur factorization A = Z*T*ZH.\nOptionally, it also orders the eigenvalues on the diagonal of the real-Schur/Schur form so that selected\neigenvalues are at the top left; computes a reciprocal condition number for the average of the selected\neigenvalues (rconde); and computes a reciprocal condition number for the right invariant subspace\ncorresponding to the selected eigenvalues (rcondv). The leading columns of Z form an orthonormal basis for\nthis invariant subspace.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1064\n\n\nFor further explanation of the reciprocal condition numbers rconde and rcondv, see [LUG], Section 4.10\n(where these quantities are called s and sep respectively).\nA real matrix is in real-Schur form if it is upper quasi-triangular with 1-by-1 and 2-by-2 blocks. 2-by-2 blocks\nwill be standardized in the form\nwhere b*c < 0. The eigenvalues of such a block are \nA complex matrix is in Schur form if it is upper triangular.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\njobvs\nMust be 'N' or 'V'.\nIf jobvs = 'N', then Schur vectors are not computed.\nIf jobvs = 'V', then Schur vectors are computed.\nsort\nMust be 'N' or 'S'. Specifies whether or not to order the eigenvalues on\nthe diagonal of the Schur form.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1065\n\n\nIf sort = 'N', then eigenvalues are not ordered.\nIf sort = 'S', eigenvalues are ordered (see select).\nselect\nIf sort = 'S', select is used to select eigenvalues to sort to the top left of\nthe Schur form.\nIf sort = 'N', select is not referenced.\nFor real flavors:\nAn eigenvalue wr[j]+sqrt(-1)*wi[j] is selected if select(wr[j], wi[j]) is\ntrue; that is, if either one of a complex conjugate pair of eigenvalues is\nselected, then both complex eigenvalues are selected.\nFor complex flavors:\nAn eigenvalue w[j] is selected if select(w[j]) is true.\nNote that a selected complex eigenvalue may no longer satisfy select(wr[j],\nwi[j])= 1 after ordering, since ordering may change the value of complex\neigenvalues (especially if the eigenvalue is ill-conditioned); in this case info\nmay be set to n+2 (see info below).\nsense\nMust be 'N', 'E', 'V', or 'B'. Determines which reciprocal condition\nnumber are computed.\nIf sense = 'N', none are computed;\nIf sense = 'E', computed for average of selected eigenvalues only;\nIf sense = 'V', computed for selected right invariant subspace only;\nIf sense = 'B', computed for both.\nIf sense is 'E', 'V', or 'B', then sort must equal 'S'.\nn\nThe order of the matrix A (n≥ 0).\na\nArrays:\na (size at least max(1, lda*n)) is an array containing the n-by-n matrix A.\nlda\nThe leading dimension of the array a. Must be at least max(1, n).\nldvs\nThe leading dimension of the output array vs. Constraints:\nldvs≥ 1;\nldvs≥ max(1, n)if jobvs = 'V'.\nOutput Parameters\na\nOn exit, this array is overwritten by the real-Schur/Schur form T.\nsdim\nIf sort = 'N', sdim= 0.\nIf sort = 'S', sdim is equal to the number of eigenvalues (after sorting)\nfor which select is true.\nNote that for real flavors complex conjugate pairs for which select is true for\neither eigenvalue count as 2.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1066\n\n\nwr, wi\nArrays, size at least max (1, n) each. Contain the real and imaginary parts,\nrespectively, of the computed eigenvalues, in the same order that they\nappear on the diagonal of the output real-Schur form T. Complex conjugate\npairs of eigenvalues appear consecutively with the eigenvalue having\npositive imaginary part first.\nw\nArray, size at least max(1, n). Contains the computed eigenvalues. The\neigenvalues are stored in the same order as they appear on the diagonal of\nthe output Schur form T.\nvs\nArray vs (size at least max(1, ldvs*n))\nIf jobvs = 'V', vs contains the orthogonal/unitary matrix Z of Schur\nvectors.\nIf jobvs = 'N', vs is not referenced.\nrconde, rcondv\nIf sense = 'E' or 'B', rconde contains the reciprocal condition number for\nthe average of the selected eigenvalues.\nIf sense = 'N' or 'V', rconde is not referenced.\nIf sense = 'V' or 'B', rcondv contains the reciprocal condition number for\nthe selected right invariant subspace.\nIf sense = 'N' or 'E', rcondv is not referenced.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nIf info = i, and\ni≤n:\nthe QR algorithm failed to compute all the eigenvalues; elements 1:ilo-1 and i+1:n of wr and wi (for real\nflavors) or w (for complex flavors) contain those eigenvalues which have converged; if jobvs = 'V', vs\ncontains the transformation which reduces A to its partially converged Schur form;\ni = n+1:\nthe eigenvalues could not be reordered because some eigenvalues were too close to separate (the problem is\nvery ill-conditioned);\ni = n+2:\nafter reordering, roundoff changed values of some complex eigenvalues so that leading eigenvalues in the\nSchur form no longer satisfy select = 1. This could also be caused by underflow due to scaling.\n?geev\nComputes the eigenvalues and left and right\neigenvectors of a general matrix.\nSyntax\nlapack_int LAPACKE_sgeev( int matrix_layout, char jobvl, char jobvr, lapack_int n,\nfloat* a, lapack_int lda, float* wr, float* wi, float* vl, lapack_int ldvl, float* vr,\nlapack_int ldvr );\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1067\n\n\nlapack_int LAPACKE_dgeev( int matrix_layout, char jobvl, char jobvr, lapack_int n,\ndouble* a, lapack_int lda, double* wr, double* wi, double* vl, lapack_int ldvl, double*\nvr, lapack_int ldvr );\nlapack_int LAPACKE_cgeev( int matrix_layout, char jobvl, char jobvr, lapack_int n,\nlapack_complex_float* a, lapack_int lda, lapack_complex_float* w, lapack_complex_float*\nvl, lapack_int ldvl, lapack_complex_float* vr, lapack_int ldvr );\nlapack_int LAPACKE_zgeev( int matrix_layout, char jobvl, char jobvr, lapack_int n,\nlapack_complex_double* a, lapack_int lda, lapack_complex_double* w,\nlapack_complex_double* vl, lapack_int ldvl, lapack_complex_double* vr, lapack_int\nldvr );\nInclude Files\n•\nmkl.h\nDescription\nThe routine computes for an n-by-n real/complex nonsymmetric matrix A, the eigenvalues and, optionally,\nthe left and/or right eigenvectors. The right eigenvector v of A satisfies\nA*v = λ*v\nwhere λ is its eigenvalue.\nThe left eigenvector u of A satisfies\nuH*A = λ*uH\nwhere uH denotes the conjugate transpose of u. The computed eigenvectors are normalized to have\nEuclidean norm equal to 1 and largest component real.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\njobvl\nMust be 'N' or 'V'.\nIf jobvl = 'N', then left eigenvectors of A are not computed.\nIf jobvl = 'V', then left eigenvectors of A are computed.\njobvr\nMust be 'N' or 'V'.\nIf jobvr = 'N', then right eigenvectors of A are not computed.\nIf jobvr = 'V', then right eigenvectors of A are computed.\nn\nThe order of the matrix A (n≥ 0).\na\na (size at least max(1, lda*n)) is an array containing the n-by-n matrix A.\nlda\nThe leading dimension of the array a. Must be at least max(1, n).\nldvl, ldvr\nThe leading dimensions of the output arrays vl and vr, respectively.\nConstraints:\nldvl≥ 1; ldvr≥ 1.\nIf jobvl = 'V', ldvl≥ max(1, n);\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1068\n\n\nIf jobvr = 'V', ldvr≥ max(1, n).\nOutput Parameters\na\nOn exit, this array is overwritten.\nwr, wi\nArrays, size at least max (1, n) each.\nContain the real and imaginary parts, respectively, of the computed\neigenvalues. Complex conjugate pairs of eigenvalues appear consecutively\nwith the eigenvalue having positive imaginary part first.\nw\nArray, size at least max(1, n).\nContains the computed eigenvalues.\nvl, vr\nArrays:\nvl (size at least max(1, ldvl*n)) .\nIf jobvl = 'N', vl is not referenced.\nFor real flavors:\nIf the j-th eigenvalue is real,the i-th component of the j-th eigenvector uj is\nstored in vl[(i - 1) + (j - 1)*ldvl] for column major layout and in\nvl[(i - 1)*ldvl + (j - 1)] for row major layout..\nIf the j-th and (j+1)-st eigenvalues form a complex conjugate pair, then for\ni = sqrt(-1), the k-th component of the j-th eigenvector uj is vl[(k - 1)\n+ (j - 1)*ldvl] + i*vl[(k - 1) + j*ldvl] for column major layout and as\nvl[(k - 1)*ldvl + (j - 1)] + i*vl[(k-1)*ldvl + j] for row major layout.\nSimilarly, the k-th component of vector (j+1) uj + 1 is vl[(k - 1) + (j -\n1)*ldvl] - i*vl[(k - 1) + j*ldvl] for column major layout and as vl[(k -\n1)*ldvl + (j - 1)] -i*vl[(k - 1)*ldvl + j] for row major layout. .\nFor complex flavors:\nThe i-th component of the j-th eigenvector uj is stored in vl[(i - 1) +\n(j - 1)*ldvl] for column major layout and in vl[(i - 1)*ldvl+(j -\n1)] for row major layout.\nvr (size at least max(1, ldvr*n)).\nIf jobvr = 'N', vr is not referenced.\nFor real flavors:\nIf the j-th eigenvalue is real, then the i-th component of j-th eigenvector vj\nis stored in vr[(i - 1) + (j - 1)*ldvr] for column major layout and in\nvr[(i - 1)*ldvr + (j - 1)] for row major layout..\nIf the j-th and (j+1)-st eigenvalues form a complex conjugate pair, then for\ni = sqrt(-1), the k-th component of the j-th eigenvector vj is vr[(k - 1)\n+ (j - 1)*ldvr] +i*vr[(k - 1) + j*ldvr] for column major layout and as\nvr[(k - 1)*ldvr + (j - 1)] + i*vr[(k - 1)*ldvr + j] for row major layout.\nSimilarly, the k-th component of vector j + 1) vj + 1 is vr[(k - 1) + (j -\n1)*ldvr] - i*vr[(k - 1) + j*ldvr] for column major layout and as vr[(k -\n1)*ldvr + (j - 1)] - i*vr[(k - 1)*ldvr + j] for row major layout.\nFor complex flavors:\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1069\n\n\nThe i-th component of the j-th eigenvector vj is stored in vr[(i - 1) + (j\n- 1)*ldvr] for column major layout and in vr[(i - 1)*ldvr + (j -\n1)] for row major layout.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nIf info = i, the QR algorithm failed to compute all the eigenvalues, and no eigenvectors have been\ncomputed; elements i+1:n of wr and wi (for real flavors) or w (for complex flavors) contain those\neigenvalues which have converged.\n?geevx\nComputes the eigenvalues and left and right\neigenvectors of a general matrix, with preliminary\nmatrix balancing, and computes reciprocal condition\nnumbers for the eigenvalues and right eigenvectors.\nSyntax\nlapack_int LAPACKE_sgeevx( int matrix_layout, char balanc, char jobvl, char jobvr, char\nsense, lapack_int n, float* a, lapack_int lda, float* wr, float* wi, float* vl,\nlapack_int ldvl, float* vr, lapack_int ldvr, lapack_int* ilo, lapack_int* ihi, float*\nscale, float* abnrm, float* rconde, float* rcondv );\nlapack_int LAPACKE_dgeevx( int matrix_layout, char balanc, char jobvl, char jobvr, char\nsense, lapack_int n, double* a, lapack_int lda, double* wr, double* wi, double* vl,\nlapack_int ldvl, double* vr, lapack_int ldvr, lapack_int* ilo, lapack_int* ihi, double*\nscale, double* abnrm, double* rconde, double* rcondv );\nlapack_int LAPACKE_cgeevx( int matrix_layout, char balanc, char jobvl, char jobvr, char\nsense, lapack_int n, lapack_complex_float* a, lapack_int lda, lapack_complex_float* w,\nlapack_complex_float* vl, lapack_int ldvl, lapack_complex_float* vr, lapack_int ldvr,\nlapack_int* ilo, lapack_int* ihi, float* scale, float* abnrm, float* rconde, float*\nrcondv );\nlapack_int LAPACKE_zgeevx( int matrix_layout, char balanc, char jobvl, char jobvr, char\nsense, lapack_int n, lapack_complex_double* a, lapack_int lda, lapack_complex_double*\nw, lapack_complex_double* vl, lapack_int ldvl, lapack_complex_double* vr, lapack_int\nldvr, lapack_int* ilo, lapack_int* ihi, double* scale, double* abnrm, double* rconde,\ndouble* rcondv );\nInclude Files\n•\nmkl.h\nDescription\nThe routine computes for an n-by-n real/complex nonsymmetric matrix A, the eigenvalues and, optionally,\nthe left and/or right eigenvectors.\nOptionally also, it computes a balancing transformation to improve the conditioning of the eigenvalues and\neigenvectors (ilo, ihi, scale, and abnrm), reciprocal condition numbers for the eigenvalues (rconde), and\nreciprocal condition numbers for the right eigenvectors (rcondv).\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1070\n\n\nThe right eigenvector v of A satisfies\nA·v = λ·v\nwhere λ is its eigenvalue.\nThe left eigenvector u of A satisfies\nuHA = λuH\nwhere uH denotes the conjugate transpose of u. The computed eigenvectors are normalized to have Euclidean\nnorm equal to 1 and largest component real.\nBalancing a matrix means permuting the rows and columns to make it more nearly upper triangular, and\napplying a diagonal similarity transformation D*A*inv(D), where D is a diagonal matrix, to make its rows and\ncolumns closer in norm and the condition numbers of its eigenvalues and eigenvectors smaller. The computed\nreciprocal condition numbers correspond to the balanced matrix. Permuting rows and columns will not\nchange the condition numbers in exact arithmetic) but diagonal scaling will. For further explanation of\nbalancing, see [LUG], Section 4.10.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nbalanc\nMust be 'N', 'P', 'S', or 'B'. Indicates how the input matrix should be\ndiagonally scaled and/or permuted to improve the conditioning of its\neigenvalues.\nIf balanc = 'N', do not diagonally scale or permute;\nIf balanc = 'P', perform permutations to make the matrix more nearly\nupper triangular. Do not diagonally scale;\nIf balanc = 'S', diagonally scale the matrix, i.e. replace A by\nD*A*inv(D), where D is a diagonal matrix chosen to make the rows and\ncolumns of A more equal in norm. Do not permute;\nIf balanc = 'B', both diagonally scale and permute A.\nComputed reciprocal condition numbers will be for the matrix after\nbalancing and/or permuting. Permuting does not change condition numbers\n(in exact arithmetic), but balancing does.\njobvl\nMust be 'N' or 'V'.\nIf jobvl = 'N', left eigenvectors of A are not computed;\nIf jobvl = 'V', left eigenvectors of A are computed.\nIf sense = 'E' or 'B', then jobvl must be 'V'.\njobvr\nMust be 'N' or 'V'.\nIf jobvr = 'N', right eigenvectors of A are not computed;\nIf jobvr = 'V', right eigenvectors of A are computed.\nIf sense = 'E' or 'B', then jobvr must be 'V'.\nsense\nMust be 'N', 'E', 'V', or 'B'. Determines which reciprocal condition\nnumber are computed.\nIf sense = 'N', none are computed;\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1071\n\n\nIf sense = 'E', computed for eigenvalues only;\nIf sense = 'V', computed for right eigenvectors only;\nIf sense = 'B', computed for eigenvalues and right eigenvectors.\nIf sense is 'E' or 'B', both left and right eigenvectors must also be\ncomputed (jobvl = 'V' and jobvr = 'V').\nn\nThe order of the matrix A (n≥ 0).\na\nArrays:\na (size at least max(1, lda*n)) is an array containing the n-by-n matrix A.\nlda\nThe leading dimension of the array a. Must be at least max(1, n).\nldvl, ldvr\nThe leading dimensions of the output arrays vl and vr, respectively.\nConstraints:\nldvl≥ 1; ldvr≥ 1.\nIf jobvl = 'V', ldvl≥ max(1, n);\nIf jobvr = 'V', ldvr≥ max(1, n).\nOutput Parameters\na\nOn exit, this array is overwritten.\nIf jobvl = 'V' or jobvr = 'V', it contains the real-Schur/Schur form of\nthe balanced version of the input matrix A.\nwr, wi\nArrays, size at least max (1, n) each. Contain the real and imaginary parts,\nrespectively, of the computed eigenvalues. Complex conjugate pairs of\neigenvalues appear consecutively with the eigenvalue having positive\nimaginary part first.\nw\nArray, size at least max(1, n). Contains the computed eigenvalues.\nvl, vr\nArrays:\nvl (size at least max(1, ldvl*n)) .\nIf jobvl = 'N', vl is not referenced.\nFor real flavors:\nIf the j-th eigenvalue is real,the i-th component of the j-th eigenvector uj is\nstored in vl[(i - 1) + (j - 1)*ldvl] for column major layout and in\nvl[(i - 1)*ldvl + (j - 1)] for row major layout..\nIf the j-th and (j+1)-st eigenvalues form a complex conjugate pair, then for\ni = sqrt(-1), the k-th component of the j-th eigenvector uj is vl[(k - 1)\n+ (j - 1)*ldvl] + i*vl[(k - 1) + j*ldvl] for column major layout and as\nvl[(k - 1)*ldvl + (j - 1)] + i*vl[(k-1)*ldvl + j] for row major layout.\nSimilarly, the k-th component of vector (j+1) uj + 1 is vl[(k - 1) + (j -\n1)*ldvl] - i*vl[(k - 1) + j*ldvl] for column major layout and as vl[(k -\n1)*ldvl + (j - 1)] -i*vl[(k - 1)*ldvl + j] for row major layout. .\nFor complex flavors:\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1072\n\n\nThe i-th component of the j-th eigenvector uj is stored in vl[(i - 1) +\n(j - 1)*ldvl] for column major layout and in vl[(i - 1)*ldvl+(j -\n1)] for row major layout.\nvr (size at least max(1, ldvr*n)).\nIf jobvr = 'N', vr is not referenced.\nFor real flavors:\nIf the j-th eigenvalue is real, then the i-th component of j-th eigenvector vj\nis stored in vr[(i - 1) + (j - 1)*ldvr] for column major layout and in\nvr[(i - 1)*ldvr + (j - 1)] for row major layout..\nIf the j-th and (j+1)-st eigenvalues form a complex conjugate pair, then for\ni = sqrt(-1), the k-th component of the j-th eigenvector vj is vr[(k - 1)\n+ (j - 1)*ldvr] +i*vr[(k - 1) + j*ldvr] for column major layout and as\nvr[(k - 1)*ldvr + (j - 1)] + i*vr[(k - 1)*ldvr + j] for row major layout.\nSimilarly, the k-th component of vector j + 1) vj + 1 is vr[(k - 1) + (j -\n1)*ldvr] - i*vr[(k - 1) + j*ldvr] for column major layout and as vr[(k -\n1)*ldvr + (j - 1)] - i*vr[(k - 1)*ldvr + j] for row major layout.\nFor complex flavors:\nThe i-th component of the j-th eigenvector vj is stored in vr[(i - 1) + (j\n- 1)*ldvr] for column major layout and in vr[(i - 1)*ldvr + (j -\n1)] for row major layout.\nilo, ihi\nilo and ihi are integer values determined when A was balanced.\nThe balanced A(i,j) = 0 if i > j and j = 1,..., ilo-1 or i = ihi\n+1,..., n.\nIf balanc = 'N' or 'S', ilo = 1 and ihi = n.\nscale\nArray, size at least max(1, n). Details of the permutations and scaling\nfactors applied when balancing A.\nIf P[j - 1] is the index of the row and column interchanged with row and\ncolumn j, and D[j - 1] is the scaling factor applied to row and column j,\nthen\nscale[j - 1] = P[j - 1], for j = 1,...,ilo-1\n= D[j - 1], for j = ilo,...,ihi\n= P[j - 1] for j = ihi+1,..., n.\nThe order in which the interchanges are made is n to ihi+1, then 1 to ilo-1.\nabnrm\nThe one-norm of the balanced matrix (the maximum of the sum of absolute\nvalues of elements of any column).\nrconde, rcondv\nArrays, size at least max(1, n) each.\nrconde[j - 1] is the reciprocal condition number of the j-th eigenvalue.\nrcondv[j - 1] is the reciprocal condition number of the j-th right eigenvector.\nReturn Values\nThis function returns a value info.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1073\n\n\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nIf info = i, the QR algorithm failed to compute all the eigenvalues, and no eigenvectors or condition\nnumbers have been computed; elements 1:ilo-1 and i+1:n of wr and wi (for real flavors) or w (for complex\nflavors) contain eigenvalues which have converged.\nSingular Value Decomposition: LAPACK Driver Routines\nTable \"Driver Routines for Singular Value Decomposition\" lists the LAPACK driver routines that perform\nsingular value decomposition .\nDriver Routines for Singular Value Decomposition\nRoutine Name\nOperation performed\n?gesvd\nComputes the singular value decomposition of a general rectangular matrix.\n?gesdd\nComputes the singular value decomposition of a general rectangular matrix using a\ndivide and conquer method.\n?gejsv\nComputes the singular value decomposition of a real matrix using a preconditioned\nJacobi SVD method.\n?gesvj\nComputes the singular value decomposition of a real matrix using Jacobi plane\nrotations.\n?ggsvd\nComputes the generalized singular value decomposition of a pair of general\nrectangular matrices.\n?gesvdx\nComputes the SVD and left and right singular vectors for a matrix.\n?bdsvdx\nComputes the SVD of a bidiagonal matrix.\n?\ngesvda_batch_stri\nded\nComputes the truncated SVD of a group of general m-by-n matrices that are stored\nat a constant stride from each other in a contiguous block of memory.\nSingular Value Decomposition - LAPACK Computational Routines\n?gesvd\nComputes the singular value decomposition of a\ngeneral rectangular matrix.\nSyntax\nlapack_int LAPACKE_sgesvd( int matrix_layout, char jobu, char jobvt, lapack_int m,\nlapack_int n, float* a, lapack_int lda, float* s, float* u, lapack_int ldu, float* vt,\nlapack_int ldvt, float* superb );\nlapack_int LAPACKE_dgesvd( int matrix_layout, char jobu, char jobvt, lapack_int m,\nlapack_int n, double* a, lapack_int lda, double* s, double* u, lapack_int ldu, double*\nvt, lapack_int ldvt, double* superb );\nlapack_int LAPACKE_cgesvd( int matrix_layout, char jobu, char jobvt, lapack_int m,\nlapack_int n, lapack_complex_float* a, lapack_int lda, float* s, lapack_complex_float*\nu, lapack_int ldu, lapack_complex_float* vt, lapack_int ldvt, float* superb );\nlapack_int LAPACKE_zgesvd( int matrix_layout, char jobu, char jobvt, lapack_int m,\nlapack_int n, lapack_complex_double* a, lapack_int lda, double* s,\nlapack_complex_double* u, lapack_int ldu, lapack_complex_double* vt, lapack_int ldvt,\ndouble* superb );\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1074\n\n\nInclude Files\n•\nmkl.h\nDescription\nThe routine computes the singular value decomposition (SVD) of a real/complex m-by-n matrix A, optionally\ncomputing the left and/or right singular vectors. The SVD is written as\nA = U*Σ*VT for real routines\nA = U*Σ*VH for complex routines\nwhere Σ is an m-by-n matrix which is zero except for its min(m,n) diagonal elements, U is an m-by-m\northogonal/unitary matrix, and V is an n-by-n orthogonal/unitary matrix. The diagonal elements of Σ are the\nsingular values of A; they are real and non-negative, and are returned in descending order. The first min(m,\nn) columns of U and V are the left and right singular vectors of A.\nThe routine returns VT (for real flavors) or VH (for complex flavors), not V.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\njobu\nMust be 'A', 'S', 'O', or 'N'. Specifies options for computing all or part\nof the matrix U.\nIf jobu = 'A', all m columns of U are returned in the array u;\nif jobu = 'S', the first min(m, n) columns of U (the left singular vectors)\nare returned in the array u;\nif jobu = 'O', the first min(m, n) columns of U (the left singular vectors)\nare overwritten on the array a;\nif jobu = 'N', no columns of U (no left singular vectors) are computed.\njobvt\nMust be 'A', 'S', 'O', or 'N'. Specifies options for computing all or part\nof the matrix VT/VH.\nIf jobvt = 'A', all n rows of VT/VH are returned in the array vt;\nif jobvt = 'S', the first min(m,n) rows of VT/VH (the right singular\nvectors) are returned in the array vt;\nif jobvt = 'O', the first min(m,n) rows of VT/VH) (the right singular\nvectors) are overwritten on the array a;\nif jobvt = 'N', no rows of VT/VH (no right singular vectors) are computed.\njobvt and jobu cannot both be 'O'.\nm\nThe number of rows of the matrix A (m≥ 0).\nn\nThe number of columns in A (n≥ 0).\na\nArrays:\na(size at least max(1, lda*n) for column major layout and max(1, lda*m)\nfor row major layout) is an array containing the m-by-n matrix A.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1075\n\n\nlda\nThe leading dimension of the array a.\nMust be at least max(1, m) for column major layout and at least max(1, n)\nfor row major layout .\nldu, ldvt\nThe leading dimensions of the output arrays u and vt, respectively.\nConstraints:\nldu≥ 1; ldvt≥ 1.\nIf jobu = 'A', ldu≥m;\nIf jobu = 'S', ldu≥m for column major layout and ldu≥ min(m, n) for row\nmajor layout;\nIf jobvt = 'A', ldvt≥n;\nIf jobvt = 'S', ldvt≥ min(m, n) for column major layout and ldvt≥n for\nrow major layout .\nOutput Parameters\na\nOn exit,\nIf jobu = 'O', a is overwritten with the first min(m,n) columns of U (the\nleft singular vectors stored columnwise);\nIf jobvt = 'O', a is overwritten with the first min(m, n) rows of VT/VH (the\nright singular vectors stored rowwise);\nIf jobu≠'O' and jobvt≠'O', the contents of a are destroyed.\ns\nArray, size at least max(1, min(m,n)). Contains the singular values of A\nsorted so that s[i] ≥ s[i + 1].\nu, vt\nArrays:\nArray u minimum size:\nColumn major\nlayout\nRow major layout\njobu = 'A'\nmax(1, ldu*m)\nmax(1, ldu*m)\njobu = 'S'\nmax(1, ldu*min(m,\nn))\nmax(1, ldu*m)\nIf jobu = 'A', u contains the m-by-m orthogonal/unitary matrix U.\nIf jobu = 'S', u contains the first min(m, n) columns of U (the left\nsingular vectors stored column-wise).\nIf jobu = 'N' or 'O', u is not referenced.\nArray v minimum size:\nColumn major\nlayout\nRow major layout\njobvt = 'A'\nmax(1, ldvt*n)\nmax(1, ldvt*n)\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1076\n\n\nColumn major\nlayout\nRow major layout\njobvt = 'S'\nmax(1, ldvt*min(m,\nn))\nmax(1, ldvt*n)\nIf jobvt = 'A', vt contains the n-by-n orthogonal/unitary matrix VT/VH.\nIf jobvt = 'S', vt contains the first min(m, n) rows of VT/VH (the right\nsingular vectors stored row-wise).\nIf jobvt = 'N'or 'O', vt is not referenced.\nsuperb\nIf ?bdsqr does not converge (indicated by the return value info > 0), on\nexit superb(0:min(m,n)-2) contains the unconverged superdiagonal\nelements of an upper bidiagonal matrix B whose diagonal is in s (not\nnecessarily sorted). B satisfies A = u*B*VT (real flavors) or A = u*B*VH\n(complex flavors), so it has the same singular values as A, and singular\nvectors related by u and vt.\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nIf info = i, then if ?bdsqr did not converge, i specifies how many superdiagonals of the intermediate\nbidiagonal form B did not converge to zero (see the description of the superb parameter for details).\n?gesdd\nComputes the singular value decomposition of a\ngeneral rectangular matrix using a divide and conquer\nmethod.\nSyntax\nlapack_int LAPACKE_sgesdd( int matrix_layout, char jobz, lapack_int m, lapack_int n,\nfloat* a, lapack_int lda, float* s, float* u, lapack_int ldu, float* vt, lapack_int\nldvt );\nlapack_int LAPACKE_dgesdd( int matrix_layout, char jobz, lapack_int m, lapack_int n,\ndouble* a, lapack_int lda, double* s, double* u, lapack_int ldu, double* vt, lapack_int\nldvt );\nlapack_int LAPACKE_cgesdd( int matrix_layout, char jobz, lapack_int m, lapack_int n,\nlapack_complex_float* a, lapack_int lda, float* s, lapack_complex_float* u, lapack_int\nldu, lapack_complex_float* vt, lapack_int ldvt );\nlapack_int LAPACKE_zgesdd( int matrix_layout, char jobz, lapack_int m, lapack_int n,\nlapack_complex_double* a, lapack_int lda, double* s, lapack_complex_double* u,\nlapack_int ldu, lapack_complex_double* vt, lapack_int ldvt );\nInclude Files\n•\nmkl.h\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1077\n\n\nDescription\nThe routine computes the singular value decomposition (SVD) of a real/complex m-by-n matrix A, optionally\ncomputing the left and/or right singular vectors.\nIf singular vectors are desired, it uses a divide-and-conquer algorithm. The SVD is written\nA = U*Σ*VT for real routines,\nA = U*Σ*VH for complex routines,\nwhere Σ is an m-by-n matrix which is zero except for its min(m,n) diagonal elements, U is an m-by-m\northogonal/unitary matrix, and V is an n-by-n orthogonal/unitary matrix. The diagonal elements of Σ are the\nsingular values of A; they are real and non-negative, and are returned in descending order. The first min(m,\nn) columns of U and V are the left and right singular vectors of A.\nNote that the routine returns vt = VT (for real flavors) or vt =VH (for complex flavors), not V.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\njobz\nMust be 'A', 'S', 'O', or 'N'.\nSpecifies options for computing all or part of the matrices U and V.\nIf jobz = 'A', all m columns of U and all n rows of VT or VH are returned\nin the arrays u and vt;\nif jobz = 'S', the first min(m, n) columns of U and the first min(m, n)\nrows of VT or VH are returned in the arrays u and vt;\nif jobz = 'O', then\nif m≥  n, the first n columns of U are overwritten in the array a and all rows\nof VT or VH are returned in the array vt;\nif m<n, all columns of U are returned in the array u and the first m rows of\nVT or VH are overwritten in the array a;\nif jobz = 'N', no columns of U or rows of VT or VH are computed.\nm\nThe number of rows of the matrix A (m≥ 0).\nn\nThe number of columns in A (n≥ 0).\na\na(size max(1, lda*n) for column major layout and max(1, lda*m) for row\nmajor layout) is an array containing the m-by-n matrix A.\nlda\nThe leading dimension of the array a. Must be at least max(1, m) for\ncolumn major layout and at least max(1, n) for row major layout.\nldu, ldvt\nThe leading dimensions of the output arrays u and vt, respectively.\nThe minimum size of ldu is\njobz\nm≥n\nm < n\n'N'\n1\n1\n'A'\nm\nm\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1078\n\n\njobz\nm≥n\nm < n\n'S'\nm for column major\nlayout; n for row\nmajor layout\nm\n'O'\n1\nm\nThe minimum size of ldvt is\njobz\nm≥n\nm < n\n'N'\n1\n1\n'A'\nn\nn\n'S'\nn\nm for column major\nlayout; n for row\nmajor layout\n'O'\nn\n1\nOutput Parameters\na\nOn exit:\nIf jobz = 'O', then if m≥ n, a is overwritten with the first n columns of U\n(the left singular vectors, stored columnwise). If m < n, a is overwritten\nwith the first m rows of VT (the right singular vectors, stored rowwise);\nIf jobz≠'O', the contents of a are destroyed.\ns\nArray, size at least max(1, min(m,n)). Contains the singular values of A\nsorted so that s(i) ≥ s(i+1).\nu, vt\nArrays:\nArray u is of size:\njobz\nm≥n\nm < n\n'N'\n1\n1\n'A'\nmax(1, ldu*m)\nmax(1, ldu*m)\n'S'\nmax(1, ldu*n) for\ncolumn major layout;\nmax(1, ldu*m) for\nrow major layout\nmax(1, ldu*m)\n'O'\n1\nmax(1, ldu*m)\nIf jobz = 'A'or jobz = 'O' and m < n, u contains the m-by-m\northogonal/unitary matrix U.\nIf jobz = 'S', u contains the first min(m, n) columns of U (the left\nsingular vectors, stored columnwise).\nIf jobz = 'O' and m≥n, or jobz = 'N', u is not referenced.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1079\n\n\nArray vt is of size:\njobz\nm≥n\nm < n\n'N'\n1\n1\n'A'\nmax(1, ldvt*n)\nmax(1, ldvt*n)\n'S'\nmax(1, ldvt*n)\nmax(1, ldvt*n ) for\ncolumn major layout;\nmax(1, ldvt*m ) for\nrow major layout;\n'O'\nmax(1, ldvt*n)\n1\nIf jobz = 'A'or jobz = 'O' and m≥n, vt contains the n-by-n orthogonal/\nunitary matrix VT.\nIf jobz = 'S', vt contains the first min(m, n) rows of VT (the right singular\nvectors, stored rowwise).\nIf jobz = 'O' and m < n, or jobz = 'N', vt is not referenced.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nIf info = -4, A had a NAN entry.\nIf info = i, then ?bdsdc did not converge, updating process failed.\n?gejsv\nComputes the singular value decomposition using a\npreconditioned Jacobi SVD method.\nSyntax\nlapack_int LAPACKE_sgejsv (int matrix_layout, char joba, char jobu, char jobv, char\njobr, char jobt, char jobp, lapack_int m, lapack_int n, float * a, lapack_int lda, float\n* sva, float * u, lapack_int ldu, float * v, lapack_int ldv, float * stat, lapack_int *\nistat);\nlapack_int LAPACKE_dgejsv (int matrix_layout, char joba, char jobu, char jobv, char\njobr, char jobt, char jobp, lapack_int m, lapack_int n, double * a, lapack_int lda,\ndouble * sva, double * u, lapack_int ldu, double * v, lapack_int ldv, double * stat,\nlapack_int * istat);\nlapack_int LAPACKE_cgejsv (int matrix_layout, char joba, char jobu, char jobv, char\njobr, char jobt, char jobp, lapack_int m, lapack_int n, lapack_complex_float * a,\nlapack_int lda, float * sva, lapack_complex_float * u, lapack_int ldu,\nlapack_complex_float * v, lapack_int ldv, float * stat, lapack_int * istat);\nlapack_int LAPACKE_zgejsv (int matrix_layout, char joba, char jobu, char jobv, char\njobr, char jobt, char jobp, lapack_int m, lapack_int n, lapack_complex_double * a,\nlapack_int lda, double * sva, lapack_complex_double * u, lapack_int ldu,\nlapack_complex_double * v, lapack_int ldv, double * stat, lapack_int * istat);\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1080\n\n\nInclude Files\n•\nmkl.h\nDescription\nThe routine computes the singular value decomposition (SVD) of a real/complex m-by-n matrix A, where m≥n.\nThe SVD is written as\nA = U*Σ*VT, for real routines\nA = U*Σ*VH, for complex routines\nwhere Σ is an m-by-n matrix which is zero except for its n diagonal elements, U is an m-by-n (or m-by-m)\northonormal matrix, and V is an n-by-n orthogonal matrix. The diagonal elements of Σ are the singular values\nof A; the columns of U and V are the left and right singular vectors of A, respectively. The matrices U and V\nare computed and stored in the arrays u and v, respectively. The diagonal of Σ is computed and stored in the\narray sva.\nThe ?gejsv routine can sometimes compute tiny singular values and their singular vectors much more\naccurately than other SVD routines.\nThe routine implements a preconditioned Jacobi SVD algorithm. It uses ?geqp3, ?geqrf, and ?gelqf as\npreprocessors and preconditioners. Optionally, an additional row pivoting can be used as a preprocessor,\nwhich in some cases results in much higher accuracy. An example is matrix A with the structure A = D1 * C\n* D2, where D1, D2 are arbitrarily ill-conditioned diagonal matrices and C is a well-conditioned matrix. In that\ncase, complete pivoting in the first QR factorizations provides accuracy dependent on the condition number\nof C, and independent of D1, D2. Such higher accuracy is not completely understood theoretically, but it\nworks well in practice.\nIf A can be written as A = B*D, with well-conditioned B and some diagonal D, then the high accuracy is\nguaranteed, both theoretically and in software, independent of D. For more details see [Drmac08-1],\n[Drmac08-2].\nThe computational range for the singular values can be the full range ( UNDERFLOW,OVERFLOW ), provided\nthat the machine arithmetic and the BLAS and LAPACK routines called by ?gejsv are implemented to work in\nthat range. If that is not the case, the restriction for safe computation with the singular values in the range\nof normalized IEEE numbers is that the spectral condition number kappa(A)=sigma_max(A)/sigma_min(A)\ndoes not overflow. This code (?gejsv) is best used in this restricted range, meaning that singular values of\nmagnitude below ||A||_2 / slamch('O') (for single precision) or ||A||_2 / dlamch('O') (for double\nprecision) are returned as zeros. See jobr for details on this.\nThis implementation is slower than the one described in [Drmac08-1], [Drmac08-2] due to replacement of\nsome non-LAPACK components, and because the choice of some tuning parameters in the iterative part\n(?gesvj) is left to the implementer on a particular machine.\nThe rank revealing QR factorization (in this code: ?geqp3) should be implemented as in [Drmac08-3].\nIf m is much larger than n, it is obvious that the inital QRF with column pivoting can be preprocessed by the\nQRF without pivoting. That well known trick is not used in ?gejsv because in some cases heavy row\nweighting can be treated with complete pivoting. The overhead in cases m much larger than n is then only\ndue to pivoting, but the benefits in accuracy have prevailed. You can incorporate this extra QRF step easily\nand also improve data movement (matrix transpose, matrix copy, matrix transposed copy) - this\nimplementation of ?gejsv uses only the simplest, naive data movement.\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1081\n\n\nProduct and Performance Information\nNotice revision #20201201\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\njoba\nMust be 'C', 'E', 'F', 'G', 'A', or 'R'.\nSpecifies the level of accuracy:\nIf joba = 'C', high relative accuracy is achieved if A = B*D with well-\nconditioned B and arbitrary diagonal matrix D. The accuracy cannot be\nspoiled by column scaling. The accuracy of the computed output depends\non the condition of B, and the procedure aims at the best theoretical\naccuracy. The relative error max_{i=1:N}|d sigma_i| / sigma_i is\nbounded by f(M,N)*epsilon* cond(B), independent of D. The input\nmatrix is preprocessed with the QRF with column pivoting. This initial\npreprocessing and preconditioning by a rank revealing QR factorization is\ncommon for all values of joba. Additional actions are specified as follows:\nIf joba = 'E', computation as with 'C' with an additional estimate of the\ncondition number of B. It provides a realistic error bound.\nIf joba = 'F', accuracy higher than in the 'C' option is achieved, if A =\nD1*C*D2 with ill-conditioned diagonal scalings D1, D2, and a well-\nconditioned matrix C. This option is advisable, if the structure of the input\nmatrix is not known and relative accuracy is desirable. The input matrix A is\npreprocessed with QR factorization with full (row and column) pivoting.\nIf joba = 'G', computation as with 'F' with an additional estimate of the\ncondition number of B, where A = B*D. If A has heavily weighted rows,\nusing this condition number gives too pessimistic error bound.\nIf joba = 'A', small singular values are the noise and the matrix is treated\nas numerically rank defficient. The error in the computed singular values is\nbounded by f(m,n)*epsilon*||A||. The computed SVD A = U*S*V**t\n(for real flavors) or A = U*S*V**H (for complex flavors) restores A up to\nf(m,n)*epsilon*||A||. This enables the procedure to set all singular\nvalues below n*epsilon*||A|| to zero.\nIf joba = 'R', the procedure is similar to the 'A' option. Rank revealing\nproperty of the initial QR factorization is used to reveal (using triangular\nfactor) a gap sigma_{r+1} < epsilon * sigma_r, in which case the\nnumerical rank is declared to be r. The SVD is computed with absolute error\nbounds, but more accurately than with 'A'.\njobu\nMust be 'U', 'F', 'W', or 'N'.\nSpecifies whether to compute the columns of the matrix U:\nIf jobu = 'U', n columns of U are returned in the array u\nIf jobu = 'F', a full set of m left singular vectors is returned in the array u.\nIf jobu = 'W', u may be used as workspace of length m*n. See the\ndescription of u.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1082\n\n\nIf jobu = 'N', u is not computed.\njobv\nMust be 'V', 'J', 'W', or 'N'.\nSpecifies whether to compute the matrix V:\nIf jobv = 'V', n columns of V are returned in the array v; Jacobi rotations\nare not explicitly accumulated.\nIf jobv = 'J', n columns of V are returned in the array v but they are\ncomputed as the product of Jacobi rotations. This option is allowed only if\njobu≠'N'\nIf jobv = 'W', v may be used as workspace of length n*n. See the\ndescription of v.\nIf jobv = 'N', v is not computed.\njobr\nMust be 'N' or 'R'.\nSpecifies the range for the singular values. If small positive singular values\nare outside the specified range, they may be set to zero. If A is scaled so\nthat the largest singular value of the scaled matrix is around sqrt(big),\nbig = ?lamch('O'), the function can remove columns of A whose norm in\nthe scaled matrix is less than sqrt(?lamch('S')) (for jobr = 'R'), or\nless than small = ?lamch('S')/?lamch('E').\nIf jobr = 'N', the function does not remove small columns of the scaled\nmatrix. This option assumes that BLAS and QR factorizations and triangular\nsolvers are implemented to work in that range. If the condition of A if\ngreater that big, use ?gesvj.\nIf jobr = 'R', restricted range for singular values of the scaled matrix A is\n[sqrt(?lamch('S'), sqrt(big)], roughly as described above. This\noption is recommended.\nFor computing the singular values in the full range [?lamch('S'),big],\nuse ?gesvj.\njobt\nMust be 'T' or 'N'.\nIf the matrix is square, the procedure may determine to use a transposed A\nif AT (for real flavors) or AH (for complex flavors) seems to be better with\nrespect to convergence. If the matrix is not square, jobt is ignored.\nThe decision is based on two values of entropy over the adjoint orbit of AT *\nA (for real flavors) or AH * A (for complex flavors). See the descriptions of\nstat[5] and stat[6].\nIf jobt = 'T', the function performs transposition if the entropy test\nindicates possibly faster convergence of the Jacobi process, if A is taken as\ninput. If A is replaced with AT or AH, the row pivoting is included\nautomatically.\nIf jobt = 'N', the functions attempts no speculations. This option can be\nused to compute only the singular values, or the full SVD (u, sigma, and v).\nFor only one set of singular vectors (u or v), the caller should provide both\nu and v, as one of the arrays is used as workspace if the matrix A is\ntransposed. The implementer can easily remove this constraint and make\nthe code more complicated. See the descriptions of u and v.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1083\n\n\nCaution\nThe jobt = 'T' option is experimental and its effect might not\nbe the same in subsequent releases. Consider using the jobt =\n'N' instead.\njobp\nMust be 'P' or 'N'.\nEnables structured perturbations of denormalized numbers. This option\nshould be active if the denormals are poorly implemented, causing slow\ncomputation, especially in cases of fast convergence. For details, see\n[Drmac08-1], [Drmac08-2] . For simplicity, such perturbations are included\nonly when the full SVD or only the singular values are requested. You can\nadd the perturbation for the cases of computing one set of singular vectors.\nIf jobp = 'P', the function introduces perturbation.\nIf jobp = 'N', the function introduces no perturbation.\nm\nThe number of rows of the input matrix A; m≥ 0.\nn\nThe number of columns in the input matrix A; m≥n≥ 0.\na, u, v\nArray a(size lda*n for column major layout and lda*m for row major\nlayout) is an array containing the m-by-n matrix A.\nu is a workspace array, its size for column major layout is ldu*n for\njobu='U' or 'W' and ldu*m for jobu='F'; for row major layout its size is at\nleast ldu*m. When jobt = 'T' and m = n, u must be provided even though\njobu = 'N'.\nv is a workspace array, its size is ldv*n. When jobt = 'T' and m = n, v\nmust be provided even though jobv = 'N'.\nlda\nThe leading dimension of the array a. Must be at least max(1, m) for\ncolumn major layout and at least max(1, n) for row major layout .\nsva\nsva is a workspace array, its size is n.\nldu\nThe leading dimension of the array u; ldu≥ 1.\njobu = 'U' or 'F' or 'W', ldu≥m for column major layout; for row major\nlayout if jobu = 'U' or jobu = 'W'ldu≥n and if jobu = 'F'ldu≥m.\nldv\nThe leading dimension of the array v; ldv≥ 1.\njobv = 'V' or 'J' or 'W', ldv≥n.\ncwork\ncwork is a workspace array of size max(2, lwork).\nrwork\nrwork is an array of size at least max(7, lrwork) for real flavors and at\nleast max(7, lwork) for complex flavors.\nOutput Parameters\nsva\nOn exit:\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1084\n\n\nFor stat[0]/stat[1] = 1: the singular values of A. During the\ncomputation sva contains Euclidean column norms of the iterated matrices\nin the array a.\nFor stat[0]≠stat[1]: the singular values of A are (stat[0]/stat[1]) *\nsva[0:n - 1]. This factored form is used if sigma_max(A) overflows or if\nsmall singular values have been saved from underflow by scaling the input\nmatrix A.\njobr = 'R', some of the singular values may be returned as exact zeros\nobtained by 'setting to zero' because they are below the numerical rank\nthreshold or are denormalized numbers.\nu\nOn exit:\nIf jobu = 'U', contains the m-by-n matrix of the left singular vectors.\nIf jobu = 'F', contains the m-by-m matrix of the left singular vectors,\nincluding an orthonormal basis of the orthogonal complement of the range\nof A.\nIf jobu = 'W' and jobv = 'V', jobt = 'T', and m = n, then u is used\nas workspace if the procedure replaces A with AT (for real flavors) or AH (for\ncomplex flavors). In that case, v is computed in u as left singular vectors of\nAT or AH and copied back to the v array. This 'W' option is just a reminder\nto the caller that in this case u is reserved as workspace of length n*n.\nIf jobu = 'N', u is not referenced.\nv\nOn exit:\nIf jobv = 'V' or 'J', contains the n-by-n matrix of the right singular\nvectors.\nIf jobv = 'W' and jobu = 'U', jobt = 'T', and m = n, then v is used\nas workspace if the procedure replaces A with AT (for real flavors) or AH (for\ncomplex flavors). In that case, u is computed in v as right singular vectors\nof AT or AH and copied back to the u array. This 'W' option is just a\nreminder to the caller that in this case v is reserved as workspace of length\nn*n.\nIf jobv = 'N', v is not referenced.\nstat\nOn exit,\nstat[0] = scale = stat[1]/stat[0] is the scaling factor such that\nscale*sva(1:n) are the computed singular values of A. See the\ndescription of sva.\nstat[1] = see the description of stat[0].\nstat[2] = sconda is an estimate for the condition number of column\nequilibrated A. If joba = 'E' or 'G', sconda is an estimate of sqrt(||(RT\n* R)-1||_1). It is computed using ?pocon. It holds n-1/4 * sconda≤ ||\nR-1||_2 ≤n-1/4 * sconda, where R is the triangular factor from the QRF of\nA. However, if R is truncated and the numerical rank is determined to be\nstrictly smaller than n, sconda is returned as -1, indicating that the smallest\nsingular values might be lost.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1085\n\n\nIf full SVD is needed, the following two condition numbers are useful for the\nanalysis of the algorithm. They are provied for a user who is familiar with\nthe details of the method.\nstat[3] = an estimate of the scaled condition number of the triangular\nfactor in the first QR factorization.\nstat[4] = an estimate of the scaled condition number of the triangular\nfactor in the second QR factorization.\nThe following two parameters are computed if jobt = 'T'. They are\nprovided for a user who is familiar with the details of the method.\nstat[5] = the entropy of AT*A :: this is the Shannon entropy of\ndiag(AT*A) / Trace(AT*A) taken as point in the probability simplex.\nstat[6] = the entropy of A*A**t.\nistat\nOn exit,\nistat[0] = the numerical rank determined after the initial QR factorization\nwith pivoting. See the descriptions of joba and jobr.\nistat[1] = the number of the computed nonzero singular value.\nistat[2] = if nonzero, a warning message. If istat[2]=1, some of the\ncolumn norms of A were denormalized floats. The requested high accuracy\nis not warranted by the data.\nFor complex flavors, istat[3] = 1 or -1. If istat[3] = 1, then the\nprocedure used AH to do the job as specified by the job parameters.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nIf info > 0, the function did not converge in the maximal number of sweeps. The computed values may be\ninaccurate.\nSee Also\n?geqp3\n?geqrf\n?gelqf\n?gesvj\n?lamch\n?pocon\n?ormlq\n?gesvj\nComputes the singular value decomposition of a real\nmatrix using Jacobi plane rotations.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1086\n\n\nSyntax\nlapack_int LAPACKE_sgesvj (int matrix_layout, char joba, char jobu, char jobv,\nlapack_int m, lapack_int n, float * a, lapack_int lda, float * sva, lapack_int mv, float\n* v, lapack_int ldv, float * stat);\nlapack_int LAPACKE_dgesvj (int matrix_layout, char joba, char jobu, char jobv,\nlapack_int m, lapack_int n, double * a, lapack_int lda, double * sva, lapack_int mv,\ndouble * v, lapack_int ldv, double * stat);\nlapack_int LAPACKE_cgesvj (int matrix_layout, char joba, char jobu, char jobv,\nlapack_int m, lapack_int n, lapack_complex_float * a, lapack_int lda, float * sva,\nlapack_int mv, lapack_complex_float * v, lapack_int ldv, float * stat);\nlapack_int LAPACKE_zgesvj (int matrix_layout, char joba, char jobu, char jobv,\nlapack_int m, lapack_int n, lapack_complex_double * a, lapack_int lda, double * sva,\nlapack_int mv, lapack_complex_double * v, lapack_int ldv, double * stat);\nInclude Files\n•\nmkl.h\nDescription\nThe routine computes the singular value decomposition (SVD) of a real or complex m-by-n matrix A, where\nm≥n.\nThe SVD of A is written as\nA = U*Σ*VT for real flavors, or\nA = U*Σ*VH for complex flavors,\nwhere Σ is an m-by-n diagonal matrix, U is an m-by-n orthonormal matrix, and V is an n-by-n orthogonal/\nunitary matrix. The diagonal elements of Σ are the singular values of A; the columns of U and V are the left\nand right singular vectors of A, respectively. The matrices U and V are computed and stored in the arrays u\nand v, respectively. The diagonal of Σ is computed and stored in the array sva.\nThe ?gesvj routine can sometimes compute tiny singular values and their singular vectors much more\naccurately than other SVD routines.\nThe n-by-n orthogonal matrix V is obtained as a product of Jacobi plane rotations. The rotations are\nimplemented as fast scaled rotations of Anda and Park [AndaPark94]. In the case of underflow of the Jacobi\nangle, a modified Jacobi transformation of Drmac ([Drmac08-4]) is used. Pivot strategy uses column\ninterchanges of de Rijk ([deRijk98]). The relative accuracy of the computed singular values and the accuracy\nof the computed singular vectors (in angle metric) is as guaranteed by the theory of Demmel and Veselic\n[Demmel92]. The condition number that determines the accuracy in the full rank case is essentially\nwhere κ(.) is the spectral condition number. The best performance of this Jacobi SVD procedure is achieved if\nused in an accelerated version of Drmac and Veselic [Drmac08-1], [Drmac08-2].\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1087\n\n\nThe computational range for the nonzero singular values is the machine number interval\n( UNDERFLOW,OVERFLOW ). In extreme cases, even denormalized singular values can be computed with the\ncorresponding gradual loss of accurate digit.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\njoba\nMust be 'L', 'U' or 'G'.\nSpecifies the structure of A:\nIf joba = 'L', the input matrix A is lower triangular.\nIf joba = 'U', the input matrix A is upper triangular.\nIf joba = 'G', the input matrix A is a general m-by-n, m≥n.\njobu\nMust be 'U', 'C' or 'N'.\nSpecifies whether to compute the left singular vectors (columns of U):\nIf jobu = 'U', the left singular vectors corresponding to the nonzero\nsingular values are computed and returned in the leading columns of A. See\nmore details in the description of a. The default numerical orthogonality\nthreshold is set to approximately TOL=CTOL*EPS, CTOL=sqrt(m), EPS\n= ?lamch('E')\nIf jobu = 'C', analogous to jobu = 'U', except that you can control the\nlevel of numerical orthogonality of the computed left singular vectors. TOL\ncan be set to TOL=CTOL*EPS, where CTOL is given on input in the array\nstat. No CTOL smaller than ONE is allowed. CTOL greater than 1 / EPS is\nmeaningless. The option 'C' can be used if m*EPS is satisfactory\northogonality of the computed left singular vectors, so CTOL=m could save a\nfew sweeps of Jacobi rotations. See the descriptions of a and stat[0].\nIf jobu = 'N', u is not computed. However, see the description of a.\njobv\nMust be 'V', 'A' or 'N'.\nSpecifies whether to compute the right singular vectors, that is, the matrix\nV:\nIf jobv = 'V', the matrix V is computed and returned in the array v.\nIf jobv = 'A', the Jacobi rotations are applied to the mv-byn array v. In\nother words, the right singular vector matrix V is not computed explicitly,\ninstead it is applied to an mv-byn matrix initially stored in the first mv rows\nof V.\nIf jobv = 'N', the matrix V is not computed and the array v is not\nreferenced.\nm\nThe number of rows of the input matrix A.\n1/slamch('E')> m≥ 0 for sgesvj.\n1/dlamch('E')> m≥ 0 for dgesvj.\nn\nThe number of columns in the input matrix A; m≥n≥ 0.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1088\n\n\na, v\nArray a(size at least lda*n for column major layout andlda*m for row\nmajor layout) is an array containing the m-by-n matrix A.\nArray v(size at least max(1, ldv*n)) contains, if jobv = 'A' the mv-by-n\nmatrix to be post-multiplied by Jacobi rotations.\nlda\nThe leading dimension of the array a. Must be at least max(1, m) for\ncolumn major layout and at least max(1, n) for row major layout .\nmv\nIfjobv = 'A', the product of Jacobi rotations in ?gesvj is applied to the\nfirst mv rows of v. See the description of jobv. 0 ≤mv≤ldv.\nldv\nThe leading dimension of the array v; ldv≥ 1.\njobv = 'V', ldv≥ max(1, n).\njobv = 'A', ldv≥ max(1, mv) for column major layout and ldv≥ max(1,\nn) for row major layout.\nstat\nArray size 6. If jobu = 'C', stat[0] = CTOL, where CTOL defines the\nthreshold for convergence. The process stops if all columns of A are\nmutually orthogonal up to CTOL*EPS, where EPS = ?lamch('E'). It is\nrequired that CTOL≥ 1 - that is, it is not allowed to force the routine to\nobtain orthogonality below ε.\nOutput Parameters\na\nOn exit:\nIf jobu = 'U' or jobu = 'C':\n•\nif info = 0, the leading columns of A contain left singular vectors\ncorresponding to the computed singular values of a that are above the\nunderflow threshold ?lamch('S'), that is, non-zero singular values. The\nnumber of the computed non-zero singular values is returned in\nstat[1]. Also see the descriptions of sva and stat. The computed\ncolumns of u are mutually numerically orthogonal up to approximately\nTOL=sqrt(m)*EPS (default); or TOL=CTOL*EPSjobu = 'C', see the\ndescription of jobu.\n•\nif info > 0, the procedure ?gesvj did not converge in the given\nnumber of iterations (sweeps). In that case, the computed columns of u\nmay not be orthogonal up to TOL. The output u (stored in a), sigma\n(given by the computed singular values in sva(1:n)) and v is still a\ndecomposition of the input matrix A in the sense that the residual ||A-\nscale*U*sigma*VT||2 / ||A||2 for real flavors or ||A-\nscale*U*sigma*VH||2 / ||A||2 for complex flavors (where scale =\nstat[0]) is small.\nIf jobu = 'N':\n•\nif info = 0, note that the left singular vectors are 'for free' in the one-\nsided Jacobi SVD algorithm. However, if only the singular values are\nneeded, the level of numerical orthogonality of u is not an issue and\niterations are stopped when the columns of the iterated matrix are\nnumerically orthogonal up to approximately m*EPS. Thus, on exit, a\ncontains the columns of u scaled with the corresponding singular values.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1089\n\n\n•\nif info > 0, the procedure ?gesvj did not converge in the given\nnumber of iterations (sweeps).\nsva\nArray size n.\nIf info = 0, depending on the value scale =stat[0], where scale is the\nscaling factor:\n•\nif scale = 1, sva[0:n - 1] contains the computed singular values of\na.\n•\nif scale≠ 1, the singular values of a are scale*sva(1:n), and this\nfactored representation is due to the fact that some of the singular\nvalues of a might underflow or overflow.\nIf info > 0, the procedure ?gesvj did not converge in the given number\nof iterations (sweeps) and scale*sva(1:n) may not be accurate.\nv\nOn exit:\nIf jobv = 'V', contains the n-by-n matrix of the right singular vectors.\nIf jobv = 'A', then v contains the product of the computed right singular\nvector matrix and the initial matrix in the array v.\nIf jobv = 'N', v is not referenced.\nstat\nOn exit,\nstat[0] = scale is the scaling factor such that scale*sva(1:n) are the\ncomputed singular values of A. See the description of sva.\nstat[1] is the number of the computed nonzero singular values.\nstat[2] is the number of the computed singular values that are larger than\nthe underflow threshold.\nstat[3] is the number of sweeps of Jacobi rotations needed for numerical\nconvergence.\nstat[4] = max_{i≠j} |COS(A(:,i),A(:,j))| in the last sweep. This is\nuseful information in cases when ?gesvj did not converge, as it can be\nused to estimate whether the output is still useful and for post festum\nanalysis.\nstat[5] is the largest absolute value over all sines of the Jacobi rotation\nangles in the last sweep. It can be useful in a post festum analysis.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nIf info > 0, the function did not converge in the maximal number (30) of sweeps. The output may still be\nuseful. See the description of stat.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1090\n\n\n?ggsvd\nComputes the generalized singular value\ndecomposition of a pair of general rectangular\nmatrices (deprecated).\nSyntax\nlapack_int LAPACKE_sggsvd( int matrix_layout, char jobu, char jobv, char jobq,\nlapack_int m, lapack_int n, lapack_int p, lapack_int* k, lapack_int* l, float* a,\nlapack_int lda, float* b, lapack_int ldb, float* alpha, float* beta, float* u,\nlapack_int ldu, float* v, lapack_int ldv, float* q, lapack_int ldq, lapack_int* iwork );\nlapack_int LAPACKE_dggsvd( int matrix_layout, char jobu, char jobv, char jobq,\nlapack_int m, lapack_int n, lapack_int p, lapack_int* k, lapack_int* l, double* a,\nlapack_int lda, double* b, lapack_int ldb, double* alpha, double* beta, double* u,\nlapack_int ldu, double* v, lapack_int ldv, double* q, lapack_int ldq, lapack_int*\niwork );\nlapack_int LAPACKE_cggsvd( int matrix_layout, char jobu, char jobv, char jobq,\nlapack_int m, lapack_int n, lapack_int p, lapack_int* k, lapack_int* l,\nlapack_complex_float* a, lapack_int lda, lapack_complex_float* b, lapack_int ldb,\nfloat* alpha, float* beta, lapack_complex_float* u, lapack_int ldu,\nlapack_complex_float* v, lapack_int ldv, lapack_complex_float* q, lapack_int ldq,\nlapack_int* iwork );\nlapack_int LAPACKE_zggsvd( int matrix_layout, char jobu, char jobv, char jobq,\nlapack_int m, lapack_int n, lapack_int p, lapack_int* k, lapack_int* l,\nlapack_complex_double* a, lapack_int lda, lapack_complex_double* b, lapack_int ldb,\ndouble* alpha, double* beta, lapack_complex_double* u, lapack_int ldu,\nlapack_complex_double* v, lapack_int ldv, lapack_complex_double* q, lapack_int ldq,\nlapack_int* iwork );\nInclude Files\n•\nmkl.h\nDescription\nThis routine is deprecated; use ggsvd3.\nThe routine computes the generalized singular value decomposition (GSVD) of an m-by-n real/complex\nmatrix A and p-by-n real/complex matrix B:\nU'*A*Q = D1*(0 R), V'*B*Q = D2*(0 R),\nwhere U, V and Q are orthogonal/unitary matrices and U', V' mean transpose/conjugate transpose of U and V\nrespectively.\nLet k+l = the effective numerical rank of the matrix (A', B')', then R is a (k+l)-by-(k+l) nonsingular upper\ntriangular matrix, D1 and D2 are m-by-(k+l) and p-by-(k+l) \"diagonal\" matrices and of the following\nstructures, respectively:\nIf m-k-l≥0,\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1091\n\n\nwhere\nC = diag(alpha[k],..., alpha[k + l - 1])\nS = diag(beta[k],...,beta[k + l - 1])\nC2 + S2 = I\nNonzero element ri j (1 ≤i≤j≤k + l) of R is stored in a[(i - 1) + (n - k - l + j - 1)*lda] for column\nmajor layout and in a[(i - 1)*lda + (n - k - l + j - 1)] for row major layout.\nIf m-k-l < 0,\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1092\n\n\nwhere\nC = diag(alpha[k],..., alpha(m)),\nS = diag(beta[k],...,beta[m - 1]),\nC2 + S2 = I\nOn exit, the location of nonzero element ri j (1 ≤i≤j≤k + l) of R depends on the value of i. For i≤m this element\nis stored in a[(i - 1) + (n - k - l + j - 1)*lda] for column major layout and in a[(i - 1)*lda +\n(n - k - l + j - 1)] for row major layout. For m < i≤k + l it is stored in b[(i - k - 1) + (n - k -\nl + j - 1)*ldb] for column major layout and in b[(i - k - 1)*ldb + (n - k - l + j - 1)] for row\nmajor layout.\nThe routine computes C, S, R, and optionally the orthogonal/unitary transformation matrices U, V and Q.\nIn particular, if B is an n-by-n nonsingular matrix, then the GSVD of A and B implicitly gives the SVD of\nA*B-1:\nA*B-1 = U*(D1*D2-1)*V'.\nIf (A', B')' has orthonormal columns, then the GSVD of A and B is also equal to the CS decomposition of A\nand B. Furthermore, the GSVD can be used to derive the solution of the eigenvalue problem:\nA'**A*x = λ*B'*B*x.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1093\n\n\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\njobu\nMust be 'U' or 'N'.\nIf jobu = 'U', orthogonal/unitary matrix U is computed.\nIf jobu = 'N', U is not computed.\njobv\nMust be 'V' or 'N'.\nIf jobv = 'V', orthogonal/unitary matrix V is computed.\nIf jobv = 'N', V is not computed.\njobq\nMust be 'Q' or 'N'.\nIf jobq = 'Q', orthogonal/unitary matrix Q is computed.\nIf jobq = 'N', Q is not computed.\nm\nThe number of rows of the matrix A (m≥ 0).\nn\nThe number of columns of the matrices A and B (n≥ 0).\np\nThe number of rows of the matrix B (p≥ 0).\na, b\nArrays:\na(size at least max(1, lda*n) for column major layout and max(1, lda*m)\nfor row major layout) contains the m-by-n matrix A.\nb(size at least max(1, ldb*n) for column major layout and max(1, ldb*p)\nfor row major layout) contains the p-by-n matrix B.\nlda\nThe leading dimension of a; at least max(1, m)for column major layout and\nmax(1, n) for row major layout.\nldb\nThe leading dimension of b; at least max(1, p)for column major layout and\nmax(1, n) for row major layout.\nldu\nThe leading dimension of the array u .\nldu≥ max(1, m) if jobu = 'U'; ldu≥ 1 otherwise.\nldv\nThe leading dimension of the array v .\nldv≥ max(1, p) if jobv = 'V'; ldv≥ 1 otherwise.\nldq\nThe leading dimension of the array q .\nldq≥ max(1, n) if jobq = 'Q'; ldq≥ 1 otherwise.\nOutput Parameters\nk, l\nOn exit, k and l specify the dimension of the subblocks. The sum k+l is\nequal to the effective numerical rank of (A', B')'.\na\nOn exit, a contains the triangular matrix R or part of R.\nb\nOn exit, b contains part of the triangular matrix R if m-k-l < 0.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1094\n\n\nalpha, beta\nArrays, size at least max(1, n) each.\nContain the generalized singular value pairs of A and B:\nalpha(1:k) = 1,\nbeta(1:k) = 0,\nand if m-k-l≥ 0,\nalpha(k+1:k+l) = C,\nbeta(k+1:k+l) = S,\nor if m-k-l < 0,\nalpha(k+1:m)= C, alpha(m+1:k+l)=0\nbeta(k+1:m) = S, beta(m+1:k+l) = 1\nand\nalpha(k+l+1:n) = 0\nbeta(k+l+1:n) = 0.\nu, v, q\nArrays:\nu, size at least max(1, ldu*m).\nIf jobu = 'U', u contains the m-by-m orthogonal/unitary matrix U.\nIf jobu = 'N', u is not referenced.\nv, size at least max(1, ldv*p).\nIf jobv = 'V', v contains the p-by-p orthogonal/unitary matrix V.\nIf jobv = 'N', v is not referenced.\nq, size at least max(1, ldq*n).\nIf jobq = 'Q', q contains the n-by-n orthogonal/unitary matrix Q.\nIf jobq = 'N', q is not referenced.\niwork\nOn exit, iwork stores the sorting information.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nIf info = 1, the Jacobi-type procedure failed to converge. For further details, see subroutine tgsja.\n?gesvdx\nComputes the SVD and left and right singular vectors\nfor a matrix.\nSyntax\nlapack_int LAPACKE_sgesvdx (int matrix_layout, char jobu, char jobvt, char range,\nlapack_int m, lapack_int n, float * a, lapack_int lda, float vl, float vu, lapack_int\nil, lapack_int iu, lapack_int * ns, float * s, float * u, lapack_int ldu, float * vt,\nlapack_int ldvt, lapack_int * superb);\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1095\n\n\nlapack_int LAPACKE_dgesvdx (int matrix_layout, char jobu, char jobvt, char range,\nlapack_int m, lapack_int n, double * a, lapack_int lda, double vl, double vu, lapack_int\nil, lapack_int iu, lapack_int *ns, double * s, double * u, lapack_int ldu, double * vt,\nlapack_int ldvt, lapack_int * superb);\nlapack_int LAPACKE_cgesvdx (int matrix_layout, char jobu, char jobvt, char range,\nlapack_int m, lapack_int n, lapack_complex_float * a, lapack_int lda, float vl, float\nvu, lapack_int il, lapack_int iu, lapack_int * ns, float * s, lapack_complex_float * u,\nlapack_int ldu, lapack_complex_float * vt, lapack_int ldvt, lapack_int * superb);\nlapack_int LAPACKE_zgesvdx (int matrix_layout, char jobu, char jobvt, char range,\nlapack_int m, lapack_int n, lapack_complex_double * a, lapack_int lda, double vl,\ndouble vu, lapack_int il, lapack_int iu, lapack_int * ns, double * s,\nlapack_complex_double * u, lapack_int ldu, lapack_complex_double * vt, lapack_int ldvt,\nlapack_int * superb);\nInclude Files\n•\nmkl.h\nDescription\n?gesvdx computes the singular value decomposition (SVD) of a real or complex m-by-n matrix A, optionally\ncomputing the left and right singular vectors. The SVD is written\nA = U * Σ * transpose(V)\nwhere Σ is an m-by-n matrix which is zero except for its min(m,n) diagonal elements, U is an m-by-m matrix,\nand V is an n-by-n matrix. The matrices U and V are orthogonal for real A, and unitary for complex A. The\ndiagonal elements of Σ are the singular values of A; they are real and non-negative, and are returned in\ndescending order. The first min(m,n) columns of U and V are the left and right singular vectors of A.\n?gesvdx uses an eigenvalue problem for obtaining the SVD, which allows for the computation of a subset of\nsingular values and vectors. See ?bdsvdx for details.\nNote that the routine returns VT, not V.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\njobu\nSpecifies options for computing all or part of the matrix U:\n= 'V': the first min(m,n) columns of U (the left singular vectors) or as\nspecified by range are returned in the array u;\n= 'N': no columns of U (no left singular vectors) are computed.\njobvt\nSpecifies options for computing all or part of the matrix VT:\n= 'V': the first min(m,n) rows of VT (the right singular vectors) or as\nspecified by range are returned in the array vt;\n= 'N': no rows of VT (no right singular vectors) are computed.\nrange\n= 'A': find all singular values.\n= 'V': all singular values in the half-open interval (vl,vu] are found.\n= 'I': the il-th through iu-th singular values are found.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1096\n\n\nm\nThe number of rows of the input matrix A. m≥ 0.\nn\nThe number of columns of the input matrix A. n≥ 0.\na\nArray, size lda*n\nOn entry, the m-by-n matrix A.\nlda\nThe leading dimension of the array a.\nlda≥ max(1,m).\nvl\nvl≥0.\nvu\nIf range='V', the lower and upper bounds of the interval to be searched for\nsingular values. vu > vl. Not referenced if range = 'A' or 'I'.\nil\niu\nIf range='I', the indices (in ascending order) of the smallest and largest\nsingular values to be returned. 1 ≤il≤iu≤ min(m,n), if min(m,n) > 0. Not\nreferenced if range = 'A' or 'V'.\nldu\nThe leading dimension of the array u. ldu≥ 1; if jobu = 'V', ldu≥m.\nldvt\nThe leading dimension of the array vt. ldvt≥ 1; if jobvt = 'V', ldvt≥ns\n(see above).\nOutput Parameters\na\nOn exit, the contents of a are destroyed.\nns\nThe total number of singular values found,\n0 ≤ns≤ min(m, n).\nIf range = 'A', ns = min(m, n); if range = 'I', ns = iu - il + 1.\ns\nArray, size (min(m,n))\nThe singular values of A, sorted so that s[i]≥s[i + 1].\nu\nArray, size ldu*ucol\nIf jobu = 'V', u contains columns of U (the left singular vectors,\nstored columnwise) as specified by range; if jobu = 'N', u is not\nreferenced.\nNOTE\nMake sure that ucol≥ns; if range = 'V', the exact value of ns\nis not known in advance and an upper bound must be used.\nvt\nArray, size ldvt*n\nIf jobvt = 'V', vt contains the rows of VT (the right singular vectors,\nstored rowwise) as specified by range; if jobvt = 'N', vt is not\nreferenced.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1097\n\n\nNOTE\nMake sure that ldvt≥ns; if range = 'V', the exact value of\nns is not known in advance and an upper bound must be\nused.\nsuperb\nArray, size (12*min(m, n)).\nIf info = 0, the first ns elements of superb are zero. If info > 0,\nthen superb contains the indices of the eigenvectors that failed to\nconverge in ?bdsvdx/?stevx.\nReturn Values\nThis function returns a value info.\n= 0: successful exit.\n< 0: if info = -i, the i-th argument had an illegal value.\n> 0: if info = i, then i eigenvectors failed to converge in ?bdsvdx/?stevx. if info = n*2 + 1, an internal error\noccurred in ?bdsvdx.\n?bdsvdx\nComputes the SVD of a bidiagonal matrix.\nSyntax\nlapack_int LAPACKE_sbdsvdx (int matrix_layout, char uplo, char jobz, char range,\nlapack_int n, float * d, float * e, float vl, float vu, lapack_int il, lapack_int iu,\nlapack_int * ns, float * s, float * z, lapack_int ldz, lapack_int * superb);\nlapack_int LAPACKE_dbdsvdx (int matrix_layout, char uplo, char jobz, char range,\nlapack_int n, double * d, double * e, double vl, double vu, lapack_int il, lapack_int\niu, lapack_int * ns, double * s, double * z, lapack_int ldz, lapack_int * superb);\nInclude Files\n•\nmkl.h\nDescription\n?bdsvdx computes the singular value decomposition (SVD) of a real n-by-n (upper or lower) bidiagonal\nmatrix B, B = U * S * VT, where S is a diagonal matrix with non-negative diagonal elements (the singular\nvalues of B), and U and VT are orthogonal matrices of left and right singular vectors, respectively.\nGiven an upper bidiagonal B with diagonal d = [d1d2 ... dn] and superdiagonal e = [e1e2 ... en - 1], ?bdsvdx\ncomputes the singular value decompositon of B through the eigenvalues and eigenvectors of the n*2-by-n*2\ntridiagonal matrix\nTGK =\n0 d1\nd1 0 e1\ne1 0 d2\nd2 ⋱⋱\n⋱⋱\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1098\n\n\nIf (s,u,v) is a singular triplet of B with ||u|| = ||v|| = 1, then (±s,q), ||q|| = 1, are eigenpairs of TGK, with\nq = P * u′ ± v′\n2\n=\nv1 u1 v2 u2 ⋯vn un\n2\n, and P = en + 1 e1 en + 2 e2 ⋯.\nGiven a TGK matrix, one can either\n1.\ncompute -s, -v and change signs so that the singular values (and corresponding vectors) are already in\ndescending order (as in ?gesvd/?gesdd) or\n2.\ncompute s, v and reorder the values (and corresponding vectors).\n?bdsvdx implements (1) by calling ?stevx (bisection plus inverse iteration, to be replaced with a version of\nthe Multiple Relative Robust Representation algorithm. (See P. Willems and B. Lang, A framework for the\nMR^3 algorithm: theory and implementation, SIAM J. Sci. Comput., 35:740-766, 2013.)\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nuplo\n= 'U': B is upper bidiagonal;\n= 'L': B is lower bidiagonal.\njobz\n= 'N': Compute singular values only;\n= 'V': Compute singular values and singular vectors.\nrange\n= 'A': Find all singular values.\n= 'V': all singular values in the half-open interval [vl,vu) are found.\n= 'I': the il-th through iu-th singular values are found.\nn\nThe order of the bidiagonal matrix.\nn >= 0.\nd\nArray, size n.\nThe n diagonal elements of the bidiagonal matrix B.\ne\nArray, size (max(1,n - 1))\nThe (n - 1) superdiagonal elements of the bidiagonal matrix B in elements 1\nto n - 1.\nvl\nvl≥ 0.\nvu\nIf range='V', the lower and upper bounds of the interval to be searched for\nsingular values. vu > vl.\nNot referenced if range = 'A' or 'I'.\nil, iu\nIf range='I', the indices (in ascending order) of the smallest and largest\nsingular values to be returned.\n1 ≤il≤iu≤ min(m,n), if min(m,n) > 0.\nNot referenced if range = 'A' or 'V'.\nldz\nThe leading dimension of the array z.\nldz≥ 1, and if jobz = 'V', ldz≥ max(2,n*2).\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1099\n\n\nOutput Parameters\nns\nThe total number of singular values found. 0 ≤ns≤n.\nIf range = 'A', ns = n, and if range = 'I', ns = iu - il + 1.\ns\nArray, size (n)\nThe first ns elements contain the selected singular values in ascending\norder.\nz\nArray, size 2*n*k\nIf jobz = 'V', then if info = 0 the first ns columns of z contain the\nsingular vectors of the matrix B corresponding to the selected singular\nvalues, with U in rows 1 to n and V in rows n+1 to n*2, i.e.\nz = U\nV\nIf jobz = 'N', then z is not referenced.\nNOTE\nMake sure that at least k = ns+1 columns are supplied in\nthe array z; if range = 'V', the exact value of ns is not\nknown in advance and an upper bound must be used.\nsuperb\nArray, size (12*n).\nIf jobz = 'V', then if info = 0, the first ns elements of iwork are\nzero. If info > 0, then iwork contains the indices of the eigenvectors\nthat failed to converge in ?stevx.\nReturn Values\nThis function returns a value info.\n= 0: successful exit.\n< 0: if info = -i, the i-th argument had an illegal value.\n> 0:\nif info = i, then i eigenvectors failed to converge in ?stevx. The indices of the eigenvectors (as returned\nby ?stevx) are stored in the array iwork.\nif info = n*2 + 1, an internal error occurred.\n?gesvda_batch_strided\nComputes the truncated SVD of a group of general m-\nby-n matrices that are stored at a constant stride from\neach other in a contiguous block of memory.\nSyntax\nvoid sgesvda_batch_strided(\n    const MKL_INT* iparm, MKL_INT* irank,\n    const MKL_INT* m,  const MKL_INT* n,\n    float* a, const MKL_INT* lda, const MKL_INT* stride_a,\n    float* s, const MKL_INT* stride_s,\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1100\n\n\n    float* u, const MKL_INT* ldu, const MKL_INT* stride_u,\n    float* vt, const MKL_INT* ldvt, const MKL_INT* stride_vt,\n    const float* tolerance, float *residual,\n    float* work, const MKL_INT* lwork,\n    const MKL_INT* batch_size, MKL_INT* info\n)\nvoid dgesvda_batch_strided(\n    const MKL_INT* iparm, MKL_INT* irank,\n    const MKL_INT* m,  const MKL_INT* n,\n    double* a, const MKL_INT* lda, const MKL_INT* stride_a,\n    double* s, const MKL_INT* stride_s,\n    double* u, const MKL_INT* ldu, const MKL_INT* stride_u,\n    double* vt, const MKL_INT* ldvt, const MKL_INT* stride_vt,\n    const double* tolerance, double *residual,\n    double* work, const MKL_INT* lwork,\n    const MKL_INT* batch_size, MKL_INT* info\n)\nvoid cgesvda_batch_strided(\n    const MKL_INT* iparm, MKL_INT* irank,\n    const MKL_INT* m,  const MKL_INT* n,\n    MKL_Complex8* a, const MKL_INT* lda, const MKL_INT* stride_a,\n    float* s, const MKL_INT* stride_s,\n    MKL_Complex8* u, const MKL_INT* ldu, const MKL_INT* stride_u,\n    MKL_Complex8* vt, const MKL_INT* ldvt, const MKL_INT* stride_vt,\n    const float* tolerance, float *residual,\n    MKL_Complex8* work, const MKL_INT* lwork,\n    const MKL_INT* batch_size, MKL_INT* info\n)\nvoid zgesvda_batch_strided(\n    const MKL_INT* iparm, MKL_INT* irank,\n    const MKL_INT* m,  const MKL_INT* n,\n    MKL_Complex16* a, const MKL_INT* lda, const MKL_INT* stride_a,\n    double* s, const MKL_INT* stride_s,\n    MKL_Complex16* u, const MKL_INT* ldu, const MKL_INT* stride_u,\n    MKL_Complex16* vt, const MKL_INT* ldvt, const MKL_INT* stride_vt,\n    const double* tolerance, double *residual,\n    MKL_Complex16* work, const MKL_INT* lwork,\n    const MKL_INT* batch_size, MKL_INT* info\n)\nInclude Files\nmkl.h\nDescription\nThe ?gesvda_batch_strided routines compute the truncated SVD for a group of general m-by-n matrices.\nAll matrices have the same parameters (matrix size, leading dimension) and are stored at constant\nstride_a from each other in a contiguous block of memory. The operation is defined as\nfor i = 0 … batch_size-1\n    Ai is a matrix at offset i * stride_a from A\n    Ai := Ui * Si*ViT\n    Ai := Ui * Si *\nend for\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1101\n\n\nwhere Ui and Vi are orthogonal matrices, and Si is a diagonal matrix with singular values on the diagonal.\nSingular values are nonnegative and listed in decreasing order. A truncated SVD of a given mxn matrix\nproduces matrices with the specified number of columns, where the number of columns is defined by the\nuser or determined at runtime with the help of the user-defined tolerance threshold.\nAn approximation of each matrix can be also obtained as a product of two low-rank matrices (low-rank\nproduct):\nAi=Pi×Qi\nwhere Pi=Ui×Si , Qi=ViT if m≥n, and Pi=Ui , Qi=Si × ViT otherwise.\nThe routines provide three possible ways to compute truncated SVD:\n•\nCompute truncated SVD with the help of the input array rank where rank(i) specifies the number of\nsingular values and vectors to be computed in parameters Ui ,Vi and Si for each matrix Ai.\n•\nCompute truncated SVD using a tolerance threshold. While computing SVD, singular values that are less\nthan the user-defined tolerance are treated as zero, and they are not computed but set to zero.\n•\nCompute truncated SVD using the effective rank. The effective rank of A is determined by treating as zero\nthose singular values that are less than the user-defined tolerance threshold times the largest singular\nvalue.\nThe routines can be also used for computing singular values only.\nInput Parameters\niparm\nArray of dimension 16 specifying options to compute truncated SVD. Also\nspecifies the type of returned SVD decomposition form. The individual\ncomponents of the iparm parameter appear below. Default values are\ndenoted with an asterisk (*).\niparm[0]\nSpecifies a criterion for treating singular values\nas zeros.\n-1\nUse default iparm values\n(iparm(0-2)=0, iparm[3]=1) . All\nother iparm settings are ignored.\n= 0*\nComputes the truncated SVD with the\nhelp of the input array irank.\n= 1\nComputes the truncated SVD using the\nparameter tolerance.\n= 2\nComputes the truncated SVD using the\neffective rank. The effective rank of A\nis determined by treating as zero those\nsingular values that are less than the\nuser-defined tolerance multiplied by\nthe largest singular value.\niparm[1]\nSpecifies the option for computing singular\nvectors.\n0*\nBoth singular values and singular vectors\nare computed.\n1\nOnly singular values are computed.\niparm[2]\nSpecifies the type of the returned SVD\ndecomposition.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1102\n\n\n0*\nComputes the truncated SVD as a product\nof three matrices:\nAi=Ui×Si×ViT\n1\nComputes the truncated SVD as a low-\nrank product:\nAi=Pi×Qi\niparm[3]\nSpecifies the option for computing the residual\nvector.\n0\nThe residual vector is not computed.\n1*\nComputes the residual vector.\nNOTE\niparm[4]–iparm[15] are reserved for future use.\nirank\nArray with size at least batch_size. If iparm[0]=0 or iparm[0]=-1,\nelement irank[i] specifies the number of singular values and/or singular\nvectors to be computed in Ui , ViT, and Si for each matrix Ai.\nm\nThe number of rows in the matrices Ai (m ≥ 0).\nn\nThe number of columns in the matrices Ai (n ≥ 0).\na\nArray of size at least stride_a * batch_size holding input matrices Ai.\nlda\nSpecifies the leading dimension of the Ai matrices: lda ≥ max(1, m).\nstrde_a\nStride between two consecutive Ai matrices: stride_a ≥max(1, lda *\nn).\nstride_s\nThe stride between two consecutive Si matrices: stride_s ≥ max(1,\nmin(m,n)).\nldu\nSpecifies the leading dimension of the Ui matrices: ldu ≥ max(1, m).\nstride_u\nThe stride between two consecutive Ui matrices: stride_u ≥ max(1, ldu\n* m).\nldvt\nSpecifies the leading dimension of the ViT matrices: ldvt ≥ max(1, n).\nstride_vt\nThe stride between two consecutive ViT matrices: stride_vt ≥ max(1,\nldvt * n).\ntolerance\nSpecifies the tolerance threshold for computing truncated SVD in the cases\nof iparm[0]=1 and iparm[0]=2. Not used otherwise.\nbatch_size\nThe number of problems in a batch. Must be at least 0.\nwork\nWorkspace array with dimension max(1, lwork).\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1103\n\n\nlwork\nThe dimension of the array work.\nIf lwork = -1, a workspace query is assumed: the routine only calculates\nthe optimal size of the work array and returns this value as the first entry of\nthe work array, and no error message related to lwork is issued by xerbla. If\nlwork is less than the required minimum size but is positive, the routine\ninternally allocates the needed memory.\nOutput Parameters\nirank\nOn exit, if iparm[0]=1 or iparm[0]=2, element irank[0] is the\nnumber of computed singular values and/or singular vectors for matrix\nAi.\na\nUnchanged on exit if the residual vector is not required. Otherwise,\ncontains the residual matrix\nAi:=Ai- Ui×Si×ViT\nif iparm[2]=0, and\nA_i:=Ai-Pi×Qi\notherwise.\ns\nArray of size at least min(m,n)*batch_size to store a batch of\nsingular values Si.\nu\nArray of size at least stride_u*batch_size to store a batch of Ui if\niparm[2]=0, or to store a batch of Pi if iparm[2]=1.\nvt\nArray of size at least stride_vt*batch_size to store a batch of ViT if\niparm[2]=0, or to store a batch of Qi if iparm[2]=1.\nresidual\nArray of dimension batch_size. If iparm[3]=1, residual[i] is the\nFrobenius norm of the matrix ||Ai - Ui×Si×ViT|| if iparm[2]=0,\nand ||Ai - Pi×Qi|| if iparm[2]=1.\ninfo\nArray of size at least batch_size, which reports the status for each\nmatrix.\nIf info[i] = 0, the execution is successful for Ai.\nIf info[0] = -j, the j-th parameter had an illegal value.\nIf info[0]= 1, an internal memory allocation failed.\nIf info[i] = 2, an input parameter contains an invalid value.\nIf info[i] = 3, an error in algorithm while computing singular values\nof Ai occurred.\nIf info[0] = 4, the routine encountered an empty structure or\nmatrix array.\nCosine-Sine Decomposition: LAPACK Driver Routines\nThis topic describes LAPACK driver routines for computing the cosine-sine decomposition (CS\ndecomposition). You can also call the corresponding computational routines to perform the same task.\nThe computation has the following phases:\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1104\n\n\n1.\nThe matrix is reduced to a bidiagonal block form.\n2.\nThe blocks are simultaneously diagonalized using techniques from the bidiagonal SVD algorithms.\nTable \"Driver Routines for Cosine-Sine Decomposition (CSD)\" lists LAPACK routines that perform CS\ndecomposition of matrices.\nComputational Routines for Cosine-Sine Decomposition (CSD)\nOperation\nReal matrices\nComplex matrices\nCompute the CS decomposition of a block-\npartitioned orthogonal matrix\norcsd uncsd\nCompute the CS decomposition of a block-\npartitioned unitary matrix\norcsd uncsd\nSee Also\nCS Computational Routines \n?orcsd/?uncsd\nComputes the CS decomposition of a block-partitioned\northogonal/unitary matrix.\nSyntax\nlapack_int LAPACKE_sorcsd( int matrix_layout, char jobu1, char jobu2, char jobv1t, char\njobv2t, char trans, char signs, lapack_int m, lapack_int p, lapack_int q, float* x11,\nlapack_int ldx11, float* x12, lapack_int ldx12, float* x21, lapack_int ldx21, float*\nx22, lapack_int ldx22, float* theta, float* u1, lapack_int ldu1, float* u2, lapack_int\nldu2, float* v1t, lapack_int ldv1t, float* v2t, lapack_int ldv2t );\nlapack_int LAPACKE_dorcsd( int matrix_layout, char jobu1, char jobu2, char jobv1t, char\njobv2t, char trans, char signs, lapack_int m, lapack_int p, lapack_int q, double* x11,\nlapack_int ldx11, double* x12, lapack_int ldx12, double* x21, lapack_int ldx21, double*\nx22, lapack_int ldx22, double* theta, double* u1, lapack_int ldu1, double* u2,\nlapack_int ldu2, double* v1t, lapack_int ldv1t, double* v2t, lapack_int ldv2t );\nlapack_int LAPACKE_cuncsd( int matrix_layout, char jobu1, char jobu2, char jobv1t, char\njobv2t, char trans, char signs, lapack_int m, lapack_int p, lapack_int q,\nlapack_complex_float* x11, lapack_int ldx11, lapack_complex_float* x12, lapack_int\nldx12, lapack_complex_float* x21, lapack_int ldx21, lapack_complex_float* x22,\nlapack_int ldx22, float* theta, lapack_complex_float* u1, lapack_int ldu1,\nlapack_complex_float* u2, lapack_int ldu2, lapack_complex_float* v1t, lapack_int ldv1t,\nlapack_complex_float* v2t, lapack_int ldv2t );\nlapack_int LAPACKE_zuncsd( int matrix_layout, char jobu1, char jobu2, char jobv1t, char\njobv2t, char trans, char signs, lapack_int m, lapack_int p, lapack_int q,\nlapack_complex_double* x11, lapack_int ldx11, lapack_complex_double* x12, lapack_int\nldx12, lapack_complex_double* x21, lapack_int ldx21, lapack_complex_double* x22,\nlapack_int ldx22, double* theta, lapack_complex_double* u1, lapack_int ldu1,\nlapack_complex_double* u2, lapack_int ldu2, lapack_complex_double* v1t, lapack_int\nldv1t, lapack_complex_double* v2t, lapack_int ldv2t );\nInclude Files\n•\nmkl.h\nDescription\nThe routines ?orcsd/?uncsd compute the CS decomposition of an m-by-m partitioned orthogonal matrix X:\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1105\n\n\nor unitary matrix:\nx11 is p-by-q. The orthogonal/unitary matrices u1, u2, v1, and v2 are p-by-p, (m-p)-by-(m-p), q-by-q, (m-q)-\nby-(m-q), respectively. C and S are r-by-r nonnegative diagonal matrices satisfying C2 + S2 = I, in which r\n= min(p,m-p,q,m-q).\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\njobu1\nIf equals Y, then u1 is computed. Otherwise, u1 is not computed.\njobu2\nIf equals Y, then u2 is computed. Otherwise, u2 is not computed.\njobv1t\nIf equals Y, then v1t is computed. Otherwise, v1t is not computed.\njobv2t\nIf equals Y, then v2t is computed. Otherwise, v2t is not computed.\ntrans\n= 'T':\nx, u1, u2, v1t, v2t are stored in row-major order.\notherwise\nx, u1, u2, v1t, v2t are stored in column-major\norder.\nsigns\n= 'O':\nThe lower-left block is made nonpositive (the\n\"other\" convention).\notherwise\nThe upper-right block is made nonpositive (the\n\"default\" convention).\nm\nThe number of rows and columns of the matrix X.\np\nThe number of rows in x11 and x12. 0 ≤p≤m.\nq\nThe number of columns in x11 and x21. 0 ≤q≤m.\nx11, x12, x21, x22\nArrays of size x11 (ldx11,q), x12 (ldx12,m - q), x21 (ldx21,q), and x22\n(ldx22,m - q).\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1106\n\n\nContain the parts of the orthogonal/unitary matrix whose CSD is desired.\nldx11, ldx12, ldx21, ldx22\nThe leading dimensions of the parts of array X. ldx11≥ max(1, p), ldx12≥\nmax(1, p), ldx21≥ max(1, m - p), ldx22≥ max(1, m - p).\nldu1\nThe leading dimension of the array u1. If jobu1 = 'Y', ldu1≥ max(1,p).\nldu2\nThe leading dimension of the array u2. If jobu2 = 'Y', ldu2≥ max(1,m-p).\nldv1t\nThe leading dimension of the array v1t. If jobv1t = 'Y', ldv1t≥\nmax(1,q).\nldv2t\nThe leading dimension of the array v2t. If jobv2t = 'Y', ldv2t≥ max(1,m-\nq).\nOutput Parameters\ntheta\nArray, size r, in which r = min(p,m-p,q,m-q).\nC = diag( cos(theta[0]), ..., cos(theta[r - 1]) ), and\nS = diag( sin(theta[0]), ..., sin(theta[r - 1]) ).\nu1\nArray, size at least max(1, ldu1*p).\nIf jobu1 = 'Y', u1 contains the p-by-p orthogonal/unitary matrix u1.\nu2\nArray, size at least max(1, ldu2*(m - p)).\nIf jobu2 = 'Y', u2 contains the (m-p)-by-(m-p) orthogonal/unitary matrix\nu2.\nv1t\nArray, size at least max(1, ldv1t*q) .\nIf jobv1t = 'Y', v1t contains the q-by-q orthogonal matrix v1T or unitary\nmatrix v1H.\nv2t\nArray, size at least max(1, ldv2t*(m - q)).\nIf jobv2t = 'Y', v2t contains the (m-q)-by-(m-q) orthogonal matrix v2T or\nunitary matrix v2H.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\n> 0: ?orcsd/?uncsd did not converge.\nSee Also\n?bbcsd\nxerbla\n?orcsd2by1/?uncsd2by1\nComputes the CS decomposition of a block-partitioned\northogonal/unitary matrix.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1107\n\n\nSyntax\nlapack_int LAPACKE_sorcsd2by1 (int matrix_layout, char jobu1, char jobu2, char jobv1t,\nlapack_int m, lapack_int p, lapack_int q, float * x11, lapack_int ldx11, float * x21,\nlapack_int ldx21, float * theta, float * u1, lapack_int ldu1, float * u2, lapack_int\nldu2, float * v1t, lapack_int ldv1t);\nlapack_int LAPACKE_dorcsd2by1 (int matrix_layout, char jobu1, char jobu2, char jobv1t,\nlapack_int m, lapack_int p, lapack_int q, double * x11, lapack_int ldx11, double * x21,\nlapack_int ldx21, double * theta, double * u1, lapack_int ldu1, double * u2, lapack_int\nldu2, double * v1t, lapack_int ldv1t);\nlapack_int LAPACKE_cuncsd2by1 (int matrix_layout, char jobu1, char jobu2, char jobv1t,\nlapack_int m, lapack_int p, lapack_int q, lapack_complex_float * x11, lapack_int ldx11,\nlapack_complex_float * x21, lapack_int ldx21, float * theta, lapack_complex_float * u1,\nlapack_int ldu1, lapack_complex_float * u2, lapack_int ldu2, lapack_complex_float *\nv1t, lapack_int ldv1t);\nlapack_int LAPACKE_zuncsd2by1 (int matrix_layout, char jobu1, char jobu2, char jobv1t,\nlapack_int m, lapack_int p, lapack_int q, lapack_complex_double * x11, lapack_int\nldx11, lapack_complex_double * x21, lapack_int ldx21, double * theta,\nlapack_complex_double * u1, lapack_int ldu1, lapack_complex_double * u2, lapack_int\nldu2, lapack_complex_double * v1t, lapack_int ldv1t);\nInclude Files\n•\nmkl.h\nDescription\nThe routines ?orcsd2by1/?uncsd2by1 compute the CS decomposition of an m-by-q matrix X with\northonormal columns that has been partitioned into a 2-by-1 block structure:\nx11 is p-by-q. The orthogonal/unitary matrices u1, u2, v1, and v2 are p-by-p, (m-p)-by-(m-p), q-by-q, (m-q)-\nby-(m-q), respectively. C and S are r-by-r nonnegative diagonal matrices satisfying C2 + S2 = I, in which r\n= min(p,m-p,q,m-q).\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1108\n\n\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\njobu1\nIf equal to 'Y', then u1 is computed. Otherwise, u1 is not computed.\njobu2\nIf equal to 'Y', then u2 is computed. Otherwise, u2 is not computed.\njobv1t\nIf equal to 'Y', then v1t is computed. Otherwise, v1t is not computed.\nm\nThe number of rows and columns of the matrix X.\np\nThe number of rows in x11. 0 ≤p≤m.\nq\nThe number of columns in x11 . 0 ≤q≤m.\nx11\nArray, size (ldx11*q).\nOn entry, the part of the orthogonal matrix whose CSD is desired.\nldx11\nThe leading dimension of the array x11. ldx11≥ max(1,p).\nx21\nArray, size (ldx21*q).\nOn entry, the part of the orthogonal matrix whose CSD is desired.\nldx21\nThe leading dimension of the array X. ldx21≥ max(1,m - p).\nldu1\nThe leading dimension of the array u1. If jobu1 = 'Y', ldu1≥ max(1,p).\nldu2\nThe leading dimension of the array u2. If jobu2 = 'Y', ldu2≥ max(1,m-p).\nldv1t\nThe leading dimension of the array v1t. If jobv1t = 'Y', ldv1t≥\nmax(1,q).\nOutput Parameters\ntheta\nArray, size r, in which r = min(p,m-p,q,m-q).\nC = diag( cos(theta(1)), ..., cos(theta(r)) ), and\nS = diag( sin(theta(1)), ..., sin(theta(r)) ).\nu1\nArray, size (ldu1*p) .\nIf jobu1 = 'Y', u1 contains the p-by-p orthogonal/unitary matrix u1.\nu2\nArray, size (ldu2*(m - p)) .\nIf jobu2 = 'Y', u2 contains the (m-p)-by-(m-p) orthogonal/unitary matrix\nu2.\nv1t\nArray, size (ldv1t*q) .\nIf jobv1t = 'Y', v1t contains the q-by-q orthogonal matrix v1T or unitary\nmatrix v1H.\nReturn Values\nThis function returns a value info.\n= 0: successful exit\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1109\n\n\n< 0: if info = -i, the i-th argument has an illegal value\n> 0: ?orcsd2by1/?uncsd2by1 did not converge.\nSee Also\n?bbcsd\nxerbla\nGeneralized Symmetric Definite Eigenvalue Problems: LAPACK Driver Routines\nThis topic describes LAPACK driver routines used for solving generalized symmetric definite eigenproblems.\nSee also computational routines that can be called to solve these problems. Table \"Driver Routines for\nSolving Generalized Symmetric Definite Eigenproblems\" lists all such driver routines.\nDriver Routines for Solving Generalized Symmetric Definite Eigenproblems\nRoutine Name\nOperation performed\nsygv/hegv\nComputes all eigenvalues and, optionally, eigenvectors of a real / complex\ngeneralized symmetric /Hermitian positive-definite eigenproblem.\nsygvd/hegvd\nComputes all eigenvalues and, optionally, eigenvectors of a real / complex\ngeneralized symmetric /Hermitian positive-definite eigenproblem. If eigenvectors\nare desired, it uses a divide and conquer method.\nsygvx/hegvx\nComputes selected eigenvalues and, optionally, eigenvectors of a real / complex\ngeneralized symmetric /Hermitian positive-definite eigenproblem.\nspgv/hpgv\nComputes all eigenvalues and, optionally, eigenvectors of a real / complex\ngeneralized symmetric /Hermitian positive-definite eigenproblem with matrices in\npacked storage.\nspgvd/hpgvd\nComputes all eigenvalues and, optionally, eigenvectors of a real / complex\ngeneralized symmetric /Hermitian positive-definite eigenproblem with matrices in\npacked storage. If eigenvectors are desired, it uses a divide and conquer method.\nspgvx/hpgvx\nComputes selected eigenvalues and, optionally, eigenvectors of a real / complex\ngeneralized symmetric /Hermitian positive-definite eigenproblem with matrices in\npacked storage.\nsbgv/hbgv\nComputes all eigenvalues and, optionally, eigenvectors of a real / complex\ngeneralized symmetric /Hermitian positive-definite eigenproblem with banded\nmatrices.\nsbgvd/hbgvd\nComputes all eigenvalues and, optionally, eigenvectors of a real / complex\ngeneralized symmetric /Hermitian positive-definite eigenproblem with banded\nmatrices. If eigenvectors are desired, it uses a divide and conquer method.\nsbgvx/hbgvx\nComputes selected eigenvalues and, optionally, eigenvectors of a real / complex\ngeneralized symmetric /Hermitian positive-definite eigenproblem with banded\nmatrices.\n?sygv\nComputes all eigenvalues and, optionally,\neigenvectors of a real generalized symmetric definite\neigenproblem.\nSyntax\nlapack_int LAPACKE_ssygv (int matrix_layout, lapack_int itype, char jobz, char uplo,\nlapack_int n, float* a, lapack_int lda, float* b, lapack_int ldb, float* w);\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1110\n\n\nlapack_int LAPACKE_dsygv (int matrix_layout, lapack_int itype, char jobz, char uplo,\nlapack_int n, double* a, lapack_int lda, double* b, lapack_int ldb, double* w);\nInclude Files\n•\nmkl.h\nDescription\nThe routine computes all the eigenvalues, and optionally, the eigenvectors of a real generalized symmetric-\ndefinite eigenproblem, of the form\nA*x = λ*B*x, A*B*x = λ*x, or B*A*x = λ*x.\nHere A and B are assumed to be symmetric and B is also positive definite.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nitype\nMust be 1 or 2 or 3.\nSpecifies the problem type to be solved:\nif itype = 1, the problem type is A*x = lambda*B*x;\nif itype = 2, the problem type is A*B*x = lambda*x;\nif itype = 3, the problem type is B*A*x = lambda*x.\njobz\nMust be 'N' or 'V'.\nIf jobz = 'N', then compute eigenvalues only.\nIf jobz = 'V', then compute eigenvalues and eigenvectors.\nuplo\nMust be 'U' or 'L'.\nIf uplo = 'U', arrays a and b store the upper triangles of A and B;\nIf uplo = 'L', arrays a and b store the lower triangles of A and B.\nn\nThe order of the matrices A and B (n≥ 0).\na, b\nArrays:\na (size at least max(1, lda*n)) contains the upper or lower triangle of the\nsymmetric matrix A, as specified by uplo.\nb (size at least max(1, ldb*n)) contains the upper or lower triangle of the\nsymmetric positive definite matrix B, as specified by uplo.\nlda\nThe leading dimension of a; at least max(1, n).\nldb\nThe leading dimension of b; at least max(1, n).\nOutput Parameters\na\nOn exit, if jobz = 'V', then if info = 0, a contains the matrix Z of\neigenvectors. The eigenvectors are normalized as follows:\nif itype = 1 or 2, ZT*B*Z = I;\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1111\n\n\nif itype = 3, ZT*inv(B)*Z = I;\nIf jobz = 'N', then on exit the upper triangle (if uplo = 'U') or the\nlower triangle (if uplo = 'L') of A, including the diagonal, is destroyed.\nb\nOn exit, if info≤n, the part of b containing the matrix is overwritten by the\ntriangular factor U or L from the Cholesky factorization B = UT*U or B =\nL*LT.\nw\nArray, size at least max(1, n).\nIf info = 0, contains the eigenvalues in ascending order.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nIf info > 0, spotrf/dpotrf or ssyev/dsyev returned an error code:\nIf info = i≤n, ssyev/dsyev failed to converge, and i off-diagonal elements of an intermediate tridiagonal\ndid not converge to zero;\nIf info = n + i, for 1 ≤i≤n, then the leading minor of order i of B is not positive-definite. The factorization\nof B could not be completed and no eigenvalues or eigenvectors were computed.\n?hegv\nComputes all eigenvalues and, optionally,\neigenvectors of a complex generalized Hermitian\npositive-definite eigenproblem.\nSyntax\nlapack_int LAPACKE_chegv( int matrix_layout, lapack_int itype, char jobz, char uplo,\nlapack_int n, lapack_complex_float* a, lapack_int lda, lapack_complex_float* b,\nlapack_int ldb, float* w );\nlapack_int LAPACKE_zhegv( int matrix_layout, lapack_int itype, char jobz, char uplo,\nlapack_int n, lapack_complex_double* a, lapack_int lda, lapack_complex_double* b,\nlapack_int ldb, double* w );\nInclude Files\n•\nmkl.h\nDescription\nThe routine computes all the eigenvalues, and optionally, the eigenvectors of a complex generalized\nHermitian positive-definite eigenproblem, of the form\nA*x = λ*B*x, A*B*x = λ*x, or B*A*x = λ*x.\nHere A and B are assumed to be Hermitian and B is also positive definite.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1112\n\n\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nitype\nMust be 1 or 2 or 3. Specifies the problem type to be solved:\nif itype = 1, the problem type is A*x = lambda*B*x;\nif itype = 2, the problem type is A*B*x = lambda*x;\nif itype = 3, the problem type is B*A*x = lambda*x.\njobz\nMust be 'N' or 'V'.\nIf jobz = 'N', then compute eigenvalues only.\nIf jobz = 'V', then compute eigenvalues and eigenvectors.\nuplo\nMust be 'U' or 'L'.\nIf uplo = 'U', arrays a and b store the upper triangles of A and B;\nIf uplo = 'L', arrays a and b store the lower triangles of A and B.\nn\nThe order of the matrices A and B (n≥ 0).\na, b\nArrays:\na (size at least max(1, lda*n)) contains the upper or lower triangle of the\nHermitian matrix A, as specified by uplo.\nb (size at least max(1, ldb*n)) contains the upper or lower triangle of the\nHermitian positive definite matrix B, as specified by uplo.\nlda\nThe leading dimension of a; at least max(1, n).\nldb\nThe leading dimension of b; at least max(1, n).\nOutput Parameters\na\nOn exit, if jobz = 'V', then if info = 0, a contains the matrix Z of\neigenvectors. The eigenvectors are normalized as follows:\nif itype = 1 or 2, ZH*B*Z = I;\nif itype = 3, ZH*inv(B)*Z = I;\nIf jobz = 'N', then on exit the upper triangle (if uplo = 'U') or the\nlower triangle (if uplo = 'L') of A, including the diagonal, is destroyed.\nb\nOn exit, if info≤n, the part of b containing the matrix is overwritten by the\ntriangular factor U or L from the Cholesky factorization B = UH*U or B =\nL*LH.\nw\nArray, size at least max(1, n).\nIf info = 0, contains the eigenvalues in ascending order.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1113\n\n\nIf info = -i, the i-th parameter had an illegal value.\nIf info > 0, cpotrf/zpotrf or cheev/zheev return an error code:\nIf info = i≤n, cheev/zheev fails to converge, and i off-diagonal elements of an intermediate tridiagonal do\nnot converge to zero;\nIf info = n + i, for 1 ≤i≤n, then the leading minor of order i of B is not positive-definite. The factorization\nof B can not be completed and no eigenvalues or eigenvectors are computed.\n?sygvd\nComputes all eigenvalues and, optionally,\neigenvectors of a real generalized symmetric definite\neigenproblem using a divide and conquer method.\nSyntax\nlapack_int LAPACKE_ssygvd (int matrix_layout, lapack_int itype, char jobz, char uplo,\nlapack_int n, float* a, lapack_int lda, float* b, lapack_int ldb, float* w);\nlapack_int LAPACKE_dsygvd (int matrix_layout, lapack_int itype, char jobz, char uplo,\nlapack_int n, double* a, lapack_int lda, double* b, lapack_int ldb, double* w);\nInclude Files\n•\nmkl.h\nDescription\nThe routine computes all the eigenvalues, and optionally, the eigenvectors of a real generalized symmetric-\ndefinite eigenproblem, of the form\nA*x = λ*B*x, A*B*x = λ*x, or B*A*x = λ*x .\nHere A and B are assumed to be symmetric and B is also positive definite.\nIt uses a divide and conquer algorithm.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nitype\nMust be 1 or 2 or 3. Specifies the problem type to be solved:\nif itype = 1, the problem type is A*x = lambda*B*x;\nif itype = 2, the problem type is A*B*x = lambda*x;\nif itype = 3, the problem type is B*A*x = lambda*x.\njobz\nMust be 'N' or 'V'.\nIf jobz = 'N', then compute eigenvalues only.\nIf jobz = 'V', then compute eigenvalues and eigenvectors.\nuplo\nMust be 'U' or 'L'.\nIf uplo = 'U', arrays a and b store the upper triangles of A and B;\nIf uplo = 'L', arrays a and b store the lower triangles of A and B.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1114\n\n\nn\nThe order of the matrices A and B (n≥ 0).\na, b\nArrays:\na (size at least lda*n) contains the upper or lower triangle of the\nsymmetric matrix A, as specified by uplo.\nb (size at least ldb*n) contains the upper or lower triangle of the\nsymmetric positive definite matrix B, as specified by uplo.\nlda\nThe leading dimension of a; at least max(1, n).\nldb\nThe leading dimension of b; at least max(1, n).\nOutput Parameters\na\nOn exit, if jobz = 'V', then if info = 0, a contains the matrix Z of\neigenvectors. The eigenvectors are normalized as follows:\nif itype = 1 or 2, ZT*B*Z = I;\nif itype = 3, ZT*inv(B)*Z = I;\nIf jobz = 'N', then on exit the upper triangle (if uplo = 'U') or the\nlower triangle (if uplo = 'L') of A, including the diagonal, is destroyed.\nb\nOn exit, if info≤n, the part of b containing the matrix is overwritten by the\ntriangular factor U or L from the Cholesky factorization B = UT*U or B =\nL*LT.\nw\nArray, size at least max(1, n).\nIf info = 0, contains the eigenvalues in ascending order.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nIf info > 0, an error code is returned as specified below.\n•\nFor info≤n:\n•\nIf info = i and jobz = 'N', then the algorithm failed to converge; i off-diagonal elements of an\nintermediate tridiagonal form did not converge to zero.\n•\nIf jobz = 'V', then the algorithm failed to compute an eigenvalue while working on the submatrix\nlying in rows and columns info/(n+1) through mod(info,n+1).\n•\nFor info > n:\n•\nIf info = n + i, for 1 ≤i≤n, then the leading minor of order i of B is not positive-definite. The\nfactorization of B could not be completed and no eigenvalues or eigenvectors were computed.\n?hegvd\nComputes all the eigenvalues, and optionally, the\neigenvectors of a complex generalized Hermitian\npositive-definite eigenproblem using a divide and\nconquer method.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1115\n\n\nSyntax\nlapack_int LAPACKE_chegvd( int matrix_layout, lapack_int itype, char jobz, char uplo,\nlapack_int n, lapack_complex_float* a, lapack_int lda, lapack_complex_float* b,\nlapack_int ldb, float* w );\nlapack_int LAPACKE_zhegvd( int matrix_layout, lapack_int itype, char jobz, char uplo,\nlapack_int n, lapack_complex_double* a, lapack_int lda, lapack_complex_double* b,\nlapack_int ldb, double* w );\nInclude Files\n•\nmkl.h\nDescription\nThe routine computes all the eigenvalues, and optionally, the eigenvectors of a complex generalized\nHermitian positive-definite eigenproblem, of the form\nA*x = λ*B*x, A*B*x = λ*x, or B*A*x = λ*x.\nHere A and B are assumed to be Hermitian and B is also positive definite.\nIt uses a divide and conquer algorithm.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nitype\nMust be 1 or 2 or 3. Specifies the problem type to be solved:\nif itype = 1, the problem type is A*x = lambda*B*x;\nif itype = 2, the problem type is A*B*x = lambda*x;\nif itype = 3, the problem type is B*A*x = lambda*x.\njobz\nMust be 'N' or 'V'.\nIf jobz = 'N', then compute eigenvalues only.\nIf jobz = 'V', then compute eigenvalues and eigenvectors.\nuplo\nMust be 'U' or 'L'.\nIf uplo = 'U', arrays a and b store the upper triangles of A and B;\nIf uplo = 'L', arrays a and b store the lower triangles of A and B.\nn\nThe order of the matrices A and B (n≥ 0).\na, b\nArrays:\na (size at least max(1, lda*n)) contains the upper or lower triangle of the\nHermitian matrix A, as specified by uplo.\nb (size at least max(1, ldb*n)) contains the upper or lower triangle of the\nHermitian positive definite matrix B, as specified by uplo.\nlda\nThe leading dimension of a; at least max(1, n).\nldb\nThe leading dimension of b; at least max(1, n).\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1116\n\n\nOutput Parameters\na\nOn exit, if jobz = 'V', then if info = 0, a contains the matrix Z of\neigenvectors. The eigenvectors are normalized as follows:\nif itype = 1 or 2, ZH* B*Z = I;\nif itype = 3, ZH*inv(B)*Z = I;\nIf jobz = 'N', then on exit the upper triangle (if uplo = 'U') or the\nlower triangle (if uplo = 'L') of A, including the diagonal, is destroyed.\nb\nOn exit, if info≤n, the part of b containing the matrix is overwritten by the\ntriangular factor U or L from the Cholesky factorization B = UH*U or B =\nL*LH.\nw\nArray, size at least max(1, n).\nIf info = 0, contains the eigenvalues in ascending order.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nIf info = i, and jobz = 'N', then the algorithm failed to converge; i off-diagonal elements of an\nintermediate tridiagonal form did not converge to zero;\nif info = i, and jobz = 'V', then the algorithm failed to compute an eigenvalue while working on the\nsubmatrix lying in rows and columns info/(n+1) through mod(info, n+1).\nIf info = n + i, for 1 ≤i≤n, then the leading minor of order i of B is not positive-definite. The factorization\nof B could not be completed and no eigenvalues or eigenvectors were computed.\n?sygvx\nComputes selected eigenvalues and, optionally,\neigenvectors of a real generalized symmetric definite\neigenproblem.\nSyntax\nlapack_int LAPACKE_ssygvx (int matrix_layout, lapack_int itype, char jobz, char range,\nchar uplo, lapack_int n, float* a, lapack_int lda, float* b, lapack_int ldb, float vl,\nfloat vu, lapack_int il, lapack_int iu, float abstol, lapack_int* m, float* w, float* z,\nlapack_int ldz, lapack_int* ifail);\nlapack_int LAPACKE_dsygvx (int matrix_layout, lapack_int itype, char jobz, char range,\nchar uplo, lapack_int n, double* a, lapack_int lda, double* b, lapack_int ldb, double\nvl, double vu, lapack_int il, lapack_int iu, double abstol, lapack_int* m, double* w,\ndouble* z, lapack_int ldz, lapack_int* ifail);\nInclude Files\n•\nmkl.h\nDescription\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1117\n\n\nThe routine computes selected eigenvalues, and optionally, the eigenvectors of a real generalized symmetric-\ndefinite eigenproblem, of the form\nA*x = λ*B*x, A*B*x = λ*x, or B*A*x = λ*x.\nHere A and B are assumed to be symmetric and B is also positive definite. Eigenvalues and eigenvectors can\nbe selected by specifying either a range of values or a range of indices for the desired eigenvalues.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nitype\nMust be 1 or 2 or 3. Specifies the problem type to be solved:\nif itype = 1, the problem type is A*x = λ*B*x;\nif itype = 2, the problem type is A*B*x = λ*x;\nif itype = 3, the problem type is B*A*x = λ*x.\njobz\nMust be 'N' or 'V'.\nIf jobz = 'N', then compute eigenvalues only.\nIf jobz = 'V', then compute eigenvalues and eigenvectors.\nrange\nMust be 'A' or 'V' or 'I'.\nIf range = 'A', the routine computes all eigenvalues.\nIf range = 'V', the routine computes eigenvalues w[i] in the half-open\ninterval:\nvl<w[i]≤vu.\nIf range = 'I', the routine computes eigenvalues with indices il to iu.\nuplo\nMust be 'U' or 'L'.\nIf uplo = 'U', arrays a and b store the upper triangles of A and B;\nIf uplo = 'L', arrays a and b store the lower triangles of A and B.\nn\nThe order of the matrices A and B (n≥ 0).\na, b\nArrays:\na (size at least max(1, lda*n)) contains the upper or lower triangle of the\nsymmetric matrix A, as specified by uplo.\nb (size at least max(1, ldb*n)) contains the upper or lower triangle of the\nsymmetric positive definite matrix B, as specified by uplo.\nlda\nThe leading dimension of a; at least max(1, n).\nldb\nThe leading dimension of b; at least max(1, n).\nvl, vu\nIf range = 'V', the lower and upper bounds of the interval to be searched\nfor eigenvalues.\nConstraint: vl< vu.\nIf range = 'A' or 'I', vl and vu are not referenced.\nil, iu\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1118\n\n\nIf range = 'I', the indices in ascending order of the smallest and largest\neigenvalues to be returned.\nConstraint: 1 ≤il≤iu≤n, if n > 0; il=1 and iu=0\nif n = 0.\nIf range = 'A' or 'V', il and iu are not referenced.\nabstol\nldz\nThe leading dimension of the output array z. Constraints:\nldz≥ 1; if jobz = 'V', ldz≥ max(1, n) for column major layout and ldz≥\nmax(1, m) for row major layout .\nOutput Parameters\na\nOn exit, the upper triangle (if uplo = 'U') or the lower triangle (if uplo =\n'L') of A, including the diagonal, is overwritten.\nb\nOn exit, if info≤n, the part of b containing the matrix is overwritten by the\ntriangular factor U or L from the Cholesky factorization B = UT*U or B =\nL*LT.\nm\nThe total number of eigenvalues found,\n0 ≤m≤n. If range = 'A', m = n, and if range = 'I',\nm = iu-il+1.\nw, z\nArrays:\nw, size at least max(1, n).\nThe first m elements of w contain the selected eigenvalues in ascending\norder.\nz(size at least max(1, ldz*m) for column major layout and max(1, ldz*n)\nfor row major layout) .\nIf jobz = 'V', then if info = 0, the first m columns of z contain the\northonormal eigenvectors of the matrix A corresponding to the selected\neigenvalues, with the i-th column of z holding the eigenvector associated\nwith w[i - 1]. The eigenvectors are normalized as follows:\nif itype = 1 or 2, ZT*B*Z = I;\nif itype = 3, ZT*inv(B)*Z = I;\nIf jobz = 'N', then z is not referenced.\nIf an eigenvector fails to converge, then that column of z contains the latest\napproximation to the eigenvector, and the index of the eigenvector is\nreturned in ifail.\nNote: you must ensure that at least max(1,m) columns are supplied in the\narray z; if range = 'V', the exact value of m is not known in advance and\nan upper bound must be used.\nifail\nArray, size at least max(1, n).\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1119\n\n\nIf jobz = 'V', then if info = 0, the first m elements of ifail are zero; if\ninfo > 0, the ifail contains the indices of the eigenvectors that failed to\nconverge.\nIf jobz = 'N', then ifail is not referenced.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nIf info > 0, spotrf/dpotrf and ssyevx/dsyevx returned an error code:\nIf info = i≤n, ssyevx/dsyevx failed to converge, and i eigenvectors failed to converge. Their indices are\nstored in the array ifail;\nIf info = n + i, for 1 ≤i≤n, then the leading minor of order i of B is not positive-definite. The factorization\nof B could not be completed and no eigenvalues or eigenvectors were computed.\nApplication Notes\nAn approximate eigenvalue is accepted as converged when it is determined to lie in an interval [a,b] of\nwidth less than or equal to abstol+ε*max(|a|,|b|), where ε is the machine precision.\nIf abstol is less than or equal to zero, then ε*||T||1 is used as tolerance, where T is the tridiagonal matrix\nobtained by reducing C to tridiagonal form, where C is the symmetric matrix of the standard symmetric\nproblem to which the generalized problem is transformed. Eigenvalues will be computed most accurately\nwhen abstol is set to twice the underflow threshold 2*?lamch('S'), not zero.\nIf this routine returns with info > 0, indicating that some eigenvectors did not converge, set abstol to\n2*?lamch('S').\n?hegvx\nComputes selected eigenvalues and, optionally,\neigenvectors of a complex generalized Hermitian\npositive-definite eigenproblem.\nSyntax\nlapack_int LAPACKE_chegvx( int matrix_layout, lapack_int itype, char jobz, char range,\nchar uplo, lapack_int n, lapack_complex_float* a, lapack_int lda, lapack_complex_float*\nb, lapack_int ldb, float vl, float vu, lapack_int il, lapack_int iu, float abstol,\nlapack_int* m, float* w, lapack_complex_float* z, lapack_int ldz, lapack_int* ifail );\nlapack_int LAPACKE_zhegvx( int matrix_layout, lapack_int itype, char jobz, char range,\nchar uplo, lapack_int n, lapack_complex_double* a, lapack_int lda,\nlapack_complex_double* b, lapack_int ldb, double vl, double vu, lapack_int il,\nlapack_int iu, double abstol, lapack_int* m, double* w, lapack_complex_double* z,\nlapack_int ldz, lapack_int* ifail );\nInclude Files\n•\nmkl.h\nDescription\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1120\n\n\nThe routine computes selected eigenvalues, and optionally, the eigenvectors of a complex generalized\nHermitian positive-definite eigenproblem, of the form\nA*x = λ*B*x, A*B*x = λ*x, or B*A*x = λ*x.\nHere A and B are assumed to be Hermitian and B is also positive definite. Eigenvalues and eigenvectors can\nbe selected by specifying either a range of values or a range of indices for the desired eigenvalues.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nitype\nMust be 1 or 2 or 3. Specifies the problem type to be solved:\nif itype = 1, the problem type is A*x = λ*B*x;\nif itype = 2, the problem type is A*B*x = λ*x;\nif itype = 3, the problem type is B*A*x = λ*x.\njobz\nMust be 'N' or 'V'.\nIf jobz = 'N', then compute eigenvalues only.\nIf jobz = 'V', then compute eigenvalues and eigenvectors.\nrange\nMust be 'A' or 'V' or 'I'.\nIf range = 'A', the routine computes all eigenvalues.\nIf range = 'V', the routine computes eigenvalues w[i] in the half-open\ninterval:\nvl<w[i]≤vu.\nIf range = 'I', the routine computes eigenvalues with indices il to iu.\nuplo\nMust be 'U' or 'L'.\nIf uplo = 'U', arrays a and b store the upper triangles of A and B;\nIf uplo = 'L', arrays a and b store the lower triangles of A and B.\nn\nThe order of the matrices A and B (n≥ 0).\na, b\nArrays:\na (size at least max(1, lda*n)) contains the upper or lower triangle of the\nHermitian matrix A, as specified by uplo.\nb (size at least max(1, ldb*n)) contains the upper or lower triangle of the\nHermitian positive definite matrix B, as specified by uplo.\nlda\nThe leading dimension of a; at least max(1, n).\nldb\nThe leading dimension of b; at least max(1, n).\nvl, vu\nIf range = 'V', the lower and upper bounds of the interval to be searched\nfor eigenvalues.\nConstraint: vl< vu.\nIf range = 'A' or 'I', vl and vu are not referenced.\nil, iu\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1121\n\n\nIf range = 'I', the indices in ascending order of the smallest and largest\neigenvalues to be returned.\nConstraint: 1 ≤il≤iu≤n, if n > 0; il=1 and iu=0\nif n = 0.\nIf range = 'A' or 'V', il and iu are not referenced.\nabstol\nThe absolute error tolerance for the eigenvalues. See Application Notes for\nmore information.\nldz\nThe leading dimension of the output array z. Constraints:\nldz≥ 1; if jobz = 'V', ldz≥ max(1, n) for column major layout and ldz≥\nmax(1, m) for row major layout.\nOutput Parameters\na\nOn exit, the upper triangle (if uplo = 'U') or the lower triangle (if uplo =\n'L') of A, including the diagonal, is overwritten.\nb\nOn exit, if info≤n, the part of b containing the matrix is overwritten by the\ntriangular factor U or L from the Cholesky factorization B = UH*U or B =\nL*LH.\nm\nThe total number of eigenvalues found,\n0 ≤m≤n. If range = 'A', m = n, and if range = 'I',\nm = iu-il+1.\nw\nArray, size at least max(1, n).\nThe first m elements of w contain the selected eigenvalues in ascending\norder.\nz\nArray z(size at least max(1, ldz*m) for column major layout and max(1,\nldz*n) for row major layout).\nIf jobz = 'V', then if info = 0, the first m columns of z contain the\northonormal eigenvectors of the matrix A corresponding to the selected\neigenvalues, with the i-th column of z holding the eigenvector associated\nwith w[i - 1]. The eigenvectors are normalized as follows:\nif itype = 1 or 2, ZH*B*Z = I;\nif itype = 3, ZH*inv(B)*Z = I;\nIf jobz = 'N', then z is not referenced.\nIf an eigenvector fails to converge, then that column of z contains the latest\napproximation to the eigenvector, and the index of the eigenvector is\nreturned in ifail.\nNote: you must ensure that at least max(1,m) columns are supplied in the\narray z; if range = 'V', the exact value of m is not known in advance and\nan upper bound must be used.\nifail\nArray, size at least max(1, n).\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1122\n\n\nIf jobz = 'V', then if info = 0, the first m elements of ifail are zero; if\ninfo > 0, the ifail contains the indices of the eigenvectors that failed to\nconverge.\nIf jobz = 'N', then ifail is not referenced.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nIf info > 0, cpotrf/zpotrf and cheevx/zheevx returned an error code:\nIf info = i≤n, cheevx/zheevx failed to converge, and i eigenvectors failed to converge. Their indices are\nstored in the array ifail;\nIf info = n + i, for 1 ≤i≤n, then the leading minor of order i of B is not positive-definite. The factorization\nof B could not be completed and no eigenvalues or eigenvectors were computed.\nApplication Notes\nAn approximate eigenvalue is accepted as converged when it is determined to lie in an interval [a,b] of width\nless than or equal to abstol+ε*max(|a|,|b|), where ε is the machine precision.\nIf abstol is less than or equal to zero, then ε*||T||1 will be used in its place, where T is the tridiagonal\nmatrix obtained by reducing C to tridiagonal form, where C is the symmetric matrix of the standard\nsymmetric problem to which the generalized problem is transformed. Eigenvalues will be computed most\naccurately when abstol is set to twice the underflow threshold 2*?lamch('S'), not zero.\nIf this routine returns with info > 0, indicating that some eigenvectors did not converge, try setting abstol\nto 2*?lamch('S').\n?spgv\nComputes all eigenvalues and, optionally,\neigenvectors of a real generalized symmetric definite\neigenproblem with matrices in packed storage.\nSyntax\nlapack_int LAPACKE_sspgv (int matrix_layout, lapack_int itype, char jobz, char uplo,\nlapack_int n, float* ap, float* bp, float* w, float* z, lapack_int ldz);\nlapack_int LAPACKE_dspgv (int matrix_layout, lapack_int itype, char jobz, char uplo,\nlapack_int n, double* ap, double* bp, double* w, double* z, lapack_int ldz);\nInclude Files\n•\nmkl.h\nDescription\nThe routine computes all the eigenvalues, and optionally, the eigenvectors of a real generalized symmetric-\ndefinite eigenproblem, of the form\nA*x = λ*B*x, A*B*x = λ*x, or B*A*x = λ*x.\nHere A and B are assumed to be symmetric, stored in packed format, and B is also positive definite.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1123\n\n\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nitype\nMust be 1 or 2 or 3. Specifies the problem type to be solved:\nif itype = 1, the problem type is A*x = lambda*B*x;\nif itype = 2, the problem type is A*B*x = lambda*x;\nif itype = 3, the problem type is B*A*x = lambda*x.\njobz\nMust be 'N' or 'V'.\nIf jobz = 'N', then compute eigenvalues only.\nIf jobz = 'V', then compute eigenvalues and eigenvectors.\nuplo\nMust be 'U' or 'L'.\nIf uplo = 'U', arrays ap and bp store the upper triangles of A and B;\nIf uplo = 'L', arrays ap and bp store the lower triangles of A and B.\nn\nThe order of the matrices A and B (n≥ 0).\nap, bp\nArrays:\nap contains the packed upper or lower triangle of the symmetric matrix A,\nas specified by uplo.\nThe dimension of ap must be at least max(1, n*(n+1)/2).\nbp contains the packed upper or lower triangle of the symmetric matrix B,\nas specified by uplo.\nThe dimension of bp must be at least max(1, n*(n+1)/2).\nldz\nThe leading dimension of the output array z; ldz≥ 1. If jobz = 'V', ldz≥\nmax(1, n).\nOutput Parameters\nap\nOn exit, the contents of ap are overwritten.\nbp\nOn exit, contains the triangular factor U or L from the Cholesky factorization\nB = UT*U or B = L*LT, in the same storage format as B.\nw, z\nArrays:\nw, size at least max(1, n).\nIf info = 0, contains the eigenvalues in ascending order.\nz (size max(1, ldz*n)) .\nIf jobz = 'V', then if info = 0, z contains the matrix Z of eigenvectors.\nThe eigenvectors are normalized as follows:\nif itype = 1 or 2, ZT*B*Z = I;\nif itype = 3, ZT*inv(B)*Z = I;\nIf jobz = 'N', then z is not referenced.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1124\n\n\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nIf info > 0, spptrf/dpptrf and sspev/dspev returned an error code:\nIf info = i≤n, sspev/dspev failed to converge, and i off-diagonal elements of an intermediate tridiagonal\ndid not converge to zero;\nIf info = n + i, for 1 ≤i≤n, then the leading minor of order i of B is not positive-definite. The factorization\nof B could not be completed and no eigenvalues or eigenvectors were computed.\n?hpgv\nComputes all eigenvalues and, optionally,\neigenvectors of a complex generalized Hermitian\npositive-definite eigenproblem with matrices in packed\nstorage.\nSyntax\nlapack_int LAPACKE_chpgv( int matrix_layout, lapack_int itype, char jobz, char uplo,\nlapack_int n, lapack_complex_float* ap, lapack_complex_float* bp, float* w,\nlapack_complex_float* z, lapack_int ldz );\nlapack_int LAPACKE_zhpgv( int matrix_layout, lapack_int itype, char jobz, char uplo,\nlapack_int n, lapack_complex_double* ap, lapack_complex_double* bp, double* w,\nlapack_complex_double* z, lapack_int ldz );\nInclude Files\n•\nmkl.h\nDescription\nThe routine computes all the eigenvalues, and optionally, the eigenvectors of a complex generalized\nHermitian positive-definite eigenproblem, of the form\nA*x = λ*B*x, A*B*x = λ*x, or B*A*x = λ*x.\nHere A and B are assumed to be Hermitian, stored in packed format, and B is also positive definite.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nitype\nMust be 1 or 2 or 3. Specifies the problem type to be solved:\nif itype = 1, the problem type is A*x = lambda*B*x;\nif itype = 2, the problem type is A*B*x = lambda*x;\nif itype = 3, the problem type is B*A*x = lambda*x.\njobz\nMust be 'N' or 'V'.\nIf jobz = 'N', then compute eigenvalues only.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1125\n\n\nIf jobz = 'V', then compute eigenvalues and eigenvectors.\nuplo\nMust be 'U' or 'L'.\nIf uplo = 'U', arrays ap and bp store the upper triangles of A and B;\nIf uplo = 'L', arrays ap and bp store the lower triangles of A and B.\nn\nThe order of the matrices A and B (n≥ 0).\nap, bp\nArrays:\nap contains the packed upper or lower triangle of the Hermitian matrix A, as\nspecified by uplo.\nThe dimension of ap must be at least max(1, n*(n+1)/2).\nbp contains the packed upper or lower triangle of the Hermitian matrix B,\nas specified by uplo.\nThe dimension of bp must be at least max(1, n*(n+1)/2).\nldz\nThe leading dimension of the output array z; ldz≥ 1. If jobz = 'V', ldz≥\nmax(1, n).\nOutput Parameters\nap\nOn exit, the contents of ap are overwritten.\nbp\nOn exit, contains the triangular factor U or L from the Cholesky factorization\nB = UH*U or B = L*LH, in the same storage format as B.\nw\nArray, size at least max(1, n).\nIf info = 0, contains the eigenvalues in ascending order.\nz\nArray z (size max(1, ldz*n)).\nIf jobz = 'V', then if info = 0, z contains the matrix Z of eigenvectors.\nThe eigenvectors are normalized as follows:\nif itype = 1 or 2, ZH*B*Z = I;\nif itype = 3, ZH*inv(B)*Z = I;\nIf jobz = 'N', then z is not referenced.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nIf info > 0, cpptrf/zpptrf and chpev/zhpev returned an error code:\nIf info = i≤n, chpev/zhpev failed to converge, and i off-diagonal elements of an intermediate tridiagonal\ndid not converge to zero;\nIf info = n + i, for 1 ≤i≤n, then the leading minor of order i of B is not positive-definite. The factorization\nof B could not be completed and no eigenvalues or eigenvectors were computed.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1126\n\n\n?spgvd\nComputes all eigenvalues and, optionally,\neigenvectors of a real generalized symmetric definite\neigenproblem with matrices in packed storage using a\ndivide and conquer method.\nSyntax\nlapack_int LAPACKE_sspgvd (int matrix_layout, lapack_int itype, char jobz, char uplo,\nlapack_int n, float* ap, float* bp, float* w, float* z, lapack_int ldz);\nlapack_int LAPACKE_dspgvd (int matrix_layout, lapack_int itype, char jobz, char uplo,\nlapack_int n, double* ap, double* bp, double* w, double* z, lapack_int ldz);\nInclude Files\n•\nmkl.h\nDescription\nThe routine computes all the eigenvalues, and optionally, the eigenvectors of a real generalized symmetric-\ndefinite eigenproblem, of the form\nA*x = λ*B*x, A*B*x = λ*x, or B*A*x = λ*x.\nHere A and B are assumed to be symmetric, stored in packed format, and B is also positive definite.\nIf eigenvectors are desired, it uses a divide and conquer algorithm.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nitype\nMust be 1 or 2 or 3. Specifies the problem type to be solved:\nif itype = 1, the problem type is A*x = lambda*B*x;\nif itype = 2, the problem type is A*B*x = lambda*x;\nif itype = 3, the problem type is B*A*x = lambda*x.\njobz\nMust be 'N' or 'V'.\nIf jobz = 'N', then compute eigenvalues only.\nIf jobz = 'V', then compute eigenvalues and eigenvectors.\nuplo\nMust be 'U' or 'L'.\nIf uplo = 'U', arrays ap and bp store the upper triangles of A and B;\nIf uplo = 'L', arrays ap and bp store the lower triangles of A and B.\nn\nThe order of the matrices A and B (n≥ 0).\nap, bp\nArrays:\nap contains the packed upper or lower triangle of the symmetric matrix A,\nas specified by uplo.\nThe dimension of ap must be at least max(1, n*(n+1)/2).\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1127\n\n\nbp contains the packed upper or lower triangle of the symmetric matrix B,\nas specified by uplo.\nThe dimension of bp must be at least max(1, n*(n+1)/2).\nldz\nThe leading dimension of the output array z; ldz≥ 1. If jobz = 'V', ldz≥\nmax(1, n).\nOutput Parameters\nap\nOn exit, the contents of ap are overwritten.\nbp\nOn exit, contains the triangular factor U or L from the Cholesky factorization\nB = UT*U or B = L*LT, in the same storage format as B.\nw, z\nArrays:\nw, size at least max(1, n).\nIf info = 0, contains the eigenvalues in ascending order.\nz (size at least max(1, ldz*n)).\nIf jobz = 'V', then if info = 0, z contains the matrix Z of eigenvectors.\nThe eigenvectors are normalized as follows:\nif itype = 1 or 2, ZT*B*Z = I;\nif itype = 3, ZT*inv(B)*Z = I;\nIf jobz = 'N', then z is not referenced.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nIf info > 0, spptrf/dpptrf and sspevd/dspevd returned an error code:\nIf info = i≤n, sspevd/dspevd failed to converge, and i off-diagonal elements of an intermediate tridiagonal\ndid not converge to zero;\nIf info = n + i, for 1 ≤i≤n, then the leading minor of order i of B is not positive-definite. The factorization\nof B could not be completed and no eigenvalues or eigenvectors were computed.\n?hpgvd\nComputes all eigenvalues and, optionally,\neigenvectors of a complex generalized Hermitian\npositive-definite eigenproblem with matrices in packed\nstorage using a divide and conquer method.\nSyntax\nlapack_int LAPACKE_chpgvd( int matrix_layout, lapack_int itype, char jobz, char uplo,\nlapack_int n, lapack_complex_float* ap, lapack_complex_float* bp, float* w,\nlapack_complex_float* z, lapack_int ldz );\nlapack_int LAPACKE_zhpgvd( int matrix_layout, lapack_int itype, char jobz, char uplo,\nlapack_int n, lapack_complex_double* ap, lapack_complex_double* bp, double* w,\nlapack_complex_double* z, lapack_int ldz );\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1128\n\n\nInclude Files\n•\nmkl.h\nDescription\nThe routine computes all the eigenvalues, and optionally, the eigenvectors of a complex generalized\nHermitian positive-definite eigenproblem, of the form\nA*x = λ*B*x, A*B*x = λ*x, or B*A*x = λ*x.\nHere A and B are assumed to be Hermitian, stored in packed format, and B is also positive definite.\nIf eigenvectors are desired, it uses a divide and conquer algorithm.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nitype\nMust be 1 or 2 or 3. Specifies the problem type to be solved:\nif itype = 1, the problem type is A*x = lambda*B*x;\nif itype = 2, the problem type is A*B*x = lambda*x;\nif itype = 3, the problem type is B*A*x = lambda*x.\njobz\nMust be 'N' or 'V'.\nIf jobz = 'N', then compute eigenvalues only.\nIf jobz = 'V', then compute eigenvalues and eigenvectors.\nuplo\nMust be 'U' or 'L'.\nIf uplo = 'U', arrays ap and bp store the upper triangles of A and B;\nIf uplo = 'L', arrays ap and bp store the lower triangles of A and B.\nn\nThe order of the matrices A and B (n≥ 0).\nap, bp\nArrays:\nap contains the packed upper or lower triangle of the Hermitian matrix A, as\nspecified by uplo.\nThe dimension of ap must be at least max(1, n*(n+1)/2).\nbp contains the packed upper or lower triangle of the Hermitian matrix B,\nas specified by uplo.\nThe dimension of bp must be at least max(1, n*(n+1)/2).\nldz\nThe leading dimension of the output array z; ldz≥ 1. If jobz = 'V', ldz≥\nmax(1, n).\nOutput Parameters\nap\nOn exit, the contents of ap are overwritten.\nbp\nOn exit, contains the triangular factor U or L from the Cholesky factorization\nB = UH*U or B = L*LH, in the same storage format as B.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1129\n\n\nw\nArray, size at least max(1, n).\nIf info = 0, contains the eigenvalues in ascending order.\nz\nArray z (size at least max(1, ldz*n)).\nIf jobz = 'V', then if info = 0, z contains the matrix Z of eigenvectors.\nThe eigenvectors are normalized as follows:\nif itype = 1 or 2, ZH*B*Z = I;\nif itype = 3, ZH*inv(B)*Z = I;\nIf jobz = 'N', then z is not referenced.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nIf info > 0, cpptrf/zpptrf and chpevd/zhpevd returned an error code:\nIf info = i≤n, chpevd/zhpevd failed to converge, and i off-diagonal elements of an intermediate tridiagonal\ndid not converge to zero;\nIf info = n + i, for 1 ≤i≤n, then the leading minor of order i of B is not positive-definite. The factorization\nof B could not be completed and no eigenvalues or eigenvectors were computed.\n?spgvx\nComputes selected eigenvalues and, optionally,\neigenvectors of a real generalized symmetric definite\neigenproblem with matrices in packed storage.\nSyntax\nlapack_int LAPACKE_sspgvx (int matrix_layout, lapack_int itype, char jobz, char range,\nchar uplo, lapack_int n, float* ap, float* bp, float vl, float vu, lapack_int il,\nlapack_int iu, float abstol, lapack_int* m, float* w, float* z, lapack_int ldz,\nlapack_int* ifail);\nlapack_int LAPACKE_dspgvx (int matrix_layout, lapack_int itype, char jobz, char range,\nchar uplo, lapack_int n, double* ap, double* bp, double vl, double vu, lapack_int il,\nlapack_int iu, double abstol, lapack_int* m, double* w, double* z, lapack_int ldz,\nlapack_int* ifail);\nInclude Files\n•\nmkl.h\nDescription\nThe routine computes selected eigenvalues, and optionally, the eigenvectors of a real generalized symmetric-\ndefinite eigenproblem, of the form\nA*x = λ*B*x, A*B*x = λ*x, or B*A*x = λ*x.\nHere A and B are assumed to be symmetric, stored in packed format, and B is also positive definite.\nEigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for\nthe desired eigenvalues.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1130\n\n\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nitype\nMust be 1 or 2 or 3. Specifies the problem type to be solved:\nif itype = 1, the problem type is A*x = lambda*B*x;\nif itype = 2, the problem type is A*B*x = lambda*x;\nif itype = 3, the problem type is B*A*x = lambda*x.\njobz\nMust be 'N' or 'V'.\nIf jobz = 'N', then compute eigenvalues only.\nIf jobz = 'V', then compute eigenvalues and eigenvectors.\nrange\nMust be 'A' or 'V' or 'I'.\nIf range = 'A', the routine computes all eigenvalues.\nIf range = 'V', the routine computes eigenvalues w[i] in the half-open\ninterval:\nvl<w[i]≤vu.\nIf range = 'I', the routine computes eigenvalues with indices il to iu.\nuplo\nMust be 'U' or 'L'.\nIf uplo = 'U', arrays ap and bp store the upper triangles of A and B;\nIf uplo = 'L', arrays ap and bp store the lower triangles of A and B.\nn\nThe order of the matrices A and B (n≥ 0).\nap, bp\nArrays:\nap contains the packed upper or lower triangle of the symmetric matrix A,\nas specified by uplo.\nThe size of ap must be at least max(1, n*(n+1)/2).\nbp contains the packed upper or lower triangle of the symmetric matrix B,\nas specified by uplo.\nThe size of bp must be at least max(1, n*(n+1)/2).\nvl, vu\nIf range = 'V', the lower and upper bounds of the interval to be searched\nfor eigenvalues.\nConstraint: vl< vu.\nIf range = 'A' or 'I', vl and vu are not referenced.\nil, iu\nIf range = 'I', the indices in ascending order of the smallest and largest\neigenvalues to be returned.\nConstraint: 1 ≤il≤iu≤n, if n > 0; il=1 and iu=0\nif n = 0.\nIf range = 'A' or 'V', il and iu are not referenced.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1131\n\n\nabstol\nThe absolute error tolerance for the eigenvalues. See Application Notes for\nmore information.\nldz\nThe leading dimension of the output array z. Constraints:\nldz≥ 1; if jobz = 'V', ldz≥ max(1, n) for column major layout and ldz≥\nmax(1, m) for row major layout .\nOutput Parameters\nap\nOn exit, the contents of ap are overwritten.\nbp\nOn exit, contains the triangular factor U or L from the Cholesky factorization\nB = UT*U or B = L*LT, in the same storage format as B.\nm\nThe total number of eigenvalues found,\n0 ≤m≤n. If range = 'A', m = n, and if range = 'I',\nm = iu-il+1.\nw, z\nArrays:\nw, size at least max(1, n).\nIf info = 0, contains the eigenvalues in ascending order.\nz(size at least max(1, ldz*m) for column major layout and max(1, ldz*n)\nfor row major layout) .\nIf jobz = 'V', then if info = 0, the first m columns of z contain the\northonormal eigenvectors of the matrix A corresponding to the selected\neigenvalues, with the i-th column of z holding the eigenvector associated\nwith w(i). The eigenvectors are normalized as follows:\nif itype = 1 or 2, ZT*B*Z = I;\nif itype = 3, ZT*inv(B)*Z = I;\nIf jobz = 'N', then z is not referenced.\nIf an eigenvector fails to converge, then that column of z contains the latest\napproximation to the eigenvector, and the index of the eigenvector is\nreturned in ifail.\nNote: you must ensure that at least max(1,m) columns are supplied in the\narray z; if range = 'V', the exact value of m is not known in advance and\nan upper bound must be used.\nifail\nArray, size at least max(1, n).\nIf jobz = 'V', then if info = 0, the first m elements of ifail are zero; if\ninfo > 0, the ifail contains the indices of the eigenvectors that failed to\nconverge.\nIf jobz = 'N', then ifail is not referenced.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1132\n\n\nIf info = -i, the i-th parameter had an illegal value.\nIf info > 0, spptrf/dpptrf and sspevx/dspevx returned an error code:\nIf info = i≤n, sspevx/dspevx failed to converge, and i eigenvectors failed to converge. Their indices are\nstored in the array ifail;\nIf info = n + i, for 1 ≤i≤n, then the leading minor of order i of B is not positive-definite. The factorization\nof B could not be completed and no eigenvalues or eigenvectors were computed.\nApplication Notes\nAn approximate eigenvalue is accepted as converged when it is determined to lie in an interval [a,b] of width\nless than or equal to abstol+ε*max(|a|,|b|), where ε is the machine precision.\nIf abstol is less than or equal to zero, then ε*||T||1 is used instead, where T is the tridiagonal matrix\nobtained by reducing A to tridiagonal form. Eigenvalues are computed most accurately when abstol is set to\ntwice the underflow threshold 2*?lamch('S'), not zero.\nIf this routine returns with info > 0, indicating that some eigenvectors did not converge, set abstol to\n2*?lamch('S').\n?hpgvx\nComputes selected eigenvalues and, optionally,\neigenvectors of a generalized Hermitian positive-\ndefinite eigenproblem with matrices in packed\nstorage.\nSyntax\nlapack_int LAPACKE_chpgvx( int matrix_layout, lapack_int itype, char jobz, char range,\nchar uplo, lapack_int n, lapack_complex_float* ap, lapack_complex_float* bp, float vl,\nfloat vu, lapack_int il, lapack_int iu, float abstol, lapack_int* m, float* w,\nlapack_complex_float* z, lapack_int ldz, lapack_int* ifail );\nlapack_int LAPACKE_zhpgvx( int matrix_layout, lapack_int itype, char jobz, char range,\nchar uplo, lapack_int n, lapack_complex_double* ap, lapack_complex_double* bp, double\nvl, double vu, lapack_int il, lapack_int iu, double abstol, lapack_int* m, double* w,\nlapack_complex_double* z, lapack_int ldz, lapack_int* ifail );\nInclude Files\n•\nmkl.h\nDescription\nThe routine computes selected eigenvalues, and optionally, the eigenvectors of a complex generalized\nHermitian positive-definite eigenproblem, of the form\nA*x = λ*B*x, A*B*x = λ*x, or B*A*x = λ*x.\nHere A and B are assumed to be Hermitian, stored in packed format, and B is also positive definite.\nEigenvalues and eigenvectors can be selected by specifying either a range of values or a range of indices for\nthe desired eigenvalues.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1133\n\n\nitype\nMust be 1 or 2 or 3. Specifies the problem type to be solved:\nif itype = 1, the problem type is A*x = lambda*B*x;\nif itype = 2, the problem type is A*B*x = lambda*x;\nif itype = 3, the problem type is B*A*x = lambda*x.\njobz\nMust be 'N' or 'V'.\nIf jobz = 'N', then compute eigenvalues only.\nIf jobz = 'V', then compute eigenvalues and eigenvectors.\nrange\nMust be 'A' or 'V' or 'I'.\nIf range = 'A', the routine computes all eigenvalues.\nIf range = 'V', the routine computes eigenvalues w[i] in the half-open\ninterval:\nvl<w[i]≤vu.\nIf range = 'I', the routine computes eigenvalues with indices il to iu.\nuplo\nMust be 'U' or 'L'.\nIf uplo = 'U', arrays ap and bp store the upper triangles of A and B;\nIf uplo = 'L', arrays ap and bp store the lower triangles of A and B.\nn\nThe order of the matrices A and B (n≥ 0).\nap, bp\nArrays:\nap contains the packed upper or lower triangle of the Hermitian matrix A, as\nspecified by uplo.\nThe dimension of ap must be at least max(1, n*(n+1)/2).\nbp contains the packed upper or lower triangle of the Hermitian matrix B,\nas specified by uplo.\nThe dimension of bp must be at least max(1, n*(n+1)/2).\nvl, vu\nIf range = 'V', the lower and upper bounds of the interval to be searched\nfor eigenvalues.\nConstraint: vl< vu.\nIf range = 'A' or 'I', vl and vu are not referenced.\nil, iu\nIf range = 'I', the indices in ascending order of the smallest and largest\neigenvalues to be returned.\nConstraint: 1 ≤il≤iu≤n, if n > 0; il=1 and iu=0\nif n = 0.\nIf range = 'A' or 'V', il and iu are not referenced.\nabstol\nThe absolute error tolerance for the eigenvalues.\nSee Application Notes for more information.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1134\n\n\nldz\nThe leading dimension of the output array z; ldz≥ 1. If jobz = 'V', ldz≥\nmax(1, n) for column major layout and ldz≥ max(1, m) for row major\nlayout.\nOutput Parameters\nap\nOn exit, the contents of ap are overwritten.\nbp\nOn exit, contains the triangular factor U or L from the Cholesky factorization\nB = UH*U or B = L*LH, in the same storage format as B.\nm\nThe total number of eigenvalues found,\n0 ≤m≤n. If range = 'A', m = n, and if range = 'I',\nm = iu-il+1.\nw\nArray, size at least max(1, n).\nIf info = 0, contains the eigenvalues in ascending order.\nz\nArray z(size at least max(1, ldz*m) for column major layout and max(1,\nldz*n) for row major layout).\nIf jobz = 'V', then if info = 0, the first m columns of z contain the\northonormal eigenvectors of the matrix A corresponding to the selected\neigenvalues, with the i-th column of z holding the eigenvector associated\nwith w(i). The eigenvectors are normalized as follows:\nif itype = 1 or 2, ZH*B*Z = I;\nif itype = 3, ZH*inv(B)*Z = I;\nIf jobz = 'N', then z is not referenced.\nIf an eigenvector fails to converge, then that column of z contains the latest\napproximation to the eigenvector, and the index of the eigenvector is\nreturned in ifail.\nNote: you must ensure that at least max(1,m) columns are supplied in the\narray z; if range = 'V', the exact value of m is not known in advance and\nan upper bound must be used.\nifail\nArray, size at least max(1, n).\nIf jobz = 'V', then if info = 0, the first m elements of ifail are zero; if\ninfo > 0, the ifail contains the indices of the eigenvectors that failed to\nconverge.\nIf jobz = 'N', then ifail is not referenced.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nIf info > 0, cpptrf/zpptrf and chpevx/zhpevx returned an error code:\nIf info = i≤n, chpevx/zhpevx failed to converge, and i eigenvectors failed to converge. Their indices are\nstored in the array ifail;\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1135\n\n\nIf info = n + i, for 1 ≤i≤n, then the leading minor of order i of B is not positive-definite. The factorization\nof B could not be completed and no eigenvalues or eigenvectors were computed.\nApplication Notes\nAn approximate eigenvalue is accepted as converged when it is determined to lie in an interval [a,b] of width\nless than or equal to abstol+ε*max(|a|,|b|), where ε is the machine precision.\nIf abstol is less than or equal to zero, then ε*||T||1 is used as tolerance, where T is the tridiagonal matrix\nobtained by reducing A to tridiagonal form. Eigenvalues will be computed most accurately when abstol is set\nto twice the underflow threshold 2*?lamch('S'), not zero.\nIf this routine returns with info > 0, indicating that some eigenvectors did not converge, try setting abstol\nto 2*?lamch('S').\n?sbgv\nComputes all eigenvalues and, optionally,\neigenvectors of a real generalized symmetric definite\neigenproblem with banded matrices.\nSyntax\nlapack_int LAPACKE_ssbgv (int matrix_layout, char jobz, char uplo, lapack_int n,\nlapack_int ka, lapack_int kb, float* ab, lapack_int ldab, float* bb, lapack_int ldbb,\nfloat* w, float* z, lapack_int ldz);\nlapack_int LAPACKE_dsbgv (int matrix_layout, char jobz, char uplo, lapack_int n,\nlapack_int ka, lapack_int kb, double* ab, lapack_int ldab, double* bb, lapack_int ldbb,\ndouble* w, double* z, lapack_int ldz);\nInclude Files\n•\nmkl.h\nDescription\nThe routine computes all the eigenvalues, and optionally, the eigenvectors of a real generalized symmetric-\ndefinite banded eigenproblem, of the form A*x = λ*B*x. Here A and B are assumed to be symmetric and\nbanded, and B is also positive definite.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\njobz\nMust be 'N' or 'V'.\nIf jobz = 'N', then compute eigenvalues only.\nIf jobz = 'V', then compute eigenvalues and eigenvectors.\nuplo\nMust be 'U' or 'L'.\nIf uplo = 'U', arrays ab and bb store the upper triangles of A and B;\nIf uplo = 'L', arrays ab and bb store the lower triangles of A and B.\nn\nThe order of the matrices A and B (n≥ 0).\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1136\n\n\nka\nThe number of super- or sub-diagonals in A\n(ka≥ 0).\nkb\nThe number of super- or sub-diagonals in B (kb≥ 0).\nab, bb\nArrays:\nab(size at least max(1, ldab*n) for column major layout and max(1,\nldab*(ka + 1)) for row major layout) is an array containing either upper or\nlower triangular part of the symmetric matrix A (as specified by uplo) in\nband storage format.\nbb(size at least max(1, ldbb*n) for column major layout and max(1,\nldbb*(kb + 1)) for row major layout) is an array containing either upper or\nlower triangular part of the symmetric matrix B (as specified by uplo) in\nband storage format.\nldab\nThe leading dimension of the array ab; must be at least ka+1 for column\nmajor layout and at least max(1, n) for row major layout .\nldbb\nThe leading dimension of the array bb; must be at least kb+1 for column\nmajor layout and at least max(1, n) for row major layout.\nldz\nThe leading dimension of the output array z; ldz≥ 1. If jobz = 'V', ldz≥\nmax(1, n).\nOutput Parameters\nab\nOn exit, the contents of ab are overwritten.\nbb\nOn exit, contains the factor S from the split Cholesky factorization B =\nST*S, as returned by pbstf/pbstf.\nw, z\nArrays:\nw, size at least max(1, n).\nIf info = 0, contains the eigenvalues in ascending order.\nz (size at least max(1, ldz*n)) .\nIf jobz = 'V', then if info = 0, z contains the matrix Z of eigenvectors,\nwith the i-th column of z holding the eigenvector associated with w(i). The\neigenvectors are normalized so that ZT*B*Z = I.\nIf jobz = 'N', then z is not referenced.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nIf info > 0, and\nif i≤n, the algorithm failed to converge, and i off-diagonal elements of an intermediate tridiagonal did not\nconverge to zero;\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1137\n\n\nif info = n + i, for 1 ≤i≤n, then pbstf/pbstf returned info = i and B is not positive-definite. The\nfactorization of B could not be completed and no eigenvalues or eigenvectors were computed.\n?hbgv\nComputes all eigenvalues and, optionally,\neigenvectors of a complex generalized Hermitian\npositive-definite eigenproblem with banded matrices.\nSyntax\nlapack_int LAPACKE_chbgv( int matrix_layout, char jobz, char uplo, lapack_int n,\nlapack_int ka, lapack_int kb, lapack_complex_float* ab, lapack_int ldab,\nlapack_complex_float* bb, lapack_int ldbb, float* w, lapack_complex_float* z,\nlapack_int ldz );\nlapack_int LAPACKE_zhbgv( int matrix_layout, char jobz, char uplo, lapack_int n,\nlapack_int ka, lapack_int kb, lapack_complex_double* ab, lapack_int ldab,\nlapack_complex_double* bb, lapack_int ldbb, double* w, lapack_complex_double* z,\nlapack_int ldz );\nInclude Files\n•\nmkl.h\nDescription\nThe routine computes all the eigenvalues, and optionally, the eigenvectors of a complex generalized\nHermitian positive-definite banded eigenproblem, of the form A*x = λ*B*x. Here A and B are Hermitian and\nbanded matrices, and matrix B is also positive definite.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\njobz\nMust be 'N' or 'V'.\nIf jobz = 'N', then compute eigenvalues only.\nIf jobz = 'V', then compute eigenvalues and eigenvectors.\nuplo\nMust be 'U' or 'L'.\nIf uplo = 'U', arrays ab and bb store the upper triangles of A and B;\nIf uplo = 'L', arrays ab and bb store the lower triangles of A and B.\nn\nThe order of the matrices A and B (n≥ 0).\nka\nThe number of super- or sub-diagonals in A\n(ka≥ 0).\nkb\nThe number of super- or sub-diagonals in B (kb≥ 0).\nab, bb\nArrays:\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1138\n\n\nab(size at least max(1, ldab*n) for column major layout and max(1,\nldab*(ka + 1)) for row major layout) is an array containing either upper or\nlower triangular part of the Hermitian matrix A (as specified by uplo) in\nband storage format.\nbb(size at least max(1, ldbb*n) for column major layout and max(1,\nldbb*(kb + 1)) for row major layout) is an array containing either upper or\nlower triangular part of the Hermitian matrix B (as specified by uplo) in\nband storage format.\nldab\nThe leading dimension of the array ab; must be at least ka+1 for column\nmajor layout and at least max(1, n for row major layout.\nldbb\nThe leading dimension of the array bb; must be at least kb+1 for column\nmajor layout and at least max(1, n for row major layout.\nldz\nThe leading dimension of the output array z; ldz≥ 1. If jobz = 'V', ldz≥\nmax(1, n).\nOutput Parameters\nab\nOn exit, the contents of ab are overwritten.\nbb\nOn exit, contains the factor S from the split Cholesky factorization B =\nSH*S, as returned by pbstf/pbstf.\nw\nArray, size at least max(1, n).\nIf info = 0, contains the eigenvalues in ascending order.\nz\nArray z (size at least max(1, ldz*n)).\nIf jobz = 'V', then if info = 0, z contains the matrix Z of eigenvectors,\nwith the i-th column of z holding the eigenvector associated with w(i). The\neigenvectors are normalized so that ZH*B*Z = I.\nIf jobz = 'N', then z is not referenced.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nIf info > 0, and\nif i≤n, the algorithm failed to converge, and i off-diagonal elements of an intermediate tridiagonal did not\nconverge to zero;\nif info = n + i, for 1 ≤i≤n, then pbstf/pbstf returned info = i and B is not positive-definite. The\nfactorization of B could not be completed and no eigenvalues or eigenvectors were computed.\n?sbgvd\nComputes all eigenvalues and, optionally,\neigenvectors of a real generalized symmetric definite\neigenproblem with banded matrices. If eigenvectors\nare desired, it uses a divide and conquer method.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1139\n\n\nSyntax\nlapack_int LAPACKE_ssbgvd (int matrix_layout, char jobz, char uplo, lapack_int n,\nlapack_int ka, lapack_int kb, float* ab, lapack_int ldab, float* bb, lapack_int ldbb,\nfloat* w, float* z, lapack_int ldz);\nlapack_int LAPACKE_dsbgvd (int matrix_layout, char jobz, char uplo, lapack_int n,\nlapack_int ka, lapack_int kb, double* ab, lapack_int ldab, double* bb, lapack_int ldbb,\ndouble* w, double* z, lapack_int ldz);\nInclude Files\n•\nmkl.h\nDescription\nThe routine computes all the eigenvalues, and optionally, the eigenvectors of a real generalized symmetric-\ndefinite banded eigenproblem, of the form A*x = λ*B*x. Here A and B are assumed to be symmetric and\nbanded, and B is also positive definite.\nIf eigenvectors are desired, it uses a divide and conquer algorithm.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\njobz\nMust be 'N' or 'V'.\nIf jobz = 'N', then compute eigenvalues only.\nIf jobz = 'V', then compute eigenvalues and eigenvectors.\nuplo\nMust be 'U' or 'L'.\nIf uplo = 'U', arrays ab and bb store the upper triangles of A and B;\nIf uplo = 'L', arrays ab and bb store the lower triangles of A and B.\nn\nThe order of the matrices A and B (n≥ 0).\nka\nThe number of super- or sub-diagonals in A\n(ka≥ 0).\nkb\nThe number of super- or sub-diagonals in B (kb≥ 0).\nab, bb\nArrays:\nab(size at least max(1, ldab*n) for column major layout and max(1,\nldab*(ka + 1)) for row major layout) is an array containing either upper or\nlower triangular part of the symmetric matrix A (as specified by uplo) in\nband storage format.\nbb(size at least max(1, ldbb*n) for column major layout and max(1,\nldbb*(kb + 1)) for row major layout) is an array containing either upper or\nlower triangular part of the symmetric matrix B (as specified by uplo) in\nband storage format.\nldab\nThe leading dimension of the array ab; must be at least ka+1 for column\nmajor layout and at least max(1, n) for row major layout.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1140\n\n\nldbb\nThe leading dimension of the array bb; must be at least kb+1 for column\nmajor layout and at least max(1, n) for row major layout.\nldz\nThe leading dimension of the output array z; ldz≥ 1. If jobz = 'V', ldz≥\nmax(1, n).\nOutput Parameters\nab\nOn exit, the contents of ab are overwritten.\nbb\nOn exit, contains the factor S from the split Cholesky factorization B =\nST*S, as returned by pbstf/pbstf.\nw, z\nArrays:\nw, size at least max(1, n).\nIf info = 0, contains the eigenvalues in ascending order.\nz (size at least max(1, ldz*n)).\nIf jobz = 'V', then if info = 0, z contains the matrix Z of eigenvectors,\nwith the i-th column of z holding the eigenvector associated with w[i -\n1]. The eigenvectors are normalized so that ZT*B*Z = I.\nIf jobz = 'N', then z is not referenced.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nIf info > 0, and\nif i≤n, the algorithm failed to converge, and i off-diagonal elements of an intermediate tridiagonal did not\nconverge to zero;\nif info = n + i, for 1 ≤i≤n, then pbstf/pbstf returned info = i and B is not positive-definite. The\nfactorization of B could not be completed and no eigenvalues or eigenvectors were computed.\n?hbgvd\nComputes all eigenvalues and, optionally,\neigenvectors of a complex generalized Hermitian\npositive-definite eigenproblem with banded matrices.\nIf eigenvectors are desired, it uses a divide and\nconquer method.\nSyntax\nlapack_int LAPACKE_chbgvd( int matrix_layout, char jobz, char uplo, lapack_int n,\nlapack_int ka, lapack_int kb, lapack_complex_float* ab, lapack_int ldab,\nlapack_complex_float* bb, lapack_int ldbb, float* w, lapack_complex_float* z,\nlapack_int ldz );\nlapack_int LAPACKE_zhbgvd( int matrix_layout, char jobz, char uplo, lapack_int n,\nlapack_int ka, lapack_int kb, lapack_complex_double* ab, lapack_int ldab,\nlapack_complex_double* bb, lapack_int ldbb, double* w, lapack_complex_double* z,\nlapack_int ldz );\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1141\n\n\nInclude Files\n•\nmkl.h\nDescription\nThe routine computes all the eigenvalues, and optionally, the eigenvectors of a complex generalized\nHermitian positive-definite banded eigenproblem, of the form A*x = λ*B*x. Here A and B are assumed to be\nHermitian and banded, and B is also positive definite.\nIf eigenvectors are desired, it uses a divide and conquer algorithm.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\njobz\nMust be 'N' or 'V'.\nIf jobz = 'N', then compute eigenvalues only.\nIf jobz = 'V', then compute eigenvalues and eigenvectors.\nuplo\nMust be 'U' or 'L'.\nIf uplo = 'U', arrays ab and bb store the upper triangles of A and B;\nIf uplo = 'L', arrays ab and bb store the lower triangles of A and B.\nn\nThe order of the matrices A and B (n≥ 0).\nka\nThe number of super- or sub-diagonals in A\n(ka≥0).\nkb\nThe number of super- or sub-diagonals in B (kb≥ 0).\nab, bb\nArrays:\nab(size at least max(1, ldab*n) for column major layout and max(1,\nldab*(ka + 1)) for row major layout) is an array containing either upper or\nlower triangular part of the Hermitian matrix A (as specified by uplo) in\nband storage format.\nbb(size at least max(1, ldbb*n) for column major layout and max(1,\nldbb*(kb + 1)) for row major layout) is an array containing either upper or\nlower triangular part of the Hermitian matrix B (as specified by uplo) in\nband storage format.\nldab\nThe leading dimension of the array ab; must be at least ka+1.\nldbb\nThe leading dimension of the array bb; must be at least kb+1.\nldz\nThe leading dimension of the output array z; ldz≥ 1. If jobz = 'V', ldz≥\nmax(1, n).\nOutput Parameters\nab\nOn exit, the contents of ab are overwritten.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1142\n\n\nbb\nOn exit, contains the factor S from the split Cholesky factorization B =\nSH*S, as returned by pbstf/pbstf.\nw\nArray, size at least max(1, n) .\nIf info = 0, contains the eigenvalues in ascending order.\nz\nArray z (size at least max(1, ldz*n)).\nIf jobz = 'V', then if info = 0, z contains the matrix Z of eigenvectors,\nwith the i-th column of z holding the eigenvector associated with w(i). The\neigenvectors are normalized so that ZH*B*Z = I.\nIf jobz = 'N', then z is not referenced.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nIf info > 0, and\nif i≤n, the algorithm failed to converge, and i off-diagonal elements of an intermediate tridiagonal did not\nconverge to zero;\nif info = n + i, for 1 ≤i≤n, then pbstf/pbstf returned info = i and B is not positive-definite. The\nfactorization of B could not be completed and no eigenvalues or eigenvectors were computed.\n?sbgvx\nComputes selected eigenvalues and, optionally,\neigenvectors of a real generalized symmetric definite\neigenproblem with banded matrices.\nSyntax\nlapack_int LAPACKE_ssbgvx (int matrix_layout, char jobz, char range, char uplo,\nlapack_int n, lapack_int ka, lapack_int kb, float* ab, lapack_int ldab, float* bb,\nlapack_int ldbb, float* q, lapack_int ldq, float vl, float vu, lapack_int il, lapack_int\niu, float abstol, lapack_int* m, float* w, float* z, lapack_int ldz, lapack_int* ifail);\nlapack_int LAPACKE_dsbgvx (int matrix_layout, char jobz, char range, char uplo,\nlapack_int n, lapack_int ka, lapack_int kb, double* ab, lapack_int ldab, double* bb,\nlapack_int ldbb, double* q, lapack_int ldq, double vl, double vu, lapack_int il,\nlapack_int iu, double abstol, lapack_int* m, double* w, double* z, lapack_int ldz,\nlapack_int* ifail);\nInclude Files\n•\nmkl.h\nDescription\nThe routine computes selected eigenvalues, and optionally, the eigenvectors of a real generalized symmetric-\ndefinite banded eigenproblem, of the form A*x = λ*B*x. Here A and B are assumed to be symmetric and\nbanded, and B is also positive definite. Eigenvalues and eigenvectors can be selected by specifying either all\neigenvalues, a range of values or a range of indices for the desired eigenvalues.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1143\n\n\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\njobz\nMust be 'N' or 'V'.\nIf jobz = 'N', then compute eigenvalues only.\nIf jobz = 'V', then compute eigenvalues and eigenvectors.\nrange\nMust be 'A' or 'V' or 'I'.\nIf range = 'A', the routine computes all eigenvalues.\nIf range = 'V', the routine computes eigenvalues w[i] in the half-open\ninterval:\nvl<w[i]≤vu.\nIf range = 'I', the routine computes eigenvalues in range il to iu.\nuplo\nMust be 'U' or 'L'.\nIf uplo = 'U', arrays ab and bb store the upper triangles of A and B;\nIf uplo = 'L', arrays ab and bb store the lower triangles of A and B.\nn\nThe order of the matrices A and B (n≥ 0).\nka\nThe number of super- or sub-diagonals in A\n(ka≥ 0).\nkb\nThe number of super- or sub-diagonals in B (kb≥ 0).\nab, bb\nArrays:\nab(size at least max(1, ldab*n) for column major layout and max(1,\nldab*(ka + 1)) for row major layout) is an array containing either upper or\nlower triangular part of the symmetric matrix A (as specified by uplo) in\nband storage format.\nbb(size at least max(1, ldbb*n) for column major layout and max(1,\nldbb*(kb + 1)) for row major layout) is an array containing either upper or\nlower triangular part of the symmetric matrix B (as specified by uplo) in\nband storage format.\nldab\nThe leading dimension of the array ab; must be at least ka+1 for column\nmajor layout and at least max(1, n) for row major layout.\nldbb\nThe leading dimension of the array bb; must be at least kb+1 for column\nmajor layout and at least max(1, n) for row major layout.\nvl, vu\nIf range = 'V', the lower and upper bounds of the interval to be searched\nfor eigenvalues.\nConstraint: vl< vu.\nIf range = 'A' or 'I', vl and vu are not referenced.\nil, iu\nIf range = 'I', the indices in ascending order of the smallest and largest\neigenvalues to be returned.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1144\n\n\nConstraint: 1 ≤il≤iu≤n, if n > 0; il=1 and iu=0\nif n = 0.\nIf range = 'A' or 'V', il and iu are not referenced.\nabstol\nThe absolute error tolerance for the eigenvalues. See Application Notes for\nmore information.\nldz\nThe leading dimension of the output array z; ldz≥ 1. If jobz = 'V', ldz≥\nmax(1, n).\nldq\nThe leading dimension of the output array q; ldq < 1.\nIf jobz = 'V', ldq < max(1, n).\nOutput Parameters\nab\nOn exit, the contents of ab are overwritten.\nbb\nOn exit, contains the factor S from the split Cholesky factorization B =\nST*S, as returned by pbstf/pbstf.\nm\nThe total number of eigenvalues found,\n0 ≤m≤n. If range = 'A', m = n, and if range = 'I',\nm = iu-il+1.\nw, z, q\nArrays:\nw, size at least max(1, n) .\nIf info = 0, contains the eigenvalues in ascending order.\nz(size max(1, ldz*m) for column major layout and max(1, ldz*n) for row\nmajor layout) .\nIf jobz = 'V', then if info = 0, z contains the matrix Z of eigenvectors,\nwith the i-th column of z holding the eigenvector associated with w(i). The\neigenvectors are normalized so that ZT*B*Z = I.\nIf jobz = 'N', then z is not referenced.\nq (size max(1, ldq*n)) .\nIf jobz = 'V', then q contains the n-by-n matrix used in the reduction of\nA*x = lambda*B*x to standard form, that is, C*x= lambda*x and\nconsequently C to tridiagonal form.\nIf jobz = 'N', then q is not referenced.\nifail\nArray, size m.\nIf jobz = 'V', then if info = 0, the first m elements of ifail are zero; if\ninfo > 0, the ifail contains the indices of the eigenvectors that failed to\nconverge.\nIf jobz = 'N', then ifail is not referenced.\nReturn Values\nThis function returns a value info.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1145\n\n\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nIf info > 0, and\nif i≤n, the algorithm failed to converge, and i off-diagonal elements of an intermediate tridiagonal did not\nconverge to zero;\nif info = n + i, for 1 ≤i≤n, then pbstf/pbstf returned info = i and B is not positive-definite. The\nfactorization of B could not be completed and no eigenvalues or eigenvectors were computed.\nApplication Notes\nAn approximate eigenvalue is accepted as converged when it is determined to lie in an interval [a,b] of width\nless than or equal to abstol+ε*max(|a|,|b|), where ε is the machine precision.\nIf abstol is less than or equal to zero, then ε*||T||1 is used as tolerance, where T is the tridiagonal matrix\nobtained by reducing A to tridiagonal form. Eigenvalues will be computed most accurately when abstol is set\nto twice the underflow threshold 2*?lamch('S'), not zero.\nIf this routine returns with info > 0, indicating that some eigenvectors did not converge, try setting abstol\nto 2*?lamch('S').\n?hbgvx\nComputes selected eigenvalues and, optionally,\neigenvectors of a complex generalized Hermitian\npositive-definite eigenproblem with banded matrices.\nSyntax\nlapack_int LAPACKE_chbgvx( int matrix_layout, char jobz, char range, char uplo,\nlapack_int n, lapack_int ka, lapack_int kb, lapack_complex_float* ab, lapack_int ldab,\nlapack_complex_float* bb, lapack_int ldbb, lapack_complex_float* q, lapack_int ldq,\nfloat vl, float vu, lapack_int il, lapack_int iu, float abstol, lapack_int* m, float* w,\nlapack_complex_float* z, lapack_int ldz, lapack_int* ifail );\nlapack_int LAPACKE_zhbgvx( int matrix_layout, char jobz, char range, char uplo,\nlapack_int n, lapack_int ka, lapack_int kb, lapack_complex_double* ab, lapack_int ldab,\nlapack_complex_double* bb, lapack_int ldbb, lapack_complex_double* q, lapack_int ldq,\ndouble vl, double vu, lapack_int il, lapack_int iu, double abstol, lapack_int* m,\ndouble* w, lapack_complex_double* z, lapack_int ldz, lapack_int* ifail );\nInclude Files\n•\nmkl.h\nDescription\nThe routine computes selected eigenvalues, and optionally, the eigenvectors of a complex generalized\nHermitian positive-definite banded eigenproblem, of the form A*x = λ*B*x. Here A and B are assumed to be\nHermitian and banded, and B is also positive definite. Eigenvalues and eigenvectors can be selected by\nspecifying either all eigenvalues, a range of values or a range of indices for the desired eigenvalues.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1146\n\n\njobz\nMust be 'N' or 'V'.\nIf jobz = 'N', then compute eigenvalues only.\nIf jobz = 'V', then compute eigenvalues and eigenvectors.\nrange\nMust be 'A' or 'V' or 'I'.\nIf range = 'A', the routine computes all eigenvalues.\nIf range = 'V', the routine computes eigenvalues w[i] in the half-open\ninterval:\nvl< w[i]≤vu.\nIf range = 'I', the routine computes eigenvalues with indices il to iu.\nuplo\nMust be 'U' or 'L'.\nIf uplo = 'U', arrays ab and bb store the upper triangles of A and B;\nIf uplo = 'L', arrays ab and bb store the lower triangles of A and B.\nn\nThe order of the matrices A and B (n≥ 0).\nka\nThe number of super- or sub-diagonals in A\n(ka≥ 0).\nkb\nThe number of super- or sub-diagonals in B (kb≥ 0).\nab, bb\nArrays:\nab(size at least max(1, ldab*n) for column major layout and max(1,\nldab*(ka + 1)) for row major layout) is an array containing either upper or\nlower triangular part of the Hermitian matrix A (as specified by uplo) in\nband storage format.\nbb(size at least max(1, ldbb*n) for column major layout and max(1,\nldbb*(kb + 1)) for row major layout) is an array containing either upper or\nlower triangular part of the Hermitian matrix B (as specified by uplo) in\nband storage format.\nldab\nThe leading dimension of the array ab; must be at least ka+1 for column\nmajor layout and at least max(1, n) for row major layout.\nldbb\nThe leading dimension of the array bb; must be at least kb+1 for column\nmajor layout and at least max(1, n) for row major layout.\nvl, vu\nIf range = 'V', the lower and upper bounds of the interval to be searched\nfor eigenvalues.\nConstraint: vl< vu.\nIf range = 'A' or 'I', vl and vu are not referenced.\nil, iu\nIf range = 'I', the indices in ascending order of the smallest and largest\neigenvalues to be returned.\nConstraint: 1 ≤il≤iu≤n, if n > 0; il=1 and iu=0\nif n = 0.\nIf range = 'A' or 'V', il and iu are not referenced.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1147\n\n\nabstol\nThe absolute error tolerance for the eigenvalues. See Application Notes for\nmore information.\nldz\nThe leading dimension of the output array z; ldz≥ 1. If jobz = 'V', ldz≥\nmax(1, n) for column major layout and at least max(1, m) for row major\nlayout.\nldq\nThe leading dimension of the output array q; ldq≥ 1. If jobz = 'V', ldq≥\nmax(1, n).\nOutput Parameters\nab\nOn exit, the contents of ab are overwritten.\nbb\nOn exit, contains the factor S from the split Cholesky factorization B =\nSH*S, as returned by pbstf/pbstf.\nm\nThe total number of eigenvalues found,\n0 ≤m≤n. If range = 'A', m = n, and if range = 'I',\nm = iu-il+1.\nw\nArray w, size at least max(1, n).\nIf info = 0, contains the eigenvalues in ascending order.\nz, q\nArrays:\nz(size max(1, ldz*m) for column major layout and max(1, ldz*n) for row\nmajor layout).\nIf jobz = 'V', then if info = 0, z contains the matrix Z of eigenvectors,\nwith the i-th column of z holding the eigenvector associated with w[i -\n1]. The eigenvectors are normalized so that ZH*B*Z = I.\nIf jobz = 'N', then z is not referenced.\nq (size max(1, ldq*n)).\nIf jobz = 'V', then q contains the n-by-n matrix used in the reduction of\nAx = λBx to standard form, that is, Cx = λx and consequently C to\ntridiagonal form.\nIf jobz = 'N', then q is not referenced.\nifail\nArray, size at least max(1, n).\nIf jobz = 'V', then if info = 0, the first m elements of ifail are zero; if\ninfo > 0, the ifail contains the indices of the eigenvectors that failed to\nconverge.\nIf jobz = 'N', then ifail is not referenced.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1148\n\n\nIf info > 0, and\nif i≤n, the algorithm failed to converge, and i off-diagonal elements of an intermediate tridiagonal did not\nconverge to zero;\nif info = n + i, for 1 ≤i≤n, then pbstf/pbstf returned info = i and B is not positive-definite. The\nfactorization of B could not be completed and no eigenvalues or eigenvectors were computed.\nApplication Notes\nAn approximate eigenvalue is accepted as converged when it is determined to lie in an interval [a,b] of width\nless than or equal to abstol+ε*max(|a|,|b|), where ε is the machine precision.\nIf abstol is less than or equal to zero, then ε*||T||1 will be used in its place, where T is the tridiagonal\nmatrix obtained by reducing A to tridiagonal form. Eigenvalues will be computed most accurately when abstol\nis set to twice the underflow threshold 2*?lamch('S'), not zero.\nIf this routine returns with info > 0, indicating that some eigenvectors did not converge, try setting abstol\nto 2*?lamch('S').\nGeneralized Nonsymmetric Eigenvalue Problems: LAPACK Driver Routines\nThis topic describes LAPACK driver routines used for solving generalized nonsymmetric eigenproblems. See\nalso computational routines that can be called to solve these problems. Table \"Driver Routines for Solving\nGeneralized Nonsymmetric Eigenproblems\" lists all such driver routines.\nDriver Routines for Solving Generalized Nonsymmetric Eigenproblems\nRoutine Name\nOperation performed\ngges\nComputes the generalized eigenvalues, Schur form, and the left and/or right Schur\nvectors for a pair of nonsymmetric matrices.\nggesx\nComputes the generalized eigenvalues, Schur form, and, optionally, the left and/or\nright matrices of Schur vectors.\ngges3\nComputes generalized Schur factorization for a pair of matrices.\nggev\nComputes the generalized eigenvalues, and the left and/or right generalized\neigenvectors for a pair of nonsymmetric matrices.\nggevx\nComputes the generalized eigenvalues, and, optionally, the left and/or right\ngeneralized eigenvectors.\nggev3\nComputes generalized Schur factorization for a pair of matrices.\n?gges\nComputes the generalized eigenvalues, Schur form,\nand the left and/or right Schur vectors for a pair of\nnonsymmetric matrices.\nSyntax\nlapack_int LAPACKE_sgges( int matrix_layout, char jobvsl, char jobvsr, char sort,\nLAPACK_S_SELECT3 select, lapack_int n, float* a, lapack_int lda, float* b, lapack_int\nldb, lapack_int* sdim, float* alphar, float* alphai, float* beta, float* vsl, lapack_int\nldvsl, float* vsr, lapack_int ldvsr );\nlapack_int LAPACKE_dgges( int matrix_layout, char jobvsl, char jobvsr, char sort,\nLAPACK_D_SELECT3 select, lapack_int n, double* a, lapack_int lda, double* b, lapack_int\nldb, lapack_int* sdim, double* alphar, double* alphai, double* beta, double* vsl,\nlapack_int ldvsl, double* vsr, lapack_int ldvsr );\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1149\n\n\nlapack_int LAPACKE_cgges( int matrix_layout, char jobvsl, char jobvsr, char sort,\nLAPACK_C_SELECT2 select, lapack_int n, lapack_complex_float* a, lapack_int lda,\nlapack_complex_float* b, lapack_int ldb, lapack_int* sdim, lapack_complex_float* alpha,\nlapack_complex_float* beta, lapack_complex_float* vsl, lapack_int ldvsl,\nlapack_complex_float* vsr, lapack_int ldvsr );\nlapack_int LAPACKE_zgges( int matrix_layout, char jobvsl, char jobvsr, char sort,\nLAPACK_Z_SELECT2 select, lapack_int n, lapack_complex_double* a, lapack_int lda,\nlapack_complex_double* b, lapack_int ldb, lapack_int* sdim, lapack_complex_double*\nalpha, lapack_complex_double* beta, lapack_complex_double* vsl, lapack_int ldvsl,\nlapack_complex_double* vsr, lapack_int ldvsr );\nInclude Files\n•\nmkl.h\nDescription\nThe ?gges routine computes the generalized eigenvalues, the generalized real/complex Schur form (S,T),\noptionally, the left and/or right matrices of Schur vectors (vsl and vsr) for a pair of n-by-n real/complex\nnonsymmetric matrices (A,B). This gives the generalized Schur factorization\n(A,B) = ( vsl*S *vsrH, vsl*T*vsrH )\nOptionally, it also orders the eigenvalues so that a selected cluster of eigenvalues appears in the leading\ndiagonal blocks of the upper quasi-triangular matrix S and the upper triangular matrix T. The leading\ncolumns of vsl and vsr then form an orthonormal/unitary basis for the corresponding left and right\neigenspaces (deflating subspaces).\nIf only the generalized eigenvalues are needed, use the driver ggev instead, which is faster.\nA generalized eigenvalue for a pair of matrices (A,B) is a scalar w or a ratio alpha / beta = w, such that A -\nw*B is singular. It is usually represented as the pair (alpha, beta), as there is a reasonable interpretation\nfor beta=0 or for both being zero. A pair of matrices (S,T) is in the generalized real Schur form if T is upper\ntriangular with non-negative diagonal and S is block upper triangular with 1-by-1 and 2-by-2 blocks. 1-by-1\nblocks correspond to real generalized eigenvalues, while 2-by-2 blocks of S are \"standardized\" by making the\ncorresponding elements of T have the form:\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1150\n\n\nand the pair of corresponding 2-by-2 blocks in S and T will have a complex conjugate pair of generalized\neigenvalues. A pair of matrices (S,T) is in generalized complex Schur form if S and T are upper triangular\nand, in addition, the diagonal of T are non-negative real numbers.\nThe ?gges routine replaces the deprecated ?gegs routine.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\njobvsl\nMust be 'N' or 'V'.\nIf jobvsl = 'N', then the left Schur vectors are not computed.\nIf jobvsl = 'V', then the left Schur vectors are computed.\njobvsr\nMust be 'N' or 'V'.\nIf jobvsr = 'N', then the right Schur vectors are not computed.\nIf jobvsr = 'V', then the right Schur vectors are computed.\nsort\nMust be 'N' or 'S'. Specifies whether or not to order the eigenvalues on\nthe diagonal of the generalized Schur form.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1151\n\n\nIf sort = 'N', then eigenvalues are not ordered.\nIf sort = 'S', eigenvalues are ordered (see select).\nselect\nThe select parameter is a pointer to a function returning a value of\nlapack_logical type. For different flavors the function has different\narguments:\nLAPACKE_sgges: lapack_logical (*LAPACK_S_SELECT3) ( const\nfloat*, const float*, const float* );\nLAPACKE_dgges: lapack_logical (*LAPACK_D_SELECT3) ( const\ndouble*, const double*, const double* );\nLAPACKE_cgges: lapack_logical (*LAPACK_C_SELECT2) ( const\nlapack_complex_float*, const lapack_complex_float* );\nLAPACKE_zgges: lapack_logical (*LAPACK_Z_SELECT2) ( const\nlapack_complex_double*, const lapack_complex_double* );\nIf sort = 'S', select is used to select eigenvalues to sort to the top left\nof the Schur form.\nIf sort = 'N', select is not referenced.\nFor real flavors:\nAn eigenvalue (alphar[j] + alphai[j])/beta[j] is selected if select(alphar[j],\nalphai[j], beta[j]) is true; that is, if either one of a complex conjugate pair\nof eigenvalues is selected, then both complex eigenvalues are selected.\nNote that in the ill-conditioned case, a selected complex eigenvalue may no\nlonger satisfy select(alphar[j], alphai[j], beta[j]) = 1 after\nordering. In this case info is set to n+2 .\nFor complex flavors:\nAn eigenvalue alpha[j] / beta[j] is selected if select(alpha[j], beta[j])\nis true.\nNote that a selected complex eigenvalue may no longer satisfy\nselect(alpha[j], beta[j]) = 1 after ordering, since ordering may\nchange the value of complex eigenvalues (especially if the eigenvalue is ill-\nconditioned); in this case info is set to n+2 (see info below).\nn\nThe order of the matrices A, B, vsl, and vsr (n≥ 0).\na, b\nArrays:\na (size at least max(1, lda*n)) is an array containing the n-by-n matrix A\n(first of the pair of matrices).\nb (size at least max(1, ldb*n)) is an array containing the n-by-n matrix B\n(second of the pair of matrices).\nlda\nThe leading dimension of the array a. Must be at least max(1, n).\nldb\nThe leading dimension of the array b. Must be at least max(1, n).\nldvsl, ldvsr\nThe leading dimensions of the output matrices vsl and vsr, respectively.\nConstraints:\nldvsl≥ 1. If jobvsl = 'V', ldvsl≥ max(1, n).\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1152\n\n\nldvsr≥ 1. If jobvsr = 'V', ldvsr≥ max(1, n).\nOutput Parameters\na\nOn exit, this array has been overwritten by its generalized Schur form S.\nb\nOn exit, this array has been overwritten by its generalized Schur form T.\nsdim\nIf sort = 'N', sdim= 0.\nIf sort = 'S', sdim is equal to the number of eigenvalues (after sorting)\nfor which select is true.\nNote that for real flavors complex conjugate pairs for which select is true\nfor either eigenvalue count as 2.\nalphar, alphai\nArrays, size at least max(1, n) each. Contain values that form generalized\neigenvalues in real flavors.\nSee beta.\nalpha\nArray, size at least max(1, n). Contain values that form generalized\neigenvalues in complex flavors. See beta.\nbeta\nArray, size at least max(1, n).\nFor real flavors:\nOn exit, (alphar[j] + alphai[j]*i)/beta[j], j=0,..., n - 1, will be the\ngeneralized eigenvalues.\nalphar[j] + alphai[j]*i and beta[j], j=0,..., n - 1 are the diagonals of the\ncomplex Schur form (S,T) that would result if the 2-by-2 diagonal blocks of\nthe real generalized Schur form of (A,B) were further reduced to triangular\nform using complex unitary transformations. If alphai[j] is zero, then the j-\nth eigenvalue is real; if positive, then the j-th and (j+1)-st eigenvalues are\na complex conjugate pair, with alphai[j+1] negative.\nFor complex flavors:\nOn exit, alpha[j]/beta[j], j=0,..., n - 1, will be the generalized eigenvalues.\nalpha[j] and beta[j], j=0,..., n - 1 are the diagonals of the complex Schur\nform (S,T) output by cgges/zgges. The beta[j] will be non-negative real.\nSee also Application Notes below.\nvsl, vsr\nArrays:\nvsl (size at least max(1, ldvsl*n)).\nIf jobvsl = 'V', this array will contain the left Schur vectors.\nIf jobvsl = 'N', vsl is not referenced.\nvsr (size at least max(1, ldvsr*n)).\nIf jobvsr = 'V', this array will contain the right Schur vectors.\nIf jobvsr = 'N', vsr is not referenced.\nReturn Values\nThis function returns a value info.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1153\n\n\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nIf info = i, and\ni≤n:\nthe QZ iteration failed. (A, B) is not in Schur form, but alphar[j], alphai[j] (for real flavors), or alpha[j] (for\ncomplex flavors), and beta[j], j = info,..., n - 1 should be correct.\ni > n: errors that usually indicate LAPACK problems:\ni = n+1: other than QZ iteration failed in hgeqz;\ni = n+2: after reordering, roundoff changed values of some complex eigenvalues so that leading\neigenvalues in the generalized Schur form no longer satisfy select = 1. This could also be caused due to\nscaling;\ni = n+3: reordering failed in tgsen.\nApplication Notes\nThe quotients alphar[j]/beta[j] and alphai[j]/beta[j] may easily over- or underflow, and beta[j] may even be\nzero. Thus, you should avoid simply computing the ratio. However, alphar and alphai will be always less than\nand usually comparable with norm(A) in magnitude, and beta always less than and usually comparable with\nnorm(B).\n?ggesx\nComputes the generalized eigenvalues, Schur form,\nand, optionally, the left and/or right matrices of Schur\nvectors.\nSyntax\nlapack_int LAPACKE_sggesx( int matrix_layout, char jobvsl, char jobvsr, char sort,\nLAPACK_S_SELECT3 select, char sense, lapack_int n, float* a, lapack_int lda, float* b,\nlapack_int ldb, lapack_int* sdim, float* alphar, float* alphai, float* beta, float* vsl,\nlapack_int ldvsl, float* vsr, lapack_int ldvsr, float* rconde, float* rcondv );\nlapack_int LAPACKE_dggesx( int matrix_layout, char jobvsl, char jobvsr, char sort,\nLAPACK_D_SELECT3 select, char sense, lapack_int n, double* a, lapack_int lda, double* b,\nlapack_int ldb, lapack_int* sdim, double* alphar, double* alphai, double* beta, double*\nvsl, lapack_int ldvsl, double* vsr, lapack_int ldvsr, double* rconde, double* rcondv );\nlapack_int LAPACKE_cggesx( int matrix_layout, char jobvsl, char jobvsr, char sort,\nLAPACK_C_SELECT2 select, char sense, lapack_int n, lapack_complex_float* a, lapack_int\nlda, lapack_complex_float* b, lapack_int ldb, lapack_int* sdim, lapack_complex_float*\nalpha, lapack_complex_float* beta, lapack_complex_float* vsl, lapack_int ldvsl,\nlapack_complex_float* vsr, lapack_int ldvsr, float* rconde, float* rcondv );\nlapack_int LAPACKE_zggesx( int matrix_layout, char jobvsl, char jobvsr, char sort,\nLAPACK_Z_SELECT2 select, char sense, lapack_int n, lapack_complex_double* a, lapack_int\nlda, lapack_complex_double* b, lapack_int ldb, lapack_int* sdim, lapack_complex_double*\nalpha, lapack_complex_double* beta, lapack_complex_double* vsl, lapack_int ldvsl,\nlapack_complex_double* vsr, lapack_int ldvsr, double* rconde, double* rcondv );\nInclude Files\n•\nmkl.h\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1154\n\n\nDescription\nThe routine computes for a pair of n-by-n real/complex nonsymmetric matrices (A,B), the generalized\neigenvalues, the generalized real/complex Schur form (S,T), optionally, the left and/or right matrices of\nSchur vectors (vsl and vsr). This gives the generalized Schur factorization\n(A,B) = ( vsl*S *vsrH, vsl*T*vsrH )\nOptionally, it also orders the eigenvalues so that a selected cluster of eigenvalues appears in the leading\ndiagonal blocks of the upper quasi-triangular matrix S and the upper triangular matrix T; computes a\nreciprocal condition number for the average of the selected eigenvalues (rconde); and computes a reciprocal\ncondition number for the right and left deflating subspaces corresponding to the selected eigenvalues\n(rcondv). The leading columns of vsl and vsr then form an orthonormal/unitary basis for the corresponding\nleft and right eigenspaces (deflating subspaces).\nA generalized eigenvalue for a pair of matrices (A,B) is a scalar w or a ratio alpha / beta = w, such that A\n- w*B is singular. It is usually represented as the pair (alpha, beta), as there is a reasonable interpretation\nfor beta=0 or for both being zero. A pair of matrices (S,T) is in generalized real Schur form if T is upper\ntriangular with non-negative diagonal and S is block upper triangular with 1-by-1 and 2-by-2 blocks. 1-by-1\nblocks correspond to real generalized eigenvalues, while 2-by-2 blocks of S will be \"standardized\" by making\nthe corresponding elements of T have the form:\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1155\n\n\nand the pair of corresponding 2-by-2 blocks in S and T will have a complex conjugate pair of generalized\neigenvalues. A pair of matrices (S,T) is in generalized complex Schur form if S and T are upper triangular\nand, in addition, the diagonal of T are non-negative real numbers.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\njobvsl\nMust be 'N' or 'V'.\nIf jobvsl = 'N', then the left Schur vectors are not computed.\nIf jobvsl = 'V', then the left Schur vectors are computed.\njobvsr\nMust be 'N' or 'V'.\nIf jobvsr = 'N', then the right Schur vectors are not computed.\nIf jobvsr = 'V', then the right Schur vectors are computed.\nsort\nMust be 'N' or 'S'. Specifies whether or not to order the eigenvalues on\nthe diagonal of the generalized Schur form.\nIf sort = 'N', then eigenvalues are not ordered.\nIf sort = 'S', eigenvalues are ordered (see select).\nselect\nThe select parameter is a pointer to a function returning a value of\nlapack_logical type. For different flavors the function has different\narguments:\nLAPACKE_sggesx: lapack_logical (*LAPACK_S_SELECT3) ( const\nfloat*, const float*, const float* );\nLAPACKE_dggesx: lapack_logical (*LAPACK_D_SELECT3) ( const\ndouble*, const double*, const double* );\nLAPACKE_cggesx: lapack_logical (*LAPACK_C_SELECT2) ( const\nlapack_complex_float*, const lapack_complex_float* );\nLAPACKE_zggesx: lapack_logical (*LAPACK_Z_SELECT2) ( const\nlapack_complex_double*, const lapack_complex_double* );\nIf sort = 'S', select is used to select eigenvalues to sort to the top left\nof the Schur form.\nIf sort = 'N', select is not referenced.\nFor real flavors:\nAn eigenvalue (alphar[j] + alphai[j])/beta[j] is selected if select(alphar[j],\nalphai[j], beta[j]) is true; that is, if either one of a complex conjugate pair\nof eigenvalues is selected, then both complex eigenvalues are selected.\nNote that in the ill-conditioned case, a selected complex eigenvalue may no\nlonger satisfy select(alphar[j], alphai[j], beta[j]) = 1 after\nordering. In this case info is set to n+2.\nFor complex flavors:\nAn eigenvalue alpha[j] / beta[j] is selected if select(alpha[j], beta[j])\nis true.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1156\n\n\nNote that a selected complex eigenvalue may no longer satisfy\nselect(alpha[j], beta[j]) = 1 after ordering, since ordering may\nchange the value of complex eigenvalues (especially if the eigenvalue is ill-\nconditioned); in this case info is set to n+2 (see info below).\nsense\nMust be 'N', 'E', 'V', or 'B'. Determines which reciprocal condition\nnumber are computed.\nIf sense = 'N', none are computed;\nIf sense = 'E', computed for average of selected eigenvalues only;\nIf sense = 'V', computed for selected deflating subspaces only;\nIf sense = 'B', computed for both.\nIf sense is 'E', 'V', or 'B', then sort must equal 'S'.\nn\nThe order of the matrices A, B, vsl, and vsr (n≥ 0).\na, b\nArrays:\na (size at least max(1, lda*n)) is an array containing the n-by-n matrix A\n(first of the pair of matrices).\nb (size at least max(1, ldb*n)) is an array containing the n-by-n matrix B\n(second of the pair of matrices).\nlda\nThe leading dimension of the array a.\nMust be at least max(1, n).\nldb\nThe leading dimension of the array b.\nMust be at least max(1, n).\nldvsl, ldvsr\nThe leading dimensions of the output matrices vsl and vsr, respectively.\nConstraints:\nldvsl≥ 1. If jobvsl = 'V', ldvsl≥ max(1, n).\nldvsr≥ 1. If jobvsr = 'V', ldvsr≥ max(1, n).\nOutput Parameters\na\nOn exit, this array has been overwritten by its generalized Schur form S.\nb\nOn exit, this array has been overwritten by its generalized Schur form T.\nsdim\nIf sort = 'N', sdim= 0.\nIf sort = 'S', sdim is equal to the number of eigenvalues (after sorting)\nfor which select is true.\nNote that for real flavors complex conjugate pairs for which select is true\nfor either eigenvalue count as 2.\nalphar, alphai\nArrays, size at least max(1, n) each. Contain values that form generalized\neigenvalues in real flavors.\nSee beta.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1157\n\n\nalpha\nArray, size at least max(1, n). Contain values that form generalized\neigenvalues in complex flavors. See beta.\nbeta\nArray, size at least max(1, n).\nFor real flavors:\nOn exit, (alphar[j] + alphai[j]*i)/beta[j], j=0,..., n - 1 will be the\ngeneralized eigenvalues.\nalphar[j] + alphai[j]*i and beta[j], j=0,..., n - 1 are the diagonals of the\ncomplex Schur form (S,T) that would result if the 2-by-2 diagonal blocks of\nthe real generalized Schur form of (A,B) were further reduced to triangular\nform using complex unitary transformations. If alphai[j] is zero, then the j-\nth eigenvalue is real; if positive, then the j-th and (j+1)-st eigenvalues are\na complex conjugate pair, with alphai[j+1] negative.\nFor complex flavors:\nOn exit, alpha[j]/beta[j], j=0,..., n - 1 will be the generalized eigenvalues.\nalpha[j] and beta[j], j=0,..., n - 1 are the diagonals of the complex Schur\nform (S,T) output by cggesx/zggesx. The beta[j] will be non-negative real.\nSee also Application Notes below.\nvsl, vsr\nArrays:\nvsl (size at least max(1, ldvsl*n)).\nIf jobvsl = 'V', this array will contain the left Schur vectors.\nIf jobvsl = 'N', vsl is not referenced.\nvsr (size at least max(1, ldvsr*n)).\nIf jobvsr = 'V', this array will contain the right Schur vectors.\nIf jobvsr = 'N', vsr is not referenced.\nrconde, rcondv\nArrays, size 2 each\nIf sense = 'E' or 'B', rconde(1) and rconde(2) contain the reciprocal\ncondition numbers for the average of the selected eigenvalues.\nNot referenced if sense = 'N' or 'V'.\nIf sense = 'V' or 'B', rcondv[0] and rcondv[1] contain the reciprocal\ncondition numbers for the selected deflating subspaces.\nNot referenced if sense = 'N' or 'E'.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nIf info = i, and\ni≤n:\nthe QZ iteration failed. (A, B) is not in Schur form, but alphar[j], alphai[j] (for real flavors), or alpha[j] (for\ncomplex flavors), and beta[j], j = info,..., n - 1 should be correct.\ni > n: errors that usually indicate LAPACK problems:\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1158\n\n\ni = n+1: other than QZ iteration failed in hgeqz;\ni = n+2: after reordering, roundoff changed values of some complex eigenvalues so that leading\neigenvalues in the generalized Schur form no longer satisfy select = 1. This could also be caused due to\nscaling;\ni = n+3: reordering failed in tgsen.\nApplication Notes\nThe quotients alphar[j]/beta[j] and alphai[j]/beta[j] may easily over- or underflow, and beta[j] may even be\nzero. Thus, you should avoid simply computing the ratio. However, alphar and alphai will be always less than\nand usually comparable with norm(A) in magnitude, and beta always less than and usually comparable with\nnorm(B).\n?gges3\nComputes generalized Schur factorization for a pair of\nmatrices.\nSyntax\nlapack_int LAPACKE_sgges3 (int matrix_layout, char jobvsl, char jobvsr, char sort,\nLAPACK_S_SELECT3 selctg, lapack_int n, float * a, lapack_int lda, float * b, lapack_int\nldb, lapack_int * sdim, float * alphar, float * alphai, float * beta, float * vsl,\nlapack_int ldvsl, float * vsr, lapack_int ldvsr);\nlapack_int LAPACKE_dgges3 (int matrix_layout, char jobvsl, char jobvsr, char sort,\nLAPACK_D_SELECT3 selctg, lapack_int n, double * a, lapack_int lda, double * b,\nlapack_int ldb, lapack_int * sdim, double * alphar, double * alphai, double * beta,\ndouble * vsl, lapack_int ldvsl, double * vsr, lapack_int ldvsr);\nlapack_int LAPACKE_cgges3 (int matrix_layout, char jobvsl, char jobvsr, char sort,\nLAPACK_C_SELECT2 selctg, lapack_int n, lapack_complex_float * a, lapack_int lda,\nlapack_complex_float * b, lapack_int ldb, lapack_int * sdim, lapack_complex_float *\nalpha, lapack_complex_float * beta, lapack_complex_float * vsl, lapack_int ldvsl,\nlapack_complex_float * vsr, lapack_int ldvsr);\nlapack_int LAPACKE_zgges3 (int matrix_layout, char jobvsl, char jobvsr, char sort,\nLAPACK_Z_SELECT2 selctg, lapack_int n, lapack_complex_double * a, lapack_int lda,\nlapack_complex_double * b, lapack_int ldb, lapack_int * sdim, lapack_complex_double *\nalpha, lapack_complex_double * beta, lapack_complex_double * vsl, lapack_int ldvsl,\nlapack_complex_double * vsr, lapack_int ldvsr);\nInclude Files\n•\nmkl.h\nDescription\nFor a pair of n-by-n real or complex nonsymmetric matrices (A,B), ?gges3 computes the generalized\neigenvalues, the generalized real or complex Schur form (S,T), and optionally the left or right matrices of\nSchur vectors (VSL and VSR). This gives the generalized Schur factorization\n(A,B) = ( (VSL)*S*(VSR)T, (VSL)*T*(VSR)T ) for real (A,B)\nor\n(A,B) = ( (VSL)*S*(VSR)H, (VSL)*T*(VSR)H ) for complex (A,B)\nwhere (VSR)H is the conjugate-transpose of VSR.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1159\n\n\nOptionally, it also orders the eigenvalues so that a selected cluster of eigenvalues appears in the leading\ndiagonal blocks of the upper quasi-triangular matrix S and the upper triangular matrix T. The leading\ncolumns of VSL and VSR then form an orthonormal basis for the corresponding left and right eigenspaces\n(deflating subspaces).\nNOTE\nIf only the generalized eigenvalues are needed, use the driver ?ggev instead, which is faster.\nA generalized eigenvalue for a pair of matrices (A,B) is a scalar w or a ratio alpha/beta = w, such that A -\nw*B is singular. It is usually represented as the pair (alpha,beta), as there is a reasonable interpretation for\nbeta=0 or both being zero.\nFor real flavors:\nA pair of matrices (S,T) is in generalized real Schur form if T is upper triangular with non-negative diagonal\nand S is block upper triangular with 1-by-1 and 2-by-2 blocks. 1-by-1 blocks correspond to real generalized\neigenvalues, while 2-by-2 blocks of S will be \"standardized\" by making the corresponding elements of T have\nthe form:\na 0\n0 b\nand the pair of corresponding 2-by-2 blocks in S and T have a complex conjugate pair of generalized\neigenvalues.\nFor complex flavors:\nA pair of matrices (S,T) is in generalized complex Schur form if S and T are upper triangular and, in addition,\nthe diagonal elements of T are non-negative real numbers.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\njobvsl\n= 'N': do not compute the left Schur vectors;\njobvsr\n= 'N': do not compute the right Schur vectors;\n= 'V': compute the right Schur vectors.\nsort\nSpecifies whether or not to order the eigenvalues on the diagonal of the\ngeneralized Schur form.\n= 'N': Eigenvalues are not ordered;\n= 'S': Eigenvalues are ordered (see selctg).\nselctg\nselctg is a function of three arguments for real flavors or two arguments\nfor complex flavors. selctg must be declared EXTERNAL in the calling\nsubroutine. If sort = 'N', selctg is not referenced. If sort = 'S', selctg\nis used to select eigenvalues to sort to the top left of the Schur form.\nFor real flavors:\nAn eigenvalue (alphar[j - 1] + alphai[j - 1])/beta[j - 1] is\nselected if selctg(alphar[j - 1],alphai[j - 1],beta[j - 1]) is true.\nIn other words, if either one of a complex conjugate pair of eigenvalues is\nselected, then both complex eigenvalues are selected.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1160\n\n\nNote that in the ill-conditioned case, a selected complex eigenvalue may no\nlonger satisfy selctg(alphar[j - 1],alphai[j - 1], beta[j - 1]) ≠ 0\nafter ordering. info is to be set to n+2 in this case.\nFor complex flavors:\nAn eigenvalue alpha[j - 1]/beta[j - 1] is selected if selctg(alpha[j\n- 1],beta[j - 1]) is true.\nNote that a selected complex eigenvalue may no longer satisfy\nselctg(alpha[j - 1],beta[j - 1])≠ 0 after ordering, since ordering\nmay change the value of complex eigenvalues (especially if the eigenvalue\nis ill-conditioned), in this case ?gges3 returns n + 2.\nn\nThe order of the matrices A, B, VSL, and VSR. n≥ 0.\na\nArray, size (lda*n). On entry, the first of the pair of matrices.\nlda\nThe leading dimension of a. lda≥ max(1,n).\nb\nArray, size (ldb*n). On entry, the second of the pair of matrices.\nldb\nThe leading dimension of b. ldb≥ max(1,n).\nldvsl\nThe leading dimension of the matrix VSL. ldvsl≥ 1, and if jobvsl = 'V',\nldvsl≥ n.\nldvsr\nThe leading dimension of the matrix VSR. ldvsr≥ 1, and if jobvsr = 'V',\nldvsr≥ n.\nOutput Parameters\na\nOn exit, a is overwritten by its generalized Schur form S.\nb\nOn exit, b is overwritten by its generalized Schur form T.\nsdim\nIf sort = 'N', sdim = 0. If sort = 'S', sdim = number of eigenvalues\n(after sorting) for which selctg is true.\nalpha\nArray, size (n).\nalphar\nArray, size (n).\nalphai\nArray, size (n).\nbeta\nArray, size (n).\nFor real flavors:\nOn exit, (alphar[j - 1] + alphai[j - 1]*i)/beta[j - 1],\nj=1,...,n, are the generalized eigenvalues. alphar[j - 1] +\nalphai[j - 1]*i, and beta[j - 1],j=1,...,n are the diagonals of\nthe complex Schur form (S,T) that would result if the 2-by-2 diagonal\nblocks of the real Schur form of (a,b) were further reduced to\ntriangular form using 2-by-2 complex unitary transformations. If\nalphai[j - 1] is zero, then the j-th eigenvalue is real; if positive,\nthen the j-th and (j+1)-st eigenvalues are a complex conjugate pair,\nwith alphai[j] negative.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1161\n\n\nNote: the quotients alphar[j - 1]/beta[j - 1] and alphai[j -\n1]/beta[j - 1] can easily over- or underflow, and beta[j - 1]\nmight even be zero. Thus, you should avoid computing the ratio\nalpha/beta by simply dividing alpha by beta. However, alphar and\nalphai is always less than and usually comparable with norm(a) in\nmagnitude, and beta is always less than and usually comparable with\nnorm(b).\nFor complex flavors:\nOn exit, alpha[j - 1][j - 1]/beta[j - 1], j=1,...,n, are the\ngeneralized eigenvalues. alpha[j - 1], j=1,...,n and beta[j - 1],\nj=1,...,n are the diagonals of the complex Schur form (a,b) output\nby ?gges3. The beta[j - 1] is non-negative real.\nNote: the quotient alpha[j - 1]/beta[j - 1] can easily over- or\nunderflow, and beta[j - 1] might even be zero. Thus, you should\navoid computing the ratio alpha/beta by simply dividing alpha by\nbeta. However, alpha is always less than and usually comparable\nwith norm(a) in magnitude, and beta is always less than and usually\ncomparable with norm(b).\nvsl\nArray, size (ldvsl*n).\nIf jobvsl = 'V', vsl contains the left Schur vectors. Not referenced if\njobvsl = 'N'.\nvsr\nArray, size (ldvsr*n).\nIf jobvsr = 'V', vsr contains the right Schur vectors. Not referenced\nif jobvsr = 'N'.\nReturn Values\nThis function returns a value info.\n= 0: successful exit < 0: if info = -i, the i-th argument had an illegal value.\n=1,...,n:\nfor real flavors:\nThe QZ iteration failed. (a,b) are not in Schur form, but alphar[j], alphai[j] and beta[j] should be\ncorrect for j=info,...,n - 1.\nThe QZ iteration failed. (a,b) are not in Schur form, but alpha[j] and beta[j] should be correct for\nj=info,...,n - 1.\nfor complex flavors:\n> n:\n=n+1: other than QZ iteration failed in ?hgeqz.\n=n+2: after reordering, roundoff changed values of some complex eigenvalues so that leading eigenvalues in\nthe Generalized Schur form no longer satisfy selctg≠ 0 This could also be caused due to scaling.\n=n+3: reordering failed in ?tgsen.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1162\n\n\n?ggev\nComputes the generalized eigenvalues, and the left\nand/or right generalized eigenvectors for a pair of\nnonsymmetric matrices.\nSyntax\nlapack_int LAPACKE_sggev( int matrix_layout, char jobvl, char jobvr, lapack_int n,\nfloat* a, lapack_int lda, float* b, lapack_int ldb, float* alphar, float* alphai, float*\nbeta, float* vl, lapack_int ldvl, float* vr, lapack_int ldvr );\nlapack_int LAPACKE_dggev( int matrix_layout, char jobvl, char jobvr, lapack_int n,\ndouble* a, lapack_int lda, double* b, lapack_int ldb, double* alphar, double* alphai,\ndouble* beta, double* vl, lapack_int ldvl, double* vr, lapack_int ldvr );\nlapack_int LAPACKE_cggev( int matrix_layout, char jobvl, char jobvr, lapack_int n,\nlapack_complex_float* a, lapack_int lda, lapack_complex_float* b, lapack_int ldb,\nlapack_complex_float* alpha, lapack_complex_float* beta, lapack_complex_float* vl,\nlapack_int ldvl, lapack_complex_float* vr, lapack_int ldvr );\nlapack_int LAPACKE_zggev( int matrix_layout, char jobvl, char jobvr, lapack_int n,\nlapack_complex_double* a, lapack_int lda, lapack_complex_double* b, lapack_int ldb,\nlapack_complex_double* alpha, lapack_complex_double* beta, lapack_complex_double* vl,\nlapack_int ldvl, lapack_complex_double* vr, lapack_int ldvr );\nInclude Files\n•\nmkl.h\nDescription\nThe ?ggev routine computes the generalized eigenvalues, and optionally, the left and/or right generalized\neigenvectors for a pair of n-by-n real/complex nonsymmetric matrices (A,B).\nA generalized eigenvalue for a pair of matrices (A,B) is a scalar λ or a ratio alpha / beta = λ, such that A -\nλ*B is singular. It is usually represented as the pair (alpha, beta), as there is a reasonable interpretation for\nbeta =0 and even for both being zero.\nThe right generalized eigenvector v(j) corresponding to the generalized eigenvalue λ(j) of (A,B) satisfies\nA*v(j) = λ(j)*B*v(j).\nThe left generalized eigenvector u(j) corresponding to the generalized eigenvalue λ(j) of (A,B) satisfies\nu(j)H*A = λ(j)*u(j)H*B\nwhere u(j)H denotes the conjugate transpose of u(j).\nThe ?ggev routine replaces the deprecated ?gegv routine.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\njobvl\nMust be 'N' or 'V'.\nIf jobvl = 'N', the left generalized eigenvectors are not computed;\nIf jobvl = 'V', the left generalized eigenvectors are computed.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1163\n\n\njobvr\nMust be 'N' or 'V'.\nIf jobvr = 'N', the right generalized eigenvectors are not computed;\nIf jobvr = 'V', the right generalized eigenvectors are computed.\nn\nThe order of the matrices A, B, vl, and vr (n≥ 0).\na, b\nArrays:\na (size at least max(1, lda*n)) is an array containing the n-by-n matrix A\n(first of the pair of matrices).\nb (size at least max(1, ldb*n)) is an array containing the n-by-n matrix B\n(second of the pair of matrices).\nlda\nThe leading dimension of the array a. Must be at least max(1, n).\nldb\nThe leading dimension of the array b. Must be at least max(1, n).\nldvl, ldvr\nThe leading dimensions of the output matrices vl and vr, respectively.\nConstraints:\nldvl≥ 1. If jobvl = 'V', ldvl≥ max(1, n).\nldvr≥ 1. If jobvr = 'V', ldvr≥ max(1, n).\nOutput Parameters\na, b\nOn exit, these arrays have been overwritten.\nalphar, alphai\nArrays, size at least max(1, n) each. Contain values that form generalized\neigenvalues in real flavors.\nSee beta.\nalpha\nArray, size at least max(1, n). Contain values that form generalized\neigenvalues in complex flavors. See beta.\nbeta\nArray, size at least max(1, n).\nFor real flavors:\nOn exit, (alphar[j] + alphai[j]*i)/beta[j], j=0,..., n - 1, are the generalized\neigenvalues.\nIf alphai[j] is zero, then the j-th eigenvalue is real; if positive, then the j-th\nand (j+1)-st eigenvalues are a complex conjugate pair, with alphai[j+1]\nnegative.\nFor complex flavors:\nOn exit, alpha[j]/beta[j], j=0,..., n - 1, are the generalized eigenvalues.\nSee also Application Notes below.\nvl, vr\nArrays:\nvl (size at least max(1, ldvl*n)). Contains the matrix of left generalized\neigenvectors VL.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1164\n\n\nIf jobvl = 'V', the left generalized eigenvectors uj are stored one after\nanother in the columns of VL, in the same order as their eigenvalues. Each\neigenvector is scaled so the largest component has abs(Re) + abs(Im) =\n1.\nIf jobvl = 'N', vl is not referenced.\nFor real flavors:\nIf the j-th eigenvalue is real,then the k-th component of the j-th left\neigenvector uj is stored in vl[(k - 1) + (j - 1)*ldvl] for column\nmajor layout and in vl[(k - 1)*ldvl + (j - 1)] for row major layout..\nIf the j-th and (j+1)-st eigenvalues form a complex conjugate pair, then for\ni = sqrt(-1), the k-th components of the j-th left eigenvector ujare\nvl[(k - 1) + (j - 1)*ldvl] + i*vl[(k - 1) + j*ldvl] for column\nmajor layout and vl[(k - 1)*ldvl + (j - 1)] + i*vl[(k - 1)*ldvl\n+ j] for row major layout. Similarly, the k-th components of left\neigenvector j+1 uj+1 are vl[(k - 1) + (j - 1)*ldvl] - i*vl[(k - 1)\n+ j*ldvl] for column major layout and vl[(k - 1)*ldvl + (j - 1)] -\ni*vl[(k - 1)*ldvl + j] for row major layout..\nFor complex flavors:\nThe k-th component of the j-th left eigenvector uj is stored in vl[(k - 1)\n+ (j - 1)*ldvl] for column major layout and in vl[(k - 1)*ldvl + (j\n- 1)] for row major layout.\nvr (size at least max(1, ldvr*n)). Contains the matrix of right generalized\neigenvectors VR.\nIf jobvr = 'V', the right generalized eigenvectors vj are stored one after\nanother in the columns of VR, in the same order as their eigenvalues. Each\neigenvector is scaled so the largest component has abs(Re) + abs(Im) = 1.\nIf jobvr = 'N', vr is not referenced.\nFor real flavors:\nIf the j-th eigenvalue is real, then The k-th component of the j-th right\neigenvector vj is stored in vr[(k - 1) + (j - 1)*ldvr] for column\nmajor layout and in vr[(k - 1)*ldvr + (j - 1)] for row major layout..\nIf the j-th and (j+1)-st eigenvalues form a complex conjugate pair, then the\nk-th components of thej-th right eigenvector vj can be computed as vr[(k\n- 1) + (j - 1)*ldvr] + i*vr[(k - 1) + j*ldvr] for column major\nlayout and vr[(k - 1)*ldvr + (j - 1)] + i*vr[(k - 1)*ldvr + j]\nfor row major layout. Similarly, the k-th components of the right\neigenvector j+1 v{j+1} can be computed as vr[(k - 1) + (j - 1)*ldvr]\n- i*vr[(k - 1) + j*ldvr] for column major layout and vr[(k -\n1)*ldvr + (j - 1)] - i*vr[(k - 1)*ldvr + j] for row major layout..\nFor complex flavors:\nThe k-th component of the j-th right eigenvector vj is stored in vr[(k - 1)\n+ (j - 1)*ldvr] for column major layout and in vr[(k - 1)*ldvr + (j\n- 1)] for row major layout.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1165\n\n\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nIf info = i, and\ni≤n: the QZ iteration failed. No eigenvectors have been calculated, but alphar[j], alphai[j] (for real flavors),\nor alpha[j] (for complex flavors), and beta[j], j=info,..., n - 1 should be correct.\ni > n: errors that usually indicate LAPACK problems:\ni = n+1: other than QZ iteration failed in hgeqz;\ni = n+2: error return from tgevc.\nApplication Notes\nThe quotients alphar[j]/beta[j] and alphai[j]/beta[j] may easily over- or underflow, and beta[j] may even be\nzero. Thus, you should avoid simply computing the ratio. However, alphar and alphai (for real flavors) or\nalpha (for complex flavors) will be always less than and usually comparable with norm(A) in magnitude, and\nbeta always less than and usually comparable with norm(B).\n?ggevx\nComputes the generalized eigenvalues, and,\noptionally, the left and/or right generalized\neigenvectors.\nSyntax\nlapack_int LAPACKE_sggevx( int matrix_layout, char balanc, char jobvl, char jobvr, char\nsense, lapack_int n, float* a, lapack_int lda, float* b, lapack_int ldb, float* alphar,\nfloat* alphai, float* beta, float* vl, lapack_int ldvl, float* vr, lapack_int ldvr,\nlapack_int* ilo, lapack_int* ihi, float* lscale, float* rscale, float* abnrm, float*\nbbnrm, float* rconde, float* rcondv );\nlapack_int LAPACKE_dggevx( int matrix_layout, char balanc, char jobvl, char jobvr, char\nsense, lapack_int n, double* a, lapack_int lda, double* b, lapack_int ldb, double*\nalphar, double* alphai, double* beta, double* vl, lapack_int ldvl, double* vr,\nlapack_int ldvr, lapack_int* ilo, lapack_int* ihi, double* lscale, double* rscale,\ndouble* abnrm, double* bbnrm, double* rconde, double* rcondv );\nlapack_int LAPACKE_cggevx( int matrix_layout, char balanc, char jobvl, char jobvr, char\nsense, lapack_int n, lapack_complex_float* a, lapack_int lda, lapack_complex_float* b,\nlapack_int ldb, lapack_complex_float* alpha, lapack_complex_float* beta,\nlapack_complex_float* vl, lapack_int ldvl, lapack_complex_float* vr, lapack_int ldvr,\nlapack_int* ilo, lapack_int* ihi, float* lscale, float* rscale, float* abnrm, float*\nbbnrm, float* rconde, float* rcondv );\nlapack_int LAPACKE_zggevx( int matrix_layout, char balanc, char jobvl, char jobvr, char\nsense, lapack_int n, lapack_complex_double* a, lapack_int lda, lapack_complex_double*\nb, lapack_int ldb, lapack_complex_double* alpha, lapack_complex_double* beta,\nlapack_complex_double* vl, lapack_int ldvl, lapack_complex_double* vr, lapack_int ldvr,\nlapack_int* ilo, lapack_int* ihi, double* lscale, double* rscale, double* abnrm,\ndouble* bbnrm, double* rconde, double* rcondv );\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1166\n\n\nInclude Files\n•\nmkl.h\nDescription\nThe routine computes for a pair of n-by-n real/complex nonsymmetric matrices (A,B), the generalized\neigenvalues, and optionally, the left and/or right generalized eigenvectors.\nOptionally also, it computes a balancing transformation to improve the conditioning of the eigenvalues and\neigenvectors (ilo, ihi, lscale, rscale, abnrm, and bbnrm), reciprocal condition numbers for the eigenvalues\n(rconde), and reciprocal condition numbers for the right eigenvectors (rcondv).\nA generalized eigenvalue for a pair of matrices (A,B) is a scalar λ or a ratio alpha / beta = λ, such that A -\nλ*B is singular. It is usually represented as the pair (alpha, beta), as there is a reasonable interpretation for\nbeta=0 and even for both being zero. The right generalized eigenvector v(j) corresponding to the\ngeneralized eigenvalue λ(j) of (A,B) satisfies\nA*v(j) = λ(j)*B*v(j).\nThe left generalized eigenvector u(j) corresponding to the generalized eigenvalue λ(j) of (A,B) satisfies\nu(j)H*A = λ(j)*u(j)H*B\nwhere u(j)H denotes the conjugate transpose of u(j).\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nbalanc\nMust be 'N', 'P', 'S', or 'B'. Specifies the balance option to be\nperformed.\nIf balanc = 'N', do not diagonally scale or permute;\nIf balanc = 'P', permute only;\nIf balanc = 'S', scale only;\nIf balanc = 'B', both permute and scale.\nComputed reciprocal condition numbers will be for the matrices after\nbalancing and/or permuting. Permuting does not change condition numbers\n(in exact arithmetic), but balancing does.\njobvl\nMust be 'N' or 'V'.\nIf jobvl = 'N', the left generalized eigenvectors are not computed;\nIf jobvl = 'V', the left generalized eigenvectors are computed.\njobvr\nMust be 'N' or 'V'.\nIf jobvr = 'N', the right generalized eigenvectors are not computed;\nIf jobvr = 'V', the right generalized eigenvectors are computed.\nsense\nMust be 'N', 'E', 'V', or 'B'. Determines which reciprocal condition\nnumber are computed.\nIf sense = 'N', none are computed;\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1167\n\n\nIf sense = 'E', computed for eigenvalues only;\nIf sense = 'V', computed for eigenvectors only;\nIf sense = 'B', computed for eigenvalues and eigenvectors.\nn\nThe order of the matrices A, B, vl, and vr (n≥ 0).\na, b\nArrays:\na (size at least max(1, lda*n)) is an array containing the n-by-n matrix A\n(first of the pair of matrices).\nb (size at least max(1, ldb*n)) is an array containing the n-by-n matrix B\n(second of the pair of matrices).\nlda\nThe leading dimension of the array a.\nMust be at least max(1, n).\nldb\nThe leading dimension of the array b.\nMust be at least max(1, n).\nldvl, ldvr\nThe leading dimensions of the output matrices vl and vr, respectively.\nConstraints:\nldvl≥ 1. If jobvl = 'V', ldvl≥ max(1, n).\nldvr≥ 1. If jobvr = 'V', ldvr≥ max(1, n).\nOutput Parameters\na, b\nOn exit, these arrays have been overwritten.\nIf jobvl = 'V' or jobvr = 'V' or both, then a contains the first part of\nthe real Schur form of the \"balanced\" versions of the input A and B, and b\ncontains its second part.\nalphar, alphai\nArrays, size at least max(1, n) each. Contain values that form generalized\neigenvalues in real flavors.\nSee beta.\nalpha\nArray, size at least max(1, n). Contain values that form generalized\neigenvalues in complex flavors. See beta.\nbeta\nArray, size at least max(1, n).\nFor real flavors:\nOn exit, (alphar[j] + alphai[j]*i)/beta[j], j=0,..., n - 1, will be the\ngeneralized eigenvalues.\nIf alphai[j] is zero, then the j-th eigenvalue is real; if positive, then the j-th\nand (j+1)-st eigenvalues are a complex conjugate pair, with alphai[j+1]\nnegative.\nFor complex flavors:\nOn exit, alpha[j]/beta[j], j=0,..., n - 1, will be the generalized eigenvalues.\nSee also Application Notes below.\nvl, vr\nArrays:\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1168\n\n\nvl (size at least max(1, ldvl*n)).\nIf jobvl = 'V', the left generalized eigenvectors u(j) are stored one after\nanother in the columns of vl, in the same order as their eigenvalues. Each\neigenvector will be scaled so the largest component have abs(Re) +\nabs(Im) = 1.\nIf jobvl = 'N', vl is not referenced.\nFor real flavors:\nIf the j-th eigenvalue is real, then k-th component of j-th left eigenvector uj\nis stored in vl[(k - 1) + (j - 1)*ldvl] for column major layout and in\nvl[(k - 1)*ldvl + (j - 1)] for row major layout..\nIf the j-th and (j+1)-st eigenvalues form a complex conjugate pair, then for\ni = sqrt(-1), the k-th components of the j-th left eigenvector uj can be\ncomputed as vl[(k - 1) + (j - 1)*ldvl] + i*vl[(k - 1) + j*ldvl]\nfor column major layout and vl[(k - 1)*ldvl + (j - 1)] + i*vl[(k -\n1)*ldvl + j] for row major layout. Similarly, the k-th components of the\nleft eigenvector j+1 uj+1 can be computed as vl[(k - 1) + (j -\n1)*ldvl] - i*vl[(k - 1) + j*ldvl] for column major layout and vl[(k\n- 1)*ldvl + (j - 1)] - i*vl[(k - 1)*ldvl + j] for row major\nlayout..\nFor complex flavors:\nThe k-th component of the j-th left eigenvector uj is stored in vl[(k - 1)\n+ (j - 1)*ldvl] for column major layout and in vl[(k - 1)*ldvl + (j\n- 1)] for row major layout.\nvr (size at least max(1, ldvr*n)).\nIf jobvr = 'V', the right generalized eigenvectors v(j) are stored one after\nanother in the columns of vr, in the same order as their eigenvalues. Each\neigenvector will be scaled so the largest component have abs(Re) +\nabs(Im) = 1.\nIf jobvr = 'N', vr is not referenced.\nFor real flavors:\nIf the j-th eigenvalue is real, then the k-th component of the j-th right\neigenvector vj is stored in vr[(k - 1) + (j - 1)*ldvr] for column\nmajor layout and in vr[(k - 1)*ldvr + (j - 1)] for row major layout..\nIf the j-th and (j+1)-st eigenvalues form a complex conjugate pair, then\nThe k-th components of the j-th right eigenvector vj can be computed as\nvr[(k - 1) + (j - 1)*ldvr] + i*vr[(k - 1) + j*ldvr] for column\nmajor layout and vr[(k - 1)*ldvr + (j - 1)] + i*vr[(k - 1)*ldvr\n+ j] for row major layout. Respectively, the k-th components of right\neigenvector j+1 vj + 1 can be computed as vr[(k - 1) + (j - 1)*ldvr]\n- i*vr[(k - 1) + j*ldvr] for column major layout and vr[(k -\n1)*ldvr + (j - 1)] - i*vr[(k - 1)*ldvr + j] for row major layout..\nFor complex flavors:\nThe k-th component of the j-th right eigenvector vj is stored in vr[(k - 1)\n+ (j - 1)*ldvr] for column major layout and in vr[(k - 1)*ldvr + (j\n- 1)] for row major layout.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1169\n\n\nilo, ihi\nilo and ihi are integer values such that on exit Ai j = 0 and Bi j = 0 if i >\nj and j = 1,..., ilo-1 or i = ihi+1,..., n.\nIf balanc = 'N' or 'S', ilo = 1 and ihi = n.\nlscale, rscale\nArrays, size at least max(1, n) each.\nlscale contains details of the permutations and scaling factors applied to the\nleft side of A and B.\nIf PL(j) is the index of the row interchanged with row j, and DL(j) is the\nscaling factor applied to row j, then\nlscale[j - 1] = PL(j), for j = 1,..., ilo-1\n= DL(j), for j = ilo,...,ihi\n= PL(j) for j = ihi+1,..., n.\nThe order in which the interchanges are made is n to ihi+1, then 1 to ilo-1.\nrscale contains details of the permutations and scaling factors applied to the\nright side of A and B.\nIf PR(j) is the index of the column interchanged with column j, and DR(j)\nis the scaling factor applied to column j, then\nrscale[j - 1] = PR(j), for j = 1,..., ilo-1\n= DR(j), for j = ilo,...,ihi\n= PR(j) for j = ihi+1,..., n.\nThe order in which the interchanges are made is n to ihi+1, then 1 to ilo-1.\nabnrm, bbnrm\nThe one-norms of the balanced matrices A and B, respectively.\nrconde, rcondv\nArrays, size at least max(1, n) each.\nIf sense = 'E', or 'B', rconde contains the reciprocal condition numbers\nof the eigenvalues, stored in consecutive elements of the array. For a\ncomplex conjugate pair of eigenvalues two consecutive elements of rconde\nare set to the same value. Thus rconde[j], rcondv[j], and the j-th columns\nof vl and vr all correspond to the same eigenpair (but not in general the j-th\neigenpair, unless all eigenpairs are selected).\nIf sense = 'N', or 'V', rconde is not referenced.\nIf sense = 'V', or 'B', rcondv contains the estimated reciprocal condition\nnumbers of the eigenvectors, stored in consecutive elements of the array.\nFor a complex eigenvector two consecutive elements of rcondv are set to\nthe same value.\nIf the eigenvalues cannot be reordered to compute , rcondv[j] is set to 0;\nthis can only occur when the true value would be very small anyway.\nIf sense = 'N', or 'E', rcondv is not referenced.\nReturn Values\nThis function returns a value info.\nIf info=0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1170\n\n\nIf info = i, and\ni≤n: the QZ iteration failed. No eigenvectors have been calculated, but alphar[j], alphai[j] (for real flavors),\nor alpha[j] (for complex flavors), and beta[j], j=info,..., n - 1 should be correct.\ni > n: errors that usually indicate LAPACK problems:\ni = n+1: other than QZ iteration failed in hgeqz;\ni = n+2: error return from tgevc.\nApplication Notes\nThe quotients alphar[j]/beta[j] and alphai[j]/beta[j] may easily over- or underflow, and beta[j] may even be\nzero. Thus, you should avoid simply computing the ratio. However, alphar and alphai (for real flavors) or\nalpha (for complex flavors) will be always less than and usually comparable with norm(A) in magnitude, and\nbeta always less than and usually comparable with norm(B).\n?ggev3\nComputes the generalized eigenvalues and the left\nand right generalized eigenvectors for a pair of\nmatrices.\nSyntax\nlapack_int LAPACKE_sggev3 (int matrix_layout, char jobvl, char jobvr, lapack_int n,\nfloat * a, lapack_int lda, float * b, lapack_int ldb, float * alphar, float * alphai,\nfloat * beta, float * vl, lapack_int ldvl, float * vr, lapack_int ldvr);\nlapack_int LAPACKE_dggev3 (int matrix_layout, char jobvl, char jobvr, lapack_int n,\ndouble * a, lapack_int lda, double * b, lapack_int ldb, double * alphar, double *\nalphai, double * beta, double * vl, lapack_int ldvl, double * vr, lapack_int ldvr);\nlapack_int LAPACKE_cggev3 (int matrix_layout, char jobvl, char jobvr, lapack_int n,\nlapack_complex_float * a, lapack_int lda, lapack_complex_float * b, lapack_int ldb,\nlapack_complex_float * alpha, lapack_complex_float * beta, lapack_complex_float * vl,\nlapack_int ldvl, lapack_complex_float * vr, lapack_int ldvr);\nlapack_int LAPACKE_zggev3 (int matrix_layout, char jobvl, char jobvr, lapack_int n,\nlapack_complex_double * a, lapack_int lda, lapack_complex_double * b, lapack_int ldb,\nlapack_complex_double * alpha, lapack_complex_double * beta, lapack_complex_double *\nvl, lapack_int ldvl, lapack_complex_double * vr, lapack_int ldvr);\nInclude Files\n•\nmkl.h\nDescription\nFor a pair of n-by-n real or complex nonsymmetric matrices (A, B), ?ggev3 computes the generalized\neigenvalues, and optionally, the left and right generalized eigenvectors.\nA generalized eigenvalue for a pair of matrices (A, B) is a scalar λ or a ratio alpha/beta = λ, such that A -\nλ*B is singular. It is usually represented as the pair (alpha,beta), as there is a reasonable interpretation for\nbeta=0, and even for both being zero.\nFor real flavors:\nThe right eigenvector vj corresponding to the eigenvalue λj of (A, B) satisfies\nA * vj = λj * B * vj.\nThe left eigenvector uj corresponding to the eigenvalue λj of (A, B) satisfies\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1171\n\n\nujH * A = λj * ujH * B\nwhere ujH is the conjugate-transpose of uj.\nFor complex flavors:\nThe right generalized eigenvector vj corresponding to the generalized eigenvalue λj of (A, B) satisfies\nA * vj = λj * B * vj.\nThe left generalized eigenvector uj corresponding to the generalized eigenvalues λj of (A, B) satisfies\nujH * A = λj * ujH * B\nwhere ujH is the conjugate-transpose of uj.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\njobvl\n= 'N': do not compute the left generalized eigenvectors;\n= 'V': compute the left generalized eigenvectors.\njobvr\n= 'N': do not compute the right generalized eigenvectors;\n= 'V': compute the right generalized eigenvectors.\nn\nThe order of the matrices A, B, VL, and VR.\nn≥ 0.\na\nArray, size (lda*n).\nOn entry, the matrix A in the pair (A, B).\nlda\nThe leading dimension of a.\nlda≥ max(1,n).\nb\nArray, size (ldb*n).\nOn entry, the matrix B in the pair (A, B).\nldb\nThe leading dimension of b.\nldb≥ max(1,n).\nldvl\nThe leading dimension of the matrix VL.\nldvl≥ 1, and if jobvl = 'V', ldvl≥n.\nldvr\nThe leading dimension of the matrix VR.\nldvr≥ 1, and if jobvr = 'V', ldvr≥n.\nOutput Parameters\na\nOn exit, a is overwritten.\nb\nOn exit, b is overwritten.\nalphar\nArray, size (n).\nalphai\nArray, size (n).\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1172\n\n\nalpha\nArray, size (n).\nbeta\nArray, size (n).\nFor real flavors:\nOn exit, (alphar[j] + alphai[j]*i)/beta[j], j=0,...,n - 1, are the\ngeneralized eigenvalues. If alphai[j - 1] is zero, then the j-th\neigenvalue is real; if positive, then the j-th and (j+1)-st eigenvalues\nare a complex conjugate pair, with alphai[j] negative.\nNote: the quotients alphar[j - 1]/beta[j - 1] and alphai[j -\n1]/beta[j - 1] can easily over- or underflow, and beta(j) might\neven be zero. Thus, you should avoid computing the ratio alpha/beta\nby simply dividing alpha by beta. However, alphar and alphai are\nalways less than and usually comparable with norm(A) in magnitude,\nand beta is always less than and usually comparable with norm(B).\nFor complex flavors:\nOn exit, alpha[j]/beta[j], j=0,...,n - 1, are the generalized\neigenvalues.\nNote: the quotients alpha[j - 1]/beta[j - 1] may easily over- or\nunderflow, and beta(j) can even be zero. Thus, you should avoid\ncomputing the ratio alpha/beta by simply dividing alpha by beta.\nHowever, alpha is always less than and usually comparable with\nnorm(A) in magnitude, and betais always less than and usually\ncomparable with norm(B).\nvl\nArray, size (ldvl*n).\nFor real flavors:\nIf jobvl = 'V', the left eigenvectors uj are stored one after another in\nthe columns of vl, in the same order as their eigenvalues. If the j-th\neigenvalue is real, then uj = the j-th column of vl. If the j-th and (j\n+1)-st eigenvalues form a complex conjugate pair, then the real part\nof uj = the j-th column of vl and the imaginary part of vj = the (j +\n1)-st column of vl.\nEach eigenvector is scaled so the largest component has abs(real\npart)+abs(imag. part)=1.\nNot referenced if jobvl = 'N'.\nFor complex flavors:\nIf jobvl = 'V', the left generalized eigenvectors uj are stored one\nafter another in the columns of vl, in the same order as their\neigenvalues.\nEach eigenvector is scaled so the largest component has abs(real\npart) + abs(imag. part) = 1.\nNot referenced if jobvl = 'N'.\nvr\nArray, size (ldvr*n).\nFor real flavors:\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1173\n\n\nIf jobvr = 'V', the right eigenvectors vj are stored one after another\nin the columns of vr, in the same order as their eigenvalues. If the j-\nth eigenvalue is real, then vj = the j-th column of vr. If the j-th and (j\n+ 1)-st eigenvalues form a complex conjugate pair, then the real part\nof vj = the j-th column of vr and the imaginary part of vj = the (j +\n1)-st column of vr.\nEach eigenvector is scaled so the largest component has abs(real\npart)+abs(imag. part)=1.\nNot referenced if jobvr = 'N'.\nFor complex flavors:\nIf jobvr = 'V', the right generalized eigenvectors vj are stored one\nafter another in the columns of vr, in the same order as their\neigenvalues. Each eigenvector is scaled so the largest component has\nabs(real part) + abs(imag. part) = 1.\nNot referenced if jobvr = 'N'.\nReturn Values\nThis function returns a value info.\n= 0: successful exit\n< 0: if info = -i, the i-th argument had an illegal value.\n=1,...,n:\nfor real flavors:\nThe QZ iteration failed. No eigenvectors have been calculated, but alphar[j], alphar[j] and beta[j] should\nbe correct for j=info,...,n - 1.\nfor complex flavors:\nThe QZ iteration failed. No eigenvectors have been calculated, but alpha[j] and beta[j] should be correct for\nj=info,...,n - 1.\n> n:\n=n + 1: other than QZ iteration failed in ?hgeqz,\n=n + 2: error return from ?tgevc.\nLAPACK Auxiliary Routines\nRoutine naming conventions, mathematical notation, and matrix storage schemes used for LAPACK auxiliary\nroutines are the same as for the driver and computational routines described in previous chapters.\n?lacgv\nConjugates a complex vector.\nSyntax\nlapack_int LAPACKE_clacgv (lapack_int n, lapack_complex_float* x, lapack_int incx);\nlapack_int LAPACKE_zlacgv (lapack_int n, lapack_complex_double* x, lapack_int incx);\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1174\n\n\nInclude Files\n•\nmkl.h\nDescription\nThe routine conjugates a complex vector x of length n and increment incx (see \"Vector Arguments in BLAS\"\nin Appendix B).\nInput Parameters\nA <datatype> placeholder, if present, is used for the C interface data types in the C interface section above.\nSee C Interface Conventions for the C interface principal conventions and type definitions.\nn\nThe length of the vector x (n≥ 0).\nx\nArray, dimension (1+(n-1)* |incx|).\nContains the vector of length n to be conjugated.\nincx\nThe spacing between successive elements of x.\nOutput Parameters\nx\nOn exit, overwritten with conjg(x).\n?lacrm\nMultiplies a complex matrix by a square real matrix.\nSyntax\ncall clacrm( m, n, a, lda, b, ldb, c, ldc, rwork )\ncall zlacrm( m, n, a, lda, b, ldb, c, ldc, rwork )\nInclude Files\n•\nmkl.h\nDescription\nThe routine performs a simple matrix-matrix multiplication of the form\nC = A*B,\nwhere A is m-by-n and complex, B is n-by-n and real, C is m-by-n and complex.\nInput Parameters\nm\nINTEGER. The number of rows of the matrix A and of the matrix C (m≥ 0).\nn\nINTEGER. The number of columns and rows of the matrix B and the number\nof columns of the matrix C\n(n≥ 0).\na\nCOMPLEX for clacrm\nDOUBLE COMPLEX for zlacrm\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1175\n\n\nArray, DIMENSION(lda, n). Contains the m-by-n matrix A.\nlda\nINTEGER. The leading dimension of the array a, lda≥max(1, m).\nb\nREAL for clacrm\nDOUBLE PRECISION for zlacrm\nArray, DIMENSION(ldb, n). Contains the n-by-n matrix B.\nldb\nINTEGER. The leading dimension of the array b, ldb≥max(1, n).\nldc\nINTEGER. The leading dimension of the output array c, ldc≥max(1, n).\nrwork\nREAL for clacrm\nDOUBLE PRECISION for zlacrm\nWorkspace array, DIMENSION(2*m*n).\nOutput Parameters\nc\nCOMPLEX for clacrm\nDOUBLE COMPLEX for zlacrm\nArray, DIMENSION (ldc, n). Contains the m-by-n matrix C.\n?syconv\nConverts a symmetric matrix given by a triangular\nmatrix factorization into two matrices and vice versa.\nSyntax\nlapack_int LAPACKE_ssyconv (int matrix_layout, char uplo, char way, lapack_int n, float\n* a, lapack_int lda, const lapack_int * ipiv, float * e);\nlapack_int LAPACKE_dsyconv (int matrix_layout, char uplo, char way, lapack_int n,\ndouble* a, lapack_int lda, const lapack_int * ipiv, double * e);\nlapack_int LAPACKE_csyconv (int matrix_layout, char uplo, char way, lapack_int n,\nlapack_complex_float * a, lapack_int lda, const lapack_int * ipiv, lapack_complex_float\n* e);\nlapack_int LAPACKE_zsyconv (int matrix_layout, char uplo, char way, lapack_int n,\nlapack_complex_double* a, lapack_int lda, const lapack_int * ipiv,\nlapack_complex_double * e);\nInclude Files\n•\nmkl.h\nDescription\nThe routine converts matrix A, which results from a triangular matrix factorization, into matrices L and D and\nvice versa. The routine returns non-diagonalized elements of D and applies or reverses permutation done\nwith the triangular matrix factorization.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1176\n\n\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major ( LAPACK_COL_MAJOR ).\nuplo\nMust be 'U' or 'L'.\nIndicates whether the details of the factorization are stored as an upper or\nlower triangular matrix:\nIf uplo = 'U': the upper triangular, A = U*D*UT.\nIf uplo = 'L': the lower triangular, A = L*D*LT.\nway\nMust be 'C' or 'R'.\nn\nThe order of matrix A; n≥ 0.\na\nArray of size max(1,lda *n).\nThe block diagonal matrix D and the multipliers used to obtain the factor U\nor L as computed by ?sytrf.\nlda\nThe leading dimension of a; lda≥ max(1, n).\nipiv\nArray, size at least max(1, n).\nDetails of the interchanges and the block structure of D, as returned\nby ?sytrf.\nOutput Parameters\ne\nArray of size max(1, n) containing the superdiagonal/subdiagonal of the\nsymmetric 1-by-1 or 2-by-2 block diagonal matrix D in L*D*LT.\nReturn Values\ninfo\nIf info = 0, the execution is successful.\nIf info < 0, the i-th parameter had an illegal value.\nIf info = -1011, memory allocation error occurred.\nSee Also\n?sytrf\n?syr\nPerforms the symmetric rank-1 update of a complex\nsymmetric matrix.\nSyntax\nlapack_int LAPACKE_csyr (int matrix_layout, char uplo, lapack_int n,\nlapack_complex_float alpha, const lapack_complex_float * x, lapack_int incx,\nlapack_complex_float * a, lapack_int lda);\nlapack_int LAPACKE_zsyr (int matrix_layout, char uplo, lapack_int n,\nlapack_complex_double alpha, const lapack_complex_double * x, lapack_int incx,\nlapack_complex_double * a, lapack_int lda);\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1177\n\n\nInclude Files\n•\nmkl.h\nDescription\nThe routine performs the symmetric rank 1 operation defined as\na := alpha*x*xH + a,\nwhere:\n•\nalpha is a complex scalar.\n•\nx is an n-element complex vector.\n•\na is an n-by-n complex symmetric matrix.\nThese routines have their real equivalents in BLAS (see ?syr in Chapter \"BLAS and Sparse BLAS Routines\").\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nuplo\nSpecifies whether the upper or lower triangular part of the array a is used:\nIf uplo = 'U' or 'u', then the upper triangular part of the array a is used.\nIf uplo = 'L' or 'l', then the lower triangular part of the array a is used.\nn\nSpecifies the order of the matrix a. The value of n must be at least zero.\nalpha\nSpecifies the scalar alpha.\nx\nArray, size at least (1 + (n - 1)*abs(incx)). Before entry, the\nincremented array x must contain the n-element vector x.\nincx\nSpecifies the increment for the elements of x. The value of incx must not be\nzero.\na\nArray, size max(1, lda*n). Before entry with uplo = 'U' or 'u', the\nleading n-by-n upper triangular part of the array a must contain the upper\ntriangular part of the symmetric matrix and the strictly lower triangular part\nof a is not referenced.\nBefore entry with uplo = 'L' or 'l', the leading n-by-n lower triangular\npart of the array a must contain the lower triangular part of the symmetric\nmatrix and the strictly upper triangular part of a is not referenced.\nlda\nSpecifies the leading dimension of a as declared in the calling\n(sub)program. The value of lda must be at least max(1,n).\nOutput Parameters\na\nWith uplo = 'U' or 'u', the upper triangular part of the array a is\noverwritten by the upper triangular part of the updated matrix.\nWith uplo = 'L' or 'l', the lower triangular part of the array a is\noverwritten by the lower triangular part of the updated matrix.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1178\n\n\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info < 0, the i-th parameter had an illegal value.\nIf info = -1011, memory allocation error occurred.\ni?max1\nFinds the index of the vector element whose real part\nhas maximum absolute value.\nSyntax\nMKL_INT icmax1(const MKL_INT*n, const MKL_Complex8*cx, const MKL_INT*incx)\nMKL_INT izmax1(const MKL_INT*n, const MKL_Complex16*cx, const MKL_INT*incx)\nInclude Files\n•\nmkl.h\nDescription\nGiven a complex vector cx, the i?max1 functions return the index of the first vector element of maximum\nabsolute value. These functions are based on the BLAS functions icamax/izamax, but using the absolute\nvalue of components. They are designed for use with clacon/zlacon.\nInput Parameters\nn\nSpecifies the number of elements in the vector cx.\ncx\nArray, size at least (1+(n-1)*abs(incx)).\nContains the input vector.\nincx\nSpecifies the spacing between successive elements of cx.\nReturn Values\nIndex of the vector element of maximum absolute value.\n?sum1\nForms the 1-norm of the complex vector using the\ntrue absolute value.\nSyntax\nfloat scsum1(const MKL_INT*n, const MKL_Complex8*cx, const MKL_INT*incx)\ndouble dzsum1(const MKL_INT*n, const MKL_Complex16*cx, const MKL_INT*incx)\nInclude Files\n•\nmkl.h\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1179\n\n\nDescription\nGiven a complex vector cx, scsum1/dzsum1 functions take the sum of the absolute values of vector elements\nand return a single/double precision result, respectively. These functions are based on scasum/dzasum from\nLevel 1 BLAS, but use the true absolute value and were designed for use with clacon/zlacon.\nInput Parameters\nn\nSpecifies the number of elements in the vector cx.\ncx\nArray, size at least (1+(n-1)*abs(incx)).\nContains the input vector whose elements will be summed.\nincx\nSpecifies the spacing between successive elements of cx (incx > 0).\nReturn Values\nSum of absolute values.\n?gelq2\nComputes the LQ factorization of a general\nrectangular matrix using an unblocked algorithm.\nSyntax\nlapack_int LAPACKE_sgelq2 (int matrix_layout, lapack_int m, lapack_int n, float* a,\nlapack_int lda, float* tau);\nlapack_int LAPACKE_dgelq2 (int matrix_layout, lapack_int m, lapack_int n, double* a,\nlapack_int lda, double * tau);\nlapack_int LAPACKE_cgelq2 (int matrix_layout, lapack_int m, lapack_int n,\nlapack_complex_float* a, lapack_int lda, lapack_complex_float* tau);\nlapack_int LAPACKE_zgelq2 (int matrix_layout, lapack_int m, lapack_int n,\nlapack_complex_double* a, lapack_int lda, lapack_complex_double* tau);\nInclude Files\n•\nmkl.h\nDescription\nThe routine computes an LQ factorization of a real/complex m-by-n matrix A as A = L*Q.\nThe routine does not form the matrix Q explicitly. Instead, Q is represented as a product of min(m, n)\nelementary reflectors :\nQ = H(k) ... H(2) H(1) (or Q = H(k)H ... H(2)HH(1)H for complex flavors), where k = min(m, n)\nEach H(i) has the form\nH(i) = I - tau*v*vT for real flavors, or\nH(i) = I - tau*v*vH for complex flavors,\nwhere tau is a real/complex scalar stored in tau(i), and v is a real/complex vector with v1:i-1 = 0 and vi =\n1.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1180\n\n\nOn exit, the j-th (i+1 ≤j≤n) component of vector v (for real functions) or its conjugate (for complex functions)\nis stored in a[i - 1 + lda*(j - 1)] for column major layout or in a[j - 1 + lda*(i - 1)] for row\nmajor layout.\nInput Parameters\nA <datatype> placeholder, if present, is used for the C interface data types in the C interface section above.\nSee C Interface Conventions for the C interface principal conventions and type definitions.\nm\nThe number of rows in the matrix A (m≥ 0).\nn\nThe number of columns in A (n≥ 0).\na\nArray, size at least max(1, lda*n) for column major and max(1, lda*m)\nfor row major layout. Array a contains the m-by-n matrix A.\nlda\nThe leading dimension of a; at least max(1, m) for column major layout and\nmax(1,n) for row major layout.\nOutput Parameters\na\nOverwritten by the factorization data as follows:\non exit, the elements on and below the diagonal of the array a contain the\nm-by-min(n,m) lower trapezoidal matrix L (L is lower triangular if n≥m); the\nelements above the diagonal, with the array tau, represent the orthogonal/\nunitary matrix Q as a product of min(n,m) elementary reflectors.\ntau\nArray, size at least max(1, min(m, n)).\nContains scalar factors of the elementary reflectors.\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nIf info = -1011, memory allocation error occurred.\n?geqr2\nComputes the QR factorization of a general\nrectangular matrix using an unblocked algorithm.\nSyntax\nlapack_int LAPACKE_sgeqr2 (int matrix_layout, lapack_int m, lapack_int n, float* a,\nlapack_int lda, float* tau);\nlapack_int LAPACKE_dgeqr2 (int matrix_layout, lapack_int m, lapack_int n, double* a,\nlapack_int lda, double* tau);\nlapack_int LAPACKE_cgeqr2 (int matrix_layout, lapack_int m, lapack_int n,\nlapack_complex_float* a, lapack_int lda, lapack_complex_float* tau);\nlapack_int LAPACKE_zgeqr2 (int matrix_layout, lapack_int m, lapack_int n,\nlapack_complex_double* a, lapack_int lda, lapack_complex_double* tau);\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1181\n\n\nInclude Files\n•\nmkl.h\nDescription\nThe routine computes a QR factorization of a real/complex m-by-n matrix A as A = Q*R.\nThe routine does not form the matrix Q explicitly. Instead, Q is represented as a product of min(m, n)\nelementary reflectors :\nQ = H(1)*H(2)* ... *H(k), where k = min(m, n)\nEach H(i) has the form\nH(i) = I - tau*v*vT for real flavors, or\nH(i) = I - tau*v*vH for complex flavors\nwhere tau is a real/complex scalar stored in tau[i], and v is a real/complex vector with v1:i-1 = 0 and vi =\n1.\nOn exit, vi+1:m is stored in a(i+1:m, i).\nInput Parameters\nA <datatype> placeholder, if present, is used for the C interface data types in the C interface section above.\nSee C Interface Conventions for the C interface principal conventions and type definitions.\nm\nThe number of rows in the matrix A (m≥ 0).\nn\nThe number of columns in A (n≥ 0).\na\nArray, size at least max(1, lda*n) for column major and max(1, lda*m)\nfor row major layout. Array a contains the m-by-n matrix A.\nlda\nThe leading dimension of a; at least max(1, m) for column major layout and\nmax(1,n) for row major layout.\nOutput Parameters\na\nOverwritten by the factorization data as follows:\non exit, the elements on and above the diagonal of the array a contain the\nmin(n,m)-by-n upper trapezoidal matrix R (R is upper triangular if m≥n); the\nelements below the diagonal, with the array tau, represent the orthogonal/\nunitary matrix Q as a product of elementary reflectors.\ntau\nArray, size at least max(1, min(m, n)).\nContains scalar factors of the elementary reflectors.\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nIf info = -1011, memory allocation error occurred.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1182\n\n\n?geqrt2\nComputes a QR factorization of a general real or\ncomplex matrix using the compact WY representation\nof Q.\nSyntax\nlapack_int LAPACKE_sgeqrt2 (int matrix_layout, lapack_int m, lapack_int n, float * a,\nlapack_int lda, float * t, lapack_int ldt );\nlapack_int LAPACKE_dgeqrt2 (int matrix_layout, lapack_int m, lapack_int n, double * a,\nlapack_int lda, double * t, lapack_int ldt );\nlapack_int LAPACKE_cgeqrt2 (int matrix_layout, lapack_int m, lapack_int n,\nlapack_complex_float * a, lapack_int lda, lapack_complex_float * t, lapack_int ldt );\nlapack_int LAPACKE_zgeqrt2 (int matrix_layout, lapack_int m, lapack_int n,\nlapack_complex_double * a, lapack_int lda, lapack_complex_double * t, lapack_int ldt );\nInclude Files\n•\nmkl.h\nDescription\nThe strictly lower triangular matrix V contains the elementary reflectors H(i) in the ith column below the\ndiagonal. For example, if m=5 and n=3, the matrix V is\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1183\n\n\nwhere vi represents the vector that defines H(i). The vectors are returned in the lower triangular part of array\na.\nNOTE\nThe 1s along the diagonal of V are not stored in a.\nThe block reflector H is then given by\nH = I - V*T*VT for real flavors, and\nH = I - V*T*VH for complex flavors,\nwhere VT is the transpose and VH is the conjugate transpose of V.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1184\n\n\nInput Parameters\nm\nThe number of rows in the matrix A (m ≥ n).\nn\nThe number of columns in A (n ≥ 0).\na\nArray, size at least max(1, lda*n) for column major and max(1, lda*m)\nfor row major layout. Array a contains the m-by-n matrix A.\nlda\nThe leading dimension of a; at least max(1, m) for column major layout and\nmax(1,n) for row major layout.\nldt\nThe leading dimension of t; at least max(1, n).\nOutput Parameters\na\nOverwritten by the factorization data as follows:\nThe elements on and above the diagonal of the array contain the n-by-n\nupper triangular matrix R. The elements below the diagonal are the\ncolumns of V.\nt\nArray, size at least max(1, ldt*n).\nThe n-by-n upper triangular factor of the block reflector. The elements on\nand above the diagonal contain the block reflector T. The elements below\nthe diagonal are not used.\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info < 0 and info = -i, the ith argument had an illegal value.\nIf info = -1011, memory allocation error occurred.\n?geqrt3\nRecursively computes a QR factorization of a general\nreal or complex matrix using the compact WY\nrepresentation of Q.\nSyntax\nlapack_int LAPACKE_sgeqrt3 (int matrix_layout , lapack_int m , lapack_int n , float *\na , lapack_int lda , float * t , lapack_int ldt );\nlapack_int LAPACKE_dgeqrt3 (int matrix_layout , lapack_int m , lapack_int n , double *\na , lapack_int lda , double * t , lapack_int ldt );\nlapack_int LAPACKE_cgeqrt3 (int matrix_layout , lapack_int m , lapack_int n ,\nlapack_complex_float * a , lapack_int lda , lapack_complex_float * t , lapack_int\nldt );\nlapack_int LAPACKE_zgeqrt3 (int matrix_layout , lapack_int m , lapack_int n ,\nlapack_complex_double * a , lapack_int lda , lapack_complex_double * t , lapack_int\nldt );\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1185\n\n\nInclude Files\n•\nmkl.h\nDescription\nThe strictly lower triangular matrix V contains the elementary reflectors H(i) in the ith column below the\ndiagonal. For example, if m=5 and n=3, the matrix V is\nwhere vi represents one of the vectors that define H(i). The vectors are returned in the lower part of\ntriangular array a.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1186\n\n\nNOTE\nThe 1s along the diagonal of V are not stored in a.\nThe block reflector H is then given by\nH = I - V*T*VT for real flavors, and\nH = I - V*T*VH for complex flavors,\nwhere VT is the transpose and VHis the conjugate transpose of V.\nInput Parameters\nm\nThe number of rows in the matrix A (m ≥ n).\nn\nThe number of columns in A (n ≥ 0).\na\nArray, size at least max(1, lda*n) for column major and max(1, lda*m)\nfor row major layout. Array a contains the m-by-n matrix A.\nlda\nThe leading dimension of a; at least max(1, m) for column major layout and\nmax(1,n) for row major layout.\nldt\nThe leading dimension of t; at least max(1, n).\nOutput Parameters\na\nThe elements on and above the diagonal of the array contain the n-by-n\nupper triangular matrix R. The elements below the diagonal are the\ncolumns of V.\nt\nArray, size ldt by n.\nThe n-by-n upper triangular factor of the block reflector. The elements on\nand above the diagonal contain the block reflector T. The elements below\nthe diagonal are not used.\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info < 0 and info = -i, the ith argument had an illegal value.\nIf info = -1011, memory allocation error occurred.\n?getf2\nComputes the LU factorization of a general m-by-n\nmatrix using partial pivoting with row interchanges\n(unblocked algorithm).\nSyntax\nlapack_int LAPACKE_sgetf2 (int matrix_layout, lapack_int m, lapack_int n, float* a,\nlapack_int lda, lapack_int * ipiv);\nlapack_int LAPACKE_dgetf2 (int matrix_layout, lapack_int m, lapack_int n, double* a,\nlapack_int lda, lapack_int * ipiv);\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1187\n\n\nlapack_int LAPACKE_cgetf2 (int matrix_layout, lapack_int m, lapack_int n,\nlapack_complex_float* a, lapack_int lda, lapack_int * ipiv);\nlapack_int LAPACKE_zgetf2 (int matrix_layout, lapack_int m, lapack_int n,\nlapack_complex_double* a, lapack_int lda, lapack_int * ipiv);\nInclude Files\n•\nmkl.h\nDescription\nThe routine computes the LU factorization of a general m-by-n matrix A using partial pivoting with row\ninterchanges. The factorization has the form\nA = P*L*U\nwhere p is a permutation matrix, L is lower triangular with unit diagonal elements (lower trapezoidal if m >\nn) and U is upper triangular (upper trapezoidal if m < n).\nInput Parameters\nA <datatype> placeholder, if present, is used for the C interface data types in the C interface section above.\nSee C Interface Conventions for the C interface principal conventions and type definitions.\nm\nThe number of rows in the matrix A (m≥ 0).\nn\nThe number of columns in A (n≥ 0).\na\nArray, size at least max(1, lda*n) for column major and max(1, lda*m)\nfor row major layout. Array a contains the m-by-n matrix A.\nlda\nThe leading dimension of a; at least max(1, m) for column major layout and\nmax(1,n) for row major layout.\nOutput Parameters\na\nOverwritten by L and U. The unit diagonal elements of L are not stored.\nipiv\nArray, size at least max(1,min(m,n)).\nThe pivot indices: for 1 ≤ i ≤ n, row i was interchanged with row ipiv(i).\nReturn Values\nThis function returns a value info.\nIf info = -i, the i-th parameter had an illegal value.\nIf info = i >0, uii is 0. The factorization has been completed, but U is exactly singular. Division by 0 will\noccur if you use the factor U for solving a system of linear equations.\nIf info = -1011, memory allocation error occurred.\n?lacn2\nEstimates the 1-norm of a square matrix, using\nreverse communication for evaluating matrix-vector\nproducts.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1188\n\n\nSyntax\nC:\nlapack_int LAPACKE_slacn2 (lapack_int n, float * v, float * x, lapack_int * isgn, float\n* est, lapack_int * kase, lapack_int * isave);\nlapack_int LAPACKE_clacn2 (lapack_int n, lapack_complex_float * v, lapack_complex_float\n* x, float * est, lapack_int * kase, lapack_int * isave);\nlapack_int LAPACKE_dlacn2 (lapack_int n, double * v, double * x, lapack_int * isgn,\ndouble * est, lapack_int * kase, lapack_int * isave);\nlapack_int LAPACKE_zlacn2 (lapack_int n, lapack_complex_double * v,\nlapack_complex_double * x, double * est, lapack_int * kase, lapack_int * isave);\nInclude Files\n•\nmkl.h\nDescription\nThe routine estimates the 1-norm of a square, real or complex matrix A. Reverse communication is used for\nevaluating matrix-vector products.\nInput Parameters\nn\nThe order of the matrix A (n≥ 1).\nv, x\nArrays, size (n) each.\nv is a workspace array.\nx is used as input after an intermediate return.\nisgn\nWorkspace array, size (n), used with real flavors only.\nest\nOn entry with kase set to 1 or 2, and isave(1) = 1, est must be\nunchanged from the previous call to the routine.\nkase\nOn the initial call to the routine, kase must be set to 0.\nisave\nArray, size (3).\nContains variables from the previous call to the routine.\nOutput Parameters\nest\nAn estimate (a lower bound) for norm(A).\nkase\nOn an intermediate return, kase is set to 1 or 2, indicating whether x is\noverwritten by A*x or AT*x for real flavors and A*x or AH*x for complex\nflavors.\nOn the final return, kase is set to 0.\nv\nOn the final return, v = A*w, where est = norm(v)/norm(w) (w is not\nreturned).\nx\nOn an intermediate return, x is overwritten by\nA*x, if kase = 1,\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1189\n\n\nAT*x, if kase = 2 (for real flavors),\nAH*x, if kase = 2 (for complex flavors),\nand the routine must be re-called with all the other parameters unchanged.\nisave\nThis parameter is used to save variables between calls to the routine.\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\n?lacpy\nCopies all or part of one two-dimensional array to\nanother.\nSyntax\nlapack_int LAPACKE_slacpy (int matrix_layout, char uplo, lapack_int m, lapack_int n,\nconst float* a, lapack_int lda, float* b, lapack_int ldb);\nlapack_int LAPACKE_dlacpy (int matrix_layout, char uplo, lapack_int m, lapack_int n,\nconst double* a, lapack_int lda, double* b, lapack_int ldb);\nlapack_int LAPACKE_clacpy (int matrix_layout, char uplo, lapack_int m, lapack_int n,\nconst lapack_complex_float* a, lapack_int lda, lapack_complex_float* b, lapack_int\nldb);\nlapack_int LAPACKE_zlacpy (int matrix_layout, char uplo, lapack_int m, lapack_int n,\nconst lapack_complex_double* a, lapack_int lda, lapack_complex_double* b, lapack_int\nldb);\nInclude Files\n•\nmkl.h\nDescription\nThe routine copies all or part of a two-dimensional matrix A to another matrix B.\nInput Parameters\nA <datatype> placeholder, if present, is used for the C interface data types in the C interface section above.\nSee C Interface Conventions for the C interface principal conventions and type definitions.\nuplo\nSpecifies the part of the matrix A to be copied to B.\nIf uplo = 'U', the upper triangular part of A;\nif uplo = 'L', the lower triangular part of A.\nOtherwise, all of the matrix A is copied.\nm\nThe number of rows in the matrix A (m≥ 0).\nn\nThe number of columns in A (n≥ 0).\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1190\n\n\na\nArray, size at least max(1, lda*n) for column major and max(1, lda*m)\nfor row major layout. A contains the m-by-n matrix A.\nIf uplo = 'U', only the upper triangle or trapezoid is accessed; if uplo =\n'L', only the lower triangle or trapezoid is accessed.\nlda\nThe leading dimension of a; lda≥max(1,m) for column major layout\nand max(1,n) for row major layout.\nldb\nThe leading dimension of the output array b; ldb≥ max(1, m)for column\nmajor layout and max(1,n) for row major layout.\nOutput Parameters\nb\nArray, size at least max(1, ldb*n) for column major and max(1, ldb*m)\nfor row major layout. Array a contains the m-by-n matrix B.\nOn exit, B = A in the locations specified by uplo.\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\n?lakf2\nForms a matrix containing Kronecker products\nbetween the given matrices.\nSyntax\nvoid slakf2 (lapack_int *m, lapack_int *n, float *a, lapack_int *lda, float *b, float\n*d, float *e, float *z, lapack_int *ldz);\nvoid dlakf2 (lapack_int *m, lapack_int *n, double *a, lapack_int *lda, double *b, double\n*d, double *e, double *z, lapack_int *ldz);\nvoid clakf2 (lapack_int *m, lapack_int *n, lapack_complex *a, lapack_int *lda,\nlapack_complex *b, lapack_complex *d, lapack_complex *e, lapack_complex *z, lapack_int\n*ldz);\nvoid zlakf2 (lapack_int *m, lapack_int *n, lapack_complex_double *a, lapack_int *lda,\nlapack_complex_double *b, lapack_complex_double *d, lapack_complex_double *e,\nlapack_complex_double *z, lapack_int *ldz);\nInclude Files\n•\nmkl.h\nDescription\nThe routine ?lakf2 forms the 2*m*n by 2*m*n matrix Z.\n,\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1191\n\n\nwhere In is the identity matrix of size n and XT is the transpose of X. kron(X, Y) is the Kronecker product\nbetween the matrices X and Y.\nInput Parameters\nm\nSize of matrix, m≥ 1\nn\nSize of matrix, n≥ 1\na\nArray, size lda-by-n. The matrix A in the output matrix Z.\nlda\nThe leading dimension of a, b, d, and e. lda≥m+n.\nb\nArray, size lda by n. Matrix used in forming the output matrix Z.\nd\nArray, size lda by m. Matrix used in forming the output matrix Z.\ne\nArray, size lda by n. Matrix used in forming the output matrix Z.\nldz\nThe leading dimension of Z. ldz≥ 2* m*n.\nOutput Parameters\nz\nArray, size ldz-by-2*m*n. The resultant Kronecker m*n*2 -by-m*n*2\nmatrix.\n?lange\nReturns the value of the 1-norm, Frobenius norm,\ninfinity-norm, or the largest absolute value of any\nelement of a general rectangular matrix.\nSyntax\nfloat LAPACKE_slange (int matrix_layout, char norm, lapack_int m, lapack_int n, const\nfloat * a, lapack_int lda);\ndouble LAPACKE_dlange (int matrix_layout, char norm, lapack_int m, lapack_int n, const\ndouble * a, lapack_int lda);\nfloat LAPACKE_clange (int matrix_layout, char norm, lapack_int m, lapack_int n, const\nlapack_complex_float * a, lapack_int lda);\ndouble LAPACKE_zlange (int matrix_layout, char norm, lapack_int m, lapack_int n, const\nlapack_complex_double * a, lapack_int lda);\nInclude Files\n•\nmkl.h\nDescription\nThe function ?lange returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the\nelement of largest absolute value of a real/complex matrix A.\nInput Parameters\nA <datatype> placeholder, if present, is used for the C interface data types in the C interface section above.\nSee C Interface Conventions for the C interface principal conventions and type definitions.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1192\n\n\nnorm\nSpecifies the value to be returned by the routine:\n= 'M' or 'm': val = max(abs(Aij)), largest absolute value of the matrix\nA.\n= '1' or 'O' or 'o': val = norm1(A), 1-norm of the matrix A\n(maximum column sum),\n= 'I' or 'i': val = normI(A), infinity norm of the matrix A (maximum\nrow sum),\n= 'F', 'f', 'E' or 'e': val = normF(A), Frobenius norm of the matrix\nA (square root of sum of squares).\nm\nThe number of rows of the matrix A.\nm≥ 0. When m = 0, ?lange is set to zero.\nn\nThe number of columns of the matrix A.\nn≥ 0. When n = 0, ?lange is set to zero.\na\nArray, size at least max(1, lda*n) for column major and max(1, lda*m)\nfor row major layout. Array a contains the m-by-n matrix A.\nlda\nThe leading dimension of the array a.\nlda≥ max(n,1) for column major layout and max(1,n) for row major\nlayout.\n?lansy\nReturns the value of the 1-norm, or the Frobenius\nnorm, or the infinity norm, or the element of largest\nabsolute value of a real/complex symmetric matrix.\nSyntax\nfloat LAPACKE_slansy (int matrix_layout, char norm, char uplo, lapack_int n, const\nfloat * a, lapack_int lda);\ndouble LAPACKE_dlansy (int matrix_layout, char norm, char uplo, lapack_int n, const\ndouble * a, lapack_int lda);\nfloat LAPACKE_clansy (int matrix_layout, char norm, char uplo, lapack_int n, const\nlapack_complex_float * a, lapack_int lda);\ndouble LAPACKE_zlansy (int matrix_layout, char norm, char uplo, lapack_int n, const\nlapack_complex_double * a, lapack_int lda);\nInclude Files\n•\nmkl.h\nDescription\nThe function ?lansy returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the\nelement of largest absolute value of a real/complex symmetric matrix A.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1193\n\n\nInput Parameters\nA <datatype> placeholder, if present, is used for the C interface data types in the C interface section above.\nSee C Interface Conventions for the C interface principal conventions and type definitions.\nnorm\nSpecifies the value to be returned by the routine:\n= 'M' or 'm': val = max(abs(Aij)), largest absolute value of the matrix\nA.\n= '1' or 'O' or 'o': val = norm1(A), 1-norm of the matrix A\n(maximum column sum),\n= 'I' or 'i': val = normI(A), infinity norm of the matrix A (maximum\nrow sum),\n= 'F', 'f', 'E' or 'e': val = normF(A), Frobenius norm of the matrix\nA (square root of sum of squares).\nuplo\nSpecifies whether the upper or lower triangular part of the symmetric\nmatrix A is to be referenced.\n= 'U': Upper triangular part of A is referenced.\n= 'L': Lower triangular part of A is referenced\nn\nThe order of the matrix A. n≥ 0. When n = 0, ?lansy is set to zero.\na\nArray, size at least max(1,lda*n). The symmetric matrix A.\nIf uplo = 'U', the leading n-by-n upper triangular part of a contains the\nupper triangular part of the matrix A, and the strictly lower triangular part\nof a is not referenced.\nIf uplo = 'L', the leading n-by-n lower triangular part of a contains the\nlower triangular part of the matrix A, and the strictly upper triangular part\nof a is not referenced.\nlda\nThe leading dimension of the array a.\nlda≥ max(n,1).\n?lanhe\nReturns the value of the 1-norm, or the Frobenius\nnorm, or the infinity norm, or the element of largest\nabsolute value of a complex Hermitian matrix.\nSyntax\nfloat LAPACKE_clanhe (int matrix_layout, char norm, char uplo, lapack_int n, const\nlapack_complex_float * a, lapack_int lda);\ndouble LAPACKE_zlanhe (int matrix_layout, char norm, char uplo, lapack_int n, const\nlapack_complex_double * a, lapack_int lda);\nInclude Files\n•\nmkl.h\nDescription\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1194\n\n\nThe function ?lanhe returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the\nelement of largest absolute value of a complex Hermitian matrix A.\nInput Parameters\nA <datatype> placeholder, if present, is used for the C interface data types in the C interface section above.\nSee C Interface Conventions for the C interface principal conventions and type definitions.\nnorm\nSpecifies the value to be returned by the routine:\n= 'M' or 'm': val = max(abs(Aij)), largest absolute value of the matrix A.\n= '1' or 'O' or 'o': val = norm1(A), 1-norm of the matrix A (maximum\ncolumn sum),\n= 'I' or 'i': val = normI(A), infinity norm of the matrix A (maximum\nrow sum),\n= 'F', 'f', 'E' or 'e': val = normF(A), Frobenius norm of the matrix A\n(square root of sum of squares).\nuplo\nSpecifies whether the upper or lower triangular part of the Hermitian matrix\nA is to be referenced.\n= 'U': Upper triangular part of A is referenced.\n= 'L': Lower triangular part of A is referenced\nn\nThe order of the matrix A. n≥ 0. When n = 0, ?lanhe is set to zero.\na\nArray, size at least max(1, lda*n). The Hermitian matrix A.\nIf uplo = 'U', the leading n-by-n upper triangular part of a contains the\nupper triangular part of the matrix A, and the strictly lower triangular part\nof a is not referenced.\nIf uplo = 'L', the leading n-by-n lower triangular part of a contains the\nlower triangular part of the matrix A, and the strictly upper triangular part\nof a is not referenced.\nlda\nThe leading dimension of the array a.\nlda≥ max(n,1).\n?lantr\nReturns the value of the 1-norm, or the Frobenius\nnorm, or the infinity norm, or the element of largest\nabsolute value of a trapezoidal or triangular matrix.\nSyntax\nfloat LAPACKE_slantr (char * norm, char * uplo, char * diag, lapack_int * m, lapack_int\n* n, const float * a, lapack_int * lda, float * work);\ndouble LAPACKE_dlantr (char * norm, char * uplo, char * diag, lapack_int * m,\nlapack_int * n, const double * a, lapack_int * lda, double * work);\nfloat LAPACKE_clantr (char * norm, char * uplo, char * diag, lapack_int * m, lapack_int\n* n, const lapack_complex_float * a, lapack_int * lda, float * work);\ndouble LAPACKE_zlantr (char * norm, char * uplo, char * diag, lapack_int * m,\nlapack_int * n, const lapack_complex_double * a, lapack_int * lda, double * work);\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1195\n\n\nInclude Files\n•\nmkl.h\nDescription\nThe function ?lantr returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the\nelement of largest absolute value of a trapezoidal or triangular matrix A.\nInput Parameters\nA <datatype> placeholder, if present, is used for the C interface data types in the C interface section above.\nSee C Interface Conventions for the C interface principal conventions and type definitions.\nnorm\nSpecifies the value to be returned by the routine:\n= 'M' or 'm': val = max(abs(Aij)), largest absolute value of the matrix\nA.\n= '1' or 'O' or 'o': val = norm1(A), 1-norm of the matrix A\n(maximum column sum),\n= 'I' or 'i': val = normI(A), infinity norm of the matrix A (maximum\nrow sum),\n= 'F', 'f', 'E' or 'e': val = normF(A), Frobenius norm of the matrix\nA (square root of sum of squares).\nuplo\nSpecifies whether the matrix A is upper or lower trapezoidal.\n= 'U': Upper trapezoidal\n= 'L': Lower trapezoidal.\nNote that A is triangular instead of trapezoidal if m = n.\ndiag\nSpecifies whether or not the matrix A has unit diagonal.\n= 'N': Non-unit diagonal\n= 'U': Unit diagonal.\nm\nThe number of rows of the matrix A. m≥ 0, and if uplo = 'U', m ≤ n.\nWhen m = 0, ?lantr is set to zero.\nn\nThe number of columns of the matrix A. n≥ 0, and if uplo = 'L', n ≤ m.\nWhen n = 0, ?lantr is set to zero.\na\nArray, size at least max(1, lda*n) for column major and max(1, lda*m)\nfor row major layout.\nThe trapezoidal matrix A (A is triangular if m = n).\nIf uplo = 'U', the leading m-by-n upper trapezoidal part of the array a\ncontains the upper trapezoidal matrix, and the strictly lower triangular part\nof A is not referenced.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1196\n\n\nIf uplo = 'L', the leading m-by-n lower trapezoidal part of the array a\ncontains the lower trapezoidal matrix, and the strictly upper triangular part\nof A is not referenced. Note that when diag = 'U', the diagonal elements\nof A are not referenced and are assumed to be one.\nlda\nThe leading dimension of the array a.\nlda≥ max(m,1)for column major layout and ≥max(1,n) for row major\nlayout.\nLAPACKE_set_nancheck\nTurns NaN checking off or on\nLAPACKE_set_nancheck(int flag);\nDescription\nThe routine sets a value for the LAPACKE NaN checking flag, which indicates whether or not LAPACKE\nroutines check input matrices for NaNs.\nInput Parameters\nflag\nIf flag= 0, NaN checking is turned OFF. Otherwise, it is turned ON.\nLAPACKE_get_nancheck\nGets the current NaN checking flag, which indicates\nwhether NaN checking has been turned off or on.\nint flag = LAPACKE_get_nancheck ();\nDescription\nThe function returns the current value for the LAPACKE NaN checking flag, which indicates whether or not\nLAPACKE routines check input matrices for NaNs.\nReturn Value\nAn integer value is returned which indicates the current NaN checking status.\nThe returned flag value is either 0 (OFF) or 1 (ON), even though any integer value can be used as an input\nparameter for LAPACKE_set_nancheck.\nFor example, the following code turns on NaN checking:\nLAPACKE_set_nancheck(100);\nint flag = LAPACKE_get_nancheck(); // flag==1, not 100.\n?lapmr\nRearranges rows of a matrix as specified by a\npermutation vector.\nSyntax\nlapack_int LAPACKE_slapmr (int matrix_layout, lapack_logical forwrd, lapack_int m,\nlapack_int n, float* x, lapack_int ldx, lapack_int * k);\nlapack_int LAPACKE_dlapmr (int matrix_layout, lapack_logical forwrd, lapack_int m,\nlapack_int n, double* x, lapack_int ldx, lapack_int * k);\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1197\n\n\nlapack_int LAPACKE_clapmr (int matrix_layout, lapack_logical forwrd, lapack_int m,\nlapack_int n, lapack_complex_float* x, lapack_int ldx, lapack_int * k);\nlapack_int LAPACKE_zlapmr (int matrix_layout, lapack_logical forwrd, lapack_int m,\nlapack_int n, lapack_complex_double* x, lapack_int ldx, lapack_int * k);\nInclude Files\n•\nmkl.h\nDescription\nThe ?lapmr routine rearranges the rows of the m-by-n matrix X as specified by the permutation k[0],\nk[1], ... , k[m-1] of the integers 1,...,m.\nIf forwrd is true, forward permutation:\nX(k[i-1],:) is moved to X{i,:) for i= 1,2,...,m.\nIf forwrd is false, backward permutation:\nX{i,:) is moved to X(k[i-1,:) for i = 1,2,...,m.\nInput Parameters\nA <datatype> placeholder, if present, is used for the C interface data types in the C interface section above.\nSee C Interface Conventions for the C interface principal conventions and type definitions.\nforwrd\nIf forwrd is true, forward permutation.\nIf forwrd is false, backward permutation.\nm\nThe number of rows of the matrix X. m≥ 0.\nn\nThe number of columns of the matrix X. n≥ 0.\nx\nArray, size at least max(1, ldx*n) for column major and max(1, ldx*m)\nfor row major layout. On entry, the m-by-n matrix X.\nldx\nThe leading dimension of the array X, ldx≥ max(1,m)for column major\nlayout and ldx≥ max(1,n) for row major layout.\nk\nArray, size (m). On entry, k contains the permutation vector and is used as\ninternal workspace.\nOutput Parameters\nx\nOn exit, x contains the permuted matrix X.\nk\nOn exit, k is reset to its original value.\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nIf info = -1011, memory allocation error occurred.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1198\n\n\nSee Also\n?lapmt\nPerforms a forward or backward permutation of the\ncolumns of a matrix.\nSyntax\nlapack_int LAPACKE_slapmt (int matrix_layout, lapack_logical forwrd, lapack_int m,\nlapack_int n, float * x, lapack_int ldx, lapack_int * k);\nlapack_int LAPACKE_dlapmt (int matrix_layout, lapack_logical forwrd, lapack_int m,\nlapack_int n, double * x, lapack_int ldx, lapack_int * k);\nlapack_int LAPACKE_clapmt (int matrix_layout, lapack_logical forwrd, lapack_int m,\nlapack_int n, lapack_complex_float * x, lapack_int ldx, lapack_int * k);\nlapack_int LAPACKE_zlapmt (int matrix_layout, lapack_logical forwrd, lapack_int m,\nlapack_int n, lapack_complex_double * x, lapack_int ldx, lapack_int * k);\nInclude Files\n•\nmkl.h\nDescription\nThe routine ?lapmt rearranges the columns of the m-by-n matrix X as specified by the permutation k[i -\n1]for i = 1,...,n.\nIf forwrd≠ 0, forward permutation:\nX(*,k(j)) is moved to X(*,j) for j=1,2,...,n.\nIf forwrd = 0, backward permutation:\nX(*,j) is moved to X(*,k(j)) for j = 1,2,...,n.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nforwrd\nIf forwrd≠ 0, forward permutation\nIf forwrd = 0, backward permutation\nm\nThe number of rows of the matrix X. m≥ 0.\nn\nThe number of columns of the matrix X. n≥ 0.\nx\nArray, size ldx*n. On entry, the m-by-n matrix X.\nldx\nThe leading dimension of the array x, ldx≥ max(1,m).\nk\nArray, size (n). On entry, k contains the permutation vector and is used as\ninternal workspace.\nOutput Parameters\nx\nOn exit, x contains the permuted matrix X.\nk\nOn exit, k is reset to its original value.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1199\n\n\nSee Also\n?lapmr\n?lapy2\nReturns sqrt(x2+y2).\nSyntax\nfloat LAPACKE_slapy2 (floatx, floaty);\ndouble LAPACKE_dlapy2 (doublex, doubley);\nInclude Files\n•\nmkl.h\nDescription\nThe function ?lapy2 returns sqrt(x2+y2), avoiding unnecessary overflow or harmful underflow.\nInput Parameters\nA <datatype> placeholder, if present, is used for the C interface data types in the C interface section above.\nSee C Interface Conventions for the C interface principal conventions and type definitions.\nx, y\nSpecify the input values x and y.\nReturn Values\nThe function returns a value val.\nIf val=-1D0, the first argument was NaN.\nIf val=-2D0, the second argument was NaN.\n?lapy3\nReturns sqrt(x2+y2+z2).\nSyntax\nfloat LAPACKE_slapy3 (floatx, floaty, floatz);\ndouble LAPACKE_dlapy3 (double x, doubley, doublez);\nInclude Files\n•\nmkl.h\nDescription\nThe function ?lapy3 returns sqrt(x2+y2+z2), avoiding unnecessary overflow or harmful underflow.\nInput Parameters\nA <datatype> placeholder, if present, is used for the C interface data types in the C interface section above.\nSee C Interface Conventions for the C interface principal conventions and type definitions.\nx, y, z\nSpecify the input values x, y and z.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1200\n\n\nReturn Values\nThis function returns a value val.\nIf val = -1D0, the first argument was NaN.\nIf val = -2D0, the second argument was NaN.\nIf val = -3D0, the third argument was NaN.\n?laran\nReturns a random real number from a uniform\ndistribution.\nSyntax\nfloat slaran (lapack_int *iseed);\ndouble dlaran (lapack_int *iseed);\nDescription\nThe ?laran routine returns a random real number from a uniform (0,1) distribution. This routine uses a\nmultiplicative congruential method with modulus 248 and multiplier 33952834046453. 48-bit integers are\nstored in four integer array elements with 12 bits per element. Hence the routine is portable across machines\nwith integers of 32 bits or more.\nInput Parameters\niseed\nArray, size 4. On entry, the seed of the random number generator. The\narray elements must be between 0 and 4095, and iseed[3] must be odd.\nOutput Parameters\niseed\nOn exit, the seed is updated.\nReturn Values\nThe function returns a random number.\n?larfb\nApplies a block reflector or its transpose/conjugate-\ntranspose to a general rectangular matrix.\nSyntax\nlapack_int LAPACKE_slarfb (int matrix_layout , char side , char trans , char direct ,\nchar storev , lapack_int m , lapack_int n , lapack_int k , const float * v , lapack_int\nldv , const float * t , lapack_int ldt , float * c , lapack_int ldc );\nlapack_int LAPACKE_dlarfb (int matrix_layout , char side , char trans , char direct ,\nchar storev , lapack_int m , lapack_int n , lapack_int k , const double * v ,\nlapack_int ldv , const double * t , lapack_int ldt , double * c , lapack_int\nldc );lapack_int LAPACKE_clarfb (int matrix_layout , char side , char trans , char\ndirect , char storev , lapack_int m , lapack_int n , lapack_int k , const\nlapack_complex_float * v , lapack_int ldv , const lapack_complex_float * t , lapack_int\nldt , lapack_complex_float * c , lapack_int ldc );\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1201\n\n\nlapack_int LAPACKE_zlarfb (int matrix_layout , char side , char trans , char direct ,\nchar storev , lapack_int m , lapack_int n , lapack_int k , const lapack_complex_double\n* v , lapack_int ldv , const lapack_complex_double * t , lapack_int ldt ,\nlapack_complex_double * c , lapack_int ldc );\nInclude Files\n•\nmkl.h\nDescription\nThe real flavors of the routine ?larfb apply a real block reflector H or its transpose HT to a real m-by-n\nmatrix C from either left or right.\nThe complex flavors of the routine ?larfb apply a complex block reflector H or its conjugate transpose HH to\na complex m-by-n matrix C from either left or right.\nInput Parameters\nA <datatype> placeholder, if present, is used for the C interface data types in the C interface section above.\nSee C Interface Conventions for the C interface principal conventions and type definitions.\nside\nIf side = 'L': apply H or HT for real flavors and H or HH for complex\nflavors from the left.\nIf side = 'R': apply H or HT for real flavors and H or HH for complex\nflavors from the right.\ntrans\nIf trans = 'N': apply H (No transpose).\nIf trans = 'C': apply HH (Conjugate transpose).\nIf trans = 'T': apply HT (Transpose).\ndirect\nIndicates how H is formed from a product of elementary reflectors\nIf direct = 'F': H = H(1)*H(2)*. . . *H(k) (forward)\nIf direct = 'B': H = H(k)* . . . H(2)*H(1) (backward)\nstorev\nIndicates how the vectors which define the elementary reflectors are\nstored:\nIf storev = 'C': Column-wise\nIf storev = 'R': Row-wise\nm\nThe number of rows of the matrix C.\nn\nThe number of columns of the matrix C.\nk\nThe order of the matrix T (equal to the number of elementary reflectors\nwhose product defines the block reflector).\nv\nThe size limitations depend on values of parameters storev and side as\ndescribed in the following table:\nstorev = C\nstorev = R\nside = L\nside = R\nside = L\nside = R\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1202\n\n\nColumn\nmajor\nmax(1,ldv*\nk)\nmax(1,ldv*\nk)\nmax(1,ldv*\nm)\nmax(1,ldv*\nn)\nRow major\nmax(1,ldv*\nm)\nmax(1,ldv*\nn)\nmax(1,ldv*\nk)\nmax(1,ldv*\nk)\nThe matrix v. See Application Notes below.\nldv\nThe leading dimension of the array v.It should satisfy the following\nconditions:\nstorev = C\nstorev = R\nside = L\nside = R\nside = L\nside = R\nColumn\nmajor\nmax(1,m)\nmax(1,n)\nmax(1,k)\nmax(1,k)\nRow major\nmax(1,k)\nmax(1,k)\nmax(1,m)\nmax(1,n)\nt\nArray, size at least max(1,ldt * k).\nContains the triangular k-by-k matrix T in the representation of the block\nreflector.\nldt\nThe leading dimension of the array t.\nldt≥k.\nc\nArray, size at least max(1, ldc * n) for column major layout and max(1, ldc\n* m) for row major layout.\nOn entry, the m-by-n matrix C.\nldc\nThe leading dimension of the array c.\nldc≥ max(1,m) for column major layout and ldc≥ max(1,n) for row\nmajor layout.\nOutput Parameters\nc\nOn exit, c is overwritten by the product of the following:\n•\nH*C, or HT*C, or C*H, or C*HT for real flavors\n•\nH*C, or HH*C, or C*H, or C*HH for complex flavors\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nIf info = -1011, memory allocation error occurred.\nApplication Notes\nThe shape of the matrix V and the storage of the vectors which define the H(i) is best illustrated by the\nfollowing example with n = 5 and k = 3. The elements equal to 1 are not stored; the corresponding array\nelements are modified but restored on exit. The rest of the array is not used.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1203\n\n\n?larfg\nGenerates an elementary reflector (Householder\nmatrix).\nSyntax\nlapack_int LAPACKE_slarfg (lapack_int n , float * alpha , float * x , lapack_int incx ,\nfloat * tau );\nlapack_int LAPACKE_dlarfg (lapack_int n , double * alpha , double * x , lapack_int\nincx , double * tau );\nlapack_int LAPACKE_clarfg (lapack_int n , lapack_complex_float * alpha ,\nlapack_complex_float * x , lapack_int incx , lapack_complex_float * tau );\nlapack_int LAPACKE_zlarfg (lapack_int n , lapack_complex_double * alpha ,\nlapack_complex_double * x , lapack_int incx , lapack_complex_double * tau );\nInclude Files\n•\nmkl.h\nDescription\nThe routine ?larfg generates a real/complex elementary reflector H of order n, such that\nfor real flavors and\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1204\n\n\nfor complex flavors,\nwhere alpha and beta are scalars (with beta real for all flavors), and x is an (n-1)-element real/complex\nvector. H is represented in the form\nfor real flavors and\nfor complex flavors,\nwhere tau is a real/complex scalar and v is a real/complex (n-1)-element vector, respectively. Note that for\nclarfg/zlarfg, H is not Hermitian.\nIf the elements of x are all zero (and, for complex flavors, alpha is real), then tau = 0 and H is taken to be\nthe unit matrix.\nOtherwise, 1 ≤ tau ≤ 2 (for real flavors), or\n1 ≤ Re(tau) ≤ 2 and abs(tau-1) ≤ 1 (for complex flavors).\nInput Parameters\nA <datatype> placeholder, if present, is used for the C interface data types in the C interface section above.\nSee C Interface Conventions for the C interface principal conventions and type definitions.\nn\nThe order of the elementary reflector.\nalpha\nx\nArray, size (1+(n-2)*abs(incx)).\nOn entry, the vector x.\nincx\nThe increment between elements of x. incx > 0.\nOutput Parameters\nalpha\nOn exit, it is overwritten with the value beta.\nx\nOn exit, it is overwritten with the vector v.\ntau\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info = -2, alpha is NaN\nIf info = -3, array x contains NaN components.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1205\n\n\n?larft\nForms the triangular factor T of a block reflector H = I\n- V*T*V**H.\nSyntax\nlapack_int LAPACKE_slarft (int matrix_layout , char direct , char storev , lapack_int\nn , lapack_int k , const float * v , lapack_int ldv , const float * tau , float * t ,\nlapack_int ldt );\nlapack_int LAPACKE_dlarft (int matrix_layout , char direct , char storev , lapack_int\nn , lapack_int k , const double * v , lapack_int ldv , const double * tau , double * t ,\nlapack_int ldt );\nlapack_int LAPACKE_clarft (int matrix_layout , char direct , char storev , lapack_int\nn , lapack_int k , const lapack_complex_float * v , lapack_int ldv , const\nlapack_complex_float * tau , lapack_complex_float * t , lapack_int ldt );\nlapack_int LAPACKE_zlarft (int matrix_layout , char direct , char storev , lapack_int\nn , lapack_int k , const lapack_complex_double * v , lapack_int ldv , const\nlapack_complex_double * tau , lapack_complex_double * t , lapack_int ldt );\nInclude Files\n•\nmkl.h\nDescription\nThe routine ?larft forms the triangular factor T of a real/complex block reflector H of order n, which is\ndefined as a product of k elementary reflectors.\nIf direct = 'F', H = H(1)*H(2)* . . .*H(k) and T is upper triangular;\nIf direct = 'B', H = H(k)*. . .*H(2)*H(1) and T is lower triangular.\nIf storev = 'C', the vector which defines the elementary reflector H(i) is stored in the i-th column of the\narray v, and H = I - V*T*VT (for real flavors) or H = I - V*T*VH (for complex flavors) .\nIf storev = 'R', the vector which defines the elementary reflector H(i) is stored in the i-th row of the array\nv, and H = I - VT*T*V (for real flavors) or H = I - VH*T*V (for complex flavors).\nInput Parameters\nA <datatype> placeholder, if present, is used for the C interface data types in the C interface section above.\nSee C Interface Conventions for the C interface principal conventions and type definitions.\ndirect\nSpecifies the order in which the elementary reflectors are multiplied to form\nthe block reflector:\n= 'F': H = H(1)*H(2)*. . . *H(k) (forward)\n= 'B': H = H(k)*. . .*H(2)*H(1) (backward)\nstorev\nSpecifies how the vectors which define the elementary reflectors are stored\n(see also Application Notes below):\n= 'C': column-wise\n= 'R': row-wise.\nn\nThe order of the block reflector H. n≥ 0.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1206\n\n\nk\nThe order of the triangular factor T (equal to the number of elementary\nreflectors). k≥ 1.\nv\nThe size limitations depend on values of parameters storev and side as\ndescribed in the following table:\nstorev = C\nstorev = R\nColumn major\nmax(1,ldv*k)\nmax(1,ldv*n)\nRow major\nmax(1,ldv*n)\nmax(1,ldv*k)\nThe matrix v. See Application Notes below.\nldv\nThe leading dimension of the array v.\nIf storev = 'C', ldv≥ max(1,n) for column major and ldv≥max(1,k) for\nrow major;\nif storev = 'R', ldv≥k for column major and ldv≥max(1,n) for row\nmajor.\ntau\nArray, size (k). tau[i-1] must contain the scalar factor of the elementary\nreflector H(i).\nldt\nThe leading dimension of the output array t. ldt≥k.\nOutput Parameters\nt\nArray, size ldt * k. The k-by-k triangular factor T of the block reflector. If\ndirect = 'F', T is upper triangular; if direct = 'B', T is lower\ntriangular. The rest of the array is not used.\nv\nThe matrix V.\nApplication Notes\nThe shape of the matrix V and the storage of the vectors which define the H(i) is best illustrated by the\nfollowing example with n = 5 and k = 3. The elements equal to 1 are not stored; the corresponding array\nelements are modified but restored on exit. The rest of the array is not used.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1207\n\n\n?larfx\nApplies an elementary reflector to a general\nrectangular matrix, with loop unrolling when the\nreflector has order less than or equal to 10.\nSyntax\nlapack_int LAPACKE_slarfx (int matrix_layout , char side , lapack_int m , lapack_int\nn , const float * v , float tau , float * c , lapack_int ldc , float * work );\nlapack_int LAPACKE_dlarfx (int matrix_layout , char side , lapack_int m , lapack_int\nn , const double * v , double tau , double * c , lapack_int ldc , double * work );\nlapack_int LAPACKE_clarfx (int matrix_layout , char side , lapack_int m , lapack_int\nn , const lapack_complex_float * v , lapack_complex_float tau , lapack_complex_float *\nc , lapack_int ldc , lapack_complex_float * work );\nlapack_int LAPACKE_zlarfx (int matrix_layout , char side , lapack_int m , lapack_int\nn , const lapack_complex_double * v , lapack_complex_double tau , lapack_complex_double\n* c , lapack_int ldc , lapack_complex_double * work );\nInclude Files\n•\nmkl.h\nDescription\nThe routine ?larfx applies a real/complex elementary reflector H to a real/complex m-by-n matrix C, from\neither the left or the right.\nH is represented in the following forms:\n•\nH = I - tau*v*vT, where tau is a real scalar and v is a real vector.\n•\nH = I - tau*v*vH, where tau is a complex scalar and v is a complex vector.\nIf tau = 0, then H is taken to be the unit matrix.\nInput Parameters\nA <datatype> placeholder, if present, is used for the C interface data types in the C interface section above.\nSee C Interface Conventions for the C interface principal conventions and type definitions.\nside\nIf side = 'L': form H*C\nIf side = 'R': form C*H.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1208\n\n\nm\nThe number of rows of the matrix C.\nn\nThe number of columns of the matrix C.\nv\nArray, size\n(m) if side = 'L' or\n(n) if side = 'R'.\nThe vector v in the representation of H.\ntau\nThe value tau in the representation of H.\nc\nArray, size at least max(1, ldc*n) for column major layout and max (1,\nldc*m) for row major layout. On entry, the m-by-n matrix C.\nldc\nThe leading dimension of the array c. lda≥ (1,m).\nwork\nWorkspace array, size\n(n) if side = 'L' or\n(m) if side = 'R'.\nwork is not referenced if H has order < 11.\nOutput Parameters\nc\nOn exit, C is overwritten by the matrix H*C if side = 'L', or C*H if side =\n'R'.\n?large\nPre- and post-multiplies a real general matrix with a\nrandom orthogonal matrix.\nSyntax\nvoid slarge (lapack_int *n, float *a, lapack_int *lda, lapack_int *iseed, float * work,\nlapack_int *info);\nvoid dlarge (lapack_int *n, double *a, lapack_int *lda, lapack_int *iseed, double *\nwork, lapack_int *info);\nvoid clarge (lapack_int *n, lapack_complex *a, lapack_int *lda, lapack_int *iseed,\nlapack_complex * work, lapack_int *info);\nvoid zlarge (lapack_int *n, lapack_complex_double *a, lapack_int *lda, lapack_int\n*iseed, lapack_complex_double * work, lapack_int *info);\nInclude Files\n•\nmkl.h\nDescription\nThe routine ?large pre- and post-multiplies a general n-by-n matrix A with a random orthogonal or unitary\nmatrix: A = U*D*UT .\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1209\n\n\nInput Parameters\nn\nThe order of the matrix A. n≥0\na\nArray, size lda by n.\nOn entry, the original n-by-n matrix A.\nlda\nThe leading dimension of the array a. lda≥n.\niseed\nArray, size 4.\nOn entry, the seed of the random number generator. The array elements\nmust be between 0 and 4095, and iseed[3] must be odd.\nwork\nWorkspace array, size 2*n.\nOutput Parameters\na\nOn exit, A is overwritten by U*A*U' for some random orthogonal matrix U.\niseed\nOn exit, the seed is updated.\ninfo\nIf info = 0, the execution is successful.\nIf info < 0, the i -th parameter had an illegal value.\n?larnd\nReturns a random real number from a uniform or\nnormal distribution.\nSyntax\nfloat slarnd (lapack_int *idist, lapack_int *iseed);\ndouble dlarnd (lapack_int *idist, lapack_int *iseed);\nThe data types for complex variations depend on whether or not the application links with Gnu Fortran\n(gfortran) libraries.\nFor non-gfortran (libmkl_intel_*) interface libraries:\nvoid clarnd (lapack_complex_float *res, lapack_int *idist, lapack_int *iseed);\nvoid zlarnd (lapack_complex_double *res, lapack_int *idist, lapack_int *iseed);\nFor gfortran (libmkl_gf_*) interface libraries:\nlapack_complex_float clarnd (lapack_int *idist, lapack_int *iseed);\nlapack_complex_double zlarnd (lapack_int *idist, lapack_int *iseed);\nTo understand the difference between the non-gfortran and gfortran interfaces and when to use each of\nthem, see Dynamic Libraries in the lib/intel64 Directory in the oneAPI Math Kernel Library Developer Guide.\nInclude Files\n•\nmkl.h\nDescription\nThe routine ?larnd returns a random number from a uniform or normal distribution.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1210\n\n\nInput Parameters\nidist\nSpecifies the distribution of the random numbers. For slarnd and dlanrd:\n= 1: uniform (0,1)\n= 2: uniform (-1,1)\n= 3: normal (0,1).\nFor clarnd and zlanrd:\n= 1: real and imaginary parts each uniform (0,1)\n= 2: real and imaginary parts each uniform (-1,1)\n= 3: real and imaginary parts each normal (0,1)\n= 4: uniformly distributed on the disc abs(z) ≤ 1\n= 5: uniformly distributed on the circle abs(z) = 1\niseed\nArray, size 4.\nOn entry, the seed of the random number generator. The array elements\nmust be between 0 and 4095, and iseed[3] must be odd.\nOutput Parameters\niseed\nOn exit, the seed is updated.\nReturn Values\nThe function returns a random number (for complex variations libmkl_gf_* interface layer/libraries return\nthe result as the parameter res).\n?larnv\nReturns a vector of random numbers from a uniform\nor normal distribution.\nSyntax\nlapack_int LAPACKE_slarnv (lapack_int idist , lapack_int * iseed , lapack_int n , float\n* x );\nlapack_int LAPACKE_dlarnv (lapack_int idist , lapack_int * iseed , lapack_int n ,\ndouble * x );\nlapack_int LAPACKE_clarnv (lapack_int idist , lapack_int * iseed , lapack_int n ,\nlapack_complex_float * x );\nlapack_int LAPACKE_zlarnv (lapack_int idist , lapack_int * iseed , lapack_int n ,\nlapack_complex_double * x );\nInclude Files\n•\nmkl.h\nDescription\nThe routine ?larnv returns a vector of n random real/complex numbers from a uniform or normal\ndistribution.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1211\n\n\nThis routine calls the auxiliary routine ?laruv to generate random real numbers from a uniform (0,1)\ndistribution, in batches of up to 128 using vectorisable code. The Box-Muller method is used to transform\nnumbers from a uniform to a normal distribution.\nInput Parameters\nA <datatype> placeholder, if present, is used for the C interface data types in the C interface section above.\nSee C Interface Conventions for the C interface principal conventions and type definitions.\nidist\nSpecifies the distribution of the random numbers: for slarnv and dlarnv:\n= 1: uniform (0,1)\n= 2: uniform (-1,1)\n= 3: normal (0,1).\nfor clarnv and zlarnv:\n= 1: real and imaginary parts each uniform (0,1)\n= 2: real and imaginary parts each uniform (-1,1)\n= 3: real and imaginary parts each normal (0,1)\n= 4: uniformly distributed on the disc abs(z) < 1\n= 5: uniformly distributed on the circle abs(z) = 1\niseed\nArray, size (4).\nOn entry, the seed of the random number generator; the array elements\nmust be between 0 and 4095, and iseed(4) must be odd.\nn\nThe number of random numbers to be generated.\nOutput Parameters\nx\nArray, size (n). The generated random numbers.\niseed\nOn exit, the seed is updated.\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\n?laror\nPre- or post-multiplies an m-by-n matrix by a random\northogonal/unitary matrix.\nSyntax\nvoid slaror (char *side, char *init, lapack_int *m, lapack_int *n, float *a, lapack_int\n*lda, lapack_int *iseed, float *x, lapack_int *info);\nvoid dlaror (char *side, char *init, lapack_int *m, lapack_int *n, double *a, lapack_int\n*lda, lapack_int *iseed, double *x, lapack_int *info);\nvoid claror (char *side, char *init, lapack_int *m, lapack_int *n, lapack_complex *a,\nlapack_int *lda, lapack_int *iseed, lapack_complex *x, lapack_int *info);\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1212\n\n\nvoid zlaror (char *side, char *init, lapack_int *m, lapack_int *n,\nlapack_complex_double *a, lapack_int *lda, lapack_int *iseed, lapack_complex_double *x,\nlapack_int *info);\nInclude Files\n•\nmkl.h\nDescription\nThe routine ?laror pre- or post-multiplies an m-by-n matrix A by a random orthogonal or unitary matrix U,\noverwriting A. A may optionally be initialized to the identity matrix before multiplying by U. U is generated\nusing the method of G.W. Stewart (SIAM J. Numer. Anal. 17, 1980, 403-409).\nInput Parameters\nside\nSpecifies whether A is multiplied by U on the left or right.\nfor slaror and dlaror:\nIf side = 'L', multiply A on the left (premultiply) by U.\nIf side = 'R', multiply A on the right (postmultiply) by UT.\nIf side = 'C' or 'T', multiply A on the left by U and the right by UT.\nfor claror and zlaror:\nIf side = 'L', multiply A on the left (premultiply) by U.\nIf side = 'R', multiply A on the right (postmultiply) by UC>.\nIf side = 'C', multiply A on the left by U and the right by UC>\nIfside = 'T', multiply A on the left by U and the right by UT.\ninit\nSpecifies whether or not a should be initialized to the identity matrix.\nIf init = 'I', initialize a to (a section of) the identity matrix before\napplying U.\nIf init = 'N', no initialization. Apply U to the input matrix A.\ninit = 'I' generates square or rectangular orthogonal matrices:\nFor m = n and side = 'L' or 'R', the rows and the columns are\northogonal to each other.\nFor rectangular matrices where m < n:\n•\nIf side = 'R', ?laror produces a dense matrix in which rows are\northogonal and columns are not.\n•\nIf side= 'L', ?laror produces a matrix in which rows are orthogonal,\nfirst m columns are orthogonal, and remaining columns are zero.\nFor rectangular matrices where m > n:\n•\nIf side = 'L', ?laror produces a dense matrix in which columns are\northogonal and rows are not.\n•\nIf side = 'R', ?laror produces a matrix in which columns are\northogonal, first m rows are orthogonal, and remaining rows are zero.\nm\nThe number of rows of A.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1213\n\n\nn\nThe number of columns of A.\na\nArray, size lda by n.\nlda\nThe leading dimension of the array a.\nlda≥ max(1, m).\niseed\nArray, size (4).\nOn entry, specifies the seed of the random number generator. The array\nelements must be between 0 and 4095; if not they are reduced mod 4096.\nAlso, iseed[3] must be odd.\nx\nWorkspace array, size (3*max( m, n )) .\nValue of side\nLength of workspace\n'L'\n2*m + n\n'R'\n2*n + m\n'C' or 'T'\n3*n\nOutput Parameters\na\nOn exit, overwritten\nby UA ( if side = 'L' ),\nby AU ( if side = 'R' ),\nby UAUT ( if side = 'C' or 'T').\niseed\nThe values of iseed are changed on exit, and can be used in the next call\nto continue the same random number sequence.\ninfo\nArray, size (4).\nFor slaror and dlaror:\nIf info = 0, the execution is successful.\nIf info < 0, the i -th parameter had an illegal value.\nIf info = 1, the random numbers generated by ?laror are bad.\nFor claror and zlaror:\nIf info = 0, the execution is successful.\nIf info = -1, side is not 'L', 'R', 'C', or 'T'.\nIf info = -3, if m is negative.\nIf info = -4, if m is negative or if side is 'C' or 'T' and n is not equal to\nm .\nIf info = -6, if lda is less than m .\n?larot\nApplies a Givens rotation to two adjacent rows or\ncolumns.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1214\n\n\nSyntax\nvoid slarot (lapack_logical *lrows, lapack_logical *ileft, lapack_logical *iright,\nlapack_int *nl, float *c, float *s, float *a, lapack_int *lda, float *xleft, float\n*xright);\nvoid dlarot (lapack_logical *lrows, lapack_logical *ileft, lapack_logical *iright,\nlapack_int *nl, double *c, double *s, double *a, lapack_int *lda, double *xleft, double\n*xright);\nvoid clarot (lapack_logical *lrows, lapack_logical *ileft, lapack_logical *iright,\nlapack_int *nl, lapack_complex *c, lapack_complex *s, lapack_complex *a, lapack_int\n*lda, lapack_complex *xleft, lapack_complex *xright);\nvoid zlarot (lapack_logical *lrows, lapack_logical *ileft, lapack_logical *iright,\nlapack_int *nl, lapack_complex_double *c, lapack_complex_double *s,\nlapack_complex_double *a, lapack_int *lda, lapack_complex_double *xleft,\nlapack_complex_double *xright);\nInclude Files\n•\nmkl.h\nDescription\nThe routine ?larot applies a Givens rotation to two adjacent rows or columns, where one element of the\nfirst or last column or row is stored in some format other than GE so that elements of the matrix may be\nused or modified for which no array element is provided.\nOne example is a symmetric matrix in SB format (bandwidth = 4), for which uplo = 'L'. Two adjacent rows\nwill have the format:\nrow j :      C > C > C > C > C > . . . . \nrow j + 1 :  C > C > C > C > C > . . . .\n'*' indicates elements for which storage is provided.\n'.' indicates elements for which no storage is provided, but are not necessarily zero; their values are\ndetermined by symmetry.\n' ' indicates elements which are required to be zero, and have no storage provided.\nThose columns which have two '*' entries can be handled by srot (for slarot and clarot), or by\ndrot( for dlarot and zlarot).\nThose columns which have no '*' entries can be ignored, since as long as the Givens rotations are carefully\napplied to preserve symmetry, their values are determined.\nThose columns which have one '*' have to be handled separately, by using separate variables p and q :\nrow j :      C > C > C > C > C > p. . . . \nrow j + 1 :  q  C > C > C > C > C > . . . . \nIf element p is set correctly, ?larot rotates the column and sets p to its new value. The next call to ?larot\nrotates columns j and j +1, and restore symmetry. The element q is zero at the beginning, and non-zero\nafter the rotation. Later, rotations would presumably be chosen to zero q out.\nTypical Calling Sequences: rotating the i -th and (i +1)-st rows.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1215\n\n\nInput Parameters\nlrows\nIf lrows = 1, ?larot rotates two rows.\nIf lrows = 0, ?larot rotates two columns.\nlleft\nIf lleft = 1, xleft is used instead of the corresponding element of a for\nthe first element in the second row (if lrows = 0) or column (if lrows=1).\nIf lleft = 0, the corresponding element of a is used.\nlright\nIf lleft = 1, xright is used instead of the corresponding element of a for\nthe first element in the second row (if lrows = 0) or column (if lrows=1).\nIf lright = 0, the corresponding element of a is used.\nnl\nThe length of the rows (if lrows=1) or columns (if lrows=1) to be rotated.\nIf xleft or xright are used, the columns or rows they are in should be\nincluded in nl, e.g., if lleft = lright = 1, then nl must be at least 2.\nThe number of rows or columns to be rotated exclusive of those involving\nxleft and/or xright may not be negative, i.e., nl minus how many of\nlleft and lright are 1 must be at least zero; if not, xerbla is called.\nc, s\nSpecify the Givens rotation to be applied.\nIf lrows = 1, then the matrix\nis applied from the left.\nIf lrows = 0, then the transpose thereof is applied from the right.\na\nThe array containing the rows or columns to be rotated. The first element of\na should be the upper left element to be rotated.\nlda\nThe \"effective\" leading dimension of a.\nIf a contains a matrix stored in GE or SY format, then this is just the\nleading dimension of A.\nIf a contains a matrix stored in band (GB or SB) format, then this should be\none less than the leading dimension used in the calling routine. Thus, if a\nin ?larot is of size lda*n, then a[(j - 1)*lda] would be the j -th\nelement in the first of the two rows to be rotated, and a[(j - 1)*lda +\n1] would be the j -th in the second, regardless of how the array may be\nstored in the calling routine. a cannot be dimensioned, because for band\nformat the row number may exceed lda, which is not legal FORTRAN.\nIf lrows = 1, then lda must be at least 1, otherwise it must be at least nl\nminus the number of 1 values in xleft and xright.\nxleft\nIf lrows = 1, xleft is used and modified instead of a[1] (if lrows = 1)\nor a[lda + 1] (if lrows = 0).\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1216\n\n\nxright\nIf lright = 1, xright is used and modified instead of a[(nl - 1)*lda]\n(if lrows = 1) or a[nl - 1] (if lrows = 0).\nOutput Parameters\na\nOn exit, modified array A.\n?lartgp\nGenerates a plane rotation.\nSyntax\nlapack_int LAPACKE_slartgp (float f, floatg, float* cs, float* sn, float* r);\nlapack_int LAPACKE_dlartgp (doublef, doubleg, double* cs, double* sn, double* r);\nInclude Files\n•\nmkl.h\nDescription\nThe routine generates a plane rotation so that\nwhere cs2 + sn2 = 1\nThis is a slower, more accurate version of the BLAS Level 1 routine ?rotg, except for the following\ndifferences:\n•\nf and g are unchanged on return.\n•\nIf g=0, then cs=(+/-)1 and sn=0.\n•\nIf f=0 and g≠ 0, then cs=0 and sn=(+/-)1.\nThe sign is chosen so that r≥ 0.\nInput Parameters\nA <datatype> placeholder, if present, is used for the C interface data types in the C interface section above.\nSee C Interface Conventions for the C interface principal conventions and type definitions.\nf, g\nThe first and second component of the vector to be rotated.\nOutput Parameters\ncs\nThe cosine of the rotation.\nsn\nThe sine of the rotation.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1217\n\n\nr\nThe nonzero component of the rotated vector.\nReturn Values\nIf info = 0, the execution is successful.\nIf info =-1,f is NaN.\nIf info = -2, g is NaN.\nSee Also\n?rotg\n?lartgs\n?lartgs\nGenerates a plane rotation designed to introduce a\nbulge in implicit QR iteration for the bidiagonal SVD\nproblem.\nSyntax\nlapack_int LAPACKE_slartgs (floatx, floaty, floatsigma, float* cs, float* sn);\nlapack_int LAPACKE_dlartgs (doublex, doubley, doublesigma, double* cs, double* sn);\nInclude Files\n•\nmkl.h\nDescription\nThe routine generates a plane rotation designed to introduce a bulge in Golub-Reinsch-style implicit QR\niteration for the bidiagonal SVD problem. x and y are the top-row entries, and sigma is the shift. The\ncomputed cs and sn define a plane rotation that satisfies the following:\nwith r nonnegative.\nIf x2 - sigma and x * y are 0, the rotation is by π/2\nInput Parameters\nA <datatype> placeholder, if present, is used for the C interface data types in the C interface section above.\nSee C Interface Conventions for the C interface principal conventions and type definitions.\nx, y\nThe (1,1) and (1,2) entries of an upper bidiagonal matrix, respectively.\nsigma\nShift\nOutput Parameters\ncs\nThe cosine of the rotation.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1218\n\n\nsn\nThe sine of the rotation.\nReturn Values\nIf info = 0, the execution is successful.\nIf info = - 1, x is NaN.\nIf info = - 2, y is NaN.\nIf info = - 3, sigma is NaN.\nSee Also\n?lartgp\n?lascl\nMultiplies a general rectangular matrix by a real scalar\ndefined as cto/cfrom.\nSyntax\nlapack_int LAPACKE_slascl (int matrix_layout, char type, lapack_int kl, lapack_int ku,\nfloat cfrom, float cto, lapack_int m, lapack_int n, float * a, lapack_int lda);\nlapack_int LAPACKE_dlascl (int matrix_layout, char type, lapack_int kl, lapack_int ku,\ndouble cfrom, double cto, lapack_int m, lapack_int n, double * a, lapack_int lda);\nlapack_int LAPACKE_clascl (int matrix_layout, char type, lapack_int kl, lapack_int ku,\nfloat cfrom, float cto, lapack_int m, lapack_int n, lapack_complex_float * a,\nlapack_int lda);\nlapack_int LAPACKE_zlascl (int matrix_layout, char type, lapack_int kl, lapack_int ku,\ndouble cfrom, double cto, lapack_int m, lapack_int n, lapack_complex_double * a,\nlapack_int lda);\nInclude Files\n•\nmkl.h\nDescription\nThe routine ?lascl multiplies the m-by-n real/complex matrix A by the real scalar cto/cfrom. The operation\nis performed without over/underflow as long as the final result cto*A(i,j)/cfrom does not over/underflow.\ntype specifies that A may be full, upper triangular, lower triangular, upper Hessenberg, or banded.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\ntype\nThis parameter specifies the storage type of the input matrix.\n= 'G': A is a full matrix.\n= 'L': A is a lower triangular matrix.\n= 'U': A is an upper triangular matrix.\n= 'H': A is an upper Hessenberg matrix.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1219\n\n\n= 'B': A is a symmetric band matrix with lower bandwidth kl and upper\nbandwidth ku and with the only the lower half stored\n= 'Q': A is a symmetric band matrix with lower bandwidth kl and upper\nbandwidth ku and with the only the upper half stored.\n= 'Z': A is a band matrix with lower bandwidth kl and upper bandwidth ku.\nSee description of the ?gbtrf function for storage details.\nkl\nThe lower bandwidth of A. Referenced only if type = 'B', 'Q' or 'Z'.\nku\nThe upper bandwidth of A. Referenced only if type = 'B', 'Q' or 'Z'.\ncfrom, cto\nThe matrix A is multiplied by cto/cfrom. A(i,j) is computed without over/\nunderflow if the final result cto*A(i,j)/cfrom can be represented without\nover/underflow. cfrom must be nonzero.\nm\nThe number of rows of the matrix A. m≥ 0.\nn\nThe number of columns of the matrix A. n≥ 0.\na\nArray, size (lda*n). The matrix to be multiplied by cto/cfrom. See type for\nthe storage type.\nlda\nThe leading dimension of the array a.\nlda≥ max(1,m).\nOutput Parameters\na\nThe multiplied matrix A.\ninfo\nIf info = 0 - successful exit\nIf info = -i < 0, the i-th argument had an illegal value.\nSee Also\n?gbtrf\n?lasd0\nComputes the singular values of a real upper\nbidiagonal n-by-m matrix B with diagonal d and off-\ndiagonal e. Used by ?bdsdc.\nSyntax\nvoid slasd0( lapack_int *n, lapack_int *sqre, float *d, float *e, float *u, lapack_int\n*ldu, float *vt, lapack_int *ldvt, lapack_int *smlsiz, lapack_int *iwork, float *work,\nlapack_int *info );\nvoid dlasd0( lapack_int *n, lapack_int *sqre, double *d, double *e, double *u,\nlapack_int *ldu, double *vt, lapack_int *ldvt, lapack_int *smlsiz, lapack_int *iwork,\ndouble *work, lapack_int *info );\nInclude Files\n•\nmkl.h\nDescription\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1220\n\n\nUsing a divide and conquer approach, the routine ?lasd0 computes the singular value decomposition (SVD)\nof a real upper bidiagonal n-by-m matrix B with diagonal d and offdiagonal e, where m = n + sqre.\nThe algorithm computes orthogonal matrices U and VT such that B = U*S*VT. The singular values S are\noverwritten on d.\nThe related subroutine ?lasda computes only the singular values, and optionally, the singular vectors in\ncompact form.\nInput Parameters\nn\nOn entry, the row dimension of the upper bidiagonal matrix. This is also the\ndimension of the main diagonal array d.\nsqre\nSpecifies the column dimension of the bidiagonal matrix.\nIf sqre = 0: the bidiagonal matrix has column dimension m = n.\nIf sqre = 1: the bidiagonal matrix has column dimension m = n+1.\nd\nArray, DIMENSION (n). On entry, d contains the main diagonal of the\nbidiagonal matrix.\ne\nArray, DIMENSION (m-1). Contains the subdiagonal entries of the bidiagonal\nmatrix. On exit, e is destroyed.\nldu\nOn entry, leading dimension of the output array u.\nldvt\nOn entry, leading dimension of the output array vt.\nsmlsiz\nOn entry, maximum size of the subproblems at the bottom of the\ncomputation tree.\niwork\nWorkspace array, dimension must be at least (8n).\nwork\nWorkspace array, dimension must be at least (3m2+2m).\nOutput Parameters\nd\nOn exit d, If info = 0, contains singular values of the bidiagonal matrix.\nu\nArray, DIMENSION at least (ldq, n). On exit, u contains the left singular\nvectors.\nvt\nArray, DIMENSION at least (ldvt, m). On exit, vtT contains the right singular\nvectors.\ninfo\nIf info = 0: successful exit.\nIf info = -i < 0, the i-th argument had an illegal value.\nIf info = 1, a singular value did not converge.\n?lasd1\nComputes the SVD of an upper bidiagonal matrix B of\nthe specified size. Used by ?bdsdc.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1221\n\n\nSyntax\nvoid slasd1( lapack_int *nl, lapack_int *nr, lapack_int *sqre, float *d, float *alpha,\nfloat *beta, float *u, lapack_int *ldu, float *vt, lapack_int *ldvt, lapack_int *idxq,\nlapack_int *iwork, float *work, lapack_int *info );\nvoid dlasd1( lapack_int *nl, lapack_int *nr, lapack_int *sqre, double *d, double *alpha,\ndouble *beta, double *u, lapack_int *ldu, double *vt, lapack_int *ldvt, lapack_int\n*idxq, lapack_int *iwork, double *work, lapack_int *info );\nInclude Files\n•\nmkl.h\nDescription\nThe routine computes the SVD of an upper bidiagonal n-by-m matrix B, where n = nl + nr + 1 and m = n\n+ sqre.\nThe routine ?lasd1 is called from ?lasd0.\nA related subroutine ?lasd7 handles the case in which the singular values (and the singular vectors in\nfactored form) are desired.\n?lasd1 computes the SVD as follows:\n= U(out)*(D(out) 0)*VT(out)\nwhereZT = (Z1TaZ2Tb) = uT*VTT, and u is a vector of dimension m with alpha and beta in the nl+1 and nl\n+2-th entries and zeros elsewhere; and the entry b is empty if sqre = 0.\nThe left singular vectors of the original matrix are stored in u, and the transpose of the right singular vectors\nare stored in vt, and the singular values are in d. The algorithm consists of three stages:\n1.\nThe first stage consists of deflating the size of the problem when there are multiple singular values or\nwhen there are zeros in the Z vector. For each such occurrence the dimension of the secular equation\nproblem is reduced by one. This stage is performed by the routine ?lasd2.\n2.\nThe second stage consists of calculating the updated singular values. This is done by finding the square\nroots of the roots of the secular equation via the routine ?lasd4 (as called by ?lasd3). This routine\nalso calculates the singular vectors of the current problem.\n3.\nThe final stage consists of computing the updated singular vectors directly using the updated singular\nvalues. The singular vectors for the current problem are multiplied with the singular vectors from the\noverall problem.\nInput Parameters\nnl\nThe row dimension of the upper block.\nnl≥ 1.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1222\n\n\nnr\nThe row dimension of the lower block.\nnr≥ 1.\nsqre\nIf sqre = 0: the lower block is an nr-by-nr square matrix.\nIf sqre = 1: the lower block is an nr-by-(nr+1) rectangular matrix. The\nbidiagonal matrix has row dimension n = nl + nr + 1, and column\ndimension m = n + sqre.\nd\nArray, DIMENSION (nl+nr+1). n = nl+nr+1. On entry d(1:nl,1:nl)\ncontains the singular values of the upper block; and d(nl+2:n) contains\nthe singular values of the lower block.\nalpha\nContains the diagonal element associated with the added row.\nbeta\nContains the off-diagonal element associated with the added row.\nu\nArray, DIMENSION (ldu, n). On entry u(1:nl, 1:nl) contains the left\nsingular vectors of the upper block; u(nl+2:n, nl+2:n) contains the left\nsingular vectors of the lower block.\nldu\nThe leading dimension of the array U.\nldu≥ max(1, n).\nvt\nArray, DIMENSION (ldvt, m), where m = n + sqre.\nOn entry vt(1:nl+1, 1:nl+1)T contains the right singular vectors of the\nupper block; vt(nl+2:m, nl+2:m)T contains the right singular vectors of\nthe lower block.\nldvt\nThe leading dimension of the array vt.\nldvt≥ max(1, M).\niwork\nWorkspace array, DIMENSION (4n).\nwork\nWorkspace array, DIMENSION (3m2 + 2m).\nOutput Parameters\nd\nOn exit d(1:n) contains the singular values of the modified matrix.\nalpha\nOn exit, the diagonal element associated with the added row deflated by\nmax( abs( alpha ), abs( beta ), abs( D(I) ) ), I = 1,n.\nbeta\nOn exit, the off-diagonal element associated with the added row deflated by\nmax( abs( alpha ), abs( beta ), abs( D(I) ) ), I = 1,n.\nu\nOn exit u contains the left singular vectors of the bidiagonal matrix.\nvt\nOn exit vtT contains the right singular vectors of the bidiagonal matrix.\nidxq\nArray, DIMENSION (n). Contains the permutation which will reintegrate the\nsubproblem just solved back into sorted order, that is, d(idxq( i = 1,\nn )) will be in ascending order.\ninfo\nIf info = 0: successful exit.\nIf info = -i < 0, the i-th argument had an illegal value.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1223\n\n\nIf info = 1, a singular value did not converge.\n?lasd2\nMerges the two sets of singular values together into a\nsingle sorted set. Used by ?bdsdc.\nSyntax\nvoid slasd2( lapack_int *nl, lapack_int *nr, lapack_int *sqre, lapack_int *k, float *d,\nfloat *z, float *alpha, float *beta, float *u, lapack_int *ldu, float *vt, lapack_int\n*ldvt, float *dsigma, float *u2, lapack_int *ldu2, float *vt2, lapack_int *ldvt2,\nlapack_int *idxp, lapack_int *idx, lapack_int *idxq, lapack_int *coltyp, lapack_int\n*info );\nvoid dlasd2( lapack_int *nl, lapack_int *nr, lapack_int *sqre, lapack_int *k, double *d,\ndouble *z, double *alpha, double *beta, double *u, lapack_int *ldu, double *vt,\nlapack_int *ldvt, double *dsigma, double *u2, lapack_int *ldu2, double *vt2, lapack_int\n*ldvt2, lapack_int *idxp, lapack_int *idx, lapack_int *idxq, lapack_int *coltyp,\nlapack_int *info );\nInclude Files\n•\nmkl.h\nDescription\nThe routine ?lasd2 merges the two sets of singular values together into a single sorted set. Then it tries to\ndeflate the size of the problem. There are two ways in which deflation can occur: when two or more singular\nvalues are close together or if there is a tiny entry in the Z vector. For each such occurrence the order of the\nrelated secular equation problem is reduced by one.\nThe routine ?lasd2 is called from ?lasd1.\nInput Parameters\nnl\nThe row dimension of the upper block.\nnl≥ 1.\nnr\nThe row dimension of the lower block.\nnr≥ 1.\nsqre\nIf sqre = 0): the lower block is an nr-by-nr square matrix\nIf sqre = 1): the lower block is an nr-by-(nr+1) rectangular matrix. The\nbidiagonal matrix has n = nl + nr + 1 rows and m = n + sqre≥n\ncolumns.\nd\nArray, DIMENSION (n). On entry d contains the singular values of the two\nsubmatrices to be combined.\nalpha\nContains the diagonal element associated with the added row.\nbeta\nContains the off-diagonal element associated with the added row.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1224\n\n\nu\nArray, DIMENSION (ldu, n). On entry u contains the left singular vectors of\ntwo submatrices in the two square blocks with corners at (1,1), (nl, nl), and\n(nl+2, nl+2), (n,n).\nldu\nThe leading dimension of the array u.\nldu≥n.\nldu2\nThe leading dimension of the output array u2. ldu2≥n.\nvt\nArray, DIMENSION (ldvt, m). On entry, vtT contains the right singular\nvectors of two submatrices in the two square blocks with corners at (1,1),\n(nl+1, nl+1), and (nl+2, nl+2), (m, m).\nldvt\nThe leading dimension of the array vt. ldvt≥m.\nldvt2\nThe leading dimension of the output array vt2. ldvt2≥m.\nidxp\nWorkspace array, DIMENSION (n). This will contain the permutation used to\nplace deflated values of D at the end of the array. On output idxp(2:k)\npoints to the nondeflated d-values and idxp(k+1:n) points to the deflated\nsingular values.\nidx\nWorkspace array, DIMENSION (n). This will contain the permutation used to\nsort the contents of d into ascending order.\ncoltyp\nWorkspace array, DIMENSION (n). As workspace, this array contains a label\nthat indicates which of the following types a column in the u2 matrix or a\nrow in the vt2 matrix is:\n1 : non-zero in the upper half only\n2 : non-zero in the lower half only\n3 : dense\n4 : deflated.\nidxq\nArray, DIMENSION (n). This parameter contains the permutation that\nseparately sorts the two sub-problems in D in the ascending order. Note\nthat entries in the first half of this permutation must first be moved one\nposition backwards and entries in the second half must have nl+1 added to\ntheir values.\nOutput Parameters\nk\nContains the dimension of the non-deflated matrix, This is the order of the\nrelated secular equation. 1 ≤ k ≤ n.\nd\nOn exit D contains the trailing (n-k) updated singular values (those which\nwere deflated) sorted into increasing order.\nu\nOn exit u contains the trailing (n-k) updated left singular vectors (those\nwhich were deflated) in its last n-k columns.\nz\nArray, DIMENSION (n). On exit, z contains the updating row vector in the\nsecular equation.\ndsigma\nArray, DIMENSION (n). Contains a copy of the diagonal elements (k-1\nsingular values and one zero) in the secular equation.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1225\n\n\nu2\nArray, DIMENSION (ldu2, n). Contains a copy of the first k-1 left singular\nvectors which will be used by ?lasd3 in a matrix multiply (?gemm) to solve\nfor the new left singular vectors. u2 is arranged into four blocks. The first\nblock contains a column with 1 at nl+1 and zero everywhere else; the\nsecond block contains non-zero entries only at and above nl; the third\ncontains non-zero entries only below nl+1; and the fourth is dense.\nvt\nOn exit, vtT contains the trailing (n-k) updated right singular vectors (those\nwhich were deflated) in its last n-k columns. In case sqre =1, the last row\nof vt spans the right null space.\nvt2\nArray, DIMENSION (ldvt2, n). vt2T contains a copy of the first k right\nsingular vectors which will be used by ?lasd3 in a matrix multiply (?gemm)\nto solve for the new right singular vectors. vt2 is arranged into three blocks.\nThe first block contains a row that corresponds to the special 0 diagonal\nelement in sigma; the second block contains non-zeros only at and before\nnl +1; the third block contains non-zeros only at and after nl +2.\nidxc\nArray, DIMENSION (n). This will contain the permutation used to arrange the\ncolumns of the deflated u matrix into three groups: the first group contains\nnon-zero entries only at and above nl, the second contains non-zero entries\nonly below nl+2, and the third is dense.\ncoltyp\nOn exit, it is an array of dimension 4, with coltyp(i) being the dimension of\nthe i-th type columns.\ninfo\nIf info = 0): successful exit\nIf info = -i < 0, the i-th argument had an illegal value.\n?lasd3\nFinds all square roots of the roots of the secular\nequation, as defined by the values in D and Z, and\nthen updates the singular vectors by matrix\nmultiplication. Used by ?bdsdc.\nSyntax\nvoid slasd3( lapack_int *nl, lapack_int *nr, lapack_int *sqre, lapack_int *k, float *d,\nfloat *q, lapack_int *ldq, float *dsigma, float *u, lapack_int *ldu, float *u2,\nlapack_int *ldu2, float *vt, lapack_int *ldvt, float *vt2, lapack_int *ldvt2,\nlapack_int *idxc, lapack_int *ctot, float *z, lapack_int *info );\nvoid dlasd3( lapack_int *nl, lapack_int *nr, lapack_int *sqre, lapack_int *k, double *d,\ndouble *q, lapack_int *ldq, double *dsigma, double *u, lapack_int *ldu, double *u2,\nlapack_int *ldu2, double *vt, lapack_int *ldvt, double *vt2, lapack_int *ldvt2,\nlapack_int *idxc, lapack_int *ctot, double *z, lapack_int *info );\nInclude Files\n•\nmkl.h\nDescription\nThe routine ?lasd3 finds all the square roots of the roots of the secular equation, as defined by the values in\nD and Z.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1226\n\n\nIt makes the appropriate calls to ?lasd4 and then updates the singular vectors by matrix multiplication.\nThe routine ?lasd3 is called from ?lasd1.\nInput Parameters\nnl\nThe row dimension of the upper block.\nnl≥ 1.\nnr\nThe row dimension of the lower block.\nnr≥ 1.\nsqre\nIf sqre = 0): the lower block is an nr-by-nr square matrix.\nIf sqre = 1): the lower block is an nr-by-(nr+1) rectangular matrix. The\nbidiagonal matrix has n = nl + nr + 1 rows and m = n + sqre≥n\ncolumns.\nk\nThe size of the secular equation, 1 ≤ k ≤ n.\nq\nWorkspace array, DIMENSION at least (ldq, k).\nldq\nThe leading dimension of the array Q.\nldq≥k.\ndsigma\nArray, DIMENSION (k). The first k elements of this array contain the old\nroots of the deflated updating problem. These are the poles of the secular\nequation.\nldu\nThe leading dimension of the array u.\nldu≥n.\nu2\nArray, DIMENSION (ldu2, n).\nThe first k columns of this matrix contain the non-deflated left singular\nvectors for the split problem.\nldu2\nThe leading dimension of the array u2.\nldu2≥n.\nldvt\nThe leading dimension of the array vt.\nldvt≥n.\nvt2\nArray, DIMENSION (ldvt2, n).\nThe first k columns of vt2' contain the non-deflated right singular vectors\nfor the split problem.\nldvt2\nThe leading dimension of the array vt2.\nldvt2≥n.\nidxc\nArray, DIMENSION (n).\nThe permutation used to arrange the columns of u (and rows of vt) into\nthree groups: the first group contains non-zero entries only at and above\n(or before) nl +1; the second contains non-zero entries only at and below\n(or after) nl+2; and the third is dense. The first column of u and the row of\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1227\n\n\nvt are treated separately, however. The rows of the singular vectors found\nby ?lasd4 must be likewise permuted before the matrix multiplies can take\nplace.\nctot\nArray, DIMENSION (4). A count of the total number of the various types of\ncolumns in u (or rows in vt), as described in idxc.\nThe fourth column type is any column which has been deflated.\nz\nArray, DIMENSION (k). The first k elements of this array contain the\ncomponents of the deflation-adjusted updating row vector.\nOutput Parameters\nd\nArray, DIMENSION (k). On exit the square roots of the roots of the secular\nequation, in ascending order.\nu\nArray, DIMENSION (ldu, n).\nThe last n - k columns of this matrix contain the deflated left singular\nvectors.\nvt\nArray, DIMENSION (ldvt, m).\nThe last m - k columns of vt' contain the deflated right singular vectors.\nvt2\nDestroyed on exit.\nz\nDestroyed on exit.\ninfo\nIf info = 0): successful exit.\nIf info = -i < 0, the i-th argument had an illegal value.\nIf info = 1, an singular value did not converge.\nApplication Notes\nThis code makes very mild assumptions about floating point arithmetic. It will work on machines with a guard\ndigit in add/subtract, or on those binary machines without guard digits which subtract like the Cray XMP, Cray\nYMP, Cray C 90, or Cray 2. It could conceivably fail on hexadecimal or decimal machines without guard digits,\nbut we know of none.\n?lasd4\nComputes the square root of the i-th updated\neigenvalue of a positive symmetric rank-one\nmodification to a positive diagonal matrix. Used\nby ?bdsdc.\nSyntax\nvoid slasd4( lapack_int *n, lapack_int *i, float *d, float *z, float *delta, float *rho,\nfloat *sigma, float *work, lapack_int *info);\nvoid dlasd4( lapack_int *n, lapack_int *i, double *d, double *z, double *delta, double\n*rho, double *sigma, double *work, lapack_int *info);\nInclude Files\n•\nmkl.h\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1228\n\n\nDescription\nThe routine computes the square root of the i-th updated eigenvalue of a positive symmetric rank-one\nmodification to a positive diagonal matrix whose entries are given as the squares of the corresponding\nentries in the array d, and that 0 ≤ d(i) < d(j) for i < j and that rho > 0. This is arranged by the\ncalling routine, and is no loss in generality. The rank-one modified system is thus\ndiag(d)*diag(d) + rho*Z*ZT,\nwhere the Euclidean norm of Z is equal to 1.The method consists of approximating the rational functions in\nthe secular equation by simpler interpolating rational functions.\nInput Parameters\nn\nThe length of all arrays.\ni\nThe index of the eigenvalue to be computed. 1 ≤ i ≤ n.\nd\nArray, DIMENSION (n).\nThe original eigenvalues. They must be in order, 0 ≤ d(i) < d(j) for i <\nj.\nz\nArray, DIMENSION (n).\nThe components of the updating vector.\nrho\nThe scalar in the symmetric updating formula.\nwork\nWorkspace array, DIMENSION (n ).\nIf n≠ 1, work contains (d(j) + sigma_i) in its j-th component.\nIf n = 1, then work( 1 ) = 1.\nOutput Parameters\ndelta\nArray, DIMENSION (n).\nIf n≠ 1, delta contains (d(j) - sigma_i) in its j-th component.\nIf n = 1, then delta (1) = 1. The vector delta contains the information\nnecessary to construct the (singular) eigenvectors.\nsigma\nThe computed sigma_i, the i-th updated eigenvalue.\ninfo\n= 0: successful exit\n> 0: If info = 1, the updating process failed.\n?lasd5\nComputes the square root of the i-th eigenvalue of a\npositive symmetric rank-one modification of a 2-by-2\ndiagonal matrix.Used by ?bdsdc.\nSyntax\nvoid slasd5( lapack_int *i, float *d, float *z, float *delta, float *rho, float *dsigma,\nfloat *work );\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1229\n\n\nvoid dlasd5( lapack_int *i, double *d, double *z, double *delta, double *rho, double\n*dsigma, double *work );\nInclude Files\n•\nmkl.h\nDescription\nThe routine computes the square root of the i-th eigenvalue of a positive symmetric rank-one modification of\na 2-by-2 diagonal matrix diag(d)*diag(d)+rho*Z*ZT\nThe diagonal entries in the array d must satisfy 0 ≤ d(i) < d(j) for i<i, rho mustbe greater than 0, and\nthat the Euclidean norm of the vector Z is equal to 1.\nInput Parameters\ni\nThe index of the eigenvalue to be computed. i = 1 or i = 2.\nd\nArray, dimension (2 ).\nThe original eigenvalues, 0 ≤ d(1) < d(2).\nz\nArray, dimension ( 2 ).\nThe components of the updating vector.\nrho\nThe scalar in the symmetric updating formula.\nwork\nWorkspace array, dimension ( 2 ). Contains (d(j) + sigma_i) in its j-th\ncomponent.\nOutput Parameters\ndelta\nArray, dimension ( 2 ).\nContains (d(j) - sigma_i) in its j-th component. The vector delta\ncontains the information necessary to construct the eigenvectors.\ndsigma\nThe computed sigma_i, the i-th updated eigenvalue.\n?lasd6\nComputes the SVD of an updated upper bidiagonal\nmatrix obtained by merging two smaller ones by\nappending a row. Used by ?bdsdc.\nSyntax\nvoid slasd6( lapack_int *icompq, lapack_int *nl, lapack_int *nr, lapack_int *sqre,\nfloat *d, float *vf, float *vl, float *alpha, float *beta, lapack_int *idxq, lapack_int\n*perm, lapack_int *givptr, lapack_int *givcol, lapack_int *ldgcol, float *givnum,\nlapack_int *ldgnum, float *poles, float *difl, float *difr, float *z, lapack_int *k,\nfloat *c, float *s, float *work, lapack_int *iwork, lapack_int *info );\nvoid dlasd6( lapack_int *icompq, lapack_int *nl, lapack_int *nr, lapack_int *sqre,\ndouble *d, double *vf, double *vl, double *alpha, double *beta, lapack_int *idxq,\nlapack_int *perm, lapack_int *givptr, lapack_int *givcol, lapack_int *ldgcol, double\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1230\n\n\n*givnum, lapack_int *ldgnum, double *poles, double *difl, double *difr, double *z,\nlapack_int *k, double *c, double *s, double *work, lapack_int *iwork, lapack_int\n*info );\nInclude Files\n•\nmkl.h\nDescription\nThe routine ?lasd6 computes the SVD of an updated upper bidiagonal matrix B obtained by merging two\nsmaller ones by appending a row. This routine is used only for the problem which requires all singular values\nand optionally singular vector matrices in factored form. B is an n-by-m matrix with n = nl + nr + 1 and m\n= n + sqre. A related subroutine, ?lasd1, handles the case in which all singular values and singular vectors\nof the bidiagonal matrix are desired. ?lasd6 computes the SVD as follows:\n= U(out)*(D(out)*VT(out)\nwhere Z' = (Z1' aZ2' b) = u'*VT', and u is a vector of dimension m with alpha and beta in the nl+1\nand nl+2-th entries and zeros elsewhere; and the entry b is empty if sqre = 0.\nThe singular values of B can be computed using D1, D2, the first components of all the right singular vectors\nof the lower block, and the last components of all the right singular vectors of the upper block. These\ncomponents are stored and updated in vf and vl, respectively, in ?lasd6. Hence U and VT are not explicitly\nreferenced.\nThe singular values are stored in D. The algorithm consists of two stages:\n1.\nThe first stage consists of deflating the size of the problem when there are multiple singular values or if\nthere is a zero in the Z vector. For each such occurrence the dimension of the secular equation problem\nis reduced by one. This stage is performed by the routine ?lasd7.\n2.\nThe second stage consists of calculating the updated singular values. This is done by finding the roots\nof the secular equation via the routine ?lasd4 (as called by ?lasd8). This routine also updates vf and\nvl and computes the distances between the updated singular values and the old singular\nvalues. ?lasd6 is called from ?lasda.\nInput Parameters\nicompq\nSpecifies whether singular vectors are to be computed in factored form:\n= 0: Compute singular values only\n= 1: Compute singular vectors in factored form as well.\nnl\nThe row dimension of the upper block.\nnl≥ 1.\nnr\nThe row dimension of the lower block.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1231\n\n\nnr≥ 1.\nsqre\n= 0: the lower block is an nr-by-nr square matrix.\n= 1: the lower block is an nr-by-(nr+1) rectangular matrix.\nThe bidiagonal matrix has row dimension n=nl+nr+1, and column\ndimension m = n + sqre.\nd\nArray, dimension ( nl+nr+1 ). On entry d(1:nl,1:nl) contains the singular\nvalues of the upper block, and d(nl+2:n) contains the singular values of the\nlower block.\nvf\nArray, dimension ( m ).\nOn entry, vf(1:nl+1) contains the first components of all right singular\nvectors of the upper block; and vf(nl+2:m)\ncontains the first components of all right singular vectors of the lower block.\nvl\nArray, dimension ( m ).\nOn entry, vl(1:nl+1) contains the last components of all right singular\nvectors of the upper block; and vl(nl+2:m) contains the last components of\nall right singular vectors of the lower block.\nalpha\nContains the diagonal element associated with the added row.\nbeta\nContains the off-diagonal element associated with the added row.\nldgcol\nThe leading dimension of the output array givcol, must be at least n.\nldgnum\nThe leading dimension of the output arrays givnum and poles, must be at\nleast n.\nwork\nWorkspace array, dimension ( 4m ).\niwork\nWorkspace array, dimension ( 3n ).\nOutput Parameters\nd\nOn exit d(1:n) contains the singular values of the modified matrix.\nvf\nOn exit, vf contains the first components of all right singular vectors of the\nbidiagonal matrix.\nvl\nOn exit, vl contains the last components of all right singular vectors of the\nbidiagonal matrix.\nalpha\nOn exit, the diagonal element associated with the added row deflated by\nmax(abs(alpha), abs(beta), abs(D(I))), I = 1,n.\nbeta\nOn exit, the off-diagonal element associated with the added row deflated by\nmax(abs(alpha), abs(beta), abs(D(I))), I = 1,n.\nidxq\nArray, dimension (n). This contains the permutation which will reintegrate\nthe subproblem just solved back into sorted order, that is, d( idxq( i =\n1, n ) ) will be in ascending order.\nperm\nArray, dimension (n). The permutations (from deflation and sorting) to be\napplied to each block. Not referenced if icompq = 0.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1232\n\n\ngivptr\nThe number of Givens rotations which took place in this subproblem. Not\nreferenced if icompq = 0.\ngivcol\nArray, dimension ( ldgcol, 2 ). Each pair of numbers indicates a pair of\ncolumns to take place in a Givens rotation. Not referenced if icompq = 0.\ngivnum\nArray, dimension ( ldgnum, 2 ). Each number indicates the C or S value to\nbe used in the corresponding Givens rotation. Not referenced if icompq =\n0.\npoles\nArray, dimension ( ldgnum, 2 ). On exit, poles(1,*) is an array containing\nthe new singular values obtained from solving the secular equation, and\npoles(2,*) is an array containing the poles in the secular equation. Not\nreferenced if icompq = 0.\ndifl\nArray, dimension (n). On exit, difl(i) is the distance between i-th updated\n(undeflated) singular value and the i-th (undeflated) old singular value.\ndifr\nArray, dimension (ldgnum, 2 ) if icompq = 1 and dimension (n) if\nicompq = 0.\nOn exit, difr(i, 1) is the distance between i-th updated (undeflated) singular\nvalue and the i+1-th (undeflated) old singular value. If icompq = 1,\ndifr(1: k, 2) is an array containing the normalizing factors for the right\nsingular vector matrix.\nSee ?lasd8 for details on difl and difr.\nz\nArray, dimension ( m ).\nThe first elements of this array contain the components of the deflation-\nadjusted updating row vector.\nk\nContains the dimension of the non-deflated matrix. This is the order of the\nrelated secular equation. 1 ≤ k ≤ n.\nc\nc contains garbage if sqre =0 and the C-value of a Givens rotation related\nto the right null space if sqre = 1.\ns\ns contains garbage if sqre =0 and the S-value of a Givens rotation related\nto the right null space if sqre = 1.\ninfo\n= 0: successful exit.\n< 0: if info = -i, the i-th argument had an illegal value.\n> 0: if info = 1, an singular value did not converge\n?lasd7\nMerges the two sets of singular values together into a\nsingle sorted set. Then it tries to deflate the size of\nthe problem. Used by ?bdsdc.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1233\n\n\nSyntax\nvoid slasd7( lapack_int *icompq, lapack_int *nl, lapack_int *nr, lapack_int *sqre,\nlapack_int *k, float *d, float *z, float *zw, float *vf, float *vfw, float *vl, float\n*vlw, float *alpha, float *beta, float *dsigma, lapack_int *idx, lapack_int *idxp,\nlapack_int *idxq, lapack_int *perm, lapack_int *givptr, lapack_int *givcol, lapack_int\n*ldgcol, float *givnum, lapack_int *ldgnum, float *c, float *s, lapack_int *info );\nvoid dlasd7( lapack_int *icompq, lapack_int *nl, lapack_int *nr, lapack_int *sqre,\nlapack_int *k, double *d, double *z, double *zw, double *vf, double *vfw, double *vl,\ndouble *vlw, double *alpha, double *beta, double *dsigma, lapack_int *idx, lapack_int\n*idxp, lapack_int *idxq, lapack_int *perm, lapack_int *givptr, lapack_int *givcol,\nlapack_int *ldgcol, double *givnum, lapack_int *ldgnum, double *c, double *s,\nlapack_int *info );\nInclude Files\n•\nmkl.h\nDescription\nThe routine ?lasd7 merges the two sets of singular values together into a single sorted set. Then it tries to\ndeflate the size of the problem. There are two ways in which deflation can occur: when two or more singular\nvalues are close together or if there is a tiny entry in the Z vector. For each such occurrence the order of the\nrelated secular equation problem is reduced by one. ?lasd7 is called from ?lasd6.\nInput Parameters\nicompq\nSpecifies whether singular vectors are to be computed in compact form, as\nfollows:\n= 0: Compute singular values only.\n= 1: Compute singular vectors of upper bidiagonal matrix in compact form.\nnl\nThe row dimension of the upper block.\nnl≥ 1.\nnr\nThe row dimension of the lower block.\nnr≥ 1.\nsqre\n= 0: the lower block is an nr-by-nr square matrix.\n= 1: the lower block is an nr-by-(nr+1) rectangular matrix. The bidiagonal\nmatrix has n = nl + nr + 1 rows and m = n + sqre≥n columns.\nd\nArray, DIMENSION (n). On entry d contains the singular values of the two\nsubmatrices to be combined.\nzw\nArray, DIMENSION ( m ).\nWorkspace for z.\nvf\nArray, DIMENSION ( m ). On entry, vf(1:nl+1) contains the first\ncomponents of all right singular vectors of the upper block; and vf(nl\n+2:m) contains the first components of all right singular vectors of the\nlower block.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1234\n\n\nvfw\nArray, DIMENSION ( m ).\nWorkspace for vf.\nvl\nArray, DIMENSION ( m ).\nOn entry, vl(1:nl+1) contains the last components of all right singular\nvectors of the upper block; and vl(nl+2:m) contains the last components\nof all right singular vectors of the lower block.\nVLW\nArray, DIMENSION ( m ).\nWorkspace for VL.\nalpha\nREAL for slasd7\nDOUBLE PRECISION for dlasd7.\nContains the diagonal element associated with the added row.\nbeta\nContains the off-diagonal element associated with the added row.\nidx\nWorkspace array, DIMENSION (n). This will contain the permutation used to\nsort the contents of d into ascending order.\nidxp\nWorkspace array, DIMENSION (n). This will contain the permutation used to\nplace deflated values of d at the end of the array.\nidxq\nArray, DIMENSION (n).\nThis contains the permutation which separately sorts the two sub-problems\nin d into ascending order. Note that entries in the first half of this\npermutation must first be moved one position backward; and entries in the\nsecond half must first have nl+1 added to their values.\nldgcol\nThe leading dimension of the output array givcol, must be at least n.\nldgnum\nThe leading dimension of the output array givnum, must be at least n.\nOutput Parameters\nk\nContains the dimension of the non-deflated matrix, this is the order of the\nrelated secular equation.\n1 ≤ k ≤ n.\nd\nOn exit, d contains the trailing (n-k) updated singular values (those which\nwere deflated) sorted into increasing order.\nz\nArray, DIMENSION (m).\nOn exit, Z contains the updating row vector in the secular equation.\nvf\nOn exit, vf contains the first components of all right singular vectors of the\nbidiagonal matrix.\nvl\nOn exit, vl contains the last components of all right singular vectors of the\nbidiagonal matrix.\ndsigma\nArray, DIMENSION (n). Contains a copy of the diagonal elements (k-1\nsingular values and one zero) in the secular equation.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1235\n\n\nidxp\nOn output, idxp(2: k) points to the nondeflated d-values and idxp( k+1:n)\npoints to the deflated singular values.\nperm\nArray, DIMENSION (n).\nThe permutations (from deflation and sorting) to be applied to each singular\nblock. Not referenced if icompq = 0.\ngivptr\nThe number of Givens rotations which took place in this subproblem. Not\nreferenced if icompq = 0.\ngivcol\nArray, DIMENSION ( ldgcol, 2 ). Each pair of numbers indicates a pair of\ncolumns to take place in a Givens rotation. Not referenced if icompq = 0.\ngivnum\nArray, DIMENSION ( ldgnum, 2 ). Each number indicates the C or S value to\nbe used in the corresponding Givens rotation. Not referenced if icompq =\n0.\nc\nIf sqre =0, then c contains garbage, and if sqre = 1, then c contains C-\nvalue of a Givens rotation related to the right null space.\nS\nIf sqre =0, then s contains garbage, and if sqre = 1, then s contains S-\nvalue of a Givens rotation related to the right null space.\ninfo\n= 0: successful exit.\n< 0: if info = -i, the i-th argument had an illegal value.\n?lasd8\nFinds the square roots of the roots of the secular\nequation, and stores, for each element in D, the\ndistance to its two nearest poles. Used by ?bdsdc.\nSyntax\nvoid slasd8( lapack_int *icompq, lapack_int *k, float *d, float *z, float *vf, float\n*vl, float *difl, float *difr, lapack_int *lddifr, float *dsigma, float *work,\nlapack_int *info );\nvoid dlasd8( lapack_int *icompq, lapack_int *k, double *d, double *z, double *vf, double\n*vl, double *difl, double *difr, lapack_int *lddifr, double *dsigma, double *work,\nlapack_int *info );\nInclude Files\n•\nmkl.h\nDescription\nThe routine ?lasd8 finds the square roots of the roots of the secular equation, as defined by the values in\ndsigma and z. It makes the appropriate calls to ?lasd4, and stores, for each element in d, the distance to its\ntwo nearest poles (elements in dsigma). It also updates the arrays vf and vl, the first and last components of\nall the right singular vectors of the original bidiagonal matrix. ?lasd8 is called from ?lasd6.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1236\n\n\nInput Parameters\nicompq\nSpecifies whether singular vectors are to be computed in factored form in\nthe calling routine:\n= 0: Compute singular values only.\n= 1: Compute singular vectors in factored form as well.\nk\nThe number of terms in the rational function to be solved by ?lasd4. k≥ 1.\nz\nArray, DIMENSION ( k ).\nThe first k elements of this array contain the components of the deflation-\nadjusted updating row vector.\nvf\nArray, DIMENSION ( k ).\nOn entry, vf contains information passed through dbede8.\nvl\nArray, DIMENSION ( k ). On entry, vl contains information passed through\ndbede8.\nlddifr\nThe leading dimension of the output array difr, must be at least k.\ndsigma\nArray, DIMENSION ( k ).\nThe first k elements of this array contain the old roots of the deflated\nupdating problem. These are the poles of the secular equation.\nwork\nWorkspace array, DIMENSION at least (3k).\nOutput Parameters\nd\nArray, DIMENSION ( k ).\nOn output, D contains the updated singular values.\nz\nUpdated on exit.\nvf\nOn exit, vf contains the first k components of the first components of all\nright singular vectors of the bidiagonal matrix.\nvl\nOn exit, vl contains the first k components of the last components of all\nright singular vectors of the bidiagonal matrix.\ndifl\nArray, DIMENSION ( k ). On exit, difl(i) = d(i) - dsigma(i).\ndifr\nArray,\nDIMENSION ( lddifr, 2 ) if icompq = 1 and\nDIMENSION ( k ) if icompq = 0.\nOn exit, difr(i,1) = d(i) - dsigma(i+1), difr(k,1) is not defined\nand will not be referenced. If icompq = 1, difr(1:k,2) is an array\ncontaining the normalizing factors for the right singular vector matrix.\ndsigma\nThe elements of this array may be very slightly altered in value.\ninfo\n= 0: successful exit.\n< 0: if info = -i, the i-th argument had an illegal value.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1237\n\n\n> 0: If info = 1, an singular value did not converge.\n?lasd9\nFinds the square roots of the roots of the secular\nequation, and stores, for each element in D, the\ndistance to its two nearest poles. Used by ?bdsdc.\nSyntax\nvoid slasd9( lapack_int *icompq, lapack_int *k, float *d, float *z, float *vf, float\n*vl, float *difl, float *difr, float *dsigma, float *work, lapack_int *info );\nvoid dlasd9( lapack_int *icompq, lapack_int *k, double *d, double *z, double *vf, double\n*vl, double *difl, double *difr, double *dsigma, double *work, lapack_int *info );\nInclude Files\n•\nmkl.h\nDescription\nThe routine ?lasd9 finds the square roots of the roots of the secular equation, as defined by the values in\ndsigma and z. It makes the appropriate calls to ?lasd4, and stores, for each element in d, the distance to its\ntwo nearest poles (elements in dsigma). It also updates the arrays vf and vl, the first and last components of\nall the right singular vectors of the original bidiagonal matrix. ?lasd9 is called from ?lasd7.\nInput Parameters\nicompq\nSpecifies whether singular vectors are to be computed in factored form in\nthe calling routine:\nIf icompq = 0, compute singular values only;\nIf icompq = 1, compute singular vector matrices in factored form also.\nk\nThe number of terms in the rational function to be solved by slasd4. k≥ 1.\ndsigma\nArray, DIMENSION(k).\nThe first k elements of this array contain the old roots of the deflated\nupdating problem. These are the poles of the secular equation.\nz\nArray, DIMENSION (k). The first k elements of this array contain the\ncomponents of the deflation-adjusted updating row vector.\nvf\nArray, DIMENSION(k). On entry, vf contains information passed through\nsbede8.\nvl\nArray, DIMENSION(k). On entry, vl contains information passed through\nsbede8.\nwork\nWorkspace array, DIMENSION at least (3k).\nOutput Parameters\nd\nArray, DIMENSION(k). d(i) contains the updated singular values.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1238\n\n\nvf\nOn exit, vf contains the first k components of the first components of all\nright singular vectors of the bidiagonal matrix.\nvl\nOn exit, vl contains the first k components of the last components of all\nright singular vectors of the bidiagonal matrix.\ndifl\nArray, DIMENSION (k).\nOn exit, difl(i) = d(i) - dsigma(i).\ndifr\nArray,\nDIMENSION (ldu, 2) if icompq =1 and\nDIMENSION (k) if icompq = 0.\nOn exit, difr(i, 1) = d(i) - dsigma(i+1), difr(k, 1) is not defined\nand will not be referenced.\nIf icompq = 1, difr(1:k, 2) is an array containing the normalizing\nfactors for the right singular vector matrix.\ninfo\n= 0: successful exit.\n< 0: if info = -i, the i-th argument had an illegal value.\n> 0: If info = 1, an singular value did not converge\n?lasda\nComputes the singular value decomposition (SVD) of a\nreal upper bidiagonal matrix with diagonal d and off-\ndiagonal e. Used by ?bdsdc.\nSyntax\nvoid slasda( lapack_int *icompq, lapack_int *smlsiz, lapack_int *n, lapack_int *sqre,\nfloat *d, float *e, float *u, lapack_int *ldu, float *vt, lapack_int *k, float *difl,\nfloat *difr, float *z, float *poles, lapack_int *givptr, lapack_int *givcol, lapack_int\n*ldgcol, lapack_int *perm, float *givnum, float *c, float *s, float *work, lapack_int\n*iwork, lapack_int *info );\nvoid dlasda( lapack_int *icompq, lapack_int *smlsiz, lapack_int *n, lapack_int *sqre,\ndouble *d, double *e, double *u, lapack_int *ldu, double *vt, lapack_int *k, double\n*difl, double *difr, double *z, double *poles, lapack_int *givptr, lapack_int *givcol,\nlapack_int *ldgcol, lapack_int *perm, double *givnum, double *c, double *s, double\n*work, lapack_int *iwork, lapack_int *info );\nInclude Files\n•\nmkl.h\nDescription\nUsing a divide and conquer approach, ?lasda computes the singular value decomposition (SVD) of a real\nupper bidiagonal n-by-m matrix B with diagonal d and off-diagonal e, where m = n + sqre.\nThe algorithm computes the singular values in the SVDB = U*S*VT. The orthogonal matrices U and VT are\noptionally computed in compact form. A related subroutine ?lasd0 computes the singular values and the\nsingular vectors in explicit form.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1239\n\n\nInput Parameters\nicompq\nSpecifies whether singular vectors are to be computed in compact form, as\nfollows:\n= 0: Compute singular values only.\n= 1: Compute singular vectors of upper bidiagonal matrix in compact form.\nsmlsiz\nThe maximum size of the subproblems at the bottom of the computation\ntree.\nn\nThe row dimension of the upper bidiagonal matrix. This is also the\ndimension of the main diagonal array d.\nsqre\nSpecifies the column dimension of the bidiagonal matrix.\nIf sqre = 0: the bidiagonal matrix has column dimension m = n\nIf sqre = 1: the bidiagonal matrix has column dimension m = n + 1.\nd\nArray, DIMENSION (n). On entry, d contains the main diagonal of the\nbidiagonal matrix.\ne\nArray, DIMENSION ( m - 1 ). Contains the subdiagonal entries of the\nbidiagonal matrix. On exit, e is destroyed.\nldu\nThe leading dimension of arrays u, vt, difl, difr, poles, givnum, and z.\nldu≥n.\nldgcol\nThe leading dimension of arrays givcol and perm. ldgcol≥n.\nwork\nWorkspace array, DIMENSION (6n+(smlsiz+1)2).\niwork\nWorkspace array, Dimension must be at least (7n).\nOutput Parameters\nd\nOn exit d, if info = 0, contains the singular values of the bidiagonal\nmatrix.\nu\nArray, DIMENSION (ldu, smlsiz) if icompq =1.\nNot referenced if icompq = 0.\nIf icompq = 1, on exit, u contains the left singular vector matrices of all\nsubproblems at the bottom level.\nvt\nArray, DIMENSION ( ldu, smlsiz+1 ) if icompq = 1, and not referenced if\nicompq = 0. If icompq = 1, on exit, vt' contains the right singular vector\nmatrices of all subproblems at the bottom level.\nk\nArray, DIMENSION (n) if icompq = 1 and\nDIMENSION (1) if icompq = 0.\nIf icompq = 1, on exit, k(i) is the dimension of the i-th secular equation on\nthe computation tree.\ndifl\nREAL for slasda\nDOUBLE PRECISION for dlasda.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1240\n\n\nArray, DIMENSION ( ldu, nlvl ),\nwhere nlvl = floor(log2(n/smlsiz)).\ndifr\nArray,\nDIMENSION ( ldu, 2 nlvl ) if icompq = 1 and\nDIMENSION (n) if icompq = 0.\nIf icompq = 1, on exit, difl(1:n, i) and difr(1:n,2i -1) record distances\nbetween singular values on the i-th level and singular values on the (i -1)-\nth level, and difr(1:n, 2i ) contains the normalizing factors for the right\nsingular vector matrix. See ?lasd8 for details.\nz\nArray,\nDIMENSION ( ldu, nlvl ) if icompq = 1 and\nDIMENSION (n) if icompq = 0. The first k elements of z(1, i) contain the\ncomponents of the deflation-adjusted updating row vector for subproblems\non the i-th level.\npoles\nArray, DIMENSION(ldu, 2*nlvl)\nif icompq = 1, and not referenced if icompq = 0. If icompq = 1, on exit,\npoles(1, 2i - 1) and poles(1, 2i) contain the new and old singular values\ninvolved in the secular equations on the i-th level.\ngivptr\nArray, DIMENSION (n) if icompq = 1, and not referenced if icompq = 0. If\nicompq = 1, on exit, givptr( i ) records the number of Givens rotations\nperformed on the i-th problem on the computation tree.\ngivcol\nArray, DIMENSION(ldgcol, 2*nlvl) if icompq = 1, and not referenced if\nicompq = 0. If icompq = 1, on exit, for each i, givcol(1, 2 i - 1) and\ngivcol(1, 2 i) record the locations of Givens rotations performed on the i-th\nlevel on the computation tree.\nperm\nArray, DIMENSION ( ldgcol, nlvl ) if icompq = 1, and not referenced if\nicompq = 0. If icompq = 1, on exit, perm (1, i) records permutations\ndone on the i-th level of the computation tree.\ngivnum\nArray DIMENSION ( ldu, 2*nlvl ) if icompq = 1, and not referenced if\nicompq = 0. If icompq = 1, on exit, for each i, givnum(1, 2 i - 1) and\ngivnum(1, 2 i) record the C- and S-values of Givens rotations performed on\nthe i-th level on the computation tree.\nc\nArray,\nDIMENSION (n) if icompq = 1, and\nDIMENSION (1) if icompq = 0.\nIf icompq = 1 and the i-th subproblem is not square, on exit, c(i) contains\nthe C-value of a Givens rotation related to the right null space of the i-th\nsubproblem.\ns\nArray,\nDIMENSION (n) icompq = 1, and\nDIMENSION (1) if icompq = 0.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1241\n\n\nIf icompq = 1 and the i-th subproblem is not square, on exit, s(i) contains\nthe S-value of a Givens rotation related to the right null space of the i-th\nsubproblem.\ninfo\n= 0: successful exit.\n< 0: if info = -i, the i-th argument had an illegal value\n> 0: If info = 1, an singular value did not converge\n?lasdq\nComputes the SVD of a real bidiagonal matrix with\ndiagonal d and off-diagonal e. Used by ?bdsdc.\nSyntax\nvoid slasdq( char *uplo, lapack_int *sqre, lapack_int *n, lapack_int *ncvt, lapack_int\n*nru, lapack_int *ncc, float *d, float *e, float *vt, lapack_int *ldvt, float *u,\nlapack_int *ldu, float *c, lapack_int *ldc, float *work, lapack_int *info );\nvoid dlasdq( char *uplo, lapack_int *sqre, lapack_int *n, lapack_int *ncvt, lapack_int\n*nru, lapack_int *ncc, double *d, double *e, double *vt, lapack_int *ldvt, double *u,\nlapack_int *ldu, double *c, lapack_int *ldc, double *work, lapack_int *info );\nInclude Files\n•\nmkl.h\nDescription\nThe routine ?lasdq computes the singular value decomposition (SVD) of a real (upper or lower) bidiagonal\nmatrix with diagonal d and off-diagonal e, accumulating the transformations if desired. If B is the input\nbidiagonal matrix, the algorithm computes orthogonal matrices Q and P such that B = Q*S*PT. The singular\nvalues S are overwritten on d.\nThe input matrix U is changed to U*Q if desired.\nThe input matrix VT is changed to PT*VT if desired.\nThe input matrix C is changed to QT*C if desired.\nInput Parameters\nuplo\nOn entry, uplo specifies whether the input bidiagonal matrix is upper or\nlower bidiagonal.\nIf uplo = 'U' or 'u', B is upper bidiagonal;\nIf uplo = 'L' or 'l', B is lower bidiagonal.\nsqre\n= 0: then the input matrix is n-by-n.\n= 1: then the input matrix is n-by-(n+1) if uplu = 'U' and (n+1)-by-n if\nuplu\n= 'L'. The bidiagonal matrix has n = nl + nr + 1 rows and m = n +\nsqre≥n columns.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1242\n\n\nn\nOn entry, n specifies the number of rows and columns in the matrix. n must\nbe at least 0.\nncvt\nOn entry, ncvt specifies the number of columns of the matrix VT. ncvt must\nbe at least 0.\nnru\nOn entry, nru specifies the number of rows of the matrix U. nru must be at\nleast 0.\nncc\nOn entry, ncc specifies the number of columns of the matrix C. ncc must be\nat least 0.\nd\nArray, DIMENSION (n). On entry, d contains the diagonal entries of the\nbidiagonal matrix.\ne\nArray, DIMENSION is (n-1) if sqre = 0 and n if sqre = 1. On entry, the\nentries of e contain the off-diagonal entries of the bidiagonal matrix.\nvt\nArray, DIMENSION (ldvt, ncvt). On entry, contains a matrix which on exit\nhas been premultiplied by PT, dimension n-by-ncvt if sqre = 0 and (n+1)-\nby-ncvt if sqre = 1 (not referenced if ncvt=0).\nldvt\nOn entry, ldvt specifies the leading dimension of vt as declared in the calling\n(sub) program. ldvt must be at least 1. If ncvt is nonzero, ldvt must also be\nat least n.\nu\nArray, DIMENSION (ldu, n). On entry, contains a matrix which on exit has\nbeen postmultiplied by Q, dimension nru-by-n if sqre = 0 and nru-by-(n\n+1) if sqre = 1 (not referenced if nru=0).\nldu\nOn entry, ldu specifies the leading dimension of u as declared in the calling\n(sub) program. ldu must be at least max(1, nru ) .\nc\nArray, DIMENSION (ldc, ncc). On entry, contains an n-by-ncc matrix which\non exit has been premultiplied by Q', dimension n-by-ncc if sqre = 0 and\n(n+1)-by-ncc if sqre = 1 (not referenced if ncc=0).\nldc\nOn entry, ldc specifies the leading dimension of C as declared in the calling\n(sub) program. ldc must be at least 1. If ncc is non-zero, ldc must also be\nat least n.\nwork\nArray, DIMENSION (4n). This is a workspace array. Only referenced if one of\nncvt, nru, or ncc is nonzero, and if n is at least 2.\nOutput Parameters\nd\nOn normal exit, d contains the singular values in ascending order.\ne\nOn normal exit, e will contain 0. If the algorithm does not converge, d and e\nwill contain the diagonal and superdiagonal entries of a bidiagonal matrix\northogonally equivalent to the one given as input.\nvt\nOn exit, the matrix has been premultiplied by P'.\nu\nOn exit, the matrix has been postmultiplied by Q.\nc\nOn exit, the matrix has been premultiplied by Q'.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1243\n\n\ninfo\nOn exit, a value of 0 indicates a successful exit. If info < 0, argument\nnumber -info is illegal. If info > 0, the algorithm did not converge, and\ninfo specifies how many superdiagonals did not converge.\n?lasdt\nCreates a tree of subproblems for bidiagonal divide\nand conquer. Used by ?bdsdc.\nSyntax\nvoid slasdt( lapack_int *n, lapack_int *lvl, lapack_int *nd, lapack_int *inode,\nlapack_int *ndiml, lapack_int *ndimr, lapack_int *msub );\nvoid dlasdt( lapack_int *n, lapack_int *lvl, lapack_int *nd, lapack_int *inode,\nlapack_int *ndiml, lapack_int *ndimr, lapack_int *msub );\nInclude Files\n•\nmkl.h\nDescription\nThe routine creates a tree of subproblems for bidiagonal divide and conquer.\nInput Parameters\nn\nOn entry, the number of diagonal elements of the bidiagonal matrix.\nmsub\nOn entry, the maximum row dimension each subproblem at the bottom of\nthe tree can be of.\nOutput Parameters\nlvl\nOn exit, the number of levels on the computation tree.\nnd\nOn exit, the number of nodes on the tree.\ninode\nArray, DIMENSION (n). On exit, centers of subproblems.\nndiml\nArray, DIMENSION (n). On exit, row dimensions of left children.\nndimr\nArray, DIMENSION (n). On exit, row dimensions of right children.\n?laset\nInitializes the off-diagonal elements and the diagonal\nelements of a matrix to given values.\nSyntax\nlapack_int LAPACKE_slaset (int matrix_layout , char uplo , lapack_int m , lapack_int\nn , float alpha , float beta , float * a , lapack_int lda );\nlapack_int LAPACKE_dlaset (int matrix_layout , char uplo , lapack_int m , lapack_int\nn , double alpha , double beta , double * a , lapack_int lda );\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1244\n\n\nlapack_int LAPACKE_claset (int matrix_layout , char uplo , lapack_int m , lapack_int\nn , lapack_complex_float alpha , lapack_complex_float beta , lapack_complex_float * a ,\nlapack_int lda );\nlapack_int LAPACKE_zlaset (int matrix_layout , char uplo , lapack_int m , lapack_int\nn , lapack_complex_double alpha , lapack_complex_double beta , lapack_complex_double *\na , lapack_int lda );\nInclude Files\n•\nmkl.h\nDescription\nThe routine initializes an m-by-n matrix A to beta on the diagonal and alpha on the off-diagonals.\nInput Parameters\nA <datatype> placeholder, if present, is used for the C interface data types in the C interface section above.\nSee C Interface Conventions for the C interface principal conventions and type definitions.\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major ( LAPACK_COL_MAJOR ).\nuplo\nSpecifies the part of the matrix A to be set.\nIf uplo = 'U', upper triangular part is set; the strictly lower triangular\npart of A is not changed.\nIf uplo = 'L': lower triangular part is set; the strictly upper triangular\npart of A is not changed.\nOtherwise: All of the matrix A is set.\nm\nThe number of rows of the matrix A. m≥ 0.\nn\nThe number of columns of the matrix A.\nn≥ 0.\nalpha, beta\nThe constants to which the off-diagonal and diagonal elements are to be\nset, respectively.\na\nArray, size at least max(1, lda*n) for column major and max(1, lda*m)\nfor row major layout.\nThe array a contains the m-by-n matrix A.\nlda\nThe leading dimension of the array a.\nlda≥ max(1,m) for column major layout and lda ≥ max(1,n) for row major\nlayout.\nOutput Parameters\na\nOn exit, the leading m-by-n submatrix of A is set as follows:\nif uplo = 'U', Aij = alpha, 1≤i≤j-1, 1≤j≤n,\nif uplo = 'L', Aij = alpha, j+1≤i≤m, 1≤j≤n,\notherwise, Aij = alpha, 1≤i≤m, 1≤j≤n, i≠j,\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1245\n\n\nand, for all uplo, Aii = beta, 1≤i≤min(m, n).\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info = i< 0, the i-th parameter had an illegal value.\nIf info = -1011, memory allocation error occurred.\n?lasrt\nSorts numbers in increasing or decreasing order.\nSyntax\nlapack_int LAPACKE_slasrt (char id , lapack_int n , float * d );\nlapack_int LAPACKE_dlasrt (char id , lapack_int n , double * d );\nInclude Files\n•\nmkl.h\nDescription\nThe routine ?lasrt sorts the numbers in d in increasing order (if id = 'I') or in decreasing order (if id =\n'D'). It uses Quick Sort, reverting to Insertion Sort on arrays of size ≤ 20. Dimension of stack limits n to\nabout 232.\nInput Parameters\nA <datatype> placeholder, if present, is used for the C interface data types in the C interface section above.\nSee C Interface Conventions for the C interface principal conventions and type definitions.\nid\n= 'I': sort d in increasing order;\n= 'D': sort d in decreasing order.\nn\nThe length of the array d.\nd\nOn entry, the array to be sorted.\nOutput Parameters\nd\nOn exit, d has been sorted into increasing order\n(d[0]≤d[1]≤ ... ≤ d[n-1]) or into decreasing order\n(d[0] ≥ d[1] ≥ ... ≥d[n-1]), depending on id.\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info < 0, the i-th parameter had an illegal value.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1246\n\n\n?laswp\nPerforms a series of row interchanges on a general\nrectangular matrix.\nSyntax\nlapack_int LAPACKE_slaswp (int matrix_layout , lapack_int n , float * a , lapack_int\nlda , lapack_int k1 , lapack_int k2 , const lapack_int * ipiv , lapack_int incx );\nlapack_int LAPACKE_dlaswp (int matrix_layout , lapack_int n , double * a , lapack_int\nlda , lapack_int k1 , lapack_int k2 , const lapack_int * ipiv , lapack_int incx );\nlapack_int LAPACKE_claswp (int matrix_layout , lapack_int n , lapack_complex_float *\na , lapack_int lda , lapack_int k1 , lapack_int k2 , const lapack_int * ipiv ,\nlapack_int incx );\nlapack_int LAPACKE_zlaswp (int matrix_layout , lapack_int n , lapack_complex_double *\na , lapack_int lda , lapack_int k1 , lapack_int k2 , const lapack_int * ipiv ,\nlapack_int incx );\nInclude Files\n•\nmkl.h\nDescription\nThe routine performs a series of row interchanges on the matrix A. One row interchange is initiated for each\nof rows k1 through k2 of A.\nInput Parameters\nA <datatype> placeholder, if present, is used for the C interface data types in the C interface section above.\nSee C Interface Conventions for the C interface principal conventions and type definitions.\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major ( LAPACK_COL_MAJOR ).\nn\nThe number of columns of the matrix A.\na\nArray, size max(1, lda*n) for column major and max(1, lda*mm) for row\nmajor layout. Here mm is not less than maximum of values\nipiv[k1-1+j*|incx|], 0≤j<k2-k1.\nArray a contains the m-by-n matrix A.\nlda\nThe leading dimension of the array a.\nk1\nThe first element of ipiv for which a row interchange will be done.\nk2\nThe last element of ipiv for which a row interchange will be done.\nipiv\nArray, size k1+(k2-k1)*|incx|).\nThe vector of pivot indices. Only the elements in positions k1 through k2 of\nipiv are accessed.\nipiv(k) = l implies rows k and l are to be interchanged.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1247\n\n\nincx\nThe increment between successive values of ipiv. If ipiv is negative, the\npivots are applied in reverse order.\nOutput Parameters\na\nOn exit, the permuted matrix.\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nIf info = -1011, memory allocation error occurred.\n?latm1\nComputes the entries of a matrix as specified.\nSyntax\nvoid slatm1 (lapack_int *mode, *cond, lapack_int *irsign, lapack_int *idist, lapack_int\n*iseed, float *d, lapack_int *n, lapack_int *info);\nvoid dlatm1 (lapack_int *mode, *cond, lapack_int *irsign, lapack_int *idist, lapack_int\n*iseed, double *d, lapack_int *n, lapack_int *info);\nvoid clatm1 (lapack_int *mode, *cond, lapack_int *irsign, lapack_int *idist, lapack_int\n*iseed, lapack_complex *d, lapack_int *n, lapack_int *info);\nvoid zlatm1 (lapack_int *mode, *cond, lapack_int *irsign, lapack_int *idist, lapack_int\n*iseed, lapack_complex_double *d, lapack_int *n, lapack_int *info);\nInclude Files\n•\nmkl.h\nDescription\nThe ?latm1 routine computes the entries of D(1..n) as specified by mode, cond and irsign. idist and\niseed determine the generation of random numbers.\n?latm1 is called by slatmr (for slatm1 and dlatm1), and by clatmr(for clatm1 and zlatm1) to generate\nrandom test matrices for LAPACK programs.\nInput Parameters\nmode\nOn entry describes how d is to be computed:\nmode = 0 means do not change d.\nmode = 1 sets d[0] = 1 and d[1:n - 1] = 1.0/cond\nmode = 2 sets d[0:n - 2] = 1 and d[n - 1]=1.0/cond\nmode = 3 sets d[i - 1]=cond**(-(i-1)/(n-1))\nmode = 4 sets d[i - 1]= 1 - (i-1)/(n-1)*(1 - 1/cond)\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1248\n\n\nmode = 5 sets d to random numbers in the range ( 1/cond , 1 ) such\nthat their logarithms are uniformly distributed.\nmode = 6 sets d to random numbers from same distribution as the rest of\nthe matrix.\nmode < 0 has the same meaning as abs(mode), except that the order of\nthe elements of d is reversed.\nThus if mode is positive, d has entries ranging from 1 to 1/cond, if\nnegative, from 1/cond to 1.\ncond\nOn entry, used as described under mode above. If used, it must be ≥ 1.\nirsign\nOn entry, if mode is not -6, 0, or 6, determines sign of entries of d.\nIf irsign = 0, entries of d are unchanged.\nIf irsign = 1, each entry of d is multiplied by a random complex number\nuniformly distributed with absolute value 1.\nidist\nSpecifies the distribution of the random numbers.\nFor slatm1 and dlatm1:\n= 1: uniform (0,1)\n= 2: uniform (-1,1)\n= 3: normal (0,1)\nFor clatm1 and zlatm1:\n= 1: real and imaginary parts each uniform (0,1)\n= 2: real and imaginary parts each uniform (-1,1)\n= 3: real and imaginary parts each normal (0,1)\n= 4: complex number uniform in disk(0, 1)\niseed\nArray, size (4).\nSpecifies the seed of the random number generator. The random number\ngenerator uses a linear congruential sequence limited to small integers, and\nso should produce machine independent random numbers. The values of\niseed[3] are changed on exit, and can be used in the next call to ?latm1\nto continue the same random number sequence.\nd\nArray, size n.\nn\nNumber of entries of d.\nOutput Parameters\niseed\nOn exit, the seed is updated.\nd\nOn exit, d is updated, unless mode = 0.\ninfo\nIf info = 0, the execution is successful.\nIf info = -1, mode is not in range -6 to 6.\nIf info = -2, mode is neither -6, 0 nor 6, and irsign is neither 0 nor 1.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1249\n\n\nIf info = -3, mode is neither -6, 0 nor 6 and cond is less than 1.\nIf info = -4, mode equals 6 or -6 and idist is not in range 1 to 4.\nIf info = -7, n is negative.\n?latm2\nReturns an entry of a random matrix.\nSyntax\nfloat slatm2 (lapack_int *m, lapack_int *n, lapack_int *i, lapack_int *j, lapack_int \n*kl, lapack_int *ku, lapack_int *idist, lapack_int *iseed, float *d, lapack_int *igrade,\nfloat *dl, float *dr, lapack_int *ipvtng, lapack_int *iwork, float *sparse);\ndouble dlatm2 (lapack_int *m, lapack_int *n, lapack_int *i, lapack_int *j, lapack_int \n*kl, lapack_int *ku, lapack_int *idist, lapack_int *iseed, double *d, lapack_int \n*igrade, double *dl, double *dr, lapack_int *ipvtng, lapack_int *iwork, double *sparse);\nThe data types for complex variations depend on whether or not the application links with Gnu Fortran\n(gfortran) libraries.\nFor non-gfortran (libmkl_intel_*) interface libraries:\nvoid clatm2 (lapack_complex_float *res, lapack_int *m, lapack_int *n, lapack_int *i,\nlapack_int *j, lapack_int *kl, lapack_int *ku, lapack_int *idist, lapack_int *iseed,\nlapack_complex_float *d, lapack_int *igrade, lapack_complex_float *dl,\nlapack_complex_float *dr, lapack_int *ipvtng, lapack_int *iwork, float *sparse);\nvoid zlatm2 (lapack_complex_double *res, lapack_int *m, lapack_int *n, lapack_int *i,\nlapack_int *j, lapack_int *kl, lapack_int *ku, lapack_int *idist, lapack_int *iseed,\nlapack_complex_double *d, lapack_int *igrade, lapack_complex_double *dl,\nlapack_complex_double *dr, lapack_int *ipvtng, lapack_int *iwork, double *sparse);\nFor gfortran (libmkl_gf_*) interface libraries:\nlapack_complex_float clatm2 (lapack_int *m, lapack_int *n, lapack_int *i, lapack_int \n*j, lapack_int *kl, lapack_int *ku, lapack_int *idist, lapack_int *iseed,\nlapack_complex_float *d, lapack_int *igrade, lapack_complex_float *dl,\nlapack_complex_float *dr, lapack_int *ipvtng, lapack_int *iwork, float *sparse);\nlapack_complex_double zlatm2 (lapack_int *m, lapack_int *n, lapack_int *i, lapack_int \n*j, lapack_int *kl, lapack_int *ku, lapack_int *idist, lapack_int *iseed,\nlapack_complex_double *d, lapack_int *igrade, lapack_complex_double *dl,\nlapack_complex_double *dr, lapack_int *ipvtng, lapack_int *iwork, double *sparse);\nTo understand the difference between the non-gfortran and gfortran interfaces and when to use each of\nthem, see Dynamic Libraries in the lib/intel64 Directory in the oneAPI Math Kernel Library Developer Guide.\nInclude Files\n•\nmkl.h\nDescription\nThe ?latm2 routine returns entry (i , j ) of a random matrix of dimension (m, n). It is called by the ?latmr\nroutine in order to build random test matrices. No error checking on parameters is done, because this routine\nis called in a tight loop by ?latmr which has already checked the parameters.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1250\n\n\nUse of ?latm2 differs from ?latm3 in the order in which the random number generator is called to fill in\nrandom matrix entries. With ?latm2, the generator is called to fill in the pivoted matrix columnwise.\nWith ?latm2, the generator is called to fill in the matrix columnwise, after which it is pivoted. Thus, ?latm3\ncan be used to construct random matrices which differ only in their order of rows and/or columns. ?latm2 is\nused to construct band matrices while avoiding calling the random number generator for entries outside the\nband (and therefore generating random numbers).\nThe matrix whose (i , j ) entry is returned is constructed as follows (this routine only computes one entry):\n•\nIf i is outside (1..m) or j is outside (1..n), returns zero (this is convenient for generating matrices in\nband format).\n•\nGenerate a matrix A with random entries of distribution idist.\n•\nSet the diagonal to D.\n•\nGrade the matrix, if desired, from the left (by dl) and/or from the right (by dr or dl) as specified by\nigrade.\n•\nPermute, if desired, the rows and/or columns as specified by ipvtng and iwork.\n•\nBand the matrix to have lower bandwidth kl and upper bandwidth ku.\n•\nSet random entries to zero as specified by sparse.\nInput Parameters\nm\nNumber of rows of the matrix.\nn\nNumber of columns of the matrix.\ni\nRow of the entry to be returned.\nj\nColumn of the entry to be returned.\nkl\nLower bandwidth.\nku\nUpper bandwidth.\nidist\nOn entry, idist specifies the type of distribution to be used to generate a\nrandom matrix .\nfor slatm2 and dlatm2:\n= 1: uniform (0,1)\n= 2: uniform (-1,1)\n= 3: normal (0,1)\nfor clatm2 and zlatm2:\n= 1: real and imaginary parts each uniform (0,1)\n= 2: real and imaginary parts each uniform (-1,1)\n= 3: real and imaginary parts each normal (0,1)\n= 4: complex number uniform in disk (0, 1)\niseed\nArray, size 4.\nSeed for the random number generator.\nd\nArray, size (min(i, j)). Diagonal entries of matrix.\nigrade\nSpecifies grading of matrix as follows:\n= 0: no grading\n= 1: matrix premultiplied by diag( dl )\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1251\n\n\n= 2: matrix postmultiplied by diag( dr )\n= 3: matrix premultiplied by diag( dl ) and postmultiplied by diag( dr)\n= 4: matrix premultiplied by diag( dl ) and postmultiplied by\ninv( diag( dl ) )\nFor slatm2 and slatm2:\n= 5: matrix premultiplied by diag( dl ) and postmultiplied by diag( dl)\nFor clatm2 and zlatm2:\n= 5: matrix premultiplied by diag( dl ) and postmultiplied by\ndiag( conjg( dl ) )\n= 6: matrix premultiplied by diag( dl ) and postmultiplied by diag( dl)\ndl\nArray, size (i or j), as appropriate.\nLeft scale factors for grading matrix.\ndr\nArray, size (i or j), as appropriate.\nRight scale factors for grading matrix.\nipvtng\nOn entry specifies pivoting permutations as follows:\n= 0: none\n= 1: row pivoting\n= 2: column pivoting\n= 3: full pivoting, i.e., on both sides\niwork\nArray, size (i or j), as appropriate. This array specifies the permutation\nused. The row (or column) in position k was originally in position iwork[k\n- 1]. This differs from iwork for ?latm3.\nsparse\nSpecifies the sparsity of the matrix. If sparse matrix is to be generated,\nsparse should lie between 0 and 1. A uniform ( 0, 1 ) random number x is\ngenerated and compared to sparse. If x is larger the matrix entry is\nunchanged and if x is smaller the entry is set to zero. Thus on the average\na fraction sparse of the entries will be set to zero.\nOutput Parameters\niseed\nOn exit, the seed is updated.\nReturn Values\nThe function returns an entry of a random matrix (for complex variations libmkl_gf_* interface layer/\nlibraries return the result as the parameter res).\n?latm3\nReturns set entry of a random matrix.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1252\n\n\nSyntax\nfloat slatm3 (lapack_int *m, lapack_int *n, lapack_int *i, lapack_int *j, lapack_int \n*isub, lapack_int *jsub, lapack_int *kl, lapack_int *ku, lapack_int *idist, lapack_int\n*iseed, float *d, lapack_int *igrade, float *dl, float *dr, lapack_int *ipvtng,\nlapack_int *iwork, float *sparse);\ndouble dlatm3 (lapack_int *m, lapack_int *n, lapack_int *i, lapack_int *j, lapack_int \n*isub, lapack_int *jsub, lapack_int *kl, lapack_int *ku, lapack_int *idist, lapack_int\n*iseed, double *d, lapack_int *igrade, double *dl, double *dr, lapack_int *ipvtng,\nlapack_int *iwork, double *sparse);\nThe data types for complex variations depend on whether or not the application links with Gnu Fortran\n(gfortran) libraries.\nFor non-gfortran (libmkl_intel_*) interface libraries:\nvoid clatm3 (lapack_complex_float *res, lapack_int *m, lapack_int *n, lapack_int *i,\nlapack_int *j, lapack_int *isub, lapack_int *jsub, lapack_int *kl, lapack_int *ku,\nlapack_int *idist, lapack_int *iseed, lapack_complex_float *d, lapack_int *igrade,\nlapack_complex_float *dl, lapack_complex_float *dr, lapack_int *ipvtng, lapack_int\n*iwork, float *sparse);\nvoid zlatm3 (lapack_complex_double *res, lapack_int *m, lapack_int *n, lapack_int *i,\nlapack_int *j, lapack_int *isub, lapack_int *jsub, lapack_int *kl, lapack_int *ku,\nlapack_int *idist, lapack_int *iseed, lapack_complex_double *d, lapack_int *igrade,\nlapack_complex_double *dl, lapack_complex_double *dr, lapack_int *ipvtng, lapack_int\n*iwork, double *sparse);\nFor gfortran (libmkl_gf_*) interface libraries:\nlapack_complex_float clatm3 (lapack_int *m, lapack_int *n, lapack_int *i, lapack_int \n*j, lapack_int *isub, lapack_int *jsub, lapack_int *kl, lapack_int *ku, lapack_int \n*idist, lapack_int *iseed, lapack_complex_float *d, lapack_int *igrade,\nlapack_complex_float *dl, lapack_complex_float *dr, lapack_int *ipvtng, lapack_int\n*iwork, float *sparse);\nlapack_complex_double zlatm3 (lapack_int *m, lapack_int *n, lapack_int *i, lapack_int \n*j, lapack_int *isub, lapack_int *jsub, lapack_int *kl, lapack_int *ku, lapack_int \n*idist, lapack_int *iseed, lapack_complex_double *d, lapack_int *igrade,\nlapack_complex_double *dl, lapack_complex_double *dr, lapack_int *ipvtng, lapack_int\n*iwork, double *sparse);\nTo understand the difference between the non-gfortran and gfortran interfaces and when to use each of\nthem, see Dynamic Libraries in the lib/intel64 Directory in the oneAPI Math Kernel Library Developer Guide.\nInclude Files\n•\nmkl.h\nDescription\nThe ?latm3 routine returns the (isub, jsub) entry of a random matrix of dimension (m, n) described by the\nother parameters. (isub, jsub) is the final position of the (i ,j ) entry after pivoting according to ipvtng and\niwork. ?latm3 is called by the ?latmr routine in order to build random test matrices. No error checking on\nparameters is done, because this routine is called in a tight loop by ?latmr which has already checked the\nparameters.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1253\n\n\nUse of ?latm3 differs from ?latm2 in the order in which the random number generator is called to fill in\nrandom matrix entries. With ?latm2, the generator is called to fill in the pivoted matrix columnwise.\nWith ?latm3, the generator is called to fill in the matrix columnwise, after which it is pivoted. Thus, ?latm3\ncan be used to construct random matrices which differ only in their order of rows and/or columns. ?latm2 is\nused to construct band matrices while avoiding calling the random number generator for entries outside the\nband (and therefore generating random numbers in different orders for different pivot orders).\nThe matrix whose (isub, jsub ) entry is returned is constructed as follows (this routine only computes one\nentry):\n•\nIf isub is outside (1..m) or jsub is outside (1..n), returns zero (this is convenient for generating\nmatrices in band format).\n•\nGenerate a matrix A with random entries of distribution idist.\n•\nSet the diagonal to D.\n•\nGrade the matrix, if desired, from the left (by dl) and/or from the right (by dr or dl) as specified by\nigrade.\n•\nPermute, if desired, the rows and/or columns as specified by ipvtng and iwork.\n•\nBand the matrix to have lower bandwidth kl and upper bandwidth ku.\n•\nSet random entries to zero as specified by sparse.\nInput Parameters\nm\nNumber of rows of matrix.\nn\nNumber of columns of matrix.\ni\nRow of unpivoted entry to be returned.\nj\nColumn of unpivoted entry to be returned.\nisub\nRow of pivoted entry to be returned.\njsub\nColumn of pivoted entry to be returned.\nkl\nLower bandwidth.\nku\nUpper bandwidth.\nidist\nOn entry, idist specifies the type of distribution to be used to generate a\nrandom matrix.\nfor slatm2 and dlatm2:\n= 1: uniform (0,1)\n= 2: uniform (-1,1)\n= 3: normal (0,1)\nfor clatm2 and zlatm2:\n= 1: real and imaginary parts each uniform (0,1)\n= 2: real and imaginary parts each uniform (-1,1)\n= 3: real and imaginary parts each normal (0,1)\n= 4: complex number uniform in disk(0, 1)\niseed\nArray, size 4.\nSeed for random number generator.\nd\nArray, size (min(i, j)). Diagonal entries of matrix.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1254\n\n\nigrade\nSpecifies grading of matrix as follows:\n= 0: no grading\n= 1: matrix premultiplied by diag( dl )\n= 2: matrix postmultiplied by diag( dr )\n= 3: matrix premultiplied by diag( dl ) and postmultiplied by diag( dr)\n= 4: matrix premultiplied by diag( dl ) and postmultiplied by\ninv( diag( dl ) )\nFor slatm2 and slatm2:\n= 5: matrix premultiplied by diag( dl ) and postmultiplied by diag( dl)\nFor clatm2 and zlatm2:\n= 5: matrix premultiplied by diag( dl ) and postmultiplied by\ndiag( conjg( dl ) )\n= 6: matrix premultiplied by diag( dl ) and postmultiplied by diag( dl)\ndl\nArray, size (i or j, as appropriate).\nLeft scale factors for grading matrix.\ndr\nArray, size (i or j, as appropriate).\nRight scale factors for grading matrix.\nipvtng\nOn entry specifies pivoting permutations as follows:\nIf ipvtng = 0: none.\nIf ipvtng = 1: row pivoting.\nIf ipvtng = 2: column pivoting.\nIf ipvtng = 3: full pivoting, i.e., on both sides.\nsparse\nOn entry, specifies the sparsity of the matrix if sparse matrix is to be\ngenerated. sparse should lie between 0 and 1. A uniform( 0, 1 ) random\nnumber x is generated and compared to sparse; if x is larger the matrix\nentry is unchanged and if x is smaller the entry is set to zero. Thus on the\naverage a fraction sparse of the entries will be set to zero.\niwork\nArray, size (i or j, as appropriate). This array specifies the permutation\nused. The row (or column) originally in position k is in position iwork[k -\n1] after pivoting. This differs from iwork for ?latm2.\nOutput Parameters\nisub\nOn exit, row of pivoted entry is updated.\njsub\nOn exit, column of pivoted entry is updated.\niseed\nOn exit, the seed is updated.\nReturn Values\nThe function returns an entry of a random matrix (for complex variations libmkl_gf_* interface layer/\nlibraries return the result as the parameter res).\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1255\n\n\n?latm5\nGenerates matrices involved in the Generalized\nSylvester equation.\nSyntax\nvoid slatm5 (*prtype, lapack_int *m, lapack_int *n, float *a, lapack_int *lda, float *b,\nlapack_int *ldb, float *c, lapack_int *ldc, float *d, lapack_int *ldd, float *e,\nlapack_int *lde, float *f, lapack_int *ldf, float *r, lapack_int *ldr, float *l,\nlapack_int *ldl, float *alpha, lapack_int *qblcka, lapack_int *qblckb);\nvoid dlatm5 (*prtype, lapack_int *m, lapack_int *n, double *a, lapack_int *lda, double\n*b, lapack_int *ldb, double *c, lapack_int *ldc, double *d, lapack_int *ldd, double *e,\nlapack_int *lde, double *f, lapack_int *ldf, double *r, lapack_int *ldr, double *l,\nlapack_int *ldl, double *alpha, lapack_int *qblcka, lapack_int *qblckb);\nvoid clatm5 (*prtype, lapack_int *m, lapack_int *n, lapack_complex_float *a, lapack_int\n*lda, lapack_complex_float *b, lapack_int *ldb, lapack_complex_float *c, lapack_int \n*ldc, lapack_complex_float *d, lapack_int *ldd, lapack_complex_float *e, lapack_int \n*lde, lapack_complex_float *f, lapack_int *ldf, lapack_complex_float *r, lapack_int\n*ldr, lapack_complex_float *l, lapack_int *ldl, float *alpha, lapack_int *qblcka,\nlapack_int *qblckb);\nvoid zlatm5 (*prtype, lapack_int *m, lapack_int *n, lapack_complex_double *a,\nlapack_int *lda, lapack_complex_double *b, lapack_int *ldb, lapack_complex_double *c,\nlapack_int *ldc, lapack_complex_double *d, lapack_int *ldd, lapack_complex_double *e,\nlapack_int *lde, lapack_complex_double *f, lapack_int *ldf, lapack_complex_double *r,\nlapack_int *ldr, lapack_complex_double *l, lapack_int *ldl, float *alpha, lapack_int\n*qblcka, lapack_int *qblckb);\nInclude Files\n•\nmkl.h\nDescription\nThe ?latm5 routine generates matrices involved in the Generalized Sylvester equation:\nA * R - L * B = C\nD * R - L * E = F\nThey also satisfy the diagonalization condition:\nInput Parameters\nprtype\nSpecifies the type of matrices to generate.\n•\nIf prtype = 1, A and B are Jordan blocks, D and E are identity\nmatrices.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1256\n\n\nA:\nIf (i == j) then Ai, j = 1.0.\nIf (j == i + 1) then Ai, j = -1.0.\nOtherwise Ai, j = 0.0, i, j = 1...m\nB:\nIf (i == j) then Bi, j = 1.0 - alpha.\nIf (j == i + 1) then Bi, j = 1.0 .\nOtherwise Bi, j = 0.0, i, j = 1...n.\nD:\nIf (i == j) then Di, j = 1.0.\nOtherwise Di, j = 0.0, i, j = 1...m.\nE:\nIf (i == j) then Ei, j = 1.0\nOtherwise Ei, j = 0.0, i, j = 1...n.\nL = R are chosen from [-10...10], which specifies the right hand sides\n(C, F).\n•\nIf prtype = 2 or 3: Triangular and/or quasi- triangular.\nA:\nIf (i ≤ j) then Ai, j = [-1...1].\nOtherwise Ai, j = 0.0, i, j = 1...M.\nIf (prtype = 3) then Ak + 1, k + 1 = Ak, k;\nAk + 1, k = [-1...1];\nsign(Ak, k + 1) = -(sign(Ak + 1, k).\nk = 1, m- 1, qblcka\nB :\nIf (i ≤ j) then Bi, j = [-1...1].\nOtherwise Bi, j = 0.0, i, j = 1...n.\nIf (prtype = 3) thenBk + 1, k + 1 = Bk, k\nBk + 1, k = [-1...1]\nsign(Bk, k + 1)= -(sign(Bk + 1, k) \nk = 1, n - 1, qblckb.\nD:\nIf (i ≤ j) then Di, j = [-1...1].\nOtherwise Di, j = 0.0, i, j = 1...m.\nE:\nIf (i <= j) then Ei, j = [-1...1].\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1257\n\n\nOtherwise Ei, j = 0.0, i, j = 1...N.\nL, R are chosen from [-10...10], which specifies the right hand sides (C,\nF).\n•\nIf prtype = 4 Full\n   Ai, j = [-10...10]\n   Di, j = [-1...1] i,j = 1...m\n   Bi, j = [-10...10]\n   Ei, j = [-1...1] i,j = 1...n\n   Ri, j = [-10...10]\n   Li, j = [-1...1] i = 1..m ,j = 1...n\nL and R specifies the right hand sides (C, F).\n•\nIf prtype = 5 special case common and/or close eigs.\nm\nSpecifies the order of A and D and the number of rows in C, F, R and L.\nn\nSpecifies the order of B and E and the number of columns in C, F, R and L.\nlda\nThe leading dimension of a.\nldb\nThe leading dimension of b.\nldc\nThe leading dimension of c.\nldd\nThe leading dimension of d.\nlde\nThe leading dimension of e.\nldf\nThe leading dimension of f.\nldr\nThe leading dimension of r.\nldl\nThe leading dimension of l.\nalpha\nParameter used in generating prtype = 1 and 5 matrices.\nqblcka\nWhen prtype = 3, specifies the distance between 2-by-2 blocks on the\ndiagonal in A. Otherwise, qblcka is not referenced. qblcka > 1.\nqblckb\nWhen prtype = 3, specifies the distance between 2-by-2 blocks on the\ndiagonal in B. Otherwise, qblckb is not referenced. qblckb > 1.\nOutput Parameters\na\nArray, size lda*m. On exit a contains them-by-m array A initialized\naccording to prtype.\nb\nArray, size ldb*n. On exit b contains the n-by-n array B initialized\naccording to prtype.\nc\nArray, size ldc*n. On exit c contains the m-by-n array C initialized\naccording to prtype.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1258\n\n\nd\nArray, size ldd*m. On exit d contains the m-by-m array D initialized\naccording to prtype.\ne\nArray, size lde*n. On exit e contains the n-by-n array E initialized according\nto prtype.\nf\nArray, size ldf*n. On exit f contains the m-by-n array F initialized\naccording to prtype.\nr\nArray, size ldr*n. On exit R contains the m-by-n array R initialized\naccording to prtype.\nl\nArray, size ldl*n. On exit l contains the m-by-narray L initialized according\nto prtype.\n?latm6\nGenerates test matrices for the generalized eigenvalue\nproblem, their corresponding right and left\neigenvector matrices, and also reciprocal condition\nnumbers for all eigenvalues and the reciprocal\ncondition numbers of eigenvectors corresponding to\nthe 1th and 5th eigenvalues.\nSyntax\nvoid slatm6 (lapack_int *type, lapack_int *n, float *a, lapack_int *lda, float *b, float\n*x, lapack_int *ldx, float *y, lapack_int *ldy, float *alpha, float *beta, float *wx,\nfloat *wy, float *s, float *dif);\nvoid dlatm6 (lapack_int *type, lapack_int *n, double *a, lapack_int *lda, double *b,\ndouble *x, lapack_int *ldx, double *y, lapack_int *ldy, double *alpha, double *beta,\ndouble *wx, double *wy, double *s, double *dif);\nvoid clatm6 (lapack_int *type, lapack_int *n, lapack_complex_float *a, lapack_int *lda,\nlapack_complex_float *b, lapack_complex_float *x, lapack_int *ldx, lapack_complex_float\n*y, lapack_int *ldy, lapack_complex_float *alpha, lapack_complex_float *beta,\nlapack_complex_float *wx, lapack_complex_float *wy, float *s, float *dif);\nvoid zlatm6 (lapack_int *type, lapack_int *n, lapack_complex_double *a, lapack_int\n*lda, lapack_complex_double *b, lapack_complex_double *x, lapack_int *ldx,\nlapack_complex_double *y, lapack_int *ldy, lapack_complex_double *alpha,\nlapack_complex_double *beta, lapack_complex_double *wx, lapack_complex_double *wy,\ndouble *s, double *dif);\nInclude Files\n•\nmkl.h\nDescription\nThe ?latm6 routine generates test matrices for the generalized eigenvalue problem, their corresponding right\nand left eigenvector matrices, and also reciprocal condition numbers for all eigenvalues and the reciprocal\ncondition numbers of eigenvectors corresponding to the 1th and 5th eigenvalues.\nThere two kinds of test matrix pairs:\n       (A, B)= inverse(YH) * (Da, Db) * inverse(X)\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1259\n\n\nType 1:\nType 2:\nIn both cases the same inverse(YH) and inverse(X) are used to compute (A, B), giving the exact eigenvectors\nto (A,B) as (YH, X):\n,\nwhere a, b, x and y will have all values independently of each other.\nInput Parameters\ntype\nSpecifies the problem type.\nn\nSize of the matrices A and B.\nlda\nThe leading dimension of a and of b.\nldx\nThe leading dimension of x.\nldy\nThe leading dimension of y.\nalpha, beta\nWeighting constants for matrix A.\nwx\nConstant for right eigenvector matrix.\nwy\nConstant for left eigenvector matrix.\nOutput Parameters\na\nArray, size lda*n. On exit, a contains the n-by-n matrix initialized\naccording to type.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1260\n\n\nb\nArray, size lda*n. On exit, b contains the n-by-n matrix initialized\naccording to type.\nx\nArray, size ldx*n. On exit, x contains the n-by-n matrix of right\neigenvectors.\ny\nArray, size ldy*n. On exit, y is the n-by-n matrix of left eigenvectors.\ns\nArray, size (n). s[i - 1] is the reciprocal condition number for eigenvalue\ni .\ndif\nArray, size(n). dif[i - 1] is the reciprocal condition number for\neigenvector i .\n?latme\nGenerates random non-symmetric square matrices\nwith specified eigenvalues.\nSyntax\nvoid slatme (lapack_int *n, char *dist, lapack_int *iseed, float *d, lapack_int *mode,\nfloat *cond, float *dmax, char *ei, char *rsign, char *upper, char *sim, float *ds,\nlapack_int *modes, float *conds, lapack_int *kl, lapack_int *ku, float *anorm, float *a,\nlapack_int *lda, float *work, lapack_int *info);void dlatme (lapack_int *n, char *dist,\nlapack_int *iseed, double *d, lapack_int *mode, double *cond, double *dmax, char *ei,\nchar *rsign, char *upper, char *sim, double *ds, lapack_int *modes, double *conds,\nlapack_int *kl, lapack_int *ku, double *anorm, double *a, lapack_int *lda, double *work,\nlapack_int *info);void clatme (lapack_int *n, char *dist, lapack_int *iseed,\nlapack_complex_float *d, lapack_int *mode, float *cond, lapack_complex_float *dmax,\nchar *ei, char *rsign, char *upper, char *sim, float *ds, lapack_int *modes, float\n*conds, lapack_int *kl, lapack_int *ku, float *anorm, lapack_complex_float *a,\nlapack_int *lda, lapack_complex_float *work, lapack_int *info);void zlatme (lapack_int\n*n, char *dist, lapack_int *iseed, lapack_complex_double *d, lapack_int *mode, double\n*cond, lapack_complex_double *dmax, char *ei, char *rsign, char *upper, char *sim,\ndouble *ds, lapack_int *modes, double *conds, lapack_int *kl, lapack_int *ku, double\n*anorm, lapack_complex_double *a, lapack_int *lda, lapack_complex_double *work,\nlapack_int *info);\nInclude Files\n•\nmkl.h\nDescription\nThe ?latme routine generates random non-symmetric square matrices with specified eigenvalues. ?latme\noperates by applying the following sequence of operations:\n1.\nSet the diagonal to d, where d may be input or computed according to mode, cond, dmax, and rsign as\ndescribed below.\n2.\nIf upper = 'T', the upper triangle of a is set to random values out of distribution dist.\n3.\nIf sim='T', a is multiplied on the left by a random matrix X, whose singular values are specified by ds,\nmodes, and conds, and on the right by X inverse.\n4.\nIf kl < n-1, the lower bandwidth is reduced to kl using Householder transformations. If ku < n-1,\nthe upper bandwidth is reduced to ku.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1261\n\n\n5.\nIf anorm is not negative, the matrix is scaled to have maximum-element-norm anorm.\nNOTE\nSince the matrix cannot be reduced beyond Hessenberg form, no packing options are\navailable.\nInput Parameters\nn\nThe number of columns (or rows) of A.\ndist\nOn entry, dist specifies the type of distribution to be used to generate the\nrandom eigen-/singular values, and on the upper triangle (see upper).\nIf dist = 'U': uniform( 0, 1 )\nIf dist = 'S': uniform( -1, 1 )\nIf dist = 'N': normal( 0, 1 )\nIf dist = 'D': uniform on the complex disc |z| < 1.\niseed\nArray, size 4.\nOn entry iseed specifies the seed of the random number generator. The\nelements should lie between 0 and 4095 inclusive, and iseed[3] should be\nodd. The random number generator uses a linear congruential sequence\nlimited to small integers, and so should produce machine independent\nrandom numbers.\nd\nArray, size (n). This array is used to specify the eigenvalues of A.\nIf mode = 0, then d is assumed to contain the eigenvalues. Otherwise they\nare computed according to mode, cond, dmax, and rsign and placed in d.\nmode\nOn entry mode describes how the eigenvalues are to be specified:\nmode = 0 means use d (with ei for slatme and dlatme) as input.\nmode = 1 sets d[0] = 1 and d(2:n]=1.0/cond.\nmode = 2 sets d[0:n - 2] = 1 and d[n - 1]=1.0/cond.\nmode = 3 sets d[i - 1] = cond**(-(i-1)/(n-1)).\nmode = 4 sets d[i - 1] = 1 - (i-1)/(n-1)*(1 - 1/cond).\nmode = 5 sets d to random numbers in the range ( 1/cond , 1 ) such\nthat their logarithms are uniformly distributed.\nmode = 6 sets d to random numbers from same distribution as the rest of\nthe matrix.\nmode < 0 has the same meaning as abs(mode), except that the order of\nthe elements of d is reversed.\nThus if mode is between 1 and 4, d has entries ranging from 1 to 1/cond, if\nbetween -1 and -4, d has entries ranging from 1/cond to 1.\ncond\nOn entry, this is used as described under mode above. If used, it must be ≥\n1.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1262\n\n\ndmax\nIf mode is not -6, 0 or 6, the contents of d as computed according to mode\nand cond are scaled by dmax / max(abs(d[i - 1])). Note that dmax\nneeds not be positive or real: if dmax is negative or complex (or zero), d will\nbe scaled by a negative or complex number (or zero). If rsign='F' then\nthe largest (absolute) eigenvalue will be equal to dmax.\nei\nUsed by slatme and dlatme only.\nArray, size (n).\nIf mode = 0, and ei[0]is not ' ' (space character), this array specifies\nwhich elements of d (on input) are real eigenvalues and which are the real\nand imaginary parts of a complex conjugate pair of eigenvalues. The\nelements of ei may then only have the values 'R' and 'I'.\nIf ei[j - 1] = 'R' and ei[j] = 'I', then the j -th eigenvalue is\ncmplx( d[j - 1] , d[j] ), and the (j +1)-th is the complex conjugate\nthereof.\nIf ei[j - 1] = ei[j]='R', then the j-th eigenvalue is d[j - 1] (i.e.,\nreal). ei[0] may not be 'I', nor may two adjacent elements of ei both\nhave the value 'I'.\nIf mode is not 0, then ei is ignored. If mode is 0 and ei[0] = ' ', then\nthe eigenvalues will all be real.\nrsign\nIf mode is not 0, 6, or -6, and rsign = 'T', then the elements of d, as\ncomputed according to mode and cond, are multiplied by a random sign (+1\nor -1) for slatme and dlatme or by a complex number from the unit circle\n|z| = 1 for clatme and zlatme.\nIf rsign = 'F', the elements of d are not multiplied. rsign may only have\nthe values 'T' or 'F'.\nupper\nIf upper = 'T', then the elements of a above the diagonal will be set to\nrandom numbers out of dist.\nIf upper = 'F', they will not. upper may only have the values 'T' or 'F'.\nsim\nIf sim = 'T', then a will be operated on by a \"similarity transform\", i.e.,\nmultiplied on the left by a matrix X and on the right by X inverse. X = USV,\nwhere U and V are random unitary matrices and S is a (diagonal) matrix of\nsingular values specified by ds, modes, and conds.\nIf sim = 'F', then a will not be transformed.\nds\nThis array is used to specify the singular values of X, in the same way that\nd specifies the eigenvalues of a. If mode = 0, the ds contains the singular\nvalues, which may not be zero.\nmodes\nSimilar to mode, but for specifying the diagonal of S. modes = -6 and +6\nare not allowed (since they would result in randomly ill-conditioned\neigenvalues.)\nconds\nSimilar to cond, but for specifying the diagonal of S.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1263\n\n\nkl\nThis specifies the lower bandwidth of the matrix. kl = 1 specifies upper\nHessenberg form. If kl is at least n-1, then A will have full lower\nbandwidth.\nku\nThis specifies the upper bandwidth of the matrix. ku = 1 specifies lower\nHessenberg form.\nIf ku is at least n-1, then a will have full upper bandwidth.\nIf ku and ku are both at least n-1, then a will be dense. Only one of ku and\nkl may be less than n-1.\nanorm\nIf anorm is not negative, then a is scaled by a non-negative real number to\nmake the maximum-element-norm of a to be anorm.\nlda\nNumber of rows of matrix A.\nwork\nArray, size (3*n). Workspace.\nOutput Parameters\niseed\nOn exit, the seed is updated.\nd\nModified if mode is nonzero.\nds\nModified if mode is nonzero.\na\nArray, size lda*n. On exit, a is the desired test matrix.\ninfo\nIf info = 0, execution is successful.\nIf info = -1, n is negative .\nIf info = -2, dist is an illegal string.\nIf info = -5, mode is not in range -6 to 6.\nIf info = -6, cond is less than 1.0, and mode is not -6, 0, or 6 .\nIf info = -9, rsign is not 'T' or 'F' .\nIf info = -10, upper is not 'T' or 'F'.\nIf info = -11, sim is not 'T' or 'F'.\nIf info = -12, modes = 0 and ds has a zero singular value.\nIf info = -13, modes is not in the range -5 to 5.\nIf info = -14, modes is nonzero and conds is less than 1. .\nIf info = -15, kl is less than 1.\nIf info = -16, ku is less than 1, or kl and ku are both less than n-1.\nIf info = -19, lda is less than m.\nIf info = 1, error return from ?latm1 (computing d) .\nIf info = 2, cannot scale to dmax (max. eigenvalue is 0) .\nIf info = 3, error return from slatm1(for slatme and clatme), dlatm1\n(for dlatme and zlatme) .\nIf info = 4, error return from ?large.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1264\n\n\nIf info = 5, zero singular value from slatm1(for slatme and clatme),\ndlatm1(for dlatme and zlatme).\n?latmr\nGenerates random matrices of various types.\nSyntax\nvoid slatmr (lapack_int *m, lapack_int *n, char *dist, lapack_int *iseed, char *sym,\nfloat *d, lapack_int *mode, float *cond, float *dmax, char *rsign, char *grade, float\n*dl, lapack_int *model, float *condl, float *dr, lapack_int *moder, float *condr, char\n*pivtng, lapack_int *ipivot, lapack_int *kl, lapack_int *ku, float *sparse, float\n*anorm, char *pack, float *a, lapack_int *lda, lapack_int *iwork, lapack_int *info);\nvoid dlatmr (lapack_int *m, lapack_int *n, char *dist, lapack_int *iseed, char *sym,\ndouble *d, lapack_int *mode, double *cond, double *dmax, char *rsign, char *grade,\ndouble *dl, lapack_int *model, double *condl, double *dr, lapack_int *moder, double\n*condr, char *pivtng, lapack_int *ipivot, lapack_int *kl, lapack_int *ku, double\n*sparse, double *anorm, char *pack, double *a, lapack_int *lda, lapack_int *iwork,\nlapack_int *info);\nvoid clatmr (lapack_int *m, lapack_int *n, char *dist, lapack_int *iseed, char *sym,\nlapack_complex *d, lapack_int *mode, float *cond, lapack_complex *dmax, char *rsign,\nchar *grade, lapack_complex *dl, lapack_int *model, float *condl, lapack_complex *dr,\nlapack_int *moder, float *condr, char *pivtng, lapack_int *ipivot, lapack_int *kl,\nlapack_int *ku, float *sparse, float *anorm, char *pack, float *a, lapack_int *lda,\nlapack_int *iwork, lapack_int *info);\nvoid zlatmr (lapack_int *m, lapack_int *n, char *dist, lapack_int *iseed, char *sym,\nlapack_complex_double *d, lapack_int *mode, float *cond, lapack_complex_double *dmax,\nchar *rsign, char *grade, lapack_complex_double *dl, lapack_int *model, float *condl,\nlapack_complex_double *dr, lapack_int *moder, float *condr, char *pivtng, lapack_int\n*ipivot, lapack_int *kl, lapack_int *ku, float *sparse, float *anorm, char *pack, float\n*a, lapack_int *lda, lapack_int *iwork, lapack_int *info);\nDescription\nThe ?latmr routine operates by applying the following sequence of operations:\n1.\nGenerate a matrix A with random entries of distribution dist:\nIf sym = 'S', the matrix is symmetric,\nIf sym = 'H', the matrix is Hermitian,\nIf sym = 'N', the matrix is nonsymmetric.\n2.\nSet the diagonal to D, where D may be input or computed according to mode, cond, dmax and rsign as\ndescribed below.\n3.\nGrade the matrix, if desired, from the left or right as specified by grade. The inputs dl, model, condl,\ndr, moder and condr also determine the grading as described below.\n4.\nPermute, if desired, the rows and/or columns as specified by pivtng and ipivot.\n5.\nSet random entries to zero, if desired, to get a random sparse matrix as specified by sparse.\n6.\nMake A a band matrix, if desired, by zeroing out the matrix outside a band of lower bandwidth kl and\nupper bandwidth ku.\n7.\nScale A, if desired, to have maximum entry anorm.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1265\n\n\n8.\nPack the matrix if desired. See options specified by the pack parameter.\nNOTE\nIf two calls to ?latmr differ only in the pack parameter, they generate mathematically equivalent\nmatrices. If two calls to ?latmr both have full bandwidth (kl = m-1 and ku = n-1), and differ only in\nthe pivtng and pack parameters, then the matrices generated differ only in the order of the rows and\ncolumns, and otherwise contain the same data. This consistency cannot be and is not maintained with\nless than full bandwidth.\nInput Parameters\nm\nNumber of rows of A.\nn\nNumber of columns of A.\ndist\nOn entry, dist specifies the type of distribution to be used to generate a\nrandom matrix .\nIf dist = 'U', real and imaginary parts are independent uniform( 0, 1 ).\nIf dist = 'S', real and imaginary parts are independent uniform( -1, 1 ).\nIf dist = 'N', real and imaginary parts are independent normal( 0, 1 ).\nIf dist = 'D', distribution is uniform on interior of unit disk.\niseed\nArray, size 4.\nOn entry, iseed specifies the seed of the random number generator. They\nshould lie between 0 and 4095 inclusive, and iseed[3] should be odd. The\nrandom number generator uses a linear congruential sequence limited to\nsmall integers, and so should produce machine independent random\nnumbers.\nsym\nIf sym = 'S', generated matrix is symmetric.\nIf sym = 'H', generated matrix is Hermitian.\nIf sym = 'N', generated matrix is nonsymmetric.\nd\nOn entry this array specifies the diagonal entries of the diagonal of A. d\nmay either be specified on entry, or set according to mode and cond as\ndescribed below. If the matrix is Hermitian, the real part of d is taken. May\nbe changed on exit if mode is nonzero.\nmode\nOn entry describes how d is to be used:\nmode = 0 means use d as input.\nmode = 1 sets d[0]=1 and d[1:n - 1]=1.0/cond.\nmode = 2 sets d[0:n - 2]=1 and d[n - 1]=1.0/cond.\nmode = 3 sets d[i - 1]=cond**(-(i-1)/(n-1)).\nmode = 4 sets d[i - 1]=1 - (i-1)/(n-1)*(1 - 1/cond).\nmode = 5 sets d to random numbers in the range ( 1/cond , 1 ) such\nthat their logarithms are uniformly distributed.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1266\n\n\nmode = 6 sets d to random numbers from same distribution as the rest of\nthe matrix.\nmode < 0 has the same meaning as abs(mode), except that the order of\nthe elements of d is reversed.\nThus if mode is between 1 and 4, d has entries ranging from 1 to 1/cond, if\nbetween -1 and -4, D has entries ranging from 1/cond to 1.\ncond\nOn entry, used as described under mode above. If used, cond must be ≥ 1.\ndmax\nIf mode is not -6, 0, or 6, the diagonal is scaled by dmax /\nmax(abs(d[i])), so that maximum absolute entry of diagonal is\nabs(dmax). If dmax is complex (or zero), the diagonal is scaled by a\ncomplex number (or zero).\nrsign\nIf mode is not -6, 0, or 6, specifies the sign of the diagonal as follows:\nFor slatmr and dlatmr, if rsign = 'T', diagonal entries are multiplied 1\nor -1 with a probability of 0.5.\nFor clatmr and zlatmr, if rsign = 'T', diagonal entries are multiplied by\na random complex number uniformly distributed with absolute value 1.\nIf rsign = 'F', diagonal entries are unchanged.\ngrade\nSpecifies grading of matrix as follows:\nIf grade = 'N', there is no grading\nIf grade = 'L', matrix is premultiplied by diag( dl) (only if matrix is\nnonsymmetric)\nIf grade = 'R', matrix is postmultiplied by diag( dr ) (only if matrix is\nnonsymmetric)\nIf grade = 'B', matrix is premultiplied by diag( dl ) and postmultiplied by\ndiag( dr ) (only if matrix is nonsymmetric)\nIf grade = 'H', matrix is premultiplied by diag( dl ) and postmultiplied by\ndiag( conjg(dl) ) (only if matrix is Hermitian or nonsymmetric)\nIf grade = 'S', matrix is premultiplied by diag(dl ) and postmultiplied by\ndiag( dl ) (only if matrix is symmetric or nonsymmetric)\nIf grade = 'E', matrix is premultiplied by diag( dl ) and postmultiplied by\ninv( diag( dl ) ) (only if matrix is nonsymmetric)\nNOTE\nif grade = 'E', then m must equal n.\ndl\nArray, size (m).\nIf model = 0, then on entry this array specifies the diagonal entries of a\ndiagonal matrix used as described under grade above.\nIf model is not zero, then dl is set according to model and condl,\nanalogous to the way D is set according to mode and cond (except there is\nno dmax parameter for dl).\nIf grade = 'E', then dl cannot have zero entries.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1267\n\n\nNot referenced if grade = 'N' or 'R'. Changed on exit.\nmodel\nThis specifies how the diagonal array dl is computed, just as mode specifies\nhow D is computed.\ncondl\nWhen model is not zero, this specifies the condition number of the\ncomputed dl.\ndr\nIf moder = 0, then on entry this array specifies the diagonal entries of a\ndiagonal matrix used as described under grade above.\nIf moder is not zero, then dr is set according to moder and condr,\nanalogous to the way d is set according to mode and cond (except there is\nno dmax parameter for dr).\nNot referenced if grade = 'N', 'L', 'H''S' or 'E'.\nmoder\nThis specifies how the diagonal array dr is to be computed, just as mode\nspecifies how d is to be computed.\ncondr\nWhen moder is not zero, this specifies the condition number of the\ncomputed dr.\npivtng\nOn entry specifies pivoting permutations as follows:\nIf pivtng = 'N' or ' ': no pivoting permutation.\nIf pivtng = 'L': left or row pivoting (matrix must be nonsymmetric).\nIf pivtng = 'R': right or column pivoting (matrix must be nonsymmetric).\nIf pivtng = 'B' or 'F': both or full pivoting, i.e., on both sides. In this\ncase, m must equal n.\nIf two calls to ?latmr both have full bandwidth (kl = m - 1 and ku =\nn-1), and differ only in the pivtng and pack parameters, then the matrices\ngenerated differs only in the order of the rows and columns, and otherwise\ncontain the same data. This consistency cannot be maintained with less\nthan full bandwidth.\nipivot\nArray, size (n or m) This array specifies the permutation used. After the\nbasic matrix is generated, the rows, columns, or both are permuted.\nIf row pivoting is selected, ?latmr starts with the last row and interchanges\nrow m and row ipivot[m - 1], then moves to the next-to-last row,\ninterchanging rows [m - 2] and row ipivot[m - 2], and so on. In terms\nof \"2-cycles\", the permutation is (1 ipivot[0]) (2 ipivot[1]) ...\n(mipivot[m - 1]) where the rightmost cycle is applied first. This is the\ninverse of the effect of pivoting in LINPACK. The idea is that factoring (with\npivoting) an identity matrix which has been inverse-pivoted in this way\nshould result in a pivot vector identical to ipivot. Not referenced if pivtng\n= 'N'.\nsparse\nOn entry, specifies the sparsity of the matrix if a sparse matrix is to be\ngenerated. sparse should lie between 0 and 1. To generate a sparse\nmatrix, for each matrix entry a uniform ( 0, 1 ) random number x is\ngenerated and compared to sparse; if x is larger the matrix entry is\nunchanged and if x is smaller the entry is set to zero. Thus on the average\na fraction sparse of the entries is set to zero.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1268\n\n\nkl\nOn entry, specifies the lower bandwidth of the matrix. For example, kl = 0\nimplies upper triangular, kl = 1 implies upper Hessenberg, and kl at least\nm-1 implies the matrix is not banded. Must equal ku if matrix is symmetric\nor Hermitian.\nku\nOn entry, specifies the upper bandwidth of the matrix. For example, ku = 0\nimplies lower triangular, ku = 1 implies lower Hessenberg, and kuat least\nn-1 implies the matrix is not banded. Must equal kl if matrix is symmetric\nor Hermitian.\nanorm\nOn entry, specifies maximum entry of output matrix (output matrix is\nmultiplied by a constant so that its largest absolute entry equal anorm) if\nanorm is nonnegative. If anorm is negative no scaling is done.\npack\nOn entry, specifies packing of matrix as follows:\nIf pack = 'N': no packing\nIf pack = 'U': zero out all subdiagonal entries (if symmetric or Hermitian)\nIf pack = 'L': zero out all superdiagonal entries (if symmetric or\nHermitian)\nIf pack = 'C': store the upper triangle columnwise (only if matrix\nsymmetric or Hermitian or square upper triangular)\nIf pack = 'R': store the lower triangle columnwise (only if matrix\nsymmetric or Hermitian or square lower triangular) (same as upper half\nrowwise if symmetric) (same as conjugate upper half rowwise if Hermitian)\nIf pack = 'B': store the lower triangle in band storage scheme (only if\nmatrix symmetric or Hermitian)\nIf pack = 'Q': store the upper triangle in band storage scheme (only if\nmatrix symmetric or Hermitian)\nIf pack = 'Z': store the entire matrix in band storage scheme (pivoting\ncan be provided for by using this option to store A in the trailing rows of the\nallocated storage)\nUsing these options, the various LAPACK packed and banded storage\nschemes can be obtained:\nLAPACK storage scheme\nValue of pack\nGB\n'Z'\nPB, HB or TB\n'B' or 'Q'\nPP, HP or TP\n'C' or 'R'\nIf two calls to ?latmr differ only in the pack parameter, they generate\nmathematically equivalent matrices.\nlda\nOn entry, lda specifies the first dimension of a as declared in the calling\nprogram.\nIf pack = 'N', 'U' or 'L', lda must be at least max( 1, m ).\nIf pack = 'C' or 'R', lda must be at least 1.\nIf pack = 'B', or 'Q', lda must be min( ku + 1, n ).\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1269\n\n\nIf pack = 'Z', lda must be at least kuu + kll + 1, where kuu =\nmin( ku, n-1 ) and kll = min( kl, n-1 ).\niwork\nArray, size (n or m). Workspace. Not referenced if pivtng = 'N'. Changed\non exit.\nOutput Parameters\niseed\nOn exit, the seed is changed.\nd\nMay be changed on exit if mode is nonzero.\ndl\nOn exit, array is changed.\ndr\nOn exit, array is changed.\na\nOn exit, a is the desired test matrix. Only those entries of a which are\nsignificant on output is referenced (even if a is in packed or band\nstorage format). The unoccupied corners of a in band format are\nzeroed out.\ninfo\nIf info = 0, the execution is successful.\nIf info = -1, m is negative or unequal to n and sym = 'S' or 'H'.\nIf info = -2, n is negative .\nIf info = -3, dist is an illegal string.\nIf info = -5, sym is an illegal string..\nIf info = -7, mode is not in range -6 to 6.\nIf info = -8, cond is less than 1.0, and mode is neither -6, 0 nor 6.\nIf info = -10, mode is neither -6, 0 nor 6 and rsign is an illegal\nstring.\nIf info = -11, grade is an illegal string, or grade = 'E' and m is\nnot equal to n, or grade='L', 'R', 'B', 'S' or 'E' and sym =\n'H', or grade = 'L', 'R', 'B', 'H' or 'E' and sym = 'S'\nIf info = -12,grade = 'E'and dl contains zero .\nIf info = -13, model is not in range -6 to 6 and grade = 'L',\n'B', 'H', 'S' or 'E' .\nIf info = -14, condl is less than 1.0, grade = 'L', 'B', 'H',\n'S' or 'E', and model is neither -6, 0 nor 6.\nIf info = -16, moder is not in range -6 to 6 and grade = 'R' or\n'B' .\nIf info = -17, condr is less than 1.0, grade = 'R' or 'B', and\nmoder is neither -6, 0 nor 6 .\nIf info = -18, pivtng is an illegal string, or pivtng = 'B' or 'F'\nand m is not equal to n, or pivtng = 'L' or 'R' and sym = 'S' or\n'H'.\nIf info = -19, ipivot contains out of range number and pivtng is\nnot equal to 'N' .\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1270\n\n\nIf info = -20, kl is negative.\nIf info = -21, ku is negative, or sym = 'S' or 'H' and ku not\nequal to kl .\nIf info = -22, sparse is not in range 0 to 1.\nIf info = -24, pack is an illegal string, or pack = 'U', 'L', 'B'\nor 'Q' and sym = 'N', or pack = 'C' and sym = 'N' and either\nkl is not equal to 0 or n is not equal to m, or pack = 'R' and sym =\n'N', and either ku is not equal to 0 or n is not equal to m .\nIf info = -26, lda is too small .\nIf info = 1, error return from ?latm1 (computing D ) .\nIf info = 2, cannot scale to dmax (max. entry is 0) .\nIf info = 3, error return from ?latm1(computing dl) .\nIf info = 4, error return from ?latm1(computing dr) .\nIf info = 5, anorm is positive, but matrix constructed prior to\nattempting to scale it to have norm anorm, is zero .\n?lauum\nComputes the product U*UT(U*UH) or LT*L (LH*L),\nwhere U and L are upper or lower triangular matrices\n(blocked algorithm).\nSyntax\nlapack_int LAPACKE_slauum (int matrix_layout , char uplo , lapack_int n , float * a ,\nlapack_int lda );\nlapack_int LAPACKE_dlauum (int matrix_layout , char uplo , lapack_int n , double * a ,\nlapack_int lda );\nlapack_int LAPACKE_clauum (int matrix_layout , char uplo , lapack_int n ,\nlapack_complex_float * a , lapack_int lda );\nlapack_int LAPACKE_zlauum (int matrix_layout , char uplo , lapack_int n ,\nlapack_complex_double * a , lapack_int lda );\nInclude Files\n•\nmkl.h\nDescription\nThe routine ?lauum computes the product U*UT or LT*L for real flavors, and U*UH or LH*L for complex\nflavors. Here the triangular factor U or L is stored in the upper or lower triangular part of the array a.\nIf uplo = 'U' or 'u', then the upper triangle of the result is stored, overwriting the factor U in A.\nIf uplo = 'L' or 'l', then the lower triangle of the result is stored, overwriting the factor L in A.\nThis is the blocked form of the algorithm, calling BLAS Level 3 Routines.\nInput Parameters\nA <datatype> placeholder, if present, is used for the C interface data types in the C interface section above.\nSee C Interface Conventions for the C interface principal conventions and type definitions.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1271\n\n\nuplo\nSpecifies whether the triangular factor stored in the array a is upper or\nlower triangular:\n= 'U': Upper triangular\n= 'L': Lower triangular\nn\nThe order of the triangular factor U or L. n≥ 0.\na\nArray of size max(1,lda *n).\nOn entry, the triangular factor U or L.\nlda\nThe leading dimension of the array a. lda≥ max(1,n).\nOutput Parameters\na\nOn exit,\nif uplo = 'U', then the upper triangle of a is overwritten with the upper\ntriangle of the product U*UT(U*UH);\nif uplo = 'L', then the lower triangle of a is overwritten with the lower\ntriangle of the product LT*L (LH*L).\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info = -k, the k-th parameter had an illegal value.\nIf info = -1011, memory allocation error occurred.\n?syswapr\nApplies an elementary permutation on the rows and\ncolumns of a symmetric matrix.\nSyntax\nlapack_int LAPACKE_ssyswapr (int matrix_layout , char uplo , lapack_int n , float * a ,\nlapack_int i1 , lapack_int i2 );\nlapack_int LAPACKE_dsyswapr (int matrix_layout , char uplo , lapack_int n , double *\na , lapack_int i1 , lapack_int i2 );\nlapack_int LAPACKE_csyswapr (int matrix_layout , char uplo , lapack_int n ,\nlapack_complex_float * a , lapack_int i1 , lapack_int i2 );\nlapack_int LAPACKE_zsyswapr (int matrix_layout , char uplo , lapack_int n ,\nlapack_complex_double * a , lapack_int i1 , lapack_int i2 );\nInclude Files\n•\nmkl.h\nDescription\nThe routine applies an elementary permutation on the rows and columns of a symmetric matrix.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1272\n\n\nInput Parameters\nA <datatype> placeholder, if present, is used for the C interface data types in the C interface section above.\nSee C Interface Conventions for the C interface principal conventions and type definitions.\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR) or\ncolumn major ( LAPACK_COL_MAJOR ).\nuplo\nMust be 'U' or 'L'.\nIndicates how the input matrix A has been factored:\nIf uplo = 'U', the array a stores the upper triangular factor U of the\nfactorization A = U*D*UT.\nIf uplo = 'L', the array a stores the lower triangular factor L of the\nfactorization A = L*D*LT.\nn\nThe order of matrix A; n≥ 0.\nnrhs\nThe number of right-hand sides; nrhs≥ 0.\na\nArray of size at least max(1,lda*n).\nThe array a contains the block diagonal matrix D and the multipliers used to\nobtain the factor U or L as computed by ?sytrf.\ni1\nIndex of the first row to swap.\ni2\nIndex of the second row to swap.\nOutput Parameters\na\nIf info = 0, the symmetric inverse of the original matrix.\nIf info = 'U', the upper triangular part of the inverse is formed and the part of\nA below the diagonal is not referenced.\nIf info = 'L', the lower triangular part of the inverse is formed and the part of\nA above the diagonal is not referenced.\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nIf info = -1011, memory allocation error occurred.\nSee Also\n?sytrf\n?heswapr\nApplies an elementary permutation on the rows and\ncolumns of a Hermitian matrix.\nSyntax\nlapack_int LAPACKE_cheswapr (int matrix_layout, char uplo, lapack_int n,\nlapack_complex_float* a, lapack_int i1, lapack_int i2);\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1273\n\n\nlapack_int LAPACKE_zheswapr (int matrix_layout, char uplo, lapack_int n,\nlapack_complex_double* a, lapack_int i1, lapack_int i2);\nInclude Files\n•\nmkl.h\nDescription\nThe routine applies an elementary permutation on the rows and columns of a Hermitian matrix.\nInput Parameters\nA <datatype> placeholder, if present, is used for the C interface data types in the C interface section above.\nSee C Interface Conventions for the C interface principal conventions and type definitions.\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR) or\ncolumn major ( LAPACK_COL_MAJOR ).\nuplo\nMust be 'U' or 'L'.\nIndicates how the input matrix A has been factored:\nIf uplo = 'U', the array a stores the upper triangular factor U of the\nfactorization A = U*D*UH.\nIf uplo = 'L', the array a stores the lower triangular factor L of the\nfactorization A = L*D*LH.\nn\nThe order of matrix A; n≥ 0.\nnrhs\nThe number of right-hand sides; nrhs≥ 0.\na\nArray of size at least max(1,lda*n).\nThe array a contains the block diagonal matrix D and the multipliers used to\nobtain the factor U or L as computed by ?hetrf.\ni1\nIndex of the first row to swap.\ni2\nIndex of the second row to swap.\nOutput Parameters\na\nIf info = 0, the inverse of the original matrix.\nIf info = 'U', the upper triangular part of the inverse is formed and the part of\nA below the diagonal is not referenced.\nIf info = 'L', the lower triangular part of the inverse is formed and the part of\nA above the diagonal is not referenced.\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nIf info = -1011, memory allocation error occurred.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1274\n\n\nSee Also\n?hetrf\n?sfrk\nPerforms a symmetric rank-k operation for matrix in\nRFP format.\nSyntax\nlapack_int LAPACKE_ssfrk (int matrix_layout , char transr , char uplo , char trans ,\nlapack_int n , lapack_int k , float alpha , const float * a , lapack_int lda , float\nbeta , float * c );\nlapack_int LAPACKE_dsfrk (int matrix_layout , char transr , char uplo , char trans ,\nlapack_int n , lapack_int k , double alpha , const double * a , lapack_int lda , double\nbeta , double * c );\nInclude Files\n•\nmkl.h\nDescription\nThe ?sfrk routines perform a matrix-matrix operation using symmetric matrices. The operation is defined as\nC := alpha*A*AT + beta*C,\nor\nC := alpha*AT*A + beta*C,\nwhere:\nalpha and beta are scalars,\nC is an n-by-n symmetric matrix in rectangular full packed (RFP) format,\nA is an n-by-k matrix in the first case and a k-by-n matrix in the second case.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major ( LAPACK_COL_MAJOR ).\ntransr\nif transr = 'N' or 'n', the normal form of RFP C is stored;\nif transr= 'T' or 't', the transpose form of RFP C is stored.\nuplo\nSpecifies whether the upper or lower triangular part of the array c is used.\nIf uplo = 'U' or 'u', then the upper triangular part of the array c is used.\nIf uplo = 'L' or 'l', then the low triangular part of the array c is used.\ntrans\nSpecifies the operation:\nif trans = 'N' or 'n', then C := alpha*A*AT + beta*C;\nif trans = 'T' or 't', then C := alpha*AT*A + beta*C;\nn\nSpecifies the order of the matrix C. The value of n must be at least zero.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1275\n\n\nk\nOn entry with trans = 'N' or 'n', k specifies the number of columns of\nthe matrix A, and on entry with trans = 'T' or 't', k specifies the\nnumber of rows of the matrix A.\nThe value of k must be at least zero.\nalpha\nSpecifies the scalar alpha.\na\nArray, size max(1,lda*ka), where ka is in the following table:\nCol_major\nRow_major\ntrans = 'N'\nk\nn\ntrans = 'T'\nn\nk\nBefore entry with trans = 'N' or 'n', the leading n-by-k part of the array\na must contain the matrix A, otherwise the leading k-by-n part of the array\na must contain the matrix A.\nlda\nSpecifies the leading dimension of a as declared in the calling\n(sub)program. lda is defined by the following table:\nCol_major\nRow_major\ntrans = 'N'\nmax(1,n)\nmax(1,k)\ntrans = 'T'\nmax(1,k)\nmax(1,n)\nbeta\nSpecifies the scalar beta.\nc\nArray, size (n*(n+1)/2 ). Before entry contains the symmetric matrix C in \nRFP format.\nOutput Parameters\nc\nIf trans = 'N' or 'n', then c contains C := alpha*A*A' + beta*C;\nif trans = 'T' or 't', then c contains C := alpha*A'*A + beta*C;\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info < 0, the i-th parameter had an illegal value.\nIf info = -1011, memory allocation error occurred.\n?hfrk\nPerforms a Hermitian rank-k operation for matrix in\nRFP format.\nSyntax\nlapack_int LAPACKE_chfrk( int matrix_layout, char transr, char uplo, char trans,\nlapack_int n, lapack_int k, float alpha, const lapack_complex_float* a, lapack_int lda,\nfloat beta, lapack_complex_float* c );\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1276\n\n\nlapack_int LAPACKE_zhfrk( int matrix_layout, char transr, char uplo, char trans,\nlapack_int n, lapack_int k, double alpha, const lapack_complex_double* a, lapack_int\nlda, double beta, lapack_complex_double* c );\nInclude Files\n•\nmkl.h\nDescription\nThe ?hfrk routines perform a matrix-matrix operation using Hermitian matrices. The operation is defined as\nC := alpha*A*AH + beta*C,\nor\nC := alpha*AH*A + beta*C,\nwhere:\nalpha and beta are real scalars,\nC is an n-by-n Hermitian matrix in RFP format,\nA is an n-by-k matrix in the first case and a k-by-n matrix in the second case.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major ( LAPACK_COL_MAJOR ).\ntransr\nif transr = 'N' or 'n', the normal form of RFP C is stored;\nif transr = 'C' or 'c', the conjugate-transpose form of RFP C is stored.\nuplo\nSpecifies whether the upper or lower triangular part of the array c is used.\nIf uplo = 'U' or 'u', then the upper triangular part of the array c is used.\nIf uplo = 'L' or 'l', then the low triangular part of the array c is used.\ntrans\nSpecifies the operation:\nif trans = 'N' or 'n', then C := alpha*A*AH + beta*C;\nif trans = 'C' or 'c', then C := alpha*AH*A + beta*C.\nn\nSpecifies the order of the matrix C. The value of n must be at least zero.\nk\nOn entry with trans = 'N' or 'n', k specifies the number of columns of\nthe matrix a, and on entry with trans = 'T' or 't' or 'C' or 'c', k\nspecifies the number of rows of the matrix a.\nThe value of k must be at least zero.\nalpha\nSpecifies the scalar alpha.\na\nArray, size max(1,lda*ka), where ka is in the following table:\nCol_major\nRow_major\ntrans = 'N'\nk\nn\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1277\n\n\ntrans = 'T'\nn\nk\nBefore entry with trans = 'N' or 'n', the leading n-by-k part of the array\na must contain the matrix A, otherwise the leading k-by-n part of the array\na must contain the matrix A.\nlda\nSpecifies the leading dimension of a as declared in the calling\n(sub)program. lda is defined by the following table:\nCol_major\nRow_major\ntrans = 'N'\nmax(1,n)\nmax(1,k)\ntrans = 'T'\nmax(1,k)\nmax(1,n)\nbeta\nSpecifies the scalar beta.\nc\nArray, size (n*(n+1)/2 ). Before entry contains the Hermitian matrix C in\nin RFP format.\nOutput Parameters\nc\nIf trans = 'N' or 'n', then c contains C := alpha*A*AH + beta*C;\nif trans = 'C' or 'c', then c contains C := alpha*AH*A + beta*C ;\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info < 0, the i-th parameter had an illegal value.\nIf info = -1011, memory allocation error occurred.\n?tfsm\nSolves a matrix equation (one operand is a triangular\nmatrix in RFP format).\nSyntax\nlapack_int LAPACKE_stfsm (int matrix_layout , char transr , char side , char uplo ,\nchar trans , char diag , lapack_int m , lapack_int n , float alpha , const float * a ,\nfloat * b , lapack_int ldb );\nlapack_int LAPACKE_dtfsm (int matrix_layout , char transr , char side , char uplo ,\nchar trans , char diag , lapack_int m , lapack_int n , double alpha , const double * a ,\ndouble * b , lapack_int ldb );\nlapack_int LAPACKE_ctfsm (int matrix_layout , char transr , char side , char uplo ,\nchar trans , char diag , lapack_int m , lapack_int n , lapack_complex_float alpha ,\nconst lapack_complex_float * a , lapack_complex_float * b , lapack_int ldb );\nlapack_int LAPACKE_ztfsm (int matrix_layout , char transr , char side , char uplo ,\nchar trans , char diag , lapack_int m , lapack_int n , lapack_complex_double alpha ,\nconst lapack_complex_double * a , lapack_complex_double * b , lapack_int ldb );\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1278\n\n\nInclude Files\n•\nmkl.h\nDescription\nThe ?tfsm routines solve one of the following matrix equations:\nop(A)*X = alpha*B,\nor\nX*op(A) = alpha*B,\nwhere:\nalpha is a scalar,\nX and B are m-by-n matrices,\nA is a unit, or non-unit, upper or lower triangular matrix in rectangular full packed (RFP) format.\nop(A) can be one of the following:\n•\nop(A) = A or op(A) = AT for real flavors\n•\nop(A) = A or op(A) = AH for complex flavors\nThe matrix B is overwritten by the solution matrix X.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major ( LAPACK_COL_MAJOR ).\ntransr\nif transr = 'N' or 'n', the normal form of RFP A is stored;\nif transr = 'T' or 't', the transpose form of RFP A is stored;\nif transr = 'C' or 'c', the conjugate-transpose form of RFP A is stored.\nside\nSpecifies whether op(A) appears on the left or right of X in the equation:\nif side = 'L' or 'l', then op(A)*X = alpha*B;\nif side = 'R' or 'r', then X*op(A) = alpha*B.\nuplo\nSpecifies whether the RFP matrix A is upper or lower triangular:\nif uplo = 'U' or 'u', then the matrix is upper triangular;\nif uplo = 'L' or 'l', then the matrix is low triangular.\ntrans\nSpecifies the form of op(A) used in the matrix multiplication:\nif trans = 'N' or 'n', then op(A) = A;\nif trans = 'T' or 't', then op(A) = A';\nif trans = 'C' or 'c', then op(A) = conjg(A').\ndiag\nSpecifies whether the RFP matrix A is unit triangular:\nif diag = 'U' or 'u' then the matrix is unit triangular;\nif diag = 'N' or 'n', then the matrix is not unit triangular.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1279\n\n\nm\nSpecifies the number of rows of B. The value of m must be at least zero.\nn\nSpecifies the number of columns of B. The value of n must be at least zero.\nalpha\nSpecifies the scalar alpha.\nWhen alpha is zero, then a is not referenced and b need not be set before\nentry.\na\nArray, size (n*(n+1)/2). Contains the matrix A in RFP format.\nb\nArray, size max(1, ldb*n) for column major and max(1, ldb*m) for row\nmajor.\nBefore entry, the leading m-by-n part of the array b must contain the right-\nhand side matrix B.\nldb\nSpecifies the leading dimension of b as declared in the calling\n(sub)program. The value of ldb must be at least max(1, m) for column\nmajor and max(1,n) for row major.\nOutput Parameters\nb\nOverwritten by the solution matrix X.\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info < 0, the i-th parameter had an illegal value.\nIf info = -1011, memory allocation error occurred.\n?tfttp\nCopies a triangular matrix from the rectangular full\npacked format (TF) to the standard packed format\n(TP) .\nSyntax\nlapack_int LAPACKE_stfttp (int matrix_layout , char transr , char uplo , lapack_int n ,\nconst float * arf , float * ap );\nlapack_int LAPACKE_dtfttp (int matrix_layout , char transr , char uplo , lapack_int n ,\nconst double * arf , double * ap );\nlapack_int LAPACKE_ctfttp (int matrix_layout , char transr , char uplo , lapack_int n ,\nconst lapack_complex_float * arf , lapack_complex_float * ap );\nlapack_int LAPACKE_ztfttp (int matrix_layout , char transr , char uplo , lapack_int n ,\nconst lapack_complex_double * arf , lapack_complex_double * ap );\nInclude Files\n•\nmkl.h\nDescription\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1280\n\n\nThe routine copies a triangular matrix A from the Rectangular Full Packed (RFP) format to the standard\npacked format. For the description of the RFP format, see Matrix Storage Schemes.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major ( LAPACK_COL_MAJOR ).\ntransr\n= 'N': arf is in the Normal format,\n= 'T': arf is in the Transpose format (for stfttp and dtfttp),\n= 'C': arf is in the Conjugate-transpose format (for ctfttp and ztfttp).\nuplo\nSpecifies whether A is upper or lower triangular:\n= 'U': A is upper triangular,\n= 'L': A is lower triangular.\nn\nThe order of the matrix A. n≥ 0.\narf\nArray, size at least max (1, n*(n+1)/2).\nOn entry, the upper or lower triangular matrix A stored in the RFP format.\nOutput Parameters\nap\nArray, size at least max (1, n*(n+1)/2).\nOn exit, the upper or lower triangular matrix A, packed columnwise in a\nlinear array.\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info < 0, the i-th parameter had an illegal value.\nIf info = -1011, memory allocation error occurred.\n?tfttr\nCopies a triangular matrix from the rectangular full\npacked format (TF) to the standard full format (TR) .\nSyntax\nlapack_int LAPACKE_stfttr (int matrix_layout , char transr , char uplo , lapack_int n ,\nconst float * arf , float * a , lapack_int lda );\nlapack_int LAPACKE_dtfttr (int matrix_layout , char transr , char uplo , lapack_int n ,\nconst double * arf , double * a , lapack_int lda );\nlapack_int LAPACKE_ctfttr (int matrix_layout , char transr , char uplo , lapack_int n ,\nconst lapack_complex_float * arf , lapack_complex_float * a , lapack_int lda );\nlapack_int LAPACKE_ztfttr (int matrix_layout , char transr , char uplo , lapack_int n ,\nconst lapack_complex_double * arf , lapack_complex_double * a , lapack_int lda );\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1281\n\n\nInclude Files\n•\nmkl.h\nDescription\nThe routine copies a triangular matrix A from the Rectangular Full Packed (RFP) format to the standard full\nformat. For the description of the RFP format, see Matrix Storage Schemes.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major ( LAPACK_COL_MAJOR ).\ntransr\n= 'N': arf is in the Normal format,\n= 'T': arf is in the Transpose format (for stfttr and dtfttr),\n= 'C': arf is in the Conjugate-transpose format (for ctfttr and ztfttr).\nuplo\nSpecifies whether A is upper or lower triangular:\n= 'U': A is upper triangular,\n= 'L': A is lower triangular.\nn\nThe order of the matrices arf and a. n≥ 0.\narf\nArray, size at least max (1, n*(n+1)/2).\nOn entry, the upper or lower triangular matrix A stored in the RFP\nformat.\nlda\nThe leading dimension of the array a. lda ≥ max(1,n).\nOutput Parameters\na\nArray, size max(1,lda *n).\nOn exit, the triangular matrix A. If uplo = 'U', the leading n-by-n upper\ntriangular part of the array a contains the upper triangular matrix, and the\nstrictly lower triangular part of a is not referenced. If uplo = 'L', the leading\nn-by-n lower triangular part of the array a contains the lower triangular\nmatrix, and the strictly upper triangular part of a is not referenced.\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info < 0, the i-th parameter had an illegal value.\nIf info = -1011, memory allocation error occurred.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1282\n\n\n?tpqrt2\nComputes a QR factorization of a real or complex\n\"triangular-pentagonal\" matrix, which is composed of\na triangular block and a pentagonal block, using the\ncompact WY representation for Q.\nSyntax\nlapack_int LAPACKE_stpqrt2 (int matrix_layout, lapack_int m, lapack_int n, lapack_int\nl, float * a, lapack_int lda, float * b, lapack_int ldb, float * t, lapack_int ldt);\nlapack_int LAPACKE_dtpqrt2 (int matrix_layout, lapack_int m, lapack_int n, lapack_int\nl, double * a, lapack_int lda, double * b, lapack_int ldb, double * t, lapack_int ldt);\nlapack_int LAPACKE_ctpqrt2 (int matrix_layout, lapack_int m, lapack_int n, lapack_int\nl, lapack_complex_float * a, lapack_int lda, lapack_complex_float * b, lapack_int ldb,\nlapack_complex_float * t, lapack_int ldt );\nlapack_int LAPACKE_ztpqrt2 (int matrix_layout, lapack_int m, lapack_int n, lapack_int\nl, lapack_complex_double * a, lapack_int lda, lapack_complex_double * b, lapack_int\nldb, lapack_complex_double * t, lapack_int ldt );\nInclude Files\n•\nmkl.h\nDescription\nThe input matrix C is an (n+m)-by-n matrix\nwhere A is an n-by-n upper triangular matrix, and B is an m-by-n pentagonal matrix consisting of an (m-l)-\nby-n rectangular matrix B1 on top of an l-by-n upper trapezoidal matrix B2:\nThe upper trapezoidal matrix B2 consists of the first l rows of an n-by-n upper triangular matrix, where 0 ≤\nl ≤ min(m,n). If l=0, B is an m-by-n rectangular matrix. If m=l=n, B is upper triangular. The matrix W\ncontains the elementary reflectors H(i) in the ith column below the diagonal (of A) in the (n+m)-by-n input\nmatrix C so that W can be represented as\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1283\n\n\nThus, V contains all of the information needed for W, and is returned in array b.\nNOTE\nV has the same form as B:\nThe columns of V represent the vectors which define the H(i)s.\nThe (m+n)-by-(m+n) block reflector H is then given by\nH = I - W*T*WT for real flavors, and\nH = I - W*T*WH for complex flavors\nwhere WT is the transpose of W, WH is the conjugate transpose of W, and T is the upper triangular factor of\nthe block reflector.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major ( LAPACK_COL_MAJOR ).\nm\nThe total number of rows in the matrix B (m ≥ 0).\nn\nThe number of columns in B and the order of the triangular matrix A (n ≥\n0).\nl\nThe number of rows of the upper trapezoidal part of B (min(m, n) ≥ l ≥ 0).\na, b\nArrays: a, size max(1, lda *n) contains the n-by-n upper triangular matrix\nA.\nb, size max(1,ldb* n) for column major and max(1,ldb*m) for row major,\nthe pentagonal m-by-n matrix B. The first (m-l) rows contain the\nrectangular B1 matrix, and the next l rows contain the upper trapezoidal\nB2 matrix.\nlda\nThe leading dimension of a; at least max(1, n).\nldb\nThe leading dimension of b; at least max(1, m) for column major and\nmax(1,n) for row major.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1284\n\n\nldt\nThe leading dimension of t; at least max(1, n).\nOutput Parameters\na\nThe elements on and above the diagonal of the array contain the upper\ntriangular matrix R.\nb\nThe pentagonal matrix V.\nt\nArray, size max(1, ldt *n).\nThe upper n-by-n upper triangular factor T of the block reflector.\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info < 0 and info = -i, the ith argument had an illegal value.\nIf info = -1011, memory allocation error occurred.\n?tprfb\nApplies a real or complex \"triangular-pentagonal\"\nblocked reflector to a real or complex matrix, which is\ncomposed of two blocks.\nSyntax\nlapack_int LAPACKE_stprfb (int matrix_layout, char side, char trans, char direct, char\nstorev, lapack_int m, lapack_int n, lapack_int k, lapack_int l, const float * v,\nlapack_int ldv, const float * t, lapack_int ldt, float * a, lapack_int lda, float * b,\nlapack_int ldb);\nlapack_int LAPACKE_dtprfb (int matrix_layout, char side, char trans, char direct, char\nstorev, lapack_int m, lapack_int n, lapack_int k, lapack_int l, const double * v,\nlapack_int ldv, const double * t, lapack_int ldt, double * a, lapack_int lda, double *\nb, lapack_int ldb);\nlapack_int LAPACKE_ctprfb (int matrix_layout, char side, char trans, char direct, char\nstorev, lapack_int m, lapack_int n, lapack_int k, lapack_int l, const\nlapack_complex_float * v, lapack_int ldv, const lapack_complex_float * t, lapack_int\nldt, lapack_complex_float * a, lapack_int lda, lapack_complex_float * b, lapack_int\nldb);\nlapack_int LAPACKE_ztprfb (int matrix_layout, char side, char trans, char direct, char\nstorev, lapack_int m, lapack_int n, lapack_int k, lapack_int l, const\nlapack_complex_double * v, lapack_int ldv, const lapack_complex_double * t, lapack_int\nldt, lapack_complex_double * a, lapack_int lda, lapack_complex_double * b, lapack_int\nldb);\nInclude Files\n•\nmkl.h\nDescription\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1285\n\n\nThe ?tprfb routine applies a real or complex \"triangular-pentagonal\" block reflector H, HT, or HH from either\nthe left or the right to a real or complex matrix C, which is composed of two blocks A and B.\nThe block B is m-by-n. If side = 'R', A is m-by-k, and if side = 'L', A is of size k-by-n.\nThe pentagonal matrix V is composed of a rectangular block V1 and a trapezoidal block V2. The size of the\ntrapezoidal block is determined by the parameter l, where 0≤l≤k. if l=k, the V2 block of V is triangular; if\nl=0, there is no trapezoidal block, thus V = V1 is rectangular.\ndirect='F'\ndirect='B'\nstorev='C'\nV2 is upper trapezoidal (first l rows of k-by-k\nupper triangular)\nV2 is lower trapezoidal (last l rows of k-by-k\nlower triangular matrix)\nstorev='R'\nV2 is lower trapezoidal (first l columns of k-\nby-k lower triangular matrix)\nV2 is upper trapezoidal (last l columns of k-\nby-k upper triangular matrix)\nside='L'\nside='R'\nstorev='C'\nV is m-by-k\nV2 is l-by-k\nV is n-by-k\nV2 is l-by-k\nstorev='R'\nV is k-by-m\nV2 is k-by-l\nV is k-by-n\nV2 is k-by-l\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1286\n\n\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major ( LAPACK_COL_MAJOR ).\nside\n= 'L': apply H, HT, or HH from the left,\n= 'R': apply H, HT, or HH from the right.\ntrans\n= 'N': apply H (no transpose),\n= 'T': apply HT (transpose),\n= 'C': apply HH (conjugate transpose).\ndirect\nIndicates how H is formed from a product of elementary reflectors:\n= 'F': H = H(1) H(2) . . . H(k) (Forward),\n= 'B': H = H(k) . . . H(2) H(1) (Backward).\nstorev\nIndicates how the vectors that define the elementary reflectors are stored:\n= 'C': Columns,\n= 'R': Rows.\nm\nThe total number of rows in the matrix B (m ≥ 0).\nn\nThe number of columns in B (n ≥ 0).\nk\nThe order of the matrix T, which is the number of elementary reflectors\nwhose product defines the block reflector. (k ≥ 0)\nl\nThe order of the trapezoidal part of V. (k ≥ l ≥ 0).\nv\nAn array containing the pentagonal matrix V (the elementary reflectors\nH(1), H(2), …, H(k). The size limitations depend on values of\nparameters storev and side as described in the following table\nstorev = C\nstorev = R\nside = L\nside = R\nside = L\nside = R\nColumn\nmajor\nmax(1,ldv*\nk)\nmax(1,ldv*\nk)\nmax(1,ldv*\nm)\nmax(1,ldv*\nn)\nRow major\nmax(1,ldv*\nm)\nmax(1,ldv*\nn)\nmax(1,ldv*\nk)\nmax(1,ldv*\nk)\nldv\nThe leading dimension of the array v.It should satisfy the following\nconditions:\nstorev = C\nstorev = R\nside = L\nside = R\nside = L\nside = R\nColumn\nmajor\nmax(1,m)\nmax(1,n)\nmax(1,k)\nmax(1,k)\nRow major\nmax(1,k)\nmax(1,k)\nmax(1,m)\nmax(1,n)\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1287\n\n\nt\nArray size max(1,ldt * k). The triangular k-by-k matrix T in the\nrepresentation of the block reflector.\nldt\nThe leading dimension of the array t (ldt ≥ k).\na\nsize should satisfy the following conditions:\nk if side = 'R'.\nside = L\nside = R\nColumn major\nmax(1,lda*n)\nmax(1,lda*k)\nRow major\nmax(1,lda*k)\nmax(1,lda*m)\nThe k-by-n or m-by-k matrix A.\nlda\nThe leading dimension of the array a should satisfy the following conditions:\nside = L\nside = R\nColumn major\nmax(1,k)\nmax(1,m)\nRow major\nmax(1,n)\nmax(1,k)\nb\nArray size at least max(1, ldb *n) for column major layout and max(1, ldb\n*m) for row major layout, the m-by-n matrix B.\nldb\nThe leading dimension of the array b (ldb ≥ max(1, m) for column major\nlayout and ldb ≥ max(1, n) for row major layout).\nOutput Parameters\na\nContains the corresponding block of H*C, HT*C, HH*C, C*H, C*HT, or C*HH.\nb\nContains the corresponding block of H*C, HT*C, HH*C, C*H, C*HT, or C*HH.\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info < 0, the i-th parameter had an illegal value.\nIf info = -1011, memory allocation error occurred.\n?tpttf\nCopies a triangular matrix from the standard packed\nformat (TP) to the rectangular full packed format (TF).\nSyntax\nlapack_int LAPACKE_stpttf (int matrix_layout , char transr , char uplo , lapack_int n ,\nconst float * ap , float * arf );\nlapack_int LAPACKE_dtpttf (int matrix_layout , char transr , char uplo , lapack_int n ,\nconst double * ap , double * arf );\nlapack_int LAPACKE_ctpttf (int matrix_layout , char transr , char uplo , lapack_int n ,\nconst lapack_complex_float * ap , lapack_complex_float * arf );\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1288\n\n\nlapack_int LAPACKE_ztpttf (int matrix_layout , char transr , char uplo , lapack_int n ,\nconst lapack_complex_double * ap , lapack_complex_double * arf );\nInclude Files\n•\nmkl.h\nDescription\nThe routine copies a triangular matrix A from the standard packed format to the Rectangular Full Packed\n(RFP) format. For the description of the RFP format, see Matrix Storage Schemes.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major ( LAPACK_COL_MAJOR).\ntransr\n= 'N': arf must be in the Normal format,\n= 'T': arf must be in the Transpose format (for stpttf and dtpttf),\n= 'C': arf must be in the Conjugate-transpose format (for ctpttf and\nztpttf).\nuplo\nSpecifies whether A is upper or lower triangular:\n= 'U': A is upper triangular,\n= 'L': A is lower triangular.\nn\nThe order of the matrix A. n≥ 0.\nap\nArray, size at least max (1, n*(n+1)/2).\nOn entry, the upper or lower triangular matrix A, packed in a linear array.\nSee Matrix Storage Schemes for more information.\nOutput Parameters\narf\nArray, size at least max (1, n*(n+1)/2).\nOn exit, the upper or lower triangular matrix A stored in the RFP\nformat.\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\n< 0: if info = -i, the i-th parameter had an illegal value.\nIf info = -1011, memory allocation error occurred.\n?tpttr\nCopies a triangular matrix from the standard packed\nformat (TP) to the standard full format (TR) .\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1289\n\n\nSyntax\nlapack_int LAPACKE_stpttr (int matrix_layout , char uplo , lapack_int n , const float *\nap , float * a , lapack_int lda );\nlapack_int LAPACKE_dtpttr (int matrix_layout , char uplo , lapack_int n , const double\n* ap , double * a , lapack_int lda );\nlapack_int LAPACKE_ctpttr (int matrix_layout , char uplo , lapack_int n , const\nlapack_complex_float * ap , lapack_complex_float * a , lapack_int lda );\nlapack_int LAPACKE_ztpttr (int matrix_layout , char uplo , lapack_int n , const\nlapack_complex_double * ap , lapack_complex_double * a , lapack_int lda );\nInclude Files\n•\nmkl.h\nDescription\nThe routine copies a triangular matrix A from the standard packed format to the standard full format.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major ( LAPACK_COL_MAJOR ).\nuplo\nSpecifies whether A is upper or lower triangular:\n= 'U': A is upper triangular,\n= 'L': A is lower triangular.\nn\nThe order of the matrices ap and a. n≥ 0.\nap\nArray, size at least max (1, n*(n+1)/2). (see Matrix Storage Schemes).\nlda\nThe leading dimension of the array a. lda ≥ max(1,n).\nOutput Parameters\na\nArray, size max(1,lda*n).\nOn exit, the triangular matrix A. If uplo = 'U', the leading n-by-n upper\ntriangular part of the array a contains the upper triangular part of the\nmatrix A, and the strictly lower triangular part of a is not referenced. If\nuplo = 'L', the leading n-by-n lower triangular part of the array a contains\nthe lower triangular part of the matrix A, and the strictly upper triangular\npart of a is not referenced.\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info = -i, the i-th parameter had an illegal value.\nIf info = -1011, memory allocation error occurred.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1290\n\n\n?trttf\nCopies a triangular matrix from the standard full\nformat (TR) to the rectangular full packed format (TF).\nSyntax\nlapack_int LAPACKE_strttf (int matrix_layout , char transr , char uplo , lapack_int n ,\nconst float * a , lapack_int lda , float * arf );\nlapack_int LAPACKE_dtrttf (int matrix_layout , char transr , char uplo , lapack_int n ,\nconst double * a , lapack_int lda , double * arf );\nlapack_int LAPACKE_ctrttf (int matrix_layout , char transr , char uplo , lapack_int n ,\nconst lapack_complex_float * a , lapack_int lda , lapack_complex_float * arf );\nlapack_int LAPACKE_ztrttf (int matrix_layout , char transr , char uplo , lapack_int n ,\nconst lapack_complex_double * a , lapack_int lda , lapack_complex_double * arf );\nInclude Files\n•\nmkl.h\nDescription\nThe routine copies a triangular matrix A from the standard full format to the Rectangular Full Packed (RFP)\nformat. For the description of the RFP format, see Matrix Storage Schemes.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major ( LAPACK_COL_MAJOR ).\ntransr\n= 'N': arf must be in the Normal format,\n= 'T': arf must be in the Transpose format (for strttf and dtrttf),\n= 'C': arf must be in the Conjugate-transpose format (for ctrttf and\nztrttf).\nuplo\nSpecifies whether A is upper or lower triangular:\n= 'U': A is upper triangular,\n= 'L': A is lower triangular.\nn\nThe order of the matrix A. n≥ 0.\na\nArray, size max(1,(lda*n)).\nOn entry, the triangular matrix A. If uplo = 'U', the leading n-by-n upper\ntriangular part of the array a contains the upper triangular matrix, and the\nstrictly lower triangular part of a is not referenced. If uplo = 'L', the leading\nn-by-n lower triangular part of the array a contains the lower triangular\nmatrix, and the strictly upper triangular part of a is not referenced.\nlda\nThe leading dimension of the array a. lda ≥ max(1,n).\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1291\n\n\nOutput Parameters\narf\nArray, size at least max (1, n*(n+1)/2).\nOn exit, the upper or lower triangular matrix A stored in the RFP\nformat.\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info < 0, the i-th parameter had an illegal value.\nIf info = -1011, memory allocation error occurred.\n?trttp\nCopies a triangular matrix from the standard full\nformat (TR) to the standard packed format (TP) .\nSyntax\nlapack_int LAPACKE_strttp (int matrix_layout , char uplo , lapack_int n , const float *\na , lapack_int lda , float * ap );\nlapack_int LAPACKE_dtrttp (int matrix_layout , char uplo , lapack_int n , const double\n* a , lapack_int lda , double * ap );\nlapack_int LAPACKE_ctrttp (int matrix_layout , char uplo , lapack_int n , const\nlapack_complex_float * a , lapack_int lda , lapack_complex_float * ap );\nlapack_int LAPACKE_ztrttp (int matrix_layout , char uplo , lapack_int n , const\nlapack_complex_double * a , lapack_int lda , lapack_complex_double * ap );\nInclude Files\n•\nmkl.h\nDescription\nThe routine copies a triangular matrix A from the standard full format to the standard packed format.\nInput Parameters\nuplo\nSpecifies whether A is upper or lower triangular:\n= 'U': A is upper triangular,\n= 'L': A is lower triangular.\nn\nThe order of the matrix A, n≥ 0.\na\nArray, size max(1, lda *n).\nOn entry, the triangular matrix A. If uplo = 'U', the leading n-by-n upper\ntriangular part of the array a contains the upper triangular matrix, and the\nstrictly lower triangular part of a is not referenced. If uplo = 'L', the leading\nn-by-n lower triangular part of the array a contains the lower triangular\nmatrix, and the strictly upper triangular part of a is not referenced.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1292\n\n\nlda\nThe leading dimension of the array a. lda ≥ max(1,n).\nOutput Parameters\nap\nArray, size at least max (1, n*(n+1)/2).\nOn exit, the upper or lower triangular matrix A, packed columnwise in a\nlinear array. (see Matrix Storage Schemes)\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info < 0, the i-th parameter had an illegal value.\nIf info = -1011, memory allocation error occurred.\n?lacp2\nCopies all or part of a real two-dimensional array to a\ncomplex array.\nSyntax\nlapack_int LAPACKE_clacp2 (int matrix_layout , char uplo , lapack_int m , lapack_int\nn , const float * a , lapack_int lda , lapack_complex_float * b , lapack_int ldb );\nlapack_int LAPACKE_zlacp2 (int matrix_layout , char uplo , lapack_int m , lapack_int\nn , const double * a , lapack_int lda , lapack_complex_double * b , lapack_int ldb );\nInclude Files\n•\nmkl.h\nDescription\nThe routine copies all or part of a real matrix A to another matrix B.\nInput Parameters\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major (LAPACK_COL_MAJOR).\nuplo\nSpecifies the part of the matrix A to be copied to B.\nIf uplo = 'U', the upper triangular part of A;\nif uplo = 'L', the lower triangular part of A.\nOtherwise, all of the matrix A is copied.\nm\nThe number of rows in the matrix A (m≥ 0).\nn\nThe number of columns in A (n≥ 0).\na\nArray, size at least max(1,lda*n) for column major and max(1,lda*m)\nfor row major, contains the m-by-n matrix A.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1293\n\n\nIf uplo = 'U', only the upper triangle or trapezoid is accessed; if uplo =\n'L', only the lower triangle or trapezoid is accessed.\nlda\nThe leading dimension of a; lda≥ max(1, m) for column major and lda≥\nmax(1, n) for row major.\nldb\nThe leading dimension of the output array b; ldb≥ max(1, m) for column\nmajor and ldb≥ max(1, n) for row major.\nOutput Parameters\nb\nArray, size at least max(1,ldb*n) for column major layout and\nmax(1,ldb*m) for row major layout, contains the m-by-n matrix B.\nOn exit, B = A in the locations specified by uplo.\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info < 0, the i-th parameter had an illegal value.\nIf info = -1011, memory allocation error occurred.\n?larcm\nMultiplies a square real matrix by a complex matrix.\nSyntax\nlapack_int LAPACKE_clarcm(int matrix_layout,lapack_int m,lapack_int n,const float\n*a,lapack_int lda,const lapack_complex_float * b,lapack_int ldb,lapack_complex_float *\nc,lapack_int ldc);\nlapack_int LAPACKE_zlarcm(int matrix_layout,lapack_int m,lapack_int n,const double *\na,lapack_int lda,const lapack_complex_double *b,lapack_int ldb,lapack_complex_double\n*c ,lapack_int ldc);\nDescription\nThe routine performs a simple matrix-matrix multiplication of the form\nC = A*B,\nwhere A is m-by-m and real, B is m-by-n and complex, and C is m-by-n and complex.\nInput Parameters\nm\nThe number of rows and columns of matrix A and the number of rows of\nmatrix C (m≥ 0).\nn\nThe number of columns of matrix B and the number of columns of matrix C\n(n≥ 0).\na\nArray, size [lda* m]. Contains the m-by-m matrix A.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1294\n\n\nlda\nThe leading dimension of the array a, lda≥max(1, m).\nb\nArray, size(ldb, n). Contains the m-by-n matrix B.\nldb\nThe leading dimension of the array b, ldb≥max(1, m) for column-major\nlayout; ldb≥max(1, n) for row-major layout .\nldc\nThe leading dimension of the array c, ldc≥max(1, m) for column-major\nlayout; ldc≥max(1, n) for row-major layout .\nOutput Parameters\nc\nArray, size (ldc, n). Contains the m-by-n matrix C.\nReturn Values\nThis function returns a value info. If info = 0, the execution is successful. If info = -i, parameter i had\nan illegal value.\nmkl_?tppack\nCopies a triangular/symmetric matrix or submatrix\nfrom standard full format to standard packed format.\nSyntax\nlapack_int LAPACKE_mkl_stppack (int matrix_layout, char uplo, char trans, lapack_int n,\nfloat* ap, lapack_int i, lapack_int j, lapack_int rows, lapack_int cols, const float* a,\nlapack_int lda);\nlapack_int LAPACKE_mkl_dtppack (int matrix_layout, char uplo, char trans, lapack_int n,\ndouble* ap, lapack_int i, lapack_int j, lapack_int rows, lapack_int cols, const double*\na, lapack_int lda);\nlapack_int LAPACKE_mkl_ctppack (int matrix_layout, char uplo, char trans, lapack_int n,\nMKL_Complex8* ap, lapack_int i, lapack_int j, lapack_int rows, lapack_int cols, const\nMKL_Complex8* a, lapack_int lda);\nlapack_int LAPACKE_mkl_ztppack (int matrix_layout, char uplo, char trans, lapack_int n,\nMKL_Complex16* ap, lapack_int i, lapack_int j, lapack_int rows, lapack_int cols, const\nMKL_Complex16* a, lapack_int lda);\nInclude Files\n•\nmkl.h\nDescription\nThe routine copies a triangular or symmetric matrix or its submatrix from standard full format to packed\nformat\nAPi:i+rows-1, j:j+cols-1 := op(A)\nStandard packed formats include:\n•\nTP: triangular packed storage\n•\nSP: symmetric indefinite packed storage\n•\nHP: Hermitian indefinite packed storage\n•\nPP: symmetric or Hermitian positive definite packed storage\nFull formats include:\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1295\n\n\n•\nGE: general\n•\nTR: triangular\n•\nSY: symmetric indefinite\n•\nHE: Hermitian indefinite\n•\nPO: symmetric or Hermitian positive definite\nNOTE\nAny elements of the copied submatrix rectangular outside of the triangular part of the\nmatrix AP are skipped.\nInput Parameters\nA <datatype> placeholder, if present, is used for the C interface data types in the C interface section above.\nSee C Interface Conventions for the C interface principal conventions and type definitions.\nuplo\nSpecifies whether the matrix AP is upper or lower triangular.\nIf uplo = 'U', AP is upper triangular.\nIf uplo = 'L': AP is lower triangular.\ntrans\nSpecifies whether or not the copied block of A is transposed or not.\nIf trans = 'N', no transpose: op(A) = A.\nIf trans = 'T',transpose: op(A) = AT.\nIf trans = 'C',conjugate transpose: op(A) = AH. For real data this is the\nsame as trans = 'T'.\nn\nThe order of the matrix AP; n ≥ 0\ni, j\nCoordinates of the left upper corner of the destination submatrix in AP.\nIf uplo=’U’, 1 ≤i≤j≤n.\nIf uplo=’L’, 1 ≤j≤i≤n.\nrows\nNumber of rows in the destination submatrix. 0 ≤rows≤n - i + 1.\ncols\nNumber of columns in the destination submatrix. 0 ≤cols≤n - j + 1.\na\nPointer to the source submatrix.\nArray a contains the rows-by-cols submatrix stored as unpacked rows-by-\ncolumns if trans = ’N’, or unpacked columns-by-rows if trans = ’T’ or\ntrans = ’C’.\nThe size of a is\ntrans = 'N'\ntrans='T' or\ntrans='C'\nmatrix_layout =\nLAPACK_COL_MAJOR\nlda*cols\nlda*rows\nmatrix_layout =\nLAPACK_ROW_MAJOR\nlda*rows\nlda*cols\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1296\n\n\nNOTE\nIf there are elements outside of the triangular part of AP, they\nare skipped and are not copied from a.\nlda\nThe leading dimension of the array a.\ntrans = 'N'\ntrans='T' or\ntrans='C'\nmatrix_layout =\nLAPACK_COL_MAJOR\nlda≥ max(1,\nrows)\nlda≥ max(1, cols)\nmatrix_layout =\nLAPACK_ROW_MAJOR\nlda≥ max(1,\ncols)\nlda≥ max(1, rows)\nOutput Parameters\nap\nArray of size at least max(1, n(n+1)/2). The array ap contains either\nthe upper or the lower triangular part of the matrix AP (as specified by\nuplo) in packed storage (see Matrix Storage Schemes). The submatrix\nof ap from row i to row i + rows - 1 and column j to column j +\ncols - 1 is overwritten with a copy of the source matrix.\nReturn Values\nThis function returns a value info. If info=0, the execution is successful. If info = -i, the i-th parameter\nhad an illegal value.\nmkl_?tpunpack\nCopies a triangular/symmetric matrix or submatrix\nfrom standard packed format to full format.\nSyntax\nlapack_int LAPACKE_mkl_stpunpack ( int matrix_layout,  char uplo, char trans,\n lapack_int n,  const float* ap, lapack_int i, lapack_int j, lapack_int rows,\nlapack_int cols,  float* a,  lapack_int lda );\nlapack_int LAPACKE_mkl_dtpunpack ( int matrix_layout,  char uplo, char trans,\n lapack_int n,  const double* ap, lapack_int i, lapack_int j, lapack_int rows,\nlapack_int cols,  double* a,  lapack_int lda );\nlapack_int LAPACKE_mkl_ctpunpack ( int matrix_layout,  char uplo, char trans,\n lapack_int n,  const MKL_Complex8* ap, lapack_int i, lapack_int j, lapack_int rows,\nlapack_int cols,   MKL_Complex8* a,  lapack_int lda );\nlapack_int LAPACKE_mkl_ztpunpack ( int matrix_layout,  char uplo, char trans,\n lapack_int n,  const MKL_Complex16* ap, lapack_int i, lapack_int j, lapack_int rows,\nlapack_int cols,  MKL_Complex16* a,  lapack_int lda );\nInclude Files\n•\nmkl.h\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1297\n\n\nDescription\nThe routine copies a triangular or symmetric matrix or its submatrix from standard packed format to full\nformat.\nA := op(APi:i+rows-1, j:j+cols-1)\nStandard packed formats include:\n•\nTP: triangular packed storage\n•\nSP: symmetric indefinite packed storage\n•\nHP: Hermitian indefinite packed storage\n•\nPP: symmetric or Hermitian positive definite packed storage\nFull formats include:\n•\nGE: general\n•\nTR: triangular\n•\nSY: symmetric indefinite\n•\nHE: Hermitian indefinite\n•\nPO: symmetric or Hermitian positive definite\nNOTE\nAny elements of the copied submatrix rectangular outside of the triangular part of AP are\nskipped.\nInput Parameters\nA <datatype> placeholder, if present, is used for the C interface data types in the C interface section above.\nSee C Interface Conventions for the C interface principal conventions and type definitions.\nmatrix_layout\nSpecifies whether matrix storage layout is row major\n(LAPACK_ROW_MAJOR) or column major (LAPACK_COL_MAJOR).\nuplo\nSpecifies whether matrix AP is upper or lower triangular.\nIf uplo = 'U', AP is upper triangular.\nIf uplo = 'L': AP is lower triangular.\ntrans\nSpecifies whether or not the copied block of AP is transposed.\nIf trans = 'N', no transpose: op(AP) = AP.\nIf trans = 'T',transpose: op(AP) = APT.\nIf trans = 'C',conjugate transpose: op(AP) = APH. For real data this\nis the same as trans = 'T'.\nn\nThe order of the matrix AP; n ≥ 0.\nap\nArray, size at least max(1, n(n+1)/2). The array ap contains either\nthe upper or the lower triangular part of the matrix AP (as specified by\nuplo) in packed storage (see Matrix Storage Schemes). It is the\nsource for the submatrix of AP from row i to row i + rows - 1 and\ncolumn j to column j + cols - 1 to be copied.\ni, j\nCoordinates of left upper corner of the submatrix in AP to copy.\nIf uplo=’U’, 1 ≤i≤j≤n.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1298\n\n\nIf uplo=’L’, 1 ≤j≤i≤n.\nrows\nNumber of rows to copy. 0 ≤rows≤n - i + 1.\ncols\nNumber of columns to copy. 0 ≤cols≤n - j + 1.\nlda\nThe leading dimension of array a.\ntrans = 'N'\ntrans='T' or\ntrans='C'\nmatrix_layout =\nLAPACK_COL_MAJOR\nlda≥ max(1,rows)\nlda≥ max(1,cols)\nmatrix_layout =\nLAPACK_ROW_MAJOR\nlda≥ max(1,cols)\nlda≥ max(1,rows)\nOutput Parameters\na\nPointer to the destination matrix. On exit, array a is overwritten with a\ncopy of the unpacked rows-by-cols submatrix of ap unpacked rows-\nby-columns if trans = ’N’, or unpacked columns-by-rows if trans\n= ’T’ or trans = ’C’.\nThe size of a is\ntrans = 'N'\ntrans='T' or\ntrans='C'\nmatrix_layout =\nLAPACK_COL_MAJOR\nlda*cols\nlda*rows\nmatrix_layout =\nLAPACK_ROW_MAJOR\nlda*rows\nlda*cols\nNOTE\nIf there are elements outside of the triangular part of ap\nindicated by uplo, they are skipped and are not copied to\na.\nReturn Values\nThis function returns a value info. If info=0, the execution is successful. If info = -i, the i-th parameter\nhad an illegal value.\nLAPACK Utility Functions and Routines\nThis section describes LAPACK utility functions and routines.\nSummary information about these routines is given in the following table:\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1299\n\n\nLAPACK Utility Routines\nRoutine Name\nData\nTypes\nDescription\nilaver\n \nReturns the version of the Lapack library.\nilaenv\n \nEnvironmental enquiry function which returns values for tuning\nalgorithmic performance.\n?lamch\ns, d\nDetermines machine parameters for floating-point arithmetic.\nSee Also\nlsame  Tests two characters for equality regardless of the case.\nlsamen  Tests two character strings for equality regardless of the case.\nsecond/dsecnd  Returns elapsed time in seconds. Use to estimate real time between two calls to\nthis function.\nxerbla Error handling function called by BLAS, LAPACK, Vector Math, and Vector Statistics\nfunctions.\nilaver\nReturns the version of the LAPACK library.\nSyntax\nvoid LAPACKE_ilaver (lapack_int * vers_major, lapack_int * vers_minor, lapack_int *\nvers_patch);\nInclude Files\n•\nmkl.h\nDescription\nThis routine returns the version of the LAPACK library.\nOutput Parameters\nvers_major\nReturns the major version of the LAPACK library.\nvers_minor\nReturns the minor version from the major version of the LAPACK library.\nvers_patch\nReturns the patch version from the minor version of the LAPACK library.\nilaenv\nEnvironmental enquiry function that returns values for\ntuning algorithmic performance.\nSyntax\nMKL_INT ilaenv (const MKL_INT *ispec, const char *name, const char *opts, const MKL_INT\n*n1, const MKL_INT *n2, const MKL_INT *n3, const MKL_INT *n4);\nInclude Files\n•\nmkl.h\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1300\n\n\nDescription\nThe enquiry function ilaenv is called from the LAPACK routines to choose problem-dependent parameters\nfor the local environment. See ispec below for a description of the parameters.\nThis version provides a set of parameters that should give good, but not optimal, performance on many of\nthe currently available computers.\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nInput Parameters\nispec\nSpecifies the parameter to be returned as the value of ilaenv:\n= 1: the optimal blocksize; if this value is 1, an unblocked algorithm will\ngive the best performance.\n= 2: the minimum block size for which the block routine should be used; if\nthe usable block size is less than this value, an unblocked routine should be\nused.\n= 3: the crossover point (in a block routine, for n less than this value, an\nunblocked routine should be used)\n= 4: the number of shifts, used in the nonsymmetric eigenvalue routines\n(deprecated)\n= 5: the minimum column dimension for blocking to be used; rectangular\nblocks must have dimension at least k-by-m, where k is given by\nilaenv(2,...) and m by ilaenv(5,...)\n= 6: the crossover point for the SVD (when reducing an m-by-n matrix to\nbidiagonal form, if max(m,n)/min(m,n) exceeds this value, a QR\nfactorization is used first to reduce the matrix to a triangular form.)\n= 7: the number of processors\n= 8: the crossover point for the multishift QR and QZ methods for\nnonsymmetric eigenvalue problems (deprecated).\n= 9: maximum size of the subproblems at the bottom of the computation\ntree in the divide-and-conquer algorithm (used by ?gelsd and ?gesdd)\n=10: ieee NaN arithmetic can be trusted not to trap\n=11: infinity arithmetic can be trusted not to trap\n12 ≤ ispec ≤ 16: ?hseqr or one of its subroutines, see iparmq for detailed\nexplanation.\nname\nThe name of the calling subroutine, in either upper case or lower case.\nopts\nThe character options to the subroutine name, concatenated into a single\ncharacter string. For example, uplo = 'U', trans = 'T', and diag =\n'N' for a triangular routine would be specified as opts = 'UTN'.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1301\n\n\nNOTE\nUse only uppercase characters for the opts string.\nn1, n2, n3, n4\nProblem dimensions for the subroutine name; these may not all be\nrequired.\nOutput Parameters\nvalue\nIf value≥ 0: the value of the parameter specified by ispec;\nIf value = -k < 0: the k-th argument had an illegal value.\nReturn Values\nilaenv returns value.\nIf value≥ 0: the value of the parameter specified by ispec;\nIf value = -k < 0: the k-th argument had an illegal value.\nApplication Notes\nThe following conventions have been used when calling ilaenv from the LAPACK routines:\n1.\nopts is a concatenation of all of the character options to subroutine name, in the same order that they\nappear in the argument list for name, even if they are not used in determining the value of the\nparameter specified by ispec.\n2.\nThe problem dimensions n1, n2, n3, n4 are specified in the order that they appear in the argument list\nfor name. n1 is used first, n2 second, and so on, and unused problem dimensions are passed a value of\n-1.\n3.\nThe parameter value returned by ilaenv is checked for validity in the calling subroutine. For example,\nilaenv is used to retrieve the optimal blocksize for strtri as follows:\n  nb := ilaenv( 1, 'strtri', strcat (uplo, diag), n, -1, -1, -1> );\n  if( nb <= 1 ) {\n      nb := max( 1, n );\n  }\nBelow is an example of ilaenv usage in C language:\n#include <stdio.h> \n#include \"mkl.h\"\nint main(void) \n{\n   int size = 1000;\n   int ispec = 1;\n   int dummy = -1;\n   int blockSize1 = ilaenv(&ispec, \"dsytrd\", \"U\", &size, &dummy, &dummy, &dummy);\n   int blockSize2 = ilaenv(&ispec, \"dormtr\", \"LUN\", &size, &size, &dummy, &dummy);\n   printf(\"DSYTRD blocksize = %d\\n\", blockSize1);\n   printf(\"DORMTR blocksize = %d\\n\", blockSize2);\n   return 0; \n}\nSee Also\n?hseqr\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1302\n\n\n?lamch\nDetermines machine parameters for floating-point\narithmetic.\nSyntax\nfloat LAPACKE_slamch (char cmach );\ndouble LAPACKE_dlamch (char cmach );\nInclude Files\n•\nmkl.h\nDescription\nThe function ?lamch determines single precision and double precision machine parameters.\nInput Parameters\ncmach\nSpecifies the value to be returned by ?lamch:\n= 'E' or 'e', val = eps\n= 'S' or 's', val = sfmin\n= 'B' or 'b', val = base\n= 'P' or 'p', val = eps*base\n= 'n' or 'n', val = t\n= 'R' or 'r', val = rnd\n= 'M' or 'm', val = emin\n= 'U' or 'u', val = rmin\n= 'L' or 'l', val = emax\n= 'O' or 'o', val = rmax\nwhere\neps = relative machine precision;\nsfmin = safe minimum, such that 1/sfmin does not overflow;\nbase = base of the machine;\nprec = eps*base;\nt = number of (base) digits in the mantissa;\nrnd = 1.0 when rounding occurs in addition, 0.0 otherwise;\nemin = minimum exponent before (gradual) underflow;\nrmin = underflow_threshold - base**(emin-1);\nemax = largest exponent before overflow;\nrmax = overflow_threshold - (base**emax)*(1-eps).\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1303\n\n\nNOTE\nYou can use a character string for cmach instead of a single\ncharacter in order to make your code more readable. The first\ncharacter of the string determines the value to be returned. For\nexample, 'Precision' is interpreted as 'p'.\nOutput Parameters\nval\nValue returned by the function.\nLAPACK Test Functions and Routines\nThis section describes LAPACK test functions and routines.\n?lagge\nGenerates a general m-by-n matrix .\nSyntax\nlapack_int LAPACKE_slagge (int matrix_layout , lapack_int m , lapack_int n , lapack_int\nkl , lapack_int ku , const float * d , float * a , lapack_int lda , lapack_int *\niseed );\nlapack_int LAPACKE_dlagge (int matrix_layout , lapack_int m , lapack_int n , lapack_int\nkl , lapack_int ku , const double * d , double * a , lapack_int lda , lapack_int *\niseed );\nlapack_int LAPACKE_clagge (int matrix_layout , lapack_int m , lapack_int n , lapack_int\nkl , lapack_int ku , const float * d , lapack_complex_float * a , lapack_int lda ,\nlapack_int * iseed );\nlapack_int LAPACKE_zlagge (int matrix_layout , lapack_int m , lapack_int n , lapack_int\nkl , lapack_int ku , const double * d , lapack_complex_double * a , lapack_int lda ,\nlapack_int * iseed );\nInclude Files\n•\nmkl.h\nDescription\nThe routine generates a general m-by-n matrix A, by pre- and post- multiplying a real diagonal matrix D with\nrandom matrices U and V:\nA := U*D*V,\nwhere U and V are orthogonal for real flavors and unitary for complex flavors. The lower and upper\nbandwidths may then be reduced to kl and ku by additional orthogonal transformations.\nInput Parameters\nA <datatype> placeholder, if present, is used for the C interface data types in the C interface section above.\nSee C Interface Conventions for the C interface principal conventions and type definitions.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1304\n\n\nm\nThe number of rows of the matrix A (m≥ 0).\nn\nThe number of columns of the matrix A (n≥ 0).\nkl\nThe number of nonzero subdiagonals within the band of A (0 ≤kl≤m-1).\nku\nThe number of nonzero superdiagonals within the band of A (0 ≤ku≤n-1).\nd\nThe array d with the dimension of (min(m, n)) contains the diagonal\nelements of the diagonal matrix D.\nlda\nThe leading dimension of the array a (lda≥m) for column major layout and\n(lda≥n) for row major layout.\niseed\nThe array iseed with the dimension of 4 contains the seed of the random\nnumber generator. The elements must be between 0 and 4095 and iseed\nmust be odd.\nOutput Parameters\na\nThe array a with size at least max(1,lda*n) for column major layout and\nmax(1,lda*m) for row major layout contains the generated m-by-n matrix\nA.\niseed\nThe array iseed contains the updated seed on exit.\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info < 0, the i-th parameter had an illegal value.\nIf info = -1011, memory allocation error occurred.\n?laghe\nGenerates a complex Hermitian matrix .\nSyntax\nlapack_int LAPACKE_claghe (int matrix_layout , lapack_int n , lapack_int k , const\nfloat * d , lapack_complex_float * a , lapack_int lda , lapack_int * iseed );\nlapack_int LAPACKE_zlaghe (int matrix_layout , lapack_int n , lapack_int k , const\ndouble * d , lapack_complex_double * a , lapack_int lda , lapack_int * iseed );\nInclude Files\n•\nmkl.h\nDescription\nThe routine generates a complex Hermitian matrix A, by pre- and post- multiplying a real diagonal matrix D\nwith random unitary matrix:\nA := U*D*UH\nThe semi-bandwidth may then be reduced to k by additional unitary transformations.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1305\n\n\nInput Parameters\nA <datatype> placeholder, if present, is used for the C interface data types in the C interface section above.\nSee C Interface Conventions for the C interface principal conventions and type definitions.\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major ( LAPACK_COL_MAJOR ).\nn\nThe order of the matrix A (n≥ 0).\nk\nThe number of nonzero subdiagonals within the band of A (0 ≤k≤n-1).\nd\nThe array d with the dimension of (n) contains the diagonal elements of the\ndiagonal matrix D.\nlda\nThe leading dimension of the array a (lda≥n).\niseed\nThe array iseed with the dimension of 4 contains the seed of the random\nnumber generator. The elements must be between 0 and 4095 and\niseed[3] must be odd.\nOutput Parameters\na\nThe array a of size at least max (1,lda*n) contains the generated n-by-n\nHermitian matrix D.\niseed\nThe array iseed contains the updated seed on exit.\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info < 0, the i-th parameter had an illegal value.\nIf info = -1011, memory allocation error occurred.\n?lagsy\nGenerates a symmetric matrix by pre- and post-\nmultiplying a real diagonal matrix with a random\nunitary matrix .\nSyntax\nlapack_int LAPACKE_slagsy (int matrix_layout , lapack_int n , lapack_int k , const\nfloat * d , float * a , lapack_int lda , lapack_int * iseed );\nlapack_int LAPACKE_dlagsy (int matrix_layout , lapack_int n , lapack_int k , const\ndouble * d , double * a , lapack_int lda , lapack_int * iseed );\nlapack_int LAPACKE_clagsy (int matrix_layout , lapack_int n , lapack_int k , const\nfloat * d , lapack_complex_float * a , lapack_int lda , lapack_int * iseed );\nlapack_int LAPACKE_zlagsy (int matrix_layout , lapack_int n , lapack_int k , const\ndouble * d , lapack_complex_double * a , lapack_int lda , lapack_int * iseed );\nInclude Files\n•\nmkl.h\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1306\n\n\nDescription\nThe ?lagsy routine generates a symmetric matrix A by pre- and post- multiplying a real diagonal matrix D\nwith a random matrix U:\nA := U*D*UT,\nwhere U is orthogonal for real flavors and unitary for complex flavors. The semi-bandwidth may then be\nreduced to k by additional unitary transformations.\nInput Parameters\nA <datatype> placeholder, if present, is used for the C interface data types in the C interface section above.\nSee C Interface Conventions for the C interface principal conventions and type definitions.\nn\nThe order of the matrix A (n≥ 0).\nk\nThe number of nonzero subdiagonals within the band of A (0 ≤k≤n-1).\nd\nThe array d with the dimension of (n) contains the diagonal elements of the\ndiagonal matrix D.\nlda\nThe leading dimension of the array a (lda≥n).\niseed\nThe array iseed with the dimension of 4 contains the seed of the random\nnumber generator. The elements must be between 0 and 4095 and\niseed[3] must be odd.\nOutput Parameters\na\nThe array aof size max (1,lda*n) contains the generated symmetric n-by-n\nmatrix D.\niseed\nThe array iseed contains the updated seed on exit.\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info < 0, the i-th parameter had an illegal value.\nIf info = -1011, memory allocation error occurred.\n?latms\nGenerates a general m-by-n matrix with specific\nsingular values.\nSyntax\nlapack_int LAPACKE_slatms (int matrix_layout, lapack_int m, lapack_int n, char dist,\nlapack_int * iseed, char sym, float * d, lapack_int mode, float cond, float dmax,\nlapack_int kl, lapack_int ku, char pack, float * a, lapack_int lda);\nlapack_int LAPACKE_dlatms (int matrix_layout, lapack_int m, lapack_int n, char dist,\nlapack_int * iseed, char sym, double * d, lapack_int mode, double cond, double dmax,\nlapack_int kl, lapack_int ku, char pack, double * a, lapack_int lda);\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1307\n\n\nlapack_int LAPACKE_clatms (int matrix_layout, lapack_int m, lapack_int n, char dist,\nlapack_int * iseed, char sym, float * d, lapack_int mode, float cond, float dmax,\nlapack_int kl, lapack_int ku, char pack, lapack_complex_float * a, lapack_int lda);\nlapack_int LAPACKE_zlatms (int matrix_layout, lapack_int m, lapack_int n, char dist,\nlapack_int * iseed, char sym, double * d, lapack_int mode, double cond, double dmax,\nlapack_int kl, lapack_int ku, char pack, lapack_complex_double * a, lapack_int lda);\nInclude Files\n•\nmkl.h\nDescription\nThe ?latms routine generates random matrices with specified singular values, or symmetric/Hermitian\nmatrices with specified eigenvalues for testing LAPACK programs.\nIt applies this sequence of operations:\n1.\nSet the diagonal to d, where d is input or computed according to mode, cond, dmax, and sym as\ndescribed in Input Parameters.\n2.\nGenerate a matrix with the appropriate band structure, by one of two methods:\nMethod A\n1.\nGenerate a dense m-by-n matrix by multiplying d on the left\nand the right by random unitary matrices, then:\n2.\nReduce the bandwidth according to kl and ku, using\nHouseholder transformations.\nMethod B:\nConvert the bandwidth-0 (i.e., diagonal) matrix to a bandwidth-1\nmatrix using Givens rotations, \"chasing\" out-of-band elements\nback, much as in QR; then convert the bandwidth-1 to a\nbandwidth-2 matrix, etc.\nNote that for reasonably small bandwidths (relative to m and n)\nthis requires less storage, as a dense matrix is not generated.\nAlso, for symmetric or Hermitian matrices, only one triangle is\ngenerated.\nMethod A is chosen if the bandwidth is a large fraction of the order of the matrix, and lda is at least m (so a\ndense matrix can be stored.) Method B is chosen if the bandwidth is small (less than (1/2)*n for symmetric\nor Hermitian or less than .3*n+m for nonsymmetric), or lda is less than m and not less than the bandwidth.\nPack the matrix if desired, using one of the methods specified by the pack parameter.\nIf Method B is chosen and band format is specified, then the matrix is generated in the band format and no\nrepacking is necessary.\nInput Parameters\nA <datatype> placeholder, if present, is used for the C interface data types in the C interface section above.\nSee C Interface Conventions for the C interface principal conventions and type definitions.\nmatrix_layout\nSpecifies whether matrix storage layout is row major (LAPACK_ROW_MAJOR)\nor column major ( LAPACK_COL_MAJOR ).\nm\nThe number of rows of the matrix A (m≥ 0).\nn\nThe number of columns of the matrix A (n≥ 0).\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1308\n\n\ndist\nSpecifies the type of distribution to be used to generate the random\nsingular values or eigenvalues:\n•\n'U': uniform distribution (0, 1)\n•\n'S': symmetric uniform distribution (-1, 1)\n•\n'N': normal distribution (0, 1)\niseed\nArray with size 4.\nSpecifies the seed of the random number generator. Values should lie\nbetween 0 and 4095 inclusive, and iseed[3] should be odd. The random\nnumber generator uses a linear congruential sequence limited to small\nintegers, and so should produce machine independent random numbers.\nThe values of the array are modified, and can be used in the next call\nto ?latms to continue the same random number sequence.\nsym\nIf sym='S' or 'H', the generated matrix is symmetric or Hermitian, with\neigenvalues specified by d, cond, mode, and dmax; they can be positive,\nnegative, or zero.\nIf sym='P', the generated matrix is symmetric or Hermitian, with\neigenvalues (which are singular, non-negative values) specified by d, cond,\nmode, and dmax.\nIf sym='N', the generated matrix is nonsymmetric, with singular, non-\nnegative values specified by d, cond, mode, and dmax.\nd\nArray, size (MIN(m , n))\nThis array is used to specify the singular values or eigenvalues of A (see the\ndescription of sym). If mode=0, then d is assumed to contain the\neigenvalues or singular values, otherwise elements of d are computed\naccording to mode, cond, and dmax.\nmode\nDescribes how the singular/eigenvalues are specified.\n•\nmode = 0: use d as input\n•\nmode = 1: set d[0] = 1 and d[1:n - 1] = 1.0/cond\n•\nmode = 2: set d[0:n - 2] = 1 and d[n - 1] = 1.0/cond\n•\nmode = 3: set d[i] = cond-i/(n - 1)\n•\nmode = 4: set d[i] = 1 - i/(n - 1)*(1 - 1/cond)\n•\nmode = 5: set elements of d to random numbers in the range (1/cond ,\n1) such that their logarithms are uniformly distributed.\n•\nmode = 6: set elements of d to random numbers from same distribution\nas the rest of the matrix.\nmode < 0 has the same meaning as ABS(mode), except that the order of the\nelements of d is reversed. Thus, if mode is positive, d has entries ranging\nfrom 1 to 1/cond, if negative, from 1/cond to 1.\nIf sym='S' or 'H', and mode is not 0, 6, nor -6, then the elements of d are\nalso given a random sign (multiplied by +1 or -1).\ncond\nUsed in setting d as described for the mode parameter. If used, cond≥ 1.\ndmax\nIf mode is not -6, 0 nor 6, the contents of d, as computed according to mode\nand cond, are scaled by dmax / max(abs(d[i-1])); thus, the maximum\nabsolute eigenvalue or singular value (the norm) is abs(dmax).\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1309\n\n\nNOTE\ndmax need not be positive: if dmax is negative (or zero), d will be\nscaled by a negative number (or zero).\nkl\nSpecifies the lower bandwidth of the matrix. For example, kl=0 implies\nupper triangular, kl=1 implies upper Hessenberg, and kl being at least m -\n1 means that the matrix has full lower bandwidth. kl must equal ku if the\nmatrix is symmetric or Hermitian.\nku\nSpecifies the upper bandwidth of the matrix. For example, ku=0 implies\nlower triangular, ku=1 implies lower Hessenberg, and ku being at least n -\n1 means that the matrix has full upper bandwidth. kl must equal ku if the\nmatrix is symmetric or Hermitian.\npack\nSpecifies packing of matrix:\n•\n'N': no packing\n•\n'U': zero out all subdiagonal entries (if symmetric or Hermitian)\n•\n'L': zero out all superdiagonal entries (if symmetric or Hermitian)\n•\n'B': store the lower triangle in band storage scheme (only if matrix\nsymmetric, Hermitian, or lower triangular)\n•\n'Q': store the upper triangle in band storage scheme (only if matrix\nsymmetric, Hermitian, or upper triangular)\n•\n'Z': store the entire matrix in band storage scheme (pivoting can be\nprovided for by using this option to store A in the trailing rows of the\nallocated storage)\nUsing these options, the various LAPACK packed and banded storage\nschemes can be obtained:\n'Z'\n'B'\n'Q'\n'C'\n'R'\nGB: general band\nx\nPB: symmetric positive definite band\nx\nx\nSB: symmetric band\nx\nx\nHB: Hermitian band\nx\nx\nTB: triangular band\nx\nx\nPP: symmetric positive definite packed\nx\nx\nSP: symmetric packed\nx\nx\nHP: Hermitian packed\nx\nx\nTP: triangular packed\nx\nx\nIf two calls to ?latms differ only in the pack parameter, they generate\nmathematically equivalent matrices.\nlda\nlda specifies the first dimension of a as declared in the calling program.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1310\n\n\nIf pack='N', 'U', 'L', 'C', or 'R', then lda must be at least m for column major\nor at least n for row major.\nIf pack='B' or 'Q', then lda must be at least MIN(kl, m - 1) (which is\nequal to MIN(ku,n - 1)).\nIf pack='Z', lda must be large enough to hold the packed array: MIN( ku,\nn - 1) + MIN( kl, m - 1) + 1.\nOutput Parameters\niseed\nThe array iseed contains the updated seed.\nd\nThe array d contains the updated seed.\nNOTE\nThe array d is not modified if mode = 0.\na\nArray of size lda by n.\nThe array a contains the generated m-by-n matrix A.\na is first generated in full (unpacked) form, and then packed, if so specified\nby pack. Thus, the first m elements of the first n columns are always\nmodified. If pack specifies a packed or banded storage scheme, all lda\nelements of the first n columns are modified; the elements of the array\nwhich do not correspond to elements of the generated matrix are set to\nzero.\nReturn Values\nThis function returns a value info.\nIf info = 0, the execution is successful.\nIf info < 0, the i-th parameter had an illegal value.\nIf info = -1011, memory allocation error occurred.\nIf info = 2, cannot scale to dmax (maximum singular value is 0).\nIf info = 3, error return from lagge, ?laghe, or lagsy.\nAdditional LAPACK Routines (Included for Compatibility with Netlib LAPACK)\nLAPACK_DECL lapack_int LAPACKE_chesv_aa_2stage (int matrix_layout , char uplo ,\nlapack_int n , lapack_int nrhs , lapack_complex_float * a , lapack_int lda ,\nlapack_complex_float * tb , lapack_int ltb , lapack_int * ipiv , lapack_int * ipiv2 ,\nlapack_complex_float * b , lapack_int ldb );\nLAPACK_DECL lapack_int LAPACKE_dsysv_aa_2stage (int matrix_layout , char uplo ,\nlapack_int n , lapack_int nrhs , double * a , lapack_int lda , double * tb , lapack_int\nltb , lapack_int * ipiv , lapack_int * ipiv2 , double * b , lapack_int ldb );\nLAPACK_DECL lapack_int LAPACKE_ssysv_aa_2stage (int matrix_layout , char uplo ,\nlapack_int n , lapack_int nrhs , float * a , lapack_int lda , float * tb , lapack_int\nltb , lapack_int * ipiv , lapack_int * ipiv2 , float * b , lapack_int ldb );\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1311\n\n\nLAPACK_DECL lapack_int LAPACKE_zhesv_aa_2stage (int matrix_layout , char uplo ,\nlapack_int n , lapack_int nrhs , lapack_complex_double * a , lapack_int lda ,\nlapack_complex_double * tb , lapack_int ltb , lapack_int * ipiv , lapack_int * ipiv2 ,\nlapack_complex_double * b , lapack_int ldb );\nLAPACK_DECL lapack_int LAPACKE_chetrf_aa_2stage (int matrix_layout , char uplo ,\nlapack_int n , lapack_complex_float * a , lapack_int lda , lapack_complex_float * tb ,\nlapack_int ltb , lapack_int * ipiv , lapack_int * ipiv2 );\nLAPACK_DECL lapack_int LAPACKE_dsytrf_aa_2stage (int matrix_layout , char uplo ,\nlapack_int n , double * a , lapack_int lda , double * tb , lapack_int ltb , lapack_int\n* ipiv , lapack_int * ipiv2 );\nLAPACK_DECL lapack_int LAPACKE_ssytrf_aa_2stage (int matrix_layout , char uplo ,\nlapack_int n , float * a , lapack_int lda , float * tb , lapack_int ltb , lapack_int *\nipiv , lapack_int * ipiv2 );\nLAPACK_DECL lapack_int LAPACKE_zhetrf_aa_2stage (int matrix_layout , char uplo ,\nlapack_int n , lapack_complex_double * a , lapack_int lda , lapack_complex_double *\ntb , lapack_int ltb , lapack_int * ipiv , lapack_int * ipiv2 );\nLAPACK_DECL lapack_int LAPACKE_chetrs_aa_2stage (int matrix_layout , char uplo ,\nlapack_int n , lapack_int nrhs , lapack_complex_float * a , lapack_int lda ,\nlapack_complex_float * tb , lapack_int ltb , lapack_int * ipiv , lapack_int * ipiv2 ,\nlapack_complex_float * b , lapack_int ldb );\nLAPACK_DECL lapack_int LAPACKE_dsytrs_aa_2stage (int matrix_layout , char uplo ,\nlapack_int n , lapack_int nrhs , double * a , lapack_int lda , double * tb , lapack_int\nltb , lapack_int * ipiv , lapack_int * ipiv2 , double * b , lapack_int ldb );\nLAPACK_DECL lapack_int LAPACKE_ssytrs_aa_2stage (int matrix_layout , char uplo ,\nlapack_int n , lapack_int nrhs , float * a , lapack_int lda , float * tb , lapack_int\nltb , lapack_int * ipiv , lapack_int * ipiv2 , float * b , lapack_int ldb );\nLAPACK_DECL lapack_int LAPACKE_zhetrs_aa_2stage (int matrix_layout , char uplo ,\nlapack_int n , lapack_int nrhs , lapack_complex_double * a , lapack_int lda ,\nlapack_complex_double * tb , lapack_int ltb , lapack_int * ipiv , lapack_int * ipiv2 ,\nlapack_complex_double * b , lapack_int ldb );\ncall csysv_aa_2stage (uplo , n , nrhs , a , lda , tb , ltb , ipiv , ipiv2 , b , ldb ,\ninfo);\nLAPACK_DECL lapack_int LAPACKE_csysv_aa_2stage (int matrix_layout , char uplo ,\nlapack_int n , lapack_int nrhs , lapack_complex_float * a , lapack_int lda ,\nlapack_complex_float * tb , lapack_int ltb , lapack_int * ipiv , lapack_int * ipiv2 ,\nlapack_complex_float * b , lapack_int ldb );\ncall zsysv_aa_2stage (uplo , n , nrhs , a , lda , tb , ltb , ipiv , ipiv2 , b , ldb ,\ninfo);\nLAPACK_DECL lapack_int LAPACKE_zsysv_aa_2stage (int matrix_layout , char uplo ,\nlapack_int n , lapack_int nrhs , lapack_complex_double * a , lapack_int lda ,\nlapack_complex_double * tb , lapack_int ltb , lapack_int * ipiv , lapack_int * ipiv2 ,\nlapack_complex_double * b , lapack_int ldb );\nLAPACK_DECL lapack_int LAPACKE_csytrf_aa_2stage (int matrix_layout , char uplo ,\nlapack_int n , lapack_complex_float * a , lapack_int lda , lapack_complex_float * tb ,\nlapack_int ltb , lapack_int * ipiv , lapack_int * ipiv2 );\nLAPACK_DECL lapack_int LAPACKE_zsytrf_aa_2stage (int matrix_layout , char uplo ,\nlapack_int n , lapack_complex_double * a , lapack_int lda , lapack_complex_double *\ntb , lapack_int ltb , lapack_int * ipiv , lapack_int * ipiv2 );\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1312\n\n\nLAPACK_DECL lapack_int LAPACKE_csytrs_aa_2stage (int matrix_layout , char uplo ,\nlapack_int n , lapack_int nrhs , lapack_complex_float * a , lapack_int lda ,\nlapack_complex_float * tb , lapack_int ltb , lapack_int * ipiv , lapack_int * ipiv2 ,\nlapack_complex_float * b , lapack_int ldb );\nLAPACK_DECL lapack_int LAPACKE_zsytrs_aa_2stage (int matrix_layout , char uplo ,\nlapack_int n , lapack_int nrhs , lapack_complex_double * a , lapack_int lda ,\nlapack_complex_double * tb , lapack_int ltb , lapack_int * ipiv , lapack_int * ipiv2 ,\nlapack_complex_double * b , lapack_int ldb );\nLAPACK_DECL lapack_int LAPACKE_ssyev_2stage (int matrix_layout, char jobz, char uplo,\nlapack_int n, float * a, lapack_int lda, float * w);\nLAPACK_DECL lapack_int LAPACKE_dsyev_2stage (int matrix_layout, char jobz, char uplo,\nlapack_int n, double * a, lapack_int lda, double * w);\nLAPACK_DECL lapack_int LAPACKE_ssyevd_2stage (int matrix_layout, char jobz, char uplo,\nlapack_int n, float * a, lapack_int lda, float * w);\nLAPACK_DECL lapack_int LAPACKE_dsyevd_2stage (int matrix_layout, char jobz, char uplo,\nlapack_int n, double * a, lapack_int lda, double * w);\nLAPACK_DECL lapack_int LAPACKE_ssyevr_2stage (int matrix_layout, char jobz, char range,\nchar uplo, lapack_int n, float * a, lapack_int lda, float vl, float vu, lapack_int il,\nlapack_int iu, float abstol, lapack_int * m, float * w, float * z, lapack_int ldz,\nlapack_int * isuppz);\nLAPACK_DECL lapack_int LAPACKE_dsyevr_2stage (int matrix_layout, char jobz, char range,\nchar uplo, lapack_int n, double * a, lapack_int lda, double vl, double vu, lapack_int\nil, lapack_int iu, double abstol, lapack_int * m, double * w, double * z, lapack_int\nldz, lapack_int * isuppz);\nLAPACK_DECL lapack_int LAPACKE_ssyevx_2stage (int matrix_layout, char jobz, char range,\nchar uplo, lapack_int n, float * a, lapack_int lda, float vl, float vu, lapack_int il,\nlapack_int iu, float abstol, lapack_int * m, float * w, float * z, lapack_int ldz,\nlapack_int * ifail);\nLAPACK_DECL lapack_int LAPACKE_dsyevx_2stage (int matrix_layout, char jobz, char range,\nchar uplo, lapack_int n, double * a, lapack_int lda, double vl, double vu, lapack_int\nil, lapack_int iu, double abstol, lapack_int * m, double * w, double * z, lapack_int\nldz, lapack_int * ifail);\nLAPACK_DECL lapack_int LAPACKE_ssygv_2stage (int matrix_layout, lapack_int itype, char\njobz, char uplo, lapack_int n, float * a, lapack_int lda, float * b, lapack_int ldb,\nfloat * w);\nLAPACK_DECL lapack_int LAPACKE_dsygv_2stage (int matrix_layout, lapack_int itype, char\njobz, char uplo, lapack_int n, double * a, lapack_int lda, double * b, lapack_int ldb,\ndouble * w);\nLAPACK_DECL lapack_int LAPACKE_cheev_2stage (int matrix_layout, char jobz, char uplo,\nlapack_int n, lapack_complex_float * a, lapack_int lda, float * w);\nLAPACK_DECL lapack_int LAPACKE_zheev_2stage (int matrix_layout, char jobz, char uplo,\nlapack_int n, lapack_complex_double * a, lapack_int lda, double * w);\nLAPACK_DECL lapack_int LAPACKE_cheevd_2stage (int matrix_layout, char jobz, char uplo,\nlapack_int n, lapack_complex_float * a, lapack_int lda, float * w);\nLAPACK_DECL lapack_int LAPACKE_zheevd_2stage (int matrix_layout, char jobz, char uplo,\nlapack_int n, lapack_complex_double * a, lapack_int lda, double * w);\nLAPACK_DECL lapack_int LAPACKE_cheevr_2stage (int matrix_layout, char jobz, char range,\nchar uplo, lapack_int n, lapack_complex_float * a, lapack_int lda, float vl, float vu,\nlapack_int il, lapack_int iu, float abstol, lapack_int * m, float * w,\nlapack_complex_float * z, lapack_int ldz, lapack_int * isuppz);\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1313\n\n\nLAPACK_DECL lapack_int LAPACKE_zheevr_2stage (int matrix_layout, char jobz, char range,\nchar uplo, lapack_int n, lapack_complex_double * a, lapack_int lda, double vl, double\nvu, lapack_int il, lapack_int iu, double abstol, lapack_int * m, double * w,\nlapack_complex_double * z, lapack_int ldz, lapack_int * isuppz);\nLAPACK_DECL lapack_int LAPACKE_cheevx_2stage (int matrix_layout, char jobz, char range,\nchar uplo, lapack_int n, lapack_complex_float * a, lapack_int lda, float vl, float vu,\nlapack_int il, lapack_int iu, float abstol, lapack_int * m, float * w,\nlapack_complex_float * z, lapack_int ldz, lapack_int * ifail);\nLAPACK_DECL lapack_int LAPACKE_zheevx_2stage (int matrix_layout, char jobz, char range,\nchar uplo, lapack_int n, lapack_complex_double * a, lapack_int lda, double vl, double\nvu, lapack_int il, lapack_int iu, double abstol, lapack_int * m, double * w,\nlapack_complex_double * z, lapack_int ldz, lapack_int * ifail);\nLAPACK_DECL lapack_int LAPACKE_chegv_2stage (int matrix_layout, lapack_int itype, char\njobz, char uplo, lapack_int n, lapack_complex_float * a, lapack_int lda,\nlapack_complex_float * b, lapack_int ldb, float * w);\nLAPACK_DECL lapack_int LAPACKE_zhegv_2stage (int matrix_layout, lapack_int itype, char\njobz, char uplo, lapack_int n, lapack_complex_double * a, lapack_int lda,\nlapack_complex_double * b, lapack_int ldb, double * w);\nLAPACK_DECL lapack_int LAPACKE_ssbev_2stage (int matrix_layout, char jobz, char uplo,\nlapack_int n, lapack_int kd, float * ab, lapack_int ldab, float * w, float * z,\nlapack_int ldz);\nLAPACK_DECL lapack_int LAPACKE_dsbev_2stage (int matrix_layout, char jobz, char uplo,\nlapack_int n, lapack_int kd, double * ab, lapack_int ldab, double * w, double * z,\nlapack_int ldz);\nLAPACK_DECL lapack_int LAPACKE_ssbevd_2stage (int matrix_layout, char jobz, char uplo,\nlapack_int n, lapack_int kd, float * ab, lapack_int ldab, float * w, float * z,\nlapack_int ldz);\nLAPACK_DECL lapack_int LAPACKE_dsbevd_2stage (int matrix_layout, char jobz, char uplo,\nlapack_int n, lapack_int kd, double * ab, lapack_int ldab, double * w, double * z,\nlapack_int ldz);\nLAPACK_DECL lapack_int LAPACKE_ssbevx_2stage (int matrix_layout, char jobz, char range,\nchar uplo, lapack_int n, lapack_int kd, float * ab, lapack_int ldab, float * q,\nlapack_int ldq, float vl, float vu, lapack_int il, lapack_int iu, float abstol,\nlapack_int * m, float * w, float * z, lapack_int ldz, lapack_int * ifail);\nLAPACK_DECL lapack_int LAPACKE_dsbevx_2stage (int matrix_layout, char jobz, char range,\nchar uplo, lapack_int n, lapack_int kd, double * ab, lapack_int ldab, double * q,\nlapack_int ldq, double vl, double vu, lapack_int il, lapack_int iu, double abstol,\nlapack_int * m, double * w, double * z, lapack_int ldz, lapack_int * ifail);\nLAPACK_DECL lapack_int LAPACKE_chbev_2stage (int matrix_layout, char jobz, char uplo,\nlapack_int n, lapack_int kd, lapack_complex_float * ab, lapack_int ldab, float * w,\nlapack_complex_float * z, lapack_int ldz);\nLAPACK_DECL lapack_int LAPACKE_zhbev_2stage (int matrix_layout, char jobz, char uplo,\nlapack_int n, lapack_int kd, lapack_complex_double * ab, lapack_int ldab, double * w,\nlapack_complex_double * z, lapack_int ldz);\nLAPACK_DECL lapack_int LAPACKE_chbevd_2stage (int matrix_layout, char jobz, char uplo,\nlapack_int n, lapack_int kd, lapack_complex_float * ab, lapack_int ldab, float * w,\nlapack_complex_float * z, lapack_int ldz);\nLAPACK_DECL lapack_int LAPACKE_zhbevd_2stage (int matrix_layout, char jobz, char uplo,\nlapack_int n, lapack_int kd, lapack_complex_double * ab, lapack_int ldab, double * w,\nlapack_complex_double * z, lapack_int ldz);\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1314\n\n\nLAPACK_DECL lapack_int LAPACKE_chbevx_2stage (int matrix_layout, char jobz, char range,\nchar uplo, lapack_int n, lapack_int kd, lapack_complex_float * ab, lapack_int ldab,\nlapack_complex_float * q, lapack_int ldq, float vl, float vu, lapack_int il, lapack_int\niu, float abstol, lapack_int * m, float * w, lapack_complex_float * z, lapack_int ldz,\nlapack_int * ifail);\nLAPACK_DECL lapack_int LAPACKE_zhbevx_2stage (int matrix_layout, char jobz, char range,\nchar uplo, lapack_int n, lapack_int kd, lapack_complex_double * ab, lapack_int ldab,\nlapack_complex_double * q, lapack_int ldq, double vl, double vu, lapack_int il,\nlapack_int iu, double abstol, lapack_int * m, double * w, lapack_complex_double * z,\nlapack_int ldz, lapack_int * ifail);\nFor descriptions of these functions, please see https://www.netlib.org/lapack/explore-html/files.html.\nScaLAPACK Routines\nIntel® oneAPI Math Kernel Library implements routines from the ScaLAPACK package for distributed-memory\narchitectures. Routines are supported for both real and complex dense and band matrices to perform the\ntasks of solving systems of linear equations, solving linear least-squares problems, eigenvalue and singular\nvalue problems, as well as performing a number of related computational tasks.\nIntel® oneAPI Math Kernel Library (oneMKL) ScaLAPACK routines are written in FORTRAN 77 with exception\nof a few utility routines written in C to exploit the IEEE arithmetic. All routines are available in all precision\ntypes: single precision, double precision, complexm, and double complex precision. See\nthemkl_scalapack.h header file for C declarations of ScaLAPACK routines.\nNOTE\nScaLAPACK routines are provided only for Intel® 64 or Intel® Many Integrated Core architectures.\nSee descriptions of ScaLAPACK computational routines that perform distinct computational tasks, as well as \ndriver routinesfor solving standard types of problems in one call. Additionally, Intel® oneAPI Math Kernel\nLibrary implements ScaLAPACKAuxiliary Routines, Utility Functions and Routines, and Matrix Redistribution/\nCopy Routines. The library includes routines for both real and complex data.\nThe <install_directory>/examples/scalapackf directory contains sample code demonstrating the use\nof ScaLAPACK routines.\nGenerally, ScaLAPACK runs on a network of computers using MPI as a message-passing layer and a set of\nprebuilt communication subprograms (BLACS), as well as a set of BLAS optimized for the target architecture.\nIntel® oneAPI Math Kernel Library (oneMKL) version of ScaLAPACK is optimized for Intel® processors. For the\ndetailed system and environment requirements, seeIntel® oneAPI Math Kernel Library (oneMKL) Release\nNotes and Intel® oneAPI Math Kernel Library (oneMKL) Developer Guide.\nFor full reference on ScaLAPACK routines and related information, see [SLUG].\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nOverview of ScaLAPACK Routines\nThe model of the computing environment for ScaLAPACK is represented as a one-dimensional array of\nprocesses (for operations on band or tridiagonal matrices) or also a two-dimensional process grid (for\noperations on dense matrices). To use ScaLAPACK, all global matrices or vectors should be distributed on this\narray or grid prior to calling the ScaLAPACK routines.\nScaLAPACK is closely tied to other components, including BLAS, BLACS, LAPACK, and PBLAS.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1315\n\n\nScaLAPACK Array Descriptors\nScaLAPACK uses two-dimensional block-cyclic data distribution as a layout for dense matrix computations.\nThis distribution provides good work balance between available processors, and also allows use of BLAS Level\n3 routines for optimal local computations. Information about the data distribution that is required to establish\nthe mapping between each global matrix and its corresponding process and memory location is contained in\nthe array called the array descriptor associated with each global matrix. The size of the array descriptor is\ndenoted as dlen_.\nLet A be a two-dimensional block cyclicly distributed matrix with the array descriptor array desca. The\nmeaning of each array descriptor element depends on the type of the matrix A. The tables \"Array descriptor\nfor dense matrices\" and \"Array descriptor for narrow-band and tridiagonal matrices\" describe the meaning of\neach element for the different types of matrices.\nArray descriptor for dense matrices (dlen_=9)\nElement\nName\nStored in\nDescription\nElement Index\nNumber\ndtype_a\ndesca[dtype_]\nDescriptor type ( =1 for dense matrices).\n0\nctxt_a\ndesca[ctxt_]\nBLACS context handle for the process grid.\n1\nm_a\ndesca[m_]\nNumber of rows in the global matrix A.\n2\nn_a\ndesca[n_]\nNumber of columns in the global matrix A.\n3\nmb_a\ndesca[mb_]\nRow blocking factor.\n4\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1316\n\n\nElement\nName\nStored in\nDescription\nElement Index\nNumber\nnb_a\ndesca[nb_]\nColumn blocking factor.\n5\nrsrc_a\ndesca[rsrc_]\nProcess row over which the first row of the\nglobal matrix A is distributed.\n6\ncsrc_a\ndesca[csrc_]\nProcess column over which the first column of\nthe global matrix A is distributed.\n7\nlld_a\ndesca[lld_]\nLeading dimension of the local matrix A.\n8\nArray descriptor for narrow-band and tridiagonal matrices (dlen_=7)\nElement\nName\nStored in\nDescription\nElement Index\nNumber\ndtype_a\ndesca[dtype_]\nDescriptor type\n•\ndtype_a=501: 1-by-P grid,\n•\ndtype_a=502: P-by-1 grid.\n0\nctxt_a\ndesca[ctxt_]\nBLACS context handle indicating the BLACS\nprocess grid over which the global matrix A is\ndistributed. The context itself is global, but the\nhandle (the integer value) can vary.\n1\nn_a\ndesca[n_]\nThe size of the matrix dimension being\ndistributed.\n2\nnb_a\ndesca[nb_]\nThe blocking factor used to distribute the\ndistributed dimension of the matrix A.\n3\nsrc_a\ndesca[src_]\nThe process row or column over which the first\nrow or column of the matrix A is distributed.\n4\nlld_a\ndesca[lld_]\nThe leading dimension of the local matrix\nstoring the local blocks of the distributed\nmatrix A. The minimum value of lld_a depends\non dtype_a.\n•\ndtype_a=501: lld_a≥ max(size of\nundistributed dimension, 1),\n•\ndtype_a=502: lld_a≥ max(nb_a, 1).\n5\nNot\napplicable\nReserved for future use.\n6\nSimilar notations are used for different matrices. For example: lld_b is the leading dimension of the local\nmatrix storing the local blocks of the distributed matrix B and dtype_z is the type of the global matrix Z.\nThe number of rows and columns of a global dense matrix that a particular process in a grid receives after\ndata distributing is denoted by LOCr() and LOCc(), respectively. To compute these numbers, you can use the\nScaLAPACK tool routine numroc.\nAfter the block-cyclic distribution of global data is done, you may choose to perform an operation on a\nsubmatrix sub(A) of the global matrix A defined by the following 6 values (for dense matrices):\nm\nThe number of rows of sub(A)\nn\nThe number of columns of sub(A)\na\nA pointer to the local matrix containing the entire global matrix A\nia\nThe row index of sub(A) in the global matrix A\nja\nThe column index of sub(A) in the global matrix A\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1317\n\n\ndesca\nThe array descriptor for the global matrix A\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nNaming Conventions for ScaLAPACK Routines\nFor each routine introduced in this chapter, you can use the ScaLAPACK name. The naming convention for\nScaLAPACK routines is similar to that used for LAPACK routines. A general rule is that each routine name in\nScaLAPACK, which has an LAPACK equivalent, is simply the LAPACK name prefixed by initial letter p.\nScaLAPACK names have the structure p?yyzzz or p?yyzz, which is described below.\nThe initial letter p is a distinctive prefix of ScaLAPACK routines and is present in each such routine.\nThe second symbol ? indicates the data type:\ns\nreal, single precision\nd\nreal, double precision\nc\ncomplex, single precision\nz\ncomplex, double precision\nThe second and third letters yy indicate the matrix type as:\nge\ngeneral\ngb\ngeneral band\ngg\na pair of general matrices (for a generalized problem)\ndt\ngeneral tridiagonal (diagonally dominant-like)\ndb\ngeneral band (diagonally dominant-like)\npo\nsymmetric or Hermitian positive-definite\npb\nsymmetric or Hermitian positive-definite band\npt\nsymmetric or Hermitian positive-definite tridiagonal\nsy\nsymmetric\nst\nsymmetric tridiagonal (real)\nhe\nHermitian\nor\northogonal\ntr\ntriangular (or quasi-triangular)\ntz\ntrapezoidal\nun\nunitary\nFor computational routines, the last three letters zzz indicate the computation performed and have the same\nmeaning as for LAPACK routines.\nFor driver routines, the last two letters zz or three letters zzz have the following meaning:\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1318\n\n\nsv\na simple driver for solving a linear system\nsvx\nan expert driver for solving a linear system\nls\na driver for solving a linear least squares problem\nev\na simple driver for solving a symmetric eigenvalue problem\nevd\na simple driver for solving an eigenvalue problem using a divide and conquer\nalgorithm\nevx\nan expert driver for solving a symmetric eigenvalue problem\nsvd\na driver for computing a singular value decomposition\ngvx\nan expert driver for solving a generalized symmetric definite eigenvalue problem\nSimple driver here means that the driver just solves the general problem, whereas an expert driver is more\nversatile and can also optionally perform some related computations (such, for example, as refining the\nsolution and computing error bounds after the linear system is solved).\nScaLAPACK Computational Routines\nIn the sections that follow, the descriptions of ScaLAPACK computational routines are given. These routines\nperform distinct computational tasks that can be used for:\n•\nSolving Systems of Linear Equations\n•\nOrthogonal Factorizations and LLS Problems\n•\nSymmetric Eigenproblems\n•\nNonsymmetric Eigenproblems\n•\nSingular Value Decomposition\n•\nGeneralized Symmetric-Definite Eigenproblems\nSee also the respective driver routines.\nSystems of Linear Equations: ScaLAPACK Computational Routines\nScaLAPACK supports routines for the systems of equations with the following types of matrices:\n•\ngeneral\n•\ngeneral banded\n•\ngeneral diagonally dominant-like banded (including general tridiagonal)\n•\nsymmetric or Hermitian positive-definite\n•\nsymmetric or Hermitian positive-definite banded\n•\nsymmetric or Hermitian positive-definite tridiagonal\nA diagonally dominant-like matrix is defined as a matrix for which it is known in advance that pivoting is not\nrequired in the LU factorization of this matrix.\nFor the above matrix types, the library includes routines for performing the following computations: factoring\nthe matrix; equilibrating the matrix; solving a system of linear equations; estimating the condition number of\na matrix; refining the solution of linear equations and computing its error bounds; inverting the matrix. Note\nthat for some of the listed matrix types only part of the computational routines are provided (for example,\nroutines that refine the solution are not provided for band or tridiagonal matrices). See Table “Computational\nRoutines for Systems of Linear Equations” for full list of available routines.\nTo solve a particular problem, you can either call two or more computational routines or call a corresponding \ndriver routine that combines several tasks in one call. Thus, to solve a system of linear equations with a\ngeneral matrix, you can first call p?getrf(LU factorization) and then p?getrs(computing the solution).\nThen, you might wish to call p?gerfs to refine the solution and get the error bounds. Alternatively, you can\njust use the driver routine p?gesvx which performs all these tasks in one call.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1319\n\n\nTable “Computational Routines for Systems of Linear Equations” lists the ScaLAPACK computational routines\nfor factorizing, equilibrating, and inverting matrices, estimating their condition numbers, solving systems of\nequations with real matrices, refining the solution, and estimating its error.\nComputational Routines for Systems of Linear Equations\nMatrix type, storage\nscheme\nFactorize\nmatrix\nEquilibrate\nmatrix\nSolve\nsystem\nCondition\nnumber\nEstimate\nerror\nInvert\nmatrix\ngeneral (partial pivoting)\np?getrf\np?geequ\np?getrs\np?gecon\np?gerfs\np?getri\ngeneral band (partial\npivoting)\np?gbtrf\n \np?gbtrs\n \n \n \ngeneral band (no\npivoting)\np?dbtrf\n \np?dbtrs\n \n \n \ngeneral tridiagonal (no\npivoting)\np?dttrf\n \np?dttrs\n \n \n \nsymmetric/Hermitian\npositive-definite\np?potrf\np?poequ\np?potrs\np?pocon\np?porfs\np?potri\nsymmetric/Hermitian\npositive-definite, band\np?pbtrf\n \np?pbtrs\n \n \n \nsymmetric/Hermitian\npositive-definite,\ntridiagonal\np?pttrf\n \np?pttrs\n \n \n \ntriangular\n \n \np?trtrs\np?trcon\np?trrfs\np?trtri\nIn this table ? stands for s (single precision real), d (double precision real), c (single precision complex), or z\n(double precision complex).\nMatrix Factorization: ScaLAPACK Computational Routines\nThis section describes the ScaLAPACK routines for matrix factorization. The following factorizations are\nsupported:\n•\nLU factorization of general matrices\n•\nLU factorization of diagonally dominant-like matrices\n•\nCholesky factorization of real symmetric or complex Hermitian positive-definite matrices\nYou can compute the factorizations using full and band storage of matrices.\np?getrf\nComputes the LU factorization of a general m-by-n\ndistributed matrix.\nSyntax\nvoid psgetrf (MKL_INT *m , MKL_INT *n , float *a , MKL_INT *ia , MKL_INT *ja , MKL_INT\n*desca , MKL_INT *ipiv , MKL_INT *info );\nvoid pdgetrf (MKL_INT *m , MKL_INT *n , double *a , MKL_INT *ia , MKL_INT *ja , MKL_INT\n*desca , MKL_INT *ipiv , MKL_INT *info );\nvoid pcgetrf (MKL_INT *m , MKL_INT *n , MKL_Complex8 *a , MKL_INT *ia , MKL_INT *ja ,\nMKL_INT *desca , MKL_INT *ipiv , MKL_INT *info );\nvoid pzgetrf (MKL_INT *m , MKL_INT *n , MKL_Complex16 *a , MKL_INT *ia , MKL_INT *ja ,\nMKL_INT *desca , MKL_INT *ipiv , MKL_INT *info );\nInclude Files\n•\nmkl_scalapack.h\nDescription\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1320\n\n\nThe p?getrffunction forms the LU factorization of a general m-by-n distributed matrix sub(A) = A(ia:ia\n+m-1, ja:ja+n-1) as\nA = P*L*U\nwhere P is a permutation matrix, L is lower triangular with unit diagonal elements (lower trapezoidal if m>n)\nand U is upper triangular (upper trapezoidal if m < n). L and U are stored in sub(A).\nThe function uses partial pivoting, with row interchanges.\nNOTE\nThis function supports the Progress Routine feature. See mkl_progress for details.\nInput Parameters\nm\n(global) The number of rows in the distributed matrix sub(A); m≥0.\nn\n(global) The number of columns in the distributed matrix sub(A); n≥0.\na\n(local)\nPointer into the local memory to an array of local size lld_a*LOCc(ja+n-1).\nContains the local pieces of the distributed matrix sub(A) to be factored.\nia, ja\n(global) The row and column indices in the global matrix A indicating the\nfirst row and the first column of the matrix sub(A), respectively.\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\nOutput Parameters\na\nOverwritten by local pieces of the factors L and U from the factorization A =\nP*L*U. The unit diagonal elements of L are not stored.\nipiv\n(local) Array of size LOCr(m_a)+ mb_a.\nContains the pivoting information: local row i was interchanged with global\nrow ipiv[i-1]. This array is tied to the distributed matrix A.\ninfo\n(global)\nIf info=0, the execution is successful.\ninfo < 0: if the i-th argument is an array and the j-th entry, indexed j - 1,\nhad an illegal value, then info = -(i*100+j); if the i-th argument is a\nscalar and had an illegal value, then info = -i.\nIf info = i > 0, uia+i, ja+j-1 is 0. The factorization has been completed,\nbut the factor U is exactly singular. Division by zero will occur if you use the\nfactor U for solving a system of linear equations.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?gbtrf\nComputes the LU factorization of a general n-by-n\nbanded distributed matrix.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1321\n\n\nSyntax\nvoid psgbtrf (MKL_INT *n , MKL_INT *bwl , MKL_INT *bwu , float *a , MKL_INT *ja ,\nMKL_INT *desca , MKL_INT *ipiv , float *af , MKL_INT *laf , float *work , MKL_INT\n*lwork , MKL_INT *info );\nvoid pdgbtrf (MKL_INT *n , MKL_INT *bwl , MKL_INT *bwu , double *a , MKL_INT *ja ,\nMKL_INT *desca , MKL_INT *ipiv , double *af , MKL_INT *laf , double *work , MKL_INT\n*lwork , MKL_INT *info );\nvoid pcgbtrf (MKL_INT *n , MKL_INT *bwl , MKL_INT *bwu , MKL_Complex8 *a , MKL_INT\n*ja , MKL_INT *desca , MKL_INT *ipiv , MKL_Complex8 *af , MKL_INT *laf , MKL_Complex8\n*work , MKL_INT *lwork , MKL_INT *info );\nvoid pzgbtrf (MKL_INT *n , MKL_INT *bwl , MKL_INT *bwu , MKL_Complex16 *a , MKL_INT\n*ja , MKL_INT *desca , MKL_INT *ipiv , MKL_Complex16 *af , MKL_INT *laf , MKL_Complex16\n*work , MKL_INT *lwork , MKL_INT *info );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?gbtrf function computes the LU factorization of a general n-by-n real/complex banded distributed\nmatrix A(1:n, ja:ja+n-1) using partial pivoting with row interchanges.\nThe resulting factorization is not the same factorization as returned from the LAPACK function ?gbtrf.\nAdditional permutations are performed on the matrix for the sake of parallelism.\nThe factorization has the form\nA(1:n, ja:ja+n-1) = P*L*U*Q\nwhere P and Q are permutation matrices, and L and U are banded lower and upper triangular matrices,\nrespectively. The matrix Q represents reordering of columns for the sake of parallelism, while P represents\nreordering of rows for numerical stability using classic partial pivoting.\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nInput Parameters\nn\n(global) The number of rows and columns in the distributed submatrix\nA(1:n, ja:ja+n-1); n≥ 0.\nbwl\n(global) The number of sub-diagonals within the band of A\n( 0 ≤ bwl ≤ n-1 ).\nbwu\n(global) The number of super-diagonals within the band of A\n( 0 ≤ bwu ≤ n-1 ).\na\n(local)\nPointer into the local memory to an array of local size lld_a*LOCc(ja+n-1)\nwhere\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1322\n\n\nlld_a≥ 2*bwl + 2*bwu +1.\nContains the local pieces of the n-by-n distributed banded matrix A(1:n,\nja:ja+n-1) to be factored.\nja\n(global) The index in the global matrix A indicating the start of the matrix to\nbe operated on (which may be either all of A or a submatrix of A).\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\nIf dtype_a = 501, then dlen_≥ 7;\nelse if dtype_a = 1, then dlen_≥ 9.\nlaf\n(local) The size of the array af.\nMust be laf≥ (nb_a+bwu)*(bwl+bwu)+6*(bwl+bwu)*(bwl+2*bwu).\nIf laf is not large enough, an error code will be returned and the minimum\nacceptable size will be returned in af[0].\nwork\n(local) Same type as a. Workspace array of size lwork.\nlwork\n(local or global) The size of the work array (lwork≥ 1). If lwork is too\nsmall, the minimal acceptable size will be returned in work[0] and an error\ncode is returned.\nOutput Parameters\na\nOn exit, this array contains details of the factorization. Note that additional\npermutations are performed on the matrix, so that the factors returned are\ndifferent from those returned by LAPACK.\nipiv\n(local) array.\nThe size of ipiv must be ≥nb_a.\nContains pivot indices for local factorizations. Note that you should not alter\nthe contents of this array between factorization and solve.\naf\n(local)\nArray of size laf.\nAuxiliary fill-in space. The fill-in space is created in a call to the factorization\nfunction p?gbtrf and is stored in af.\nNote that if a linear system is to be solved using p?gbtrs after the\nfactorization function,af must not be altered after the factorization.\nwork[0]\nOn exit, work[0] contains the minimum value of lwork required.\ninfo\n(global)\nIf info=0, the execution is successful.\ninfo < 0:\nIf the i-th argument is an array and the j-th entry, indexed j - 1, had an\nillegal value, then info = -(i*100+j); if the i-th argument is a scalar and\nhad an illegal value, then info = -i.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1323\n\n\ninfo> 0:\nIf info = k ≤ NPROCS, the submatrix stored on processor info and\nfactored locally was not nonsingular, and the factorization was not\ncompleted.\nIf info = k > NPROCS, the submatrix stored on processor info-NPROCS\nrepresenting interactions with other processors was not nonsingular, and\nthe factorization was not completed.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?dbtrf\nComputes the LU factorization of a n-by-n diagonally\ndominant-like banded distributed matrix.\nSyntax\nvoid psdbtrf (MKL_INT *n , MKL_INT *bwl , MKL_INT *bwu , float *a , MKL_INT *ja ,\nMKL_INT *desca , float *af , MKL_INT *laf , float *work , MKL_INT *lwork , MKL_INT\n*info );\nvoid pddbtrf (MKL_INT *n , MKL_INT *bwl , MKL_INT *bwu , double *a , MKL_INT *ja ,\nMKL_INT *desca , double *af , MKL_INT *laf , double *work , MKL_INT *lwork , MKL_INT\n*info );\nvoid pcdbtrf (MKL_INT *n , MKL_INT *bwl , MKL_INT *bwu , MKL_Complex8 *a , MKL_INT\n*ja , MKL_INT *desca , MKL_Complex8 *af , MKL_INT *laf , MKL_Complex8 *work , MKL_INT\n*lwork , MKL_INT *info );\nvoid pzdbtrf (MKL_INT *n , MKL_INT *bwl , MKL_INT *bwu , MKL_Complex16 *a , MKL_INT\n*ja , MKL_INT *desca , MKL_Complex16 *af , MKL_INT *laf , MKL_Complex16 *work , MKL_INT\n*lwork , MKL_INT *info );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?dbtrffunction computes the LU factorization of a n-by-n real/complex diagonally dominant-like\nbanded distributed matrix A(1:n, ja:ja+n-1) without pivoting.\nNOTE\nA matrix is called diagonally dominant-like if pivoting is not required for LU to be\nnumerically stable.\nNote that the resulting factorization is not the same factorization as returned from LAPACK. Additional\npermutations are performed on the matrix for the sake of parallelism.\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1324\n\n\nInput Parameters\nn\n(global) The number of rows and columns in the distributed submatrix\nA(1:n, ja:ja+n-1); n≥ 0.\nbwl\n(global) The number of sub-diagonals within the band of A\n(0 ≤ bwl ≤ n-1).\nbwu\n(global) The number of super-diagonals within the band of A\n(0 ≤ bwu ≤ n-1).\na\n(local)\nPointer into the local memory to an array of local size lld_a*LOCc(ja+n-1).\nContains the local pieces of the n-by-n distributed banded matrix A(1:n,\nja:ja+n-1) to be factored.\nja\n(global) The index in the global matrix A indicating the start of the matrix to\nbe operated on (which may be either all of A or a submatrix of A).\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\nIf dtype_a = 501, then dlen_≥ 7;\nelse if dtype_a = 1, then dlen_≥ 9.\nlaf\n(local) The size of the array af.\nMust be laf≥NB*(bwl+bwu)+6*(max(bwl,bwu))2 .\nIf laf is not large enough, an error code will be returned and the minimum\nacceptable size will be returned in af[0].\nwork\n(local) Workspace array of size lwork.\nlwork\n(local or global) The size of the work array, must be lwork≥\n(max(bwl,bwu))2. If lwork is too small, the minimal acceptable size will\nbe returned in work[0] and an error code is returned.\nOutput Parameters\na\nOn exit, this array contains details of the factorization. Note that additional\npermutations are performed on the matrix, so that the factors returned are\ndifferent from those returned by LAPACK.\naf\n(local)\nArray of size laf.\nAuxiliary fill-in space. The fill-in space is created in a call to the factorization\nfunction p?dbtrf and is stored in af.\nNote that if a linear system is to be solved using p?dbtrs after the\nfactorization function,af must not be altered after the factorization.\nwork[0]\nOn exit, work[0] contains the minimum value of lwork required for\noptimum performance.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1325\n\n\ninfo\n(global)\nIf info=0, the execution is successful.\ninfo < 0:\nIf the i-th argument is an array and the j-th entry, indexed j - 1, had an\nillegal value, then info = -(i*100+j); if the i-th argument is a scalar and\nhad an illegal value, then info = -i.\ninfo> 0:\nIf info = k ≤ NPROCS, the submatrix stored on processor info and\nfactored locally was not diagonally dominant-like, and the factorization was\nnot completed.\nIf info = k > NPROCS, the submatrix stored on processor info-NPROCS\nrepresenting interactions with other processors was not nonsingular, and\nthe factorization was not completed.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?dttrf\nComputes the LU factorization of a diagonally\ndominant-like tridiagonal distributed matrix.\nSyntax\nvoid psdttrf (MKL_INT *n , float *dl , float *d , float *du , MKL_INT *ja , MKL_INT\n*desca , float *af , MKL_INT *laf , float *work , MKL_INT *lwork , MKL_INT *info );\nvoid pddttrf (MKL_INT *n , double *dl , double *d , double *du , MKL_INT *ja , MKL_INT\n*desca , double *af , MKL_INT *laf , double *work , MKL_INT *lwork , MKL_INT *info );\nvoid pcdttrf (MKL_INT *n , MKL_Complex8 *dl , MKL_Complex8 *d , MKL_Complex8 *du ,\nMKL_INT *ja , MKL_INT *desca , MKL_Complex8 *af , MKL_INT *laf , MKL_Complex8 *work ,\nMKL_INT *lwork , MKL_INT *info );\nvoid pzdttrf (MKL_INT *n , MKL_Complex16 *dl , MKL_Complex16 *d , MKL_Complex16 *du ,\nMKL_INT *ja , MKL_INT *desca , MKL_Complex16 *af , MKL_INT *laf , MKL_Complex16 *work ,\nMKL_INT *lwork , MKL_INT *info );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?dttrffunction computes the LU factorization of an n-by-n real/complex diagonally dominant-like\ntridiagonal distributed matrix A(1:n, ja:ja+n-1) without pivoting for stability.\nThe resulting factorization is not the same factorization as returned from LAPACK. Additional permutations\nare performed on the matrix for the sake of parallelism.\nThe factorization has the form:\nA(1:n, ja:ja+n-1) = P*L*U*PT,\nwhere P is a permutation matrix, and L and U are banded lower and upper triangular matrices, respectively.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1326\n\n\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nInput Parameters\nn\n(global) The number of rows and columns to be operated on, that is, the\norder of the distributed submatrix A(1:n, ja:ja+n-1) (n≥ 0).\ndl, d, du\n(local)\nPointers to the local arrays of size nb_a each.\nOn entry, the array dl contains the local part of the global vector storing\nthe subdiagonal elements of the matrix. Globally, dl[0] is not referenced,\nand dl must be aligned with d.\nOn entry, the array d contains the local part of the global vector storing the\ndiagonal elements of the matrix.\nOn entry, the array du contains the local part of the global vector storing\nthe super-diagonal elements of the matrix. du[n-1] is not referenced, and\ndu must be aligned with d.\nja\n(global) The index in the global matrix A indicating the start of the matrix to\nbe operated on (which may be either all of A or a submatrix of A).\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\nIf dtype_a = 501, then dlen_≥ 7;\nelse if dtype_a = 1, then dlen_≥ 9.\nlaf\n(local) The size of the array af.\nMust be laf≥ 2*(NB+2) .\nIf laf is not large enough, an error code will be returned and the minimum\nacceptable size will be returned in af[0].\nwork\n(local) Same type as d. Workspace array of size lwork.\nlwork\n(local or global) The size of the work array, must be at least lwork≥\n8*NPCOL.\nOutput Parameters\ndl, d, du\nOn exit, overwritten by the information containing the factors of the matrix.\naf\n(local)\nArray of size laf.\nAuxiliary fill-in space. The fill-in space is created in a call to the factorization\nfunction p?dttrf and is stored in af.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1327\n\n\nNote that if a linear system is to be solved using p?dttrs after the\nfactorization function,af must not be altered.\nwork[0]\nOn exit, work[0] contains the minimum value of lwork required for\noptimum performance.\ninfo\n(global)\nIf info=0, the execution is successful.\ninfo < 0:\nIf the i-th argument is an array and the j-th entry, indexed j - 1, had an\nillegal value, then info = -(i*100+j); if the i-th argument is a scalar and\nhad an illegal value, then info = -i.\ninfo> 0:\nIf info = k ≤ NPROCS, the submatrix stored on processor info and\nfactored locally was not diagonally dominant-like, and the factorization was\nnot completed.\nIf info = k > NPROCS, the submatrix stored on processor info-NPROCS\nrepresenting interactions with other processors was not nonsingular, and\nthe factorization was not completed.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?potrf\nComputes the Cholesky factorization of a symmetric\n(Hermitian) positive-definite distributed matrix.\nSyntax\nvoid pspotrf (char *uplo , MKL_INT *n , float *a , MKL_INT *ia , MKL_INT *ja , MKL_INT\n*desca , MKL_INT *info );\nvoid pdpotrf (char *uplo , MKL_INT *n , double *a , MKL_INT *ia , MKL_INT *ja , MKL_INT\n*desca , MKL_INT *info );\nvoid pcpotrf (char *uplo , MKL_INT *n , MKL_Complex8 *a , MKL_INT *ia , MKL_INT *ja ,\nMKL_INT *desca , MKL_INT *info );\nvoid pzpotrf (char *uplo , MKL_INT *n , MKL_Complex16 *a , MKL_INT *ia , MKL_INT *ja ,\nMKL_INT *desca , MKL_INT *info );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?potrffunction computes the Cholesky factorization of a real symmetric or complex Hermitian positive-\ndefinite distributed n-by-n matrix A(ia:ia+n-1, ja:ja+n-1), denoted below as sub(A).\nThe factorization has the form\nsub(A) = UH*U if uplo='U', or\nsub(A) = L*LH if uplo='L'\nwhere L is a lower triangular matrix and U is upper triangular.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1328\n\n\nInput Parameters\nuplo\n(global)\nIndicates whether the upper or lower triangular part of sub(A) is stored.\nMust be 'U' or 'L'.\nIf uplo = 'U', the array a stores the upper triangular part of the matrix\nsub(A) that is factored as UH*U.\nIf uplo = 'L', the array a stores the lower triangular part of the\nmatrix sub(A) that is factored as L*LH.\nn\n(global) The order of the distributed matrix sub(A) (n≥0).\na\n(local)\nPointer into the local memory to an array of size lld_a*LOCc(ja+n-1).\nOn entry, this array contains the local pieces of the n-by-n symmetric/\nHermitian distributed matrix sub(A) to be factored.\nDepending on uplo, the array a contains either the upper or the lower\ntriangular part of the matrix sub(A) (see uplo).\nia, ja\n(global) The row and column indices in the global matrix A indicating the\nfirst row and the first column of the matrix sub(A), respectively.\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\nOutput Parameters\na\nThe upper or lower triangular part of a is overwritten by the Cholesky factor\nU or L, as specified by uplo.\ninfo\n(global) .\nIf info=0, the execution is successful;\ninfo < 0: if the i-th argument is an array, and the j-th entry, indexed j -\n1, had an illegal value, then info = -(i*100+j); if the i-th argument is a\nscalar and had an illegal value, then info = -i.\nIf info = k >0, the leading minor of order k, A(ia:ia+k-1, ja:ja+k-1), is\nnot positive-definite, and the factorization could not be completed.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?pbtrf\nComputes the Cholesky factorization of a symmetric\n(Hermitian) positive-definite banded distributed\nmatrix.\nSyntax\nvoid pspbtrf (char *uplo , MKL_INT *n , MKL_INT *bw , float *a , MKL_INT *ja , MKL_INT\n*desca , float *af , MKL_INT *laf , float *work , MKL_INT *lwork , MKL_INT *info );\nvoid pdpbtrf (char *uplo , MKL_INT *n , MKL_INT *bw , double *a , MKL_INT *ja , MKL_INT\n*desca , double *af , MKL_INT *laf , double *work , MKL_INT *lwork , MKL_INT *info );\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1329\n\n\nvoid pcpbtrf (char *uplo , MKL_INT *n , MKL_INT *bw , MKL_Complex8 *a , MKL_INT *ja ,\nMKL_INT *desca , MKL_Complex8 *af , MKL_INT *laf , MKL_Complex8 *work , MKL_INT\n*lwork , MKL_INT *info );\nvoid pzpbtrf (char *uplo , MKL_INT *n , MKL_INT *bw , MKL_Complex16 *a , MKL_INT *ja ,\nMKL_INT *desca , MKL_Complex16 *af , MKL_INT *laf , MKL_Complex16 *work , MKL_INT\n*lwork , MKL_INT *info );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?pbtrffunction computes the Cholesky factorization of an n-by-n real symmetric or complex Hermitian\npositive-definite banded distributed matrix A(1:n, ja:ja+n-1).\nThe resulting factorization is not the same factorization as returned from LAPACK. Additional permutations\nare performed on the matrix for the sake of parallelism.\nThe factorization has the form:\nA(1:n, ja:ja+n-1) = P*UH*U*PT, if uplo='U', or\nA(1:n, ja:ja+n-1) = P*L*LH*PT, if uplo='L',\nwhere P is a permutation matrix and U and L are banded upper and lower triangular matrices, respectively.\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nInput Parameters\nuplo\n(global) Must be 'U' or 'L'.\nIf uplo = 'U', upper triangle of A(1:n, ja:ja+n-1) is stored;\nIf uplo = 'L', lower triangle of A(1:n, ja:ja+n-1) is stored.\nn\n(global) The order of the distributed submatrix A(1:n, ja:ja+n-1).\n(n≥0).\nbw\n(global)\nThe number of superdiagonals of the distributed matrix if uplo = 'U', or\nthe number of subdiagonals if uplo = 'L' (bw≥0).\na\n(local)\nPointer into the local memory to an array of size lld_a*LOCc(ja+n-1).\nOn entry, this array contains the local pieces of the upper or lower triangle\nof the symmetric/Hermitian band distributed matrix A(1:n, ja:ja+n-1) to\nbe factored.\nja\n(global) The index in the global matrix A indicating the start of the matrix to\nbe operated on (which may be either all of A or a submatrix of A).\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1330\n\n\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\nIf dtype_a = 501, then dlen_≥ 7;\nelse if dtype_a = 1, then dlen_≥ 9.\nlaf\n(local) The size of the array af.\nMust be laf≥ (NB+2*bw)*bw.\nIf laf is not large enough, an error code will be returned and the minimum\nacceptable size will be returned in af[0].\nwork\n(local) Workspace array of size lwork.\nlwork\n(local or global) The size of the work array, must be lwork≥bw2.\nOutput Parameters\na\nOn exit, if info=0, contains the permuted triangular factor U or L from the\nCholesky factorization of the band matrix A(1:n, ja:ja+n-1), as specified\nby uplo.\naf\n(local)\nArray of size laf. Auxiliary fill-in space. The fill-in space is created in a call\nto the factorization function p?pbtrf and stored in af. Note that if a linear\nsystem is to be solved using p?pbtrs after the factorization function,af\nmust not be altered.\nwork[0]\nOn exit, work[0] contains the minimum value of lwork required for\noptimum performance.\ninfo\n(global)\nIf info=0, the execution is successful.\ninfo < 0:\nIf the i-th argument is an array and the j-th entry, indexed j - 1, had an\nillegal value, then info = -(i*100+j); if the i-th argument is a scalar and\nhad an illegal value, then info = -i.\ninfo>0:\nIf info = k ≤ NPROCS, the submatrix stored on processor info and\nfactored locally was not positive definite, and the factorization was not\ncompleted.\nIf info = k > NPROCS, the submatrix stored on processor info-NPROCS\nrepresenting interactions with other processors was not nonsingular, and\nthe factorization was not completed.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1331\n\n\np?pttrf\nComputes the Cholesky factorization of a symmetric\n(Hermitian) positive-definite tridiagonal distributed\nmatrix.\nSyntax\nvoid pspttrf (MKL_INT *n , float *d , float *e , MKL_INT *ja , MKL_INT *desca , float\n*af , MKL_INT *laf , float *work , MKL_INT *lwork , MKL_INT *info );\nvoid pdpttrf (MKL_INT *n , double *d , double *e , MKL_INT *ja , MKL_INT *desca , double\n*af , MKL_INT *laf , double *work , MKL_INT *lwork , MKL_INT *info );\nvoid pcpttrf (MKL_INT *n , float *d , MKL_Complex8 *e , MKL_INT *ja , MKL_INT *desca ,\nMKL_Complex8 *af , MKL_INT *laf , MKL_Complex8 *work , MKL_INT *lwork , MKL_INT\n*info );\nvoid pzpttrf (MKL_INT *n , double *d , MKL_Complex16 *e , MKL_INT *ja , MKL_INT\n*desca , MKL_Complex16 *af , MKL_INT *laf , MKL_Complex16 *work , MKL_INT *lwork ,\nMKL_INT *info );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?pttrffunction computes the Cholesky factorization of an n-by-n real symmetric or complex hermitian\npositive-definite tridiagonal distributed matrix A(1:n, ja:ja+n-1).\nThe resulting factorization is not the same factorization as returned from LAPACK. Additional permutations\nare performed on the matrix for the sake of parallelism.\nThe factorization has the form:\nA(1:n, ja:ja+n-1) = P*L*D*LH*PT, or\nA(1:n, ja:ja+n-1) = P*UH*D*U*PT,\nwhere P is a permutation matrix, and U and L are tridiagonal upper and lower triangular matrices,\nrespectively.\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nInput Parameters\nn\n(global) The order of the distributed submatrix A(1:n, ja:ja+n-1)\n(n≥ 0).\nd, e\n(local)\nPointers into the local memory to arrays of size nb_a each.\nOn entry, the array d contains the local part of the global vector storing the\nmain diagonal of the distributed matrix A.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1332\n\n\nOn entry, the array e contains the local part of the global vector storing the\nupper diagonal of the distributed matrix A.\nja\n(global) The index in the global matrix A indicating the start of the matrix to\nbe operated on (which may be either all of A or a submatrix of A).\ndesca\n(global and local ) array of size dlen_. The array descriptor for the\ndistributed matrix A.\nIf dtype_a = 501, then dlen_≥ 7;\nelse if dtype_a = 1, then dlen_≥ 9.\nlaf\n(local) The size of the array af.\nMust be laf≥nb_a+2.\nIf laf is not large enough, an error code will be returned and the minimum\nacceptable size will be returned in af[0].\nwork\n(local) Workspace array of size lwork .\nlwork\n(local or global) The size of the work array, must be at least\nlwork≥ 8*NPCOL.\nOutput Parameters\nd, e\nOn exit, overwritten by the details of the factorization.\naf\n(local)\nArray of size laf.\nAuxiliary fill-in space. The fill-in space is created in a call to the factorization\nfunction p?pttrf and stored in af.\nNote that if a linear system is to be solved using p?pttrs after the\nfactorization function,af must not be altered.\nwork[0]\nOn exit, work[0] contains the minimum value of lwork required for\noptimum performance.\ninfo\n(global)\nIf info=0, the execution is successful.\ninfo < 0:\nIf the i-th argument is an array and the j-th entry, indexed j - 1, had an\nillegal value, then info = -(i*100+j); if the i-th argument is a scalar and\nhad an illegal value, then info = -i.\ninfo> 0:\nIf info = k ≤ NPROCS, the submatrix stored on processor info and\nfactored locally was not positive definite, and the factorization was not\ncompleted.\nIf info = k > NPROCS, the submatrix stored on processor info-NPROCS\nrepresenting interactions with other processors was not nonsingular, and\nthe factorization was not completed.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1333\n\n\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\nSolving Systems of Linear Equations: ScaLAPACK Computational Routines\nThis section describes the ScaLAPACK routines for solving systems of linear equations. Before calling most of\nthese routines, you need to factorize the matrix of your system of equations (see Routines for Matrix\nFactorization in this chapter). However, the factorization is not necessary if your system of equations has a\ntriangular matrix.\np?getrs\nSolves a system of distributed linear equations with a\ngeneral square matrix, using the LU factorization\ncomputed by p?getrf.\nSyntax\nvoid psgetrs (char *trans , MKL_INT *n , MKL_INT *nrhs , float *a , MKL_INT *ia ,\nMKL_INT *ja , MKL_INT *desca , MKL_INT *ipiv , float *b , MKL_INT *ib , MKL_INT *jb ,\nMKL_INT *descb , MKL_INT *info );\nvoid pdgetrs (char *trans , MKL_INT *n , MKL_INT *nrhs , double *a , MKL_INT *ia ,\nMKL_INT *ja , MKL_INT *desca , MKL_INT *ipiv , double *b , MKL_INT *ib , MKL_INT *jb ,\nMKL_INT *descb , MKL_INT *info );\nvoid pcgetrs (char *trans , MKL_INT *n , MKL_INT *nrhs , MKL_Complex8 *a , MKL_INT\n*ia , MKL_INT *ja , MKL_INT *desca , MKL_INT *ipiv , MKL_Complex8 *b , MKL_INT *ib ,\nMKL_INT *jb , MKL_INT *descb , MKL_INT *info );\nvoid pzgetrs (char *trans , MKL_INT *n , MKL_INT *nrhs , MKL_Complex16 *a , MKL_INT\n*ia , MKL_INT *ja , MKL_INT *desca , MKL_INT *ipiv , MKL_Complex16 *b , MKL_INT *ib ,\nMKL_INT *jb , MKL_INT *descb , MKL_INT *info );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?getrsfunction solves a system of distributed linear equations with a general n-by-n distributed matrix\nsub(A) = A(ia:ia+n-1, ja:ja+n-1) using the LU factorization computed by p?getrf.\nThe system has one of the following forms specified by trans:\nsub(A)*X = sub(B) (no transpose),\nsub(A)T*X = sub(B) (transpose),\nsub(A)H*X = sub(B) (conjugate transpose),\nwhere sub(B) = B(ib:ib+n-1, jb:jb+nrhs-1).\nBefore calling this function,you must call p?getrf to compute the LU factorization of sub(A).\nInput Parameters\ntrans\n(global) Must be 'N' or 'T' or 'C'.\nIndicates the form of the equations:\nIf trans = 'N', then sub(A)*X = sub(B) is solved for X.\nIf trans = 'T', then sub(A)T*X = sub(B) is solved for X.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1334\n\n\nIf trans = 'C', then sub(A)H *X = sub(B) is solved for X.\nn\n(global) The number of linear equations; the order of the matrix sub(A)\n(n≥0).\nnrhs\n(global) The number of right hand sides; the number of columns of the\ndistributed matrix sub(B) (nrhs≥0).\na, b\n(local)\nPointers into the local memory to arrays of local sizes lld_a*LOCc(ja+n-1)\nand lld_b*LOCc(jb+nrhs-1), respectively.\nOn entry, the array a contains the local pieces of the factors L and U from\nthe factorization sub(A) = P*L*U; the unit diagonal elements of L are not\nstored. On entry, the array b contains the right hand sides sub(B).\nia, ja\n(global) The row and column indices in the global matrix A indicating the\nfirst row and the first column of the matrix sub(A), respectively.\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\nipiv\n(local) Array of size of LOCr(m_a) + mb_a. Contains the pivoting\ninformation: local row i of the matrix was interchanged with the global row\nipiv[i-1].\nThis array is tied to the distributed matrix A.\nib, jb\n(global) The row and column indices in the global matrix B indicating the\nfirst row and the first column of the matrix sub(B), respectively.\ndescb\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix B.\nOutput Parameters\nb\nOn exit, overwritten by the solution distributed matrix X.\ninfo\nIf info=0, the execution is successful. info < 0:\nIf the i-th argument is an array and the j-th entry, indexed j - 1, had an\nillegal value, then info = -(i*100+j); if the i-th argument is a scalar and\nhad an illegal value, then info = -i.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?gbtrs\nSolves a system of distributed linear equations with a\ngeneral band matrix, using the LU factorization\ncomputed by p?gbtrf.\nSyntax\nvoid psgbtrs (char *trans , MKL_INT *n , MKL_INT *bwl , MKL_INT *bwu , MKL_INT *nrhs ,\nfloat *a , MKL_INT *ja , MKL_INT *desca , MKL_INT *ipiv , float *b , MKL_INT *ib ,\nMKL_INT *descb , float *af , MKL_INT *laf , float *work , MKL_INT *lwork , MKL_INT\n*info );\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1335\n\n\nvoid pdgbtrs (char *trans , MKL_INT *n , MKL_INT *bwl , MKL_INT *bwu , MKL_INT *nrhs ,\ndouble *a , MKL_INT *ja , MKL_INT *desca , MKL_INT *ipiv , double *b , MKL_INT *ib ,\nMKL_INT *descb , double *af , MKL_INT *laf , double *work , MKL_INT *lwork , MKL_INT\n*info );\nvoid pcgbtrs (char *trans , MKL_INT *n , MKL_INT *bwl , MKL_INT *bwu , MKL_INT *nrhs ,\nMKL_Complex8 *a , MKL_INT *ja , MKL_INT *desca , MKL_INT *ipiv , MKL_Complex8 *b ,\nMKL_INT *ib , MKL_INT *descb , MKL_Complex8 *af , MKL_INT *laf , MKL_Complex8 *work ,\nMKL_INT *lwork , MKL_INT *info );\nvoid pzgbtrs (char *trans , MKL_INT *n , MKL_INT *bwl , MKL_INT *bwu , MKL_INT *nrhs ,\nMKL_Complex16 *a , MKL_INT *ja , MKL_INT *desca , MKL_INT *ipiv , MKL_Complex16 *b ,\nMKL_INT *ib , MKL_INT *descb , MKL_Complex16 *af , MKL_INT *laf , MKL_Complex16 *work ,\nMKL_INT *lwork , MKL_INT *info );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?gbtrs function solves a system of distributed linear equations with a general band distributed matrix\nsub(A) = A(1:n, ja:ja+n-1) using the LU factorization computed by p?gbtrf.\nThe system has one of the following forms specified by trans:\nsub(A)*X = sub(B) (no transpose),\nsub(A)T*X = sub(B) (transpose),\nsub(A)H*X = sub(B) (conjugate transpose),\nwhere sub(B) = B(ib:ib+n-1, 1:nrhs).\nBefore calling this function,you must call p?gbtrf to compute the LU factorization of sub(A).\nInput Parameters\ntrans\n(global) Must be 'N' or 'T' or 'C'.\nIndicates the form of the equations:\nIf trans = 'N', then sub(A)*X = sub(B) is solved for X.\nIf trans = 'T', then sub(A)T*X = sub(B) is solved for X.\nIf trans = 'C', then sub(A)H *X = sub(B) is solved for X.\nn\n(global) The number of linear equations; the order of the distributed matrix\nsub(A) (n≥ 0).\nbwl\n(global) The number of sub-diagonals within the band of A( 0 ≤ bwl ≤\nn-1 ).\nbwu\n(global) The number of super-diagonals within the band of A( 0 ≤ bwu ≤\nn-1 ).\nnrhs\n(global) The number of right hand sides; the number of columns of the\ndistributed matrix sub(B) (nrhs≥ 0).\na, b\n(local)\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1336\n\n\nPointers into the local memory to arrays of local sizes lld_a*LOCc(ja+n-1)\nand lld_b*LOCc(nrhs), respectively.\nThe array a contains details of the LU factorization of the distributed band\nmatrix A.\nOn entry, the array b contains the local pieces of the right hand sides\nB(ib:ib+n-1, 1:nrhs).\nja\n(global) The index in the global matrix A indicating the start of the matrix to\nbe operated on ( which may be either all of A or a submatrix of A).\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\nIf dtype_a = 501, then dlen_≥ 7;\nelse if dtype_a = 1, then dlen_≥ 9.\nib\n(global) The index in the global matrix A indicating the start of the matrix to\nbe operated on (which may be either all of A or a submatrix of A).\ndescb\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\nIf dtype_b = 502, then dlen_≥ 7;\nelse if dtype_b = 1, then dlen_≥ 9.\nlaf\n(local) The size of the array af.\nMust be laf≥nb_a*(bwl+bwu)+6*(bwl+bwu)*(bwl+2*bwu).\nIf laf is not large enough, an error code will be returned and the minimum\nacceptable size will be returned in af[0].\nwork\n(local) Same type as a. Workspace array of size lwork.\nlwork\n(local or global) The size of the work array, must be at least\nlwork≥nrhs*(nb_a+2*bwl+4*bwu).\nOutput Parameters\nipiv\n(local) array.\nThe size of ipiv must be ≥nb_a.\nContains pivot indices for local factorizations. Note that you should not alter\nthe contents of this array between factorization and solve.\nb\nOn exit, overwritten by the local pieces of the solution distributed matrix X.\naf\n(local)\nArray of size laf.\nAuxiliary Fill-in space. The fill-in space is created in a call to the\nfactorization function p?gbtrf and is stored in af.\nNote that if a linear system is to be solved using p?gbtrs after the\nfactorization function,af must not be altered after the factorization.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1337\n\n\nwork[0]\nOn exit, work[0] contains the minimum value of lwork required for\noptimum performance.\ninfo\nIf info=0, the execution is successful.\ninfo < 0:\nIf the i-th argument is an array and the j-th entry, indexed j - 1, had an\nillegal value, then info = -(i*100+j); if the i-th argument is a scalar and\nhad an illegal value, then info = -i.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?dbtrs\nSolves a system of linear equations with a diagonally\ndominant-like banded distributed matrix using the\nfactorization computed by p?dbtrf.\nSyntax\nvoid psdbtrs (char *trans , MKL_INT *n , MKL_INT *bwl , MKL_INT *bwu , MKL_INT *nrhs ,\nfloat *a , MKL_INT *ja , MKL_INT *desca , float *b , MKL_INT *ib , MKL_INT *descb ,\nfloat *af , MKL_INT *laf , float *work , MKL_INT *lwork , MKL_INT *info );\nvoid pddbtrs (char *trans , MKL_INT *n , MKL_INT *bwl , MKL_INT *bwu , MKL_INT *nrhs ,\ndouble *a , MKL_INT *ja , MKL_INT *desca , double *b , MKL_INT *ib , MKL_INT *descb ,\ndouble *af , MKL_INT *laf , double *work , MKL_INT *lwork , MKL_INT *info );\nvoid pcdbtrs (char *trans , MKL_INT *n , MKL_INT *bwl , MKL_INT *bwu , MKL_INT *nrhs ,\nMKL_Complex8 *a , MKL_INT *ja , MKL_INT *desca , MKL_Complex8 *b , MKL_INT *ib ,\nMKL_INT *descb , MKL_Complex8 *af , MKL_INT *laf , MKL_Complex8 *work , MKL_INT\n*lwork , MKL_INT *info );\nvoid pzdbtrs (char *trans , MKL_INT *n , MKL_INT *bwl , MKL_INT *bwu , MKL_INT *nrhs ,\nMKL_Complex16 *a , MKL_INT *ja , MKL_INT *desca , MKL_Complex16 *b , MKL_INT *ib ,\nMKL_INT *descb , MKL_Complex16 *af , MKL_INT *laf , MKL_Complex16 *work , MKL_INT\n*lwork , MKL_INT *info );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?dbtrsfunction solves for X one of the systems of equations:\nsub(A)*X = sub(B),\n(sub(A))T*X = sub(B), or\n(sub(A))H*X = sub(B),\nwhere sub(A) = A(1:n, ja:ja+n-1) is a diagonally dominant-like banded distributed matrix, and sub(B)\ndenotes the distributed matrix B(ib:ib+n-1, 1:nrhs).\nThis function uses the LU factorization computed by p?dbtrf.\nInput Parameters\ntrans\n(global) Must be 'N' or 'T' or 'C'.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1338\n\n\nIndicates the form of the equations:\nIf trans = 'N', then sub(A)*X = sub(B) is solved for X.\nIf trans = 'T', then (sub(A))T*X = sub(B) is solved for X.\nIf trans = 'C', then (sub(A))H*X = sub(B) is solved for X.\nn\n(global) The order of the distributed matrix sub(A) (n≥ 0).\nbwl\n(global) The number of subdiagonals within the band of A\n( 0 ≤ bwl ≤ n-1 ).\nbwu\n(global) The number of superdiagonals within the band of A\n( 0 ≤ bwu ≤ n-1 ).\nnrhs\n(global) The number of right hand sides; the number of columns of the\ndistributed matrix sub(B) (nrhs≥ 0).\na, b\n(local)\nPointers into the local memory to arrays of local sizes lld_a*LOCc(ja+n-1)\nand lld_b*LOCc(nrhs), respectively.\nOn entry, the array a contains details of the LU factorization of the band\nmatrix A, as computed by p?dbtrf.\nOn entry, the array b contains the local pieces of the right hand side\ndistributed matrix sub(B).\nja\n(global) The index in the global matrix A indicating the start of the matrix to\nbe operated on (which may be either all of A or a submatrix of A).\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\nIf dtype_a = 501, then dlen_≥ 7;\nelse if dtype_a = 1, then dlen_≥ 9.\nib\n(global) The row index in the global matrix B indicating the first row of the\nmatrix to be operated on (which may be either all of B or a submatrix of B).\ndescb\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix B.\nIf dtype_b = 502, then dlen_≥ 7;\nelse if dtype_b = 1, then dlen_≥ 9.\naf, work\n(local)\nArrays of size laf and lwork, respectively The array af contains auxiliary\nfill-in space. The fill-in space is created in a call to the factorization function\np?dbtrf and is stored in af.\nThe array work is a workspace array.\nlaf\n(local) The size of the array af.\nMust be laf≥NB*(bwl+bwu)+6*(max(bwl,bwu))2 .\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1339\n\n\nIf laf is not large enough, an error code will be returned and the minimum\nacceptable size will be returned in af[0].\nlwork\n(local or global) The size of the array work, must be at least\nlwork≥ (max(bwl,bwu))2.\nOutput Parameters\nb\nOn exit, this array contains the local pieces of the solution distributed\nmatrix X.\nwork[0]\nOn exit, work[0] contains the minimum value of lwork required for\noptimum performance.\ninfo\nIf info=0, the execution is successful. info < 0:\nIf the i-th argument is an array and the j-th entry, indexed j - 1, had an\nillegal value, then info = -(i*100+j); if the i-th argument is a scalar and\nhad an illegal value, then info = -i.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?dttrs\nSolves a system of linear equations with a diagonally\ndominant-like tridiagonal distributed matrix using the\nfactorization computed by p?dttrf.\nSyntax\nvoid psdttrs (char *trans , MKL_INT *n , MKL_INT *nrhs , float *dl , float *d , float\n*du , MKL_INT *ja , MKL_INT *desca , float *b , MKL_INT *ib , MKL_INT *descb , float\n*af , MKL_INT *laf , float *work , MKL_INT *lwork , MKL_INT *info );\nvoid pddttrs (char *trans , MKL_INT *n , MKL_INT *nrhs , double *dl , double *d , double\n*du , MKL_INT *ja , MKL_INT *desca , double *b , MKL_INT *ib , MKL_INT *descb , double\n*af , MKL_INT *laf , double *work , MKL_INT *lwork , MKL_INT *info );\nvoid pcdttrs (char *trans , MKL_INT *n , MKL_INT *nrhs , MKL_Complex8 *dl ,\nMKL_Complex8 *d , MKL_Complex8 *du , MKL_INT *ja , MKL_INT *desca , MKL_Complex8 *b ,\nMKL_INT *ib , MKL_INT *descb , MKL_Complex8 *af , MKL_INT *laf , MKL_Complex8 *work ,\nMKL_INT *lwork , MKL_INT *info );\nvoid pzdttrs (char *trans , MKL_INT *n , MKL_INT *nrhs , MKL_Complex16 *dl ,\nMKL_Complex16 *d , MKL_Complex16 *du , MKL_INT *ja , MKL_INT *desca , MKL_Complex16\n*b , MKL_INT *ib , MKL_INT *descb , MKL_Complex16 *af , MKL_INT *laf , MKL_Complex16\n*work , MKL_INT *lwork , MKL_INT *info );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?dttrsfunction solves for X one of the systems of equations:\nsub(A)*X = sub(B),\n(sub(A))T*X = sub(B), or\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1340\n\n\n(sub(A))H*X = sub(B),\nwhere sub(A) =A(1:n, ja:ja+n-1) is a diagonally dominant-like tridiagonal distributed matrix, and sub(B)\ndenotes the distributed matrix B(ib:ib+n-1, 1:nrhs).\nThis function uses the LU factorization computed by p?dttrf.\nInput Parameters\ntrans\n(global) Must be 'N' or 'T' or 'C'.\nIndicates the form of the equations:\nIf trans = 'N', then sub(A)*X = sub(B) is solved for X.\nIf trans = 'T', then (sub(A))T*X = sub(B) is solved for X.\nIf trans = 'C', then (sub(A))H*X = sub(B) is solved for X.\nn\n(global) The order of the distributed matrix sub(A) (n≥ 0).\nnrhs\n(global) The number of right hand sides; the number of columns of the\ndistributed matrix sub(B) (nrhs≥ 0).\ndl, d, du\n(local)\nPointers to the local arrays of size nb_a each.\nOn entry, these arrays contain details of the factorization. Globally, dl[0]\nand du[n-1] are not referenced; dl and du must be aligned with d.\nja\n(global) The index in the global matrix A indicating the start of the matrix to\nbe operated on (which may be either all of A or a submatrix of A).\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\nIf dtype_a = 501 or dtype_a = 502, then dlen_≥ 7;\nelse if dtype_a = 1, then dlen_≥ 9.\nb\n(local) Same type as d.\nPointer into the local memory to an array of local size lld_b*LOCc(nrhs)\nOn entry, the array b contains the local pieces of the n-by-nrhs right hand\nside distributed matrix sub(B).\nib\n(global) The row index in the global matrix B indicating the first row of the\nmatrix to be operated on (which may be either all of B or a submatrix of B).\ndescb\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix B.\nIf dtype_b = 502, then dlen_≥ 7;\nelse if dtype_b = 1, then dlen_≥ 9.\naf, work\n(local)\nArrays of size laf and (lwork), respectively.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1341\n\n\nThe array af contains auxiliary fill-in space. The fill-in space is created in a\ncall to the factorization function p?dttrf and is stored in af. If a linear\nsystem is to be solved using p?dttrs after the factorization function,af\nmust not be altered.\nThe array work is a workspace array.\nlaf\n(local) The size of the array af.\nMust be laf≥NB*(bwl+bwu)+6*(bwl+bwu)*(bwl+2*bwu).\nIf laf is not large enough, an error code will be returned and the minimum\nacceptable size will be returned in af[0].\nlwork\n(local or global) The size of the array work, must be at least lwork≥\n10*NPCOL+4*nrhs.\nOutput Parameters\nb\nOn exit, this array contains the local pieces of the solution distributed\nmatrix X.\nwork[0]\nOn exit, work[0] contains the minimum value of lwork required for\noptimum performance.\ninfo\nIf info=0, the execution is successful. info < 0:\nIf the i-th argument is an array and the j-th entry, indexed j - 1, had an\nillegal value, then info = -(i*100+j); if the i-th argument is a scalar and\nhad an illegal value, then info = -i.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?potrs\nSolves a system of linear equations with a Cholesky-\nfactored symmetric/Hermitian distributed positive-\ndefinite matrix.\nSyntax\nvoid pspotrs (char *uplo , MKL_INT *n , MKL_INT *nrhs , float *a , MKL_INT *ia , MKL_INT\n*ja , MKL_INT *desca , float *b , MKL_INT *ib , MKL_INT *jb , MKL_INT *descb , MKL_INT\n*info );\nvoid pdpotrs (char *uplo , MKL_INT *n , MKL_INT *nrhs , double *a , MKL_INT *ia ,\nMKL_INT *ja , MKL_INT *desca , double *b , MKL_INT *ib , MKL_INT *jb , MKL_INT *descb ,\nMKL_INT *info );\nvoid pcpotrs (char *uplo , MKL_INT *n , MKL_INT *nrhs , MKL_Complex8 *a , MKL_INT *ia ,\nMKL_INT *ja , MKL_INT *desca , MKL_Complex8 *b , MKL_INT *ib , MKL_INT *jb , MKL_INT\n*descb , MKL_INT *info );\nvoid pzpotrs (char *uplo , MKL_INT *n , MKL_INT *nrhs , MKL_Complex16 *a , MKL_INT\n*ia , MKL_INT *ja , MKL_INT *desca , MKL_Complex16 *b , MKL_INT *ib , MKL_INT *jb ,\nMKL_INT *descb , MKL_INT *info );\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1342\n\n\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?potrsfunction solves for X a system of distributed linear equations in the form:\nsub(A)*X = sub(B) ,\nwhere sub(A) = A(ia:ia+n-1, ja:ja+n-1) is an n-by-n real symmetric or complex Hermitian positive\ndefinite distributed matrix, and sub(B) denotes the distributed matrix B(ib:ib+n-1, jb:jb+nrhs-1).\nThis function uses Cholesky factorization\nsub(A) = UH*U, or sub(A) = L*LH\ncomputed by p?potrf.\nInput Parameters\nuplo\n(global) Must be 'U' or 'L'.\nIf uplo = 'U', upper triangle of sub(A) is stored;\nIf uplo = 'L', lower triangle of sub(A) is stored.\nn\n(global) The order of the distributed matrix sub(A) (n≥0).\nnrhs\n(global) The number of right hand sides; the number of columns of the\ndistributed matrix sub(B) (nrhs≥0).\na, b\n(local)\nPointers into the local memory to arrays of local sizes\nlld_a*LOCc(ja+n-1) and lld_b*LOCc(jb+nrhs-1), respectively.\nThe array a contains the factors L or U from the Cholesky factorization\nsub(A) = L*LH or sub(A) = UH*U, as computed by p?potrf.\nOn entry, the array b contains the local pieces of the right hand sides\nsub(B).\nia, ja\n(global) The row and column indices in the global matrix A indicating the\nfirst row and the first column of the matrix sub(A), respectively.\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\nib, jb\n(global) The row and column indices in the global matrix B indicating the\nfirst row and the first column of the matrix sub(B), respectively.\ndescb\n(local) array of size dlen_. The array descriptor for the distributed matrix B.\nOutput Parameters\nb\nOverwritten by the local pieces of the solution matrix X.\ninfo\nIf info=0, the execution is successful. \ninfo < 0: if the i-th argument is an array and the j-th entry, indexed j - 1,\nhad an illegal value, then info = -(i*100+j); if the i-th argument is a\nscalar and had an illegal value, then info = -i.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1343\n\n\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?pbtrs\nSolves a system of linear equations with a Cholesky-\nfactored symmetric/Hermitian positive-definite band\nmatrix.\nSyntax\nvoid pspbtrs (char *uplo , MKL_INT *n , MKL_INT *bw , MKL_INT *nrhs , float *a , MKL_INT\n*ja , MKL_INT *desca , float *b , MKL_INT *ib , MKL_INT *descb , float *af , MKL_INT\n*laf , float *work , MKL_INT *lwork , MKL_INT *info );\nvoid pdpbtrs (char *uplo , MKL_INT *n , MKL_INT *bw , MKL_INT *nrhs , double *a ,\nMKL_INT *ja , MKL_INT *desca , double *b , MKL_INT *ib , MKL_INT *descb , double *af ,\nMKL_INT *laf , double *work , MKL_INT *lwork , MKL_INT *info );\nvoid pcpbtrs (char *uplo , MKL_INT *n , MKL_INT *bw , MKL_INT *nrhs , MKL_Complex8 *a ,\nMKL_INT *ja , MKL_INT *desca , MKL_Complex8 *b , MKL_INT *ib , MKL_INT *descb ,\nMKL_Complex8 *af , MKL_INT *laf , MKL_Complex8 *work , MKL_INT *lwork , MKL_INT\n*info );\nvoid pzpbtrs (char *uplo , MKL_INT *n , MKL_INT *bw , MKL_INT *nrhs , MKL_Complex16\n*a , MKL_INT *ja , MKL_INT *desca , MKL_Complex16 *b , MKL_INT *ib , MKL_INT *descb ,\nMKL_Complex16 *af , MKL_INT *laf , MKL_Complex16 *work , MKL_INT *lwork , MKL_INT\n*info );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?pbtrsfunction solves for X a system of distributed linear equations in the form:\nsub(A)*X = sub(B) ,\nwhere sub(A) = A(1:n, ja:ja+n-1) is an n-by-n real symmetric or complex Hermitian positive definite\ndistributed band matrix, and sub(B) denotes the distributed matrix B(ib:ib+n-1, 1:nrhs).\nThis function uses Cholesky factorization\nsub(A) = P*UH*U*PT, or sub(A) = P*L*LH*PT\ncomputed by p?pbtrf.\nInput Parameters\nuplo\n(global) Must be 'U' or 'L'.\nIf uplo = 'U', upper triangle of sub(A) is stored;\nIf uplo = 'L', lower triangle of sub(A) is stored.\nn\n(global) The order of the distributed matrix sub(A) (n≥0).\nbw\n(global) The number of superdiagonals of the distributed matrix if uplo =\n'U', or the number of subdiagonals if uplo = 'L' (bw≥0).\nnrhs\n(global) The number of right hand sides; the number of columns of the\ndistributed matrix sub(B) (nrhs≥0).\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1344\n\n\na, b\n(local)\nPointers into the local memory to arrays of local sizes lld_a*LOCc(ja+n-1)\nand lld_b*LOCc(nrhs-1), respectively.\nThe array a contains the permuted triangular factor U or L from the\nCholesky factorization sub(A) = P*UH*U*PT, or sub(A) = P*L*LH*PT of the\nband matrix A, as returned by p?pbtrf.\nOn entry, the array b contains the local pieces of the n-by-nrhs right hand\nside distributed matrix sub(B).\nja\n(global) The index in the global matrix A indicating the start of the matrix to\nbe operated on (which may be either all of A or a submatrix of A).\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\nIf dtype_a = 501, then dlen_≥ 7;\nelse if dtype_a = 1, then dlen_≥ 9.\nib\n(global) The row index in the global matrix B indicating the first row of the\nmatrix sub(B).\ndescb\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix B.\nIf dtype_b = 502, then dlen_≥ 7;\nelse if dtype_b = 1, then dlen_≥ 9.\naf, work\n(local) Arrays, same type as a.\nThe array af is of size laf. It contains auxiliary fill-in space. The fill-in\nspace is created in a call to the factorization function p?dbtrf and is stored\nin af.\nThe array work is a workspace array of size lwork.\nlaf\n(local) The size of the array af.\nMust be laf≥nrhs*bw.\nIf laf is not large enough, an error code will be returned and the minimum\nacceptable size will be returned in af[0].\nlwork\n(local or global) The size of the array work, must be at least lwork≥bw2.\nOutput Parameters\nb\nOn exit, if info=0, this array contains the local pieces of the n-by-nrhs\nsolution distributed matrix X.\nwork[0]\nOn exit, work[0] contains the minimum value of lwork required for\noptimum performance.\ninfo\nIf info=0, the execution is successful.\ninfo < 0:\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1345\n\n\nIf the i-th argument is an array and the j-th entry, indexed j - 1, had an\nillegal value, then info = -(i*100+j); if the i-th argument is a scalar and\nhad an illegal value, then info = -i.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?pttrs\nSolves a system of linear equations with a symmetric\n(Hermitian) positive-definite tridiagonal distributed\nmatrix using the factorization computed by p?pttrf.\nSyntax\nvoid pspttrs (MKL_INT *n , MKL_INT *nrhs , float *d , float *e , MKL_INT *ja , MKL_INT\n*desca , float *b , MKL_INT *ib , MKL_INT *descb , float *af , MKL_INT *laf , float\n*work , MKL_INT *lwork , MKL_INT *info );\nvoid pdpttrs (MKL_INT *n , MKL_INT *nrhs , double *d , double *e , MKL_INT *ja , MKL_INT\n*desca , double *b , MKL_INT *ib , MKL_INT *descb , double *af , MKL_INT *laf , double\n*work , MKL_INT *lwork , MKL_INT *info );\nvoid pcpttrs (char *uplo , MKL_INT *n , MKL_INT *nrhs , float *d , MKL_Complex8 *e ,\nMKL_INT *ja , MKL_INT *desca , MKL_Complex8 *b , MKL_INT *ib , MKL_INT *descb ,\nMKL_Complex8 *af , MKL_INT *laf , MKL_Complex8 *work , MKL_INT *lwork , MKL_INT\n*info );\nvoid pzpttrs (char *uplo , MKL_INT *n , MKL_INT *nrhs , double *d , MKL_Complex16 *e ,\nMKL_INT *ja , MKL_INT *desca , MKL_Complex16 *b , MKL_INT *ib , MKL_INT *descb ,\nMKL_Complex16 *af , MKL_INT *laf , MKL_Complex16 *work , MKL_INT *lwork , MKL_INT\n*info );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?pttrsfunction solves for X a system of distributed linear equations in the form:\nsub(A)*X = sub(B) ,\nwhere sub(A) = A(1:n, ja:ja+n-1) is an n-by-n real symmetric or complex Hermitian positive definite\ntridiagonal distributed matrix, and sub(B) denotes the distributed matrix B(ib:ib+n-1, 1:nrhs).\nThis function uses the factorization\nsub(A) = P*L*D*LH*PT, or sub(A) = P*UH*D*U*PT\ncomputed by p?pttrf.\nInput Parameters\nuplo\n(global, used in complex flavors only)\nMust be 'U' or 'L'.\nIf uplo = 'U', upper triangle of sub(A) is stored;\nIf uplo = 'L', lower triangle of sub(A) is stored.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1346\n\n\nn\n(global) The order of the distributed matrix sub(A) (n≥0).\nnrhs\n(global) The number of right hand sides; the number of columns of the\ndistributed matrix sub(B) (nrhs≥0).\nd, e\n(local)\nPointers into the local memory to arrays of size nb_a each.\nThese arrays contain details of the factorization as returned by p?pttrf\nja\n(global) The index in the global matrix A indicating the start of the matrix to\nbe operated on (which may be either all of A or a submatrix of A).\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\nIf dtype_a = 501 or dtype_a = 502, then dlen_≥ 7;\nelse if dtype_a = 1, then dlen_≥ 9.\nb\n(local) Same type as d, e.\nPointer into the local memory to an array of local size\nlld_b*LOCc(nrhs).\nOn entry, the array b contains the local pieces of the n-by-nrhsright hand\nside distributed matrix sub(B).\nib\n(global) The row index in the global matrix B indicating the first row of the\nmatrix to be operated on (which may be either all of B or a submatrix of B).\ndescb\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix B.\nIf dtype_b = 502, then dlen_≥ 7;\nelse if dtype_b = 1, then dlen_≥ 9.\naf, work\n(local)\nArrays of size laf and (lwork), respectively. The array af contains\nauxiliary fill-in space. The fill-in space is created in a call to the factorization\nfunction p?pttrf and is stored in af.\nThe array work is a workspace array.\nlaf\n(local) The size of the array af.\nMust be laf≥nb_a+2.\nIf laf is not large enough, an error code is returned and the minimum\nacceptable size will be returned in af[0].\nlwork\n(local or global) The size of the array work, must be at least\nlwork≥ (10+2*min(100,nrhs))*NPCOL+4*nrhs.\nOutput Parameters\nb\nOn exit, this array contains the local pieces of the solution distributed\nmatrix X.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1347\n\n\nwork[0])\nOn exit, work[0] contains the minimum value of lwork required for\noptimum performance.\ninfo\nIf info=0, the execution is successful. \ninfo < 0:\nif the i-th argument is an array and the j-th entry, indexed j - 1, had\nan illegal value, then info = -(i*100+j); if the i-th argument is a\nscalar and had an illegal value, then info = -i.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?trtrs\nSolves a system of linear equations with a triangular\ndistributed matrix.\nSyntax\nvoid pstrtrs (char *uplo , char *trans , char *diag , MKL_INT *n , MKL_INT *nrhs , float\n*a , MKL_INT *ia , MKL_INT *ja , MKL_INT *desca , float *b , MKL_INT *ib , MKL_INT *jb ,\nMKL_INT *descb , MKL_INT *info );\nvoid pdtrtrs (char *uplo , char *trans , char *diag , MKL_INT *n , MKL_INT *nrhs ,\ndouble *a , MKL_INT *ia , MKL_INT *ja , MKL_INT *desca , double *b , MKL_INT *ib ,\nMKL_INT *jb , MKL_INT *descb , MKL_INT *info );\nvoid pctrtrs (char *uplo , char *trans , char *diag , MKL_INT *n , MKL_INT *nrhs ,\nMKL_Complex8 *a , MKL_INT *ia , MKL_INT *ja , MKL_INT *desca , MKL_Complex8 *b ,\nMKL_INT *ib , MKL_INT *jb , MKL_INT *descb , MKL_INT *info );\nvoid pztrtrs (char *uplo , char *trans , char *diag , MKL_INT *n , MKL_INT *nrhs ,\nMKL_Complex16 *a , MKL_INT *ia , MKL_INT *ja , MKL_INT *desca , MKL_Complex16 *b ,\nMKL_INT *ib , MKL_INT *jb , MKL_INT *descb , MKL_INT *info );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?trtrsfunction solves for X one of the following systems of linear equations:\nsub(A)*X = sub(B),\n(sub(A))T*X = sub(B), or\n(sub(A))H*X = sub(B),\nwhere sub(A) = A(ia:ia+n-1, ja:ja+n-1) is a triangular distributed matrix of order n, and sub(B) denotes\nthe distributed matrix B(ib:ib+n-1, jb:jb+nrhs-1).\nA check is made to verify that sub(A) is nonsingular.\nInput Parameters\nuplo\n(global) Must be 'U' or 'L'.\nIndicates whether sub(A) is upper or lower triangular:\nIf uplo = 'U', then sub(A) is upper triangular.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1348\n\n\nIf uplo = 'L', then sub(A) is lower triangular.\ntrans\n(global) Must be 'N' or 'T' or 'C'.\nIndicates the form of the equations:\nIf trans = 'N', then sub(A)*X = sub(B) is solved for X.\nIf trans = 'T', then sub(A)T*X = sub(B) is solved for X.\nIf trans = 'C', then sub(A)H*X = sub(B) is solved for X.\ndiag\n(global) Must be 'N' or 'U'.\nIf diag = 'N', then sub(A) is not a unit triangular matrix.\nIf diag = 'U', then sub(A) is unit triangular.\nn\n(global) The order of the distributed matrix sub(A) (n≥0).\nnrhs\n(global) The number of right-hand sides; i.e., the number of columns of the\ndistributed matrix sub(B) (nrhs≥0).\na, b\n(local)\nPointers into the local memory to arrays of local sizes lld_a*LOCc(ja+n-1)\nand lld_b*LOCc(jb+nrhs-1), respectively.\nThe array a contains the local pieces of the distributed triangular matrix\nsub(A).\nIf uplo = 'U', the leading n-by-n upper triangular part of sub(A) contains\nthe upper triangular matrix, and the strictly lower triangular part of sub(A)\nis not referenced.\nIf uplo = 'L', the leading n-by-n lower triangular part of sub(A) contains\nthe lower triangular matrix, and the strictly upper triangular part of sub(A)\nis not referenced.\nIf diag = 'U', the diagonal elements of sub(A) are also not referenced\nand are assumed to be 1.\nOn entry, the array b contains the local pieces of the right hand side\ndistributed matrix sub(B).\nia, ja\n(global) The row and column indices in the global matrix A indicating the\nfirst row and the first column of the matrix sub(A), respectively.\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\nib, jb\n(global) The row and column indices in the global matrix B indicating the\nfirst row and the first column of the matrix sub(B), respectively.\ndescb\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix B.\nOutput Parameters\nb\nOn exit, if info=0, sub(B) is overwritten by the solution matrix X.\ninfo\nIf info=0, the execution is successful.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1349\n\n\ninfo < 0:\nif the i-th argument is an array and the j-th entry, indexed j - 1, had an\nillegal value, then info = -(i*100+j); if the i-th argument is a scalar and\nhad an illegal value, then info = -i.\ninfo> 0:\nif info = i, the i-th diagonal element of sub(A) is zero, indicating that the\nsubmatrix is singular and the solutions X have not been computed.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\nEstimating the Condition Number: ScaLAPACK Computational Routines\nThis section describes the ScaLAPACK routines for estimating the condition number of a matrix. The condition\nnumber is used for analyzing the errors in the solution of a system of linear equations. Since the condition\nnumber may be arbitrarily large when the matrix is nearly singular, the routines actually compute the\nreciprocal condition number.\np?gecon\nEstimates the reciprocal of the condition number of a\ngeneral distributed matrix in either the 1-norm or the\ninfinity-norm.\nSyntax\nvoid psgecon (char *norm , MKL_INT *n , float *a , MKL_INT *ia , MKL_INT *ja , MKL_INT\n*desca , float *anorm , float *rcond , float *work , MKL_INT *lwork , MKL_INT *iwork ,\nMKL_INT *liwork , MKL_INT *info );\nvoid pdgecon (char *norm , MKL_INT *n , double *a , MKL_INT *ia , MKL_INT *ja , MKL_INT\n*desca , double *anorm , double *rcond , double *work , MKL_INT *lwork , MKL_INT\n*iwork , MKL_INT *liwork , MKL_INT *info );\nvoid pcgecon (char *norm , MKL_INT *n , MKL_Complex8 *a , MKL_INT *ia , MKL_INT *ja ,\nMKL_INT *desca , float *anorm , float *rcond , MKL_Complex8 *work , MKL_INT *lwork ,\nfloat *rwork , MKL_INT *lrwork , MKL_INT *info );\nvoid pzgecon (char *norm , MKL_INT *n , MKL_Complex16 *a , MKL_INT *ia , MKL_INT *ja ,\nMKL_INT *desca , double *anorm , double *rcond , MKL_Complex16 *work , MKL_INT *lwork ,\ndouble *rwork , MKL_INT *lrwork , MKL_INT *info );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?gecon function estimates the reciprocal of the condition number of a general distributed real/complex\nmatrix sub(A) = A(ia:ia+n-1, ja:ja+n-1) in either the 1-norm or infinity-norm, using the LU factorization\ncomputed by p?getrf.\nAn estimate is obtained for ||(sub(A))-1||, and the reciprocal of the condition number is computed as\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1350\n\n\nInput Parameters\nnorm\n(global) Must be '1' or 'O' or 'I'.\nSpecifies whether the 1-norm condition number or the infinity-norm\ncondition number is required.\nIf norm = '1' or 'O', then the 1-norm is used;\nIf norm = 'I', then the infinity-norm is used.\nn\n(global) The order of the distributed matrix sub(A) (n≥ 0).\na\n(local)\nPointer into the local memory to an array of size lld_a*LOCc(ja+n-1).\nThe array a contains the local pieces of the factors L and U from the\nfactorization sub(A) = P*L*U; the unit diagonal elements of L are not\nstored.\nia, ja\n(global) The row and column indices in the global matrix A indicating the\nfirst row and the first column of the matrix sub(A), respectively.\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\nanorm\n(global)\nIf norm = '1' or 'O', the 1-norm of the original distributed matrix sub(A);\nIf norm = 'I', the infinity-norm of the original distributed matrix sub(A).\nwork\n(local)\nThe array work of size lwork is a workspace array.\nlwork\n(local or global) The size of the array work.\nFor real flavors:\nlwork must be at least\nlwork≥ 2*LOCr(n+mod(ia-1,mb_a))+2*LOCc(n+mod(ja-1,nb_a))\n+max(2, max(nb_a*max(1, iceil(NPROW-1, NPCOL)), LOCc(n\n+mod(ja-1,nb_a)) + nb_a*max(1, iceil(NPCOL-1, NPROW)))).\nFor complex flavors:\nlwork must be at least\nlwork≥ 2*LOCr(n+mod(ia-1,mb_a))+max(2,\nmax(nb_a*iceil(NPROW-1, NPCOL), LOCc(n+mod(ja-1,nb_a))+\nnb_a*iceil(NPCOL-1, NPROW))).\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1351\n\n\nLOCr and LOCc values can be computed using the ScaLAPACK tool function\nnumroc; NPROW and NPCOL can be determined by calling the function\nblacs_gridinfo.\nNOTE\niceil(x,y) is the ceiling of x/y, and mod(x,y) is the integer\nremainder of x/y.\niwork\n(local) Workspace array of size liwork. Used in real flavors only.\nliwork\n(local or global) The size of the array iwork; used in real flavors only. Must\nbe at least\nliwork≥LOCr(n+mod(ia-1,mb_a)).\nrwork\n(local)\nWorkspace array of size lrwork. Used in complex flavors only.\nlrwork\n(local or global) The size of the array rwork; used in complex flavors only.\nMust be at least\nlrwork≥ max(1, 2*LOCc(n+mod(ja-1,nb_a))).\nOutput Parameters\nrcond\n(global)\nThe reciprocal of the condition number of the distributed matrix sub(A). See\nDescription.\nwork[0]\nOn exit, work[0] contains the minimum value of lwork required for\noptimum performance.\niwork[0]\nOn exit, iwork[0] contains the minimum value of liwork required for\noptimum performance (for real flavors).\nrwork[0]\nOn exit, rwork[0] contains the minimum value of lrwork required for\noptimum performance (for complex flavors).\ninfo\n(global) If info=0, the execution is successful.\ninfo < 0:\nIf the i-th argument is an array and the j-th entry, indexed j - 1, had an\nillegal value, then info = -(i*100+j); if the i-th argument is a scalar and\nhad an illegal value, then info = -i.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?pocon\nEstimates the reciprocal of the condition number (in\nthe 1 - norm) of a symmetric / Hermitian positive-\ndefinite distributed matrix.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1352\n\n\nSyntax\nvoid pspocon (char *uplo , MKL_INT *n , float *a , MKL_INT *ia , MKL_INT *ja , MKL_INT\n*desca , float *anorm , float *rcond , float *work , MKL_INT *lwork , MKL_INT *iwork ,\nMKL_INT *liwork , MKL_INT *info );\nvoid pdpocon (char *uplo , MKL_INT *n , double *a , MKL_INT *ia , MKL_INT *ja , MKL_INT\n*desca , double *anorm , double *rcond , double *work , MKL_INT *lwork , MKL_INT\n*iwork , MKL_INT *liwork , MKL_INT *info );\nvoid pcpocon (char *uplo , MKL_INT *n , MKL_Complex8 *a , MKL_INT *ia , MKL_INT *ja ,\nMKL_INT *desca , float *anorm , float *rcond , MKL_Complex8 *work , MKL_INT *lwork ,\nfloat *rwork , MKL_INT *lrwork , MKL_INT *info );\nvoid pzpocon (char *uplo , MKL_INT *n , MKL_Complex16 *a , MKL_INT *ia , MKL_INT *ja ,\nMKL_INT *desca , double *anorm , double *rcond , MKL_Complex16 *work , MKL_INT *lwork ,\ndouble *rwork , MKL_INT *lrwork , MKL_INT *info );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?poconfunction estimates the reciprocal of the condition number (in the 1 - norm) of a real symmetric\nor complex Hermitian positive definite distributed matrix sub(A) = A(ia:ia+n-1, ja:ja+n-1), using the\nCholesky factorization sub(A) = UH*U or sub(A) = L*LH computed by p?potrf.\nAn estimate is obtained for ||(sub(A))-1||, and the reciprocal of the condition number is computed as\nInput Parameters\nuplo\n(global) Must be 'U' or 'L'.\nSpecifies whether the factor stored in sub(A) is upper or lower triangular.\nIf uplo = 'U', sub(A) stores the upper triangular factor U of the Cholesky\nfactorization sub(A) = UH*U.\nIf uplo = 'L', sub(A) stores the lower triangular factor L of the Cholesky\nfactorization sub(A) = L*LH.\nn\n(global) The order of the distributed matrix sub(A) (n≥0).\na\n(local)\nPointer into the local memory to an array of size lld_a*LOCc(ja+n-1).\nThe array a contains the local pieces of the factors L or U from the Cholesky\nfactorization sub(A) = UH*U, or sub(A) = L*LH, as computed by p?potrf.\nia, ja\n(global) The row and column indices in the global matrix A indicating the\nfirst row and the first column of the matrix sub(A), respectively.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1353\n\n\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\nanorm\n(global)\nThe 1-norm of the symmetric/Hermitian distributed matrix sub(A).\nwork\n(local)\nThe array work of size lwork is a workspace array.\nlwork\n(local or global) The size of the array work.\nFor real flavors:\nlwork must be at least\nlwork≥ 2*LOCr(n+mod(ia-1,mb_a))+2*LOCc(n+mod(ja-1,nb_a))\n+max(2, max(nb_a*iceil(NPROW-1, NPCOL), LOCc(n\n+mod(ja-1,nb_a))+nb_a*iceil(NPCOL-1, NPROW))).\nFor complex flavors:\nlwork must be at least\nlwork≥ 2*LOCr(n+mod(ia-1,mb_a))+max(2,\nmax(nb_a*max(1,iceil(NPROW-1, NPCOL)), LOCc(n+mod(ja-1,nb_a))\n+nb_a*max(1,iceil(NPCOL-1, NPROW)))).\nIf lwork = -1, then lwork is a global input and a workspace query is\nassumed. The routine only calculates the minimum and optimal size for all\nwork arrays. Each value is returned in the first entry of the corresponding\nwork array, and no error message is issued by pxerbla.\nNOTE\niceil(x,y) is the ceiling of x/y, and mod(x,y) is the integer\nremainder of x/y.\niwork\n(local) Workspace array of size liwork. Used in real flavors only.\nliwork\n(local or global) The size of the array iwork; used in real flavors only. Must\nbe at least liwork≥LOCr(n+mod(ia-1,mb_a)).\nIf liwork = -1, then liwork is a global input and a workspace query is\nassumed. The routine only calculates the minimum and optimal size for all\nwork arrays. Each value is returned in the first entry of the corresponding\nwork array, and no error message is issued by pxerbla.\nrwork\n(local)\nWorkspace array of size lrwork. Used in complex flavors only.\nlrwork\n(local or global) The size of the array rwork; used in complex flavors only.\nMust be at least lrwork≥ 2*LOCc(n+mod(ja-1,nb_a)).\nIf lrwork = -1, then lrwork is a global input and a workspace query is\nassumed. The routine only calculates the minimum and optimal size for all\nwork arrays. Each value is returned in the first entry of the corresponding\nwork array, and no error message is issued by pxerbla.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1354\n\n\nOutput Parameters\nrcond\n(global)\nThe reciprocal of the condition number of the distributed matrix sub(A).\nwork[0]\nOn exit, work[0] contains the minimum value of lwork required for\noptimum performance.\niwork[0]\nOn exit, iwork[0] contains the minimum value of liwork required for\noptimum performance (for real flavors).\nrwork[0]\nOn exit, rwork[0] contains the minimum value of lrwork required for\noptimum performance (for complex flavors).\ninfo\n(global) If info=0, the execution is successful.\ninfo < 0:\nIf the i-th argument is an array and the j-th entry, indexed j - 1, had an\nillegal value, then info = -(i*100+j); if the i-th argument is a scalar and\nhad an illegal value, then info = -i.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?trcon\nEstimates the reciprocal of the condition number of a\ntriangular distributed matrix in either 1-norm or\ninfinity-norm.\nSyntax\nvoid pstrcon (char *norm , char *uplo , char *diag , MKL_INT *n , float *a , MKL_INT\n*ia , MKL_INT *ja , MKL_INT *desca , float *rcond , float *work , MKL_INT *lwork ,\nMKL_INT *iwork , MKL_INT *liwork , MKL_INT *info );\nvoid pdtrcon (char *norm , char *uplo , char *diag , MKL_INT *n , double *a , MKL_INT\n*ia , MKL_INT *ja , MKL_INT *desca , double *rcond , double *work , MKL_INT *lwork ,\nMKL_INT *iwork , MKL_INT *liwork , MKL_INT *info );\nvoid pctrcon (char *norm , char *uplo , char *diag , MKL_INT *n , MKL_Complex8 *a ,\nMKL_INT *ia , MKL_INT *ja , MKL_INT *desca , float *rcond , MKL_Complex8 *work ,\nMKL_INT *lwork , float *rwork , MKL_INT *lrwork , MKL_INT *info );\nvoid pztrcon (char *norm , char *uplo , char *diag , MKL_INT *n , MKL_Complex16 *a ,\nMKL_INT *ia , MKL_INT *ja , MKL_INT *desca , double *rcond , MKL_Complex16 *work ,\nMKL_INT *lwork , double *rwork , MKL_INT *lrwork , MKL_INT *info );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?trconfunction estimates the reciprocal of the condition number of a triangular distributed matrix\nsub(A) = A(ia:ia+n-1, ja:ja+n-1), in either the 1-norm or the infinity-norm.\nThe norm of sub(A) is computed and an estimate is obtained for ||(sub(A))-1||, then the reciprocal of the\ncondition number is computed as\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1355\n\n\nInput Parameters\nnorm\n(global) Must be '1' or 'O' or 'I'.\nSpecifies whether the 1-norm condition number or the infinity-norm\ncondition number is required.\nIf norm = '1' or 'O', then the 1-norm is used;\nIf norm = 'I', then the infinity-norm is used.\nuplo\n(global) Must be 'U' or 'L'.\nIf uplo = 'U', sub(A) is upper triangular. If uplo = 'L', sub(A) is lower\ntriangular.\ndiag\n(global) Must be 'N' or 'U'.\nIf diag = 'N', sub(A) is non-unit triangular. If diag = 'U', sub(A) is unit\ntriangular.\nn\n(global) The order of the distributed matrix sub(A), (n≥0).\na\n(local)\nPointer into the local memory to an array of size lld_a*LOCc(ja+n-1).\nThe array a contains the local pieces of the triangular distributed matrix\nsub(A).\nIf uplo = 'U', the leading n-by-n upper triangular part of this distributed\nmatrix contains the upper triangular matrix, and its strictly lower triangular\npart is not referenced.\nIf uplo = 'L', the leading n-by-n lower triangular part of this distributed\nmatrix contains the lower triangular matrix, and its strictly upper triangular\npart is not referenced.\nIf diag = 'U', the diagonal elements of sub(A) are also not referenced\nand are assumed to be 1.\nia, ja\n(global) The row and column indices in the global matrix A indicating the\nfirst row and the first column of the matrix sub(A), respectively.\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\nwork\n(local)\nThe array work of size lwork is a workspace array.\nlwork\n(local or global) The size of the array work.\nFor real flavors:\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1356\n\n\nlwork must be at least\nlwork≥ 2*LOCr(n+mod(ia-1,mb_a))+LOCc(n+mod(ja-1,nb_a))\n+max(2, max(nb_a*max(1,iceil(NPROW-1, NPCOL)),\nLOCc(n+mod(ja-1,nb_a))+nb_a*max(1,iceil(NPCOL-1, NPROW)))).\nFor complex flavors:\nlwork must be at least\nlwork≥ 2*LOCr(n+mod(ia-1,mb_a))+max(2,\nmax(nb_a*iceil(NPROW-1, NPCOL),\nLOCc(n+mod(ja-1,nb_a))+nb_a*iceil(NPCOL-1, NPROW))).\nNOTE\niceil(x,y) is the ceiling of x/y, and mod(x,y) is the integer\nremainder of x/y.\niwork\n(local) Workspace array of size liwork. Used in real flavors only.\nliwork\n(local or global) The size of the array iwork; used in real flavors only. Must\nbe at least\nliwork≥LOCr(n+mod(ia-1,mb_a)).\nrwork\n(local)\nWorkspace array of size lrwork. Used in complex flavors only.\nlrwork\n(local or global) The size of the array rwork; used in complex flavors only.\nMust be at least \nlrwork≥LOCc(n+mod(ja-1,nb_a)).\nOutput Parameters\nrcond\n(global)\nThe reciprocal of the condition number of the distributed matrix sub(A).\nwork[0]\nOn exit, work[0] contains the minimum value of lwork required for\noptimum performance.\niwork[0]\nOn exit, iwork[0] contains the minimum value of liwork required for\noptimum performance (for real flavors).\nrwork[0]\nOn exit, rwork[0] contains the minimum value of lrwork required for\noptimum performance (for complex flavors).\ninfo\n(global) If info=0, the execution is successful.\ninfo < 0:\nIf the i-th argument is an array and the j-th entry, indexed j - 1, had an\nillegal value, then info = -(i*100+j); if the i-th argument is a scalar and\nhad an illegal value, then info = -i.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1357\n\n\nRefining the Solution and Estimating Its Error: ScaLAPACK Computational Routines\nThis section describes the ScaLAPACK routines for refining the computed solution of a system of linear\nequations and estimating the solution error. You can call these routines after factorizing the matrix of the\nsystem of equations and computing the solution (see Routines for Matrix Factorization and Solving Systems\nof Linear Equations).\np?gerfs\nImproves the computed solution to a system of linear\nequations and provides error bounds and backward\nerror estimates for the solution.\nSyntax\nvoid psgerfs (char *trans , MKL_INT *n , MKL_INT *nrhs , float *a , MKL_INT *ia ,\nMKL_INT *ja , MKL_INT *desca , float *af , MKL_INT *iaf , MKL_INT *jaf , MKL_INT\n*descaf , MKL_INT *ipiv , float *b , MKL_INT *ib , MKL_INT *jb , MKL_INT *descb , float\n*x , MKL_INT *ix , MKL_INT *jx , MKL_INT *descx , float *ferr , float *berr , float\n*work , MKL_INT *lwork , MKL_INT *iwork , MKL_INT *liwork , MKL_INT *info );\nvoid pdgerfs (char *trans , MKL_INT *n , MKL_INT *nrhs , double *a , MKL_INT *ia ,\nMKL_INT *ja , MKL_INT *desca , double *af , MKL_INT *iaf , MKL_INT *jaf , MKL_INT\n*descaf , MKL_INT *ipiv , double *b , MKL_INT *ib , MKL_INT *jb , MKL_INT *descb ,\ndouble *x , MKL_INT *ix , MKL_INT *jx , MKL_INT *descx , double *ferr , double *berr ,\ndouble *work , MKL_INT *lwork , MKL_INT *iwork , MKL_INT *liwork , MKL_INT *info );\nvoid pcgerfs (char *trans , MKL_INT *n , MKL_INT *nrhs , MKL_Complex8 *a , MKL_INT\n*ia , MKL_INT *ja , MKL_INT *desca , MKL_Complex8 *af , MKL_INT *iaf , MKL_INT *jaf ,\nMKL_INT *descaf , MKL_INT *ipiv , MKL_Complex8 *b , MKL_INT *ib , MKL_INT *jb , MKL_INT\n*descb , MKL_Complex8 *x , MKL_INT *ix , MKL_INT *jx , MKL_INT *descx , float *ferr ,\nfloat *berr , MKL_Complex8 *work , MKL_INT *lwork , float *rwork , MKL_INT *lrwork ,\nMKL_INT *info );\nvoid pzgerfs (char *trans , MKL_INT *n , MKL_INT *nrhs , MKL_Complex16 *a , MKL_INT\n*ia , MKL_INT *ja , MKL_INT *desca , MKL_Complex16 *af , MKL_INT *iaf , MKL_INT *jaf ,\nMKL_INT *descaf , MKL_INT *ipiv , MKL_Complex16 *b , MKL_INT *ib , MKL_INT *jb ,\nMKL_INT *descb , MKL_Complex16 *x , MKL_INT *ix , MKL_INT *jx , MKL_INT *descx , double\n*ferr , double *berr , MKL_Complex16 *work , MKL_INT *lwork , double *rwork , MKL_INT\n*lrwork , MKL_INT *info );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?gerfs function improves the computed solution to one of the systems of linear equations\nsub(A)*sub(X) = sub(B),\nsub(A)T*sub(X) = sub(B), or\nsub(A)H*sub(X) = sub(B) and provides error bounds and backward error estimates for the solution.\nHere sub(A) = A(ia:ia+n-1, ja:ja+n-1), sub(B) = B(ib:ib+n-1, jb:jb+nrhs-1), and sub(X) = X(ix:ix\n+n-1, jx:jx+nrhs-1).\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1358\n\n\nInput Parameters\ntrans\n(global) Must be 'N' or 'T' or 'C'.\nSpecifies the form of the system of equations:\nIf trans = 'N', the system has the form sub(A)*sub(X) = sub(B) (No\ntranspose);\nIf trans = 'T', the system has the form sub(A)T*sub(X) = sub(B)\n(Transpose);\nIf trans = 'C', the system has the form sub(A)H*sub(X) = sub(B)\n(Conjugate transpose).\nn\n(global) The order of the distributed matrix sub(A) (n≥ 0).\nnrhs\n(global) The number of right-hand sides, i.e., the number of columns of the\nmatrices sub(B) and sub(X) (nrhs≥ 0).\na, af, b, x\n(local)\nPointers into the local memory to arrays of local sizes\na: lld_a * LOCc(ja+n-1),\naf: lld_af * LOCc(jaf+n-1), \nb: lld_b * LOCc(jb+nrhs-1),\nx: lld_x * LOCc(jx+nrhs-1).\nThe array a contains the local pieces of the distributed matrix sub(A).\nThe array af contains the local pieces of the distributed factors of the\nmatrix sub(A) = P*L*U as computed by p?getrf.\nThe array b contains the local pieces of the distributed matrix of right hand\nsides sub(B).\nOn entry, the array x contains the local pieces of the distributed solution\nmatrix sub(X).\nia, ja\n(global) The row and column indices in the global matrix A indicating the\nfirst row and the first column of the matrix sub(A), respectively.\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\niaf, jaf\n(global) The row and column indices in the global matrix AF indicating the\nfirst row and the first column of the matrix sub(AF), respectively.\ndescaf\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix AF.\nib, jb\n(global) The row and column indices in the global matrix B indicating the\nfirst row and the first column of the matrix sub(B), respectively.\ndescb\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix B.\nix, jx\n(global) The row and column indices in the global matrix X indicating the\nfirst row and the first column of the matrix sub(X), respectively.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1359\n\n\ndescx\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix X.\nipiv\n(local)\nArray of size LOCr(m_af) + mb_af.\nThis array contains pivoting information as computed by p?getrf. If\nipiv[i]=j, then the local row i+1 was swapped with the global row jwhere\ni=0, ... , LOCr(m_af) + mb_af- 1.\nThis array is tied to the distributed matrix A.\nwork\n(local)\nThe array work of size lwork is a workspace array.\nlwork\n(local or global) The size of the array work.\nFor real flavors:\nlwork must be at least\nlwork≥ 3*LOCr(n+mod(ia-1,mb_a))\nFor complex flavors:\nlwork must be at least\nlwork≥ 2*LOCr(n+mod(ia-1,mb_a))\nNOTE\nmod(x,y) is the integer remainder of x/y.\niwork\n(local) Workspace array, size liwork. Used in real flavors only.\nliwork\n(local or global) The size of the array iwork; used in real flavors only. Must\nbe at least\nliwork≥LOCr(n+mod(ib-1,mb_b)).\nrwork\n(local)\nWorkspace array, size lrwork. Used in complex flavors only.\nlrwork\n(local or global) The size of the array rwork; used in complex flavors only.\nMust be at least lrwork≥LOCr(n+mod(ib-1,mb_b))).\nOutput Parameters\nx\nOn exit, contains the improved solution vectors.\nferr, berr\nArrays of size LOCc(jb+nrhs-1) each.\nThe array ferr contains the estimated forward error bound for each\nsolution vector of sub(X).\nIf XTRUE is the true solution corresponding to sub(X), ferr is an estimated\nupper bound for the magnitude of the largest element in (sub(X) - XTRUE)\ndivided by the magnitude of the largest element in sub(X). The estimate is\nas reliable as the estimate for rcond, and is almost always a slight\noverestimate of the true error.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1360\n\n\nThis array is tied to the distributed matrix X.\nThe array berr contains the component-wise relative backward error of\neach solution vector (that is, the smallest relative change in any entry of\nsub(A) or sub(B) that makes sub(X) an exact solution). This array is tied to\nthe distributed matrix X.\nwork[0]\nOn exit, work[0] contains the minimum value of lwork required for\noptimum performance.\niwork[0]\nOn exit, iwork[0] contains the minimum value of liwork required for\noptimum performance (for real flavors).\nrwork[0]\nOn exit, rwork[0] contains the minimum value of lrwork required for\noptimum performance (for complex flavors).\ninfo\n(global) If info=0, the execution is successful.\ninfo < 0:\nIf the i-th argument is an array and the j-th entry, indexed j - 1, had an\nillegal value, then info = -(i*100+j); if the i-th argument is a scalar and\nhad an illegal value, then info = -i.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?porfs\nImproves the computed solution to a system of linear\nequations with symmetric/Hermitian positive definite\ndistributed matrix and provides error bounds and\nbackward error estimates for the solution.\nSyntax\nvoid psporfs (char *uplo , MKL_INT *n , MKL_INT *nrhs , float *a , MKL_INT *ia , MKL_INT\n*ja , MKL_INT *desca , float *af , MKL_INT *iaf , MKL_INT *jaf , MKL_INT *descaf , float\n*b , MKL_INT *ib , MKL_INT *jb , MKL_INT *descb , float *x , MKL_INT *ix , MKL_INT *jx ,\nMKL_INT *descx , float *ferr , float *berr , float *work , MKL_INT *lwork , MKL_INT\n*iwork , MKL_INT *liwork , MKL_INT *info );\nvoid pdporfs (char *uplo , MKL_INT *n , MKL_INT *nrhs , double *a , MKL_INT *ia ,\nMKL_INT *ja , MKL_INT *desca , double *af , MKL_INT *iaf , MKL_INT *jaf , MKL_INT\n*descaf , double *b , MKL_INT *ib , MKL_INT *jb , MKL_INT *descb , double *x , MKL_INT\n*ix , MKL_INT *jx , MKL_INT *descx , double *ferr , double *berr , double *work ,\nMKL_INT *lwork , MKL_INT *iwork , MKL_INT *liwork , MKL_INT *info );\nvoid pcporfs (char *uplo , MKL_INT *n , MKL_INT *nrhs , MKL_Complex8 *a , MKL_INT *ia ,\nMKL_INT *ja , MKL_INT *desca , MKL_Complex8 *af , MKL_INT *iaf , MKL_INT *jaf , MKL_INT\n*descaf , MKL_Complex8 *b , MKL_INT *ib , MKL_INT *jb , MKL_INT *descb , MKL_Complex8\n*x , MKL_INT *ix , MKL_INT *jx , MKL_INT *descx , float *ferr , float *berr ,\nMKL_Complex8 *work , MKL_INT *lwork , float *rwork , MKL_INT *lrwork , MKL_INT *info );\nvoid pzporfs (char *uplo , MKL_INT *n , MKL_INT *nrhs , MKL_Complex16 *a , MKL_INT\n*ia , MKL_INT *ja , MKL_INT *desca , MKL_Complex16 *af , MKL_INT *iaf , MKL_INT *jaf ,\nMKL_INT *descaf , MKL_Complex16 *b , MKL_INT *ib , MKL_INT *jb , MKL_INT *descb ,\nMKL_Complex16 *x , MKL_INT *ix , MKL_INT *jx , MKL_INT *descx , double *ferr , double\n*berr , MKL_Complex16 *work , MKL_INT *lwork , double *rwork , MKL_INT *lrwork ,\nMKL_INT *info );\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1361\n\n\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?porfsfunction improves the computed solution to the system of linear equations\nsub(A)*sub(X) = sub(B),\nwhere sub(A) = A(ia:ia+n-1, ja:ja+n-1) is a real symmetric or complex Hermitian positive definite\ndistributed matrix and\nsub(B) = B(ib:ib+n-1, jb:jb+nrhs-1),\nsub(X) = X(ix:ix+n-1, jx:jx+nrhs-1)\nare right-hand side and solution submatrices, respectively. This function also provides error bounds and\nbackward error estimates for the solution.\nInput Parameters\nuplo\n(global) Must be 'U' or 'L'.\nSpecifies whether the upper or lower triangular part of the symmetric/\nHermitian matrix sub(A) is stored.\nIf uplo = 'U', sub(A) is upper triangular. If uplo = 'L', sub(A) is lower\ntriangular.\nn\n(global) The order of the distributed matrix sub(A) (n≥0).\nnrhs\n(global) The number of right-hand sides, i.e., the number of columns of the\nmatrices sub(B) and sub(X) (nrhs≥0).\na, af, b, x\n(local)\nPointers into the local memory to arrays of local sizes\na: lld_a * LOCc(ja+n-1),\naf: lld_af * LOCc(jaf+n-1), \nb: lld_b * LOCc(jb+nrhs-1),\nx: lld_x * LOCc(jx+nrhs-1).\nThe array a contains the local pieces of the n-by-n symmetric/Hermitian\ndistributed matrix sub(A).\nIf uplo = 'U', the leading n-by-n upper triangular part of sub(A) contains\nthe upper triangular part of the matrix, and its strictly lower triangular part\nis not referenced.\nIf uplo = 'L', the leading n-by-n lower triangular part of sub(A) contains\nthe lower triangular part of the distributed matrix, and its strictly upper\ntriangular part is not referenced.\nThe array af contains the factors L or U from the Cholesky factorization\nsub(A) = L*LH or sub(A) = UH*U, as computed by p?potrf.\nOn entry, the array b contains the local pieces of the distributed matrix of\nright hand sides sub(B).\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1362\n\n\nOn entry, the array x contains the local pieces of the solution vectors\nsub(X).\nia, ja\n(global) The row and column indices in the global matrix A indicating the\nfirst row and the first column of the matrix sub(A), respectively.\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\niaf, jaf\n(global) The row and column indices in the global matrix AF indicating the\nfirst row and the first column of the matrix sub(AF), respectively.\ndescaf\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix AF.\nib, jb\n(global) The row and column indices in the global matrix B indicating the\nfirst row and the first column of the matrix sub(B), respectively.\ndescb\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix B.\nix, jx\n(global) The row and column indices in the global matrix X indicating the\nfirst row and the first column of the matrix sub(X), respectively.\ndescx\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix X.\nwork\n(local)\nThe array work of size lwork is a workspace array.\nlwork\n(local) The size of the array work.\nFor real flavors:\nlwork must be at least\nlwork≥ 3*LOCr(n+mod(ia-1,mb_a))\nFor complex flavors:\nlwork must be at least\nlwork≥ 2*LOCr(n+mod(ia-1,mb_a))\nNOTE\nmod(x,y) is the integer remainder of x/y.\niwork\n(local) Workspace array of size liwork. Used in real flavors only.\nliwork\n(local or global) The size of the array iwork; used in real flavors only. Must\nbe at least\nliwork≥LOCr(n+mod(ib-1,mb_b)).\nrwork\n(local)\nWorkspace array of size lrwork. Used in complex flavors only.\nlrwork\n(local or global) The size of the array rwork; used in complex flavors only.\nMust be at least lrwork≥LOCr(n+mod(ib-1,mb_b))).\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1363\n\n\nOutput Parameters\nx\nOn exit, contains the improved solution vectors.\nferr, berr\nArrays of size LOCc(jb+nrhs-1) each.\nThe array ferr contains the estimated forward error bound for each\nsolution vector of sub(X).\nIf XTRUE is the true solution corresponding to sub(X), ferr is an estimated\nupper bound for the magnitude of the largest element in (sub(X) - XTRUE)\ndivided by the magnitude of the largest element in sub(X). The estimate is\nas reliable as the estimate for rcond, and is almost always a slight\noverestimate of the true error.\nThis array is tied to the distributed matrix X.\nThe array berr contains the component-wise relative backward error of\neach solution vector (that is, the smallest relative change in any entry of\nsub(A) or sub(B) that makes sub(X) an exact solution). This array is tied to\nthe distributed matrix X.\nwork[0]\nOn exit, work[0] contains the minimum value of lwork required for\noptimum performance.\niwork[0]\nOn exit, iwork[0] contains the minimum value of liwork required for\noptimum performance (for real flavors).\nrwork[0]\nOn exit, rwork[0] contains the minimum value of lrwork required for\noptimum performance (for complex flavors).\ninfo\n(global) If info=0, the execution is successful.\ninfo < 0:\nIf the i-th argument is an array and the j-th entry, indexed j - 1, had an\nillegal value, then info = -(i*100+j); if the i-th argument is a scalar and\nhad an illegal value, then info = -i.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?trrfs\nProvides error bounds and backward error estimates\nfor the solution to a system of linear equations with a\ndistributed triangular coefficient matrix.\nSyntax\nvoid pstrrfs (char *uplo , char *trans , char *diag , MKL_INT *n , MKL_INT *nrhs , float\n*a , MKL_INT *ia , MKL_INT *ja , MKL_INT *desca , float *b , MKL_INT *ib , MKL_INT *jb ,\nMKL_INT *descb , float *x , MKL_INT *ix , MKL_INT *jx , MKL_INT *descx , float *ferr ,\nfloat *berr , float *work , MKL_INT *lwork , MKL_INT *iwork , MKL_INT *liwork , MKL_INT\n*info );\nvoid pdtrrfs (char *uplo , char *trans , char *diag , MKL_INT *n , MKL_INT *nrhs ,\ndouble *a , MKL_INT *ia , MKL_INT *ja , MKL_INT *desca , double *b , MKL_INT *ib ,\nMKL_INT *jb , MKL_INT *descb , double *x , MKL_INT *ix , MKL_INT *jx , MKL_INT *descx ,\ndouble *ferr , double *berr , double *work , MKL_INT *lwork , MKL_INT *iwork , MKL_INT\n*liwork , MKL_INT *info );\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1364\n\n\nvoid pctrrfs (char *uplo , char *trans , char *diag , MKL_INT *n , MKL_INT *nrhs ,\nMKL_Complex8 *a , MKL_INT *ia , MKL_INT *ja , MKL_INT *desca , MKL_Complex8 *b ,\nMKL_INT *ib , MKL_INT *jb , MKL_INT *descb , MKL_Complex8 *x , MKL_INT *ix , MKL_INT\n*jx , MKL_INT *descx , float *ferr , float *berr , MKL_Complex8 *work , MKL_INT *lwork ,\nfloat *rwork , MKL_INT *lrwork , MKL_INT *info );\nvoid pztrrfs (char *uplo , char *trans , char *diag , MKL_INT *n , MKL_INT *nrhs ,\nMKL_Complex16 *a , MKL_INT *ia , MKL_INT *ja , MKL_INT *desca , MKL_Complex16 *b ,\nMKL_INT *ib , MKL_INT *jb , MKL_INT *descb , MKL_Complex16 *x , MKL_INT *ix , MKL_INT\n*jx , MKL_INT *descx , double *ferr , double *berr , MKL_Complex16 *work , MKL_INT\n*lwork , double *rwork , MKL_INT *lrwork , MKL_INT *info );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?trrfsfunction provides error bounds and backward error estimates for the solution to one of the\nsystems of linear equations\nsub(A)*sub(X) = sub(B),\nsub(A)T*sub(X) = sub(B), or\nsub(A)H*sub(X) = sub(B) ,\nwhere sub(A) = A(ia:ia+n-1, ja:ja+n-1) is a triangular matrix,\nsub(B) = B(ib:ib+n-1, jb:jb+nrhs-1), and\nsub(X) = X(ix:ix+n-1, jx:jx+nrhs-1).\nThe solution matrix X must be computed by p?trtrs or some other means before entering this function. The\nfunction p?trrfs does not do iterative refinement because doing so cannot improve the backward error.\nInput Parameters\nuplo\n(global) Must be 'U' or 'L'.\nIf uplo = 'U', sub(A) is upper triangular. If uplo = 'L', sub(A) is lower\ntriangular.\ntrans\n(global) Must be 'N' or 'T' or 'C'.\nSpecifies the form of the system of equations:\nIf trans = 'N', the system has the form sub(A)*sub(X) = sub(B) (No\ntranspose);\nIf trans = 'T', the system has the form sub(A)T*sub(X) = sub(B)\n(Transpose);\nIf trans = 'C', the system has the form sub(A)H*sub(X) = sub(B)\n(Conjugate transpose).\ndiag\nMust be 'N' or 'U'.\nIf diag = 'N', then sub(A) is non-unit triangular.\nIf diag = 'U', then sub(A) is unit triangular.\nn\n(global) The order of the distributed matrix sub(A) (n≥0).\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1365\n\n\nnrhs\n(global) The number of right-hand sides, that is, the number of columns of\nthe matrices sub(B) and sub(X) (nrhs≥0).\na, b, x\n(local)\nPointers into the local memory to arrays of local sizes\na: lld_a * LOCc(ja+n-1),\nb: lld_b * LOCc(jb+nrhs-1),\nx: lld_x * LOCc(jx+nrhs-1).\nThe array a contains the local pieces of the original triangular distributed\nmatrix sub(A).\nIf uplo = 'U', the leading n-by-n upper triangular part of sub(A) contains\nthe upper triangular part of the matrix, and its strictly lower triangular part\nis not referenced.\nIf uplo = 'L', the leading n-by-n lower triangular part of sub(A) contains\nthe lower triangular part of the distributed matrix, and its strictly upper\ntriangular part is not referenced.\nIf diag = 'U', the diagonal elements of sub(A) are also not referenced\nand are assumed to be 1.\nOn entry, the array b contains the local pieces of the distributed matrix of\nright hand sides sub(B).\nOn entry, the array x contains the local pieces of the solution vectors\nsub(X).\nia, ja\n(global) The row and column indices in the global matrix A indicating the\nfirst row and the first column of the matrix sub(A), respectively.\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\nib, jb\n(global) The row and column indices in the global matrix B indicating the\nfirst row and the first column of the matrix sub(B), respectively.\ndescb\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix B.\nix, jx\n(global) The row and column indices in the global matrix X indicating the\nfirst row and the first column of the matrix sub(X), respectively.\ndescx\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix X.\nwork\n(local)\nThe array work of size lwork is a workspace array.\nlwork\n(local) The size of the array work.\nFor real flavors:\nlwork must be at least lwork≥ 3*LOCr(n+mod(ia-1,mb_a))\nFor complex flavors:\nlwork must be at least\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1366\n\n\nlwork≥ 2*LOCr(n+mod(ia-1,mb_a))\nNOTE\nmod(x,y) is the integer remainder of x/y.\niwork\n(local) Workspace array of size liwork. Used in real flavors only.\nliwork\n(local or global) The size of the array iwork; used in real flavors only. Must\nbe at least\nliwork≥LOCr(n+mod(ib-1,mb_b)).\nrwork\n(local)\nWorkspace array of size lrwork. Used in complex flavors only.\nlrwork\n(local or global) The size of the array rwork; used in complex flavors only.\nMust be at least lrwork≥LOCr(n+mod(ib-1,mb_b))).\nOutput Parameters\nferr, berr\nArrays of size LOCc(jb+nrhs-1) each.\nThe array ferr contains the estimated forward error bound for each\nsolution vector of sub(X).\nIf XTRUE is the true solution corresponding to sub(X), ferr is an estimated\nupper bound for the magnitude of the largest element in (sub(X) - XTRUE)\ndivided by the magnitude of the largest element in sub(X). The estimate is\nas reliable as the estimate for rcond, and is almost always a slight\noverestimate of the true error.\nThis array is tied to the distributed matrix X.\nThe array berr contains the component-wise relative backward error of\neach solution vector (that is, the smallest relative change in any entry of\nsub(A) or sub(B) that makes sub(X) an exact solution). This array is tied to\nthe distributed matrix X.\nwork[0]\nOn exit, work[0] contains the minimum value of lwork required for\noptimum performance.\niwork[0]\nOn exit, iwork[0] contains the minimum value of liwork required for\noptimum performance (for real flavors).\nrwork[0]\nOn exit, rwork[0] contains the minimum value of lrwork required for\noptimum performance (for complex flavors).\ninfo\n(global) If info=0, the execution is successful.\ninfo < 0:\nIf the i-th argument is an array and the j-th entry, indexed j - 1, had an\nillegal value, then info = -(i*100+j); if the i-th argument is a scalar and\nhad an illegal value, then info = -i.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1367\n\n\nMatrix Inversion: ScaLAPACK Computational Routines\nThis sections describes ScaLAPACK routines that compute the inverse of a matrix based on the previously\nobtained factorization. Note that it is not recommended to solve a system of equations Ax = b by first\ncomputing A-1 and then forming the matrix-vector product x = A-1b. Call a solver routine instead (see \nSolving Systems of Linear Equations); this is more efficient and more accurate.\np?getri\nComputes the inverse of a LU-factored distributed\nmatrix.\nSyntax\nvoid psgetri (MKL_INT *n , float *a , MKL_INT *ia , MKL_INT *ja , MKL_INT *desca ,\nMKL_INT *ipiv , float *work , MKL_INT *lwork , MKL_INT *iwork , MKL_INT *liwork ,\nMKL_INT *info );\nvoid pdgetri (MKL_INT *n , double *a , MKL_INT *ia , MKL_INT *ja , MKL_INT *desca ,\nMKL_INT *ipiv , double *work , MKL_INT *lwork , MKL_INT *iwork , MKL_INT *liwork ,\nMKL_INT *info );\nvoid pcgetri (MKL_INT *n , MKL_Complex8 *a , MKL_INT *ia , MKL_INT *ja , MKL_INT\n*desca , MKL_INT *ipiv , MKL_Complex8 *work , MKL_INT *lwork , MKL_INT *iwork , MKL_INT\n*liwork , MKL_INT *info );\nvoid pzgetri (MKL_INT *n , MKL_Complex16 *a , MKL_INT *ia , MKL_INT *ja , MKL_INT\n*desca , MKL_INT *ipiv , MKL_Complex16 *work , MKL_INT *lwork , MKL_INT *iwork ,\nMKL_INT *liwork , MKL_INT *info );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?getrifunction computes the inverse of a general distributed matrix sub(A) = A(ia:ia+n-1, ja:ja\n+n-1) using the LU factorization computed by p?getrf. This method inverts U and then computes the\ninverse of sub(A) by solving the system\ninv(sub(A))*L = inv(U)\nfor inv(sub(A)).\nInput Parameters\nn\n(global) The number of rows and columns to be operated on, that is, the\norder of the distributed matrix sub(A) (n≥0).\na\n(local)\nPointer into the local memory to an array of local size lld_a*LOCc(ja+n-1).\nOn entry, the array a contains the local pieces of the L and U obtained by\nthe factorization sub(A) = P*L*U computed by p?getrf.\nia, ja\n(global) The row and column indices in the global matrix A indicating the\nfirst row and the first column of the matrix sub(A), respectively.\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1368\n\n\nwork\n(local)\nThe array work of size lwork is a workspace array.\nlwork\n(local) The size of the array work. lwork must be at least\nlwork≥LOCr(n+mod(ia-1,mb_a))*nb_a.\nNOTE\nmod(x,y) is the integer remainder of x/y.\nThe array work is used to keep at most an entire column block of sub(A).\niwork\n(local) Workspace array used for physically transposing the pivots, size\nliwork.\nliwork\n(local or global) The size of the array iwork.\nThe minimal value liwork of is determined by the following code:\nif NPROW == NPCOL then\nliwork = LOCc(n_a + mod(ja-1,nb_a))+ nb_a \nelse \nliwork = LOCc(n_a + mod(ja-1,nb_a)) + \nmax(ceil(ceil(LOCr(m_a)/mb_a)/(lcm/NPROW)),nb_a)\nend if\nwhere lcm is the least common multiple of process rows and columns\n(NPROW and NPCOL).\nOutput Parameters\nipiv\n(local)\nArray of size LOCr(m_a)+ mb_a.\nThis array contains the pivoting information.\nIf ipiv[i]=j, then the local row i+1 was swapped with the global row\njwhere i=0, ... , LOCr(m_a) + mb_a- 1.\nThis array is tied to the distributed matrix A.\nwork[0]\nOn exit, work[0] contains the minimum value of lwork required for\noptimum performance.\niwork[0]\nOn exit, iwork[0] contains the minimum value of liwork required for\noptimum performance.\ninfo\n(global) If info=0, the execution is successful.\ninfo < 0:\nIf the i-th argument is an array and the j-th entry, indexed j - 1, had an\nillegal value, then info = -(i*100+j); if the i-th argument is a scalar and\nhad an illegal value, then info = -i.\ninfo> 0:\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1369\n\n\nIf info = i, the matrix element U(i,i) is exactly zero. The factorization has\nbeen completed, but the factor U is exactly singular, and division by zero\nwill occur if it is used to solve a system of equations.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?potri\nComputes the inverse of a symmetric/Hermitian\npositive definite distributed matrix.\nSyntax\nvoid pspotri (char *uplo , MKL_INT *n , float *a , MKL_INT *ia , MKL_INT *ja , MKL_INT\n*desca , MKL_INT *info );\nvoid pdpotri (char *uplo , MKL_INT *n , double *a , MKL_INT *ia , MKL_INT *ja , MKL_INT\n*desca , MKL_INT *info );\nvoid pcpotri (char *uplo , MKL_INT *n , MKL_Complex8 *a , MKL_INT *ia , MKL_INT *ja ,\nMKL_INT *desca , MKL_INT *info );\nvoid pzpotri (char *uplo , MKL_INT *n , MKL_Complex16 *a , MKL_INT *ia , MKL_INT *ja ,\nMKL_INT *desca , MKL_INT *info );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?potrifunction computes the inverse of a real symmetric or complex Hermitian positive definite\ndistributed matrix sub(A) = A(ia:ia+n-1, ja:ja+n-1) using the Cholesky factorization sub(A) = UH*U or\nsub(A) = L*LH computed by p?potrf.\nInput Parameters\nuplo\n(global) Must be 'U' or 'L'.\nSpecifies whether the upper or lower triangular part of the symmetric/\nHermitian matrix sub(A) is stored.\nIf uplo = 'U', upper triangle of sub(A) is stored. If uplo = 'L', lower\ntriangle of sub(A) is stored.\nn\n(global) The number of rows and columns to be operated on, that is, the\norder of the distributed matrix sub(A) (n≥0).\na\n(local)\nPointer into the local memory to an array of local size lld_a*LOCc(ja+n-1).\nOn entry, the array a contains the local pieces of the triangular factor U or L\nfrom the Cholesky factorization sub(A) = UH*U, or sub(A) = L*LH, as\ncomputed by p?potrf.\nia, ja\n(global) The row and column indices in the global matrix A indicating the\nfirst row and the first column of the matrix sub(A), respectively.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1370\n\n\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\nOutput Parameters\na\nOn exit, overwritten by the local pieces of the upper or lower triangle of the\n(symmetric/Hermitian) inverse of sub(A).\ninfo\n(global) If info=0, the execution is successful.\ninfo < 0:\nIf the i-th argument is an array and the j-th entry, indexed j - 1, had an\nillegal value, then info = -(i*100+j); if the i-th argument is a scalar and\nhad an illegal value, then info = -i.\ninfo> 0:\nIf info = i, the element (i, i) of the factor U or L is zero, and the inverse\ncould not be computed.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?trtri\nComputes the inverse of a triangular distributed\nmatrix.\nSyntax\nvoid pstrtri (char *uplo , char *diag , MKL_INT *n , float *a , MKL_INT *ia , MKL_INT\n*ja , MKL_INT *desca , MKL_INT *info );\nvoid pdtrtri (char *uplo , char *diag , MKL_INT *n , double *a , MKL_INT *ia , MKL_INT\n*ja , MKL_INT *desca , MKL_INT *info );\nvoid pctrtri (char *uplo , char *diag , MKL_INT *n , MKL_Complex8 *a , MKL_INT *ia ,\nMKL_INT *ja , MKL_INT *desca , MKL_INT *info );\nvoid pztrtri (char *uplo , char *diag , MKL_INT *n , MKL_Complex16 *a , MKL_INT *ia ,\nMKL_INT *ja , MKL_INT *desca , MKL_INT *info );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?trtrifunction computes the inverse of a real or complex upper or lower triangular distributed matrix\nsub(A) = A(ia:ia+n-1, ja:ja+n-1).\nInput Parameters\nuplo\n(global) Must be 'U' or 'L'.\nSpecifies whether the distributed matrix sub(A) is upper or lower triangular.\nIf uplo = 'U', sub(A) is upper triangular.\nIf uplo = 'L', sub(A) is lower triangular.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1371\n\n\ndiag\nMust be 'N' or 'U'.\nSpecifies whether or not the distributed matrix sub(A) is unit triangular.\nIf diag = 'N', then sub(A) is non-unit triangular.\nIf diag = 'U', then sub(A) is unit triangular.\nn\n(global) The number of rows and columns to be operated on, that is, the\norder of the distributed matrix sub(A) (n≥0).\na\n(local)\nPointer into the local memory to an array of local size lld_a*LOCc(ja+n-1).\nThe array a contains the local pieces of the triangular distributed matrix\nsub(A).\nIf uplo = 'U', the leading n-by-n upper triangular part of sub(A) contains\nthe upper triangular matrix to be inverted, and the strictly lower triangular\npart of sub(A) is not referenced.\nIf uplo = 'L', the leading n-by-n lower triangular part of sub(A) contains\nthe lower triangular matrix, and the strictly upper triangular part of sub(A)\nis not referenced.\nia, ja\n(global) The row and column indices in the global matrix A indicating the\nfirst row and the first column of the matrix sub(A), respectively.\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\nOutput Parameters\na\nOn exit, overwritten by the (triangular) inverse of the original matrix.\ninfo\n(global) If info=0, the execution is successful.\ninfo < 0:\nIf the i-th argument is an array and the j-th entry, indexed j - 1, had an\nillegal value, then info = -(i*100+j); if the i-th argument is a scalar and\nhad an illegal value, then info = -i.\ninfo> 0:\nIf info = k, the matrix element A(ia+k-1, ja+k-1) is exactly zero. The\ntriangular matrix sub(A) is singular and its inverse cannot be computed.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\nMatrix Equilibration: ScaLAPACK Computational Routines\nScaLAPACK routines described in this section are used to compute scaling factors needed to equilibrate a\nmatrix. Note that these routines do not actually scale the matrices.\np?geequ\nComputes row and column scaling factors intended to\nequilibrate a general rectangular distributed matrix\nand reduce its condition number.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1372\n\n\nSyntax\nvoid psgeequ (MKL_INT *m , MKL_INT *n , float *a , MKL_INT *ia , MKL_INT *ja , MKL_INT\n*desca , float *r , float *c , float *rowcnd , float *colcnd , float *amax , MKL_INT\n*info );\nvoid pdgeequ (MKL_INT *m , MKL_INT *n , double *a , MKL_INT *ia , MKL_INT *ja , MKL_INT\n*desca , double *r , double *c , double *rowcnd , double *colcnd , double *amax ,\nMKL_INT *info );\nvoid pcgeequ (MKL_INT *m , MKL_INT *n , MKL_Complex8 *a , MKL_INT *ia , MKL_INT *ja ,\nMKL_INT *desca , float *r , float *c , float *rowcnd , float *colcnd , float *amax ,\nMKL_INT *info );\nvoid pzgeequ (MKL_INT *m , MKL_INT *n , MKL_Complex16 *a , MKL_INT *ia , MKL_INT *ja ,\nMKL_INT *desca , double *r , double *c , double *rowcnd , double *colcnd , double\n*amax , MKL_INT *info );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?geequfunction computes row and column scalings intended to equilibrate an m-by-n distributed matrix\nsub(A) = A(ia:ia+m-1, ja:ja+n-1) and reduce its condition number. The output array r returns the row\nscale factors ri , and the array c returns the column scale factors cj . These factors are chosen to try to make\nthe largest element in each row and column of the matrix B with elements bij=ri*aij*cj have absolute value 1.\nri and cj are restricted to be between SMLNUM = smallest safe number and BIGNUM = largest safe number.\nUse of these scaling factors is not guaranteed to reduce the condition number of sub(A) but works well in\npractice.\nSMLNUM and BIGNUM are parameters representing machine precision. You can use the ?lamch routines to\ncompute them. For example, compute single precision values of SMLNUM and BIGNUM as follows:\nSMLNUM = slamch ('s')\nBIGNUM = 1 / SMLNUM\nThe auxiliary function p?laqge uses scaling factors computed by p?geequ to scale a general rectangular\nmatrix.\nInput Parameters\nm\n(global) The number of rows to be operated on, that is, the number of rows\nof the distributed matrix sub(A) (m≥ 0).\nn\n(global) The number of columns to be operated on, that is, the number of\ncolumns of the distributed matrix sub(A) (n≥ 0).\na\n(local)\nPointer into the local memory to an array of local size lld_a*LOCc(ja+n-1).\nThe array a contains the local pieces of the m-by-n distributed matrix whose\nequilibration factors are to be computed.\nia, ja\n(global) The row and column indices in the global matrix A indicating the\nfirst row and the first column of the matrix sub(A), respectively.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1373\n\n\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\nOutput Parameters\nr, c\n(local)\nArrays of sizes LOCr(m_a) and LOCc(n_a), respectively.\nIf info = 0, or info>ia+m-1, r[i] contain the row scale factors for sub(A)\nfor ia-1≤ i<ia+m-1. r is aligned with the distributed matrix A, and\nreplicated across every process column. r is tied to the distributed matrix\nA.\nIf info = 0, c[i] contain the column scale factors for sub(A) for ja-1≤\ni<ja+n-1. c is aligned with the distributed matrix A, and replicated down\nevery process row. c is tied to the distributed matrix A.\nrowcnd, colcnd\n(global)\nIf info = 0 or info>ia+m-1, rowcnd contains the ratio of the smallest ri\nto the largest ri (ia ≤ i ≤ ia+m-1). If rowcnd≥ 0.1 and amax is neither too\nlarge nor too small, it is not worth scaling by ri.\nIf info = 0, colcnd contains the ratio of the smallest cj to the largest cj\n(ja ≤ j ≤ ja+n-1).\nIf colcnd≥ 0.1, it is not worth scaling by cj.\namax\n(global)\nAbsolute value of the largest matrix element. If amax is very close to\noverflow or very close to underflow, the matrix should be scaled.\ninfo\n(global) If info=0, the execution is successful.\ninfo < 0:\nIf the i-th argument is an array and the j-th entry, indexed j - 1, had an\nillegal value, then info = -(i*100+j); if the i-th argument is a scalar and\nhad an illegal value, then info = -i.\ninfo> 0:\nIf info = i and\ni ≤ m, the i-th row of the distributed matrix\nsub(A) is exactly zero;\ni>m, the (i - m)-th column of the distributed\nmatrix sub(A) is exactly zero.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?poequ\nComputes row and column scaling factors intended to\nequilibrate a symmetric (Hermitian) positive definite\ndistributed matrix and reduce its condition number.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1374\n\n\nSyntax\nvoid pspoequ (MKL_INT *n , float *a , MKL_INT *ia , MKL_INT *ja , MKL_INT *desca , float\n*sr , float *sc , float *scond , float *amax , MKL_INT *info );\nvoid pdpoequ (MKL_INT *n , double *a , MKL_INT *ia , MKL_INT *ja , MKL_INT *desca ,\ndouble *sr , double *sc , double *scond , double *amax , MKL_INT *info );\nvoid pcpoequ (MKL_INT *n , MKL_Complex8 *a , MKL_INT *ia , MKL_INT *ja , MKL_INT\n*desca , float *sr , float *sc , float *scond , float *amax , MKL_INT *info );\nvoid pzpoequ (MKL_INT *n , MKL_Complex16 *a , MKL_INT *ia , MKL_INT *ja , MKL_INT\n*desca , double *sr , double *sc , double *scond , double *amax , MKL_INT *info );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?poequ function computes row and column scalings intended to equilibrate a real symmetric or\ncomplex Hermitian positive definite distributed matrix sub(A) = A(ia:ia+n-1, ja:ja+n-1) and reduce its\ncondition number (with respect to the two-norm). The output arrays sr and sc return the row and column\nscale factors\nThese factors are chosen so that the scaled distributed matrix B with elements bij=s(i)*aij*s(j) has ones on\nthe diagonal.\nThis choice of sr and sc puts the condition number of B within a factor n of the smallest possible condition\nnumber over all possible diagonal scalings.\nThe auxiliary function p?laqsy uses scaling factors computed by p?geequ to scale a general rectangular\nmatrix.\nInput Parameters\nn\n(global) The number of rows and columns to be operated on, that is, the\norder of the distributed matrix sub(A) (n≥0).\na\n(local)\nPointer into the local memory to an array of local size lld_a*LOCc(ja+n-1).\nThe array a contains the n-by-n symmetric/Hermitian positive definite\ndistributed matrix sub(A) whose scaling factors are to be computed. Only\nthe diagonal elements of sub(A) are referenced.\nia, ja\n(global) The row and column indices in the global matrix A indicating the\nfirst row and the first column of the matrix sub(A), respectively.\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\nOutput Parameters\nsr, sc\n(local) \nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1375\n\n\nArrays of sizes LOCr(m_a) and LOCc(n_a), respectively.\nIf info = 0, the array sr(ia:ia+n-1) contains the row scale factors for\nsub(A). sr is aligned with the distributed matrix A, and replicated across\nevery process column. sr is tied to the distributed matrix A.\nIf info = 0, the array sc(ja:ja+n-1) contains the column scale factors\nfor sub(A). sc is aligned with the distributed matrix A, and replicated down\nevery process row. sc is tied to the distributed matrix A.\nscond\n(global) \nIf info = 0, scond contains the ratio of the smallest sr[i] ( or sc[j]) to\nthe largest sr[i] ( or sc[j]), with\nia-1≤i<ia+n-1 and ja-1≤j<ja+n-1.\nIf scond≥ 0.1 and amax is neither too large nor too small, it is not worth\nscaling by sr ( or sc ).\namax\n(global)\nAbsolute value of the largest matrix element. If amax is very close to\noverflow or very close to underflow, the matrix should be scaled.\ninfo\n(global)\nIf info=0, the execution is successful.\ninfo < 0:\nIf the i-th argument is an array and the j-th entry, indexed j - 1, had an\nillegal value, then info = -(i*100+j); if the i-th argument is a scalar and\nhad an illegal value, then info = -i.\ninfo> 0:\nIf info = k, the k-th diagonal entry of sub(A) is nonpositive.\n \nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\nOrthogonal Factorizations: ScaLAPACK Computational Routines\nThis section describes the ScaLAPACK routines for the QR(RQ) and LQ(QL) factorization of matrices. Routines\nfor the RZ factorization as well as for generalized QR and RQ factorizations are also included. For the\nmathematical definition of the factorizations, see the respective LAPACK sections or refer to [SLUG].\nTable \"Computational Routines for Orthogonal Factorizations\" lists ScaLAPACK routines that perform\northogonal factorization of matrices.\nComputational Routines for Orthogonal Factorizations\nMatrix type,\nfactorization\nFactorize\nwithout\npivoting\nFactorize with\npivoting\nGenerate matrix\nQ\nApply matrix Q\ngeneral matrices, QR\nfactorization\np?geqrf\np?geqpf\np?orgqr\np?ungqr\np?ormqr\np?unmqr\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1376\n\n\nMatrix type,\nfactorization\nFactorize\nwithout\npivoting\nFactorize with\npivoting\nGenerate matrix\nQ\nApply matrix Q\ngeneral matrices, RQ\nfactorization\np?gerqf\n \np?orgrq\np?ungrq\np?ormrq\np?unmrq\ngeneral matrices, LQ\nfactorization\np?gelqf\n \np?orglq\np?unglq\np?ormlq\np?unmlq\ngeneral matrices, QL\nfactorization\np?geqlf\np?orgql\np?ungql\np?ormql\np?unmql\ntrapezoidal matrices,\nRZ factorization\np?tzrzf\n \n \np?ormrz\np?unmrz\npair of matrices,\ngeneralized QR\nfactorization\np?ggqrf\n \npair of matrices,\ngeneralized RQ\nfactorization\np?ggrqf\n \n \n \np?geqrf\nComputes the QR factorization of a general m-by-n\nmatrix.\nSyntax\nvoid psgeqrf (MKL_INT *m , MKL_INT *n , float *a , MKL_INT *ia , MKL_INT *ja , MKL_INT\n*desca , float *tau , float *work , MKL_INT *lwork , MKL_INT *info );\nvoid pdgeqrf (MKL_INT *m , MKL_INT *n , double *a , MKL_INT *ia , MKL_INT *ja , MKL_INT\n*desca , double *tau , double *work , MKL_INT *lwork , MKL_INT *info );\nvoid pcgeqrf (MKL_INT *m , MKL_INT *n , MKL_Complex8 *a , MKL_INT *ia , MKL_INT *ja ,\nMKL_INT *desca , MKL_Complex8 *tau , MKL_Complex8 *work , MKL_INT *lwork , MKL_INT\n*info );\nvoid pzgeqrf (MKL_INT *m , MKL_INT *n , MKL_Complex16 *a , MKL_INT *ia , MKL_INT *ja ,\nMKL_INT *desca , MKL_Complex16 *tau , MKL_Complex16 *work , MKL_INT *lwork , MKL_INT\n*info );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?geqrf function forms the QR factorization of a general m-by-n distributed matrix sub(A)= A(ia:ia\n+m-1, ja:ja+n-1) as\nA=Q*R.\nInput Parameters\nm\n(global) The number of rows in the distributed matrix sub(A); (m≥ 0).\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1377\n\n\nn\n(global) The number of columns in the distributed matrix sub(A); (n≥ 0).\na\n(local)\nPointer into the local memory to an array of local size lld_a*LOCc(ja+n-1).\nContains the local pieces of the distributed matrix sub(A) to be factored.\nia, ja\n(global) The row and column indices in the global matrix A indicating the\nfirst row and the first column of the submatrix A(ia:ia+m-1, ja:ja+n-1),\nrespectively.\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A\nwork\n(local).\nWorkspace array of size lwork.\nlwork\n(local or global) size of work, must be at least lwork≥nb_a *\n(mp0+nq0+nb_a), where\niroff = mod(ia-1, mb_a), icoff = mod(ja-1, nb_a),\niarow = indxg2p(ia, mb_a, MYROW, rsrc_a, NPROW),\niacol = indxg2p(ja, nb_a, MYCOL, csrc_a, NPCOL),\nmp0 = numroc(m+iroff, mb_a, MYROW, iarow, NPROW),\nnq0 = numroc(n+icoff, nb_a, MYCOL, iacol, NPCOL), and numroc,\nindxg2p are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL\ncan be determined by calling the function blacs_gridinfo.\nIf lwork = -1, then lwork is global input and a workspace query is\nassumed; the function only calculates the minimum and optimal size for all\nwork arrays. Each of these values is returned in the first entry of the\ncorresponding work array, and no error message is issued by pxerbla.\nOutput Parameters\na\nThe elements on and above the diagonal of sub(A) contain the min(m,n)-by-\nn upper trapezoidal matrix R (R is upper triangular if m≥n); the elements\nbelow the diagonal, with the array tau, represent the orthogonal/unitary\nmatrix Q as a product of elementary reflectors (see Application Notes\nbelow).\ntau\n(local)\nArray of size LOCc(ja+min(m,n)-1).\nContains the scalar factor of elementary reflectors. tau is tied to the\ndistributed matrix A.\nwork[0]\nOn exit, work[0] contains the minimum value of lwork required for\noptimum performance.\ninfo\n(global)\n= 0, the execution is successful.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1378\n\n\n< 0, if the i-th argument is an array and the j-th entry, indexed j - 1, had\nan illegal value, then info = -(i*100+j); if the i-th argument is a scalar\nand had an illegal value, then info = -i.\nApplication Notes\nThe matrix Q is represented as a product of elementary reflectors\nQ = H(ja)*H(ja+1)*...*H(ja+k-1),\nwhere k = min(m,n).\nEach H(i) has the form\nH(i) = I - tau*v*v'\nwhere tau is a real/complex scalar, and v is a real/complex vector with v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is\nstored on exit in A(ia+i:ia+m-1, ja+i-1), and tau in tau[ja+i-2].\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?geqpf\nComputes the QR factorization of a general m-by-n\nmatrix with pivoting.\nSyntax\nvoid psgeqpf (MKL_INT *m , MKL_INT *n , float *a , MKL_INT *ia , MKL_INT *ja , MKL_INT\n*desca , MKL_INT *ipiv , float *tau , float *work , MKL_INT *lwork , MKL_INT *info );\nvoid pdgeqpf (MKL_INT *m , MKL_INT *n , double *a , MKL_INT *ia , MKL_INT *ja , MKL_INT\n*desca , MKL_INT *ipiv , double *tau , double *work , MKL_INT *lwork , MKL_INT *info );\nvoid pcgeqpf (MKL_INT *m , MKL_INT *n , MKL_Complex8 *a , MKL_INT *ia , MKL_INT *ja ,\nMKL_INT *desca , MKL_INT *ipiv , MKL_Complex8 *tau , MKL_Complex8 *work , MKL_INT\n*lwork , float *rwork , MKL_INT *lrwork , MKL_INT *info );\nvoid pzgeqpf (MKL_INT *m , MKL_INT *n , MKL_Complex16 *a , MKL_INT *ia , MKL_INT *ja ,\nMKL_INT *desca , MKL_INT *ipiv , MKL_Complex16 *tau , MKL_Complex16 *work , MKL_INT\n*lwork , double *rwork , MKL_INT *lrwork , MKL_INT *info );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?geqpf function forms the QR factorization with column pivoting of a general m-by-n distributed matrix\nsub(A)= A(ia:ia+m-1, ja:ja+n-1) as\nsub(A)*P=Q*R.\nInput Parameters\nm\n(global) The number of rows in the matrix sub(A) (m≥ 0).\nn\n(global) The number of columns in the matrix sub(A) (n≥ 0).\na\n(local)\nPointer into the local memory to an array of local size lld_a*LOCc(ja+n-1).\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1379\n\n\nContains the local pieces of the distributed matrix sub(A) to be factored.\nia, ja\n(global) The row and column indices in the global matrix A indicating the\nfirst row and the first column of the submatrix A(ia:ia+m-1, ja:ja+n-1),\nrespectively.\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\nwork\n(local).\nWorkspace array of size lwork.\nlwork\n(local or global) size of work, must be at least\nFor real flavors:\nlwork≥max(3,mp0+nq0) + LOCc (ja+n-1) + nq0.\nFor complex flavors:\nlwork≥max(3,mp0+nq0) .\nHere\niroff = mod(ia-1, mb_a), icoff = mod(ja-1, nb_a),\niarow = indxg2p(ia, mb_a, MYROW, rsrc_a, NPROW),\niacol = indxg2p(ja, nb_a, MYCOL, csrc_a, NPCOL),\nmp0 = numroc(m+iroff, mb_a, MYROW, iarow, NPROW ),\nnq0 = numroc(n+icoff, nb_a, MYCOL, iacol, NPCOL),\nLOCc (ja+n-1) = numroc(ja+n-1, nb_a, MYCOL,csrc_a, NPCOL),\nand numroc, indxg2p are ScaLAPACK tool functions.\nYou can determine MYROW, MYCOL, NPROW and NPCOL by calling the\nblacs_gridinfofunction.\nIf lwork = -1, then lwork is global input and a workspace query is\nassumed; the function only calculates the minimum and optimal size for all\nwork arrays. Each of these values is returned in the first entry of the\ncorresponding work array, and no error message is issued by pxerbla.\nrwork\n(local).\nWorkspace array of size lrwork (complex flavors only).\nlrwork\n(local or global) size of rwork (complex flavors only). The value of lrwork\nmust be at least\nlwork≥LOCc (ja+n-1) + nq0 .\nHere\niroff = mod(ia-1, mb_a), icoff = mod(ja-1, nb_a),\niarow = indxg2p(ia, mb_a, MYROW, rsrc_a, NPROW),\niacol = indxg2p(ja, nb_a, MYCOL, csrc_a, NPCOL),\nmp0 = numroc(m+iroff, mb_a, MYROW, iarow, NPROW ),\nnq0 = numroc(n+icoff, nb_a, MYCOL, iacol, NPCOL),\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1380\n\n\nLOCc (ja+n-1) = numroc(ja+n-1, nb_a, MYCOL,csrc_a, NPCOL),\nand numroc, indxg2p are ScaLAPACK tool functions.\nYou can determine MYROW, MYCOL, NPROW and NPCOL by calling the\nblacs_gridinfofunction.\nIf lrwork = -1, then lrwork is global input and a workspace query is\nassumed; the function only calculates the minimum and optimal size for all\nwork arrays. Each of these values is returned in the first entry of the\ncorresponding work array, and no error message is issued by pxerbla.\nOutput Parameters\na\nThe elements on and above the diagonal of sub(A)contain the min(m,n)-by-\nn upper trapezoidal matrix R (R is upper triangular if m≥n); the elements\nbelow the diagonal, with the array tau, represent the orthogonal/unitary\nmatrix Q as a product of elementary reflectors (see Application Notes\nbelow).\nipiv\n(local) Array of size LOCc(ja+n-1).\nipiv[i] = k, the local (i+1)-th column of sub(A)*P was the global k-th\ncolumn of sub(A) (0 ≤ i < LOCc(ja+n-1). ipiv is tied to the distributed\nmatrix A.\ntau\n(local)\nArray of size LOCc(ja+min(m, n)-1).\nContains the scalar factor tau of elementary reflectors. tau is tied to the\ndistributed matrix A.\nwork[0]\nOn exit, work[0] contains the minimum value of lwork required for\noptimum performance.\nrwork[0]\nOn exit, rwork[0] contains the minimum value of lrwork required for\noptimum performance.\ninfo\n(global)\n= 0, the execution is successful.\n< 0, if the i-th argument is an array and the j-th entry, indexed j - 1, had\nan illegal value, then info = -(i*100+j); if the i-th argument is a scalar\nand had an illegal value, then info = -i.\nApplication Notes\nThe matrix Q is represented as a product of elementary reflectors\nQ = H(1)*H(2)*...*H(k)\nwhere k = min(m,n).\nEach H(i) has the form\nH = I - tau*v*v'\nwhere tau is a real/complex scalar, and v is a real/complex vector with v(1:i-1) = 0 and v(i) = 1; v(i+1:m) is\nstored on exit in A(ia+i:ia+m-1, ja+i-1).\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1381\n\n\nThe matrix P is represented in ipiv as follows: if ipiv[j]= i then the (j+1)-th column of P is the i-th\ncanonical unit vector (0 ≤ j < LOCc(ja+n-1).\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?orgqr\nGenerates the orthogonal matrix Q of the QR\nfactorization formed by p?geqrf.\nSyntax\nvoid psorgqr (MKL_INT *m , MKL_INT *n , MKL_INT *k , float *a , MKL_INT *ia , MKL_INT\n*ja , MKL_INT *desca , float *tau , float *work , MKL_INT *lwork , MKL_INT *info );\nvoid pdorgqr (MKL_INT *m , MKL_INT *n , MKL_INT *k , double *a , MKL_INT *ia , MKL_INT\n*ja , MKL_INT *desca , double *tau , double *work , MKL_INT *lwork , MKL_INT *info );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?orgqrfunction generates the whole or part of m-by-n real distributed matrix Q denoting A(ia:ia+m-1,\nja:ja+n-1) with orthonormal columns, which is defined as the first n columns of a product of k elementary\nreflectors of order m\nQ= H(1)*H(2)*...*H(k)\nas returned by p?geqrf.\nInput Parameters\nm\n(global) The number of rows in the matrix sub(Q) (m≥ 0).\nn\n(global) The number of columns in the matrix sub(Q) (m≥n≥ 0).\nk\n(global) The number of elementary reflectors whose product defines the\nmatrix Q(n≥k≥ 0).\na\n(local)\nPointer into the local memory to an array of local size lld_a*LOCc(ja+n-1).\nThe j-th column of the matrix stored in amust contain the vector that\ndefines the elementary reflector H(j), ja≤ j ≤ ja +k-1, as returned by \np?geqrf in the k columns of its distributed matrix argument A(ia:*, ja:ja\n+k-1).\nia, ja\n(global) The row and column indices in the global matrix A indicating the\nfirst row and the first column of the submatrix A(ia:ia+m-1, ja:ja+n-1),\nrespectively.\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\ntau\n(local)\nArray of size LOCc(ja+k-1).\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1382\n\n\nContains the scalar factor tau[j] of elementary reflectors H(j+1) as\nreturned by p?geqrf (0 ≤ j < LOCc(ja+k-1)). tau is tied to the\ndistributed matrix A.\nwork\n(local)\nWorkspace array of size of lwork.\nlwork\n(local or global) size of work.\nMust be at least lwork≥nb_a*(nqa0 + mpa0 + nb_a), where\niroffa = mod(ia-1, mb_a), icoffa = mod(ja-1, nb_a),\niarow = indxg2p(ia, mb_a, MYROW, rsrc_a, NPROW),\niacol = indxg2p(ja, nb_a, MYCOL, csrc_a, NPCOL),\nmpa0 = numroc(m+iroffa, mb_a, MYROW, iarow, NPROW),\nnqa0 = numroc(n+icoffa, nb_a, MYCOL, iacol, NPCOL);\nindxg2p and numroc are ScaLAPACK tool functions; MYROW, MYCOL, NPROW\nand NPCOL can be determined by calling the function blacs_gridinfo.\nIf lwork = -1, then lwork is global input and a workspace query is\nassumed; the function only calculates the minimum and optimal size for all\nwork arrays. Each of these values is returned in the first entry of the\ncorresponding work array, and no error message is issued by pxerbla.\nOutput Parameters\na\nContains the local pieces of the m-by-n distributed matrix Q.\nwork[0]\nOn exit, [0] contains the minimum value of lwork required for optimum\nperformance.\ninfo\n(global)\n= 0: the execution is successful.\n< 0: if the i-th argument is an array and the j-th entry, indexed j - 1, had\nan illegal value, then info = -(i*100+j); if the i-th argument is a scalar\nand had an illegal value, then info = -i.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?ungqr\nGenerates the complex unitary matrix Q of the QR\nfactorization formed by p?geqrf.\nSyntax\nvoid pcungqr (MKL_INT *m , MKL_INT *n , MKL_INT *k , MKL_Complex8 *a , MKL_INT *ia ,\nMKL_INT *ja , MKL_INT *desca , MKL_Complex8 *tau , MKL_Complex8 *work , MKL_INT\n*lwork , MKL_INT *info );\nvoid pzungqr (MKL_INT *m , MKL_INT *n , MKL_INT *k , MKL_Complex16 *a , MKL_INT *ia ,\nMKL_INT *ja , MKL_INT *desca , MKL_Complex16 *tau , MKL_Complex16 *work , MKL_INT\n*lwork , MKL_INT *info );\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1383\n\n\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThis function generates the whole or part of m-by-n complex distributed matrix Q denoting A(ia:ia+m-1,\nja:ja+n-1) with orthonormal columns, which is defined as the first n columns of a product of k elementary\nreflectors of order m\nQ = H(1)*H(2)*...*H(k)\nas returned by p?geqrf.\nInput Parameters\nm\n(global) The number of rows in the matrix sub(Q); (m≥0).\nn\n(global) The number of columns in the matrix sub(Q) (m≥n≥0).\nk\n(global) The number of elementary reflectors whose product defines the\nmatrix Q (n≥k≥0).\na\n(local)\nPointer into the local memory to an array of size lld_a*LOCc(ja+n-1). The\nj-th column of the matrix stored in amust contain the vector that defines\nthe elementary reflector H(j), ja≤ j≤ ja +k-1, as returned by p?geqrf in\nthe k columns of its distributed matrix argument A(ia:*, ja:ja+k-1).\nia, ja\n(global) The row and column indices in the global matrix A indicating the\nfirst row and the first column of the submatrix A, respectively.\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\ntau\n(local)\nArray of size LOCc(ja+k-1).\nContains the scalar factor tau[j] of elementary reflectors H(j+1) as\nreturned by p?geqrf (0 ≤ j < LOCc(ja+k-1)). tau is tied to the\ndistributed matrix A.\nwork\n(local)\nWorkspace array of size of lwork.\nlwork\n(local or global) size of work, must be at least lwork≥nb_a*(nqa0 + mpa0\n+ nb_a), where\niroffa = mod(ia-1, mb_a),\nicoffa = mod(ja-1, nb_a),\niarow = indxg2p(ia, mb_a, MYROW, rsrc_a, NPROW),\niacol = indxg2p(ja, nb_a, MYCOL, csrc_a, NPCOL),\nmpa0 = numroc(m+iroffa, mb_a, MYROW, iarow, NPROW),\nnqa0 = numroc(n+icoffa, nb_a, MYCOL, iacol, NPCOL)\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1384\n\n\nindxg2p and numroc are ScaLAPACK tool functions; MYROW, MYCOL, NPROW\nand NPCOL can be determined by calling the function blacs_gridinfo.\nIf lwork = -1, then lwork is global input and a workspace query is\nassumed; the function only calculates the minimum and optimal size for all\nwork arrays. Each of these values is returned in the first entry of the\ncorresponding work array, and no error message is issued by pxerbla.\nOutput Parameters\na\nContains the local pieces of the m-by-n distributed matrix Q.\nwork[0]\nOn exit work[0] contains the minimum value of lwork required for\noptimum performance.\ninfo\n(global)\n= 0: the execution is successful.\n< 0: if the i-th argument is an array and the j-th entry, indexed j - 1, had\nan illegal value, then info = -(i*100+j); if the i-th argument is a scalar\nand had an illegal value, then info = -i.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?ormqr\nMultiplies a general matrix by the orthogonal matrix Q\nof the QR factorization formed by p?geqrf.\nSyntax\nvoid psormqr (char *side , char *trans , MKL_INT *m , MKL_INT *n , MKL_INT *k , float\n*a , MKL_INT *ia , MKL_INT *ja , MKL_INT *desca , float *tau , float *c , MKL_INT *ic ,\nMKL_INT *jc , MKL_INT *descc , float *work , MKL_INT *lwork , MKL_INT *info );\nvoid pdormqr (char *side , char *trans , MKL_INT *m , MKL_INT *n , MKL_INT *k , double\n*a , MKL_INT *ia , MKL_INT *ja , MKL_INT *desca , double *tau , double *c , MKL_INT\n*ic , MKL_INT *jc , MKL_INT *descc , double *work , MKL_INT *lwork , MKL_INT *info );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?ormqrfunction overwrites the general real m-by-n distributed matrix sub (C) = C(iс:iс+m-1,jс:jс\n+n-1) with\nside ='L'\nside ='R'\ntrans = 'N':\nQ*sub(C)\nsub(C)*Q\ntrans = 'T':\nQT*sub(C)\nsub(C)*QT\nwhere Q is a real orthogonal distributed matrix defined as the product of k elementary reflectors\nQ = H(1) H(2)... H(k)\nas returned by p?geqrf. Q is of order m if side = 'L' and of order n if side = 'R'.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1385\n\n\nInput Parameters\nside\n(global)\n='L':Q or QT is applied from the left.\n='R':Q or QT is applied from the right.\ntrans\n(global)\n='N', no transpose, Q is applied.\n='T', transpose, QT is applied.\nm\n(global) The number of rows in the distributed matrix sub(C) (m≥0).\nn\n(global) The number of columns in the distributed matrix sub(C) (n≥0).\nk\n(global) The number of elementary reflectors whose product defines the\nmatrix Q. Constraints:\nIf side = 'L', m≥k≥0\nIf side = 'R', n≥k≥0.\na\n(local)\nPointer into the local memory to an array of size lld_a*LOCc(ja+n-1). The\nj-th column of the matrix stored in amust contain the vector that defines\nthe elementary reflector H(j), ja≤j≤ja+k-1, as returned by p?geqrf in the\nk columns of its distributed matrix argument A(ia:*, ja:ja+k-1). A(ia:*,\nja:ja+k-1) is modified by the function but restored on exit.\nIf side = 'L', lld_a ≥ max(1, LOCr(ia+m-1))\nIf side = 'R', lld_a ≥ max(1, LOCr(ia+n-1))\nia, ja\n(global) The row and column indices in the global matrix A indicating the\nfirst row and the first column of the submatrix A, respectively.\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\ntau\n(local)\nArray of size LOCc(ja+k-1).\nContains the scalar factor tau[j] of elementary reflectors H(j+1) as\nreturned by p?geqrf (0 ≤ j < LOCc(ja+k-1)). tau is tied to the\ndistributed matrix A.\nc\n(local)\nPointer into the local memory to an array of local size lld_c*LOCc(jc+n-1).\nContains the local pieces of the distributed matrix sub(C) to be factored.\nic, jc\n(global) The row and column indices in the global matrix C indicating the\nfirst row and the first column of the matrix sub(C), respectively.\ndescc\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix C.\nwork\n(local)\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1386\n\n\nWorkspace array of size of lwork.\nlwork\n(local or global) size of work, must be at least:\nif side = 'L',\nlwork≥max((nb_a*(nb_a-1))/2, (nqc0+mpc0)*nb_a) + nb_a*nb_a\nelse if side = 'R', \nlwork≥max((nb_a*(nb_a-1))/2, (nqc0+max(npa0+numroc(numroc(n\n+icoffc, nb_a, 0, 0, NPCOL), nb_a, 0, 0, lcmq), mpc0))*nb_a)\n+ nb_a*nb_a\nend if\nwhere\nlcmq = lcm/NPCOL with lcm = ilcm(NPROW, NPCOL),\niroffa = mod(ia-1, mb_a),\nicoffa = mod(ja-1, nb_a),\niarow = indxg2p(ia, mb_a, MYROW, rsrc_a, NPROW),\nnpa0= numroc(n+iroffa, mb_a, MYROW, iarow, NPROW),\niroffc = mod(ic-1, mb_c),\nicoffc = mod(jc-1, nb_c),\nicrow = indxg2p(ic, mb_c, MYROW, rsrc_c, NPROW),\niccol = indxg2p(jc, nb_c, MYCOL, csrc_c, NPCOL),\nmpc0= numroc(m+iroffc, mb_c, MYROW, icrow, NPROW),\nnqc0= numroc(n+icoffc, nb_c, MYCOL, iccol, NPCOL),\nilcm, indxg2p and numroc are ScaLAPACK tool functions; MYROW, MYCOL,\nNPROW and NPCOL can be determined by calling the function\nblacs_gridinfo.\nIf lwork = -1, then lwork is global input and a workspace query is\nassumed; the function only calculates the minimum and optimal size for all\nwork arrays. Each of these values is returned in the first entry of the\ncorresponding work array, and no error message is issued by pxerbla.\nOutput Parameters\nc\nOverwritten by the product Q*sub(C), or QT*sub(C), or sub(C)*QT, or\nsub(C)*Q.\nwork[0]\nOn exit work[0] contains the minimum value of lwork required for\noptimum performance.\ninfo\n(global)\n= 0: the execution is successful.\n< 0: if the i-th argument is an array and the j-th entry, indexed j - 1, had\nan illegal value, then info = -(i*100+j); if the i-th argument is a scalar\nand had an illegal value, then info = -i.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1387\n\n\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?unmqr\nMultiplies a complex matrix by the unitary matrix Q of\nthe QR factorization formed by p?geqrf.\nSyntax\nvoid pcunmqr (char *side , char *trans , MKL_INT *m , MKL_INT *n , MKL_INT *k ,\nMKL_Complex8 *a , MKL_INT *ia , MKL_INT *ja , MKL_INT *desca , MKL_Complex8 *tau ,\nMKL_Complex8 *c , MKL_INT *ic , MKL_INT *jc , MKL_INT *descc , MKL_Complex8 *work ,\nMKL_INT *lwork , MKL_INT *info );\nvoid pzunmqr (char *side , char *trans , MKL_INT *m , MKL_INT *n , MKL_INT *k ,\nMKL_Complex16 *a , MKL_INT *ia , MKL_INT *ja , MKL_INT *desca , MKL_Complex16 *tau ,\nMKL_Complex16 *c , MKL_INT *ic , MKL_INT *jc , MKL_INT *descc , MKL_Complex16 *work ,\nMKL_INT *lwork , MKL_INT *info );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThis function overwrites the general complex m-by-n distributed matrix sub (C) = C(iс:iс+m-1,jс:jс+n-1)\nwith\nside ='L'\nside ='R'\ntrans = 'N':\nQ*sub(C)\nsub(C)*Q\ntrans = 'T':\nQH*sub(C)\nsub(C)*QH\nwhere Q is a complex unitary distributed matrix defined as the product of k elementary reflectors\nQ = H(1) H(2)... H(k) as returned by p?geqrf. Q is of order m if side = 'L' and of order n if side ='R'.\nInput Parameters\nside\n(global)\n='L': Q or QH is applied from the left.\n='R': Q or QH is applied from the right.\ntrans\n(global)\n='N', no transpose, Q is applied.\n='C', conjugate transpose, QH is applied.\nm\n(global) The number of rows in the distributed matrix sub(C) (m≥0).\nn\n(global) The number of columns in the distributed matrix sub(C) (n≥0).\nk\n(global) The number of elementary reflectors whose product defines the\nmatrix Q. Constraints:\nIf side = 'L', m≥k≥0\nIf side = 'R', n≥k≥0.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1388\n\n\na\n(local)\nPointer into the local memory to an array of size lld_a*LOCc(ja+k-1). The\nj-th column of the matrix stored in amust contain the vector that defines\nthe elementary reflector H(j), ja≤j≤ja+k-1, as returned by p?geqrf in the\nk columns of its distributed matrix argument A(ia:*, ja:ja+k-1). A(ia:*,\nja:ja+k-1) is modified by the function but restored on exit.\nIf side = 'L', lld_a ≥ max(1, LOCr(ia+m-1))\nIf side = 'R', lld_a ≥ max(1, LOCr(ia+n-1))\nia, ja\n(global) The row and column indices in the global matrix A indicating the\nfirst row and the first column of the submatrix A, respectively.\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\ntau\n(local)\nArray of size LOCc(ja+k-1).\nContains the scalar factor tau[j] of elementary reflectors H(j+1) as\nreturned by p?geqrf (0 ≤ j < LOCc(ja+k-1)). tau is tied to the\ndistributed matrix A.\nc\n(local)\nPointer into the local memory to an array of local size lld_c*LOCc(jc+n-1).\nContains the local pieces of the distributed matrix sub(C) to be factored.\nic, jc\n(global) The row and column indices in the global matrix C indicating the\nfirst row and the first column of the submatrix C, respectively.\ndescc\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix C.\nwork\n(local)\nWorkspace array of size of lwork.\nlwork\n(local or global) size of work, must be at least:\nIf side = 'L',\nlwork≥max((nb_a*(nb_a-1))/2, (nqc0 + mpc0)*nb_a) + nb_a*nb_a\nelse if side = 'R',\nlwork≥max((nb_a*(nb_a-1))/2, (nqc0 + max(npa0 +\nnumroc(numroc(n+icoffc, nb_a, 0, 0, NPCOL), nb_a, 0, 0,\nlcmq), mpc0))*nb_a) + nb_a*nb_a\nend if\nwhere\nlcmq = lcm/NPCOL with lcm = ilcm (NPROW, NPCOL),\niroffa = mod(ia-1, mb_a),\nicoffa = mod(ja-1, nb_a),\niarow = indxg2p(ia, mb_a, MYROW, rsrc_a, NPROW),\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1389\n\n\nnpa0 = numroc(n+iroffa, mb_a, MYROW, iarow, NPROW),\niroffc = mod(ic-1, mb_c),\nicoffc = mod(jc-1, nb_c),\nicrow = indxg2p(ic, mb_c, MYROW, rsrc_c, NPROW),\niccol = indxg2p(jc, nb_c, MYCOL, csrc_c, NPCOL),\nmpc0 = numroc(m+iroffc, mb_c, MYROW, icrow, NPROW),\nnqc0 = numroc(n+icoffc, nb_c, MYCOL, iccol, NPCOL),\nilcm, indxg2p and numroc are ScaLAPACK tool functions; MYROW, MYCOL,\nNPROW and NPCOL can be determined by calling the function\nblacs_gridinfo.\nIf lwork = -1, then lwork is global input and a workspace query is\nassumed; the function only calculates the minimum and optimal size for all\nwork arrays. Each of these values is returned in the first entry of the\ncorresponding work array, and no error message is issued by pxerbla.\nOutput Parameters\nc\nOverwritten by the product Q*sub(C), or QH*sub(C), or sub(C)*QH, or\nsub(C)*Q .\nwork[0]\nOn exit work[0] contains the minimum value of lwork required for\noptimum performance.\ninfo\n(global)\n= 0: the execution is successful.\n< 0: if the i-th argument is an array and the j-th entry, indexed j - 1, had\nan illegal value, then info = -(i*100+j); if the i-th argument is a scalar\nand had an illegal value, then info = -i.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?gelqf\nComputes the LQ factorization of a general\nrectangular matrix.\nSyntax\nvoid psgelqf (MKL_INT *m , MKL_INT *n , float *a , MKL_INT *ia , MKL_INT *ja , MKL_INT\n*desca , float *tau , float *work , MKL_INT *lwork , MKL_INT *info );\nvoid pdgelqf (MKL_INT *m , MKL_INT *n , double *a , MKL_INT *ia , MKL_INT *ja , MKL_INT\n*desca , double *tau , double *work , MKL_INT *lwork , MKL_INT *info );\nvoid pcgelqf (MKL_INT *m , MKL_INT *n , MKL_Complex8 *a , MKL_INT *ia , MKL_INT *ja ,\nMKL_INT *desca , MKL_Complex8 *tau , MKL_Complex8 *work , MKL_INT *lwork , MKL_INT\n*info );\nvoid pzgelqf (MKL_INT *m , MKL_INT *n , MKL_Complex16 *a , MKL_INT *ia , MKL_INT *ja ,\nMKL_INT *desca , MKL_Complex16 *tau , MKL_Complex16 *work , MKL_INT *lwork , MKL_INT\n*info );\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1390\n\n\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?gelqf function computes the LQ factorization of a real/complex distributed m-by-n matrix sub(A)=\nA(ia:ia+m-1,ja:ja+n-1) = L*Q.\nInput Parameters\nm\n(global) The number of rows in the distributed submatrix sub(A) (m≥ 0).\nn\n(global) The number of columns in the distributed submatrix sub(A) (n≥\n0).\na\n(local)\nPointer into the local memory to an array of local size lld_a*LOCc(ja+n-1).\nContains the local pieces of the distributed matrix sub(A) to be factored.\nia, ja\n(global) The row and column indices in the global array A indicating the first\nrow and the first column of the submatrix A(ia:ia+m-1,ja:ja+n-1),\nrespectively.\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\nwork\n(local)\nWorkspace array of size of lwork.\nlwork\n(local or global) size of work, must be at least lwork≥mb_a*(mp0 + nq0 +\nmb_a), where\niroff = mod(ia-1, mb_a),\nicoff = mod(ja-1, nb_a),\niarow = indxg2p(ia, mb_a, MYROW, rsrc_a, NPROW),\niacol = indxg2p(ja, nb_a, MYCOL, csrc_a, NPCOL),\nmp0 = numroc(m+iroff, mb_a, MYROW, iarow, NPROW),\nnq0 = numroc(n+icoff, nb_a, MYCOL, iacol, NPCOL)\nindxg2p and numroc are ScaLAPACK tool functions; MYROW, MYCOL, NPROW\nand NPCOL can be determined by calling the function blacs_gridinfo.\nNOTE\nmod(x,y) is the integer remainder of x/y.\nIf lwork = -1, then lwork is global input and a workspace query is\nassumed; the function only calculates the minimum and optimal size for all\nwork arrays. Each of these values is returned in the first entry of the\ncorresponding work array, and no error message is issued by pxerbla.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1391\n\n\nOutput Parameters\na\nThe elements on and below the diagonal of sub(A) contain the m-by-\nmin(m,n) lower trapezoidal matrix L (L is lower trapezoidal if m ≤ n); the\nelements above the diagonal, with the array tau, represent the orthogonal/\nunitary matrix Q as a product of elementary reflectors (see Application\nNotes below).\ntau\n(local)\nArray of size LOCr(ia+min(m, n)-1).\nContains the scalar factors of elementary reflectors. tau is tied to the\ndistributed matrix A.\nwork[0]\nOn exit, work[0] contains the minimum value of lwork required for\noptimum performance.\ninfo\n(global)\n= 0: the execution is successful.\n< 0: if the i-th argument is an array and the j-th entry, indexed j - 1, had\nan illegal value, then info = -(i*100+j); if the i-th argument is a scalar\nand had an illegal value, then info = -i.\nApplication Notes\nThe matrix Q is represented as a product of elementary reflectors\nQ = H(ia+k-1)*H(ia+k-2)*...*H(ia),\nwhere k = min(m,n)\nEach H(i) has the form\nH(i) = I - tau*v*v'\nwhere tau is a real/complex scalar, and v is a real/complex vector with v(1:i-1) = 0 and v(i) = 1; v(i+1:n) is\nstored on exit in A(ia+i-1,ja+i:ja+n-1), and tau in tau[ia+i-2].\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?orglq\nGenerates the real orthogonal matrix Q of the LQ\nfactorization formed by p?gelqf.\nSyntax\nvoid psorglq (MKL_INT *m , MKL_INT *n , MKL_INT *k , float *a , MKL_INT *ia , MKL_INT\n*ja , MKL_INT *desca , float *tau , float *work , MKL_INT *lwork , MKL_INT *info );\nvoid pdorglq (MKL_INT *m , MKL_INT *n , MKL_INT *k , double *a , MKL_INT *ia , MKL_INT\n*ja , MKL_INT *desca , double *tau , double *work , MKL_INT *lwork , MKL_INT *info );\nInclude Files\n•\nmkl_scalapack.h\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1392\n\n\nDescription\nThe p?orglq function generates the whole or part of m-by-n real distributed matrix Q denoting A(ia:ia\n+m-1,ja:ja+n-1) with orthonormal rows, which is defined as the first m rows of a product of k elementary\nreflectors of order n\nQ = H(k)*...* H(2)* H(1)\nas returned by p?gelqf.\nInput Parameters\nm\n(global) The number of rows in the matrix sub(Q); (m≥0).\nn\n(global) The number of columns in the matrix sub(Q) (n≥m≥0).\nk\n(global) The number of elementary reflectors whose product defines the\nmatrix Q(m≥k≥0).\na\n(local)\nPointer into the local memory to an array of local size lld_a*LOCc(ja+n-1).\nOn entry, the i-th row of the matrix stored in amust contain the vector that\ndefines the elementary reflector H(i), ia≤i≤ia+k-1, as returned by \np?gelqf in the k rows of its distributed matrix argument A(ia:ia+k-1,\nja:*).\nia, ja\n(global) The row and column indices in the global matrix A indicating the\nfirst row and the first column of the submatrix A(ia:ia+m-1,ja:ja+n-1),\nrespectively.\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\nwork\n(local)\nWorkspace array of size of lwork.\nlwork\n(local or global) size of work, must be at least\nlwork≥mb_a*(mpa0+nqa0+mb_a), where\niroffa = mod(ia-1, mb_a),\nicoffa = mod(ja-1, nb_a),\niarow = indxg2p(ia, mb_a, MYROW, rsrc_a, NPROW),\niacol = indxg2p(ja, nb_a, MYCOL, csrc_a, NPCOL),\nmpa0 = numroc(m+iroffa, mb_a, MYROW, iarow, NPROW),\nnqa0 = numroc(n+icoffa, nb_a, MYCOL, iacol, NPCOL)\nNOTE\nmod(x,y) is the integer remainder of x/y.\nindxg2p and numroc are ScaLAPACK tool functions; MYROW, MYCOL, NPROW\nand NPCOL can be determined by calling the function blacs_gridinfo.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1393\n\n\nIf lwork = -1, then lwork is global input and a workspace query is\nassumed; the function only calculates the minimum and optimal size for all\nwork arrays. Each of these values is returned in the first entry of the\ncorresponding work array, and no error message is issued by pxerbla.\nOutput Parameters\na\nContains the local pieces of the m-by-n distributed matrix Q to be factored.\ntau\n(local)\nArray of size LOCr(ia+k-1).\nContains the scalar factors tau[j] of elementary reflectors H(j+1), 0 ≤ j <\nLOCr(ia+k-1). tau is tied to the distributed matrix A.\nwork[0]\nOn exit, work[0] contains the minimum value of lwork required for\noptimum performance.\ninfo\n(global)\n= 0: the execution is successful.\n< 0: if the i-th argument is an array and the j-th entry, indexed j - 1, had\nan illegal value, then info = -(i*100+j); if the i-th argument is a scalar\nand had an illegal value, then info = -i.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?unglq\nGenerates the unitary matrix Q of the LQ factorization\nformed by p?gelqf.\nSyntax\nvoid pcunglq (MKL_INT *m , MKL_INT *n , MKL_INT *k , MKL_Complex8 *a , MKL_INT *ia ,\nMKL_INT *ja , MKL_INT *desca , MKL_Complex8 *tau , MKL_Complex8 *work , MKL_INT\n*lwork , MKL_INT *info );\nvoid pzunglq (MKL_INT *m , MKL_INT *n , MKL_INT *k , MKL_Complex16 *a , MKL_INT *ia ,\nMKL_INT *ja , MKL_INT *desca , MKL_Complex16 *tau , MKL_Complex16 *work , MKL_INT\n*lwork , MKL_INT *info );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThis function generates the whole or part of m-by-n complex distributed matrix Q denoting A(ia:ia\n+m-1,ja:ja+n-1) with orthonormal rows, which is defined as the first m rows of a product of k elementary\nreflectors of order n\nQ = (H(k))H...*(H(2))H*(H(1))H as returned by p?gelqf.\nInput Parameters\nm\n(global) The number of rows in the matrix sub(Q) (m≥0).\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1394\n\n\nn\n(global) The number of columns in the matrix sub(Q) (n≥m≥0).\nk\n(global) The number of elementary reflectors whose product defines the\nmatrix Q(m≥k≥0).\na\n(local)\nPointer into the local memory to an array of local size lld_a*LOCc(ja+n-1).\nOn entry, the i-th row of the matrix stored in amust contain the vector that\ndefines the elementary reflector H(i), ia≤i≤ia+k-1, as returned by \np?gelqf in the k rows of its distributed matrix argument A(ia:ia+k-1,\nja:*).\nia, ja\n(global) The row and column indices in the global matrix A indicating the\nfirst row and the first column of the submatrix A(ia:ia+m-1,ja:ja+n-1),\nrespectively.\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\ntau\n(local)\nArray of size LOCr(ia+k-1).\nContains the scalar factors tau[j] of elementary reflectors H(j+1), 0 ≤ j <\nLOCr(ia+k-1). tau is tied to the distributed matrix A.\nwork\n(local)\nWorkspace array of size of lwork.\nlwork\n(local or global) size of work, must be at least\nlwork≥mb_a*(mpa0+nqa0+mb_a), where\niroffa = mod(ia-1, mb_a),\nicoffa = mod(ja-1, nb_a),\niarow = indxg2p(ia, mb_a, MYROW, rsrc_a, NPROW),\niacol = indxg2p(ja, nb_a, MYCOL, csrc_a, NPCOL),\nmpa0 = numroc(m+iroffa, mb_a, MYROW, iarow, NPROW),\nnqa0 = numroc(n+icoffa, nb_a, MYCOL, iacol, NPCOL)\nindxg2p and numroc are ScaLAPACK tool functions; MYROW, MYCOL, NPROW\nand NPCOL can be determined by calling the function blacs_gridinfo.\nNOTE\nmod(x,y) is the integer remainder of x/y.\nIf lwork = -1, then lwork is global input and a workspace query is\nassumed; the function only calculates the minimum and optimal size for all\nwork arrays. Each of these values is returned in the first entry of the\ncorresponding work array, and no error message is issued by pxerbla.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1395\n\n\nOutput Parameters\na\nContains the local pieces of the m-by-n distributed matrix Q to be factored.\nwork[0]\nOn exit, work[0] contains the minimum value of lwork required for\noptimum performance.\ninfo\n(global)\n= 0: the execution is successful.\n< 0: if the i-th argument is an array and the j-th entry, indexed j - 1, had\nan illegal value, then info = -(i*100+j); if the i-th argument is a scalar\nand had an illegal value, then info = -i.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?ormlq\nMultiplies a general matrix by the orthogonal matrix Q\nof the LQ factorization formed by p?gelqf.\nSyntax\nvoid psormlq (char *side , char *trans , MKL_INT *m , MKL_INT *n , MKL_INT *k , float\n*a , MKL_INT *ia , MKL_INT *ja , MKL_INT *desca , float *tau , float *c , MKL_INT *ic ,\nMKL_INT *jc , MKL_INT *descc , float *work , MKL_INT *lwork , MKL_INT *info );\nvoid pdormlq (char *side , char *trans , MKL_INT *m , MKL_INT *n , MKL_INT *k , double\n*a , MKL_INT *ia , MKL_INT *ja , MKL_INT *desca , double *tau , double *c , MKL_INT\n*ic , MKL_INT *jc , MKL_INT *descc , double *work , MKL_INT *lwork , MKL_INT *info );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?ormlq function overwrites the general real m-by-n distributed matrix sub(C) = C(iс:iс+m-1,jс:jс\n+n-1) with\nside ='L'\nside ='R'\ntrans = 'N':\nQ*sub(C)\nsub(C)*Q\ntrans = 'T':\nQT*sub(C)\nsub(C)*QT\nwhere Q is a real orthogonal distributed matrix defined as the product of k elementary reflectors\nQ = H(k)...H(2) H(1)\nas returned by p?gelqf. Q is of order m if side = 'L' and of order n if side = 'R'.\nInput Parameters\nside\n(global)\n='L': Q or QT is applied from the left.\n='R': Q or QT is applied from the right.\ntrans\n(global)\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1396\n\n\n='N', no transpose, Q is applied.\n='T', transpose, QT is applied.\nm\n(global) The number of rows in the distributed matrix sub(C) (m≥0).\nn\n(global) The number of columns in the distributed matrix sub(C) (n≥0).\nk\n(global) The number of elementary reflectors whose product defines the\nmatrix Q. Constraints:\nIf side = 'L', m≥k≥0\nIf side = 'R', n≥k≥0.\na\n(local)\nPointer into the local memory to an array of size lld_a*LOCc(ja+m-1), if\nside = 'L' and lld_a*LOCc(ja+n-1), if side = 'R'. The i-th row of the\nmatrix stored in amust contain the vector that defines the elementary\nreflector H(i), ia≤i≤ia+k-1, as returned by p?gelqf in the k rows of its\ndistributed matrix argument A(ia:ia+k-1, ja:*).\nA(ia:ia+k-1, ja:*) is modified by the function but restored on exit.\nia, ja\n(global) The row and column indices in the global matrix A indicating the\nfirst row and the first column of the submatrix A, respectively.\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\ntau\n(local)\nArray of size LOCc(ja+k-1).\nContains the scalar factor tau[j] of elementary reflectors H(j+1) as\nreturned by p?gelqf (0 ≤ j < LOCc(ja+k-1)). tau is tied to the\ndistributed matrix A.\nc\n(local)\nPointer into the local memory to an array of local size lld_c*LOCc(jc+n-1).\nContains the local pieces of the distributed matrix sub(C) to be factored.\nic, jc\n(global) The row and column indices in the global matrix C indicating the\nfirst row and the first column of the submatrix C, respectively.\ndescc\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix C.\nwork\n(local)\nWorkspace array of size of lwork.\nlwork\n(local or global) size of the array work; must be at least:\nIf side = 'L',\nlwork≥max((mb_a*(mb_a-1))/2, (mpc0+maxmqa0)+ numroc(numroc(m\n+ iroffc, mb_a, 0, 0, NPROW), mb_a, 0, 0, lcmp), nqc0))*\nmb_a) + mb_a*mb_a\nelse if side = 'R',\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1397\n\n\nlwork≥max((mb_a* (mb_a-1))/2, (mpc0+nqc0)*mb_a + mb_a*mb_a\nend if\nwhere\nlcmp = lcm/NPROW with lcm = ilcm (NPROW, NPCOL),\niroffa = mod(ia-1, mb_a),\nicoffa = mod(ja-1, nb_a),\niacol = indxg2p(ja, nb_a, MYCOL, csrc_a, NPCOL),\nmqa0 = numroc(m+icoffa, nb_a, MYCOL, iacol, NPCOL),\niroffc = mod(ic-1, mb_c),\nicoffc = mod(jc-1, nb_c),\nicrow = indxg2p(ic, mb_c, MYROW, rsrc_c, NPROW),\niccol = indxg2p(jc, nb_c, MYCOL, csrc_c, NPCOL),\nmpc0 = numroc(m+iroffc, mb_c, MYROW, icrow, NPROW),\nnqc0 = numroc(n+icoffc, nb_c, MYCOL, iccol, NPCOL),\nNOTE\nmod(x,y) is the integer remainder of x/y.\nilcm, indxg2p and numroc are ScaLAPACK tool functions; MYROW, MYCOL,\nNPROW and NPCOL can be determined by calling the function\nblacs_gridinfo.\nIf lwork = -1, then lwork is global input and a workspace query is\nassumed; the function only calculates the minimum and optimal size for all\nwork arrays. Each of these values is returned in the first entry of the\ncorresponding work array, and no error message is issued by pxerbla.\nOutput Parameters\nc\nOverwritten by the product Q*sub(C), or Q' *sub (C), or sub(C)*Q', or\nsub(C)*Q\nwork[0]\nOn exit work[0] contains the minimum value of lwork required for\noptimum performance.\ninfo\n(global)\n= 0: the execution is successful.\n< 0: if the i-th argument is an array and the j-th entry, indexed j - 1, had\nan illegal value, then info = -(i*100+j); if the i-th argument is a scalar\nand had an illegal value, then info = -i.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1398\n\n\np?unmlq\nMultiplies a general matrix by the unitary matrix Q of\nthe LQ factorization formed by p?gelqf.\nSyntax\nvoid pcunmlq (char *side , char *trans , MKL_INT *m , MKL_INT *n , MKL_INT *k ,\nMKL_Complex8 *a , MKL_INT *ia , MKL_INT *ja , MKL_INT *desca , MKL_Complex8 *tau ,\nMKL_Complex8 *c , MKL_INT *ic , MKL_INT *jc , MKL_INT *descc , MKL_Complex8 *work ,\nMKL_INT *lwork , MKL_INT *info );\nvoid pzunmlq (char *side , char *trans , MKL_INT *m , MKL_INT *n , MKL_INT *k ,\nMKL_Complex16 *a , MKL_INT *ia , MKL_INT *ja , MKL_INT *desca , MKL_Complex16 *tau ,\nMKL_Complex16 *c , MKL_INT *ic , MKL_INT *jc , MKL_INT *descc , MKL_Complex16 *work ,\nMKL_INT *lwork , MKL_INT *info );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThis function overwrites the general complex m-by-n distributed matrix sub(C) = C(iс:iс+m-1,jс:jс+n-1)\nwith\nside ='L'\nside ='R'\ntrans = 'N':\nQ*sub(C)\nsub(C)*Q\ntrans = 'T':\nQH*sub(C)\nsub(C)*QH\nwhere Q is a complex unitary distributed matrix defined as the product of k elementary reflectors\nQ = H(k)' ... H(2)' H(1)'\nas returned by p?gelqf. Q is of order m if side = 'L' and of order n if side = 'R'.\nInput Parameters\nside\n(global)\n='L': Q or QH is applied from the left.\n='R': Q or QH is applied from the right.\ntrans\n(global)\n='N', no transpose, Q is applied.\n='C', conjugate transpose, QH is applied.\nm\n(global) The number of rows in the distributed matrix sub(C) (m≥0).\nn\n(global) The number of columns in the distributed matrix sub(C)(n≥0).\nk\n(global) The number of elementary reflectors whose product defines the\nmatrix Q. Constraints:\nIf side = 'L', m≥k≥0\nIf side = 'R', n≥k≥0.\na\n(local)\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1399\n\n\nPointer into the local memory to an array of size lld_a*LOCc(ja+m-1), if\nside = 'L' and lld_a*LOCc(ja+n-1), if side = 'R', where lld_a≥\nmax(1, LOCr (ia+k-1)). The i-th column of the matrix stored in amust\ncontain the vector that defines the elementary reflector H(i), ia≤i≤ia+k-1,\nas returned by p?gelqf in the k rows of its distributed matrix argument\nA( ia:ia+k-1, ja:*). A( ia:ia+k-1, ja:*) is modified by the function but\nrestored on exit.\nia, ja\n(global) The row and column indices in the global matrix A indicating the\nfirst row and the first column of the submatrix A, respectively.\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\ntau\n(local)\nArray of size LOCc(ia+k-1).\nContains the scalar factor tau[j] of elementary reflectors H(j+1) as\nreturned by p?gelqf (0 ≤ j < LOCc(ia+k-1)). tau is tied to the\ndistributed matrix A.\nc\n(local)\nPointer into the local memory to an array of local size lld_c*LOCc(jc+n-1).\nContains the local pieces of the distributed matrix sub(C) to be factored.\nic, jc\n(global) The row and column indices in the global matrix C indicating the\nfirst row and the first column of the submatrix C, respectively.\ndescc\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix C.\nwork\n(local)\nWorkspace array of size of lwork.\nlwork\n(local or global) size of the array work; must be at least:\nIf side = 'L',\nlwork≥max((mb_a*(mb_a-1))/2, (mpc0 + maxmqa0)+\nnumroc(numroc(m + iroffc, mb_a, 0, 0, NPROW), mb_a, 0, 0,\nlcmp), nqc0))*mb_a) + mb_a*mb_a\nelse if side = 'R',\nlwork≥max((mb_a* (mb_a-1))/2, (mpc0 + nqc0)*mb_a + mb_a*mb_a\nend if\nwhere\nlcmp = lcm/NPROW with lcm = ilcm (NPROW, NPCOL),\niroffa = mod(ia-1, mb_a),\nicoffa = mod(ja-1, nb_a),\niacol = indxg2p(ja, nb_a, MYCOL, csrc_a, NPCOL),\nmqa0 = numroc(m + icoffa, nb_a, MYCOL, iacol, NPCOL),\niroffc = mod(ic-1, mb_c),\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1400\n\n\nicoffc = mod(jc-1, nb_c),\nicrow = indxg2p(ic, mb_c, MYROW, rsrc_c, NPROW),\niccol = indxg2p(jc, nb_c, MYCOL, csrc_c, NPCOL),\nmpc0 = numroc(m+iroffc, mb_c, MYROW, icrow, NPROW),\nnqc0 = numroc(n+icoffc, nb_c, MYCOL, iccol, NPCOL),\nNOTE\nmod(x,y) is the integer remainder of x/y.\nilcm, indxg2p and numroc are ScaLAPACK tool functions; MYROW, MYCOL,\nNPROW and NPCOL can be determined by calling the function\nblacs_gridinfo.\nIf lwork = -1, then lwork is global input and a workspace query is\nassumed; the function only calculates the minimum and optimal size for all\nwork arrays. Each of these values is returned in the first entry of the\ncorresponding work array, and no error message is issued by pxerbla.\nOutput Parameters\nc\nOverwritten by the product Q*sub(C), or Q'*sub (C), or sub(C)*Q', or\nsub(C)*Q\nwork[0]\nOn exit work[0] contains the minimum value of lwork required for\noptimum performance.\ninfo\n(global)\n= 0: the execution is successful.\n< 0: if the i-th argument is an array and the j-th entry, indexed j - 1, had\nan illegal value, then info = -(i*100+j); if the i-th argument is a scalar\nand had an illegal value, then info = -i.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?geqlf\nComputes the QL factorization of a general matrix.\nSyntax\nvoid psgeqlf (MKL_INT *m , MKL_INT *n , float *a , MKL_INT *ia , MKL_INT *ja , MKL_INT\n*desca , float *tau , float *work , MKL_INT *lwork , MKL_INT *info );\nvoid pdgeqlf (MKL_INT *m , MKL_INT *n , double *a , MKL_INT *ia , MKL_INT *ja , MKL_INT\n*desca , double *tau , double *work , MKL_INT *lwork , MKL_INT *info );\nvoid pcgeqlf (MKL_INT *m , MKL_INT *n , MKL_Complex8 *a , MKL_INT *ia , MKL_INT *ja ,\nMKL_INT *desca , MKL_Complex8 *tau , MKL_Complex8 *work , MKL_INT *lwork , MKL_INT\n*info );\nvoid pzgeqlf (MKL_INT *m , MKL_INT *n , MKL_Complex16 *a , MKL_INT *ia , MKL_INT *ja ,\nMKL_INT *desca , MKL_Complex16 *tau , MKL_Complex16 *work , MKL_INT *lwork , MKL_INT\n*info );\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1401\n\n\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?geqlf function forms the QL factorization of a real/complex distributed m-by-n matrix sub(A)=\nA(ia:ia+m-1, ja:ja+n-1) = Q*L.\nInput Parameters\nm\n(global) The number of rows in the matrix sub(Q); (m≥ 0).\nn\n(global) The number of columns in the matrix sub(Q) (n≥ 0).\na\n(local)\nPointer into the local memory to an array of local size lld_a*LOCc(ja+n-1).\nContains the local pieces of the distributed matrix sub(A) to be factored.\nia, ja\n(global) The row and column indices in the global matrix A indicating the\nfirst row and the first column of the submatrix A(ia:ia+m-1, ja:ja+n-1),\nrespectively.\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\nwork\n(local)\nWorkspace array of size of lwork.\nlwork\n(local or global) size of work, must be at least lwork≥nb_a*(mp0 + nq0 +\nnb_a), where\niroff = mod(ia-1, mb_a),\nicoff = mod(ja-1, nb_a),\niarow = indxg2p(ia, mb_a, MYROW, rsrc_a, NPROW),\niacol = indxg2p(ja, nb_a, MYCOL, csrc_a, NPCOL),\nmp0 = numroc(m+iroff, mb_a, MYROW, iarow, NPROW),\nnq0 = numroc(n+icoff, nb_a, MYCOL, iacol, NPCOL)\nNOTE\nmod(x,y) is the integer remainder of x/y.\nnumroc and indxg2p are ScaLAPACK tool functions; MYROW, MYCOL, NPROW\nand NPCOL can be determined by calling the function blacs_gridinfo.\nIf lwork = -1, then lwork is global input and a workspace query is\nassumed; the function only calculates the minimum and optimal size for all\nwork arrays. Each of these values is returned in the first entry of the\ncorresponding work array, and no error message is issued by pxerbla.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1402\n\n\nOutput Parameters\na\nOn exit, if m≥n, the lower triangle of the distributed submatrix A(ia+m-n:ia\n+m-1, ja:ja+n-1) contains the n-by-n lower triangular matrix L; if m≤n, the\nelements on and below the (n - m)-th superdiagonal contain the m-by-n\nlower trapezoidal matrix L; the remaining elements, with the array tau,\nrepresent the orthogonal/unitary matrix Q as a product of elementary\nreflectors (see Application Notes below).\ntau\n(local)\nArray of size LOCc(ja+n-1).\nContains the scalar factors of elementary reflectors. tau is tied to the\ndistributed matrix A.\nwork[0]\nOn exit, work[0] contains the minimum value of lwork required for\noptimum performance.\ninfo\n(global)\n= 0: the execution is successful.\n< 0: if the i-th argument is an array and the j-th entry, indexed j - 1, had\nan illegal value, then info = -(i*100+j); if the i-th argument is a scalar\nand had an illegal value, then info = -i.\nApplication Notes\nThe matrix Q is represented as a product of elementary reflectors\nQ = H(ja+k-1)*...*H(ja+1)*H(ja)\nwhere k = min(m,n)\nEach H(i) has the form\nH(i) = I - tau*v*v'\nwhere tau is a real/complex scalar, and v is a real/complex vector with v(m-k+i+1:m) = 0 and v(m-k+i) = 1;\nv(1:m-k+i-1) is stored on exit in A(ia:ia+m-k+i-2, ja+n-k+i-1), and tau in tau[ja+n-k+i-2].\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?orgql\nGenerates the orthogonal matrix Q of the QL\nfactorization formed by p?geqlf.\nSyntax\nvoid psorgql (MKL_INT *m , MKL_INT *n , MKL_INT *k , float *a , MKL_INT *ia , MKL_INT\n*ja , MKL_INT *desca , float *tau , float *work , MKL_INT *lwork , MKL_INT *info );\nvoid pdorgql (MKL_INT *m , MKL_INT *n , MKL_INT *k , double *a , MKL_INT *ia , MKL_INT\n*ja , MKL_INT *desca , double *tau , double *work , MKL_INT *lwork , MKL_INT *info );\nInclude Files\n•\nmkl_scalapack.h\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1403\n\n\nDescription\nThe p?orgql function generates the whole or part of m-by-n real distributed matrix Q denoting A(ia:ia\n+m-1,ja:ja+n-1) with orthonormal rows, which is defined as the first m rows of a product of k elementary\nreflectors of order n\nQ = H(k)*...*H(2)*H(1)\nas returned by p?geqlf.\nInput Parameters\nm\n(global) The number of rows in the matrix sub(Q), (m≥0).\nn\n(global) The number of columns in the matrix sub(Q),(m≥n≥0).\nk\n(global) The number of elementary reflectors whose product defines the\nmatrix Q(n≥k≥0).\na\n(local)\nPointer into the local memory to an array of local size lld_a*LOCc(ja+n-1).\nOn entry, the j-th column of the matrix stored in amust contain the vector\nthat defines the elementary reflector H(j),ja+n-k≤j≤ja+n-1, as returned\nby p?geqlf in the k columns of its distributed matrix argument A(ia:*,ja\n+n-k:ja+n-1).\nia, ja\n(global) The row and column indices in the global matrix A indicating the\nfirst row and the first column of the submatrix A(ia:ia+m-1,ja:ja+n-1),\nrespectively.\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\ntau\n(local)\nArray of size LOCc(ja+n-1).\nContains the scalar factors tau[j] of elementary reflectors H(j+1), 0 ≤ j <\nLOCr(ia+n-1). tau is tied to the distributed matrix A.\nwork\n(local)\nWorkspace array of size of lwork.\nlwork\n(local or global) size of work, must be at least\nlwork≥nb_a*(nqa0+mpa0+nb_a), where\niroffa = mod(ia-1, mb_a),\nicoffa = mod(ja-1, nb_a),\niarow = indxg2p(ia, mb_a, MYROW, rsrc_a, NPROW),\niacol = indxg2p(ja, nb_a, MYCOL, csrc_a, NPCOL),\nmpa0 = numroc(m+iroffa, mb_a, MYROW, iarow, NPROW),\nnqa0 = numroc(n+icoffa, nb_a, MYCOL, iacol, NPCOL)\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1404\n\n\nNOTE\nmod(x,y) is the integer remainder of x/y.\nindxg2p and numroc are ScaLAPACK tool functions; MYROW, MYCOL, NPROW\nand NPCOL can be determined by calling the function blacs_gridinfo.\nIf lwork = -1, then lwork is global input and a workspace query is\nassumed; the function only calculates the minimum and optimal size for all\nwork arrays. Each of these values is returned in the first entry of the\ncorresponding work array, and no error message is issued by pxerbla.\nOutput Parameters\na\nContains the local pieces of the m-by-n distributed matrix Q to be factored.\nwork[0]\nOn exit, work[0] contains the minimum value of lwork required for\noptimum performance.\ninfo\n(global)\n= 0: the execution is successful.\n< 0: if the i-th argument is an array and the j-th entry, indexed j - 1, had\nan illegal value, then info = -(i*100+j); if the i-th argument is a scalar\nand had an illegal value, then info = -i.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?ungql\nGenerates the unitary matrix Q of the QL factorization\nformed by p?geqlf.\nSyntax\nvoid pcungql (const MKL_INT *m , const MKL_INT *n , const MKL_INT *k , MKL_Complex8\n*a , const MKL_INT *ia , const MKL_INT *ja , const MKL_INT *desca , const MKL_Complex8\n*tau , MKL_Complex8 *work , const MKL_INT *lwork , MKL_INT *info );\nvoid pzungql (const MKL_INT *m , const MKL_INT *n , const MKL_INT *k , MKL_Complex16\n*a , const MKL_INT *ia , const MKL_INT *ja , const MKL_INT *desca , const MKL_Complex16\n*tau , MKL_Complex16 *work , const MKL_INT *lwork , MKL_INT *info );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThis function generates the whole or part of m-by-n complex distributed matrix Q denoting A(ia:ia\n+m-1,ja:ja+n-1) with orthonormal rows, which is defined as the first n columns of a product of k\nelementary reflectors of order m\nQ = (H(k))H...*(H(2))H*(H(1))H as returned by p?geqlf.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1405\n\n\nInput Parameters\nm\n(global) The number of rows in the matrix sub(Q) (m≥0).\nn\n(global) The number of columns in the matrix sub(Q) (m≥n≥0).\nk\n(global) The number of elementary reflectors whose product defines the\nmatrix Q(n≥k≥0).\na\n(local)\nPointer into the local memory to an array of local size lld_a*LOCc(ja\n+n-1). On entry, the j-th columnof the matrix stored in a must\ncontain the vector that defines the elementary reflector H(j), ja+n-\nk≤ j≤ ja+n-1, as returned by p?geqlf in the k columns of its\ndistributed matrix argument A(ia:*, ja+n-k: ja+n-1).\nia, ja\n(global) The row and column indices in the global matrix A indicating the\nfirst row and the first column of the submatrix A(ia:ia+m-1,ja:ja+n-1),\nrespectively.\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\ntau\n(local)\nArray of size LOCr(ia+n-1).\nContains the scalar factors tau[j] of elementary reflectors H(j+1), 0 ≤ j <\nLOCr(ia+n-1). tau is tied to the distributed matrix A.\nwork\n(local)\nWorkspace array of size of lwork.\nlwork\n(local or global) size of work, must be at least lwork≥nb_a*(nqa0 + mpa0\n+ nb_a), where\niroffa = mod(ia-1, mb_a),\nicoffa = mod(ja-1, nb_a),\niarow = indxg2p(ia, mb_a, MYROW, rsrc_a, NPROW),\niacol = indxg2p(ja, nb_a, MYCOL, csrc_a, NPCOL),\nmpa0 = numroc(m+iroffa, mb_a, MYROW, iarow, NPROW),\nnqa0 = numroc(n+icoffa, nb_a, MYCOL, iacol, NPCOL)\nindxg2p and numroc are ScaLAPACK tool functions; MYROW, MYCOL, NPROW\nand NPCOL can be determined by calling the function blacs_gridinfo.\nIf lwork = -1, then lwork is global input and a workspace query is\nassumed; the function only calculates the minimum and optimal size for all\nwork arrays. Each of these values is returned in the first entry of the\ncorresponding work array, and no error message is issued by pxerbla.\nOutput Parameters\na\nContains the local pieces of the m-by-n distributed matrix Q to be factored.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1406\n\n\nwork[0]\nOn exit, work[0] contains the minimum value of lwork required for\noptimum performance.\ninfo\n(global)\n= 0: the execution is successful.\n< 0: if the i-th argument is an array and the j-th entry, indexed j - 1, had\nan illegal value, then info = -(i*100+j); if the i-th argument is a scalar\nand had an illegal value, then info = -i.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?ormql\nMultiplies a general matrix by the orthogonal matrix Q\nof the QL factorization formed by p?geqlf.\nSyntax\nvoid psormql (char *side , char *trans , MKL_INT *m , MKL_INT *n , MKL_INT *k , float\n*a , MKL_INT *ia , MKL_INT *ja , MKL_INT *desca , float *tau , float *c , MKL_INT *ic ,\nMKL_INT *jc , MKL_INT *descc , float *work , MKL_INT *lwork , MKL_INT *info );\nvoid pdormql (char *side , char *trans , MKL_INT *m , MKL_INT *n , MKL_INT *k , double\n*a , MKL_INT *ia , MKL_INT *ja , MKL_INT *desca , double *tau , double *c , MKL_INT\n*ic , MKL_INT *jc , MKL_INT *descc , double *work , MKL_INT *lwork , MKL_INT *info );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?ormqlfunction overwrites the general real m-by-n distributed matrix sub(C) = C(iс:iс+m-1,jс:jс\n+n-1) with\nside ='L'\nside ='R'\ntrans = 'N':\nQ*sub(C)\nsub(C)*Q\ntrans = 'T':\nQT*sub(C)\nsub(C)*QT\nwhere Q is a real orthogonal distributed matrix defined as the product of k elementary reflectors\nQ = H(k)' ... H(2)' H(1)'\nas returned by p?geqlf. Q is of order m if side = 'L' and of order n if side = 'R'.\nInput Parameters\nside\n(global)\n='L': Q or QT is applied from the left.\n='R': Q or QT is applied from the right.\ntrans\n(global)\n='N', no transpose, Q is applied.\n='T', transpose, QT is applied.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1407\n\n\nm\n(global) The number of rows in the distributed matrix sub(C), (m≥0).\nn\n(global) The number of columns in the distributed matrix sub(C), (n≥0).\nk\n(global) The number of elementary reflectors whose product defines the\nmatrix Q. Constraints:\nIf side = 'L', m≥k≥0\nIf side = 'R', n≥k≥0.\na\n(local)\nPointer into the local memory to an array of size lld_a*LOCc(ja+k-1). The\nj-th column of the matrix stored in amust contain the vector that defines\nthe elementary reflector H(j), ja≤j≤ja+k-1, as returned by p?gelqf in the\nk columns of its distributed matrix argument A(ia:*, ja:ja+k-1). A(ia:*,\nja:ja+k-1) is modified by the function but restored on exit.\nIf side = 'L',lld_a ≥ max(1, LOCr(ia+m-1)),\nIf side = 'R', lld_a ≥ max(1, LOCr(ia+n-1)).\nia, ja\n(global) The row and column indices in the global matrix A indicating the\nfirst row and the first column of the submatrix A, respectively.\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\ntau\n(local)\nArray of size LOCc(ja+n-1).\nContains the scalar factor tau[j] of elementary reflectors H(j+1) as\nreturned by p?geqlf (0 ≤ j < LOCc(ja+k-1)). tau is tied to the\ndistributed matrix A.\nc\n(local)\nPointer into the local memory to an array of local size lld_c*LOCc(jc+n-1).\nContains the local pieces of the distributed matrix sub(C) to be factored.\nic, jc\n(global) The row and column indices in the global matrix C indicating the\nfirst row and the first column of the submatrix C, respectively.\ndescc\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix C.\nwork\n(local)\nWorkspace array of size of lwork.\nlwork\n(local or global) dimension of work, must be at least:\nIf side = 'L',\nlwork≥max((nb_a*(nb_a-1))/2, (nqc0+mpc0)*nb_a + nb_a*nb_a\nelse if side ='R',\nlwork≥max((nb_a*(nb_a-1))/2, (nqc0+max(npa0 +\nnumroc(numroc(n+icoffc, nb_a, 0, 0, NPCOL), nb_a, 0, 0,\nlcmq), mpc0))*nb_a) + nb_a*nb_a\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1408\n\n\nend if\nwhere\nlcmq = lcm/NPCOL with lcm = ilcm (NPROW, NPCOL),\niroffa = mod(ia-1, mb_a),\nicoffa = mod(ja-1, nb_a),\niarow = indxg2p(ia, mb_a, MYROW, rsrc_a, NPROW),\nnpa0= numroc(n + iroffa, mb_a, MYROW, iarow, NPROW),\niroffc = mod(ic-1, mb_c),\nicoffc = mod(jc-1, nb_c),\nicrow = indxg2p(ic, mb_c, MYROW, rsrc_c, NPROW),\niccol = indxg2p(jc, nb_c, MYCOL, csrc_c, NPCOL),\nmpc0 = numroc(m+iroffc, mb_c, MYROW, icrow, NPROW),\nnqc0 = numroc(n+icoffc, nb_c, MYCOL, iccol, NPCOL),\nNOTE\nmod(x,y) is the integer remainder of x/y.\nilcm, indxg2p and numroc are ScaLAPACK tool functions; MYROW, MYCOL,\nNPROW and NPCOL can be determined by calling the function\nblacs_gridinfo.\nIf lwork = -1, then lwork is global input and a workspace query is\nassumed; the function only calculates the minimum and optimal size for all\nwork arrays. Each of these values is returned in the first entry of the\ncorresponding work array, and no error message is issued by pxerbla.\nOutput Parameters\nc\nOverwritten by the product Q* sub(C), or Q'*sub (C), or sub(C)* Q', or\nsub(C)* Q\nwork[0]\nOn exit work[0] contains the minimum value of lwork required for\noptimum performance.\ninfo\n(global)\n= 0: the execution is successful.\n< 0: if the i-th argument is an array and the j-th entry, indexed j - 1, had\nan illegal value, then info = -(i*100+j); if the i-th argument is a scalar\nand had an illegal value, then info = -i.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?unmql\nMultiplies a general matrix by the unitary matrix Q of\nthe QL factorization formed by p?geqlf.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1409\n\n\nSyntax\nvoid pcunmql (char *side , char *trans , MKL_INT *m , MKL_INT *n , MKL_INT *k ,\nMKL_Complex8 *a , MKL_INT *ia , MKL_INT *ja , MKL_INT *desca , MKL_Complex8 *tau ,\nMKL_Complex8 *c , MKL_INT *ic , MKL_INT *jc , MKL_INT *descc , MKL_Complex8 *work ,\nMKL_INT *lwork , MKL_INT *info );\nvoid pzunmql (char *side , char *trans , MKL_INT *m , MKL_INT *n , MKL_INT *k ,\nMKL_Complex16 *a , MKL_INT *ia , MKL_INT *ja , MKL_INT *desca , MKL_Complex16 *tau ,\nMKL_Complex16 *c , MKL_INT *ic , MKL_INT *jc , MKL_INT *descc , MKL_Complex16 *work ,\nMKL_INT *lwork , MKL_INT *info );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThis function overwrites the general complex m-by-n distributed matrix sub(C) = C(iс:iс+m-1,jс:jс+n-1)\nwith\nside ='L'\nside ='R'\ntrans = 'N':\nQ*sub(C)\nsub(C)*Q\ntrans = 'C':\nQH*sub(C)\nsub(C)*QH\nwhere Q is a complex unitary distributed matrix defined as the product of k elementary reflectors\nQ = H(k)' ... H(2)' H(1)'\nas returned by p?geqlf. Q is of order m if side = 'L' and of order n if side = 'R'.\nInput Parameters\nside\n(global)\n='L': Q or QH is applied from the left.\n='R': Q or QH is applied from the right.\ntrans\n(global)\n='N', no transpose, Q is applied.\n='C', conjugate transpose, QH is applied.\nm\n(global) The number of rows in the distributed matrix sub(C) (m≥0).\nn\n(global) The number of columns in the distributed matrix sub(C)(n≥0).\nk\n(global) The number of elementary reflectors whose product defines the\nmatrix Q. Constraints:\nIf side = 'L', m≥k≥0\nIf side = 'R', n≥k≥0.\na\n(local)\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1410\n\n\nPointer into the local memory to an array of size lld_a*LOCc(ja+k-1). The\nj-th column of the matrix stored in amust contain the vector that defines\nthe elementary reflector H(j), ja≤j≤ja+k-1, as returned by p?geqlf in the\nk columns of its distributed matrix argument A(ia:*, ja:ja+k-1). A(ia:*,\nja:ja+k-1) is modified by the function but restored on exit.\nIf side = 'L',lld_a ≥ max(1, LOCr(ia+m-1)),\nIf side = 'R', lld_a ≥ max(1, LOCr(ia+n-1)).\nia, ja\n(global) The row and column indices in the global matrix A indicating the\nfirst row and the first column of the submatrix A, respectively.\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\ntau\n(local)\nArray of size LOCc(ia+n-1).\nContains the scalar factor tau[j] of elementary reflectors H(j+1) as\nreturned by p?geqlf (0 ≤ j < LOCc(ia+n-1)). tau is tied to the\ndistributed matrix A.\nc\n(local)\nPointer into the local memory to an array of local size lld_c*LOCc(jc+n-1).\nContains the local pieces of the distributed matrix sub(C) to be factored.\nic, jc\n(global) The row and column indices in the global matrix C indicating the\nfirst row and the first column of the submatrix C, respectively.\ndescc\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix C.\nwork\n(local)\nWorkspace array of size of lwork.\nlwork\n(local or global) size of work, must be at least:\nIf side = 'L',\nlwork≥max((nb_a* (nb_a-1))/2, (nqc0+mpc0)*nb_a + nb_a*nb_a\nelse if side ='R',\nlwork≥max((nb_a*(nb_a-1))/2, (nqc0+maxnpa0)+ numroc(numroc(n\n+icoffc, nb_a, 0, 0, NPCOL), nb_a, 0, 0, lcmq), mpc0))*nb_a)\n+ nb_a*nb_a\nend if\nwhere\nlcmp = lcm/NPCOL with lcm = ilcm (NPROW, NPCOL),\niroffa = mod(ia-1, mb_a),\nicoffa = mod(ja-1, nb_a),\niarow = indxg2p(ia, mb_a, MYROW, rsrc_a, NPROW),\nnpa0 = numroc (n + iroffa, mb_a, MYROW, iarow, NPROW),\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1411\n\n\niroffc = mod(ic-1, mb_c),\nicoffc = mod(jc-1, nb_c),\nicrow = indxg2p(ic, mb_c, MYROW, rsrc_c, NPROW),\niccol = indxg2p(jc, nb_c, MYCOL, csrc_c, NPCOL),\nmpc0 = numroc(m+iroffc, mb_c, MYROW, icrow, NPROW),\nnqc0 = numroc(n+icoffc, nb_c, MYCOL, iccol, NPCOL),\nNOTE\nmod(x,y) is the integer remainder of x/y.\nilcm, indxg2p and numroc are ScaLAPACK tool functions; MYROW, MYCOL,\nNPROW and NPCOL can be determined by calling the function\nblacs_gridinfo.\nNOTE\nmod(x,y) is the integer remainder of x/y.\nIf lwork = -1, then lwork is global input and a workspace query is\nassumed; the function only calculates the minimum and optimal size for all\nwork arrays. Each of these values is returned in the first entry of the\ncorresponding work array, and no error message is issued by pxerbla.\nOutput Parameters\nc\nOverwritten by the product Q* sub(C), or Q' sub (C), or sub(C)* Q', or\nsub(C)* Q\nwork[0]\nOn exit work[0] contains the minimum value of lwork required for\noptimum performance.\ninfo\n(global)\n= 0: the execution is successful.\n< 0: if the i-th argument is an array and the j-th entry, indexed j - 1, had\nan illegal value, then info = -(i*100+j); if the i-th argument is a scalar\nand had an illegal value, then info = -i.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?gerqf\nComputes the RQ factorization of a general\nrectangular matrix.\nSyntax\nvoid psgerqf (MKL_INT *m , MKL_INT *n , float *a , MKL_INT *ia , MKL_INT *ja , MKL_INT\n*desca , float *tau , float *work , MKL_INT *lwork , MKL_INT *info );\nvoid pdgerqf (MKL_INT *m , MKL_INT *n , double *a , MKL_INT *ia , MKL_INT *ja , MKL_INT\n*desca , double *tau , double *work , MKL_INT *lwork , MKL_INT *info );\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1412\n\n\nvoid pcgerqf (MKL_INT *m , MKL_INT *n , MKL_Complex8 *a , MKL_INT *ia , MKL_INT *ja ,\nMKL_INT *desca , MKL_Complex8 *tau , MKL_Complex8 *work , MKL_INT *lwork , MKL_INT\n*info );\nvoid pzgerqf (MKL_INT *m , MKL_INT *n , MKL_Complex16 *a , MKL_INT *ia , MKL_INT *ja ,\nMKL_INT *desca , MKL_Complex16 *tau , MKL_Complex16 *work , MKL_INT *lwork , MKL_INT\n*info );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?gerqf function forms the QR factorization of a general m-by-n distributed matrix sub(A)= A(ia:ia\n+m-1, ja:ja+n-1) as\nA= R*Q\nInput Parameters\nm\n(global) The number of rows in the distributed matrix sub(A); (m≥0).\nn\n(global) The number of columns in the distributed matrix sub(A); (n≥0).\na\n(local)\nPointer into the local memory to an array of local size lld_a*LOCc(ja+n-1).\nContains the local pieces of the distributed matrix sub(A) to be factored.\nia, ja\n(global) The row and column indices in the global matrix A indicating the\nfirst row and the first column of the submatrix A(ia:ia+m-1, ja:ja+n-1),\nrespectively.\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A\nwork\n(local).\nWorkspace array of size lwork.\nlwork\n(local or global) size of work, must be at least\nlwork≥mb_a*(mp0+nq0+mb_a), where\niroff = mod(ia-1, mb_a),\nicoff = mod(ja-1, nb_a),\niarow = indxg2p(ia, mb_a, MYROW, rsrc_a, NPROW),\niacol = indxg2p(ja, nb_a, MYCOL, csrc_a, NPCOL),\nmp0 = numroc(m+iroff, mb_a, MYROW, iarow, NPROW),\nNOTE\nmod(x,y) is the integer remainder of x/y.\nnq0 = numroc(n+icoff, nb_a, MYCOL, iacol, NPCOL) and numroc,\nindxg2p are ScaLAPACK tool functions; MYROW, MYCOL, NPROW and NPCOL\ncan be determined by calling the function blacs_gridinfo.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1413\n\n\nIf lwork = -1, then lwork is global input and a workspace query is\nassumed; the function only calculates the minimum and optimal size for all\nwork arrays. Each of these values is returned in the first entry of the\ncorresponding work array, and no error message is issued by pxerbla.\nOutput Parameters\na\nOn exit, if m≤n, the upper triangle of A(ia:ia+m-1, ja:ja+n-1) contains the\nm-by-m upper triangular matrix R; if m≥n, the elements on and above the (m\n- n)-th subdiagonal contain the m-by-n upper trapezoidal matrix R; the\nremaining elements, with the array tau, represent the orthogonal/unitary\nmatrix Q as a product of elementary reflectors (see Application Notes\nbelow).\ntau\n(local)\nArray of size LOCr(ia+m-1).\nContains the scalar factor of elementary reflectors. tau is tied to the\ndistributed matrix A.\nwork[0]\nOn exit, work[0] contains the minimum value of lwork required for\noptimum performance.\ninfo\n(global)\n= 0, the execution is successful.\n< 0, if the i-th argument is an array and the j-th entry, indexed j - 1, had\nan illegal value, then info = -(i*100+j); if the i-th argument is a scalar\nand had an illegal value, then info = -i.\nApplication Notes\nThe matrix Q is represented as a product of elementary reflectors\nQ = H(ia)*H(ia+1)*...*H(ia+k-1),\nwhere k = min(m,n).\nEach H(i) has the form\nH(i) = I - tau*v*v'\nwhere tau is a real/complex scalar, and v is a real/complex vector with v(n-k+i+1:n) = 0 and v(n-k+i) = 1;\nv(1:n-k+i-1) is stored on exit in A(ia+m-k+i-1,ja:ja+n-k+i-2), and tau in tau[ia+m-k+i-2].\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?orgrq\nGenerates the orthogonal matrix Q of the RQ\nfactorization formed by p?gerqf.\nSyntax\nvoid psorgrq (MKL_INT *m , MKL_INT *n , MKL_INT *k , float *a , MKL_INT *ia , MKL_INT\n*ja , MKL_INT *desca , float *tau , float *work , MKL_INT *lwork , MKL_INT *info );\nvoid pdorgrq (MKL_INT *m , MKL_INT *n , MKL_INT *k , double *a , MKL_INT *ia , MKL_INT\n*ja , MKL_INT *desca , double *tau , double *work , MKL_INT *lwork , MKL_INT *info );\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1414\n\n\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?orgrqfunction generates the whole or part of m-by-n real distributed matrix Q denoting A(ia:ia\n+m-1,ja:ja+n-1) with orthonormal rows that is defined as the last m rows of a product of k elementary\nreflectors of order n\nQ= H(1)*H(2)*...*H(k)\nas returned by p?gerqf.\nInput Parameters\nm\n(global) The number of rows in the matrix sub(Q), (m≥0).\nn\n(global) The number of columns in the matrix sub(Q), (n≥m≥0).\nk\n(global) The number of elementary reflectors whose product defines the\nmatrix Q(m≥k≥0).\na\n(local)\nPointer into the local memory to an array of local size lld_a*LOCc(ja+n-1).\nThe i-th row of the matrix stored in amust contain the vector that defines\nthe elementary reflector H(i), ia≤i≤ia+m-1, as returned by p?gerqf in the\nk rows of its distributed matrix argument A(ia+m-k:ia+m-1, ja:*).\nia, ja\n(global) The row and column indices in the global matrix A indicating the\nfirst row and the first column of the submatrix A, respectively.\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\ntau\n(local)\nArray of size LOCc(ja+k-1).\nContains the scalar factor tau[i] of elementary reflectors H(i+1) as\nreturned by p?gerqf, 0 ≤ i < LOCr(ja+k-1). tau is tied to the distributed\nmatrix A.\nwork\n(local)\nWorkspace array of size of lwork.\nlwork\n(local or global) size of work, must be at least lwork≥mb_a*(mpa0 + nqa0\n+ mb_a), where\niroffa = mod(ia-1, mb_a),\nicoffa = mod(ja-1, nb_a),\niarow = indxg2p(ia, mb_a, MYROW, rsrc_a, NPROW),\niacol = indxg2p(ja, nb_a, MYCOL, csrc_a, NPCOL),\nmpa0 = numroc(m+iroffa, mb_a, MYROW, iarow, NPROW),\nnqa0 = numroc(n+icoffa, nb_a, MYCOL, iacol, NPCOL)\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1415\n\n\nindxg2p and numroc are ScaLAPACK tool functions; MYROW, MYCOL, NPROW\nand NPCOL can be determined by calling the function blacs_gridinfo.\nNOTE\nmod(x,y) is the integer remainder of x/y.\nIf lwork = -1, then lwork is global input and a workspace query is\nassumed; the function only calculates the minimum and optimal size for all\nwork arrays. Each of these values is returned in the first entry of the\ncorresponding work array, and no error message is issued by pxerbla.\nOutput Parameters\na\nContains the local pieces of the m-by-n distributed matrix Q.\nwork[0]\nOn exit, work[0] contains the minimum value of lwork required for\noptimum performance.\ninfo\n(global)\n= 0: the execution is successful.\n< 0: if the i-th argument is an array and the j-th entry, indexed j - 1, had\nan illegal value, then info = -(i*100+j); if the i-th argument is a scalar\nand had an illegal value, then info = -i.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?ungrq\nGenerates the unitary matrix Q of the RQ factorization\nformed by p?gerqf.\nSyntax\nvoid pcungrq (MKL_INT *m , MKL_INT *n , MKL_INT *k , MKL_Complex8 *a , MKL_INT *ia ,\nMKL_INT *ja , MKL_INT *desca , MKL_Complex8 *tau , MKL_Complex8 *work , MKL_INT\n*lwork , MKL_INT *info );\nvoid pzungrq (MKL_INT *m , MKL_INT *n , MKL_INT *k , MKL_Complex16 *a , MKL_INT *ia ,\nMKL_INT *ja , MKL_INT *desca , MKL_Complex16 *tau , MKL_Complex16 *work , MKL_INT\n*lwork , MKL_INT *info );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThis function generates the m-by-n complex distributed matrix Q denoting A(ia:ia+m-1,ja:ja+n-1) with\northonormal rows, which is defined as the last m rows of a product of k elementary reflectors of order n\nQ = (H(1))H*(H(2))H*...*(H(k))H as returned by p?gerqf.\nInput Parameters\nm\n(global) The number of rows in the matrix sub(Q); (m≥0).\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1416\n\n\nn\n(global) The number of columns in the matrix sub(Q) (n≥m≥0).\nk\n(global) The number of elementary reflectors whose product defines the\nmatrix Q(m≥k≥0).\na\n(local)\nPointer into the local memory to an array of size lld_a*LOCc(ja+n-1). The\ni-th row of the matrix stored in amust contain the vector that defines the\nelementary reflector H(i), ia+m-k≤i≤ia+m-1, as returned by p?gerqf in\nthe k rows of its distributed matrix argument A(ia+m-k:ia+m-1, ja:*).\nia, ja\n(global) The row and column indices in the global matrix A indicating the\nfirst row and the first column of the submatrix A, respectively.\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\ntau\n(local)\nArray of size LOCr(ia+m-1).\nContains the scalar factor tau[i] of elementary reflectors H(i+1) as\nreturned by p?gerqf, 0 ≤ i < LOCr(ia+m-1). tau is tied to the distributed\nmatrix A.\nwork\n(local)\nWorkspace array of size of lwork.\nlwork\n(local or global) size of work, must be at least lwork≥mb_a*(mpa0\n+nqa0+mb_a), where\niroffa = mod(ia-1, mb_a),\nicoffa = mod(ja-1, nb_a),\niarow = indxg2p(ia, mb_a, MYROW, rsrc_a, NPROW),\niacol = indxg2p(ja, nb_a, MYCOL, csrc_a, NPCOL),\nmpa0 = numroc(m+iroffa, mb_a, MYROW, iarow, NPROW),\nnqa0 = numroc(n+icoffa, nb_a, MYCOL, iacol, NPCOL)\nNOTE\nmod(x,y) is the integer remainder of x/y.\nindxg2p and numroc are ScaLAPACK tool functions; MYROW, MYCOL, NPROW\nand NPCOL can be determined by calling the function blacs_gridinfo.\nIf lwork = -1, then lwork is global input and a workspace query is\nassumed; the function only calculates the minimum and optimal size for all\nwork arrays. Each of these values is returned in the first entry of the\ncorresponding work array, and no error message is issued by pxerbla.\nOutput Parameters\na\nContains the local pieces of the m-by-n distributed matrix Q.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1417\n\n\nwork[0]\nOn exit work[0] contains the minimum value of lwork required for\noptimum performance.\ninfo\n(global)\n= 0: the execution is successful.\n< 0: if the i-th argument is an array and the j-th entry, indexed j - 1, had\nan illegal value, then info = -(i*100+j); if the i-th argument is a scalar\nand had an illegal value, then info = -i.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?ormr3\nApplies an orthogonal distributed matrix to a general\nm-by-n distributed matrix.\nSyntax\nvoid psormr3 (const char* side, const char* trans, const MKL_INT* m, const MKL_INT* n,\nconst MKL_INT* k, const MKL_INT* l, const float* a, const MKL_INT* ia, const MKL_INT*\nja, const MKL_INT* desca, const float* tau, float* c, const MKL_INT* ic, const MKL_INT*\njc, const MKL_INT* descc, float* work, const MKL_INT* lwork, MKL_INT* info);\nvoid pdormr3 (const char* side, const char* trans, const MKL_INT* m, const MKL_INT* n,\nconst MKL_INT* k, const MKL_INT* l, const double* a, const MKL_INT* ia, const MKL_INT*\nja, const MKL_INT* desca, const double* tau, double* c, const MKL_INT* ic, const\nMKL_INT* jc, const MKL_INT* descc, double* work, const MKL_INT* lwork, MKL_INT* info);\nInclude Files\n•\nmkl_scalapack.h\nDescription\np?ormr3 overwrites the general real m-by-n distributed matrix sub( C ) = C(ic:ic+m-1,jc:jc+n-1) with\nside = 'L'\nside = 'R'\ntrans = 'N'\nQ * sub( C )\nsub( C ) * Q\ntrans = 'T'\nQT * sub( C )\nQ * sub( C )\nsub( C ) * QT\nwhere Q is a real orthogonal distributed matrix defined as the product of k elementary reflectors\nQ = H(1) H(2) . . . H(k)\nas returned by p?tzrzf. Q is of order m if side = 'L' and of order n if side = 'R'.\nInput Parameters\nside\n(global)\n= 'L': apply Q or QT from the Left;\n= 'R': apply Q or QT from the Right.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1418\n\n\ntrans\n(global)\n= 'N': No transpose, apply Q;\n= 'T': Transpose, apply QT.\nm\n(global)\nThe number of rows to be operated on i.e the number of rows of the\ndistributed submatrix sub( C ). m >= 0.\nn\n(global)\nThe number of columns to be operated on i.e the number of columns of the\ndistributed submatrix sub( C ). n >= 0.\nk\n(global)\nThe number of elementary reflectors whose product defines the matrix Q.\nIf side = 'L', m >= k >= 0,\nif side = 'R', n >= k >= 0.\nl\n(global)\nThe columns of the distributed submatrix sub( A ) containing the\nmeaningful part of the Householder reflectors.\nIf side = 'L', m >= l >= 0,\nif side = 'R', n >= l >= 0.\na\n(local)\nPointer into the local memory to an array of size lld_a*LOCc(ja+m-1) if\nside='L', and lld_a*LOCc(ja+n-1) if side='R', where lld_a >=\nMAX(1,LOCr(ia+k-1));\nOn entry, the i-th row must contain the vector which defines the elementary\nreflector H(i), ia <= i <= ia+k-1, as returned by p?tzrzf in the k rows of\nits distributed matrix argument A(ia:ia+k-1,ja:*).\nA(ia:ia+k-1,ja:*) is modified by the routine but restored on exit.\nia\n(global)\nThe row index in the global array a indicating the first row of sub( A ).\nja\n(global)\nThe column index in the global array a indicating the first column of\nsub( A ).\ndesca\n(global and local)\nArray of size dlen_.\nThe array descriptor for the distributed matrix A.\ntau\n(local)\nArray, size LOCc(ia+k-1).\nThis array contains the scalar factors tau(i) of the elementary reflectors\nH(i) as returned by p?tzrzf. tau is tied to the distributed matrix A.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1419\n\n\nc\n(local)\nPointer into the local memory to an array of size lld_c*LOCc(jc+n-1) .\nOn entry, the local pieces of the distributed matrix sub( C ).\nic\n(global)\nThe row index in the global array c indicating the first row of sub( C ).\njc\n(global)\nThe column index in the global array c indicating the first column of\nsub( C ).\ndescc\n(global and local)\nArray of size dlen_.\nThe array descriptor for the distributed matrix C.\nwork\n(local)\nArray, size (lwork)\nlwork\n(local)\nThe size of the array work.\nlwork is local input and must be at least\nIf side = 'L', lwork >= MpC0 + MAX( MAX( 1, NqC0 ), numroc( numroc( m\n+IROFFC,mb_a,0,0,NPROW ),mb_a,0,0,NqC0 ) );\nif side = 'R', lwork >= NqC0 + MAX( 1, MpC0 );\nwhere LCMP = LCM / NPROW\nLCM = iclm( NPROW, NPCOL ),\nIROFFC = MOD( ic-1, mb_c ),\nICOFFC = MOD( jc-1, nb_c),\nICROW = indxg2p( ic, mb_c, MYROW, rsrc_c, NPROW ),\nICCOL = indxg2p( jc, nb_c, MYCOL, csrc_c, NPCOL ),\nMpC0 = numroc( m+IROFFC, mb_c, MYROW, ICROW, NPROW ),\nNqC0 = numroc( n+ICOFFC, nb_c, MYCOL, ICCOL, NPCOL ),\nilcm, indxg2p, and numroc are ScaLAPACK tool functions;\nMYROW, MYCOL, NPROW and NPCOL can be determined by calling the\nsubroutine blacs_gridinfo.\nIf lwork = -1, then lwork is global input and a workspace query is\nassumed; the routine only calculates the minimum and optimal size for all\nwork arrays. Each of these values is returned in the first entry of the\ncorresponding work array, and no error message is issued by pxerbla.\nOutput Parameters\nc\nOn exit, sub( C ) is overwritten by Q*sub( C ) or Q'*sub( C ) or\nsub( C )*Q' or sub( C )*Q.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1420\n\n\nwork\nOn exit, work[0] returns the minimal and optimal lwork.\ninfo\n(local)\n= 0: successful exit\n< 0: If the i-th argument is an array and the j-th entry had an illegal\nvalue, then info = -(i*100+j), if the i-th argument is a scalar and\nhad an illegal value, then info = -i.\nApplication Notes\nAlignment requirements\nThe distributed submatrices A(ia:*, ja:*) and C(ic:ic+m-1,jc:jc+n-1) must verify some alignment\nproperties, namely the following expressions should be true:\nIf side = 'L',\n( nb_a = mb_c .AND. ICOFFA = IROFFC )\nIf side = 'R',\n( nb_a = nb_c .AND. ICOFFA = ICOFFC .AND. IACOL = ICCOL )\np?unmr3\nApplies an orthogonal distributed matrix to a general\nm-by-n distributed matrix.\nSyntax\nvoid pcunmr3 (const char* side, const char* trans, const MKL_INT* m, const MKL_INT* n,\nconst MKL_INT* k, const MKL_INT* l, const MKL_Complex8* a, const MKL_INT* ia, const\nMKL_INT* ja, const MKL_INT* desca, const MKL_Complex8* tau, MKL_Complex8* c, const\nMKL_INT* ic, const MKL_INT* jc, const MKL_INT* descc, MKL_Complex8* work, const\nMKL_INT* lwork, MKL_INT* info);\nvoid pzunmr3 (const char* side, const char* trans, const MKL_INT* m, const MKL_INT* n,\nconst MKL_INT* k, const MKL_INT* l, const MKL_Complex16* a, const MKL_INT* ia, const\nMKL_INT* ja, const MKL_INT* desca, const MKL_Complex16* tau, MKL_Complex16* c, const\nMKL_INT* ic, const MKL_INT* jc, const MKL_INT* descc, MKL_Complex16* work, const\nMKL_INT* lwork, MKL_INT* info);\nInclude Files\n•\nmkl_scalapack.h\nDescription\np?unmr3 overwrites the general complex m-by-n distributed matrix sub( C ) = C(ic:ic+m-1,jc:jc+n-1) with\n                      side = 'L'             side = 'R'\ntrans = 'N': Q * sub( C )         sub( C ) * Q\ntrans = 'C': QH * sub( C )       sub( C ) * QH\nwhere Q is a complex unitary distributed matrix defined as the product of k elementary reflectors\nQ = H(1)' H(2)' . . . H(k)'\nas returned by p?tzrzf. Q is of order m if side = 'L' and of order n if side = 'R'.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1421\n\n\nInput Parameters\nside\n(global)\n= 'L': apply Q or QH from the Left;\n= 'R': apply Q or QH from the Right.\ntrans\n(global)\n= 'N': No transpose, apply Q;\n= 'C': Conjugate transpose, apply QH.\nm\n(global)\nThe number of rows to be operated on i.e the number of rows of the\ndistributed submatrix sub( C ). m >= 0.\nn\n(global)\nThe number of columns to be operated on i.e the number of columns of the\ndistributed submatrix sub( C ). n >= 0.\nk\n(global)\nThe number of elementary reflectors whose product defines the matrix Q.\nIf side = 'L', m >= k >= 0, if side = 'R', n >= k >= 0.\nl\n(global)\nThe columns of the distributed submatrix sub( A ) containing the\nmeaningful part of the Householder reflectors.\nIf side = 'L', m >= l >= 0, if side = 'R', n >= l >= 0.\na\n(local)\nPointer into the local memory to an array of size lld_a*LOCc(ja+m-1) if\nside='L', and lld_a*LOCc(ja+n-1) if side='R', where lld_a >=\nMAX(1,LOCr(ia+k-1));\nOn entry, the i-th row must contain the vector which defines the elementary\nreflector H(i), ia <= i <= ia+k-1, as returned by p?tzrzf in the k rows of\nits distributed matrix argument A(ia:ia+k-1,ja:*).\nA(ia:ia+k-1,ja:*) is modified by the routine but restored on exit.\nia\n(global)\nThe row index in the global array a indicating the first row of sub( A ).\nja\n(global)\nThe column index in the global array a indicating the first column of\nsub( A ).\ndesca\n(global and local)\nArray of size dlen_.\nThe array descriptor for the distributed matrix A.\ntau\n(local)\nArray, size LOCc(ia+k-1).\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1422\n\n\nThis array contains the scalar factors tau(i) of the elementary reflectors\nH(i) as returned by p?tzrzf. tau is tied to the distributed matrix A.\nc\n(local)\nPointer into the local memory to an array of size lld_c*LOCc(jc+n-1) .\nOn entry, the local pieces of the distributed matrix sub( C ).\nic\n(global)\nThe row index in the global array c indicating the first row of sub( C ).\njc\n(global)\nThe column index in the global array c indicating the first column of\nsub( C ).\ndescc\n(global and local)\nArray of size dlen_.\nThe array descriptor for the distributed matrix C.\nwork\n(local)\nArray, size (lwork)\nOn exit, work(1) returns the minimal and optimal lwork.\nlwork\n(local or global)\nThe size of the array work.\nlwork is local input and must be at least\nIf side = 'L', lwork >= MpC0 + MAX( MAX( 1, NqC0 ), numroc( numroc( m\n+IROFFC,mb_a,0,0,NPROW ),mb_a,0,0,LCMP ) );\nif side = 'R', lwork >= NqC0 + MAX( 1, MpC0 );\nwhere LCMP = LCM / NPROW with LCM = ICLM( NPROW, NPCOL ),\nIROFFC = MOD( ic-1, MB_C ), ICOFFC = MOD( jc-1, nb_c ),\nICROW = indxg2p( ic, MB_C, MYROW, rsrc_c, NPROW ),\nICCOL = indxg2p( jc, nb_c, MYCOL, csrc_c, NPCOL ),\nMpC0 = numroc( m+IROFFC, MB_C, MYROW, ICROW, NPROW ),\nNqC0 = numroc( n+ICOFFC, nb_c, MYCOL, ICCOL, NPCOL ),\nilcm, indxg2p, and numroc are ScaLAPACK tool functions;\nMYROW, MYCOL, NPROW and NPCOL can be determined by calling the\nsubroutine blacs_gridinfo.\nIf lwork = -1, then lwork is global input and a workspace query is\nassumed; the routine only calculates the minimum and optimal size for all\nwork arrays. Each of these values is returned in the first entry of the\ncorresponding work array, and no error message is issued by pxerbla.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1423\n\n\nOutput Parameters\nc\nOn exit, sub( C ) is overwritten by Q*sub( C ) or Q'*sub( C ) or\nsub( C )*Q' or sub( C )*Q.\nwork\n(local)\nArray, size (lwork)\nOn exit, work[0] returns the minimal and optimal lwork.\ninfo\n(local)\n= 0: successful exit\n< 0: If the i-th argument is an array and the j-th entry had an illegal\nvalue, then info = -(i*100+j), if the i-th argument is a scalar and\nhad an illegal value, then info = -i.\nApplication Notes\nAlignment requirements\nThe distributed submatrices A(ia:*, ja:*) and C(ic:ic+m-1,jc:jc+n-1) must verify some alignment\nproperties, namely the following expressions should be true:\nIf side = 'L', ( nb_a = MB_C and ICOFFA = IROFFC )\nIf side = 'R', ( nb_a = nb_c and ICOFFA = ICOFFC and IACOL = ICCOL )\np?ormrq\nMultiplies a general matrix by the orthogonal matrix Q\nof the RQ factorization formed by p?gerqf.\nSyntax\nvoid psormrq (char *side , char *trans , MKL_INT *m , MKL_INT *n , MKL_INT *k , float\n*a , MKL_INT *ia , MKL_INT *ja , MKL_INT *desca , float *tau , float *c , MKL_INT *ic ,\nMKL_INT *jc , MKL_INT *descc , float *work , MKL_INT *lwork , MKL_INT *info );\nvoid pdormrq (char *side , char *trans , MKL_INT *m , MKL_INT *n , MKL_INT *k , double\n*a , MKL_INT *ia , MKL_INT *ja , MKL_INT *desca , double *tau , double *c , MKL_INT\n*ic , MKL_INT *jc , MKL_INT *descc , double *work , MKL_INT *lwork , MKL_INT *info );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?ormrqfunction overwrites the general real m-by-n distributed matrix sub (C) = C(iс:iс+m-1,jс:jс\n+n-1) with\nside ='L'\nside ='R'\ntrans = 'N':\nQ*sub(C)\nsub(C)*Q\ntrans = 'T':\nQT*sub(C)\nsub(C)*QT\nwhere Q is a real orthogonal distributed matrix defined as the product of k elementary reflectors\nQ = H(1) H(2)... H(k)\nas returned by p?gerqf. Q is of order m if side = 'L' and of order n if side = 'R'.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1424\n\n\nInput Parameters\nside\n(global)\n='L': Q or QT is applied from the left.\n='R': Q or QT is applied from the right.\ntrans\n(global)\n='N', no transpose, Q is applied.\n='T', transpose, QT is applied.\nm\n(global) The number of rows in the distributed matrix sub(C) (m≥0).\nn\n(global) The number of columns in the distributed matrix sub(C) (n≥0).\nk\n(global) The number of elementary reflectors whose product defines the\nmatrix Q. Constraints:\nIf side = 'L', m≥k≥0\nIf side = 'R', n≥k≥0.\na\n(local)\nPointer into the local memory to an array of size lld_a*LOCc(ja+m-1) if\nside = 'L', and lld_a*LOCc(ja+n-1) if side = 'R'.\nThe i-th row of the matrix stored in a must contain the vector that defines\nthe elementary reflector H(i), ia≤i≤ia+k-1, as returned by p?gerqf in the\nk rows of its distributed matrix argument A(ia:ia+k-1, ja:*). A(ia:ia\n+k-1, ja:*) is modified by the function but restored on exit.\nia, ja\n(global) The row and column indices in the global matrix A indicating the\nfirst row and the first column of the submatrix A, respectively.\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\ntau\n(local)\nArray of size LOCc(ja+k-1).\nContains the scalar factor tau[i] of elementary reflectors H(i+1) as\nreturned by p?gerqf (0 ≤ i < LOCc(ja+k-1)). tau is tied to the distributed\nmatrix A.\nc\n(local)\nPointer into the local memory to an array of local size lld_c*LOCc(jc+n-1).\nContains the local pieces of the distributed matrix sub(C) to be factored.\nic, jc\n(global) The row and column indices in the global matrix C indicating the\nfirst row and the first column of the matrix sub(C), respectively.\ndescc\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix C.\nwork\n(local)\nWorkspace array of size of lwork.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1425\n\n\nlwork\n(local or global) size of work, must be at least:\nIf side = 'L',\nlwork≥max((mb_a*(mb_a-1))/2, (mpc0 + max(mqa0 +\nnumroc(numroc(n+iroffc, mb_a, 0, 0, NPROW), mb_a, 0, 0,\nlcmp), nqc0))*mb_a) + mb_a*mb_a\nelse if side ='R',\nlwork≥max((mb_a*(mb_a-1))/2, (mpc0 + nqc0)*mb_a) + mb_a*mb_a\nend if\nwhere\nlcmp = lcm/NPROW with lcm = ilcm (NPROW, NPCOL),\niroffa = mod(ia-1, mb_a),\nicoffa = mod(ja-1, nb_a),\niacol = indxg2p(ja, nb_a, MYCOL, csrc_a, NPCOL),\nmqa0 = numroc(n+icoffa, nb_a, MYCOL, iacol, NPCOL),\niroffc = mod(ic-1, mb_c),\nicoffc = mod(jc-1, nb_c),\nicrow = indxg2p(ic, mb_c, MYROW, rsrc_c, NPROW),\niccol = indxg2p(jc, nb_c, MYCOL, csrc_c, NPCOL),\nmpc0 = numroc(m+iroffc, mb_c, MYROW, icrow, NPROW),\nnqc0 = numroc(n+icoffc, nb_c, MYCOL, iccol, NPCOL),\nNOTE\nmod(x,y) is the integer remainder of x/y.\nilcm, indxg2p and numroc are ScaLAPACK tool functions; MYROW, MYCOL,\nNPROW and NPCOL can be determined by calling the function\nblacs_gridinfo.\nIf lwork = -1, then lwork is global input and a workspace query is\nassumed; the function only calculates the minimum and optimal size for all\nwork arrays. Each of these values is returned in the first entry of the\ncorresponding work array, and no error message is issued by pxerbla.\nOutput Parameters\nc\nOverwritten by the product Q* sub(C), or Q'*sub (C), or sub(C)* Q', or\nsub(C)* Q\nwork[0]\nOn exit work[0] contains the minimum value of lwork required for\noptimum performance.\ninfo\n(global)\n= 0: the execution is successful.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1426\n\n\n< 0: if the i-th argument is an array and the j-th entry, indexed j - 1, had\nan illegal value, then info = -(i*100+j); if the i-th argument is a scalar\nand had an illegal value, then info = -i.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?unmrq\nMultiplies a general matrix by the unitary matrix Q of\nthe RQ factorization formed by p?gerqf.\nSyntax\nvoid pcunmrq (char *side , char *trans , MKL_INT *m , MKL_INT *n , MKL_INT *k ,\nMKL_Complex8 *a , MKL_INT *ia , MKL_INT *ja , MKL_INT *desca , MKL_Complex8 *tau ,\nMKL_Complex8 *c , MKL_INT *ic , MKL_INT *jc , MKL_INT *descc , MKL_Complex8 *work ,\nMKL_INT *lwork , MKL_INT *info );\nvoid pzunmrq (char *side , char *trans , MKL_INT *m , MKL_INT *n , MKL_INT *k ,\nMKL_Complex16 *a , MKL_INT *ia , MKL_INT *ja , MKL_INT *desca , MKL_Complex16 *tau ,\nMKL_Complex16 *c , MKL_INT *ic , MKL_INT *jc , MKL_INT *descc , MKL_Complex16 *work ,\nMKL_INT *lwork , MKL_INT *info );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThis function overwrites the general complex m-by-n distributed matrix sub (C) = C(iс:iс+m-1,jс:jс+n-1)\nwith\nside ='L'\nside ='R'\ntrans = 'N':\nQ*sub(C)\nsub(C)*Q\ntrans = 'C':\nQH*sub(C)\nsub(C)*QH\nwhere Q is a complex unitary distributed matrix defined as the product of k elementary reflectors\nQ = H(1)' H(2)'... H(k)'\nas returned by p?gerqf. Q is of order m if side = 'L' and of order n if side = 'R'.\nInput Parameters\nside\n(global)\n='L': Q or QH is applied from the left.\n='R': Q or QH is applied from the right.\ntrans\n(global)\n='N', no transpose, Q is applied.\n='C', conjugate transpose, QH is applied.\nm\n(global) The number of rows in the distributed matrix sub(C) , (m≥0).\nn\n(global) The number of columns in the distributed matrix sub(C), (n≥0).\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1427\n\n\nk\n(global) The number of elementary reflectors whose product defines the\nmatrix Q. Constraints:\nIf side = 'L', m≥k≥0\nIf side = 'R', n≥k≥0.\na\n(local)\nPointer into the local memory to an array of size lld_a*LOCc(ja+m-1) if\nside = 'L', and lld_a*LOCc(ja+n-1) if side = 'R'. The i-th row of the\nmatrix stored in amust contain the vector that defines the elementary\nreflector H(i), ia≤i≤ia+k-1, as returned by p?gerqf in the k rows of its\ndistributed matrix argument A(ia:ia+k-1, ja:*). A(ia:ia+k-1, ja:*) is\nmodified by the function but restored on exit.\nia, ja\n(global) The row and column indices in the global matrix A indicating the\nfirst row and the first column of the submatrix A, respectively.\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\ntau\n(local)\nArray of size LOCc(ja+k-1).\nContains the scalar factor tau[i] of elementary reflectors H(i+1) as\nreturned by p?gerqf (0 ≤ i < LOCc(ja+k-1)). tau is tied to the distributed\nmatrix A.\nc\n(local)\nPointer into the local memory to an array of local size lld_c*LOCc(jc+n-1).\nContains the local pieces of the distributed matrix sub(C) to be factored.\nic, jc\n(global) The row and column indices in the global matrix C indicating the\nfirst row and the first column of the submatrix C, respectively.\ndescc\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix C.\nwork\n(local)\nWorkspace array of size of lwork.\nlwork\n(local or global) size of work, must be at least:\nIf side = 'L',\nlwork≥max((mb_a*(mb_a-1))/2, (mpc0 +\nmax(mqa0+numroc(numroc(n+iroffc, mb_a, 0, 0, NPROW), mb_a,\n0, 0, lcmp), nqc0))*mb_a) + mb_a*mb_a\nelse if side = 'R',\nlwork≥max((mb_a*(mb_a-1))/2, (mpc0 + nqc0)*mb_a) + mb_a*mb_a\nend if\nwhere\nlcmp = lcm/NPROW with lcm = ilcm(NPROW, NPCOL),\niroffa = mod(ia-1, mb_a),\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1428\n\n\nicoffa = mod(ja-1, nb_a),\niacol = indxg2p(ja, nb_a, MYCOL, csrc_a, NPCOL),\nmqa0 = numroc(m+icoffa, nb_a, MYCOL, iacol, NPCOL),\niroffc = mod(ic-1, mb_c),\nicoffc = mod(jc-1, nb_c),\nicrow = indxg2p(ic, mb_c, MYROW, rsrc_c, NPROW),\niccol = indxg2p(jc, nb_c, MYCOL, csrc_c, NPCOL),\nmpc0 = numroc(m+iroffc, mb_c, MYROW, icrow, NPROW),\nnqc0 = numroc(n+icoffc, nb_c, MYCOL, iccol, NPCOL),\nNOTE\nmod(x,y) is the integer remainder of x/y.\nilcm, indxg2p and numroc are ScaLAPACK tool functions; MYROW, MYCOL,\nNPROW and NPCOL can be determined by calling the function\nblacs_gridinfo.\nIf lwork = -1, then lwork is global input and a workspace query is\nassumed; the function only calculates the minimum and optimal size for all\nwork arrays. Each of these values is returned in the first entry of the\ncorresponding work array, and no error message is issued by pxerbla.\nOutput Parameters\nc\nOverwritten by the product Q* sub(C) or Q'*sub (C), or sub(C)* Q', or\nsub(C)* Q\nwork[0]\nOn exit work[0] contains the minimum value of lwork required for\noptimum performance.\ninfo\n(global)\n= 0: the execution is successful.\n< 0: if the i-th argument is an array and the j-th entry, indexed j - 1, had\nan illegal value, then info = -(i*100+j); if the i-th argument is a scalar\nand had an illegal value, then info = -i.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?tzrzf\nReduces the upper trapezoidal matrix A to upper\ntriangular form.\nSyntax\nvoid pstzrzf (MKL_INT *m , MKL_INT *n , float *a , MKL_INT *ia , MKL_INT *ja , MKL_INT\n*desca , float *tau , float *work , MKL_INT *lwork , MKL_INT *info );\nvoid pdtzrzf (MKL_INT *m , MKL_INT *n , double *a , MKL_INT *ia , MKL_INT *ja , MKL_INT\n*desca , double *tau , double *work , MKL_INT *lwork , MKL_INT *info );\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1429\n\n\nvoid pctzrzf (MKL_INT *m , MKL_INT *n , MKL_Complex8 *a , MKL_INT *ia , MKL_INT *ja ,\nMKL_INT *desca , MKL_Complex8 *tau , MKL_Complex8 *work , MKL_INT *lwork , MKL_INT\n*info );\nvoid pztzrzf (MKL_INT *m , MKL_INT *n , MKL_Complex16 *a , MKL_INT *ia , MKL_INT *ja ,\nMKL_INT *desca , MKL_Complex16 *tau , MKL_Complex16 *work , MKL_INT *lwork , MKL_INT\n*info );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?tzrzffunction reduces the m-by-n (m ≤ n) real/complex upper trapezoidal matrix sub(A)= A(ia:ia\n+m-1, ja:ja+n-1) to upper triangular form by means of orthogonal/unitary transformations. The upper\ntrapezoidal matrix A is factored as\nA = (R 0)*Z,\nwhere Z is an n-by-n orthogonal/unitary matrix and R is an m-by-m upper triangular matrix.\nInput Parameters\nm\n(global) The number of rows in the matrix sub(A); (m≥0).\nn\n(global) The number of columns in the matrix sub(A) (n≥0).\na\n(local)\nPointer into the local memory to an array of size lld_a*LOCc(ja+n-1).\nContains the local pieces of the m-by-n distributed matrix sub (A) to be\nfactored.\nia, ja\n(global) The row and column indices in the global matrix A indicating the\nfirst row and the first column of the submatrix A, respectively.\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\nwork\n(local)\nWorkspace array of size of lwork.\nlwork\n(local or global) size of work, must be at least\nlwork≥mb_a*(mp0+nq0+mb_a), where\niroff = mod(ia-1, mb_a),\nicoff = mod(ja-1, nb_a),\niarow = indxg2p(ia, mb_a, MYROW, rsrc_a, NPROW),\niacol = indxg2p(ja, nb_a, MYCOL, csrc_a, NPCOL),\nmp0 = numroc (m+iroff, mb_a, MYROW, iarow, NPROW),\nnq0 = numroc (n+icoff, nb_a, MYCOL, iacol, NPCOL)\nNOTE\nmod(x,y) is the integer remainder of x/y.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1430\n\n\nindxg2p and numroc are ScaLAPACK tool functions; MYROW, MYCOL, NPROW\nand NPCOL can be determined by calling the function blacs_gridinfo.\nIf lwork = -1, then lwork is global input and a workspace query is\nassumed; the function only calculates the minimum and optimal size for all\nwork arrays. Each of these values is returned in the first entry of the\ncorresponding work array, and no error message is issued by pxerbla.\nOutput Parameters\na\nOn exit, the leading m-by-m upper triangular part of sub(A) contains the\nupper triangular matrix R, and elements m+1 to n of the first m rows of sub\n(A), with the array tau, represent the orthogonal/unitary matrix Z as a\nproduct of m elementary reflectors.\nwork[0]\nOn exit work[0] contains the minimum value of lwork required for\noptimum performance.\ntau\n(local)\nArray of size LOCr(ia+m-1).\nContains the scalar factor of elementary reflectors. tau is tied to the\ndistributed matrix A.\ninfo\n(global)\n= 0: the execution is successful.\n< 0:if the i-th argument is an array and the j-th entry, indexed j - 1, had\nan illegal value, then info = -(i*100+j); if the i-th argument is a scalar\nand had an illegal value, then info = -i.\nApplication Notes\nThe factorization is obtained by the Householder's method. The k-th transformation matrix, Z(k), which is or\nwhose conjugate transpose is used to introduce zeros into the (m - k +1)-th row of sub(A), is given in the\nform\nwhere\nT(k) = i - tau*u(k)*u(k)',\ntau is a scalar and Z(k) is an (n - m) element vector. tau and Z(k) are chosen to annihilate the elements of\nthe k-th row of sub(A). The scalar tau is returned in the k-th element of tau, indexed k-1, and the vector\nu(k) in the k-th row of sub(A), such that the elements of Z(k) are in a(k, m + 1),..., a(k, n). The\nelements of R are returned in the upper triangular part of sub(A). Z is given by\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1431\n\n\nZ = Z(1) * Z(2) *... * Z(m).\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?ormrz\nMultiplies a general matrix by the orthogonal matrix\nfrom a reduction to upper triangular form formed by\np?tzrzf.\nSyntax\nvoid psormrz (char *side , char *trans , MKL_INT *m , MKL_INT *n , MKL_INT *k , MKL_INT\n*l , float *a , MKL_INT *ia , MKL_INT *ja , MKL_INT *desca , float *tau , float *c ,\nMKL_INT *ic , MKL_INT *jc , MKL_INT *descc , float *work , MKL_INT *lwork , MKL_INT\n*info );\nvoid pdormrz (char *side , char *trans , MKL_INT *m , MKL_INT *n , MKL_INT *k , MKL_INT\n*l , double *a , MKL_INT *ia , MKL_INT *ja , MKL_INT *desca , double *tau , double *c ,\nMKL_INT *ic , MKL_INT *jc , MKL_INT *descc , double *work , MKL_INT *lwork , MKL_INT\n*info );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThis function overwrites the general real m-by-n distributed matrix sub(C) = C(iс:iс+m-1,jс:jс+n-1) with\nside ='L'\nside ='R'\ntrans = 'N':\nQ*sub(C)\nsub(C)*Q\ntrans = 'T':\nQT*sub(C)\nsub(C)*QT\nwhere Q is a real orthogonal distributed matrix defined as the product of k elementary reflectors\nQ = H(1) H(2)... H(k)\nas returned by p?tzrzf. Q is of order m if side = 'L' and of order n if side = 'R'.\nInput Parameters\nside\n(global)\n='L': Q or QT is applied from the left.\n='R': Q or QT is applied from the right.\ntrans\n(global)\n='N', no transpose, Q is applied.\n='T', transpose, QT is applied.\nm\n(global) The number of rows in the distributed matrix sub(C)(m≥0).\nn\n(global) The number of columns in the distributed matrix sub(C)(n≥0).\nk\n(global) The number of elementary reflectors whose product defines the\nmatrix Q. Constraints:\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1432\n\n\nIf side = 'L', m ≥ k ≥0\nIf side = 'R', n ≥ k ≥0.\nl\n(global)\nThe columns of the distributed matrix sub(A) containing the meaningful\npart of the Householder reflectors.\nIf side = 'L', m ≥ l ≥0\nIf side = 'R', n ≥ l ≥0.\na\n(local)\nPointer into the local memory to an array of size lld_a*LOCc(ja+m-1) if\nside = 'L', and lld_a*LOCc(ja+n-1) if side = 'R', where lld_a ≥\nmax(1,LOCr(ia+k-1)).\nThe i-th row of the matrix stored in amust contain the vector that defines\nthe elementary reflector H(i), ia≤i≤ia+k-1, as returned by p?tzrzf in the\nk rows of its distributed matrix argument A(ia:ia+k-1, ja:*). A(ia:ia\n+k-1, ja:*) is modified by the function but restored on exit.\nia, ja\n(global) The row and column indices in the global matrix A indicating the\nfirst row and the first column of the submatrix A, respectively.\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\ntau\n(local)\nArray of size LOCc(ia+k-1).\nContains the scalar factor tau[i] of elementary reflectors H(i+1) as\nreturned by p?tzrzf (0 ≤ i < LOCc(ia+k-1)). tau is tied to the distributed\nmatrix A.\nc\n(local)\nPointer into the local memory to an array of local size lld_c*LOCc(jc+n-1).\nContains the local pieces of the distributed matrix sub(C) to be factored.\nic, jc\n(global) The row and column indices in the global matrix C indicating the\nfirst row and the first column of the submatrix C, respectively.\ndescc\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix C.\nwork\n(local)\nWorkspace array of size of lwork.\nlwork\n(local or global) size of work, must be at least:\nIf side = 'L',\nlwork≥max((mb_a*(mb_a-1))/2, (mpc0 + max(mqa0 +\nnumroc(numroc(n+iroffc, mb_a, 0, 0, NPROW), mb_a, 0, 0,\nlcmp), nqc0))*mb_a) + mb_a*mb_a\nelse if side ='R',\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1433\n\n\nlwork≥max((mb_a*(mb_a-1))/2, (mpc0 + nqc0)*mb_a) + mb_a*mb_a\nend if\nwhere\nlcmp = lcm/NPROW with lcm = ilcm (NPROW, NPCOL),\niroffa = mod(ia-1, mb_a), icoffa = mod(ja-1, nb_a),\niacol = indxg2p(ja, nb_a, MYCOL, csrc_a, NPCOL),\nmqa0 = numroc(n+icoffa, nb_a, MYCOL, iacol, NPCOL),\niroffc = mod(ic-1, mb_c),\nicoffc = mod(jc-1, nb_c),\nicrow = indxg2p(ic, mb_c, MYROW, rsrc_c, NPROW),\niccol = indxg2p(jc, nb_c, MYCOL, csrc_c, NPCOL),\nmpc0 = numroc(m+iroffc, mb_c, MYROW, icrow, NPROW),\nnqc0 = numroc(n+icoffc, nb_c, MYCOL, iccol, NPCOL),\nNOTE\nmod(x,y) is the integer remainder of x/y.\nilcm, indxg2p and numroc are ScaLAPACK tool functions; MYROW, MYCOL,\nNPROW and NPCOL can be determined by calling the function\nblacs_gridinfo.\nIf lwork = -1, then lwork is global input and a workspace query is\nassumed; the function only calculates the minimum and optimal size for all\nwork arrays. Each of these values is returned in the first entry of the\ncorresponding work array, and no error message is issued by pxerbla.\nOutput Parameters\nc\nOverwritten by the product Q*sub(C), or Q'*sub (C), or sub(C)*Q', or\nsub(C)*Q\nwork[0]\nOn exit work[0] contains the minimum value of lwork required for\noptimum performance.\ninfo\n(global)\n= 0: the execution is successful.\n< 0: if the i-th argument is an array and the j-th entry, indexed j - 1, had\nan illegal value, then info = -(i*100+j); if the i-th argument is a scalar\nand had an illegal value, then info = -i.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1434\n\n\np?unmrz\nMultiplies a general matrix by the unitary\ntransformation matrix from a reduction to upper\ntriangular form determined by p?tzrzf.\nSyntax\nvoid pcunmrz (char *side , char *trans , MKL_INT *m , MKL_INT *n , MKL_INT *k , MKL_INT\n*l , MKL_Complex8 *a , MKL_INT *ia , MKL_INT *ja , MKL_INT *desca , MKL_Complex8 *tau ,\nMKL_Complex8 *c , MKL_INT *ic , MKL_INT *jc , MKL_INT *descc , MKL_Complex8 *work ,\nMKL_INT *lwork , MKL_INT *info );\nvoid pzunmrz (char *side , char *trans , MKL_INT *m , MKL_INT *n , MKL_INT *k , MKL_INT\n*l , MKL_Complex16 *a , MKL_INT *ia , MKL_INT *ja , MKL_INT *desca , MKL_Complex16\n*tau , MKL_Complex16 *c , MKL_INT *ic , MKL_INT *jc , MKL_INT *descc , MKL_Complex16\n*work , MKL_INT *lwork , MKL_INT *info );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThis function overwrites the general complex m-by-n distributed matrix sub (C) = C(iс:iс+m-1,jс:jс+n-1)\nwith\nside ='L'\nside ='R'\ntrans = 'N':\nQ*sub(C)\nsub(C)*Q\ntrans = 'C':\nQH*sub(C)\nsub(C)*QH\nwhere Q is a complex unitary distributed matrix defined as the product of k elementary reflectors\nQ = H(1)' H(2)'... H(k)'\nas returned by pctzrzf/pztzrzf. Q is of order m if side = 'L' and of order n if side = 'R'.\nInput Parameters\nside\n(global)\n='L': Q or QH is applied from the left.\n='R': Q or QH is applied from the right.\ntrans\n(global)\n='N', no transpose, Q is applied.\n='C', conjugate transpose, QH is applied.\nm\n(global) The number of rows in the distributed matrix sub(C), (m≥0).\nn\n(global) The number of columns in the distributed matrix sub(C), (n≥0).\nk\n(global) The number of elementary reflectors whose product defines the\nmatrix Q. Constraints:\nIf side = 'L', m≥k≥0\nIf side = 'R', n≥k≥0.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1435\n\n\nl\n(global) The columns of the distributed matrix sub(A) containing the\nmeaningful part of the Householder reflectors.\nIf side = 'L', m≥l≥0\nIf side = 'R', n≥l≥0.\na\n(local)\nPointer into the local memory to an array of size lld_a*LOCc(ja+m-1) if\nside = 'L', and lld_a*LOCc(ja+n-1) if side = 'R', where lld_a ≥\nmax(1, LOCr(ja+k-1)). The i-th row of the matrix stored in amust\ncontain the vector that defines the elementary reflector H(i), ia≤i≤ia+k-1,\nas returned by p?gerqf in the k rows of its distributed matrix argument\nA(ia:ia+k-1, ja:*). A(ia:ia+k-1, ja:*) is modified by the function but\nrestored on exit.\nia, ja\n(global) The row and column indices in the global matrix A indicating the\nfirst row and the first column of the submatrix A, respectively.\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\ntau\n(local)\nArray of size LOCc(ia+k-1).\nContains the scalar factor tau[i] of elementary reflectors H(i+1) as\nreturned by p?gerqf (0 ≤ i < LOCc(ia+k-1)). tau is tied to the distributed\nmatrix A.\nc\n(local)\nPointer into the local memory to an array of local size lld_c*LOCc(jc+n-1).\nContains the local pieces of the distributed matrix sub(C) to be factored.\nic, jc\n(global) The row and column indices in the global matrix C indicating the\nfirst row and the first column of the submatrix C, respectively.\ndescc\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix C.\nwork\n(local)\nWorkspace array of size lwork.\nlwork\n(local or global) size of work, must be at least:\nIf side = 'L',\nlwork≥max((mb_a*(mb_a-1))/2, (mpc0+max(mqa0+numroc(numroc(n\n+iroffc, mb_a, 0, 0, NPROW), mb_a, 0, 0, lcmp), nqc0))*mb_a)\n+ mb_a*mb_a\nelse if side ='R',\nlwork≥max((mb_a*(mb_a-1))/2, (mpc0+nqc0)*mb_a) + mb_a*mb_a\nend if\nwhere\nlcmp = lcm/NPROW with lcm = ilcm(NPROW, NPCOL),\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1436\n\n\niroffa = mod(ia-1, mb_a),\nicoffa = mod(ja-1, nb_a),\niacol = indxg2p(ja, nb_a, MYCOL, csrc_a, NPCOL),\nmqa0 = numroc(m+icoffa, nb_a, MYCOL, iacol, NPCOL),\niroffc = mod(ic-1, mb_c),\nicoffc = mod(jc-1, nb_c),\nicrow = indxg2p(ic, mb_c, MYROW, rsrc_c, NPROW),\niccol = indxg2p(jc, nb_c, MYCOL, csrc_c, NPCOL),\nmpc0 = numroc(m+iroffc, mb_c, MYROW, icrow, NPROW),\nnqc0 = numroc(n+icoffc, nb_c, MYCOL, iccol, NPCOL),\nNOTE\nmod(x,y) is the integer remainder of x/y.\nilcm, indxg2p and numroc are ScaLAPACK tool functions; MYROW, MYCOL,\nNPROW and NPCOL can be determined by calling the function\nblacs_gridinfo.\nIf lwork = -1, then lwork is global input and a workspace query is\nassumed; the function only calculates the minimum and optimal size for all\nwork arrays. Each of these values is returned in the first entry of the\ncorresponding work array, and no error message is issued by pxerbla.\nOutput Parameters\nc\nOverwritten by the product Q* sub(C), or Q'*sub (C), or sub(C)*Q', or\nsub(C)*Q\nwork[0]\nOn exit work[0] contains the minimum value of lwork required for\noptimum performance.\ninfo\n(global)\n= 0: the execution is successful.\n< 0: if the i-th argument is an array and the j-th entry, indexed j - 1, had\nan illegal value, then info = -(i*100+j); if the i-th argument is a scalar\nand had an illegal value, then info = -i.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?ggqrf\nComputes the generalized QR factorization.\nSyntax\nvoid psggqrf (MKL_INT *n , MKL_INT *m , MKL_INT *p , float *a , MKL_INT *ia , MKL_INT\n*ja , MKL_INT *desca , float *taua , float *b , MKL_INT *ib , MKL_INT *jb , MKL_INT\n*descb , float *taub , float *work , MKL_INT *lwork , MKL_INT *info );\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1437\n\n\nvoid pdggqrf (MKL_INT *n , MKL_INT *m , MKL_INT *p , double *a , MKL_INT *ia , MKL_INT\n*ja , MKL_INT *desca , double *taua , double *b , MKL_INT *ib , MKL_INT *jb , MKL_INT\n*descb , double *taub , double *work , MKL_INT *lwork , MKL_INT *info );\nvoid pcggqrf (MKL_INT *n , MKL_INT *m , MKL_INT *p , MKL_Complex8 *a , MKL_INT *ia ,\nMKL_INT *ja , MKL_INT *desca , MKL_Complex8 *taua , MKL_Complex8 *b , MKL_INT *ib ,\nMKL_INT *jb , MKL_INT *descb , MKL_Complex8 *taub , MKL_Complex8 *work , MKL_INT\n*lwork , MKL_INT *info );\nvoid pzggqrf (MKL_INT *n , MKL_INT *m , MKL_INT *p , MKL_Complex16 *a , MKL_INT *ia ,\nMKL_INT *ja , MKL_INT *desca , MKL_Complex16 *taua , MKL_Complex16 *b , MKL_INT *ib ,\nMKL_INT *jb , MKL_INT *descb , MKL_Complex16 *taub , MKL_Complex16 *work , MKL_INT\n*lwork , MKL_INT *info );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?ggqrffunction forms the generalized QR factorization of an n-by-m matrix\nsub(A) = A(ia:ia+n-1, ja:ja+m-1)\nand an n-by-p matrix\nsub(B) = B(ib:ib+n-1, jb:jb+p-1):\nas\nsub(A) = Q*R, sub(B) = Q*T*Z,\nwhere Q is an n-by-n orthogonal/unitary matrix, Z is a p-by-p orthogonal/unitary matrix, and R and T\nassume one of the forms:\nIf n ≥ m\nor if n < m\nwhere R11 is upper triangular, and\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1438\n\n\nwhere T12 or T21 is an upper triangular matrix.\nIn particular, if sub(B) is square and nonsingular, the GQR factorization of sub(A) and sub(B) implicitly gives\nthe QR factorization of inv (sub(B))* sub (A):\ninv(sub(B))*sub(A) = ZH*(inv(T)*R)\nInput Parameters\nn\n(global) The number of rows in the distributed matrices sub (A) and sub(B)\n(n≥0).\nm\n(global) The number of columns in the distributed matrix sub(A) (m≥0).\np\nThe number of columns in the distributed matrix sub(B) (p≥0).\na\n(local)\nPointer into the local memory to an array of size lld_a*LOCc(ja+m-1).\nContains the local pieces of the n-by-m matrix sub(A) to be factored.\nia, ja\n(global) The row and column indices in the global matrix A indicating the\nfirst row and the first column of the submatrix A, respectively.\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\nb\n(local)\nPointer into the local memory to an array of size lld_b*LOCc(jb+p-1).\nContains the local pieces of the n-by-p matrix sub(B) to be factored.\nib, jb\n(global) The row and column indices in the global matrix B\nindicating the first row and the first column of the submatrix B,\nrespectively.\ndescb\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix B.\nwork\n(local)\nWorkspace array of size of lwork.\nlwork\n(local or global) Sze of work, must be at least\nlwork≥max(nb_a*(npa0+mqa0+nb_a), max((nb_a*(nb_a-1))/2,\n(pqb0+npb0)*nb_a)+nb_a*nb_a, mb_b*(npb0+pqb0+mb_b)),\nwhere\niroffa = mod(ia-1, mb_A),\nicoffa = mod(ja-1, nb_a),\niarow = indxg2p(ia, mb_a, MYROW, rsrc_a, NPROW),\niacol = indxg2p(ja, nb_a, MYCOL, csrc_a, NPCOL),\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1439\n\n\nnpa0 = numroc (n+iroffa, mb_a, MYROW, iarow, NPROW),\nmqa0 = numroc (m+icoffa, nb_a, MYCOL, iacol, NPCOL)\niroffb = mod(ib-1, mb_b),\nicoffb = mod(jb-1, nb_b),\nibrow = indxg2p(ib, mb_b, MYROW, rsrc_b, NPROW),\nibcol = indxg2p(jb, nb_b, MYCOL, csrc_b, NPCOL),\nnpb0 = numroc (n+iroffa, mb_b, MYROW, Ibrow, NPROW),\npqb0 = numroc(m+icoffb, nb_b, MYCOL, ibcol, NPCOL)\nNOTE\nmod(x,y) is the integer remainder of x/y.\nand numroc, indxg2p are ScaLAPACK tool functions; MYROW, MYCOL, NPROW\nand NPCOL can be determined by calling the function blacs_gridinfo.\nIf lwork = -1, then lwork is global input and a workspace query is\nassumed; the function only calculates the minimum and optimal size for all\nwork arrays. Each of these values is returned in the first entry of the\ncorresponding work array, and no error message is issued by pxerbla.\nOutput Parameters\na\nOn exit, the elements on and above the diagonal of sub (A) contain the\nmin(n, m)-by-m upper trapezoidal matrix R (R is upper triangular if n≥m); the\nelements below the diagonal, with the array taua, represent the\northogonal/unitary matrix Q as a product of min(n, m) elementary\nreflectors. (See Application Notes below).\ntaua, taub\n(local)\nArrays of size LOCc(ja+min(n,m)-1) for taua and LOCr(ib+n-1) for\ntaub.\nThe array taua contains the scalar factors of the elementary reflectors\nwhich represent the orthogonal/unitary matrix Q. taua is tied to the\ndistributed matrix A. (See Application Notes below).\nThe array taub contains the scalar factors of the elementary reflectors\nwhich represent the orthogonal/unitary matrix Z. taub is tied to the\ndistributed matrix B. (See Application Notes below).\nwork[0]\nOn exit work[0] contains the minimum value of lwork required for\noptimum performance.\ninfo\n(global)\n= 0: the execution is successful.\n< 0: if the i-th argument is an array and the j-th entry, indexed j - 1, had\nan illegal value, then info = -(i*100+j); if the i-th argument is a scalar\nand had an illegal value, then info = -i.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1440\n\n\nApplication Notes\nThe matrix Q is represented as a product of elementary reflectors\nQ = H(ja)*H(ja+1)*...*H(ja+k-1),\nwhere k= min(n,m).\nEach H(i) has the form\nH(i) = i - taua*v*v'\nwhere taua is a real/complex scalar, and v is a real/complex vector with v(1:i-1) = 0 and v(i) = 1; v(i+1:n)\nis stored on exit in A(ia+i:ia+n-1, ja+i-1) , and taua in taua[ja+i-2].To form Q explicitly, use ScaLAPACK\nfunction p?orgqr/p?ungqr. To use Q to update another matrix, use ScaLAPACK function p?ormqr/p?unmqr.\nThe matrix Z is represented as a product of elementary reflectors\nZ = H(ib)*H(ib+1)*...*H(ib+k-1), where k= min(n,p).\nEach H(i) has the form\nH(i) = i - taub*v*v'\nwhere taub is a real/complex scalar, and v is a real/complex vector with v(p-k+i+1:p) = 0 and v(p-k+i) = 1;\nv(1:p-k+i-1) is stored on exit in B(ib+n-k+i-1,jb:jb+p-k+i-2), and taub in taub[ib+n-k+i-2]. To form Z\nexplicitly, use ScaLAPACK function p?orgrq/p?ungrq. To use Z to update another matrix, use ScaLAPACK\nfunction p?ormrq/p?unmrq.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?ggrqf\nComputes the generalized RQ factorization.\nSyntax\nvoid psggrqf (MKL_INT *m , MKL_INT *p , MKL_INT *n , float *a , MKL_INT *ia , MKL_INT\n*ja , MKL_INT *desca , float *taua , float *b , MKL_INT *ib , MKL_INT *jb , MKL_INT\n*descb , float *taub , float *work , MKL_INT *lwork , MKL_INT *info );\nvoid pdggrqf (MKL_INT *m , MKL_INT *p , MKL_INT *n , double *a , MKL_INT *ia , MKL_INT\n*ja , MKL_INT *desca , double *taua , double *b , MKL_INT *ib , MKL_INT *jb , MKL_INT\n*descb , double *taub , double *work , MKL_INT *lwork , MKL_INT *info );\nvoid pcggrqf (MKL_INT *m , MKL_INT *p , MKL_INT *n , MKL_Complex8 *a , MKL_INT *ia ,\nMKL_INT *ja , MKL_INT *desca , MKL_Complex8 *taua , MKL_Complex8 *b , MKL_INT *ib ,\nMKL_INT *jb , MKL_INT *descb , MKL_Complex8 *taub , MKL_Complex8 *work , MKL_INT\n*lwork , MKL_INT *info );\nvoid pzggrqf (MKL_INT *m , MKL_INT *p , MKL_INT *n , MKL_Complex16 *a , MKL_INT *ia ,\nMKL_INT *ja , MKL_INT *desca , MKL_Complex16 *taua , MKL_Complex16 *b , MKL_INT *ib ,\nMKL_INT *jb , MKL_INT *descb , MKL_Complex16 *taub , MKL_Complex16 *work , MKL_INT\n*lwork , MKL_INT *info );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?ggrqffunction forms the generalized RQ factorization of an m-by-n matrix sub(A) = A(ia:ia+m-1,\nja:ja+n-1) and a p-by-n matrix sub(B) = B(ib:ib+p-1, jb:jb+n-1):\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1441\n\n\nsub(A) = R*Q, sub(B) = Z*T*Q,\nwhere Q is an n-by-n orthogonal/unitary matrix, Z is a p-by-p orthogonal/unitary matrix, and R and T\nassume one of the forms:\nor\nwhere R11 or R21 is upper triangular, and\nor\nwhere T11 is upper triangular.\nIn particular, if sub(B) is square and nonsingular, the GRQ factorization of sub(A) and sub(B) implicitly gives\nthe RQ factorization of sub (A)*inv(sub(B)):\nsub(A)*inv(sub(B))= (R*inv(T))*Z'\nwhere inv(sub(B)) denotes the inverse of the matrix sub(B), and Z' denotes the transpose (conjugate\ntranspose) of matrix Z.\nInput Parameters\nm\n(global) The number of rows in the distributed matrices sub (A) (m≥0).\np\nThe number of rows in the distributed matrix sub(B) (p≥0).\nn\n(global) The number of columns in the distributed matrices sub(A) and\nsub(B) (n≥0).\na\n(local)\nPointer into the local memory to an array of size lld_a*LOCc(ja+n-1).\nContains the local pieces of the m-by-n distributed matrix sub(A) to be\nfactored.\nia, ja\n(global) The row and column indices in the global matrix A indicating the\nfirst row and the first column of the submatrix A, respectively.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1442\n\n\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\nb\n(local)\nPointer into the local memory to an array of size lld_b*LOCc(jb+n-1).\nContains the local pieces of the p-by-n matrix sub(B) to be factored.\nib, jb\n(global) The row and column indices in the global matrix B indicating the\nfirst row and the first column of the submatrix B, respectively.\ndescb\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix B.\nwork\n(local)\nWorkspace array of size of lwork.\nlwork\n(local or global)\nSize of work, must be at least lwork≥max(mb_a*(mpa0+nqa0+mb_a),\nmax((mb_a*(mb_a-1))/2, (ppb0+nqb0)*mb_a) + mb_a*mb_a,\nnb_b*(ppb0+nqb0+nb_b)), where\niroffa = mod(ia-1, mb_A),\nicoffa = mod(ja-1, nb_a),\niarow = indxg2p(ia, mb_a, MYROW, rsrc_a, NPROW),\niacol = indxg2p(ja, nb_a, MYCOL, csrc_a, NPCOL),\nmpa0 = numroc (m+iroffa, mb_a, MYROW, iarow, NPROW),\nnqa0 = numroc (m+icoffa, nb_a, MYCOL, iacol, NPCOL)\niroffb = mod(ib-1, mb_b),\nicoffb = mod(jb-1, nb_b),\nibrow = indxg2p(ib, mb_b, MYROW, rsrc_b, NPROW ),\nibcol = indxg2p(jb, nb_b, MYCOL, csrc_b, NPCOL ),\nppb0 = numroc (p+iroffb, mb_b, MYROW, ibrow,NPROW),\nnqb0 = numroc (n+icoffb, nb_b, MYCOL, ibcol,NPCOL)\nNOTE\nmod(x,y) is the integer remainder of x/y.\nand numroc, indxg2p are ScaLAPACK tool functions; MYROW, MYCOL, NPROW\nand NPCOL can be determined by calling the function blacs_gridinfo.\nIf lwork = -1, then lwork is global input and a workspace query is\nassumed; the function only calculates the minimum and optimal size for all\nwork arrays. Each of these values is returned in the first entry of the\ncorresponding work array, and no error message is issued by pxerbla.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1443\n\n\nOutput Parameters\na\nOn exit, if m≤n, the upper triangle of A(ia:ia+m-1, ja+n-m:ja+n-1)\ncontains the m-by-m upper triangular matrix R; if m≥n, the elements on and\nabove the (m-n)-th subdiagonal contain the m-by-n upper trapezoidal matrix\nR; the remaining elements, with the array taua, represent the orthogonal/\nunitary matrix Q as a product of min(n,m) elementary reflectors (see\nApplication Notes below).\ntaua, taub\n(local)\nArrays of size LOCr(ia+m-1)for taua and LOCc(jb+min(p,n)-1) for\ntaub.\nThe array taua contains the scalar factors of the elementary reflectors\nwhich represent the orthogonal/unitary matrix Q. taua is tied to the\ndistributed matrix A.(See Application Notes below).\nThe array taub contains the scalar factors of the elementary reflectors\nwhich represent the orthogonal/unitary matrix Z. taub is tied to the\ndistributed matrix B. (See Application Notes below).\nwork[0]\nOn exit work[0] contains the minimum value of lwork required for\noptimum performance.\ninfo\n(global)\n= 0: the execution is successful.\n< 0: if the i-th argument is an array and the j-th entry, indexed j - 1, had\nan illegal value, then info = -(i*100+j); if the i-th argument is a scalar\nand had an illegal value, then info = -i.\nApplication Notes\nThe matrix Q is represented as a product of elementary reflectors\nQ = H(ia)*H(ia+1)*...*H(ia+k-1),\nwhere k= min(m,n).\nEach H(i) has the form\nH(i) = i - taua*v*v'\nwhere taua is a real/complex scalar, and v is a real/complex vector with v(n-k+i+1:n) = 0 and v(n-k+i) = 1;\nv(1:n-k+i-1) is stored on exit in A(ia+m-k+i-1, ja:ja+n-k+i-2), and taua in taua[ia+m-k+i-2]. To form Q\nexplicitly, use ScaLAPACK function p?orgrq/p?ungrq. To use Q to update another matrix, use ScaLAPACK\nfunction p?ormrq/p?unmrq.\nThe matrix Z is represented as a product of elementary reflectors\nZ = H(jb)*H(jb+1)*...*H(jb+k-1), where k= min(p,n).\nEach H(i) has the form\nH(i) = i - taub*v*v'\nwhere taub is a real/complex scalar, and v is a real/complex vector with v(1:i-1) = 0 and v(i)= 1; v(i+1:p) is\nstored on exit in B(ib+i:ib+p-1,jb+i-1), and taub in taub[jb+i-2]. To form Z explicitly, use ScaLAPACK\nfunction p?orgqr/p?ungqr. To use Z to update another matrix, use ScaLAPACK function p?ormqr/p?unmqr.\n \n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1444\n\n\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\nSymmetric Eigenvalue Problems: ScaLAPACK Computational Routines\nTo solve a symmetric eigenproblem with ScaLAPACK, you usually need to reduce the matrix to real\ntridiagonal form T and then find the eigenvalues and eigenvectors of the tridiagonal matrix T. ScaLAPACK\nincludes routines for reducing the matrix to a tridiagonal form by an orthogonal (or unitary) similarity\ntransformation A = QTQH as well as for solving tridiagonal symmetric eigenvalue problems. These routines\nare listed in Table \"Computational Routines for Solving Symmetric Eigenproblems\".\nThere are different routines for symmetric eigenproblems, depending on whether you need eigenvalues only\nor eigenvectors as well, and on the algorithm used (either the QTQ algorithm, or bisection followed by\ninverse iteration).\nComputational Routines for Solving Symmetric Eigenproblems\nOperation\nDense symmetric/\nHermitian matrix\nOrthogonal/unitary\nmatrix\nSymmetric\ntridiagonal\nmatrix\nReduce to tridiagonal form A = QTQH\np?sytrd/p?hetrd\n \n \nMultiply matrix after reduction\n \np?ormtr/p?unmtr\n \nFind all eigenvalues and eigenvectors\nof a tridiagonal matrix T by a QTQ\nmethod\n \n \nsteqr2*\nFind selected eigenvalues of a\ntridiagonal matrix T via bisection\n \n \np?stebz\nFind selected eigenvectors of a\ntridiagonal matrix T by inverse\niteration\n \n \np?stein\n* This routine is described as part of auxiliary ScaLAPACK routines.\np?syngst\nReduces a complex Hermitian-definite generalized\neigenproblem to standard form.\nSyntax\nvoid pssyngst (const MKL_INT* ibtype, const char* uplo, const MKL_INT* n, float* a,\nconst MKL_INT* ia, const MKL_INT* ja, const MKL_INT* desca, const float* b, const\nMKL_INT* ib, const MKL_INT* jb, const MKL_INT* descb, float* scale, float* work, const\nMKL_INT* lwork, MKL_INT* info);\nvoid pdsyngst (const MKL_INT* ibtype, const char* uplo, const MKL_INT* n, double* a,\nconst MKL_INT* ia, const MKL_INT* ja, const MKL_INT* desca, const double* b, const\nMKL_INT* ib, const MKL_INT* jb, const MKL_INT* descb, double* scale, double* work,\nconst MKL_INT* lwork, MKL_INT* info);\nInclude Files\n•\nmkl_scalapack.h\nDescription\np?syngst reduces a complex Hermitian-definite generalized eigenproblem to standard form.\np?syngst performs the same function as p?hegst, but is based on rank 2K updates, which are faster and\nmore scalable than triangular solves (the basis of p?syngst).\np?syngst calls p?hegst when uplo='U', hence p?hengst provides improved performance only when\nuplo='L', ibtype=1.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1445\n\n\np?syngst also calls p?hegst when insufficient workspace is provided, hence p?syngst provides improved\nperformance only when lwork >= 2 * NP0 * NB + NQ0 * NB + NB * NB\nIn the following sub( A ) denotes A( ia:ia+n-1, ja:ja+n-1 ) and sub( B ) denotes B( ib:ib+n-1, jb:jb\n+n-1 ).\nIf ibtype = 1, the problem is sub( A )*x = lambda*sub( B )*x, and sub( A ) is overwritten by\ninv(UH)*sub( A )*inv(U) or inv(L)*sub( A )*inv(LH)\nIf ibtype = 2 or 3, the problem is sub( A )*sub( B )*x = lambda*x or sub( B )*sub( A )*x = lambda*x, and\nsub( A ) is overwritten by U*sub( A )*UH or LH*sub( A )*L.\nsub( B ) must have been previously factorized as UH*U or L*LH by p?potrf.\nInput Parameters\nibtype\n(global)\n= 1: compute inv(UH)*sub( A )*inv(U) or inv(L)*sub( A )*inv(LH);\n= 2 or 3: compute U*sub( A )*UH or LH*sub( A )*L.\nuplo\n(global)\n= 'U': Upper triangle of sub( A ) is stored and sub( B ) is factored as UH*U;\n= 'L': Lower triangle of sub( A ) is stored and sub( B ) is factored as L*LH.\nn\n(global)\nThe order of the matrices sub( A ) and sub( B ). n >= 0.\na\n(local)\nPointer into the local memory to an array of size lld_a*LOCc(ja+n-1).\nOn entry, this array contains the local pieces of the n-by-n Hermitian\ndistributed matrix sub( A ). If uplo = 'U', the leading n-by-n upper\ntriangular part of sub( A ) contains the upper triangular part of the matrix,\nand its strictly lower triangular part is not referenced. If uplo = 'L', the\nleading n-by-n lower triangular part of sub( A ) contains the lower\ntriangular part of the matrix, and its strictly upper triangular part is not\nreferenced.\nia\n(global)\nA's global row index, which points to the beginning of the submatrix which\nis to be operated on.\nja\n(global)\nA's global column index, which points to the beginning of the submatrix\nwhich is to be operated on.\ndesca\n(global and local)\nArray of size dlen_.\nThe array descriptor for the distributed matrix A.\nb\n(local)\nPointer into the local memory to an array of size lld_b*LOCc(jb+n-1).\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1446\n\n\nOn entry, this array contains the local pieces of the triangular factor from\nthe Cholesky factorization of sub( B ), as returned by p?potrf.\nib\n(global)\nB's global row index, which points to the beginning of the submatrix which\nis to be operated on.\njb\n(global)\nB's global column index, which points to the beginning of the submatrix\nwhich is to be operated on.\ndescb\n(global and local)\nArray of size dlen_.\nThe array descriptor for the distributed matrix B.\nwork\n(local)\nArray, size (lwork)\nlwork\n(local or global)\nThe size of the array work.\nlwork is local input and must be at least lwork >= MAX( NB * ( NP0 +1 ),\n3 * NB )\nWhen ibtype = 1 and uplo = 'L', p?syngst provides improved\nperformance when lwork >= 2 * NP0 * NB + NQ0 * NB + NB * NB,\nwhere NB = mb_a = nb_a,\nNP0 = numroc( n, NB, 0, 0, NPROW ),\nNQ0 = numroc( n, NB, 0, 0, NPROW ),\nnumroc is a ScaLAPACK tool functions\nMYROW, MYCOL, NPROW and NPCOL can be determined by calling the\nsubroutine blacs_gridinfo.\nIf lwork = -1, then lwork is global input and a workspace query is\nassumed; the routine only calculates the optimal size for all work arrays.\nEach of these values is returned in the first entry of the corresponding work\narray, and no error message is issued by pxerbla.\nOutput Parameters\na\nOn exit, if info = 0, the transformed matrix, stored in the same\nformat as sub( A ).\nscale\n(global)\nAmount by which the eigenvalues should be scaled to compensate for\nthe scaling performed in this routine. At present, scale is always\nreturned as 1.0, it is returned here to allow for future enhancement.\nwork\n(local)\nArray, size (lwork)\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1447\n\n\nOn exit, work[0] returns the minimal and optimal lwork.\ninfo\n(global)\n= 0: successful exit\n< 0: If the i-th argument is an array and the j-th entry had an illegal\nvalue, then info = -(i*100+j), if the i-th argument is a scalar and\nhad an illegal value, then info = -i.\np?syntrd\nReduces a real symmetric matrix to symmetric\ntridiagonal form.\nSyntax\nvoid pssyntrd (const char* uplo, const MKL_INT* n, float* a, const MKL_INT* ia, const\nMKL_INT* ja, const MKL_INT* desca, float* d, float* e, float* tau, float* work, const\nMKL_INT* lwork, MKL_INT* info);\nvoid pdsyntrd (const char* uplo, const MKL_INT* n, double* a, const MKL_INT* ia, const\nMKL_INT* ja, const MKL_INT* desca, double* d, double* e, double* tau, double* work,\nconst MKL_INT* lwork, MKL_INT* info);\nInclude Files\n•\nmkl_scalapack.h\nDescription\np?syntrd is a prototype version of p?sytrd which uses tailored codes (either the serial, ?sytrd, or the\nparallel code, p?syttrd) when the workspace provided by the user is adequate.\np?syntrd reduces a real symmetric matrix sub( A ) to symmetric tridiagonal form T by an orthogonal\nsimilarity transformation:\nQ' * sub( A ) * Q = T, where sub( A ) = A(ia:ia+n-1,ja:ja+n-1).\nFeatures\np?syntrd is faster than p?sytrd on almost all matrices, particularly small ones (i.e. n < 500 * sqrt(P) ),\nprovided that enough workspace is available to use the tailored codes.\nThe tailored codes provide performance that is essentially independent of the input data layout.\nThe tailored codes place no restrictions on ia, ja, MB or NB. At present, ia, ja, MB and NB are restricted to\nthose values allowed by p?hetrd to keep the interface simple (see the Application Notes section for more\ninformation about the restrictions).\nInput Parameters\nuplo\n(global)\nSpecifies whether the upper or lower triangular part of the symmetric\nmatrix sub( A ) is stored:\n= 'U': Upper triangular\n= 'L': Lower triangular\nn\n(global)\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1448\n\n\nThe number of rows and columns to be operated on, i.e. the order of the\ndistributed submatrix sub( A ). n >= 0.\na\n(local)\nPointer into the local memory to an array of size lld_a*LOCc(ja+n-1).\nOn entry, this array contains the local pieces of the symmetric distributed\nmatrix sub( A ). If uplo = 'U', the leading n-by-n upper triangular part of\nsub( A ) contains the upper triangular part of the matrix, and its strictly\nlower triangular part is not referenced. If uplo = 'L', the leading n-by-n\nlower triangular part of sub( A ) contains the lower triangular part of the\nmatrix, and its strictly upper triangular part is not referenced.\nia\n(global)\nThe row index in the global array a indicating the first row of sub( A ).\nja\n(global)\nThe column index in the global array a indicating the first column of\nsub( A ).\ndesca\n(global and local)\nArray of size dlen_.\nThe array descriptor for the distributed matrix A.\nwork\n(local)\nArray, size (lwork)\nlwork\n(local or global)\nThe size of the array work.\nlwork is local input and must be at least lwork >= MAX( NB * ( NP +1 ), 3\n* NB )\nFor optimal performance, greater workspace is needed, i.e.\nlwork >= 2*( ANB+1 )*( 4*NPS+2 ) + ( NPS + 4 ) * NPS\nANB = pjlaenv( ICTXT, 3, 'p?syttrd', 'L', 0, 0, 0, 0 )\nICTXT = desca( ctxt_ )\nSQNPC = INT( sqrt( REAL( NPROW * NPCOL ) ) )\nnumroc is a ScaLAPACK tool function.\npjlaenv is a ScaLAPACK environmental inquiry function.\nNPROW and NPCOL can be determined by calling the subroutine\nblacs_gridinfo.\nOutput Parameters\na\nOn exit, if uplo = 'U', the diagonal and first superdiagonal of sub( A )\nare overwritten by the corresponding elements of the tridiagonal\nmatrix T, and the elements above the first superdiagonal, with the\narray tau, represent the orthogonal matrix Q as a product of\nelementary reflectors; if uplo = 'L', the diagonal and first subdiagonal\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1449\n\n\nof sub( A ) are overwritten by the corresponding elements of the\ntridiagonal matrix T, and the elements below the first subdiagonal,\nwith the array tau, represent the orthogonal matrix Q as a product of\nelementary reflectors. See Further Details.\nd\n(local)\nArray, size LOCc(ja+n-1)\nThe diagonal elements of the tridiagonal matrix T: d(i) = A(i,i). d is\ntied to the distributed matrix A.\ne\n(local)\nArray, size LOCc(ja+n-1) if uplo = 'U', LOCc(ja+n-2) otherwise.\nThe off-diagonal elements of the tridiagonal matrix T: e(i) = A(i,i+1) if\nuplo = 'U', e(i) = A(i+1,i) if uplo = 'L'. e is tied to the distributed\nmatrix A.\ntau\n(local)\nArray, size LOCc(ja+n-1).\nThis array contains the scalar factors tau of the elementary reflectors.\ntau is tied to the distributed matrix A.\nwork\n(local)\nArray, size (lwork)\nOn exit, work[0] returns the optimal lwork.\ninfo\n(global)\n= 0: successful exit\n< 0: If the i-th argument is an array and the j-th entry had an illegal\nvalue, then info = -(i*100+j), if the i-th argument is a scalar and\nhad an illegal value, then info = -i.\nApplication Notes\nIf uplo = 'U', the matrix Q is represented as a product of elementary reflectors\nQ = H(n-1) . . . H(2) H(1).\nEach H(i) has the form\nH(i) = I - tau * v * v', where tau is a complex scalar, and v is a complex vector with v(i+1:n) = 0 and v(i) =\n1; v(1:i-1) is stored on exit in A(ia:ia+i-2,ja+i), and tau in tau(ja+i-1).\nIf uplo = 'L', the matrix Q is represented as a product of elementary reflectors\nQ = H(1) H(2) . . . H(n-1).\nEach H(i) has the form\nH(i) = I - tau * v * v', where tau is a complex scalar, and v is a complex vector with v(1:i) = 0 and v(i+1) =\n1; v(i+2:n) is stored on exit in A(ia+i+1:ia+n-1,ja+i-1), and tau in tau(ja+i-1).\nThe contents of sub( A ) on exit are illustrated by the following examples with n = 5:\nif uplo = 'U':         \n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1450\n\n\nd e v2 v3 v4\nd e v3 v4\nd\ne v3\nd\ne\nd\nif uplo = 'L':\nd\ne\nd\nv1 e\nd\nv1 v2 e d\nv1 v2 v3 e d\nwhere d and e denote diagonal and off-diagonal elements of T, and vi denotes an element of the vector\ndefining H(i).\nAlignment requirements\nThe distributed submatrix sub( A ) must verify some alignment properties, namely the following expression\nshould be true:\n( mb_a = nb_a and IROFFA = ICOFFA and IROFFA = 0 ) with IROFFA = mod( ia-1, mb_a), and ICOFFA =\nmod( ja-1, nb_a ).\np?sytrd\nReduces a symmetric matrix to real symmetric\ntridiagonal form by an orthogonal similarity\ntransformation.\nSyntax\nvoid pssytrd (char *uplo , MKL_INT *n , float *a , MKL_INT *ia , MKL_INT *ja , MKL_INT\n*desca , float *d , float *e , float *tau , float *work , MKL_INT *lwork , MKL_INT\n*info );\nvoid pdsytrd (char *uplo , MKL_INT *n , double *a , MKL_INT *ia , MKL_INT *ja , MKL_INT\n*desca , double *d , double *e , double *tau , double *work , MKL_INT *lwork , MKL_INT\n*info );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?sytrd function reduces a real symmetric matrix sub(A) to symmetric tridiagonal form T by an\northogonal similarity transformation:\nQ'*sub(A)*Q = T,\nwhere sub(A) = A(ia:ia+n-1,ja:ja+n-1).\nInput Parameters\nuplo\n(global)\nSpecifies whether the upper or lower triangular part of the symmetric\nmatrix sub(A) is stored:\nIf uplo = 'U', upper triangular\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1451\n\n\nIf uplo = 'L', lower triangular\nn\n(global) The order of the distributed matrix sub(A) (n≥0).\na\n(local)\nPointer into the local memory to an array of size lld_a*LOCc(ja+n-1). On\nentry, this array contains the local pieces of the symmetric distributed\nmatrix sub(A).\nIf uplo = 'U', the leading n-by-n upper triangular part of sub(A) contains\nthe upper triangular part of the matrix, and its strictly lower triangular part\nis not referenced.\nIf uplo = 'L', the leading n-by-n lower triangular part of sub(A) contains\nthe lower triangular part of the matrix, and its strictly upper triangular part\nis not referenced. See Application Notes below.\nia, ja\n(global) The row and column indices in the global matrix A indicating the\nfirst row and the first column of the submatrix A, respectively.\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\nwork\n(local)\nWorkspace array of size lwork.\nlwork\n(local or global) size of work, must be at least:\nlwork ≥ max(NB*(np +1), 3*NB),\nwhere NB = mb_a = nb_a,\nnp = numroc(n, NB, MYROW, iarow, NPROW),\niarow = indxg2p(ia, NB, MYROW, rsrc_a, NPROW).\nindxg2p and numroc are ScaLAPACK tool functions; MYROW, MYCOL, NPROW\nand NPCOL can be determined by calling the function blacs_gridinfo.\nIf lwork = -1, then lwork is global input and a workspace query is\nassumed; the function only calculates the minimum and optimal size for all\nwork arrays. Each of these values is returned in the first entry of the\ncorresponding work array, and no error message is issued by pxerbla.\nOutput Parameters\na\nOn exit, if uplo = 'U', the diagonal and first superdiagonal of sub(A) are\noverwritten by the corresponding elements of the tridiagonal matrix T, and\nthe elements above the first superdiagonal, with the array tau, represent\nthe orthogonal matrix Q as a product of elementary reflectors; if uplo =\n'L', the diagonal and first subdiagonal of sub(A) are overwritten by the\ncorresponding elements of the tridiagonal matrix T, and the elements below\nthe first subdiagonal, with the array tau, represent the orthogonal matrix Q\nas a product of elementary reflectors. See Application Notes below.\nd\n(local)\nArrays of size LOCc(ja+n-1) .The diagonal elements of the tridiagonal\nmatrix T:\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1452\n\n\nd[i]= A(i+1,i+1), 0 ≤i < LOCc(ja+n-1).\nd is tied to the distributed matrix A.\ne\n(local)\nArrays of size LOCc(ja+n-1) if uplo = 'U', LOCc(ja+n-2) otherwise.\nThe off-diagonal elements of the tridiagonal matrix T:\ne[i]= A(i+1,i+2), 0 ≤i < LOCc(ja+n-1) if uplo = 'U',\ne[i] = A(i+2,i+1) if uplo = 'L'.\ne is tied to the distributed matrix A.\ntau\n(local)\nArrays of size LOCc(ja+n-1). This array contains the scalar factors of the\nelementary reflectors. tau is tied to the distributed matrix A.\nwork[0]\nOn exit work[0] contains the minimum value of lwork required for\noptimum performance.\ninfo\n(global)\n= 0: the execution is successful.\n< 0: if the i-th argument is an array and the j-th entry, indexed j - 1, had\nan illegal value, then info = -(i*100+j); if the i-th argument is a scalar\nand had an illegal value, then info = -i.\nApplication Notes\nIf uplo = 'U', the matrix Q is represented as a product of elementary reflectors\nQ = H(n-1)... H(2) H(1).\nEach H(i) has the form\nH(i) = i - tau * v * v',\nwhere tau is a real scalar, and v is a real vector with v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on exit in\nA(ia:ia+i-2, ja+i), and tau in tau[ja+i-2].\nIf uplo = 'L', the matrix Q is represented as a product of elementary reflectors\nQ = H(1) H(2)... H(n-1).\nEach H(i) has the form\nH(i) = i - tau * v * v',\nwhere tau is a real scalar, and v is a real vector with v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored on exit in\nA(ia+i+1:ia+n-1,ja+i-1), and tau in tau[ja+i-2].\nThe contents of sub(A) on exit are illustrated by the following examples with n = 5:\nIf uplo = 'U':\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1453\n\n\nIf uplo = 'L':\nwhere d and e denote diagonal and off-diagonal elements of T, and vi denotes an element of the vector\ndefining H(i).\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?ormtr\nMultiplies a general matrix by the orthogonal\ntransformation matrix from a reduction to tridiagonal\nform determined by p?sytrd.\nSyntax\nvoid psormtr (char *side , char *uplo , char *trans , MKL_INT *m , MKL_INT *n , float\n*a , MKL_INT *ia , MKL_INT *ja , MKL_INT *desca , float *tau , float *c , MKL_INT *ic ,\nMKL_INT *jc , MKL_INT *descc , float *work , MKL_INT *lwork , MKL_INT *info );\nvoid pdormtr (char *side , char *uplo , char *trans , MKL_INT *m , MKL_INT *n , double\n*a , MKL_INT *ia , MKL_INT *ja , MKL_INT *desca , double *tau , double *c , MKL_INT\n*ic , MKL_INT *jc , MKL_INT *descc , double *work , MKL_INT *lwork , MKL_INT *info );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThis function overwrites the general real distributed m-by-n matrix sub(C) = C(iс:iс+m-1,jс:jс+n-1) with\nside ='L'\nside ='R'\ntrans = 'N':\nQ*sub(C)\nsub(C)*Q\ntrans = 'T':\nQT*sub(C)\nsub(C)*QT\nwhere Q is a real orthogonal distributed matrix of order nq, with nq = m if side = 'L' and nq = n if side =\n'R'.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1454\n\n\nQ is defined as the product of nq elementary reflectors, as returned by p?sytrd.\nIf uplo = 'U', Q = H(nq-1)... H(2) H(1);\nIf uplo = 'L', Q = H(1) H(2)... H(nq-1).\nInput Parameters\nside\n(global)\n='L': Q or QT is applied from the left.\n='R': Q or QT is applied from the right.\ntrans\n(global)\n='N', no transpose, Q is applied.\n='T', transpose, QT is applied.\nuplo\n(global)\n= 'U': Upper triangle of A(ia:*, ja:*) contains elementary reflectors\nfrom p?sytrd;\n= 'L': Lower triangle of A(ia:*,ja:*) contains elementary reflectors\nfrom p?sytrd\nm\n(global) The number of rows in the distributed matrix sub(C) (m≥0).\nn\n(global) The number of columns in the distributed matrix sub(C) (n≥0).\na\n(local)\nPointer into the local memory to an array of size lld_a*LOCc(ja+m-1) if\nside = 'L', and lld_a*LOCc(ja+n-1) if side = 'R'.\nContains the vectors that define the elementary reflectors, as returned by\np?sytrd.\nIf side='L', lld_a ≥ max(1,LOCr(ia+m-1));\nIf side ='R', lld_a ≥ max(1, LOCr(ia+n-1)).\nia, ja\n(global) The row and column indices in the global matrix A indicating the\nfirst row and the first column of the submatrix A, respectively.\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\ntau\n(local)\nArray of size of ltau where\nif side = 'L' and uplo = 'U', ltau = LOCc(m_a),\nif side = 'L' and uplo = 'L', ltau = LOCc(ja+m-2),\nif side = 'R' and uplo = 'U', ltau = LOCc(n_a),\nif side = 'R' and uplo = 'L', ltau = LOCc(ja+n-2).\ntau[i] must contain the scalar factor of the elementary reflector H(i+1), as\nreturned by p?sytrd (0 ≤ i < ltau). tau is tied to the distributed matrix A.\nc\n(local)\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1455\n\n\nPointer into the local memory to an array of size lld_c*LOCc(jc+n-1).\nContains the local pieces of the distributed matrix sub (C).\nic, jc\n(global) The row and column indices in the global matrix C indicating the\nfirst row and the first column of the submatrix C, respectively.\ndescc\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix C.\nwork\n(local)\nWorkspace array of size lwork.\nlwork\n(local or global) size of work, must be at least:\nif uplo = 'U',\niaa= ia; jaa= ja+1, icc= ic; jcc= jc;\nelse uplo = 'L',\niaa= ia+1, jaa= ja;\nIf side = 'L',\nicc= ic+1; jcc= jc;\nelse icc= ic; jcc= jc+1;\nend if\nend if\nIf side = 'L',\nmi= m-1; ni= n\nlwork ≥ max((nb_a*(nb_a-1))/2, (nqc0 + mpc0)*nb_a) +\nnb_a*nb_a\nelse\nIf side = 'R',\nmi= m; mi = n-1;\nlwork≥max((nb_a*(nb_a-1))/2, (nqc0 +\nmax(npa0+numroc(numroc(ni+icoffc, nb_a, 0, 0, NPCOL), nb_a,\n0, 0, lcmq), mpc0))*nb_a)+ nb_a*nb_a\nend if\nwhere lcmq = lcm/NPCOL with lcm = ilcm(NPROW, NPCOL),\niroffa = mod(iaa-1, mb_a),\nicoffa = mod(jaa-1, nb_a),\niarow = indxg2p(iaa, mb_a, MYROW, rsrc_a, NPROW),\nnpa0 = numroc(ni+iroffa, mb_a, MYROW, iarow, NPROW),\niroffc = mod(icc-1, mb_c),\nicoffc = mod(jcc-1, nb_c),\nicrow = indxg2p(icc, mb_c, MYROW, rsrc_c, NPROW),\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1456\n\n\niccol = indxg2p(jcc, nb_c, MYCOL, csrc_c, NPCOL),\nmpc0 = numroc(mi+iroffc, mb_c, MYROW, icrow, NPROW),\nnqc0 = numroc(ni+icoffc, nb_c, MYCOL, iccol, NPCOL),\nNOTE\nmod(x,y) is the integer remainder of x/y.\nilcm, indxg2p and numroc are ScaLAPACK tool functions; MYROW, MYCOL,\nNPROW and NPCOL can be determined by calling the function\nblacs_gridinfo. If lwork = -1, then lwork is global input and a\nworkspace query is assumed; the function only calculates the minimum and\noptimal size for all work arrays. Each of these values is returned in the first\nentry of the corresponding work array, and no error message is issued by\npxerbla.\nOutput Parameters\nc\nOverwritten by the product Q*sub(C), or Q'*sub(C), or sub(C)*Q', or\nsub(C)*Q.\nwork[0]\nOn exit work[0] contains the minimum value of lwork required for\noptimum performance.\ninfo\n(global)\n= 0: the execution is successful.\n< 0: if the i-th argument is an array and the j-th entry, indexed j - 1, had\nan illegal value, then info = -(i*100+j); if the i-th argument is a scalar\nand had an illegal value, then info = -i.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?hengst\nReduces a complex Hermitian-definite generalized\neigenproblem to standard form.\nSyntax\nvoid pchengst (const MKL_INT* ibtype, const char* uplo, const MKL_INT* n, MKL_Complex8*\na, const MKL_INT* ia, const MKL_INT* ja, const MKL_INT* desca, const MKL_Complex8* b,\nconst MKL_INT* ib, const MKL_INT* jb, const MKL_INT* descb, float* scale, MKL_Complex8*\nwork, const MKL_INT* lwork, MKL_INT* info);\nvoid pzhengst (const MKL_INT* ibtype, const char* uplo, const MKL_INT* n,\nMKL_Complex16* a, const MKL_INT* ia, const MKL_INT* ja, const MKL_INT* desca, const\nMKL_Complex16* b, const MKL_INT* ib, const MKL_INT* jb, const MKL_INT* descb, double*\nscale, MKL_Complex16* work, const MKL_INT* lwork, MKL_INT* info);\nInclude Files\n•\nmkl_scalapack.h\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1457\n\n\nDescription\np?hengst reduces a complex Hermitian-definite generalized eigenproblem to standard form.\np?hengst performs the same function as p?hegst, but is based on rank 2K updates, which are faster and\nmore scalable than triangular solves (the basis of p?hengst).\np?hengst calls p?hegst when uplo='U', hence p?hengst provides improved performance only when\nuplo='L' and ibtype=1.\np?hengst also calls p?hegst when insufficient workspace is provided, hence p?hengst provides improved\nperformance only when lwork is sufficient (as described in the parameter descriptions).\nIn the following sub( A ) denotes the submatrix A( ia:ia+n-1, ja:ja+n-1 ) and sub( B ) denotes the\nsubmatrix B( ib:ib+n-1, jb:jb+n-1 ).\nIf ibtype = 1, the problem is sub( A )*x = lambda*sub( B )*x, and sub( A ) is overwritten by\ninv(UH)*sub( A )*inv(U) or inv(L)*sub( A )*inv(LH)\nIf ibtype = 2 or 3, the problem is sub( A )*sub( B )*x = lambda*x or sub( B )*sub( A )*x = lambda*x, and\nsub( A ) is overwritten by U*sub( A )*UH or LH*sub( A )*L.\nsub( B ) must have been previously factorized as UH*U or L*LH by p?potrf.\nInput Parameters\nibtype\n(global)\n= 1: compute inv(UH)*sub( A )*inv(U) or inv(L)*sub( A )*inv(LH);\n= 2 or 3: compute U*sub( A )*UH or LH*sub( A )*L.\nuplo\n(global)\n= 'U': Upper triangle of sub( A ) is stored and sub( B ) is factored as UH*U;\n= 'L': Lower triangle of sub( A ) is stored and sub( B ) is factored as L*LH.\nn\n(global)\nThe order of the matrices sub( A ) and sub( B ). n >= 0.\na\n(local)\nPointer into the local memory to an array of size lld_a*LOCc(ja+n-1).\nOn entry, this array contains the local pieces of the n-by-n Hermitian\ndistributed matrix sub( A ). If uplo = 'U', the leading n-by-n upper\ntriangular part of sub( A ) contains the upper triangular part of the matrix,\nand its strictly lower triangular part is not referenced. If uplo = 'L', the\nleading n-by-n lower triangular part of sub( A ) contains the lower\ntriangular part of the matrix, and its strictly upper triangular part is not\nreferenced.\nia\n(global)\nGlobal row index of matrix A, which points to the beginning of the\nsubmatrix on which to operate.\nja\n(global)\nGlobal column index of matrix A, which points to the beginning of the\nsubmatrix on which to operate.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1458\n\n\ndesca\n(global and local)\nArray of size dlen_.\nThe array descriptor for the distributed matrix A.\nb\n(local)\nPointer into the local memory to an array of size lld_b*LOCc(jb+n-1).\nib\n(global)\nGlobal row index of matrix B, which points to the beginning of the\nsubmatrix on which to operate.\njb\n(global)\nGlobal column index of matrix B, which points to the beginning of the\nsubmatrix on which to operate.\ndescb\n(global and local)\nArray of size dlen_.\nThe array descriptor for the distributed matrix B.\nwork\n(local)\nArray, size (lwork)\nOn exit, work( 1 ) returns the minimal and optimal lwork.\nlwork\n(local)\nThe size of the array work.\nlwork is local input and must be at least lwork >= MAX( NB * ( NP0\n+1 ), 3 * NB ).\nWhen ibtype = 1 and uplo = 'L', p?hengst provides improved\nperformance when lwork >= 2 * NP0 * NB + NQ0 * NB + NB * NB, where\nNB = mb_a = nb_a, NP0 = numroc( n, NB, 0, 0, NPROW ), NQ0 =\nnumroc( n, NB, 0, 0, NPROW ), and numroc is a ScaLAPACK tool function.\nMYROW, MYCOL, NPROW and NPCOL can be determined by calling the\nsubroutine blacs_gridinfo.\nIf lwork = -1, then lwork is global input and a workspace query is\nassumed; the routine only calculates the optimal size for all work arrays.\nEach of these values is returned in the first entry of the corresponding work\narray, and no error message is issued by pxerbla.\nOutput Parameters\na\nOn exit, if info = 0, the transformed matrix, stored in the same\nformat as sub( A ).\nscale\n(global)\nAmount by which the eigenvalues should be scaled to compensate for\nthe scaling performed in this routine.\nscale is always returned as 1.0.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1459\n\n\nwork\nOn exit, work[0] returns the minimal and optimal lwork.\ninfo\n(global)\n= 0: successful exit\n< 0: If the i-th argument is an array and the j-entry had an illegal\nvalue, then info = -(i*100+j), if the i-th argument is a scalar and\nhad an illegal value, then info = -i.\np?hentrd\nReduces a complex Hermitian matrix to Hermitian\ntridiagonal form.\nSyntax\nvoid pchentrd (const char* uplo, const MKL_INT* n, MKL_Complex8* a, const MKL_INT* ia,\nconst MKL_INT* ja, const MKL_INT* desca, float* d, float* e, MKL_Complex8* tau,\nMKL_Complex8* work, const MKL_INT* lwork, float* rwork, const MKL_INT* lrwork, MKL_INT*\ninfo);\nvoid pzhentrd (const char* uplo, const MKL_INT* n, MKL_Complex16* a, const MKL_INT* ia,\nconst MKL_INT* ja, const MKL_INT* desca, double* d, double* e, MKL_Complex16* tau,\nMKL_Complex16* work, const MKL_INT* lwork, double* rwork, const MKL_INT* lrwork,\nMKL_INT* info);\nInclude Files\n•\nmkl_scalapack.h\nDescription\np?hentrd is a prototype version of p?hetrd which uses tailored codes (either the serial, ?hetrd, or the\nparallel code, p?hettrd) when adequate workspace is provided.\np?hentrd reduces a complex Hermitian matrix sub( A ) to Hermitian tridiagonal form T by an unitary\nsimilarity transformation:\nQ' * sub( A ) * Q = T, where sub( A ) = A(ia:ia+n-1,ja:ja+n-1).\np?hentrd is faster than p?hetrd on almost all matrices, particularly small ones (i.e. n < 500 * sqrt(P) ),\nprovided that enough workspace is available to use the tailored codes.\nThe tailored codes provide performance that is essentially independent of the input data layout.\nThe tailored codes place no restrictions on ia, ja, MB or NB. At present, ia, ja, MB and NB are restricted to\nthose values allowed by p?hetrd to keep the interface simple (see the Application Notes section for more\ninformation about the restrictions).\nInput Parameters\nuplo\n(global)\nSpecifies whether the upper or lower triangular part of the Hermitian matrix\nsub( A ) is stored:\n= 'U': Upper triangular\n= 'L': Lower triangular\nn\n(global)\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1460\n\n\nThe number of rows and columns to be operated on, i.e. the order of the\ndistributed submatrix sub( A ). n >= 0.\na\n(local)\nPointer into the local memory to an array of size lld_a*LOCc(ja+n-1).\nOn entry, this array contains the local pieces of the Hermitian distributed\nmatrix sub( A ). If uplo = 'U', the leading n-by-n upper triangular part of\nsub( A ) contains the upper triangular part of the matrix, and its strictly\nlower triangular part is not referenced. If uplo = 'L', the leading n-by-n\nlower triangular part of sub( A ) contains the lower triangular part of the\nmatrix, and its strictly upper triangular part is not referenced.\nia\n(global)\nThe row index in the global array a indicating the first row of sub( A ).\nja\n(global)\nThe column index in the global array a indicating the first column of\nsub( A ).\ndesca\n(global and local)\nArray of size dlen_.\nThe array descriptor for the distributed matrix A.\nwork\n(local)\nArray, size (lwork)\nlwork\n(local or global)\nThe size of the array work.\nlwork is local input and must be at least lwork >= MAX( NB * ( NP +1 ), 3\n* NB ).\nFor optimal performance, greater workspace is needed:\nlwork >= 2*( ANB+1 )*( 4*NPS+2 ) + ( NPS + 4 ) * NPS\nANB = pjlaenv( ICTXT, 3, 'p?hettrd', 'L', 0, 0, 0, 0 )\nICTXT = desca( ctxt_ )\nSQNPC = INT( sqrt( REAL( NPROW * NPCOL ) ) )\nNPS = MAX( numroc( n, 1, 0, 0, SQNPC ), 2*ANB )\nnumroc is a ScaLAPACK tool function.\npjlaenv is a ScaLAPACK environmental inquiry function.\nNPROW and NPCOL can be determined by calling the subroutine\nblacs_gridinfo.\nrwork\n(local)\nArray, size (lrwork)\nlrwork\n(local or global)\nThe size of the array rwork.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1461\n\n\nlrwork is local input and must be at least lrwork >= 1.\nFor optimal performance, greater workspace is needed, i.e. lrwork >=\nMAX( 2 * n )\nOutput Parameters\na\nOn exit, if uplo = 'U', the diagonal and first superdiagonal of sub( A )\nare overwritten by the corresponding elements of the tridiagonal\nmatrix T, and the elements above the first superdiagonal, with the\narray tau, represent the unitary matrix Q as a product of elementary\nreflectors; if uplo = 'L', the diagonal and first subdiagonal of sub( A )\nare overwritten by the corresponding elements of the tridiagonal\nmatrix T, and the elements below the first subdiagonal, with the array\ntau, represent the unitary matrix Q as a product of elementary\nreflectors. See Application Notes.\nd\n(local)\nArray, size LOCc(ja+n-1)\nThe diagonal elements of the tridiagonal matrix T: d[i - 1] = A(i,i).\nd is tied to the distributed matrix A.\ne\n(local)\nArray, size LOCc(ja+n-1) if uplo = 'U', LOCc(ja+n-2) otherwise.\nThe off-diagonal elements of the tridiagonal matrix T: e[i - 1] =\nA(i,i+1) if uplo = 'U', e[i - 1] = A(i+1,i) if uplo = 'L'. e is tied to\nthe distributed matrix A.\ntau\n(local)\nArray, size LOCc(ja+n-1).\nThis array contains the scalar factors tau of the elementary reflectors.\ntau is tied to the distributed matrix A.\nwork\nOn exit, work[0] returns the optimal lwork.\nrwork\nOn exit, rwork[0] returns the optimal lrwork.\ninfo\n(global)\n= 0: successful exit\n< 0: If the i-th argument is an array and the j-th entry had an illegal\nvalue, then info = -(i*100+j), if the i-th argument is a scalar and\nhad an illegal value, then info = -i.\nApplication Notes\nIf uplo = 'U', the matrix Q is represented as a product of elementary reflectors\nQ = H(n-1) . . . H(2) H(1).\nEach H(i) has the form\nH(i) = I - tau * v * v', where tau is a complex scalar, and v is a complex vector with v(i+1:n) = 0 and v(i) =\n1; v(1:i-1) is stored on exit in A(ia:ia+i-2,ja+i), and tau in tau(ja+i-1).\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1462\n\n\nIf uplo = 'L', the matrix Q is represented as a product of elementary reflectors\nQ = H(1) H(2) . . . H(n-1).\nEach H(i) has the form\nH(i) = I - tau * v * v', where tau is a complex scalar, and v is a complex vector with v(1:i) = 0 and v(i+1) =\n1; v(i+2:n) is stored on exit in A(ia+i+1:ia+n-1,ja+i-1), and tau in tau(ja+i-1).\nThe contents of sub( A ) on exit are illustrated by the following examples with n = 5:\nif uplo = 'U':         \nd e v2 v3 v4\nd e v3 v4\nd\ne v3\nd\ne\nd\nif uplo = 'L':\nd\ne\nd\nv1 e\nd\nv1 v2 e d\nv1 v2 v3 e d\nwhere d and e denote diagonal and off-diagonal elements of T, and vi denotes an element of the vector\ndefining H(i).\nAlignment requirements\nThe distributed submatrix sub( A ) must verify some alignment properties, namely the following expression\nshould be true:\n( mb_a = nb_a and IROFFA = ICOFFA and IROFFA = 0 ) with IROFFA = mod( ia-1, mb_a), and ICOFFA =\nmod( ja-1, nb_a ).\np?hetrd\nReduces a Hermitian matrix to Hermitian tridiagonal\nform by a unitary similarity transformation.\nSyntax\nvoid pchetrd (char *uplo , MKL_INT *n , MKL_Complex8 *a , MKL_INT *ia , MKL_INT *ja ,\nMKL_INT *desca , float *d , float *e , MKL_Complex8 *tau , MKL_Complex8 *work , MKL_INT\n*lwork , MKL_INT *info );\nvoid pzhetrd (char *uplo , MKL_INT *n , MKL_Complex16 *a , MKL_INT *ia , MKL_INT *ja ,\nMKL_INT *desca , double *d , double *e , MKL_Complex16 *tau , MKL_Complex16 *work ,\nMKL_INT *lwork , MKL_INT *info );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?hetrd function reduces a complex Hermitian matrix sub(A) to Hermitian tridiagonal form T by a\nunitary similarity transformation:\nQ'*sub(A)*Q = T\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1463\n\n\nwhere sub(A) = A(ia:ia+n-1,ja:ja+n-1).\nInput Parameters\nuplo\n(global)\nSpecifies whether the upper or lower triangular part of the Hermitian matrix\nsub(A) is stored:\nIf uplo = 'U', upper triangular\nIf uplo = 'L', lower triangular\nn\n(global) The order of the distributed matrix sub(A) (n≥0).\na\n(local)\nPointer into the local memory to an array of size lld_a*LOCc(ja+n-1). On\nentry, this array contains the local pieces of the Hermitian distributed\nmatrix sub(A).\nIf uplo = 'U', the leading n-by-n upper triangular part of sub(A) contains\nthe upper triangular part of the matrix, and its strictly lower triangular part\nis not referenced.\nIf uplo = 'L', the leading n-by-n lower triangular part of sub(A) contains\nthe lower triangular part of the matrix, and its strictly upper triangular part\nis not referenced. (see Application Notes below).\nia, ja\n(global) The row and column indices in the global matrix A indicating the\nfirst row and the first column of the submatrix A, respectively.\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\nwork\n(local)\nWorkspace array of size lwork.\nlwork\n(local or global) size of work, must be at least:\nlwork≥max(NB*(np +1), 3*NB)\nwhere NB = mb_a = nb_a,\nnp = numroc(n, NB, MYROW, iarow, NPROW),\niarow = indxg2p(ia, NB, MYROW, rsrc_a, NPROW).\nindxg2p and numroc are ScaLAPACK tool functions; MYROW, MYCOL, NPROW\nand NPCOL can be determined by calling the function blacs_gridinfo.\nIf lwork = -1, then lwork is global input and a workspace query is\nassumed; the function only calculates the minimum and optimal size for all\nwork arrays. Each of these values is returned in the first entry of the\ncorresponding work array, and no error message is issued by pxerbla.\nOutput Parameters\na\nOn exit,\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1464\n\n\nIf uplo = 'U', the diagonal and first superdiagonal of sub(A) are\noverwritten by the corresponding elements of the tridiagonal matrix T, and\nthe elements above the first superdiagonal, with the array tau, represent\nthe unitary matrix Q as a product of elementary reflectors;if uplo = 'L',\nthe diagonal and first subdiagonal of sub(A) are overwritten by the\ncorresponding elements of the tridiagonal matrix T, and the elements below\nthe first subdiagonal, with the array tau, represent the unitary matrix Q as\na product of elementary reflectors (see Application Notes below).\nd\n(local)\nArrays of size LOCc(ja+n-1). The diagonal elements of the tridiagonal\nmatrix T:\nd[i]= A(i+1,i+1), 0 ≤i < LOCc(ja+n-1).\nd is tied to the distributed matrix A.\ne\n(local)\nArrays of size LOCc(ja+n-1) if uplo = 'U'; LOCc(ja+n-2) - otherwise.\nThe off-diagonal elements of the tridiagonal matrix T:\ne[i]= A(i+1,i+2), 0 ≤i < LOCc(ja+n-1) if uplo = 'U',\ne[i] = A(i+2,i+1) if uplo = 'L'.\ne is tied to the distributed matrix A.\ntau\n(local)\nArray of size LOCc(ja+n-1). This array contains the scalar factors of the\nelementary reflectors. tau is tied to the distributed matrix A.\nwork[0]\nOn exit work[0] contains the minimum value of lwork required for\noptimum performance.\ninfo\n(global)\n= 0: the execution is successful.\n< 0: if the i-th argument is an array and the j-th entry, indexed j - 1, had\nan illegal value, then info = -(i*100+j); if the i-th argument is a scalar\nand had an illegal value, then info = -i.\nApplication Notes\nIf uplo = 'U', the matrix Q is represented as a product of elementary reflectors\nQ = H(n-1)*...*H(2)*H(1).\nEach H(i) has the form\nH(i) = i - tau*v*v',\nwhere tau is a complex scalar, and v is a complex vector with v(i+1:n) = 0 and v(i) = 1; v(1:i-1) is stored on\nexit in A(ia:ia+i-2, ja+i), and tau in tau[ja+i-2].\nIf uplo = 'L', the matrix Q is represented as a product of elementary reflectors\nQ = H(1)*H(2)*...*H(n-1).\nEach H(i) has the form\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1465\n\n\nH(i) = i - tau*v*v',\nwhere tau is a complex scalar, and v is a complex vector with v(1:i) = 0 and v(i+1) = 1; v(i+2:n) is stored\non exit in A(ia+i+1:ia+n-1,ja+i-1), and tau in tau[ja+i-2].\nThe contents of sub(A) on exit are illustrated by the following examples with n = 5:\nIf uplo = 'U':\nIf uplo = 'L':\nwhere d and e denote diagonal and off-diagonal elements of T, and vi denotes an element of the vector\ndefining H(i).\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?unmtr\nMultiplies a general matrix by the unitary\ntransformation matrix from a reduction to tridiagonal\nform determined by p?hetrd.\nSyntax\nvoid pcunmtr (char *side , char *uplo , char *trans , MKL_INT *m , MKL_INT *n ,\nMKL_Complex8 *a , MKL_INT *ia , MKL_INT *ja , MKL_INT *desca , MKL_Complex8 *tau ,\nMKL_Complex8 *c , MKL_INT *ic , MKL_INT *jc , MKL_INT *descc , MKL_Complex8 *work ,\nMKL_INT *lwork , MKL_INT *info );\nvoid pzunmtr (char *side , char *uplo , char *trans , MKL_INT *m , MKL_INT *n ,\nMKL_Complex16 *a , MKL_INT *ia , MKL_INT *ja , MKL_INT *desca , MKL_Complex16 *tau ,\nMKL_Complex16 *c , MKL_INT *ic , MKL_INT *jc , MKL_INT *descc , MKL_Complex16 *work ,\nMKL_INT *lwork , MKL_INT *info );\nInclude Files\n•\nmkl_scalapack.h\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1466\n\n\nDescription\nThis function overwrites the general complex distributed m-by-n matrix sub(C) = C(iс:iс+m-1,jс:jс+n-1)\nwith\nside ='L'\nside ='R'\ntrans = 'N':\nQ*sub(C)\nsub(C)*Q\ntrans = 'C':\nQH*sub(C)\nsub(C)*QH\nwhere Q is a complex unitary distributed matrix of order nq, with nq =m if side = 'L' and nq =n if side =\n'R'.\nQ is defined as the product of nq-1 elementary reflectors, as returned by p?hetrd.\nIf uplo = 'U', Q = H(nq-1)... H(2) H(1);\nIf uplo = 'L', Q = H(1) H(2)... H(nq-1).\nInput Parameters\nside\n(global)\n='L': Q or QH is applied from the left.\n='R': Q or QH is applied from the right.\ntrans\n(global)\n='N', no transpose, Q is applied.\n='C', conjugate transpose, QH is applied.\nuplo\n(global)\n= 'U': Upper triangle of A(ia:*, ja:*) contains elementary reflectors\nfrom p?hetrd;\n= 'L': Lower triangle of A(ia:*,ja:*) contains elementary reflectors\nfrom p?hetrd\nm\n(global) The number of rows in the distributed matrix sub(C) (m≥0).\nn\n(global) The number of columns in the distributed matrix sub(C) (n≥0).\na\n(local)\nPointer into the local memory to an array of size lld_a*LOCc(ja+m-1) if\nside = 'L', and lld_a*LOCc(ja+n-1) if side = 'R'.\nContains the vectors which define the elementary reflectors, as returned by\np?hetrd.\nIf side='L', lld_a ≥ max(1,LOCr(ia+m-1));\nIf side ='R', lld_a ≥ max(1,LOCr(ia+n-1)).\nia, ja\n(global) The row and column indices in the global matrix A indicating the\nfirst row and the first column of the submatrix A, respectively.\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\ntau\n(local)\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1467\n\n\nArray of size of ltau where\nIf side = 'L' and uplo = 'U', ltau = LOCc(m_a),\nif side = 'L' and uplo = 'L', ltau = LOCc(ja+m-2),\nif side = 'R' and uplo = 'U', ltau = LOCc(n_a),\nif side = 'R' and uplo = 'L', ltau = LOCc(ja+n-2).\ntau[i] must contain the scalar factor of the elementary reflector H(i+1), as\nreturned by p?hetrd (0 ≤ i < ltau). tau is tied to the distributed matrix A.\nc\n(local)\nPointer into the local memory to an array of size lld_c*LOCc(jc+n-1).\nContains the local pieces of the distributed matrix sub (C).\nic, jc\n(global) The row and column indices in the global matrix C indicating the\nfirst row and the first column of the submatrix C, respectively.\ndescc\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix C.\nwork\n(local)\nWorkspace array of size lwork.\nlwork\n(local or global) size of work, must be at least:\nIf uplo = 'U',\niaa= ia; jaa= ja+1, icc= ic; jcc= jc;\nelse uplo = 'L',\niaa= ia+1, jaa= ja;\nIf side = 'L',\nicc= ic+1; jcc= jc;\nelse icc= ic; jcc= jc+1;\nend if\nend if\nIf side = 'L',\nmi= m-1; ni= n\nlwork ≥ max((nb_a*(nb_a-1))/2, (nqc0 + mpc0)*nb_a) +\nnb_a*nb_a\nelse\nIf side = 'R',\nmi= m; mi = n-1;\nlwork ≥ max((nb_a*(nb_a-1))/2, (nqc0 +\nmax(npa0+numroc(numroc(ni+icoffc, nb_a, 0, 0, NPCOL), nb_a,\n0, 0, lcmq), mpc0))*nb_a) + nb_a*nb_a\nend if\nwhere lcmq = lcm/NPCOL with lcm = ilcm(NPROW, NPCOL),\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1468\n\n\niroffa = mod(iaa-1, mb_a),\nicoffa = mod(jaa-1, nb_a),\niarow = indxg2p(iaa, mb_a, MYROW, rsrc_a, NPROW),\nnpa0 = numroc(ni+iroffa, mb_a, MYROW, iarow, NPROW),\niroffc = mod(icc-1, mb_c),\nicoffc = mod(jcc-1, nb_c),\nicrow = indxg2p(icc, mb_c, MYROW, rsrc_c, NPROW),\niccol = indxg2p(jcc, nb_c, MYCOL, csrc_c, NPCOL),\nmpc0 = numroc(mi+iroffc, mb_c, MYROW, icrow, NPROW),\nnqc0 = numroc(ni+icoffc, nb_c, MYCOL, iccol, NPCOL),\nNOTE\nmod(x,y) is the integer remainder of x/y.\nilcm, indxg2p and numroc are ScaLAPACK tool functions; MYROW, MYCOL,\nNPROW and NPCOL can be determined by calling the function\nblacs_gridinfo. If lwork = -1, then lwork is global input and a\nworkspace query is assumed; the function only calculates the minimum and\noptimal size for all work arrays. Each of these values is returned in the first\nentry of the corresponding work array, and no error message is issued by\npxerbla.\nOutput Parameters\nc\nOverwritten by the product Q*sub(C), or Q'*sub(C), or sub(C)*Q', or\nsub(C)*Q.\nwork[0]\nOn exit work[0] contains the minimum value of lwork required for\noptimum performance.\ninfo\n(global)\n= 0: the execution is successful.\n< 0: if the i-th argument is an array and the j-th entry, indexed j - 1, had\nan illegal value, then info = -(i*100+j); if the i-th argument is a scalar\nand had an illegal value, then info = -i.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?stebz\nComputes the eigenvalues of a symmetric tridiagonal\nmatrix by bisection.\nSyntax\nvoid psstebz (MKL_INT *ictxt , char *range , char *order , MKL_INT *n , float *vl ,\nfloat *vu , MKL_INT *il , MKL_INT *iu , float *abstol , float *d , float *e , MKL_INT\n*m , MKL_INT *nsplit , float *w , MKL_INT *iblock , MKL_INT *isplit , float *work ,\nMKL_INT *lwork , MKL_INT *iwork , MKL_INT *liwork , MKL_INT *info );\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1469\n\n\nvoid pdstebz (MKL_INT *ictxt , char *range , char *order , MKL_INT *n , double *vl ,\ndouble *vu , MKL_INT *il , MKL_INT *iu , double *abstol , double *d , double *e ,\nMKL_INT *m , MKL_INT *nsplit , double *w , MKL_INT *iblock , MKL_INT *isplit , double\n*work , MKL_INT *lwork , MKL_INT *iwork , MKL_INT *liwork , MKL_INT *info );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?stebz function computes the eigenvalues of a symmetric tridiagonal matrix in parallel. These may be\nall eigenvalues, all eigenvalues in the interval [vlvu], or the eigenvalues il through iu. A static partitioning\nof work is done at the beginning of p?stebz which results in all processes finding an (almost) equal number\nof eigenvalues.\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nInput Parameters\nictxt\n(global) The BLACS context handle.\nrange\n(global) Must be 'A' or 'V' or 'I'.\nIf range = 'A', the function computes all eigenvalues.\nIf range = 'V', the function computes eigenvalues in the interval [vl,\nvu].\nIf range ='I', the function computes eigenvalues il through iu.\norder\n(global) Must be 'B' or 'E'.\nIf order = 'B', the eigenvalues are to be ordered from smallest to largest\nwithin each split-off block.\nIf order = 'E', the eigenvalues for the entire matrix are to be ordered\nfrom smallest to largest.\nn\n(global) The order of the tridiagonal matrix T(n≥0).\nvl, vu\n(global)\nIf range = 'V', the function computes the lower and the upper bounds for\nthe eigenvalues on the interval [1, vu].\nIf range = 'A' or 'I', vl and vu are not referenced.\nil, iu\n(global)\nConstraint: 1≤il≤iu≤n.\nIf range = 'I', the index of the smallest eigenvalue is returned for il and\nof the largest eigenvalue for iu (assuming that the eigenvalues are in\nascending order) must be returned.\nIf range = 'A' or 'V', il and iu are not referenced.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1470\n\n\nabstol\n(global)\nThe absolute tolerance to which each eigenvalue is required. An eigenvalue\n(or cluster) is considered to have converged if it lies in an interval of width\nabstol. If abstol≤0, then the tolerance is taken as ulp||T||, where ulp is\nthe machine precision, and ||T|| means the 1-norm of T\nEigenvalues will be computed most accurately when abstol is set to the\nunderflow threshold slamch('U'), not 0. Note that if eigenvectors are\ndesired later by inverse iteration (p?stein), abstol should be set to\n2*p?lamch('S').\nd\n(global) \nArray of size n.\nContains n diagonal elements of the tridiagonal matrix T. To avoid overflow,\nthe matrix must be scaled so that its largest entry is no greater than the\noverflow(1/2) * underflow(1/4) in absolute value, and for greatest\naccuracy, it should not be much smaller than that.\ne\n(global)\nArray of size n - 1.\nContains (n-1) off-diagonal elements of the tridiagonal matrix T. To avoid\noverflow, the matrix must be scaled so that its largest entry is no greater\nthan overflow(1/2) * underflow(1/4) in absolute value, and for greatest\naccuracy, it should not be much smaller than that.\nwork\n(local)\nArray of size max(5n, 7). This is a workspace array.\nlwork\n(local) The size of the work array must be ≥ max(5n, 7).\nIf lwork = -1, then lwork is global input and a workspace query is\nassumed; the function only calculates the minimum and optimal size for all\nwork arrays. Each of these values is returned in the first entry of the\ncorresponding work array, and no error message is issued by pxerbla.\niwork\n(local) Array of size max(4n, 14). This is a workspace array.\nliwork\n(local) The size of the iwork array must ≥max(4n, 14, NPROCS).\nIf liwork = -1, then liwork is global input and a workspace query is\nassumed; the function only calculates the minimum and optimal size for all\nwork arrays. Each of these values is returned in the first entry of the\ncorresponding work array, and no error message is issued by pxerbla.\nOutput Parameters\nm\n(global) The actual number of eigenvalues found. 0≤m≤n\nnsplit\n(global) The number of diagonal blocks detected in T. 1≤nsplit≤n\nw\n(global)\nArray of size n. On exit, the first m elements of w contain the eigenvalues on\nall processes.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1471\n\n\niblock\n(global)\nArray of size n. At each row/column j where e[j-1] is zero or small, the\nmatrix T is considered to split into a block diagonal matrix. On exit\niblock[i] specifies which block (from 1 to the number of blocks) the\neigenvalue w[i] belongs to.\nNOTE\nIn the (theoretically impossible) event that bisection does not\nconverge for some or all eigenvalues, info is set to 1 and the\nones for which it did not are identified by a negative block\nnumber.\nisplit\n(global)\nArray of size n.\nContains the splitting points, at which T breaks up into submatrices. The\nfirst submatrix consists of rows/columns 1 to isplit[0], the second of\nrows/columns isplit[0]+1 through isplit[1], and so on, and the\nnsplit-th submatrix consists of rows/columns isplit[nsplit-2]+1\nthrough isplit[nsplit-1]=n. (Only the first nsplit elements are used,\nbut since the nsplit values are not known, n words must be reserved for\nisplit.)\ninfo\n(global)\nIf info = 0, the execution is successful.\nIf info < 0, if info = -i, the i-th argument has an illegal value.\nIf info> 0, some or all of the eigenvalues fail to converge or are not\ncomputed.\nIf info = 1, bisection fails to converge for some eigenvalues; these\neigenvalues are flagged by a negative block number. The effect is that the\neigenvalues may not be as accurate as the absolute and relative tolerances.\nIf info = 2, mismatch between the number of eigenvalues output and the\nnumber desired.\nIf info = 3: range='I', and the Gershgorin interval initially used is\nincorrect. No eigenvalues are computed. Probable cause: the machine has a\nsloppy floating-point arithmetic. Increase the fudge parameter, recompile,\nand try again.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?stedc\nComputes all eigenvalues and eigenvectors of a\nsymmetric tridiagonal matrix in parallel.\nSyntax\nvoid psstedc (const char* compz, const MKL_INT* n, float* d, float* e, float* q, const\nMKL_INT* iq, const MKL_INT* jq, const MKL_INT* descq, float* work, MKL_INT* lwork,\nMKL_INT* iwork, const MKL_INT* liwork, MKL_INT* info);\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1472\n\n\nvoid pdstedc (const char* compz, const MKL_INT* n, double* d, double* e, double* q,\nconst MKL_INT* iq, const MKL_INT* jq, const MKL_INT* descq, double* work, MKL_INT*\nlwork, MKL_INT* iwork, const MKL_INT* liwork, MKL_INT* info);\nInclude Files\n•\nmkl_scalapack.h\nDescription\np?stedc computes all eigenvalues and eigenvectors of a symmetric tridiagonal matrix in parallel, using the\ndivide and conquer algorithm.\nInput Parameters\ncompz\n= 'N': Compute eigenvalues only. (NOT IMPLEMENTED YET)\n= 'I': Compute eigenvectors of tridiagonal matrix also.\n= 'V': Compute eigenvectors of original dense symmetric matrix also. On\nentry, Z contains the orthogonal matrix used to reduce the original matrix\nto tridiagonal form. (NOT IMPLEMENTED YET)\nn\n(global)\nThe order of the tridiagonal matrix T. n >= 0.\nd\n(global)\nArray, size (n)\nOn entry, the diagonal elements of the tridiagonal matrix.\ne\n(global)\nArray, size (n-1).\nOn entry, the subdiagonal elements of the tridiagonal matrix.\niq\n(global)\nQ's global row index, which points to the beginning of the submatrix which\nis to be operated on.\njq\n(global)\nQ's global column index, which points to the beginning of the submatrix\nwhich is to be operated on.\ndescq\n(global and local)\nArray of size dlen_.\nThe array descriptor for the distributed matrix Q.\nwork\n(local)\nArray, size (lwork)\nlwork\n(local)\nThe size of the array work.\nlwork = 6*n + 2*NP*NQ\nNP = numroc( n, NB, MYROW, DESCQ( rsrc_ ), NPROW )\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1473\n\n\nNQ = numroc( n, NB, MYCOL, DESCQ( csrc_ ), NPCOL )\nnumroc is a ScaLAPACK tool function.\nIf lwork = -1, the lwork is global input and a workspace query is\nassumed; the routine only calculates the minimum size for the work array.\nThe required workspace is returned as the first element of work and no\nerror message is issued by pxerbla.\niwork\n(local)\nArray, size (liwork)\nliwork\nThe size of the array iwork.\nliwork = 2 + 7*n + 8*NPCOL\nOutput Parameters\nd\nOn exit, if info = 0, the eigenvalues in descending order.\nq\n(local)\nArray, local size ( lld_q, LOCc(jq+n-1))\nq contains the orthonormal eigenvectors of the symmetric tridiagonal\nmatrix.\nOn output, q is distributed across the P processes in block cyclic\nformat.\nwork\nOn output, work[0] returns the workspace needed.\niwork\nOn exit, if liwork > 0, iwork[0] returns the optimal liwork.\ninfo\n(global)\n= 0: successful exit.\n< 0: If the i-th argument is an array and the j-th entry had an illegal\nvalue, then info = -(i*100+j), if the i-th argument is a scalar and\nhad an illegal value, then info = -i.\n> 0: The algorithm failed to compute the info/(n+1)-th eigenvalue\nwhile working on the submatrix lying in global rows and columns\nmod(info,n+1).\np?stein\nComputes the eigenvectors of a tridiagonal matrix\nusing inverse iteration.\nSyntax\nvoid psstein (MKL_INT *n , float *d , float *e , MKL_INT *m , float *w , MKL_INT\n*iblock , MKL_INT *isplit , float *orfac , float *z , MKL_INT *iz , MKL_INT *jz ,\nMKL_INT *descz , float *work , MKL_INT *lwork , MKL_INT *iwork , MKL_INT *liwork ,\nMKL_INT *ifail , MKL_INT *iclustr , float *gap , MKL_INT *info );\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1474\n\n\nvoid pdstein (MKL_INT *n , double *d , double *e , MKL_INT *m , double *w , MKL_INT\n*iblock , MKL_INT *isplit , double *orfac , double *z , MKL_INT *iz , MKL_INT *jz ,\nMKL_INT *descz , double *work , MKL_INT *lwork , MKL_INT *iwork , MKL_INT *liwork ,\nMKL_INT *ifail , MKL_INT *iclustr , double *gap , MKL_INT *info );\nvoid pcstein (MKL_INT *n , float *d , float *e , MKL_INT *m , float *w , MKL_INT\n*iblock , MKL_INT *isplit , float *orfac , MKL_Complex8 *z , MKL_INT *iz , MKL_INT *jz ,\nMKL_INT *descz , float *work , MKL_INT *lwork , MKL_INT *iwork , MKL_INT *liwork ,\nMKL_INT *ifail , MKL_INT *iclustr , float *gap , MKL_INT *info );\nvoid pzstein (MKL_INT *n , double *d , double *e , MKL_INT *m , double *w , MKL_INT\n*iblock , MKL_INT *isplit , double *orfac , MKL_Complex16 *z , MKL_INT *iz , MKL_INT\n*jz , MKL_INT *descz , double *work , MKL_INT *lwork , MKL_INT *iwork , MKL_INT\n*liwork , MKL_INT *ifail , MKL_INT *iclustr , double *gap , MKL_INT *info );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?stein function computes the eigenvectors of a symmetric tridiagonal matrix T corresponding to\nspecified eigenvalues, by inverse iteration. p?stein does not orthogonalize vectors that are on different\nprocesses. The extent of orthogonalization is controlled by the input parameter lwork. Eigenvectors that are\nto be orthogonalized are computed by the same process. p?stein decides on the allocation of work among\nthe processes and then calls ?stein2 (modified LAPACK function) on each individual process. If insufficient\nworkspace is allocated, the expected orthogonalization may not be done.\nNOTE\nIf the eigenvectors obtained are not orthogonal, increase lwork and run the code again.\np = NPROW*NPCOL is the total number of processes.\nInput Parameters\nn\n(global) The order of the matrix T(n≥ 0).\nm\n(global) The number of eigenvectors to be returned.\nd, e, w\n(global)\nArrays: \nd of size n contains the diagonal elements of T.\ne of size n-1 contains the off-diagonal elements of T.\nw of size m contains all the eigenvalues grouped by split-off block. The\neigenvalues are supplied from smallest to largest within the block. (Here\nthe output array w from p?stebz with order = 'B' is expected. The array\nshould be replicated in all processes.)\niblock\n(global)\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1475\n\n\nArray of size n. The submatrix indices associated with the corresponding\neigenvalues in w: 1 for eigenvalues belonging to the first submatrix from\nthe top, 2 for those belonging to the second submatrix, etc. (The output\narray iblock from p?stebz is expected here).\nisplit\n(global)\nArray of size n. The splitting points at which T breaks up into submatrices.\nThe first submatrix consists of rows/columns 1 to isplit[0], the second of\nrows/columns isplit[0]+1 through isplit[1], and so on, and the\nnsplit-th submatrix consists of rows/columns isplit[nsplit-2]+1\nthrough isplit[nsplit-1]=n. (The output array isplit from p?stebz is\nexpected here.)\norfac\n(global)\norfac specifies which eigenvectors should be orthogonalized. Eigenvectors\nthat correspond to eigenvalues within orfac*||T|| of each other are to be\northogonalized. However, if the workspace is insufficient (see lwork), this\ntolerance may be decreased until all eigenvectors can be stored in one\nprocess. No orthogonalization is done if orfac is equal to zero. A default\nvalue of 1000 is used if orfac is negative. orfac should be identical on all\nprocesses\niz, jz\n(global) The row and column indices in the global matrix Z indicating the\nfirst row and the first column of the submatrix Z, respectively.\ndescz\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix Z.\nwork\n(local).\nWorkspace array of size lwork.\nlwork\n(local)\nlwork controls the extent of orthogonalization which can be done. The\nnumber of eigenvectors for which storage is allocated on each process is\nnvec = floor((lwork-max(5*n,np00*mq00))/n). Eigenvectors\ncorresponding to eigenvalue clusters of size (nvec - ceil(m/p) + 1) are\nguaranteed to be orthogonal (the orthogonality is similar to that obtained\nfrom ?stein2).\nNOTE\nlwork must be no smaller than max(5*n,np00*mq00) + ceil(m/\np)*n and should have the same input value on all processes.\nIt is the minimum value of lwork input on different processes that is\nsignificant.\nIf lwork = -1, then lwork is global input and a workspace query is\nassumed; the function only calculates the minimum and optimal size for all\nwork arrays. Each of these values is returned in the first entry of the\ncorresponding work array, and no error message is issued by pxerbla.\niwork\n(local)\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1476\n\n\nWorkspace array of size 3n+p+1.\nliwork\n(local) The size of the array iwork. It must be greater than 3*n+p+1.\nIf liwork = -1, then liwork is global input and a workspace query is\nassumed; the function only calculates the minimum and optimal size for all\nwork arrays. Each of these values is returned in the first entry of the\ncorresponding work array, and no error message is issued by pxerbla.\nOutput Parameters\nz\n(local)\nArray of size descz[dlen_-1], n/NPCOL + NB). z contains the computed\neigenvectors associated with the specified eigenvalues. Any vector which\nfails to converge is set to its current iterate after MAXIT iterations\n(See ?stein2). On output, z is distributed across the p processes in block\ncyclic format.\nwork\nOn exit, work[0] gives a lower bound on the workspace (lwork) that\nguarantees the user desired orthogonalization (see orfac). Note that this\nmay overestimate the minimum workspace needed.\niwork\nOn exit, iwork[0] contains the amount of integer workspace required.\nOn exit, the iwork[1] through iwork[p+1] indicate the eigenvectors\ncomputed by each process. Process i computes eigenvectors indexed\niwork[i+1]+1 through iwork[i+2].\nifail\n(global) Array of size m. On normal exit, all elements of ifail are zero. If\none or more eigenvectors fail to converge after MAXIT iterations (as\nin ?stein), then info > 0 is returned. If mod(info, m+1)>0, then for i=1\nto mod(info,m+1), the eigenvector corresponding to the eigenvalue\nw[ifail[i-1]-1] failed to converge (w refers to the array of eigenvalues\non output).\nNOTE\nmod(x,y) is the integer remainder of x/y.\niclustr\n(global) Array of size 2*p.\nThis output array contains indices of eigenvectors corresponding to a cluster\nof eigenvalues that could not be orthogonalized due to insufficient\nworkspace (see lwork, orfac and info). Eigenvectors corresponding to\nclusters of eigenvalues indexed iclustr(2*I-1) to iclustr(2*I), i = 1\nto info/(m+1), could not be orthogonalized due to lack of workspace.\nHence the eigenvectors corresponding to these clusters may not be\northogonal. iclustr is a zero terminated array: iclustr[2*k-1]≠ 0 and\niclustr[2*k] = 0 if and only if k is the number of clusters.\ngap\n(global)\nThis output array contains the gap between eigenvalues whose\neigenvectors could not be orthogonalized. The info/m output values\nin this array correspond to the info/(m+1) clusters indicated by the\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1477\n\n\narray iclustr. As a result, the dot product between eigenvectors\ncorresponding to the i-th cluster may be as high as\n(O(n)*macheps)/gap[i-1].\ninfo\n(global)\nIf info = 0, the execution is successful.\nIf info < 0: If the i-th argument is an array and the j-th entry, indexed\nj-1, had an illegal value, then info = -(i*100+j),\nIf the i-th argument is a scalar and had an illegal value, then info = -i.\nIf info < 0: if info = -i, the i-th argument had an illegal value.\nIf info > 0: if mod(info, m+1) = i, then i eigenvectors failed to converge\nin MAXIT iterations. Their indices are stored in the array ifail. If info/(m\n+1) = i, then eigenvectors corresponding to i clusters of eigenvalues could\nnot be orthogonalized due to insufficient workspace. The indices of the\nclusters are stored in the array iclustr.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\nNonsymmetric Eigenvalue Problems: ScaLAPACK Computational Routines\nThis section describes ScaLAPACK routines for solving nonsymmetric eigenvalue problems, computing the\nSchur factorization of general matrices, as well as performing a number of related computational tasks.\nTo solve a nonsymmetric eigenvalue problem with ScaLAPACK, you usually need to reduce the matrix to the\nupper Hessenberg form and then solve the eigenvalue problem with the Hessenberg matrix obtained.\nTable \"Computational Routines for Solving Nonsymmetric Eigenproblems\"lists ScaLAPACK routines for\nreducing the matrix to the upper Hessenberg form by an orthogonal (or unitary) similarity transformation A=\nQHQH, as well as routines for solving eigenproblems with Hessenberg matrices, and multiplying the matrix\nafter reduction.\nComputational Routines for Solving Nonsymmetric Eigenproblems\nOperation performed\nGeneral matrix\nOrthogonal/Unitary\nmatrix\nHessenberg matrix\nReduce to Hessenberg form A= QHQH\np?gehrd\n \n \nMultiply the matrix after reduction\n \np?ormhr/ p?unmhr\n \nFind eigenvalues and Schur\nfactorization\n \n \np?lahqr\np?gehrd\nReduces a general matrix to upper Hessenberg form.\nSyntax\nvoid psgehrd (MKL_INT *n , MKL_INT *ilo , MKL_INT *ihi , float *a , MKL_INT *ia ,\nMKL_INT *ja , MKL_INT *desca , float *tau , float *work , MKL_INT *lwork , MKL_INT\n*info );\nvoid pdgehrd (MKL_INT *n , MKL_INT *ilo , MKL_INT *ihi , double *a , MKL_INT *ia ,\nMKL_INT *ja , MKL_INT *desca , double *tau , double *work , MKL_INT *lwork , MKL_INT\n*info );\nvoid pcgehrd (MKL_INT *n , MKL_INT *ilo , MKL_INT *ihi , MKL_Complex8 *a , MKL_INT\n*ia , MKL_INT *ja , MKL_INT *desca , MKL_Complex8 *tau , MKL_Complex8 *work , MKL_INT\n*lwork , MKL_INT *info );\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1478\n\n\nvoid pzgehrd (MKL_INT *n , MKL_INT *ilo , MKL_INT *ihi , MKL_Complex16 *a , MKL_INT\n*ia , MKL_INT *ja , MKL_INT *desca , MKL_Complex16 *tau , MKL_Complex16 *work , MKL_INT\n*lwork , MKL_INT *info );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?gehrd function reduces a real/complex general distributed matrix sub(A) to upper Hessenberg form H\nby an orthogonal or unitary similarity transformation\nQ'*sub(A)*Q = H,\nwhere sub(A) = A(ia:ia+n-1, ja:ja+n-1).\nInput Parameters\nn\n(global). The order of the distributed matrix sub(A) (n≥0).\nilo, ihi\n(global).\nIt is assumed that sub(A) is already upper triangular in rows ia:ia+ilo-2\nand ia+ihi:ia+n-1 and columns ja:ja+ilo-2 and ja+ihi:ja+n-1. (See\nApplication Notes below).\nIf n > 0, 1≤ilo≤ihi≤n; otherwise set ilo = 1, ihi = n.\na\n(local)\nPointer into the local memory to an array of size lld_a*LOCc(ja+n-1). On\nentry, this array contains the local pieces of the n-by-n general distributed\nmatrix sub(A) to be reduced.\nia, ja\n(global) The row and column indices in the global matrix A indicating the\nfirst row and the first column of the submatrix A, respectively.\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\nwork\n(local)\nWorkspace array of size lwork.\nlwork\n(local or global) size of the array work. lwork is local input and must be at\nleast\nlwork≥NB*NB + NB*max(ihip+1, ihlp+inlq)\nwhere NB = mb_a = nb_a,\niroffa = mod(ia-1, NB),\nicoffa = mod(ja-1, NB),\nioff = mod(ia+ilo-2, NB), iarow = indxg2p(ia, NB, MYROW,\nrsrc_a, NPROW), ihip = numroc(ihi+iroffa, NB, MYROW, iarow,\nNPROW),\nilrow = indxg2p(ia+ilo-1, NB, MYROW, rsrc_a, NPROW),\nihlp = numroc(ihi-ilo+ioff+1, NB, MYROW, ilrow, NPROW),\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1479\n\n\nilcol = indxg2p(ja+ilo-1, NB, MYCOL, csrc_a, NPCOL),\ninlq = numroc(n-ilo+ioff+1, NB, MYCOL, ilcol, NPCOL),\nNOTE\nmod(x,y) is the integer remainder of x/y.\nindxg2p and numroc are ScaLAPACK tool functions; MYROW, MYCOL, NPROW\nand NPCOL can be determined by calling the function blacs_gridinfo.\nIf lwork = -1, then lwork is global input and a workspace query is\nassumed; the function only calculates the minimum and optimal size for all\nwork arrays. Each of these values is returned in the first entry of the\ncorresponding work array, and no error message is issued by pxerbla.\nOutput Parameters\na\nOn exit, the upper triangle and the first subdiagonal of sub(A) are\noverwritten with the upper Hessenberg matrix H, and the elements below\nthe first subdiagonal, with the array tau, represent the orthogonal/unitary\nmatrix Q as a product of elementary reflectors (see Application Notes\nbelow).\ntau\n(local).\nArray of size at least max(ja+n-2).\nThe scalar factors of the elementary reflectors (see Application Notes\nbelow). Elements ja:ja+ilo-2 and ja+ihi:ja+n-2 of the global vector\ntau are set to zero. tau is tied to the distributed matrix A.\nwork[0]\nOn exit work[0] contains the minimum value of lwork required for\noptimum performance.\ninfo\n(global)\n= 0: the execution is successful.\n< 0: if the i-th argument is an array and the j-th entry, indexed j - 1, had\nan illegal value, then info = -(i*100+j); if the i-th argument is a scalar\nand had an illegal value, then info = -i.\nApplication Notes\nThe matrix Q is represented as a product of (ihi-ilo) elementary reflectors\nQ = H(ilo)*H(ilo+1)*...*H(ihi-1).\nEach H(i) has the form\nH(i)= i - tau*v*v'\nwhere tau is a real/complex scalar, and v is a real/complex vector with v(1:i)= 0, v(i+1)= 1 and v(ihi\n+1:n)= 0; v(i+2:ihi) is stored on exit in A(ia+ilo+i:ia+ihi-1,ja+ilo+i-2), and tau in tau[ja+ilo\n+i-3]. The contents of A\n(ia:ia+n-1,ja:ja+n-1) are illustrated by the following example, with n = 7, ilo = 2 and ihi = 6:\non entry\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1480\n\n\non exit\nwhere a denotes an element of the original matrix sub(A), H denotes a modified element of the upper\nHessenberg matrix H, and vi denotes an element of the vector defining H(ja+ilo+i-2).\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?ormhr\nMultiplies a general matrix by the orthogonal\ntransformation matrix from a reduction to Hessenberg\nform determined by p?gehrd.\nSyntax\nvoid psormhr (char *side , char *trans , MKL_INT *m , MKL_INT *n , MKL_INT *ilo ,\nMKL_INT *ihi , float *a , MKL_INT *ia , MKL_INT *ja , MKL_INT *desca , float *tau ,\nfloat *c , MKL_INT *ic , MKL_INT *jc , MKL_INT *descc , float *work , MKL_INT *lwork ,\nMKL_INT *info );\nvoid pdormhr (char *side , char *trans , MKL_INT *m , MKL_INT *n , MKL_INT *ilo ,\nMKL_INT *ihi , double *a , MKL_INT *ia , MKL_INT *ja , MKL_INT *desca , double *tau ,\ndouble *c , MKL_INT *ic , MKL_INT *jc , MKL_INT *descc , double *work , MKL_INT *lwork ,\nMKL_INT *info );\nInclude Files\n•\nmkl_scalapack.h\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1481\n\n\nDescription\nThe p?ormhr function overwrites the general real distributed m-by-n matrix sub(C)= C(iс:iс+m-1,jс:jс\n+n-1) with\nside ='L'\nside ='R'\ntrans = 'N':\nQ*sub(C)\nsub(C)*Q\ntrans = 'T':\nQT*sub(C)\nsub(C)*QT\nwhere Q is a real orthogonal distributed matrix of order nq, with nq = m if side = 'L' and nq = n if side =\n'R'.\nQ is defined as the product of ihi-ilo elementary reflectors, as returned by p?gehrd.\nQ = H(ilo) H(ilo+1)... H(ihi-1).\nInput Parameters\nside\n(global)\n='L': Q or QT is applied from the left.\n='R': Q or QT is applied from the right.\ntrans\n(global)\n='N', no transpose, Q is applied.\n='T', transpose, QT is applied.\nm\n(global) The number of rows in the distributed matrix sub (C) (m≥0).\nn\n(global) The number of columns in he distributed matrix sub (C) (n≥0).\nilo, ihi\n(global)\nilo and ihi must have the same values as in the previous call of p?gehrd.\nQ is equal to the unit matrix except for the distributed submatrix Q(ia\n+ilo:ia+ihi-1,ja+ilo:ja+ihi-1).\nIf side = 'L', 1≤ilo≤ihi≤max(1,m);\nIf side = 'R', 1≤ilo≤ihi≤max(1,n);\nilo and ihi are relative indexes.\na\n(local)\nPointer into the local memory to an array of size lld_a*LOCc(ja+m-1) if\nside = 'L', and lld_a*LOCc(ja+n-1) if side = 'R'.\nContains the vectors which define the elementary reflectors, as returned by\np?gehrd.\nia, ja\n(global) The row and column indices in the global matrix A indicating the\nfirst row and the first column of the submatrix A, respectively.\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\ntau\n(local)\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1482\n\n\nArray of size LOCc(ja+m-2) if side = 'L', and LOCc(ja+n-2) if side =\n'R'.\ntau[j] contains the scalar factor of the elementary reflector H(j+1) as\nreturned by p?gehrd (0 ≤ j < size(tau)). tau is tied to the distributed\nmatrix A.\nc\n(local)\nPointer into the local memory to an array of size lld_c*LOCc(jc+n-1).\nContains the local pieces of the distributed matrix sub(C).\nic, jc\n(global) The row and column indices in the global matrix C indicating the\nfirst row and the first column of the submatrix C, respectively.\ndescc\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix C.\nwork\n(local)\nWorkspace array with size lwork.\nlwork\n(local or global)\nThe size of the array work.\nlwork must be at least iaa = ia + ilo; jaa = ja+ilo-1;\nIf side = 'L',\nmi = ihi-ilo; ni = n; icc = ic + ilo; jcc = jc; lwork ≥\nmax((nb_a*(nb_a-1))/2, (nqc0+mpc0)*nb_a) + nb_a*nb_a\nelse if side = 'R',\nmi = m; ni = ihi-ilo; icc = ic; jcc = jc + ilo; lwork ≥\nmax((nb_a*(nb_a-1))/2, (nqc0+max(npa0+numroc(numroc(ni\n+icoffc, nb_a, 0, 0, NPCOL), nb_a, 0, 0, lcmq), mpc0))*nb_a)\n+ nb_a*nb_a\nend if\nwhere lcmq = lcm/NPCOL with lcm = ilcm(NPROW, NPCOL),\niroffa = mod(iaa-1, mb_a),\nicoffa = mod(jaa-1, nb_a),\niarow = indxg2p(iaa, mb_a, MYROW, rsrc_a, NPROW),\nnpa0 = numroc(ni+iroffa, mb_a, MYROW, iarow, NPROW),\niroffc = mod(icc-1, mb_c), icoffc = mod(jcc-1, nb_c),\nicrow = indxg2p(icc, mb_c, MYROW, rsrc_c, NPROW),\niccol = indxg2p(jcc, nb_c, MYCOL, csrc_c, NPCOL),\nmpc0 = numroc(mi+iroffc, mb_c, MYROW, icrow, NPROW),\nnqc0 = numroc(ni+icoffc, nb_c, MYCOL, iccol, NPCOL),\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1483\n\n\nNOTE\nmod(x,y) is the integer remainder of x/y.\nilcm, indxg2p and numroc are ScaLAPACK tool functions; MYROW, MYCOL,\nNPROW and NPCOL can be determined by calling the function\nblacs_gridinfo.\nIf lwork = -1, then lwork is global input and a workspace query is\nassumed; the function only calculates the minimum and optimal size for all\nwork arrays. Each of these values is returned in the first entry of the\ncorresponding work array, and no error message is issued by pxerbla.\nOutput Parameters\nc\nsub(C) is overwritten by Q*sub(C), or Q'*sub(C), or sub(C)*Q', or\nsub(C)*Q.\nwork[0]\nOn exit work[0] contains the minimum value of lwork required for\noptimum performance.\ninfo\n(global)\n= 0: the execution is successful.\n< 0: if the i-th argument is an array and the j-th entry, indexed j - 1, had\nan illegal value, then info = -(i*100+j); if the i-th argument is a scalar\nand had an illegal value, then info = -i.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?unmhr\nMultiplies a general matrix by the unitary\ntransformation matrix from a reduction to Hessenberg\nform determined by p?gehrd.\nSyntax\nvoid pcunmhr (char *side , char *trans , MKL_INT *m , MKL_INT *n , MKL_INT *ilo ,\nMKL_INT *ihi , MKL_Complex8 *a , MKL_INT *ia , MKL_INT *ja , MKL_INT *desca ,\nMKL_Complex8 *tau , MKL_Complex8 *c , MKL_INT *ic , MKL_INT *jc , MKL_INT *descc ,\nMKL_Complex8 *work , MKL_INT *lwork , MKL_INT *info );\nvoid pzunmhr (char *side , char *trans , MKL_INT *m , MKL_INT *n , MKL_INT *ilo ,\nMKL_INT *ihi , MKL_Complex16 *a , MKL_INT *ia , MKL_INT *ja , MKL_INT *desca ,\nMKL_Complex16 *tau , MKL_Complex16 *c , MKL_INT *ic , MKL_INT *jc , MKL_INT *descc ,\nMKL_Complex16 *work , MKL_INT *lwork , MKL_INT *info );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThis function overwrites the general complex distributed m-by-n matrix sub(C) = C(iс:iс+m-1,jс:jс+n-1)\nwith\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1484\n\n\nside ='L'\nside ='R'\ntrans = 'N':\nQ*sub(C)\nsub(C)*Q\ntrans = 'H':\nQH*sub(C)\nsub(C)*QH\nwhere Q is a complex unitary distributed matrix of order nq, with nq = m if side = 'L' and nq = n if side\n= 'R'.\nQ is defined as the product of ihi-ilo elementary reflectors, as returned by p?gehrd.\nQ = H(ilo) H(ilo+1)... H(ihi-1).\nInput Parameters\nside\n(global)\n='L': Q or QH is applied from the left.\n='R': Q or QH is applied from the right.\ntrans\n(global)\n='N', no transpose, Q is applied.\n='C', conjugate transpose, QH is applied.\nm\n(global) The number of rows in the distributed matrix sub (C) (m≥0).\nn\n(global) The number of columns in the distributed matrix sub (C) (n≥0).\nilo, ihi\n(global)\nThese must be the same parameters ilo and ihi, respectively, as supplied\nto p?gehrd. Q is equal to the unit matrix except in the distributed\nsubmatrixQ(ia+ilo:ia+ihi-1,ja+ilo:ja+ihi-1).\nIf side ='L', then 1≤ilo≤ihi≤max(1,m).\nIf side = 'R', then 1≤ilo≤ihi≤max(1,n)\nilo and ihi are relative indexes.\na\n(local)\nPointer into the local memory to an array of size lld_a*LOCc(ja+m-1) if\nside = 'L', and lld_a*LOCc(ja+n-1) if side = 'R'.\nContains the vectors which define the elementary reflectors, as returned by\np?gehrd.\nia, ja\n(global) The row and column indices in the global matrix A indicating the\nfirst row and the first column of the submatrix A, respectively.\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\ntau\n(local)\nArray of size LOCc(ja+m-2), if side = 'L', and LOCc(ja+n-2) if side =\n'R'.\ntau[j] contains the scalar factor of the elementary reflector H(j+1) as\nreturned by p?gehrd (0 ≤ j < size(tau)). tau is tied to the distributed\nmatrix A.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1485\n\n\nc\n(local)\nPointer into the local memory to an array of size lld_c*LOCc(jc+n-1).\nContains the local pieces of the distributed matrix sub(C).\nic, jc\n(global) The row and column indices in the global matrix C indicating the\nfirst row and the first column of the submatrix C, respectively.\ndescc\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix C.\nwork\n(local)\nWorkspace array with size lwork.\nlwork\n(local or global)\nThe size of the array work.\nlwork must be at least iaa = ia + ilo;jaa = ja+ilo-1;\nIf side = 'L', mi = ihi-ilo; ni = n; icc = ic + ilo; jcc = jc;\nlwork ≥ max((nb_a*(nb_a-1))/2, (nqc0+mpc0)*nb_a) + nb_a*nb_a\nelse if side = 'R',\nmi = m; ni = ihi-ilo; icc = ic; jcc = jc + ilo; lwork ≥\nmax((nb_a*(nb_a-1))/2, (nqc0 + max(npa0+numroc(numroc(ni\n+icoffc, nb_a, 0, 0, NPCOL), nb_a, 0, 0, lcmq ),\nmpc0))*nb_a) + nb_a*nb_a\nend if\nwhere lcmq = lcm/NPCOL with lcm = ilcm(NPROW, NPCOL),\niroffa = mod(iaa-1, mb_a),\nicoffa = mod(jaa-1, nb_a),\niarow = indxg2p(iaa, mb_a, MYROW, rsrc_a, NPROW),\nnpa0 = numroc(ni+iroffa, mb_a, MYROW, iarow, NPROW),\niroffc = mod(icc-1, mb_c),\nicoffc = mod(jcc-1, nb_c),\nicrow = indxg2p(icc, mb_c, MYROW, rsrc_c, NPROW),\niccol = indxg2p(jcc, nb_c, MYCOL, csrc_c, NPCOL),\nmpc0 = numroc(mi+iroffc, mb_c, MYROW, icrow, NPROW),\nnqc0 = numroc(ni+icoffc, nb_c, MYCOL, iccol, NPCOL),\nNOTE\nmod(x,y) is the integer remainder of x/y.\nilcm, indxg2p and numroc are ScaLAPACK tool functions; MYROW, MYCOL,\nNPROW and NPCOL can be determined by calling the function\nblacs_gridinfo.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1486\n\n\nIf lwork = -1, then lwork is global input and a workspace query is\nassumed; the function only calculates the minimum and optimal size for all\nwork arrays. Each of these values is returned in the first entry of the\ncorresponding work array, and no error message is issued by pxerbla.\nOutput Parameters\nc\nC is overwritten by Q* sub(C) or Q'*sub(C) or sub(C)*Q' or sub(C)*Q.\nwork[0])\nOn exit work[0] contains the minimum value of lwork required for\noptimum performance.\ninfo\n(global)\n= 0: the execution is successful.\n< 0: if the i-th argument is an array and the j-th entry, indexed j - 1, had\nan illegal value, then info = -(i*100+j); if the i-th argument is a scalar\nand had an illegal value, then info = -i.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?lahqr\nComputes the Schur decomposition and/or\neigenvalues of a matrix already in Hessenberg form.\nSyntax\nvoid pslahqr (MKL_INT *wantt, MKL_INT *wantz, MKL_INT *n, MKL_INT *ilo, MKL_INT *ihi,\nfloat *a, MKL_INT *desca, float *wr, float *wi, MKL_INT *iloz, MKL_INT *ihiz, float *z,\nMKL_INT *descz, float *work, MKL_INT *lwork, MKL_INT *iwork, MKL_INT *ilwork, MKL_INT\n*info );\nvoid pdlahqr (MKL_INT *wantt, MKL_INT *wantz, MKL_INT *n, MKL_INT *ilo, MKL_INT *ihi,\ndouble *a, MKL_INT *desca, double *wr, double *wi, MKL_INT *iloz, MKL_INT *ihiz, double\n*z, MKL_INT *descz, double *work, MKL_INT *lwork, MKL_INT *iwork, MKL_INT *ilwork,\nMKL_INT *info );\nvoid pclahqr (const MKL_INT *wantt, const MKL_INT *wantz, const MKL_INT *n, const\nMKL_INT *ilo, const MKL_INT *ihi, MKL_Complex8 *a, const MKL_INT *desca, MKL_Complex8\n*w, const MKL_INT *iloz, const MKL_INT *ihiz, MKL_Complex8 *z, const MKL_INT *descz,\nMKL_Complex8 *work, const MKL_INT *lwork, const MKL_INT *iwork, const MKL_INT *ilwork,\nMKL_INT *info );\nvoid pzlahqr (const MKL_INT *wantt, const MKL_INT *wantz, const MKL_INT *n, const\nMKL_INT *ilo, const MKL_INT *ihi, MKL_Complex16 *a, const MKL_INT *desca, MKL_Complex16\n*w, const MKL_INT *iloz, const MKL_INT *ihiz, MKL_Complex16 *z, const MKL_INT *descz,\nMKL_Complex16 *work, const MKL_INT *lwork, const MKL_INT *iwork, const MKL_INT *ilwork,\nMKL_INT *info );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThis is an auxiliary function used to find the Schur decomposition and/or eigenvalues of a matrix already in\nHessenberg form from columns ilo and ihi.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1487\n\n\nNOTE\nThese restrictions apply to the use of p?lahqr:\n•\nThe code requires the distributed block size to be square and at least 6.\n•\nThe code requires A and Z to be distributed identically and have identical contexts.\n•\nThe matrix A must be in upper Hessenberg form. If elements below the subdiagonal are non-zero,\nthe resulting transformations can be nonsimilar.\n•\nAll eigenvalues are distributed to all the nodes.\nInput Parameters\nwantt\n(global)\nIf wantt≠ 0, the full Schur form T is required;\nIf wantt = 0, only eigenvalues are required.\nwantz\n(global)\nIf wantz≠ 0, the matrix of Schur vectors Z is required;\nIf wantz = 0, Schur vectors are not required.\nn\n(global) The order of the Hessenberg matrix A (and z if wantz is non-zero).\nn≥0.\nilo, ihi\n(global)\nIt is assumed that A is already upper quasi-triangular in rows and columns\nihi+1:n, and that A(ilo, ilo-1) = 0 (unless ilo = 1). p?lahqr works\nprimarily with the Hessenberg submatrix in rows and columns ilo to ihi,\nbut applies transformations to all of H if wantt is non-zero.\n1≤ilo≤max(1,ihi); ihi ≤ n.\na\n(global)\nArray, of size lld_a * LOCc(n) . On entry, the upper Hessenberg matrix A.\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\niloz, ihiz\n(global) Specify the rows of the matrix Z to which transformations must be\napplied if wantz is non-zero. 1≤iloz≤ilo; ihi≤ihiz≤n.\nz\n(global )\nArray. If wantz is non-zero, on entry z must contain the current matrix Z of\ntransformations accumulated by pdhseqr. If wantz is zero, z is not\nreferenced.\ndescz\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix Z.\nwork\n(local)\nWorkspace array with size lwork.\nlwork\n(local) The size of work. lwork is assumed big enough so that lwork≥3*n\n+ max(2*max(lld_z,lld_a) + 2*LOCq(n), 7*ceil(n/hbl)/\nlcm(NPROW,NPCOL))).\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1488\n\n\nIf lwork = -1, then work[0] gets set to the above number and the code\nreturns immediately.\niwork\n(global and local) array of size ilwork. Not referenced and can be NULL\npointer.\nilwork\n(local) This holds some of the iblk integer arrays. Not referenced and can be\nNULL pointer.\nOutput Parameters\na\nOn exit, if wantt is non-zero, A is upper quasi-triangular in rows and\ncolumns ilo:ihi, with any 2-by-2 or larger diagonal blocks not yet in\nstandard form. If wantt is zero, the contents of A are unspecified on exit.\nwork[0]\nOn exit work[0] contains the minimum value of lwork required for\noptimum performance.\nwr, wi\n(global replicated output)\nArrays of size n each. The real and imaginary parts, respectively, of the\ncomputed eigenvalues ilo to ihiare stored in the corresponding elements\nof wr and wi. If two eigenvalues are computed as a complex conjugate pair,\nthey are stored in consecutive elements of wr and wi, say the i-th and (i\n+1)-th, with wi[i-1]> 0 and wi[i] < 0. If wantt is zero, the eigenvalues are\nstored in the same order as on the diagonal of the Schur form returned in\nA. A may be returned with larger diagonal blocks until the next release.\nw\n(global replicated output)\nArray of size n. The computed eigenvalues ilo to ihi are stored in the\ncorresponding elements of w. If two eigenvalues are computed as a complex\nconjugate pair, they are stored in consecutive elements of w, say the i-th\nand (i+1)-th, with w[i-1]> 0 and w[i] < 0. If wantt is zero, the eigenvalues\nare stored in the same order as on the diagonal of the Schur form returned\nin A. A may be returned with larger diagonal blocks until the next release.\nz\nOn exit z has been updated; transformations are applied only to the\nsubmatrix Z(iloz:ihiz, ilo:ihi).\ninfo\n(global)\n= 0: the execution is successful.\n< 0: the parameter number - info is incorrect or inconsistent\n> 0: p?lahqr failed to compute all the eigenvalues ilo to ihi in a total of\n30*(ihi-ilo+1) iterations; if info = i, elements i+1: ihi of wr and wi\ncontain the eigenvalues that have been successfully computed.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?trevc\nComputes right and/or left eigenvectors of a complex\nupper triangular matrix in parallel.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1489\n\n\nSyntax\nvoid pctrevc (const char* side, const char* howmny, const MKL_INT* select, const\nMKL_INT* n, MKL_Complex8* t, const MKL_INT* desct, MKL_Complex8* vl, const MKL_INT*\ndescvl, MKL_Complex8* vr, const MKL_INT* descvr, const MKL_INT* mm, MKL_INT* m,\nMKL_Complex8* work, float* rwork, MKL_INT* info);\nvoid pztrevc (const char* side, const char* howmny, const MKL_INT* select, const\nMKL_INT* n, MKL_Complex16* t, const MKL_INT* desct, MKL_Complex16* vl, const MKL_INT*\ndescvl, MKL_Complex16* vr, const MKL_INT* descvr, const MKL_INT* mm, MKL_INT* m,\nMKL_Complex16* work, double* rwork, MKL_INT* info);\nvoid pdtrevc (const char* side, const char* howmny, const MKL_INT* select, const\nMKL_INT* n, double* t, const MKL_INT* desct, double* vl, const MKL_INT* descvl, double*\nvr, const MKL_INT* descvr, const MKL_INT* mm, MKL_INT* m, double* work, MKL_INT* info);\nvoid pstrevc (const char* side, const char* howmny, const MKL_INT* select, const\nMKL_INT* n, float* t, const MKL_INT* desct, float* vl, const MKL_INT* descvl, float* vr,\nconst MKL_INT* descvr, const MKL_INT* mm, MKL_INT* m, float* work, MKL_INT* info);\nInclude Files\n•\nmkl_scalapack.h\nDescription\np?trevc computes some or all of the right and/or left eigenvectors of a complex upper triangular matrix T in\nparallel.\nThe right eigenvector x and the left eigenvector y of T corresponding to an eigenvalue w are defined by:\nT*x = w*x,\ny'*T = w*y'\nwhere y' denotes the conjugate transpose of the vector y.\nIf all eigenvectors are requested, the routine may either return the matrices X and/or Y of right or left\neigenvectors of T, or the products Q*X and/or Q*Y, where Q is an input unitary matrix. If T was obtained\nfrom the Schur factorization of an original matrix A = Q*T*Q', then Q*X and Q*Y are the matrices of right or\nleft eigenvectors of A.\nInput Parameters\nside\n(global)\n= 'R': compute right eigenvectors only;\n= 'L': compute left eigenvectors only;\n= 'B': compute both right and left eigenvectors.\nhowmny\n(global)\n= 'A': compute all right and/or left eigenvectors;\n= 'B': compute all right and/or left eigenvectors, and backtransform them\nusing the input matrices supplied in vr and/or vl;\n= 'S': compute selected right and/or left eigenvectors, specified by the\nlogical array select.\nselect\n(global)\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1490\n\n\nArray, size (n)\nIf howmny = 'S', select specifies the eigenvectors to be computed.\nIf howmny = 'A' or 'B', select is not referenced. To select the eigenvector\ncorresponding to the j-th eigenvalue, select[j - 1] must be set to non-\nzero.\nn\n(global)\nThe order of the matrix T. n >= 0.\nt\n(local)\nArray, size lld_t*LOCc(n).\nThe upper triangular matrix T. T is modified, but restored on exit.\ndesct\n(global and local)\nArray of size dlen_.\nThe array descriptor for the distributed matrix T.\nvl\n(local)\nArray, size (descvl(lld_),mm)\nOn entry, if side = 'L' or 'B' and howmny = 'B', vl must contain an n-by-n\nmatrix Q (usually the unitary matrix Q of Schur vectors returned\nby ?hseqr).\ndescvl\n(global and local)\nArray of size dlen_.\nThe array descriptor for the distributed matrix VL.\nvr\n(local)\nArray, size descvr(lld_)*mm.\nOn entry, if side = 'R' or 'B' and howmny = 'B', vr must contain an n-by-n\nmatrix Q (usually the unitary matrix Q of Schur vectors returned\nby ?hseqr).\ndescvr\n(global and local)\nArray of size dlen_.\nThe array descriptor for the distributed matrix VR.\nmm\n(global)\nThe number of columns in the arrays vl and/or vr. mm >= m.\nwork\n(local)\nArray, size ( 2*desct(lld_) )\nAdditional workspace may be required if p?lattrs is updated to use work.\nrwork\nArray, size ( desct(lld_) )\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1491\n\n\nOutput Parameters\nt\nThe upper triangular matrix T. T is modified, but restored on exit.\nvl\nOn exit, if side = 'L' or 'B', vl contains:\nif howmny = 'A', the matrix Y of left eigenvectors of T;\nif howmny = 'B', the matrix Q*Y;\nif howmny = 'S', the left eigenvectors of T specified by select, stored\nconsecutively in the columns of vl, in the same order as their\neigenvalues. If side = 'R', vl is not referenced.\nvr\nOn exit, if side = 'R' or 'B', vr contains:\nif howmny = 'A', the matrix X of right eigenvectors of T;\nif howmny = 'B', the matrix Q*X;\nif howmny = 'S', the right eigenvectors of T specified by select,\nstored consecutively in the columns of vr, in the same order as their\neigenvalues. If side = 'L', vr is not referenced.\nm\n(global)\nThe number of columns in the arrays vl and/or vr actually used to\nstore the eigenvectors. If howmny = 'A' or 'B', m is set to n. Each\nselected eigenvector occupies one column.\ninfo\n(global)\n= 0: successful exit\n< 0: if info = -i, the i-th argument had an illegal value\nApplication Notes\nThe algorithm used in this program is basically backward (forward) substitution. Scaling should be used to\nmake the code robust against possible overflow. But scaling has not yet been implemented in p?lattrs\nwhich is called by this routine to solve the triangular systems. p?lattrs just calls p?trsv.\nEach eigenvector is normalized so that the element of largest magnitude has magnitude 1; here the\nmagnitude of a complex number (x,y) is taken to be |x| + |y|.\nSingular Value Decomposition: ScaLAPACK Driver Routines\nThis section describes ScaLAPACK routines for computing the singular value decomposition (SVD) of a\ngeneral m-by-n matrix A (see LAPACK\"Singular Value Decomposition\" ).\nTo find the SVD of a general matrix A, this matrix is first reduced to a bidiagonal matrix B by a unitary\n(orthogonal) transformation, and then SVD of the bidiagonal matrix is computed. Note that the SVD of B is\ncomputed using the LAPACK routine ?bdsqr .\nTable \"Computational Routines for Singular Value Decomposition (SVD)\" lists ScaLAPACK computational\nroutines for performing this decomposition.\nComputational Routines for Singular Value Decomposition (SVD)\nOperation\nGeneral matrix\nOrthogonal/unitary matrix\nReduce A to a bidiagonal matrix\np?gebrd\n \nMultiply matrix after reduction\n \np?ormbr/p?unmbr\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1492\n\n\np?gebrd\nReduces a general matrix to bidiagonal form.\nSyntax\nvoid psgebrd (MKL_INT *m , MKL_INT *n , float *a , MKL_INT *ia , MKL_INT *ja , MKL_INT\n*desca , float *d , float *e , float *tauq , float *taup , float *work , MKL_INT\n*lwork , MKL_INT *info );\nvoid pdgebrd (MKL_INT *m , MKL_INT *n , double *a , MKL_INT *ia , MKL_INT *ja , MKL_INT\n*desca , double *d , double *e , double *tauq , double *taup , double *work , MKL_INT\n*lwork , MKL_INT *info );\nvoid pcgebrd (MKL_INT *m , MKL_INT *n , MKL_Complex8 *a , MKL_INT *ia , MKL_INT *ja ,\nMKL_INT *desca , float *d , float *e , MKL_Complex8 *tauq , MKL_Complex8 *taup ,\nMKL_Complex8 *work , MKL_INT *lwork , MKL_INT *info );\nvoid pzgebrd (MKL_INT *m , MKL_INT *n , MKL_Complex16 *a , MKL_INT *ia , MKL_INT *ja ,\nMKL_INT *desca , double *d , double *e , MKL_Complex16 *tauq , MKL_Complex16 *taup ,\nMKL_Complex16 *work , MKL_INT *lwork , MKL_INT *info );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?gebrd function reduces a real/complex general m-by-n distributed matrix sub(A)= A(ia:ia+m-1,\nja:ja+n-1) to upper or lower bidiagonal form B by an orthogonal/unitary transformation:\nQ'*sub(A)*P = B.\nIf m≥ n, B is upper bidiagonal; if m < n, B is lower bidiagonal.\nInput Parameters\nm\n(global) The number of rows in the distributed matrix sub(A) (m≥0).\nn\n(global) The number of columns in the distributed matrix sub(A) (n≥0).\na\n(local)\nReal pointer into the local memory to an array of size lld_a*LOCc(ja+n-1).\nOn entry, this array contains the distributed matrix sub (A).\nia, ja\n(global) The row and column indices in the global matrix A indicating the\nfirst row and the first column of the submatrix A, respectively.\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\nwork\n(local)\nWorkspace array of size lwork.\nlwork\n(local or global) size of work, must be at least:\nlwork≥nb*(mpa0 + nqa0+1)+ nqa0\nwhere nb = mb_a = nb_a,\niroffa = mod(ia-1, nb),\nicoffa = mod(ja-1, nb),\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1493\n\n\niarow = indxg2p(ia, nb, MYROW, rsrc_a, NPROW),\niacol = indxg2p (ja, nb, MYCOL, csrc_a, NPCOL),\nmpa0 = numroc(m +iroffa, nb, MYROW, iarow, NPROW),\nnqa0 = numroc(n +icoffa, nb, MYCOL, iacol, NPCOL),\nNOTE\nmod(x,y) is the integer remainder of x/y.\nindxg2p and numroc are ScaLAPACK tool functions; MYROW, MYCOL, NPROW\nand NPCOL can be determined by calling the function blacs_gridinfo.\nIf lwork = -1, then lwork is global input and a workspace query is\nassumed; the function only calculates the minimum and optimal size for all\nwork arrays. Each of these values is returned in the first entry of the\ncorresponding work array, and no error message is issued by pxerbla.\nOutput Parameters\na\nOn exit, if m≥n, the diagonal and the first superdiagonal of sub(A) are\noverwritten with the upper bidiagonal matrix B; the elements below the\ndiagonal, with the array tauq, represent the orthogonal/unitary matrix Q as\na product of elementary reflectors, and the elements above the first\nsuperdiagonal, with the array taup, represent the orthogonal matrix P as a\nproduct of elementary reflectors. If m < n, the diagonal and the first\nsubdiagonal are overwritten with the lower bidiagonal matrix B; the\nelements below the first subdiagonal, with the array tauq, represent the\northogonal/unitary matrix Q as a product of elementary reflectors, and the\nelements above the diagonal, with the array taup, represent the orthogonal\nmatrix P as a product of elementary reflectors. See Application Notes below.\nd\n(local)\nArray of size LOCc(ja+min(m,n)-1) if m≥n and LOCr(ia+min(m,n)-1)\notherwise. The distributed diagonal elements of the bidiagonal matrix B: \nd[i] = A(i+1,i+1), 0 ≤ i < size (d).\nd is tied to the distributed matrix A.\ne\n(local)\nArray of size LOCr(ia+min(m,n)-1) if m≥n; LOCc(ja+min(m,n)-2)\notherwise. The distributed off-diagonal elements of the bidiagonal\ndistributed matrix B:\nIf m≥n, e[i] = A(i+1,i+2) for i = 0,1,..., n-2; if m < n, e[i] = A(i+2,i+1)\nfor i = 0,1,..., m-2. e is tied to the distributed matrix A.\ntauq, taup\n(local)\nArrays of size LOCc(ja+min(m,n)-1) for tauq and LOCr(ia\n+min(m,n)-1) for taup. Contain the scalar factors of the elementary\nreflectors that represent the orthogonal/unitary matrices Q and P,\nrespectively. tauq and taup are tied to the distributed matrix A. See\nApplication Notes below.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1494\n\n\nwork[0]\nOn exit work[0] contains the minimum value of lwork required for\noptimum performance.\ninfo\n(global)\n= 0: the execution is successful.\n< 0: if the i-th argument is an array and the j-th entry, indexed j - 1, had\nan illegal value, then info = -(i*100+j); if the i-th argument is a scalar\nand had an illegal value, then info = -i.\nApplication Notes\nThe matrices Q and P are represented as products of elementary reflectors:\nIf m≥n,\nQ = H(1)*H(2)*...*H(n), and P = G(1)*G(2)*...*G(n-1).\nEach H(i) and G(i) has the form:\nH(i)= i - tauq * v * v' and G(i) = i - taup*u*u'\nwhere tauq and taup are real/complex scalars, and v and u are real/complex vectors;\nv(1:i-1) = 0, v(i) = 1, and v(i+1:m) is stored on exit in A(ia+i:ia+m-1,ja+i-1);\nu(1:i) = 0, u(i+1) = 1, and u(i+2:n) is stored on exit in A (ia+i-1,ja+i+1:ja+n-1);\ntauq is stored in tauq[ja+i-2] and taup in taup[ia+i-2].\nIf m < n,\nQ = H(1)*H(2)*...*H(m-1), and P = G(1)* G(2)*...* G(m)\nEach H (i) and G(i) has the form:\nH(i)= i-tauq*v*v' and G(i)= i-taup*u*u'\nhere tauq and taup are real/complex scalars, and v and u are real/complex vectors;\nv(1:i) = 0, v(i+1) = 1, and v(i+2:m) is stored on exit in A (ia+i:ia+m-1,ja+i-1); u(1:i-1) = 0, u(i) = 1, and\nu(i+1:n) is stored on exit in A(ia+i-1,ja+i+1:ja+n-1);\ntauq is stored in tauq[ja+i-2] and taup in taup[ia+i-2].\nThe contents of sub(A) on exit are illustrated by the following examples:\nm = 6 and n = 5(m > n):\nm = 5 and n = 6(m < n):\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1495\n\n\nwhere d and e denote diagonal and off-diagonal elements of B, vi denotes an element of the vector defining\nH(i), and ui an element of the vector defining G(i).\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?ormbr\nMultiplies a general matrix by one of the orthogonal\nmatrices from a reduction to bidiagonal form\ndetermined by p?gebrd.\nSyntax\nvoid psormbr (char *vect , char *side , char *trans , MKL_INT *m , MKL_INT *n , MKL_INT\n*k , float *a , MKL_INT *ia , MKL_INT *ja , MKL_INT *desca , float *tau , float *c ,\nMKL_INT *ic , MKL_INT *jc , MKL_INT *descc , float *work , MKL_INT *lwork , MKL_INT\n*info );\nvoid pdormbr (char *vect , char *side , char *trans , MKL_INT *m , MKL_INT *n , MKL_INT\n*k , double *a , MKL_INT *ia , MKL_INT *ja , MKL_INT *desca , double *tau , double *c ,\nMKL_INT *ic , MKL_INT *jc , MKL_INT *descc , double *work , MKL_INT *lwork , MKL_INT\n*info );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nIf vect = 'Q', the p?ormbr function overwrites the general real distributed m-by-n matrix sub(C) = C(iс:iс\n+m-1,jс:jс+n-1) with\nside ='L'\nside ='R'\ntrans = 'N':\nQ sub(C)\nsub(C) Q\ntrans = 'T':\nQT sub(C)\nsub(C) QT\nIf vect = 'P', the function overwrites sub(C) with\nside ='L'\nside ='R'\ntrans = 'N':\nP sub(C)\nsub(C) P\ntrans = 'T':\nPT sub(C)\nsub(C) PT\nHere Q and PT are the orthogonal distributed matrices determined by p?gebrd when reducing a real\ndistributed matrix A(ia:*, ja:*) to bidiagonal form: A(ia:*, ja:*) = Q*B*PT. Q and PT are defined as\nproducts of elementary reflectors H(i) and G(i) respectively.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1496\n\n\nLet nq = m if side = 'L' and nq = n if side = 'R'. Therefore nq is the order of the orthogonal matrix Q or\nPT that is applied.\nIf vect = 'Q', A(ia:*, ja:*) is assumed to have been an nq-by-k matrix:\nIf nq ≥ k, Q = H(1) H(2)...H(k);\nIf nq < k, Q = H(1) H(2)...H(nq-1).\nIf vect = 'P', A(ia:*, ja:*) is assumed to have been a k-by-nq matrix:\nIf k < nq, P = G(1) G(2)...G(k);\nIf k ≥ nq, P = G(1) G(2)...G(nq-1).\nInput Parameters\nvect\n(global)\nIf vect ='Q', then Q or QT is applied.\nIf vect ='P', then P or PT is applied.\nside\n(global)\nIf side ='L', then Q or QT, P or PT is applied from the left.\nIf side ='R', then Q or QT, P or PT is applied from the right.\ntrans\n(global)\nIf trans = 'N', no transpose, Q or P is applied.\nIf trans = 'T', then QT or PT is applied.\nm\n(global) The number of rows in the distributed matrix sub (C).\nn\n(global) The number of columns in the distributed matrix sub (C).\nk\n(global)\nIf vect = 'Q', the number of columns in the original distributed matrix\nreduced by p?gebrd;\nIf vect = 'P', the number of rows in the original distributed matrix\nreduced by p?gebrd.\nConstraints: k≥ 0.\na\n(local)\nPointer into the local memory to an array of size lld_a * LOCc(ja\n+min(nq,k)-1) if vect='Q', and lld_a * LOCc(ja+nq-1) if vect = 'P'.\nnq = m if side = 'L', and nq = n otherwise.\nThe vectors that define the elementary reflectors H(i) and G(i), whose\nproducts determine the matrices Q and P, as returned by p?gebrd.\nIf vect = 'Q', lld_a≥max(1, LOCr(ia+nq-1));\nIf vect = 'P', lld_a≥max(1, LOCr(ia+min(nq, k)-1)).\nia, ja\n(global) The row and column indices in the global matrix A indicating the\nfirst row and the first column of the submatrix A, respectively.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1497\n\n\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\ntau\n(local)\nArray of size LOCc(ja+min(nq, k)-1), if vect = 'Q', and LOCr(ia\n+min(nq, k)-1), if vect = 'P'.\ntau[i] must contain the scalar factor of the elementary reflector H(i+1) or\nG (i+1)\nwhich determines Q or P, as returned by pdgebrd in its array argument\ntauq or taup. tau is tied to the distributed matrix A.\nc\n(local)\nPointer into the local memory to an array of size lld_c*LOCc(jc+n-1).\nContains the local pieces of the distributed matrix sub (C).\nic, jc\n(global) The row and column indices in the global matrix C indicating the\nfirst row and the first column of the submatrix C, respectively.\ndescc\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix C.\nwork\n(local)\nWorkspace array of size lwork.\nlwork\n(local or global) size of work, must be at least:\nIf side = 'L'\nnq = m;\nif ((vect = 'Q' and nq≥k) or (vect is not equal to 'Q' and nq>k)),\niaa=ia; jaa=ja; mi=m; ni=n; icc=ic; jcc=jc;\nelse\niaa= ia+1; jaa=ja; mi=m-1; ni=n; icc=ic+1; jcc= jc;\nend if\nelse\nIf side = 'R', nq = n;\nif((vect = 'Q' and nq≥k) or (vect is not equal to 'Q' and\nnq>k)),\niaa=ia; jaa=ja; mi=m; ni=n; icc=ic; jcc=jc;\nelse\niaa= ia; jaa= ja+1; mi= m; ni= n-1; icc= ic; jcc= jc+1;\nend if\nend if\nIf vect = 'Q',\nIf side = 'L', lwork≥max((nb_a*(nb_a-1))/2, (nqc0 + mpc0)*nb_a) +\nnb_a * nb_a\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1498\n\n\nelse if side = 'R',\nlwork≥max((nb_a*(nb_a-1))/2, (nqc0 + max(npa0 +\nnumroc(numroc(ni+icoffc, nb_a, 0, 0, NPCOL), nb_a, 0, 0,\nlcmq), mpc0))*nb_a) + nb_a*nb_a\nend if\nelse if vect is not equal to 'Q', if side = 'L',\nlwork≥max((mb_a*(mb_a-1))/2, (mpc0 + max(mqa0 +\nnumroc(numroc(mi+iroffc, mb_a, 0, 0, NPROW), mb_a, 0, 0,\nlcmp), nqc0))*mb_a) + mb_a*mb_a\nelse if side = 'R',\nlwork≥max((mb_a*(mb_a-1))/2, (mpc0 + nqc0)*mb_a) + mb_a*mb_a\nend if\nend if\nwhere lcmp = lcm/NPROW, lcmq = lcm/NPCOL, with lcm =\nilcm(NPROW, NPCOL),\niroffa = mod(iaa-1, mb_a),\nicoffa = mod(jaa-1, nb_a),\niarow = indxg2p(iaa, mb_a, MYROW, rsrc_a, NPROW),\niacol = indxg2p(jaa, nb_a, MYCOL, csrc_a, NPCOL),\nmqa0 = numroc(mi+icoffa, nb_a, MYCOL, iacol, NPCOL),\nnpa0 = numroc(ni+iroffa, mb_a, MYROW, iarow, NPROW),\niroffc = mod(icc-1, mb_c),\nicoffc = mod(jcc-1, nb_c),\nicrow = indxg2p(icc, mb_c, MYROW, rsrc_c, NPROW),\niccol = indxg2p(jcc, nb_c, MYCOL, csrc_c, NPCOL),\nmpc0 = numroc(mi+iroffc, mb_c, MYROW, icrow, NPROW),\nnqc0 = numroc(ni+icoffc, nb_c, MYCOL, iccol, NPCOL),\nNOTE\nmod(x,y) is the integer remainder of x/y.\nindxg2p and numroc are ScaLAPACK tool functions; MYROW, MYCOL, NPROW\nand NPCOL can be determined by calling the function blacs_gridinfo.\nIf lwork = -1, then lwork is global input and a workspace query is\nassumed; the function only calculates the minimum and optimal size for all\nwork arrays. Each of these values is returned in the first entry of the\ncorresponding work array, and no error message is issued by pxerbla.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1499\n\n\nOutput Parameters\nc\nOn exit, if vect='Q', sub(C) is overwritten by Q*sub(C), or Q'*sub(C), or\nsub(C)*Q', or sub(C)*Q; if vect='P', sub(C) is overwritten by P*sub(C),\nor P'*sub(C), or sub(C)*P, or sub(C)*P'.\nwork[0]\nOn exit work[0] contains the minimum value of lwork required for\noptimum performance.\ninfo\n(global)\n= 0: the execution is successful.\n< 0: if the i-th argument is an array and the j-th entry, indexed j - 1, had\nan illegal value, then info = -(i*100+j); if the i-th argument is a scalar\nand had an illegal value, then info = -i.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?unmbr\nMultiplies a general matrix by one of the unitary\ntransformation matrices from a reduction to bidiagonal\nform determined by p?gebrd.\nSyntax\nvoid pcunmbr (char *vect , char *side , char *trans , MKL_INT *m , MKL_INT *n , MKL_INT\n*k , MKL_Complex8 *a , MKL_INT *ia , MKL_INT *ja , MKL_INT *desca , MKL_Complex8 *tau ,\nMKL_Complex8 *c , MKL_INT *ic , MKL_INT *jc , MKL_INT *descc , MKL_Complex8 *work ,\nMKL_INT *lwork , MKL_INT *info );\nvoid pzunmbr (char *vect , char *side , char *trans , MKL_INT *m , MKL_INT *n , MKL_INT\n*k , MKL_Complex16 *a , MKL_INT *ia , MKL_INT *ja , MKL_INT *desca , MKL_Complex16\n*tau , MKL_Complex16 *c , MKL_INT *ic , MKL_INT *jc , MKL_INT *descc , MKL_Complex16\n*work , MKL_INT *lwork , MKL_INT *info );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nIf vect = 'Q', the p?unmbr function overwrites the general complex distributed m-by-n matrix sub(C) =\nC(iс:iс+m-1,jс:jс+n-1) with\nside ='L'\nside ='R'\ntrans = 'N':\nQ*sub(C)\nsub(C)*Q\ntrans = 'C':\nQH*sub(C)\nsub(C)*QH\nIf vect = 'P', the function overwrites sub(C) with\nside ='L'\nside ='R'\ntrans = 'N':\nP*sub(C)\nsub(C)*P\ntrans = 'C':\nPH*sub(C)\nsub(C)*PH\nHere Q and PH are the unitary distributed matrices determined by p?gebrd when reducing a complex\ndistributed matrix A(ia:*, ja:*) to bidiagonal form: A(ia:*, ja:*) = Q*B*PH.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1500\n\n\nQ and PH are defined as products of elementary reflectors H(i) and G(i) respectively.\nLet nq = m if side = 'L' and nq = n if side = 'R'. Therefore nq is the order of the unitary matrix Q or PH\nthat is applied.\nIf vect = 'Q', A(ia:*, ja:*) is assumed to have been an nq-by-k matrix:\nIf nq ≥ k, Q = H(1) H(2)... H(k);\nIf nq < k, Q = H(1) H(2)... H(nq-1).\nIf vect = 'P', A(ia:*, ja:*) is assumed to have been a k-by-nq matrix:\nIf k < nq, P = G(1) G(2)... G(k);\nIf k ≥ nq, P = G(1) G(2)... G(nq-1).\nInput Parameters\nvect\n(global)\nIf vect ='Q', then Q or QH is applied.\nIf vect ='P', then P or PH is applied.\nside\n(global)\nIf side ='L', then Q or QH, P or PH is applied from the left.\nIf side ='R', then Q or QH, P or PH is applied from the right.\ntrans\n(global)\nIf trans = 'N', no transpose, Q or P is applied.\nIf trans = 'C', conjugate transpose, QH or PH is applied.\nm\n(global) The number of rows in the distributed matrix sub (C) m≥0.\nn\n(global) The number of columns in the distributed matrix sub (C) n≥0.\nk\n(global)\nIf vect = 'Q', the number of columns in the original distributed matrix\nreduced by p?gebrd;\nIf vect = 'P', the number of rows in the original distributed matrix\nreduced by p?gebrd.\nConstraints: k≥ 0.\na\n(local)\nPointer into the local memory to an array of size lld_a * LOCc(ja\n+min(nq,k)-1) if vect='Q', and lld_a * LOCc(ja+nq-1) if vect = 'P'.\nnq = m if side = 'L', and nq = n otherwise.\nThe vectors that define the elementary reflectors H(i) and G(i), whose\nproducts determine the matrices Q and P, as returned by p?gebrd.\nIf vect = 'Q', lld_a ≥ max(1, LOCr(ia+nq-1));\nIf vect = 'P', lld_a ≥ max(1, LOCr(ia+min(nq, k)-1)).\nia, ja\n(global) The row and column indices in the global matrix A indicating the\nfirst row and the first column of the submatrix A, respectively.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1501\n\n\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\ntau\n(local)\nArray of size LOCc(ja+min(nq, k)-1), if vect = 'Q', and LOCr(ia\n+min(nq, k)-1), if vect = 'P'.\ntau[i] must contain the scalar factor of the elementary reflector H(i+1) or\nG (i+1), which determines Q or P, as returned by p?gebrd in its array\nargument tauq or taup. tau is tied to the distributed matrix A.\nc\n(local)\nPointer into the local memory to an array of size lld_c*LOCc(jc+n-1).\nContains the local pieces of the distributed matrix sub (C).\nic, jc\n(global) The row and column indices in the global matrix C indicating the\nfirst row and the first column of the submatrix C, respectively.\ndescc\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix C.\nwork\n(local)\nWorkspace array of size lwork.\nlwork\n(local or global) size of work, must be at least:\nIf side = 'L'\nnq = m;\nif ((vect = 'Q' and nq ≥ k) or (vect is not equal to 'Q' and\nnq>k)), iaa= ia; jaa= ja; mi= m; ni= n; icc= ic; jcc= jc;\nelse\niaa= ia+1; jaa= ja; mi= m-1; ni= n; icc= ic+1; jcc= jc;\nend if\nelse\nIf side = 'R', nq = n;\nif ((vect = 'Q' and nq ≥ k) or (vect is not equal to 'Q' and\nnq≥k)),\niaa= ia; jaa= ja; mi= m; ni= n; icc= ic; jcc= jc;\nelse\niaa= ia; jaa= ja+1; mi= m; ni= n-1; icc= ic; jcc= jc+1;\nend if\nend if\nIf vect = 'Q',\nIf side = 'L', lwork ≥ max((nb_a*(nb_a-1))/2,\n(nqc0+mpc0)*nb_a) + nb_a*nb_a\nelse if side = 'R',\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1502\n\n\nlwork ≥ max((nb_a*(nb_a-1))/2, (nqc0 +\nmax(npa0+numroc(numroc(ni+icoffc, nb_a, 0, 0, NPCOL), nb_a,\n0, 0, lcmq), mpc0))*nb_a) + nb_a*nb_a\nend if\nelse if vect is not equal to 'Q',\nif side = 'L',\nlwork ≥ max((mb_a*(mb_a-1))/2, (mpc0 +\nmax(mqa0+numroc(numroc(mi+iroffc, mb_a, 0, 0, NPROW), mb_a,\n0, 0, lcmp), nqc0))*mb_a) + mb_a*mb_a\nelse if side = 'R',\nlwork ≥ max((mb_a*(mb_a-1))/2, (mpc0 + nqc0)*mb_a) +\nmb_a*mb_a\nend if\nend if\nwhere lcmp = lcm/NPROW, lcmq = lcm/NPCOL, with lcm =\nilcm(NPROW, NPCOL),\niroffa = mod(iaa-1, mb_a),\nicoffa = mod(jaa-1, nb_a),\niarow = indxg2p(iaa, mb_a, MYROW, rsrc_a, NPROW),\niacol = indxg2p(jaa, nb_a, MYCOL, csrc_a, NPCOL),\nmqa0 = numroc(mi+icoffa, nb_a, MYCOL, iacol, NPCOL),\nnpa0 = numroc(ni+iroffa, mb_a, MYROW, iarow, NPROW),\niroffc = mod(icc-1, mb_c),\nicoffc = mod(jcc-1, nb_c),\nicrow = indxg2p(icc, mb_c, MYROW, rsrc_c, NPROW),\niccol = indxg2p(jcc, nb_c, MYCOL, csrc_c, NPCOL),\nmpc0 = numroc(mi+iroffc, mb_c, MYROW, icrow, NPROW),\nnqc0 = numroc(ni+icoffc, nb_c, MYCOL, iccol, NPCOL),\nNOTE\nmod(x,y) is the integer remainder of x/y.\nindxg2p and numroc are ScaLAPACK tool functions; MYROW, MYCOL, NPROW\nand NPCOL can be determined by calling the function blacs_gridinfo.\nIf lwork = -1, then lwork is global input and a workspace query is\nassumed; the function only calculates the minimum and optimal size for all\nwork arrays. Each of these values is returned in the first entry of the\ncorresponding work array, and no error message is issued by pxerbla.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1503\n\n\nOutput Parameters\nc\nOn exit, if vect='Q', sub(C) is overwritten by Q*sub(C), or Q'*sub(C), or\nsub(C)*Q', or sub(C)*Q; if vect='P', sub(C) is overwritten by P*sub(C), or\nP'*sub(C), or sub(C)*P, or sub(C)*P'.\nwork[0]\nOn exit work[0] contains the minimum value of lwork required for\noptimum performance.\ninfo\n(global)\n= 0: the execution is successful.\n< 0: if the i-th argument is an array and the j-th entry, indexed j - 1, had\nan illegal value, then info = -(i*100+j); if the i-th argument is a scalar\nand had an illegal value, then info = -i.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\nGeneralized Symmetric-Definite Eigenvalue Problems: ScaLAPACK Computational Routines\nThis section describes ScaLAPACK routines that allow you to reduce the generalized symmetric-definite\neigenvalue problems (see LAPACKGeneralized Symmetric-Definite Eigenvalue Problems ) to standard\nsymmetric eigenvalue problem Cy = λy, which you can solve by calling ScaLAPACK routines (see Symmetric\nEigenproblems).\nTable \"Computational Routines for Reducing Generalized Eigenproblems to Standard Problems\" lists these\nroutines.\nComputational Routines for Reducing Generalized Eigenproblems to Standard Problems\nOperation\nReal symmetric matrices\nComplex Hermitian matrices\nReduce to standard problems\np?sygst\np?hegst\np?sygst\nReduces a real symmetric-definite generalized\neigenvalue problem to the standard form.\nSyntax\nvoid pssygst (MKL_INT *ibtype , char *uplo , MKL_INT *n , float *a , MKL_INT *ia ,\nMKL_INT *ja , MKL_INT *desca , float *b , MKL_INT *ib , MKL_INT *jb , MKL_INT *descb ,\nfloat *scale , MKL_INT *info );\nvoid pdsygst (MKL_INT *ibtype , char *uplo , MKL_INT *n , double *a , MKL_INT *ia ,\nMKL_INT *ja , MKL_INT *desca , double *b , MKL_INT *ib , MKL_INT *jb , MKL_INT *descb ,\ndouble *scale , MKL_INT *info );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?sygstfunction reduces real symmetric-definite generalized eigenproblems to the standard form.\nIn the following sub(A) denotes A(ia:ia+n-1, ja:ja+n-1) and sub(B) denotes B(ib:ib+n-1, jb:jb+n-1).\nIf ibtype = 1, the problem is\nsub(A)*x = λ*sub(B)*x,\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1504\n\n\nand sub(A) is overwritten by inv(UT)*sub(A)*inv(U), or inv(L)*sub(A)*inv(LT).\nIf ibtype = 2 or 3, the problem is\nsub(A)*sub(B)*x = λ*x, or sub(B)*sub(A)*x = λ*x,\nand sub(A) is overwritten by U*sub(A)*UT, or LT*sub(A)*L.\nsub(B) must have been previously factorized as UT*U or L*LT by p?potrf.\nInput Parameters\nibtype\n(global) Must be 1 or 2 or 3.\nIf itype = 1, compute inv(UT)*sub(A)*inv(U), or inv(L)*sub(A)*inv(LT);\nIf itype = 2 or 3, compute U*sub(A)*UT, or LT*sub(A)*L.\nuplo\n(global) Must be 'U' or 'L'.\nIf uplo = 'U', the upper triangle of sub(A) is stored and sub (B) is\nfactored as UT*U.\nIf uplo = 'L', the lower triangle of sub(A) is stored and sub (B) is\nfactored as L*LT.\nn\n(global) The order of the matrices sub (A) and sub (B) (n≥ 0).\na\n(local)\nPointer into the local memory to an array of size lld_a*LOCc(ja+n-1). On\nentry, the array contains the local pieces of the n-by-n symmetric\ndistributed matrix sub(A).\nIf uplo = 'U', the leading n-by-n upper triangular part of sub(A) contains\nthe upper triangular part of the matrix, and its strictly lower triangular part\nis not referenced.\nIf uplo = 'L', the leading n-by-n lower triangular part of sub(A) contains\nthe lower triangular part of the matrix, and its strictly upper triangular part\nis not referenced.\nia, ja\n(global) The row and column indices in the global matrix A indicating the\nfirst row and the first column of the submatrix A, respectively.\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\nb\n(local)\nPointer into the local memory to an array of size lld_b*LOCc(jb+n-1). On\nentry, the array contains the local pieces of the triangular factor from the\nCholesky factorization of sub (B) as returned by p?potrf.\nib, jb\n(global) The row and column indices in the global matrix B indicating the\nfirst row and the first column of the submatrix B, respectively.\ndescb\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix B.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1505\n\n\nOutput Parameters\na\nOn exit, if info = 0, the transformed matrix, stored in the same format as\nsub(A).\nscale\n(global)\nAmount by which the eigenvalues should be scaled to compensate for the\nscaling performed in this function. At present, scale is always returned as\n1.0, it is returned here to allow for future enhancement.\ninfo\n(global)\nIf info = 0, the execution is successful. If info < 0, if the i-th argument\nis an array and the j-th entry, indexed j - 1, had an illegal value, then info\n= -(i*100+j); if the i-th argument is a scalar and had an illegal value, then\ninfo = -i.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?hegst\nReduces a Hermitian positive-definite generalized\neigenvalue problem to the standard form.\nSyntax\nvoid pchegst (MKL_INT *ibtype , char *uplo , MKL_INT *n , MKL_Complex8 *a , MKL_INT\n*ia , MKL_INT *ja , MKL_INT *desca , MKL_Complex8 *b , MKL_INT *ib , MKL_INT *jb ,\nMKL_INT *descb , float *scale , MKL_INT *info );\nvoid pzhegst (MKL_INT *ibtype , char *uplo , MKL_INT *n , MKL_Complex16 *a , MKL_INT\n*ia , MKL_INT *ja , MKL_INT *desca , MKL_Complex16 *b , MKL_INT *ib , MKL_INT *jb ,\nMKL_INT *descb , double *scale , MKL_INT *info );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?hegst function reduces complex Hermitian positive-definite generalized eigenproblems to the\nstandard form.\nIn the following sub(A) denotes A(ia:ia+n-1, ja:ja+n-1) and sub(B) denotes B(ib:ib+n-1, jb:jb+n-1).\nIf ibtype = 1, the problem is\nsub(A)*x = λ*sub(B)*x,\nand sub(A) is overwritten by inv(UH)*sub(A)*inv(U), or inv(L)*sub(A)*inv(LH).\nIf ibtype = 2 or 3, the problem is\nsub(A)*sub(B)*x = λ*x, or sub(B)*sub(A)*x = λ*x,\nand sub(A) is overwritten by U*sub(A)*UH, or LH*sub(A)*L.\nsub(B) must have been previously factorized as UH*U or L*LH by p?potrf.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1506\n\n\nInput Parameters\nibtype\n(global) Must be 1 or 2 or 3.\nIf itype = 1, compute inv(UH)*sub(A)*inv(U), or inv(L)*sub(A)*inv(LH);\nIf itype = 2 or 3, compute U*sub(A)*UH, or LH*sub(A)*L.\nuplo\n(global) Must be 'U' or 'L'.\nIf uplo = 'U', the upper triangle of sub(A) is stored and sub (B) is\nfactored as UH*U.\nIf uplo = 'L', the lower triangle of sub(A) is stored and sub (B) is\nfactored as L*LH.\nn\n(global) The order of the matrices sub (A) and sub (B) (n≥0).\na\n(local)\nPointer into the local memory to an array of size lld_a*LOCc(ja+n-1). On\nentry, the array contains the local pieces of the n-by-n Hermitian distributed\nmatrix sub(A). If uplo = 'U', the leading n-by-n upper triangular part of\nsub(A) contains the upper triangular part of the matrix, and its strictly\nlower triangular part is not referenced. If uplo = 'L', the leading n-by-n\nlower triangular part of sub(A) contains the lower triangular part of the\nmatrix, and its strictly upper triangular part is not referenced.\nia, ja\n(global) The row and column indices in the global matrix A indicating the\nfirst row and the first column of the submatrix A, respectively.\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\nb\n(local)\nPointer into the local memory to an array of size lld_b*LOCc(jb+n-1). On\nentry, the array contains the local pieces of the triangular factor from the\nCholesky factorization of sub (B) as returned by p?potrf.\nib, jb\n(global) The row and column indices in the global matrix B indicating the\nfirst row and the first column of the submatrix B, respectively.\ndescb\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix B.\nOutput Parameters\na\nOn exit, if info = 0, the transformed matrix, stored in the same format as\nsub(A).\nscale\n(global)\nAmount by which the eigenvalues should be scaled to compensate for the\nscaling performed in this function. At present, scale is always returned as\n1.0, it is returned here to allow for future enhancement.\ninfo\n(global)\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1507\n\n\nIf info = 0, the execution is successful. If info <0, if the i-th argument is\nan array and the j-th entry, indexed j - 1, had an illegal value, then info =\n-(i*100+j); if the i-th argument is a scalar and had an illegal value, then\ninfo = -i.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\nScaLAPACK Driver Routines\nTable \"ScaLAPACK Driver Routines\" lists ScaLAPACK driver routines available for solving systems of linear\nequations, linear least-squares problems, standard eigenvalue and singular value problems, and generalized\nsymmetric definite eigenproblems.\nScaLAPACK Driver Routines\nType of Problem\nMatrix type, storage scheme\nDriver\nLinear equations\ngeneral (partial pivoting)\np?gesv (simple driver) / p?gesvx\n(expert driver)\n \ngeneral band (partial pivoting)\np?gbsv (simple driver)\n \ngeneral band (no pivoting)\np?dbsv (simple driver)\n \ngeneral tridiagonal (no pivoting)\np?dtsv (simple driver)\n \nsymmetric/Hermitian positive-definite\np?posv (simple driver) / p?posvx\n(expert driver)\n \nsymmetric/Hermitian positive-definite,\nband\np?pbsv (simple driver)\n \nsymmetric/Hermitian positive-definite,\ntridiagonal\np?ptsv (simple driver)\nLinear least squares problem\ngeneral m-by-n\np?gels\nNon-symmetric eigenvalue\nproblem\ngeneral\np?geevx (expert driver)\nSymmetric eigenvalue problem\nsymmetric/Hermitian\np?syev / p?heev (simple driver); \np?syevd / p?heevd (simple driver with\na divide and conquer algorithm); \np?syevx / p?heevx (expert driver); \np?syevr / p?heevr (simple driver with\nMRRR algorithm)\nSingular value decomposition\ngeneral m-by-n\np?gesvd\nGeneralized symmetric definite\neigenvalue problem\nsymmetric/Hermitian, one matrix also\npositive-definite\np?sygvx / p?hegvx (expert driver)\np?geevx\nComputes for an n-by-n real/complex non-symmetric\nmatrix A, the eigenvalues and, optionally, the left\nand/or right eigenvectors.\nSyntax\nvoid psgeevx (const char *balanc, const char *jobvl, const char *jobvr, const char\n*sense, const MKL_INT *n, float *a, const MKL_INT *desca, float *wr, float *wi, float\n*vl, const MKL_INT *descvl, float *vr, const MKL_INT *descvr, MKL_INT *ilo, MKL_INT\n*ihi, float *scale, float *abnrm, float *rconde, float *rcondv, float *work, const\nMKL_INT *lwork, MKL_INT *info);\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1508\n\n\nvoid pdgeevx (const char *balanc, const char *jobvl, const char *jobvr, const char\n*sense, const MKL_INT *n, double *a, const MKL_INT *desca, double *wr, double *wi,\ndouble *vl, const MKL_INT *descvl, double *vr, const MKL_INT *descvr, MKL_INT *ilo,\nMKL_INT *ihi, double *scale, double *abnrm, double *rconde, double *rcondv, double\n*work, const MKL_INT *lwork, MKL_INT *info);\nvoid pcgeevx (const char *balanc, const char *jobvl, const char *jobvr, const char\n*sense, const MKL_INT *n, MKL_Complex8 *a, const MKL_INT *desca, MKL_Complex8 *w,\nMKL_Complex8 *vl, const MKL_INT *descvl, MKL_Complex8 *vr, const MKL_INT *descvr,\nMKL_INT *ilo, MKL_INT *ihi, float *scale, float *abnrm, float *rconde, float *rcondv,\nMKL_Complex8 *work, const MKL_INT *lwork, MKL_INT *info);\nvoid pzgeevx (const char *balanc, const char *jobvl, const char *jobvr, const char\n*sense, const MKL_INT *n, MKL_Complex16 *a, const MKL_INT *desca, MKL_Complex16 *w,\nMKL_Complex16 *vl, const MKL_INT *descvl, MKL_Complex16 *vr, const MKL_INT *descvr,\nMKL_INT *ilo, MKL_INT *ihi, double *scale, double *abnrm, double *rconde, double\n*rcondv, MKL_Complex16 *work, const MKL_INT *lwork, MKL_INT *info);\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?geevx function computes for an n-by-n real/complex non-symmetric matrix A, the eigenvalues and,\noptionally, the left and/or right eigenvectors.\nOptionally also, it computes a balancing transformation to improve the conditioning of the eigenvalues and\neigenvectors (ilo, ihi, scale, and abnrm), reciprocal condition numbers for the eigenvalues (rconde).\nThe right eigenvector v of A satisfies\nA ⋅v = λ ⋅v\nwhere ƛ is its eigenvalue.\nThe left eigenvector u of A satisfies.\nuHA = ƛuH\nwhere uH denotes the conjugate transpose of u. The computed eigenvectors are normalized to have\nEuclidean norm equal to 1 and largest component real.\nBalancing a matrix means permuting the rows and columns to make it more nearly upper triangular, and\napplying a diagonal similarity transformation D*A*inv(D), where D is a diagonal matrix, to make its rows and\ncolumns closer in norm and the condition number of its eigenvalues smaller. The computed reciprocal\ncondition numbers correspond to the balanced matrix. Permuting rows and columns will not change the\ncondition numbers in exact arithmetic, but diagonal scaling will.\nNOTE\nThe current version doesn’t support computation of the reciprocal condition numbers for the\nright eigenvectors.\nCurrent Notes and Restrictions\nAll the p?geevx interfaces call p?lahqr for computing eigenvalues and eigenvectors of the Hessenberg\nmatrices. There are several restrictions for the usage of p?lahqr, which include:\n•\nThe current implementation of p?lahqr requires the distributed block size to be square and at least six\n(6); unlike simpler codes like LU, this algorithm is extremely sensitive to block size.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1509\n\n\n•\nThe current implementation of p?lahqr requires that input matrix A, the left and right eigenvector\nmatrices VR and/or VL to be distributed identically and have identical context.\nParameters\nbalanc\n(global). Must be 'N', 'P', 'S', or 'B'. Indicates how the input matrix should\nbe diagonally scaled and/or permuted to improve the conditioning of its\neigenvalues.\nIf balanc = 'N', do not diagonally scale or permute;\nIf balanc = 'P', perform permutations to make the matrix more nearly upper\ntriangular. Do not diagonally scale;\nIf balanc = 'S', diagonally scale the matrix, that is, replace A by\nD*A*inv(D), where D is a diagonal matrix chosen to make the rows and\ncolumns of A more equal in norm. Do not permute;\nIf balanc = 'B', both diagonally scale and permute A.\nComputed reciprocal condition numbers will be for the matrix after\nbalancing and/or permuting. Permuting does not change condition numbers\n(in exact arithmetic), but balancing does.\njobvl\n(global). Must be 'N' or 'V.\nIf jobvl = 'N', left eigenvectors of A are not computed;\nIf jobvl = 'V', left eigenvectors of A are computed.\nIf sense = 'E', then jobvl must be 'V'.\njobvr\n(global). Must be 'N' or 'V.\nIf jobvr = 'N', right eigenvectors of A are not computed;\nIf jobvr = 'V', right eigenvectors of A are computed.\nIf sense = 'E', then jobvr must be 'V'.\nsense\n(global). Must be 'N' or 'E. Determines which reciprocal condition numbers\nare computed.\nIf sense = 'N', none are computed.\nIf sense = 'E', computed for eigenvalues only.\nn\n(global) The order of the distributed matrix A (n≥0).\na\n(local)\nPointer into the local memory to an array of size lld_a*LOCc(n). On entry,\nthis array contains the local pieces of the n-by-n general distributed matrix\nA to be reduced.\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\nwr, wi\n(global output) Arrays, size at least max (1, n) each. Contain the real and\nimaginary parts, respectively, of the computed eigenvalues. Complex\nconjugate pairs of eigenvalues appear consecutively with the eigenvalue\nhaving positive imaginary part first.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1510\n\n\nw\n(global output) Array, size at least max(1, n). Contains the computed\neigenvalues.\nvl\n(local output)\nPointer into the local memory to an array of size (DESCVL(LLD_),LOCc(n)).\nIf jobvl = 'N', vl is not referenced. If jobvl = 'V', the vl parameter contains\nthe local pieces of the left eigenvectors of the matrix A.\ndescvl\n(global and local input) array of size dlen_. The array descriptor for the\ndistributed matrix vl.\nvr\n(local output)\nPointer into the local memory to an array of size (DESCVR(LLD_),LOCc(n)).\nIf jobvr = 'N', vr is not referenced. If jobvr = 'V', the vr parameter contains\nthe local pieces of the right eigenvectors of the matrix A.\ndescvr\n(global and local input) array of size dlen_. The array descriptor for the\ndistributed matrix vr.\nilo, ihi\n(global output)\nilo and ihi are integer values determined when A was balanced.\nThe balanced A(i,j) = 0 if i > j and j = 1,..., ilo-1 or i= ihi+1,..., n.\nIf balanc = 'N' or 'S', ilo = 1 and ihi = n.\nscale\n(global output)\nArray, size at least max(1, n). Details of the permutations and scaling\nfactors applied when balancing A.\nIf P[j - 1] is the index of the row and column interchanged with row and\ncolumn j, and D[j - 1] is the scaling factor applied to row and column j, then\nscale[j - 1] = P[j - 1], for j = 1,...,ilo-1\n= D[j - 1], for j = ilo,...,ihi\n= P[j - 1] for j = ihi+1,..., n.\nThe order in which the interchanges are made is n to ihi+1, then 1 to ilo-1.\nabnrm\nThe one-norm of the balanced matrix (the maximum of the sum of absolute\nvalues of elements of any column).\nrconde\nArray, size at least max(1, n).\nrconde[j - 1] is the reciprocal condition number of the j-th eigenvalue.\nrcondv\nNot supported in the current version. It could be null pointer.\nwork\n(local)\nWorkspace array of size lwork.\nlwork\n(local or global) size of the array work.\nIf lwork = -1, then lwork is global input and a workspace query is assumed;\nthe function only calculates the minimum size for the work array. These\nvalues are returned in the first entry of the work array, and no error\nmessage is issued by pxerbla.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1511\n\n\ninfo\n(global)\n= 0: the execution is successful.\n< 0: if the i-th argument is an array and the j-th entry, indexed j- 1, had an\nillegal value, then info = -(i*100+j); if the i-th argument is a scalar and had\nan illegal value, then info = -i.\np?gesv\nComputes the solution to the system of linear\nequations with a square distributed matrix and\nmultiple right-hand sides.\nSyntax\nvoid psgesv (MKL_INT *n , MKL_INT *nrhs , float *a , MKL_INT *ia , MKL_INT *ja , MKL_INT\n*desca , MKL_INT *ipiv , float *b , MKL_INT *ib , MKL_INT *jb , MKL_INT *descb , MKL_INT\n*info );\nvoid pdgesv (MKL_INT *n , MKL_INT *nrhs , double *a , MKL_INT *ia , MKL_INT *ja ,\nMKL_INT *desca , MKL_INT *ipiv , double *b , MKL_INT *ib , MKL_INT *jb , MKL_INT\n*descb , MKL_INT *info );\nvoid pcgesv (MKL_INT *n , MKL_INT *nrhs , MKL_Complex8 *a , MKL_INT *ia , MKL_INT *ja ,\nMKL_INT *desca , MKL_INT *ipiv , MKL_Complex8 *b , MKL_INT *ib , MKL_INT *jb , MKL_INT\n*descb , MKL_INT *info );\nvoid pzgesv (MKL_INT *n , MKL_INT *nrhs , MKL_Complex16 *a , MKL_INT *ia , MKL_INT\n*ja , MKL_INT *desca , MKL_INT *ipiv , MKL_Complex16 *b , MKL_INT *ib , MKL_INT *jb ,\nMKL_INT *descb , MKL_INT *info );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?gesvfunction computes the solution to a real or complex system of linear equations sub(A)*X =\nsub(B), where sub(A) = A(ia:ia+n-1, ja:ja+n-1) is an n-by-n distributed matrix and X and sub(B) =\nB(ib:ib+n-1, jb:jb+nrhs-1) are n-by-nrhs distributed matrices.\nThe LU decomposition with partial pivoting and row interchanges is used to factor sub(A) as sub(A) =\nP*L*U, where P is a permutation matrix, L is unit lower triangular, and U is upper triangular. L and U are\nstored in sub(A). The factored form of sub(A) is then used to solve the system of equations sub(A)*X =\nsub(B).\nInput Parameters\nn\n(global) The number of rows and columns to be operated on, that is, the\norder of the distributed submatrix sub(A) (n≥ 0).\nnrhs\n(global) The number of right hand sides, that is, the number of columns of\nthe distributed submatrices B and X(nrhs≥ 0).\na, b\n(local)\nPointers into the local memory to arrays of local size a: lld_a*LOCc(ja\n+n-1) and b: lld_b*LOCc(jb+nrhs-1), respectively.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1512\n\n\nOn entry, the array a contains the local pieces of the n-by-n distributed\nmatrix sub(A) to be factored.\nOn entry, the array b contains the right hand side distributed matrix sub(B).\nia, ja\n(global) The row and column indices in the global matrix A indicating the\nfirst row and the first column of sub(A), respectively.\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\nib, jb\n(global) The row and column indices in the global matrix B indicating the\nfirst row and the first column of sub(B), respectively.\ndescb\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix B.\nOutput Parameters\na\nOverwritten by the factors L and U from the factorization sub(A) = P*L*U;\nthe unit diagonal elements of L are not stored .\nb\nOverwritten by the solution distributed matrix X.\nipiv\n(local) Array of size LOCr(m_a)+mb_a. This array contains the pivoting\ninformation. The (local) row i of the matrix was interchanged with the\n(global) row ipiv[i - 1].\nThis array is tied to the distributed matrix A.\ninfo\n(global) If info=0, the execution is successful.\ninfo < 0:\nIf the i-th argument is an array and the j-th entry had an illegal value, then\ninfo = -(i*100+j); if the i-th argument is a scalar and had an illegal\nvalue, then info = -i.\ninfo> 0:\nIf info = k, U(ia+k-1,ja+k-1) is exactly zero. The factorization has been\ncompleted, but the factor U is exactly singular, so the solution could not be\ncomputed.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?gesvx\nUses the LU factorization to compute the solution to\nthe system of linear equations with a square matrix A\nand multiple right-hand sides, and provides error\nbounds on the solution.\nSyntax\nvoid psgesvx (char *fact , char *trans , MKL_INT *n , MKL_INT *nrhs , float *a , MKL_INT\n*ia , MKL_INT *ja , MKL_INT *desca , float *af , MKL_INT *iaf , MKL_INT *jaf , MKL_INT\n*descaf , MKL_INT *ipiv , char *equed , float *r , float *c , float *b , MKL_INT *ib ,\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1513\n\n\nMKL_INT *jb , MKL_INT *descb , float *x , MKL_INT *ix , MKL_INT *jx , MKL_INT *descx ,\nfloat *rcond , float *ferr , float *berr , float *work , MKL_INT *lwork , MKL_INT\n*iwork , MKL_INT *liwork , MKL_INT *info );\nvoid pdgesvx (char *fact , char *trans , MKL_INT *n , MKL_INT *nrhs , double *a ,\nMKL_INT *ia , MKL_INT *ja , MKL_INT *desca , double *af , MKL_INT *iaf , MKL_INT *jaf ,\nMKL_INT *descaf , MKL_INT *ipiv , char *equed , double *r , double *c , double *b ,\nMKL_INT *ib , MKL_INT *jb , MKL_INT *descb , double *x , MKL_INT *ix , MKL_INT *jx ,\nMKL_INT *descx , double *rcond , double *ferr , double *berr , double *work , MKL_INT\n*lwork , MKL_INT *iwork , MKL_INT *liwork , MKL_INT *info );\nvoid pcgesvx (char *fact , char *trans , MKL_INT *n , MKL_INT *nrhs , MKL_Complex8 *a ,\nMKL_INT *ia , MKL_INT *ja , MKL_INT *desca , MKL_Complex8 *af , MKL_INT *iaf , MKL_INT\n*jaf , MKL_INT *descaf , MKL_INT *ipiv , char *equed , float *r , float *c ,\nMKL_Complex8 *b , MKL_INT *ib , MKL_INT *jb , MKL_INT *descb , MKL_Complex8 *x ,\nMKL_INT *ix , MKL_INT *jx , MKL_INT *descx , float *rcond , float *ferr , float *berr ,\nMKL_Complex8 *work , MKL_INT *lwork , float *rwork , MKL_INT *lrwork , MKL_INT *info );\nvoid pzgesvx (char *fact , char *trans , MKL_INT *n , MKL_INT *nrhs , MKL_Complex16\n*a , MKL_INT *ia , MKL_INT *ja , MKL_INT *desca , MKL_Complex16 *af , MKL_INT *iaf ,\nMKL_INT *jaf , MKL_INT *descaf , MKL_INT *ipiv , char *equed , double *r , double *c ,\nMKL_Complex16 *b , MKL_INT *ib , MKL_INT *jb , MKL_INT *descb , MKL_Complex16 *x ,\nMKL_INT *ix , MKL_INT *jx , MKL_INT *descx , double *rcond , double *ferr , double\n*berr , MKL_Complex16 *work , MKL_INT *lwork , double *rwork , MKL_INT *lrwork ,\nMKL_INT *info );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?gesvx function uses the LU factorization to compute the solution to a real or complex system of linear\nequations AX = B, where A denotes the n-by-n submatrix A(ia:ia+n-1, ja:ja+n-1), B denotes the n-by-\nnrhs submatrix B(ib:ib+n-1, jb:jb+nrhs-1) and X denotes the n-by-nrhs submatrix X(ix:ix+n-1,\njx:jx+nrhs-1).\nError bounds on the solution and a condition estimate are also provided.\nIn the following description, af stands for the subarray of af from row iaf and column jaf to row iaf+n-1 and\ncolumn jaf+n-1.\nThe function p?gesvx performs the following steps:\n1.\nIf fact = 'E', real scaling factors R and C are computed to equilibrate the system:\ntrans = 'N': diag(R)*A*diag(C) *diag(C)-1*X = diag(R)*B\ntrans = 'T': (diag(R)*A*diag(C))T *diag(R)-1*X = diag(C)*B\ntrans = 'C': (diag(R)*A*diag(C))H *diag(R)-1*X = diag(C)*B\nWhether or not the system will be equilibrated depends on the scaling of the matrix A, but if\nequilibration is used, A is overwritten by diag(R)*A*diag(C) and B by diag(R)*B (if trans='N') or\ndiag(c)*B (if trans = 'T' or 'C').\n2.\nIf fact = 'N' or 'E', the LU decomposition is used to factor the matrix A (after equilibration if fact\n= 'E') as A = PLU, where P is a permutation matrix, L is a unit lower triangular matrix, and U is\nupper triangular.\n3.\nThe factored form of A is used to estimate the condition number of the matrix A. If the reciprocal of the\ncondition number is less than relative machine precision, steps 4 - 6 are skipped.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1514\n\n\n4.\nThe system of equations is solved for X using the factored form of A.\n5.\nIterative refinement is applied to improve the computed solution matrix and calculate error bounds and\nbackward error estimates for it.\n6.\nIf equilibration was used, the matrix X is premultiplied by diag(C) (if trans = 'N') or diag(R) (if\ntrans = 'T' or 'C') so that it solves the original system before equilibration.\nInput Parameters\nfact\n(global) Must be 'F', 'N', or 'E'.\nSpecifies whether or not the factored form of the matrix A is supplied on\nentry, and if not, whether the matrix A should be equilibrated before it is\nfactored.\nIf fact = 'F' then, on entry, af and ipiv contain the factored form of A. If\nequed is not 'N', the matrix A has been equilibrated with scaling factors\ngiven by r and c. Arrays a, af, and ipiv are not modified.\nIf fact = 'N', the matrix A is copied to af and factored.\nIf fact = 'E', the matrix A is equilibrated if necessary, then copied to af\nand factored.\ntrans\n(global) Must be 'N', 'T', or 'C'.\nSpecifies the form of the system of equations:\nIf trans = 'N', the system has the form A*X = B (No transpose);\nIf trans = 'T', the system has the form AT*X = B (Transpose);\nIf trans = 'C', the system has the form AH*X = B (Conjugate transpose);\nn\n(global) The number of linear equations; the order of the submatrix A(n≥\n0).\nnrhs\n(global) The number of right hand sides; the number of columns of the\ndistributed submatrices B and X(nrhs≥ 0).\na, af, b, work\n(local)\nPointers into the local memory to arrays of local size a: lld_a*LOCc(ja\n+n-1), af: lld_af*LOCc(ja+n-1), b: lld_b*LOCc(jb+nrhs-1), work:\nlwork.\nThe array a contains the matrix A. If fact = 'F' and equed is not 'N',\nthen A must have been equilibrated by the scaling factors in r and/or c.\nThe array af is an input argument if fact = 'F'. In this case it contains on\nentry the factored form of the matrix A, that is, the factors L and U from\nthe factorization A = P*L*U as computed by p?getrf. If equed is not 'N',\nthen af is the factored form of the equilibrated matrix A.\nThe array b contains on entry the matrix B whose columns are the right-\nhand sides for the systems of equations.\nwork is a workspace array. The size of work is (lwork).\nia, ja\n(global) The row and column indices in the global matrix A indicating the\nfirst row and the first column of the submatrix A(ia:ia+n-1, ja:ja\n+n-1), respectively.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1515\n\n\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\niaf, jaf\n(global) The row and column indices in the global matrix AF indicating the\nfirst row and the first column of the subarray af, respectively.\ndescaf\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix AF.\nib, jb\n(global) The row and column indices in the global matrix B indicating the\nfirst row and the first column of the submatrix B(ib:ib+n-1, jb:jb\n+nrhs-1), respectively.\ndescb\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix B.\nipiv\n(local) Array of size LOCr(m_a)+mb_a.\nThe array ipiv is an input argument if fact = 'F' .\nOn entry, it contains the pivot indices from the factorization A = P*L*U as\ncomputed by p?getrf; (local) row i of the matrix was interchanged with\nthe (global) row ipiv[i - 1].\nThis array must be aligned with A(ia:ia+n-1, *).\nequed\n(global) Must be 'N', 'R', 'C', or 'B'. equed is an input argument if fact\n= 'F' . It specifies the form of equilibration that was done:\nIf equed = 'N', no equilibration was done (always true if fact = 'N');\nIf equed = 'R', row equilibration was done, that is, A has been\npremultiplied by diag(r);\nIf equed = 'C', column equilibration was done, that is, A has been\npostmultiplied by diag(c);\nIf equed = 'B', both row and column equilibration was done; A has been\nreplaced by diag(r)*A*diag(c).\nr, c\n(local)\nArrays of size LOCr(m_a) and LOCc(n_a), respectively.\nThe array r contains the row scale factors for A, and the array c contains\nthe column scale factors for A. These arrays are input arguments if fact =\n'F' only; otherwise they are output arguments. If equed = 'R' or 'B', A\nis multiplied on the left by diag(r); if equed = 'N' or 'C', r is not\naccessed.\nIf fact = 'F' and equed = 'R' or 'B', each element of r must be\npositive.\nIf equed = 'C' or 'B', A is multiplied on the right by diag(c); if equed =\n'N' or 'R', c is not accessed.\nIf fact = 'F' and equed = 'C' or 'B', each element of c must be\npositive. Array r is replicated in every process column, and is aligned with\nthe distributed matrix A. Array c is replicated in every process row, and is\naligned with the distributed matrix A.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1516\n\n\nix, jx\n(global) The row and column indices in the global matrix X indicating the\nfirst row and the first column of the submatrix X(ix:ix+n-1, jx:jx\n+nrhs-1), respectively.\ndescx\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix X.\nlwork\n(local or global) The size of the array work ; must be at least\nmax(p?gecon(lwork), p?gerfs(lwork))+LOCr(n_a).\niwork\n(local, psgesvx/pdgesvx only). Workspace array. The size of iwork is\n(liwork).\nliwork\n(local, psgesvx/pdgesvx only). The size of the array iwork , must be at\nleast LOCr(n_a).\nrwork\n(local)\nWorkspace array, used in complex flavors only.\nThe size of rwork is (lrwork).\nlrwork\n(local or global, pcgesvx/pzgesvx only). The size of the array rwork;must\nbe at least 2*LOCc(n_a) .\nOutput Parameters\nx\n(local)\nPointer into the local memory to an array of local size lld_x*LOCc(jx\n+nrhs-1).\nIf info = 0, the array x contains the solution matrix X to the original\nsystem of equations. Note that A and B are modified on exit if equed≠'N',\nand the solution to the equilibrated system is:\ndiag(C)-1*X, if trans = 'N' and equed = 'C' or 'B'; and\ndiag(R)-1*X, if trans = 'T' or 'C' and equed = 'R' or 'B'.\na\nArray a is not modified on exit if fact = 'F' or 'N', or if fact = 'E' and\nequed = 'N'.\nIf equed≠'N', A is scaled on exit as follows:\nequed = 'R': A = diag(R)*A\nequed = 'C': A = A*diag(c)\nequed = 'B': A = diag(R)*A*diag(c)\naf\nIf fact = 'N' or 'E', then af is an output argument and on exit returns\nthe factors L and U from the factorization A = P*L*U of the original matrix\nA (if fact = 'N') or of the equilibrated matrix A (if fact = 'E'). See the\ndescription of a for the form of the equilibrated matrix.\nb\nOverwritten by diag(R)*B if trans = 'N' and equed = 'R' or 'B';\noverwritten by diag(c)*B if trans = 'T' and equed = 'C' or 'B'; not\nchanged if equed = 'N'.\nr, c\nThese arrays are output arguments if fact≠'F'.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1517\n\n\nSee the description of r, c in Input Arguments section.\nrcond\n(global).\nAn estimate of the reciprocal condition number of the matrix A after\nequilibration (if done). The function sets rcond =0 if the estimate\nunderflows; in this case the matrix is singular (to working precision).\nHowever, anytime rcond is small compared to 1.0, for the working\nprecision, the matrix may be poorly conditioned or even singular.\nferr, berr\n(local)\nArrays of size LOCc(n_b) each. Contain the component-wise forward and\nrelative backward errors, respectively, for each solution vector.\nArrays ferr and berr are both replicated in every process row, and are\naligned with the matrices B and X.\nipiv\nIf fact = 'N' or 'E', then ipiv is an output argument and on exit contains\nthe pivot indices from the factorization A = P*L*U of the original matrix A\n(if fact = 'N') or of the equilibrated matrix A (if fact = 'E').\nequed\nIf fact≠'F' , then equed is an output argument. It specifies the form of\nequilibration that was done (see the description of equed in Input\nArguments section).\nwork[0]\nIf info=0, on exit work[0] returns the minimum value of lwork required\nfor optimum performance.\niwork[0]\nIf info=0, on exit iwork[0] returns the minimum value of liwork required\nfor optimum performance.\nrwork[0]\nIf info=0, on exit rwork[0] returns the minimum value of lrwork required\nfor optimum performance.\ninfo\nIf info=0, the execution is successful.\ninfo < 0: if the ith argument is an array and the jth entry had an illegal\nvalue, then info = -(i*100+j); if the ith argument is a scalar and had an\nillegal value, then info = -i. If info = i, and i ≤ n, then U(i,i) is\nexactly zero. The factorization has been completed, but the factor U is\nexactly singular, so the solution and error bounds could not be computed. If\ninfo = i, and i = n +1, then U is nonsingular, but rcond is less than\nmachine precision. The factorization has been completed, but the matrix is\nsingular to working precision and the solution and error bounds have not\nbeen computed.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?gbsv\nComputes the solution to the system of linear\nequations with a general banded distributed matrix\nand multiple right-hand sides.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1518\n\n\nSyntax\nvoid psgbsv (MKL_INT *n , MKL_INT *bwl , MKL_INT *bwu , MKL_INT *nrhs , float *a ,\nMKL_INT *ja , MKL_INT *desca , MKL_INT *ipiv , float *b , MKL_INT *ib , MKL_INT *descb ,\nfloat *work , MKL_INT *lwork , MKL_INT *info );\nvoid pdgbsv (MKL_INT *n , MKL_INT *bwl , MKL_INT *bwu , MKL_INT *nrhs , double *a ,\nMKL_INT *ja , MKL_INT *desca , MKL_INT *ipiv , double *b , MKL_INT *ib , MKL_INT\n*descb , double *work , MKL_INT *lwork , MKL_INT *info );\nvoid pcgbsv (MKL_INT *n , MKL_INT *bwl , MKL_INT *bwu , MKL_INT *nrhs , MKL_Complex8\n*a , MKL_INT *ja , MKL_INT *desca , MKL_INT *ipiv , MKL_Complex8 *b , MKL_INT *ib ,\nMKL_INT *descb , MKL_Complex8 *work , MKL_INT *lwork , MKL_INT *info );\nvoid pzgbsv (MKL_INT *n , MKL_INT *bwl , MKL_INT *bwu , MKL_INT *nrhs , MKL_Complex16\n*a , MKL_INT *ja , MKL_INT *desca , MKL_INT *ipiv , MKL_Complex16 *b , MKL_INT *ib ,\nMKL_INT *descb , MKL_Complex16 *work , MKL_INT *lwork , MKL_INT *info );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?gbsvfunction computes the solution to a real or complex system of linear equations\nsub(A)*X = sub(B),\nwhere sub(A) = A(1:n, ja:ja+n-1) is an n-by-n real/complex general banded distributed matrix with bwl\nsubdiagonals and bwu superdiagonals, and X and sub(B)= B(ib:ib+n-1, 1:rhs) are n-by-nrhs distributed\nmatrices.\nThe LU decomposition with partial pivoting and row interchanges is used to factor sub(A) as sub(A) =\nP*L*U*Q, where P and Q are permutation matrices, and L and U are banded lower and upper triangular\nmatrices, respectively. The matrix Q represents reordering of columns for the sake of parallelism, while P\nrepresents reordering of rows for numerical stability using classic partial pivoting.\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nInput Parameters\nn\n(global) The number of rows and columns to be operated on, that is, the\norder of the distributed matrix sub(A) (n≥ 0).\nbwl\n(global) The number of subdiagonals within the band of A (0≤ bwl ≤ n-1 ).\nbwu\n(global) The number of superdiagonals within the band of A (0≤ bwu ≤\nn-1 ).\nnrhs\n(global) The number of right hand sides; the number of columns of the\ndistributed matrix sub(B) (nrhs≥ 0).\na, b\n(local)\nPointers into the local memory to arrays of local size a: lld_a*LOCc(ja\n+n-1) and b: lld_b*LOCc(nrhs).\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1519\n\n\nOn entry, the array a contains the local pieces of the global array A.\nOn entry, the array b contains the right hand side distributed matrix sub(B).\nja\n(global) The index in the global matrix A indicating the start of the matrix to\nbe operated on (which may be either all of A or a submatrix of A).\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\nIf desca[dtype_ - 1] = 501, then dlen_≥ 7;\nelse if desca[dtype_ - 1] = 1, then dlen_≥ 9.\nib\n(global) The row index in the global matrix B indicating the first row of the\nmatrix to be operated on (which may be either all of B or a submatrix of B).\ndescb\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix B.\nIf descb[dtype_-1] = 502, then dlen_≥ 7;\nelse if descb[dtype_-1] = 1, then dlen_≥ 9.\nwork\n(local)\nWorkspace array of size lwork.\nlwork\n(local or global) The size of the array work, must be at least lwork≥ (NB\n+bwu)*(bwl+bwu)+6*(bwl+bwu)*(bwl+2*bwu) +\n+ max(nrhs *(NB+2*bwl+4*bwu), 1).\nOutput Parameters\na\nOn exit, contains details of the factorization. Note that the resulting\nfactorization is not the same factorization as returned from LAPACK.\nAdditional permutations are performed on the matrix for the sake of\nparallelism.\nb\nOn exit, this array contains the local pieces of the solution distributed\nmatrix X.\nipiv\n(local) array.\nThe size of ipiv must be at least desca[NB - 1]. This array contains pivot\nindices for local factorizations. You should not alter the contents between\nfactorization and solve.\nwork[0]\nOn exit, work[0] contains the minimum value of lwork required for\noptimum performance.\ninfo\nIf info=0, the execution is successful. info < 0:\nIf the ith argument is an array and the j-th entry had an illegal value, then\ninfo = -(i*100+j); if the ith argument is a scalar and had an illegal\nvalue, then info = -i.\ninfo> 0:\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1520\n\n\nIf info = k ≤ NPROCS, the submatrix stored on processor info and\nfactored locally was not nonsingular, and the factorization was not\ncompleted. If info = k > NPROCS, the submatrix stored on processor\ninfo-NPROCS representing interactions with other processors was not\nnonsingular, and the factorization was not completed.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?dbsv\nSolves a general band system of linear equations.\nSyntax\nvoid psdbsv (MKL_INT *n , MKL_INT *bwl , MKL_INT *bwu , MKL_INT *nrhs , float *a ,\nMKL_INT *ja , MKL_INT *desca , float *b , MKL_INT *ib , MKL_INT *descb , float *work ,\nMKL_INT *lwork , MKL_INT *info );\nvoid pddbsv (MKL_INT *n , MKL_INT *bwl , MKL_INT *bwu , MKL_INT *nrhs , double *a ,\nMKL_INT *ja , MKL_INT *desca , double *b , MKL_INT *ib , MKL_INT *descb , double *work ,\nMKL_INT *lwork , MKL_INT *info );\nvoid pcdbsv (MKL_INT *n , MKL_INT *bwl , MKL_INT *bwu , MKL_INT *nrhs , MKL_Complex8\n*a , MKL_INT *ja , MKL_INT *desca , MKL_Complex8 *b , MKL_INT *ib , MKL_INT *descb ,\nMKL_Complex8 *work , MKL_INT *lwork , MKL_INT *info );\nvoid pzdbsv (MKL_INT *n , MKL_INT *bwl , MKL_INT *bwu , MKL_INT *nrhs , MKL_Complex16\n*a , MKL_INT *ja , MKL_INT *desca , MKL_Complex16 *b , MKL_INT *ib , MKL_INT *descb ,\nMKL_Complex16 *work , MKL_INT *lwork , MKL_INT *info );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?dbsvfunction solves the following system of linear equations:\nA(1:n, ja:ja+n-1)* X = B(ib:ib+n-1, 1:nrhs),\nwhere A(1:n, ja:ja+n-1) is an n-by-n real/complex banded diagonally dominant-like distributed matrix\nwith bandwidth bwl, bwu.\nGaussian elimination without pivoting is used to factor a reordering of the matrix into LU.\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nInput Parameters\nn\n(global) The order of the distributed submatrix A, (n≥ 0).\nbwl\n(global) Number of subdiagonals. 0 ≤ bwl ≤ n-1.\nbwu\n(global) Number of subdiagonals. 0 ≤ bwu ≤ n-1.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1521\n\n\nnrhs\n(global) The number of right-hand sides; the number of columns of the\ndistributed submatrix B, (nrhs ≥ 0).\na\n(local).\nPointer into the local memory to an array with leading size lld_a ≥ (bwl\n+bwu+1) (stored in desca). On entry, this array contains the local pieces of\nthe distributed matrix.\nja\n(global) The index in the global matrix A indicating the start of the matrix to\nbe operated on (which may be either all of A or a submatrix of A).\ndesca\n(global and local) array of size dlen.\nIf 1d type (dtype_a=501 or 502), dlen ≥ 7;\nIf 2d type (dtype_a=1), dlen ≥ 9.\nThe array descriptor for the distributed matrix A.\nContains information of mapping of A to memory.\nb\n(local)\nPointer into the local memory to an array of local lead size lld_b ≥ nb. On\nentry, this array contains the local pieces of the right hand sides B(ib:ib\n+n-1, 1:nrhs).\nib\n(global) The row index in the global matrix B indicating the first row of the\nmatrix to be operated on (which may be either all of b or a submatrix of B).\ndescb\n(global and local) array of size dlen.\nIf 1d type (dtype_b =502), dlen ≥ 7;\nIf 2d type (dtype_b =1), dlen ≥ 9.\nThe array descriptor for the distributed matrix B.\nContains information of mapping of B to memory.\nwork\n(local).\nTemporary workspace. This space may be overwritten in between calls to\nfunctions. work must be the size given in lwork.\nlwork\n(local or global) Size of user-input workspace work. If lwork is too small,\nthe minimal acceptable size will be returned in work[0] and an error code\nis returned.\nlwork ≥ nb(bwl+bwu)+6max(bwl,bwu)*max(bwl,bwu)\n+max((max(bwl,bwu)nrhs), max(bwl,bwu)*max(bwl,bwu))\nOutput Parameters\na\nOn exit, this array contains information containing details of the\nfactorization.\nNote that permutations are performed on the matrix, so that the factors\nreturned are different from those returned by LAPACK.\nb\nOn exit, this contains the local piece of the solutions distributed matrix X.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1522\n\n\nwork\nOn exit, work[0] contains the minimal lwork.\ninfo\n(local) If info=0, the execution is successful.\n< 0: If the i-th argument is an array and the j-entry had an illegal value,\nthen info = -(i*100+j), if the i-th argument is a scalar and had an\nillegal value, then info = -i.\n> 0: If info = k < NPROCS, the submatrix stored on processor info and\nfactored locally was not positive definite, and the factorization was not\ncompleted.\nIf info = k > NPROCS, the submatrix stored on processor info-NPROCS\nrepresenting interactions with other processors was not positive definite,\nand the factorization was not completed.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?dtsv\nSolves a general tridiagonal system of linear\nequations.\nSyntax\nvoid psdtsv (MKL_INT *n , MKL_INT *nrhs , float *dl , float *d , float *du , MKL_INT\n*ja , MKL_INT *desca , float *b , MKL_INT *ib , MKL_INT *descb , float *work , MKL_INT\n*lwork , MKL_INT *info );\nvoid pddtsv (MKL_INT *n , MKL_INT *nrhs , double *dl , double *d , double *du , MKL_INT\n*ja , MKL_INT *desca , double *b , MKL_INT *ib , MKL_INT *descb , double *work , MKL_INT\n*lwork , MKL_INT *info );\nvoid pcdtsv (MKL_INT *n , MKL_INT *nrhs , MKL_Complex8 *dl , MKL_Complex8 *d ,\nMKL_Complex8 *du , MKL_INT *ja , MKL_INT *desca , MKL_Complex8 *b , MKL_INT *ib ,\nMKL_INT *descb , MKL_Complex8 *work , MKL_INT *lwork , MKL_INT *info );\nvoid pzdtsv (MKL_INT *n , MKL_INT *nrhs , MKL_Complex16 *dl , MKL_Complex16 *d ,\nMKL_Complex16 *du , MKL_INT *ja , MKL_INT *desca , MKL_Complex16 *b , MKL_INT *ib ,\nMKL_INT *descb , MKL_Complex16 *work , MKL_INT *lwork , MKL_INT *info );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe function solves a system of linear equations\nA(1:n, ja:ja+n-1) * X = B(ib:ib+n-1, 1:nrhs),\nwhere A(1:n, ja:ja+n-1) is an n-by-n complex tridiagonal diagonally dominant-like distributed matrix.\nGaussian elimination without pivoting is used to factor a reordering of the matrix into L U.\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1523\n\n\nProduct and Performance Information\nNotice revision #20201201\nInput Parameters\nn\n(global) The order of the distributed submatrix A(n≥ 0).\nnrhs\nThe number of right hand sides; the number of columns of the distributed\nmatrix B(nrhs≥ 0).\ndl\n(local).\nPointer to local part of global vector storing the lower diagonal of the\nmatrix. Globally, dl[0] is not referenced, and dl must be aligned with d.\nMust be of size > desca[nb_ - 1].\nd\n(local).\nPointer to local part of global vector storing the main diagonal of the matrix.\ndu\n(local).\nPointer to local part of global vector storing the upper diagonal of the\nmatrix. Globally, du[n - 1] is not referenced, and du must be aligned with\nd.\nja\n(global) The index in the global matrix A indicating the start of the matrix to\nbe operated on (which may be either all of A or a submatrix of A).\ndesca\n(global and local) array of size dlen.\nIf 1d type (dtype_a=501 or 502), dlen ≥ 7;\nIf 2d type (dtype_a=1), dlen ≥ 9.\nThe array descriptor for the distributed matrix A.\nContains information of mapping of A to memory.\nb\n(local)\nPointer into the local memory to an array of local lead size lld_b > nb. On\nentry, this array contains the local pieces of the right hand sides B(ib:ib\n+n-1, 1:nrhs).\nib\n(global) The row index in the global matrix B indicating the first row of the\nmatrix to be operated on (which may be either all of b or a submatrix of B).\ndescb\n(global and local) array of size dlen.\nIf 1d type (dtype_b =502), dlen ≥ 7;\nIf 2d type (dtype_b =1), dlen ≥ 9.\nThe array descriptor for the distributed matrix B.\nContains information of mapping of B to memory.\nwork\n(local).\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1524\n\n\nlwork\n(local or global) Size of user-input workspace work. If lwork is too small,\nthe minimal acceptable size will be returned in work[0] and an error code\nis returned. lwork > (12*NPCOL+3*nb)+max((10+2*min(100,\nnrhs))*NPCOL+4*nrhs, 8*NPCOL)\nOutput Parameters\ndl\nOn exit, this array contains information containing the * factors of the\nmatrix.\nd\nOn exit, this array contains information containing the * factors of the\nmatrix. Must be of size > desca[nb_ - 1].\ndu\nOn exit, this array contains information containing the * factors of the\nmatrix. Must be of size > desca[nb_ - 1].\nb\nOn exit, this contains the local piece of the solutions distributed matrix X.\nwork\nOn exit, work[0] contains the minimal lwork.\ninfo\n(local) If info=0, the execution is successful.\n< 0: If the i-th argument is an array and the j-entry had an illegal value,\nthen info = -(i*100+j), if the i-th argument is a scalar and had an\nillegal value, then info = -i.\n> 0: If info = k<NPROCS, the submatrix stored on processor info and\nfactored locally was not positive definite, and the factorization was not\ncompleted.\nIf info = k > NPROCS, the submatrix stored on processor info-NPROCS\nrepresenting interactions with other processors was not positive definite,\nand the factorization was not completed.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?posv\nSolves a symmetric positive definite system of linear\nequations.\nSyntax\nvoid psposv (char *uplo , MKL_INT *n , MKL_INT *nrhs , float *a , MKL_INT *ia , MKL_INT\n*ja , MKL_INT *desca , float *b , MKL_INT *ib , MKL_INT *jb , MKL_INT *descb , MKL_INT\n*info );\nvoid pdposv (char *uplo , MKL_INT *n , MKL_INT *nrhs , double *a , MKL_INT *ia , MKL_INT\n*ja , MKL_INT *desca , double *b , MKL_INT *ib , MKL_INT *jb , MKL_INT *descb , MKL_INT\n*info );\nvoid pcposv (char *uplo , MKL_INT *n , MKL_INT *nrhs , MKL_Complex8 *a , MKL_INT *ia ,\nMKL_INT *ja , MKL_INT *desca , MKL_Complex8 *b , MKL_INT *ib , MKL_INT *jb , MKL_INT\n*descb , MKL_INT *info );\nvoid pzposv (char *uplo , MKL_INT *n , MKL_INT *nrhs , MKL_Complex16 *a , MKL_INT *ia ,\nMKL_INT *ja , MKL_INT *desca , MKL_Complex16 *b , MKL_INT *ib , MKL_INT *jb , MKL_INT\n*descb , MKL_INT *info );\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1525\n\n\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?posvfunction computes the solution to a real/complex system of linear equations\nsub(A)*X = sub(B),\nwhere sub(A) denotes A(ia:ia+n-1,ja:ja+n-1) and is an n-by-n symmetric/Hermitian distributed positive\ndefinite matrix and X and sub(B) denoting B(ib:ib+n-1,jb:jb+nrhs-1) are n-by-nrhs distributed\nmatrices. The Cholesky decomposition is used to factor sub(A) as\nsub(A) = UT*U, if uplo = 'U', or\nsub(A) = L*LT, if uplo = 'L',\nwhere U is an upper triangular matrix and L is a lower triangular matrix. The factored form of sub(A) is then\nused to solve the system of equations.\nInput Parameters\nuplo\n(global) Must be 'U' or 'L'.\nIndicates whether the upper or lower triangular part of sub(A) is stored.\nn\n(global) The order of the distributed matrix sub(A) (n≥ 0).\nnrhs\nThe number of right-hand sides; the number of columns of the distributed\nmatrix sub(B) (nrhs≥ 0).\na\n(local)\nPointer into the local memory to an array of size lld_a*LOCc(ja+n-1). On\nentry, this array contains the local pieces of the n-by-n symmetric\ndistributed matrix sub(A) to be factored.\nIf uplo = 'U', the leading n-by-n upper triangular part of sub(A) contains\nthe upper triangular part of the matrix, and its strictly lower triangular part\nis not referenced.\nIf uplo = 'L', the leading n-by-n lower triangular part of sub(A) contains\nthe lower triangular part of the distributed matrix, and its strictly upper\ntriangular part is not referenced.\nia, ja\n(global) The row and column indices in the global matrix A indicating the\nfirst row and the first column of the submatrix A, respectively.\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\nb\n(local)\nPointer into the local memory to an array of size lld_b*LOCc(jb+nrhs-1).\nOn entry, the local pieces of the right hand sides distributed matrix sub(B).\nib, jb\n(global) The row and column indices in the global matrix B indicating the\nfirst row and the first column of the submatrix B, respectively.\ndescb\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix B.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1526\n\n\nOutput Parameters\na\nOn exit, if info = 0, this array contains the local pieces of the factor U or L\nfrom the Cholesky factorization sub(A) = UH*U, or L*LH.\nb\nOn exit, if info = 0, sub(B) is overwritten by the solution distributed\nmatrix X.\ninfo\n(global)\nIf info =0, the execution is successful.\nIf info < 0: If the i-th argument is an array and the j-th entry, indexed\nj-1, had an illegal value, then info = -(i*100+j), if the i-th argument is\na scalar and had an illegal value, then info = -i.\nIf info > 0: If info = k, the leading minor of order k, A(ia:ia+k-1,\nja:ja+k-1) is not positive definite, and the factorization could not be\ncompleted, and the solution has not been computed.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?posvx\nSolves a symmetric or Hermitian positive definite\nsystem of linear equations.\nSyntax\nvoid psposvx (char *fact , char *uplo , MKL_INT *n , MKL_INT *nrhs , float *a , MKL_INT\n*ia , MKL_INT *ja , MKL_INT *desca , float *af , MKL_INT *iaf , MKL_INT *jaf , MKL_INT\n*descaf , char *equed , float *sr , float *sc , float *b , MKL_INT *ib , MKL_INT *jb ,\nMKL_INT *descb , float *x , MKL_INT *ix , MKL_INT *jx , MKL_INT *descx , float *rcond ,\nfloat *ferr , float *berr , float *work , MKL_INT *lwork , MKL_INT *iwork , MKL_INT\n*liwork , MKL_INT *info );\nvoid pdposvx (char *fact , char *uplo , MKL_INT *n , MKL_INT *nrhs , double *a , MKL_INT\n*ia , MKL_INT *ja , MKL_INT *desca , double *af , MKL_INT *iaf , MKL_INT *jaf , MKL_INT\n*descaf , char *equed , double *sr , double *sc , double *b , MKL_INT *ib , MKL_INT\n*jb , MKL_INT *descb , double *x , MKL_INT *ix , MKL_INT *jx , MKL_INT *descx , double\n*rcond , double *ferr , double *berr , double *work , MKL_INT *lwork , MKL_INT *iwork ,\nMKL_INT *liwork , MKL_INT *info );\nvoid pcposvx (char *fact , char *uplo , MKL_INT *n , MKL_INT *nrhs , MKL_Complex8 *a ,\nMKL_INT *ia , MKL_INT *ja , MKL_INT *desca , MKL_Complex8 *af , MKL_INT *iaf , MKL_INT\n*jaf , MKL_INT *descaf , char *equed , float *sr , float *sc , MKL_Complex8 *b , MKL_INT\n*ib , MKL_INT *jb , MKL_INT *descb , MKL_Complex8 *x , MKL_INT *ix , MKL_INT *jx ,\nMKL_INT *descx , float *rcond , float *ferr , float *berr , MKL_Complex8 *work ,\nMKL_INT *lwork , float *rwork , MKL_INT *lrwork , MKL_INT *info );\nvoid pzposvx (char *fact , char *uplo , MKL_INT *n , MKL_INT *nrhs , MKL_Complex16 *a ,\nMKL_INT *ia , MKL_INT *ja , MKL_INT *desca , MKL_Complex16 *af , MKL_INT *iaf , MKL_INT\n*jaf , MKL_INT *descaf , char *equed , double *sr , double *sc , MKL_Complex16 *b ,\nMKL_INT *ib , MKL_INT *jb , MKL_INT *descb , MKL_Complex16 *x , MKL_INT *ix , MKL_INT\n*jx , MKL_INT *descx , double *rcond , double *ferr , double *berr , MKL_Complex16\n*work , MKL_INT *lwork , double *rwork , MKL_INT *lrwork , MKL_INT *info );\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1527\n\n\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?posvxfunction uses the Cholesky factorization A=UT*U or A=L*LT to compute the solution to a real or\ncomplex system of linear equations\nA(ia:ia+n-1, ja:ja+n-1)*X = B(ib:ib+n-1, jb:jb+nrhs-1),\nwhere A(ia:ia+n-1, ja:ja+n-1) is a n-by-n matrix and X and B(ib:ib+n-1,jb:jb+nrhs-1) are n-by-\nnrhs matrices.\nError bounds on the solution and a condition estimate are also provided.\nIn the following comments y denotes Y(iy:iy+m-1, jy:jy+k-1), an m-by-k matrix where y can be a, af, b\nand x.\nThe function p?posvx performs the following steps:\n1.\nIf fact = 'E', real scaling factors s are computed to equilibrate the system:\ndiag(sr)*A*diag(sc)*inv(diag(sc))*X = diag(sr)*B\nWhether or not the system will be equilibrated depends on the scaling of the matrix A, but if\nequilibration is used, A is overwritten by diag(sr)*A*diag(sc) and B by diag(sr)*B .\n2.\nIf fact = 'N' or 'E', the Cholesky decomposition is used to factor the matrix A (after equilibration if\nfact = 'E') as\nA = UT*U, if uplo = 'U', or\nA = L*LT, if uplo = 'L',\nwhere U is an upper triangular matrix and L is a lower triangular matrix.\n3.\nThe factored form of A is used to estimate the condition number of the matrix A. If the reciprocal of the\ncondition number is less than machine precision, steps 4-6 are skipped\n4.\nThe system of equations is solved for X using the factored form of A.\n5.\nIterative refinement is applied to improve the computed solution matrix and calculate error bounds and\nbackward error estimates for it.\n6.\nIf equilibration was used, the matrix X is premultiplied by diag(sr) so that it solves the original system\nbefore equilibration.\nInput Parameters\nfact\n(global) Must be 'F', 'N', or 'E'.\nSpecifies whether or not the factored form of the matrix A is supplied on\nentry, and if not, whether the matrix A should be equilibrated before it is\nfactored.\nIf fact = 'F': on entry, af contains the factored form of A. If equed =\n'Y', the matrix A has been equilibrated with scaling factors given by s. a\nand af will not be modified.\nIf fact = 'N', the matrix A will be copied to af and factored.\nIf fact = 'E', the matrix A will be equilibrated if necessary, then copied to\naf and factored.\nuplo\n(global) Must be 'U' or 'L'.\nIndicates whether the upper or lower triangular part of A is stored.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1528\n\n\nn\n(global) The order of the distributed matrix sub(A) (n≥ 0).\nnrhs\n(global) The number of right-hand sides; the number of columns of the\ndistributed submatrices B and X. (nrhs≥ 0).\na\n(local)\nPointer into the local memory to an array of local size lld_a*LOCc(ja\n+n-1). On entry, the symmetric/Hermitian matrix A, except if fact = 'F'\nand equed = 'Y', then A must contain the equilibrated matrix\ndiag(sr)*A*diag(sc).\nIf uplo = 'U', the leading n-by-n upper triangular part of A contains the\nupper triangular part of the matrix A, and the strictly lower triangular part\nof A is not referenced.\nIf uplo = 'L', the leading n-by-n lower triangular part of A contains the\nlower triangular part of the matrix A, and the strictly upper triangular part\nof A is not referenced. A is not modified if fact = 'F' or 'N', or if fact =\n'E' and equed = 'N' on exit.\nia, ja\n(global) The row and column indices in the global matrix A indicating the\nfirst row and the first column of the submatrix A, respectively.\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\naf\n(local)\nPointer into the local memory to an array of local size lld_af*LOCc(ja\n+n-1).\nIf fact = 'F', then af is an input argument and on entry contains the\ntriangular factor U or L from the Cholesky factorization A = UT*U or A =\nL*LT, in the same storage format as A. If equed ≠ 'N', then af is the\nfactored form of the equilibrated matrix diag(sr)*A*diag(sc).\niaf, jaf\n(global) The row and column indices in the global matrix AF indicating the\nfirst row and the first column of the submatrix AF, respectively.\ndescaf\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix AF.\nequed\n(global) Must be 'N' or 'Y'.\nequed is an input argument if fact = 'F'. It specifies the form of\nequilibration that was done:\nIf equed = 'N', no equilibration was done (always true if fact = 'N');\nIf equed = 'Y', equilibration was done and A has been replaced by\ndiag(sr)*A*diag(sc).\nsr\n(local)\nArray of size lld_a.\nThe array s contains the scale factors for A. This array is an input argument\nif fact = 'F' only; otherwise it is an output argument.\nIf equed = 'N', s is not accessed.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1529\n\n\nIf fact = 'F' and equed = 'Y', each element of s must be positive.\nb\n(local)\nPointer into the local memory to an array of local size lld_b*LOCc(jb\n+nrhs-1). On entry, the n-by-nrhs right-hand side matrix B.\nib, jb\n(global) The row and column indices in the global matrix B indicating the\nfirst row and the first column of the submatrix B, respectively.\ndescb\n(global and local) Array of size dlen_. The array descriptor for the\ndistributed matrix B.\nx\n(local)\nPointer into the local memory to an array of local size lld_x*LOCc(jx\n+nrhs-1).\nix, jx\n(global) The row and column indices in the global matrix X indicating the\nfirst row and the first column of the submatrix X, respectively.\ndescx\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix X.\nwork\n(local)\nWorkspace array of size lwork.\nlwork\n(local or global)\nThe size of the array work. lwork is local input and must be at least lwork\n= max(p?pocon(lwork), p?porfs(lwork)) + LOCr(n_a).\nlwork = 3*desca[lld_ - 1].\nIf lwork = -1, then lwork is global input and a workspace query is\nassumed; the function only calculates the minimum and optimal size for all\nwork arrays. Each of these values is returned in the first entry of the\ncorresponding work array, and no error message is issued by pxerbla.\niwork\n(local) Workspace array of size liwork.\nliwork\n(local or global)\nThe size of the array iwork. liwork is local input and must be at least\nliwork = desca[lld_ - 1]liwork = LOCr(n_a).\nIf liwork = -1, then liwork is global input and a workspace query is\nassumed; the function only calculates the minimum and optimal size for all\nwork arrays. Each of these values is returned in the first entry of the\ncorresponding work array, and no error message is issued by pxerbla.\nOutput Parameters\na\nOn exit, if fact = 'E' and equed = 'Y', a is overwritten by\ndiag(sr)*a*diag(sc).\naf\nIf fact = 'N', then af is an output argument and on exit returns the\ntriangular factor U or L from the Cholesky factorization A = UT*U or A =\nL*LT of the original matrix A.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1530\n\n\nIf fact = 'E', then af is an output argument and on exit returns the\ntriangular factor U or L from the Cholesky factorization A = UT*U or A =\nL*LT of the equilibrated matrix A (see the description of A for the form of\nthe equilibrated matrix).\nequed\nIf fact≠'F' , then equed is an output argument. It specifies the form of\nequilibration that was done (see the description of equed in Input\nArguments section).\nsr\nThis array is an output argument if fact≠'F'.\nSee the description of sr in Input Arguments section.\nsc\nThis array is an output argument if fact≠'F'.\nSee the description of sc in Input Arguments section.\nb\nOn exit, if equed = 'N', b is not modified; if trans = 'N' and equed =\n'R' or 'B', b is overwritten by diag(r)*b; if trans = 'T' or 'C' and\nequed = 'C' or 'B', b is overwritten by diag(c)*b.\nx\n(local)\nIf info = 0 the n-by-nrhs solution matrix X to the original system of\nequations.\nNote that A and B are modified on exit if equed≠'N', and the solution to\nthe equilibrated system is\ninv(diag(sc))*X if trans = 'N' and equed = 'C' or 'B', or\ninv(diag(sr))*X if trans = 'T' or 'C' and equed = 'R' or 'B'.\nrcond\n(global)\nAn estimate of the reciprocal condition number of the matrix A after\nequilibration (if done). If rcond is less than the machine precision (in\nparticular, if rcond=0), the matrix is singular to working precision. This\ncondition is indicated by a return code of info > 0.\nferr\nArrays of size at least max(LOC,n_b). The estimated forward error bounds\nfor each solution vector X(j) (the j-th column of the solution matrix X). If\nxtrue is the true solution, ferr[j - 1] bounds the magnitude of the largest\nentry in (X(j) - xtrue) divided by the magnitude of the largest entry in\nX(j). The quality of the error bound depends on the quality of the estimate\nof norm(inv(A)) computed in the code; if the estimate of norm(inv(A))\nis accurate, the error bound is guaranteed.\nberr\n(local)\nArrays of size at least max(LOC,n_b). The componentwise relative\nbackward error of each solution vector X(j) (the smallest relative change in\nany entry of A or B that makes X(j) an exact solution).\nwork[0]\n(local) On exit, work[0] returns the minimal and optimal liwork.\ninfo\n(global)\nIf info=0, the execution is successful.\n< 0: if info = -i, the i-th argument had an illegal value\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1531\n\n\n> 0: if info = i, and i is ≤ n: if info = i, the leading minor of order i of\na is not positive definite, so the factorization could not be completed, and\nthe solution and error bounds could not be computed.\n= n+1: rcond is less than machine precision. The factorization has been\ncompleted, but the matrix is singular to working precision, and the solution\nand error bounds have not been computed.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?pbsv\nSolves a symmetric/Hermitian positive definite banded\nsystem of linear equations.\nSyntax\nvoid pspbsv (char *uplo , MKL_INT *n , MKL_INT *bw , MKL_INT *nrhs , float *a , MKL_INT\n*ja , MKL_INT *desca , float *b , MKL_INT *ib , MKL_INT *descb , float *work , MKL_INT\n*lwork , MKL_INT *info );\nvoid pdpbsv (char *uplo , MKL_INT *n , MKL_INT *bw , MKL_INT *nrhs , double *a , MKL_INT\n*ja , MKL_INT *desca , double *b , MKL_INT *ib , MKL_INT *descb , double *work , MKL_INT\n*lwork , MKL_INT *info );\nvoid pcpbsv (char *uplo , MKL_INT *n , MKL_INT *bw , MKL_INT *nrhs , MKL_Complex8 *a ,\nMKL_INT *ja , MKL_INT *desca , MKL_Complex8 *b , MKL_INT *ib , MKL_INT *descb ,\nMKL_Complex8 *work , MKL_INT *lwork , MKL_INT *info );\nvoid pzpbsv (char *uplo , MKL_INT *n , MKL_INT *bw , MKL_INT *nrhs , MKL_Complex16 *a ,\nMKL_INT *ja , MKL_INT *desca , MKL_Complex16 *b , MKL_INT *ib , MKL_INT *descb ,\nMKL_Complex16 *work , MKL_INT *lwork , MKL_INT *info );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?pbsvfunction solves a system of linear equations\nA(1:n, ja:ja+n-1)*X = B(ib:ib+n-1, 1:nrhs),\nwhere A(1:n, ja:ja+n-1) is an n-by-n real/complex banded symmetric positive definite distributed matrix\nwith bandwidth bw.\nCholesky factorization is used to factor a reordering of the matrix into L*L'.\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nInput Parameters\nuplo\n(global) Must be 'U' or 'L'.\nIndicates whether the upper or lower triangular of A is stored.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1532\n\n\nIf uplo = 'U', the upper triangular A is stored\nIf uplo = 'L', the lower triangular of A is stored.\nn\n(global) The order of the distributed matrix A(n≥ 0).\nbw\n(global) The number of subdiagonals in L or U. 0 ≤ bw ≤ n-1.\nnrhs\n(global) The number of right-hand sides; the number of columns in\nB(nrhs≥ 0).\na\n(local).\nPointer into the local memory to an array with leading size lld_a ≥ (bw\n+1) (stored in desca). On entry, this array contains the local pieces of the\ndistributed matrix sub(A) to be factored.\nja\n(global) The index in the global matrix A indicating the start of the matrix to\nbe operated on (which may be either all of A or a submatrix of A).\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\nb\n(local)\nPointer into the local memory to an array of local lead size lld_b ≥ nb. On\nentry, this array contains the local pieces of the right hand sides B(ib:ib\n+n-1, 1:nrhs).\nib\n(global) The row index in the global matrix B indicating the first row of the\nmatrix to be operated on (which may be either all of b or a submatrix of B).\ndescb\n(global and local) array of size dlen.\nIf 1D type (dtype_b =502), dlen ≥ 7;\nIf 2D type (dtype_b =1), dlen ≥ 9.\nThe array descriptor for the distributed matrix B.\nContains information of mapping of B to memory.\nwork\n(local).\nTemporary workspace. This space may be overwritten in between calls to\nfunctions. work must be the size given in lwork.\nlwork\n(local or global) Size of user-input workspace work. If lwork is too small,\nthe minimal acceptable size will be returned in work[0] and an error code\nis returned. lwork ≥ (nb+2*bw)*bw +max((bw*nrhs), bw*bw)\nOutput Parameters\na\nOn exit, this array contains information containing details of the\nfactorization. Note that permutations are performed on the matrix, so that\nthe factors returned are different from those returned by LAPACK.\nb\nOn exit, contains the local piece of the solutions distributed matrix X.\nwork\nOn exit, work[0] contains the minimal lwork.\ninfo\n(global) If info=0, the execution is successful.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1533\n\n\n< 0: If the i-th argument is an array and the j-entry had an illegal value,\nthen info = -(i*100+j), if the i-th argument is a scalar and had an\nillegal value, then info = -i.\n> 0: If info = k ≤ NPROCS, the submatrix stored on processor info and\nfactored locally was not positive definite, and the factorization was not\ncompleted.\nIf info = k > NPROCS, the submatrix stored on processor info-NPROCS\nrepresenting interactions with other processors was not positive definite,\nand the factorization was not completed.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?ptsv\nSyntax\nSolves a symmetric or Hermitian positive definite tridiagonal system of linear equations.\nvoid psptsv (MKL_INT *n , MKL_INT *nrhs , float *d , float *e , MKL_INT *ja , MKL_INT\n*desca , float *b , MKL_INT *ib , MKL_INT *descb , float *work , MKL_INT *lwork ,\nMKL_INT *info );\nvoid pdptsv (MKL_INT *n , MKL_INT *nrhs , double *d , double *e , MKL_INT *ja , MKL_INT\n*desca , double *b , MKL_INT *ib , MKL_INT *descb , double *work , MKL_INT *lwork ,\nMKL_INT *info );\nvoid pcptsv (char *uplo , MKL_INT *n , MKL_INT *nrhs , float *d , MKL_Complex8 *e ,\nMKL_INT *ja , MKL_INT *desca , MKL_Complex8 *b , MKL_INT *ib , MKL_INT *descb ,\nMKL_Complex8 *work , MKL_INT *lwork , MKL_INT *info );\nvoid pzptsv (char *uplo , MKL_INT *n , MKL_INT *nrhs , double *d , MKL_Complex16 *e ,\nMKL_INT *ja , MKL_INT *desca , MKL_Complex16 *b , MKL_INT *ib , MKL_INT *descb ,\nMKL_Complex16 *work , MKL_INT *lwork , MKL_INT *info );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?ptsvfunction solves a system of linear equations\nA(1:n, ja:ja+n-1)*X = B(ib:ib+n-1, 1:nrhs),\nwhere A(1:n, ja:ja+n-1) is an n-by-n real tridiagonal symmetric positive definite distributed matrix.\nCholesky factorization is used to factor a reordering of the matrix into L*L'.\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1534\n\n\nInput Parameters\nn\n(global) The order of matrix A(n≥ 0).\nnrhs\n(global) The number of right-hand sides; the number of columns of the\ndistributed submatrix B(nrhs≥ 0).\nd\n(local)\nPointer to local part of global vector storing the main diagonal of the matrix.\ne\n(local)\nPointer to local part of global vector storing the upper diagonal of the\nmatrix. Globally, du(n) is not referenced, and du must be aligned with d.\nja\n(global) The index in the global matrix A indicating the start of the matrix to\nbe operated on (which may be either all of A or a submatrix of A).\ndesca\n(global and local) array of size dlen.\nIf 1d type (dtype_a=501 or 502), dlen ≥ 7;\nIf 2d type (dtype_a=1), dlen ≥ 9.\nThe array descriptor for the distributed matrix A.\nContains information of mapping of A to memory.\nb\n(local)\nPointer into the local memory to an array of local lead size lld_b ≥ nb.\nOn entry, this array contains the local pieces of the right hand sides\nB(ib:ib+n-1, 1:nrhs).\nib\n(global) The row index in the global matrix B indicating the first row of the\nmatrix to be operated on (which may be either all of b or a submatrix of B).\ndescb\n(global and local) array of size dlen.\nIf 1d type (dtype_b = 502), dlen ≥ 7;\nIf 2d type (dtype_b = 1), dlen ≥ 9.\nThe array descriptor for the distributed matrix B.\nContains information of mapping of B to memory.\nwork\n(local).\nTemporary workspace. This space may be overwritten in between calls to\nfunctions. work must be the size given in lwork.\nlwork\n(local or global) Size of user-input workspace work. If lwork is too small,\nthe minimal acceptable size will be returned in work[0] and an error code\nis returned. lwork > (12*NPCOL+3*nb)+max((10+2*min(100,\nnrhs))*NPCOL+4*nrhs, 8*NPCOL).\nOutput Parameters\nd\nOn exit, this array contains information containing the factors of the matrix.\nMust be of size greater than or equal to desca[nb_ - 1].\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1535\n\n\ne\nOn exit, this array contains information containing the factors of the matrix.\nMust be of size greater than or equal to desca[nb_ - 1].\nb\nOn exit, this contains the local piece of the solutions distributed matrix X.\nwork\nOn exit, work[0] contains the minimal lwork.\ninfo\n(local) If info=0, the execution is successful.\n< 0: If the i-th argument is an array and the j-entry had an illegal value,\nthen info = -(i*100+j), if the i-th argument is a scalar and had an\nillegal value, then info = -i.\n> 0: If info = k ≤ NPROCS, the submatrix stored on processor info and\nfactored locally was not positive definite, and the factorization was not\ncompleted.\nIf info = k > NPROCS, the submatrix stored on processor info-NPROCS\nrepresenting interactions with other processors was not positive definite,\nand the factorization was not completed.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?gels\nSolves overdetermined or underdetermined linear\nsystems involving a matrix of full rank.\nSyntax\nvoid psgels (char *trans , MKL_INT *m , MKL_INT *n , MKL_INT *nrhs , float *a , MKL_INT\n*ia , MKL_INT *ja , MKL_INT *desca , float *b , MKL_INT *ib , MKL_INT *jb , MKL_INT\n*descb , float *work , MKL_INT *lwork , MKL_INT *info );\nvoid pdgels (char *trans , MKL_INT *m , MKL_INT *n , MKL_INT *nrhs , double *a , MKL_INT\n*ia , MKL_INT *ja , MKL_INT *desca , double *b , MKL_INT *ib , MKL_INT *jb , MKL_INT\n*descb , double *work , MKL_INT *lwork , MKL_INT *info );\nvoid pcgels (char *trans , MKL_INT *m , MKL_INT *n , MKL_INT *nrhs , MKL_Complex8 *a ,\nMKL_INT *ia , MKL_INT *ja , MKL_INT *desca , MKL_Complex8 *b , MKL_INT *ib , MKL_INT\n*jb , MKL_INT *descb , MKL_Complex8 *work , MKL_INT *lwork , MKL_INT *info );\nvoid pzgels (char *trans , MKL_INT *m , MKL_INT *n , MKL_INT *nrhs , MKL_Complex16 *a ,\nMKL_INT *ia , MKL_INT *ja , MKL_INT *desca , MKL_Complex16 *b , MKL_INT *ib , MKL_INT\n*jb , MKL_INT *descb , MKL_Complex16 *work , MKL_INT *lwork , MKL_INT *info );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?gels function solves overdetermined or underdetermined real/ complex linear systems involving an\nm-by-n matrix sub(A) = A(ia:ia+m-1,ja:ja+n-1), or its transpose/ conjugate-transpose, using a QTQ or\nLQ factorization of sub(A). It is assumed that sub(A) has full rank.\nThe following options are provided:\n1.\nIf trans = 'N' and m≥n: find the least squares solution of an overdetermined system, that is, solve\nthe least squares problem\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1536\n\n\nminimize ||sub(B) - sub(A)*X||\n2.\nIf trans = 'N' and m < n: find the minimum norm solution of an underdetermined system sub(A)*X\n= sub(B).\n3.\nIf trans = 'T' and m≥n: find the minimum norm solution of an undetermined system sub(A)T*X =\nsub(B).\n4.\nIf trans = 'T' and m < n: find the least squares solution of an overdetermined system, that is, solve\nthe least squares problem\nminimize ||sub(B) - sub(A)T*X||,\nwhere sub(B) denotes B(ib:ib+m-1, jb:jb+nrhs-1) when trans = 'N' and B(ib:ib+n-1,\njb:jb+nrhs-1) otherwise. Several right hand side vectors b and solution vectors x can be handled in a\nsingle call; when trans = 'N', the solution vectors are stored as the columns of the n-by-nrhs right\nhand side matrix sub(B) and the m-by-nrhs right hand side matrix sub(B) otherwise.\nInput Parameters\ntrans\n(global) Must be 'N', or 'T'.\nIf trans = 'N', the linear system involves matrix sub(A);\nIf trans = 'T', the linear system involves the transposed matrix AT (for\nreal flavors only).\nm\n(global) The number of rows in the distributed matrix sub (A) (m≥ 0).\nn\n(global) The number of columns in the distributed matrix sub (A) (n≥ 0).\nnrhs\n(global) The number of right-hand sides; the number of columns in the\ndistributed submatrices sub(B) and X. (nrhs≥ 0).\na\n(local)\nPointer into the local memory to an array of size lld_a*LOCc(ja+n-1). On\nentry, contains the m-by-n matrix A.\nia, ja\n(global) The row and column indices in the global matrix A indicating the\nfirst row and the first column of the submatrix A, respectively.\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\nb\n(local)\nPointer into the local memory to an array of local size lld_b*LOCc(jb\n+nrhs-1). On entry, this array contains the local pieces of the distributed\nmatrix B of right-hand side vectors, stored columnwise; sub(B) is m-by-\nnrhs if trans='N', and n-by-nrhs otherwise.\nib, jb\n(global) The row and column indices in the global matrix B indicating the\nfirst row and the first column of the submatrix B, respectively.\ndescb\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix B.\nwork\n(local)\nWorkspace array with size lwork.\nlwork\n(local or global) .\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1537\n\n\nThe size of the array worklwork is local input and must be at least lwork ≥\nltau + max(lwf, lws), where if m > n, then\nltau = numroc(ja+min(m,n)-1, nb_a, MYCOL, csrc_a, NPCOL),\nlwf = nb_a*(mpa0 + nqa0 + nb_a)\nlws = max((nb_a*(nb_a-1))/2, (nrhsqb0 + mpb0)*nb_a) +\nnb_a*nb_a\nelse\nltau = numroc(ia+min(m,n)-1, mb_a, MYROW, rsrc_a, NPROW),\nlwf = mb_a * (mpa0 + nqa0 + mb_a)\nlws = max((mb_a*(mb_a-1))/2, (npb0 + max(nqa0 +\nnumroc(numroc(n+iroffb, mb_a, 0, 0, NPROW), mb_a, 0, 0,\nlcmp), nrhsqb0))*mb_a) + mb_a*mb_a\nend if,\nwhere lcmp = lcm/NPROW with lcm = ilcm(NPROW, NPCOL),\niroffa = mod(ia-1, mb_a),\nicoffa = mod(ja-1, nb_a),\niarow = indxg2p(ia, mb_a, MYROW, rsrc_a, NPROW),\niacol= indxg2p(ja, nb_a, MYROW, rsrc_a, NPROW)\nmpa0 = numroc(m+iroffa, mb_a, MYROW, iarow, NPROW),\nnqa0 = numroc(n+icoffa, nb_a, MYCOL, iacol, NPCOL),\niroffb = mod(ib-1, mb_b),\nicoffb = mod(jb-1, nb_b),\nibrow = indxg2p(ib, mb_b, MYROW, rsrc_b, NPROW),\nibcol = indxg2p(jb, nb_b, MYCOL, csrc_b, NPCOL),\nmpb0 = numroc(m+iroffb, mb_b, MYROW, icrow, NPROW),\nnqb0 = numroc(n+icoffb, nb_b, MYCOL, ibcol, NPCOL),\nNOTE\nmod(x,y) is the integer remainder of x/y.\nilcm, indxg2p and numroc are ScaLAPACK tool functions; MYROW, MYCOL,\nNPROW, and NPCOL can be determined by calling the function\nblacs_gridinfo.\nIf lwork = -1, then lwork is global input and a workspace query is\nassumed; the function only calculates the minimum and optimal size for all\nwork arrays. Each of these values is returned in the first entry of the\ncorresponding work array, and no error message is issued by pxerbla.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1538\n\n\nOutput Parameters\na\nOn exit, If m≥n, sub(A) is overwritten by the details of its QR factorization as\nreturned by p?geqrf; if m < n, sub(A) is overwritten by details of its LQ\nfactorization as returned by p?gelqf.\nb\nOn exit, sub(B) is overwritten by the solution vectors, stored columnwise: if\ntrans = 'N' and m≥n, rows 1 to n of sub(B) contain the least squares\nsolution vectors; the residual sum of squares for the solution in each\ncolumn is given by the sum of squares of elements n+1 to m in that\ncolumn;\nIf trans = 'N' and m < n, rows 1 to n of sub(B) contain the minimum\nnorm solution vectors;\nIf trans = 'T' and m≥n, rows 1 to m of sub(B) contain the minimum norm\nsolution vectors; if trans = 'T' and m < n, rows 1 to m of sub(B) contain\nthe least squares solution vectors; the residual sum of squares for the\nsolution in each column is given by the sum of squares of elements m+1 to n\nin that column.\nwork[0]\nOn exit, work[0] contains the minimum value of lwork required for\noptimum performance.\ninfo\n(global)\n= 0: the execution is successful.\n< 0: if the i-th argument is an array and the j-entry had an illegal value,\nthen info = - (i* 100+j), if the i-th argument is a scalar and had an\nillegal value, then info = -i.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?syev\nComputes all eigenvalues and, optionally,\neigenvectors of a symmetric matrix.\nSyntax\nvoid pssyev (char *jobz , char *uplo , MKL_INT *n , float *a , MKL_INT *ia , MKL_INT\n*ja , MKL_INT *desca , float *w , float *z , MKL_INT *iz , MKL_INT *jz , MKL_INT\n*descz , float *work , MKL_INT *lwork , MKL_INT *info );\nvoid pdsyev (char *jobz , char *uplo , MKL_INT *n , double *a , MKL_INT *ia , MKL_INT\n*ja , MKL_INT *desca , double *w , double *z , MKL_INT *iz , MKL_INT *jz , MKL_INT\n*descz , double *work , MKL_INT *lwork , MKL_INT *info );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?syevfunction computes all eigenvalues and, optionally, eigenvectors of a real symmetric matrix A by\ncalling the recommended sequence of ScaLAPACK functions.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1539\n\n\nIn its present form, the function assumes a homogeneous system and makes no checks for consistency of\nthe eigenvalues or eigenvectors across the different processes. Because of this, it is possible that a\nheterogeneous system may return incorrect results without any error messages.\nInput Parameters\nnp = the number of rows local to a given process.\nnq = the number of columns local to a given process.\njobz\n(global) Must be 'N' or 'V'. Specifies if it is necessary to compute the\neigenvectors:\nIf jobz ='N', then only eigenvalues are computed.\nIf jobz ='V', then eigenvalues and eigenvectors are computed.\nuplo\n(global) Must be 'U' or 'L'. Specifies whether the upper or lower\ntriangular part of the symmetric matrix A is stored:\nIf uplo = 'U', a stores the upper triangular part of A.\nIf uplo = 'L', a stores the lower triangular part of A.\nn\n(global) The number of rows and columns of the matrix A(n≥ 0).\na\n(local)\nBlock cyclic array of global size n*n and local size lld_a*LOCc(ja+n-1).\nOn entry, the symmetric matrix A.\nIf uplo = 'U', only the upper triangular part of A is used to define the\nelements of the symmetric matrix.\nIf uplo = 'L', only the lower triangular part of A is used to define the\nelements of the symmetric matrix.\nia, ja\n(global) The row and column indices in the global matrix A indicating the\nfirst row and the first column of the submatrix A, respectively.\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\niz, jz\n(global) The row and column indices in the global matrix Z indicating the\nfirst row and the first column of the submatrix Z, respectively.\ndescz\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix Z.\nwork\n(local)\nArray of size lwork.\nlwork\n(local) See below for definitions of variables used to define lwork.\nIf no eigenvectors are requested (jobz = 'N'), then lwork ≥ 5*n +\nsizesytrd + 1,\nwhere sizesytrdis the workspace for p?sytrd and is max(NB*(np +1),\n3*NB).\nIf eigenvectors are requested (jobz = 'V') then the amount of workspace\nrequired to guarantee that all eigenvectors are computed is:\nqrmem = 2*n-2\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1540\n\n\nlwmin = 5*n + n*ldc + max(sizemqrleft, qrmem) + 1\nVariable definitions:\nnb = desca[mb_ - 1] = desca[nb_ - 1] = descz[mb_ - 1] =\ndescz[nb_ - 1];\nnn = max(n, nb, 2);\ndesca[rsrc_ - 1] = desca[rsrc_ - 1] = descz[rsrc_ - 1] =\ndescz[csrc_ - 1] = 0\nnp = numroc(nn, nb, 0, 0, NPROW)\nnq = numroc(max(n, nb, 2), nb, 0, 0, NPCOL)\nnrc = numroc(n, nb, myprowc, 0, NPROCS)\nldc = max(1, nrc)\nsizemqrleft is the workspace for p?ormtr when its side argument is 'L'.\nmyprowc is defined when a new context is created as follows:\ncall blacs_get(desca[ctxt_ - 1], 0, contextc)\ncall blacs_gridinit(contextc, 'R', NPROCS, 1)\ncall blacs_gridinfo(contextc, nprowc, npcolc, myprowc,\nmypcolc)\nIf lwork = -1, then lwork is global input and a workspace query is\nassumed; the function only calculates the minimum and optimal size for all\nwork arrays. Each of these values is returned in the first entry of the\ncorresponding work array, and no error message is issued by pxerbla.\nOutput Parameters\na\nOn exit, the lower triangle (if uplo='L') or the upper triangle (if\nuplo='U') of A, including the diagonal, is destroyed.\nw\n(global).\nArray of size n.\nOn normal exit, the first entries contain the selected eigenvalues in\nascending order.\nz\n(local).\nArray, global size n*n, local size lld_z*LOCc(jz+n-1). If jobz = 'V',\nthen on normal exit the first columns of z contain the orthonormal\neigenvectors of the matrix corresponding to the selected eigenvalues.\nIf jobz = 'N', then z is not referenced.\nwork[0]\nOn output, work[0] returns the workspace needed to guarantee\ncompletion. If the input parameters are incorrect, work[0] may also be\nincorrect.\nIf jobz = 'N'work[0] = minimal (optimal) amount of workspace\nIf jobz = 'V'work[0] = minimal workspace required to generate all the\neigenvectors.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1541\n\n\ninfo\n(global)\nIf info = 0, the execution is successful.\nIf info < 0: If the i-th argument is an array and the j-entry had an illegal\nvalue, then info = -(i*100+j), if the i-th argument is a scalar and had\nan illegal value, then info = -i.\nIf info > 0:\nIf info= 1 through n, the i-th eigenvalue did not converge in ?steqr2\nafter a total of 30n iterations.\nIf info= n+1, then p?syev has detected heterogeneity by finding that\neigenvalues were not identical across the process grid. In this case, the\naccuracy of the results from p?syev cannot be guaranteed.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?syevd\nComputes all eigenvalues and eigenvectors of a real\nsymmetric matrix by using a divide and conquer\nalgorithm.\nSyntax\nvoid pssyevd (char *jobz , char *uplo , MKL_INT *n , float *a , MKL_INT *ia , MKL_INT\n*ja , MKL_INT *desca , float *w , float *z , MKL_INT *iz , MKL_INT *jz , MKL_INT\n*descz , float *work , MKL_INT *lwork , MKL_INT *iwork , MKL_INT *liwork , MKL_INT\n*info );\nvoid pdsyevd (char *jobz , char *uplo , MKL_INT *n , double *a , MKL_INT *ia , MKL_INT\n*ja , MKL_INT *desca , double *w , double *z , MKL_INT *iz , MKL_INT *jz , MKL_INT\n*descz , double *work , MKL_INT *lwork , MKL_INT *iwork , MKL_INT *liwork , MKL_INT\n*info );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?syevd function computes all eigenvalues and eigenvectors of a real symmetric matrix A by using a\ndivide and conquer algorithm.\nInput Parameters\nnp = the number of rows local to a given process.\nnq = the number of columns local to a given process.\njobz\n(global) Must be 'N' or 'V'.\nSpecifies whether it is necessary to compute the eigenvectors:\nIf jobz = 'N', then only eigenvalues are computed (not yet\nimplemented).\nIf jobz = 'V', then eigenvalues and eigenvectors are computed.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1542\n\n\nuplo\n(global) Must be 'U' or 'L'.\nSpecifies whether the upper or lower triangular part of the Hermitian matrix\nA is stored:\nIf uplo = 'U', a stores the upper triangular part of A.\nIf uplo = 'L', a stores the lower triangular part of A.\nn\n(global) The number of rows and columns of the matrix A(n≥ 0).\na\n(local).\nBlock cyclic array of global size n*n and local size lld_a*LOCc(ja+n-1).\nOn entry, the symmetric matrix A.\nIf uplo = 'U', only the upper triangular part of A is used to define the\nelements of the symmetric matrix.\nIf uplo = 'L', only the lower triangular part of A is used to define the\nelements of the symmetric matrix.\nia, ja\n(global) The row and column indices in the global matrix A indicating the\nfirst row and the first column of the submatrix A, respectively.\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A. If desca[ctxt_ - 1] is incorrect, p?syevd cannot\nguarantee correct error reporting.\niz, jz\n(global) The row and column indices in the global matrix Z indicating the\nfirst row and the first column of the submatrix Z, respectively.\ndescz\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix Z. descz[ctxt_ - 1] must equal desca[ctxt_ - 1].\nwork\n(local).\nArray of size lwork.\nlwork\n(local) The size of the array work.\nIf eigenvalues are requested:\nlwork≥ max( 1+6*n + 2*np*nq, trilwmin) + 2*n\nwith trilwmin = 3*n + max( nb*( np + 1), 3*nb )\nnp = numroc( n, nb, myrow, iarow, NPROW)\nnq = numroc( n, nb, mycol, iacol, NPCOL)\nIf lwork = -1, then lwork is global input and a workspace query is\nassumed; the function only calculates the size required for optimal\nperformance for all work arrays. The required workspace is returned as the\nfirst element of the corresponding work arrays, and no error message is\nissued by pxerbla.\niwork\n(local) Workspace array of size liwork.\nliwork\n(local) , size of iwork.\nliwork = 7*n + 8*npcol + 2.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1543\n\n\nOutput Parameters\na\nOn exit, the lower triangle (if uplo = 'L'), or the upper triangle (if uplo =\n'U') of A, including the diagonal, is overwritten.\nw\n(global).\nArray of size n. If info = 0, w contains the eigenvalues in the ascending\norder.\nz\n(local).\nArray, global size (n, n), local size lld_z*LOCc(jz+n-1).\nThe z parameter contains the orthonormal eigenvectors of the matrix A.\nwork[0]\nOn exit, returns adequate workspace to allow optimal performance.\niwork[0]\n(local).\nOn exit, if liwork > 0, iwork[0] returns the optimal liwork.\ninfo\n(global)\nIf info = 0, the execution is successful.\nIf info < 0:\nIf the i-th argument is an array and the j-entry had an illegal value, then\ninfo = -(i*100+j). If the i-th argument is a scalar and had an illegal\nvalue, then info = -i.\nIf info> 0:\nThe algorithm failed to compute the info/(n+1)-th eigenvalue while\nworking on the submatrix lying in global rows and columns mod(info,n\n+1).\nNOTE\nmod(x,y) is the integer remainder of x/y.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?syevr\nComputes selected eigenvalues and, optionally,\neigenvectors of a real symmetric matrix using\nRelatively Robust Representation.\nSyntax\nvoid pssyevr(char* jobz, char* range, char* uplo, MKL_INT* n, float* a, MKL_INT* ia,\nMKL_INT* ja, MKL_INT* desca, float* vl, float* vu, MKL_INT* il, MKL_INT* iu, MKL_INT* m,\nMKL_INT* nz, float* w, float* z, MKL_INT* iz, MKL_INT* jz, MKL_INT* descz, float* work,\nMKL_INT* lwork, MKL_INT* iwork, MKL_INT* liwork, MKL_INT* info);\nvoid pdsyevr(char* jobz, char* range, char* uplo, MKL_INT* n, double* a, MKL_INT* ia,\nMKL_INT* ja, MKL_INT* desca, double* vl, double* vu, MKL_INT* il, MKL_INT* iu, MKL_INT*\nm, MKL_INT* nz, double* w, double* z, MKL_INT* iz, MKL_INT* jz, MKL_INT* descz, double*\nwork, MKL_INT* lwork, MKL_INT* iwork, MKL_INT* liwork, MKL_INT* info);\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1544\n\n\nInclude Files\n•\nmkl_scalapack.h\nDescription\np?syevr computes selected eigenvalues and, optionally, eigenvectors of a real symmetric matrix A\ndistributed in 2D blockcyclic format by calling the recommended sequence of ScaLAPACK functions.\nFirst, the matrix A is reduced to real symmetric tridiagonal form. Then, the eigenproblem is solved using the\nparallel MRRR algorithm. Last, if eigenvectors have been computed, a backtransformation is done.\nUpon successful completion, each processor stores a copy of all computed eigenvalues in w. The eigenvector\nmatrix z is stored in 2D block-cyclic format distributed over all processors.\nNote that subsets of eigenvalues/vectors can be selected by specifying a range of values or a range of indices\nfor the desired eigenvalues.\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nInput Parameters\njobz\n(global)\nSpecifies whether or not to compute the eigenvectors:\n= 'N': Compute eigenvalues only.\n= 'V': Compute eigenvalues and eigenvectors.\nrange\n(global)\n= 'A': all eigenvalues will be found.\n= 'V': all eigenvalues in the interval [vl,vu] will be found.\n= 'I': the il-th through iu-th eigenvalues will be found.\nuplo\n(global)\nSpecifies whether the upper or lower triangular part of the symmetric\nmatrix A is stored:\n= 'U': Upper triangular\n= 'L': Lower triangular\nn\n(global )\nThe number of rows and columns of the matrix a. n≥ 0\na\nBlock cyclic array of global size n * n), local size lld_a * LOCc(ja+n-1).\nThis array contains the local pieces of the symmetric distributed matrix A. If\nuplo = 'U', only the upper triangular part of a is used to define the\nelements of the symmetric matrix. If uplo = 'L', only the lower triangular\npart of a is used to define the elements of the symmetric matrix.\nOn exit, the lower triangle (if uplo='L') or the upper triangle (if uplo='U')\nof a, including the diagonal, is destroyed.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1545\n\n\nia\n(global )\nGlobal row index in the global matrix A that points to the beginning of the\nsubmatrix which is to be operated on. It should be set to 1 when operating\non a full matrix.\nja\n(global )\nGlobal column index in the global matrix A that points to the beginning of\nthe submatrix which is to be operated on. It should be set to 1 when\noperating on a full matrix.\ndesca\n(global and local) array of size dlen_=9.\nThe array descriptor for the distributed matrix a.\nvl\n(global )\nIf range='V', the lower bound of the interval to be searched for\neigenvalues. Not referenced if range = 'A' or 'I'.\nvu\n(global )\nIf range='V', the upper bound of the interval to be searched for\neigenvalues. Not referenced if range = 'A' or 'I'.\nil\n(global )\nIf range='I', the index (from smallest to largest) of the smallest eigenvalue\nto be returned. il≥ 1.\nNot referenced if range = 'A'.\niu\n(global )\nIf range='I', the index (from smallest to largest) of the largest eigenvalue\nto be returned. min(il,n) ≤iu≤n.\nNot referenced if range = 'A'.\niz\n(global )\nGlobal row index in the global matrix Z that points to the beginning of the\nsubmatrix which is to be operated on. It should be set to 1 when operating\non a full matrix.\njz\n(global )\nGlobal column index in the global matrix Z that points to the beginning of\nthe submatrix which is to be operated on. It should be set to 1 when\noperating on a full matrix.\ndescz\narray of size dlen_.\nThe array descriptor for the distributed matrix z.\nThe context descz[ctxt_ - 1] must equal desca[ctxt_ - 1]. Also note the\narray alignment requirements specified below.\nwork\n(local workspace) array of size lwork\nlwork\n(local )\nSize of work, must be at least 3.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1546\n\n\nSee below for definitions of variables used to define lwork.\nIf no eigenvectors are requested (jobz = 'N') then\nlwork≥ 2 + 5*n + max( 12 * nn, neig * ( np0 + 1 ) )\nIf eigenvectors are requested (jobz = 'V' ) then the amount of workspace\nrequired is:\nlwork≥ 2 + 5*n + max( 18*nn, np0 * mq0 + 2 * neig * neig ) + (2 +\niceil( neig, nprow*npcol))*nn\nVariable definitions:\nneig = number of eigenvectors requested\nnb = desca[ mb_ - 1] = desca( nb_ ) = descz[ mb_ - 1] = descz( nb_ )\nnn = max( n, neig, 2 )\ndesca[ rsrc_ - 1] = desca[ csrc_nb_ - 1] = descz[rsrc_ - 1] =\ndescz[csrc_ - 1] = 0\nnp0 = numroc( nn, neig, 0, 0, nprow )\nmq0 = numroc( max( neig, neig, 2 ), neig, 0, 0, npcol )\niceil( x, y ) is a ScaLAPACK function returning ceiling(x/y), and nprow and\nnpcol can be determined by calling the function blacs_gridinfo.\nIf lwork = -1, then lwork is global input and a workspace query is\nassumed; the function only calculates the size required for optimal\nperformance for all work arrays. Each of these values is returned in the first\nentry of the corresponding work arrays, and no error message is issued by \npxerbla.\nliwork\n(local )\nsize of iwork\nLet nnp = max( n, nprow*npcol + 1, 4 ). Then:\nliwork≥ 12*nnp + 2*n when the eigenvectors are desired\nliwork≥ 10*nnp + 2*n when only the eigenvalues have to be computed\nIf liwork = -1, then liwork is global input and a workspace query is\nassumed; the function only calculates the minimum and optimal size for all\nwork arrays. Each of these values is returned in the first entry of the\ncorresponding work array, and no error message is issued by pxerbla.\nOUTPUT Parameters\nm\n(global )\nTotal number of eigenvalues found. 0 ≤m≤n.\nnz\n(global )\nTotal number of eigenvectors computed. 0 ≤nz≤m.\nThe number of columns of z that are filled.\nIf jobz≠ 'V', nz is not referenced.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1547\n\n\nIf jobz = 'V', nz = m\nw\n(global ) array of size n\nUpon successful exit, the first m entries contain the selected eigenvalues in\nascending order.\nz\nBlock-cyclic array, global sizen*n, local size lld_z*LOCc(jz+n-1).\nOn exit, contains local pieces of distributed matrix Z.\nwork\nOn return, work[0] contains the optimal amount of workspace required for\nefficient execution. If jobz='N' work[0] = optimal amount of workspace\nrequired to compute the eigenvalues. If jobz='V' work[0] = optimal\namount of workspace required to compute eigenvalues and eigenvectors.\niwork\n(local workspace) array\nOn return, iwork[0] contains the amount of integer workspace required.\ninfo\n(global )\n= 0: successful exit\n< 0: If the i-th argument is an array and the jth-entry had an illegal value,\nthen info = -(i*100+j), if the i-th argument is a scalar and had an illegal\nvalue, then info = -i.\nApplication Notes\nThe distributed submatrices a(ia:*, ja:*) and z(iz:iz+m-1,jz:jz+n-1) must satisfy the following\nalignment properties:\n1.\nIdentical (quadratic) dimension: desca[m_ - 1] = descz[m_ - 1] = desca[n_ - 1] = descz[n_ - 1]\n2.\nQuadratic conformal blocking: desca[mb_ - 1] = desca[nb_ - 1] = descz[mb_ - 1] = descz[nb_ - 1],\ndesca[rsrc_ - 1] = descz[rsrc_ - 1]\n3.\nmod( ia-1, mb_a ) = mod( iz-1, mb_z ) = 0\nNOTE\nmod(x,y) is the integer remainder of x/y.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?syevx\nComputes selected eigenvalues and, optionally,\neigenvectors of a symmetric matrix.\nSyntax\nvoid pssyevx (char *jobz , char *range , char *uplo , MKL_INT *n , float *a , MKL_INT\n*ia , MKL_INT *ja , MKL_INT *desca , float *vl , float *vu , MKL_INT *il , MKL_INT *iu ,\nfloat *abstol , MKL_INT *m , MKL_INT *nz , float *w , float *orfac , float *z , MKL_INT\n*iz , MKL_INT *jz , MKL_INT *descz , float *work , MKL_INT *lwork , MKL_INT *iwork ,\nMKL_INT *liwork , MKL_INT *ifail , MKL_INT *iclustr , float *gap , MKL_INT *info );\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1548\n\n\nvoid pdsyevx (char *jobz , char *range , char *uplo , MKL_INT *n , double *a , MKL_INT\n*ia , MKL_INT *ja , MKL_INT *desca , double *vl , double *vu , MKL_INT *il , MKL_INT\n*iu , double *abstol , MKL_INT *m , MKL_INT *nz , double *w , double *orfac , double\n*z , MKL_INT *iz , MKL_INT *jz , MKL_INT *descz , double *work , MKL_INT *lwork ,\nMKL_INT *iwork , MKL_INT *liwork , MKL_INT *ifail , MKL_INT *iclustr , double *gap ,\nMKL_INT *info );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?syevxfunction computes selected eigenvalues and, optionally, eigenvectors of a real symmetric matrix\nA by calling the recommended sequence of ScaLAPACK functions. Eigenvalues and eigenvectors can be\nselected by specifying either a range of values or a range of indices for the desired eigenvalues.\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nInput Parameters\nnp = the number of rows local to a given process.\nnq = the number of columns local to a given process.\njobz\n(global) Must be 'N' or 'V'. Specifies if it is necessary to compute the\neigenvectors:\nIf jobz ='N', then only eigenvalues are computed.\nIf jobz ='V', then eigenvalues and eigenvectors are computed.\nrange\n(global) Must be 'A', 'V', or 'I'.\nIf range = 'A', all eigenvalues will be found.\nIf range = 'V', all eigenvalues in the half-open interval [vl, vu] will be\nfound.\nIf range = 'I', the eigenvalues with indices il through iu will be found.\nuplo\n(global) Must be 'U' or 'L'.\nSpecifies whether the upper or lower triangular part of the symmetric\nmatrix A is stored:\nIf uplo = 'U', a stores the upper triangular part of A.\nIf uplo = 'L', a stores the lower triangular part of A.\nn\n(global) The number of rows and columns of the matrix A(n≥ 0).\na\n(local).\nBlock cyclic array of global size n*n and local size lld_a*LOCc(ja+n-1).\nOn entry, the symmetric matrix A.\nIf uplo = 'U', only the upper triangular part of A is used to define the\nelements of the symmetric matrix.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1549\n\n\nIf uplo = 'L', only the lower triangular part of A is used to define the\nelements of the symmetric matrix.\nia, ja\n(global) The row and column indices in the global matrix A indicating the\nfirst row and the first column of the submatrix A, respectively.\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\nvl, vu\n(global)\nIf range = 'V', the lower and upper bounds of the interval to be searched\nfor eigenvalues; vl ≤ vu. Not referenced if range = 'A' or 'I'.\nil, iu\n(global)\nIf range ='I', the indices of the smallest and largest eigenvalues to be\nreturned.\nConstraints: il ≥ 1\nmin(il,n) ≤ iu ≤ n\nNot referenced if range = 'A' or 'V'.\nabstol\n(global).\nIf jobz='V', setting abstol to p?lamch(context, 'U') yields the most\northogonal eigenvectors.\nThe absolute error tolerance for the eigenvalues. An approximate\neigenvalue is accepted as converged when it is determined to lie in an\ninterval [a, b] of width less than or equal to\nabstol + eps * max(|a|,|b|),\nwhere eps is the machine precision. If abstol is less than or equal to zero,\nthen eps*norm(T) will be used in its place, where norm(T) is the 1-norm of\nthe tridiagonal matrix obtained by reducing A to tridiagonal form.\nEigenvalues will be computed most accurately when abstol is set to twice\nthe underflow threshold 2*p?lamch('S') not zero. If this function returns\nwith (mod(info,2) ≠ 0) or (mod(info/8,2) ≠ 0)), indicating that some\neigenvalues or eigenvectors did not converge, try setting abstol to\n2*p?lamch('S').\norfac\n(global).\nSpecifies which eigenvectors should be reorthogonalized. Eigenvectors that\ncorrespond to eigenvalues which are within tol=orfac*norm(A)of each\nother are to be reorthogonalized. However, if the workspace is insufficient\n(see lwork), tol may be decreased until all eigenvectors to be\nreorthogonalized can be stored in one process. No reorthogonalization will\nbe done if orfac equals zero. A default value of 1.0e-3 is used if orfac is\nnegative. orfac should be identical on all processes.\niz, jz\n(global) The row and column indices in the global matrix Z indicating the\nfirst row and the first column of the submatrix Z, respectively.\ndescz\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix Z.descz[ctxt_ - 1] must equal desca[ctxt_ - 1].\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1550\n\n\nwork\n(local)\nArray of size lwork.\nlwork\n(local) The size of the array work.\nSee below for definitions of variables used to define lwork.\nIf no eigenvectors are requested (jobz = 'N'), then lwork ≥ 5*n +\nmax(5*nn, NB*(np0 + 1)).\nIf eigenvectors are requested (jobz = 'V'), then the amount of workspace\nrequired to guarantee that all eigenvectors are computed is:\nlwork ≥ 5*n + max(5*nn, np0*mq0 + 2*NB*NB) + iceil(neig,\nNPROW*NPCOL)*nn\nThe computed eigenvectors may not be orthogonal if the minimal\nworkspace is supplied and orfac is too small. If you want to guarantee\northogonality (at the cost of potentially poor performance) you should add\nthe following to lwork:\n(clustersize-1)*n,\nwhere clustersize is the number of eigenvalues in the largest cluster, where\na cluster is defined as a set of close eigenvalues:\n{w[k - 1],..., w[k+clustersize-2]|w[j] ≤ w[j-1]) +\norfac*2*norm(A)},\nwhere\nneig = number of eigenvectors requested\nnb = desca[mb_ - 1] = desca[nb_ - 1] = descz[mb_ - 1] =\ndescz[nb_ - 1];\nnn = max(n, nb, 2);\ndesca[rsrc_ - 1] = desca[nb_ - 1] = descz[rsrc_ - 1] =\ndescz[csrc_ - 1] = 0;\nnp0 = numroc(nn, nb, 0, 0, NPROW);\nmq0 = numroc(max(neig, nb, 2), nb, 0, 0, NPCOL) \niceil(x, y) is a ScaLAPACK function returning ceiling(x/y)\nIf lwork is too small to guarantee orthogonality, p?syevx attempts to\nmaintain orthogonality in the clusters with the smallest spacing between the\neigenvalues.\nIf lwork is too small to compute all the eigenvectors requested, no\ncomputation is performed and info= -23 is returned.\nNote that when range='V', number of requested eigenvectors are not\nknown until the eigenvalues are computed. In this case and if lwork is large\nenough to compute the eigenvalues, p?sygvx computes the eigenvalues\nand as many eigenvectors as possible.\nRelationship between workspace, orthogonality & performance:\nGreater performance can be achieved if adequate workspace is provided. In\nsome situations, performance can decrease as the provided workspace\nincreases above the workspace amount shown below:\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1551\n\n\nlwork ≥ max(lwork, 5*n + nsytrd_lwopt),\nwhere lwork, as defined previously, depends upon the number of\neigenvectors requested, and\nnsytrd_lwopt = n + 2*(anb+1)*(4*nps+2) + (nps + 3)*nps;\nanb = pjlaenv(desca[ctxt_ - 1], 3, 'p?syttrd', 'L', 0, 0, 0,\n0);\nsqnpc = int(sqrt(dble(NPROW * NPCOL)));\nnps = max(numroc(n, 1, 0, 0, sqnpc), 2*anb);\nnumroc is a ScaLAPACK tool functions; \npjlaenv is a ScaLAPACK environmental inquiry function\nMYROW, MYCOL, NPROW and NPCOL can be determined by calling the function\nblacs_gridinfo.\nFor large n, no extra workspace is needed, however the biggest boost in\nperformance comes for small n, so it is wise to provide the extra workspace\n(typically less than a megabyte per process).\nIf clustersize > n/sqrt(NPROW*NPCOL), then providing enough space\nto compute all the eigenvectors orthogonally will cause serious degradation\nin performance. At the limit (that is, clustersize = n-1) p?stein will\nperform no better than ?stein on single processor.\nFor clustersize = n/sqrt(NPROW*NPCOL) reorthogonalizing all\neigenvectors will increase the total execution time by a factor of 2 or more.\nFor clustersize>n/sqrt(NPROW*NPCOL) execution time will grow as the\nsquare of the cluster size, all other factors remaining equal and assuming\nenough workspace. Less workspace means less reorthogonalization but\nfaster execution.\nIf lwork = -1, then lwork is global input and a workspace query is\nassumed; the function only calculates the size required for optimal\nperformance for all work arrays. Each of these values is returned in the first\nentry of the corresponding work arrays, and no error message is issued by\npxerbla.\niwork\n(local) Workspace array.\nliwork\n(local) , size of iwork. liwork ≥ 6*nnp\nWhere: nnp = max(n, NPROW*NPCOL + 1, 4)\nIf liwork = -1, then liwork is global input and a workspace query is\nassumed; the function only calculates the minimum and optimal size for all\nwork arrays. Each of these values is returned in the first entry of the\ncorresponding work array, and no error message is issued by pxerbla.\nOutput Parameters\na\nOn exit, the lower triangle (if uplo = 'L') or the upper triangle (if uplo =\n'U')of A, including the diagonal, is overwritten.\nm\n(global) The total number of eigenvalues found; 0 ≤ m ≤ n.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1552\n\n\nnz\n(global) Total number of eigenvectors computed. 0 ≤ nz ≤ m.\nThe number of columns of z that are filled.\nIf jobz ≠ 'V', nz is not referenced.\nIf jobz = 'V', nz = m unless the user supplies insufficient space and\np?syevx is not able to detect this before beginning computation. To get all\nthe eigenvectors requested, the user must supply both sufficient space to\nhold the eigenvectors in z (m≤descz[n_ - 1]) and sufficient workspace to\ncompute them. (See lwork). p?syevx is always able to detect insufficient\nspace without computation unless range = 'V'.\nw\n(global).\nArray of size n. The first m elements contain the selected eigenvalues in\nascending order.\nz\n(local).\nArray, global size n*n, local size lld_z*LOCc(jz+n-1).\nIf jobz = 'V', then on normal exit the first m columns of z contain the\northonormal eigenvectors of the matrix corresponding to the selected\neigenvalues. If an eigenvector fails to converge, then that column of z\ncontains the latest approximation to the eigenvector, and the index of the\neigenvector is returned in ifail.\nIf jobz = 'N', then z is not referenced.\nwork[0]\nOn exit, returns workspace adequate workspace to allow optimal\nperformance.\niwork[0]\nOn return, iwork[0] contains the amount of integer workspace required.\nifail\n(global).\nArray of size n.\nIf jobz = 'V', then on normal exit, the first m elements of ifail are zero. If\n(mod(info,2) ≠ 0) on exit, then ifail contains the indices of the\neigenvectors that failed to converge.\nIf jobz = 'N', then ifail is not referenced.\niclustr\n(global) Array of size (2*NPROW*NPCOL)\nThis array contains indices of eigenvectors corresponding to a cluster of\neigenvalues that could not be reorthogonalized due to insufficient\nworkspace (see lwork, orfac and info). Eigenvectors corresponding to\nclusters of eigenvalues indexed iclustr(2*i-1) to iclustr(2*i), could\nnot be reorthogonalized due to lack of workspace. Hence the eigenvectors\ncorresponding to these clusters may not be orthogonal. iclustr is a zero\nterminated array. iclustr[2*k - 1] ≠ 0 and iclustr[2*k] = 0 if and\nonly if k is the number of clusters.\niclustr is not referenced if jobz = 'N'.\ngap\n(global)\nArray of size NPROW*NPCOL\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1553\n\n\nThis array contains the gap between eigenvalues whose eigenvectors could\nnot be reorthogonalized. The output values in this array correspond to the\nclusters indicated by the array iclustr. As a result, the dot product between\neigenvectors corresponding to the ith cluster may be as high as (C*n)/\ngap[i - 1] where C is a small constant.\ninfo\n(global)\nIf info = 0, the execution is successful.\nIf info < 0:\nIf the i-th argument is an array and the j-entry had an illegal value, then\ninfo = -(i*100+j), if the i-th argument is a scalar and had an illegal\nvalue, then info = -i.\nIf info> 0: if (mod(info,2)≠0), then one or more eigenvectors failed to\nconverge. Their indices are stored in ifail. Ensure\nabstol=2.0*p?lamch('U').\nIf (mod(info/2,2)≠0), then eigenvectors corresponding to one or more\nclusters of eigenvalues could not be reorthogonalized because of insufficient\nworkspace.The indices of the clusters are stored in the array iclustr.\nIf (mod(info/4,2)≠0), then space limit prevented p?syevxf rom\ncomputing all of the eigenvectors between vl and vu. The number of\neigenvectors computed is returned in nz.\nIf (mod(info/8,2)≠0), then p?stebz failed to compute eigenvalues.\nEnsure abstol=2.0*p?lamch('U').\nNOTE\nmod(x,y) is the integer remainder of x/y.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?heev\nComputes all eigenvalues and, optionally,\neigenvectors of a complex Hermitian matrix.\nSyntax\nvoid pcheev (char *jobz , char *uplo , MKL_INT *n , MKL_Complex8 *a , MKL_INT *ia ,\nMKL_INT *ja , MKL_INT *desca , float *w , MKL_Complex8 *z , MKL_INT *iz , MKL_INT *jz ,\nMKL_INT *descz , MKL_Complex8 *work , MKL_INT *lwork , float *rwork , MKL_INT *lrwork ,\nMKL_INT *info );\nvoid pzheev (char *jobz , char *uplo , MKL_INT *n , MKL_Complex16 *a , MKL_INT *ia ,\nMKL_INT *ja , MKL_INT *desca , double *w , MKL_Complex16 *z , MKL_INT *iz , MKL_INT\n*jz , MKL_INT *descz , MKL_Complex16 *work , MKL_INT *lwork , double *rwork , MKL_INT\n*lrwork , MKL_INT *info );\nInclude Files\n•\nmkl_scalapack.h\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1554\n\n\nDescription\nThe p?heev function computes all eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix A\nby calling the recommended sequence of ScaLAPACK functions. The function assumes a homogeneous\nsystem and makes spot checks of the consistency of the eigenvalues across the different processes. A\nheterogeneous system may return incorrect results without any error messages.\nInput Parameters\nnp = the number of rows local to a given process.\nnq = the number of columns local to a given process.\njobz\n(global) Must be 'N' or 'V'.\nSpecifies if it is necessary to compute the eigenvectors:\nIf jobz = 'N', then only eigenvalues are computed.\nIf jobz = 'V', then eigenvalues and eigenvectors are computed.\nuplo\n(global) Must be 'U' or 'L'.\nSpecifies whether the upper or lower triangular part of the Hermitian matrix\nA is stored:\nIf uplo = 'U', a stores the upper triangular part of A.\nIf uplo = 'L', a stores the lower triangular part of A.\nn\n(global) The number of rows and columns of the matrix A(n≥ 0).\na\n(local).\nBlock cyclic array of global size n*n and local size lld_a*LOCc(ja+n-1).\nOn entry, the Hermitian matrix A.\nIf uplo = 'U', only the upper triangular part of A is used to define the\nelements of the Hermitian matrix.\nIf uplo = 'L', only the lower triangular part of A is used to define the\nelements of the Hermitian matrix.\nia, ja\n(global) The row and column indices in the global matrix A indicating the\nfirst row and the first column of the submatrix A, respectively.\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A. If desca[ctxt_ - 1] is incorrect, p?heev cannot\nguarantee correct error reporting.\niz, jz\n(global) The row and column indices in the global matrix Z indicating the\nfirst row and the first column of the submatrix Z, respectively.\ndescz\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix Z. descz[ctxt_ - 1] must equal desca[ctxt_ - 1].\nwork\n(local).\nArray of size lwork.\nlwork\n(local) The size of the array work.\nIf only eigenvalues are requested (jobz = 'N'):\nlwork≥max(nb*(np0 + 1), 3) + 3*n\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1555\n\n\nIf eigenvectors are requested (jobz = 'V'), then the amount of workspace\nrequired:\nlwork≥ (np0+nq0+nb)*nb + 3*n + n2\nwith nb = desca[mb_ - 1] = desca[ nb_ - 1] = nb = descz[mb_ -\n1] = descz[ nb_ - 1]\nnp0 = numroc(nn, nb, 0, 0, NPROW).\nnq0 = numroc( max( n, nb, 2 ), nb, 0, 0, NPCOL).\nIf lwork = -1, then lwork is global input and a workspace query is\nassumed; the function only calculates the size required for optimal\nperformance for all work arrays. The required workspace is returned as the\nfirst element of the corresponding work arrays, and no error message is\nissued by pxerbla.\nrwork\n(local).\nWorkspace array of size lrwork.\nlrwork\n(local) The size of the array rwork.\nSee below for definitions of variables used to define lrwork.\nIf no eigenvectors are requested (jobz = 'N'), then lrwork≥ 2*n.\nIf eigenvectors are requested (jobz = 'V'), then lrwork≥ 2*n + 2*n-2.\nIf lrwork = -1, then lrwork is global input and a workspace query is\nassumed; the function only calculates the minimum size required for the\nrwork array. The required workspace is returned as the first element of\nrwork, and no error message is issued by pxerbla.\nOutput Parameters\na\nOn exit, the lower triangle (if uplo = 'L'), or the upper triangle (if uplo =\n'U') of A, including the diagonal, is overwritten.\nw\n(global).\nArray of size n. The first m elements contain the selected eigenvalues in\nascending order.\nz\n(local).\nArray, global size n*n, local size lld_z*LOCc(jz+n-1).\nIf jobz ='V', then on normal exit the first columns of z contain the\northonormal eigenvectors of the matrix corresponding to the selected\neigenvalues. If an eigenvector fails to converge, then that column of z\ncontains the latest approximation to the eigenvector, and the index of the\neigenvector is returned in ifail.\nIf jobz = 'N', then z is not referenced.\nwork[0]\nOn exit, returns adequate workspace to allow optimal performance.\nIf jobz ='N', then work[0] = minimal workspace only for eigenvalues.\nIf jobz ='V', then work[0] = minimal workspace required to generate all\nthe eigenvectors.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1556\n\n\nrwork[0]\n(local)\nOn output, rwork[0] returns workspace required to guarantee completion.\ninfo\n(global)\nIf info = 0, the execution is successful.\nIf info < 0:\nIf the i-th argument is an array and the j-entry had an illegal value, then\ninfo = -(i*100+j). If the i-th argument is a scalar and had an illegal\nvalue, then info = -i.\nIf info> 0:\nIf info = 1 through n, the i-th eigenvalue did not converge in ?steqr2\nafter a total of 30*n iterations.\nIf info = n+1, then p?heev detected heterogeneity, and the accuracy of\nthe results cannot be guaranteed.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?heevd\nComputes all eigenvalues and eigenvectors of a\ncomplex Hermitian matrix by using a divide and\nconquer algorithm.\nSyntax\nvoid pcheevd (char *jobz , char *uplo , MKL_INT *n , MKL_Complex8 *a , MKL_INT *ia ,\nMKL_INT *ja , MKL_INT *desca , float *w , MKL_Complex8 *z , MKL_INT *iz , MKL_INT *jz ,\nMKL_INT *descz , MKL_Complex8 *work , MKL_INT *lwork , float *rwork , MKL_INT *lrwork ,\nMKL_INT *iwork , MKL_INT *liwork , MKL_INT *info );\nvoid pzheevd (char *jobz , char *uplo , MKL_INT *n , MKL_Complex16 *a , MKL_INT *ia ,\nMKL_INT *ja , MKL_INT *desca , double *w , MKL_Complex16 *z , MKL_INT *iz , MKL_INT\n*jz , MKL_INT *descz , MKL_Complex16 *work , MKL_INT *lwork , double *rwork , MKL_INT\n*lrwork , MKL_INT *iwork , MKL_INT *liwork , MKL_INT *info );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?heevd function computes all eigenvalues and eigenvectors of a complex Hermitian matrix A by using a\ndivide and conquer algorithm.\nInput Parameters\nnp = the number of rows local to a given process.\nnq = the number of columns local to a given process.\njobz\n(global) Must be 'N' or 'V'.\nSpecifies whether it is necessary to compute the eigenvectors:\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1557\n\n\nIf jobz = 'N', then only eigenvalues are computed (not yet\nimplemented).\nIf jobz = 'V', then eigenvalues and eigenvectors are computed.\nuplo\n(global) Must be 'U' or 'L'.\nSpecifies whether the upper or lower triangular part of the Hermitian matrix\nA is stored:\nIf uplo = 'U', a stores the upper triangular part of A.\nIf uplo = 'L', a stores the lower triangular part of A.\nn\n(global) The number of rows and columns of the matrix A(n≥ 0).\na\n(local).\nBlock cyclic array of global size n*n and local size lld_a*LOCc(ja+n-1).\nOn entry, the Hermitian matrix A.\nIf uplo = 'U', only the upper triangular part of A is used to define the\nelements of the Hermitian matrix.\nIf uplo = 'L', only the lower triangular part of A is used to define the\nelements of the Hermitian matrix.\nia, ja\n(global) The row and column indices in the global matrix A indicating the\nfirst row and the first column of the submatrix A, respectively.\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A. If desca[ctxt_ - 1] is incorrect, p?heevd cannot\nguarantee correct error reporting.\niz, jz\n(global) The row and column indices in the global matrix Z indicating the\nfirst row and the first column of the submatrix Z, respectively.\ndescz\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix Z. descz[ctxt_ - 1] must equal desca[ctxt_ - 1].\nwork\n(local).\nArray of size lwork.\nlwork\n(local) The size of the array work.\nIf eigenvalues are requested:\nlwork = n + (nb0 + mq0 + nb)*nb\nwith np0 = numroc( max( n, nb, 2 ), nb, 0, 0, NPROW)\nmq0 = numroc( max( n, nb, 2 ), nb, 0, 0, NPCOL)\nIf lwork = -1, then lwork is global input and a workspace query is\nassumed; the function only calculates the size required for optimal\nperformance for all work arrays. The required workspace is returned as the\nfirst element of the corresponding work arrays, and no error message is\nissued by pxerbla.\nrwork\n(local).\nWorkspace array of size lrwork.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1558\n\n\nlrwork\n(local) The size of the array rwork.\nlrwork≥ 1 + 9*n + 3*np*nq,\nwith np = numroc( n, nb, myrow, iarow, NPROW)\nnq = numroc( n, nb, mycol, iacol, NPCOL)\niwork\n(local) Workspace array of size liwork.\nliwork\n(local) , size of iwork.\nliwork = 7*n + 8*npcol + 2.\nOutput Parameters\na\nOn exit, the lower triangle (if uplo = 'L'), or the upper triangle (if uplo =\n'U') of A, including the diagonal, is overwritten.\nw\n(global).\nArray of size n. If info = 0, w contains the eigenvalues in the ascending\norder.\nz\n(local).\nArray, global size n*n, local size lld_z*LOCc(jz+n-1).\nThe z parameter contains the orthonormal eigenvectors of the matrix A.\nwork[0]\nOn exit, returns adequate workspace to allow optimal performance.\nrwork[0]\n(local)\nOn output, rwork[0] returns workspace required to guarantee completion.\niwork[0]\n(local).\nOn return, iwork[0] contains the amount of integer workspace required.\ninfo\n(global)\nIf info = 0, the execution is successful.\nIf info < 0:\nIf the i-th argument is an array and the j-entry had an illegal value, then\ninfo = -(i*100+j). If the i-th argument is a scalar and had an illegal\nvalue, then info = -i.\nIf info> 0:\nIf info = 1 through n, the i-th eigenvalue did not converge.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?heevr\nComputes selected eigenvalues and, optionally,\neigenvectors of a Hermitian matrix using Relatively\nRobust Representation.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1559\n\n\nSyntax\nvoid pcheevr(char* jobz, char* range, char* uplo, MKL_INT* n, MKL_Complex8* a, MKL_INT*\nia, MKL_INT* ja, MKL_INT* desca, float* vl, float* vu, MKL_INT* il, MKL_INT* iu,\nMKL_INT* m, MKL_INT* nz, float* w, MKL_Complex8* z, MKL_INT* iz, MKL_INT* jz, MKL_INT*\ndescz, MKL_Complex8* work, MKL_INT* lwork, float* rwork, MKL_INT* lrwork, MKL_INT*\niwork, MKL_INT* liwork, MKL_INT* info);\nvoid pzheevr(char* jobz, char* range, char* uplo, MKL_INT* n, MKL_Complex16* a, MKL_INT*\nia, MKL_INT* ja, MKL_INT* desca, double* vl, double* vu, MKL_INT* il, MKL_INT* iu,\nMKL_INT* m, MKL_INT* nz, double* w, MKL_Complex16* z, MKL_INT* iz, MKL_INT* jz, MKL_INT*\ndescz, MKL_Complex16* work, MKL_INT* lwork, double* rwork, MKL_INT* lrwork, MKL_INT*\niwork, MKL_INT* liwork, MKL_INT* info);\nInclude Files\n•\nmkl_scalapack.h\nDescription\np?heevr computes selected eigenvalues and, optionally, eigenvectors of a complex Hermitian matrix A\ndistributed in 2D blockcyclic format by calling the recommended sequence of ScaLAPACK functions.\nFirst, the matrix A is reduced to complex Hermitian tridiagonal form. Then, the eigenproblem is solved using\nthe parallel MRRR algorithm. Last, if eigenvectors have been computed, a backtransformation is done.\nUpon successful completion, each processor stores a copy of all computed eigenvalues in w. The eigenvector\nmatrix Z is stored in 2D block-cyclic format distributed over all processors.\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nInput Parameters\njobz\n(global)\nSpecifies whether or not to compute the eigenvectors:\n= 'N': Compute eigenvalues only.\n= 'V': Compute eigenvalues and eigenvectors.\nrange\n(global)\n= 'A': all eigenvalues will be found.\n= 'V': all eigenvalues in the interval [vl,vu] will be found.\n= 'I': the il-th through iu-th eigenvalues will be found.\nuplo\n(global)\nSpecifies whether the upper or lower triangular part of the Hermitian matrix\nA is stored:\n= 'U': Upper triangular\n= 'L': Lower triangular\nn\n(global )\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1560\n\n\nThe number of rows and columns of the matrix A. n≥ 0\na\nBlock-cyclic array, global size n * n), local size lld_a * LOCc(ja+n-1)\nContains the local pieces of the Hermitian distributed matrix A. If uplo =\n'U', only the upper triangular part of a is used to define the elements of the\nHermitian matrix. If uplo = 'L', only the lower triangular part of a is used to\ndefine the elements of the Hermitian matrix.\nia\n(global )\nGlobal row index in the global matrix A that points to the beginning of the\nsubmatrix which is to be operated on. It should be set to 1 when operating\non a full matrix.\nja\n(global )\nGlobal column index in the global matrix A that points to the beginning of\nthe submatrix which is to be operated on. It should be set to 1 when\noperating on a full matrix.\ndesca\n(global and local) array of size dlen_. (The ScaLAPACK descriptor length is\ndlen_ = 9.)\nThe array descriptor for the distributed matrix a. The descriptor stores\ndetails about the 2D block-cyclic storage, see the notes below. If desca is\nincorrect, p?heevr cannot work correctly.\nAlso note the array alignment requirements specified below\nvl\n(global)\nIf range='V', the lower bound of the interval to be searched for\neigenvalues. Not referenced if range = 'A' or 'I'.\nvu\n(global)\nIf range='V', the upper bound of the interval to be searched for\neigenvalues. Not referenced if range = 'A' or 'I'.\nil\n(global )\nIf range='I', the index (from smallest to largest) of the smallest eigenvalue\nto be returned. il≥ 1.\nNot referenced if range = 'A'.\niu\n(global )\nIf range='I', the index (from smallest to largest) of the largest eigenvalue\nto be returned. min(il,n) ≤iu≤n.\nNot referenced if range = 'A'.\niz\n(global )\nGlobal row index in the global matrix Z that points to the beginning of the\nsubmatrix which is to be operated on. It should be set to 1 when operating\non a full matrix.\njz\n(global )\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1561\n\n\nGlobal column index in the global matrix Z that points to the beginning of\nthe submatrix which is to be operated on. It should be set to 1 when\noperating on a full matrix.\ndescz\n(global and local) array of size dlen_.\nThe array descriptor for the distributed matrix z. descz[ctxt_ - 1] must\nequal desca[ctxt_ - 1]\nwork\n(local workspace) array of size lwork\nlwork\n(local )\nSize of work array, must be at least 3.\nIf only eigenvalues are requested:\nlwork≥n + max( nb * ( np00 + 1 ), nb * 3 )\nIf eigenvectors are requested:\nlwork≥n + ( np00 + mq00 + nb ) * nb\nFor definitions of np00 and mq00, see lrwork.\nFor optimal performance, greater workspace is needed, i.e.\nlwork≥ max( lwork, nhetrd_lwork )\nWhere lwork is as defined above, and\nnhetrd_lwork = n + 2*( anb+1 )*( 4*nps+2 ) + ( nps + 1 ) * nps\nictxt = desca[ctxt_ - 1]\nanb = pjlaenv( ictxt, 3, 'PCHETTRD', 'L', 0, 0, 0, 0 )\nsqnpc = sqrt( real( nprow * npcol ) )\nnps = max( numroc( n, 1, 0, 0, sqnpc ), 2*anb )\nIf lwork = -1, then lwork is global input and a workspace query is\nassumed; the function only calculates the optimal size for all work arrays.\nEach of these values is returned in the first entry of the corresponding work\narray, and no error message is issued by pxerbla.\nrwork\n(local workspace) array of size lrwork\nlrwork\n(local )\nSize of rwork, must be at least 3.\nSee below for definitions of variables used to define lrwork.\nIf no eigenvectors are requested (jobz = 'N') then\nlrwork≥ 2 + 5 * n + max( 12 * n, nb * ( np00 + 1 ) )\nIf eigenvectors are requested (jobz = 'V' ) then the amount of workspace\nrequired is:\nlrwork≥ 2 + 5 * n + max( 18*n, np00 * mq00 + 2 * nb * nb ) +\n(2 + iceil( neig, nprow*npcol))*n\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1562\n\n\nNOTE\niceil(x,y) is the ceiling of x/y.\nVariable definitions:\nneig = number of eigenvectors requested\nnb = desca[ mb_ - 1] = desca[ nb_ - 1] = descz[ mb_ - 1] = descz[nb_\n- 1]\nnn = max( n, nb, 2 )\ndesca[ rsrc_ - 1] = desca[csrc_ - 1] = descz[ rsrc_ - 1] = descz[csrc_ -\n1] = 0\nnp00 = numroc( nn, nb, 0, 0, nprow )\nmq00 = numroc( max( neig, nb, 2 ), nb, 0, 0, npcol )\niceil( x, y ) is a ScaLAPACK function returning ceiling(x/y), and nprow and\nnpcol can be determined by calling the function blacs_gridinfo.\nIf lrwork = -1, then lrwork is global input and a workspace query is\nassumed; the function only calculates the size required for optimal\nperformance for all work arrays. Each of these values is returned in the first\nentry of the corresponding work arrays, and no error message is issued by\npxerbla\niwork\n(local workspace) array of size liwork\nliwork\n(local )\nsize of iwork\nLet nnp = max( n, nprow*npcol + 1, 4 ). Then:\nliwork≥ 12*nnp + 2*n when the eigenvectors are desired\nliwork≥ 10*nnp + 2*n when only the eigenvalues have to be computed\nIf liwork = -1, then liwork is global input and a workspace query is\nassumed; the function only calculates the minimum and optimal size for all\nwork arrays. Each of these values is returned in the first entry of the\ncorresponding work array, and no error message is issued by pxerbla\nOUTPUT Parameters\na\nThe lower triangle (if uplo='L') or the upper triangle (if uplo='U') of a,\nincluding the diagonal, is destroyed.\nm\n(global )\nTotal number of eigenvalues found. 0 ≤m≤n.\nnz\n(global )\nTotal number of eigenvectors computed. 0 ≤nz≤m.\nThe number of columns of z that are filled.\nIf jobz≠ 'V', nz is not referenced.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1563\n\n\nIf jobz = 'V', nz = m\nw\n(global ) array of size n\nOn normal exit, the first m entries contain the selected eigenvalues in\nascending order.\nz\n(local ) array, global size n * n), local size lld_z*LOCc(jz+n-1)\nIf jobz = 'V', then on normal exit the first m columns of z contain the\northonormal eigenvectors of the matrix corresponding to the selected\neigenvalues.\nIf jobz = 'N', then z is not referenced.\nwork\nwork[0] returns workspace adequate workspace to allow optimal\nperformance.\nrwork\nOn return, rwork[0] contains the optimal amount of workspace required\nfor efficient execution. if jobz='N' rwork[0] = optimal amount of\nworkspace required to compute the eigenvalues. if jobz='V' rwork[0] =\noptimal amount of workspace required to compute eigenvalues and\neigenvectors.\niwork\nOn return, iwork[0] contains the amount of integer workspace required.\ninfo\n(global )\n= 0: successful exit\n< 0: If the i-th argument is an array and the j-th entry had an illegal value,\nthen info = -(i*100+j), if the i-th argument is a scalar and had an illegal\nvalue, then info = -i.\nApplication Notes\nThe distributed submatrices a(ia:*, ja:*) and z(iz:iz+m-1,jz:jz+n-1) must satisfy the following\nalignment properties:\n1.\nIdentical (quadratic) dimension: desca[m_ - 1] = descz[m_ - 1] = desca[n_ - 1] = descz[n_ - 1]\n2.\nQuadratic conformal blocking: desca[mb_ - 1] = desca[nb_ - 1] = descz[mb_ - 1] = descz[nb_ - 1],\ndesca[rsrc_ - 1] = descz[rsrc_ - 1]\n3.\nmod( ia-1, mb_a ) = mod( iz-1, mb_z ) = 0\nNOTE\nmod(x,y) is the integer remainder of x/y.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?heevx\nComputes selected eigenvalues and, optionally,\neigenvectors of a Hermitian matrix.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1564\n\n\nSyntax\nvoid pcheevx (char *jobz , char *range , char *uplo , MKL_INT *n , MKL_Complex8 *a ,\nMKL_INT *ia , MKL_INT *ja , MKL_INT *desca , float *vl , float *vu , MKL_INT *il ,\nMKL_INT *iu , float *abstol , MKL_INT *m , MKL_INT *nz , float *w , float *orfac ,\nMKL_Complex8 *z , MKL_INT *iz , MKL_INT *jz , MKL_INT *descz , MKL_Complex8 *work ,\nMKL_INT *lwork , float *rwork , MKL_INT *lrwork , MKL_INT *iwork , MKL_INT *liwork ,\nMKL_INT *ifail , MKL_INT *iclustr , float *gap , MKL_INT *info );\nvoid pzheevx (char *jobz , char *range , char *uplo , MKL_INT *n , MKL_Complex16 *a ,\nMKL_INT *ia , MKL_INT *ja , MKL_INT *desca , double *vl , double *vu , MKL_INT *il ,\nMKL_INT *iu , double *abstol , MKL_INT *m , MKL_INT *nz , double *w , double *orfac ,\nMKL_Complex16 *z , MKL_INT *iz , MKL_INT *jz , MKL_INT *descz , MKL_Complex16 *work ,\nMKL_INT *lwork , double *rwork , MKL_INT *lrwork , MKL_INT *iwork , MKL_INT *liwork ,\nMKL_INT *ifail , MKL_INT *iclustr , double *gap , MKL_INT *info );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?heevx function computes selected eigenvalues and, optionally, eigenvectors of a complex Hermitian\nmatrix A by calling the recommended sequence of ScaLAPACK functions. Eigenvalues and eigenvectors can\nbe selected by specifying either a range of values or a range of indices for the desired eigenvalues.\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nInput Parameters\nnp = the number of rows local to a given process.\nnq = the number of columns local to a given process.\njobz\n(global) Must be 'N' or 'V'.\nSpecifies if it is necessary to compute the eigenvectors:\nIf jobz = 'N', then only eigenvalues are computed.\nIf jobz = 'V', then eigenvalues and eigenvectors are computed.\nrange\n(global) Must be 'A', 'V', or 'I'.\nIf range = 'A', all eigenvalues will be found.\nIf range = 'V', all eigenvalues in the half-open interval [vl, vu] will be\nfound.\nIf range = 'I', the eigenvalues with indices il through iu will be found.\nuplo\n(global) Must be 'U' or 'L'.\nSpecifies whether the upper or lower triangular part of the Hermitian matrix\nA is stored:\nIf uplo = 'U', a stores the upper triangular part of A.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1565\n\n\nIf uplo = 'L', a stores the lower triangular part of A.\nn\n(global) The number of rows and columns of the matrix A(n≥ 0).\na\n(local).\nBlock cyclic array of global size n*n and local size lld_a*LOCc(ja+n-1).\nOn entry, the Hermitian matrix A.\nIf uplo = 'U', only the upper triangular part of A is used to define the\nelements of the Hermitian matrix.\nIf uplo = 'L', only the lower triangular part of A is used to define the\nelements of the Hermitian matrix.\nia, ja\n(global) The row and column indices in the global matrix A indicating the\nfirst row and the first column of the submatrix A, respectively.\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A. If desca[ctxt_ - 1] is incorrect, p?heevx cannot\nguarantee correct error reporting.\nvl, vu\n(global)\nIf range = 'V', the lower and upper bounds of the interval to be searched\nfor eigenvalues; not referenced if range = 'A' or 'I'.\nil, iu\n(global)\nIf range ='I', the indices of the smallest and largest eigenvalues to be\nreturned.\nConstraints:\nil ≥ 1; min(il,n) ≤ iu ≤ n.\nNot referenced if range = 'A' or 'V'.\nabstol\n(global).\nIf jobz='V', setting abstol to p?lamch(context, 'U') yields the most\northogonal eigenvectors.\nThe absolute error tolerance for the eigenvalues. An approximate\neigenvalue is accepted as converged when it is determined to lie in an\ninterval [a, b] of width less than or equal to abstol+eps*max(|a|,|b|),\nwhere eps is the machine precision. If abstol is less than or equal to zero,\nthen eps*norm(T) will be used in its place, where norm(T) is the 1-norm\nof the tridiagonal matrix obtained by reducing A to tridiagonal form.\nEigenvalues are computed most accurately when abstol is set to twice the\nunderflow threshold 2*p?lamch('S'), not zero. If this function returns\nwith ((mod(info,2)≠0).or.(mod(info/8,2)≠0)), indicating that some\neigenvalues or eigenvectors did not converge, try setting abstol to\n2*p?lamch('S').\nNOTE\nmod(x,y) is the integer remainder of x/y.\norfac\n(global).\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1566\n\n\nSpecifies which eigenvectors should be reorthogonalized. Eigenvectors that\ncorrespond to eigenvalues which are within tol=orfac*norm(A) of each\nother are to be reorthogonalized. However, if the workspace is insufficient\n(see lwork), tol may be decreased until all eigenvectors to be\nreorthogonalized can be stored in one process. No reorthogonalization will\nbe done if orfac equals zero. A default value of 1.0e-3 is used if orfac is\nnegative.\norfac should be identical on all processes.\niz, jz\n(global) The row and column indices in the global matrix Z indicating the\nfirst row and the first column of the submatrix Z, respectively.\ndescz\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix Z. descz[ctxt_ - 1] must equal desca[ctxt_ - 1].\nwork\n(local).\nArray of size lwork.\nlwork\n(local) The size of the array work.\nIf only eigenvalues are requested:\nlwork≥n + max(nb*(np0 + 1), 3)\nIf eigenvectors are requested:\nlwork≥n + (np0+mq0+nb)*nb\nwith nq0 = numroc(nn, nb, 0, 0, NPCOL).\nlwork≥ 5*n + max(5*nn, np0*mq0+2*nb*nb) + iceil(neig,\nNPROW*NPCOL)*nn\nFor optimal performance, greater workspace is needed, that is\nlwork≥max(lwork, nhetrd_lwork)\nwhere lwork is as defined above, and nhetrd_lwork = n + 2*(anb\n+1)*(4*nps+2) + (nps+1)*nps\nictxt = desca[ctxt_ - 1]\nanb = pjlaenv(ictxt, 3, 'pchettrd', 'L', 0, 0, 0, 0)\nsqnpc = sqrt(dble(NPROW * NPCOL))\nnps = max(numroc(n, 1, 0, 0, sqnpc), 2*anb)\nIf lwork = -1, then lwork is global input and a workspace query is\nassumed; the function only calculates the size required for optimal\nperformance for all work arrays. Each of these values is returned in the first\nentry of the corresponding work arrays, and no error message is issued by\npxerbla.\nrwork\n(local)\nWorkspace array of size lrwork.\nlrwork\n(local) The size of the array work.\nSee below for definitions of variables used to define lwork.\nIf no eigenvectors are requested (jobz = 'N'), then lrwork≥ 5*nn+4*n.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1567\n\n\nIf eigenvectors are requested (jobz = 'V'), then the amount of workspace\nrequired to guarantee that all eigenvectors are computed is:\nlrwork≥ 4*n + max(5*nn, np0*mq0+2*nb*nb) + iceil(neig,\nNPROW*NPCOL)*nn\nThe computed eigenvectors may not be orthogonal if the minimal\nworkspace is supplied and orfac is too small. If you want to guarantee\northogonality (at the cost of potentially poor performance) you should add\nthe following values to lrwork:\n(clustersize-1)*n,\nwhere clustersize is the number of eigenvalues in the largest cluster, where\na cluster is defined as a set of close eigenvalues:\n{w[k - 1],..., w[k+clustersize-2]|w[j] ≤\nw[j-1]+orfac*2*norm(A)}.\nVariable definitions:\nneig = number of eigenvectors requested;\nnb = desca[mb_ - 1] = desca[nb_ - 1] = descz[mb_ - 1] =\ndescz[nb_ - 1];\nnn = max(n, NB, 2);\ndesca[rsrc_ - 1] = desca[nb_ - 1] = descz[rsrc_ - 1] =\ndescz[csrc_ - 1] = 0;\nnp0 = numroc(nn, nb, 0, 0, NPROW);\nmq0 = numroc(max(neig, nb, 2), nb, 0, 0, NPCOL);\niceil(x, y) is a ScaLAPACK function returning ceiling(x/y)\nWhen lrwork is too small:\nIf lwork is too small to guarantee orthogonality, p?heevx attempts to\nmaintain orthogonality in the clusters with the smallest spacing between the\neigenvalues. If lwork is too small to compute all the eigenvectors requested,\nno computation is performed and info= -23 is returned. Note that when\nrange='V', p?heevx does not know how many eigenvectors are requested\nuntil the eigenvalues are computed. Therefore, when range='V' and as\nlong as lwork is large enough to allow p?heevx to compute the eigenvalues,\np?heevx will compute the eigenvalues and as many eigenvectors as it can.\nRelationship between workspace, orthogonality and performance:\nIf clustersize ≥ n/sqrt(NPROW*NPCOL), then providing enough space\nto compute all the eigenvectors orthogonally will cause serious degradation\nin performance. In the limit (that is, clustersize = n-1)p?stein will\nperform no better than ?stein on 1 processor.\nFor clustersize = n/sqrt(NPROW*NPCOL) reorthogonalizing all\neigenvectors will increase the total execution time by a factor of 2 or more.\nFor clustersize>n/sqrt(NPROW*NPCOL) execution time will grow as the\nsquare of the cluster size, all other factors remaining equal and assuming\nenough workspace. Less workspace means less reorthogonalization but\nfaster execution.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1568\n\n\nIf lwork = -1, then lwork is global input and a workspace query is\nassumed; the function only calculates the size required for optimal\nperformance for all work arrays. Each of these values is returned in the first\nentry of the corresponding work arrays, and no error message is issued by\npxerbla.\niwork\n(local) Workspace array.\nliwork\n(local), size of iwork.\nliwork ≥ 6*nnp\nWhere: nnp = max(n, NPROW*NPCOL+1, 4)\nIf liwork = -1, then liwork is global input and a workspace query is\nassumed; the function only calculates the minimum and optimal size for all\nwork arrays. Each of these values is returned in the first entry of the\ncorresponding work array, and no error message is issued by pxerbla.\nOutput Parameters\na\nOn exit, the lower triangle (if uplo = 'L'), or the upper triangle (if uplo =\n'U') of A, including the diagonal, is overwritten.\nm\n(global) The total number of eigenvalues found; 0 ≤ m ≤ n.\nnz\n(global) Total number of eigenvectors computed. 0 ≤ nz ≤ m.\nThe number of columns of z that are filled.\nIf jobz ≠ 'V', nz is not referenced.\nIf jobz = 'V', nz = m unless the user supplies insufficient space and\np?heevx is not able to detect this before beginning computation. To get all\nthe eigenvectors requested, the user must supply both sufficient space to\nhold the eigenvectors in z (m≤descz[n_ - 1]) and sufficient workspace to\ncompute them. (See lwork). p?heevx is always able to detect insufficient\nspace without computation unless range='V'.\nw\n(global).\nArray of size n. The first m elements contain the selected eigenvalues in\nascending order.\nz\n(local).\nArray, global size n*n, local size lld_z*LOCc(jz+n-1).\nIf jobz ='V', then on normal exit the first m columns of z contain the\northonormal eigenvectors of the matrix corresponding to the selected\neigenvalues. If an eigenvector fails to converge, then that column of z\ncontains the latest approximation to the eigenvector, and the index of the\neigenvector is returned in ifail.\nIf jobz = 'N', then z is not referenced.\nwork[0]\nOn exit, returns adequate workspace to allow optimal performance.\nrwork\n(local).\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1569\n\n\nArray of size lrwork. On return, rwork[0] contains the optimal amount of\nworkspace required for efficient execution.\nIf jobz='N'rwork[0] = optimal amount of workspace required to compute\neigenvalues efficiently.\nIf jobz='V'rwork[0] = optimal amount of workspace required to compute\neigenvalues and eigenvectors efficiently with no guarantee on orthogonality.\nIf range='V', it is assumed that all eigenvectors may be required.\niwork[0]\n(local)\nOn return, iwork[0] contains the amount of integer workspace required.\nifail\n(global)\nArray of size n.\nIf jobz ='V', then on normal exit, the first m elements of ifail are zero. If\n(mod(info,2)≠0) on exit, then ifail contains the indices of the eigenvectors\nthat failed to converge.\nIf jobz = 'N', then ifail is not referenced.\niclustr\n(global)\nArray of size 2*NPROW*NPCOL.\nThis array contains indices of eigenvectors corresponding to a cluster of\neigenvalues that could not be reorthogonalized due to insufficient\nworkspace (see lwork, orfac and info). Eigenvectors corresponding to\nclusters of eigenvalues indexed iclustr[2*i - 2]) to iclustr[2*i -\n1], could not be reorthogonalized due to lack of workspace. Hence the\neigenvectors corresponding to these clusters may not be orthogonal.\niclustr is a zero terminated array. (iclustr[2*k - 1]≠0 and\niclustr[2*k]=0) if and only if k is the number of clusters. iclustr is not\nreferenced if jobz = 'N'.\ngap\n(global)\nArray of size (NPROW*NPCOL)\nThis array contains the gap between eigenvalues whose eigenvectors could\nnot be reorthogonalized. The output values in this array correspond to the\nclusters indicated by the array iclustr. As a result, the dot product between\neigenvectors corresponding to the i-th cluster may be as high as (C*n)/\ngap(i) where C is a small constant.\ninfo\n(global)\nIf info = 0, the execution is successful.\nIf info < 0:\nIf the i-th argument is an array and the j-entry had an illegal value, then\ninfo = -(i*100+j). If the i-th argument is a scalar and had an illegal\nvalue, then info = -i.\nIf info> 0:\nIf (mod(info,2)≠0), then one or more eigenvectors failed to converge.\nTheir indices are stored in ifail. Ensure abstol=2.0*p?lamch('U')\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1570\n\n\nIf (mod(info/2,2)≠0), then eigenvectors corresponding to one or more\nclusters of eigenvalues could not be reorthogonalized because of insufficient\nworkspace.The indices of the clusters are stored in the array iclustr.\nIf (mod(info/4,2)≠0), then space limit prevented p?syevx from\ncomputing all of the eigenvectors between vl and vu. The number of\neigenvectors computed is returned in nz.\nIf (mod(info/8,2)≠0), then p?stebz failed to compute eigenvalues.\nEnsure abstol=2.0*p?lamch('U').\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?gesvd\nComputes the singular value decomposition of a\ngeneral matrix, optionally computing the left and/or\nright singular vectors.\nSyntax\nvoid psgesvd (char *jobu , char *jobvt , MKL_INT *m , MKL_INT *n , float *a , MKL_INT\n*ia , MKL_INT *ja , MKL_INT *desca , float *s , float *u , MKL_INT *iu , MKL_INT *ju ,\nMKL_INT *descu , float *vt , MKL_INT *ivt , MKL_INT *jvt , MKL_INT *descvt , float\n*work , MKL_INT *lwork , float *rwork , MKL_INT *info );\nvoid pdgesvd (char *jobu , char *jobvt , MKL_INT *m , MKL_INT *n , double *a , MKL_INT\n*ia , MKL_INT *ja , MKL_INT *desca , double *s , double *u , MKL_INT *iu , MKL_INT *ju ,\nMKL_INT *descu , double *vt , MKL_INT *ivt , MKL_INT *jvt , MKL_INT *descvt , double\n*work , MKL_INT *lwork , double *rwork , MKL_INT *info );\nvoid pcgesvd (char *jobu , char *jobvt , MKL_INT *m , MKL_INT *n , MKL_Complex8 *a ,\nMKL_INT *ia , MKL_INT *ja , MKL_INT *desca , float *s , MKL_Complex8 *u , MKL_INT *iu ,\nMKL_INT *ju , MKL_INT *descu , MKL_Complex8 *vt , MKL_INT *ivt , MKL_INT *jvt , MKL_INT\n*descvt , MKL_Complex8 *work , MKL_INT *lwork , float *rwork , MKL_INT *info );\nvoid pzgesvd (char *jobu , char *jobvt , MKL_INT *m , MKL_INT *n , MKL_Complex16 *a ,\nMKL_INT *ia , MKL_INT *ja , MKL_INT *desca , double *s , MKL_Complex16 *u , MKL_INT\n*iu , MKL_INT *ju , MKL_INT *descu , MKL_Complex16 *vt , MKL_INT *ivt , MKL_INT *jvt ,\nMKL_INT *descvt , MKL_Complex16 *work , MKL_INT *lwork , double *rwork , MKL_INT\n*info );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?gesvd function computes the singular value decomposition (SVD) of an m-by-n matrix A, optionally\ncomputing the left and/or right singular vectors. The SVD is written\nA = U*Σ*VT,\nwhere Σ is an m-by-n matrix that is zero except for its min(m, n) diagonal elements, U is an m-by-m\northogonal matrix, and V is an n-by-n orthogonal matrix. The diagonal elements of Σ are the singular values\nof A and the columns of U and V are the corresponding right and left singular vectors, respectively. The\nsingular values are returned in array s in decreasing order and only the first min(m,n) columns of U and rows\nof vt = VT are computed.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1571\n\n\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nNOTE\nThe distributed submatrix sub(A) must verify certain alignment properties. These\nexpressions must be true:\n•\nmb_a = nb_a = nb\n•\niroffa = icoffa\nwhere:\n•\niroffa = mod(ia-1, nb )\n•\nicoffa = mod(ja-1, nb )\nInput Parameters\nmp = number of local rows in A and U\nnq = number of local columns in A and VT\nsize = min(m, n)\nsizeq = number of local columns in U\nsizep = number of local rows in VT\njobu\n(global) Specifies options for computing all or part of the matrix U.\nIf jobu = 'V', the first size columns of U (the left singular vectors) are\nreturned in the array u;\nIf jobu ='N', no columns of U (no left singular vectors)are computed.\njobvt\n(global)\nSpecifies options for computing all or part of the matrix VT.\nIf jobvt = 'V', the first size rows of VT (the right singular vectors) are\nreturned in the array vt;\nIf jobvt = 'N', no rows of VT(no right singular vectors) are computed.\nm\n(global) The number of rows of the matrix A(m≥ 0).\nn\n(global) The number of columns in A(n≥ 0).\na\n(local).\nBlock cyclic array, global size (m, n), local size (mp, nq).\nia, ja\n(global) The row and column indices in the global matrix A indicating the\nfirst row and the first column of the submatrix A, respectively.\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1572\n\n\niu, ju\n(global) The row and column indices in the global matrix U indicating the\nfirst row and the first column of the submatrix U, respectively.\ndescu\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix U.\nivt, jvt\n(global) The row and column indices in the global matrix VT indicating the\nfirst row and the first column of the submatrix VT, respectively.\ndescvt\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix VT.\nwork\n(local).\nWorkspace array of size lwork\nlwork\n(local) The size of the array work;\nlwork > 2 + 6*sizeb + max(watobd, wbdtosvd),\nwhere sizeb = max(m, n), and watobd and wbdtosvd refer, respectively,\nto the workspace required to bidiagonalize the matrix A and to go from the\nbidiagonal matrix to the singular value decomposition USVT.\nFor watobd, the following holds:\nwatobd = max(max(wp?lange,wp?gebrd), max(wp?lared2d, wp?\nlared1d)),\nwhere wp?lange, wp?lared1d, wp?lared2d, wp?gebrd are the workspaces\nrequired respectively for the subprograms p?lange, p?lared1d,\np?lared2d, p?gebrd. Using the standard notation\nmp = numroc(m, mb, MYROW, desca[ctxt_ - 1], NPROW),\nnq = numroc(n, nb, MYCOL, desca[lld_ - 1], NPCOL),\nthe workspaces required for the above subprograms are\nwp?lange = mp,\nwp?lared1d = nq0,\nwp?lared2d = mp0,\nwp?gebrd = nb*(mp + nq + 1) + nq,\nwhere nq0 and mp0 refer, respectively, to the values obtained at MYCOL =\n0 and MYROW = 0. In general, the upper limit for the workspace is given by\na workspace required on processor (0,0):\nwatobd ≤ nb*(mp0 + nq0 + 1) + nq0.\nIn case of a homogeneous process grid this upper limit can be used as an\nestimate of the minimum workspace for every processor.\nFor wbdtosvd, the following holds:\nwbdtosvd = size*(wantu*nru + wantvt*ncvt) + max(w?bdsqr,\nmax(wantu*wp?ormbrqln, wantvt*wp?ormbrprt)),\nwhere\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1573\n\n\nwantu(wantvt) = 1, if left/right singular vectors are wanted, and\nwantu(wantvt) = 0, otherwise. w?bdsqr, wp?ormbrqln, and wp?ormbrprt\nrefer respectively to the workspace required for the subprograms ?bdsqr,\np?ormbr(qln), and p?ormbr(prt), where qln and prt are the values of the\narguments vect, side, and trans in the call to p?ormbr. nru is equal to the\nlocal number of rows of the matrix U when distributed 1-dimensional\n\"column\" of processes. Analogously, ncvt is equal to the local number of\ncolumns of the matrix VT when distributed across 1-dimensional \"row\" of\nprocesses. Calling the LAPACK procedure ?bdsqr requires\nw?bdsqr = max(1, 2*size + (2*size - 4)* max(wantu, wantvt))\non every processor. Finally,\nwp?ormbrqln = max((nb*(nb-1))/2, (sizeq+mp)*nb)+nb*nb,\nwp?ormbrprt = max((mb*(mb-1))/2, (sizep+nq)*mb)+mb*mb,\nIf lwork = -1, then lwork is global input and a workspace query is\nassumed; the function only calculates the minimum size for the work array.\nThe required workspace is returned as the first element of work and no\nerror message is issued by pxerbla.\nrwork\nWorkspace array of size 1 + 4*sizeb. Not used for psgesvd and pdgesvd.\nOutput Parameters\na\nOn exit, the contents of a are destroyed.\ns\n(global).\nArray of size size.\nContains the singular values of A sorted so that s(i) ≥s(i+1).\nu\n(local).\nlocal size mp*sizeq, global size m*size)\nIf jobu = 'V', u contains the first min(m, n) columns of U.\nIf jobu = 'N' or 'O', u is not referenced.\nvt\n(local).\nlocal size (sizep, nq), global size (size, n)\nIf jobvt = 'V', vt contains the first size rows of VTif jobu = 'N', vt is\nnot referenced.\nwork\nOn exit, if info = 0, then work[0] returns the required minimal size of\nlwork.\nrwork\nOn exit, if info = 0, then rwork[0] returns the required size of rwork.\ninfo\n(global)\nIf info = 0, the execution is successful.\nIf info < 0, If info = -i, the ith parameter had an illegal value.\nIf info > 0 i, then if ?bdsqr did not converge,\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1574\n\n\nIf info = min(m,n) + 1, then p?gesvd has detected heterogeneity by\nfinding that eigenvalues were not identical across the process grid. In this\ncase, the accuracy of the results from p?gesvd cannot be guaranteed.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?sygvx\nComputes selected eigenvalues and, optionally,\neigenvectors of a real generalized symmetric definite\neigenproblem.\nSyntax\nvoid pssygvx (MKL_INT *ibtype , char *jobz , char *range , char *uplo , MKL_INT *n ,\nfloat *a , MKL_INT *ia , MKL_INT *ja , MKL_INT *desca , float *b , MKL_INT *ib , MKL_INT\n*jb , MKL_INT *descb , float *vl , float *vu , MKL_INT *il , MKL_INT *iu , float\n*abstol , MKL_INT *m , MKL_INT *nz , float *w , float *orfac , float *z , MKL_INT *iz ,\nMKL_INT *jz , MKL_INT *descz , float *work , MKL_INT *lwork , MKL_INT *iwork , MKL_INT\n*liwork , MKL_INT *ifail , MKL_INT *iclustr , float *gap , MKL_INT *info );\nvoid pdsygvx (MKL_INT *ibtype , char *jobz , char *range , char *uplo , MKL_INT *n ,\ndouble *a , MKL_INT *ia , MKL_INT *ja , MKL_INT *desca , double *b , MKL_INT *ib ,\nMKL_INT *jb , MKL_INT *descb , double *vl , double *vu , MKL_INT *il , MKL_INT *iu ,\ndouble *abstol , MKL_INT *m , MKL_INT *nz , double *w , double *orfac , double *z ,\nMKL_INT *iz , MKL_INT *jz , MKL_INT *descz , double *work , MKL_INT *lwork , MKL_INT\n*iwork , MKL_INT *liwork , MKL_INT *ifail , MKL_INT *iclustr , double *gap , MKL_INT\n*info );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?sygvxfunction computes all the eigenvalues, and optionally, the eigenvectors of a real generalized\nsymmetric-definite eigenproblem, of the form\nsub(A)*x = λ*sub(B)*x, sub(A) sub(B)*x = λ*x, or sub(B)*sub(A)*x = λ*x.\nHere x denotes eigen vectors, λ (lambda) denotes eigenvalues, sub(A) denoting A(ia:ia+n-1, ja:ja\n+n-1) is assumed to symmetric, and sub(B) denoting B(ib:ib+n-1, jb:jb+n-1) is also positive definite.\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nInput Parameters\nibtype\n(global) Must be 1 or 2 or 3.\nSpecifies the problem type to be solved:\nIf ibtype = 1, the problem type is sub(A)*x = lambda*sub(B)*x;\nIf ibtype = 2, the problem type is sub(A)*sub(B)*x = lambda*x;\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1575\n\n\nIf ibtype = 3, the problem type is sub(B)*sub(A)*x = lambda*x.\njobz\n(global) Must be 'N' or 'V'.\nIf jobz ='N', then compute eigenvalues only.\nIf jobz ='V', then compute eigenvalues and eigenvectors.\nrange\n(global) Must be 'A' or 'V' or 'I'.\nIf range = 'A', the function computes all eigenvalues.\nIf range = 'V', the function computes eigenvalues in the interval: [vl,\nvu]\nIf range = 'I', the function computes eigenvalues with indices il through\niu.\nuplo\n(global) Must be 'U' or 'L'.\nIf uplo = 'U', arrays a and b store the upper triangles of sub(A) and sub\n(B);\nIf uplo = 'L', arrays a and b store the lower triangles of sub(A) and sub\n(B).\nn\n(global) The order of the matrices sub(A) and sub (B), n≥ 0.\na\n(local)\nPointer into the local memory to an array of size lld_a*LOCc(ja+n-1). On\nentry, this array contains the local pieces of the n-by-n symmetric\ndistributed matrix sub(A).\nIf uplo = 'U', the leading n-by-n upper triangular part of sub(A) contains\nthe upper triangular part of the matrix.\nIf uplo = 'L', the leading n-by-n lower triangular part of sub(A) contains\nthe lower triangular part of the matrix.\nia, ja\n(global) The row and column indices in the global matrix A indicating the\nfirst row and the first column of the submatrix A, respectively.\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A. If desca[ctxt_ - 1] is incorrect, p?sygvx cannot\nguarantee correct error reporting.\nb\n(local).\nPointer into the local memory to an array of size lld_b*LOCc(jb+n-1). On\nentry, this array contains the local pieces of the n-by-n symmetric\ndistributed matrix sub(B).\nIf uplo = 'U', the leading n-by-n upper triangular part of sub(B) contains\nthe upper triangular part of the matrix.\nIf uplo = 'L', the leading n-by-n lower triangular part of sub(A) contains\nthe lower triangular part of the matrix.\nib, jb\n(global) The row and column indices in the global matrix B indicating the\nfirst row and the first column of the submatrix B, respectively.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1576\n\n\ndescb\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix B. descb[ctxt_ - 1] must be equal to desca[ctxt_ -\n1].\nvl, vu\n(global)\nIf range = 'V', the lower and upper bounds of the interval to be searched\nfor eigenvalues.\nIf range = 'A' or 'I', vl and vu are not referenced.\nil, iu\n(global)\nIf range = 'I', the indices in ascending order of the smallest and largest\neigenvalues to be returned. Constraint: il ≥ 1, min(il, n)≤ iu ≤ n\nIf range = 'A' or 'V', il and iu are not referenced.\nabstol\n(global)\nIf jobz='V', setting abstol to p?lamch(context, 'U') yields the most\northogonal eigenvectors.\nThe absolute error tolerance for the eigenvalues. An approximate\neigenvalue is accepted as converged when it is determined to lie in an\ninterval [a,b] of width less than or equal to\nabstol + eps*max(|a|,|b|),\nwhere eps is the machine precision. If abstol is less than or equal to zero,\nthen eps*norm(T) will be used in its place, where norm(T) is the 1-norm\nof the tridiagonal matrix obtained by reducing A to tridiagonal form.\nEigenvalues will be computed most accurately when abstol is set to twice\nthe underflow threshold 2*p?lamch('S') not zero. If this function returns\nwith ((mod(info,2)≠0) or (mod(info/8,2)≠0)), indicating that some\neigenvalues or eigenvectors did not converge, try setting abstol to\n2*p?lamch('S').\nNOTE\nmod(x,y) is the integer remainder of x/y.\norfac\n(global).\nSpecifies which eigenvectors should be reorthogonalized. Eigenvectors that\ncorrespond to eigenvalues which are within tol=orfac*norm(A) of each\nother are to be reorthogonalized. However, if the workspace is insufficient\n(see lwork), tol may be decreased until all eigenvectors to be\nreorthogonalized can be stored in one process. No reorthogonalization will\nbe done if orfac equals zero. A default value of 1.0e-3 is used if orfac is\nnegative. orfac should be identical on all processes.\niz, jz\n(global) The row and column indices in the global matrix Z indicating the\nfirst row and the first column of the submatrix Z, respectively.\ndescz\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix Z.descz[ctxt_ - 1] must equal desca[ctxt_ - 1].\nwork\n(local)\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1577\n\n\nWorkspace array of size lwork\nlwork\n(local)\nSize of the array work. See below for definitions of variables used to define\nlwork.\nIf no eigenvectors are requested (jobz = 'N'), then lwork ≥ 5*n +\nmax(5*nn, NB*(np0 + 1)).\nIf eigenvectors are requested (jobz = 'V'), then the amount of workspace\nrequired to guarantee that all eigenvectors are computed is:\nlwork ≥ 5*n + max(5*nn, np0*mq0 + 2*nb*nb) + iceil(neig,\nNPROW*NPCOL)*nn.\nThe computed eigenvectors may not be orthogonal if the minimal\nworkspace is supplied and orfac is too small. If you want to guarantee\northogonality at the cost of potentially poor performance you should add\nthe following to lwork:\n(clustersize-1)*n,\nwhere clustersize is the number of eigenvalues in the largest cluster, where\na cluster is defined as a set of close eigenvalues:\n{w[k - 1],..., w[k+clustersize - 2]|w[j] ≤ w[j - 1] +\norfac*2*norm(A)}\nVariable definitions:\nneig = number of eigenvectors requested,\nnb = desca[mb_ - 1] = desca[nb_ - 1] = descz[mb_ - 1] =\ndescz[nb_ - 1],\nnn = max(n, nb, 2),\ndesca[rsrc_ - 1] = desca[nb_ - 1] = descz[rsrc_ - 1] =\ndescz[csrc_ - 1] = 0,\nnp0 = numroc(nn, nb, 0, 0, NPROW),\nmq0 = numroc(max(neig, nb, 2), nb, 0, 0, NPCOL)\niceil(x, y) is a ScaLAPACK function returning ceiling(x/y)\nIf lwork is too small to guarantee orthogonality, p?syevx attempts to\nmaintain orthogonality in the clusters with the smallest spacing between the\neigenvalues.\nIf lwork is too small to compute all the eigenvectors requested, no\ncomputation is performed and info= -23 is returned.\nNote that when range='V', number of requested eigenvectors are not\nknown until the eigenvalues are computed. In this case and if lwork is large\nenough to compute the eigenvalues, p?sygvx computes the eigenvalues\nand as many eigenvectors as possible.\nGreater performance can be achieved if adequate workspace is provided. In\nsome situations, performance can decrease as the provided workspace\nincreases above the workspace amount shown below:\nlwork ≥ max(lwork, 5*n + nsytrd_lwopt, nsygst_lwopt), where\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1578\n\n\nlwork, as defined previously, depends upon the number of eigenvectors\nrequested, and\nnsytrd_lwopt = n + 2*(anb+1)*(4*nps+2) + (nps+3)*nps\nnsygst_lwopt = 2*np0*nb + nq0*nb + nb*nb\nanb = pjlaenv(desca[ctxt_ - 1], 3, p?syttrd ', 'L', 0, 0, 0,\n0)\nsqnpc = int(sqrt(dble(NPROW * NPCOL)))\nnps = max(numroc(n, 1, 0, 0, sqnpc), 2*anb)\nNB = desca[mb_ - 1]\nnp0 = numroc(n, nb, 0, 0, NPROW)\nnq0 = numroc(n, nb, 0, 0, NPCOL)\nnumroc is a ScaLAPACK tool functions;\npjlaenv is a ScaLAPACK environmental inquiry function\nMYROW, MYCOL, NPROW and NPCOL can be determined by calling the function\nblacs_gridinfo.\nFor large n, no extra workspace is needed, however the biggest boost in\nperformance comes for small n, so it is wise to provide the extra workspace\n(typically less than a Megabyte per process).\nIf clustersize ≥ n/sqrt(NPROW*NPCOL), then providing enough space\nto compute all the eigenvectors orthogonally will cause serious degradation\nin performance. At the limit (that is, clustersize = n-1) p?stein will\nperform no better than ?stein on a single processor.\nFor clustersize = n/sqrt(NPROW*NPCOL) reorthogonalizing all\neigenvectors will increase the total execution time by a factor of 2 or more.\nFor clustersize>n/sqrt(NPROW*NPCOL) execution time will grow as the\nsquare of the cluster size, all other factors remaining equal and assuming\nenough workspace. Less workspace means less reorthogonalization but\nfaster execution.\nIf lwork = -1, then lwork is global input and a workspace query is\nassumed; the function only calculates the size required for optimal\nperformance for all work arrays. Each of these values is returned in the first\nentry of the corresponding work arrays, and no error message is issued by\npxerbla.\niwork\n(local) Workspace array.\nliwork\n(local) , size of iwork.\nliwork ≥ 6*nnp\nWhere:\nnnp = max(n, NPROW*NPCOL + 1, 4)\nIf liwork = -1, then liwork is global input and a workspace query is\nassumed; the function only calculates the minimum and optimal size for all\nwork arrays. Each of these values is returned in the first entry of the\ncorresponding work array, and no error message is issued by pxerbla.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1579\n\n\nOutput Parameters\na\nOn exit,\nIf jobz = 'V', and if info = 0, sub(A) contains the distributed matrix Z\nof eigenvectors. The eigenvectors are normalized as follows:\nfor ibtype = 1 or 2, ZT*sub(B)*Z = i;\nfor ibtype = 3, ZT*inv(sub(B))*Z = i.\nIf jobz = 'N', then on exit the upper triangle (if uplo='U') or the lower\ntriangle (if uplo='L') of sub(A), including the diagonal, is destroyed.\nb\nOn exit, if info ≤ n, the part of sub(B) containing the matrix is overwritten\nby the triangular factor U or L from the Cholesky factorization sub(B) =\nUT*U or sub(B) = L*LT.\nm\n(global) The total number of eigenvalues found, 0 ≤ m ≤ n.\nnz\n(global)\nTotal number of eigenvectors computed. 0 ≤ nz ≤ m. The number of\ncolumns of z that are filled.\nIf jobz ≠ 'V', nz is not referenced.\nIf jobz = 'V', nz = m unless the user supplies insufficient space and\np?sygvx is not able to detect this before beginning computation. To get all\nthe eigenvectors requested, the user must supply both sufficient space to\nhold the eigenvectors in z (m≤descz(n_)) and sufficient workspace to\ncompute them. (See lwork below.) p?sygvx is always able to detect\ninsufficient space without computation unless range='V'.\nw\n(global)\nArray of size n. On normal exit, the first m entries contain the selected\neigenvalues in ascending order.\nz\n(local). \nglobal size n*n, local size lld_z*LOCc(jz+n-1).\nIf jobz = 'V', then on normal exit the first m columns of z contain the\northonormal eigenvectors of the matrix corresponding to the selected\neigenvalues. If an eigenvector fails to converge, then that column of z\ncontains the latest approximation to the eigenvector, and the index of the\neigenvector is returned in ifail.\nIf jobz = 'N', then z is not referenced.\nwork\nIf jobz='N'work[0] = optimal amount of workspace required to compute\neigenvalues efficiently\nIf jobz = 'V'work[0] = optimal amount of workspace required to\ncompute eigenvalues and eigenvectors efficiently with no guarantee on\northogonality.\nIf range='V', it is assumed that all eigenvectors may be required.\nifail\n(global)\nArray of size n.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1580\n\n\nifail provides additional information when info≠0\nIf (mod(info/16,2)≠0) then ifail[0] indicates the order of the smallest\nminor which is not positive definite. If (mod(info,2)≠0) on exit, then ifail\ncontains the indices of the eigenvectors that failed to converge.\nIf neither of the above error conditions hold and jobz = 'V', then the first\nm elements of ifail are set to zero.\niclustr\n(global)\nArray of size (2*NPROW*NPCOL). This array contains indices of eigenvectors\ncorresponding to a cluster of eigenvalues that could not be reorthogonalized\ndue to insufficient workspace (see lwork, orfac and info). Eigenvectors\ncorresponding to clusters of eigenvalues indexed iclustr[2*i - 2] to\niclustr[2*i - 1], could not be reorthogonalized due to lack of\nworkspace. Hence the eigenvectors corresponding to these clusters may not\nbe orthogonal. iclustr is a zero terminated array.\n(iclustr[2*k - 1]≠0.and. iclustr[2*k]=0) if and only if k is the\nnumber of clusters iclustr is not referenced if jobz = 'N'.\ngap\n(global)\nArray of size NPROW*NPCOL. This array contains the gap between\neigenvalues whose eigenvectors could not be reorthogonalized. The output\nvalues in this array correspond to the clusters indicated by the array iclustr.\nAs a result, the dot product between eigenvectors corresponding to the i-th\ncluster may be as high as (C*n)/gap[i - 1], where C is a small constant.\ninfo\n(global)\nIf info = 0, the execution is successful.\nIf info <0: the i-th argument is an array and the j-entry had an illegal\nvalue, then info = -(i*100+j), if the i-th argument is a scalar and had\nan illegal value, then info = -i.\nIf info> 0:\nIf (mod(info,2)≠0), then one or more eigenvectors failed to converge.\nTheir indices are stored in ifail.\nIf (mod(info,2,2)≠0), then eigenvectors corresponding to one or more\nclusters of eigenvalues could not be reorthogonalized because of insufficient\nworkspace. The indices of the clusters are stored in the array iclustr.\nIf (mod(info/4,2)≠0), then space limit prevented p?sygvx from\ncomputing all of the eigenvectors between vl and vu. The number of\neigenvectors computed is returned in nz.\nIf (mod(info/8,2)≠0), then p?stebz failed to compute eigenvalues.\nIf (mod(info/16,2)≠0), then B was not positive definite. ifail(1) indicates\nthe order of the smallest minor which is not positive definite.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1581\n\n\np?hegvx\nComputes selected eigenvalues and, optionally,\neigenvectors of a complex generalized Hermitian\npositive-definite eigenproblem.\nSyntax\nvoid pchegvx (MKL_INT *ibtype , char *jobz , char *range , char *uplo , MKL_INT *n ,\nMKL_Complex8 *a , MKL_INT *ia , MKL_INT *ja , MKL_INT *desca , MKL_Complex8 *b ,\nMKL_INT *ib , MKL_INT *jb , MKL_INT *descb , float *vl , float *vu , MKL_INT *il ,\nMKL_INT *iu , float *abstol , MKL_INT *m , MKL_INT *nz , float *w , float *orfac ,\nMKL_Complex8 *z , MKL_INT *iz , MKL_INT *jz , MKL_INT *descz , MKL_Complex8 *work ,\nMKL_INT *lwork , float *rwork , MKL_INT *lrwork , MKL_INT *iwork , MKL_INT *liwork ,\nMKL_INT *ifail , MKL_INT *iclustr , float *gap , MKL_INT *info );\nvoid pzhegvx (MKL_INT *ibtype , char *jobz , char *range , char *uplo , MKL_INT *n ,\nMKL_Complex16 *a , MKL_INT *ia , MKL_INT *ja , MKL_INT *desca , MKL_Complex16 *b ,\nMKL_INT *ib , MKL_INT *jb , MKL_INT *descb , double *vl , double *vu , MKL_INT *il ,\nMKL_INT *iu , double *abstol , MKL_INT *m , MKL_INT *nz , double *w , double *orfac ,\nMKL_Complex16 *z , MKL_INT *iz , MKL_INT *jz , MKL_INT *descz , MKL_Complex16 *work ,\nMKL_INT *lwork , double *rwork , MKL_INT *lrwork , MKL_INT *iwork , MKL_INT *liwork ,\nMKL_INT *ifail , MKL_INT *iclustr , double *gap , MKL_INT *info );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?hegvx function computes all the eigenvalues, and optionally, the eigenvectors of a complex\ngeneralized Hermitian positive-definite eigenproblem, of the form\nsub(A)*x = λ*sub(B)*x, sub(A)*sub(B)*x = λ*x, or sub(B)*sub(A)*x = λ*x.\nHere sub (A) denoting A(ia:ia+n-1, ja:ja+n-1) and sub(B) are assumed to be Hermitian and sub(B)\ndenoting B(ib:ib+n-1, jb:jb+n-1) is also positive definite.\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nInput Parameters\nibtype\n(global) Must be 1 or 2 or 3.\nSpecifies the problem type to be solved:\nIf ibtype = 1, the problem type is\nsub(A)*x = lambda*sub(B)*x;\nIf ibtype = 2, the problem type is\nsub(A)*sub(B)*x = lambda*x;\nIf ibtype = 3, the problem type is\nsub(B)*sub(A)*x = lambda*x.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1582\n\n\njobz\n(global) Must be 'N' or 'V'.\nIf jobz ='N', then compute eigenvalues only.\nIf jobz ='V', then compute eigenvalues and eigenvectors.\nrange\n(global) Must be 'A' or 'V' or 'I'.\nIf range = 'A', the function computes all eigenvalues.\nIf range = 'V', the function computes eigenvalues in the interval: [vl,\nvu]\nIf range = 'I', the function computes eigenvalues with indices il through\niu.\nuplo\n(global) Must be 'U' or 'L'.\nIf uplo = 'U', arrays a and b store the upper triangles of sub(A) and sub\n(B);\nIf uplo = 'L', arrays a and b store the lower triangles of sub(A) and sub\n(B).\nn\n(global)\nThe order of the matrices sub(A) and sub (B) (n≥ 0).\na\n(local)\nPointer into the local memory to an array of size lld_a*LOCc(ja+n-1). On\nentry, this array contains the local pieces of the n-by-n Hermitian\ndistributed matrix sub(A). If uplo = 'U', the leading n-by-n upper\ntriangular part of sub(A) contains the upper triangular part of the matrix. If\nuplo = 'L', the leading n-by-n lower triangular part of sub(A) contains\nthe lower triangular part of the matrix.\nia, ja\n(global)\nThe row and column indices in the global matrix A indicating the first row\nand the first column of the submatrix A, respectively.\ndesca\n(global and local) array of size dlen_.\nThe array descriptor for the distributed matrix A. If desca[ctxt_ - 1] is\nincorrect, p?hegvx cannot guarantee correct error reporting.\nb\n(local).\nPointer into the local memory to an array of size lld_b*LOCc(jb+n-1). On\nentry, this array contains the local pieces of the n-by-n Hermitian\ndistributed matrix sub(B).\nIf uplo = 'U', the leading n-by-n upper triangular part of sub(B) contains\nthe upper triangular part of the matrix.\nIf uplo = 'L', the leading n-by-n lower triangular part of sub(B) contains\nthe lower triangular part of the matrix.\nib, jb\n(global)\nThe row and column indices in the global matrix B indicating the first row\nand the first column of the submatrix B, respectively.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1583\n\n\ndescb\n(global and local) array of size dlen_.\nThe array descriptor for the distributed matrix B.descb[ctxt_ - 1] must\nbe equal to desca[ctxt_ - 1].\nvl, vu\n(global)\nIf range = 'V', the lower and upper bounds of the interval to be searched\nfor eigenvalues.\nIf range = 'A' or 'I', vl and vu are not referenced.\nil, iu\n(global)\nIf range = 'I', the indices in ascending order of the smallest and largest\neigenvalues to be returned. Constraint: il≥ 1, min(il, n) ≤ iu ≤ n\nIf range = 'A' or 'V', il and iu are not referenced.\nabstol\n(global)\nIf jobz='V', setting abstol to p?lamch(context, 'U') yields the most\northogonal eigenvectors.\nThe absolute error tolerance for the eigenvalues. An approximate\neigenvalue is accepted as converged when it is determined to lie in an\ninterval [a,b] of width less than or equal to\nabstol + eps*max(|a|,|b|),\nwhere eps is the machine precision. If abstol is less than or equal to zero,\nthen eps*norm(T) will be used in its place, where norm(T) is the 1-norm of\nthe tridiagonal matrix obtained by reducing A to tridiagonal form.\nEigenvalues will be computed most accurately when abstol is set to twice\nthe underflow threshold 2*p?lamch('S') not zero. If this function returns\nwith ((mod(info,2)≠0).or. * (mod(info/8,2)≠0)), indicating that\nsome eigenvalues or eigenvectors did not converge, try setting abstol to\n2*p?lamch('S').\nNOTE\nmod(x,y) is the integer remainder of x/y.\norfac\n(global).\nSpecifies which eigenvectors should be reorthogonalized. Eigenvectors that\ncorrespond to eigenvalues which are within tol=orfac*norm(A) of each\nother are to be reorthogonalized. However, if the workspace is insufficient\n(see lwork), tol may be decreased until all eigenvectors to be\nreorthogonalized can be stored in one process. No reorthogonalization will\nbe done if orfac equals zero. A default value of 1.0E-3 is used if orfac is\nnegative. orfac should be identical on all processes.\niz, jz\n(global) The row and column indices in the global matrix Z indicating the\nfirst row and the first column of the submatrix Z, respectively.\ndescz\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix Z.descz[ctxt_ - 1] must equal desca[ctxt_ - 1].\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1584\n\n\nwork\n(local)\nWorkspace array of size lwork\nlwork\n(local).\nThe size of the array work.\nIf only eigenvalues are requested:\nlwork ≥ n+ max(NB*(np0 + 1), 3)\nIf eigenvectors are requested:\nlwork ≥ n + (np0+ mq0 + NB)*NB\nwith nq0 = numroc(nn, NB, 0, 0, NPCOL).\nFor optimal performance, greater workspace is needed, that is\nlwork ≥ max(lwork, n, nhetrd_lwopt, nhegst_lwopt)\nwhere lwork is as defined above, and\nnhetrd_lwork = 2*(anb+1)*(4*nps+2) + (nps + 1)*nps;\nnhegst_lwopt = 2*np0*nb + nq0*nb + nb*nb\nnb = desca[mb_ - 1]\nnp0 = numroc(n, nb, 0, 0, NPROW)\nnq0 = numroc(n, nb, 0, 0, NPCOL)\nictxt = desca[ctxt_ - 1]\nanb = pjlaenv(ictxt, 3, 'p?hettrd', 'L', 0, 0, 0, 0)\nsqnpc = sqrt(dble(NPROW * NPCOL))\nnps = max(numroc(n, 1, 0, 0, sqnpc), 2*anb)\nnumroc is a ScaLAPACK tool functions;\npjlaenv is a ScaLAPACK environmental inquiry function MYROW, MYCOL,\nNPROW and NPCOL can be determined by calling the function\nblacs_gridinfo.\nIf lwork = -1, then lwork is global input and a workspace query is\nassumed; the function only calculates the size required for optimal\nperformance for all work arrays. Each of these values is returned in the first\nentry of the corresponding work arrays, and no error message is issued by\npxerbla.\nrwork\n(local)\nWorkspace array of size lrwork.\nlrwork\n(local) The size of the array rwork.\nSee below for definitions of variables used to define lrwork.\nIf no eigenvectors are requested (jobz = 'N'), then lrwork ≥ 5*nn+4*n\nIf eigenvectors are requested (jobz = 'V'), then the amount of workspace\nrequired to guarantee that all eigenvectors are computed is:\nlrwork ≥ 4*n + max(5*nn, np0*mq0)+iceil(neig,\nNPROW*NPCOL)*nn\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1585\n\n\nThe computed eigenvectors may not be orthogonal if the minimal\nworkspace is supplied and orfac is too small. If you want to guarantee\northogonality (at the cost of potentially poor performance) you should add\nthe following value to lrwork:\n(clustersize-1)*n,\nwhere clustersize is the number of eigenvalues in the largest cluster, where\na cluster is defined as a set of close eigenvalues:\n{w]k - 1],..., w[k+clustersize - 2]|w[j] ≤ w[j -\n1]+orfac*2*norm(A)}\nVariable definitions:\nneig = number of eigenvectors requested;\nnb = desca[mb_ - 1] = desca[nb_ - 1] = descz[mb_ - 1] =\ndescz[nb_ - 1];\nnn = max(n, nb, 2);\ndesca[rsrc_ - 1] = desca[nb_ - 1] = descz[rsrc_ - 1] =\ndescz[csrc_ - 1] = 0 ;\nnp0 = numroc(nn, nb, 0, 0, NPROW);\nmq0 = numroc(max(neig, nb, 2), nb, 0, 0, NPCOL); \niceil(x, y) is a ScaLAPACK function returning ceiling(x/y).\nWhen lrwork is too small:\nIf lwork is too small to guarantee orthogonality, p?hegvx attempts to\nmaintain orthogonality in the clusters with the smallest spacing between the\neigenvalues.\nIf lwork is too small to compute all the eigenvectors requested, no\ncomputation is performed and info= -25 is returned. Note that when\nrange='V', p?hegvx does not know how many eigenvectors are requested\nuntil the eigenvalues are computed. Therefore, when range='V' and as\nlong as lwork is large enough to allow p?hegvx to compute the eigenvalues,\np?hegvx will compute the eigenvalues and as many eigenvectors as it can.\nRelationship between workspace, orthogonality & performance:\nIf clustersize > n/sqrt(NPROW*NPCOL), then providing enough space\nto compute all the eigenvectors orthogonally will cause serious degradation\nin performance. In the limit (that is, clustersize = n-1) p?stein will\nperform no better than ?stein on 1 processor.\nFor clustersize = n/sqrt(NPROW*NPCOL) reorthogonalizing all\neigenvectors will increase the total execution time by a factor of 2 or more.\nFor clustersize>n/sqrt(NPROW*NPCOL) execution time will grow as the\nsquare of the cluster size, all other factors remaining equal and assuming\nenough workspace. Less workspace means less reorthogonalization but\nfaster execution.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1586\n\n\nIf lwork = -1, then lrwork is global input and a workspace query is\nassumed; the function only calculates the size required for optimal\nperformance for all work arrays. Each of these values is returned in the first\nentry of the corresponding work arrays, and no error message is issued by\npxerbla.\niwork\n(local) Workspace array.\nliwork\n(local) , size of iwork.\nliwork ≥ 6*nnp\nWhere: nnp = max(n, NPROW*NPCOL + 1, 4)\nIf liwork = -1, then liwork is global input and a workspace query is\nassumed; the function only calculates the minimum and optimal size for all\nwork arrays. Each of these values is returned in the first entry of the\ncorresponding work array, and no error message is issued by pxerbla.\nOutput Parameters\na\nOn exit, if jobz = 'V', then if info = 0, sub(A) contains the distributed\nmatrix Z of eigenvectors.\nThe eigenvectors are normalized as follows:\nIf ibtype = 1 or 2, then ZH*sub(B)*Z = i;\nIf ibtype = 3, then ZH*inv(sub(B))*Z = i.\nIf jobz = 'N', then on exit the upper triangle (if uplo='U') or the lower\ntriangle (if uplo='L') of sub(A), including the diagonal, is destroyed.\nb\nOn exit, if info ≤ n, the part of sub(B) containing the matrix is overwritten\nby the triangular factor U or L from the Cholesky factorization sub(B) =\nUH*U, or sub(B) = L*LH.\nm\n(global) The total number of eigenvalues found, 0 ≤ m ≤ n.\nnz\n(global) Total number of eigenvectors computed. 0 < nz < m. The number\nof columns of z that are filled.\nIf jobz ≠ 'V', nz is not referenced.\nIf jobz = 'V', nz = m unless the user supplies insufficient space and\np?hegvx is not able to detect this before beginning computation. To get all\nthe eigenvectors requested, the user must supply both sufficient space to\nhold the eigenvectors in z (m≤descz[n_ - 1]) and sufficient workspace to\ncompute them. (See lwork below.) The function p?hegvx is always able to\ndetect insufficient space without computation unless range = 'V'.\nw\n(global)\nArray of size n. On normal exit, the first m entries contain the selected\neigenvalues in ascending order.\nz\n(local).\nglobal size n*n, local size lld_z*LOCc(jz+n-1).\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1587\n\n\nIf jobz = 'V', then on normal exit the first m columns of z contain the\northonormal eigenvectors of the matrix corresponding to the selected\neigenvalues. If an eigenvector fails to converge, then that column of z\ncontains the latest approximation to the eigenvector, and the index of the\neigenvector is returned in ifail.\nIf jobz = 'N', then z is not referenced.\nwork\nOn exit, work[0] returns the optimal amount of workspace.\nrwork\nOn exit, rwork[0] contains the amount of workspace required for optimal\nefficiency\nIf jobz='N'rwork[0] = optimal amount of workspace required to compute\neigenvalues efficiently\nIf jobz='V'rwork[0] = optimal amount of workspace required to compute\neigenvalues and eigenvectors efficiently with no guarantee on orthogonality.\nIf range='V', it is assumed that all eigenvectors may be required when\ncomputing optimal workspace.\nifail\n(global)\nArray of size n.\nifail provides additional information when info≠0\nIf (mod(info/16,2)≠0), then ifail[0] indicates the order of the\nsmallest minor which is not positive definite.\nIf (mod(info,2)≠0) on exit, then ifail[0] contains the indices of the\neigenvectors that failed to converge.\nIf neither of the above error conditions are held, and jobz = 'V', then the\nfirst m elements of ifail are set to zero.\niclustr\n(global)\nArray of size (2*NPROW*NPCOL). This array contains indices of eigenvectors\ncorresponding to a cluster of eigenvalues that could not be reorthogonalized\ndue to insufficient workspace (see lwork, orfac and info). Eigenvectors\ncorresponding to clusters of eigenvalues indexed iclustr(2*i-1) to\niclustr(2*i), could not be reorthogonalized due to lack of workspace.\nHence the eigenvectors corresponding to these clusters may not be\northogonal.\niclustr() is a zero terminated array. (iclustr(2*k)\n≠0.and.clustr(2*k+1)=0) if and only if k is the number of clusters.\niclustr is not referenced if jobz = 'N'.\ngap\n(global)\nArray of size NPROW*NPCOL.\nThis array contains the gap between eigenvalues whose eigenvectors could\nnot be reorthogonalized. The output values in this array correspond to the\nclusters indicated by the array iclustr. As a result, the dot product between\neigenvectors corresponding to the i-th cluster may be as high as (C*n)/\ngap(i), where C is a small constant.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1588\n\n\ninfo\n(global)\nIf info = 0, the execution is successful.\nIf info <0: the i-th argument is an array and the j-entry had an illegal\nvalue, then info = -(i*100+j), if the i-th argument is a scalar and had\nan illegal value, then info = -i.\nIf info> 0:\nIf (mod(info,2)≠0), then one or more eigenvectors failed to converge.\nTheir indices are stored in ifail.\nIf (mod(info,2,2)≠0), then eigenvectors corresponding to one or more\nclusters of eigenvalues could not be reorthogonalized because of insufficient\nworkspace. The indices of the clusters are stored in the array iclustr.\nIf (mod(info/4,2)≠0), then space limit prevented p?sygvx from\ncomputing all of the eigenvectors between vl and vu. The number of\neigenvectors computed is returned in nz.\nIf (mod(info/8,2)≠0), then p?stebz failed to compute eigenvalues.\nIf (mod(info/16,2)≠0), then B was not positive definite. ifail(1) indicates\nthe order of the smallest minor which is not positive definite.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\nScaLAPACK Auxiliary Routines\nScaLAPACK Auxiliary Routines\nRoutine Name\nData\nTypes\nDescription\np?lacgv\nc,z\nConjugates a complex vector.\np?max1\nc,z\nFinds the index of the element whose real part has maximum\nabsolute value (similar to the Level 1 PBLAS p?amax, but using the\nabsolute value to the real part).\npmpcol\ns,d\nFinds the collaborators of a process.\npmpim2\ns,d\nComputes the eigenpair range assignments for all processes.\n?combamax1\nc,z\nFinds the element with maximum real part absolute value and its\ncorresponding global index.\np?sum1\nsc,dz\nForms the 1-norm of a complex vector similar to Level 1 PBLAS\np?asum, but using the true absolute value.\np?dbtrsv\ns,d,c,z\nComputes an LU factorization of a general tridiagonal matrix with\nno pivoting. The routine is called by p?dbtrs.\np?dttrsv\ns,d,c,z\nComputes an LU factorization of a general band matrix, using\npartial pivoting with row interchanges. The routine is called by\np?dttrs.\np?gebal\ns,d\nBalances a general real/complex matrix.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1589\n\n\nRoutine Name\nData\nTypes\nDescription\np?gebd2\ns,d,c,z\nReduces a general rectangular matrix to real bidiagonal form by an\northogonal/unitary transformation (unblocked algorithm).\np?gehd2\ns,d,c,z\nReduces a general matrix to upper Hessenberg form by an\northogonal/unitary similarity transformation (unblocked algorithm).\np?gelq2\ns,d,c,z\nComputes an LQ factorization of a general rectangular matrix\n(unblocked algorithm).\np?geql2\ns,d,c,z\nComputes a QL factorization of a general rectangular matrix\n(unblocked algorithm).\np?geqr2\ns,d,c,z\nComputes a QR factorization of a general rectangular matrix\n(unblocked algorithm).\np?gerq2\ns,d,c,z\nComputes an RQ factorization of a general rectangular matrix\n(unblocked algorithm).\np?getf2\ns,d,c,z\nComputes an LU factorization of a general matrix, using partial\npivoting with row interchanges (local blocked algorithm).\np?labrd\ns,d,c,z\nReduces the first nb rows and columns of a general rectangular\nmatrix A to real bidiagonal form by an orthogonal/unitary\ntransformation, and returns auxiliary matrices that are needed to\napply the transformation to the unreduced part of A.\np?lacon\ns,d,c,z\nEstimates the 1-norm of a square matrix, using the reverse\ncommunication for evaluating matrix-vector products.\np?laconsb\ns,d\nLooks for two consecutive small subdiagonal elements.\np?lacp2\ns,d,c,z\nCopies all or part of a distributed matrix to another distributed\nmatrix.\np?lacp3\ns,d\nCopies from a global parallel array into a local replicated array or\nvice versa.\np?lacpy\ns,d,c,z\nCopies all or part of one two-dimensional array to another.\np?laevswp\ns,d,c,z\nMoves the eigenvectors from where they are computed to\nScaLAPACK standard block cyclic array.\np?lahrd\ns,d,c,z\nReduces the first nb columns of a general rectangular matrix A so\nthat elements below the kth subdiagonal are zero, by an\northogonal/unitary transformation, and returns auxiliary matrices\nthat are needed to apply the transformation to the unreduced part\nof A.\np?laiect\ns,d,c,z\nExploits IEEE arithmetic to accelerate the computations of\neigenvalues.\np?lamve\ns, d\nCopies all or part of one two-dimensional distributed array to\nanother.\np?lange\ns,d,c,z\nReturns the value of the 1-norm, Frobenius norm, infinity-norm, or\nthe largest absolute value of any element, of a general rectangular\nmatrix.\np?lanhs\ns,d,c,z\nReturns the value of the 1-norm, Frobenius norm, infinity-norm, or\nthe largest absolute value of any element, of an upper Hessenberg\nmatrix.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1590\n\n\nRoutine Name\nData\nTypes\nDescription\np?lansy, p?lanhe\ns,d,c,z/c\n,z\nReturns the value of the 1-norm, Frobenius norm, infinity-norm, or\nthe largest absolute value of any element of a real symmetric or\ncomplex Hermitian matrix.\np?lantr\ns,d,c,z\nReturns the value of the 1-norm, Frobenius norm, infinity-norm, or\nthe largest absolute value of any element, of a triangular matrix.\np?lapiv\ns,d,c,z\nApplies a permutation matrix to a general distributed matrix,\nresulting in row or column pivoting.\np?laqge\ns,d,c,z\nScales a general rectangular matrix, using row and column scaling\nfactors computed by p?geequ.\np?laqr0\ns,d\nComputes the eigenvalues of a Hessenberg matrix and optionally\nreturns the matrices from the Schur decomposition.\np?laqr1\ns,d\nSets a scalar multiple of the first column of the product of a 2-by-2\nor 3-by-3 matrix and specified shifts.\np?laqr2\ns,d\nPerforms the orthogonal/unitary similarity transformation of a\nHessenberg matrix to detect and deflate fully converged\neigenvalues from a trailing principal submatrix (aggressive early\ndeflation).\np?laqr3\ns,d\nPerforms the orthogonal/unitary similarity transformation of a\nHessenberg matrix to detect and deflate fully converged\neigenvalues from a trailing principal submatrix (aggressive early\ndeflation).\np?laqr5\ns,d\nPerforms a single small-bulge multi-shift QR sweep.\np?laqsy\ns,d,c,z\nScales a symmetric/Hermitian matrix, using scaling factors\ncomputed by p?poequ.\np?lared1d\ns,d\nRedistributes an array assuming that the input array bycol is\ndistributed across rows and that all process columns contain the\nsame copy of bycol.\np?lared2d\ns,d\nRedistributes an array assuming that the input array byrow is\ndistributed across columns and that all process rows contain the\nsame copy of byrow .\np?larf\ns,d,c,z\nApplies an elementary reflector to a general rectangular matrix.\np?larfb\ns,d,c,z\nApplies a block reflector or its transpose/conjugate-transpose to a\ngeneral rectangular matrix.\np?larfc\nc,z\nApplies the conjugate transpose of an elementary reflector to a\ngeneral matrix.\np?larfg\ns,d,c,z\nGenerates an elementary reflector (Householder matrix).\np?larft\ns,d,c,z\nForms the triangular vector T of a block reflector H=I-VTVH\np?larz\ns,d,c,z\nApplies an elementary reflector as returned by p?tzrzf to a\ngeneral matrix.\np?larzb\ns,d,c,z\nApplies a block reflector or its transpose/conjugate-transpose as\nreturned by p?tzrzf to a general matrix.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1591\n\n\nRoutine Name\nData\nTypes\nDescription\np?larzc\nc,z\nApplies (multiplies by) the conjugate transpose of an elementary\nreflector as returned by p?tzrzf to a general matrix.\np?larzt\ns,d,c,z\nForms the triangular factor T of a block reflector H=I-VTVH as\nreturned by p?tzrzf.\np?lascl\ns,d,c,z\nMultiplies a general rectangular matrix by a real scalar defined as\nCto/Cfrom.\np?laset\ns,d,c,z\nInitializes the off-diagonal elements of a matrix to α and the\ndiagonal elements to β.\np?lasmsub\ns,d\nLooks for a small subdiagonal element from the bottom of the\nmatrix that it can safely set to zero.\np?lassq\ns,d,c,z\nUpdates a sum of squares represented in scaled form.\np?laswp\ns,d,c,z\nPerforms a series of row interchanges on a general rectangular\nmatrix.\np?latra\ns,d,c,z\nComputes the trace of a general square distributed matrix.\np?latrd\ns,d,c,z\nReduces the first nb rows and columns of a symmetric/Hermitian\nmatrix A to real tridiagonal form by an orthogonal/unitary similarity\ntransformation.\np?latrz\ns,d,c,z\nReduces an upper trapezoidal matrix to upper triangular form by\nmeans of orthogonal/unitary transformations.\np?lauu2\ns,d,c,z\nComputes the product UUH or LHL, where U and L are upper or\nlower triangular matrices (local unblocked algorithm).\np?lauum\ns,d,c,z\nComputes the product UUH or LHL, where U and L are upper or\nlower triangular matrices.\np?lawil\ns,d\nForms the Wilkinson transform.\np?org2l/p?ung2l\ns,d,c,z\nGenerates all or part of the orthogonal/unitary matrix Q from a QL\nfactorization determined by p?geqlf (unblocked algorithm).\np?org2r/p?ung2r\ns,d,c,z\nGenerates all or part of the orthogonal/unitary matrix Q from a QR\nfactorization determined by p?geqrf (unblocked algorithm).\np?orgl2/p?ungl2\ns,d,c,z\nGenerates all or part of the orthogonal/unitary matrix Q from an LQ\nfactorization determined by p?gelqf (unblocked algorithm).\np?orgr2/p?ungr2\ns,d,c,z\nGenerates all or part of the orthogonal/unitary matrix Q from an\nRQ factorization determined by p?gerqf (unblocked algorithm).\np?orm2l/p?unm2l\ns,d,c,z\nMultiplies a general matrix by the orthogonal/unitary matrix from a\nQL factorization determined by p?geqlf (unblocked algorithm).\np?orm2r/p?unm2r\ns,d,c,z\nMultiplies a general matrix by the orthogonal/unitary matrix from a\nQR factorization determined by p?geqrf (unblocked algorithm).\np?orml2/p?unml2\ns,d,c,z\nMultiplies a general matrix by the orthogonal/unitary matrix from\nan LQ factorization determined by p?gelqf (unblocked algorithm).\np?ormr2/p?unmr2\ns,d,c,z\nMultiplies a general matrix by the orthogonal/unitary matrix from\nan RQ factorization determined by p?gerqf (unblocked algorithm).\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1592\n\n\nRoutine Name\nData\nTypes\nDescription\np?pbtrsv\ns,d,c,z\nSolves a single triangular linear system via frontsolve or backsolve\nwhere the triangular matrix is a factor of a banded matrix\ncomputed by p?pbtrf.\np?pttrsv\ns,d,c,z\nSolves a single triangular linear system via frontsolve or backsolve\nwhere the triangular matrix is a factor of a tridiagonal matrix\ncomputed by p?pttrf.\np?potf2\ns,d,c,z\nComputes the Cholesky factorization of a symmetric/Hermitian\npositive definite matrix (local unblocked algorithm).\np?rot\ns,d\nApplies a planar rotation to two distributed vectors.\np?rscl\ns,d,cs,zd\nMultiplies a vector by the reciprocal of a real scalar.\np?sygs2/p?hegs2\ns,d,c,z\nReduces a symmetric/Hermitian positive-definite generalized\neigenproblem to standard form, using the factorization results\nobtained from p?potrf (local unblocked algorithm).\np?sytd2/p?hetd2\ns,d,c,z\nReduces a symmetric/Hermitian matrix to real symmetric\ntridiagonal form by an orthogonal/unitary similarity transformation\n(local unblocked algorithm).\np?trord\ns,d\nReorders the Schur factorization of a general matrix.\np?trsen\ns,d\nReorders the Schur factorization of a matrix and (optionally)\ncomputes the reciprocal condition numbers and invariant subspace\nfor the selected cluster of eigenvalues.\np?trti2\ns,d,c,z\nComputes the inverse of a triangular matrix (local unblocked\nalgorithm).\n?lamsh\ns,d\nSends multiple shifts through a small (single node) matrix to\nmaximize the number of bulges that can be sent through.\n?laqr6\ns,d\nPerforms a single small-bulge multi-shift QR sweep collecting the\ntransformations.\n?lar1va\ns,d\nComputes scaled eigenvector corresponding to given eigenvalue.\n?laref\ns,d\nApplies Householder reflectors to matrices on either their rows or\ncolumns.\n?larrb2\ns,d\nProvides limited bisection to locate eigenvalues for more accuracy.\n?larrd2\ns,d\nComputes the eigenvalues of a symmetric tridiagonal matrix to\nsuitable accuracy.\n?larre2\ns,d\nGiven a tridiagonal matrix, sets small off-diagonal elements to zero\nand for each unreduced block, finds base representations and\neigenvalues.\n?larre2a\ns,d\nGiven a tridiagonal matrix, sets small off-diagonal elements to zero\nand for each unreduced block, finds base representations and\neigenvalues.\n?larrf2\ns,d\nFinds a new relatively robust representation such that at least one\nof the eigenvalues is relatively isolated.\n?larrv2\ns,d\nComputes the eigenvectors of the tridiagonal matrix T = L*D*LT\ngiven L, D and the eigenvalues of L*D*LT.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1593\n\n\nRoutine Name\nData\nTypes\nDescription\n?lasorte\ns,d\nSorts eigenpairs by real and complex data types.\n?lasrt2\ns,d\nSorts numbers in increasing or decreasing order.\n?stegr2\ns,d\nComputes selected eigenvalues and eigenvectors of a real\nsymmetric tridiagonal matrix.\n?stegr2a\ns,d\nComputes selected eigenvalues and initial representations needed\nfor eigenvector computations.\n?stegr2b\ns,d\nFrom eigenvalues and initial representations computes the selected\neigenvalues and eigenvectors of the real symmetric tridiagonal\nmatrix in parallel on multiple processors.\n?stein2\ns,d\nComputes the eigenvectors corresponding to specified eigenvalues\nof a real symmetric tridiagonal matrix, using inverse iteration.\n?dbtf2\ns,d,c,z\nComputes an LU factorization of a general band matrix with no\npivoting (local unblocked algorithm).\n?dbtrf\ns,d,c,z\nComputes an LU factorization of a general band matrix with no\npivoting (local blocked algorithm).\n?dttrf\ns,d,c,z\nComputes an LU factorization of a general tridiagonal matrix with\nno pivoting (local blocked algorithm).\n?dttrsv\ns,d,c,z\nSolves a general tridiagonal system of linear equations using the LU\nfactorization computed by ?dttrf.\n?pttrsv\ns,d,c,z\nSolves a symmetric (Hermitian) positive-definite tridiagonal system\nof linear equations, using the LDLH factorization computed\nby ?pttrf.\n?steqr2\ns,d\nComputes all eigenvalues and, optionally, eigenvectors of a\nsymmetric tridiagonal matrix using the implicit QL or QR method.\n?trmvt\ns,d,c,z\nPerforms matrix-vector operations.\npilaenv\nNA\nReturns the positive integer value of the logical blocking size.\npilaenvx\nNA\nCalled from the ScaLAPACK routines to choose problem-dependent\nparameters for the local environment.\npjlaenv\nNA\nCalled from the ScaLAPACK symmetric and Hermitian tailored\neigen-routines to choose problem-dependent parameters for the\nlocal environment.\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\np?lacgv\nConjugates a complex vector.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1594\n\n\nSyntax\nvoid pclacgv (MKL_INT *n , MKL_Complex8 *x , MKL_INT *ix , MKL_INT *jx , MKL_INT\n*descx , MKL_INT *incx );\nvoid pzlacgv (MKL_INT *n , MKL_Complex16 *x , MKL_INT *ix , MKL_INT *jx , MKL_INT\n*descx , MKL_INT *incx );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?lacgvfunction conjugates a complex vector sub(X) of length n, where sub(X) denotes X(ix, jx:jx\n+n-1) if incx = m_x, and X(ix:ix+n-1, jx) if incx = 1.\nInput Parameters\nn\n(global) The length of the distributed vector sub(X).\nx\n(local).\nPointer into the local memory to an array of size lld_x * LOCc(n_x). On\nentry the vector to be conjugated x[i] = X(ix+(jx-1)*m_x+i*incx), 0\n≤ i < n.\nix\n(global) The row index in the global matrix X indicating the first row of\nsub(X).\njx\n(global) The column index in the global matrix X indicating the first column\nof sub(X).\ndescx\n(global and local) Array of size dlen_=9. The array descriptor for the\ndistributed matrix X.\nincx\n(global) The global increment for the elements of X. Only two values of\nincx are supported in this version, namely 1 and m_x. incx must not be\nzero.\nOutput Parameters\nx\n(local).\nOn exit, the local pieces of conjugated distributed vector sub(X).\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?max1\nFinds the index of the element whose real part has\nmaximum absolute value (similar to the Level 1 PBLAS\np?amax, but using the absolute value to the real part).\nSyntax\nvoid pcmax1 (MKL_INT *n , MKL_Complex8 *amax , MKL_INT *indx , MKL_Complex8 *x ,\nMKL_INT *ix , MKL_INT *jx , MKL_INT *descx , MKL_INT *incx );\nvoid pzmax1 (MKL_INT *n , MKL_Complex16 *amax , MKL_INT *indx , MKL_Complex16 *x ,\nMKL_INT *ix , MKL_INT *jx , MKL_INT *descx , MKL_INT *incx );\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1595\n\n\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?max1function computes the global index of the maximum element in absolute value of a distributed\nvector sub(X). The global index is returned in indx and the value is returned in amax, where sub(X) denotes\nX(ix:ix+n-1, jx) if incx = 1, X(ix, jx:jx+n-1) if incx = m_x.\nInput Parameters\nn\n(global). The number of components of the distributed vector sub(X). n ≥ 0.\nx\n(local)\nPointer into the local memory to an array of size lld_x * LOCc(jx+n-1). On\nentry this array contains the local pieces of the distributed vector sub(X).\nix\n(global) The row index in the global matrix X indicating the first row of\nsub(X).\njx\n(global) The column index in the global matrix X indicating the first column\nof sub(X).\ndescx\n(global and local) Array of size dlen_. The array descriptor for the\ndistributed matrix X.\nincx\n(global).The global increment for the elements of X. Only two values of incx\nare supported in this version, namely 1 and m_x. incx must not be zero.\nOutput Parameters\namax\n(global output).The absolute value of the largest entry of the distributed\nvector sub(X) only in the scope of sub(X).\nindx\n(global output).The global index of the element of the distributed vector\nsub(X) whose real part has maximum absolute value.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\npilaver\nReturns the ScaLAPACK version.\nSyntax\nvoid pilaver (MKL_INT* vers_major, MKL_INT* vers_minor, MKL_INT* vers_patch);\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThis function returns the ScaLAPACK version.\nOutput Parameters\nvers_major\nReturn the ScaLAPACK major version.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1596\n\n\nvers_minor\nReturn the ScaLAPACK minor version from the major version.\nvers_patch\nReturn the ScaLAPACK patch version from the minor version.\npmpcol\nFinds the collaborators of a process.\nSyntax\nvoid pmpcol(MKL_INT* myproc, MKL_INT* nprocs, MKL_INT* iil, MKL_INT* needil, MKL_INT*\nneediu, MKL_INT* pmyils, MKL_INT* pmyius, MKL_INT* colbrt, MKL_INT* frstcl, MKL_INT*\nlastcl);\nInclude Files\n•\nmkl_scalapack.h\nDescription\nUsing the output from pmpim2 and given the information on eigenvalue clusters, pmpcol finds the\ncollaborators of myproc.\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nInput Parameters\nmyproc\nThe processor number, 0 ≤myproc < nprocs.\nnprocs\nThe total number of processors available.\niil\nThe index of the leftmost eigenvalue in the eigenvalue cluster.\nneedil\nThe leftmost position in the eigenvalue cluster needed by myproc.\nneediu\nThe rightmost position in the eigenvalue cluster needed by myproc.\npmyils\narray\nFor each processor p, 0 < p≤nprocs, pmyils[p-1] is the index of the first\neigenvalue in the eigenvalue cluster to be computed.\npmyils[p-1] equals zero if p stays idle.\npmyius\narray\nFor each processor p, pmyius[p-1] is the index of the last eigenvalue in the\neigenvalue cluster to be computed.\npmyius[p-1] equals zero if p stays idle.\nOUTPUT Parameters\ncolbrt\nNon-zero if myproc collaborates.\nfrstcl, lastcl\nFirst and last collaborator of myproc .\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1597\n\n\nmyproc collaborates with:\nfrstcl, ..., myproc-1, myproc+1, ...,lastcl\nIf myproc = frstcl, there are no collaborators on the left. If myproc =\nlastcl, there are no collaborators on the right.\nIf frstcl = 0 and lastcl = nprocs-1, then myproc collaborates with\neverybody\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\npmpim2\nComputes the eigenpair range assignments for all\nprocesses.\nSyntax\nvoid pmpim2(MKL_INT* il, MKL_INT* iu, MKL_INT* nprocs, MKL_INT* pmyils, MKL_INT*\npmyius);\nInclude Files\n•\nmkl_scalapack.h\nDescription\npmpim2 is the scheduling function. It computes for all processors the eigenpair range assignments.\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nInput Parameters\nil, iu\nThe range of eigenpairs to be computed.\nnprocs\nThe total number of processors available.\nOutput Parameters\npmyils\narray\nFor each processor p, pmyils[p-1] is the index of the first eigenvalue\nin a cluster to be computed.\npmyils[p-1] equals zero if p stays idle.\npmyius\narray\nFor each processor p, pmyius[p-1] is the index of the last eigenvalue\nin a cluster to be computed.\npmyius[p-1] equals zero if p stays idle.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1598\n\n\n?combamax1\nFinds the element with maximum real part absolute\nvalue and its corresponding global index.\nSyntax\nvoid ccombamax1 (MKL_Complex8 *v1 , MKL_Complex8 *v2 );\nvoid zcombamax1 (MKL_Complex16 *v1 , MKL_Complex16 *v2 );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe ?combamax1function finds the element having maximum real part absolute value as well as its\ncorresponding global index.\nInput Parameters\nv1\n(local)\nArray of size 2. The first maximum absolute value element and its global\nindex. v1[0]=amax, v1[1]=indx.\nv2\n(local)\nArray of size 2. The second maximum absolute value element and its global\nindex. v2[0]=amax, v2[1]=indx.\nOutput Parameters\nv1\n(local).\nThe first maximum absolute value element and its global index.\nv1[0]=amax, v1[1]=indx.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?sum1\nForms the 1-norm of a complex vector similar to Level\n1 PBLAS p?asum, but using the true absolute value.\nSyntax\nvoid pscsum1 (MKL_INT *n , float *asum , MKL_Complex8 *x , MKL_INT *ix , MKL_INT *jx ,\nMKL_INT *descx , MKL_INT *incx );\nvoid pdzsum1 (MKL_INT *n , double *asum , MKL_Complex16 *x , MKL_INT *ix , MKL_INT\n*jx , MKL_INT *descx , MKL_INT *incx );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?sum1function returns the sum of absolute values of a complex distributed vector sub(x) in asum,\nwhere sub(x) denotes X(ix:ix+n-1, jx:jx), if incx = 1, X(ix:ix, jx:jx+n-1), if incx = m_x.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1599\n\n\nBased on p?asum from the Level 1 PBLAS. The change is to use the 'genuine' absolute value.\nInput Parameters\nn\n(global). The number of components of the distributed vector sub(x). n ≥ 0.\nx\n(local )\nPointer into the local memory to an array of size lld_x * LOCc(jx+n-1). This\narray contains the local pieces of the distributed vector sub(X).\nix\n(global) The row index in the global matrix X indicating the first row of\nsub(X).\njx\n(global) The column index in the global matrix X indicating the first column\nof sub(X)\ndescx\n(local) Array of size dlen_=9. The array descriptor for the distributed matrix\nX.\nincx\n(global) The global increment for the elements of X. Only two values of\nincx are supported in this version, namely 1 and m_x.\nOutput Parameters\nasum\n(local)\nThe sum of absolute values of the distributed vector sub(X) only in its\nscope.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?dbtrsv\nComputes an LU factorization of a general triangular\nmatrix with no pivoting. The function is called by\np?dbtrs.\nSyntax\nvoid psdbtrsv (char *uplo , char *trans , MKL_INT *n , MKL_INT *bwl , MKL_INT *bwu ,\nMKL_INT *nrhs , float *a , MKL_INT *ja , MKL_INT *desca , float *b , MKL_INT *ib ,\nMKL_INT *descb , float *af , MKL_INT *laf , float *work , MKL_INT *lwork , MKL_INT\n*info );\nvoid pddbtrsv (char *uplo , char *trans , MKL_INT *n , MKL_INT *bwl , MKL_INT *bwu ,\nMKL_INT *nrhs , double *a , MKL_INT *ja , MKL_INT *desca , double *b , MKL_INT *ib ,\nMKL_INT *descb , double *af , MKL_INT *laf , double *work , MKL_INT *lwork , MKL_INT\n*info );\nvoid pcdbtrsv (char *uplo , char *trans , MKL_INT *n , MKL_INT *bwl , MKL_INT *bwu ,\nMKL_INT *nrhs , MKL_Complex8 *a , MKL_INT *ja , MKL_INT *desca , MKL_Complex8 *b ,\nMKL_INT *ib , MKL_INT *descb , MKL_Complex8 *af , MKL_INT *laf , MKL_Complex8 *work ,\nMKL_INT *lwork , MKL_INT *info );\nvoid pzdbtrsv (char *uplo , char *trans , MKL_INT *n , MKL_INT *bwl , MKL_INT *bwu ,\nMKL_INT *nrhs , MKL_Complex16 *a , MKL_INT *ja , MKL_INT *desca , MKL_Complex16 *b ,\nMKL_INT *ib , MKL_INT *descb , MKL_Complex16 *af , MKL_INT *laf , MKL_Complex16 *work ,\nMKL_INT *lwork , MKL_INT *info );\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1600\n\n\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?dbtrsvfunction solves a banded triangular system of linear equations\nA(1 :n, ja:ja+n-1) * X = B(ib:ib+n-1, 1 :nrhs) or\nA(1 :n, ja:ja+n-1)T * X = B(ib:ib+n-1, 1 :nrhs) (for real flavors); A(1 :n, ja:ja+n-1)H* X = B(ib:ib+n-1,\n1 :nrhs) (for complex flavors),\nwhere A(1 :n, ja:ja+n-1) is a banded triangular matrix factor produced by the Gaussian elimination code of \np?dbtrf and is stored in A(1 :n, ja:ja+n-1) and af. The matrix stored in A(1 :n, ja:ja+n-1) is either\nupper or lower triangular according to uplo, and the choice of solving A(1 :n, ja:ja+n-1) or A(1 :n, ja:ja\n+n-1)T is dictated by the user by the parameter trans.\nThe function p?dbtrf must be called first.\nInput Parameters\nuplo\n(global)\nIf uplo='U', the upper triangle of A(1:n, ja:ja+n-1) is stored,\nif uplo = 'L', the lower triangle of A(1:n, ja:ja+n-1) is stored.\ntrans\n(global)\nIf trans = 'N', solve with A(1:n, ja:ja+n-1),\nif trans = 'C', solve with conjugate transpose A(1:n, ja:ja+n-1).\nn\n(global) The order of the distributed submatrix A;(n≥ 0).\nbwl\n(global) Number of subdiagonals. 0 ≤ bwl ≤ n-1.\nbwu\n(global) Number of subdiagonals. 0 ≤ bwu ≤ n-1.\nnrhs\n(global) The number of right-hand sides; the number of columns of the\ndistributed submatrix B (nrhs≥ 0).\na\n(local).\nPointer into the local memory to an array of size lld_a * LOCc(ja+n-1),\nwhere lld_a≥(bwl+bwu+1). On entry, this array contains the local pieces of\nthe n-by-n unsymmetric banded distributed Cholesky factor L or LT,\nrepresented in global A as A(1 :n, ja:ja+n-1). This local portion is stored\nin the packed banded format used in LAPACK. See the Application Notes\nbelow and the ScaLAPACK manual for more detail on the format of\ndistributed matrices.\nja\n(global) The index in the global matrix A that points to the start of the\nmatrix to be operated on (which may be either all of A or a submatrix of A).\ndesca\n(global and local) array of size dlen_.\nif 1d type (dtype_a = 501 or 502), dlen≥ 7;\nif 2d type (dtype_a = 1), dlen≥ 9. The array descriptor for the distributed\nmatrix A. Contains information of mapping of A to memory.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1601\n\n\nb\n(local)\nPointer into the local memory to an array of local lead dimension lld_b≥nb.\nOn entry, this array contains the local pieces of the right-hand sides\nB(ib:ib+n-1, 1:nrhs).\nib\n(global) The row index in the global matrix B that points to the first row of\nthe matrix to be operated on (which may be either all of B or a submatrix of\nB).\ndescb\n(global and local) array of size dlen_.\nif 1d type (dtype_b =502), dlen≥7;\nif 2d type (dtype_b =1), dlen≥9. The array descriptor for the distributed\nmatrix B. Contains information of mapping B to memory.\nlaf\n(local)\nSize of user-input auxiliary fill-in space af.\nlaf≥nb*(bwl+bwu)+6*max(bwl, bwu)*max(bwl, bwu). If laf is not\nlarge enough, an error code is returned and the minimum acceptable size\nwill be returned in af[0].\nwork\n(local).\nTemporary workspace. This space may be overwritten in between function\ncalls.\nwork must be the size given in lwork.\nlwork\n(local or global)\nSize of user-input workspace work. If lwork is too small, the minimal\nacceptable size will be returned in work[0] and an error code is returned.\nlwork≥ max(bwl, bwu)*nrhs.\nOutput Parameters\na\n(local).\nThis local portion is stored in the packed banded format used in LAPACK.\nPlease see the ScaLAPACK manual for more detail on the format of\ndistributed matrices.\nb\nOn exit, this contains the local piece of the solutions distributed matrix X.\naf\n(local).\nauxiliary fill-in space. The fill-in space is created in a call to the factorization\nfunction p?dbtrf and is stored in af. If a linear system is to be solved\nusing p?dbtrf after the factorization function, af must not be altered after\nthe factorization.\nwork\nOn exit, work[0] contains the minimal lwork.\ninfo\n(local).\nIf info = 0, the execution is successful.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1602\n\n\n< 0: If the i-th argument is an array and the j-th entry, indexed j-1, had an\nillegal value, then info= - (i*100+j), if the i-th argument is a scalar and\nhad an illegal value, then info = -i.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?dttrsv\nComputes an LU factorization of a general band\nmatrix, using partial pivoting with row interchanges.\nThe function is called by p?dttrs.\nSyntax\nvoid psdttrsv (char *uplo , char *trans , MKL_INT *n , MKL_INT *nrhs , float *dl , float\n*d , float *du , MKL_INT *ja , MKL_INT *desca , float *b , MKL_INT *ib , MKL_INT\n*descb , float *af , MKL_INT *laf , float *work , MKL_INT *lwork , MKL_INT *info );\nvoid pddttrsv (char *uplo , char *trans , MKL_INT *n , MKL_INT *nrhs , double *dl ,\ndouble *d , double *du , MKL_INT *ja , MKL_INT *desca , double *b , MKL_INT *ib ,\nMKL_INT *descb , double *af , MKL_INT *laf , double *work , MKL_INT *lwork , MKL_INT\n*info );\nvoid pcdttrsv (char *uplo , char *trans , MKL_INT *n , MKL_INT *nrhs , MKL_Complex8\n*dl , MKL_Complex8 *d , MKL_Complex8 *du , MKL_INT *ja , MKL_INT *desca , MKL_Complex8\n*b , MKL_INT *ib , MKL_INT *descb , MKL_Complex8 *af , MKL_INT *laf , MKL_Complex8\n*work , MKL_INT *lwork , MKL_INT *info );\nvoid pzdttrsv (char *uplo , char *trans , MKL_INT *n , MKL_INT *nrhs , MKL_Complex16\n*dl , MKL_Complex16 *d , MKL_Complex16 *du , MKL_INT *ja , MKL_INT *desca ,\nMKL_Complex16 *b , MKL_INT *ib , MKL_INT *descb , MKL_Complex16 *af , MKL_INT *laf ,\nMKL_Complex16 *work , MKL_INT *lwork , MKL_INT *info );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?dttrsvfunction solves a tridiagonal triangular system of linear equations\nA(1 :n, ja:ja+n-1)*X = B(ib:ib+n-1, 1 :nrhs) or\nA(1 :n, ja:ja+n-1)T * X = B(ib:ib+n-1, 1 :nrhs) for real flavors; A(1 :n, ja:ja+n-1)H* X =\nB(ib:ib+n-1, 1 :nrhs) for complex flavors,\nwhere A(1 :n, ja:ja+n-1) is a tridiagonal matrix factor produced by the Gaussian elimination code of \np?dttrf and is stored in A(1 :n, ja:ja+n-1) and af.\nThe matrix stored in A(1 :n, ja:ja+n-1) is either upper or lower triangular according to uplo, and the\nchoice of solving A(1 :n, ja:ja+n-1) or A(1 :n, ja:ja+n-1)T is dictated by the user by the parameter\ntrans.\nThe function p?dttrf must be called first.\nInput Parameters\nuplo\n(global)\nIf uplo='U', the upper triangle of A(1:n, ja:ja+n-1) is stored,\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1603\n\n\nif uplo = 'L', the lower triangle of A(1:n, ja:ja+n-1) is stored.\ntrans\n(global)\nIf trans = 'N', solve with A(1:n, ja:ja+n-1),\nif trans = 'C', solve with conjugate transpose A(1:n, ja:ja+n-1).\nn\n(global) The order of the distributed submatrix A;(n≥ 0).\nnrhs\n(global) The number of right-hand sides; the number of columns of the\ndistributed submatrix B(ib:ib+n-1, 1:nrhs). (nrhs≥ 0).\ndl\n(local).\nPointer to local part of global vector storing the lower diagonal of the\nmatrix.\nGlobally, dl[0] is not referenced, and dl must be aligned with d.\nMust be of size ≥nb_a.\nd\n(local).\nPointer to local part of global vector storing the main diagonal of the matrix.\ndu\n(local).\nPointer to local part of global vector storing the upper diagonal of the\nmatrix.\nGlobally, du[n-1] is not referenced, and du must be aligned with d.\nja\n(global) The index in the global matrix A that points to the start of the\nmatrix to be operated on (which may be either all of A or a submatrix of A).\ndesca\n(global and local) array of size dlen_.\nif 1d type (dtype_a = 501 or 502), dlen≥ 7;\nif 2d type (dtype_a = 1), dlen≥ 9.\nThe array descriptor for the distributed matrix A. Contains information of\nmapping of A to memory.\nb\n(local)\nPointer into the local memory to an array of local lead dimension lld_b≥nb.\nOn entry, this array contains the local pieces of the right-hand sides\nB(ib:ib+n-1, 1 :nrhs).\nib\n(global) The row index in the global matrix B that points to the first row of\nthe matrix to be operated on (which may be either all of B or a submatrix of\nB).\ndescb\n(global and local) array of size dlen_.\nif 1d type (dtype_b = 502), dlen≥7;\nif 2d type (dtype_b = 1), dlen≥ 9.\nThe array descriptor for the distributed matrix B. Contains information of\nmapping B to memory.\nlaf\n(local).\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1604\n\n\nSize of user-input auxiliary fill-in space af.\nlaf≥ 2*(nb+2). If laf is not large enough, an error code is returned and\nthe minimum acceptable size will be returned in af[0].\nwork\n(local).\nTemporary workspace. This space may be overwritten in between function\ncalls.\nwork must be the size given in lwork.\nlwork\n(local or global)\nSize of user-input workspace work. If lwork is too small, the minimal\nacceptable size will be returned in work[0] and an error code is returned.\nlwork≥ 10*npcol+4*nrhs.\nOutput Parameters\ndl\n(local).\nOn exit, this array contains information containing the factors of the matrix.\nd\nOn exit, this array contains information containing the factors of the matrix.\nMust be of size ≥nb_a.\nb\nOn exit, this contains the local piece of the solutions distributed matrix X.\naf\n(local).\nAuxiliary fill-in space. The fill-in space is created in a call to the factorization\nfunction p?dttrf and is stored in af. If a linear system is to be solved\nusing p?dttrs after the factorization function, af must not be altered after\nthe factorization.\nwork\nOn exit, work[0] contains the minimal lwork.\ninfo\n(local).\nIf info=0, the execution is successful.\nif info< 0: If the i-th argument is an array and the j-th entry,\nindexed j-1, had an illegal value, then info = - (i*100+j), if the i-th\nargument is a scalar and had an illegal value, then info = -i.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?gebal\nBalances a general real/complex matrix.\nSyntax\nvoid psgebal(char* job, MKL_INT* n, float* a, MKL_INT* desca, MKL_INT* ilo, MKL_INT*\nihi, float* scale, MKL_INT* info);\nvoid pdgebal(char* job, MKL_INT* n, double* a, MKL_INT* desca, MKL_INT* ilo, MKL_INT*\nihi, double* scale, MKL_INT* info);\nvoid pcgebal(char* job, MKL_INT* n, complex float* a, MKL_INT* desca, MKL_INT* ilo,\nMKL_INT* ihi, float* scale, MKL_INT* info);\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1605\n\n\nvoid pzgebal(char* job, MKL_INT* n, complex double* a, MKL_INT* desca, MKL_INT* ilo,\nMKL_INT* ihi, double* scale, MKL_INT* info);\nInclude Files\n•\nmkl_scalapack.h\nDescription\np?gebal balances a general real/complex matrix A. This involves, first, permuting A by a similarity\ntransformation to isolate eigenvalues in the first 1 to ilo-1 and last ihi+1 to n elements on the diagonal;\nand second, applying a diagonal similarity transformation to rows and columns ilo to ihi to make the rows\nand columns as close in norm as possible. Both steps are optional.\nBalancing may reduce the 1-norm of the matrix, and improve the accuracy of the computed eigenvalues\nand/or eigenvectors.\nInput Parameters\njob\n(global )\nSpecifies the operations to be performed on a:\n= 'N': none: simply set ilo = 1, ihi = n, scale[i] = 1.0 for i = 0,...,n-1;\n= 'P': permute only;\n= 'S': scale only;\n= 'B': both permute and scale.\nn\n(global )\nThe order of the matrix A (n≥ 0).\na\n(local ) Pointer into the local memory to an array of size lld_a * LOCc(n)\nThis array contains the local pieces of global input matrix A.\ndesca\n(global and local) array of size dlen_.\nThe array descriptor for the distributed matrix A.\nOUTPUT Parameters\na\nOn exit, a is overwritten by the balanced matrixA.\nIf job = 'N', a is not referenced.\nSee Notes for further details.\nilo, ihi\n(global )\nilo and ihi are set to integers such that on exit matrix elements A(i,j) are\nzero if i > j and j = 1,...,ilo-1 or i = ihi+1,...,n.\nIf job = 'N' or 'S', ilo = 1 and ihi = n.\nscale\n(global ) array of size n.\nDetails of the permutations and scaling factors applied to a. If pj is the\nindex of the row and column interchanged with row and column j and dj is\nthe scaling factor applied to row and column j, then\nscale[j-1] = pj for j = 1,...,ilo-1, ihi+1,..., n\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1606\n\n\nscale[j-1] = dj for j = ilo,...,ihi\nThe order in which the interchanges are made is n to ihi+1, then 1 to\nilo-1.\ninfo\n(global )\n= 0: successful exit.\n< 0: if info = -i, the i-th argument had an illegal value.\nApplication Notes\nThe permutations consist of row and column interchanges which put the matrix in the form\nwhere T1 and T2 are upper triangular matrices whose eigenvalues lie along the diagonal. The column indices\nilo and ihi mark the starting and ending columns of the submatrix B. Balancing consists of applying a\ndiagonal similarity transformation D-1BD to make the 1-norms of each row of B and its corresponding column\nnearly equal. The output matrix is\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1607\n\n\nInformation about the permutations P and the diagonal matrix D is returned in the vector scale.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?gebd2\nReduces a general rectangular matrix to real\nbidiagonal form by an orthogonal/unitary\ntransformation (unblocked algorithm).\nSyntax\nvoid psgebd2 (MKL_INT *m , MKL_INT *n , float *a , MKL_INT *ia , MKL_INT *ja , MKL_INT\n*desca , float *d , float *e , float *tauq , float *taup , float *work , MKL_INT\n*lwork , MKL_INT *info );\nvoid pdgebd2 (MKL_INT *m , MKL_INT *n , double *a , MKL_INT *ia , MKL_INT *ja , MKL_INT\n*desca , double *d , double *e , double *tauq , double *taup , double *work , MKL_INT\n*lwork , MKL_INT *info );\nvoid pcgebd2 (MKL_INT *m , MKL_INT *n , MKL_Complex8 *a , MKL_INT *ia , MKL_INT *ja ,\nMKL_INT *desca , float *d , float *e , MKL_Complex8 *tauq , MKL_Complex8 *taup ,\nMKL_Complex8 *work , MKL_INT *lwork , MKL_INT *info );\nvoid pzgebd2 (MKL_INT *m , MKL_INT *n , MKL_Complex16 *a , MKL_INT *ia , MKL_INT *ja ,\nMKL_INT *desca , double *d , double *e , MKL_Complex16 *tauq , MKL_Complex16 *taup ,\nMKL_Complex16 *work , MKL_INT *lwork , MKL_INT *info );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?gebd2function reduces a real/complex general m-by-n distributed matrix sub(A) = A(ia:ia+m-1,\nja:ja+n-1) to upper or lower bidiagonal form B by an orthogonal/unitary transformation:\nQ'*sub(A)*P = B.\nIf m ≥ n, B is the upper bidiagonal; if m<n, B is the lower bidiagonal.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1608\n\n\nInput Parameters\nm\n(global)\nThe number of rows of the distributed matrix sub(A). (m≥0).\nn\n(global)\nThe number of columns in the distributed matrix sub(A). (n≥0).\na\n(local).\nPointer into the local memory to an array of sizelld_a * LOCc(ja+n-1).\nOn entry, this array contains the local pieces of the general distributed\nmatrix sub(A).\nia, ja\n(global) The row and column indices in the global matrix A indicating the\nfirst row and the first column of the matrix sub(A), respectively.\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\nwork\n(local).\nThis is a workspace array of size lwork.\nlwork\n(local or global)\nThe size of the array work.\nlwork is local input and must be at least lwork≥ max(mpa0, nqa0),\nwhere nb = mb_a = nb_a, iroffa = mod(ia-1, nb),\niarow = indxg2p(ia, nb, myrow, rsrc_a, nprow),\niacol = indxg2p(ja, nb, mycol, csrc_a, npcol),\nmpa0 = numroc(m+iroffa, nb, myrow, iarow, nprow),\nnqa0 = numroc(n+icoffa, nb, mycol, iacol, npcol).\nindxg2p and numroc are ScaLAPACK tool functions; myrow, mycol, nprow,\nand npcol can be determined by calling the function blacs_gridinfo.\nIf lwork = -1, then lwork is global input and a workspace query is\nassumed; the function only calculates the minimum and optimal size for all\nwork arrays. Each of these values is returned in the first entry of the\ncorresponding work array, and no error message is issued by pxerbla.\nOutput Parameters\na\n(local).\nOn exit, if m ≥ n, the diagonal and the first superdiagonal of sub(A) are\noverwritten with the upper bidiagonal matrix B; the elements below the\ndiagonal, with the array tauq, represent the orthogonal/unitary matrix Q as\na product of elementary reflectors, and the elements above the first\nsuperdiagonal, with the array taup, represent the orthogonal matrix P as a\nproduct of elementary reflectors. If m < n, the diagonal and the first\nsubdiagonal are overwritten with the lower bidiagonal matrix B; the\nelements below the first subdiagonal, with the array tauq, represent the\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1609\n\n\northogonal/unitary matrix Q as a product of elementary reflectors, and the\nelements above the diagonal, with the array taup, represent the orthogonal\nmatrix P as a product of elementary reflectors. See Applications Notes\nbelow.\nd\n(local)\nArray of size LOCc(ja+min(m,n)-1) if m ≥ n; LOCr(ia+min(m,n)-1)\notherwise. The distributed diagonal elements of the bidiagonal matrix B: \nd[i] = A(i+1,i+1), i=0, 1,..., size (d) - 1 . d is tied to the distributed matrix\nA.\ne\n(local)\nArray of size LOCc(ja+min(m,n)-1) if m≥ n; LOCr(ia+min(m,n)-2)\notherwise. The distributed diagonal elements of the bidiagonal matrix B:\nif m ≥ n, e[i] = A(i+1,i+2) for i = 0, 1, ... , n-2;\nif m < n, e[i] = A(i+2,i+1) for i = 0, 1, ..., m-2. e is tied to the distributed\nmatrix A.\ntauq\n(local).\nArray of size LOCc(ja+min(m,n)-1). The scalar factors of the elementary\nreflectors which represent the orthogonal/unitary matrix Q. tauq is tied to\nthe distributed matrix A.\ntaup\n(local).\nArray of size LOCr(ia+min(m,n)-1). The scalar factors of the elementary\nreflectors which represent the orthogonal/unitary matrix P. taup is tied to\nthe distributed matrix A.\nwork\nOn exit, work[0] returns the minimal and optimal lwork.\ninfo\n(local)\nIf info = 0, the execution is successful.\nif info < 0: If the i-th argument is an array and the j-th entry, indexed\nj-1, had an illegal value, then info = - (i*100+j), if the i-th argument is a\nscalar and had an illegal value, then info = -i.\nApplication Notes\nThe matrices Q and P are represented as products of elementary reflectors:\nIf m≥n,\nQ = H(1)*H(2)*...*H(n), and P = G(1)*G(2)*...*G(n-1)\nEach H(i) and G(i) has the form:\nH(i) = I - tauq*v*v', and G(i) = I - taup*u*u',\nwhere tauq and taup are real/complex scalars, and v and u are real/complex vectors. v(1: i-1) = 0, v(i)\n= 1, and v(i+i:m) is stored on exit in\nA(ia+i-ia+m-1, ja+i-1);\nu(1:i) = 0, u(i+1) = 1, and u(i+2:n) is stored on exit in A(ia+i-1, ja+i+1:ja+n-1);\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1610\n\n\ntauq is stored in tauq[ja+i-2] and taup in taup[ia+i-2].\nIf m < n,\nv(1: i) = 0, v(i+1) = 1, and v(i+2:m) is stored on exit in A(ia+i+1: ia+m-1, ja+i-1);\nu(1: i-1) = 0, u(i) = 1, and u(i+1 :n) is stored on exit in A(ia+i-1,ja+i:ja+n-1);\ntauq is stored in tauq[ja+i-2] and taup in taup[ia+i-2].\nThe contents of sub(A) on exit are illustrated by the following examples:\nwhere d and e denote diagonal and off-diagonal elements of B, vi denotes an element of the vector defining\nH(i), and ui an element of the vector defining G(i).\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?gehd2\nReduces a general matrix to upper Hessenberg form\nby an orthogonal/unitary similarity transformation\n(unblocked algorithm).\nSyntax\nvoid psgehd2 (MKL_INT *n , MKL_INT *ilo , MKL_INT *ihi , float *a , MKL_INT *ia ,\nMKL_INT *ja , MKL_INT *desca , float *tau , float *work , MKL_INT *lwork , MKL_INT\n*info );\nvoid pdgehd2 (MKL_INT *n , MKL_INT *ilo , MKL_INT *ihi , double *a , MKL_INT *ia ,\nMKL_INT *ja , MKL_INT *desca , double *tau , double *work , MKL_INT *lwork , MKL_INT\n*info );\nvoid pcgehd2 (MKL_INT *n , MKL_INT *ilo , MKL_INT *ihi , MKL_Complex8 *a , MKL_INT\n*ia , MKL_INT *ja , MKL_INT *desca , MKL_Complex8 *tau , MKL_Complex8 *work , MKL_INT\n*lwork , MKL_INT *info );\nvoid pzgehd2 (MKL_INT *n , MKL_INT *ilo , MKL_INT *ihi , MKL_Complex16 *a , MKL_INT\n*ia , MKL_INT *ja , MKL_INT *desca , MKL_Complex16 *tau , MKL_Complex16 *work , MKL_INT\n*lwork , MKL_INT *info );\nInclude Files\n•\nmkl_scalapack.h\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1611\n\n\nDescription\nThe p?gehd2function reduces a real/complex general distributed matrix sub(A) to upper Hessenberg form H\nby an orthogonal/unitary similarity transformation: Q'*sub(A)*Q = H, where sub(A) = A(ia+n-1 :ia\n+n-1, ja+n-1 :ja+n-1).\nInput Parameters\nn\n(global) The order of the distributed submatrix A. (n≥ 0).\nilo, ihi\n(global) It is assumed that the matrix sub(A) is already upper triangular in\nrows ia:ia+ilo-2 and ia+ihi:ia+n-1 and columns ja:ja+jlo-2 and ja\n+jhi:ja+n-1. See Application Notes for further information.\nIf n≥ 0, 1 ≤ ilo ≤ ihi ≤ n; otherwise set ilo = 1, ihi = n.\na\n(local).\nPointer into the local memory to an array of sizelld_a * LOCc(ja+n-1).\nOn entry, this array contains the local pieces of the n-by-n general\ndistributed matrix sub(A) to be reduced.\nia, ja\n(global) The row and column indices in the global matrix A indicating the\nfirst row and the first column of sub(A), respectively.\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\nwork\n(local).\nThis is a workspace array of size lwork.\nlwork\n(local or global)\nThe size of the array work.\nlwork is local input and must be at least lwork≥nb + max( npa0, nb ),\nwhere nb = mb_a = nb_a, iroffa = mod( ia-1, nb ), iarow =\nindxg2p ( ia, nb, myrow, rsrc_a, nprow ),npa0 = numroc(ihi\n+iroffa, nb, myrow, iarow, nprow ).\nindxg2p and numroc are ScaLAPACK tool functions;myrow, mycol, nprow,\nand npcol can be determined by calling the function blacs_gridinfo.\nIf lwork = -1, then lwork is global input and a workspace query is\nassumed; the function only calculates the minimum and optimal size for all\nwork arrays. Each of these values is returned in the first entry of the\ncorresponding work array, and no error message is issued by pxerbla.\nOutput Parameters\na\n(local). On exit, the upper triangle and the first subdiagonal of sub(A) are\noverwritten with the upper Hessenberg matrix H, and the elements below\nthe first subdiagonal, with the array tau, represent the orthogonal/unitary\nmatrix Q as a product of elementary reflectors. (see Application Notes\nbelow).\ntau\n(local).\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1612\n\n\nArray of size LOCc(ja+n-2) The scalar factors of the elementary reflectors\n(see Application Notes below). Elements ja:ja+ilo-2 and ja+ihi:ja+n-2\nof the global vector tau are set to zero. tau is tied to the distributed matrix\nA.\nwork\nOn exit, work[0] returns the minimal and optimal lwork.\ninfo\n(local)\nIf info = 0, the execution is successful.\nif info < 0: If the i-th argument is an array and the j-th entry, indexed j-1,\nhad an illegal value, then info = - (i*100+j), if the i-th argument is a\nscalar and had an illegal value, then info = -i.\nApplication Notes\nThe matrix Q is represented as a product of (ihi-ilo) elementary reflectors\nQ = H(ilo)*H(ilo+1)*...*H(ihi-1).\nEach H(i) has the form\nH(i) = I - tau*v*v',\nwhere tau is a real/complex scalar, and v is a real/complex vector with v(1: i)=0, v(i+1)=1 and v(ihi\n+1:n)=0; v(i+2:ihi) is stored on exit in A(ia+ilo+i:ia+ihi-1, ia+ilo+i-2), and tau in tau[ja+ilo\n+i-3].\nThe contents of A(ia:ia+n-1, ja:ja+n-1) are illustrated by the following example, with n = 7, ilo = 2\nand ihi = 6:\nwhere a denotes an element of the original matrix sub(A), h denotes a modified element of the upper\nHessenberg matrix H, and vi denotes an element of the vector defining H(ja+ilo+i-2).\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?gelq2\nComputes an LQ factorization of a general rectangular\nmatrix (unblocked algorithm).\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1613\n\n\nSyntax\nvoid psgelq2 (MKL_INT *m , MKL_INT *n , float *a , MKL_INT *ia , MKL_INT *ja , MKL_INT\n*desca , float *tau , float *work , MKL_INT *lwork , MKL_INT *info );\nvoid pdgelq2 (MKL_INT *m , MKL_INT *n , double *a , MKL_INT *ia , MKL_INT *ja , MKL_INT\n*desca , double *tau , double *work , MKL_INT *lwork , MKL_INT *info );\nvoid pcgelq2 (MKL_INT *m , MKL_INT *n , MKL_Complex8 *a , MKL_INT *ia , MKL_INT *ja ,\nMKL_INT *desca , MKL_Complex8 *tau , MKL_Complex8 *work , MKL_INT *lwork , MKL_INT\n*info );\nvoid pzgelq2 (MKL_INT *m , MKL_INT *n , MKL_Complex16 *a , MKL_INT *ia , MKL_INT *ja ,\nMKL_INT *desca , MKL_Complex16 *tau , MKL_Complex16 *work , MKL_INT *lwork , MKL_INT\n*info );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?gelq2function computes an LQ factorization of a real/complex distributed m-by-n matrix sub(A) =\nA(ia:ia+m-1, ja:ja+n-1) = L*Q.\nInput Parameters\nm\n(global)\nThe number of rows of the distributed matrix sub(A). (m≥0).\nn\n(global)\nThe number of columns of the distributed matrix sub(A). (n≥0).\na\n(local).\nPointer into the local memory to an array of size lld_a * LOCc(ja+n-1).\nOn entry, this array contains the local pieces of the m-by-n distributed\nmatrix sub(A) which is to be factored.\nia, ja\n(global) The row and column indices in the global matrix A indicating the\nfirst row and the first column of sub(A), respectively.\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\nwork\n(local).\nThis is a workspace array of size lwork.\nlwork\n(local or global)\nThe size of the array work.\nlwork is local input and must be at least lwork≥nq0 + max( 1, mp0 ),\nwhere iroff = mod(ia-1, mb_a), icoff = mod(ja-1, nb_a),\niarow = indxg2p(ia, mb_a, myrow, rsrc_a, nprow),\niacol = indxg2p(ja, nb_a, mycol, csrc_a, npcol),\nmp0 = numroc(m+iroff, mb_a, myrow, iarow, nprow),\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1614\n\n\nnq0 = numroc(n+icoff, nb_a, mycol, iacol, npcol),\nindxg2p and numroc are ScaLAPACK tool functions; myrow, mycol, nprow,\nand npcol can be determined by calling the function blacs_gridinfo.\nIf lwork = -1, then lwork is global input and a workspace query is\nassumed; the function only calculates the minimum and optimal size\nfor all work arrays. Each of these values is returned in the first entry\nof the corresponding work array, and no error message is issued by \npxerbla.\nOutput Parameters\na\n(local).\nOn exit, the elements on and below the diagonal of sub(A) contain the m by\nmin(m,n) lower trapezoidal matrix L (L is lower triangular if m ≤ n); the\nelements above the diagonal, with the array tau, represent the orthogonal/\nunitary matrix Q as a product of elementary reflectors (see Application\nNotes below).\ntau\n(local).\nArray of size LOCr(ia+min(m, n)-1). This array contains the scalar\nfactors of the elementary reflectors. tau is tied to the distributed matrix A.\nwork\nOn exit, work[0] returns the minimal and optimal lwork.\ninfo\n(local) If info = 0, the execution is successful. if info < 0: If the i-th\nargument is an array and the j-th entry, indexed j-1, had an illegal value,\nthen info = - (i*100+j), if the i-th argument is a scalar and had an illegal\nvalue, then info = -i.\nApplication Notes\nThe matrix Q is represented as a product of elementary reflectors\nQ =H(ia+k-1)*H(ia+k-2)*. . . *H(ia) for real flavors, Q =(H(ia+k-1))H*(H(ia\n+k-2))H...*(H(ia))H for complex flavors,\nwhere k = min(m,n).\nEach H(i) has the form\nH(i) = I - tau*v*v'\nwhere tau is a real/complex scalar, and v is a real/complex vector with v(1: i-1) = 0 and v(i) = 1; v(i\n+1: n) (for real flavors) or conjg(v(i+1: n)) (for complex flavors) is stored on exit in A(ia+i-1,ja+i:ja\n+n-1), and tau in tau[ia+i-2].\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?geql2\nComputes a QL factorization of a general rectangular\nmatrix (unblocked algorithm).\nSyntax\nvoid psgeql2 (MKL_INT *m , MKL_INT *n , float *a , MKL_INT *ia , MKL_INT *ja , MKL_INT\n*desca , float *tau , float *work , MKL_INT *lwork , MKL_INT *info );\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1615\n\n\nvoid pdgeql2 (MKL_INT *m , MKL_INT *n , double *a , MKL_INT *ia , MKL_INT *ja , MKL_INT\n*desca , double *tau , double *work , MKL_INT *lwork , MKL_INT *info );\nvoid pcgeql2 (MKL_INT *m , MKL_INT *n , MKL_Complex8 *a , MKL_INT *ia , MKL_INT *ja ,\nMKL_INT *desca , MKL_Complex8 *tau , MKL_Complex8 *work , MKL_INT *lwork , MKL_INT\n*info );\nvoid pzgeql2 (MKL_INT *m , MKL_INT *n , MKL_Complex16 *a , MKL_INT *ia , MKL_INT *ja ,\nMKL_INT *desca , MKL_Complex16 *tau , MKL_Complex16 *work , MKL_INT *lwork , MKL_INT\n*info );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?geql2function computes a QL factorization of a real/complex distributed m-by-n matrix sub(A) =\nA(ia:ia+m-1, ja:ja+n-1)= Q *L.\nInput Parameters\nm\n(global)\nThe number of rows in the distributed matrix sub(A). (m≥ 0).\nn\n(global)\nThe number of columns in the distributed matrix sub(A). (n≥ 0).\na\n(local).\nPointer into the local memory to an array of size lld_a * LOCc(ja+n-1).\nOn entry, this array contains the local pieces of the m-by-n distributed\nmatrix sub(A) which is to be factored.\nia, ja\n(global) The row and column indices in the global matrix A indicating the\nfirst row and the first column of sub(A), respectively.\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\nwork\n(local).\nThis is a workspace array of size lwork.\nlwork\n(local or global)\nThe size of the array work.\nlwork is local input and must be at least lwork≥mp0 + max(1, nq0),\nwhere iroff = mod(ia-1, mb_a), icoff = mod(ja-1, nb_a),\niarow = indxg2p(ia, mb_a, myrow, rsrc_a, nprow),\niacol = indxg2p(ja, nb_a, mycol, csrc_a, npcol),\nmp0 = numroc(m+iroff, mb_a, myrow, iarow, nprow),\nnq0 = numroc(n+icoff, nb_a, mycol, iacol, npcol),\nindxg2p and numroc are ScaLAPACK tool functions; myrow, mycol, nprow,\nand npcol can be determined by calling the function blacs_gridinfo.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1616\n\n\nIf lwork = -1, then lwork is global input and a workspace query is\nassumed; the function only calculates the minimum and optimal size for all\nwork arrays. Each of these values is returned in the first entry of the\ncorresponding work array, and no error message is issued by pxerbla.\nOutput Parameters\na\n(local).\nOn exit,\nif m ≥ n, the lower triangle of the distributed submatrix A(ia+m-n:ia+m-1,\nja:ja+n-1) contains the n-by-n lower triangular matrix L;\nif m ≤ n, the elements on and below the (n-m)-th superdiagonal contain\nthe m-by-n lower trapezoidal matrix L; the remaining elements, with the\narray tau, represent the orthogonal/ unitary matrix Q as a product of\nelementary reflectors (see Application Notes below).\ntau\n(local).\nArray of size LOCc(ja+n-1). This array contains the scalar factors of the\nelementary reflectors. tau is tied to the distributed matrix A.\nwork\nOn exit, work[0] returns the minimal and optimal lwork.\ninfo\n(local).\nIf info = 0, the execution is successful. if info < 0: If the i-th argument\nis an array and the j-th entry, indexed j-1, had an illegal value, then info\n= - (i*100+j), if the i-th argument is a scalar and had an illegal value,\nthen info = -i.\nApplication Notes\nThe matrix Q is represented as a product of elementary reflectors\nQ = H(ja+k-1)*...*H(ja+1)*H(ja), where k = min(m,n).\nEach H(i) has the form\nH(i) = I- tau *v*v'\nwhere tau is a real/complex scalar, and v is a real/complex vector with v(m-k+i+1: m) = 0 and v(m-k+i) =\n1; v(1: m-k+i-1) is stored on exit in A(ia:ia+m-k+i-2, ja+n-k+i-1), and tau in tau[ja+n-k+i-2].\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?geqr2\nComputes a QR factorization of a general rectangular\nmatrix (unblocked algorithm).\nSyntax\nvoid psgeqr2 (MKL_INT *m , MKL_INT *n , float *a , MKL_INT *ia , MKL_INT *ja , MKL_INT\n*desca , float *tau , float *work , MKL_INT *lwork , MKL_INT *info );\nvoid pdgeqr2 (MKL_INT *m , MKL_INT *n , double *a , MKL_INT *ia , MKL_INT *ja , MKL_INT\n*desca , double *tau , double *work , MKL_INT *lwork , MKL_INT *info );\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1617\n\n\nvoid pcgeqr2 (MKL_INT *m , MKL_INT *n , MKL_Complex8 *a , MKL_INT *ia , MKL_INT *ja ,\nMKL_INT *desca , MKL_Complex8 *tau , MKL_Complex8 *work , MKL_INT *lwork , MKL_INT\n*info );\nvoid pzgeqr2 (MKL_INT *m , MKL_INT *n , MKL_Complex16 *a , MKL_INT *ia , MKL_INT *ja ,\nMKL_INT *desca , MKL_Complex16 *tau , MKL_Complex16 *work , MKL_INT *lwork , MKL_INT\n*info );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?geqr2function computes a QR factorization of a real/complex distributed m-by-n matrix sub(A) =\nA(ia:ia+m-1, ja:ja+n-1)= Q*R.\nInput Parameters\nm\n(global)\nThe number of rows in the distributed matrix sub(A). (m≥0).\nn\n(global) The number of columns in the distributed matrix sub(A). (n≥0).\na\n(local).\nPointer into the local memory to an array of size lld_a * LOCc(ja+n-1).\nOn entry, this array contains the local pieces of the m-by-n distributed\nmatrix sub(A) which is to be factored.\nia, ja\n(global) The row and column indices in the global matrix A indicating the\nfirst row and the first column of sub(A), respectively.\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\nwork\n(local).\nThis is a workspace array of size lwork.\nlwork\n(local or global)\nThe size of the array work.\nlwork is local input and must be at least lwork≥mp0+max(1, nq0),\nwhere iroff = mod(ia-1, mb_a), icoff = mod(ja-1, nb_a),\niarow = indxg2p(ia, mb_a, myrow, rsrc_a, nprow),\niacol = indxg2p(ja, nb_a, mycol, csrc_a, npcol),\nmp0 = numroc(m+iroff, mb_a, myrow, iarow, nprow),\nnq0 = numroc(n+icoff, nb_a, mycol, iacol, npcol).\nindxg2p and numroc are ScaLAPACK tool functions; myrow, mycol, nprow,\nand npcol can be determined by calling the function blacs_gridinfo.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1618\n\n\nIf lwork = -1, then lwork is global input and a workspace query is\nassumed; the function only calculates the minimum and optimal size for all\nwork arrays. Each of these values is returned in the first entry of the\ncorresponding work array, and no error message is issued by pxerbla.\nOutput Parameters\na\n(local).\nOn exit, the elements on and above the diagonal of sub(A) contain the\nmin(m,n) by n upper trapezoidal matrix R (R is upper triangular if m≥n); the\nelements below the diagonal, with the array tau, represent the orthogonal/\nunitary matrix Q as a product of elementary reflectors (see Application\nNotes below).\ntau\n(local).\nArray of size LOCc(ja+min(m,n)-1). This array contains the scalar factors of\nthe elementary reflectors. tau is tied to the distributed matrix A.\nwork\nOn exit, work[0] returns the minimal and optimal lwork.\ninfo\n(local)\nIf info = 0, the execution is successful. if info < 0:\nIf the i-th argument is an array and the j-th entry, indexed j-1, had an\nillegal value, then info = - (i*100+j),\nif the i-th argument is a scalar and had an illegal value, then info = -i.\nApplication Notes\nThe matrix Q is represented as a product of elementary reflectors\nQ = H(ja)*H(ja+1)*. . .* H(ja+k-1), where k = min(m,n).\nEach H(i) has the form\nH(j)= I - tau*v*v',\nwhere tau is a real/complex scalar, and v is a real/complex vector with v(1: i-1) = 0 and v(i) = 1; v(i+1: m)\nis stored on exit in A(ia+i:ia+m-1, ja+i-1), and tau in tau[ja+i-2].\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?gerq2\nComputes an RQ factorization of a general rectangular\nmatrix (unblocked algorithm).\nSyntax\nvoid psgerq2 (MKL_INT *m , MKL_INT *n , float *a , MKL_INT *ia , MKL_INT *ja , MKL_INT\n*desca , float *tau , float *work , MKL_INT *lwork , MKL_INT *info );\nvoid pdgerq2 (MKL_INT *m , MKL_INT *n , double *a , MKL_INT *ia , MKL_INT *ja , MKL_INT\n*desca , double *tau , double *work , MKL_INT *lwork , MKL_INT *info );\nvoid pcgerq2 (MKL_INT *m , MKL_INT *n , MKL_Complex8 *a , MKL_INT *ia , MKL_INT *ja ,\nMKL_INT *desca , MKL_Complex8 *tau , MKL_Complex8 *work , MKL_INT *lwork , MKL_INT\n*info );\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1619\n\n\nvoid pzgerq2 (MKL_INT *m , MKL_INT *n , MKL_Complex16 *a , MKL_INT *ia , MKL_INT *ja ,\nMKL_INT *desca , MKL_Complex16 *tau , MKL_Complex16 *work , MKL_INT *lwork , MKL_INT\n*info );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?gerq2function computes an RQ factorization of a real/complex distributed m-by-n matrix sub(A) =\nA(ia:ia+m-1, ja:ja+n-1) = R*Q.\nInput Parameters\nm\n(global) The number of rows in the distributed matrix sub(A). (m≥0).\nn\n(global) The number of columns in the distributed matrix sub(A). (n≥0).\na\n(local).\nPointer into the local memory to an array of size lld_a * LOCc(ja+n-1).\nOn entry, this array contains the local pieces of the m-by-n distributed\nmatrix sub(A) which is to be factored.\nia, ja\n(global) The row and column indices in the global matrix A indicating the\nfirst row and the first column of sub(A), respectively.\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\nwork\n(local).\nThis is a workspace array of size lwork.\nlwork\n(local or global)\nThe size of the array work.\nlwork is local input and must be at least lwork≥nq0 + max(1, mp0),\nwhere\niroff = mod(ia-1, mb_a), icoff = mod(ja-1, nb_a),\niarow = indxg2p(ia, mb_a, myrow, rsrc_a, nprow),\niacol = indxg2p(ja, nb_a, mycol, csrc_a, npcol), mp0 =\nnumroc( m+iroff, mb_a, myrow, iarow, nprow),\nnq0 = numroc(n+icoff, nb_a, mycol, iacol, npcol),\nindxg2p and numroc are ScaLAPACK tool functions; myrow, mycol, nprow,\nand npcol can be determined by calling the function blacs_gridinfo.\nIf lwork = -1, then lwork is global input and a workspace query is\nassumed; the function only calculates the minimum and optimal size for all\nwork arrays. Each of these values is returned in the first entry of the\ncorresponding work array, and no error message is issued by pxerbla.\nOutput Parameters\na\n(local).\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1620\n\n\nOn exit,\nif m ≤ n, the upper triangle of A(ia+m-n:ia+m-1, ja:ja+n-1) contains the\nm-by-m upper triangular matrix R;\nif m ≥ n, the elements on and above the (m-n)-th subdiagonal contain the\nm-by-n upper trapezoidal matrix R; the remaining elements, with the array\ntau, represent the orthogonal/ unitary matrix Q as a product of elementary\nreflectors (see Application Notes below).\ntau\n(local).\nArray of size LOCr(ia+m -1). This array contains the scalar factors of the\nelementary reflectors. tau is tied to the distributed matrix A.\nwork\nOn exit, work[0] returns the minimal and optimal lwork.\ninfo\n(local)\nIf info = 0, the execution is successful.\nif info < 0: If the i-th argument is an array and the j-th entry, indexed j-1,\nhad an illegal value, then info = - (i*100+j), if the i-th argument is a\nscalar and had an illegal value, then info = -i.\nApplication Notes\nThe matrix Q is represented as a product of elementary reflectors\nQ = H(ia)*H(ia+1)*...*H(ia+k-1) for real flavors,\nQ = (H(ia))H*(H(ia+1))H...*(H(ia+k-1))H for complex flavors,\nwhere k = min(m, n).\nEach H(i) has the form\nH(i) = I - tau*v*v',\nwhere tau is a real/complex scalar, and v is a real/complex vector with v(n-k+i+1:n) = 0 and v(n-k+i) =\n1; v(1:n-k+i-1) for real flavors or conjg(v(1:n-k+i-1)) for complex flavors is stored on exit in A(ia+m-\nk+i-1, ja:ja+n-k+i-2), and tau in tau[ia+m-k+i-2].\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?getf2\nComputes an LU factorization of a general matrix,\nusing partial pivoting with row interchanges (local\nblocked algorithm).\nSyntax\nvoid psgetf2 (MKL_INT *m , MKL_INT *n , float *a , MKL_INT *ia , MKL_INT *ja , MKL_INT\n*desca , MKL_INT *ipiv , MKL_INT *info );\nvoid pdgetf2 (MKL_INT *m , MKL_INT *n , double *a , MKL_INT *ia , MKL_INT *ja , MKL_INT\n*desca , MKL_INT *ipiv , MKL_INT *info );\nvoid pcgetf2 (MKL_INT *m , MKL_INT *n , MKL_Complex8 *a , MKL_INT *ia , MKL_INT *ja ,\nMKL_INT *desca , MKL_INT *ipiv , MKL_INT *info );\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1621\n\n\nvoid pzgetf2 (MKL_INT *m , MKL_INT *n , MKL_Complex16 *a , MKL_INT *ia , MKL_INT *ja ,\nMKL_INT *desca , MKL_INT *ipiv , MKL_INT *info );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?getf2function computes an LU factorization of a general m-by-n distributed matrix sub(A) = A(ia:ia\n+m-1, ja:ja+n-1) using partial pivoting with row interchanges.\nThe factorization has the form sub(A) = P * L* U, where P is a permutation matrix, L is lower triangular\nwith unit diagonal elements (lower trapezoidal if m>n), and U is upper triangular (upper trapezoidal if m < n).\nThis is the right-looking Parallel Level 2 BLAS version of the algorithm.\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nInput Parameters\nm\n(global)\nThe number of rows in the distributed matrix sub(A). (m≥0).\nn\n(global) The number of columns in the distributed matrix sub(A). (nb_a -\nmod(ja-1, nb_a)≥n≥0).\na\n(local).\nPointer into the local memory to an array of size lld_a * LOCc(ja+n-1).\nOn entry, this array contains the local pieces of the m-by-n distributed\nmatrix sub(A).\nia, ja\n(global) The row and column indices in the global matrix A indicating the\nfirst row and the first column of the matrix sub(A), respectively.\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\nOutput Parameters\nipiv\n(local)\nArray of size(LOCr(m_a) + mb_a). This array contains the pivoting\ninformation. ipiv[i] -> The global row that local row (i +1) was swapped\nwith, i = 0, 1, ... , LOCr(m_a) + mb_a - 1. This array is tied to the\ndistributed matrix A.\ninfo\n(local).\nIf info = 0: successful exit.\nIf info < 0:\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1622\n\n\n•\nif the i-th argument is an array and the j-th entry, indexed j-1, had an\nillegal value, then info = -(i*100+j),\n•\nif the i-th argument is a scalar and had an illegal value, then info = -\ni.\nIf info > 0: If info = k, the matrix element U(ia+k-1, ja+k-1) is\nexactly zero. The factorization has been completed, but the factor U is\nexactly singular, and division by zero will occur if it is used to solve a\nsystem of equations.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?labrd\nReduces the first nb rows and columns of a general\nrectangular matrix A to real bidiagonal form by an\northogonal/unitary transformation, and returns\nauxiliary matrices that are needed to apply the\ntransformation to the unreduced part of A.\nSyntax\nvoid pslabrd (MKL_INT *m , MKL_INT *n , MKL_INT *nb , float *a , MKL_INT *ia , MKL_INT\n*ja , MKL_INT *desca , float *d , float *e , float *tauq , float *taup , float *x ,\nMKL_INT *ix , MKL_INT *jx , MKL_INT *descx , float *y , MKL_INT *iy , MKL_INT *jy ,\nMKL_INT *descy , float *work );\nvoid pdlabrd (MKL_INT *m , MKL_INT *n , MKL_INT *nb , double *a , MKL_INT *ia , MKL_INT\n*ja , MKL_INT *desca , double *d , double *e , double *tauq , double *taup , double *x ,\nMKL_INT *ix , MKL_INT *jx , MKL_INT *descx , double *y , MKL_INT *iy , MKL_INT *jy ,\nMKL_INT *descy , double *work );\nvoid pclabrd (MKL_INT *m , MKL_INT *n , MKL_INT *nb , MKL_Complex8 *a , MKL_INT *ia ,\nMKL_INT *ja , MKL_INT *desca , float *d , float *e , MKL_Complex8 *tauq , MKL_Complex8\n*taup , MKL_Complex8 *x , MKL_INT *ix , MKL_INT *jx , MKL_INT *descx , MKL_Complex8 *y ,\nMKL_INT *iy , MKL_INT *jy , MKL_INT *descy , MKL_Complex8 *work );\nvoid pzlabrd (MKL_INT *m , MKL_INT *n , MKL_INT *nb , MKL_Complex16 *a , MKL_INT *ia ,\nMKL_INT *ja , MKL_INT *desca , double *d , double *e , MKL_Complex16 *tauq ,\nMKL_Complex16 *taup , MKL_Complex16 *x , MKL_INT *ix , MKL_INT *jx , MKL_INT *descx ,\nMKL_Complex16 *y , MKL_INT *iy , MKL_INT *jy , MKL_INT *descy , MKL_Complex16 *work );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?labrdfunction reduces the first nb rows and columns of a real/complex general m-by-n distributed\nmatrix sub(A) = A(ia:ia+m-1, ja:ja+n-1) to upper or lower bidiagonal form by an orthogonal/unitary\ntransformation Q'* A * P, and returns the matrices X and Y necessary to apply the transformation to the\nunreduced part of sub(A).\nIf m ≥n, sub(A) is reduced to upper bidiagonal form; if m < n, sub(A) is reduced to lower bidiagonal form.\nThis is an auxiliary function called by p?gebrd.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1623\n\n\nInput Parameters\nm\n(global) The number of rows in the distributed matrix sub(A). (m≥ 0).\nn\n(global) The number of columns in the distributed matrix sub(A). (n ≥ 0).\nnb\n(global)\nThe number of leading rows and columns of sub(A) to be reduced.\na\n(local).\nPointer into the local memory to an array of size lld_a * LOCc(ja+n-1).\nOn entry, this array contains the local pieces of the general distributed\nmatrix sub(A).\nia, ja\n(global) The row and column indices in the global matrix A indicating the\nfirst row and the first column of the matrix sub(A), respectively.\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\nix, jx\n(global) The row and column indices in the global matrix X indicating the\nfirst row and the first column of the matrix sub(X), respectively.\ndescx\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix X.\niy, jy\n(global) The row and column indices in the global matrix Y indicating the\nfirst row and the first column of the matrix sub(Y), respectively.\ndescy\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix Y.\nwork\n(local).\nWorkspace array of sizelwork.\nlwork ≥ nb_a + nq,\nwith nq = numroc(n+mod(ia-1, nb_y), nb_y, mycol, iacol,\nnpcol)\niacol = indxg2p (ja, nb_a, mycol, csrc_a, npcol)\nindxg2p and numroc are ScaLAPACK tool functions; myrow, mycol, nprow,\nand npcol can be determined by calling the function blacs_gridinfo.\nOutput Parameters\na\n(local)\nOn exit, the first nb rows and columns of the matrix are overwritten; the\nrest of the distributed matrix sub(A) is unchanged.\nIf m ≥ n, elements on and below the diagonal in the first nb columns, with\nthe array tauq, represent the orthogonal/unitary matrix Q as a product of\nelementary reflectors; and elements above the diagonal in the first nb rows,\nwith the array taup, represent the orthogonal/unitary matrix P as a product\nof elementary reflectors.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1624\n\n\nIf m < n, elements below the diagonal in the first nb columns, with the\narray tauq, represent the orthogonal/unitary matrix Q as a product of\nelementary reflectors, and elements on and above the diagonal in the first\nnb rows, with the array taup, represent the orthogonal/unitary matrix P as\na product of elementary reflectors. See Application Notes below.\nd\n(local).\nArray of size LOCr(ia+min(m,n)-1) if m ≥ n; LOCc(ja+min(m,n)-1)\notherwise. The distributed diagonal elements of the bidiagonal distributed\nmatrix B:\nd[i] = A(ia+i, ja+i), i= 0, 1, ..., size (d)-1\nd is tied to the distributed matrix A.\ne\n(local).\nArray of size LOCr(ia+min(m,n)-1) if m ≥ n; LOCc(ja+min(m,n)-2)\notherwise. The distributed off-diagonal elements of the bidiagonal\ndistributed matrix B:\nif m ≥ n, e[i] = A(ia+i, ja+i+1) for i = 0, 1, ..., n-2;\nif m<n, e[i] = A(ia+i+1, ja+i) for i = 0, 1, ..., m-2.\ne is tied to the distributed matrix A.\ntauq, taup\n(local).\nArray size LOCc(ja+min(m, n)-1) for tauq, size LOCr(ia+min(m, n)-1) for\ntaup. The scalar factors of the elementary reflectors which represent the\northogonal/unitary matrix Q for tauq, P for taup. tauq and taup are tied to\nthe distributed matrix A. See Application Notes below.\nx\n(local)\nPointer into the local memory to an array of size lld_x* nb. On exit, the\nlocal pieces of the distributed m-by-nb matrix X(ix:ix+m-1, jx:jx+nb-1)\nrequired to update the unreduced part of sub(A).\ny\n(local).\nPointer into the local memory to an array of size lld_y* nb. On exit, the\nlocal pieces of the distributed n-by-nb matrix Y(iy:iy+n-1, jy:jy+nb-1)\nrequired to update the unreduced part of sub(A).\nApplication Notes\nThe matrices Q and P are represented as products of elementary reflectors:\nQ = H(1)*H(2)*...*H(nb), and P = G(1)*G(2)*...*G(nb)\nEach H(i) and G(i) has the form:\nH(i) = I - tauq*v*v' , and G(i) = I - taup*u*u',\nwhere tauq and taup are real/complex scalars, and v and u are real/complex vectors.\nIf m ≥ n, v(1: i-1 ) = 0, v(i) = 1, and v(i:m) is stored on exit in\nA(ia+i-1:ia+m-1, ja+i-1); u(1:i) = 0, u(i+1 ) = 1, and u(i+1:n) is stored on exit in A(ia+i-1,\nja+i:ja+n-1); tauq is stored in tauq[ja+i-2] and taup in taup[ia+i-2].\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1625\n\n\nIf m < n, v(1: i) = 0, v(i+1 ) = 1, and v(i+1:m) is stored on exit in\nA(ia+i+1:ia+m-1, ja+i-1); u(1:i-1 ) = 0, u(i) = 1, and u(i:n) is stored on exit in A(ia+i-1, ja\n+i:ja+n-1); tauq is stored in tauq[ja+i-2] and taup in taup[ia+i-2]. The elements of the vectors v and\nu together form the m-by-nb matrix V and the nb-by-n matrix U' which are necessary, with X and Y, to apply\nthe transformation to the unreduced part of the matrix, using a block update of the form: sub(A):= sub(A)\n- V*Y' - X*U'. The contents of sub(A) on exit are illustrated by the following examples with nb = 2:\nwhere a denotes an element of the original matrix which is unchanged, vi denotes an element of the vector\ndefining H(i), and ui an element of the vector defining G(i).\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?lacon\nEstimates the 1-norm of a square matrix, using the\nreverse communication for evaluating matrix-vector\nproducts.\nSyntax\nvoid pslacon (MKL_INT *n , float *v , MKL_INT *iv , MKL_INT *jv , MKL_INT *descv , float\n*x , MKL_INT *ix , MKL_INT *jx , MKL_INT *descx , MKL_INT *isgn , float *est , MKL_INT\n*kase );\nvoid pdlacon (MKL_INT *n , double *v , MKL_INT *iv , MKL_INT *jv , MKL_INT *descv ,\ndouble *x , MKL_INT *ix , MKL_INT *jx , MKL_INT *descx , MKL_INT *isgn , double *est ,\nMKL_INT *kase );\nvoid pclacon (MKL_INT *n , MKL_Complex8 *v , MKL_INT *iv , MKL_INT *jv , MKL_INT\n*descv , MKL_Complex8 *x , MKL_INT *ix , MKL_INT *jx , MKL_INT *descx , float *est ,\nMKL_INT *kase );\nvoid pzlacon (MKL_INT *n , MKL_Complex16 *v , MKL_INT *iv , MKL_INT *jv , MKL_INT\n*descv , MKL_Complex16 *x , MKL_INT *ix , MKL_INT *jx , MKL_INT *descx , double *est ,\nMKL_INT *kase );\nInclude Files\n•\nmkl_scalapack.h\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1626\n\n\nDescription\nThe p?laconfunction estimates the 1-norm of a square, real/unitary distributed matrix A. Reverse\ncommunication is used for evaluating matrix-vector products. x and v are aligned with the distributed matrix\nA, this information is implicitly contained within iv, ix, descv, and descx.\nInput Parameters\nn\n(global) The length of the distributed vectors v and x. n ≥ 0.\nv\n(local).\nPointer into the local memory to an array of size LOCr(n+mod(iv-1, mb_v)).\nOn the final return, v = a*w, where est = norm(v)/norm(w) (w is not\nreturned).\niv, jv\n(global) The row and column indices in the global matrix V indicating the\nfirst row and the first column of the submatrix V, respectively.\ndescv\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix V.\nx\n(local).\nPointer into the local memory to an array of size LOCr(n+mod(ix-1, mb_x)).\nix, jx\n(global) The row and column indices in the global matrix X indicating the\nfirst row and the first column of the submatrix X, respectively.\ndescx\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix X.\nisgn\n(local).\nArray of size LOCr(n+mod(ix-1, mb_x)). isgn is aligned with x and v.\nkase\n(local).\nOn the initial call to p?lacon, kase should be 0.\nOutput Parameters\nx\n(local).\nOn an intermediate return, X should be overwritten by A*X, if kase=1, A'\n*X, if kase=2,\np?lacon must be re-called with all the other parameters unchanged.\nest\n(global).\nkase\n(local)\nOn an intermediate return, kase is 1 or 2, indicating whether X should be\noverwritten by A*X, or A'*X. On the final return from p?lacon, kase is\nagain 0.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1627\n\n\np?laconsb\nLooks for two consecutive small subdiagonal elements.\nSyntax\nvoid pslaconsb (const float *a, const MKL_INT *desca, const MKL_INT *i, const MKL_INT\n*l, MKL_INT *m, const float *h44, const float *h33, const float *h43h34, float *buf,\nconst MKL_INT *lwork );\nvoid pdlaconsb (const double *a, const MKL_INT *desca, const MKL_INT *i, const MKL_INT\n*l, MKL_INT *m, const double *h44, const double *h33, const double *h43h34, double *buf,\nconst MKL_INT *lwork );\nvoid pclaconsb (const MKL_Complex8 *a , const MKL_INT *desca , const MKL_INT *i , const\nMKL_INT *l , MKL_INT *m , const MKL_Complex8 *h44 , const MKL_Complex8 *h33 , const\nMKL_Complex8 *h43h34 , MKL_Complex8 *buf , const MKL_INT *lwork );\nvoid pzlaconsb (const MKL_Complex16 *a , const MKL_INT *desca , const MKL_INT *i ,\nconst MKL_INT *l , MKL_INT *m , const MKL_Complex16 *h44 , const MKL_Complex16 *h33 ,\nconst MKL_Complex16 *h43h34 , MKL_Complex16 *buf , const MKL_INT *lwork );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?laconsbfunction looks for two consecutive small subdiagonal elements by analyzing the effect of\nstarting a double shift QR iteration given by h44, h33, and h43h34 to see if this process makes a subdiagonal\nnegligible.\nInput Parameters\na\n(local)\nArray of size lld_a*LOCc(n_a). On entry, the Hessenberg matrix whose\ntridiagonal part is being scanned. Unchanged on exit.\ndesca\n(global and local)\nArray of size dlen_. The array descriptor for the distributed matrix A.\ni\n(global)\nThe global location of the bottom of the unreduced submatrix of A.\nUnchanged on exit.\nl\n(global)\nThe global location of the top of the unreduced submatrix of A. Unchanged\non exit.\nh44, h33, h43h34\n(global).\nThese three values are for the double shift QR iteration.\nlwork\n(local)\nThis must be at least 7*ceil(ceil( (i-l)/mb_a )/lcm(nprow,\nnpcol)). Here lcm is the least common multiple and nprow*npcol is the\nlogical grid size.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1628\n\n\nOutput Parameters\nm\n(global). On exit, this yields the starting location of the QR double shift.\nThis will satisfy: \nl ≤ m ≤ i-2.\nbuf\n(local).\nArray of size lwork.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?lacp2\nCopies all or part of a distributed matrix to another\ndistributed matrix.\nSyntax\nvoid pslacp2 (char *uplo , MKL_INT *m , MKL_INT *n , float *a , MKL_INT *ia , MKL_INT\n*ja , MKL_INT *desca , float *b , MKL_INT *ib , MKL_INT *jb , MKL_INT *descb );\nvoid pdlacp2 (char *uplo , MKL_INT *m , MKL_INT *n , double *a , MKL_INT *ia , MKL_INT\n*ja , MKL_INT *desca , double *b , MKL_INT *ib , MKL_INT *jb , MKL_INT *descb );\nvoid pclacp2 (char *uplo , MKL_INT *m , MKL_INT *n , MKL_Complex8 *a , MKL_INT *ia ,\nMKL_INT *ja , MKL_INT *desca , MKL_Complex8 *b , MKL_INT *ib , MKL_INT *jb , MKL_INT\n*descb );\nvoid pzlacp2 (char *uplo , MKL_INT *m , MKL_INT *n , MKL_Complex16 *a , MKL_INT *ia ,\nMKL_INT *ja , MKL_INT *desca , MKL_Complex16 *b , MKL_INT *ib , MKL_INT *jb , MKL_INT\n*descb );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?lacp2function copies all or part of a distributed matrix A to another distributed matrix B. No\ncommunication is performed, p?lacp2 performs a local copy sub(A):= sub(B), where sub(A) denotes\nA(ia:ia+m-1, a:ja+n-1) and sub(B) denotes B(ib:ib+m-1, jb:jb+n-1).\np?lacp2 requires that only dimension of the matrix operands is distributed.\nInput Parameters\nuplo\n(global) Specifies the part of the distributed matrix sub(A) to be copied:\n= 'U': Upper triangular part is copied; the strictly lower triangular part of\nsub(A) is not referenced;\n= 'L': Lower triangular part is copied; the strictly upper triangular part of\nsub(A) is not referenced.\nOtherwise: all of the matrix sub(A) is copied.\nm\n(global)\nThe number of rows in the distributed matrix sub(A). (m ≥ 0).\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1629\n\n\nn\n(global)\nThe number of columns in the distributed matrix sub(A). (n ≥ 0).\na\n(local).\nPointer into the local memory to an array of sizelld_a * LOCc(ja+n-1).\nOn entry, this array contains the local pieces of the m-by-n distributed\nmatrix sub(A).\nia, ja\n(global) The row and column indices in the global matrix A indicating the\nfirst row and the first column of sub(A), respectively.\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\nib, jb\n(global) The row and column indices in the global matrix B indicating the\nfirst row and the first column of sub(B), respectively.\ndescb\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix B.\nOutput Parameters\nb\n(local).\nPointer into the local memory to an array of size lld_b * LOCc(jb+n-1).\nThis array contains on exit the local pieces of the distributed matrix sub( B )\nset as follows:\nif uplo = 'U', B(ib+i-1, jb+j-1) = A(ia+i-1, ja+j-1), 1≤i≤j, 1≤j≤n;\nif uplo = 'L', B(ib+i-1, jb+j-1) = A(ia+i-1, ja+j-1), j≤i≤m, 1≤j≤n;\notherwise, B(ib+i-1, jb+j-1) = A(ia+i-1, ja+j-1), 1≤i≤m, 1≤j≤n.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?lacp3\nCopies from a global parallel array into a local\nreplicated array or vice versa.\nSyntax\nvoid pslacp3 (const MKL_INT *m, const MKL_INT *i, float *a, const MKL_INT *desca, float\n*b, const MKL_INT *ldb, const MKL_INT *ii, const MKL_INT *jj, const MKL_INT *rev );\nvoid pdlacp3 (const MKL_INT *m, const MKL_INT *i, double *a, const MKL_INT *desca,\ndouble *b, const MKL_INT *ldb, const MKL_INT *ii, const MKL_INT *jj, const MKL_INT\n*rev );\nvoid pclacp3 (const MKL_INT *m, const MKL_INT *i, MKL_Complex8 *a, const MKL_INT\n*desca, MKL_Complex8 *b, const MKL_INT *ldb, const MKL_INT *ii, const MKL_INT *jj,\nconst MKL_INT *rev);\nvoid pzlacp3 (const MKL_INT *m, const MKL_INT *i, MKL_Complex16 *a, const MKL_INT\n*desca, MKL_Complex16 *b, const MKL_INT *ldb, const MKL_INT *ii, const MKL_INT *jj,\nconst MKL_INT *rev);\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1630\n\n\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThis is an auxiliary function that copies from a global parallel array into a local replicated array or vise versa.\nNote that the entire submatrix that is copied gets placed on one node or more. The receiving node can be\nspecified precisely, or all nodes can receive, or just one row or column of nodes.\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nInput Parameters\nm\n(global)\nm is the order of the square submatrix that is copied.\nm≥ 0. Unchanged on exit.\ni\n(global) The matrix element A(i, i) is the global location that the copying\nstarts from. Unchanged on exit.\na\n(local)\nArray of size lld_a*LOCc(n_a). On entry, the parallel matrix to be copied\ninto or from.\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\nb\n(local)\nArray of size ldb*LOCc(m). If rev = 0, this is the global portion of the\nmatrix A(i:i+m-1, i:i+m-1). If rev = 1, this is unchanged on exit.\nldb\n(local)\nThe leading dimension of B.\nii, jj\n(global) By using rev 0 and 1, data can be sent out and returned again. If\nrev = 0, then ii is destination row index and jj is destination column index\nfor the node(s) receiving the replicated matrixB. If ii ≥ 0, jj ≥ 0, then node\n(ii, jj) receives the data. If ii = -1, jj ≥ 0, then all rows in column jj receive\nthe data. If ii ≥ 0, jj = -1, then all cols in row ii receive the data. If ii = -1, jj\n= -1, then all nodes receive the data. If rev !=0, then ii is the source row\nindex for the node(s) sending the replicated B.\nrev\n(global) Use rev = 0 to send global matrixA into locally replicated matrixB\n(on node (ii, jj)). Use rev != 0 to send locally replicated B from node (ii, jj)\nto its owner (which changes depending on its location in A) into the global\nA.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1631\n\n\nOutput Parameters\na\nOn exit, if rev = 1, the copied data. Unchanged on exit if rev = 0.\nb\nIf rev = 1, this is unchanged on exit.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?lacpy\nCopies all or part of one two-dimensional array to\nanother.\nSyntax\nvoid pslacpy (char *uplo , MKL_INT *m , MKL_INT *n , float *a , MKL_INT *ia , MKL_INT\n*ja , MKL_INT *desca , float *b , MKL_INT *ib , MKL_INT *jb , MKL_INT *descb );\nvoid pdlacpy (char *uplo , MKL_INT *m , MKL_INT *n , double *a , MKL_INT *ia , MKL_INT\n*ja , MKL_INT *desca , double *b , MKL_INT *ib , MKL_INT *jb , MKL_INT *descb );\nvoid pclacpy (char *uplo , MKL_INT *m , MKL_INT *n , MKL_Complex8 *a , MKL_INT *ia ,\nMKL_INT *ja , MKL_INT *desca , MKL_Complex8 *b , MKL_INT *ib , MKL_INT *jb , MKL_INT\n*descb );\nvoid pzlacpy (char *uplo , MKL_INT *m , MKL_INT *n , MKL_Complex16 *a , MKL_INT *ia ,\nMKL_INT *ja , MKL_INT *desca , MKL_Complex16 *b , MKL_INT *ib , MKL_INT *jb , MKL_INT\n*descb );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?lacpyfunction copies all or part of a distributed matrix A to another distributed matrix B. No\ncommunication is performed, p?lacpy performs a local copy sub(B):= sub(A), where sub(A) denotes\nA(ia:ia+m-1,ja:ja+n-1) and sub(B) denotes B(ib:ib+m-1,jb:jb+n-1).\nInput Parameters\nuplo\n(global) Specifies the part of the distributed matrix sub(A) to be copied:\n= 'U': Upper triangular part; the strictly lower triangular part of sub(A) is\nnot referenced;\n= 'L': Lower triangular part; the strictly upper triangular part of sub(A) is\nnot referenced.\nOtherwise: all of the matrix sub(A) is copied.\nm\n(global)\nThe number of rows in the distributed matrix sub(A). (m≥0).\nn\n(global)\nThe number of columns in the distributed matrix sub(A). (n≥0).\na\n(local).\nPointer into the local memory to an array of size lld_a * LOCc(ja+n-1).\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1632\n\n\nOn entry, this array contains the local pieces of the distributed matrix\nsub(A).\nia, ja\n(global) The row and column indices in the global matrix A indicating the\nfirst row and the first column of the matrix sub(A), respectively.\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\nib, jb\n(global) The row and column indices in the global matrix B indicating the\nfirst row and the first column of sub(B) respectively.\ndescb\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\nOutput Parameters\nb\n(local).\nPointer into the local memory to an array of size lld_b * LOCc(jb+n-1).\nThis array contains on exit the local pieces of the distributed matrix sub(B)\nset as follows:\nif uplo = 'U', B(ib+i-1, jb+j-1) = A(ia+i-1, ja+j-1), 1≤i≤j, 1≤j≤n;\nif uplo = 'L', B(ib+i-1, jb+j-1) = A(ia+i-1, ja+j-1), j≤i≤m, 1≤j≤n;\notherwise, B(ib+i-1, jb+j-1) = A(ia+i-1, ja+j-1), 1≤i≤m, 1≤j≤n.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?laevswp\nMoves the eigenvectors from where they are\ncomputed to ScaLAPACK standard block cyclic array.\nSyntax\nvoid pslaevswp (MKL_INT *n , float *zin , MKL_INT *ldzi , float *z , MKL_INT *iz ,\nMKL_INT *jz , MKL_INT *descz , MKL_INT *nvs , MKL_INT *key , float *work , MKL_INT\n*lwork );\nvoid pdlaevswp (MKL_INT *n , double *zin , MKL_INT *ldzi , double *z , MKL_INT *iz ,\nMKL_INT *jz , MKL_INT *descz , MKL_INT *nvs , MKL_INT *key , double *work , MKL_INT\n*lwork );\nvoid pclaevswp (MKL_INT *n , float *zin , MKL_INT *ldzi , MKL_Complex8 *z , MKL_INT\n*iz , MKL_INT *jz , MKL_INT *descz , MKL_INT *nvs , MKL_INT *key , float *rwork ,\nMKL_INT *lrwork );\nvoid pzlaevswp (MKL_INT *n , double *zin , MKL_INT *ldzi , MKL_Complex16 *z , MKL_INT\n*iz , MKL_INT *jz , MKL_INT *descz , MKL_INT *nvs , MKL_INT *key , double *rwork ,\nMKL_INT *lrwork );\nInclude Files\n•\nmkl_scalapack.h\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1633\n\n\nDescription\nThe p?laevswpfunction moves the eigenvectors (potentially unsorted) from where they are computed, to a\nScaLAPACK standard block cyclic array, sorted so that the corresponding eigenvalues are sorted.\nInput Parameters\nnp = the number of rows local to a given process.\nnq = the number of columns local to a given process.\nn\n(global)\nThe order of the matrix A. n ≥ 0.\nzin\n(local).\nArray of size ldzi * nvs[iam+1]. The eigenvectors on input. iam is a process\nrank from [0, nprocs) interval. Each eigenvector resides entirely in one\nprocess. Each process holds a contiguous set of nvs[iam+1] eigenvectors.\nThe global number of the first eigenvector that the process holds is: ((sum\nfor i=[0, iam] of nvs[i])+1).\nldzi\n(local)\nThe leading dimension of the zin array.\niz, jz\n(global) The row and column indices in the global matrix Z indicating the\nfirst row and the first column of the submatrix Z, respectively.\ndescz\n(global and local)\nArray of size dlen_. The array descriptor for the distributed matrix Z.\nnvs\n(global)\nArray of size nprocs+1\nnvs[i] = number of eigenvectors held by processes [0, i)\nnvs[0] = number of eigenvectors held by processes [0, 0) = 0\nnvs[nprocs]= number of eigenvectors held by [0, nprocs)= total number of\neigenvectors.\nkey\n(global)\nArray of size n. Indicates the actual index (after sorting) for each of the\neigenvectors.\nrwork\n(local).\nArray of size lrwork.\nlrwork\n(local)\nSize of work.\nOutput Parameters\nz\n(local).\nArray of global size n* n and of local size lld_z * nq. The eigenvectors on\noutput. The eigenvectors are distributed in a block cyclic manner in both\ndimensions, with a block size of nb.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1634\n\n\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?lahrd\nReduces the first nb columns of a general rectangular\nmatrix A so that elements below the k-th subdiagonal\nare zero, by an orthogonal/unitary transformation,\nand returns auxiliary matrices that are needed to\napply the transformation to the unreduced part of A.\nSyntax\nvoid pslahrd (MKL_INT *n , MKL_INT *k , MKL_INT *nb , float *a , MKL_INT *ia , MKL_INT\n*ja , MKL_INT *desca , float *tau , float *t , float *y , MKL_INT *iy , MKL_INT *jy ,\nMKL_INT *descy , float *work );\nvoid pdlahrd (MKL_INT *n , MKL_INT *k , MKL_INT *nb , double *a , MKL_INT *ia , MKL_INT\n*ja , MKL_INT *desca , double *tau , double *t , double *y , MKL_INT *iy , MKL_INT *jy ,\nMKL_INT *descy , double *work );\nvoid pclahrd (MKL_INT *n , MKL_INT *k , MKL_INT *nb , MKL_Complex8 *a , MKL_INT *ia ,\nMKL_INT *ja , MKL_INT *desca , MKL_Complex8 *tau , MKL_Complex8 *t , MKL_Complex8 *y ,\nMKL_INT *iy , MKL_INT *jy , MKL_INT *descy , MKL_Complex8 *work );\nvoid pzlahrd (MKL_INT *n , MKL_INT *k , MKL_INT *nb , MKL_Complex16 *a , MKL_INT *ia ,\nMKL_INT *ja , MKL_INT *desca , MKL_Complex16 *tau , MKL_Complex16 *t , MKL_Complex16\n*y , MKL_INT *iy , MKL_INT *jy , MKL_INT *descy , MKL_Complex16 *work );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?lahrdfunction reduces the first nb columns of a real general n-by-(n-k+1) distributed matrix A(ia:ia\n+n-1 , ja:ja+n-k) so that elements below the k-th subdiagonal are zero. The reduction is performed by\nan orthogonal/unitary similarity transformation Q'*A*Q. The function returns the matrices V and T which\ndetermine Q as a block reflector I-V*T*V', and also the matrix Y = A*V*T.\nThis is an auxiliary function called by p?gehrd. In the following comments sub(A) denotes A(ia:ia+n-1,\nja:ja+n-1).\nInput Parameters\nn\n(global)\nThe order of the distributed matrix sub(A). n ≥ 0.\nk\n(global)\nThe offset for the reduction. Elements below the k-th subdiagonal in the\nfirst nb columns are reduced to zero.\nnb\n(global)\nThe number of columns to be reduced.\na\n(local).\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1635\n\n\nPointer into the local memory to an array of size lld_a * LOCc(ja+n-k). On\nentry, this array contains the local pieces of the n-by-(n-k+1) general\ndistributed matrix A(ia:ia+n-1, ja:ja+n-k).\nia, ja\n(global) The row and column indices in the global matrix A indicating the\nfirst row and the first column of the matrix sub(A), respectively.\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\niy, jy\n(global) The row and column indices in the global matrix Y indicating the\nfirst row and the first column of the matrix sub(Y), respectively.\ndescy\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix Y.\nwork\n(local).\nArray of size nb.\nOutput Parameters\na\n(local).\nOn exit, the elements on and above the k-th subdiagonal in the first nb\ncolumns are overwritten with the corresponding elements of the reduced\ndistributed matrix; the elements below the k-th subdiagonal, with the array\ntau, represent the matrix Q as a product of elementary reflectors. The other\ncolumns of the matrix A(ia:ia+n-1, ja:ja+n-k) are unchanged. (See\nApplication Notes below.)\ntau\n(local)\nArray of size LOCc(ja+n-2). The scalar factors of the elementary reflectors\n(see Application Notes below). tau is tied to the distributed matrix A.\nt\n(local)\nArray of size nb_a* nb_a. The upper triangular matrix T.\ny\n(local).\nPointer into the local memory to an array of size lld_y* nb_a. On exit, this\narray contains the local pieces of the n-by-nb distributed matrix Y. lld_y ≥\nLOCr(ia+n-1).\nApplication Notes\nThe matrix Q is represented as a product of nb elementary reflectors\nQ = H(1)*H(2)*...*H(nb).\nEach H(i) has the form\nH(i) = i-tau*v*v',\nwhere tau is a real/complex scalar, and v is a real/complex vector with v(1: i+k-1)= 0, v(i+k)= 1; v(i+k\n+1:n) is stored on exit in A(ia+i+k:ia+n-1, ja+i-1), and tau in tau[ja+i-2].\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1636\n\n\nThe elements of the vectors v together form the (n-k+1)-by-nb matrix V which is needed, with T and Y, to\napply the transformation to the unreduced part of the matrix, using an update of the form: A(ia:ia+n-1,\nja:ja+n-k) := (I-V*T*V')*(A(ia:ia+n-1, ja:ja+n-k)-Y*V'). The contents of A(ia:ia+n-1, ja:ja+n-k) on exit\nare illustrated by the following example with n = 7, k = 3, and nb = 2:\nwhere a denotes an element of the original matrix A(ia:ia+n-1, ja:ja+n-k), h denotes a modified element\nof the upper Hessenberg matrix H, and vi denotes an element of the vector defining H(i).\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?laiect\nExploits IEEE arithmetic to accelerate the\ncomputations of eigenvalues.\nSyntax\nvoid pslaiect (float *sigma , MKL_INT *n , float *d , MKL_INT *count );\nvoid pdlaiectb (float *sigma , MKL_INT *n , float *d , MKL_INT *count );\nvoid pdlaiectl (float *sigma , MKL_INT *n , float *d , MKL_INT *count );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?laiectfunction computes the number of negative eigenvalues of (A- σI). This implementation of the\nSturm Sequence loop exploits IEEE arithmetic and has no conditionals in the innermost loop. The signbit for\nreal function pslaiect is assumed to be bit 32. Double-precision functions pdlaiectb and pdlaiectl differ\nin the order of the double precision word storage and, consequently, in the signbit location. For pdlaiectb,\nthe double precision word is stored in the big-endian word order and the signbit is assumed to be bit 32. For\npdlaiectl, the double precision word is stored in the little-endian word order and the signbit is assumed to\nbe bit 64.\nThis is a ScaLAPACK internal function and arguments are not checked for unreasonable values.\nInput Parameters\nsigma\nThe shift. p?laiect finds the number of eigenvalues less than equal to\nsigma.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1637\n\n\nn\nThe order of the tridiagonal matrix T. n≥ 1.\nd\nArray of size 2n-1.\nOn entry, this array contains the diagonals and the squares of the off-\ndiagonal elements of the tridiagonal matrix T. These elements are assumed\nto be interleaved in memory for better cache performance. The diagonal\nentries of T are in the entries d[0], d[2],..., d[2n-2], while the\nsquares of the off-diagonal entries are d[1], d[3], ..., d[2n-3]. To\navoid overflow, the matrix must be scaled so that its largest entry is no\ngreater than overflow(1/2) * underflow(1/4) in absolute value, and for\ngreatest accuracy, it should not be much smaller than that.\nOutput Parameters\nn\nThe count of the number of eigenvalues of T less than or equal to sigma.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?lamve\nCopies all or part of one two-dimensional distributed\narray to another.\nSyntax\nvoid pslamve(char* uplo, MKL_INT* m, MKL_INT* n, float* a, MKL_INT* ia, MKL_INT* ja,\nMKL_INT* desca, float* b, MKL_INT* ib, MKL_INT* jb, MKL_INT* descb, float* dwork);\nvoid pdlamve(char* uplo, MKL_INT* m, MKL_INT* n, double* a, MKL_INT* ia, MKL_INT* ja,\nMKL_INT* desca, double* b, MKL_INT* ib, MKL_INT* jb, MKL_INT* descb, double* dwork);\nInclude Files\n•\nmkl_scalapack.h\nDescription\np?lamve copies all or part of a distributed matrix A to another distributed matrix B. There is no alignment\nassumptions at all except that A and B are of the same size.\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nInput Parameters\nuplo\n(global )\nSpecifies the part of the distributed matrix sub( A ) to be copied:\n= 'U': Upper triangular part is copied; the strictly lower triangular part of\nsub( A ) is not referenced;\n= 'L': Lower triangular part is copied; the strictly upper triangular part of\nsub( A ) is not referenced;\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1638\n\n\nOtherwise: All of the matrix sub( A ) is copied.\nm\n(global )\nThe number of rows to be operated on, which is the number of rows of the\ndistributed matrix sub( A ). m≥ 0.\nn\n(global )\nThe number of columns to be operated on, which is the number of columns\nof the distributed matrix sub( A ). n≥ 0.\na\n(local ) pointer into the local memory to an array of size lld_a * LOCc(ja\n+n-1) . This array contains the local pieces of the distributed matrix\nsub( A ) to be copied from.\nia\n(global )\nThe row index in the global matrix A indicating the first row of sub( A ).\nja\n(global )\nThe column index in the global matrix A indicating the first column of\nsub( A ).\ndesca\n(global and local) array of size dlen_.\nThe array descriptor for the distributed matrix A.\nib\n(global )\nThe row index in the global matrix B indicating the first row of sub( B ).\njb\n(global )\nThe column index in the global matrix B indicating the first column of\nsub( B ).\ndescb\n(global and local) array of size dlen_.\nThe array descriptor for the distributed matrix B.\ndwork\n(local workspace) array\nIf uplo = 'U' or uplo = 'L' and number of processors > 1, the length of\ndwork is at least as large as the length of b.\nOtherwise, dwork is not referenced.\nOUTPUT Parameters\nb\n(local ) pointer into the local memory to an array of size lld_b * LOCc(jb\n+n-1) . This array contains on exit the local pieces of the distributed matrix\nsub( B ).\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?lange\nReturns the value of the 1-norm, Frobenius norm,\ninfinity-norm, or the largest absolute value of any\nelement, of a general rectangular matrix.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1639\n\n\nSyntax\nfloat pslange (char *norm , MKL_INT *m , MKL_INT *n , float *a , MKL_INT *ia , MKL_INT\n*ja , MKL_INT *desca , float *work );\ndouble pdlange (char *norm , MKL_INT *m , MKL_INT *n , double *a , MKL_INT *ia , MKL_INT\n*ja , MKL_INT *desca , double *work );\nfloat pclange (char *norm , MKL_INT *m , MKL_INT *n , MKL_Complex8 *a , MKL_INT *ia ,\nMKL_INT *ja , MKL_INT *desca , float *work );\ndouble pzlange (char *norm , MKL_INT *m , MKL_INT *n , MKL_Complex16 *a , MKL_INT *ia ,\nMKL_INT *ja , MKL_INT *desca , double *work );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?langefunction returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the\nelement of largest absolute value of a distributed matrix sub(A) = A(ia:ia+m-1, ja:ja+n-1).\nInput Parameters\nnorm\n(global) Specifies what value is returned by the function:\n= 'M' or 'm': val = max(abs(Aij)), largest absolute value of the matrix\nA, it s not a matrix norm.\n= '1' or 'O' or 'o': val = norm1(A), 1-norm of the matrix A\n(maximum column sum),\n= 'I' or 'i': val = normI(A), infinity norm of the matrix A (maximum\nrow sum),\n= 'F', 'f', 'E' or 'e': val = normF(A), Frobenius norm of the matrix\nA (square root of sum of squares).\nm\n(global)\nThe number of rows in the distributed matrix sub(A). When m = 0,\np?lange is set to zero. m ≥ 0.\nn\n(global)\nThe number of columns in the distributed matrix sub(A). When n = 0,\np?lange is set to zero. n ≥ 0.\na\n(local).\nPointer into the local memory to an array of size lld_a * LOCc(ja+n-1)\ncontaining the local pieces of the distributed matrix sub(A).\nia, ja\n(global) The row and column indices in the global matrix A indicating the\nfirst row and the first column of the matrix sub(A), respectively.\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\nwork\n(local).\nArray size lwork.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1640\n\n\nlwork ≥ 0 if norm = 'M' or 'm' (not referenced),\nnq0 if norm = '1', 'O' or 'o',\nmp0 if norm = 'I' or 'i',\n0 if norm = 'F', 'f', 'E' or 'e' (not referenced),\nwhere\niroffa = mod(ia-1, mb_a), icoffa = mod(ja-1, nb_a),\niarow = indxg2p(ia, mb_a, myrow, rsrc_a, nprow),\niacol = indxg2p(ja, nb_a, mycol, csrc_a, npcol),\nmp0 = numroc(m+iroffa, mb_a, myrow, iarow, nprow),\nnq0 = numroc(n+icoffa, nb_a, mycol, iacol, npcol),\nindxg2p and numroc are ScaLAPACK tool functions; myrow, mycol, nprow,\nand npcol can be determined by calling the function blacs_gridinfo.\nOutput Parameters\nval\nThe value returned by the function.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?lanhs\nReturns the value of the 1-norm, Frobenius norm,\ninfinity-norm, or the largest absolute value of any\nelement, of an upper Hessenberg matrix.\nSyntax\nfloat pslanhs (char *norm , MKL_INT *n , float *a , MKL_INT *ia , MKL_INT *ja , MKL_INT\n*desca , float *work );\ndouble pdlanhs (char *norm , MKL_INT *n , double *a , MKL_INT *ia , MKL_INT *ja ,\nMKL_INT *desca , double *work );\nfloat pclanhs (char *norm , MKL_INT *n , MKL_Complex8 *a , MKL_INT *ia , MKL_INT *ja ,\nMKL_INT *desca , float *work );\ndouble pzlanhs (char *norm , MKL_INT *n , MKL_Complex16 *a , MKL_INT *ia , MKL_INT\n*ja , MKL_INT *desca , double *work );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?lanhsfunction returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the\nelement of largest absolute value of an upper Hessenberg distributed matrix sub(A) = A(ia:ia+m-1,\nja:ja+n-1).\nInput Parameters\nnorm\nSpecifies the value to be returned by the function:\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1641\n\n\n= 'M' or 'm': val = max(abs(Aij)), largest absolute value of the matrix\nA.\n= '1' or 'O' or 'o': val = norm1(A), 1-norm of the matrix A\n(maximum column sum),\n= 'I' or 'i': val = normI(A), infinity norm of the matrix A (maximum\nrow sum),\n= 'F', 'f', 'E' or 'e': val = normF(A), Frobenius norm of the matrix\nA (square root of sum of squares).\nn\n(global)\nThe number of columns in the distributed matrix sub(A). When n =\n0, p?lanhs is set to zero. n≥ 0.\na\n(local).\nPointer into the local memory to an array of size lld_a * LOCc(ja+n-1)\ncontaining the local pieces of the distributed matrix sub(A).\nia, ja\n(global)\nThe row and column indices in the global matrix A indicating the first row\nand the first column of the matrix sub(A), respectively.\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\nwork\n(local).\nArray of size lwork.\nlwork ≥ 0 if norm = 'M' or 'm' (not referenced),\nnq0 if norm = '1', 'O' or 'o',\nmp0 if norm = 'I' or 'i',\n0 if norm = 'F', 'f', 'E' or 'e' (not referenced),\nwhere\niroffa = mod( ia-1, mb_a ), icoffa = mod( ja-1, nb_a ),\niarow = indxg2p( ia, mb_a, myrow, rsrc_a, nprow ),\niacol = indxg2p( ja, nb_a, mycol, csrc_a, npcol ),\nmp0 = numroc( m+iroffa, mb_a, myrow, iarow, nprow ),\nnq0 = numroc( n+icoffa, nb_a, mycol, iacol, npcol ),\nindxg2p and numroc are ScaLAPACK tool functions; myrow, imycol, nprow,\nand npcol can be determined by calling the function blacs_gridinfo.\nOutput Parameters\nval\nThe value returned by the function.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1642\n\n\np?lansy, p?lanhe\nReturns the value of the 1-norm, Frobenius norm,\ninfinity-norm, or the largest absolute value of any\nelement, of a real symmetric or a complex Hermitian\nmatrix.\nSyntax\nfloat pslansy (char *norm , char *uplo , MKL_INT *n , float *a , MKL_INT *ia , MKL_INT\n*ja , MKL_INT *desca , float *work );\ndouble pdlansy (char *norm , char *uplo , MKL_INT *n , double *a , MKL_INT *ia , MKL_INT\n*ja , MKL_INT *desca , double *work );\nfloat pclansy (char *norm , char *uplo , MKL_INT *n , MKL_Complex8 *a , MKL_INT *ia ,\nMKL_INT *ja , MKL_INT *desca , float *work );\ndouble pzlansy (char *norm , char *uplo , MKL_INT *n , MKL_Complex16 *a , MKL_INT *ia ,\nMKL_INT *ja , MKL_INT *desca , double *work );\nfloat pclanhe (char *norm , char *uplo , MKL_INT *n , MKL_Complex8 *a , MKL_INT *ia ,\nMKL_INT *ja , MKL_INT *desca , float *work );\ndouble pzlanhe (char *norm , char *uplo , MKL_INT *n , MKL_Complex16 *a , MKL_INT *ia ,\nMKL_INT *ja , MKL_INT *desca , double *work );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?lansy and p?lanhefunctions return the value of the 1-norm, or the Frobenius norm, or the infinity\nnorm, or the element of largest absolute value of a distributed matrix sub(A) = A(ia:ia+m-1, ja:ja\n+n-1).\nInput Parameters\nnorm\n(global) Specifies what value is returned by the function:\n= 'M' or 'm': val = max(abs(Aij)), largest absolute value of the matrix\nA, it s not a matrix norm.\n= '1' or 'O' or 'o': val = norm1(A), 1-norm of the matrix A\n(maximum column sum),\n= 'I' or 'i': val = normI(A), infinity norm of the matrix A (maximum\nrow sum),\n= 'F', 'f', 'E' or 'e': val = normF(A), Frobenius norm of the matrix\nA (square root of sum of squares).\nuplo\n(global) Specifies whether the upper or lower triangular part of the\nsymmetric matrix sub(A) is to be referenced.\n= 'U': Upper triangular part of sub(A) is referenced,\n= 'L': Lower triangular part of sub(A) is referenced.\nn\n(global)\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1643\n\n\nThe number of columns in the distributed matrix sub(A). When n = 0,\np?lansy is set to zero. n ≥ 0.\na\n(local).\nPointer into the local memory to an array of size lld_a * LOCc(ja+n-1)\ncontaining the local pieces of the distributed matrix sub(A).\nIf uplo = 'U', the leading n-by-n upper triangular part of sub(A) contains\nthe upper triangular matrix whose norm is to be computed, and the strictly\nlower triangular part of this matrix is not referenced. If uplo = 'L', the\nleading n-by-n lower triangular part of sub(A) contains the lower triangular\nmatrix whose norm is to be computed, and the strictly upper triangular part\nof sub(A) is not referenced.\nia, ja\n(global) The row and column indices in the global matrix A indicating the\nfirst row and the first column of the matrix sub(A), respectively.\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\nwork\n(local).\nArray of size lwork.\nlwork ≥ 0 if norm = 'M' or 'm' (not referenced),\n2*nq0+mp0+ldw if norm = '1', 'O' or 'o', 'I' or 'i',\nwhere ldw is given by:\nif( nprow≠npcol ) then\nldw = mb_a*iceil(iceil(np0,mb_a),(lcm/nprow))\nelse\nldw = 0\nend if\n0 if norm = 'F', 'f', 'E' or 'e' (not referenced),\nwhere lcm is the least common multiple of nprow and npcol, lcm =\nilcm( nprow, npcol ) and iceil(x,y) is a ScaLAPACK function that\nreturns ceiling (x/y).\niroffa = mod(ia-1, mb_a ), icoffa = mod( ja-1, nb_a),\niarow = indxg2p(ia, mb_a, myrow, rsrc_a, nprow),\niacol = indxg2p(ja, nb_a, mycol, csrc_a, npcol),\nmp0 = numroc(m+iroffa, mb_a, myrow, iarow, nprow),\nnq0 = numroc(n+icoffa, nb_a, mycol, iacol, npcol),\nilcm, iceil, indxg2p, and numroc are ScaLAPACK tool functions; myrow,\nmycol, nprow, and npcol can be determined by calling the function\nblacs_gridinfo.\nOutput Parameters\nval\nThe value returned by the function.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1644\n\n\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?lantr\nReturns the value of the 1-norm, Frobenius norm,\ninfinity-norm, or the largest absolute value of any\nelement, of a triangular matrix.\nSyntax\nfloat pslantr (char *norm , char *uplo , char *diag , MKL_INT *m , MKL_INT *n , float\n*a , MKL_INT *ia , MKL_INT *ja , MKL_INT *desca , float *work );\ndouble pdlantr (char *norm , char *uplo , char *diag , MKL_INT *m , MKL_INT *n , double\n*a , MKL_INT *ia , MKL_INT *ja , MKL_INT *desca , double *work );\nfloat pclantr (char *norm , char *uplo , char *diag , MKL_INT *m , MKL_INT *n ,\nMKL_Complex8 *a , MKL_INT *ia , MKL_INT *ja , MKL_INT *desca , float *work );\ndouble pzlantr (char *norm , char *uplo , char *diag , MKL_INT *m , MKL_INT *n ,\nMKL_Complex16 *a , MKL_INT *ia , MKL_INT *ja , MKL_INT *desca , double *work );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?lantrfunction returns the value of the 1-norm, or the Frobenius norm, or the infinity norm, or the\nelement of largest absolute value of a trapezoidal or triangular distributed matrix sub(A) = A(ia:ia+m-1,\nja:ja+n-1).\nInput Parameters\nnorm\n(global) Specifies what value is returned by the function:\n= 'M' or 'm': val = max(abs(Aij)), largest absolute value of the matrix\nA, it s not a matrix norm.\n= '1' or 'O' or 'o': val = norm1(A), 1-norm of the matrix A\n(maximum column sum),\n= 'I' or 'i': val = normI(A), infinity norm of the matrix A (maximum\nrow sum),\n= 'F', 'f', 'E' or 'e': val = normF(A), Frobenius norm of the matrix\nA (square root of sum of squares).\nuplo\n(global)\nSpecifies whether the upper or lower triangular part of the symmetric\nmatrix sub(A) is to be referenced.\n= 'U': Upper trapezoidal,\n= 'L': Lower trapezoidal.\nNote that sub(A) is triangular instead of trapezoidal if m = n.\ndiag\n(global)\nSpecifies whether the distributed matrix sub(A) has unit diagonal.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1645\n\n\n= 'N': Non-unit diagonal.\n= 'U': Unit diagonal.\nm\n(global)\nThe number of rows in the distributed matrix sub(A). When m = 0,\np?lantr is set to zero. m ≥ 0.\nn\n(global)\nThe number of columns in the distributed matrix sub(A). When n = 0,\np?lantr is set to zero. n ≥ 0.\na\n(local).\nPointer into the local memory to an array of sizelld_a * LOCc(ja+n-1)\ncontaining the local pieces of the distributed matrix sub(A).\nia, ja\n(global)\nThe row and column indices in the global matrix A indicating the first row\nand the first column of the matrix sub(A), respectively.\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\nwork\n(local).\nArray size lwork.\nlwork ≥ 0 if norm = 'M' or 'm' (not referenced),\nnq0 if norm = '1', 'O' or 'o',\nmp0 if norm = 'I' or 'i',\n0 if norm = 'F', 'f', 'E' or 'e' (not referenced),\niroffa = mod(ia-1, mb_a ), icoffa = mod( ja-1, nb_a),\niarow = indxg2p(ia, mb_a, myrow, rsrc_a, nprow),\niacol = indxg2p(ja, nb_a, mycol, csrc_a, npcol),\nmp0 = numroc(m+iroffa, mb_a, myrow, iarow, nprow),\nnq0 = numroc(n+icoffa, nb_a, mycol, iacol, npcol),\nindxg2p and numroc are ScaLAPACK tool functions; myrow, mycol, nprow,\nand npcol can be determined by calling the function blacs_gridinfo.\nOutput Parameters\nval\nThe value returned by the function.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?lapiv\nApplies a permutation matrix to a general distributed\nmatrix, resulting in row or column pivoting.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1646\n\n\nSyntax\nvoid pslapiv (char *direc , char *rowcol , char *pivroc , MKL_INT *m , MKL_INT *n ,\nfloat *a , MKL_INT *ia , MKL_INT *ja , MKL_INT *desca , MKL_INT *ipiv , MKL_INT *ip ,\nMKL_INT *jp , MKL_INT *descip , MKL_INT *iwork );\nvoid pdlapiv (char *direc , char *rowcol , char *pivroc , MKL_INT *m , MKL_INT *n ,\ndouble *a , MKL_INT *ia , MKL_INT *ja , MKL_INT *desca , MKL_INT *ipiv , MKL_INT *ip ,\nMKL_INT *jp , MKL_INT *descip , MKL_INT *iwork );\nvoid pclapiv (char *direc , char *rowcol , char *pivroc , MKL_INT *m , MKL_INT *n ,\nMKL_Complex8 *a , MKL_INT *ia , MKL_INT *ja , MKL_INT *desca , MKL_INT *ipiv , MKL_INT\n*ip , MKL_INT *jp , MKL_INT *descip , MKL_INT *iwork );\nvoid pzlapiv (char *direc , char *rowcol , char *pivroc , MKL_INT *m , MKL_INT *n ,\nMKL_Complex16 *a , MKL_INT *ia , MKL_INT *ja , MKL_INT *desca , MKL_INT *ipiv , MKL_INT\n*ip , MKL_INT *jp , MKL_INT *descip , MKL_INT *iwork );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?lapivfunction applies either P (permutation matrix indicated by ipiv) or inv(P) to a general m-by-n\ndistributed matrix sub(A) = A(ia:ia+m-1, ja:ja+n-1), resulting in row or column pivoting. The pivot\nvector may be distributed across a process row or a column. The pivot vector should be aligned with the\ndistributed matrix A. This function will transpose the pivot vector, if necessary.\nFor example, if the row pivots should be applied to the columns of sub(A), pass rowcol='C' and\npivroc='C'.\nInput Parameters\ndirec\n(global)\nSpecifies in which order the permutation is applied:\n= 'F' (Forward): Applies pivots forward from top of matrix. Computes\nP*sub(A).\n= 'B' (Backward): Applies pivots backward from bottom of matrix.\nComputes inv(P)*sub(A).\nrowcol\n(global)\nSpecifies if the rows or columns are to be permuted:\n= 'R': Rows will be permuted,\n= 'C': Columns will be permuted.\npivroc\n(global)\nSpecifies whether ipiv is distributed over a process row or column:\n= 'R': ipiv is distributed over a process row,\n= 'C': ipiv is distributed over a process column.\nm\n(global)\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1647\n\n\nThe number of rows in the distributed matrix sub(A). When m = 0,\np?lapiv is set to zero. m ≥ 0.\nn\n(global)\nThe number of columns in the distributed matrix sub(A). When n = 0,\np?lapiv is set to zero. n ≥ 0.\na\n(local).\nPointer into the local memory to an array of size lld_a * LOCc(ja+n-1)\ncontaining the local pieces of the distributed matrix sub(A).\nia, ja\n(global)\nThe row and column indices in the global matrix A indicating the first row\nand the first column of the matrix sub(A), respectively.\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\nipiv\n(local)\nArray of size lipiv ;\nwhen rowcol='R' or 'r':\nlipiv≥LOCr(ia+m-1) + mb_a if pivroc='C' or 'c',\nlipiv≥LOCc(m + mod(jp-1, nb_p)) if pivroc='R' or 'r', and,\nwhen rowcol='C' or 'c':\nlipiv≥LOCr(n + mod(ip-1, mb_p)) if pivroc='C' or 'c',\nlipiv≥LOCc(ja+n-1) + nb_a if pivroc='R' or 'r'.\nThis array contains the pivoting information. ipiv(i) is the global row\n(column), local row (column) i was swapped with. When rowcol='R' or\n'r' and pivroc='C' or 'c', or rowcol='C' or 'c' and pivroc='R' or\n'r', the last piece of this array of size mb_a (resp. nb_a) is used as\nworkspace. In those cases, this array is tied to the distributed matrix A.\nip, jp\n(global) The row and column indices in the global matrix P indicating the\nfirst row and the first column of the matrix sub(P), respectively.\ndescip\n(global and local) array of size dlen_. The array descriptor for the\ndistributed vector ipiv.\niwork\n(local).\nArray of size ldw, where ldw is equal to the workspace necessary for\ntransposition, and the storage of the transposed ipiv:\nLet lcm be the least common multiple of nprow and npcol.\nif( *rowcol == 'r' &&  *pivroc == 'r') {\n      if( nprow == npcol) {\nldw = LOCr( n_p + (*jp-1)%nb_p ) + nb_p;\n     } else {\nldw = LOCr( n_p + (*jp-1)%nb_p )+\nnb_p * ceil( ceil(LOCc(n_p)/nb_p) / (lcm/npcol) );\n     }\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1648\n\n\n} else if( *rowcol == 'c' &&  *pivroc == 'c') {\n       if( nprow == npcol ) {\nldw = LOCc( m_p + (*ip-1)%mb_p ) + mb_p;\n       } else {\nldw = LOCc( m_p + (*ip-1)%mb_p ) +\nmb_p *ceil(ceil(LOCr(m_p)/mb_p) / (lcm/nprow) );\n      }\n} else {\n// iwork is not referenced.\nOutput Parameters\na\n(local).\nOn exit, the local pieces of the permuted distributed submatrix.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?lapv2\nApplies a permutation to an m-by-n distributed matrix.\nSyntax\nvoid pslapv2 (const char* direc, const char* rowcol, const MKL_INT* m, const MKL_INT*\nn, float* a, const MKL_INT* ia, const MKL_INT* ja, const MKL_INT* desca, const MKL_INT*\nipiv, const MKL_INT* ip, const MKL_INT* jp, const MKL_INT* descip);\nvoid pdlapv2 (const char* direc, const char* rowcol, const MKL_INT* m, const MKL_INT*\nn, double* a, const MKL_INT* ia, const MKL_INT* ja, const MKL_INT* desca, const\nMKL_INT* ipiv, const MKL_INT* ip, const MKL_INT* jp, const MKL_INT* descip);\nvoid pclapv2 (const char* direc, const char* rowcol, const MKL_INT* m, const MKL_INT*\nn, MKL_Complex8* a, const MKL_INT* ia, const MKL_INT* ja, const MKL_INT* desca, const\nMKL_INT* ipiv, const MKL_INT* ip, const MKL_INT* jp, const MKL_INT* descip);\nvoid pzlapv2 (const char* direc, const char* rowcol, const MKL_INT* m, const MKL_INT*\nn, MKL_Complex16* a, const MKL_INT* ia, const MKL_INT* ja, const MKL_INT* desca, const\nMKL_INT* ipiv, const MKL_INT* ip, const MKL_INT* jp, const MKL_INT* descip);\nInclude Files\n•\nmkl_scalapack.h\nDescription\np?lapv2 applies either P (permutation matrix indicated by ipiv) or inv( P ) to an m-by-n distributed matrix\nsub( A ) denoting A(ia:ia+m-1,ja:ja+n-1), resulting in row or column pivoting. The pivot vector should be\naligned with the distributed matrix A. For pivoting the rows of sub( A ), ipiv should be distributed along a\nprocess column and replicated over all process rows. Similarly, ipiv should be distributed along a process\nrow and replicated over all process columns for column pivoting.\nInput Parameters\ndirec\n(global)\nSpecifies in which order the permutation is applied:\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1649\n\n\n= 'F' (Forward) Applies pivots Forward from top of matrix. Computes P *\nsub( A );\n= 'B' (Backward) Applies pivots Backward from bottom of matrix. Computes\ninv( P ) * sub( A ).\nrowcol\n(global)\nSpecifies if the rows or columns are to be permuted:\n= 'R' Rows will be permuted,\n= 'C' Columns will be permuted.\nm\n(global)\nThe number of rows to be operated on, i.e. the number of rows of the\ndistributed submatrix sub( A ). m >= 0.\nn\n(global)\nThe number of columns to be operated on, i.e. the number of columns of\nthe distributed submatrix sub( A ). n >= 0.\na\nPointer into local memory to an array of size lld_a*LOCc(ja+n-1) .\nOn entry, this local array contains the local pieces of the distributed matrix\nsub( A ) to which the row or columns interchanges will be applied.\nia\n(global)\nThe row index in the global array a indicating the first row of sub( A ).\nja\n(global)\nThe column index in the global array a indicating the first column of\nsub( A ).\ndesca\n(global and local)\nArray of size dlen_.\nThe array descriptor for the distributed matrix A.\nipiv\nArray, size >= LOCr(m_a)+mb_a if rowcol = 'R', LOCc(n_a)+nb_a\notherwise.\nIt contains the pivoting information. ipiv[i - 1] is the global row (column),\nlocal row (column) i was swapped with. The last piece of the array of size\nmb_a or nb_a is used as workspace. ipiv is tied to the distributed matrix\nA.\nip\n(global)\nThe global row index of ipiv, which points to the beginning of the\nsubmatrix on which to operate.\njp\n(global)\nThe global column index of ipiv, which points to the beginning of the\nsubmatrix on which to operate.\ndescip\n(global and local)\nArray of size 8.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1650\n\n\nThe array descriptor for the distributed matrix ipiv.\nOutput Parameters\na\nOn exit, this array contains the local pieces of the permuted\ndistributed matrix.\np?laqge\nScales a general rectangular matrix, using row and\ncolumn scaling factors computed by p?geequ .\nSyntax\nvoid pslaqge (MKL_INT *m , MKL_INT *n , float *a , MKL_INT *ia , MKL_INT *ja , MKL_INT\n*desca , float *r , float *c , float *rowcnd , float *colcnd , float *amax , char\n*equed );\nvoid pdlaqge (MKL_INT *m , MKL_INT *n , double *a , MKL_INT *ia , MKL_INT *ja , MKL_INT\n*desca , double *r , double *c , double *rowcnd , double *colcnd , double *amax , char\n*equed );\nvoid pclaqge (MKL_INT *m , MKL_INT *n , MKL_Complex8 *a , MKL_INT *ia , MKL_INT *ja ,\nMKL_INT *desca , float *r , float *c , float *rowcnd , float *colcnd , float *amax ,\nchar *equed );\nvoid pzlaqge (MKL_INT *m , MKL_INT *n , MKL_Complex16 *a , MKL_INT *ia , MKL_INT *ja ,\nMKL_INT *desca , double *r , double *c , double *rowcnd , double *colcnd , double\n*amax , char *equed );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?laqgefunction equilibrates a general m-by-n distributed matrix sub(A) = A(ia:ia+m-1, ja:ja+n-1)\nusing the row and scaling factors in the vectors r and c computed by p?geequ.\nInput Parameters\nm\n(global)\nThe number of rows in the distributed matrix sub(A). (m ≥0).\nn\n(global)\nThe number of columns in the distributed matrix sub(A). (n ≥0).\na\n(local).\nPointer into the local memory to an array of size lld_a * LOCc(ja+n-1).\nOn entry, this array contains the distributed matrix sub(A).\nia, ja\n(global) The row and column indices in the global matrix A indicating the\nfirst row and the first column of the matrix sub(A), respectively.\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1651\n\n\nr\n(local).\nArray of size LOCr(m_a). The row scale factors for sub(A). r is aligned with\nthe distributed matrix A, and replicated across every process column. r is\ntied to the distributed matrix A.\nc\n(local).\nArray of size LOCc(n_a). The row scale factors for sub(A). c is aligned with\nthe distributed matrix A, and replicated across every process column. c is\ntied to the distributed matrix A.\nrowcnd\n(local).\nThe global ratio of the smallest r[i] to the largest r[i] , ia-1 ≤ i ≤ ia\n+m-2.\ncolcnd\n(local).\nThe global ratio of the smallest c[i] to the largest c[i], ia-1 ≤ i ≤ ia+n-2.\namax\n(global).\nAbsolute value of largest distributed submatrix entry.\nOutput Parameters\na\n(local).\nOn exit, the equilibrated distributed matrix. See equed for the form of the\nequilibrated distributed submatrix.\nequed\n(global)\nSpecifies the form of equilibration that was done.\n= 'N': No equilibration\n= 'R': Row equilibration, that is, sub(A) has been pre-multiplied by\ndiag(r[ia-1:ia+m-2]),\n= 'C': column equilibration, that is, sub(A) has been post-multiplied by\ndiag(c[ja-1:ja+n-2]),\n= 'B': Both row and column equilibration, that is, sub(A) has been\nreplaced by diag(r[ia-1:ia+m-2])* sub(A) * diag(c[ja-1:ja\n+n-2]).\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?laqr0\nComputes the eigenvalues of a Hessenberg matrix and\noptionally returns the matrices from the Schur\ndecomposition.\nSyntax\nvoid pslaqr0(MKL_INT* wantt, MKL_INT* wantz, MKL_INT* n, MKL_INT* ilo, MKL_INT* ihi,\nfloat* h, MKL_INT* desch, float* wr, float* wi, MKL_INT* iloz, MKL_INT* ihiz, float* z,\nMKL_INT* descz, float* work, MKL_INT* lwork, MKL_INT* iwork, MKL_INT* liwork, MKL_INT*\ninfo, MKL_INT* reclevel);\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1652\n\n\nvoid pdlaqr0(MKL_INT* wantt, MKL_INT* wantz, MKL_INT* n, MKL_INT* ilo, MKL_INT* ihi,\ndouble* h, MKL_INT* desch, double* wr, double* wi, MKL_INT* iloz, MKL_INT* ihiz, double*\nz, MKL_INT* descz, double* work, MKL_INT* lwork, MKL_INT* iwork, MKL_INT* liwork,\nMKL_INT* info, MKL_INT* reclevel);\nInclude Files\n•\nmkl_scalapack.h\nDescription\np?laqr0 computes the eigenvalues of a Hessenberg matrix H and, optionally, the matrices T and Z from the\nSchur decomposition H = Z*T*ZT, where T is an upper quasi-triangular matrix (the Schur form), and Z is the\northogonal matrix of Schur vectors.\nOptionally Z may be postmultiplied into an input orthogonal matrix Q so that this function can give the Schur\nfactorization of a matrix A which has been reduced to the Hessenberg form H by the orthogonal matrix Q: A\n= Q * H * QT = (QZ) * T * (QZ)T.\nInput Parameters\nwantt\n(global )\nNon-zero : the full Schur form T is required;\nZero : only eigenvalues are required.\nwantz\n(global )\nNon-zero : the matrix of Schur vectors Z is required;\nZero: Schur vectors are not required.\nn\n(global )\nThe order of the Hessenberg matrix H (and Z if wantzis non-zero). n≥ 0.\nilo, ihi\n(global )\nIt is assumed that the matrix H is already upper triangular in rows and\ncolumns 1:ilo-1 and ihi+1:n. ilo and ihi are normally set by a previous\ncall to p?gebal, and then passed to p?gehrd when the matrix output by\nihi is reduced to Hessenberg form. Otherwise ilo and ihi should be set\nto 1 and n, respectively. If n > 0, then 1 ≤ilo≤ihi≤n.\nIf n = 0, then ilo = 1 and ihi = 0.\nh\n(global ) array of size lld_h * LOCc(n)\nThe upper Hessenberg matrix H.\ndesch\n(global and local )\nArray of size dlen_.\nThe array descriptor for the distributed matrix H.\niloz, ihiz\nSpecify the rows of the matrix Z to which transformations must be applied if\nwantz is non-zero, 1 ≤iloz≤ilo; ihi≤ihiz≤n.\nz\nArray of size lld_z * LOCc(n).\nIf wantz is non-zero, contains the matrix Z.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1653\n\n\nIf wantzequals zero, z is not referenced.\ndescz\n(global and local ) array of size dlen_.\nThe array descriptor for the distributed matrix Z.\nwork\n(local workspace) array of size lwork\nlwork\n(local )\nThe length of the workspace array work.\niwork\n(local workspace) array of size liwork\nliwork\n(local )\nThe length of the workspace array iwork.\nreclevel\n(local )\nLevel of recursion. reclevel = 0 must hold on entry.\nOUTPUT Parameters\nh\nOn exit, if wantt is non-zero, the matrix H is upper quasi-triangular in rows\nand columns ilo:ihi, with 1-by-1 and 2-by-2 blocks on the main diagonal.\nThe 2-by-2 diagonal blocks (corresponding to complex conjugate pairs of\neigenvalues) are returned in standard form, with H(i,i) = H(i+1,i+1) and H(i\n+1,i)*H(i,i+1) < 0. If info = 0 and wanttequals zero, the contents of h\nare unspecified on exit.\nwr, wi\nThe real and imaginary parts, respectively, of the computed eigenvalues\nilo to ihi are stored in the corresponding elements of wr and wi. If two\neigenvalues are computed as a complex conjugate pair, they are stored in\nconsecutive elements of wr and wi, say the i-th and (i+1)th, with wi[i-1] >\n0 and wi[i] < 0. If wantt is non-zero, the eigenvalues are stored in the\nsame order as on the diagonal of the Schur form returned in h.\nz\nUpdated matrix with transformations applied only to the submatrix \nZ(ilo:ihi,ilo:ihi).\nIf COMPZ = 'I', on exit, if info = 0, z contains the orthogonal matrix Z of\nthe Schur vectors of H.\nIf wantz is non-zero, then Z(ilo:ihi,iloz:ihiz) is replaced by\nZ(ilo:ihi,iloz:ihiz)*U, when U is the orthogonal/unitary Schur factor of\nH(ilo:ihi,ilo:ihi).\nIf wantzequals zero, then z is not defined.\nwork[0]\nOn exit, if info = 0, work[0] returns the optimal lwork.\niwork[0]\nOn exit, if info = 0, iwork[0] returns the optimal liwork.\ninfo\n> 0: if info = i, then the function failed to compute all the eigenvalues.\nElements 0:ilo-2 and i:n-1 of wr and wi contain those eigenvalues which\nhave been successfully computed.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1654\n\n\n> 0: if wanttequals zero, then the remaining unconverged eigenvalues are\nthe eigenvalues of the upper Hessenberg matrix rows and columns ilo\nthrough ihi of the final output value of H.\n> 0: if wantt is non-zero, then (initial value of H)*U = U*(final value of H),\nwhere U is an orthogonal/unitary matrix. The final value of H is upper\nHessenberg and quasi-triangular/triangular in rows and columns info+1\nthrough ihi.\n> 0: if wantz is non-zero, then (final value of \nZ(ilo:ihi,iloz:ihiz))=(initial value of Z(ilo:ihi,iloz:ihiz))*U, where\nU is the orthogonal/unitary matrix in the previous expression (regardless of\nthe value of wantt).\n> 0: if wantzequals zero, then z is not accessed.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?laqr1\nSets a scalar multiple of the first column of the\nproduct of a 2-by-2 or 3-by-3 matrix and specified\nshifts.\nSyntax\nvoid pslaqr1(MKL_INT* wantt, MKL_INT* wantz, MKL_INT* n, MKL_INT* ilo, MKL_INT* ihi,\nfloat* a, MKL_INT* desca, float* wr, float* wi, MKL_INT* iloz, MKL_INT* ihiz, float* z,\nMKL_INT* descz, float* work, MKL_INT* lwork, MKL_INT* iwork, MKL_INT* ilwork, MKL_INT*\ninfo);\nvoid pdlaqr1(MKL_INT* wantt, MKL_INT* wantz, MKL_INT* n, MKL_INT* ilo, MKL_INT* ihi,\ndouble* a, MKL_INT* desca, double* wr, double* wi, MKL_INT* iloz, MKL_INT* ihiz, double*\nz, MKL_INT* descz, double* work, MKL_INT* lwork, MKL_INT* iwork, MKL_INT* ilwork,\nMKL_INT* info);\nInclude Files\n•\nmkl_scalapack.h\nDescription\np?laqr1 is an auxiliary function used to find the Schur decomposition and/or eigenvalues of a matrix already\nin Hessenberg form from columns ilo to ihi.\nThis is a modified version of p?lahqr from ScaLAPACK version 1.7.3. The following modifications were\nmade:\n•\nWorkspace query functionality was added.\n•\nAggressive early deflation is implemented.\n•\nAggressive deflation (looking for two consecutive small subdiagonal elements by PSLACONSB) is\nabandoned.\n•\nThe returned Schur form is now in canonical form, i.e., the returned 2-by-2 blocks really correspond to\ncomplex conjugate pairs of eigenvalues.\n•\nFor some reason, the original version of p?lahqr sometimes did not read out the converged eigenvalues\ncorrectly. This is now fixed.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1655\n\n\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nInput Parameters\nwantt\n(global )\nNon-zero : the full Schur form T is required;\nZero: only eigenvalues are required.\nwantz\nNon-zero : the matrix of Schur vectors Z is required;\nZero: Schur vectors are not required.\nn\n(global )\nThe order of the Hessenberg matrix A (and Z if wantzis non-zero). n≥ 0.\nilo, ihi\n(global )\nIt is assumed that the matrix A is already upper quasi-triangular in rows\nand columns ihi+1:n, and that A(ilo,ilo-1) = 0 (unless ilo = 1).\np?laqr1 works primarily with the Hessenberg submatrix in rows and\ncolumns ilo to ihi, but applies transformations to all of H if wantt is non-\nzero.\n1 ≤ilo≤ max(1,ihi); ihi≤n.\na\n(global ) array of size lld_a * LOCc(n)\nOn entry, the upper Hessenberg matrix A.\ndesca\n(global and local ) array of size dlen_.\nThe array descriptor for the distributed matrix A.\niloz, ihiz\n(global )\nSpecify the rows of the matrix Z to which transformations must be applied if\nwantz is non-zero.\n1 ≤iloz≤ilo; ihi≤ihiz≤n.\nz\n(global ) array of size lld_z * LOCc(n).\nIf wantz is non-zero, on entry z must contain the current matrix Z of\ntransformations accumulated by p?hseqr\nIf wantz is zero, z is not referenced.\ndescz\n(global and local ) array of size dlen_.\nThe array descriptor for the distributed matrix Z.\nwork\n(local output) array of size lwork\nlwork\n(local )\nThe size of the work array (lwork>=1).\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1656\n\n\nIf lwork=-1, then a workspace query is assumed.\niwork\n(global and local ) array of size ilwork\nThis holds the some of the IBLK integer arrays.\nilwork\n(local )\nThe size of the iwork array (ilwork≥ 3 ).\nOUTPUT Parameters\na\nIf wantt is non-zero, the matrix A is upper quasi-triangular in rows and\ncolumns ilo:ihi, with any 2-by-2 or larger diagonal blocks not yet in\nstandard form. If wanttequals zero, the contents of a are unspecified on\nexit.\nwr, wi\n(global replicated ) array of size n\nThe real and imaginary parts, respectively, of the computed eigenvalues\nilo to ihi are stored in the corresponding elements of wr and wi. If two\neigenvalues are computed as a complex conjugate pair, they are stored in\nconsecutive elements of wr and wi, say the i-th and (i+1)th, with wi[i-1] >\n0 and wi[i] < 0. If wantt is non-zero, the eigenvalues are stored in the\nsame order as on the diagonal of the Schur form returned in a. a may be\nreturned with larger diagonal blocks until the next release.\nz\nOn exit z is updated; transformations are applied only to the submatrix\nZ(iloz:ihiz,ilo:ihi).\nIf wantzequals zero, z is not referenced.\nwork[0]\nOn exit, if info = 0, work[0] returns the optimal lwork.\ninfo\n(global )\n< 0: parameter number -info incorrect or inconsistent\n= 0: successful exit\n> 0: p?laqr1 failed to compute all the eigenvalues ilo to ihi in a total of\n30*(ihi-ilo+1) iterations; if info = i, elements i:ihi-1 of wr and wi\ncontain those eigenvalues which have been successfully computed.\nApplication Notes\nThis algorithm is very similar to p?ahqr. Unlike p?lahqr, instead of sending one double shift through the\nlargest unreduced submatrix, this algorithm sends multiple double shifts and spaces them apart so that there\ncan be parallelism across several processor row/columns. Another critical difference is that this algorithm\naggregrates multiple transforms together in order to apply them in a block fashion.\nCurrent Notes and/or Restrictions:\n•\nThis code requires the distributed block size to be square and at least six (6); unlike simpler codes like\nLU, this algorithm is extremely sensitive to block size. Unwise choices of too small a block size can lead to\nbad performance.\n•\nThis code requires a and z to be distributed identically and have identical contxts.\n•\nThis release currently does not have a function for resolving the Schur blocks into regular 2x2 form after\nthis code is completed. Because of this, a significant performance impact is required while the deflation is\ndone by sometimes a single column of processors.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1657\n\n\n•\nThis code does not currently block the initial transforms so that none of the rows or columns for any bulge\nare completed until all are started. To offset pipeline start-up it is recommended that at least\n2*LCM(NPROW,NPCOL) bulges are used (if possible)\n•\nThe maximum number of bulges currently supported is fixed at 32. In future versions this will be limited\nonly by the incoming work array.\n•\nThe matrix A must be in upper Hessenberg form. If elements below the subdiagonal are nonzero, the\nresulting transforms may be nonsimilar. This is also true with the LAPACK function.\n•\nFor this release, it is assumed rsrc_=csrc_=0\n•\nCurrently, all the eigenvalues are distributed to all the nodes. Future releases will probably distribute the\neigenvalues by the column partitioning.\n•\nThe internals of this function are subject to change.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?laqr2\nPerforms the orthogonal/unitary similarity\ntransformation of a Hessenberg matrix to detect and\ndeflate fully converged eigenvalues from a trailing\nprincipal submatrix (aggressive early deflation).\nSyntax\nvoid pslaqr2(MKL_INT* wantt, MKL_INT* wantz, MKL_INT* n, MKL_INT* ktop, MKL_INT* kbot,\nMKL_INT* nw, float* a, MKL_INT* desca, MKL_INT* iloz, MKL_INT* ihiz, float* z, MKL_INT*\ndescz, MKL_INT* ns, MKL_INT* nd, float* sr, float* si, float* t, MKL_INT* ldt, float* v,\nMKL_INT* ldv, float* wr, float* wi, float* work, MKL_INT* lwork);\nvoid pdlaqr2(MKL_INT* wantt, MKL_INT* wantz, MKL_INT* n, MKL_INT* ktop, MKL_INT* kbot,\nMKL_INT* nw, double* a, MKL_INT* desca, MKL_INT* iloz, MKL_INT* ihiz, double* z,\nMKL_INT* descz, MKL_INT* ns, MKL_INT* nd, double* sr, double* si, double* t, MKL_INT*\nldt, double* v, MKL_INT* ldv, double* wr, double* wi, double* work, MKL_INT* lwork);\nInclude Files\n•\nmkl_scalapack.h\nDescription\np?laqr2 accepts as input an upper Hessenberg matrix A and performs an orthogonal similarity\ntransformation designed to detect and deflate fully converged eigenvalues from a trailing principal submatrix.\nOn output Ais overwritten by a new Hessenberg matrix that is a perturbation of an orthogonal similarity\ntransformation of A. It is to be hoped that the final version of A has many zero subdiagonal entries.\nThis function handles small deflation windows which is affordable by one processor. Normally, it is called by \np?laqr1. All the inputs are assumed to be valid without checking.\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nInput Parameters\nwantt\n(global )\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1658\n\n\nIf wantt is non-zero, then the Hessenberg matrix A is fully updated so that\nthe quasi-triangular Schur factor may be computed (in cooperation with the\ncalling function).\nIf wantt equals zero, then only enough of A is updated to preserve the\neigenvalues.\nwantz\n(global )\nIf wantz is non-zero, then the orthogonal matrix Z is updated so that the\northogonal Schur factor may be computed (in cooperation with the calling\nfunction).\nIf wantz equals zero, then z is not referenced.\nn\n(global )\nThe order of the matrix A and (if wantz is non-zero) the order of the\northogonal matrix Z.\nktop, kbot\n(global )\nIt is assumed without a check that either kbot = n or A(kbot+1,kbot)=0.\nkbot and ktop together determine an isolated block along the diagonal of\nthe Hessenberg matrix. However, A(ktop,ktop-1)=0 is not essentially\nnecessary if wantt is non-zero .\nnw\n(global )\nDeflation window size. 1 ≤nw≤ (kbot-ktop+1). Normally nw≥ 3 if p?laqr2 is\ncalled by p?laqr1.\na\n(local ) array of size lld_a * LOCc(n)\nThe initial n-by-n section of a stores the Hessenberg matrix undergoing\naggressive early deflation.\ndesca\n(global and local) array of size dlen_.\nThe array descriptor for the distributed matrix A.\niloz, ihiz\n(global )\nSpecify the rows of the matrix Zto which transformations must be applied if\nwantz is non-zero. 1 ≤iloz≤ihiz≤n.\nz\nArray of size lld_z * LOCc(n)\nIf wantz is non-zero, then on output, the orthogonal similarity\ntransformation mentioned above has been accumulated into the matrix\nZ(iloz:ihiz,\nkbot:ktop), stored in z, from the right.\nIf wantz is zero, then z is unreferenced.\ndescz\n(global and local) array of size dlen_.\nThe array descriptor for the distributed matrix Z.\nt\n(local workspace) array of size ldt * nw.\nldt\n(local )\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1659\n\n\nThe leading dimension of the array t. ldt≥nw.\nv\n(local workspace) array of size ldv * nw.\nldv\n(local )\nThe leading dimension of the array v. ldv≥nw.\nwr, wi\n(local workspace) array of size kbot.\nwork\n(local workspace) array of size lwork.\nlwork\n(local )\nwork(lwork) is a local array and lwork is assumed big enough so that\nlwork≥nw*nw.\nOUTPUT Parameters\na\nOn output a has been transformed by an orthogonal similarity\ntransformation, perturbed, and returned to Hessenberg form that (it is to be\nhoped) has some zero subdiagonal entries.\nz\nns\n(global )\nThe number of unconverged (that is, approximate) eigenvalues returned in\nsr and si that may be used as shifts by the calling function.\nnd\n(global )\nThe number of converged eigenvalues uncovered by this function.\nsr, si\n(global ) array of size kbot\nOn output, the real and imaginary parts of approximate eigenvalues that\nmay be used for shifts are stored in sr[kbot-nd-ns] through sr[kbot-\nnd-1] and si[kbot-nd-ns] through si[kbot-nd-1], respectively.\nOn processor #0, the real and imaginary parts of converged eigenvalues\nare stored in sr[kbot-nd] through sr[kbot-1] and si[kbot-nd] through\nsi[kbot-1], respectively. On other processors, these entries are set to\nzero.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?laqr3\nPerforms the orthogonal/unitary similarity\ntransformation of a Hessenberg matrix to detect and\ndeflate fully converged eigenvalues from a trailing\nprincipal submatrix (aggressive early deflation).\nSyntax\nvoid pslaqr3(MKL_INT* wantt, MKL_INT* wantz, MKL_INT* n, MKL_INT* ktop, MKL_INT* kbot,\nMKL_INT* nw, float* h, MKL_INT* desch, MKL_INT* iloz, MKL_INT* ihiz, float* z, MKL_INT*\ndescz, MKL_INT* ns, MKL_INT* nd, float* sr, float* si, float* v, MKL_INT* descv,\nMKL_INT* nh, float* t, MKL_INT* desct, MKL_INT* nv, float* wv, MKL_INT* descw, float*\nwork, MKL_INT* lwork, MKL_INT* iwork, MKL_INT* liwork, MKL_INT* reclevel);\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1660\n\n\nvoid pdlaqr3(MKL_INT* wantt, MKL_INT* wantz, MKL_INT* n, MKL_INT* ktop, MKL_INT* kbot,\nMKL_INT* nw, double* h, MKL_INT* desch, MKL_INT* iloz, MKL_INT* ihiz, double* z,\nMKL_INT* descz, MKL_INT* ns, MKL_INT* nd, double* sr, double* si, double* v, MKL_INT*\ndescv, MKL_INT* nh, double* t, MKL_INT* desct, MKL_INT* nv, double* wv, MKL_INT* descw,\ndouble* work, MKL_INT* lwork, MKL_INT* iwork, MKL_INT* liwork, MKL_INT* reclevel);\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThis function accepts as input an upper Hessenberg matrix H and performs an orthogonal similarity\ntransformation designed to detect and deflate fully converged eigenvalues from a trailing principal submatrix.\nOn output H is overwritten by a new Hessenberg matrix that is a perturbation of an orthogonal similarity\ntransformation of H. It is to be hoped that the final version of H has many zero subdiagonal entries.\nInput Parameters\nwantt\n(global )\nIf wantt is non-zero, then the Hessenberg matrix H is fully updated so that\nthe quasi-triangular Schur factor may be computed (in cooperation with the\ncalling function).\nIf wantt equals zero, then only enough of H is updated to preserve the\neigenvalues.\nwantz\n(global )\nIf wantz is non-zero, then the orthogonal matrix Z is updated so that the\northogonal Schur factor may be computed (in cooperation with the calling\nfunction).\nIf wantz equals zero, then z is not referenced.\nn\n(global )\nThe order of the matrix H and (if wantz is non-zero), the order of the\northogonal matrix Z.\nktop\n(global )\nIt is assumed that either ktop = 1 or H (ktop,ktop-1)=0. kbot and ktop\ntogether determine an isolated block along the diagonal of the Hessenberg\nmatrix.\nkbot\n(global )\nIt is assumed without a check that either kbot = n or H (kbot+1,kbot)=0.\nkbot and ktop together determine an isolated block along the diagonal of\nthe Hessenberg matrix.\nnw\n(global )\nDeflation window size. 1 ≤nw≤ (kbot-ktop+1).\nh\n(local ) array of size lld_h * LOCc(n)\nThe initial n-by-n section of H stores the Hessenberg matrix undergoing\naggressive early deflation.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1661\n\n\ndesch\n(global and local) array of size dlen_.\nThe array descriptor for the distributed matrix H.\niloz, ihiz\n(global )\nSpecify the rows of the matrix Z to which transformations must be applied if\nwantz is non-zero. 1 ≤iloz≤ihiz≤n.\nz\nArray of size lld_z * LOCc(n)\nIf wantz is non-zero, then on output, the orthogonal similarity\ntransformation mentioned above has been accumulated into the matrix\nZ(iloz:ihiz,kbot:ktop) from the right.\nIf wantz is zero, then z is unreferenced.\ndescz\n(global and local) array of size dlen_.\nThe array descriptor for the distributed matrix Z.\nv\n(global workspace) array of size lld_v * LOCcnw)\nAn nw-by-nw distributed work array.\ndescv\n(global and local) array of size dlen_.\nThe array descriptor for the distributed matrix V.\nnh\nThe number of columns of t. nh≥nw.\nt\n(global workspace) array of size lld_t * LOCc(nh)\ndesct\n(global and local) array of size dlen_.\nThe array descriptor for the distributed matrix T.\nnv\n(global )\nThe number of rows of work array wv available for workspace. nv≥nw.\nwv\n(global workspace) array of size lld_w *LOCc(nw)\ndescw\n(global and local) array of size dlen_.\nThe array descriptor for the distributed matrix wv.\nwork\n(local workspace) array of size lwork.\nlwork\n(local )\nThe size of the work array work (lwork≥1). lwork = 2*nw suffices, but\ngreater efficiency may result from larger values of lwork.\nIf lwork = -1, then a workspace query is assumed; p?laqr3 only\nestimates the optimal workspace size for the given values of n, nw, ktop\nand kbot. The estimate is returned in work[0]. No error message related to\nlwork is issued by xerbla. Neither h nor z are accessed.\niwork\n(local workspace) array of size liwork\nliwork\n(local )\nThe length of the workspace array iwork (liwork≥1).\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1662\n\n\nIf liwork=-1, then a workspace query is assumed.\nOUTPUT Parameters\nh\nOn output h has been transformed by an orthogonal similarity\ntransformation, perturbed, and the returned to Hessenberg form that (it is\nto be hoped) has some zero subdiagonal entries.\nz\nIF wantz is non-zero, then on output, the orthogonal similarity\ntransformation mentioned above has been accumulated into the matrix\nZ(iloz:ihiz,kbot:ktop) from the right.\nIf wantz is zero, then z is unreferenced.\nns\n(global )\nThe number of unconverged (that is, approximate) eigenvalues returned in\nsr and si that may be used as shifts by the calling function.\nnd\n(global )\nThe number of converged eigenvalues uncovered by this function.\nsr, si\n(global ) array of size kbot. The real and imaginary parts of approximate\neigenvalues that may be used for shifts are stored in sr[kbot-nd-ns]\nthrough sr[kbot-nd-1] and si[kbot-nd-ns] through si[kbot-nd-1],\nrespectively. The real and imaginary parts of converged eigenvalues are\nstored in sr[kbot-nd] through sr[kbot-1] and si[kbot-nd] through\nsi[kbot-1], respectively.\nwork[0]\nOn exit, if info = 0, work[0] returns the optimal lwork\niwork[0]\nOn exit, if info = 0, iwork[0] returns the optimal liwork\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?laqr5\nPerforms a single small-bulge multi-shift QR sweep.\nSyntax\nvoid pslaqr5(MKL_INT* wantt, MKL_INT* wantz, MKL_INT* kacc22, MKL_INT* n, MKL_INT*\nktop, MKL_INT* kbot, MKL_INT* nshfts, float* sr, float* si, float* h, MKL_INT* desch,\nMKL_INT* iloz, MKL_INT* ihiz, float* z, MKL_INT* descz, float* work, MKL_INT* lwork,\nMKL_INT* iwork, MKL_INT* liwork);\nvoid pdlaqr5(MKL_INT* wantt, MKL_INT* wantz, MKL_INT* kacc22, MKL_INT* n, MKL_INT*\nktop, MKL_INT* kbot, MKL_INT* nshfts, double* sr, double* si, double* h, MKL_INT* desch,\nMKL_INT* iloz, MKL_INT* ihiz, double* z, MKL_INT* descz, double* work, MKL_INT* lwork,\nMKL_INT* iwork, MKL_INT* liwork);\nInclude Files\n•\nmkl_scalapack.h\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1663\n\n\nDescription\nThis auxiliary function called by p?laqr0 performs a single small-bulge multi-shift QR sweep by chasing\nseparated groups of bulges along the main block diagonal of a Hessenberg matrix H.\nInput Parameters\nwantt\n(global) scalar\nwanttis non-zero if the quasi-triangular Schur factor is being\ncomputed. wantt is set to zero otherwise.\nwantz\n(global) scalar\nwantzis non-zero if the orthogonal Schur factor is being computed.\nwantz is set to zero otherwise.\nkacc22\n(global)\nValue 0, 1, or 2. Specifies the computation mode of far-from-diagonal\northogonal updates.\n= 0: p?laqr5 does not accumulate reflections and does not use\nmatrix-matrix multiply to update far-from-diagonal matrix entries.\n= 1: p?laqr5 accumulates reflections and uses matrix-matrix multiply\nto update the far-from-diagonal matrix entries.\n= 2: p?laqr5 accumulates reflections, uses matrix-matrix multiply to\nupdate the far-from-diagonal matrix entries, and takes advantage of\n2-by-2 block structure during matrix multiplies.\nn\n(global) scalar\nThe order of the Hessenberg matrix H and, if wantzis non-zero, the\norder of the orthogonal matrix Z.\nktop, kbot\n(global) scalar\nThese are the first and last rows and columns of an isolated diagonal\nblock upon which the QR sweep is to be applied. It is assumed without\na check that either ktop = 1 or H(ktop,ktop-1) = 0 and either kbot\n= n or H(kbot+1,kbot) = 0.\nnshfts\n(global) scalar\nnshfts gives the number of simultaneous shifts. nshfts must be\npositive and even.\nsr, si\n(global) Array of size nshfts\nsr contains the real parts and si contains the imaginary parts of the\nnshfts shifts of origin that define the multi-shift QR sweep.\nh\n(local) Array of size lld_h * LOCc(n)\nOn input h contains a Hessenberg matrix H.\ndesch\n(global and local)\narray of size dlen_.\nThe array descriptor for the distributed matrix H .\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1664\n\n\niloz, ihiz\n(global)\nSpecify the rows of the matrix Z to which transformations must be\napplied if wantzis non-zero. 1 ≤iloz≤ihiz≤n\nz\n(local) array of size lld_z * LOCc(n)\nIf wantzis non-zero, then the QR Sweep orthogonal similarity\ntransformation is accumulated into the matrix\nZ(iloz:ihiz,kbot:ktop) from the right. If wantzequals zero, then z\nis unreferenced.\ndescz\n(global and local) array of size dlen_.\nThe array descriptor for the distributed matrix Z.\nwork\n(local workspace) array of size lwork\nlwork\n(local)\nThe size of the work array (lwork≥1).\nIf lwork=-1, then a workspace query is assumed.\niwork\n(local workspace) array of size liwork\nliwork\n(local)\nThe size of the iwork array (liwork≥1).\nIf liwork=-1, then a workspace query is assumed.\nOutput Parameters\nh\nA multi-shift QR sweep with shifts sr(j)+i*si(j) is applied to the\nisolated diagonal block in rows and columns ktop through kbot of the\nmatrix H.\nz\nIf wantzis non-zero, z is updated with transformations applied only to\nthe submatrix Z(iloz:ihiz,kbot:ktop).\nwork[0]\nOn exit, if info = 0, work[0] returns the optimal lwork.\niwork[0]\nOn exit, if info = 0, iwork[0] returns the optimal liwork.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?laqsy\nScales a symmetric/Hermitian matrix, using scaling\nfactors computed by p?poequ .\nSyntax\nvoid pslaqsy (char *uplo , MKL_INT *n , float *a , MKL_INT *ia , MKL_INT *ja , MKL_INT\n*desca , float *sr , float *sc , float *scond , float *amax , char *equed );\nvoid pdlaqsy (char *uplo , MKL_INT *n , double *a , MKL_INT *ia , MKL_INT *ja , MKL_INT\n*desca , double *sr , double *sc , double *scond , double *amax , char *equed );\nvoid pclaqsy (char *uplo , MKL_INT *n , MKL_Complex8 *a , MKL_INT *ia , MKL_INT *ja ,\nMKL_INT *desca , float *sr , float *sc , float *scond , float *amax , char *equed );\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1665\n\n\nvoid pzlaqsy (char *uplo , MKL_INT *n , MKL_Complex16 *a , MKL_INT *ia , MKL_INT *ja ,\nMKL_INT *desca , double *sr , double *sc , double *scond , double *amax , char *equed );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?laqsyfunction equilibrates a symmetric distributed matrix sub(A) = A(ia:ia+n-1, ja:ja+n-1) using\nthe scaling factors in the vectors sr and sc. The scaling factors are computed by p?poequ.\nInput Parameters\nuplo\n(global) Specifies the upper or lower triangular part of the symmetric\ndistributed matrix sub(A) is to be referenced:\n= 'U': Upper triangular part;\n= 'L': Lower triangular part.\nn\n(global)\nThe order of the distributed matrix sub(A). n ≥ 0.\na\n(local).\nPointer into the local memory to an array of size lld_a * LOCc(ja+n-1).\nOn entry, this array contains the local pieces of the distributed matrix\nsub(A). On entry, the local pieces of the distributed symmetric matrix\nsub(A).\nIf uplo = 'U', the leading n-by-n upper triangular part of sub(A) contains\nthe upper triangular part of the matrix, and the strictly lower triangular part\nof sub(A) is not referenced.\nIf uplo = 'L', the leading n-by-n lower triangular part of sub(A) contains\nthe lower triangular part of the matrix, and the strictly upper triangular part\nof sub(A) is not referenced.\nia, ja\n(global)\nThe row and column indices in the global matrix A indicating the first row\nand the first column of the matrix sub(A), respectively.\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\nsr\n(local)\nArray of size LOCr(m_a). The scale factors for the matrix A(ia:ia+m-1, ja:ja\n+n-1). sr is aligned with the distributed matrix A, and replicated across\nevery process column. sr is tied to the distributed matrix A.\nsc\n(local)\nArray of size LOCc(m_a). The scale factors for the matrix A (ia:ia+m-1,\nja:ja+n-1). sc is aligned with the distributed matrix A, and replicated\nacross every process column. sc is tied to the distributed matrix A.\nscond\n(global).\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1666\n\n\nRatio of the smallest sr[i] (respectively sc[j]) to the largest sr[i]\n(respectively sc[j]), with ia -1 ≤ i < ia+n-1 and ja -1 ≤ j < ja+n-1.\namax\n(global).\nAbsolute value of largest distributed submatrix entry.\nOutput Parameters\na\nOn exit,\nif equed = 'Y', the equilibrated matrix:\ndiag(sr ia, ..., sr ia+n-1) * sub(A) * diag(sc ja, ..., sc ja+n-1).\nequed\n(global).\nSpecifies whether or not equilibration was done.\n= 'N': No equilibration.\n= 'Y': Equilibration was done, that is, sub(A) has been replaced by:\ndiag(sr ia, ..., sr ia+n-1) * sub(A) * diag(sc ja, ..., sc ja+n-1).\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?lared1d\nRedistributes an array assuming that the input array,\nbycol, is distributed across rows and that all process\ncolumns contain the same copy of bycol.\nSyntax\nvoid pslared1d (MKL_INT *n , MKL_INT *ia , MKL_INT *ja , MKL_INT *desc , float *bycol ,\nfloat *byall , float *work , MKL_INT *lwork );\nvoid pdlared1d (MKL_INT *n , MKL_INT *ia , MKL_INT *ja , MKL_INT *desc , double\n*bycol , double *byall , double *work , MKL_INT *lwork );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?lared1dfunction redistributes a 1D array. It assumes that the input array bycol is distributed across\nrows and that all process column contain the same copy of bycol. The output array byall is identical on all\nprocesses and contains the entire array.\nInput Parameters\nnp = Number of local rows in bycol()\nn\n(global)\nThe number of elements to be redistributed. n≥ 0.\nia, ja\n(global) ia, ja must be equal to 1.\ndesc\n(local) array of size 9. A 2D array descriptor, which describes bycol.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1667\n\n\nbycol\n(local).\nDistributed block cyclic array of global size n and of local size np. bycol is\ndistributed across the process rows. All process columns are assumed to\ncontain the same value.\nwork\n(local).\nsize lwork. Used to hold the buffers sent from one process to another.\nlwork\n(local)\nThe size of the work array. lwork ≥ numroc(n, desc[nb_], 0, 0,\nnpcol).\nOutput Parameters\nbyall\n(global).\nGlobal size n, local size n. byall is exactly duplicated on all processes. It\ncontains the same values as bycol, but it is replicated across all processes\nrather than being distributed.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?lared2d\nRedistributes an array assuming that the input array\nbyrow is distributed across columns and that all\nprocess rows contain the same copy of byrow.\nSyntax\nvoid pslared2d (MKL_INT *n , MKL_INT *ia , MKL_INT *ja , MKL_INT *desc , float *byrow ,\nfloat *byall , float *work , MKL_INT *lwork );\nvoid pdlared2d (MKL_INT *n , MKL_INT *ia , MKL_INT *ja , MKL_INT *desc , double\n*byrow , double *byall , double *work , MKL_INT *lwork );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?lared2dfunction redistributes a 1D array. It assumes that the input array byrow is distributed across\ncolumns and that all process rows contain the same copy of byrow. The output array byall will be identical\non all processes and will contain the entire array.\nInput Parameters\nnp = Number of local rows in byrow()\nn\n(global)\nThe number of elements to be redistributed. n≥ 0.\nia, ja\n(global) ia, ja must be equal to 1.\ndesc\n(local) array of size dlen_. A 2D array descriptor, which describes byrow.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1668\n\n\nbyrow\n(local).\nDistributed block cyclic array of global size n and of local size np.\nbyrow is distributed across the process columns. All process rows\nare assumed to contain the same value.\nwork\n(local).\nsize lwork. Used to hold the buffers sent from one process to another.\nlwork\n(local) The size of the work array. lwork ≥ numroc(n, desc[nb_], 0, 0,\nnpcol).\nOutput Parameters\nbyall\n(global).\nGlobal size n, local size n. byall is exactly duplicated on all processes. It\ncontains the same values as byrow, but it is replicated across all processes\nrather than being distributed.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?larf\nApplies an elementary reflector to a general\nrectangular matrix.\nSyntax\nvoid pslarf (char *side , MKL_INT *m , MKL_INT *n , float *v , MKL_INT *iv , MKL_INT\n*jv , MKL_INT *descv , MKL_INT *incv , float *tau , float *c , MKL_INT *ic , MKL_INT\n*jc , MKL_INT *descc , float *work );\nvoid pdlarf (char *side , MKL_INT *m , MKL_INT *n , double *v , MKL_INT *iv , MKL_INT\n*jv , MKL_INT *descv , MKL_INT *incv , double *tau , double *c , MKL_INT *ic , MKL_INT\n*jc , MKL_INT *descc , double *work );\nvoid pclarf (char *side , MKL_INT *m , MKL_INT *n , MKL_Complex8 *v , MKL_INT *iv ,\nMKL_INT *jv , MKL_INT *descv , MKL_INT *incv , MKL_Complex8 *tau , MKL_Complex8 *c ,\nMKL_INT *ic , MKL_INT *jc , MKL_INT *descc , MKL_Complex8 *work );\nvoid pzlarf (char *side , MKL_INT *m , MKL_INT *n , MKL_Complex16 *v , MKL_INT *iv ,\nMKL_INT *jv , MKL_INT *descv , MKL_INT *incv , MKL_Complex16 *tau , MKL_Complex16 *c ,\nMKL_INT *ic , MKL_INT *jc , MKL_INT *descc , MKL_Complex16 *work );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?larffunction applies a real/complex elementary reflector Q (or QT) to a real/complex m-by-n\ndistributed matrix sub(C) = C(ic:ic+m-1, jc:jc+n-1), from either the left or the right. Q is represented\nin the form\nQ = I-tau*v*v',\nwhere tau is a real/complex scalar and v is a real/complex vector.\nIf tau = 0, then Q is taken to be the unit matrix.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1669\n\n\nInput Parameters\nside\n(global).\n= 'L': form Q*sub(C),\n= 'R': form sub(C)*Q, Q=QT.\nm\n(global)\nThe number of rows in the distributed submatrix sub(A). (m≥ 0).\nn\n(global)\nThe number of columns in the distributed submatrix sub(A). (n ≥ 0).\nv\n(local).\nPointer into the local memory to an array of size lld_v * LOCc(n_v),\ncontaining the local pieces of the global distributed matrix V representing\nthe Householder transformation Q,\nV(iv:iv+m-1, jv) if side = 'L' and incv = 1,\nV(iv, jv:jv+m-1) if side = 'L' and incv = m_v,\nV(iv:iv+n-1, jv) if side = 'R' and incv = 1,\nV(iv, jv:jv+n-1) if side = 'R' and incv = m_v.\nThe array v is the representation of Q. v is not used if tau = 0.\niv, jv\n(global) The row and column indices in the global matrix V indicating the\nfirst row and the first column of the matrix sub(V), respectively.\ndescv\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix V.\nincv\n(global)\nThe global increment for the elements of V. Only two values of incv are\nsupported in this version, namely 1 and m_v.\nincv must not be zero.\ntau\n(local).\nArray of size LOCc(jv) if incv = 1, and LOCr(iv) otherwise. This array\ncontains the Householder scalars related to the Householder vectors.\ntau is tied to the distributed matrix V.\nc\n(local).\nPointer into the local memory to an array of size lld_c * LOCc(jc+n-1),\ncontaining the local pieces of sub(C).\nic, jc\n(global)\nThe row and column indices in the global matrix C indicating the first row\nand the first column of the matrix sub(C), respectively.\ndescc\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix C.\nwork\n(local).\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1670\n\n\nArray of size lwork.\nIf incv = 1,\n   if side = 'L',\n    if ivcol = iccol,\n      lwork≥nqc0\n    else\n      lwork≥mpc0 + max( 1, nqc0 )\n    end if\n  else if side = 'R' ,\n    lwork≥nqc0 + max( max( 1, mpc0), numroc(numroc( n+\n      icoffc,nb_v,0,0,npcol),nb_v,0,0,lcmq ) )\n  end if\nelse if incv = m_v,\n  if side = 'L',\n    lwork≥mpc0 + max( max( 1, nqc0 ), numroc(\n      numroc(m+iroffc,mb_v,0,0,nprow ),mb_v,0,0, lcmp ) )\n  else if side = 'R',\n    if ivrow = icrow,\n      lwork≥mpc0\n    else\n      lwork≥nqc0 + max( 1, mpc0 )\n    end if\n  end if\nend if,\nwhere lcm is the least common multiple of nprow and npcol and lcm =\nilcm( nprow, npcol ), lcmp = lcm/nprow, lcmq = lcm/npcol,\niroffc = mod( ic-1, mb_c ), icoffc = mod( jc-1, nb_c ),\nicrow = indxg2p( ic, mb_c, myrow, rsrc_c, nprow ),\niccol = indxg2p( jc, nb_c, mycol, csrc_c, npcol ),\nmpc0 = numroc( m+iroffc, mb_c, myrow, icrow, nprow ),\nnqc0 = numroc( n+icoffc, nb_c, mycol, iccol, npcol ),\nilcm, indxg2p, and numroc are ScaLAPACK tool functions; myrow, mycol,\nnprow, and npcol can be determined by calling the function\nblacs_gridinfo.\nOutput Parameters\nc\n(local).\nOn exit, sub(C) is overwritten by the Q*sub(C) if side = 'L',\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1671\n\n\nor sub(C) * Q if side = 'R'.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?larfb\nApplies a block reflector or its transpose/conjugate-\ntranspose to a general rectangular matrix.\nSyntax\nvoid pslarfb (char *side , char *trans , char *direct , char *storev , MKL_INT *m ,\nMKL_INT *n , MKL_INT *k , float *v , MKL_INT *iv , MKL_INT *jv , MKL_INT *descv , float\n*t , float *c , MKL_INT *ic , MKL_INT *jc , MKL_INT *descc , float *work );\nvoid pdlarfb (char *side , char *trans , char *direct , char *storev , MKL_INT *m ,\nMKL_INT *n , MKL_INT *k , double *v , MKL_INT *iv , MKL_INT *jv , MKL_INT *descv ,\ndouble *t , double *c , MKL_INT *ic , MKL_INT *jc , MKL_INT *descc , double *work );\nvoid pclarfb (char *side , char *trans , char *direct , char *storev , MKL_INT *m ,\nMKL_INT *n , MKL_INT *k , MKL_Complex8 *v , MKL_INT *iv , MKL_INT *jv , MKL_INT *descv ,\nMKL_Complex8 *t , MKL_Complex8 *c , MKL_INT *ic , MKL_INT *jc , MKL_INT *descc ,\nMKL_Complex8 *work );\nvoid pzlarfb (char *side , char *trans , char *direct , char *storev , MKL_INT *m ,\nMKL_INT *n , MKL_INT *k , MKL_Complex16 *v , MKL_INT *iv , MKL_INT *jv , MKL_INT\n*descv , MKL_Complex16 *t , MKL_Complex16 *c , MKL_INT *ic , MKL_INT *jc , MKL_INT\n*descc , MKL_Complex16 *work );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?larfbfunction applies a real/complex block reflector Q or its transpose QT/conjugate transpose QH to a\nreal/complex distributed m-by-n matrix sub(C) = C(ic:ic+m-1, jc:jc+n-1) from the left or the right.\nInput Parameters\nside\n(global)\nif side = 'L': apply Q or QT for real flavors (QH for complex flavors) from\nthe Left;\nif side = 'R': apply Q or QTfor real flavors (QH for complex flavors) from\nthe Right.\ntrans\n(global)\nif trans = 'N': no transpose, apply Q;\nfor real flavors, if trans='T': transpose, apply QT\nfor complex flavors, if trans = 'C': conjugate transpose, apply QH;\ndirect\n(global) Indicates how Q is formed from a product of elementary reflectors.\nif direct = 'F': Q = H(1)*H(2)*...*H(k) (Forward)\nif direct = 'B': Q = H(k)*...*H(2)*H(1) (Backward)\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1672\n\n\nstorev\n(global)\nIndicates how the vectors that define the elementary reflectors are stored:\nif storev = 'C': Columnwise\nif storev = 'R': Rowwise.\nm\n(global)\nThe number of rows in the distributed matrix sub(C). (m ≥ 0).\nn\n(global)\nThe number of columns in the distributed matrix sub(C). (n ≥ 0).\nk\n(global)\nThe order of the matrix T.\nv\n(local).\nPointer into the local memory to an array of size \nlld_v * LOCc(jv+k-1) if storev = 'C',\nlld_v * LOCc(jv+m-1) if storev = 'R' and side = 'L',\nlld_v * LOCc(jv+n-1) if storev = 'R' and side = 'R'.\nIt contains the local pieces of the distributed vectors V representing the\nHouseholder transformation.\nif storev = 'C' and side = 'L', lld_v ≥ max(1,LOCr(iv+m-1));\nif storev = 'C' and side = 'R', lld_v ≥ max(1,LOCr(iv+n-1));\nif storev = 'R', lld_v≥LOCr(jv+k-1).\niv, jv\n(global)\nThe row and column indices in the global matrix V indicating the first row\nand the first column of the matrix sub(V), respectively.\ndescv\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix V.\nc\n(local).\nPointer into the local memory to an array of size lld_c * LOCc(jc+n-1),\ncontaining the local pieces of sub(C).\nic, jc\n(global) The row and column indices in the global matrix C indicating the\nfirst row and the first column of the matrix sub(C), respectively.\ndescc\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix C.\nwork\n(local).\nWorkspace array of size lwork.\nIf storev = 'C',\n  if side = 'L',\n    lwork≥ ( nqc0 + mpc0 ) * k\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1673\n\n\n  else if side = 'R',\n    lwork ≥ ( nqc0 + max( npv0 + numroc( numroc( n +\n      icoffc, nb_v, 0, 0, npcol ), nb_v, 0, 0, lcmq ),\n      mpc0 ) ) * k\n  end if\nelse if storev = 'R' ,\n  if side = 'L' ,\n    lwork≥ ( mpc0 + max( mqv0 + numroc( numroc( m +\n    iroffc, mb_v, 0, 0, nprow ), mb_v, 0, 0, lcmp ),\n    nqc0 ) ) * k\n  else if side = 'R',\n    lwork ≥ ( mpc0 + nqc0 ) * k\n  end if\nend if,\nwhere\nlcmq = lcm / npcol with lcm = iclm( nprow, npcol ),\niroffv = mod( iv-1, mb_v ), icoffv = mod( jv-1, nb_v ),\nivrow = indxg2p( iv, mb_v, myrow, rsrc_v, nprow ),\nivcol = indxg2p( jv, nb_v, mycol, csrc_v, npcol ),\nMqV0 = numroc( m+icoffv, nb_v, mycol, ivcol, npcol ),\nNpV0 = numroc( n+iroffv, mb_v, myrow, ivrow, nprow ),\niroffc = mod( ic-1, mb_c ), icoffc = mod( jc-1, nb_c ),\nicrow = indxg2p( ic, mb_c, myrow, rsrc_c, nprow ),\niccol = indxg2p( jc, nb_c, mycol, csrc_c, npcol ),\nMpC0 = numroc( m+iroffc, mb_c, myrow, icrow, nprow ),\nNpC0 = numroc( n+icoffc, mb_c, myrow, icrow, nprow ),\nNqC0 = numroc( n+icoffc, nb_c, mycol, iccol, npcol ),\nilcm, indxg2p, and numroc are ScaLAPACK tool functions; myrow, mycol,\nnprow, and npcol can be determined by calling the function\nblacs_gridinfo.\nOutput Parameters\nt\n(local).\nArray of size mb_v * mb_vif storev = 'R', and nb_v * nb_vif storev =\n'C'. The triangular matrix t is the representation of the block reflector.\nc\n(local).\nOn exit, sub(C) is overwritten by the Q*sub(C), or Q'*sub(C), or\nsub(C)*Q, or sub(C)*Q'. Q' is transpose (conjugate transpose) of Q.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1674\n\n\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?larfc\nApplies the conjugate transpose of an elementary\nreflector to a general matrix.\nSyntax\nvoid pclarfc (char *side , MKL_INT *m , MKL_INT *n , MKL_Complex8 *v , MKL_INT *iv ,\nMKL_INT *jv , MKL_INT *descv , MKL_INT *incv , MKL_Complex8 *tau , MKL_Complex8 *c ,\nMKL_INT *ic , MKL_INT *jc , MKL_INT *descc , MKL_Complex8 *work );\nvoid pzlarfc (char *side , MKL_INT *m , MKL_INT *n , MKL_Complex16 *v , MKL_INT *iv ,\nMKL_INT *jv , MKL_INT *descv , MKL_INT *incv , MKL_Complex16 *tau , MKL_Complex16 *c ,\nMKL_INT *ic , MKL_INT *jc , MKL_INT *descc , MKL_Complex16 *work );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?larfcfunction applies a complex elementary reflector QH to a complex m-by-n distributed matrix\nsub(C) = C(ic:ic+m-1, jc:jc+n-1), from either the left or the right. Q is represented in the form\nQ = i-tau*v*v',\nwhere tau is a complex scalar and v is a complex vector.\nIf tau = 0, then Q is taken to be the unit matrix.\nInput Parameters\nside\n(global)\nif side = 'L': form QH*sub(C) ;\nif side = 'R': form sub (C)*QH.\nm\n(global)\nThe number of rows in the distributed matrix sub(C). (m ≥ 0).\nn\n(global)\nThe number of columns in the distributed matrix sub(C). (n ≥ 0).\nv\n(local).\nPointer into the local memory to an array of size lld_v * LOCc(n_v),\ncontaining the local pieces of the global distributed matrix V representing\nthe Householder transformation Q,\nV(iv:iv+m-1, jv) if side = 'L' and incv = 1,\nV(iv, jv:jv+m-1) if side = 'L' and incv = m_v,\nV(iv:iv+n-1, jv) if side = 'R' and incv = 1,\nV(iv, jv:jv+n-1) if side = 'R' and incv = m_v.\nThe array v is the representation of Q. v is not used if tau = 0.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1675\n\n\niv, jv\n(global)\nThe row and column indices in the global matrix V indicating the first row\nand the first column of the matrix sub(V), respectively.\ndescv\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix V.\nincv\n(global)\nThe global increment for the elements of v. Only two values of incv are\nsupported in this version, namely 1 and m_v.\nincv must not be zero.\ntau\n(local)\nArray of size LOCc(jv) if incv = 1, and LOCr(iv) otherwise. This array\ncontains the Householder scalars related to the Householder vectors.\ntau is tied to the distributed matrix V.\nc\n(local).\nPointer into the local memory to an array of size lld_c * LOCc(jc+n-1),\ncontaining the local pieces of sub(C).\nic, jc\n(global)\nThe row and column indices in the global matrix C indicating the first row\nand the first column of the matrix sub(C), respectively.\ndescc\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix C.\nwork\n(local).\nWorkspace array of size lwork.\nIf incv = 1,\n  if side = 'L' ,\n    if ivcol = iccol,\n      lwork ≥ nqc0\n    else\n      lwork ≥ mpc0 + max( 1, nqc0 )\n    end if\n  else if side = 'R',\n    lwork ≥ nqc0 + max( max( 1, mpc0 ), numroc( numroc(\n      n+icoffc,nb_v,0,0,npcol ), nb_v,0,0,lcmq ) )\n  end if\nelse if incv = m_v,\n  if side = 'L',\n    lwork ≥ mpc0 + max( max( 1, nqc0 ), numroc( numroc(\n      m+iroffc,mb_v,0,0,nprow ),mb_v,0,0,lcmp ) )\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1676\n\n\n  else if side = 'R' ,\n    if ivrow = icrow,\n      lwork ≥ mpc0\n    else\n      lwork ≥ nqc0 + max( 1, mpc0 )\n    end if\n  end if\nend if,\nwhere lcm is the least common multiple of nprow and npcol and lcm =\nilcm(nprow, npcol),\nlcmp = lcm/nprow, lcmq = lcm/npcol,\niroffc = mod(ic-1, mb_c), icoffc = mod(jc-1, nb_c),\nicrow = indxg2p(ic, mb_c, myrow, rsrc_c, nprow),\niccol = indxg2p(jc, nb_c, mycol, csrc_c, npcol),\nmpc0 = numroc(m+iroffc, mb_c, myrow, icrow, nprow),\nnqc0 = numroc(n+icoffc, nb_c, mycol, iccol, npcol),\nilcm, indxg2p, and numroc are ScaLAPACK tool functions;myrow, mycol,\nnprow, and npcol can be determined by calling the function\nblacs_gridinfo.\nOutput Parameters\nc\n(local).\nOn exit, sub(C) is overwritten by the QH*sub(C) if side = 'L', or sub(C)\n* QH if side = 'R'.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?larfg\nGenerates an elementary reflector (Householder\nmatrix).\nSyntax\nvoid pslarfg (MKL_INT *n , float *alpha , MKL_INT *iax , MKL_INT *jax , float *x ,\nMKL_INT *ix , MKL_INT *jx , MKL_INT *descx , MKL_INT *incx , float *tau );\nvoid pdlarfg (MKL_INT *n , double *alpha , MKL_INT *iax , MKL_INT *jax , double *x ,\nMKL_INT *ix , MKL_INT *jx , MKL_INT *descx , MKL_INT *incx , double *tau );\nvoid pclarfg (MKL_INT *n , MKL_Complex8 *alpha , MKL_INT *iax , MKL_INT *jax ,\nMKL_Complex8 *x , MKL_INT *ix , MKL_INT *jx , MKL_INT *descx , MKL_INT *incx ,\nMKL_Complex8 *tau );\nvoid pzlarfg (MKL_INT *n , MKL_Complex16 *alpha , MKL_INT *iax , MKL_INT *jax ,\nMKL_Complex16 *x , MKL_INT *ix , MKL_INT *jx , MKL_INT *descx , MKL_INT *incx ,\nMKL_Complex16 *tau );\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1677\n\n\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?larfgfunction generates a real/complex elementary reflector H of order n, such that\nwhere alpha is a scalar (a real scalar - for complex flavors), and sub(X) is an (n-1)-element real/complex\ndistributed vector X(ix:ix+n-2, jx) if incx = 1 and X(ix, jx:jx+n-2) if incx = m_x. H is represented\nin the form\nwhere tau is a real/complex scalar and v is a real/complex (n-1)-element vector. Note that H is not\nHermitian.\nIf the elements of sub(X) are all zero (and X(iax, jax) is real for complex flavors), then tau = 0 and H is\ntaken to be the unit matrix.\nOtherwise 1 ≤ real(tau) ≤ 2 and abs(tau-1) ≤ 1.\nInput Parameters\nn\n(global)\nThe global order of the elementary reflector. n ≥ 0.\niax, jax\n(global)\nThe global row and column indices of X(iax, jax) in the global matrix X.\nx\n(local).\nPointer into the local memory to an array of size lld_x * LOCc(n_x). This\narray contains the local pieces of the distributed vector sub(X). Before\nentry, the incremented array sub(X) must contain vector x.\nix, jx\n(global)\nThe row and column indices in the global matrix X indicating the first row\nand the first column of sub(X), respectively.\ndescx\n(global and local)\nArray of size dlen_. The array descriptor for the distributed matrix X.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1678\n\n\nincx\n(global)\nThe global increment for the elements of x. Only two values of incx are\nsupported in this version, namely 1 and m_x. incx must not be zero.\nOutput Parameters\nalpha\n(local)\nOn exit, alpha is computed in the process scope having the vector sub(X).\nx\n(local).\nOn exit, it is overwritten with the vector v.\ntau\n(local).\nArray of size LOCc(jx) if incx = 1, and LOCr(ix) otherwise. This array\ncontains the Householder scalars related to the Householder vectors.\ntau is tied to the distributed matrix X.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?larft\nForms the triangular vector T of a block reflector H=I-\nV*T*VH.\nSyntax\nvoid pslarft (char *direct , char *storev , MKL_INT *n , MKL_INT *k , float *v , MKL_INT\n*iv , MKL_INT *jv , MKL_INT *descv , float *tau , float *t , float *work );\nvoid pdlarft (char *direct , char *storev , MKL_INT *n , MKL_INT *k , double *v ,\nMKL_INT *iv , MKL_INT *jv , MKL_INT *descv , double *tau , double *t , double *work );\nvoid pclarft (char *direct , char *storev , MKL_INT *n , MKL_INT *k , MKL_Complex8 *v ,\nMKL_INT *iv , MKL_INT *jv , MKL_INT *descv , MKL_Complex8 *tau , MKL_Complex8 *t ,\nMKL_Complex8 *work );\nvoid pzlarft (char *direct , char *storev , MKL_INT *n , MKL_INT *k , MKL_Complex16\n*v , MKL_INT *iv , MKL_INT *jv , MKL_INT *descv , MKL_Complex16 *tau , MKL_Complex16\n*t , MKL_Complex16 *work );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?larftfunction forms the triangular factor T of a real/complex block reflector H of order n, which is\ndefined as a product of k elementary reflectors.\nIf direct = 'F', H = H(1)*H(2)...*H(k), and T is upper triangular;\nIf direct = 'B', H = H(k)*...*H(2)*H(1), and T is lower triangular.\nIf storev = 'C', the vector which defines the elementary reflector H(i) is stored in the i-th column of the\ndistributed matrix V, and\nH = I-V*T*V'\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1679\n\n\nIf storev = 'R', the vector which defines the elementary reflector H(i) is stored in the i-th row of the\ndistributed matrix V, and\nH = I-V'*T*V.\nInput Parameters\ndirect\n(global)\nSpecifies the order in which the elementary reflectors are multiplied to form\nthe block reflector:\nif direct = 'F': H = H(1)*H(2)*...*H(k) (forward)\nif direct = 'B': H = H(k)*...*H(2)*H(1) (backward).\nstorev\n(global)\nSpecifies how the vectors that define the elementary reflectors are stored\n(See Application Notes below):\nif storev = 'C': columnwise;\nif storev = 'R': rowwise.\nn\n(global)\nThe order of the block reflector H. n ≥ 0.\nk\n(global)\nThe order of the triangular factor T, is equal to the number of elementary\nreflectors. \n1 ≤ k ≤ mb_v (= nb_v).\nv\nPointer into the local memory to an array of local size\nLOCr(iv+n-1) * LOCc(jv+k-1) if storev = 'C', and \nLOCr(iv+k-1) * LOCc(jv+n-1) if storev = 'R'.\nThe distributed matrix V contains the Householder vectors. (See Application\nNotes below).\niv, jv\n(global)\nThe row and column indices in the global matrix V indicating the first row\nand the first column of the matrix sub(V), respectively.\ndescv\n(local) array of size dlen_. The array descriptor for the distributed matrix V.\ntau\n(local)\nArray of size LOCr(iv+k-1) if incv = m_v, and LOCc(jv+k-1) otherwise.\nThis array contains the Householder scalars related to the Householder\nvectors.\ntau is tied to the distributed matrix V.\nwork\n(local).\nWorkspace array of size k*(k -1)/2.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1680\n\n\nOutput Parameters\nv\nt\n(local)\nArray of size nb_v * nb_v if storev = 'C', and mb_v * mb_v otherwise. It\ncontains the k-by-k triangular factor of the block reflector associated with v.\nIf direct = 'F', t is upper triangular;\nif direct = 'B', t is lower triangular.\nApplication Notes\nThe shape of the matrix V and the storage of the vectors that define the H(i) is best illustrated by the\nfollowing example with n = 5 and k = 3. The elements equal to 1 are not stored; the corresponding array\nelements are modified but restored on exit. The rest of the array is not used.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?larz\nApplies an elementary reflector as returned by\np?tzrzf to a general matrix.\nSyntax\nvoid pslarz (char *side , MKL_INT *m , MKL_INT *n , MKL_INT *l , float *v , MKL_INT\n*iv , MKL_INT *jv , MKL_INT *descv , MKL_INT *incv , float *tau , float *c , MKL_INT\n*ic , MKL_INT *jc , MKL_INT *descc , float *work );\nvoid pdlarz (char *side , MKL_INT *m , MKL_INT *n , MKL_INT *l , double *v , MKL_INT\n*iv , MKL_INT *jv , MKL_INT *descv , MKL_INT *incv , double *tau , double *c , MKL_INT\n*ic , MKL_INT *jc , MKL_INT *descc , double *work );\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1681\n\n\nvoid pclarz (char *side , MKL_INT *m , MKL_INT *n , MKL_INT *l , MKL_Complex8 *v ,\nMKL_INT *iv , MKL_INT *jv , MKL_INT *descv , MKL_INT *incv , MKL_Complex8 *tau ,\nMKL_Complex8 *c , MKL_INT *ic , MKL_INT *jc , MKL_INT *descc , MKL_Complex8 *work );\nvoid pzlarz (char *side , MKL_INT *m , MKL_INT *n , MKL_INT *l , MKL_Complex16 *v ,\nMKL_INT *iv , MKL_INT *jv , MKL_INT *descv , MKL_INT *incv , MKL_Complex16 *tau ,\nMKL_Complex16 *c , MKL_INT *ic , MKL_INT *jc , MKL_INT *descc , MKL_Complex16 *work );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?larzfunction applies a real/complex elementary reflector Q (or QT) to a real/complex m-by-n\ndistributed matrix sub(C) = C(ic:ic+m-1, jc:jc+n-1), from either the left or the right. Q is represented\nin the form\nQ = I-tau*v*v',\nwhere tau is a real/complex scalar and v is a real/complex vector.\nIf tau = 0, then Q is taken to be the unit matrix.\nQ is a product of k elementary reflectors as returned by p?tzrzf.\nInput Parameters\nside\n(global)\nif side = 'L': form Q*sub(C),\nif side = 'R': form sub(C)*Q, Q = QT (for real flavors).\nm\n(global)\nThe number of rows in the distributed matrix sub(C). (m ≥ 0).\nn\n(global)\nThe number of columns in the distributed matrix sub(C). (n ≥ 0).\nl\n(global)\nThe columns of the distributed matrix sub(A) containing the meaningful\npart of the Householder reflectors. If side = 'L', m ≥ l ≥ 0,\nif side = 'R', n ≥ l ≥ 0.\nv\n(local).\nPointer into the local memory to an array of size lld_v * LOCc(n_v)\ncontaining the local pieces of the global distributed matrix V representing\nthe Householder transformation Q,\nV(iv:iv+l-1, jv) if side = 'L' and incv = 1,\nV(iv, jv:jv+l-1) if side = 'L' and incv = m_v,\nV(iv:iv+l-1, jv) if side = 'R' and incv = 1,\nV(iv, jv:jv+l-1) if side = 'R' and incv = m_v.\nThe vector v in the representation of Q. v is not used if tau = 0.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1682\n\n\niv, jv\n(global) The row and column indices in the global distributed matrix V\nindicating the first row and the first column of the matrix sub(V),\nrespectively.\ndescv\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix V.\nincv\n(global)\nThe global increment for the elements of V. Only two values of incv are\nsupported in this version, namely 1 and m_v.\nincv must not be zero.\ntau\n(local)\nArray of size LOCc(jv) if incv = 1, and LOCr(iv) otherwise. This array\ncontains the Householder scalars related to the Householder vectors.\ntau is tied to the distributed matrix V.\nc\n(local).\nPointer into the local memory to an array of size lld_c * LOCc(jc+n-1),\ncontaining the local pieces of sub(C).\nic, jc\n(global)\nThe row and column indices in the global matrix C indicating the first row\nand the first column of the matrix sub(C), respectively.\ndescc\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix C.\nwork\n(local).\nArray of size lwork\nIf incv = 1,\n  if side = 'L' ,\n    if ivcol = iccol,\n      lwork ≥ NqC0\n    else\n      lwork ≥ MpC0 + max(1, NqC0)\n    end if\n  else if side = 'R' ,\n    lwork ≥ NqC0 + max(max(1, MpC0), numroc(numroc(n\n+icoffc,nb_v,0,0,npcol),nb_v,0,0,lcmq))\n  end if\nelse if incv = m_v,\n  if side = 'L' ,\n    lwork ≥ MpC0 + max(max(1, NqC0), numroc(numroc(m\n+iroffc,mb_v,0,0,nprow),mb_v,0,0,lcmp))\n  else if side = 'R' ,\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1683\n\n\n    if ivrow = icrow,\n      lwork ≥ MpC0\n    else\n      lwork ≥ NqC0 + max(1, MpC0)\n    end if\n  end if\nend if.\nHere lcm is the least common multiple of nprow and npcol and\nlcm = ilcm( nprow, npcol ), lcmp = lcm / nprow,\nlcmq = lcm / npcol,\niroffc = mod( ic-1, mb_c ), icoffc = mod( jc-1, nb_c ),\nicrow = indxg2p( ic, mb_c, myrow, rsrc_c, nprow ),\niccol = indxg2p( jc, nb_c, mycol, csrc_c, npcol ),\nmpc0 = numroc( m+iroffc, mb_c, myrow, icrow, nprow ),\nnqc0 = numroc( n+icoffc, nb_c, mycol, iccol, npcol ),\nilcm, indxg2p, and numroc are ScaLAPACK tool functions; myrow, mycol,\nnprow, and npcol can be determined by calling the function\nblacs_gridinfo.\nOutput Parameters\nc\n(local).\nOn exit, sub(C) is overwritten by the Q*sub(C) if side = 'L', or\nsub(C)*Q if side = 'R'.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?larzb\nApplies a block reflector or its transpose/conjugate-\ntranspose as returned by p?tzrzf to a general\nmatrix.\nSyntax\nvoid pslarzb (char *side , char *trans , char *direct , char *storev , MKL_INT *m ,\nMKL_INT *n , MKL_INT *k , MKL_INT *l , float *v , MKL_INT *iv , MKL_INT *jv , MKL_INT\n*descv , float *t , float *c , MKL_INT *ic , MKL_INT *jc , MKL_INT *descc , float\n*work );\nvoid pdlarzb (char *side , char *trans , char *direct , char *storev , MKL_INT *m ,\nMKL_INT *n , MKL_INT *k , MKL_INT *l , double *v , MKL_INT *iv , MKL_INT *jv , MKL_INT\n*descv , double *t , double *c , MKL_INT *ic , MKL_INT *jc , MKL_INT *descc , double\n*work );\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1684\n\n\nvoid pclarzb (char *side , char *trans , char *direct , char *storev , MKL_INT *m ,\nMKL_INT *n , MKL_INT *k , MKL_INT *l , MKL_Complex8 *v , MKL_INT *iv , MKL_INT *jv ,\nMKL_INT *descv , MKL_Complex8 *t , MKL_Complex8 *c , MKL_INT *ic , MKL_INT *jc ,\nMKL_INT *descc , MKL_Complex8 *work );\nvoid pzlarzb (char *side , char *trans , char *direct , char *storev , MKL_INT *m ,\nMKL_INT *n , MKL_INT *k , MKL_INT *l , MKL_Complex16 *v , MKL_INT *iv , MKL_INT *jv ,\nMKL_INT *descv , MKL_Complex16 *t , MKL_Complex16 *c , MKL_INT *ic , MKL_INT *jc ,\nMKL_INT *descc , MKL_Complex16 *work );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?larzbfunction applies a real/complex block reflector Q or its transpose QT (conjugate transpose QH for\ncomplex flavors) to a real/complex distributed m-by-n matrix sub(C) = C(ic:ic+m-1, jc:jc+n-1) from the\nleft or the right.\nQ is a product of k elementary reflectors as returned by p?tzrzf.\nCurrently, only storev = 'R' and direct = 'B' are supported.\nInput Parameters\nside\n(global)\nif side = 'L': apply Q or QT (QH for complex flavors) from the Left;\nif side = 'R': apply Q or QT (QH for complex flavors) from the Right.\ntrans\n(global)\nif trans = 'N': No transpose, apply Q;\nIf trans='T': Transpose, apply QT (real flavors);\nIf trans='C': Conjugate transpose, apply QH (complex flavors).\ndirect\n(global)\nIndicates how H is formed from a product of elementary reflectors.\nif direct = 'F': H = H(1)*H(2)*...*H(k) - forward (not supported) ;\nif direct = 'B': H = H(k)*...*H(2)*H(1) - backward.\nstorev\n(global)\nIndicates how the vectors that define the elementary reflectors are stored:\nif storev = 'C': columnwise (not supported ).\nif storev = 'R': rowwise.\nm\n(global)\nThe number of rows in the distributed submatrix sub(C). (m ≥ 0).\nn\n(global)\nThe number of columns in the distributed submatrix sub(C). (n ≥ 0).\nk\n(global)\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1685\n\n\nThe order of the matrix T. (= the number of elementary reflectors whose\nproduct defines the block reflector).\nl\n(global)\nThe columns of the distributed submatrix sub(A) containing the meaningful\npart of the Householder reflectors.\nIf side = 'L', m ≥ l ≥ 0,\nif side = 'R', n ≥ l ≥ 0.\nv\n(local).\nPointer into the local memory to an array of size lld_v * LOCc(jv+m-1) if\nside = 'L', lld_v * LOCc(jv+m-1) if side = 'R'.\nIt contains the local pieces of the distributed vectors V representing the\nHouseholder transformation as returned by p?tzrzf.\nlld_v ≥ LOCr(iv+k-1).\niv, jv\n(global)\nThe row and column indices in the global matrix V indicating the first row\nand the first column of the submatrix sub(V), respectively.\ndescv\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix V.\nt\n(local)\nArray of size mb_v* mb_v.\nThe lower triangular matrix T in the representation of the block reflector.\nc\n(local).\nPointer into the local memory to an array of size lld_c * LOCc(jc+n-1).\nOn entry, the m-by-n distributed matrix sub(C).\nic, jc\n(global)\nThe row and column indices in the global matrix C indicating the first row\nand the first column of the submatrix sub(C), respectively.\ndescc\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix C.\nwork\n(local).\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1686\n\n\nArray of size lwork.\nIf storev = 'C' ,\n  if side = 'L' ,\n    lwork ≥(nqc0 + mpc0)* k\n  else if side = 'R' ,\n    lwork ≥ (nqc0 + max(npv0 + numroc(numroc(n+icoffc, nb_v, 0, 0, \nnpcol),\n           nb_v, 0, 0, lcmq), mpc0))* k\n  end if\nelse if storev = 'R' ,\n  if side = 'L' ,\n    lwork ≥ (mpc0 + max(mqv0 + numroc(numroc(m+iroffc, mb_v, 0, 0, \nnprow),\n             mb_v, 0, 0, lcmp), nqc0))* k\n  else if side = 'R' ,\n    lwork ≥ (mpc0 + nqc0) * k\n  end if\n  end if.\nHere lcmq = lcm/npcol with lcm = iclm(nprow, npcol),\niroffv = mod(iv-1, mb_v), icoffv = mod( jv-1, nb_v),\nivrow = indxg2p(iv, mb_v, myrow, rsrc_v, nprow),\nivcol = indxg2p(jv, nb_v, mycol, csrc_v, npcol),\nmqv0 = numroc(m+icoffv, nb_v, mycol, ivcol, npcol),\nnpv0 = numroc(n+iroffv, mb_v, myrow, ivrow, nprow),\niroffc = mod(ic-1, mb_c ), icoffc= mod( jc-1, nb_c),\nicrow= indxg2p(ic, mb_c, myrow, rsrc_c, nprow),\niccol= indxg2p(jc, nb_c, mycol, csrc_c, npcol),\nmpc0 = numroc(m+iroffc, mb_c, myrow, icrow, nprow),\nnpc0 = numroc(n+icoffc, mb_c, myrow, icrow, nprow),\nnqc0 = numroc(n+icoffc, nb_c, mycol, iccol, npcol),\nilcm, indxg2p, and numroc are ScaLAPACK tool functions; myrow, mycol,\nnprow, and npcol can be determined by calling the function\nblacs_gridinfo.\nOutput Parameters\nc\n(local).\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1687\n\n\nOn exit, sub(C) is overwritten by the Q*sub(C), or Q'*sub(C), or\nsub(C)*Q, or sub(C)*Q', where Q' is the transpose (conjugate transpose)\nof Q.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?larzc\nApplies (multiplies by) the conjugate transpose of an\nelementary reflector as returned by p?tzrzf to a\ngeneral matrix.\nSyntax\nvoid pclarzc (char *side , MKL_INT *m , MKL_INT *n , MKL_INT *l , MKL_Complex8 *v ,\nMKL_INT *iv , MKL_INT *jv , MKL_INT *descv , MKL_INT *incv , MKL_Complex8 *tau ,\nMKL_Complex8 *c , MKL_INT *ic , MKL_INT *jc , MKL_INT *descc , MKL_Complex8 *work );\nvoid pzlarzc (char *side , MKL_INT *m , MKL_INT *n , MKL_INT *l , MKL_Complex16 *v ,\nMKL_INT *iv , MKL_INT *jv , MKL_INT *descv , MKL_INT *incv , MKL_Complex16 *tau ,\nMKL_Complex16 *c , MKL_INT *ic , MKL_INT *jc , MKL_INT *descc , MKL_Complex16 *work );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?larzcfunction applies a complex elementary reflector QH to a complex m-by-n distributed matrix\nsub(C) = C(ic:ic+m-1, jc:jc+n-1), from either the left or the right. Q is represented in the form\nQ = i-tau*v*v',\nwhere tau is a complex scalar and v is a complex vector.\nIf tau = 0, then Q is taken to be the unit matrix.\nQ is a product of k elementary reflectors as returned by p?tzrzf.\nInput Parameters\nside\n(global)\nif side = 'L': form QH*sub(C);\nif side = 'R': form sub(C)*QH .\nm\n(global)\nThe number of rows in the distributed matrix sub(C). (m ≥ 0).\nn\n(global)\nThe number of columns in the distributed matrix sub(C). (n ≥ 0).\nl\n(global)\nThe columns of the distributed matrix sub(A) containing the meaningful\npart of the Householder reflectors.\nIf side = 'L', m ≥ l ≥ 0,\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1688\n\n\nif side = 'R', n ≥ l ≥ 0.\nv\n(local).\nPointer into the local memory to an array of size lld_v * LOCc(n_v)\ncontaining the local pieces of the global distributed matrix V representing\nthe Householder transformation Q,\nV(iv:iv+l-1, jv) if side = 'L' and incv = 1,\nV(iv, jv:jv+l-1) if side = 'L' and incv = m_v,\nV(iv:iv+l-1, jv) if side = 'R' and incv = 1,\nV(iv, jv:jv+l-1) if side = 'R' and incv = m_v.\nThe vector v in the representation of Q. v is not used if tau = 0.\niv, jv\n(global)\nThe row and column indices in the global matrix V indicating the first row\nand the first column of the matrix sub(V), respectively.\ndescv\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix V.\nincv\n(global)\nThe global increment for the elements of V. Only two values of incv are\nsupported in this version, namely 1 and m_v.\nincv must not be zero.\ntau\n(local)\nArray of size LOCc(jv) if incv = 1, and LOCr(iv) otherwise. This array\ncontains the Householder scalars related to the Householder vectors.\ntau is tied to the distributed matrix V.\nc\n(local).\nPointer into the local memory to an array of size lld_c * LOCc(jc+n-1),\ncontaining the local pieces of sub(C).\nic, jc\n(global)\nThe row and column indices in the global matrix C indicating the first row\nand the first column of the matrix sub(C), respectively.\ndescc\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix C.\nwork\n(local).\nIf incv = 1,\n  if side = 'L' ,\n    if ivcol = iccol,\n      lwork ≥ nqc0\n    else\n      lwork ≥ mpc0 + max(1, nqc0)\n    end if\n  else if side = 'R' ,\n    lwork ≥ nqc0 + max(max(1, mpc0), numroc(numroc(n+icoffc, nb_v, \nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1689\n\n\n0, 0, npcol),\n           nb_v, 0, 0, lcmq))  end if\nelse if incv = m_v,\n  if side = 'L' ,\n    lwork ≥ mpc0 + max(max(1, nqc0), numroc(numroc(m+iroffc, mb_v, \n0, 0, nprow),\n           mb_v, 0, 0, lcmp))\n  else if side = 'R',\n    if ivrow = icrow,\n      lwork ≥ mpc0\n    else\n      lwork ≥ nqc0 + max(1, mpc0)\n    end if\n  end if\n         end if\nHere lcm is the least common multiple of nprow and npcol;\nlcm = ilcm(nprow, npcol), lcmp = lcm/nprow, lcmq= lcm/npcol,\niroffc = mod(ic-1, mb_c), icoffc= mod(jc-1, nb_c),\nicrow = indxg2p(ic, mb_c, myrow, rsrc_c, nprow),\niccol = indxg2p(jc, nb_c, mycol, csrc_c, npcol),\nmpc0 = numroc(m+iroffc, mb_c, myrow, icrow, nprow),\nnqc0 = numroc(n+icoffc, nb_c, mycol, iccol, npcol),\nilcm, indxg2p, and numroc are ScaLAPACK tool functions;\nmyrow, mycol, nprow, and npcol can be determined by calling the function\nblacs_gridinfo.\nOutput Parameters\nc\n(local).\nOn exit, sub(C) is overwritten by the QH*sub(C) if side = 'L', or\nsub(C)*QH if side = 'R'.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?larzt\nForms the triangular factor T of a block reflector H=I-\nV*T*VH as returned by p?tzrzf.\nSyntax\nvoid pslarzt (char *direct , char *storev , MKL_INT *n , MKL_INT *k , float *v , MKL_INT\n*iv , MKL_INT *jv , MKL_INT *descv , float *tau , float *t , float *work );\nvoid pdlarzt (char *direct , char *storev , MKL_INT *n , MKL_INT *k , double *v ,\nMKL_INT *iv , MKL_INT *jv , MKL_INT *descv , double *tau , double *t , double *work );\nvoid pclarzt (char *direct , char *storev , MKL_INT *n , MKL_INT *k , MKL_Complex8 *v ,\nMKL_INT *iv , MKL_INT *jv , MKL_INT *descv , MKL_Complex8 *tau , MKL_Complex8 *t ,\nMKL_Complex8 *work );\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1690\n\n\nvoid pzlarzt (char *direct , char *storev , MKL_INT *n , MKL_INT *k , MKL_Complex16\n*v , MKL_INT *iv , MKL_INT *jv , MKL_INT *descv , MKL_Complex16 *tau , MKL_Complex16\n*t , MKL_Complex16 *work );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?larztfunction forms the triangular factor T of a real/complex block reflector H of order greater than\nn, which is defined as a product of k elementary reflectors as returned by p?tzrzf.\nIf direct = 'F', H = H(1)*H(2)*...*H(k), and T is upper triangular;\nIf direct = 'B', H = H(k)*...*H(2)*H(1), and T is lower triangular.\nIf storev = 'C', the vector which defines the elementary reflector H(i), is stored in the i-th column of the\narray v, and\nH = i-v*t*v'.\nIf storev = 'R', the vector, which defines the elementary reflector H(i), is stored in the i-th row of the\narray v, and\nH = i-v'*t*v\nCurrently, only storev = 'R' and direct = 'B' are supported.\nInput Parameters\ndirect\n(global)\nSpecifies the order in which the elementary reflectors are multiplied to form\nthe block reflector:\nif direct = 'F': H = H(1)*H(2)*...*H(k) (Forward, not supported)\nif direct = 'B': H = H(k)*...*H(2)*H(1) (Backward).\nstorev\n(global)\nSpecifies how the vectors which defines the elementary reflectors are\nstored:\nif storev = 'C': columnwise (not supported);\nif storev = 'R': rowwise.\nn\n(global)\nThe order of the block reflector H. n ≥ 0.\nk\n(global)\nThe order of the triangular factor T (= the number of elementary\nreflectors).\n1≤k≤mb_v(= nb_v).\nv\nPointer into the local memory to an array of local size LOCr(iv+k-1) *\nLOCc(jv+n-1).\nThe distributed matrix V contains the Householder vectors. See Application\nNotes below.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1691\n\n\niv, jv\n(global) The row and column indices in the global matrix V indicating the\nfirst row and the first column of the matrix sub(V), respectively.\ndescv\n(local) array of size dlen_. The array descriptor for the distributed matrix V.\ntau\n(local)\nArray of size LOCr(iv+k-1) if incv = m_v, and LOCc(jv+k-1) otherwise.\nThis array contains the Householder scalars related to the Householder\nvectors.\ntau is tied to the distributed matrix V.\nwork\n(local).\nWorkspace array of size(k*(k-1)/2).\nOutput Parameters\nv\nt\n(local)\nArray of size mb_v* mb_v. It contains the k-by-k triangular factor of the\nblock reflector associated with v. t is lower triangular.\nApplication Notes\nThe shape of the matrix V and the storage of the vectors which define the H(i) is best illustrated by the\nfollowing example with n = 5 and k = 3. The elements equal to 1 are not stored; the corresponding array\nelements are modified but restored on exit. The rest of the array is not used.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1692\n\n\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?lascl\nMultiplies a general rectangular matrix by a real scalar\ndefined as Cto/Cfrom.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1693\n\n\nSyntax\nvoid pslascl (char *type , float *cfrom , float *cto , MKL_INT *m , MKL_INT *n , float\n*a , MKL_INT *ia , MKL_INT *ja , MKL_INT *desca , MKL_INT *info );\nvoid pdlascl (char *type , double *cfrom , double *cto , MKL_INT *m , MKL_INT *n ,\ndouble *a , MKL_INT *ia , MKL_INT *ja , MKL_INT *desca , MKL_INT *info );\nvoid pclascl (char *type , float *cfrom , float *cto , MKL_INT *m , MKL_INT *n ,\nMKL_Complex8 *a , MKL_INT *ia , MKL_INT *ja , MKL_INT *desca , MKL_INT *info );\nvoid pzlascl (char *type , double *cfrom , double *cto , MKL_INT *m , MKL_INT *n ,\nMKL_Complex16 *a , MKL_INT *ia , MKL_INT *ja , MKL_INT *desca , MKL_INT *info );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?lasclfunction multiplies the m-by-n real/complex distributed matrix sub(A) denoting A(ia:ia+m-1,\nja:ja+n-1) by the real/complex scalar cto/cfrom. This is done without over/underflow as long as the final\nresult cto*A(i,j)/cfrom does not over/underflow. type specifies that sub(A) may be full, upper triangular,\nlower triangular or upper Hessenberg.\nInput Parameters\ntype\n(global)\ntype indicates the storage type of the input distributed matrix.\nif type = 'G': sub(A) is a full matrix,\nif type = 'L': sub(A) is a lower triangular matrix,\nif type = 'U': sub(A) is an upper triangular matrix,\nif type = 'H': sub(A) is an upper Hessenberg matrix.\ncfrom, cto\n(global)\nThe distributed matrix sub(A) is multiplied by cto/cfrom. A(i,j) is\ncomputed without over/underflow if the final result cto*A(i,j)/cfrom can be\nrepresented without over/underflow. cfrom must be nonzero.\nm\n(global)\nThe number of rows in the distributed matrix sub(A). (m≥0).\nn\n(global)\nThe number of columns in the distributed matrix sub(A). (n≥0).\na\n(local input/local output)\nPointer into the local memory to an array of size lld_a * LOCc(ja+n-1).\nThis array contains the local pieces of the distributed matrix sub(A).\nia, ja\n(global)\nThe column and row indices in the global matrix A indicating the first row\nand column of the matrix sub(A), respectively.\ndesca\n(global and local)\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1694\n\n\nArray of size dlen_.The array descriptor for the distributed matrix A.\nOutput Parameters\na\n(local).\nOn exit, this array contains the local pieces of the distributed matrix\nmultiplied by cto/cfrom.\ninfo\n(local)\nif info = 0: the execution is successful.\nif info < 0: If the i-th argument is an array and the j-th entry, indexed\nj-1, had an illegal value, then info = -(i*100+j),\nif the i-th argument is a scalar and had an illegal value, then info = -i.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?lase2\nInitializes an m-by-n distributed matrix.\nSyntax\nvoid pslase2 (const char* uplo, const MKL_INT* m, const MKL_INT* n, const float* alpha,\nconst float* beta, float* a, const MKL_INT* ia, const MKL_INT* ja, const MKL_INT*\ndesca);\nvoid pdlase2 (const char* uplo, const MKL_INT* m, const MKL_INT* n, const double*\nalpha, const double* beta, double* a, const MKL_INT* ia, const MKL_INT* ja, const\nMKL_INT* desca);\nvoid pclase2 (const char* uplo, const MKL_INT* m, const MKL_INT* n, const MKL_Complex8*\nalpha, const MKL_Complex8* beta, MKL_Complex8* a, const MKL_INT* ia, const MKL_INT* ja,\nconst MKL_INT* desca);\nvoid pzlase2 (const char* uplo, const MKL_INT* m, const MKL_INT* n, const\nMKL_Complex16* alpha, const MKL_Complex16* beta, MKL_Complex16* a, const MKL_INT* ia,\nconst MKL_INT* ja, const MKL_INT* desca);\nInclude Files\n•\nmkl_scalapack.h\nDescription\np?lase2 initializes an m-by-n distributed matrix sub( A ) denoting A(ia:ia+m-1,ja:ja+n-1) to beta on the\ndiagonal and alpha on the off-diagonals. p?lase2 requires that only the dimension of the matrix operand is\ndistributed.\nInput Parameters\nuplo\n(global)\nSpecifies the part of the distributed matrix sub( A ) to be set:\n= 'U': Upper triangular part is set; the strictly lower triangular part of\nsub( A ) is not changed;\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1695\n\n\n= 'L': Lower triangular part is set; the strictly upper triangular part of\nsub( A ) is not changed;\nOtherwise: All of the matrix sub( A ) is set.\nm\n(global)\nThe number of rows to be operated on i.e the number of rows of the\ndistributed submatrix sub( A ). m >= 0.\nn\n(global)\nThe number of columns to be operated on i.e the number of columns of the\ndistributed submatrix sub( A ). n >= 0.\nalpha\n(global)\nThe constant to which the off-diagonal elements are to be set.\nbeta\n(global)\nThe constant to which the diagonal elements are to be set.\nia\n(global)\nThe row index in the global array a indicating the first row of sub( A ).\nja\n(global)\nThe column index in the global array a indicating the first column of\nsub( A ).\ndesca\n(global and local)\nArray of size dlen_.\nThe array descriptor for the distributed matrix A.\nOutput Parameters\na\n(local)\nPointer into the local memory to an array of size lld_a*LOCc(ja+n-1).\nThis array contains the local pieces of the distributed matrix sub( A )\nto be set.\nOn exit, the leading m-by-n submatrix sub( A ) is set as follows:\nif uplo = 'U', A(ia+i-1,ja+j-1) = alpha, 1<=i<=j-1, 1<=j<=n,\nif uplo = 'L', A(ia+i-1,ja+j-1) = alpha, j+1<=i<=m, 1<=j<=n,\notherwise, A(ia+i-1,ja+j-1) = alpha, 1<=i<=m, 1<=j<=n, ia+i !=\nja+j,\nand, for all uplo, A(ia+i-1,ja+i-1) = beta, 1<=i<=min(m,n).\np?laset\nInitializes the offdiagonal elements of a matrix to alpha\nand the diagonal elements to beta.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1696\n\n\nSyntax\nvoid pslaset (char *uplo , MKL_INT *m , MKL_INT *n , float *alpha , float *beta , float\n*a , MKL_INT *ia , MKL_INT *ja , MKL_INT *desca );\nvoid pdlaset (char *uplo , MKL_INT *m , MKL_INT *n , double *alpha , double *beta ,\ndouble *a , MKL_INT *ia , MKL_INT *ja , MKL_INT *desca );\nvoid pclaset (char *uplo , MKL_INT *m , MKL_INT *n , MKL_Complex8 *alpha , MKL_Complex8\n*beta , MKL_Complex8 *a , MKL_INT *ia , MKL_INT *ja , MKL_INT *desca );\nvoid pzlaset (char *uplo , MKL_INT *m , MKL_INT *n , MKL_Complex16 *alpha ,\nMKL_Complex16 *beta , MKL_Complex16 *a , MKL_INT *ia , MKL_INT *ja , MKL_INT *desca );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?lasetfunction initializes an m-by-n distributed matrix sub(A) denoting A(ia:ia+m-1, ja:ja+n-1) to\nbeta on the diagonal and alpha on the offdiagonals.\nInput Parameters\nuplo\n(global)\nSpecifies the part of the distributed matrix sub(A) to be set:\nif uplo = 'U': upper triangular part; the strictly lower triangular part of\nsub(A) is not changed;\nif uplo = 'L': lower triangular part; the strictly upper triangular part of\nsub(A) is not changed.\nOtherwise: all of the matrix sub(A) is set.\nm\n(global)\nThe number of rows in the distributed matrix sub(A). (m≥0).\nn\n(global)\nThe number of columns in the distributed matrix sub(A). (n≥0).\nalpha\n(global).\nThe constant to which the offdiagonal elements are to be set.\nbeta\n(global).\nThe constant to which the diagonal elements are to be set.\nOutput Parameters\na\n(local).\nPointer into the local memory to an array of size lld_a * LOCc(ja+n-1).\nThis array contains the local pieces of the distributed matrix sub(A) to be\nset. On exit, the leading m-by-n matrix sub(A) is set as follows:\nif uplo = 'U', A(ia+i-1, ja+j-1) = alpha, 1≤i≤j-1, 1≤j≤n,\nif uplo = 'L', A(ia+i-1, ja+j-1) = alpha, j+1≤i≤ m, 1≤j≤n,\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1697\n\n\notherwise, A(ia+i-1, ja+j-1) = alpha, 1≤i≤m, 1≤j≤n, ia+i≠ja+j, and, for\nall uplo, A(ia+i-1, ja+i-1) = beta, 1≤i≤min(m,n).\nia, ja\n(global)\nThe column and row indices in the distributed matrix A indicating the first\nrow and column of the matrix sub(A), respectively.\ndesca\n(global and local)\nArray of size dlen_. The array descriptor for the distributed matrix A.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?lasmsub\nLooks for a small subdiagonal element from the\nbottom of the matrix that it can safely set to zero.\nSyntax\nvoid pslasmsub (const float *a, const MKL_INT *desca, const MKL_INT *i, const MKL_INT\n*l, MKL_INT *k, const float *smlnum, float *buf, const MKL_INT *lwork );\nvoid pdlasmsub (const double *a, const MKL_INT *desca, const MKL_INT *i, const MKL_INT\n*l, MKL_INT *k, const double *smlnum, double *buf, const MKL_INT *lwork );\nvoid pclasmsub (const MKL_Complex8 *a , const MKL_INT *desca , const MKL_INT *i , const\nMKL_INT *l , MKL_INT *k , const float *smlnum , MKL_Complex8 *buf , const MKL_INT\n*lwork );\nvoid pzlasmsub (const MKL_Complex16 *a , const MKL_INT *desca , const MKL_INT *i ,\nconst MKL_INT *l , MKL_INT *k , const double *smlnum , MKL_Complex16 *buf , const\nMKL_INT *lwork );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?lasmsubfunction looks for a small subdiagonal element from the bottom of the matrix that it can\nsafely set to zero. This function performs a global maximum and must be called by all processes.\nInput Parameters\na\n(local)\nArray of size lld_a*LOCc(n_a).\nOn entry, the Hessenberg matrix whose tridiagonal part is being scanned.\nUnchanged on exit.\ndesca\n(global and local)\nArray of size dlen_. The array descriptor for the distributed matrix A.\ni\n(global)\nThe global location of the bottom of the unreduced submatrix of A.\nUnchanged on exit.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1698\n\n\nl\n(global)\nThe global location of the top of the unreduced submatrix of A.\nUnchanged on exit.\nsmlnum\n(global)\nOn entry, a \"small number\" for the given matrix. Unchanged on exit. The\nmachine-dependent constants for the stopping criterion.\nlwork\n(local)\nThis must be at least 2*ceil(ceil((i-l)/mb_a )/ lcm(nprow,npcol)).\nHere lcm is least common multiple and nprowxnpcol is the logical grid size.\nOutput Parameters\nk\n(global)\nOn exit, this yields the bottom portion of the unreduced submatrix. This will\nsatisfy: l ≤ k ≤ i-1.\nbuf\n(local).\nArray of size lwork.\nApplication Notes\nThis routine parallelizes the code from ?lahqr that looks for a single small subdiagonal element.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?lasrt\nSorts the numbers in an array and the corresponding\nvectors in increasing order.\nSyntax\nvoid pslasrt (const char* id, const MKL_INT* n, float* d, const float* q, const\nMKL_INT* iq, const MKL_INT* jq, const MKL_INT* descq, float* work, const MKL_INT*\nlwork, MKL_INT* iwork, const MKL_INT* liwork, MKL_INT* info);\nvoid pdlasrt (const char* id, const MKL_INT* n, double* d, const double* q, const\nMKL_INT* iq, const MKL_INT* jq, const MKL_INT* descq, double* work, const MKL_INT*\nlwork, MKL_INT* iwork, const MKL_INT* liwork, MKL_INT* info);\nInclude Files\n•\nmkl_scalapack.h\nDescription\np?lasrt sorts the numbers in d and the corresponding vectors in q in increasing order.\nInput Parameters\nid\n(global)\n= 'I': sort d in increasing order;\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1699\n\n\n= 'D': sort d in decreasing order. (NOT IMPLEMENTED YET)\nn\n(global)\nThe number of columns to be operated on i.e the number of columns of the\ndistributed submatrix sub( Q ). n >= 0.\nd\n(global)\nArray, size (n)\nq\n(local)\nPointer into the local memory to an array of size lld_q*LOCc(jq+n-1) .\nThis array contains the local pieces of the distributed matrix sub( A ) to be\ncopied from.\niq\n(global)\nThe row index in the global array A indicating the first row of sub( Q ).\njq\n(global)\nThe column index in the global array A indicating the first column of\nsub( Q ).\ndescq\n(global and local)\nArray of size dlen_.\nThe array descriptor for the distributed matrix A.\nwork\n(local)\nArray, size (lwork)\nlwork\n(local)\nThe size of the array work.\nlwork = MAX( n, NP * ( NB + NQ )), where NP = numroc( n, NB, MYROW,\nIAROW, NPROW ), NQ = numroc( n, NB, MYCOL, DESCQ( csrc_ ), NPCOL ).\nnumroc is a ScaLAPACK tool function.\niwork\n(local)\nArray, size (liwork)\nliwork\n(local)\nThe size of the array iwork.\nliwork = n + 2*NB + 2*NPCOL\nOutput Parameters\nd\nOn exit, the numbers in d are sorted in increasing order.\ninfo\n(global)\n= 0: successful exit\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1700\n\n\n< 0: If the i-th argument is an array and the j-th entry had an illegal\nvalue, then info = -(i*100+j), if the i-th argument is a scalar and\nhad an illegal value, then info = -i.\np?lassq\nUpdates a sum of squares represented in scaled form.\nSyntax\nvoid pslassq (MKL_INT *n , float *x , MKL_INT *ix , MKL_INT *jx , MKL_INT *descx ,\nMKL_INT *incx , float *scale , float *sumsq );\nvoid pdlassq (MKL_INT *n , double *x , MKL_INT *ix , MKL_INT *jx , MKL_INT *descx ,\nMKL_INT *incx , double *scale , double *sumsq );\nvoid pclassq (MKL_INT *n , MKL_Complex8 *x , MKL_INT *ix , MKL_INT *jx , MKL_INT\n*descx , MKL_INT *incx , float *scale , float *sumsq );\nvoid pzlassq (MKL_INT *n , MKL_Complex16 *x , MKL_INT *ix , MKL_INT *jx , MKL_INT\n*descx , MKL_INT *incx , double *scale , double *sumsq );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?lassqfunction returns the values scl and smsq such that\nscl 2 * smsq = x12 + ... + xn2 + scale 2*sumsq,\nwhere\nxi= sub(X) = X(ix + (jx-1)*m_x + (i - 1)*incx) for pslassq/pdlassq ,\nxi= sub(X) = abs(X(ix + (jx-1)*m_x + (i - 1)*incx) for pclassq/pzlassq.\nFor real functions pslassq/pdlassq the value of sumsq is assumed to be non-negative and scl returns the\nvalue\nscl = max(scale, abs(xi)).\nFor complex functions pclassq/pzlassq the value of sumsq is assumed to be at least unity and the value of\nssq will then satisfy\n1.0 ≤ ssq ≤sumsq +2n\nValue scale is assumed to be non-negative and scl returns the value\nFor all functions p?lassq values scale and sumsq must be supplied in scale and sumsq respectively, and\nscale and sumsq are overwritten by scl and ssq respectively.\nAll functions p?lassq make only one pass through the vector sub(X).\nInput Parameters\nn\n(global)\nThe length of the distributed vector sub(x ).\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1701\n\n\nx\nThe array that stores the vector for which a scaled sum of squares is\ncomputed:\nx[ix + (jx-1)*m_x + i*incx], 0 ≤ i < n.\nix\n(global)\nThe row index in the global matrix X indicating the first row of sub(X).\njx\n(global)\nThe column index in the global matrix X indicating the first column of\nsub(X).\ndescx\n(global and local) array of size dlen_.\nThe array descriptor for the distributed matrix X.\nincx\n(global)\nThe global increment for the elements of X. Only two values of incx are\nsupported in this version, namely 1 and m_x. The argument incx must not\nequal zero.\nscale\n(local).\nOn entry, the value scale in the equation above.\nsumsq\n(local)\nOn entry, the value sumsq in the equation above.\nOutput Parameters\nscale\n(local).\nOn exit, scale is overwritten with scl , the scaling factor for the sum of\nsquares.\nsumsq\n(local).\nOn exit, sumsq is overwritten with the value smsq, the basic sum of squares\nfrom which scl has been factored out.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?laswp\nPerforms a series of row interchanges on a general\nrectangular matrix.\nSyntax\nvoid pslaswp (char *direc , char *rowcol , MKL_INT *n , float *a , MKL_INT *ia , MKL_INT\n*ja , MKL_INT *desca , MKL_INT *k1 , MKL_INT *k2 , MKL_INT *ipiv );\nvoid pdlaswp (char *direc , char *rowcol , MKL_INT *n , double *a , MKL_INT *ia ,\nMKL_INT *ja , MKL_INT *desca , MKL_INT *k1 , MKL_INT *k2 , MKL_INT *ipiv );\nvoid pclaswp (char *direc , char *rowcol , MKL_INT *n , MKL_Complex8 *a , MKL_INT *ia ,\nMKL_INT *ja , MKL_INT *desca , MKL_INT *k1 , MKL_INT *k2 , MKL_INT *ipiv );\nvoid pzlaswp (char *direc , char *rowcol , MKL_INT *n , MKL_Complex16 *a , MKL_INT\n*ia , MKL_INT *ja , MKL_INT *desca , MKL_INT *k1 , MKL_INT *k2 , MKL_INT *ipiv );\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1702\n\n\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?laswpfunction performs a series of row or column interchanges on the distributed matrix\nsub(A)=A(ia:ia+n-1, ja:ja+n-1). One interchange is initiated for each of rows or columns k1 through k2\nof sub(A). This function assumes that the pivoting information has already been broadcast along the process\nrow or column. Also note that this function will only work for k1-k2 being in the same mb (or nb) block. If\nyou want to pivot a full matrix, use p?lapiv.\nInput Parameters\ndirec\n(global)\nSpecifies in which order the permutation is applied:\n= 'F' - forward,\n= 'B' - backward.\nrowcol\n(global)\nSpecifies if the rows or columns are permuted:\n= 'R' - rows,\n= 'C' - columns.\nn\n(global)\nIf rowcol='R', the length of the rows of the distributed matrix A(*,\nja:ja+n-1) to be permuted;\nIf rowcol='C', the length of the columns of the distributed matrix A(ia:ia\n+n-1 , *) to be permuted;\na\n(local)\nPointer into the local memory to an array of size lld_a * LOCc(n_a). On\nentry, this array contains the local pieces of the distributed matrix to which\nthe row/columns interchanges will be applied.\nia\n(global)\nThe row index in the global matrix A indicating the first row of sub(A).\nja\n(global)\nThe column index in the global matrix A indicating the first column of\nsub(A).\ndesca\n(global and local) array of size dlen_.\nThe array descriptor for the distributed matrix A.\nk1\n(global)\nThe first element of ipiv for which a row or column interchange will be done.\nk2\n(global)\nThe last element of ipiv for which a row or column interchange will be done.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1703\n\n\nipiv\n(local)\nArray of size LOCr(m_a)+mb_a for row pivoting and LOCr(n_a)+nb_a for\ncolumn pivoting. This array is tied to the matrix A, ipiv[k]=l implies rows\n(or columns) k+1 and l are to be interchanged, k = 0, 1, ..., size (ipiv) -1.\nOutput Parameters\nA\n(local)\nOn exit, the permuted distributed matrix.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?latra\nComputes the trace of a general square distributed\nmatrix.\nSyntax\nfloat pslatra (MKL_INT *n , float *a , MKL_INT *ia , MKL_INT *ja , MKL_INT *desca );\ndouble pdlatra (MKL_INT *n , double *a , MKL_INT *ia , MKL_INT *ja , MKL_INT *desca );\nvoid pclatra (MKL_Complex8 * , MKL_INT *n , MKL_Complex8 *a , MKL_INT *ia , MKL_INT\n*ja , MKL_INT *desca );\nvoid pzlatra (MKL_Complex16 * , MKL_INT *n , MKL_Complex16 *a , MKL_INT *ia , MKL_INT\n*ja , MKL_INT *desca );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThis function computes the trace of an n-by-n distributed matrix sub(A) denoting A(ia:ia+n-1, ja:ja\n+n-1). The result is left on every process of the grid.\nInput Parameters\nn\n(global)\nThe number of rows and columns to be operated on, that is, the order of\nthe distributed matrix sub(A). n ≥0.\na\n(local).\nPointer into the local memory to an array of size lld_a * LOCc(ja+n-1)\ncontaining the local pieces of the distributed matrix, the trace of which is to\nbe computed.\nia, ja\n(global) The row and column indices respectively in the global matrix A\nindicating the first row and the first column of the matrix sub(A),\nrespectively.\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1704\n\n\nOutput Parameters\nval\nThe value returned by the function.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?latrd\nReduces the first nb rows and columns of a\nsymmetric/Hermitian matrix A to real tridiagonal form\nby an orthogonal/unitary similarity transformation.\nSyntax\nvoid pslatrd (char *uplo , MKL_INT *n , MKL_INT *nb , float *a , MKL_INT *ia , MKL_INT\n*ja , MKL_INT *desca , float *d , float *e , float *tau , float *w , MKL_INT *iw ,\nMKL_INT *jw , MKL_INT *descw , float *work );\nvoid pdlatrd (char *uplo , MKL_INT *n , MKL_INT *nb , double *a , MKL_INT *ia , MKL_INT\n*ja , MKL_INT *desca , double *d , double *e , double *tau , double *w , MKL_INT *iw ,\nMKL_INT *jw , MKL_INT *descw , double *work );\nvoid pclatrd (char *uplo , MKL_INT *n , MKL_INT *nb , MKL_Complex8 *a , MKL_INT *ia ,\nMKL_INT *ja , MKL_INT *desca , float *d , float *e , MKL_Complex8 *tau , MKL_Complex8\n*w , MKL_INT *iw , MKL_INT *jw , MKL_INT *descw , MKL_Complex8 *work );\nvoid pzlatrd (char *uplo , MKL_INT *n , MKL_INT *nb , MKL_Complex16 *a , MKL_INT *ia ,\nMKL_INT *ja , MKL_INT *desca , double *d , double *e , MKL_Complex16 *tau ,\nMKL_Complex16 *w , MKL_INT *iw , MKL_INT *jw , MKL_INT *descw , MKL_Complex16 *work );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?latrdfunction reduces nb rows and columns of a real symmetric or complex Hermitian matrix\nsub(A)= A(ia:ia+n-1, ja:ja+n-1) to symmetric/complex tridiagonal form by an orthogonal/unitary\nsimilarity transformation Q'*sub(A)*Q, and returns the matrices V and W, which are needed to apply the\ntransformation to the unreduced part of sub(A).\nIf uplo = U, p?latrd reduces the last nb rows and columns of a matrix, of which the upper triangle is\nsupplied;\nif uplo = L, p?latrd reduces the first nb rows and columns of a matrix, of which the lower triangle is\nsupplied.\nThis is an auxiliary function called by p?sytrd/p?hetrd.\nInput Parameters\nuplo\n(global)\nSpecifies whether the upper or lower triangular part of the symmetric/\nHermitian matrix sub(A) is stored:\n= 'U': Upper triangular\n= L: Lower triangular.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1705\n\n\nn\n(global)\nThe number of rows and columns to be operated on, that is, the order of\nthe distributed matrix sub(A). n ≥ 0.\nnb\n(global)\nThe number of rows and columns to be reduced.\na\nPointer into the local memory to an array of size lld_a * LOCc(ja+n-1).\nOn entry, this array contains the local pieces of the symmetric/Hermitian\ndistributed matrix sub(A).\nIf uplo = U, the leading n-by-n upper triangular part of sub(A) contains\nthe upper triangular part of the matrix, and its strictly lower triangular part\nis not referenced.\nIf uplo = L, the leading n-by-n lower triangular part of sub(A) contains the\nlower triangular part of the matrix, and its strictly upper triangular part is\nnot referenced.\nia\n(global)\nThe row index in the global matrix A indicating the first row of sub(A).\nja\n(global)\nThe column index in the global matrix A indicating the first column of\nsub(A).\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\niw\n(global)\nThe row index in the global matrix W indicating the first row of sub(W).\njw\n(global)\nThe column index in the global matrix W indicating the first column of\nsub(W).\ndescw\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix W.\nwork\n(local)\nWorkspace array of size nb_a.\nOutput Parameters\na\n(local)\nOn exit, if uplo = 'U', the last nb columns have been reduced to\ntridiagonal form, with the diagonal elements overwriting the diagonal\nelements of sub(A); the elements above the diagonal with the array tau\nrepresent the orthogonal/unitary matrix Q as a product of elementary\nreflectors;\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1706\n\n\nif uplo = 'L', the first nb columns have been reduced to tridiagonal form,\nwith the diagonal elements overwriting the diagonal elements of sub(A);\nthe elements below the diagonal with the array tau represent the\northogonal/unitary matrix Q as a product of elementary reflectors.\nd\n(local)\nArray of size LOCc(ja+n-1).\nThe diagonal elements of the tridiagonal matrix T: d[i] = A(i+1,i+1), i = 0,\n1, ..., LOCc(ja+n-1)-1. d is tied to the distributed matrix A.\ne\n(local)\nArray of size LOCc(ja+n-1) if uplo = 'U', LOCc(ja+n-2) otherwise.\nThe off-diagonal elements of the tridiagonal matrix T:\ne[i] = A(i + 1, i + 2) if uplo = 'U',\ne[i] = A(i + 2, i + 1) if uplo = 'L',\ni = 0, 1, ..., LOCc(ja+n-1)-1.\ne is tied to the distributed matrix A.\ntau\n(local)\nArray of size LOCc(ja+n-1). This array contains the scalar factors of the\nelementary reflectors. tau is tied to the distributed matrix A.\nw\n(local)\nPointer into the local memory to an array of size lld_w* nb_w. This array\ncontains the local pieces of the n-by-nb_w matrix w required to update the\nunreduced part of sub(A).\nApplication Notes\nIf uplo = 'U', the matrix Q is represented as a product of elementary reflectors\nQ = H(n)*H(n-1)*...*H(n-nb+1)\nEach H(i) has the form\nH(i) = I - tau*v*v' ,\nwhere tau is a real/complex scalar, and v is a real/complex vector with v(i:n) = 0 and v(i-1) = 1;\nv(1:i-1) is stored on exit in A(ia:ia+i-1, ja+i), and tau in tau[ja+i-2].\nIf uplo = L, the matrix Q is represented as a product of elementary reflectors\nQ = H(1)*H(2)*...*H(nb)\nEach H(i) has the form\nH(i) = I - tau*v*v' ,\nwhere tau is a real/complex scalar, and v is a real/complex vector with v(1:i) = 0 and v(i+1) = 1; v(i\n+2: n) is stored on exit in A(ia+i+1: ia+n-1, ja+i-1), and tau in tau[ja+i-2].\nThe elements of the vectors v together form the n-by-nb matrix V which is needed, with W, to apply the\ntransformation to the unreduced part of the matrix, using a symmetric/Hermitian rank-2k update of the\nform:\nsub(A) := sub(A)-vw'-wv'.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1707\n\n\nThe contents of a on exit are illustrated by the following examples with\nn = 5 and nb = 2:\nwhere d denotes a diagonal element of the reduced matrix, a denotes an element of the original matrix that\nis unchanged, and vi denotes an element of the vector defining H(i).\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?latrs\nSolves a triangular system of equations with the scale\nfactor set to prevent overflow.\nSyntax\nvoid pslatrs (char *uplo , char *trans , char *diag , char *normin , MKL_INT *n , float\n*a , MKL_INT *ia , MKL_INT *ja , MKL_INT *desca , float *x , MKL_INT *ix , MKL_INT *jx ,\nMKL_INT *descx , float *scale , float *cnorm , float *work );\nvoid pdlatrs (char *uplo , char *trans , char *diag , char *normin , MKL_INT *n , double\n*a , MKL_INT *ia , MKL_INT *ja , MKL_INT *desca , double *x , MKL_INT *ix , MKL_INT\n*jx , MKL_INT *descx , double *scale , double *cnorm , double *work );\nvoid pclatrs (char *uplo , char *trans , char *diag , char *normin , MKL_INT *n ,\nMKL_Complex8 *a , MKL_INT *ia , MKL_INT *ja , MKL_INT *desca , MKL_Complex8 *x ,\nMKL_INT *ix , MKL_INT *jx , MKL_INT *descx , float *scale , float *cnorm , MKL_Complex8\n*work );\nvoid pzlatrs (char *uplo , char *trans , char *diag , char *normin , MKL_INT *n ,\nMKL_Complex16 *a , MKL_INT *ia , MKL_INT *ja , MKL_INT *desca , MKL_Complex16 *x ,\nMKL_INT *ix , MKL_INT *jx , MKL_INT *descx , double *scale , double *cnorm ,\nMKL_Complex16 *work );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?latrsfunction solves a triangular system of equations Ax = sb, ATx = sb or AHx = sb, where s is a\nscale factor set to prevent overflow. The description of the function will be extended in the future releases.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1708\n\n\nInput Parameters\nuplo\nSpecifies whether the matrix A is upper or lower triangular.\n= 'U': Upper triangular\n= 'L': Lower triangular\ntrans\nSpecifies the operation applied to Ax.\n= 'N': Solve Ax = s*b (no transpose)\n= 'T': Solve ATx = s*b (transpose)\n= 'C': Solve AHx = s*b (conjugate transpose),\nwhere s - is a scale factor\ndiag\nSpecifies whether or not the matrix A is unit triangular.\n= 'N': Non-unit triangular\n= 'U': Unit triangular\nnormin\nSpecifies whether cnorm has been set or not.\n= 'Y': cnorm contains the column norms on entry;\n= 'N': cnorm is not set on entry. On exit, the norms will be computed and\nstored in cnorm.\nn\nThe order of the matrix A. n ≥ 0\na\nArray of size lda* n. Contains the triangular matrix A.\nIf uplo = U, the leading n-by-n upper triangular part of the array a\ncontains the upper triangular matrix, and the strictly lower triangular part\nof a is not referenced.\nIf uplo = 'L', the leading n-by-n lower triangular part of the array a\ncontains the lower triangular matrix, and the strictly upper triangular part\nof a is not referenced.\nIf diag = 'U', the diagonal elements of a are also not referenced and are\nassumed to be 1.\nia, ja\n(global) The row and column indices in the global matrix A indicating the\nfirst row and the first column of the submatrix A, respectively.\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\nx\nArray of size n. On entry, the right hand side b of the triangular system.\nix\n(global).The row index in the global matrix X indicating the first row of\nsub(x).\njx\n(global)\nThe column index in the global matrix X indicating the first column of\nsub(X).\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1709\n\n\ndescx\n(global and local)\nArray of size dlen_. The array descriptor for the distributed matrix X.\ncnorm\nArray of size n. If normin = 'Y', cnorm is an input argument and cnorm[j]\ncontains the norm of the off-diagonal part of the (j+1)-th column of the\nmatrix A, j=0, 1, ..., n-1. If trans = 'N', cnorm[j] must be greater than or\nequal to the infinity-norm, and if trans = 'T' or 'C', cnorm[j] must be\ngreater than or equal to the 1-norm.\nwork\n(local).\nTemporary workspace.\nOutput Parameters\nX\nOn exit, x is overwritten by the solution vector x.\nscale\nArray of size lda* n. The scaling factor s for the triangular system as\ndescribed above.\nIf scale = 0, the matrix A is singular or badly scaled, and the vector x is\nan exact or approximate solution to Ax = 0.\ncnorm\nIf normin = 'N', cnorm is an output argument and cnorm[j] returns the 1-\nnorm of the off-diagonal part of the (j+1)-th column of A, j=0, 1, ..., n-1.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?latrz\nReduces an upper trapezoidal matrix to upper\ntriangular form by means of orthogonal/unitary\ntransformations.\nSyntax\nvoid pslatrz (MKL_INT *m , MKL_INT *n , MKL_INT *l , float *a , MKL_INT *ia , MKL_INT\n*ja , MKL_INT *desca , float *tau , float *work );\nvoid pdlatrz (MKL_INT *m , MKL_INT *n , MKL_INT *l , double *a , MKL_INT *ia , MKL_INT\n*ja , MKL_INT *desca , double *tau , double *work );\nvoid pclatrz (MKL_INT *m , MKL_INT *n , MKL_INT *l , MKL_Complex8 *a , MKL_INT *ia ,\nMKL_INT *ja , MKL_INT *desca , MKL_Complex8 *tau , MKL_Complex8 *work );\nvoid pzlatrz (MKL_INT *m , MKL_INT *n , MKL_INT *l , MKL_Complex16 *a , MKL_INT *ia ,\nMKL_INT *ja , MKL_INT *desca , MKL_Complex16 *tau , MKL_Complex16 *work );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?latrzfunction reduces the m-by-n(m ≤ n) real/complex upper trapezoidal matrix sub(A) =\n[A(ia:ia+m-1, ja:ja+m-1)A(ia:ia+m-1, ja+n-l:ja+n-1)] to upper triangular form by means of\northogonal/unitary transformations.\nThe upper trapezoidal matrix sub(A) is factored as\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1710\n\n\nsub(A) = ( R 0 )*Z,\nwhere Z is an n-by-n orthogonal/unitary matrix and R is an m-by-m upper triangular matrix.\nInput Parameters\nm\n(global)\nThe number of rows in the distributed matrix sub(A). m ≥ 0.\nn\n(global)\nThe number of columns in the distributed matrix sub(A). n ≥ 0.\nl\n(global)\nThe number of columns of the distributed matrix sub(A) containing the\nmeaningful part of the Householder reflectors. l > 0.\na\n(local)\nPointer into the local memory to an array of size lld_a * LOCc(ja+n-1). On\nentry, the local pieces of the m-by-n distributed matrix sub(A), which is to\nbe factored.\nia\n(global)\nThe row index in the global matrix A indicating the first row of sub(A).\nja\n(global)\nThe column index in the global matrix A indicating the first column of\nsub(A).\ndesca\n(global and local) array of size dlen_.\nThe array descriptor for the distributed matrix A.\nwork\n(local)\nWorkspace array of size lwork.\nlwork ≥ nq0 + max(1, mp0), where\niroff = mod(ia-1, mb_a),\nicoff = mod(ja-1, nb_a),\niarow = indxg2p(ia, mb_a, myrow, rsrc_a, nprow),\niacol = indxg2p(ja, nb_a, mycol, csrc_a, npcol),\nmp0 = numroc(m+iroff, mb_a, myrow, iarow, nprow),\nnq0 = numroc(n+icoff, nb_a, mycol, iacol, npcol),\nnumroc, indxg2p, and numroc are ScaLAPACK tool functions; myrow,\nmycol, nprow, and npcol can be determined by calling the function\nblacs_gridinfo.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1711\n\n\nOutput Parameters\na\nOn exit, the leading m-by-m upper triangular part of sub(A) contains the\nupper triangular matrix R, and elements n-l+1 to n of the first m rows of\nsub(A), with the array tau, represent the orthogonal/unitary matrix Z as a\nproduct of m elementary reflectors.\ntau\n(local)\nArray of sizeLOCr(ja+m-1). This array contains the scalar factors of the\nelementary reflectors. tau is tied to the distributed matrix A.\nApplication Notes\nThe factorization is obtained by Householder's method. The k-th transformation matrix, Z(k), which is used\n(or, in case of complex functions, whose conjugate transpose is used) to introduce zeros into the (m - k +\n1)-th row of sub(A), is given in the form\nwhere\ntau is a scalar and z( k ) is an (n-m)-element vector. tau and z( k ) are chosen to annihilate the elements\nof the k-th row of sub(A). The scalar tau is returned in the k-th element of tau, indexed k-1, and the vector\nu( k ) in the k-th row of sub(A), such that the elements of z(k ) are in A( k, m + 1 ), ..., A( k,\nn ). The elements of R are returned in the upper triangular part of sub(A).\nZ is given by\nZ =  Z(1)Z(2)...Z(m).\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?lauu2\nComputes the product U*U' or L'*L, where U and L\nare upper or lower triangular matrices (local\nunblocked algorithm).\nSyntax\nvoid pslauu2 (char *uplo , MKL_INT *n , float *a , MKL_INT *ia , MKL_INT *ja , MKL_INT\n*desca );\nvoid pdlauu2 (char *uplo , MKL_INT *n , double *a , MKL_INT *ia , MKL_INT *ja , MKL_INT\n*desca );\nvoid pclauu2 (char *uplo , MKL_INT *n , MKL_Complex8 *a , MKL_INT *ia , MKL_INT *ja ,\nMKL_INT *desca );\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1712\n\n\nvoid pzlauu2 (char *uplo , MKL_INT *n , MKL_Complex16 *a , MKL_INT *ia , MKL_INT *ja ,\nMKL_INT *desca );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?lauu2function computes the product U*U' or L'*L, where the triangular factor U or L is stored in the\nupper or lower triangular part of the distributed matrix\nsub(A)= A(ia:ia+n-1, ja:ja+n-1).\nIf uplo = 'U' or 'u', then the upper triangle of the result is stored, overwriting the factor U in sub(A).\nIf uplo = 'L' or 'l', then the lower triangle of the result is stored, overwriting the factor L in sub(A).\nThis is the unblocked form of the algorithm, calling BLAS Level 2 Routines. No communication is performed\nby this function, the matrix to operate on should be strictly local to one process.\nInput Parameters\nuplo\n(global)\nSpecifies whether the triangular factor stored in the matrix sub(A) is upper\nor lower triangular:\n= U: upper triangular\n= L: lower triangular.\nn\n(global)\nThe number of rows and columns to be operated on, that is, the order of\nthe triangular factor U or L. n ≥ 0.\na\n(local)\nPointer into the local memory to an array of size lld_a * LOCc(ja+n-1). On\nentry, the local pieces of the triangular factor U or L.\nia\n(global)\nThe row index in the global matrix A indicating the first row of sub(A).\nja\n(global)\nThe column index in the global matrix A indicating the first column of\nsub(A).\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\nOutput Parameters\na\n(local)\nOn exit, if uplo = 'U', the upper triangle of the distributed matrix sub(A)\nis overwritten with the upper triangle of the product U*U'; if uplo = 'L',\nthe lower triangle of sub(A) is overwritten with the lower triangle of the\nproduct L'*L.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1713\n\n\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?lauum\nComputes the product U*U' or L'*L, where U and L\nare upper or lower triangular matrices.\nSyntax\nvoid pslauum (char *uplo , MKL_INT *n , float *a , MKL_INT *ia , MKL_INT *ja , MKL_INT\n*desca );\nvoid pdlauum (char *uplo , MKL_INT *n , double *a , MKL_INT *ia , MKL_INT *ja , MKL_INT\n*desca );\nvoid pclauum (char *uplo , MKL_INT *n , MKL_Complex8 *a , MKL_INT *ia , MKL_INT *ja ,\nMKL_INT *desca );\nvoid pzlauum (char *uplo , MKL_INT *n , MKL_Complex16 *a , MKL_INT *ia , MKL_INT *ja ,\nMKL_INT *desca );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?lauumfunction computes the product U*U' or L'*L, where the triangular factor U or L is stored in the\nupper or lower triangular part of the matrix sub(A)= A(ia:ia+n-1, ja:ja+n-1).\nIf uplo = 'U' or 'u', then the upper triangle of the result is stored, overwriting the factor U in sub(A). If\nuplo = 'L' or 'l', then the lower triangle of the result is stored, overwriting the factor L in sub(A).\nThis is the blocked form of the algorithm, calling Level 3 PBLAS.\nInput Parameters\nuplo\n(global)\nSpecifies whether the triangular factor stored in the matrix sub(A) is upper\nor lower triangular:\n= 'U': upper triangular\n= 'L': lower triangular.\nn\n(global)\nThe number of rows and columns to be operated on, that is, the order of\nthe triangular factor U or L. n ≥ 0.\na\n(local)\nPointer into the local memory to an array of size lld_a * LOCc(ja+n-1). On\nentry, the local pieces of the triangular factor U or L.\nia\n(global)\nThe row index in the global matrix A indicating the first row of sub(A).\nja\n(global)\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1714\n\n\nThe column index in the global matrix A indicating the first column of\nsub(A).\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\nOutput Parameters\na\n(local)\nOn exit, if uplo = 'U', the upper triangle of the distributed matrix sub(A)\nis overwritten with the upper triangle of the product U*U' ; if uplo = 'L',\nthe lower triangle of sub(A) is overwritten with the lower triangle of the\nproduct L'*L.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?lawil\nForms the Wilkinson transform.\nSyntax\nvoid pslawil (const MKL_INT *ii, const MKL_INT *jj, const MKL_INT *m, const float *a,\nconst MKL_INT *desca, const float *h44, const float *h33, const float *h43h34, float\n*v );\nvoid pdlawil (const MKL_INT *ii, const MKL_INT *jj, const MKL_INT *m, const double *a,\nconst MKL_INT *desca, const double *h44, const double *h33, const double *h43h34,\ndouble *v );\nvoid pclawil (const MKL_INT *ii , const MKL_INT *jj , const MKL_INT *m , const\nMKL_Complex8 *a , const MKL_INT *desca , const MKL_Complex8 *h44 , const MKL_Complex8\n*h33 , const MKL_Complex8 *h43h34 , MKL_Complex8 *v );\nvoid pzlawil (const MKL_INT *ii , const MKL_INT *jj , const MKL_INT *m , const\nMKL_Complex16 *a , const MKL_INT *desca , const MKL_Complex16 *h44 , const\nMKL_Complex16 *h33 , const MKL_Complex16 *h43h34 , MKL_Complex16 *v );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?lawilfunction gets the transform given by h44, h33, and h43h34 into v starting at row m.\nInput Parameters\nii\n(global)\nNumber of the process row which owns the matrix element A(m+2, m+2).\njj\n(global)\nNumber of the process column which owns the matrix element A(m+2, m\n+2).\nm\n(global)\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1715\n\n\nOn entry, the location from where the transform starts (row m). Unchanged\non exit.\na\n(local)\nArray of size lld_a*LOCc(n_a).\nOn entry, the Hessenberg matrix. Unchanged on exit.\ndesca\n(global and local)\nArray of size dlen_. The array descriptor for the distributed matrix A.\nUnchanged on exit.\nh43h34\n(global)\nThese three values are for the double shift QR iteration. Unchanged on exit.\nOutput Parameters\nv\n(global)\nArray of size 3 that contains the transform on output.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?org2l/p?ung2l\nGenerates all or part of the orthogonal/unitary matrix\nQ from a QL factorization determined by p?geqlf\n(unblocked algorithm).\nSyntax\nvoid psorg2l (MKL_INT *m , MKL_INT *n , MKL_INT *k , float *a , MKL_INT *ia , MKL_INT\n*ja , MKL_INT *desca , float *tau , float *work , MKL_INT *lwork , MKL_INT *info );\nvoid pdorg2l (MKL_INT *m , MKL_INT *n , MKL_INT *k , double *a , MKL_INT *ia , MKL_INT\n*ja , MKL_INT *desca , double *tau , double *work , MKL_INT *lwork , MKL_INT *info );\nvoid pcung2l (MKL_INT *m , MKL_INT *n , MKL_INT *k , MKL_Complex8 *a , MKL_INT *ia ,\nMKL_INT *ja , MKL_INT *desca , MKL_Complex8 *tau , MKL_Complex8 *work , MKL_INT\n*lwork , MKL_INT *info );\nvoid pzung2l (MKL_INT *m , MKL_INT *n , MKL_INT *k , MKL_Complex16 *a , MKL_INT *ia ,\nMKL_INT *ja , MKL_INT *desca , MKL_Complex16 *tau , MKL_Complex16 *work , MKL_INT\n*lwork , MKL_INT *info );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?org2l/p?ung2lfunction generates an m-by-n real/complex distributed matrix Q denoting A(ia:ia\n+m-1, ja:ja+n-1) with orthonormal columns, which is defined as the last n columns of a product of k\nelementary reflectors of order m:\nQ = H(k)*...*H(2)*H(1) as returned by p?geqlf.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1716\n\n\nInput Parameters\nm\n(global)\nThe number of rows in the distributed submatrix Q. m ≥ 0.\nn\n(global)\nThe number of columns in the distributed submatrix Q. m ≥ n ≥ 0.\nk\n(global)\nThe number of elementary reflectors whose product defines the matrix Q.\nn≥ k ≥ 0.\na\nPointer into the local memory to an array of size lld_a * LOCc(ja+n-1).\nOn entry, the j-th column of the matrix stored in amust contain the vector\nthat defines the elementary reflector H(j), ja+n-k ≤ j ≤ ja+n-k, as\nreturned by p?geqlf in the k columns of its distributed matrix argument\nA(ia:*,ja+n-k:ja+n-1).\nia\n(global)\nThe row index in the global matrix A indicating the first row of sub(A).\nja\n(global)\nThe column index in the global matrix A indicating the first column of\nsub(A).\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\ntau\n(local)\nArray of size LOCc(ja+n-1).\ntau[j] contains the scalar factor of the elementary reflector H(j+1), j =\n0, 1, ..., LOCc(ja+n-1)-1, as returned by p?geqlf.\nwork\n(local)\nWorkspace array of size lwork.\nlwork\n(local or global)\nThe size of the array work.\nlwork is local input and must be at least lwork ≥ mpa0 + max(1, nqa0),\nwhere\niroffa = mod(ia-1, mb_a), \nicoffa = mod(ja-1, nb_a),\niarow = indxg2p(ia, mb_a, myrow, rsrc_a, nprow),\niacol = indxg2p(ja, nb_a, mycol, csrc_a, npcol),\nmpa0 = numroc(m+iroffa, mb_a, myrow, iarow, nprow),\nnqa0 = numroc(n+icoffa, nb_a, mycol, iacol, npcol).\nindxg2p and numroc are ScaLAPACK tool functions; myrow, mycol, nprow,\nand npcol can be determined by calling the function blacs_gridinfo.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1717\n\n\nIf lwork = -1, then lwork is global input and a workspace query is\nassumed; the function only calculates the minimum and optimal size for all\nwork arrays. Each of these values is returned in the first entry of the\ncorresponding work array, and no error message is issued by pxerbla.\nOutput Parameters\na\nOn exit, this array contains the local pieces of the m-by-n distributed matrix\nQ.\nwork\nOn exit, work[0] returns the minimal and optimal lwork.\ninfo\n(local).\n= 0: successful exit\n< 0: if the i-th argument is an array and the j-th entry, indexed j-1, had an\nillegal value,\nthen info = - (i*100 +j),\nif the i-th argument is a scalar and had an illegal value,\nthen info = -i.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?org2r/p?ung2r\nGenerates all or part of the orthogonal/unitary matrix\nQ from a QR factorization determined by p?geqrf\n(unblocked algorithm).\nSyntax\nvoid psorg2r (MKL_INT *m , MKL_INT *n , MKL_INT *k , float *a , MKL_INT *ia , MKL_INT\n*ja , MKL_INT *desca , float *tau , float *work , MKL_INT *lwork , MKL_INT *info );\nvoid pdorg2r (MKL_INT *m , MKL_INT *n , MKL_INT *k , double *a , MKL_INT *ia , MKL_INT\n*ja , MKL_INT *desca , double *tau , double *work , MKL_INT *lwork , MKL_INT *info );\nvoid pcung2r (MKL_INT *m , MKL_INT *n , MKL_INT *k , MKL_Complex8 *a , MKL_INT *ia ,\nMKL_INT *ja , MKL_INT *desca , MKL_Complex8 *tau , MKL_Complex8 *work , MKL_INT\n*lwork , MKL_INT *info );\nvoid pzung2r (MKL_INT *m , MKL_INT *n , MKL_INT *k , MKL_Complex16 *a , MKL_INT *ia ,\nMKL_INT *ja , MKL_INT *desca , MKL_Complex16 *tau , MKL_Complex16 *work , MKL_INT\n*lwork , MKL_INT *info );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?org2r/p?ung2rfunction generates an m-by-n real/complex matrix Q denoting A(ia:ia+m-1, ja:ja\n+n-1) with orthonormal columns, which is defined as the first n columns of a product of k elementary\nreflectors of order m:\nQ = H(1)*H(2)*...*H(k)\nas returned by p?geqrf.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1718\n\n\nInput Parameters\nm\n(global)\nThe number of rows in the distributed submatrix Q.m ≥ 0.\nn\n(global)\nThe number of columns in the distributed submatrix Q. m ≥ n ≥ 0.\nk\n(global)\nThe number of elementary reflectors whose product defines the matrix Q. n\n≥ k ≥ 0.\na\nPointer into the local memory to an array of sizelld_a * LOCc(ja+n-1)\nOn entry, the j-th column of the matrix stored in amust contain the vector\nthat defines the elementary reflector H(j), ja ≤ j ≤ ja+k-1, as returned\nby p?geqrf in the k columns of its distributed matrix argument\nA(ia:*,ja:ja+k-1).\nia\n(global)\nThe row index in the global matrix A indicating the first row of sub(A).\nja\n(global)\nThe column index in the global matrix A indicating the first column of\nsub(A).\ndesca\n(global and local) array of size dlen_.\nThe array descriptor for the distributed matrix A.\ntau\n(local)\nArray of size LOCc(ja+k-1).\ntau[j] contains the scalar factor of the elementary reflector H(j+1), j =\n0, 1, ..., LOCc(ja+k-1)-1, as returned by p?geqrf. This array is tied\nto the distributed matrix A.\nwork\n(local)\nWorkspace array of size lwork.\nlwork\n(local or global)\nThe size of the array work.\nlwork is local input and must be at least lwork ≥ mpa0 + max(1, nqa0),\nwhere\niroffa = mod(ia-1, mb_a , icoffa = mod(ja-1, nb_a),\niarow = indxg2p(ia, mb_a, myrow, rsrc_a, nprow),\niacol = indxg2p(ja, nb_a, mycol, csrc_a, npcol),\nmpa0 = numroc(m+iroffa, mb_a, myrow, iarow, nprow),\nnqa0 = numroc(n+icoffa, nb_a, mycol, iacol, npcol).\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1719\n\n\nindxg2p and numroc are ScaLAPACK tool functions; myrow, mycol, nprow,\nand npcol can be determined by calling the function blacs_gridinfo.\nIf lwork = -1, then lwork is global input and a workspace query is\nassumed; the function only calculates the minimum and optimal size for all\nwork arrays. Each of these values is returned in the first entry of the\ncorresponding work array, and no error message is issued by pxerbla.\nOutput Parameters\na\nOn exit, this array contains the local pieces of the m-by-n distributed matrix\nQ.\nwork\nOn exit, work[0] returns the minimal and optimal lwork.\ninfo\n(local).\n= 0: successful exit\n< 0: if the i-th argument is an array and the j-th entry, indexed j-1, had an\nillegal value,\nthen info = - (i*100 +j),\nif the i-th argument is a scalar and had an illegal value,\nthen info = -i.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?orgl2/p?ungl2\nGenerates all or part of the orthogonal/unitary matrix\nQ from an LQ factorization determined by p?gelqf\n(unblocked algorithm).\nSyntax\nvoid psorgl2 (MKL_INT *m , MKL_INT *n , MKL_INT *k , float *a , MKL_INT *ia , MKL_INT\n*ja , MKL_INT *desca , float *tau , float *work , MKL_INT *lwork , MKL_INT *info );\nvoid pdorgl2 (MKL_INT *m , MKL_INT *n , MKL_INT *k , double *a , MKL_INT *ia , MKL_INT\n*ja , MKL_INT *desca , double *tau , double *work , MKL_INT *lwork , MKL_INT *info );\nvoid pcungl2 (MKL_INT *m , MKL_INT *n , MKL_INT *k , MKL_Complex8 *a , MKL_INT *ia ,\nMKL_INT *ja , MKL_INT *desca , MKL_Complex8 *tau , MKL_Complex8 *work , MKL_INT\n*lwork , MKL_INT *info );\nvoid pzungl2 (MKL_INT *m , MKL_INT *n , MKL_INT *k , MKL_Complex16 *a , MKL_INT *ia ,\nMKL_INT *ja , MKL_INT *desca , MKL_Complex16 *tau , MKL_Complex16 *work , MKL_INT\n*lwork , MKL_INT *info );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?orgl2/p?ungl2function generates a m-by-n real/complex matrix Q denoting A(ia:ia+m-1, ja:ja\n+n-1) with orthonormal rows, which is defined as the first m rows of a product of k elementary reflectors of\norder n\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1720\n\n\nQ = H(k)*...*H(2)*H(1) (for real flavors),\nQ = (H(k))H*...*(H(2))H*(H(1))H (for complex flavors) as returned by p?gelqf.\nInput Parameters\nm\n(global)\nThe number of rows in the distributed submatrix Q. m ≥ 0.\nn\n(global)\nThe number of columns in the distributed submatrix Q. n ≥ m ≥ 0.\nk\n(global)\nThe number of elementary reflectors whose product defines the matrix Q. m\n≥ k ≥ 0.\na\nPointer into the local memory to an array of size lld_a * LOCc(ja+n-1).\nOn entry, the i-th row of the matrix stored in amust contain the vector that\ndefines the elementary reflector H(i), ia ≤ i ≤ ia+k-1, as returned by \np?gelqf in the k rows of its distributed matrix argument A(ia:ia+k-1,\nja:*).\nia\n(global)\nThe row index in the global matrix A indicating the first row of sub(A).\nja\n(global)\nThe column index in the global matrix A indicating the first column of\nsub(A).\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\ntau\n(local)\nArray of size LOCr(ja+k-1). tau[j] contains the scalar factor of the\nelementary reflectors H(j+1), j = 0, 1, ..., LOCr(ja+k-1)-1, as returned by \np?gelqf. This array is tied to the distributed matrix A.\nWORK\n(local)\nWorkspace array of size lwork.\nlwork\n(local or global)\nThe size of the array work.\nlwork is local input and must be at least lwork ≥ nqa0 + max(1, mpa0),\nwhere \niroffa = mod(ia-1, mb_a), \nicoffa = mod(ja-1, nb_a),\niarow = indxg2p(ia, mb_a, myrow, rsrc_a, nprow),\niacol = indxg2p(ja, nb_a, mycol, csrc_a, npcol),\nmpa0 = numroc(m+iroffa, mb_a, myrow, iarow, nprow),\nnqa0 = numroc(n+icoffa, nb_a, mycol, iacol, npcol).\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1721\n\n\nindxg2p and numroc are ScaLAPACK tool functions; myrow, mycol, nprow,\nand npcol can be determined by calling the function blacs_gridinfo.\nIf lwork = -1, then lwork is global input and a workspace query is\nassumed; the function only calculates the minimum and optimal size for all\nwork arrays. Each of these values is returned in the first entry of the\ncorresponding work array, and no error message is issued by pxerbla.\nOutput Parameters\na\nOn exit, this array contains the local pieces of the m-by-n distributed matrix\nQ.\nwork\nOn exit, work[0] returns the minimal and optimal lwork.\ninfo\n(local)\n= 0: successful exit\n< 0: if the i-th argument is an array and the j-th entry, indexed j-1, had an\nillegal value,\nthen info = - (i*100 +j),\nif the i-th argument is a scalar and had an illegal value,\nthen info = -i.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?orgr2/p?ungr2\nGenerates all or part of the orthogonal/unitary matrix\nQ from an RQ factorization determined by p?gerqf\n(unblocked algorithm).\nSyntax\nvoid psorgr2 (MKL_INT *m , MKL_INT *n , MKL_INT *k , float *a , MKL_INT *ia , MKL_INT\n*ja , MKL_INT *desca , float *tau , float *work , MKL_INT *lwork , MKL_INT *info );\nvoid pdorgr2 (MKL_INT *m , MKL_INT *n , MKL_INT *k , double *a , MKL_INT *ia , MKL_INT\n*ja , MKL_INT *desca , double *tau , double *work , MKL_INT *lwork , MKL_INT *info );\nvoid pcungr2 (MKL_INT *m , MKL_INT *n , MKL_INT *k , MKL_Complex8 *a , MKL_INT *ia ,\nMKL_INT *ja , MKL_INT *desca , MKL_Complex8 *tau , MKL_Complex8 *work , MKL_INT\n*lwork , MKL_INT *info );\nvoid pzungr2 (MKL_INT *m , MKL_INT *n , MKL_INT *k , MKL_Complex16 *a , MKL_INT *ia ,\nMKL_INT *ja , MKL_INT *desca , MKL_Complex16 *tau , MKL_Complex16 *work , MKL_INT\n*lwork , MKL_INT *info );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?orgr2/p?ungr2function generates an m-by-n real/complex matrix Q denoting A(ia:ia+m-1, ja:ja\n+n-1) with orthonormal rows, which is defined as the last m rows of a product of k elementary reflectors of\norder n\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1722\n\n\nQ = H(1)*H(2)*...*H(k) (for real flavors);\nQ = (H(1))H*(H(2))H...*(H(k))H (for complex flavors) as returned by p?gerqf.\nInput Parameters\nm\n(global)\nThe number of rows in the distributed submatrix Q. m ≥ 0.\nn\n(global)\nThe number of columns in the distributed submatrix Q. n ≥ m ≥ 0.\nk\n(global)\nThe number of elementary reflectors whose product defines the matrix Q. m\n≥ k ≥ 0.\na\nPointer into the local memory to an array of size lld_a * LOCc(ja+n-1).\nOn entry, the i-th row of the matrix stored in amust contain the vector that\ndefines the elementary reflector H(i), ia+m-k ≤ i ≤ ia+m-1, as returned by \np?gerqf in the k rows of its distributed matrix argument A(ia+m-k:ia\n+m-1, ja:*).\nia\n(global)\nThe row index in the global matrix A indicating the first row of sub(A).\nja\n(global)\nThe column index in the global matrix A indicating the first column of\nsub(A).\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\ntau\n(local)\nArray of size LOCr(ja+m-1). tau[j] contains the scalar factor of the\nelementary reflectors H(j+1), j = 0, 1, ..., LOCr(ja+m-1)-1, as returned by \np?gerqf. This array is tied to the distributed matrix A.\nwork\n(local)\nWorkspace array of size lwork.\nlwork\n(local or global)\nThe size of the array work.\nlwork is local input and must be at least lwork ≥ nqa0 + max(1, mpa0 ),\nwhere iroffa = mod( ia-1, mb_a ), icoffa = mod( ja-1, nb_a ),\niarow = indxg2p( ia, mb_a, myrow, rsrc_a, nprow ),\niacol = indxg2p( ja, nb_a, mycol, csrc_a, npcol ),\nmpa0 = numroc( m+iroffa, mb_a, myrow, iarow, nprow ),\nnqa0 = numroc( n+icoffa, nb_a, mycol, iacol, npcol ).\nindxg2p and numroc are ScaLAPACK tool functions; myrow, mycol, nprow,\nand npcol can be determined by calling the function blacs_gridinfo.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1723\n\n\nIf lwork = -1, then lwork is global input and a workspace query is\nassumed; the function only calculates the minimum and optimal size for all\nwork arrays. Each of these values is returned in the first entry of the\ncorresponding work array, and no error message is issued by pxerbla.\nOutput Parameters\na\nOn exit, this array contains the local pieces of the m-by-n distributed matrix\nQ.\nwork\nOn exit, work[0] returns the minimal and optimal lwork.\ninfo\n(local)\n= 0: successful exit\n< 0: if the i-th argument is an array and the j-th entry, indexed j-1, had an\nillegal value,\nthen info = - (i*100 +j),\nif the i-th argument is a scalar and had an illegal value,\nthen info = -i.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?orm2l/p?unm2l\nMultiplies a general matrix by the orthogonal/unitary\nmatrix from a QL factorization determined by p?geqlf\n(unblocked algorithm).\nSyntax\nvoid psorm2l (char *side , char *trans , MKL_INT *m , MKL_INT *n , MKL_INT *k , float\n*a , MKL_INT *ia , MKL_INT *ja , MKL_INT *desca , float *tau , float *c , MKL_INT *ic ,\nMKL_INT *jc , MKL_INT *descc , float *work , MKL_INT *lwork , MKL_INT *info );\nvoid pdorm2l (char *side , char *trans , MKL_INT *m , MKL_INT *n , MKL_INT *k , double\n*a , MKL_INT *ia , MKL_INT *ja , MKL_INT *desca , double *tau , double *c , MKL_INT\n*ic , MKL_INT *jc , MKL_INT *descc , double *work , MKL_INT *lwork , MKL_INT *info );\nvoid pcunm2l (char *side , char *trans , MKL_INT *m , MKL_INT *n , MKL_INT *k ,\nMKL_Complex8 *a , MKL_INT *ia , MKL_INT *ja , MKL_INT *desca , MKL_Complex8 *tau ,\nMKL_Complex8 *c , MKL_INT *ic , MKL_INT *jc , MKL_INT *descc , MKL_Complex8 *work ,\nMKL_INT *lwork , MKL_INT *info );\nvoid pzunm2l (char *side , char *trans , MKL_INT *m , MKL_INT *n , MKL_INT *k ,\nMKL_Complex16 *a , MKL_INT *ia , MKL_INT *ja , MKL_INT *desca , MKL_Complex16 *tau ,\nMKL_Complex16 *c , MKL_INT *ic , MKL_INT *jc , MKL_INT *descc , MKL_Complex16 *work ,\nMKL_INT *lwork , MKL_INT *info );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?orm2l/p?unm2lfunction overwrites the general real/complex m-by-n distributed matrix sub\n(C)=C(ic:ic+m-1,jc:jc+n-1) with\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1724\n\n\nQ*sub(C) if side = 'L' and trans = 'N', or\nQT*sub(C) / QH*sub(C) if side = 'L' and trans = 'T' (for real flavors) or trans = 'C' (for complex\nflavors), or\nsub(C)*Q if side = 'R' and trans = 'N', or\nsub(C)*QT / sub(C)*QH if side = 'R' and trans = 'T' (for real flavors) or trans = 'C' (for complex\nflavors).\nwhere Q is a real orthogonal or complex unitary distributed matrix defined as the product of k elementary\nreflectors\nQ = H(k)*...*H(2)*H(1) as returned by p?geqlf . Q is of order m if side = 'L' and of order n if side =\n'R'.\nInput Parameters\nside\n(global)\n= 'L': apply Q or QT for real flavors (QH for complex flavors) from the left,\n= 'R': apply Q or QT for real flavors (QH for complex flavors) from the\nright.\ntrans\n(global)\n= 'N': apply Q (no transpose)\n= 'T': apply QT (transpose, for real flavors)\n= 'C': apply QH (conjugate transpose, for complex flavors)\nm\n(global)\nThe number of rows in the distributed matrix sub(C). m ≥ 0.\nn\n(global)\nThe number of columns in the distributed matrix sub(C). n ≥ 0.\nk\n(global)\nThe number of elementary reflectors whose product defines the matrix Q.\nIf side = 'L', m ≥ k ≥ 0;\nif side = 'R', n ≥ k ≥ 0.\na\n(local)\nPointer into the local memory to an array of size lld_a * LOCc(ja+k-1).\nOn entry, the j-th row of the matrix stored in amust contain the vector that\ndefines the elementary reflector H(j), ja ≤ j ≤ ja+k-1, as returned by \np?geqlf in the k columns of its distributed matrix argument A(ia:*,ja:ja\n+k-1). The argument A(ia:*,ja:ja+k-1) is modified by the function but\nrestored on exit.\nIf side = 'L', lld_a ≥ max(1, LOCr(ia+m-1)),\nif side = 'R', lld_a ≥ max(1, LOCr(ia+n-1)).\nia\n(global)\nThe row index in the global matrix A indicating the first row of sub(A).\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1725\n\n\nja\n(global)\nThe column index in the global matrix A indicating the first column of\nsub(A).\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\ntau\n(local)\nArray of size LOCc(ja+n-1). tau[j] contains the scalar factor of the\nelementary reflector H(j+1), j = 0, 1, ..., LOCc(ja+n-1)-1, as returned by \np?geqlf. This array is tied to the distributed matrix A.\nc\n(local)\nPointer into the local memory to an array of size lld_c * LOCc(jc+n-1).On\nentry, the local pieces of the distributed matrix sub (C).\nic\n(global)\nThe row index in the global matrix C indicating the first row of sub(C).\njc\n(global)\nThe column index in the global matrix C indicating the first column of\nsub(C).\ndescc\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix C.\nwork\n(local)\nWorkspace array of size lwork.\nOn exit, work(1) returns the minimal and optimal lwork.\nlwork\n(local or global)\nThe size of the array work.\nlwork is local input and must be at least\nif side = 'L', lwork ≥ mpc0 + max(1, nqc0),\nif side = 'R', lwork ≥ nqc0 + max(max(1, mpc0), numroc(numroc(n\n+icoffc, nb_a, 0, 0, npcol), nb_a, 0, 0, lcmq)),\nwhere \nlcmq = lcm/npcol, \nlcm = iclm(nprow, npcol),\niroffc = mod(ic-1, mb_c),\nicoffc = mod(jc-1, nb_c),\nicrow = indxg2p(ic, mb_c, myrow, rsrc_c, nprow),\niccol = indxg2p(jc, nb_c, mycol, csrc_c, npcol),\nMqc0 = numroc(m+icoffc, nb_c, mycol, icrow, nprow),\nNpc0 = numroc(n+iroffc, mb_c, myrow, iccol, npcol),\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1726\n\n\nilcm, indxg2p, and numroc are ScaLAPACK tool functions; myrow, mycol,\nnprow, and npcol can be determined by calling the function\nblacs_gridinfo.\nIf lwork = -1, then lwork is global input and a workspace query is\nassumed; the function only calculates the minimum and optimal size for all\nwork arrays. Each of these values is returned in the first entry of the\ncorresponding work array, and no error message is issued by pxerbla.\nOutput Parameters\nc\nOn exit, c is overwritten by Q*sub(C), or QT*sub(C)/ QH*sub(C), or\nsub(C)*Q, or sub(C)*QT / sub(C)*QH\nwork\nOn exit, work[0] returns the minimal and optimal lwork.\ninfo\n(local)\n= 0: successful exit\n< 0: if the i-th argument is an array and the j-th entry, indexed j-1, had an\nillegal value,\nthen info = - (i*100 +j),\nif the i-th argument is a scalar and had an illegal value,\nthen info = -i.\nNOTE\nThe distributed submatrices A(ia:*, ja:*) and C(ic:ic+m-1,jc:jc+n-1) must verify some\nalignment properties, namely the following expressions should be true:\nIf side = 'L', ( mb_a == mb_c && iroffa == iroffc && iarow == icrow )\nIf side = 'R', ( mb_a == nb_c && iroffa == iroffc ).\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?orm2r/p?unm2r\nMultiplies a general matrix by the orthogonal/unitary\nmatrix from a QR factorization determined by\np?geqrf (unblocked algorithm).\nSyntax\nvoid psorm2r (char *side , char *trans , MKL_INT *m , MKL_INT *n , MKL_INT *k , float\n*a , MKL_INT *ia , MKL_INT *ja , MKL_INT *desca , float *tau , float *c , MKL_INT *ic ,\nMKL_INT *jc , MKL_INT *descc , float *work , MKL_INT *lwork , MKL_INT *info );\nvoid pdorm2r (char *side , char *trans , MKL_INT *m , MKL_INT *n , MKL_INT *k , double\n*a , MKL_INT *ia , MKL_INT *ja , MKL_INT *desca , double *tau , double *c , MKL_INT\n*ic , MKL_INT *jc , MKL_INT *descc , double *work , MKL_INT *lwork , MKL_INT *info );\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1727\n\n\nvoid pcunm2r (char *side , char *trans , MKL_INT *m , MKL_INT *n , MKL_INT *k ,\nMKL_Complex8 *a , MKL_INT *ia , MKL_INT *ja , MKL_INT *desca , MKL_Complex8 *tau ,\nMKL_Complex8 *c , MKL_INT *ic , MKL_INT *jc , MKL_INT *descc , MKL_Complex8 *work ,\nMKL_INT *lwork , MKL_INT *info );\nvoid pzunm2r (char *side , char *trans , MKL_INT *m , MKL_INT *n , MKL_INT *k ,\nMKL_Complex16 *a , MKL_INT *ia , MKL_INT *ja , MKL_INT *desca , MKL_Complex16 *tau ,\nMKL_Complex16 *c , MKL_INT *ic , MKL_INT *jc , MKL_INT *descc , MKL_Complex16 *work ,\nMKL_INT *lwork , MKL_INT *info );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?orm2r/p?unm2rfunction overwrites the general real/complex m-by-n distributed matrix sub\n(C)=C(ic:ic+m-1, jc:jc+n-1) with\nQ*sub(C) if side = 'L' and trans = 'N', or\nQT*sub(C) / QH*sub(C) if side = 'L' and trans = 'T' (for real flavors) or trans = 'C' (for complex\nflavors), or\nsub(C)*Q if side = 'R' and trans = 'N', or\nsub(C)*QT / sub(C)*QH if side = 'R' and trans = 'T' (for real flavors) or trans = 'C' (for complex\nflavors).\nwhere Q is a real orthogonal or complex unitary matrix defined as the product of k elementary reflectors\nQ = H(k)*...*H(2)*H(1) as returned by p?geqrf . Q is of order m if side = 'L' and of order n if side =\n'R'.\nInput Parameters\nside\n(global)\n= 'L': apply Q or QT for real flavors (QH for complex flavors) from the left,\n= 'R': apply Q or QT for real flavors (QH for complex flavors) from the\nright.\ntrans\n(global)\n= 'N': apply Q (no transpose)\n= 'T': apply QT (transpose, for real flavors)\n= 'C': apply QH (conjugate transpose, for complex flavors)\nm\n(global)\nThe number of rows in the distributed matrix sub(C). m ≥ 0.\nn\n(global)\nThe number of columns in the distributed matrix sub(C). n ≥ 0.\nk\n(global)\nThe number of elementary reflectors whose product defines the matrix Q.\nIf side = 'L', m ≥ k ≥ 0;\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1728\n\n\nif side = 'R', n ≥ k ≥ 0.\na\n(local)\nPointer into the local memory to an array of size lld_a * LOCc(ja+k-1).\nOn entry, the j-th column of the matrix stored in amust contain the vector\nthat defines the elementary reflector H(j), ja ≤ j ≤ja+k-1, as returned by \np?geqrf in the k columns of its distributed matrix argument A(ia:*,ja:ja\n+k-1). The argument A(ia:*,ja:ja+k-1) is modified by the function but\nrestored on exit.\nIf side = 'L', lld_a ≥ max(1, LOCr(ia+m-1)),\nif side = 'R', lld_a ≥ max(1, LOCr(ia+n-1)).\nia\n(global)\nThe row index in the global matrix A indicating the first row of sub(A).\nja\n(global)\nThe column index in the global matrix A indicating the first column of\nsub(A).\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\ntau\n(local)\nArray of size LOCc(ja+k-1). tau[j] contains the scalar factor of the\nelementary reflector H(j+1), j = 0, 1, ..., LOCc(ja+k-1)-1, as returned by \np?geqrf. This array is tied to the distributed matrix A.\nc\n(local)\nPointer into the local memory to an array of size lld_c * LOCc(jc+n-1).\nOn entry, the local pieces of the distributed matrix sub (C).\nic\n(global)\nThe row index in the global matrix C indicating the first row of sub(C).\njc\n(global)\nThe column index in the global matrix C indicating the first column of\nsub(C).\ndescc\n(global and local) array of size dlen_.\nThe array descriptor for the distributed matrix C.\nwork\n(local)\nWorkspace array of size lwork.\nlwork\n(local or global)\nThe size of the array work.\nlwork is local input and must be at least\nif side = 'L', lwork ≥ mpc0 + max(1, nqc0),\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1729\n\n\nif side = 'R', lwork ≥ nqc0 + max(max(1, mpc0), numroc(numroc(n\n+icoffc, nb_a, 0, 0, npcol), nb_a, 0, 0, lcmq)),\nwhere \nlcmq = lcm/npcol ,\nlcm = iclm(nprow, npcol),\niroffc = mod(ic-1, mb_c), \nicoffc = mod(jc-1, nb_c),\nicrow = indxg2p(ic, mb_c, myrow, rsrc_c, nprow),\niccol = indxg2p(jc, nb_c, mycol, csrc_c, npcol),\nMqc0 = numroc(m+icoffc, nb_c, mycol, icrow, nprow),\nNpc0 = numroc(n+iroffc, mb_c, myrow, iccol, npcol),\nilcm, indxg2p and numroc are ScaLAPACK tool functions; myrow, mycol,\nnprow, and npcol can be determined by calling the function\nblacs_gridinfo.\nIf lwork = -1, then lwork is global input and a workspace query is\nassumed; the function only calculates the minimum and optimal size for all\nwork arrays. Each of these values is returned in the first entry of the\ncorresponding work array, and no error message is issued by pxerbla.\nOutput Parameters\nc\nOn exit, c is overwritten by Q*sub(C), or QT*sub(C)/ QH*sub(C), or\nsub(C)*Q, or sub(C)*QT / sub(C)*QH\nwork\nOn exit, work[0] returns the minimal and optimal lwork.\ninfo\n(local)\n= 0: successful exit\n< 0: if the i-th argument is an array and the j-th entry, indexed j-1, had an\nillegal value,\nthen info = - (i*100 +j),\nif the i-th argument is a scalar and had an illegal value,\nthen info = -i.\nNOTE\nThe distributed submatrices A(ia:*, ja:*) and C(ic:ic+m-1, jc:jc+n-1) must verify some\nalignment properties, namely the following expressions should be true:\nIf side = 'L', (mb_a == mb_c) && (iroffa == iroffc) && (iarow == icrow).\nIf side = 'R', (mb_a == nb_c) && (iroffa == iroffc).\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1730\n\n\np?orml2/p?unml2\nMultiplies a general matrix by the orthogonal/unitary\nmatrix from an LQ factorization determined by\np?gelqf (unblocked algorithm).\nSyntax\nvoid psorml2 (char *side , char *trans , MKL_INT *m , MKL_INT *n , MKL_INT *k , float\n*a , MKL_INT *ia , MKL_INT *ja , MKL_INT *desca , float *tau , float *c , MKL_INT *ic ,\nMKL_INT *jc , MKL_INT *descc , float *work , MKL_INT *lwork , MKL_INT *info );\nvoid pdorml2 (char *side , char *trans , MKL_INT *m , MKL_INT *n , MKL_INT *k , double\n*a , MKL_INT *ia , MKL_INT *ja , MKL_INT *desca , double *tau , double *c , MKL_INT\n*ic , MKL_INT *jc , MKL_INT *descc , double *work , MKL_INT *lwork , MKL_INT *info );\nvoid pcunml2 (char *side , char *trans , MKL_INT *m , MKL_INT *n , MKL_INT *k ,\nMKL_Complex8 *a , MKL_INT *ia , MKL_INT *ja , MKL_INT *desca , MKL_Complex8 *tau ,\nMKL_Complex8 *c , MKL_INT *ic , MKL_INT *jc , MKL_INT *descc , MKL_Complex8 *work ,\nMKL_INT *lwork , MKL_INT *info );\nvoid pzunml2 (char *side , char *trans , MKL_INT *m , MKL_INT *n , MKL_INT *k ,\nMKL_Complex16 *a , MKL_INT *ia , MKL_INT *ja , MKL_INT *desca , MKL_Complex16 *tau ,\nMKL_Complex16 *c , MKL_INT *ic , MKL_INT *jc , MKL_INT *descc , MKL_Complex16 *work ,\nMKL_INT *lwork , MKL_INT *info );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?orml2/p?unml2function overwrites the general real/complex m-by-n distributed matrix sub\n(C)=C(ic:ic+m-1, jc:jc+n-1) with\nQ*sub(C) if side = 'L' and trans = 'N', or\nQT*sub(C) / QH*sub(C) if side = 'L' and trans = 'T' (for real flavors) or trans = 'C' (for complex\nflavors), or\nsub(C)*Q if side = 'R' and trans = 'N', or\nsub(C)*QT / sub(C)*QH if side = 'R' and trans = 'T' (for real flavors) or trans = 'C' (for complex\nflavors).\nwhere Q is a real orthogonal or complex unitary distributed matrix defined as the product of k elementary\nreflectors\nQ = H(k)*...*H(2)*H(1) (for real flavors)\nQ = (H(k))H*...*(H(2))H*(H(1))H (for complex flavors)\nas returned by p?gelqf . Q is of order m if side = 'L' and of order n if side = 'R'.\nInput Parameters\nside\n(global)\n= 'L': apply Q or QT for real flavors (QH for complex flavors) from the left,\n= 'R': apply Q or QT for real flavors (QH for complex flavors) from the\nright.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1731\n\n\ntrans\n(global)\n= 'N': apply Q (no transpose)\n= 'T': apply QT (transpose, for real flavors)\n= 'C': apply QH (conjugate transpose, for complex flavors)\nm\n(global)\nThe number of rows in the distributed matrix sub(C). m ≥ 0.\nn\n(global)\nThe number of columns in the distributed matrix sub(C). n ≥ 0.\nk\n(global)\nThe number of elementary reflectors whose product defines the matrix Q.\nIf side = 'L', m ≥ k ≥ 0;\nif side = 'R', n ≥ k ≥ 0.\na\n(local)\nPointer into the local memory to an array of size\nlld_a * LOCc(ja+m-1) if side='L',\nlld_a * LOCc(ja+n-1) if side='R',\nwhere lld_a ≥ max (1, LOCr(ia+k-1)).\nOn entry, the i-th row of the matrix stored in amust contain the vector that\ndefines the elementary reflector H(i), ia ≤ i ≤ ia+k-1, as returned by \np?gelqf in the k rows of its distributed matrix argument A(ia:ia+k-1,\nja:*). The argument A(ia:ia+k-1, ja:*) is modified by the function but\nrestored on exit.\nia\n(global)\nThe row index in the global matrix A indicating the first row of sub(A).\nja\n(global)\nThe column index in the global matrix A indicating the first column of\nsub(A).\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\ntau\n(local)\nArray of size LOCc(ia+k-1). tau[i] contains the scalar factor of the\nelementary reflector H(i+1), i = 0, 1, ..., LOCc(ja+k-1)-1, as returned by \np?gelqf. This array is tied to the distributed matrix A.\nc\n(local)\nPointer into the local memory to an array of size lld_c * LOCc(jc+n-1). On\nentry, the local pieces of the distributed matrix sub (C).\nic\n(global)\nThe row index in the global matrix C indicating the first row of sub(C).\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1732\n\n\njc\n(global)\nThe column index in the global matrix C indicating the first column of\nsub(C).\ndescc\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix C.\nwork\n(local)\nWorkspace array of size lwork.\nlwork\n(local or global)\nThe size of the array work.\nlwork is local input and must be at least\nif side = 'L', lwork ≥ mqc0 + max(max( 1, npc0), numroc(numroc(m\n+icoffc, mb_a, 0, 0, nprow), mb_a, 0, 0, lcmp)),\nif side = 'R', lwork ≥ npc0 + max(1, mqc0),\nwhere \nlcmp = lcm / nprow,\nlcm = iclm(nprow, npcol),\niroffc = mod(ic-1, mb_c), \nicoffc = mod(jc-1, nb_c),\nicrow = indxg2p(ic, mb_c, myrow, rsrc_c, nprow),\niccol = indxg2p(jc, nb_c, mycol, csrc_c, npcol),\nMpc0 = numroc(m+icoffc, mb_c, mycol, icrow, nprow),\nNqc0 = numroc(n+iroffc, nb_c, myrow, iccol, npcol),\nilcm, indxg2p and numroc are ScaLAPACK tool functions; myrow, mycol,\nnprow, and npcol can be determined by calling the function\nblacs_gridinfo.\nIf lwork = -1, then lwork is global input and a workspace query is\nassumed; the function only calculates the minimum and optimal size for all\nwork arrays. Each of these values is returned in the first entry of the\ncorresponding work array, and no error message is issued by pxerbla.\nOutput Parameters\nc\nOn exit, c is overwritten by Q*sub(C), or QT*sub(C)/ QH*sub(C), or\nsub(C)*Q, or sub(C)*QT / sub(C)*QH\nwork\nOn exit, work[0] returns the minimal and optimal lwork.\ninfo\n(local)\n= 0: successful exit\n< 0: if the i-th argument is an array and the j-th entry, indexed j-1, had an\nillegal value,\nthen info = - (i*100 +j),\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1733\n\n\nif the i-th argument is a scalar and had an illegal value,\nthen info = -i.\nNOTE\nThe distributed submatrices A(ia:*, ja:*) and C(ic:ic+m-1, jc:jc+n-1) must verify some\nalignment properties, namely the following expressions should be true:\nIf side = 'L', (nb_a == mb_c && icoffa == iroffc)\nIf side = 'R', (nb_a == nb_c && icoffa == icoffc && iacol == iccol).\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?ormr2/p?unmr2\nMultiplies a general matrix by the orthogonal/unitary\nmatrix from an RQ factorization determined by\np?gerqf (unblocked algorithm).\nSyntax\nvoid psormr2 (char *side , char *trans , MKL_INT *m , MKL_INT *n , MKL_INT *k , float\n*a , MKL_INT *ia , MKL_INT *ja , MKL_INT *desca , float *tau , float *c , MKL_INT *ic ,\nMKL_INT *jc , MKL_INT *descc , float *work , MKL_INT *lwork , MKL_INT *info );\nvoid pdormr2 (char *side , char *trans , MKL_INT *m , MKL_INT *n , MKL_INT *k , double\n*a , MKL_INT *ia , MKL_INT *ja , MKL_INT *desca , double *tau , double *c , MKL_INT\n*ic , MKL_INT *jc , MKL_INT *descc , double *work , MKL_INT *lwork , MKL_INT *info );\nvoid pcunmr2 (char *side , char *trans , MKL_INT *m , MKL_INT *n , MKL_INT *k ,\nMKL_Complex8 *a , MKL_INT *ia , MKL_INT *ja , MKL_INT *desca , MKL_Complex8 *tau ,\nMKL_Complex8 *c , MKL_INT *ic , MKL_INT *jc , MKL_INT *descc , MKL_Complex8 *work ,\nMKL_INT *lwork , MKL_INT *info );\nvoid pzunmr2 (char *side , char *trans , MKL_INT *m , MKL_INT *n , MKL_INT *k ,\nMKL_Complex16 *a , MKL_INT *ia , MKL_INT *ja , MKL_INT *desca , MKL_Complex16 *tau ,\nMKL_Complex16 *c , MKL_INT *ic , MKL_INT *jc , MKL_INT *descc , MKL_Complex16 *work ,\nMKL_INT *lwork , MKL_INT *info );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?ormr2/p?unmr2function overwrites the general real/complex m-by-n distributed matrix sub\n(C)=C(ic:ic+m-1, jc:jc+n-1) with\nQ*sub(C) if side = 'L' and trans = 'N', or\nQT*sub(C) / QH*sub(C) if side = 'L' and trans = 'T' (for real flavors) or trans = 'C' (for complex\nflavors), or\nsub(C)*Q if side = 'R' and trans = 'N', or\nsub(C)*QT / sub(C)*QH if side = 'R' and trans = 'T' (for real flavors) or trans = 'C' (for complex\nflavors).\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1734\n\n\nwhere Q is a real orthogonal or complex unitary distributed matrix defined as the product of k elementary\nreflectors\nQ = H(1)*H(2)*...*H(k) (for real flavors)\nQ = (H(1))H*(H(2))H*...*(H(k))H (for complex flavors)\nas returned by p?gerqf . Q is of order m if side = 'L' and of order n if side = 'R'.\nInput Parameters\nside\n(global)\n= 'L': apply Q or QT for real flavors (QH for complex flavors) from the left,\n= 'R': apply Q or QT for real flavors (QH for complex flavors) from the\nright.\ntrans\n(global)\n= 'N': apply Q (no transpose)\n= 'T': apply QT (transpose, for real flavors)\n= 'C': apply QH(conjugate transpose, for complex flavors)\nm\n(global)\nThe number of rows in the distributed matrix sub(C). m ≥ 0.\nn\n(global)\nThe number of columns in the distributed matrix sub(C). n ≥ 0.\nk\n(global)\nThe number of elementary reflectors whose product defines the matrix Q.\nIf side = 'L', m ≥ k ≥ 0;\nif side = 'R', n ≥ k ≥ 0.\na\n(local)\nPointer into the local memory to an array of size\nlld_a * LOCc(ja+m-1) if side='L',\nlld_a * LOCc(ja+n-1) if side='R',\nwhere lld_a ≥ max (1, LOCr(ia+k-1)).\nOn entry, the i-th row of the matrix stored in amust contain the vector that\ndefines the elementary reflector H(i), ia ≤ i ≤ ia+k-1, as returned by \np?gerqf in the k rows of its distributed matrix argument A(ia:ia+k-1,\nja:*).\nThe argument A(ia:ia+k-1, ja:*) is modified by the function but\nrestored on exit.\nia\n(global)\nThe row index in the global matrix A indicating the first row of sub(A).\nja\n(global)\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1735\n\n\nThe column index in the global matrix A indicating the first column of\nsub(A).\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\ntau\n(local)\nArray of size LOCc(ia+k-1). tau[j] contains the scalar factor of the\nelementary reflector H(j+1), j = 0, 1, ..., LOCc(ja+k-1)-1, as returned by \np?gerqf. This array is tied to the distributed matrix A.\nc\n(local)\nPointer into the local memory to an array of size lld_c * LOCc(jc+n-1). On\nentry, the local pieces of the distributed matrix sub (C).\nic\n(global)\nThe row index in the global matrix C indicating the first row of sub(C).\njc\n(global)\nThe column index in the global matrix C indicating the first column of\nsub(C).\ndescc\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix C.\nwork\n(local)\nWorkspace array of size lwork.\nlwork\n(local or global)\nThe size of the array work.\nlwork is local input and must be at least\nif side = 'L', lwork ≥ mpc0 + max(max(1, nqc0), numroc(numroc(m\n+iroffc, mb_a, 0, 0, nprow), mb_a, 0, 0, lcmp)),\nif side = 'R', lwork ≥ nqc0 + max(1, mpc0),\nwhere lcmp = lcm/nprow,\nlcm = iclm(nprow, npcol),\niroffc = mod(ic-1, mb_c),\nicoffc = mod(jc-1, nb_c),\nicrow = indxg2p(ic, mb_c, myrow, rsrc_c, nprow),\niccol = indxg2p(jc, nb_c, mycol, csrc_c, npcol),\nMpc0 = numroc(m+iroffc, mb_c, myrow, icrow, nprow),\nNqc0 = numroc(n+icoffc, nb_c, mycol, iccol, npcol),\nilcm, indxg2p and numroc are ScaLAPACK tool functions; myrow, mycol,\nnprow, and npcol can be determined by calling the function\nblacs_gridinfo.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1736\n\n\nIf lwork = -1, then lwork is global input and a workspace query is\nassumed; the function only calculates the minimum and optimal size for all\nwork arrays. Each of these values is returned in the first entry of the\ncorresponding work array, and no error message is issued by pxerbla.\nOutput Parameters\nc\nOn exit, c is overwritten by Q*sub(C), or QT*sub(C)/ QH*sub(C), or\nsub(C)*Q, or sub(C)*QT / sub(C)*QH\nwork\nOn exit, work[0] returns the minimal and optimal lwork.\ninfo\n(local)\n= 0: successful exit\n< 0: if the i-th argument is an array and the j-th entry, indexed j-1, had an\nillegal value,\nthen info = - (i*100 +j),\nif the i-th argument is a scalar and had an illegal value,\nthen info = -i.\nNOTE\nThe distributed submatrices A(ia:*, ja:*) and C(ic:ic+m-1,jc:jc+n-1) must verify some\nalignment properties, namely the following expressions should be true:\nIf side = 'L', (nb_a == mb_c) && (icoffa == iroffc).\nIf side = 'R', (nb_a == nb_c) && (icoffa == icoffc) && (iacol == iccol).\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?pbtrsv\nSolves a single triangular linear system via frontsolve\nor backsolve where the triangular matrix is a factor of\na banded matrix computed by p?pbtrf.\nSyntax\nvoid pspbtrsv (char *uplo , char *trans , MKL_INT *n , MKL_INT *bw , MKL_INT *nrhs ,\nfloat *a , MKL_INT *ja , MKL_INT *desca , float *b , MKL_INT *ib , MKL_INT *descb ,\nfloat *af , MKL_INT *laf , float *work , MKL_INT *lwork , MKL_INT *info );\nvoid pdpbtrsv (char *uplo , char *trans , MKL_INT *n , MKL_INT *bw , MKL_INT *nrhs ,\ndouble *a , MKL_INT *ja , MKL_INT *desca , double *b , MKL_INT *ib , MKL_INT *descb ,\ndouble *af , MKL_INT *laf , double *work , MKL_INT *lwork , MKL_INT *info );\nvoid pcpbtrsv (char *uplo , char *trans , MKL_INT *n , MKL_INT *bw , MKL_INT *nrhs ,\nMKL_Complex8 *a , MKL_INT *ja , MKL_INT *desca , MKL_Complex8 *b , MKL_INT *ib ,\nMKL_INT *descb , MKL_Complex8 *af , MKL_INT *laf , MKL_Complex8 *work , MKL_INT\n*lwork , MKL_INT *info );\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1737\n\n\nvoid pzpbtrsv (char *uplo , char *trans , MKL_INT *n , MKL_INT *bw , MKL_INT *nrhs ,\nMKL_Complex16 *a , MKL_INT *ja , MKL_INT *desca , MKL_Complex16 *b , MKL_INT *ib ,\nMKL_INT *descb , MKL_Complex16 *af , MKL_INT *laf , MKL_Complex16 *work , MKL_INT\n*lwork , MKL_INT *info );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?pbtrsvfunction solves a banded triangular system of linear equations\nA(1:n, ja:ja+n-1)*X = B(jb:jb+n-1, 1:nrhs)\nor\nA(1:n, ja:ja+n-1)T*X = B(jb:jb+n-1, 1:nrhs) for real flavors,\nA(1:n, ja:ja+n-1)H*X = B(jb:jb+n-1, 1:nrhs) for complex flavors,\nwhere A(1:n, ja:ja+n-1) is a banded triangular matrix factor produced by the Cholesky factorization code \np?pbtrf and is stored in A(1:n, ja:ja+n-1) and af. The matrix stored in A(1:n, ja:ja+n-1) is either\nupper or lower triangular according to uplo.\nThe function p?pbtrf must be called first.\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nInput Parameters\nuplo\n(global) Must be 'U' or 'L'.\nIf uplo = 'U', upper triangle of A(1:n, ja:ja+n-1) is stored;\nIf uplo = 'L', lower triangle of A(1:n, ja:ja+n-1) is stored.\ntrans\n(global) Must be 'N' or 'T' or 'C'.\nIf trans = 'N', solve with A(1:n, ja:ja+n-1);\nIf trans = 'T' or 'C' for real flavors, solve with A(1:n, ja:ja+n-1)T.\nIf trans = 'C' for complex flavors, solve with conjugate transpose\n(A(1:n, ja:ja+n-1)H.\nn\n(global)\nThe number of rows and columns to be operated on, that is, the order of\nthe distributed submatrix A(1:n, ja:ja+n-1). n ≥ 0.\nbw\n(global)\nThe number of subdiagonals in 'L' or 'U', 0 ≤bw≤n-1.\nnrhs\n(global)\nThe number of right hand sides; the number of columns of the distributed\nsubmatrix B(jb:jb+n-1, 1:nrhs); nrhs≥ 0.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1738\n\n\na\n(local)\nPointer into the local memory to an array with the first size lld_a ≥ (bw\n+1), stored in desca.\nOn entry, this array contains the local pieces of the n-by-n symmetric\nbanded distributed Cholesky factor L or LT*A(1:n, ja:ja+n-1).\nThis local portion is stored in the packed banded format used in LAPACK.\nSee the Application Notes below and the ScaLAPACK manual for more detail\non the format of distributed matrices.\nja\n(global) The index in the global in the global matrix A that points to the\nstart of the matrix to be operated on (which may be either all of A or a\nsubmatrix of A).\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\nIf 1D type (dtype_a = 501), then dlen≥ 7;\nIf 2D type (dtype_a = 1), then dlen ≥ 9.\nContains information on mapping of A to memory. (See ScaLAPACK manual\nfor full description and options.)\nb\n(local)\nPointer into the local memory to an array of local lead size lld_b ≥nb.\nOn entry, this array contains the local pieces of the right hand sides\nB(jb:jb+n-1, 1:nrhs).\nib\n(global) The row index in the global matrix B that points to the first row of\nthe matrix to be operated on (which may be either all of B or a submatrix of\nB).\ndescb\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix B.\nIf 1D type (dtype_b = 502), then dlen≥ 7;\nIf 2D type (dtype_b = 1), then dlen≥ 9.\nContains information on mapping of B to memory. Please, see ScaLAPACK\nmanual for full description and options.\nlaf\n(local)\nThe size of user-input auxiliary fill-in space af. Must be laf ≥ (nb\n+2*bw)*bw . If laf is not large enough, an error code will be returned and\nthe minimum acceptable size will be returned in af[0].\nwork\n(local)\nThe array work is a temporary workspace array of size lwork. This space\nmay be overwritten in between function calls.\nlwork\n(local or global) The size of the user-input workspace work, must be at\nleast lwork ≥bw*nrhs. If lwork is too small, the minimal acceptable size\nwill be returned in work[0] and an error code is returned.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1739\n\n\nOutput Parameters\naf\n(local)\nThe array af is of size laf. It contains auxiliary fill-in space. The fill-in\nspace is created in a call to the factorization function p?pbtrf and is stored\nin af. If a linear system is to be solved using p?pbtrs after the\nfactorization function, af must not be altered after the factorization.\nb\nOn exit, this array contains the local piece of the solutions distributed\nmatrix X.\nwork[0]\nOn exit, work[0] contains the minimum value of lwork.\ninfo\n(local)\n= 0: successful exit\n< 0: if the i-th argument is an array and the j-th entry, indexed j-1, had an\nillegal value,\nthen info = - (i*100 +j),\nif the i-th argument is a scalar and had an illegal value,\nthen info = -i.\nApplication Notes\nIf the factorization function and the solve function are to be called separately to solve various sets of right-\nhand sides using the same coefficient matrix, the auxiliary space af must not be altered between calls to the\nfactorization function and the solve function.\nThe best algorithm for solving banded and tridiagonal linear systems depends on a variety of parameters,\nespecially the bandwidth. Currently, only algorithms designed for the case N/P>>bw are implemented. These\nalgorithms go by many names, including Divide and Conquer, Partitioning, domain decomposition-type, etc.\nThe Divide and Conquer algorithm assumes the matrix is narrowly banded compared with the number of\nequations. In this situation, it is best to distribute the input matrix A one-dimensionally, with columns atomic\nand rows divided amongst the processes. The basic algorithm divides the banded matrix up into P pieces with\none stored on each processor, and then proceeds in 2 phases for the factorization or 3 for the solution of a\nlinear system.\n1.\nLocal Phase: The individual pieces are factored independently and in parallel. These factors are\napplied to the matrix creating fill-in, which is stored in a non-inspectable way in auxiliary space af.\nMathematically, this is equivalent to reordering the matrix A as PAPT and then factoring the principal\nleading submatrix of size equal to the sum of the sizes of the matrices factored on each processor. The\nfactors of these submatrices overwrite the corresponding parts of A in memory.\n2.\nReduced System Phase: A small (bw*(P-1)) system is formed representing interaction of the larger\nblocks and is stored (as are its factors) in the space af. A parallel Block Cyclic Reduction algorithm is\nused. For a linear system, a parallel front solve followed by an analogous backsolve, both using the\nstructure of the factored matrix, are performed.\n3.\nBack Subsitution Phase: For a linear system, a local backsubstitution is performed on each processor\nin parallel.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1740\n\n\np?pttrsv\nSolves a single triangular linear system via frontsolve\nor backsolve where the triangular matrix is a factor of\na tridiagonal matrix computed by p?pttrf .\nSyntax\nvoid pspttrsv (char *uplo , MKL_INT *n , MKL_INT *nrhs , float *d , float *e , MKL_INT\n*ja , MKL_INT *desca , float *b , MKL_INT *ib , MKL_INT *descb , float *af , MKL_INT\n*laf , float *work , MKL_INT *lwork , MKL_INT *info );\nvoid pdpttrsv (char *uplo , MKL_INT *n , MKL_INT *nrhs , double *d , double *e , MKL_INT\n*ja , MKL_INT *desca , double *b , MKL_INT *ib , MKL_INT *descb , double *af , MKL_INT\n*laf , double *work , MKL_INT *lwork , MKL_INT *info );\nvoid pcpttrsv (char *uplo , char *trans , MKL_INT *n , MKL_INT *nrhs , float *d ,\nMKL_Complex8 *e , MKL_INT *ja , MKL_INT *desca , MKL_Complex8 *b , MKL_INT *ib ,\nMKL_INT *descb , MKL_Complex8 *af , MKL_INT *laf , MKL_Complex8 *work , MKL_INT\n*lwork , MKL_INT *info );\nvoid pzpttrsv (char *uplo , char *trans , MKL_INT *n , MKL_INT *nrhs , double *d ,\nMKL_Complex16 *e , MKL_INT *ja , MKL_INT *desca , MKL_Complex16 *b , MKL_INT *ib ,\nMKL_INT *descb , MKL_Complex16 *af , MKL_INT *laf , MKL_Complex16 *work , MKL_INT\n*lwork , MKL_INT *info );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?pttrsvfunction solves a tridiagonal triangular system of linear equations\nA(1:n, ja:ja+n-1)*X = B(jb:jb+n-1, 1:nrhs)\nor\nA(1:n, ja:ja+n-1)T*X = B(jb:jb+n-1, 1:nrhs) for real flavors,\nA(1:n, ja:ja+n-1)H*X = B(jb:jb+n-1, 1:nrhs) for complex flavors,\nwhere A(1:n, ja:ja+n-1) is a tridiagonal triangular matrix factor produced by the Cholesky factorization\ncode p?pttrf and is stored in A(1:n, ja:ja+n-1) and af. The matrix stored in A(1:n, ja:ja+n-1) is\neither upper or lower triangular according to uplo.\nThe function p?pttrf must be called first.\nInput Parameters\nuplo\n(global) Must be 'U' or 'L'.\nIf uplo = 'U', upper triangle of A(1:n, ja:ja+n-1) is stored;\nIf uplo = 'L', lower triangle of A(1:n, ja:ja+n-1) is stored.\ntrans\n(global) Must be 'N' or 'C'.\nIf trans = 'N', solve with A(1:n, ja:ja+n-1);\nIf trans = 'C' (for complex flavors), solve with conjugate transpose\n(A(1:n, ja:ja+n-1))H.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1741\n\n\nn\n(global)\nThe number of rows and columns to be operated on, that is, the order of\nthe distributed submatrix A(1:n, ja:ja+n-1). n ≥ 0.\nnrhs\n(global)\nThe number of right hand sides; the number of columns of the distributed\nsubmatrix B(jb:jb+n-1, 1:nrhs); nrhs ≥ 0.\nd\n(local)\nPointer to the local part of the global vector storing the main diagonal of the\nmatrix; must be of size ≥nb_a.\ne\n(local)\nPointer to the local part of the global vector du storing the upper diagonal of\nthe matrix; must be of size ≥nb_a. Globally, du(n) is not referenced, and du\nmust be aligned with d.\nja\n(global) The index in the global matrix A that points to the start of the\nmatrix to be operated on (which may be either all of A or a submatrix of A).\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\nIf 1D type (dtype_a = 501 or 502), then dlen ≥ 7;\nIf 2D type (dtype_a = 1), then dlen ≥ 9.\nContains information on mapping of A to memory. See ScaLAPACK manual\nfor full description and options.\nb\n(local)\nPointer into the local memory to an array of local lead size lld_b ≥ nb.\nOn entry, this array contains the local pieces of the right hand sides\nB(jb:jb+n-1, 1:nrhs).\nib\n(global) The row index in the global matrix B that points to the first row of\nthe matrix to be operated on (which may be either all of B or a submatrix of\nB).\ndescb\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix B.\nIf 1D type (dtype_b = 502), then dlen ≥ 7;\nIf 2D type (dtype_b = 1), then dlen ≥ 9.\nContains information on mapping of B to memory. See ScaLAPACK manual\nfor full description and options.\nlaf\n(local)\nThe size of user-input auxiliary fill-in space af. Must be laf ≥ (nb\n+2*bw)*bw.\nIf laf is not large enough, an error code will be returned and the minimum\nacceptable size will be returned in af[0].\nwork\n(local)\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1742\n\n\nThe array work is a temporary workspace array of size lwork. This space\nmay be overwritten in between function calls.\nlwork\n(local or global) The size of the user-input workspace work, must be at\nleast lwork ≥(10+2*min(100, nrhs))*npcol+4*nrhs. If lwork is too\nsmall, the minimal acceptable size will be returned in work[0] and an error\ncode is returned.\nOutput Parameters\nd, e\n(local).\nOn exit, these arrays contain information on the factors of the matrix.\naf\n(local)\nThe array af is of size laf. It contains auxiliary fill-in space. The fill-in\nspace is created in a call to the factorization function p?pbtrf and is stored\nin af. If a linear system is to be solved using p?pttrs after the\nfactorization function, af must not be altered after the factorization.\nb\nOn exit, this array contains the local piece of the solutions distributed\nmatrix X.\nwork[0]\nOn exit, work[0] contains the minimum value of lwork.\ninfo\n(local)\n= 0: successful exit\n< 0: if the i-th argument is an array and the j-th entry, indexed j-1, had an\nillegal value,\nthen info = - (i*100 +j),\nif the i-th argument is a scalar and had an illegal value,\nthen info = -i.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?potf2\nComputes the Cholesky factorization of a symmetric/\nHermitian positive definite matrix (local unblocked\nalgorithm).\nSyntax\nvoid pspotf2 (char *uplo , MKL_INT *n , float *a , MKL_INT *ia , MKL_INT *ja , MKL_INT\n*desca , MKL_INT *info );\nvoid pdpotf2 (char *uplo , MKL_INT *n , double *a , MKL_INT *ia , MKL_INT *ja , MKL_INT\n*desca , MKL_INT *info );\nvoid pcpotf2 (char *uplo , MKL_INT *n , MKL_Complex8 *a , MKL_INT *ia , MKL_INT *ja ,\nMKL_INT *desca , MKL_INT *info );\nvoid pzpotf2 (char *uplo , MKL_INT *n , MKL_Complex16 *a , MKL_INT *ia , MKL_INT *ja ,\nMKL_INT *desca , MKL_INT *info );\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1743\n\n\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?potf2function computes the Cholesky factorization of a real symmetric or complex Hermitian positive\ndefinite distributed matrix sub (A)=A(ia:ia+n-1, ja:ja+n-1).\nThe factorization has the form\nsub(A) = U'*U, if uplo = 'U', or sub(A) = L*L', if uplo = 'L',\nwhere U is an upper triangular matrix, L is lower triangular. X' denotes transpose (conjugate transpose) of X.\nInput Parameters\nuplo\n(global)\nSpecifies whether the upper or lower triangular part of the symmetric/\nHermitian matrix A is stored.\n= 'U': upper triangle of sub (A) is stored;\n= 'L': lower triangle of sub (A) is stored.\nn\n(global)\nThe number of rows and columns to be operated on, that is, the order of\nthe distributed matrix sub (A). n ≥ 0.\na\n(local)\nPointer into the local memory to an array of size lld_a * LOCc(ja+n-1)\ncontaining the local pieces of the n-by-n symmetric distributed matrix\nsub(A) to be factored.\nIf uplo = 'U', the leading n-by-n upper triangular part of sub(A) contains\nthe upper triangular matrix and the strictly lower triangular part of this\nmatrix is not referenced.\nIf uplo = 'L', the leading n-by-n lower triangular part of sub(A) contains\nthe lower triangular matrix and the strictly upper triangular part of sub(A) is\nnot referenced.\nia, ja\n(global)\nThe row and column indices in the global matrix A indicating the first row\nand the first column of the sub(A), respectively.\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\nOutput Parameters\na\n(local)\nOn exit,\nif uplo = 'U', the upper triangular part of the distributed matrix contains\nthe Cholesky factor U;\nif uplo = 'L', the lower triangular part of the distributed matrix contains\nthe Cholesky factor L.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1744\n\n\ninfo\n(local)\n= 0: successful exit\n< 0: if the i-th argument is an array and the j-th entry, indexed j-1, had an\nillegal value,\nthen info = - (i*100 +j),\nif the i-th argument is a scalar and had an illegal value,\nthen info = -i.\n> 0: if info = k, the leading minor of order k is not positive definite, and\nthe factorization could not be completed.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?rot\nApplies a planar rotation to two distributed vectors.\nSyntax\nvoid psrot(MKL_INT* n, float* x, MKL_INT* ix, MKL_INT* jx, MKL_INT* descx, MKL_INT*\nincx, float* y, MKL_INT* iy, MKL_INT* jy, MKL_INT* descy, MKL_INT* incy, float* cs,\nfloat* sn, float* work, MKL_INT* lwork, MKL_INT* info);\nvoid pdrot(MKL_INT* n, double* x, MKL_INT* ix, MKL_INT* jx, MKL_INT* descx, MKL_INT*\nincx, double* y, MKL_INT* iy, MKL_INT* jy, MKL_INT* descy, MKL_INT* incy, double* cs,\ndouble* sn, double* work, MKL_INT* lwork, MKL_INT* info);\nInclude Files\n•\nmkl_scalapack.h\nDescription\np?rot applies a planar rotation defined by cs and sn to the two distributed vectors sub(x) and sub(y).\nInput Parameters\nn\n(global )\nThe number of elements to operate on when applying the planar rotation to\nx and y (n≥0).\nx\n(local) array of size ( (jx-1)*m_x + ix + ( n - 1 )*abs( incx ) )\nThis array contains the entries of the distributed vector sub( x ).\nix\n(global )\nThe global row index of the submatrix of the distributed matrix x to operate\non. If incx = 1, then it is required that ix = iy. 1 ≤ix≤m_x.\njx\n(global )\nThe global column index of the submatrix of the distributed matrix x to\noperate on. If incx = m_x, then it is required that jx = jy. 1 ≤ix≤n_x.\ndescx\n(global and local) array of size 9\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1745\n\n\nThe array descriptor of the distributed matrix x.\nincx\n(global )\nThe global increment for the elements of x. Only two values of incx are\nsupported in this version, namely 1 and m_x. Moreover, it must hold that\nincx = m_x if incy =m_y and that incx = 1 if incy = 1.\ny\n(local) array of size ( (jy-1)*m_y + iy + ( n - 1 )*abs( incy ) )\nThis array contains the entries of the distributed vector sub( y ).\niy\n(global )\nThe global row index of the submatrix of the distributed matrix y to operate\non. If incy = 1, then it is required that iy = ix. 1 ≤iy≤m_y.\njy\n(global )\nThe global column index of the submatrix of the distributed matrix y to\noperate on. If incy = m_x, then it is required that jy = jx. 1 ≤jy≤m_y.\ndescy\n(global and local) array of size 9\nThe array descriptor of the distributed matrix y.\nincy\n(global )\nThe global increment for the elements of y. Only two values of incy are\nsupported in this version, namely 1 and m_y. Moreover, it must hold that\nincy = m_y if incx = m_x and that incy = 1 if incx = 1.\ncs, sn\n(global)\nThe parameters defining the properties of the planar rotation. It must hold\nthat 0 ≤cs,sn≤ 1 and that sn2 + cs2 = 1. The latter is hardly checked in\nfinite precision arithmetics.\nwork\n(local workspace) array of size lwork\nlwork\n(local )\nThe length of the workspace array work.\nIf incx = 1 and incy = 1, then lwork = 2*m_x\nIf lwork = -1, then a workspace query is assumed; the function only\ncalculates the optimal size of the work array, returns this value as the first\nentry of the IWORK array, and no error message related to LIWORK is\nissued by pxerbla.\nOUTPUT Parameters\nx\ny\nwork[0]\nOn exit, if info = 0, work[0] returns the optimal lwork\ninfo\n(global )\n= 0: successful exit\n< 0: if info = -i, the i-th argument had an illegal value.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1746\n\n\nIf the i-th argument is an array and the j-th entry, indexed j-1, had an\nillegal value, then info = -(i*100+j), if the i-th argument is a scalar and\nhad an illegal value, then info = -i.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?rscl\nMultiplies a vector by the reciprocal of a real scalar.\nSyntax\nvoid psrscl (MKL_INT *n , float *sa , float *sx , MKL_INT *ix , MKL_INT *jx , MKL_INT\n*descx , MKL_INT *incx );\nvoid pdrscl (MKL_INT *n , double *sa , double *sx , MKL_INT *ix , MKL_INT *jx , MKL_INT\n*descx , MKL_INT *incx );\nvoid pcsrscl (MKL_INT *n , float *sa , MKL_Complex8 *sx , MKL_INT *ix , MKL_INT *jx ,\nMKL_INT *descx , MKL_INT *incx );\nvoid pzdrscl (MKL_INT *n , double *sa , MKL_Complex16 *sx , MKL_INT *ix , MKL_INT *jx ,\nMKL_INT *descx , MKL_INT *incx );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?rsclfunction multiplies an n-element real/complex vector sub(X) by the real scalar 1/a. This is done\nwithout overflow or underflow as long as the final result sub(X)/a does not overflow or underflow.\nsub(X) denotes X(ix:ix+n-1, jx:jx), if incx = 1,\nand X(ix:ix, jx:jx+n-1), if incx = m_x.\nInput Parameters\nn\n(global)\nThe number of components of the distributed vector sub(X). n ≥ 0.\nsa\nThe scalar a that is used to divide each component of the vector sub(X).\nThis parameter must be ≥ 0.\nsx\nArray containing the local pieces of a distributed matrix of size of at least\n((jx-1)*m_x + ix + (n-1)*abs(incx)). This array contains the entries\nof the distributed vector sub(X).\nix\n(global) The row index of the submatrix of the distributed matrix X to\noperate on.\njx\n(global)\nThe column index of the submatrix of the distributed matrix X to operate\non.\ndescx\n(global and local)\nArray of size 9. The array descriptor for the distributed matrix X.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1747\n\n\nincx\n(global)\nThe increment for the elements of X. This version supports only two values\nof incx, namely 1 and m_x.\nOutput Parameters\nsx\nOn exit, the result x/a.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?sygs2/p?hegs2\nReduces a symmetric/Hermitian positive-definite\ngeneralized eigenproblem to standard form, using the\nfactorization results obtained from p?potrf (local\nunblocked algorithm).\nSyntax\nvoid pssygs2 (MKL_INT *ibtype , char *uplo , MKL_INT *n , float *a , MKL_INT *ia ,\nMKL_INT *ja , MKL_INT *desca , float *b , MKL_INT *ib , MKL_INT *jb , MKL_INT *descb ,\nMKL_INT *info );\nvoid pdsygs2 (MKL_INT *ibtype , char *uplo , MKL_INT *n , double *a , MKL_INT *ia ,\nMKL_INT *ja , MKL_INT *desca , double *b , MKL_INT *ib , MKL_INT *jb , MKL_INT *descb ,\nMKL_INT *info );\nvoid pchegs2 (MKL_INT *ibtype , char *uplo , MKL_INT *n , MKL_Complex8 *a , MKL_INT\n*ia , MKL_INT *ja , MKL_INT *desca , MKL_Complex8 *b , MKL_INT *ib , MKL_INT *jb ,\nMKL_INT *descb , MKL_INT *info );\nvoid pzhegs2 (MKL_INT *ibtype , char *uplo , MKL_INT *n , MKL_Complex16 *a , MKL_INT\n*ia , MKL_INT *ja , MKL_INT *desca , MKL_Complex16 *b , MKL_INT *ib , MKL_INT *jb ,\nMKL_INT *descb , MKL_INT *info );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?sygs2/p?hegs2function reduces a real symmetric-definite or a complex Hermitian positive-definite\ngeneralized eigenproblem to standard form.\nHere sub(A) denotes A(ia:ia+n-1, ja:ja+n-1), and sub(B) denotes B(ib:ib+n-1, jb:jb+n-1).\nIf ibtype = 1, the problem is\nsub(A)*x = λ*sub(B)*x\nand sub(A) is overwritten by\ninv(UT)*sub(A)*inv(U) or inv(L)*sub(A)*inv(LT) - for real flavors, and\ninv(UH)*sub(A)*inv(U) or inv(L)*sub(A)*inv(LH) - for complex flavors.\nIf ibtype = 2 or 3, the problem is\nsub(A)*sub(B)x = λ*x or sub(B)*sub(A)x =λ*x\nand sub(A) is overwritten by\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1748\n\n\nU*sub(A)*UT or L**T*sub(A)*L- for real flavors and\nU*sub(A)*UH or L**H*sub(A)*L- for complex flavors.\nThe matrix sub(B) must have been previously factorized as UT*U or L*LT (for real flavors), or as UH*U or\nL*LH (for complex flavors) by p?potrf.\nInput Parameters\nibtype\n(global)\n= 1:\ncompute inv(UT)*sub(A)*inv(U), or inv(L)*sub(A)*inv(LT) for real\nfunctions,\nand inv(UH)*sub(A)*inv(U), or inv(L)*sub(A)*inv(LH) for complex\nfunctions;\n= 2 or 3:\ncompute U*sub(A)*UT, or LT*sub(A)*L for real functions,\nand U*sub(A)*UH or LH*sub(A)*L for complex functions.\nuplo\n(global)\nSpecifies whether the upper or lower triangular part of the symmetric/\nHermitian matrix sub(A) is stored, and how sub(B) is factorized.\n= 'U': Upper triangular of sub(A) is stored and sub(B) is factorized as UT*U\n(for real functions) or as UH*U (for complex functions).\n= 'L': Lower triangular of sub(A) is stored and sub(B) is factorized as L*LT\n(for real functions) or as L*LH (for complex functions)\nn\n(global)\nThe order of the matrices sub(A) and sub(B). n ≥ 0.\na\n(local)\nPointer into the local memory to an array of size lld_a * LOCc(ja+n-1).\nOn entry, this array contains the local pieces of the n-by-n symmetric/\nHermitian distributed matrix sub(A).\nIf uplo = 'U', the leading n-by-n upper triangular part of sub(A) contains\nthe upper triangular part of the matrix, and the strictly lower triangular part\nof sub(A) is not referenced.\nIf uplo = 'L', the leading n-by-n lower triangular part of sub(A) contains\nthe lower triangular part of the matrix, and the strictly upper triangular part\nof sub(A) is not referenced.\nia, ja\n(global)\nThe row and column indices in the global matrix A indicating the first row\nand the first column of the sub(A), respectively.\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\nB\n(local)\nPointer into the local memory to an array of size lld_b * LOCc(jb+n-1).\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1749\n\n\nOn entry, this array contains the local pieces of the triangular factor from\nthe Cholesky factorization of sub(B) as returned by p?potrf.\nib, jb\n(global)\nThe row and column indices in the global matrix B indicating the first row\nand the first column of the sub(B), respectively.\ndescb\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix B.\nOutput Parameters\na\n(local)\nOn exit, if info = 0, the transformed matrix is stored in the same format\nas sub(A).\ninfo\n= 0: successful exit.\n< 0: if the i-th argument is an array and the j-th entry, indexed j-1, had an\nillegal value,\nthen info = - (i*100+ j),\nif the i-th argument is a scalar and had an illegal value,\nthen info = -i.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?sytd2/p?hetd2\nReduces a symmetric/Hermitian matrix to real\nsymmetric tridiagonal form by an orthogonal/unitary\nsimilarity transformation (local unblocked algorithm).\nSyntax\nvoid pssytd2 (char *uplo, MKL_INT *n, float *a, MKL_INT *ia, MKL_INT *ja, MKL_INT\n*desca, float *d, float *e, float *tau, float *work, MKL_INT *lwork, MKL_INT *info);\nvoid pdsytd2 (char *uplo, MKL_INT *n, double *a, MKL_INT *ia, MKL_INT *ja, MKL_INT\n*desca, double *d, double *e, double *tau, double *work, MKL_INT *lwork, MKL_INT *info);\nvoid pchetd2 (char *uplo, MKL_INT *n, MKL_Complex8 *a, MKL_INT *ia, MKL_INT *ja, MKL_INT\n*desca, float *d, float *e, MKL_Complex8 *tau, MKL_Complex8 *work, MKL_INT *lwork,\nMKL_INT *info);\nvoid pzhetd2 (char *uplo, MKL_INT *n, MKL_Complex16 *a, MKL_INT *ia, MKL_INT *ja,\nMKL_INT *desca, double *d, double *e, MKL_Complex16 *tau, MKL_Complex16 *work, MKL_INT\n*lwork, MKL_INT *info);\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?sytd2/p?hetd2function reduces a real symmetric/complex Hermitian matrix sub(A) to symmetric/\nHermitian tridiagonal form T by an orthogonal/unitary similarity transformation:\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1750\n\n\nQ'*sub(A)*Q = T, where sub(A) = A(ia:ia+n-1, ja:ja+n-1).\nInput Parameters\nuplo\n(global)\nSpecifies whether the upper or lower triangular part of the symmetric/\nHermitian matrix sub(A) is stored:\n= 'U': upper triangular\n= 'L': lower triangular\nn\n(global)\nThe number of rows and columns to be operated on, that is, the order of\nthe distributed matrix sub(A). n ≥ 0.\na\n(local)\nPointer into the local memory to an array of size lld_a * LOCc(ja+n-1).\nOn entry, this array contains the local pieces of the n-by-n symmetric/\nHermitian distributed matrix sub(A).\nIf uplo = 'U', the leading n-by-n upper triangular part of sub(A) contains\nthe upper triangular part of the matrix, and the strictly lower triangular part\nof sub(A) is not referenced.\nIf uplo = 'L', the leading n-by-n lower triangular part of sub(A) contains\nthe lower triangular part of the matrix, and the strictly upper triangular part\nof sub(A) is not referenced.\nia, ja\n(global)\nThe row and column indices in the global matrix A indicating the first row\nand the first column of the sub(A), respectively.\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\nwork\n(local)\nThe array work is a temporary workspace array of size lwork.\nOutput Parameters\na\nOn exit, if uplo = 'U', the diagonal and first superdiagonal of sub(A) are\noverwritten by the corresponding elements of the tridiagonal matrix T, and\nthe elements above the first superdiagonal, with the array tau, represent\nthe orthogonal/unitary matrix Q as a product of elementary reflectors;\nif uplo = 'L', the diagonal and first subdiagonal of A are overwritten by\nthe corresponding elements of the tridiagonal matrix T, and the elements\nbelow the first subdiagonal, with the array tau, represent the orthogonal/\nunitary matrix Q as a product of elementary reflectors. See the Application\nNotes below.\nd\n(local)\nArray of sizeLOCc(ja+n-1). The diagonal elements of the tridiagonal matrix\nT:\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1751\n\n\nd[i] = A(i+1,i+1), where i=0,1, ..., LOCc(ja+n-1) -1 ; d is tied to the\ndistributed matrix A.\ne\n(local)\nArray of size LOCc(ja+n-1),\nif uplo = 'U', LOCc(ja+n-2) otherwise.\nThe off-diagonal elements of the tridiagonal matrix T:\ne[i] = A(i+1,i+2) if uplo = 'U',\ne[i] = A(i+2,i+1) if uplo = 'L',\nwhere i=0,1, ..., LOCc(ja+n-1) -1.\ne is tied to the distributed matrix A.\ntau\n(local)\nArray of size LOCc(ja+n-1).\nThe scalar factors of the elementary reflectors. tau is tied to the distributed\nmatrix A.\nwork[0]\nOn exit, work[0] returns the minimal and optimal value of lwork.\nlwork\n(local or global)\nThe size of the workspace array work.\nlwork is local input and must be at least lwork ≥ 3n.\nIf lwork = -1, then lwork is global input and a workspace query is\nassumed; the function only calculates the minimum and optimal size for all\nwork arrays. Each of these values is returned in the first entry of the\ncorresponding work array, and no error message is issued by pxerbla.\ninfo\n(local)\n= 0: successful exit\n< 0: if the i-th argument, indexed i-1, is an array and the j-th entry had an\nillegal value,\nthen info = -(i*100+j),\nif the i-th argument is a scalar and had an illegal value,\nthen info = -i.\nApplication Notes\nIf uplo = 'U', the matrix Q is represented as a product of elementary reflectors\nQ = H(n-1)*...*H(2)*H(1)\nEach H(i) has the form\nH(i) = I - tau*v*v',\nwhere tau is a real/complex scalar, and v is a real/complex vector with v(i+1:n) = 0 and v(i) = 1;\nv(1:i-1) is stored on exit in A(ia:ia+i-2, ja+i), and tau in tau[ja+i-2].\nIf uplo = 'L', the matrix Q is represented as a product of elementary reflectors\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1752\n\n\nQ = H(1)*H(2)*...*H(n-1).\nEach H(i) has the form\nH(i) = I - tau*v*v' ,\nwhere tau is a real/complex scalar, and v is a real/complex vector with v(1:i) = 0 and v(i+1) = 1; v(i\n+2:n) is stored on exit in A(ia+i+1:ia+n-1, ja+i-1), and tau in tau[ja+i-2].\nThe contents of sub (A) on exit are illustrated by the following examples with n = 5:\nwhere d and e denotes diagonal and off-diagonal elements of T, and vi denotes an element of the vector\ndefining H(i).\nNOTE\nThe distributed matrix sub(A) must verify some alignment properties, namely the following\nexpression should be true:\n( mb_a==nb_a && iroffa==icoffa )where iroffa = mod(ia - 1, mb_a) and icoffa =\nmod(ja -1, nb_a).\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?trord\nReorders the Schur factorization of a general matrix.\nSyntax\nvoid pstrord( char* compq, MKL_INT* select, MKL_INT* para, MKL_INT* n, float* t,\nMKL_INT* it, MKL_INT* jt, MKL_INT* desct, float* q, MKL_INT* iq, MKL_INT* jq, MKL_INT*\ndescq, float* wr, float* wi, MKL_INT* m, float* work, MKL_INT* lwork, MKL_INT* iwork,\nMKL_INT* liwork, MKL_INT* info);\nvoid pdtrord(char* compq, MKL_INT* select, MKL_INT* para, MKL_INT* n, double* t,\nMKL_INT* it, MKL_INT* jt, MKL_INT* desct, double* q, MKL_INT* iq, MKL_INT* jq, MKL_INT*\ndescq, double* wr, double* wi, MKL_INT* m, double* work, MKL_INT* lwork, MKL_INT* iwork,\nMKL_INT* liwork, MKL_INT* info);\nInclude Files\n•\nmkl_scalapack.h\nDescription\np?trord reorders the real Schur factorization of a real matrix A = Q*T*QT, so that a selected cluster of\neigenvalues appears in the leading diagonal blocks of the upper quasi-triangular matrix T, and the leading\ncolumns of Q form an orthonormal basis of the corresponding right invariant subspace.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1753\n\n\nT must be in Schur form (as returned by p?lahqr), that is, block upper triangular with 1-by-1 and 2-by-2\ndiagonal blocks.\nThis function uses a delay and accumulate procedure for performing the off-diagonal updates.\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nInput Parameters\ncompq\n(global)\n= 'V': update the matrix q of Schur vectors;\n= 'N': do not update q.\nselect\n(global) array of size n\nselect specifies the eigenvalues in the selected cluster. To select a real\neigenvalue w(j), select[j-1] must be set to 1. To select a complex\nconjugate pair of eigenvalues w(j) and w(j+1), corresponding to a 2-by-2\ndiagonal block, either select[j-1] or select[j] or both must be set to 1; a\ncomplex conjugate pair of eigenvalues must be either both included in the\ncluster or both excluded.\npara\n(global)\nBlock parameters:\npara[0]\nmaximum number of concurrent computational\nwindows allowed in the algorithm; 0 < para[0]≤\nmin(nprow, npcol) must hold;\npara[1]\nnumber of eigenvalues in each window; 0 <\npara[1] < para[2] must hold;\npara[2]\nwindow size; para[1] < para[2] < mb_t must\nhold;\npara[3]\nminimal percentage of FLOPS required for\nperforming matrix-matrix multiplications instead\nof pipelined orthogonal transformations; 0\n≤para[3]≤ 100 must hold;\npara[4]\nwidth of block column slabs for row-wise\napplication of pipelined orthogonal\ntransformations in their factorized form; 0 <\npara[4]≤mb_t must hold.\npara[5]\nthe maximum number of eigenvalues moved\ntogether over a process border; in practice, this\nwill be approximately half of the cross border\nwindow size; 0 < para[5]≤para[1] must hold.\nn\n(global)\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1754\n\n\nThe order of the globally distributed matrix t. n≥ 0.\nt\n(local) array of size lld_t * LOCc(n).\nThe local pieces of the global distributed upper quasi-triangular matrix T, in\nSchur form.\nit, jt\n(global)\nThe row and column index in the global matrix T indicating the first column\nof T. it = jt = 1 must hold (see Application Notes).\ndesct\n(global and local) array of size dlen_.\nThe array descriptor for the global distributed matrix T.\nq\n(local) array of size lld_q * LOCc(n).\nOn entry, if compq = 'V', the local pieces of the global distributed matrix Q\nof Schur vectors.\nIf compq = 'N', q is not referenced.\niq, jq\n(global)\nThe column index in the global matrix Q indicating the first column of Q. iq\n= jq = 1 must hold (see Application Notes).\ndescq\n(global and local) array of size dlen_.\nThe array descriptor for the global distributed matrix Q.\nwork\n(local workspace) array of size lwork\nlwork\n(local)\nThe size of the array work.\nIf lwork = -1, then a workspace query is assumed; the function only\ncalculates the optimal size of the work array, returns this value as the first\nentry of the work array, and no error message related to lwork is issued by \npxerbla.\niwork\n(local workspace) array of size liwork\nliwork\n(local)\nThe size of the array iwork.\nIf liwork = -1, then a workspace query is assumed; the function only\ncalculates the optimal size of the iwork array, returns this value as the first\nentry of the iwork array, and no error message related to liwork is issued\nby pxerbla\nOUTPUT Parameters\nselect\n(global) array of size n\nThe (partial) reordering is displayed.\nt\nOn exit, t is overwritten by the local pieces of the reordered matrix T, again\nin Schur form, with the selected eigenvalues in the globally leading diagonal\nblocks.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1755\n\n\nq\nOn exit, if compq = 'V', q has been postmultiplied by the global orthogonal\ntransformation matrix which reorders t; the leading m columns of q form an\northonormal basis for the specified invariant subspace.\nIf compq = 'N', q is not referenced.\nwr, wi\n(global ) array of size n\nThe real and imaginary parts, respectively, of the reordered eigenvalues of\nthe matrix T. The eigenvalues are in principle stored in the same order as\non the diagonal of T, with wr[i] = T(i+1,i+1) and, if T(i:i+1,i:i+1) is a 2-\nby-2 diagonal block, wi[i-1] > 0 and wi[i] = -wi[i-1].\nNote also that if a complex eigenvalue is sufficiently ill-conditioned, then its\nvalue may differ significantly from its value before reordering.\nm\n(global )\nThe size of the specified invariant subspace.\n0 ≤m≤n.\nwork[0]\nOn exit, if info = 0, work[0] returns the optimal lwork.\niwork[0]\nOn exit, if info = 0, iwork[0] returns the optimal liwork.\ninfo\n(global)\n= 0: successful exit\n< 0: if info = -i, the i-th argument had an illegal value. If the i-th\nargument is an array and the j-th entry, indexed j-1, had an illegal value,\nthen info = -(i*1000+j), if the i-th argument is a scalar and had an illegal\nvalue, then info = -i.\n> 0: here we have several possibilities\n•\nReordering of t failed because some eigenvalues are too close to\nseparate (the problem is very ill-conditioned);\nt may have been partially reordered, and wr and wi contain the\neigenvalues in the same order as in t.\nOn exit, info = {the index of t where the swap failed (indexing starts\nat 1)}.\n•\nA 2-by-2 block to be reordered split into two 1-by-1 blocks and the\nsecond block failed to swap with an adjacent block.\nOn exit, info = {the index of t where the swap failed}.\n•\nIf info = n+1, there is no valid BLACS context (see the BLACS\ndocumentation for details).\nApplication Notes\nThe following alignment requirements must hold:\n•\nmb_t = nb_t = mb_q = nb_q\n•\nrsrc_t = rsrc_q\n•\ncsrc_t = csrc_q\nAll matrices must be blocked by a block factor larger than or equal to two (3). This is to simplify reordering\nacross processor borders in the presence of 2-by-2 blocks.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1756\n\n\nThis algorithm cannot work on submatrices of t and q, i.e., it = jt = iq = jq = 1 must hold. This is\nhowever no limitation since p?lahqr does not compute Schur forms of submatrices anyway.\nParallel execution recommendations:\n•\nUse a square grid, if possible, for maximum performance. The block parameters in para should be kept\nwell below the data distribution block size.\n•\nIn general, the parallel algorithm strives to perform as much work as possible without crossing the block\nborders on the main block diagonal.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?trsen\nReorders the Schur factorization of a matrix and\n(optionally) computes the reciprocal condition\nnumbers and invariant subspace for the selected\ncluster of eigenvalues.\nSyntax\nvoid pstrsen(char* job, char* compq, MKL_INT* select, MKL_INT* para, MKL_INT* n, float*\nt, MKL_INT* it, MKL_INT* jt, MKL_INT* desct, float* q, MKL_INT* iq, MKL_INT* jq,\nMKL_INT* descq, float* wr, float* wi, MKL_INT* m, float* s, float* sep, float* work,\nMKL_INT* lwork, MKL_INT* iwork, MKL_INT* liwork, MKL_INT* info);\nvoid pdtrsen(char* job, char* compq, MKL_INT* select, MKL_INT* para, MKL_INT* n, double*\nt, MKL_INT* it, MKL_INT* jt, MKL_INT* desct, double* q, MKL_INT* iq, MKL_INT* jq,\nMKL_INT* descq, double* wr, double* wi, MKL_INT* m, double* s, double* sep, double*\nwork, MKL_INT* lwork, MKL_INT* iwork, MKL_INT* liwork, MKL_INT* info);\nInclude Files\n•\nmkl_scalapack.h\nDescription\np?trsen reorders the real Schur factorization of a real matrix A = Q*T*QT, so that a selected cluster of\neigenvalues appears in the leading diagonal blocks of the upper quasi-triangular matrix T, and the leading\ncolumns of Q form an orthonormal basis of the corresponding right invariant subspace. The reordering is\nperformed by p?trord.\nOptionally the function computes the reciprocal condition numbers of the cluster of eigenvalues and/or the\ninvariant subspace.\nT must be in Schur form (as returned by p?lahqr), that is, block upper triangular with 1-by-1 and 2-by-2\ndiagonal blocks.\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nInput Parameters\njob\n(global )\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1757\n\n\nSpecifies whether condition numbers are required for the cluster of\neigenvalues (s) or the invariant subspace (sep):\n= 'N': no condition numbers are required;\n= 'E': only the condition number for the cluster of eigenvalues is computed\n(s);\n= 'V': only the condition number for the invariant subspace is computed\n(sep);\n= 'B': condition numbers for both the cluster and the invariant subspace are\ncomputed (s and sep).\ncompq\n(global )\n= 'V': update the matrix q of Schur vectors;\n= 'N': do not update q.\nselect\n(global ) array of size n\nselect specifies the eigenvalues in the selected cluster. To select a real\neigenvalue w(j), select[j-1] must be set to a non-zero number. To select a\ncomplex conjugate pair of eigenvalues w(j) and w(j+1), corresponding to a\n2-by-2 diagonal block, either select[j-1] or select[j] or both must be set\nto a non-zero number; a complex conjugate pair of eigenvalues must be\neither both included in the cluster or both excluded.\npara\n(global )\nBlock parameters:\npara[0]\nmaximum number of concurrent computational\nwindows allowed in the algorithm; 0 < para[0]≤\nmin(NPROW,NPCOL) must hold;\npara[1]\nnumber of eigenvalues in each window; 0 <\npara[1] < para[2] must hold;\npara[2]\nwindow size; para[1] < para[2] < mb_t must\nhold;\npara[3]\nminimal percentage of flops required for\nperforming matrix-matrix multiplications instead\nof pipelined orthogonal transformations; 0\n≤para[3]≤ 100 must hold;\npara[4]\nwidth of block column slabs for row-wise\napplication of pipelined orthogonal\ntransformations in their factorized form; 0 <\npara[4]≤mb_t must hold.\npara[5]\nthe maximum number of eigenvalues moved\ntogether over a process border; in practice, this\nwill be approximately half of the cross border\nwindow size 0 < para[5]≤para[1] must hold;\nn\n(global )\nThe order of the globally distributed matrix t. n≥ 0.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1758\n\n\nt\n(local ) array of size lld_t * LOCc(n).\nThe local pieces of the global distributed upper quasi-triangular matrix T, in\nSchur form.\nit, jt\n(global )\nThe row and column index in the global matrix T indicating the first column\nof T. it = jt = 1 must hold (see Application Notes).\ndesct\n(global and local) array of size dlen_.\nThe array descriptor for the global distributed matrix T.\nq\n(local ) array of size lld_q * LOCc(n).\nOn entry, if compq = 'V', the local pieces of the global distributed matrix Q\nof Schur vectors.\nIf compq = 'N', q is not referenced.\niq, jq\n(global )\nThe column index in the global matrix Q indicating the first column of Q. iq\n= jq = 1 must hold (see Application Notes).\ndescq\n(global and local) array of size dlen_.\nThe array descriptor for the global distributed matrix Q.\nwork\n(local workspace) array of size lwork\nlwork\n(local )\nThe size of the array work.\nIf lwork = -1, then a workspace query is assumed; the function only\ncalculates the optimal size of the work array, returns this value as the first\nentry of the work array, and no error message related to lwork is issued by \npxerbla.\niwork\n(local workspace) array of size liwork\nliwork\n(local )\nThe size of the array iwork.\nIf liwork = -1, then a workspace query is assumed; the function only\ncalculates the optimal size of the iwork array, returns this value as the first\nentry of the iwork array, and no error message related to liwork is issued\nby pxerbla.\nOUTPUT Parameters\nt\nt is overwritten by the local pieces of the reordered matrix T, again in\nSchur form, with the selected eigenvalues in the globally leading diagonal\nblocks.\nq\nOn exit, if compq = 'V', q has been postmultiplied by the global orthogonal\ntransformation matrix which reorders t; the leading m columns of q form an\northonormal basis for the specified invariant subspace.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1759\n\n\nIf compq = 'N', q is not referenced.\nwr, wi\n(global ) array of size n\nThe real and imaginary parts, respectively, of the reordered eigenvalues of\nthe matrix T. The eigenvalues are in principle stored in the same order as\non the diagonal of T, with wr[i] = T(i+1,i+1) and, if T(i:i+1,i:i+1) is a 2-\nby-2 diagonal block, wi[i-1] > 0 and wi[i] = -wi[i-1].\nNote also that if a complex eigenvalue is sufficiently ill-conditioned, then its\nvalue may differ significantly from its value before reordering.\nm\n(global )\nThe size of the specified invariant subspace. 0 ≤m≤n.\ns\n(global )\nIf job = 'E' or 'B', s is a lower bound on the reciprocal condition number for\nthe selected cluster of eigenvalues. s cannot underestimate the true\nreciprocal condition number by more than a factor of sqrt(n). If m = 0 or n,\ns = 1.\nIf job = 'N' or 'V', s is not referenced.\nsep\n(global )\nIf job = 'V' or 'B', sep is the estimated reciprocal condition number of the\nspecified invariant subspace. If\nm = 0 or n, sep = norm(t).\nIf job = 'N' or 'E', sep is not referenced.\nwork[0]\nOn exit, if info = 0, work[0] returns the optimal lwork.\niwork[0]\nOn exit, if info = 0, iwork[0] returns the optimal liwork.\ninfo\n(global )\n= 0: successful exit\n< 0: if info = -i, the i-th argument had an illegal value.\nIf the i-th argument is an array and the j-th entry, indexed j-1, had an\nillegal value, then info = -(i*1000+j), if the i-th argument is a scalar and\nhad an illegal value, then info = -i.\n> 0: here we have several possibilities\n•\nReordering of t failed because some eigenvalues are too close to\nseparate (the problem is very ill-conditioned); t may have been partially\nreordered, and wr and wi contain the eigenvalues in the same order as\nin t.\nOn exit, info = {the index of t where the swap failed (indexing starts\nat 1)}.\n•\nA 2-by-2 block to be reordered split into two 1-by-1 blocks and the\nsecond block failed to swap with an adjacent block.\nOn exit, info = {the index of t where the swap failed}.\n•\nIf info = n+1, there is no valid BLACS context (see the BLACS\ndocumentation for details).\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1760\n\n\nApplication Notes\nThe following alignment requirements must hold:\n•\nmb_t = nb_t = mb_q = nb_q\n•\nrsrc_t = rsrc_q\n•\ncsrc_t = csrc_q\nAll matrices must be blocked by a block factor larger than or equal to two (3). This to simplify reordering\nacross processor borders in the presence of 2-by-2 blocks.\nThis algorithm cannot work on submatrices of t and q, i.e., it = jt = iq = jq = 1 must hold. This is\nhowever no limitation since p?lahqr does not compute Schur forms of submatrices anyway.\nFor parallel execution, use a square grid, if possible, for maximum performance. The block parameters in\npara should be kept well below the data distribution block size.\nIn general, the parallel algorithm strives to perform as much work as possible without crossing the block\nborders on the main block diagonal.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?trti2\nComputes the inverse of a triangular matrix (local\nunblocked algorithm).\nSyntax\nvoid pstrti2 (char *uplo , char *diag , MKL_INT *n , float *a , MKL_INT *ia , MKL_INT\n*ja , MKL_INT *desca , MKL_INT *info );\nvoid pdtrti2 (char *uplo , char *diag , MKL_INT *n , double *a , MKL_INT *ia , MKL_INT\n*ja , MKL_INT *desca , MKL_INT *info );\nvoid pctrti2 (char *uplo , char *diag , MKL_INT *n , MKL_Complex8 *a , MKL_INT *ia ,\nMKL_INT *ja , MKL_INT *desca , MKL_INT *info );\nvoid pztrti2 (char *uplo , char *diag , MKL_INT *n , MKL_Complex16 *a , MKL_INT *ia ,\nMKL_INT *ja , MKL_INT *desca , MKL_INT *info );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?trti2function computes the inverse of a real/complex upper or lower triangular block matrix sub (A)\n= A(ia:ia+n-1, ja:ja+n-1).\nThis matrix should be contained in one and only one process memory space (local operation).\nInput Parameters\nuplo\n(global)\nSpecifies whether the matrix sub (A) is upper or lower triangular.\n= 'U': sub (A) is upper triangular\n= 'L': sub (A) is lower triangular.\ndiag\n(global)\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1761\n\n\nSpecifies whether or not the matrix A is unit triangular.\n= 'N': sub (A) is non-unit triangular\n= 'U': sub (A) is unit triangular.\nn\n(global)\nThe number of rows and columns to be operated on, i.e., the order of the\ndistributed submatrix sub(A). n ≥ 0.\na\n(local)\nPointer into the local memory to an array, size lld_a * LOCc(ja+n-1).\nOn entry, this array contains the local pieces of the triangular matrix\nsub(A).\nIf uplo = 'U', the leading n-by-n upper triangular part of the matrix\nsub(A) contains the upper triangular part of the matrix, and the strictly\nlower triangular part of sub(A) is not referenced.\nIf uplo = 'L', the leading n-by-n lower triangular part of the matrix\nsub(A) contains the lower triangular part of the matrix, and the strictly\nupper triangular part of sub(A) is not referenced. If diag = 'U', the\ndiagonal elements of sub(A) are not referenced either and are assumed to\nbe 1.\nia, ja\n(global)\nThe row and column indices in the global matrix A indicating the first row\nand the first column of the sub(A), respectively.\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\nOutput Parameters\na\nOn exit, the (triangular) inverse of the original matrix, in the same storage\nformat.\ninfo\n= 0: successful exit\n< 0: if the i-th argument is an array and the j-th entry, indexed j-1, had an\nillegal value,\nthen info = - (i*100+j),\nif the i-th argument is a scalar and had an illegal value,\nthen info = -i.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\n?lahqr2\nUpdates the eigenvalues and Schur decomposition.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1762\n\n\nSyntax\nvoid clahqr2 (const MKL_INT* wantt, const MKL_INT* wantz, const MKL_INT* n, const\nMKL_INT* ilo, const MKL_INT* ihi, MKL_Complex8* h, const MKL_INT* ldh, MKL_Complex8* w,\nconst MKL_INT* iloz, const MKL_INT* ihiz, MKL_Complex8* z, const MKL_INT* ldz, MKL_INT*\ninfo);\nvoid zlahqr2 (const MKL_INT* wantt, const MKL_INT* wantz, const MKL_INT* n, const\nMKL_INT* ilo, const MKL_INT* ihi, MKL_Complex16* h, const MKL_INT* ldh, MKL_Complex16*\nw, const MKL_INT* iloz, const MKL_INT* ihiz, MKL_Complex16* z, const MKL_INT* ldz,\nMKL_INT* info);\nInclude Files\n•\nmkl_scalapack.h\nDescription\n?lahqr2 is an auxiliary routine called by ?hseqr to update the eigenvalues and Schur decomposition already\ncomputed by ?hseqr, by dealing with the Hessenberg submatrix in rows and columns ilo to ihi. This\nversion of ?lahqr (not the standard LAPACK version) uses a double-shift algorithm (like LAPACK's ?lahqr).\nUnlike the standard LAPACK convention, this does not assume the subdiagonal is real, nor does it work to\npreserve this quality if given.\nInput Parameters\nwantt\n≠ 0: the full Schur form T is required;\n= 0: only eigenvalues are required.\nwantz\n≠ 0: the matrix of Schur vectors Z is required;\n= 0: Schur vectors are not required.\nn\nThe order of the matrix H. n >= 0.\nilo, ihi\nIt is assumed that the matrix H is upper triangular in rows and columns ihi\n+1 :n, and that matrix element H(ilo,ilo-1) = 0 (unless ilo =\n1). ?lahqr works primarily with the Hessenberg submatrix in rows and\ncolumns ilo to ihi, but applies transformations to all of h if wantt is\nnonzero.\n1 <= ilo <= max(1,ihi); ihi <= n.\nh\nArray, size ldh*n.\nOn entry, the upper Hessenberg matrix H.\nldh\nThe leading dimension of the array h. ldh >= max(1,n).\niloz, ihiz\nSpecify the rows of Z to which transformations must be applied if wantz≠ 0.\n1 <= iloz <= ilo; ihi <= ihiz <= n.\nz\nArray, size ldz*n.\nIf wantz≠ 0, on entry z must contain the current matrix Z of\ntransformations. If wantz= 0, z is not referenced.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1763\n\n\nldz\nThe leading dimension of the array z. ldz >= max(1,n).\nOutput Parameters\nh\nOn exit, if wantt≠ 0, h is upper triangular in rows and columns\nilo:ihi. If wantt= 0, the contents of h are unspecified on exit.\nw\nArray, size (n)\nThe computed eigenvalues ilo to ihi are stored in the corresponding\nelements of w. If wantt≠ 0, the eigenvalues are stored in the same\norder as on the diagonal of the Schur form returned in h, with w[i] =\nH(i, i).\nz\nIf wantz≠ 0, on exit z has been updated; transformations are applied\nonly to the submatrix Z(iloz:ihiz,ilo:ihi). If wantz= 0, z is not\nreferenced.\ninfo\n= 0: successful exit\n> 0: if info = i, ?lahqr failed to compute all the eigenvalues ilo to\nihi in a total of 30*(ihi-ilo+1) iterations; elements w[i:ihi - 1]\ncontain those eigenvalues which have been successfully computed.\n?lamsh\nSends multiple shifts through a small (single node)\nmatrix to maximize the number of bulges that can be\nsent through.\nSyntax\nvoid slamsh (float *s, const MKL_INT *lds, MKL_INT *nbulge, const MKL_INT *jblk, float\n*h, const MKL_INT *ldh, const MKL_INT *n, const float *ulp );\nvoid dlamsh (double *s, const MKL_INT *lds, MKL_INT *nbulge, const MKL_INT *jblk,\ndouble *h, const MKL_INT *ldh, const MKL_INT *n, const double *ulp );\nvoid clamsh (MKL_Complex8 *s , const MKL_INT *lds , MKL_INT *nbulge , const MKL_INT\n*jblk , MKL_Complex8 *h , const MKL_INT *ldh , const MKL_INT *n , const float *ulp );\nvoid zlamsh (MKL_Complex16 *s , const MKL_INT *lds , MKL_INT *nbulge , const MKL_INT\n*jblk , MKL_Complex16 *h , const MKL_INT *ldh , const MKL_INT *n , const double *ulp );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe ?lamshfunction sends multiple shifts through a small (single node) matrix to see how small consecutive\nsubdiagonal elements are modified by subsequent shifts in an effort to maximize the number of bulges that\ncan be sent through. The function should only be called when there are multiple shifts/bulges (nbulge > 1)\nand the first shift is starting in the middle of an unreduced Hessenberg matrix because of two or more small\nconsecutive subdiagonal elements.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1764\n\n\nInput Parameters\ns\n(local)\nArray of size lds*2*jblk.\nOn entry, the matrix of shifts. Only the 2x2 diagonal of s is referenced. It is\nassumed that s has jblk double shifts (size 2).\nlds\n(local)\nOn entry, the leading dimension of S; unchanged on exit. 1<nbulge ≤ jblk\n≤ lds/2.\nnbulge\n(local)\nOn entry, the number of bulges to send through h (>1). nbulge should be\nless than the maximum determined (jblk). 1<nbulge ≤ jblk ≤ lds/2.\njblk\n(local)\nOn entry, the number of double shifts determined for S; unchanged on exit.\nh\n(local)\nArray of size ldh*n.\nOn entry, the local matrix to apply the shifts on.\nh should be aligned so that the starting row is 2.\nldh\n(local)\nOn entry, the leading dimension of H; unchanged on exit.\nn\n(local)\nOn entry, the size of H. If all the bulges are expected to go through, n\nshould be at least 4nbulge+2. Otherwise, nbulge may be reduced by this\nfunction.\nulp\n(local)\nOn entry, machine precision. Unchanged on exit.\nOutput Parameters\ns\nOn exit, the data is rearranged in the best order for applying.\nnbulge\nOn exit, the maximum number of bulges that can be sent through.\nh\nOn exit, the data is destroyed.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\n?lapst\nSorts the numbers in increasing or decreasing order.\nSyntax\nvoid slapst (const char* id, const MKL_INT* n, const float* d, MKL_INT* indx, MKL_INT*\ninfo);\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1765\n\n\nvoid dlapst (const char* id, const MKL_INT* n, const double* d, MKL_INT* indx, MKL_INT*\ninfo);\nInclude Files\n•\nmkl_scalapack.h\nDescription\n?lapst is a modified version of the LAPACK routine ?lasrt.\nDefine a permutation indx that sorts the numbers in d in increasing order (if id = 'I') or in decreasing order\n(if id = 'D' ).\nUse Quick Sort, reverting to Insertion sort on arrays of size <= 20. Dimension of STACK limits n to about\n232.\nInput Parameters\nid\n= 'I': sort d in increasing order;\n= 'D': sort d in decreasing order.\nn\nThe length of the array d.\nd\nArray, size (n)\nThe array to be sorted.\nOutput Parameters\nindx\nArray, size (n).\nThe permutation which sorts the array d.\ninfo\n= 0: successful exit\n< 0: if info = -i, the i-th argument had an illegal value\n?laqr6\nPerforms a single small-bulge multi-shift QR sweep\ncollecting the transformations.\nSyntax\nvoid slaqr6(char* job, MKL_INT* wantt, MKL_INT* wantz, MKL_INT* kacc22, MKL_INT* n,\nMKL_INT* ktop, MKL_INT* kbot, MKL_INT* nshfts, float* sr, float* si, float* h, MKL_INT*\nldh, MKL_INT* iloz, MKL_INT* ihiz, float* z, MKL_INT* ldz, float* v, MKL_INT* ldv,\nfloat* u, MKL_INT* ldu, MKL_INT* nv, float* wv, MKL_INT* ldwv, MKL_INT* nh, float* wh,\nMKL_INT* ldwh);\nvoid dlaqr6(char* job, MKL_INT* wantt, MKL_INT* wantz, MKL_INT* kacc22, MKL_INT* n,\nMKL_INT* ktop, MKL_INT* kbot, MKL_INT* nshfts, double* sr, double* si, double* h,\nMKL_INT* ldh, MKL_INT* iloz, MKL_INT* ihiz, double* z, MKL_INT* ldz, double* v, MKL_INT*\nldv, double* u, MKL_INT* ldu, MKL_INT* nv, double* wv, MKL_INT* ldwv, MKL_INT* nh,\ndouble* wh, MKL_INT* ldwh);\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1766\n\n\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThis auxiliary function performs a single small-bulge multi-shift QR sweep, moving the chain of bulges from\ntop to bottom in the submatrix H(ktop:kbot,ktop:kbot), collecting the transformations in the matrix V or\naccumulating the transformations in the matrix Z (see below).\nThis is a modified version of ?laqr5 from LAPACK 3.1.\nInput Parameters\njob\nSet the kind of job to do in ?laqr6, as follows:\njob = 'I': Introduce and chase bulges in submatrix\njob = 'C': Chase bulges from top to bottom of submatrix\njob = 'O': Chase bulges off submatrix\nwantt\nwanttis non-zero if the quasi-triangular Schur factor is being computed.\nwantt is set to zero otherwise.\nwantz\nwantzis non-zero if the orthogonal Schur factor is being computed. wantz\nis set to zero otherwise.\nkacc22\nSpecifies the computation mode of far-from-diagonal orthogonal updates.\n= 0: ?laqr6 does not accumulate reflections and does not use matrix-\nmatrix multiply to update far-from-diagonal matrix entries.\n= 1: ?laqr6 accumulates reflections and uses matrix-matrix multiply to\nupdate the far-from-diagonal matrix entries.\n= 2: ?laqr6 accumulates reflections, uses matrix-matrix multiply to update\nthe far-from-diagonal matrix entries, and takes advantage of 2-by-2 block\nstructure during matrix multiplies.\nn\nn is the order of the Hessenberg matrix H upon which this function\noperates.\nktop, kbot\nThese are the first and last rows and columns of an isolated diagonal block\nupon which the QR sweep is to be applied. It is assumed without a check\nthat either ktop = 1 or H(ktop,ktop-1) = 0 and either kbot = n or H(kbot\n+1,kbot) = 0.\nnshfts\nnshfts gives the number of simultaneous shifts. nshfts must be positive\nand even.\nsr, si\nArray of size nshfts\nsr contains the real parts and si contains the imaginary parts of the\nnshfts shifts of origin that define the multi-shift QR sweep.\nh\nArray of size ldh * n\nOn input h contains a Hessenberg matrix H.\nldh\nldh is the leading dimension of H just as declared in the calling function.\nldh≥ max(1,n).\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1767\n\n\niloz, ihiz\nSpecify the rows of the matrix Zto which transformations must be applied if\nwantzis non-zero. 1≤iloz≤ihiz≤n\nz\nArray of size ldz * ktop\nIf wantzis non-zero, then the QR sweep orthogonal similarity\ntransformation is accumulated into the matrix Z(iloz:ihiz,kbot:ktop),\nstored in the array z, from the right.\nIf wantzequals zero, then z is unreferenced.\nldz\nldz is the leading dimension of z just as declared in the calling function.\nldz≥n.\nv\n(workspace) array of size ldv * nshfts/2\nldv\nldv is the leading dimension of v as declared in the calling function. ldv≥3.\nu\n(workspace) array of size ldu * (3*nshfts-3)\nldu\nldu is the leading dimension of u just as declared in the calling function.\nldu≥3*nshfts-3.\nnh\nnh is the number of columns in array wh available for workspace. nh≥1 is\nrequired for usage of this workspace, otherwise the updates of the far-\nfrom-diagonal elements will be updated without level 3 BLAS.\nwh\n(workspace) array of size ldwh * nh\nldwh\nLeading dimension of wh just as declared in the calling function.\nldwh≥3*nshfts-3.\nnv\nnv is the number of rows in wv available for workspace. nv≥1 is required for\nusage of this workspace, otherwise the updates of the far-from-diagonal\nelements will be updated without level 3 BLAS.\nwv\n(workspace) array of size ldwv * 3*nshfts\nldwv\nscalar\nldwv is the leading dimension of wv as declared in the in the calling\nfunction. ldwv≥nv.\nOUTPUT Parameters\nh\nA multi-shift QR sweep with shifts sr[j]+i*si[j] is applied to the isolated\ndiagonal block in matrix rows and columns ktop through kbot.\nz\nIf wantzis non-zero, then the QR sweep orthogonal/unitary similarity\ntransformation is accumulated into the matrix Z(iloz:ihiz,kbot:ktop)\nfrom the right.\nIf wantzequals zero, then z is unreferenced.\nApplication Notes\nNotes\nBased on contributions by Karen Braman and Ralph Byers, Department of Mathematics, University of Kansas,\nUSA Robert Granat, Department of Computing Science and HPC2N, Umea University, Sweden\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1768\n\n\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\n?lar1va\nComputes scaled eigenvector corresponding to given\neigenvalue.\nSyntax\nvoid slar1va(MKL_INT* n, MKL_INT* b1, MKL_INT* bn, float* lambda, float* d, float* l,\nfloat* ld, float* lld, float* pivmin, float* gaptol, float* z, MKL_INT* wantnc, MKL_INT*\nnegcnt, float* ztz, float* mingma, MKL_INT* r, MKL_INT* isuppz, float* nrminv, float*\nresid, float* rqcorr, float* work);\nvoid dlar1va(MKL_INT* n, MKL_INT* b1, MKL_INT* bn, double* lambda, double* d, double* l,\ndouble* ld, double* lld, double* pivmin, double* gaptol, double* z, MKL_INT* wantnc,\nMKL_INT* negcnt, double* ztz, double* mingma, MKL_INT* r, MKL_INT* isuppz, double*\nnrminv, double* resid, double* rqcorr, double* work);\nInclude Files\n•\nmkl_scalapack.h\nDescription\n?slar1va computes the (scaled) r-th column of the inverse of the submatrix in rows b1 through bn of the\ntridiagonal matrix LDLT - λI. When λ is close to an eigenvalue, the computed vector is an accurate\neigenvector. Usually, r corresponds to the index where the eigenvector is largest in magnitude. The following\nsteps accomplish this computation :\n1.\nStationary qd transform, LDLT - λI = L+D+L+T,\n2.\nProgressive qd transform, LDLT - λI = U-D-U-T,\n3.\nComputation of the diagonal elements of the inverse of LDLT - λI by combining the above transforms,\nand choosing r as the index where the diagonal of the inverse is (one of the) largest in magnitude.\n4.\nComputation of the (scaled) r-th column of the inverse using the twisted factorization obtained by\ncombining the top part of the stationary and the bottom part of the progressive transform.\nInput Parameters\nn\nThe order of the matrix LDLT.\nb1\nFirst index of the submatrix of LDLT.\nbn\nLast index of the submatrix of LDLT.\nlambda\nThe shift λ. In order to compute an accurate eigenvector, lambda should be\na good approximation to an eigenvalue of LDLT.\nl\nArray of size n-1\nThe (n-1) subdiagonal elements of the unit bidiagonal matrix L, in elements\n0 to n-2.\nd\nArray of size n\nThe n diagonal elements of the diagonal matrix D.\nld\nArray of size n-1\nThe n-1 elements l[i]*d[i], i=0,...,n-2.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1769\n\n\nlld\nArray of size n-1\nThe n-1 elements l[i]*l]i]*d[i], i=0,...,n-2.\npivmin\nThe minimum pivot in the Sturm sequence.\ngaptol\nTolerance that indicates when eigenvector entries are negligible with respect\nto their contribution to the residual.\nz\nArray of size n\nOn input, all entries of z must be set to 0.\nwantnc\nSpecifies whether negcnt has to be computed.\nr\nThe twist index for the twisted factorization used to compute z.\nOn input, 0 ≤r≤n. If r is input as 0, r is set to the index where (LDLT - σI)-1\nis largest in magnitude. If 1 ≤r≤n, r is unchanged.\nIdeally, r designates the position of the maximum entry in the eigenvector.\nwork\n(Workspace) array of size 4*n\nOUTPUT Parameters\nz\nOn output, z contains the (scaled) r-th column of the inverse. The scaling is\nsuch that z[r-1] equals 1.\nnegcnt\nIf wantncis non-zero then negcnt = the number of pivots < pivmin in the\nmatrix factorization LDLT, and negcnt = -1 otherwise.\nztz\nThe square of the 2-norm of z.\nmingma\nThe reciprocal of the largest (in magnitude) diagonal element of the inverse\nof LDLT - σI.\nr\nOn output, r contains the twist index used to compute z.\nisuppz\narray of size 2\nThe support of the vector in z, i.e., the vector z is non-zero only in\nelements isuppz[0] and isuppz[1].\nnrminv\nnrminv = 1/SQRT( ztz )\nresid\nThe residual of the FP vector.\nresid = ABS( mingma )/SQRT( ztz )\nrqcorr\nThe Rayleigh Quotient correction to lambda.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\n?laref\nApplies Householder reflectors to matrices on their\nrows or columns.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1770\n\n\nSyntax\nvoid slaref (const char* type, float* a, const MKL_INT* lda, const MKL_INT* wantz,\nfloat* z, const MKL_INT* ldz, const MKL_INT* block, MKL_INT* irow1, MKL_INT* icol1,\nconst MKL_INT* istart, const MKL_INT* istop, const MKL_INT* itmp1, const MKL_INT*\nitmp2, const MKL_INT* liloz, const MKL_INT* lihiz, const float* vecs, float* v2, float*\nv3, float* t1, float* t2, float* t3);\nvoid dlaref (const char* type, double* a, const MKL_INT* lda, const MKL_INT* wantz,\ndouble* z, const MKL_INT* ldz, const MKL_INT* block, MKL_INT* irow1, MKL_INT* icol1,\nconst MKL_INT* istart, const MKL_INT* istop, const MKL_INT* itmp1, const MKL_INT*\nitmp2, const MKL_INT* liloz, const MKL_INT* lihiz, const double* vecs, double* v2,\ndouble* v3, double* t1, double* t2, double* t3);\nvoid claref (const char* type, MKL_Complex8* a, const MKL_INT* lda, const MKL_INT*\nwantz, MKL_Complex8* z, const MKL_INT* ldz, const MKL_INT* block, MKL_INT* irow1,\nMKL_INT* icol1, const MKL_INT* istart, const MKL_INT* istop, const MKL_INT* itmp1,\nconst MKL_INT* itmp2, const MKL_INT* liloz, const MKL_INT* lihiz, const MKL_Complex8*\nvecs, MKL_Complex8* v2, MKL_Complex8* v3, MKL_Complex8* t1, MKL_Complex8* t2,\nMKL_Complex8* t3);\nvoid zlaref (const char* type, MKL_Complex16* a, const MKL_INT* lda, const MKL_INT*\nwantz, MKL_Complex16* z, const MKL_INT* ldz, const MKL_INT* block, MKL_INT* irow1,\nMKL_INT* icol1, const MKL_INT* istart, const MKL_INT* istop, const MKL_INT* itmp1,\nconst MKL_INT* itmp2, const MKL_INT* liloz, const MKL_INT* lihiz, const MKL_Complex16*\nvecs, MKL_Complex16* v2, MKL_Complex16* v3, MKL_Complex16* t1, MKL_Complex16* t2,\nMKL_Complex16* t3);\nInclude Files\n•\nmkl_scalapack.h\nDescription\n?laref applies one or several Householder reflectors of size 3 to one or two matrices (if column is specified)\non either their rows or columns.\nInput Parameters\ntype\n(local)\nIf 'R': Apply reflectors to the rows of the matrix (apply from left)\nOtherwise: Apply reflectors to the columns of the matrix\nUnchanged on exit.\na\n(local)\nArray, lld_a*LOCc(ja+n-1)\nOn entry, the matrix to receive the reflections.\nlda\n(local)\nOn entry, the leading dimension of a.\nUnchanged on exit.\nwantz\n(local)\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1771\n\n\nIf wantz≠ 0, then apply any column reflections to z as well.\nIf wantz = 0, then do no additional work on z.\nz\n(local)\nArray, ldz*ncols, where the value ncols depends on other arguments. If\nwantzwantz≠ 0 and type≠ 'R' then ncols = icol1 + 3*(lihiz - liloz +\n1). Otherwise, ncols is unused.\nOn entry, the second matrix to receive column reflections.\nThis is changed only if wantz is set.\nldz\n(local)\nOn entry, the leading dimension of z.\nUnchanged on exit.\nblock\n(local)\nIf nonzero, then apply several reflectors at once and read their data from\nthe vecs array.\nIf zero, apply the single reflector given by v2, v3, t1, t2, and t3.\nirow1\n(local)\nOn entry, the local row element of a.\nicol1\n(local)\nOn entry, the local column element of a.\nistart\n(local)\nSpecifies the \"number\" of the first reflector. This is used as an index into\nvecs if block is set. istart is ignored if block is zero.\nistop\n(local)\nSpecifies the \"number\" of the last reflector. This is used as an index into\nvecs if block is set. istop is ignored if block is zero.\nitmp1\n(local)\nStarting range into a. For rows, this is the local first column. For columns,\nthis is the local first row.\nitmp2\n(local)\nEnding range into a. For rows, this is the local last column. For columns,\nthis is the local last row.\nliloz, lihiz\n(local)\nThese serve the same purpose as itmp1, itmp2 but for z when wantz is\nset.\nvecs\n(local)\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1772\n\n\nArray of size 3*N (matrix size)\nThis holds the size 3 reflectors one after another and this is only accessed\nwhen block is nonzero\nv2, v3, t1, t2, t3\n(local)\nThis holds information on a single size 3 Householder reflector and is read\nwhen block is zero, and overwritten when block is nonzero\nOutput Parameters\na\nThe updated matrix on exit.\nz\nThis is changed only if wantz is set.\nirow1\nUndefined on output.\nicol1\nUndefined on output.\nv2, v3, t1, t2, t3\nOverwritten when block is nonzero.\n?larrb2\nProvides limited bisection to locate eigenvalues for\nmore accuracy.\nSyntax\nvoid slarrb2(MKL_INT* n, float* d, float* lld, MKL_INT* ifirst, MKL_INT* ilast, float*\nrtol1, float* rtol2, MKL_INT* offset, float* w, float* wgap, float* werr, float* work,\nMKL_INT* iwork, float* pivmin, float* lgpvmn, float* lgspdm, MKL_INT* twist, MKL_INT*\ninfo);\nvoid dlarrb2(MKL_INT* n, double* d, double* lld, MKL_INT* ifirst, MKL_INT* ilast,\ndouble* rtol1, double* rtol2, MKL_INT* offset, double* w, double* wgap, double* werr,\ndouble* work, MKL_INT* iwork, double* pivmin, double* lgpvmn, double* lgspdm, MKL_INT*\ntwist, MKL_INT* info);\nInclude Files\n•\nmkl_scalapack.h\nDescription\nGiven the relatively robust representation (RRR) LDLT, ?larrb2 does \"limited\" bisection to refine the\neigenvalues of LDLT with indices in a given range to more accuracy. Initial guesses for these eigenvalues are\ninput in w, the corresponding estimate of the error in these guesses and their gaps are input in werr and\nwgap, respectively. During bisection, intervals [left, right] are maintained by storing their mid-points and\nsemi-widths in the arrays w and werr respectively. The range of indices is specified by the ifirst, ilast,\nand offset parameters, as explained in Input Parameters.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1773\n\n\nNOTE\nThere are very few minor differences between larrb from LAPACK and this current\nfunction ?larrb2. The most important reason for creating this nearly identical copy is\nprofiling: in the ScaLAPACK MRRR algorithm, eigenvalue computation using ?larrb2 is used\nfor refinement in the construction of the representation tree, as opposed to the initial\ncomputation of the eigenvalues for the root RRR which uses ?larrb. When profiling, this\nallows an easy quantification of refinement work vs. computing eigenvalues of the root.\nInput Parameters\nn\nThe order of the matrix.\nd\nArray of size n.\nThe n diagonal elements of the diagonal matrix D.\nlld\nArray of size n-1.\nThe (n-1) elements li+1*li+1*d[i], i=0, ..., n-2.\nifirst\nThe index of the first eigenvalue to be computed.\nilast\nThe index of the last eigenvalue to be computed.\nrtol1, rtol2\nTolerance for the convergence of the bisection intervals.\nAn interval [left, right] has converged if right - left < max (rtol1 * gap,\nrtol2 * max(|left|, |right|)) where gap is the (estimated) distance to the\nnearest eigenvalue.\noffset\nOffset for the arrays w, wgap and werr, i.e., the elements indexed ifirst -\noffset - 1 through ilast - offset -1 of these arrays are to be used.\nw\nArray of size n\nOn input, w[ifirst - offset - 1] through w[ilast - offset - 1] are\nestimates of the eigenvalues of LDLT indexed ifirst through ilast.\nwgap\nArray of size n-1.\nOn input, the (estimated) gaps between consecutive eigenvalues of LDLT,\ni.e., wgap[I - offset - 1] is the gap between eigenvalues I and I + 1. Note\nthat if ifirst = ilast then wgap[ifirst - offset - 1] must be set to\nzero.\nwerr\nArray of size n.\nOn input, werr[ifirst - offset - 1] through werr[ilast - offset - 1]\nare the errors in the estimates of the corresponding elements in w.\nwork\n(workspace) array of size 4*n.\nWorkspace.\niwork\n(workspace) array of size 2*n.\nWorkspace.\npivmin\nThe minimum pivot in the Sturm sequence.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1774\n\n\nlgpvmn\nLogarithm of pivmin, precomputed.\nlgspdm\nLogarithm of the spectral diameter, precomputed.\ntwist\nThe twist index for the twisted factorization that is used for the negcount.\ntwist = n: Compute negcount from LDLT - λI = L+D+L+T\ntwist = 1: Compute negcount from LDLT - λI = U-D-U-T\ntwist = r, 1 < r < n: Compute negcount from LDLT - λI = Nr Δr NrT\nOUTPUT Parameters\nw\nOn output, the eigenvalue estimates in w are refined.\nwgap\nOn output, the eigenvalue gaps in wgap are refined.\nwerr\nOn output, the errors in werr are refined.\ninfo\nError flag.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\n?larrd2\nComputes the eigenvalues of a symmetric tridiagonal\nmatrix to suitable accuracy.\nSyntax\nvoid slarrd2(char* range, char* order, MKL_INT* n, float* vl, float* vu, MKL_INT* il,\nMKL_INT* iu, float* gers, float* reltol, float* d, float* e, float* e2, float* pivmin,\nMKL_INT* nsplit, MKL_INT* isplit, MKL_INT* m, float* w, float* werr, float* wl, float*\nwu, MKL_INT* iblock, MKL_INT* indexw, float* work, MKL_INT* iwork, MKL_INT* dol,\nMKL_INT* dou, MKL_INT* info);\nvoid dlarrd2(char* range, char* order, MKL_INT* n, double* vl, double* vu, MKL_INT* il,\nMKL_INT* iu, double* gers, double* reltol, double* d, double* e, double* e2, double*\npivmin, MKL_INT* nsplit, MKL_INT* isplit, MKL_INT* m, double* w, double* werr, double*\nwl, double* wu, MKL_INT* iblock, MKL_INT* indexw, double* work, MKL_INT* iwork, MKL_INT*\ndol, MKL_INT* dou, MKL_INT* info);\nInclude Files\n•\nmkl_scalapack.h\nDescription\n?larrd2 computes the eigenvalues of a symmetric tridiagonal matrix T to limited initial accuracy. This is an\nauxiliary code to be called from larre2a.\n?larrd2 has been created using the LAPACK code larrd which itself stems from stebz. The motivation for\ncreating ?larrd2 is efficiency: When computing eigenvalues in parallel and the input tridiagonal matrix splits\ninto blocks, ?larrd2 can skip over blocks which contain none of the eigenvalues from DOL to DOU for which\nthe processor responsible. In extreme cases (such as large matrices consisting of many blocks of small size\nlike 2x2), the gain can be substantial.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1775\n\n\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nInput Parameters\nrange\n= 'A': (\"All\") all eigenvalues will be found.\n= 'V': (\"Value\") all eigenvalues in the half-open interval (vl, vu] will be\nfound.\n= 'I': (\"Index\") eigenvalues of the entire matrix with the indices in a given\nrange will be found.\norder\n= 'B': (\"By Block\") the eigenvalues will be grouped by split-off block (see\niblock, isplit) and ordered from smallest to largest within the block.\n= 'E': (\"Entire matrix\") the eigenvalues for the entire matrix will be ordered\nfrom smallest to largest.\nn\nThe order of the tridiagonal matrix T. n >= 0.\nvl, vu\nIf range='V', the lower and upper bounds of the interval to be searched for\neigenvalues. Eigenvalues less than or equal to vl, or greater than vu, will\nnot be returned. vl < vu.\nNot referenced if range = 'A' or 'I'.\nil, iu\nIf range='I', the indices (in ascending order) of the smallest eigenvalue, to\nbe returned in w[il-1], and largest eigenvalue, to be returned in w[iu-1].\n1 ≤il≤iu≤=n, if n > 0; il = 1 and iu = 0 if n = 0.\nNot referenced if range = 'A' or 'V'.\ngers\nArray of size 2*n\nThe n Gerschgorin intervals (the i-th Gerschgorin interval is (gers[2*i-2],\ngers[2*i-1])).\nreltol\nThe minimum relative width of an interval. When an interval is narrower\nthan reltol times the larger (in magnitude) endpoint, then it is considered\nto be sufficiently small, i.e., converged. Note: this should always be at least\nradix*machine epsilon.\nd\nArray of size n\nThe n diagonal elements of the tridiagonal matrix T.\ne\nArray of size n-1\nThe (n-1) off-diagonal elements of the tridiagonal matrix T.\ne2\nArray of size n-1\nThe (n-1) squared off-diagonal elements of the tridiagonal matrix T.\npivmin\nThe minimum pivot allowed in the sturm sequence for T.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1776\n\n\nnsplit\nThe number of diagonal blocks in the matrix T.\n1 ≤nsplit≤n.\nisplit\nArray of size n\nThe splitting points, at which T breaks up into submatrices.\nThe first submatrix consists of rows/columns 1 to isplit[0], the second of\nrows/columns isplit[0]+1 through isplit[1], etc., and the nsplit-th\nsubmatrix consists of rows/columns isplit[nsplit-2]+1 through\nisplit[nsplit-1]=n.\n(Only the first nsplit elements will actually be used, but since the user\ncannot know a priori what value nsplit will have, n words must be\nreserved for isplit.)\nwork\n(workspace) Array of size 4*n\niwork\n(workspace) Array of size 3*n\ndol, dou\nSpecifying an index range dol:dou allows the user to work on only a\nselected part of the representation tree.\nOtherwise, the setting dol=1, dou=n should be applied.\nNote that dol and dou refer to the order in which the eigenvalues are\nstored in W.\nOUTPUT Parameters\nm\nThe actual number of eigenvalues found. 0 ≤m≤n.\n(See also the description of info=2,3.)\nw\nArray of size n\nOn exit, the first m elements of w will contain the eigenvalue\napproximations. ?larrd2 computes an interval Ij = (aj, bj] that includes\neigenvalue j. The eigenvalue approximation is given as the interval midpoint\nw[j-1]= (aj + bj)/2. The corresponding error is bounded by werr[j-1] =\nabs(aj - bj)/2.\nwerr\nArray of size n\nThe error bound on the corresponding eigenvalue approximation in w.\nwl, wu\nThe interval (wl, wu] contains all the wanted eigenvalues.\nIf range='V', then wl=vl and wu=vu.\nIf range='A', then wl and wu are the global Gerschgorin bounds\non the spectrum.\nIf range='I', then wl and wu are computed by SLAEBZ from the\nindex range specified.\niblock\nArray of size n\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1777\n\n\nAt each row/column j where e[j-1] is zero or small, the matrix T is\nconsidered to split into a block diagonal matrix. On exit, if info = 0,\niblock[i] specifies to which block (from 0 to the number of blocks minus\none) the eigenvalue w[i] belongs. (?larrd2 may use the remaining n-m\nelements as workspace.)\nindexw\nArray of size n\nThe indices of the eigenvalues within each block (submatrix); for example,\nindexw[i]= j and iblock[i]=k imply that the (i+1)-th eigenvalue w[i] is the\nj-th eigenvalue in block k.\ninfo\n= 0: successful exit\n< 0: if info = -i, the i-th argument had an illegal value\n> 0: some or all of the eigenvalues failed to converge or were not\ncomputed:\n•\n=1 or 3: Bisection failed to converge for some eigenvalues; these\neigenvalues are flagged by a negative block number. The effect is that\nthe eigenvalues may not be as accurate as the absolute and relative\ntolerances.\n•\n=2 or 3: range='I' only: Not all of the eigenvalues il:iu were found.\n•\n= 4: range='I', and the Gershgorin interval initially used was too small.\nNo eigenvalues were computed.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\n?larre2\nGiven a tridiagonal matrix, sets small off-diagonal\nelements to zero and for each unreduced block, finds\nbase representations and eigenvalues.\nSyntax\nvoid slarre2(char* range, MKL_INT* n, float* vl, float* vu, MKL_INT* il, MKL_INT* iu,\nfloat* d, float* e, float* e2, float* rtol1, float* rtol2, float* spltol, MKL_INT*\nnsplit, MKL_INT* isplit, MKL_INT* m, MKL_INT* dol, MKL_INT* dou, float* w, float* werr,\nfloat* wgap, MKL_INT* iblock, MKL_INT* indexw, float* gers, float* pivmin, float* work,\nMKL_INT* iwork, MKL_INT* info);\nvoid dlarre2(char* range, MKL_INT* n, double* vl, double* vu, MKL_INT* il, MKL_INT* iu,\ndouble* d, double* e, double* e2, double* rtol1, double* rtol2, double* spltol, MKL_INT*\nnsplit, MKL_INT* isplit, MKL_INT* m, MKL_INT* dol, MKL_INT* dou, double* w, double*\nwerr, double* wgap, MKL_INT* iblock, MKL_INT* indexw, double* gers, double* pivmin,\ndouble* work, MKL_INT* iwork, MKL_INT* info);\nInclude Files\n•\nmkl_scalapack.h\nDescription\nTo find the desired eigenvalues of a given real symmetric tridiagonal matrix T, ?larre2 sets, via ?larra,\n\"small\" off-diagonal elements to zero. For each block Ti, it finds\n•\na suitable shift at one end of the block's spectrum,\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1778\n\n\n•\nthe root RRR, Ti - σiI = LiDiLiT, and\n•\neigenvalues of each LiDiLiT.\nThe representations and eigenvalues found are then returned to ?stegr2 to compute the eigenvectors T.\n?larre2 is more suitable for parallel computation than the original LAPACK code for computing the root RRR\nand its eigenvalues. When computing eigenvalues in parallel and the input tridiagonal matrix splits into\nblocks, ?larre2 can skip over blocks which contain none of the eigenvalues from dol to dou for which the\nprocessor is responsible. In extreme cases (such as large matrices consisting of many blocks of small size,\ne.g. 2x2), the gain can be substantial.\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nInput Parameters\nrange\n= 'A': (\"All\") all eigenvalues will be found.\n= 'V': (\"Value\") all eigenvalues in the half-open interval (vl, vu] will be\nfound.\n= 'I': (\"Index\") eigenvalues of the entire matrix with the indices in a given\nrange will be found.\nn\nThe order of the matrix. n > 0.\nvl, vu\nIf range='V', the lower and upper bounds for the eigenvalues.\nEigenvalues less than or equal to vl, or greater than vu, will not be\nreturned. vl < vu.\nil, iu\nIf range='I', the indices (in ascending order) of the smallest eigenvalue, to\nbe returned in w[il-1], and largest eigenvalue, to be returned in w[iu-1].\n1 ≤il≤iu≤n.\nd\nArray of size n\nThe n diagonal elements of the tridiagonal matrix T.\ne\nArray of size n\nThe first (n-1) entries contain the subdiagonal elements of the tridiagonal\nmatrix T; e[n-1] need not be set.\ne2\nArray of size n\nThe first (n-1) entries contain the squares of the subdiagonal elements of\nthe tridiagonal matrix T; e2[n-1] need not be set.\nrtol1, rtol2\nParameters for bisection.\nAn interval [left, right] has converged if right-left<max( rtol1*gap,\nrtol2*max(|left|,|right|) )\nspltol\nThe threshold for splitting.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1779\n\n\ndol, dou\nSpecifying an index range dol:dou allows the user to work on only a\nselected part of the representation tree. Otherwise, the setting dol=1,\ndou=n should be applied.\nNote that dol and dou refer to the order in which the eigenvalues are\nstored in w.\nwork\nWorkspace array of size 6*n\niwork\nWorkspace array of size 5*n\nOUTPUT Parameters\nvl, vu\nIf range='I' or ='A', ?larre2 contains bounds on the desired part of the\nspectrum.\nd\nThe n diagonal elements of the diagonal matrices Di.\ne\ne contains the subdiagonal elements of the unit bidiagonal matrices Li. The\nentries e[isplit[i]], 0 ≤i<nsplit, contain the base points σi+1 on output.\ne2\nThe entries e2[isplit[i]], 0≤i<nsplit, are set to zero.\nnsplit\nThe number of blocks T splits into. 1 ≤nsplit≤n.\nisplit\nArray of size n\nThe splitting points, at which T breaks up into blocks.\nThe first block consists of rows/columns 1 to isplit[0], the second of\nrows/columns isplit[0]+1 through isplit[1], etc., and the nsplit-th\nblock consists of rows/columns isplit[nsplit-2]+1 through\nisplit[nsplit-1]=n.\nm\nThe total number of eigenvalues (of all LiDiLiT) found.\nw\nArray of size n\nThe first m elements contain the eigenvalues. The eigenvalues of each of the\nblocks, LiDiLiT, are sorted in ascending order (?larre2 may use the\nremaining n-m elements as workspace).\nNote that immediately after exiting this function, only the eigenvalues in\nwwith indices in range dol-1:dou-1 might rely on this processor when the\neigenvalue computation is done in parallel.\nwerr\nArray of size n\nThe error bound on the corresponding eigenvalue in w.\nNote that immediately after exiting this function, only the uncertainties in\nwerrwith indices in range dol-1:dou-1 might rely on this processor when\nthe eigenvalue computation is done in parallel.\nwgap\nArray of size n\nThe separation from the right neighbor eigenvalue in w.\nThe gap is only with respect to the eigenvalues of the same block as each\nblock has its own representation tree.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1780\n\n\nException: at the right end of a block we store the left gap\nNote that immediately after exiting this function, only the gaps in wgapwith\nindices in range dol-1:dou-1 might rely on this processor when the\neigenvalue computation is done in parallel.\niblock\nArray of size n\nThe indices of the blocks (submatrices) associated with the corresponding\neigenvalues in w; iblock[i]=1 if eigenvalue w[i] belongs to the first block\nfrom the top, iblock[i]=2 if w[i] belongs to the second block, and so on.\nindexw\nArray of size n\nThe indices of the eigenvalues within each block (submatrix); for example,\nindexw[i]= 10 and iblock[i]=2 imply that the (i+1)-th eigenvalue w[i] is\nthe 10th eigenvalue in block 2.\ngers\nArray of size 2*n\nThe n Gerschgorin intervals (the i-th Gerschgorin interval is (gers[2*i-2],\ngers[2*i-1])).\npivmin\nThe minimum pivot in the sturm sequence for T.\ninfo\n= 0: successful exit\n> 0: A problem occurred in ?larre2.\n< 0: One of the called functions signaled an internal problem.\nNeeds inspection of the corresponding parameter info for further\ninformation.\n=-1: Problem in ?larrd.\n=-2: Not enough internal iterations to find the base representation.\n=-3: Problem in ?larrb when computing the refined root representation\nfor ?lasq2.\n=-4: Problem in ?larrb when preforming bisection on the desired part of\nthe spectrum.\n=-5: Problem in ?lasq2\n=-6: Problem in ?lasq2\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\n?larre2a\nGiven a tridiagonal matrix, sets small off-diagonal\nelements to zero and for each unreduced block, finds\nbase representations and eigenvalues.\nSyntax\nvoid slarre2a(char* range, MKL_INT* n, float* vl, float* vu, MKL_INT* il, MKL_INT* iu,\nfloat* d, float* e, float* e2, float* rtol1, float* rtol2, float* spltol, MKL_INT*\nnsplit, MKL_INT* isplit, MKL_INT* m, MKL_INT* dol, MKL_INT* dou, MKL_INT* needil,\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1781\n\n\nMKL_INT* neediu, float* w, float* werr, float* wgap, MKL_INT* iblock, MKL_INT* indexw,\nfloat* gers, float* sdiam, float* pivmin, float* work, MKL_INT* iwork, float* minrgp,\nMKL_INT* info);\nvoid dlarre2a(char* range, MKL_INT* n, double* vl, double* vu, MKL_INT* il, MKL_INT* iu,\ndouble* d, double* e, double* e2, double* rtol1, double* rtol2, double* spltol, MKL_INT*\nnsplit, MKL_INT* isplit, MKL_INT* m, MKL_INT* dol, MKL_INT* dou, MKL_INT* needil,\nMKL_INT* neediu, double* w, double* werr, double* wgap, MKL_INT* iblock, MKL_INT*\nindexw, double* gers, double* sdiam, double* pivmin, double* work, MKL_INT* iwork,\ndouble* minrgp, MKL_INT* info);\nInclude Files\n•\nmkl_scalapack.h\nDescription\nTo find the desired eigenvalues of a given real symmetric tridiagonal matrix T, ?larre2a sets any \"small\" off-\ndiagonal elements to zero, and for each unreduced block Ti, it finds\n•\na suitable shift at one end of the block's spectrum,\n•\nthe base representation, Ti - σiI = LiDiLiT, and\n•\neigenvalues of each LiDiLiT.\nNOTE\nThe algorithm obtains a crude picture of all the wanted eigenvalues (as selected by range).\nHowever, to reduce work and improve scalability, only the eigenvalues dol to dou are\nrefined. Furthermore, if the matrix splits into blocks, RRRs for blocks that do not contain\neigenvalues from dol to dou are skipped. The DQDS algorithm (function ?lasq2) is not used,\nunlike in the sequential case. Instead, eigenvalues are computed in parallel to some figures\nusing bisection.\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nInput Parameters\nrange\n= 'A': (\"All\") all eigenvalues will be found.\n= 'V': (\"Value\") all eigenvalues in the half-open interval (vl, vu] will be\nfound.\n= 'I': (\"Index\") eigenvalues of the entire matrix with the indices in a given\nrange will be found.\nn\nThe order of the matrix. n > 0.\nvl, vu\nIf range='V', the lower and upper bounds for the eigenvalues. Eigenvalues\nless than or equal to vl, or greater than vu, will not be returned. vl < vu.\nIf range='I' or ='A', ?larre2a computes bounds on the desired part of the\nspectrum.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1782\n\n\nil, iu\nIf range='I', the indices (in ascending order) of the smallest eigenvalue, to\nbe returned in w[il-1], and largest eigenvalue, to be returned in w[iu-1].\n1 ≤il≤iu≤n.\nd\nArray of size n\nOn entry, the n diagonal elements of the tridiagonal matrix T.\ne\nArray of size n\nThe first (n-1) entries contain the subdiagonal elements of the tridiagonal\nmatrix T; e[n-1] need not be set.\ne2\nArray of size n\nThe first (n-1) entries contain the squares of the subdiagonal elements of\nthe tridiagonal matrix T; e2[n-1] need not be set.\nrtol1, rtol2\nParameters for bisection.\nAn interval [left,right] has converged if right - left < max( rtol1*gap,\nrtol2*max(|left|,|right|) )\nspltol\nThe threshold for splitting.\ndol, dou\nIf the user wants to work on only a selected part of the representation tree,\nhe can specify an index range dol:dou.\nOtherwise, the setting dol=1, dou=n should be applied.\nNote that dol and dou refer to the order in which the eigenvalues are\nstored in w.\nwork\nWorkspace array of size 6*n\niwork\nWorkspace array of size 5*n\nminrgp\nThe minimum relative gap threshold to decide whether an eigenvalue or a\ncluster boundary is reached.\nOUTPUT Parameters\nvl, vu\nIf range='V', the lower and upper bounds for the eigenvalues. Eigenvalues\nless than or equal to vl, or greater than vu, are not returned. vl < vu.\nIf range='I' or range='A', ?larre2a computes bounds on the desired part\nof the spectrum.\nd\nThe n diagonal elements of the diagonal matrices Di.\ne\ne contains the subdiagonal elements of the unit bidiagonal matrices Li. The\nentries e[isplit[i]], 0 ≤i<nsplit, contain the base points σi+1 on output.\ne2\nThe entries e2[isplit[i ]], 0≤i<nsplit have been set to zero.\nnsplit\nThe number of blocks T splits into. 1 ≤nsplit≤n.\nisplit\nArray of size n\nThe splitting points, at which T breaks up into blocks.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1783\n\n\nThe first block consists of rows/columns 1 to isplit[0], the second of\nrows/columns isplit[0]+1 through isplit[1], etc., and the nsplit-th\nblock consists of rows/columns isplit[nsplit-2]+1 through\nisplit[nsplit-1]=n.\nm\nThe total number of eigenvalues (of all LiDiLiT) found.\nneedil, neediu\nThe indices of the leftmost and rightmost eigenvalues of the root node RRR\nwhich are needed to accurately compute the relevant part of the\nrepresentation tree.\nw\nArray of size n\nThe first m elements contain the eigenvalues. The eigenvalues of each of the\nblocks, LiDiLiT, are sorted in ascending order ( ?larre2a may use the\nremaining n-m elements as workspace).\nNote that immediately after exiting this function, only the eigenvalues in\nwwith indices in range dol-1:dou-1 rely on this processor because the\neigenvalue computation is done in parallel.\nwerr\nArray of size n\nThe error bound on the corresponding eigenvalue in w.\nNote that immediately after exiting this function, only the uncertainties in\nwerrwith indices in range dol-1:dou-1 are reliable on this processor\nbecause the eigenvalue computation is done in parallel.\nwgap\nArray of size n\nThe separation from the right neighbor eigenvalue in w. The gap is only with\nrespect to the eigenvalues of the same block as each block has its own\nrepresentation tree.\nException: at the right end of a block we store the left gap\nNote that immediately after exiting this function, only the gaps in wgapwith\nindices in range dol-1:dou-1 are reliable on this processor because the\neigenvalue computation is done in parallel.\niblock\nArray of size n\nThe indices of the blocks (submatrices) associated with the corresponding\neigenvalues in w; iblock[i]=1 if eigenvalue w[i] belongs to the first block\nfrom the top, iblock[i]=2 if w[i] belongs to the second block, and so on.\nindexw\nArray of size n\nThe indices of the eigenvalues within each block (submatrix); for example,\nindexw[i]= 10 and iblock[i]=2 imply that the (i+1)-th eigenvalue w[i] is\nthe 10th eigenvalue in block 2.\ngers\nArray of size 2*n\nThe n Gerschgorin intervals (the i-th Gerschgorin interval is (gers[2*i-2],\ngers[2*i-1])).\npivmin\nThe minimum pivot in the sturm sequence for T.\ninfo\n= 0: successful exit\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1784\n\n\n> 0: A problem occurred in ?larre2a.\n< 0: One of the called functions signaled an internal problem. Needs\ninspection of the corresponding parameter info for further information.\n=-1: Problem in ?larrd2.\n=-2: Not enough internal iterations to find base representation.\n=-3: Problem in ?larrb2 when computing the refined root representation.\n=-4: Problem in ?larrb2 when preforming bisection on the desired part of\nthe spectrum.\n= -9 Problem: m < dou-dol+1, that is the code found fewer eigenvalues\nthan it was supposed to.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\n?larrf2\nFinds a new relatively robust representation such that\nat least one of the eigenvalues is relatively isolated.\nSyntax\nvoid slarrf2(MKL_INT* n, float* d, float* l, float* ld, MKL_INT* clstrt, MKL_INT* clend,\nMKL_INT* clmid1, MKL_INT* clmid2, float* w, float* wgap, float* werr, MKL_INT* trymid,\nfloat* spdiam, float* clgapl, float* clgapr, float* pivmin, float* sigma, float* dplus,\nfloat* lplus, float* work, MKL_INT* info);\nvoid dlarrf2(MKL_INT* n, double* d, double* l, double* ld, MKL_INT* clstrt, MKL_INT*\nclend, MKL_INT* clmid1, MKL_INT* clmid2, double* w, double* wgap, double* werr, MKL_INT*\ntrymid, double* spdiam, double* clgapl, double* clgapr, double* pivmin, double* sigma,\ndouble* dplus, double* lplus, double* work, MKL_INT* info);\nInclude Files\n•\nmkl_scalapack.h\nDescription\nGiven the initial representation LDLT and its cluster of close eigenvalues (in a relative measure), defined by\nthe indices of the first and last eigenvalues in the cluster, ?larrf2 finds a new relatively robust\nrepresentation LDLT - σ I = L+D+L+T such that at least one of the eigenvalues of L+D+L+T is relatively\nisolated.\nThis is an enhanced version of ?larrf that also tries shifts in the middle of the cluster, should there be a\nlarge gap, in order to break large clusters into at least two pieces.\nInput Parameters\nn\nThe order of the matrix (subblock, if the matrix was split).\nd\nArray of size n\nThe n diagonal elements of the diagonal matrix D.\nl\nArray of size n-1\nThe (n-1) subdiagonal elements of the unit bidiagonal matrix L.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1785\n\n\nld\nArray of size n-1\nThe (n-1) elements l[i]*d[i].\nclstrt\nThe index of the first eigenvalue in the cluster.\nclend\nThe index of the last eigenvalue in the cluster.\nclmid1, clmid2\nThe index of a middle eigenvalue pair with large gap.\nw\nArray of size ≥ (clend-clstrt+1)\nThe eigenvalue approximations of LD LT in ascending order. w[clstrt - 1]\nthrough w[clend - 1] form the cluster of relatively close eigenalues.\nwgap\nArray of size ≥ (clend-clstrt+1)\nThe separation from the right neighbor eigenvalue in w.\nwerr\nArray of size ≥ (clend-clstrt+1)\nwerr contains the semiwidth of the uncertainty interval of the\ncorresponding eigenvalue approximation in w.\nspdiam\nEstimate of the spectral diameter obtained from the Gerschgorin intervals\nclgapl, clgapr\nAbsolute gap on each end of the cluster.\nSet by the calling function to protect against shifts too close to eigenvalues\noutside the cluster.\npivmin\nThe minimum pivot allowed in the Sturm sequence.\nwork\nWorkspace array of size 2*n\nOUTPUT Parameters\nwgap\nContains refined values of its input approximations. Very small gaps are\nunchanged.\nsigma\nThe shift (σ) used to form L+D+L+T.\ndplus\nArray of size n\nThe n diagonal elements of the diagonal matrix D+.\nlplus\nArray of size n-1\nThe first (n-1) elements of lplus contain the subdiagonal elements of the\nunit bidiagonal matrix L+.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\n?larrv2\nComputes the eigenvectors of the tridiagonal matrix T\n= L*D*LT given L, D and the eigenvalues of L*D*LT.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1786\n\n\nSyntax\nvoid slarrv2(MKL_INT* n, float* vl, float* vu, float* d, float* l, float* pivmin,\nMKL_INT* isplit, MKL_INT* m, MKL_INT* dol, MKL_INT* dou, MKL_INT* needil, MKL_INT*\nneediu, float* minrgp, float* rtol1, float* rtol2, float* w, float* werr, float* wgap,\nMKL_INT* iblock, MKL_INT* indexw, float* gers, float* sdiam, float* z, MKL_INT* ldz,\nMKL_INT* isuppz, float* work, MKL_INT* iwork, MKL_INT* vstart, MKL_INT* finish,\nMKL_INT* maxcls, MKL_INT* ndepth, MKL_INT* parity, MKL_INT* zoffset, MKL_INT* info);\nvoid dlarrv2(MKL_INT* n, double* vl, double* vu, double* d, double* l, double* pivmin,\nMKL_INT* isplit, MKL_INT* m, MKL_INT* dol, MKL_INT* dou, MKL_INT* needil, MKL_INT*\nneediu, double* minrgp, double* rtol1, double* rtol2, double* w, double* werr, double*\nwgap, MKL_INT* iblock, MKL_INT* indexw, double* gers, double* sdiam, double* z, MKL_INT*\nldz, MKL_INT* isuppz, double* work, MKL_INT* iwork, MKL_INT* vstart, MKL_INT* finish,\nMKL_INT* maxcls, MKL_INT* ndepth, MKL_INT* parity, MKL_INT* zoffset, MKL_INT* info);\nInclude Files\n•\nmkl_scalapack.h\nDescription\n?larrv2 computes the eigenvectors of the tridiagonal matrix T = LDLT given L, D and approximations to the\neigenvalues of LDLT. The input eigenvalues should have been computed by larre2a or by previous calls\nto ?larrv2.\nThe major difference between the parallel and the sequential construction of the representation tree is that in\nthe parallel case, not all eigenvalues of a given cluster might be computed locally. Other processors might\n\"own\" and refine part of an eigenvalue cluster. This is crucial for scalability. Thus there might be\ncommunication necessary before the current level of the representation tree can be parsed.\nPlease note:\n•\nThe calling sequence has two additional integer parameters, dol and dou, that should satisfy\nm≥dou≥dol≥1. These parameters are only relevant when both eigenvalues and eigenvectors are computed\n(stegr2b parameter jobz = 'V'). ?larrv2 only computes the eigenvectors corresponding to eigenvalues\ndol through dou in w. (That is, instead of computing the eigenvectors belonging to w[0] through w[m-1],\nonly the eigenvectors belonging to eigenvalues w[dol - 1] through w[dou -1] are computed. In this case,\nonly the eigenvalues dol:dou are guaranteed to be accurately refined to all figures by Rayleigh-Quotient\niteration.\n•\nThe additional arguments vstart, finish, ndepth, parity, zoffset are included as a thread-safe\nimplementation equivalent to save variables. These variables store details about the local representation\ntree which is computed layerwise. For scalability reasons, eigenvalues belonging to the locally relevant\nrepresentation tree might be computed on other processors. These need to be communicated before the\ninspection of the RRRs can proceed on any given layer. Note that only when the variable finish is non-\nzero, the computation has ended. All eigenpairs between dol and dou have been computed. m is set to\ndou - dol + 1.\n•\n?larrv2 needs more workspace in z than the sequential slarrv. It is used to store the conformal\nembedding of the local representation tree.\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1787\n\n\nInput Parameters\nn\nThe order of the matrix. n≥ 0.\nvl, vu\nLower and upper bounds of the interval that contains the desired\neigenvalues. vl < vu. Needed to compute gaps on the left or right end of\nthe extremal eigenvalues in the desired range. vu is currently not used but\nkept as parameter in case needed.\nd\nArray of size n\nThe n diagonal elements of the diagonal matrix d. On exit, d is overwritten.\nl\nArray of size n\nThe (n-1) subdiagonal elements of the unit bidiagonal matrix L are in\nelements 0 to n-2 of l (if the matrix is not split.) At the end of each block is\nstored the corresponding shift as given by ?larre. On exit, l is\noverwritten.\npivmin\nThe minimum pivot allowed in the sturm sequence.\nisplit\nArray of size n\nThe splitting points, at which the matrix T breaks up into blocks. The first\nblock consists of rows/columns 1 to isplit[ 0 ], the second of rows/\ncolumns isplit[ 0 ] + 1 through isplit[ 1 ], etc.\nm\nThe total number of input eigenvalues. 0 ≤m≤n.\ndol, dou\nIf you want to compute only selected eigenvectors from all the eigenvalues\nsupplied, you can specify an index range dol:dou. Or else the setting\ndol=1, dou=m should be applied. Note that dol and dou refer to the order\nin which the eigenvalues are stored in w. If you want to compute only\nselected eigenpairs, the columns dol-1 to dou+1 of the eigenvector space\nZ contain the computed eigenvectors. All other columns of Z are set to\nzero.\nIf dol > 1, then Z(:,dol-1-zoffset) is used.\nIf dou < m, then Z(:,dou+1-zoffset) is used.\nneedil, neediu\nDescribe which are the left and right outermost eigenvalues that still need\nto be included in the computation. These indices indicate whether\neigenvalues from other processors are needed to correctly compute the\nconformally embedded representation tree.\nWhen dol≤needil≤neediu≤dou, all required eigenvalues are local to the\nprocessor and no communication is required to compute its part of the\nrepresentation tree.\nminrgp\nThe minimum relative gap threshold to decide whether an eigenvalue or a\ncluster boundary is reached.\nrtol1, rtol2\nParameters for bisection. An interval [left,right] has converged if right-left <\nmax( rtol1*gap, rtol2*max(|left|,|right|) )\nw\nArray of size n\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1788\n\n\nThe first m elements of w contain the approximate eigenvalues for which\neigenvectors are to be computed. The eigenvalues should be grouped by\nsplit-off block and ordered from smallest to largest within the block. (The\noutput array w from ?stegr2a is expected here.) Furthermore, they are\nwith respect to the shift of the corresponding root representation for their\nblock.\nwerr\nArray of size n\nThe first m elements contain the semiwidth of the uncertainty interval of the\ncorresponding eigenvalue in w.\nwgap\nArray of size n\nThe separation from the right neighbor eigenvalue in w.\niblock\nArray of size n\nThe indices of the blocks (submatrices) associated with the corresponding\neigenvalues in w; iblock[i]=1 if eigenvalue w[i] belongs to the first block\nfrom the top, iblock[i]=2 if w[i] belongs to the second block, and so on.\nindexw\nArray of size n\nThe indices of the eigenvalues within each block (submatrix). For example:\nindexw[i]= 10 and iblock[i]=2 imply that the (i+1)-th eigenvalue w[i] is\nthe 10th eigenvalue in block 2.\ngers\nArray of size 2*n\nThe n Gerschgorin intervals (the i-th Gerschgorin interval is (gers[2*i-2],\ngers[2*i-1])). The Gerschgorin intervals should be computed from the\noriginal unshifted matrix.\nNot used but kept as parameter for possible future use.\nsdiam\nArray of size n\nThe spectral diameters for all unreduced blocks.\nldz\nThe leading dimension of the array z. ldz≥ 1, and if stegr2b parameter\njobz = 'V', ldz≥ max(1,n).\nwork\n(workspace) array of size 12*n\niwork\n(workspace) Array of size 7*n\nvstart\nNon-zero on initialization, set to zero afterwards.\nfinish\nA flag that indicates whether all eigenpairs have been computed.\nmaxcls\nThe largest cluster worked on by this processor in the representation tree.\nndepth\nThe current depth of the representation tree. Set to zero on initial pass,\nchanged when the deeper levels of the representation tree are generated.\nparity\nAn internal parameter needed for the storage of the clusters on the current\nlevel of the representation tree.\nzoffset\nOffset for storing the eigenpairs when z is distributed in 1D-cyclic fashion.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1789\n\n\nOUTPUT Parameters\nneedil, neediu\nw\nUnshifted eigenvalues for which eigenvectors have already been computed.\nwerr\nContains refined values of its input approximations.\nwgap\nContains refined values of its input approximations. Very small gaps are\nchanged.\nz\nArray of size ldz * max(1,m)\nIf info = 0, the first m columns of the matrix Z, stored in the array z,\ncontain the orthonormal eigenvectors of the matrix T corresponding to the\ninput eigenvalues, with the i-th column of Z holding the eigenvector\nassociated with w[i - 1].\nIn the distributed version, only a subset of columns is accessed, see dol,\ndou, and zoffset.\nisuppz\nArray of size 2*max(1,m)\nThe support of the eigenvectors in z, i.e., the indices indicating the non-\nzero elements in z. The i-th eigenvector is non-zero only in elements\nisuppz[ 2*i-2 ] through isuppz[ 2*i-1 ].\nvstart\nNon-zero on initialization, set to zero afterwards.\nfinish\nA flag that indicates whether all eigenpairs have been computed.\nmaxcls\nThe largest cluster worked on by this processor in the representation tree.\nndepth\nThe current depth of the representation tree. Set to zero on initial pass,\nchanged when the deeper levels of the representation tree are generated.\nparity\nAn internal parameter needed for the storage of the clusters on the current\nlevel of the representation tree.\ninfo\n= 0: successful exit\n> 0: A problem occured in ?larrv2.\n< 0: One of the called functions signaled an internal problem.\nNeeds inspection of the corresponding parameter info for further\ninformation.\n=-1: Problem in ?larrb2 when refining a child's eigenvalues.\n=-2: Problem in ?larrf2 when computing the RRR of a child. When a child\nis inside a tight cluster, it can be difficult to find an RRR. A partial remedy\nfrom the user's point of view is to make the parameter minrgp smaller and\nrecompile. However, as the orthogonality of the computed vectors is\nproportional to 1/minrgp, be aware that decreasing minrgp might be\nreduce precision.\n=-3: Problem in ?larrb2 when refining a single eigenvalue after the\nRayleigh correction was rejected.\n= 5: The Rayleigh Quotient Iteration failed to converge to full accuracy.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1790\n\n\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\n?lasorte\nSorts eigenpairs by real and complex data types.\nSyntax\nvoid slasorte (float *s , MKL_INT *lds , MKL_INT *j , float *out , MKL_INT *info );\nvoid dlasorte (double *s , MKL_INT *lds , MKL_INT *j , double *out , MKL_INT *info );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe ?lasortefunction sorts eigenpairs so that real eigenpairs are together and complex eigenpairs are\ntogether. This helps to employ 2x2 shifts easily since every second subdiagonal is guaranteed to be zero. This\nfunction does no parallel work and makes no calls.\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nInput Parameters\ns\n(local)\nArray of size lds.\nOn entry, a matrix already in Schur form.\nlds\n(local)\nOn entry, the leading dimension of the array s; unchanged on exit.\nj\n(local)\nOn entry, the order of the matrix S; unchanged on exit.\nout\n(local)\nArray of size 2*j. The work buffer required by the function.\ninfo\n(local)\nSet, if the input matrix had an odd number of real eigenvalues and things\ncould not be paired or if the input matrix S was not originally in Schur form.\n0 indicates successful completion.\nOutput Parameters\ns\nOn exit, the diagonal blocks of S have been rewritten to pair the\neigenvalues. The resulting matrix is no longer similar to the input.\nout\nWork buffer.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1791\n\n\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\n?lasrt2\nSorts numbers in increasing or decreasing order.\nSyntax\nvoid slasrt2 (char *id , MKL_INT *n , float *d , MKL_INT *key , MKL_INT *info );\nvoid dlasrt2 (char *id , MKL_INT *n , double *d , MKL_INT *key , MKL_INT *info );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe ?lasrt2function is modified LAPACK function ?lasrt, which sorts the numbers in d in increasing order\n(if id = 'I') or in decreasing order (if id = 'D' ). It uses Quick Sort, reverting to Insertion Sort on arrays\nof size ≤ 20. The size of STACK limits n to about 232.\nInput Parameters\nid\n= 'I': sort d in increasing order;\n= 'D': sort d in decreasing order.\nn\nThe length of the array d.\nd\nArray of size n.\nOn entry, the array to be sorted.\nkey\nArray of size n.\nOn entry, key contains a key to each of the entries in d.\nTypically, key[i]= i+1 for all i = 0, ..., n-1.\nOutput Parameters\nd\nOn exit, d has been sorted into increasing order\n(d[0] ≤ ... ≤ d[n - 1] )\nor into decreasing order\n(d[0] ≥ ... ≥ d[n - 1] ),\ndepending on id.\ninfo\n= 0: successful exit\n< 0: if info = -i, the i-th argument had an illegal value.\nkey\nOn exit, key is permuted in exactly the same manner as d was permuted\nfrom input to output. Therefore, if key[i] = i+1 for all i =0, ..., n-1 on input,\nd[i] on output equals d[key[i]-1] on input.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1792\n\n\n?stegr2\nComputes selected eigenvalues and eigenvectors of a\nreal symmetric tridiagonal matrix.\nSyntax\nvoid sstegr2(char* jobz, char* range, MKL_INT* n, float* d, float* e, float* vl, float*\nvu, MKL_INT* il, MKL_INT* iu, MKL_INT* m, float* w, float* z, MKL_INT* ldz, MKL_INT*\nnzc, MKL_INT* isuppz, float* work, MKL_INT* lwork, MKL_INT* iwork, MKL_INT* liwork,\nMKL_INT* dol, MKL_INT* dou, MKL_INT* zoffset, MKL_INT* info);\nvoid dstegr2(char* jobz, char* range, MKL_INT* n, double* d, double* e, double* vl,\ndouble* vu, MKL_INT* il, MKL_INT* iu, MKL_INT* m, double* w, double* z, MKL_INT* ldz,\nMKL_INT* nzc, MKL_INT* isuppz, double* work, MKL_INT* lwork, MKL_INT* iwork, MKL_INT*\nliwork, MKL_INT* dol, MKL_INT* dou, MKL_INT* zoffset, MKL_INT* info);\nInclude Files\n•\nmkl_scalapack.h\nDescription\n?stegr2 computes selected eigenvalues and, optionally, eigenvectors of a real symmetric tridiagonal matrix\nT. It is invoked in the ScaLAPACK MRRR driver p?syevr and the corresponding Hermitian version either when\nonly eigenvalues are to be computed, or when only a single processor is used (the sequential-like case).\n?stegr2 has been adapted from LAPACK's ?stegr. Please note the following crucial changes.\n1.\nThe calling sequence has two additional integer parameters, dol and dou, that should satisfy\nm≥dou≥dol≥1. ?stegr2only computes the eigenpairs corresponding to eigenvalues dol through dou in\nw, indexed dol-1 through dou-1. (That is, instead of computing the eigenpairs belonging to w[0]\nthrough w[m-1], only the eigenvectors belonging to eigenvalues w[dol-1] through w[dou-1] are\ncomputed. In this case, only the eigenvalues dol through dou are guaranteed to be fully accurate.\n2.\nm is not the number of eigenvalues specified by range, but is m = dou - dol + 1. This concerns the\ncase where only eigenvalues are computed, but on more than one processor. Thus, in this case m refers\nto the number of eigenvalues computed on this processor.\n3.\nThe arrays w and z might not contain all the wanted eigenpairs locally, instead this information is\ndistributed over other processors.\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nInput Parameters\njobz\n= 'N': Compute eigenvalues only;\n= 'V': Compute eigenvalues and eigenvectors.\nrange\n= 'A': all eigenvalues will be found.\n= 'V': all eigenvalues in the half-open interval (vl,vu] will be found.\n= 'I': eigenvalues of the entire matrix with the indices in a given range will\nbe found.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1793\n\n\nn\nThe order of the matrix. n≥ 0.\nd\nArray of size n\nOn entry, the n diagonal elements of the tridiagonal matrix T. Overwritten\non exit.\ne\nArray of size n\nOn entry, the (n-1) subdiagonal elements of the tridiagonal matrix T in\nelements 0 to n-2 of e. e[n-1] need not be set on input, but is used\ninternally as workspace. Overwritten on exit.\nvl\nvu\nIf range='V', the lower and upper bounds of the interval to be searched for\neigenvalues. vl < vu.\nNot referenced if range = 'A' or 'I'.\nil, iu\nIf range='I', the indices (in ascending order) of the smallest eigenvalue, to\nbe returned in w[il-1], and largest eigenvalue, to be returned in w[iu-1].\n1 ≤il≤iu≤n, if n > 0.\nNot referenced if range = 'A' or 'V'.\nldz\nThe leading dimension of the array z. ldz≥ 1, and if jobz = 'V', then ldz≥\nmax(1,n).\nnzc\nThe number of eigenvectors to be held in the array z, storing the matrix Z.\nIf range = 'A', then nzc≥ max(1,n).\nIf range = 'V', then nzc≥ the number of eigenvalues in (vl,vu].\nIf range = 'I', then nzc≥iu-il+1.\nIf nzc = -1, then a workspace query is assumed; the function calculates the\nnumber of columns of the matrix Z that are needed to hold the\neigenvectors. This value is returned as the first entry of the z array, and no\nerror message related to nzc is issued.\nlwork\nThe size of the array work. lwork≥ max(1,18*n)\nif jobz = 'V', and lwork≥ max(1,12*n) if jobz = 'N'. If lwork = -1, then a\nworkspace query is assumed; the function only calculates the optimal size\nof the work array, returns this value as the first entry of the work array,\nand no error message related to lwork is issued.\nliwork\nThe size of the array iwork. liwork≥ max(1,10*n) if the eigenvectors are\ndesired, and liwork≥ max(1,8*n) if only the eigenvalues are to be\ncomputed.\nIf liwork = -1, then a workspace query is assumed; the function only\ncalculates the optimal size of the iwork array, returns this value as the first\nentry of the iwork array, and no error message related to liwork is issued.\ndol, dou\nFrom the eigenvalues w[0] through w[m-1], only eigenvectors Z(:,dol) to\nZ(:,dou) are computed.\nIf dol > 1, then Z(:,dol-1-zoffset) is used and overwritten.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1794\n\n\nIf dou < m, then Z(:,dou+1-zoffset) is used and overwritten.\nzoffset\nOffset for storing the eigenpairs when z is distributed in 1D-cyclic fashion\nOUTPUT Parameters\nm\nGlobally summed over all processors, m equals the total number of\neigenvalues found. 0 ≤m≤n. If range = 'A', m = n, and if range = 'I', m = iu-\nil+1. The local output equals m = dou - dol + 1.\nw\nArray of size n\nThe first m elements contain the selected eigenvalues in ascending order.\nNote that immediately after exiting this function, only the eigenvalues\nindexed dol-1 through dou-1 are reliable on this processor because the\neigenvalue computation is done in parallel. Other processors will hold\nreliable information on other parts of the w array. This information is\ncommunicated in the ScaLAPACK driver.\nz\nArray of size ldz * max(1,m).\nIf jobz = 'V', and if info = 0, then the first m columns of the matrix Z\nstored in z contain some of the orthonormal eigenvectors of the matrix T\ncorresponding to the selected eigenvalues, with the i-th column of Z holding\nthe eigenvector associated with w[i-1].\nIf jobz = 'N', then z is not referenced.\nNote: the user must ensure that at least max(1,m) columns of the matrix\nare supplied in the array z; if range = 'V', the exact value of m is not known\nin advance and can be computed with a workspace query by setting nzc =\n-1, see below.\nisuppz\narray of size 2*max(1,m)\nThe support of the eigenvectors in z, i.e., the indices indicating the nonzero\nelements in z. The i-th computed eigenvector is nonzero only in elements\nisuppz[ 2*i-2 ] through isuppz[ 2*i -1]. This is relevant in the case when\nthe matrix is split. isuppz is only set if n>2.\nwork\nOn exit, if info = 0, work[0] returns the optimal (and minimal) lwork.\niwork\nOn exit, if info = 0, iwork[0] returns the optimal liwork.\ninfo\nOn exit, info\n= 0: successful exit\nother:if info = -i, the i-th argument had an illegal value\nif info = 10X, internal error in ?larre2,\nif info = 20X, internal error in ?larrv.\nHere, the digit X = ABS( iinfo ) < 10, where iinfo is the nonzero error\ncode returned by ?larre2 or ?larrv, respectively.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1795\n\n\n?stegr2a\nComputes selected eigenvalues and initial\nrepresentations needed for eigenvector computations.\nSyntax\nvoid sstegr2a(char* jobz, char* range, MKL_INT* n, float* d, float* e, float* vl, float*\nvu, MKL_INT* il, MKL_INT* iu, MKL_INT* m, float* w, float* z, MKL_INT* ldz, MKL_INT*\nnzc, float* work, MKL_INT* lwork, MKL_INT* iwork, MKL_INT* liwork, MKL_INT* dol,\nMKL_INT* dou, MKL_INT* needil, MKL_INT* neediu, MKL_INT* inderr, MKL_INT* nsplit,\nfloat* pivmin, float* scale, float* wl, float* wu, MKL_INT* info);\nvoid dstegr2a(char* jobz, char* range, MKL_INT* n, double* d, double* e, double* vl,\ndouble* vu, MKL_INT* il, MKL_INT* iu, MKL_INT* m, double* w, double* z, MKL_INT* ldz,\nMKL_INT* nzc, double* work, MKL_INT* lwork, MKL_INT* iwork, MKL_INT* liwork, MKL_INT*\ndol, MKL_INT* dou, MKL_INT* needil, MKL_INT* neediu, MKL_INT* inderr, MKL_INT* nsplit,\ndouble* pivmin, double* scale, double* wl, double* wu, MKL_INT* info);\nInclude Files\n•\nmkl_scalapack.h\nDescription\n?stegr2a computes selected eigenvalues and initial representations needed for eigenvector computations\nin ?stegr2b. It is invoked in the ScaLAPACK MRRR driver p?syevr and the corresponding Hermitian version\nwhen both eigenvalues and eigenvectors are computed in parallel on multiple processors. For this\ncase, ?stegr2a implements the first part of the MRRR algorithm, parallel eigenvalue computation and finding\nthe root RRR. At the end of ?stegr2a, other processors might have a part of the spectrum that is needed to\ncontinue the computation locally. Once this eigenvalue information has been received by the processor, the\ncomputation can then proceed by calling the second part of the parallel MRRR algorithm, ?stegr2b.\nPlease note:\n•\nThe calling sequence has two additional integer parameters, (compared to LAPACK's stegr), these are\ndol and dou and should satisfy m≥dou≥dol≥1. These parameters are only relevant for the case jobz = 'V'.\nGlobally invoked over all processors, ?stegr2a computes all the eigenvalues specified by range.\n?stegr2a locally only computes the eigenvalues corresponding to eigenvalues dol through dou in w,\nindexed dol-1 through dou-1. (That is, instead of computing the eigenvectors belonging to w([0] through\nw[m-1], only the eigenvectors belonging to eigenvalues w[dol-1] through w[dou-1] are computed. In this\ncase, only the eigenvalues dol through dou are guaranteed to be fully accurate.\n•\nm is not the number of eigenvalues specified by range, but it is m = dou - dol + 1. Instead, m refers to\nthe number of eigenvalues computed on this processor.\n•\nWhile no eigenvectors are computed in ?stegr2a itself (this is done later in ?stegr2b), the interface\nIf jobz = 'V' then, depending on range and dol, dou, ?stegr2a might need more workspace in z then\nthe original ?stegr. In particular, the arrays w and z might not contain all the wanted eigenpairs locally,\ninstead this information is distributed over other processors.\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1796\n\n\nInput Parameters\njobz\n= 'N': Compute eigenvalues only;\n= 'V': Compute eigenvalues and eigenvectors.\nrange\n= 'A': all eigenvalues will be found.\n= 'V': all eigenvalues in the half-open interval (vl,vu] will be found.\n= 'I': eigenvalues of the entire matrix with the indices in a given range will\nbe found.\nn\nThe order of the matrix. n≥ 0.\nd\nArray of size n\nThe n diagonal elements of the tridiagonal matrix T. Overwritten on exit.\ne\nArray of size n\nOn entry, the (n-1) subdiagonal elements of the tridiagonal matrix T in\nelements 0 to n-2 of e. e[n-1] need not be set on input, but is used\ninternally as workspace. Overwritten on exit.\nvl, vu\nIf range='V', the lower and upper bounds of the interval to be searched for\neigenvalues. vl < vu.\nNot referenced if range = 'A' or 'I'.\nil, iu\nIf range='I', the indices (in ascending order) of the smallest eigenvalue, to\nbe returned in w[il-1], and largest eigenvalue, to be returned in w[iu-1]. 1\n≤il≤iu≤n, if n > 0.\nNot referenced if range = 'A' or 'V'.\nldz\nThe leading dimension of the array z. ldz≥ 1, and if jobz = 'V', then ldz≥\nmax(1,n).\nnzc\nThe number of eigenvectors to be held in the array z.\nIf range = 'A', then nzc≥ max(1,n).\nIf range = 'V', then nzc≥ the number of eigenvalues in (vl,vu].\nIf range = 'I', then nzc≥iu-il+1.\nIf nzc = -1, then a workspace query is assumed; the function calculates the\nnumber of columns of the matrix stored in array z that are needed to hold\nthe eigenvectors. This value is returned as the first entry of the z array, and\nno error message related to nzc is issued.\nlwork\nThe size of the array work. lwork≥ max(1,18*n) if jobz = 'V', and lwork≥\nmax(1,12*n) if jobz = 'N'.\nIf lwork = -1, then a workspace query is assumed; the function only\ncalculates the optimal size of the work array, returns this value as the first\nentry of the work array, and no error message related to lwork is issued.\nliwork\nThe size of the array iwork. liwork≥ max(1,10*n) if the eigenvectors are\ndesired, and liwork≥ max(1,8*n) if only the eigenvalues are to be\ncomputed.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1797\n\n\nIf liwork = -1, then a workspace query is assumed; the function only\ncalculates the optimal size of the iwork array, returns this value as the first\nentry of the iwork array, and no error message related to liwork is issued.\ndol, dou\nFrom all the eigenvalues w[0] through w[m-1], only eigenvalues w[dol-1]\nthrough w[dou-1] are computed.\nOUTPUT Parameters\nm\nGlobally summed over all processors, m equals the total number of\neigenvalues found. 0 ≤m≤n.\nIf range = 'A', m = n, and if range = 'I', m = iu-il+1.\nThe local output equals m = dou - dol + 1.\nw\nArray of size n\nThe first m elements contain approximations to the selected eigenvalues in\nascending order. Note that immediately after exiting this function, only the\neigenvalues indexed dol-1 through dou-1 are reliable on this processor\nbecause the eigenvalue computation is done in parallel. The other entries\nare very crude preliminary approximations. Other processors hold reliable\ninformation on these other parts of the w array.\nThis information is communicated in the ScaLAPACK driver.\nz\nArray of size ldz * max(1,m).\n?stegr2a does not compute eigenvectors, this is done in ?stegr2b. The\nargument z as well as all related other arguments only appear to keep the\ninterface consistent and to signal to the user that this function is meant to\nbe used when eigenvectors are computed.\nwork\nOn exit, if info = 0, work[0] returns the optimal (and minimal) lwork.\niwork\nOn exit, if info = 0, iwork[0] returns the optimal liwork.\nneedil, neediu\nThe indices of the leftmost and rightmost eigenvalues needed to accurately\ncompute the relevant part of the representation tree. This information can\nbe used to find out which processors have the relevant eigenvalue\ninformation needed so that it can be communicated.\ninderr\ninderr points to the place in the work space where the eigenvalue\nuncertainties (errors) are stored.\nnsplit\nThe number of blocks into which T splits. 1 ≤nsplit≤n.\npivmin\nThe minimum pivot in the sturm sequence for T.\nscale\nThe scaling factor for the tridiagonal T.\nwl, wu\nThe interval (wl, wu] contains all the wanted eigenvalues.\nIt is either given by the user or computed in ?larre2a.\ninfo\nOn exit, info = 0: successful exit\nother: if info = -i, the i-th argument had an illegal value\nif info = 10x, internal error in ?larre2a,\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1798\n\n\nHere, the digit x = abs( iinfo ) < 10, where iinfo is the nonzero error\ncode returned by ?larre2a.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\n?stegr2b\nFrom eigenvalues and initial representations computes\nthe selected eigenvalues and eigenvectors of the real\nsymmetric tridiagonal matrix in parallel on multiple\nprocessors.\nSyntax\nvoid sstegr2b(char* jobz, MKL_INT* n, float* d, float* e, MKL_INT* m, float* w, float*\nz, MKL_INT* ldz, MKL_INT* nzc, MKL_INT* isuppz, float* work, MKL_INT* lwork, MKL_INT*\niwork, MKL_INT* liwork, MKL_INT* dol, MKL_INT* dou, MKL_INT* needil, MKL_INT* neediu,\nMKL_INT* indwlc, float* pivmin, float* scale, float* wl, float* wu, MKL_INT* vstart,\nMKL_INT* finish, MKL_INT* maxcls, MKL_INT* ndepth, MKL_INT* parity, MKL_INT* zoffset,\nMKL_INT* info);\nvoid dstegr2b(char* jobz, MKL_INT* n, double* d, double* e, MKL_INT* m, double* w,\ndouble* z, MKL_INT* ldz, MKL_INT* nzc, MKL_INT* isuppz, double* work, MKL_INT* lwork,\nMKL_INT* iwork, MKL_INT* liwork, MKL_INT* dol, MKL_INT* dou, MKL_INT* needil, MKL_INT*\nneediu, MKL_INT* indwlc, double* pivmin, double* scale, double* wl, double* wu, MKL_INT*\nvstart, MKL_INT* finish, MKL_INT* maxcls, MKL_INT* ndepth, MKL_INT* parity, MKL_INT*\nzoffset, MKL_INT* info);\nInclude Files\n•\nmkl_scalapack.h\nDescription\n?stegr2b should only be called after a call to ?stegr2a. From eigenvalues and initial representations\ncomputed by ?stegr2a, ?stegr2b computes the selected eigenvalues and eigenvectors of the real\nsymmetric tridiagonal matrix in parallel on multiple processors. It is potentially invoked multiple times on a\ngiven processor because the locally relevant representation tree might depend on spectral information that is\n\"owned\" by other processors and might need to be communicated.\nPlease note:\n•\nThe calling sequence has two additional integer parameters, dol and dou, that should satisfy\nm≥dou≥dol≥1. These parameters are only relevant for the case jobz = 'V'. ?stegr2b only computes the\neigenvectors corresponding to eigenvalues dol through dou in w, indexed dol-1 through dou-1. (That is,\ninstead of computing the eigenvectors belonging to w([0] through w[m-1], only the eigenvectors belonging\nto eigenvalues w[dol-1] through w[dou-1] are computed. In this case, only the eigenvalues dol through\ndou are guaranteed to be accurately refined to all figures by Rayleigh-Quotient iteration.\n•\nThe additional arguments vstart, finish, ndepth, parity, zoffset are included as a thread-safe\nimplementation equivalent to save variables. These variables store details about the local representation\ntree which is computed layerwise. For scalability reasons, eigenvalues belonging to the locally relevant\nrepresentation tree might be computed on other processors. These need to be communicated before the\ninspection of the RRRs can proceed on any given layer. Note that only when the variable finishis non-\nzero, the computation has ended. All eigenpairs between dol and dou have been computed. m is set to\ndou - dol + 1.\n•\n?stegr2b needs more workspace in z than the sequential ?stegr. It is used to store the conformal\nembedding of the local representation tree.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1799\n\n\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nInput Parameters\njobz\n= 'N': Compute eigenvalues only;\n= 'V': Compute eigenvalues and eigenvectors.\nn\nThe order of the matrix. n≥ 0.\nd\nArray of size n\nThe n diagonal elements of the tridiagonal matrix T. Overwritten on exit.\ne\nArray of size n\nThe (n-1) subdiagonal elements of the tridiagonal matrix T in elements 0 to\nn-2 of e. e[n-1] need not be set on input, but is used internally as\nworkspace. Overwritten on exit.\nm\nThe total number of eigenvalues found in ?stegr2a. 0 ≤m≤n.\nw\nArray of size n\nThe first m elements contain approximations to the selected eigenvalues in\nascending order. Note that only the eigenvalues from the locally relevant\npart of the representation tree, that is all the clusters that include\neigenvalues from dol through dou, are reliable on this processor. (It does\nnot need to know about any others anyway.)\nldz\nThe leading dimension of the array z. ldz≥ 1, and if jobz = 'V', then ldz≥\nmax(1,n).\nnzc\nThe number of eigenvectors to be held in the array z, storing the matrix Z.\nlwork\nThe size of the array work. lwork≥ max(1,18*n)\nif jobz = 'V', and lwork≥ max(1,12*n) if jobz = 'N'.\nIf lwork = -1, then a workspace query is assumed; the function only\ncalculates the optimal size of the work array, returns this value as the first\nentry of the work array, and no error message related to lwork is issued.\nliwork\nThe size of the array iwork. liwork≥ max(1,10*n) if the eigenvectors are\ndesired, and liwork≥ max(1,8*n) if only the eigenvalues are to be\ncomputed.\nIf liwork = -1, then a workspace query is assumed; the function only\ncalculates the optimal size of the iwork array, returns this value as the first\nentry of the iwork array, and no error message related to liwork is issued.\ndol, dou\nFrom the eigenvalues w[0] through w[m-1], only eigenvectors Z(:,dol) to\nZ(:,dou) are computed.\nIf dol > 1, then Z(:,dol-1-zoffset) is used and overwritten.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1800\n\n\nIf dou < m, then Z(:,dou+1-zoffset) is used and overwritten.\nneedil, neediu\nDescribes which are the left and right outermost eigenvalues still to be\ncomputed. Initially computed by ?larre2a, modified in the course of the\nalgorithm.\npivmin\nThe minimum pivot in the sturm sequence for T.\nscale\nThe scaling factor for T. Used for unscaling the eigenvalues at the very end\nof the algorithm.\nwl, wu\nThe interval (wl, wu] contains all the wanted eigenvalues.\nvstart\nNon-zero on initialization, set to zero afterwards.\nfinish\nIndicates whether all eigenpairs have been computed.\nmaxcls\nThe largest cluster worked on by this processor in the representation tree.\nndepth\nThe current depth of the representation tree. Set to zero on initial pass,\nchanged when the deeper levels of the representation tree are generated.\nparity\nAn internal parameter needed for the storage of the clusters on the current\nlevel of the representation tree.\nzoffset\nOffset for storing the eigenpairs when z is distributed in 1D-cyclic fashion.\nOUTPUT Parameters\nz\nArray of size ldz * max(1,m)\nIf jobz = 'V', and if info = 0, then a subset of the first m columns of the\nmatrix Z, stored in z, contain the orthonormal eigenvectors of the matrix T\ncorresponding to the selected eigenvalues, with the i-th column of Z holding\nthe eigenvector associated with w[i-1].\nSee dol, dou for more information.\nisuppz\narray of size 2*max(1,m).\nThe support of the eigenvectors in z, i.e., the indices indicating the nonzero\nelements in z. The i-th computed eigenvector is nonzero only in elements\nisuppz[ 2*i-2 ] through isuppz[ 2*i -1]. This is relevant in the case when\nthe matrix is split. isuppz is only set if n>2.\nwork\nOn exit, if info = 0, work[0] returns the optimal (and minimal) lwork.\niwork\nOn exit, if info = 0, iwork[0] returns the optimal liwork.\nneedil, neediu\nModified in the course of the algorithm.\nindwlc\nPointer into the workspace location where the local eigenvalue\nrepresentations are stored. (\"Local eigenvalues\" are those relative to the\nindividual shifts of the RRRs.)\nvstart\nNon-zero on initialization, set to zero afterwards.\nfinish\nIndicates whether all eigenpairs have been computed\nmaxcls\nThe largest cluster worked on by this processor in the representation tree.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1801\n\n\nndepth\nThe current depth of the representation tree. Set to zero on initial pass,\nchanged when the deeper levels of the representation tree are generated.\nparity\nAn internal parameter needed for the storage of the clusters on the current\nlevel of the representation tree.\ninfo\nOn exit, info\n= 0: successful exit\nother:if info = -i, the i-th argument had an illegal value\nif info = 20x, internal error in ?larrv2.\nHere, the digit x = abs( iinfo ) < 10, where iinfo is the nonzero error\ncode returned by ?larrv2\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\n?stein2\nComputes the eigenvectors corresponding to specified\neigenvalues of a real symmetric tridiagonal matrix,\nusing inverse iteration.\nSyntax\nvoid sstein2 (MKL_INT *n , float *d , float *e , MKL_INT *m , float *w , MKL_INT\n*iblock , MKL_INT *isplit , float *orfac , float *z , MKL_INT *ldz , float *work ,\nMKL_INT *iwork , MKL_INT *ifail , MKL_INT *info );\nvoid dstein2 (MKL_INT *n , double *d , double *e , MKL_INT *m , double *w , MKL_INT\n*iblock , MKL_INT *isplit , double *orfac , double *z , MKL_INT *ldz , double *work ,\nMKL_INT *iwork , MKL_INT *ifail , MKL_INT *info );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe ?stein2function is a modified LAPACK function ?stein. It computes the eigenvectors of a real\nsymmetric tridiagonal matrix T corresponding to specified eigenvalues, using inverse iteration.\nThe maximum number of iterations allowed for each eigenvector is specified by an internal parameter maxits\n(currently set to 5).\nInput Parameters\nn\nThe order of the matrix T (n ≥ 0).\nm\nThe number of eigenvectors to be found (0 ≤ m ≤ n).\nd, e, w\nArrays:\nd, of size n. The n diagonal elements of the tridiagonal matrix T.\ne, of size n.\nThe (n-1) subdiagonal elements of the tridiagonal matrix T, in elements 1\nto n-1. e[n-1] need not be set.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1802\n\n\nw, of size n.\nThe first m elements of w contain the eigenvalues for which eigenvectors are\nto be computed. The eigenvalues should be grouped by split-off block and\nordered from smallest to largest within the block. (The output array w\nfrom ?stebz with ORDER = 'B' is expected here).\nThe size of w must be at least max(1, n).\niblock\nArray of size n.\nThe submatrix indices associated with the corresponding eigenvalues in w;\niblock[i] = 1, if eigenvalue w[i] belongs to the first submatrix from the\ntop,\niblock[i] = 2, if eigenvalue w[i] belongs to the second submatrix, etc. (The\noutput array iblock from ?stebz is expected here).\nisplit\nArray of size n.\nThe splitting points, at which T breaks up into submatrices. The first\nsubmatrix consists of rows/columns 1 to isplit[0], the second submatrix\nconsists of rows/columns isplit[0]+1 through isplit[1], etc. (The\noutput array isplit from ?stebz is expected here).\norfac\norfac specifies which eigenvectors should be orthogonalized. Eigenvectors\nthat correspond to eigenvalues which are within orfac*||T|| of each other\nare to be orthogonalized.\nldz\nThe leading dimension of the output array z; ldz ≥ max(1, n).\nwork\nWorkspace array of size 5n.\niwork\nWorkspace array of size n.\nOutput Parameters\nz\nArray of size ldz * m.\nThe computed eigenvectors. The eigenvector associated with the eigenvalue\nw[i] is stored in the (i+1)-th column of the matrix Z represented by z,\ni=0, ..., m-1. Any vector that fails to converge is set to its current iterate\nafter maxits iterations.\nifail\nArray of size m.\nOn normal exit, all elements of ifail are zero. If one or more eigenvectors\nfail to converge after maxits iterations, then their indices are stored in the\narray ifail.\ninfo\ninfo = 0, the exit is successful.\ninfo < 0: if info = -i, the i-th had an illegal value.\ninfo > 0: if info = i, then i eigenvectors failed to converge in maxits\niterations. Their indices are stored in the array ifail.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1803\n\n\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\n?dbtf2\nComputes an LU factorization of a general band matrix\nwith no pivoting (local unblocked algorithm).\nSyntax\nvoid sdbtf2 (MKL_INT *m , MKL_INT *n , MKL_INT *kl , MKL_INT *ku , float *ab , MKL_INT\n*ldab , MKL_INT *info );\nvoid ddbtf2 (MKL_INT *m , MKL_INT *n , MKL_INT *kl , MKL_INT *ku , double *ab , MKL_INT\n*ldab , MKL_INT *info );\nvoid cdbtf2 (MKL_INT *m , MKL_INT *n , MKL_INT *kl , MKL_INT *ku , MKL_Complex8 *ab ,\nMKL_INT *ldab , MKL_INT *info );\nvoid zdbtf2 (MKL_INT *m , MKL_INT *n , MKL_INT *kl , MKL_INT *ku , MKL_Complex16 *ab ,\nMKL_INT *ldab , MKL_INT *info );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe ?dbtf2function computes an LU factorization of a general real/complex m-by-n band matrix A without\nusing partial pivoting with row interchanges.\nThis is the unblocked version of the algorithm, calling BLAS Routines and Functions.\nInput Parameters\nm\nThe number of rows of the matrix A(m ≥ 0).\nn\nThe number of columns in A(n ≥ 0).\nkl\nThe number of sub-diagonals within the band of A(kl ≥ 0).\nku\nThe number of super-diagonals within the band of A(ku ≥ 0).\nab\nArray of size ldab * n.\nThe matrix A in band storage, in rows kl+1 to 2kl+ku+1; rows 1 to kl\nof the matrix need not be set. The j-th column of A is stored in the\narray ab as follows: ab[kl+ku+i-j+(j-1)*ldab] = A(i,j) for max(1,j-\nku) ≤ i ≤ min(m,j+kl).\nldab\nThe leading dimension of the array ab.\n(ldab ≥ 2kl + ku +1)\nOutput Parameters\nab\nOn exit, details of the factorization: U is stored as an upper triangular band\nmatrix with kl+ku superdiagonals in rows 1 to kl+ku+1, and the multipliers\nused during the factorization are stored in rows kl+ku+2 to 2*kl+ku+1.\nSee the Application Notes below for further details.\ninfo\n= 0: successful exit\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1804\n\n\n< 0: if info = - i, the i-th argument had an illegal value,\n> 0: if info = + i, the matrix elementU(i,i) is 0. The factorization has been\ncompleted, but the factor U is exactly singular. Division by 0 will occur if\nyou use the factor U for solving a system of linear equations.\nApplication Notes\nThe band storage scheme is illustrated by the following example, when m = n = 6, kl = 2, ku = 1:\nThe function does not use array elements marked *; elements marked + need not be set on entry, but the\nfunction requires them to store elements of U, because of fill-in resulting from the row interchanges.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\n?dbtrf\nComputes an LU factorization of a general band matrix\nwith no pivoting (local blocked algorithm).\nSyntax\nvoid sdbtrf (MKL_INT *m , MKL_INT *n , MKL_INT *kl , MKL_INT *ku , float *ab , MKL_INT\n*ldab , MKL_INT *info );\nvoid ddbtrf (MKL_INT *m , MKL_INT *n , MKL_INT *kl , MKL_INT *ku , double *ab , MKL_INT\n*ldab , MKL_INT *info );\nvoid cdbtrf (MKL_INT *m , MKL_INT *n , MKL_INT *kl , MKL_INT *ku , MKL_Complex8 *ab ,\nMKL_INT *ldab , MKL_INT *info );\nvoid zdbtrf (MKL_INT *m , MKL_INT *n , MKL_INT *kl , MKL_INT *ku , MKL_Complex16 *ab ,\nMKL_INT *ldab , MKL_INT *info );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThis function computes an LU factorization of a real m-by-n band matrix A without using partial pivoting or\nrow interchanges.\nThis is the blocked version of the algorithm, calling BLAS Routines and Functions.\nInput Parameters\nm\nThe number of rows of the matrix A (m ≥ 0).\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1805\n\n\nn\nThe number of columns in A(n ≥ 0).\nkl\nThe number of sub-diagonals within the band of A(kl ≥ 0).\nku\nThe number of super-diagonals within the band of A(ku ≥ 0).\nab\nArray of size ldab * n.\nThe matrix A in band storage, in rows kl+1 to 2kl+ku+1; rows 1 to kl need\nnot be set. The j-th column of A is stored in the array ab as follows: ab[kl\n+ku+i-j+(j-1)*ldab] = A(i,j) for max(1,j-ku) ≤ i ≤ min(m,j\n+kl).\nldab\nThe leading dimension of the array ab.\n(ldab ≥ 2kl + ku +1)\nOutput Parameters\nab\nOn exit, details of the factorization: U is stored as an upper triangular band\nmatrix with kl+ku superdiagonals in rows 1 to kl+ku+1, and the multipliers\nused during the factorization are stored in rows kl+ku+2 to 2*kl+ku+1.\nSee the Application Notes below for further details.\ninfo\n= 0: successful exit\n< 0: if info = - i, the i-th argument had an illegal value,\n> 0: if info = + i, the matrix element U(i,i) is 0. The factorization\nhas been completed, but the factor U is exactly singular. Division by\n0 will occur if you use the factor U for solving a system of linear\nequations.\nApplication Notes\nThe band storage scheme is illustrated by the following example, when m = n = 6, kl = 2, ku = 1:\nThe function does not use array elements marked *.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\n?dttrf\nComputes an LU factorization of a general tridiagonal\nmatrix with no pivoting (local blocked algorithm).\nSyntax\nvoid sdttrf (MKL_INT *n , float *dl , float *d , float *du , MKL_INT *info );\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1806\n\n\nvoid ddttrf (MKL_INT *n , double *dl , double *d , double *du , MKL_INT *info );\nvoid cdttrf (MKL_INT *n , MKL_Complex8 *dl , MKL_Complex8 *d , MKL_Complex8 *du ,\nMKL_INT *info );\nvoid zdttrf (MKL_INT *n , MKL_Complex16 *dl , MKL_Complex16 *d , MKL_Complex16 *du ,\nMKL_INT *info );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe ?dttrffunction computes an LU factorization of a real or complex tridiagonal matrix A using elimination\nwithout partial pivoting.\nThe factorization has the form A = L*U, where L is a product of unit lower bidiagonal matrices and U is upper\ntriangular with nonzeros only in the main diagonal and first superdiagonal.\nInput Parameters\nn\nThe order of the matrix A(n ≥ 0).\ndl, d, du\nArrays containing elements of A.\nThe array dl of size (n-1) contains the sub-diagonal elements of A.\nThe array d of size n contains the diagonal elements of A.\nThe array du of size (n-1) contains the super-diagonal elements of A.\nOutput Parameters\ndl\nOverwritten by the (n-1) multipliers that define the matrix L from the LU\nfactorization of A.\nd\nOverwritten by the n diagonal elements of the upper triangular matrix U\nfrom the LU factorization of A.\ndu\nOverwritten by the (n-1) elements of the first super-diagonal of U.\ninfo\n= 0: successful exit\n< 0: if info = - i, the i-th argument had an illegal value,\n> 0: if info = i, the matrix element U(i,i) is exactly 0. The factorization has\nbeen completed, but the factor U is exactly singular. Division by 0 will occur\nif you use the factor U for solving a system of linear equations.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\n?dttrsv\nSolves a general tridiagonal system of linear equations\nusing the LU factorization computed by ?dttrf.\nSyntax\nvoid sdttrsv (char *uplo , char *trans , MKL_INT *n , MKL_INT *nrhs , float *dl , float\n*d , float *du , float *b , MKL_INT *ldb , MKL_INT *info );\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1807\n\n\nvoid ddttrsv (char *uplo , char *trans , MKL_INT *n , MKL_INT *nrhs , double *dl ,\ndouble *d , double *du , double *b , MKL_INT *ldb , MKL_INT *info );\nvoid cdttrsv (char *uplo , char *trans , MKL_INT *n , MKL_INT *nrhs , MKL_Complex8\n*dl , MKL_Complex8 *d , MKL_Complex8 *du , MKL_Complex8 *b , MKL_INT *ldb , MKL_INT\n*info );\nvoid zdttrsv (char *uplo , char *trans , MKL_INT *n , MKL_INT *nrhs , MKL_Complex16\n*dl , MKL_Complex16 *d , MKL_Complex16 *du , MKL_Complex16 *b , MKL_INT *ldb , MKL_INT\n*info );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe ?dttrsvfunction solves one of the following systems of linear equations:\nL*X = B, LT*X = B, or LH*X = B,\nU*X = B, UT*X = B, or UH*X = B\nwith factors of the tridiagonal matrix A from the LU factorization computed by ?dttrf.\nInput Parameters\nuplo\nSpecifies whether to solve with L or U.\ntrans\nMust be 'N' or 'T' or 'C'.\nIndicates the form of the equations:\nIf trans = 'N', then A*X=B is solved for X (no transpose).\nIf trans = 'T', then AT*X = B is solved for X (transpose).\nIf trans = 'C', then AH*X = B is solved for X (conjugate transpose).\nn\nThe order of the matrix A(n ≥ 0).\nnrhs\nThe number of right-hand sides, that is, the number of columns in the\nmatrix B(nrhs ≥ 0).\ndl,d,du,b\nThe array dl of size (n - 1) contains the (n - 1) multipliers that define the\nmatrix L from the LU factorization of A.\nThe array d of size n contains n diagonal elements of the upper triangular\nmatrix U from the LU factorization of A.\nThe array du of size (n - 1) contains the (n - 1) elements of the first super-\ndiagonal of U.\nOn entry, the array b of size ldb * nrhs contains the right-hand side of\nmatrix B.\nldb\nThe leading dimension of the array b; ldb ≥ max(1, n).\nOutput Parameters\nb\nOverwritten by the solution matrix X.\ninfo\nIf info=0, the execution is successful.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1808\n\n\nIf info = -i, the i-th parameter had an illegal value.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\n?pttrsv\nSolves a symmetric (Hermitian) positive-definite\ntridiagonal system of linear equations, using the\nL*D*LH factorization computed by ?pttrf.\nSyntax\nvoid spttrsv (char *trans , MKL_INT *n , MKL_INT *nrhs , float *d , float *e , float\n*b , MKL_INT *ldb , MKL_INT *info );\nvoid dpttrsv (char *trans , MKL_INT *n , MKL_INT *nrhs , double *d , double *e , double\n*b , MKL_INT *ldb , MKL_INT *info );\nvoid cpttrsv (char *uplo , char *trans , MKL_INT *n , MKL_INT *nrhs , float *d ,\nMKL_Complex8 *e , MKL_Complex8 *b , MKL_INT *ldb , MKL_INT *info );\nvoid zpttrsv (char *uplo , char *trans , MKL_INT *n , MKL_INT *nrhs , double *d ,\nMKL_Complex16 *e , MKL_Complex16 *b , MKL_INT *ldb , MKL_INT *info );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe ?pttrsvfunction solves one of the triangular systems:\nLT*X = B, or L*X = B for real flavors,\nor\nL*X = B, or LH*X = B,\nU*X = B, or UH*X = B for complex flavors,\nwhere L (or U for complex flavors) is the Cholesky factor of a Hermitian positive-definite tridiagonal matrix A\nsuch that\nA = L*D*LH (computed by spttrf/dpttrf)\nor\nA = UH*D*U or A = L*D*LH (computed by cpttrf/zpttrf).\nInput Parameters\nuplo\nMust be 'U' or 'L'.\nSpecifies whether the superdiagonal or the subdiagonal of the tridiagonal\nmatrix A is stored and the form of the factorization:\nIf uplo = 'U', e is the superdiagonal of U, and A = UH*D*U or A =\nL*D*LH;\nif uplo = 'L', e is the subdiagonal of L, and A = L*D*LH.\nThe two forms are equivalent, if A is real.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1809\n\n\ntrans\nSpecifies the form of the system of equations:\nfor real flavors:\nif trans = 'N': L*X = B (no transpose)\nif trans = 'T': LT*X = B (transpose)\nfor complex flavors:\nif trans = 'N': U*X = B or L*X = B (no transpose)\nif trans = 'C': UH*X = B or LH*X = B (conjugate transpose).\nn\nThe order of the tridiagonal matrix A. n ≥ 0.\nnrhs\nThe number of right hand sides, that is, the number of columns of the\nmatrix B. nrhs ≥ 0.\nd\narray of size n. The n diagonal elements of the diagonal matrix D from the\nfactorization computed by ?pttrf.\ne\narray of size (n-1). The (n-1) off-diagonal elements of the unit bidiagonal\nfactor U or L from the factorization computed by ?pttrf. See uplo.\nb\narray of size ldb* nrhs.\nOn entry, the right hand side matrix B.\nldb\nThe leading dimension of the array b.\nldb ≥ max(1, n).\nOutput Parameters\nb\nOn exit, the solution matrix X.\ninfo\n= 0: successful exit\n< 0: if info = -i, the i-th argument had an illegal value.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\n?steqr2\nComputes all eigenvalues and, optionally,\neigenvectors of a symmetric tridiagonal matrix using\nthe implicit QL or QR method.\nSyntax\nvoid ssteqr2 (char *compz , MKL_INT *n , float *d , float *e , float *z , MKL_INT *ldz ,\nMKL_INT *nr , float *work , MKL_INT *info );\nvoid dsteqr2 (char *compz , MKL_INT *n , double *d , double *e , double *z , MKL_INT\n*ldz , MKL_INT *nr , double *work , MKL_INT *info );\nInclude Files\n•\nmkl_scalapack.h\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1810\n\n\nDescription\nThe ?steqr2function is a modified version of LAPACK function ?steqr. The ?steqr2function computes all\neigenvalues and, optionally, eigenvectors of a symmetric tridiagonal matrix using the implicit QL or QR\nmethod. ?steqr2 is modified from ?steqr to allow each ScaLAPACK process running ?steqr2 to perform\nupdates on a distributed matrix Q. Proper usage of ?steqr2 can be gleaned from examination of ScaLAPACK\nfunction p?syev.\nInput Parameters\ncompz\nMust be 'N' or 'I'.\nIf compz = 'N', the function computes eigenvalues only. If compz = 'I',\nthe function computes the eigenvalues and eigenvectors of the tridiagonal\nmatrix T.\nz must be initialized to the identity matrix by p?laset or ?laset prior to\nentering this function.\nn\nThe order of the matrix T(n ≥ 0).\nd, e, work\nArrays: \nd contains the diagonal elements of T. The size of d must be at least\nmax(1, n).\ne contains the (n-1) subdiagonal elements of T. The size of e must be at\nleast max(1, n-1).\nwork is a workspace array. The size of work is max(1, 2*n-2). If compz =\n'N', then work is not referenced.\nz\n(local)\nArray of global size n* n and of local size ldz* nr.\nIf compz = 'V', then z contains the orthogonal matrix used in the\nreduction to tridiagonal form.\nldz\nThe leading dimension of the array z. Constraints:\nldz ≥ 1,\nldz ≥ max(1, n), if eigenvectors are desired.\nnr\nnr = max(1, numroc(n, nb, myprow, 0, nprocs)).\nIf compz = 'N', then nr is not referenced.\nOutput Parameters\nd\nOn exit, the eigenvalues in ascending order, if info = 0.\nSee also info.\ne\nOn exit, e has been destroyed.\nz\nOn exit, if info = 0, then,\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1811\n\n\nif compz = 'V', z contains the orthonormal eigenvectors of the original\nsymmetric matrix, and if compz = 'I', z contains the orthonormal\neigenvectors of the symmetric tridiagonal matrix. If compz = 'N', then z is\nnot referenced.\ninfo\ninfo = 0, the exit is successful.\ninfo < 0: if info = -i, the i-th had an illegal value.\ninfo > 0: the algorithm has failed to find all the eigenvalues in a total of\n30n iterations;\nif info = i, then i elements of e have not converged to zero; on exit, d\nand e contain the elements of a symmetric tridiagonal matrix, which is\northogonally similar to the original matrix.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\n?trmvt\nPerforms matrix-vector operations.\nSyntax\nvoid strmvt (const char* uplo, const MKL_INT* n, const float* t, const MKL_INT* ldt,\nfloat* x, const MKL_INT* incx, const float* y, const MKL_INT* incy, float* w, const\nMKL_INT* incw, const float* z, const MKL_INT* incz);\nvoid dtrmvt (const char* uplo, const MKL_INT* n, const double* t, const MKL_INT* ldt,\ndouble* x, const MKL_INT* incx, const double* y, const MKL_INT* incy, double* w, const\nMKL_INT* incw, const double* z, const MKL_INT* incz);\nvoid ctrmvt (const char* uplo, const MKL_INT* n, const MKL_Complex8* t, const MKL_INT*\nldt, MKL_Complex8* x, const MKL_INT* incx, const MKL_Complex8* y, const MKL_INT* incy,\nMKL_Complex8* w, const MKL_INT* incw, const MKL_Complex8* z, const MKL_INT* incz);\nvoid ztrmvt (const char* uplo, const MKL_INT* n, const MKL_Complex16* t, const MKL_INT*\nldt, MKL_Complex16* x, const MKL_INT* incx, const MKL_Complex16* y, const MKL_INT*\nincy, MKL_Complex16* w, const MKL_INT* incw, const MKL_Complex16* z, const MKL_INT*\nincz);\nInclude Files\n•\nmkl_scalapack.h\nDescription\n?trmvt performs the matrix-vector operations as follows:\nstrmvt and dtrmvt:   x := T' *y, and w := T *z\nctrmvt and ztrmvt:   x := conjg( T' ) *y, and w := T *z,\nwhere x is an n element vector and T is an n-by-n upper or lower triangular matrix.\nInput Parameters\nuplo\nOn entry, uplo specifies whether the matrix is an upper or lower triangular\nmatrix as follows:\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1812\n\n\nuplo = 'U' or 'u'\nA is an upper triangular matrix.\nuplo = 'L' or 'l'\nA is a lower triangular matrix.\nUnchanged on exit.\nn\nOn entry, n specifies the order of the matrix A. n must be at least zero.\nUnchanged on exit.\nt\nArray of size ( ldt, n ).\nBefore entry with uplo = 'U' or 'u', the leading n-by-n upper triangular part\nof the array t must contain the upper triangular matrix and the strictly\nlower triangular part of t is not referenced.\nBefore entry with uplo = 'L' or 'l', the leading n-by-n lower triangular part\nof the array t must contain the lower triangular matrix and the strictly\nupper triangular part of t is not referenced.\nldt\nOn entry, lda specifies the first dimension of A as declared in the calling\n(sub) program. lda must be at least max( 1, n ).\nUnchanged on exit.\nincx\nOn entry, incx specifies the increment for the elements of x. incx must\nnot be zero.\nUnchanged on exit.\ny\nArray of size at least ( 1 + ( n - 1 )*abs( incy ) ).\nBefore entry, the incremented array y must contain the n element vector y.\nUnchanged on exit.\nincy\nOn entry, incy specifies the increment for the elements of y. incy must\nnot be zero.\nUnchanged on exit.\nincw\nOn entry, incw specifies the increment for the elements of w. incw must\nnot be zero.\nUnchanged on exit.\nz\nArray of size at least ( 1 + ( n - 1 )*abs( incz ) ).\nBefore entry, the incremented array z must contain the n element vector z.\nUnchanged on exit.\nincz\nOn entry, incz specifies the increment for the elements of z. incz must\nnot be zero.\nUnchanged on exit.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1813\n\n\nOutput Parameters\nt\nBefore entry with uplo = 'U' or 'u', the leading n-by-n upper\ntriangular part of the array t must contain the upper triangular matrix\nand the strictly lower triangular part of t is not referenced.\nBefore entry with uplo = 'L' or 'l', the leading n-by-n lower triangular\npart of the array t must contain the lower triangular matrix and the\nstrictly upper triangular part of t is not referenced.\nx\nArray of size at least ( 1 + ( n - 1 )*abs( incx ) ).\nOn exit, x = T' * y.\nw\nArray of size at least ( 1 + ( n - 1 )*abs( incw ) ).\nOn exit, w = T * z.\npilaenv\nReturns the positive integer value of the logical\nblocking size.\nSyntax\nMKL_INT pilaenv (const MKL_INT *ictxt , const char *prec);\nInclude Files\n•\nmkl_pblas.h\nDescription\npilaenv returns the positive integer value of the logical blocking size. This value is machine and precision\nspecific. This version provides a logical blocking size which should give good though not optimal performance\non many of the currently available distributed-memory concurrent computers. You are encouraged to modify\nthis subroutine to set this tuning parameter for your particular machine.\nInput Parameters\nictxt\nOn entry, ictxt specifies the BLACS context handle, indicating the global\ncontext of the operation. The context itself is global, but the value of ictxt\nis local.\nprec\nOn input, prec specifies the precision for which the logical block size should\nbe returned as follows:\nprec = 'S' or 's' single precision real,\nprec = 'D' or 'd' double precision real,\nprec = 'C' or 'c' single precision complex,\nprec = 'Z' or 'z' double precision complex,\nprec = 'I' or 'i' integer.\nApplication Notes\nBefore modifying this routine to tune the library performance on your system, be aware of the following:\n1.\nThe value this function returns must be strictly larger than zero,\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1814\n\n\n2.\nIf you are planning to link your program with different instances of the library (for example, on a\nheterogeneous machine), you must compile each instance of the library with exactly the same version\nof this routine for obvious interoperability reasons.\npilaenvx\nCalled from the ScaLAPACK routines to choose\nproblem-dependent parameters for the local\nenvironment.\nSyntax\nMKL_INT pilaenvx (const MKL_INT* ictxt, const MKL_INT* ispec, const char* name, const\nchar* opts, const MKL_INT* n1, const MKL_INT* n2, const MKL_INT* n3, const MKL_INT*\nInclude Files\n•\nmkl.h\nDescription\npilaenvx is called from the ScaLAPACK routines to choose problem-dependent parameters for the local\nenvironment. See ispec for a description of the parameters. This version provides a set of parameters which\nshould give good, though not optimal, performance on many of the currently available computers. You are\nencouraged to modify this subroutine to set the tuning parameters for your particular machine using the\noption and problem size information in the arguments.\nInput Parameters\nictxt\n(local input)On entry, ictxt specifies the BLACS context handle, indicating\nthe global context of the operation. The context itself is global, but the\nvalue of ictxt is local.\nispec\n(global input)\nSpecifies the parameter to be returned as the value of pilaenvx.\n= 1: the optimal blocksize; if this value is 1, an unblocked algorithm will\ngive the best performance (unlikely).\n= 2: the minimum block size for which the block routine should be used; if\nthe usable block size is less than this value, an unblocked routine should be\nused.\n= 3: the crossover point (in a block routine, for N less than this value, an\nunblocked routine should be used).\n= 4: the number of shifts, used in the nonsymmetric eigenvalue routines\n(DEPRECATED).\n= 5: the minimum column dimension for blocking to be used; rectangular\nblocks must have dimension at least k by m, where k is given by\npilaenvx(2,...) and m by pilaenvx(5,...).\n= 6: the crossover point for the SVD (when reducing an m by n matrix to\nbidiagonal form, if max(m,n)/min(m,n) exceeds this value, a QR\nfactorization is used first to reduce the matrix to a triangular form).\n= 7: the number of processors.\n= 8: the crossover point for the multishift QR method for nonsymmetric\neigenvalue problems (DEPRECATED).\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1815\n\n\n= 9: maximum size of the subproblems at the bottom of the computation\ntree in the divide-and-conquer algorithm (used by ?gelsd and ?gesdd).\n=10: IEEE NaN arithmetic can be trusted not to trap.\n=11: infinity arithmetic can be trusted not to trap.\n12 <= ispec <= 16:\np?hseqr or one of its subroutines, see piparmq for detailed explanation.\n17 <= ispec <= 22:\nParameters for pb?trord/p?hseqr (not all), as follows:\n=17: maximum number of concurrent computational windows;\n=18: number of eigenvalues/bulges in each window;\n=19: computational window size;\n=20: minimal percentage of FLOPS required for performing matrix-matrix\nmultiplications instead of pipelined orthogonal transformations;\n=21: width of block column slabs for row-wise application of pipelined\northogonal transformations in their factorized form;\n=22: the maximum number of eigenvalues moved together over a process\nborder;\n=23: the number of processors involved in Aggressive Early Deflation\n(AED);\n=99: Maximum iteration chunksize in OpenMP parallelization.\nname\n(global input)\nThe name of the calling subroutine, in either upper case or lower case.\nopts\n(global input) The character options to the subroutine name, concatenated\ninto a single character string. For example, uplo = 'U', trans = 'T',\nand diag = 'N' for a triangular routine would be specified as opts =\n'UTN'.\nn1, n2, n3, and n4\n(global input) Problem dimensions for the subroutine name; these may not\nall be required.\nOutput Parameters\nresult\n(global output)\n>= 0: the value of the parameter specified by ispec.\n< 0: if pilaenvx = -k, the k-th argument had an illegal value.\nApplication Notes\nThe following conventions have been used when calling ilaenv from the LAPACK routines:\n1.\nopts is a concatenation of all of the character options to subroutine name, in the same order that they\nappear in the argument list for name, even if they are not used in determining the value of the\nparameter specified by ispec.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1816\n\n\n2.\nThe problem dimensions n1, n2, n3, and n4 are specified in the order that they appear in the argument\nlist for name. n1 is used first, n2 second, and so on, and unused problem dimensions are passed a value\nof -1.\n3.\nThe parameter value returned by ilaenv is checked for validity in the calling subroutine. For example,\nilaenv is used to retrieve the optimal block size for strtri as follows:\n NB = ilaenv( 1, 'STRTRI', UPLO // DIAG, N, -1, -1, -1 );\nif( NB<=1 ) {\n   NB = MAX( 1, N );\n}\nThe same conventions hold for this ScaLAPACK-style variant.\npjlaenv\nCalled from the ScaLAPACK symmetric and Hermitian\ntailored eigen-routines to choose problem-dependent\nparameters for the local environment.\nSyntax\nMKL_INT pjlaenv (const MKL_INT* ictxt, const MKL_INT* ispec, const char* name, const\nchar* opts, const MKL_INT* n1, const MKL_INT* n2, const MKL_INT* n3, const MKL_INT*\nn4);\nInclude Files\n•\nmkl.h\nDescription\npjlaenv is called from the ScaLAPACK symmetric and Hermitian tailored eigen-routines to choose problem-\ndependent parameters for the local environment. See ispec for a description of the parameters. This version\nprovides a set of parameters which should give good, though not optimal, performance on many of the\ncurrently available computers. You are encouraged to modify this subroutine to set the tuning parameters for\nyour particular machine using the option and problem size information in the arguments.\nInput Parameters\nispec\n(global input) Specifies the parameter to be returned as the value of\npjlaenv.\n= 1: the data layout blocksize;\n= 2: the panel blocking factor;\n= 3: the algorithmic blocking factor;\n= 4: execution path control;\n= 5: maximum size for direct call to the LAPACK routine.\nname\n(global input) The name of the calling subroutine, in either upper case or\nlower case.\nopts\n(global input) The character options to the subroutine name, concatenated\ninto a single character string. For example, uplo = 'U', trans = 'T',\nand diag = 'N' for a triangular routine would be specified as opts =\n'UTN'.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1817\n\n\nn1, n2, n3, and n4\n(global input) Problem dimensions for the subroutine name; these may not\nall be required. At present, only n1 is used, and it (n1) is used only for\n'TTRD'.\nOutput Parameters\nresult\n(global or local output)\n>= 0: the value of the parameter specified by ispec.\n< 0: if pjlaenv = -k, the k-th argument had an illegal value. Most\nparameters set via a call to pjlaenv must be identical on all\nprocessors and hence pjlaenv will return the same value to all\nprocesors (i.e. global output). However some, in particular, the panel\nblocking factor can be different on each processor and hence pjlaenv\ncan return different values on different processors (i.e. local output).\nApplication Notes\nThe following conventions have been used when calling pjlaenv from the ScaLAPACK routines:\n1.\nopts is a concatenation of all of the character options to subroutine name, in the same order that they\nappear in the argument list for name, even if they are not used in determining the value of the\nparameter specified by ispec.\n2.\nThe problem dimensions n1, n2, n3, and n4 are specified in the order that they appear in the argument\nlist for name. n1 is used first, n2 second, and so on, and unused problem dimensions are passed a\nvalue of -1.\na.\nThe parameter value returned by pjlaenv is checked for validity in the calling subroutine. For\nexample, pjlaenv is used to retrieve the optimal blocksize for STRTRI as follows:\nNB = pjlaenv( 1, 'STRTRI', UPLO // DIAG, N, -1, -1, -1 );\nIF( NB>=1 ) {\n   NB = MAX( 1, N );\n}\npjlaenv is patterned after ilaenv and keeps the same interface in anticipation of future needs, even though\npjlaenv is only sparsely used at present in ScaLAPACK. Most ScaLAPACK codes use the input data layout\nblocking factor as the algorithmic blocking factor - hence there is no need or opportunity to set the\nalgorithmic or data decomposition blocking factor. pXYYtevx.f and pXYYtgvx.f and pXYYttrd.f are the\nonly codes which call pjlaenv. pXYYtevx.f and pXYYtgvx.f redistribute the data to the best data layout\nfor each transformation. pXYYttrd.f uses a data layout blocking factor of 1.\nAdditional ScaLAPACK Routines\nvoid pchettrd (const char *uplo , const MKL_INT *n , MKL_Complex8 *a , const MKL_INT\n*ia , const MKL_INT *ja , const MKL_INT *desca , float *d , float *e , MKL_Complex8\n*tau , MKL_Complex8 *work , const MKL_INT *lwork , MKL_INT *info );\nvoid pzhettrd (const char *uplo , const MKL_INT *n , MKL_Complex16 *a , const MKL_INT\n*ia , const MKL_INT *ja , const MKL_INT *desca , double *d , double *e , MKL_Complex16\n*tau , MKL_Complex16 *work , const MKL_INT *lwork , MKL_INT *info );\nvoid pslaed0 (const MKL_INT *n , float *d , float *e , float *q , const MKL_INT *iq ,\nconst MKL_INT *jq , const MKL_INT *descq , float *work , MKL_INT *iwork , MKL_INT\n*info );\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1818\n\n\nvoid pdlaed0 (const MKL_INT *n , double *d , double *e , double *q , const MKL_INT\n*iq , const MKL_INT *jq , const MKL_INT *descq , double *work , MKL_INT *iwork ,\nMKL_INT *info );\nvoid pslaed1 (const MKL_INT *n , const MKL_INT *n1 , float *d , const MKL_INT *id ,\nfloat *q , const MKL_INT *iq , const MKL_INT *jq , const MKL_INT *descq , const float\n*rho , float *work , MKL_INT *iwork , MKL_INT *info );\nvoid pdlaed1 (const MKL_INT *n , const MKL_INT *n1 , double *d , const MKL_INT *id ,\ndouble *q , const MKL_INT *iq , const MKL_INT *jq , const MKL_INT *descq , const double\n*rho , double *work , MKL_INT *iwork , MKL_INT *info );\nvoid pslaed2 (const MKL_INT *ictxt , MKL_INT *k , const MKL_INT *n , const MKL_INT\n*n1 , const MKL_INT *nb , float *d , const MKL_INT *drow , const MKL_INT *dcol , float\n*q , const MKL_INT *ldq , float *rho , const float *z , float *w , float *dlamda , float\n*q2 , const MKL_INT *ldq2 , float *qbuf , MKL_INT *ctot , MKL_INT *psm , const MKL_INT\n*npcol , MKL_INT *indx , MKL_INT *indxc , MKL_INT *indxp , MKL_INT *indcol , MKL_INT\n*coltyp , MKL_INT *nn , MKL_INT *nn1 , MKL_INT *nn2 , MKL_INT *ib1 , MKL_INT *ib2 );\nvoid pdlaed2 (const MKL_INT *ictxt , MKL_INT *k , const MKL_INT *n , const MKL_INT\n*n1 , const MKL_INT *nb , double *d , const MKL_INT *drow , const MKL_INT *dcol ,\ndouble *q , const MKL_INT *ldq , double *rho , const double *z , double *w , double\n*dlamda , double *q2 , const MKL_INT *ldq2 , double *qbuf , MKL_INT *ctot , MKL_INT\n*psm , const MKL_INT *npcol , MKL_INT *indx , MKL_INT *indxc , MKL_INT *indxp , MKL_INT\n*indcol , MKL_INT *coltyp , MKL_INT *nn , MKL_INT *nn1 , MKL_INT *nn2 , MKL_INT *ib1 ,\nMKL_INT *ib2 );\nvoid pslaed3 (const MKL_INT *ictxt , MKL_INT *k , const MKL_INT *n , const MKL_INT\n*nb , float *d , const MKL_INT *drow , const MKL_INT *dcol , float *rho , float\n*dlamda , float *w , const float *z , float *u , const MKL_INT *ldu , float *buf ,\nMKL_INT *indx , MKL_INT *indcol , MKL_INT *indrow , MKL_INT *indxr , MKL_INT *indxc ,\nMKL_INT *ctot , const MKL_INT *npcol , MKL_INT *info );\nvoid pdlaed3 (const MKL_INT *ictxt , MKL_INT *k , const MKL_INT *n , const MKL_INT\n*nb , double *d , const MKL_INT *drow , const MKL_INT *dcol , double *rho , double\n*dlamda , double *w , const double *z , double *u , const MKL_INT *ldu , double *buf ,\nMKL_INT *indx , MKL_INT *indcol , MKL_INT *indrow , MKL_INT *indxr , MKL_INT *indxc ,\nMKL_INT *ctot , const MKL_INT *npcol , MKL_INT *info );\nvoid pslaedz (const MKL_INT *n , const MKL_INT *n1 , const MKL_INT *id , const float\n*q , const MKL_INT *iq , const MKL_INT *jq , const MKL_INT *ldq , const MKL_INT\n*descq , float *z , float *work );\nvoid pdlaedz (const MKL_INT *n , const MKL_INT *n1 , const MKL_INT *id , const double\n*q , const MKL_INT *iq , const MKL_INT *jq , const MKL_INT *ldq , const MKL_INT\n*descq , double *z , double *work );\nvoid pdlaiectb (const double *sigma , const MKL_INT *n , const double *d , MKL_INT\n*count );\nvoid pdlaiectl (const double *sigma , const MKL_INT *n , const double *d , MKL_INT\n*count );\nvoid slamov (const char *UPLO , const MKL_INT *M , const MKL_INT *N , const float *A ,\nconst MKL_INT *LDA , float *B , const MKL_INT *LDB );\nvoid dlamov (const char *UPLO , const MKL_INT *M , const MKL_INT *N , const double *A ,\nconst MKL_INT *LDA , double *B , const MKL_INT *LDB );\nvoid clamov (const char *UPLO , const MKL_INT *M , const MKL_INT *N , const\nMKL_Complex8 *A , const MKL_INT *LDA , MKL_Complex8 *B , const MKL_INT *LDB );\nvoid zlamov (const char *UPLO , const MKL_INT *M , const MKL_INT *N , const\nMKL_Complex16 *A , const MKL_INT *LDA , MKL_Complex16 *B , const MKL_INT *LDB );\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1819\n\n\nvoid pslamr1d (const MKL_INT *n , float *a , const MKL_INT *ia , const MKL_INT *ja ,\nconst MKL_INT *desca , float *b , const MKL_INT *ib , const MKL_INT *jb , const MKL_INT\n*descb );\nvoid pdlamr1d (const MKL_INT *n , double *a , const MKL_INT *ia , const MKL_INT *ja ,\nconst MKL_INT *desca , double *b , const MKL_INT *ib , const MKL_INT *jb , const\nMKL_INT *descb );\nvoid pclamr1d (const MKL_INT *n , MKL_Complex8 *a , const MKL_INT *ia , const MKL_INT\n*ja , const MKL_INT *desca , MKL_Complex8 *b , const MKL_INT *ib , const MKL_INT *jb ,\nconst MKL_INT *descb );\nvoid pzlamr1d (const MKL_INT *n , MKL_Complex16 *a , const MKL_INT *ia , const MKL_INT\n*ja , const MKL_INT *desca , MKL_Complex16 *b , const MKL_INT *ib , const MKL_INT *jb ,\nconst MKL_INT *descb );\nvoid clanv2 (MKL_Complex8 *a , MKL_Complex8 *b , MKL_Complex8 *c , MKL_Complex8 *d ,\nMKL_Complex8 *rt1 , MKL_Complex8 *rt2 , float *cs , MKL_Complex8 *sn );\nvoid zlanv2 (MKL_Complex16 *a , MKL_Complex16 *b , MKL_Complex16 *c , MKL_Complex16\n*d , MKL_Complex16 *rt1 , MKL_Complex16 *rt2 , double *cs , MKL_Complex16 *sn );\nvoid pclattrs (const char *uplo , const char *trans , const char *diag , const char\n*normin , const MKL_INT *n , const MKL_Complex8 *a , const MKL_INT *ia , const MKL_INT\n*ja , const MKL_INT *desca , MKL_Complex8 *x , const MKL_INT *ix , const MKL_INT *jx ,\nconst MKL_INT *descx , float *scale , float *cnorm , MKL_INT *info );\nvoid pzlattrs (const char *uplo , const char *trans , const char *diag , const char\n*normin , const MKL_INT *n , const MKL_Complex16 *a , const MKL_INT *ia , const MKL_INT\n*ja , const MKL_INT *desca , MKL_Complex16 *x , const MKL_INT *ix , const MKL_INT *jx ,\nconst MKL_INT *descx , double *scale , double *cnorm , MKL_INT *info );\nvoid pssyttrd (const char *uplo , const MKL_INT *n , float *a , const MKL_INT *ia ,\nconst MKL_INT *ja , const MKL_INT *desca , float *d , float *e , float *tau , float\n*work , const MKL_INT *lwork , MKL_INT *info );\nvoid pdsyttrd (const char *uplo , const MKL_INT *n , double *a , const MKL_INT *ia ,\nconst MKL_INT *ja , const MKL_INT *desca , double *d , double *e , double *tau , double\n*work , const MKL_INT *lwork , MKL_INT *info );\nMKL_INT piparmq (const MKL_INT *ictxt , const MKL_INT *ispec , const char *name , const\nchar *opts , const MKL_INT *n , const MKL_INT *ilo , const MKL_INT *ihi , const MKL_INT\n*lworknb );\nFor descriptions of these functions, please see http://www.netlib.org/scalapack/explore-html/files.html.\nScaLAPACK Utility Functions and Routines\nThis section describes ScaLAPACK utility functions and routines. Summary information about these routines is\ngiven in the following table:\nScaLAPACK Utility Functions and Routines\nRoutine Name\nData Types\nDescription\np?labad\ns,d\nReturns the square root of the underflow and overflow thresholds if the\nexponent-range is very large.\np?lachkieee\ns,d\nPerforms a simple check for the features of the IEEE standard.\np?lamch\ns,d\nDetermines machine parameters for floating-point arithmetic.\np?lasnbt\ns,d\nComputes the position of the sign bit of a floating-point number.\ndescinit\nN/A\nInitializes the array descriptor for distributed matrix.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1820\n\n\nRoutine Name\nData Types\nDescription\nnumroc\nN/A\nComputes the number of rows or columns of a distributed matrix\nowned by the process.\nSee Also\npxerbla  Error handling routine called by ScaLAPACK routines.\np?labad\nReturns the square root of the underflow and overflow\nthresholds if the exponent-range is very large.\nSyntax\nvoid pslabad (MKL_INT *ictxt , float *small , float *large );\nvoid pdlabad (MKL_INT *ictxt , double *small , double *large );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?labadfunction takes as input the values computed by p?lamch for underflow and overflow, and\nreturns the square root of each of these values if the log of large is sufficiently large. This function is\nintended to identify machines with a large exponent range, such as the Crays, and redefine the underflow\nand overflow limits to be the square roots of the values computed by p?lamch. This function is needed\nbecause p?lamch does not compensate for poor arithmetic in the upper half of the exponent range, as is\nfound on a Cray.\nIn addition, this function performs a global minimization and maximization on these values, to support\nheterogeneous computing networks.\nInput Parameters\nictxt\n(global)\nThe BLACS context handle in which the computation takes place.\nsmall\n(local).\nOn entry, the underflow threshold as computed by p?lamch.\nlarge\n(local).\nOn entry, the overflow threshold as computed by p?lamch.\nOutput Parameters\nsmall\n(local).\nOn exit, if log10(large) is sufficiently large, the square root of small,\notherwise unchanged.\nlarge\n(local).\nOn exit, if log10(large) is sufficiently large, the square root of large,\notherwise unchanged.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1821\n\n\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?lachkieee\nPerforms a simple check for the features of the IEEE\nstandard.\nSyntax\nvoid pslachkieee (MKL_INT *isieee , float *rmax , float *rmin );\nvoid pdlachkieee (MKL_INT *isieee , float *rmax , float *rmin );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?lachkieeefunction performs a simple check to make sure that the features of the IEEE standard are\nimplemented. In some implementations, p?lachkieee may not return.\nThis is a ScaLAPACK internal function and arguments are not checked for unreasonable values.\nInput Parameters\nrmax\n(local).\nThe overflow threshold(= ?lamch ('O')).\nrmin\n(local).\nThe underflow threshold(= ?lamch ('U')).\nOutput Parameters\nisieee\n(local).\nOn exit, isieee = 1 implies that all the features of the IEEE standard that\nwe rely on are implemented. On exit, isieee = 0 implies that some the\nfeatures of the IEEE standard that we rely on are missing.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?lamch\nDetermines machine parameters for floating-point\narithmetic.\nSyntax\nfloat pslamch (MKL_INT *ictxt , char *cmach );\ndouble pdlamch (MKL_INT *ictxt , char *cmach );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?lamchfunction determines single precision machine parameters.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1822\n\n\nInput Parameters\nictxt\n(global). The BLACS context handle in which the computation takes place.\ncmach\n(global)\nSpecifies the value to be returned by p?lamch:\n= 'E' or 'e', p?lamch := eps\n= 'S' or 's' , p?lamch := sfmin\n= 'B' or 'b', p?lamch := base\n= 'P' or 'p', p?lamch := eps*base\n= 'N' or 'n', p?lamch := t\n= 'R' or 'r', p?lamch := rnd\n= 'M' or 'm', p?lamch := emin\n= 'U' or 'u', p?lamch := rmin\n= 'L' or 'l', p?lamch := emax\n= 'O' or 'o', p?lamch := rmax,\nwhere\neps = relative machine precision\nsfmin = safe minimum, such that 1/sfmin does not overflow\nbase = base of the machine\nprec = eps*base\nt = number of (base) digits in the mantissa\nrnd = 1.0 when rounding occurs in addition, 0.0 otherwise\nemin = minimum exponent before (gradual) underflow\nrmin = underflow threshold - base(emin-1)\nemax = largest exponent before overflow\nrmax = overflow threshold - (baseemax)*(1-eps)\nOutput Parameters\nval\nValue returned by the function.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?lasnbt\nComputes the position of the sign bit of a floating-\npoint number.\nSyntax\nvoid pslasnbt (MKL_INT *ieflag );\nvoid pdlasnbt (MKL_INT *ieflag );\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1823\n\n\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?lasnbtfunction finds the position of the signbit of a single/double precision floating point number. This\nfunction assumes IEEE arithmetic, and hence, tests only the 32-nd bit (for single precision) or 32-nd and 64-\nth bits (for double precision) as a possibility for the signbit. sizeof(int) is assumed equal to 4 bytes.\nIf a compile time flag (NO_IEEE) indicates that the machine does not have IEEE arithmetic, ieflag = 0 is\nreturned.\nOutput Parameters\nieflag\nThis flag indicates the position of the signbit of any single/double precision\nfloating point number.\nieflag = 0, if the compile time flag NO_IEEE indicates that the machine\ndoes not have IEEE arithmetic, or if sizeof(int) is different from 4 bytes.\nieflag = 1 indicates that the signbit is the 32-nd bit for a single precision\nfunction.\nIn the case of a double precision function:\nieflag = 1 indicates that the signbit is the 32-nd bit (Big Endian).\nieflag = 2 indicates that the signbit is the 64-th bit (Little Endian).\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\ndescinit\nInitializes the array descriptor for distributed matrix.\nSyntax\nvoid descinit (MKL_INT *desc, const MKL_INT *m, const MKL_INT *n, const MKL_INT *mb,\nconst MKL_INT *nb, const MKL_INT *irsrc, const MKL_INT *icsrc, const MKL_INT *ictxt,\nconst MKL_INT *lld, MKL_INT *info);\nDescription\nThe descintfunction initializes the array descriptor for distributed matrix.\nInput Parameters\ndesc\n(global) array of dimension DLEN_. The array descriptor of a distributed\nmatrix to be set.\nm\n(global input) The number of rows in the distributed matrix. M >=0.\nn\n(global input) The number of columns in the distributed matrix. N >=0.\nmb\n(global input) The blocking factor used to distribute the rows of the matrix.\nMB >= 1.\nnb\n(global input) The blocking factor used to distribute the columns of the\nmatrix. NB >= 1.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1824\n\n\nlrsrc\n(global input) The process row over which the first row of the matrix is\ndistributed. 0 <= IRSRC < NPROW.\nlcsrc\n(global input) The process column over which the first column of the matrix\nis distributed. 0 <= ICSRC < NPCOL.\nictxt\n(global input) The BLACS context handle, indicating the global context of\nthe operation on the matrix. The context itself is global.\nlld\n(local input) The leading dimension of the local array storing the local\nblocks of the distributed matrix. LLD >= MAX(1,LOCr(M)). LOCr() denotes\nthe number of rows of a global dense matrix that the process in a grid\nreceives after data distributing.\nOutput Parameters\ninfo\n(output)\n= 0: successful exit\n< 0: if INFO = -i, the i-th argument had an illegal value\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\nnumroc\nComputes the number of rows or columns of a\ndistributed matrix owned by the process.\nSyntax\nMKL_INT numroc (const MKL_INT *n, const MKL_INT *nb, const MKL_INT *iproc, const\nMKL_INT *srcproc, const MKL_INT *nprocs);\nDescription\nThe numrocfunction computes the number of rows or columns of a distributed matrix owned by the process.\nInput Parameters\nn\n(global input) The number of rows/columns in distributed matrix.\nnb\n(global input) Block size, size of the blocks the distributed matrix is split\ninto.\niproc\n(local input) The coordinate of the process whose local array row or column\nis to be determined.\nsrcproc\n(global input) The coordinate of the process that possesses the first row or\ncolumn of the distributed matrix.\nnprocs\n(global input) The total number processes over which the matrix is\ndistributed.\nOutput Parameters\ninfo\n(output) Value returned by the function.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1825\n\n\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\nScaLAPACK Redistribution/Copy Routines\nThis section describes ScaLAPACK redistribution/copy routines. Summary information about these routines is\ngiven in the following table:\nScaLAPACK Redistribution/Copy Routines\nRoutine Name\nData Types\nDescription\np?gemr2d\ns,d,c,z,i\nCopies a submatrix from one general rectangular matrix to another.\np?trmr2d\ns,d,c,z,i\nCopies a submatrix from one trapezoidal matrix to another.\nSee Also\npxerbla  Error handling routine called by ScaLAPACK routines.\np?gemr2d\nCopies a submatrix from one general rectangular\nmatrix to another.\nSyntax\nvoid psgemr2d (MKL_INT *m, MKL_INT *n, float *a , MKL_INT *ia , MKL_INT *ja , MKL_INT\n*desca , float *b , MKL_INT *ib , MKL_INT *jb , MKL_INT *descb , MKL_INT *ictxt );\nvoid pdgemr2d (MKL_INT *m , MKL_INT *n , double *a , MKL_INT *ia , MKL_INT *ja , MKL_INT\n*desca , double *b , MKL_INT *ib , MKL_INT *jb , MKL_INT *descb , MKL_INT *ictxt );\nvoid pcgemr2d (MKL_INT *m , MKL_INT *n MKL_Complex8 *a , MKL_INT *ia , MKL_INT *ja ,\nMKL_INT *desca , MKL_Complex8 *b , MKL_INT *ib , MKL_INT *jb , MKL_INT *descb , MKL_INT\n*ictxt );\nvoid pzgemr2d (MKL_INT *m , MKL_INT *n , MKL_Complex16 *a , MKL_INT *ia , MKL_INT *ja ,\nMKL_INT *desca , MKL_Complex16 *b , MKL_INT *ib , MKL_INT *jb , MKL_INT *descb ,\nMKL_INT *ictxt );\nvoid pigemr2d (MKL_INT *m , MKL_INT *n , MKL_INT *a , MKL_INT *ia , MKL_INT *ja ,\nMKL_INT *desca , MKL_INT *b , MKL_INT *ib , MKL_INT *jb , MKL_INT *descb , MKL_INT\n*ictxt );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?gemr2dfunction copies the indicated matrix or submatrix of A to the indicated matrix or submatrix of\nB. It provides a truly general copy from any block cyclicly-distributed matrix or submatrix to any other block\ncyclicly-distributed matrix or submatrix. With p?trmr2d, these functions are the only ones in the ScaLAPACK\nlibrary which provide inter-context operations: they can take a matrix or submatrix A in context A\n(distributed over process grid A) and copy it to a matrix or submatrix B in context B (distributed over process\ngrid B).\nThere does not need to be a relationship between the two operand matrices or submatrices other than their\nglobal size and the fact that they are both legal block cyclicly-distributed matrices or submatrices. This\nmeans that they can, for example, be distributed across different process grids, have varying block sizes and\ndiffering matrix starting points, or be contained in different sized distributed matrices.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1826\n\n\nTake care when context A is disjoint from context B. The general rules for which parameters need to be set\nare:\n•\nAll calling processes must have the correct m and n.\n•\nProcesses in context A must correctly define all parameters describing A.\n•\nProcesses in context B must correctly define all parameters describing B.\n•\nProcesses which are not members of context A must pass ctxt_a = -1 and need not set other parameters\ndescribing A.\n•\nProcesses which are not members of contextB must pass ctxt_b = -1 and need not set other parameters\ndescribing B.\nBecause of its generality, p?gemr2d can be used for many operations not usually associated with copy\nfunctions. For instance, it can be used to a take a matrix on one process and distribute it across a process\ngrid, or the reverse. If a supercomputer is grouped into a virtual parallel machine with a workstation, for\ninstance, this function can be used to move the matrix from the workstation to the supercomputer and back.\nIn ScaLAPACK, it is called to copy matrices from a two-dimensional process grid to a one-dimensional\nprocess grid. It can be used to redistribute matrices so that distributions providing maximal performance can\nbe used by various component libraries, as well.\nNote that this function requires an array descriptor with dtype_ = 1.\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nInput Parameters\nm\n(global) The number of rows of matrix A to be copied (m≥0).\nn\n(global) The number of columns of matrix A to be copied (n≥0).\na\n(local)\nPointer into the local memory to array of size lld_a* LOCc(ja+n-1)\ncontaining the source matrix A.\nia, ja\n(global) The row and column indices in the array A indicating the first row\nand the first column, respectively, of the submatrix of A) to copy. 1\n≤ia≤total_rows_in_a - m +1, 1 ≤ja≤total_columns_in_a - n +1.\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\nOnly dtype_a = 1 is supported, so dlen_ = 9.\nIf the calling process is not part of the context of A, ctxt_a must be equal to\n-1.\nib, jb\n(global) The row and column indices in the array B indicating the first row\nand the first column, respectively, of the submatrix B to which to copy the\nmatrix. 1 ≤ib≤total_rows_in_b - m +1, 1 ≤jb≤total_columns_in_b - n +1.\ndescb\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix B.\nOnly dtype_b = 1 is supported, so dlen_ = 9.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1827\n\n\nIf the calling process is not part of the context of B, ctxt_b must be equal to\n-1.\nictxt\n(global).\nThe context encompassing at least the union of all processes in context A\nand context B. All processes in the context ictxt must call this function,\neven if they do not own a piece of either matrix.\nOutput Parameters\nb\nPointer into the local memory to array of size lld_b*LOCc(jb+n-1).\nOverwritten by the submatrix from A.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\np?trmr2d\nCopies a submatrix from one trapezoidal matrix to\nanother.\nSyntax\nvoid pstrmr2d (char *uplo , char *diag , MKL_INT *m , MKL_INT *n , float *a , MKL_INT\n*ia , MKL_INT *ja , MKL_INT *desca , float *b , MKL_INT *ib , MKL_INT *jb , MKL_INT\n*descb , MKL_INT *ictxt );\nvoid pdtrmr2d (char *uplo , char *diag , MKL_INT *m , MKL_INT *n , MKL_INT *nrhs ,\ndouble *a , MKL_INT *ia , MKL_INT *ja , MKL_INT *desca , double *b , MKL_INT *ib ,\nMKL_INT *jb , MKL_INT *descb , MKL_INT *ictxt );\nvoid pctrmr2d (char *uplo , char *diag , MKL_INT *m , MKL_INT *n , MKL_INT *nrhs ,\nMKL_Complex8 *a , MKL_INT *ia , MKL_INT *ja , MKL_INT *desca , MKL_Complex8 *b ,\nMKL_INT *ib , MKL_INT *jb , MKL_INT *descb , MKL_INT *ictxt );\nvoid pztrmr2d (char *uplo , char *diag , MKL_INT *m , MKL_INT *n , MKL_INT *nrhs ,\nMKL_Complex16 *a , MKL_INT *ia , MKL_INT *ja , MKL_INT *desca , MKL_Complex16 *b ,\nMKL_INT *ib , MKL_INT *jb , MKL_INT *descb , MKL_INT *ictxt );\nvoid pitrmr2d (char *uplo , char *diag , MKL_INT *m , MKL_INT *n , MKL_INT *a , MKL_INT\n*ia , MKL_INT *ja , MKL_INT *desca , MKL_INT *b , MKL_INT *ib , MKL_INT *jb , MKL_INT\n*descb , MKL_INT *ictxt );\nInclude Files\n•\nmkl_scalapack.h\nDescription\nThe p?trmr2dfunction copies the indicated matrix or submatrix of A to the indicated matrix or submatrix of\nB. It provides a truly general copy from any block cyclicly-distributed matrix or submatrix to any other block\ncyclicly-distributed matrix or submatrix. With p?gemr2d, these functions are the only ones in the ScaLAPACK\nlibrary which provide inter-context operations: they can take a matrix or submatrix A in context A\n(distributed over process grid A) and copy it to a matrix or submatrix B in context B (distributed over process\ngrid B).\nThe p?trmr2dfunction assumes the matrix or submatrix to be trapezoidal. Only the upper or lower part is\ncopied, and the other part is unchanged.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1828\n\n\nThere does not need to be a relationship between the two operand matrices or submatrices other than their\nglobal size and the fact that they are both legal block cyclicly-distributed matrices or submatrices. This\nmeans that they can, for example, be distributed across different process grids, have varying block sizes and\ndiffering matrix starting points, or be contained in different sized distributed matrices.\nTake care when context A is disjoint from context B. The general rules for which parameters need to be set\nare:\n•\nAll calling processes must have the correct m and n.\n•\nProcesses in context A must correctly define all parameters describing A.\n•\nProcesses in context B must correctly define all parameters describing B.\n•\nProcesses which are not members of context A must pass ctxt_a = -1 and need not set other parameters\ndescribing A.\n•\nProcesses which are not members of contextB must pass ctxt_b = -1 and need not set other parameters\ndescribing B.\nBecause of its generality, p?trmr2d can be used for many operations not usually associated with copy\nfunctions. For instance, it can be used to a take a matrix on one process and distribute it across a process\ngrid, or the reverse. If a supercomputer is grouped into a virtual parallel machine with a workstation, for\ninstance, this function can be used to move the matrix from the workstation to the supercomputer and back.\nIn ScaLAPACK, it is called to copy matrices from a two-dimensional process grid to a one-dimensional\nprocess grid. It can be used to redistribute matrices so that distributions providing maximal performance can\nbe used by various component libraries, as well.\nNote that this function requires an array descriptor with dtype_ = 1.\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nInput Parameters\nuplo\n(global) Specifies whether to copy the upper or lower part of the matrix or\nsubmatrix.\nuplo = 'U'\nCopy the upper triangular part.\nuplo = 'L'\nCopy the lower triangular part.\ndiag\n(global) Specifies whether to copy the diagonal of the matrix or submatrix.\ndiag = 'U'\nDo not copy the diagonal.\ndiag = 'N'\nCopy the diagonal.\nm\n(global) The number of rows of matrix A to be copied (m≥0).\nn\n(global) The number of columns of matrix A to be copied (n≥0).\na\n(local)\nPointer into the local memory to array of size lld_a* LOCc(ja+n-1)\ncontaining the source matrix A.\nia, ja\n(global) The row and column indices in the array A indicating the first row\nand the first column, respectively, of the submatrix of A) to copy. 1\n≤ia≤total_rows_in_a - m +1, 1 ≤ja≤total_columns_in_a - n +1.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1829\n\n\ndesca\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix A.\nOnly dtype_a = 1 is supported, so dlen_ = 9.\nIf the calling process is not part of the context of A, ctxt_a must be equal to\n-1.\nib, jb\n(global) The row and column indices in the array B indicating the first row\nand the first column, respectively, of the submatrix B to which to copy the\nmatrix. 1 ≤ib≤total_rows_in_b - m +1, 1 ≤jb≤total_columns_in_b - n +1.\ndescb\n(global and local) array of size dlen_. The array descriptor for the\ndistributed matrix B.\nOnly dtype_b = 1 is supported, so dlen_ = 9.\nIf the calling process is not part of the context of B, ctxt_b must be equal to\n-1.\nictxt\n(global).\nThe context encompassing at least the union of all processes in context A\nand context B. All processes in the context ictxt must call this function,\neven if they do not own a piece of either matrix.\nOutput Parameters\nb\nPointer into the local memory to array of size lld_b*LOCc(jb+n-1).\nOverwritten by the submatrix from A.\nSee Also\nOverview  for details of ScaLAPACK array descriptor structures and related notations.\nSparse Solver Routines\nIntel® oneAPI Math Kernel Library (oneMKL) sparse solver algorithms for solving real or complex, symmetric,\nstructurally symmetric or nonsymmetric, positive definite, indefinite or Hermitian square sparse linear system\nof algebraic equations.\nThe terms and concepts required to understand the use of the Intel® oneAPI Math Kernel Library (oneMKL)\nsparse solver routines are discussed in the Appendix \"Linear Solvers Basics\". If you are familiar with linear\nsparse solvers and sparse matrix storage schemes, you can skip these sections and go directly to the\ninterface descriptions.\nSee the description of\n•\nthe direct sparse solver based on PARDISO*, which is referred to here as Intel MKL PARDISO;\n•\nthe alternative interface for the direct sparse solver, which is referred to here as the DSS interface;\n•\niterative sparse solvers (ISS) based on the reverse communication interface (RCI);\n•\npreconditioners based on the incomplete LU factorization technique.\n•\na direct sparse solver based on QR decomposition.\noneMKL PARDISO - Parallel Direct Sparse Solver Interface\nThis section describes the interface to the shared-memory multiprocessing parallel direct sparse solver\nknown as the Intel® oneAPI Math Kernel Library (oneMKL) PARDISO solver.\nThe Intel® oneAPI Math Kernel Library (oneMKL) PARDISO package is a high-performance, robust, memory\nefficient, and easy to use software package for solving large sparse linear systems of equations on shared\nmemory multiprocessors. The solver uses a combination of left- and right-looking Level-3 BLAS supernode\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1830\n\n\ntechniques [Schenk00-2]. To improve sequential and parallel sparse numerical factorization performance, the\nalgorithms are based on a Level-3 BLAS update and pipelining parallelism is used with a combination of left-\nand right-looking supernode techniques [Schenk00, Schenk01, Schenk02, Schenk03]. The parallel pivoting\nmethods allow complete supernode pivoting to compromise numerical stability and scalability during the\nfactorization process. For sufficiently large problem sizes, numerical experiments demonstrate that the\nscalability of the parallel algorithm is nearly independent of the shared-memory multiprocessing architecture.\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nThe following table lists the names of the Intel® oneAPI Math Kernel Library (oneMKL) PARDISO routines and\ndescribes their general use.\noneMKL PARDISO Routines\nRoutine\nDescription\npardisoinit\nInitializes Intel® oneAPI Math Kernel Library (oneMKL)\nPARDISO with default parameters depending on the\nmatrix type.\npardiso\nCalculates the solution of a set of sparse linear equations\nwith single or multiple right-hand sides.\npardiso_64\nCalculates the solution of a set of sparse linear equations\nwith single or multiple right-hand sides, 64-bit integer\nversion.\nmkl_pardiso_pivot\nReplaces routine which handles Intel® oneAPI Math Kernel\nLibrary (oneMKL) PARDISO pivots with user-defined\nroutine.\npardiso_getdiag\nReturns diagonal elements of initial and factorized matrix.\npardiso_export\nPlaces pointers dedicated for sparse representation of\nrequested matrix into MKL PARDISO.\npardiso_handle_store\nStore internal structures from pardiso to a file.\npardiso_handle_restore\nRestore pardiso internal structures from a file.\npardiso_handle_delete\nDelete files with pardiso internal structure data.\npardiso_handle_store_64\nStore internal structures from pardiso_64 to a file.\npardiso_handle_restore_64\nRestore pardiso_64 internal structures from a file.\npardiso_handle_delete_64\nDelete files with pardiso_64 internal structure data.\nThe Intel® oneAPI Math Kernel Library (oneMKL) PARDISO solver supports a wide range of real and complex\nsparse matrix types (seethe figure below).\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1831\n\n\n__border__top\nSparse Matrices That Can Be Solved with the oneMKL PARDISO Solver\nThe Intel® oneAPI Math Kernel Library (oneMKL) PARDISO solver performs four tasks:\n•\nanalysis and symbolic factorization\n•\nnumerical factorization\n•\nforward and backward substitution including iterative refinement\n•\ntermination to release all internal solver memory.\nTo find code examples that use Intel® oneAPI Math Kernel Library (oneMKL) PARDISO routines to solve\nsystems of linear equations, unzip theC archive file in the examplesfolder of the Intel® oneAPI Math Kernel\nLibrary (oneMKL) installation directory. Code examples will be in theexamples/solverc/source folder.\nSupported Matrix Types\nThe analysis steps performed by Intel® oneAPI Math Kernel Library (oneMKL) PARDISO depend on the\nstructure of the input matrixA.\nSymmetric Matrices\nThe solver first computes a symmetric fill-in reducing permutation P based on\neither the minimum degree algorithm [Liu85] or the nested dissection algorithm\nfrom the METIS package [Karypis98] (both included with Intel® oneAPI Math\nKernel Library (oneMKL)), followed by the parallel left-right looking numerical\nCholesky factorization [Schenk00-2] of PAPT = LLT for symmetric positive-\ndefinite matrices, or PAPT = LDLT for symmetric indefinite matrices. The solver\nuses diagonal pivoting, or 1x1 and 2x2 Bunch-Kaufman pivoting for symmetric\nindefinite matrices. An approximation of X is found by forward and backward\nsubstitution and optional iterative refinement.\nWhenever numerically acceptable 1x1 and 2x2 pivots cannot be found within the\ndiagonal supernode block, the coefficient matrix is perturbed. One or two passes\nof iterative refinement may be required to correct the effect of the perturbations.\nThis restricting notion of pivoting with iterative refinement is effective for highly\nindefinite symmetric systems. Furthermore, for a large set of matrices from\ndifferent applications areas, this method is as accurate as a direct factorization\nmethod that uses complete sparse pivoting techniques [Schenk04].\nAnother method of improving the pivoting accuracy is to use symmetric weighted\nmatching algorithms. These algorithms identify large entries in the coefficient\nmatrix A that, if permuted close to the diagonal, permit the factorization process\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1832\n\n\nto identify more acceptable pivots and proceed with fewer pivot perturbations.\nThese algorithms are based on maximum weighted matchings and improve the\nquality of the factor in a complementary way to the alternative of using more\ncomplete pivoting techniques.\nThe inertia is also computed for real symmetric indefinite matrices.\nStructurally Symmetric\nMatrices\nThe solver first computes a symmetric fill-in reducing permutation P followed by\nthe parallel numerical factorization of PAPT = QLUT. The solver uses partial\npivoting in the supernodes and an approximation of X is found by forward and\nbackward substitution and optional iterative refinement.\nNonsymmetric Matrices\nThe solver first computes a nonsymmetric permutation PMPS and scaling matrices\nDr and Dc with the aim of placing large entries on the diagonal to enhance\nreliability of the numerical factorization process [Duff99]. In the next step the\nsolver computes a fill-in reducing permutation P based on the matrix PMPSA +\n(PMPSA)T followed by the parallel numerical factorization\nQLUR = PPMPSDrADcP\nwith supernode pivoting matrices Q and R. When the factorization algorithm\nreaches a point where it cannot factor the supernodes with this pivoting strategy,\nit uses a pivoting perturbation strategy similar to [Li99]. The magnitude of the\npotential pivot is tested against a constant threshold of\nalpha = eps*||A2||inf,\nwhere eps is the machine precision, A2 = P*PMPS*Dr*A*Dc*P, and ||A2||inf is\nthe infinity norm of A. Any tiny pivots encountered during elimination are set to\nthe sign (lII)*eps*||A2||inf, which trades off some numerical stability for the\nability to keep pivots from getting too small. Although many failures could render\nthe factorization well-defined but essentially useless, in practice the diagonal\nelements are rarely modified for a large class of matrices. The result of this\npivoting approach is that the factorization is, in general, not exact and iterative\nrefinement may be needed.\nSparse Data Storage\nIntel® oneAPI Math Kernel Library (oneMKL) PARDISO stores sparse data in several formats:\n•\nCSR3: The 3-array variation of the compressed sparse row format described in Three Array Variation of\nCSR Format.\n•\nBSR3: The three-array variation of the block compressed sparse row format described in Three Array\nVariation of BSR Format. Use iparm[36] to specify the block size.\n•\nVBSR: Variable BSR format. Intel® oneAPI Math Kernel Library (oneMKL) PARDISO analyzes the matrix\nprovided in CSR3 format and converts it into an internal structure which can improve performance for\nmatrices with a block structure. Useiparm[36] = -t (0 < t≤ 100) to specify use of internal VBSR format\nand to set the degree of similarity required to combine elements of the matrix. For example, if you set\niparm[36] = -80, two rows of the input matrix are combined when their non-zero patterns are 80% or\nmore similar.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1833\n\n\nNOTE\nIntel® oneAPI Math Kernel Library (oneMKL) supports only the VBSR format for real and symmetric\npositive definite or indefinite matrices (mtype = 2 or mtype = -2).\nIntel® oneAPI Math Kernel Library (oneMKL) supports these features for all matrix types as long\nasiparm[23]=1:\n•\niparm[30] > 0: Partial solution\n•\niparm[35] > 0: Schur complement\n•\niparm[59] > 0: OOC Intel® oneAPI Math Kernel Library (oneMKL) PARDISO\nFor all storage formats, the Intel® oneAPI Math Kernel Library (oneMKL) PARDISO parameterja is used for\nthe columns array, ia is used for rowIndex, and a is used for values. The algorithms in Intel® oneAPI Math\nKernel Library (oneMKL) PARDISO require column indicesja to be in increasing order per row and that the\ndiagonal element in each row be present for any structurally symmetric matrix. For symmetric or\nnonsymmetric matrices the diagonal elements which are equal to zero are not necessary.\nCaution\nIntel® oneAPI Math Kernel Library (oneMKL) PARDISO column indicesja must be in increasing order\nper row. You can validate the sparse matrix structure with the matrix checker (iparm[26])\nNOTE\nWhile the presence of zero diagonal elements for symmetric matrices is not required, you should\nexplicitly set zero diagonal elements for symmetric matrices. Otherwise, Intel® oneAPI Math Kernel\nLibrary (oneMKL) PARDISO creates internal copies of arraysia, ja, and a full of diagonal elements,\nwhich require additional memory and computational time. However, the memory and time required the\ndiagonal elements in internal arrays is usually not significant compared to the memory and the time\nrequired to factor and solve the matrix.\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nStorage of Matrices\nBy default, Intel® oneAPI Math Kernel Library (oneMKL) PARDISO stores data in RAM. This is referred to as\nIn-Core (IC) mode. However, you can specify that Intel® oneAPI Math Kernel Library (oneMKL) PARDISO\nstore matrices on disk by settingiparm[59]. This mode is called the Out-of-Core (OOC) mode.\nYou can set the following parameters for the OOC mode.\nParameter/Environment Variable\nName\nDescription\nMKL_PARDISO_OOC_PATH\nDirectory for storing data created in the OOC mode.\nMKL_PARDISO_OOC_FILE_NAME\nFull file name (incl. path) which will be used for the OOC files\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1834\n\n\nParameter/Environment Variable\nName\nDescription\nMKL_PARDISO_OOC_MAX_CORE_SIZE\nMaximum size of RAM (in megabytes) available for Intel®\noneAPI Math Kernel Library (oneMKL) PARDISO\nMKL_PARDISO_OOC_MAX_SWAP_SIZE\nMaximum swap size (in megabytes) available for Intel® oneAPI\nMath Kernel Library (oneMKL) PARDISO\nMKL_PARDISO_OOC_KEEP_FILE\nA flag which determines whether temporary data files will be\ndeleted or stored\nBy default, the current working directory is used in the OOC mode as a directory path for storing data. All\nwork arrays will be stored in files named ooc_temp with different extensions. When\nMKL_PARDISO_OOC_FILE_NAME is not set and MKL_PARDISO_OOC_PATH is set, the names for the created files\nwill contain <path>/mkl_pardiso or <path>\\mkl_pardiso depending on the OS. Setting\nMKL_PARDISO_OOC_FILE_NAME=<filename> will override the path which could have been set in\nMKL_PARDISO_OOC_PATH. In this case <filename> will be used for naming the OOC files.\nBy default, MKL_PARDISO_OOC_MAX_CORE_SIZE is 2000 (MB) and MKL_PARDISO_OOC_MAX_SWAP_SIZE is 0.\nNOTE\nDo not set the sum of MKL_PARDISO_OOC_MAX_CORE_SIZE and MKL_PARDISO_OOC_MAX_SWAP_SIZE\ngreater than the size of the RAM plus the size of the swap memory. Be sure to allow enough free\nmemory for the operating system and any other processes which need to be running.\nBy default, all temporary data files will be deleted. For keeping them it is required to set\nMKL_PARDISO_OOC_KEEP_FILE to 0.\nOOC parameters can be set in a configuration file. You can set the path to this file and its name using\nenvironmental variables MKL_PARDISO_OOC_CFG_PATH and MKL_PARDISO_OOC_CFG_FILE_NAME.\nFor setting parameters of OOC mode either environment variables or a configuration file can be used. When\nthe last option is chosen, by default the name of the file is pardiso_ooc.cfg and it should be placed in the\nworking directory. If needed, the user can set the path to the configuration file using environmental variables\nMKL_PARDISO_OOC_CFG_PATH and MKL_PARDISO_OOC_CFG_FILE_NAME. These variables specify the path and\nfilename as follows:\n•\nLinux* OS and OS X*: <MKL_PARDISO_OOC_CFG_PATH>/ <MKL_PARDISO_OOC_CFG_FILE_NAME>\n•\nWindows* OS: <MKL_PARDISO_OOC_CFG_PATH>\\<MKL_PARDISO_OOC_CFG_FILE_NAME>\nAn example of the configuration file:\nMKL_PARDISO_OOC_PATH = <path>\nMKL_PARDISO_OOC_MAX_CORE_SIZE = N\nMKL_PARDISO_OOC_MAX_SWAP_SIZE = K\nMKL_PARDISO_OOC_KEEP_FILE = 0 (or 1)\nCaution\nThe maximum length of the path lines in the configuration files is 1000 characters.\nAlternatively, the OOC parameters can be set as environment variables via command line.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1835\n\n\nFor Linux* OS and OS X*:\nexport MKL_PARDISO_OOC_PATH = <path>\nexport MKL_PARDISO_OOC_MAX_CORE_SIZE = N\nexport MKL_PARDISO_OOC_MAX_SWAP_SIZE = K\nexport MKL_PARDISO_OOC_KEEP_FILE = 0 (or 1)\nFor Windows* OS:\nset MKL_PARDISO_OOC_PATH = <path>\nset MKL_PARDISO_OOC_MAX_CORE_SIZE = N\nset MKL_PARDISO_OOC_MAX_SWAP_SIZE = K\nset MKL_PARDISO_OOC_KEEP_FILE = 0 (or 1)\nwhere <path> should follow the OS naming convention.\nDirect-Iterative Preconditioning for Nonsymmetric Linear Systems\nThe solver uses a combination of direct and iterative methods [Sonn89] to accelerate the linear solution\nprocess for transient simulation. Most applications of sparse solvers require solutions of systems with\ngradually changing values of the nonzero coefficient matrix, but with an identical sparsity pattern. In these\napplications, the analysis phase of the solvers has to be performed only once and the numerical\nfactorizations are the important time-consuming steps during the simulation. Intel® oneAPI Math Kernel\nLibrary (oneMKL) PARDISO uses a numerical factorization and applies the factors in a preconditioned Krylov\nSubspace iteration. If the iteration does not converge, the solver automatically switches back to the\nnumerical factorization. This method can be applied to nonsymmetric matrices in Intel® oneAPI Math Kernel\nLibrary (oneMKL) PARDISO. You can select the method using theiparm[3] input parameter. The\niparm[19]parameter returns the error status after running Intel® oneAPI Math Kernel Library (oneMKL)\nPARDISO.\nSingle and Double Precision Computations\nIntel® oneAPI Math Kernel Library (oneMKL) PARDISO solves tasks using single or double precision. Each\nprecision has its benefits and drawbacks. Double precision variables have more digits to store value, so the\nsolver uses more memory for keeping data. But this mode solves matrices with better accuracy, which is\nespecially important for input matrices with large condition numbers.\nSingle precision variables have fewer digits to store values, so the solver uses less memory than in the\ndouble precision mode. Additionally this mode usually takes less time. But as computations are made less\nprecisely, only some systems of equations can be solved accurately enough using single precision.\nSeparate Forward and Backward Substitution\nThe solver execution step (see parameterphase = 33 below) can be divided into two or three separate\nsubstitutions: forward, backward, and possible diagonal. This separation can be explained by the examples of\nsolving systems with different matrix types.\nA real symmetric positive definite matrix A (mtype = 2) is factored by Intel® oneAPI Math Kernel Library\n(oneMKL) PARDISO asA = L*LT . In this case the solution of the system A*x=b can be found as sequence of\nsubstitutions: L*y=b (forward substitution, phase =331) andLT*x=y (backward substitution, phase =333).\nA real nonsymmetric matrix A (mtype = 11) is factored by Intel® oneAPI Math Kernel Library (oneMKL)\nPARDISO asA = L*U . In this case the solution of the system A*x=b can be found by the following sequence:\nL*y=b (forward substitution, phase =331) andU*x=y (backward substitution, phase =333).\nSolving a system with a real symmetric indefinite matrix A (mtype = -2) is slightly different from the cases\nabove. Intel® oneAPI Math Kernel Library (oneMKL) PARDISO factors this matrix asA=LDLT, and the solution\nof the system A*x=b can be calculated as the following sequence of substitutions: L*y=b (forward\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1836\n\n\nsubstitution, phase =331), D*v=y (diagonal substitution, phase =332), and finally LT*x=v (backward\nsubstitution, phase =333). Diagonal substitution makes sense only for symmetric indefinite matrices (mtype\n= -2, -4, 6). For matrices of other types a solution can be found as described in the first two examples.\nCaution\nThe number of refinement steps (iparm[7]) must be set to zero if a solution is calculated with\nseparate substitutions (phase = 331, 332, 333), otherwise Intel® oneAPI Math Kernel Library\n(oneMKL) PARDISO produces the wrong result.\nNOTE\nDifferent pivoting (iparm[20]) produces different LDLT factorization. Therefore results of forward,\ndiagonal and backward substitutions with diagonal pivoting can differ from results of the same steps\nwith Bunch-Kaufman pivoting. Of course, the final results of sequential execution of forward, diagonal\nand backward substitution are equal to the results of the full solving step (phase=33) regardless of the\npivoting used.\nCallback Function for Pivoting Control\nIn-core Intel® oneAPI Math Kernel Library (oneMKL) PARDISO allows you to control pivoting with a callback\nroutine,mkl_pardiso_pivot. You can then use the pardiso_getdiag routine to access the diagonal elements.\nSet iparm[55] to 1 in order to use the callback functionality.\nLow Rank Update\nUse low rank update to accelerate the factorization step in Intel® oneAPI Math Kernel Library (oneMKL)\nPARDISO when you use multiple matrices with identical structure and similar values. After callingpardiso in\nthe usual manner for factorization (phase = 12, 13, 22, or 23) for some matrix A1, low rank update can be\napplied to the factorization step (phase = 22 or 23) of some matrix A2 with identical structure.\nTo use the low rank update feature, set iparm[38] = 1 while also setting iparm[23] = 10. Additionally,\nsupply an array that lists the values in A2 that are different from A1 using the perm parameter as outlined in\nthe pardiso perm parameter description.\nImportant\nLow rank update can only be called for matrices with the exact same pattern of nonzero values. As\nsuch, the value of the mtype, ia, ja, and iparm[23] parameters should also be identical. In general,\nthe low rank factorization should be called with the same parameters as the preceding factorization\nstep for the same internal data structure handle (except for array a, iparm[38], and perm).\nLow rank update does not currently support Intel TBB threading. In this case, Intel® oneAPI Math\nKernel Library (oneMKL) PARDISO defaults to full factorization instead.\nLow rank update cannot be used in combination with a user-supplied permutation vector - in other\nwords, you must use the default values of iparm[4] = 0, iparm[30] = 0, and iparm[35] = 0).\nAdditionally, iparm[3], iparm[5], iparm[27], iparm[36], iparm[55], and iparm[59] must all be\nset to the default value of 0.\npardiso\nCalculates the solution of a set of sparse linear\nequations with single or multiple right-hand sides.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1837\n\n\nSyntax\nvoid pardiso (_MKL_DSS_HANDLE_t pt, const MKL_INT *maxfct, const MKL_INT *mnum, const\nMKL_INT *mtype, const MKL_INT *phase, const MKL_INT *n, const void *a, const MKL_INT\n*ia, const MKL_INT *ja, MKL_INT *perm, const MKL_INT *nrhs, MKL_INT *iparm, const\nMKL_INT *msglvl, void *b, void *x, MKL_INT *error);\nInclude Files\n•\nmkl.h\nDescription\nThe pardiso routine calculates the solution of a set of sparse linear equations\nA*X = B\nwith single or multiple right-hand sides, using a parallel LU, LDL, or LLT factorization, where A is an n-by-n\nmatrix, and X and B are n-by-nrhs vectors or matrices.\nNotes\n•\nThis routine supports usage of the mkl_progress with OpenMP, TBB, and sequential threading. See \nmkl_progress for details. The case of iparm[23]=10 does not support this feature.\n•\nIf iparm[26] is set to 1 (Matrix checker), Intel® oneAPI Math Kernel Library PARDISO uses the\nauxiliary routine sparse_matrix_checker to check integer arrays ia and ja.\nsparse_matrix_checker has its own set of error values (from 21 to 24) that are returned in the\nevent of an unsuccessful matrix check. For more details, refer to the sparse_matrix_checker\ndocumentation.\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nInput Parameters\npt\nArray with size of 64.\nHandle to internal data structure. The entries must be set to zero prior to\nthe first call to pardiso. Unique for factorization.\nCaution\nAfter the first call to pardiso do not directly modify pt, as that\ncould cause a serious memory leak.\nUse the pardiso_handle_store or pardiso_handle_store_64 routine to\nstore the content of pt to a file. Restore the contents of pt from the file\nusing pardiso_handle_restore or pardiso_handle_restore_64. Use\npardiso_handle_store and pardiso_handle_restore with pardiso,\nand pardiso_handle_store_64 and pardiso_handle_restore_64 with\npardiso_64.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1838\n\n\nmaxfct\nMaximum number of factors with identical sparsity structure that must be\nkept in memory at the same time. In most applications this value is equal\nto 1. It is possible to store several different factorizations with the same\nnonzero structure at the same time in the internal data structure\nmanagement of the solver.\npardiso can process several matrices with an identical matrix sparsity\npattern and it can store the factors of these matrices at the same time.\nMatrices with a different sparsity structure can be kept in memory with\ndifferent memory address pointers pt.\nmnum\nIndicates the actual matrix for the solution phase. With this scalar you can\ndefine which matrix to factorize. The value must be: 1 ≤mnum≤maxfct.\nIn most applications this value is 1.\nmtype\nDefines the matrix type, which influences the pivoting method. The Intel®\noneAPI Math Kernel Library (oneMKL) PARDISO solver supports the\nfollowing matrices:\n1\nreal and structurally symmetric\n2\nreal and symmetric positive definite\n-2\nreal and symmetric indefinite\n3\ncomplex and structurally symmetric\n4\ncomplex and Hermitian positive definite\n-4\ncomplex and Hermitian indefinite\n6\ncomplex and symmetric\n11\nreal and nonsymmetric\n13\ncomplex and nonsymmetric\nphase\nControls the execution of the solver. Usually it is a two- or three-digit\ninteger. The first digit indicates the starting phase of execution and the\nsecond digit indicates the ending phase. Intel® oneAPI Math Kernel Library\n(oneMKL) PARDISO has the following phases of execution:\n•\nPhase 1: Fill-reduction analysis and symbolic factorization\n•\nPhase 2: Numerical factorization\n•\nPhase 3: Forward and Backward solve including optional iterative\nrefinement\nThis phase can be divided into two or three separate substitutions:\nforward, backward, and diagonal (see Separate Forward and Backward\nSubstitution).\n•\nMemory release phase (phase= 0 or phase= -1)\nIf a previous call to the routine has computed information from previous\nphases, execution may start at any phase. The phase parameter can have\nthe following values:\nphase\nSolver Execution Steps\n11\nAnalysis\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1839\n\n\nphase\nSolver Execution Steps\n12\nAnalysis, numerical factorization\n13\nAnalysis, numerical factorization, solve, iterative\nrefinement\n22\nNumerical factorization\n23\nNumerical factorization, solve, iterative refinement\n33\nSolve, iterative refinement\n331\nlike phase=33, but only forward substitution\n332\nlike phase=33, but only diagonal substitution (if\navailable)\n333\nlike phase=33, but only backward substitution\n0\nRelease internal memory for L and U matrix number\nmnum\n-1\nRelease all internal memory for all matrices\nIf iparm[35] = 0, phases 331, 332, and 333 perform this decomposition:\nA =\nL11\n0\nL12 L22\nD11\n0\n0\nD22\nU11 U21\n0\nU22\nIf iparm[35] = 2, phases 331, 332, and 333 perform a different\ndecomposition:\nA =\nL11 0\nL12 I\nI 0\n0 S\nU11 U21\n0\nI\nYou can supply a custom implementation for phase 332 instead of calling\npardiso. For example, it can be implemented with dense LAPACK\nfunctionality. Custom implementation also allows you to substitute the\nmatrix S with your own.\nNOTE\nFor very large Schur complement matrices use LAPACK\nfunctionality to compute the Schur complement vector instead\nof the Intel® oneAPI Math Kernel Library (oneMKL) PARDISO\nphase 332 implementation.\nn\nNumber of equations in the sparse linear systems of equations A*X = B.\nConstraint: n > 0.\na\nArray. Contains the non-zero elements of the coefficient matrix A\ncorresponding to the indices in ja. The coefficient matrix can be either real\nor complex. The matrix must be stored in the three-array variant of the\ncompressed sparse row (CSR3) or in the three-array variant of the block\ncompressed sparse row (BSR3) format, and the matrix must be stored with\nincreasing values of ja for each row.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1840\n\n\nFor CSR3 format, the size of a is the same as that of ja. Refer to the\nvalues array description in Three Array Variation of CSR Format for more\ndetails.\nFor BSR3 format the size of a is the size of ja multiplied by the square of\nthe block size. Refer to the values array description in Three Array\nVariation of BSR Format for more details.\nNOTE\nIf you set iparm[36]to a negative value, Intel® oneAPI Math\nKernel Library (oneMKL) PARDISO converts the data from CSR3\nformat to an internal variable BSR (VBSR) format. SeeSparse\nData Storage.\nia\nArray, size (n+1).\nFor CSR3 format, ia[i] (i<n) points to the first column index of row i in\nthe array ja. That is, ia[i] gives the index of the element in array a that\ncontains the first non-zero element from row i of A. The last element ia[n]\nis taken to be equal to the number of non-zero elements in A, plus one.\nRefer to rowIndex array description in Three Array Variation of CSR Format\nfor more details.\nFor BSR3 format, ia[i] (i<n) points to the first column index of row i in\nthe array ja. That is, ia[i] gives the index of the element in array a that\ncontains the first non-zero block from row i of A. The last element ia[n] is\ntaken to be equal to the number of non-zero blcoks in A, plus one. Refer to\nrowIndex array description in Three Array Variation of BSR Format for more\ndetails.\nThe array ia is accessed in all phases of the solution process.\nIndexing of ia is one-based by default, but it can be changed to zero-based\nby setting the appropriate value to the parameter iparm[34].\nja\nFor CSR3 format, array ja contains column indices of the sparse matrix A.\nIt is important that the indices are in increasing order per row. For\nstructurally symmetric matrices it is assumed that all diagonal elements are\nstored (even if they are zeros) in the list of non-zero elements in a and ja.\nFor symmetric matrices, the solver needs only the upper triangular part of\nthe system as is shown for columns array in Three Array Variation of CSR\nFormat.\nFor BSR3 format, array ja contains column indices of the sparse matrix A.\nIt is important that the indices are in increasing order per row. For\nstructurally symmetric matrices it is assumed that all diagonal blocks are\nstored (even if they are zeros) in the list of non-zero blocks in a and ja. For\nsymmetric matrices, the solver needs only the upper triangular part of the\nsystem as is shown for columns array in Three Array Variation of BSR\nFormat.\nThe array ja is accessed in all phases of the solution process.\nIndexing of ja is one-based by default, but it can be changed to zero-based\nby setting the appropriate value to the parameter iparm[34].\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1841\n\n\nperm\nArray, size (n). Depending on the value of iparm[4] and iparm[30], holds\nthe permutation vector of size n, specifies elements used for computing a\npartial solution, or specifies differing values of the input matrices for low\nrank update.\n•\nIf iparm[4] = 1, iparm[30] = 0, and iparm[35] = 0, perm specifies\nthe fill-in reducing ordering to the solver. Let A be the original matrix\nand C = P*A*PT be the permuted matrix. Row (column) i of C is the\nperm[i] row (column) of A. The array perm is also used to return the\npermutation vector calculated during fill-in reducing ordering stage.\nNOTE\nBe aware that setting iparm[4] = 1 prevents use of a parallel\nalgorithm for the solve step.\n•\nIf iparm[4] = 2, iparm[30] = 0, and iparm[35] = 0, the permutation\nvector computed in phase 11 is returned in the perm array.\n•\nIf iparm[4] = 0, iparm[30] > 0, and iparm[35] = 0, perm specifies\nelements of the right-hand side to use or of the solution to compute for\na partial solution.\n•\nIf iparm[4] = 0, iparm[30] = 0, and iparm[35] > 0, perm specifies\nelements for a Schur complement.\n•\nIf iparm[38] = 1, perm specifies values that differ in A for low rank\nupdate (see Low Rank Update). The size of the array must be at least\n2*ndiff + 1, where ndiff is the number of values of A that are different.\nThe values of perm should be:\nperm = {ndiff, row_index1, column_index1, row_index2,\ncolumn_index2, ...., row_index_ndiff, column_index_ndiff}\nwhere row_index_m and column_index_m are the row and column\nindices of the m-th differing non-zero value in matrix A. The row and\ncolumn index pairs can be in any order, but must use zero-based\nindexing regardless of the value of iparm[34].\nSee iparm[4], iparm[30], and iparm[38] for more details.\nIndexing of perm is one-based by default, but unless iparm[38] = 1 it can\nbe changed to zero-based by setting the appropriate value to the parameter \niparm[34].\nnrhs\nNumber of right-hand sides that need to be solved for.\niparm\nArray, size (64). This array is used to pass various parameters to Intel®\noneAPI Math Kernel Library (oneMKL) PARDISO and to return some useful\ninformation after execution of the solver.\nSee pardiso iparm Parameter for more details about the iparm parameters.\nmsglvl\nMessage level information. If msglvl = 0 then pardiso generates no\noutput, if msglvl = 1 the solver prints statistical information to the screen.\nb\nArray, size (n*nrhs). On entry, contains the right-hand side vector/matrix\nB, which is placed in memory contiguously. The b[+k*nrhs] element must\nhold the i-th component of k-th right-hand side vector. Note that b is only\naccessed in the solution phase.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1842\n\n\nOutput Parameters\n(See also Intel MKL PARDISO Parameters in Tabular Form.)\npt\nHandle to internal data structure.\nperm\nSee the Input Parameter description of the perm array.\niparm\nOn output, some iparm values report information such as the numbers of\nnon-zero elements in the factors.\nSee pardiso iparm Parameter for more details about the iparm parameters.\nb\nOn output, the array is replaced with the solution if iparm[5] = 1.\nx\nArray, size (n*nrhs). If iparm[5]=0 it contains solution vector/matrix X,\nwhich is placed contiguously in memory. The x[i + k*n] element must\nhold the i-th component of the k-th solution vector. Note that x is only\naccessed in the solution phase.\nerror\nThe error indicator according to the below table:\nerror\nInformation\n0\nno error\n-1\ninput inconsistent\n-2\nnot enough memory\n-3\nreordering problem\n-4\nZero pivot, numerical factorization or iterative\nrefinement problem. If the error appears during the\nsolution phase, try to change the pivoting perturbation\n(iparm[9]) and also increase the number of iterative\nrefinement steps. If it does not help, consider changing\nthe scaling, matching and pivoting options (iparm[10],\niparm[12], iparm[20])\n-5\nunclassified (internal) error\n-6\nreordering failed (matrix types 11 and 13 only)\n-7\ndiagonal matrix is singular\n-8\n32-bit integer overflow problem\n-9\nnot enough memory for OOC\n-10\nerror opening OOC files\n-11\nread/write error with OOC files\n-12\n(pardiso_64 only) pardiso_64 called from 32-bit\nlibrary\n-13\ninterrupted by the (user-defined) mkl_progress function\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1843\n\n\nerror\nInformation\n-15\ninternal error which can appear for iparm[23]=10 and \niparm[12]=1. Try switch matching off (set \niparm[12]=0 and rerun.)\npardisoinit\nInitialize Intel® oneAPI Math Kernel Library (oneMKL)\nPARDISO with default parameters in accordance with\nthe matrix type.\nSyntax\nvoid pardisoinit (_MKL_DSS_HANDLE_t pt, const MKL_INT *mtype, MKL_INT *iparm );\nInclude Files\n•\nmkl.h\nDescription\nThis function initializes the solver handle pt for Intel® oneAPI Math Kernel Library (oneMKL) PARDISO with\nzero values (as needed for the very first call of pardiso) and sets default iparm values in accordance with\nthe matrix type mtype.\nThe recommended way is to avoid using pardisoinit and to initialize pt and set the values of the iparm\narray manually as the default parameters might not be the best for a particular use case.\nAn alternative method to set default iparm values is to call pardiso in the analysis phase with iparm(1)=0.\nIn this case, the solver handle pt must be initialized with zero values.\nThe pardisoinit routine initializes only the in-core version of Intel® oneAPI Math Kernel Library (oneMKL)\nPARDISO. Switching to the out-of-core version of Intel® oneAPI Math Kernel Library (oneMKL) PARDISO as\nwell as changing default iparm values can be done after the call to pardisoinit but before the first call to\npardiso.\nThe pardisoinit routine cannot be used together with the pardiso_64 routine.\nInput Parameters\nmtype\nMatrix type. Based on this value pardisoinit chooses default values for\nthe iparm array. Refer to the section oneMKL PARDISO Parameters in\nTabular Formfor more details about the default values of Intel® oneAPI Math\nKernel Library (oneMKL) PARDISO.\nOutput Parameters\npt\nArray of size 64. Handle to internal data structure. The pardisoinit\nroutine nullifies the array pt.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1844\n\n\nNOTE\nIt is very important that pt is initialized with zero before the\nfirst call of Intel® oneAPI Math Kernel Library (oneMKL)\nPARDISO. After that first call you must never modify the array,\nbecause it could cause a serious memory leak or a crash.\niparm\nArray of size 64. This array is used to set various options for Intel® oneAPI\nMath Kernel Library (oneMKL) PARDISO and to return some useful\ninformation after execution of the solver. Thepardisoinit routine fills in\nthe iparm array with the default values. Refer to the section oneMKL\nPARDISO Parameters in Tabular Form for more details about the default\nvalues of Intel® oneAPI Math Kernel Library (oneMKL) PARDISO.\npardiso_64\nCalculates the solution of a set of sparse linear\nequations with single or multiple right-hand sides, 64-\nbit integer version.\nSyntax\nvoid pardiso_64 (_MKL_DSS_HANDLE_t pt, const long long int *maxfct, const long long int\n*mnum, const long long int *mtype, const long long int *phase, const long long int *n,\nconst void *a, const long long int *ia, const long long int *ja, long long int *perm,\nconst long long int *nrhs, long long int *iparm, const long long int *msglvl, void *b,\nvoid *x, long long int *error);\nInclude Files\n•\nmkl.h\nDescription\npardiso_64 is an alternative ILP64 (64-bit integer) version of the pardiso routine (see Description section\nfor more details). The interface of pardiso_64 is the same as the interface of pardiso, but it accepts and\nreturns all integer data as long long int.\nUse pardiso_64 when pardisofor solving large matrices (with the number of non-zero elements on the\norder of 500 million or more). You can use it together with the usual LP64 interfaces for the rest of Intel®\noneAPI Math Kernel Library (oneMKL) functionality. In other words, if you use 64-bit integer version\n(pardiso_64), you do not need to re-link your applications with ILP64 libraries. Take into account that\npardiso_64 may perform slower than regular pardiso on the reordering and symbolic factorization phase.\nNOTE\npardiso_64 is supported only in the 64-bit libraries. If pardiso_64 is called from the 32-bit libraries,\nit returns error =-12.\nNOTE\nThis routine supports the Progress Routine feature. See Progress Function for details.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1845\n\n\nInput Parameters\nThe input parameters of pardiso_64 are the same as the input parameters of pardiso, but pardiso_64\naccepts all integer data as long long int.\nOutput Parameters\nThe output parameters of pardiso_64 are the same as the output parameters of pardiso, but pardiso_64\nreturns all integer data as long long int.\nmkl_pardiso_pivot\nReplaces routine which handles Intel® oneAPI Math\nKernel Library (oneMKL) PARDISO pivots with user-\ndefined routine.\nSyntax\nvoid mkl_pardiso_pivot (const void *ai, void *bi, const void *eps);\nInclude Files\n•\nmkl.h\nDescription\nThe mkl_pardiso_pivotroutine allows you to handle diagonal elements which arise during numerical\nfactorization that are zero or near zero. By default, Intel® oneAPI Math Kernel Library (oneMKL) PARDISO\ndetermines that a diagonal elementbi is a pivot if bi < eps, and if so, replaces it with eps. But you can\nprovide your own routine to modify the resulting factorized matrix in case there are small elements on the\ndiagonal during the factorization step.\nNOTE\nTo use this routine, you must set iparm[55] to 1 before the main pardiso loop.\nNOTE\nThe matrix types mtype=2 (symmetric positive-definite matrix) and mtype=4 (complex and\nHermitian positive definite) are not supported, because the Cholesky factorization without\npivoting is used for these matrix types.\nInput Parameters\nai\nDiagonal element of initial matrix corresponding to pivot element.\nbi\nDiagonal element of factorized matrix that could be chosen as a pivot\nelement.\neps\nScalar to compare with diagonal of factorized matrix. On input equal to\nparameter described by iparm[9].\nOutput Parameters\nbi\nIn case element is chosen as a pivot, value with which to replace the pivot.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1846\n\n\npardiso_getdiag\nReturns diagonal elements of initial and factorized\nmatrix.\nSyntax\nvoid pardiso_getdiag (const _MKL_DSS_HANDLE_t pt, void *df, void *da, const MKL_INT\n*mnum, MKL_INT *error);\nInclude Files\n•\nmkl.h\nDescription\nThis routine returns the diagonal elements of the initial and factorized matrix for a real or Hermitian matrix.\nNOTE\nIn order to use this routine, you must set iparm[55] to 1 before the main pardiso loop.\nIf iparm[23] is set to 10 (an improved two-level factorization algorithm for nonsymmetric matrices),\nIntel® oneAPI Math Kernel Library PARDISO will automatically use the classic algorithm for\nfactorization.\nInput Parameters\npt\nArray with a size of 64. Handle to internal data structure for the Intel®\noneAPI Math Kernel Library (oneMKL) PARDISO solver. The entries must be\nset to zero prior to the first call topardiso. Unique for factorization.\nmnum\nIndicates the actual matrix for the solution phase of the Intel® oneAPI Math\nKernel Library (oneMKL) PARDISO solver. With this scalar you can define the\ndiagonal elements of the factorized matrix that you want to obtain. The\nvalue must be: 1 ≤mnum ≤ maxfct. In most applications this value is 1.\nOutput Parameters\ndf\nArray with a dimension of n. Contains diagonal elements of the factorized\nmatrix after factorization.\nNOTE\nElements of df correspond to diagonal elements of matrix\nLcomputed during phase 22. Because during phase 22 Intel®\noneAPI Math Kernel Library (oneMKL) PARDISO makes\nadditional permutations to improve stability, it is possible that\narraydf is not in line with the perm array computed during phase\n11.\nda\nArray with a dimension of n. Contains diagonal elements of the initial\nmatrix.\nerror\nThe error indicator.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1847\n\n\nerror\nInformation\n0\nno error\n-1\nDiagonal information not turned on before pardiso\nmain loop (iparm[55]=0).\npardiso_export\nPlaces pointers dedicated for sparse representation of\na requested matrix (values, rows, and columns) into\nMKL PARDISO\nSyntax\nvoid pardiso_export (const _MKL_DSS_HANDLE_t pt, void* values, MKL_INT* rows, MKL_INT*\ncolumns, MKL_INT* step, MKL_INT* iparm, MKL_INT* error);\nInclude Files\n•\nmkl.h\nDescription\nThis auxiliary routine places pointers dedicated for sparse representation of a requested matrix (values,\nrows, and columns) into MKL PARDISO. The matrix will be stored in the three-array variant of the\ncompressed sparse row (CSR3 format) with 0-based indexing.\nNOTE\nCurrently, this routine can be used only for a sparse Schur complement matrix. All\nparameters related to the Schur complement matrix (perm, iparm) must be set before the\nreordering stage of MKL PARDISO (phase = 11) is called.\nInput Parameters\npt\nArray with a size of 64. Handle to internal data structure for the Intel\n®\nMKL PARDISO solver. The entries must be set to zero prior to the first\ncall to pardiso. Unique for factorization.\niparm\nThis array is used to pass various parameters to Intel\n® MKL PARDISO\nand to return some useful information after execution of the solver.\nstep\nStage indicator. These are the currently supported values:\nStep\nvalue\nNotes\n1\nUsed to place pointers related to a Schur complement\nmatrix in MKL PARDISO. The routine with step equal to\n1 must be called between the reordering and\nfactorization phases of MKL PARDISO.\n−1\nUsed to clean the internal handle.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1848\n\n\nInput/Output Parameters\nvalues\nParameter type: input/output parameter.\nThis array contains the non-zero elements of the requested matrix.\nrows\nParameter type: input/output parameter.\nArray of size (size + 1)\nFor CSR3 format, rows[i] ( i < size ) points to the first column\nindex of row i in the array columns; that is, rows[i] gives the index\nof the element in the array values that contains the first non-zero\nelement from row i of the sparse matrix. The last element,\nrows[size], is equal to the number of non-zero elements in the\nsparse matrix.\ncolumns\nParameter type: input/output parameter.\nThis array contains the column indices for the non-zero elements of\nthe requested matrix.\nerror\nParameter type: output parameter.\nThe error status:\n•\n0 indicates no error.\n•\n1 indicates inconsistent input data.\nUsage Example\nThe following C-style example demonstrates how to use the pardiso_export routine to get the sparse\nrepresentation (that is, three-array CSR format) of a Schur complement matrix.\n#include \"mkl.h\"\n/*\n * Call the reordering phase of MKL PARDISO with iparm[35] set to -1 in\n * order to compute the Schur complement matrix only, or -2 to compute all\n * factorization arrays.  perm array indices related to the Schur complement\n * matrix must be set to 1.\n */\nphase = 11;\nfor ( i = 0; i < schur_size; i++ ) { perm[i] = 1.; }\niparm[35] = -1;\npardiso(pt, &maxfct, &mnum, &mtype, &phase, &n, a, ia, ja, perm, &nrhs,\n  iparm, &msglvl, b, x, &error);\n/*\n * After the reordering phase, iparm[35] will contain the number of non-zero\n * elements for the Schur complement matrix.  Arrays dedicated to the sparse\n * representation of the Schur complement matrix must be allocated before\n * the factorization stage of MKL PARDISO is called.\n */\nschur_nnz     = iparm[35];\nschur_rows    = (MKL_INT   *) mkl_malloc(schur_size+1, ALIGNMENT);\nschur_columns = (MKL_INT   *) mkl_malloc(schur_nnz , ALIGNMENT);\nschur_values  = (DATA_TYPE *) mkl_malloc(schur_nnz , ALIGNMENT);\n/*\n * Call to the pardiso_export routine with step equal to 1 in order to put\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1849\n\n\n * pointers related to the three-array CSR format into MKL PARDISO:\n */\npardiso_export(pt, schur_values, schur_ia, schur_ja, &step, iparm, &error);\n/*\n * Call the factorization phase of PARDISO with iparm[35] equal to -1 or -2\n * to compute the Schur complement matrix:\n */\nphase = 22;\niparm[35] = -1;\npardiso(pt, &maxfct, &mnum, &mtype, &phase, &n, a, ia, ja, perm, &nrhs,\n  iparm, &msglvl, b, x, &error);\n/*\n * After the factorization stage, schur_values, schur_rows, and\n * schur_columns will contain the Schur complement matrix in CSR3 format.\n */\npardiso_handle_store\nStore internal structures from pardiso to a file.\nSyntax\nvoid pardiso_handle_store (_MKL_DSS_HANDLE_t pt, const char *dirname, MKL_INT *error);\nInclude Files\n•\nmkl.h\nDescription\nThis function stores Intel® oneAPI Math Kernel Library (oneMKL) PARDISO structures to a file, allowing you to\nstore Intel® oneAPI Math Kernel Library (oneMKL) PARDISO internal structures between the stages of\nthepardiso routine. The pardiso_handle_restoreroutine can restore the Intel® oneAPI Math Kernel Library\n(oneMKL) PARDISO internal structures from the file.\nInput Parameters\npt\nArray with a size of 64. Handle to internal data structure.\ndirname\nString containing the name of the directory to which to write the files with\nthe content of the internal structures. Use an empty string (\"\") to specify\nthe current directory. The routine creates a file named handle.pds in the\ndirectory.\nOutput Parameters\npt\nHandle to internal data structure.\nerror\nThe error indicator.\nerror\nInformation\n0\nNo error.\n-2\nNot enough memory.\n-10\nCannot open file for writing.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1850\n\n\nerror\nInformation\n-11\nError while writing to file.\n-13\nWrong file format.\npardiso_handle_restore\nRestore pardiso internal structures from a file.\nSyntax\nvoid pardiso_handle_restore (_MKL_DSS_HANDLE_t pt, const char *dirname, MKL_INT\n*error);\nInclude Files\n•\nmkl.h\nDescription\nThis function restores Intel® oneAPI Math Kernel Library (oneMKL) PARDISO structures from a file. This\nallows you to restore Intel® oneAPI Math Kernel Library (oneMKL) PARDISO internal structures stored\nbypardiso_handle_store after a phase of the pardiso routine and continue execution of the next phase.\nInput Parameters\ndirname\nString containing the name of the directory in which the file with the\ncontent of the internal structures are located. Use an empty string (\"\") to\nspecify the current directory.\nOutput Parameters\npt\nArray with a dimension of 64. Handle to internal data structure.\nerror\nThe error indicator.\nerror\nInformation\n0\nNo error.\n-2\nNot enough memory.\n-10\nCannot open file for reading.\n-11\nError while reading from file.\n-13\nWrong file format.\npardiso_handle_delete\nDelete files with pardiso internal structure data.\nSyntax\nvoid pardiso_handle_delete (const char *dirname, MKL_INT *error);\nInclude Files\n•\nmkl.h\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1851\n\n\nDescription\nThis function deletes files generated with pardiso_handle_storethat contain Intel® oneAPI Math Kernel\nLibrary (oneMKL) PARDISO internal structures.\nInput Parameters\ndirname\nString containing the name of the directory in which the file with the\ncontent of the internal structures are located. Use an empty string (\"\") to\nspecify the current directory.\nOutput Parameters\nerror\nThe error indicator.\nerror\nInformation\n0\nNo error.\n-10\nCannot delete files.\npardiso_handle_store_64\nStore internal structures from pardiso_64 to a file.\nSyntax\nvoid pardiso_handle_store_64 (_MKL_DSS_HANDLE_t pt, const char *dirname, MKL_INT\n*error);\nInclude Files\n•\nmkl.h\nDescription\nThis function stores Intel® oneAPI Math Kernel Library (oneMKL) PARDISO structures to a file, allowing you to\nstore Intel® oneAPI Math Kernel Library (oneMKL) PARDISO internal structures between the stages of\nthepardiso_64 routine. The pardiso_handle_restore_64routine can restore the Intel® oneAPI Math Kernel\nLibrary (oneMKL) PARDISO internal structures from the file.\nInput Parameters\npt\nArray with a dimension of 64. Handle to internal data structure.\ndirname\nString containing the name of the directory to which to write the files with\nthe content of the internal structures. Use an empty string (\"\") to specify\nthe current directory. The routine creates a file named handle.pds in the\ndirectory.\nOutput Parameters\npt\nHandle to internal data structure.\nerror\nThe error indicator.\nerror\nInformation\n0\nNo error.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1852\n\n\nerror\nInformation\n-2\nNot enough memory.\n-10\nCannot open file for writing.\n-11\nError while writing to file.\n-12\nNot supported in 32-bit library - routine is only\nsupported in 64-bit libraries.\n-13\nWrong file format.\npardiso_handle_restore_64\nRestore pardiso_64 internal structures from a file.\nSyntax\nvoid pardiso_handle_restore_64 (_MKL_DSS_HANDLE_t pt, const char *dirname, MKL_INT\n*error);\nInclude Files\n•\nmkl.h\nDescription\nThis function restores Intel® oneAPI Math Kernel Library (oneMKL) PARDISO structures from a file. This\nallows you to restore Intel® oneAPI Math Kernel Library (oneMKL) PARDISO internal structures stored\nbypardiso_handle_store_64 after a phase of the pardiso_64 routine and continue execution of the next\nphase.\nInput Parameters\ndirname\nString containing the name of the directory in which the file with the\ncontent of the internal structures are located. Use an empty string (\"\") to\nspecify the current directory.\nInput Parameters\npt\nArray with a dimension of 64. Handle to internal data structure.\nerror\nThe error indicator.\nerror\nInformation\n0\nNo error.\n-2\nNot enough memory.\n-10\nCannot open file for reading.\n-11\nError while reading from file.\n-13\nWrong file format.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1853\n\n\npardiso_handle_delete_64\nSyntax\nDelete files with pardiso_64 internal structure data.\nvoid pardiso_handle_delete_64 (const char *dirname, MKL_INT *error);\nInclude Files\n•\nmkl.h\nDescription\nThis function deletes files generated with pardiso_handle_store_64that contain Intel® oneAPI Math Kernel\nLibrary (oneMKL) PARDISO internal structures.\nInput Parameters\ndirname\nString containing the name of the directory in which the file with the\ncontent of the internal structures are located. Use an empty string (\"\") to\nspecify the current directory.\nOutput Parameters\nerror\nThe error indicator.\nerror\nInformation\n0\nNo error.\n-10\nCannot delete files.\n-12\nNot supported in 32-bit library - routine is only\nsupported in 64-bit libraries.\noneMKL PARDISO Parameters in Tabular Form\nThe following table lists all parameters of Intel® oneAPI Math Kernel Library (oneMKL) PARDISO and gives\ntheir brief descriptions.\nParameter\nType\nDescription\nValues\nComments\nIn/\nOut\npt\nvoid*\nSolver internal\ndata address\npointer\n0\nMust be initialized with\nzeros and never be\nmodified later\nin/o\nut\nmaxfct\nMKL_INT*\nMaximal number of\nfactors in memory\n>0\nGenerally used value is 1\nin\nmnum\nMKL_INT*\nThe number of\nmatrix (from 1 to\nmaxfct) to solve\n[1:\nmaxfct]\nGenerally used value is 1\nin\nmtype\nMKL_INT*\nMatrix type\n1\nReal and structurally\nsymmetric\nin\n2\nReal and symmetric\npositive definite\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1854\n\n\nParameter\nType\nDescription\nValues\nComments\nIn/\nOut\n-2\nReal and symmetric\nindefinite\n3\nComplex and structurally\nsymmetric\n4\nComplex and Hermitian\npositive definite\n-4\nComplex and Hermitian\nindefinite\n6\nComplex and symmetric\nmatrix\n11\nReal and nonsymmetric\nmatrix\n13\nComplex and\nnonsymmetric matrix\nphase\nMKL_INT*\nControls the\nexecution of the\nsolver\nFor iparm[35] >\n0, phases 331,\n332, and 333\nperform a different\ndecomposition. See\nthe phase\nparameter of \npardiso for details.\n11\nAnalysis\nin\n12\nAnalysis, numerical\nfactorization\n13\nAnalysis, numerical\nfactorization, solve\n22\nNumerical factorization\n23\nNumerical factorization,\nsolve\n33\nSolve, iterative refinement\n331\nphase=33, but only\nforward substitution\n332\nphase=33, but only\ndiagonal substitution\n333\nphase=33, but only\nbackward substitution\n0\nRelease internal memory\nfor L and U of the matrix\nnumber mnum\n-1\nRelease all internal\nmemory for all matrices\nn\nMKL_INT*\nNumber of\nequations in the\nsparse linear\nsystem A*X = B\n>0\nin\na\nvoid*\nContains the non-\nzero elements of\nthe coefficient\nmatrix A\n*\nThe size of a is the same\nas that of ja, and the\ncoefficient matrix can be\neither real or complex. The\nin\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1855\n\n\nParameter\nType\nDescription\nValues\nComments\nIn/\nOut\nmatrix must be stored in\nthe 3-array variation of\ncompressed sparse row\n(CSR3) format with\nincreasing values of ja for\neach row\nia[n ]\nMKL_INT*\nrowIndex array in\nCSR3 format\n>=0\nia[i] gives the index of\nthe element in array a that\ncontains the first non-zero\nelement from row i of A.\nThe last element ia(n) is\ntaken to be equal to the\nnumber of non-zero\nelements in A.\nNote: iparm[34] indicates\nwhether row/column\nindexing starts from 1 or\n0.\nin\nja\nMKL_INT*\ncolumns array in\nCSR3 format\n>=0\nThe column indices for\neach row of A must be\nsorted in increasing order.\nFor structurally symmetric\nmatrices zero diagonal\nelements must be stored\nin a and ja. Zero diagonal\nelements should be stored\nfor symmetric matrices,\nalthough they are not\nrequired. For symmetric\nmatrices, the solver needs\nonly the upper triangular\npart of the system.\nNote: iparm[34] indicates\nwhether row/column\nindexing starts from 1 or\n0.\nin\nperm[n ]\nMKL_INT*\nHolds the\npermutation vector\nof size n, specifies\nelements used for\ncomputing a partial\nsolution, or\nspecifies differing\nvalues of the input\nmatrices for low\nrank update\n>=0\nYou can apply your own\nfill-in reducing ordering\n(iparm[4]= 1) or return\nthe permutation from the\nsolver (iparm[4]= 2 ).\nLet C = P*A*PT be the\npermuted matrix. Row\n(column) i of C is the\nperm(i) row (column) of\nA. The numbering of the\narray must describe a\npermutation.\nin/o\nut\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1856\n\n\nParameter\nType\nDescription\nValues\nComments\nIn/\nOut\nTo specify elements for a\npartial solution, set\niparm[4]= 0, \niparm[30]> 0, and \niparm[35]= 0.\nTo specify elements for a\nSchur complement, set\niparm[4]= 0, \niparm[30]= 0, and \niparm[35]> 0.\nTo specify values that\ndiffer in A for low rank\nupdate (see Low Rank\nUpdate), set iparm[38] =\n1. The size of the array\nmust be at least 2*ndiff +\n1, where ndiff is the\nnumber of values of A that\nare different. The values of\nperm should be:\nperm = {ndiff,\nrow_index1,\ncolumn_index1,\nrow_index2,\ncolumn_index2, ....,\nrow_index_ndiff,\ncolumn_index_ndiff}\nwhere row_index_m and\ncolumn_index_m are the\nrow and column indices of\nthe m-th differing non-\nzero value in matrix A. The\nrow and column index\npairs can be in any order,\nbut must use zero-based\nindexing regardless of the\nvalue of iparm[34].\nNOTE\nUnless you have specified\nlow rank update, \niparm[34] indicates\nwhether row/column\nindexing starts from 1 or 0.\nnrhs\nMKL_INT*\nNumber of right-\nhand sides that\nneed to be solved\nfor\n>=0\nGenerally used value is 1\nTo obtain better Intel®\noneAPI Math Kernel\nLibrary (oneMKL) PARDISO\nperformance, during the\nnumerical factorization\nphase you can provide the\nmaximum number of\nin\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1857\n\n\nParameter\nType\nDescription\nValues\nComments\nIn/\nOut\nright-hand sides, which\ncan be used further during\nthe solving phase.\niparm[64]\nMKL_INT*\nThis array is used\nto pass various\nparameters to\nIntel® oneAPI Math\nKernel Library\n(oneMKL) PARDISO\nand to return some\nuseful information\nafter execution of\nthe solver (see \npardiso iparm\nParameter for more\ndetails)\n*\nIf iparm[0]=0, Intel®\noneAPI Math Kernel\nLibrary (oneMKL) PARDISO\nfillsiparm[1] through\niparm[63] with default\nvalues and uses them.\nin/o\nut\nmsglvl\nMKL_INT*\nMessage level\ninformation\n0\nIntel® oneAPI Math Kernel\nLibrary (oneMKL) PARDISO\ngenerates no output\nin\n1\nIntel® oneAPI Math Kernel\nLibrary (oneMKL) PARDISO\nprints statistical\ninformation\nb[n*nrhs]\nvoid*\nRight-hand side\nvectors\n*\nOn entry, contains the\nright-hand side vector/\nmatrix B, which is placed\ncontiguously in memory.\nThe b[i+k*n] element\nmust hold the i-th\ncomponent of k-th right-\nhand side vector. Note that\nb is only accessed in the\nsolution phase.\nOn output, the array is\nreplaced with the solution\nif iparm[5]=1.\nin/o\nut\nx[n*nrhs]\nvoid*\nSolution vectors\n*\nOn output, if iparm[5]=0,\ncontains solution vector/\nmatrix X which is placed\ncontiguously in memory.\nThe x[i+k*n] element\nmust hold the i-th\ncomponent of k-th solution\nvector. Note that x is only\naccessed in the solution\nphase.\nout\nerror\nMKL_INT*\nError indicator\n0\nNo error\nout\n-1\nInput inconsistent\n-2\nNot enough memory\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1858\n\n\nParameter\nType\nDescription\nValues\nComments\nIn/\nOut\n-3\nReordering problem\n-4\nZero pivot, numerical\nfactorization or iterative\nrefinement problem\n-5\nUnclassified (internal)\nerror\n-6\nReordering failed (matrix\ntypes 11 and 13 only)\n-7\nDiagonal matrix is singular\n-8\n32-bit integer overflow\nproblem\n-9\nNot enough memory for\nOOC\n-10\nProblems with opening\nOOC temporary files\n-11\nRead/write problems with\nthe OOC data file\n1) See description of PARDISO_DATA_TYPE in PARDISO_DATA_TYPE.\npardiso iparm Parameter\nThis table describes all individual components of the Intel® oneAPI Math Kernel Library (oneMKL)\nPARDISOiparm parameter. Components which are not used must be initialized with 0. Default values are\ndenoted with an asterisk (*).\nComponent\nDescription\niparm[0]\ninput\nUse default values.\n0\niparm[1] - iparm[63] are filled with default values.\n≠0\nYou must supply all values in components iparm[1] - iparm[63].\niparm[1]\ninput\nFill-in reducing ordering for the input matrix.\nCaution\nYou can control the parallel execution of the solver by explicitly setting the\nMKL_NUM_THREADS environment variable. If fewer OpenMP threads are available than\nspecified, the execution may slow down instead of speeding up. If MKL_NUM_THREADS is\nnot defined, then the solver uses all available processors.\n0\nThe minimum degree algorithm [Li99].\n2*\nThe nested dissection algorithm from the METIS package [Karypis98].\n3\nThe parallel (OpenMP) version of the nested dissection algorithm. It can decrease the\ntime of computations on multi-core computers, especially when Intel® oneAPI Math\nKernel Library (oneMKL) PARDISO Phase 1 takes significant time.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1859\n\n\nComponent\nDescription\nNOTE\nSetting iparm[1] = 3 prevents the use of CNR mode (iparm[33] > 0)\nbecause Intel® oneAPI Math Kernel Library (oneMKL) PARDISO uses dynamic\nparallelism.\niparm[2]\nReserved. Set to zero.\niparm[3]\ninput\nPreconditioned CGS/CG.\nThis parameter controls preconditioned CGS [Sonn89] for nonsymmetric or structurally\nsymmetric matrices and Conjugate-Gradients for symmetric matrices. iparm[3] has\nthe form iparm[3]= 10*L+K.\nK=0\nThe factorization is always computed as required by phase.\nK=1\nCGS iteration replaces the computation of LU. The preconditioner is LU that\nwas computed at a previous step (the first step or last step with a failure) in a\nsequence of solutions needed for identical sparsity patterns.\nK=2\nCGS iteration for symmetric positive definite matrices replaces the computation\nof LLT. The preconditioner is LLT that was computed at a previous step (the\nfirst step or last step with a failure) in a sequence of solutions needed for\nidentical sparsity patterns.\nThe value L controls the stopping criterion of the Krylov Subspace iteration:\nepsCGS = 10-L is used in the stopping criterion\n||dxi|| / ||dx0|| < epsCGS\nwhere ||dxi|| = ||inv(L*U)*ri|| for K = 1 or ||dxi|| = ||inv(L*LT)*ri|| for\nK = 2 and ri is the residue at iteration i of the preconditioned Krylov Subspace\niteration.\nA maximum number of 150 iterations is fixed with the assumption that the iteration\nwill converge before consuming half the factorization time. Intermediate convergence\nrates and residue excursions are checked and can terminate the iteration process. If\nphase =23, then the factorization for a given A is automatically recomputed in cases\nwhere the Krylov Subspace iteration failed, and the corresponding direct solution is\nreturned. Otherwise the solution from the preconditioned Krylov Subspace iteration is\nreturned. Using phase =33 results in an error message (error=-4) if the stopping\ncriteria for the Krylov Subspace iteration can not be reached. More information on the\nfailure can be obtained from iparm[19].\nThe default is iparm[3]=0, and other values are only recommended for an advanced\nuser. iparm[3] must be greater than or equal to zero.\nExamples:\niparm[3]\nDescription\n31\nLU-preconditioned CGS iteration with a stopping criterion of 1.0E-3 for\nnonsymmetric matrices\n61\nLU-preconditioned CGS iteration with a stopping criterion of 1.0E-6 for\nnonsymmetric matrices\n62\nLLT-preconditioned CGS iteration with a stopping criterion of 1.0E-6 for\nsymmetric positive definite matrices\niparm[4]\ninput\nUser permutation.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1860\n\n\nComponent\nDescription\nThis parameter controls whether user supplied fill-in reducing permutation is used\ninstead of the integrated multiple-minimum degree or nested dissection algorithms.\nAnother use of this parameter is to control obtaining the fill-in reducing permutation\nvector calculated during the reordering stage of Intel® oneAPI Math Kernel Library\n(oneMKL) PARDISO.\nThis option is useful for testing reordering algorithms, adapting the code to special\napplications problems (for instance, to move zero diagonal elements to the end of\nP*A*PT), or for using the permutation vector more than once for matrices with\nidentical sparsity structures. For definition of the permutation, see the description of\nthe perm parameter.\nCaution\nYou can only set one of iparm[4], iparm[30], and iparm[35], so be sure that the\niparm[30] (partial solution) and the iparm[35] (Schur complement) parameters are 0\nif you set iparm[4].\n0*\nUser permutation in the perm array is ignored.\n1\nIntel® oneAPI Math Kernel Library (oneMKL) PARDISO uses the user supplied\nfill-in reducing permutation from theperm array. iparm[1] is ignored.\nNOTE\nSetting iparm[4] = 1 prevents use of a parallel algorithm for the solve step.\n2\nIntel® oneAPI Math Kernel Library (oneMKL) PARDISO returns the permutation\nvector computed at phase 1 in theperm array.\niparm[5]\ninput\nWrite solution on x.\nNOTE\nThe array x is always used.\n0*\nThe array x contains the solution; right-hand side vector b is kept unchanged.\n1\nThe solver stores the solution on the right-hand side b.\niparm[6]\noutput\nNumber of iterative refinement steps performed.\nReports the number of iterative refinement steps that were actually performed during\nthe solve step.\niparm[7]\ninput\nIterative refinement step.\nOn entry to the solve and iterative refinement step, iparm[7] must be set to the\nmaximum number of iterative refinement steps that the solver performs.\nNOTE Perturbed pivots result in iterative refinement (independent of the value of\niparm[7]) and the number of executed iterations is reported in iparm[6].\n0*\nThe solver automatically performs two steps of iterative refinement when\nperturbed pivots are obtained during the numerical factorization.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1861\n\n\nComponent\nDescription\n>0\nMaximum number of iterative refinement steps that the solver performs. The\nsolver performs not more than the absolute value of iparm[7] steps of\niterative refinement. The solver might stop the process before the maximum\nnumber of steps if\n•\na satisfactory level of accuracy of the solution in terms of backward error is\nachieved,\n•\nor if it determines that the required accuracy cannot be reached. In this\ncase the solver returns -4 in the error parameter.\nThe number of executed iterations is reported in iparm[6].\n<0\nMaximum number of iterative refinement steps with a negative sign. Unlike the\ncase above the accumulation of the residuum uses extended precision real and\ncomplex data types.\nNOTE Currently, this feature is only supported for sequential and OpenMP\nthreading.\niparm[8] input\nTolerance level for the relative residual in the iterative refinement process. If set to a\nnon-zero value, an additional criterion is used for stopping the iterative refinement:\n∥r ∥\n∥b ∥< 10−iparm 8\nIf set to zero, default checks are used to determine when to stop the iterations (see\niparm[7] description).\nNOTE Currently it is only used for iparm[23]=1 or 10 and OpenMP threading.\niparm[9]\ninput\nPivoting perturbation.\nThis parameter instructs Intel® oneAPI Math Kernel Library (oneMKL) PARDISO how to\nhandle small pivots or zero pivots for nonsymmetric matrices (mtype =11 or mtype\n=13) and symmetric matrices (mtype =-2, mtype =-4, or mtype =6). For these\nmatrices the solver uses a complete supernode pivoting approach. When the\nfactorization algorithm reaches a point where it cannot factor the supernodes with this\npivoting strategy, it uses a pivoting perturbation strategy similar to [Li99], \n[Schenk04].\nSmall pivots are perturbed with eps = 10-iparm[9].\nThe magnitude of the potential pivot is tested against a constant threshold of\nalpha = eps*||A2||inf,\nwhere eps = 10(-iparm[9]), A2 = P*PMPS*Dr*A*Dc*P, and ||A2||inf is the infinity\nnorm of the scaled and permuted matrix A. Any tiny pivots encountered during\nelimination are set to the sign (lII)*eps*||A2||inf, which trades off some\nnumerical stability for the ability to keep pivots from getting too small. Small pivots\nare therefore perturbed with eps = 10(-iparm[9]).\n13*\nThe default value for nonsymmetric matrices(mtype =11, mtype=13), eps =\n10-13.\n8*\nThe default value for symmetric indefinite matrices (mtype =-2, mtype=-4,\nmtype=6), eps = 10-8.\niparm[10]\ninput\nScaling vectors.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1862\n\n\nComponent\nDescription\nIntel® oneAPI Math Kernel Library (oneMKL) PARDISO uses a maximum weight\nmatching algorithm to permute large elements on the diagonal and to scale so that the\ndiagonal elements are equal to 1 and the absolute values of the off-diagonal entries\nare less than or equal to 1. This scaling method is applied only to nonsymmetric\nmatrices (mtype = 11 or mtype = 13). The scaling can also be used for symmetric\nindefinite matrices (mtype = -2, mtype =-4, or mtype = 6) when the symmetric\nweighted matchings are applied (iparm[12] = 1).\nUse iparm[10] = 1 (scaling) and iparm[12] = 1 (matching) for highly indefinite\nsymmetric matrices, for example, from interior point optimizations or saddle point\nproblems. Note that in the analysis phase (phase=11) you must provide the numerical\nvalues of the matrix A in array a in case of scaling and symmetric weighted matching.\n0*\nDisable scaling. Default for symmetric indefinite matrices.\n1*\nEnable scaling. Default for nonsymmetric matrices.\nScale the matrix so that the diagonal elements are equal to 1 and the absolute\nvalues of the off-diagonal entries are less or equal to 1. This scaling method is\napplied to nonsymmetric matrices (mtype = 11, mtype = 13). The scaling can\nalso be used for symmetric indefinite matrices (mtype = -2, mtype = -4,\nmtype = 6) when the symmetric weighted matchings are applied (iparm[12]\n= 1).\nNote that in the analysis phase (phase=11) you must provide the numerical\nvalues of the matrix A in case of scaling.\niparm[11]\ninput\nSolve with transposed or conjugate transposed matrix A.\nNOTE\nFor real matrices, the terms transposed and conjugate transposed are equivalent.\n0*\nSolve a linear system AX = B.\n1\nSolve a conjugate transposed system AHX = B based on the factorization of the\nmatrix A.\n2\nSolve a transposed system ATX = B based on the factorization of the matrix A.\niparm[12]\ninput\nImproved accuracy using (non-) symmetric weighted matching.\nIntel® oneAPI Math Kernel Library (oneMKL) PARDISO can use a maximum weighted\nmatching algorithm to permute large elements close the diagonal. This strategy adds\nan additional level of reliability to the factorization methods and complements the\nalternative of using more complete pivoting techniques during the numerical\nfactorization.\n  \n0*\nDisable matching. Default for symmetric indefinite matrices.\n1*\nEnable matching. Default for nonsymmetric matrices.\nMaximum weighted matching algorithm to permute large elements close to the\ndiagonal.\nIt is recommended to use iparm[10] = 1 (scaling) and iparm[12]= 1\n(matching) for highly indefinite symmetric matrices, for example from interior\npoint optimizations or saddle point problems.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1863\n\n\nComponent\nDescription\nNote that in the analysis phase (phase=11) you must provide the numerical\nvalues of the matrix A in case of symmetric weighted matching.\niparm[13]\noutput\nNumber of perturbed pivots.\nAfter factorization, contains the number of perturbed pivots for the matrix types: 1, 3,\n11, 13, -2, -4 and 6.\niparm[14]\noutput\nPeak memory on symbolic factorization.\nThe total peak memory in kilobytes that the solver needs during the analysis and\nsymbolic factorization phase.\nThis value is only computed in phase 1.\niparm[15]\noutput\nPermanent memory on symbolic factorization.\nPermanent memory from the analysis and symbolic factorization phase in kilobytes\nthat the solver needs in the factorization and solve phases.\nThis value is only computed in phase 1.\niparm[16]\noutput\nSize of factors/Peak memory on numerical factorization and solution.\nThis parameter provides the size in kilobytes of the total memory consumed by in-core\nIntel® oneAPI Math Kernel Library (oneMKL) PARDISO for internal floating point arrays.\nThis parameter is computed in phase 1. Seeiparm[62] for the OOC mode.\nThe total peak memory consumed by Intel® oneAPI Math Kernel Library (oneMKL)\nPARDISO ismax(iparm[14], iparm[15]+iparm[16])\niparm[17]\ninput/output\nReport the number of non-zero elements in the factors.\n<0\nEnable reporting if iparm[17] < 0 on entry. The default value is -1.\n>=0\nDisable reporting.\niparm[18]\ninput/output\nReport number of floating point operations (in 106 floating point operations) that are\nnecessary to factor the matrix A.\n<0\nEnable report if iparm[18] < 0 on entry. This increases the reordering time.\n>=0\n*\nDisable report.\niparm[19]\noutput\nReport CG/CGS diagnostics.\n>0\nCGS succeeded, reports the number of completed iterations.\n<0\nCG/CGS failed (error=-4 after the solution phase).\nIf phase= 23, then the factors L and U are recomputed for the matrix A and\nthe error flag error=0 in case of a successful factorization. If phase = 33,\nthen error = -4 signals failure.\niparm[19]= - it_cgs*10 - cgs_error.\nPossible values of cgs_error:\n1 - fluctuations of the residuum are too large\n2 - ||dxmax_it_cgs/2|| is too large (slow convergence)\n3 - stopping criterion is not reached at max_it_cgs\n4 - perturbed pivots caused iterative refinement\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1864\n\n\nComponent\nDescription\n5 - factorization is too fast for this matrix. It is better to use the factorization\nmethod with iparm[3] = 0\niparm[20]\ninput\nPivoting for symmetric indefinite matrices.\nNOTE\nUse iparm[10] = 1 (scaling) and iparm[12] = 1 (matchings) for highly indefinite\nsymmetric matrices, for example from interior point optimizations or saddle point problems.\n0\nApply 1x1 diagonal pivoting during the factorization process.\n1*\nApply 1x1 and 2x2 Bunch-Kaufman pivoting during the factorization process.\nBunch-Kaufman pivoting is available for matrices of mtype=-2, mtype=-4, or\nmtype=6.\n2\nApply 1x1 diagonal pivoting during the factorization process. Using this value is\nthe same as using iparm[20] = 0 except that the solve step does not\nautomatically make iterative refinements when perturbed pivots are obtained\nduring numerical factorization. The number of iterations is limited to the\nnumber of iterative refinements specified by iparm[7] (0 by default).\n3\nApply 1x1 and 2x2 Bunch-Kaufman pivoting during the factorization process.\nBunch-Kaufman pivoting is available for matrices of mtype=-2, mtype=-4, or\nmtype=6. Using this value is the same as using iparm[20] = 1 except that the\nsolve step does not automatically make iterative refinements when perturbed\npivots are obtained during numerical factorization. The number of iterations is\nlimited to the number of iterative refinements specified by iparm[7] (0 by\ndefault).\niparm[21]\noutput\nInertia: number of positive eigenvalues.\nIntel® oneAPI Math Kernel Library (oneMKL) PARDISO reports the number of positive\neigenvalues for symmetric indefinite matrices.\niparm[22]\noutput\nInertia: number of negative eigenvalues.\nIntel® oneAPI Math Kernel Library (oneMKL) PARDISO reports the number of negative\neigenvalues for symmetric indefinite matrices.\niparm[23]\ninput\nParallel factorization control.\n0*\nIntel® oneAPI Math Kernel Library (oneMKL) PARDISO uses the classic\nalgorithm for factorization.\n1\nIntel® oneAPI Math Kernel Library (oneMKL) PARDISO uses a two-level\nfactorization algorithm. This algorithm generally improves scalability in case of\nparallel factorization on many OpenMP threads (more than eight).\nNOTE Disable iparm[10] (scaling) and iparm[12]= 1 (matching) when using\nthe two-level factorization algorithm. Otherwise Intel® oneAPI Math Kernel Library\n(oneMKL) PARDISO uses the classic factorization algorithm.\n10\nIntel® oneAPI Math Kernel Library (oneMKL) PARDISO uses an improved two-\nlevel factorization algorithm for nonsymmetric matrices.\niparm[24]\ninput\nParallel forward/backward solve control.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1865\n\n\nComponent\nDescription\n0*\nIntel® oneAPI Math Kernel Library (oneMKL) PARDISO uses the following\nstrategy for parallelizing the solving step:\nIn the case of the one right-hand side, the parallelization will be performed by\npartitioning the matrix.\nOtherwise, the parallelization will be over the right-hand sides.\nThis feature is available only for in-core Intel® oneAPI Math Kernel Library\n(oneMKL) PARDISO (seeiparm[59]).\n1\nIntel® oneAPI Math Kernel Library (oneMKL) PARDISO uses the sequential\nforward and backward solve.\n2\nIndependent from the number of the right-hand sides, Intel® oneAPI Math\nKernel Library (oneMKL) PARDISO uses the parallel algorithm based on the\nmatrix partitioning.\nThis feature is available only for in-core Intel® oneAPI Math Kernel Library\n(oneMKL) PARDISO (seeiparm[59]).\niparm[25]\nReserved. Set to zero.\niparm[26]\ninput\nMatrix checker.\n0*\nIntel® oneAPI Math Kernel Library (oneMKL) PARDISO does not check the\nsparse matrix representation for errors.\n1\nIntel® oneAPI Math Kernel Library (oneMKL) PARDISO checks integer arraysia\nand ja. In particular, Intel® oneAPI Math Kernel Library (oneMKL) PARDISO\nchecks whether column indices are sorted in increasing order within each row.\niparm[27]\ninput\nSingle or double precision Intel® oneAPI Math Kernel Library (oneMKL) PARDISO.\nSee iparm[7] for information on controlling the precision of the refinement steps.\nImportant\nThe iparm[27]value is stored in the Intel® oneAPI Math Kernel Library (oneMKL)\nPARDISO handle between Intel® oneAPI Math Kernel Library (oneMKL) PARDISO calls, so\nthe precision mode can be changed only during phase 1.\n0*\nInput arrays (a, x and b) and all internal arrays must be presented in double\nprecision.\n1\nInput arrays (a, x and b) must be presented in single precision.\nIn this case all internal computations are performed in single precision.\niparm[28]\nReserved. Set to zero.\niparm[29]\noutput\nNumber of zero or negative pivots.\nIf Intel® oneAPI Math Kernel Library (oneMKL) PARDISO detects zero or negative pivot\nformtype=2 or mtype=4 matrix types, the factorization is stopped. Intel® oneAPI Math\nKernel Library (oneMKL) PARDISO returns immediately with anerror = -4, and\niparm[29] reports the number of the equation where the zero or negative pivot is\ndetected.\nNote: The returned value can be different for the parallel and sequential version in\ncase of several zero/negative pivots.\niparm[30]\nPartial solve and computing selected components of the solution vectors.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1866\n\n\nComponent\nDescription\ninput\nThis parameter controls the solve step of Intel® oneAPI Math Kernel Library (oneMKL)\nPARDISO. It can be used if only a few components of the solution vectors are needed\nor if you want to reduce the computation cost at the solve step by utilizing the sparsity\nof the right-hand sides. To use this option the input permutation vector defineperm so\nthat when perm(i) = 1 it means that either the i-th component in the right-hand\nsides is nonzero, or the i-th component in the solution vectors is computed, or both,\ndepending on the value of iparm[30].\nThe permutation vector permmust be present in all phases of Intel® oneAPI Math\nKernel Library (oneMKL) PARDISO software. At the reordering step, the software\noverwrites the input vectorperm by a permutation vector used by the software at the\nfactorization and solver step. If m is the number of components such that perm(i) =\n1, then the last m components of the output vector perm are a set of the indices i\nsatisfying the condition perm(i) = 1 on input.\nNOTE\nTurning on this option often increases the time used by Intel® oneAPI Math Kernel Library\n(oneMKL) PARDISO for factorization and reordering steps, but it can reduce the time\nrequired for the solver step.\nImportant\nYou can use this feature for both in-core and out-of-core Intel® oneAPI Math Kernel Library\n(oneMKL) PARDISO as long as iparm[23]=1. Otherwise, you cannot use partial solve for\nout-of-core mode and you will need to setiparm[59]=0 for in-core mode. Set the\nparameters iparm[7] (iterative refinement steps), iparm[3] (preconditioned CGS), \niparm[4] (user permutation), and iparm[35] (Schur complement) to 0 as well.\n0*\nDisables this option.\n1\nit is assumed that the right-hand sides have only a few non-zero components*\nand the input permutation vector perm is defined so that perm(i) = 1 means\nthat the (i)-th component in the right-hand sides is nonzero. In this case Intel®\noneAPI Math Kernel Library (oneMKL) PARDISO only uses the non-zero\ncomponents of the right-hand side vectors and computes only corresponding\ncomponents in the solution vectors. That means thei-th component in the\nsolution vectors is only computed if perm(i) = 1.\n2\nIt is assumed that the right-hand sides have only a few non-zero components*\nand the input permutation vector perm is defined so that perm(i) = 1 means\nthat the i-th component in the right-hand sides is nonzero.\nUnlike for iparm[30]=1, all components of the solution vector are computed\nfor this setting and all components of the right-hand sides are used. Because\nall components are used, for iparm[30]=2 you must set the i-th component\nof the right-hand sides to zero explicitly if perm(i) is not equal to 1.\n3\nSelected components of the solution vectors are computed. The perm array is\nnot related to the right-hand sides and it only indicates which components of\nthe solution vectors should be computed. In this case perm(i) = 1 means that\nthe i-th component in the solution vectors is computed.\niparm[31] -\niparm[32]\nReserved. Set to zero.\niparm[33]\ninput\nOptimal number of OpenMP threads for conditional numerical reproducibility (CNR)\nmode.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1867\n\n\nComponent\nDescription\nIntel® oneAPI Math Kernel Library (oneMKL) PARDISO reads the value ofiparm[33]\nduring the analysis phase (phase 1), so you cannot change it later.\nBecause Intel® oneAPI Math Kernel Library (oneMKL) PARDISO uses C random number\ngenerator facilities during the analysis phase (phase 1) you must take these\nprecautions to get numerically reproducible results:\n•\nDo not alter the states of the random number generators.\n•\nDo not run multiple instances of Intel® oneAPI Math Kernel Library (oneMKL)\nPARDISO in parallel in the analysis phase (phase 1).\nNOTE\nCNR is only available for the in-core version of Intel® oneAPI Math Kernel Library (oneMKL)\nPARDISO and the non-parallel version of the nested dissection algorithm. You must also:\n•\nset iparm[59] to 0 in order to use the in-core version,\n•\nnot set iparm[1] to 3 in order to not use the parallel version of the nested\ndissection algorithm.\nOtherwise Intel® oneAPI Math Kernel Library (oneMKL) PARDISO does not\nproduce numerically repeatable results even if CNR is enabled for Intel® oneAPI\nMath Kernel Library (oneMKL) using the functionality described inSupport\nFunctions for CNR.\n0*\nCNR mode for Intel® oneAPI Math Kernel Library (oneMKL) PARDISO is enabled\nonly if it is enabled for Intel® oneAPI Math Kernel Library (oneMKL) using the\nfunctionality described inSupport Functions for CNRand the in-core version is\nused. Intel® oneAPI Math Kernel Library (oneMKL) PARDISO determines the\noptimal number of OpenMP threads automatically, and produces numerically\nreproducible results regardless of the number of threads.\n>0\nCNR mode is enabled for Intel® oneAPI Math Kernel Library (oneMKL) PARDISO\nif in-core version is used and the optimal number of OpenMP threads for Intel®\noneAPI Math Kernel Library (oneMKL) PARDISO to rely on is defined by the\nvalue ofiparm[33]. You can use iparm[33]to enable CNR mode independent\nfrom other Intel® oneAPI Math Kernel Library (oneMKL) domains. To get the\nbest performance, setiparm[33]to the actual number of hardware threads\ndedicated for Intel® oneAPI Math Kernel Library (oneMKL) PARDISO.\nSettingiparm[33] to fewer OpenMP threads than the maximum number of\nthem in use reduces the scalability of the problem being solved. Setting\niparm[33]to more threads than are available can reduce the performance of\nIntel® oneAPI Math Kernel Library (oneMKL) PARDISO.\niparm[34]\ninput\nOne- or zero-based indexing of columns and rows.\nNOTE\nSchur complement may be inaccurate or incorrect if pivots are detected.\nPlease, check the output of iparm[28] .\n0*\nOne-based indexing: columns and rows indexing in arrays ia, ja, and perm\nstarts from 1 (Fortran-style indexing).\n1\nZero-based indexing: columns and rows indexing in arrays ia, ja, and perm\nstarts from 0 (C-style indexing).\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1868\n\n\nComponent\nDescription\niparm[35]\ninput/output\nSchur complement matrix computation control. To calculate this matrix, you must set\nthe input permutation vector perm to a set of indexes such that when perm(i) = 1,\nthe i-th element of the initial matrix is an element of the Schur matrix.\nCaution\nYou can only set one of iparm[4], iparm[30], and iparm[35], so be sure that the\niparm[4] (user permutation) and the iparm[30] (partial solution) parameters are 0 if\nyou set iparm[35].\n0*\nDo not compute Schur complement.\n1\nCompute Schur complement matrix as part of Intel® oneAPI Math Kernel\nLibrary (oneMKL) PARDISO factorization step and return it in the solution\nvector.\nNOTE\nThis option only computes the Schur complement matrix, and does not calculate\nfactorization arrays.\n2\nCompute Schur complement matrix as part of Intel® oneAPI Math Kernel\nLibrary (oneMKL) PARDISO factorization step and return it in the solution\nvector. Since this option calculates factorization arrays you can use it to launch\npartial or full solution of the entire problem after the factorization step.\n-1\nSame as iparm[35] equals 1, but the Schur complement matrix is provided in\n3-array CSR sparse format. Use in combination with pardiso_export. After\nreordering stage of MKL PARDISO, iparm[35] contains number of nonzero\nelements for Schur complement matrix. Set it once again before calling the\nfactorization phase.\nNOTE\nThis option is available only when iparm[23]is not equal to 0.\n-2\nSame as iparm[35] equals 2, but the Schur complement matrix is provided in\n3-array CSR sparse format. Use in combination with pardiso_export. After\nreordering stage of MKL PARDISO, iparm[35] contains number of nonzero\nelements for Schur complement matrix. Set it once again before calling the\nfactorization phase.\nNOTE\nThis option is available only when iparm[23]is not equal to 0.\niparm[36]\ninput\nFormat for matrix storage.\n0*\nUse CSR format (see Three Array Variation of CSR Format) for matrix storage.\n> 0\nUse BSR format (see Three Array Variation of BSR Format) for matrix storage\nwith blocks of size iparm[36].\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1869\n\n\nComponent\nDescription\nNOTE\nIntel® oneAPI Math Kernel Library (oneMKL) does not support BSR format in these\ncases:\n•\niparm[10] > 0: Scaling vectors\n•\niparm[12] > 0: Weighted matching\n•\niparm[30] > 0: Partial solution\n•\niparm[35] > 0: Schur complement\n•\niparm[55] > 0: Pivoting control\n•\niparm[59] > 0: OOC Intel® oneAPI Math Kernel Library (oneMKL)\nPARDISO\n< 0\nConvert supplied matrix to variable BSR (VBSR) format (see Sparse Data\nStorage) for matrix storage. Intel® oneAPI Math Kernel Library (oneMKL)\nPARDISO analyzes the matrix provided in CSR3 format and converts it to an\ninternal VBSR format. Setiparm[36] = -t, 0 < t≤ 100.\nNOTE\nIntel® oneAPI Math Kernel Library (oneMKL) supports only the VBSR format for real\nand symmetric positive definite or indefinite matrices (mtype = 2 or mtype= -2).\nIntel® oneAPI Math Kernel Library (oneMKL) does not support VBSR format in these\ncases:\n•\niparm[10] > 0: Scaling vectors\n•\niparm[12] > 0: Weighted matching\n•\niparm[55] > 0: Pivoting control\nNOTEIntel® oneAPI Math Kernel Library (oneMKL) supports these features for all\nmatrix types as long asiparm[23]=1:\n•\niparm[30] > 0: Partial solution\n•\niparm[35] > 0: Schur complement\n•\niparm[59] > 0: OOC Intel® oneAPI Math Kernel Library (oneMKL)\nPARDISO\niparm[37]\nReserved. Set to zero.\niparm[38]\nEnable low rank update (see Low Rank Update) to accelerate factorization for multiple\nmatrices with identical structure and similar values.\n0*\nDo not use low rank update functionality.\n1\nUse low rank update functionality. You must also set iparm[23] = 10 and\nprovide a list of changed values in the perm array.\nThis option requires the default settings of iparm[3], iparm[4], iparm[5],\niparm[27], iparm[30], iparm[35], iparm[36], iparm[55], and iparm[59]\nas well.\niparm[39] -\niparm[41]\nReserved. Set to zero.\niparm[42]\nControl parameter for the computation of the diagonal of inverse matrix.\n0*\nDo not compute the diagonal of inverse matrix.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1870\n\n\nComponent\nDescription\n1\nIntel® oneAPI Math Kernel Library (oneMKL) PARDISO computes the diagonal\nof the inverse matrix during the factorization phase. This feature is only\navailable with two-level factorization algorithm (iparm[23] = 1) and real\nsymmetric matrices (mtype = 2 or mtype = -2). The diagonal is returned in\nthe solution vector.\niparm[43] -\niparm[54]\nReserved. Set to zero.\niparm[55]\nDiagonal and pivoting control.\n0*\nInternal function used to work with pivot and calculation of diagonal arrays\nturned off.\n1\nYou can use the mkl_pardiso_pivot callback routine to control pivot elements\nwhich appear during numerical factorization. Additionally, you can obtain the\nelements of initial matrix and factorized matrices after the pardiso\nfactorization step diagonal using the pardiso_getdiagroutine. This parameter\ncan be turned on only in the in-core version of Intel® oneAPI Math Kernel\nLibrary (oneMKL) PARDISO.\nNOTE oneMKL PARDISO uses the Cholesky factorization without pivoting for\nmtype=2 (symmetric positive-definite matrix) and mtype=4 (complex and\nHermitian positive definite). Accordingly, setting iparm[55] = 1 is ignored for\nthese matrix types.\niparm[56] -\niparm[58]\nReserved. Set to zero.\niparm[59]\ninput\nIntel® oneAPI Math Kernel Library (oneMKL) PARDISO mode.\niparm[59]switches between in-core (IC) and out-of-core (OOC) Intel® oneAPI Math\nKernel Library (oneMKL) PARDISO. OOC can solve very large problems by holding the\nmatrix factors in files on the disk, which requires a reduced amount of main memory\ncompared to IC.\nUnless you are operating in sequential mode, you can switch between IC and OOC\nmodes after the reordering phase. However, you can get better Intel® oneAPI Math\nKernel Library (oneMKL) PARDISO performance by settingiparm[59] before the\nreordering phase.\nThe amount of memory used in OOC mode depends on the number of OpenMP\nthreads.\nNOTE\nWhen iparm[59] > 0, use the MKL_PARDISO_OOC_FILE_NAME environment variable\nto store factors.\nWarning\nDo not increase the number of OpenMP threads used for cluster_sparse_solver between the\nfirst call and the factorization or solution phase. Because the minimum amount of memory\nrequired for out-of-core execution depends on the number of OpenMP threads, increasing it\nafter the initial call can cause incorrect results.\n0*\nIC mode.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1871\n\n\nComponent\nDescription\n1\nIC mode is used if the total amount of RAM (in megabytes) needed for storing\nthe matrix factors is less than sum of two values of the environment variables:\nMKL_PARDISO_OOC_MAX_CORE_SIZE (default value 2000 MB) and\nMKL_PARDISO_OOC_MAX_SWAP_SIZE (default value 0 MB); otherwise OOC\nmode is used. In this case amount of RAM used by OOC mode cannot exceed\nthe value of MKL_PARDISO_OOC_MAX_CORE_SIZE.\nIf the total peak memory needed for storing the local arrays is more than\nMKL_PARDISO_OOC_MAX_CORE_SIZE, increase\nMKL_PARDISO_OOC_MAX_CORE_SIZE if possible.\nNOTE\nConditional numerical reproducibility (CNR) is not supported for this mode.\n2\nOOC mode.\nThe OOC mode can solve very large problems by holding the matrix factors in\nfiles on the disk. Hence the amount of RAM required by OOC mode is\nsignificantly reduced compared to IC mode.\nIf the total peak memory needed for storing the local arrays is more than\nMKL_PARDISO_OOC_MAX_CORE_SIZE, increase\nMKL_PARDISO_OOC_MAX_CORE_SIZE if possible.\nTo obtain better Intel® oneAPI Math Kernel Library (oneMKL) PARDISO\nperformance, during the numerical factorization phase you can provide the\nmaximum number of right-hand sides, which can be used further during the\nsolving phase.\nRefer to How to use Intel® MKL OOC PARDISO and Storage of Matrices for\nmore details about OOC.\niparm[60] -\niparm[61]\nReserved. Set to zero.\niparm[62]\noutput\nSize of the minimum OOC memory for numerical factorization and solution.\nThis parameter provides the size in kilobytes of the minimum memory required by\nOOC Intel® oneAPI Math Kernel Library (oneMKL) PARDISO for internal floating point\narrays. This parameter is computed in phase 1.\nTotal peak memory consumption of OOC Intel® oneAPI Math Kernel Library (oneMKL)\nPARDISO can be estimated asmax(iparm[14], iparm[15] + iparm[62]).\niparm[63]\nRese\nrved\n. Set\nto\nzero.\nNOTE\nGenerally in sparse matrices, components which are equal to zero can be considered non-zero if\nnecessary. For example, in order to make a matrix structurally symmetric, elements which are zero\ncan be considered non-zero. See Sparse Matrix Storage Formats for an example.\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1872\n\n\nProduct and Performance Information\nNotice revision #20201201\nPARDISO_DATA_TYPE\nThe following table lists the values of PARDISO_DATA_TYPE depending on the matrix types and values of the\nparameter iparm[27].\nData type value\nMatrix type mtype\niparm[27]\ncomments\ndouble\n1, 2, -2, 11\n0\nReal matrices, do\nfloat\n1\nReal matrices, sin\nMKL_Complex16\n3, 6, 13, 4, -4\n0\nComplex matrices\nprecision\nMKL_Complex8\n1\nComplex matrices\nprecision\nParallel Direct Sparse Solver for Clusters Interface\nThe Parallel Direct Sparse Solver for Clusters Interface solves large linear systems of equations with sparse\nmatrices on clusters. It is\n•\nhigh performing\n•\nrobust\n•\nmemory efficient\n•\neasy to use\nA hybrid implementation combines Message Passing Interface (MPI) technology for data exchange between\nparallel tasks (processes) running on different nodes, and OpenMP* technology for parallelism inside each\nnode of the cluster. This approach effectively uses modern hardware resources such as clusters consisting of\nnodes with multi-core processors. The solver code is optimized for the latest Intel processors, but also\nperforms well on clusters consisting of non-Intel processors.\nCode examples are available in the Intel® oneAPI Math Kernel Library (oneMKL) installationexamples\ndirectory.\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nParallel Direct Sparse Solver for Clusters Interface Algorithm\nParallel Direct Sparse Solver for Clusters Interface solves a set of sparse linear equations\nA*X = B\nwith multiple right-hand sides using a distributed LU, LLT , LDLT or LDL* factorization, where A is an n-by-n\nmatrix, and X and B are n-by-nrhs matrices.\nThe solution comprises four tasks:\n•\nanalysis and symbolic factorization;\n•\nnumerical factorization;\n•\nforward and backward substitution including iterative refinement;\n•\ntermination to release all internal solver memory.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1873\n\n\nThe solver first computes a symmetric fill-in reducing permutation P based on the nested dissection\nalgorithm from the METIS package [Karypis98](included with Intel® oneAPI Math Kernel Library (oneMKL)),\nfollowed by the Cholesky or other type of factorization (depending on matrix type)[Schenk00-2] of PAPT. The\nsolver uses either diagonal pivoting, or 1x1 and 2x2 Bunch and Kaufman pivoting for symmetric indefinite or\nHermitian matrices before finding an approximation of X by forward and backward substitution and iterative\nrefinement.\nThe initial matrix A is perturbed whenever numerically acceptable 1x1 and 2x2 pivots cannot be found within\nthe diagonal blocks. One or two passes of iterative refinement may be required to correct the effect of the\nperturbations. This restricted notion of pivoting with iterative refinement is effective for highly indefinite\nsymmetric systems. For a large set of matrices from different application areas, the accuracy of this method\nis comparable to a direct factorization method that uses complete sparse pivoting techniques [Schenk04].\nParallel Direct Sparse Solver for Clusters additionally improves the pivoting accuracy by applying symmetric\nweighted matching algorithms. These methods identify large entries in the coefficient matrix A that, if\npermuted close to the diagonal, enable the factorization process to identify more acceptable pivots and\nproceed with fewer pivot perturbations. The methods are based on maximum weighted matching and\nimprove the quality of the factor in a complementary way to the alternative idea of using more complete\npivoting techniques.\nParallel Direct Sparse Solver for Clusters Interface Matrix Storage\nThe sparse data storage in the Parallel Direct Sparse Solver for Clusters Interface follows the scheme\ndescribed in the Sparse Matrix Storage Formats section using the variable ja for columns, ia for rowIndex,\nand a for values. Column indices ja must be in increasing order per row.\nWhen an input data structure is not accessed in a call, a NULL pointer or any valid address can be passed as\na placeholder for that argument.\nAlgorithm Parallelization and Data Distribution\nIntel® oneAPI Math Kernel Library (oneMKL) Parallel Direct Sparse Solver for Clusters enables parallel\nexecution of the solution algorithm with efficient data distribution.\nThe master MPI process performs the symbolic factorization phase to represent matrix A as computational\ntree. Then matrix A is divided among all MPI processes in a one-dimensional manner. The same distribution\nis used for L-factor (the lower triangular matrix in Cholesky decomposition). Matrix A and all required internal\ndata are broadcast to subordinate MPI processes. Each MPI process fills in its own parts of L-factor with initial\nvalues of the matrix A.\nParallel Direct Sparse Solver for Clusters Interface computes all independent parts of L-factor completely in\nparallel. When a block of the factor must be updated by other blocks, these updates are independently\npassed to a temporary array on each updating MPI process. It further gathers the result into an updated\nblock using the MPI_Reduce()routine. The computations within an MPI process are dynamically divided\namong OpenMP threads using pipelining parallelism with a combination of left- and right-looking techniques\nsimilar to those of the PARDISO* software. Level 3 BLAS operations from Intel® oneAPI Math Kernel Library\n(oneMKL) ensure highly efficient performance of block-to-block update operations.\nDuring forward/backward substitutions, respective Right Hand Side (RHS) parts are distributed among all\nMPI processes. All these processes participate in the computation of the solution. Finally, the solution is\ngathered on the master MPI process.\nThis approach demonstrates good scalability on clusters with Infiniband* technology. Another advantage of\nthe approach is the effective distribution of L-factor among cluster nodes. This enables the solution of tasks\nwith a much higher number of non-zero elements than it is possible with any Symmetric Multiprocessing\n(SMP) in-core direct solver.\nThe algorithm ensures that the memory required to keep internal data on each MPI process is decreased\nwhen the number of MPI processes in a run increases. However, the solver requires that matrix A and some\nother internal arrays completely fit into the memory of each MPI process.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1874\n\n\nTo get the best performance, run one MPI process per physical node and set the number of OpenMP* threads\nper node equal to the number of physical cores on the node.\nNOTE\nInstead of calling MPI_Init(), initialize MPI with MPI_Init_thread() and set the MPI threading level\nto MPI_THREAD_FUNNELED or higher. For details, see the code examples in <install_dir>/\nexamples.\ncluster_sparse_solver\nCalculates the solution of a set of sparse linear\nequations with single or multiple right-hand sides.\nSyntax\nvoid cluster_sparse_solver (_MKL_DSS_HANDLE_t pt, const MKL_INT *maxfct, const MKL_INT\n*mnum, const MKL_INT *mtype, const MKL_INT *phase, const MKL_INT *n, const void *a,\nconst MKL_INT *ia, const MKL_INT *ja, MKL_INT *perm, const MKL_INT *nrhs, MKL_INT\n*iparm, const MKL_INT *msglvl, void *b, void *x, const int *comm, MKL_INT *error);\nInclude Files\n•\nmkl_cluster_sparse_solver.h\nDescription\nThe routine cluster_sparse_solver calculates the solution of a set of sparse linear equations\nA*X = B\nwith single or multiple right-hand sides, using a parallel LU, LDL, or LLT factorization, where A is an n-by-n\nmatrix, and X and B are n-by-nrhs vectors or matrices.\nNOTE\nThis routine supports the Progress Routine feature. See Progress Function for details.\nInput Parameters\nNOTE\nMost of the input parameters (except for the pt, phase, and comm parameters and, for the\ndistributed format, the a, ia, and ja arrays) must be set on the master MPI process only, and\nignored on other processes. Other MPI processes get all required data from the master MPI\nprocess using the MPI communicator, comm.\npt\nArray of size 64.\nHandle to internal data structure. The entries must be set to zero before the\nfirst call to cluster_sparse_solver.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1875\n\n\nCaution\nAfter the first call to cluster_sparse_solver do not modify pt,\nas that could cause a serious memory leak.\nmaxfct\nIgnored; assumed equal to 1.\nmnum\nIgnored; assumed equal to 1.\nmtype\nDefines the matrix type, which influences the pivoting method. The Parallel\nDirect Sparse Solver for Clusters solver supports the following matrices:\n1\nreal and structurally symmetric\n2\nreal and symmetric positive definite\n-2\nreal and symmetric indefinite\n3\ncomplex and structurally symmetric\n4\ncomplex and Hermitian positive definite\n-4\ncomplex and Hermitian indefinite\n6\ncomplex and symmetric\n11\nreal and nonsymmetric\n13\ncomplex and nonsymmetric\nphase\nControls the execution of the solver. Usually it is a two- or three-digit\ninteger. The first digit indicates the starting phase of execution and the\nsecond digit indicates the ending phase. Parallel Direct Sparse Solver for\nClusters has the following phases of execution:\n•\nPhase 1: Fill-reduction analysis and symbolic factorization\n•\nPhase 2: Numerical factorization\n•\nPhase 3: Forward and Backward solve including optional iterative\nrefinement\n•\nMemory release (phase= -1)\nIf a previous call to the routine has computed information from previous\nphases, execution may start at any phase. The phase parameter can have\nthe following values:\nphase\nSolver Execution Steps\n11\nAnalysis\n12\nAnalysis, numerical factorization\n13\nAnalysis, numerical factorization, solve, iterative\nrefinement\n22\nNumerical factorization\n23\nNumerical factorization, solve, iterative refinement\n33\nSolve, iterative refinement\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1876\n\n\nphase\nSolver Execution Steps\n-1\nRelease all internal memory for all matrices\nn\nNumber of equations in the sparse linear systems of equations A*X = B.\nConstraint: n > 0.\na\nArray. Contains the non-zero elements of the coefficient matrix A\ncorresponding to the indices in ja. The coefficient matrix can be either real\nor complex. The matrix must be stored in the three-array variant of the\ncompressed sparse row (CSR3) or in the three-array variant of the block\ncompressed sparse row (BSR3) format, and the matrix must be stored with\nincreasing values of ja for each row.\nFor CSR3 format, the size of a is the same as that of ja. Refer to the\nvalues array description in Three Array Variation of CSR Format for more\ndetails.\nFor BSR3 format the size of a is the size of ja multiplied by the square of\nthe block size. Refer to the values array description in Three Array\nVariation of BSR Format for more details.\nNOTE\nFor centralized input (iparm[39]=0), provide the a array for the\nmaster MPI process only. For distributed assembled input\n(iparm[39]=1 or iparm[39]=2), provide it for all MPI processes.\nImportant\nThe column indices of non-zero elements of each row of the\nmatrix A must be stored in increasing order.\nia\nFor CSR3 format, ia[i] (i<n) points to the first column index of row i in\nthe array ja. That is, ia[i] gives the index of the element in array a that\ncontains the first non-zero element from row i of A. The last element ia[n]\nis taken to be equal to the number of non-zero elements in A, plus one.\nRefer to rowIndex array description in Three Array Variation of CSR Format\nfor more details.\nFor BSR3 format, ia[i] (i<n) points to the first column index of row i in\nthe array ja. That is, ia[i] gives the index of the element in array a that\ncontains the first non-zero block from row i of A. The last element ia[n] is\ntaken to be equal to the number of non-zero blcoks in A, plus one. Refer to\nrowIndex array description in Three Array Variation of BSR Format for more\ndetails.\nThe array ia is accessed in all phases of the solution process.\nIndexing of ia is one-based by default, but it can be changed to zero-based\nby setting the appropriate value to the parameter iparm[34]. For zero-\nbased indexing, the last element ia[n] is assumed to be equal to the\nnumber of non-zero elements in matrix A.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1877\n\n\nNOTE\nFor centralized input (iparm[39]=0), provide the ia array at the\nmaster MPI process only. For distributed assembled input\n(iparm[39]=1 or iparm[39]=2), provide it at all MPI processes.\nja\nFor CSR3 format, array ja contains column indices of the sparse matrix A.\nIt is important that the indices are in increasing order per row. For\nsymmetric matrices, the solver needs only the upper triangular part of the\nsystem as is shown for columns array in Three Array Variation of CSR\nFormat.\nFor BSR3 format, array ja contains column indices of the sparse matrix A.\nIt is important that the indices are in increasing order per row. For\nsymmetric matrices, the solver needs only the upper triangular part of the\nsystem as is shown for columns array in Three Array Variation of BSR\nFormat.\nThe array ja is accessed in all phases of the solution process.\nIndexing of ja is one-based by default, but it can be changed to zero-based\nby setting the appropriate value to the parameter iparm(35).\nNOTE\nFor centralized input (iparm(40)=0), provide the ja array at the\nmaster MPI process only. For distributed assembled input\n(iparm(40)=1 or iparm(40)=2), provide it at all MPI processes.\nperm\nIgnored.\nnrhs\nNumber of right-hand sides that need to be solved for.\niparm\nArray, size 64. This array is used to pass various parameters to Parallel\nDirect Sparse Solver for Clusters Interface and to return some useful\ninformation after execution of the solver.\nSee cluster_sparse_solver iparm Parameter for more details about the\niparm parameters.\nmsglvl\nMessage level information. If msglvl = 0 then cluster_sparse_solver\ngenerates no output, if msglvl = 1 the solver prints statistical information\nto the screen.\nStatistics include information such as the number of non-zero elements in\nL-factor and the timing for each phase.\nSet msglvl = 1 if you report a problem with the solver, since the additional\ninformation provided can facilitate a solution.\nb\nArray, size n*nrhs. On entry, contains the right-hand side vector/matrix B,\nwhich is placed in memory contiguously. The b[i+k*n] must hold the i-th\ncomponent of k-th right-hand side vector. Note that b is only accessed in\nthe solution phase.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1878\n\n\ncomm\nMPI communicator. The solver uses the Fortran MPI communicator\ninternally. Convert the MPI communicator to Fortran using the\nMPI_Comm_c2f() function. See the examples in the <install_dir>/\nexamples directory.\nOutput Parameters\npt\nHandle to internal data structure.\nperm\nIgnored.\niparm\nOn output, some iparm values report information such as the numbers of\nnon-zero elements in the factors.\nSee cluster_sparse_solver iparm Parameter for more details about the\niparm parameters.\nb\nOn output, the array is replaced with the solution if iparm[5] = 1.\nx\nArray, size (n*nrhs). If iparm[5]=0 it contains solution vector/matrix X,\nwhich is placed contiguously in memory. The x[i+k*n] element must hold\nthe i-th component of the k-th solution vector. Note that x is only accessed\nin the solution phase.\nerror\nThe error indicator according to the below table:\nerror\nInformation\n0\nno error\n-1\ninput inconsistent\n-2\nnot enough memory\n-3\nreordering problem\n-4\nZero pivot, numerical factorization or iterative\nrefinement problem. If the error appears during the\nsolution phase, try to change the pivoting perturbation\n(iparm[9]) and also increase the number of iterative\nrefinement steps. If it does not help, consider changing\nthe scaling, matching and pivoting options (iparm[10],\niparm[12], iparm[20])\n-5\nunclassified (internal) error\n-6\nreordering failed (matrix types 11 and 13 only)\n-7\ndiagonal matrix is singular\n-8\n32-bit integer overflow problem\n-9\nnot enough memory for OOC\n-10\nerror opening OOC files\n-11\nread/write error with OOC files\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1879\n\n\ncluster_sparse_solver_64\nCalculates the solution of a set of sparse linear\nequations with single or multiple right-hand sides.\nSyntax\nvoid cluster_sparse_solver_64 (_MKL_DSS_HANDLE_t pt, const long long int *maxfct, const\nlong long int *mnum, const long long int *mtype, const long long int *phase, const long\nlong int *n, const void *a, const long long int *ia, const long long int *ja, long long\nint *perm, const long long int *nrhs, long long int *iparm, const long long int\n*msglvl, void *b, void *x, const int *comm, long long int *error);\nInclude Files\n•\nmkl_cluster_sparse_solver.h\nDescription\nThe routine cluster_sparse_solver_64 is an alternative ILP64 (64-bit integer) version of the \ncluster_sparse_solver routine (see the Description section for more details). The interface of\ncluster_sparse_solver_64 is the same as the interface of cluster_sparse_solver, but it accepts and\nreturns all integer data as long long int.\nUse cluster_sparse_solver_64 when cluster_sparse_solverfor solving large matrices (with the\nnumber of non-zero elements on the order of 500 million or more). You can use it together with the usual\nLP64 interfaces for the rest of Intel® oneAPI Math Kernel Library (oneMKL) functionality. In other words, if\nyou use 64-bit integer version (cluster_sparse_solver_64), you do not need to re-link your applications\nwith ILP64 libraries. Take into account that cluster_sparse_solver_64 may perform slower than regular\ncluster_sparse_solver on the reordering and symbolic factorization phase.\nNOTE\ncluster_sparse_solver_64 is supported only in the 64-bit libraries. If\ncluster_sparse_solver_64 is called from the 32-bit libraries, it returns error =-12.\nNOTE\nThis routine supports the Progress Routine feature. See Progress Function for details.\nInput Parameters\nThe input parameters of cluster_sparse_solver_64 are the same as the input parameters of\ncluster_sparse_solver, but cluster_sparse_solver_64 accepts all integer data as long long int.\nOutput Parameters\nThe output parameters of cluster_sparse_solver_64 are the same as the output parameters of\ncluster_sparse_solver, but cluster_sparse_solver_64 returns all integer data as long long int.\ncluster_sparse_solver_get_csr_size\nComputes the (local) number of rows and (local)\nnumber of nonzero entries for (distributed) CSR data\ncorresponding to the provided name.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1880\n\n\nSyntax\nvoid cluster_sparse_solver_get_csr_size (_MKL_DSS_HANDLE_t pt, _MKL_DSS_EXPORT_DATA\nname , MKL_INT *local_nrows, MKL_INT *local_nnz, const int*comm, MKL_INT *error);\nInclude Files\n•\nmkl_cluster_sparse_solver.h\nDescription\nThis routine uses the internal data created during the factorization phase of cluster_sparse_solver for\nmatrix A. The routine then:\n•\nComputes the local number of rows and the local number of nonzeros for CSR data that correspond to the\nprovided name\n•\nReturns the computed values in local_nrows and local_nnz\nIt is assumed that the CSR data defined by the name will be distributed in the same way as the matrix A (as\ndefined by iparm[39]) used in cluster_sparse_solver.\nThe returned values can be used for allocating CSR arrays for factors L and U, and also for allocating arrays\nfor permutations P and Q, or scaling matrix D which can then be used with\ncluster_sparse_solver_set_csr_ptrs or cluster_sparse_solver_set_ptr for exporting\ncorresponding data via cluster_sparse_solver_export.\nNOTE\nOnly call this routine after the factorization phase (phase=22) of the cluster_sparse_solver has\nbeen called. Neither pt, nor iparm should be changed after the preceding call to\ncluster_sparse_solver.\nInput Parameters\npt\nArray with size of 64.\nHandle to internal data structure used in the prior calls to\ncluster_sparse_solver.\nCaution\nDo not modify pt after the calls to cluster_sparse_solver.\nname\nSpecifies CSR data for which the output values are computed.\nSPARSE_PTLUQT_L\nFactor L from P*A*Q=L*U.\nSPARSE_PTLUQT_U\nFactor U from P*A*Q=L*U.\nSPARSE_DPTLUQT_L\nFactor L from P* (D-1A)*Q=L*U.\nSPARSE_DPTLUQT_U\nFactor U from P* (D-1A)*Q=L*U.\nlocal_nrows\nOn entry, an array of size 1.\nlocal_nnz\nOn entry, an array of size 1.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1881\n\n\ncomm\nMPI communicator. The solver uses the Fortran MPI communicator\ninternally. Convert the MPI communicator to Fortran using the\nMPI_Comm_c2f() function. See the examples in the <install_dir>/\nexamples directory.\nOutput Parameters\nlocal_nrows\nOn output, the local number of rows for the CSR data which correspond to\nthe name.\nlocal_nnz\nOn output, the local number of nonzero entries for the CSR data which\ncorrespond to the name.\nerror\nThe error indicator:\nerror\nInformation\n0\nno error\n-1\npt is a null pointer\n-2\ninvalid pt\n-3\ninvalid name\n-4\nunsupported name\n-9\nunsupported internal code path, consider switching off\nnon-default iparm parameters for\ncluster_sparse_solver\n-10\nunsupported case when the matrix A is distributed\namong processes with overlap in the preceding calls to\ncluster_sparse_solver\n-12\ninternal memory error\nNOTE Refer to cl_solver_export_c.c for an example using this functionality.\ncluster_sparse_solver_set_csr_ptrs\nSaves internally-provided pointers to the 3-array CSR\ndata corresponding to the specified name.\nSyntax\nvoid cluster_sparse_solver_set_csr_ptrs (_MKL_DSS_HANDLE_t pt, _MKL_DSS_EXPORT_DATA\nname, MKL_INT *rowptr, MKL_INT *colindx, void *vals, const int *comm, MKL_INT *error);\nInclude Files\n•\nmkl_cluster_sparse_solver.h\nDescription\nThis routine internally saves the input pointers, rowptr, colindx, and vals, of the 3-array CSR data, which\ncorrespond to the provided name. It is assumed that the exported data will be distributed in the same way as\nthe matrix A (as defined by iparm[39]) used in cluster_sparse_solver.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1882\n\n\nThe saved pointers can then be used for exporting corresponding data by means of\ncluster_sparse_solver_export.\nNOTE\nOnly call this routine after the factorization phase (phase=22) of the cluster_sparse_solver has\nbeen called. Neither pt, nor iparm should be changed after the preceding call to\ncluster_sparse_solver.\nInput Parameters\npt\nArray with size of 64.\nHandle to internal data structure used in the prior calls to\ncluster_sparse_solver.\nCaution\nDo not modify pt after the calls to cluster_sparse_solver.\nname\nSpecifies for which CSR data the pointers are provided.\nSPARSE_PTLUQT_L\nFactor L from P*A*Q=L*U.\nSPARSE_PTLUQT_U\nFactor U from P*A*Q=L*U.\nSPARSE_DPTLUQT_L\nFactor L from P* (D-1A)*Q=L*U.\nSPARSE_DPTLUQT_U\nFactor U from P* (D-1A)*Q=L*U.\nrowptr\nArray of length at least (local_nrows+1) where local_nrows is the local\nnumber of rows, which can be obtained by calling\ncluster_sparse_solver_get_csr_size. This array contains row indices,\nsuch that rowptr[i] - indexing is the first index of row i in the array's\nvals and colindx. Here, the value of indexing is 0 for zero-based\nindexing and 1 for one-based indexing, and must be the same as it was for\nthe matrix A used in the preceding calls to cluster_sparse_solver (also\nstored in iparm[34] ).\nRefer to pointerB array description in CSR Format for more details.\ncolindx\nArray of length at least rowptr[local_nrows] – rowptr[0]. Indexing\n(zero- or one-based) must be the same as for rowptr. For one-based\nindexing, the array contains the column indices plus one for each non-zero\nelement of the matrix which corresponds to the name. For zero-based\nindexing, the array contains the column indices for each non-zero element\nof the matrix.\nvals\nArray containing non-zero elements of the matrix which corresponds to the\nname. Its length is equal to length of the colindx array. Refer to values\narray description in CSR Format for more details.\nIt will be interpreted internally as float*/double*(MKL_Complex8*/\nMKL_Complex16*) depending on mtype (type of the matrix A) and\niparm[27] (precision) specified in the preceding call to\ncluster_sparse_solver.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1883\n\n\ncomm\nMPI communicator. The solver uses the Fortran MPI communicator\ninternally. Convert the MPI communicator to Fortran using the\nMPI_Comm_c2f() function. See the examples in the <install_dir>/\nexamples directory.\nOutput Parameters\nerror\nThe error indicator:\nerror\nInformation\n0\nno error\n-1\npt is a null pointer\n-2\ninvalid pt\n-3\ninvalid name\n-4\nunsupported name\n-9\nunsupported internal code path, consider switching off\nnon-default iparm parameters for\ncluster_sparse_solver\n-12\ninternal memory error\nNOTE Refer to cl_solver_export_c.c for an example using this functionality.\ncluster_sparse_solver_set_ptr\nInternally saves a provided pointer to the data\ncorresponding to the specified name.\nSyntax\nvoid cluster_sparse_solver_set_ptr (_MKL_DSS_HANDLE_t pt, _MKL_DSS_EXPORT_DATA name,\nvoid *ptr, const int *comm, MKL_INT *error);\nInclude Files\n•\nmkl_cluster_sparse_solver.h\nDescription\nThis routine internally saves the input pointer, ptr, of the data which correspond to the provided name. The\nsaved pointer can then be used for exporting corresponding data by means of\ncluster_sparse_solver_export.\nNOTE\nOnly call this routine after the factorization phase (phase=22) of the cluster_sparse_solver has\nbeen called. Neither pt, nor iparm should be changed after the preceding call to\ncluster_sparse_solver.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1884\n\n\nInput Parameters\npt\nArray with size of 64.\nHandle to internal data structure used in the prior calls to\ncluster_sparse_solver.\nCaution\nDo not modify pt after the calls to cluster_sparse_solver.\nname\nSpecifies the data for which the pointer is provided.\nSPARSE_PTLUQT_P\nPermutation P from P*A*Q=L*U.\nSPARSE_PTLUQT_Q\nPermutation Q from P*A*Q=L*U.\nSPARSE_DPTLUQT_P\nPermutation P from P* (D-1A)*Q=L*U.\nSPARSE_DPTLUQT_Q\nPermutation Q from P* (D-1A)*Q=L*U.\nSPARSE_DPTLUQT_D\nScaling (diagonal) D from P* (D-1A)*Q=L*U.\nvals\nArray containing elements of the vector representation for the data which\ncorresponds to the name. Its length should be at least local_nrows, where\nlocal_nrows is the local number of rows in a corresponding matrix\n(obtained from cluster_sparse_solver_get_csr_size, for example).\nFor permutations P and Q, vals is interpreted as MKL_INT* , while for the\nscaling it is interpreted as float*/double*(MKL_Complex8*/\nMKL_Complex16*) depending on mtype (type of the matrix A) and\niparm[27] (precision) specified in the preceding call to\ncluster_sparse_solver.\ncomm\nMPI communicator. The solver uses the Fortran MPI communicator\ninternally. Convert the MPI communicator to Fortran using the\nMPI_Comm_c2f() function. See the examples in the <install_dir>/\nexamples directory.\nOutput Parameters\nerror\nThe error indicator:\nerror\nInformation\n0\nno error\n-1\npt is a null pointer\n-2\ninvalid pt\n-3\ninvalid name\n-4\nunsupported name\n-9\nunsupported internal code path, consider switching off\nnon-default iparm parameters for\ncluster_sparse_solver\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1885\n\n\nerror\nInformation\n-12\ninternal memory error\nNOTE Refer to cl_solver_export_c.c for an example using this functionality.\ncluster_sparse_solver_export\nComputes data corresponding to the specified\ndecomposition (defined by export operation) and fills\nthe pointers provided by calls to\ncluster_sparse_solver_set_ptr and/or\ncluster_sparse_solver_set_csr_ptrs.\nSyntax\nvoid cluster_sparse_solver_export (_MKL_DSS_HANDLE_t pt, _MKL_DSS_EXPORT_OPERATION\noperation, const int *comm, MKL_INT *error);\nInclude Files\n•\nmkl_cluster_sparse_solver.h\nDescription\nThis routine computes the data for the pointers of the (distributed) data to be exported (as defined the\nspecified operation). It is assumed that the exported data will be distributed in the same way as the matrix A\n(as defined by iparm[39]) used in cluster_sparse_solver.\nNOTE\nOnly call this routine after the factorization phase (phase=22) of the cluster_sparse_solver has\nbeen called. Neither pt, nor iparm should be changed after the preceding call to\ncluster_sparse_solver.\nNOTE\nOnly call this routine after all pointers to the data required for the specified operation have been\nprovided by means of calling cluster_sparse_solver_set_ptr and/or\ncluster_sparse_solver_set_csr_ptrs.\nInput Parameters\npt\nArray with size of 64.\nHandle to internal data structure used in the prior calls to\ncluster_sparse_solver.\nCaution\nDo not modify pt after the calls to cluster_sparse_solver.\noperation\nSpecifies a particular operation which defines what data are exported\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1886\n\n\nSPARSE_PTLUQT\nExporting data from decomposition P*A*Q=L*U.\nSPARSE_DPTLUQT\nExporting data from decomposition from P*\n(D-1A)*Q=L*U.\nNOTE\nCurrently, for operation=SPARSE_DPTLUQT a real (complex) unit\nvector is provided for the scaling matrix D. Do not turn on\nscaling(iparm[10]>0) or matching(iparm[12]>0) in the iparm\nduring the call to cluster_sparse_solver for this value of\noperation.\ncomm\nMPI communicator. The solver uses the Fortran MPI communicator\ninternally. Convert the MPI communicator to Fortran using the\nMPI_Comm_c2f() function. See the examples in the <install_dir>/\nexamples directory.\nOutput Parameters\nerror\nThe error indicator:\nerror\nInformation\n0\nno error\n-1\npt is a null pointer\n-2\ninvalid pt\n-5\ninvalid operation\n-6\npointers to some of the data required for the specified\noperation were not provided prior to calling\ncluster_sparse_solver_export\n-9\nunsupported internal code path, consider switching off\nnon-default iparm parameters for\ncluster_sparse_solver\n-10\nunsupported case when the matrix A is distributed\namong processes with overlap in the preceding calls to\ncluster_sparse_solver\n-12\ninternal memory error\nNOTE Refer to cl_solver_export_c.c for an example using this functionality.\ncluster_sparse_solver iparm Parameter\nThe following table describes all individual components of the Parallel Direct Sparse Solver for Clusters\nInterface iparm parameter. Components which are not used must be initialized with 0. Default values are\ndenoted with an asterisk (*).\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1887\n\n\nComponent\nDescription\niparm[0]\ninput\nUse default values.\n0\niparm[1] - iparm(64) are filled with default values.\n!=0\nYou must supply all values in components iparm[1] - iparm(64).\niparm[1]\ninput\nFill-in reducing ordering for the input matrix.\n2*\nThe nested dissection algorithm from the METIS package [Karypis98].\n3\nThe parallel version of the nested dissection algorithm. It can decrease the time of\ncomputations on multi-core computers, especially when Phase 1 takes significant time.\n10\nThe MPI version of the nested dissection and symbolic factorization algorithms for the\nmatrix in distributed assembled matrix input format (iparm[39] > 0) . The input matrix\nfor the reordering must be distributed among different MPI processes without any\nintersection and all MPI ranks must have at least one row of the input matrix. Use\niparm[40] and iparm[41] to set the bounds of the domain. During all of Phase 1, the\nentire matrix is not gathered on any one process, which can decrease computation time\n(especially when Phase 1 takes significant time) and decrease memory usage for each MPI\nprocess on the cluster.\nNOTE Distributed reordering does not work if any of matching(iparm[12]=1)/\nscaling(iparm[10]=1)/BSR format(iparm[36]>1)/Schur complement\nmatrix computation control(iparm[35]>0)/Partial\nsolve(iparm[30] > 0) is turned on, or if the distributed input matrix has\noverlapping distribution of rows across MPI processes.\nNOTE\nIf you set iparm[1] = 10, comm = -1 (MPI communicator), and if there is one\nMPI process, optimization and full parallelization with the OpenMP version of the\nnested dissection and symbolic factorization algorithms proceeds. This can decrease\ncomputation time on multi-core computers. In this case, set iparm[40] = 1 and\niparm[41] = n for one-based indexing, or to 0 and n - 1, respectively, for zero-\nbased indexing.\niparm[2]\nReserved. Set to zero.\niparm[3]\nReserved. Set to zero.\niparm[4]\ninput\nUser permutation.\nThis parameter controls whether user supplied fill-in reducing permutation is used instead\nof the integrated multiple-minimum degree or nested dissection algorithms. Another use\nof this parameter is to control obtaining the fill-in reducing permutation vector calculated\nduring the reordering stage of Intel® oneAPI Math Kernel Library (oneMKL) PARDISO.\nThis option is useful for testing reordering algorithms, adapting the code to special\napplications problems (for instance, to move zero diagonal elements to the end of\nP*A*PT), or for using the permutation vector more than once for matrices with identical\nsparsity structures. For definition of the permutation, see the description of the perm\nparameter.\nCaution\nYou can only set one of iparm[4], iparm[30], and iparm[35], so be sure that the\niparm[30] (partial solution) and the iparm[35] (Schur complement) parameters are 0 if\nyou set iparm[4].\n0\nUser permutation in the perm array is ignored.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1888\n\n\nComponent\nDescription\n1\nIntel® oneAPI Math Kernel Library (oneMKL) PARDISO uses the user supplied fill-in reducing\npermutation from theperm array. iparm[1] is ignored.\n2\nIntel® oneAPI Math Kernel Library (oneMKL) PARDISO returns the permutation vector\ncomputed at phase 1 in theperm array.\niparm[5]\ninput\nWrite solution on x.\nNOTE\nThe array x is always used.\n0*\nThe array x contains the solution; right-hand side vector b is kept unchanged.\n1\nThe solver stores the solution on the right-hand side b.\niparm[6]\noutput\nNumber of iterative refinement steps performed.\nReports the number of iterative refinement steps that were actually performed during the\nsolve step.\niparm[7]\ninput\nIterative refinement step.\nOn entry to the solve and iterative refinement step, iparm[7] must be set to the\nmaximum number of iterative refinement steps that the solver performs.\n0*\nThe solver automatically performs two steps of iterative refinement when\nperturbed pivots are obtained during the numerical factorization.\n>0\nMaximum number of iterative refinement steps that the solver performs. The\nsolver performs not more than the absolute value of iparm[7] steps of iterative\nrefinement. The solver might stop the process before the maximum number of\nsteps if\n•\na satisfactory level of accuracy of the solution in terms of backward error is\nachieved,\n•\nor if it determines that the required accuracy cannot be reached. In this case\nParallel Direct Sparse Solver for Clusters Interface returns -4 in the error\nparameter.\nThe number of executed iterations is reported in iparm[6].\n<0\nSame as above, but the accumulation of the residuum uses extended precision\nreal and complex data types.\nPerturbed pivots result in iterative refinement (independent of iparm[7]=0) and\nthe number of executed iterations is reported in iparm[6].\niparm[8]\nReserved. Set to zero.\niparm[9]\ninput\nPivoting perturbation.\nThis parameter instructs Parallel Direct Sparse Solver for Clusters Interface how to handle\nsmall pivots or zero pivots for nonsymmetric matrices (mtype =11 or mtype =13) and\nsymmetric matrices (mtype =-2, mtype =-4, or mtype =6). For these matrices the\nsolver uses a complete supernode pivoting approach. When the factorization algorithm\nreaches a point where it cannot factor the supernodes with this pivoting strategy, it uses a\npivoting perturbation strategy similar to [Li99], [Schenk04].\nSmall pivots are perturbed with eps = 10-iparm[9].\nThe magnitude of the potential pivot is tested against a constant threshold of\nalpha = eps*||A2||inf,\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1889\n\n\nComponent\nDescription\nwhere eps = 10(-iparm[9]), A2 = P*PMPS*Dr*A*Dc*P, and ||A2||inf is the infinity\nnorm of the scaled and permuted matrix A. Any tiny pivots encountered during elimination\nare set to the sign (lII)*eps*||A2||inf, which trades off some numerical stability for\nthe ability to keep pivots from getting too small. Small pivots are therefore perturbed with\neps = 10(-iparm[9]).\n13*\nThe default value for nonsymmetric matrices(mtype =11, mtype=13), eps =\n10-13.\n8*\nThe default value for symmetric indefinite matrices (mtype =-2, mtype=-4,\nmtype=6), eps = 10-8.\niparm[10]\ninput\nScaling vectors.\nParallel Direct Sparse Solver for Clusters Interface uses a maximum weight matching\nalgorithm to permute large elements on the diagonal and to scale.\nUse iparm[10] = 1 (scaling) and iparm[12] = 1 (matching) for highly indefinite\nsymmetric matrices, for example, from interior point optimizations or saddle point\nproblems. Note that in the analysis phase (phase=11) you must provide the numerical\nvalues of the matrix A in array a in case of scaling and symmetric weighted matching.\n0*\nDisable scaling. Default for symmetric indefinite matrices.\n1*\nEnable scaling. Default for nonsymmetric matrices.\nScale the matrix so that the diagonal elements are equal to 1 and the absolute\nvalues of the off-diagonal entries are less or equal to 1. This scaling method is\napplied to nonsymmetric matrices (mtype = 11, mtype = 13). The scaling can\nalso be used for symmetric indefinite matrices (mtype = -2, mtype = -4, mtype\n= 6) when the symmetric weighted matchings are applied (iparm[12] = 1).\nNote that in the analysis phase (phase=11) you must provide the numerical\nvalues of the matrix A in case of scaling.\niparm[11]\nSolve with transposed or conjugate transposed matrix A.\nNOTE\nFor real matrices, the terms transposed and conjugate transposed are equivalent.\n0*\nSolve a linear system AX = B.\n1\nSolve a conjugate transposed system AHX = B based on the factorization of the\nmatrix A.\n2\nSolve a transposed system ATX = B based on the factorization of the matrix A.\niparm[12]\ninput\nImproved accuracy using (non-) symmetric weighted matching.\nParallel Direct Sparse Solver for Clusters Interface can use a maximum weighted matching\nalgorithm to permute large elements close the diagonal. This strategy adds an additional\nlevel of reliability to the factorization methods and complements the alternative of using\nmore complete pivoting techniques during the numerical factorization.\n  \n0*\nDisable matching. Default for symmetric indefinite matrices.\n1*\nEnable matching. Default for nonsymmetric matrices.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1890\n\n\nComponent\nDescription\nMaximum weighted matching algorithm to permute large elements close to the\ndiagonal.\nIt is recommended to use iparm[10] = 1 (scaling) and iparm[12]= 1\n(matching) for highly indefinite symmetric matrices, for example from interior\npoint optimizations or saddle point problems.\nNote that in the analysis phase (phase=11) you must provide the numerical\nvalues of the matrix A in case of symmetric weighted matching.\niparm[13]\noutput\nNumber of perturbed pivots.\nAfter factorization, contains the number of perturbed pivots for the matrix types: 1, 3, 11,\n13, -2, -4 and 6.\niparm[14]\noutput\nPeak memory on symbolic factorization.\nThe total peak memory in kilobytes that the solver needs during the analysis and symbolic\nfactorization phase.\nThis value is only computed in phase 1.\niparm[15]\noutput\nPermanent memory on symbolic factorization.\nPermanent memory from the analysis and symbolic factorization phase in kilobytes that\nthe solver needs in the factorization and solve phases.\nThis value is only computed in phase 1.\niparm[16]\noutput\nSize of factors/Peak memory on numerical factorization and solution.\nThis parameter provides the size in kilobytes of the total memory consumed by in-core\nIntel® oneAPI Math Kernel Library (oneMKL) PARDISO for internal floating point arrays.\nThis parameter is computed in phase 1. Seeiparm[62] for the OOC mode.\nThe total peak memory consumed by Intel® oneAPI Math Kernel Library (oneMKL)\nPARDISO ismax(iparm[14], iparm[15]+iparm[16])\niparm[17]\ninput/output\nReport the number of non-zero elements in the factors.\n<0\nEnable reporting if iparm[17] < 0 on entry. The default value is -1.\n>=0\nDisable reporting.\niparm[18] -\niparm[19]\nReserved. Set to zero.\niparm[20]\ninput\nPivoting for symmetric indefinite matrices.\n0\nApply 1x1 diagonal pivoting during the factorization process.\n1*\nApply 1x1 and 2x2 Bunch-Kaufman pivoting during the factorization process.\nBunch-Kaufman pivoting is available for matrices of mtype=-2, mtype=-4, or\nmtype=6.\niparm[21]\noutput\nInertia: number of positive eigenvalues.\nIntel® oneAPI Math Kernel Library (oneMKL) PARDISO reports the number of positive\neigenvalues for symmetric indefinite matrices.\niparm[22]\noutput\nInertia: number of negative eigenvalues.\nIntel® oneAPI Math Kernel Library (oneMKL) PARDISO reports the number of negative\neigenvalues for symmetric indefinite matrices.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1891\n\n\nComponent\nDescription\niparm[23] -\niparm[25]\nReserved. Set to zero.\niparm[26]\ninput\nMatrix checker.\n0*\nDo not check the sparse matrix representation for errors.\n1\nCheck integer arrays ia and ja. In particular, check whether the column indices\nare sorted in increasing order within each row.\niparm[27]\ninput\nSingle or double precision Parallel Direct Sparse Solver for Clusters Interface.\nSee iparm[7] for information on controlling the precision of the refinement steps.\n0*\nInput arrays (a, x and b) and all internal arrays must be presented in double\nprecision.\n1\nInput arrays (a, x and b) must be presented in single precision.\nIn this case all internal computations are performed in single precision.\niparm[28]\nReserved. Set to zero.\niparm[29]\noutput\nNumber of zero or negative pivots.\nIf Intel® oneAPI Math Kernel Library (oneMKL) PARDISO detects zero or negative pivot\nformtype=2 or mtype=4 matrix types, the factorization is stopped. Intel® oneAPI Math\nKernel Library (oneMKL) PARDISO returns immediately with anerror = -4, and\niparm[29] reports the number of the equation where the zero or negative pivot is\ndetected.\nNote: The returned value can be different for the parallel and sequential version in case of\nseveral zero/negative pivots.\niparm[30]\ninput\nPartial solve and computing selected components of the solution vectors.\nThis parameter controls the solve step of Intel® oneAPI Math Kernel Library (oneMKL)\nPARDISO. It can be used if only a few components of the solution vectors are needed or if\nyou want to reduce the computation cost at the solve step by utilizing the sparsity of the\nright-hand sides. To use this option the input permutation vector defineperm so that when\nperm(i) = 1 it means that either the i-th component in the right-hand sides is nonzero,\nor the i-th component in the solution vectors is computed, or both, depending on the\nvalue of iparm[30].\nThe permutation vector permmust be present in all phases of Intel® oneAPI Math Kernel\nLibrary (oneMKL) PARDISO software. At the reordering step, the software overwrites the\ninput vectorperm by a permutation vector used by the software at the factorization and\nsolver step. If m is the number of components such that perm(i) = 1, then the last m\ncomponents of the output vector perm are a set of the indices i satisfying the condition\nperm(i) = 1 on input.\nNOTE\nTurning on this option often increases the time used by Intel® oneAPI Math Kernel Library\n(oneMKL) PARDISO for factorization and reordering steps, but it can reduce the time required\nfor the solver step.\nImportant\nSet the parameters iparm[7] (iterative refinement steps), iparm[3] (preconditioned CGS), \niparm[4] (user permutation), and iparm[35] (Schur complement) to 0 as well.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1892\n\n\nComponent\nDescription\n0*\nDisables this option.\n1\nit is assumed that the right-hand sides have only a few non-zero components*\nand the input permutation vector perm is defined so that perm(i) = 1 means\nthat the (i)-th component in the right-hand sides is nonzero. In this case Intel®\noneAPI Math Kernel Library (oneMKL) PARDISO only uses the non-zero\ncomponents of the right-hand side vectors and computes only corresponding\ncomponents in the solution vectors. That means thei-th component in the\nsolution vectors is only computed if perm(i) = 1.\n2\nIt is assumed that the right-hand sides have only a few non-zero components*\nand the input permutation vector perm is defined so that perm(i) = 1 means\nthat the i-th component in the right-hand sides is nonzero.\nUnlike for iparm[30]=1, all components of the solution vector are computed for\nthis setting and all components of the right-hand sides are used. Because all\ncomponents are used, for iparm[30]=2 you must set the i-th component of the\nright-hand sides to zero explicitly if perm(i) is not equal to 1.\n3\nSelected components of the solution vectors are computed. The perm array is not\nrelated to the right-hand sides and it only indicates which components of the\nsolution vectors should be computed. In this case perm(i) = 1 means that the i-\nth component in the solution vectors is computed.\niparm[31] -\niparm[33]\nReserved. Set to zero.\niparm[34]\ninput\nOne- or zero-based indexing of columns and rows.\n0*\nOne-based indexing: columns and rows indexing in arrays ia, ja, and perm\nstarts from 1 (Fortran-style indexing).\n1\nZero-based indexing: columns and rows indexing in arrays ia, ja, and perm\nstarts from 0 (C-style indexing).\niparm[35]\ninput\nSchur complement matrix computation control. To calculate this matrix, you must set the\ninput permutation vector perm to a set of indexes such that when perm(i) = 1, the i-th\nelement of the initial matrix is an element of the Schur matrix.\nCaution\nYou can only set one of iparm[4], iparm[30], and iparm[35], so be sure that the\niparm[4] (user permutation) and the iparm[30] (partial solution) parameters are 0 if you\nset iparm[35].\n0*\nDo not compute Schur complement.\n1\nCompute Schur complement matrix as part of Intel® oneAPI Math Kernel Library\n(oneMKL) PARDISO factorization step and return it in the solution vector.\nNOTE\nThis option only computes the Schur complement matrix, and does not calculate\nfactorization arrays.\n2\nCompute Schur complement matrix as part of Intel® oneAPI Math Kernel Library\n(oneMKL) PARDISO factorization step and return it in the solution vector. Since\nthis option calculates factorization arrays you can use it to launch partial or full\nsolution of the entire problem after the factorization step.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1893\n\n\nComponent\nDescription\niparm[36]\ninput\nFormat for matrix storage.\n0*\nUse CSR format (see Three Array Variation of BSR Format) for matrix storage.\n1\nUse CSR format (see Three Array Variation of BSR Format) for matrix storage.\n< 0\nConvert supplied matrix to variable BSR (VBSR) format (see Sparse Data\nStorage) for matrix storage. Intel® oneAPI Math Kernel Library (oneMKL)\nPARDISO analyzes the matrix provided in CSR3 format and converts it to an\ninternal VBSR format. Setiparm[36] = -t, 0 < t≤ 100.\niparm[37] -\niparm[38]\nReserved. Set to zero.\niparm[39]\ninput\nMatrix input format.\nNOTE\nPerformance of the reordering step of the Parallel Direct Sparse Solver for Clusters Interface is\nslightly better for assembled format (CSR, iparm[39] = 0) than for distributed format\n(DCSR, iparm[39] > 0) for the same matrices, so if the matrix is assembled on one node do\nnot distribute it before calling cluster_sparse_solver.\n0*\nProvide the matrix in usual centralized input format: the master MPI process\nstores all data from matrix A, with rank=0.\n1\nProvide the matrix in distributed assembled matrix input format. In this case,\neach MPI process stores only a part (or domain) of the matrix A data. Set the\nbounds of the domain using iparm[40] and iparm[41]. The solution vector is\nplaced on the master process.\n2\nProvide the matrix in distributed assembled matrix input format. In this case,\neach MPI process stores only a part (or domain) of the matrix A data. Set the\nbounds of the domain using iparm[40] and iparm[41]. The solution vector, A,\nand RHS elements are distributed between processes in same manner.\n3\nProvide the matrix in distributed assembled matrix input format. In this case,\neach MPI process stores only a part (or domain) of the matrix A data. Set the\nbounds of the domain using iparm[40] and iparm[41]. The A and RHS\nelements are distributed between processes in same manner and the solution\nvector is the same on each process\niparm[40]\ninput\nBeginning of input domain.\nThe number of the matrix A row, RHS element, and, for iparm[39]=2, solution vector\nthat begins the input domain belonging to this MPI process.\nOnly applicable to the distributed assembled matrix input format (iparm[39]> 0).\nSee Sparse Matrix Storage Formats for more details.\niparm[41]\ninput\nEnd of input domain.\nThe number of the matrix A row, RHS element, and, for iparm[39]=2, solution vector\nthat ends the input domain belonging to this MPI process.\nOnly applicable to the distributed assembled matrix input format (iparm[39]> 0).\nSee Sparse Matrix Storage Formats for more details.\niparm[42] -\niparm[58]\nReserved. Set to zero.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1894\n\n\nComponent\nDescription\ninput\niparm[59]\ninput\ncluster_sparse_solver mode.\niparm[59] switches between in-core (IC) and out-of-core (OOC) of cluster_sparse_solver.\nOOC can solve very large problems by holding the matrix factors in files on the disk,\nwhich requires a reduced amount of main memory compared to IC.\nUnless you are operating in sequential mode, you can switch between IC and OOC modes\nafter the reordering phase. However, you can get better cluster_sparse_solver\nperformance by setting iparm[59] before the reordering phase.\nThe amount of memory used in OOC mode depends on the number of OpenMP threads.\nNOTE\nWhen iparm[59] > 0, use the MKL_PARDISO_OOC_FILE_NAME environment variable to\nstore factors.\nWarning\nDo not increase the number of OpenMP threads used for cluster_sparse_solver between the\nfirst call and the factorization or solution phase. Because the minimum amount of memory\nrequired for out-of-core execution depends on the number of OpenMP threads, increasing it\nafter the initial call can cause incorrect results.\n0*\nIC mode.\n1\nIC mode is used if the total amount of RAM (in megabytes) needed for storing\nthe matrix factors is less than sum of two values of the environment variables:\nMKL_PARDISO_OOC_MAX_CORE_SIZE (default value 2000 MB) and\nMKL_PARDISO_OOC_MAX_SWAP_SIZE (default value 0 MB); otherwise OOC mode\nis used. In this case amount of RAM used by OOC mode cannot exceed the value\nof MKL_PARDISO_OOC_MAX_CORE_SIZE.\nIf the total peak memory needed for storing the local arrays is more than\nMKL_PARDISO_OOC_MAX_CORE_SIZE, increase\nMKL_PARDISO_OOC_MAX_CORE_SIZE if possible.\nNOTE\nConditional numerical reproducibility (CNR) is not supported for this mode.\n2\nOOC mode.\nThe OOC mode can solve very large problems by holding the matrix factors in\nfiles on the disk. Hence the amount of RAM required by OOC mode is significantly\nreduced compared to IC mode.\nIf the total peak memory needed for storing the local arrays is more than\nMKL_PARDISO_OOC_MAX_CORE_SIZE, increase\nMKL_PARDISO_OOC_MAX_CORE_SIZE if possible.\nTo obtain better cluster_sparse_solver performance, during the numerical\nfactorization phase you can provide the maximum number of right-hand sides,\nwhich can be used further during the solving phase.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1895\n\n\nComponent\nDescription\nNOTE To use OOC mode, you must disable iparm[10] (scaling) and iparm[12] =\n1 (matching).\niparm[60] -\niparm[61]\ninput\nReserved. Set to zero.\niparm[62]\noutput\nSize of the minimum OOC memory for numerical factorization and solution.\nThis parameter provides the size in kilobytes of the minimum memory required by OOC\nIntel® oneAPI Math Kernel Library (oneMKL) PARDISO for internal floating point arrays.\nThis parameter is computed in phase 1.\nTotal peak memory consumption of OOC Intel® oneAPI Math Kernel Library (oneMKL)\nPARDISO can be estimated asmax(iparm[14], iparm[15] + iparm[62]).\niparm[63]\ninput\nReser\nved.\nSet to\nzero.\nNOTE\nGenerally in sparse matrices, components which are equal to zero can be considered non-zero if\nnecessary. For example, in order to make a matrix structurally symmetric, elements which are zero\ncan be considered non-zero. See Sparse Matrix Storage Formats for an example.\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nDirect Sparse Solver (DSS) Interface Routines\nIntel® oneAPI Math Kernel Library (oneMKL) supports the DSS interface, an alternative to the Intel® oneAPI\nMath Kernel Library (oneMKL) PARDISO interface for the direct sparse solver. The DSS interface implements\na group of user-callable routines that are used in the step-by-step solving process and utilizes the general\nscheme described inAppendix A Linear Solvers Basics for solving sparse systems of linear equations. This\ninterface also includes one routine for gathering statistics related to the solving process.\nThe DSS interface also supports the out-of-core (OOC) mode.\nTable \"DSS Interface Routines\" lists the names of the routines and describes their general use.\nDSS Interface Routines\nRoutine\nDescription\ndss_create\nInitializes the solver and creates the basic data structures\nnecessary for the solver. This routine must be called\nbefore any other DSS routine.\ndss_define_structure\nInforms the solver of the locations of the non-zero\nelements of the matrix.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1896\n\n\nRoutine\nDescription\ndss_reorder\nBased on the non-zero structure of the matrix, computes\na permutation vector to reduce fill-in during the factoring\nprocess.\ndss_factor_real, dss_factor_complex\nComputes the LU, LDLT or LLT factorization of a real or\ncomplex matrix.\ndss_solve_real, dss_solve_complex\nComputes the solution vector for a system of equations\nbased on the factorization computed in the previous\nphase.\ndss_delete\nDeletes all data structures created during the solving\nprocess.\ndss_statistics\nReturns statistics about various phases of the solving\nprocess.\nTo find a single solution vector for a single system of equations with a single right-hand side, invoke the\nIntel® oneAPI Math Kernel Library (oneMKL) DSS interface routines in this order:\n1.\ndss_create\n2.\ndss_define_structure\n3.\ndss_reorder\n4.\ndss_factor_real, dss_factor_complex\n5.\ndss_solve_real, dss_solve_complex\n6.\ndss_delete\nHowever, in certain applications it is necessary to produce solution vectors for multiple right-hand sides for a\ngiven factorization and/or factor several matrices with the same non-zero structure. Consequently, it is\nsometimes necessary to invoke the Intel® oneAPI Math Kernel Library (oneMKL) sparse routines in an order\nother than that listed, which is possible using the DSS interface. The solving process is conceptually divided\ninto six phases.Figure \"Typical order for invoking DSS interface routines\" indicates the typical order in which\nthe DSS interface routines can be invoked.\n__border__top\nTypical order for invoking DSS interface routines\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1897\n\n\nSee the code examples that use the DSS interface routines to solve systems of linear equations in the Intel®\noneAPI Math Kernel Library (oneMKL) installation directory ( dss_*.c).\n•\nexamples/solverc/source\nDSS Interface Description\nEach DSS routine reads from or writes to a data object called a handle. Refer to Memory Allocation and\nHandles to determine the correct method for declaring a handle argument for each language. For simplicity,\nthe descriptions in DSS routines refer to the data type as MKL_DSS_HANDLE.\nRoutine Options\nThe DSS routines have an integer argument (referred below to as opt) for passing various options to the\nroutines. The permissible values for opt should be specified using only the symbol constants defined in the\nlanguage-specific header files (see Implementation Details). The routines accept options for setting the\nmessage and termination levels as described in Table \"Symbolic Names for the Message and Termination\nLevels Options\". Additionally, each routine accepts the option MKL_DSS_DEFAULTS that sets the default\nvalues (as documented) for opt to the routine.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1898\n\n\nSymbolic Names for the Message and Termination Levels Options\nMessage Level\nTermination Level\nMKL_DSS_MSG_LVL_SUCCESS\nMKL_DSS_TERM_LVL_SUCCESS\nMKL_DSS_MSG_LVL_INFO\nMKL_DSS_TERM_LVL_INFO\nMKL_DSS_MSG_LVL_WARNING\nMKL_DSS_TERM_LVL_WARNING\nMKL_DSS_MSG_LVL_ERROR\nMKL_DSS_TERM_LVL_ERROR\nMKL_DSS_MSG_LVL_FATAL\nMKL_DSS_TERM_LVL_FATAL\nThe settings for message and termination levels can be set on any call to a DSS routine. However, once set\nto a particular level, they remain at that level until they are changed in another call to a DSS routine.\nYou can specify both message and termination level for a DSS routine by adding the options together. For\nexample, to set the message level to debug and the termination level to error for all the DSS routines, use\nthe following call:\ndss_create( handle, MKL_DSS_MSG_LVL_INFO + MKL_DSS_TERM_LVL_ERROR)\nUser Data Arrays\nMany of the DSS routines take arrays of user data as input. For example, pointers to user arrays are passed\nto the routine dss_define_structure to describe the location of the non-zero entries in the matrix.\nCaution\nDo not modify the contents of these arrays after they are passed to one of the solver routines.\nDSS Implementation Details\nTo promote portability across platforms and ease of use across different languages, use the mkl_dss.h\nheader file.\nThe header file defines symbolic constants for returned error values, function options, certain defined data\ntypes, and function prototypes.\nNOTE\nConstants for options, returned error values, and message severities must be referred only by the\nsymbolic names that are defined in these header files. Use of the Intel® oneAPI Math Kernel Library\n(oneMKL) DSS software without including one of the above header files is not supported.\nMemory Allocation and Handles\nYou do not need to allocate any temporary working storage in order to use the Intel® oneAPI Math Kernel\nLibrary (oneMKL) DSS routines, because the solver itself allocates any required storage. To enable multiple\nusers to access the solver simultaneously, the solver keeps track of the storage allocated for a particular\napplication by using ahandle data object.\nEach of the Intel® oneAPI Math Kernel Library (oneMKL) DSS routines creates, uses, or deletes a handle.\nConsequently, any program calling an Intel® oneAPI Math Kernel Library (oneMKL) DSS routine must be able\nto allocate storage for a handle. The exact syntax for allocating storage for a handle varies from language to\nlanguage. To standardize the handle declarations, the language-specific header files declare constants and\ndefined data types that must be used when declaring a handle object in your code.\n#include \"mkl_dss.h\" \n_MKL_DSS_HANDLE_t handle;\nIn addition to the definition for the correct declaration of a handle, the include file also defines the following:\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1899\n\n\n•\nfunction prototypes for languages that support prototypes\n•\nsymbolic constants that are used for the returned error values\n•\nuser options for the solver routines\n•\nconstants indicating message severity.\nDSS Routines\ndss_create\nInitializes the solver.\nSyntax\nMKL_INT dss_create(_MKL_DSS_HANDLE_t *handle, MKL_INT const *opt)\nInclude Files\n•\nmkl.h\nDescription\nThe dss_create routine initializes the solver. After the call to dss_create, all subsequent invocations of the\nIntel® oneAPI Math Kernel Library (oneMKL) DSS routines must use the value of the handle returned\nbydss_create.\nWARNING\nDo not write the value of handle directly.\nThe default value of the parameter opt is\nMKL_DSS_MSG_LVL_WARNING + MKL_DSS_TERM_LVL_ERROR.\nBy default, the DSS routines use double precision for solving systems of linear equations. The precision used\nby the DSS routines can be set to single mode by adding the following value to the opt parameter:\nMKL_DSS_SINGLE_PRECISION.\nInput data and internal arrays are required to have single precision.\nBy default, the DSS routines use Fortran style (one-based) indexing for input arrays of integer types (the\nfirst value is referenced as array element 1). To set indexing to C style (the first value is referenced as array\nelement 0), add the following value to the opt parameter:\nMKL_DSS_ZERO_BASED_INDEXING.\nThe opt parameter can also control number of refinement steps used on the solution stage by specifying the\ntwo following values:\nMKL_DSS_REFINEMENT_OFF - maximum number of refinement steps is set to zero;\nMKL_DSS_REFINEMENT_ON (default value) - maximum number of refinement steps is set to 2.\nBy default, DSS uses in-core computations. To launch the out-of-core version of DSS (OOC DSS) you can add\nto this parameter one of two possible values: MKL_DSS_OOC_STRONG and MKL_DSS_OOC_VARIABLE.\nMKL_DSS_OOC_STRONG - OOC DSS is used.\nMKL_DSS_OOC_VARIABLE - if the memory needed for the matrix factors is less than the value of the\nenvironment variable MKL_PARDISO_OOC_MAX_CORE_SIZE, then the OOC DSS uses the in-core kernels of\nIntel® oneAPI Math Kernel Library (oneMKL) PARDISO, otherwise it uses the OOC computations.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1900\n\n\nThe variable MKL_PARDISO_OOC_MAX_CORE_SIZE defines the maximum size of RAM allowed for storing work\narrays associated with the matrix factors. It is ignored if MKL_DSS_OOC_STRONG is set. The default value of\nMKL_PARDISO_OOC_MAX_CORE_SIZE is 2000 MB. This value and default path and file name for storing\ntemporary data can be changed using the configuration file pardiso_ooc.cfg or command line (See more\ndetails in the description of the pardiso routine).\nWARNING\nOther than message and termination level options, do not change the OOC DSS settings\nafter they are specified in the routine dss_create.\nInput Parameters\nopt\nParameter to pass the DSS options. The default value is\nMKL_DSS_MSG_LVL_WARNING + MKL_DSS_TERM_LVL_ERROR.\nOutput Parameters\nhandle\nPointer to the data structure storing internal DSS results\n(MKL_DSS_HANDLE).\nReturn Values\nMKL_DSS_SUCCESS\nMKL_DSS_INVALID_OPTION\nMKL_DSS_OUT_OF_MEMORY\nMKL_DSS_MSG_LVL_ERR\nMKL_DSS_TERM_LVL_ERR\ndss_define_structure\nCommunicates locations of non-zero elements in the\nmatrix to the solver.\nSyntax\nMKL_INT dss_define_structure(_MKL_DSS_HANDLE_t *handle, MKL_INT const *opt, MKL_INT\nconst *rowIndex, MKL_INT const *nRows, MKL_INT const *nCols, MKL_INT const *columns,\nMKL_INT const *nNonZeros);\nInclude Files\n•\nmkl.h\nDescription\nThe routine dss_define_structure communicates the locations of the nNonZeros number of non-zero\nelements in a matrix of nRows * nCols size to the solver.\nNOTE\nThe Intel® oneAPI Math Kernel Library (oneMKL) DSS software operates only on square\nmatrices, sonRows must be equal to nCols.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1901\n\n\nTo communicate the locations of non-zero elements in the matrix, do the following:\n1.\nDefine the general non-zero structure of the matrix by specifying the value for the options argument\nopt. You can set the following values for real matrices:\n•\nMKL_DSS_SYMMETRIC_STRUCTURE\n•\nMKL_DSS_SYMMETRIC\n•\nMKL_DSS_NON_SYMMETRIC\nand for complex matrices:\n•\nMKL_DSS_SYMMETRIC_STRUCTURE_COMPLEX\n•\nMKL_DSS_SYMMETRIC_COMPLEX\n•\nMKL_DSS_NON_SYMMETRIC_COMPLEX\nThe information about the matrix type must be defined in dss_define_structure.\n2.\nProvide the actual locations of the non-zeros by means of the arrays rowIndex and columns (see \nSparse Matrix Storage Format).\nNOTE No diagonal element can be omitted from the values array. If there is a zero value on\nthe diagonal, for example, that element nonetheless must be explicitly represented.\nInput Parameters\nopt\nParameter to pass the DSS options. The default value for the matrix\nstructure is MKL_DSS_SYMMETRIC.\nrowIndex\nArray of size nRows+1. Defines the location of non-zero entries in the\nmatrix.\nnRows\nNumber of rows in the matrix.\nnCols\nNumber of columns in the matrix; must be equal to nRows.\ncolumns\nArray of size nNonZeros. Defines the column location of non-zero\nentries in the matrix.\nnNonZeros\nNumber of non-zero elements in the matrix.\nOutput Parameters\nhandle\nPointer to the data structure storing internal DSS results\n(MKL_DSS_HANDLE).\nReturn Values\nMKL_DSS_SUCCESS\nMKL_DSS_STATE_ERR\nMKL_DSS_INVALID_OPTION\nMKL_DSS_STRUCTURE_ERR\nMKL_DSS_ROW_ERR\nMKL_DSS_COL_ERR\nMKL_DSS_NOT_SQUARE\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1902\n\n\nMKL_DSS_TOO_FEW_VALUES\nMKL_DSS_TOO_MANY_VALUES\nMKL_DSS_OUT_OF_MEMORY\nMKL_DSS_MSG_LVL_ERR\nMKL_DSS_TERM_LVL_ERR\ndss_reorder\nComputes or sets a permutation vector that minimizes\nthe fill-in during the factorization phase.\nSyntax\nMKL_INT dss_reorder(_MKL_DSS_HANDLE_t *handle, MKL_INT const *opt, MKL_INT const *perm)\nInclude Files\n•\nmkl.h\nDescription\nIf opt contains the option MKL_DSS_AUTO_ORDER, then the routine dss_reorder computes a permutation\nvector that minimizes the fill-in during the factorization phase. For this option, the routine ignores the\ncontents of the perm array.\nIf opt contains the option MKL_DSS_METIS_OPENMP_ORDER, then the routine dss_reorder computes\npermutation vector using the parallel nested dissections algorithm to minimize the fill-in during the\nfactorization phase. This option can be used to decrease the time of dss_reorder call on multi-core\ncomputers. For this option, the routine ignores the contents of the perm array.\nIf opt contains the option MKL_DSS_MY_ORDER, then you must supply a permutation vector in the array\nperm. In this case, the array perm is of length nRows, where nRows is the number of rows in the matrix as\ndefined by the previous call to dss_define_structure.\nIf opt contains the option MKL_DSS_GET_ORDER, then the permutation vector computed during the\ndss_reorder call is copied to the array perm. In this case you must allocate the array perm beforehand.\nThe permutation vector is computed in the same way as if the option MKL_DSS_AUTO_ORDER is set.\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nInput Parameters\nopt\nParameter to pass the DSS options. The default value for the\npermutation type is MKL_DSS_AUTO_ORDER.\nperm\nArray of length nRows. Contains a user-defined permutation vector\n(accessed only if opt contains MKL_DSS_MY_ORDER or\nMKL_DSS_GET_ORDER).\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1903\n\n\nOutput Parameters\nhandle\nPointer to the data structure storing internal DSS results\n(MKL_DSS_HANDLE).\nReturn Values\nMKL_DSS_SUCCESS\nMKL_DSS_STATE_ERR\nMKL_DSS_INVALID_OPTION\nMKL_DSS_REORDER_ERR\nMKL_DSS_REORDER1_ERR\nMKL_DSS_I32BIT_ERR\nMKL_DSS_FAILURE\nMKL_DSS_OUT_OF_MEMORY\nMKL_DSS_MSG_LVL_ERR\nMKL_DSS_TERM_LVL_ERR\ndss_factor_real, dss_factor_complex\nCompute factorization of the matrix with previously\nspecified location of non-zero elements.\nSyntax\nMKL_INT dss_factor_real(_MKL_DSS_HANDLE_t *handle, MKL_INT const *opt, void const\n*rValues)\nMKL_INT dss_factor_complex(_MKL_DSS_HANDLE_t *handle, MKL_INT const *opt, void const\n*cValues)\nInclude Files\n•\nmkl.h\nDescription\nThese routines compute factorization of the matrix whose non-zero locations were previously specified by a\ncall to dss_define_structure and whose non-zero values are given in the array rValues, cValues or\nValues. Data type These arrays must be of length nNonZeros as defined in a previous call to\ndss_define_structure.\nNOTE\nThe data type (single or double precision) of rValues, cValues, Values must be in\ncorrespondence with precision specified by the parameter opt in the routine dss_create.\nThe opt argument can contain one of the following options:\n•\nMKL_DSS_POSITIVE_DEFINITE\n•\nMKL_DSS_INDEFINITE\n•\nMKL_DSS_HERMITIAN_POSITIVE_DEFINITE\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1904\n\n\n•\nMKL_DSS_HERMITIAN_INDEFINITE\ndepending on your matrix's type.\nNOTE\nThis routine supports the Progress Routine feature. See Progress Function for details.\nInput Parameters\nhandle\nPointer to the data structure storing internal DSS results\n(MKL_DSS_HANDLE).\nopt\nParameter to pass the DSS options. The default value is\nMKL_DSS_POSITIVE_DEFINITE.\nrValues\nArray of elements of the matrix A. Real data, single or double\nprecision as it is specified by the parameter opt in the routine\ndss_create.\ncValues\nArray of elements of the matrix A. Complex data, single or double\nprecision as it is specified by the parameter opt in the routine\ndss_create.\nReturn Values\nMKL_DSS_SUCCESS\nMKL_DSS_STATE_ERR\nMKL_DSS_INVALID_OPTION\nMKL_DSS_OPTION_CONFLICT\nMKL_DSS_VALUES_ERR\nMKL_DSS_OUT_OF_MEMORY\nMKL_DSS_ZERO_PIVOT\nMKL_DSS_FAILURE\nMKL_DSS_MSG_LVL_ERR\nMKL_DSS_TERM_LVL_ERR\nMKL_DSS_OOC_MEM_ERR\nMKL_DSS_OOC_OC_ERR\nMKL_DSS_OOC_RW_ERR\ndss_solve_real, dss_solve_complex\nCompute the corresponding solution vector and place\nit in the output array.\nSyntax\nMKL_INT dss_solve_real(_MKL_DSS_HANDLE_t *handle, MKL_INT const *opt, void const\n*rRhsValues, MKL_INT const *nRhs, void *rSolValues)\nMKL_INT dss_solve_complex(_MKL_DSS_HANDLE_t *handle, MKL_INT const *opt, void const\n*cRhsValues, MKL_INT const *nRhs, void *cSolValues)\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1905\n\n\nInclude Files\n•\nmkl.h\nDescription\nFor each right-hand side column vector defined in the arrays rRhsValues, cRhsValues, or RhsValues,\nthese routines compute the corresponding solution vector and place it in the arrays rSolValues,\ncSolValues, or SolValues respectively.\nNOTE\nThe data type (single or double precision) of all arrays must be in correspondence with\nprecision specified by the parameter opt in the routine dss_create.\nThe lengths of the right-hand side and solution vectors, nRows and nCols respectively, must be defined in a\nprevious call to dss_define_structure.\nBy default, both routines perform the full solution step (it corresponds to phase = 33in Intel® oneAPI Math\nKernel Library (oneMKL) PARDISO). The parameteropt enables you to calculate the final solution step-by-\nstep, calling forward and backward substitutions.\nIf it is set to MKL_DSS_FORWARD_SOLVE, the forward substitution (corresponding to phase = 331in Intel®\noneAPI Math Kernel Library (oneMKL) PARDISO) is performed;\nif it is set to MKL_DSS_DIAGONAL_SOLVE, the diagonal substitution (corresponding to phase = 332in Intel®\noneAPI Math Kernel Library (oneMKL) PARDISO) is performed, if possible;\nif it is set to MKL_DSS_BACKWARD_SOLVE, the backward substitution (corresponding to phase = 333in Intel®\noneAPI Math Kernel Library (oneMKL) PARDISO) is performed.\nFor more details about using these substitutions for different types of matrices, see Separate Forward and\nBackward Substitutionin the Intel® oneAPI Math Kernel Library (oneMKL) PARDISO solver description.\nThis parameter also can control the number of refinement steps that is used on the solution stage: if it is set\nto MKL_DSS_REFINEMENT_OFF, the maximum number of refinement steps equal to zero, and if it is set to\nMKL_DSS_REFINEMENT_ON (default value), the maximum number of refinement steps is equal to 2.\nMKL_DSS_CONJUGATE_SOLVE option added to the parameter opt enables solving a conjugate transposed\nsystem AHX = B based on the factorization of the matrix A. This option is equivalent to the parameter\niparm[11]= 1in Intel® oneAPI Math Kernel Library (oneMKL) PARDISO.\nMKL_DSS_TRANSPOSE_SOLVE option added to the parameter opt enables solving a transposed system ATX =\nB based on the factorization of the matrix A. This option is equivalent to the parameter iparm[11]= 2in\nIntel® oneAPI Math Kernel Library (oneMKL) PARDISO.\nInput Parameters\nhandle\nPointer to the data structure storing internal DSS results\n(MKL_DSS_HANDLE).\nopt\nParameter to pass the DSS options.\nnRhs\nNumber of the right-hand sides in the system of linear equations.\nrRhsValues\nArray of size nRows * nRhs. Contains real right-hand side vectors.\nReal data, single or double precision as it is specified by the parameter\nopt in the routine dss_create.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1906\n\n\ncRhsValues\nArray of size nRows * nRhs. Contains complex right-hand side\nvectors. Complex data, single or double precision as it is specified by\nthe parameter opt in the routine dss_create.\nRhsValues\nArray of size nRows * nRhs. Contains right-hand side vectors. Real or\ncomplex data, single or double precision as it is specified by the\nparameter opt in the routine dss_create.\nOutput Parameters\nrSolValues\nArray of size nCols * nRhs. Contains real solution vectors. Real data,\nsingle or double precision as it is specified by the parameter opt in\nthe routine dss_create.\ncSolValues\nArray of size nCols * nRhs. Contains complex solution vectors.\nComplex data, single or double precision as it is specified by the\nparameter opt in the routine dss_create.\nReturn Values\nMKL_DSS_SUCCESS\nMKL_DSS_STATE_ERR\nMKL_DSS_INVALID_OPTION\nMKL_DSS_OUT_OF_MEMORY\nMKL_DSS_DIAG_ERR\nMKL_DSS_FAILURE\nMKL_DSS_MSG_LVL_ERR\nMKL_DSS_TERM_LVL_ERR\nMKL_DSS_OOC_MEM_ERR\nMKL_DSS_OOC_OC_ERR\nMKL_DSS_OOC_RW_ERR\ndss_delete\nDeletes all of the data structures created during the\nsolutions process.\nSyntax\nMKL_INT dss_delete(_MKL_DSS_HANDLE_t const *handle, MKL_INT const *opt)\nInclude Files\n•\nmkl.h\nDescription\nThe routine dss_delete deletes all data structures created during the solving process.\nInput Parameters\nopt\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1907\n\n\nParameter to pass the DSS options. The default value is\nMKL_DSS_MSG_LVL_WARNING + MKL_DSS_TERM_LVL_ERROR.\nOutput Parameters\nhandle\nPointer to the data structure storing internal DSS results\n(MKL_DSS_HANDLE).\nReturn Values\nMKL_DSS_SUCCESS\nMKL_DSS_STATE_ERR\nMKL_DSS_INVALID_OPTION\nMKL_DSS_OUT_OF_MEMORY\nMKL_DSS_MSG_LVL_ERR\nMKL_DSS_TERM_LVL_ERR\ndss_statistics\nReturns statistics about various phases of the solving\nprocess.\nSyntax\nMKL_INT dss_statistics(_MKL_DSS_HANDLE_t *handle, MKL_INT const *opt, _CHARACTER_STR_t\nconst *statArr, _DOUBLE_PRECISION_t *retValues)\nInclude Files\n•\nmkl.h\nDescription\nThe dss_statistics routine returns statistics about various phases of the solving process. This routine\ngathers the following statistics:\n–\ntime taken to do reordering,\n–\ntime taken to do factorization,\n–\nduration of problem solving,\n–\ndeterminant of the symmetric indefinite input matrix,\n–\ninertia of the symmetric indefinite input matrix,\n–\nnumber of floating point operations taken during factorization,\n–\ntotal peak memory needed during the analysis and symbolic factorization,\n–\npermanent memory needed from the analysis and symbolic factorization,\n–\nmemory consumption for the factorization and solve phases.\nStatistics are returned in accordance with the input string specified by the parameter statArr. The value of\nthe statistics is returned in double precision in a return array, which you must allocate beforehand.\nFor multiple statistics, multiple string constants separated by commas can be used as input. Return values\nare put into the return array in the same order as specified in the input string.\nStatistics can only be requested at the appropriate stages of the solving process. For example, requesting\nFactorTime before a matrix is factored leads to an error.\nThe following table shows the point at which each individual statistics item can be requested:\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1908\n\n\nStatistics Calling Sequences\nType of Statistics\nWhen to Call\nReorderTime\nAfter dss_reorder is completed successfully.\nFactorTime\nAfter dss_factor_real or dss_factor_complex is completed successfully.\nSolveTime\nAfter dss_solve_real or dss_solve_complex is completed successfully.\nDeterminant\nAfter dss_factor_real or dss_factor_complex is completed successfully.\nInertia\nAfter dss_factor_real is completed successfully and the matrix is real, symmetric, and\nindefinite.\nFlops\nAfter dss_factor_real or dss_factor_complex is completed successfully.\nPeakmem\nAfter dss_reorder is completed successfully.\nFactormem\nAfter dss_reorder is completed successfully.\nSolvemem\nAfter dss_factor_real or dss_factor_complex is completed successfully.\nInput Parameters\nhandle\nPointer to the data structure storing internal DSS results\n(MKL_DSS_HANDLE).\nopt\nParameter to pass the DSS options.\nstatArr\nInput string that defines the type of the returned statistics. The\nparameter can include one or more of the following string constants\n(case of the input string has no effect):\nReorderTime\nAmount of time taken to do the reordering.\nFactorTime\nAmount of time taken to do the factorization.\nSolveTime\nAmount of time taken to solve the problem after\nfactorization.\nDeterminant\nDeterminant of the matrix A.\nFor real matrices: the determinant is returned as\ndet_pow, det_base in two consecutive return\narray locations, where 1.0 ≤ abs(det_base)\n< 10.0 and determinant =\ndet_base*10(det_pow).\nFor complex matrices: the determinant is\nreturned as det_pow, det_re, det_im in three\nconsecutive return array locations, where 1.0\n≤abs(det_re) + abs(det_im) < 10.0 and\ndeterminant = (det_re,\ndet_im)*10(det_pow).\nInertia\nInertia of a real symmetric matrix is defined as a\ntriplet of nonnegative integers (p,n,z), where\np is the number of positive eigenvalues, n is the\nnumber of negative eigenvalues, and z is the\nnumber of zero eigenvalues.\nInertia is returned as three consecutive return\narray locations p, n, z.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1909\n\n\nComputing inertia can lead to incorrect results\nfor matrixes with a cluster of eigenvalues which\nare near 0.\nInertia of a k-by-k real symmetric positive\ndefinite matrix is always (k, 0, 0). Therefore\nInertia is returned only in cases of real\nsymmetric indefinite matrices. For all other\nmatrix types, an error message is returned.\nFlops\nNumber of floating point operations performed\nduring the factorization.\nPeakmem\nTotal peak memory in kilobytes that the solver\nneeds during the analysis and symbolic\nfactorization phase.\nFactormem\nPermanent memory in kilobytes that the solver\nneeds from the analysis and symbolic\nfactorization phase in the factorization and solve\nphases.\nSolvemem\nTotal double precision memory consumption\n(kilobytes) of the solver for the factorization and\nsolve phases.\nOutput Parameters\nretValues\nValue of the statistics returned.\nFinding 'time used to reorder' and 'inertia' of a matrix\nThe example below illustrates the use of the dss_statistics routine.\nTo find the above values, call dss_statistics(handle, opt, statArr, retValue), where staArr is\n\"ReorderTime,Inertia\"\nIn this example, retValue has the following values:\nretValue[0]\nTime to reorder.\nretValue[1]\nPositive Eigenvalues.\nretValue[2]\nNegative Eigenvalues.\nretValue[3]\nZero Eigenvalues.\nReturn Values\nMKL_DSS_SUCCESS\nMKL_DSS_INVALID_OPTION\nMKL_DSS_STATISTICS_INVALID_MATRIX\nMKL_DSS_STATISTICS_INVALID_STATE\nMKL_DSS_STATISTICS_INVALID_STRING\nMKL_DSS_MSG_LVL_ERR\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1910\n\n\nMKL_DSS_TERM_LVL_ERR\nIterative Sparse Solvers based on Reverse Communication Interface (RCI ISS)\nIntel® oneAPI Math Kernel Library (oneMKL) supports iterative sparse solvers (ISS) based on the reverse\ncommunication interface (RCI), referred to here as the RCI ISS interface. The RCI ISS interface implements\na group of user-callable routines that are used in the step-by-step solving process of a symmetric positive\ndefinite system (RCI conjugate gradient solver, or RCI CG), and of a non-symmetric indefinite (non-\ndegenerate) system (RCI flexible generalized minimal residual solver, or RCI FGMRES) of linear algebraic\nequations. This interface uses the general RCI scheme described in [Dong95].\nSee the Appendix A Linear Solvers Basics for discussion of terms and concepts related to the ISS routines.\nThe term RCI indicates that when the solver needs the results of certain operations (for example, matrix-\nvector multiplications), the user performs them and passes the result to the solver. This makes the solver\nmore universal as it is independent of the specific implementation of the operations like the matrix-vector\nmultiplication. To perform such operations, the user can use the built-in sparse matrix-vector multiplications\nand triangular solvers routines described in Sparse BLAS Level 2 and Level 3 Routines.\nNOTE\nThe RCI CG solver is implemented in two versions: for system of equations with a single right-hand\nside, and for systems of equations with multiple right-hand sides.\nThe CG method may fail to compute the solution or compute the wrong solution if the matrix of the\nsystem is not symmetric and not positive definite.\nThe FGMRES method may fail if the matrix is degenerate.\nTable \"RCI CG Interface Routines\" lists the names of the routines, and describes their general use.\nRCI ISS Interface Routines\nRoutine\nDescription\ndcg_init, dcgmrhs_init, dfgmres_init\nInitializes the solver.\ndcg_check, dcgmrhs_check, dfgmres_check\nChecks the consistency and correctness of the user defined data.\ndcg, dcgmrhs, dfgmres\nComputes the approximate solution vector.\ndcg_get, dcgmrhs_get, dfgmres_get\nRetrieves the number of the current iteration.\nThe Intel® oneAPI Math Kernel Library (oneMKL) RCI ISS interface routines are normally invoked in this\norder:\n1.\n<system_type>_init\n2.\n<system_type>_check\n3.\n<system_type>\n4.\n<system_type>_get\nAdvanced users can change that order if they need it. Others should follow the above order of calls.\nThe following diagram indicates the typical order in which the RCI ISS interface routines are invoked.\n__border__top\nTypical Order for Invoking RCI ISS interface Routines\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1911\n\n\nSee the code examples that use the RCI ISS interface routines to solve systems of linear equations in the\nIntel® oneAPI Math Kernel Library (oneMKL) installation directory.\n•\nexamples/solverc/source\nCG Interface Description\nEach routine for the RCI CG solver is implemented in two versions: for a system of equations with a single\nright-hand side (SRHS), and for a system of equations with multiple right-hand sides (MRHS). The names of\nroutines for a system with MRHS contain the suffix mrhs.\nRoutine Options\nAll of the RCI CG routines have common parameters for passing various options to the routines (see CG\nCommon Parameters). The values for these parameters can be changed during computations.\nUser Data Arrays\nMany of the RCI CG routines take arrays of user data as input. For example, user arrays are passed to the\nroutine dcgto compute the solution of a system of linear algebraic equations. The Intel® oneAPI Math Kernel\nLibrary (oneMKL) RCI CG routines do not make copies of the user input arrays to minimize storage\nrequirements and improve overall run-time efficiency.\nCG Common Parameters\nNOTE\nThe default and initial values listed below are assigned to the parameters by calling the dcg_init/\ndcgmrhs_init routine.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1912\n\n\nn\nMKL_INT, this parameter sets the size of the problem in the dcg_init/\ndcgmrhs_init routine. All the other routines use the ipar[0] parameter\ninstead. Note that the coefficient matrix A is a square matrix of size n*n.\nx\ndouble array of size n for SRHS, or matrix of size (n*nrhs) for MRHS. This\nparameter contains the current approximation to the solution. Before the\nfirst call to the dcg/dcgmrhs routine, it contains the initial approximation to\nthe solution.\nnrhs\nMKL_INT, this parameter sets the number of right-hand sides for MRHS\nroutines.\nb\ndouble array containing a single right-hand side vector, or matrix of size\nn*nrhs containing right-hand side vectors.\nRCI_request\nMKL_INT, this parameter gives information about the result of work of the\nRCI CG routines. Negative values of the parameter indicate that the routine\ncompleted with errors or warnings. The 0 value indicates successful\ncompletion of the task. Positive values mean that you must perform specific\nactions:\nRCI_request= 1\nmultiply the matrix by tmp [0:n - 1), put the\nresult in tmp[n:2*n - 1), and return the\ncontrol to the dcg/dcgmrhs routine;\nRCI_request= 2\nto perform the stopping tests. If they fail, return\nthe control to the dcg/dcgmrhs routine. If the\nstopping tests succeed, it indicates that the\nsolution is found and stored in the x array;\nRCI_request= 3\nfor SRHS: apply the preconditioner to\ntmp[2*n:3*n - 1], put the result in\ntmp[3*n:4*n - 1], and return the control to\nthe dcg routine;\nfor MRHS: apply the preconditioner to\ntmp[2+ipar[2]*n:(3 + ipar[2])*n - 1],\nput the result in tmp[3*n:4*n - 1], and return\nthe control to the dcgmrhs routine.\nNote that the dcg_get/dcgmrhs_get routine does not change the\nparameter RCI_request. This enables use of this routine inside the reverse\ncommunication computations.\nipar\nMKL_INT array, of size 128 for SRHS, and of size (128+2*nrhs) for MRHS.\nThis parameter specifies the integer set of data for the RCI CG\ncomputations:\nipar[0]\nspecifies the size of the problem. The dcg_init/\ndcgmrhs_init routine assigns ipar[0]=n. All\nthe other routines use this parameter instead of\nn. There is no default value for this parameter.\nipar[1]\nspecifies the type of output for error and\nwarning messages generated by the RCI CG\nroutines. The default value 6 means that all\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1913\n\n\nmessages are displayed on the screen.\nOtherwise, the error and warning messages are\nwritten to the newly created files\ndcg_errors.txt and\ndcg_check_warnings.txt, respectively. Note\nthat if ipar[5] and ipar[6] parameters are set\nto 0, error and warning messages are not\ngenerated at all.\nipar[2]\nfor SRHS: contains the current stage of the RCI\nCG computations. The initial value is 1;\nfor MRHS: contains the number of the right-\nhand side for which the calculations are\ncurrently performed.\nWARNING\nAvoid altering this variable during\ncomputations.\nipar[3]\ncontains the current iteration number. The initial\nvalue is 0.\nipar[4]\nspecifies the maximum number of iterations.\nThe default value is min(150, n).\nipar[5]\nif the value is not equal to 0, the routines output\nerror messages in accordance with the\nparameter ipar[1]. Otherwise, the routines do\nnot output error messages at all, but return a\nnegative value of the parameter RCI_request.\nThe default value is 1.\nipar[6]\nif the value is not equal to 0, the routines output\nwarning messages in accordance with the\nparameter ipar[1]. Otherwise, the routines do\nnot output warning messages at all, but they\nreturn a negative value of the parameter\nRCI_request. The default value is 1.\nipar[7]\nif the value is not equal to 0, the dcg/dcgmrhs\nroutine performs the stopping test for the\nmaximum number of iterations: ipar[3] ≤\nipar[4]. Otherwise, the method is stopped and\nthe corresponding value is assigned to the\nRCI_request. If the value is 0, the routine does\nnot perform this stopping test. The default value\nis 1.\nipar[8]\nif the value is not equal to 0, the dcg/dcgmrhs\nroutine performs the residual stopping test:\ndpar(5) ≤ dpar(4)= dpar(1)*dpar(3)+\ndpar(2). Otherwise, the method is stopped and\ncorresponding value is assigned to the\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1914\n\n\nRCI_request. If the value is 0, the routine does\nnot perform this stopping test. The default value\nis 0.\nipar[9]\nif the value is not equal to 0, the dcg/dcgmrhs\nroutine requests a user-defined stopping test by\nsetting the output parameter RCI_request=2.\nIf the value is 0, the routine does not perform\nthe user defined stopping test. The default value\nis 1.\nNOTE\nAt least one of the parameters ipar[7]-\nipar[9] must be set to 1.\nipar[10]\nif the value is equal to 0, the dcg/dcgmrhs\nroutine runs the non-preconditioned version of\nthe corresponding CG method. Otherwise, the\nroutine runs the preconditioned version of the\nCG method, and by setting the output\nparameter RCI_request=3, indicates that you\nmust perform the preconditioning step. The\ndefault value is 0.\nipar[11:127]\nare reserved and not used in the current RCI CG\nSRHS and MRHS routines.\nNOTE\nFor future compatibility, you must declare the\narray ipar with length 128 for a single right-\nhand side.\nipar[11:127 +\n2*nrhs]\nare reserved for internal use in the current RCI\nCG SRHS and MRHS routines.\nNOTE\nFor future compatibility, you must declare the\narray ipar with length 128+2*nrhs for\nmultiple right-hand sides.\ndpar\ndouble array, for SRHS of size 128, for MRHS of size (128+2*nrhs); this\nparameter is used to specify the double precision set of data for the RCI CG\ncomputations, specifically:\ndpar[0]\nspecifies the relative tolerance. The default\nvalue is 1.0X10-6.\ndpar[1]\nspecifies the absolute tolerance. The default\nvalue is 0.0.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1915\n\n\ndpar[2]\nspecifies the square norm of the initial residual\n(if it is computed in the dcg/dcgmrhs routine).\nThe initial value is 0.0.\ndpar[3]\nservice variable equal to\ndpar[0]*dpar[2]+dpar[1] (if it is computed\nin the dcg/dcgmrhs routine). The initial value is\n0.0.\ndpar[4]\nspecifies the square norm of the current\nresidual. The initial value is 0.0.\ndpar[5]\nspecifies the square norm of residual from the\nprevious iteration step (if available). The initial\nvalue is 0.0.\ndpar[6]\ncontains the alpha parameter of the CG method.\nThe initial value is 0.0.\ndpar[7]\ncontains the beta parameter of the CG method,\nit is equal to dpar[4]/dpar[5] The initial value\nis 0.0.\ndpar[8:127]\nare reserved and not used in the current RCI CG\nSRHS and MRHS routines.\nNOTE\nFor future compatibility, you must declare the\narray dpar with length 128 for a single right-\nhand side.\ndpar(9:128+2*nrhs)\n[8:127 + 2*nrhs]\nare reserved for internal use in the current RCI\nCG SRHS and MRHS routines.\nNOTE\nFor future compatibility, you must declare the\narray dpar with length 128+2*nrhs for\nmultiple right-hand sides.\ntmp\ndouble array of size (n*4)for SRHS, and (n*(3+nrhs))for MRHS. This\nparameter is used to supply the double precision temporary space for the\nRCI CG computations, specifically:\ntmp[0:n - 1]\nspecifies the current search direction. The initial\nvalue is 0.0.\ntmp[n:2*n - 1]\ncontains the matrix multiplied by the current\nsearch direction. The initial value is 0.0.\ntmp[2*n:3*n - 1]\ncontains the current residual. The initial value is\n0.0.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1916\n\n\ntmp[3*n:4*n - 1]\ncontains the inverse of the preconditioner\napplied to the current residual for the SRHS\nversion of CG. There is no initial value for this\nparameter.\ntmp[4*n:(4 + nrhs)*n\n- 1]\ncontains the inverse of the preconditioner\napplied to the current residual for the MRHS\nversion of CG. There is no initial value for this\nparameter.\nNOTE\nYou can define this array in the code using RCI CG SRHS as\ndoubletmp[3*n] if you run only non-preconditioned CG iterations.\nFGMRES Interface Description\nRoutine Options\nAll of the RCI FGMRES routines have common parameters for passing various options to the routines (see \nFGMRES Common Parameters). The values for these parameters can be changed during computations.\nUser Data Arrays\nMany of the RCI FGMRES routines take arrays of user data as input. For example, user arrays are passed to\nthe routine dfgmresto compute the solution of a system of linear algebraic equations. To minimize storage\nrequirements and improve overall run-time efficiency, the Intel® oneAPI Math Kernel Library (oneMKL) RCI\nFGMRES routines do not make copies of the user input arrays.\nFGMRES Common Parameters\nNOTE\nThe default and initial values listed below are assigned to the parameters by calling the dfgmres_init\nroutine.\nn\nMKL_INT, this parameter sets the size of the problem in the dfgmres_init\nroutine. All the other routines use the ipar[0] parameter instead. Note\nthat the coefficient matrix A is a square matrix of size n*n.\nx\ndouble array, this parameter contains the current approximation to the\nsolution vector. Before the first call to the dfgmres routine, it contains the\ninitial approximation to the solution vector.\nb\ndouble array, this parameter contains the right-hand side vector.\nDepending on user requests (see the parameter ipar[12]), it might\ncontain the approximate solution after execution.\nRCI_request\nMKL_INT, this parameter gives information about the result of work of the\nRCI FGMRES routines. Negative values of the parameter indicate that the\nroutine completed with errors or warnings. The 0 value indicates successful\ncompletion of the task. Positive values mean that you must perform specific\nactions:\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1917\n\n\nRCI_request= 1\nmultiply the matrix by tmp[ipar[21] -\n1:ipar[21] + n - 2], put the result in\ntmp[ipar[22] - 1:ipar[22] + n - 2], and\nreturn the control to the dfgmres routine;\nRCI_request= 2\nperform the stopping tests. If they fail, return\nthe control to the dfgres routine. Otherwise,\nthe solution can be updated by a subsequent call\nto dfgmres_get routine;\nRCI_request= 3\napply the preconditioner to tmp[ipar[21] -\n1:ipar[21] + n - 2], put the result in\ntmp[ipar[22] - 1:ipar[22] + n - 2], and\nreturn the control to the dfgmres routine.\nRCI_request= 4\ncheck if the norm of the current orthogonal\nvector is zero, within the rounding or\ncomputational errors. Return the control to the\ndfgmres routine if it is not zero, otherwise\ncomplete the solution process by calling\ndfgmres_get routine.\nipar[128]\nMKL_INT array, this parameter specifies the integer set of data for the RCI\nFGMRES computations:\nipar[0]\nspecifies the size of the problem. The\ndfgmres_init routine assigns ipar[0]=n. All\nthe other routines uses this parameter instead\nof n. There is no default value for this\nparameter.\nipar[1]\nspecifies the type of output for error and\nwarning messages that are generated by the\nRCI FGMRES routines. The default value 6\nmeans that all messages are displayed on the\nscreen. Otherwise the error and warning\nmessages are written to the newly created file\nMKL_RCI_FGMRES_Log.txt. Note that if\nipar[5] and ipar[6] parameters are set to 0,\nerror and warning messages are not generated\nat all.\nipar[2]\ncontains the current stage of the RCI FGMRES\ncomputations. The initial value is 1.\nWARNING\nAvoid altering this variable during\ncomputations.\nipar[3]\ncontains the current iteration number. The initial\nvalue is 0.\nipar[4]\nspecifies the maximum number of iterations.\nThe default value is min (150,n).\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1918\n\n\nipar[5]\nif the value is not 0, the routines output error\nmessages in accordance with the parameter\nipar[1]. If it is 0, the routines do not output\nerror messages at all, but return a negative\nvalue of the parameter RCI_request. The\ndefault value is 1.\nipar[6]\nif the value is not 0, the routines output warning\nmessages in accordance with the parameter\nipar[1]. Otherwise, the routines do not output\nwarning messages at all, but they return a\nnegative value of the parameter RCI_request.\nThe default value is 1.\nipar[7]\nif the value is not equal to 0, the dfgmres\nroutine performs the stopping test for the\nmaximum number of iterations: ipar[3]\n≤ipar[4]. If the value is 0, the dfgmres routine\ndoes not perform this stopping test. The default\nvalue is 1.\nipar[8]\nif the value is not 0, the dfgmres routine\nperforms the residual stopping test: dpar[4]\n≤dpar[3].If the value is 0, the dfgmres routine\ndoes not perform this stopping test. The default\nvalue is 0.\nipar[9]\nif the value is not 0, the dfgmres routine\nindicates that the user-defined stopping test\nshould be performed by setting\nRCI_request=2. If the value is 0, the dfgmres\nroutine does not perform the user-defined\nstopping test. The default value is 1.\nNOTE\nAt least one of the parameters ipar[7]-\nipar[9] must be set to 1.\nipar[10]\nif the value is 0, the dfgmres routine runs the\nnon-preconditioned version of the FGMRES\nmethod. Otherwise, the routine runs the\npreconditioned version of the FGMRES method,\nand requests that you perform the\npreconditioning step by setting the output\nparameter RCI_request=3. The default value is\n0.\nipar[11]\nif the value is not equal to 0, the dfgmres\nroutine performs the automatic test for zero\nnorm of the currently generated vector:\ndpar[6]≤dpar[7], where dpar[7] contains the\ntolerance value. Otherwise, the routine indicates\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1919\n\n\nthat you must perform this check by setting the\noutput parameter RCI_request=4. The default\nvalue is 0.\nipar[12]\nif the value is equal to 0, the dfgmres_get\nroutine updates the solution to the vector x\naccording to the computations done by the\ndfgmres routine. If the value is positive, the\nroutine writes the solution to the right-hand side\nvector b. If the value is negative, the routine\nreturns only the number of the current iteration,\nand does not update the solution. The default\nvalue is 0.\nNOTE\nIt is possible to call the dfgmres_get routine\nat any place in the code, but you must pay\nspecial attention to the parameter ipar[12].\nThe RCI FGMRES iterations can be continued\nafter the call to dfgmres_get routine only if\nthe parameter ipar[12] is not equal to zero.\nIf ipar[12] is positive, then the updated\nsolution overwrites the right-hand side in the\nvector b. If you want to run the restarted\nversion of FGMRES with the same right-hand\nside, then it must be saved in a different\nmemory location before the first call to the\ndfgmres_get routine with positive\nipar[12].\nipar[13]\ncontains the internal iteration counter that\ncounts the number of iterations before the\nrestart takes place. The initial value is 0.\nWARNING\nDo not alter this variable during\ncomputations.\nipar[14]\nspecifies the number of the non-restarted\nFGMRES iterations. To run the restarted version\nof the FGMRES method, assign the number of\niterations to ipar[14] before the restart. The\ndefault value is min(150, n), which means that\nby default the non-restarted version of FGMRES\nmethod is used.\nipar[15]\nservice variable specifying the location of the\nrotated Hessenberg matrix from which the\nmatrix stored in the packed format (see Matrix\nArguments in the Appendix B for details) is\nstarted in the tmp array.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1920\n\n\nipar[16]\nservice variable specifying the location of the\nrotation cosines from which the vector of cosines\nis started in the tmp array.\nipar[17]\nservice variable specifying the location of the\nrotation sines from which the vector of sines is\nstarted in the tmp array.\nipar[18]\nservice variable specifying the location of the\nrotated residual vector from which the vector is\nstarted in the tmp array.\nipar[19]\nservice variable, specifies the location of the\nleast squares solution vector from which the\nvector is started in the tmp array.\nipar[20]\nservice variable specifying the location of the set\nof preconditioned vectors from which the set is\nstarted in the tmp array. The memory locations\nin the tmp array starting from ipar[20] are\nused only for the preconditioned FGMRES\nmethod.\nipar[21]\nspecifies the memory location from which the\nfirst vector (source) used in operations\nrequested via RCI_request is started in the tmp\narray.\nipar[22]\nspecifies the memory location from which the\nsecond vector (output) used in operations\nrequested via RCI_request is started in the tmp\narray.\nipar[23:127]\nare reserved and not used in the current RCI\nFGMRES routines.\nNOTE\nYou must declare the array ipar with length\n128. While defining the array in the code as\nipar[23]works, there is no guarantee of\nfuture compatibility with Intel® oneAPI Math\nKernel Library (oneMKL).\ndpar(128)\ndouble array, this parameter specifies the double precision set of data for\nthe RCI CG computations, specifically:\ndpar[0]\nspecifies the relative tolerance. The default\nvalue is 1.0e-6.\ndpar[1]\nspecifies the absolute tolerance. The default\nvalue is 0.0e-0.\ndpar[2]\nspecifies the Euclidean norm of the initial\nresidual (if it is computed in the dfgmres\nroutine). The initial value is 0.0.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1921\n\n\ndpar[3]\nservice variable equal to\ndpar[0]*dpar[2]+dpar[1] (if it is computed\nin the dfgmres routine). The initial value is 0.0.\ndpar[4]\nspecifies the Euclidean norm of the current\nresidual. The initial value is 0.0.\ndpar[5]\nspecifies the Euclidean norm of residual from the\nprevious iteration step (if available). The initial\nvalue is 0.0.\ndpar[6]\ncontains the norm of the generated vector. The\ninitial value is 0.0.\nNOTE\nIn terms of [Saad03] this parameter is the\ncoefficient hk+1,k of the Hessenberg matrix.\ndpar[7]\ncontains the tolerance for the zero norm of the\ncurrently generated vector. The default value is\n1.0e-12.\ndpar[8:127]\nare reserved and not used in the current RCI\nFGMRES routines.\nNOTE\nYou must declare the array dpar with length\n128. While defining the array in the code as\ndouble dpar[8]works, there is no\nguarantee of future compatibility with Intel®\noneAPI Math Kernel Library (oneMKL).\ntmp\ndouble array of size ((2*ipar[14] + 1)*n + ipar[14]*(ipar[14] +\n9)/2 + 1)) used to supply the double precision temporary space for the\nRCI FGMRES computations, specifically:\ntmp[0:ipar[15] - 2]\ncontains the sequence of vectors generated by\nthe FGMRES method. The initial value is 0.0.\ntmp[ipar[15] -\n1:ipar[16] - 2]\ncontains the rotated Hessenberg matrix\ngenerated by the FGMRES method; the matrix is\nstored in the packed format. There is no initial\nvalue for this part of tmp array.\ntmp[ipar[16] -\n1:ipar[17] - 2]\ncontains the rotation cosines vector generated\nby the FGMRES method. There is no initial value\nfor this part of tmp array.\ntmp[ipar[17] -\n1:ipar[18] - 2]\ncontains the rotation sines vector generated by\nthe FGMRES method. There is no initial value for\nthis part of tmp array.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1922\n\n\ntmp[ipar[18] -\n1:ipar[19] - 2]\ncontains the rotated residual vector generated\nby the FGMRES method. There is no initial value\nfor this part of tmp array.\ntmp[ipar[19] -\n1:ipar[20] - 2]\ncontains the solution vector to the least squares\nproblem generated by the FGMRES method.\nThere is no initial value for this part of tmp\narray.\ntmp[ipar[20] - 1:*]\ncontains the set of preconditioned vectors\ngenerated for the FGMRES method by the user.\nThis part of tmp array is not used if the non-\npreconditioned version of FGMRES method is\ncalled. There is no initial value for this part of\ntmp array.\nNOTE\nYou can define this array in the code as\ndouble tmp[(2*ipar[14] + 1)*n +\nipar[14]*(ipar[14] + 9)/2 + 1)] if you\nrun only non-preconditioned FGMRES\niterations.\nRCI ISS Routines\ndcg_init\nInitializes the solver.\nSyntax\nvoid dcg_init (const MKL_INT *n , const double *x , const double *b , MKL_INT\n*RCI_request , MKL_INT *ipar , double *dpar , double *tmp );\nInclude Files\n•\nmkl.h\nDescription\nThe routine dcg_initinitializes the solver. After initialization, all subsequent invocations of the Intel® oneAPI\nMath Kernel Library (oneMKL) RCI CG routines use the values of all parameters returned by the\nroutinedcg_init. Advanced users can skip this step and set the values in the ipar and dpar arrays directly.\nCaution\nYou can modify the contents of these arrays after they are passed to the solver routine only\nif you are sure that the values are correct and consistent. You can perform a basic check for\ncorrectness and consistency by calling the dcg_check routine, but it does not guarantee that\nthe method will work correctly.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1923\n\n\nInput Parameters\nn\nSets the size of the problem.\nx\nArray of size n. Contains the initial approximation to the solution vector.\nNormally it is equal to 0 or to b.\nb\nArray of size n. Contains the right-hand side vector.\nOutput Parameters\nRCI_request\nGives information about the result of the routine.\nipar\nArray of size 128. Refer to the CG Common Parameters.\ndpar\nArray of size 128. Refer to the CG Common Parameters.\ntmp\nArray of size (n*4). Refer to the CG Common Parameters.\nReturn Values\nRCI_request= 0\nIndicates that the task completed normally.\nRCI_request= -10000\nIndicates failure to complete the task.\ndcg_check\nChecks consistency and correctness of the user\ndefined data.\nSyntax\nvoid dcg_check (const MKL_INT *n , const double *x , const double *b , MKL_INT\n*RCI_request , MKL_INT *ipar , double *dpar , double *tmp );\nInclude Files\n•\nmkl.h\nDescription\nThe routine dcg_check checks consistency and correctness of the parameters to be passed to the solver\nroutine dcg. However this operation does not guarantee that the solver returns the correct result. It only\nreduces the chance of making a mistake in the parameters of the method. Skip this operation only if you are\nsure that the correct data is specified in the solver parameters.\nThe lengths of all vectors must be defined in a previous call to the dcg_init routine.\nIf none of the stopping criteria (ipar[7]-ipar[9]) has been enabled, both ipar[7] and ipar[8] will be set to\n1.\nInput Parameters\nipar\nArray of size 128. Refer to the FGMRES Common Parameters.\nn\nSets the size of the problem.\nx\nArray of size n. Contains the initial approximation to the solution vector.\nNormally it is equal to 0 or to b.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1924\n\n\nb\nArray of size n. Contains the right-hand side vector.\nOutput Parameters\nRCI_request\nGives information about result of the routine.\nipar\nArray of size 128. Refer to the CG Common Parameters. Only ipar[7]-\nipar[8] might be changed\ndpar\nArray of size 128. Refer to the CG Common Parameters.\ntmp\nArray of size (n*4). Refer to the CG Common Parameters.\nReturn Values\nRCI_request= 0\nIndicates that the task completed normally.\nRCI_request= -1100\nIndicates that the task is interrupted and the errors occur.\nRCI_request= -1001\nIndicates that there are some warning messages.\nRCI_request= -1010\nIndicates that the routine changed some parameters to\nmake them consistent or correct.\nRCI_request= -1011\nIndicates that there are some warning messages and that\nthe routine changed some parameters.\ndcg\nComputes the approximate solution vector.\nSyntax\nvoid dcg (const MKL_INT *n , double *x , const double *b , MKL_INT *RCI_request ,\nMKL_INT *ipar , double *dpar , double *tmp );\nInclude Files\n•\nmkl.h\nDescription\nThe dcg routine computes the approximate solution vector using the CG method [Young71]. The routine dcg\nuses the vector in the array x before the first call as an initial approximation to the solution. The parameter\nRCI_request gives you information about the task completion and requests results of certain operations that\nare required by the solver.\nNote that lengths of all vectors must be defined in a previous call to the dcg_init routine.\nInput Parameters\nn\nSets the size of the problem.\nx\nArray of size n. Contains the initial approximation to the solution vector.\nb\nArray of size n. Contains the right-hand side vector.\ntmp\nArray of size (n*4). Refer to the CG Common Parameters.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1925\n\n\nOutput Parameters\nRCI_request\nGives information about result of work of the routine.\nx\nArray of size n. Contains the updated approximation to the solution vector.\nipar\nArray of size 128. Refer to the CG Common Parameters.\ndpar\nArray of size 128. Refer to the CG Common Parameters.\ntmp\nArray of size (n*4). Refer to the CG Common Parameters.\nReturn Values\nRCI_request=0\nIndicates that the task completed normally and the solution\nis found and stored in the vector x. This occurs only if the\nstopping tests are fully automatic. For the user defined\nstopping tests, see the description of the RCI_request= 2.\nRCI_request=-1\nIndicates that the routine was interrupted because the\nmaximum number of iterations was reached, but the\nrelative stopping criterion was not met. This situation\noccurs only if you request both tests.\nRCI_request=-2\nIndicates that the routine was interrupted because of an\nattempt to divide by zero. This situation happens if the\nmatrix is non-positive definite or almost non-positive\ndefinite.\nRCI_request=- 10\nIndicates that the routine was interrupted because the\nresidual norm is invalid. This usually happens because the\nvalue dpar[5] was altered outside of the routine, or the\ndcg_check routine was not called.\nRCI_request=-11\nIndicates that the routine was interrupted because it enters\nthe infinite cycle. This usually happens because the values\nipar[7], ipar[8], ipar[9] were altered outside of the\nroutine, or the dcg_check routine was not called.\nRCI_request= 1\nIndicates that you must multiply the matrix by tmp[0:n -\n1], put the result in the tmp[n:2*n - 1], and return\ncontrol back to the routine dcg.\nRCI_request= 2\nIndicates that you must perform the stopping tests. If they\nfail, return control back to the dcg routine. Otherwise, the\nsolution is found and stored in the vector x.\nRCI_request= 3\nIndicates that you must apply the preconditioner to\n[2*n:3*n - 1], put the result in the [3*n:4*n - 1], and\nreturn control back to the routine dcg.\ndcg_get\nRetrieves the number of the current iteration.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1926\n\n\nSyntax\nvoid dcg_get (const MKL_INT *n , const double *x , const double *b , const MKL_INT\n*RCI_request , const MKL_INT *ipar , const double *dpar , const double *tmp , MKL_INT\n*itercount );\nInclude Files\n•\nmkl.h\nDescription\nThe routine dcg_get retrieves the current iteration number of the solutions process.\nInput Parameters\nn\nSets the size of the problem.\nx\nArray of size n. Contains the approximation vector to the solution.\nb\nArray of size n. Contains the right-hand side vector.\nRCI_request\nThis parameter is not used.\nipar\nArray of size 128. Refer to the CG Common Parameters.\ndpar\nArray of size 128. Refer to the CG Common Parameters.\ntmp\nArray of size (n*4). Refer to the CG Common Parameters.\nOutput Parameters\nitercount\nReturns the current iteration number.\nReturn Values\nThe routine dcg_get has no return values.\ndcgmrhs_init\nInitializes the RCI CG solver with MHRS.\nSyntax\nvoid dcgmrhs_init (const MKL_INT *n , const double *x , const MKL_INT *nrhs , const\ndouble *b , const MKL_INT *method , MKL_INT *RCI_request , MKL_INT *ipar , double\n*dpar , double *tmp );\nInclude Files\n•\nmkl.h\nDescription\nThe routine dcgmrhs_initinitializes the solver. After initialization all subsequent invocations of the Intel®\noneAPI Math Kernel Library (oneMKL) RCI CG with multiple right-hand sides (MRHS) routines use the values\nof all parameters that are returned bydcgmrhs_init. Advanced users may skip this step and set the values\nto these parameters directly in the appropriate routines.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1927\n\n\nWARNING\nYou can modify the contents of these arrays after they are passed to the solver routine only\nif you are sure that the values are correct and consistent. You can perform a basic check for\ncorrectness and consistency by calling the dcgmrhs_check routine, but it does not guarantee\nthat the method will work correctly.\nInput Parameters\nn\nSets the size of the problem.\nx\nArray of size n*nrhs. Contains the initial approximation to the solution\nvectors. Normally it is equal to 0 or to b.\nnrhs\nSets the number of right-hand sides.\nb\nArray of size n*nrhs. Contains the right-hand side vectors.\nmethod\nSpecifies the method of solution:\nA value of 1 indicates CG with multiple right-hand sides (default value)\nOutput Parameters\nRCI_request\nGives information about the result of the routine.\nipar\nArray of size (128+2*nrhs). Refer to the CG Common Parameters.\ndpar\nArray of size (128+2*nrhs). Refer to the CG Common Parameters.\ntmp\nArray of size (n*(3+nrhs)). Refer to the CG Common Parameters.\nReturn Values\nRCI_request= 0\nIndicates that the task completed normally.\nRCI_request= -10000\nIndicates failure to complete the task.\ndcgmrhs_check\nChecks consistency and correctness of the user\ndefined data.\nSyntax\nvoid dcgmrhs_check (const MKL_INT *n , const double *x , const MKL_INT *nrhs , const\ndouble *b , MKL_INT *RCI_request , MKL_INT *ipar , double *dpar , double *tmp );\nInclude Files\n•\nmkl.h\nDescription\nThe routine dcgmrhs_check checks the consistency and correctness of the parameters to be passed to the\nsolver routine dcgmrhs. While this operation reduces the chance of making a mistake in the parameters, it\ndoes not guarantee that the solver returns the correct result.\nIf you are sure that the correct data is specified in the solver parameters, you can skip this operation.\nThe lengths of all vectors must be defined in a previous call to the dcgmrhs_init routine.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1928\n\n\nIf none of the stopping criteria (ipar[7]-ipar[9]) has been enabled, both ipar[7] and ipar[8] will be set to\n1.\nInput Parameters\nn\nSets the size of the problem.\nx\nArray of size n*nrhs. Contains the initial approximation to the solution\nvectors. Normally it is equal to 0 or to b.\nnrhs\nThis parameter sets the number of right-hand sides.\nb\nArray of size n*nrhs. Contains the right-hand side vectors.\nOutput Parameters\nRCI_request\nReturns information about the results of the routine.\nipar\nArray of size (128+2*nrhs). Refer to the CG Common Parameters. Only\nipar[7]-ipar[8] might be changed.\ndpar\nArray of size (128+2*nrhs). Refer to the CG Common Parameters.\ntmp\nArray of size (n*(3+nrhs)). Refer to the CG Common Parameters.\nReturn Values\nRCI_request= 0\nIndicates that the task completed normally.\nRCI_request= -1100\nIndicates that the task is interrupted and the errors occur.\nRCI_request= -1001\nIndicates that there are some warning messages.\nRCI_request= -1010\nIndicates that the routine changed some parameters to\nmake them consistent or correct.\nRCI_request= -1011\nIndicates that there are some warning messages and that\nthe routine changed some parameters.\ndcgmrhs\nComputes the approximate solution vectors.\nSyntax\nvoid dcgmrhs (const MKL_INT *n , double *x , const MKL_INT *nrhs , const double *b ,\nMKL_INT *RCI_request , MKL_INT *ipar , double *dpar , double *tmp );\nInclude Files\n•\nmkl.h\nDescription\nThe routine dcgmrhs computes approximate solution vectors using the CG with multiple right-hand sides\n(MRHS) method [Young71]. The routine dcgmrhs uses the value that was in the x before the first call as an\ninitial approximation to the solution. The parameter RCI_request gives information about task completion\nstatus and requests results of certain operations that are required by the solver.\nNote that lengths of all vectors are assumed to have been defined in a previous call to the dcgmrhs_init\nroutine.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1929\n\n\nInput Parameters\nn\nSets the size of the problem, and the sizes of arrays x and b.\nx\nArray of size n*nrhs. Contains the initial approximation to the solution\nvectors.\nnrhs\nSets the number of right-hand sides.\nb\nArray of size n*nrhs. Contains the right-hand side vectors.\ntmp\nArray of size (n, 3+nrhs). Refer to the CG Common Parameters.\nOutput Parameters\nRCI_request\nGives information about result of work of the routine.\nx\nArray of size (n*nrhs). Contains the updated approximation to the solution\nvectors.\nipar\nArray of size (128+2*nrhs). Refer to the CG Common Parameters.\ndpar\nArray of size (128+2*nrhs). Refer to the CG Common Parameters.\ntmp\nArray of size (n*(3+nrhs)). Refer to the CG Common Parameters.\nReturn Values\nRCI_request=0\nIndicates that the task completed normally and the solution\nis found and stored in the vector x. This occurs only if the\nstopping tests are fully automatic. For the user defined\nstopping tests, see the description of the RCI_request= 2.\nRCI_request=-1\nIndicates that the routine was interrupted because the\nmaximum number of iterations was reached, but the\nrelative stopping criterion was not met. This situation\noccurs only if both tests are requested by the user.\nRCI_request=-2\nThe routine was interrupted because of an attempt to\ndivide by zero. This situation happens if the matrix is non-\npositive definite or almost non-positive definite.\nRCI_request=- 10\nIndicates that the routine was interrupted because the\nresidual norm is invalid. This usually happens because the\nvalue dpar[5] was altered outside of the routine, or the\ndcg_check routine was not called.\nRCI_request=-11\nIndicates that the routine was interrupted because it enters\nthe infinite cycle. This usually happens because the values\nipar[7], ipar[8], ipar[9] were altered outside of the\nroutine, or the dcg_check routine was not called.\nRCI_request= 1\nIndicates that you must multiply the matrix by tmp[0:n -\n1], put the result in the tmp[n:2*n - 1], and return\ncontrol back to the routine dcg.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1930\n\n\nRCI_request= 2\nIndicates that you must perform the stopping tests. If they\nfail, return control back to the dcg routine. Otherwise, the\nsolution is found and stored in the vector x.\nRCI_request= 3\nIndicates that you must apply the preconditioner to\ntmp[2*n:3*n - 1], put the result in the tmp[3*n:4*n -\n1], and return control back to the routine dcg.\ndcgmrhs_get\nRetrieves the number of the current iteration.\nSyntax\nvoid dcgmrhs_get (const MKL_INT *n , const double *x , const MKL_INT *nrhs , const\ndouble *b , const MKL_INT *RCI_request , const MKL_INT *ipar , const double *dpar ,\nconst double *tmp , MKL_INT *itercount );\nInclude Files\n•\nmkl.h\nDescription\nThe routine dcgmrhs_get retrieves the current iteration number of the solving process.\nInput Parameters\nn\nSets the size of the problem.\nx\nArray of size n*nrhs. Contains the initial approximation to the solution\nvectors.\nnrhs\nSets the number of right-hand sides.\nb\nArray of size n*nrhs. Contains the right-hand side .\nRCI_request\nThis parameter is not used.\nipar\nArray of size (128+2*nrhs). Refer to the CG Common Parameters.\ndpar\nArray of size (128+2*nrhs). Refer to the CG Common Parameters.\ntmp\nArray of size (n*(3+nrhs)). Refer to the CG Common Parameters.\nOutput Parameters\nitercount\nArray of size nrhs. Returns the current iteration number for each right-hand\nside.\nReturn Values\nThe routine dcgmrhs_get has no return values.\ndfgmres_init\nInitializes the solver.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1931\n\n\nSyntax\nvoid dfgmres_init (const MKL_INT *n , const double *x , const double *b , MKL_INT\n*RCI_request , MKL_INT *ipar , double *dpar , double *tmp );\nInclude Files\n•\nmkl.h\nDescription\nThe routine dfgmres_initinitializes the solver. After initialization all subsequent invocations of Intel® oneAPI\nMath Kernel Library (oneMKL) RCI FGMRES routines use the values of all parameters that are returned\nbydfgmres_init. Advanced users can skip this step and set the values in the ipar and dpar arrays directly.\nWARNING\nYou can modify the contents of these arrays after they are passed to the solver routine only\nif you are sure that the values are correct and consistent. You can perform a basic check for\ncorrectness and consistency by calling the dfgmres_check routine, but it does not guarantee\nthat the method will work correctly.\nInput Parameters\nn\nSets the size of the problem.\nx\nArray of size n. Contains the initial approximation to the solution vector.\nNormally it is equal to 0 or to b.\nb\nArray of size n. Contains the right-hand side vector.\nOutput Parameters\nRCI_request\nGives information about the result of the routine.\nipar\nArray of size 128. Refer to the FGMRES Common Parameters.\ndpar\nArray of size 128. Refer to the FGMRES Common Parameters.\ntmp\nArray of size ((2*ipar[14] + 1)*n + ipar[14]*(ipar[14] + 9)/2 +\n1). Refer to the FGMRES Common Parameters.\nReturn Values\nRCI_request= 0\nIndicates that the task completed normally.\nRCI_request= -10000\nIndicates failure to complete the task.\ndfgmres_check\nChecks consistency and correctness of the user\ndefined data.\nSyntax\nvoid dfgmres_check (const MKL_INT *n, const double *x, const double *b, MKL_INT\n*RCI_request, MKL_INT *ipar, double *dpar, double *tmp );\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1932\n\n\nInclude Files\n•\nmkl.h\nDescription\nThe routine dfgmres_check checks consistency and correctness of the parameters to be passed to the solver\nroutine dfgmres. However, this operation does not guarantee that the method gives the correct result. It\nonly reduces the chance of making a mistake in the parameters of the routine. Skip this operation only if you\nare sure that the correct data is specified in the solver parameters.\nThe lengths of all vectors are assumed to have been defined in a previous call to the dfgmres_init routine.\nIn particular, the routine checks the consistency of ipar[15]-ipar[20] and ipar[0], ipar[14]. If the values\ndo not agree, the routine emits a warning and modifies ipar[15]-ipar[20] to comply with the values of\nipar[0], ipar[14]. A possible use case for this modification is a non-default value (not the one set by a\npossible call to dfgmres_init) of ipar[14].\nAlso, if none of the stopping criteria (ipar[7]-ipar[9]) has been enabled, both ipar[7] and ipar[9] will be\nset to 1.\nNOTE: It is not strictly necessary to call the dfgmres_check routine unless the values of ipar[14] or\nipar[0] are changed after the last call to dfgmres_init.\nInput Parameters\nipar\nArray of size 128. Refer to the FGMRES Common Parameters.\nn\nSets the size of the problem.\nx\nArray of size n. Contains the initial approximation to the solution vector.\nNormally it is equal to 0 or to b.\nb\nArray of size n. Contains the right-hand side vector.\nOutput Parameters\nRCI_request\nGives information about result of the routine.\nipar\nArray of size 128. Refer to the FGMRES Common Parameters. Only ipar[7]-\nipar[8] and ipar[15]-ipar[20] might be changed.\ndpar\nArray of size 128. Refer to the FGMRES Common Parameters.\ntmp\nArray of size ((2*ipar[14] + 1)*n + ipar[14]*(ipar[14] + 9)/2 +\n1). Refer to the FGMRES Common Parameters.\nReturn Values\nRCI_request= 0\nIndicates that the task completed normally.\nRCI_request= -1100\nIndicates that the task is interrupted and the errors occur.\nRCI_request= -1001\nIndicates that there are some warning messages.\nRCI_request= -1010\nIndicates that the routine changed some parameters to\nmake them consistent or correct.\nRCI_request= -1011\nIndicates that there are some warning messages and that\nthe routine changed some parameters.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1933\n\n\ndfgmres\nMakes the FGMRES iterations.\nSyntax\nvoid dfgmres (const MKL_INT *n, double *x, double *b, MKL_INT *RCI_request, MKL_INT\n*ipar, double *dpar, double *tmp );\nInclude Files\n•\nmkl.h\nDescription\nThe routine dfgmres performs the FGMRES iterations [Saad03], using the value that was in the array x\nbefore the first call as an initial approximation of the solution vector. To update the current approximation to\nthe solution, the dfgmres_get routine must be called. The RCI FGMRES iterations can be continued after the\ncall to the dfgmres_get routine only if the value of the parameter ipar[12] is not equal to 0 (default\nvalue). Note that the updated solution overwrites the right-hand side in the vector b if the parameter\nipar[12] is positive, and the restarted version of the FGMRES method can not be run. If you want to keep\nthe right-hand side, you must be save it in a different memory location before the first call to the\ndfgmres_get routine with a positive ipar[12].\nThe parameter RCI_request gives information about the task completion and requests results of certain\noperations that the solver requires.\nThe lengths of all the vectors must be defined in a previous call to the dfgmres_init routine.\nInput Parameters\nn\nSets the size of the problem.\nx\nArray of size n. Contains the initial approximation to the solution vector.\nb\nArray of size n. Contains the right-hand side vector.\ntmp\nArray of size [12]. Refer to the FGMRES Common Parameters.\nOutput Parameters\nRCI_request\nInforms about result of work of the routine.\nipar\nArray of size 128. Refer to the FGMRES Common Parameters.\ndpar\nArray of size 128. Refer to the FGMRES Common Parameters.\ntmp\nArray of size ((2*ipar[14]+1)*n+ipar[14]*ipar[14]+9)/2 + 1. Refer\nto the FGMRES Common Parameters.\nReturn Values\nRCI_request=0\nIndicates that the task completed normally and the solution\nis found and stored in the vector x. This occurs only if the\nstopping tests are fully automatic. For the user defined\nstopping tests, see the description of the RCI_request= 2\nor 4.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1934\n\n\nRCI_request=-1\nIndicates that the routine was interrupted because the\nmaximum number of iterations was reached, but the\nrelative stopping criterion was not met. This situation\noccurs only if you request both tests.\nRCI_request= -10\nIndicates that the routine was interrupted because of an\nattempt to divide by zero. Usually this happens if the\nmatrix is degenerate or almost degenerate. However, it\nmay happen if the parameter dpar is altered, or if the\nmethod is not stopped when the solution is found.\nRCI_request= -11\nIndicates that the routine was interrupted because it\nentered an infinite cycle. Usually this happens because the\nvalues ipar[7], ipar[8], ipar[9] were altered outside of\nthe routine, or the dfgmres_check routine was not called.\nRCI_request= -12\nIndicates that the routine was interrupted because errors\nwere found in the method parameters. Usually this happens\nif the parameters ipar and dpar were altered by mistake\noutside the routine.\nRCI_request= 1\nIndicates that you must multiply the matrix by\ntmp[ipar[21] - 1:ipar[21] + n - 2], put the result\nin the tmp[ipar[22] - 1:ipar[22] + n - 2], and\nreturn control back to the routine dfgmres.\nRCI_request= 2\nIndicates that you must perform the stopping tests. If they\nfail, return control to the dfgmres routine. Otherwise, the\nFGMRES solution is found, and you can run the\nfgmres_get routine to update the computed solution in the\nvector x.\nRCI_request= 3\nIndicates that you must apply the inverse preconditioner to\ntmp[ipar[21] - 1:ipar[21] + n - 2], put the result\nin the tmp[ipar[22] - 1:ipar[22] + n - 2], and\nreturn control back to the routine dfgmres.\nRCI_request= 4\nIndicates that you must check the norm of the currently\ngenerated vector. If it is not zero within the computational/\nrounding errors, return control to the dfgmres routine.\nOtherwise, the FGMRES solution is found, and you can run\nthe dfgmres_get routine to update the computed solution\nin the vector x.\ndfgmres_get\nRetrieves the number of the current iteration and\nupdates the solution.\nSyntax\nvoid dfgmres_get (const MKL_INT *n, double *x, double *b, MKL_INT *RCI_request, const\nMKL_INT *ipar, const double *dpar, double *tmp, MKL_INT *itercount );\nInclude Files\n•\nmkl.h\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1935\n\n\nDescription\nThe routine dfgmres_get retrieves the current iteration number of the solution process and updates the\nsolution according to the computations performed by the dfgmres routine. To retrieve the current iteration\nnumber only, set the parameter ipar[12]= -1 beforehand. Normally, you should do this before proceeding\nfurther with the computations. If the intermediate solution is needed, the method parameters must be set\nproperly. For details see FGMRES Common Parametersand the Iterative Sparse Solver code examples in the\nIntel® oneAPI Math Kernel Library (oneMKL) installation directory:\n•\nexamples/solverc/source\nInput Parameters\nn\nSets the size of the problem.\nipar\nArray of size 128. Refer to the FGMRES Common Parameters.\ndpar\nArray of size 128. Refer to the FGMRES Common Parameters.\ntmp\nArray of size ((2*ipar[14]+1)*n+ipar[14]*ipar[14]+9)/2 + 1). Refer\nto the FGMRES Common Parameters.\nOutput Parameters\nx\nArray of size n. If ipar[12]= 0, it contains the updated approximation to\nthe solution according to the computations done in dfgmres routine.\nOtherwise, it is not changed.\nb\nArray of size n. If ipar(13)> 0, it contains the updated approximation to\nthe solution according to the computations done in dfgmres routine.\nOtherwise, it is not changed.\nRCI_request\nGives information about result of the routine.\nitercount\nContains the value of the current iteration number.\nReturn Values\nRCI_request= 0\nIndicates that the task completed normally.\nRCI_request= -12\nIndicates that the routine was interrupted because errors\nwere found in the routine parameters. Usually this happens\nif the parameters ipar and dpar were altered by mistake\noutside of the routine.\nRCI_request= -10000\nIndicates that the routine failed to complete the task.\nRCI ISS Implementation Details\nSeveral aspects of the Intel® oneAPI Math Kernel Library (oneMKL) RCI ISS interface are platform-specific\nand language-specific. To promote portability across platforms and ease of use across different languages,\ninclude one of the Intel® oneAPI Math Kernel Library (oneMKL) RCI ISS language-specific header files.\nNOTE\nIntel® oneAPI Math Kernel Library (oneMKL) does not support the RCI ISS interface unless you include\nthe language-specific header file.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1936\n\n\nPreconditioners based on Incomplete LU Factorization Technique\nPreconditioners, or accelerators are used to accelerate an iterative solution process. In some cases, their use\ncan reduce the number of iterations dramatically and thus lead to better solver performance. Although the\nterms preconditioner and accelerator are synonyms, hereafter only preconditioner is used.\nIntel® oneAPI Math Kernel Library (oneMKL) provides two preconditioners, ILU0 and ILUT, for sparse matrices\npresented in the format accepted in the Intel® oneAPI Math Kernel Library (oneMKL) direct sparse solvers\n(three-array variation of the CSR storage format described inSparse Matrix Storage Format ). The algorithms\nused are described in [Saad03].\nThe ILU0 preconditioner is based on a well-known factorization of the original matrix into a product of two\ntriangular matrices: lower and upper triangular matrices. Usually, such decomposition leads to some fill-in in\nthe resulting matrix structure in comparison with the original matrix. The distinctive feature of the ILU0\npreconditioner is that it preserves the structure of the original matrix in the result.\nUnlike the ILU0 preconditioner, the ILUT preconditioner preserves some resulting fill-in in the preconditioner\nmatrix structure. The distinctive feature of the ILUT algorithm is that it calculates each element of the\npreconditioner and saves each one if it satisfies two conditions simultaneously: its value is greater than the\nproduct of the given tolerance and matrix row norm, and its value is in the given bandwidth of the resulting\npreconditioner matrix.\nBoth ILU0 and ILUT preconditioners can apply to any non-degenerate matrix. They can be used alone or\ntogether with the Intel® oneAPI Math Kernel Library (oneMKL) RCI FGMRES solver (seeSparse Solver\nRoutines). Avoid using these preconditioners with MKL RCI CG solver because in general, they produce a\nnon-symmetric resulting matrix even if the original matrix is symmetric. Usually, an inverse of the\npreconditioner is required in this case. To do this the Intel® oneAPI Math Kernel Library (oneMKL) triangular\nsolver routinemkl_dcsrtrsv must be applied twice: for the lower triangular part of the preconditioner, and\nthen for its upper triangular part.\nNOTE\nAlthough ILU0 and ILUT preconditioners apply to any non-degenerate matrix, in some cases the\nalgorithm may fail to ensure successful termination and the required result. Whether or not the\npreconditioner produces an acceptable result can only be determined in practice.\nA preconditioner may increase the number of iterations for an arbitrary case of the system and the\ninitial solution, and even ruin the convergence. It is your responsibility as a user to choose a suitable\npreconditioner.\nGeneral Scheme of Using ILUT and RCI FGMRES Routines\nThe general scheme for use is the same for both preconditioners. Some differences exist in the calling\nparameters of the preconditioners and in the subsequent call of two triangular solvers. You can see all these\ndifferences in the preconditioner code examples (dcsrilu*.*) in the examplesfolder of the Intel® oneAPI\nMath Kernel Library (oneMKL) installation directory:\n•\nexamples/solverc/source\nILU0 and ILUT Preconditioners Interface Description\nThe concepts required to understand the use of the Intel® oneAPI Math Kernel Library (oneMKL)\npreconditioner routines are discussed in theAppendix A Linear Solvers Basics.\nUser Data Arrays\nThe preconditioner routines take arrays of user data as input. To minimize storage requirements and improve\noverall run-time efficiency, the Intel® oneAPI Math Kernel Library (oneMKL) preconditioner routines do not\nmake copies of the user input arrays.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1937\n\n\nCommon Parameters\nSome parameters of the preconditioners are common with the FGMRES Common Parameters. The routine \ndfgmres_init specifies their default and initial values. However, some parameters can be redefined with\nother values. These parameters are listed below.\nFor the ILU0 preconditioner:\nipar[1] - specifies the destination of error messages generated by the ILU0 routine. The default value 6\nmeans that all error messages are displayed on the screen. Otherwise routine creates a log file called\nMKL_PREC_log.txt and writes error messages to it. Note if the parameter ipar[5] is set to 0, then error\nmessages are not generated at all.\nipar[5] - specifies whether error messages are generated. If its value is not equal to 0, the ILU0 routine\nreturns error messages as specified by the parameter ipar[1]. Otherwise, the routine does not generate\nerror messages at all, but returns a negative value for the parameter ierr. The default value is 1.\nFor the ILUT preconditioner:\nipar[1] - specifies the destination of error messages generated by the ILUT routine. The default value 6\nmeans that all messages are displayed on the screen. Otherwise routine creates a log file called\nMKL_PREC_log.txt and writes error messages to it. Note if the parameter ipar[5] is set to 0, then error\nmessages are not generated at all.\nipar[5] - specifies whether error messages are generated. If its value is not equal to 0, the ILUT routine\nreturns error messages as specified by the parameter ipar[1]. Otherwise, the routine does not generate\nerror messages at all, but returns a negative value for the parameter ierr. The default value is 1.\nipar[6] - if its value is greater than 0, the ILUT routine generates warning messages as specified by the\nparameter ipar[1] and continues calculations. If its value is equal to 0, the routine returns a positive value\nof the parameter ierr. If its value is less than 0, the routine generates a warning message as specified by\nthe parameter ipar[1] and returns a positive value of the parameter ierr. The default value is 1.\ndcsrilu0\nILU0 preconditioner based on incomplete LU\nfactorization of a sparse matrix.\nSyntax\nvoid dcsrilu0 (const MKL_INT *n , const double *a , const MKL_INT *ia , const MKL_INT\n*ja , double *bilu0 , const MKL_INT *ipar , const double *dpar , MKL_INT *ierr );\nInclude Files\n•\nmkl.h\nDescription\nThe routine dcsrilu0 computes a preconditioner B [Saad03] of a given sparse matrix A stored in the format\naccepted in the direct sparse solvers:\nA~B=L*U , where L is a lower triangular matrix with a unit diagonal, U is an upper triangular matrix with a\nnon-unit diagonal, and the portrait of the original matrix A is used to store the incomplete factors L and U.\nCaution\nThis routine supports only one-based indexing of the array parameters.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1938\n\n\nInput Parameters\nn\nSize (number of rows or columns) of the original square n-by-n matrix A.\na\nArray containing the set of elements of the matrix A. Its length is equal to\nthe number of non-zero elements in the matrix A. Refer to the values\narray description in the Sparse Matrix Storage Format for more details.\nia\nArray of size (n+1) containing begin indices of rows of the matrix A such\nthat ia[i] is the index in the array a of the first non-zero element from the\nrow i. The value of the last element ia[n] is equal to the number of non-\nzero elements in the matrix A, plus one. Refer to the rowIndex array\ndescription in the Sparse Matrix Storage Format for more details.\nja\nArray containing the column indices for each non-zero element of the\nmatrix A. It is important that the indices are in increasing order per row.\nThe matrix size is equal to the size of the array a. Refer to the columns\narray description in the Sparse Matrix Storage Format for more details.\nCaution\nIf column indices are not stored in ascending order for each row\nof matrix, the result of the routine might not be correct.\nipar\nArray of size 128. This parameter specifies the integer set of data for both\nthe ILU0 and RCI FGMRES computations. Refer to the ipar array\ndescription in the FGMRES Common Parameters for more details on\nFGMRES parameter entries. The entries that are specific to ILU0 are listed\nbelow.\nipar[30]\nspecifies how the routine operates when a zero\ndiagonal element occurs during calculation. If this\nparameter is set to 0 (the default value set by the\nroutine dfgmres_init), then the calculations are\nstopped and the routine returns a non-zero error\nvalue. Otherwise, the diagonal element is set to the\nvalue of dpar[31] and the calculations continue.\nNOTE\nYou can declare the ipar array with a size of 32. However, for\nfuture compatibility you must declare the array ipar with length\n128.\ndpar\nArray of size 128. This parameter specifies the double precision set of data\nfor both the ILU0 and RCI FGMRES computations. Refer to the dpar array\ndescription in the FGMRES Common Parameters for more details on\nFGMRES parameter entries. The entries specific to ILU0 are listed below.\ndpar[30]\nspecifies a small value, which is compared with the\ncomputed diagonal elements. When ipar[30] is not\n0, then diagonal elements less than dpar[30] are\nset to dpar[31]. The default value is 1.0e-16.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1939\n\n\nNOTE\nThis parameter can be set to the negative\nvalue, because the calculation uses its\nabsolute value.\nIf this parameter is set to 0, the comparison with\nthe diagonal element is not performed.\ndpar[31]\nspecifies the value that is assigned to the diagonal\nelement if its value is less than dpar[30] (see\nabove). The default value is 1.0e-10.\nNOTE\nYou can declare the dpar array with a size of 32. However, for\nfuture compatibility you must declare the array dpar with length\n128.\nOutput Parameters\nbilu0\nArray B containing non-zero elements of the resulting preconditioning\nmatrix B, stored in the format accepted in direct sparse solvers. Its size is\nequal to the number of non-zero elements in the matrix A. Refer to the\nvalues array description in the Sparse Matrix Storage Format section for\nmore details.\nierr\nError flag, gives information about the routine completion.\nNOTE\nTo present the resulting preconditioning matrix in the CSR3 format the arrays ia (row\nindices) and ja (column indices) of the input matrix must be used.\nReturn Values\nierr=0\nIndicates that the task completed normally.\nierr=-101\nIndicates that the routine was interrupted and that error\noccurred: at least one diagonal element is omitted from the\nmatrix in CSR3 format (see Sparse Matrix Storage Format).\nierr=-102\nIndicates that the routine was interrupted because the\nmatrix contains a diagonal element with the value of zero.\nierr=-103\nIndicates that the routine was interrupted because the\nmatrix contains a diagonal element which is so small that it\ncould cause an overflow, or that it would cause a bad\napproximation to ILU0.\nierr=-104\nIndicates that the routine was interrupted because the\nmemory is insufficient for the internal work array.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1940\n\n\nierr=-105\nIndicates that the routine was interrupted because the\ninput matrix size n is less than or equal to 0.\nierr=-106\nIndicates that the routine was interrupted because the\ncolumn indices ja are not in the ascending order.\ndcsrilut\nILUT preconditioner based on the incomplete LU\nfactorization with a threshold of a sparse matrix.\nSyntax\nvoid dcsrilut (const MKL_INT *n, const double *a, const MKL_INT *ia, const MKL_INT *ja,\ndouble *bilut, MKL_INT *ibilut, MKL_INT *jbilut, const double *tol, const MKL_INT\n*maxfil, const MKL_INT *ipar, const double *dpar, MKL_INT *ierr);\nInclude Files\n•\nmkl.h\nDescription\nThe routine dcsrilut computes a preconditioner B [Saad03] of a given sparse matrix A stored in the format\naccepted in the direct sparse solvers:\nA~B=L*U , where L is a lower triangular matrix with unit diagonal and U is an upper triangular matrix with\nnon-unit diagonal.\nThe following threshold criteria are used to generate the incomplete factors L and U:\n1) the resulting entry must be greater than the matrix current row norm multiplied by the parameter tol,\nand\n2) the number of the non-zero elements in each row of the resulting L and U factors must not be greater\nthan the value of the parameter maxfil.\nCaution\nThis routine supports only one-based indexing of the array parameters.\nInput Parameters\nn\nSize (number of rows or columns) of the original square n-by-n matrix A.\na\nArray containing all non-zero elements of the matrix A. The length of the\narray is equal to their number. Refer to values array description in the \nSparse Matrix Storage Format section for more details.\nia\nArray of size (n+1) containing indices of non-zero elements in the array a.\nia[i] is the index of the first non-zero element from the row i. The value\nof the last element ia[n] is equal to the number of non-zeros in the matrix\nA, plus one. Refer to the rowIndex array description in the Sparse Matrix\nStorage Format for more details.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1941\n\n\nja\nArray of size equal to the size of the array a. This array contains the column\nnumbers for each non-zero element of the matrix A. It is important that the\nindices are in increasing order per row. Refer to the columns array\ndescription in the Sparse Matrix Storage Format for more details.\nCaution\nIf column indices are not stored in ascending order for each row\nof matrix, the result of the routine might not be correct.\ntol\nTolerance for threshold criterion for the resulting entries of the\npreconditioner.\nmaxfil\nMaximum fill-in, which is half of the preconditioner bandwidth. The number\nof non-zero elements in the rows of the preconditioner cannot exceed\n(2*maxfil+1).\nipar\nArray of size 128. This parameter is used to specify the integer set of data\nfor both the ILUT and RCI FGMRES computations. Refer to the ipar array\ndescription in the FGMRES Common Parameters for more details on\nFGMRES parameter entries. The entries specific to ILUT are listed below.\nipar[30]\nspecifies how the routine operates if the value of the\ncomputed diagonal element is less than the current\nmatrix row norm multiplied by the value of the\nparameter tol. If ipar[30] = 0, then the\ncalculation is stopped and the routine returns non-\nzero error value. Otherwise, the value of the\ndiagonal element is set to a value determined by\ndpar[30] (see its description below), and the\ncalculations continue.\nNOTE\nThere is no default value for ipar[30] even\nif the preconditioner is used within the RCI\nISS context. Always set the value of this\nentry.\nNOTE\nYou must declare the array ipar with length 128. While defining the\narray in the code as ipar[30]works, there is no guarantee of future\ncompatibility with Intel® oneAPI Math Kernel Library (oneMKL).\ndpar\nArray of size 128. This parameter specifies the double precision set of data\nfor both ILUT and RCI FGMRES computations. Refer to the dpar array\ndescription in the FGMRES Common Parameters for more details on\nFGMRES parameter entries. The entries that are specific to ILUT are listed\nbelow.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1942\n\n\ndpar[30]\nused to adjust the value of small diagonal elements.\nDiagonal elements with a value less than the current\nmatrix row norm multiplied by tol are replaced with\nthe value of dpar[30] multiplied by the matrix row\nnorm.\nNOTE\nThere is no default value for dpar[30] entry\neven if the preconditioner is used within RCI\nISS context. Always set the value of this\nentry.\nNOTE\nYou must declare the array dpar with length 128. While defining the\narray in the code as ipar[30]works, there is no guarantee of future\ncompatibility with Intel® oneAPI Math Kernel Library (oneMKL).\nOutput Parameters\nbilut\nArray containing non-zero elements of the resulting preconditioning matrix\nB, stored in the format accepted in the direct sparse solvers. Refer to the\nvalues array description in the Sparse Matrix Storage Format for more\ndetails. The size of the array is equal to (2*maxfil+1)*n-\nmaxfil*(maxfil+1)+1.\nNOTE\nProvide enough memory for this array before calling the routine.\nOtherwise, the routine may fail to complete successfully with a\ncorrect result.\nibilut\nArray of size (n+1) containing indices of non-zero elements in the array\nbilut. ibilut[i] is the index of the first non-zero element from the row i.\nThe value of the last element ibilut[n] is equal to the number of non-\nzeros in the matrix B, plus one. Refer to the rowIndex array description in\nthe Sparse Matrix Storage Format for more details.\njbilut\nArray, its size is equal to the size of the array bilut. This array contains\nthe column numbers for each non-zero element of the matrix B. Refer to\nthe columns array description in the Sparse Matrix Storage Format for\nmore details.\nierr\nError flag, gives information about the routine completion.\nReturn Values\nierr=0\nIndicates that the task completed normally.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1943\n\n\nierr=-101\nIndicates that the routine was interrupted because of an\nerror: the number of elements in some matrix row specified\nin the sparse format is equal to or less than 0.\nierr=-102\nIndicates that the routine was interrupted because the\nvalue of the computed diagonal element is less than the\nproduct of the given tolerance and the current matrix row\nnorm, and it cannot be replaced as ipar[30]=0.\nierr=-103\nIndicates that the routine was interrupted because the\nelement ia[i] is less than or equal to the element ia[i -\n1] (see Sparse Matrix Storage Format).\nierr=-104\nIndicates that the routine was interrupted because the\nmemory is insufficient for the internal work arrays.\nierr=-105\nIndicates that the routine was interrupted because the\ninput value of maxfil is less than 0.\nierr=-106\nIndicates that the routine was interrupted because the size\nn of the input matrix is less than 0.\nierr=-107\nIndicates that the routine was interrupted because an\nelement of the array ja is less than 1, or greater than n\n(see Sparse Matrix Storage Format).\nierr=101\nThe value of maxfil is greater than or equal to n. The\ncalculation is performed with the value of maxfil set to\n(n-1).\nierr=102\nThe value of tol is less than 0. The calculation is\nperformed with the value of the parameter set to (-tol)\nierr=103\nThe absolute value of tol is greater than value of\ndpar[30]; it can result in instability of the calculation.\nierr=104\nThe value of dpar[30] is equal to 0. It can cause\ncalculations to fail.\nSparse Matrix Checker Routines\nIntel® oneAPI Math Kernel Library (oneMKL) provides a sparse matrix checker so that you can find errors in\nthe storage of sparse matrices before calling Intel® oneAPI Math Kernel Library (oneMKL) PARDISO, DSS, or\nSparse BLAS routines.\nsparse_matrix_checker\nChecks the correctness of a sparse matrix.\nSyntax\nMKL_INT sparse_matrix_checker (sparse_struct* handle);\nInclude Files\n•\nmkl.h\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1944\n\n\nDescription\nThe sparse_matrix_checker routine checks a user-defined array used to store a sparse matrix in order to\ndetect issues which could cause problems in routines that require sparse input matrices, such as Intel®\noneAPI Math Kernel Library (oneMKL) PARDISO, DSS, or Sparse BLAS.\nInput Parameters\nhandle\nPointer to the data structure describing the sparse array to check.\nReturn Values\nThe routine returns a value error. Additionally, the check_result parameter returns information about\nwhere the error occurred, which can be used when message_level is MKL_NO_PRINT.\nSparse Matrix Checker Error Values\nerror value\nMeaning\nLocation\nMKL_SPARSE_CHECKER_SUC\nCESS\nThe input array successfully\npassed all checks.\nMKL_SPARSE_CHECKER_NON\n_MONOTONIC\nThe input array is not 0 or 1\nbased (, ia[0] is not 0 or\n1) or elements of ia are not\nin non-decreasing order as\nrequired.\nC:\nia[i] and ia[i + 1] are incompatible.\ncheck_result[0] = i\ncheck_result[1] = ia[i]\ncheck_result[2] = ia[i + 1]\nMKL_SPARSE_CHECKER_OUT\n_OF_RANGE\nThe value of the ja array is\nlower than the number of\nthe first column or greater\nthan the number of the last\ncolumn.\nC:\nia[i] and ia[i + 1] are incompatible.\ncheck_result[0] = i\ncheck_result[1] = ia[i]\ncheck_result[2] = ia[i + 1]\nMKL_SPARSE_CHECKER_NON\nTRIANGULAR\nThe matrix_structure\nparameter is\nMKL_UPPER_TRIANGULAR\nand both ia and ja are not\nupper triangular, or the\nmatrix_structure\nparameter is\nMKL_LOWER_TRIANGULAR\nand both ia and ja are not\nlower triangular\nC:\nia[i] and ja[j] are incompatible.\ncheck_result[0] = i\ncheck_result[1] = ia[i] = j\ncheck_result[2] = ja[j]\nMKL_SPARSE_CHECKER_NON\nORDERED\nThe elements of the ja\narray are not in non-\ndecreasing order in each\nrow as required.\nC:\nia[i] and ia[i + 1] are incompatible.\ncheck_result[0] = j\ncheck_result[1] = ja[j]\ncheck_result[2] = ja[j + 1]\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1945\n\n\nSee Also\nsparse_matrix_checker_init Initializes handle for sparse matrix checker.\nIntel® oneAPI Math Kernel Library (oneMKL) PARDISO - Parallel Direct Sparse Solver Interface\nSparse BLAS Level 2 and Level 3 Routines\nSparse Matrix Storage Formats\nsparse_matrix_checker_init\nInitializes handle for sparse matrix checker.\nSyntax\nvoid sparse_matrix_checker_init (sparse_struct* handle);\nInclude Files\n•\nmkl.h\nDescription\nThe sparse_matrix_checker_init routine initializes the handle for the sparse_matrix_checker routine.\nThe handle variable contains this data:\nDescription of sparse_matrix_checkerhandle Data\nField\nType\nPossible Values\nMeaning\nn\nMKL_INT\nOrder of the matrix\nstored in sparse array.\ncsr_ia\nMKL_INT*\nPointer to ia array for\nmatrix_format =\nMKL_CSR\ncsr_ja\nMKL_INT*\nPointer to ja array for\nmatrix_format =\nMKL_CSR\ncheck_result[3]\nMKL_INT\nSee Sparse Matrix\nChecker Error Values for\na description of the\nvalues returned in\ncheck_result.\nIndicates location of\nproblem in array when\nmessage_level =\nMKL_NO_PRINT.\nindexing\nsparse_matrix_index\ning\nMKL_ZERO_BASED\nMKL_ONE_BASED\nIndexing style used in\narray.\nmatrix_structure\nsparse_matrix_struc\ntures\nMKL_GENERAL_STRUCTU\nRE\nMKL_UPPER_TRIANGULA\nR\nMKL_LOWER_TRIANGULA\nR\nMKL_STRUCTURAL_SYMM\nETRIC\nType of sparse matrix\nstored in array.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1946\n\n\nField\nType\nPossible Values\nMeaning\nmatrix_format\nsparse_matrix_forma\nts\nMKL_CSR\nFormat of array used for\nsparse matrix storage.\nmessage_level\nsparse_matrix_messa\nge_levels\nMKL_NO_PRINT\nMKL_PRINT\nDetermines whether or\nnot feedback is provided\non the screen.\nprint_style\nsparse_matrix_print\n_styles\nMKL_C_STYLE\nMKL_FORTRAN_STYLE\nDetermines style of\nmessages when\nmessage_level =\nMKL_PRINT.\nInput Parameters\nhandle\nPointer to the data structure describing the sparse array to check.\nOutput Parameters\nhandle\nPointer to the initialized data structure.\nSee Also\nsparse_matrix_checker Checks the correctness of a sparse matrix.\nIntel® oneAPI Math Kernel Library (oneMKL) PARDISO - Parallel Direct Sparse Solver Interface\nSparse BLAS Level 2 and Level 3 Routines\nSparse Matrix Storage Formats\nExtended Eigensolver Routines\nNOTE\nIntel® oneAPI Math Kernel Library (oneMKL) only supports the shared memory programming (SMP)\nversion of the eigenvalue solver.\n•\nThe FEAST Algorithm gives a brief description of the algorithm underlying the Extended Eigensolver.\n•\nExtended Eigensolver Functionality describes the problems that can and cannot be solved with the\nExtended Eigensolver and how to get the best results from the routines.\n•\nExtended Eigensolver Interfaces gives a reference for calling Extended Eigensolver routines.\n•\nThe FEAST Algorithm\nThe Extended Eigensolver functionality is a set of high-performance numerical routines for solving symmetric\nstandard eigenvalue problems, Ax=λx, or generalized symmetric-definite eigenvalue problems, Ax=λBx. It\nyields all the eigenvalues (λ) and eigenvectors (x) within a given search interval [λ min , λ max]. It is based on\nthe FEAST algorithm, an innovative fast and stable numerical algorithm presented in [Polizzi09], which\nfundamentally differs from the traditional Krylov subspace iteration based techniques (Arnoldi and Lanczos\nalgorithms [Bai00]) or other Davidson-Jacobi techniques [Sleijpen96]. The FEAST algorithm is inspired by the\ndensity-matrix representation and contour integration techniques in quantum mechanics.\nThe FEAST numerical algorithm obtains eigenpair solutions using a numerically efficient contour integration\ntechnique. The main computational tasks in the FEAST algorithm consist of solving a few independent linear\nsystems along the contour and solving a reduced eigenvalue problem. Consider a circle centered in the\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1947\n\n\nmiddle of the search interval [λ min , λ max]. The numerical integration over the circle in the current version of\nFEAST is performed using Ne-point Gauss-Legendre quadrature with xe the e-th Gauss node associated with\nthe weight ωe. For example, for the case Ne = 8:\n( x1, ω1 ) = (0.183434642495649 , 0.362683783378361),\n( x2, ω2 ) = (-0.183434642495649 , 0.362683783378361),\n( x3, ω3 ) = (0.525532409916328 , 0.313706645877887),\n( x4, ω4 ) = (-0.525532409916328 , 0.313706645877887),\n( x5, ω5 ) = (0.796666477413626 , 0.222381034453374),\n( x6, ω6 ) = (-0.796666477413626 , 0.222381034453374),\n( x7, ω7 ) = (0.960289856497536 , 0.101228536290376), and\n( x8, ω8 ) = (-0.960289856497536 , 0.101228536290376).\nThe figure FEAST Pseudocode shows the basic pseudocode for the FEAST algorithm for the case of real\nsymmetric (left pane) and complex Hermitian (right pane) generalized eigenvalue problems, using N for the\nsize of the system and M for the number of eigenvalues in the search interval (see [Polizzi09]).\nNOTE\nThe pseudocode presents a simplified version of the actual algorithm. Refer to http://arxiv.org/abs/\n1302.0432 for an in-depth presentation and mathematical proof of convergence of FEAST.\nFEAST Pseudocode\nA: real symmetric\nB: symmetric positive definite (SPD)\nℜ{x}: real part of x\nA: complex Hermitian\nB: Hermitian positive definite (HPD)\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1948\n\n\nExtended Eigensolver Functionality\nThe eigenvalue problems covered are as follows:\n•\nstandard, Ax = λx\n•\nA complex Hermitian\n•\nA real symmetric\n•\ngeneralized, Ax = λBx\n•\nA complex Hermitian, B Hermitian positive definite (hpd)\n•\nA real symmetric and B real symmetric positive definite (spd)\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1949\n\n\nThe Extended Eigensolver functionality offers:\n•\nReal/Complex and Single/Double precisions: double precision is recommended to provide better accuracy\nof eigenpairs.\n•\nReverse communication interfaces (RCI) provide maximum flexibility for specific applications. RCI are\nindependent of matrix format and inner system solvers, so you must provide your own linear system\nsolvers (direct or iterative) and matrix-matrix multiply routines.\n•\nPredefined driver interfaces for dense, LAPACK banded, and sparse (CSR) formats are less flexible but are\noptimized and easy to use:\n•\nThe Extended Eigensolver interfaces for dense matrices are likely to be slower than the comparable\nLAPACK routines because the FEAST algorithm has a higher computational cost.\n•\nThe Extended Eigensolver interfaces for banded matrices support banded LAPACK-type storage.\n•\nThe Extended Eigensolver sparse interfaces support compressed sparse row format and use the Intel®\noneAPI Math Kernel Library (oneMKL) PARDISO solver.\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nParallelism in Extended Eigensolver Routines\nHow you achieve parallelism in Extended Eigensolver routines depends on which interface you use.\nParallelism (via shared memory programming) is not explicitly implemented in Extended Eigensolver routines\nwithin one node: the inner linear systems are currently solved one after another.\n•\nUsing the Extended Eigensolver RCI interfaces, you can achieve parallelism by providing a threaded inner\nsystem solver and a matrix-matrix multiplication routine. When using the RCI interfaces, you are\nresponsible for activating the threaded capabilities of your BLAS and LAPACK libraries most likely using\nthe shell variable OMP_NUM_THREADS.\n•\nUsing the predefined Extended Eigensolver interfaces, parallelism can be implicitly obtained within the\nshared memory version of BLAS, LAPACK or Intel® oneAPI Math Kernel Library (oneMKL) PARDISO. The\nshell variableMKL_NUM_THREADScan be used for automatically setting the number of OpenMP threads\n(cores) for BLAS, LAPACK, and Intel® oneAPI Math Kernel Library (oneMKL) PARDISO.\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nAchieving Performance With Extended Eigensolver Routines\nIn order to use the Extended Eigensolver Routines , you need to provide\n•\nthe search interval and the size of the subspace M0 (overestimation of the number of eigenvalues M within\na given search interval);\n•\nthe system matrix in dense, banded, or sparse CSR format if the Extended Eigensolver predefined\ninterfaces are used, or a high-performance complex direct or iterative system solver and matrix-vector\nmultiplication routine if RCI interfaces are used.\nIn return, you can expect\n•\nfast convergence with very high accuracy when seeking up to 1000 eigenpairs (in two to four iterations\nusing M0 = 1.5M, and Ne = 8 or at most using Ne = 16 contour points);\n•\nan extremely robust approach.\nThe performance of the basic FEAST algorithm depends on a trade-off between the choices of the number of\nGauss quadrature points Ne, the size of the subspace M0, and the number of outer refinement loops to reach\nthe desired accuracy. In practice you should use M0 > 1.5 M, Ne = 8, and at most two refinement loops.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1950\n\n\nFor better performance:\n•\nM0 should be much smaller than the size of the eigenvalue problem, so that the arithmetic complexity\ndepends mainly on the inner system solver (O(NM) for narrow-banded or sparse systems).\n•\nParallel scalability performance depends on the shared memory capabilities of the of the inner system\nsolver.\n•\nFor very large sparse and challenging systems, application users should make use of the Extended\nEigensolver RCI interfaces with customized highly-efficient iterative systems solvers and preconditioners.\n•\nFor the Extended Eigensolver interfaces for banded matrices, the parallel performance scalability is\nlimited.\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nExtended Eigensolver Interfaces for Eigenvalues within Interval\nExtended Eigensolver Naming Conventions\nThere are two different types of interfaces available in the Extended Eigensolver routines:\n1.\nThe reverse communication interfaces (RCI):\n?feast_<matrix type>_rci\nThese interfaces are matrix free format (the interfaces are independent of the matrix data formats).\nYou must provide matrix-vector multiply and direct/iterative linear system solvers for your own explicit\nor implicit data format.\n2.\nThe predefined interfaces:\n?feast_<matrix type><type of eigenvalue problem>\nare predefined drivers for ?feast reverse communication interface that act on commonly used matrix\ndata storage (dense, banded and compressed sparse row representation), using internal matrix-vector\nroutines and selected inner linear system solvers.\nFor these interfaces:\n•\n? indicates the data type of matrix A (and matrix B if any) defined as follows:\ns\nfloat\nd\ndouble\nc\nMKL_Complex8\nz\nMKL_Complex16\n•\n<matrix type> defined as follows:\nValue of <matrix type>\nMatrix format\nInner linear system solver used by\nExtended Eigensolver\nsy\n(symmetric real)\nDense\nLAPACK dense solvers\nhe\n(Hermitian\ncomplex)\nsb\n(symmetric\nbanded real)\nBanded-LAPACK\nInternal banded solver\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1951\n\n\nValue of <matrix type>\nMatrix format\nInner linear system solver used by\nExtended Eigensolver\nhb\n(Hermitian\nbanded complex)\nscsr\n(symmetric real)\nCompressed sparse row\nPARDISO solver\nhcsr\n(Hermitian\ncomplex)\ns\n(symmetric real)\nReverse\ncommunications\ninterfaces\nUser defined\nh\n(Hermitian\ncomplex)\n•\n<type of eigenvalue problem> is:\ngv\ngeneralized eigenvalue problem\nev\nstandard eigenvalue problem\nFor example, sfeast_scsrev is a single-precision routine with a symmetric real matrix stored in sparse\ncompressed-row format for a standard eigenvalue problem, and zfeast_hrci is a complex double-precision\nroutine with a Hermitian matrix using the reverse communication interface.\nNote that:\n•\n? can be s or d if a matrix is real symmetric: <matrix type> is sy, sb, or scsr.\n•\n? can be c or z if a matrix is complex Hermitian: <matrix type> is he, hb, or hcsr.\n•\n? can be c or z if the Extended Eigensolver RCI interface is used for solving a complex Hermitian problem.\n•\n? can be s or d if the Extended Eigensolver RCI interface is used for solving a real symmetric problem.\nfeastinit\nInitialize Extended Eigensolver input parameters with\ndefault values.\nSyntax\nfeastinit (MKL_INT* fpm);\nInclude Files\n•\nmkl.h\nDescription\nThis routine sets all Extended Eigensolver parameters to their default values.\nOutput Parameters\nfpm\nArray, size 128. This array is used to pass various parameters to Extended\nEigensolver routines. See Extended Eigensolver Input Parameters for a\ncomplete description of the parameters and their default values.\nExtended Eigensolver Input Parameters\nThe input parameters for Extended Eigensolver routines are contained in an MKL_INT array named fpm. To\ncall the Extended Eigensolver interfaces, this array should be initialized using the routine feastinit.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1952\n\n\nParamet\ner\nDefault\nDescription\nfpm[0]\n0\nSpecifies whether Extended Eigensolver routines print runtime status.\nfpm[0]=0\nExtended Eigensolver routines do not generate runtime\nmessages at all.\nfpm[0]=1\nExtended Eigensolver routines print runtime status to the\nscreen.\nfpm[1]\n8\nThe number of contour points Ne = 8 (see the description of FEAST algorithm).\nMust be one of {3,4,5,6,8,10,12,16,20,24,32,40,48}.\nfpm[2]\n12\nError trace double precision stopping criteria ε (ε = 10-fpm[2]) .\nfpm[3]\n20\nMaximum number of Extended Eigensolver refinement loops allowed. If no\nconvergence is reached within fpm[3] refinement loops, Extended Eigensolver\nroutines return info=2.\nfpm[4]\n0\nUser initial subspace. If fpm[4]=0 then Extended Eigensolver routines generate\ninitial subspace, if fpm[4]=1 the user supplied initial subspace is used.\nfpm[5]\n0\nExtended Eigensolver stopping test.\nfpm[5]=0\nExtended Eigensolvers are stopped if this residual stopping\ntest is satisfied:\n•\ngeneralized eigenvalue problem: \n•\nstandard eigenvalue problem: \nwhere mode is the total number of eigenvalues found in the\nsearch interval and ε = 10-fpm[6] for real and complex or ε\n= 10-fpm[2] for double precision and double complex.\nfpm[5]=1\nExtended Eigensolvers are stopped if this trace stopping test\nis satisfied:\n,\nwhere tracej denotes the sum of all eigenvalues found in the\nsearch interval [emin, emax] at the j-th Extended\nEigensolver iteration:\n.\nfpm[6]\n5\nError trace single precision stopping criteria (10-fpm[6]) .\nfpm[13]\n0\nfpm[13]=0\nStandard use for Extended Eigensolver routines.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1953\n\n\nParamet\ner\nDefault\nDescription\nfpm[13]=1\nNon-standard use for Extended Eigensolver routines: return\nthe computed eigenvectors subspace after one single\ncontour integration.\nfpm[26]\n0\nSpecifies whether Extended Eigensolver routines check input matrices (applies to\nCSR format only).\nfpm[26]=0\nExtended Eigensolver routines do not check input matrices.\nfpm[26]=1\nExtended Eigensolver routines check input matrices.\nfpm[27]\n0\nCheck if matrix B is positive definite. Set fpm[27] = 1 to check if B is positive\ndefinite.\nfpm[29]\nto\nfpm[62]\n-\nReserved for future use.\nfpm[63]\n0\nUse the Intel® oneAPI Math Kernel Library (oneMKL) PARDISO solver with the user-\ndefined PARDISOiparm array settings.\nNOTE\nThis option can only be used by Extended Eigensolver Predefined Interfaces for Sparse\nMatrices.\nfpm[63]=0\nExtended Eigensolver routines use the Intel® oneAPI Math\nKernel Library (oneMKL) PARDISO defaultiparm settings\ndefined by calling the pardisoinit subroutine.\nfpm[63]=1\nThe values from fpm[64] to fpm[127] correspond to\niparm[0] to iparm[63] respectively according to the\nformula fpm[64 + i]= iparm[i] for i = 0, 1, ..., 63.\nExtended Eigensolver Output Details\nErrors and warnings encountered during a run of the Extended Eigensolver routines are stored in an integer\nvariable, info. If the value of the output info parameter is not 0, either an error or warning was\nencountered. The possible return values for the info parameter along with the error code descriptions are\ngiven in the following table.\nReturn Codes for info Parameter\ninfo\nClassification\nDescription\n202\nError\nProblem with size of the system n (n≤0)\n201\nError\nProblem with size of initial subspace m0 (m0≤0 or m0>n)\n200\nError\nProblem with emin,emax (emin≥emax)\n(100+i)\nError\nProblem with i-th value of the input Extended Eigensolver\nparameter (fpm[i - 1]). Only the parameters in use are checked.\n4\nWarning\nSuccessful return of only the computed subspace after call with\nfpm[13] = 1\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1954\n\n\ninfo\nClassification\nDescription\n3\nWarning\nSize of the subspace m0 is too small (m0<m)\n2\nWarning\nNo Convergence (number of iteration loops >fpm[3])\n1\nWarning\nNo eigenvalue found in the search interval. See remark below for\nfurther details.\n0\nSuccessful exit\n-1\nError\nInternal error for allocation memory.\n-2\nError\nInternal error of the inner system solver. Possible reasons: not\nenough memory for inner linear system solver or inconsistent\ninput.\n-3\nError\nInternal error of the reduced eigenvalue solver\nPossible cause: matrix B may not be positive definite. It can be\nchecked by setting fpm[27] = 1 before calling an Extended\nEigensolver routine, or by using LAPACK routines.\n-4\nError\nMatrix B is not positive definite.\n-(100+i)\nError\nProblem with the i-th argument of the Extended Eigensolver\ninterface.\nIn some extreme cases the return value info=1 may indicate that the Extended Eigensolver routine has\nfailed to find the eigenvalues in the search interval. This situation could arise if a very large search interval is\nused to locate a small and isolated cluster of eigenvalues (i.e. the dimension of the search interval is many\norders of magnitude larger than the number of contour points. It is then either recommended to increase the\nnumber of contour points fpm[1] or simply rescale more appropriately the search interval. Rescaling means\nthe initial problem of finding all eigenvalues the search interval [λmin,λmax] for the standard eigenvalue\nproblem A x=λx is replaced with the problem of finding all eigenvalues in the search interval [λmin/ t, λmax/ t]\nfor the standard eigenvalue problem (A/t) x=(λ/t) x where t is a scaling factor.\nExtended Eigensolver RCI Routines\nIf you do not require specific linear system solvers or matrix storage schemes, you can skip this section and\ngo directly to Extended Eigensolver Predefined Interfaces.\nExtended Eigensolver RCI Interface Description\nThe Extended Eigensolver RCI interfaces can be used to solve standard or generalized eigenvalue problems,\nand are independent of the format of the matrices. As mentioned earlier, the Extended Eigensolver algorithm\nis based on the contour integration techniques of the matrix resolvent G(σ )= (σB - A)-1 over a circle. For\nsolving a generalized eigenvalue problem, Extended Eigensolver has to perform one or more of the following\noperations at each contour point denoted below by Ze :\n•\nFactorize the matrix (Ze *B - A)\n•\nSolve the linear system (Ze *B - A)X = Y or (Ze *B - A)HX = Y with multiple right hand sides, where H\nmeans transpose conjugate\n•\nMatrix-matrix multiply BX = Y or AX = Y\nFor solving a standard eigenvalue problem, replace the matrix B with the identity matrix I.\nThe primary aim of RCI interfaces is to isolate these operations: the linear system solver, factorization of the\nmatrix resolvent at each contour point, and matrix-matrix multiplication. This gives universality to RCI\ninterfaces as they are independent of data structures and the specific implementation of the operations like\nmatrix-vector multiplication or inner system solvers. However, this approach requires some additional effort\nwhen calling the interface. In particular, operations listed above are performed by routines that you supply on\ndata structures that you find most appropriate for the problem at hand.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1955\n\n\nTo initialize an Extended Eigensolver RCI routine, set the job indicator (ijob) parameter to the value -1.\nWhen the routine requires the results of an operation, it generates a special value of ijob to indicate the\noperation that needs to be performed. The routine also returns ze, the coordinate along the complex contour,\nthe values of array work or workc, and the number of columns to be used. Your subroutine then must\nperform the operation at the given contour point ze, store the results in prescribed array, and return control\nto the Extended Eigensolver RCI routine.\nThe following pseudocode shows the general scheme for using the Extended Eigensolver RCI functionality for\na real symmetric problem:\n    ijob=-1; // initialization\n    do while (ijob!=0) {\n       ?feast_srci(&ijob, &N, &Ze, work, workc, Aq,  Bq,\n                   fpm, &epsout, &loop, &Emin, &Emax, &M0, E, lambda, &q, res, &info);\n       switch(ijob) {\n       case 10:  // Factorize the complex matrix (ZeB-A)\n                 break;\n     \n       case 11: // Solve the complex linear system (ZeB-A)x=workc\n                // Put result in workc\n                 break;\n     \n       case 30: // Perform multiplication A by Qi..Qj columns of QNxM0\n                // where i = fpm[23] and j = fpm[23]+fpm[24]−1\n                // Qi..Qj located in q starting from q+N*(i-1)\n                 break;\n     \n       case 40: // Perform multiplication B by Qi..Qj columns of QNxM0\n                // where i = fpm[23] and j = fpm[23]+fpm[24]−1\n                // Qi..Qj located in q starting from q+N*(i-1)\n                // Result is stored in work+N*(i-1)\n                 break;\n       }\n     }\nNOTE\nThe ? option in ?feast in the pseudocode given above should be replaced by either s or d, depending\non the matrix data type of the eigenvalue system.\nThe next pseudocode shows the general scheme for using the Extended Eigensolver RCI functionality for a\ncomplex Hermitian problem:\n    ijob=-1; // initialization\n    while (ijob!=0) {\n       ?feast_hrci(&ijob, &N, &Ze, work, workc, Aq,  Bq,\n                   fpm, &epsout, &loop, &Emin, &Emax, &M0, E, lambda, &q, res, &info);\n       switch (ijob) {\n       case 10: // Factorize the complex matrix (ZeB-A)\n       break;\n       case 11: // Solve the linear system (ZeB−A)y=workc\n                // Put result in workc\n       break;\n       case 20: // Factorize (if needed by case 21) the complex matrix     (ZeB−A)ˆH\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1956\n\n\n                   // ATTENTION: This option requires additional memory storage\n                   // (i.e . the resulting matrix from case 10 cannot be overwritten)\n       break;\n       case 21: // Solve the linear system (ZeB−A)ˆHy=workc\n                // Put result in  workc\n                   // REMARK: case 20 becomes obsolete if this solve can be performed\n                   // using the factorization in case 10\n       break;\n       case 30: // Multiply A by Qi..Qj columns of QNxM0,\n                // where i = fpm[23] and j = fpm[23]+fpm[24]−1\n                // Qi..Qj located in q starting from q+N*(i-1)\n                // Result is stored in work+N*(i-1) \n       break;\n       case 40: // Perform multiplication B by Qi..Qj columns of QNxM0\n                // where i = fpm[23] and j = fpm[23]+fpm[24]−1\n                // Qi..Qj located in q starting from q+N*(i-1)\n                // Result is stored in work+N*(i-1)\n       break;\n       }\n    }\nend do\nNOTE\nThe ? option in ?feast in the pseudocode given above should be replaced by either c or z, depending\non the matrix data type of the eigenvalue system.\nIf case 20 can be avoided, performance could be up to twice as fast, and Extended Eigensolver\nfunctionality would use half of the memory.\nIf an iterative solver is used along with a preconditioner, the factorization of the preconditioner could be\nperformed with ijob = 10 (and ijob = 20 if applicable) for a given value of Ze, and the associated iterative\nsolve would then be performed with ijob = 11 (and ijob = 21 if applicable).\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\n?feast_srci/?feast_hrci\nExtended Eigensolver RCI interface.\nSyntax\nvoid sfeast_srci (MKL_INT* ijob, const MKL_INT* n, MKL_Complex8* ze, float* work,\nMKL_Complex8* workc, float* aq, float* sq, MKL_INT* fpm, float* epsout, MKL_INT* loop,\nconst float* emin, const float* emax, MKL_INT* m0, float* lambda, float* q, MKL_INT* m,\nfloat* res, MKL_INT* info);\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1957\n\n\nvoid dfeast_srci (MKL_INT* ijob, const MKL_INT* n, MKL_Complex16* ze, double* work,\nMKL_Complex16* workc, double* aq, double* sq, MKL_INT* fpm, double* epsout, MKL_INT*\nloop, const double* emin, const double* emax, MKL_INT* m0, double* lambda, double* q,\nMKL_INT* m, double* res, MKL_INT* info);\nvoid cfeast_hrci (MKL_INT* ijob, const MKL_INT* n, MKL_Complex8* ze, MKL_Complex8*\nwork, MKL_Complex8* workc, MKL_Complex8* aq, MKL_Complex8* sq, MKL_INT* fpm, float*\nepsout, MKL_INT* loop, const float* emin, const float* emax, MKL_INT* m0, float* lambda,\nMKL_Complex8* q, MKL_INT* m, float* res, MKL_INT* info);\nvoid zfeast_hrci (MKL_INT* ijob, const MKL_INT* n, MKL_Complex16* ze, MKL_Complex16*\nwork, MKL_Complex16* workc, MKL_Complex16* aq, MKL_Complex16* sq, MKL_INT* fpm, double*\nepsout, MKL_INT* loop, const double* emin, const double* emax, MKL_INT* m0, double*\nlambda, MKL_Complex16* q, MKL_INT* m, double* res, MKL_INT* info);\nInclude Files\n•\nmkl.h\nDescription\nCompute eigenvalues as described in Extended Eigensolver RCI Interface Description.\nInput Parameters\nijob\nJob indicator variable. On entry, a call to ?feast_srci/?feast_hrci with\nijob=-1 initializes the eigensolver.\nn\nSets the size of the problem. n > 0.\nwork\nWorkspace array of size n by m0.\nworkc\nWorkspace array of size n by m0.\naq, sq\nWorkspace arrays of size m0 by m0.\nfpm\nArray, size of 128. This array is used to pass various parameters to\nExtended Eigensolver routines. See Extended Eigensolver Input Parameters\nfor a complete description of the parameters and their default values.\nemin, emax\nThe lower and upper bounds of the interval to be searched for eigenvalues;\nemin ≤ emax.\nNOTE Users are advised to avoid situations in which\neigenvalues nearly coincide with the interval endpoints. This\nmay lead to unpredictable selection or omission of such\neigenvalues. Users should instead specify a slightly larger\ninterval than needed and, if required, pick valid eigenvalues and\ntheir corresponding eigenvectors for subsequent use.\nm0\nOn entry, specifies the initial guess for subspace size to be used, 0 < m0≤n.\nSet m0 ≥ m where m is the total number of eigenvalues located in the\ninterval [emin, emax]. If the initial guess is wrong, Extended Eigensolver\nroutines return info=3.\nq\nOn entry, if fpm[4]=1, the array q of size n by m contains a basis of guess\nsubspace where n is the order of the input matrix.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1958\n\n\nOutput Parameters\nijob\nOn exit, the parameter carries the status flag that indicates the condition of\nthe return. The status information is divided into three categories:\n1.\nA zero value indicates successful completion of the task.\n2.\nA positive value indicates that the solver requires a matrix-vector\nmultiplication or solving a specific system with a complex coefficient.\n3.\nA negative value indicates successful initiation.\nA non-zero value of ijob specifically means the following:\n•\nijob = 10 - factorize the complex matrix Ze*B - A at a given contour\npoint Ze and return the control to the ?feast_srci/?feast_hrci\nroutine where Ze is a complex number meaning contour point and its\nvalue is defined internally in ?feast_srci/?feast_hrci.\n•\nijob =11 - solve the complex linear system (Ze*B - A)*y = workc, put\nthe solution in workc and return the control to\nthe ?feast_srci/?feast_hrci routine.\n•\nijob =20 - factorize the complex matrix (Ze*B - A)H at a given contour\npoint Ze and return the control to the ?feast_srci/?feast_hrci\nroutine where Ze is a complex number meaning contour point and its\nvalue is defined internally in ?feast_srci/?feast_hrci.\nThe symbol XH means transpose conjugate of matrix X.\n•\nijob = 21 - solve the complex linear system(Ze*B - A)H*y = workc, put\nthe solution in workc and return the control to\nthe ?feast_srci/?feast_hrci routine. The case ijob=20 becomes\nobsolete if the solve can be performed using the factorization computed\nfor ijob=10.\nThe symbol XH mean transpose conjugate of matrix X.\n•\nijob = 30 - multiply matrix A by Qj..Qi, put the result in work + N*(i -\n1), and return the control to the ?feast_srci/?feast_hrci routine.\ni is fpm[24], and j is fpm[23] + fpm[24] - 1.\n•\nijob = 40 - multiply matrix B by Qj..Qi, put the result in work + N*(i -\n1) and return the control to the ?feast_srci/?feast_hrci routine. If a\nstandard eigenvalue problem is solved, just return work = q.\ni is fpm[24], and j is fpm[23] + fpm[24] - 1.\n•\nijob = -2 - rerun the ?feast_srci/?feast_hrci task with the same\nparameters.\nze\nDefines the coordinate along the complex contour. All values of ze are\ngenerated by ?feast_srci/?feast_hrci internally.\nfpm\nOn output, contains coordinates of columns of work array needed for\niterative refinement. (See Extended Eigensolver RCI Interface Description.)\nepsout\nOn output, contains the relative error on the trace: |tracei - tracei-1| /max(|\nemin|, |emax|)\nloop\nOn output, contains the number of refinement loop executed. Ignored on\ninput.\nlambda\nArray of length m0. On output, the first m entries of lambda are eigenvalues\nfound in the interval.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1959\n\n\nq\nOn output, q contains all eigenvectors corresponding to lambda.\nm\nThe total number of eigenvalues found in the interval [emin, emax]: 0 ≤ m\n≤ m0.\nres\nArray of length m0. On exit, the first m components contain the relative\nresidual vector:\n•\ngeneralized eigenvalue problem:\n•\nstandard eigenvalue problem:\nfor i=0, 1, …, m - 1, and where m is the total number of eigenvalues found in\nthe search interval.\ninfo\nIf info=0, the execution is successful. If info ≠ 0, see Output Eigensolver\ninfo Details.\nExtended Eigensolver Predefined Interfaces\nThe predefined interfaces include routines for standard and generalized eigenvalue problems, and for dense,\nbanded, and sparse matrices.\nMatrix Type\nStandard Eigenvalue Problem\nGeneralized Eigenvalue\nProblem\nDense\n?feast_syev\n?feast_heev\n?feast_sygv\n?feast_hegv\nBanded\n?feast_sbev\n?feast_hbev\n?feast_sbgv\n?feast_hbgv\nSparse\n?feast_scsrev\n?feast_hcsrev\n?feast_scsrgv\n?feast_hcsrgv\nMatrix Storage\nThe symmetric and Hermitian matrices used in Extended Eigensolvers predefined interfaces can be stored in\nfull, band, and sparse formats.\n•\nIn the full storage format (described in Full Storage in additional detail) you store all elements, all of the\nelements in the upper triangle of the matrix, or all of the elements in the lower triangle of the matrix.\n•\nIn the band storage format (described in Band storage in additional detail), you store only the elements\nalong a diagonal band of the matrix.\n•\nIn the sparse format (described in Storage Arrays for a Matrix in CSR Format (3-Array Variation)), you\nstore only the non-zero elements of the matrix.\nIn generalized eigenvalue systems you must use the same family of storage format for both matrices A and\nB. The bandwidth can be different for the banded format (klb can be different from kla), and the position of\nthe non-zero elements can also be different for the sparse format (CSR coordinates ib and jb can be\ndifferent from ia and ja).\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1960\n\n\n?feast_syev/?feast_heev\nExtended Eigensolver interface for standard\neigenvalue problem with dense matrices.\nSyntax\nvoid sfeast_syev (const char * uplo, const MKL_INT * n, const float * a, const MKL_INT\n* lda, MKL_INT * fpm, float * epsout, MKL_INT * loop, const float * emin, const float *\nemax, MKL_INT * m0, float * e, float * x, MKL_INT * m, float * res, MKL_INT * info);\nvoid dfeast_syev (const char * uplo, const MKL_INT * n, const double * a, const MKL_INT\n* lda, MKL_INT * fpm, double * epsout, MKL_INT * loop, const double * emin, const\ndouble * emax, MKL_INT * m0, double * e, double * x, MKL_INT * m, double * res, MKL_INT\n* info);\nvoid cfeast_heev (const char * uplo, const MKL_INT * n, const MKL_Complex8 * a, const\nMKL_INT * lda, MKL_INT * fpm, float * epsout, MKL_INT * loop, const float * emin, const\nfloat * emax, MKL_INT * m0, float * e, MKL_Complex8 * x, MKL_INT * m, float * res,\nMKL_INT * info);\nvoid zfeast_heev (const char * uplo, const MKL_INT * n, const MKL_Complex16 * a, const\nMKL_INT * lda, MKL_INT * fpm, double * epsout, MKL_INT * loop, const double * emin,\nconst double * emax, MKL_INT * m0, double * e, MKL_Complex16 * x, MKL_INT * m, double *\nres, MKL_INT * info);\nInclude Files\n•\nmkl.h\nDescription\nThe routines compute all the eigenvalues and eigenvectors for standard eigenvalue problems, Ax = λx, within\na given search interval.\nInput Parameters\nuplo\nMust be 'U' or 'L' or 'F' .\nIf uplo = 'U', a stores the upper triangular parts of A.\nIf uplo = 'L', a stores the lower triangular parts of A.\nIf uplo= 'F' , a stores the full matrix A.\nn\nSets the size of the problem. n > 0.\na\nArray of dimension lda by n, contains either full matrix A or upper or lower\ntriangular part of the matrix A, as specified by uplo\nlda\nThe leading dimension of the array a. Must be at least max(1, n).\nfpm\nArray, dimension of 128. This array is used to pass various parameters to\nExtended Eigensolver routines. See Extended Eigensolver Input Parameters\nfor a complete description of the parameters and their default values.\nemin, emax\nThe lower and upper bounds of the interval to be searched for eigenvalues;\nemin ≤ emax.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1961\n\n\nNOTE Users are advised to avoid situations in which\neigenvalues nearly coincide with the interval endpoints. This\nmay lead to unpredictable selection or omission of such\neigenvalues. Users should instead specify a slightly larger\ninterval than needed and, if required, pick valid eigenvalues and\ntheir corresponding eigenvectors for subsequent use.\nm0\nOn entry, specifies the initial guess for subspace dimension to be used, 0 <\nm0≤n. Set m0 ≥ m where m is the total number of eigenvalues located in the\ninterval [emin, emax]. If the initial guess is wrong, Extended Eigensolver\nroutines return info=3.\nx\nOn entry, if fpm[4]=1, the array x of size n by m contains a basis of guess\nsubspace where n is the order of the input matrix.\nOutput Parameters\nepsout\nOn output, contains the relative error on the trace: |tracei - tracei-1| /max(|\nemin|, |emax|)\nloop\nOn output, contains the number of refinement loop executed. Ignored on\ninput.\ne\nArray of length m0. On output, the first m entries of e are eigenvalues found\nin the interval.\nx\nOn output, the first m columns of x contain the orthonormal eigenvectors\ncorresponding to the computed eigenvalues e, with the i-th column of x\nholding the eigenvector associated with e[i].\nm\nThe total number of eigenvalues found in the interval [emin, emax]: 0 ≤ m\n≤ m0.\nres\nArray of length m0. On exit, the first m components contain the relative\nresidual vector:\nfor i=1, 2, …, m, and where m is the total number of eigenvalues found in\nthe search interval.\ninfo\nIf info=0, the execution is successful. If info ≠ 0, see Output Eigensolver\ninfo Details.\n?feast_sygv/?feast_hegv\nExtended Eigensolver interface for generalized\neigenvalue problem with dense matrices.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1962\n\n\nSyntax\nvoid sfeast_sygv (const char * uplo, const MKL_INT * n, const float * a, const MKL_INT\n* lda, const float * b, const MKL_INT * ldb, MKL_INT * fpm, float * epsout, MKL_INT *\nloop, const float * emin, const float * emax, MKL_INT * m0, float * e, float * x,\nMKL_INT * m, float * res, MKL_INT * info);\nvoid dfeast_sygv (const char * uplo, const MKL_INT * n, const double * a, const MKL_INT\n* lda, const double * b, const MKL_INT * ldb, MKL_INT * fpm, double * epsout, MKL_INT *\nloop, const double * emin, const double * emax, MKL_INT * m0, double * e, double * x,\nMKL_INT * m, double * res, MKL_INT * info);\nvoid cfeast_hegv (const char * uplo, const MKL_INT * n, const MKL_Complex8 * a, const\nMKL_INT * lda, const MKL_Complex8 * b, const MKL_INT * ldb, MKL_INT * fpm, float *\nepsout, MKL_INT * loop, const float * emin, const float * emax, MKL_INT * m0, float * e,\nMKL_Complex8 * x, MKL_INT * m, float * res, MKL_INT * info);\nvoid zfeast_hegv (const char * uplo, const MKL_INT * n, const MKL_Complex16 * a, const\nMKL_INT * lda, const MKL_Complex16 * b, const MKL_INT * ldb, MKL_INT * fpm, double *\nepsout, MKL_INT * loop, const double * emin, const double * emax, MKL_INT * m0, double\n* e, MKL_Complex16 * x, MKL_INT * m, double * res, MKL_INT * info);\nInclude Files\n•\nmkl.h\nDescription\nThe routines compute all the eigenvalues and eigenvectors for generalized eigenvalue problems, Ax = λBx,\nwithin a given search interval.\nInput Parameters\nuplo\nMust be 'U' or 'L' or 'F' .\nIf UPLO = 'U', a and b store the upper triangular parts of A and B\nrespectively.\nIf UPLO = 'L', a and b store the lower triangular parts of A and B\nrespectively.\nIf UPLO= 'F', a and b store the full matrices A and B respectively.\nn\nSets the size of the problem. n > 0.\na\nArray of dimension lda by n, contains either full matrix A or upper or lower\ntriangular part of the matrix A, as specified by uplo\nlda\nThe leading dimension of the array a. Must be at least max(1, n).\nb\nArray of dimension ldb by n, contains either full matrix B or upper or lower\ntriangular part of the matrix B, as specified by uplo\nldb\nThe leading dimension of the array B. Must be at least max(1, n).\nfpm\nArray, dimension of 128. This array is used to pass various parameters to\nExtended Eigensolver routines. See Extended Eigensolver Input Parameters\nfor a complete description of the parameters and their default values.\nemin, emax\nThe lower and upper bounds of the interval to be searched for eigenvalues;\nemin ≤ emax.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1963\n\n\nNOTE Users are advised to avoid situations in which\neigenvalues nearly coincide with the interval endpoints. This\nmay lead to unpredictable selection or omission of such\neigenvalues. Users should instead specify a slightly larger\ninterval than needed and, if required, pick valid eigenvalues and\ntheir corresponding eigenvectors for subsequent use.\nm0\nOn entry, specifies the initial guess for subspace dimension to be used, 0 <\nm0≤n. Set m0 ≥ m where m is the total number of eigenvalues located in the\ninterval [emin, emax]. If the initial guess is wrong, Extended Eigensolver\nroutines return info=3.\nx\nOn entry, if fpm[4]=1, the array x of size n by m contains a basis of guess\nsubspace where n is the order of the input matrix.\nOutput Parameters\nepsout\nOn output, contains the relative error on the trace: |tracei - tracei-1| /max(|\nemin|, |emax|)\nloop\nOn output, contains the number of refinement loop executed. Ignored on\ninput.\ne\nArray of length m0. On output, the first m entries of e are eigenvalues found\nin the interval.\nx\nOn output, the first m columns of x contain the orthonormal eigenvectors\ncorresponding to the computed eigenvalues e, with the i-th column of x\nholding the eigenvector associated with e[i].\nm\nThe total number of eigenvalues found in the interval [emin, emax]: 0 ≤ m\n≤ m0.\nres\nArray of length m0. On exit, the first m components contain the relative\nresidual vector:\nfor i=1, 2, …, m, and where m is the total number of eigenvalues found in\nthe search interval.\ninfo\nIf info=0, the execution is successful. If info ≠ 0, see Output Eigensolver\ninfo Details.\n?feast_sbev/?feast_hbev\nExtended Eigensolver interface for standard\neigenvalue problem with banded matrices.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1964\n\n\nSyntax\nvoid sfeast_sbev (const char * uplo, const MKL_INT * n, const MKL_INT * kla, const\nfloat * a, const MKL_INT * lda, MKL_INT * fpm, float * epsout, MKL_INT * loop, const\nfloat * emin, const float * emax, MKL_INT * m0, float * e, float * x, MKL_INT * m, float\n* res, MKL_INT * info);\nvoid dfeast_sbev (const char * uplo, const MKL_INT * n, const MKL_INT * kla, const\ndouble * a, const MKL_INT * lda, MKL_INT * fpm, double * epsout, MKL_INT * loop, const\ndouble * emin, const double * emax, MKL_INT * m0, double * e, double * x, MKL_INT * m,\ndouble * res, MKL_INT * info);\nvoid cfeast_hbev (const char * uplo, const MKL_INT * n, const MKL_INT * kla, const\nMKL_Complex8 * a, const MKL_INT * lda, MKL_INT * fpm, float * epsout, MKL_INT * loop,\nconst float * emin, const float * emax, MKL_INT * m0, float * e, MKL_Complex8 * x,\nMKL_INT * m, float * res, MKL_INT * info);\nvoid zfeast_hbev (const char * uplo, const MKL_INT * n, const MKL_INT * kla, const\nMKL_Complex16 * a, const MKL_INT * lda, MKL_INT * fpm, double * epsout, MKL_INT * loop,\nconst double * emin, const double * emax, MKL_INT * m0, double * e, MKL_Complex16 * x,\nMKL_INT * m, double * res, MKL_INT * info);\nInclude Files\n•\nmkl.h\nDescription\nThe routines compute all the eigenvalues and eigenvectors for standard eigenvalue problems, Ax = λx, within\na given search interval.\nInput Parameters\nuplo\nMust be 'U' or 'L' or 'F' .\nIf uplo = 'U', a stores the upper triangular parts of A.\nIf uplo = 'L', a stores the lower triangular parts of A.\nIf uplo= 'F' , a stores the full matrix A.\nn\nSets the size of the problem. n > 0.\nkla\nThe number of super- or sub-diagonals within the band in A (kla≥ 0).\na\nArray of dimension lda by n, contains either full matrix A or upper or lower\ntriangular part of the matrix A, as specified by uplo\nlda\nThe leading dimension of the array a. Must be at least max(1, n).\nfpm\nArray, dimension of 128. This array is used to pass various parameters to\nExtended Eigensolver routines. See Extended Eigensolver Input Parameters\nfor a complete description of the parameters and their default values.\nemin, emax\nThe lower and upper bounds of the interval to be searched for eigenvalues;\nemin ≤ emax.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1965\n\n\nNOTE Users are advised to avoid situations in which\neigenvalues nearly coincide with the interval endpoints. This\nmay lead to unpredictable selection or omission of such\neigenvalues. Users should instead specify a slightly larger\ninterval than needed and, if required, pick valid eigenvalues and\ntheir corresponding eigenvectors for subsequent use.\nm0\nOn entry, specifies the initial guess for subspace dimension to be used, 0 <\nm0≤n. Set m0 ≥ m where m is the total number of eigenvalues located in the\ninterval [emin, emax]. If the initial guess is wrong, Extended Eigensolver\nroutines return info=3.\nx\nOn entry, if fpm[4]=1, the array x of size n by m contains a basis of guess\nsubspace where n is the order of the input matrix.\nOutput Parameters\nepsout\nOn output, contains the relative error on the trace: |tracei - tracei-1| /max(|\nemin|, |emax|)\nloop\nOn output, contains the number of refinement loop executed. Ignored on\ninput.\ne\nArray of length m0. On output, the first m entries of e are eigenvalues found\nin the interval.\nx\nOn output, the first m columns of x contain the orthonormal eigenvectors\ncorresponding to the computed eigenvalues e, with the i-th column of x\nholding the eigenvector associated with e[i].\nm\nThe total number of eigenvalues found in the interval [emin, emax]: 0 ≤ m\n≤ m0.\nres\nArray of length m0. On exit, the first m components contain the relative\nresidual vector:\nfor i=1, 2, …, m, and where m is the total number of eigenvalues found in\nthe search interval.\ninfo\nIf info=0, the execution is successful. If info ≠ 0, see Output Eigensolver\ninfo Details.\n?feast_sbgv/?feast_hbgv\nExtended Eigensolver interface for generalized\neigenvalue problem with banded matrices.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1966\n\n\nSyntax\nvoid sfeast_sbgv (const char * uplo, const MKL_INT * n, const MKL_INT * kla, const\nfloat * a, const MKL_INT * lda, const MKL_INT * klb, const float * b, const MKL_INT *\nldb, MKL_INT * fpm, float * epsout, MKL_INT * loop, const float * emin, const float *\nemax, MKL_INT * m0, float * e, float * x, MKL_INT * m, float * res, MKL_INT * info);\nvoid dfeast_sbgv (const char * uplo, const MKL_INT * n, const MKL_INT * kla, const\ndouble * a, const MKL_INT * lda, const MKL_INT * klb, const double * b, const MKL_INT *\nldb, MKL_INT * fpm, double * epsout, MKL_INT * loop, const double * emin, const double\n* emax, MKL_INT * m0, double * e, double * x, MKL_INT * m, double * res, MKL_INT *\ninfo);\nvoid cfeast_hbgv (const char * uplo, const MKL_INT * n, const MKL_INT * kla, const\nMKL_Complex8 * a, const MKL_INT * lda, const MKL_INT * klb, const MKL_Complex8 * b,\nconst MKL_INT * ldb, MKL_INT * fpm, float * epsout, MKL_INT * loop, const float * emin,\nconst float * emax, MKL_INT * m0, float * e, MKL_Complex8 * x, MKL_INT * m, float * res,\nMKL_INT * info);\nvoid zfeast_hbgv (const char * uplo, const MKL_INT * n, const MKL_INT * kla, const\nMKL_Complex16 * a, const MKL_INT * lda, const MKL_INT * klb, const MKL_Complex16 * b,\nconst MKL_INT * ldb, MKL_INT * fpm, double * epsout, MKL_INT * loop, const double *\nemin, const double * emax, MKL_INT * m0, double * e, MKL_Complex16 * x, MKL_INT * m,\ndouble * res, MKL_INT * info);\nInclude Files\n•\nmkl.h\nDescription\nThe routines compute all the eigenvalues and eigenvectors for generalized eigenvalue problems, Ax = λBx,\nwithin a given search interval.\nNOTE\nBoth matrices A and B must use the same family of storage format. The bandwidth,\nhowever, can be different (klb can be different from kla).\nInput Parameters\nuplo\nMust be 'U' or 'L' or 'F' .\nIf UPLO = 'U', a and b store the upper triangular parts of A and B\nrespectively.\nIf UPLO = 'L', a and b store the lower triangular parts of A and B\nrespectively.\nIf UPLO= 'F', a and b store the full matrices A and B respectively.\nn\nSets the size of the problem. n > 0.\nkla\nThe number of super- or sub-diagonals within the band in A (kla≥ 0).\na\nArray of dimension lda by n, contains either full matrix A or upper or lower\ntriangular part of the matrix A, as specified by uplo\nlda\nThe leading dimension of the array a. Must be at least max(1, n).\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1967\n\n\nklb\nThe number of super- or sub-diagonals within the band in B (klb≥ 0).\nb\nArray of dimension ldb by n, contains either full matrix B or upper or lower\ntriangular part of the matrix B, as specified by uplo\nldb\nThe leading dimension of the array B. Must be at least max(1, n).\nfpm\nArray, dimension of 128. This array is used to pass various parameters to\nExtended Eigensolver routines. See Extended Eigensolver Input Parameters\nfor a complete description of the parameters and their default values.\nemin, emax\nThe lower and upper bounds of the interval to be searched for eigenvalues;\nemin ≤ emax.\nNOTE Users are advised to avoid situations in which\neigenvalues nearly coincide with the interval endpoints. This\nmay lead to unpredictable selection or omission of such\neigenvalues. Users should instead specify a slightly larger\ninterval than needed and, if required, pick valid eigenvalues and\ntheir corresponding eigenvectors for subsequent use.\nm0\nOn entry, specifies the initial guess for subspace dimension to be used, 0 <\nm0≤n. Set m0 ≥ m where m is the total number of eigenvalues located in the\ninterval [emin, emax]. If the initial guess is wrong, Extended Eigensolver\nroutines return info=3.\nx\nOn entry, if fpm[4]=1, the array x of size n by m contains a basis of guess\nsubspace where n is the order of the input matrix.\nOutput Parameters\nepsout\nOn output, contains the relative error on the trace: |tracei - tracei-1| /max(|\nemin|, |emax|)\nloop\nOn output, contains the number of refinement loop executed. Ignored on\ninput.\ne\nArray of length m0. On output, the first m entries of e are eigenvalues found\nin the interval.\nx\nOn output, the first m columns of x contain the orthonormal eigenvectors\ncorresponding to the computed eigenvalues e, with the i-th column of x\nholding the eigenvector associated with e[i].\nm\nThe total number of eigenvalues found in the interval [emin, emax]: 0 ≤ m\n≤ m0.\nres\nArray of length m0. On exit, the first m components contain the relative\nresidual vector:\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1968\n\n\nfor i=1, 2, …, m, and where m is the total number of eigenvalues found in\nthe search interval.\ninfo\nIf info=0, the execution is successful. If info ≠ 0, see Output Eigensolver\ninfo Details.\n?feast_scsrev/?feast_hcsrev\nExtended Eigensolver interface for standard\neigenvalue problem with sparse matrices.\nSyntax\nvoid sfeast_scsrev (const char * uplo, const MKL_INT * n, const float * a, const\nMKL_INT * ia, const MKL_INT * ja, MKL_INT * fpm, float * epsout, MKL_INT * loop, const\nfloat * emin, const float * emax, MKL_INT * m0, float * e, float * x, MKL_INT * m, float\n* res, MKL_INT * info);\nvoid dfeast_scsrev (const char * uplo, const MKL_INT * n, const double * a, const\nMKL_INT * ia, const MKL_INT * ja, MKL_INT * fpm, double * epsout, MKL_INT * loop, const\ndouble * emin, const double * emax, MKL_INT * m0, double * e, double * x, MKL_INT * m,\ndouble * res, MKL_INT * info);\nvoid cfeast_hcsrev (const char * uplo, const MKL_INT * n, const MKL_Complex8 * a, const\nMKL_INT * ia, const MKL_INT * ja, MKL_INT * fpm, float * epsout, MKL_INT * loop, const\nfloat * emin, const float * emax, MKL_INT * m0, float * e, MKL_Complex8 * x, MKL_INT *\nm, float * res, MKL_INT * info);\nvoid zfeast_hcsrev (const char * uplo, const MKL_INT * n, const MKL_Complex16 * a,\nconst MKL_INT * ia, const MKL_INT * ja, MKL_INT * fpm, double * epsout, MKL_INT * loop,\nconst double * emin, const double * emax, MKL_INT * m0, double * e, MKL_Complex16 * x,\nMKL_INT * m, double * res, MKL_INT * info);\nInclude Files\n•\nmkl.h\nDescription\nThe routines compute all the eigenvalues and eigenvectors for standard eigenvalue problems, Ax = λx, within\na given search interval.\nInput Parameters\nuplo\nMust be 'U' or 'L' or 'F' .\nIf uplo = 'U', a stores the upper triangular parts of A.\nIf uplo = 'L', a stores the lower triangular parts of A.\nIf uplo= 'F' , a stores the full matrix A.\nn\nSets the size of the problem. n > 0.\na\nArray containing the nonzero elements of either the full matrix A or the\nupper or lower triangular part of the matrix A, as specified by uplo.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1969\n\n\nia\nArray of length n + 1, containing indices of elements in the array a , such\nthat ia[i] is the index in the array a of the first non-zero element from the\nrow i . The value of the last element ia[n] is equal to the number of non-\nzeros plus one.\nja\nArray containing the column indices for each non-zero element of the\nmatrix A being represented in the array a . Its length is equal to the length\nof the array a.\nfpm\nArray, dimension of 128. This array is used to pass various parameters to\nExtended Eigensolver routines. See Extended Eigensolver Input Parameters\nfor a complete description of the parameters and their default values.\nemin, emax\nThe lower and upper bounds of the interval to be searched for eigenvalues;\nemin ≤ emax.\nNOTE Users are advised to avoid situations in which\neigenvalues nearly coincide with the interval endpoints. This\nmay lead to unpredictable selection or omission of such\neigenvalues. Users should instead specify a slightly larger\ninterval than needed and, if required, pick valid eigenvalues and\ntheir corresponding eigenvectors for subsequent use.\nm0\nOn entry, specifies the initial guess for subspace dimension to be used, 0 <\nm0≤n. Set m0 ≥ m where m is the total number of eigenvalues located in the\ninterval [emin, emax]. If the initial guess is wrong, Extended Eigensolver\nroutines return info=3.\nx\nOn entry, if fpm[4]=1, the array x of size n by m contains a basis of guess\nsubspace where n is the order of the input matrix.\nOutput Parameters\nfpm\nOn output, the last 64 values correspond to Intel® oneAPI Math Kernel\nLibrary (oneMKL) PARDISOiparm[0] to iparm[63] (regardless of the value\nof fpm[63] on input).\nepsout\nOn output, contains the relative error on the trace: |tracei - tracei-1| /max(|\nemin|, |emax|)\nloop\nOn output, contains the number of refinement loop executed. Ignored on\ninput.\ne\nArray of length m0. On output, the first m entries of e are eigenvalues found\nin the interval.\nx\nOn output, the first m columns of x contain the orthonormal eigenvectors\ncorresponding to the computed eigenvalues e, with the i-th column of x\nholding the eigenvector associated with e[i].\nm\nThe total number of eigenvalues found in the interval [emin, emax]: 0 ≤ m\n≤ m0.\nres\nArray of length m0. On exit, the first m components contain the relative\nresidual vector:\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1970\n\n\nfor i=1, 2, …, m, and where m is the total number of eigenvalues found in\nthe search interval.\ninfo\nIf info=0, the execution is successful. If info ≠ 0, see Output Eigensolver\ninfo Details.\n?feast_scsrgv/?feast_hcsrgv\nExtended Eigensolver interface for generalized\neigenvalue problem with sparse matrices.\nSyntax\nvoid sfeast_scsrgv (const char * uplo, const MKL_INT * n, const float * a, const\nMKL_INT * ia, const MKL_INT * ja, const float * b, const MKL_INT * ib, const MKL_INT *\njb, MKL_INT * fpm, float * epsout, MKL_INT * loop, const float * emin, const float *\nemax, MKL_INT * m0, float * e, float * x, MKL_INT * m, float * res, MKL_INT * info);\nvoid dfeast_scsrgv (const char * uplo, const MKL_INT * n, const double * a, const\nMKL_INT * ia, const MKL_INT * ja, const double * b, const MKL_INT * ib, const MKL_INT *\njb, MKL_INT * fpm, double * epsout, MKL_INT * loop, const double * emin, const double *\nemax, MKL_INT * m0, double * e, double * x, MKL_INT * m, double * res, MKL_INT * info);\nvoid cfeast_hcsrgv (const char * uplo, const MKL_INT * n, const MKL_Complex8 * a, const\nMKL_INT * ia, const MKL_INT * ja, const MKL_Complex8 * b, const MKL_INT * ib, const\nMKL_INT * jb, MKL_INT * fpm, float * epsout, MKL_INT * loop, const float * emin, const\nfloat * emax, MKL_INT * m0, float * e, MKL_Complex8 * x, MKL_INT * m, float * res,\nMKL_INT * info);\nvoid zfeast_hcsrgv (const char * uplo, const MKL_INT * n, const MKL_Complex16 * a,\nconst MKL_INT * ia, const MKL_INT * ja, const MKL_Complex16 * b, const MKL_INT * ib,\nconst MKL_INT * jb, MKL_INT * fpm, double * epsout, MKL_INT * loop, const double *\nemin, const double * emax, MKL_INT * m0, double * e, MKL_Complex16 * x, MKL_INT * m,\ndouble * res, MKL_INT * info);\nInclude Files\n•\nmkl.h\nDescription\nThe routines compute all the eigenvalues and eigenvectors for generalized eigenvalue problems, Ax = λBx,\nwithin a given search interval.\nNOTE\nBoth matrices A and B must use the same family of storage format. The position of the non-\nzero elements can be different (CSR coordinates ib and jb can be different from ia and ja).\nInput Parameters\nuplo\nMust be 'U' or 'L' or 'F' .\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1971\n\n\nIf UPLO = 'U', a and b store the upper triangular parts of A and B\nrespectively.\nIf UPLO = 'L', a and b store the lower triangular parts of A and B\nrespectively.\nIf UPLO= 'F', a and b store the full matrices A and B respectively.\nn\nSets the size of the problem. n > 0.\na\nArray containing the nonzero elements of either the full matrix A or the\nupper or lower triangular part of the matrix A, as specified by uplo.\nia\nArray of length n + 1, containing indices of elements in the array a , such\nthat ia[i - 1] is the index in the array a of the first non-zero element\nfrom the row i . The value of the last element ia[n] is equal to the number\nof non-zeros plus one.\nja\nArray containing the column indices for each non-zero element of the\nmatrix A being represented in the array a . Its length is equal to the length\nof the array a.\nb\nArray of dimension ldb by *, contains the nonzero elements of either the\nfull matrix B or the upper or lower triangular part of the matrix B, as\nspecified by uplo.\nib\nArray of length n + 1, containing indices of elements in the array b , such\nthat ib[i - 1] is the index in the array b of the first non-zero element\nfrom the row i . The value of the last element ib[n] is equal to the number\nof non-zeros plus one.\njb\nArray containing the column indices for each non-zero element of the\nmatrix B being represented in the array b . Its length is equal to the length\nof the array b.\nfpm\nArray, dimension of 128. This array is used to pass various parameters to\nExtended Eigensolver routines. See Extended Eigensolver Input Parameters\nfor a complete description of the parameters and their default values.\nemin, emax\nThe lower and upper bounds of the interval to be searched for eigenvalues;\nemin ≤ emax.\nNOTE Users are advised to avoid situations in which\neigenvalues nearly coincide with the interval endpoints. This\nmay lead to unpredictable selection or omission of such\neigenvalues. Users should instead specify a slightly larger\ninterval than needed and, if required, pick valid eigenvalues and\ntheir corresponding eigenvectors for subsequent use.\nm0\nOn entry, specifies the initial guess for subspace dimension to be used, 0 <\nm0≤n. Set m0 ≥ m where m is the total number of eigenvalues located in the\ninterval [emin, emax]. If the initial guess is wrong, Extended Eigensolver\nroutines return info=3.\nx\nOn entry, if fpm[4]=1, the array x of size n by m contains a basis of guess\nsubspace where n is the order of the input matrix.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1972\n\n\nOutput Parameters\nfpm\nOn output, the last 64 values correspond to Intel® oneAPI Math Kernel\nLibrary (oneMKL) PARDISOiparm[0] to iparm[63] (regardless of the value\nof fpm[63] on input).\nepsout\nOn output, contains the relative error on the trace: |tracei - tracei-1| /max(|\nemin|, |emax|)\nloop\nOn output, contains the number of refinement loop executed. Ignored on\ninput.\ne\nArray of length m0. On output, the first m entries of e are eigenvalues found\nin the interval.\nx\nOn output, the first m columns of x contain the orthonormal eigenvectors\ncorresponding to the computed eigenvalues e, with the i-th column of x\nholding the eigenvector associated with e[i].\nm\nThe total number of eigenvalues found in the interval [emin, emax]: 0 ≤ m\n≤ m0.\nres\nArray of length m0. On exit, the first m components contain the relative\nresidual vector:\nAxi −λiBxi 1\nmax Emin , Emax\nBxi 1\nfor i=1, 2, …, m, and where m is the total number of eigenvalues found in\nthe search interval.\ninfo\nIf info=0, the execution is successful. If info ≠ 0, see Output Eigensolver\ninfo Details.\nExtended Eigensolver Interfaces for Extremal Eigenvalues/Singular Values\nThe topics in this section discuss Extended Eigensolver interfaces to find extremal eigenvalues as well as\nsingular values.\nExtended Eigensolver Interfaces to find largest/smallest eigenvalues\nThe predefined interfaces include routines for standard and generalized eigenvalue problems and sparse\nmatrices.\nMatrix Type\nStandard Eigenvalue Problem\nGeneralized Eigenvalue\nProblem\nSparse\nmkl_sparse_?_ev\nmkl_sparse_?_gv\nmkl_sparse_?_ev\nComputes the largest/smallest eigenvalues and\ncorresponding eigenvectors of a standard eigenvalue\nproblem\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1973\n\n\nSyntax\nsparse_status_t mkl_sparse_s_ev (char *which, MKL_INT *pm, sparse_matrix_t A, struct\nmatrix_descr descrA, MKL_INT k0, MKL_INT *k, float *E, float *X, float *res);\nsparse_status_t mkl_sparse_d_ev (char *which, MKL_INT *pm, sparse_matrix_t A, struct\nmatrix_descr descrA, MKL_INT k0, MKL_INT *k, double *E, double *X, double *res);\nInclude Files\n•\nmkl_solvers_ee.h\nDescription\nThe mkl_sparse_?_ev routine computes the largest/smallest eigenvalues and corresponding eigenvectors of\na standard eigenvalue problem.\nAx = lambda x\nwhere A is the real symmetric matrix.\nInput Parameters\nwhich\nIndicates eigenvalues for which to search:\n•\nwhich = 'L' indicates the largest eigenvalues.\n•\nwhich = 'S' indicates the smallest eigenvalues.\npm\nArray of size 128. This array is used to pass various parameters to\nExtended Eigensolver routines. See • Extended Eigensolver Input\nParameters for Extremal Eigenvalue Problem for a complete\ndescription of the parameters and their default values.\nA\nHandle containing sparse matrix in internal data structure.\ndescrA\nStructure specifying sparse matrix properties.\nsparse_matrix_type_t\ntype\nSpecifies the type of a sparse matrix:\n•SPARSE_MATRIX_TYPE_GENERAL\nThe matrix is processed as-is.\n•SPARSE_MATRIX_TYPE_SYMMETRIC\nThe matrix is symmetric (only the\nrequested triangle is processed).\nsparse_fill_mode_t\nmode\nSpecifies the triangular matrix part for\nsymmetric, Hermitian, triangular, and block-\ntriangular matrices:\n•SPARSE_FILL_MODE_LOWER\nThe lower triangular matrix part is\nprocessed.\n•SPARSE_FILL_MODE_UPPER\nThe upper triangular matrix part is\nprocessed.\nsparse_diag_type_t\ndiag\nSpecifies the diagonal type for non-general\nmatrices:\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1974\n\n\n•SPARSE_DIAG_NON_UNIT\nDiagonal elements might not be equal to\none.\n•SPARSE_DIAG_UNIT\nDiagonal elements are equal to one\nk0\nThe desired number of the largest/smallest eigenvalues to find.\nOutput Parameters\nk\nNumber of eigenvalues found.\nE\nArray of size k0. Contains k largest/smallest eigenvalues.\nX\nArray of size k0*Number of columns of the matrix A. Contains k\neigenvectors.\nRes\nArray of size k0. Contains k residuals.\nReturn Values\nThe function returns a value indicating whether the operation was successful or not, and why.\nSPARSE_STATUS_SUCCESS\nThe operation was successful.\nSPARSE_STATUS_NOT_INITIALIZED\nThe routine encountered an empty handle or matrix array.\nSPARSE_STATUS_ALLOC_FAILED\nInternal memory allocation failed.\nSPARSE_STATUS_INVALID_VALUE\nThe input parameters contain an invalid value.\nSPARSE_STATUS_EXECUTION_FAILED\nExecution failed.\nSPARSE_STATUS_INTERNAL_ERROR\nAn error in algorithm implementation occurred.\nSPARSE_STATUS_NOT_SUPPORTED\nThe requested operation is not supported.\nmkl_sparse_?_gv\nComputes the largest/smallest eigenvalues and\ncorresponding eigenvectors of a generalized\neigenvalue problem\nSyntax\nsparse_status_t mkl_sparse_s_gv (char *which, MKL_INT *pm, sparse_matrix_t A, struct\nmatrix_descr descrA, sparse_matrix_t B, struct matrix_descr descrB, MKL_INT k0, MKL_INT\n*k, float *E, float *X, float *res);\nsparse_status_t mkl_sparse_d_gv (char *which, MKL_INT *pm, sparse_matrix_t A, struct\nmatrix_descr descrA, sparse_matrix_t B, struct matrix_descr descrB, MKL_INT k0, MKL_INT\n*k, double *E, double *X, double *res);\nInclude Files\n•\nmkl_solvers_ee.h\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1975\n\n\nDescription\nThe mkl_sparse_?_gv routine computes the largest/smallest eigenvalues and corresponding eigenvectors of\na generalized eigenvalue problem.\nAx = lambda Bx\nwhere A is the real symmetric matrix and B is the real symmetric positive definite matrix.\nInput Parameters\nwhich\nIndicates eigenvalues for which to search:\n•\nwhich = 'L' indicates the largest eigenvalues.\n•\nwhich = 'S' indicates the smallest eigenvalues.\npm\nArray of size 128. This array is used to pass various parameters to\nExtended Eigensolver routines. See • Extended Eigensolver Input\nParameters for Extremal Eigenvalue Problem for a complete\ndescription of the parameters and their default values.\nA\nHandle containing sparse matrix in internal data structure.\ndescrA\nStructure specifying sparse matrix properties.\nsparse_matrix_type_t\ntype\nSpecifies the type of a sparse matrix:\n•SPARSE_MATRIX_TYPE_GENERAL\nThe matrix is processed as-is.\n•SPARSE_MATRIX_TYPE_SYMMETRIC\nThe matrix is symmetric (only the\nrequested triangle is processed).\nsparse_fill_mode_t\nmode\nSpecifies the triangular matrix part for\nsymmetric, Hermitian, triangular, and block-\ntriangular matrices:\n•SPARSE_FILL_MODE_LOWER\nThe lower triangular matrix part is\nprocessed.\n•SPARSE_FILL_MODE_UPPER\nThe upper triangular matrix part is\nprocessed.\nsparse_diag_type_t\ndiag\nSpecifies the diagonal type for non-general\nmatrices:\n•SPARSE_DIAG_NON_UNIT\nDiagonal elements might not be equal to\none.\n•SPARSE_DIAG_UNIT\nDiagonal elements are equal to one\nB\nHandle containing sparse matrix in internal data structure.\ndescrB\nStructure specifying sparse matrix properties.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1976\n\n\nsparse_matrix_type_t\ntype\nSpecifies the type of a sparse matrix:\n•SPARSE_MATRIX_TYPE_GENERAL\nThe matrix is processed as-is.\n•SPARSE_MATRIX_TYPE_SYMMETRIC\nThe matrix is symmetric (only the\nrequested triangle is processed).\nsparse_fill_mode_t\nmode\nSpecifies the triangular matrix part for\nsymmetric, Hermitian, triangular, and block-\ntriangular matrices:\n•SPARSE_FILL_MODE_LOWER\nThe lower triangular matrix part is\nprocessed.\n•SPARSE_FILL_MODE_UPPER\nThe upper triangular matrix part is\nprocessed.\nsparse_diag_type_t\ndiag\nSpecifies the diagonal type for non-general\nmatrices:\n•SPARSE_DIAG_NON_UNIT\nDiagonal elements might not be equal to\none.\n•SPARSE_DIAG_UNIT\nDiagonal elements are equal to one\nk0\nThe desired number of the largest/smallest eigenvalues to find.\nOutput Parameters\nk\nNumber of eigenvalues found.\nE\nArray of size k0. Contains k largest/smallest eigenvalues.\nX\nArray of size k0*Number of columns of matrix A. Contains k\neigenvectors.\nRes\nArray of size k0. Contains k residuals.\nReturn Values\nThe function returns a value indicating whether the operation was successful or not, and why.\nSPARSE_STATUS_SUCCESS\nThe operation was successful.\nSPARSE_STATUS_NOT_INITIALIZED\nThe routine encountered an empty handle or matrix array.\nSPARSE_STATUS_ALLOC_FAILED\nInternal memory allocation failed.\nSPARSE_STATUS_INVALID_VALUE\nThe input parameters contain an invalid value.\nSPARSE_STATUS_EXECUTION_FAILED\nExecution failed.\nSPARSE_STATUS_INTERNAL_ERROR\nAn error in algorithm implementation occurred.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1977\n\n\nSPARSE_STATUS_NOT_SUPPORTED\nThe requested operation is not supported.\nExtended Eigensolver Interfaces to find largest/smallest singular values\nThe predefined interfaces include routines to find the largest and smallest singular values and the\ncorresponding singular vectors of sparse matrices.\nMatrix Type\nStandard singular value\nproblem\nSparse\nmkl_sparse_?_svd\nmkl_sparse_?_svd\nComputes the largest/smallest singular values of a\nsingular-value problem\nSyntax\nsparse_status_t mkl_sparse_s_svd (char *whichS, char *whichV, MKL_INT *pm,\nsparse_matrix_t A, struct matrix_descr descrA, MKL_INT k0, MKL_INT *k, float *E, float\n*XL, float *XR, float *res);\nsparse_status_t mkl_sparse_d_svd (char *whichS, char *whichV, MKL_INT *pm,\nsparse_matrix_t A, struct matrix_descr descrA, MKL_INT k0, MKL_INT *k, double *E,\ndouble *XL, double *XR, double *res);\nInclude Files\n•\nmkl_solvers_ee.h\nDescription\nThe mkl_sparse_?_svd routine computes the largest/smallest singular values of a singular-value problem.\nAATx = σx or ATAx = σx, where A is the real rectangular matrix.\nInput Parameters\nwhichS\nIndicates eigenvalues for which to search:\n•\nwhichS = 'L' indicates the largest eigenvalues.\n•\nwhichS = 'S' indicates the smallest eigenvalues.\nwhichV\nIndicates singular vectors for which to search:\n•\nwhichV = 'R' indicates right singular vectors.\n•\nwhichV = 'L' indicates left singular vectors.\npm\nArray of size 128. This array is used to pass various parameters to\nExtended Eigensolver routines. See • Extended Eigensolver Input\nParameters for Extremal Eigenvalue Problem for a complete\ndescription of the parameters and their default values.\nA\nHandle containing sparse matrix in internal data structure.\ndescrA\nStructure specifying sparse matrix properties.\nsparse_matrix_type_t\ntype\nSpecifies the type of a sparse matrix:\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1978\n\n\n•SPARSE_MATRIX_TYPE_GENERAL\nThe matrix is processed as-is.\n•SPARSE_MATRIX_TYPE_SYMMETRIC\nThe matrix is symmetric (only the\nrequested triangle is processed).\nsparse_fill_mode_t\nmode\nSpecifies the triangular matrix part for\nsymmetric, Hermitian, triangular, and block-\ntriangular matrices:\n•SPARSE_FILL_MODE_LOWER\nThe lower triangular matrix part is\nprocessed.\n•SPARSE_FILL_MODE_UPPER\nThe upper triangular matrix part is\nprocessed.\nsparse_diag_type_t\ndiag\nSpecifies the diagonal type for non-general\nmatrices:\n•SPARSE_DIAG_NON_UNIT\nDiagonal elements might not be equal to\none.\n•SPARSE_DIAG_UNIT\nDiagonal elements are equal to one\nk0\nThe desired number of the largest/smallest eigenvalues to find.\nOutput Parameters\nk\nNumber of eigenvalues found.\nE\nArray of size k0. Contains k largest/smallest eigenvalues.\nXL\nArray of size k0*Number of rows of matrix A. Contains k left singular\nvectors.\nXR\nArray of size k0*Number of columns of matrix A. Contains k right\nsingular vectors.\nRes\nArray that contains k residuals.\nReturn Values\nThe function returns a value indicating whether the operation was successful or not, and why.\nSPARSE_STATUS_SUCCESS\nThe operation was successful.\nSPARSE_STATUS_NOT_INITIALIZED\nThe routine encountered an empty handle or matrix array.\nSPARSE_STATUS_ALLOC_FAILED\nInternal memory allocation failed.\nSPARSE_STATUS_INVALID_VALUE\nThe input parameters contain an invalid value.\nSPARSE_STATUS_EXECUTION_FAILED\nExecution failed.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1979\n\n\nSPARSE_STATUS_INTERNAL_ERROR\nAn error in algorithm implementation occurred.\nSPARSE_STATUS_NOT_SUPPORTED\nThe requested operation is not supported.\nmkl_sparse_ee_init\nInitializes Extended Eigensolver input parameters with\ndefault values\nSyntax\nsparse_status_t mkl_sparse_ee_init (MKL_INT* pm);\nInclude Files\n•\nmkl_solvers_ee.h\nDescription\nThis routine sets all Extended Eigensolver parameters to their default values.\nOutput Parameters\npm\nArray of size 128. This array is used to pass various parameters to\nExtended Eigensolver routines. See • Extended Eigensolver Input\nParameters for Extremal Eigenvalue Problem for a complete\ndescription of the parameters and their default values.\nExtended Eigensolver Input Parameters for Extremal Eigenvalue Problem\nThe input parameters for Extended Eigensolver routines are contained in an MKL_INT array named pm. To call\nthe Extended Eigensolver interfaces, initialize this array using the mkl_sparse_ee_init routine.\nParameter\nDefa\nult\nDescription\npm[0]\n0\nReserved for future use.\npm[1]\n6\nDefines the tolerance for the stopping criteria:\n∈= 10−pm 1 + 1\npm[2]\n0\nSpecifies the algorithm to use:\n•\n0 - Decided at runtime\n•\n1 - The Krylov-Schur method\n•\n2 - Subspace Iteration technique based on FEAST algorithm\npm[3]\n*\nThis parameter is referenced only for Krylov-Schur Method. It indicates the\nnumber of Lanczos/Arnoldi vectors (NCV) generated at each iteration.\nThis parameter must be less than or equal to size of matrix and greater than\nnumber of eigenvalues (k0) to be computed. If unspecified, NCV is set to be at\nleast 1.5 times larger than k0.\npm[4]\n*\nMaximum number of iterations. If unspecified, this parameter is set to 10000 for\nthe Krylov-Schur method and 60 for the subspace iteration method.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1980\n\n\nParameter\nDefa\nult\nDescription\npm[5]\n0\nPower of Chebychev expansion for approximate spectral projector. Only referenced\nwhen pm[2]=1\npm[6]\n1\nUsed only for Krylov-Schur Method.\nIf 0, then the method only computes eigenvalues.\nIf 1, then the method computes eigenvalues and eigenvectors. The subspace\niteration method always computes eigenvectors/singular vectors. You must allocate\nthe required memory space.\npm[7]\n0\nConvergence stopping criteria.\nDefines whether the stopping criteria for the iterations with respect to the true\nresiduals (used if pm[8] is not zero) and residual norm estimates are relative to\nthe eigenvalues/singular values or not.\nIf 0, the stopping criteria with respect to the true residuals is:\nAx −λx\nλ\n< 10−pm 1 + 1\nor\nAx −λBx\nλ\n< 10−pm 1 + 1\nIf 1, the stopping criteria with respect to the true residuals is:\nAx −λx\n< 10−pm 1 + 1\nor\nAx −λBx\n< 10−pm 1 + 1\nfor a generalized eigenproblem.\nThe residual norm estimates are based on the magnitude of the last eigenvector of\nthe Schur decomposition matrix and the exact formula can be found in the\nliterature. When pm[7]=0, the residual norm estimate is additionally divided by the\nmagnitude of the computed eigenvalue and compared to 10(-pm[1]+1).\npm[8]\n0\nSpecifies if for detecting convergence the solver must compute the true residuals\nfor eigenpairs for the Krylov-Schur method or it can only use the residual norm\nestimates.\nIf 0, only residual norm estimates are used.\nIf 1, the solver computes not just residual norm estimates but also the true\nresiduals as defined in the description of pm[7].\npm[9]\n0\nUsed only for the Krylov-Schur method and only as an output parameter.\nReports the reason for exiting the iteration loop of the method:\n•\nIf 0, the iterations stopped since convergence has been detected.\n•\nIf -1, maximum number of iterations has been reached and even the residual\nnorm estimates have not converged.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1981\n\n\nParameter\nDefa\nult\nDescription\n•\nIf -2, maximum number of iterations has been reached despite the residual\nnorm estimates have converged (but the true residuals for eigenpairs have\nnot).\n•\nIf -3, the iterations stagnated and even the residual norm estimates have not\nconverged.\n•\nIf -4, the iterations stagnated while the eigenvalues have converged (but the\ntrue residuals for eigenpairs do not).\npm[10] to\npm[128]\n-\nReserved for future use.\nVector Mathematical Functions\nIntel® oneAPI Math Kernel Library (oneMKL) Vector Mathematics functions (VM) compute a mathematical\nfunction of each of the vector elements. VM includes a set of highly optimized functions (arithmetic, power,\ntrigonometric, exponential, hyperbolic, special, and rounding) that operate on vectors of real and complex\nnumbers.\nApplication programs that improve performance with VM include nonlinear programming software,\ncomputation of integrals, financial calculations, computer graphics, and many others.\nVM functions fall into the following groups according to the operations they perform:\n•\nVM Mathematical Functions compute values of mathematical functions, such as sine, cosine, exponential,\nor logarithm, on vectors stored contiguously in memory.\n•\nVM Pack/Unpack Functions convert to and from vectors with positive increment indexing, vector indexing,\nand mask indexing (see Appendix \"Vector Arguments in VM\" for details on vector indexing methods).\n•\nVM Service Functions set/get the accuracy modes and the error codes, and free memory.\nThe VM mathematical functions take an input vector as an argument, compute values of the respective\nfunction element-wise, and return the results in an output vector. All the VM mathematical functions can\nperform in-place operations, where the input and output arrays are at the same memory locations. For VM\nmathematical functions with positive increment indexing, in-place operations are supported only when the\ninput and output increments have the same value.\nThe Intel® oneAPI Math Kernel Library (oneMKL) interfaces are given in mkl_vml_functions.h.\nNOTE On 64-bit platforms, oneMKL provides VML C interfaces with the _64 suffix to support large data\narrays in the LP64 interface library. For more interface library details, see \"Using the ILP64 Interface\nvs. LP64 Interface\" in the developer guide.\nExamples that demonstrate how to use the VM functions are located in:\n${MKL}/examples/vmlc/source\nSee VM performance and accuracy data in the online VM Performance and Accuracy Data document available\nat https://www.intel.com/content/www/us/en/developer/tools/oneapi/onemkl-documentation.html.\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1982\n\n\nVM Data Types, Accuracy Modes, and Performance Tips\nVM includes mathematical and pack/unpack vector functions for single and double precision vector\narguments of real and compex types. Intel® oneAPI Math Kernel Library (oneMKL) provides Fortran and C\ninterfaces for all VM functions, including the associated service functions. The Function Naming Conventions\ntopic shows how to call these functions.\nPerformance depends on a number of factors, including vectorization and threading overhead. The\nrecommended usage is as follows:\n•\nUse VM for vector lengths larger than 40 elements.\n•\nUse the Intel® Compiler for vector lengths less than 40 elements.\nAll VM vector functions support the following accuracy modes:\n•\nHigh Accuracy (HA), the default mode\n•\nLow Accuracy (LA), which improves performance by reducing accuracy of the two least significant bits\n•\nEnhanced Performance (EP), which provides better performance at the cost of significantly reduced\naccuracy. Approximately half of the bits in the mantissa are correct.\nNote that using the EP mode does not guarantee accurate processing of corner cases and special values.\nAlthough the default accuracy is HA, LA is sufficient in most cases. For applications that require less accuracy\n(for example, media applications, some Monte Carlo simulations, etc.), the EP mode may be sufficient.\nVM handles special values in accordance with the C99 standard [C99].\nIntel® oneAPI Math Kernel Library (oneMKL) offers both functions and environment variables to switch\nbetween modes for VM. See the Intel® oneAPI Math Kernel Library (oneMKL) Developer Guide for details\nabout the environment variables. Use the vmlSetMode(mode) function (see Table \"Values of the mode\nParameter\") to switch between the HA, LA, and EP modes. The vmlGetMode() function returns the current\nmode.\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nSee Also\nFunction Naming Conventions \nVM Naming Conventions\nThe VM function names are of mixed (lower and upper) case.\nThe VM mathematical and pack/unpack function names have the following structure:\nv[m]<?><name><mod>\nwhere\n•\nv is a prefix indicating vector operations.\n•\n[m] is an optional prefix for mathematical functions that indicates additional argument to specify a VM\nmode for a given function call (see vmlSetMode for possible values and their description).\n•\n<?> is a precision prefix that indicates one of the following data types:\ns\nfloat.\nd\ndouble.\nc\nMKL_Complex8.\nz\nMKL_Complex16.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1983\n\n\n•\n<name> indicates the function short name, with some of its letters in uppercase. See examples in Table\n\"VM Mathematical Functions\".\n•\n<mod> field (written in uppercase) is present only in the pack/unpack functions and indicates the\nindexing method used:\ni\nindexing with a positive increment\nv\nindexing with an index vector\nm\nindexing with a mask vector.\nThe VM service function names have the following structure:\nvml<name>\nwhere\n<name> indicates the function short name, with some of its letters in uppercase. See examples in Table \"VM\nService Functions\".\nTo call VM functions from an application program, use conventional function calls. For example, call the\nvector single precision real exponential function as\nvsExp ( n, a, y );\nVM Function Interfaces\nVM interfaces include the function names and argument lists. The following sections describe the interfaces\nfor the VM functions. Note that some of the functions have multiple input and output arguments\nSome VM functions may also take scalar arguments as input. See the function description for the naming\nconventions of such arguments.\nVM Mathematical Function Interfaces\nv<?><name>( n, a, [scalar input arguments,]y );\nv<?><name>I( n, a, inca, [scalar input arguments,] y, incy );\nv<?><name>( n, a, b, [scalar input arguments,]y );\nv<?><name>I( n, a, inca, b, incb, [scalar input arguments,] y, incy );\nv<?><name>( n, a, y, z );\nv<?><name>I( n, a, inca, y, incy, z, incz );\nvm<?><name>( n, a, [scalar input arguments,]y, mode );\nvm<?><name>I( n, a, inca, [scalar input arguments,] y, incy, mode );\nvm<?><name>( n, a, b, [scalar input arguments,]y, mode );\nvm<?><name>I( n, a, inca, b, incb, [scalar input arguments,] y, incy, mode );\nvm<?><name>( n, a, y, z, mode );\nvm<?><name>I( n, a, inca, y, incy, z, incz, mode);\nVM Mathematical Functions\nVM Pack Function Interfaces\nv<?>PackI( n, a, inca, y );\nv<?>PackV( n, a, ia, y );\nv<?>PackM( n, a, ma, y );\nVM Unpack Function Interfaces\nv<?>UnpackI( n, a, y, incy );\nv<?>UnpackV( n, a, y, iy );\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1984\n\n\nv<?>UnpackM( n, a, y, my );\nVM Service Function Interfaces\noldmode = vmlSetMode( mode );\nmode = vmlGetMode( void );\nolderr = vmlSetErrStatus ( err );\nerr = vmlGetErrStatus( void );\nolderr = vmlClearErrStatus( void );\noldcallback = vmlSetErrorCallBack( callback );\ncallback = vmlGetErrorCallBack( void );\noldcallback = vmlClearErrorCallBack( void );\nNote that oldmode, olderr, and oldcallback refer to settings prior to the call.\nVM Input Parameters\nn\nnumber of elements to be calculated\na\nfirst input vector\nb\nsecond input vector\ninca\nvector increment for the input vector a\nincb\nvector increment for the input vector b\nia\nindex vector for the input vector a\nma\nmask vector for the input vector a\nincy\nvector increment for the output vector y\nincz\nvector increment for the output vector z\niy\nindex vector for the output vector y\nmy\nmask vector for the output vector y\nerr\nerror code\nmode\nVM mode\ncallback\naddress of the callback function\nVM Output Parameters\ny\nfirst output vector\nz\nsecond output vector\nerr\nerror code\nmode\nVM mode\nolderr\nformer error code\noldmode\nformer VM mode\ncallback\naddress of the callback function\noldcallback\naddress of the former callback function\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1985\n\n\nSee the data types of the parameters used in each function in the respective function description section. All\nIntel® oneAPI Math Kernel Library (oneMKL) VM mathematical functions can perform in-place operations. For\nVM mathematical functions with positive increment indexing, (for example, v?PowI), in-place operations are\nsupported only when the input and output increments have the same value.\nVector Indexing Methods\nClassic VM mathematical functions work with unit stride. Strided VM mathematical functions (names with “I”\nsuffix) work with arbitrary integer increments. Increments may be positive, negative or equal to zero. For\nexample:\nvsExpI (n, a, inca, r, incr)\nis equivalent to:\nfor (i=0; i<n; i++)\n{\n    r[i * incr] = exp (a[i * inca]);\n}\nwhere\ni – current index,\ninca – input index increment,\nincr – output index increment.\nn – the number of elements to be computed (important: n is not the maximum array size).\nSo, when calling vsExpI(n, a, inca, r, incr) be sure that the input vector a is allocated at least for 1\n+ (n-1)*inca elements and the result vector r has a space for 1 + (n-1)*incr elements.\nNOTE The order of computations is not guaranteed and no array bounds-checking is performed;\ntherefore, the results for overlapped and in-place arrays are not generally deterministic for increments\nother than 1.\nFor output index increment, equal to 0, the result is not deterministic and generally nonsensical.\nUse negative increments to step from base pointers in reverse order.\nFor example:\nvsExpI (n, a, -2, r, -3)\nis equivalent to:\nfor (i=0; i<n; i++)\n{\n    r[- i*3] = exp (a[-i*2]);\n}\nNOTE Pass pointers to the desired ending array element in memory as an argument for negative\nstrides.\nFor example:\nvsExpI (n, a, 2, r + 1000, -3).\nUse a zero increment for one fixed argument rather than an array.\nFor example:\nvsMulI (n, a, 1, b, 0, r, 1)\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1986\n\n\nis equivalent to:\nfor (i=0; i<n; i++)\n{\n    r[i] = a[i] * b[0];\n}\nVM Pack/Unpack functions use the following indexing methods to do this task:\n•\npositive increment\n•\nindex vector\n•\nmask vector\nThe indexing method used in a particular function is indicated by the indexing modifier (see the description of\nthe <mod> field in Function Naming Conventions). For more information on the indexing methods, see Vector\nArguments in VM.\nVM Pack/Unpack Functions\nVM Error Diagnostics\nThe VM mathematical functions incorporate the error handling mechanism, which is controlled by the\nfollowing service functions:\n \nvmlGetErrStatus,\nvmlSetErrStatus,\nvmlClearErrStatus\nThese functions operate with a global variable called VM Error\nStatus. The VM Error Status flags an error, a warning, or a\nsuccessful execution of a VM function.\nvmlGetErrCallBack,\nvmlSetErrCallBack,\nvmlClearErrCallBack\nThese functions enable you to customize the error handling. For\nexample, you can identify a particular argument in a vector where\nan error occurred or that caused a warning.\nvmlSetMode, vmlGetMode\nThese functions get and set a VM mode. If you set a new VM mode\nusing the vmlSetMode function, you can store the previous VM\nmode returned by the routine and restore it at any point of your\napplication.\nIf both an error and a warning situation occur during the function call, the VM Error Status variable keeps\nonly the value of the error code. See Table \"Values of the VM Error Status\" for possible values. If a VM\nfunction does not encounter errors or warnings, it sets the VM Error Status to VML_STATUS_OK.\nIf you use incorrect input arguments to a VM function (VML_STATUS_BADSIZE and VML_STATUS_BADMEM), the\nfunction calls xerbla to report errors. See Table \"Values of the VM Error Status\" for details.\nYou can use the vmlSetMode and vmlGetMode functions to modify error handling behavior. Depending on the\nVM mode, the error handling behavior includes the following operations:\n•\nsetting the VM Error Status to a value corresponding to the observed error or warning\n•\nsetting the errno variable to one of the values described in Table \"Set Values of the errno Variable\"\n•\nwriting error text information to the stderr stream\n•\nraising the appropriate exception on an error, if necessary\n•\ncalling the additional error handler callback function that is set by vmlSetErrorCallBack.\nSet Values of the errno Variable\nValue of errno\nDescription\n0\nNo errors are detected.\nEINVAL\nThe array dimension is not positive.\nEACCES\nNULL pointer is passed.\nEDOM\nAt least one of array values is out of a range of definition.\nERANGE\nAt least one of array values caused a singularity, overflow or\nunderflow.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1987\n\n\nSee Also\nvmlGetErrStatus Gets the VM Error Status.\nvmlSetErrStatus Sets the new VM Error Status according to err and stores the previous VM Error\nStatus to olderrSets the global VM Status according to new values and returns the previous VM\nStatus.\nvmlClearErrStatus Sets the VM Error Status to VML_STATUS_OK and stores the previous VM Error\nStatus to olderr.\nvmlSetErrorCallBack Sets the additional error handler callback function and gets the old callback\nfunction.\nvmlGetErrorCallBack Gets the additional error handler callback function.\nvmlClearErrorCallBack Deletes the additional error handler callback function and retrieves the\nformer callback function.\nvmlGetMode Gets the VM mode.\nvmlSetMode Sets a new mode for VM functions according to the mode parameter and stores the\nprevious VM mode to oldmode.\nVM Mathematical Functions\nThis section describes VM functions that compute values of mathematical functions on real and complex\nvector arguments.\nEach function is introduced by its short name, a brief description of its purpose, and the calling sequence for\neach type of data, as well as a description of the input/output arguments.\nThe input range of parameters is equal to the mathematical range of the input data type, unless the function\ndescription specifies input threshold values, which mark off the precision overflow, as follows:\n•\nFLT_MAX denotes the maximum number representable in single precision real data type\n•\nDBL_MAX denotes the maximum number representable in double precision real data type\nTable \"VM Mathematical Functions\" lists available mathematical functions and associated data types.\nVM Mathematical Functions\nFunction\nData Types\nDescription\nArithmetic Functions\nv?Add\ns, d, c, z\nAdds vector elements\nv?Sub\ns, d, c, z\nSubtracts vector elements\nv?Sqr\ns, d\nSquares vector elements\nv?Mul\ns, d, c, z\nMultiplies vector elements\nv?MulByConj\nc, z\nMultiplies elements of one vector by conjugated elements of the\nsecond vector\nv?Conj\nc, z\nConjugates vector elements\nv?Abs\ns, d, c, z\nComputes the absolute value of vector elements\nv?Arg\nc, z\nComputes the argument of vector elements\nv?LinearFrac\ns, d\nPerforms linear fraction transformation of vectors\nv?Fmod\ns, d\nPerforms element by element computation of the modulus function of\nvector a with respect to vector b\nv?Remainder\ns, d\nPerforms element by element computation of the remainder function\non the elements of vector a and the corresponding elements of vector\nb\nPower and Root Functions\nv?Inv\ns, d\nInverts vector elements\nv?Div\ns, d, c, z\nDivides elements of one vector by elements of the second vector\nv?Sqrt\ns, d, c, z\nComputes the square root of vector elements\nv?InvSqrt\ns, d\nComputes the inverse square root of vector elements\nv?Cbrt\ns, d\nComputes the cube root of vector elements\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1988\n\n\nFunction\nData Types\nDescription\nv?InvCbrt\ns, d\nComputes the inverse cube root of vector elements\nv?Pow2o3\ns, d\nComputes the cube root of the square of each vector element\nv?Pow3o2\ns, d\nComputes the square root of the cube of each vector element\nv?Pow\ns, d, c, z\nRaises each vector element to the specified power\nv?Powx\ns, d, c, z\nRaises each vector element to the constant power\nv?Powr\ns, d\nComputes a to the power b for elements of two vectors, where the\nelements of vector argument a are all non-negative\nv?Hypot\ns, d\nComputes the square root of sum of squares\nExponential and Logarithmic Functions\nv?Exp\ns, d, c, z\nComputes the base e exponential of vector elements\nv?Exp2\ns, d\nComputes the base 2 exponential of vector elements\nv?Exp10\ns, d\nComputes the base 10 exponential of vector elements\nv?Expm1\ns, d\nComputes the base e exponential of vector elements decreased by 1\nv?Ln\ns, d, c, z\nComputes the natural logarithm of vector elements\nv?Log2\ns, d\nComputes the base 2 logarithm of vector elements\nv?Log10\ns, d, c, z\nComputes the base 10 logarithm of vector elements\nv?Log1p\ns, d\nComputes the natural logarithm of vector elements that are\nincreased by 1\nv?Logb\ns, d\nComputes the exponents of the elements of input vector a\nTrigonometric Functions\nv?Cos\ns, d, c, z\nComputes the cosine of vector elements\nv?Sin\ns, d, c, z\nComputes the sine of vector elements\nv?SinCos\ns, d\nComputes the sine and cosine of vector elements\nv?CIS\nc, z\nComputes the complex exponent of vector elements (cosine and sine\ncombined to complex value)\nv?Tan\ns, d, c, z\nComputes the tangent of vector elements\nv?Acos\ns, d, c, z\nComputes the inverse cosine of vector elements\nv?Asin\ns, d, c, z\nComputes the inverse sine of vector elements\nv?Atan\ns, d, c, z\nComputes the inverse tangent of vector elements\nv?Atan2\ns, d\nComputes the four-quadrant inverse tangent of ratios of the elements\nof two vectors\nv?Cospi\ns, d\nComputes the cosine of vector elements multiplied by π\nv?Sinpi\ns, d\nComputes the sine of vector elements multiplied by π\nv?Tanpi\ns, d\nComputes the tangent of vector elements multiplied by π\nv?Acospi\ns, d\nComputes the inverse cosine of vector elements divided by π\nv?Asinpi\ns, d\nComputes the inverse sine of vector elements divided by π\nv?Atanpi\ns, d\nComputes the inverse tangent of vector elements divided by π\nv?Atan2pi\ns, d\nComputes the four-quadrant inverse tangent of the ratios of the\ncorresponding elementss of two vectors divided by π\nv?Cosd\ns, d\nComputes the cosine of vector elements multiplied by π/180\nv?Sind\ns, d\nComputes the sine of vector elements multiplied by π/180\nv?Tand\ns, d\nComputes the tangent of vector elements multiplied by π/180\nHyperbolic Functions\nv?Cosh\ns, d, c, z\nComputes the hyperbolic cosine of vector elements\nv?Sinh\ns, d, c, z\nComputes the hyperbolic sine of vector elements\nv?Tanh\ns, d, c, z\nComputes the hyperbolic tangent of vector elements\nv?Acosh\ns, d, c, z\nComputes the inverse hyperbolic cosine of vector elements\nv?Asinh\ns, d, c, z\nComputes the inverse hyperbolic sine of vector elements\nv?Atanh\ns, d, c, z\nComputes the inverse hyperbolic tangent of vector elements.\nSpecial Functions\nv?Erf\ns, d\nComputes the error function value of vector elements\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1989\n\n\nFunction\nData Types\nDescription\nv?Erfc\ns, d\nComputes the complementary error function value of vector elements\nv?CdfNorm\ns, d\nComputes the cumulative normal distribution function value of vector\nelements\nv?ErfInv\ns, d\nComputes the inverse error function value of vector elements\nv?ErfcInv\ns, d\nComputes the inverse complementary error function value of vector\nelements\nv?CdfNormInv\ns, d\nComputes the inverse cumulative normal distribution function value\nof vector elements\nv?LGamma\ns, d\nComputes the natural logarithm for the absolute value of the gamma\nfunction of vector elements\nv?TGamma\ns, d\nComputes the gamma function of vector elements\nv?ExpInt1\ns, d\nComputes the exponential integral of vector elements\nRounding Functions\nv?Floor\ns, d\nRounds towards minus infinity\nv?Ceil\ns, d\nRounds towards plus infinity\nv?Trunc\ns, d\nRounds towards zero infinity\nv?Round\ns, d\nRounds to nearest integer\nv?NearbyInt\ns, d\nRounds according to current mode\nv?Rint\ns, d\nRounds according to current mode and raising inexact result\nexception\nv?Modf\ns, d\nComputes the integer and fractional parts\nv?Frac\ns, d\nComputes the fractional part\nMiscellaneous Functions\nv?CopySign\ns, d\nReturns vector of elements of one argument with signs changed to\nmatch other argument elements\nv?NextAfter\ns, d\nReturns vector of elements containing the next representable\nfloating-point values following the values from the elements of one\nvector in the direction of the corresponding elements of another\nvector\nv?Fdim\ns, d\nReturns vector containing the differences of the corresponding\nelements of the vector arguments if the first is larger and +0\notherwise\nv?Fmax\ns, d\nReturns the larger of each pair of elements of the two vector\narguments\nv?Fmin\ns, d\nReturns the smaller of each pair of elements of the two vector\narguments\nv?MaxMag\ns, d\nReturns the element with the larger magnitude between each pair of\nelements of the two vector arguments\nv?MinMag\ns, d\nReturns the element with the smaller magnitude between each pair\nof elements of the two vector arguments\nSpecial Value Notations\nThis topic defines notations of special values for complex functions. The definitions are provided in text,\ntables, or formulas.\n•\nz, z1, z2, etc. denote complex numbers.\n•\ni, i2=-1 is the imaginary unit.\n•\nx, X, x1, x2, etc. denote real imaginary parts.\n•\ny, Y, y1, y2, etc. denote imaginary parts.\n•\nX and Y represent any finite positive IEEE-754 floating point values, if not stated otherwise.\n•\nQuiet NaN and signaling NaN are denoted with QNAN and SNAN, respectively.\n•\nThe IEEE-754 positive infinities or floating-point numbers are denoted with a + sign before X, Y, etc.\n•\nThe IEEE-754 negative infinities or floating-point numbers are denoted with a - sign before X, Y, etc.\nCONJ(z) and CIS(z) are defined as follows:\nCONJ(x+i·y)=x-i·y\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1990\n\n\nCIS(y)=cos(y)+i·sin(y).\nThe special value tables show the result of the function for the z argument at the intersection of the RE(z)\ncolumn and the i*IM(z) row. If the function raises an exception on the argument z, the lower part of this\ncell shows the raised exception and the VM Error Status. An empty cell indicates that this argument is normal\nand the result is defined mathematically.\nArithmetic Functions\nArithmetic functions perform the basic mathematical operations like addition, subtraction, multiplication or\ncomputation of the absolute value of the vector elements.\nv?Add\nPerforms element by element addition of vector a and\nvector b.\nSyntax\nvsAdd( n, a, b, y );\nvsAddI(n, a, inca, b, incb, y, incy);\nvmsAdd( n, a, b, y, mode );\nvmsAddI(n, a, inca, b, incb, y, incy, mode);\nvdAdd( n, a, b, y );\nvdAddI(n, a, inca, b, incb, y, incy);\nvmdAdd( n, a, b, y, mode );\nvmdAddI(n, a, inca, b, incb, y, incy, mode);\nvcAdd( n, a, b, y );\nvcAddI(n, a, inca, b, incb, y, incy);\nvmcAdd( n, a, b, y, mode );\nvmcAddI(n, a, inca, b, incb, y, incy, mode);\nvzAdd( n, a, b, y );\nvzAddI(n, a, inca, b, incb, y, incy);\nvmzAdd( n, a, b, y, mode );\nvmzAddI(n, a, inca, b, incb, y, incy, mode);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nn\nconst MKL_INT\nSpecifies the number of elements to be\ncalculated.\na, b\nconst float* for vsAdd, vmsAdd\nconst double* for vdAdd, vmdAdd\nPointers to arrays that contain the input vectors\na and b.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1991\n\n\nName\nType\nDescription\nconst MKL_Complex8* for vcAdd,\nvmcAdd\nconst MKL_Complex16* for vzAdd,\nvmzAdd\ninca, incb,\nincy\nconst MKL_INT\nSpecifies increments for the elements of a, b,\nand y.\nmode\nconst MKL_INT64\nOverrides global VM mode setting for this\nfunction call. See vmlSetMode for possible\nvalues and their description.\nOutput Parameters\nName\nType\nDescription\ny\nfloat* for vsAdd, vmsAdd\ndouble* for vdAdd, vmdAdd\nMKL_Complex8* for vcAdd, vmcAdd\nMKL_Complex16* for vzAdd, vmzAdd\nPointer to an array that contains the output\nvector y.\nDescription\nThe v?Add function performs element by element addition of vector a and vector b.\nSpecial values for Real Function v?Add\nArgument 1\nArgument 2\nResult\nException\n+0\n+0\n+0\n \n+0\n-0\n+0\n \n-0\n+0\n+0\n \n-0\n-0\n-0\n \n+∞\n+∞\n+∞\n \n+∞\n-∞\nQNAN\nINVALID\n-∞\n+∞\nQNAN\nINVALID\n-∞\n-∞\n-∞\n \nSNAN\nany value\nQNAN\nINVALID\nany value\nSNAN\nQNAN\nINVALID\nQNAN\nnon-SNAN\nQNAN\n \nnon-SNAN\nQNAN\nQNAN\n \nSpecifications for special values of the complex functions are defined according to the following formula\nAdd(x1+i*y1,x2+i*y2) = (x1+x2) + i*(y1+y2)\nOverflow in a complex function occurs (supported in the HA/LA accuracy modes only) when all RE(x), RE(y),\nIM(x), IM(y) arguments are finite numbers, but the real or imaginary part of the computed result is so\nlarge that it does not fit the target precision. In this case, the function returns ∞ in that part of the result,\nraises the OVERFLOW exception, and sets the VM Error Status to VML_STATUS_OVERFLOW (overriding any\npossible VML_STATUS_ACCURACYWARNING status).\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1992\n\n\nv?Sub\nPerforms element by element subtraction of vector b\nfrom vector a.\nSyntax\nvsSub( n, a, b, y );\nvsSubI(n, a, inca, b, incb, y, incy);\nvmsSub( n, a, b, y, mode );\nvmsSubI(n, a, inca, b, incb, y, incy, mode);\nvdSub( n, a, b, y );\nvdSubI(n, a, inca, b, incb, y, incy);\nvmdSub( n, a, b, y, mode );\nvmdSubI(n, a, inca, b, incb, y, incy, mode);\nvcSub( n, a, b, y );\nvcSubI(n, a, inca, b, incb, y, incy);\nvmcSub( n, a, b, y, mode );\nvmcSubI(n, a, inca, b, incb, y, incy, mode);\nvzSub( n, a, b, y );\nvzSubI(n, a, inca, b, incb, y, incy);\nvmzSub( n, a, b, y, mode );\nvmzSubI(n, a, inca, b, incb, y, incy, mode);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nn\nconst MKL_INT\nSpecifies the number of elements to be\ncalculated.\na, b\nconst float* for vsSub, vmsSub\nconst double* for vdSub, vmdSub\nconst MKL_Complex8* for vcSub,\nvmcSub\nconst MKL_Complex16* for vzSub,\nvmzSub\nPointers to arrays that contain the input vectors\na and b.\ninca, incb,\nincy\nconst MKL_INT\nSpecifies increments for the elements of a, b,\nand y.\nmode\nconst MKL_INT64\nOverrides global VM mode setting for this\nfunction call. See vmlSetMode for possible\nvalues and their description.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1993\n\n\nOutput Parameters\nName\nType\nDescription\ny\nfloat* for vsSub, vmsSub\ndouble* for vdSub, vmdSub\nMKL_Complex8* for vcSub, vmcSub\nMKL_Complex16* for vzSub, vmzSub\nPointer to an array that contains the output\nvector y.\nDescription\nThe v?Sub function performs element by element subtraction of vector b from vector a.\nSpecial values for Real Function v?Sub(x)\nArgument 1\nArgument 2\nResult\nException\n+0\n+0\n+0\n \n+0\n-0\n+0\n \n-0\n+0\n-0\n \n-0\n-0\n+0\n \n+∞\n+∞\nQNAN\nINVALID\n+∞\n-∞\n+∞\n \n-∞\n+∞\n-∞\n \n-∞\n-∞\nQNAN\nINVALID\nSNAN\nany value\nQNAN\nINVALID\nany value\nSNAN\nQNAN\nINVALID\nQNAN\nnon-SNAN\nQNAN\n \nnon-SNAN\nQNAN\nQNAN\n \nSpecifications for special values of the complex functions are defined according to the following formula\nSub(x1+i*y1,x2+i*y2) = (x1-x2) + i*(y1-y2).\nOverflow in a complex function occurs (supported in the HA/LA accuracy modes only) when all RE(x), RE(y),\nIM(x), IM(y) arguments are finite numbers, but the real or imaginary part of the computed result is so\nlarge that it does not fit the target precision. In this case, the function returns ∞ in that part of the result,\nraises the OVERFLOW exception, and sets the VM Error Status to VML_STATUS_OVERFLOW (overriding any\npossible VML_STATUS_ACCURACYWARNING status).\nv?Sqr\nPerforms element by element squaring of the vector.\nSyntax\nvsSqr( n, a, y );\nvsSqrI(n, a, inca, y, incy);\nvmsSqr( n, a, y, mode );\nvmsSqrI(n, a, inca, y, incy, mode);\nvdSqr( n, a, y );\nvdSqrI(n, a, inca, y, incy);\nvmdSqr( n, a, y, mode );\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1994\n\n\nvmdSqrI(n, a, inca, y, incy, mode);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nn\nconst MKL_INT\nSpecifies the number of elements to be calculated.\na\nconst float* for vsSqr,\nvmsSqr\nconst double* for vdSqr,\nvmdSqr\nPointer to an array that contains the input vector a.\ninca, incy\nconst MKL_INT\nSpecifies increments for the elements of a and y.\nmode\nconst MKL_INT64\nOverrides global VM mode setting for this function call. See \nvmlSetMode for possible values and their description.\nOutput Parameters\nName\nType\nDescription\ny\nfloat* for vsSqr, vmsSqr\ndouble* for vdSqr, vmdSqr\nPointer to an array that contains the output\nvector y.\nDescription\nThe v?Sqr function performs element by element squaring of the vector.\nSpecial Values for Real Function v?Sqr(x)\nArgument\nResult\nException\n+0\n+0\n \n-0\n+0\n \n+∞\n+∞\n \n-∞\n+∞\n \nQNAN\nQNAN\n \nSNAN\nQNAN\nINVALID\nv?Mul\nPerforms element by element multiplication of vector\na and vector b.\nSyntax\nvsMul( n, a, b, y );\nvsMulI(n, a, inca, b, incb, y, incy);\nvmsMul( n, a, b, y, mode );\nvmsMulI(n, a, inca, b, incb, y, incy, mode);\nvdMul( n, a, b, y );\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1995\n\n\nvdMulI(n, a, inca, b, incb, y, incy);vdMulI(n, a, inca, b, incb, y, incy);\nvmdMul( n, a, b, y, mode );\nvmdMulI(n, a, inca, b, incb, y, incy, mode);\nvcMul( n, a, b, y );\nvcMulI(n, a, inca, b, incb, y, incy);\nvmcMul( n, a, b, y, mode );\nvmcMulI(n, a, inca, b, incb, y, incy, mode);\nvzMul( n, a, b, y );\nvzMulI(n, a, inca, b, incb, y, incy);\nvmzMul( n, a, b, y, mode );\nvmzMulI(n, a, inca, b, incb, y, incy, mode);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nn\nconst MKL_INT\nSpecifies the number of elements to be\ncalculated.\na, b\nconst float* for vsMul, vmsMul\nconst double* for vdMul, vmdMul\nconst MKL_Complex8* for vcMul,\nvmcMul\nconst MKL_Complex16* for vzMul,\nvmzMul\nPointers to arrays that contain the input vectors\na and b.\ninca, incb,\nincy\nconst MKL_INT\nSpecifies increments for the elements of a, b,\nand y.\nmode\nconst MKL_INT64\nOverrides global VM mode setting for this\nfunction call. See vmlSetMode for possible\nvalues and their description.\nOutput Parameters\nName\nType\nDescription\ny\nfloat* for vsMul, vmsMul\ndouble* for vdMul, vmdMul\nMKL_Complex8* for vcMul, vmcMul\nMKL_Complex16* for vzMul, vmzMul\nPointer to an array that contains the output\nvector y.\nDescription\nThe v?Mul function performs element by element multiplication of vector a and vector b.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1996\n\n\nSpecial values for Real Function v?Mul(x)\nArgument 1\nArgument 2\nResult\nException\n+0\n+0\n+0\n \n+0\n-0\n-0\n \n-0\n+0\n-0\n \n-0\n-0\n+0\n \n+0\n+∞\nQNAN\nINVALID\n+0\n-∞\nQNAN\nINVALID\n-0\n+∞\nQNAN\nINVALID\n-0\n-∞\nQNAN\nINVALID\n+∞\n+0\nQNAN\nINVALID\n+∞\n-0\nQNAN\nINVALID\n-∞\n+0\nQNAN\nINVALID\n-∞\n-0\nQNAN\nINVALID\n+∞\n+∞\n+∞\n \n+∞\n-∞\n-∞\n \n-∞\n+∞\n-∞\n \n-∞\n-∞\n+∞\n \nSNAN\nany value\nQNAN\nINVALID\nany value\nSNAN\nQNAN\nINVALID\nQNAN\nnon-SNAN\nQNAN\n \nnon-SNAN\nQNAN\nQNAN\n \nSpecifications for special values of the complex functions are defined according to the following formula\nMul(x1+i*y1,x2+i*y2) = (x1*x2-y1*y2) + i*(x1*y2+y1*x2).\nOverflow in a complex function occurs (supported in the HA/LA accuracy modes only) when all RE(x), RE(y),\nIM(x), IM(y) arguments are finite numbers, but the real or imaginary part of the computed result is so\nlarge that it does not fit the target precision. In this case, the function returns ∞ in that part of the result,\nraises the OVERFLOW exception, and sets the VM Error Status to VML_STATUS_OVERFLOW (overriding any\npossible VML_STATUS_ACCURACYWARNING status).\nv?MulByConj\nPerforms element by element multiplication of vector\na element and conjugated vector b element.\nSyntax\nvcMulByConj( n, a, b, y );\nvsMulByConjI(n, a, inca, b, incb, y, incy);\nvmcMulByConj( n, a, b, y, mode );\nvmsMulByConjI(n, a, inca, b, incb, y, incy, mode);\nvzMulByConj( n, a, b, y );\nvdMulByConjI(n, a, inca, b, incb, y, incy);\nvmzMulByConj( n, a, b, y, mode );\nvmdMulByConjI(n, a, inca, b, incb, y, incy, mode);\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1997\n\n\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nn\nconst MKL_INT\nSpecifies the number of elements to be\ncalculated.\na, b\nconst MKL_Complex8* for\nvcMulByConj, vmcMulByConj\nconst MKL_Complex16* for\nvzMulByConj, vmzMulByConj\nPointers to arrays that contain the input vectors\na and b.\ninca, incb,\nincy\nconst MKL_INT\nSpecifies increments for the elements of a, b,\nand y.\nmode\nconst MKL_INT64\nOverrides global VM mode setting for this\nfunction call. See vmlSetMode for possible\nvalues and their description.\nOutput Parameters\nName\nType\nDescription\ny\nMKL_Complex8* for vcMulByConj,\nvmcMulByConj\nMKL_Complex16* for vzMulByConj,\nvmzMulByConj\nPointer to an array that contains the output\nvector y.\nDescription\nThe v?MulByConj function performs element by element multiplication of vector a element and conjugated\nvector b element.\nSpecifications for special values of the functions are found according to the formula\nMulByConj(x1+i*y1,x2+i*y2) = Mul(x1+i*y1,x2-i*y2).\nOverflow in a complex function occurs (supported in the HA/LA accuracy modes only) when all RE(x), RE(y),\nIM(x), IM(y) arguments are finite numbers, but the real or imaginary part of the computed result is so\nlarge that it does not fit the target precision. In this case, the function returns ∞ in that part of the result,\nraises the OVERFLOW exception, and sets the VM Error Status to VML_STATUS_OVERFLOW (overriding any\npossible VML_STATUS_ACCURACYWARNING status).\nv?Conj\nPerforms element by element conjugation of the\nvector.\nSyntax\nvcConj( n, a, y );\nvcConjI(n, a, inca, y, incy);\nvmcConj( n, a, y, mode );\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n1998\n\n\nvmcConjI(n, a, inca, y, incy, mode);\nvzConj( n, a, y );\nvzConjI(n, a, inca, y, incy);\nvmzConj( n, a, y, mode );\nvmzConjI(n, a, inca, y, incy, mode);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nn\nconst MKL_INT\nSpecifies the number of elements to be\ncalculated.\na\nconst MKL_Complex8* for vcConj,\nvmcConj\nconst MKL_Complex16* for vzConj,\nvmzConj\nPointer to an array that contains the input vector\na.\ninca, incy\nconst MKL_INT\nSpecifies increments for the elements of a and y.\nmode\nconst MKL_INT64\nOverrides global VM mode setting for this\nfunction call. See vmlSetMode for possible\nvalues and their description.\nOutput Parameters\nName\nType\nDescription\ny\nMKL_Complex8* for vcConj, vmcConj\nMKL_Complex16* for vzConj,\nvmzConj\nPointer to an array that contains the output\nvector y.\nDescription\nThe v?Conj function performs element by element conjugation of the vector.\nNo special values are specified. The function does not raise floating-point exceptions.\nv?Abs\nComputes absolute value of vector elements.\nSyntax\nvsAbs( n, a, y );\nvsAbsI(n, a, inca, y, incy);\nvmsAbs( n, a, y, mode );\nvmsAbsI(n, a, inca, y, incy, mode);\nvdAbs( n, a, y );\nvdAbsI(n, a, inca, y, incy);\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n1999\n\n\nvmdAbs( n, a, y, mode );\nvmdAbsI(n, a, inca, y, incy, mode);\nvcAbs( n, a, y );\nvcAbsI(n, a, inca, y, incy);\nvmcAbs( n, a, y, mode );\nvmcAbsI(n, a, inca, y, incy, mode);\nvzAbs( n, a, y );\nvzAbsI(n, a, inca, y, incy);\nvmzAbs( n, a, y, mode );\nvmzAbsI(n, a, inca, y, incy, mode);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nn\nconst MKL_INT\nSpecifies the number of elements to be\ncalculated.\na\nconst float* for vsAbs, vmsAbs\nconst double* for vdAbs, vmdAbs\nconst MKL_Complex8* for vcAbs,\nvmcAbs\nconst MKL_Complex16* for vzAbs,\nvmzAbs\nPointer to an array that contains the input vector\na.\ninca, incy\nconst MKL_INT\nSpecifies increments for the elements of a and y.\nmode\nconst MKL_INT64\nOverrides global VM mode setting for this\nfunction call. See vmlSetMode for possible\nvalues and their description.\nOutput Parameters\nName\nType\nDescription\ny\nfloat* for vsAbs, vmsAbs, vcAbs,\nvmcAbs\ndouble* for vdAbs, vmdAbs, vzAbs,\nvmzAbs\nPointer to an array that contains the output\nvector y.\nDescription\nThe v?Abs function computes an absolute value of vector elements.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2000\n\n\nSpecial Values for Real Function v?Abs(x)\nArgument\nResult\nException\n+0\n+0\n \n-0\n+0\n \n+∞\n+∞\n \n-∞\n+∞\n \nQNAN\nQNAN\n \nSNAN\nQNAN\nINVALID\nSpecifications for special values of the complex functions are defined according to the following formula\nAbs(z) = Hypot(RE(z),IM(z)).\nv?Arg\nComputes argument of vector elements.\nSyntax\nvcArg( n, a, y );\nvcArgI(n, a, inca, y, incy);\nvmcArg( n, a, y, mode );\nvmcArgI(n, a, inca, y, incy, mode);\nvzArg( n, a, y );\nvzArgI(n, a, inca, y, incy);\nvmzArg( n, a, y, mode );\nvmzArgI(n, a, inca, y, incy, mode);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nn\nconst MKL_INT\nSpecifies the number of elements to be\ncalculated.\na\nconst MKL_Complex8* for vcArg,\nvmcArg\nconst MKL_Complex16* for vzArg,\nvmcArg\nPointer to an array that contains the input vector\na.\ninca, incy\nconst MKL_INT\nSpecifies increments for the elements of a and y.\nmode\nconst MKL_INT64\nOverrides global VM mode setting for this\nfunction call. See vmlSetMode for possible\nvalues and their description.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2001\n\n\nOutput Parameters\nName\nType\nDescription\ny\nfloat* for vcArg, vmcArg\ndouble* for vzArg, vmcArg\nPointer to an array that contains the output\nvector y.\nDescription\nThe v?Arg function computes argument of vector elements.\nSee Special Value Notations for the conventions used in the table below.\nSpecial Values for Complex Function v?Arg(z)\nRE(z)\ni·IM(z\n)\n-∞\n \n-X\n \n-0\n \n+0\n \n+X\n \n+∞\n \nNAN\n \n+i·∞\n+3·π/4\n+π/2\n+π/2\n+π/2\n+π/2\n+π/4\nNAN\n+i·Y\n+π\n+π/2\n+π/2\n+0\nNAN\n+i·0\n+π\n+π\n+π\n+0\n+0\n+0\nNAN\n-i·0\n-π\n-π\n-π\n-0\n-0\n-0\nNAN\n-i·Y\n-π\n-π/2\n-π/2\n-0\nNAN\n-i·∞\n-3·π/4\n-π/2\n-π/2\n-π/2\n-π/2\n-π/4\nNAN\n+i·NAN\nNAN\nNAN\nNAN\nNAN\nNAN\nNAN\nNAN\nNotes:\n•\nraises INVALID exception when real or imaginary part of the argument is SNAN\n•\nArg(z)=Atan2(IM(z),RE(z)).\nv?LinearFrac\nPerforms linear fraction transformation of vectors a\nand b with scalar parameters.\nSyntax\nvsLinearFrac( n, a, b, scalea, shifta, scaleb, shiftb, y );\nvsLinearFracI(n, a, inca, b, incb, scalea, shifta, scaleb, shiftb, y, incy);\nvmsLinearFrac( n, a, b, scalea, shifta, scaleb, shiftb, y, mode );\nvmsLinearFracI(n, a, inca, b, incb, scalea, shifta, scaleb, shiftb, y, incy, mode);\nvdLinearFrac( n, a, b, scalea, shifta, scaleb, shiftb, y )\nvdLinearFracI(n, a, inca, b, incb, scalea, shifta, scaleb, shiftb, y, incy);\nvmdLinearFrac( n, a, b, scalea, shifta, scaleb, shiftb, y, mode );\nvmdLinearFracI(n, a, inca, b, incb, scalea, shifta, scaleb, shiftb, y, incy, mode);\nInclude Files\n•\nmkl.h\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2002\n\n\nInput Parameters\nName\nType\nDescription\nn\nconst MKL_INT\nSpecifies the number of elements to be\ncalculated.\na, b\nconst float* for vsLinearFrac,\nvmsLinearFrac\nconst double* for vdLinearFrac,\nvmdLinearFrac\nPointers to arrays that contain the input vectors\na and b.\ninca, incb,\nincy\nconst MKL_INT\nSpecifies increments for the elements of a, b,\nand y.\nscalea, scaleb const float for vsLinearFrac,\nvmsLinearFrac\nconst double for vdLinearFrac,\nvmdLinearFrac\nConstant values for scaling multipliers of vectors\na and b.\nshifta, shiftb const float for vsLinearFrac,\nvmsLinearFrac\nconst double for vdLinearFrac,\nvmdLinearFrac\nConstant values for shifting addends of vectors a\nand b.\nmode\nconst MKL_INT64\nOverrides global VM mode setting for this\nfunction call. See vmlSetMode for possible\nvalues and their description.\nOutput Parameters\nName\nType\nDescription\ny\nfloat* for vsLinearFrac,\nvmsLinearFrac\ndouble* for vdLinearFrac,\nvmdLinearFrac\nPointer to an array that contains the output\nvector y.\nDescription\nThe v?LinearFrac function performs a linear fraction transformation of vector a by vector b with scalar\nparameters: scaling multipliers scalea, scaleb and shifting addends shifta, shiftb:\ny[i]=(scalea·a[i]+shifta)/(scaleb·b[i]+shiftb), i=1,2 … n\nThe v?LinearFrac function is implemented in the EP accuracy mode only, therefore no special values are\ndefined for this function. If used in HA or LA mode, v?LinearFrac sets the VM Error Status to\nVML_STATUS_ACCURACYWARNING (see the Values of the VM Status table). Correctness is guaranteed within\nthe threshold limitations defined for each input parameter (see the table below); otherwise, the behavior is\nunspecified.\n \nThreshold Limitations on Input Parameters\n2EMIN/2≤ |scalea| ≤ 2(EMAX-2)/2\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2003\n\n\nThreshold Limitations on Input Parameters\n2EMIN/2≤ |scaleb| ≤ 2(EMAX-2)/2\n|shifta| ≤ 2EMAX-2\n|shiftb| ≤ 2EMAX-2\n2EMIN/2≤a[i] ≤ 2(EMAX-2)/2\n2EMIN/2≤b[i] ≤ 2(EMAX-2)/2\na[i] ≠ - (shifta/scalea)*(1-δ1), |δ1| ≤ 21-(p-1)/2\nb[i] ≠ - (shiftb/scaleb)*(1-δ2), |δ2| ≤ 21-(p-1)/2\nEMIN and EMAX are the minimum and maximum exponents and p is the number of significant bits (precision)\nfor the corresponding data type according to the ANSI/IEEE Standard 754-2008 ([IEEE754]):\n•\nfor single precision EMIN = -126, EMAX = 127, p = 24\n•\nfor double precision EMIN = -1022, EMAX = 1023, p = 53\nThe thresholds become less strict for common cases with scalea=0 and/or scaleb=0:\n•\nif scalea=0, there are no limitations for the values of a[i] and shifta.\n•\nif scaleb=0, there are no limitations for the values of b[i] and shiftb.\nExample\nTo use the v?LinearFrac to shift vector a by a scalar value, set scaleb to 0. Note that even if scaleb is 0,\nb must be declared.\n#include <stdio.h>\n#include \"mkl_vml.h\"\nint main()\n{\n  double a[10], *b;\n  double r[10];\n  double scalea = 1.0, scaleb = 0.0;\n  double shifta = -1.0, shiftb = 1.0;\n  MKL_INT i=0,n=10;\n  a[0]=-10000.0000;\n  a[1]=-7777.7777;\n  a[2]=-5555.5555;\n  a[3]=-3333.3333;\n  a[4]=-1111.1111;\n  a[5]=1111.1111;\n  a[6]=3333.3333;\n  a[7]=5555.5555;\n  a[8]=7777.7777;\n  a[9]=10000.0000;\n \n  vdLinearFrac( n, a, b, scalea, shifta, scaleb, shiftb, r );\n  for(i=0;i<10;i++) {\n    printf(\"%25.14f %25.14f\\n\",a[i],r[i]);\n  }\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2004\n\n\n  return 0;\n}\nTo use the v?LinearFrac to compute shifta/(scaleb·b[i]+shiftb), set scalea to 0. Note that even if\nscalea is 0, a must be declared.\nv?Fmod\nThe v?Fmod function performs element by element\ncomputation of the modulus function of vector a with\nrespect to vector b.\nSyntax\nvsFmod (n, a, b, y);\nvsFmodI(n, a, inca, b, incb, y, incy);\nvmsFmod (n, a, b, y, mode);\nvmsFmodI(n, a, inca, b, incb, y, incy, mode);\nvdFmod (n, a, b, y);\nvdFmodI(n, a, inca, b, incb, y, incy);\nvmdFmod (n, a, b, y, mode);\nvmdFmodI(n, a, inca, b, incb, y, incy, mode);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nn\nconst MKL_INT\nSpecifies the number of elements to be\ncalculated.\na, b\nconst float* for vsFmod\nconst float* for vmsFmod\nconst double* for vdFmod\nconst double* for vmdFmod\nPointers to arrays containing the input vectors a\nand b.\ninca, incb,\nincy\nconst MKL_INT\nSpecifies increments for the elements of a, b,\nand y.\nmode\nconst MKL_INT64\nOverrides the global VM mode setting for this\nfunction call. See vmlSetMode for possible\nvalues and their description.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2005\n\n\nOutput Parameters\nName\nType\nDescription\ny\nfloat* for vsFmod\nfloat* for vmsFmod\ndouble* for vdFmod\ndouble* for vmdFmod\nPointer to an array containing the output vector\ny.\nDescription\nThe v?Fmod function computes the modulus function of each element of vector a, with respect to the\ncorresponding elements of vector b:\nai - bi*trunc(ai/bi)\nIn general, the modulus function fmod (ai, bi) returns the value ai - n*bi for some integer n such that if\nbi is nonzero, the result has the same sign as ai and a magnitude less than the magnitude of bi.\nSpecial values for Real Function v?Fmod(x, y)\nArgument 1\nArgument 2\nResult\nVM Error Status\nException\nx not NAN\n±0\nNAN\nVML_STATUS_SING\nINVALID\n±∞\ny not NAN\nNAN\nVML_STATUS_SING\nINVALID\n±0\ny≠ 0, not NAN\n±0\n \nx finite\n±∞\nx\nUNDERFLOW if x is\nsubnormal\nNAN\ny\nNAN\nx\nNAN\nNAN\nNOTE\nIf element i in the result of v?Fmod is 0, its sign is that of ai.\nSee Also\nDiv Performs element by element division of vector a by vector b\nRemainder Performs element by element computation of the remainder function on the elements\nof vector a and the corresponding elements of vector b.\nv?Remainder\nPerforms element by element computation of the\nremainder function on the elements of vector a and\nthe corresponding elements of vector b.\nSyntax\nvsRemainder (n, a, b, y);\nvsRemainderI(n, a, inca, b, incb, y, incy);\nvmsRemainder (n, a, b, y, mode);\nvmsRemainderI(n, a, inca, b, incb, y, incy, mode);\nvdRemainder (n, a, b, y);\nvdRemainderI(n, a, inca, b, incb, y, incy);\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2006\n\n\nvmdRemainder (n, a, b, y, mode);\nvmdRemainderI(n, a, inca, b, incb, y, incy, mode);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nn\nconst MKL_INT\nSpecifies the number of elements to be\ncalculated.\na, b\nconst float* for vsRemainder\nconst float* for vmsRemainder\nconst double* for vdRemainder\nconst double* for vmdRemainder\nPointers to arrays containing the input vectors a\nand b.\ninca, incb,\nincy\nconst MKL_INT\nSpecifies increments for the elements of a, b,\nand y.\nmode\nconst MKL_INT64\nOverrides the global VM mode setting for this\nfunction call. See vmlSetMode for possible\nvalues and their description.\nOutput Parameters\nName\nType\nDescription\ny\nfloat* for vsRemainder\nfloat* for vmsRemainder\ndouble* for vdRemainder\ndouble* for vmdRemainder\nPointer to an array containing the output vector\ny.\nDescription\nComputes the remainder of each element of vector a, with respect to the corresponding elements of vector\nb: compute the values of n such that\nn = ai - n*bi\nwhere n is the integer nearest to the exact value of ai/bi. If two integers are equally close to ai/bi, n is the\neven one. If n is zero, it has the same sign as ai.\nSpecial values for Real Function v?Remainder(x, y)\nArgument 1\nArgument 2\nResult\nVM Error Status\nException\nx not NAN\n±0\nNAN\nVML_STATUS_DOM\nINVALID\n±∞\ny not NAN\nNAN\nINVALID\n±0\ny≠ 0, not NAN\n±0\n \nx finite\n±∞\nx\nUNDERFLOW if x is\nsubnormal\nNAN\ny\nNAN\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2007\n\n\nArgument 1\nArgument 2\nResult\nVM Error Status\nException\nx\nNAN\nNAN\nNOTE\nIf element i in the result of v?Remainder is 0, its sign is that of ai.\nSee Also\nDiv Performs element by element division of vector a by vector b\nFmod The v?Fmod function performs element by element computation of the modulus function of\nvector a with respect to vector b.\nPower and Root Functions\nv?Inv\nPerforms element by element inversion of the vector.\nSyntax\nvsInv( n, a, y );\nvsInvI(n, a, inca, y, incy);\nvmsInv( n, a, y, mode );\nvmsInvI(n, a, inca, y, incy, mode);\nvdInv( n, a, y );\nvdInvI(n, a, inca, y, incy);\nvmdInv( n, a, y, mode );\nvmdInvI(n, a, inca, y, incy, mode);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nn\nconst MKL_INT\nSpecifies the number of elements to be calculated.\na\nconst float* for vsInv,\nvmsInv\nconst double* for vdInv,\nvmdInv\nPointer to an array that contains the input vector a.\ninca, incy\nconst MKL_INT\nSpecifies increments for the elements of a and y.\nmode\nconst MKL_INT64\nOverrides global VM mode setting for this function call. See \nvmlSetMode for possible values and their description.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2008\n\n\nOutput Parameters\nName\nType\nDescription\ny\nfloat* for vsInv, vmsInv\ndouble* for vdInv, vmdInv\nPointer to an array that contains the output\nvector y.\nDescription\nThe v?Inv function performs element by element inversion of the vector.\nSpecial Values for Real Function v?Inv(x)\nArgument\nResult\nVM Error Status\nException\n+0\n+∞\nVML_STATUS_SING\nZERODIVIDE\n-0\n-∞\nVML_STATUS_SING\nZERODIVIDE\n+∞\n+0\n \n-∞\n-0\n \nQNAN\nQNAN\n \nSNAN\nQNAN\nINVALID\nv?Div\nPerforms element by element division of vector a by\nvector b\nSyntax\nvsDiv( n, a, b, y );\nvsDivI(n, a, inca, b, incb, y, incy);\nvmsDiv( n, a, b, y, mode );\nvmsDivI(n, a, inca, b, incb, y, incy, mode);\nvdDiv( n, a, b, y );\nvdDivI(n, a, inca, b, incb, y, incy);\nvmdDiv( n, a, b, y, mode );\nvmdDivI(n, a, inca, b, incb, y, incy, mode);\nvcDiv( n, a, b, y );\nvcDivI(n, a, inca, b, incb, y, incy);\nvmcDiv( n, a, b, y, mode );\nvmcDivI(n, a, inca, b, incb, y, incy, mode);\nvzDiv( n, a, b, y );\nvzDivI(n, a, inca, b, incb, y, incy);\nvmzDiv( n, a, b, y, mode );\nvmzDivI(n, a, inca, b, incb, y, incy, mode);\nInclude Files\n•\nmkl.h\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2009\n\n\nInput Parameters\nName\nType\nDescription\nn\nconst MKL_INT\nSpecifies the number of elements to be\ncalculated.\na, b\nconst float* for vsDiv, vmsDiv\nconst double* for vdDiv, vmdDiv\nconst MKL_Complex8* for vcDiv,\nvmcDiv\nconst MKL_Complex16* for vzDiv,\nvmzDiv\nPointers to arrays that contain the input vectors\na and b.\ninca, incb,\nincy\nconst MKL_INT\nSpecifies increments for the elements of a, b,\nand y.\nmode\nconst MKL_INT64\nOverrides global VM mode setting for this\nfunction call. See vmlSetMode for possible\nvalues and their description.\nPrecision Overflow Thresholds for Real v?Div Function\nData Type\nThreshold Limitations on Input Parameters\nsingle precision\nabs(a[i]) < abs(b[i]) * FLT_MAX\ndouble precision\nabs(a[i]) < abs(b[i]) * DBL_MAX\nPrecision overflow thresholds for the complex v?Div function are beyond the scope of this document.\nOutput Parameters\nName\nType\nDescription\ny\nfloat* for vsDiv, vmsDiv\ndouble* for vdDiv, vmdDiv\nMKL_Complex8* for vcDiv, vmcDiv\nMKL_Complex16* for vzDiv, vmzDiv\nPointer to an array that contains the output\nvector y.\nDescription\nThe v?Div function performs element by element division of vector a by vector b.\nSpecial values for Real Function v?Div(x)\nArgument 1\nArgument 2\nResult\nVM Error Status\nException\nX > +0\n+0\n+∞\nVML_STATUS_SING\nZERODIVIDE\nX > +0\n-0\n-∞\nVML_STATUS_SING\nZERODIVIDE\nX < +0\n+0\n-∞\nVML_STATUS_SING\nZERODIVIDE\nX < +0\n-0\n+∞\nVML_STATUS_SING\nZERODIVIDE\n+0\n+0\nQNAN\nVML_STATUS_SING\n \n-0\n-0\nQNAN\nVML_STATUS_SING\n \nX > +0\n+∞\n+0\n \nX > +0\n-∞\n-0\n \n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2010\n\n\nArgument 1\nArgument 2\nResult\nVM Error Status\nException\n+∞\n+∞\nQNAN\n \n-∞\n-∞\nQNAN\n \nQNAN\nQNAN\nQNAN\n \nSNAN\nSNAN\nQNAN\nINVALID\nSpecifications for special values of the complex functions are defined according to the following formula\nDiv(x1+i*y1,x2+i*y2) = (x1+i*y1)*(x2-i*y2)/(x2*x2+y2*y2).\nOverflow in a complex function occurs when x2+i*y2 is not zero, x1, x2, y1, y2 are finite numbers, but the\nreal or imaginary part of the exact result is so large that it does not fit the target precision. In that case, the\nfunction returns ∞ in that part of the result, raises the OVERFLOW exception, and sets the VM Error Status to\nVML_STATUS_OVERFLOW.\nv?Sqrt\nComputes a square root of vector elements.\nSyntax\nvsSqrt( n, a, y );\nvsSqrtI(n, a, inca, y, incy);\nvmsSqrt( n, a, y, mode );\nvmsSqrtI(n, a, inca, y, incy, mode);\nvdSqrt( n, a, y );\nvdSqrtI(n, a, inca, y, incy);\nvmdSqrt( n, a, y, mode );\nvmdSqrtI(n, a, inca, y, incy, mode);\nvcSqrt( n, a, y );\nvcSqrtI(n, a, inca, y, incy);\nvmcSqrt( n, a, y, mode );\nvmcSqrtI(n, a, inca, y, incy, mode);\nvzSqrt( n, a, y );\nvzSqrtI(n, a, inca, y, incy);\nvmzSqrt( n, a, y, mode );\nvmzSqrtI(n, a, inca, y, incy, mode);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nn\nconst MKL_INT\nSpecifies the number of elements to be\ncalculated.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2011\n\n\nName\nType\nDescription\na\nconst float* for vsSqrt, vmsSqrt\nconst double* for vdSqrt, vmdSqrt\nconst MKL_Complex8* for vcSqrt,\nvmcSqrt\nconst MKL_Complex16* for vzSqrt,\nvmzSqrt\nPointer to an array that contains the input vector\na.\ninca, incy\nconst MKL_INT\nSpecifies increments for the elements of a and y.\nmode\nconst MKL_INT64\nOverrides global VM mode setting for this\nfunction call. See vmlSetMode for possible\nvalues and their description.\nOutput Parameters\nName\nType\nDescription\ny\nfloat* for vsSqrt, vmsSqrt\ndouble* for vdSqrt, vmdSqrt\nMKL_Complex8* for vcSqrt, vmcSqrt\nMKL_Complex16* for vzSqrt,\nvmzSqrt\nPointer to an array that contains the output\nvector y.\nDescription\nThe v?Sqrt function computes a square root of vector elements.\nSpecial Values for Real Function v?Sqrt(x)\nArgument\nResult\nVM Error Status\nException\nX < +0\nQNAN\nVML_STATUS_ERRDOM\nINVALID\n+0\n+0\n \n \n-0\n-0\n \n \n-∞\nQNAN\nVML_STATUS_ERRDOM\nINVALID\n+∞\n+∞\n \n \nQNAN\nQNAN\n \n \nSNAN\nQNAN\n \nINVALID\nSee Special Value Notations for the conventions used in the table below.\nSpecial Values for Complex Function v?Sqrt(z)\nRE(z)\ni·IM(z)\n-∞\n \n-X\n \n-0\n \n+0\n \n+X\n \n+∞\n \nNAN\n \n+i·∞\n+∞+i·∞\n+∞+i·∞\n+∞+i·∞\n+∞+i·∞\n+∞+i·∞\n+∞+i·∞\n+∞+i·∞\n+i·Y\n+0+i·∞\n+∞+i·0\nQNAN+i·QNAN\n+i·0\n+0+i·∞\n+0+i·0\n+0+i·0\n+∞+i·0\nQNAN+i·QNAN\n-i·0\n+0-i·∞\n+0-i·0\n+0-i·0\n+∞-i·0\nQNAN+i·QNAN\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2012\n\n\nRE(z)\ni·IM(z)\n-∞\n \n-X\n \n-0\n \n+0\n \n+X\n \n+∞\n \nNAN\n \n-i·Y\n+0-i·∞\n+∞-i·0\nQNAN+i·QNAN\n-i·∞\n+∞-i·∞\n+∞-i·∞\n+∞-i·∞\n+∞-i·∞\n+∞-i·∞\n+∞-i·∞\n+∞-i·∞\n+i·NAN\nQNAN+i·QNAN\nQNAN+i·QNAN\nQNAN+i·QNAN\nQNAN+i·QNAN\nQNAN+i·QNAN\n+∞+i·QNAN\nQNAN+i·QNAN\nNotes:\n•\nraises INVALID exception when the real or imaginary part of the argument is SNAN\n•\nSqrt(CONJ(z))=CONJ(Sqrt(z)).\nv?InvSqrt\nComputes an inverse square root of vector elements.\nSyntax\nvsInvSqrt( n, a, y );\nvsInvSqrtI(n, a, inca, y, incy);\nvmsInvSqrt( n, a, y, mode );\nvmsInvSqrtI(n, a, inca, y, incy, mode);\nvdInvSqrt( n, a, y );\nvdInvSqrtI(n, a, inca, y, incy);\nvmdInvSqrt( n, a, y, mode );\nvmdInvSqrtI(n, a, inca, y, incy, mode);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nn\nconst MKL_INT\nSpecifies the number of elements to be\ncalculated.\na\nconst float* for vsInvSqrt,\nvmsInvSqrt\nconst double* for vdInvSqrt,\nvmdInvSqrt\nPointer to an array that contains the input vector\na.\ninca, incy\nconst MKL_INT\nSpecifies increments for the elements of a and y.\nmode\nconst MKL_INT64\nOverrides global VM mode setting for this\nfunction call. See vmlSetMode for possible\nvalues and their description.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2013\n\n\nOutput Parameters\nName\nType\nDescription\ny\nfloat* for vsInvSqrt, vmsInvSqrt\ndouble* for vdInvSqrt, vmdInvSqrt\nPointer to an array that contains the output\nvector y.\nDescription\nThe v?InvSqrt function computes an inverse square root of vector elements.\nSpecial Values for Real Function v?InvSqrt(x)\nArgument\nResult\nVM Error Status\nException\nX < +0\nQNAN\nVML_STATUS_ERRDOM\nINVALID\n+0\n+∞\nVML_STATUS_SING\nZERODIVIDE\n-0\n-∞\nVML_STATUS_SING\nZERODIVIDE\n-∞\nQNAN\nVML_STATUS_ERRDOM\nINVALID\n+∞\n+0\n \n \nQNAN\nQNAN\n \n \nSNAN\nQNAN\n \nINVALID\nv?Cbrt\nComputes a cube root of vector elements.\nSyntax\nvsCbrt( n, a, y );\nvsCbrtI(n, a, inca, y, incy);\nvmsCbrt( n, a, y, mode );\nvmsCbrtI(n, a, inca, y, incy, mode);\nvdCbrt( n, a, y );\nvdCbrtI(n, a, inca, y, incy);\nvmdCbrt( n, a, y, mode );\nvmdCbrtI(n, a, inca, y, incy, mode);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nn\nconst MKL_INT\nSpecifies the number of elements to be\ncalculated.\na\nconst float* for vsCbrt, vmsCbrt\nconst double* for vdCbrt, vmdCbrt\nPointer to an array that contains the input vector\na.\ninca, incy\nconst MKL_INT\nSpecifies increments for the elements of a and y.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2014\n\n\nName\nType\nDescription\nmode\nconst MKL_INT64\nOverrides global VM mode setting for this\nfunction call. See vmlSetMode for possible\nvalues and their description.\nOutput Parameters\nName\nType\nDescription\ny\nfloat* for vsCbrt, vmsCbrt\ndouble* for vdCbrt, vmdCbrt\nPointer to an array that contains the output\nvector y.\nDescription\nThe v?Cbrt function computes a cube root of vector elements.\nSpecial Values for Real Function v?Cbrt(x)\nArgument\nResult\nException\n+0\n+0\n \n-0\n-0\n \n+∞\n+∞\n \n-∞\n-∞\n \nQNAN\nQNAN\n \nSNAN\nQNAN\nINVALID\nv?InvCbrt\nComputes an inverse cube root of vector elements.\nSyntax\nvsInvCbrt( n, a, y );\nvsInvCbrtI(n, a, inca, y, incy);\nvmsInvCbrt( n, a, y, mode );\nvmsInvCbrtI(n, a, inca, y, incy, mode);\nvdInvCbrt( n, a, y );\nvdInvCbrtI(n, a, inca, y, incy);\nvmdInvCbrt( n, a, y, mode );\nvmdInvCbrtI(n, a, inca, y, incy, mode);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nn\nconst MKL_INT\nSpecifies the number of elements to be\ncalculated.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2015\n\n\nName\nType\nDescription\na\nconst float* for vsInvCbrt,\nvmsInvCbrt\nconst double* for vdInvCbrt,\nvmdInvCbrt\nPointer to an array that contains the input vector\na.\ninca, incy\nconst MKL_INT\nSpecifies increments for the elements of a and y.\nmode\nconst MKL_INT64\nOverrides global VM mode setting for this\nfunction call. See vmlSetMode for possible\nvalues and their description.\nOutput Parameters\nName\nType\nDescription\ny\nfloat* for vsInvCbrt, vmsInvCbrt\ndouble* for vdInvCbrt, vmdInvCbrt\nPointer to an array that contains the output\nvector y.\nDescription\nThe v?InvCbrt function computes an inverse cube root of vector elements.\nSpecial Values for Real Function v?InvCbrt(x)\nArgument\nResult\nVM Error Status\nException\n+0\n+∞\nVML_STATUS_SING\nZERODIVIDE\n-0\n-∞\nVML_STATUS_SING\nZERODIVIDE\n+∞\n+0\n \n-∞\n-0\n \nQNAN\nQNAN\n \nSNAN\nQNAN\nINVALID\nv?Pow2o3\nComputes the cube root of the square of each vector\nelement.\nSyntax\nvsPow2o3( n, a, y );\nvsPow2o3I(n, a, inca, y, incy);\nvmsPow2o3( n, a, y, mode );\nvmsPow2o3I(n, a, inca, y, incy, mode);\nvdPow2o3( n, a, y );\nvdPow2o3I(n, a, inca, y, incy);\nvmdPow2o3( n, a, y, mode );\nvmdPow2o3I(n, a, inca, y, incy, mode);\nInclude Files\n•\nmkl.h\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2016\n\n\nInput Parameters\nName\nType\nDescription\nn\nconst MKL_INT\nSpecifies the number of elements to be\ncalculated.\na\nconst float* for vsPow2o3,\nvmsPow2o3\nconst double* for vdPow2o3,\nvmdPow2o3\nPointers to arrays that contain the input vector\na.\ninca, incy\nconst MKL_INT\nSpecifies increments for the elements of a and y.\nmode\nconst MKL_INT64\nOverrides global VM mode setting for this\nfunction call. See vmlSetMode for possible\nvalues and their description.\nOutput Parameters\nName\nType\nDescription\ny\nfloat* for vsPow2o3, vmsPow2o3\ndouble* for vdPow2o3, vmdPow2o3\nPointer to an array that contains the output\nvector y.\nDescription\nThe v?Pow2o3 function computes the cube root of the square of each vector element.\nSpecial Values for Real Function v?Pow2o3(x)\nArgument\nResult\nException\n+0\n+0\n \n-0\n+0\n \n+∞\n+∞\n \n-∞\n+∞\n \nQNAN\nQNAN\n \nSNAN\nQNAN\nINVALID\nv?Pow3o2\nComputes the square root of the cube of each vector\nelement.\nSyntax\nvsPow3o2( n, a, y );\nvsPow3o2I(n, a, inca, y, incy);\nvmsPow3o2( n, a, y, mode );\nvmsPow3o2I(n, a, inca, y, incy, mode);\nvdPow3o2( n, a, y );\nvdPow3o2I(n, a, inca, y, incy);\nvmdPow3o2( n, a, y, mode );\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2017\n\n\nvmdPow3o2I(n, a, inca, y, incy, mode);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nn\nconst MKL_INT\nSpecifies the number of elements to be\ncalculated.\na\nconst float* for vsPow3o2,\nvmsPow3o2\nconst double* for vdPow3o2,\nvmdPow3o2\nPointers to arrays that contain the input vector\na.\ninca, incy\nconst MKL_INT\nSpecifies increments for the elements of a and y.\nmode\nconst MKL_INT64\nOverrides global VM mode setting for this\nfunction call. See vmlSetMode for possible\nvalues and their description.\nPrecision Overflow Thresholds for Pow3o2 Function\nData Type\nThreshold Limitations on Input Parameters\nsingle precision\nabs(a[i]) < ( FLT_MAX )2/3\ndouble precision\nabs(a[i]) < ( DBL_MAX )2/3\nOutput Parameters\nName\nType\nDescription\ny\nfloat* for vsPow3o2, vmsPow3o2\ndouble* for vdPow3o2, vmdPow3o2\nPointer to an array that contains the output\nvector y.\nDescription\nThe v?Pow3o2 function computes the square root of the cube of each vector element.\nSpecial Values for Real Function v?Pow3o2(x)\nArgument\nResult\nVM Error Status\nException\nX < +0\nQNAN\nVML_STATUS_ERRDOM\nINVALID\n+0\n+0\n \n \n-0\n-0\n \n \n-∞\nQNAN\nVML_STATUS_ERRDOM\nINVALID\n+∞\n+∞\n \n \nQNAN\nQNAN\n \n \nSNAN\nQNAN\n \nINVALID\nv?Pow\nComputes a to the power b for elements of two\nvectors.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2018\n\n\nSyntax\nvsPow( n, a, b, y );\nvsPowI(n, a, inca, b, incb, y, incy);\nvmsPow( n, a, b, y, mode );\nvmsPowI(n, a, inca, b, incb, y, incy, mode);\nvdPow( n, a, b, y );\nvdPowI(n, a, inca, b, incb, y, incy);\nvmdPow( n, a, b, y, mode );\nvmdPowI(n, a, inca, b, incb, y, incy, mode);\nvcPow( n, a, b, y );\nvcPowI(n, a, inca, b, incb, y, incy);\nvmcPow( n, a, b, y, mode );\nvmcPowI(n, a, inca, b, incb, y, incy, mode);\nvzPow( n, a, b, y );\nvzPowI(n, a, inca, b, incb, y, incy);\nvmzPow( n, a, b, y, mode );\nvmzPowI(n, a, inca, b, incb, y, incy, mode);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nn\nconst MKL_INT\nSpecifies the number of elements to be\ncalculated.\na, b\nconst float* for vsPow, vmsPow\nconst double* for vdPow, vmdPow\nconst MKL_Complex8* for vcPow,\nvmcPow\nconst MKL_Complex16* for vzPow,\nvmzPow\nPointers to arrays that contain the input vectors\na and b.\ninca, incb,\nincy\nconst MKL_INT\nSpecifies increments for the elements of a, b,\nand y.\nmode\nconst MKL_INT64\nOverrides global VM mode setting for this\nfunction call. See vmlSetMode for possible\nvalues and their description.\nPrecision Overflow Thresholds for Real v?Pow Function\nData Type\nThreshold Limitations on Input Parameters\nsingle precision\nabs(a[i]) < ( FLT_MAX )1/b[i]\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2019\n\n\nData Type\nThreshold Limitations on Input Parameters\ndouble precision\nabs(a[i]) < ( DBL_MAX )1/b[i]\nPrecision overflow thresholds for the complex v?Pow function are beyond the scope of this document.\nOutput Parameters\nName\nType\nDescription\ny\nfloat* for vsPow, vmsPow\ndouble* for vdPow, vmdPow\nMKL_Complex8* for vcPow, vmcPow\nMKL_Complex16* for vzPow, vmzPow\nPointer to an array that contains the output\nvector y.\nDescription\nThe v?Pow function computes a to the power b for elements of two vectors.\nThe real function v(s/d)Pow has certain limitations on the input range of a and b parameters. Specifically, if\na[i] is positive, then b[i] may be arbitrary. For negative a[i], the value of b[i] must be an integer\n(either positive or negative).\nThe complex function v(c/z)Pow has no input range limitations.\nSpecial values for Real Function v?Pow(x,y)\nArgument 1\n(X)\nArgument 2\n(Y)\nResult\nVM Error Status\nException\n+0\nneg. odd integer\n+∞\nVML_STATUS_ERRDOM\nZERODIVIDE\n-0\nneg. odd integer\n-∞\nVML_STATUS_ERRDOM\nZERODIVIDE\n+0\nneg. even integer\n+∞\nVML_STATUS_ERRDOM\nZERODIVIDE\n-0\nneg. even integer\n+∞\nVML_STATUS_ERRDOM\nZERODIVIDE\n+0\nneg. non-integer\n+∞\nVML_STATUS_ERRDOM\nZERODIVIDE\n-0\nneg. non-integer\n+∞\nVML_STATUS_ERRDOM\nZERODIVIDE\n-0\npos. odd integer\n+0\n \n \n-0\npos. odd integer\n-0\n \n \n+0\npos. even integer\n+0\n \n \n-0\npos. even integer\n+0\n \n \n+0\npos. non-integer\n+0\n \n \n-0\npos. non-integer\n+0\n \n \n-1\n+∞\n+1\n \n \n-1\n-∞\n+1\n \n \n+1\nany value\n+1\n \n \n+1\n+0\n+1\n \n \n+1\n-0\n+1\n \n \n+1\n+∞\n+1\n \n \n+1\n-∞\n+1\n \n \n+1\nQNAN\n+1\n \n \nany value\n+0\n+1\n \n \n+0\n+0\n+1\n \n \n-0\n+0\n+1\n \n \n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2020\n\n\nArgument 1\n(X)\nArgument 2\n(Y)\nResult\nVM Error Status\nException\n+∞\n+0\n+1\n \n \n-∞\n+0\n+1\n \n \nQNAN\n+0\n+1\n \n \nany value\n-0\n+1\n \n \n+0\n-0\n+1\n \n \n-0\n-0\n+1\n \n \n+∞\n-0\n+1\n \n \n-∞\n-0\n+1\n \n \nQNAN\n-0\n+1\n \n \nX < +0\nnon-integer\nQNAN\nVML_STATUS_ERRDOM\nINVALID\n|X| < 1\n-∞\n+∞\n \n \n+0\n-∞\n+∞\nVML_STATUS_ERRDOM\nZERODIVIDE\n-0\n-∞\n+∞\nVML_STATUS_ERRDOM\nZERODIVIDE\n|X| > 1\n-∞\n+0\n \n \n+∞\n-∞\n+0\n \n \n-∞\n-∞\n+0\n \n \n|X| < 1\n+∞\n+0\n \n \n+0\n+∞\n+0\n \n \n-0\n+∞\n+0\n \n \n|X| > 1\n+∞\n+∞\n \n \n+∞\n+∞\n+∞\n \n \n-∞\n+∞\n+∞\n \n \n-∞\nneg. odd integer\n-0\n \n \n-∞\nneg. even integer\n+0\n \n \n-∞\nneg. non-integer\n+0\n \n \n-∞\npos. odd integer\n-∞\n \n \n-∞\npos. even integer\n+∞\n \n \n-∞\npos. non-integer\n+∞\n \n \n+∞\nX < +0\n+0\n \n \n+∞\nX > +0\n+∞\n \n \nBig finite value*\nBig finite value*\n+/-∞\nVML_STATUS_OVERFLOW\nOVERFLOW\nQNAN\nQNAN\nQNAN\n \n \nQNAN\nSNAN\nQNAN\n \nINVALID\nSNAN\nQNAN\nQNAN\n \nINVALID\nSNAN\nSNAN\nQNAN\n \nINVALID\nThe complex double precision versions of this function, vzPow and vmzPow, are implemented in the EP\naccuracy mode only. If used in HA or LA mode, vzPow and vmzPow set the VM Error Status to\nVML_STATUS_ACCURACYWARNING (see the Values of the VM Status table).\n* Overflow in a real function is supported only in the HA/LA accuracy modes. The overflow occurs when x and\ny are finite numbers, but the result is too large to fit the target precision. In this case, the function:\n1.\nReturns ∞ in the result.\n2.\nRaises the OVERFLOW exception.\n3.\nSets the VM Error Status to VML_STATUS_OVERFLOW.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2021\n\n\nOverflow in a complex function occurs (supported in the HA/LA accuracy modes only) when all RE(x), RE(y),\nIM(x), IM(y) arguments are finite numbers, but the real or imaginary part of the computed result is so\nlarge that it does not fit the target precision. In this case, the function returns ∞ in that part of the result,\nraises the OVERFLOW exception, and sets the VM Error Status to VML_STATUS_OVERFLOW (overriding any\npossible VML_STATUS_ACCURACYWARNING status).\nv?Powx\nComputes vector a to the scalar power b.\nSyntax\nvsPowx( n, a, b, y );\nvsPowxI(n, a, inca, b, y, incy);\nvmsPowx( n, a, b, y, mode );\nvmsPowxI(n, a, inca, b, y, incy, mode);\nvdPowx( n, a, b, y );\nvdPowxI(n, a, inca, b, y, incy);\nvmdPowx( n, a, b, y, mode );\nvmdPowxI(n, a, inca, b, y, incy, mode);\nvcPowx( n, a, b, y );\nvcPowxI(n, a, inca, b, y, incy);\nvmcPowx( n, a, b, y, mode );\nvmcPowxI(n, a, inca, b, y, incy, mode);\nvzPowx( n, a, b, y );\nvzPowxI(n, a, inca, b, y, incy);\nvmzPowx( n, a, b, y, mode );\nvmzPowxI(n, a, inca, b, y, incy, mode);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nn\nconst MKL_INT\nNumber of elements to be calculated.\na\nconst float* for vsPowx, vmsPowx\nconst double* for vdPowx, vmdPowx\nconst MKL_Complex8* for vcPowx,\nvmcPowx\nconst MKL_Complex16* for vzPowx,\nvmzPowx\nPointer to an array that contains the input vector\na.\nb\nconst float for vsPowx, vmsPowx\nConstant value for power b.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2022\n\n\nName\nType\nDescription\nconst double for vdPowx, vmdPowx\nconst MKL_Complex8 for vcPowx,\nvmcPowx\nconst MKL_Complex16 for vzPowx,\nvmzPowx\ninca, incy\nconst MKL_INT\nSpecifies increments for the elements of a and y.\nmode\nconst MKL_INT64\nOverrides global VM mode setting for this\nfunction call. See vmlSetMode for possible\nvalues and their description.\nPrecision Overflow Thresholds for Real v?Powx Function\nData Type\nThreshold Limitations on Input Parameters\nsingle precision\nabs(a[i]) < ( FLT_MAX )1/b\ndouble precision\nabs(a[i]) < ( DBL_MAX )1/b\nPrecision overflow thresholds for the complex v?Powx function are beyond the scope of this document.\nOutput Parameters\nName\nType\nDescription\ny\nfloat* for vsPowx, vmsPowx\ndouble* for vdPowx, vmdPowx\nMKL_Complex8* for vcPowx, vmcPowx\nMKL_Complex16* for vzPowx,\nvmzPowx\nPointer to an array that contains the output\nvector y.\nDescription\nThe v?Powx function computes a to the power b for a vector a and a scalar b.\nThe real function v(s/d)Powx has certain limitations on the input range of a and b parameters. Specifically,\nif a[i] is positive, then b may be arbitrary. For negative a[i], the value of b must be an integer (either\npositive or negative).\nThe complex function v(c/z)Powx has no input range limitations.\nSpecial values and VM Error Status treatment are the same as for the v?Pow function.\nv?Powr\nComputes a to the power b for elements of two\nvectors, where the elements of vector argument a are\nall non-negative.\nSyntax\nvsPowr (n, a, b, y);\nvsPowrI(n, a, inca, b, incb, y, incy);\nvmsPowr (n, a, b, y, mode);\nvmsPowrI(n, a, inca, b, incb, y, incy, mode);\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2023\n\n\nvdPowr (n, a, b, y);\nvdPowrI(n, a, inca, b, incb, y, incy);\nvmdPowr (n, a, b, y, mode);\nvmdPowrI(n, a, inca, b, incb, y, incy, mode);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nn\nconst MKL_INT\nSpecifies the number of elements to be\ncalculated.\na, b\nconst float* for vsPowr\nconst float* for vmsPowr\nconst double* for vdPowr\nconst double* for vmdPowr\nPointers to arrays containing the input vectors a\nand b.\ninca, incb,\nincy\nconst MKL_INT\nSpecifies increments for the elements of a, b,\nand y.\nmode\nconst MKL_INT64\nOverrides the global VM mode setting for this\nfunction call. See vmlSetMode for possible\nvalues and their description.\nOutput Parameters\nName\nType\nDescription\ny\nfloat* for vsPowr\nfloat* for vmsPowr\ndouble* for vdPowr\ndouble* for vmdPowr\nPointer to an array containing the output vector\ny.\nDescription\nThe v?Powr function raises each element of vector a by the corresponding element of vector b. The elements\nof a are all nonnegative (ai≥ 0).\nPrecision Overflow Thresholds for Real Function v?Powr\nData Type\nThreshold Limitations on Input Parameters\nsingle precision\nai < (FLT_MAX)1/bi\ndouble precision\nai < (DBL_MAX)1/bi\nSpecial values and VM Error Status treatment for v?Powr function are the same as for v?Pow, unless\notherwise indicated in this table:\nSpecial values for Real Function v?Powr(x)\nArgument 1\nArgument 2\nResult\nVM Error Status\nException\nx < 0\nany value y\nNAN\nVML_STATUS_ERRDOM\nINVALID\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2024\n\n\nArgument 1\nArgument 2\nResult\nVM Error Status\nException\n0 < x < ∞\n±0\n1\n±0\n-∞ < y < 0\n+∞\n±0\n-∞\n+∞\n±0\ny > 0\n+0\n1\n-∞ < y < ∞\n1\n±0\n±0\nNAN\n+∞\n±0\nNAN\n1\n+∞\nNAN\nx≥ 0\nNAN\nNAN\nNAN\nany value y\nNAN\n0 < x <1\n-∞\n+∞\nx > 1\n-∞\n+0\n0 ≤x < 1\n+∞\n+0\nx > 1\n+∞\n+∞\n+∞\nx < +0\n+0\n+∞\nx > +0\n+∞\nQNAN\nQNAN\nQNAN\nVML_STATUS_ERRDOM\nQNAN\nSNAN\nQNAN\nVML_STATUS_ERRDOM\nINVALID\nSNAN\nQNAN\nQNAN\nVML_STATUS_ERRDOM\nINVALID\nSNAN\nSNAN\nQNAN\nVML_STATUS_ERRDOM\nINVALID\nSee Also\nPow Computes a to the power b for elements of two vectors.\nPowx Computes vector a to the scalar power b.\nv?Hypot\nComputes a square root of sum of two squared\nelements.\nSyntax\nvsHypot( n, a, b, y );\nvsHypotI(n, a, inca, b, incb, y, incy);\nvmsHypot( n, a, b, y, mode );\nvmsHypotI(n, a, inca, b, incb, y, incy, mode);\nvdHypot( n, a, b, y );\nvdHypotI(n, a, inca, b, incb, y, incy);\nvmdHypot( n, a, b, y, mode );\nvmdHypotI(n, a, inca, b, incb, y, incy, mode);\nInclude Files\n•\nmkl.h\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2025\n\n\nInput Parameters\nName\nType\nDescription\nn\nconst MKL_INT\nNumber of elements to be calculated.\na, b\nconst float* for vsHypot,\nvmsHypot\nconst double* for vdHypot,\nvmdHypot\nPointers to arrays that contain the input vectors\na and b.\ninca, incb,\nincy\nconst MKL_INT\nSpecifies increments for the elements of a, b,\nand y.\n \n \n \nmode\nconst MKL_INT64\nOverrides global VM mode setting for this\nfunction call. See vmlSetMode for possible\nvalues and their description.\nPrecision Overflow Thresholds for Hypot Function\nData Type\nThreshold Limitations on Input Parameters\nsingle precision\nabs(a[i]) < sqrt(FLT_MAX)\nabs(b[i]) < sqrt(FLT_MAX)\ndouble precision\nabs(a[i]) < sqrt(DBL_MAX)\nabs(b[i]) < sqrt(DBL_MAX)\nOutput Parameters\nName\nType\nDescription\ny\nfloat* for vsHypot, vmsHypot\ndouble* for vdHypot, vmdHypot\nPointer to an array that contains the output\nvector y.\nDescription\nThe function v?Hypot computes a square root of sum of two squared elements.\nSpecial values for Real Function v?Hypot(x)\nArgument 1\nArgument 2\nResult\nException\n+0\n+0\n+0\n \n-0\n-0\n+0\n \n+∞\nany value\n+∞\n \nany value\n+∞\n+∞\n \nSNAN\nany value\nQNAN\nINVALID\nany value\nSNAN\nQNAN\nINVALID\nQNAN\nany value\nQNAN\n \nany value\nQNAN\nQNAN\n \nExponential and Logarithmic Functions\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2026\n\n\nv?Exp\nComputes an exponential of vector elements.\nSyntax\nvsExp( n, a, y );\nvsExpI(n, a, inca, y, incy);\nvmsExp( n, a, y, mode );\nvmsExpI(n, a, inca, y, incy, mode);\nvdExp( n, a, y );\nvdExpI(n, a, inca, y, incy);\nvmdExp( n, a, y, mode );\nvmdExpI(n, a, inca, y, incy, mode);\nvcExp( n, a, y );\nvcExpI(n, a, inca, y, incy);\nvmcExp( n, a, y, mode );\nvmcExpI(n, a, inca, y, incy, mode);\nvzExp( n, a, y );\nvzExpI(n, a, inca, y, incy);\nvmzExp( n, a, y, mode );\nvmzExpI(n, a, inca, y, incy, mode);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nn\nconst MKL_INT\nSpecifies the number of elements to be\ncalculated.\na\nconst float* for vsExp, vmsExp\nconst double* for vdExp, vmdExp\nconst MKL_Complex8* for vcExp,\nvmcExp\nconst MKL_Complex16* for vzExp,\nvmzExp\nPointer to an array that contains the input vector\na.\ninca, incy\nconst MKL_INT\nSpecifies increments for the elements of a and y.\nmode\nconst MKL_INT64\nOverrides global VM mode setting for this\nfunction call. See vmlSetMode for possible\nvalues and their description.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2027\n\n\nPrecision Overflow Thresholds for Real v?Exp Function\nData Type\nThreshold Limitations on Input Parameters\nsingle precision\na[i] < Ln( FLT_MAX )\ndouble precision\na[i] < Ln( DBL_MAX )\nPrecision overflow thresholds for the complex v?Exp function are beyond the scope of this document.\nOutput Parameters\nName\nType\nDescription\ny\nfloat* for vsExp, vmsExp\ndouble* for vdExp, vmdExp\nMKL_Complex8* for vcExp, vmcExp\nMKL_Complex16* for vzExp, vmzExp\nPointer to an array that contains the output\nvector y.\nDescription\nThe v?Exp function computes an exponential of vector elements.\nSpecial Values for Real Function v?Exp(x)\nArgument\nResult\nVM Error Status\nException\n+0\n+1\n \n \n-0\n+1\n \n \nX > overflow\n+∞\nVML_STATUS_OVERFLOW\nOVERFLOW\nX < underflow\n+0\nVML_STATUS_UNDERFLOW\nUNDERFLOW\n+∞\n+∞\n \n \n-∞\n+0\n \n \nQNAN\nQNAN\n \n \nSNAN\nQNAN\n \nINVALID\nSee Special Value Notations for the conventions used in the table below.\nSpecial Values for Complex Function v?Exp(z)\nRE(z)\ni·IM(z)\n-∞\n \n-X\n \n-0\n \n+0\n \n+X\n \n+∞\n \nNAN\n \n+i·∞\n \n+0+i·0\n \nQNAN\n+i·QNAN\nINVALID\nQNAN\n+i·QNAN\nINVALID\nQNAN\n+i·QNAN\nINVALID\nQNAN\n+i·QNAN\nINVALID\n+∞\n+i·QNAN\nINVALID\nQNAN\n+i·QNAN\nINVALID\n+i·Y\n+0·CIS(Y)\n+∞·CIS(Y)\nQNAN\n+i·QNAN\n+i·0\n+0·CIS(0)\n+1+i·0\n+1+i·0\n+∞+i·0\nQNAN+i·0\n-i·0\n+0·CIS(0)\n+1-i·0\n+1-i·0\n+∞-i·0\nQNAN-i·0\n-i·Y\n+0·CIS(Y)\n+∞·CIS(Y)\nQNAN\n+i·QNAN\n-i·∞\n \n+0-i·0\n \nQNAN\n+i·QNAN\nINVALID\nQNAN\n+i·QNAN\nINVALID\nQNAN\n+i·QNAN\nINVALID\nQNAN\n+i·QNAN\nINVALID\n+∞\n+i·QNAN\nINVALID\nQNAN\n+i·QNAN\nINVALID\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2028\n\n\nRE(z)\ni·IM(z)\n-∞\n \n-X\n \n-0\n \n+0\n \n+X\n \n+∞\n \nNAN\n \n+i·NA\nN\n+0+i·0\nQNAN\n+i·QNAN\nINVALID\nQNAN\n+i·QNAN\nINVALID\nQNAN\n+i·QNAN\nINVALID\nQNAN\n+i·QNAN\nINVALID\n+∞\n+i·QNAN\nQNAN\n+i·QNAN\nNotes:\n•\nraises the INVALID exception when real or imaginary part of the argument is SNAN\n•\nraises the INVALID exception on argument z=-∞+i·QNAN\n•\nraises the OVERFLOW exception and sets the VM Error Status to VML_STATUS_OVERFLOW in the case of\noverflow, that is, when both RE(z) and IM(z) are finite non-zero numbers, but the real or imaginary part\nof the exact result is so large that it does not meet the target precision.\nv?Exp2\nComputes the base 2 exponential of vector elements.\nSyntax\nvsExp2 (n, a, y);\nvsExp2I(n, a, inca, y, incy);\nvmsExp2 (n, a, y, mode);\nvmsExp2I(n, a, inca, y, incy, mode);\nvdExp2 (n, a, y);\nvdExp2I(n, a, inca, y, incy);\nvmdExp2 (n, a, y, mode);\nvmdExp2I(n, a, inca, y, incy, mode);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nn\nconst MKL_INT\nSpecifies the number of elements to be\ncalculated.\na\nconst float* for vsExp2\nconst float* for vmsExp2\nconst double* for vdExp2\nconst double* for vmdExp2\nPointer to the array containing the input vector\na.\ninca, incy\nconst MKL_INT\nSpecifies increments for the elements of a and y.\nmode\nconst MKL_INT64\nOverrides the global VM mode setting for this\nfunction call. See vmlSetMode for possible\nvalues and their description.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2029\n\n\nOutput Parameters\nName\nType\nDescription\ny\nfloat* for vsExp2\nfloat* for vmsExp2\ndouble* for vdExp2\ndouble* for vmdExp2\nPointer to an array containing the output vector\ny.\nDescription\nThe v?Exp2 function computes the base 2 exponential of vector elements.\nPrecision Overflow Thresholds for Real Function v?Exp2\nData Type\nThreshold Limitations on Input Parameters\nsingle precision\nai < log2(FLT_MAX)\ndouble precision\nai < log2(DBL_MAX)\nSee Special Value Notations for the conventions used in this table:\nSpecial values for Real Function v?Exp2(x)\nArgument\nResult\nVM Error Status\nException\n+0\n+1\n-0\n+1\nx > overflow\n+∞\nVML_STATUS_OVERFLOW\nOVERFLOW\nx < underflow\n+0\nVML_STATUS_UNDERFLOW\nUNDERFLOW\n+∞\n+∞\n-∞\n+0\nQNAN\nQNAN\nSNAN\nQNAN\nINVALID\nSee Also\nExp Computes an exponential of vector elements.\nExp10 Computes the base 10 exponential of vector elements.\nv?Exp10\nComputes the base 10 exponential of vector elements.\nSyntax\nvsExp10 (n, a, y);\nvsExp10I(n, a, inca, y, incy);\nvmsExp10 (n, a, y, mode);\nvmsExp10I(n, a, inca, y, incy, mode);\nvdExp10 (n, a, y);\nvdExp10I(n, a, inca, y, incy);\nvmdExp10 (n, a, y, mode);\nvmdExp10I(n, a, inca, y, incy, mode);\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2030\n\n\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nn\nconst MKL_INT\nSpecifies the number of elements to be\ncalculated.\na\nconst float* for vsExp10\nconst float* for vmsExp10\nconst double* for vdExp10\nconst double* for vmdExp10\nPointer to the array containing the input vector\na.\ninca, incy\nconst MKL_INT\nSpecifies increments for the elements of a and y.\nmode\nconst MKL_INT64\nOverrides the global VM mode setting for this\nfunction call. See vmlSetMode for possible\nvalues and their description.\nOutput Parameters\nName\nType\nDescription\ny\nfloat* for vsExp10\nfloat* for vmsExp10\ndouble* for vdExp10\ndouble* for vmdExp10\nPointer to an array containing the output vector\ny.\nDescription\nThe v?Exp10 function computes the base 10 exponential of vector elements.\nPrecision Overflow Thresholds for Real Function v?Exp10\nData Type\nThreshold Limitations on Input Parameters\nsingle precision\nai < log10(FLT_MAX)\ndouble precision\nai < log10(DBL_MAX)\nSee Special Value Notations for the conventions used in this table:\nSpecial values for Real Function v?Pow(x)\nArgument\nResult\nVM Error Status\nException\n+0\n+1\n-0\n+1\nx > overflow\n+∞\nVML_STATUS_OVERFLOW\nOVERFLOW\nx < underflow\n+0\nVML_STATUS_UNDERFLOW\nUNDERFLOW\n+∞\n+∞\n-∞\n+0\nQNAN\nQNAN\nSNAN\nQNAN\nINVALID\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2031\n\n\nSee Also\nExp Computes an exponential of vector elements.\nExp2 Computes the base 2 exponential of vector elements.\nv?Expm1\nComputes an exponential of vector elements\ndecreased by 1.\nSyntax\nvsExpm1( n, a, y );\nvsExpm1I(n, a, inca, y, incy);\nvmsExpm1( n, a, y, mode );\nvmsExpm1I(n, a, inca, y, incy, mode);\nvdExpm1( n, a, y );\nvdExpm1I(n, a, inca, y, incy);\nvmdExpm1( n, a, y, mode );\nvmdExpm1I(n, a, inca, y, incy, mode);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nn\nconst MKL_INT\nSpecifies the number of elements to be\ncalculated.\na\nconst float* for vsExpm1,\nvmsExpm1\nconst double* for vdExpm1,\nvmdExpm1\nPointer to an array that contains the input vector\na.\ninca, incy\nconst MKL_INT\nSpecifies increments for the elements of a and y.\nmode\nconst MKL_INT64\nOverrides global VM mode setting for this\nfunction call. See vmlSetMode for possible\nvalues and their description.\nPrecision Overflow Thresholds for Expm1 Function\nData Type\nThreshold Limitations on Input Parameters\nsingle precision\na[i] < Ln( FLT_MAX )\ndouble precision\na[i] < Ln( DBL_MAX )\nOutput Parameters\nName\nType\nDescription\ny\nfloat* for vsExpm1, vmsExpm1\ndouble* for vdExpm1, vmdExpm1\nPointer to an array that contains the output\nvector y.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2032\n\n\nDescription\nThe v?Expm1 function computes an exponential of vector elements decreased by 1.\nSpecial Values for Real Function v?Expm1(x)\nArgument\nResult\nVM Error Status\nException\n+0\n+0\n \n \n-0\n+0\n \n \nX > overflow\n+∞\nVML_STATUS_OVERFLOW\nOVERFLOW\n+∞\n+∞\n \n \n-∞\n-1\n \n \nQNAN\nQNAN\n \n \nSNAN\nQNAN\n \nINVALID\nv?Ln\nComputes natural logarithm of vector elements.\nSyntax\nvsLn( n, a, y );\nvsLnI(n, a, inca, y, incy);\nvmsLn( n, a, y, mode );\nvmsLnI(n, a, inca, y, incy, mode);\nvdLn( n, a, y );\nvdLnI(n, a, inca, y, incy);\nvmdLn( n, a, y, mode );\nvmdLnI(n, a, inca, y, incy, mode);\nvcLn( n, a, y );\nvcLnI(n, a, inca, y, incy);\nvmcLn( n, a, y, mode );\nvmcLnI(n, a, inca, y, incy, mode);\nvzLn( n, a, y );\nvzLnI(n, a, inca, y, incy);\nvmzLn( n, a, y, mode );\nvmzLnI(n, a, inca, y, incy, mode);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nn\nconst MKL_INT\nSpecifies the number of elements to be\ncalculated.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2033\n\n\nName\nType\nDescription\na\nconst float* for vsLn, vmsLn\nconst double* for vdLn, vmdLn\nconst MKL_Complex8* for vcLn,\nvmcLn\nconst MKL_Complex16* for vzLn,\nvmzLn\nPointer to an array that contains the input vector\na.\ninca, incy\nconst MKL_INT\nSpecifies increments for the elements of a and y.\nmode\nconst MKL_INT64\nOverrides global VM mode setting for this\nfunction call. See vmlSetMode for possible\nvalues and their description.\nOutput Parameters\nName\nType\nDescription\ny\nfloat* for vsLn, vmsLn\ndouble* for vdLn, vmdLn\nMKL_Complex8* for vcLn, vmcLn\nMKL_Complex16* for vzLn, vmzLn\nPointer to an array that contains the output\nvector y.\nDescription\nThe v?Ln function computes natural logarithm of vector elements.\nSpecial Values for Real Function v?Ln(x)\nArgument\nResult\nVM Error Status\nException\n+1\n+0\n \n \nX < +0\nQNAN\nVML_STATUS_ERRDOM\nINVALID\n+0\n-∞\nVML_STATUS_SING\nZERODIVIDE\n-0\n-∞\nVML_STATUS_SING\nZERODIVIDE\n-∞\nQNAN\nVML_STATUS_ERRDOM\nINVALID\n+∞\n+∞\n \n \nQNAN\nQNAN\n \n \nSNAN\nQNAN\n \nINVALID\nSee Special Value Notations for the conventions used in the table below.\nSpecial Values for Complex Function v?Ln(z)\nRE(z)\ni·IM(z\n)\n-∞\n \n-X\n \n-0\n \n+0\n \n+X\n \n+∞\n \nNAN\n \n+i·∞\n+∞+i·π/2\n+∞+i·π/2\n+∞+i·π/2\n+∞+i·π/2\n+∞+i·π/4\n+∞+i·QNAN\n+i·Y\n+∞+i·π\n+∞+i·0\nQNAN\n+i·QNAN\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2034\n\n\nRE(z)\ni·IM(z\n)\n-∞\n \n-X\n \n-0\n \n+0\n \n+X\n \n+∞\n \nNAN\n \nINVALID\n+i·0\n+∞+i·π\n-∞+i·π\nZERODIVID\nE\n-∞+i·0\nZERODIVID\nE\n+∞+i·0\nQNAN\n+i·QNAN\nINVALID\n-i·0\n+∞-i·π\n-∞-i·π\nZERODIVID\nE\n-∞-i·0\nZERODIVID\nE\n+∞-i·0\nQNAN\n+i·QNAN\nINVALID\n-i·Y\n+∞-i·π\n+∞-i·0\nQNAN\n+i·QNAN\nINVALID\n-i·∞\n+∞-i·π/2\n+∞-i·π/2\n+∞-i·π/2\n+∞-i·π/2\n+∞-i·π/4\n+∞+i·QNAN\n+i·NAN\n+∞\n+i·QNAN\nQNAN\n+i·QNAN\nINVALID\nQNAN\n+i·QNAN\nINVALID\nQNAN\n+i·QNAN\nINVALID\nQNAN\n+i·QNAN\nINVALID\n+∞\n+i·QNAN\nQNAN\n+i·QNAN\nINVALID\nNotes:\n•\nraises INVALID exception when real or imaginary part of the argument is SNAN\nv?Log2\nComputes the base 2 logarithm of vector elements.\nSyntax\nvsLog2 (n, a, y);\nvsLog2I(n, a, inca, y, incy);\nvmsLog2 (n, a, y, mode);\nvmsLog2I(n, a, inca, y, incy, mode);\nvdLog2 (n, a, y);\nvdLog2I(n, a, inca, y, incy);\nvmdLog2 (n, a, y, mode);\nvmdLog2I(n, a, inca, y, incy, mode);\nInclude Files\n•\nmkl.h\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2035\n\n\nInput Parameters\nName\nType\nDescription\nn\nconst MKL_INT\nSpecifies the number of elements to be\ncalculated.\na\nconst float* for vsLog2\nconst float* for vmsLog2\nconst double* for vdLog2\nconst double* for vmdLog2\nPointer to the array containing the input vector\na.\ninca, incy\nconst MKL_INT\nSpecifies increments for the elements of a and y.\nmode\nconst MKL_INT64\nOverrides the global VM mode setting for this\nfunction call. See vmlSetMode for possible\nvalues and their description.\nOutput Parameters\nName\nType\nDescription\ny\nfloat* for vsLog2\nfloat* for vmsLog2\ndouble* for vdLog2\ndouble* for vmdLog2\nPointer to an array containing the output vector\ny.\nDescription\nThe v?Log2 function computes the base 2 logarithm of vector elements.\nSee Special Value Notations for the conventions used in this table:\nSpecial values for Real Function v?Log2(x)\nArgument\nResult\nVM Error Status\nException\n+1\n+0\nx < +0\nQNAN\nVML_STATUS_ERRDOM\nINVALID\n+0\n-∞\nVML_STATUS_SING\nZERODIVIDE\n-0\n-∞\nVML_STATUS_SING\nZERODIVIDE\n-∞\nQNAN\nVML_STATUS_ERRDOM\nINVALID\n+∞\n+∞\nQNAN\nQNAN\nSNAN\nQNAN\nINVALID\nSee Also\nLn Computes natural logarithm of vector elements.\nLog10 Computes the base 10 logarithm of vector elements.\nv?Log10\nComputes the base 10 logarithm of vector elements.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2036\n\n\nSyntax\nvsLog10( n, a, y );\nvsLog10I(n, a, inca, y, incy);\nvmsLog10( n, a, y, mode );\nvmsLog10I(n, a, inca, y, incy, mode);\nvdLog10( n, a, y );\nvdLog10I(n, a, inca, y, incy);\nvmdLog10( n, a, y, mode );\nvmdLog10I(n, a, inca, y, incy, mode);\nvcLog10( n, a, y );\nvcLog10I(n, a, inca, y, incy);\nvmcLog10( n, a, y, mode );\nvmcLog10I(n, a, inca, y, incy, mode);\nvzLog10( n, a, y );\nvzLog10I(n, a, inca, y, incy);\nvmzLog10( n, a, y, mode );\nvmzLog10I(n, a, inca, y, incy, mode);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nn\nconst MKL_INT\nSpecifies the number of elements to be\ncalculated.\na\nconst float* for vsLog10,\nvmsLog10\nconst double* for vdLog10,\nvmdLog10\nconst MKL_Complex8* for vcLog10,\nvmcLog10\nconst MKL_Complex16* for vzLog10,\nvmzLog10\nPointer to an array that contains the input vector\na.\ninca, incy\nconst MKL_INT\nSpecifies increments for the elements of a and y.\nmode\nconst MKL_INT64\nOverrides global VM mode setting for this\nfunction call. See vmlSetMode for possible\nvalues and their description.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2037\n\n\nOutput Parameters\nName\nType\nDescription\ny\nfloat* for vsLog10, vmsLog10\ndouble* for vdLog10, vmdLog10\nMKL_Complex8* for vcLog10,\nvmcLog10\nMKL_Complex16* for vzLog10,\nvmzLog10\nPointer to an array that contains the output\nvector y.\nDescription\nThe v?Log10 function computes the base 10 logarithm of vector elements.\nSpecial Values for Real Function v?Log10(x)\nArgument\nResult\nVM Error Status\nException\n+1\n+0\n \n \nX < +0\nQNAN\nVML_STATUS_ERRDOM\nINVALID\n+0\n-∞\nVML_STATUS_SING\nZERODIVIDE\n-0\n-∞\nVML_STATUS_SING\nZERODIVIDE\n-∞\nQNAN\nVML_STATUS_ERRDOM\nINVALID\n+∞\n+∞\n \n \nQNAN\nQNAN\n \n \nSNAN\nQNAN\n \nINVALID\nSee Special Value Notations for the conventions used in the table below.\nSpecial Values for Complex Function v?Log10(z)\nRE(z)\ni·IM(z\n)\n-∞\n \n-X\n \n-0\n \n+0\n \n+X\n \n+∞\n \nNAN\n \n+i·∞\n+∞+i·QNAN\nINVALID\n+i·Y\n+∞+i·0\nQNAN\n+i·QNAN\nINVALID\n+i·0\nZERODIVID\nE\n-∞+i·0\nZERODIVID\nE\n+∞+i·0\nQNAN\n+i·QNAN\nINVALID\n-i·0\nZERODIVID\nE\n-∞-i·0\nZERODIVID\nE\n+∞-i·0\nQNAN-\ni·QNAN\nINVALID\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2038\n\n\nRE(z)\ni·IM(z\n)\n-∞\n \n-X\n \n-0\n \n+0\n \n+X\n \n+∞\n \nNAN\n \n-i·Y\n+∞-i·0\nQNAN\n+i·QNAN\nINVALID\n-i·∞\n+∞+i·QNAN\n+i·NAN\n+∞\n+i·QNAN\nQNAN\n+i·QNAN\nINVALID\nQNAN\n+i·QNAN\nINVALID\nQNAN\n+i·QNAN\nINVALID\nQNAN\n+i·QNAN\nINVALID\n+∞\n+i·QNAN\nQNAN\n+i·QNAN\nINVALID\nNotes:\n•\nraises INVALID exception when real or imaginary part of the argument is SNAN\nv?Log1p\nComputes a natural logarithm of vector elements that\nare increased by 1.\nSyntax\nvsLog1p( n, a, y );\nvsLog1pI(n, a, inca, y, incy);\nvmsLog1p( n, a, y, mode );\nvmsLog1pI(n, a, inca, y, incy, mode);\nvdLog1p( n, a, y );\nvdLog1pI(n, a, inca, y, incy);\nvmdLog1p( n, a, y, mode );\nvmdLog1pI(n, a, inca, y, incy, mode);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nn\nconst MKL_INT\nSpecifies the number of elements to be\ncalculated.\na\nconst float* for vsLog1p,\nvmsLog1p\nconst double* for vdLog1p,\nvmdLog1p\nPointer to an array that contains the input vector\na.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2039\n\n\nName\nType\nDescription\ninca, incy\nconst MKL_INT\nSpecifies increments for the elements of a and y.\nmode\nconst MKL_INT64\nOverrides global VM mode setting for this\nfunction call. See vmlSetMode for possible\nvalues and their description.\nOutput Parameters\nName\nType\nDescription\ny\nfloat* for vsLog1p, vmsLog1p\ndouble* for vdLog1p, vmdLog1p\nPointer to an array that contains the output\nvector y.\nDescription\nThe v?Log1p function computes a natural logarithm of vector elements that are increased by 1.\nSpecial Values for Real Function v?Log1p(x)\nArgument\nResult\nVM Error Status\nException\n-1\n-∞\nVML_STATUS_SING\nZERODIVIDE\nX < -1\nQNAN\nVML_STATUS_ERRDOM\nINVALID\n+0\n+0\n \n \n-0\n-0\n \n \n-∞\nQNAN\nVML_STATUS_ERRDOM\nINVALID\n+∞\n+∞\n \n \nQNAN\nQNAN\n \n \nSNAN\nQNAN\n \nINVALID\nv?Logb\nComputes the exponents of the elements of input\nvector a.\nSyntax\nvsLogb (n, a, y);\nvsLogbI(n, a, inca, y, incy);\nvmsLogb (n, a, y, mode);\nvmsLogbI(n, a, inca, y, incy, mode);\nvdLogb (n, a, y);\nvdLogbI(n, a, inca, y, incy);\nvmdLogb (n, a, y, mode);\nvmdLogbI(n, a, inca, y, incy, mode);\nInclude Files\n•\nmkl.h\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2040\n\n\nInput Parameters\nName\nType\nDescription\nn\nconst MKL_INT\nSpecifies the number of elements to be\ncalculated.\na\nconst float* for vsLogb\nconst float* for vmsLogb\nconst double* for vdLogb\nconst double* for vmdLogb\nPointer to the array containing the input vector\na.\ninca, incy\nconst MKL_INT\nSpecifies increments for the elements of a and y.\nmode\nconst MKL_INT64\nOverrides the global VM mode setting for this\nfunction call. See vmlSetMode for possible\nvalues and their description.\nOutput Parameters\nName\nType\nDescription\ny\nfloat* for vsLogb\nfloat* for vmsLogb\ndouble* for vdLogb\ndouble* for vmdLogb\nPointer to an array containing the output vector\ny.\nDescription\nThe v?Logb function computes the exponents of the elements of the input vector a. For each element ai of\nvector a, this is the integral part of log2|ai|. The returned value is exact and is independent of the current\nrounding direction mode.\nSee Special Value Notations for the conventions used in this table:\nSpecial values for Real Function v?Logb(x)\nArgument\nResult\nVM Error Status\nException\n+0\n-∞\nVML_STATUS_ERRDOM\nZERODIVIDE\n-0\n-∞\nVML_STATUS_ERRDOM\nZERODIVIDE\n-∞\n+∞\n+∞\n+∞\nQNAN\nQNAN\nSNAN\nQNAN\nINVALID\nTrigonometric Functions\nv?Cos\nComputes cosine of vector elements.\nSyntax\nvsCos( n, a, y );\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2041\n\n\nvsCosI(n, a, inca, y, incy);\nvmsCos( n, a, y, mode );\nvmsCosI(n, a, inca, y, incy, mode);\nvdCos( n, a, y );\nvdCosI(n, a, inca, y, incy);\nvmdCos( n, a, y, mode );\nvmdCosI(n, a, inca, y, incy, mode);\nvcCos( n, a, y );\nvcCosI(n, a, inca, y, incy);\nvmcCos( n, a, y, mode );\nvmcCosI(n, a, inca, y, incy, mode);\nvzCos( n, a, y );\nvzCosI(n, a, inca, y, incy);\nvmzCos( n, a, y, mode );\nvmzCosI(n, a, inca, y, incy, mode);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nn\nconst MKL_INT\nSpecifies the number of elements to be\ncalculated.\na\nconst float* for vsCos, vmsCos\nconst double* for vdCos, vmdCos\nconst MKL_Complex8* for vcCos,\nvmcCos\nconst MKL_Complex16* for vzCos,\nvmzCos\nPointer to an array that contains the input vector\na.\ninca, incy\nconst MKL_INT\nSpecifies increments for the elements of a and y.\nmode\nconst MKL_INT64\nOverrides global VM mode setting for this\nfunction call. See vmlSetMode for possible\nvalues and their description.\nOutput Parameters\nName\nType\nDescription\ny\nfloat* for vsCos, vmsCos\ndouble* for vdCos, vmdCos\nMKL_Complex8* for vcCos, vmcCos\nPointer to an array that contains the output\nvector y.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2042\n\n\nName\nType\nDescription\nMKL_Complex16* for vzCos, vmzCos\nDescription\nThe v?Cos function computes cosine of vector elements.\nNote that arguments abs(a[i]) ≤ 213 and abs(a[i]) ≤ 216 for single and double precisions respectively\nare called fast computational path. These are trigonometric function arguments for which VM provides the\nbest possible performance. Avoid arguments that do not belong to the fast computational path in the VM\nHigh Accuracy (HA) and Low Accuracy (LA) functions. Alternatively, you can use VM Enhanced Performance\n(EP) functions that are fast on the entire function domain. However, these functions provide less accuracy.\nSpecial Values for Real Function v?Cos(x)\nArgument\nResult\nVM Error Status\nException\n+0\n+1\n \n \n-0\n+1\n \n \n+∞\nQNAN\nVML_STATUS_ERRDOM\nINVALID\n-∞\nQNAN\nVML_STATUS_ERRDOM\nINVALID\nQNAN\nQNAN\n \n \nSNAN\nQNAN\n \nINVALID\nSpecifications for special values of the complex functions are defined according to the following formula\nCos(z) = Cosh(i*z).\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nv?Sin\nComputes sine of vector elements.\nSyntax\nvsSin( n, a, y );\nvsSinI(n, a, inca, y, incy);\nvmsSin( n, a, y, mode );\nvmsSinI(n, a, inca, y, incy, mode);\nvdSin( n, a, y );\nvdSinI(n, a, inca, y, incy);\nvmdSin( n, a, y, mode );\nvmdSinI(n, a, inca, y, incy, mode);\nvcSin( n, a, y );\nvcSinI(n, a, inca, y, incy);\nvmcSin( n, a, y, mode );\nvmcSinI(n, a, inca, y, incy, mode);\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2043\n\n\nvzSin( n, a, y );\nvzSinI(n, a, inca, y, incy);\nvmzSin( n, a, y, mode );\nvmzSinI(n, a, inca, y, incy, mode);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nn\nconst MKL_INT\nSpecifies the number of elements to be\ncalculated.\na\nconst float* for vsSin, vmsSin\nconst double* for vdSin, vmdSin\nconst MKL_Complex8* for vcSin,\nvmcSin\nconst MKL_Complex16* for vzSin,\nvmzSin\nPointer to an array that contains the input vector\na.\ninca, incy\nconst MKL_INT\nSpecifies increments for the elements of a and y.\nmode\nconst MKL_INT64\nOverrides global VM mode setting for this\nfunction call. See vmlSetMode for possible\nvalues and their description.\nOutput Parameters\nName\nType\nDescription\ny\nfloat* for vsSin, vmsSin\ndouble* for vdSin, vmdSin\nMKL_Complex8* for vcSin, vmcSin\nMKL_Complex16* for vzSin, vmzSin\nPointer to an array that contains the output\nvector y.\nDescription\nThe function computes sine of vector elements.\nNote that arguments abs(a[i]) ≤ 213 and abs(a[i]) ≤ 216 for single and double precisions respectively\nare called fast computational path. These are trigonometric function arguments for which VM provides the\nbest possible performance. Avoid arguments that do not belong to the fast computational path in the VM\nHigh Accuracy (HA) and Low Accuracy (LA) functions. Alternatively, you can use VM Enhanced Performance\n(EP) functions that are fast on the entire function domain. However, these functions provide less accuracy.\nSpecial Values for Real Function v?Sin(x)\nArgument\nResult\nVM Error Status\nException\n+0\n+0\n \n \n-0\n-0\n \n \n+∞\nQNAN\nVML_STATUS_ERRDOM\nINVALID\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2044\n\n\nArgument\nResult\nVM Error Status\nException\n-∞\nQNAN\nVML_STATUS_ERRDOM\nINVALID\nQNAN\nQNAN\n \n \nSNAN\nQNAN\n \nINVALID\nSpecifications for special values of the complex functions are defined according to the following formula\nSin(z) = -i*Sinh(i*z).\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nv?SinCos\nComputes sine and cosine of vector elements.\nSyntax\nvsSinCos( n, a, y, z );\nvsSinCosI(n, a, inca, y, incy, z, incz);\nvmsSinCos( n, a, y, z, mode );\nvmsSinCosI(n, a, inca, y, incy, z, incz, mode);\nvdSinCos( n, a, y, z );\nvdSinCosI(n, a, inca, y, incy, z, incz);\nvmdSinCos( n, a, y, z, mode );\nvmdSinCosI(n, a, inca, y, incy, z, incz, mode);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nn\nconst MKL_INT\nSpecifies the number of elements to be\ncalculated.\na\nconst float* for vsSinCos,\nvmsSinCos\nconst double* for vdSinCos,\nvmdSinCos\nPointer to an array that contains the input vector\na.\ninca, incy,\nincz\nconst MKL_INT\nSpecifies increments for the elements of a, y,\nand z.\nmode\nconst MKL_INT64\nOverrides global VM mode setting for this\nfunction call. See vmlSetMode for possible\nvalues and their description.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2045\n\n\nOutput Parameters\nName\nType\nDescription\ny, z\nfloat* for vsSinCos, vmsSinCos\ndouble* for vdSinCos, vmdSinCos\nPointers to arrays that contain the output vectors\ny (for sinevalues) and z(for cosine values).\nDescription\nThe function computes sine and cosine of vector elements.\nNote that arguments abs(a[i]) ≤ 213 and abs(a[i]) ≤ 216 for single and double precisions respectively\nare called fast computational path. These are trigonometric function arguments for which VM provides the\nbest possible performance. Avoid arguments that do not belong to the fast computational path in the VM\nHigh Accuracy (HA) and Low Accuracy (LA) functions. Alternatively, you can use VM Enhanced Performance\n(EP) functions that are fast on the entire function domain. However, these functions provide less accuracy.\nSpecial Values for Real Function v?SinCos(x)\nArgument\nResult 1\nResult 2\nVM Error Status\nException\n+0\n+0\n+1\n \n \n-0\n-0\n+1\n \n \n+∞\nQNAN\nQNAN\nVML_STATUS_ERRDOM\nINVALID\n-∞\nQNAN\nQNAN\nVML_STATUS_ERRDOM\nINVALID\nQNAN\nQNAN\nQNAN\n \n \nSNAN\nQNAN\nQNAN\n \nINVALID\nSpecifications for special values of the complex functions are defined according to the following formula\nSin(z) = -i*Sinh(i*z).\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nv?CIS\nComputes complex exponent of real vector elements\n(cosine and sine of real vector elements combined to\ncomplex value).\nSyntax\nvcCIS( n, a, y );\nvcCISI(n, a, inca, y, incy);\nvmcCIS( n, a, y, mode );\nvmcCISI(n, a, inca, y, incy, mode);\nvzCIS( n, a, y );\nvzCISI(n, a, inca, y, incy);\nvmzCIS( n, a, y, mode );\nvmzCISI(n, a, inca, y, incy, mode);\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2046\n\n\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nn\nconst MKL_INT\nSpecifies the number of elements to be\ncalculated.\na\nconst float* for vcCIS, vmcCIS\nconst double* for vzCIS, vmzCIS\nPointer to an array that contains the input vector\na.\ninca, incy\nconst MKL_INT\nSpecifies increments for the elements of a and y.\nmode\nconst MKL_INT64\nOverrides global VM mode setting for this\nfunction call. See vmlSetMode for possible\nvalues and their description.\nOutput Parameters\nName\nType\nDescription\ny\nMKL_Complex8* for vcCIS, vmcCIS\nMKL_Complex16* for vzCIS, vmzCIS\nPointer to an array that contains the output\nvector y.\nDescription\nThe v?CIS function computes complex exponent of real vector elements (cosine and sine of real vector\nelements combined to complex value).\nSee Special Value Notations for the conventions used in the table below.\nSpecial Values for Complex Function v?CIS(x)\nx\nCIS(x)\n+ ∞\nQNAN+i·QNAN\nINVALID\n+ 0\n+1+i·0\n- 0\n+1-i·0\n- ∞\nQNAN+i·QNAN\nINVALID\nNAN\nQNAN+i·QNAN\nNotes:\n•\nraises INVALID exception when the argument is SNAN\n•\nraises INVALID exception and sets the VM Error Status to VML_STATUS_ERRDOM for x=+∞, x=-∞\nv?Tan\nComputes tangent of vector elements.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2047\n\n\nSyntax\nvsTan( n, a, y );\nvsTanI(n, a, inca, y, incy);\nvmsTan( n, a, y, mode );\nvmsTanI(n, a, inca, y, incy, mode);\nvdTan( n, a, y );\nvdTanI(n, a, inca, y, incy);\nvmdTan( n, a, y, mode );\nvmdTanI(n, a, inca, y, incy, mode);\nvcTan( n, a, y );\nvcTanI(n, a, inca, y, incy);\nvmcTan( n, a, y, mode );\nvmcTanI(n, a, inca, y, incy, mode);\nvzTan( n, a, y );\nvzTanI(n, a, inca, y, incy);\nvmzTan( n, a, y, mode );\nvmzTanI(n, a, inca, y, incy, mode);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nn\nconst MKL_INT\nSpecifies the number of elements to be\ncalculated.\na\nconst float* for vsTan, vmsTan\nconst double* for vdTan, vmdTan\nconst MKL_Complex8* for vcTan,\nvmcTan\nconst MKL_Complex16* for vzTan,\nvmzTan\nPointer to an array that contains the input vector\na.\ninca, incy\nconst MKL_INT\nSpecifies increments for the elements of a and y.\nmode\nconst MKL_INT64\nOverrides global VM mode setting for this\nfunction call. See vmlSetMode for possible\nvalues and their description.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2048\n\n\nOutput Parameters\nName\nType\nDescription\ny\nfloat* for vsTan, vmsTan\ndouble* for vdTan, vmdTan\nMKL_Complex8* for vcTan, vmcTan\nMKL_Complex16* for vzTan, vmzTan\nPointer to an array that contains the output\nvector y.\nDescription\nThe v?Tan function computes tangent of vector elements.\nNote that arguments abs(a[i]) ≤ 213 and abs(a[i]) ≤ 216 for single and double precisions respectively\nare called fast computational path. These are trigonometric function arguments for which VM provides the\nbest possible performance. Avoid arguments that do not belong to the fast computational path in the VM\nHigh Accuracy (HA) and Low Accuracy (LA) functions. Alternatively, you can use VM Enhanced Performance\n(EP) functions that are fast on the entire function domain. However, these functions provide less accuracy.\nSpecial Values for Real Function v?Tan(x)\nArgument\nResult\nVM Error Status\nException\n+0\n+0\n \n \n-0\n-0\n \n \n+∞\nQNAN\nVML_STATUS_ERRDOM\nINVALID\n-∞\nQNAN\nVML_STATUS_ERRDOM\nINVALID\nQNAN\nQNAN\n \n \nSNAN\nQNAN\n \nINVALID\nSpecifications for special values of the complex functions are defined according to the following formula\nTan(z) = -i*Tanh(i*z).\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nv?Acos\nComputes inverse cosine of vector elements.\nSyntax\nvsAcos( n, a, y );\nvsAcosI(n, a, inca, y, incy);\nvmsAcos( n, a, y, mode );\nvmsAcosI(n, a, inca, y, incy, mode);\nvdAcos( n, a, y );\nvdAcosI(n, a, inca, y, incy);\nvmdAcos( n, a, y, mode );\nvmdAcosI(n, a, inca, y, incy, mode);\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2049\n\n\nvcAcos( n, a, y );\nvcAcosI(n, a, inca, y, incy);\nvmcAcos( n, a, y, mode );\nvmcAcosI(n, a, inca, y, incy, mode);\nvzAcos( n, a, y );\nvzAcosI(n, a, inca, y, incy);\nvmzAcos( n, a, y, mode );\nvmzAcosI(n, a, inca, y, incy, mode);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nn\nconst int\nSpecifies the number of elements to be\ncalculated.\na\nconst float* for vsAcos, vmsAcos\nconst double* for vdAcos, vmdAcos\nconst MKL_Complex8* for vcAcos,\nvmcAcos\nconst MKL_Complex16* for vzAcos,\nvmzAcos\nPointer to an array that contains the input vector\na.\ninca, incy\nconst MKL_INT\nSpecifies increments for the elements of a and y.\nmode\nconst MKL_INT64\nOverrides global VM mode setting for this\nfunction call. See vmlSetMode for possible\nvalues and their description.\nOutput Parameters\nName\nType\nDescription\ny\nfloat* for vsAcos, vmsAcos\ndouble* for vdAcos, vmdAcos\nMKL_Complex8* for vcAcos, vmcAcos\nMKL_Complex16* for vzAcos,\nvmzAcos\nPointer to an array that contains the output\nvector y.\nDescription\nThe v?Acos function computes inverse cosine of vector elements.\nSpecial Values for Real Function v?Acos(x)\nArgument\nResult\nVM Error Status\nException\n+0\n+π/2\n \n \n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2050\n\n\nArgument\nResult\nVM Error Status\nException\n-0\n+π/2\n \n \n+1\n+0\n \n \n-1\n+π\n \n \n|X| > 1\nQNAN\nVML_STATUS_ERRDOM\nINVALID\n+∞\nQNAN\nVML_STATUS_ERRDOM\nINVALID\n-∞\nQNAN\nVML_STATUS_ERRDOM\nINVALID\nQNAN\nQNAN\n \n \nSNAN\nQNAN\n \nINVALID\nSee Special Value Notations for the conventions used in the table below.\nSpecial Values for Complex Function v?Acos(z)\nRE(z)\ni·IM(z\n)\n-∞\n \n-X\n \n-0\n \n+0\n \n+X\n \n+∞\n \nNAN\n \n+i·∞\n+3·π/4-i·∞\n+π/2-i·∞\n+π/2-i·∞\n+π/2-i·∞\n+π/2-i·∞\n+π/4-i·∞\nQNAN-i·∞\n+i·Y\n+π-i·∞\n+0-i·∞\nQNAN\n+i·QNAN\n+i·0\n+π-i·∞\n+π/2-i·0\n+π/2-i·0\n+0-i·∞\nQNAN\n+i·QNAN\n-i·0\n+π+i·∞\n+π/2+i·0\n+π/2+i·0\n+0+i·∞\nQNAN\n+i·QNAN\n-i·Y\n+π+i·∞\n+0+i·∞\nQNAN\n+i·QNAN\n-i·∞\n+3π/4+i·∞\n+π/2+i·∞\n+π/2+i·∞\n+π/2+i·∞\n+π/2+i·∞\n+π/4+i·∞\nQNAN+i·∞\n+i·NAN\nQNAN+i·∞\nQNAN\n+i·QNAN\n+π/\n2+i·QNAN\n+π/\n2+i·QNAN\nQNAN\n+i·QNAN\nQNAN+i·∞\nQNAN\n+i·QNAN\nNotes:\n•\nraises INVALID exception when real or imaginary part of the argument is SNAN\n•\nAcos(CONJ(z))=CONJ(Acos(z)).\nv?Asin\nComputes inverse sine of vector elements.\nSyntax\nvsAsin( n, a, y );\nvsAsinI(n, a, inca, y, incy);\nvmsAsin( n, a, y, mode );\nvmsAsinI(n, a, inca, y, incy, mode);\nvdAsin( n, a, y );\nvdAsinI(n, a, inca, y, incy);\nvmdAsin( n, a, y, mode );\nvmdAsinI(n, a, inca, y, incy, mode);\nvcAsin( n, a, y );\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2051\n\n\nvcAsinI(n, a, inca, y, incy);\nvmcAsin( n, a, y, mode );\nvmcAsinI(n, a, inca, y, incy, mode);\nvzAsin( n, a, y );\nvzAsinI(n, a, inca, y, incy);\nvmzAsin( n, a, y, mode );\nvmzAsinI(n, a, inca, y, incy, mode);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nn\nconst MKL_INT\nSpecifies the number of elements to be\ncalculated.\na\nconst float* for vsAsin, vmsAsin\nconst double* for vdAsin, vmdAsin\nconst MKL_Complex8* for vcAsin,\nvmcAsin\nconst MKL_Complex16* for vzAsin,\nvmzAsin\nPointer to an array that contains the input vector\na.\ninca, incy\nconst MKL_INT\nSpecifies increments for the elements of a and y.\nmode\nconst MKL_INT64\nOverrides global VM mode setting for this\nfunction call. See vmlSetMode for possible\nvalues and their description.\nOutput Parameters\nName\nType\nDescription\ny\nfloat* for vsAsin, vmsAsin\ndouble* for vdAsin, vmdAsin\nMKL_Complex8* for vcAsin, vmcAsin\nMKL_Complex16* for vzAsin,\nvmzAsin\nPointer to an array that contains the output\nvector y.\nDescription\nThe v?Asin function computes inverse sine of vector elements.\nSpecial Values for Real Function v?Asin(x)\nArgument\nResult\nVM Error Status\nException\n+0\n+0\n \n \n-0\n-0\n \n \n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2052\n\n\nArgument\nResult\nVM Error Status\nException\n+1\n+π/2\n \n \n-1\n-π/2\n \n \n|X| > 1\nQNAN\nVML_STATUS_ERRDOM\nINVALID\n+∞\nQNAN\nVML_STATUS_ERRDOM\nINVALID\n-∞\nQNAN\nVML_STATUS_ERRDOM\nINVALID\nQNAN\nQNAN\n \n \nSNAN\nQNAN\n \nINVALID\nSpecifications for special values of the complex functions are defined according to the following formula\nAsin(z) = -i*Asinh(i*z).\nv?Atan\nComputes inverse tangent of vector elements.\nSyntax\nvsAtan( n, a, y );\nvsAtanI(n, a, inca, y, incy);\nvmsAtan( n, a, y, mode );\nvmsAtanI(n, a, inca, y, incy, mode);\nvdAtan( n, a, y );\nvdAtanI(n, a, inca, y, incy);\nvmdAtan( n, a, y, mode );\nvmdAtanI(n, a, inca, y, incy, mode);\nvcAtan( n, a, y );\nvcAtanI(n, a, inca, y, incy);\nvmcAtan( n, a, y, mode );\nvmcAtanI(n, a, inca, y, incy, mode);\nvzAtan( n, a, y );\nvzAtanI(n, a, inca, y, incy);\nvmzAtan( n, a, y, mode );\nvmzAtanI(n, a, inca, y, incy, mode);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nn\nconst MKL_INT\nSpecifies the number of elements to be\ncalculated.\na\nconst float* for vsAtan, vmsAtan\nPointer to an array that contains the input vector\na.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2053\n\n\nName\nType\nDescription\nconst double* for vdAtan, vmdAtan\nconst MKL_Complex8* for vcAtan,\nvmcAtan\nconst MKL_Complex16* for vzAtan,\nvmzAtan\ninca, incy\nconst MKL_INT\nSpecifies increments for the elements of a and y.\nmode\nconst MKL_INT64\nOverrides global VM mode setting for this\nfunction call. See vmlSetMode for possible\nvalues and their description.\nOutput Parameters\nName\nType\nDescription\ny\nfloat* for vsAtan, vmsAtan\ndouble* for vdAtan, vmdAtan\nMKL_Complex8* for vcAtan, vmcAtan\nMKL_Complex16* for vzAtan,\nvmzAtan\nPointer to an array that contains the output\nvector y.\nDescription\nThe v?Atan function computes inverse tangent of vector elements.\nSpecial Values for Real Function v?Atan(x)\nArgument\nResult\nVM Error Status\nException\n+0\n+0\n \n \n-0\n-0\n \n \n+∞\n+π/2\n \n \n-∞\n-π/2\n \n \nQNAN\nQNAN\n \n \nSNAN\nQNAN\n \nINVALID\nSpecifications for special values of the complex functions are defined according to the following formula\nAtan(z) = -i*Atanh(i*z).\nv?Atan2\nComputes four-quadrant inverse tangent of elements\nof two vectors.\nSyntax\nvsAtan2( n, a, b, y );\nvsAtan2I(n, a, inca, b, incb, y, incy);\nvmsAtan2( n, a, b, y, mode );\nvmsAtan2I(n, a, inca, b, incb, y, incy, mode);\nvdAtan2( n, a, b, y );\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2054\n\n\nvdAtan2I(n, a, inca, b, incb, y, incy);\nvmdAtan2( n, a, b, y, mode );\nvmdAtan2I(n, a, inca, b, incb, y, incy, mode);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nn\nconst MKL_INT\nSpecifies the number of elements to be\ncalculated.\na, b\nconst float* for vsAtan2,\nvmsAtan2\nconst double* for vdAtan2,\nvmdAtan2\nPointers to arrays that contain the input vectors\na and b.\ninca, incb,\nincy\nconst MKL_INT\nSpecifies increments for the elements of a, b and\ny.\nmode\nconst MKL_INT64\nOverrides global VM mode setting for this\nfunction call. See vmlSetMode for possible\nvalues and their description.\nOutput Parameters\nName\nType\nDescription\ny\nfloat* for vsAtan2, vmsAtan2\ndouble* for vdAtan2, vmdAtan2\nPointer to an array that contains the output\nvector y.\nDescription\nThe v?Atan2 function computes four-quadrant inverse tangent of elements of two vectors.\nThe elements of the output vectory are computed as the four-quadrant arctangent of a[i] / b[i].\nSpecial values for Real Function v?Atan2(x)\nArgument 1\nArgument 2\nResult\nException\n-∞\n-∞\n-3*π/4\n \n-∞\nX < +0\n-π/2\n \n-∞\n-0\n-π/2\n \n-∞\n+0\n-π/2\n \n-∞\nX > +0\n-π/2\n \n-∞\n+∞\n-π/4\n \nX < +0\n-∞\n-π\n \nX < +0\n-0\n-π/2\n \nX < +0\n+0\n-π/2\n \nX < +0\n+∞\n-0\n \n-0\n-∞\n-π\n \nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2055\n\n\nArgument 1\nArgument 2\nResult\nException\n-0\nX < +0\n-π\n \n-0\n-0\n-π\n \n-0\n+0\n-0\n \n-0\nX > +0\n-0\n \n-0\n+∞\n-0\n \n+0\n-∞\n+π\n \n+0\nX < +0\n+π\n \n+0\n-0\n+π\n \n+0\n+0\n+0\n \n+0\nX > +0\n+0\n \n+0\n+∞\n+0\n \nX > +0\n-∞\n+π\n \nX > +0\n-0\n+π/2\n \nX > +0\n+0\n+π/2\n \nX > +0\n+∞\n+0\n \n+∞\n-∞\n+3*π/4\n \n+∞\nX < +0\n+π/2\n \n+∞\n-0\n+π/2\n \n+∞\n+0\n+π/2\n \n+∞\nX > +0\n+π/2\n \n+∞\n+∞\n+π/4\n \nX > +0\nQNAN\nQNAN\n \nX > +0\nSNAN\nQNAN\nINVALID\nQNAN\nX > +0\nQNAN\n \nSNAN\nX > +0\nQNAN\nINVALID\nQNAN\nQNAN\nQNAN\n \nQNAN\nSNAN\nQNAN\nINVALID\nSNAN\nQNAN\nQNAN\nINVALID\nSNAN\nSNAN\nQNAN\nINVALID\nv?Cospi\nComputes the cosine of vector elements multiplied by\nπ.\nSyntax\nvsCospi (n, a, y);\nvsCospiI(n, a, inca, y, incy);\nvmsCospi (n, a, y, mode);\nvmsCospiI(n, a, inca, y, incy, mode);\nvdCospi (n, a, y);\nvdCospiI(n, a, inca, y, incy);\nvmdCospi (n, a, y, mode);\nvmdCospiI(n, a, inca, y, incy, mode);\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2056\n\n\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nn\nconst MKL_INT\nSpecifies the number of elements to be\ncalculated.\na\nconst float* for vsCospi\nconst float* for vmsCospi\nconst double* for vdCospi\nconst double* for vmdCospi\nPointer to the array containing the input vector\na.\ninca, incy\nconst MKL_INT\nSpecifies increments for the elements of a and y.\nmode\nconst MKL_INT64\nOverrides the global VM mode setting for this\nfunction call. See vmlSetMode for possible\nvalues and their description.\nOutput Parameters\nName\nType\nDescription\ny\nfloat* for vsCospi\nfloat* for vmsCospi\ndouble* for vdCospi\ndouble* for vmdCospi\nPointer to an array containing the output vector\ny.\nDescription\nThe v?Cospi function computes the cosine of vector elements multiplied by π. For an argument x, the\nfunction computes cos(π*x).\nSpecial values for Real Function v?Cospi(x)\nArgument\nResult\nVM Error Status\nException\n+0\n+1\n-0\n+1\nn + 0.5, for any integer n\nwhere n + 0.5 is\nrepresentable\n+0\n±∞\nQNAN\nVML_STATUS_ERRDOM\nINVALID\nQNAN\nQNAN\nSNAN\nQNAN\nINVALID\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2057\n\n\nApplication Notes\nIf arguments abs(ai) ≤ 222 for single precision or abs(ai ) ≤ 243 for double precision, they belong to the fast\ncomputational path: arguments for which VM provides the best possible performance. Avoid arguments with\ndo not belong to the fast computational path in VM High Accuracy (HA) or Low Accuracy (LA) functions. For\narguments which do not belong to the fast computational path you can use VM Enhanced Performance (EP)\nfunctions, which are fast on the entire function domain. However, these functions provide lower accuracy.\nSee Also\nCos Computes cosine of vector elements.\nCosd Computes the cosine of vector elements multiplied by π/180.\nv?Sinpi\nComputes the sine of vector elements multiplied by π.\nSyntax\nvsSinpi (n, a, y);\nvsSinpiI(n, a, inca, y, incy);\nvmsSinpi (n, a, y, mode);\nvmsSinpiI(n, a, inca, y, incy, mode);\nvdSinpi (n, a, y);\nvdSinpiI(n, a, inca, y, incy);\nvmdSinpi (n, a, y, mode);\nvmdSinpiI(n, a, inca, y, incy, mode);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nn\nconst MKL_INT\nSpecifies the number of elements to be\ncalculated.\na\nconst float* for vsSinpi\nconst float* for vmsSinpi\nconst double* for vdSinpi\nconst double* for vmdSinpi\nPointer to the array containing the input vector\na.\ninca, incy\nconst MKL_INT\nSpecifies increments for the elements of a and y.\nmode\nconst MKL_INT64\nOverrides the global VM mode setting for this\nfunction call. See vmlSetMode for possible\nvalues and their description.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2058\n\n\nOutput Parameters\nName\nType\nDescription\ny\nfloat* for vsSinpi\nfloat* for vmsSinpi\ndouble* for vdSinpi\ndouble* for vmdSinpi\nPointer to an array containing the output vector\ny.\nDescription\nThe v?Sinpi function computes the sine of vector elements multiplied by π. For an argument x, the function\ncomputes sin(π*x).\nSpecial values for Real Function v?Sinpi(x)\nArgument\nResult\nVM Error Status\nException\n+0\n+0\n-0\n-0\n+n, positive integer\n+0\n-n, negative integer\n-0\n±∞\nQNAN\nVML_STATUS_ERRDOM\nINVALID\nQNAN\nQNAN\nSNAN\nQNAN\nINVALID\nApplication Notes\nIf arguments abs(ai) ≤ 222 for single precision or abs(ai) ≤ 251 for double precision, they belong to the fast\ncomputational path: arguments for which VM provides the best possible performance. Avoid arguments with\ndo not belong to the fast computational path in VM High Accuracy (HA) or Low Accuracy (LA) functions. For\narguments which do not belong to the fast computational path you can use VM Enhanced Performance (EP)\nfunctions, which are fast on the entire function domain. However, these functions provide lower accuracy.\nSee Also\nSin  Computes sine of vector elements.\nSind Computes the sine of vector elements multiplied by π/180.\nv?Tanpi\nComputes the tangent of vector elements multiplied\nby π.\nSyntax\nvsTanpi (n, a, y);\nvsTanpiI(n, a, inca, y, incy);\nvmsTanpi (n, a, y, mode);\nvmsTanpiI(n, a, inca, y, incy, mode);\nvdTanpi (n, a, y);\nvdTanpiI(n, a, inca, y, incy);\nvmdTanpi (n, a, y, mode);\nvmdTanpiI(n, a, inca, y, incy, mode);\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2059\n\n\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nn\nconst MKL_INT\nSpecifies the number of elements to be\ncalculated.\na\nconst float* for vsTanpi\nconst float* for vmsTanpi\nconst double* for vdTanpi\nconst double* for vmdTanpi\nPointer to the array containing the input vector\na.\ninca, incy\nconst MKL_INT\nSpecifies increments for the elements of a and y.\nmode\nconst MKL_INT64\nOverrides the global VM mode setting for this\nfunction call. See vmlSetMode for possible\nvalues and their description.\nOutput Parameters\nName\nType\nDescription\ny\nfloat* for vsTanpi\nfloat* for vmsTanpi\ndouble* for vdTanpi\ndouble* for vmdTanpi\nPointer to an array containing the output vector\ny.\nDescription\nThe v?Tanpi function computes the tangent of vector elements multiplied by π. For an argument x, the\nfunction computes tan(π*x).\nSpecial values for Real Function v?Tanpi(x)\nArgument\nResult\nVM Error Status\nException\n+0\n+1\n-0\n+1\n±∞\nQNAN\nVML_STATUS_ERRDOM\nINVALID\nn, even integer\ncopysign(0.0, n)\nn, odd integer\ncopysign(0.0, -n)\nn + 0.5, for n even integer\nand n + 0.5 representable\n+∞\nn + 0.5, for n odd integer\nand n + 0.5 representable\n-∞\nQNAN\nQNAN\nSNAN\nQNAN\nINVALID\nThe copysign(x, y) function returns the first vector argument x with the sign changed to match that of the\nsecond argument y.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2060\n\n\nApplication Notes\nIf arguments abs(ai) ≤ 2 13 for single precision or abs(ai ) ≤ 2 67 for double precision, they belong to the fast\ncomputational path: arguments for which VM provides the best possible performance. Avoid arguments with\ndo not belong to the fast computational path in VM High Accuracy (HA) or Low Accuracy (LA) functions. For\narguments which do not belong to the fast computational path you can use VM Enhanced Performance (EP)\nfunctions, which are fast on the entire function domain. However, these functions provide lower accuracy.\nSee Also\nTan Computes tangent of vector elements.\nTand Computes the tangent of vector elements multiplied by π/180.\nv?Acospi\nComputes the inverse cosine of vector elements\ndivided by π.\nSyntax\nvsAcospi (n, a, y);\nvsAcospiI(n, a, inca, y, incy);\nvmsAcospi (n, a, y, mode);\nvmsAcospiI(n, a, inca, y, incy, mode);\nvdAcospi (n, a, y);\nvdAcospiI(n, a, inca, y, incy);\nvmdAcospi (n, a, y, mode);\nvmdAcospiI(n, a, inca, y, incy, mode);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nn\nconst MKL_INT\nSpecifies the number of elements to be\ncalculated.\na\nconst float* for vsAcospi\nconst float* for vmsAcospi\nconst double* for vdAcospi\nconst double* for vmdAcospi\nPointer to the array containing the input vector\na.\ninca, incy\nconst MKL_INT\nSpecifies increments for the elements of a and y.\nmode\nconst MKL_INT64\nOverrides the global VM mode setting for this\nfunction call. See vmlSetMode for possible\nvalues and their description.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2061\n\n\nOutput Parameters\nName\nType\nDescription\ny\nfloat* for vsAcospi\nfloat* for vmsAcospi\ndouble* for vdAcospi\ndouble* for vmdAcospi\nPointer to an array containing the output vector\ny.\nDescription\nThe v?Acospi function computes the inverse cosine of vector elements divided by π. For an argument x, the\nfunction computes acos(x)/π.\nSee Special Value Notations for the conventions used in this table:\nSpecial values for Real Function v?Acospi(x)\nArgument\nResult\nVM Error Status\nException\n+0\n+1/2\n-0\n+1/2\n+1\n+0\n-1\n+1\n|x| > 1\nQNAN\nVML_STATUS_ERRDOM\nINVALID\n+∞\nQNAN\nVML_STATUS_ERRDOM\nINVALID\n- ∞\nQNAN\nVML_STATUS_ERRDOM\nINVALID\nQNAN\nQNAN\nSNAN\nQNAN\nINVALID\nSee Also\nAcos Computes inverse cosine of vector elements.\nv?Asinpi\nComputes the inverse sine of vector elements divided\nby π.\nSyntax\nvsAsinpi (n, a, y);\nvsAsinpiI(n, a, inca, y, incy);\nvmsAsinpi (n, a, y, mode);\nvmsAsinpiI(n, a, inca, y, incy, mode);\nvdAsinpi (n, a, y);\nvdAsinpiI(n, a, inca, y, incy);\nvmdAsinpi (n, a, y, mode);\nvmdAsinpiI(n, a, inca, y, incy, mode);\nInclude Files\n•\nmkl.h\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2062\n\n\nInput Parameters\nName\nType\nDescription\nn\nconst MKL_INT\nSpecifies the number of elements to be\ncalculated.\na\nconst float* for vsAsinpi\nconst float* for vmsAsinpi\nconst double* for vdAsinpi\nconst double* for vmdAsinpi\nPointer to the array containing the input vector\na.\ninca, incy\nconst MKL_INT\nSpecifies increments for the elements of a and y.\nmode\nconst MKL_INT64\nOverrides the global VM mode setting for this\nfunction call. See vmlSetMode for possible\nvalues and their description.\nOutput Parameters\nName\nType\nDescription\ny\nfloat* for vsAsinpi\nfloat* for vmsAsinpi\ndouble* for vdAsinpi\ndouble* for vmdAsinpi\nPointer to an array containing the output vector\ny.\nDescription\nThe v?Asinpi function computes the inverse sine of vector elements divided by π. For an argument x, the\nfunction computes asin(x)/π.\nSpecial values for Real Function v?Asinpi(x)\nArgument\nResult\nVM Error Status\nException\n+0\n+0\n-0\n-0\n+1\n+1/2\n-1\n-1/2\n|x| > 1\nQNAN\nVML_STATUS_ERRDOM\nINVALID\n+∞\nQNAN\nVML_STATUS_ERRDOM\nINVALID\n-∞\nQNAN\nVML_STATUS_ERRDOM\nINVALID\nQNAN\nQNAN\nSNAN\nQNAN\nINVALID\nSee Also\nAsin Computes inverse sine of vector elements.\nv?Atanpi\nComputes the inverse tangent of vector elements\ndivided by π.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2063\n\n\nSyntax\nvsAtanpi (n, a, y);\nvsAtanpiI(n, a, inca, y, incy);\nvmsAtanpi (n, a, y, mode);\nvmsAtanpiI(n, a, inca, y, incy, mode);\nvdAtanpi (n, a, y);\nvdAtanpiI(n, a, inca, y, incy);\nvmdAtanpi (n, a, y, mode);\nvmdAtanpiI(n, a, inca, y, incy, mode);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nn\nconst MKL_INT\nSpecifies the number of elements to be\ncalculated.\na\nconst float* for vsAtanpi\nconst float* for vmsAtanpi\nconst double* for vdAtanpi\nconst double* for vmdAtanpi\nPointer to the array containing the input vector\na.\ninca, incy\nconst MKL_INT\nSpecifies increments for the elements of a and y.\nmode\nconst MKL_INT64\nOverrides the global VM mode setting for this\nfunction call. See vmlSetMode for possible\nvalues and their description.\nOutput Parameters\nName\nType\nDescription\ny\nfloat* for vsAtanpi\nfloat* for vmsAtanpi\ndouble* for vdAtanpi\ndouble* for vmdAtanpi\nPointer to an array containing the output vector\ny.\nDescription\nThe v?Atanpi function computes the inverse tangent of vector elements divided by π. For an argument x,\nthe function computes atan(x)/π.\nSpecial values for Real Function v?Atanpi(x)\nArgument\nResult\nVM Error Status\nException\n+0\n+0\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2064\n\n\nArgument\nResult\nVM Error Status\nException\n-0\n-0\n+∞\n+1/2\n-∞\n-1/2\nQNAN\nQNAN\nSNAN\nQNAN\nINVALID\nSee Also\nAtan Computes inverse tangent of vector elements.\nv?Atan2pi\nComputes the four-quadrant inverse tangent of the\nratios of the corresponding elements of two vectors\ndivided by π.\nSyntax\nvsAtan2pi (n, a, b, y);\nvsAtan2piI(n, a, inca, b, incb, y, incy);\nvmsAtan2pi (n, a, b, y, mode);\nvmsAtan2piI(n, a, inca, b, incb, y, incy, mode);\nvdAtan2pi (n, a, b, y);\nvdAtan2piI(n, a, inca, b, incb, y, incy);\nvmdAtan2pi (n, a, b, y, mode);\nvmdAtan2piI(n, a, inca, b, incb, y, incy, mode);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nn\nconst MKL_INT\nSpecifies the number of elements to be\ncalculated.\na, b\nconst float* for vsAtan2pi\nconst float* for vmsAtan2pi\nconst double* for vdAtan2pi\nconst double* for vmdAtan2pi\nPointers to the arrays containing the input\nvectors a and b.\ninca, incb,\nincy\nconst MKL_INT\nSpecifies increments for the elements of a, b and\ny.\nmode\nconst MKL_INT64\nOverrides the global VM mode setting for this\nfunction call. See vmlSetMode for possible\nvalues and their description.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2065\n\n\nOutput Parameters\nName\nType\nDescription\ny\nfloat* for vsAtan2pi\nfloat* for vmsAtan2pi\ndouble* for vdAtan2pi\ndouble* for vmdAtan2pi\nPointer to an array containing the output vector\ny.\nDescription\nThe v?Atan2pi function computes the four-quadrant inverse tangent of the ratios of the corresponding\nelements of two vectors divided by π.\nFor the elements of the output vector y, the function computers the four-quadrant arctangent of ai/bi, with\nthe result divided by π.\nSpecial values for Real Function v?Atan2pi(x, y)\nArgument 1\nArgument 2\nResult\nException\n-∞\n-∞\n-3/4\n-∞\nx < +0\n-1/2\n-∞\n-0\n+1/2\n-∞\n+0\n-1/2\n-∞\nx > +0\n-1/2\n-∞\n+∞\n-1/4\nx < +0\n-∞\n-1\nx < +0\n-0\n-1/2\nx < +0\n+0\n-1/2\nx < +0\n+∞\n-0\n-0\n-∞\n-1\n-0\nx < +0\n-1\n-0\n-0\n-1\n-0\n+0\n-0\n-0\nx > +0\n-0\n-0\n+∞\n-0\n+0\n-∞\n+1\n+0\nx < +0\n+1\n+0\n-0\n+1\n+0\n+0\n+0\n+0\nx > +0\n+0\n+0\n+∞\n+0\nx > +0\n-∞\n+1\nx > +0\n-0\n+1/2\nx > +0\n+0\n+1/2\nx > +0\n+∞\n+1/4\n+∞\n-∞\n+3/4\n+∞\nx < +0\n+1/2\n+∞\n-0\n+1/2\n+∞\n+0\n+1/2\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2066\n\n\nArgument 1\nArgument 2\nResult\nException\n+∞\nx > +0\n+1/2\n+∞\n+∞\n+1/4\nx > +0\nQNAN\nQNAN\nx > +0\nSNAN\nQNAN\nINVALID\nQNAN\nx > +0\nQNAN\nSNAN\nx > +0\nQNAN\nINVALID\nQNAN\nQNAN\nQNAN\nQNAN\nSNAN\nQNAN\nINVALID\nSNAN\nQNAN\nQNAN\nINVALID\nSNAN\nSNAN\nQNAN\nINVALID\nSee Also\nAtan2 Computes four-quadrant inverse tangent of elements of two vectors.\nv?Cosd\nComputes the cosine of vector elements multiplied by\nπ/180.\nSyntax\nvsCosd (n, a, y);\nvsCosdI(n, a, inca, y, incy);\nvmsCosd (n, a, y, mode);\nvmsCosdI(n, a, inca, y, incy, mode);\nvdCosd (n, a, y);\nvdCosdI(n, a, inca, y, incy);\nvmdCosd (n, a, y, mode);\nvmdCosdI(n, a, inca, y, incy, mode);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nn\nconst MKL_INT\nSpecifies the number of elements to be\ncalculated.\na\nconst float* for vsCosd\nconst float* for vmsCosd\nconst double* for vdCosd\nconst double* for vmdCosd\nPointer to the array containing the input vector\na.\ninca, incy\nconst MKL_INT\nSpecifies increments for the elements of a and y.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2067\n\n\nName\nType\nDescription\nmode\nconst MKL_INT64\nOverrides the global VM mode setting for this\nfunction call. See vmlSetMode for possible\nvalues and their description.\nOutput Parameters\nName\nType\nDescription\ny\nfloat* for vsCosd\nfloat* for vmsCosd\ndouble* for vdCosd\ndouble* for vmdCosd\nPointer to an array containing the output vector\ny.\nDescription\nThe v?Cosd function computes the cosine of vector elements multiplied by π/180. For an argument x, the\nfunction computes cos(π*x/180).\nSpecial values for Real Function v?Cosd(x)\nArgument\nResult\nVM Error Status\nException\n+0\n+1\n-0\n+1\n±∞\nQNAN\nVML_STATUS_ERRDOM\nINVALID\nQNAN\nQNAN\nSNAN\nQNAN\nINVALID\nApplication Notes\nIf arguments abs(ai) ≤ 224 for single precision or abs(ai ) ≤ 252 for double precision, they belong to the fast\ncomputational path: arguments for which VM provides the best possible performance. Avoid arguments with\ndo not belong to the fast computational path in VM High Accuracy (HA) or Low Accuracy (LA) functions. For\narguments which do not belong to the fast computational path you can use VM Enhanced Performance (EP)\nfunctions, which are fast on the entire function domain. However, these functions provide lower accuracy.\nSee Also\nCos Computes cosine of vector elements.\nCospi Computes the cosine of vector elements multiplied by π.\nv?Sind\nComputes the sine of vector elements multiplied by π/\n180.\nSyntax\nvsSind (n, a, y);\nvsSindI(n, a, inca, y, incy);\nvmsSind (n, a, y, mode);\nvmsSindI(n, a, inca, y, incy, mode);\nvdSind (n, a, y);\nvdSindI(n, a, inca, y, incy);\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2068\n\n\nvmdSind (n, a, y, mode);\nvmdSindI(n, a, inca, y, incy, mode);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nn\nconst MKL_INT\nSpecifies the number of elements to be\ncalculated.\na\nconst float* for vsSind\nconst float* for vmsSind\nconst double* for vdSind\nconst double* for vmdSind\nPointer to the array containing the input vector\na.\ninca, incy\nconst MKL_INT\nSpecifies increments for the elements of a and y.\nmode\nconst MKL_INT64\nOverrides the global VM mode setting for this\nfunction call. See vmlSetMode for possible\nvalues and their description.\nOutput Parameters\nName\nType\nDescription\ny\nfloat* for vsSind\nfloat* for vmsSind\ndouble* for vdSind\ndouble* for vmdSind\nPointer to an array containing the output vector\ny.\nDescription\nThe v?Sind function computes the sine of vector elements multiplied by π/180. For an argument x, the\nfunction computes sin(π*x/180).\nSpecial values for Real Function v?Sind(x)\nArgument\nResult\nVM Error Status\nException\n+0\n+0\n-0\n-0\n±∞\nQNAN\nVML_STATUS_ERRDOM\nINVALID\nQNAN\nQNAN\nSNAN\nQNAN\nINVALID\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2069\n\n\nApplication Notes\nIf arguments abs(ai) ≤ 224 for single precision or abs(ai) ≤ 252 for double precision, they belong to the fast\ncomputational path: arguments for which VM provides the best possible performance. Avoid arguments with\ndo not belong to the fast computational path in VM High Accuracy (HA) or Low Accuracy (LA) functions. For\narguments which do not belong to the fast computational path you can use VM Enhanced Performance (EP)\nfunctions, which are fast on the entire function domain. However, these functions provide lower accuracy.\nSee Also\nSin  Computes sine of vector elements.\nSinpi Computes the sine of vector elements multiplied by π.\nv?Tand\nComputes the tangent of vector elements multiplied\nby π/180.\nSyntax\nvsTand (n, a, y);\nvsTandI(n, a, inca, y, incy);\nvmsTand (n, a, y, mode);\nvmsTandI(n, a, inca, y, incy, mode);\nvdTand (n, a, y);\nvdTandI(n, a, inca, y, incy);\nvmdTand (n, a, y, mode);\nvmdTandI(n, a, inca, y, incy, mode);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nn\nconst MKL_INT\nSpecifies the number of elements to be\ncalculated.\na\nconst float* for vsTand\nconst float* for vmsTand\nconst double* for vdTand\nconst double* for vmdTand\nPointer to the array containing the input vector\na.\ninca, incy\nconst MKL_INT\nSpecifies increments for the elements of a and y.\nmode\nconst MKL_INT64\nOverrides the global VM mode setting for this\nfunction call. See vmlSetMode for possible\nvalues and their description.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2070\n\n\nOutput Parameters\nName\nType\nDescription\ny\nfloat* for vsTand\nfloat* for vmsTand\ndouble* for vdTand\ndouble* for vmdTand\nPointer to an array containing the output vector\ny.\nDescription\nThe v?Tand function computes the tangent of vector elements multiplied by π/180. For an argument x, the\nfunction computes tan(π*x/180).\nSpecial values for Real Function v?Tand(x)\nArgument\nResult\nVM Error Status\nException\n+0\n+1\n-0\n+1\n±∞\nQNAN\nVML_STATUS_ERRDOM\nINVALID\nQNAN\nQNAN\nSNAN\nQNAN\nINVALID\nThe copysign(x, y) function returns the first vector argument x with the sign changed to match that of the\nsecond argument y.\nApplication Notes\nIf arguments abs(ai) ≤ 2 38 for single precision or abs(ai )≤ 2 67 for double precision, they belong to the fast\ncomputational path: arguments for which VM provides the best possible performance. Avoid arguments with\ndo not belong to the fast computational path in VM High Accuracy (HA) or Low Accuracy (LA) functions. For\narguments which do not belong to the fast computational path you can use VM Enhanced Performance (EP)\nfunctions, which are fast on the entire function domain. However, these functions provide lower accuracy.\nSee Also\nTan Computes tangent of vector elements.\nTanpi Computes the tangent of vector elements multiplied by π.\nHyperbolic Functions\nv?Cosh\nComputes hyperbolic cosine of vector elements.\nSyntax\nvsCosh( n, a, y );\nvsCoshI(n, a, inca, y, incy);\nvmsCosh( n, a, y, mode );\nvmsCoshI(n, a, inca, y, incy, mode);\nvdCosh( n, a, y );\nvdCoshI(n, a, inca, y, incy);\nvmdCosh( n, a, y, mode );\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2071\n\n\nvmdCoshI(n, a, inca, y, incy, mode);\nvcCosh( n, a, y );\nvcCoshI(n, a, inca, y, incy);\nvmcCosh( n, a, y, mode );\nvmcCoshI(n, a, inca, y, incy, mode);\nvzCosh( n, a, y );\nvzCoshI(n, a, inca, y, incy);\nvmzCosh( n, a, y, mode );\nvmzCoshI(n, a, inca, y, incy, mode);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nn\nconst MKL_INT\nSpecifies the number of elements to be\ncalculated.\na\nconst float* for vsCosh, vmsCosh\nconst double* for vdCosh, vmdCosh\nconst MKL_Complex8* for vcCosh,\nvmcCosh\nconst MKL_Complex16* for vzCosh,\nvmzCosh\nPointer to an array that contains the input vector\na.\ninca, incy\nconst MKL_INT\nSpecifies increments for the elements of a and y.\nmode\nconst MKL_INT64\nOverrides global VM mode setting for this\nfunction call. See vmlSetMode for possible\nvalues and their description.\nPrecision Overflow Thresholds for Real v?Cosh Function\nData Type\nThreshold Limitations on Input Parameters\nsingle precision\n-Ln(FLT_MAX)-Ln2 <a[i] < Ln(FLT_MAX)+Ln2\ndouble precision\n-Ln(DBL_MAX)-Ln2 <a[i] < Ln(DBL_MAX)+Ln2\nPrecision overflow thresholds for the complex v?Cosh function are beyond the scope of this document.\nOutput Parameters\nName\nType\nDescription\ny\nfloat* for vsCosh, vmsCosh\ndouble* for vdCosh, vmdCosh\nMKL_Complex8* for vcCosh, vmcCosh\nPointer to an array that contains the output\nvector y.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2072\n\n\nName\nType\nDescription\nMKL_Complex16* for vzCosh,\nvmzCosh\nDescription\nThe v?Cosh function computes hyperbolic cosine of vector elements.\nSpecial Values for Real Function v?Cosh(x)\nArgument\nResult\nVM Error Status\nException\n+0\n+1\n \n \n-0\n+1\n \n \nX > overflow\n+∞\nVML_STATUS_OVERFLOW\nOVERFLOW\nX < -overflow\n+∞\nVML_STATUS_OVERFLOW\nOVERFLOW\n+∞\n+∞\n \n \n-∞\n+∞\n \n \nQNAN\nQNAN\n \n \nSNAN\nQNAN\n \nINVALID\nSee Special Value Notations for the conventions used in the table below.\nSpecial Values for Complex Function v?Cosh(z)\nRE(z)\ni·IM(z)\n-∞\n \n-X\n \n-0\n \n+0\n \n+X\n \n+∞\n \nNAN\n \n+i·∞\n+∞\n+i·QNAN\nINVALID\nQNAN\n+i·QNAN\nINVALID\nQNAN-i·0\nINVALID\nQNAN+i·0\nINVALID\nQNAN\n+i·QNAN\nINVALID\n+∞\n+i·QNAN\nINVALID\nQNAN\n+i·QNAN\n+i·Y\n+∞·Cos(Y)-\ni·∞·Sin(Y)\n+∞·CIS(Y)\nQNAN\n+i·QNAN\n+i·0\n+∞-i·0\n+1-i·0\n+1+i·0\n+∞+i·0\nQNAN+i·0\n-i·0\n+∞+i·0\n+1+i·0\n+1-i·0\n+∞-i·0\nQNAN-i·0\n-i·Y\n+∞·Cos(Y)-\ni·∞·Sin(Y)\n+∞·CIS(Y)\nQNAN\n+i·QNAN\n-i·∞\n+∞\n+i·QNAN\nINVALID\nQNAN\n+i·QNAN\nINVALID\nQNAN+i·0\nINVALID\nQNAN-i·0\nINVALID\nQNAN\n+i·QNAN\nINVALID\n+∞\n+i·QNAN\nINVALID\nQNAN\n+i·QNAN\n+i·NAN\n+∞\n+i·QNAN\nQNAN\n+i·QNAN\nQNAN\n+i·QNAN\nQNAN-\ni·QNAN\nQNAN\n+i·QNAN\n+∞\n+i·QNAN\nQNAN\n+i·QNAN\nNotes:\n•\nraises the INVALID exception when the real or imaginary part of the argument is SNAN\n•\nraises the OVERFLOW exception and sets the VM Error Status to VML_STATUS_OVERFLOW in the case of\noverflow, that is, when RE(z), IM(z) are finite non-zero numbers, but the real or imaginary part of the\nexact result is so large that it does not meet the target precision.\n•\nCosh(CONJ(z))=CONJ(Cosh(z))\n•\nCosh(-z)=Cosh(z).\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2073\n\n\nv?Sinh\nComputes hyperbolic sine of vector elements.\nSyntax\nvsSinh( n, a, y );\nvsSinhI(n, a, inca, y, incy);\nvmsSinh( n, a, y, mode );\nvmsSinhI(n, a, inca, y, incy, mode);\nvdSinh( n, a, y );\nvdSinhI(n, a, inca, y, incy);\nvmdSinh( n, a, y, mode );\nvmdSinhI(n, a, inca, y, incy, mode);\nvcSinh( n, a, y );\nvcSinhI(n, a, inca, y, incy);\nvmcSinh( n, a, y, mode );\nvmcSinhI(n, a, inca, y, incy, mode);\nvzSinh( n, a, y );\nvzSinhI(n, a, inca, y, incy);\nvmzSinh( n, a, y, mode );\nvmzSinhI(n, a, inca, y, incy, mode);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nn\nconst MKL_INT\nSpecifies the number of elements to be\ncalculated.\na\nconst float* for vsSinh, vmsSinh\nconst double* for vdSinh, vmdSinh\nconst MKL_Complex8* for vcSinh,\nvmcSinh\nconst MKL_Complex16* for vzSinh,\nvmzSinh\nPointer to an array that contains the input vector\na.\ninca, incy\nconst MKL_INT\nSpecifies increments for the elements of a and y.\nmode\nconst MKL_INT64\nOverrides global VM mode setting for this\nfunction call. See vmlSetMode for possible\nvalues and their description.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2074\n\n\nPrecision Overflow Thresholds for Real v?Sinh Function\nData Type\nThreshold Limitations on Input Parameters\nsingle precision\n-Ln(FLT_MAX)-Ln2 <a[i] < Ln(FLT_MAX)+Ln2\ndouble precision\n-Ln(DBL_MAX)-Ln2 <a[i] < Ln(DBL_MAX)+Ln2\nPrecision overflow thresholds for the complex v?Sinh function are beyond the scope of this document.\nOutput Parameters\nName\nType\nDescription\ny\nfloat* for vsSinh, vmsSinh\ndouble* for vdSinh, vmdSinh\nMKL_Complex8* for vcSinh, vmcSinh\nMKL_Complex16* for vzSinh,\nvmzSinh\nPointer to an array that contains the output\nvector y.\nDescription\nThe v?Sinh function computes hyperbolic sine of vector elements.\nSpecial Values for Real Function v?Sinh(x)\nArgument\nResult\nVM Error Status\nException\n+0\n+0\n \n \n-0\n-0\n \n \nX > overflow\n+∞\nVML_STATUS_OVERFLOW\nOVERFLOW\nX < -overflow\n-∞\nVML_STATUS_OVERFLOW\nOVERFLOW\n+∞\n+∞\n \n \n-∞\n-∞\n \n \nQNAN\nQNAN\n \n \nSNAN\nQNAN\n \nINVALID\nSee Special Value Notations for the conventions used in the table below.\nSpecial Values for Complex Function v?Sinh(z)\nRE(z)\ni·IM(z)\n-∞\n \n-X\n \n-0\n \n+0\n \n+X\n \n+∞\n \nNAN\n \n+i·∞\n-∞+i·QNAN\nINVALID\nQNAN\n+i·QNAN\nINVALID\n-0+i·QNAN\nINVALID\n+0+i·QNA\nN\nINVALID\nQNAN\n+i·QNAN\nINVALID\n+∞\n+i·QNAN\nINVALID\nQNAN\n+i·QNAN\n+i·Y\n-∞·Cos(Y)+\ni·∞·Sin(Y)\n+∞·CIS(Y)\nQNAN\n+i·QNAN\n+i·0\n-∞+i·0\n-0+i·0\n+0+i·0\n+∞+i·0\nQNAN+i·0\n-i·0\n-∞-i·0\n-0-i·0\n+0-i·0\n+∞-i·0\nQNAN-i·0\n-i·Y\n-∞·Cos(Y)+\ni·∞·Sin(Y)\n+∞·CIS(Y)\nQNAN\n+i·QNAN\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2075\n\n\nRE(z)\ni·IM(z)\n-∞\n \n-X\n \n-0\n \n+0\n \n+X\n \n+∞\n \nNAN\n \n-i·∞\n-∞+i·QNAN\nINVALID\nQNAN\n+i·QNAN\nINVALID\n-0+i·QNAN\nINVALID\n+0+i·QNA\nN\nINVALID\nQNAN\n+i·QNAN\nINVALID\n+∞\n+i·QNAN\nINVALID\nQNAN\n+i·QNAN\n+i·NAN\n-∞+i·QNAN\nQNAN\n+i·QNAN\n-0+i·QNAN\n+0+i·QNA\nN\nQNAN\n+i·QNAN\n+∞\n+i·QNAN\nQNAN\n+i·QNAN\nNotes:\n•\nraises the INVALID exception when the real or imaginary part of the argument is SNAN\n•\nraises the OVERFLOW exception and sets the VM Error Status to VML_STATUS_OVERFLOW in the case of\noverflow, that is, when RE(z), IM(z) are finite non-zero numbers, but the real or imaginary part of the\nexact result is so large that it does not meet the target precision.\n•\nSinh(CONJ(z))=CONJ(Sinh(z))\n•\nSinh(-z)=-Sinh(z).\nv?Tanh\nComputes hyperbolic tangent of vector elements.\nSyntax\nvsTanh( n, a, y );\nvsTanhI(n, a, inca, y, incy);\nvmsTanh( n, a, y, mode );\nvmsTanhI(n, a, inca, y, incy, mode);\nvdTanh( n, a, y );\nvdTanhI(n, a, inca, y, incy);\nvmdTanh( n, a, y, mode );\nvmdTanhI(n, a, inca, y, incy, mode);\nvcTanh( n, a, y );\nvcTanhI(n, a, inca, y, incy);\nvmcTanh( n, a, y, mode );\nvmcTanhI(n, a, inca, y, incy, mode);\nvzTanh( n, a, y );\nvzTanhI(n, a, inca, y, incy);\nvmzTanh( n, a, y, mode );\nvmzTanhI(n, a, inca, y, incy, mode);\nInclude Files\n•\nmkl.h\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2076\n\n\nInput Parameters\nName\nType\nDescription\nn\nconst MKL_INT\nSpecifies the number of elements to be\ncalculated.\na\nconst float* for vsTanh, vmsTanh\nconst double* for vdTanh, vmdTanh\nconst MKL_Complex8* for vcTanh,\nvmcTanh\nconst MKL_Complex16* for vzTanh,\nvmzTanh\nPointer to an array that contains the input vector\na.\ninca, incy\nconst MKL_INT\nSpecifies increments for the elements of a and y.\nmode\nconst MKL_INT64\nOverrides global VM mode setting for this\nfunction call. See vmlSetMode for possible\nvalues and their description.\nOutput Parameters\nName\nType\nDescription\ny\nfloat* for vsTanh, vmsTanh\ndouble* for vdTanh, vmdTanh\nMKL_Complex8* for vcTanh, vmcTanh\nMKL_Complex16* for vzTanh,\nvmzTanh\nPointer to an array that contains the output\nvector y.\nDescription\nThe v?Tanh function computes hyperbolic tangent of vector elements.\nSpecial Values for Real Function v?Tanh(x)\nArgument\nResult\nException\n+0\n+0\n \n-0\n-0\n \n+∞\n+1\n \n-∞\n-1\n \nQNAN\nQNAN\n \nSNAN\nQNAN\nINVALID\nSee Special Value Notations for the conventions used in the table below.\nSpecial Values for Complex Function v?Tanh(z)\nRE(z)\ni·IM(z)\n-∞\n \n-X\n \n-0\n \n+0\n \n+X\n \n+∞\n \nNAN\n \n+i·∞\n-1+i·0\nQNAN\n+i·QNAN\nINVALID\nQNAN\n+i·QNAN\nINVALID\nQNAN\n+i·QNAN\nINVALID\nQNAN\n+i·QNAN\nINVALID\n+1+i·0\nQNAN\n+i·QNAN\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2077\n\n\nRE(z)\ni·IM(z)\n-∞\n \n-X\n \n-0\n \n+0\n \n+X\n \n+∞\n \nNAN\n \n+i·Y\n-1+i·0·Tan(\nY)\n+1+i·0·Ta\nn(Y)\nQNAN\n+i·QNAN\n+i·0\n-1+i·0\n-0+i·0\n+0+i·0\n+1+i·0\nQNAN+i·0\n-i·0\n-1-i·0\n-0-i·0\n+0-i·0\n+1-i·0\nQNAN-i·0\n-i·Y\n-1+i·0·Tan(\nY)\n+1+i·0·Ta\nn(Y)\nQNAN\n+i·QNAN\n-i·∞\n-1-i·0\nQNAN\n+i·QNAN\nINVALID\nQNAN\n+i·QNAN\nINVALID\nQNAN\n+i·QNAN\nINVALID\nQNAN\n+i·QNAN\nINVALID\n+1-i·0\nQNAN\n+i·QNAN\n+i·NAN\n-1+i·0\nQNAN\n+i·QNAN\nQNAN\n+i·QNAN\nQNAN\n+i·QNAN\nQNAN\n+i·QNAN\n+1+i·0\nQNAN\n+i·QNAN\nNotes:\n•\nraises INVALID exception when real or imaginary part of the argument is SNAN\n•\nTanh(CONJ(z))=CONJ(Tanh(z))\n•\nTanh(-z)=-Tanh(z).\nv?Acosh\nComputes inverse hyperbolic cosine (nonnegative) of\nvector elements.\nSyntax\nvsAcosh( n, a, y );\nvsAcoshI(n, a, inca, y, incy);\nvmsAcosh( n, a, y, mode );\nvmsAcoshI(n, a, inca, y, incy, mode);\nvdAcosh( n, a, y );\nvdAcoshI(n, a, inca, y, incy);\nvmdAcosh( n, a, y, mode );\nvmdAcoshI(n, a, inca, y, incy, mode);\nvcAcosh( n, a, y );\nvcAcoshI(n, a, inca, y, incy);\nvmcAcosh( n, a, y, mode );\nvmcAcoshI(n, a, inca, y, incy, mode);\nvzAcosh( n, a, y );\nvzAcoshI(n, a, inca, y, incy);\nvmzAcosh( n, a, y, mode );\nvmzAcoshI(n, a, inca, y, incy, mode);\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2078\n\n\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nn\nconst MKL_INT\nSpecifies the number of elements to be\ncalculated.\na\nconst float* for vsAcosh,\nvmsAcosh\nconst double* for vdAcosh,\nvmdAcosh\nconst MKL_Complex8* for vcAcosh,\nvmcAcosh\nconst MKL_Complex16* for vzAcosh,\nvmzAcosh\nPointer to an array that contains the input vector\na.\ninca, incy\ncnst MKL_INT\nSpecifies increments for the elements of a and y.\nmode\nconst MKL_INT64\nOverrides global VM mode setting for this\nfunction call. See vmlSetMode for possible\nvalues and their description.\nOutput Parameters\nName\nType\nDescription\ny\nfloat* for vsAcosh, vmsAcosh\ndouble* for vdAcosh, vmdAcosh\nMKL_Complex8* for vcAcosh,\nvmcAcosh\nMKL_Complex16* for vzAcosh,\nvmzAcosh\nPointer to an array that contains the output\nvector y.\nDescription\nThe v?Acosh function computes inverse hyperbolic cosine (nonnegative) of vector elements.\nSpecial Values for Real Function v?Acosh(x)\nArgument\nResult\nVM Error Status\nException\n+1\n+0\n \n \nX < +1\nQNAN\nVML_STATUS_ERRDOM\nINVALID\n-∞\nQNAN\nVML_STATUS_ERRDOM\nINVALID\n+∞\n+∞\n \n \nQNAN\nQNAN\n \n \nSNAN\nQNAN\n \nINVALID\nSee Special Value Notations for the conventions used in the table below.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2079\n\n\nSpecial Values for Complex Function v?Acosh(z)\nRE(z)\ni·IM(z\n)\n-∞\n \n-X\n \n-0\n \n+0\n \n+X\n \n+∞\n \nNAN\n \n+i·∞\n+∞+i·π/2\n+∞+i·π/2\n+∞+i·π/2\n+∞+i·π/2\n+∞+i·π/4\n+∞+i·QNAN\n+i·Y\n+∞+i·π\n+∞+i·0\nQNAN\n+i·QNAN\n+i·0\n+∞+i·π\n+0+i·π/2\n+0+i·π/2\n+∞+i·0\nQNAN\n+i·QNAN\n-i·0\n+∞+i·π\n+0+i·π/2\n+0+i·π/2\n+∞+i·0\nQNAN\n+i·QNAN\n-i·Y\n+∞+i·π\n+∞+i·0\nQNAN\n+i·QNAN\n-i·∞\n+∞-i·π/2\n+∞-i·π/2\n+∞-i·π/2\n+∞-i·π/2\n+∞-i·π/4\n+∞+i·QNAN\n+i·NAN\n+∞\n+i·QNAN\nQNAN\n+i·QNAN\nQNAN\n+i·QNAN\nQNAN\n+i·QNAN\nQNAN\n+i·QNAN\n+∞\n+i·QNAN\nQNAN\n+i·QNAN\nNotes:\n•\nraises INVALID exception when real or imaginary part of the argument is SNAN\n•\nAcosh(CONJ(z))=CONJ(Acosh(z)).\nv?Asinh\nComputes inverse hyperbolic sine of vector elements.\nSyntax\nvsAsinh( n, a, y );\nvsAsinhI(n, a, inca, y, incy);\nvmsAsinh( n, a, y, mode );\nvmsAsinhI(n, a, inca, y, incy, mode);\nvdAsinh( n, a, y );\nvdAsinhI(n, a, inca, y, incy);\nvmdAsinh( n, a, y, mode );\nvmdAsinhI(n, a, inca, y, incy, mode);\nvcAsinh( n, a, y );\nvcAsinhI(n, a, inca, y, incy);\nvmcAsinh( n, a, y, mode );\nvmcAsinhI(n, a, inca, y, incy, mode);\nvzAsinh( n, a, y );\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2080\n\n\nvzAsinhI(n, a, inca, y, incy);\nvmzAsinh( n, a, y, mode );\nvmzAsinhI(n, a, inca, y, incy, mode);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nn\nconst MKL_INT\nSpecifies the number of elements to be\ncalculated.\na\nconst float* for vsAsinh,\nvmsAsinh\nconst double* for vdAsinh,\nvmdAsinh\nconst MKL_Complex8* for vcAsinh,\nvmcAsinh\nconst MKL_Complex16* for vzAsinh,\nvmzAsinh\nPointer to an array that contains the input vector\na.\ninca, incy\nconst MKL_INT\nSpecifies increments for the elements of a and y.\nmode\nconst MKL_INT64\nOverrides global VM mode setting for this\nfunction call. See vmlSetMode for possible\nvalues and their description.\nOutput Parameters\nName\nType\nDescription\ny\nfloat* for vsAsinh, vmsAsinh\ndouble* for vdAsinh, vmdAsinh\nMKL_Complex8* for vcAsinh,\nvmcAsinh\nMKL_Complex16* for vzAsinh,\nvmzAsinh\nPointer to an array that contains the output\nvector y.\nDescription\nThe v?Asinh function computes inverse hyperbolic sine of vector elements.\nSpecial Values for Real Function v?Asinh(x)\nArgument\nResult\nException\n+0\n+0\n \n-0\n-0\n \n+∞\n+∞\n \n-∞\n-∞\n \nQNAN\nQNAN\n \nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2081\n\n\nArgument\nResult\nException\nSNAN\nQNAN\nINVALID\nSee Special Value Notations for the conventions used in the table below.\nSpecial Values for Complex Function v?Asinh(z)\nRE(z)\ni·IM(z\n)\n-∞\n \n-X\n \n-0\n \n+0\n \n+X\n \n+∞\n \nNAN\n \n+i·∞\n-∞+i·π/4\n-∞+i·π/2\n+∞+i·π/2\n+∞+i·π/2\n+∞+i·π/2\n+∞+i·π/4\n+∞+i·QNAN\n+i·Y\n-∞+i·0\n+∞+i·0\nQNAN\n+i·QNAN\n+i·0\n+∞+i·0\n+0+i·0\n+0+i·0\n+∞+i·0\nQNAN\n+i·QNAN\n-i·0\n-∞-i·0\n-0-i·0\n+0-i·0\n+∞-i·0\nQNAN-\ni·QNAN\n-i·Y\n-∞-i·0\n+∞-i·0\nQNAN\n+i·QNAN\n-i·∞\n-∞-i·π/4\n-∞-i·π/2\n-∞-i·π/2\n+∞-i·π/2\n+∞-i·π/2\n+∞-i·π/4\n+∞+i·QNAN\n+i·NAN\n-∞+i·QNAN\nQNAN\n+i·QNAN\nQNAN\n+i·QNAN\nQNAN\n+i·QNAN\nQNAN\n+i·QNAN\n+∞+i·QNAN\nQNAN\n+i·QNAN\nNotes:\n•\nraises INVALID exception when real or imaginary part of the argument is SNAN\n•\nAsinh(CONJ(z))=CONJ(Asinh(z))\n•\nAsinh(-z)=-Asinh(z).\nv?Atanh\nComputes inverse hyperbolic tangent of vector\nelements.\nSyntax\nvsAtanh( n, a, y );\nvsAtanhI(n, a, inca, y, incy);\nvmsAtanh( n, a, y, mode );\nvmsAtanhI(n, a, inca, y, incy, mode);\nvdAtanh( n, a, y );\nvdAtanhI(n, a, inca, y, incy);\nvmdAtanh( n, a, y, mode );\nvmdAtanhI(n, a, inca, y, incy, mode);\nvcAtanh( n, a, y );\nvcAtanhI(n, a, inca, y, incy);\nvmcAtanh( n, a, y, mode );\nvmcAtanhI(n, a, inca, y, incy, mode);\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2082\n\n\nvzAtanh( n, a, y );\nvzAtanhI(n, a, inca, y, incy);\nvmzAtanh( n, a, y, mode );\nvmzAtanhI(n, a, inca, y, incy, mode);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nn\nconst MKL_INT\nSpecifies the number of elements to be\ncalculated.\na\nconst float* for vsAtanh,\nvmsAtanh\nconst double* for vdAtanh,\nvmdAtanh\nconst MKL_Complex8* for vcAtanh,\nvmcAtanh\nconst MKL_Complex16* for vzAtanh,\nvmzAtanh\nPointer to an array that contains the input vector\na.\ninca, incy\nconst MKL_INT\nSpecifies increments for the elements of a and y.\nmode\nconst MKL_INT64\nOverrides global VM mode setting for this\nfunction call. See vmlSetMode for possible\nvalues and their description.\nOutput Parameters\nName\nType\nDescription\ny\nfloat* for vsAtanh, vmsAtanh\ndouble* for vdAtanh, vmdAtanh\nMKL_Complex8* for vcAtanh,\nvmcAtanh\nMKL_Complex16* for vzAtanh,\nvmzAtanh\nPointer to an array that contains the output\nvector y.\nDescription\nThe v?Atanh function computes inverse hyperbolic tangent of vector elements.\nSpecial Values for Real Function v?Atanh(x)\nArgument\nResult\nVM Error Status\nException\n+1\n+∞\nVML_STATUS_SING\nZERODIVIDE\n-1\n-∞\nVML_STATUS_SING\nZERODIVIDE\n|X| > 1\nQNAN\nVML_STATUS_ERRDOM\nINVALID\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2083\n\n\nArgument\nResult\nVM Error Status\nException\n+∞\nQNAN\nVML_STATUS_ERRDOM\nINVALID\n-∞\nQNAN\nVML_STATUS_ERRDOM\nINVALID\nQNAN\nQNAN\n \n \nSNAN\nQNAN\n \nINVALID\nSee Special Value Notations for the conventions used in the table below.\nSpecial Values for Complex Function v?Atanh(z)\nRE(z)\ni·IM(z\n)\n-∞\n \n-X\n \n-0\n \n+0\n \n+X\n \n+∞\n \nNAN\n \n+i·∞\n-0+i·π/2\n-0+i·π/2\n-0+i·π/2\n+0+i·π/2\n+0+i·π/2\n+0+i·π/2\n+0+i·π/2\n+i·Y\n-0+i·π/2\n+0+i·π/2\nQNAN\n+i·QNAN\n+i·0\n-0+i·π/2\n-0+i·0\n+0+i·0\n+0+i·π/2\nQNAN\n+i·QNAN\n-i·0\n-0-i·π/2\n-0-i·0\n+0-i·0\n+0-i·π/2\nQNAN-\ni·QNAN\n-i·Y\n-0-i·π/2\n+0-i·π/2\nQNAN\n+i·QNAN\n-i·∞\n-0-i·π/2\n-0-i·π/2\n-0-i·π/2\n+0-i·π/2\n+0-i·π/2\n+0-i·π/2\n+0-i·π/2\n+i·NAN\n-0+i·QNAN\nQNAN\n+i·QNAN\n-0+i·QNAN\n+0+i·QNA\nN\nQNAN\n+i·QNAN\n+0+i·QNA\nN\nQNAN\n+i·QNAN\nNotes:\n•\nAtanh(+-1+-i*0)=+-∞+-i*0, and ZERODIVIDE exception is raised\n•\nraises INVALID exception when real or imaginary part of the argument is SNAN\n•\nAtanh(CONJ(z))=CONJ(Atanh(z))\n•\nAtanh(-z)=-Atanh(z).\nSpecial Functions\nv?Erf\nComputes the error function value of vector elements.\nSyntax\nvsErf( n, a, y );\nvsErfI(n, a, inca, y, incy);\nvmsErf( n, a, y, mode );\nvmsErfI(n, a, inca, y, incy, mode);\nvdErf( n, a, y );\nvdErfI(n, a, inca, y, incy);\nvmdErf( n, a, y, mode );\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2084\n\n\nvmdErfI(n, a, inca, y, incy, mode);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nn\nconst MKL_INT\nSpecifies the number of elements to be\ncalculated.\na\nconst float* for vsErf, vmsErf\nconst double* for vdErf, vmdErf\nPointer to an array that contains the input vector\na.\ninca, incy\nconst MKL_INT\nSpecifies increments for the elements of a and y.\nmode\nconst MKL_INT64\nOverrides global VM mode setting for this\nfunction call. See vmlSetMode for possible\nvalues and their description.\nOutput Parameters\nName\nType\nDescription\ny\nfloat* for vsErf, vmsErf\ndouble* for vdErf, vmdErf\nPointer to an array that contains the output\nvector y.\nDescription\nThe Erf function computes the error function values for elements of the input vector a and writes them to\nthe output vector y.\nThe error function is defined as given by:\nUseful relations:\nwhere erfc is the complementary error function.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2085\n\n\nwhere\nis the cumulative normal distribution function.\nwhere Φ-1(x) and erf-1(x) are the inverses to Φ(x) and erf(x) respectively.\nThe following figure illustrates the relationships among Erf family functions (Erf, Erfc, CdfNorm).\n__border__top\nErf Family Functions Relationship\nUseful relations for these functions:\nSpecial Values for Real Function v?Erf(x)\nArgument\nResult\nException\n+∞\n+1\n \n-∞\n-1\n \nQNAN\nQNAN\n \n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2086\n\n\nArgument\nResult\nException\nSNAN\nQNAN\nINVALID\nSee Also\nErfc\nCdfNorm\nv?Erfc\nComputes the complementary error function value of\nvector elements.\nSyntax\nvsErfc( n, a, y );\nvsErfcI(n, a, inca, y, incy);\nvmsErfc( n, a, y, mode );\nvmsErfcI(n, a, inca, y, incy, mode);\nvdErfc( n, a, y );\nvdErfcI(n, a, inca, y, incy);\nvmdErfc( n, a, y, mode );\nvmdErfcI(n, a, inca, y, incy, mode);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nn\nconst MKL_INT\nSpecifies the number of elements to be\ncalculated.\na\nconst float* for vsErfc, vmsErfc\nconst double* for vdErfc, vmdErfc\nPointer to an array that contains the input vector\na.\ninca, incy\nconst MKL_INT\nSpecifies increments for the elements of a and y.\nmode\nconst MKL_INT64\nOverrides global VM mode setting for this\nfunction call. See vmlSetMode for possible\nvalues and their description.\nOutput Parameters\nName\nType\nDescription\ny\nfloat* for vsErfc, vmsErfc\ndouble* for vdErfc, vmdErfc\nPointer to an array that contains the output\nvector y.\nDescription\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2087\n\n\nThe Erfc function computes the complementary error function values for elements of the input vector a and\nwrites them to the output vector y.\nThe complementary error function is defined as follows:\nUseful relations:\nwhere\nis the cumulative normal distribution function.\nwhere Φ-1(x) and erf-1(x) are the inverses to Φ(x) and erf(x) respectively.\nSee also Figure \"Erf Family Functions Relationship\" in Erf function description for Erfc function relationship\nwith the other functions of Erf family.\nSpecial Values for Real Function v?Erfc(x)\nArgument\nResult\nVM Error Status\nException\nX > underflow\n+0\nVML_STATUS_UNDERFLOW\nUNDERFLOW\n+∞\n+0\n \n \n-∞\n+2\n \n \nQNAN\nQNAN\n \n \nSNAN\nQNAN\n \nINVALID\nSee Also\nErf\nCdfNorm\nv?CdfNorm\nComputes the cumulative normal distribution function\nvalues of vector elements.\nSyntax\nvsCdfNorm( n, a, y );\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2088\n\n\nvsCdfNormI(n, a, inca, y, incy);\nvmsCdfNorm( n, a, y, mode );\nvmsCdfNormI(n, a, inca, y, incy, mode);\nvdCdfNorm( n, a, y );\nvdCdfNormI(n, a, inca, y, incy);\nvmdCdfNorm( n, a, y, mode );\nvmdCdfNormI(n, a, inca, y, incy, mode);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nn\nconst MKL_INT\nSpecifies the number of elements to be\ncalculated.\na\nconst float* for vsCdfNorm,\nvmsCdfNorm\nconst double* for vdCdfNorm,\nvmdCdfNorm\nPointer to an array that contains the input vector\na.\ninca, incy\nconst MKL_INT\nSpecifies increments for the elements of a and y.\nmode\nconst MKL_INT64\nOverrides global VM mode setting for this\nfunction call. See vmlSetMode for possible\nvalues and their description.\nOutput Parameters\nName\nType\nDescription\ny\nfloat* for vsCdfNorm, vmsCdfNorm\ndouble* for vdCdfNorm, vmdCdfNorm\nPointer to an array that contains the output\nvector y.\nDescription\nThe CdfNorm function computes the cumulative normal distribution function values for elements of the input\nvector a and writes them to the output vector y.\nThe cumulative normal distribution function is defined as given by:\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2089\n\n\nUseful relations:\nwhere Erf and Erfc are the error and complementary error functions.\nSee also Figure \"Erf Family Functions Relationship\" in Erf function description for CdfNorm function\nrelationship with the other functions of Erf family.\nSpecial Values for Real Function v?CdfNorm(x)\nArgument\nResult\nVM Error Status\nException\nX < underflow\n+0\nVML_STATUS_UNDERFLOW\nUNDERFLOW\n+∞\n+1\n \n \n-∞\n+0\n \n \nQNAN\nQNAN\n \n \nSNAN\nQNAN\n \nINVALID\nSee Also\nErf\nErfc\nv?ErfInv\nComputes inverse error function value of vector\nelements.\nSyntax\nvsErfInv( n, a, y );\nvsErfInvI(n, a, inca, y, incy);\nvmsErfInv( n, a, y, mode );\nvmsErfInvI(n, a, inca, y, incy, mode);\nvdErfInv( n, a, y );\nvdErfInvI(n, a, inca, y, incy);\nvmdErfInv( n, a, y, mode );\nvmdErfInvI(n, a, inca, y, incy, mode);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nn\nconst MKL_INT\nSpecifies the number of elements to be\ncalculated.\na\nconst float* for vsErfInv,\nvmsErfInv\nPointer to an array that contains the input vector\na.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2090\n\n\nName\nType\nDescription\nconst double* for vdErfInv,\nvmdErfInv\ninca, incy\nconst MKL_INT\nSpecifies increments for the elements of a and y.\nmode\nconst MKL_INT64\nOverrides global VM mode setting for this\nfunction call. See vmlSetMode for possible\nvalues and their description.\nOutput Parameters\nName\nType\nDescription\ny\nfloat* for vsErfInv, vmsErfInv\ndouble* for vdErfInv, vmdErfInv\nPointer to an array that contains the output\nvector y.\nDescription\nThe ErfInv function computes the inverse error function values for elements of the input vector a and writes\nthem to the output vector y\ny = erf-1(a),\nwhere erf(x) is the error function defined as given by:\nUseful relations:\nwhere erfc is the complementary error function.\nwhere\nis the cumulative normal distribution function.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2091\n\n\nwhere Φ-1(x) and erf-1(x) are the inverses to Φ(x) and erf(x) respectively.\nFigure \"ErfInv Family Functions Relationship\" illustrates the relationships among ErfInv family functions\n(ErfInv, ErfcInv, CdfNormInv).\n__border__top\nErfInv Family Functions Relationship\nUseful relations for these functions:\nSpecial Values for Real Function v?ErfInv(x)\nArgument\nResult\nVM Error Status\nException\n+0\n+0\n \n \n-0\n-0\n \n \n+1\n+∞\nVML_STATUS_SING\nZERODIVIDE\n-1\n-∞\nVML_STATUS_SING\nZERODIVIDE\n|X| > 1\nQNAN\nVML_STATUS_ERRDOM\nINVALID\n+∞\nQNAN\nVML_STATUS_ERRDOM\nINVALID\n-∞\nQNAN\nVML_STATUS_ERRDOM\nINVALID\nQNAN\nQNAN\n \n \nSNAN\nQNAN\n \nINVALID\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2092\n\n\nSee Also\nErfcInv\nCdfNormInv\nv?ErfcInv\nComputes the inverse complementary error function\nvalue of vector elements.\nSyntax\nvsErfcInv( n, a, y );\nvsErfcInvI(n, a, inca, y, incy);\nvmsErfcInv( n, a, y, mode );\nvmsErfcInvI(n, a, inca, y, incy, mode);\nvdErfcInv( n, a, y );\nvdErfcInvI(n, a, inca, y, incy);\nvmdErfcInv( n, a, y, mode );\nvmdErfcInvI(n, a, inca, y, incy, mode);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nn\nconst MKL_INT\nSpecifies the number of elements to be\ncalculated.\na\nconst float* for vsErfcInv,\nvmsErfcInv\nconst double* for vdErfcInv,\nvmdErfcInv\nPointer to an array that contains the input vector\na.\ninca, incy\nconst MKL_INT\nSpecifies increments for the elements of a and y.\nmode\nconst MKL_INT64\nOverrides global VM mode setting for this\nfunction call. See vmlSetMode for possible\nvalues and their description.\nOutput Parameters\nName\nType\nDescription\ny\nfloat* for vsErfcInv, vmsErfcInv\ndouble* for vdErfcInv, vmdErfcInv\nPointer to an array that contains the output\nvector y.\nDescription\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2093\n\n\nThe ErfcInv function computes the inverse complimentary error function values for elements of the input\nvector a and writes them to the output vector y.\nThe inverse complementary error function is defined as given by:\nwhere erf(x) denotes the error function and erfinv(x) denotes the inverse error function.\nSee also Figure \"ErfInv Family Functions Relationship\" in ErfInv function description for ErfcInv function\nrelationship with the other functions of ErfInv family.\nSpecial Values for Real Function v?ErfcInv(x)\nArgument\nResult\nVM Error Status\nException\n+1\n+0\n \n \n+2\n-∞\nVML_STATUS_SING\nZERODIVIDE\n-0\n+∞\nVML_STATUS_SING\nZERODIVIDE\n+0\n+∞\nVML_STATUS_SING\nZERODIVIDE\nX < -0\nQNAN\nVML_STATUS_ERRDOM\nINVALID\nX > +2\nQNAN\nVML_STATUS_ERRDOM\nINVALID\n+∞\nQNAN\nVML_STATUS_ERRDOM\nINVALID\n-∞\nQNAN\nVML_STATUS_ERRDOM\nINVALID\nQNAN\nQNAN\n \n \nSNAN\nQNAN\n \nINVALID\nSee Also\nErfInv\nCdfNormInv\nv?CdfNormInv\nComputes the inverse cumulative normal distribution\nfunction values of vector elements.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2094\n\n\nSyntax\nvsCdfNormInv( n, a, y );\nvsCdfNormInvI(n, a, inca, y, incy);\nvmsCdfNormInv( n, a, y, mode );\nvmsCdfNormInvI(n, a, inca, y, incy, mode);\nvdCdfNormInv( n, a, y );\nvdCdfNormInvI(n, a, inca, y, incy);\nvmdCdfNormInv( n, a, y, mode );\nvmdCdfNormInvI(n, a, inca, y, incy, mode);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nn\nconst MKL_INT\nSpecifies the number of elements to be\ncalculated.\na\nconst float* for vsCdfNormInv,\nvmsCdfNormInv\nconst double* for vdCdfNormInv,\nvmdCdfNormInv\nPointer to an array that contains the input vector\na.\ninca, incy\nconst MKL_INT\nSpecifies increments for the elements of a and y.\nmode\nconst MKL_INT64\nOverrides global VM mode setting for this\nfunction call. See vmlSetMode for possible\nvalues and their description.\nOutput Parameters\nName\nType\nDescription\ny\nfloat* for vsCdfNormInv,\nvmsCdfNormInv\ndouble* for vdCdfNormInv,\nvmdCdfNormInv\nPointer to an array that contains the output\nvector y.\nDescription\nThe CdfNormInv function computes the inverse cumulative normal distribution function values for elements\nof the input vector a and writes them to the output vector y.\nThe inverse cumulative normal distribution function is defined as given by:\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2095\n\n\nwhere CdfNorm(x) denotes the cumulative normal distribution function.\nUseful relations:\nwhere erfinv(x) denotes the inverse error function and erfcinv(x) denotes the inverse complementary\nerror functions.\nSee also Figure \"ErfInv Family Functions Relationship\" in ErfInv function description for CdfNormInv\nfunction relationship with the other functions of ErfInv family.\nSpecial Values for Real Function v?CdfNormInv(x)\nArgument\nResult\nVM Error Status\nException\n+0.5\n+0\n \n \n+1\n+∞\nVML_STATUS_SING\nZERODIVIDE\n-0\n-∞\nVML_STATUS_SING\nZERODIVIDE\n+0\n-∞\nVML_STATUS_SING\nZERODIVIDE\nX < -0\nQNAN\nVML_STATUS_ERRDOM\nINVALID\nX > +1\nQNAN\nVML_STATUS_ERRDOM\nINVALID\n+∞\nQNAN\nVML_STATUS_ERRDOM\nINVALID\n-∞\nQNAN\nVML_STATUS_ERRDOM\nINVALID\nQNAN\nQNAN\n \n \nSNAN\nQNAN\n \nINVALID\nSee Also\nErfInv\nErfcInv\nv?LGamma\nComputes the natural logarithm of the absolute value\nof gamma function for vector elements.\nSyntax\nvsLGamma( n, a, y );\nvsLGammaI(n, a, inca, y, incy);\nvmsLGamma( n, a, y, mode );\nvmsLGammaI(n, a, inca, y, incy, mode);\nvdLGamma( n, a, y );\nvdLGammaI(n, a, inca, y, incy);\nvmdLGamma( n, a, y, mode );\nvmdLGammaI(n, a, inca, y, incy, mode);\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2096\n\n\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nn\nconst MKL_INT\nSpecifies the number of elements to be\ncalculated.\na\nconst float* for vsLGamma,\nvmsLGamma\nconst double* for vdLGamma,\nvmdLGamma\nPointer to an array that contains the input vector\na.\ninca, incy\nconst MKL_INT\nSpecifies increments for the elements of a and y.\nmode\nconst MKL_INT64\nOverrides global VM mode setting for this\nfunction call. See vmlSetMode for possible\nvalues and their description.\nOutput Parameters\nName\nType\nDescription\ny\nfloat* for vsLGamma, vmsLGamma\ndouble* for vdLGamma, vmdLGamma\nPointer to an array that contains the output\nvector y.\nDescription\nThe v?LGamma function computes the natural logarithm of the absolute value of gamma function for elements\nof the input vector a and writes them to the output vector y. Precision overflow thresholds for the v?LGamma\nfunction are beyond the scope of this document. If the result does not meet the target precision, the function\nraises the OVERFLOW exception and sets the VM Error Status to VML_STATUS_OVERFLOW.\nSpecial Values for Real Function v?LGamma(x)\nArgument\nResult\nVM Error Status\nException\n+1\n+0\n \n \n+2\n+0\n \n \n+0\n+∞\nVML_STATUS_SING\nZERODIVIDE\n-0\n+∞\nVML_STATUS_SING\nZERODIVIDE\nnegative integer\n+∞\nVML_STATUS_SING\nZERODIVIDE\n-∞\n+∞\n \n \n+∞\n+∞\n \n \nX > overflow\n+∞\nVML_STATUS_OVERFLOW\nOVERFLOW\nQNAN\nQNAN\n \n \nSNAN\nQNAN\n \nINVALID\nv?TGamma\nComputes the gamma function of vector elements.\nSyntax\nvsTGamma( n, a, y );\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2097\n\n\nvsTGammaI(n, a, inca, y, incy);\nvmsTGamma( n, a, y, mode );\nvmsTGammaI(n, a, inca, y, incy, mode);\nvdTGamma( n, a, y );\nvdTGammaI(n, a, inca, y, incy);\nvmdTGamma( n, a, y, mode );\nvmdTGammaI(n, a, inca, y, incy, mode);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nn\nconst MKL_INT\nSpecifies the number of elements to be\ncalculated.\na\nconst float* for vsTGamma,\nvmsTGamma\nconst double* for vdTGamma,\nvmdTGamma\nPointer to an array that contains the input vector\na.\ninca, incy\nconst MKL_INT\nSpecifies increments for the elements of a and y.\nmode\nconst MKL_INT64\nOverrides global VM mode setting for this\nfunction call. See vmlSetMode for possible\nvalues and their description.\nOutput Parameters\nName\nType\nDescription\ny\nfloat* for vsTGamma, vmsTGamma\ndouble* for vdTGamma, vmdTGamma\nPointer to an array that contains the output\nvector y.\nDescription\nThe v?TGamma function computes the gamma function for elements of the input vector a and writes them to\nthe output vector y. Precision overflow thresholds for the v?TGamma function are beyond the scope of this\ndocument. If the result does not meet the target precision, the function raises the OVERFLOW exception and\nsets the VM Error Status to VML_STATUS_OVERFLOW.\nSpecial Values for Real Function v?TGamma(x)\nArgument\nResult\nVM Error Status\nException\n+0\n+∞\nVML_STATUS_SING\nZERODIVIDE\n-0\n-∞\nVML_STATUS_SING\nZERODIVIDE\nnegative integer\nQNAN\nVML_STATUS_ERRDOM\nINVALID\n-∞\nQNAN\nVML_STATUS_ERRDOM\nINVALID\n+∞\n+∞\n \n \nX > overflow\n+∞\nVML_STATUS_OVERFLOW\nOVERFLOW\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2098\n\n\nArgument\nResult\nVM Error Status\nException\nQNAN\nQNAN\n \n \nSNAN\nQNAN\n \nINVALID\nv?ExpInt1\nComputes the exponential integral of vector elements.\nSyntax\nvsExpInt1( n, a, y );\nvsExpInt1I(n, a, inca, y, incy);\nvmsExpInt1( n, a, y, mode );\nvmsExpInt1I(n, a, inca, y, incy, mode);\nvdExpInt1( n, a, y );\nvdExpInt1I(n, a, inca, y, incy);\nvmdExpInt1( n, a, y, mode );\nvmdExpInt1I(n, a, inca, y, incy, mode);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nn\nconst MKL_INT\nSpecifies the number of elements to be\ncalculated.\na\nconst float* for vsExpInt1,\nvmsExpInt1\nconst double* for vdExpInt1,\nvmdExpInt1\nPointer to an array that contains the input vector\na.\ninca, incy\nconst MKL_INT\nSpecifies increments for the elements of a and y.\nmode\nconst MKL_INT64\nOverrides global VM mode setting for this\nfunction call. See vmlSetMode for possible\nvalues and their description.\nOutput Parameters\nName\nType\nDescription\ny\nfloat* for vsExpInt1, vmsExpInt1\ndouble* for vdExpInt1, vmdExpInt1\nPointer to an array that contains the output\nvector y.\nDescription\nThe v?ExpInt1 function computes the exponential integral E1 of vector elements.\nFor positive real values x, this can be written as:\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2099\n\n\nE1 x = ∫x\n∞e−t\nt\ndt = ∫1\n∞e−xt\nt\ndt.\nFor negative real values x, the result is defined as NAN.\nSpecial Values for Real Function v?ExpInt1(x)\nArgument\nResult\nVM Error Status\nException\nx < +0\nQNAN\nVML_STATUS_ERRDOM\nINVALID\n+0\n+∞\nVML_STATUS_SING\nZERODIVIDE\n-0\n+∞\nVML_STATUS_SING\nZERODIVIDE\n+∞\n+0\n \n \n-∞\nQNAN\nVML_STATUS_ERRDOM\nINVALID\nQNAN\nQNAN\n \n \nSNAN\nQNAN\n \nINVALID\nRounding Functions\nv?Floor\nComputes an integer value rounded towards minus\ninfinity for each vector element.\nSyntax\nvsFloor( n, a, y );\nvsFloorI(n, a, inca, y, incy);\nvmsFloor( n, a, y, mode );\nvmsFloorI(n, a, inca, y, incy, mode);\nvdFloor( n, a, y );\nvdFloorI(n, a, inca, y, incy);\nvmdFloor( n, a, y, mode );\nvmdFloorI(n, a, inca, y, incy, mode);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nn\nconst MKL_INT\nSpecifies the number of elements to be\ncalculated.\na\nconst float* for vsFloor,\nvmsFloor\nconst double* for vdFloor,\nvmdFloor\nPointer to an array that contains the input vector\na.\ninca, incy\nconst MKL_INT\nSpecifies increments for the elements of a and y.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2100\n\n\nName\nType\nDescription\nmode\nconst MKL_INT64\nOverrides global VM mode setting for this\nfunction call. See vmlSetMode for possible\nvalues and their description.\nOutput Parameters\nName\nType\nDescription\ny\nfloat* for vsFloor, vmsFloor\ndouble* for vdFloor, vmdFloor\nPointer to an array that contains the output\nvector y.\nDescription\nThe function computes an integer value rounded towards minus infinity for each vector element.\nSpecial Values for Real Function v?Floor(x)\nArgument\nResult\nException\n+0\n+0\n \n-0\n-0\n \n+∞\n+∞\n \n-∞\n-∞\n \nSNAN\nQNAN\nINVALID\nQNAN\nQNAN\n \nv?Ceil\nComputes an integer value rounded towards plus\ninfinity for each vector element.\nSyntax\nvsCeil( n, a, y );\nvsCeilI(n, a, inca, y, incy);\nvmsCeil( n, a, y, mode );\nvmsCeilI(n, a, inca, y, incy, mode);\nvdCeil( n, a, y );\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2101\n\n\nvdCeilI(n, a, inca, y, incy);\nvmdCeil( n, a, y, mode );\nvmdCeilI(n, a, inca, y, incy, mode);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nn\nconst MKL_INT\nSpecifies the number of elements to be\ncalculated.\na\nconst float* for vsCeil, vmsCeil\nconst double* for vdCeil, vmdCeil\nPointer to an array that contains the input vector\na.\ninca, incy\nconst MKL_INT\nSpecifies increments for the elements of a and y.\nmode\nconst MKL_INT64\nOverrides global VM mode setting for this\nfunction call. See vmlSetMode for possible\nvalues and their description.\nOutput Parameters\nName\nType\nDescription\ny\nfloat* for vsCeil, vmsCeil\ndouble* for vdCeil, vmdCeil\nPointer to an array that contains the output\nvector y.\nDescription\nThe function computes an integer value rounded towards plus infinity for each vector element.\nSpecial Values for Real Function v?Ceil(x)\nArgument\nResult\nException\n+0\n+0\n \n-0\n-0\n \n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2102\n\n\nArgument\nResult\nException\n+∞\n+∞\n \n-∞\n-∞\n \nSNAN\nQNAN\nINVALID\nQNAN\nQNAN\n \nv?Trunc\nComputes an integer value rounded towards zero for\neach vector element.\nSyntax\nvsTrunc( n, a, y );\nvsTruncI(n, a, inca, y, incy);\nvmsTrunc( n, a, y, mode );\nvmsTruncI(n, a, inca, y, incy, mode);\nvdTrunc( n, a, y );\nvdTruncI(n, a, inca, y, incy);\nvmdTrunc( n, a, y, mode );\nvmdTruncI(n, a, inca, y, incy, mode);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nn\nconst MKL_INT\nSpecifies the number of elements to be\ncalculated.\na\nconst float* for vsTrunc,\nvmsTrunc\nconst double* for vdTrunc,\nvmdTrunc\nPointer to an array that contains the input vector\na.\ninca, incy\nconst MKL_INT\nSpecifies increments for the elements of a and y.\nmode\nconst MKL_INT64\nOverrides global VM mode setting for this\nfunction call. See vmlSetMode for possible\nvalues and their description.\nOutput Parameters\nName\nType\nDescription\ny\nfloat* for vsTrunc, vmsTrunc\ndouble* for vdTrunc, vmdTrunc\nPointer to an array that contains the output\nvector y.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2103\n\n\nDescription\nThe function computes an integer value rounded towards zero for each vector element.\nSpecial Values for Real Function v?Trunc(x)\nArgument\nResult\nException\n+0\n+0\n \n-0\n-0\n \n+∞\n+∞\n \n-∞\n-∞\n \nSNAN\nQNAN\nINVALID\nQNAN\nQNAN\n \nv?Round\nComputes a value rounded to the nearest integer for\neach vector element.\nSyntax\nvsRound( n, a, y );\nvsRoundI(n, a, inca, y, incy);\nvmsRound( n, a, y, mode );\nvmsRoundI(n, a, inca, y, incy, mode);\nvdRound( n, a, y );\nvdRoundI(n, a, inca, y, incy);\nvmdRound( n, a, y, mode );\nvmdRoundI(n, a, inca, y, incy, mode);\nInclude Files\n•\nmkl.h\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2104\n\n\nInput Parameters\nName\nType\nDescription\nn\nconst MKL_INT\nSpecifies the number of elements to be\ncalculated.\na\nconst float* for vsRound,\nvmsRound\nconst double* for vdRound,\nvmdRound\nPointer to an array that contains the input vector\na.\ninca, incy\nconst MKL_INT\nSpecifies increments for the elements of a and y.\nmode\nconst MKL_INT64\nOverrides global VM mode setting for this\nfunction call. See vmlSetMode for possible\nvalues and their description.\nOutput Parameters\nName\nType\nDescription\ny\nfloat* for vsRound, vmsRound\ndouble* for vdRound, vmdRound\nPointer to an array that contains the output\nvector y.\nDescription\nThe function computes a value rounded to the nearest integer for each vector element. Input elements that\nare halfway between two consecutive integers are always rounded away from zero regardless of the rounding\nmode.\nSpecial Values for Real Function v?Round(x)\nArgument\nResult\nException\n+0\n+0\n \n-0\n-0\n \n+∞\n+∞\n \n-∞\n-∞\n \nSNAN\nQNAN\nINVALID\nQNAN\nQNAN\n \nv?NearbyInt\nComputes a rounded integer value in the current\nrounding mode for each vector element.\nSyntax\nvsNearbyInt( n, a, y );\nvsNearbyIntI(n, a, inca, y, incy);\nvmsNearbyInt( n, a, y, mode );\nvmsNearbyIntI(n, a, inca, y, incy, mode);\nvdNearbyInt( n, a, y );\nvdNearbyIntI(n, a, inca, y, incy);\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2105\n\n\nvmdNearbyInt( n, a, y, mode );\nvmdNearbyIntI(n, a, inca, y, incy, mode);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nn\nconst MKL_INT\nSpecifies the number of elements to be\ncalculated.\na\nconst float* for vsNearbyInt,\nvmsNearbyInt\nconst double* for vdNearbyInt,\nvmdNearbyInt\nPointer to an array that contains the input vector\na.\ninca, incy\nconst MKL_INT\nSpecifies increments for the elements of a and y.\nmode\nconst MKL_INT64\nOverrides global VM mode setting for this\nfunction call. See vmlSetMode for possible\nvalues and their description.\nOutput Parameters\nName\nType\nDescription\ny\nfloat* for vsNearbyInt,\nvmsNearbyInt\ndouble* for vdNearbyInt,\nvmdNearbyInt\nPointer to an array that contains the output\nvector y.\nDescription\nThe v?NearbyInt function computes a rounded integer value in a current rounding mode for each vector\nelement.\nSpecial Values for Real Function v?NearbyInt(x)\nArgument\nResult\nException\n+0\n+0\n \n-0\n-0\n \n+∞\n+∞\n \n-∞\n-∞\n \nSNAN\nQNAN\nINVALID\nQNAN\nQNAN\n \nv?Rint\nComputes a rounded integer value in the current\nrounding mode.\nSyntax\nvsRint( n, a, y );\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2106\n\n\nvsRintI(n, a, inca, y, incy);\nvmsRint( n, a, y, mode );\nvmsRintI(n, a, inca, y, incy, mode);\nvdRint( n, a, y );\nvdRintI(n, a, inca, y, incy);\nvmdRint( n, a, y, mode );\nvmdRintI(n, a, inca, y, incy, mode);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nn\nconst MKL_INT\nSpecifies the number of elements to be\ncalculated.\na\nconst float* for vsRint, vmsRint\nconst double* for vdRint, vmdRint\nPointer to an array that contains the input vector\na.\ninca, incy\nconst MKL_INT\nSpecifies increments for the elements of a and y.\nmode\nconst MKL_INT64\nOverrides global VM mode setting for this\nfunction call. See vmlSetMode for possible\nvalues and their description.\nOutput Parameters\nName\nType\nDescription\ny\nfloat* for vsRint, vmsRint\ndouble* for vdRint, vmdRint\nPointer to an array that contains the output\nvector y.\nDescription\nThe v?Rint function computes a rounded floating-point integer value using the current rounding mode for\neach vector element.\nThe rounding mode affects the results computed for inputs that fall between consecutive integers. For\nexample:\n•\nf(0.5) = 0, for rounding modes set to round to nearest round toward zero or to minus infinity.\n•\nf(0.5) = 1, for rounding modes set to plus infinity.\n•\nf(-1.5) = -2, for rounding modes set to round to nearest or to minus infinity.\n•\nf(-1.5) = -1, for rounding modes set to round toward zero or to plus infinity.\nSpecial Values for Real Function v?Rint(x)\nArgument\nResult\nException\n+0\n+0\n \n-0\n-0\n \n+∞\n+∞\n \nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2107\n\n\nArgument\nResult\nException\n-∞\n-∞\n \nSNAN\nQNAN\nINVALID\nQNAN\nQNAN\n \nv?Modf\nComputes a truncated integer value and the remaining\nfraction part for each vector element.\nSyntax\nvsModf( n, a, y, z );\nvsModfI(n, a, inca, y, incy, z, incz);\nvmsModf( n, a, y, z, mode );\nvmsModfI(n, a, inca, y, incy, z, incz, mode);\nvdModf( n, a, y, z );\nvdModfI(n, a, inca, y, incy, z, incz);\nvmdModf( n, a, y, z, mode );\nvmdModfI(n, a, inca, y, incy, z, incz, mode);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nn\nconst MKL_INT\nSpecifies the number of elements to be\ncalculated.\na\nconst float* for vsModf, vmsModf\nconst double* for vdModf, vmdModf\nPointer to an array that contains the input vector\na.\ninca, incy,\nincz\nconst MKL_INT\nSpecifies increments for the elements of a, y,\nand z.\nmode\nconst MKL_INT64\nOverrides global VM mode setting for this\nfunction call. See vmlSetMode for possible\nvalues and their description.\nOutput Parameters\nName\nType\nDescription\ny, z\nfloat* for vsModf, vmsModf\ndouble* for vdModf, vmdModf\nPointer to an array that contains the output\nvector y and z.\nDescription\nThe function computes a truncated integer value and the remaining fraction part for each vector element.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2108\n\n\nSpecial Values for Real Function v?Modf(x)\nArgument\nResult: y(i)\nResult: z(i)\nException\n+0\n+0\n+0\n \n-0\n-0\n-0\n \n+∞\n+∞\n+0\n \n-∞\n-∞\n-0\n \nSNAN\nQNAN\nQNAN\nINVALID\nQNAN\nQNAN\nQNAN\n \nv?Frac\nComputes a signed fractional part for each vector\nelement.\nSyntax\nvsFrac( n, a, y );\nvsFracI(n, a, inca, y, incy);\nvmsFrac( n, a, y, mode );\nvmsFracI(n, a, inca, y, incy, mode);\nvdFrac( n, a, y );\nvdFracI(n, a, inca, y, incy);\nvmdFrac( n, a, y, mode );\nvmdFracI(n, a, inca, y, incy, mode);\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2109\n\n\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nn\nconst MKL_INT\nSpecifies the number of elements to be\ncalculated.\na\nconst float* for vsFrac, vmsFrac\nconst double* for vdFrac, vmdFrac\nPointer to an array that contains the input vector\na.\ninca, incy\nconst MKL_INT\nSpecifies increments for the elements of a and y.\nmode\nconst MKL_INT64\nOverrides global VM mode setting for this\nfunction call. See vmlSetMode for possible\nvalues and their description.\nOutput Parameters\nName\nType\nDescription\ny\nfloat* for vsFrac, vmsFrac\ndouble* for vdFrac, vmdFrac\nPointer to an array that contains the output\nvector y.\nDescription\nThe function computes a signed fractional part for each vector element.\nSpecial Values for Real Function v?Frac(x)\nArgument\nResult\nException\n+0\n+0\n \n-0\n-0\n \n+∞\n+0\n \n-∞\n-0\n \nSNAN\nQNAN\nINVALID\nQNAN\nQNAN\n \n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2110\n\n\nVM Pack/Unpack Functions\nThis section describes VM functions that convert vectors with unit increment to and from vectors with positive\nincrement indexing, vector indexing, and mask indexing (see Appendix \"Vector Arguments in VM\" for details\non vector indexing methods).\nThe table below lists available VM Pack/Unpack functions, together with data types and indexing methods\nassociated with them.\nVM Pack/Unpack Functions\nFunction Short Name\nData\nTypes\nIndexing\nMethods\nDescription\nv?Pack\ns, d, c,\nz\nI,V,M\nGathers elements of arrays, indexed by different methods.\nv?Unpack\ns, d, c,\nz\nI,V,M\nScatters vector elements to arrays with different indexing.\nSee Also\nAppendix \"Vector Arguments in VM\" \nv?Pack\nCopies elements of an array with specified indexing to\na vector with unit increment.\nSyntax\nvsPackI( n, a, inca, y );\nvsPackV( n, a, ia, y );\nvsPackM( n, a, ma, y );\nvdPackI( n, a, inca, y );\nvdPackV( n, a, ia, y );\nvdPackM( n, a, ma, y );\nvcPackI( n, a, inca, y );\nvcPackV( n, a, ia, y );\nvcPackM( n, a, ma, y );\nvzPackI( n, a, inca, y );\nvzPackV( n, a, ia, y );\nvzPackM( n, a, ma, y );\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nn\nconst MKL_INT\nSpecifies the number of elements to be calculated.\na\nconst float* for vsPackI,\nvsPackV, vsPackM\nSpecifies pointer to an array that contains the input\nvector a. The arrays must be:\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2111\n\n\nName\nType\nDescription\nconst double* for vdPackI,\nvdPackV, vdPackM\nconst MKL_Complex8* for vcPackI,\nvcPackV, vcPackM\nconst MKL_Complex16* for\nvzPackI, vzPackV, vzPackM\nfor v?PackI, at least(1 + (n-1)*inca)\nfor v?PackV, at least max( n,max(ia[j]) ), j=0,\n…, n-1\nfor v?PackM, at least n.\ninca\nconst MKL_INT for vsPackI,\nvdPackI, vcPackI, vzPackI\nSpecifies the increment for the elements of a.\nia\nconst int* for vsPackV, vdPackV,\nvcPackV, vzPackV\nSpecifies the pointer to an array of size at least n\nthat contains the index vector for the elements of a.\nma\nconst int* for vsPackM, vdPackM,\nvcPackM, vzPackM\nSpecifies the pointer to an array of size at least n\nthat contains the mask vector for the elements of a.\nOutput Parameters\nName\nType\nDescription\ny\nfloat* for vsPackI, vsPackV,\nvsPackM\ndouble* for vdPackI, vdPackV,\nvdPackM\nconst MKL_Complex8* for vcPackI,\nvcPackV, vcPackM\nconst MKL_Complex16* for vzPackI,\nvzPackV, vzPackM\nPointer to an array of size at least n that\ncontains the output vector y.\nv?Unpack\nCopies elements of a vector with unit increment to an\narray with specified indexing.\nSyntax\nvsUnpackI( n, a, y, incy );\nvsUnpackV( n, a, y, iy );\nvsUnpackM( n, a, y, my );\nvdUnpackI( n, a, y, incy );\nvdUnpackV( n, a, y, iy );\nvdUnpackM( n, a, y, my );\nvcUnpackI( n, a, y, incy );\nvcUnpackV( n, a, y, iy );\nvcUnpackM( n, a, y, my );\nvzUnpackI( n, a, y, incy );\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2112\n\n\nvzUnpackV( n, a, y, iy );\nvzUnpackM( n, a, y, my );\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nn\nconst MKL_INT\nSpecifies the number of elements to be\ncalculated.\na\nconst float* for vsUnpackI,\nvsUnpackV, vsUnpackM\nconst double* for vdUnpackI,\nvdUnpackV, vdUnpackM\nconst MKL_Complex8* for\nvcUnpackI, vcUnpackV, vcUnpackM\nconst MKL_Complex16* for\nvzUnpackI, vzUnpackV, vzUnpackM\nSpecifies the pointer to an array of size at least n\nthat contains the input vector a.\nincy\nconst MKL_INT for vsUnpackI,\nvdUnpackI, vcUnpackI, vzUnpackI\nSpecifies the increment for the elements of y.\niy\nconst int* for vsUnpackV,\nvdUnpackV, vcUnpackV, vzUnpackV\nSpecifies the pointer to an array of size at least n\nthat contains the index vector for the elements\nof a.\nmy\nconst int* for vsUnpackM,\nvdUnpackM, vcUnpackM, vzUnpackM\nSpecifies the pointer to an array of size at least n\nthat contains the mask vector for the elements\nof a.\nOutput Parameters\nName\nType\nDescription\ny\nfloat* for vsUnpackI, vsUnpackV,\nvsUnpackM\ndouble* for vdUnpackI,\nvdUnpackV, vdUnpackM\nconst MKL_Complex8* for\nvcUnpackI, vcUnpackV, vcUnpackM\nconst MKL_Complex16* for\nvzUnpackI, vzUnpackV, vzUnpackM\nSpecifies the pointer to an array that contains the\noutput vector y.\nThe array must be:\n \nfor v?UnpackI, at least (1 + (n-1)*incy)\n \nfor v?UnpackV, at least\nmax( n,max(ia[j]) ),j=0,..., n-1,\n \nfor v?UnpackM, at least n.\nVM Service Functions\nThe VM Service functions enable you to set/get the accuracy mode and error code. These functions are\navailable both in the Fortran and C interfaces. The table below lists available VM Service functions and their\nshort description.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2113\n\n\nVM Service Functions\nFunction Short Name\nDescription\nvmlSetMode\nSets the VM mode\nvmlGetMode\nGets the VM mode\nMKLFreeTls\nFrees allocated VM/VS thread local storage memory from within DllMain\nroutine (Windows* OS only)\nvmlSetErrStatus\nSets the VM Error Status\nvmlGetErrStatus\nGets the VM Error Status\nvmlClearErrStatus\nClears the VM Error Status\nvmlSetErrorCallBack\nSets the additional error handler callback function\nvmlGetErrorCallBack\nGets the additional error handler callback function\nvmlClearErrorCallBack\nDeletes the additional error handler callback function\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nvmlSetMode\nSets a new mode for VM functions according to the\nmode parameter and stores the previous VM mode to\noldmode.\nSyntax\noldmode = vmlSetMode( mode );\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nmode\nconst MKL_UINT\nSpecifies the VM mode to be set.\nOutput Parameters\nName\nType\nDescription\noldmode\nunsigned int\nSpecifies the former VM mode.\nDescription\nThe vmlSetMode function sets a new mode for VM functions according to the mode parameter and stores the\nprevious VM mode to oldmode. The mode change has a global effect on all the VM functions within a thread.\nNOTE\nYou can override the global mode setting and change the mode for a given VM function call\nby using a respective vm[s,d]<Func> variant of the function.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2114\n\n\nThe mode parameter is designed to control accuracy, handling of denormalized numbers, and error handling. \nTable \"Values of the mode Parameter\" lists values of the mode parameter. You can obtain all other possible\nvalues of the mode parameter from the mode parameter values by a using bitwise OR ( | ) operation to\ncombine one value for accuracy, one value for handling of denormalized numbers, and one value for error\ncontrol options. The default value of the mode parameter is VML_HA | VML_FTZDAZ_CURRENT |\nVML_ERRMODE_DEFAULT.\nThe VML_FTZDAZ_ON mode is specifically designed to improve the performance of computations that involve\ndenormalized numbers at the cost of reasonable accuracy loss. This mode changes the numeric behavior of\nthe functions: denormalized input values are treated as zeros (DAZ = denormals-are-zero) and denormalized\nresults are flushed to zero (FTZ = flush-to-zero). Accuracy loss may occur if input and/or output values are\nclose to denormal range.\nValues of the mode Parameter\nValue of mode\nDescription\nAccuracy Control\nVML_HA\nhigh accuracy versions of VM functions\nVML_LA\nlow accuracy versions of VM functions\nVML_EP\nenhanced performance accuracy versions of VM functions\nDenormalized Numbers Handling Control\nVML_FTZDAZ_ON\nFaster processing of denormalized inputs is enabled.\nVML_FTZDAZ_OFF\nFaster processing of denormalized inputs is disabled.\nVML_FTZDAZ_CURRENT\nKeep the current CPU settings for denormalized inputs.\nError Mode Control\nVML_ERRMODE_IGNORE\nOn computation error, VM Error status is updated, but otherwise no\naction is set. Cannot be combined with other VML_ERRMODE\nsettings.\nVML_ERRMODE_NOERR\nOn computation error, VM Error status is not updated and no action is\nset. Cannot be combined with other VML_ERRMODE settings.\nVML_ERRMODE_ERRNO\nOn error, the errno variable is set.\nVML_ERRMODE_STDERR\nOn error, the error text information is written to stderr.\nVML_ERRMODE_EXCEPT\nOn error, an exception is raised.\nVML_ERRMODE_CALLBACK\nOn error, an additional error handler function is called.\nVML_ERRMODE_DEFAULT\nOn error, the errno variable is set, an exception is raised, and an\nadditional error handler function is called.\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nExamples\nThe following example shows how to set low accuracy, fast processing for denormalized numbers and stderr\nerror mode:\nvmlSetMode( VML_LA ); \nvmlSetMode( VML_LA | VML_FTZDAZ_ON | VML_ERRMODE_STDERR );\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2115\n\n\nvmlGetMode\nGets the VM mode.\nSyntax\nmod = vmlGetMode( void );\nInclude Files\n•\nmkl.h\nOutput Parameters\nName\nType\nDescription\nmod\nunsigned int\nSpecifies the packed mode parameter.\nDescription\nThe function vmlGetMode returns the VM mode parameter that controls accuracy, handling of denormalized\nnumbers, and error handling options. The mod variable value is a combination of the values listed in the\ntable \"Values of the mode Parameter\". You can obtain these values using the respective mask from the table \n\"Values of Mask for the mode Parameter\".\nValues of Mask for the mode Parameter\nValue of mask\nDescription\nVML_ACCURACY_MASK\nSpecifies mask for accuracy mode selection.\nVML_FTZDAZ_MASK\nSpecifies mask for FTZDAZ mode selection.\nVML_ERRMODE_MASK\nSpecifies mask for error mode selection.\nSee example below:\nExamples\naccm = vmlGetMode(void )& VML_ACCURACY_MASK; \ndenm = vmlGetMode(void )& VML_FTZDAZ_MASK; \nerrm = vmlGetMode(void )& VML_ERRMODE_MASK;\nMKLFreeTls\nFrees allocated VM/VS thread local storage memory\nfrom within DllMain routine. Use on Windows* OS\nonly.\nSyntax\nvoid MKLFreeTls( const MKL_UINT fdwReason );\nInclude Files\n•\nmkl_vml_functions_win.h\nDescription\nThe MKLFreeTls routine frees thread local storage (TLS) memory which has been allocated when using MKL\nstatic libraries to link into a DLL on the Windows* OS. The routine should only be used within DllMain.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2116\n\n\nNOTE\nIt is only necessary to use MKLFreeTls for TLS data in the VM and VS domains.\nInput Parameters\nName\nType\nDescription\nfdwReason\nconst MKL_UINT\nReason code from the DllMain call.\nExample\nBOOL WINAPI DllMain(HINSTANCE hInst, DWORD fdwReason, LPVOID lpvReserved)\n{\n    /*customer code*/\n    MKLFreeTls(fdwReason);\n    /*customer code*/\n}\nvmlSetErrStatus\nSets the new VM Error Status according to err and\nstores the previous VM Error Status to olderrSets the\nglobal VM Status according to new values and returns\nthe previous VM Status.\nSyntax\nolderr = vmlSetErrStatus( status );\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nstatus\nconst MKL_INT\nSpecifies the VM error status to be set.\nOutput Parameters\nName\nType\nDescription\nolderr\nint\nSpecifies the former VM error status.\nDescription\nTable \"Values of the VM Status\" lists possible values of the err parameter.\nValues of the VM Status\nStatus\nDescription\nSuccessful Execution\nVML_STATUS_OK\nThe execution was completed successfully.\nWarnings\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2117\n\n\nStatus\nDescription\nVML_STATUS_ACCURACYWARNING\nThe execution was completed successfully in a different accuracy\nmode.\nErrors\nVML_STATUS_BADSIZE\nThe function does not support the preset accuracy mode. The Low\nAccuracy mode is used instead.\nVML_STATUS_BADMEM\nNULL pointer is passed.\nVML_STATUS_ERRDOM\nAt least one of array values is out of a range of definition.\nVML_STATUS_SING\nAt least one of the input array values causes a divide-by-zero\nexception or produces an invalid (QNaN) result.\nVML_STATUS_OVERFLOW\nAn overflow has happened during the calculation process.\nVML_STATUS_UNDERFLOW\nAn underflow has happened during the calculation process.\nExamples\nolderr = vmlSetErrStatus( VML_STATUS_OK );\nolderr = vmlSetErrStatus( VML_STATUS_ERRDOM );\nolderr = vmlSetErrStatus( VML_STATUS_UNDERFLOW );\nvmlGetErrStatus\nGets the VM Error Status.\nSyntax\nerr = vmlGetErrStatus( void );\nInclude Files\n•\nmkl.h\nOutput Parameters\nName\nType\nDescription\nerr\nint\nSpecifies the VM error status.\nvmlClearErrStatus\nSets the VM Error Status to VML_STATUS_OK and\nstores the previous VM Error Status to olderr.\nSyntax\nolderr = vmlClearErrStatus( void );\nInclude Files\n•\nmkl.h\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2118\n\n\nOutput Parameters\nName\nType\nDescription\nolderr\nint\nSpecifies the former VM error status.\nvmlSetErrorCallBack\nSets the additional error handler callback function and\ngets the old callback function.\nSyntax\noldcallback = vmlSetErrorCallBack( callback );\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nDescription\ncallback\nPointer to the callback function.\nThe callback function has the following format:\nstatic int __cdecl \nMyHandler(DefVmlErrorContext*\npContext)\n{\n   /* Handler body */\n};\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2119\n\n\nName\nDescription\nThe passed error structure is defined as follows:\ntypedef struct _DefVmlErrorContext\n{\nint iCode;/* Error status value */\nint iIndex;/* Index for bad array\n   element, or bad array\n   dimension, or bad\n   array pointer */\ndouble dbA1; /* Error argument 1 */\ndouble dbA2; /* Error argument 2 */\ndouble dbR1; /* Error result 1 */\ndouble dbR2; /* Error result 2 */\nchar cFuncName[64]; /* Function name */\nint iFuncNameLen; /* Length of \nfunctionname*/\ndouble dbA1Im; /* Error argument 1, imag \npart*/\ndouble dbA2Im; /* Error argument 2, imag \npart*/\ndouble dbR1Im; /* Error result 1, imag \npart*/\ndouble dbR2Im; /* Error result 2, imag \npart*/\n} DefVmlErrorContext;\nOutput Parameters\nName\nType\nDescription\noldcallback\nint\nPointer to the former callback function.\nDescription\nThe callback function is called on each VM mathematical function error if VML_ERRMODE_CALLBACK error\nmode is set (see \"Values of the mode Parameter\").\nUse the vmlSetErrorCallBack() function if you need to define your own callback function instead of\ndefault empty callback function.\nThe input structure for a callback function contains the following information about the error encountered:\n•\nthe input value that caused an error\n•\nlocation (array index) of this value\n•\nthe computed result value\n•\nerror code\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2120\n\n\n•\nname of the function in which the error occurred.\nYou can insert your own error processing into the callback function. This may include correcting the passed\nresult values in order to pass them back and resume computation. The standard error handler is called after\nthe callback function only if it returns 0.\nvmlGetErrorCallBack\nGets the additional error handler callback function.\nSyntax\ncallback = vmlGetErrorCallBack( void );\nInclude Files\n•\nmkl.h\nOutput Parameters\nName\nDescription\ncallback\nPointer to the callback function\nvmlClearErrorCallBack\nDeletes the additional error handler callback function\nand retrieves the former callback function.\nSyntax\noldcallback = vmlClearErrorCallBack( void );\nInclude Files\n•\nmkl.h\nOutput Parameters\nName\nType\nDescription\noldcallback\nint\nPointer to the former callback function\nMiscellaneous VM Functions\nv?CopySign\nReturns vector of elements of one argument with\nsigns changed to match other argument elements.\nSyntax\nvsCopySign (n, a, y);\nvsCopySignI(n, a, inca, b, incb, y, incy);\nvmsCopySign (n, a, y, mode);\nvmsCopySignI(n, a, inca, b, incb, y, incy, mode);\nvdCopySign (n, a, y);\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2121\n\n\nvdCopySignI(n, a, inca, b, incb, y, incy);\nvmdCopySign (n, a, y, mode);\nvmdCopySignI(n, a, inca, b, incb, y, incy, mode);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nn\nconst MKL_INT\nSpecifies the number of elements to be\ncalculated.\na\nconst float* for vsCopySign\nconst float* for vmsCopySign\nconst double* for vdCopySign\nconst double* for vmdCopySign\nPointer to the array containing the input vector\na.\ninca, incb,\nincy\nconst MKL_INT\nSpecifies increments for the elements of a, b,\nand y.\nmode\nconst MKL_INT64\nOverrides the global VM mode setting for this\nfunction call. See vmlSetMode for possible\nvalues and their description.\nOutput Parameters\nName\nType\nDescription\ny\nfloat* for vsCopySign\nfloat* for vmsCopySign\ndouble* for vdCopySign\ndouble* for vmdCopySign\nPointer to an array containing the output vector\ny.\nDescription\nThe v?CopySign function returns the first vector argument elements with the sign changed to match the\nsign of the second vector argument's corresponding elements.\nv?NextAfter\nReturns vector of elements containing the next\nrepresentable floating-point values following the\nvalues from the elements of one vector in the\ndirection of the corresponding elements of another\nvector.\nSyntax\nvsNextAfter (n, a, b, y);\nvsNextAfterI(n, a, inca, b, incb, y, incy);\nvmsNextAfter (n, a, b, y, mode);\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2122\n\n\nvmsNextAfterI(n, a, inca, b, incb, y, incy, mode);\nvdNextAfter (n, a, b, y);\nvdNextAfterI(n, a, inca, b, incb, y, incy);\nvmdNextAfter (n, a, b, y, mode);\nvmdNextAfterI(n, a, inca, b, incb, y, incy, mode);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nn\nconst MKL_INT\nSpecifies the number of elements to be\ncalculated.\na, b\nconst float* for vsNextAfter\nconst float* for vmsNextAfter\nconst double* for vdNextAfter\nconst double* for vmdNextAfter\nPointers to the arrays containing the input\nvectors a and b.\ninca, incb,\nincy\nconst MKL_INT\nSpecifies increments for the elements of a, b,\nand y.\nmode\nconst MKL_INT64\nOverrides the global VM mode setting for this\nfunction call. See vmlSetMode for possible\nvalues and their description.\nOutput Parameters\nName\nType\nDescription\ny\nfloat* for vsNextAfter\nfloat* for vmsNextAfter\ndouble* for vdNextAfter\ndouble* for vmdNextAfter\nPointer to an array containing the output vector\ny.\nDescription\nThe v?NextAfter function returns a vector containing the next representable floating-point values following\nthe first vector argument elements in the direction of the second vector argument's corresponding elements.\nSpecial cases:\nOverflow\nThe function raises overflow and inexact floating-point exceptions and\nsets VML_STATUS_OVERFLOW if an input vector argument element is\nfinite and the corresponding result vector element value is infinite.\nUnderflow\nThe function raises underflow and inexact floating-point exceptions\nand sets VML_STATUS_UNDERFLOW if a result vector element value is\nsubnormal or zero, and different from the corresponding input vector\nargument element.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2123\n\n\nEven though underflow or overflow can occur, the returned value is independent of the current rounding\ndirection mode.\nv?Fdim\nReturns vector containing the differences of the\ncorresponding elements of the vector arguments if the\nfirst is larger and +0 otherwise.\nSyntax\nvsFdim (n, a, b, y);\nvsFdimI(n, a, inca, b, incb, y, incy);\nvmsFdim (n, a, b, y, mode);\nvmsFdimI(n, a, inca, b, incb, y, incy, mode);\nvdFdim (n, a, b, y);\nvdFdimI(n, a, inca, b, incb, y, incy);\nvmdFdim (n, a, b, y, mode);\nvmdFdimI(n, a, inca, b, incb, y, incy, mode);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nn\nconst MKL_INT\nSpecifies the number of elements to be\ncalculated.\na, b\nconst float* for vsFdim\nconst float* for vmsFdim\nconst double* for vdFdim\nconst double* for vmdFdim\nPointers to the arrays containing the input\nvectors a and b.\ninca, incb,\nincy\nconst MKL_INT\nSpecifies increments for the elements of a, b,\nand y.\nmode\nconst MKL_INT64\nOverrides the global VM mode setting for this\nfunction call. See vmlSetMode for possible\nvalues and their description.\nOutput Parameters\nName\nType\nDescription\ny\nfloat* for vsFdim\nfloat* for vmsFdim\ndouble* for vdFdim\ndouble* for vmdFdim\nPointer to an array containing the output vector\ny.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2124\n\n\nDescription\nThe v?Fdim function returns a vector containing the differences of the corresponding elements of the first\nand second vector arguments if the first element is larger, and +0 otherwise.\nSpecial values for Real Function v?Fdim(x, y)\nArgument 1\nArgument 2\nResult\nVM Error Status\nException\nany\nQNAN\nQNAN\nany\nSNAN\nQNAN\nINVALID\nQNAN\nany\nQNAN\nSNAN\nany\nQNAN\nINVALID\nv?Fmax\nReturns the larger of each pair of elements of the two\nvector arguments.\nSyntax\nvsFmax (n, a, b, y);\nvsFmaxI(n, a, inca, b, incb, y, incy);\nvmsFmax (n, a, b, y, mode);\nvmsFmaxI(n, a, inca, b, incb, y, incy, mode);\nvdFmax (n, a, b, y);\nvdFmaxI(n, a, inca, b, incb, y, incy);\nvmdFmax (n, a, b, y, mode);\nvmdFmaxI(n, a, inca, b, incb, y, incy, mode);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nn\nconst MKL_INT\nSpecifies the number of elements to be\ncalculated.\na, b\nconst float* for vsFmax\nconst float* for vmsFmax\nconst double* for vdFmax\nconst double* for vmdFmax\nPointers to the arrays containing the input\nvectors a and b.\ninca, incb,\nincy\nconst MKL_INT\nSpecifies increments for the elements of a, b,\nand y.\nmode\nconst MKL_INT64\nOverrides the global VM mode setting for this\nfunction call. See vmlSetMode for possible\nvalues and their description.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2125\n\n\nOutput Parameters\nName\nType\nDescription\ny\nfloat* for vsFmax\nfloat* for vmsFmax\ndouble* for vdFmax\ndouble* for vmdFmax\nPointer to an array containing the output vector\ny.\nDescription\nThe v?Fmax function returns a vector with element values equal to the larger value from each pair of\ncorresponding elements of the two vectors a and b: if ai < biv?Fmax returns bi, otherwise v?Fmax returns ai.\nSpecial values for Real Function v?Fmax(x, y)\nArgument 1\nArgument 2\nResult\nVM Error Status\nException\nai not NAN\nNAN\nai\nNAN\nbi not NAN\nbi\nNAN\nNAN\nNAN\nSee Also\nFmin Returns the smaller of each pair of elements of the two vector arguments.\nMaxMag Returns the element with the larger magnitude between each pair of elements of the two\nvector arguments.\nv?Fmin\nReturns the smaller of each pair of elements of the\ntwo vector arguments.\nSyntax\nvsFmin (n, a, b, y);\nvsFminI(n, a, inca, b, incb, y, incy);\nvmsFmin (n, a, b, y, mode);\nvmsFminI(n, a, inca, b, incb, y, incy, mode);\nvdFmin (n, a, b, y);\nvdFminI(n, a, inca, b, incb, y, incy);\nvmdFmin (n, a, b, y, mode);\nvmdFminI(n, a, inca, b, incb, y, incy, mode);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nn\nconst MKL_INT\nSpecifies the number of elements to be\ncalculated.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2126\n\n\nName\nType\nDescription\na, b\nconst float* for vsFmin\nconst float* for vmsFmin\nconst double* for vdFmin\nconst double* for vmdFmin\nPointers to the arrays containing the input\nvectors a and b.\ninca, incb,\nincy\nconst MKL_INT\nSpecifies increments for the elements of a, b,\nand y.\nmode\nconst MKL_INT64\nOverrides the global VM mode setting for this\nfunction call. See vmlSetMode for possible\nvalues and their description.\nOutput Parameters\nName\nType\nDescription\ny\nfloat* for vsFmin\nfloat* for vmsFmin\ndouble* for vdFmin\ndouble* for vmdFmin\nPointer to an array containing the output vector\ny.\nDescription\nThe v?Fmin function returns a vector with element values equal to the smaller value from each pair of\ncorresponding elements of the two vectors a and b: if bi < aiv?Fmin returns bi, otherwise v?Fmin returns ai.\nSpecial values for Real Function v?Fmin(x, y)\nArgument 1\nArgument 2\nResult\nVM Error Status\nException\nai not NAN\nNAN\nai\nNAN\nbi not NAN\nbi\nNAN\nNAN\nNAN\nSee Also\nFmax Returns the larger of each pair of elements of the two vector arguments.\nMinMag Returns the element with the smaller magnitude between each pair of elements of the\ntwo vector arguments.\nv?MaxMag\nReturns the element with the larger magnitude\nbetween each pair of elements of the two vector\narguments.\nSyntax\nvsMaxMag (n, a, b, y);\nvsMaxMagI(n, a, inca, b, incb, y, incy);\nvmsMaxMag (n, a, b, y, mode);\nvmsMaxMagI(n, a, inca, b, incb, y, incy, mode);\nvdMaxMag (n, a, b, y);\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2127\n\n\nvdMaxMagI(n, a, inca, b, incb, y, incy);\nvmdMaxMag (n, a, b, y, mode);\nvmdMaxMagI(n, a, inca, b, incb, y, incy, mode);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nn\nconst MKL_INT\nSpecifies the number of elements to be\ncalculated.\na, b\nconst float* for vsMaxMag\nconst float* for vmsMaxMag\nconst double* for vdMaxMag\nconst double* for vmdMaxMag\nPointers to the arrays containing the input\nvectors a and b.\ninca, incb,\nincy\nconst MKL_INT\nSpecifies increments for the elements of a, b,\nand y.\nmode\nconst MKL_INT64\nOverrides the global VM mode setting for this\nfunction call. See vmlSetMode for possible\nvalues and their description.\nOutput Parameters\nName\nType\nDescription\ny\nfloat* for vsMaxMag\nfloat* for vmsMaxMag\ndouble* for vdMaxMag\ndouble* for vmdMaxMag\nPointer to an array containing the output vector\ny.\nDescription\nThe v?MaxMag function returns a vector with element values equal to the element with the larger magnitude\nfrom each pair of corresponding elements of the two vectors a and b:\n•\nIf |ai| > |bi| v?MaxMag returns ai, otherwise v?MaxMag returns ai.\n•\nIf |bi| > |ai| v?MaxMag returns bi, otherwise v?MaxMag returns ai.\n•\nOtherwise v?MaxMag behaves like v?Fmax.\nSpecial values for Real Function v?MaxMag(x, y)\nArgument 1\nArgument 2\nResult\nVM Error Status\nException\nai not NAN\nNAN\nai\nNAN\nbi not NAN\nbi\nNAN\nNAN\nNAN\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2128\n\n\nSee Also\nMinMag Returns the element with the smaller magnitude between each pair of elements of the\ntwo vector arguments.\nFmax Returns the larger of each pair of elements of the two vector arguments.\nv?MinMag\nReturns the element with the smaller magnitude\nbetween each pair of elements of the two vector\narguments.\nSyntax\nvsMinMag (n, a, b, y);\nvsMinMagI(n, a, inca, b, incb, y, incy);\nvmsMinMag (n, a, b, y, mode);\nvmsMinMagI(n, a, inca, b, incb, y, incy, mode);\nvdMinMag (n, a, b, y);\nvdMinMagI(n, a, inca, b, incb, y, incy);\nvmdMinMag (n, a, b, y, mode);\nvmdMinMagI(n, a, inca, b, incb, y, incy, mode);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nn\nconst MKL_INT\nSpecifies the number of elements to be\ncalculated.\na, b\nconst float* for vsMinMag\nconst float* for vmsMinMag\nconst double* for vdMinMag\nconst double* for vmdMinMag\nPointers to the arrays containing the input\nvectors a and b.\ninca, incb,\nincy\nconst MKL_INT\nSpecifies increments for the elements of a, b,\nand y.\nmode\nconst MKL_INT64\nOverrides the global VM mode setting for this\nfunction call. See vmlSetMode for possible\nvalues and their description.\nOutput Parameters\nName\nType\nDescription\ny\nfloat* for vsMinMag\nfloat* for vmsMinMag\ndouble* for vdMinMag\nPointer to an array containing the output vector\ny.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2129\n\n\nName\nType\nDescription\ndouble* for vmdMinMag\nDescription\nThe v?MinMag function returns a vector with element values equal to the element with the smaller\nmagnitude from each pair of corresponding elements of the two vectors a and b:\n•\nIf |ai| < |bi| v?MaxMag returns ai, otherwise v?MaxMag returns ai.\n•\nIf |bi| < |ai| v?MaxMag returns bi, otherwise v?MaxMag returns ai.\n•\nOtherwise v?MaxMag behaves like v?Fmin.\nSpecial values for Real Function v?MinMag(x, y)\nArgument 1\nArgument 2\nResult\nVM Error Status\nException\nai not NAN\nNAN\nai\nNAN\nbi not NAN\nbi\nNAN\nNAN\nNAN\nSee Also\nMaxMag Returns the element with the larger magnitude between each pair of elements of the two\nvector arguments.\nFmin Returns the smaller of each pair of elements of the two vector arguments.\nStatistical Functions\nStatistical functions in Intel® oneAPI Math Kernel Library (oneMKL) are known as the Vector Statistics (VS).\nThey are designed for the purpose of\n•\ngenerating vectors of pseudorandom, quasi-random, and non-deterministic random numbers\n•\nperforming mathematical operations of convolution and correlation\n•\ncomputing basic statistical estimates for single and double precision multi-dimensional datasets\nThe corresponding functionality is described in the respective Random Number Generators, Convolution and\nCorrelation, and Summary Statistics topics.\nSee VS performance data in the online VS Performance Data document available at https://www.intel.com/\ncontent/www/us/en/developer/tools/oneapi/onemkl-documentation.html.\nThe basic notion in VS is a task. The task object is a data structure or descriptor that holds the parameters\nrelated to a specific statistical operation: random number generation, convolution and correlation, or\nsummary statistics estimation. Such parameters can be an identifier of a random number generator, its\ninternal state and parameters, data arrays, their shape and dimensions, an identifier of the operation and so\nforth. You can modify the VS task parameters using the VS service functions.\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2130\n\n\nRandom Number Generators\nIntel® oneAPI Math Kernel Library (oneMKL) VS provides a set of routines implementing commonly used\npseudorandom, quasi-random, or non-deterministic random number generators with continuous and discrete\ndistribution. To improve performance, all these routines were developed using the calls to the highly\noptimizedBasic Random Number Generators (BRNGs) and vector mathematical functions (VM, see \"Vector\nMathematical Functions\").\nVS provides interfaces both for Fortran and C languages. For users of the C and C++ languages the\nmkl_vsl.h header file is provided. All header files are found in the following directory:\n${MKL}/include\nAll VS routines can be classified into three major categories:\n•\nTransformation routines for different types of statistical distributions, for example, uniform, normal\n(Gaussian), binomial, etc. These routines indirectly call basic random number generators, which are\npseudorandom, quasi-random, or non-deterministic random number generators. Detailed description of\nthe generators can be found in Distribution Generators.\n•\nService routines to handle random number streams: create, initialize, delete, copy, save to a binary file,\nload from a binary file, get the index of a basic generator. The description of these routines can be found\nin Service Routines.\n•\nRegistration routines for basic pseudorandom generators and routines that obtain properties of the\nregistered generators (see Advanced Service Routines).\nThe last two categories are referred to as service routines.\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nRandom Number Generators Conventions\nThis document makes no specific differentiation between random, pseudorandom, and quasi-random\nnumbers, nor between random, pseudorandom, and quasi-random number generators unless the context\nrequires otherwise. For details, refer to the ‘Random Numbers’ section in VS Notesdocument provided at the\nIntel® oneAPI Math Kernel Library (oneMKL) web page.\nAll generators of nonuniform distributions, both discrete and continuous, are built on the basis of the uniform\ndistribution generators, called Basic Random Number Generators (BRNGs). The pseudorandom numbers with\nnonuniform distribution are obtained through an appropriate transformation of the uniformly distributed\npseudorandom numbers. Such transformations are referred to as generation methods. For a given\ndistribution, several generation methods can be used. See VS Notes for the description of methods available\nfor each generator.\nAn RNG task determines environment in which random number generation is performed, in particular\nparameters of the BRNG and its internal state. Output of VS generators is a stream of random numbers that\nare used in Monte Carlo simulations. A random stream descriptor and a random stream are used as\nsynonyms of an RNG task in the document unless the context requires otherwise.\nThe random stream descriptor specifies which BRNG should be used in a given transformation method. See\nthe Random Streams and RNGs in Parallel Computation section of VS Notes.\nThe term computational node means a logical or physical unit that can process data in parallel.\nRandom Number Generators Mathematical Notation\nThe following notation is used throughout the text:\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2131\n\n\nN\nThe set of natural numbers N = {1, 2, 3 ...}.\nZ\nThe set of integers Z = {... -3, -2, -1, 0, 1, 2, 3 ...}.\nR\nThe set of real numbers.\nThe floor of a (the largest integer less than or equal to a).\n⊕ or xor\nBitwise exclusive OR.\nBinomial coefficient or combination (α∈R, α≥ 0; k∈N∪{0}).\nFor α≥k binomial coefficient is defined as\nIf α < k, then\nΦ(x)\nCumulative Gaussian distribution function\ndefined over - ∞ < x < + ∞.\nΦ(-∞) = 0, Φ(+∞) = 1.\nΓ(α)\nThe complete gamma function\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2132\n\n\nwhere α > 0.\nB(p, q)\nThe complete beta function\nwhere p>0 and q>0.\nLCG(a,c, m)\nLinear Congruential Generator xn+1 = (axn + c) mod m, where a is\ncalled the multiplier, c is called the increment, and m is called the\nmodulus of the generator.\nMCG(a,m)\nMultiplicative Congruential Generator xn+1 = (axn) mod m is a special\ncase of Linear Congruential Generator, where the increment c is taken\nto be 0.\nGFSR(p, q)\nGeneralized Feedback Shift Register Generator\nxn  = xn-p⊕xn-q.\nRandom Number Generators Naming Conventions\nThe names of the routines, types, and constants in VS random number generators are case-sensitive and can\ncontain lowercase and uppercase characters (viRngUniform).\nThe names of generator routines have the following structure:\nv<type of result>Rng<distribution>   \nwhere\n•\nv is the prefix of a VS vector function.\n•\n<type of result> is either s, d, or i and specifies one of the following types:\ns\nfloat\nd\ndouble\ni\nint\nPrefixes s and d apply to continuous distributions only, prefix i applies\nonly to discrete case.\n•\nrng indicates that the routine is a random generator.\n•\n<distribution> specifies the type of statistical distribution.\nOn 64-bit platforms, routines with the _64 suffix support large data arrays in the LP64 interface library and\nenable you to mix integer types in one application. For more interface library details, see \"Using the ILP64\nInterface vs. LP64 Interface\" in the developer guide.\nNames of service routines follow the template below:\nvsl<name>\nwhere\n•\nvsl is the prefix of a VS service function.\n•\n<name> contains a short function name.\nFor a more detailed description of service routines, refer to Service Routines and Advanced Service Routines.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2133\n\n\nThe prototype of each generator routine corresponding to a given probability distribution fits the following\nstructure:\nstatus = <function name>( method, stream, n, r, [<distribution parameters>] )\nwhere\n•\nmethod defines the method of generation. A detailed description of this parameter can be found in table \n\"Values of <method> in method parameter\". See below, where the structure of the method parameter\nname is explained.\n•\nstream defines the descriptor of the random stream and must have a non-zero value. Random streams,\ndescriptors, and their usage are discussed further in Random Streams and Service Routines.\n•\nn defines the number of random values to be generated. If n is less than or equal to zero, no values are\ngenerated. Furthermore, if n is negative, an error condition is set.\n•\nr defines the destination array for the generated numbers. The dimension of the array must be large\nenough to store at least n random numbers.\n•\nstatus defines the error status of a VS routine. See Error Reporting for a detailed description of error\nstatus values.\nAdditional parameters included into <distribution parameters> field are individual for each generator routine\nand are described in detail in Distribution Generators.\nTo invoke a distribution generator, use a call to the respective VS routine. For example, to obtain a vector r,\ncomposed of n independent and identically distributed random numbers with normal (Gaussian) distribution,\nthat have the mean value a and standard deviation sigma, write the following:\nstatus = vsRngGaussian( method, stream, n, r, a, sigma )\nThe name of a method parameter has the following structure:\nVSL_RNG_METHOD_method<distribution>_<method>\nVSL_RNG_METHOD_<distribution>_<method>_ACCURATE\nwhere\n•\n<distribution> is the probability distribution.\n•\n<method> is the method name.\nType of the name structure for the method parameter corresponds to fast and accurate modes of random\nnumber generation (see \"Distribution Generators\" and VS Notes for details).\nMethod names VSL_RNG_METHOD_<distribution>_<method>\nand\nVSL_RNG_METHOD_<distribution>_<method>_ACCURATE\nshould be used with\nv<precision>Rng<distribution>\nfunction only, where\n•\n<precision> is\ns\nfor single precision continuous distribution\nd\nfor double precision continuous distribution\ni\nfor discrete distribution\n•\n<distribution> is the probability distribution.\nis the probability distribution.Table \"Values of <method> in method parameter\" provides specific predefined\nvalues of the method name. The third column contains names of the functions that use the given method.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2134\n\n\nValues of <method> in method parameter\nMethod\nShort Description\nFunctions\nSTD\nStandard method. Currently there is only one method for these\nfunctions.\nUniform\n(continuous),\nUniform\n(discrete), \nUniformBits, \nUniformBits32, \nUniformBits64\nBOXMULLER\nBOXMULLER generates normally distributed random number x\nthru the pair of uniformly distributed numbers u1 and u2\naccording to the formula:\nGaussian, \nGaussianMV\nBOXMULLER2\nBOXMULLER2 generates normally distributed random numbers x1\nand x2 thru the pair of uniformly distributed numbers u1 and u2\naccording to the formulas:\nGaussian, \nGaussianMV, \nLognormal\nICDF\nInverse cumulative distribution function method.\nExponential, \nLaplace, \nWeibull, Cauchy, \nRayleigh, \nGumbel, \nBernoulli, \nGeometric, \nGaussian, \nGaussianMV, \nLognormal\nGNORM\nFor α > 1, a gamma distributed random number is generated as\na cube of properly scaled normal random number; for 0.6 ≤α <\n1, a gamma distributed random number is generated using\nrejection from Weibull distribution; for α < 0.6, a gamma\ndistributed random number is obtained using transformation of\nexponential power distribution; for α = 1, gamma distribution is\nreduced to exponential distribution.\nGamma\nCJA\nFor min(p, q) > 1, Cheng method is used; for min(p, q) <\n1, Johnk method is used, if q + K·p2+ C≤ 0 (K = 0.852...,\nC=-0.956...) otherwise, Atkinson switching algorithm is used;\nfor max(p, q) < 1, method of Johnk is used; for min(p, q) <\n1, max(p, q)> 1, Atkinson switching algorithm is used (CJA\nstands for the first letters of Cheng, Johnk, Atkinson); for p =\n1 or q = 1, inverse cumulative distribution function method is\nused;for p = 1 and q = 1, beta distribution is reduced to\nuniform distribution.\nBeta\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2135\n\n\nMethod\nShort Description\nFunctions\nBTPE\nAcceptance/rejection method for\nntrial·min(p,1 - p)≥ 30\nwith decomposition into 4 regions:\n–\n2 parallelograms\n–\ntriangle\n–\nleft exponential tail\n–\nright exponential tail\nBinomial\nH2PE\nAcceptance/rejection method for large mode of distribution with\ndecomposition into 3 regions:\n–\nrectangular\n–\nleft exponential tail\n–\nright exponential tail\nHypergeometric\nPTPE\nAcceptance/rejection method for λ≥ 27 with decomposition into\n4 regions:\n–\n2 parallelograms\n–\ntriangle\n–\nleft exponential tail\n–\nright exponential tail;\notherwise, table lookup method is used.\nPoisson\nPOISNORM\nfor λ≥ 1, method based on Poisson inverse CDF approximation\nby Gaussian inverse CDF;\nfor λ < 1, table lookup method is used.\nPoisson, \nPoissonV\nNBAR\nAcceptance/rejection method for ,\nwith decomposition into 5 regions:\n–\nrectangular\n–\n2 trapezoid\n–\nleft exponential tail\n–\nright exponential tail\nNegBinomial\nCHI2GAMMA\nRandom number generator of chi-square distribution with ν\ndegrees of freedom. To generate any successive random\nnumber x of the chi-square distribution:\n•\nIf ν is 1 or 3, a chi-square distributed random number is\ngenerated as a sum of squares of ν independent normal\nrandom numbers with mean value a = 0 and standard\ndeviation σ =1.\n•\nIf ν is even and 2 ≤ ν ≤ 16, a chi-square distributed random\nnumber is generated using the formula:\nx = −2ln ∏i = 1\nv/2 ui\nChiSquare\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2136\n\n\nMethod\nShort Description\nFunctions\nwhere ui are successive random numbers uniformly\ndistributed over the interval (0, 1)\n•\nIf ν ≥ 17 or ν is odd and 5 ≤ ν ≤ 15, a chi-square distribution\nis reduced to a Gamma distribution with these parameters:\n•\nShape α = ν / 2\n•\nOffset a = 0\n•\nScale factor β = 2\nThe random numbers of the Gamma distribution are\ngenerated using the VSL_RNG_METHOD_GAMMA_GNORM\nmethod.\nNOTE\nIn this document, routines are often referred to by their base name (Gaussian) when this does not\nlead to ambiguity. In the routine reference, the full name (vsrnggaussian, vsRngGaussian) is\nalways used in prototypes and code examples.\nBasic Generators\nVS provides pseudorandom, quasi-random, and non-deterministic random number generators. This includes\nthe following BRNGs, which differ in speed and other properties:\n•\nthe 31-bit multiplicative congruential pseudorandom number generator MCG(1132489760, 231 -1)\n[L'Ecuyer99]\n•\nthe 32-bit generalized feedback shift register pseudorandom number generator GFSR(250,103)\n[Kirkpatrick81]\n•\nthe combined multiple recursive pseudorandom number generator MRG32k3a [L'Ecuyer99a]\n•\nthe 59-bit multiplicative congruential pseudorandom number generator MCG(1313, 259) from NAG\nNumerical Libraries [NAG]\n•\nWichmann-Hill pseudorandom number generator (a set of 273 basic generators) from NAG Numerical\nLibraries [NAG]\n•\nMersenne Twister pseudorandom number generator MT19937 [Matsumoto98] with period length 219937-1\nof the produced sequence\n•\nSet of 6024 Mersenne Twister pseudorandom number generators MT2203 [Matsumoto98],\n[Matsumoto00]. Each of them generates a sequence of period length equal to 22203-1. Parameters of the\ngenerators provide mutual independence of the corresponding sequences.\n•\nSIMD-oriented Fast Mersenne Twister pseudorandom number generator SFMT19937 [Saito08] with a\nperiod length equal to 219937-1 of the produced sequence.\n•\nSobol quasi-random number generator [Sobol76], [Bratley88], which works in arbitrary dimension. For\ndimensions greater than 40 the user should supply initialization parameters (initial direction numbers and\nprimitive polynomials or direction numbers) by using vslNewStreamEx function. See additional details on\ninterface for registration of the parameters in the library in VS Notes.\n•\nNiederreiter quasi-random number generator [Bratley92], which works in arbitrary dimension. For\ndimensions greater than 318 the user should supply initialization parameters (irreducible polynomials or\ndirection numbers) by using vslNewStreamEx function. See additional details on interface for registration\nof the parameters in the library in VS Notes.\n•\nNon-deterministic random number generator (RDRAND-based generators only) [AVX], [IntelSWMan].\nNOTE\nYou can use a non-deterministic random number generator only if the underlying hardware supports it.\nFor instructions on how to detect if an Intel CPU supports a non-deterministic random number\ngenerator see, for example, Chapter 8: Post-32nm Processor Instructions in [AVX] or Chapter 4:\nRdRand Instruction Usage in [BMT].\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2137\n\n\nNOTE\nThe time required by some non-deterministic sources to generate a random number is not constant,\nso you might have to make multiple requests before the next random number is available. VS limits\nthe number of retries for requests to the non-deterministic source to 10. You can redefine the\nmaximum number of retries during the initialization of the non-deterministic random number\ngenerator with the vslNewStreamEx function.\nFor more details on the non-deterministic source implementation for Intel CPUs please refer to Section\n7.3.17, Volume 1, Random Number Generator Instruction in [IntelSWMan] and Section 4.2.2, RdRand\nRetry Loop in [BMT].\n•\nPhilox4x32-10 counter-based pseudorandom number generator with a period of\n2128PHILOX4X32X10[Salmon11].\n•\nARS-5 counter-based pseudorandom number generator with a period of 2128, which uses instructions from\nthe AES-NI set ARS5[Salmon11].\nSee some testing results for the generators in VS Notes and comparative performance data at https://\nwww.intel.com/content/www/us/en/developer/tools/oneapi/onemkl-documentation.html.\nVS provides means of registration of such user-designed generators through the steps described in Advanced\nService Routines.\nFor some basic generators, VS provides two methods of creating independent random streams in\nmultiprocessor computations, which are the leapfrog method and the block-splitting method. These sequence\nsplitting methods are also useful in sequential Monte Carlo.\nIn addition, MT2203 pseudorandom number generator is a set of 6024 generators designed to create up to\n6024 independent random sequences, which might be used in parallel Monte Carlo simulations. Another\ngenerator that has the same feature is Wichmann-Hill. It allows creating up to 273 independent random\nstreams. The properties of the generators designed for parallel computations are discussed in detail in\n[Coddington94].\nYou may want to design and use your own basic generators. VS provides means of registration of such user-\ndesigned generators through the steps described in Advanced Service Routines.\nThere is also an option to utilize externally generated random numbers in VS distribution generator routines.\nFor this purpose VS provides three additional basic random number generators:\n–\nfor external random data packed in 32-bit integer array\n–\nfor external random data stored in double precision floating-point array; data is supposed to be uniformly\ndistributed over (a,b) interval\n–\nfor external random data stored in single precision floating-point array; data is supposed to be uniformly\ndistributed over (a,b) interval.\nSuch basic generators are called the abstract basic random number generators.\nSee VS Notes for a more detailed description of the generator properties.\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nBRNG Parameter Definition\nPredefined values for the brng input parameter are as follows:\nValues of brng parameter\nValue\nShort Description\nVSL_BRNG_MCG31\nA 31-bit multiplicative congruential generator.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2138\n\n\nValue\nShort Description\nVSL_BRNG_R250\nA generalized feedback shift register generator.\nVSL_BRNG_MRG32K3A\nA combined multiple recursive generator with two components\nof order 3.\nVSL_BRNG_MCG59\nA 59-bit multiplicative congruential generator.\nVSL_BRNG_WH\nA set of 273 Wichmann-Hill combined multiplicative\ncongruential generators.\nVSL_BRNG_MT19937\nA Mersenne Twister pseudorandom number generator.\nVSL_BRNG_MT2203\nA set of 6024 Mersenne Twister pseudorandom number\ngenerators.\nVSL_BRNG_SFMT19937\nA SIMD-oriented Fast Mersenne Twister pseudorandom number\ngenerator.\nVSL_BRNG_SOBOL\nA 32-bit Gray code-based generator producing low-discrepancy\nsequences for dimensions 1 ≤ s ≤ 40; user-defined\ndimensions are also available.\nVSL_BRNG_NIEDERR\nA 32-bit Gray code-based generator producing low-discrepancy\nsequences for dimensions 1 ≤ s ≤ 318; user-defined\ndimensions are also available.\nVSL_BRNG_IABSTRACT\nAn abstract random number generator for integer arrays.\nVSL_BRNG_DABSTRACT\nAn abstract random number generator for double precision\nfloating-point arrays.\nVSL_BRNG_SABSTRACT\nAn abstract random number generator for single precision\nfloating-point arrays.\nVSL_BRNG_NONDETERM\nA non-deterministic random number generator.\nVSL_BRNG_PHILOX4X32X10\nA Philox4x32-10 counter-based pseudorandom number\ngenerator.\nVSL_BRNG_ARS5\nAn ARS-5 counter-based pseudorandom number generator\nthat uses instructions from the AES-NI set.\nSee VS Notes for detailed description.\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nRandom Streams\nRandom stream (or stream) is an abstract source of pseudo- and quasi-random sequences of uniform\ndistribution. You can operate with stream state descriptors only. A stream state descriptor, which holds state\ndescriptive information for a particular BRNG, is a necessary parameter in each routine of a distribution\ngenerator. Only the distribution generator routines operate with random streams directly. See VS Notes for\ndetails.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2139\n\n\nNOTE\nRandom streams associated with abstract basic random number generator are called the abstract\nrandom streams. See VS Notes for detailed description of abstract streams and their use.\nYou can create unlimited number of random streams by VS Service Routines like NewStream and utilize them\nin any distribution generator to get the sequence of numbers of given probability distribution. When they are\nno longer needed, the streams should be deleted calling service routine DeleteStream.\nVS provides service functions SaveStreamF and LoadStreamF to save random stream descriptive data to a\nbinary file and to read this data from a binary file respectively. See VS Notes for detailed description.\nBRNG Data Types\ntypedef(void*)VSLStreamStatePtr;\nSee Advanced Service Routines for the format of the stream state structure for user-designed generators.\nError Reporting\nVS RNG routines return status codes of the performed operation to report errors to the calling program. The\napplication should perform error-related actions and/or recover from the error. The status codes are of\ninteger type and have the following format:\nVSL_ERROR_<ERROR_NAME> - indicates VS errors common for all VS domains.\nVSL_RNG_ERROR_<ERROR_NAME> - indicates VS RNG errors.\nVS RNG errors are of negative values while warnings are of positive values. The status code of zero value\nindicates successful completion of the operation: VSL_ERROR_OK (or synonymic VSL_STATUS_OK).\nStatus Codes\nStatus Code\nDescription\nCommon VSL\nVSL_ERROR_OK, VSL_STATUS_OK\nNo error, execution is successful.\nVSL_ERROR_BADARGS\nInput argument value is not valid.\nVSL_ERROR_CPU_NOT_SUPPORTED\nCPU version is not supported.\nVSL_ERROR_FEATURE_NOT_IMPLEMENTED\nFeature invoked is not implemented.\nVSL_ERROR_MEM_FAILURE\nSystem cannot allocate memory.\nVSL_ERROR_NULL_PTR\nInput pointer argument is NULL.\nVSL_ERROR_UNKNOWN\nUnknown error.\n \nVS RNG Specific\nVSL_RNG_ERROR_BAD_FILE_FORMAT\nFile format is unknown.\nVSL_RNG_ERROR_BAD_MEM_FORMAT\nDescriptive random stream format is\nunknown.\nVSL_RNG_ERROR_BAD_NBITS\nThe value in NBits field is bad.\nVSL_RNG_ERROR_BAD_NSEEDS\nThe value in NSeeds field is bad.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2140\n\n\nStatus Code\nDescription\nVSL_RNG_ERROR_BAD_STREAM\nThe random stream is invalid.\nVSL_RNG_ERROR_BAD_STREAM_STATE_SIZE\nThe value in StreamStateSize field is bad.\nVSL_RNG_ERROR_BAD_UPDATE\nCallback function for an abstract BRNG returns\nan invalid number of updated entries in a\nbuffer, that is, < 0 or >nmax.\nVSL_RNG_ERROR_BAD_WORD_SIZE\nThe value in WordSize field is bad.\nVSL_RNG_ERROR_BRNG_NOT_SUPPORTED\nBRNG is not supported by the function.\nVSL_RNG_ERROR_BRNG_TABLE_FULL\nRegistration cannot be completed due to lack\nof free entries in the table of registered\nBRNGs.\nVSL_RNG_ERROR_BRNGS_INCOMPATIBLE\nTwo BRNGs are not compatible for the\noperation.\nVSL_RNG_ERROR_FILE_CLOSE\nError in closing the file.\nVSL_RNG_ERROR_FILE_OPEN\nError in opening the file.\nVSL_RNG_ERROR_FILE_READ\nError in reading the file.\nVSL_RNG_ERROR_FILE_WRITE\nError in writing the file.\nVSL_RNG_ERROR_INVALID_ABSTRACT_STREAM\nThe abstract random stream is invalid.\nVSL_RNG_ERROR_INVALID_BRNG_INDEX\nBRNG index is not valid.\nVSL_RNG_ERROR_LEAPFROG_UNSUPPORTED\nBRNG does not support Leapfrog method.\nVSL_RNG_ERROR_NO_NUMBERS\nCallback function for an abstract BRNG returns\nzero as the number of updated entries in a\nbuffer.\nVSL_RNG_ERROR_QRNG_PERIOD_ELAPSED\nPeriod of the generator is exceeded.\nVSL_RNG_ERROR_SKIPAHEAD_UNSUPPORTED\nBRNG does not support Skip-Ahead method.\nVSL_RNG_ERROR_SKIPAHEADEX_UNSUPPORTED\nBRNG does not support advanced Skip-Ahead\nmethod.\nVSL_RNG_ERROR_UNSUPPORTED_FILE_VER\nFile format version is not supported.\nVSL_RNG_ERROR_NONDETERM_NOT_SUPPORTED\nNon-deterministic random number generator\nis not supported on the CPU running the\napplication.\nVSL_RNG_ERROR_NONDETERM_ NRETRIES_EXCEEDED\nNumber of retries to generate a random\nnumber using non-deterministic random\nnumber generator exceeds threshold (see\nSection 7.2.1.12 Non-deterministic in [VS\nNotes] for more details)\nVSL_RNG_ERROR_ARS5_NOT_SUPPORTED\nARS-5 random number generator is not\nsupported on the CPU running the application.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2141\n\n\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nVS RNG Usage ModelIntel® oneMKL RNG Usage Model\nA typical algorithm for VSoneMKL random number generators is as follows:\n1.\nCreate and initialize stream/streams. Functions vslNewStream, vslNewStreamEx, vslCopyStream,\nvslCopyStreamState, vslLeapfrogStream, vslSkipAheadStream, vslSkipAheadStreamEx.\n2.\nCall one or more RNGs.\n3.\nProcess the output.\n4.\nDelete the stream or streams with the function vslDeleteStream.\nNOTE\nYou may reiterate steps 2-3. Random number streams may be generated for different threads.\nThe following example demonstrates generation of a random stream that is output of basic generator\nMT19937. The seed is equal to 777. The stream is used to generate 10,000 normally distributed random\nnumbers in blocks of 1,000 random numbers with parameters a = 5 and sigma = 2. Delete the streams after\ncompleting the generation. The purpose of the example is to calculate the sample mean for normal\ndistribution with the given parameters.\nExample of VS RNG Usage\n#include <stdio.h>\n#include \"mkl_vsl.h\"\n \nint main()\n{\n   double r[1000]; /* buffer for random numbers */\n   double s; /* average */\n   VSLStreamStatePtr stream;\n   int i, j;\n    \n   /* Initializing */        \n   s = 0.0;\n   vslNewStream( &stream, VSL_BRNG_MT19937, 777 );\n    \n   /* Generating */        \n   for ( i=0; i<10; i++ ) {\n      vdRngGaussian( VSL_RNG_METHOD_GAUSSIAN_ICDF, stream, 1000, r, 5.0, 2.0 );\n      for ( j=0; j<1000; j++ ) {\n         s += r[j];\n      }\n   }\n   s /= 10000.0;\n    \n   /* Deleting the stream */        \n   vslDeleteStream( &stream );\n    \n   /* Printing results */        \n   printf( \"Sample mean of normal distribution = %f\\n\", s );\n    \n   return 0;\n}\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2142\n\n\nAdditionally, examples that demonstrate usage of VS random number generators are available in:\n${MKL}/examples/vslc/source\nService Routines\nStream handling comprises routines for creating, deleting, or copying the streams and getting the index of a\nbasic generator. A random stream can also be saved to and then read from a binary file. Table \"Service\nRoutines\" lists all available service routines\nService Routines\nRoutine\nShort Description\nvslNewStream\nCreates and initializes a random stream.\nvslNewStreamEx\nCreates and initializes a random stream for the generators\nwith multiple initial conditions.\nvsliNewAbstractStream\nCreates and initializes an abstract random stream for integer\narrays.\nvsldNewAbstractStream\nCreates and initializes an abstract random stream for double\nprecision floating-point arrays.\nvslsNewAbstractStream\nCreates and initializes an abstract random stream for single\nprecision floating-point arrays.\nvslDeleteStream\nDeletes previously created stream.\nvslCopyStream\nCopies a stream to another stream.\nvslCopyStreamState\nCreates a copy of a random stream state.\nvslSaveStreamF\nWrites a stream to a binary file.\nvslLoadStreamF\nReads a stream from a binary file.\nvslSaveStreamM\nWrites a random stream descriptive data, including state, to a\nmemory buffer.\nvslLoadStreamM\nCreates a new stream and reads stream descriptive data,\nincluding state, from the memory buffer.\nvslGetStreamSize\nComputes size of memory necessary to hold the random\nstream.\nvslLeapfrogStream\nInitializes the stream by the leapfrog method to generate a\nsubsequence of the original sequence.\nvslSkipAheadStream\nInitializes the stream by the skip-ahead method.\nvslSkipAheadStreamEx\nInitializes the stream by the advanced skip-ahead method.\nvslGetStreamStateBrng\nObtains the index of the basic generator responsible for the\ngeneration of a given random stream.\nvslGetNumRegBrngs\nObtains the number of currently registered basic generators.\nMost of the generator-based work comprises three basic steps:\n1.\nCreating and initializing a stream (vslNewStream, vslNewStreamEx, vslCopyStream, \nvslCopyStreamState, vslLeapfrogStream, vslSkipAheadStream, vslSkipAheadStreamEx).\n2.\nGenerating random numbers with given distribution, see Distribution Generators.\n3.\nDeleting the stream (vslDeleteStream).\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2143\n\n\nNote that you can concurrently create multiple streams and obtain random data from one or several\ngenerators by using the stream state. You must use the vslDeleteStream function to delete all the streams\nafterwards.\nvslNewStream\nCreates and initializes a random stream.\nSyntax\nstatus = vslNewStream( &stream, brng, seed );\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nbrng\nconst MKL_INT\nIndex of the basic generator to initialize the stream. See \nTable Values of brng parameter for specific value.\nseed\nconst MKL_UINT\nInitial condition of the stream. In the case of a quasi-\nrandom number generator seed parameter is used to set\nthe dimension. If the dimension is greater than the\ndimension that brng can support or is less than 1, then the\ndimension is assumed to be equal to 1.\nOutput Parameters\nName\nType\nDescription\nstream\nVSLStreamStatePtr*\nStream state descriptor\nDescription\nFor a basic generator with number brng, this function creates a new stream and initializes it with a 32-bit\nseed. The seed is an initial value used to select a particular sequence generated by the basic generator brng.\nThe function is also applicable for generators with multiple initial conditions. Use this function to create and\ninitialize a new stream with a 32-bit seed only. If you need to provide multiple initial conditions such as\nseveral 32-bit or wider seeds, use the function vslNewStreamEx. See VS Notes for a more detailed\ndescription of stream initialization for different basic generators.\nNOTE\nThis function is not applicable for abstract basic random number generators. Please use\nvsliNewAbstractStream, vslsNewAbstractStream or vsldNewAbstractStream to utilize\ninteger, single-precision or double-precision external random data respectively.\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2144\n\n\nReturn Values\nVSL_ERROR_OK, VSL_STATUS_OK\nIndicates no error, execution is successful.\nVSL_RNG_ERROR_INVALID_BRNG_INDEX\nBRNG index is invalid.\nVSL_ERROR_MEM_FAILURE\nSystem cannot allocate memory for stream.\nVSL_RNG_ERROR_NONDETERMINISTIC_NOT_SUPP\nORTED\nNon-deterministic random number generator is not\nsupported.\nVSL_RNG_ERROR_ARS5_NOT_SUPPORTED\nARS-5 random number generator is not supported on the\nCPU running the application.\nvslNewStreamEx\nCreates and initializes a random stream for generators\nwith multiple initial conditions.\nSyntax\nstatus = vslNewStreamEx( &stream, brng, n, params );\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nbrng\nconst MKL_INT\nIndex of the basic generator to initialize the stream. See \nTable \"Values of brng parameter\" for specific value.\nn\nconst MKL_INT\nNumber of initial conditions contained in params\nparams\nconst unsigned int\nArray of initial conditions necessary for the basic generator\nbrng to initialize the stream. In the case of a quasi-random\nnumber generator only the first element in params\nparameter is used to set the dimension. If the dimension is\ngreater than the dimension that brng can support or is less\nthan 1, then the dimension is assumed to be equal to 1.\nOutput Parameters\nName\nType\nDescription\nstream\nVSLStreamStatePtr*\nStream state descriptor\nDescription\nThe vslNewStreamEx function provides an advanced tool to set the initial conditions for a basic generator if\nits input arguments imply several initialization parameters. Initial values are used to select a particular\nsequence generated by the basic generator brng. Whenever possible, use vslNewStream, which is analogous\nto vslNewStreamEx except that it takes only one 32-bit initial condition. In particular, vslNewStreamEx may\nbe used to initialize the state table in Generalized Feedback Shift Register Generators (GFSRs). A more\ndetailed description of this issue can be found in VS Notes.\nThis function is also used to pass user-defined initialization parameters of quasi-random number generators\ninto the library. See VS Notes for the format for their passing and registration in VS.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2145\n\n\nNOTE\nThis function is not applicable for abstract basic random number generators. Please use \nvsliNewAbstractStream, vslsNewAbstractStream or vsldNewAbstractStream to utilize\ninteger, single-precision or double-precision external random data respectively.\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nReturn Values\nVSL_ERROR_OK, VSL_STATUS_OK\nIndicates no error, execution is successful.\nVSL_RNG_ERROR_INVALID_BRNG_INDEX\nBRNG index is invalid.\nVSL_ERROR_MEM_FAILURE\nSystem cannot allocate memory for stream.\nVSL_RNG_ERROR_NONDETERMINISTIC_NOT_SUPP\nORTED\nNon-deterministic random number generator is not\nsupported.\nVSL_RNG_ERROR_ARS5_NOT_SUPPORTED\nARS-5 random number generator is not supported on the\nCPU running the application.\nvsliNewAbstractStream\nCreates and initializes an abstract random stream for\ninteger arrays.\nSyntax\nstatus = vsliNewAbstractStream( &stream, n, ibuf, icallback );\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nn\nconst MKL_INT\nSize of the array ibuf\nibuf\nconst unsigned int\nArray of n 32-bit integers\nicallback\nPointer to the callback function used for ibuf update\nNOTE\nFormat of the callback function:\nint iUpdateFunc( VSLStreamStatePtrstream, int* n, unsigned int ibuf[], int* nmin, int* \nnmax, int* idx );\nThe callback function returns the number of elements in the array actually updated by the function.Table\nicallback Callback Function Parameters gives the description of the callback function parameters.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2146\n\n\nicallback Callback Function Parameters\nParameters\nShort Description\nstream\nAbstract random stream descriptor\nn\nSize of ibuf\nibuf\nArray of random numbers associated with the stream stream\nnmin\nMinimal quantity of numbers to update\nnmax\nMaximal quantity of numbers that can be updated\nidx\nPosition in cyclic buffer ibuf to start update 0≤idx<n.\nOutput Parameters\nName\nType\nDescription\nstream\nVSLStreamStatePtr*\nDescriptor of the stream state structure\nDescription\nThe vsliNewAbstractStream function creates a new abstract stream and associates it with an integer array\nibuf and your callback function icallback that is intended for updating of ibuf content.\nReturn Values\nVSL_ERROR_OK, VSL_STATUS_OK\nIndicates no error, execution is successful.\nVSL_ERROR_BADARGS\nParameter n is not positive.\nVSL_ERROR_MEM_FAILURE\nSystem cannot allocate memory for stream.\nVSL_ERROR_NULL_PTR\nEither buffer or callback function parameter is a NULL\npointer.\nvsldNewAbstractStream\nCreates and initializes an abstract random stream for\ndouble precision floating-point arrays.\nSyntax\nstatus = vsldNewAbstractStream( &stream, n, dbuf, a, b, dcallback );\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nn\nconst MKL_INT\nSize of the array dbuf\ndbuf\nconst double\nArray of n double precision floating-point random numbers\nwith uniform distribution over interval (a,b)\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2147\n\n\nName\nType\nDescription\na\nconst double\nLeft boundary a\nb\nconst double\nRight boundary b\ndcallback\nSee Note below\nPointer to the callback function used for update of the array\ndbuf\nOutput Parameters\nName\nType\nDescription\nstream\nVSLStreamStatePtr*\nDescriptor of the stream state structure\nNOTE\nFormat of the callback function:\nint dUpdateFunc( VSLStreamStatePtr stream, int* n, double dbuf[], int* nmin, int* nmax, \nint* idx );\nThe callback function returns the number of elements in the array actually updated by the function.Table\ndcallback Callback Function Parameters gives the description of the callback function parameters.\ndcallback Callback Function Parameters\nParameters\nShort Description\nstream\nAbstract random stream descriptor\nn\nSize of dbuf\ndbuf\nArray of random numbers associated with the stream stream\nnmin\nMinimal quantity of numbers to update\nnmax\nMaximal quantity of numbers that can be updated\nidx\nPosition in cyclic buffer dbuf to start update 0≤idx<n.\nDescription\nThe vsldNewAbstractStream function creates a new abstract stream for double precision floating-point\narrays with random numbers of the uniform distribution over interval (a,b). The function associates the\nstream with a double precision array dbuf and your callback function dcallback that is intended for updating\nof dbuf content.\nReturn Values\nVSL_ERROR_OK, VSL_STATUS_OK\nIndicates no error, execution is successful.\nVSL_ERROR_BADARGS\nParameter n is not positive.\nVSL_ERROR_MEM_FAILURE\nSystem cannot allocate memory for stream.\nVSL_ERROR_NULL_PTR\nEither buffer or callback function parameter is a NULL\npointer.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2148\n\n\nvslsNewAbstractStream\nCreates and initializes an abstract random stream for\nsingle precision floating-point arrays.\nSyntax\nstatus = vslsNewAbstractStream( &stream, n, sbuf, a, b, scallback );\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nn\nconst MKL_INT\nSize of the array sbuf\nsbuf\nconst float\nArray of n single precision floating-point random numbers\nwith uniform distribution over interval (a,b)\na\nconst float\nLeft boundary a\nb\nconst float\nRight boundary b\nscallback\nSee Note below\nPointer to the callback function used for update of the array\nsbuf\nOutput Parameters\nName\nType\nDescription\nstream\nVSLStreamStatePtr*\nDescriptor of the stream state structure\nNOTE\nFormat of the callback function in C:\nint sUpdateFunc( VSLStreamStatePtr stream, int* n, float sbuf[], int* nmin, int* nmax, \nint* idx );\nThe callback function returns the number of elements in the array actually updated by the function.Table\nscallback Callback Function Parameters gives the description of the callback function parameters.\nscallback Callback Function Parameters\nParameters\nShort Description\nstream\nAbstract random stream descriptor\nn\nSize of sbuf\nsbuf\nArray of random numbers associated with the stream stream\nnmin\nMinimal quantity of numbers to update\nnmax\nMaximal quantity of numbers that can be updated\nidx\nPosition in cyclic buffer sbuf to start update 0≤idx<n.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2149\n\n\nDescription\nThe vslsNewAbstractStream function creates a new abstract stream for single precision floating-point\narrays with random numbers of the uniform distribution over interval (a,b). The function associates the\nstream with a single precision array sbuf and your callback function scallback that is intended for updating of\nsbuf content.\nReturn Values\nVSL_ERROR_OK, VSL_STATUS_OK\nIndicates no error, execution is successful.\nVSL_ERROR_BADARGS\nParameter n is not positive.\nVSL_ERROR_MEM_FAILURE\nSystem cannot allocate memory for stream.\nVSL_ERROR_NULL_PTR\nEither buffer or callback function parameter is a NULL\npointer.\nvslDeleteStream\nDeletes a random stream.\nSyntax\nstatus = vslDeleteStream( &stream );\nInclude Files\n•\nmkl.h\nInput/Output Parameters\nName\nType\nDescription\nstream\nVSLStreamStatePtr*\nStream state descriptor. Must have non-zero value. After\nthe stream is successfully deleted, the pointer is set to\nNULL.\nDescription\nThe function deletes the random stream created by one of the initialization functions.\nReturn Values\nVSL_ERROR_OK, VSL_STATUS_OK\nIndicates no error, execution is successful.\nVSL_ERROR_NULL_PTR\nstream parameter is a NULL pointer.\nVSL_RNG_ERROR_BAD_STREAM\nstream is not a valid random stream.\nvslCopyStream\nCreates a copy of a random stream.\nSyntax\nstatus = vslCopyStream( &newstream, srcstream );\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2150\n\n\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nsrcstream\nconst VSLStreamStatePtr\nPointer to the stream state structure to be copied\nOutput Parameters\nName\nType\nDescription\nnewstream\nVSLStreamStatePtr*\nCopied random stream descriptor\nDescription\nThe function creates an exact copy of srcstream and stores its descriptor to newstream.\nReturn Values\nVSL_ERROR_OK, VSL_STATUS_OK\nIndicates no error, execution is successful.\nVSL_ERROR_NULL_PTR\nsrcstream parameter is a NULL pointer.\nVSL_RNG_ERROR_BAD_STREAM\nsrcstream is not a valid random stream.\nVSL_ERROR_MEM_FAILURE\nSystem cannot allocate memory for newstream.\nvslCopyStreamState\nCreates a copy of a random stream state.\nSyntax\nstatus = vslCopyStreamState( deststream, srcstream );\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nsrcstream\nconst VSLStreamStatePtr\nPointer to the stream state structure, from which the state\nstructure is copied\nOutput Parameters\nName\nType\nDescription\ndeststrea\nm\nVSLStreamStatePtr\nPointer to the stream state structure where the stream\nstate is copied\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2151\n\n\nDescription\nThe vslCopyStreamState function copies a stream state from srcstream to the existing deststream stream.\nBoth the streams should be generated by the same basic generator. An error message is generated when the\nindex of the BRNG that produced deststream stream differs from the index of the BRNG that generated\nsrcstream stream.\nUnlike vslCopyStream function, which creates a new stream and copies both the stream state and other\ndata from srcstream, the function vslCopyStreamState copies only srcstream stream state data to the\ngenerated deststream stream.\nReturn Values\nVSL_ERROR_OK, VSL_STATUS_OK\nIndicates no error, execution is successful.\nVSL_ERROR_NULL_PTR\nEither srcstream or deststream is a NULL pointer.\nVSL_RNG_ERROR_BAD_STREAM\nEither srcstream or deststream is not a valid random\nstream.\nVSL_RNG_ERROR_BRNGS_INCOMPATIBLE\nBRNG associated with srcstream is not compatible with\nBRNG associated with deststream.\nvslSaveStreamF\nWrites random stream descriptive data, including\nstream state, to binary file.\nSyntax\nerrstatus = vslSaveStreamF( stream, fname );\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nstream\nconst VSLStreamStatePtr\nRandom stream to be written to the file\nfname\nconst char*\nFile name specified as a null-terminated string\nOutput Parameters\nName\nType\nDescription\nerrstatus\nint\nError status of the operation\nDescription\nThe vslSaveStreamF function writes the random stream descriptive data, including the stream state, to the\nbinary file. Random stream descriptive data is saved to the binary file with the name fname. The random\nstream stream must be a valid stream created by vslNewStream-like or vslCopyStream-like service\nroutines. If the stream cannot be saved to the file, errstatus has a non-zero value. The random stream can\nbe read from the binary file using the vslLoadStreamF function.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2152\n\n\nReturn Values\nVSL_ERROR_OK, VSL_STATUS_OK\nIndicates no error, execution is successful.\nVSL_ERROR_NULL_PTR\nEither fname or stream is a NULL pointer.\nVSL_RNG_ERROR_BAD_STREAM\nstream is not a valid random stream.\nVSL_RNG_ERROR_FILE_OPEN\nIndicates an error in opening the file.\nVSL_RNG_ERROR_FILE_WRITE\nIndicates an error in writing the file.\nVSL_RNG_ERROR_FILE_CLOSE\nIndicates an error in closing the file.\nVSL_ERROR_MEM_FAILURE\nSystem cannot allocate memory for internal needs.\nvslLoadStreamF\nCreates new stream and reads stream descriptive\ndata, including stream state, from binary file.\nSyntax\nerrstatus = vslLoadStreamF( &stream, fname );\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nfname\nconst char*\nFile name specified as a null-terminated string\nOutput Parameters\nName\nType\nDescription\nstream\nVSLStreamStatePtr*\nPointer to a new random stream\nerrstatus\nint\nError status of the operation\nDescription\nThe vslLoadStreamF function creates a new stream and reads stream descriptive data, including the stream\nstate, from the binary file. A new random stream is created using the stream descriptive data from the\nbinary file with the name fname. If the stream cannot be read (for example, an I/O error occurs or the file\nformat is invalid), errstatus has a non-zero value. To save random stream to the file, use vslSaveStreamF\nfunction.\nCaution\nCalling vslLoadStreamF with a previously initialized stream pointer can have unintended\nconsequences such as a memory leak. To initialize a stream which has been in use until\ncalling vslLoadStreamF, you should call the vslDeleteStream function first to deallocate the\nresources.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2153\n\n\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nReturn Values\nVSL_ERROR_OK, VSL_STATUS_OK\nIndicates no error, execution is successful.\nVSL_ERROR_NULL_PTR\nfname is a NULL pointer.\nVSL_RNG_ERROR_FILE_OPEN\nIndicates an error in opening the file.\nVSL_RNG_ERROR_FILE_WRITE\nIndicates an error in writing the file.\nVSL_RNG_ERROR_FILE_CLOSE\nIndicates an error in closing the file.\nVSL_ERROR_MEM_FAILURE\nSystem cannot allocate memory for internal needs.\nVSL_RNG_ERROR_BAD_FILE_FORMAT\nUnknown file format.\nVSL_RNG_ERROR_UNSUPPORTED_FILE_VER\nFile format version is unsupported.\nVSL_RNG_ERROR_NONDETERMINISTIC_NOT_SUPP\nORTED\nNon-deterministic random number generator is not\nsupported.\nVSL_RNG_ERROR_ARS5_NOT_SUPPORTED\nARS-5 random number generator is not supported on the\nCPU running the application.\nvslSaveStreamM\nWrites random stream descriptive data, including\nstream state, to a memory buffer.\nSyntax\nerrstatus = vslSaveStreamM( stream, memptr );\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nstream\nconst VSLStreamStatePtr\nRandom stream to be written to the memory\nmemptr\nchar*\nMemory buffer to save random stream descriptive data to\nOutput Parameters\nName\nType\nDescription\nerrstatus\nint\nError status of the operation\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2154\n\n\nDescription\nThe vslSaveStreamM function writes the random stream descriptive data, including the stream state, to the\nmemory at memptr. Random stream stream must be a valid stream created by vslNewStream-like or \nvslCopyStream-like service routines. The memptr parameter must be a valid pointer to the memory of size\nsufficient to hold the random stream stream. Use the service routine vslGetStreamSize to determine this\namount of memory.\nIf the stream cannot be saved to the memory, errstatus has a non-zero value. The random stream can be\nread from the memory pointed by memptr using the vslLoadStreamM function.\nReturn Values\nVSL_ERROR_OK, VSL_STATUS_OK\nIndicates no error, execution is successful.\nVSL_ERROR_NULL_PTR\nEither memptr or stream is a NULL pointer.\nVSL_RNG_ERROR_BAD_STREAM\nstream is a NULL pointer.\nvslLoadStreamM\nCreates a new stream and reads stream descriptive\ndata, including stream state, from the memory buffer.\nSyntax\nerrstatus = vslLoadStreamM( &stream, memptr );\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nmemptr\nconst char*\nMemory buffer to load random stream descriptive data from\nOutput Parameters\nName\nType\nDescription\nstream\nVSLStreamStatePtr*\nPointer to a new random stream\nerrstatus\nint\nError status of the operation\nDescription\nThe vslLoadStreamM function creates a new stream and reads stream descriptive data, including the stream\nstate, from the memory buffer. A new random stream is created using the stream descriptive data from the\nmemory pointer by memptr. If the stream cannot be read (for example, memptr is invalid), errstatus has a\nnon-zero value. To save random stream to the memory, use vslSaveStreamM function. Use the service\nroutine vslGetStreamSize to determine the amount of memory sufficient to hold the random stream.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2155\n\n\nCaution\nCalling LoadStreamM with a previously initialized stream pointer can have unintended\nconsequences such as a memory leak. To initialize a stream which has been in use until\ncalling vslLoadStreamM, you should call the vslDeleteStream function first to deallocate the\nresources.\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nReturn Values\nVSL_ERROR_OK, VSL_STATUS_OK\nIndicates no error, execution is successful.\nVSL_ERROR_NULL_PTR\nmemptr is a NULL pointer.\nVSL_ERROR_MEM_FAILURE\nSystem cannot allocate memory for internal needs.\nVSL_RNG_ERROR_BAD_MEM_FORMAT\nDescriptive random stream format is unknown.\nVSL_RNG_ERROR_NONDETERMINISTIC_NOT_SUPP\nORTED\nNon-deterministic random number generator is not\nsupported.\nVSL_RNG_ERROR_ARS5_NOT_SUPPORTED\nARS-5 random number generator is not supported on the\nCPU running the application.\nvslGetStreamSize\nComputes size of memory necessary to hold the\nrandom stream.\nSyntax\nmemsize = vslGetStreamSize( stream );\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nstream\nconst VSLStreamStatePtr\nRandom stream\nOutput Parameters\nName\nType\nDescription\nmemsize\nint\nAmount of memory in bytes necessary to hold descriptive\ndata of random stream stream\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2156\n\n\nDescription\nThe vslGetStreamSize function returns the size of memory in bytes which is necessary to hold the given\nrandom stream. Use the output of the function to allocate the buffer to which you will save the random\nstream by means of the vslSaveStreamM function.\nReturn Values\nVSL_ERROR_OK, VSL_STATUS_OK\nIndicates no error, execution is successful.\nVSL_RNG_ERROR_BAD_STREAM\nstream is a NULL pointer.\nvslLeapfrogStream\nInitializes a stream using the leapfrog method.\nSyntax\nstatus = vslLeapfrogStream( stream, k, nstreams );\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nstream\nVSLStreamStatePtr\nPointer to the stream state structure to which leapfrog\nmethod is applied\nk\nconst MKL_INT\nIndex of the computational node, or stream number\nnstreams\nconst MKL_INT\nLargest number of computational nodes, or stride\nDescription\nThe vslLeapfrogStream function generates random numbers in a random stream with non-unit stride. This\nfeature is particularly useful in distributing random numbers from the original stream across the nstreams\nbuffers without generating the original random sequence with subsequent manual distribution.\nOne of the important applications of the leapfrog method is splitting the original sequence into non-\noverlapping subsequences across nstreams computational nodes. The function initializes the original random\nstream (see Figure \"Leapfrog Method\") to generate random numbers for the computational node k, 0 ≤k <\nnstreams, where nstreams is the largest number of computational nodes used.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2157\n\n\n__border__top\nLeapfrog Method\nThe leapfrog method is supported only for those basic generators that allow splitting elements by the\nleapfrog method, which is more efficient than simply generating them by a generator with subsequent\nmanual distribution across computational nodes. See VS Notes for details.\nFor quasi-random basic generators, the leapfrog method allows generating individual components of quasi-\nrandom vectors instead of whole quasi-random vectors. In this case nstreams parameter should be equal to\nthe dimension of the quasi-random vector while k parameter should be the index of a component to be\ngenerated (0 ≤k < nstreams). Other parameters values are not allowed.\nThe following code illustrates the initialization of three independent streams using the leapfrog method:\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2158\n\n\nCode for Leapfrog Method\n... \nVSLStreamStatePtr stream1; \nVSLStreamStatePtr stream2; \nVSLStreamStatePtr stream3;\n/* Creating 3 identical streams */ \nstatus = vslNewStream(&stream1, VSL_BRNG_MCG31, 174); \nstatus = vslCopyStream(&stream2, stream1); \nstatus = vslCopyStream(&stream3, stream1);\n/* Leapfrogging the streams \n*/ \nstatus = vslLeapfrogStream(stream1, 0, 3); \nstatus = vslLeapfrogStream(stream2, 1, 3); \nstatus = vslLeapfrogStream(stream3, 2, 3);\n/* Generating random numbers \n*/ \n... \n/* Deleting the streams \n*/ \nstatus = vslDeleteStream(&stream1); \nstatus = vslDeleteStream(&stream2); \nstatus = vslDeleteStream(&stream3); \n...\nReturn Values\nVSL_ERROR_OK, VSL_STATUS_OK\nIndicates no error, execution is successful.\nVSL_ERROR_NULL_PTR\nstream is a NULL pointer.\nVSL_RNG_ERROR_BAD_STREAM\nstream is not a valid random stream.\nVSL_RNG_ERROR_LEAPFROG_UNSUPPORTED\nBRNG does not support Leapfrog method.\nvslSkipAheadStream\nInitializes a stream using the block-splitting method.\nSyntax\nstatus = vslSkipAheadStream( stream, nskip);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nstream\nVSLStreamStatePtr\nPointer to the stream state structure to which block-\nsplitting method is applied\nnskip\nconst long long int\nNumber of skipped elements\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2159\n\n\nDescription\nThe vslSkipAheadStream function skips a given number of elements in a random stream. This feature is\nparticularly useful in distributing random numbers from original random stream across different\ncomputational nodes. If the largest number of random numbers used by a computational node is nskip, then\nthe original random sequence may be split by vslSkipAheadStream into non-overlapping blocks of nskip\nsize so that each block corresponds to the respective computational node. The number of computational\nnodes is unlimited. This method is known as the block-splitting method or as the skip-ahead method. (see \nFigure \"Block-Splitting Method\").\n__border__top\nBlock-Splitting Method\nThe skip-ahead method is supported only for those basic generators that allow skipping elements by the\nskip-ahead method, which is more efficient than simply generating them by generator with subsequent\nmanual skipping. See VS Notes for details.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2160\n\n\nPlease note that for quasi-random basic generators the skip-ahead method works with components of quasi-\nrandom vectors rather than with whole quasi-random vectors. Therefore, to skip NS quasi-random vectors,\nset the nskip parameter equal to the NS*DIMEN, where DIMEN is the dimension of the quasi-random vector.\nIf this operation results in exceeding the period of the quasi-random number generator, which is 232-1, the\nlibrary returns the VSL_RNG_ERROR_QRNG_PERIOD_ELAPSED error code.\nThe following code illustrates how to initialize three independent streams using the vslSkipAheadStream\nfunction:\nCode for Block-Splitting Method\nVSLStreamStatePtr stream1;\nVSLStreamStatePtr stream2;\nVSLStreamStatePtr stream3;\n/* Creating the 1st stream\n*/\nstatus = vslNewStream(&stream1, VSL_BRNG_MCG31, 174);\n/* Skipping ahead by 7 elements the 2nd stream */\nstatus = vslCopyStream(&stream2, stream1);\nstatus = vslSkipAheadStream(stream2, 7);\n/* Skipping ahead by 7 elements the 3rd stream */\nstatus = vslCopyStream(&stream3, stream2);\nstatus = vslSkipAheadStream(stream3, 7);\n/* Generating random numbers\n*/\n...\n/* Deleting the streams\n*/\nstatus = vslDeleteStream(&stream1);\nstatus = vslDeleteStream(&stream2);\nstatus = vslDeleteStream(&stream3);\n...\nReturn Values\nVSL_ERROR_OK, VSL_STATUS_OK\nIndicates no error, execution is successful.\nVSL_ERROR_NULL_PTR\nstream is a NULL pointer.\nVSL_RNG_ERROR_BAD_STREAM\nstream is not a valid random stream.\nVSL_RNG_ERROR_SKIPAHEAD_UNSUPPORTED\nBRNG does not support the Skip-Ahead method.\nVSL_RNG_ERROR_QRNG_PERIOD_ELAPSED\nPeriod of the quasi-random number generator is exceeded.\nvslSkipAheadStreamEx\nInitializes a stream using the block-splitting method\nwith partitioned number of skipped elements.\nSyntax\nstatus = vslSkipAheadStreamEx( stream, n, nskip);\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2161\n\n\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nstream\nVSLStreamStatePtr\nPointer to the stream state structure to which block-\nsplitting method is applied\nn\nconst MKL_INT\nNumber of summands in nskip\nnskip\nconst MKL_UINT64[]\nPartitioned number of skipped elements\nDescription\nThe vslSkipAheadStreamEx function skips a given number of elements in a random stream. This feature is\nparticularly useful in distributing random numbers from original random stream across different\ncomputational nodes. If the largest number of random numbers used by a computational node is nskip, then\nthe original random sequence may be split by vslSkipAheadStreamEx into non-overlapping blocks of nskip\nsize so that each block corresponds to the respective computational node. The number of computational\nnodes is unlimited. This method is known as the block-splitting method or as the skip-ahead method.\nUse this function when the number of elements to skip in a random stream is greater than 263. Prior calls to\nthe function represent the number of skipped elements with array of size n as shown below:\nnskip[0]+ nskip[1]*264+nskip[2]* 2128+ … +nskip[n-1]*2(64*(n-1) );\nWhen the number of skipped elements is less than 263 you can use either vslSkipAheadtreamEx or\nvslSkipAheadStream. The following code illustrates how to initialize three independent streams using the\nvslSkipAheadStreamEx function:\nVSLStreamStatePtr stream1; VSLStreamStatePtr stream2; VSLStreamStatePtr stream3;\n/* Creating the 1st stream\n*/\nstatus = vslNewStream(&stream1, VSL_BRNG_MCG31, 174);\n/* To skip 2^64 elements in the random stream SkipAheadStreamEx(nskip) function should be called \nwith nskip represented as nskip = 2^64 = 0 + 1 * 2^64\n*/\nMKL_UINT64 nskip[2];\nnskip[0]=0;\nnskip[1]=1;\n/* Skipping ahead by 2^64 elements the 2nd stream\n/*\nstatus = vslCopyStream(&stream2, stream1); \nstatus = vslSkipAheadStreamEx (stream2, 2, nskip);\n/* Skipping ahead by 2^64 elements the 3rd stream\n/*\nstatus = vslCopyStream(&stream3, stream2); \nstatus = vslSkipAheadStreamEx (stream3, 2, nskip);\n/* Generating random numbers\n*/\n...\n/* Deleting the streams\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2162\n\n\n*/\nstatus = vslDeleteStream(&stream1);\nstatus = vslDeleteStream(&stream2);\nstatus = vslDeleteStream(&stream3);\n...\nReturn Values\nVSL_ERROR_OK, VSL_STATUS_OK\nIndicates no error, execution is successful.\nVSL_ERROR_NULL_PTR\nstream is a NULL pointer.\nVSL_RNG_ERROR_BAD_STREAM\nstream is not a valid random stream.\nVSL_RNG_ERROR_SKIPAHEADEX_UNSUPPORTED\nBRNG does not support the advanced Skip-Ahead method.\nvslGetStreamStateBrng\nReturns index of a basic generator used for generation\nof a given random stream.\nSyntax\nbrng = vslGetStreamStateBrng( stream );\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nstream\nconst VSLStreamStatePtr\nPointer to the stream state structure\nOutput Parameters\nName\nType\nDescription\nbrng\nint\nIndex of the basic generator assigned for the generation of\nstream ; negative in case of an error\nDescription\nThe vslGetStreamStateBrng function retrieves the index of a basic generator used for generation of a\ngiven random stream.\nReturn Values\nVSL_ERROR_NULL_PTR\nstream is a NULL pointer.\nVSL_RNG_ERROR_BAD_STREAM\nstream is not a valid random stream.\nvslGetNumRegBrngs\nObtains the number of currently registered basic\ngenerators.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2163\n\n\nSyntax\nnregbrngs = vslGetNumRegBrngs( void );\nInclude Files\n•\nmkl.h\nOutput Parameters\nName\nType\nDescription\nnregbrngs\nint\nNumber of basic generators registered at the moment of\nthe function call\nDescription\nThe vslGetNumRegBrngs function obtains the number of currently registered basic generators. Whenever\nuser registers a user-designed basic generator, the number of registered basic generators is incremented.\nThe maximum number of basic generators that can be registered is determined by the VSL_MAX_REG_BRNGS\nparameter.\nDistribution Generators\noneMKLVS routines are used to generate random numbers with different types of distribution. Each function\ngroup is introduced below by the type of underlying distribution and contains a short description of its\nfunctionality, as well as specifications of the call sequence and the explanation of input and output\nparameters. Table \"Continuous Distribution Generators\" and Table \"Discrete Distribution Generators\" list the\nrandom number generator routines with data types and output distributions, and sets correspondence\nbetween data types of the generator routines and the basic random number generators.\nContinuous Distribution Generators\nType of Distribution\nData\nTypes\nBRNG Data\nType\nDescription\nvRngUniform\ns, d\ns, d\nUniform continuous distribution on the\ninterval [a,b)\nvRngGaussian\ns, d\ns, d\nNormal (Gaussian) distribution\nvRngGaussianMV\ns, d\ns, d\nNormal (Gaussian) multivariate distribution\nvRngExponential\ns, d\ns, d\nExponential distribution\nvRngLaplace\ns, d\ns, d\nLaplace distribution (double exponential\ndistribution)\nvRngWeibull\ns, d\ns, d\nWeibull distribution\nvRngCauchy\ns, d\ns, d\nCauchy distribution\nvRngRayleigh\ns, d\ns, d\nRayleigh distribution\nvRngLognormal\ns, d\ns, d\nLognormal distribution\nvRngGumbel\ns, d\ns, d\nGumbel (extreme value) distribution\nvRngGamma\ns, d\ns, d\nGamma distribution\nvRngBeta\ns, d\ns, d\nBeta distribution\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2164\n\n\nType of Distribution\nData\nTypes\nBRNG Data\nType\nDescription\nvRngChiSquare\ns, d\ns, d\nChi-Square distribution\n \nDiscrete Distribution Generators\nType of Distribution\nData Types\nBRNG Data Type\nDescription\nvRngUniform\ni\nd\nUniform discrete\ndistribution on the\ninterval [a,b)\nvRngUniformBits\ni\ni\nUnderlying BRNG integer\nrecurrence\nvRngUniformBits32\ni\ni\nUniformly distributed\nbits in 32-bit chunks\nvRngUniformBits64\ni\ni\nUniformly distributed\nbits in 64-bit chunks\nvRngBernoulli\ni\ns\nBernoulli distribution\nvRngGeometric\ni\ns\nGeometric distribution\nvRngBinomial\ni\nd\nBinomial distribution\nvRngHypergeometric\ni\nd\nHypergeometric\ndistribution\nvRngPoisson\ni\ns (for\nVSL_RNG_METHOD_POIS\nSON_POISNORM)\ns (for distribution\nparameter λ≥ 27) and d\n(for λ < 27) (for\nVSL_RNG_METHOD_POIS\nSON_PTPE)\nPoisson distribution\nvRngPoisson\ni\ns\nPoisson distribution with\nvarying mean\nvRngNegBinomial\ni\nd\nNegative binomial\ndistribution, or Pascal\ndistribution\nvRngMultinomial\ni\nd\nMultinomial distribution\nModes of random number generation\nThe library provides two modes of random number generation, accurate and fast. Accurate generation mode\nis intended for the applications that are highly demanding to accuracy of calculations. When used in this\nmode, the generators produce random numbers lying completely within definitional domain for all values of\nthe distribution parameters. For example, random numbers obtained from the generator of continuous\ndistribution that is uniform on interval [a,b] belong to this interval irrespective of what a and b values may\nbe. Fast mode provides high performance of generation and also guarantees that generated random numbers\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2165\n\n\nbelong to the definitional domain except for some specific values of distribution parameters. The generation\nmode is set by specifying relevant value of the method parameter in generator routines. List of distributions\nthat support accurate mode of generation is given in the table below.\nDistribution Generators Supporting Accurate Mode\nType of Distribution\nData Types\nvRngUniform\ns, d\nvRngExponential\ns, d\nvRngWeibull\ns, d\nvRngRayleigh\ns, d\nvRngLognormal\ns, d\nvRngGamma\ns, d\nvRngBeta\ns, d\nSee additional details about accurate and fast mode of random number generation in VS Notes.\nNew method names\nThe current version of oneMKL has a modified structure of VS RNG method names. (SeeRNG Naming\nConventions for details.) The old names are kept for backward compatibility. The tables below set\ncorrespondence between the new and legacy method names for VS random number generators.\nMethod Names for Continuous Distribution Generators\nRNG\nLegacy Method Name\nNew Method Name\nvRngUniform\nVSL_METHOD_SUNIFORM_STD,\nVSL_METHOD_DUNIFORM_STD,\nVSL_METHOD_SUNIFORM_STD_ACCURATE,\nVSL_METHOD_DUNIFORM_STD_ACCURATE\nVSL_RNG_METHOD_UNIFORM_STD,\nVSL_RNG_METHOD_UNIFORM_STD_ACCURATE\nvRngGaussian VSL_METHOD_SGAUSSIAN_BOXMULLER,\nVSL_METHOD_SGAUSSIAN_BOXMULLER2,\nVSL_METHOD_SGAUSSIAN_ICDF,\nVSL_METHOD_DGAUSSIAN_BOXMULLER,\nVSL_METHOD_DGAUSSIAN_BOXMULLER2,\nVSL_METHOD_DGAUSSIAN_ICDF\nVSL_RNG_METHOD_GAUSSIAN_BOXMULLER,\nVSL_RNG_METHOD_GAUSSIAN_BOXMULLER2,\nVSL_RNG_METHOD_GAUSSIAN_ICDF\nvRngGaussianMV\nVSL_METHOD_SGAUSSIANMV_BOXMULLER,\nVSL_METHOD_SGAUSSIANMV_BOXMULLER2,\nVSL_METHOD_SGAUSSIANMV_ICDF,\nVSL_METHOD_DGAUSSIANMV_BOXMULLER,\nVSL_METHOD_DGAUSSIANMV_BOXMULLER2,\nVSL_METHOD_DGAUSSIANMV_ICDF\nVSL_RNG_METHOD_GAUSSIANMV_BOXMULLER\n,\nVSL_RNG_METHOD_GAUSSIANMV_BOXMULLER\n2, VSL_RNG_METHOD_GAUSSIANMV_ICDF\nvRngExponential\nVSL_METHOD_SEXPONENTIAL_ICDF,\nVSL_METHOD_DEXPONENTIAL_ICDF,\nVSL_METHOD_SEXPONENTIAL_ICDF_ACCUR\nATE,\nVSL_METHOD_DEXPONENTIAL_ICDF_ACCUR\nATE\nVSL_RNG_METHOD_EXPONENTIAL_ICDF,\nVSL_RNG_METHOD_EXPONENTIAL_ICDF_ACC\nURATE\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2166\n\n\nRNG\nLegacy Method Name\nNew Method Name\nvRngLaplace\nVSL_METHOD_SLAPLACE_ICDF,\nVSL_METHOD_DLAPLACEL_ICDF\nVSL_RNG_METHOD_LAPLACE_ICDF\nvRngWeibull\nVSL_METHOD_SWEIBULL_ICDF,\nVSL_METHOD_DWEIBULL_ICDF,\nVSL_METHOD_SWEIBULL_ICDF_ACCURATE,\nVSL_METHOD_DWEIBULL_ICDF_ACCURATE\nVSL_RNG_METHOD_WEIBULL_ICDF,\nVSL_RNG_METHOD_WEIBULL_ICDF_ACCURAT\nE\nvRngCauchy\nVSL_METHOD_SCAUCHY_ICDF,\nVSL_METHOD_DCAUCHY_ICDF\nVSL_RNG_METHOD_CAUCHY_ICDF\nvRngRayleigh VSL_METHOD_SRAYLEIGH_ICDF,\nVSL_METHOD_DRAYLEIGH_ICDF,\nVSL_METHOD_SRAYLEIGH_ICDF_ACCURATE,\nVSL_METHOD_DRAYLEIGH_ICDF_ACCURATE\nVSL_RNG_METHOD_RAYLEIGH_ICDF,\nVSL_RNG_METHOD_RAYLEIGH_ICDF_ACCURA\nTE\nvRngLognormalVSL_METHOD_SLOGNORMAL_BOXMULLER2,\nVSL_METHOD_DLOGNORMAL_BOXMULLER2,\nVSL_METHOD_SLOGNORMAL_BOXMULLER2_A\nCCURATE,\nVSL_METHOD_DLOGNORMAL_BOXMULLER2_A\nCCURATE\nVSL_RNG_METHOD_LOGNORMAL_BOXMULLER2\n,\nVSL_RNG_METHOD_LOGNORMAL_BOXMULLER2\n_ACCURATE\nVSL_METHOD_SLOGNORMAL_ICDF,\nVSL_METHOD_DLOGNORMAL_ICDF,\nVSL_METHOD_SLOGNORMAL_ICDF_ACCURAT\nE,\nVSL_METHOD_DLOGNORMAL_ICDF_ACCURAT\nE\nVSL_RNG_METHOD_LOGNORMAL_ICDF,\nVSL_RNG_METHOD_LOGNORMAL_ICDF_ACCUR\nATE\nvRngGumbel\nVSL_METHOD_SGUMBEL_ICDF,\nVSL_METHOD_DGUMBEL_ICDF\nVSL_RNG_METHOD_GUMBEL_ICDF\nvRngGamma\nVSL_METHOD_SGAMMA_GNORM,\nVSL_METHOD_DGAMMA_GNORM,\nVSL_METHOD_SGAMMA_GNORM_ACCURATE,\nVSL_METHOD_DGAMMA_GNORM_ACCURATE\nVSL_RNG_METHOD_GAMMA_GNORM,\nVSL_RNG_METHOD_GAMMA_GNORM_ACCURATE\nvRngBeta\nVSL_METHOD_SBETA_CJA,\nVSL_METHOD_DBETA_CJA,\nVSL_METHOD_SBETA_CJA_ACCURATE,\nVSL_METHOD_DBETA_CJA_ACCURATE\nVSL_RNG_METHOD_BETA_CJA,\nVSL_RNG_METHOD_BETA_CJA_ACCURATE\n \nMethod Names for Discrete Distribution Generators\nRNG\nLegacy Method Name\nNew Method Name\nvRngUniform\nVSL_METHOD_IUNIFORM_STD\nVSL_RNG_METHOD_UNIFORM_STD\nvRngUniformBitsVSL_METHOD_IUNIFORMBITS_STD\nVSL_RNG_METHOD_UNIFORMBITS_STD\nvRngBernoulli\nVSL_METHOD_IBERNOULLI_ICDF\nVSL_RNG_METHOD_BERNOULLI_ICDF\nvRngGeometric\nVSL_METHOD_IGEOMETRIC_ICDF\nVSL_RNG_METHOD_GEOMETRIC_ICDF\nvRngBinomial\nVSL_METHOD_IBINOMIAL_BTPE\nVSL_RNG_METHOD_BINOMIAL_BTPE\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2167\n\n\nRNG\nLegacy Method Name\nNew Method Name\nvRngHypergeometric\nVSL_METHOD_IHYPERGEOMETRIC_H2PE\nVSL_RNG_METHOD_HYPERGEOMETRIC_H2PE\nvRngPoisson\nVSL_METHOD_IPOISSON_PTPE,\nVSL_METHOD_IPOISSON_POISNORM\nVSL_RNG_METHOD_POISSON_PTPE,\nVSL_RNG_METHOD_POISSON_POISNORM\nvRngPoissonV\nVSL_METHOD_IPOISSONV_POISNORM\nVSL_RNG_METHOD_POISSONV_POISNORM\nvRngNegBinomialVSL_METHOD_INEGBINOMIAL_NBAR\nVSL_RNG_METHOD_NEGBINOMIAL_NBAR\nContinuous Distributions\nThis section describes routines for generating random numbers with continuous distribution.\nvRngUniform Continuous Distribution Generators\nGenerates random numbers with uniform distribution.\nSyntax\nstatus = vsRngUniform( method, stream, n, r, a, b );\nstatus = vdRngUniform( method, stream, n, r, a, b );\nInclude Files\n•\nmkl.h\nDescription\nThe vRngUniform function generates random numbers uniformly distributed over the interval [a, b), where\na, b are the left and right bounds of the interval, respectively, and a, b∈R ; a < b.\nThe probability density function is given by:\nThe cumulative distribution function is as follows:\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2168\n\n\nProduct and Performance Information\nNotice revision #20201201\nInput Parameters\nName\nType\nDescription\nmethod\nconst MKL_INT\nGeneration method; the specific values are as follows:\nVSL_RNG_METHOD_UNIFORM_STD\nVSL_RNG_METHOD_UNIFORM_STD_ACCURATE\nStandard method.\nstream\nVSLStreamStatePtr\nPointer to the stream state structure\nn\nconst MKL_INT\nNumber of random values to be generated.\na\nconst float for vsRngUniform\nconst double for\nvdRngUniform\nLeft bound a.\nb\nconst float for\nvsRngUniform\nconst double for\nvdRngUniform\nRight bound b.\nOutput Parameters\nName\nType\nDescription\nr\nfloat* for vsRngUniform\ndouble* for vdRngUniform\nVector of n random numbers uniformly distributed over the\ninterval [a,b)\nReturn Values\nVSL_ERROR_OK, VSL_STATUS_OK\nIndicates no error, execution is successful.\nVSL_ERROR_NULL_PTR\nstream is a NULL pointer.\nVSL_RNG_ERROR_BAD_STREAM\nstream is not a valid random stream.\nVSL_RNG_ERROR_BAD_UPDATE\nCallback function for an abstract BRNG returns an invalid\nnumber of updated entries in a buffer, that is, < 0 or >\nnmax .\nVSL_RNG_ERROR_NO_NUMBERS\nCallback function for an abstract BRNG returns 0 as the\nnumber of updated entries in a buffer.\nVSL_RNG_ERROR_QRNG_PERIOD_ELAPSED\nPeriod of the generator has been exceeded.\nVSL_RNG_ERROR_NONDETERM_NRETRIES_EXCEED\nED\nNumber of retries to generate a random number by using\nnon-deterministic random number generator exceeds\nthreshold.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2169\n\n\nVSL_RNG_ERROR_ARS5_NOT_SUPPORTED\nARS-5 random number generator is not supported on the\nCPU running the application.\nvRngGaussian\nGenerates normally distributed random numbers.\nSyntax\nstatus = vsRngGaussian( method, stream, n, r, a, sigma );\nstatus = vdRngGaussian( method, stream, n, r, a, sigma );\nInclude Files\n•\nmkl.h\nDescription\nThe vRngGaussian function generates random numbers with normal (Gaussian) distribution with mean value\na and standard deviation σ, where\na, σ∈R ; σ > 0.\nThe probability density function is given by:\nThe cumulative distribution function is as follows:\nThe cumulative distribution function Fa,σ(x) can be expressed in terms of standard normal distribution Φ(x)\nas\nFa,σ(x) = Φ((x - a)/σ)\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2170\n\n\nInput Parameters\nName\nType\nDescription\nmethod\nconst MKL_INT\nGeneration method. The specific values are as follows:\n                                \nVSL_RNG_METHOD_GAUSSIAN_BOXMULLER\n                            \n                                \nVSL_RNG_METHOD_GAUSSIAN_BOXMULLER2\n                            \n                                \nVSL_RNG_METHOD_GAUSSIAN_ICDF\n                            \nSee brief description of the methods BOXMULLER,\nBOXMULLER2, and ICDF in Table \"Values of <method> in\nmethod parameter\"\nstream\nVSLStreamStatePtr\nPointer to the stream state structure\nn\nconst MKL_INT\nNumber of random values to be generated.\na\nconst float for\nvsRngGaussian\nconst double for\nvdRngGaussian\nMean value a.\nsigma\nconst float for\nvsRngGaussian\nconst double for\nvdRngGaussian\nStandard deviation σ.\nOutput Parameters\nName\nType\nDescription\nr\nfloat* for vsRngGaussian\ndouble* for vdRngGaussian\nVector of n normally distributed random numbers.\nReturn Values\nVSL_ERROR_OK, VSL_STATUS_OK\nIndicates no error, execution is successful.\nVSL_ERROR_NULL_PTR\nstream is a NULL pointer.\nVSL_RNG_ERROR_BAD_STREAM\nstream is not a valid random stream.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2171\n\n\nVSL_RNG_ERROR_BAD_UPDATE\nCallback function for an abstract BRNG returns an invalid\nnumber of updated entries in a buffer, that is, < 0 or >\nnmax.\nVSL_RNG_ERROR_NO_NUMBERS\nCallback function for an abstract BRNG returns 0 as the\nnumber of updated entries in a buffer.\nVSL_RNG_ERROR_QRNG_PERIOD_ELAPSED\nPeriod of the generator has been exceeded.\nVSL_RNG_ERROR_NONDETERM_NRETRIES_EXCEED\nED\nNumber of retries to generate a random number by using\nnon-deterministic random number generator exceeds\nthreshold.\nVSL_RNG_ERROR_ARS5_NOT_SUPPORTED\nARS-5 random number generator is not supported on the\nCPU running the application.\nvRngGaussianMV\nGenerates random numbers from multivariate normal\ndistribution.\nSyntax\nstatus = vsRngGaussianMV( method, stream, n, r, dimen, mstorage, a, t );\nstatus = vdRngGaussianMV( method, stream, n, r, dimen, mstorage, a, t );\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nmethod\nconst MKL_INT\nGeneration method. The specific values are as follows:\nVSL_RNG_METHOD_GAUSSIANMV_BOXMULLER\nVSL_RNG_METHOD_GAUSSIANMV_BOXMULLER2\nVSL_RNG_METHOD_GAUSSIANMV_ICDF\nSee brief description of the methods BOXMULLER,\nBOXMULLER2, and ICDF in Table \"Values of <method> in\nmethod parameter\"\nstream\nVSLStreamStatePtr\nPointer to the stream state structure\nn\nconst MKL_INT\nNumber of d-dimensional vectors to be generated\ndimen\nconst MKL_INT\nDimension d ( d ≥ 1) of output random vectors\nmstorage\nconst MKL_INT\nMatrix storage scheme for lower triangular matrix T. The\nroutine supports three matrix storage schemes:\n•\nVSL_MATRIX_STORAGE_FULL— all d x d elements of the\nmatrix T are passed, however, only the lower triangle\npart is actually used in the routine.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2172\n\n\nName\nType\nDescription\n•\nVSL_MATRIX_STORAGE_PACKED— lower triangle\nelements of T are packed by rows into a one-\ndimensional array.\n•\nVSL_MATRIX_STORAGE_DIAGONAL— only diagonal\nelements of T are passed.\na\nconst float* for\nvsRngGaussianMV\nconst double* for\nvdRngGaussianMV\nMean vector a of dimension d\nt\nconst float* for\nvsRngGaussianMV\nconst double* for\nvdRngGaussianMV\nElements of the lower triangular matrix passed according to\nthe matrix T storage scheme mstorage.\nOutput Parameters\nName\nType\nDescription\nr\nfloat* for vsRngGaussianMV\ndouble* for vdRngGaussianMV\nArray of n random vectors of dimension dimen\nDescription\nThe vRngGaussianMV function generates random numbers with d-variate normal (Gaussian) distribution with\nmean value a and variance-covariance matrix C, where a∈Rd; C is a d×d symmetric positive-definite matrix.\nThe probability density function is given by:\nwhere x∈Rd .\nMatrix C can be represented as C = TTT, where T is a lower triangular matrix - Cholesky factor of C.\nInstead of variance-covariance matrix C the generation routines require Cholesky factor of C in input. To\ncompute Cholesky factor of matrix C, the user may call Intel® oneAPI Math Kernel Library (oneMKL) LAPACK\nroutines for matrix factorization:?potrf or ?pptrf for v?RngGaussianMV/v?rnggaussianmv routines (?\nmeans either s or d for single and double precision respectively). See Application Notes for more details.\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2173\n\n\nApplication Notes\nSince matrices are stored in Fortran by columns, while in C they are stored by rows, the usage of Intel®\noneAPI Math Kernel Library (oneMKL) factorization routines (assuming Fortran matrices storage) in\ncombination with multivariate normal RNG (assuming C matrix storage) is slightly different in C and Fortran.\nThe following tables help in using these routines in C and Fortran. For further information please refer to the\nappropriate VS example file.\nUsing Cholesky Factorization Routines in C\nMatrix Storage Scheme\nVariance-\nCovariance Matrix\nArgument\nFactorization\nRoutine\nUPLO\nParameter\nin\nFactorizati\non Routine\nResult of\nFactorizatio\nn as Input\nArgument\nfor RNG\nVSL_MATRIX_STORAGE_FULL\nC in C two-\ndimensional array\nspotrf for\nvsRngGaussianMV\ndpotrf for\nvdRngGaussianMV\n‘U’\nLower\ntriangle of T.\nUpper\ntriangle is not\nused.\nVSL_MATRIX_STORAGE_PACK\nED\nLower triangle of C\npacked by columns\ninto one-\ndimensional array\nspptrf for\nvsRngGaussianMV\ndpptrf for\nvdRngGaussianMV\n‘L’\nLower\ntriangle of T\npacked by\ncolumns into\na one-\ndimensional\narray.\nReturn Values\nVSL_ERROR_OK, VSL_STATUS_OK\nIndicates no error, execution is successful.\nVSL_ERROR_NULL_PTR\nstream is a NULL pointer.\nVSL_RNG_ERROR_BAD_STREAM\nstream is not a valid random stream.\nVSL_RNG_ERROR_BAD_UPDATE\nCallback function for an abstract BRNG returns an invalid\nnumber of updated entries in a buffer, that is, < 0 or >\nnmax.\nVSL_RNG_ERROR_NO_NUMBERS\nCallback function for an abstract BRNG returns 0 as the\nnumber of updated entries in a buffer.\nVSL_RNG_ERROR_QRNG_PERIOD_ELAPSED\nPeriod of the generator has been exceeded.\nVSL_RNG_ERROR_NONDETERM_NRETRIES_EXCEED\nED\nNumber of retries to generate a random number by using\nnon-deterministic random number generator exceeds\nthreshold.\nVSL_RNG_ERROR_ARS5_NOT_SUPPORTED\nARS-5 random number generator is not supported on the\nCPU running the application.\nvRngExponential\nGenerates exponentially distributed random numbers.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2174\n\n\nSyntax\nstatus = vsRngExponential( method, stream, n, r, a, beta );\nstatus = vdRngExponential( method, stream, n, r, a, beta );\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nmethod\nconst MKL_INT\nGeneration method. The specific values are as follows:\nVSL_RNG_METHOD_EXPONENTIAL_ICDF\nVSL_RNG_METHOD_EXPONENTIAL_ICDF_ACCURATE\nInverse cumulative distribution function method\nstream\nVSLStreamStatePtr\nPointer to the stream state structure\n \n \n \nn\nconst MKL_INT\nNumber of random values to be generated\na\nconst float for\nvsRngExponential\nconst double for\nvdRngExponential\nDisplacement a\n \n \n \nbeta\nconst float for\nvsRngExponential\nconst double for\nvdRngExponential\nScalefactor β.\nOutput Parameters\nName\nType\nDescription\nr\nfloat* for vsRngExponential\ndouble* for vdRngExponential\nVector of n exponentially distributed random numbers\nDescription\nThe vRngExponential function generates random numbers with exponential distribution that has\ndisplacement a and scalefactor β, where a, β∈R ; β > 0.\nThe probability density function is given by:\nThe cumulative distribution function is as follows:\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2175\n\n\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nReturn Values\nVSL_ERROR_OK, VSL_STATUS_OK\nIndicates no error, execution is successful.\nVSL_ERROR_NULL_PTR\nstream is a NULL pointer.\nVSL_RNG_ERROR_BAD_STREAM\nstream is not a valid random stream.\nVSL_RNG_ERROR_BAD_UPDATE\nCallback function for an abstract BRNG returns an invalid\nnumber of updated entries in a buffer, that is, < 0 or >\nnmax.\nVSL_RNG_ERROR_NO_NUMBERS\nCallback function for an abstract BRNG returns 0 as the\nnumber of updated entries in a buffer.\nVSL_RNG_ERROR_QRNG_PERIOD_ELAPSED\nPeriod of the generator has been exceeded.\nVSL_RNG_ERROR_NONDETERM_NRETRIES_EXCEED\nED\nNumber of retries to generate a random number by using\nnon-deterministic random number generator exceeds\nthreshold.\nVSL_RNG_ERROR_ARS5_NOT_SUPPORTED\nARS-5 random number generator is not supported on the\nCPU running the application.\nvRngLaplace\nGenerates random numbers with Laplace distribution.\nSyntax\nstatus = vsRngLaplace( method, stream, n, r, a, beta );\nstatus = vdRngLaplace( method, stream, n, r, a, beta );\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nmethod\nconst MKL_INT\nGeneration method. The specific values are as follows:\nVSL_RNG_METHOD_LAPLACE_ICDF\nInverse cumulative distribution function method\nstream\nVSLStreamStatePtr\nPointer to the stream state structure\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2176\n\n\nName\nType\nDescription\nn\nconst MKL_INT\nNumber of random values to be generated\na\nconst float for vsRngLaplace\nconst double for\nvdRngLaplace\nMean value a\nbeta\nconst float for vsRngLaplace\nconst double for\nvdRngLaplace\nScalefactor β.\nOutput Parameters\nName\nType\nDescription\nr\nfloat* for vsRngLaplace\ndouble* for vdRngLaplace\nVector of n Laplace distributed random numbers\nDescription\nThe vRngLaplace function generates random numbers with Laplace distribution with mean value (or\naverage) a and scalefactor β, where a, β∈R ; β > 0. The scalefactor value determines the standard\ndeviation as\nThe probability density function is given by:\nThe cumulative distribution function is as follows:\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nReturn Values\nVSL_ERROR_OK, VSL_STATUS_OK\nIndicates no error, execution is successful.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2177\n\n\nVSL_ERROR_NULL_PTR\nstream is a NULL pointer.\nVSL_RNG_ERROR_BAD_STREAM\nstream is not a valid random stream.\nVSL_RNG_ERROR_BAD_UPDATE\nCallback function for an abstract BRNG returns an invalid\nnumber of updated entries in a buffer, that is, < 0 or >\nnmax.\nVSL_RNG_ERROR_NO_NUMBERS\nCallback function for an abstract BRNG returns 0 as the\nnumber of updated entries in a buffer.\nVSL_RNG_ERROR_QRNG_PERIOD_ELAPSED\nPeriod of the generator has been exceeded.\nVSL_RNG_ERROR_NONDETERM_NRETRIES_EXCEED\nED\nNumber of retries to generate a random number by using\nnon-deterministic random number generator exceeds\nthreshold.\nVSL_RNG_ERROR_ARS5_NOT_SUPPORTED\nARS-5 random number generator is not supported on the\nCPU running the application.\nvRngWeibull\nGenerates Weibull distributed random numbers.\nSyntax\nstatus = vsRngWeibull( method, stream, n, r, alpha, a, beta );\nstatus = vdRngWeibull( method, stream, n, r, alpha, a, beta );\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nmethod\nconst MKL_INT\nGeneration method. The specific values are as follows:\nVSL_RNG_METHOD_WEIBULL_ICDF\nVSL_RNG_METHOD_WEIBULL_ICDF_ACCURATE\nInverse cumulative distribution function method\nstream\nVSLStreamStatePtr\nPointer to the stream state structure\nn\nconst MKL_INT\nNumber of random values to be generated\nalpha\nconst float for vsRngWeibull\nconst double for\nvdRngWeibull\nShape α.\na\nconst float for vsRngWeibull\nconst double for\nvdRngWeibull\nDisplacement a\nbeta\nconst float for vsRngWeibull Scalefactor β.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2178\n\n\nName\nType\nDescription\nconst double for\nvdRngWeibull\nOutput Parameters\nName\nType\nDescription\nr\nfloat* for vsRngWeibull\ndouble* for vdRngWeibull\nVector of n Weibull distributed random numbers\nDescription\nThe vRngWeibull function generates Weibull distributed random numbers with displacement a, scalefactor β,\nand shape α, where α, β, a∈R ; α > 0, β > 0.\nThe probability density function is given by:\nThe cumulative distribution function is as follows:\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nReturn Values\nVSL_ERROR_OK, VSL_STATUS_OK\nIndicates no error, execution is successful.\nVSL_ERROR_NULL_PTR\nstream is a NULL pointer.\nVSL_RNG_ERROR_BAD_STREAM\nstream is not a valid random stream.\nVSL_RNG_ERROR_BAD_UPDATE\nCallback function for an abstract BRNG returns an invalid\nnumber of updated entries in a buffer, that is, < 0 or >\nnmax.\nVSL_RNG_ERROR_NO_NUMBERS\nCallback function for an abstract BRNG returns 0 as the\nnumber of updated entries in a buffer.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2179\n\n\nVSL_RNG_ERROR_QRNG_PERIOD_ELAPSED\nPeriod of the generator has been exceeded.\nVSL_RNG_ERROR_NONDETERM_NRETRIES_EXCEED\nED\nNumber of retries to generate a random number by using\nnon-deterministic random number generator exceeds\nthreshold.\nVSL_RNG_ERROR_ARS5_NOT_SUPPORTED\nARS-5 random number generator is not supported on the\nCPU running the application.\nvRngCauchy\nGenerates Cauchy distributed random values.\nSyntax\nstatus = vsRngCauchy( method, stream, n, r, a, beta );\nstatus = vdRngCauchy( method, stream, n, r, a, beta );\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nmethod\nconst MKL_INT\nGeneration method. The specific values are as follows:\nVSL_RNG_METHOD_CAUCHY_ICDF\nInverse cumulative distribution function method\nstream\nVSLStreamStatePtr\nPointer to the stream state structure\nn\nconst MKL_INT\nNumber of random values to be generated\na\nconst float for vsRngCauchy\nconst double for vdRngCauchy\nDisplacementa.\nbeta\nconst float for vsRngCauchy\nconst double for vdRngCauchy\nScalefactor β.\nOutput Parameters\nName\nType\nDescription\nr\nfloat* for vsRngCauchy\ndouble* for vdRngCauchy\nVector of n Cauchy distributed random numbers\nDescription\nThe function generates Cauchy distributed random numbers with displacement a and scalefactor β, where a,\nβ∈R ; β > 0.\nThe probability density function is given by:\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2180\n\n\nThe cumulative distribution function is as follows:\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nReturn Values\nVSL_ERROR_OK, VSL_STATUS_OK\nIndicates no error, execution is successful.\nVSL_ERROR_NULL_PTR\nstream is a NULL pointer.\nVSL_RNG_ERROR_BAD_STREAM\nstream is not a valid random stream.\nVSL_RNG_ERROR_BAD_UPDATE\nCallback function for an abstract BRNG returns an invalid\nnumber of updated entries in a buffer, that is, < 0 or >\nnmax.\nVSL_RNG_ERROR_NO_NUMBERS\nCallback function for an abstract BRNG returns 0 as the\nnumber of updated entries in a buffer.\nVSL_RNG_ERROR_QRNG_PERIOD_ELAPSED\nPeriod of the generator has been exceeded.\nVSL_RNG_ERROR_NONDETERM_NRETRIES_EXCEED\nED\nNumber of retries to generate a random number by using\nnon-deterministic random number generator exceeds\nthreshold.\nVSL_RNG_ERROR_ARS5_NOT_SUPPORTED\nARS-5 random number generator is not supported on the\nCPU running the application.\nvRngRayleigh\nGenerates Rayleigh distributed random values.\nSyntax\nstatus = vsRngRayleigh( method, stream, n, r, a, beta );\nstatus = vdRngRayleigh( method, stream, n, r, a, beta );\nInclude Files\n•\nmkl.h\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2181\n\n\nInput Parameters\nName\nType\nDescription\nmethod\nconst MKL_INT\nGeneration method. The specific values are as follows:\nVSL_RNG_METHOD_RAYLEIGH_ICDF\nVSL_RNG_METHOD_RAYLEIGH_ICDF_ACCURATE\nInverse cumulative distribution function method\nstream\nVSLStreamStatePtr\nPointer to the stream state structure\nn\nconst MKL_INT\nNumber of random values to be generated\na\nconst float for\nvsRngRayleigh\nconst double for\nvdRngRayleigh\nDisplacement a\nbeta\nconst float for\nvsRngRayleigh\nconst double for\nvdRngRayleigh\nScalefactor β.\nOutput Parameters\nName\nType\nDescription\nr\nfloat* for vsRngRayleigh\ndouble* for vdRngRayleigh\nVector of n Rayleigh distributed random numbers\nDescription\nThe vRngRayleigh function generates Rayleigh distributed random numbers with displacement a and\nscalefactor β, where a, β∈R ; β > 0.\nThe Rayleigh distribution is a special case of the Weibull distribution, where the shape parameter α = 2.\nThe probability density function is given by:\nThe cumulative distribution function is as follows:\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2182\n\n\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nReturn Values\nVSL_ERROR_OK, VSL_STATUS_OK\nIndicates no error, execution is successful.\nVSL_ERROR_NULL_PTR\nstream is a NULL pointer.\nVSL_RNG_ERROR_BAD_STREAM\nstream is not a valid random stream.\nVSL_RNG_ERROR_BAD_UPDATE\nCallback function for an abstract BRNG returns an invalid\nnumber of updated entries in a buffer, that is, < 0 or >\nnmax.\nVSL_RNG_ERROR_NO_NUMBERS\nCallback function for an abstract BRNG returns 0 as the\nnumber of updated entries in a buffer.\nVSL_RNG_ERROR_QRNG_PERIOD_ELAPSED\nPeriod of the generator has been exceeded.\nVSL_RNG_ERROR_NONDETERM_NRETRIES_EXCEED\nED\nNumber of retries to generate a random number by using\nnon-deterministic random number generator exceeds\nthreshold.\nVSL_RNG_ERROR_ARS5_NOT_SUPPORTED\nARS-5 random number generator is not supported on the\nCPU running the application.\nvRngLognormal\nGenerates lognormally distributed random numbers.\nSyntax\nstatus = vsRngLognormal( method, stream, n, r, a, sigma, b, beta );\nstatus = vdRngLognormal( method, stream, n, r, a, sigma, b, beta );\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nmethod\nconst MKL_INT\nGeneration method. The specific values are as follows:\nVSL_RNG_METHOD_LOGNORMAL_BOXMULLER2\nVSL_RNG_METHOD_LOGNORMAL_BOXMULLER2_ACCURATE\nBox Muller 2 based method\n VSL_RNG_METHOD_LOGNORMAL_ICDF\n VSL_RNG_METHOD_LOGNORMAL_ICDF_ACCURATE\nInverse cumulative distribution function based method\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2183\n\n\nName\nType\nDescription\nstream\nVSLStreamStatePtr\nPointer to the stream state structure\nn\nconst MKL_INT\nNumber of random values to be generated\na\nconst float for\nvsRngLognormal\nconst double for\nvdRngLognormal\nAverage a of the subject normal distribution\nsigma\nconst float for\nvsRngLognormal\nconst double for\nvdRngLognormal\nStandard deviation σ of the subject normal distribution\nb\nconst float for\nvsRngLognormal\nconst double for\nvdRngLognormal\nDisplacement b\nbeta\nconst float for\nvsRngLognormal\nconst double for\nvdRngLognormal\nScalefactor β.\nOutput Parameters\nName\nType\nDescription\nr\nfloat* for vsRngLognormal\ndouble* for vdRngLognormal\nVector of n lognormally distributed random numbers\nDescription\nThe vRngLognormal function generates lognormally distributed random numbers with average of distribution\na and standard deviation σ of subject normal distribution, displacement b, and scalefactor β, where a, σ, b,\nβ∈R ; σ > 0 , β > 0.\nThe probability density function is given by:\nThe cumulative distribution function is as follows:\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2184\n\n\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nReturn Values\nVSL_ERROR_OK, VSL_STATUS_OK\nIndicates no error, execution is successful.\nVSL_ERROR_NULL_PTR\nstream is a NULL pointer.\nVSL_RNG_ERROR_BAD_STREAM\nstream is not a valid random stream.\nVSL_RNG_ERROR_BAD_UPDATE\nCallback function for an abstract BRNG returns an invalid\nnumber of updated entries in a buffer, that is, < 0 or >\nnmax.\nVSL_RNG_ERROR_NO_NUMBERS\nCallback function for an abstract BRNG returns 0 as the\nnumber of updated entries in a buffer.\nVSL_RNG_ERROR_QRNG_PERIOD_ELAPSED\nPeriod of the generator has been exceeded.\nVSL_RNG_ERROR_NONDETERM_NRETRIES_EXCEED\nED\nNumber of retries to generate a random number by using\nnon-deterministic random number generator exceeds\nthreshold.\nVSL_RNG_ERROR_ARS5_NOT_SUPPORTED\nARS-5 random number generator is not supported on the\nCPU running the application.\nvRngGumbel\nGenerates Gumbel distributed random values.\nSyntax\nstatus = vsRngGumbel( method, stream, n, r, a, beta );\nstatus = vdRngGumbel( method, stream, n, r, a, beta );\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nmethod\nconst MKL_INT\nGeneration method. The specific values are as follows:\nVSL_RNG_METHOD_GUMBEL_ICDF\nInverse cumulative distribution function method\nstream\nVSLStreamStatePtr\nPointer to the stream state structure\nn\nconst MKL_INT\nNumber of random values to be generated\na\nconst float for vsRngGumbel\nDisplacementa.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2185\n\n\nName\nType\nDescription\nconst double for vdRngGumbel\nbeta\nconst float for vsRngGumbel\nconst double for vdRngGumbel\nScalefactor β.\nOutput Parameters\nName\nType\nDescription\nr\nfloat* for vsRngGumbel\ndouble* for vdRngGumbel\nVector of n random numbers with Gumbel distribution\nDescription\nThe vRngGumbel function generates Gumbel distributed random numbers with displacement a and\nscalefactor β, where a, β∈R ; β > 0.\nThe probability density function is given by:\nThe cumulative distribution function is as follows:\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nReturn Values\nVSL_ERROR_OK, VSL_STATUS_OK\nIndicates no error, execution is successful.\nVSL_ERROR_NULL_PTR\nstream is a NULL pointer.\nVSL_RNG_ERROR_BAD_STREAM\nstream is not a valid random stream.\nVSL_RNG_ERROR_BAD_UPDATE\nCallback function for an abstract BRNG returns an invalid\nnumber of updated entries in a buffer, that is, < 0 or >\nnmax.\nVSL_RNG_ERROR_NO_NUMBERS\nCallback function for an abstract BRNG returns 0 as the\nnumber of updated entries in a buffer.\nVSL_RNG_ERROR_QRNG_PERIOD_ELAPSED\nPeriod of the generator has been exceeded.\nVSL_RNG_ERROR_NONDETERM_NRETRIES_EXCEED\nED\nNumber of retries to generate a random number by using\nnon-deterministic random number generator exceeds\nthreshold.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2186\n\n\nVSL_RNG_ERROR_ARS5_NOT_SUPPORTED\nARS-5 random number generator is not supported on the\nCPU running the application.\nvRngGamma\nGenerates gamma distributed random values.\nSyntax\nstatus = vsRngGamma( method, stream, n, r, alpha, a, beta );\nstatus = vdRngGamma( method, stream, n, r, alpha, a, beta );\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nmethod\nconst MKL_INT\nGeneration method. The specific values are as follows:\nVSL_RNG_METHOD_GAMMA_GNORM\nVSL_RNG_METHOD_GAMMA_GNORM_ACCURATE\nAcceptance/rejection method using random numbers with\nGaussian distribution. See brief description of the method\nGNORM in Table \"Values of <method> in method parameter\"\nstream\nVSLStreamStatePtr\nPointer to the stream state structure\nn\nconst MKL_INT\nNumber of random values to be generated\nalpha\nconst float for vsRngGamma\nconst double for vdRngGamma\nShape α.\na\nconst float for vsRngGamma\nconst double for vdRngGamma\nDisplacement a.\nbeta\nconst float for vsRngGamma\nconst double for vdRngGamma\nScalefactor β.\nOutput Parameters\nName\nType\nDescription\nr\nfloat* for vsRngGamma\ndouble* for vdRngGamma\nVector of n random numbers with gamma distribution\nDescription\nThe vRngGamma function generates random numbers with gamma distribution that has shape parameter α,\ndisplacement a, and scale parameter β, where α, β, and a∈R ; α > 0, β > 0.\nThe probability density function is given by:\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2187\n\n\nwhere Γ(α) is the complete gamma function.\nThe cumulative distribution function is as follows:\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nReturn Values\nVSL_ERROR_OK, VSL_STATUS_OK\nIndicates no error, execution is successful.\nVSL_ERROR_NULL_PTR\nstream is a NULL pointer.\nVSL_RNG_ERROR_BAD_STREAM\nstream is not a valid random stream.\nVSL_RNG_ERROR_BAD_UPDATE\nCallback function for an abstract BRNG returns an invalid\nnumber of updated entries in a buffer, that is, < 0 or >\nnmax.\nVSL_RNG_ERROR_NO_NUMBERS\nCallback function for an abstract BRNG returns 0 as the\nnumber of updated entries in a buffer.\nVSL_RNG_ERROR_QRNG_PERIOD_ELAPSED\nPeriod of the generator has been exceeded.\nVSL_RNG_ERROR_NONDETERM_NRETRIES_EXCEED\nED\nNumber of retries to generate a random number by using\nnon-deterministic random number generator exceeds\nthreshold.\nVSL_RNG_ERROR_ARS5_NOT_SUPPORTED\nARS-5 random number generator is not supported on the\nCPU running the application.\nvRngBeta\nGenerates beta distributed random values.\nSyntax\nstatus = vsRngBeta( method, stream, n, r, p, q, a, beta );\nstatus = vdRngBeta( method, stream, n, r, p, q, a, beta );\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2188\n\n\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nmethod\nconst MKL_INT\nGeneration method. The specific values are as follows:\nVSL_RNG_METHOD_BETA_CJA\nVSL_RNG_METHOD_BETA_CJA_ACCURATE\nSee brief description of the method CJA in Table \"Values of\n<method> in method parameter\"\nstream\nVSLStreamStatePtr\nPointer to the stream state structure\nn\nconst MKL_INT\nNumber of random values to be generated\np\nconst float for vsRngBeta\nconst double for vdRngBeta\nShape p\nq\nconst float for vsRngBeta\nconst double for vdRngBeta\nShape q\na\nconst float for vsRngBeta\nconst double for vdRngBeta\nDisplacementa.\nbeta\nconst float for vsRngBeta\nconst double for vdRngBeta\nScalefactor β.\nOutput Parameters\nName\nType\nDescription\nr\nfloat* for vsRngBeta\ndouble* for vdRngBeta\nVector of n random numbers with beta distribution\nDescription\nThe vRngBeta function generates random numbers with beta distribution that has shape parameters p and\nq, displacement a, and scale parameter β, where p, q, a, and β∈R ; p > 0, q > 0, β > 0.\nThe probability density function is given by:\nwhere B(p, q) is the complete beta function.\nThe cumulative distribution function is as follows:\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2189\n\n\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nReturn Values\nVSL_ERROR_OK, VSL_STATUS_OK\nIndicates no error, execution is successful.\nVSL_ERROR_NULL_PTR\nstream is a NULL pointer.\nVSL_RNG_ERROR_BAD_STREAM\nstream is not a valid random stream.\nVSL_RNG_ERROR_BAD_UPDATE\nCallback function for an abstract BRNG returns an invalid\nnumber of updated entries in a buffer, that is, < 0 or >\nnmax.\nVSL_RNG_ERROR_NO_NUMBERS\nCallback function for an abstract BRNG returns 0 as the\nnumber of updated entries in a buffer.\nVSL_RNG_ERROR_QRNG_PERIOD_ELAPSED\nPeriod of the generator has been exceeded.\nVSL_RNG_ERROR_NONDETERM_NRETRIES_EXCEED\nED\nNumber of retries to generate a random number by using\nnon-deterministic random number generator exceeds\nthreshold.\nVSL_RNG_ERROR_ARS5_NOT_SUPPORTED\nARS-5 random number generator is not supported on the\nCPU running the application.\nvRngChiSquare\nGenerates chi-square distributed random values.\nSyntax\nstatus = vsRngChiSquare( method, stream, n, r, v );\nstatus = vdRngChiSquare( method, stream, n, r, v );\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nmethod\nconst MKL_INT\nGeneration method. The specific value is:\nVSL_RNG_METHOD_CHISQUARE_CHI2GAMMA\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2190\n\n\nName\nType\nDescription\nFor a description of\nVSL_RNG_METHOD_CHISQUARE_CHI2GAMMA, see Random\nNumber Generators Naming Conventions.\nstream\nVSLStreamStatePtr\nPointer to the stream state structure\nn\nconst MKL_INT\nNumber of random values to be generated\n \n \n \nv\nconst MKL_INT\nDegrees of freedom\n \n \n \nOutput Parameters\nName\nType\nDescription\nr\nfloat* for vsRngChiSquare\ndouble* for vdRngChiSquare\nVector of n random numbers with chi-square distribution\nDescription\nThe vRngChiSquare function generates random numbers with chi-square distribution and ν degrees of\nfreedom, ν ∈N, ν > 0.\nThe probability density function is:\nfv x =\nx\nv −2\n2\n e−x\n2\n2v/2Γ v\n2\n ,   x ≥0\n0,                           x < 0\nThe cumulative distribution function is:\nFv x =\n∫0\nx y\nv −2\n2\ne−y\n2\n2v/2Γ v\n2\ndy,   x ≥0\n0,                                      x < 0\nReturn Values\nVSL_ERROR_OK, VSL_STATUS_OK\nIndicates no error, execution is successful.\nVSL_ERROR_NULL_PTR\nstream is a NULL pointer.\nVSL_RNG_ERROR_BAD_STREAM\nstream is not a valid random stream.\nVSL_RNG_ERROR_BAD_UPDATE\nCallback function for an abstract BRNG returns an invalid\nnumber of updated entries in a buffer, that is, < 0 or >\nnmax.\nVSL_RNG_ERROR_NO_NUMBERS\nCallback function for an abstract BRNG returns 0 as the\nnumber of updated entries in a buffer.\nVSL_RNG_ERROR_QRNG_PERIOD_ELAPSED\nPeriod of the generator has been exceeded.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2191\n\n\nVSL_RNG_ERROR_NONDETERM_NRETRIES_EXCEED\nED\nNumber of retries to generate a random number by using\nnon-deterministic random number generator exceeds\nthreshold.\nVSL_RNG_ERROR_ARS5_NOT_SUPPORTED\nARS-5 random number generator is not supported on the\nCPU running the application.\nDiscrete Distributions\nThis section describes routines for generating random numbers with discrete distribution.\nvRngUniform Discrete Distribution Generators\nGenerates random numbers uniformly distributed over\nthe interval [a, b).\nSyntax\nstatus = viRngUniform( method, stream, n, r, a, b );\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nmethod\nconst MKL_INT\nGeneration method; the specific value is as follows:\nVSL_RNG_METHOD_UNIFORM_STD\nStandard method. Currently there is only one method for\nthis distribution generator.\nstream\nVSLStreamStatePtr\nPointer to the stream state structure\nn\nconst MKL_INT\nNumber of random values to be generated\na\nconst int\nLeft interval bound a\nb\nconst int\nRight interval bound b\nOutput Parameters\nName\nType\nDescription\nr\nint*\nVector of n random numbers uniformly distributed over the\ninterval [a,b)\nDescription\nThe vRngUniform function generates random numbers uniformly distributed over the interval [a, b), where\na, b are the left and right bounds of the interval respectively, and a, b∈Z; a < b.\nThe probability distribution is given by:\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2192\n\n\nThe cumulative distribution function is as follows:\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nReturn Values\nVSL_ERROR_OK, VSL_STATUS_OK\nIndicates no error, execution is successful.\nVSL_ERROR_NULL_PTR\nstream is a NULL pointer.\nVSL_RNG_ERROR_BAD_STREAM\nstream is not a valid random stream.\nVSL_RNG_ERROR_BAD_UPDATE\nCallback function for an abstract BRNG returns an invalid\nnumber of updated entries in a buffer, that is, < 0 or >\nnmax.\nVSL_RNG_ERROR_NO_NUMBERS\nCallback function for an abstract BRNG returns 0 as the\nnumber of updated entries in a buffer.\nVSL_RNG_ERROR_QRNG_PERIOD_ELAPSED\nPeriod of the generator has been exceeded.\nVSL_RNG_ERROR_NONDETERM_NRETRIES_EXCEED\nED\nNumber of retries to generate a random number by using\nnon-deterministic random number generator exceeds\nthreshold.\nVSL_RNG_ERROR_ARS5_NOT_SUPPORTED\nARS-5 random number generator is not supported on the\nCPU running the application.\nvRngUniformBits\nGenerates bits of underlying BRNG integer recurrence.\nSyntax\nstatus = viRngUniformBits( method, stream, n, r );\nInclude Files\n•\nmkl.h\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2193\n\n\nInput Parameters\nName\nType\nDescription\nmethod\nconst MKL_INT\nGeneration method; the specific value is\nVSL_RNG_METHOD_UNIFORMBITS_STD\nstream\nVSLStreamStatePtr\nPointer to the stream state structure\nn\nconst MKL_INT\nNumber of random values to be generated\nOutput Parameters\nName\nType\nDescription\nr\nunsigned int*\nVector of n random integer numbers. If the stream was\ngenerated by a 64 or a 128-bit generator, each integer\nvalue is represented by two or four elements of r\nrespectively. The number of bytes occupied by each integer\nis contained in the field WordSize of the structure\nVSLBRngProperties. The total number of bits that are\nactually used to store the value are contained in the field\nNBits of the same structure.See Advanced Service Routines\nfor a more detailed discussion of VSLBRngProperties.\nDescription\nThe vRngUniformBits function generates integer random values with uniform bit distribution. The\ngenerators of uniformly distributed numbers can be represented as recurrence relations over integer values\nin modular arithmetic. Apparently, each integer can be treated as a vector of several bits. In a truly random\ngenerator, these bits are random, while in pseudorandom generators this randomness can be violated. For\nexample, a well known drawback of linear congruential generators is that lower bits are less random than\nhigher bits (for example, see [Knuth81]). For this reason, care should be taken when using this function.\nTypically, in a 32-bit LCG only 24 higher bits of an integer value can be considered random. See VS Notes for\ndetails.\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nReturn Values\nVSL_ERROR_OK, VSL_STATUS_OK\nIndicates no error, execution is successful.\nVSL_ERROR_NULL_PTR\nstream is a NULL pointer.\nVSL_RNG_ERROR_BAD_STREAM\nstream is not a valid random stream.\nVSL_RNG_ERROR_BAD_UPDATE\nCallback function for an abstract BRNG returns an invalid\nnumber of updated entries in a buffer, that is, < 0 or >\nnmax.\nVSL_RNG_ERROR_NO_NUMBERS\nCallback function for an abstract BRNG returns 0 as the\nnumber of updated entries in a buffer.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2194\n\n\nVSL_RNG_ERROR_QRNG_PERIOD_ELAPSED\nPeriod of the generator has been exceeded.\nVSL_RNG_ERROR_NONDETERM_NRETRIES_EXCEED\nED\nNumber of retries to generate a random number by using\nnon-deterministic random number generator exceeds\nthreshold.\nVSL_RNG_ERROR_ARS5_NOT_SUPPORTED\nARS-5 random number generator is not supported on the\nCPU running the application.\nvRngUniformBits32\nGenerates uniformly distributed bits in 32-bit chunks.\nSyntax\nstatus = viRngUniformBits32( method, stream, n, r );\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nmethod\nconst MKL_INT\nGeneration method; the specific value is\nVSL_RNG_METHOD_UNIFORMBITS32_STD\nstream\nVSLStreamStatePtr\nPointer to the stream state structure\nn\nconst MKL_INT\nNumber of random values to be generated\nOutput Parameters\nName\nType\nDescription\nr\nunsigned int*\nVector of n 32-bit random integer numbers with uniform bit\ndistribution.\nDescription\nThe vRngUniformBits32 function generates uniformly distributed bits in 32-bit chunks. Unlike\nvRngUniformBits, which provides the output of underlying integer recurrence and does not guarantee\nuniform distribution across bits, vRngUniformBits32 is designed to ensure each bit in the 32-bit chunk is\nuniformly distributed. See VS Notes for details.\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nReturn Values\nVSL_ERROR_OK, VSL_STATUS_OK\nIndicates no error, execution is successful.\nVSL_ERROR_NULL_PTR\nstream is a NULL pointer.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2195\n\n\nVSL_RNG_ERROR_BAD_STREAM\nstream is not a valid random stream.\nVSL_RNG_ERROR_BRNG_NOT_SUPPORTED\nBRNG is not supported by the function.\nVSL_RNG_ERROR_NONDETERM_NRETRIES_EXCEED\nED\nNumber of retries to generate a random number by using\nnon-deterministic random number generator exceeds\nthreshold.\nVSL_RNG_ERROR_ARS5_NOT_SUPPORTED\nARS-5 random number generator is not supported on the\nCPU running the application.\nvRngUniformBits64\nGenerates uniformly distributed bits in 64-bit chunks.\nSyntax\nstatus = viRngUniformBits64( method, stream, n, r );\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nmethod\nconst MKL_INT\nGeneration method; the specific value is\nVSL_RNG_METHOD_UNIFORMBITS64_STD\nstream\nVSLStreamStatePtr\nPointer to the stream state structure\nn\nconst MKL_INT\nNumber of random values to be generated\nOutput Parameters\nName\nType\nDescription\nr\nunsigned MKL_INT64*\nVector of n 64-bit random integer numbers with uniform bit\ndistribution.\nDescription\nThe vRngUniformBits64 function generates uniformly distributed bits in 64-bit chunks. Unlike\nvRngUniformBits, which provides the output of underlying integer recurrence and does not guarantee\nuniform distribution across bits, vRngUniformBits64 is designed to ensure each bit in the 64-bit chunk is\nuniformly distributed. See VS Notes for details.\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nReturn Values\nVSL_ERROR_OK, VSL_STATUS_OK\nIndicates no error, execution is successful.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2196\n\n\nVSL_ERROR_NULL_PTR\nstream is a NULL pointer.\nVSL_RNG_ERROR_BAD_STREAM\nstream is not a valid random stream.\nVSL_RNG_ERROR_BRNG_NOT_SUPPORTED\nBRNG is not supported by the function.\nVSL_RNG_ERROR_NONDETERM_NRETRIES_EXCEED\nED\nNumber of retries to generate a random number by using\nnon-deterministic random number generator exceeds\nthreshold.\nVSL_RNG_ERROR_ARS5_NOT_SUPPORTED\nARS-5 random number generator is not supported on the\nCPU running the application.\nvRngBernoulli\nGenerates Bernoulli distributed random values.\nSyntax\nstatus = viRngBernoulli( method, stream, n, r, p );\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nmethod\nconst MKL_INT\nGeneration method. The specific value is as follows:\nVSL_RNG_METHOD_BERNOULLI_ICDF\nInverse cumulative distribution function method.\nstream\nVSLStreamStatePtr\nPointer to the stream state structure\nn\nconst MKL_INT\nNumber of random values to be generated\np\nconst double\nSuccess probability p of a trial\nOutput Parameters\nName\nType\nDescription\nr\nint*\nVector of n Bernoulli distributed random values\nDescription\nThe vRngBernoulli function generates Bernoulli distributed random numbers with probability p of a single\ntrial success, where\np∈R; 0 ≤p≤ 1.\nA variate is called Bernoulli distributed, if after a trial it is equal to 1 with probability of success p, and to 0\nwith probability 1 - p.\nThe probability distribution is given by:\nP(X = 1) = p\nP(X = 0) = 1 - p\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2197\n\n\nThe cumulative distribution function is as follows:\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nReturn Values\nVSL_ERROR_OK, VSL_STATUS_OK\nIndicates no error, execution is successful.\nVSL_ERROR_NULL_PTR\nstream is a NULL pointer.\nVSL_RNG_ERROR_BAD_STREAM\nstream is not a valid random stream.\nVSL_RNG_ERROR_BAD_UPDATE\nCallback function for an abstract BRNG returns an invalid\nnumber of updated entries in a buffer, that is, < 0 or >\nnmax.\nVSL_RNG_ERROR_NO_NUMBERS\nCallback function for an abstract BRNG returns 0 as the\nnumber of updated entries in a buffer.\nVSL_RNG_ERROR_QRNG_PERIOD_ELAPSED\nPeriod of the generator has been exceeded.\nVSL_RNG_ERROR_ARS5_NOT_SUPPORTED\nARS-5 random number generator is not supported on the\nCPU running the application.\nvRngGeometric\nGenerates geometrically distributed random values.\nSyntax\nstatus = viRngGeometric( method, stream, n, r, p );\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nmethod\nconst MKL_INT\nGeneration method. The specific value is as follows:\nVSL_RNG_METHOD_GEOMETRIC_ICDF\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2198\n\n\nName\nType\nDescription\nInverse cumulative distribution function method.\nstream\nVSLStreamStatePtr\nPointer to the stream state structure\nn\nconst MKL_INT\nNumber of random values to be generated\np\nconst double\nSuccess probability p of a trial\nOutput Parameters\nName\nType\nDescription\nr\nint*\nVector of n geometrically distributed random values\nDescription\nThe vRngGeometric function generates geometrically distributed random numbers with probability p of a\nsingle trial success, where p∈R; 0 < p < 1.\nA geometrically distributed variate represents the number of independent Bernoulli trials preceding the first\nsuccess. The probability of a single Bernoulli trial success is p.\nThe probability distribution is given by:\nP(X = k) = p·(1 - p)k, k∈ {0,1,2, ... }.\nThe cumulative distribution function is as follows:\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nReturn Values\nVSL_ERROR_OK, VSL_STATUS_OK\nIndicates no error, execution is successful.\nVSL_ERROR_NULL_PTR\nstream is a NULL pointer.\nVSL_RNG_ERROR_BAD_STREAM\nstream is not a valid random stream.\nVSL_RNG_ERROR_BAD_UPDATE\nCallback function for an abstract BRNG returns an invalid\nnumber of updated entries in a buffer, that is, < 0 or >\nnmax.\nVSL_RNG_ERROR_NO_NUMBERS\nCallback function for an abstract BRNG returns 0 as the\nnumber of updated entries in a buffer.\nVSL_RNG_ERROR_QRNG_PERIOD_ELAPSED\nPeriod of the generator has been exceeded.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2199\n\n\nVSL_RNG_ERROR_NONDETERM_NRETRIES_EXCEED\nED\nNumber of retries to generate a random number by using\nnon-deterministic random number generator exceeds\nthreshold.\nVSL_RNG_ERROR_ARS5_NOT_SUPPORTED\nARS-5 random number generator is not supported on the\nCPU running the application.\nvRngBinomial\nGenerates binomially distributed random numbers.\nSyntax\nstatus = viRngBinomial( method, stream, n, r, ntrial, p );\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nmethod\nconst MKL_INT\nGeneration method. The specific value is as follows:\nVSL_RNG_METHOD_BINOMIAL_BTPE\nSee brief description of the BTPE method in Table \"Values of\n<method> in method parameter\".\nstream\nVSLStreamStatePtr\nPointer to the stream state structure\nn\nconst MKL_INT\nNumber of random values to be generated\nntrial\nconst int\nNumber of independent trials m\np\nconst double\nSuccess probability p of a single trial\nOutput Parameters\nName\nType\nDescription\nr\nint*\nVector of n binomially distributed random values\nDescription\nThe vRngBinomial function generates binomially distributed random numbers with number of independent\nBernoulli trials m, and with probability p of a single trial success, where p∈R; 0 ≤p≤ 1, m∈N.\nA binomially distributed variate represents the number of successes in m independent Bernoulli trials with\nprobability of a single trial success p.\nThe probability distribution is given by:\nThe cumulative distribution function is as follows:\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2200\n\n\n \nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nReturn Values\nVSL_ERROR_OK, VSL_STATUS_OK\nIndicates no error, execution is successful.\nVSL_ERROR_NULL_PTR\nstream is a NULL pointer.\nVSL_RNG_ERROR_BAD_STREAM\nstream is not a valid random stream.\nVSL_RNG_ERROR_BAD_UPDATE\nCallback function for an abstract BRNG returns an invalid\nnumber of updated entries in a buffer, that is, < 0 or >\nnmax.\nVSL_RNG_ERROR_NO_NUMBERS\nCallback function for an abstract BRNG returns 0 as the\nnumber of updated entries in a buffer.\nVSL_RNG_ERROR_QRNG_PERIOD_ELAPSED\nPeriod of the generator has been exceeded.\nVSL_RNG_ERROR_NONDETERM_NRETRIES_EXCEED\nED\nNumber of retries to generate a random number by using\nnon-deterministic random number generator exceeds\nthreshold.\nVSL_RNG_ERROR_ARS5_NOT_SUPPORTED\nARS-5 random number generator is not supported on the\nCPU running the application.\nvRngHypergeometric\nGenerates hypergeometrically distributed random\nvalues.\nSyntax\nstatus = viRngHypergeometric( method, stream, n, r, l, s, m );\nInclude Files\n•\nmkl.h\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2201\n\n\nInput Parameters\nName\nType\nDescription\nmethod\nconst MKL_INT\nGeneration method. The specific value is as follows:\nVSL_RNG_METHOD_HYPERGEOMETRIC_H2PE\nSee brief description of the H2PE method in Table \"Values of\n<method> in method parameter\"\nstream\nVSLStreamStatePtr\nPointer to the stream state structure\nn\nconst MKL_INT\nNumber of random values to be generated\nl\nconst int\nLot size l\ns\nconst int\nSize of sampling without replacement s\nm\nconst int\nNumber of marked elements m\nOutput Parameters\nName\nType\nDescription\nr\nint*\nVector of n hypergeometrically distributed random values\nDescription\nThe vRngHypergeometric function generates hypergeometrically distributed random values with lot size l,\nsize of sampling s, and number of marked elements in the lot m, where l, m, s∈N∪{0}; l≥ max(s, m).\nConsider a lot of l elements comprising m \"marked\" and l-m \"unmarked\" elements. A trial sampling without\nreplacement of exactly s elements from this lot helps to define the hypergeometric distribution, which is the\nprobability that the group of s elements contains exactly k marked elements.\nThe probability distribution is given by:)\n, k∈ {max(0, s + m - l), ..., min(s, m)}\nThe cumulative distribution function is as follows:\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2202\n\n\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nReturn Values\nVSL_ERROR_OK, VSL_STATUS_OK\nIndicates no error, execution is successful.\nVSL_ERROR_NULL_PTR\nstream is a NULL pointer.\nVSL_RNG_ERROR_BAD_STREAM\nstream is not a valid random stream.\nVSL_RNG_ERROR_BAD_UPDATE\nCallback function for an abstract BRNG returns an invalid\nnumber of updated entries in a buffer, that is, < 0 or >\nnmax.\nVSL_RNG_ERROR_NO_NUMBERS\nCallback function for an abstract BRNG returns 0 as the\nnumber of updated entries in a buffer.\nVSL_RNG_ERROR_QRNG_PERIOD_ELAPSED\nPeriod of the generator has been exceeded.\nVSL_RNG_ERROR_NONDETERM_NRETRIES_EXCEED\nED\nNumber of retries to generate a random number by using\nnon-deterministic random number generator exceeds\nthreshold.\nVSL_RNG_ERROR_ARS5_NOT_SUPPORTED\nARS-5 random number generator is not supported on the\nCPU running the application.\nvRngPoisson\nGenerates Poisson distributed random values.\nSyntax\nstatus = viRngPoisson( method, stream, n, r, lambda );\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nmethod\nconst MKL_INT\nGeneration method. The specific values are as follows:\nVSL_RNG_METHOD_POISSON_PTPE\nVSL_RNG_METHOD_POISSON_POISNORM\nSee brief description of the PTPE and POISNORM methods in \nTable \"Values of <method> in method parameter\".\nstream\nVSLStreamStatePtr\nPointer to the stream state structure\nn\nconst MKL_INT\nNumber of random values to be generated\nlambda\nconst double\nDistribution parameter λ.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2203\n\n\nOutput Parameters\nName\nType\nDescription\nr\nint*\nVector of n Poisson distributed random values\nDescription\nThe vRng\"Poisson function generates Poisson distributed random numbers with distribution parameter λ,\nwhere λ∈R; λ > 0.\nThe probability distribution is given by:\nk∈ {0, 1, 2, ...}.\nThe cumulative distribution function is as follows:\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nReturn Values\nVSL_ERROR_OK, VSL_STATUS_OK\nIndicates no error, execution is successful.\nVSL_ERROR_NULL_PTR\nstream is a NULL pointer.\nVSL_RNG_ERROR_BAD_STREAM\nstream is not a valid random stream.\nVSL_RNG_ERROR_BAD_UPDATE\nCallback function for an abstract BRNG returns an invalid\nnumber of updated entries in a buffer, that is, < 0 or >\nnmax.\nVSL_RNG_ERROR_NO_NUMBERS\nCallback function for an abstract BRNG returns 0 as the\nnumber of updated entries in a buffer.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2204\n\n\nVSL_RNG_ERROR_QRNG_PERIOD_ELAPSED\nPeriod of the generator has been exceeded.\nVSL_RNG_ERROR_NONDETERM_NRETRIES_EXCEED\nED\nNumber of retries to generate a random number by using\nnon-deterministic random number generator exceeds\nthreshold.\nVSL_RNG_ERROR_ARS5_NOT_SUPPORTED\nARS-5 random number generator is not supported on the\nCPU running the application.\nvRngPoissonV\nGenerates Poisson distributed random values with\nvarying mean.\nSyntax\nstatus = viRngPoissonV( method, stream, n, r, lambda );\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nmethod\nconst MKL_INT\nGeneration method. The specific value is as follows:\nVSL_RNG_METHOD_POISSONV_POISNORM\nSee brief description of the POISNORM method in Table\n\"Values of <method> in method parameter\"\nstream\nVSLStreamStatePtr\nPointer to the stream state structure\nn\nconst MKL_INT\nNumber of random values to be generated\nlambda\nconst double*\nArray of n distribution parametersλi.\nOutput Parameters\nName\nType\nDescription\nr\nint*\nVector of n Poisson distributed random values\nDescription\nThe vRngPoissonV function generates n Poisson distributed random numbers xi(i = 1, ..., n) with distribution\nparameter λi, where λi∈R; λi > 0.\nThe probability distribution is given by:\nThe cumulative distribution function is as follows:\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2205\n\n\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nReturn Values\nVSL_ERROR_OK, VSL_STATUS_OK\nIndicates no error, execution is successful.\nVSL_ERROR_NULL_PTR\nstream is a NULL pointer.\nVSL_RNG_ERROR_BAD_STREAM\nstream is not a valid random stream.\nVSL_RNG_ERROR_BAD_UPDATE\nCallback function for an abstract BRNG returns an invalid\nnumber of updated entries in a buffer, that is, < 0 or >\nnmax.\nVSL_RNG_ERROR_NO_NUMBERS\nCallback function for an abstract BRNG returns 0 as the\nnumber of updated entries in a buffer.\nVSL_RNG_ERROR_QRNG_PERIOD_ELAPSED\nPeriod of the generator has been exceeded.\nVSL_RNG_ERROR_NONDETERM_NRETRIES_EXCEED\nED\nNumber of retries to generate a random number by using\nnon-deterministic random number generator exceeds\nthreshold.\nVSL_RNG_ERROR_ARS5_NOT_SUPPORTED\nARS-5 random number generator is not supported on the\nCPU running the application.\nvRngNegBinomial\nGenerates random numbers with negative binomial\ndistribution.\nSyntax\nstatus = viRngNegbinomial( method, stream, n, r, a, p );\nInclude Files\n•\nmkl.h\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2206\n\n\nInput Parameters\nName\nType\nDescription\nmethod\nconst MKL_INT\nGeneration method. The specific value is:\nVSL_RNG_METHOD_NEGBINOMIAL_NBAR\nSee brief description of the NBAR method in Table \"Values of\n<method> in method parameter\"\nstream\nVSLStreamStatePtr\npointer to the stream state structure\nn\nconst MKL_INT\nNumber of random values to be generated\na\nconst double\nThe first distribution parameter a\np\nconst double\nThe second distribution parameter p\nOutput Parameters\nName\nType\nDescription\nr\nint*\nVector of n random values with negative binomial\ndistribution.\nDescription\nThe vRngNegBinomial function generates random numbers with negative binomial distribution and\ndistribution parameters a and p, where p, a∈R; 0 < p < 1; a > 0.\nIf the first distribution parameter a∈N, this distribution is the same as Pascal distribution. If a∈N, the\ndistribution can be interpreted as the expected time of a-th success in a sequence of Bernoulli trials, when\nthe probability of success is p.\nThe probability distribution is given by:\nThe cumulative distribution function is as follows:\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2207\n\n\nReturn Values\nVSL_ERROR_OK, VSL_STATUS_OK\nIndicates no error, execution is successful.\nVSL_ERROR_NULL_PTR\nstream is a NULL pointer.\nVSL_RNG_ERROR_BAD_STREAM\nstream is not a valid random stream.\nVSL_RNG_ERROR_BAD_UPDATE\nCallback function for an abstract BRNG returns an invalid\nnumber of updated entries in a buffer, that is, < 0 or >\nnmax.\nVSL_RNG_ERROR_NO_NUMBERS\nCallback function for an abstract BRNG returns 0 as the\nnumber of updated entries in a buffer.\nVSL_RNG_ERROR_QRNG_PERIOD_ELAPSED\nPeriod of the generator has been exceeded.\nVSL_RNG_ERROR_ARS5_NOT_SUPPORTED\nARS-5 random number generator is not supported on the\nCPU running the application.\nvRngMultinomial\nGenerates multinomially distributed random numbers.\nSyntax\nstatus = viRngMultinomial( method, stream, n, r, ntrial, k, p );\nInclude Files\n•\nmkl.h\nInput Parameters\nmethod\nconst MKL_INT\nGeneration method. The specific value is as follows:\nVSL_RNG_METHOD_MULTINOMIAL_MULTPOISSON\nstream\nVSLStreamStatePtr\nPointer to the stream state structure.\nn\nconst MKL_INT\nNumber of random values to be generated.\nntrial\nconst int\nNumber of independent trials m.\nk\nconst int\nNumber of possible outcomes.\np\nconst double*\nProbability vector of k possible outcomes.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2208\n\n\nOutput Parameters\nr\nint*\nArray of n random vectors of dimension k.\nDescription\nThe vRngMultinomial function generates multinomially distributed random numbers with m independent\ntrials and k possible mutually exclusive outcomes, with corresponding probabilities pi, where pi∈R; 0 ≤pi≤\n1, m∈N, k∈N.\nThe probability distribution is given by:\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nReturn Values\nVSL_ERROR_OK,\nVSL_STATUS_OK\nIndicates no error (execution was successful).\nVSL_ERROR_NULL_PTR\nstream is a NULL pointer.\nVSL_RNG_ERROR_BAD_STR\nEAM\nstream is not a valid random stream.\nVSL_RNG_ERROR_BAD_UPD\nATE\nA callback function for an abstract BRNG returns an invalid number of updated\nentries in a buffer; that is, < 0 or > nmax.\nVSL_RNG_ERROR_NO_NUMB\nERS\nA callback function for an abstract BRNG returns 0 as the number of updated\nentries in a buffer.\nVSL_RNG_ERROR_ARS5_NO\nT_SUPPORTED\nAn ARS-5 random number generator is not supported on the CPU running the\napplication.\nVSL_DISTR_MULTINOMIAL\n_BAD_PROBABILITY_ARRA\nY\nBad multinomial distribution probability array.\nAdvanced Service Routines\nThis section describes service routines for registering a user-designed basic generator (vslRegisterBrng)\nand for obtaining properties of the previously registered basic generators (vslGetBrngProperties). See VS\nNotes (\"Basic Generators\" section of VS Structure chapter) for substantiation of the need for several basic\ngenerators including user-defined BRNGs.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2209\n\n\nAdvanced Service Routine Data Types\nThe Advanced Service routines refer to a structure defining the properties of the basic generator.\nThis structure is described as follows:\ntypedef struct _VSLBRngProperties {\n       int StreamStateSize;\n       int NSeeds;\n       int IncludesZero;\n       int WordSize;\n       int NBits;\n       InitStreamPtr InitStream;\n       sBRngPtr sBRng;\n       dBRngPtr dBRng;\n       iBRngPtr iBRng; \n} VSLBRngProperties;\nThe following table provides brief descriptions of the fields engaged in the above structure:\nField Descriptions\nField\nShort Description\nStreamStateSize\nThe size, in bytes, of the stream state structure for a given basic\ngenerator.\nNSeeds\nThe number of 32-bit initial conditions (seeds) necessary to initialize\nthe stream state structure for a given basic generator.\nIncludesZero\nFlag value indicating whether the generator can produce a random\n0.\nWordSize\nMachine word size, in bytes, used in integer-value computations.\nPossible values: 4, 8, and 16 for 32, 64, and 128-bit generators,\nrespectively.\nNBits\nThe number of bits required to represent a random value in integer\narithmetic. Note that, for instance, 48-bit random values are stored\nto 64-bit (8 byte) memory locations. In this case, wordsize/\nWordSize is equal to 8 (number of bytes used to store the random\nvalue), while nbits/NBits contains the actual number of bits\noccupied by the value (in this example, 48).\nInitStream\nContains the pointer to the initialization routine of a given basic\ngenerator.\nsBRng\nContains the pointer to the basic generator of single precision real\nnumbers uniformly distributed over the interval (a,b) (float).\ndBRng\nContains the pointer to the basic generator of double precision real\nnumbers uniformly distributed over the interval (a,b) (double).\niBRng\nContains the pointer to the basic generator of integer numbers with\nuniform bit distribution (unsigned int).\nvslRegisterBrng\nRegisters user-defined basic generator.\nSyntax\nbrng = vslRegisterBrng( &properties );\n1 A specific generator that permits operations over single bits and bit groups of random numbers.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2210\n\n\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\npropertie\ns\nconst VSLBRngProperties*\nPointer to the structure containing properties of the basic\ngenerator to be registered\nOutput Parameters\nName\nType\nDescription\nbrng\nint\nNumber (index) of the registered basic generator; used for\nidentification. Negative values indicate the registration\nerror.\nDescription\nAn example of a registration procedure can be found in the respective directory of the VS examples.\nReturn Values\nVSL_ERROR_OK, VSL_STATUS_OK\nIndicates no error, execution is successful.\nVSL_RNG_ERROR_BRNG_TABLE_FULL\nRegistration cannot be completed due to lack of free entries\nin the table of registered BRNGs.\nVSL_RNG_ERROR_BAD_STREAM_STATE_SIZE\nBad value in StreamStateSize field.\nVSL_RNG_ERROR_BAD_WORD_SIZE\nBad value in WordSize field.\nVSL_RNG_ERROR_BAD_NSEEDS\nBad value in NSeeds field.\nVSL_RNG_ERROR_BAD_NBITS\nBad value in NBits field.\nVSL_ERROR_NULL_PTR\nAt least one of the fields iBrng, dBrng, sBrng or\nInitStream is a NULL pointer.\nvslGetBrngProperties\nReturns structure with properties of a given basic\ngenerator.\nSyntax\nstatus = vslGetBrngProperties( brng, &properties );\nInclude Files\n•\nmkl.h\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2211\n\n\nInput Parameters\nName\nType\nDescription\nbrng\nconst int\nNumber (index) of the registered basic generator; used for\nidentification. See specific values in Table \"Values of brng\nparameter\". Negative values indicate the registration error.\nOutput Parameters\nName\nType\nDescription\npropertie\ns\nVSLBRngProperties*\nPointer to the structure containing properties of the\ngenerator with number brng\nDescription\nThe vslGetBrngProperties function returns a structure with properties of a given basic generator.\nReturn Values\nVSL_ERROR_OK, VSL_STATUS_OK\nIndicates no error, execution is successful.\nVSL_RNG_ERROR_INVALID_BRNG_INDEX\nBRNG index is invalid.\nFormats for User-Designed Generators\nTo register a user-designed basic generator using vslRegisterBrng function, you need to pass the pointer\niBrng to the integer-value implementation of the generator; the pointers sBrng and dBrng to the generator\nimplementations for single and double precision values, respectively; and pass the pointer InitStream to\nthe stream initialization routine. See recommendations below on defining such functions with input and\noutput arguments. An example of the registration procedure for a user-designed generator can be found in\nthe respective directory of VS examples.\nThe respective pointers are defined as follows:\ntypedef int(*InitStreamPtr)( int method, VSLStreamStatePtr stream, int n, const unsigned int \nparams[] );\ntypedef int(*sBRngPtr)( VSLStreamStatePtr stream, int n, float r[], float a, float b );\ntypedef int(*dBRngPtr)( VSLStreamStatePtr stream, int n, double r[], double a, double b );\ntypedef int(*iBRngPtr)( VSLStreamStatePtr stream, int n, unsigned int r[] );\nInitStream\nint MyBrngInitStream( int method, VSLStreamStatePtr stream, int n, const unsigned int params[] ) \n{\n      /* Initialize the stream */\n      ... \n} /* MyBrngInitStream */\nDescription\nThe initialization routine of a user-designed generator must initialize stream according to the specified\ninitialization method, initial conditions params and the argument n. The value of method determines the\ninitialization method to be used.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2212\n\n\n•\nIf method is equal to 1, the initialization is by the standard generation method, which must be supported\nby all basic generators. In this case the function assumes that the stream structure was not previously\ninitialized. The value of n is used as the actual number of 32-bit values passed as initial conditions\nthrough params. Note, that the situation when the actual number of initial conditions passed to the\nfunction is not sufficient to initialize the generator is not an error. Whenever it occurs, the basic generator\nmust initialize the missing conditions using default settings.\n•\nIf method is equal to 2, the generation is by the leapfrog method, where n specifies the number of\ncomputational nodes (independent streams). Here the function assumes that the stream was previously\ninitialized by the standard generation method. In this case params contains only one element, which\nidentifies the computational node. If the generator does not support the leapfrog method, the function\nmust return the error code VSL_RNG_ERROR_LEAPFROG_UNSUPPORTED.\n•\nIf method is equal to 3, the generation is by the block-splitting method. Same as above, the stream is\nassumed to be previously initialized by the standard generation method; params is not used, n identifies\nthe number of skipped elements. If the generator does not support the block-splitting method, the\nfunction must return the error code VSL_RNG_ERROR_SKIPAHEAD_UNSUPPORTED.\n•\nIf method is equal to 4, the generation is by the advanced block-splitting method. The stream is assumed\nto be previously initialized by the standard generation method; params is converted to MKL_UINT64[]\nand n is used as actual number of 64-bit values in params. If the generator does not support the\nadvanced block-splitting method, the function must return the error code\nVSL_RNG_ERROR_SKIPAHEADEX_UNSUPPORTED.\nFor a more detailed description of the leapfrog and the block-splitting methods, refer to the description of \nvslLeapfrogStream, vslSkipAheadStream, and vslSkipAheadStreamEx, respectively.\nStream state structure is individual for every generator. However, each structure has a number of fields that\nare the same for all the generators:\ntypedef struct \n{\n    unsigned int Reserved1[2];\n    unsigned int Reserved2[2];\n    [fields specific for the given generator] \n} MyStreamState;\nThe fields Reserved1 and Reserved2 are reserved for private needs only, and must not be modified by the\nuser. When including specific fields into the structure, follow the rules below:\n•\nThe fields must fully describe the current state of the generator. For example, the state of a linear\ncongruential generator can be identified by only one initial condition;\n•\nIf the generator can use both the leapfrog and the block-splitting methods, additional fields should be\nintroduced to identify the independent streams. For example, in LCG(a, c, m), apart from the initial\nconditions, two more fields should be specified: the value of the multiplier ak and the value of the\nincrement (ak-1)c/(a-1).\nFor a more detailed discussion, refer to [Knuth81], and [Gentle98]. An example of the registration procedure\ncan be found in the respective directory of VS examples.\niBRng\nint iMyBrng( VSLStreamStatePtr stream, int n, unsigned int r[] ) \n{\n    int i; /* Loop variable */\n    /* Generating integer random numbers */\n    /* Pay attention to word size needed to\n    store only random number */\n    for( i = 0; i < n; i++)\n    {\n        r[i] = ...;\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2213\n\n\n    }\n    /* Update stream state */\n    ...\n    return errcode; \n} /* iMyBrng */\nNOTE\nWhen using 64 and 128-bit generators, consider digit capacity to store the numbers to the random\nvector r correctly. For example, storing one 64-bit value requires two elements of r, the first to store\nthe lower 32 bits and the second to store the higher 32 bits. Similarly, use 4 elements of r to store a\n128-bit value.\nsBRng\nint sMyBrng( VSLStreamStatePtr stream, int n, float r[], float a, float b ) \n{\n    int   i;    /* Loop variable */\n    /* Generating float (a,b) random numbers */\n    for ( i = 0; i < n; i++ )\n    {\n        r[i] = ...;\n    }\n    /* Update stream state */\n    ...\n    return errcode; \n} /* sMyBrng */\ndBRng\nint dMyBrng( VSLStreamStatePtr stream, int n, double r[], double a, double b ) \n{\n    int i;   /* Loop variable */\n    /* Generating double (a,b) random numbers */\n    for ( i = 0; i < n; i++ )\n    {\n        r[i] = ...;\n    }\n    /* Update stream state */\n    ...\n    return errcode; \n} /* dMyBrng */\nConvolution and Correlation\nIntel® oneAPI Math Kernel Library (oneMKL) VS provides a set of routines intended to perform linear\nconvolution and correlation transformations for single and double precision real and complex data.\nFor correct definition of implemented operations, see the Mathematical Notation and Definitions.\nThe current implementation provides:\n•\nFourier algorithms for one-dimensional single and double precision real and complex data\n•\nFourier algorithms for multi-dimensional single and double precision real and complex data\n•\nDirect algorithms for one-dimensional single and double precision real and complex data\n•\nDirect algorithms for multi-dimensional single and double precision real and complex data\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2214\n\n\nOne-dimensional algorithms cover the following functions from the IBM* ESSL library:\nSCONF, SCORF\nSCOND, SCORD\nSDCON, SDCOR\nDDCON, DDCOR\nSDDCON, SDDCOR.\nSpecial wrappers are designed to simulate these ESSL functions. The wrappers are provided as sample\nsources:\n${MKL}/examples/vslc/essl/vsl_wrappers\nAdditionally, you can browse the examples demonstrating the calculation of the ESSL functions through the\nwrappers:\n${MKL}/examples/vslc/essl\nThe convolution and correlation API provides interfaces for Fortran 90 and C/89 languages. You can use the\nC89 interface with later versions of the C/C++.\nIntel® oneAPI Math Kernel Library (oneMKL) providesthe mkl_vsl.h header file. All header files are in the\ndirectory\n${MKL}/include\nThe convolution and correlation API is implemented through task objects, or tasks. Task object is a data\nstructure, or descriptor, which holds parameters that determine the specific convolution or correlation\noperation. Such parameters may be precision, type, and number of dimensions of user data, an identifier of\nthe computation algorithm to be used, shapes of data arrays, and so on.\nAll the Intel® oneAPI Math Kernel Library (oneMKL) VS convolution and correlation routines process task\nobjects in one way or another: either create a new task descriptor, change the parameter settings, compute\nmathematical results of the convolution or correlation using the stored parameters, or perform other\noperations. Accordingly, all routines are split into the following groups:\nTask Constructors - routines that create a new task object descriptor and set up most common parameters.\nTask Editors - routines that can set or modify some parameter settings in the existing task descriptor.\nTask Execution Routines - compute results of the convolution or correlation operation over the actual input\ndata, using the operation parameters held in the task descriptor.\nTask Copy - routines used to make several copies of the task descriptor.\nTask Destructors - routines that delete task objects and free the memory.\nWhen the task is executed or copied for the first time, a special process runs which is called task\ncommitment. During this process, consistency of task parameters is checked and the required work data are\nprepared. If the parameters are consistent, the task is tagged as committed successfully. The task remains\ncommitted until you edit its parameters. Hence, the task can be executed multiple times after a single\ncommitment process. Since the task commitment process may include costly intermediate calculations such\nas preparation of Fourier transform of input data, launching the process only once can help speed up overall\nperformance.\nConvolution and Correlation Naming Conventions\nThe names of routines, types, and constants in the convolution and correlation API are case-sensitive and\ncan contain both lowercase and uppercase characters (vslsConvExec).\nThe names of routines have the following structure:\nvsl[datatype]{Conv|Corr}<base name>   \nwhere\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2215\n\n\n•\nvsl is a prefix indicating that the routine belongs to Intel® MKL Vector Statistics.\n•\n[datatype] is optional. If present, the symbol specifies the type of the input and output data and can be\ns (for single precision real type), d (for double precision real type), c (for single precision complex type),\nor z (for double precision complex type).\n•\nConv or Corr specifies whether the routine refers to convolution or correlation task, respectively.\n•\n<base name> field specifies a particular functionality that the routine is designed for, for example,\nNewTask, DeleteTask.\nConvolution and Correlation Data Types\nAll convolution or correlation routines use the following types for specifying data objects:\nType\nData Object\nVSLConvTaskPtr\nPointer to a task descriptor for convolution\nVSLCorrTaskPtr\nPointer to a task descriptor for correlation\nfloat\nInput/output user real data in single precision\ndouble\nInput/output user real data in double precision\nMKL_Complex8\nInput/output user complex data in single precision\nMKL_Complex16\nInput/output user complex data in double precision\nint\nAll other data\nGeneric integer type (without specifying the byte size) is used for all integer data.\nNOTE\nThe actual size of the generic integer type is platform-dependent. Before you compile your application,\nset an appropriate byte size for integers. See details in the 'Using the ILP64 Interface vs. LP64\nInterface' section of the Intel® oneAPI Math Kernel Library (oneMKL) Developer Guide.\nConvolution and Correlation Parameters\nBasic parameters held by the task descriptor are assigned values when the task object is created, copied, or\nmodified by task editors. Parameters of the correlation or convolution task are initially set up by task\nconstructors when the task object is created. Parameter changes or additional settings are made by task\neditors. More parameters which define location of the data being convolved need to be specified when the\ntask execution routine is invoked.\nAccording to how the parameters are passed or assigned values, all of them can be categorized as either\nexplicit (directly passed as routine parameters when a task object is created or executed) or optional\n(assigned some default or implicit values during task construction).\nThe following table lists all applicable parameters used in the Intel® oneAPI Math Kernel Library (oneMKL)\nconvolution and correlation API.\nConvolution and Correlation Task Parameters\nName\nCategory\nType\nDefault Value\nLabel\nDescription\njob\nexplicit\ninteger\nImplied by the\nconstructor\nname\nSpecifies whether the task relates to\nconvolution or correlation\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2216\n\n\nName\nCategory\nType\nDefault Value\nLabel\nDescription\ntype\nexplicit\ninteger\nImplied by the\nconstructor\nname\nSpecifies the type (real or complex) of the\ninput/output data. Set to real in the current\nversion.\nprecision\nexplicit\ninteger\nImplied by the\nconstructor\nname\nSpecifies precision (single or double) of the\ninput/output data to be provided in arrays\nx,y,z.\nmode\nexplicit\ninteger\nNone\nSpecifies whether the convolution/\ncorrelation computation should be done via\nFourier transforms, or by a direct method,\nor by automatically choosing between the\ntwo. See SetMode for the list of named\nconstants for this parameter.\nmethod\noptional\ninteger\n\"auto\"\nHints at a particular computation method if\nseveral methods are available for the given\nmode. Setting this parameter to \"auto\"\nmeans that software will choose the best\navailable method.\ninternal_pre\ncision\noptional\ninteger\nSet equal to the\nvalue of\nprecision\nSpecifies precision of internal calculations.\nCan enforce double precision calculations\neven when input/output data are single\nprecision. See SetInternalPrecision for\nthe list of named constants for this\nparameter.\ndims\nexplicit\ninteger\nNone\nSpecifies the rank (number of dimensions)\nof the user data provided in arrays x,y,z.\nCan be in the range from 1 to 7.\nx,y\nexplicit\nreal\narrays\nNone\nSpecify input data arrays. See Data\nAllocation for more information.\nz\nexplicit\nreal\narray\nNone\nSpecifies output data array. See Data\nAllocation for more information.\nxshape,\nyshape,\nzshape\nexplicit\ninteger\narrays\nNone\nDefine shapes of the arrays x, y, z. See \nData Allocation for more information.\nxstride,\nystride,\nzstride\nexplicit\ninteger\narrays\nNone\nDefine strides within arrays x, y, z, that is\nspecify the physical location of the input\nand output data in these arrays. See Data\nAllocation for more information.\nstart\noptional\ninteger\narray\nUndefined\nDefines the first element of the\nmathematical result that will be stored to\noutput array z. See SetStart and Data\nAllocation for more information.\ndecimation\noptional\ninteger\narray\nUndefined\nDefines how to thin out the mathematical\nresult that will be stored to output array z.\nSee SetDecimation and Data Allocation for\nmore information.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2217\n\n\nUsers may pass the NULL pointer instead of either or all of the parameters xstride, ystride, or zstride\nfor multi-dimensional calculations. In this case, the software assumes the dense data allocation for the\narrays x, y, or z due to the Fortran-style \"by columns\" representation of multi-dimensional arrays.\nConvolution and Correlation Task Status and Error Reporting\nThe task status is an integer value, which is zero if no error has been detected while processing the task, or a\nspecific non-zero error code otherwise. Negative status values indicate errors, and positive values indicate\nwarnings.\nAn error can be caused by invalid parameter values, a system fault like a memory allocation failure, or can\nbe an internal error self-detected by the software.\nEach task descriptor contains the current status of the task. When creating a task object, the constructor\nassigns the VSL_STATUS_OK status to the task. When processing the task afterwards, other routines such as\neditors or executors can change the task status if an error occurs and write a corresponding error code into\nthe task status field.\nNote that at the stage of creating a task or editing its parameters, the set of parameters may be\ninconsistent. The parameter consistency check is only performed during the task commitment operation,\nwhich is implicitly invoked before task execution or task copying. If an error is detected at this stage, task\nexecution or task copying is terminated and the task descriptor saves the corresponding error code. Once an\nerror occurs, any further attempts to process that task descriptor is terminated and the task keeps the same\nerror code.\nNormally, every convolution or correlation function (except DeleteTask) returns the status assigned to the\ntask while performing the function operation.\nThe header files define symbolic names for the status codes. These names are defined as macros via the\n#define statements.\nIf there is no error, the VSL_STATUS_OK status is returned, which is defined as zero:\n#define VSL_STATUS_OK 0\nIn case of an error, a non-zero error code is returned, which indicates the origin of the failure. The following\nstatus codes for the convolution/correlation error codes are pre-defined in the header files.\nConvolution/Correlation Status Codes\nStatus Code\nDescription\nVSL_CC_ERROR_NOT_IMPLEMENTED\nRequested functionality is not implemented.\nVSL_CC_ERROR_ALLOCATION_FAILURE\nMemory allocation failure.\nVSL_CC_ERROR_BAD_DESCRIPTOR\nTask descriptor is corrupted.\nVSL_CC_ERROR_SERVICE_FAILURE\nA service function has failed.\nVSL_CC_ERROR_EDIT_FAILURE\nFailure while editing the task.\nVSL_CC_ERROR_EDIT_PROHIBITED\nYou cannot edit this parameter.\nVSL_CC_ERROR_COMMIT_FAILURE\nTask commitment has failed.\nVSL_CC_ERROR_COPY_FAILURE\nFailure while copying the task.\nVSL_CC_ERROR_DELETE_FAILURE\nFailure while deleting the task.\nVSL_CC_ERROR_BAD_ARGUMENT\nBad argument or task parameter.\nVSL_CC_ERROR_JOB\nBad parameter: job.\nSL_CC_ERROR_KIND\nBad parameter: kind.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2218\n\n\nStatus Code\nDescription\nVSL_CC_ERROR_MODE\nBad parameter: mode.\nVSL_CC_ERROR_METHOD\nBad parameter: method.\nVSL_CC_ERROR_TYPE\nBad parameter: type.\nVSL_CC_ERROR_EXTERNAL_PRECISION\nBad parameter: external_precision.\nVSL_CC_ERROR_INTERNAL_PRECISION\nBad parameter: internal_precision.\nVSL_CC_ERROR_PRECISION\nIncompatible external/internal precisions.\nVSL_CC_ERROR_DIMS\nBad parameter: dims.\nVSL_CC_ERROR_XSHAPE\nBad parameter: xshape.\nVSL_CC_ERROR_YSHAPE\nBad parameter: yshape.\nCallback function for an abstract BRNG returns an invalid\nnumber of updated entries in a buffer, that is, < 0 or\n>nmax.\nVSL_CC_ERROR_ZSHAPE\nBad parameter: zshape.\nVSL_CC_ERROR_XSTRIDE\nBad parameter: xstride.\nVSL_CC_ERROR_YSTRIDE\nBad parameter: ystride.\nVSL_CC_ERROR_ZSTRIDE\nBad parameter: zstride.\nVSL_CC_ERROR_X\nBad parameter: x.\nVSL_CC_ERROR_Y\nBad parameter: y.\nVSL_CC_ERROR_Z\nBad parameter: z.\nVSL_CC_ERROR_START\nBad parameter: start.\nVSL_CC_ERROR_DECIMATION\nBad parameter: decimation.\nVSL_CC_ERROR_OTHER\nAnother error.\nConvolution and Correlation Task Constructors\nTask constructors are routines intended for creating a new task descriptor and setting up basic parameters.\nNo additional parameter adjustment is typically required and other routines can use the task object.\nIntel® MKL implementation of the convolution and correlation API provides two different forms of\nconstructors: a general form and an X-form. X-form constructors work in the same way as the general form\nconstructors but also assign particular data to the first operand vector used in the convolution or correlation\noperation (stored in array x).\nUsing X-form constructors is recommended when you need to compute multiple convolutions or correlations\nwith the same data vector held in array x against different vectors held in array y. This helps improve\nperformance by eliminating unnecessary overhead in repeated computation of intermediate data required for\nthe operation.\nEach constructor routine has an associated one-dimensional version that provides algorithmic and\ncomputational benefits.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2219\n\n\nNOTE\nIf the constructor fails to create a task descriptor, it returns the NULL task pointer.\nThe Table \"Task Constructors\" lists available task constructors:\nTask Constructors\nRoutine\nDescription\nvslConvNewTask/vslCorrNewTask\nCreates a new convolution or correlation task descriptor for a\nmultidimensional case.\nvslConvNewTask1D/\nvslCorrNewTask1D\nCreates a new convolution or correlation task descriptor for a\none-dimensional case.\nvslConvNewTaskX/vslCorrNewTaskX\nCreates a new convolution or correlation task descriptor as an\nX-form for a multidimensional case.\nvslConvNewTaskX1D/\nvslCorrNewTaskX1D\nCreates a new convolution or correlation task descriptor as an\nX-form for a one-dimensional case.\nvslConvNewTask/vslCorrNewTask\nCreates a new convolution or correlation task\ndescriptor for multidimensional case.\nSyntax\nstatus = vslsConvNewTask(task, mode, dims, xshape, yshape, zshape);\nstatus = vsldConvNewTask(task, mode, dims, xshape, yshape, zshape);\nstatus = vslcConvNewTask(task, mode, dims, xshape, yshape, zshape);\nstatus = vslzConvNewTask(task, mode, dims, xshape, yshape, zshape);\nstatus = vslsCorrNewTask(task, mode, dims, xshape, yshape, zshape);\nstatus = vsldCorrNewTask(task, mode, dims, xshape, yshape, zshape);\nstatus = vslcCorrNewTask(task, mode, dims, xshape, yshape, zshape);\nstatus = vslzCorrNewTask(task, mode, dims, xshape, yshape, zshape);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nmode\nconst MKL_INT\nSpecifies whether convolution/correlation calculation must\nbe performed by using a direct algorithm or through Fourier\ntransform of the input data. See Table \"Values of mode\nparameter\" for a list of possible values.\ndims\nconst MKL_INT\nRank of user data. Specifies number of dimensions for the\ninput and output arrays x, y, and z used during the\nexecution stage. Must be in the range from 1 to 7. The\nvalue is explicitly assigned by the constructor.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2220\n\n\nName\nType\nDescription\nxshape\nconst int[]\nDefines the shape of the input data for the source array x.\nSee Data Allocation for more information.\nyshape\nconst int[]\nDefines the shape of the input data for the source array y.\nSee Data Allocation for more information.\nzshape\nconst int[]\nDefines the shape of the output data to be stored in array\nz. See Data Allocation for more information.\nOutput Parameters\nName\nType\nDescription\ntask\nVSLConvTaskPtr* for\nvslsConvNewTask,\nvsldConvNewTask,\nvslcConvNewTask,\nvslzConvNewTask\nVSLCorrTaskPtr* for\nvslsCorrNewTask,\nvsldCorrNewTask,\nvslcConvNewTask,\nvslzConvNewTask\nPointer to the task descriptor if created successfully or NULL\npointer otherwise.\nstatus\nint\nSet to VSL_STATUS_OK if the task is created successfully or\nset to non-zero error code otherwise.\nDescription\nEach vslConvNewTask/vslCorrNewTask constructor creates a new convolution or correlation task descriptor\nwith the user specified values for explicit parameters. The optional parameters are set to their default values\n(see Table \"Convolution and Correlation Task Parameters\").\nThe parameters xshape, yshape, and zshape define the shapes of the input and output data provided by\nthe arrays x, y, and z, respectively. Each shape parameter is an array of integers with its length equal to the\nvalue of dims. You explicitly assign the shape parameters when calling the constructor. If the value of the\nparameter dims is 1, then xshape, yshape, zshape are equal to the number of elements read from the\narrays x and y or stored to the array z. Note that values of shape parameters may differ from physical\nshapes of arrays x, y, and z if non-trivial strides are assigned.\nIf the constructor fails to create a task descriptor, it returns a NULL task pointer.\nvslConvNewTask1D/vslCorrNewTask1D\nCreates a new convolution or correlation task\ndescriptor for one-dimensional case.\nSyntax\nstatus = vslsConvNewTask1D(task, mode, xshape, yshape, zshape);\nstatus = vsldConvNewTask1D(task, mode, xshape, yshape, zshape);\nstatus = vslcConvNewTask1D(task, mode, xshape, yshape, zshape);\nstatus = vslzConvNewTask1D(task, mode, xshape, yshape, zshape);\nstatus = vslsCorrNewTask1D(task, mode, xshape, yshape, zshape);\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2221\n\n\nstatus = vsldCorrNewTask1D(task, mode, xshape, yshape, zshape);\nstatus = vslcCorrNewTask1D(task, mode, xshape, yshape, zshape);\nstatus = vslzCorrNewTask1D(task, mode, xshape, yshape, zshape);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nmode\nconst MKL_INT\nSpecifies whether convolution/correlation calculation must\nbe performed by using a direct algorithm or through Fourier\ntransform of the input data. See Table \"Values of mode\nparameter\" for a list of possible values.\nxshape\nconst MKL_INT\nDefines the length of the input data sequence for the\nsource array x. See Data Allocation for more information.\nyshape\nconst MKL_INT\nDefines the length of the input data sequence for the\nsource array y. See Data Allocation for more information.\nzshape\nconst MKL_INT\nDefines the length of the output data sequence to be stored\nin array z. See Data Allocation for more information.\nOutput Parameters\nName\nType\nDescription\ntask\nVSLConvTaskPtr* for\nvslsConvNewTask1D,\nvsldConvNewTask1D,\nvslcConvNewTask1D,\nvslzConvNewTask1D\nVSLCorrTaskPtr* for\nvslsCorrNewTask1D,\nvsldCorrNewTask1D,\nvslcCorrNewTask1D,\nvslzCorrNewTask1D\nPointer to the task descriptor if created successfully or NULL\npointer otherwise.\nstatus\nint\nSet to VSL_STATUS_OK if the task is created successfully or\nset to non-zero error code otherwise.\nDescription\nEach vslConvNewTask1D/vslCorrNewTask1D constructor creates a new convolution or correlation task\ndescriptor with the user specified values for explicit parameters. The optional parameters are set to their\ndefault values (see Table \"Convolution and Correlation Task Parameters\"). Unlike vslConvNewTask/\nvslCorrNewTask, these routines represent a special one-dimensional version of the constructor which\nassumes that the value of the parameter dims is 1. The parameters xshape, yshape, and zshape are equal\nto the number of elements read from the arrays x and y or stored to the array z. You explicitly assign the\nshape parameters when calling the constructor.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2222\n\n\nvslConvNewTaskX/vslCorrNewTaskX\nCreates a new convolution or correlation task\ndescriptor for multidimensional case and assigns\nsource data to the first operand vector.\nSyntax\nstatus = vslsConvNewTaskX(task, mode, dims, xshape, yshape, zshape, x, xstride);\nstatus = vsldConvNewTaskX(task, mode, dims, xshape, yshape, zshape, x, xstride);\nstatus = vslcConvNewTaskX(task, mode, dims, xshape, yshape, zshape, x, xstride);\nstatus = vslzConvNewTaskX(task, mode, dims, xshape, yshape, zshape, x, xstride);\nstatus = vslsCorrNewTaskX(task, mode, dims, xshape, yshape, zshape, x, xstride);\nstatus = vsldCorrNewTaskX(task, mode, dims, xshape, yshape, zshape, x, xstride);\nstatus = vslcCorrNewTaskX(task, mode, dims, xshape, yshape, zshape, x, xstride);\nstatus = vslzCorrNewTaskX(task, mode, dims, xshape, yshape, zshape, x, xstride);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nmode\nconst MKL_INT\nSpecifies whether convolution/correlation calculation must\nbe performed by using a direct algorithm or through Fourier\ntransform of the input data. See Table \"Values of mode\nparameter\" for a list of possible values.\ndims\nconst MKL_INT\nRank of user data. Specifies number of dimensions for the\ninput and output arrays x, y, and z used during the\nexecution stage. Must be in the range from 1 to 7. The\nvalue is explicitly assigned by the constructor.\nxshape\nconst int[]\nDefines the shape of the input data for the source array x.\nSee Data Allocation for more information.\nyshape\nconst int[]\nDefines the shape of the input data for the source array y.\nSee Data Allocation for more information.\nzshape\nconst int[]\nDefines the shape of the output data to be stored in array\nz.See Data Allocation for more information.\nx\nconst float[] for real data in\nsingle precision flavors,\nconst double[] for real data\nin double precision flavors,\nconst MKL_Complex8[] for\ncomplex data in single precision\nflavors,\nconst MKL_Complex16[] for\ncomplex data in double precision\nflavors\nPointer to the array containing input data for the first\noperand vector.See Data Allocation for more information.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2223\n\n\nName\nType\nDescription\nxstride\nconst int[]\nStrides for input data in the array x.\nOutput Parameters\nName\nType\nDescription\ntask\nVSLConvTaskPtr* for\nvslsConvNewTaskX,\nvsldConvNewTaskX,\nvslcConvNewTaskX,\nvslzConvNewTaskX\nVSLCorrTaskPtr* for\nvslsCorrNewTaskX,\nvsldCorrNewTaskX,\nvslcCorrNewTaskX,\nvslzCorrNewTaskX\nPointer to the task descriptor if created successfully or NULL\npointer otherwise.\nstatus\nint\nSet to VSL_STATUS_OK if the task is created successfully or\nset to non-zero error code otherwise.\nDescription\nEach vslConvNewTaskX/vslCorrNewTaskX constructor creates a new convolution or correlation task\ndescriptor with the user specified values for explicit parameters. The optional parameters are set to their\ndefault values (see Table \"Convolution and Correlation Task Parameters\").\nUnlike vslConvNewTask/vslCorrNewTask, these routines represent the so called X-form version of the\nconstructor, which means that in addition to creating the task descriptor they assign particular data to the\nfirst operand vector in array x used in convolution or correlation operation. The task descriptor created by the\nvslConvNewTaskX/vslCorrNewTaskX constructor keeps the pointer to the array x all the time, that is, until\nthe task object is deleted by one of the destructor routines (see vslConvDeleteTask/vslCorrDeleteTask).\nUsing this form of constructors is recommended when you need to compute multiple convolutions or\ncorrelations with the same data vector in array x against different vectors in array y. This helps improve\nperformance by eliminating unnecessary overhead in repeated computation of intermediate data required for\nthe operation.\nThe parameters xshape, yshape, and zshape define the shapes of the input and output data provided by\nthe arrays x, y, and z, respectively. Each shape parameter is an array of integers with its length equal to the\nvalue of dims. You explicitly assign the shape parameters when calling the constructor. If the value of the\nparameter dims is 1, then xshape, yshape, and zshape are equal to the number of elements read from the\narrays x and y or stored to the array z. Note that values of shape parameters may differ from physical\nshapes of arrays x, y, and z if non-trivial strides are assigned.\nThe stride parameter xstride specifies the physical location of the input data in the array x. In a one-\ndimensional case, stride is an interval between locations of consecutive elements of the array. For example, if\nthe value of the parameter xstride is s, then only every sth element of the array x will be used to form the\ninput sequence. The stride value must be positive or negative but not zero.\nvslConvNewTaskX1D/vslCorrNewTaskX1D\nCreates a new convolution or correlation task\ndescriptor for one-dimensional case and assigns\nsource data to the first operand vector.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2224\n\n\nSyntax\nstatus = vslsConvNewTaskX1D(task, mode, xshape, yshape, zshape, x, xstride);\nstatus = vsldConvNewTaskX1D(task, mode, xshape, yshape, zshape, x, xstride);\nstatus = vslcConvNewTaskX1D(task, mode, xshape, yshape, zshape, x, xstride);\nstatus = vslzConvNewTaskX1D(task, mode, xshape, yshape, zshape, x, xstride);\nstatus = vslsCorrNewTaskX1D(task, mode, xshape, yshape, zshape, x, xstride);\nstatus = vsldCorrNewTaskX1D(task, mode, xshape, yshape, zshape, x, xstride);\nstatus = vslcCorrNewTaskX1D(task, mode, xshape, yshape, zshape, x, xstride);\nstatus = vslzCorrNewTaskX1D(task, mode, xshape, yshape, zshape, x, xstride);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nmode\nconst MKL_INT\nSpecifies whether convolution/correlation calculation must\nbe performed by using a direct algorithm or through Fourier\ntransform of the input data. See Table \"Values of mode\nparameter\" for a list of possible values.\nxshape\nconst MKL_INT\nDefines the length of the input data sequence for the\nsource array x. See Data Allocation for more information.\nyshape\nconst MKL_INT\nDefines the length of the input data sequence for the\nsource array y. See Data Allocation for more information.\nzshape\nconst MKL_INT\nDefines the length of the output data sequence to be stored\nin array z. See Data Allocation for more information.\nx\nconst float[] for real data in\nsingle precision flavors,\nconst double[] for real data\nin double precision flavors,\nconst MKL_Complex8[] for\ncomplex data in single precision\nflavors,\nconst MKL_Complex16[] for\ncomplex data in double precision\nflavors\nPointer to the array containing input data for the first\noperand vector. See Data Allocation for more information.\nxstride\nconst MKL_INT\nStride for input data sequence in the arrayx.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2225\n\n\nOutput Parameters\nName\nType\nDescription\ntask\nVSLConvTaskPtr* for\nvslsConvNewTaskX1D,\nvsldConvNewTaskX1D,\nvslcConvNewTaskX1D,\nvslzConvNewTaskX1D\nVSLCorrTaskPtr* for\nvslsCorrNewTaskX1D,\nvsldCorrNewTaskX1D,\nvslcCorrNewTaskX1D,\nvslzCorrNewTaskX1D\nPointer to the task descriptor if created successfully or NULL\npointer otherwise.\nstatus\nint\nSet to VSL_STATUS_OK if the task is created successfully or\nset to non-zero error code otherwise.\nDescription\nEach vslConvNewTaskX1D/vslCorrNewTaskX1D constructor creates a new convolution or correlation task\ndescriptor with the user specified values for explicit parameters. The optional parameters are set to their\ndefault values (see Table \"Convolution and Correlation Task Parameters\").\nThese routines represent a special one-dimensional version of the so called X-form of the constructor. This\nassumes that the value of the parameter dims is 1 and that in addition to creating the task descriptor,\nconstructor routines assign particular data to the first operand vector in array x used in convolution or\ncorrelation operation. The task descriptor created by the vslConvNewTaskX1D/vslCorrNewTaskX1D\nconstructor keeps the pointer to the array x all the time, that is, until the task object is deleted by one of the\ndestructor routines (see vslConvDeleteTask/vslCorrDeleteTask).\nUsing this form of constructors is recommended when you need to compute multiple convolutions or\ncorrelations with the same data vector in array x against different vectors in array y. This helps improve\nperformance by eliminating unnecessary overhead in repeated computation of intermediate data required for\nthe operation.\nThe parameters xshape, yshape, and zshape are equal to the number of elements read from the arrays x\nand y or stored to the array z. You explicitly assign the shape parameters when calling the constructor.\nThe stride parameters xstride specifies the physical location of the input data in the array x and is an\ninterval between locations of consecutive elements of the array. For example, if the value of the parameter\nxstride is s, then only every sth element of the array x will be used to form the input sequence. The stride\nvalue must be positive or negative but not zero.\nConvolution and Correlation Task Editors\nTask editors in convolution and correlation API of Intel® oneAPI Math Kernel Library (oneMKL) are routines\nintended for setting up or changing the following task parameters (seeTable \"Convolution and Correlation\nTask Parameters\"):\n•\nmode\n•\ninternal_precision\n•\nstart\n•\ndecimation\nFor setting up or changing each of the above parameters, a separate routine exists.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2226\n\n\nNOTE\nFields of the task descriptor structure are accessible only through the set of task editor routines\nprovided with the software.\nThe work data computed during the last commitment process may become invalid with respect to new\nparameter settings. That is why after applying any of the editor routines to change the task descriptor\nsettings, the task loses its commitment status and goes through the full commitment process again during\nthe next execution or copy operation. For more information on task commitment, see the Introduction to\nConvolution and Correlation.\nTable \"Task Editors\" lists available task editors.\nTask Editors\nRoutine\nDescription\nvslConvSetMode/vslCorrSetMode\nChanges the value of the parameter mode for the\noperation of convolution or correlation.\nvslConvSetInternalPrecision/\nvslCorrSetInternalPrecision\nChanges the value of the parameter\ninternal_precision for the operation of convolution or\ncorrelation.\nvslConvSetStart/vslCorrSetStart\nSets the value of the parameter start for the operation\nof convolution or correlation.\nvslConvSetDecimation/\nvslCorrSetDecimation\nSets the value of the parameter decimation for the\noperation of convolution or correlation.\nNOTE\nYou can use the NULL task pointer in calls to editor routines. In this case, the routine is terminated\nand no system crash occurs.\nvslConvSetMode/vslCorrSetMode\nChanges the value of the parameter mode in the\nconvolution or correlation task descriptor.\nSyntax\nstatus = vslConvSetMode(task, newmode);\nstatus = vslCorrSetMode(task, newmode);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\ntask\nVSLConvTaskPtr for\nvslConvSetMode\nVSLCorrTaskPtr for\nvslCorrSetMode\nPointer to the task descriptor.\nnewmode\nconst MKL_INT\nNew value of the parameter mode.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2227\n\n\nOutput Parameters\nName\nType\nDescription\nstatus\nint\nCurrent status of the task.\nDescription\nThis function is declared in mkl_vsl_functions.h.\nThe function routine changes the value of the parameter mode for the operation of convolution or correlation.\nThis parameter defines whether the computation should be done via Fourier transforms of the input/output\ndata or using a direct algorithm. Initial value for mode is assigned by a task constructor.\nPredefined values for the mode parameter are as follows:\nValues of mode parameter\nValue\nPurpose\nVSL_CONV_MODE_FFT\nCompute convolution by using fast Fourier transform.\nVSL_CORR_MODE_FFT\nCompute correlation by using fast Fourier transform.\nVSL_CONV_MODE_DIRECT\nCompute convolution directly.\nVSL_CORR_MODE_DIRECT\nCompute correlation directly.\nVSL_CONV_MODE_AUTO\nAutomatically choose direct or Fourier mode for convolution.\nVSL_CORR_MODE_AUTO\nAutomatically choose direct or Fourier mode for correlation.\nvslConvSetInternalPrecision/vslCorrSetInternalPrecision\nChanges the value of the parameter internal_precision\nin the convolution or correlation task descriptor.\nSyntax\nstatus = vslConvSetInternalPrecision(task, precision);\nstatus = vslCorrSetInternalPrecision(task, precision);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\ntask\nVSLConvTaskPtr for\nvslConvSetInternalPrecisio\nn\nVSLCorrTaskPtr for\nvslCorrSetInternalPrecisio\nn\nPointer to the task descriptor.\nprecision\nconst MKL_INT\nNew value of the parameter internal_precision.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2228\n\n\nOutput Parameters\nName\nType\nDescription\nstatus\nint\nCurrent status of the task.\nDescription\nThe vslConvSetInternalPrecision/vslCorrSetInternalPrecision routine changes the value of the\nparameter internal_precision for the operation of convolution or correlation. This parameter defines\nwhether the internal computations of the convolution or correlation result should be done in single or double\nprecision. Initial value for internal_precision is assigned by a task constructor and set to either \"single\"\nor \"double\" according to the particular flavor of the constructor used.\nChanging the internal_precision can be useful if the default setting of this parameter was \"single\" but\nyou want to calculate the result with double precision even if input and output data are represented in single\nprecision.\nPredefined values for the internal_precision input parameter are as follows:\nValues of internal_precision Parameter\nValue\nPurpose\nVSL_CONV_PRECISION_SINGLE\nCompute convolution with single precision.\nVSL_CORR_PRECISION_SINGLE\nCompute correlation with single precision.\nVSL_CONV_PRECISION_DOUBLE\nCompute convolution with double precision.\nVSL_CORR_PRECISION_DOUBLE\nCompute correlation with double precision.\nvslConvSetStart/vslCorrSetStart\nChanges the value of the parameter start in the\nconvolution or correlation task descriptor.\nSyntax\nstatus = vslConvSetStart(task, start);\nstatus = vslCorrSetStart(task, start);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\ntask\nVSLConvTaskPtr for\nvslConvSetStart\nVSLCorrTaskPtr for\nvslCorrSetStart\nPointer to the task descriptor.\nstart\nconst int[]\nNew value of the parameter start.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2229\n\n\nOutput Parameters\nName\nType\nDescription\nstatus\nint\nCurrent status of the task.\nDescription\nThe vslConvSetStart/vslCorrSetStart routine sets the value of the parameter start for the operation\nof convolution or correlation. In a one-dimensional case, this parameter points to the first element in the\nmathematical result that should be stored in the output array. In a multidimensional case, start is an array of\nindices and its length is equal to the number of dimensions specified by the parameter dims. For more\ninformation about the definition and effect of this parameter, see Data Allocation.\nDuring the initial task descriptor construction, the default value for start is undefined and this parameter is\nnot used. Therefore the only way to set and use the start parameter is via assigning it some value by one\nof the vslConvSetStart/vslCorrSetStart routines.\nvslConvSetDecimation/vslCorrSetDecimation\nChanges the value of the parameter decimation in the\nconvolution or correlation task descriptor.\nSyntax\nstatus = vslConvSetDecimation(task, decimation);\nstatus = vslCorrSetDecimation(task, decimation);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\ntask\nVSLConvTaskPtr for\nvslConvSetDecimation\nVSLCorrTaskPtr for\nvslCorrSetDecimation\nPointer to the task descriptor.\ndecimatio\nn\nconst int[]\nNew value of the parameter decimation.\nOutput Parameters\nName\nType\nDescription\nstatus\nint\nCurrent status of the task.\nDescription\nThe routine sets the value of the parameter decimation for the operation of convolution or correlation. This\nparameter determines how to thin out the mathematical result of convolution or correlation before writing it\ninto the output data array. For example, in a one-dimensional case, if decimation = d > 1, only every d-th\nelement of the mathematical result is written to the output array z. In a multidimensional case, decimation is\nan array of indices and its length is equal to the number of dimensions specified by the parameter dims. For\nmore information about the definition and effect of this parameter, see Data Allocation.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2230\n\n\nDuring the initial task descriptor construction, the default value for decimation is undefined and this\nparameter is not used. Therefore the only way to set and use the decimation parameter is via assigning it\nsome value by one of the vslSetDecimation routines.\nTask Execution Routines\nTask execution routines compute convolution or correlation results based on parameters held by the task\ndescriptor and on the user data supplied for input vectors.\nAfter you create and adjust a task, you can execute it multiple times by applying to different input/output\ndata of the same type, precision, and shape.\nIntel® oneAPI Math Kernel Library (oneMKL) provides the following forms of convolution/correlation execution\nroutines:\n•\nGeneral form executors that use the task descriptor created by the general form constructor and expect\nto get two source data arrays x and y on input\n•\nX-form executors that use the task descriptor created by the X-form constructor and expect to get only\none source data array y on input because the first array x has been already specified on the construction\nstage\nWhen the task is executed for the first time, the execution routine includes a task commitment operation,\nwhich involves two basic steps: parameters consistency check and preparation of auxiliary data (for example,\nthis might be the calculation of Fourier transform for input data).\nEach execution routine has an associated one-dimensional version that provides algorithmic and\ncomputational benefits.\nNOTE\nYou can use the NULL task pointer in calls to execution routines. In this case, the routine is terminated\nand no system crash occurs.\nIf the task is executed successfully, the execution routine returns the zero status code. If an error is\ndetected, the execution routine returns an error code which signals that a specific error has occurred. In\nparticular, an error status code is returned in the following cases:\n•\nif the task pointer is NULL\n•\nif the task descriptor is corrupted\n•\nif calculation has failed for some other reason.\nNOTE\nIntel® MKL does not control floating-point errors, like overflow or gradual underflow, or operations with\nNaNs, etc.\nIf an error occurs, the task descriptor stores the error code.\nThe table below lists all task execution routines.\nTask Execution Routines\nRoutine\nDescription\nvslConvExec/vslCorrExec\nComputes convolution or correlation for a multidimensional case.\nvslConvExec1D/vslCorrExec1D\nComputes convolution or correlation for a one-dimensional case.\nvslConvExecX/vslCorrExecX\nComputes convolution or correlation as X-form for a\nmultidimensional case.\nvslConvExecX1D/vslCorrExecX1D\nComputes convolution or correlation as X-form for a one-\ndimensional case.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2231\n\n\nvslConvExec/vslCorrExec\nComputes convolution or correlation for\nmultidimensional case.\nSyntax\nstatus = vslsConvExec(task, x, xstride, y, ystride, z, zstride);\nstatus = vsldConvExec(task, x, xstride, y, ystride, z, zstride);\nstatus = vslcConvExec(task, x, xstride, y, ystride, z, zstride);\nstatus = vslzConvExec(task, x, xstride, y, ystride, z, zstride);\nstatus = vslsCorrExec(task, x, xstride, y, ystride, z, zstride);\nstatus = vsldCorrExec(task, x, xstride, y, ystride, z, zstride);\nstatus = vslcCorrExec(task, x, xstride, y, ystride, z, zstride);\nstatus = vslzCorrExec(task, x, xstride, y, ystride, z, zstride);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\ntask\nVSLConvTaskPtr for\nvslsConvExec, vsldConvExec,\nvslcConvExec, vslzConvExec\nVSLCorrTaskPtr for\nvslsCorrExec, vsldCorrExec,\nvslcCorrExec, vslzCorrExec\nPointer to the task descriptor\nx, y\nconst float[] for\nvslsConvExec and\nvslsCorrExec,\nconst double[] for\nvsldConvExec and\nvsldCorrExec,\nconst MKL_Complex8[] for\nvslcConvExec and\nvslcCorrExec,\nconst MKL_Complex16[] for\nvslzConvExec and\nvslzCorrExec\nPointers to arrays containing input data. See Data\nAllocation for more information.\nxstride,\nystride,\nzstride\nconst int[]\nStrides for input and output data. For more information, see \nstride parameters.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2232\n\n\nOutput Parameters\nName\nType\nDescription\nz\nconst float[] for\nvslsConvExec and\nvslsCorrExec,\nconst double[] for\nvsldConvExec and\nvsldCorrExec,\nconst MKL_Complex8[] for\nvslcConvExec and\nvslcCorrExec,\nconst MKL_Complex16[] for\nvslzConvExec and\nvslzCorrExec\nPointer to the array that stores output data. See Data\nAllocation for more information.\nstatus\nint\nSet to VSL_STATUS_OK if the task is executed successfully\nor set to non-zero error code otherwise.\nDescription\nEach of the vslConvExec/vslCorrExec routines computes convolution or correlation of the data provided by\nthe arrays x and y and then stores the results in the array z. Parameters of the operation are read from the\ntask descriptor created previously by a corresponding vslConvNewTask/vslCorrNewTask constructor and\npointed to by task. If task is NULL, no operation is done.\nThe stride parameters xstride, ystride, and zstride specify the physical location of the input and output\ndata in the arrays x, y, and z, respectively. In a one-dimensional case, stride is an interval between locations\nof consecutive elements of the array. For example, if the value of the parameter zstride is s, then only\nevery sth element of the array z will be used to store the output data. The stride value must be positive or\nnegative but not zero.\nvslConvExec1D/vslCorrExec1D\nComputes convolution or correlation for one-\ndimensional case.\nSyntax\nstatus = vslsConvExec1D(task, x, xstride, y, ystride, z, zstride);\nstatus = vsldConvExec1D(task, x, xstride, y, ystride, z, zstride);\nstatus = vslcConvExec1D(task, x, xstride, y, ystride, z, zstride);\nstatus = vslzConvExec1D(task, x, xstride, y, ystride, z, zstride);\nstatus = vslsCorrExec1D(task, x, xstride, y, ystride, z, zstride);\nstatus = vsldCorrExec1D(task, x, xstride, y, ystride, z, zstride);\nstatus = vslcCorrExec1D(task, x, xstride, y, ystride, z, zstride);\nstatus = vslzCorrExec1D(task, x, xstride, y, ystride, z, zstride);\nInclude Files\n•\nmkl.h\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2233\n\n\nInput Parameters\nName\nType\nDescription\ntask\nVSLConvTaskPtr for\nvslsConvExec1D,\nvsldConvExec1D,\nvslcConvExec1D,\nvslzConvExec1D\nVSLCorrTaskPtr for\nvslsCorrExec1D,\nvsldCorrExec1D,\nvslcCorrExec1D,\nvslzCorrExec1D\nPointer to the task descriptor.\nx, y\nconst float[] for\nvslsConvExec1D and\nvslsCorrExec1D,\nconst double[] for\nvsldConvExec1D and\nvsldCorrExec1D,\nconst MKL_Complex8[] for\nvslcConvExec1D and\nvslcCorrExec1D,\nconst MKL_Complex16[] for\nvslzConvExec1D and\nvslzCorrExec1D\nPointers to arrays containing input data. See Data\nAllocation for more information.\nxstride,\nystride,\nzstride\nconst MKL_INT\nStrides for input and output data. For more information, see \nstride parameters.\nOutput Parameters\nName\nType\nDescription\nz\nconst float[] for\nvslsConvExec1D and\nvslsCorrExec1D,\nconst double[] for\nvsldConvExec1D and\nvsldCorrExec1D,\nconst MKL_Complex8[] for\nvslcConvExec1D and\nvslcCorrExec1D,\nconst MKL_Complex16[] for\nvslzConvExec1D and\nvslzCorrExec1D\nPointer to the array that stores output data. See Data\nAllocation for more information.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2234\n\n\nName\nType\nDescription\nstatus\nint\nSet to VSL_STATUS_OK if the task is executed successfully\nor set to non-zero error code otherwise.\nDescription\nEach of the vslConvExec1D/vslCorrExec1D routines computes convolution or correlation of the data\nprovided by the arrays x and y and then stores the results in the array z. These routines represent a special\none-dimensional version of the operation, assuming that the value of the parameter dims is 1. Using this\nversion of execution routines can help speed up performance in case of one-dimensional data.\nParameters of the operation are read from the task descriptor created previously by a corresponding \nvslConvNewTask1D/vslCorrNewTask1D constructor and pointed to by task. If task is NULL, no operation\nis done.\nvslConvExecX/vslCorrExecX\nComputes convolution or correlation for\nmultidimensional case with the fixed first operand\nvector.\nSyntax\nstatus = vslsConvExecX(task, y, ystride, z, zstride);\nstatus = vsldConvExecX(task, y, ystride, z, zstride);\nstatus = vslcConvExecX(task, y, ystride, z, zstride);\nstatus = vslzConvExecX(task, y, ystride, z, zstride);\nstatus = vslsCorrExecX(task, y, ystride, z, zstride);\nstatus = vslcCorrExecX(task, y, ystride, z, zstride);\nstatus = vslzCorrExecX(task, y, ystride, z, zstride);\nstatus = vsldCorrExecX(task, y, ystride, z, zstride);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\ntask\nVSLConvTaskPtr for\nvslsConvExecX,\nvsldConvExecX,\nvslcConvExecX,\nvslzConvExecX\nVSLCorrTaskPtr for\nvslsCorrExecX,\nvsldCorrExecX,\nvslcCorrExecX,\nvslzCorrExecX\nPointer to the task descriptor.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2235\n\n\nName\nType\nDescription\nx ,y\nconst float[] for\nvslsConvExecX and\nvslsCorrExecX,\nconst double[] for\nvsldConvExecX and\nvsldCorrExecX,\nconst MKL_Complex8[] for\nvslcConvExecX and\nvslcCorrExecX,\nconst MKL_Complex16[] for\nvslzConvExecX and\nvslzCorrExecX\nPointer to array containing input data (for the second\noperand vector). See Data Allocation for more information.\nystride ,z\nstride\nconst int[]\nStrides for input and output data. For more information, see \nstride parameters.\nOutput Parameters\nName\nType\nDescription\nz\nconst float[] for\nvslsConvExecX and\nvslsCorrExecX,\nconst double[] for\nvsldConvExecX and\nvsldCorrExecX,\nconst MKL_Complex8[] for\nvslcConvExecX and\nvslcCorrExecX,\nconst MKL_Complex16[] for\nvslzConvExecX and\nvslzCorrExecX\nPointer to the array that stores output data. See Data\nAllocation for more information.\nstatus\nint\nSet to VSL_STATUS_OK if the task is executed successfully\nor set to non-zero error code otherwise.\nDescription\nEach of the vslConvExecX/vslCorrExecX routines computes convolution or correlation of the data provided\nby the arrays x and y and then stores the results in the array z. These routines represent a special version of\nthe operation, which assumes that the first operand vector was set on the task construction stage and the\ntask object keeps the pointer to the array x.\nParameters of the operation are read from the task descriptor created previously by a corresponding \nvslConvNewTaskX/vslCorrNewTaskX constructor and pointed to by task. If task is NULL, no operation is\ndone.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2236\n\n\nUsing this form of execution routines is recommended when you need to compute multiple convolutions or\ncorrelations with the same data vector in array x against different vectors in array y. This helps improve\nperformance by eliminating unnecessary overhead in repeated computation of intermediate data required for\nthe operation.\nvslConvExecX1D/vslCorrExecX1D\nComputes convolution or correlation for one-\ndimensional case with the fixed first operand vector.\nSyntax\nstatus = vslsConvExecX1D(task, y, ystride, z, zstride);\nstatus = vsldConvExecX1D(task, y, ystride, z, zstride);\nstatus = vslcConvExecX1D(task, y, ystride, z, zstride);\nstatus = vslzConvExecX1D(task, y, ystride, z, zstride);\nstatus = vslsCorrExecX1D(task, y, ystride, z, zstride);\nstatus = vslcCorrExecX1D(task, y, ystride, z, zstride);\nstatus = vslzCorrExecX1D(task, y, ystride, z, zstride);\nstatus = vsldCorrExecX1D(task, y, ystride, z, zstride);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\ntask\nVSLConvTaskPtr for\nvslsConvExecX1D,\nvsldConvExecX1D,\nvslcConvExecX1D,\nvslzConvExecX1D\nVSLCorrTaskPtr for\nvslsCorrExecX1D,\nvsldCorrExecX1D,\nvslcCorrExecX1D,\nvslzCorrExecX1D\nPointer to the task descriptor.\nx , y\nconst float[] for\nvslsConvExecX1D and\nvslsCorrExecX1D,\nconst double[] for\nvsldConvExecX1D and\nvsldCorrExecX1D,\nconst MKL_Complex8[] for\nvslcConvExecX1D and\nvslcCorrExecX1D,\nPointer to array containing input data (for the second\noperand vector). See Data Allocation for more information.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2237\n\n\nName\nType\nDescription\nconst MKL_Complex16[] for\nvslzConvExecX1D and\nvslzCorrExecX1D\nystride,\nzstride\nconst MKL_INT\nStrides for input and output data. For more information, see \nstride parameters.\nOutput Parameters\nName\nType\nDescription\nz\nconst float[] for\nvslsConvExecX1D and\nvslsCorrExecX1D,\nconst double[] for\nvsldConvExecX1D and\nvsldCorrExecX1D,\nconst MKL_Complex8[] for\nvslcConvExecX1D and\nvslcCorrExecX1D,\nconst MKL_Complex16[] for\nvslzConvExecX1D and\nvslzCorrExecX1D\nPointer to the array that stores output data. See Data\nAllocation for more information.\nstatus\nint\nSet to VSL_STATUS_OK if the task is executed successfully\nor set to non-zero error code otherwise.\nDescription\nEach of the vslConvExecX1D/vslCorrExecX1D routines computes convolution or correlation of one-\ndimensional (assuming that dims =1) data provided by the arrays x and y and then stores the results in the\narray z. These routines represent a special version of the operation, which expects that the first operand\nvector was set on the task construction stage.\nParameters of the operation are read from the task descriptor created previously by a corresponding \nvslConvNewTaskX1D/vslCorrNewTaskX1D constructor and pointed to by task. If task is NULL, no\noperation is done.\nUsing this form of execution routines is recommended when you need to compute multiple one-dimensional\nconvolutions or correlations with the same data vector in array x against different vectors in array y. This\nhelps improve performance by eliminating unnecessary overhead in repeated computation of intermediate\ndata required for the operation.\nConvolution and Correlation Task Destructors\nTask destructors are routines designed for deleting task objects and deallocating memory.\nvslConvDeleteTask/vslCorrDeleteTask\nDestroys the task object and frees the memory.\nSyntax\nerrcode = vslConvDeleteTask(task);\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2238\n\n\nerrcode = vslCorrDeleteTask(task);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\ntask\nVSLConvTaskPtr* for\nvslConvDeleteTask\nVSLCorrTaskPtr* for\nvslCorrDeleteTask\nPointer to the task descriptor.\nOutput Parameters\nName\nType\nDescription\nerrcode\nint\nContains 0 if the task object is deleted successfully.\nContains an error code if an error occurred.\nDescription\nThe vslConvDeleteTask/vslCorrvDeleteTask routine deletes the task descriptor object and frees any\nworking memory and the memory allocated for the data structure. The task pointer is set to NULL.\nNote that if the vslConvDeleteTask/vslCorrvDeleteTask routine does not delete the task successfully,\nthe routine returns an error code. This error code has no relation to the task status code and does not\nchange it.\nNOTE\nYou can use the NULL task pointer in calls to destructor routines. In this case, the routine\nterminates with no system crash.\nConvolution and Correlation Task Copiers\nThe routines are designed for copying convolution and correlation task descriptors.\nvslConvCopyTask/vslCorrCopyTask\nCopies a descriptor for convolution or correlation task.\nSyntax\nstatus = vslConvCopyTask(newtask, srctask);\nstatus = vslCorrCopyTask(newtask, srctask);\nInclude Files\n•\nmkl.h\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2239\n\n\nInput Parameters\nName\nType\nDescription\nsrctask\nconst VSLConvTaskPtr for\nvslConvCopyTask\nconst VSLCorrTaskPtr for\nvslCorrCopyTask\nPointer to the source task descriptor.\nOutput Parameters\nName\nType\nDescription\nnewtask\nVSLConvTaskPtr* for\nvslConvCopyTask\nVSLCorrTaskPtr* for\nvslCorrCopyTask\nPointer to the new task descriptor.\nstatus\nint\nCurrent status of the source task.\nDescription\nIf a task object srctask already exists, you can use an appropriate vslConvCopyTask/vslCorrCopyTask\nroutine to make its copy in newtask. After the copy operation, both source and new task objects will become\ncommitted (see Introduction to Convolution and Correlation for information about task commitment). If the\nsource task was not previously committed, the commitment operation for this task is implicitly invoked\nbefore copying starts. If an error occurs during source task commitment, the task stores the error code in\nthe status field. If an error occurs during copy operation, the routine returns a NULL pointer instead of a\nreference to a new task object.\nConvolution and Correlation Usage Examples\nThis section demonstrates how you can use the Intel® oneAPI Math Kernel Library (oneMKL) routines to\nperform some common convolution and correlation operations both for single-threaded and multithreaded\ncalculations. The following two sample functionsscond1 and sconf1 simulate the convolution and correlation\nfunctions SCOND and SCONF found in IBM ESSL* library. The functions assume single-threaded calculations\nand can be used with C or C++ compilers.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2240\n\n\nFunction scond1 for Single-Threaded Calculations\n#include \"mkl_vsl.h\"\nint scond1(\n    float h[], int inch,\n    float x[], int incx,\n    float y[], int incy,\n    int nh, int nx, int iy0, int ny)\n{\n    int status;\n    VSLConvTaskPtr task;\n    vslsConvNewTask1D(&task,VSL_CONV_MODE_DIRECT,nh,nx,ny);\n    vslConvSetStart(task, &iy0);\n    status = vslsConvExec1D(task, h,inch, x,incx, y,incy);\n    vslConvDeleteTask(&task);\n    return status;\n}\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2241\n\n\nFunction sconf1 for Single-Threaded Calculations\n#include \"mkl_vsl.h\"\nint sconf1(\n    int init,\n    float h[], int inc1h,\n    float x[], int inc1x, int inc2x,\n    float y[], int inc1y, int inc2y,\n    int nh, int nx, int m, int iy0, int ny,\n    void* aux1, int naux1, void* aux2, int naux2)\n{\n    int status;\n    /* assume that aux1!=0 and naux1 is big enough */\n    VSLConvTaskPtr* task = (VSLConvTaskPtr*)aux1;\n    if (init != 0)\n        /* initialization: */\n        status = vslsConvNewTaskX1D(task,VSL_CONV_MODE_FFT,\n         nh,nx,ny, h,inc1h);\n    if (init == 0) {\n        /* calculations: */\n        int i;\n        vslConvSetStart(*task, &iy0);\n        for (i=0; i<m; i++) {\n         float* xi = &x[inc2x * i];\n         float* yi = &y[inc2y * i];\n         /* task is implicitly committed at i==0 */\n         status = vslsConvExecX1D(*task, xi, inc1x, yi, inc1y);\n        };\n    };\n    vslConvDeleteTask(task);\n    return status;\n}\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2242\n\n\nUsing Multiple Threads\nFor functions such as sconf1 described in the previous example, parallel calculations may be more\npreferable instead of cycling. If m>1, you can use multiple threads for invoking the task execution against\ndifferent data sequences. For such cases, use task copy routines to create m copies of the task object before\nthe calculations stage and then run these copies with different threads. Ensure that you make all necessary\nparameter adjustments for the task (using Task Editors) before copying it.\nThe sample code in this case may look as follows:\nif (init == 0) {\n    int i, status, ss[M];\n    VSLConvTaskPtr tasks[M];\n    /* assume that M is big enough */\n    . . .\n    vslConvSetStart(*task, &iy0);\n    . . .\n    for (i=0; i<m; i++)\n        /* implicit commitment at i==0 */\n        vslConvCopyTask(&tasks[i],*task);\n    . . .\nThen, m threads may be started to execute different copies of the task:\n. . .\n        float* xi = &x[inc2x * i];\n        float* yi = &y[inc2y * i];\n        ss[i]=vslsConvExecX1D(tasks[i], xi,inc1x, yi,inc1y);\n    . . .\nAnd finally, after all threads have finished the calculations, overall status should be collected from all task\nobjects. The following code signals the first error found, if any:\n    . . .\n    for (i=0; i<m; i++) {\n        status = ss[i];\n        if (status != 0) /* 0 means \"OK\" */\n            break;\n    };\n    return status;\n}; /* end if init==0 */\nExecution routines modify the task internal state (fields of the task structure). Such modifications may\nconflict with each other if different threads work with the same task object simultaneously. That is why\ndifferent threads must use different copies of the task.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2243\n\n\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nConvolution and Correlation Mathematical Notation and Definitions\nThe following notation is necessary to explain the underlying mathematical definitions used in the text:\nR = (-∞, +∞)\nThe set of real numbers.\nZ = {0, ±1, ±2, ...}\nThe set of integer numbers.\nZN = Z× ... ×Z\nThe set of N-dimensional series of integer numbers.\np = (p1, ..., pN) ∈ZN\nN-dimensional series of integers.\nu:ZN→R\nFunction u with arguments from ZN and values from R.\nu(p) = u(p1, ..., pN)\nThe value of the function u for the argument (p1, ..., pN).\nw = u*v\nFunction w is the convolution of the functions u, v.\nw = u•v\nFunction w is the correlation of the functions u, v.\nGiven series p, q∈ZN:\n•\nseries r = p + q is defined as rn = pn + qn for every n=1,...,N\n•\nseries r = p - q is defined as rn = pn - qn for every n=1,...,N\n•\nseries r = sup{p, q} is defines as rn = max{pn, qn} for every n=1,...,N\n•\nseries r = inf{p, q} is defined as rn = min{pn, qn} for every n=1,...,N\n•\ninequality p≤q means that pn≤qn for every n=1,...,N.\nA function u(p) is called a finite function if there exist series Pmin, Pmax∈ZN such that:\nu(p) \n        ≠ 0 \n      \nimplies\n Pmin≤p≤ Pmax.\nOperations of convolution and correlation are only defined for finite functions.\nConsider functions u, v and series Pmin, PmaxQmin, Qmax∈ZN such that:\nu(p) ≠ 0 implies Pmin≤p≤ Pmax.\nv(q) ≠ 0 implies Qmin≤q≤ Qmax.\nDefinitions of linear correlation and linear convolution for functions u and v are given below.\nLinear Convolution\nIf function w = u*v is the convolution of u and v, then:\nw(r) ≠ 0 implies Rmin≤r≤Rmax,\nwhere Rmin = Pmin + Qmin and Rmax = Pmax + Qmax.\nIf Rmin≤r≤Rmax, then:\nw(r) = ∑u(t)·v(r−t) is the sum for all t∈ZN such that Tmin≤t≤Tmax,\nwhere Tmin = sup{Pmin, r− Qmax} and Tmax = inf{Pmax, r− Qmin}.\nLinear Correlation\nIf function w = u•v is the correlation of u and v, then:\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2244\n\n\nw(r) ≠ 0 implies Rmin≤r≤Rmax,\nwhere Rmin = Qmin - Pmax and Rmax = Qmax - Pmin.\nIf Rmin≤r≤Rmax, then:\nw(r) = ∑u(t)·v(r+t) is the sum for all t∈ZN such that Tmin≤t≤Tmax,\nwhere Tmin = sup{Pmin, Qmin−r} and Tmax = inf{Pmax, Qmax−r}.\nRepresentation of the functions u, v, was the input/output data for the Intel® oneAPI Math Kernel Library\n(oneMKL) convolution and correlation functions is described in theData Allocation.\nConvolution and Correlation Data Allocation\nThis section explains the relation between:\n•\nmathematical finite functions u, v, w introduced in Mathematical Notation and Definitions;\n•\nmulti-dimensional input and output data vectors representing the functions u, v, w;\n•\narrays u, v, w used to store the input and output data vectors in computer memory\nThe convolution and correlation routine parameters that determine the allocation of input and output data\nare the following:\n•\nData arrays x, y, z\n•\nShape arrays xshape, yshape, zshape\n•\nStrides within arrays xstride, ystride, zstride\n•\nParameters start, decimation\nFinite Functions and Data Vectors\nThe finite functions u(p), v(q), and w(r) introduced above are represented as multi-dimensional vectors of\ninput and output data:\ninputu(i1,...,idims) for u(p1,...,pN)\ninputv(j1,...,jdims) for v(q1,...,qN)\noutput(k1,...,kdims) for w(r1,...,rN).\nParameter dims represents the number of dimensions and is equal to N.\nThe parameters xshape, yshape, and zshape define the shapes of input/output vectors:\ninputu(i1,...,idims) is defined if 1 ≤in≤xshape(n) for every n=1,...,dims\ninputv(j1,...,jdims) is defined if 1 ≤jn≤yshape(n) for every n=1,...,dims\noutput(k1,...,kdims) is defined if 1 ≤kn≤zshape(n) for every n=1,...,dims.\nRelation between the input vectors and the functions u and v is defined by the following formulas:\ninputu(i1,...,idims)= u(p1,...,pN), where pn = Pnmin + (in-1) for every n\ninputv(j1,...,jdims)= v(q1,...,qN), where qn=Qnmin + (jn-1) for every n.\nThe relation between the output vector and the function w(r) is similar (but only in the case when\nparameters start and decimation are not defined):\noutput(k1,...,kdims)= w(r1,...,rN), where rn=Rnmin + (kn-1) for every n.\nIf the parameter start is defined, it must belong to the interval Rnmin≤start(n)≤Rnmax. If defined, the\nstart parameter replaces Rmin in the formula:\noutput(k1,...,kdims)=w(r1,...,rN), where rn=start(n) + (kn-1)\nIf the parameter decimation is defined, it changes the relation according to the following formula:\noutput(k1,...,kdims)=w(r1,...,rN), where rn= Rnmin + (kn-1)*decimation(n)\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2245\n\n\nIf both parameters start and decimation are defined, the formula is as follows:\noutput(k1,...,kdims)=w(r1,...,rN), where rn=start(n) + (kn-1)*decimation(n)\nThe convolution and correlation software checks the values of zshape, start, and decimation during task\ncommitment. If rn exceeds Rnmax for some kn,n=1,...,dims, an error is raised.\nAllocation of Data Vectors\nBoth parameter arrays x and y contain input data vectors in memory, while array z is intended for storing\noutput data vector. To access the memory, the convolution and correlation software uses only pointers to\nthese arrays and ignores the array shapes.\nFor parameters x, y, and z, you can provide one-dimensional arrays with the requirement that actual length\nof these arrays be sufficient to store the data vectors.\nThe allocation of the input and output data inside the arrays x, y, and z is described below assuming that the\narrays are one-dimensional. Given multi-dimensional indices i, j, k∈ZN, one-dimensional indices e, f, g∈Z are\ndefined such that:\ninputu(i1,...,idims) is allocated at x(e)\ninputv(j1,...,jdims) is allocated at y(f)\noutput(k1,...,kdims) is allocated at z(g).\nThe indices e, f, and g are defined as follows:\ne = 1 + ∑xstride(n)·dx(n) (the sum is for all n=1,...,dims)\nf = 1 + ∑ystride(n)·dy(n) (the sum is for all n=1,...,dims)\ng = 1 + ∑zstride(n)·dz(n) (the sum is for all n=1,...,dims)\nThe distances dx(n), dy(n), and dz(n) depend on the signum of the stride:\ndx(n) = in-1 if xstride(n)>0, or dx(n) = in-xshape(n) if xstride(n)<0\ndy(n) = jn-1 if ystride(n)>0, or dy(n) = jn-yshape(n) if ystride(n)<0\ndz(n) = kn-1 if zstride(n)>0, or dz(n) = kn-zshape(n) if zstride(n)<0\nThe definitions of indices e, f, and g assume that indexes for arrays x, y, and z are started from unity:\nx(e) is defined for e=1,...,length(x)\ny(f) is defined for f=1,...,length(y)\nz(g) is defined for g=1,...,length(z)\nBelow is a detailed explanation about how elements of the multi-dimensional output vector are stored in the\narray z for one-dimensional and two-dimensional cases.\nOne-dimensional case. If dims=1, then zshape is the number of the output values to be stored in the\narray z. The actual length of array z may be greater than zshape elements.\nIf zstride>1, output values are stored with the stride: output(1) is stored to z(1), output(2) is stored to\nz(1+zstride), and so on. Hence, the actual length of z must be at least 1+zstride*(zshape-1) elements\nor more.\nIf zstride<0, it still defines the stride between elements of array z. However, the order of the used elements\nis the opposite. For the k-th output value, output(k) is stored in z(1+|zstride|*(zshape-k)), where |\nzstride| is the absolute value of zstride. The actual length of the array z must be at least 1+|zstride|\n*(zshape - 1) elements.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2246\n\n\nTwo-dimensional case. If dims=2, the output data is a two-dimensional matrix. The value zstride(1)\ndefines the stride inside matrix columns, that is, the stride between the output(k1, k2) and output(k1+1,\nk2) for every pair of indices k1, k2. On the other hand, zstride(2) defines the stride between columns,\nthat is, the stride between output(k1,k2) and output(k1,k2+1).\nIf zstride(2) is greater than zshape(1), this causes sparse allocation of columns. If the value of\nzstride(2) is smaller than zshape(1), this may result in the transposition of the output matrix. For\nexample, if zshape = (2,3), you can define zstride = (3,1) to allocate output values like transposed\nmatrix of the shape 3x2.\nWhether zstride assumes this kind of transformations or not, you need to ensure that different elements\noutput (k1, ...,kdims) will be stored in different locations z(g).\nSummary Statistics\nThe Summary Statistics domain provides routines that compute basic statistical estimates for single and\ndouble precision multi-dimensional datasets.\nThe Summary Statistics routines calculate:\n•\nraw and central moments up to the fourth order\n•\nskewness and excess kurtosis (further referred to as kurtosis for brevity)\n•\nvariation coefficient\n•\nquantiles and order statistics\n•\nminimum and maximum\n•\nvariance-covariance/correlation matrix\n•\npooled/group variance-covariance matrix and mean\n•\npartial variance-covariance/correlation matrix\n•\nrobust estimators for variance-covariance matrix and mean in presence of outliers\n•\nraw/central partial sums up to the fourth order (for brevity referred to as raw/central sums)\n•\nmatrix of cross-products and sums of squares (for brevity referred to as cross-product matrix)\n•\nmedian absolute deviation, mean absolute deviation\nThe library also contains functions to perform the following tasks:\n•\nDetect outliers in datasets\n•\nSupport missing values in datasets\n•\nParameterize correlation matrices\n•\nCompute quantiles for streaming data\nMathematical Notation and Definitions defines the supported operations in the Summary Statistics routines.\nYou can access the Summary Statistics routines through the Fortran 90 and C89 language interfaces. You can\nuse the C89 interface with later versions of the C/C++.\nThe mkl_vsl.h header file is in the ${MKL}/include directory.\nYou can find examples that demonstrate calculation of the Summary Statistics estimates in the ${MKL}/\nexamples/vslc example directory.\nThe Summary Statistics API is implemented through task objects, or tasks. A task object is a data structure,\nor a descriptor, holding parameters that determine a specific Summary Statistics operation. For example,\nsuch parameters may be precision, dimensions of user data, the matrix of the observations, or shapes of\ndata arrays.\nAll the Summary Statistics routines process a task object as follows:\n1.\nCreate a task.\n2.\nModify settings of the task parameters.\n3.\nCompute statistical estimates.\n4.\nDestroy the task.\nThe Summary Statistics functions fall into the following categories:\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2247\n\n\nTask Constructors - routines that create a new task object descriptor and set up most common parameters\n(dimension, number of observations, and matrix of the observations).\nTask Editors - routines that can set or modify some parameter settings in the existing task descriptor.\nTask Computation Routine - a routine that computes specified statistical estimates.\nTask Destructor - a routine that deletes the task object and frees the memory.\nA Summary Statistics task object contains a series of pointers to the input and output data arrays. You can\nread and modify the datasets and estimates at any time but you should allocate and release memory for\nsuch data.\nSee detailed information on the algorithms, API, and their usage in the Intel® oneAPI Math Kernel Library\n(oneMKL) Summary Statistics Application Notes [SS Notes].\nSummary Statistics Naming Conventions\nThe names of Summary Statistics routines, types, and constants are case-sensitive and can contain\nlowercase and uppercase characters (vslsSSEditQuantiles).\nThe names of routines have the following structure:\nvsl[datatype]SS<base name>   \nwhere\n•\nvslis a prefix indicating that the routine belongs to Intel® oneAPI Math Kernel Library (oneMKL) Vector\nStatistics.\n•\n[datatype] specifies the type of the input and/or output data and can be s (single precision real type), d\n(double precision real type), or i (integer type).\n•\nSS/ss indicates that the routine is intended for calculations of the Summary Statistics estimates.\n•\n<base name> specifies a particular functionality that the routine is designed for, for example, NewTask,\nCompute, DeleteTask.\nNOTE\nThe Summary Statistics routine vslDeleteTask for deletion of the task is independent of the data\ntype and its name omits the [datatype] field.\nOn 64-bit platforms, routines with the _64 suffix support large data arrays in the LP64 interface library and\nenable you to mix integer types in one application. For more interface library details, see \"Using the ILP64\nInterface vs. LP64 Interface\" in the developer guide.\nSummary Statistics Data Types\nThe Summary Statistics routines use the following data types for calculations:\nType\nData Object\nVSLSSTaskPtr\nPointer to a Summary Statistics task\nfloat\nInput/output user data in single precision\ndouble\nInput/output user data in double precision\nMKL_INT or long long\nOther data\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2248\n\n\nNOTE\nThe actual size of the generic integer type is platform-specific and can be 32 or 64 bits in length.\nBefore you compile your application, set an appropriate size for integers. See details in the 'Using the\nILP64 Interface vs. LP64 Interface' section of the Intel® oneAPI Math Kernel Library (oneMKL)\nDeveloper Guide.\nSummary Statistics Parameters\nThe basic parameters in the task descriptor (addresses of dimensions, number of observations, and datasets)\nare assigned values when the task editors create or modify the task object. Other parameters are determined\nby the specific task and changed by the task editors.\nSummary Statistics Task Status and Error Reporting\nThe task status is an integer value, which is zero if no error is detected, or a specific non-zero error code\notherwise. Negative status values indicate errors, and positive values indicate warnings. An error can be\ncaused by invalid parameter values or a memory allocation failure.\nThe header files define symbolic names for the status codes. These names are defined as macros via\n#define statements.\nThe header files define the following status codes for the Summary Statistics error codes:\nSummary Statistics Status Codes\nStatus Code\nDescription\nVSL_STATUS_OK\nOperation is successfully completed.\nVSL_SS_ERROR_ALLOCATION_FAILURE\nMemory allocation has failed.\nVSL_SS_ERROR_BAD_DIMEN\nDimension value is invalid.\nVSL_SS_ERROR_BAD_OBSERV_N\nInvalid number (zero or negative) of\nobservations was obtained.\nVSL_SS_ERROR_STORAGE_NOT_SUPPORTED\nStorage format is not supported.\nVSL_SS_ERROR_BAD_INDC_ADDR\nArray of indices is not defined.\nVSL_SS_ERROR_BAD_WEIGHTS\nArray of weights contains negative values.\nVSL_SS_ERROR_BAD_MEAN_ADDR\nArray of means is not defined.\nVSL_SS_ERROR_BAD_2R_MOM_ADDR\nArray of the second order raw moments is not\ndefined.\nVSL_SS_ERROR_BAD_3R_MOM_ADDR\nArray of the third order raw moments is not\ndefined.\nVSL_SS_ERROR_BAD_4R_MOM_ADDR\nArray of the fourth order raw moments is not\ndefined.\nVSL_SS_ERROR_BAD_2C_MOM_ADDR\nArray of the second order central moments is\nnot defined.\nVSL_SS_ERROR_BAD_3C_MOM_ADDR\nArray of the third order central moments is\nnot defined.\nVSL_SS_ERROR_BAD_4C_MOM_ADDR\nArray of the fourth order central moments is\nnot defined.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2249\n\n\nStatus Code\nDescription\nVSL_SS_ERROR_BAD_KURTOSIS_ADDR\nArray of kurtosis values is not defined.\nVSL_SS_ERROR_BAD_SKEWNESS_ADDR\nArray of skewness values is not defined.\nVSL_SS_ERROR_BAD_MIN_ADDR\nArray of minimum values is not defined.\nVSL_SS_ERROR_BAD_MAX_ADDR\nArray of maximum values is not defined.\nVSL_SS_ERROR_BAD_VARIATION_ADDR\nArray of variation coefficients is not defined.\nVSL_SS_ERROR_BAD_COV_ADDR\nCovariance matrix is not defined.\nVSL_SS_ERROR_BAD_COR_ADDR\nCorrelation matrix is not defined.\nVSL_SS_ERROR_BAD_QUANT_ORDER_ADDR\nArray of quantile orders is not defined.\nVSL_SS_ERROR_BAD_QUANT_ORDER\nQuantile order value is invalid.\nVSL_SS_ERROR_BAD_QUANT_ADDR\nArray of quantiles is not defined.\nVSL_SS_ERROR_BAD_ORDER_STATS_ADDR\nArray of order statistics is not defined.\nVSL_SS_ERROR_MOMORDER_NOT_SUPPORTED\nMoment of requested order is not supported.\nVSL_SS_NOT_FULL_RANK_MATRIX\nCorrelation matrix is not of full rank.\nVSL_SS_ERROR_ALL_OBSERVS_OUTLIERS\nAll observations are outliers. (At least one\nobservation must not be an outlier.)\nVSL_SS_ERROR_BAD_ROBUST_COV_ADDR\nRobust covariance matrix is not defined.\nVSL_SS_ERROR_BAD_ROBUST_MEAN_ADDR\nArray of robust means is not defined.\nVSL_SS_ERROR_METHOD_NOT_SUPPORTED\nRequested method is not supported.\nVSL_SS_ERROR_NULL_TASK_DESCRIPTOR\nTask descriptor is null.\nVSL_SS_ERROR_BAD_OBSERV_ADDR\nDataset matrix is not defined.\nVSL_SS_ERROR_BAD_ACCUM_WEIGHT_ADDR\nPointer to the variable that holds the value of\naccumulated weight is not defined.\nVSL_SS_ERROR_SINGULAR_COV\nCovariance matrix is singular.\nVSL_SS_ERROR_BAD_POOLED_COV_ADDR\nPooled covariance matrix is not defined.\nVSL_SS_ERROR_BAD_POOLED_MEAN_ADDR\nArray of pooled means is not defined.\nVSL_SS_ERROR_BAD_GROUP_COV_ADDR\nGroup covariance matrix is not defined.\nVSL_SS_ERROR_BAD_GROUP_MEAN_ADDR\nArray of group means is not defined.\nVSL_SS_ERROR_BAD_GROUP_INDC_ADDR\nArray of group indices is not defined.\nVSL_SS_ERROR_BAD_GROUP_INDC\nGroup indices have improper values.\nVSL_SS_ERROR_BAD_OUTLIERS_PARAMS_ADDR\nArray of parameters for the outlier detection\nalgorithm is not defined.\nVSL_SS_ERROR_BAD_OUTLIERS_PARAMS_N_ADDR\nPointer to size of the parameter array for the\noutlier detection algorithm is not defined.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2250\n\n\nStatus Code\nDescription\nVSL_SS_ERROR_BAD_OUTLIERS_WEIGHTS_ADDR\nOutput of the outlier detection algorithm is not\ndefined.\nVSL_SS_ERROR_BAD_ROBUST_COV_PARAMS_ADDR\nArray of parameters of the robust covariance\nestimation algorithm is not defined.\nVSL_SS_ERROR_BAD_ROBUST_COV_PARAMS_N_ADDR\nPointer to the number of parameters of the\nalgorithm for robust covariance is not defined.\nVSL_SS_ERROR_BAD_STORAGE_ADDR\nPointer to the variable that holds the storage\nformat is not defined.\nVSL_SS_ERROR_BAD_PARTIAL_COV_IDX_ADDR\nArray that encodes sub-components of a\nrandom vector for the partial covariance\nalgorithm is not defined.\nVSL_SS_ERROR_BAD_PARTIAL_COV_IDX\nArray that encodes sub-components of a\nrandom vector for partial covariance has\nimproper values.\nVSL_SS_ERROR_BAD_PARTIAL_COV_ADDR\nPartial covariance matrix is not defined.\nVSL_SS_ERROR_BAD_PARTIAL_COR_ADDR\nPartial correlation matrix is not defined.\nVSL_SS_ERROR_BAD_MI_PARAMS_ADDR\nArray of parameters for the Multiple\nImputation method is not defined.\nVSL_SS_ERROR_BAD_MI_PARAMS_N_ADDR\nPointer to number of parameters for the\nMultiple Imputation method is not defined.\nVSL_SS_ERROR_BAD_MI_BAD_PARAMS_N\nSize of the parameter array of the Multiple\nImputation method is invalid.\nVSL_SS_ERROR_BAD_MI_PARAMS\nParameters of the Multiple Imputation method\nare invalid.\nVSL_SS_ERROR_BAD_MI_INIT_ESTIMATES_N_ADDR\nPointer to the number of initial estimates in\nthe Multiple Imputation method is not defined.\nVSL_SS_ERROR_BAD_MI_INIT_ESTIMATES_ADDR\nArray of initial estimates for the Multiple\nImputation method is not defined.\nVSL_SS_ERROR_BAD_MI_SIMUL_VALS_ADDR\nArray of simulated missing values in the\nMultiple Imputation method is not defined.\nVSL_SS_ERROR_BAD_MI_SIMUL_VALS_N_ADDR\nPointer to the size of the array of simulated\nmissing values in the Multiple Imputation\nmethod is not defined.\nVSL_SS_ERROR_BAD_MI_ESTIMATES_N_ADDR\nPointer to the number of parameter estimates\nin the Multiple Imputation method is not\ndefined.\nVSL_SS_ERROR_BAD_MI_ESTIMATES_ADDR\nArray of parameter estimates in the Multiple\nImputation method is not defined.\nVSL_SS_ERROR_BAD_MI_SIMUL_VALS_N\nInvalid size of the array of simulated values in\nthe Multiple Imputation method.\nVSL_SS_ERROR_BAD_MI_ESTIMATES_N\nInvalid size of an array to hold parameter\nestimates obtained using the Multiple\nImputation method.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2251\n\n\nStatus Code\nDescription\nVSL_SS_ERROR_BAD_MI_OUTPUT_PARAMS\nArray of output parameters in the Multiple\nImputation method is not defined.\nVSL_SS_ERROR_BAD_MI_PRIOR_N_ADDR\nPointer to the number of prior parameters is\nnot defined.\nVSL_SS_ERROR_BAD_MI_PRIOR_ADDR\nArray of prior parameters is not defined.\nVSL_SS_ERROR_BAD_MI_MISSING_VALS_N\nInvalid number of missing values was\nobtained.\nVSL_SS_SEMIDEFINITE_COR\nCorrelation matrix passed into the\nparameterization function is semi-definite.\nVSL_SS_ERROR_BAD_PARAMTR_COR_ADDR\nCorrelation matrix to be parameterized is not\ndefined.\nVSL_SS_ERROR_BAD_COR\nAll eigenvalues of the correlation matrix to be\nparameterized are non-positive.\nVSL_SS_ERROR_BAD_STREAM_QUANT_PARAMS_N_ADDR\nPointer to the number of parameters for the\nquantile computation algorithm for streaming\ndata is not defined.\nVSL_SS_ERROR_BAD_STREAM_QUANT_PARAMS_ADDR\nArray of parameters of the quantile\ncomputation algorithm for streaming data is\nnot defined.\nVSL_SS_ERROR_BAD_STREAM_QUANT_PARAMS_N\nInvalid number of parameters of the quantile\ncomputation algorithm for streaming data has\nbeen obtained.\nVSL_SS_ERROR_BAD_STREAM_QUANT_PARAMS\nInvalid parameters of the quantile\ncomputation algorithm for streaming data\nhave been passed.\nVSL_SS_ERROR_BAD_STREAM_QUANT_ORDER_ADDR\nArray of the quantile orders for streaming\ndata is not defined.\nVSL_SS_ERROR_BAD_STREAM_QUANT_ORDER\nInvalid quantile order for streaming data is\ndefined.\nVSL_SS_ERROR_BAD_STREAM_QUANT_ADDR\nArray of quantiles for streaming data is not\ndefined.\nVSL_SS_ERROR_BAD_SUM_ADDR\nArray of sums is not defined.\nVSL_SS_ERROR_BAD_2R_SUM_ADDR\nArray of raw sums of 2nd order is not defined.\nVSL_SS_ERROR_BAD_3R_SUM_ADDR\nArray of raw sums of 3rd order is not defined.\nVSL_SS_ERROR_BAD_4R_SUM_ADDR\nArray of raw sums of 4th order is not defined.\nVSL_SS_ERROR_BAD_2C_SUM_ADDR\nArray of central sums of 2nd order is not\ndefined.\nVSL_SS_ERROR_BAD_3C_SUM_ADDR\nArray of central sums of 3rd order is not\ndefined.\nVSL_SS_ERROR_BAD_4C_SUM_ADDR\nArray of central sums of 4th order is not\ndefined.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2252\n\n\nStatus Code\nDescription\nVSL_SS_ERROR_BAD_CP_SUM_ADDR\nCross-product matrix is not defined.\nVSL_SS_ERROR_BAD_MDAD_ADDR\nArray of median absolute deviations is not\ndefined.\nVSL_SS_ERROR_BAD_MNAD_ADDR\nArray of mean absolute deviations is not\ndefined.\nVSL_SS_ERROR_BAD_SORTED_OBSERV_ADDR\nArray for storing observation sorting results is\nnot defined.\nVSL_SS_ERROR_ERROR_INDICES_NOT_SUPPORTED\nArray of indices is not supported.\nRoutines for robust covariance estimation, outlier detection, partial covariance estimation, multiple\nimputation, and parameterization of a correlation matrix can return internal error codes that are related to a\nspecific implementation. Such error codes indicate invalid input data or other bugs in the Intel® oneAPI Math\nKernel Library (oneMKL) routines other than the Summary Statistics routines.\nSummary Statistics Task Constructors\nTask constructors are routines intended for creating a new task descriptor and setting up basic parameters.\nNOTE\nIf the constructor fails to create a task descriptor, it returns the NULL task pointer.\nvslSSNewTask\nCreates and initializes a new summary statistics task\ndescriptor.\nSyntax\nstatus = vslsSSNewTask(&task, p, n, xstorage, x, w, indices);\nstatus = vsldSSNewTask(&task, p, n, xstorage, x, w, indices);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\np\nconst MKL_INT*\nDimension of the task, number of\nvariables\nn\nconst MKL_INT*\nNumber of observations\nxstorage\nconst MKL_INT*\nStorage format of matrix of observations\nx\nconst float* for vslsSSNewTask\nconst double* for vsldSSNewTask\nMatrix of observations\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2253\n\n\nName\nType\nDescription\nw\nconst float* for vslsSSNewTask\nconst double* for vsldSSNewTask\nArray of weights of size n. Elements of the\narrays are non-negative numbers. If a\nNULL pointer is passed, each observation\nis assigned weight equal to 1.\nindices\nconst MKL_INT*\nArray of vector components that will be\nprocessed. Size of array is p. If a NULL\npointer is passed, all components of\nrandom vector are processed.\nOutput Parameters\nName\nType\nDescription\ntask\nVSLSSTaskPtr*\nDescriptor of the task\nstatus\nint\nSet to VSL_STATUS_OK if the task is created\nsuccessfully, otherwise a non-zero error code is\nreturned.\nDescription\nEach vslSSNewTask constructor routine creates a new summary statistics task descriptor with the user-\nspecified value for a required parameter, dimension of the task. The optional parameters (matrix of\nobservations, its storage format, number of observations, weights of observations, and indices of the random\nvector components) are set to their default values.\nThe observations of random p-dimensional vector ξ = (ξ1, ..., ξi, ..., ξp), which are n vectors of dimension p,\nare passed as a one-dimensional array x. The parameter xstorage defines the storage format of the\nobservations and takes one of the possible values listed in Table \"Storage format of matrix of observations\nand order statistics\".\nStorage format of matrix of observations, order statistics, and matrix of sorted observations\nParameter\nDescription\nVSL_SS_MATRIX_STORAGE_ROWS\nThe observations of random vector ξ are packed by rows:\nn data points for the vector component ξ1 come first, n\ndata points for the vector component ξ2 come second,\nand so forth.\nVSL_SS_MATRIX_STORAGE_COLS\nThe observations of random vector ξ are packed by\ncolumns: the first p-dimensional observation of the\nvector ξ comes first, the second p-dimensional\nobservation of the vector comes second, and so forth.\nA one-dimensional array w of size n contains non-negative weights assigned to the observations. You can\npass a NULL array into the constructor. In this case, each observation is assigned the default value of the\nweight.\nYou can choose vector components for which you wish to compute statistical estimates. If an element of the\nvector indices of size p contains 0, the observations that correspond to this component are excluded from the\ncalculations. If you pass the NULL value of the parameter into the constructor, statistical estimates for all\nrandom variables are computed.\nIf the constructor fails to create a task descriptor, it returns the NULL task pointer.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2254\n\n\nSummary Statistics Task Editors\nTask editors are intended to set up or change the task parameters listed in Table \"Parameters of Summary\nStatistics Task to Be Initialized or Modified\". As an example, to compute the sample mean for a one-\ndimensional dataset, initialize a variable for the mean value, and pass its address into the task as shown in\nthe example below:\n#define DIM    1\n#define N   1000\nint main()\n{\n     VSLSSTaskPtr task;\n     double x[N];\n     double mean;\n     MKL_INT p, n, xstorage;\n     int status;\n     /* initialize variables used in the computations of sample mean */\n     p = DIM;\n     n = N;\n     xstorage = VSL_SS_MATRIX_STORAGE_ROWS;\n     mean = 0.0;\n     /* create task */\n     status = vsldSSNewTask( &task, &p, &n, &xstorage, x, 0, 0 );\n     /* initialize task parameters */\n     status = vsldSSEditTask( task, VSL_SS_ED_MEAN, &mean );\n     /* compute mean using SS fast method */ \n     status = vsldSSCompute(task, VSL_SS_MEAN, VSL_SS_METHOD_FAST );\n     /* deallocate task resources */\n     status = vslSSDeleteTask( &task ); \n     return 0;\n}\nUse the single (vslsssedittask) or double (vsldssedittask) version of an editor, to initialize single or\ndouble precision version task parameters, respectively. Use an integer version of an editor\n(vslissedittask) to initialize parameters of the integer type.\nTable \"Summary Statistics Task Editors\" lists the task editors for Summary Statistics. Each of them initializes\nand/or modifies a respective group of related parameters.\nSummary Statistics Task Editors\nEditor\nDescription\nvslSSEditTask\nChanges a pointer in the task descriptor.\nvslSSEditMoments\nChanges pointers to arrays associated with raw and central\nmoments.\nvslSSEditSums\nModifies the pointers to arrays that hold sum estimates.\nvslSSEditCovCor\nChanges pointers to arrays associated with covariance and/or\ncorrelation matrices.\nvslSSEditCP\nModifies the pointers to cross-product matrix parameters.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2255\n\n\nEditor\nDescription\nvslSSEditPartialCovCor\nChanges pointers to arrays associated with partial covariance\nand/or correlation matrices.\nvslSSEditQuantiles\nChanges pointers to arrays associated with quantile/order statistics\ncalculations.\nvslSSEditStreamQuantiles\nChanges pointers to arrays for quantile related calculations for\nstreaming data.\nvslSSEditPooledCovariance\nChanges pointers to arrays associated with algorithms related to a\npooled covariance matrix.\nvslSSEditRobustCovariance\nChanges pointers to arrays for robust estimation of a covariance\nmatrix and mean.\nvslSSEditOutliersDetection\nChanges pointers to arrays for detection of outliers.\nvslSSEditMissingValues\nChanges pointers to arrays associated with the method of\nsupporting missing values in a dataset.\nvslSSEditCorParameterization\nChanges pointers to arrays associated with the algorithm for\nparameterization of a correlation matrix.\nNOTE\nYou can use the NULL task pointer in calls to editor routines. In this case, the routine is terminated\nand no system crash occurs.\nvslSSEditTask\nModifies address of an input/output parameter in the\ntask descriptor.\nSyntax\nstatus = vslsSSEditTask(task, parameter, par_addr);\nstatus = vsldSSEditTask(task, parameter, par_addr);\nstatus = vsliSSEditTask(task, parameter, par_addr);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\ntask\nVSLSSTaskPtr\nDescriptor of the task\nparameter\nconst MKL_INT\nParameter to change\npar_addr\nconst float* for vslsSSEditTask\nconst double* for vsldSSEditTask\nconst MKL_INT* for vsliSSEditTask\nAddress of the new parameter\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2256\n\n\nOutput Parameters\nName\nType\nDescription\nstatus\nint\nCurrent status of the task\nDescription\nThe vslSSEditTask routine replaces the pointer to the parameter stored in the Summary Statistics task\ndescriptor with the par_addr pointer. If you pass the NULL pointer to the editor, no changes take place in the\ntask and a corresponding error code is returned. See Table \"Parameters of Summary Statistics Task to Be\nInitialized or Modified\" for the predefined values of the parameter.\nUse the single (vslsssedittask) or double (vsldssedittask) version of the editor, to initialize single or\ndouble precision version task parameters, respectively. Use an integer version of the editor\n(vslissedittask) to initialize parameters of the integer type.\nParameters of Summary Statistics Task to Be Initialized or Modified\nParameter Value\nType\nPurpose\nInitialization\nVSL_SS_ED_DIMEN\ni\nAddress of a variable that\nholds the task dimension\nRequired. Positive integer value.\nVSL_SS_ED_OBSERV_N\ni\nAddress of a variable that\nholds the number of\nobservations\nRequired. Positive integer value.\nVSL_SS_ED_OBSERV\nd, s\nAddress of the observation\nmatrix\nRequired. Provide the matrix\ncontaining your observations.\nVSL_SS_ED_OBSERV_STORAGE\ni\nAddress of a variable that\nholds the storage format for\nthe observation matrix\nRequired. Provide a storage\nformat supported by the library\nwhenever you pass a matrix of\nobservations.1\nVSL_SS_ED_INDC\ni\nAddress of the array of\nindices\nOptional. Provide this array if you\nneed to process individual\ncomponents of the random vector.\nSet entry i of the array to one to\ninclude the ith coordinate in the\nanalysis. Set entry i of the array\nto zero to exclude the ith\ncoordinate from the analysis.\nVSL_SS_ED_WEIGHTS\nd, s\nAddress of the array of\nobservation weights\nOptional. If the observations have\nweights different from the default\nweight (one), set entries of the\narray to non-negative floating\npoint values.\nVSL_SS_ED_MEAN\nd, s\nAddress of the array of\nmeans\nOptional. Set entries of the array\nto meaningful values (typically\nzero) if you intend to compute a\nprogressive estimate. Otherwise,\ndo not initialize the array.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2257\n\n\nParameter Value\nType\nPurpose\nInitialization\nVSL_SS_ED_2R_MOM\nd, s\nAddress of an array of raw\nmoments of the second\norder\nOptional. Set entries of the array\nto meaningful values (typically\nzero) if you intend to compute a\nprogressive estimate. Otherwise,\ndo not initialize the array.\nVSL_SS_ED_3R_MOM\nd, s\nAddress of an array of raw\nmoments of the third order\nOptional. Set entries of the array\nto meaningful values (typically\nzero) if you intend to compute a\nprogressive estimate. Otherwise,\ndo not initialize the array.\nVSL_SS_ED_4R_MOM\nd, s\nAddress of an array of raw\nmoments of the fourth order\nOptional. Set entries of the array\nto meaningful values (typically\nzero) if you intend to compute a\nprogressive estimate. Otherwise,\ndo not initialize the array.\nVSL_SS_ED_2C_MOM\nd, s\nAddress of an array of\ncentral moments of the\nsecond order\nOptional. Set entries of the array\nto meaningful values (typically\nzero) if you intend to compute a\nprogressive estimate. Otherwise,\ndo not initialize the array. Make\nsure you also provide arrays for\nraw moments of the first and\nsecond order.\nVSL_SS_ED_3C_MOM\nd, s\nAddress of an array of\ncentral moments of the\nthird order\nOptional. Set entries of the array\nto meaningful values (typically\nzero) if you intend to compute a\nprogressive estimate. Otherwise,\ndo not initialize the array. Make\nsure you also provide arrays for\nraw moments of the first, second,\nand third order.\nVSL_SS_ED_4C_MOM\nd, s\nAddress of an array of\ncentral moments of the\nfourth order\nOptional. Set entries of the array\nto meaningful values (typically\nzero) if you intend to compute a\nprogressive estimate. Otherwise,\ndo not initialize the array. Make\nsure you also provide arrays for\nraw moments of the first, second,\nthird, and fourth order.\nVSL_SS_ED_KURTOSIS\nd, s\nAddress of the array of\nkurtosis estimates\nOptional. Set entries of the array\nto meaningful values (typically\nzero) if you intend to compute a\nprogressive estimate. Otherwise,\ndo not initialize the array. Make\nsure you also provide arrays for\nraw moments of the first, second,\nthird, and fourth order.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2258\n\n\nParameter Value\nType\nPurpose\nInitialization\nVSL_SS_ED_SKEWNESS\nd, s\nAddress of the array of\nskewness estimates\nOptional. Set entries of the array\nto meaningful values (typically\nzero) if you intend to compute a\nprogressive estimate. Otherwise,\ndo not initialize the array. Make\nsure you also provide arrays for\nraw moments of the first, second,\nand third order.\nVSL_SS_ED_MIN\nd, s\nAddress of the array of\nminimum estimates\nOptional. Set entries of array to\nmeaningful values, such as the\nvalues of the first observation.\nVSL_SS_ED_MAX\nd, s\nAddress of the array of\nmaximum estimates\nOptional. Set entries of array to\nmeaningful values, such as the\nvalues of the first observation.\nVSL_SS_ED_VARIATION\nd, s\nAddress of the array of\nvariation coefficients\nOptional. Set entries of the array\nto meaningful values (typically\nzero) if you intend to compute a\nprogressive estimate. Otherwise,\ndo not initialize the array. Make\nsure you also provide arrays for\nraw moments of the first and\nsecond order.\nVSL_SS_ED_COV\nd, s\nAddress of a covariance\nmatrix\nOptional. Set entries of the array\nto meaningful values (typically\nzero) if you intend to compute a\nprogressive estimate. Make sure\nyou also provide an array for the\nmean.\nVSL_SS_ED_COV_STORAGE\ni\nAddress of the variable that\nholds the storage format for\na covariance matrix\nRequired. Provide a storage\nformat supported by the library\nwhenever you intend to compute\nthe covariance matrix.2\nVSL_SS_ED_COR\nd, s\nAddress of a correlation\nmatrix\nOptional. Set entries of the array\nto meaningful values (typically\nzero) if you intend to compute a\nprogressive estimate. If you\ninitialize the matrix in non-trivial\nway, make sure that the main\ndiagonal contains variance values.\nAlso, provide an array for the\nmean.\nVSL_SS_ED_COR_STORAGE\ni\nAddress of the variable that\nholds the correlation storage\nformat for a correlation\nmatrix\nRequired. Provide a storage\nformat supported by the library\nwhenever you intend to compute\nthe correlation matrix.2\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2259\n\n\nParameter Value\nType\nPurpose\nInitialization\nVSL_SS_ED_ACCUM_WEIGHT\nd, s\nAddress of the array of size\n2 that holds the\naccumulated weight (sum of\nweights) in the first position\nand the sum of weights\nsquared in the second\nposition\nOptional. Set the entries of the\nmatrix to meaningful values\n(typically zero) if you intend to do\nprogressive processing of the\ndataset or need the sum of\nweights and sum of squared\nweights assigned to observations.\nVSL_SS_ED_QUANT_ORDER_N\ni\nAddress of the variable that\nholds the number of\nquantile orders\nRequired. Positive integer value.\nProvide the number of quantile\norders whenever you compute\nquantiles.\nVSL_SS_ED_QUANT_ORDER\nd, s\nAddress of the array of\nquantile orders\nRequired. Set entries of array to\nvalues from the interval (0,1).\nProvide this parameter whenever\nyou compute quantiles.\nVSL_SS_ED_QUANT_QUANTILE\nS\nd, s\nAddress of the array of\nquantiles\nNone.\nVSL_SS_ED_ORDER_STATS\nd, s\nAddress of the array of\norder statistics\nNone.\nVSL_SS_ED_GROUP_INDC\ni\nAddress of the array of\ngroup indices used in\ncomputation of a pooled\ncovariance matrix\nRequired. Set entry i to integer\nvalue k if the observation belongs\nto group k. Values of k take values\nin the range [0, g-1], where g is\nthe number of groups.\nVSL_SS_ED_POOLED_COV_STO\nRAGE\ni\nAddress of a variable that\nholds the storage format for\na pooled covariance matrix\nRequired. Provide a storage\nformat supported by the library\nwhenever you intend to compute\npooled covariance.2\nVSL_SS_ED_POOLED_MEAN\nd, s\nAddress of an array of\npooled means\nNone.\nVSL_SS_ED_POOLED_COV\nd, s\nAddress of pooled\ncovariance matrices\nNone.\nVSL_SS_ED_GROUP_COV_INDC\ni\nAddress of an array of\nindices for which\ncovariance/means should be\ncomputed\nOptional. Set the kth entry of the\narray to 1 if you need group\ncovariance and mean for group k;\notherwise set it to zero.\nVSL_SS_ED_REQ_GROUP_INDC\ni\nAddress of an array of\nindices for which group\nestimates such as\ncovariance or means are\nrequested\nOptional. Set the kth entry of the\narray to 1 if you need an estimate\nfor group k; otherwise set it to\nzero.\nVSL_SS_ED_GROUP_MEANS\ni\nAddress of an array of group\nmeans\nNone.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2260\n\n\nParameter Value\nType\nPurpose\nInitialization\nVSL_SS_ED_GROUP_COV_STOR\nAGE\nd, s\nAddress of a variable that\nholds the storage format for\na group covariance matrix\nRequired. Provide a storage\nformat supported by the library\nwhenever you intend to get group\ncovariance.2\nVSL_SS_ED_GROUP_COV\nd, s\nAddress of group covariance\nmatrices\nNone.\nVSL_SS_ED_ROBUST_COV_STO\nRAGE\nd, s\nAddress of a variable that\nholds the storage format for\na robust covariance matrix\nRequired. Provide a storage\nformat supported by the library\nwhenever you compute robust\ncovariance2.\nVSL_SS_ED_ROBUST_COV_PAR\nAMS_N\ni\nAddress of a variable that\nholds the number of\nalgorithmic parameters of\nthe method for robust\ncovariance estimation\nRequired. Set to the number of\nTBS parameters,\nVSL_SS_TBS_PARAMS_N.\nVSL_SS_ED_ROBUST_COV_PAR\nAMS\nd, s\nAddress of an array of\nparameters of the method\nfor robust estimation of a\ncovariance\nRequired. Set the entries of the\narray according to the description\nin vslSSEditRobustCovariance.\nVSL_SS_ED_ROBUST_MEAN\ni\nAddress of an array of\nrobust means\nNone.\nVSL_SS_ED_ROBUST_COV\nd, s\nAddress of a robust\ncovariance matrix\nNone.\nVSL_SS_ED_OUTLIERS_PARAM\nS_N\nd, s\nAddress of a variable that\nholds the number of\nparameters of the outlier\ndetection method\nRequired. Set to the number of\noutlier detection parameters,\nVSL_SS_BACON_PARAMS_N.\nVSL_SS_ED_OUTLIERS_PARAM\nS\ni\nAddress of an array of\nalgorithmic parameters for\nthe outlier detection method\nRequired. Set the entries of the\narray according to the description\nin vslSSEditOutliersDetection.\nVSL_SS_ED_OUTLIERS_WEIGH\nT\nd, s\nAddress of an array of\nweights assigned to\nobservations by the outlier\ndetection method\nNone.\nVSL_SS_ED_ORDER_STATS_ST\nORAGE\nd, s\nAddress of a variable that\nholds the storage format of\nan order statistics matrix\nRequired. Provide a storage\nformat supported by the library\nwhenever you compute a matrix\nof order statistics.1\nVSL_SS_ED_PARTIAL_COV_ID\nX\ni\nAddress of an array that\nencodes subcomponents of\na random vector\nRequired. Set the entries of the\narray according to the description\nin vslSSEditPartialCovCor.\nVSL_SS_ED_PARTIAL_COV\nd, s\nAddress of a partial\ncovariance matrix\nNone.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2261\n\n\nParameter Value\nType\nPurpose\nInitialization\nVSL_SS_ED_PARTIAL_COV_ST\nORAGE\ni\nAddress of a variable that\nholds the storage format of\na partial covariance matrix\nRequired. Provide a storage\nformat supported by the library\nwhenever you compute the partial\ncovariance.2\nVSL_SS_ED_PARTIAL_COR\nd, s\nAddress of a partial\ncorrelation matrix\nNone.\nVSL_SS_ED_PARTIAL_COR_ST\nORAGE\ni\nAddress of a variable that\nholds the storage format for\na partial correlation matrix\nRequired. Provide a storage\nformat supported by the library\nwhenever you compute the partial\ncorrelation.2\nVSL_SS_ED_MI_PARAMS_N\ni\nAddress of a variable that\nholds the number of\nalgorithmic parameters for\nthe Multiple Imputation\nmethod\nRequired. Set to the number of MI\nparameters,\nVSL_SS_MI_PARAMS_SIZE.\nVSL_SS_ED_MI_PARAMS\nd, s\nAddress of an array of\nalgorithmic parameters for\nthe Multiple Imputation\nmethod\nRequired. Set entries of the array\naccording to the description in \nvslSSEditMissingValues.\nVSL_SS_ED_MI_INIT_ESTIMA\nTES_N\ni\nAddress of a variable that\nholds the number of initial\nestimates for the Multiple\nImputation method\nOptional. Set to p+p*(p+1)/2,\nwhere p is the task dimension.\nVSL_SS_ED_MI_INIT_ESTIMA\nTES\nd, s\nAddress of an array of initial\nestimates for the Multiple\nImputation method\nOptional. Set the values of the\narray according to the description\nin \"Basic Components of the\nMultiple Imputation Function in\nSummary Statistics\" in the Intel®\noneAPI Math Kernel Library\n(oneMKL) Summary Statistics\nApplication Notes document [SS\nNotes].\nVSL_SS_ED_MI_SIMUL_VALS_\nN\ni\nAddress of a variable that\nholds the number of\nsimulated values in the\nMultiple Imputation method\nOptional. Positive integer\nindicating the number of missing\npoints in the observation matrix.\nVSL_SS_ED_MI_SIMUL_VALS\nd, s\nAddress of an array of\nsimulated values in the\nMultiple Imputation method\nNone.\nVSL_SS_ED_MI_ESTIMATES_N\ni\nAddress of a variable that\nholds the number of\nestimates obtained as a\nresult of the Multiple\nImputation method\nOptional. Positive integer number\ndefined according to the\ndescription in \"Basic Components\nof the Multiple Imputation\nFunction in Summary Statistics\" in\nthe Intel® oneAPI Math Kernel\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2262\n\n\nParameter Value\nType\nPurpose\nInitialization\nLibrary (oneMKL) Summary\nStatistics Application Notes\ndocument [SS Notes].\nVSL_SS_ED_MI_ESTIMATES\nd, s\nAddress of an array of\nestimates obtained as a\nresult of the Multiple\nImputation method\nNone.\nVSL_SS_ED_MI_PRIOR_N\ni\nAddress of a variable that\nholds the number of prior\nparameters for the Multiple\nImputation method\nOptional. If you pass a user-\ndefined array of prior parameters,\nset this parameter to (p2+3*p\n+4)/2, where p is the task\ndimension.\nVSL_SS_ED_MI_PRIOR\nd, s\nAddress of an array of prior\nparameters for the Multiple\nImputation method\nOptional. Set entries of the array\nof prior parameters according to\nthe description in \"Basic\nComponents of the Multiple\nImputation Function in Summary\nStatistics\" in the Intel® oneAPI\nMath Kernel Library (oneMKL)\nSummary Statistics Application\nNotes document [SS Notes].\nVSL_SS_ED_PARAMTR_COR\nd, s\nAddress of a parameterized\ncorrelation matrix\nNone.\nVSL_SS_ED_PARAMTR_COR_ST\nORAGE\ni\nAddress of a variable that\nholds the storage format of\na parameterized correlation\nmatrix\nRequired. Provide a storage\nformat supported by the library\nwhenever you compute the\nparameterized correlation matrix.2\nVSL_SS_ED_STREAM_QUANT_P\nARAMS_N\ni\nAddress of a variable that\nholds the number of\nparameters of a quantile\ncomputation method for\nstreaming data\nRequired. Set to the number of\nquantile computation parameters,\nVSL_SS_SQUANTS_ZW_PARAMS_N.\nVSL_SS_ED_STREAM_QUANT_P\nARAMS\nd, s\nAddress of an array of\nparameters of a quantile\ncomputation method for\nstreaming data\nRequired. Set the entries of the\narray according to the description\nin \"Computing Quantiles for\nStreaming Data\" in the Intel®\noneAPI Math Kernel Library\n(oneMKL) Summary Statistics\nApplication Notes document [SS\nNotes].\nVSL_SS_ED_STREAM_QUANT_O\nRDER_N\ni\nAddress of a variable that\nholds the number of\nquantile orders for\nstreaming data\nRequired. Positive integer value.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2263\n\n\nParameter Value\nType\nPurpose\nInitialization\nVSL_SS_ED_STREAM_QUANT_O\nRDER\nd, s\nAddress of an array of\nquantile orders for\nstreaming data\nRequired. Set entries of the array\nto values from the interval (0,1).\nProvide this parameter whenever\nyou compute quantiles.\nVSL_SS_ED_STREAM_QUANT_Q\nUANTILES\nd, s\nAddress of an array of\nquantiles for streaming data\nNone.\nVSL_SS_ED_SUM\nd, s\nAddress of array of sums\nOptional. Set entries of the array\nto meaningful values (typically\nzero) if you intend to compute a\nprogressive estimate. Otherwise,\ndo not initialize the array.\nVSL_SS_ED_2R_SUM\nd, s\nAddress of array of raw\nsums of 2nd order\nOptional. Set entries of the array\nto meaningful values (typically\nzero) if you intend to compute a\nprogressive estimate. Otherwise,\ndo not initialize the array.\nVSL_SS_ED_3R_SUM\nd, s\nAddress of array of raw\nsums of 3rd order\nOptional. Set entries of the array\nto meaningful values (typically\nzero) if you intend to compute a\nprogressive estimate. Otherwise,\ndo not initialize the array.\nVSL_SS_ED_4R_SUM\nd, s\nAddress of array of raw\nsums of 4th order\nOptional. Set entries of the array\nto meaningful values (typically\nzero) if you intend to compute a\nprogressive estimate. Otherwise,\ndo not initialize the array.\nVSL_SS_ED_2C_SUM\nd, s\nAddress of array of central\nsums of 2nd order\nOptional. Set entries of the array\nto meaningful values (typically\nzero) if you intend to compute a\nprogressive estimate. Otherwise,\ndo not initialize the array.\nVSL_SS_ED_3C_SUM\nd, s\nAddress of array of central\nsums of 3rd order\nOptional. Set entries of the array\nto meaningful values (typically\nzero) if you intend to compute a\nprogressive estimate. Otherwise,\ndo not initialize the array.\nVSL_SS_ED_4C_SUM\nd, s\nAddress of array of central\nsums of 4th order\nOptional. Set entries of the array\nto meaningful values (typically\nzero) if you intend to compute a\nprogressive estimate. Otherwise,\ndo not initialize the array.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2264\n\n\nParameter Value\nType\nPurpose\nInitialization\nVSL_SS_ED_CP\nd, s\nAddress of cross-product\nmatrix\nOptional. Set entries of the array\nto meaningful values (typically\nzero) if you intend to compute a\nprogressive estimate. Otherwise,\ndo not initialize the array.\nVSL_SS_ED_MDAD\nd, s\nAddress of array of median\nabsolute deviations\nNone.\nVSL_SS_ED_MNAD\nd, s\nAddress of array of mean\nabsolute deviations\nNone.\nVSL_SS_ED_SORTED_OBSERV\nd, s\nAddress of the array that\nstores sorted results\nNone.\nVSL_SS_ED_SORTED_OBSERV_\nSTORAGE\ni\nAddress of a variable that\nholds the storage format of\nan output matrix\nRequired. Provide a supported\nstorage format whenever you\nspecify sorting of the observation\nmatrix.\n1.\nSee Table: \"Storage format of matrix of observations and order statistics\" for storage formats.\n2.\nSee Table: \"Storage formats of a variance-covariance/correlation matrix\" for storage formats.\nvslSSEditMoments\nModifies the pointers to arrays that hold moment\nestimates.\nSyntax\nstatus = vslsSSEditMoments(task, mean, r2m, r3m, r4m, c2m, c3m, c4m);\nstatus = vsldSSEditMoments(task, mean, r2m, r3m, r4m, c2m, c3m, c4m);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\ntask\nVSLSSTaskPtr\nDescriptor of the task\nmean\nfloat* for vslsSSEditMoments\ndouble* for vsldSSEditMoments\nPointer to the array of means\nr2m\nfloat* for vslsSSEditMoments\ndouble* for vsldSSEditMoments\nPointer to the array of raw moments of\nthe 2nd order\nr3m\nfloat* for vslsSSEditMoments\ndouble* for vsldSSEditMoments\nPointer to the array of raw moments of\nthe 3rd order\nr4m\nfloat* for vslsSSEditMoments\ndouble* for vsldSSEditMoments\nPointer to the array of raw moments of\nthe 4th order\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2265\n\n\nName\nType\nDescription\nc2m\nfloat* for vslsSSEditMoments\ndouble* for vsldSSEditMoments\nPointer to the array of central moments of\nthe 2nd order\nc3m\nfloat* for vslsSSEditMoments\ndouble* for vsldSSEditMoments\nPointer to the array of central moments of\nthe 3rd order\nc4m\nfloat* for vslsSSEditMoments\ndouble* for vsldSSEditMoments\nPointer to the array of central moments of\nthe 4th order\nOutput Parameters\nName\nType\nDescription\nstatus\nint\nCurrent status of the task\nDescription\nThe vslSSEditMoments routine replaces pointers to the arrays that hold estimates of raw and central\nmoments with values passed as corresponding parameters of the routine. If you pass a value of NULL for a\nspecific input parameter, the value of that parameter in the task descriptor is unchanged.\nvslSSEditSums\nModifies the pointers to arrays that hold sum\nestimates.\nSyntax\nstatus = vslsSSEditSums(task, sum, r2s, r3s, r4s, c2s, c3s, c4s);\nstatus = vsldSSEditSums(task, sum, r2s, r3s, r4s, c2s, c3s, c4s);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\ntask\nVSLSSTaskPtr\nDescriptor of the task\nsum\nfloat* for vslsSSEditSums\ndouble* for vsldSSEditSums\nPointer to the array of sums\nr2s\nfloat* for vslsSSEditSums\ndouble* for vsldSSEditSums\nPointer to the array of raw sums of the\nsecond order\nr3s\nfloat* for vslsSSEditSums\ndouble* for vsldSSEditSums\nPointer to the array of raw sums of the\nthird order\nr4s\nfloat* for vslsSSEditSums\ndouble* for vsldSSEditSums\nPointer to the array of raw sums of the\nfourth order\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2266\n\n\nName\nType\nDescription\nc2s\nfloat* for vslsSSEditSums\ndouble* for vsldSSEditSums\nPointer to the array of central sums of the\nsecond order\nc3s\nfloat* for vslsSSEditSums\ndouble* for vsldSSEditSums\nPointer to the array of central sums of the\nthird order\nc4s\nfloat* for vslsSSEditSums\ndouble* for vsldSSEditSums\nPointer to the array of central sums of the\nfourth order\nOutput Parameters\nName\nType\nDescription\nstatus\nint\nCurrent status of the task\nDescription\nThe vslSSEditSums routine replaces pointers to the arrays that hold estimates of raw and central sums with\nvalues passed as corresponding parameters of the routine. If you pass a value of NULL for a specific input\nparameter, the value of that parameter in the task descriptor is unchanged.\nvslSSEditCovCor\nModifies the pointers to covariance/correlation/cross-\nproduct parameters.\nSyntax\nstatus = vslsSSEditCovCor(task, mean, cov, cov_storage, cor, cor_storage);\nstatus = vsldSSEditCovCor(task, mean, cov, cov_storage, cor, cor_storage);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\ntask\nVSLSSTaskPtr\nDescriptor of the task\nmean\nfloat* for vslsSSEditCovCor\ndouble* for vsldSSEditCovCor\nPointer to the array of means\ncov\nfloat* for vslsSSEditCovCor\ndouble* for vsldSSEditCovCor\nPointer to a covariance matrix\ncov_storage\nconst MKL_INT*\nPointer to the storage format of the\ncovariance matrix\ncor\nfloat* for vslsSSEditCovCor\ndouble* for vsldSSEditCovCor\nPointer to a correlation matrix\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2267\n\n\nName\nType\nDescription\ncor_storage\nconst MKL_INT*\nPointer to the storage format of the\ncorrelation matrix\nOutput Parameters\nName\nType\nDescription\nstatus\nint\nCurrent status of the task\nDescription\nThe vslSSEditCovCor routine replaces pointers to the array of means, covariance/correlation arrays, and\ntheir storage format with values passed as corresponding parameters of the routine. If you pass a value of\nNULL for a specific input parameter, the value of that parameter in the task descriptor is unchanged.\nThe storage parameters, cov_storage and cor_storage, describe the storage format used for the p-by-p\nsymmetric variance-covariance/correlation/cross-product matrix C. The matrix C can be described as\nC =\nc1, 1 c1, 2 ⋯⋯⋯c1, p\nc2, 1 c2, 2 ⋯⋯⋯c2, p\n⋮\n⋮\n⋱\n⋮\n⋮\n⋮\nci, j\n⋮\n⋮\n⋮\n⋱\n⋮\ncp, 1 cp, 2 ⋯⋯⋯cp, p\nTable \"Storage formats of a variance-covariance/correlation/cross-product matrix\" shows how the matrix is\nstored in a one-dimensional array cp for different values of the storage parameters.\nStorage formats of variance-covariance/correlation/cross-product matrices\nParameter\nDescription\nVSL_SS_MATRIX_STORAGE_FULL\nThe array cp contains all elements of the matrix stored\nsequentially, row-by-row:\ncp[0] contains c1, 1\ncp[1] contains c1, 2\ncp[p - 1] contains c1,p\ncp[p] contains c2,1\ncp[p*p - 1] contains cp,p\nThe size of array cp is p*p.\nVSL_SS_MATRIX_STORAGE_L_PACKED\nThe array cp contains the lower triangular part of the\nsymmetric matrix stored sequentially, row-by-row:\ncp[0] contains c1, 1\ncp[1] contains c2, 1\ncp[2] contains c2, 2\nand so on.\nThe size of the array is p*(p+ 1)/2.\nVSL_SS_MATRIX_STORAGE_U_PACKED\nThe array cp contains the upper triangular part of the\nsymmetric matrix stored sequentially, row-by-row:\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2268\n\n\nParameter\nDescription\ncp[0] contains c1, 1\ncp[1] contains c1, 2\ncp[3] contains c1, 3\nand so on.\nThe size of the array is p*(p+ 1)/2.\nvslSSEditCP\nModifies the pointers to cross-product matrix\nparameters.\nSyntax\nstatus = vslsSSEditCP(task, mean, sum, cp, cp_storage);\nstatus = vsldSSEditCP(task, mean, sum, cp, cp_storage);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\ntask\nVSLSSTaskPtr\nDescriptor of the task\nmean\nfloat* for vslsSSEditCP\ndouble* for vsldSSEditCP\nPointer to array of means\nsum\nfloat* for vslsSSEditCP\ndouble* for vsldSSEditCP\nPointer to array of sums\ncp\nfloat* for vslsSSEditCP\ndouble* for vsldSSEditCP\nPointer to a cross-product matrix\ncp_storage\nconst MKL_INT*\nPointer to the storage format of the cross-\nproduct matrix\nOutput Parameters\nName\nType\nDescription\nstatus\nint\nCurrent status of the task\nDescription\nThe vslSSEditCP routine replaces pointers to the array of means, array of sums, cross-product matrix, and\nits storage format with values passed as corresponding parameters of the routine. See Table: \"Storage\nformats of a variance-covariance/correlation/cross-product matrix\" for possible values of the cp_storage\nparameter. If you pass a value of NULL for a specific input parameter, the value of that parameter in the task\ndescriptor is unchanged.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2269\n\n\nStorage formats of variance-covariance/correlation/cross-product matrices\nParameter\nDescription\nVSL_SS_MATRIX_STORAGE_FULL\nThe array cp contains all elements of the matrix stored\nsequentially, row-by-row:\ncp[0] contains c1, 1\ncp[1] contains c1, 2\ncp[p - 1] contains c1,p\ncp[p] contains c2,1\ncp[p*p - 1] contains cp,p\nThe size of array cp is p*p.\nVSL_SS_MATRIX_STORAGE_L_PACKED\nThe array cp contains the lower triangular part of the\nsymmetric matrix stored sequentially, row-by-row:\ncp[0] contains c1, 1\ncp[1] contains c2, 1\ncp[2] contains c2, 2\nand so on.\nThe size of the array is p*(p+ 1)/2.\nVSL_SS_MATRIX_STORAGE_U_PACKED\nThe array cp contains the upper triangular part of the\nsymmetric matrix stored sequentially, row-by-row:\ncp[0] contains c1, 1\ncp[1] contains c1, 2\ncp[3] contains c1, 3\nand so on.\nThe size of the array is p*(p+ 1)/2.\nvslSSEditPartialCovCor\nModifies the pointers to partial covariance/correlation\nparameters.\nSyntax\nstatus = vslsSSEditPartialCovCor(task, p_idx_array, cov, cov_storage, cor, cor_storage,\np_cov, p_cov_storage, p_cor, p_cor_storage);\nstatus = vsldSSEditPartialCovCor(task, p_idx_array, cov, cov_storage, cor, cor_storage,\np_cov, p_cov_storage, p_cor, p_cor_storage);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\ntask\nVSLSSTaskPtr\nDescriptor of the task\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2270\n\n\nName\nType\nDescription\np_idx_array\nconst MKL_INT*\nPointer to the array that encodes indices\nof subcomponents Z and Y of the random\nvector as described in section \nMathematical Notation and Definitions.\np_idx_array[i] equals to\n-1 if the i-th component of the random\nvector belongs to Z\n1, if the i-th component of the random\nvector belongs to Y.\ncov\nconst float* for\nvslsSSEditPartialCovCor\nconst double* for\nvsldSSEditPartialCovCor\nPointer to a covariance matrix\ncov_storage\nconst MKL_INT*\nPointer to the storage format of the\ncovariance matrix\ncor\nconst float* for\nvslsSSEditPartialCovCor\nconst double* for\nvsldSSEditPartialCovCor\nPointer to a correlation matrix\ncor_storage\nconst MKL_INT*\nPointer to the storage format of the\ncorrelation matrix\np_cov\nfloat* for vslsSSEditPartialCovCor\ndouble* for vsldSSEditPartialCovCor\nPointer to a partial covariance matrix\np_cov_storage\nconst MKL_INT*\nPointer to the storage format of the\npartial covariance matrix\np_cor\nfloat* for vslsSSEditPartialCovCor\ndouble* for vsldSSEditPartialCovCor\nPointer to a partial correlation matrix\np_cor_storage\nconst MKL_INT*\nPointer to the storage format of the\npartial correlation matrix\nOutput Parameters\nName\nType\nDescription\nstatus\nint\nCurrent status of the task\nDescription\nThe vslSSEditPartialCovCor routine replaces pointers to covariance/correlation arrays, partial covariance/\ncorrelation arrays, and their storage format with values passed as corresponding parameters of the routine.\nSee Table \"Storage formats of a variance-covariance/correlation matrix\" for possible values of the\ncov_storage, cor_storage, p_cov_storage, and p_cor_storage parameters. If you pass a value of NULL\nfor a specific input parameter, the value of that parameter in the task descriptor is unchanged.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2271\n\n\nvslSSEditQuantiles\nModifies the pointers to parameters related to quantile\ncomputations.\nSyntax\nstatus = vslsSSEditQuantiles(task, quant_order_n, quant_order, quants, order_stats,\norder_stats_storage);\nstatus = vsldSSEditQuantiles(task, quant_order_n, quant_order, quants, order_stats,\norder_stats_storage);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\ntask\nVSLSSTaskPtr\nDescriptor of the task\nquant_order_n\nconst MKL_INT*\nPointer to the number of quantile\norders\nquant_order\nconst float* for\nvslsSSEditQuantiles\nconst double* for\nvsldSSEditQuantiles\nPointer to the array of quantile\norders\nquants\nfloat* for vslsSSEditQuantiles\ndouble* for vsldSSEditQuantiles\nPointer to the array of quantiles\norder_stats\nfloat* for vslsSSEditQuantiles\ndouble* for vsldSSEditQuantiles\nPointer to the array of order\nstatistics\norder_stats_storage\nconst MKL_INT*\nPointer to the storage format of the\norder statistics array\nOutput Parameters\nName\nType\nDescription\nstatus\nint\nCurrent status of the task\nDescription\nThe vslSSEditQuantiles routine replaces pointers to the number of quantile orders, the array of quantile\norders, the array of quantiles, the array that holds order statistics, and the storage format for the order\nstatistics with values passed into the routine. See Table \"Storage format of matrix of observations and order\nstatistics\" for possible values of the order_statistics_storage parameter. If you pass a value of NULL for\na specific input parameter, the value of that parameter in the task descriptor is unchanged.\nvslSSEditStreamQuantiles\nModifies the pointers to parameters related to quantile\ncomputations for streaming data.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2272\n\n\nSyntax\nstatus = vslsSSEditStreamQuantiles(task, quant_order_n, quant_order, quants, nparams,\nparams);\nstatus = vsldSSEditStreamQuantiles(task, quant_order_n, quant_order, quants, nparams,\nparams);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\ntask\nVSLSSTaskPtr\nDescriptor of the task\nquant_order_n\nconst MKL_INT*\nPointer to the number of quantile orders\nquant_order\nconst float* for\nvslsSSEditStreamQuantiles\nconst double* for\nvsldSSEditStreamQuantiles\nPointer to the array of quantile orders\nquants\nfloat* for\nvslsSSEditStreamQuantiles\ndouble* for\nvsldSSEditStreamQuantiles\nPointer to the array of quantiles\nnparams\nconst MKL_INT*\nPointer to the number of the algorithm\nparameters\nparams\nconst float*\nfor vslsSSEditStreamQuantiles\nconst double*\nfor vsldSSEditStreamQuantiles\nPointer to the array of the algorithm\nparameters\nOutput Parameters\nName\nType\nDescription\nstatus\nint\nCurrent status of the task\nDescription\nThe vslSSEditStreamQuantiles routine replaces pointers to the number of quantile orders, the array of\nquantile orders, the array of quantiles, the number of the algorithm parameters, and the array of the\nalgorithm parameters with values passed into the routine. If you pass a value of NULL for a specific input\nparameter, the value of that parameter in the task descriptor is unchanged.\nvslSSEditPooledCovariance\nModifies pooled/group covariance matrix array\npointers.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2273\n\n\nSyntax\nstatus = vslsSSEditPooledCovariance(task, grp_indices, pld_mean, pld_cov,\nreq_grp_indices, grp_means, grp_cov);\nstatus = vsldSSEditPooledCovariance(task, grp_indices, pld_mean, pld_cov,\nreq_grp_indices, grp_means, grp_cov);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\ntask\nVSLSSTaskPtr\nDescriptor of the task\ngrp_indices\nconst MKL_INT*\nPointer to an array of size n. The i-th\nelement of the array contains the number\nof the group the observation belongs to.\npld_mean\nfloat* for\nvslsSSEditPooledCovariance\ndouble* for\nvsldSSEditPooledCovariance\nPointer to the array of pooled means\npld_cov\nfloat* for\nvslsSSEditPooledCovariance\ndouble* for\nvsldSSEditPooledCovariance\nPointer to the array that holds a pooled\ncovariance matrix\nreq_grp_indices\nconst MKL_INT*\nPointer to the array that contains indices\nof groups for which estimates to return\n(such as covariance and mean)\ngrp_means\nfloat* for\nvslsSSEditPooledCovariance\ndouble* for\nvsldSSEditPooledCovariance\nPointer to the array of group means\ngrp_cov\nfloat* for\nvslsSSEditPooledCovariance\ndouble* for\nvsldSSEditPooledCovariance\nPointer to the array that holds group\ncovariance matrices\nOutput Parameters\nName\nType\nDescription\nstatus\nint\nCurrent status of the task\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2274\n\n\nDescription\nThe vslSSEditPooledCovariance routine replaces pointers to the array of group indices, the array of\npooled means, the array for a pooled covariance matrix, and pointers to the array of indices of group\nmatrices, the array of group means, and the array for group covariance matrices with values passed in the\neditors. If you pass a value of NULL for a specific input parameter, the value of that parameter in the task\ndescriptor is unchanged.. Use the vslSSEditTask routine to replace the storage format for pooled and\ngroup covariance matrices.\nvslSSEditRobustCovariance\nModifies pointers to arrays related to a robust\ncovariance matrix.\nSyntax\nstatus = vslsSSEditRobustCovariance(task, rcov_storage, nparams, params, rmean, rcov);\nstatus = vsldSSEditRobustCovariance(task, rcov_storage, nparams, params, rmean, rcov);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\ntask\nVSLSSTaskPtr\nDescriptor of the task\nrcov_storage\nconst MKL_INT*\nPointer to the storage format of a robust\ncovariance matrix\nnparams\nconst MKL_INT*\nPointer to the number of method\nparameters\nparams\nconst float* for\nvslsSSEditRobustCovariance\nconst double* for\nvsldSSEditRobustCovariance\nPointer to the array of method parameters\n \n \n \nrmean\nfloat* for\nvslsSSEditRobustCovariance\ndouble* for\nvsldSSEditRobustCovariance\nPointer to the array of robust means\nrcov\nfloat* for\nvslsSSEditRobustCovariance\ndouble* for\nvsldSSEditRobustCovariance\nPointer to a robust covariance matrix\nOutput Parameters\nName\nType\nDescription\nstatus\nint\nCurrent status of the task\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2275\n\n\nDescription\nThe vslSSEditRobustCovariance routine uses values passed as parameters of the routine to replace:\n•\npointers to covariance matrix storage\n•\npointers to the number of method parameters and to the array of the method parameters of size nparams\n•\npointers to the arrays that hold robust means and covariance\nSee Table \"Storage formats of a variance-covariance/correlation matrix\" for possible values of the\nrcov_storage parameter. If you pass a value of NULL for a specific input parameter, the value of that\nparameter in the task descriptor is unchanged.\nIntel® oneAPI Math Kernel Library (oneMKL) provides a Translated Biweight S-estimator (TBS) for robust\nestimation of a variance-covariance matrix and mean [Rocke96]. Use one iteration of the Maronna algorithm\nwith the reweighting step [Maronna02] to compute the initial point of the algorithm. Pack the parameters of\nthe TBS algorithm into the params array and pass them into the editor. Table \"Structure of the Array of TBS\nParameters\" describes the params structure.\nStructure of the Array of TBS Parameters\nArray Position\nAlgorithm\nParameter\nDescription\n0\nε\nBreakdown point, the number of outliers the algorithm can\nhold. By default, the value is (n-p)/(2n).\n1\nα\nAsymptotic rejection probability, see details in [Rocke96]. By\ndefault, the value is 0.001.\n2\nδ\nStopping criterion: the algorithm is terminated if weights are\nchanged less than δ. By default, the value is 0.001.\n3\nmax_iter\nMaximum number of iterations. The algorithm terminates after\nmax_iter iterations. By default, the value is 10.\nIf you set this parameter to zero, the function returns a robust\nestimate of the variance-covariance matrix computed using\nthe Maronna method [Maronna02] only.\nThe robust estimator of variance-covariance implementation in Intel® oneAPI Math Kernel Library (oneMKL)\nrequires that the number of observationsn be greater than twice the number of variables: n > 2p.\nSee additional details of the algorithm usage model in the Intel® oneAPI Math Kernel Library (oneMKL)\nSummary Statistics Application Notes document [SS Notes].\nvslSSEditOutliersDetection\nModifies array pointers related to multivariate outliers\ndetection.\nSyntax\nstatus = vslsSSEditOutliersDetection(task, nparams, params, w);\nstatus = vsldSSEditOutliersDetection(task, nparams, params, w);\nInclude Files\n•\nmkl.h\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2276\n\n\nInput Parameters\nName\nType\nDescription\ntask\nVSLSSTaskPtr\nDescriptor of the task\nnparams\nconst MKL_INT*\nPointer to the number of method\nparameters\nparams\nconst float* for\nvslsSSEditOutliersDetection\nconst double* for\nvsldSSEditOutliersDetection\nPointer to the array of method parameters\n \n \n \nw\nfloat* for\nvslsSSEditOutliersDetection\ndouble* for\nvsldSSEditOutliersDetection\nPointer to an array of size n. The array\nholds the weights of observations to be\nmarked as outliers.\nOutput Parameters\nName\nType\nDescription\nstatus\nint\nCurrent status of the task\nDescription\nThe vslSSEditOutliersDetection routine uses the parameters passed to replace\n•\nthe pointers to the number of method parameters and to the array of the method parameters of size\nnparams\n•\nthe pointer to the array that holds the calculated weights of the observations\nIf you pass a value of NULL for a specific input parameter, the value of that parameter in the task descriptor\nis unchanged.\nIntel® oneAPI Math Kernel Library (oneMKL) provides the BACON algorithm ([Billor00]) for the detection of\nmultivariate outliers. Pack the parameters of the BACON algorithm into the params array and pass them into\nthe editor. Table \"Structure of the Array of BACON Parameters\" describes the params structure.\nStructure of the Array of BACON Parameters\nArray Position\nAlgorithm\nParameter\nDescription\n0\nMethod to start the\nalgorithm\nThe parameter takes one of the following possible values:\nVSL_SS_METHOD_BACON_MEDIAN_INIT, if the algorithm is\nstarted using the median estimate. This is the default value of\nthe parameter.\nVSL_SS_METHOD_BACON_MAHALANOBIS_INIT, if the algorithm\nis started using the Mahalanobis distances.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2277\n\n\nArray Position\nAlgorithm\nParameter\nDescription\n1\nα\nOne-tailed probability that defines the (1 - α) quantile of χ2\ndistribution with p degrees of freedom. The recommended\nvalue is α/ n, where n is the number of observations. By\ndefault, the value is 0.05.\n2\nδ\nStopping criterion; the algorithm is terminated if the size of\nthe basic subset is changed less than δ. By default, the value is\n0.005.\nOutput of the algorithm is the vector of weights, BaconWeights, such that BaconWeights(i) = 0 if i-th\nobservation is detected as an outlier. Otherwise BaconWeights(i) = w(i), where w is the vector of input\nweights. If you do not provide the vector of input weights, BaconWeights(i) is set to 1 if the i-th observation\nis not detected as an outlier.\nSee additional details about usage model of the algorithm in the Intel® oneAPI Math Kernel Library (oneMKL)\nSummary Statistics Application Notes document [SS Notes].\nvslSSEditMissingValues\nModifies pointers to arrays associated with the method\nof supporting missing values in a dataset.\nSyntax\nstatus = vslsSSEditMissingValues(task, nparams, params, init_estimates_n,\ninit_estimates, prior_n, prior, simul_missing_vals_n, simul_missing_vals, estimates_n,\nestimates);\nstatus = vsldSSEditMissingValues(task, nparams, params, init_estimates_n,\ninit_estimates, prior_n, prior, simul_missing_vals_n, simul_missing_vals, estimates_n,\nestimates);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\ntask\nVSLSSTaskPtr\nDescriptor of the task\nnparams\nconst MKL_INT*\nPointer to the number of method\nparameters\nparams\nconst float* for\nvslsSSEditMissingValues\nconst double* for\nvsldSSEditMissingValues\nPointer to the array of method\nparameters\ninit_estimates_n\nconst MKL_INT*\nPointer to the number of initial\nestimates for mean and a variance-\ncovariance matrix\n \n \n \n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2278\n\n\nName\nType\nDescription\ninit_estimates\nconst float* for\nvslsSSEditMissingValues\nconst double* for\nvsldSSEditMissingValues\nPointer to the array that holds initial\nestimates for mean and a variance-\ncovariance matrix\nprior_n\nconst MKL_INT*\nPointer to the number of prior\nparameters\nprior\nconst float* for\nvslsSSEditMissingValues\nconst double* for\nvsldSSEditMissingValues\nPointer to the array of prior\nparameters\nsimul_missing_vals_n\nconst MKL_INT*\nPointer to the size of the array that\nholds output of the Multiple\nImputation method\nsimul_missing_vals\nfloat* for vslsSSEditMissingValues\ndouble* for vsldSSEditMissingValues\nPointer to the array of size k*m,\nwhere k is the total number of\nmissing values, and m is number of\ncopies of missing values. The array\nholds m sets of simulated missing\nvalues for the matrix of\nobservations.\nestimates_n\nconst MKL_INT*\nPointer to the number of estimates\nto be returned by the routine\nestimates\nfloat* for vslsSSEditMissingValues\ndouble* for vsldSSEditMissingValues\nPointer to the array that holds\nestimates of the mean and a\nvariance-covariance matrix.\nOutput Parameters\nName\nType\nDescription\nstatus\nint\nCurrent status of the task\nDescription\nThe vslSSEditMissingValues routine uses values passed as parameters of the routine to replace pointers\nto the number and the array of the method parameters, pointers to the number and the array of initial\nmean/variance-covariance estimates, the pointer to the number and the array of prior parameters, pointers\nto the number and the array of simulated missing values, and pointers to the number and the array of the\nintermediate mean/covariance estimates. If you pass a value of NULL for a specific input parameter, the\nvalue of that parameter in the task descriptor is unchanged.\nBefore you call the Summary Statistics routines to process missing values, preprocess the dataset and\ndenote missing observations with one of the following predefined constants:\n•\nVSL_SS_SNAN, if the dataset is stored in single precision floating-point arithmetic\n•\nVSL_SS_DNAN, if the dataset is stored in double precision floating-point arithmetic\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2279\n\n\nIntel® oneAPI Math Kernel Library (oneMKL) provides theVSL_SS_METHOD_MI method to support missing\nvalues in the dataset based on the Multiple Imputation (MI) approach described in [Schafer97]. The following\ncomponents support Multiple Imputation:\n•\nExpectation Maximization (EM) algorithm to compute the start point for the Data Augmentation (DA)\nprocedure\n•\nDA function\nNOTE\nThe DA component of the MI procedure is simulation-based and uses the VSL_BRNG_MCG59\nbasic random number generator with predefined seed = 250 and the Gaussian distribution\ngenerator (ICDFmethod) available in Intel® oneAPI Math Kernel Library (oneMKL)\n[Gaussian].\nPack the parameters of the MI algorithm into the params array. Table \"Structure of the Array of MI\nParameters\" describes the params structure.\nStructure of the Array of MI Parameters\nArray Position\nAlgorithm Parameter\nDescription\n0\nem_iter_num\nMaximal number of iterations for the EM algorithm.\nBy default, this value is 50.\n1\nda_iter_num\nMaximal number of iterations for the DA algorithm.\nBy default, this value is 30.\n2\nε\nStopping criterion for the EM algorithm. The\nalgorithm terminates if the maximal module of the\nelement-wise difference between the previous and\ncurrent parameter values is less than ε. By default,\nthis value is 0.001.\n3\nm\nNumber of sets to impute\n4\nmissing_vals_num\nTotal number of missing values in the datasets\nYou can also pass initial estimates into the EM algorithm by packing both the vector of means and the\nvariance-covariance matrix as a one-dimensional array init_estimates. The size of the array should be at\nleast p + p(p + 1)/2. For i=0, .., p-1, the init_estimates[i] array contains the initial estimate of means.\nThe remaining positions of the array are occupied by the upper triangular part of the variance-covariance\nmatrix.\nIf you provide no initial estimates for the EM algorithm, the editor uses the default values, that is, the vector\nof zero means and the unitary matrix as a variance-covariance matrix. You can also pass prior parameters\nfor μ and Σ into the library: μ0, τ, m, and Λ-1. Pack these parameters as a one-dimensional array prior with\na size of at least\n(p2 + 3p + 4)/2.\nThe storage format is as follows:\n•\nprior[0], ..., prior[p-1] contain the elements of the vector μ0.\n•\nprior[p] contains the parameter τ.\n•\nprior[p+1] contains the parameter m.\n•\nThe remaining positions are occupied by the upper-triangular part of the inverted matrix Λ-1.\nIf you provide no prior parameters, the editor uses their default values:\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2280\n\n\n•\nThe array of p zeros is used as μ0.\n•\nτ is set to 0.\n•\nm is set to p.\n•\nThe zero matrix is used as an initial approximate of Λ-1.\nThe EditMissingValues editor returns m sets of imputed values and/or a sequence of parameter estimates\ndrawn during the DA procedure.\nThe editor returns the imputed values as the simul_missing_vals array. The size of the array should be\nsufficient to hold m sets each of the missing_vals_num size, that is, at least m*missing_vals_num in total.\nThe editor packs the imputed values one by one in the order of their appearance in the matrix of\nobservations.\nFor example, consider a task of dimension 4. The total number of observations n is 10. The second\nobservation vector misses variables 1 and 2, and the seventh observation vector lacks variable 1. The\nnumber of sets to impute is m=2. Then, simul_missing_vals[0] and simul_missing_vals[1] contains\nthe first and the second points for the second observation vector, and simul_missing_vals[2] holds the\nfirst point for the seventh observation. Positions 3, 4, and 5 are formed similarly.\nTo estimate convergence of the DA algorithm and choose a proper value of the number of DA iterations,\nrequest the sequence of parameter estimates that are produced during the DA procedure. The editor returns\nthe sequence of parameters as a single array. The size of the array is\nm*da_iter_num*(p+(p2+p)/2)\nwhere\n•\nm is the number of sets of values to impute.\n•\nda_iter_num is the number of DA iterations.\n•\nThe value p+(p2+p)/2 determines the size of the memory to hold one set of the parameter estimates.\nIn each set of the parameters, the vector of means occupies the first p positions and the remaining (p2+p)/2\npositions are intended for the upper triangular part of the variance-covariance matrix.\nUpon successful generation of m sets of imputed values, you can place them in cells of the data matrix with\nmissing values and use the Summary Statistics routines to analyze and get estimates for each of the m\ncomplete datasets.\nNOTE\nIntel® oneAPI Math Kernel Library (oneMKL) implementation of the MI algorithm rewrites\ncells of the dataset that contain theVSL_SS_SNAN/VSL_SS_DNAN values. If you want to use the\nSummary Statistics routines to process the data with missing values again, mask the\npositions of the empty cells.\nSee additional details of the algorithm usage model in the Intel® oneAPI Math Kernel Library (oneMKL)\nSummary Statistics Application Notes document [SS Notes].\nvslSSEditCorParameterization\nModifies pointers to arrays related to the algorithm of\ncorrelation matrix parameterization.\nSyntax\nstatus = vslsSSEditCorParameterization(task, cor, cor_storage, pcor, pcor_storage);\nstatus = vsldSSEditCorParameterization(task, cor, cor_storage, pcor, pcor_storage);\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2281\n\n\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\ntask\nVSLSSTaskPtr\nDescriptor of the task\ncor\nconst float* for\nvslsSSEditCorParameterization\nconst double* for\nvsldSSEditCorParameterization\nPointer to the correlation matrix\ncor_storage\nconst MKL_INT*\nPointer to the storage format of the\ncorrelation matrix\npcor\nfloat* for\nvslsSSEditCorParameterization\ndouble* for\nvsldSSEditCorParameterization\nPointer to the parameterized correlation\nmatrix\npor_storage\nconst MKL_INT*\nPointer to the storage format of the\nparameterized correlation matrix\nOutput Parameters\nName\nType\nDescription\nstatus\nint\nCurrent status of the task\nDescription\nThe vslSSEditCorParameterization routine uses values passed as parameters of the routine to replace\npointers to the correlation matrix, pointers to the correlation matrix storage format, a pointer to the\nparameterized correlation matrix, and a pointer to the parameterized correlation matrix storage format. See \nTable \"Storage formats of a variance-covariance/correlation matrix\" for possible values of the cor_storage\nand pcor_storage parameters. If you pass a value of NULL for a specific input parameter, the value of that\nparameter in the task descriptor is unchanged.\nSummary Statistics Task Computation Routines\nTask computation routines calculate statistical estimates on the data provided and parameters held in the\ntask descriptor. After you create the task and initialize its parameters, you can call the computation routines\nas many times as necessary. Table \"Summary Statistics Estimates Obtained with vslSSCompute Routine\"\nlists the respective statistical estimates.\nNOTE\nThe Summary Statistics computation routines do not signal floating-point errors, such as overflow or\ngradual underflow, or operations with NaNs (except for the missing values in the observations).\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2282\n\n\nSummary Statistics Estimates Obtained with vslSSCompute Routine\nEstimate\nSupport of\nObservations\nAvailable in Blocks\nDescription\nVSL_SS_MEAN\nYes\nComputes the array of means.\nVSL_SS_SUM\nYes\nComputes the array of sums.\nVSL_SS_2R_MOM\nYes\nComputes the array of the 2nd order raw\nmoments.\nVSL_SS_2R_SUM\nYes\nComputes the array of raw sums of the 2nd\norder.\nVSL_SS_3R_MOM\nYes\nComputes the array of the 3rd order raw\nmoments.\nVSL_SS_3R_SUM\nYes\nComputes the array of raw sums of the 3rd\norder.\nVSL_SS_4R_MOM\nYes\nComputes the array of the 4th order raw\nmoments.\nVSL_SS_4R_SUM\nYes\nComputes the array of raw sums of the 4th\norder.\nVSL_SS_2C_MOM\nYes\nComputes the array of the 2nd order central\nmoments.\nVSL_SS_2C_SUM\nYes\nComputes the array of central sums of the 2nd\norder.\nVSL_SS_3C_MOM\nYes\nComputes the array of the 3rd order central\nmoments.\nVSL_SS_3C_SUM\nYes\nComputes the array of central sums of the 3rd\norder.\nVSL_SS_4C_MOM\nYes\nComputes the array of the 4th order central\nmoments.\nVSL_SS_4C_SUM\nYes\nComputes the array of central sums of the 4th\norder.\nVSL_SS_KURTOSIS\nYes\nComputes the array of kurtosis values.\nVSL_SS_SKEWNESS\nYes\nComputes the array of skewness values.\nVSL_SS_MIN\nYes\nComputes the array of minimum values.\nVSL_SS_MAX\nYes\nComputes the array of maximum values.\nVSL_SS_VARIATION\nYes\nComputes the array of variation coefficients.\nVSL_SS_COV\nYes\nComputes a covariance matrix.\nVSL_SS_COR\nYes\nComputes a correlation matrix. The main\ndiagonal of the correlation matrix holds\nvariances of the random vector components.\nVSL_SS_CP\nYes\nComputes a cross-product matrix.\nVSL_SS_POOLED_COV\nNo\nComputes a pooled covariance matrix.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2283\n\n\nEstimate\nSupport of\nObservations\nAvailable in Blocks\nDescription\nVSL_SS_POOLED_MEAN\nNo\nComputes an array of pooled means.\nVSL_SS_GROUP_COV\nNo\nComputes group covariance matrices.\nVSL_SS_GROUP_MEAN\nNo\nComputes group means.\nVSL_SS_QUANTS\nNo\nComputes quantiles.\nVSL_SS_ORDER_STATS\nNo\nComputes order statistics.\nVSL_SS_ROBUST_COV\nNo\nComputes a robust covariance matrix.\nVSL_SS_OUTLIERS\nNo\nDetects outliers in the dataset.\nVSL_SS_PARTIAL_COV\nNo\nComputes a partial covariance matrix.\nVSL_SS_PARTIAL_COR\nNo\nComputes a partial correlation matrix.\nVSL_SS_MISSING_VALS\nNo\nSupports missing values in datasets.\nVSL_SS_PARAMTR_COR\nNo\nComputes a parameterized correlation matrix.\nVSL_SS_STREAM_QUANTS\nYes\nComputes quantiles for streaming data.\nVSL_SS_MDAD\nNo\nComputes median absolute deviation.\nVSL_SS_MNAD\nNo\nComputes mean absolute deviation.\nVSL_SS_SORTED_OBSERV\nNo\nSorts the dataset by the components of the\nrandom vector ξ.\nTable \"Summary Statistics Computation Method\"lists estimate calculation methods supported by Intel®\noneAPI Math Kernel Library (oneMKL). See theIntel® oneAPI Math Kernel Library (oneMKL) Summary\nStatistics Application Notes document [SS Notes] for a detailed description of the methods.\nSummary Statistics Computation Method\nMethod\nDescription\nVSL_SS_METHOD_FAST\nFast method for calculation of the estimates:\n•\nraw/central moments/sums, skewness, kurtosis,\nvariation, variance-covariance/correlation/cross-\nproduct matrix\n•\nmin/max/quantile/order statistics\n•\npartial variance-covariance\n•\nmedian/mean absolute deviation\nVSL_SS_METHOD_FAST_USER_MEAN\nFast method for calculation of the estimates given user-\ndefined mean:\n•\ncentral moments/sums of 2-4 order, skewness,\nkurtosis, variation, variance-covariance/correlation/\ncross-product matrix, mean absolute deviation\nVSL_SS_METHOD_1PASS\nOne-pass method for calculation of estimates:\n•\nraw/central moments/sums, skewness, kurtosis,\nvariation, variance-covariance/correlation/cross-\nproduct matrix\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2284\n\n\nMethod\nDescription\n•\npooled/group covariance matrix\nVSL_SS_METHOD_TBS\nTBS method for robust estimation of covariance and\nmean\nVSL_SS_METHOD_BACON\nBACON method for detection of multivariate outliers\nVSL_SS_METHOD_MI\nMultiple imputation method for support of missing values\nVSL_SS_METHOD_SD\nSpectral decomposition method for parameterization of a\ncorrelation matrix\nVSL_SS_METHOD_SQUANTS_ZW\nZhang-Wang (ZW) method for quantile estimation for\nstreaming data\nVSL_SS_METHOD_SQUANTS_ZW_FAST\nFast ZW method for quantile estimation for streaming\ndata\nVSL_SS_METHOD_RADIX\nRadix method for dataset sorting\nYou can calculate all requested estimates in one call of the routine. For example, to compute a kurtosis and\ncovariance matrix using a fast method, pass a combination of the pre-defined parameters into the Compute\nroutine as shown in the example below:\n...\nmethod = VSL_SS_METHOD_FAST;\ntask_params = VSL_SS_KURTOSIS|VSL_SS_COV;\n…\nstatus = vsldSSCompute( task, task_params, method );\nTo compute statistical estimates for the next block of observations, you can do one of the following:\n•\ncopy the observations to memory, starting with the address available to the task\n•\nuse one of the appropriate Editors to modify the pointer to the new dataset in the task.\nThe library does not detect your changes of the dataset and computed statistical estimates. To obtain\nstatistical estimates for a new matrix, change the observations and initialize relevant arrays. You can follow\nthis procedure to compute statistical estimates for observations that come in portions. See Table \"Summary\nStatistics Estimates Obtained with vslSSCompute Routine\"for information on such observations supported by\nthe Intel® oneAPI Math Kernel Library (oneMKL) Summary Statistics estimators.\nTo modify parameters of the task using the Task Editors, set the address of the targeted matrix of the\nobservations or change the respective vector component indices. After you complete editing the task\nparameters, you can compute statistical estimates in the modified environment.\nIf the task completes successfully, the computation routine returns the zero status code. If an error is\ndetected, the computation routine returns an error code. In particular, an error status code is returned in the\nfollowing cases:\n•\nthe task pointer is NULL\n•\nmemory allocation has failed\n•\nthe calculation has failed for some other reason\nNOTE\nYou can use the NULL task pointer in calls to editor routines. In this case, the routine is terminated\nand no system crash occurs.\nvslSSCompute\nComputes Summary Statistics estimates.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2285\n\n\nSyntax\nstatus = vslsSSCompute(task, estimates, method);\nstatus = vsldSSCompute(task, estimates, method);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\ntask\nVSLSSTaskPtr\nDescriptor of the task\nestimates\nconst MKL_INT64\nList of statistical estimates to compute\nmethod\nconst MKL_INT\nMethod to be used in calculations\nOutput Parameters\nName\nType\nDescription\nstatus\nint\nCurrent status of the task\nDescription\nThe vslSSCompute routine calculates statistical estimates passed as the estimates parameter using the\nalgorithms passed as the method parameter of the routine. The computations are done in the context of the\ntask descriptor that contains pointers to all required and optional, if necessary, properly initialized arrays. In\none call of the function, you can compute several estimates using proper methods for their calculation. See \nTable \"Summary Statistics Estimates Obtained with Compute Routine\" for the list of the estimates that you\ncan calculate with the vslSSCompute routine. See Table \"Summary Statistics Computation Methods\" for the\nlist of possible values of the method parameter.\nTo initialize single or double precision version task parameters, use the single (vslssscompute) or double\n(vsldsscompute) version of the editor, respectively. To initialize parameters of the integer type, use an\ninteger version of the editor (vslisscompute).\nNOTE\nRequesting a combination of the VSL_SS_MISSING_VALS value and any other estimate\nparameter in the Compute function results in processing only the missing values.\nApplication Notes\nBe aware that when computing a correlation matrix, the vslSSCompute routine allocates an additional array\nfor each thread which is running the task. If you are running on a large number of threads vslSSCompute\nmight consume large amounts of memory.\nWhen calculating covariance, correlation, or cross product, the number of bytes of memory required is at\nleast (P*P*T + P*T)*b, where P is the dimension of the task or number of variables, T is the number of\nthreads, and b is the number of bytes required for each unit of data. If observation is weighted and the\nmethod is VSL_SS_METHOD_FAST, then the memory required is at least (P*P*T + P*T + N*P)*b, where N is\nthe number of observations.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2286\n\n\nSummary Statistics Task Destructor\nTask destructor is the vslSSDeleteTask routine intended to delete task objects and release memory.\nvslSSDeleteTask\nDestroys the task object and releases the memory.\nSyntax\nstatus = vslSSDeleteTask(&task);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\ntask\nVSLSSTaskPtr*\nDescriptor of the task to destroy\nOutput Parameters\nName\nType\nDescription\nstatus\nint\nSets to VSL_STATUS_OK if the task is\ndeleted; otherwise a non-zero code is\nreturned.\nDescription\nThe vslSSDeleteTask routine deletes the task descriptor object, releases the memory allocated for the\nstructure, and sets the task pointer to NULL. If vslSSDeleteTask fails to delete the task successfully, it\nreturns an error code.\nNOTE\nCall of the destructor with the NULL pointer as the parameter results in termination of the\nfunction with no system crash.\nSummary Statistics Usage Examples\nThe following examples show various standard operations with Summary Statistics routines.\nCalculating Fixed Estimates for Fixed Data\nThe example shows recurrent calculation of the same estimates with a given set of variables for the complete\nlife cycle of the task in the case of a variance-covariance matrix. The set of vector components to process\nremains unchanged, and the data comes in blocks. Before you call the vslSSCompute routine, initialize\npointers to arrays for mean and covariance and set buffers.\n….\ndouble w[2];\ndouble indices[DIM] = {1, 0, 1};\n        \n/* calculating mean for 1st and 3d random vector components */\n        \nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2287\n\n\n/* Initialize parameters of the task */\np = DIM;\nn = N;\n        \nxstorage   = VSL_SS_MATRIX_STORAGE_ROWS;\ncovstorage = VSL_SS_MATRIX_STORAGE_FULL;\n        \nw[0] = 0.0; w[1] = 0.0;\n        \nfor ( i = 0; i < p; i++ ) mean[i] = 0.0;\nfor ( i = 0; i < p*p; i++ ) cov[i] = 0.0;\n        \nstatus = vsldSSNewTask( &task, &p, &n, &xstorage, x, 0, indices );\n        \nstatus = vsldSSEditTask  ( task, VSL_SS_ED_ACCUM_WEIGHT, w    );\nstatus = vsldSSEditCovCor( task, mean, cov, &covstorage, 0, 0 );\n    \nYou can process data arrays that come in blocks as follows:\nfor ( i = 0; i < num_of_blocks; i++ )\n{\n    status = vsldSSCompute( task, VSL_SS_COV, VSL_SS_METHOD_FAST );\n    /* Read new data block into array x */\n}\n…\nCalculating Different Estimates for Variable Data\nThe context of your calculation may change in the process of data analysis. The example below shows the\ndata that comes in two blocks. You need to estimate a covariance matrix for the complete data, and the third\ncentral moment for the second block of the data using the weights that were accumulated for the previous\ndatasets. The second block of the data is stored in another array. You can proceed as follows:\n/* Set parameters for the task */\np = DIM;\nn = N;\nxstorage   = VSL_SS_MATRIX_STORAGE_ROWS;\ncovstorage = VSL_SS_MATRIX_STORAGE_FULL;\nw[0] = 0.0; w[1] = 0.0;\nfor ( i = 0; i < p; i++ ) mean[i] = 0.0;\nfor ( i = 0; i < p*p; i++ ) cov[i] = 0.0;\n/* Create task */\nstatus = vsldSSNewTask( &task, &p, &n, &xstorage, x1, 0, indices );\n/* Initialize the task parameters */\nstatus = vsldSSEditTask( task, VSL_SS_ED_ACCUM_WEIGHT, w );\nstatus = vsldSSEditCovCor( task, mean, cov, &covstorage, 0, 0 );\n/* Calculate covariance for the x1 data */\nstatus = vsldSSCompute( task, VSL_SS_COV, VSL_SS_METHOD_FAST );\n/* Initialize array of the 3d central moments and pass the pointer to the task */\nfor ( i = 0; i < p; i++ ) c3_m[i] = 0.0;\n/* Modify task context */\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2288\n\n\nstatus = vsldSSEditTask( task, VSL_SS_ED_3C_MOM, c3_m );\nstatus = vsldSSEditTask( task, VSL_SS_ED_OBSERV, x2 );\n/* Calculate covariance for the x1 & x2 data block */\n/* Calculate the 3d central moment for the 2nd data block using earlier accumulated weight */\nstatus = vsldSSCompute(task, VSL_SS_COV|VSL_SS_3C_MOM, VSL_SS_METHOD_FAST );\n…\nstatus = vslSSDeleteTask( &task );\nSimilarly, you can modify indices of the variables to be processed for the next data block.\nSummary Statistics Mathematical Notation and Definitions\nThe following notations are used in the mathematical definitions and the description of the Intel® oneAPI\nMath Kernel Library (oneMKL) Summary Statistics functions.\nMatrix and Weights of Observations\nFor a random p-dimensional vector ξ = (ξ1,..., ξi,..., ξp), this manual denotes the following:\n•\n(X)i=(xij)j=1..n is the result of n independent observations for the i-th component ξi of the vector ξ.\n•\nThe two-dimensional array X=(xij)n x p is the matrix of observations.\n•\nThe column [X]j=(xij)i=1..p of the matrix X is the j-th observation of the random vector ξ.\nEach observation [X]j is assigned a non-negative weight wj , where\n•\nThe vector (wj)j=1..n is a vector of weights corresponding to n observations of the random vector ξ.\n•\nW = ∑\ni = 1\nn\nwi\nis the accumulated weight corresponding to observations X.\nVector of sample means\nM X = M1 X , …, Mp X\n with Mi X = 1\nw ∑\nj = 1\nn\nwjxij\nfor all i = 1, ..., p.\nVector of sample partial sums\nS X = S1 X , …, Sp X\n with Si X = ∑\nj = 1\nn\nwjxij\nfor all i = 1, ..., p.\nVector of sample variances\nV X = V1 X , …, Vp X\n with Vi X = 1\nB ∑\nj = 1\nn\nwj xij −Mi X\n2, B = W −∑\nj = 1\nn\nwj\n2/W\nfor all i = 1, ..., p.\nVector of sample raw/algebraic moments of k-th order, k≥ 1\nR k X = R1\nk X , …, Rp\nk X\n with Ri\nk X = 1\nW ∑\nj = 1\nn\nwjxij\nk\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2289\n\n\nfor all i = 1, ..., p.\nVector of sample raw/algebraic partial sums of k-th order, k= 2, 3, 4 (raw/algebraic partial sums\nof squares/cubes/fourth powers)\nSk X = S1\nk X , …, Sp\nk X\n with Si\nk X = ∑\nj = 1\nn\nwjxij\nk\nfor all i = 1, ..., p.\nVector of sample central moments of the third and the fourth order\nC(k)(X) = C1\n(k)(X), …, Cp\n(k)(X)  with Ci\n(k)(X) = 1\nB ∑\nj = 1\nn\nwj xij −Mi(X)\nk\n, B = ∑\nj = 1\nn\nwj\nfor all i = 1, ..., p and k = 3, 4.\nVector of sample central partial sums of k-th order, k= 2, 3, 4 (central partial sums of squares/\ncubes/fourth powers)\nSk X = S1\nk X , …, Sp\nk X\n with Si\nk X = ∑\nj = 1\nn\nwj xij −Si X\nk\nfor all i = 1, ..., p.\nVector of sample excess kurtosis values\nB X = B1 X , …, Bp X\n with Bi(X) =\nCi\n(4)(X)\nVi\n2(X)\n−3\nfor all i = 1, ..., p.\nVector of sample skewness values\nΓ X = Γ1 X , …, Γp X\n with Γi(X) =\nCi\n(3)(X)\nVi\n1.5(X)\nfor all i = 1, ..., p.\nVector of sample variation coefficients\nVC X = VC1 X , …, VCp X\n with VCi(X) =\nVi\n0.5(X)\nMi(X)\nfor all i = 1, ..., p.\nMatrix of order statistics\nMatrix Y = (yij)pxn, in which the i-th row (Y)i = (yij)j=1..n is obtained as a result of sorting in the\nascending order of row (X)i = (xij)j=1..n in the original matrix of observations.\nVector of sample minimum values\nMin X = Min1 X , …, Minp X\n, where Mini X = yi1\nfor all i = 1, ..., p.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2290\n\n\nVector of sample maximum values\nMax X = Max1 X , …, Maxp X\n, where Maxi X = yin\nfor all i = 1, ..., p.\nVector of sample median values\nMed X = Med1 X , …, Medp X\n, where Medi X =\nyi, (n + 1)/2, if  n is odd\nyi, n/2 + yi, n/2 + 1 /2, if n is even\nfor all i = 1, ..., p.\nVector of sample median absolute deviations\nMDAD X = MDAD1 X , …, MDADp X\n, where MDADi X = Medi Z  with Z = zij i = 1…p, j = 1…n,\nzij = xij −Medi X\nfor all i = 1, ..., p.\nVector of sample mean absolute deviations\nMNAD X = MNAD1 X , …, MNADp X\n, where MNADi X = Mi Z  with Z = zij i = 1…p, j = 1…n,\nzij = xij −Mi X\nfor all i = 1, ..., p.\nVector of sample quantile values\nFor a positive integer number q and k belonging to the interval [0, q-1], point zi is the k-th q quantile of the\nrandom variable ξi if P{ξi≤zi} ≥β and P{ξi≤zi} ≥ 1 - β, where\n•\nP is the probability measure.\n•\nβ = k/n is the quantile order.\nThe calculation of quantiles is as follows:\nj = [(n-1)β] and f = {(n-1)β} as integer and fractional parts of the number (n-1)β, respectively, and the\nvector of sample quantile values is\nQ(X,β) = (Q1(X,β), ..., Qp(X,β))\nwhere\n(Qi(X,β) = yi,j+1 + f(yi,j+2 - yi,j+1)\nfor all i = 1, ..., p.\nVariance-covariance matrix\nC(X) = (cij(X))p x p\nwhere\ncij X = 1\nB ∑\nk = 1\nn\nwk xik −Mi X\nxjk −Mj X\n, B = W −∑\nj = 1\nn\nwj\n2/W\nCross-product matrix (matrix of cross-products and sums of squares)\nCP(X) = (cpij(X))p x p\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2291\n\n\nwhere\ncpij X = ∑\nk = 1\nn\nwk xik −Mi X\nxjk −Mj X\nPooled and group variance-covariance matrices\nThe set N = {1, ..., n} is partitioned into non-intersecting subsets\nGi, i = 1..g, N =\n∪\ni = 1\ng\nGi\nThe observation [X]j = (xij)i=1..p belongs to the group r if j∈Gr. One observation belongs to one group\nonly. The group mean and variance-covariance matrices are calculated similarly to the formulas above:\nM(r)(X) = M1\n(r)(X), …, Mp\n(r)(X)  with Mi\nr X =\n1\nW(r) ∑\nj ∈Gr\nwjxij, W(r) = ∑\nj ∈Gr\nwj\nfor all i = 1, ..., p,\nC r X = cij\nr X\np × p\nwhere\ncij\nr X =\n1\nB r\n∑\nk ∈Gr\nwk xik −Mi\nr X\nxjk −Mj\nr X\n, B r = W r −∑\nj ∈Gr\nwj\n2/W r\nfor all i = 1, ..., p and j = 1, ..., p.\nA pooled variance-covariance matrix and a pooled mean are computed as weighted mean over group\ncovariance matrices and group means, correspondingly:\nMpooled(X) = M1\npooled(X), …, Mp\npooled(X)  with Mi\npooled X =\n1\nW 1 + … + W g ∑\nr = 1\ng\nW r Mi\nr X\nfor all i = 1, ..., p,\nCpooled X = cij\npooled X\np × p, cij\npooled X =\n1\nB 1 + … + B g ∑\nr = 1\ng\nB r cij\nr X\nfor all i = 1, ..., p and j = 1, ..., p.\nCorrelation matrix\nR X = rij X\np × p, where rij X =\ncij\nciicjj\nfor all i = 1, ..., p and j = 1, ..., p.\nPartial variance-covariance matrix\nFor a random vector ξ partitioned into two components Z and Y, a variance-covariance matrix C describes the\nstructure of dependencies in the vector ξ:\nC X =\nCZ X\nCZY X\nCYZ X\nCY X\n.\nThe partial covariance matrix P(X) =(pij(X))kxk is defined as\nP X = CY X −CYZ X CZ\n−1CZY X .\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2292\n\n\nwhere k is the dimension of Y.\nPartial correlation matrix\nThe following is a partial correlation matrix for all i = 1, ..., k and j = 1, ..., k:\nRP X = rpij X\nk × k, where rpij X =\npij X\npii X pjj X\nwhere\n•\nk is the dimension of Y.\n•\npij(X) are elements of the partial variance-covariance matrix.\nSorted dataset\nMatrix Y = (yij)pxn, in which the i-th row (Y)i is obtained as a result of sorting in ascending order the row (X)i\n= (xij)j = 1..n in the original matrix of observations.\nFourier Transform Functions\nThe general form of the discrete Fourier transform is\nzk1, k2, ..., kd = σ × ∑\njd = 0\nnd −1\n... ∑\nj2 = 0\nn2 −1\n∑\nj1 = 0\nn1 −1\nwj1, j2, ..., jd exp δi2π ∑\nl = 1\nd\njlkl/nl\nfor kl = 0, ... nl-1 (l = 1, ..., d), where σ is a scale factor, δ = -1 for the forward transform, and δ = +1\nfor the inverse (backward) transform. In the forward transform, the input (periodic) sequence {wj1, j2, ...,\njd} belongs to the set of complex-valued sequences and real-valued sequences. Respective domains for the\nbackward transform are represented by complex-valued sequences and complex-valued conjugate-even\nsequences.\nThe Intel® oneAPI Math Kernel Library (oneMKL) provides an interface for computing a discrete Fourier\ntransform through the fast Fourier transform algorithm. Prefixes Dfti in function names and DFTI in the\nnames of configuration parameters stand for Discrete Fourier Transform Interface.\nThe manual describes the following implementations of the fast Fourier transform functions available in Intel®\noneAPI Math Kernel Library (oneMKL):\n•\nFast Fourier transform (FFT) functions for single-processor or shared-memory systems (see FFT\nFunctions)\n•\nCluster FFT functions for distributed-memory architectures (available only for Intel® 64 architectures)\nNOTE\nIntel® oneAPI Math Kernel Library (oneMKL) also supports the FFTW3* interfaces to the fast Fourier\ntransform functionality for shared memory paradigm (SMP) systems.\nBoth FFT and Cluster FFT functions compute an FFT in five steps:\n1.\nAllocate a fresh descriptor for the problem with a call to the DftiCreateDescriptor or \nDftiCreateDescriptorDM function. The descriptor captures the configuration of the transform, such\nas the dimensionality (or rank), sizes, number of transforms, memory layout of the input/output data\n(defined by strides), and scaling factors. Many of the configuration settings are assigned default values\nin this call which you might need to modify in your application.\n2.\nOptionally adjust the descriptor configuration with a call to the DftiSetValue or DftiSetValueDM\nfunction as needed. Typically, you must carefully define the data storage layout for an FFT or the data\ndistribution among processes for a Cluster FFT. The configuration settings of the descriptor, such as the\ndefault values, can be obtained with the DftiGetValue or DftiGetValueDM function.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2293\n\n\n3.\nCommit the descriptor with a call to the DftiCommitDescriptor or DftiCommitDescriptorDM\nfunction, that is, make the descriptor ready for the transform computation. Once the descriptor is\ncommitted, the parameters of the transform, such as the type and number of transforms, strides and\ndistances, the type and storage layout of the data, and so on, are \"frozen\" in the descriptor.\n4.\nCompute the transform with a call to the DftiComputeForward/DftiComputeBackward or \nDftiComputeForwardDM/DftiComputeBackwardDM functions as many times as needed. Because the\ndescriptor is defined and committed separately, all that the compute functions do is take the input and\noutput data and compute the transform as defined. To modify any configuration parameters for another\ncall to a compute function, use DftiSetValue followed by DftiCommitDescriptor (DftiSetValueDM\nfollowed by DftiCommitDescriptorDM) or create and commit another descriptor.\n5.\nDeallocate the descriptor with a call to the DftiFreeDescriptor or DftiFreeDescriptorDM function.\nThis returns the memory internally consumed by the descriptor to the operating system.\nAll the above functions return an integer status value, which is zero upon successful completion of the\noperation. You can interpret a non-zero status with the help of the DftiErrorClass or DftiErrorMessage\nfunction.\nThe FFT functions support lengths with arbitrary factors. You can improve performance of the Intel® oneAPI\nMath Kernel Library (oneMKL) FFT if the length of your data vector permits factorization into powers of\noptimized radices. See the Intel® oneAPI Math Kernel Library (oneMKL) Developer Guide for specific radices\nsupported efficiently.\nNOTE\nThe FFT functions assume the Cartesian representation of complex data (that is, the real and\nimaginary parts define a complex number). The Intel® oneAPI Math Kernel Library (oneMKL) Vector\nMathematical Functions provide efficient tools for conversion to and from polar representation (see \nExample \"Conversion from Cartesian to polar representation of complex data\" and Example\n\"Conversion from polar to Cartesian representation of complex data\").\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nFFT Functions\nThe fast Fourier transform function library of Intel® oneAPI Math Kernel Library (oneMKL) provides one-\ndimensional, two-dimensional, and multi-dimensional transforms (of up to seven dimensions) and offers both\nFortran and C interfaces for all transform functions.\nTable \"FFT Functions in Intel® oneAPI Math Kernel Library (oneMKL)\"lists FFT functions implemented in Intel®\noneAPI Math Kernel Library (oneMKL):\nFFT Functions in oneMKL\nFunction Name\nOperation\nDescriptor Manipulation Functions\nDftiCreateDescriptor\nAllocates the descriptor data structure and initializes it with default\nconfiguration values.\nDftiCommitDescriptor\nPerforms all initialization for the actual FFT computation.\nDftiFreeDescriptor\nFrees memory allocated for a descriptor.\nDftiCopyDescriptor\nMakes a copy of an existing descriptor.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2294\n\n\nFunction Name\nOperation\nFFT Computation Functions\nDftiComputeForward\nComputes the forward FFT.\nDftiComputeBackward\nComputes the backward FFT.\nDescriptor Configuration Functions\nDftiSetValue\nSets one particular configuration parameter with the specified\nconfiguration value.\nDftiGetValue\nGets the value of one particular configuration parameter.\nStatus Checking Functions\nDftiErrorClass\nChecks if the status reflects an error of a predefined class.\nDftiErrorMessage\nTranslates the numeric value of an error status into a message.\nFFT Interface\nThe Intel® oneAPI Math Kernel Library (oneMKL) FFT functions are provided with the Fortran and C interfaces.\nThe materials presented in this section assume the availability of native complex types in C as they are\nspecified in C9X.\nTo use the FFT functions, you need to include mkl_dfti.h in your C code.\nThe C interface provides the DFTI_DESCRIPTOR_HANDLE type, named constants of two enumeration types\nDFTI_CONFIG_PARAM and DFTI_CONFIG_VALUE, and functions, some of which accept different numbers of\ninput arguments.\nNOTE\nThe current version of the library may not support some of the FFT functions or functionality. You can\nfind the complete list of the implementation-specific exceptions in the Intel® oneAPI Math Kernel\nLibrary (oneMKL) Release Notes.\nFor the main categories of Intel® oneAPI Math Kernel Library (oneMKL) FFT functions, see FFT Functions.\nComputing an FFT\nYou can find code examples that compute transforms in the Fourier Transform Functions Code Examples.\nUsually you can compute an FFT by five function calls (refer to the usage model for details). A single data\nstructure, the descriptor, stores configuration parameters that can be changed independently.\nThe descriptor data structure, when created, contains information about the length and domain of the FFT to\nbe computed, as well as the setting of several configuration parameters. Default settings for some of these\nparameters are as follows:\n•\nScale factor: none (that is, σ = 1)\n•\nNumber of data sets: one\n•\nData storage: contiguous\n•\nPlacement of results: in-place (the computed result overwrites the input data)\nThe default settings can be changed one at a time through the function DftiSetValue as illustrated in \nExample \"Changing Default Settings (C)\".\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2295\n\n\nConfiguration Settings\nEach of the configuration parameters is identified by a named constant in the MKL_DFTI module. These\nnamed constants have the enumeration type DFTI_CONFIG_PARAM and are declared in the mkl_dfti.h\nheader file.\nThough exposed in that header file, the configuration parameters DFTI_FWD_DISTANCE and\nDFTI_BWD_DISTANCE are specific to the DPC++ DFT routines (see the Data Parallel C++ Developer\nReference): their use in another context is neither supported nor currently enabled. All other Intel® oneAPI\nMath Kernel Library (oneMKL) FFT configuration parameters are readable. Some of them are read-only, while\nothers can be set using the DftiCreateDescriptor or DftiSetValue function.\nValues of the configuration parameters fall into the following groups:\n•\nValues that have native data types. For example, the number of simultaneous transforms requested has\nan integer value, while the scale factor for a forward transform is a floating-point number.\n•\nValues that are discrete in nature and are provided in the MKL_DFTI module as named constants. For\nexample, the domain of the forward transform requires values to be named constants. The named\nconstants for configuration values have the enumeration type DFTI_CONFIG_VALUE.\nThe Table \"Configuration Parameters\" summarizes the information on configuration parameters, along with\ntheir types and values. For more details of each configuration parameter, see the subsection describing this\nparameter.\nConfiguration Parameters\nConfiguration Parameter\nType/Value\nComments\nMost common configuration parameters, no default, must be set explicitly by DftiCreateDescriptor\nDFTI_PRECISION\nNamed constant\nDFTI_SINGLE or\nDFTI_DOUBLE\nPrecision of the computation.\nDFTI_FORWARD_DOMAIN\nNamed constant\nDFTI_COMPLEX or\nDFTI_REAL\nType of the transform.\nDFTI_DIMENSION\nInteger scalar\nDimension of the transform.\nDFTI_LENGTHS\nInteger scalar/array\nLengths of each dimension.\nCommon configuration parameters, settable by DftiSetValue\nDFTI_PLACEMENT\nNamed constant\nDFTI_INPLACE or \nDFTI_NOT_INPLACE\nDefines whether the result overwrites\nthe input data. Default value:\nDFTI_INPLACE.\nDFTI_FORWARD_SCALE\nFloating-point scalar\nScale factor for the forward transform.\nDefault value: 1.0.\nPrecision of the value should be the\nsame as defined by DFTI_PRECISION.\nDFTI_BACKWARD_SCALE\nFloating-point scalar\nScale factor for the backward transform.\nDefault value: 1.0.\nPrecision of the value should be the\nsame as defined by DFTI_PRECISION.\nDFTI_NUMBER_OF_USER_THREADS\nInteger scalar\nThis configuration parameter is no longer\nused and kept for compatibility with\nprevious versions of Intel® oneAPI Math\nKernel Library (oneMKL).\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2296\n\n\nConfiguration Parameter\nType/Value\nComments\nDFTI_THREAD_LIMIT\nInteger scalar\nLimits the number of threads for the \nDftiComputeForward and \nDftiComputeBackward.\nDefault value: 0.\nDFTI_DESCRIPTOR_NAME\nCharacter string\nAssigns a name to a descriptor. Assumed\nlength of the string is\nDFTI_MAX_NAME_LENGTH.\nDefault value: empty string.\nData layout configuration parameters for single and multiple transforms. Settable by DftiSetValue\nDFTI_INPUT_STRIDES\nInteger array\nDefines the input data layout.\nNOTE The default strides are set during\ncreation of the descriptor based on the\ndesired dimension and lengths. For more\ndetails, see DFTI_INPUT_STRIDES,\nDFTI_OUTPUT_STRIDES.\nDFTI_OUTPUT_STRIDES\nInteger array\nDefines the output data layout.\nNOTE The default strides are set during\ncreation of the descriptor based on the\ndesired dimension and lengths. For more\ndetails, see DFTI_INPUT_STRIDES,\nDFTI_OUTPUT_STRIDES.\nDFTI_NUMBER_OF_TRANSFORMS\nInteger scalar\nNumber of transforms.\nDefault value: 1.\nDFTI_INPUT_DISTANCE\nInteger scalar\nDefines the distance between input data\nsets for multiple transforms.\nDefault value: 0.\nDFTI_OUTPUT_DISTANCE\nInteger scalar\nDefines the distance between output\ndata sets for multiple transforms.\nDefault value: 0.\nDFTI_COMPLEX_STORAGE\nNamed constant\nDFTI_COMPLEX_COMPLE\nX or DFTI_REAL_REAL\nDefines whether the real and imaginary\nparts of data for a complex transform\nare interleaved in one array or split in\ntwo arrays.\nDefault value: DFTI_COMPLEX_COMPLEX.\nDFTI_REAL_STORAGE\nNamed constant\nDFTI_REAL_REAL\nDefines how real data for a real\ntransform is stored. Only the\nDFTI_REAL_REAL value is supported.\nDFTI_CONJUGATE_EVEN_STORAGE\nNamed constant\nDFTI_COMPLEX_COMPLE\nX or\nDFTI_COMPLEX_REAL\nDefines whether the complex data in the\nbackward domain of a real transform is\nstored as complex elements or as real\nelements.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2297\n\n\nConfiguration Parameter\nType/Value\nComments\nDFTI_COMPLEX_REAL is supported only\nfor 1D transforms.\nThe default value is\nDFTI_COMPLEX_COMPLEX.\nDFTI_PACKED_FORMAT\nNamed constant\nDFTI_CCE_FORMAT,\nDFTI_CCS_FORMAT,\nDFTI_PACK_FORMAT, or\nDFTI_PERM_FORMAT\nDefines the layout for the elements of\nthe conjugate-even sequence in the\nbackward domain of the real transform\n(in association with the configuration\nparameter\nDFTI_CONJUGATE_EVEN_STORAGE).\nThe default value is DFTI_CCE_FORMAT.\nNOTE Transforms greater than 1D support\nonly DFTI_CCE_FORMAT.\nAdvanced configuration parameters, settable by DftiSetValue\nDFTI_WORKSPACE\nNamed constant\nDFTI_ALLOW or\nDFTI_AVOID\nDefines whether the library should prefer\nalgorithms using additional memory.\nDefault value: DFTI_ALLOW.\nDFTI_ORDERING\nNamed constant\nDFTI_ORDERED or\nDFTI_BACKWARD_SCRAM\nBLED\nDefines whether the result of a complex\ntransform is ordered or permuted.\nDefault value: DFTI_ORDERED.\nDFTI_DESTROY_INPUT\nNamed constant\nDFTI_ALLOW or\nDFTI_AVOID\nDefines whether the input data may be\noverwritten for out-of-place transforms.\nDefault value: DFTI_AVOID.\nRead-Only configuration parameters\nDFTI_COMMIT_STATUS\nNamed constant\nDFTI_UNCOMMITTED or\nDFTI_COMMITTED\nReadiness of the descriptor for\ncomputation.\nDFTI_VERSION\nString\nVersion of Intel® oneAPI Math Kernel\nLibrary (oneMKL). Assumed length of the\nstring is DFTI_VERSION_LENGTH.\nSee Also\nConfiguring and Computing an FFT in C/C++ \nDFTI_PRECISION\nThe configuration parameter DFTI_PRECISION denotes the floating-point precision in which the transform is\nto be carried out. A setting of DFTI_SINGLE stands for single precision, and a setting of DFTI_DOUBLE stands\nfor double precision. The data must be presented in this precision, the computation is carried out in this\nprecision, and the result is delivered in this precision.\nDFTI_PRECISION does not have a default value. Set it explicitly by calling the DftiCreateDescriptor\nfunction.\nTo better understand configuration of the precision of transforms, refer to these examples in your Intel®\noneAPI Math Kernel Library (oneMKL) directory:\n./examples/dftc/source/basic_sp_complex_dft_1d.c\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2298\n\n\n./examples/dftc/source/basic_dp_complex_dft_1d.c\nSee Also\nDFTI_FORWARD_DOMAIN\nDFTI_DIMENSION, DFTI_LENGTHS\nDftiCreateDescriptor\nDFTI_FORWARD_DOMAIN\nThe general form of a discrete Fourier transform is\nzk1, k2, ..., kd = σ × ∑\njd = 0\nnd −1\n... ∑\nj2 = 0\nn2 −1\n∑\nj1 = 0\nn1 −1\nwj1, j2, ..., jd exp δi2π ∑\nl = 1\nd\njlkl/nl\nfor kl = 0, ... nl-1 (l = 1, ..., d), where σ is a scale factor, δ = -1 for the forward transform, and δ = +1\nfor the backward transform.\nThe Intel® oneAPI Math Kernel Library (oneMKL) implementation of the FFT algorithm, used for fast\ncomputation of discrete Fourier transforms, supports forward transforms on input sequences of two domains,\nas specified by the DFTI_FORWARD_DOMAIN configuration parameter: general complex-valued sequences\n(DFTI_COMPLEX domain) and general real-valued sequences (DFTI_REAL domain). The forward transform\nmaps the forward domain to the corresponding backward domain, as shown in Table \"Correspondence of\nForward and Backward Domain\".\nThe conjugate-even domain covers complex-valued sequences with the symmetry property:\nx k1, k2, ..., kd = conjugate x n1−k1, n2 −k2, ..., nd −kd\nwhere the index arithmetic is performed modulo respective size, that is,\nx ..., exprs, ... ≡x ..., mod exprs, ns , ... ,\nand therefore\nx ..., ns, ... ≡x ..., 0, ... .\nDue to this property of conjugate-even sequences, only a part of such sequence is stored in the computer\nmemory, as described in DFTI_CONJUGATE_EVEN_STORAGE.\nCorrespondence of Forward and Backward Domain\nForward Domain\nImplied Backward Domain\nComplex (DFTI_COMPLEX)\nComplex (DFTI_COMPLEX)\nReal (DFTI_REAL)\nConjugate-even\nDFTI_FORWARD_DOMAIN does not have a default value. Set it explicitly by calling the\nDftiCreateDescriptor function.\nTo better understand usage of the DFTI_FORWARD_DOMAIN configuration parameter, you can refer to these\nexamples in your Intel® oneAPI Math Kernel Library (oneMKL) directory:\n./examples/dftc/source/basic_sp_complex_dft_1d.c\n./examples/dftc/source/basic_sp_real_dft_1d.c\nSee Also\nDFTI_PRECISION\nDFTI_DIMENSION, DFTI_LENGTHS\nDftiCreateDescriptor\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2299\n\n\nDFTI_DIMENSION, DFTI_LENGTHS\nThe dimension of the transform is a positive integer value represented in an integer scalar of MKL_LONG data\ntype. For a one-dimensional transform, the transform length is specified by a positive integer value\nrepresented in an integer scalar of MKL_LONG data type. For multi-dimensional (≥ 2) transform, the lengths\nof each of the dimensions are supplied in an integer array (of MKL_LONG data type).\nDFTI_DIMENSION and DFTI_LENGTHS do not have a default value. To set them, use the\nDftiCreateDescriptor function and not the DftiSetValue function.\nTo better understand usage of the DFTI_DIMENSION and DFTI_LENGTHS configuration parameters, you can\nrefer to basic examples of one-, two-, and three-dimensional transforms in your Intel® oneAPI Math Kernel\nLibrary (oneMKL) directory. Naming conventions for the examples are self-explanatory. For example, refer to\nthese examples of single-precision two-dimensional transforms:\n./examples/dftc/source/basic_sp_real_dft_2d.c\n./examples/dftc/source/basic_sp_complex_dft_2d.c\nSee Also\nDFTI_FORWARD_DOMAIN\nDFTI_PRECISION\nDftiCreateDescriptor\nDftiSetValue\nDFTI_PLACEMENT\nBy default, the computational functions overwrite the input data with the output result. That is, the default\nsetting of the configuration parameter DFTI_PLACEMENT is DFTI_INPLACE. You can change that by setting it\nto DFTI_NOT_INPLACE.\nNOTE\nWhen the configuration parameter is set to DFTI_NOT_INPLACE, the input and output data sets must\nhave no common elements.\nTo better understand usage of the DFTI_PLACEMENT configuration parameter, see this example in your Intel®\noneAPI Math Kernel Library (oneMKL) directory:\n./examples/dftc/source/config_placement.c\nSee Also\nDftiSetValue\nDFTI_FORWARD_SCALE, DFTI_BACKWARD_SCALE\nThe forward transform and backward transform are each associated with a scale factor σ of its own having\nthe default value of 1. You can specify the scale factors using one or both of the configuration parameters\nDFTI_FORWARD_SCALE and DFTI_BACKWARD_SCALE. For example, for a one-dimensional transform of length\nn, you can use the default scale of 1 for the forward transform and set the scale factor for the backward\ntransform to be 1/n, thus making the backward transform the inverse of the forward transform.\nSet the scale factor configuration parameter using a real floating-point data type of the same precision as the\nvalue for DFTI_PRECISION.\nNOTE\nFor inquiry of the scale factor with the DftiGetValue function, the config_val parameter must have\nthe same floating-point precision as the descriptor.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2300\n\n\nSee Also\nDftiSetValue\nDFTI_PRECISION\nDftiGetValue\nDFTI_NUMBER_OF_USER_THREADS\nThe DFTI_NUMBER_OF_USER_THREADS configuration parameter is no longer used and kept for compatibility\nwith previous versions of Intel® oneAPI Math Kernel Library (oneMKL).\nSee Also\nDftiSetValue\nDFTI_THREAD_LIMIT\nIn some situations you may need to limit the number of threads that the DftiComputeForward and\nDftiComputeBackward functions use. For example, if more than one thread calls Intel® oneAPI Math Kernel\nLibrary (oneMKL), it might be important that the thread calling these functions does not oversubscribe\ncomputing resources (CPU cores). Similarly, a known limit of the maximum number of threads to be used in\ncomputations might help the DftiCommitDescriptor function to select a more optimal computation\nmethod.\nSet the parameter DFTI_THREAD_LIMIT as follows:\n•\nTo a positive number, to specify the maximum number of threads to be used by the compute functions.\n•\nTo zero (the default value), to use the maximum number of threads permitted in Intel® oneAPI Math\nKernel Library (oneMKL) FFT functions. See \"Techniques to Set the Number of Threads\" in the Intel®\noneAPI Math Kernel Library (oneMKL) Developer Guide for more information.\nOn an attempt to set a negative value, the DftiSetValue function returns an error and does not update the\ndescriptor.\nThe value of the DFTI_THREAD_LIMIT configuration parameter returned by the DftiGetValue function is\ndefined as follows:\n•\n1 if Intel® oneAPI Math Kernel Library (oneMKL) runs in the sequential mode\n•\nDepends of the commit status of the descriptor if Intel® oneAPI Math Kernel Library (oneMKL) runs in a\nthreaded mode:\nCommit Status\nValue\nNot committed\nThe value of DFTI_THREAD_LIMIT set in a previous call to the\nDftiSetValue function or the default value\nCommitted\nThe upper limit on the number of threads used by the\nDftiComputeForward and DftiComputeBackward functions\nTo better understand usage of the DFTI_THREAD_LIMIT configuration parameter, see this example in your\nIntel® oneAPI Math Kernel Library (oneMKL) directory:\n./examples/dftc/source/config_thread_limit.c\nSee Also\nDftiGetValue\nDftiSetValue\nDftiCommitDescriptor\nDftiComputeForward\nDftiComputeBackward\nThreading Control Functions \nDFTI_COMMIT_STATUS\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2301\n\n\nDFTI_INPUT_STRIDES, DFTI_OUTPUT_STRIDES\nThe FFT interface provides configuration parameters that define the layout of multidimensional data in the\ncomputer memory. For d-dimensional data set X defined by dimensions N1 x N2 x ... x Nd, the layout\ndescribes where a particular element X(k1, k2, ..., kd) of the data set is located. The memory address of the\nelement X(k1, k2 , ..., kd) is expressed by the formula,\naddress of X(k1, k2, ..., kd) = the address stored in the pointer supplied to the compute function + (s0 +\nk1*s1 + k2*s2 + ...+ kd*sd) * u,\nWhere u is the number of bytes per element of the desired precision for the assumed data type in the\ncorresponding domain (see Table \"Assumed Element Types of the Input/Output Data\" below), where s0 is the\ndisplacement, and s1, ..., sd are generalized strides. The configuration parameters DFTI_INPUT_STRIDES and\nDFTI_OUTPUT_STRIDES enable you to get and set these values. The configuration value is an array of values\n(s0, s1, ..., sd) of MKL_LONG data type.\nThe DFTI_FORWARD_DOMAIN, DFTI_COMPLEX_STORAGE, and DFTI_CONJUGATE_EVEN_STORAGE configuration\nparameters define the type of the elements as shown in Table \"Assumed Element Types of the Input/Output\nData\":\nAssumed Element Types of the Input/Output Data\nDescriptor Configuration\nElement\nType in the\nForward\nDomain\nElement\nType in the\nBackward\nDomain\nDFTI_FORWARD_DOMAIN=DFTI_COMPLEX\nDFTI_COMPLEX_STORAGE=DFTI_COMPLEX_COMPLEX\nComplex\nComplex\nDFTI_FORWARD_DOMAIN=DFTI_COMPLEX\nDFTI_COMPLEX_STORAGE=DFTI_REAL_REAL\nReal\nReal\nDFTI_FORWARD_DOMAIN=DFTI_REAL\nDFTI_CONJUGATE_EVEN_STORAGE=DFTI_COMPLEX_REAL\nReal\nReal\nDFTI_FORWARD_DOMAIN=DFTI_REAL\nDFTI_CONJUGATE_EVEN_STORAGE=DFTI_COMPLEX_COMPLEX\nReal\nComplex\nThe DFTI_INPUT_STRIDES configuration parameter defines the layout of the input data, while the element\ntype is defined by the forward domain for the DftiComputeForward function and by the backward domain\nfor the DftiComputeBackward function. The DFTI_OUTPUT_STRIDES configuration parameter defines the\nlayout of the output data, while the element type is defined by the backward domain for the\nDftiComputeForward function and by the forward domain for DftiComputeBackward function.\nNOTE\nThe DFTI_INPUT_STRIDES and DFTI_OUTPUT_STRIDES configuration parameters define the layout of\ninput and output data, and not the forward-domain and backward-domain data. If the data layouts in\nforward domain and backward domain differ, set DFTI_INPUT_STRIDES and DFTI_OUTPUT_STRIDES\nexplicitly and then commit the descriptor before calling computation functions.\nFor in-place transforms (DFTI_PLACEMENT=DFTI_INPLACE), the configuration set by DFTI_OUTPUT_STRIDES\nis ignored when the element types in the forward and backward domains are the same. If they are different,\nset DFTI_OUTPUT_STRIDES explicitly (even though the transform is in-place). Ensure a consistent\nconfiguration for in-place transforms, that is, the locations of the first elements on input and output must\ncoincide in each dimension.\nThe FFT interface supports both positive and negative stride values. If you use negative strides, set the\ndisplacement of the data as follows:\ns0 = ∑\ni = 1\nd\n(Ni −1) ⋅max( −si, 0).\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2302\n\n\nThe default setting of strides in a general multi-dimensional case assumes that the array that contains the\ndata has no padding. The order of the strides depends on the programming language. For example:\nMKL_LONG dims[] = { nd, …, n2, n1 };\nDftiCreateDescriptor( &hand, precision, domain, d, dims );\n// The above call assumes data declaration:  type X[nd]…[n2][n1]\n// Default strides are { 0, nd-1*…*n2*n1, …,  n2*n1*1, n1*1, 1 }\nNote that in case of a real FFT (DFTI_FORWARD_DOMAIN=DFTI_REAL), where different data layouts in the\nbackward domain are available (see DFTI_PACKED_FORMAT), the default value of the strides is not intuitive\nfor the recommended CCE format (configuration setting\nDFTI_CONJUGATE_EVEN_STORAGE=DFTI_COMPLEX_COMPLEX). In case of an in-place real transform with the\nCCE format, set the strides explicitly, as follows:\nMKL_LONG dims[] = { nd, …, n2, n1 };\nMKL_LONG rstrides[] = { 0, 2*nd-1*…*n2*(n1/2+1), …, 2*n2*(n1/2+1), 2*(n1/2+1), 1 };\nMKL_LONG cstrides[] = { 0, nd-1*…*n2*(n1/2+1), …, n2*(n1/2+1), (n1/2+1), 1 };\nDftiCreateDescriptor( &hand, precision,  DFTI_REAL, d, dims );\nDftiSetValue(hand, DFTI_CONJUGATE_EVEN_STORAGE, DFTI_COMPLEX_COMPLEX);\n// Set the strides appropriately for forward/backward transform\nLimitation Note Transforms with the number of points N of a non-unit stride dimension exceeding\n2^(27−p) −1 for N a power-of-two, or 2^(23−p) − 1 for N not a power-of-two, are currently not\nsupported, where p=0 for single precision and p=1 for double precision. If a descriptor is created (for\nexample, using DftiCreateDescriptor) and set (for example, using DftiSetValue) to do such a\ntransform, a DFTI_1D_MEMORY_EXCEEDS_INT32 error is returned at commit time (for example, by\nDftiCommitDescriptor).\nTo better understand configuration of strides, you can also refer to these examples in your Intel® oneAPI\nMath Kernel Library (oneMKL) directory:\n./examples/dftc/source/basic_sp_complex_dft_2d.c\n./examples/dftc/source/basic_sp_complex_dft_3d.c\n./examples/dftc/source/basic_dp_complex_dft_2d.c\n./examples/dftc/source/basic_dp_complex_dft_3d.c\n./examples/dftc/source/basic_sp_real_dft_2d.c\n./examples/dftc/source/basic_sp_real_dft_3d.c\n./examples/dftc/source/basic_dp_real_dft_2d.c\n./examples/dftc/source/basic_dp_real_dft_3d.c\nSee Also\nDFTI_FORWARD_DOMAIN\nDFTI_PLACEMENT\nFFT Code Examples \nDftiSetValue\nDftiCommitDescriptor\nDftiComputeForward\nDftiComputeBackward\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2303\n\n\nDFTI_NUMBER_OF_TRANSFORMS\nIf you need to perform a large number of identical FFTs, you can do this in a single call to a DftiCompute*\nfunction with the value of the DFTI_NUMBER_OF_TRANSFORMS configuration parameter equal to the actual\nnumber of the transforms. The default value of this parameter is 1. You can set this parameter to a positive\ninteger value of the MKL_LONG data type. When setting the number of transforms to a value greater than\none, you also need to specify the distance between the input data sets and the distance between the output\ndata sets using one of the DFTI_INPUT_DISTANCE and DFTI_OUTPUT_DISTANCE configuration parameters or\nboth.\nImportant\n•\nThe data sets to be transformed must not have common elements.\n•\nAll the sets of data must be located within the same memory block.\nTo better understand usage of the DFTI_NUMBER_OF_TRANSFORMS configuration parameter, see this example\nin your Intel® oneAPI Math Kernel Library (oneMKL) directory:\n./examples/dftc/source/config_number_of_transforms.c\nSee Also\nFFT Computation Functions\nDFTI_INPUT_DISTANCE, DFTI_OUTPUT_DISTANCE\nDftiSetValue\nDFTI_INPUT_DISTANCE, DFTI_OUTPUT_DISTANCE\nThe FFT interface in Intel® oneAPI Math Kernel Library (oneMKL) enables computation of multiple transforms.\nTo compute multiple transforms, you need to specify the data distribution of the multiple sets of data. The\ndistance between the first data elements of consecutive data sets, DFTI_INPUT_DISTANCE for input data or\nDFTI_OUTPUT_DISTANCE for output data, specifies the distribution. The configuration setting is a value of\nMKL_LONG data type.\nThe default value for both configuration settings is one. You must set this parameter explicitly if the number\nof transforms is greater than one (see DFTI_NUMBER_OF_TRANSFORMS).\nThe distance is counted in elements of the data type defined by the descriptor configuration (rather than by\nthe type of the variable passed to the computation functions). Specifically, the DFTI_FORWARD_DOMAIN,\nDFTI_COMPLEX_STORAGE, and DFTI_CONJUGATE_EVEN_STORAGE configuration parameters define the type of\nthe elements as shown in Table \"Assumed Element Types of the Input/Output Data\".\nNOTE\nThe configuration parameters DFTI_INPUT_DISTANCE and DFTI_OUTPUT_DISTANCE define the\ndistance within input and output data, and not within the forward-domain and backward-domain data.\nIf the distances in the forward and backward domains differ, set DFTI_INPUT_DISTANCE and\nDFTI_OUTPUT_DISTANCE explicitly and then commit the descriptor before calling computation\nfunctions.\nFor in-place transforms (DFTI_PLACEMENT=DFTI_INPLACE), the configuration set by\nDFTI_OUTPUT_DISTANCE is ignored when the element types in the forward and backward domains are the\nsame. If they are different, set DFTI_OUTPUT_DISTANCE explicitly (even though the transform is in-place).\nEnsure a consistent configuration for in-place transforms, that is, the locations of the data sets on input and\noutput must coincide.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2304\n\n\nThis example illustrates setting of the DFTI_INPUT_DISTANCE configuration parameter:\nMKL_LONG dims[] = { nd, …, n2, n1 };\nMKL_LONG distance = nd*…*n2*n1;\nDftiCreateDescriptor( &hand, precision,  DFTI_COMPLEX, d, dims );\nDftiSetValue( hand, DFTI_NUMBER_OF_TRANSFORMS, (MKL_LONG)howmany );\nDftiSetValue( hand, DFTI_INPUT_DISTANCE,  distance );\nTo better understand configuration of the distances, see these code examples in your Intel® oneAPI Math\nKernel Library (oneMKL) directory:\n./examples/dftc/source/config_number_of_transforms.c\nSee Also\nDFTI_PLACEMENT\nDftiSetValue\nDftiCommitDescriptor\nDftiComputeForward\nDftiComputeBackward\nDFTI_COMPLEX_STORAGE, DFTI_REAL_STORAGE, DFTI_CONJUGATE_EVEN_STORAGE\nDepending on the value of the DFTI_FORWARD_DOMAIN configuration parameter, the implementation of FFT\nsupports several storage schemes for input and output data (see document [3] for the rationale behind the\ndefinition of the storage schemes). The data elements are placed within contiguous memory blocks, defined\nwith generalized strides (see DFTI_INPUT_STRIDES, DFTI_OUTPUT_STRIDES). For multiple transforms, all\nsets of data should be located within the same memory block, and the data sets should be placed at the\nsame distance from each other (see DFTI_NUMBER_OF TRANSFORMS and DFTI_INPUT DISTANCE,\nDFTI_OUTPUT_DISTANCE).\nTip\nAvoid setting up multidimensional arrays with lists of pointers to one-dimensional arrays. Instead use\na one-dimensional array with the explicit indexing to access the data elements.\nFFT Examples demonstrate the usage of storage formats.\nDFTI_COMPLEX_STORAGE: storage schemes for a complex domain\nFor the DFTI_COMPLEX forward domain, both input and output sequences belong to a complex domain. In\nthis case, the configuration parameter DFTI_COMPLEX_STORAGE can have one of the two values:\nDFTI_COMPLEX_COMPLEX (default) or DFTI_REAL_REAL.\nNOTE\nIn the Intel® oneAPI Math Kernel Library (oneMKL)FFT interface, storage schemes for a forward\ncomplex domain and the respective backward complex domain are the same.\nWith DFTI_COMPLEX_COMPLEX storage, complex-valued data sequences are referenced by a single complex\nparameter (array) AZ so that a complex-valued element zk1, k2, ..., kd of the m-th d-dimensional sequence is\nlocated at AZ[m*distance + stride0 + k1*stride1 + k2*stride2+ ... kd*strided] as a structure\nconsisting of the real and imaginary parts.\nThis code illustrates the use of the DFTI_COMPLEX_COMPLEX storage:\ncomplex *AZ = malloc( N1*N2*N3*M * sizeof(AZ[0]) );\nMKL_LONG ios[4], iodist; // input/output strides and distance\n...\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2305\n\n\n// on input:  Z{k1,k2,k3,m}\n// = AZ[ ios[0] + k1*ios[1] + k2*ios[2] + k3*ios[3] + m*iodist ]\nstatus = DftiComputeForward( desc, AZ );\n// on output: Z{k1,k2,k3,m}\n// = AZ[ ios[0] + k1*ios[1] + k2*ios[2] + k3*ios[3] + m*iodist ]\nWith the DFTI_REAL_REAL storage, complex-valued data sequences are referenced by two real parameters\nAR and AI so that a complex-valued element zk1, k2, ..., kd of the m-th sequence is computed as\nAR[m*distance + stride0 + k1*stride1 + k2*stride2+ ... kd*strided] + √(-1) *\nAI[m*distance + stride0 + k1*stride1 + k2*stride2+ ... kd*strided].\nThis code illustrates the use of the DFTI_REAL_REAL storage:\nfloat *AR = malloc( N1*N2*N3*M * sizeof(AR[0]) );\nfloat *AI = malloc( N1*N2*N3*M * sizeof(AI[0]) );\nMKL_LONG ios[4], iodist; // input/output strides and distance\n...\n// on input:  Z{k1,…,kd,m}\n// =   AR[ ios[0] + k1*ios[1] + k2*ios[2] + k3*ios[3] + m*iodist ]\n// + I*AI[ ios[0] + k1*ios[1] + k2*ios[2] + k3*ios[3] + m*iodist ]\nstatus = DftiComputeForward( desc, AR, AI );\n// on output: Z{k1,…,kd,m}  \n// =   AR[ ios[0] + k1*ios[1] + k2*ios[2] + k3*ios[3] + m*iodist ]\n// + I*AI[ ios[0] + k1*ios[1] + k2*ios[2] + k3*ios[3] + m*iodist ]\nDFTI_REAL_STORAGE: storage schemes for a real domain\nThe Intel® oneAPI Math Kernel Library (oneMKL) FFT interface supports only one configuration value for this\nstorage scheme: DFTI_REAL_REAL. With the DFTI_REAL_REAL storage, real-valued data sequences in a real\ndomain are referenced by one real parameter AR so that real-valued element of the m-th sequence is located\nas AR[m*distance + stride0 + k1*stride1 + k2*stride2+ ... kd*strided]. \nDFTI_CONJUGATE_EVEN_STORAGE: storage scheme for a conjugate-even domain\nThe Intel® oneAPI Math Kernel Library (oneMKL) FFT interface supports two configuration values for this\nparameter: DFTI_COMPLEX_COMPLEX (default) and DFTI_COMPLEX_REAL (for 1D problems only). The\nconjugate-even symmetry of the data enables storing only about a half of the whole mathematical result, so\nthat one part of it can be directly referenced in the memory while the other part can be reconstructed\ndepending on the selected storage configuration.\nWith the DFTI_COMPLEX_COMPLEX storage, the complex-valued data sequences in the conjugate-even\ndomain are referenced by one complex parameter AZ so that a complex-valued element zk1, k2, ..., kd of the m-\nth sequence can be referenced or reconstructed as described below.\nConsider a d-dimensional real-to-complex transform:\nBecause the input sequence R is real-valued, the mathematical result Z has conjugate-even symmetry:\nzk1, k2, ..., kd = conjugate (zN1-k1, N2-k2, ..., Nd-kd),\nwhere index arithmetic is performed modulo the length of the respective dimension. Obviously, the first\nelement of the result is real-valued:\nz0, 0, ..., 0 = conjugate (z0, 0, ..., 0 ).\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2306\n\n\nFor dimensions with even lengths, some of the other elements are real-valued too. For example, if Ns is even,\nz0, 0, ..., Ns /2, 0, ..., 0 = conjugate (z0, 0, ...,Ns /2, 0, ..., 0 ).\nWith the conjugate-even symmetry, approximately a half of the result suffices to fully reconstruct it. For an\narbitrary dimension h, it suffices to store elements zk1, ...,kh, ..., kd for the following indices:\n•\nkh = 0, ..., ⌊Nh /2⌋\n•\nki = 0, …, Ni -1, where i = 1, …, d and i≠h\nThe symmetry property enables reconstructing the remaining elements: for kh = ⌊Nh /2⌋ + 1, ... , Nh- 1. In\nthe Intel® oneAPI Math Kernel Library (oneMKL) FFT interface, the halved dimension is the last dimension.\nThe following code illustrates usage of the DFTI_COMPLEX_COMPLEX storage for a conjugate-even domain:\nfloat *AR = malloc( N1*N2*M * sizeof(AR[0]) );\ncomplex *AZ = malloc( N1*(N2/2+1)*M * sizeof(AZ[0]) );\nMKL_LONG is[3], os[3], idist, odist; // input and output strides and distance\n...\n// on input:  R{k1,k2,m}  \n// = AR[is[0] + k1*is[1] + k2*is[2] + m*idist]\nstatus = DftiComputeForward( desc, R, C );\n// on output:\n// for k2=0…N2/2:      Z{k1,k2,m} = AZ[os[0]+k1*os[1]+k2*os[2]+m*odist]\n// for k2=N2/2+1…N2-1: Z{k1,k2,m} = conj(AZ[os[0]+(N1-k1)%N1*os[1]\n//                                              +(N2-k2)%N2*os[2]+m*odist]) \nFor the backward transform, the input and output parameters and layouts exchange roles: set the strides\ndescribing the layout in the backward/forward domain as input/output strides, respectively. For example:\n...\nstatus = DftiSetValue( desc, DFTI_INPUT_STRIDES,  fwd_domain_strides );\nstatus = DftiSetValue( desc, DFTI_OUTPUT_STRIDES, bwd_domain_strides );\nstatus = DftiCommitDescriptor( desc );\nstatus = DftiComputeForward( desc, ... );\n...\nstatus = DftiSetValue( desc, DFTI_INPUT_STRIDES,  bwd_domain_strides );\nstatus = DftiSetValue( desc, DFTI_OUTPUT_STRIDES, fwd_domain_strides );\nstatus = DftiCommitDescriptor( desc );\nstatus = DftiComputeBackward( desc, ... );\nImportant\nFor in-place transforms, ensure the first element of the input data has the same location as the first\nelement of the output data for each dimension.\nSee Also\nDftiSetValue\nDFTI_PACKED_FORMAT\nThe result of the forward transform of real data is a conjugate-even sequence. Due to the symmetry\nproperty, only a part of the complex-valued sequence is stored in memory. The combination of the\nDFTI_PACKED_FORMAT and DFTI_CONJUGATE_EVEN_STORAGE configuration parameters defines how the\nconjugate-even sequence data is packed. If DFTI_CONJUGATE_EVEN_STORAGE is set to\nDFTI_COMPLEX_COMPLEX (default), the only possible value of DFTI_PACKED_FORMAT is DFTI_CCE_FORMAT;\nthis association of configuration parameters is supported for transforms of any dimension. For a description\nof the corresponding packed format, see DFTI_CONJUGATE_EVEN_STORAGE. For one-dimensional transforms\n(only) with DFTI_CONJUGATE_EVEN_STORAGE set to DFTI_COMPLEX_REAL, the DFTI_PACKED_FORMAT\nconfiguration parameter must be DFTI_CCS_FORMAT, DFTI_PACK_FORMAT, or DFTI_PERM_FORMAT. The\ncorresponding packed formats are explained and illustrated below.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2307\n\n\nDFTI_CCS_FORMAT for One-dimensional Transforms\nThe following figure illustrates the storage of a one-dimensional (1D) size-N conjugate-even sequence in a\nreal array for the CCS, PACK, and PERM packed formats. The CCS format requires an array of size N+2, while\nthe other formats require an array of size N. Zero-based indexing is used.\nStorage of a 1D Size-N Conjugate-even Sequence in a Real Array\nNOTE For storage of a one-dimensional conjugate-even sequence in a real array, CCS is in the same\nformat as CCE.\nThe real and imaginary parts of the complex-valued conjugate-even sequence Zk are located in a real-valued\narray AC as illustrated by figure \"Storage of a 1D Size-N Conjugate-even Sequence in a Real Array\" and can\nbe used to reconstruct the whole conjugate-even sequence as follows:\nfloat *AR; // malloc( sizeof(float)*N )\nfloat *AC; // malloc( sizeof(float)*(N+2) )\n...\nstatus = DftiSetValue( desc, DFTI_PACKED_FORMAT, DFTI_CCS_FORMAT );\n...\n// on input:  R{k} = AR[k]  \nstatus = DftiComputeForward( desc, AR, AC );  // real-to-complex FFT\n// on output:\n// for k=0…N/2:     Z{k} =   AC[2*k+0]         + I*AC[2*k+1]\n// for k=N/2+1…N-1: Z{k} =   AC[2*(N-k)%N + 0] - I*AC[2*(N-k)%N + 1] \nDFTI_PACK_FORMAT for One-dimensional Transforms\nThe real and imaginary parts of the complex-valued conjugate-even sequence Zk are located in a real-valued\narray AC as illustrated by figure \"Storage of a 1D Size-N Conjugate-even Sequence in a Real Array\" and can\nbe used to reconstruct the whole conjugate-even sequence as follows:\nfloat *AR; // malloc( sizeof(float)*N )\nfloat *AC; // malloc( sizeof(float)*N )\n...\nstatus = DftiSetValue( desc, DFTI_PACKED_FORMAT, DFTI_PACK_FORMAT );\n...\n// on input:  R{k} = AR[k]  \nstatus = DftiComputeForward( desc, AR, AC );  // real-to-complex FFT\n// on output: Z{k} = re + I*im, where\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2308\n\n\n// if (k == 0) {\n//     re =  AC[0];\n//     im =  0;\n// } else if (k == N-k) {\n//     re =  AC[2*k-1];\n//     im =  0;\n// } else if (k <= N/2) {\n//     re =  AC[2*k-1];\n//     im =  AC[2*k-0];\n// } else {\n//     re =  AC[2*(N-k)-1];\n//     im = -AC[2*(N-k)-0];\n// }\nDFTI_PERM_FORMAT for One-dimensional Transforms\nThe real and imaginary parts of the complex-valued conjugate-even sequence Zk are located in real-valued\narray AC as illustrated by figure \"Storage of a 1D Size-N Conjugate-even Sequence in a Real Array\" and can\nbe used to reconstruct the whole conjugate-even sequence as follows:\nfloat *AR; // malloc( sizeof(float)*N )\nfloat *AC; // malloc( sizeof(float)*N )\n...\nstatus = DftiSetValue( desc, DFTI_PACKED_FORMAT, DFTI_PERM_FORMAT );\n...\n// on input:  R{k} = AR[k]  \nstatus = DftiComputeForward( desc, AR, AC );  // real-to-complex FFT\n// on output: Z{k} = re + I*im, where\n// if (k == 0) {\n//     re =  AC[0];\n//     im =  0;\n// } else if (k == N-k) {\n//     re =  AC[1];\n//     im =  0;\n// } else if (k <= N/2) {\n//     re =  AC[2*k+0 - N%2];\n//     im =  AC[2*k+1 - N%2];\n// } else {\n//     re =  AC[2*(N-k)+0 - N%2];\n//     im = -AC[2*(N-k)+1 - N%2];\n// }\nSee Also\nDftiSetValue\nDFTI_WORKSPACE\nThe computation step for some FFT algorithms requires a scratch space for permutation or other purposes.\nTo manage the use of the auxiliary storage, Intel® oneAPI Math Kernel Library (oneMKL) enables you to set\nthe configuration parameterDFTI_WORKSPACE with the following values:\nDFTI_ALLOW\n(default) Permits the use of the auxiliary storage.\nDFTI_AVOID\nInstructs Intel® oneAPI Math Kernel Library (oneMKL) to avoid using\nthe auxiliary storage if possible.\nSee Also\nDftiSetValue\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2309\n\n\nDFTI_COMMIT_STATUS\nThe DFTI_COMMIT_STATUS configuration parameter indicates whether the descriptor is ready for\ncomputation. The parameter has two possible values:\nDFTI_UNCOMMITTED\nDefault value, set after a successful call of DftiCreateDescriptor.\nDFTI_COMMITTED\nThe value after a successful call to DftiCommitDescriptor.\nA computation function called with an uncommitted descriptor returns an error.\nYou cannot directly set this configuration parameter in a call to DftiSetValue, but a change in the\nconfiguration of a committed descriptor may change the commit status of the descriptor to\nDFTI_UNCOMMITTED.\nSee Also\nDftiCreateDescriptor\nDftiCommitDescriptor\nDftiSetValue\nDFTI_ORDERING\nSome FFT algorithms apply an explicit permutation stage that is time consuming [4]. The exclusion of this\nstep is similar to applying an FFT to input data whose order is scrambled, or allowing a scrambled order of\nthe FFT results. In applications such as convolution and power spectrum calculation, the order of result or\ndata is unimportant and thus using scrambled data is acceptable if it leads to better performance. The\nfollowing options are available in Intel® oneAPI Math Kernel Library (oneMKL):\n•\nDFTI_ORDERED: Forward transform data ordered, backward transform data ordered (default option).\n•\nDFTI_BACKWARD_SCRAMBLED: Forward transform data ordered, backward transform data scrambled.\nTable \"Scrambled Order Transform\" tabulates the effect of this configuration setting.\nScrambled Order Transform\nDftiComputeForward\nDftiComputeBackward\nDFTI_ORDERING\nInput → Output\nInput → Output\nDFTI_ORDERED\nordered → ordered\nordered → ordered\nDFTI_BACKWARD_SCRAMBLED\nordered → scrambled\nscrambled → ordered\nNOTE\nThe word \"scrambled\" in this table means \"permit scrambled order if possible\". In some situations\npermitting out-of-order data gives no performance advantage and an implementation may choose to\nignore the suggestion.\nSee Also\nDftiSetValue\nDFTI_DESTROY_INPUT\nAllowing the library to overwrite your input data in case of out-of-place transforms may be an appropriate\nalternative to using DFTI_WORKSPACE to reduce the memory footprint of your application. Intel® oneAPI\nMath Kernel Library (oneMKL) enables you to communicate whether this is allowed via the configuration\nparameter DFTI_DESTROY_INPUT, with the following values:\nDFTI_AVOID\nThe default. Instructs the library to leave the input\ndata unchanged.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2310\n\n\nDFTI_ALLOW\nPermits the input data to be overwritten.\nSee Also\nDftiSetValue\nFFT Descriptor Manipulation Functions\nThis category contains the following functions: create a descriptor, commit a descriptor, copy a descriptor,\nand free a descriptor.\nDftiCreateDescriptor\nAllocates the descriptor data structure and initializes it\nwith default configuration values.\nSyntax\nstatus = DftiCreateDescriptor(&desc_handle, precision, forward_domain, dimension,\nlength);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nprecision\nenum\nPrecision of the transform: DFTI_SINGLE or\nDFTI_DOUBLE.\nforward_domain\nenum\nForward domain of the transform:\nDFTI_COMPLEX or DFTI_REAL.\ndimension\nMKL_LONG\nDimension of the transform.\nlength\nMKL_LONG if dimension = 1.\nArray of type MKL_LONG otherwise.\nLength of the transform for a one-dimensional\ntransform. Lengths of each dimension for a\nmulti-dimensional transform.\nOutput Parameters\nName\nType\nDescription\ndesc_handle\nDFTI_DESCRIPTOR_HANDLE\nFFT descriptor.\nstatus\nMKL_LONG\nFunction completion status.\nDescription\nThis function allocates memory for the descriptor data structure and instantiates it with all the default\nconfiguration settings for the precision, forward domain, dimension, and length of the desired transform.\nBecause memory is allocated dynamically, the result is actually a pointer to the created descriptor. This\nfunction is slightly different from the \"initialization\" function that can be found in software packages or\nlibraries that implement more traditional algorithms for computing an FFT. This function does not perform\nany significant computational work such as computation of twiddle factors. The function \nDftiCommitDescriptor does this work after the function DftiSetValue has set values of all necessary\nparameters.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2311\n\n\nThe function returns zero when it completes successfully. See Status Checking Functions for more\ninformation on the returned status.\nPrototype\n \n/* Note that the preprocessor definition provided below only illustrates \n * that the actual function called may be determined at compile time. \n * You can rely only on the declaration of the function. \n * For precise definition of the preprocessor macro, see the include/mkl_dfti.h\n * file in the Intel MKL directory.\n */\nMKL_LONG DftiCreateDescriptor(DFTI_DESCRIPTOR_HANDLE * pHandle,\n     enum DFTI_CONFIG_VALUE precision,\n     enum DFTI_CONFIG_VALUE domain,\n     MKL_LONG dimension, ... /* length/lengths */ );\n                 \n#define DftiCreateDescriptor(desc,prec,domain,dim,sizes) \\\n     ((prec)==DFTI_SINGLE && (dim)==1) ? \\\n     some_actual_function_s1d((desc),(domain),(MKL_LONG)(sizes)) : \\\n     ...\n \nVariable length/lengths is interpreted as a scalar (MKL_LONG) or an array (MKL_LONG*), depending on the\nvalue of parameter dimension. If the value of parameter precision is known at compile time, an\noptimizing compiler retains only the call to the respective specific function, thereby reducing the size of the\nstatically linked application. Avoid direct calls to the specific functions used in the preprocessor macro\ndefinition, because their interface may change in future releases of the library. If the use of the macro is\nundesirable, you can safely undefine it after inclusion of the Intel® oneAPI Math Kernel Library (oneMKL) FFT\nheader file, as follows:\n#include \"mkl_dfti.h\"\n#undef DftiCreateDescriptor\n            \nSee Also\nDFTI_PRECISION  configuration parameter\nDFTI_FORWARD_DOMAIN  configuration parameter\nDFTI_DIMENSION, DFTI_LENGTHS  configuration parameters\nConfiguration Parameters, summary table\nDftiCommitDescriptor\nPerforms all initialization for the actual FFT\ncomputation.\nSyntax\nstatus = DftiCommitDescriptor(desc_handle);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\ndesc_handle\nDFTI_DESCRIPTOR_HANDLE\nFFT descriptor.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2312\n\n\nOutput Parameters\nName\nType\nDescription\ndesc_handle\nDFTI_DESCRIPTOR_HANDLE\nUpdated FFT descriptor.\nstatus\nMKL_LONG\nFunction completion status.\nDescription\nThis function completes initialization of a previously created descriptor, which is required before the\ndescriptor can be used for FFT computations. Typically, committing the descriptor performs all initialization\nthat is required for the actual FFT computation. The initialization done by the function may involve exploring\ndifferent factorizations of the input length to find the optimal computation method.\nIf you call the DftiSetValue function to change configuration parameters of a committed descriptor (see \nDescriptor Configuration Functions), you must re-commit the descriptor before invoking a computation\nfunction. Typically, a committal function call is immediately followed by a computation function call (see FFT\nComputation Functions).\nThe function returns zero when it completes successfully. See Status Checking Functions for more\ninformation on the returned status.\nPrototype\n \nMKL_LONG DftiCommitDescriptor( DFTI_DESCRIPTOR_HANDLE );\n \nDftiFreeDescriptor\nFrees the memory allocated for a descriptor.\nSyntax\nstatus = DftiFreeDescriptor(&desc_handle);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\ndesc_handle\nDFTI_DESCRIPTOR_HANDLE\nFFT descriptor.\n \n \n \nOutput Parameters\nName\nType\nDescription\ndesc_handle\nDFTI_DESCRIPTOR_HANDLE\nMemory for the FFT descriptor is released.\n \n \n \nstatus\nMKL_LONG\nFunction completion status.\nDescription\nThis function frees all memory allocated for a descriptor.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2313\n\n\nNOTE\nMemory allocation/deallocation inside Intel® oneAPI Math Kernel Library (oneMKL) is\nmanaged by Intel® oneAPI Math Kernel Library (oneMKL) memory management software.\nSo, even after successful completion of FreeDescriptor, the memory space may continue\nbeing allocated for the application because the memory management software sometimes\ndoes not return the memory space to the OS, but considers the space free and can reuse it\nfor future memory allocation. See Example mkl_free_buffers: Usage with FFT Functions on\nhow to use Intel® oneAPI Math Kernel Library (oneMKL) memory management software and\nrelease memory to the OS.\nThe function returns zero when it completes successfully. See Status Checking Functions for more\ninformation on the returned status.\nPrototype\n \nMKL_LONG DftiFreeDescriptor( DFTI_DESCRIPTOR_HANDLE * );\n \nDftiCopyDescriptor\nMakes a copy of an existing descriptor.\nSyntax\nstatus = DftiCopyDescriptor(desc_handle_original, &desc_handle_copy);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\ndesc_handle_original\nDFTI_DESCRIPTOR_HANDLE\nThe FFT descriptor to copy.\nDFTI_DESCRIPTOR_HANDLE\nThe FFT descriptor to copy.\n \n \n \nOutput Parameters\nName\nType\nDescription\ndfti_handle_dst\nDFTI_DESCRIPTOR_HANDLE\nThe copy of the FFT descriptor.\n \n \n \nstatus\nMKL_LONG\nFunction completion status.\nDescription\nThis function makes a copy of an existing descriptor. The resulting descriptor desc_handle_copy and the\nexisting descriptor desc_handle_original specify the same configuration of the transform, but do not have\nany memory areas in common (\"deep copy\").\nThe function returns zero when it completes successfully. See Status Checking Functions for more\ninformation on the returned status.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2314\n\n\nPrototype\n \nMKL_LONG DftiCopyDescriptor( DFTI_DESCRIPTOR_HANLDE, DFTI_DESCRIPTOR_HANDLE * );\n \nFFT Descriptor Configuration Functions\nThis category contains the following functions: the value setting function DftiSetValue sets one particular\nconfiguration parameter to an appropriate value, and the value-getting function DftiGetValue reads the\nvalue of one particular configuration parameter. While all configuration parameters are readable, you cannot\nset a few of them. Some of these contain fixed information of a particular implementation such as version\nnumber, or dynamic information, which is derived by the implementation during execution of one of the\nfunctions. See Configuration Settings for details.\nDftiSetValue\nSets one particular configuration parameter with the\nspecified configuration value.\nSyntax\nstatus = DftiSetValue(desc_handle, config_param, config_val);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\ndesc_handle\nDFTI_DESCRIPTOR_HANDLE\nFFT descriptor.\nconfig_param\nenum\nConfiguration parameter.\nconfig_val\nDepends on the configuration\nparameter.\nConfiguration value.\nOutput Parameters\nName\nType\nDescription\ndesc_handle\nDFTI_DESCRIPTOR_HANDLE\nUpdated FFT descriptor.\nstatus\nMKL_LONG\nFunction completion status.\nDescription\nThis function sets one particular configuration parameter with the specified configuration value. Each\nconfiguration parameter is a named constant, and the configuration value must have the corresponding type,\nwhich can be a named constant or a native type. For available configuration parameters and the\ncorresponding configuration values, see:\n•\nDFTI_PRECISION\n•\nDFTI_FORWARD_DOMAIN\n•\nDFTI_DIMENSION, DFTI_LENGTHS\n•\nDFTI_PLACEMENT\n•\nDFTI_FORWARD_SCALE, DFTI_BACKWARD_SCALE\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2315\n\n\n•\nDFTI_THREAD_LIMIT\n•\nDFTI_INPUT_STRIDES, DFTI_OUTPUT_STRIDES\n•\nDFTI_NUMBER_OF_TRANSFORMS\n•\nDFTI_INPUT_DISTANCE, DFTI_OUTPUT_DISTANCE\n•\nDFTI_COMPLEX_STORAGE, DFTI_REAL_STORAGE, DFTI_CONJUGATE_EVEN_STORAGE\n•\nDFTI_PACKED_FORMAT\n•\nDFTI_WORKSPACE\n•\nDFTI_ORDERING\n•\nDFTI_DESTROY_INPUT\nYou cannot use the DftiSetValue function to change configuration parameters DFTI_FORWARD_DOMAIN,\nDFTI_PRECISION, DFTI_DIMENSION, and DFTI_LENGTHS. Use the DftiCreateDescriptor function to set\nthem.\nFunction calls needed to configure an FFT descriptor for a particular call to an FFT computation function are\nsummarized in Configuring and Computing an FFT in C C++.\nThe function returns zero when it completes successfully. See Status Checking Functions for more\ninformation on the returned status.\nPrototype\n \n                MKL_LONG DftiSetValue( DFTI_DESCRIPTOR_HANDLE, DFTI_CONFIG_PARAM , ... );\n \nSee Also\nConfiguration Settings  for more information on configuration parameters.\nDftiCreateDescriptor\nDftiGetValue\nDftiGetValue\nGets the configuration value of one particular\nconfiguration parameter.\nSyntax\nstatus = DftiGetValue(desc_handle, config_param, &config_val);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\ndesc_handle\nDFTI_DESCRIPTOR_HANDLE\nFFT descriptor.\nconfig_param\nenum\nConfiguration parameter. See Table\n\"Configuration Parameters\" for allowable values\nof config_param.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2316\n\n\nOutput Parameters\nName\nType\nDescription\nconfig_val\nDepends on the configuration\nparameter.\nConfiguration value.\nstatus\nMKL_LONG\nFunction completion status.\nDescription\nThis function gets the configuration value of one particular configuration parameter. Each configuration\nparameter is a named constant, and the configuration value must have the corresponding type, which can be\na named constant or a native type. For available configuration parameters and the corresponding\nconfiguration values, see:\n•\nDFTI_PRECISION\n•\nDFTI_FORWARD_DOMAIN\n•\nDFTI_DIMENSION, DFTI_LENGTH\n•\nDFTI_PLACEMENT\n•\nDFTI_FORWARD_SCALE, DFTI_BACKWARD_SCALE\n•\nDFTI_THREAD_LIMIT\n•\nDFTI_INPUT_STRIDES, DFTI_OUTPUT_STRIDES\n•\nDFTI_NUMBER_OF_TRANSFORMS\n•\nDFTI_INPUT_DISTANCE, DFTI_OUTPUT_DISTANCE\n•\nDFTI_COMPLEX_STORAGE, DFTI_REAL_STORAGE, DFTI_CONJUGATE_EVEN_STORAGE\n•\nDFTI_PACKED_FORMAT\n•\nDFTI_WORKSPACE\n•\nDFTI_COMMIT_STATUS\n•\nDFTI_ORDERING\n•\nDFTI_DESTROY_INPUT\nThe function returns zero when it completes successfully. See Status Checking Functions for more\ninformation on the returned status.\nPrototype\n \n                MKL_LONG DftiGetValue( DFTI_DESCRIPTOR_HANDLE,\n      DFTI_CONFIG_PARAM ,\n      ... );\n \nConfiguration Settings  for more information on configuration parameters.\nDftiSetValue\nFFT Computation Functions\nThis category contains the following functions: compute the forward transform and compute the backward\ntransform.\nDftiComputeForward\nComputes the forward FFT.\nSyntax\nstatus = DftiComputeForward(desc_handle, x_inout);\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2317\n\n\nstatus = DftiComputeForward(desc_handle, x_in, y_out);\nstatus = DftiComputeForward(desc_handle, xre_inout, xim_inout);\nstatus = DftiComputeForward(desc_handle, xre_in, xim_in, yre_out, yim_out);\nInput Parameters\nName\nType\nDescription\ndesc_handle\nDFTI_DESCRIPTOR_HANDLE\nFFT descriptor.\nx_inout, x_in\nArray of type float or double depending\non the precision of the transform.\nData to be transformed in case of a\nreal forward domain or, in the case of a\ncomplex forward domain in association\nwith FTI_COMPLEX_COMPLEX, set for\nDFTI_COMPLEX_STORAGE.\nxre_inout,\nxim_inout,\nxre_in, xim_in\nArray of type float or double depending\non the precision of the transform.\nReal and imaginary parts of the data to\nbe transformed in the case of a\ncomplex forward domain.\nThe suffix in parameter names corresponds to the value of the configuration parameter DFTI_PLACEMENT as\nfollows:\n•\n_inout to DFTI_INPLACE\n•\n_in to DFTI_NOT_INPLACE\nOutput Parameters\nName\nType\nDescription\ny_out\nArray of type float or double depending\non the precision of the transform.\nThe transformed data in case of a real\nbackward domain or, in the case of a\ncomplex forward domain in association\nwith DFTI_COMPLEX_COMPLEX, set for\nDFTI_COMPLEX_STORAGE.\nxre_inout,\nxim_inout,\nyre_out, yim_out\nArray of type float or double depending\non the precision of the transform.\nReal and imaginary parts of the\ntransformed data in the case of a\ncomplex forward domain in association\nwith DFTI_REAL_REAL set for\nDFTI_COMPLEX_STORAGE.\nstatus\nMKL_LONG\nFunction completion status.\nThe suffix in parameter names corresponds to the value of the configuration parameter DFTI_PLACEMENT as\nfollows:\n•\n_inout to DFTI_INPLACE\n•\n_out to DFTI_NOT_INPLACE\nInclude Files\n•\nmkl.h\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2318\n\n\nDescription\nThe DftiComputeForward function accepts the descriptor handle parameter and one or more data\nparameters. Given a successfully configured and committed descriptor, this function computes the forward\nFFT, that is, the transform with the minus sign in the exponent, δ = -1.\nThe DFTI_COMPLEX_STORAGE, DFTI_REAL_STORAGE, and DFTI_CONJUGATE_EVEN_STORAGE configuration\nparameters define the layout of the input and output data and must be properly set in a call to the\nDftiSetValue function. The forward domain and the precision of the transform are determined by the\nconfiguration settings DFTI_FORWARD_DOMAIN and DFTI_PRECISION, which are during construction of the\ndescriptor.\nThe FFT descriptor must be properly configured prior to the function call. Function calls needed to configure\nan FFT descriptor for a particular call to an FFT computation function are summarized in Configuring and\nComputing an FFT in C C++.\nThe number and types of the data parameters that the function requires may vary depending on the\nconfiguration of the descriptor. This variation is accommodated by variable parameters.\nThe function returns zero when it completes successfully. See Status Checking Functions for more\ninformation on the returned status.\nPrototype\n \n  MKL_LONG DftiComputeForward( DFTI_DESCRIPTOR_HANDLE, void*, ... ); \nSee Also\nConfiguration Settings\nDFTI_FORWARD_DOMAIN\nDFTI_PLACEMENT\nDFTI_PACKED_FORMAT\nDFTI_COMPLEX_STORAGE, DFTI_REAL_STORAGE, DFTI_CONJUGATE_EVEN_STORAGE\nDFTI_DIMENSION, DFTI_LENGTHS\nDFTI_INPUT_DISTANCE, DFTI_OUTPUT_DISTANCE\nDFTI_INPUT_STRIDES, DFTI_OUTPUT_STRIDES\nDftiComputeBackward\nDftiSetValue\nDftiComputeBackward\nComputes the backward FFT.\nSyntax\nstatus = DftiComputeBackward(desc_handle, x_inout);\nstatus = DftiComputeBackward(desc_handle, y_in, x_out);\nstatus = DftiComputeBackward(desc_handle, xre_inout, xim_inout);\nstatus = DftiComputeBackward(desc_handle, yre_in, yim_in, xre_out, xim_out);\nInput Parameters\nName\nType\nDescription\ndesc_handle\nDFTI_DESCRIPTOR_HANDLE\nFFT descriptor.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2319\n\n\nName\nType\nDescription\nx_inout, y_in\nArray of type float or double depending\non the precision of the transform.\nData to be transformed in case of a\nreal forward domain or, in the case of a\ncomplex forward domain in association\nwith DFTI_COMPLEX_COMPLEX, set for\nDFTI_COMPLEX_STORAGE.\nxre_inout,\nxim_inout,\nyre_in, yim_in\nArray of type float or double depending\non the precision of the transform.\nReal and imaginary parts of the data to\nbe transformed in the case of a\ncomplex forward domain in association\nwith DFTI_REAL_REAL set for\nDFTI_COMPLEX_STORAGE.\nThe suffix in parameter names corresponds to the value of the configuration parameter DFTI_PLACEMENT as\nfollows:\n•\n_inout to DFTI_INPLACE\n•\n_in to DFTI_NOT_INPLACE\nOutput Parameters\nName\nType\nDescription\nx_out\nArray of type float or double depending\non the precision of the transform.\nThe transformed data in case of a real\nforward domain or, in the case of a\ncomplex forward domain in association\nwith DFTI_COMPLEX_COMPLEX, set for\nDFTI_COMPLEX_STORAGE.\nxre_inout,\nxim_inout,\nxre_out, xim_out\nArray of type float or double depending\non the precision of the transform.\nReal and imaginary parts of the\ntransformed data in the case of a\ncomplex forward domain in association\nwith DFTI_REAL_REAL set for\nDFTI_COMPLEX_STORAGE.\nstatus\nMKL_LONG\nFunction completion status.\nThe suffix in parameter names corresponds to the value of the configuration parameter DFTI_PLACEMENT as\nfollows:\n•\n_inout to DFTI_INPLACE\n•\n_out to DFTI_NOT_INPLACE\nInclude Files\n•\nmkl.h\nDescription\nThe function accepts the descriptor handle parameter and one or more data parameters. Given a successfully\nconfigured and committed descriptor, the DftiComputeBackward function computes the inverse FFT, that is,\nthe transform with the plus sign in the exponent, δ = +1.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2320\n\n\nThe DFTI_COMPLEX_STORAGE, DFTI_REAL_STORAGE, and DFTI_CONJUGATE_EVEN_STORAGE configuration\nparameters define the layout of the input and output data and must be properly set in a call to the\nDftiSetValue function. The forward domain and the precision of the transform are determined by the\nconfiguration settings DFTI_FORWARD_DOMAIN and DFTI_PRECISION, which are during construction of the\ndescriptor.\nThe FFT descriptor must be properly configured prior to the function call. Function calls needed to configure\nan FFT descriptor for a particular call to an FFT computation function are summarized in Configuring and\nComputing an FFT in C C++.\nThe number and types of the data parameters that the function requires may vary depending on the\nconfiguration of the descriptor. This variation is accommodated by variable parameters.\nThe function returns zero when it completes successfully. See Status Checking Functions for more\ninformation on the returned status.\nPrototype\n    \n  MKL_LONG DftiComputeBackward( DFTI_DESCRIPTOR_HANDLE, void *, ... );\n \nSee Also\nConfiguration Settings\nDFTI_FORWARD_DOMAIN\nDFTI_PLACEMENT\nDFTI_PACKED_FORMAT\nDFTI_COMPLEX_STORAGE, DFTI_REAL_STORAGE, DFTI_CONJUGATE_EVEN_STORAGE\nDFTI_DIMENSION, DFTI_LENGTHS\nDFTI_INPUT_DISTANCE, DFTI_OUTPUT_DISTANCE\nDFTI_INPUT_STRIDES, DFTI_OUTPUT_STRIDES\nDftiComputeForward\nDftiSetValue\nConfiguring and Computing an FFT in C/C++\nThe table below summarizes information on configuring and computing an FFT in C/C++ for all kinds of\ntransforms and possible combinations of input and output domains.\nFFT to Compute\nInput Data\nOutput Data\nRequired FFT Function Calls\nComplex-to-\ncomplex,\nin-place,\nforward or\nbackward\nInterleaved\ncomplex\nnumbers\nInterleaved\ncomplex\nnumbers\n/* Configure a Descriptor */\nstatus = DftiCreateDescriptor(&hand, \n<precision>, DFTI_COMPLEX, <dimension>, \n<sizes>);\nstatus = DftiCommitDescriptor(hand);\n/* Compute an FFT */\n/* forward FFT */\nstatus = DftiComputeForward(hand, X_inout);\n/* or backward FFT */\nstatus = DftiComputeBackward(hand, X_inout);\nComplex-to-\ncomplex,\nout-of-place,\nInterleaved\ncomplex\nnumbers\nInterleaved\ncomplex\nnumbers\n/* Configure a Descriptor */\nstatus = DftiCreateDescriptor(&hand, \n<precision>, DFTI_COMPLEX, <dimension>, \nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2321\n\n\nFFT to Compute\nInput Data\nOutput Data\nRequired FFT Function Calls\nforward or\nbackward\n<sizes>);\nstatus = DftiSetValue(hand, DFTI_PLACEMENT, \nDFTI_NOT_INPLACE);\nstatus = DftiCommitDescriptor(hand);\n/* Compute an FFT */\n/* forward FFT */\nstatus = DftiComputeForward(hand, X_in, \nY_out);\n/* or backward FFT */\nstatus = DftiComputeBackward(hand, X_in, \nY_out);\nComplex-to-\ncomplex,\nin-place,\nforward or\nbackward\nSplit-complex\nnumbers\nSplit-complex\nnumbers\n/* Configure a Descriptor */\nstatus = DftiCreateDescriptor(&hand, \n<precision>, DFTI_COMPLEX, <dimension>, \n<sizes>);\nstatus = DftiSetValue(hand, \nDFTI_COMPLEX_STORAGE, DFTI_REAL_REAL);\nstatus = DftiCommitDescriptor(hand);\n/* Compute an FFT */\n/* forward FFT */\nstatus = DftiComputeForward(hand, Xre_inout, \nXim_inout);\n/* or backward FFT */\nstatus = DftiComputeBackward(hand, Xre_inout, \nXim_inout);\nComplex-to-\ncomplex,\nout-of-place,\nforward or\nbackward\nSplit-complex\nnumbers\nSplit-complex\nnumbers\n/* Configure a Descriptor */\nstatus = DftiCreateDescriptor(&hand, \n<precision>, DFTI_COMPLEX, <dimension>, \n<sizes>);\nstatus = DftiSetValue(hand, \nDFTI_COMPLEX_STORAGE, DFTI_REAL_REAL);\nstatus = DftiSetValue(hand, DFTI_PLACEMENT, \nDFTI_NOT_INPLACE);\nstatus = DftiCommitDescriptor(hand);\n/* Compute an FFT */\n/* forward FFT  */\nstatus = DftiComputeForward(hand, Xre_in, \nXim_in, Yre_out, Yim_out);\n/* or backward FFT  */\nstatus = DftiComputeBackward(hand, Xre_in, \nXim_in, Yre_out, Yim_out);\n                  \nReal-to-complex,\nin-place,\nforward\nReal numbers\nNumbers in\nthe CCE\nformat\n/* Configure a Descriptor */\nstatus = DftiCreateDescriptor(&hand, \n<precision>, DFTI_REAL, <dimension>, <sizes>);\nstatus = DftiSetValue(hand, \nDFTI_CONJUGATE_EVEN_STORAGE,  \nDFTI_COMPLEX_COMPLEX);\nstatus = DftiSetValue(hand, \n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2322\n\n\nFFT to Compute\nInput Data\nOutput Data\nRequired FFT Function Calls\nDFTI_PACKED_FORMAT, DFTI_CCE_FORMAT);\nstatus = DftiSetValue(hand, \nDFTI_INPUT_STRIDES, <real_strides>);\nstatus = DftiSetValue(hand, \nDFTI_OUTPUT_STRIDES, <complex_strides>);\nstatus = DftiCommitDescriptor(hand);\n/* Compute an FFT */\nstatus = DftiComputeForward(hand, X_inout);\nReal-to-complex,\nout-of-place,\nforward\nReal numbers\nNumbers in\nthe CCE\nformat\n/* Configure a Descriptor */\nstatus = DftiCreateDescriptor(&hand, \n<precision>, DFTI_REAL, <dimension>, <sizes>);\nstatus = DftiSetValue(hand, \nDFTI_CONJUGATE_EVEN_STORAGE, \nDFTI_COMPLEX_COMPLEX);\nstatus = DftiSetValue(hand, \nDFTI_PACKED_FORMAT, DFTI_CCE_FORMAT);\nstatus = DftiSetValue(hand, DFTI_PLACEMENT, \nDFTI_NOT_INPLACE);\nstatus = DftiSetValue(hand, \nDFTI_INPUT_STRIDES, <real_strides>);\nstatus = DftiSetValue(hand, \nDFTI_OUTPUT_STRIDES, \n<complex_strides>);\nstatus = DftiCommitDescriptor(hand);\n/* Compute an FFT */\nstatus = DftiComputeForward(hand, X_in, \nY_out);\n                  \nComplex-to-real,\nin-place,\nbackward\nNumbers in\nthe CCE\nformat\nReal numbers\n/* Configure a Descriptor */\nstatus = DftiCreateDescriptor(&hand, \n<precision>, DFTI_REAL, <dimension>, <sizes>);\nstatus = DftiSetValue(hand, \nDFTI_CONJUGATE_EVEN_STORAGE,  \nDFTI_COMPLEX_COMPLEX);\nstatus = DftiSetValue(hand, \nDFTI_PACKED_FORMAT, DFTI_CCE_FORMAT);\nstatus = DftiSetValue(hand, \nDFTI_INPUT_STRIDES, <complex_strides>);\nstatus = DftiSetValue(hand, \nDFTI_OUTPUT_STRIDES, <real_strides>);\nstatus = DftiCommitDescriptor(hand);\n/* Compute  an FFT */\nstatus = DftiComputeBackward(hand, X_inout);\nComplex-to-real,\nout-of-place,\nbackward\nNumbers in\nthe CCE\nformat\nReal numbers\n/* Configure a Descriptor */\nstatus = DftiCreateDescriptor(&hand, \n<precision>, DFTI_REAL, <dimension>, <sizes>);\nstatus = DftiSetValue(hand, \nDFTI_CONJUGATE_EVEN_STORAGE,  \nDFTI_COMPLEX_COMPLEX);\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2323\n\n\nFFT to Compute\nInput Data\nOutput Data\nRequired FFT Function Calls\nstatus = DftiSetValue(hand, DFTI_PLACEMENT, \nDFTI_NOT_INPLACE);\nstatus = DftiSetValue(hand, \nDFTI_PACKED_FORMAT, DFTI_CCE_FORMAT);\nstatus = DftiSetValue(hand, \nDFTI_INPUT_STRIDES, <complex_strides>);\nstatus = DftiSetValue(hand, \nDFTI_OUTPUT_STRIDES, <real_strides>);\nstatus = DftiCommitDescriptor(hand);\n/* Compute an FFT */\nstatus = DftiComputeBackward(hand, X_in, \nY_out);\nYou can find C programs that illustrate configuring and computing FFTs in the examples/dftc/ subdirectory\nof your Intel® oneAPI Math Kernel Library (oneMKL) directory.\nStatus Checking Functions\nAll of the descriptor manipulation, FFT computation, and descriptor configuration functions return an integer\nvalue denoting the status of the operation. The functions in this category check that status. The first function\nis a logical function that checks whether the status reflects an error of a predefined class, and the second is\nan error message function that returns a character string.\nDftiErrorClass\nChecks whether the status reflects an error of a\npredefined class.\nSyntax\npredicate = DftiErrorClass(status, error_class);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nstatus\nMKL_LONG\nCompletion status of a fast Fourier transform\n(FFT) function.\nerror_class\nMKL_LONG\nPredefined error class.\nOutput Parameters\nName\nType\nDescription\npredicate\nMKL_LONG\nResult of checking.\n \n \n \nDescription\nThe FFT interface in Intel® oneAPI Math Kernel Library (oneMKL) provides a set of predefined error\nclasseslisted in Table \"Predefined Error Classes\". They are named constants and have the type MKL_LONG.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2324\n\n\nPredefined Error Classes\nNamed Constants\nComments\nDFTI_NO_ERROR\nNo error. The zero status belongs to this class.\nDFTI_MEMORY_ERROR\nUsually associated with memory allocation.\nDFTI_INVALID_CONFIGURATION\nInvalid settings in one or more configuration parameters.\nDFTI_INCONSISTENT_CONFIGURATION\nInconsistent configuration or input parameters.\nDFTI_NUMBER_OF_THREADS_ERROR\nNumber of OMP threads in the computation function is\nnot equal to the number of OMP threads in the\ninitialization stage (commit function).\nDFTI_MULTITHREADED_ERROR\nUsually associated with a value that OMP routines return\nin case of errors.\nDFTI_BAD_DESCRIPTOR\nDescriptor is unusable for computation.\nDFTI_UNIMPLEMENTED\nUnimplemented legitimate settings; implementation\ndependent.\nDFTI_MKL_INTERNAL_ERROR\nInternal library error.\nDFTI_1D_LENGTH_EXCEEDS_INT32\nLength of one of the dimensions exceeds 232 -1 (4\nbytes).\nDFTI_1D_MEMORY_EXCEEDS_INT32\nData size of one of the transforms exceeds 231 -1 bytes.\nNOTE\nUse DFTI_1D_MEMORY_EXCEEDS_INT32 instead of DFTI_1D_LENGTH_EXCEEDS_INT32 for better\naccuracy.\nThe DftiErrorClass function returns a non-zero value if the status belongs to the predefined error class. To\ncheck whether a function call was successful, call DftiErrorClass with a specific error class. However, the\nzero value of the status belongs to the DFTI_NO_ERROR class and thus the zero status indicates successful\ncompletion of an operation. See Example \"Using Status Checking Functions\" for an illustration of correct use\nof the status checking functions.\nNOTE\nIt is incorrect to directly compare a status with a predefined class.\nPrototype\n \nMKL_LONG DftiErrorClass( MKL_LONG , MKL_LONG ); \nDftiErrorMessage\nGenerates an error message.\nSyntax\nerror_message = DftiErrorMessage(status);\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2325\n\n\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nstatus\nMKL_LONG\nCompletion status of a function.\n \n \n \nOutput Parameters\nName\nType\nDescription\nerror_message\nArray of char\nThe character string with the\nerror message.\n \n \n \nDescription\nThe error message function generates an error message character string. The function returns a pointer to a\nconstant character string, that is, a character array with terminating '\\0' character, and you do not need to\nfree this pointer.\nExample Using Status Checking Function shows how this function can be used.\nPrototype\n \nchar *DftiErrorMessage( MKL_LONG );\n \nCluster FFT Functions\nThis section describes the cluster Fast Fourier Transform (FFT) functions implemented in Intel® oneAPI Math\nKernel Library (oneMKL).\nNOTE\nThese functions are available only for Intel® 64 architectures.\nThe cluster FFT function library was designed to perform fast Fourier transforms on a cluster, that is, a group\nof computers interconnected via a network. Each computer (node) in the cluster has its own memory and\nprocessor(s). Data interchanges between the nodes are provided by the network.\nOne or more processes may be running in parallel on each cluster node. To organize communication between\ndifferent processes, the cluster FFT function library uses the Message Passing Interface (MPI). To avoid\ndependence on a specific MPI implementation (for example, MPICH, Intel® MPI, and others), the library works\nwith MPI via a message-passing library for linear algebra called BLACS.\nCluster FFT functions of Intel® oneAPI Math Kernel Library (oneMKL) provide one-dimensional, two-\ndimensional, and multi-dimensional (up to the order of 7) functions and both Fortran and C interfaces for all\ntransform functions.\nTo develop applications using the cluster FFT functions, you should have basic skills in MPI programming.\nThe interfaces for the Intel® oneAPI Math Kernel Library (oneMKL) cluster FFT functions are similar to the\ncorresponding interfaces for the conventional Intel® oneAPI Math Kernel Library (oneMKL)FFT functions. Refer\nthere for details not explained in this section.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2326\n\n\nTable \"Cluster FFT Functions in Intel® oneAPI Math Kernel Library (oneMKL)\" lists cluster FFT functions\nimplemented in Intel® oneAPI Math Kernel Library (oneMKL):\nCluster FFT Functions in oneMKL\nFunction Name\nOperation\nDescriptor Manipulation Functions\nDftiCreateDescriptorDM\nAllocates memory for the descriptor data structure and\npreliminarily initializes it.\nDftiCommitDescriptorDM\nPerforms all initialization for the actual FFT computation.\nDftiFreeDescriptorDM\nFrees memory allocated for a descriptor.\nFFT Computation Functions\nDftiComputeForwardDM\nComputes the forward FFT.\nDftiComputeBackwardDM\nComputes the backward FFT.\nDescriptor Configuration Functions\nDftiSetValueDM\nSets one particular configuration parameter with the specified\nconfiguration value.\nDftiGetValueDM\nGets the value of one particular configuration parameter.\nComputing Cluster FFT\nThe Intel® oneAPI Math Kernel Library (oneMKL)cluster FFT functions are provided with Fortran and C\ninterfaces.\nCluster FFT computation is performed by DftiComputeForwardDM and DftiComputeBackwardDM functions,\ncalled in a program using MPI, which will be referred to as MPI program. After an MPI program starts, a\nnumber of processes are created. MPI identifies each process by its rank. The processes are independent of\none another and communicate via MPI. A function called in an MPI program is invoked in all the processes.\nEach process manipulates data according to its rank. Input or output data for a cluster FFT transform is a\nsequence of real or complex values. A cluster FFT computation function operates on the local part of the\ninput data, that is, some part of the data to be operated in a particular process, as well as generates local\npart of the output data. While each process performs its part of computations, running in parallel and\ncommunicating through MPI, the processes perform the entire FFT computation. FFT computations using the\nIntel® oneAPI Math Kernel Library (oneMKL) cluster FFT functions are typically effected by a number of steps\nlisted below:\n1.\nInitiate MPI by calling MPI_Init (the function must be called prior to calling any FFT function and any\nMPI function).\n2.\nAllocate memory for the descriptor and create it by calling DftiCreateDescriptorDM.\n3.\nSpecify one of several values of configuration parameters by one or more calls to DftiSetValueDM.\n4.\nObtain values of configuration parameters needed to create local data arrays; the values are retrieved\nby calling DftiGetValueDM.\n5.\nInitialize the descriptor for the FFT computation by calling DftiCommitDescriptorDM.\n6.\nCreate arrays for local parts of input and output data and fill the local part of input data with values.\n(For more information, see Distributing Data among Processes.)\n7.\nCompute the transform by calling DftiComputeForwardDM or DftiComputeBackwardDM.\n8.\nGather local output data into the global array using MPI functions. (This step is optional because you\nmay need to immediately employ the data differently.)\n9.\nRelease memory allocated for the descriptor by calling DftiFreeDescriptorDM.\n10.\nFinalize communication through MPI by calling MPI_Finalize (the function must be called after the\nlast call to a cluster FFT function and the last call to an MPI function).\nSeveral code examples in Examples for Cluster FFT Functions in the Code Examples appendix illustrate\ncluster FFT computations.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2327\n\n\nDistributing Data Among Processes\nThe Intel® oneAPI Math Kernel Library (oneMKL) cluster FFT functions store all input and output multi-\ndimensional arrays (matrices) in one-dimensional arrays (vectors). The arrays are stored in the row-major\norder. For example, a two-dimensional matrix A of size (m,n) is stored in a vector B of size m*n so that\nB[i*n+j]=A[i][j] (i=0, ..., m-1, j=0, ..., n-1) .\nNOTE\nOrder of FFT dimensions is the same as the order of array dimensions in the programming language.\nFor example, a 3-dimensional FFT with Lengths=(m,n,l) can be computed over an array Ar[m][n][l].\nAll MPI processes involved in cluster FFT computation operate their own portions of data. These local arrays\nmake up the virtual global array that the fast Fourier transform is applied to. It is your responsibility to\nproperly allocate local arrays (if needed), fill them with initial data and gather resulting data into an actual\nglobal array or process the resulting data differently. To be able do this, see sections below on how the\nvirtual global array is composed of the local ones.\nMulti-dimensional transforms\nIf the dimension of transform is greater than one, the cluster FFT function library splits data in the dimension\nwhose index changes most slowly, so that the parts contain all elements with several consecutive values of\nthis index. It is the first dimension in C. If the global array is two-dimensional, it gives each process several\nconsecutive rows. Local arrays are placed in memory allocated for the virtual global array consecutively, in\nthe order determined by process ranks. For example, in case of two processes, during the computation of a\nthree-dimensional transform whose matrix has size (11,15,12), the processes may store local arrays of sizes\n(6,15,12) and (5,15,12), respectively.\nIf p is the number of MPI processes and the matrix of a transform to be computed has size (m,n,l), each MPI\nprocess works with local data array of size (mq , n, l), where Σmq=m, q=0, ... , p-1. Local input arrays must\ncontain appropriate parts of the actual global input array, and then local output arrays will contain\nappropriate parts of the actual global output array. You can figure out which particular rows of the global\narray the local array must contain from the following configuration parameters of the cluster FFT interface:\nCDFT_LOCAL_NX, CDFT_LOCAL_START_X, and CDFT_LOCAL_SIZE. To retrieve values of the parameters, use\nthe DftiGetValueDM function:\n•\nCDFT_LOCAL_NX specifies how many rows of the global array the current process receives.\n•\nCDFT_LOCAL_START_X specifies which row of the global input or output array corresponds to the first row\nof the local input or output array. If A is a global array and L is the appropriate local array, then\nL[i][j][k]=A[i+cdft_local_start_x][j][k], where i=0, ..., mq-1, j=0, ..., n-1, k=0, ..., l-1.\nExample \"2D Out-of-place Cluster FFT Computation\" shows how the data is distributed among processes for\na two-dimensional cluster FFT computation.\nOne-dimensional transforms\nIn this case, input and output data are distributed among processes differently and even the numbers of\nelements stored in a particular process before and after the transform may be different. Each local array\nstores a segment of consecutive elements of the appropriate global array. Such segment is determined by\nthe number of elements and a shift with respect to the first array element. So, to specify segments of the\nglobal input and output arrays that a particular process receives, four configuration parameters are needed:\nCDFT_LOCAL_NX, CDFT_LOCAL_START_X, CDFT_LOCAL_OUT_NX, and CDFT_LOCAL_OUT_START_X. Use the \nDftiGetValueDM function to retrieve their values. The meaning of the four configuration parameters\ndepends upon the type of the transform, as shown in Table \"Data Distribution Configuration Parameters for\n1D Transforms\":\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2328\n\n\nData Distribution Configuration Parameters for 1D Transforms\nMeaning of the Parameter\nForward Transform\nBackward Transform\nNumber of elements in input\narray\nCDFT_LOCAL_NX\nCDFT_LOCAL_OUT_NX\nElements shift in input array\nCDFT_LOCAL_START_X\nCDFT_LOCAL_OUT_START_X\nNumber of elements in output\narray\nCDFT_LOCAL_OUT_NX\nCDFT_LOCAL_NX\nElements shift in output array\nCDFT_LOCAL_OUT_START_X\nCDFT_LOCAL_START_X\nMemory size for local data\nThe memory size needed for local arrays cannot be just calculated from CDFT_LOCAL_NX\n(CDFT_LOCAL_OUT_NX), because the cluster FFT functions sometimes require allocating a little bit more\nmemory for local data than just the size of the appropriate sub-array. The configuration parameter\nCDFT_LOCAL_SIZE specifies the size of the local input and output array in data elements. Each local input\nand output arrays must have size not less than CDFT_LOCAL_SIZE*size_of_element. Note that in the current\nimplementation of the cluster FFT interface, data elements can be real or complex values, each complex\nvalue consisting of the real and imaginary parts. If you employ a user-defined workspace for in-place\ntransforms (for more information, refer to Table \"Settable configuration Parameters\"), it must have the same\nsize as the local arrays. Example \"1D In-place Cluster FFT Computations\" illustrates how the cluster FFT\nfunctions distribute data among processes in case of a one-dimensional FFT computation performed with a\nuser-defined workspace.\nAvailable Auxiliary Functions\nIf a global input array is located on one MPI process and you want to obtain its local parts or you want to\ngather the global output array on one MPI process, you can use functions MKL_CDFT_ScatterData and\nMKL_CDFT_GatherData to distribute or gather data among processes, respectively. These functions are\ndefined in a file that is delivered with Intel® oneAPI Math Kernel Library (oneMKL) and located in the following\nsubdirectory of the Intel® oneAPI Math Kernel Library (oneMKL) installation directory: examples/cdftc/\nsource/cdft_example_support.c.\nRestriction on Lengths of Transforms\nThe algorithm that the Intel® oneAPI Math Kernel Library (oneMKL) cluster FFT functions use to distribute\ndata among processes imposes a restriction on lengths of transforms with respect to the number of MPI\nprocesses used for the FFT computation:\n•\nFor a multi-dimensional transform, the lengths of the first two dimensions must be not less than the\nnumber of MPI processes.\n•\nThe length of a one-dimensional transform must be the product of two integers each of which is not less\nthan the number of MPI processes.\nNon-compliance with the restriction causes an error CDFT_SPREAD_ERROR (refer to Error Codes for details).\nTo achieve the compliance, you can change the transform lengths and/or the number of MPI processes, which\nis specified at start of an MPI program. MPI-2 enables changing the number of processes during execution of\nan MPI program.\nCluster FFT Interface\nTo use the cluster FFT functions, you need to access the header file mkl_cdft.h through \"include\".\nThe C interface provides a structure type DFTI_DESCRIPTOR_DM_HANDLE and a number of functions, some of\nwhich accept a different number of input arguments.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2329\n\n\nTo provide communication between parallel processes through MPI, the following include statement must be\npresent in your code:\n•\nC/C++:\n#include \"mpi.h\"\nThere are three main categories of the cluster FFT functions in Intel® oneAPI Math Kernel Library (oneMKL):\n1.\nDescriptor Manipulation. There are three functions in this category. The DftiCreateDescriptorDM\nfunction creates an FFT descriptor whose storage is allocated dynamically. The \nDftiCommitDescriptorDM function \"commits\" the descriptor to all its settings. The \nDftiFreeDescriptorDM function frees up the memory allocated for the descriptor.\n2.\nFFT Computation. There are two functions in this category. The DftiComputeForwardDM function\nperforms the forward FFT computation, and the DftiComputeBackwardDM function performs the\nbackward FFT computation.\n3.\nDescriptor Configuration. There are two functions in this category. The DftiSetValueDM function\nsets one specific configuration value to one of the many configuration parameters. The \nDftiGetValueDM function gets the current value of any of these configuration parameters, all of which\nare readable. These parameters, though many, are handled one at a time.\nCluster FFT Descriptor Manipulation Functions\nThere are three functions in this category: create a descriptor, commit a descriptor, and free a descriptor.\nDftiCreateDescriptorDM\nAllocates memory for the descriptor data structure\nand preliminarily initializes it.\nSyntax\nstatus = DftiCreateDescriptorDM(comm, &handle, v1, v2, dim, size );\nstatus = DftiCreateDescriptorDM(comm, &handle, v1, v2, dim, sizes );\nInclude Files\n•\nmkl_cdft.h\nInput Parameters\ncomm\nMPI communicator, e.g. MPI_COMM_WORLD.\nv1\nPrecision of the transform.\nv2\nType of the forward domain. Must be DFTI_COMPLEX for complex-to-\ncomplex transforms or DFTI_REAL for real-to-complex transforms.\ndim\nDimension of the transform.\nsize\nLength of the transform in a one-dimensional case.\nsizes\nLengths of the transform in a multi-dimensional case.\nOutput Parameters\nhandle\nPointer to the descriptor handle of transform. If the function\ncompletes successfully, the pointer to the created handle is stored in\nthe variable.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2330\n\n\nDescription\nThis function allocates memory in a particular MPI process for the descriptor data structure and instantiates it\nwith default configuration settings with respect to the precision, domain, dimension, and length of the\ndesired transform. The domain is understood to be the domain of the forward transform. The result is a\npointer to the created descriptor. This function is slightly different from the \"initialization\" function \nDftiCommitDescriptorDM in a more traditional software packages or libraries used for computing the FFT.\nThis function does not perform any significant computation work, such as twiddle factors computation,\nbecause the default configuration settings can still be changed using the function DftiSetValueDM.\nThe value of the parameter v1 is specified through named constants DFTI_SINGLE and DFTI_DOUBLE. It\ncorresponds to precision of input data, output data, and computation. A setting of DFTI_SINGLE indicates\nsingle-precision floating-point data type and a setting of DFTI_DOUBLE indicates double-precision floating-\npoint data type.\nThe parameter dim is a simple positive integer indicating the dimension of the transform.\nFor one-dimensional transforms, length is a single integer value of the parameter size having type\nMKL_LONG; for multi-dimensional transforms, length is supplied with the parameter sizes, which is an array\nof integers having type MKL_LONG.\nReturn Values\nThe function returns DFTI_NO_ERROR when completes successfully. In this case, the pointer to the created\ndescriptor handle is stored in handle. If the function fails, it returns a value of another error class constant\n(for the list of constants, refer to Error Codes).\nPrototype\n   \nMKL_LONG DftiCreateDescriptorDM(MPI_Comm,DFTI_DESCRIPTOR_DM_HANDLE*,\n   enum DFTI_CONFIG_VALUE,enum DFTI_CONFIG_VALUE,MKL_LONG,...);\n   \nDftiCommitDescriptorDM\nPerforms all initialization for the actual FFT\ncomputation.\nSyntax\nstatus = DftiCommitDescriptorDM(handle);\nInclude Files\n•\nmkl_cdft.h\nInput Parameters\nhandle\nThe descriptor handle. Must be valid, that is, created in a call to \nDftiCreateDescriptorDM.\nDescription\nThe cluster FFT interface requires a function that completes initialization of a previously created descriptor\nbefore the descriptor can be used for FFT computations in a particular MPI process. The\nDftiCommitDescriptorDM function performs all initialization that facilitates the actual FFT computation. For\nthe current implementation, it may involve exploring many different factorizations of the input length to\nsearch for a highly efficient computation method.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2331\n\n\nAny changes of configuration parameters of a committed descriptor via the set value function (see Descriptor\nConfiguration Functions) requires a re-committal of the descriptor before a computation function can be\ninvoked. Typically, this committal function is called right before a computation function call (see FFT\nComputation Functions).\nReturn Values\nThe function returns DFTI_NO_ERROR when completes successfully. If the function fails, it returns a value of\nanother error class constant (for the list of constants, refer to Error Codes).\nPrototype\n   \nMKL_LONG DftiCommitDescriptorDM(DFTI_DESCRIPTOR_DM_HANDLE handle);\n   \nDftiFreeDescriptorDM\nFrees memory allocated for a descriptor.\nSyntax\nstatus = DftiFreeDescriptorDM(&handle);\nInclude Files\n•\nmkl_cdft.h\nInput Parameters\nhandle\nThe descriptor handle. Must be valid, that is, created in a call to \nDftiCreateDescriptorDM.\nOutput Parameters\nhandle\nThe descriptor handle. Memory allocated for the handle is released on\noutput.\nDescription\nThis function frees up all memory allocated for a descriptor in a particular MPI process. Call the\nDftiFreeDescriptorDM function to delete the descriptor handle. Upon successful completion of\nDftiFreeDescriptorDM the descriptor handle is no longer valid.\nReturn Values\nThe function returns DFTI_NO_ERROR when completes successfully. If the function fails, it returns a value of\nanother error class constant (for the list of constants, refer to Error Codes).\nPrototype\n   \nMKL_LONG DftiFreeDescriptorDM(DFTI_DESCRIPTOR_DM_HANDLE *handle);\n   \nCluster FFT Computation Functions\nThere are two functions in this category: compute the forward transform and compute the backward\ntransform.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2332\n\n\nDftiComputeForwardDM\nComputes the forward FFT.\nSyntax\nstatus = DftiComputeForwardDM(handle, in_X, out_X);\nstatus = DftiComputeForwardDM(handle, in_out_X);\nInclude Files\n•\nmkl_cdft.h\nInput Parameters\nhandle\nThe descriptor handle.\nin_X, in_out_X\nLocal part of input data. Array of real or complex values (depending\non the forward domain type). Refer to Distributing Data among\nProcesses on how to allocate and initialize the array.\nOutput Parameters\nout_X, in_out_X\nLocal part of output data. Array of complex values. Refer to \nDistributing Data among Processes on how to allocate the array.\nDescription\nThe DftiComputeForwardDM function computes the forward FFT. Forward FFT is the transform using the\nfactor e-i2π/n.\nBefore you call the function, the valid descriptor, created by DftiCreateDescriptorDM, must be configured\nand committed using the DftiCommitDescriptorDM function.\nThe computation is carried out by an internal call to the DftiComputeForward function. So, the functions\nhave very much in common, and details not explicitly mentioned below can be found in the description of\nDftiComputeForward.\nThe local part of input data, as well as the local part of the output data, is an appropriate sequence of real or\ncomplex values (each complex value consists of two real numbers: real part and imaginary part) that a\nparticular process stores. See Distributing Data Among Processes for details.\nRefer to Configuration Settings for the list of configuration parameters that the descriptor passes to the\nfunction.\nThe configuration parameter DFTI_PRECISION determines the precision of input data, output data, and\ntransform: a setting of DFTI_SINGLE indicates single-precision floating-point data type and a setting of\nDFTI_DOUBLE indicates double-precision floating-point data type.\nThe configuration parameter DFTI_PLACEMENT informs the function whether the computation should be in-\nplace. If the value of this parameter is DFTI_INPLACE (default), you must call the function with two\nparameters, otherwise you must supply three parameters. If DFTI_PLACEMENT = DFTI_INPLACE and three\nparameters are supplied, then the third parameter is ignored.\nCaution\nEven in case of an out-of-place transform, local array of input data in_X may be changed. To\nsave data, make its copy before calling DftiComputeForwardDM.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2333\n\n\nIn case of an in-place transform, DftiComputeForwardDM dynamically allocates and deallocates a work\nbuffer of the same size as the local input/output array requires.\nNOTE\nYou can specify your own workspace of the same size through the configuration parameter\nCDFT_WORKSPACE to avoid redundant memory allocation.\nReturn Values\nThe function returns DFTI_NO_ERROR when completes successfully. If the function fails, it returns a value of\nanother error class constant (for the list of constants, refer to Error Codes).\nPrototype\n   \nMKL_LONG DftiComputeForwardDM(DFTI_DESCRIPTOR_DM_HANDLE handle, void *in_X,...);\n   \nDftiComputeBackwardDM\nComputes the backward FFT.\nSyntax\nstatus = DftiComputeBackwardDM(handle, in_X, out_X);\nstatus = DftiComputeBackwardDM(handle, in_out_X);\nInclude Files\n•\nmkl_cdft.h\nInput Parameters\nhandle\nThe descriptor handle.\nin_X, in_out_X\nLocal part of input data. Array of complex values. Refer to \nDistributing Data among Processes on how to allocate and initialize\nthe array.\nOutput Parameters\nout_X, in_out_X\nLocal part of output data. Array of real or complex values\n(depending on the forward domain type. Refer to Distributing Data\namong Processes on how to allocate the array.\nDescription\nThe DftiComputeBackwardDM function computes the backward FFT. Backward FFT is the transform using the\nfactor ei2π/n.\nBefore you call the function, the valid descriptor, created by DftiCreateDescriptorDM, must be configured\nand committed using the DftiCommitDescriptorDM function.\nThe computation is carried out by an internal call to the DftiComputeBackward function. So, the functions\nhave very much in common, and details not explicitly mentioned below can be found in the description of\nDftiComputeBackward.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2334\n\n\nThe local part of input data, as well as the local part of the output data, is an appropriate sequence of real or\ncomplex values (each complex value consists of two real numbers: real part and imaginary part) that a\nparticular process stores. See Distributing Data among Processes for details.\nRefer to Configuration Settings for the list of configuration parameters that the descriptor passes to the\nfunction.\nThe configuration parameter DFTI_PRECISION determines the precision of input data, output data, and\ntransform: a setting of DFTI_SINGLE indicates single-precision floating-point data type and a setting of\nDFTI_DOUBLE indicates double-precision floating-point data type.\nThe configuration parameter DFTI_PLACEMENT informs the function whether the computation should be in-\nplace. If the value of this parameter is DFTI_INPLACE (default), you must call the function with two\nparameters, otherwise you must supply three parameters. If DFTI_PLACEMENT = DFTI_INPLACE and three\nparameters are supplied, then the third parameter is ignored.\nCaution\nEven in case of an out-of-place transform, local array of input data in_X may be changed. To\nsave data, make its copy before calling DftiComputeBackwardDM.\nIn case of an in-place transform, DftiComputeBackwardDM dynamically allocates and deallocates a work\nbuffer of the same size as the local input/output array requires.\nNOTE\nYou can specify your own workspace of the same size through the configuration parameter\nCDFT_WORKSPACE to avoid redundant memory allocation.\nReturn Values\nThe function returns DFTI_NO_ERROR when completes successfully. If the function fails, it returns a value of\nanother error class constant (for the list of constants, refer to Error Codes).\nPrototype\n   \nMKL_LONG DftiComputeBackwardDM(DFTI_DESCRIPTOR_DM_HANDLE handle, void *in_X,...);\n   \nCluster FFT Descriptor Configuration Functions\nThere are two functions in this category: the value-setting function DftiSetValueDM sets one particular\nconfiguration parameter to an appropriate value, and the value-getting function DftiGetValueDM reads the\nvalue of one particular configuration parameter.\nSome configuration parameters used by cluster FFT functions originate from the conventional FFT interface\n(see Configuration Settings for details).\nOther configuration parameters are specific to the cluster FFT. Integer values of these parameters have type\nMKL_LONG. The exact type of the configuration parameters being floating-point scalars is float or double\n(matching DFTI_PRECISION). The configuration parameters whose values are named constants have the\nenum type. They are defined in the mkl_cdft.h header file.\nThe names of the configuration parameters specific to the cluster FFT interface have the CDFT prefix.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2335\n\n\nDftiSetValueDM\nSets one particular configuration parameter with the\nspecified configuration value.\nSyntax\nstatus = DftiSetValueDM (handle, param, value);\nInclude Files\n•\nmkl_cdft.h\nInput Parameters\nhandle\nThe descriptor handle. Must be valid, that is, created in a call to \nDftiCreateDescriptorDM.\nparam\nName of a parameter to be set up in the descriptor handle. See \nTable \"Settable Configuration Parameters\" for the list of available\nparameters.\nvalue\nValue of the parameter.\nDescription\nThis function sets one particular configuration parameter with the specified configuration value. The\nconfiguration parameter is one of the named constants listed in the table below, and the configuration value\nmust have the corresponding type. See Configuration Settings for details of the meaning of each setting and\nfor possible values of the parameters whose values are named constants.\nSettable Configuration Parameters\nParameter Name\nData Type\nDescription\nDefault Value\nDFTI_FORWARD_SCALE\nFloating-point\nscalar\nScale factor of forward\ntransform.\n1.0\nDFTI_BACKWARD_SCALE\nFloating-point\nscalar\nScale factor of backward\ntransform.\n1.0\nDFTI_PLACEMENT\nNamed constant\nPlacement of the computation\nresult.\nDFTI_INPLACE\nDFTI_ORDERING\nNamed constant\nScrambling of data order.\nDFTI_ORDERED\nCDFT_WORKSPACE\nArray of an\nappropriate type\nAuxiliary buffer, a user-\ndefined workspace. Enables\nsaving memory during in-\nplace computations.\nNULL (allocate\nworkspace dynamically).\nDFTI_PACKED_FORMAT\nNamed constant\nPacked format for storing\nconjugate-even sequence (in\nthe case of a real forward\ndomain).\n•\nDFTI_PERM_FORMAT\n― default and the\nonly available value\nfor one-dimensional\ntransforms\n•\nDFTI_CCE_FORMAT ―\ndefault and the only\navailable value for\nmulti-dimensional\ntransforms\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2336\n\n\nParameter Name\nData Type\nDescription\nDefault Value\nDFTI_TRANSPOSE\nNamed constant\nThis parameter determines\nhow the output data is\nlocated for multi-dimensional\ntransforms. If the parameter\nvalue is DFTI_NONE, the data\nis located in a usual manner\ndescribed in this document. If\nthe value is DFTI_ALLOW, the\nlast (first) global transposition\nis not performed for a forward\n(backward) transform.\nDFTI_NONE\nReturn Values\nThe function returns DFTI_NO_ERROR when completes successfully. If the function fails, it returns a value of\nanother error class constant (for the list of constants, refer to Error Codes).\nPrototype\n   \nMKL_LONG DftiSetValueDM(DFTI_DESCRIPTOR_DM_HANDLE handle, int param,...);\n   \nDftiGetValueDM\nGets the value of one particular configuration\nparameter.\nSyntax\nstatus = DftiGetValueDM(handle, param, &value);\nInclude Files\n•\nmkl_cdft.h\nInput Parameters\nhandle\nThe descriptor handle. Must be valid, that is, created in a call to \nDftiCreateDescriptorDM.\nparam\nName of a parameter to be retrieved from the descriptor. See Table\n\"Retrievable Configuration Parameters\" for the list of available\nparameters.\nOutput Parameters\nvalue\nValue of the parameter.\nDescription\nThis function gets the configuration value of one particular configuration parameter. The configuration\nparameter is one of the named constants listed in the table below, and the configuration value is the\ncorresponding appropriate type, which can be a named constant or a native type. Possible values of the\nnamed constants can be found in Table \"Configuration Parameters\" and relevant subsections of the \nConfiguration Settings section.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2337\n\n\nRetrievable Configuration Parameters\nParameter Name\nData Type\nDescription\nDFTI_PRECISION\nNamed constant\nPrecision of computation, input data and\noutput data.\nDFTI_DIMENSION\nInteger scalar\nDimension of the transform\nDFTI_LENGTHS\nArray of integer values\nArray of lengths of the transform. Number of\nlengths corresponds to the dimension of the\ntransform.\nDFTI_FORWARD_SCALE\nFloating-point scalar\nScale factor of forward transform.\nDFTI_BACKWARD_SCALE\nFloating-point scalar\nScale factor of backward transform.\nDFTI_PLACEMENT\nNamed constant\nPlacement of the computation result.\nDFTI_COMMIT_STATUS\nNamed constant\nShows whether descriptor has been\ncommitted.\nDFTI_FORWARD_DOMAIN\nNamed constant\nForward domain of transforms, has the value\nof DFTI_COMPLEX or DFTI_REAL.\nDFTI_ORDERING\nNamed constant\nScrambling of data order.\nCDFT_MPI_COMM\nType of MPI\ncommunicator\nMPI communicator used for transforms.\nCDFT_LOCAL_SIZE\nInteger scalar\nNecessary size of input, output, and buffer\narrays in data elements.\nCDFT_LOCAL_X_START\nInteger scalar\nRow/element number of the global array that\ncorresponds to the first row/element of the\nlocal array. For more information, see \nDistributing Data among Processes.\nCDFT_LOCAL_NX\nInteger scalar\nThe number of rows/elements of the global\narray stored in the local array. For more\ninformation, see Distributing Data among\nProcesses.\nCDFT_LOCAL_OUT_X_START\nInteger scalar\nElement number of the appropriate global\narray that corresponds to the first element of\nthe input or output local array in a 1D case.\nFor details, see Distributing Data among\nProcesses.\nCDFT_LOCAL_OUT_NX\nInteger scalar\nThe number of elements of the appropriate\nglobal array that are stored in the input or\noutput local array in a 1D case. For details,\nsee Distributing Data among Processes.\nReturn Values\nThe function returns DFTI_NO_ERROR when completes successfully. If the function fails, it returns a value of\nanother error class constant (for the list of constants, refer to Error Codes).\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2338\n\n\nPrototype\n   \nMKL_LONG DftiGetValueDM(DFTI_DESCRIPTOR_DM_HANDLE handle, int param,...);\n   \nError Codes\nAll the cluster FFT functions return an integer value denoting the status of the operation. These values are\nidentified by named constants. Each function returns DFTI_NO_ERROR if no errors were encountered during\nexecution. Otherwise, a function generates an error code. In addition to FFT error codes, the cluster FFT\ninterface has its own ones. The named constants specific to the cluster FFT interface have the CDFT prefix in\ntheir names. Table \"Error Codes that Cluster FFT Functions Return\" lists error codes that the cluster FFT\nfunctions may return.\nError Codes that Cluster FFT Functions Return\nNamed Constants\nComments\nDFTI_NO_ERROR\nNo error.\nDFTI_MEMORY_ERROR\nUsually associated with memory allocation.\nDFTI_INVALID_CONFIGURATION\nInvalid settings of one or more configuration parameters.\nDFTI_INCONSISTENT_CONFIGURA\nTION\nInconsistent configuration or input parameters.\nDFTI_NUMBER_OF_THREADS_ERRO\nR\nNumber of OMP threads in the computation function is not equal to\nthe number of OMP threads in the initialization stage (commit\nfunction).\nDFTI_MULTITHREADED_ERROR\nUsually associated with a value that OMP routines return in case of\nerrors.\nDFTI_BAD_DESCRIPTOR\nDescriptor is unusable for computation.\nDFTI_UNIMPLEMENTED\nUnimplemented legitimate settings; implementation dependent.\nDFTI_MKL_INTERNAL_ERROR\nInternal library error.\nDFTI_1D_LENGTH_EXCEEDS_INT3\n2\nLength of one of the dimensions exceeds 232 -1 (4 bytes).\nCDFT_SPREAD_ERROR\nData cannot be distributed (For more information, see Distributing\nData among Processes.)\nCDFT_MPI_ERROR\nMPI error. Occurs when calling MPI.\nPBLAS Routines\nIntel® oneAPI Math Kernel Libraryimplements the PBLAS (Parallel Basic Linear Algebra Subprograms) routines\nfrom the ScaLAPACK package for distributed-memory architecture. PBLAS is intended for using in vector-\nvector, matrix-vector, and matrix-matrix operations to simplify the parallelization of linear codes. The design\nof PBLAS is as consistent as possible with that of the BLAS. The routine descriptions are arranged in several\nsections according to the PBLAS level of operation:\n•\nPBLAS Level 1 Routines (distributed vector-vector operations)\n•\nPBLAS Level 2 Routines (distributed matrix-vector operations)\n•\nPBLAS Level 3 Routines (distributed matrix-matrix operations)\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2339\n\n\nEach section presents the routine and function group descriptions in alphabetical order by the routine group\nname; for example, the p?asum group, the p?axpy group. The question mark in the group name corresponds\nto a character indicating the data type (s, d, c, and z or their combination); see Routine Naming\nConventions.\nNOTE\nPBLAS routines are provided only with Intel® oneAPI Math Kernel Library (oneMKL) versions for Linux*\nand Windows* OSs.\nGenerally, PBLAS runs on a network of computers using MPI as a message-passing layer and a set of prebuilt\ncommunication subprograms (BLACS), as well as a set of PBLAS optimized for the target architecture. The\nIntel® oneAPI Math Kernel Library (oneMKL) version of PBLAS is optimized for Intel® processors. For the\ndetailed system and environment requirements seeIntel® oneAPI Math Kernel Library (oneMKL) Release\nNotes and Intel® oneAPI Math Kernel Library (oneMKL) Developer Guide.\nFor full reference on PBLAS routines and related information, see http://www.netlib.org/scalapack/html/\npblas_qref.html.\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nPBLAS Routines Overview\nThe model of the computing environment for PBLAS is represented as a one-dimensional array of processes\nor also a two-dimensional process grid. To use PBLAS, all global matrices or vectors must be distributed on\nthis array or grid prior to calling the PBLAS routines.\nPBLAS uses the two-dimensional block-cyclic data distribution as a layout for dense matrix computations.\nThis distribution provides good work balance between available processors, as well as gives the opportunity\nto use PBLAS Level 3 routines for optimal local computations. Information about the data distribution that is\nrequired to establish the mapping between each global array and its corresponding process and memory\nlocation is contained in the so called array descriptor associated with each global array. Table \"Content of\nthe array descriptor for dense matrices\" gives an example of an array descriptor structure.\nContent of Array Descriptor for Dense Matrices\nArray Element #\nName\nDefinition\n1\ndtype\nDescriptor type ( =1 for dense matrices)\n2\nctxt\nBLACS context handle for the process grid\n3\nm\nNumber of rows in the global array\n4\nn\nNumber of columns in the global array\n5\nmb\nRow blocking factor\n6\nnb\nColumn blocking factor\n7\nrsrc\nProcess row over which the first row of the global array is distributed\n8\ncsrc\nProcess column over which the first column of the global array is\ndistributed\n9\nlld\nLeading dimension of the local array\nThe number of rows and columns of a global dense matrix that a particular process in a grid receives after\ndata distributing is denoted by LOCr() and LOCc(), respectively. To compute these numbers, you can use\nthe ScaLAPACK tool routine numroc.\nAfter the block-cyclic distribution of global data is done, you may choose to perform an operation on a\nsubmatrix of the global matrix A, which is contained in the global subarray sub(A), defined by the following\n6 values (for dense matrices):\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2340\n\n\nm\nThe number of rows of sub(A)\nn\nThe number of columns of sub(A)\na\nA pointer to the local array containing the entire global array A\nia\nThe row index of sub(A) in the global array\nja\nThe column index of sub(A) in the global array\ndesca\nThe array descriptor for the global array A\nIntel® oneAPI Math Kernel Library (oneMKL) provides the PBLAS routines with interface similar to the\ninterface used in the Netlib PBLAS (see http://www.netlib.org/scalapack/html/pblas_qref.html).\nPBLAS Routine Naming Conventions\nThe naming convention for PBLAS routines is similar to that used for BLAS routines (see Routine Naming\nConventions). A general rule is that each routine name in PBLAS, which has a BLAS equivalent, is simply the\nBLAS name prefixed by initial letter p that stands for \"parallel\".\nThe Intel® oneAPI Math Kernel Library (oneMKL) PBLAS routine names have the following structure:\n    p <character> <name> <mod> ( )\nThe <character> field indicates the Fortran data type:\ns\nreal, single precision\nc\ncomplex, single precision\nd\nreal, double precision\nz\ncomplex, double precision\ni\ninteger\nSome routines and functions can have combined character codes, such as sc or dz.\nFor example, the function pscasum uses a complex input array and returns a real value.\nThe <name> field, in PBLAS level 1, indicates the operation type. For example, the PBLAS level 1 routines\np?dot, p?swap, p?copy compute a vector dot product, vector swap, and a copy vector, respectively.\nIn PBLAS level 2 and 3, <name> reflects the matrix argument type:\nge\ngeneral matrix\nsy\nsymmetric matrix\nhe\nHermitian matrix\ntr\ntriangular matrix\nIn PBLAS level 3, the <name>=tran indicates the transposition of the matrix.\nThe <mod> field, if present, provides additional details of the operation. The PBLAS level 1 names can have\nthe following characters in the <mod> field:\nc\nconjugated vector\nu\nunconjugated vector\nThe PBLAS level 2 names can have the following additional characters in the <mod> field:\nmv\nmatrix-vector product\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2341\n\n\nsv\nsolving a system of linear equations with matrix-vector operations\nr\nrank-1 update of a matrix\nr2\nrank-2 update of a matrix.\nThe PBLAS level 3 names can have the following additional characters in the <mod> field:\nmm\nmatrix-matrix product\nsm\nsolving a system of linear equations with matrix-matrix operations\nrk\nrank-k update of a matrix\nr2k\nrank-2k update of a matrix.\nThe examples below show how to interpret PBLAS routine names:\npddot\n<p> <d> <dot>: double-precision real distributed vector-vector dot product\npcdotc\n<p> <c> <dot> <c>: complex distributed vector-vector dot product, conjugated\npscasum\n<p> <sc> <asum>: sum of magnitudes of distributed vector elements, single\nprecision real output and single precision complex input\npcdotu\n<p> <c> <dot> <u>: distributed vector-vector dot product, unconjugated,\ncomplex\npsgemv\n<p> <s> <ge> <mv>: distributed matrix-vector product, general matrix, single\nprecision\npztrmm\n<p> <z> <tr> <mm>: distributed matrix-matrix product, triangular matrix,\ndouble-precision complex.\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nPBLAS Level 1 Routines\nPBLAS Level 1 includes routines and functions that perform distributed vector-vector operations. Table\n\"PBLAS Level 1 Routine Groups and Their Data Types\" lists the PBLAS Level 1 routine groups and the data\ntypes associated with them.\nPBLAS Level 1 Routine Groups and Their Data Types\nRoutine or\nFunction Group\nData Types\nDescription\np?amax\ns, d, c, z\nCalculates an index of the distributed vector element with\nmaximum absolute value\np?asum\ns, d, sc, dz\nCalculates sum of magnitudes of a distributed vector\np?axpy\ns, d, c, z\nCalculates distributed vector-scalar product\np?copy\ns, d, c, z\nCopies a distributed vector\np?dot\ns, d\nCalculates a dot product of two distributed real vectors\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2342\n\n\nRoutine or\nFunction Group\nData Types\nDescription\np?dotc\nc, z\nCalculates a dot product of two distributed complex\nvectors, one of them is conjugated\np?dotu\nc, z\nCalculates a dot product of two distributed complex\nvectors\np?nrm2\ns, d, sc, dz\nCalculates the 2-norm (Euclidean norm) of a distributed\nvector\np?scal\ns, d, c, z, cs, zd\nCalculates a product of a distributed vector by a scalar\np?swap\ns, d, c, z\nSwaps two distributed vectors\np?amax\nComputes the global index of the element of a\ndistributed vector with maximum absolute value.\nSyntax\nvoid psamax (const MKL_INT *n , float *amax , MKL_INT *indx , const float *x , const\nMKL_INT *ix , const MKL_INT *jx , const MKL_INT *descx , const MKL_INT *incx );\nvoid pdamax (const MKL_INT *n , double *amax , MKL_INT *indx , const double *x , const\nMKL_INT *ix , const MKL_INT *jx , const MKL_INT *descx , const MKL_INT *incx );\nvoid pcamax (const MKL_INT *n , MKL_Complex8 *amax , MKL_INT *indx , const MKL_Complex8\n*x , const MKL_INT *ix , const MKL_INT *jx , const MKL_INT *descx , const MKL_INT\n*incx );\nvoid pzamax (const MKL_INT *n , MKL_Complex16 *amax , MKL_INT *indx , const\nMKL_Complex16 *x , const MKL_INT *ix , const MKL_INT *jx , const MKL_INT *descx , const\nMKL_INT *incx );\nInclude Files\n•\nmkl_pblas.h\nDescription\nThe functions p?amax compute global index of the maximum element in absolute value of a distributed\nvector sub(x),\nwhere sub(x) denotes X(ix, jx:jx+n-1) if incx=m_x, and X(ix: ix+n-1, jx) if incx= 1.\nInput Parameters\nn\n(global) The length of distributed vector sub(x), n≥0.\nx\n(local)\nArray, size (jx-1)*m_x + ix+(n-1)*abs(incx)).\nThis array contains the entries of the distributed vector sub(x).\nix, jx\n(global) The row and column indices in the distributed matrix X indicating\nthe first row and the first column of the submatrix sub(X), respectively.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2343\n\n\ndescx\n(global and local) array of dimension 9. The array descriptor of the\ndistributed matrix X.\nincx\n(global) Specifies the increment for the elements of sub(x). Only two\nvalues are supported, namely 1 and m_x. incx must not be zero.\nOutput Parameters\namax\n(global).\nMaximum absolute value (magnitude) of elements of the distributed vector\nonly in its scope.\nindx\n(global) The global index of the maximum element in absolute value of the\ndistributed vector sub(x) only in its scope.\np?asum\nComputes the sum of magnitudes of elements of a\ndistributed vector.\nSyntax\nvoid psasum (const MKL_INT *n , float *asum , const float *x , const MKL_INT *ix ,\nconst MKL_INT *jx , const MKL_INT *descx , const MKL_INT *incx );\nvoid pdasum (const MKL_INT *n , double *asum , const double *x , const MKL_INT *ix ,\nconst MKL_INT *jx , const MKL_INT *descx , const MKL_INT *incx );\nvoid pscasum (const MKL_INT *n , float *asum , const MKL_Complex8 *x , const MKL_INT\n*ix , const MKL_INT *jx , const MKL_INT *descx , const MKL_INT *incx );\nvoid pdzasum (const MKL_INT *n , double *asum , const MKL_Complex16 *x , const MKL_INT\n*ix , const MKL_INT *jx , const MKL_INT *descx , const MKL_INT *incx );\nInclude Files\n•\nmkl_pblas.h\nDescription\nThe functions p?asum compute the sum of the magnitudes of elements of a distributed vector sub(x),\nwhere sub(x) denotes X(ix, jx:jx+n-1) if incx=m_x, and X(ix: ix+n-1, jx) if incx= 1.\nInput Parameters\nn\n(global) The length of distributed vector sub(x), n≥0.\nx\n(local)\nArray, size (jx-1)*m_x + ix+(n-1)*abs(incx)).\nThis array contains the entries of the distributed vector sub(x).\nix, jx\n(global) The row and column indices in the distributed matrix X indicating\nthe first row and the first column of the submatrix sub(X), respectively.\ndescx\n(global and local) array of dimension 9. The array descriptor of the\ndistributed matrix X.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2344\n\n\nincx\n(global) Specifies the increment for the elements of sub(x). Only two\nvalues are supported, namely 1 and m_x. incx must not be zero.\nOutput Parameters\nasum\n(local) and pscasum.\nContains the sum of magnitudes of elements of the distributed vector only\nin its scope.\np?axpy\nComputes a distributed vector-scalar product and\nadds the result to a distributed vector.\nSyntax\nvoid psaxpy (const MKL_INT *n , const float *a , const float *x , const MKL_INT *ix ,\nconst MKL_INT *jx , const MKL_INT *descx , const MKL_INT *incx , float *y , const\nMKL_INT *iy , const MKL_INT *jy , const MKL_INT *descy , const MKL_INT *incy );\nvoid pdaxpy (const MKL_INT *n , const double *a , const double *x , const MKL_INT *ix ,\nconst MKL_INT *jx , const MKL_INT *descx , const MKL_INT *incx , double *y , const\nMKL_INT *iy , const MKL_INT *jy , const MKL_INT *descy , const MKL_INT *incy );\nvoid pcaxpy (const MKL_INT *n , const MKL_Complex8 *a , const MKL_Complex8 *x , const\nMKL_INT *ix , const MKL_INT *jx , const MKL_INT *descx , const MKL_INT *incx ,\nMKL_Complex8 *y , const MKL_INT *iy , const MKL_INT *jy , const MKL_INT *descy , const\nMKL_INT *incy );\nvoid pzaxpy (const MKL_INT *n , const MKL_Complex16 *a , const MKL_Complex16 *x , const\nMKL_INT *ix , const MKL_INT *jx , const MKL_INT *descx , const MKL_INT *incx ,\nMKL_Complex16 *y , const MKL_INT *iy , const MKL_INT *jy , const MKL_INT *descy , const\nMKL_INT *incy );\nInclude Files\n•\nmkl_pblas.h\nDescription\nThe p?axpy routines perform the following operation with distributed vectors:\nsub(y) := sub(y) + a*sub(x)\nwhere:\na is a scalar;\nsub(x) and sub(y) are n-element distributed vectors.\nsub(x) denotes X(ix, jx:jx+n-1) if incx=m_x, and X(ix: ix+n-1, jx) if incx= 1;\nsub(y) denotes Y(iy, jy:jy+n-1) if incy=m_y, and Y(iy: iy+n-1, jy) if incy= 1.\nInput Parameters\nn\n(global) The length of distributed vectors, n≥0.\na\n(local)\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2345\n\n\nSpecifies the scalar a.\nx\n(local)\nArray, size (jx-1)*m_x + ix+(n-1)*abs(incx)).\nThis array contains the entries of the distributed vector sub(x).\nix, jx\n(global) The row and column indices in the distributed matrix X indicating\nthe first row and the first column of the submatrix sub(X), respectively.\ndescx\n(global and local) array of dimension 9. The array descriptor of the\ndistributed matrix X.\nincx\n(global) Specifies the increment for the elements of sub(x). Only two\nvalues are supported, namely 1 and m_x. incx must not be zero.\ny\n(local)\nArray, size (jy-1)*m_y + iy+(n-1)*abs(incy)).\nThis array contains the entries of the distributed vector sub(y).\niy, jy\n(global) The row and column indices in the distributed matrix Y indicating\nthe first row and the first column of the submatrix sub(Y), respectively.\ndescy\n(global and local) array of dimension 9. The array descriptor of the\ndistributed matrix Y.\nincy\n(global) Specifies the increment for the elements of sub(y). Only two\nvalues are supported, namely 1 and m_y. incy must not be zero.\nOutput Parameters\ny\nOverwritten by sub(y) := sub(y) + a*sub(x).\np?copy\nCopies one distributed vector to another vector.\nSyntax\nvoid picopy (const MKL_INT *n , const MKL_INT *x , const MKL_INT *ix , const MKL_INT\n*jx , const MKL_INT *descx , const MKL_INT *incx , MKL_INT *y , const MKL_INT *iy ,\nconst MKL_INT *jy , const MKL_INT *descy , const MKL_INT *incy );\nvoid pscopy (const MKL_INT *n , const float *x , const MKL_INT *ix , const MKL_INT\n*jx , const MKL_INT *descx , const MKL_INT *incx , float *y , const MKL_INT *iy , const\nMKL_INT *jy , const MKL_INT *descy , const MKL_INT *incy );\nvoid pdcopy (const MKL_INT *n , const double *x , const MKL_INT *ix , const MKL_INT\n*jx , const MKL_INT *descx , const MKL_INT *incx , double *y , const MKL_INT *iy ,\nconst MKL_INT *jy , const MKL_INT *descy , const MKL_INT *incy );\nvoid pccopy (const MKL_INT *n , const MKL_Complex8 *x , const MKL_INT *ix , const\nMKL_INT *jx , const MKL_INT *descx , const MKL_INT *incx , MKL_Complex8 *y , const\nMKL_INT *iy , const MKL_INT *jy , const MKL_INT *descy , const MKL_INT *incy );\nvoid pzcopy (const MKL_INT *n , const MKL_Complex16 *x , const MKL_INT *ix , const\nMKL_INT *jx , const MKL_INT *descx , const MKL_INT *incx , MKL_Complex16 *y , const\nMKL_INT *iy , const MKL_INT *jy , const MKL_INT *descy , const MKL_INT *incy );\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2346\n\n\nInclude Files\n•\nmkl_pblas.h\nDescription\nThe p?copy routines perform a copy operation with distributed vectors defined as\nsub(y) = sub(x),\nwhere sub(x) and sub(y) are n-element distributed vectors.\nsub(x) denotes X(ix, jx:jx+n-1) if incx=m_x, and X(ix: ix+n-1, jx) if incx= 1;\nsub(y) denotes Y(iy, jy:jy+n-1) if incy=m_y, and Y(iy: iy+n-1, jy) if incy= 1.\nInput Parameters\nn\n(global) The length of distributed vectors, n≥0.\nx\n(local)\nArray, size (jx-1)*m_x + ix+(n-1)*abs(incx)).\nThis array contains the entries of the distributed vector sub(x).\nix, jx\n(global) The row and column indices in the distributed matrix X indicating\nthe first row and the first column of the submatrix sub(X), respectively.\ndescx\n(global and local) array of dimension 9. The array descriptor of the\ndistributed matrix X.\nincx\n(global) Specifies the increment for the elements of sub(x). Only two\nvalues are supported, namely 1 and m_x. incx must not be zero.\ny\n(local)\nArray, size (jy-1)*m_y + iy+(n-1)*abs(incy)).\nThis array contains the entries of the distributed vector sub(y).\niy, jy\n(global) The row and column indices in the distributed matrix Y indicating\nthe first row and the first column of the submatrix sub(Y), respectively.\ndescy\n(global and local) array of dimension 9. The array descriptor of the\ndistributed matrix Y.\nincy\n(global) Specifies the increment for the elements of sub(y). Only two\nvalues are supported, namely 1 and m_y. incy must not be zero.\nOutput Parameters\ny\nOverwritten with the distributed vector sub(x).\np?dot\nComputes the dot product of two distributed real\nvectors.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2347\n\n\nSyntax\nvoid psdot (const MKL_INT *n , float *dot , const float *x , const MKL_INT *ix , const\nMKL_INT *jx , const MKL_INT *descx , const MKL_INT *incx , const float *y , const\nMKL_INT *iy , const MKL_INT *jy , const MKL_INT *descy , const MKL_INT *incy );\nvoid pddot (const MKL_INT *n , double *dot , const double *x , const MKL_INT *ix ,\nconst MKL_INT *jx , const MKL_INT *descx , const MKL_INT *incx , const double *y ,\nconst MKL_INT *iy , const MKL_INT *jy , const MKL_INT *descy , const MKL_INT *incy );\nInclude Files\n•\nmkl_pblas.h\nDescription\nThe ?dot functions compute the dot product dot of two distributed real vectors defined as\ndot  = sub(x)'*sub(y)\nwhere sub(x) and sub(y) are n-element distributed vectors.\nsub(x) denotes X(ix, jx:jx+n-1) if incx=m_x, and X(ix: ix+n-1, jx) if incx= 1;\nsub(y) denotes Y(iy, jy:jy+n-1) if incy=m_y, and Y(iy: iy+n-1, jy) if incy= 1.\nInput Parameters\nn\n(global) The length of distributed vectors, n≥0.\nx\n(local)\nArray, size (jx-1)*m_x + ix+(n-1)*abs(incx)).\nThis array contains the entries of the distributed vector sub(x).\nix, jx\n(global) The row and column indices in the distributed matrix X indicating\nthe first row and the first column of the submatrix sub(X), respectively.\ndescx\n(global and local) array of dimension 9. The array descriptor of the\ndistributed matrix X.\nincx\n(global) Specifies the increment for the elements of sub(x). Only two\nvalues are supported, namely 1 and m_x. incx must not be zero.\ny\n(local)\nArray, size (jy-1)*m_y + iy+(n-1)*abs(incy)).\nThis array contains the entries of the distributed vector sub(y).\niy, jy\n(global) The row and column indices in the distributed matrix Y indicating\nthe first row and the first column of the submatrix sub(Y), respectively.\ndescy\n(global and local) array of dimension 9. The array descriptor of the\ndistributed matrix Y.\nincy\n(global) Specifies the increment for the elements of sub(y). Only two\nvalues are supported, namely 1 and m_y. incy must not be zero.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2348\n\n\nOutput Parameters\ndot\n(local)\nDot product of sub(x) and sub(y) only in their scope.\np?dotc\nComputes the dot product of two distributed complex\nvectors, one of them is conjugated.\nSyntax\nvoid pcdotc (const MKL_INT *n , MKL_Complex8 *dotc , const MKL_Complex8 *x , const\nMKL_INT *ix , const MKL_INT *jx , const MKL_INT *descx , const MKL_INT *incx , const\nMKL_Complex8 *y , const MKL_INT *iy , const MKL_INT *jy , const MKL_INT *descy , const\nMKL_INT *incy );\nvoid pzdotc (const MKL_INT *n , MKL_Complex16 *dotc , const MKL_Complex16 *x , const\nMKL_INT *ix , const MKL_INT *jx , const MKL_INT *descx , const MKL_INT *incx , const\nMKL_Complex16 *y , const MKL_INT *iy , const MKL_INT *jy , const MKL_INT *descy , const\nMKL_INT *incy );\nInclude Files\n•\nmkl_pblas.h\nDescription\nThe p?dotc functions compute the dot product dotc of two distributed vectors, with one vector conjugated:\ndotc  = conjg(sub(x)')*sub(y)\nwhere sub(x) and sub(y) are n-element distributed vectors.\nsub(x) denotes X(ix, jx:jx+n-1) if incx=m_x, and X(ix: ix+n-1, jx) if incx= 1;\nsub(y) denotes Y(iy, jy:jy+n-1) if incy=m_y, and Y(iy: iy+n-1, jy) if incy= 1.\nInput Parameters\nn\n(global) The length of distributed vectors, n≥0.\nx\n(local)\nArray, size (jx-1)*m_x + ix+(n-1)*abs(incx)).\nThis array contains the entries of the distributed vector sub(x).\nix, jx\n(global) The row and column indices in the distributed matrix X indicating\nthe first row and the first column of the submatrix sub(X), respectively.\ndescx\n(global and local) array of dimension 9. The array descriptor of the\ndistributed matrix X.\nincx\n(global) Specifies the increment for the elements of sub(x). Only two\nvalues are supported, namely 1 and m_x. incx must not be zero.\ny\n(local)\nArray, size (jy-1)*m_y + iy+(n-1)*abs(incy)).\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2349\n\n\nThis array contains the entries of the distributed vector sub(y).\niy, jy\n(global) The row and column indices in the distributed matrix Y indicating\nthe first row and the first column of the submatrix sub(Y), respectively.\ndescy\n(global and local) array of dimension 9. The array descriptor of the\ndistributed matrix Y.\nincy\n(global) Specifies the increment for the elements of sub(y). Only two\nvalues are supported, namely 1 and m_y. incy must not be zero.\nOutput Parameters\ndotc\n(local)\nDot product of sub(x) and sub(y) only in their scope.\np?dotu\nComputes the dot product of two distributed complex\nvectors.\nSyntax\nvoid pcdotu (const MKL_INT *n , MKL_Complex8 *dotu , const MKL_Complex8 *x , const\nMKL_INT *ix , const MKL_INT *jx , const MKL_INT *descx , const MKL_INT *incx , const\nMKL_Complex8 *y , const MKL_INT *iy , const MKL_INT *jy , const MKL_INT *descy , const\nMKL_INT *incy );\nvoid pzdotu (const MKL_INT *n , MKL_Complex16 *dotu , const MKL_Complex16 *x , const\nMKL_INT *ix , const MKL_INT *jx , const MKL_INT *descx , const MKL_INT *incx , const\nMKL_Complex16 *y , const MKL_INT *iy , const MKL_INT *jy , const MKL_INT *descy , const\nMKL_INT *incy );\nInclude Files\n•\nmkl_pblas.h\nDescription\nThe p?dotu functions compute the dot product dotu of two distributed vectors defined as\ndotu  = sub(x)'*sub(y)\nwhere sub(x) and sub(y) are n-element distributed vectors.\nsub(x) denotes X(ix, jx:jx+n-1) if incx=m_x, and X(ix: ix+n-1, jx) if incx= 1;\nsub(y) denotes Y(iy, jy:jy+n-1) if incy=m_y, and Y(iy: iy+n-1, jy) if incy= 1.\nInput Parameters\nn\n(global) The length of distributed vectors, n≥0.\nx\n(local)\nArray, size (jx-1)*m_x + ix+(n-1)*abs(incx)).\nThis array contains the entries of the distributed vector sub(x).\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2350\n\n\nix, jx\n(global) The row and column indices in the distributed matrix X indicating\nthe first row and the first column of the submatrix sub(X), respectively.\ndescx\n(global and local) array of dimension 9. The array descriptor of the\ndistributed matrix X.\nincx\n(global) Specifies the increment for the elements of sub(x). Only two\nvalues are supported, namely 1 and m_x. incx must not be zero.\ny\n(local)\nArray, size (jy-1)*m_y + iy+(n-1)*abs(incy)).\nThis array contains the entries of the distributed vector sub(y).\niy, jy\n(global) The row and column indices in the distributed matrix Y indicating\nthe first row and the first column of the submatrix sub(Y), respectively.\ndescy\n(global and local) array of dimension 9. The array descriptor of the\ndistributed matrix Y.\nincy\n(global) Specifies the increment for the elements of sub(y). Only two\nvalues are supported, namely 1 and m_y. incy must not be zero.\nOutput Parameters\ndotu\n(local)\nDot product of sub(x) and sub(y) only in their scope.\np?nrm2\nComputes the Euclidean norm of a distributed vector.\nSyntax\nvoid psnrm2 (const MKL_INT *n , float *norm2 , const float *x , const MKL_INT *ix ,\nconst MKL_INT *jx , const MKL_INT *descx , const MKL_INT *incx );\nvoid pdnrm2 (const MKL_INT *n , double *norm2 , const double *x , const MKL_INT *ix ,\nconst MKL_INT *jx , const MKL_INT *descx , const MKL_INT *incx );\nvoid pscnrm2 (const MKL_INT *n , float *norm2 , const MKL_Complex8 *x , const MKL_INT\n*ix , const MKL_INT *jx , const MKL_INT *descx , const MKL_INT *incx );\nvoid pdznrm2 (const MKL_INT *n , double *norm2 , const MKL_Complex16 *x , const MKL_INT\n*ix , const MKL_INT *jx , const MKL_INT *descx , const MKL_INT *incx );\nInclude Files\n•\nmkl_pblas.h\nDescription\nThe p?nrm2 functions compute the Euclidean norm of a distributed vector sub(x),\nwhere sub(x) is an n-element distributed vector.\nsub(x) denotes X(ix, jx:jx+n-1) if incx=m_x, and X(ix: ix+n-1, jx) if incx= 1.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2351\n\n\nInput Parameters\nn\n(global) The length of distributed vector sub(x), n≥0.\nx\n(local)\nArray, size (jx-1)*m_x + ix+(n-1)*abs(incx)).\nThis array contains the entries of the distributed vector sub(x).\nix, jx\n(global) The row and column indices in the distributed matrix X indicating\nthe first row and the first column of the submatrix sub(X), respectively.\ndescx\n(global and local) array of dimension 9. The array descriptor of the\ndistributed matrix X.\nincx\n(global) Specifies the increment for the elements of sub(x). Only two\nvalues are supported, namely 1 and m_x. incx must not be zero.\nOutput Parameters\nnorm2\n(local) and pscnrm2.\nContains the Euclidean norm of a distributed vector only in its scope.\np?scal\nComputes a product of a distributed vector by a\nscalar.\nSyntax\nvoid psscal (const MKL_INT *n , const float *a , float *x , const MKL_INT *ix , const\nMKL_INT *jx , const MKL_INT *descx , const MKL_INT *incx );\nvoid pdscal (const MKL_INT *n , const double *a , double *x , const MKL_INT *ix , const\nMKL_INT *jx , const MKL_INT *descx , const MKL_INT *incx );\nvoid pcscal (const MKL_INT *n , const MKL_Complex8 *a , MKL_Complex8 *x , const MKL_INT\n*ix , const MKL_INT *jx , const MKL_INT *descx , const MKL_INT *incx );\nvoid pzscal (const MKL_INT *n , const MKL_Complex16 *a , MKL_Complex16 *x , const\nMKL_INT *ix , const MKL_INT *jx , const MKL_INT *descx , const MKL_INT *incx );\nvoid pcsscal (const MKL_INT *n , const float *a , MKL_Complex8 *x , const MKL_INT *ix ,\nconst MKL_INT *jx , const MKL_INT *descx , const MKL_INT *incx );\nvoid pzdscal (const MKL_INT *n , const double *a , MKL_Complex16 *x , const MKL_INT\n*ix , const MKL_INT *jx , const MKL_INT *descx , const MKL_INT *incx );\nInclude Files\n•\nmkl_pblas.h\nDescription\nThe p?scal routines multiplies a n-element distributed vector sub(x) by the scalar a:\nsub(x) = a*sub(x),\nwhere sub(x) denotes X(ix, jx:jx+n-1) if incx=m_x, and X(ix: ix+n-1, jx) if incx= 1.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2352\n\n\nInput Parameters\nn\n(global) The length of distributed vector sub(x), n≥0.\na\n(global) and pcsscal\nSpecifies the scalar a.\nx\n(local)\nArray, size (jx-1)*m_x + ix+(n-1)*abs(incx)).\nThis array contains the entries of the distributed vector sub(x).\nix, jx\n(global) The row and column indices in the distributed matrix X indicating\nthe first row and the first column of the submatrix sub(X), respectively.\ndescx\n(global and local) array of dimension 9. The array descriptor of the\ndistributed matrix X.\nincx\n(global) Specifies the increment for the elements of sub(x). Only two\nvalues are supported, namely 1 and m_x. incx must not be zero.\nOutput Parameters\nx\nOverwritten by the updated distributed vector sub(x)\np?swap\nSwaps two distributed vectors.\nSyntax\nvoid psswap (const MKL_INT *n , float *x , const MKL_INT *ix , const MKL_INT *jx ,\nconst MKL_INT *descx , const MKL_INT *incx , float *y , const MKL_INT *iy , const\nMKL_INT *jy , const MKL_INT *descy , const MKL_INT *incy );\nvoid pdswap (const MKL_INT *n , double *x , const MKL_INT *ix , const MKL_INT *jx ,\nconst MKL_INT *descx , const MKL_INT *incx , double *y , const MKL_INT *iy , const\nMKL_INT *jy , const MKL_INT *descy , const MKL_INT *incy );\nvoid pcswap (const MKL_INT *n , MKL_Complex8 *x , const MKL_INT *ix , const MKL_INT\n*jx , const MKL_INT *descx , const MKL_INT *incx , MKL_Complex8 *y , const MKL_INT\n*iy , const MKL_INT *jy , const MKL_INT *descy , const MKL_INT *incy );\nvoid pzswap (const MKL_INT *n , MKL_Complex16 *x , const MKL_INT *ix , const MKL_INT\n*jx , const MKL_INT *descx , const MKL_INT *incx , MKL_Complex16 *y , const MKL_INT\n*iy , const MKL_INT *jy , const MKL_INT *descy , const MKL_INT *incy );\nInclude Files\n•\nmkl_pblas.h\nDescription\nGiven two distributed vectors sub(x) and sub(y), the p?swap routines return vectors sub(y) and sub(x)\nswapped, each replacing the other.\nHere sub(x) denotes X(ix, jx:jx+n-1) if incx=m_x, and X(ix: ix+n-1, jx) if incx= 1;\nsub(y) denotes Y(iy, jy:jy+n-1) if incy=m_y, and Y(iy: iy+n-1, jy) if incy= 1.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2353\n\n\nInput Parameters\nn\n(global) The length of distributed vectors, n≥0.\nx\n(local)\nArray, size (jx-1)*m_x + ix+(n-1)*abs(incx)).\nThis array contains the entries of the distributed vector sub(x).\nix, jx\n(global) The row and column indices in the distributed matrix X indicating\nthe first row and the first column of the submatrix sub(X), respectively.\ndescx\n(global and local) array of dimension 9. The array descriptor of the\ndistributed matrix X.\nincx\n(global) Specifies the increment for the elements of sub(x). Only two\nvalues are supported, namely 1 and m_x. incx must not be zero.\ny\n(local)\nArray, size (jy-1)*m_y + iy+(n-1)*abs(incy)).\nThis array contains the entries of the distributed vector sub(y).\niy, jy\n(global) The row and column indices in the distributed matrix Y indicating\nthe first row and the first column of the submatrix sub(Y), respectively.\ndescy\n(global and local) array of dimension 9. The array descriptor of the\ndistributed matrix Y.\nincy\n(global) Specifies the increment for the elements of sub(y). Only two\nvalues are supported, namely 1 and m_y. incy must not be zero.\nOutput Parameters\nx\nOverwritten by distributed vector sub(y).\ny\nOverwritten by distributed vector sub(x).\nPBLAS Level 2 Routines\nThis section describes PBLAS Level 2 routines, which perform distributed matrix-vector operations. Table\n\"PBLAS Level 2 Routine Groups and Their Data Types\" lists the PBLAS Level 2 routine groups and the data\ntypes associated with them.\nPBLAS Level 2 Routine Groups and Their Data Types\nRoutine Groups\nData Types\nDescription\np?gemv\ns, d, c, z\nMatrix-vector product using a distributed general matrix\np?agemv\ns, d, c, z\nMatrix-vector product using absolute values for a\ndistributed general matrix\np?ger\ns, d\nRank-1 update of a distributed general matrix\np?gerc\nc, z\nRank-1 update (conjugated) of a distributed general\nmatrix\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2354\n\n\nRoutine Groups\nData Types\nDescription\np?geru\nc, z\nRank-1 update (unconjugated) of a distributed general\nmatrix\np?hemv\nc, z\nMatrix-vector product using a distributed Hermitian matrix\np?ahemv\nc, z\nMatrix-vector product using absolute values for a\ndistributed Hermitian matrix\np?her\nc, z\nRank-1 update of a distributed Hermitian matrix\np?her2\nc, z\nRank-2 update of a distributed Hermitian matrix\np?symv\ns, d\nMatrix-vector product using a distributed symmetric\nmatrix\np?asymv\ns, d\nMatrix-vector product using absolute values for a\ndistributed symmetric matrix\np?syr\ns, d\nRank-1 update of a distributed symmetric matrix\np?syr2\ns, d\nRank-2 update of a distributed symmetric matrix\np?trmv\ns, d, c, z\nDistributed matrix-vector product using a triangular\nmatrix\np?atrmv\ns, d, c, z\nDistributed matrix-vector product using absolute values\nfor a triangular matrix\np?trsv\ns, d, c, z\nSolves a system of linear equations whose coefficients are\nin a distributed triangular matrix\np?gemv\nComputes a distributed matrix-vector product using a\ngeneral matrix.\nSyntax\nvoid psgemv (const char *trans , const MKL_INT *m , const MKL_INT *n , const float\n*alpha , const float *a , const MKL_INT *ia , const MKL_INT *ja , const MKL_INT\n*desca , const float *x , const MKL_INT *ix , const MKL_INT *jx , const MKL_INT\n*descx , const MKL_INT *incx , const float *beta , float *y , const MKL_INT *iy , const\nMKL_INT *jy , const MKL_INT *descy , const MKL_INT *incy );\nvoid pdgemv (const char *trans , const MKL_INT *m , const MKL_INT *n , const double\n*alpha , const double *a , const MKL_INT *ia , const MKL_INT *ja , const MKL_INT\n*desca , const double *x , const MKL_INT *ix , const MKL_INT *jx , const MKL_INT\n*descx , const MKL_INT *incx , const double *beta , double *y , const MKL_INT *iy ,\nconst MKL_INT *jy , const MKL_INT *descy , const MKL_INT *incy );\nvoid pcgemv (const char *trans , const MKL_INT *m , const MKL_INT *n , const\nMKL_Complex8 *alpha , const MKL_Complex8 *a , const MKL_INT *ia , const MKL_INT *ja ,\nconst MKL_INT *desca , const MKL_Complex8 *x , const MKL_INT *ix , const MKL_INT *jx ,\nconst MKL_INT *descx , const MKL_INT *incx , const MKL_Complex8 *beta , MKL_Complex8\n*y , const MKL_INT *iy , const MKL_INT *jy , const MKL_INT *descy , const MKL_INT\n*incy );\nvoid pzgemv (const char *trans , const MKL_INT *m , const MKL_INT *n , const\nMKL_Complex16 *alpha , const MKL_Complex16 *a , const MKL_INT *ia , const MKL_INT *ja ,\nconst MKL_INT *desca , const MKL_Complex16 *x , const MKL_INT *ix , const MKL_INT *jx ,\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2355\n\n\nconst MKL_INT *descx , const MKL_INT *incx , const MKL_Complex16 *beta , MKL_Complex16\n*y , const MKL_INT *iy , const MKL_INT *jy , const MKL_INT *descy , const MKL_INT\n*incy );\nInclude Files\n•\nmkl_pblas.h\nDescription\nThe p?gemv routines perform a distributed matrix-vector operation defined as\nsub(y)  := alpha*sub(A)*sub(x) + beta*sub(y),\nor\nsub(y)  := alpha*sub(A)'*sub(x) + beta*sub(y),\nor\nsub(y)  := alpha*conjg(sub(A)')*sub(x) + beta*sub(y),\nwhere\nalpha and beta are scalars,\nsub(A) is a m-by-n submatrix, sub(A) = A(ia:ia+m-1, ja:ja+n-1),\nsub(x) and sub(y) are subvectors.\nWhen trans = 'N' or 'n', sub(x) denotes X(ix, jx:jx+n-1) if incx = m_x, and X(ix: ix+n-1, jx)\nif incx = 1,sub(y) denotes Y(iy, jy:jy+m-1) if incy = m_y, and Y(iy: iy+m-1, jy) if incy = 1.\nWhen trans = 'T' or 't', or 'C', or 'c', sub(x) denotes X(ix, jx:jx+m-1) if incx = m_x, and X(ix:\nix+m-1, jx) if incx = 1,sub(y) denotes Y(iy, jy:jy+n-1) if incy = m_y, and Y(iy: iy+m-1, jy) if\nincy = 1.\nInput Parameters\ntrans\n(global) Specifies the operation:\nif trans= 'N' or 'n', then sub(y) := alpha*sub(A)'*sub(x) +\nbeta*sub(y);\nif trans= 'T' or 't', then sub(y) := alpha*sub(A)'*sub(x) +\nbeta*sub(y);\nif trans= 'C' or 'c', then sub(y) := alpha*conjg(subA)')*sub(x) +\nbeta*sub(y).\nm\n(global) Specifies the number of rows of the distributed matrix sub(A),\nm≥0.\nn\n(global) Specifies the number of columns of the distributed matrix sub(A),\nn≥0.\nalpha\n(global)\nSpecifies the scalar alpha.\na\n(local)\nArray, size (lld_a, LOCq(ja+n-1)). Before entry this array must contain\nthe local pieces of the distributed matrix sub(A).\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2356\n\n\nia, ja\n(global) The row and column indices in the distributed matrix A indicating\nthe first row and the first column of the submatrix sub(A), respectively.\ndesca\n(global and local) array of dimension 9. The array descriptor of the\ndistributed matrix A.\nx\n(local)\nArray, size (jx-1)*m_x + ix+(n-1)*abs(incx)) when trans = 'N' or\n'n', and (jx-1)*m_x + ix+(m-1)*abs(incx)) otherwise.\nThis array contains the entries of the distributed vector sub(x).\nix, jx\n(global) The row and column indices in the distributed matrix X indicating\nthe first row and the first column of the submatrix sub(x), respectively.\ndescx\n(global and local) array of dimension 9. The array descriptor of the\ndistributed matrix X.\nincx\n(global) Specifies the increment for the elements of sub(x). Only two\nvalues are supported, namely 1 and m_x. incx must not be zero.\nbeta\n(global)\nSpecifies the scalar beta. When beta is set to zero, then sub(y) need not\nbe set on input.\ny\n(local)\nArray, size (jy-1)*m_y + iy+(m-1)*abs(incy)) when trans = 'N' or\n'n', and (jy-1)*m_y + iy+(n-1)*abs(incy)) otherwise.\nThis array contains the entries of the distributed vector sub(y).\niy, jy\n(global) The row and column indices in the distributed matrix Y indicating\nthe first row and the first column of the submatrix sub(y), respectively.\ndescy\n(global and local) array of dimension 9. The array descriptor of the\ndistributed matrix Y.\nincy\n(global) Specifies the increment for the elements of sub(y). Only two\nvalues are supported, namely 1 and m_y. incy must not be zero.\nOutput Parameters\ny\nOverwritten by the updated distributed vector sub(y).\np?agemv\nComputes a distributed matrix-vector product using\nabsolute values for a general matrix.\nSyntax\nvoid psagemv (const char *trans , const MKL_INT *m , const MKL_INT *n , const float\n*alpha , const float *a , const MKL_INT *ia , const MKL_INT *ja , const MKL_INT\n*desca , const float *x , const MKL_INT *ix , const MKL_INT *jx , const MKL_INT\n*descx , const MKL_INT *incx , const float *beta , float *y , const MKL_INT *iy , const\nMKL_INT *jy , const MKL_INT *descy , const MKL_INT *incy );\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2357\n\n\nvoid pdagemv (const char *trans , const MKL_INT *m , const MKL_INT *n , const double\n*alpha , const double *a , const MKL_INT *ia , const MKL_INT *ja , const MKL_INT\n*desca , const double *x , const MKL_INT *ix , const MKL_INT *jx , const MKL_INT\n*descx , const MKL_INT *incx , const double *beta , double *y , const MKL_INT *iy ,\nconst MKL_INT *jy , const MKL_INT *descy , const MKL_INT *incy );\nvoid pcagemv (const char *trans , const MKL_INT *m , const MKL_INT *n , const\nMKL_Complex8 *alpha , const MKL_Complex8 *a , const MKL_INT *ia , const MKL_INT *ja ,\nconst MKL_INT *desca , const MKL_Complex8 *x , const MKL_INT *ix , const MKL_INT *jx ,\nconst MKL_INT *descx , const MKL_INT *incx , const MKL_Complex8 *beta , MKL_Complex8\n*y , const MKL_INT *iy , const MKL_INT *jy , const MKL_INT *descy , const MKL_INT\n*incy );\nvoid pzagemv (const char *trans , const MKL_INT *m , const MKL_INT *n , const\nMKL_Complex16 *alpha , const MKL_Complex16 *a , const MKL_INT *ia , const MKL_INT *ja ,\nconst MKL_INT *desca , const MKL_Complex16 *x , const MKL_INT *ix , const MKL_INT *jx ,\nconst MKL_INT *descx , const MKL_INT *incx , const MKL_Complex16 *beta , MKL_Complex16\n*y , const MKL_INT *iy , const MKL_INT *jy , const MKL_INT *descy , const MKL_INT\n*incy );\nInclude Files\n•\nmkl_pblas.h\nDescription\nThe p?agemv routines perform a distributed matrix-vector operation defined as\nsub(y)  := abs(alpha)*abs(sub(A)')*abs(sub(x)) + abs(beta*sub(y)),\nor\nsub(y)  := abs(alpha)*abs(sub(A)')*abs(sub(x)) + abs(beta*sub(y)),\nor\nsub(y)  := abs(alpha)*abs(conjg(sub(A)'))*abs(sub(x)) + abs(beta*sub(y)),\nwhere\nalpha and beta are scalars,\nsub(A) is a m-by-n submatrix, sub(A) = A(ia:ia+m-1, ja:ja+n-1),\nsub(x) and sub(y) are subvectors.\nWhen trans = 'N' or 'n',\nsub(x) denotes X(ix:ix, jx:jx+n-1) if incx = m_x, and\nX(ix:ix+n-1, jx:jx) if incx = 1,\nsub(y) denotes Y(iy:iy, jy:jy+m-1) if incy = m_y, and\nY(iy:iy+m-1, jy:jy) if incy = 1.\nWhen trans = 'T' or 't', or 'C', or 'c',\nsub(x) denotes X(ix:ix, jx:jx+m-1) if incx = m_x, and\nX(ix:ix+m-1, jx:jx) if incx = 1,\nsub(y) denotes Y(iy:iy, jy:jy+n-1) if incy = m_y, and\nY(iy:iy+m-1, jy:jy) if incy = 1.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2358\n\n\nInput Parameters\ntrans\n(global) Specifies the operation:\nif trans= 'N' or 'n', then sub(y) := |alpha|*|sub(A)|*|sub(x)| +\n|beta*sub(y)|\nif trans= 'T' or 't', then sub(y) := |alpha|*|sub(A)'|*|sub(x)| +\n|beta*sub(y)|\nif trans= 'C' or 'c', then sub(y) := |alpha|*|sub(A)'|*|sub(x)| +\n|beta*sub(y)|.\nm\n(global) Specifies the number of rows of the distributed matrix sub(A),\nm≥0.\nn\n(global) Specifies the number of columns of the distributed matrix sub(A),\nn≥0.\nalpha\n(global)\nSpecifies the scalar alpha.\na\n(local)\nArray, size (lld_a, LOCq(ja+n-1)). Before entry this array must contain\nthe local pieces of the distributed matrix sub(A).\nia, ja\n(global) The row and column indices in the distributed matrix A indicating\nthe first row and the first column of the submatrix sub(A), respectively.\ndesca\n(global and local) array of dimension 9. The array descriptor of the\ndistributed matrix A.\nx\n(local)\nArray, size (jx-1)*m_x + ix+(n-1)*abs(incx)) when trans = 'N' or\n'n', and (jx-1)*m_x + ix+(m-1)*abs(incx)) otherwise.\nThis array contains the entries of the distributed vector sub(x).\nix, jx\n(global) The row and column indices in the distributed matrix X indicating\nthe first row and the first column of the submatrix sub(x), respectively.\ndescx\n(global and local) array of dimension 9. The array descriptor of the\ndistributed matrix X.\nincx\n(global) Specifies the increment for the elements of sub(x). Only two\nvalues are supported, namely 1 and m_x. incx must not be zero.\nbeta\n(global)\nSpecifies the scalar beta. When beta is set to zero, then sub(y) need not\nbe set on input.\ny\n(local)\nArray, size (jy-1)*m_y + iy+(m-1)*abs(incy)) when trans = 'N' or\n'n', and (jy-1)*m_y + iy+(n-1)*abs(incy)) otherwise.\nThis array contains the entries of the distributed vector sub(y).\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2359\n\n\niy, jy\n(global) The row and column indices in the distributed matrix Y indicating\nthe first row and the first column of the submatrix sub(y), respectively.\ndescy\n(global and local) array of dimension 9. The array descriptor of the\ndistributed matrix Y.\nincy\n(global) Specifies the increment for the elements of sub(y). Only two\nvalues are supported, namely 1 and m_y. incy must not be zero.\nOutput Parameters\ny\nOverwritten by the updated distributed vector sub(y).\np?ger\nPerforms a rank-1 update of a distributed general\nmatrix.\nSyntax\nvoid psger (const MKL_INT *m , const MKL_INT *n , const float *alpha , const float *x ,\nconst MKL_INT *ix , const MKL_INT *jx , const MKL_INT *descx , const MKL_INT *incx ,\nconst float *y , const MKL_INT *iy , const MKL_INT *jy , const MKL_INT *descy , const\nMKL_INT *incy , float *a , const MKL_INT *ia , const MKL_INT *ja , const MKL_INT\n*desca );\nvoid pdger (const MKL_INT *m , const MKL_INT *n , const double *alpha , const double\n*x , const MKL_INT *ix , const MKL_INT *jx , const MKL_INT *descx , const MKL_INT\n*incx , const double *y , const MKL_INT *iy , const MKL_INT *jy , const MKL_INT\n*descy , const MKL_INT *incy , double *a , const MKL_INT *ia , const MKL_INT *ja ,\nconst MKL_INT *desca );\nInclude Files\n•\nmkl_pblas.h\nDescription\nThe p?ger routines perform a distributed matrix-vector operation defined as\nsub(A) := alpha*sub(x)*sub(y)' + sub(A),\nwhere:\nalpha is a scalar,\nsub(A) is a m-by-n distributed general matrix, sub(A)=A(ia:ia+m-1, ja:ja+n-1),\nsub(x) is an m-element distributed vector, sub(y) is an n-element distributed vector,\nsub(x) denotes X(ix, jx:jx+m-1) if incx = m_x, and X(ix: ix+m-1, jx) if incx = 1,\nsub(y) denotes Y(iy, jy:jy+n-1) if incy = m_y, and Y(iy: iy+n-1, jy) if incy = 1.\nInput Parameters\nm\n(global) Specifies the number of rows of the distributed matrix sub(A),\nm≥0.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2360\n\n\nn\n(global) Specifies the number of columns of the distributed matrix sub(A),\nn≥0.\nalpha\n(global)\nSpecifies the scalar alpha.\nx\n(local)\nArray, size at least (jx-1)*m_x + ix+(m-1)*abs(incx)).\nThis array contains the entries of the distributed vector sub(x).\nix, jx\n(global) The row and column indices in the distributed matrix X indicating\nthe first row and the first column of the submatrix sub(x), respectively.\ndescx\n(global and local) array of dimension 9. The array descriptor of the\ndistributed matrix X.\nincx\n(global) Specifies the increment for the elements of sub(x). Only two\nvalues are supported, namely 1 and m_x. incx must not be zero.\ny\n(local)\nArray, size at least (jy-1)*m_y + iy+(n-1)*abs(incy)).\nThis array contains the entries of the distributed vector sub(y).\niy, jy\n(global) The row and column indices in the distributed matrix Y indicating\nthe first row and the first column of the submatrix sub(y), respectively.\ndescy\n(global and local) array of dimension 9. The array descriptor of the\ndistributed matrix Y.\nincy\n(global) Specifies the increment for the elements of sub(y). Only two\nvalues are supported, namely 1 and m_y. incy must not be zero.\na\n(local)\nArray, size (lld_a, LOCq(ja+n-1)).\nBefore entry this array contains the local pieces of the distributed matrix\nsub(A).\nia, ja\n(global) The row and column indices in the distributed matrix A indicating\nthe first row and the first column of the submatrix sub(A), respectively.\ndesca\n(global and local) array of dimension 9. The array descriptor of the\ndistributed matrix A.\nOutput Parameters\na\nOverwritten by the updated distributed matrix sub(A).\np?gerc\nPerforms a rank-1 update (conjugated) of a\ndistributed general matrix.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2361\n\n\nSyntax\nvoid pcgerc (const MKL_INT *m , const MKL_INT *n , const MKL_Complex8 *alpha , const\nMKL_Complex8 *x , const MKL_INT *ix , const MKL_INT *jx , const MKL_INT *descx , const\nMKL_INT *incx , const MKL_Complex8 *y , const MKL_INT *iy , const MKL_INT *jy , const\nMKL_INT *descy , const MKL_INT *incy , MKL_Complex8 *a , const MKL_INT *ia , const\nMKL_INT *ja , const MKL_INT *desca );\nvoid pzgerc (const MKL_INT *m , const MKL_INT *n , const MKL_Complex16 *alpha , const\nMKL_Complex16 *x , const MKL_INT *ix , const MKL_INT *jx , const MKL_INT *descx , const\nMKL_INT *incx , const MKL_Complex16 *y , const MKL_INT *iy , const MKL_INT *jy , const\nMKL_INT *descy , const MKL_INT *incy , MKL_Complex16 *a , const MKL_INT *ia , const\nMKL_INT *ja , const MKL_INT *desca );\nInclude Files\n•\nmkl_pblas.h\nDescription\nThe p?gerc routines perform a distributed matrix-vector operation defined as\nsub(A) := alpha*sub(x)*conjg(sub(y)') + sub(A),\nwhere:\nalpha is a scalar,\nsub(A) is a m-by-n distributed general matrix, sub(A) = A(ia:ia+m-1, ja:ja+n-1),\nsub(x) is an m-element distributed vector, sub(y) is ann-element distributed vector,\nsub(x)denotes X(ix, jx:jx+m-1) if incx = m_x, and X(ix: ix+m-1, jx) if incx = 1,\nsub(y)denotes Y(iy, jy:jy+n-1) if incy = m_y, and Y(iy: iy+n-1, jy) if incy = 1.\nInput Parameters\nm\n(global) Specifies the number of rows of the distributed matrix sub(A), m≥\n0.\nn\n(global) Specifies the number of columns of the distributed matrix sub(A),\nn≥ 0.\nalpha\n(global)\nSpecifies the scalar alpha.\nx\n(local)\nArray, size at least (jx-1)*m_x + ix+(n-1)*abs(incx)).\nThis array contains the entries of the distributed vector sub(x).\nix, jx\n(global) The row and column indices in the distributed matrix X indicating\nthe first row and the first column of the submatrix sub(x), respectively.\ndescx\n(global and local) array of dimension 9. The array descriptor of the\ndistributed matrix X.\nincx\n(global) Specifies the increment for the elements of sub(x). Only two\nvalues are supported, namely 1 and m_x. incx must not be zero.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2362\n\n\ny\n(local)\nArray, size at least (jy-1)*m_y + iy+(n-1)*abs(incy)).\nThis array contains the entries of the distributed vector sub(y).\niy, jy\n(global) The row and column indices in the distributed matrix Y indicating\nthe first row and the first column of the submatrix sub(y), respectively.\ndescy\n(global and local) array of dimension 9. The array descriptor of the\ndistributed matrix Y.\nincy\n(global) Specifies the increment for the elements of sub(y). Only two\nvalues are supported, namely 1 and m_y. incy must not be zero.\na\n(local)\nArray, size at least (lld_a, LOCq(ja+n-1)). Before entry this array\ncontains the local pieces of the distributed matrix sub(A).\nia, ja\n(global) The row and column indices in the distributed matrix A indicating\nthe first row and the first column of the submatrix sub(A), respectively.\ndesca\n(global and local) array of dimension 9. The array descriptor of the\ndistributed matrix A.\nOutput Parameters\na\nOverwritten by the updated distributed matrix sub(A).\np?geru\nPerforms a rank-1 update (unconjugated) of a\ndistributed general matrix.\nSyntax\nvoid pcgeru (const MKL_INT *m , const MKL_INT *n , const MKL_Complex8 *alpha , const\nMKL_Complex8 *x , const MKL_INT *ix , const MKL_INT *jx , const MKL_INT *descx , const\nMKL_INT *incx , const MKL_Complex8 *y , const MKL_INT *iy , const MKL_INT *jy , const\nMKL_INT *descy , const MKL_INT *incy , MKL_Complex8 *a , const MKL_INT *ia , const\nMKL_INT *ja , const MKL_INT *desca );\nvoid pzgeru (const MKL_INT *m , const MKL_INT *n , const MKL_Complex16 *alpha , const\nMKL_Complex16 *x , const MKL_INT *ix , const MKL_INT *jx , const MKL_INT *descx , const\nMKL_INT *incx , const MKL_Complex16 *y , const MKL_INT *iy , const MKL_INT *jy , const\nMKL_INT *descy , const MKL_INT *incy , MKL_Complex16 *a , const MKL_INT *ia , const\nMKL_INT *ja , const MKL_INT *desca );\nInclude Files\n•\nmkl_pblas.h\nDescription\nThe p?geru routines perform a matrix-vector operation defined as\nsub(A) := alpha*sub(x)*sub(y)' + sub(A),\nwhere:\nalpha is a scalar,\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2363\n\n\nsub(A) is a m-by-n distributed general matrix, sub(A)=A(ia:ia+m-1, ja:ja+n-1),\nsub(x) is an m-element distributed vector, sub(y) is an n-element distributed vector,\nsub(x) denotes X(ix, jx:jx+m-1) if incx = m_x, and X(ix: ix+m-1, jx) if incx = 1,\nsub(y) denotes Y(iy, jy:jy+n-1) if incy = m_y, and Y(iy: iy+n-1, jy) if incy = 1.\nInput Parameters\nm\n(global) Specifies the number of rows of the distributed matrix sub(A), m≥\n0.\nn\n(global) Specifies the number of columns of the distributed matrix sub(A),\nn≥ 0.\nalpha\n(global)\nSpecifies the scalar alpha.\nx\n(local)\nArray, size at least (jx-1)*m_x + ix+(n-1)*abs(incx)).\nThis array contains the entries of the distributed vector sub(x).\nix, jx\n(global) The row and column indices in the distributed matrix X indicating\nthe first row and the first column of the submatrix sub(x), respectively.\ndescx\n(global and local) array of dimension 9. The array descriptor of the\ndistributed matrix X.\nincx\n(global) Specifies the increment for the elements of sub(x). Only two\nvalues are supported, namely 1 and m_x. incx must not be zero.\ny\n(local)\nArray, size at least (jy-1)*m_y + iy+(n-1)*abs(incy)).\nThis array contains the entries of the distributed vector sub(y).\niy, jy\n(global) The row and column indices in the distributed matrix Y indicating\nthe first row and the first column of the submatrix sub(y), respectively.\ndescy\n(global and local) array of dimension 9. The array descriptor of the\ndistributed matrix Y.\nincy\n(global) Specifies the increment for the elements of sub(y). Only two\nvalues are supported, namely 1 and m_y. incy must not be zero.\na\n(local)\nArray, size at least (lld_a, LOCq(ja+n-1)). Before entry this array\ncontains the local pieces of the distributed matrix sub(A).\nia, ja\n(global) The row and column indices in the distributed matrix A indicating\nthe first row and the first column of the submatrix sub(A), respectively.\ndesca\n(global and local) array of dimension 9. The array descriptor of the\ndistributed matrix A.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2364\n\n\nOutput Parameters\na\nOverwritten by the updated distributed matrix sub(A).\np?hemv\nComputes a distributed matrix-vector product using a\nHermitian matrix.\nSyntax\nvoid pchemv (const char *uplo , const MKL_INT *n , const MKL_Complex8 *alpha , const\nMKL_Complex8 *a , const MKL_INT *ia , const MKL_INT *ja , const MKL_INT *desca , const\nMKL_Complex8 *x , const MKL_INT *ix , const MKL_INT *jx , const MKL_INT *descx , const\nMKL_INT *incx , const MKL_Complex8 *beta , MKL_Complex8 *y , const MKL_INT *iy , const\nMKL_INT *jy , const MKL_INT *descy , const MKL_INT *incy );\nvoid pzhemv (const char *uplo , const MKL_INT *n , const MKL_Complex16 *alpha , const\nMKL_Complex16 *a , const MKL_INT *ia , const MKL_INT *ja , const MKL_INT *desca , const\nMKL_Complex16 *x , const MKL_INT *ix , const MKL_INT *jx , const MKL_INT *descx , const\nMKL_INT *incx , const MKL_Complex16 *beta , MKL_Complex16 *y , const MKL_INT *iy ,\nconst MKL_INT *jy , const MKL_INT *descy , const MKL_INT *incy );\nInclude Files\n•\nmkl_pblas.h\nDescription\nThe p?hemv routines perform a distributed matrix-vector operation defined as\nsub(y) := alpha*sub(A)*sub(x) + beta*sub(y),\nwhere:\nalpha and beta are scalars,\nsub(A) is a n-by-n Hermitian distributed matrix, sub(A)=A(ia:ia+n-1, ja:ja+n-1) ,\nsub(x) and sub(y) are distributed vectors.\nsub(x) denotes X(ix, jx:jx+n-1) if incx = m_x, and X(ix: ix+n-1, jx) if incx = 1,\nsub(y) denotes Y(iy, jy:jy+n-1) if incy = m_y, and Y(iy: iy+n-1, jy) if incy = 1.\nInput Parameters\nuplo\n(global) Specifies whether the upper or lower triangular part of the\nHermitian distributed matrix sub(A) is used:\nIf uplo = 'U' or 'u', then the upper triangular part of the sub(A) is\nused.\nIf uplo = 'L' or 'l', then the low triangular part of the sub(A) is used.\nn\n(global) Specifies the order of the distributed matrix sub(A), n≥ 0.\nalpha\n(global)\nSpecifies the scalar alpha.\na\n(local)\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2365\n\n\nArray, size (lld_a, LOCq(ja+n-1)). This array contains the local pieces of\nthe distributed matrix sub(A).\nBefore entry when uplo = 'U' or 'u', the n-by-n upper triangular part of\nthe distributed matrix sub(A) must contain the upper triangular part of the\nHermitian distributed matrix and the strictly lower triangular part of sub(A)\nis not referenced, and when uplo = 'L' or 'l', the n-by-n lower\ntriangular part of the distributed matrix sub(A) must contain the lower\ntriangular part of the Hermitian distributed matrix and the strictly upper\ntriangular part of sub(A) is not referenced.\nia, ja\n(global) The row and column indices in the distributed matrix A indicating\nthe first row and the first column of the submatrix sub(A), respectively.\ndesca\n(global and local) array of dimension 9. The array descriptor of the\ndistributed matrix A.\nx\n(local)\nArray, size at least (jx-1)*m_x + ix+(n-1)*abs(incx)).\nThis array contains the entries of the distributed vector sub(x).\nix, jx\n(global) The row and column indices in the distributed matrix X indicating\nthe first row and the first column of the submatrix sub(x), respectively.\ndescx\n(global and local) array of dimension 9. The array descriptor of the\ndistributed matrix X.\nincx\n(global) Specifies the increment for the elements of sub(x). Only two\nvalues are supported, namely 1 and m_x. incx must not be zero.\nbeta\n(global)\nSpecifies the scalar beta. When beta is set to zero, then sub(y) need not\nbe set on input.\ny\n(local)\nArray, size at least (jy-1)*m_y + iy+(n-1)*abs(incy)).\nThis array contains the entries of the distributed vector sub(y).\niy, jy\n(global) The row and column indices in the distributed matrix Y indicating\nthe first row and the first column of the submatrix sub(y), respectively.\ndescy\n(global and local) array of dimension 9. The array descriptor of the\ndistributed matrix Y.\nincy\n(global) Specifies the increment for the elements of sub(y). Only two\nvalues are supported, namely 1 and m_y. incy must not be zero.\nOutput Parameters\ny\nOverwritten by the updated distributed vector sub(y).\np?ahemv\nComputes a distributed matrix-vector product using\nabsolute values for a Hermitian matrix.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2366\n\n\nSyntax\nvoid pcahemv (const char *uplo , const MKL_INT *n , const MKL_Complex8 *alpha , const\nMKL_Complex8 *a , const MKL_INT *ia , const MKL_INT *ja , const MKL_INT *desca , const\nMKL_Complex8 *x , const MKL_INT *ix , const MKL_INT *jx , const MKL_INT *descx , const\nMKL_INT *incx , const MKL_Complex8 *beta , MKL_Complex8 *y , const MKL_INT *iy , const\nMKL_INT *jy , const MKL_INT *descy , const MKL_INT *incy );\nvoid pzahemv (const char *uplo , const MKL_INT *n , const MKL_Complex16 *alpha , const\nMKL_Complex16 *a , const MKL_INT *ia , const MKL_INT *ja , const MKL_INT *desca , const\nMKL_Complex16 *x , const MKL_INT *ix , const MKL_INT *jx , const MKL_INT *descx , const\nMKL_INT *incx , const MKL_Complex16 *beta , MKL_Complex16 *y , const MKL_INT *iy ,\nconst MKL_INT *jy , const MKL_INT *descy , const MKL_INT *incy );\nInclude Files\n•\nmkl_pblas.h\nDescription\nThe p?ahemv routines perform a distributed matrix-vector operation defined as\nsub(y) := abs(alpha)*abs(sub(A))*abs(sub(x)) + abs(beta*sub(y)),\nwhere:\nalpha and beta are scalars,\nsub(A) is a n-by-n Hermitian distributed matrix, sub(A)=A(ia:ia+n-1, ja:ja+n-1) ,\nsub(x) and sub(y) are distributed vectors.\nsub(x) denotes X(ix, jx:jx+n-1) if incx = m_x, and X(ix: ix+n-1, jx) if incx = 1,\nsub(y) denotes Y(iy, jy:jy+n-1) if incy = m_y, and Y(iy: iy+n-1, jy) if incy = 1.\nInput Parameters\nuplo\n(global) Specifies whether the upper or lower triangular part of the\nHermitian distributed matrix sub(A) is used:\nIf uplo = 'U' or 'u', then the upper triangular part of the sub(A) is\nused.\nIf uplo = 'L' or 'l', then the low triangular part of the sub(A) is used.\nn\n(global) Specifies the order of the distributed matrix sub(A), n≥ 0.\nalpha\n(global)\nSpecifies the scalar alpha.\na\n(local)\nArray, size (lld_a, LOCq(ja+n-1)). This array contains the local pieces of\nthe distributed matrix sub(A).\nBefore entry when uplo = 'U' or 'u', the n-by-n upper triangular part of\nthe distributed matrix sub(A) must contain the upper triangular part of the\nHermitian distributed matrix and the strictly lower triangular part of sub(A)\nis not referenced, and when uplo = 'L' or 'l', the n-by-n lower\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2367\n\n\ntriangular part of the distributed matrix sub(A) must contain the lower\ntriangular part of the Hermitian distributed matrix and the strictly upper\ntriangular part of sub(A) is not referenced.\nia, ja\n(global) The row and column indices in the distributed matrix A indicating\nthe first row and the first column of the submatrix sub(A), respectively.\ndesca\n(global and local) array of dimension 9. The array descriptor of the\ndistributed matrix A.\nx\n(local)\nArray, size at least (jx-1)*m_x + ix+(n-1)*abs(incx)).\nThis array contains the entries of the distributed vector sub(x).\nix, jx\n(global) The row and column indices in the distributed matrix X indicating\nthe first row and the first column of the submatrix sub(x), respectively.\ndescx\n(global and local) array of dimension 9. The array descriptor of the\ndistributed matrix X.\nincx\n(global) Specifies the increment for the elements of sub(x). Only two\nvalues are supported, namely 1 and m_x. incx must not be zero.\nbeta\n(global)\nSpecifies the scalar beta. When beta is set to zero, then sub(y) need not\nbe set on input.\ny\n(local)\nArray, size at least (jy-1)*m_y + iy+(n-1)*abs(incy)).\nThis array contains the entries of the distributed vector sub(y).\niy, jy\n(global) The row and column indices in the distributed matrix Y indicating\nthe first row and the first column of the submatrix sub(y), respectively.\ndescy\n(global and local) array of dimension 9. The array descriptor of the\ndistributed matrix Y.\nincy\n(global) Specifies the increment for the elements of sub(y). Only two\nvalues are supported, namely 1 and m_y. incy must not be zero.\nOutput Parameters\ny\nOverwritten by the updated distributed vector sub(y).\np?her\nPerforms a rank-1 update of a distributed Hermitian\nmatrix.\nSyntax\nvoid pcher (const char *uplo , const MKL_INT *n , const float *alpha , const\nMKL_Complex8 *x , const MKL_INT *ix , const MKL_INT *jx , const MKL_INT *descx , const\nMKL_INT *incx , MKL_Complex8 *a , const MKL_INT *ia , const MKL_INT *ja , const MKL_INT\n*desca );\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2368\n\n\nvoid pzher (const char *uplo , const MKL_INT *n , const double *alpha , const\nMKL_Complex16 *x , const MKL_INT *ix , const MKL_INT *jx , const MKL_INT *descx , const\nMKL_INT *incx , MKL_Complex16 *a , const MKL_INT *ia , const MKL_INT *ja , const\nMKL_INT *desca );\nInclude Files\n•\nmkl_pblas.h\nDescription\nThe p?her routines perform a distributed matrix-vector operation defined as\nsub(A) := alpha*sub(x)*conjg(sub(x)') + sub(A),\nwhere:\nalpha is a real scalar,\nsub(A) is a n-by-n distributed Hermitian matrix, sub(A)=A(ia:ia+n-1, ja:ja+n-1),\nsub(x) is distributed vector.\nsub(x) denotes X(ix, jx:jx+n-1) if incx = m_x, and X(ix: ix+n-1, jx) if incx = 1.\nInput Parameters\nuplo\n(global) Specifies whether the upper or lower triangular part of the\nHermitian distributed matrix sub(A) is used:\nIf uplo = 'U' or 'u', then the upper triangular part of the sub(A) is\nused.\nIf uplo = 'L' or 'l', then the low triangular part of the sub(A) is used.\nn\n(global) Specifies the order of the distributed matrix sub(A), n≥ 0.\nalpha\n(global)\nSpecifies the scalar alpha.\nx\n(local)\nArray, size at least (jx-1)*m_x + ix+(n-1)*abs(incx)).\nThis array contains the entries of the distributed vector sub(x).\nix, jx\n(global) The row and column indices in the distributed matrix X indicating\nthe first row and the first column of the submatrix sub(x), respectively.\ndescx\n(global and local) array of dimension 9. The array descriptor of the\ndistributed matrix X.\nincx\n(global) Specifies the increment for the elements of sub(x). Only two\nvalues are supported, namely 1 and m_x. incx must not be zero.\na\n(local)\nArray, size (lld_a, LOCq(ja+n-1)). This array contains the local pieces of\nthe distributed matrix sub(A).\nBefore entry with uplo = 'U' or 'u', the n-by-n upper triangular part of\nthe distributed matrix sub(A) must contain the upper triangular part of the\nHermitian distributed matrix and the strictly lower triangular part of sub(A)\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2369\n\n\nis not referenced, and with uplo = 'L' or 'l', the n-by-n lower triangular\npart of the distributed matrix sub(A) must contain the lower triangular part\nof the Hermitian distributed matrix and the strictly upper triangular part of\nsub(A) is not referenced.\nia, ja\n(global) The row and column indices in the distributed matrix A indicating\nthe first row and the first column of the submatrix sub(A), respectively.\ndesca\n(global and local) array of dimension 9. The array descriptor of the\ndistributed matrix A.\nOutput Parameters\na\nWith uplo = 'U' or 'u', the upper triangular part of the array a is\noverwritten by the upper triangular part of the updated distributed matrix\nsub(A).\nWith uplo = 'L' or 'l', the lower triangular part of the array a is\noverwritten by the lower triangular part of the updated distributed matrix\nsub(A).\np?her2\nPerforms a rank-2 update of a distributed Hermitian\nmatrix.\nSyntax\nvoid pcher2 (const char *uplo , const MKL_INT *n , const MKL_Complex8 *alpha , const\nMKL_Complex8 *x , const MKL_INT *ix , const MKL_INT *jx , const MKL_INT *descx , const\nMKL_INT *incx , const MKL_Complex8 *y , const MKL_INT *iy , const MKL_INT *jy , const\nMKL_INT *descy , const MKL_INT *incy , MKL_Complex8 *a , const MKL_INT *ia , const\nMKL_INT *ja , const MKL_INT *desca );\nvoid pzher2 (const char *uplo , const MKL_INT *n , const MKL_Complex16 *alpha , const\nMKL_Complex16 *x , const MKL_INT *ix , const MKL_INT *jx , const MKL_INT *descx , const\nMKL_INT *incx , const MKL_Complex16 *y , const MKL_INT *iy , const MKL_INT *jy , const\nMKL_INT *descy , const MKL_INT *incy , MKL_Complex16 *a , const MKL_INT *ia , const\nMKL_INT *ja , const MKL_INT *desca );\nInclude Files\n•\nmkl_pblas.h\nDescription\nThe p?her2 routines perform a distributed matrix-vector operation defined as\nsub(A) := alpha*sub(x)*conj(sub(y)')+ conj(alpha)*sub(y)*conj(sub(x)') + sub(A),\nwhere:\nalpha is a scalar,\nsub(A) is a n-by-n distributed Hermitian matrix, sub(A)=A(ia:ia+n-1, ja:ja+n-1),\nsub(x) and sub(y) are distributed vectors.\nsub(x) denotes X(ix, jx:jx+n-1) if incx = m_x, and X(ix: ix+n-1, jx) if incx = 1,\nsub(y) denotes Y(iy, jy:jy+n-1) if incy = m_y, and Y(iy: iy+n-1, jy) if incy = 1.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2370\n\n\nInput Parameters\nuplo\n(global) Specifies whether the upper or lower triangular part of the\ndistributed Hermitian matrix sub(A) is used:\nIf uplo = 'U' or 'u', then the upper triangular part of the sub(A) is\nused.\nIf uplo = 'L' or 'l', then the low triangular part of the sub(A) is used.\nn\n(global) Specifies the order of the distributed matrix sub(A), n≥ 0.\nalpha\n(global)\nSpecifies the scalar alpha.\nx\n(local)\nArray, size at least (jx-1)*m_x + ix+(n-1)*abs(incx)).\nThis array contains the entries of the distributed vector sub(x).\nix, jx\n(global) The row and column indices in the distributed matrix X indicating\nthe first row and the first column of the submatrix sub(x), respectively.\ndescx\n(global and local) array of dimension 9. The array descriptor of the\ndistributed matrix X.\nincx\n(global) Specifies the increment for the elements of sub(x). Only two\nvalues are supported, namely 1 and m_x. incx must not be zero.\ny\n(local)\nArray, size at least (jy-1)*m_y + iy+(n-1)*abs(incy)).\nThis array contains the entries of the distributed vector sub(y).\niy, jy\n(global) The row and column indices in the distributed matrix Y indicating\nthe first row and the first column of the submatrix sub(y), respectively.\ndescy\n(global and local) array of dimension 9. The array descriptor of the\ndistributed matrix Y.\nincy\n(global) Specifies the increment for the elements of sub(y). Only two\nvalues are supported, namely 1 and m_y. incy must not be zero.\na\n(local)\nArray, size (lld_a, LOCq(ja+n-1)). This array contains the local pieces of\nthe distributed matrix sub(A).\nBefore entry with uplo = 'U' or 'u', the n-by-n upper triangular part of\nthe distributed matrix sub(A) must contain the upper triangular part of the\nHermitian distributed matrix and the strictly lower triangular part of sub(A)\nis not referenced, and with uplo = 'L' or 'l', the n-by-n lower triangular\npart of the distributed matrix sub(A) must contain the lower triangular part\nof the Hermitian distributed matrix and the strictly upper triangular part of\nsub(A) is not referenced.\nia, ja\n(global) The row and column indices in the distributed matrix A indicating\nthe first row and the first column of the submatrix sub(A), respectively.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2371\n\n\ndesca\n(global and local) array of dimension 9. The array descriptor of the\ndistributed matrix A.\nOutput Parameters\na\nWith uplo = 'U' or 'u', the upper triangular part of the array a is\noverwritten by the upper triangular part of the updated distributed matrix\nsub(A).\nWith uplo = 'L' or 'l', the lower triangular part of the array a is\noverwritten by the lower triangular part of the updated distributed matrix\nsub(A).\np?symv\nComputes a distributed matrix-vector product using a\nsymmetric matrix.\nSyntax\nvoid pssymv (const char *uplo , const MKL_INT *n , const float *alpha , const float\n*a , const MKL_INT *ia , const MKL_INT *ja , const MKL_INT *desca , const float *x ,\nconst MKL_INT *ix , const MKL_INT *jx , const MKL_INT *descx , const MKL_INT *incx ,\nconst float *beta , float *y , const MKL_INT *iy , const MKL_INT *jy , const MKL_INT\n*descy , const MKL_INT *incy );\nvoid pdsymv (const char *uplo , const MKL_INT *n , const double *alpha , const double\n*a , const MKL_INT *ia , const MKL_INT *ja , const MKL_INT *desca , const double *x ,\nconst MKL_INT *ix , const MKL_INT *jx , const MKL_INT *descx , const MKL_INT *incx ,\nconst double *beta , double *y , const MKL_INT *iy , const MKL_INT *jy , const MKL_INT\n*descy , const MKL_INT *incy );\nInclude Files\n•\nmkl_pblas.h\nDescription\nThe p?symv routines perform a distributed matrix-vector operation defined as\nsub(y)  := alpha*sub(A)*sub(x) + beta*sub(y),\nwhere:\nalpha and beta are scalars,\nsub(A) is a n-by-n symmetric distributed matrix, sub(A)=A(ia:ia+n-1, ja:ja+n-1) ,\nsub(x) and sub(y) are distributed vectors.\nsub(x) denotes X(ix, jx:jx+n-1) if incx = m_x, and X(ix: ix+n-1, jx) if incx = 1,\nsub(y) denotes Y(iy, jy:jy+n-1) if incy = m_y, and Y(iy: iy+n-1, jy) if incy = 1.\nInput Parameters\nuplo\n(global) Specifies whether the upper or lower triangular part of the\nsymmetric distributed matrix sub(A) is used:\nIf uplo = 'U' or 'u', then the upper triangular part of the sub(A) is\nused.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2372\n\n\nIf uplo = 'L' or 'l', then the low triangular part of the sub(A) is used.\nn\n(global) Specifies the order of the distributed matrix sub(A), n≥ 0.\nalpha\n(global)\nSpecifies the scalar alpha.\na\n(local)\nArray, size (lld_a, LOCq(ja+n-1)). This array contains the local pieces of\nthe distributed matrix sub(A).\nBefore entry when uplo = 'U' or 'u', the n-by-n upper triangular part of\nthe distributed matrix sub(A) must contain the upper triangular part of the\nsymmetric distributed matrix and the strictly lower triangular part of\nsub(A) is not referenced, and when uplo = 'L' or 'l', the n-by-n lower\ntriangular part of the distributed matrix sub(A) must contain the lower\ntriangular part of the symmetric distributed matrix and the strictly upper\ntriangular part of sub(A) is not referenced.\nia, ja\n(global) The row and column indices in the distributed matrix A indicating\nthe first row and the first column of the submatrix sub(A), respectively.\ndesca\n(global and local) array of dimension 9. The array descriptor of the\ndistributed matrix A.\nx\n(local)\nArray, size at least (jx-1)*m_x + ix+(n-1)*abs(incx)).\nThis array contains the entries of the distributed vector sub(x).\nix, jx\n(global) The row and column indices in the distributed matrix X indicating\nthe first row and the first column of the submatrix sub(x), respectively.\ndescx\n(global and local) array of dimension 9. The array descriptor of the\ndistributed matrix X.\nincx\n(global) Specifies the increment for the elements of sub(x). Only two\nvalues are supported, namely 1 and m_x. incx must not be zero.\nbeta\n(global)\nSpecifies the scalar beta. When beta is set to zero, then sub(y) need not\nbe set on input.\ny\n(local)\nArray, size at least (jy-1)*m_y + iy+(n-1)*abs(incy)).\nThis array contains the entries of the distributed vector sub(y).\niy, jy\n(global) The row and column indices in the distributed matrix Y indicating\nthe first row and the first column of the submatrix sub(y), respectively.\ndescy\n(global and local) array of dimension 9. The array descriptor of the\ndistributed matrix Y.\nincy\n(global) Specifies the increment for the elements of sub(y). Only two\nvalues are supported, namely 1 and m_y. incy must not be zero.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2373\n\n\nOutput Parameters\ny\nOverwritten by the updated distributed vector sub(y).\np?asymv\nComputes a distributed matrix-vector product using\nabsolute values for a symmetric matrix.\nSyntax\nvoid psasymv (const char *uplo , const MKL_INT *n , const float *alpha , const float\n*a , const MKL_INT *ia , const MKL_INT *ja , const MKL_INT *desca , const float *x ,\nconst MKL_INT *ix , const MKL_INT *jx , const MKL_INT *descx , const MKL_INT *incx ,\nconst float *beta , float *y , const MKL_INT *iy , const MKL_INT *jy , const MKL_INT\n*descy , const MKL_INT *incy );\nvoid pdasymv (const char *uplo , const MKL_INT *n , const double *alpha , const double\n*a , const MKL_INT *ia , const MKL_INT *ja , const MKL_INT *desca , const double *x ,\nconst MKL_INT *ix , const MKL_INT *jx , const MKL_INT *descx , const MKL_INT *incx ,\nconst double *beta , double *y , const MKL_INT *iy , const MKL_INT *jy , const MKL_INT\n*descy , const MKL_INT *incy );\nInclude Files\n•\nmkl_pblas.h\nDescription\nThe p?symv routines perform a distributed matrix-vector operation defined as\nsub(y)  := abs(alpha)*abs(sub(A))*abs(sub(x)) + abs(beta*sub(y)),\nwhere:\nalpha and beta are scalars,\nsub(A) is a n-by-n symmetric distributed matrix, sub(A)=A(ia:ia+n-1, ja:ja+n-1) ,\nsub(x) and sub(y) are distributed vectors.\nsub(x) denotes X(ix, jx:jx+n-1) if incx = m_x, and X(ix: ix+n-1, jx) if incx = 1,\nsub(y) denotes Y(iy, jy:jy+n-1) if incy = m_y, and Y(iy: iy+n-1, jy) if incy = 1.\nInput Parameters\nuplo\n(global) Specifies whether the upper or lower triangular part of the\nsymmetric distributed matrix sub(A) is used:\nIf uplo = 'U' or 'u', then the upper triangular part of the sub(A) is\nused.\nIf uplo = 'L' or 'l', then the low triangular part of the sub(A) is used.\nn\n(global) Specifies the order of the distributed matrix sub(A), n≥ 0.\nalpha\n(global)\nSpecifies the scalar alpha.\na\n(local)\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2374\n\n\nArray, size (lld_a, LOCq(ja+n-1)). This array contains the local pieces of\nthe distributed matrix sub(A).\nBefore entry when uplo = 'U' or 'u', the n-by-n upper triangular part of\nthe distributed matrix sub(A) must contain the upper triangular part of the\nsymmetric distributed matrix and the strictly lower triangular part of\nsub(A) is not referenced, and when uplo = 'L' or 'l', the n-by-n lower\ntriangular part of the distributed matrix sub(A) must contain the lower\ntriangular part of the symmetric distributed matrix and the strictly upper\ntriangular part of sub(A) is not referenced.\nia, ja\n(global) The row and column indices in the distributed matrix A indicating\nthe first row and the first column of the submatrix sub(A), respectively.\ndesca\n(global and local) array of dimension 9. The array descriptor of the\ndistributed matrix A.\nx\n(local)\nArray, size at least (jx-1)*m_x + ix+(n-1)*abs(incx)).\nThis array contains the entries of the distributed vector sub(x).\nix, jx\n(global) The row and column indices in the distributed matrix X indicating\nthe first row and the first column of the submatrix sub(x), respectively.\ndescx\n(global and local) array of dimension 9. The array descriptor of the\ndistributed matrix X.\nincx\n(global) Specifies the increment for the elements of sub(x). Only two\nvalues are supported, namely 1 and m_x. incx must not be zero.\nbeta\n(global)\nSpecifies the scalar beta. When beta is set to zero, then sub(y) need not\nbe set on input.\ny\n(local)\nArray, size at least (jy-1)*m_y + iy+(n-1)*abs(incy)).\nThis array contains the entries of the distributed vector sub(y).\niy, jy\n(global) The row and column indices in the distributed matrix Y indicating\nthe first row and the first column of the submatrix sub(y), respectively.\ndescy\n(global and local) array of dimension 9. The array descriptor of the\ndistributed matrix Y.\nincy\n(global) Specifies the increment for the elements of sub(y). Only two\nvalues are supported, namely 1 and m_y. incy must not be zero.\nOutput Parameters\ny\nOverwritten by the updated distributed vector sub(y).\np?syr\nPerforms a rank-1 update of a distributed symmetric\nmatrix.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2375\n\n\nSyntax\nvoid pssyr (const char *uplo , const MKL_INT *n , const float *alpha , const float *x ,\nconst MKL_INT *ix , const MKL_INT *jx , const MKL_INT *descx , const MKL_INT *incx ,\nfloat *a , const MKL_INT *ia , const MKL_INT *ja , const MKL_INT *desca );\nvoid pdsyr (const char *uplo , const MKL_INT *n , const double *alpha , const double\n*x , const MKL_INT *ix , const MKL_INT *jx , const MKL_INT *descx , const MKL_INT\n*incx , double *a , const MKL_INT *ia , const MKL_INT *ja , const MKL_INT *desca );\nInclude Files\n•\nmkl_pblas.h\nDescription\nThe p?syr routines perform a distributed matrix-vector operation defined as\nsub(A) := alpha*sub(x)*sub(x)' + sub(A),\nwhere:\nalpha is a scalar,\nsub(A) is a n-by-n distributed symmetric matrix, sub(A)=A(ia:ia+n-1, ja:ja+n-1) ,\nsub(x) is distributed vector.\nsub(x) denotes X(ix, jx:jx+n-1) if incx = m_x, and X(ix: ix+n-1, jx) if incx = 1,\nInput Parameters\nuplo\n(global) Specifies whether the upper or lower triangular part of the\nsymmetric distributed matrix sub(A) is used:\nIf uplo = 'U' or 'u', then the upper triangular part of the sub(A) is\nused.\nIf uplo = 'L' or 'l', then the low triangular part of the sub(A) is used.\nn\n(global) Specifies the order of the distributed matrix sub(A), n≥ 0.\nalpha\n(global)\nSpecifies the scalar alpha.\nx\n(local)\nArray, size at least (jx-1)*m_x + ix+(n-1)*abs(incx)).\nThis array contains the entries of the distributed vector sub(x).\nix, jx\n(global) The row and column indices in the distributed matrix X indicating\nthe first row and the first column of the submatrix sub(x), respectively.\ndescx\n(global and local) array of dimension 9. The array descriptor of the\ndistributed matrix X.\nincx\n(global) Specifies the increment for the elements of sub(x). Only two\nvalues are supported, namely 1 and m_x. incx must not be zero.\na\n(local)\nArray, size (lld_a, LOCq(ja+n-1)). This array contains the local pieces of\nthe distributed matrix sub(A).\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2376\n\n\nBefore entry with uplo = 'U' or 'u', the n-by-n upper triangular part of\nthe distributed matrix sub(A) must contain the upper triangular part of the\nsymmetric distributed matrix and the strictly lower triangular part of\nsub(A) is not referenced, and with uplo = 'L' or 'l', the n-by-n lower\ntriangular part of the distributed matrix sub(A) must contain the lower\ntriangular part of the symmetric distributed matrix and the strictly upper\ntriangular part of sub(A) is not referenced.\nia, ja\n(global) The row and column indices in the distributed matrix A indicating\nthe first row and the first column of the submatrix sub(A), respectively.\ndesca\n(global and local) array of dimension 9. The array descriptor of the\ndistributed matrix A.\nOutput Parameters\na\nWith uplo = 'U' or 'u', the upper triangular part of the array a is\noverwritten by the upper triangular part of the updated distributed matrix\nsub(A).\nWith uplo = 'L' or 'l', the lower triangular part of the array a is\noverwritten by the lower triangular part of the updated distributed matrix\nsub(A).\np?syr2\nPerforms a rank-2 update of a distributed symmetric\nmatrix.\nSyntax\nvoid pssyr2 (const char *uplo , const MKL_INT *n , const float *alpha , const float\n*x , const MKL_INT *ix , const MKL_INT *jx , const MKL_INT *descx , const MKL_INT\n*incx , const float *y , const MKL_INT *iy , const MKL_INT *jy , const MKL_INT *descy ,\nconst MKL_INT *incy , float *a , const MKL_INT *ia , const MKL_INT *ja , const MKL_INT\n*desca );\nvoid pdsyr2 (const char *uplo , const MKL_INT *n , const double *alpha , const double\n*x , const MKL_INT *ix , const MKL_INT *jx , const MKL_INT *descx , const MKL_INT\n*incx , const double *y , const MKL_INT *iy , const MKL_INT *jy , const MKL_INT\n*descy , const MKL_INT *incy , double *a , const MKL_INT *ia , const MKL_INT *ja ,\nconst MKL_INT *desca );\nInclude Files\n•\nmkl_pblas.h\nDescription\nThe p?syr2 routines perform a distributed matrix-vector operation defined as\nsub(A) := alpha*sub(x)*sub(y)'+ alpha*sub(y)*sub(x)' + sub(A),\nwhere:\nalpha is a scalar,\nsub(A) is a n-by-n distributed symmetric matrix, sub(A)=A(ia:ia+n-1, ja:ja+n-1) ,\nsub(x) and sub(y) are distributed vectors.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2377\n\n\nsub(x) denotes X(ix, jx:jx+n-1) if incx = m_x, and X(ix: ix+n-1, jx) if incx = 1,\nsub(y) denotes Y(iy, jy:jy+n-1) if incy = m_y, and Y(iy: iy+n-1, jy) if incy = 1.\nInput Parameters\nuplo\n(global) Specifies whether the upper or lower triangular part of the\ndistributed symmetric matrix sub(A) is used:\nIf uplo = 'U' or 'u', then the upper triangular part of the sub(A) is\nused.\nIf uplo = 'L' or 'l', then the low triangular part of the sub(A) is used.\nn\n(global) Specifies the order of the distributed matrix sub(A), n≥ 0.\nalpha\n(global)\nSpecifies the scalar alpha.\nx\n(local)\nArray, size at least (jx-1)*m_x + ix+(n-1)*abs(incx)).\nThis array contains the entries of the distributed vector sub(x).\nix, jx\n(global) The row and column indices in the distributed matrix X indicating\nthe first row and the first column of the submatrix sub(x), respectively.\ndescx\n(global and local) array of dimension 9. The array descriptor of the\ndistributed matrix X.\nincx\n(global) Specifies the increment for the elements of sub(x). Only two\nvalues are supported, namely 1 and m_x. incx must not be zero.\ny\n(local)\nArray, size at least (jy-1)*m_y + iy+(n-1)*abs(incy)).\nThis array contains the entries of the distributed vector sub(y).\niy, jy\n(global) The row and column indices in the distributed matrix Y indicating\nthe first row and the first column of the submatrix sub(y), respectively.\ndescy\n(global and local) array of dimension 9. The array descriptor of the\ndistributed matrix Y.\nincy\n(global) Specifies the increment for the elements of sub(y). Only two\nvalues are supported, namely 1 and m_y. incy must not be zero.\na\n(local)\nArray, size (lld_a, LOCq(ja+n-1)). This array contains the local pieces of\nthe distributed matrix sub(A).\nBefore entry with uplo = 'U' or 'u', the n-by-n upper triangular part of\nthe distributed matrix sub(A) must contain the upper triangular part of the\ndistributed symmetric matrix and the strictly lower triangular part of\nsub(A) is not referenced, and with uplo = 'L' or 'l', the n-by-n lower\ntriangular part of the distributed matrix sub(A) must contain the lower\ntriangular part of the distributed symmetric matrix and the strictly upper\ntriangular part of sub(A) is not referenced.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2378\n\n\nia, ja\n(global) The row and column indices in the distributed matrix A indicating\nthe first row and the first column of the submatrix sub(A), respectively.\ndesca\n(global and local) array of dimension 9. The array descriptor of the\ndistributed matrix A.\nOutput Parameters\na\nWith uplo = 'U' or 'u', the upper triangular part of the array a is\noverwritten by the upper triangular part of the updated distributed matrix\nsub(A).\nWith uplo = 'L' or 'l', the lower triangular part of the array a is\noverwritten by the lower triangular part of the updated distributed matrix\nsub(A).\np?trmv\nComputes a distributed matrix-vector product using a\ntriangular matrix.\nSyntax\nvoid pstrmv (const char *uplo , const char *trans , const char *diag , const MKL_INT\n*n , const float *a , const MKL_INT *ia , const MKL_INT *ja , const MKL_INT *desca ,\nfloat *x , const MKL_INT *ix , const MKL_INT *jx , const MKL_INT *descx , const MKL_INT\n*incx );\nvoid pdtrmv (const char *uplo , const char *trans , const char *diag , const MKL_INT\n*n , const double *a , const MKL_INT *ia , const MKL_INT *ja , const MKL_INT *desca ,\ndouble *x , const MKL_INT *ix , const MKL_INT *jx , const MKL_INT *descx , const\nMKL_INT *incx );\nvoid pctrmv (const char *uplo , const char *trans , const char *diag , const MKL_INT\n*n , const MKL_Complex8 *a , const MKL_INT *ia , const MKL_INT *ja , const MKL_INT\n*desca , MKL_Complex8 *x , const MKL_INT *ix , const MKL_INT *jx , const MKL_INT\n*descx , const MKL_INT *incx );\nvoid pztrmv (const char *uplo , const char *trans , const char *diag , const MKL_INT\n*n , const MKL_Complex16 *a , const MKL_INT *ia , const MKL_INT *ja , const MKL_INT\n*desca , MKL_Complex16 *x , const MKL_INT *ix , const MKL_INT *jx , const MKL_INT\n*descx , const MKL_INT *incx );\nInclude Files\n•\nmkl_pblas.h\nDescription\nThe p?trmv routines perform one of the following distributed matrix-vector operations defined as\nsub(x) := sub(A)*sub(x), or sub(x) :=sub( A)'*sub(x), or sub(x) := conjg(sub(A)')*sub(x),\nwhere:\nsub(A) is a n-by-n unit, or non-unit, upper or lower triangular distributed matrix, sub(A) = A(ia:ia+n-1,\nja:ja+n-1),\nsub(x) is an n-element distributed vector.\nsub(x) denotes X(ix, jx:jx+n-1) if incx = m_x, and X(ix: ix+n-1, jx) if incx = 1,\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2379\n\n\nInput Parameters\nuplo\n(global) Specifies whether the distributed matrix sub(A) is upper or lower\ntriangular:\nif uplo = 'U' or 'u', then the matrix is upper triangular;\nif uplo = 'L' or 'l', then the matrix is low triangular.\ntrans\n(global) Specifies the form of op(sub(A)) used in the matrix equation:\nif transa = 'N' or 'n', then sub(x) := sub(A)*sub(x);\nif transa = 'T' or 't', then sub(x) :=sub( A)'*sub(x);\nif transa = 'C' or 'c', then sub(x) := conjg(sub(A)')*sub(x).\ndiag\n(global) Specifies whether the matrix sub(A) is unit triangular:\nif diag = 'U' or 'u' then the matrix is unit triangular;\nif diag = 'N' or 'n', then the matrix is not unit triangular.\nn\n(global) Specifies the order of the distributed matrix sub(A), n≥0.\na\n(local)\nArray, size at least (lld_a, LOCq(1, ja+n-1)).\nBefore entry with uplo = 'U' or 'u', this array contains the local entries\ncorresponding to the entries of the upper triangular distributed matrix\nsub(A), and the local entries corresponding to the entries of the strictly\nlower triangular part of the distributed matrix sub(A) is not referenced.\nBefore entry with uplo = 'L' or 'l', this array contains the local entries\ncorresponding to the entries of the lower triangular distributed matrix\nsub(A), and the local entries corresponding to the entries of the strictly\nupper triangular part of the distributed matrix sub(A) is not referenced .\nWhen diag = 'U' or 'u', the local entries corresponding to the diagonal\nelements of the submatrix sub(A) are not referenced either, but are\nassumed to be unity.\nia, ja\n(global) The row and column indices in the distributed matrix A indicating\nthe first row and the first column of the submatrix sub(A), respectively.\ndesca\n(global and local) array of dimension 9. The array descriptor of the\ndistributed matrix A.\nx\n(local)\nArray, size at least (jx-1)*m_x + ix+(n-1)*abs(incx)).\nThis array contains the entries of the distributed vector sub(x).\nix, jx\n(global) The row and column indices in the distributed matrix X indicating\nthe first row and the first column of the submatrix sub(x), respectively.\ndescx\n(global and local) array of dimension 9. The array descriptor of the\ndistributed matrix X.\nincx\n(global) Specifies the increment for the elements of sub(x). Only two\nvalues are supported, namely 1 and m_x. incx must not be zero.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2380\n\n\nOutput Parameters\nx\nOverwritten by the transformed distributed vector sub(x).\np?atrmv\nComputes a distributed matrix-vector product using\nabsolute values for a triangular matrix.\nSyntax\nvoid psatrmv (const char *uplo , const char *trans , const char *diag , const MKL_INT\n*n , const float *alpha , const float *a , const MKL_INT *ia , const MKL_INT *ja ,\nconst MKL_INT *desca , const float *x , const MKL_INT *ix , const MKL_INT *jx , const\nMKL_INT *descx , const MKL_INT *incx , const float *beta , float *y , const MKL_INT\n*iy , const MKL_INT *jy , const MKL_INT *descy , const MKL_INT *incy );\nvoid pdatrmv (const char *uplo , const char *trans , const char *diag , const MKL_INT\n*n , const double *alpha , const double *a , const MKL_INT *ia , const MKL_INT *ja ,\nconst MKL_INT *desca , const double *x , const MKL_INT *ix , const MKL_INT *jx , const\nMKL_INT *descx , const MKL_INT *incx , const double *beta , double *y , const MKL_INT\n*iy , const MKL_INT *jy , const MKL_INT *descy , const MKL_INT *incy );\nvoid pcatrmv (const char *uplo , const char *trans , const char *diag , const MKL_INT\n*n , const MKL_Complex8 *alpha , const MKL_Complex8 *a , const MKL_INT *ia , const\nMKL_INT *ja , const MKL_INT *desca , const MKL_Complex8 *x , const MKL_INT *ix , const\nMKL_INT *jx , const MKL_INT *descx , const MKL_INT *incx , const MKL_Complex8 *beta ,\nMKL_Complex8 *y , const MKL_INT *iy , const MKL_INT *jy , const MKL_INT *descy , const\nMKL_INT *incy );\nvoid pzatrmv (const char *uplo , const char *trans , const char *diag , const MKL_INT\n*n , const MKL_Complex16 *alpha , const MKL_Complex16 *a , const MKL_INT *ia , const\nMKL_INT *ja , const MKL_INT *desca , const MKL_Complex16 *x , const MKL_INT *ix , const\nMKL_INT *jx , const MKL_INT *descx , const MKL_INT *incx , const MKL_Complex16 *beta ,\nMKL_Complex16 *y , const MKL_INT *iy , const MKL_INT *jy , const MKL_INT *descy , const\nMKL_INT *incy );\nInclude Files\n•\nmkl_pblas.h\nDescription\nThe p?atrmv routines perform one of the following distributed matrix-vector operations defined as\nsub(y) := abs(alpha)*abs(sub(A))*abs(sub(x))+ abs(beta*sub(y)), or \nsub(y) := abs(alpha)*abs(sub( A)')*abs(sub(x))+ abs(beta*sub(y)), or \nsub(y) := abs(alpha)*abs(conjg(sub(A)'))*abs(sub(x))+ abs(beta*sub(y)),\nwhere:\nalpha and beta are scalars,\nsub(A) is a n-by-n unit, or non-unit, upper or lower triangular distributed matrix, sub(A) = A(ia:ia+n-1,\nja:ja+n-1),\nsub(x) is an n-element distributed vector.\nsub(x) denotes X(ix, jx:jx+n-1) if incx = m_x, and X(ix: ix+n-1, jx) if incx = 1.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2381\n\n\nInput Parameters\nuplo\n(global) Specifies whether the distributed matrix sub(A) is upper or lower\ntriangular:\nif uplo = 'U' or 'u', then the matrix is upper triangular;\nif uplo = 'L' or 'l', then the matrix is low triangular.\ntrans\n(global) Specifies the form of op(sub(A)) used in the matrix equation:\nif trans = 'N' or 'n', then sub(y) := |alpha|*|sub(A)|*|sub(x)|+|\nbeta*sub(y)|;\nif trans = 'T' or 't', then sub(y) := |alpha|*|sub(A)'|*|sub(x)|\n+|beta*sub(y)|;\nif trans = 'C' or 'c', then sub(y) := |alpha|*|conjg(sub(A)')|*|\nsub(x)|+|beta*sub(y)|.\ndiag\n(global) Specifies whether the matrix sub(A) is unit triangular:\nif diag = 'U' or 'u' then the matrix is unit triangular;\nif diag = 'N' or 'n', then the matrix is not unit triangular.\nn\n(global) Specifies the order of the distributed matrix sub(A), n≥0.\nalpha\n(global)\nSpecifies the scalar alpha.\na\n(local)\nArray, size at least (lld_a, LOCq(1, ja+n-1)).\nBefore entry with uplo = 'U' or 'u', this array contains the local entries\ncorresponding to the entries of the upper triangular distributed matrix\nsub(A), and the local entries corresponding to the entries of the strictly\nlower triangular part of the distributed matrix sub(A) is not referenced.\nBefore entry with uplo = 'L' or 'l', this array contains the local entries\ncorresponding to the entries of the lower triangular distributed matrix\nsub(A), and the local entries corresponding to the entries of the strictly\nupper triangular part of the distributed matrix sub(A) is not referenced.\nWhen diag = 'U' or 'u', the local entries corresponding to the diagonal\nelements of the submatrix sub(A) are not referenced either, but are\nassumed to be unity.\nia, ja\n(global) The row and column indices in the distributed matrix A indicating\nthe first row and the first column of the submatrix sub(A), respectively.\ndesca\n(global and local) array of dimension 9. The array descriptor of the\ndistributed matrix A.\nx\n(local)\nArray, size at least (jx-1)*m_x + ix+(n-1)*abs(incx)).\nThis array contains the entries of the distributed vector sub(x).\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2382\n\n\nix, jx\n(global) The row and column indices in the distributed matrix X indicating\nthe first row and the first column of the submatrix sub(x), respectively.\ndescx\n(global and local) array of dimension 9. The array descriptor of the\ndistributed matrix X.\nincx\n(global) Specifies the increment for the elements of sub(x). Only two\nvalues are supported, namely 1 and m_x. incx must not be zero.\nbeta\n(global)\nSpecifies the scalar beta. When beta is set to zero, then sub(y) need not\nbe set on input.\ny\n(local)\nArray, size (jy-1)*m_y + iy+(m-1)*abs(incy)) when trans = 'N' or\n'n', and (jy-1)*m_y + iy+(n-1)*abs(incy)) otherwise.\nThis array contains the entries of the distributed vector sub(y).\niy, jy\n(global) The row and column indices in the distributed matrix Y indicating\nthe first row and the first column of the submatrix sub(y), respectively.\ndescy\n(global and local) array of dimension 9. The array descriptor of the\ndistributed matrix Y.\nincy\n(global) Specifies the increment for the elements of sub(y). Only two\nvalues are supported, namely 1 and m_y. incy must not be zero.\nOutput Parameters\ny\nOverwritten by the transformed distributed vector sub(y).\np?trsv\nSolves a system of linear equations whose coefficients\nare in a distributed triangular matrix.\nSyntax\nvoid pstrsv (const char *uplo , const char *trans , const char *diag , const MKL_INT\n*n , const float *a , const MKL_INT *ia , const MKL_INT *ja , const MKL_INT *desca ,\nfloat *x , const MKL_INT *ix , const MKL_INT *jx , const MKL_INT *descx , const MKL_INT\n*incx );\nvoid pdtrsv (const char *uplo , const char *trans , const char *diag , const MKL_INT\n*n , const double *a , const MKL_INT *ia , const MKL_INT *ja , const MKL_INT *desca ,\ndouble *x , const MKL_INT *ix , const MKL_INT *jx , const MKL_INT *descx , const\nMKL_INT *incx );\nvoid pctrsv (const char *uplo , const char *trans , const char *diag , const MKL_INT\n*n , const MKL_Complex8 *a , const MKL_INT *ia , const MKL_INT *ja , const MKL_INT\n*desca , MKL_Complex8 *x , const MKL_INT *ix , const MKL_INT *jx , const MKL_INT\n*descx , const MKL_INT *incx );\nvoid pztrsv (const char *uplo , const char *trans , const char *diag , const MKL_INT\n*n , const MKL_Complex16 *a , const MKL_INT *ia , const MKL_INT *ja , const MKL_INT\n*desca , MKL_Complex16 *x , const MKL_INT *ix , const MKL_INT *jx , const MKL_INT\n*descx , const MKL_INT *incx );\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2383\n\n\nInclude Files\n•\nmkl_pblas.h\nDescription\nThe p?trsv routines solve one of the systems of equations:\nsub(A)*sub(x) = b, or sub(A)'*sub(x) = b, or conjg(sub(A)')*sub(x) = b,\nwhere:\nsub(A) is a n-by-n unit, or non-unit, upper or lower triangular distributed matrix, sub(A) = A(ia:ia+n-1,\nja:ja+n-1),\nb and sub(x) are n-element distributed vectors,\nsub(x) denotes X(ix, jx:jx+n-1) if incx = m_x, and X(ix: ix+n-1, jx) if incx = 1,.\nThe routine does not test for singularity or near-singularity. Such tests must be performed before calling this\nroutine.\nInput Parameters\nuplo\n(global) Specifies whether the distributed matrix sub(A) is upper or lower\ntriangular:\nif uplo = 'U' or 'u', then the matrix is upper triangular;\nif uplo = 'L' or 'l', then the matrix is low triangular.\ntrans\n(global) Specifies the form of the system of equations:\nif transa = 'N' or 'n', then sub(A)*sub(x) = b;\nif transa = 'T' or 't', then sub(A)'*sub(x) = b;\nif transa = 'C' or 'c', then conjg(sub(A)')*sub(x) = b.\ndiag\n(global) Specifies whether the matrix sub(A) is unit triangular:\nif diag = 'U' or 'u' then the matrix is unit triangular;\nif diag = 'N' or 'n', then the matrix is not unit triangular.\nn\n(global) Specifies the order of the distributed matrix sub(A), n≥ 0.\na\n(local)\nArray, size at least (lld_a, LOCq(1, ja+n-1)).\nBefore entry with uplo = 'U' or 'u', this array contains the local entries\ncorresponding to the entries of the upper triangular distributed matrix\nsub(A), and the local entries corresponding to the entries of the strictly\nlower triangular part of the distributed matrix sub(A) is not referenced.\nBefore entry with uplo = 'L' or 'l', this array contains the local entries\ncorresponding to the entries of the lower triangular distributed matrix\nsub(A), and the local entries corresponding to the entries of the strictly\nupper triangular part of the distributed matrix sub(A) is not referenced .\nWhen diag = 'U' or 'u', the local entries corresponding to the diagonal\nelements of the submatrix sub(A) are not referenced either, but are\nassumed to be unity.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2384\n\n\nia, ja\n(global) The row and column indices in the distributed matrix A indicating\nthe first row and the first column of the submatrix sub(A), respectively.\ndesca\n(global and local) array of dimension 9. The array descriptor of the\ndistributed matrix A.\nx\n(local)\nArray, size at least (jx-1)*m_x + ix+(n-1)*abs(incx)).\nThis array contains the entries of the distributed vector sub(x). Before\nentry, sub(x) must contain the n-element right-hand side distributed\nvector b.\nix, jx\n(global) The row and column indices in the distributed matrix X indicating\nthe first row and the first column of the submatrix sub(x), respectively.\ndescx\n(global and local) array of dimension 9. The array descriptor of the\ndistributed matrix X.\nincx\n(global) Specifies the increment for the elements of sub(x). Only two\nvalues are supported, namely 1 and m_x. incx must not be zero.\nOutput Parameters\nx\nOverwritten with the solution vector.\nPBLAS Level 3 Routines\nThe PBLAS Level 3 routines perform distributed matrix-matrix operations. Table \"PBLAS Level 3 Routine\nGroups and Their Data Types\" lists the PBLAS Level 3 routine groups and the data types associated with\nthem.\nPBLAS Level 3 Routine Groups and Their Data Types\nRoutine Group\nData Types\nDescription\np?geadd\ns, d, c, z\nDistributed matrix-matrix sum of general matrices\np?tradd\ns, d, c, z\nDistributed matrix-matrix sum of triangular matrices\np?gemm\ns, d, c, z\nDistributed matrix-matrix product of general matrices\np?hemm\nc, z\nDistributed matrix-matrix product, one matrix is Hermitian\np?herk\nc, z\nRank-k update of a distributed Hermitian matrix\np?her2k\nc, z\nRank-2k update of a distributed Hermitian matrix\np?symm\ns, d, c, z\nMatrix-matrix product of distributed symmetric matrices\np?syrk\ns, d, c, z\nRank-k update of a distributed symmetric matrix\np?syr2k\ns, d, c, z\nRank-2k update of a distributed symmetric matrix\np?tran\ns, d\nTransposition of a real distributed matrix\np?tranc\nc, z\nTransposition of a complex distributed matrix (conjugated)\np?tranu\nc, z\nTransposition of a complex distributed matrix\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2385\n\n\nRoutine Group\nData Types\nDescription\np?trmm\ns, d, c, z\nDistributed matrix-matrix product, one matrix is triangular\np?trsm\ns, d, c, z\nSolution of a distributed matrix equation, one matrix is\ntriangular\np?geadd\nPerforms sum operation for two distributed general\nmatrices.\nSyntax\nvoid psgeadd (const char *trans , const MKL_INT *m , const MKL_INT *n , const float\n*alpha , const float *a , const MKL_INT *ia , const MKL_INT *ja , const MKL_INT\n*desca , const float *beta , float *c , const MKL_INT *ic , const MKL_INT *jc , const\nMKL_INT *descc );\nvoid pdgeadd (const char *trans , const MKL_INT *m , const MKL_INT *n , const double\n*alpha , const double *a , const MKL_INT *ia , const MKL_INT *ja , const MKL_INT\n*desca , const double *beta , double *c , const MKL_INT *ic , const MKL_INT *jc , const\nMKL_INT *descc );\nvoid pcgeadd (const char *trans , const MKL_INT *m , const MKL_INT *n , const\nMKL_Complex8 *alpha , const MKL_Complex8 *a , const MKL_INT *ia , const MKL_INT *ja ,\nconst MKL_INT *desca , const MKL_Complex8 *beta , MKL_Complex8 *c , const MKL_INT *ic ,\nconst MKL_INT *jc , const MKL_INT *descc );\nvoid pzgeadd (const char *trans , const MKL_INT *m , const MKL_INT *n , const\nMKL_Complex16 *alpha , const MKL_Complex16 *a , const MKL_INT *ia , const MKL_INT *ja ,\nconst MKL_INT *desca , const MKL_Complex16 *beta , MKL_Complex16 *c , const MKL_INT\n*ic , const MKL_INT *jc , const MKL_INT *descc );\nInclude Files\n•\nmkl_pblas.h\nDescription\nThe p?geadd routines perform sum operation for two distributed general matrices. The operation is defined\nas\nsub(C):=beta*sub(C) + alpha*op(sub(A)),\nwhere:\nop(x) is one of op(x) = x, or op(x) = x',\nalpha and beta are scalars,\nsub(C) is an m-by-n distributed matrix, sub(C)=C(ic:ic+m-1, jc:jc+n-1).\nsub(A) is a distributed matrix, sub(A)=A(ia:ia+n-1, ja:ja+m-1).\nInput Parameters\ntrans\n(global) Specifies the operation:\nif trans = 'N' or 'n', then op(sub(A)) := sub(A);\nif trans = 'T' or 't', then op(sub(A)) := sub(A)';\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2386\n\n\nif trans = 'C' or 'c', then op(sub(A)) := sub(A)'.\nm\n(global) Specifies the number of rows of the distributed matrix sub(C) and\nthe number of columns of the submatrix sub(A), m≥ 0.\nn\n(global) Specifies the number of columns of the distributed matrix sub(C)\nand the number of rows of the submatrix sub(A), n≥ 0.\nalpha\n(global)\nSpecifies the scalar alpha.\na\n(local)\nArray, size (lld_a, LOCq(ja+m-1)). This array contains the local pieces of\nthe distributed matrix sub(A).\nia, ja\n(global) The row and column indices in the distributed matrix A indicating\nthe first row and the first column of the submatrix sub(A), respectively.\ndesca\n(global and local) array of dimension 9. The array descriptor of the\ndistributed matrix A.\nbeta\n(global)\nSpecifies the scalar beta.\nWhen beta is equal to zero, then sub(C) need not be set on input.\nc\n(local)\nArray, size (lld_c, LOCq(jc+n-1)).\nThis array contains the local pieces of the distributed matrix sub(C).\nic, jc\n(global) The row and column indices in the distributed matrix C indicating\nthe first row and the first column of the submatrix sub(C), respectively.\ndescc\n(global and local) array of dimension 9. The array descriptor of the\ndistributed matrix C.\nOutput Parameters\nc\nOverwritten by the updated submatrix.\np?tradd\nPerforms sum operation for two distributed triangular\nmatrices.\nSyntax\nvoid pstradd (const char *uplo , const char *trans , const MKL_INT *m , const MKL_INT\n*n , const float *alpha , const float *a , const MKL_INT *ia , const MKL_INT *ja ,\nconst MKL_INT *desca , const float *beta , float *c , const MKL_INT *ic , const MKL_INT\n*jc , const MKL_INT *descc );\nvoid pdtradd (const char *uplo , const char *trans , const MKL_INT *m , const MKL_INT\n*n , const double *alpha , const double *a , const MKL_INT *ia , const MKL_INT *ja ,\nconst MKL_INT *desca , const double *beta , double *c , const MKL_INT *ic , const\nMKL_INT *jc , const MKL_INT *descc );\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2387\n\n\nvoid pctradd (const char *uplo , const char *trans , const MKL_INT *m , const MKL_INT\n*n , const MKL_Complex8 *alpha , const MKL_Complex8 *a , const MKL_INT *ia , const\nMKL_INT *ja , const MKL_INT *desca , const MKL_Complex8 *beta , MKL_Complex8 *c , const\nMKL_INT *ic , const MKL_INT *jc , const MKL_INT *descc );\nvoid pztradd (const char *uplo , const char *trans , const MKL_INT *m , const MKL_INT\n*n , const MKL_Complex16 *alpha , const MKL_Complex16 *a , const MKL_INT *ia , const\nMKL_INT *ja , const MKL_INT *desca , const MKL_Complex16 *beta , MKL_Complex16 *c ,\nconst MKL_INT *ic , const MKL_INT *jc , const MKL_INT *descc );\nInclude Files\n•\nmkl_pblas.h\nDescription\nThe p?tradd routines perform sum operation for two distributed triangular matrices. The operation is defined\nas\nsub(C):=beta*sub(C) + alpha*op(sub(A)),\nwhere:\nop(x) is one of op(x) = x, or op(x) = x', or op(x) = conjg(x').\nalpha and beta are scalars,\nsub(C) is an m-by-n distributed matrix, sub(C)=C(ic:ic+m-1, jc:jc+n-1).\nsub(A) is a distributed matrix, sub(A)=A(ia:ia+n-1, ja:ja+m-1).\nInput Parameters\nuplo\n(global) Specifies whether the distributed matrix sub(C) is upper or lower\ntriangular:\nif uplo = 'U' or 'u', then the matrix is upper triangular;\nif uplo = 'L' or 'l', then the matrix is low triangular.\ntrans\n(global) Specifies the operation:\nif trans = 'N' or 'n', then op(sub(A)) := sub(A);\nif trans = 'T' or 't', then op(sub(A)) := sub(A)';\nif trans = 'C' or 'c', then op(sub(A)) := conjg(sub(A)').\nm\n(global) Specifies the number of rows of the distributed matrix sub(C) and\nthe number of columns of the submatrix sub(A), m≥ 0.\nn\n(global) Specifies the number of columns of the distributed matrix sub(C)\nand the number of rows of the submatrix sub(A), n≥ 0.\nalpha\n(global)\nSpecifies the scalar alpha.\na\n(local)\nArray, size (lld_a, LOCq(ja+m-1)). This array contains the local pieces of\nthe distributed matrix sub(A).\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2388\n\n\nia, ja\n(global) The row and column indices in the distributed matrix A indicating\nthe first row and the first column of the submatrix sub(A), respectively.\ndesca\n(global and local) array of dimension 9. The array descriptor of the\ndistributed matrix A.\nbeta\n(global)\nSpecifies the scalar beta.\nWhen beta is equal to zero, then sub(C) need not be set on input.\nc\n(local)\nArray, size (lld_c, LOCq(jc+n-1)).\nThis array contains the local pieces of the distributed matrix sub(C).\nic, jc\n(global) The row and column indices in the distributed matrix C indicating\nthe first row and the first column of the submatrix sub(C), respectively.\ndescc\n(global and local) array of dimension 9. The array descriptor of the\ndistributed matrix C.\nOutput Parameters\nc\nOverwritten by the updated submatrix.\np?gemm\nComputes a scalar-matrix-matrix product and adds\nthe result to a scalar-matrix product for distributed\nmatrices.\nSyntax\nvoid psgemm (const char *transa , const char *transb , const MKL_INT *m , const MKL_INT\n*n , const MKL_INT *k , const float *alpha , const float *a , const MKL_INT *ia , const\nMKL_INT *ja , const MKL_INT *desca , const float *b , const MKL_INT *ib , const MKL_INT\n*jb , const MKL_INT *descb , const float *beta , float *c , const MKL_INT *ic , const\nMKL_INT *jc , const MKL_INT *descc );\nvoid pdgemm (const char *transa , const char *transb , const MKL_INT *m , const MKL_INT\n*n , const MKL_INT *k , const double *alpha , const double *a , const MKL_INT *ia ,\nconst MKL_INT *ja , const MKL_INT *desca , const double *b , const MKL_INT *ib , const\nMKL_INT *jb , const MKL_INT *descb , const double *beta , double *c , const MKL_INT\n*ic , const MKL_INT *jc , const MKL_INT *descc );\nvoid pcgemm (const char *transa , const char *transb , const MKL_INT *m , const MKL_INT\n*n , const MKL_INT *k , const MKL_Complex8 *alpha , const MKL_Complex8 *a , const\nMKL_INT *ia , const MKL_INT *ja , const MKL_INT *desca , const MKL_Complex8 *b , const\nMKL_INT *ib , const MKL_INT *jb , const MKL_INT *descb , const MKL_Complex8 *beta ,\nMKL_Complex8 *c , const MKL_INT *ic , const MKL_INT *jc , const MKL_INT *descc );\nvoid pzgemm (const char *transa , const char *transb , const MKL_INT *m , const MKL_INT\n*n , const MKL_INT *k , const MKL_Complex16 *alpha , const MKL_Complex16 *a , const\nMKL_INT *ia , const MKL_INT *ja , const MKL_INT *desca , const MKL_Complex16 *b , const\nMKL_INT *ib , const MKL_INT *jb , const MKL_INT *descb , const MKL_Complex16 *beta ,\nMKL_Complex16 *c , const MKL_INT *ic , const MKL_INT *jc , const MKL_INT *descc );\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2389\n\n\nInclude Files\n•\nmkl_pblas.h\nDescription\nThe p?gemm routines perform a matrix-matrix operation with general distributed matrices. The operation is\ndefined as\nsub(C) := alpha*op(sub(A))*op(sub(B)) + beta*sub(C),\nwhere:\nop(x) is one of op(x) = x, or op(x) = x',\nalpha and beta are scalars,\nsub(A)=A(ia:ia+m-1, ja:ja+k-1), sub(B)=B(ib:ib+k-1, jb:jb+n-1), and sub(C)=C(ic:ic+m-1,\njc:jc+n-1), are distributed matrices.\nInput Parameters\ntransa\n(global) Specifies the form of op(sub(A)) used in the matrix\nmultiplication:\nif transa = 'N' or 'n', then op(sub(A)) = sub(A);\nif transa = 'T' or 't', then op(sub(A)) = sub(A)';\nif transa = 'C' or 'c', then op(sub(A)) = sub(A)'.\ntransb\n(global) Specifies the form of op(sub(B)) used in the matrix multiplication:\nif transb = 'N' or 'n', then op(sub(B)) = sub(B);\nif transb = 'T' or 't', then op(sub(B)) = sub(B)';\nif transb = 'C' or 'c', then op(sub(B)) = sub(B)'.\nm\n(global) Specifies the number of rows of the distributed matrices\nop(sub(A)) and sub(C), m≥ 0.\nn\n(global) Specifies the number of columns of the distributed matrices\nop(sub(B)) and sub(C), n≥ 0.\nThe value of n must be at least zero.\nk\n(global) Specifies the number of columns of the distributed matrix\nop(sub(A)) and the number of rows of the distributed matrix op(sub(B)).\nThe value of k must be greater than or equal to 0.\nalpha\n(global)\nSpecifies the scalar alpha.\nWhen alpha is equal to zero, then the local entries of the arrays a and b\ncorresponding to the entries of the submatrices sub(A) and sub(B)\nrespectively need not be set on input.\na\n(local)\nArray, size lld_a by kla, where kla is LOCc(ja+k-1) when transa =\n'N' or 'n', and is LOCq(ja+m-1) otherwise. Before entry this array must\ncontain the local pieces of the distributed matrix sub(A).\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2390\n\n\nia, ja\n(global) The row and column indices in the distributed matrix A indicating\nthe first row and the first column of the submatrix sub(A), respectively\ndesca\n(global and local) array of dimension 9. The array descriptor of the\ndistributed matrix A.\nb\n(local)\nArray, size lld_b by klb, where klb is LOCc(jb+n-1) when transb =\n'N' or 'n', and is LOCq(jb+k-1) otherwise. Before entry this array must\ncontain the local pieces of the distributed matrix sub(B).\nib, jb\n(global) The row and column indices in the distributed matrix B indicating\nthe first row and the first column of the submatrix sub(B), respectively\ndescb\n(global and local) array of dimension 9. The array descriptor of the\ndistributed matrix B.\nbeta\n(global)\nSpecifies the scalar beta.\nWhen beta is equal to zero, then sub(C) need not be set on input.\nc\n(local)\nArray, size (lld_a, LOCq(jc+n-1)). Before entry this array must contain\nthe local pieces of the distributed matrix sub(C).\nic, jc\n(global) The row and column indices in the distributed matrix C indicating\nthe first row and the first column of the submatrix sub(C), respectively\ndescc\n(global and local) array of dimension 9. The array descriptor of the\ndistributed matrix C.\nOutput Parameters\nc\nOverwritten by the m-by-n distributed matrix\nalpha*op(sub(A))*op(sub(B)) + beta*sub(C).\np?hemm\nPerforms a scalar-matrix-matrix product (one matrix\noperand is Hermitian) and adds the result to a scalar-\nmatrix product.\nSyntax\nvoid pchemm (const char *side , const char *uplo , const MKL_INT *m , const MKL_INT\n*n , const MKL_Complex8 *alpha , const MKL_Complex8 *a , const MKL_INT *ia , const\nMKL_INT *ja , const MKL_INT *desca , const MKL_Complex8 *b , const MKL_INT *ib , const\nMKL_INT *jb , const MKL_INT *descb , const MKL_Complex8 *beta , MKL_Complex8 *c , const\nMKL_INT *ic , const MKL_INT *jc , const MKL_INT *descc );\nvoid pzhemm (const char *side , const char *uplo , const MKL_INT *m , const MKL_INT\n*n , const MKL_Complex16 *alpha , const MKL_Complex16 *a , const MKL_INT *ia , const\nMKL_INT *ja , const MKL_INT *desca , const MKL_Complex16 *b , const MKL_INT *ib , const\nMKL_INT *jb , const MKL_INT *descb , const MKL_Complex16 *beta , MKL_Complex16 *c ,\nconst MKL_INT *ic , const MKL_INT *jc , const MKL_INT *descc );\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2391\n\n\nInclude Files\n•\nmkl_pblas.h\nDescription\nThe p?hemm routines perform a matrix-matrix operation with distributed matrices. The operation is defined as\nsub(C):=alpha*sub(A)*sub(B)+ beta*sub(C),\nor\nsub(C):=alpha*sub(B)*sub(A)+ beta*sub(C),\nwhere:\nalpha and beta are scalars,\nsub(A) is a Hermitian distributed matrix, sub(A)=A(ia:ia+m-1, ja:ja+m-1), if side = 'L', and\nsub(A)=A(ia:ia+n-1, ja:ja+n-1), if side = 'R'.\nsub(B) and sub(C) are m-by-n distributed matrices.\nsub(B)=B(ib:ib+m-1, jb:jb+n-1), sub(C)=C(ic:ic+m-1, jc:jc+n-1).\nInput Parameters\nside\n(global) Specifies whether the Hermitian distributed matrix sub(A) appears\non the left or right in the operation:\nif side = 'L' or 'l', then sub(C) := alpha*sub(A) *sub(B) +\nbeta*sub(C);\nif side = 'R' or 'r', then sub(C) := alpha*sub(B) *sub(A) +\nbeta*sub(C).\nuplo\n(global) Specifies whether the upper or lower triangular part of the\nHermitian distributed matrix sub(A) is used:\nif uplo = 'U' or 'u', then the upper triangular part is used;\nif uplo = 'L' or 'l', then the lower triangular part is used.\nm\n(global) Specifies the number of rows of the distribute submatrix sub(C),\nm≥ 0.\nn\n(global) Specifies the number of columns of the distribute submatrix\nsub(C), n≥ 0.\nalpha\n(global)\nSpecifies the scalar alpha.\na\n(local)\nArray, size (lld_a, LOCq(ja+na-1)).\nBefore entry this array must contain the local pieces of the symmetric\ndistributed matrix sub(A), such that when uplo = 'U' or 'u', the na-by-\nna upper triangular part of the distributed matrix sub(A) must contain the\nupper triangular part of the Hermitian distributed matrix and the strictly\nlower triangular part of sub(A) is not referenced, and when uplo = 'L' or\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2392\n\n\n'l', the na-by-na lower triangular part of the distributed matrix sub(A)\nmust contain the lower triangular part of the Hermitian distributed matrix\nand the strictly upper triangular part of sub(A) is not referenced.\nia, ja\n(global) The row and column indices in the distributed matrix A indicating\nthe first row and the first column of the submatrix sub(A), respectively\ndesca\n(global and local) array of dimension 9. The array descriptor of the\ndistributed matrix A.\nb\n(local)\nArray, size (lld_b, LOCq(jb+n-1) ). Before entry this array must contain\nthe local pieces of the distributed matrix sub(B).\nib, jb\n(global) The row and column indices in the distributed matrix B indicating\nthe first row and the first column of the submatrix sub(B), respectively.\ndescb\n(global and local) array of dimension 9. The array descriptor of the\ndistributed matrix B.\nbeta\n(global)\nSpecifies the scalar beta.\nWhen beta is set to zero, then sub(C) need not be set on input.\nc\n(local)\nArray, size (lld_c, LOCq(jc+n-1)). Before entry this array must contain\nthe local pieces of the distributed matrix sub(C).\nic, jc\n(global) The row and column indices in the distributed matrix C indicating\nthe first row and the first column of the submatrix sub(C), respectively\ndescc\n(global and local) array of dimension 9. The array descriptor of the\ndistributed matrix C.\nOutput Parameters\nc\nOverwritten by the m-by-n updated distributed matrix.\np?herk\nPerforms a rank-k update of a distributed Hermitian\nmatrix.\nSyntax\nvoid pcherk (const char *uplo , const char *trans , const MKL_INT *n , const MKL_INT\n*k , const float *alpha , const MKL_Complex8 *a , const MKL_INT *ia , const MKL_INT\n*ja , const MKL_INT *desca , const float *beta , MKL_Complex8 *c , const MKL_INT *ic ,\nconst MKL_INT *jc , const MKL_INT *descc );\nvoid pzherk (const char *uplo , const char *trans , const MKL_INT *n , const MKL_INT\n*k , const double *alpha , const MKL_Complex16 *a , const MKL_INT *ia , const MKL_INT\n*ja , const MKL_INT *desca , const double *beta , MKL_Complex16 *c , const MKL_INT\n*ic , const MKL_INT *jc , const MKL_INT *descc );\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2393\n\n\nInclude Files\n•\nmkl_pblas.h\nDescription\nThe p?herk routines perform a distributed matrix-matrix operation defined as\nsub(C):=alpha*sub(A)*conjg(sub(A)')+ beta*sub(C),\nor\nsub(C):=alpha*conjg(sub(A)')*sub(A)+ beta*sub(C),\nwhere:\nalpha and beta are scalars,\nsub(C) is an n-by-n Hermitian distributed matrix, sub(C)=C(ic:ic+n-1, jc:jc+n-1).\nsub(A) is a distributed matrix, sub(A)=A(ia:ia+n-1, ja:ja+k-1), if trans = 'N' or 'n', and\nsub(A)=A(ia:ia+k-1, ja:ja+n-1) otherwise.\nInput Parameters\nuplo\n(global) Specifies whether the upper or lower triangular part of the\nHermitian distributed matrix sub(C) is used:\nIf uplo = 'U' or 'u', then the upper triangular part of the sub(C) is\nused.\nIf uplo = 'L' or 'l', then the low triangular part of the sub(C) is used.\ntrans\n(global) Specifies the operation:\nif trans = 'N' or 'n', then sub(C) := alpha*sub(A)*conjg(sub(A)')\n+ beta*sub(C);\nif trans = 'C' or 'c', then sub(C) := alpha*conjg(sub(A)')*sub(A)\n+ beta*sub(C).\nn\n(global) Specifies the order of the distributed matrix sub(C), n≥ 0.\nk\n(global) On entry with trans = 'N' or 'n', k specifies the number of\ncolumns of the distributed matrix sub(A) , and on entry with trans = 'T'\nor 't' or 'C' or 'c', k specifies the number of rows of the distributed\nmatrix sub(A), k≥ 0.\nalpha\n(global)\nSpecifies the scalar alpha.\na\n(local)\nArray, size (lld_a, kla), where kla is LOCq(ja+k-1) when trans = 'N'\nor 'n', and is LOCq(ja+n-1) otherwise. Before entry with trans = 'N' or\n'n', this array contains the local pieces of the distributed matrix sub(A).\nia, ja\n(global) The row and column indices in the distributed matrix A indicating\nthe first row and the first column of the submatrix sub(A), respectively.\ndesca\n(global and local) array of dimension 9. The array descriptor of the\ndistributed matrix A.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2394\n\n\nbeta\n(global)\nSpecifies the scalar beta.\nc\n(local)\nArray, size (lld_c, LOCq(jc+n-1)).\nBefore entry with uplo = 'U' or 'u', this array contains n-by-n upper\ntriangular part of the symmetric distributed matrix sub(C) and its strictly\nlower triangular part is not referenced.\nBefore entry with uplo = 'L' or 'l', this array contains n-by-n lower\ntriangular part of the symmetric distributed matrix sub(C) and its strictly\nupper triangular part is not referenced.\nic, jc\n(global) The row and column indices in the distributed matrix C indicating\nthe first row and the first column of the submatrix sub(C), respectively.\ndescc\n(global and local) array of dimension 9. The array descriptor of the\ndistributed matrix C.\nOutput Parameters\nc\nWith uplo = 'U' or 'u', the upper triangular part of sub(C) is overwritten\nby the upper triangular part of the updated distributed matrix.\nWith uplo = 'L' or 'l', the lower triangular part of sub(C) is overwritten\nby the upper triangular part of the updated distributed matrix.\np?her2k\nPerforms a rank-2k update of a Hermitian distributed\nmatrix.\nSyntax\nvoid pcher2k (const char *uplo , const char *trans , const MKL_INT *n , const MKL_INT\n*k , const MKL_Complex8 *alpha , const MKL_Complex8 *a , const MKL_INT *ia , const\nMKL_INT *ja , const MKL_INT *desca , const MKL_Complex8 *b , const MKL_INT *ib , const\nMKL_INT *jb , const MKL_INT *descb , const float *beta , MKL_Complex8 *c , const\nMKL_INT *ic , const MKL_INT *jc , const MKL_INT *descc );\nvoid pzher2k (const char *uplo , const char *trans , const MKL_INT *n , const MKL_INT\n*k , const MKL_Complex16 *alpha , const MKL_Complex16 *a , const MKL_INT *ia , const\nMKL_INT *ja , const MKL_INT *desca , const MKL_Complex16 *b , const MKL_INT *ib , const\nMKL_INT *jb , const MKL_INT *descb , const double *beta , MKL_Complex16 *c , const\nMKL_INT *ic , const MKL_INT *jc , const MKL_INT *descc );\nInclude Files\n•\nmkl_pblas.h\nDescription\nThe p?her2k routines perform a distributed matrix-matrix operation defined as\nsub(C):=alpha*sub(A)*conjg(sub(B)')+ conjg(alpha)*sub(B)*conjg(sub(A)')+beta*sub(C),\nor\nsub(C):=alpha*conjg(sub(A)')*sub(A)+ conjg(alpha)*conjg(sub(B)')*sub(A) + beta*sub(C),\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2395\n\n\nwhere:\nalpha and beta are scalars,\nsub(C) is an n-by-n Hermitian distributed matrix, sub(C) = C(ic:ic+n-1, jc:jc+n-1).\nsub(A) is a distributed matrix, sub(A)=A(ia:ia+n-1, ja:ja+k-1), if trans = 'N' or 'n', and sub(A) =\nA(ia:ia+k-1, ja:ja+n-1) otherwise.\nsub(B) is a distributed matrix, sub(B) = B(ib:ib+n-1, jb:jb+k-1), if trans = 'N' or 'n', and\nsub(B)=B(ib:ib+k-1, jb:jb+n-1) otherwise.\nInput Parameters\nuplo\n(global) Specifies whether the upper or lower triangular part of the\nHermitian distributed matrix sub(C) is used:\nIf uplo = 'U' or 'u', then the upper triangular part of the sub(C) is\nused.\nIf uplo = 'L' or 'l', then the low triangular part of the sub(C) is used.\ntrans\n(global) Specifies the operation:\nif trans = 'N' or 'n', then sub(C) := alpha*sub(A)*conjg(sub(B)')\n+ conjg(alpha)*sub(B)*conjg(sub(A)') + beta*sub(C);\nif trans = 'C' or 'c', then sub(C) := alpha*conjg(sub(A)')*sub(A)\n+ conjg(alpha)*conjg(sub(B)')*sub(A) + beta*sub(C).\nn\n(global) Specifies the order of the distributed matrix sub(C), n≥ 0.\nk\n(global) On entry with trans = 'N' or 'n', k specifies the number of\ncolumns of the distributed matrices sub(A) and sub(B), and on entry with\ntrans = 'C' or 'c' , k specifies the number of rows of the distributed\nmatrices sub(A) and sub(B), k≥ 0.\nalpha\n(global)\nSpecifies the scalar alpha.\na\n(local)\nArray, size (lld_a, kla), where kla is LOCq(ja+k-1) when trans = 'N'\nor 'n', and is LOCq(ja+n-1) otherwise. Before entry with trans = 'N' or\n'n', this array contains the local pieces of the distributed matrix sub(A).\nia, ja\n(global) The row and column indices in the distributed matrix A indicating\nthe first row and the first column of the submatrix sub(A), respectively.\ndesca\n(global and local) array of dimension 9. The array descriptor of the\ndistributed matrix A.\nb\n(local)\nArray, size (lld_b, klb), where klb is LOCq(jb+k-1) when trans = 'N'\nor 'n', and is LOCq(jb+n-1) otherwise. Before entry with trans = 'N' or\n'n', this array contains the local pieces of the distributed matrix sub(B).\nib, jb\n(global) The row and column indices in the distributed matrix B indicating\nthe first row and the first column of the submatrix sub(B), respectively.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2396\n\n\ndescb\n(global and local) array of dimension 9. The array descriptor of the\ndistributed matrix B.\nbeta\n(global)\nSpecifies the scalar beta.\nc\n(local)\nArray, size (lld_c, LOCq(jc+n-1)).\nBefore entry with uplo = 'U' or 'u', this array contains n-by-n upper\ntriangular part of the symmetric distributed matrix sub(C) and its strictly\nlower triangular part is not referenced.\nBefore entry with uplo = 'L' or 'l', this array contains n-by-n lower\ntriangular part of the symmetric distributed matrix sub(C) and its strictly\nupper triangular part is not referenced.\nic, jc\n(global) The row and column indices in the distributed matrix C indicating\nthe first row and the first column of the submatrix sub(C), respectively.\ndescc\n(global and local) array of dimension 9. The array descriptor of the\ndistributed matrix C.\nOutput Parameters\nc\nWith uplo = 'U' or 'u', the upper triangular part of sub(C) is overwritten\nby the upper triangular part of the updated distributed matrix.\nWith uplo = 'L' or 'l', the lower triangular part of sub(C) is overwritten\nby the upper triangular part of the updated distributed matrix.\np?symm\nPerforms a scalar-matrix-matrix product (one matrix\noperand is symmetric) and adds the result to a scalar-\nmatrix product for distribute matrices.\nSyntax\nvoid pssymm (const char *side , const char *uplo , const MKL_INT *m , const MKL_INT\n*n , const float *alpha , const float *a , const MKL_INT *ia , const MKL_INT *ja ,\nconst MKL_INT *desca , const float *b , const MKL_INT *ib , const MKL_INT *jb , const\nMKL_INT *descb , const float *beta , float *c , const MKL_INT *ic , const MKL_INT *jc ,\nconst MKL_INT *descc );\nvoid pdsymm (const char *side , const char *uplo , const MKL_INT *m , const MKL_INT\n*n , const double *alpha , const double *a , const MKL_INT *ia , const MKL_INT *ja ,\nconst MKL_INT *desca , const double *b , const MKL_INT *ib , const MKL_INT *jb , const\nMKL_INT *descb , const double *beta , double *c , const MKL_INT *ic , const MKL_INT\n*jc , const MKL_INT *descc );\nvoid pcsymm (const char *side , const char *uplo , const MKL_INT *m , const MKL_INT\n*n , const MKL_Complex8 *alpha , const MKL_Complex8 *a , const MKL_INT *ia , const\nMKL_INT *ja , const MKL_INT *desca , const MKL_Complex8 *b , const MKL_INT *ib , const\nMKL_INT *jb , const MKL_INT *descb , const MKL_Complex8 *beta , MKL_Complex8 *c , const\nMKL_INT *ic , const MKL_INT *jc , const MKL_INT *descc );\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2397\n\n\nvoid pzsymm (const char *side , const char *uplo , const MKL_INT *m , const MKL_INT\n*n , const MKL_Complex16 *alpha , const MKL_Complex16 *a , const MKL_INT *ia , const\nMKL_INT *ja , const MKL_INT *desca , const MKL_Complex16 *b , const MKL_INT *ib , const\nMKL_INT *jb , const MKL_INT *descb , const MKL_Complex16 *beta , MKL_Complex16 *c ,\nconst MKL_INT *ic , const MKL_INT *jc , const MKL_INT *descc );\nInclude Files\n•\nmkl_pblas.h\nDescription\nThe p?symm routines perform a matrix-matrix operation with distributed matrices. The operation is defined as\nsub(C):=alpha*sub(A)*sub(B)+ beta*sub(C),\nor\nsub(C):=alpha*sub(B)*sub(A)+ beta*sub(C),\nwhere:\nalpha and beta are scalars,\nsub(A) is a symmetric distributed matrix, sub(A)=A(ia:ia+m-1, ja:ja+m-1), if side ='L', and\nsub(A)=A(ia:ia+n-1, ja:ja+n-1), if side ='R'.\nsub(B) and sub(C) are m-by-n distributed matrices.\nsub(B)=B(ib:ib+m-1, jb:jb+n-1), sub(C)=C(ic:ic+m-1, jc:jc+n-1).\nInput Parameters\nside\n(global) Specifies whether the symmetric distributed matrix sub(A)\nappears on the left or right in the operation:\nif side = 'L' or 'l', then sub(C) := alpha*sub(A) *sub(B) +\nbeta*sub(C);\nif side = 'R' or 'r', then sub(C) := alpha*sub(B) *sub(A) +\nbeta*sub(C).\nuplo\n(global) Specifies whether the upper or lower triangular part of the\nsymmetric distributed matrix sub(A) is used:\nif uplo = 'U' or 'u', then the upper triangular part is used;\nif uplo = 'L' or 'l', then the lower triangular part is used.\nm\n(global) Specifies the number of rows of the distribute submatrix sub(C),\nm≥ 0.\nn\n(global) Specifies the number of columns of the distribute submatrix\nsub(C), m≥ 0.\nalpha\n(global)\nSpecifies the scalar alpha.\na\n(local)\nArray, size (lld_a, LOCq(ja+na-1)).\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2398\n\n\nBefore entry this array must contain the local pieces of the symmetric\ndistributed matrix sub(A), such that when uplo = 'U' or 'u', the na-by-\nna upper triangular part of the distributed matrix sub(A) must contain the\nupper triangular part of the symmetric distributed matrix and the strictly\nlower triangular part of sub(A) is not referenced, and when uplo = 'L' or\n'l', the na-by-na lower triangular part of the distributed matrix sub(A)\nmust contain the lower triangular part of the symmetric distributed matrix\nand the strictly upper triangular part of sub(A) is not referenced.\nia, ja\n(global) The row and column indices in the distributed matrix A indicating\nthe first row and the first column of the submatrix sub(A), respectively.\ndesca\n(global and local) array of dimension 9. The array descriptor of the\ndistributed matrix A.\nb\n(local)\nArray, size (lld_b, LOCq(jb+n-1) ). Before entry this array must contain\nthe local pieces of the distributed matrix sub(B).\nib, jb\n(global) The row and column indices in the distributed matrix B indicating\nthe first row and the first column of the submatrix sub(B), respectively.\ndescb\n(global and local) array of dimension 9. The array descriptor of the\ndistributed matrix B.\nbeta\n(global)\nSpecifies the scalar beta.\nWhen beta is set to zero, then sub(C) need not be set on input.\nc\n(local)\nArray, size (lld_c, LOCq(jc+n-1) ). Before entry this array must contain\nthe local pieces of the distributed matrix sub(C).\nic, jc\n(global) The row and column indices in the distributed matrix C indicating\nthe first row and the first column of the submatrix sub(C), respectively.\ndescc\n(global and local) array of dimension 9. The array descriptor of the\ndistributed matrix C.\nOutput Parameters\nc\nOverwritten by the m-by-n updated matrix.\np?syrk\nPerforms a rank-k update of a symmetric distributed\nmatrix.\nSyntax\nvoid pssyrk (const char *uplo , const char *trans , const MKL_INT *n , const MKL_INT\n*k , const float *alpha , const float *a , const MKL_INT *ia , const MKL_INT *ja ,\nconst MKL_INT *desca , const float *beta , float *c , const MKL_INT *ic , const MKL_INT\n*jc , const MKL_INT *descc );\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2399\n\n\nvoid pdsyrk (const char *uplo , const char *trans , const MKL_INT *n , const MKL_INT\n*k , const double *alpha , const double *a , const MKL_INT *ia , const MKL_INT *ja ,\nconst MKL_INT *desca , const double *beta , double *c , const MKL_INT *ic , const\nMKL_INT *jc , const MKL_INT *descc );\nvoid pcsyrk (const char *uplo , const char *trans , const MKL_INT *n , const MKL_INT\n*k , const MKL_Complex8 *alpha , const MKL_Complex8 *a , const MKL_INT *ia , const\nMKL_INT *ja , const MKL_INT *desca , const MKL_Complex8 *beta , MKL_Complex8 *c , const\nMKL_INT *ic , const MKL_INT *jc , const MKL_INT *descc );\nvoid pzsyrk (const char *uplo , const char *trans , const MKL_INT *n , const MKL_INT\n*k , const MKL_Complex16 *alpha , const MKL_Complex16 *a , const MKL_INT *ia , const\nMKL_INT *ja , const MKL_INT *desca , const MKL_Complex16 *beta , MKL_Complex16 *c ,\nconst MKL_INT *ic , const MKL_INT *jc , const MKL_INT *descc );\nInclude Files\n•\nmkl_pblas.h\nDescription\nThe p?syrk routines perform a distributed matrix-matrix operation defined as\nsub(C):=alpha*sub(A)*sub(A)'+ beta*sub(C),\nor\nsub(C):=alpha*sub(A)'*sub(A)+ beta*sub(C),\nwhere:\nalpha and beta are scalars,\nsub(C) is an n-by-n symmetric distributed matrix, sub(C)=C(ic:ic+n-1, jc:jc+n-1).\nsub(A) is a distributed matrix, sub(A)=A(ia:ia+n-1, ja:ja+k-1), if trans = 'N' or 'n', and\nsub(A)=A(ia:ia+k-1, ja:ja+n-1) otherwise.\nInput Parameters\nuplo\n(global) Specifies whether the upper or lower triangular part of the\nsymmetric distributed matrix sub(C) is used:\nIf uplo = 'U' or 'u', then the upper triangular part of the sub(C) is\nused.\nIf uplo = 'L' or 'l', then the low triangular part of the sub(C) is used.\ntrans\n(global) Specifies the operation:\nif trans = 'N' or 'n', then sub(C) := alpha*sub(A)*sub(A)' +\nbeta*sub(C);\nif trans = 'T' or 't', then sub(C) := alpha*sub(A)'*sub(A) +\nbeta*sub(C).\nn\n(global) Specifies the order of the distributed matrix sub(C), n≥ 0.\nk\n(global) On entry with trans = 'N' or 'n', k specifies the number of\ncolumns of the distributed matrix sub(A) , and on entry with trans = 'T'\nor 't' , k specifies the number of rows of the distributed matrix sub(A), k≥\n0.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2400\n\n\nalpha\n(global)\nSpecifies the scalar alpha.\na\n(local)\nArray, size (lld_a, kla), where kla is LOCq(ja+k-1) when trans = 'N'\nor 'n', and is LOCq(ja+n-1) otherwise. Before entry with trans = 'N' or\n'n', this array contains the local pieces of the distributed matrix sub(A).\nia, ja\n(global) The row and column indices in the distributed matrix A indicating\nthe first row and the first column of the submatrix sub(A), respectively.\ndesca\n(global and local) array of dimension 9. The array descriptor of the\ndistributed matrix A.\nbeta\n(global)\nSpecifies the scalar beta.\nc\n(local)\nArray, size (lld_c, LOCq(jc+n-1)).\nBefore entry with uplo = 'U' or 'u', this array contains n-by-n upper\ntriangular part of the symmetric distributed matrix sub(C) and its strictly\nlower triangular part is not referenced.\nBefore entry with uplo = 'L' or 'l', this array contains n-by-n lower\ntriangular part of the symmetric distributed matrix sub(C) and its strictly\nupper triangular part is not referenced.\nic, jc\n(global) The row and column indices in the distributed matrix C indicating\nthe first row and the first column of the submatrix sub(C), respectively.\ndescc\n(global and local) array of dimension 9. The array descriptor of the\ndistributed matrix C.\nOutput Parameters\nc\nWith uplo = 'U' or 'u', the upper triangular part of sub(C) is overwritten\nby the upper triangular part of the updated distributed matrix.\nWith uplo = 'L' or 'l', the lower triangular part of sub(C) is overwritten\nby the upper triangular part of the updated distributed matrix.\np?syr2k\nPerforms a rank-2k update of a symmetric distributed\nmatrix.\nSyntax\nvoid pssyr2k (const char *uplo , const char *trans , const MKL_INT *n , const MKL_INT\n*k , const float *alpha , const float *a , const MKL_INT *ia , const MKL_INT *ja ,\nconst MKL_INT *desca , const float *b , const MKL_INT *ib , const MKL_INT *jb , const\nMKL_INT *descb , const float *beta , float *c , const MKL_INT *ic , const MKL_INT *jc ,\nconst MKL_INT *descc );\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2401\n\n\nvoid pdsyr2k (const char *uplo , const char *trans , const MKL_INT *n , const MKL_INT\n*k , const double *alpha , const double *a , const MKL_INT *ia , const MKL_INT *ja ,\nconst MKL_INT *desca , const double *b , const MKL_INT *ib , const MKL_INT *jb , const\nMKL_INT *descb , const double *beta , double *c , const MKL_INT *ic , const MKL_INT\n*jc , const MKL_INT *descc );\nvoid pcsyr2k (const char *uplo , const char *trans , const MKL_INT *n , const MKL_INT\n*k , const MKL_Complex8 *alpha , const MKL_Complex8 *a , const MKL_INT *ia , const\nMKL_INT *ja , const MKL_INT *desca , const MKL_Complex8 *b , const MKL_INT *ib , const\nMKL_INT *jb , const MKL_INT *descb , const MKL_Complex8 *beta , MKL_Complex8 *c , const\nMKL_INT *ic , const MKL_INT *jc , const MKL_INT *descc );\nvoid pzsyr2k (const char *uplo , const char *trans , const MKL_INT *n , const MKL_INT\n*k , const MKL_Complex16 *alpha , const MKL_Complex16 *a , const MKL_INT *ia , const\nMKL_INT *ja , const MKL_INT *desca , const MKL_Complex16 *b , const MKL_INT *ib , const\nMKL_INT *jb , const MKL_INT *descb , const MKL_Complex16 *beta , MKL_Complex16 *c ,\nconst MKL_INT *ic , const MKL_INT *jc , const MKL_INT *descc );\nInclude Files\n•\nmkl_pblas.h\nDescription\nThe p?syr2k routines perform a distributed matrix-matrix operation defined as\nsub(C):=alpha*sub(A)*sub(B)'+alpha*sub(B)*sub(A)'+ beta*sub(C),\nor\nsub(C):=alpha*sub(A)'*sub(B) +alpha*sub(B)'*sub(A) + beta*sub(C),\nwhere:\nalpha and beta are scalars,\nsub(C) is an n-by-n symmetric distributed matrix, sub(C)=C(ic:ic+n-1, jc:jc+n-1).\nsub(A) is a distributed matrix, sub(A)=A(ia:ia+n-1, ja:ja+k-1), if trans = 'N' or 'n', and\nsub(A)=A(ia:ia+k-1, ja:ja+n-1) otherwise.\nsub(B) is a distributed matrix, sub(B)=B(ib:ib+n-1, jb:jb+k-1), if trans = 'N' or 'n', and\nsub(B)=B(ib:ib+k-1, jb:jb+n-1) otherwise.\nInput Parameters\nuplo\n(global) Specifies whether the upper or lower triangular part of the\nsymmetric distributed matrix sub(C) is used:\nIf uplo = 'U' or 'u', then the upper triangular part of the sub(C) is\nused.\nIf uplo = 'L' or 'l', then the low triangular part of the sub(C) is used.\ntrans\n(global) Specifies the operation:\nif trans = 'N' or 'n', then sub(C) := alpha*sub(A)*sub(B)' +\nalpha*sub(B)*sub(A)' + beta*sub(C);\nif trans = 'T' or 't', then sub(C) := alpha*sub(B)'*sub(A) +\nalpha*sub(A)'*sub(B) + beta*sub(C).\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2402\n\n\nn\n(global) Specifies the order of the distributed matrix sub(C), n≥ 0.\nk\n(global) On entry with trans = 'N' or 'n', k specifies the number of\ncolumns of the distributed matrices sub(A) and sub(B), and on entry with\ntrans = 'T' or 't' , k specifies the number of rows of the distributed\nmatrices sub(A) and sub(B), k≥ 0.\nalpha\n(global)\nSpecifies the scalar alpha.\na\n(local)\nArray, size (lld_a, kla), where kla is LOCq(ja+k-1) when trans = 'N'\nor 'n', and is LOCq(ja+n-1) otherwise. Before entry with trans = 'N' or\n'n', this array contains the local pieces of the distributed matrix sub(A).\nia, ja\n(global) The row and column indices in the distributed matrix A indicating\nthe first row and the first column of the submatrix sub(A), respectively.\ndesca\n(global and local) array of dimension 9. The array descriptor of the\ndistributed matrix A.\nb\n(local)\nArray, size (lld_b, klb), where klb is LOCq(jb+k-1) when trans = 'N'\nor 'n', and is LOCq(jb+n-1) otherwise. Before entry with trans = 'N' or\n'n', this array contains the local pieces of the distributed matrix sub(B).\nib, jb\n(global) The row and column indices in the distributed matrix B indicating\nthe first row and the first column of the submatrix sub(B), respectively.\ndescb\n(global and local) array of dimension 9. The array descriptor of the\ndistributed matrix B.\nbeta\n(global)\nSpecifies the scalar beta.\nc\n(local)\nArray, size (lld_c, LOCq(jc+n-1)).\nBefore entry with uplo = 'U' or 'u', this array contains n-by-n upper\ntriangular part of the symmetric distributed matrix sub(C) and its strictly\nlower triangular part is not referenced.\nBefore entry with uplo = 'L' or 'l', this array contains n-by-n lower\ntriangular part of the symmetric distributed matrix sub(C) and its strictly\nupper triangular part is not referenced.\nic, jc\n(global) The row and column indices in the distributed matrix C indicating\nthe first row and the first column of the submatrix sub(C), respectively.\ndescc\n(global and local) array of dimension 9. The array descriptor of the\ndistributed matrix C.\nOutput Parameters\nc\nWith uplo = 'U' or 'u', the upper triangular part of sub(C) is overwritten\nby the upper triangular part of the updated distributed matrix.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2403\n\n\nWith uplo = 'L' or 'l', the lower triangular part of sub(C) is overwritten\nby the upper triangular part of the updated distributed matrix.\np?tran\nTransposes a real distributed matrix.\nSyntax\nvoid pstran (const MKL_INT *m , const MKL_INT *n , const float *alpha , const float\n*a , const MKL_INT *ia , const MKL_INT *ja , const MKL_INT *desca , const float *beta ,\nfloat *c , const MKL_INT *ic , const MKL_INT *jc , const MKL_INT *descc );\nvoid pdtran (const MKL_INT *m , const MKL_INT *n , const double *alpha , const double\n*a , const MKL_INT *ia , const MKL_INT *ja , const MKL_INT *desca , const double\n*beta , double *c , const MKL_INT *ic , const MKL_INT *jc , const MKL_INT *descc );\nInclude Files\n•\nmkl_pblas.h\nDescription\nThe p?tran routines transpose a real distributed matrix. The operation is defined as\nsub(C):=beta*sub(C) + alpha*sub(A)',\nwhere:\nalpha and beta are scalars,\nsub(C) is an m-by-n distributed matrix, sub(C)=C(ic:ic+m-1, jc:jc+n-1).\nsub(A) is a distributed matrix, sub(A)=A(ia:ia+n-1, ja:ja+m-1).\nInput Parameters\nm\n(global) Specifies the number of rows of the distributed matrix sub(C), m≥\n0.\nn\n(global) Specifies the number of columns of the distributed matrix sub(C) ,\nn≥ 0.\nalpha\n(global)\nSpecifies the scalar alpha.\na\n(local)\nArray, size (lld_a, LOCq(ja+m-1)). This array contains the local pieces of\nthe distributed matrix sub(A).\nia, ja\n(global) The row and column indices in the distributed matrix A indicating\nthe first row and the first column of the submatrix sub(A), respectively.\ndesca\n(global and local) array of dimension 9. The array descriptor of the\ndistributed matrix A.\nbeta\n(global)\nSpecifies the scalar beta.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2404\n\n\nWhen beta is equal to zero, then sub(C) need not be set on input.\nc\n(local)\nArray, size (lld_c, LOCq(jc+n-1)).\nThis array contains the local pieces of the distributed matrix sub(C).\nic, jc\n(global) The row and column indices in the distributed matrix C indicating\nthe first row and the first column of the submatrix sub(C), respectively.\ndescc\n(global and local) array of dimension 9. The array descriptor of the\ndistributed matrix C.\nOutput Parameters\nc\nOverwritten by the updated submatrix.\np?tranu\nTransposes a distributed complex matrix.\nSyntax\nvoid pctranu (const MKL_INT *m , const MKL_INT *n , const MKL_Complex8 *alpha , const\nMKL_Complex8 *a , const MKL_INT *ia , const MKL_INT *ja , const MKL_INT *desca , const\nMKL_Complex8 *beta , MKL_Complex8 *c , const MKL_INT *ic , const MKL_INT *jc , const\nMKL_INT *descc );\nvoid pztranu (const MKL_INT *m , const MKL_INT *n , const MKL_Complex16 *alpha , const\nMKL_Complex16 *a , const MKL_INT *ia , const MKL_INT *ja , const MKL_INT *desca , const\nMKL_Complex16 *beta , MKL_Complex16 *c , const MKL_INT *ic , const MKL_INT *jc , const\nMKL_INT *descc );\nInclude Files\n•\nmkl_pblas.h\nDescription\nThe p?tranu routines transpose a complex distributed matrix. The operation is defined as\nsub(C):=beta*sub(C) + alpha*sub(A)',\nwhere:\nalpha and beta are scalars,\nsub(C) is an m-by-n distributed matrix, sub(C)=C(ic:ic+m-1, jc:jc+n-1).\nsub(A) is a distributed matrix, sub(A)=A(ia:ia+n-1, ja:ja+m-1).\nInput Parameters\nm\n(global) Specifies the number of rows of the distributed matrix sub(C), m≥\n0.\nn\n(global) Specifies the number of columns of the distributed matrix sub(C) ,\nn≥ 0.\nalpha\n(global)\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2405\n\n\nSpecifies the scalar alpha.\na\n(local)\nArray, size (lld_a, LOCq(ja+m-1)). This array contains the local pieces of\nthe distributed matrix sub(A).\nia, ja\n(global) The row and column indices in the distributed matrix A indicating\nthe first row and the first column of the submatrix sub(A), respectively.\ndesca\n(global and local) array of dimension 9. The array descriptor of the\ndistributed matrix A.\nbeta\n(global)\nSpecifies the scalar beta.\nWhen beta is equal to zero, then sub(C) need not be set on input.\nc\n(local)\nArray, size (lld_c, LOCq(jc+n-1)).\nThis array contains the local pieces of the distributed matrix sub(C).\nic, jc\n(global) The row and column indices in the distributed matrix C indicating\nthe first row and the first column of the submatrix sub(C), respectively.\ndescc\n(global and local) array of dimension 9. The array descriptor of the\ndistributed matrix C.\nOutput Parameters\nc\nOverwritten by the updated submatrix.\np?tranc\nTransposes a complex distributed matrix, conjugated.\nSyntax\nvoid pctranc (const MKL_INT *m , const MKL_INT *n , const MKL_Complex8 *alpha , const\nMKL_Complex8 *a , const MKL_INT *ia , const MKL_INT *ja , const MKL_INT *desca , const\nMKL_Complex8 *beta , MKL_Complex8 *c , const MKL_INT *ic , const MKL_INT *jc , const\nMKL_INT *descc );\nvoid pztranc (const MKL_INT *m , const MKL_INT *n , const MKL_Complex16 *alpha , const\nMKL_Complex16 *a , const MKL_INT *ia , const MKL_INT *ja , const MKL_INT *desca , const\nMKL_Complex16 *beta , MKL_Complex16 *c , const MKL_INT *ic , const MKL_INT *jc , const\nMKL_INT *descc );\nInclude Files\n•\nmkl_pblas.h\nDescription\nThe p?tranc routines transpose a complex distributed matrix. The operation is defined as\nsub(C):=beta*sub(C) + alpha*conjg(sub(A)'),\nwhere:\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2406\n\n\nalpha and beta are scalars,\nsub(C) is an m-by-n distributed matrix, sub(C)=C(ic:ic+m-1, jc:jc+n-1).\nsub(A) is a distributed matrix, sub(A)=A(ia:ia+n-1, ja:ja+m-1).\nInput Parameters\nm\n(global) Specifies the number of rows of the distributed matrix sub(C), m≥\n0.\nn\n(global) Specifies the number of columns of the distributed matrix sub(C) ,\nn≥ 0.\nalpha\n(global)\nSpecifies the scalar alpha.\na\n(local)\nArray, size (lld_a, LOCq(ja+m-1)). This array contains the local pieces of\nthe distributed matrix sub(A).\nia, ja\n(global) The row and column indices in the distributed matrix A indicating\nthe first row and the first column of the submatrix sub(A), respectively.\ndesca\n(global and local) array of dimension 9. The array descriptor of the\ndistributed matrix A.\nbeta\n(global)\nSpecifies the scalar beta.\nWhen beta is equal to zero, then sub(C) need not be set on input.\nc\n(local)\nArray, size (lld_c, LOCq(jc+n-1)).\nThis array contains the local pieces of the distributed matrix sub(C).\nic, jc\n(global) The row and column indices in the distributed matrix C indicating\nthe first row and the first column of the submatrix sub(C), respectively.\ndescc\n(global and local) array of dimension 9. The array descriptor of the\ndistributed matrix C.\nOutput Parameters\nc\nOverwritten by the updated submatrix.\np?trmm\nComputes a scalar-matrix-matrix product (one matrix\noperand is triangular) for distributed matrices.\nSyntax\nvoid pstrmm (const char *side , const char *uplo , const char *transa , const char\n*diag , const MKL_INT *m , const MKL_INT *n , const float *alpha , const float *a ,\nconst MKL_INT *ia , const MKL_INT *ja , const MKL_INT *desca , float *b , const MKL_INT\n*ib , const MKL_INT *jb , const MKL_INT *descb );\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2407\n\n\nvoid pdtrmm (const char *side , const char *uplo , const char *transa , const char\n*diag , const MKL_INT *m , const MKL_INT *n , const double *alpha , const double *a ,\nconst MKL_INT *ia , const MKL_INT *ja , const MKL_INT *desca , double *b , const\nMKL_INT *ib , const MKL_INT *jb , const MKL_INT *descb );\nvoid pctrmm (const char *side , const char *uplo , const char *transa , const char\n*diag , const MKL_INT *m , const MKL_INT *n , const MKL_Complex8 *alpha , const\nMKL_Complex8 *a , const MKL_INT *ia , const MKL_INT *ja , const MKL_INT *desca ,\nMKL_Complex8 *b , const MKL_INT *ib , const MKL_INT *jb , const MKL_INT *descb );\nvoid pztrmm (const char *side , const char *uplo , const char *transa , const char\n*diag , const MKL_INT *m , const MKL_INT *n , const MKL_Complex16 *alpha , const\nMKL_Complex16 *a , const MKL_INT *ia , const MKL_INT *ja , const MKL_INT *desca ,\nMKL_Complex16 *b , const MKL_INT *ib , const MKL_INT *jb , const MKL_INT *descb );\nInclude Files\n•\nmkl_pblas.h\nDescription\nThe p?trmm routines perform a matrix-matrix operation using triangular matrices. The operation is defined as\nsub(B) := alpha*op(sub(A))*sub(B)\nor\nsub(B) := alpha*sub(B)*op(sub(A))\nwhere:\nalpha is a scalar,\nsub(B) is an m-by-n distributed matrix, sub(B)=B(ib:ib+m-1, jb:jb+n-1).\nA is a unit, or non-unit, upper or lower triangular distributed matrix, sub(A)=A(ia:ia+m-1, ja:ja+m-1), if\nside = 'L' or 'l', and sub(A)=A(ia:ia+n-1, ja:ja+n-1), if side = 'R' or 'r'.\nop(sub(A)) is one of op(sub(A)) = sub(A), or op(sub(A)) = sub(A)', or op(sub(A)) =\nconjg(sub(A)').\nInput Parameters\nside\n(global) Specifies whether op(sub(A)) appears on the left or right of\nsub(B) in the operation:\nif side = 'L' or 'l', then sub(B) := alpha*op(sub(A))*sub(B);\nif side = 'R' or 'r', then sub(B) := alpha*sub(B)*op(sub(A)).\nuplo\n(global) Specifies whether the distributed matrix sub(A) is upper or lower\ntriangular:\nif uplo = 'U' or 'u', then the matrix is upper triangular;\nif uplo = 'L' or 'l', then the matrix is low triangular.\ntransa\n(global) Specifies the form of op(sub(A)) used in the matrix multiplication:\nif transa = 'N' or 'n', then op(sub(A)) = sub(A);\nif transa = 'T' or 't', then op(sub(A)) = sub(A)' ;\nif transa = 'C' or 'c', then op(sub(A)) = conjg(sub(A)').\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2408\n\n\ndiag\n(global) Specifies whether the matrix sub(A) is unit triangular:\nif diag = 'U' or 'u' then the matrix is unit triangular;\nif diag = 'N' or 'n', then the matrix is not unit triangular.\nm\n(global) Specifies the number of rows of the distributed matrix sub(B), m≥\n0.\nn\n(global) Specifies the number of columns of the distributed matrix sub(B),\nn≥ 0.\nalpha\n(global)\nSpecifies the scalar alpha.\nWhen alpha is zero, then the arrayb need not be set before entry.\na\n(local)\nArray, size lld_a by ka, where ka is at least LOCq(1, ja+m-1) when side\n= 'L' or 'l' and is at least LOCq(1, ja+n-1) when side = 'R' or 'r'.\nBefore entry with uplo = 'U' or 'u', this array contains the local entries\ncorresponding to the entries of the upper triangular distributed matrix\nsub(A), and the local entries corresponding to the entries of the strictly\nlower triangular part of the distributed matrix sub(A) is not referenced.\nBefore entry with uplo = 'L' or 'l', this array contains the local entries\ncorresponding to the entries of the lower triangular distributed matrix\nsub(A), and the local entries corresponding to the entries of the strictly\nupper triangular part of the distributed matrix sub(A) is not referenced .\nWhen diag = 'U' or 'u', the local entries corresponding to the diagonal\nelements of the submatrix sub(A) are not referenced either, but are\nassumed to be unity.\nia, ja\n(global) The row and column indices in the distributed matrix A indicating\nthe first row and the first column of the submatrix sub(A), respectively.\ndesca\n(global and local) array of dimension 9. The array descriptor of the\ndistributed matrix A.\nb\n(local)\nArray, size (lld_b, LOCq(1, jb+n-1)).\nBefore entry, this array contains the local pieces of the distributed matrix\nsub(B).\nib, jb\n(global) The row and column indices in the distributed matrix B indicating\nthe first row and the first column of the submatrix sub(B), respectively.\ndescb\n(global and local) array of dimension 9. The array descriptor of the\ndistributed matrix B.\nOutput Parameters\nb\nOverwritten by the transformed distributed matrix.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2409\n\n\np?trsm\nSolves a distributed matrix equation (one matrix\noperand is triangular).\nSyntax\nvoid pstrsm (const char *side , const char *uplo , const char *transa , const char\n*diag , const MKL_INT *m , const MKL_INT *n , const float *alpha , const float *a ,\nconst MKL_INT *ia , const MKL_INT *ja , const MKL_INT *desca , float *b , const MKL_INT\n*ib , const MKL_INT *jb , const MKL_INT *descb );\nvoid pdtrsm (const char *side , const char *uplo , const char *transa , const char\n*diag , const MKL_INT *m , const MKL_INT *n , const double *alpha , const double *a ,\nconst MKL_INT *ia , const MKL_INT *ja , const MKL_INT *desca , double *b , const\nMKL_INT *ib , const MKL_INT *jb , const MKL_INT *descb );\nvoid pctrsm (const char *side , const char *uplo , const char *transa , const char\n*diag , const MKL_INT *m , const MKL_INT *n , const MKL_Complex8 *alpha , const\nMKL_Complex8 *a , const MKL_INT *ia , const MKL_INT *ja , const MKL_INT *desca ,\nMKL_Complex8 *b , const MKL_INT *ib , const MKL_INT *jb , const MKL_INT *descb );\nvoid pztrsm (const char *side , const char *uplo , const char *transa , const char\n*diag , const MKL_INT *m , const MKL_INT *n , const MKL_Complex16 *alpha , const\nMKL_Complex16 *a , const MKL_INT *ia , const MKL_INT *ja , const MKL_INT *desca ,\nMKL_Complex16 *b , const MKL_INT *ib , const MKL_INT *jb , const MKL_INT *descb );\nInclude Files\n•\nmkl_pblas.h\nDescription\nThe p?trsm routines solve one of the following distributed matrix equations:\nop(sub(A))*X = alpha*sub(B),\nor\nX*op(sub(A)) = alpha*sub(B),\nwhere:\nalpha is a scalar,\nX and sub(B) are m-by-n distributed matrices, sub(B)=B(ib:ib+m-1, jb:jb+n-1);\nA is a unit, or non-unit, upper or lower triangular distributed matrix, sub(A)=A(ia:ia+m-1, ja:ja+m-1), if\nside = 'L' or 'l', and sub(A)=A(ia:ia+n-1, ja:ja+n-1), if side = 'R' or 'r';\nop(sub(A)) is one of op(sub(A)) = sub(A), or op(sub(A)) = sub(A)', or op(sub(A)) =\nconjg(sub(A)').\nThe distributed matrix sub(B) is overwritten by the solution matrix X.\nInput Parameters\nside\n(global) Specifies whether op(sub(A)) appears on the left or right of X in\nthe equation:\nif side = 'L' or 'l', then op(sub(A))*X = alpha*sub(B);\nif side = 'R' or 'r', then X*op(sub(A)) = alpha*sub(B).\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2410\n\n\nuplo\n(global) Specifies whether the distributed matrix sub(A) is upper or lower\ntriangular:\nif uplo = 'U' or 'u', then the matrix is upper triangular;\nif uplo = 'L' or 'l', then the matrix is low triangular.\ntransa\n(global) Specifies the form of op(sub(A)) used in the matrix equation:\nif transa = 'N' or 'n', then op(sub(A)) = sub(A);\nif transa = 'T' or 't', then op(sub(A)) = sub(A)';\nif transa = 'C' or 'c', then op(sub(A)) = conjg(sub(A)').\ndiag\n(global) Specifies whether the matrix sub(A) is unit triangular:\nif diag = 'U' or 'u' then the matrix is unit triangular;\nif diag = 'N' or 'n', then the matrix is not unit triangular.\nm\n(global) Specifies the number of rows of the distributed matrix sub(B), m≥\n0.\nn\n(global) Specifies the number of columns of the distributed matrix sub(B),\nn≥ 0.\nalpha\n(global)\nSpecifies the scalar alpha.\nWhen alpha is zero, then a is not referenced and b need not be set before\nentry.\na\n(local)\nArray, size lld_a by ka, where ka is at least LOCq(1, ja+m-1) when side\n= 'L' or 'l' and is at least LOCq(1, ja+n-1) when side = 'R' or 'r'.\nBefore entry with uplo = 'U' or 'u', this array contains the local entries\ncorresponding to the entries of the upper triangular distributed matrix\nsub(A), and the local entries corresponding to the entries of the strictly\nlower triangular part of the distributed matrix sub(A) is not referenced.\nBefore entry with uplo = 'L' or 'l', this array contains the local entries\ncorresponding to the entries of the lower triangular distributed matrix\nsub(A), and the local entries corresponding to the entries of the strictly\nupper triangular part of the distributed matrix sub(A) is not referenced .\nWhen diag = 'U' or 'u', the local entries corresponding to the diagonal\nelements of the submatrix sub(A) are not referenced either, but are\nassumed to be unity.\nia, ja\n(global) The row and column indices in the distributed matrix A indicating\nthe first row and the first column of the submatrix sub(A), respectively.\ndesca\n(global and local) array of dimension 9. The array descriptor of the\ndistributed matrix A.\nb\n(local)\nArray, size (lld_b, LOCq(1, jb+n-1)).\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2411\n\n\nBefore entry, this array contains the local pieces of the distributed matrix\nsub(B).\nib, jb\n(global) The row and column indices in the distributed matrix B indicating\nthe first row and the first column of the submatrix sub(B), respectively.\ndescb\n(global and local) array of dimension 9. The array descriptor of the\ndistributed matrix B.\nOutput Parameters\nb\nOverwritten by the solution distributed matrix X.\nPartial Differential Equations Support\nThe Intel® oneAPI Math Kernel Library (oneMKL) provides tools for solving Partial Differential Equations\n(PDE). These tools are Trigonometric Transform interface routines (seeTrigonometric Transform Routines) and\nPoisson Solver (see Fast Poisson Solver Routines).\nPoisson Solver is designed for fast solving of simple Helmholtz, Poisson, and Laplace problems. The solver is\nbased on the Trigonometric Transform interface, which is, in turn, based on the Intel® oneAPI Math Kernel\nLibrary (oneMKL) Fast Fourier Transform (FFT) interface (refer toFourier Transform Functions), optimized for\nIntel® processors.\nDirect use of the Trigonometric Transform routines may be helpful to those who have already implemented\ntheir own solvers similar to the Intel® oneAPI Math Kernel Library (oneMKL) Poisson Solver. As it may be hard\nenough to modify the original code so as to make it work with Poisson Solver, you are encouraged to use fast\n(staggered) sine/cosine transforms implemented in the Trigonometric Transform interface to improve\nperformance of your solver.\nBoth Trigonometric Transform and Poisson Solver routines can be called from C and Fortran.\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nTrigonometric Transform Routines\nIn addition to the Fast Fourier Transform (FFT) interface, described in Fast Fourier Transforms, Intel® oneAPI\nMath Kernel Library (oneMKL) supports theReal Discrete Trigonometric Transforms (sometimes called real-to-\nreal Discrete Fourier Transforms) interface. In this document, the interface is referred to as TT interface. It\nimplements a group of routines (TT routines) used to compute sine/cosine, staggered sine/cosine, and twice\nstaggered sine/cosine transforms (referred to as staggered2 sine/cosine transforms, for brevity). The TT\ninterface provides much flexibility of use: you can adjust routines to your particular needs at the cost of\nmanually tuning routine parameters or just call routines with default parameter values. The current Intel®\noneAPI Math Kernel Library (oneMKL) implementation of the TT interface can be used in solving partial\ndifferential equations and contains routines that are helpful for Fast Poisson and similar solvers.\nFor the list of Trigonometric Transforms currently implemented in Intel® oneAPI Math Kernel Library (oneMKL)\nTT interface, seeTransforms Implemented.\nIf you have got used to the FFTW interface (www.fftw.org), you can call the TT interface functions through\nreal-to-real FFTW to Intel® oneAPI Math Kernel Library (oneMKL) wrappers without changing FFTW function\ncalls in your code (refer toFFTW to Intel® MKL Wrappers for FFTW 3.x for details). However, you are strongly\nencouraged to use the native TT interface for better performance. Another reason why you should use the\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2412\n\n\nwrappers cautiously is that TT and the real-to-real FFTW interfaces are not fully compatible and some\nfeatures of the real-to-real FFTW, such as strides and multidimensional transforms, are not available through\nwrappers.\nTrigonometric Transforms Implemented\nTT routines allow computing the following transforms:\nForward sine transform\nF(k) = 2\nn ∑\ni = 1\nn −1\nf (i) sin kiπ\nn , k = 1, ..., n −1\nBackward sine transform\nf (i) = ∑\nk = 1\nn −1\nF(k) sin kiπ\nn , i = 1, ..., n −1\nForward staggered sine transform\nF(k) = 1\nnsin 2k −1 π\n2\nf (n) + 2\nn ∑\ni = 1\nn −1\nf (i) sin 2k −1 iπ\n2n\n, k = 1, ..., n\nBackward staggered sine transform\nf (i) = ∑\nk = 1\nn\nF(k) sin 2k −1 iπ\n2n\n, i = 1, ..., n\nForward staggered2 sine transform\nF(k) = 2\nn ∑\ni = 1\nn\nf (i) sin 2k −1 2i −1 π\n4n\n, k = 1, ..., n\nBackward staggered2 sine transform\nf (i) = ∑\nk = 1\nn\nF(k) sin 2k −1 2i −1 π\n4n\n, i = 1, ..., n\nForward cosine transform\nF(k) = 1\nn[f (o) + f (n)cos kπ] + 2\nn ∑\ni = 1\nn −1\nf (i)coskiπ\nn , k = 0, ..., n\nBackward cosine transform\nf (i) = 1\n2[F(o) + F(n)cos iπ] + ∑\nk = 1\nn −1\nF(k)coskiπ\nn , i = 0, ..., n\nForward staggered cosine transform\nF(k) = 1\nn f (0) + 2\nn ∑\ni = 1\nn −1\nf (i) cos 2k + 1 iπ\n2n\n, k = 0, ..., n −1\nBackward staggered cosine transform\nf (i) = ∑\nk = 0\nn −1\nF(k) cos 2k + 1 iπ\n2n\n, i = 0, ..., n −1\nForward staggered2 cosine transform\nF(k) = 2\nn ∑\ni = 1\nn\nf (i) cos 2k −1 2i −1 π\n4n\n, k = 1, ..., n\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2413\n\n\nBackward staggered2 cosine transform\nf (i) = ∑\nk = 1\nn\nF(k) cos 2k −1 2i −1 π\n4n\n, i = 1, ..., n\nNOTE\nThe size of the transform n can be any integer greater or equal to 2.\nSequence of Invoking TT Routines\nComputation of a transform using TT interface is conceptually divided into four steps, each of which is\nperformed via a dedicated routine. Table \"TT Interface Routines\" lists the routines and briefly describes their\npurpose and use.\nMost TT routines have versions operating with single-precision and double-precision data. Names of such\nroutines begin respectively with \"s\" and \"d\". The wildcard \"?\" stands for either of these symbols in routine\nnames.\nTT Interface Routines\nRoutine\nDescription\n?_init_trig_transform\nInitializes basic data structures of Trigonometric\nTransforms.\n?_commit_trig_transform\nChecks consistency and correctness of user-defined data\nand creates a data structure to be used by Intel® oneAPI\nMath Kernel Library (oneMKL) FFT interface1.\n?_forward_trig_transform\n?_backward_trig_transform\nComputes a forward/backward Trigonometric Transform of\na specified type using the appropriate formula (see \nTransforms Implemented).\nfree_trig_transform\nReleases the memory used by a data structure needed for\ncalling FFT interface1.\n1TT routines call Intel® oneAPI Math Kernel Library (oneMKL) FFT interface for better performance.\nTo find a transformed vector for a particular input vector only once, the Intel® oneAPI Math Kernel Library\n(oneMKL) TT interface routines are normally invoked in the order in which they are listed inTable \"TT\nInterface Routines\".\nNOTE\nThough the order of invoking TT routines may be changed, it is highly recommended to follow the\nabove order of routine calls.\nThe diagram in Figure \"Typical Order of Invoking TT Interface Routines\" indicates the typical order in which\nTT interface routines can be invoked in a general case (prefixes and suffixes in routine names are omitted).\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2414\n\n\n__border__top\nTypical Order of Invoking TT Interface Routines\nA general scheme of using TT routines for double-precision computations is shown below. A similar scheme\nholds for single-precision computations with the only difference in the initial letter of routine names.\n...\n    d_init_trig_transform(&n, &tt_type, ipar, dpar, &ir);\n/* Change parameters in ipar if necessary. */\n/* Note that the result of the Transform will be in f. If you want to preserve the data stored \nin f,\nsave it to another location before the function call below */\n    d_commit_trig_transform(f, &handle, ipar, dpar, &ir);\n    d_forward_trig_transform(f, &handle, ipar, dpar, &ir);\n    d_backward_trig_transform(f, &handle, ipar, dpar, &ir);\n    free_trig_transform(&handle, ipar, &ir);\n/* here the user may clean the memory used by f, dpar, ipar */\n...\nYou can find examples of code that uses TT interface routines to solve one-dimensional Helmholtz problem in\nthe examples\\pdettc\\source folderin your Intel® oneAPI Math Kernel Library (oneMKL) directory.\nTrigonometric Transform Interface Description\nAll types in this documentation are either standard C types float and double or MKL_INT integer type. For\nmore information on the C types, refer to C Datatypes Specific to oneMKL and the Intel® oneAPI Math Kernel\nLibrary (oneMKL) Developer Guide. To better understand usage of the types, see examples in the examples\n\\pdettc\\source folderin your Intel® oneAPI Math Kernel Library (oneMKL) directory.\nRoutine Options\nAll TT routines use parameters to pass various options to one another. These parameters are arrays ipar,\ndpar and spar. Values for these parameters should be specified very carefully (see Common Parameters).\nYou can change these values during computations to meet your needs.\nWARNING\nTo avoid failure or incorrect results, you must provide correct and consistent parameters to the\nroutines.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2415\n\n\nUser Data Arrays\nTT routines take arrays of user data as input. For example, user arrays are passed to the routine\nd_forward_trig_transformto compute a forward Trigonometric Transform. To minimize storage\nrequirements and improve the overall run-time efficiency, Intel® oneAPI Math Kernel Library (oneMKL) TT\nroutines do not make copies of user input arrays.\nNOTE\nIf you need a copy of your input data arrays, you must save them yourself.\nFor better performance, align your data arrays as recommended in the Intel® oneAPI Math Kernel Library\n(oneMKL) Developer Guide (search the document for coding techniques to improve performance).\nTT Routines\nThe section gives detailed description of TT routines, their syntax, parameters and values they return.\nDouble-precision and single-precision versions of the same routine are described together.\nTT routines call Intel® oneAPI Math Kernel Library (oneMKL) FFT interface (described inFFT Functions), which\nenhances performance of the routines.\n?_init_trig_transform\nInitializes basic data structures of a Trigonometric\nTransform.\nSyntax\nvoid d_init_trig_transform(MKL_INT *n, MKL_INT *tt_type, MKL_INT ipar[], double dpar[],\nMKL_INT *stat);\nvoid s_init_trig_transform(MKL_INT *n, MKL_INT *tt_type, MKL_INT ipar[], float spar[],\nMKL_INT *stat);\nInclude Files\n•\nmkl.h\nInput Parameters\nn\nMKL_INT*. Contains the size of the problem, which should be a\npositive integer greater than 1. Note that data vector of the transform,\nwhich other TT routines will use, must have size n+1 for all but\nstaggered2 transforms. Staggered2 transforms require the vector of\nsize n.\ntt_type\nMKL_INT*. Contains the type of transform to compute, defined via a\nset of named constants. The following constants are available in the\ncurrent implementation of TT interface: MKL_SINE_TRANSFORM,\nMKL_STAGGERED_SINE_TRANSFORM,\nMKL_STAGGERED2_SINE_TRANSFORM; MKL_COSINE_TRANSFORM,\nMKL_STAGGERED_COSINE_TRANSFORM,\nMKL_STAGGERED2_COSINE_TRANSFORM.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2416\n\n\nOutput Parameters\nipar\nMKL_INT array of size 128. Contains integer data needed for\nTrigonometric Transform computations.\ndpar\ndouble array of size 5n/2+2. Contains double-precision data needed\nfor Trigonometric Transform computations.\nspar\nfloat array of size 5n/2+2. Contains single-precision data needed for\nTrigonometric Transform computations.\nstat\nMKL_INT*. Contains the routine completion status, which is also\nwritten to ipar[6]. The status should be 0 to proceed to other TT\nroutines.\nDescription\nThe ?_init_trig_transform routine initializes basic data structures for Trigonometric Transforms of\nappropriate precision. After a call to ?_init_trig_transform, all subsequently invoked TT routines use\nvalues of ipar and dpar (spar) array parameters returned by ?_init_trig_transform. The routine\ninitializes the entire array ipar. In the dpar or spar array, ?_init_trig_transform initializes elements\nthat do not depend upon the type of transform. For a detailed description of arrays ipar, dpar and spar,\nrefer to Common Parameters. You can skip a call to the initialization routine in your code. For more\ninformation, see Caveat on Parameter Modifications.\nReturn Values\nstat= 0\nThe routine successfully completed the task. In\ngeneral, to proceed with computations, the routine\nshould complete with this stat value.\nstat= -99999\nThe routine failed to complete the task.\n?_commit_trig_transform\nChecks consistency and correctness of user's data as\nwell as initializes certain data structures required to\nperform the Trigonometric Transform.\nSyntax\nvoid d_commit_trig_transform(double f[], DFTI_DESCRIPTOR_HANDLE *handle, MKL_INT\nipar[], double dpar[], MKL_INT *stat);\nvoid s_commit_trig_transform(float f[], DFTI_DESCRIPTOR_HANDLE *handle, MKL_INT ipar[],\nfloat spar[], MKL_INT *stat);\nInclude Files\n•\nmkl.h\nInput Parameters\nf\ndouble for d_commit_trig_transform,\nfloat for s_commit_trig_transform,\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2417\n\n\narray of size n for staggered2 transforms and of size n+1 for all other\ntransforms, where n is the size of the problem. Contains data vector\nto be transformed. Note that the following values should be 0.0 up to\nrounding errors:\n•\nf[0] and f[n] for sine transforms\n•\nf[n] for staggered cosine transforms\n•\nf[0] for staggered sine transforms.\nOtherwise, the routine will produce a warning, and the result of the\ncomputations for sine transforms may be wrong. These restrictions\nmeet the requirements of the Intel® oneAPI Math Kernel Library\n(oneMKL) Poisson Solver, which the TT interface is primarily designed\nfor (for details, seeFast Poisson Solver Routines).\nipar\nMKL_INT array of size 128. Contains integer data needed for\nTrigonometric Transform computations.\ndpar\ndouble array of size 5n/2+2. Contains double-precision data needed\nfor Trigonometric Transform computations. The routine initializes most\nelements of this array.\nspar\nfloat array of size 5n/2+2. Contains single-precision data needed for\nTrigonometric Transform computations. The routine initializes most\nelements of this array.\nOutput Parameters\nhandle\nDFTI_DESCRIPTOR_HANDLE*. The data structure used by Intel®\noneAPI Math Kernel Library (oneMKL) FFT interface (for details, refer\ntoFFT Functions).\nipar\nContains integer data needed for Trigonometric Transform\ncomputations. On output, ipar[6] is updated with the stat value.\ndpar\nContains double-precision data needed for Trigonometric Transform\ncomputations. On output, the entire array is initialized.\nspar\nContains single-precision data needed for Trigonometric Transform\ncomputations. On output, the entire array is initialized.\nstat\nMKL_INT*. Contains the routine completion status, which is also\nwritten to ipar[6].\nDescription\nThe routine ?_commit_trig_transform checks consistency and correctness of the parameters to be passed\nto the transform routines ?_forward_trig_transform and/or ?_backward_trig_transform. The routine\nalso initializes the following data structures: handle, dpar in case of d_commit_trig_transform, and spar\nin case of s_commit_trig_transform. The ?_commit_trig_transform routine initializes only those\nelements of dpar or spar that depend upon the type of transform, defined in the ?_init_trig_transform\nroutine and passed to ?_commit_trig_transform with the ipar array. The size of the problem n, which\ndetermines sizes of the array parameters, is also passed to the routine with the ipar array and defined in\nthe previously called ?_init_trig_transform routine. For a detailed description of arrays ipar, dpar and\nspar, refer to Common Parameters. The routine performs only a basic check for correctness and consistency\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2418\n\n\nof the parameters. If you are going to modify parameters of TT routines, see Caveat on Parameter\nModifications. Unlike ?_init_trig_transform, you must call the ?_commit_trig_transform routine in\nyour code.\nReturn Values\nstat= 11\nThe routine produced some warnings and made some\nchanges in the parameters to achieve their correctness\nand/or consistency. You may proceed with computations by\nassigning ipar[6]=0 if you are sure that the parameters\nare correct.\nstat= 10\nThe routine made some changes in the parameters to\nachieve their correctness and/or consistency. You may\nproceed with computations by assigning ipar[6]=0 if you\nare sure that the parameters are correct.\nstat= 1\nThe routine produced some warnings. You may proceed\nwith computations by assigning ipar[6]=0 if you are sure\nthat the parameters are correct.\nstat= 0\nThe routine completed the task normally.\nstat= -100\nThe routine stopped for any of the following reasons:\n•\nAn error in the user's data was encountered.\n•\nData in ipar, dpar or spar parameters became\nincorrect and/or inconsistent as a result of\nmodifications.\nstat= -1000\nThe routine stopped because of an FFT interface error.\nstat= -10000\nThe routine stopped because the initialization failed to\ncomplete or the parameter ipar[0] was altered by\nmistake.\nNOTE\nAlthough positive values of stat usually indicate minor problems with the input data and\nTrigonometric Transform computations can be continued, you are highly recommended to\ninvestigate the problem first and achieve stat=0.\n?_forward_trig_transform\nComputes the forward Trigonometric Transform of\ntype specified by the parameter.\nSyntax\nvoid d_forward_trig_transform(double f[], DFTI_DESCRIPTOR_HANDLE *handle, MKL_INT\nipar[], double dpar[], MKL_INT *stat);\nvoid s_forward_trig_transform(float f[], DFTI_DESCRIPTOR_HANDLE *handle, MKL_INT\nipar[], float spar[], MKL_INT *stat);\nInclude Files\n•\nmkl.h\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2419\n\n\nInput Parameters\nf\ndouble for d_forward_trig_transform,\nfloat for s_forward_trig_transform,\narray of size n for staggered2 transforms and of size n+1 for all other\ntransforms, where n is the size of the problem. On input, contains\ndata vector to be transformed. Note that the following values should\nbe 0.0 up to rounding errors:\n•\nf[0] and f[n] for sine transforms\n•\nf[n] for staggered cosine transforms\n•\nf[0] for staggered sine transforms.\nOtherwise, the routine will produce a warning, and the result of the\ncomputations for sine transforms may be wrong. The above\nrestrictions meet the requirements of the Intel® oneAPI Math Kernel\nLibrary (oneMKL) Poisson Solver, which the TT interface is primarily\ndesigned for (for details, seeFast Poisson Solver Routines).\nhandle\nDFTI_DESCRIPTOR_HANDLE*. The data structure used by Intel®\noneAPI Math Kernel Library (oneMKL) FFT interface (for details,\nseeFFT Functions).\nipar\nMKL_INT array of size 128. Contains integer data needed for\nTrigonometric Transform computations.\ndpar\ndouble array of size 5n/2+2. Contains double-precision data needed\nfor Trigonometric Transform computations.\nspar\nfloat array of size 5n/2+2. Contains single-precision data needed for\nTrigonometric Transform computations.\nOutput Parameters\nf\nContains the transformed vector on output.\nipar\nContains integer data needed for Trigonometric Transform\ncomputations. On output, ipar[6] is updated with the stat value.\nstat\nMKL_INT*. Contains the routine completion status, which is also\nwritten to ipar[6].\nDescription\nThe routine computes the forward Trigonometric Transform of type defined in the ?_init_trig_transform\nroutine and passed to ?_forward_trig_transform with the ipar array. The size of the problem n, which\ndetermines sizes of the array parameters, is also passed to the routine with the ipar array and defined in\nthe previously called ?_init_trig_transform routine. The other data that facilitates the computation is\ncreated by ?_commit_trig_transform and supplied in dpar or spar. For a detailed description of arrays\nipar, dpar and spar, refer to Common Parameters. The routine has a commit step, which calls\nthe ?_commit_trig_transform routine. The transform is computed according to formulas given in \nTransforms Implemented. The routine replaces the input vector f with the transformed vector.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2420\n\n\nNOTE\nIf you need a copy of the data vector f to be transformed, make the copy before calling\nthe ?_forward_trig_transform routine.\nReturn Values\nstat= 0\nThe routine completed the task normally.\nstat= -100\nThe routine stopped for any of the following reasons:\n•\nAn error in the user's data was encountered.\n•\nData in ipar, dpar or spar parameters became\nincorrect and/or inconsistent as a result of\nmodifications.\nstat= -1000\nThe routine stopped because of an FFT interface error.\nstat= -10000\nThe routine stopped because its commit step failed to\ncomplete or the parameter ipar[0] was altered by\nmistake.\n?_backward_trig_transform\nComputes the backward Trigonometric Transform of\ntype specified by the parameter.\nSyntax\nvoid d_backward_trig_transform(double f[], DFTI_DESCRIPTOR_HANDLE *handle, MKL_INT\nipar[], double dpar[], MKL_INT *stat);\nvoid s_backward_trig_transform(float f[], DFTI_DESCRIPTOR_HANDLE *handle, MKL_INT\nipar[], float spar[], MKL_INT *stat);\nInclude Files\n•\nmkl.h\nInput Parameters\nf\ndouble for d_backward_trig_transform,\nfloat for s_backward_trig_transform,\narray of size n for staggered2 transforms and of size n+1 for all other\ntransforms, where n is the size of the problem. On input, contains\ndata vector to be transformed. Note that the following values should\nbe 0.0 up to rounding errors:\n•\nf[0] and f[n] for sine transforms\n•\nf[n] for staggered cosine transforms\n•\nf[0] for staggered sine transforms.\nOtherwise, the routine will produce a warning, and the result of the\ncomputations for sine transforms may be wrong. The above\nrestrictions meet the requirements of the Intel® oneAPI Math Kernel\nLibrary (oneMKL) Poisson Solver, which the TT interface is primarily\ndesigned for (for details, seeFast Poisson Solver Routines).\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2421\n\n\nhandle\nDFTI_DESCRIPTOR_HANDLE*. The data structure used by Intel®\noneAPI Math Kernel Library (oneMKL) FFT interface (for details,\nseeFFT Functions).\nipar\nMKL_INT array of size 128. Contains integer data needed for\nTrigonometric Transform computations.\ndpar\ndouble array of size 5n/2+2. Contains double-precision data needed\nfor Trigonometric Transform computations.\nspar\nfloat array of size 5n/2+2. Contains single-precision data needed for\nTrigonometric Transform computations.\nOutput Parameters\nf\nContains the transformed vector on output.\nipar\nContains integer data needed for Trigonometric Transform\ncomputations. On output, ipar[6] is updated with the stat value.\nstat\nMKL_INT*. Contains the routine completion status, which is also\nwritten to ipar[6].\nDescription\nThe routine computes the backward Trigonometric Transform of type defined in\nthe ?_init_trig_transform routine and passed to ?_backward_trig_transform with the ipar array.\nThe size of the problem n, which determines sizes of the array parameters, is also passed to the routine with\nthe ipar array and defined in the previously called ?_init_trig_transform routine. The other data that\nfacilitates the computation is created by ?_commit_trig_transform and supplied in dpar or spar. For a\ndetailed description of arrays ipar, dpar and spar, refer to Common Parameters. The routine has a commit\nstep, which calls the ?_commit_trig_transform routine. The transform is computed according to formulas\ngiven in Transforms Implemented. The routine replaces the input vector f with the transformed vector.\nNOTE\nIf you need a copy of the data vector f to be transformed, make the copy before calling\nthe ?_backward_trig_transform routine.\nReturn Values\nstat= 0\nThe routine completed the task normally.\nstat= -100\nThe routine stopped for any of the following reasons:\n•\nAn error in the user's data was encountered.\n•\nData in ipar, dpar or spar parameters became\nincorrect and/or inconsistent as a result of\nmodifications.\nstat= -1000\nThe routine stopped because of an FFT interface error.\nstat= -10000\nThe routine stopped because its commit step failed to\ncomplete or the parameter ipar[0] was altered by\nmistake.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2422\n\n\nfree_trig_transform\nCleans the memory allocated for the data structure\nused by the FFT interface.\nSyntax\nvoid free_trig_transform(DFTI_DESCRIPTOR_HANDLE *handle, MKL_INT ipar[], MKL_INT\n*stat);\nInclude Files\n•\nmkl.h\nInput Parameters\nipar\nMKL_INT array of size 128. Contains integer data needed for\nTrigonometric Transform computations.\nhandle\nDFTI_DESCRIPTOR_HANDLE*. The data structure used by Intel®\noneAPI Math Kernel Library (oneMKL) FFT interface (for details, refer\ntoFFT Functions).\nOutput Parameters\nhandle\nThe data structure used by Intel® oneAPI Math Kernel Library\n(oneMKL) FFT interface. Memory allocated for the structure is released\non output.\nipar\nContains integer data needed for Trigonometric Transform\ncomputations. On output, ipar[6] is updated with the stat value.\nstat\nMKL_INT*. Contains the routine completion status, which is also\nwritten to ipar[6].\nDescription\nThe free_trig_transform routine cleans the memory used by the handlestructure, needed for Intel®\noneAPI Math Kernel Library (oneMKL) FFT functions. To release the memory allocated for other parameters,\ninclude cleaning of the memory in your code.\nReturn Values\nstat= 0\nThe routine completed the task normally.\nstat= -1000\nThe routine stopped because of an FFT interface error.\nstat= -99999\nThe routine failed to complete the task.\nCommon Parameters of the Trigonometric Transforms\nThis section provides description of array parameters that hold TT routine options: ipar, dpar and spar.\nNOTE\nInitial values are assigned to the array parameters by the appropriate ?_init_trig_transform\nand ?_commit_trig_transform routines.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2423\n\n\nipar\nMKL_INT array of size 128, holds integer data needed for Trigonometric\nTransform computations. Its elements are described in Table \"Elements of the\nipar Array\":\nElements of the ipar Array\nIndex\nDescription\n0\nContains the size of the problem to solve. The ?_init_trig_transform routine sets\nipar[0]=n, and all subsequently called TT routines use ipar[0] as the size of the\ntransform.\n1\nContains error messaging options:\n•\nipar[1]=-1 indicates that all error messages will be printed to the file\nMKL_Trig_Transforms_log.txt in the folder from which the routine is called. If\nthe file does not exist, the routine tries to create it. If the attempt fails, the\nroutine prints information that the file cannot be created to the standard output\ndevice.\n•\nipar[1]=0 indicates that no error messages will be printed.\n•\nipar[1]=1 (default) indicates that all error messages will be printed to the\npreconnected default output device (usually, screen).\nIn case of errors, each TT routine assigns a non-zero value to stat regardless of the\nipar[1] setting.\n2\nContains warning messaging options:\n•\nipar[2]=-1 indicates that all warning messages will be printed to the file\nMKL_Trig_Transforms_log.txt in the directory from which the routine is called.\nIf the file does not exist, the routine tries to create it. If the attempt fails, the\nroutine prints information that the file cannot be created to the standard output\ndevice.\n•\nipar[2]=0 indicates that no warning messages will be printed.\n•\nipar[2]=1 (default) indicates that all warning messages will be printed to the\npreconnected default output device (usually, screen).\nIn case of warnings, the stat parameter will acquire a non-zero value regardless of\nthe ipar[2] setting.\n3 through 4\nReserved for future use.\n5\nContains the type of the transform. The ?_init_trig_transform routine sets\nipar[5]=tt_type, and all subsequently called TT routines use ipar[5] as the type\nof the transform.\n6\nContains the stat value returned by the last completed TT routine. Used to check\nthat the previous call to a TT routine completed with stat=0.\n7\nInforms the ?_commit_trig_transform routines whether to initialize data structures\ndpar (spar) and handle. ipar[7]=0 indicates that the routine should skip the\ninitialization and only check correctness and consistency of the parameters.\nOtherwise, the routine initializes the data structures. The default value is 1.\nThe possibility to check correctness and consistency of input data without initializing\ndata structures dpar, spar and handle enables avoiding performance losses in a\nrepeated use of the same transform for different data vectors. Note that you can\nbenefit from the opportunity that ipar[7] gives only if you are sure to have supplied\nproper tolerance value in the dpar or spar array. Otherwise, avoid tuning this\nparameter.\n8\nContains message style options for TT routines. Specifically, if ipar[8] is non-zero,\nTT routines print the messages in C-style notations. The default value is 1.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2424\n\n\nIndex\nDescription\nWhen specifying message style options, be aware that by default, numbering of\nelements in C arrays starts at 0. For example, \"parameter ipar[0]=3 should be an\neven integer\" is a C-style message. The use of ipar[8] enables you to view\nmessages in a more convenient style.\n9\nSpecifies the number of OpenMP threads to run TT routines in the OpenMP\nenvironment of the Intel® oneAPI Math Kernel Library (oneMKL) Poisson Solver. The\ndefault value is 1. You are highly recommended not to alter this value. See\nalsoCaveat on Parameter Modifications.\n10\nSpecifies the mode of compatibility with FFTW. The default value is 0. Set the value to\n1 to invoke compatibility with FFTW. In the latter case, results will not be normalized,\nbecause FFTW does not do this. It is highly recommended not to alter this value, but\nrather use real-to-real FFTW to MKL wrappers, described in FFTW to Intel® MKL\nWrappers for FFTW 3.x. See also Caveat on Parameter Modifications.\n11 through 127\nReserved for future use.\nNOTE\nWhile you can declare the ipar array as MKL_INT ipar[11], for future compatibility you should\ndeclare ipar as MKL_INT ipar[128].\nArrays dpar and spar are the same except in the data precision:\ndpar\ndouble array of size 5n/2+2, holds data needed for double-precision routines to\nperform TT computations. This array is initialized in the \nd_init_trig_transform and d_commit_trig_transform routines.\nspar\nfloat array of size 5n/2+2, holds data needed for single-precision routines to\nperform TT computations. This array is initialized in the \ns_init_trig_transform and s_commit_trig_transform routines.\nAs dpar and spar have similar elements in respective positions, the elements are described together in Table\n\"Elements of the dpar and spar Arrays\":\nElements of the dpar and spar Arrays\nIndex\nDescription\n0\nContains the first absolute tolerance used by the\nappropriate ?_commit_trig_transform routine. For a staggered cosine or a sine\ntransform, f[n] should be equal to 0.0 and for a staggered sine or a sine transform,\nf[0] should be equal to 0.0. The ?_commit_trig_transform routine checks\nwhether absolute values of these parameters are below dpar[0]*n or spar[0]*n,\ndepending on the routine precision. To suppress warnings resulting from tolerance\nchecks, set dpar[0] or spar[0] to a sufficiently large number.\n1\nReserved for future use.\n2 through 5n/2+1\nContain tabulated values of trigonometric functions. Contents of the elements depend\nupon the type of transform tt_type, set up in the ?_commit_trig_transform\nroutine:\n•\nIf tt_type=MKL_SINE_TRANSFORM, the transform uses only the first n/2 array\nelements, which contain tabulated sine values.\n•\nIf tt_type=MKL_STAGGERED_SINE_TRANSFORM, the transform uses only the first\n3n/2 array elements, which contain tabulated sine and cosine values.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2425\n\n\nIndex\nDescription\n•\nIf tt_type=MKL_STAGGERED2_SINE_TRANSFORM, the transform uses all the 5n/2\narray elements, which contain tabulated sine and cosine values.\n•\nIf tt_type=MKL_COSINE_TRANSFORM, the transform uses only the first n array\nelements, which contain tabulated cosine values.\n•\nIf tt_type=MKL_STAGGERED_COSINE_TRANSFORM, the transform uses only the\nfirst 3n/2 elements, which contain tabulated sine and cosine values.\n•\nIf tt_type=MKL_STAGGERED2_COSINE_TRANSFORM, the transform uses all the\n5n/2 elements, which contain tabulated sine and cosine values.\nNOTE\nTo save memory, you can define the array size depending upon the type of transform.\nCaveat on Parameter Modifications\nFlexibility of the TT interface enables you to skip a call to the ?_init_trig_transform routine and to\ninitialize the basic data structures explicitly in your code. You may also need to modify the contents of ipar,\ndpar and spar arrays after initialization. When doing so, provide correct and consistent data in the arrays.\nMistakenly altered arrays cause errors or wrong computation. You can perform a basic check for correctness\nand consistency of parameters by calling the ?_commit_trig_transform routine; however, this does not\nensure the correct result of a transform but only reduces the chance of errors or wrong results.\nNOTE\nTo supply correct and consistent parameters to TT routines, you should have considerable experience\nin using the TT interface and good understanding of elements that the ipar, spar and dpar arrays\ncontain and dependencies between values of these elements.\nHowever, in rare occurrences, even advanced users might fail to compute a transform using TT routines after\nthe parameter modifications. In cases like these, refer for technical support at http://www.intel.com/\nsoftware/products/support/ .\nWARNING\nThe only way that ensures proper computation of the Trigonometric Transforms is to follow a typical\nsequence of invoking the routines and not change the default set of parameters. So, avoid\nmodifications of ipar, dpar and spar arrays unless a strong need arises.\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nTrigonometric Transform Implementation Details\nSeveral aspects of the Intel® oneAPI Math Kernel Library (oneMKL) TT interface are platform-specific and\nlanguage-specific. To promote portability across platforms and ease of use across different languages, Intel®\noneAPI Math Kernel Library (oneMKL) provides you with the TT language-specific header file to include in\nyour code:\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2426\n\n\n•\nmkl_trig_transforms.h, to be used together with mkl_dfti.h.\nNOTE\n•\nUse of the Intel® oneAPI Math Kernel Library (oneMKL) TT software without including the above\nlanguage-specific header files is not supported.\nHeader File\nThe header file below defines the following function prototypes:\nvoid d_init_trig_transform(MKL_INT *, MKL_INT *, MKL_INT *, double *, MKL_INT *); \nvoid d_commit_trig_transform(double *, DFTI_DESCRIPTOR_HANDLE *, MKL_INT *, double *, MKL_INT \n*);  \nvoid d_forward_trig_transform(double *, DFTI_DESCRIPTOR_HANDLE *, MKL_INT *, double *, MKL_INT \n*); \nvoid d_backward_trig_transform(double *, DFTI_DESCRIPTOR_HANDLE *, MKL_INT *, double *, MKL_INT \n*);\n        \nvoid s_init_trig_transform(MKL_INT *, MKL_INT *, MKL_INT *, float *, MKL_INT *); \nvoid s_commit_trig_transform(float *, DFTI_DESCRIPTOR_HANDLE *, MKL_INT *, float *, MKL_INT *);  \nvoid s_forward_trig_transform(float *, DFTI_DESCRIPTOR_HANDLE *, MKL_INT *, float *, MKL_INT *); \nvoid s_backward_trig_transform(float *, DFTI_DESCRIPTOR_HANDLE *, MKL_INT *, float *, MKL_INT \n*); \nvoid free_trig_transform(DFTI_DESCRIPTOR_HANDLE *, MKL_INT *, MKL_INT *); \nFast Poisson Solver Routines\nIn addition to the Real Discrete Trigonometric Transforms (TT) interface (refer to Trigonometric Transform\nRoutines), Intel® oneAPI Math Kernel Library (oneMKL) supports thethe Poisson Solver interface. This\ninterface implements a group of routines (Poisson Solver routines) used to compute a solution of Laplace,\nPoisson, and Helmholtz problems of a special kind using discrete Fourier transforms. Laplace and Poisson\nproblems are special cases of a more general Helmholtz problem. The problems that are solved by the\nPoisson Solver interface are defined more exactly in Poisson Solver Implementation. The Poisson Solver\ninterface provides much flexibility of use: you can call routines with the default parameter values or adjust\nroutines to your particular needs by manually tuning routine parameters. You can adjust the style of error\nand warning messages to a Cnotation by setting up a dedicated parameter. This adds convenience to\ndebugging, because you can read information in the way that is natural for your code. The Intel® oneAPI\nMath Kernel Library (oneMKL) Poisson Solver interface currently contains only routines that implement the\nfollowing solvers:\n•\nFast Laplace, Poisson and Helmholtz solvers in a Cartesian coordinate system\n•\nFast Poisson and Helmholtz solvers in a spherical coordinate system.\nPoisson Solver Implementation\nPoisson Solver routines enable approximate solving of certain two-dimensional and three-dimensional\nproblems. Figure \"Structure of the Poisson Solver\" shows the general structure of the Poisson Solver.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2427\n\n\n__border__top\nStructure of the Poisson Solver\nNOTE\nAlthough in the Cartesian case, both periodic and non-periodic solvers are also supported, they use\nthe same interfaces.\nSections below provide details of the problems that can be solved using Intel® oneAPI Math Kernel Library\n(oneMKL) Poisson Solver.\nTwo-Dimensional Problems\nNotational Conventions\nThe Poisson Solver interface description uses the following notation for boundaries of a rectangular domain ax\n< x < bx, ay < y < by on a Cartesian plane:\nbd_ax = {x = ax, ay ≤ y ≤ by}, bd_bx = {x = bx, ay ≤ y ≤ by}\nbd_ay = {ax ≤ x ≤ bx, y = ay}, bd_by = {ax ≤ x ≤ bx, y = by}.\nThe following figure shows these boundaries:\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2428\n\n\nThe wildcard \"+\" may stand for any of the symbols ax, bx, ay, by, so bd_+ denotes any of the above\nboundaries.\nThe Poisson Solver interface description uses the following notation for boundaries of a rectangular domain\naφ < φ < bφ, aθ < θ < bθ on a sphere 0 ≤ φ ≤ 2 π, 0 ≤ θ ≤ π:\nbd_aφ = {φ = aφ, aθ ≤ θ ≤ bθ}, bd_bφ = {φ = bφ, aθ ≤ θ ≤ bθ},\nbd_aθ = {aφ ≤ φ ≤ bφ, θ = aθ}, bd_bθ = {aφ ≤ φ ≤ bφ, θ = bθ}.\nThe wildcard \"~\" may stand for any of the symbols aφ, bφ, aθ, bθ, so bd_~ denotes any of the above\nboundaries.\nTwo-dimensional Helmholtz problem on a Cartesian plane\nThe two-dimensional (2D) Helmholtz problem is to find an approximate solution of the Helmholtz equation\n−∂2u\n∂x2 −∂2u\n∂y2 + qu = f (x, y), q = const ≥0\nin a rectangle, that is, a rectangular domain ax< x < bx, ay< y < by, with one of the following boundary\nconditions on each boundary bd_+:\n•\nThe Dirichlet boundary condition\nu x, y = G x, y\n•\nThe Neumann boundary condition\n∂u\n∂n(x, y) = g(x, y)\nwhere\nn= -x on bd_ax, n= x on bd_bx,\nn= -y on bd_ay, n= y on bd_by.\n•\nPeriodic boundary conditions\nu(ax, y) = u(bx, y),\n∂\n∂x u(ax, y) =\n∂\n∂x u(bx, y),\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2429\n\n\nu(x, ay) = u(x, by),\n∂\n∂y u(x, ay) =\n∂\n∂y u(x, by).\nTwo-dimensional Poisson problem on a Cartesian plane\nThe Poisson problem is a special case of the Helmholtz problem, when q=0. The 2D Poisson problem is to\nfind an approximate solution of the Poisson equation\n−∂2u\n∂x2 −∂2u\n∂y2 = f (x, y)\nin a rectangle ax< x < bx, ay< y < by with the Dirichlet, Neumann, or periodic boundary conditions on each\nboundary bd_+. In case of a problem with the Neumann boundary condition on the entire boundary, you can\nfind the solution of the problem only up to a constant. In this case, the Poisson Solver will compute the\nsolution that provides the minimal Euclidean norm of a residual.\nTwo-dimensional (2D) Laplace problem on a Cartesian plane\nThe Laplace problem is a special case of the Helmholtz problem, when q=0 and f(x, y)=0. The 2D Laplace\nproblem is to find an approximate solution of the Laplace equation\n∂2u\n∂x2 + ∂2u\n∂y2 = 0\nin a rectangle ax< x < bx, ay< y < by with the Dirichlet, Neumann, or periodic boundary conditions on each\nboundary bd_+.\nHelmholtz problem on a sphere\nThe Helmholtz problem on a sphere is to find an approximate solution of the Helmholtz equation\n−Δs u + qu = f , q = const ≥0,\nΔs =\n1\nsin2 θ\n∂2\n∂φ2 +\n1\nsinθ\n∂\n∂θ sin θ ∂\n∂θ\nin a domain bounded by angles aφ≤ φ ≤ bφ, aθ≤ θ ≤ bθ (spherical rectangle), with boundary conditions for\nparticular domains listed in Table \"Details of Helmholtz Problem on a Sphere\".\nDetails of Helmholtz Problem on a Sphere\nDomain on a sphere\nBoundary condition\nPeriodic/non-\nperiodic case\nRectangular, that is, bφ - aφ < 2 π and bθ\n- aθ < π\nHomogeneous Dirichlet boundary\nconditions on each boundary bd_~\nnon-periodic\nWhere aφ = 0, bφ = 2 π, and bθ - aθ < π\nHomogeneous Dirichlet boundary\nconditions on the boundaries bd_aθ and\nbd_bθ\nperiodic\nEntire sphere, that is, aφ = 0, bφ = 2 π,\naθ = 0, and bθ = π\nBoundary condition sin θ∂u\n∂θ\n= 0θ →0\nθ →π\nat the poles\nperiodic\nPoisson problem on a sphere\nThe Poisson problem is a special case of the Helmholtz problem, when q=0. The Poisson problem on a sphere\nis to find an approximate solution of the Poisson equation\n−Δ su = f , Δs =\n1\nsin2 θ\n∂2\n∂φ2 +\n1\nsinθ\n∂\n∂θ sin θ ∂\n∂θ\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2430\n\n\nin a spherical rectangle aφ≤ φ ≤ bφ, aθ≤ θ ≤ bθ in cases listed in Table \"Details of Helmholtz Problem on a\nSphere\". The solution to the Poisson problem on the entire sphere can be found up to a constant only. In this\ncase, Poisson Solver will compute the solution that provides the minimal Euclidean norm of a residual.\nApproximation of 2D problems\nTo find an approximate solution for any of the 2D problems, in the rectangular domain a uniform mesh can\nbe defined for the Cartesian case as:\nxi = ax + ihx, yj = ay + jhy ,\ni = 0, ..., nx, j = 0, ..., ny, hx =\nbx −ax\nnx\n, hy =\nby −ay\nny\nand for the spherical case as:\nφi = aφ + ihφ, θj = aθ + jhθ ,\ni = 0, ..., nφ, j = 0, ..., nθ, hφ =\nbφ −aφ\nnφ\n, hθ =\nbθ −aθ\nnθ\n.\nThe Poisson Solver uses the standard five-point finite difference approximation on this mesh to compute the\napproximation to the solution:\n•\nIn the Cartesian case, the values of the approximate solution will be computed in the mesh points (xi , yj)\nprovided that you can supply the values of the right-hand side f(x, y) in these points and the values of the\nappropriate boundary functions G(x, y) and/or g(x,y) in the mesh points laying on the boundary of the\nrectangular domain.\n•\nIn the spherical case, the values of the approximate solution will be computed in the mesh points (φi , θj)\nprovided that you can supply the values of the right-hand side f(φ, θ) in these points.\nNOTE\nThe number of mesh intervals nφ in the φ direction of a spherical mesh must be even in the periodic\ncase. The Poisson Solver does not support spherical meshes that do not meet this condition.\nThree-Dimensional Problems\nNotational Conventions\nThe Poisson Solver interface description uses the following notation for boundaries of a parallelepiped domain\nax < x < bx, ay < y <by, az < z <bz:\nbd_ax = {x = ax, ay ≤ y ≤ by, az ≤ z ≤ bz}, bd_bx = {x = bx, ay ≤ y ≤ by, az ≤ z ≤ bz},\nbd_ay = {ax ≤ x ≤ bx, y = ay, az ≤ z ≤ bz}, bd_by = {ax ≤ x ≤ bx, y = by, az ≤ z ≤ bz},\nbd_az = {ax ≤ x ≤ bx, ay ≤ y ≤ by, z = az}, bd_bx = {ax ≤ x ≤ bx, ay ≤ y ≤ by, z = bz}.\nThe following figure shows these boundaries:\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2431\n\n\nThe wildcard \"+\" may stand for any of the symbols ax, bx, ay, by, az, bz, so bd_+ denotes any of the above\nboundaries.\nThree-dimensional (3D) Helmholtz problem\nThe 3D Helmholtz problem is to find an approximate solution of the Helmholtz equation\n−∂2u\n∂x2 −∂2u\n∂y2 −∂2u\n∂z2 + qu = f (x, y, z), q = const ≥0\nin a parallelepiped, that is, a parallelepiped domain ax< x < bx, ay< y < by, az< z < bz, with one of the\nfollowing boundary conditions on each boundary bd_+:\n•\nThe Dirichlet boundary condition\nu x, y, z = G x, y, z\n•\nThe Neumann boundary condition\n∂u\n∂n(x, y, z) = g(x, y, z)\nwhere\nn= -x on bd_ax, n= x on bd_bx,\nn= -y on bd_ay, n= y on bd_by,\nn= -z on bd_az, n= z on bd_bz.\n•\nPeriodic boundary conditions\nu(ax, y, z) = u(bx, y, z),\n∂\n∂x u(ax, y, z) =\n∂\n∂x u(bx, y, z),\nu(x, ay, z) = u(x, by, z),\n∂\n∂y u(x, ay, z) =\n∂\n∂y u(x, by, z),\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2432\n\n\nu(x, y, az) = u(x, y, bz),\n∂\n∂z u(x, y, az) =\n∂\n∂z u(x, y, bz).\nThree-dimensional (3D) Poisson problem\nThe Poisson problem is a special case of the Helmholtz problem, when q=0. The 3D Poisson problem is to\nfind an approximate solution of the Poisson equation\n−∂2u\n∂x2 −∂2u\n∂y2 −∂2u\n∂z2 = f (x, y, z)\nin a parallelepiped ax< x < bx , ay< y < by, az< z < bz with the Dirichlet, Neumann, or periodic boundary\nconditions on each boundary bd_+.\nThree-dimensional (3D) Laplace problem\nThe Laplace problem is a special case of the Helmholtz problem, when q=0 and f(x, y, z)=0. The 3D Laplace\nproblem is to find an approximate solution of the Laplace equation\n∂2u\n∂x2 + ∂2u\n∂y2 + ∂2u\n∂z2 = 0\nin a parallelepiped ax< x < bx , ay< y < by, az< z < bz with the Dirichlet, Neumann, or periodic boundary\nconditions on each boundary bd_+.\nApproximation of 3D problems\nTo find an approximate solution for each of the 3D problems, a uniform mesh can be defined in the\nparallelepiped domain as:\nxi = ax + ihx, yj = ay + jhy, zk = az + jhz ,\nwhere\ni = 0, ..., nx, j = 0, ..., ny, k = 0, ..., nz,\nhx =\nbx −ax\nnx\n, hy =\nby −ay\nny\n, hz =\nbz −az\nnz\n.\nThe Poisson Solver uses the standard seven-point finite difference approximation on this mesh to compute\nthe approximation to the solution. The values of the approximate solution will be computed in the mesh\npoints (xi, yj, zk), provided that you can supply the values of the right-hand side f(x, y, z) in these points and\nthe values of the appropriate boundary functions G(x, y, z) and/or g(x, y, z) in the mesh points laying on the\nboundary of the parallelepiped domain.\nSequence of Invoking Poisson Solver Routines\nNOTE\nThis description always shows the solution process for the Helmholtz problem, because Fast Poisson\nSolvers and Fast Laplace Solvers are special cases of Fast Helmholtz Solvers (see Poisson Solver\nImplementation).\nThe Poisson Solver interface enables you to compute a solution of the Helmholtz problem in four steps. Each\nstep is performed by a dedicated routine. Table \"Poisson Solver Interface Routines\" lists the routines and\nbriefly describes their purpose.\nMost Poisson Solver routines have versions operating with single-precision and double-precision data. Names\nof such routines begin respectively with \"s\" and \"d\". The wildcard \"?\" stands for either of these symbols in\nroutine names. The routines for the Cartesian coordinate system have 2D and 3D versions. Their names end\nrespectively in \"2D\" and \"3D\". The routines for spherical coordinate system have periodic and non-periodic\nversions. Their names end respectively in \"p\" and \"np\".\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2433\n\n\nPoisson Solver Interface Routines\nRoutine\nDescription\n?_init_Helmholtz_2D/?_init_Helmholtz_3D/\n?_init_sph_p/?_init_sph_np\nInitializes basic data structures for Fast\nHelmholtz Solver in the 2D/3D/periodic/\nnon-periodic case, respectively.\n?_commit_Helmholtz_2D/?_commit_Helmholtz_3D/\n?_commit_sph_p/?_commit_sph_np\nChecks consistency and correctness of\ninput data and initializes data structures\nfor the solver, including those used by the\nIntel® oneAPI Math Kernel Library\n(oneMKL) FFT interface1.\n?_Helmholtz_2D/?_Helmholtz_3D/?_sph_p/?_sph_np\nComputes an approximate solution of the\n2D/3D/periodic/non-periodic Helmholtz\nproblem (see Poisson Solver\nImplementation) specified by the\nparameters.\nfree_Helmholtz_2D/free_Helmholtz_3D/\nfree_sph_p/free_sph_np\nReleases the memory used by the data\nstructures needed for calling the Intel®\noneAPI Math Kernel Library (oneMKL) FFT\ninterface1.\n1Poisson Solver routines call the Intel® oneAPI Math Kernel Library (oneMKL) FFT interface for better\nperformance.\nTo find an approximate solution of Helmholtz problem only once, the Intel® oneAPI Math Kernel Library\n(oneMKL) Poisson Solver interface routines are normally invoked in the order in which they are listed inTable\n\"Poisson Solver Interface Routines\".\nNOTE\nThough the order of invoking Poisson Solver routines may be changed, it is highly recommended to\nfollow the above order of routine calls.\nThe diagram in Figure \"Typical Order of Invoking Poisson Solver Routines\" indicates the typical order in which\nPoisson Solver routines can be invoked in a general case.\n__border__top\nTypical Order of Invoking Poisson Solver Routines\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2434\n\n\nA general scheme of using Poisson Solver routines for double-precision computations in a 3D Cartesian case\nis shown below. You can change this scheme to a scheme for single-precision computations by changing the\ninitial letter of the Poisson Solver routine names from \"d\" to \"s\". You can also change the scheme below from\nthe 3D to 2D case by changing the ending of the Poisson Solver routine names.\n...\nd_init_Helmholtz_3D(&ax, &bx, &ay, &by, &az, &bz, &nx, &ny, &nz, BCtype, ipar, dpar, &stat);\n/* change parameters in ipar and/or dpar if necessary. */\n/* note that the result of the Fast Helmholtz Solver will be in f. If you want to keep the data\nthat is stored in f, save it to another location before the function call below */\nd_commit_Helmholtz_3D(f, bd_ax, bd_bx, bd_ay, bd_by, bd_az, bd_bz, &xhandle, &yhandle, ipar, \ndpar, &stat);\nd_Helmholtz_3D(f, bd_ax, bd_bx, bd_ay, bd_by, bd_az, bd_bz, &xhandle, &yhandle, ipar, dpar, \n&stat);\nfree_Helmholtz_3D (&xhandle, &yhandle, ipar, &stat);\n/* here you may clean the memory used by f, dpar, ipar */\n...\nA general scheme of using Poisson Solver routines for double-precision computations in a spherical periodic\ncase is shown below. You can change this scheme to a scheme for single-precision computations by changing\nthe initial letter of the Poisson Solver routine names from \"d\" to \"s\". You can also change the scheme below\nto a scheme for a non-periodic case by changing the ending of the Poisson Solver routine names from \"p\" to\n\"np\".\n...\nd_init_sph_p(&ap,&bp,&at,&bt,&np,&nt,&q,ipar,dpar,&stat);\n/* change parameters in ipar and/or dpar if necessary. */\n/* note that the result of the Fast Helmholtz Solver will be in f. If you want to keep the data\nthat is stored in f, save it to another location before the function call below */\nd_commit_sph_p(f,&handle_s,&handle_c,ipar,dpar,&stat);\nd_sph_p(f,&handle_s,&handle_c,ipar,dpar,&stat);\nfree_sph_p(&handle_s,&handle_c,ipar,&stat);\n/* here you may clean the memory used by f, dpar, ipar */\n...\nYou can find examples of code that uses Poisson Solver routines to solve Helmholtz problem (in both\nCartesian and spherical cases) in the examples\\pdepoissonc\\source folderin your Intel® oneAPI Math\nKernel Library (oneMKL) directory.\nFast Poisson Solver Interface Description\nAll numerical types in this section are either standard C types float and double or MKL_INT integer type.\nFor more information on the C types, refer to C Datatypes Specific to oneMKL and the Intel® oneAPI Math\nKernel Library (oneMKL) Developer Guide. To better understand usage of the types, see examples in the\nexamples\\pdepoissonc\\source folderin your Intel® oneAPI Math Kernel Library (oneMKL) directory.\nRoutine Options\nAll Poisson Solver routines use parameters for passing various options to the routines. These parameters are\narrays ipar, dpar, and spar. Values for these parameters should be specified very carefully (see Common\nParameters). You can change these values during computations to meet your needs. For more details, see\nthe descriptions of specific routines.\nWARNING\nTo avoid failure or incorrect results, you must provide correct and consistent parameters to the\nroutines.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2435\n\n\nUser Data Arrays\nPoisson Solver routines take arrays of user data as input. For example, the d_Helmholtz_3Droutine takes\nuser arrays to compute an approximate solution to the 3D Helmholtz problem. To minimize storage\nrequirements and improve the overall run-time efficiency, Intel® oneAPI Math Kernel Library (oneMKL)\nPoisson Solver routines do not make copies of user input arrays.\nNOTE\nIf you need a copy of your input data arrays, you must save them yourself.\nFor better performance, align your data arrays as recommended in the Intel® oneAPI Math Kernel Library\n(oneMKL) Developer Guide (search the document for coding techniques to improve performance).\nRoutines for the Cartesian Solver\nThe section describes Poisson Solver routines for the Cartesian case, their syntax, parameters, and return\nvalues. All flavors of the same routine are described together: single- and double-precision and 2D and 3D.\nNOTE\nSome of the routine parameters are used only in the 3D Fast Helmholtz Solver.\nPoisson Solver routines call Intel® oneAPI Math Kernel Library (oneMKL) FFT routines (described inFFT\nFunctions), which enhance performance of the Poisson Solver routines.\n?_init_Helmholtz_2D/?_init_Helmholtz_3D\nInitializes basic data structures of the Fast 2D/3D\nHelmholtz Solver.\nSyntax\nvoid d_init_Helmholtz_2D (const double * ax, const double * bx, const double * ay,\nconst double * by, const MKL_INT * nx, const MKL_INT * ny, const char * BCtype, const\ndouble * q, MKL_INT * ipar, double * dpar, MKL_INT * stat);\nvoid s_init_Helmholtz_2D (const float * ax, const float * bx, const float * ay, const\nfloat * by, const MKL_INT * nx, const MKL_INT * ny, const char * BCtype, const float *\nq, MKL_INT * ipar, float * spar, MKL_INT * stat);\nvoid d_init_Helmholtz_3D (const double * ax, const double * bx, const double * ay,\nconst double * by, const double * az, const double * bz, const MKL_INT * nx, const\nMKL_INT * ny, const MKL_INT * nz, const char * BCtype, const double * q, MKL_INT *ipar,\ndouble * dpar, MKL_INT * stat);\nvoid s_init_Helmholtz_3D (const float * ax, const float * bx, const float * ay, const\nfloat * by, const float * az, const float * bz, const MKL_INT * nx, const MKL_INT * ny,\nconst MKL_INT * nz, const char * BCtype, const float * q, MKL_INT * ipar, float * spar,\nMKL_INT * stat);\nInclude Files\n•\nmkl.h\nInput Parameters\nax\ndouble* for d_init_Helmholtz_2D/d_init_Helmholtz_3D,\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2436\n\n\nfloat* for s_init_Helmholtz_2D/s_init_Helmholtz_3D.\nThe coordinate of the leftmost boundary of the domain along the x-\naxis.\nbx\ndouble* for d_init_Helmholtz_2D/d_init_Helmholtz_3D,\nfloat* for s_init_Helmholtz_2D/s_init_Helmholtz_3D.\nThe coordinate of the rightmost boundary of the domain along the x-\naxis.\nay\ndouble* for d_init_Helmholtz_2D/d_init_Helmholtz_3D,\nfloat* for s_init_Helmholtz_2D/s_init_Helmholtz_3D.\nThe coordinate of the leftmost boundary of the domain along the y-\naxis.\nby\ndouble* for d_init_Helmholtz_2D/d_init_Helmholtz_3D,\nfloat* for s_init_Helmholtz_2D/s_init_Helmholtz_3D.\nThe coordinate of the rightmost boundary of the domain along the y-\naxis.\naz\ndouble* for d_init_Helmholtz_3D,\nfloat* for s_init_Helmholtz_3D.\nThe coordinate of the leftmost boundary of the domain along the z-\naxis. This parameter is needed only for the ?_init_Helmholtz_3D\nroutine.\nbz\ndouble* for d_init_Helmholtz_3D,\nfloat* for s_init_Helmholtz_3D.\nThe coordinate of the rightmost boundary of the domain along the z-\naxis. This parameter is needed only for the ?_init_Helmholtz_3D\nroutine.\nnx\nMKL_INT*. The number of mesh intervals along the x-axis.\nny\nMKL_INT*. The number of mesh intervals along the y-axis.\nnz\nMKL_INT*. The number of mesh intervals along the z-axis. This\nparameter is needed only for the ?_init_Helmholtz_3D routine.\nBCtype\nchar*. Contains the type of boundary conditions on each boundary.\nMust contain four characters for ?_init_Helmholtz_2D and six\ncharacters for ?_init_Helmholtz_3D. Each of the characters can be\n'N' (Neumann boundary condition), 'D' (Dirichlet boundary condition),\nor 'P' (periodic boundary conditions). Specify the types of boundary\nconditions for the boundaries in the following order: bd_ax, bd_bx,\nbd_ay, bd_by, bd_az, and bd_bz. Specify periodic boundary conditions\non the respective boundaries in pairs (for example, 'PPDD' or 'NNPP' in\nthe 2D case). The types of boundary conditions for the last two\nboundaries are needed only in the 3D case.\nq\ndouble* for d_init_Helmholtz_2D/d_init_Helmholtz_3D,\nfloat* for s_init_Helmholtz_2D/s_init_Helmholtz_3D .\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2437\n\n\nThe constant Helmholtz coefficient. Note that to solve Poisson or\nLaplace problem, you should set the value of q to 0.\nOutput Parameters\nipar\nMKL_INT array of size 128. Contains integer data to be used by Fast\nHelmholtz Solver (for details, refer to ipar).\ndpar\ndouble array of size 5*nx/2+7 in the 2D case or 5*(nx+ny)/2+9 in\nthe 3D case. Contains double-precision data to be used by Fast\nHelmholtz Solver (for details, refer to dpar and spar).\nspar\nfloat array of size 5*nx/2+7 in the 2D case or 5*(nx+ny)/2+9 in the\n3D case. Contains single-precision data to be used by Fast Helmholtz\nSolver (for details, refer to dpar and spar).\nstat\nMKL_INT*. Routine completion status, which is also written to\nipar[0]. Continue to call other Poisson Solver routines only if the\nstatus is 0.\nDescription\nThe ?_init_Helmholtz_2D/?_init_Helmholtz_3D routines initialize basic data structures for Poisson\nSolver computations of the appropriate precision. All routines invoked after a call to\na ?_init_Helmholtz_2D/?_init_Helmholtz_3D routine use values of the ipar, dpar and spar array\nparameters returned by the routine. Detailed description of the array parameters can be found in Common\nParameters.\nCaution\nData structures initialized and created by 2D flavors of the routine cannot be used by 3D\nflavors of any Poisson Solver routines, and vice versa.\nYou can skip calls to these routines in your code. However, see Caveat on Parameter Modifications for\ninformation on initializing the data structures.\nReturn Values\nstat= 0\nThe routine successfully completed the task. In general, to\nproceed with computations, the routine should complete\nwith this stat value.\nstat= -99999\nThe routine failed to complete the task because of a fatal\nerror.\n_commit_Helmholtz_2D/?_commit_Helmholtz_3D\nChecks consistency and correctness of input data and\ninitializes certain data structures required to solve\n2D/3D Helmholtz problem.\nSyntax\nvoid d_commit_Helmholtz_2D (double * f, const double * bd_ax, const double * bd_bx,\nconst double * bd_ay, const double * bd_by, DFTI_DESCRIPTOR_HANDLE * xhandle, MKL_INT *\nipar, double * dpar, MKL_INT * stat );\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2438\n\n\nvoid s_commit_Helmholtz_2D (float * f, const float * bd_ax, const float * bd_bx, const\nfloat * bd_ay, const float * bd_by, DFTI_DESCRIPTOR_HANDLE * xhandle, MKL_INT * ipar,\nfloat * spar, MKL_INT * stat );\nvoid d_commit_Helmholtz_3D (double * f, const double * bd_ax, const double * bd_bx,\nconst double * bd_ay, const double * bd_by, const double * bd_az, const double * bd_bz,\nDFTI_DESCRIPTOR_HANDLE * xhandle, DFTI_DESCRIPTOR_HANDLE * yhandle, MKL_INT * ipar,\ndouble * dpar, MKL_INT * stat );\nvoid s_commit_Helmholtz_3D (float * f, const float * bd_ax, const float * bd_bx, const\nfloat * bd_ay, const float * bd_by, const float * bd_az, const float * bd_bz,\nDFTI_DESCRIPTOR_HANDLE * xhandle, DFTI_DESCRIPTOR_HANDLE * yhandle, MKL_INT * ipar,\nfloat * spar, MKL_INT * stat );\nInclude Files\n•\nmkl.h\nInput Parameters\nf\ndouble* for d_commit_Helmholtz_2D/d_commit_Helmholtz_3D,\nfloat* for s_commit_Helmholtz_2D/s_commit_Helmholtz_3D.\nContains the right-hand side of the problem packed in a single vector:\n•\n2D problem: The size of the vector for the is (nx+1)*(ny+1). The\nvalue of the right-hand side in the mesh point (i, j) is stored in f[i\n+j*(nx+1)] .\n•\n3D problem: The size of the vector for the is (nx+1)*(ny+1)*(nz\n+1). The value of the right-hand side in the mesh point (i, j, k) is\nstored in f[i+j*(nx+1)+k*(nx+1)*(ny+1)].\nNote that to solve the Laplace problem, you should set all the\nelements of the array f to 0.\nNote also that the array f may be altered by the routine. To preserve\nthe f vector, save it to another memory location.\nipar\nMKL_INT array of size 128. Contains integer data to be used by the\nFast Helmholtz Solver (for details, refer to ipar).\ndpar\ndouble array of size depending on the dimension of the problem:\n•\n2D problem: 5*nx/2+7\n•\n3D problem: 5*(nx+ny)/2+9\nContains double-precision data to be used by the Fast Helmholtz\nSolver (for details, refer to dpar and spar).\nspar\nfloat array of size depending on the dimension of the problem:\n•\n2D problem: 5*nx/2+7\n•\n3D problem: 5*(nx+ny)/2+9\nContains single-precision data to be used by the Fast Helmholtz Solver\n(for details, refer to dpar and spar).\nbd_ax\ndouble* for d_commit_Helmholtz_2D/d_commit_Helmholtz_3D, \nfloat* for s_commit_Helmholtz_2D/s_commit_Helmholtz_3D.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2439\n\n\nContains values of the boundary condition on the leftmost boundary of\nthe domain along the x-axis (for more information, refer to a detailed\ndescription of bd_ax).\nbd_bx\ndouble* for d_commit_Helmholtz_2D/d_commit_Helmholtz_3D, \nfloat* for s_commit_Helmholtz_2D/s_commit_Helmholtz_3D.\nContains values of the boundary condition on the rightmost boundary\nof the domain along the x-axis (for more information, refer to a\ndetailed description of bd_bx).\nbd_ay\ndouble* for d_commit_Helmholtz_2D/d_commit_Helmholtz_3D, \nfloat* for s_commit_Helmholtz_2D/s_commit_Helmholtz_3D.\nContains values of the boundary condition on the leftmost boundary of\nthe domain along the y-axis (for more information, refer to a detailed\ndescription of bd_ay).\nbd_by\ndouble* for d_commit_Helmholtz_2D/d_commit_Helmholtz_3D, \nfloat* for s_commit_Helmholtz_2D/s_commit_Helmholtz_3D.\nContains values of the boundary condition on the rightmost boundary\nof the domain along the y-axis (for more information, refer to a\ndetailed description of bd_by).\nbd_az\ndouble* for d_commit_Helmholtz_3D,\nfloat* for s_commit_Helmholtz_3D.\nUsed only by ?_commit_Helmholtz_3D. Contains values of the\nboundary condition on the leftmost boundary of the domain along the\nz-axis (for more information, refer to a detailed description of bd_az).\nbd_bz\ndouble* for d_commit_Helmholtz_3D, \nfloat* for s_commit_Helmholtz_3D.\nUsed only by ?_commit_Helmholtz_3D. Contains values of the\nboundary condition on the rightmost boundary of the domain along\nthe z-axis (for more information, refer to a detailed description of\nbd_bz).\nOutput Parameters\nf\nContains right-hand side of the problem, possibly altered on output.\nipar\nContains integer data to be used by Fast Helmholtz Solver. Modified on\noutput as explained in ipar.\ndpar\nContains double-precision data to be used by Fast Helmholtz Solver.\nModified on output as explained in dpar and spar.\nspar\nContains single-precision data to be used by Fast Helmholtz Solver.\nModified on output as explained in dpar and spar.\nxhandle, yhandle\nDFTI_DESCRIPTOR_HANDLE*. Data structures used by the Intel®\noneAPI Math Kernel Library (oneMKL) FFT interface (for details, refer\ntoFFT Functions). yhandle is used only by ?_commit_Helmholtz_3D.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2440\n\n\nstat\nMKL_INT*. Routine completion status, which is also written to\nipar[0]. Continue to call other Poisson Solver routines only if the\nstatus is 0.\nDescription\nThe ?_commit_Helmholtz_2D/?_commit_Helmholtz_3D routines check the consistency and correctness of\nthe parameters to be passed to the solver routines ?_Helmholtz_2D/?_Helmholtz_3D. They also initialize\nthe xhandle and yhandle data structures, ipar array, and dpar or spar array, depending upon the routine\nprecision. Refer to Common Parameters to find out which particular array elements\nthe ?_commit_Helmholtz_2D/?_commit_Helmholtz_3D routines initialize and to what values these\nelements are initialized.\nThe routines perform only a basic check for correctness and consistency. If you are going to modify\nparameters of Poisson Solver routines, see Caveat on Parameter Modifications.\nUnlike ?_init_Helmholtz_2D/?_init_Helmholtz_3D, you must call\nthe ?_commit_Helmholtz_2D/?_commit_Helmholtz_3D routines in your code. Values of ax, bx, ay, by, az,\nand bz are passed to the routines with the spar/dpar array, and values of nx, ny, nz, and BCtype are\npassed with the ipar array.\nReturn Values\nstat= 1\nThe routine completed without errors but with warnings.\nstat= 0\nThe routine successfully completed the task.\nstat= -100\nThe routine stopped because an error in the input data was\nfound, or the data in the dpar, spar, or ipar array was\naltered by mistake.\nstat= -1000\nThe routine stopped because of an Intel® oneAPI Math\nKernel Library (oneMKL) FFT or TT interface error.\nstat= -10000\nThe routine stopped because the initialization failed to\ncomplete or the parameter ipar[0] was altered by\nmistake.\nstat= -99999\nThe routine failed to complete the task because of a fatal\nerror.\n?_Helmholtz_2D/?_Helmholtz_3D\nComputes the solution of the 2D/3D Helmholtz\nproblem specified by the parameters.\nSyntax\nvoid d_Helmholtz_2D (double * f, const double * bd_ax, const double * bd_bx, const\ndouble * bd_ay, const double *bd_by, DFTI_DESCRIPTOR_HANDLE * xhandle, MKL_INT * ipar,\nconst double * dpar, MKL_INT * stat );\nvoid s_Helmholtz_2D (float * f, const float * bd_ax, const float * bd_bx, const float *\nbd_ay, const float * bd_by, DFTI_DESCRIPTOR_HANDLE * xhandle, MKL_INT * ipar, const\nfloat * spar, MKL_INT * stat );\nvoid d_Helmholtz_3D (double * f, const double * bd_ax, const double * bd_bx, const\ndouble * bd_ay, const double *bd_by, const double * bd_az, const double * bd_bz,\nDFTI_DESCRIPTOR_HANDLE * xhandle, DFTI_DESCRIPTOR_HANDLE * yhandle, MKL_INT * ipar,\nconst double * dpar, MKL_INT * stat );\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2441\n\n\nvoid s_Helmholtz_3D (float * f, const float * bd_ax, const float * bd_bx, const float *\nbd_ay, const float * bd_by, const float * bd_az, const float * bd_bz,\nDFTI_DESCRIPTOR_HANDLE * xhandle, DFTI_DESCRIPTOR_HANDLE * yhandle, MKL_INT * ipar,\nconst float * spar, MKL_INT * stat );\nInclude Files\n•\nmkl.h\nInput Parameters\nf\ndouble* for d_Helmholtz_2D/d_Helmholtz_3D,\nfloat* for s_Helmholtz_2D/s_Helmholtz_3D.\nContains the right-hand side of the problem packed in a single vector\nand modified by the\nappropriate ?_commit_Helmholtz_2D/?_commit_Helmholtz_3D\nroutine. Note that an attempt to substitute the original right-hand side\nvector, which was passed to\nthe ?_commit_Helmholtz_2D/?_commit_Helmholtz_3D routine, at\nthis point results in an incorrect solution.\n•\n2D problem: the size of the vector is (nx+1)*(ny+1). The value of\nthe modified right-hand side in the mesh point (i, j) is stored in f[i\n+j*(nx+1)].\n•\n3D problem: the size of the vector is (nx+1)*(ny+1)*(nz+1). The\nvalue of the modified right-hand side in the mesh point (i, j, k) is\nstored in f[i+j*(nx+1)+k*(nx+1)*(ny+1)].\nxhandle, yhandle\nDFTI_DESCRIPTOR_HANDLE*. Data structures used by the Intel®\noneAPI Math Kernel Library (oneMKL) FFT interface (for details, refer\ntoFFT Functions). yhandle is used only by ?_Helmholtz_3D.\nipar\nMKL_INT array of size 128. Contains integer data to be used by Fast\nHelmholtz Solver (for details, refer to ipar).\ndpar\ndouble array of size depending on the dimension of the problem:\n•\n2D problem: 5*nx/2+7\n•\n3D problem: 5*(nx+ny)/2+9\nContains double-precision data to be used by Fast Helmholtz Solver\n(for details, refer to dpar and spar).\nspar\nfloat array of size depending on the dimension of the problem:\n•\n2D problem: 5*nx/2+7\n•\n3D problem: 5*(nx+ny)/2+9\nContains single-precision data to be used by Fast Helmholtz Solver\n(for details, refer to dpar and spar).\nbd_ax\ndouble* for d_Helmholtz_2D/d_Helmholtz_3D, \nfloat* for s_Helmholtz_2D/s_Helmholtz_3D.\nContains values of the boundary condition on the leftmost boundary of\nthe domain along the x-axis (for more information, refer to a detailed\ndescription of bd_ax).\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2442\n\n\nbd_bx\ndouble* for d_Helmholtz_2D/d_Helmholtz_3D, \nfloat* for s_Helmholtz_2D/s_Helmholtz_3D.\nContains values of the boundary condition on the rightmost boundary\nof the domain along the x-axis (for more information, refer to a\ndetailed description of bd_bx).\nbd_ay\ndouble* for d_Helmholtz_2D/d_Helmholtz_3D, \nfloat* for s_Helmholtz_2D/s_Helmholtz_3D.\nContains values of the boundary condition on the leftmost boundary of\nthe domain along the y-axis for more information, refer to a detailed\ndescription of bd_ay).\nbd_by\ndouble* for d_Helmholtz_2D/d_Helmholtz_3D, \nfloat* for s_Helmholtz_2D/s_Helmholtz_3D.\nContains values of the boundary condition on the rightmost boundary\nof the domain along the y-axis (for more information, refer to a\ndetailed description of bd_by).\nbd_az\ndouble* for d_Helmholtz_3D,\nfloat* for s_Helmholtz_3D.\nUsed only by ?_Helmholtz_3D. Contains values of the boundary\ncondition on the leftmost boundary of the domain along the z-axis (for\nmore information, refer to a detailed description of bd_az).\nbd_bz\ndouble* for d_Helmholtz_3D, \nfloat* for s_Helmholtz_3D.\nUsed only by ?_Helmholtz_3D. Contains values of the boundary\ncondition on the rightmost boundary of the domain along the z-axis\n(for more information, refer to a detailed description of bd_bz).\nNOTE\nTo avoid incorrect computation results, do not change arrays bd_ax, bd_bx, bd_ay, bd_by,\nbd_az, bd_bz between a call to the ?_commit_Helmholtz_2D/?_commit_Helmholtz_3D routine\nand a subsequent call to the appropriate ?_Helmholtz_2D/?_Helmholtz_3D routine.\nOutput Parameters\nf\nOn output, contains the approximate solution to the problem packed\nthe same way as the right-hand side of the problem was packed on\ninput.\nxhandle, yhandle\nData structures used by the Intel® oneAPI Math Kernel Library\n(oneMKL) FFT interface. Although the addresses do not change, the\nstructures are modified on output.\nipar\nContains integer data to be used by Fast Helmholtz Solver. Modified on\noutput as explained in ipar.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2443\n\n\nstat\nMKL_INT*. Routine completion status, which is also written to\nipar[0]. Continue to call other Poisson Solver routines only if the\nstatus is 0.\nDescription\nThe ?_Helmholtz_2D/?_Helmholtz_3D routines compute the approximate solution of the Helmholtz\nproblem defined in the previous calls to the corresponding initialization and commit routines. The solution is\ncomputed according to formulas given in Poisson Solver Implementation. The f parameter, which initially\nholds the packed vector of the right-hand side of the problem, is replaced by the computed solution packed\nin the same way. Values of ax, bx, ay, by, az, and bz are passed to the routines with the spar/dpar array,\nand values of nx, ny, nz, and BCtype are passed with the ipar array.\nReturn Values\nstat= 1\nThe routine completed without errors but with some\nwarnings.\nstat= 0\nThe routine successfully completed the task.\nstat= -2\nThe routine stopped because division by zero occurred. It\nusually happens if the data in the dpar or spar array was\naltered by mistake.\nstat= -3\nThe routine stopped because the sufficient memory was\nunavailable for the computations.\nstat= -100\nThe routine stopped because an error in the input data was\nfound or the data in the dpar, spar, or ipar array was\naltered by mistake.\nstat= -1000\nThe routine stopped because of the Intel® oneAPI Math\nKernel Library (oneMKL) FFT or TT interface error.\nstat= -10000\nThe routine stopped because the initialization failed to\ncomplete or the parameter ipar[0] was altered by\nmistake.\nstat= -99999\nThe routine failed to complete the task because of a fatal\nerror.\nfree_Helmholtz_2D/free_Helmholtz_3D\nReleases the memory allocated for the data structures\nused by the FFT interface.\nSyntax\nvoid free_Helmholtz_2D(DFTI_DESCRIPTOR_HANDLE* xhandle, MKL_INT* ipar, MKL_INT* stat);\nvoid free_Helmholtz_3D(DFTI_DESCRIPTOR_HANDLE* xhandle, DFTI_DESCRIPTOR_HANDLE*\nyhandle, MKL_INT* ipar, MKL_INT* stat);\nInclude Files\n•\nmkl.h\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2444\n\n\nInput Parameters\nxhandle, yhandle\nDFTI_DESCRIPTOR_HANDLE*. Data structures used by the Intel®\noneAPI Math Kernel Library (oneMKL) FFT interface (for details, refer\ntoFFT Functions). The structure yhandle is used only by\nfree_Helmholtz_3D.\nipar\nMKL_INT array of size 128. Contains integer data used by Fast\nHelmholtz Solver (for details, refer to ipar).\nOutput Parameters\nxhandle, yhandle\nData structures used by the Intel® oneAPI Math Kernel Library\n(oneMKL) FFT interface. Memory allocated for the structures is\nreleased on output.\nipar\nContains integer data used by Fast Helmholtz Solver. On output, the\nstatus of the routine call is written to ipar[0].\nstat\nMKL_INT*. Routine completion status, which is also written to\nipar[0].\nDescription\nThe free_Helmholtz_2D-free_Helmholtz_3D routine releases the memory used by the xhandle and\nyhandlestructures, which are needed for calling the Intel® oneAPI Math Kernel Library (oneMKL) FFT\nfunctions. To release memory allocated for other parameters, include memory release statements in your\ncode.\nReturn Values\nstat= 0\nThe routine successfully completed the task.\nstat= -1000\nThe routine stopped because of an Intel® oneAPI Math\nKernel Library (oneMKL) FFT or TT interface error.\nstat= -99999\nThe routine failed to complete the task because of a fatal\nerror.\nRoutines for the Spherical Solver\nThe section describes Poisson Solver routines for the spherical case, their syntax, parameters, and return\nvalues. All flavors of the same routine are described together: single- and double-precision and periodic\n(having names ending in \"p\") and non-periodic (having names ending in \"np\").\nThese Poisson Solver routines also call the Intel® oneAPI Math Kernel Library (oneMKL) FFT routines\n(described inFFT Functions), which enhance the performance of the Poisson Solver routines.\n?_init_sph_p/?_init_sph_np\nInitializes basic data structures of the periodic and\nnon-periodic Fast Helmholtz Solver on a sphere.\nSyntax\nvoid d_init_sph_p (const double * ap, const double * at, const double * bp, const\ndouble * bt, const MKL_INT * np, const MKL_INT *nt, const double * q, MKL_INT * ipar,\ndouble * dpar, MKL_INT * stat );\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2445\n\n\nvoid s_init_sph_p (const float * ap, const float * at, const float * bp, const float *\nbt, const MKL_INT * np, const MKL_INT * nt, const float * q, MKL_INT * ipar, float *\nspar, MKL_INT * stat );\nvoid d_init_sph_np (const double * ap, const double * at, const double * bp, const\ndouble * bt, const MKL_INT * np, const MKL_INT *nt, const double * q, MKL_INT * ipar,\ndouble * dpar, MKL_INT * stat );\nvoid s_init_sph_np (const float * ap, const float * at, const float * bp, const float *\nbt, const MKL_INT * np, const MKL_INT * nt, const float * q, MKL_INT * ipar, float *\nspar, MKL_INT * stat );\nInclude Files\n•\nmkl.h\nInput Parameters\nap\ndouble* for d_init_sph_p/d_init_sph_np,\nfloat* for s_init_sph_p/s_init_sph_np.\nThe coordinate (angle) of the leftmost boundary of the domain along\nthe φ-axis.\nbp\ndouble* for d_init_sph_p/d_init_sph_np,\nfloat* for s_init_sph_p/s_init_sph_np.\nThe coordinate (angle) of the rightmost boundary of the domain along\nthe φ-axis.\nat\ndouble* for d_init_sph_p/d_init_sph_np,\nfloat* for s_init_sph_p/s_init_sph_np.\nThe coordinate (angle) of the leftmost boundary of the domain along\nthe θ-axis.\nbt\ndouble* for d_init_sph_p/d_init_sph_np,\nfloat* for s_init_sph_p/s_init_sph_np.\nThe coordinate (angle) of the rightmost boundary of the domain along\nthe θ-axis.\nnp\nMKL_INT*. The number of mesh intervals along the φ-axis. Must be\neven in the periodic case.\nnt\nMKL_INT*. The number of mesh intervals along the θ-axis.\nq\ndouble* for d_init_sph_p/d_init_sph_np,\nfloat* for s_init_sph_p/s_init_sph_np.\nThe constant Helmholtz coefficient. To solve the Poisson problem, set\nthe value of q to 0.\nOutput Parameters\nipar\nMKL_INT array of size 128. Contains integer data to be used by Fast\nHelmholtz Solver on a sphere (for details, refer to ipar).\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2446\n\n\ndpar\ndouble array of size 5*np/2+nt+10. Contains double-precision data\nto be used by Fast Helmholtz Solver on a sphere (for details, refer to \ndpar and spar).\nspar\nfloat array of size 5*np/2+nt+10. Contains single-precision data to\nbe used by Fast Helmholtz Solver on a sphere (for details, refer to \ndpar and spar).\nstat\nMKL_INT*. Routine completion status, which is also written to\nipar[0]. Continue to call other Poisson Solver routines only if the\nstatus is 0.\nDescription\nThe ?_init_sph_p/?_init_sph_np routines initialize basic data structures for Poisson Solver computations.\nAll routines invoked after a call to a ?_init_Helmholtz_2D/?_init_Helmholtz_3D routine use values of\nthe ipar, dpar, and spar array parameters returned by the routine. A detailed description of the array\nparameters can be found in Common Parameters.\nCaution\nData structures initialized and created by periodic flavors of the routine cannot be used by\nnon-periodic flavors of any Poisson Solver routines for Helmholtz Solver on a sphere, and\nvice versa.\nYou can skip calls to these routines in your code. However, see Caveat on Parameter Modifications for\ninformation on initializing the data structures.\nReturn Values\nstat= 0\nThe routine successfully completed the task. In general, to\nproceed with computations, the routine should complete\nwith this stat value.\nstat= -99999\nThe routine failed to complete the task because of fatal\nerror.\n?_commit_sph_p/?_commit_sph_np\nChecks consistency and correctness of input data and\ninitializes certain data structures required to solve the\nperiodic/non-periodic Helmholtz problem on a sphere.\nSyntax\nvoid d_commit_sph_p(double* f, DFTI_DESCRIPTOR_HANDLE* handle_s,\nDFTI_DESCRIPTOR_HANDLE* handle_c, MKL_INT* ipar, double* dpar, MKL_INT* stat);\nvoid s_commit_sph_p(float* f, DFTI_DESCRIPTOR_HANDLE* handle_s, DFTI_DESCRIPTOR_HANDLE*\nhandle_c, MKL_INT* ipar, float* spar, MKL_INT* stat);\nvoid d_commit_sph_np(double* f, DFTI_DESCRIPTOR_HANDLE* handle, MKL_INT* ipar, double*\ndpar, MKL_INT* stat);\nvoid s_commit_sph_np(float* f, DFTI_DESCRIPTOR_HANDLE* handle, MKL_INT* ipar, float*\nspar, MKL_INT* stat);\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2447\n\n\nInclude Files\n•\nmkl.h\nInput Parameters\nf\ndouble* for d_commit_sph_p/d_commit_sph_np,\nfloat* for s_commit_sph_p/s_commit_sph_np.\nContains the right-hand side of the problem packed in a single vector.\nThe size of the vector is (np+1)*(nt+1) and value of the right-hand\nside in the mesh point (i, j) is stored in f[i+j*(np+1)] .\nNote that the array f may be altered by the routine. Save this vector\nto another memory location if you want to preserve it.\nipar\nMKL_INT array of size 128. Contains integer data to be used by the\nFast Helmholtz Solver on a sphere (for details, refer to ipar).\ndpar\ndouble array of size 5*np/2+nt+10. Contains double-precision data\nto be used by the Fast Helmholtz Solver on a sphere (for details, refer\nto dpar and spar).\nspar\nfloat array of size 5*np/2+nt+10. Contains single-precision data to\nbe used by the Fast Helmholtz Solver on a sphere (for details, refer to \ndpar and spar).\nOutput Parameters\nf\nContains the right-hand side of the problem, possibly altered on\noutput.\nipar\nContains integer data to be used by the Fast Helmholtz Solver on a\nsphere. Modified on output as explained in ipar.\ndpar\nContains double-precision data to be used by the Fast Helmholtz\nSolver on a sphere. Modified on output as explained in dpar and\nspar.\nspar\nContains single-precision data to be used by the Fast Helmholtz Solver\non a sphere. Modified on output as explained in dpar and spar.\nhandle_s, handle_c, handle\nDFTI_DESCRIPTOR_HANDLE*. Data structures used by the Intel®\noneAPI Math Kernel Library (oneMKL) FFT interface (for details, refer\ntoFFT Functions). handle_s and handle_c are used only\nin ?_commit_sph_p and handle is used only in ?_commit_sph_np.\nstat\nMKL_INT*. Routine completion status, which is also written to\nipar[0]. Continue to call other Poisson Solver routines only if the\nstatus is 0.\nDescription\nThe ?_commit_sph_p/?_commit_sph_np routines check consistency and correctness of the parameters to\nbe passed to the solver routines ?_sph_p/?_sph_np, respectively. They also initialize certain data\nstructures. The routine ?_commit_sph_p initializes structures handle_s and handle_c,\nand ?_commit_sph_np initializes handle. The routines also initialize the ipar array and dpar or spar array,\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2448\n\n\ndepending upon the routine precision. Refer to Common Parameters to find out which particular array\nelements the ?_commit_sph_p/?_commit_sph_np routines initialize and to what values these elements are\ninitialized.\nThe routines perform only a basic check for correctness and consistency. If you are going to modify\nparameters of Poisson Solver routines, see Caveat on Parameter Modifications.\nUnlike ?_init_sph_p/?_init_sph_np, you must call the ?_commit_sph_p/?_commit_sph_np routines.\nValues of np and nt are passed to each of the routines with the ipar array.\nReturn Values\nstat= 1\nThe routine completed without errors but with warnings.\nstat= 0\nThe routine successfully completed the task.\nstat= -100\nThe routine stopped because an error in the input data was\nfound or the data in the dpar, spar, or ipar array was\naltered by mistake.\nstat= -1000\nThe routine stopped because of an Intel® oneAPI Math\nKernel Library (oneMKL) FFT or TT interface error.\nstat= -10000\nThe routine stopped because the initialization failed to\ncomplete or the parameter ipar[0] was altered by\nmistake.\nstat= -99999\nThe routine failed to complete the task because of a fatal\nerror.\n?_sph_p/?_sph_np\nComputes the solution of the spherical Helmholtz\nproblem specified by the parameters.\nSyntax\nvoid d_sph_p(double* f, DFTI_DESCRIPTOR_HANDLE* handle_s, DFTI_DESCRIPTOR_HANDLE*\nhandle_c, MKL_INT* ipar, double* dpar, MKL_INT* stat);\nvoid s_sph_p(float* f, DFTI_DESCRIPTOR_HANDLE* handle_s, DFTI_DESCRIPTOR_HANDLE*\nhandle_c, MKL_INT* ipar, float* spar, MKL_INT* stat);\nvoid d_sph_np(double* f, DFTI_DESCRIPTOR_HANDLE* handle, MKL_INT* ipar, double* dpar,\nMKL_INT* stat);\nvoid s_sph_np(float* f, DFTI_DESCRIPTOR_HANDLE* handle, MKL_INT* ipar, float* spar,\nMKL_INT* stat);\nInclude Files\n•\nmkl.h\nInput Parameters\nf\ndouble* for d_sph_p/d_sph_np,\nfloat* for s_sph_p/s_sph_np.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2449\n\n\nContains the right-hand side of the problem packed in a single vector\nand modified by the appropriate ?_commit_sph_p/?_commit_sph_np\nroutine. Note that an attempt to substitute the original right-hand side\nvector, which was passed to the ?_commit_sph_p/?_commit_sph_np\nroutine, at this point results in an incorrect solution.\nThe size of the vector is (np+1)*(nt+1) and the value of the modified\nright-hand side in the mesh point (i, j) is stored in f[i+j*(np+1)] .\nhandle_s, handle_c, handle\nDFTI_DESCRIPTOR_HANDLE*. Data structures used by Intel® oneAPI\nMath Kernel Library (oneMKL) FFT interface (for details, refer toFFT\nFunctions). handle_s and handle_c are used only in ?_sph_p and\nhandle is used only in ?_sph_np.\nipar\nMKL_INT array of size 128. Contains integer data to be used by the\nFast Helmholtz Solver on a sphere (for details, refer to ipar).\ndpar\ndouble array of size 5*np/2+nt+10. Contains double-precision data\nto be used by the Fast Helmholtz Solver on a sphere (for details, refer\nto dpar and spar).\nspar\nfloat array of size 5*np/2+nt+10. Contains single-precision data to be\nused by the Fast Helmholtz Solver on a sphere (for details, refer to \ndpar and spar).\nOutput Parameters\nf\nOn output, contains the approximate solution to the problem packed\nthe same way as the right-hand side of the problem was packed on\ninput.\nhandle_s, handle_c, handle\nData structures used by the Intel® oneAPI Math Kernel Library\n(oneMKL) FFT interface.\nipar\nContains integer data to be used by the Fast Helmholtz Solver on a\nsphere. Modified on output as explained in ipar.\ndpar\nContains double-precision data to be used by the Fast Helmholtz\nSolver on a sphere. Modified on output as explained in dpar and\nspar.\nspar\nContains single-precision data to be used by the Fast Helmholtz Solver\non a sphere. Modified on output as explained in dpar and spar.\nstat\nMKL_INT*. Routine completion status, which is also written to\nipar[0]. Continue to call other Poisson Solver routines only if the\nstatus is 0.\nDescription\nThe sph_p/sph_np routines compute the approximate solution on a sphere of the Helmholtz problem defined\nin the previous calls to the corresponding initialization and commit routines. The solution is computed\naccording to the formulas given in Poisson Solver Implementation. The f parameter, which initially holds the\npacked vector of the right-hand side of the problem, is replaced by the computed solution packed in the\nsame way. Values of np and nt are passed to each of the routines with the ipar array.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2450\n\n\nReturn Values\nstat= 1\nThe routine completed without errors but with warnings.\nstat= 0\nThe routine successfully completed the task.\nstat= -2\nThe routine stopped because division by zero occurred. It\nusually happens if the data in the dpar or spar array was\naltered by mistake.\nstat= -3\nThe routine stopped because the memory was insufficient\nto complete the computations.\nstat= -100\nThe routine stopped because an error in the input data was\nfound or the data in the dpar, spar, or ipar array was\naltered by mistake.\nstat= -1000\nThe routine stopped because of an Intel® oneAPI Math\nKernel Library (oneMKL) FFT or TT interface error.\nstat= -10000\nThe routine stopped because the initialization failed to\ncomplete or the parameter ipar[0] was altered by\nmistake.\nstat= -99999\nThe routine failed to complete the task because of a fatal\nerror.\nfree_sph_p/free_sph_np\nReleases the memory allocated for the data structures\nused by the FFT interface.\nSyntax\nvoid free_sph_p(DFTI_DESCRIPTOR_HANDLE* handle_s, DFTI_DESCRIPTOR_HANDLE* handle_c,\nMKL_INT* ipar, MKL_INT* stat);\nvoid free_sph_np(DFTI_DESCRIPTOR_HANDLE* handle, MKL_INT* ipar, MKL_INT* stat);\nInclude Files\n•\nmkl.h\nInput Parameters\nhandle_s, handle_c, handle\nDFTI_DESCRIPTOR_HANDLE*. Data structures used by the Intel®\noneAPI Math Kernel Library (oneMKL) FFT interface (for details, refer\ntoFFT Functions). The structures handle_s and handle_c are used\nonly in free_sph_p, and handle is used only in free_sph_np.\nipar\nMKL_INT array of size 128. Contains integer data to be used by Fast\nHelmholtz Solver on a sphere (for details, refer to ipar).\nOutput Parameters\nhandle_s, handle_c, handle\nData structures used by the Intel® oneAPI Math Kernel Library\n(oneMKL) FFT interface. Memory allocated for the structures is\nreleased on output.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2451\n\n\nipar\nContains integer data to be used by Fast Helmholtz Solver on a\nsphere. On output, the status of the routine call is written to ipar[0].\nstat\nMKL_INT*. Routine completion status, which is also written to\nipar[0].\nDescription\nThe free_sph_p/free_sph_np routine releases the memory used by the handle_s, handle_c or\nhandlestructures, needed for calling the Intel® oneAPI Math Kernel Library (oneMKL) FFT functions. To\nrelease memory allocated for other parameters, include memory release statements in your code.\nReturn Values\nstat= 0\nThe routine successfully completed the task.\nstat= -1000\nThe routine stopped because of an Intel® oneAPI Math\nKernel Library (oneMKL) FFT or TT interface error.\nstat= -99999\nThe routine failed to complete the task because of a fatal\nerror.\nCommon Parameters for the Poisson Solver\nipar\nipar\nMKL_INT array of size 128, holds integer data needed for Fast Helmholtz Solver\n(both for Cartesian and spherical coordinate systems). Its elements are described\nin Table \"Elements of the ipar Array\":\nNOTE\nInitial values can be assigned to the array parameters by the\nappropriate ?_init_Helmholtz_2D/?_init_Helmholtz_3D/?_init_sph_p/?_init_sph_np\nand ?_commit_Helmholtz_2D/?_commit_Helmholtz_3D/?_commit_sph_p/?_commit_sph_np\nroutines.\nElements of the ipar Array\nIndex\nDescription\n0\nContains status value of the last Poisson Solver routine called. In general, it should be 0 on\nexit from a routine to proceed with the Fast Helmholtz Solver. The element has no predefined\nvalues. This element can also be used to inform\nthe ?_commit_Helmholtz_2D/?_commit_Helmholtz_3D/\n?_commit_sph_p/?_commit_sph_np routines of how the Commit step of the computation\nshould be carried out (see Figure \"Typical Order of Invoking Poisson Solver Routines\"). A non-\nzero value of ipar[0] with decimal representation\n=100a+10b+c, where each of a, b, and c is equal to 0 or 9, indicates that some parts of the\nCommit step should be omitted.\n•\nIf c=9, the routine omits checking of parameters and initialization of the data structures.\n•\nIf b=9,\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2452\n\n\nIndex\nDescription\n•\nIn the Cartesian case, the routine omits the adjustment of the right-hand side vector f\nto the Neumann boundary condition (multiplication of boundary values by 0.5 as well\nas incorporation of the boundary function g) and/or the Dirichlet boundary condition\n(setting boundary values to 0 as well as incorporation of the boundary function G).\n•\nFor the Helmholtz solver on a sphere, the routine omits computation of the spherical\nweights for the dpar/spar array.\n•\nIf a=9, the routine omits the normalization of the right-hand side vector f. Depending on\nthe solver, the normalization means:\n•\n2D Cartesian case: multiplication by hy2, where hy is the mesh size in the y direction\n(for details, see Poisson Solver Implementation).\n•\n3D (Cartesian) case: multiplication by hz2, where hz is the mesh size in the z direction.\n•\nHelmholtz solver on a sphere: multiplication by hθ2, where hθ is the mesh size in the θ\ndirection (for details, see Poisson Solver Implementation).\nUsing ipar[0] you can adjust the routine to your needs and improve efficiency in solving\nmultiple Helmholtz problems that differ only in the right-hand side. You must be cautious\nwhen using this method, because any misunderstanding of the commit process may cause\nincorrect results or program failure (see also Caveat on Parameter Modifications).\n1\nContains error messaging options:\n•\nipar[1]=-1 indicates that all error messages are printed to the\nMKL_Poisson_Library_log.txt file in the folder from which the routine is called. If the\nfile does not exist, the routine tries to create it. If the attempt fails, the routine prints\ninformation that the file cannot be created to the standard output device (usually, screen).\n•\nipar[1]=0 indicates that no error messages will be printed.\n•\nipar[1]=1 is the default value. It indicates that all error messages are printed to the\nstandard output device.\nIn case of errors, the stat parameter contains a non-zero value on exit from a routine\nregardless of the ipar[1] setting.\n2\nContains warning messaging options:\n•\nipar[2]=-1 indicates that all warning messages are printed to the\nMKL_Poisson_Library_log.txt file in the directory from which the routine is called. If\nthe file does not exist, the routine tries to create it. If the attempt fails, the routine prints\ninformation that the file cannot be created to the standard output device.\n•\nipar[2]=0 indicates that no warning messages will be printed.\n•\nipar[2]=1 is the default value. It indicates that all warning messages are printed to the\nstandard output device.\nIn case of warnings, the stat parameter contains a non-zero value on exit from a routine\nregardless of the ipar[2] setting.\n3 through\n5\nInternal parameters.\nParameters 6 through 11 are used only in the Cartesian case.\n6\nTakes this value:\n•\n2, if BCtype[0]='P'\n•\n1, if BCtype[0]='N'\n•\n0, if BCtype[0]='D'\n•\n-1, otherwise\n7\nTakes this value:\n•\n2, if BCtype[1]='P'\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2453\n\n\nIndex\nDescription\n•\n1, if BCtype[1]='N'\n•\n0, if BCtype[1]='D'\n•\n-1, otherwise\n8\nTakes this value:\n•\n2, if BCtype[2]='P'\n•\n1, if BCtype[2]='N'\n•\n0, if BCtype[2]='D'\n•\n-1, otherwise\n9\nTakes this value:\n•\n2, if BCtype[3]='P'\n•\n1, if BCtype[3]='N'\n•\n0, if BCtype[3]='D'\n•\n-1, otherwise\n10\nTakes this value:\n•\n2, if BCtype[4]='P'\n•\n1, if BCtype[4]='N'\n•\n0, if BCtype[4]='D'\n•\n-1, otherwise\n11\nTakes this value:\n•\n2, if BCtype[5]='P'\n•\n1, if BCtype[5]='N'\n•\n0, if BCtype[5]='D'\n•\n-1, otherwise\n12\nTakes the value of\n•\nnx, that is, the number of intervals along the x-axis, in the Cartesian case.\n•\nnp, that is, the number of intervals along the φ-axis, in the spherical case.\n13\nTakes the value of\n•\nny, that is, the number of intervals along the y-axis, in the Cartesian case\n•\nnt, that is, the number of intervals along the θ-axis, in the spherical case.\n14\nTakes the value of nz, the number of intervals along the z-axis. This parameter is used only\nin the 3D case (Cartesian).\n15\nthrough\n22\nInternal parameters which define the internal partitioning of the dpar/spar array.\nThe values of ipar[21] - ipar[119] are assigned regardless of the dimension of the problem for the\nCartesian solver or of whether the solver on a sphere is periodic.\n23\nContains message style options. Specifically:\n•\nipar[21]=0 indicates that Poisson Solver routines prints the messages in Fortran-style\nnotations.\n•\nipar[21]=1 (default) indicates that Poisson Solver routines prints the messages in C-\nstyle notations.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2454\n\n\nIndex\nDescription\n24\nContains the number of OpenMP threads to be used for computations in a multithreaded\nenvironment. The default value is 1 in the serial mode, and the result returned by the \nmkl_get_max_threads function otherwise.\n25\nthrough\n28\nInternal parameters which define the internal partitioning of the dpar/spar array.\n25\nTakes the value of ipar[18]+1, which specifies the internal partitioning of the dpar/spar\narray in the periodic Cartesian case.\n26\nTakes the value of ipar[23]+3*ipar[12]/4, which specifies the internal partitioning of the\ndpar/spar array in the periodic Cartesian case.\n27\nTakes the value of ipar[20]+1, which specifies the internal partitioning of the dpar/spar\narray in the periodic 3D Cartesian case.\n28\nTakes the value of ipar[25]+3*ipar[13]/4, which specifies the internal partitioning of the\ndpar/spar array in the periodic 3D Cartesian case.\n29\nthrough\n39\nUnused.\n40\nthrough\n59\nContain the first twenty elements of the ipar array of the first Trigonometric Transform that\nthe solver uses. (For details, see Common Parameters in the \"Trigonometric Transform\nRoutines\" section.)\n60\nthrough\n79\nContain the first twenty elements of the ipar array of the second Trigonometric Transform\nthat the 3D Cartesian and periodic spherical solvers use. (For details, see Common\nParameters in the \"Trigonometric Transform Routines\" section.)\n80\nthrough\n99\nContain the first twenty elements of the ipar array of the third Trigonometric Transform that\nthe solver uses in case of periodic boundary conditions along the x-axis. (For details, see \nCommon Parameters in the \"Trigonometric Transform Routines\" section.)\n100\nthrough\n119\nContain the first twenty elements of the ipar array of the fourth Trigonometric Transform\nused by periodic spherical solvers and 3D Cartesian solvers with periodic boundary conditions\nalong the y-axis. (For details, see Common Parameters in the \"Trigonometric Transform\nRoutines\" section.)\n120\nthrough\n128\nInternal parameters used by nonuniform 3D solvers.\nNOTE\nWhile you can declare the ipar array as MKL_INT ipar[120], for future compatibility you should\ndeclare ipar as MKL_INT ipar[128].\ndpar and spar\nArrays dpar and spar are the same except in the data precision:\ndpar\nHolds data needed for double-precision Fast Helmholtz Solver computations.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2455\n\n\n•\nFor the Cartesian solver, double array of size 5*nx/2+7 in the 2D case or\n5*(nx+ny)/2+9 in the 3D case; initialized in the \nd_init_Helmholtz_2D/d_init_Helmholtz_3D and \nd_commit_Helmholtz_2D/d_commit_Helmholtz_3D routines.\n•\nFor the spherical solver, double array of size 5*np/2+nt+10; initialized in the \nd_init_sph_p/d_init_sph_np and d_commit_sph_p/d_commit_sph_np\nroutines.\nspar\nHolds data needed for single-precision Fast Helmholtz Solver computations.\n•\nFor the Cartesian solver, float array of size 5*nx/2+7 in the 2D case or\n5*(nx+ny)/2+9 in the 3D case; initialized in the \ns_init_Helmholtz_2D/s_init_Helmholtz_3D and \ns_commit_Helmholtz_2D/s_commit_Helmholtz_3D routines.\n•\nFor the spherical solver, float array of size 5*np/2+nt+10; initialized in the \ns_init_sph_p/s_init_sph_np and s_commit_sph_p/s_commit_sph_np\nroutines.\nBecause dpar and spar have similar elements in each position, the elements are described together in Table\n\"Elements of the dpar and spar Arrays\":\nElements of the dpar and spar Arrays\nIndex\nDescription\n0\nIn the Cartesian case, contains the length of the interval along the x-axis right after a\ncall to the ?_init_Helmholtz_2D/?_init_Helmholtz_3D routine or the mesh size\nhx in the x direction (for details, see Poisson Solver Implementation) after a call to\nthe ?_commit_Helmholtz_2D/?_commit_Helmholtz_3D routine.\nIn the spherical case, contains the length of the interval along the φ-axis right after a\ncall to the ?_init_sph_p/?_init_sph_np routine or the mesh size hφ in the φ\ndirection (for details, see Poisson Solver Implementation) after a call to\nthe ?_commit_sph_p/?_commit_sph_np routine.\n1\nIn the Cartesian case, contains the length of the interval along the y-axis right after a\ncall to the ?_init_Helmholtz_2D/?_init_Helmholtz_3D routine or the mesh size\nhy in the y direction (for details, see Poisson Solver Implementation) after a call to\nthe ?_commit_Helmholtz_2D/?_commit_Helmholtz_3D routine.\nIn the spherical case, contains the length of the interval along the θ-axis right after a\ncall to the ?_init_sph_p/?_init_sph_np routine or the mesh size hθ in the θ\ndirection (for details, see Poisson Solver Implementation) after a call to\nthe ?_commit_sph_p/?_commit_sph_np routine.\n2\nIn the Cartesian case, contains the length of the interval along the z-axis right after a\ncall to the ?_init_Helmholtz_2D/?_init_Helmholtz_3D routine or the mesh size\nhz in the z direction (for details, see Poisson Solver Implementation) after a call to\nthe ?_commit_Helmholtz_2D/?_commit_Helmholtz_3D routine. In the Cartesian\nsolver, this parameter is used only in the 3D case.\nIn the spherical solver, contains the coordinate of the leftmost boundary along the θ-\naxis after a call to the ?_init_sph_p/?_init_sph_np routine.\n3\nContains the value of the coefficient q after a call to\nthe\n?_init_Helmholtz_2D/?_init_Helmholtz_3D/?_init_sph_p/?_init_sph_np\nroutine.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2456\n\n\nIndex\nDescription\n4\nContains the tolerance parameter after a call to\nthe\n?_init_Helmholtz_2D/?_init_Helmholtz_3D/?_init_sph_p/?_init_sph_np\nroutine.\n•\nIn the Cartesian case, this value is used only for the pure Neumann boundary\nconditions ( BCtype=\"NNNN\" in the 2D case; BCtype=\"NNNNNN\" in the 3D case).\nThis is a special case, because the right-hand side of the problem cannot be\narbitrary if the coefficient q is zero. The Poisson Solver verifies that the classical\nsolution exists (up to rounding errors) using this tolerance. In any case, the\nPoisson Solver computes the normal solution, that is, the solution that has the\nminimal Euclidean norm of residual. Nevertheless,\nthe ?_Helmholtz_2D/?_Helmholtz_3D routine informs you that the solution may\nnot exist in a classical sense (up to rounding errors).\n•\nIn the spherical case, the value is used for the special case of a periodic problem\non the entire sphere. This special case is similar to the Cartesian case with pure\nNeumann boundary conditions. Here the Poisson Solver computes the normal\nsolution as well. The parameter is also used to detect whether the problem is\nperiodic up to rounding errors.\nThe default value for this parameter is 1.0E-10 in case of double-precision\ncomputations or 1.0E-4 in case of single-precision computations. You can increase the\nvalue of the tolerance, for instance, to avoid the warnings that may appear.\nipar[15]-1\nthrough\nipar[16]-1\nIn the Cartesian case, contain the spectrum of the one-dimensional (1D) problem\nalong the x-axis after a call to\nthe ?_commit_Helmholtz_2D/?_commit_Helmholtz_3D routine.\nIn the spherical case, contains the spectrum of the 1D problem along the φ-axis after\na call to the ?_commit_sph_p/?_commit_sph_np routine.\nipar[17]-1\nthrough\nipar[18]-1\nIn the Cartesian case, contain the spectrum of the 1D problem along the y-axis after\na call to the ?_commit_Helmholtz_3D routine. These elements are used only in the\n3D case.\nIn the spherical case, contains the spherical weights after a call to\nthe ?_commit_sph_p/?_commit_sph_np routine.\nipar[19]-1\nthrough\nipar[20]-1\nTake the values of the (staggered) sine/cosine in the mesh points:\n•\nalong the x-axis after a call to\nthe ?_commit_Helmholtz_2D/?_commit_Helmholtz_3D routine for a Cartesian\nsolver\n•\nalong the φ-axis after a call to the ?_commit_sph_p/?_commit_sph_np routine\nfor a spherical solver.\nipar[21]-1\nthrough\nipar[22]-1\nTake the values of the (staggered) sine/cosine in the mesh points:\n•\nalong the y-axis after a call to the ?_commit_Helmholtz_3D routine for a\nCartesian 3D solver\n•\nalong the φ-axis after a call to the ?_commit_sph_p routine for a spherical\nperiodic solver.\nThese elements are not used in the 2D Cartesian case and in the non-periodic\nspherical case.\nipar[25]-1\nthrough\nipar[26]-1\nTake the values of the (staggered) sine/cosine in the mesh points along the x-axis\nafter a call to the ?_commit_Helmholtz_2D/?_commit_Helmholtz_3D routine.\nThese elements are used only in the periodic Cartesian case.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2457\n\n\nIndex\nDescription\nipar[27]-1\nthrough\nipar[28]\nTake the values of the (staggered) sine/cosine in the mesh points along the x-axis\nafter a call to the ?_commit_Helmholtz_3D routine. These elements are used only in\nthe periodic 3D Cartesian case.\nNOTE\nYou may define the array size depending upon the type of the problem to solve.\nCaveat on Parameter Modifications\nFlexibility of the Poisson Solver interface enables you to skip calls to\nthe ?_init_Helmholtz_2D/?_init_Helmholtz_3D/?_init_sph_p/?_init_sph_np routine and to\ninitialize the basic data structures explicitly in your code. You may also need to modify contents of the ipar,\ndpar, and spar arrays after initialization. When doing so, provide correct and consistent data in the arrays.\nMistakenly altered arrays cause errors or incorrect results. You can perform a basic check for correctness and\nconsistency of parameters by calling the ?_commit_Helmholtz_2D/?_commit_Helmholtz_3D routine;\nhowever, this does not ensure the correct solution but only reduces the chance of errors or wrong results.\nNOTE\nTo supply correct and consistent parameters to Poisson Solver routines, you should have considerable\nexperience in using the Poisson Solver interface and good understanding of the solution process, as\nwell as elements contained in the ipar, spar, and dpar arrays and dependencies between values of\nthese elements.\nIn rare occurrences when you fail in tuning parameters for the Fast Helmholtz Solver, refer for technical\nsupport at http://www.intel.com/software/products/support/ .\nWARNING\nThe only way that ensures a proper solution of a Helmholtz problem is to follow a typical sequence of\ninvoking the routines and not change the default set of parameters. So, avoid modifications of ipar,\ndpar, and spar arrays unless it is necessary.\nParameters That Define Boundary Conditions\nPoisson Solver routines for the Cartesian solver use the following common parameters to define the boundary\nconditions.\nParameters to Define Boundary Conditions for the Cartesian Solver\nParameter\nDescription\nbd_ax\ndouble* for d_commit_Helmholtz_2D/d_commit_Helmholtz_3D and\nd_Helmholtz_2D/d_Helmholtz_3D, \nfloat* for s_commit_Helmholtz_2D/s_commit_Helmholtz_3D and\ns_Helmholtz_2D/s_Helmholtz_3D.\nContains values of the boundary condition on the leftmost boundary of the domain along\nthe x-axis.\n•\n2D problem: the size of the array is ny+1. Its contents depend on the boundary\nconditions as follows:\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2458\n\n\nParameter\nDescription\n•\nDirichlet boundary condition (value of BCtype[0] is 'D'): values of the function G(ax,\nyj), j=0, ..., ny.\n•\nNeumann boundary condition (value of BCtype[0] is 'N'): values of the function\ng(ax, yj), j=0, ..., ny.\nThe value corresponding to the index j is placed in bd_ax[j].\n•\n3D problem: the size of the array is (ny+1)*(nz+1). Its contents depend on the\nboundary conditions as follows:\n•\nDirichlet boundary condition (value of BCtype[0] is 'D'): values of the function G(ax,\nyj, zk), j=0, ..., ny, k=0, ..., nz.\n•\nNeumann boundary condition (value of BCtype[0] is 'N'): the values of the function\ng(ax, yj, zk), j=0, ..., ny, k=0, ..., nz.\nThe values are packed in the array so that the value corresponding to indices (j, k) is\nplaced in bd_ax[j+k*(ny+1)].\nFor periodic boundary conditions (the value of BCtype[0] is 'P'), this parameter is not\nused, so it can accept a dummy pointer.\nbd_bx\ndouble* for d_commit_Helmholtz_2D/d_commit_Helmholtz_3D and\nd_Helmholtz_2D/d_Helmholtz_3D, \nfloat* for s_commit_Helmholtz_2D/s_commit_Helmholtz_3D and\ns_Helmholtz_2D/s_Helmholtz_3D.\nContains values of the boundary condition on the rightmost boundary of the domain along\nthe x-axis.\n•\n2D problem: the size of the array is ny+1. Its contents depend on the boundary\nconditions as follows:\n•\nDirichlet boundary condition (value of BCtype[1] is 'D'): values of the function G(bx,\nyj), j=0, ..., ny.\n•\nNeumann boundary condition (value of BCtype[1] is 'N'): values of the function\ng(bx, yj), j=0, ..., ny.\nThe value corresponding to the index j is placed in bd_bx[j].\n•\n3D problem: the size of the array is (ny+1)*(nz+1). Its contents depend on the\nboundary conditions as follows:\n•\nDirichlet boundary condition (value of BCtype[1] is 'D'): values of the function G(bx,\nyj, zk), j=0, ..., ny, k=0, ..., nz.\n•\nNeumann boundary condition (value of BCtype[1] is 'N'): values of the function\ng(bx, yj, zk), j=0, ..., ny, k=0, ..., nz.\nThe values are packed in the array so that the value corresponding to indices (j, k) is\nplaced in bd_bx[j+k*(ny+1)].\nFor periodic boundary conditions (the value of BCtype[1] is 'P'), this parameter is not\nused, so it can accept a dummy pointer.\nbd_ay\ndouble* for d_commit_Helmholtz_2D/d_commit_Helmholtz_3D and\nd_Helmholtz_2D/d_Helmholtz_3D, \nfloat* for s_commit_Helmholtz_2D/s_commit_Helmholtz_3D and\ns_Helmholtz_2D/s_Helmholtz_3D.\nContains values of the boundary condition on the leftmost boundary of the domain along\nthe y-axis.\n•\n2D problem: the size of the array is nx+1. Its contents depend on the boundary\nconditions as follows:\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2459\n\n\nParameter\nDescription\n•\nDirichlet boundary condition (value of BCtype[2] is 'D'): values of the function G(xi,\nay), i=0, ..., nx.\n•\nNeumann boundary condition (value of BCtype[2] is 'N'): values of the function g(xi,\nay), i=0, ..., nx.\nThe value corresponding to the index i is placed in bd_ay[i].\n•\n3D problem: the size of the array is (nx+1)*(nz+1). Its contents depend on the\nboundary conditions as follows:\n•\nDirichlet boundary condition (value of BCtype[2] is 'D'): values of the function\nG(xi,ay, zk), i=0, ..., nx, k=0, ..., nz.\n•\nNeumann boundary condition (value of BCtype[2] is 'N'): values of the function\ng(xi,ay, zk), i=0, ..., nx, k=0, ..., nz.\nThe values are packed in the array so that the value corresponding to indices (i, k) is\nplaced in bd_ay[i+k*(nx+1)].\nFor periodic boundary conditions (the value of BCtype[2] is 'P'), this parameter is not\nused, so it can accept a dummy pointer.\nbd_by\ndouble* for d_commit_Helmholtz_2D/d_commit_Helmholtz_3D and\nd_Helmholtz_2D/d_Helmholtz_3D, \nfloat* for s_commit_Helmholtz_2D/s_commit_Helmholtz_3D and\ns_Helmholtz_2D/s_Helmholtz_3D.\nContains values of the boundary condition on the rightmost boundary of the domain along\nthe y-axis.\n•\n2D problem: the size of the array is nx+1. Its contents depend on the boundary\nconditions as follows:\n•\nDirichlet boundary condition (value of BCtype[3] is 'D'): values of the function G(xi,\nby), i=0, ..., nx.\n•\nNeumann boundary condition (value of BCtype[3] is 'N'): values of the function g(xi,\nby), i=0, ..., nx.\nThe value corresponding to the index i is placed in bd_by[i].\n•\n3D problem: the size of the array is (nx+1)*(nz+1). Its contents depend on the\nboundary conditions as follows:\n•\nDirichlet boundary condition (value of BCtype[3] is 'D'): values of the function\nG(xi,by, zk), i=0, ..., nx, k=0, ..., nz.\n•\nNeumann boundary condition (value of BCtype[3] is 'N'): values of the function\ng(xi,by, zk), i=0, ..., nx, k=0, ..., nz.\nThe values are packed in the array so that the value corresponding to indices (i, k) is\nplaced in bd_by[i+k*(nx+1)].\nFor periodic boundary conditions (the value of BCtype[3] is 'P'), this parameter is not\nused, so it can accept a dummy pointer.\nbd_az\ndouble* for d_commit_Helmholtz_3D and d_Helmholtz_3D,\nfloat* for s_commit_Helmholtz_3D and s_Helmholtz_3D.\nUsed only by ?_commit_Helmholtz_3D and ?_Helmholtz_3D. Contains values of the\nboundary condition on the leftmost boundary of the domain along the z-axis.\nThe size of the array is (nx+1)*(ny+1). Its contents depend on the boundary conditions as\nfollows:\n•\nDirichlet boundary condition (value of BCtype[4] is 'D'): values of the function G(xi,\nyj,az), i=0, ..., nx, j=0, ..., ny.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2460\n\n\nParameter\nDescription\n•\nNeumann boundary condition (value of BCtype[4] is 'N'), values of the function g(xi,\nyj,az), i=0, ..., nx, j=0, ..., ny.\nThe values are packed in the array so that the value corresponding to indices (i, j) is\nplaced in bd_az[i+j*(nx+1)].\nFor periodic boundary conditions (the value of BCtype[4] is 'P'), this parameter is not\nused, so it can accept a dummy pointer.\nbd_bz\ndouble* for d_commit_Helmholtz_3D and d_Helmholtz_3D, \nfloat* for s_commit_Helmholtz_3D and s_Helmholtz_3D.\nUsed only by ?_commit_Helmholtz_3D and ?_Helmholtz_3D. Contains values of the\nboundary condition on the rightmost boundary of the domain along the z-axis.\nThe size of the array is (nx+1)*(ny+1). Its contents depend on the boundary conditions as\nfollows:\n•\nDirichlet boundary condition (value of BCtype[5] is 'D'): values of the function G(xi,\nyj,bz), i=0, ..., nx, j=0, ..., ny.\n•\nNeumann boundary condition (value of BCtype[5] is 'N'): values of the function g(xi,\nyj,bz), i=0, ..., nx, j=0, ..., ny.\nThe values are packed in the array so that the value corresponding to indices (i, j) is\nplaced in bd_bz[i+j*(nx+1)].\nFor periodic boundary conditions (the value of BCtype[5] is 'P'), this parameter is not\nused, so it can accept a dummy pointer.\nSee Also\n?_commit_Helmholtz_2D/?_commit_Helmholtz_3D\n?_Helmholtz_2D/?_Helmholtz_3D\nPoisson Solver Implementation Details\nSeveral aspects of the Intel® oneAPI Math Kernel Library (oneMKL) Poisson Solver interface are platform-\nspecific and language-specific. To promote portability of the Intel® oneAPI Math Kernel Library (oneMKL)\nPoisson Solver interface across platforms and ease of use across different languages, Intel® oneAPI Math\nKernel Library (oneMKL) provides you with the Poisson Solver language-specific header file to include in your\ncode:\n•\nmkl_poisson.h, to be used together with mkl_dfti.h.\nNOTE\n•\nUse of the Intel® oneAPI Math Kernel Library (oneMKL) Poisson Solver software without including\nthe above language-specific header files is not supported.\nHeader File\nThe header file defines the function prototypes for the Cartesian and spherical solver available in specific\nfunction descriptions.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2461\n\n\nNonlinear Optimization Problem Solvers\nIntel® oneAPI Math Kernel Library (oneMKL) provides tools for solving nonlinear least squares problems using\nthe Trust-Region (TR) algorithms. The general nonlinear solver workflow and naming conventions are\ndescribed here:\n•\nNonlinear Solver Organization and Implementation\n•\nNonlinear Solver Routine Naming Conventions\nThe solver routines are grouped according to their purpose as follows:\n•\nNonlinear Least Squares Problem without Constraints\n•\nNonlinear Least Squares Problem with Linear (Boundary) Constraints\n•\nJacobian Matrix Calculation Routines\nFor more information on the key concepts required to understand the use of the Intel® oneAPI Math Kernel\nLibrary (oneMKL) nonlinear least squares problem solver routines, see [Conn00].\nNonlinear Solver Organization and Implementation\nThe Intel® oneAPI Math Kernel Library (oneMKL) solver routines for nonlinear least squares problems use\nreverse communication interfaces (RCI). That means you need to provide the solver with information\nrequired for the iteration process, for example, the corresponding Jacobian matrix, or values of the objective\nfunction. RCI removes the dependency of the solver on specific implementation of the operations. However, it\ndoes require that you organize a computational loop.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2462\n\n\n__border__top\nTypical order for invoking RCI solver routines\nThe nonlinear least squares problem solver routines, or Trust-Region (TR) solvers, are implemented with\nthreading support. You can manage the threads using Threading Control Functions. The TR solvers use BLAS\nand LAPACK routines, and offer the same parallelism as those domains. The ?jacobi and ?jacobix routines\nof Jacobi matrix calculations are parallel. These routines (?jacobi and ?jacobix) make calls to the user-\nsupplied functions with different x parameters for multiple threads.\nMemory Allocation and Handles\nTo make the TR solver routines easy to use, you are not required to allocate temporary working storage. The\nsolver allocates all temporary memory internally. To allow multiple users to access the solver simultaneously,\nthe solver keeps track of the storage allocated for a particular application by using a data object called a\nhandle. Each TR solver routine creates, uses, or deletes a handle. The handle datatype definition can be\nfound in mkl.h (or mkl_rci.h/mkl_rci_win.h) .\nInclude one of the mentioned headers and declare the handle as :\n_TRNSP_HANDLE_t handle;\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2463\n\n\nor\n_TRNSPBC_HANDLE_t handle;\nThe first declaration is used for nonlinear least squares problems without boundary constraints, and the\nsecond is used for nonlinear least squares problems with boundary constraints.\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nNonlinear Solver Routine Naming Conventions\nThe TR routine names have the following structure:\n      <character><name>_<action>( )\nwhere\n•\n<character> indicates the data type:\ns\nfloat\nd\ndouble\n•\n<name> indicates the task type:\ntrnlsp\nnonlinear least squares problem without constraints\ntrnlspbc\nnonlinear least squares problem with boundary constraints\njacobi\ncomputation of the Jacobian matrix using central differences\n•\n<action> indicates an action on the task:\ninit\ninitializes the solver\ncheck\nchecks correctness of the input parameters\nsolve\nsolves the problem\nget\nretrieves the number of iterations, the stop criterion, the initial residual,\nand the final residual\ndelete\nreleases the allocated data\nNonlinear Least Squares Problem without Constraints\nThe nonlinear least squares problem without constraints can be described as follows:\nwhere\nF(x) : Rn → Rm is a twice differentiable function in Rn.\nSolving a nonlinear least squares problem means searching for the best approximation to the vector y with\nthe model function fi(x) and nonlinear variables x. The best approximation means that the sum of squares\nof residuals yi - fi(x) is the minimum.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2464\n\n\nSee usage examples in the examples\\c\\nonlinear_solvers folderof your Intel® oneAPI Math Kernel\nLibrary (oneMKL) directory. Specifically, see ex_nlsqp_f.fex_nlsqp_c.c.\nRCI TR Routines\nRoutine Name\nOperation\n?trnlsp_init\nInitializes the solver.\n?trnlsp_check\nChecks correctness of the input parameters.\n?trnlsp_solve\nSolves a nonlinear least squares problem using the Trust-Region\nalgorithm.\n?trnlsp_get\nRetrieves the number of iterations, stop criterion, initial residual, and\nfinal residual.\n?trnlsp_delete\nReleases allocated data.\n?trnlsp_init\nInitializes the solver of a nonlinear least squares\nproblem.\nSyntax\nMKL_INT strnlsp_init (_TRNSP_HANDLE_t* handle, const MKL_INT* n, const MKL_INT* m,\nconst float* x, const float* eps, const MKL_INT* iter1, const MKL_INT* iter2, const\nfloat* rs);\nMKL_INT dtrnlsp_init (_TRNSP_HANDLE_t* handle, const MKL_INT* n, const MKL_INT* m,\nconst double* x, const double* eps, const MKL_INT* iter1, const MKL_INT* iter2, const\ndouble* rs);\nInclude Files\n•\nmkl.h\nDescription\nThe ?trnlsp_init routine initializes the solver.\nAfter initialization, all subsequent invocations of the ?trnlsp_solve routine should use the values of the\nhandle returned by ?trnlsp_init. This handle stores internal data, including pointers to the arrays x and\neps. It is important to not move or deallocate these arrays until after calling the ?trnlsp_delete routine.\nThe eps array contains a number indicating the stopping criteria:\neps Value\nDescription\n0\nΔ < eps[0]\n1\n||F(x)||2 < eps[1]\n2\nThe Jacobian matrix is singular.\n||J(x)[m*(j-1)...m*j-1]||2 < eps[2], j = 1, ..., n\n3\n||s||2 < eps[3]\n4\n||F(x)||2 - ||F(x) - J(x)s||2 < eps[4]\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2465\n\n\neps Value\nDescription\n5\nThe trial step precision. If eps[5] = 0, then the trial step meets the required\nprecision (≤ 1.0*10-10).\nNote:\n•\nJ(x) is the Jacobian matrix.\n•\nΔ is the trust-region area.\n•\nF(x) is the value of the functional.\n•\ns is the trial step.\nInput Parameters\nn\nLength of x.\nm\nLength of F(x).\nx\nArray of size n. Initial guess. A reference to this array is stored in handle for later\nuse and modification by ?trnlsp_solve\n.\neps\nArray of size 6; contains stopping criteria. See the values in the Description\nsection.\nA reference to this array is stored in handle for later use by ?trnlsp_solve\n.\niter1\nSpecifies the maximum number of iterations.\niter2\nSpecifies the maximum number of iterations of trial step calculation.\nrs\nDefinition of initial size of the trust region (boundary of the trial step). The\nrecommend minimum value is 0.1, and the recommended maximum value is\n100.0. Based on your knowledge of the objective function and initial guess you\ncan increase or decrease the initial trust region. It can influence the iteration\nprocess, for example, the direction of the iteration process and the number of\niterations. If you set rs to 0.0, the solver uses the default value, which is 100.0.\nOutput Parameters\nhandle\nType _TRNSP_HANDLE_t.\nres\nIndicates task completion status.\n•\nres = TR_SUCCESS - the routine completed the task normally.\n•\nres = TR_INVALID_OPTION - there was an error in the input parameters.\n•\nres = TR_OUT_OF_MEMORY - there was a memory error.\nTR_SUCCESS, TR_INVALID_OPTION, and TR_OUT_OF_MEMORY are defined in the\nmkl_rci.h include file.\nSee Also\n?trnlsp_solve\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2466\n\n\n?trnlsp_check\nChecks the correctness of handle and arrays\ncontaining Jacobian matrix, objective function, and\nstopping criteria.\nSyntax\nMKL_INT strnlsp_check (_TRNSP_HANDLE_t* handle, const MKL_INT* n, const MKL_INT* m,\nconst float* fjac, const float* fvec, const float* eps, MKL_INT* info);\nMKL_INT dtrnlsp_check (_TRNSP_HANDLE_t* handle, const MKL_INT* n, const MKL_INT* m,\nconst double* fjac, const double* fvec, const double* eps, MKL_INT* info);\nInclude Files\n•\nmkl.h\nDescription\nThe ?trnlsp_check routine checks the arrays passed into the solver as input parameters. If an array\ncontains any INF or NaN values, the routine sets the flag in output array info(see the description of the\nvalues returned in the Output Parameters section for the info array).\nInput Parameters\nhandle\nType _TRNSPBC_HANDLE_t.\nn\nLength of x.\nm\nLength of F(x).\nfjac\nArray of size m by n. Contains the Jacobian matrix of the function.\nfvec\nArray of size m. Contains the function values at X, where fvec[i] = (yi –\nfi(x)).\neps\nArray of size 6; contains stopping criteria. See the values in the Description\nsection of the ?trnlsp_init.\nOutput Parameters\ninfo\nArray of size 6.\nResults of input parameter checking:\nParameter\nUsed for\nVal\nue\nDescription\ninfo[0]\nFlags for\nhandle\n0\nThe handle is valid.\n1\nThe handle is not allocated.\ninfo[1]\nFlags for\nfjac\n0\nThe fjac array is valid.\n1\nThe fjac array is not allocated\n2\nThe fjac array contains NaN.\n3\nThe fjac array contains Inf.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2467\n\n\nParameter\nUsed for\nVal\nue\nDescription\ninfo[2]\nFlags for\nfvec\n0\nThe fvec array is valid.\n1\nThe fvec array is not allocated\nThe fvec array contains NaN.\n2\n3\nThe fvec array contains Inf.\ninfo[3]\nFlags for eps\n0\nThe eps array is valid.\n1\nThe eps array is not allocated\n2\nThe eps array contains NaN.\n3\nThe eps array contains Inf.\n4\nThe eps array contains a value\nless than or equal to zero.\nres\nInformation about completion of the task.\nres = TR_SUCCESS - the routine completed the task normally.\nTR_SUCCESS is defined in the mkl_rci.h include file.\n?trnlsp_solve\nSolves a nonlinear least squares problem using the TR\nalgorithm.\nSyntax\nMKL_INT strnlsp_solve (_TRNSP_HANDLE_t* handle, float* fvec, float* fjac, MKL_INT*\nRCI_Request);\nMKL_INT dtrnlsp_solve (_TRNSP_HANDLE_t* handle, double* fvec, double* fjac, MKL_INT*\nRCI_Request);\nInclude Files\n•\nmkl.h\nDescription\nThe ?trnlsp_solve routine uses the TR algorithm to solve nonlinear least squares problems.\nThe problem is stated as follows:\nwhere\n•\nF(x):Rn → Rm\n•\nm ≥ n\nFrom a current point xcurrent, the algorithm uses the trust-region approach:\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2468\n\n\nto get xnew = xcurrent + s that satisfies\nwhere\n•\nJ(x) is the Jacobian matrix\n•\ns is the trial step\n•\n||s||2 ≤ Δcurrent\n•\nΔ is the trust-region area.\nThe RCI_Request parameter provides additional information:\nRCI_Request Value\nDescription\n2\nRequest to calculate the Jacobian matrix and put the result into fjac\n1\nRequest to recalculate the function at vector X and put the result into fvec\n0\nOne successful iteration step on the current trust-region radius (that does not\nmean that the value of x has changed)\n-1\nThe algorithm has exceeded the maximum number of iterations\n-2\nΔ < eps[0]\n-3\n||F(x)||2 < eps[1]\n-4\nThe Jacobian matrix is singular.\n||J(x)[m*(j-1)...m*j-1]||2 < eps[2], j = 1, ..., n\n-5\n||s||2 < eps[3]\n-6\n||F(x)||2 - ||F(x) - J(x)s||2 < eps[4]\nNOTE\nIf it is possible to combine computations of the function and the jacobian (RCI_Request = 1\nand 2), you can do that and provide both updated values for fvec and fjac as fulfillment of\nRCI_Request =1 (and do nothing for RCI_Request = 2).\nInput Parameters\nhandle\nType _TRNSP_HANDLE_t.\nfvec\nArray of size m. Contains the function values at X, where fvec[i] = (yi –\nfi(x)).\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2469\n\n\nfjac\nArray of size m by n. Contains the Jacobian matrix of the function.\nOutput Parameters\nfvec\nArray of size m. Updated function evaluated at x.\nRCI_Request\nInforms about the task stage.\nSee the Description section for the parameter values and their meaning.\nres\nIndicates the task completion.\nres = TR_SUCCESS - the routine completed the task normally.\nTR_SUCCESS is defined in the mkl_rci.h include file.\n?trnlsp_get\nRetrieves the number of iterations, stop criterion,\ninitial residual, and final residual.\nSyntax\nMKL_INT strnlsp_get(_TRNSP_HANDLE_t* handle, MKL_INT* iter, MKL_INT* st_cr, float* r1,\nfloat* r2);\nMKL_INT dtrnlsp_get (_TRNSP_HANDLE_t* handle, MKL_INT* iter, MKL_INT* st_cr, double*\nr1, double* r2);\nInclude Files\n•\nmkl.h\nDescription\nThe routine retrieves the current number of iterations, the stop criterion, the initial residual, and final\nresidual.\nThe initial residual is the value of the functional (||y - f(x)||) of the initial x values provided by the user.\nThe final residual is the value of the functional (||y - f(x)||) of the final x resulting from the algorithm\noperation.\nThe st_cr parameter contains a number indicating the stop criterion:\nst_cr Value\nDescription\n1\nThe algorithm has exceeded the maximum number of iterations\n2\nΔ < eps[0]\n3\n||F(x)||2 < eps[1]\n4\nThe Jacobian matrix is singular.\n||J(x)[m*(j-1)...m*j-1]||2 < eps[2], j = 1, ..., n\n5\n||s||2 < eps[3]\n6\n||F(x)||2 - ||F(x) - J(x)s||2 < eps[4]\nNote:\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2470\n\n\n•\nJ(x) is the Jacobian matrix.\n•\nΔ is the trust-region area.\n•\nF(x) is the value of the functional.\n•\ns is the trial step.\nInput Parameters\nhandle\nType _TRNSP_HANDLE_t.\nOutput Parameters\niter\nContains the current number of iterations.\nst_cr\nContains the stop criterion.\nSee the Description section for the parameter values and their meanings.\nr1\nContains the residual, (||y - f(x)||) given the initial x.\nr2\nContains the final residual, that is, the value of the functional (||y - f(x)||)\nof the final x resulting from the algorithm operation.\nres\nIndicates the task completion.\nres = TR_SUCCESS - the routine completed the task normally.\nTR_SUCCESS is defined in the mkl_rci.h include file.\n?trnlsp_delete\nReleases allocated data.\nSyntax\nMKL_INT strnlsp_delete(_TRNSP_HANDLE_t* handle);\nMKL_INT dtrnlsp_delete(_TRNSP_HANDLE_t* handle);\nInclude Files\n•\nmkl.h\nDescription\nThe ?trnlsp_delete routine releases all memory allocated for the handle. Only after calling this routine is it\nsafe for the user to move or deallocate the memory referenced by x and eps.\nThis routine flags memory as not used, but to actually release all memory you must call the support function \nmkl_free_buffers.\nInput Parameters\nhandle\nType _TRNSP_HANDLE_t.\nOutput Parameters\nres\nIndicates the task completion.\nres = TR_SUCCESS means the routine completed the task normally.\nTR_SUCCESS is defined in the mkl_rci.h include file.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2471\n\n\nNonlinear Least Squares Problem with Linear (Bound) Constraints\nThe nonlinear least squares problem with linear bound constraints is very similar to the nonlinear least\nsquares problem without constraints but it has the following constraints:\nSee usage examples in the examples\\c\\nonlinear_solvers folderof your Intel® oneAPI Math Kernel\nLibrary (oneMKL) directory. Specifically, see ex_nlsqp_bc_c.c.\nNOTE There are two options for handling the boundary constraints enabled through the parameter\narray, see the description of eps[4]\nRCI TR Routines for Problem with Bound Constraints\nRoutine Name\nOperation\n?trnlspbc_init\nInitializes the solver.\n?trnlspbc_check\nChecks correctness of the input parameters.\n?trnlspbc_solve\nSolves a nonlinear least squares problem using RCI and the Trust-\nRegion algorithm.\n?trnlspbc_get\nRetrieves the number of iterations, stop criterion, initial residual, and\nfinal residual.\n?trnlspbc_delete\nReleases allocated data.\n?trnlspbc_init\nInitializes the solver of nonlinear least squares\nproblem with linear (boundary) constraints.\nSyntax\nMKL_INT strnlspbc_init (_TRNSPBC_HANDLE_t* handle, const MKL_INT* n, const MKL_INT* m,\nconst float* x, const float* LW, const float* UP, const float* eps, const MKL_INT*\niter1, const MKL_INT* iter2, const float* rs);\nMKL_INT dtrnlspbc_init (_TRNSPBC_HANDLE_t* handle, const MKL_INT* n, const MKL_INT* m,\nconst double* x, const double* LW, const double* UP, const double* eps, const MKL_INT*\niter1, const MKL_INT* iter2, const double* rs);\nDescription\nThe ?trnlspbc_init routine initializes the solver.\nAfter initialization, all subsequent invocations of the ?trnlspbc_solve routine should use the values of the\nhandle returned by ?trnlspbc_init. This handle stores internal data, including pointers to the arrays x,\nLW, UP, and eps. It is important to not move or deallocate these arrays until after calling\nthe ?trnlspbc_delete routine.\nThe eps array contains a number indicating the stopping criteria:\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2472\n\n\neps Value\nDescription\n0\nΔ < eps[0]\n1\n||F(x)||2 < eps[1]\n2\nThe Jacobian matrix is singular.\n||J(x)[m*(j-1)...m*j-1]||2 < eps[2], j = 1, ..., n\n3\n||s||2 < eps[3]\n4\n||F(x)||2 - ||F(x) - J(x)s||2 < |eps[4]|\nNOTE\nIf eps[4] > 0, an extra scaling is applied to ‘s’ after it has been selected, to ensure\nthat it does not leave the specified domain, but scales it down to not cross the\nboundary. This preserves the solution inside the boundary, but may result in getting\nstuck in a local minimum on the boundary and exiting early due to this stopping\ncriteria\nIf eps[4] < 0, extra scaling is not applied, which may result in the\nsolution, x, leaving the domain. If this occurs, try starting over with a\ndifferent initial condition.\n5\nThe trial step precision. If eps[5] = 0, then the trial step meets the required\nprecision (≤ 1.0*10-10).\nNote:\n•\nJ(x) is the Jacobian matrix.\n•\nΔ is the trust-region area.\n•\nF(x) is the value of the functional.\n•\ns is the trial step.\nInput Parameters\nn\nLength of x.\nm\nLength of F(x).\nx\nArray of size n. Initial guess. A reference to this array is stored in handle for later\nuse and modification by ?trnlspbc_solve.\nLW\nArray of size n.\nContains low bounds for x (lwi < xi ). A reference to this array is stored in handle\nfor later use by ?trnlspbc_solve.\nUP\nArray of size n.\nContains upper bounds for x (upi > xi ). A reference to this array is stored in\nhandle for later use by ?trnlspbc_solve.\neps\nArray of size 6; contains stopping criteria. See the values in the Description\nsection. A reference to this array is stored in handle for later use\nby ?trnlspbc_solve.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2473\n\n\niter1\nSpecifies the maximum number of iterations.\niter2\nSpecifies the maximum number of iterations of trial step calculation.\nrs\nDefinition of initial size of the trust region (boundary of the trial step). The\nrecommended minimum value is 0.1, and the recommended maximum value is\n100.0. Based on your knowledge of the objective function and initial guess you\ncan increase or decrease the initial trust region. It can influence the iteration\nprocess, for example, the direction of the iteration process and the number of\niterations. If you set rs to 0.0, the solver uses the default value, which is 100.0.\nOutput Parameters\nhandle\nType _TRNSPBC_HANDLE_t.\nres\nInforms about the task completion.\n•\nres = TR_SUCCESS - the routine completed the task normally.\n•\nres = TR_INVALID_OPTION - there was an error in the input parameters.\n•\nres = TR_OUT_OF_MEMORY - there was a memory error.\nTR_SUCCESS, TR_INVALID_OPTION, and TR_OUT_OF_MEMORY are defined in the\nmkl_rci.h include file.\n?trnlspbc_check\nChecks the correctness of handle and arrays\ncontaining Jacobian matrix, objective function, lower\nand upper bounds, and stopping criteria.\nSyntax\nMKL_INT strnlspbc_check (_TRNSPBC_HANDLE_t* handle, const MKL_INT* n, const MKL_INT* m,\nconst float* fjac, const float* fvec, const float* LW, const float* UP, const float*\neps, MKL_INT* info);\nMKL_INT dtrnlspbc_check (_TRNSPBC_HANDLE_t* handle, const MKL_INT* n, const MKL_INT* m,\nconst double* fjac, const double* fvec, const double* LW, const double* UP, const\ndouble* eps, MKL_INT* info);\nInclude Files\n•\nmkl.h\nDescription\nThe ?trnlspbc_check routine checks the arrays passed into the solver as input parameters. If an array\ncontains any INF or NaN values, the routine sets the flag in output array info(see the description of the\nvalues returned in the Output Parameters section for the info array).\nInput Parameters\nhandle\nType _TRNSPBC_HANDLE_t.\nn\nLength of x.\nm\nLength of F(x).\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2474\n\n\nfjac\nArray of size m by n. Contains the Jacobian matrix of the function.\nfvec\nArray of size m. Contains the function values at X, where fvec[i] = (yi –\nfi(x)).\nLW\nArray of size n.\nContains low bounds for x (lwi < xi ).\nUP\nArray of size n.\nContains upper bounds for x (upi > xi ).\neps\nArray of size 6; contains stopping criteria. See the values in the Description\nsection of the ?trnlspbc_init.\nOutput Parameters\ninfo\nArray of size 6.\nResults of input parameter checking:\nParameter\nUsed for\nVal\nue\nDescription\ninfo[0]\nFlags for\nhandle\n0\nThe handle is valid.\n1\nThe handle is not allocated.\ninfo[1]\nFlags for\nfjac\n0\nThe fjac array is valid.\n1\nThe fjac array is not allocated\n2\nThe fjac array contains NaN.\n3\nThe fjac array contains Inf.\ninfo[2]\nFlags for\nfvec\n0\nThe fvec array is valid.\n1\nThe fvec array is not allocated\n2\nThe fvec array contains NaN.\n3\nThe fvec array contains Inf.\ninfo[3]\nFlags for LW\n0\nThe LW array is valid.\n1\nThe LW array is not allocated\n2\nThe LW array contains NaN.\n3\nThe LW array contains Inf.\n4\nThe lower bound is greater\nthan the upper bound.\ninfo[4]\nFlags for up\n0\nThe up array is valid.\n1\nThe up array is not allocated\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2475\n\n\nParameter\nUsed for\nVal\nue\nDescription\n2\nThe up array contains NaN.\n3\nThe up array contains Inf.\n4\nThe upper bound is less than\nthe lower bound.\ninfo[5]\nFlags for eps\n0\nThe eps array is valid.\n1\nThe eps array is not allocated\n2\nThe eps array contains NaN.\n3\nThe eps array contains Inf.\n4\nThe eps array contains a value\nless than or equal to zero.\nres\nInformation about completion of the task.\nres = TR_SUCCESS - the routine completed the task normally.\nTR_SUCCESS is defined in the mkl_rci.h include file.\n?trnlspbc_solve\nSolves a nonlinear least squares problem with linear\n(bound) constraints using the Trust-Region algorithm.\nSyntax\nMKL_INT strnlspbc_solve (_TRNSPBC_HANDLE_t* handle, float* fvec, float* fjac, MKL_INT*\nRCI_Request);\nMKL_INT dtrnlspbc_solve (_TRNSPBC_HANDLE_t* handle, double* fvec, double* fjac,\nMKL_INT* RCI_Request);\nInclude Files\n•\nmkl.h\nDescription\nThe ?trnlspbc_solve routine, based on RCI, uses the Trust-Region algorithm to solve nonlinear least\nsquares problems with linear (bound) constraints. The problem is stated as follows:\nwhere\nli≤xi≤ui\ni = 1, ..., n.\nThe RCI_Request parameter provides additional information:\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2476\n\n\nRCI_Request Value\nDescription\n2\nRequest to calculate the Jacobian matrix and put the result into fjac\n1\nRequest to recalculate the function at vector X and put the result into fvec\n0\nOne successful iteration step on the current trust-region radius (that does not\nmean that the value of x has changed)\n-1\nThe algorithm has exceeded the maximum number of iterations\n-2\nΔ < eps[0]\n-3\n||F(x)||2 < eps[1]\n-4\nThe Jacobian matrix is singular.\n||J(x)[m*(j-1)...m*j-1]||2 < eps[2], j = 1, ..., n\n-5\n||s||2 < eps[3]\n-6\n||F(x)||2 - ||F(x) - J(x)s||2 < |eps[4]|\nNote:\n•\nJ(x) is the Jacobian matrix.\n•\nΔ is the trust-region area.\n•\nF(x) is the value of the functional.\n•\ns is the trial step.\nInput Parameters\nhandle\nType _TRNSPBC_HANDLE_t.\nfvec\nArray of size m. Contains the function values at X, where fvec[i] = (yi –\nfi(x)).\nfjac\nArray of size m by n. Contains the Jacobian matrix of the function.\nOutput Parameters\nfvec\nArray of size m. Updated function evaluated at x.\nRCI_Request\nInforms about the task stage.\nSee the Description section for the parameter values and their meaning.\nres\nInforms about the task completion.\nres = TR_SUCCESS means the routine completed the task normally.\nTR_SUCCESS is defined in the mkl_rci.h include file.\n?trnlspbc_get\nRetrieves the number of iterations, stop criterion,\ninitial residual, and final residual.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2477\n\n\nSyntax\nMKL_INT strnlspbc_get (_TRNSPBC_HANDLE_t* handle, MKL_INT* iter, MKL_INT* st_cr, float*\nr1, float* r2);\nMKL_INT dtrnlspbc_get (_TRNSPBC_HANDLE_t* handle, MKL_INT* iter, MKL_INT* st_cr,\ndouble* r1, double* r2);\nInclude Files\n•\nmkl.h\nDescription\nThe routine retrieves the current number of iterations, the stop criterion, the initial residual, and final\nresidual.\nThe st_cr parameter contains a number indicating the stop criterion:\nst_cr Value\nDescription\n1\nThe algorithm has exceeded the maximum number of iterations\n2\nΔ < eps[0]\n3\n||F(x)||2 < eps[1]\n4\nThe Jacobian matrix is singular.\n||J(x)[m*(j-1)...m*j-1]||2 < eps[2], j = 1, ..., n\n5\n||s||2 < eps[3]\n6\n||F(x)||2 - ||F(x) - J(x)s||2 < eps[4]\nNote:\n•\nJ(x) is the Jacobian matrix.\n•\nΔ is the trust-region area.\n•\nF(x) is the value of the functional.\n•\ns is the trial step.\nInput Parameters\nhandle\nType _TRNSPBC_HANDLE_t.\nOutput Parameters\niter\nContains the current number of iterations.\nst_cr\nContains the stop criterion.\nSee the Description section for the parameter values and their meanings.\nr1\nContains the residual, (||y - f(x)||) given the initial x.\nr2\nContains the final residual, that is, the value of the function (||y - f(x)||) of\nthe final x resulting from the algorithm operation.\nres\nInforms about the task completion.\nres = TR_SUCCESS - the routine completed the task normally.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2478\n\n\nTR_SUCCESS is defined in the mkl_rci.h include file.\n?trnlspbc_delete\nReleases allocated data.\nSyntax\nMKL_INT strnlspbc_delete (_TRNSPBC_HANDLE_t* handle);\nMKL_INT dtrnlspbc_delete (_TRNSPBC_HANDLE_t* handle);\nInclude Files\n•\nmkl.h\nDescription\nThe ?trnlspbc_delete routine releases all memory allocated for the handle. Only after calling this routine\nis it safe for the user to move or deallocate the memory referenced by x, LW, UP, and eps.\nNOTE\nThis routine flags memory as not used, but to actually release all memory you must call the\nsupport function mkl_free_buffers.\nInput Parameters\nhandle\nType _TRNSPBC_HANDLE_t.\nOutput Parameters\nres\nInforms about the task completion.\nres = TR_SUCCESS means the routine completed the task normally.\nTR_SUCCESS is defined in the mkl_rci.h include file.\nJacobian Matrix Calculation Routines\nThis section describes routines that compute the Jacobian matrix using the central difference algorithm.\nJacobian matrix calculation is required to solve a nonlinear least squares problem and systems of nonlinear\nequations (with or without linear bound constraints). Routines for calculation of the Jacobian matrix have the\n\"Black-Box\" interfaces, where you pass the objective function via parameters. Your objective function must\nhave a fixed interface.\nJacobian Matrix Calculation Routines\nRoutine Name\nOperation\n?jacobi_init\nInitializes the solver.\n?jacobi_solve\nComputes the Jacobian matrix of the function on the basis of RCI\nusing the central difference algorithm.\n?jacobi_delete\nRemoves data.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2479\n\n\nRoutine Name\nOperation\n?jacobi\nComputes the Jacobian matrix of the fcn function using the central\ndifference algorithm.\n?jacobix\nPresents an alternative interface for the ?jacobi function enabling\nyou to pass additional data into the objective function.\n?jacobi_init\nInitializes the solver for Jacobian calculations.\nSyntax\nMKL_INT sjacobi_init (_JACOBIMATRIX_HANDLE_t* handle, const MKL_INT* n, const MKL_INT*\nm, const float* x, const float* fjac, const float* eps);\nMKL_INT djacobi_init (_JACOBIMATRIX_HANDLE_t* handle, const MKL_INT* n, const MKL_INT*\nm, const double* x, const double* fjac, const double* eps);\nInclude Files\n•\nmkl.h\nDescription\nThe routine initializes the solver.\nInput Parameters\nn\nLength of x.\nm\nLength of F.\nx\nArray of size n. Vector, at which the function is evaluated.\nA reference to this array is stored in handle for later use and modification\nby ?jacobi_solve.\neps\nPrecision of the Jacobian matrix calculation.\nfjac\nArray of size m by n. Contains the Jacobian matrix of the function.\nA reference to this array is stored in handle for later use and modification\nby ?jacobi_solve.\nOutput Parameters\nhandle\nData object of the _JACOBIMATRIX_HANDLE_t type. Stores internal data,\nincluding pointers to the user-provided arrays x and fjac. It is important that the\nuser does not move or deallocate these arrays until after calling\nthe ?jacobi_delete routine.\nres\nIndicates task completion status.\n•\nres = TR_SUCCESS - the routine completed the task normally.\n•\nres = TR_INVALID_OPTION - there was an error in the input parameters.\n•\nres = TR_OUT_OF_MEMORY - there was a memory error.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2480\n\n\nTR_SUCCESS, TR_INVALID_OPTION, and TR_OUT_OF_MEMORY are defined in the\nmkl_rci.h include file.\n?jacobi_solve\nComputes the Jacobian matrix of the function using\nRCI and the central difference algorithm.\nSyntax\nMKL_INT sjacobi_solve (_JACOBIMATRIX_HANDLE_t* handle, float* f1, float* f2, MKL_INT*\nRCI_Request);\nMKL_INT djacobi_solve (_JACOBIMATRIX_HANDLE_t* handle, double* f1, double* f2, MKL_INT*\nRCI_Request);\nInclude Files\n•\nmkl.h\nDescription\nThe ?jacobi_solve routine computes the Jacobian matrix of the function using RCI and the central\ndifference algorothm.\nSee usage examples in the examples\\solverc\\source folderof your Intel® oneAPI Math Kernel Library\n(oneMKL) directory. Specifically, see sjacobi_rci_c.c and djacobi_rci_c.c.\nInput Parameters\nhandle\nType _JACOBIMATRIX_HANDLE_t.\nRCI_Request\nSet to 0 before the first call to ?jacobi_solve.\nOutput Parameters\nf1\nContains the updated function values at x + eps.\nf2\nArray of size m. Contains the updated function values at x - eps.\nRCI_Request\nProvides information about the task completion. When equal to 0, the\ntask has completed successfully.\nRCI_Request= 1 indicates that you should compute the function\nvalues at the current x point and put the results into f1.\nRCI_Request= 2 indicates that you should compute the function\nvalues at the current x point and put the results into f2.\nres\nIndicates the task completion status.\n•\nres = TR_SUCCESS - the routine completed the task normally.\n•\nres = TR_INVALID_OPTION - there was an error in the input\nparameters.\nTR_SUCCESS and TR_INVALID_OPTION are defined in the mkl_rci.h\ninclude file.\nSee Also\n?jacobi_init\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2481\n\n\n?jacobi_delete\nReleases allocated data.\nSyntax\nMKL_INT sjacobi_delete (_JACOBIMATRIX_HANDLE_t* handle);\nMKL_INT djacobi_delete (_JACOBIMATRIX_HANDLE_t* handle);\nInclude Files\n•\nmkl.h\nDescription\nThe ?jacobi_delete routine releases all memory allocated for the handle. Only after calling this routine is it\nsafe for the user to move or deallocate the memory referenced by x and fjac.\nThis routine flags memory as not used, but to actually release all memory you must call the support function \nmkl_free_buffers.\nInput Parameters\nhandle\nType _JACOBIMATRIX_HANDLE_t.\nOutput Parameters\nres\nInforms about the task completion.\nres = TR_SUCCESS means the routine completed the task normally.\nTR_SUCCESS is defined in the mkl_rci.h include file.\n?jacobi\nComputes the Jacobian matrix of the objective\nfunction using the central difference algorithm.\nSyntax\nMKL_INT sjacobi (USRFCNS fcn, const MKL_INT* n, const MKL_INT* m, float* fjac, float*\nx, float* eps);\nMKL_INT djacobi (USRFCND fcn, const MKL_INT* n, const MKL_INT* m, double* fjac, double*\nx, double* eps);\nInclude Files\n•\nmkl.h\nDescription\nThe ?jacobi routine computes the Jacobian matrix for function fcn using the central difference algorithm.\nThis routine has a \"Black-Box\" interface, where you input the objective function via parameters. Your\nobjective function must have a fixed interface.\nSee calling and usage examples in the examples\\solverc\\source folderof your Intel® oneAPI Math Kernel\nLibrary (oneMKL) directory. Specifically, see ex_nlsqp_c.c and ex_nlsqp_bc_c.c.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2482\n\n\nInput Parameters\nfcn\nUser-supplied subroutine to evaluate the function that defines the least squares\nproblem. Called as fcn (m, n, x, f) with the following parameters:\nParameter\nType\nDescription\nInput Parameters\nm\nPointer to the length of f.\nn\nPointer to the length of x.\nx\nArray of size n. Vector, at which the\nfunction is evaluated. The fcn function\nshould not change this parameter.\nOutput Parameters\nf\nArray of size m; contains the function\nvalues at x.\nYou need to declare fcn as extern in the calling program.\nn\nLength of X.\nm\nLength of F.\nx\nArray of size n. Vector at which the function is evaluated.\neps\nPrecision of the Jacobian matrix calculation.\nOutput Parameters\nfjac\nArray of size m by n. Contains the Jacobian matrix of the function.\nres\nIndicates task completion status.\n•\nres = TR_SUCCESS - the routine completed the task normally.\n•\nres = TR_INVALID_OPTION - there was an error in the input parameters.\n•\nres = TR_OUT_OF_MEMORY - there was a memory error.\nTR_SUCCESS, TR_INVALID_OPTION, and TR_OUT_OF_MEMORY are defined in the\nmkl_rci.h include file.\nSee Also\n?jacobix\n?jacobix\nAlternative interface for?jacobi function for passing\nadditional data into the objective function.\nSyntax\nMKL_INT sjacobix (USRFCNXS fcn, const MKL_INT* n, const MKL_INT * m, float* fjac,\nfloat* x, float* eps, void* user_data);\nMKL_INT djacobix (USRFCNXD fcn, const MKL_INT* n, const MKL_INT * m, double* fjac,\ndouble* x, double* eps, void* user_data);\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2483\n\n\nInclude Files\n•\nmkl.h\nDescription\nThe ?jacobix routine presents an alternative interface for the ?jacobi function that enables you to pass\nadditional data into the objective function fcn.\nSee calling and usage examples in the examples\\solverc\\source folderof your Intel® oneAPI Math Kernel\nLibrary (oneMKL) directory. Specifically, see ex_nlsqp_c_x.c and ex_nlsqp_bc_c_x.c.\nInput Parameters\nfcn\nUser-supplied subroutine to evaluate the function that defines the least squares\nproblem. Called as fcn (m, n, x, f, user_data) with the following parameters:\nParameter\nDescription\nInput Parameters\nm\nPointer to the length of f.\nn\nPointer to the length of x.\nx\nArray of size n. Vector, at which the\nfunction is evaluated. The fcn function\nshould not change this parameter.\nuser_data\nPointer to your additional data, if any.\nOtherwise, a dummy argument.\nOutput Parameters\nf\nArray of size m; contains the function\nvalues at x.\nYou need to declare fcn as extern in the calling program.\nn\nLength of X.\nm\nLength of F.\nx\nArray of size n. Vector at which the function is evaluated.\neps\nPrecision of the Jacobian matrix calculation.\nuser_data\nPointer to your additional data. If there is no additional data, this is a dummy\nargument.\nOutput Parameters\nfjac\nArray of size m by n). Contains the Jacobian matrix of the function.\nres\nIndicates task completion status.\n•\nres = TR_SUCCESS - the routine completed the task normally.\n•\nres = TR_INVALID_OPTION - there was an error in the input parameters.\n•\nres = TR_OUT_OF_MEMORY - there was a memory error.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2484\n\n\nTR_SUCCESS, TR_INVALID_OPTION, and TR_OUT_OF_MEMORY are defined in the\nmkl_rci.h include file.\nSee Also\n?jacobi\nSupport Functions\nIntel® oneAPI Math Kernel Library (oneMKL) support functions are subdivided into the following groups\naccording to their purpose:\nVersion Information\nThreading Control\nError Handling\nCharacter Equality Testing\nTiming\nMemory Management\nSingle Dynamic Library Control\nConditional Numerical Reproducibility Control\nMiscellaneous\nThe following table lists Intel® oneAPI Math Kernel Library (oneMKL) support functions.\noneMKL Support Functions\nFunction Name\nOperation\nVersion Information\nmkl_get_version\nReturns the Intel® oneAPI Math Kernel Library (oneMKL)\nversion.\nmkl_get_version_string\nReturns the Intel® oneAPI Math Kernel Library (oneMKL)\nversion in a character string.\nThreading Control\nmkl_set_num_threads\nSpecifies the number of OpenMP* threads to use.\nmkl_domain_set_num_threads\nSpecifies the number of OpenMP* threads for a particular\nfunction domain.\nmkl_set_num_threads_local\nSpecifies the number of OpenMP* threads for all Intel®\noneAPI Math Kernel Library (oneMKL) functions on the\ncurrent execution thread.\nmkl_set_dynamic\nEnables Intel® oneAPI Math Kernel Library (oneMKL) to\ndynamically change the number of OpenMP* threads.\nmkl_get_max_threads\nGets the number of OpenMP* threads targeted for\nparallelism.\nmkl_domain_get_max_threads\nGets the number of OpenMP* threads targeted for\nparallelism for a particular function domain.\nmkl_get_dynamic\nDetermines whether Intel® oneAPI Math Kernel Library\n(oneMKL) is enabled to dynamically change the number\nof OpenMP* threads.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2485\n\n\nFunction Name\nOperation\nmkl_set_num_stripes\nSpecifies the number of partitions along the leading\ndimension of the output matrix for parallel ?gemm\nfunctions.\nmkl_get_num_stripes\nGets the number of partitions along the leading\ndimension of the output matrix for parallel ?gemm\nfunctions.\nError Handling\nxerbla\nError handling function called by BLAS, LAPACK, Vector\nMath, and Vector Statistics functions.\npxerbla\nHandles error conditions for the ScaLAPACK routines.\nLAPACKE_xerbla\nError handling function called by the C interface to\nLAPACK functions.\nmkl_set_exit_handler\nSets the custom handler of fatal errors.\nCharacter Equality Testing\nlsame\nTests two characters for equality regardless of the case.\nlsamen\nTests two character strings for equality regardless of the\ncase.\nTiming\nsecond/dsecnd\nReturns elapsed time in seconds. Use to estimate real\ntime between two calls to this function.\nmkl_get_cpu_clocks\nReturns elapsed CPU clocks.\nmkl_get_cpu_frequency\nReturns CPU frequency value in GHz.\nmkl_get_max_cpu_frequency\nReturns the maximum CPU frequency value in GHz.\nmkl_get_clocks_frequency\nReturns the frequency value in GHz based on constant-\nrate Time Stamp Counter.\nMemory Management\nmkl_free_buffers\nFrees unused memory allocated by the Intel® oneAPI\nMath Kernel Library (oneMKL) Memory Allocator.\nmkl_thread_free_buffers\nFrees unused memory allocated by the Intel® oneAPI\nMath Kernel Library (oneMKL) Memory Allocator in the\ncurrent thread.\nmkl_mem_stat\nReports the status of the Intel® oneAPI Math Kernel\nLibrary (oneMKL) Memory Allocator.\nmkl_peak_mem_usage\nReports the peak memory allocated by the Intel® oneAPI\nMath Kernel Library (oneMKL) Memory Allocator.\nmkl_disable_fast_mm\nTurns off the Intel® oneAPI Math Kernel Library (oneMKL)\nMemory Allocator for Intel® oneAPI Math Kernel Library\n(oneMKL) functions to directly use the systemmalloc/\nfree functions.\nmkl_malloc\nAllocates an aligned memory buffer.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2486\n\n\nFunction Name\nOperation\nmkl_calloc\nAllocates and initializes an aligned memory buffer.\nmkl_realloc\nChanges the size of memory buffer allocated by\nmkl_malloc/mkl_calloc.\nmkl_free\nFrees the aligned memory buffer allocated by\nmkl_malloc/mkl_calloc.\nmkl_set_memory_limit\nOn Linux, sets the limit of memory that Intel® oneAPI\nMath Kernel Library (oneMKL) can allocate for a specified\ntype of memory.\nSingle Dynamic Library (SDL) Control\nmkl_set_interface_layer\nSets the interface layer for Intel® oneAPI Math Kernel\nLibrary (oneMKL) at run time.\nmkl_set_threading_layer\nSets the threading layer for Intel® oneAPI Math Kernel\nLibrary (oneMKL) at run time.\nmkl_set_xerbla\nReplaces the error handling routine. Use with the Single\nDynamic Library .\nmkl_set_progress\nReplaces the progress information routine.\nmkl_set_pardiso_pivot\nReplaces the routine handling Intel® oneAPI Math Kernel\nLibrary (oneMKL) PARDISO pivots with a user-defined\nroutine. Use with the Single Dynamic Library (SDL).\nConditional Numerical Reproducibility (CNR) Control\nmkl_cbwr_set\nConfigures the CNR mode of Intel® oneAPI Math Kernel\nLibrary (oneMKL).\nmkl_cbwr_get\nReturns the current CNR settings.\nmkl_cbwr_get_auto_branch\nAutomatically detects the CNR code branch for your\nplatform.\nMiscellaneous\nmkl_progress\nProvides progress information.\nmkl_enable_instructions\nmkl_set_env_mode\nSet up the mode that ignores environment settings\nspecific to Intel® oneAPI Math Kernel Library (oneMKL).\nmkl_verbose\nEnable or disable Intel® oneAPI Math Kernel Library\n(oneMKL) Verbose mode.\nmkl_verbose_output_file\nWrite output in Intel® oneAPI Math Kernel Library\n(oneMKL) Verbose mode to a file.\nmkl_set_mpi\nSets the implementation of the message-passing\ninterface to be used by Intel® oneAPI Math Kernel Library\n(oneMKL).\nmkl_finalize\nTerminates Intel® oneAPI Math Kernel Library (oneMKL)\nexecution environment and frees resources allocated by\nthe library.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2487\n\n\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nVersion Information\nIntel® oneAPI Math Kernel Library (oneMKL) providestwo methods for extracting information about the library\nversion number:\n•\nextracting a version string using the mkl_get_version_string function\n•\nusing the mkl_get_version function to obtain an MKLVersion structure that contains the version\ninformation\nA makefile is also provided to automatically build the examples and output summary files containing the\nversion information for the current library.\nmkl_get_version\nReturnsthe Intel® oneAPI Math Kernel Library\n(oneMKL) version.\nSyntax\nvoid mkl_get_version( MKLVersion* pVersion );\nInclude Files\n•\nmkl.h\nOutput Parameters\npVersion\nPointer to the MKLVersion structure.\nDescription\nThe mkl_get_versionfunction collects information about the active C version of the Intel® oneAPI Math\nKernel Library (oneMKL) software and returns this information in a structure ofMKLVersion type by the\npVersion address. The MKLVersion structure type is defined in the mkl_types.h file. The following fields of\nthe MKLVersion structure are available:\nMajorVersion\nis the major number of the current library version.\nMinorVersion\nis the minor number of the current library version.\nUpdateVersion\nis the update number of the current library version.\nProductStatus\nis the status of the current library version. Possible variants are “Beta”\nor “Product”.\nBuild\nis the string that contains the build date and the internal build\nnumber.\nPlatform\nis the string that contains the current architecture. Possible variants\nare \"IA-32 architecture\" or \"Intel(R) 64 architecture\".\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2488\n\n\nProcessor\nis the processor optimization. Normally it is targeted for the processor\ninstalled on your system and based on the detection of the Intel®\noneAPI Math Kernel Library (oneMKL) library that is optimal for the\ninstalled processor. In the Conditional Numerical Reproducibility (CNR)\nmode, the processor optimization matches the selected CNR branch.\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nmkl_get_version Usage\n----------------------------------------------------------------------------------------------\n#include <stdio.h>\n#include <stdlib.h>\n#include \"mkl.h\"\n \nint main(void)\n  {\n    MKLVersion Version;\n \n    mkl_get_version(&Version);\n \n \n    printf(\"Major version:           %d\\n\",Version.MajorVersion);\n    printf(\"Minor version:           %d\\n\",Version.MinorVersion);\n    printf(\"Update version:          %d\\n\",Version.UpdateVersion);\n    printf(\"Product status:          %s\\n\",Version.ProductStatus);\n    printf(\"Build:                   %s\\n\",Version.Build);\n    printf(\"Platform:                %s\\n\",Version.Platform);\n    printf(\"Processor optimization:  %s\\n\",Version.Processor);\n    printf(\"================================================================\\n\");\n    printf(\"\\n\");\n \n    return 0;\n  }\nOutput:\nMajor Version\n11\nMinor Version\n0\nUpdate Version\n2\nProduct status\nProduct\nBuild\n20121113\nPlatform\nIntel(R) 64 architecture\nProcessor optimization\nIntel(R) Core(TM) i7 Processor\nSee Also\nConditional Numerical Reproducibility Control\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2489\n\n\nmkl_get_version_string\nReturns the Intel® oneAPI Math Kernel Library\n(oneMKL) version in a character string.\nSyntax\nvoid mkl_get_version_string (char* buf, int len);\nInclude Files\n•\nmkl.h\nOutput Parameters\nName\nType\nDescription\nbuf\nchar*\nSource string\nlen\nint\nLength of the source string\nDescription\nThe function returns a string that contains the Intel® oneAPI Math Kernel Library (oneMKL) version.\nFor usage details, see the code example below:\nExample\n#include <stdio.h> \n#include \"mkl.h\"\nint main(void) \n{\n  int len=198;\n  char buf[198];\n  mkl_get_version_string(buf, len);\n  printf(\"%s\\n\",buf);\n  printf(\"\\n\");\n  return 0; \n}\nThreading Control\nIntel® oneAPI Math Kernel Library (oneMKL) provides functions for OpenMP* threading control, discussed in\nthis section.\nImportant\nIf Intel® oneAPI Math Kernel Library (oneMKL) operates within the Intel® Threading Building Blocks\n(Intel® TBB) execution environment, the environment variables for OpenMP* threading control, such\nasOMP_NUM_THREADS, and Intel® oneAPI Math Kernel Library (oneMKL) functions discussed in this\nsection have no effect. If the Intel TBB threading technology is used, control the number of threads\nthrough the Intel TBB application programming interface. Read the documentation for the\ntbb::task_scheduler_init class at https://www.threadingbuildingblocks.org/docs/doxygen/\na00150.html to find out how to specify the number of Intel TBB threads.\nIf Intel® oneAPI Math Kernel Library (oneMKL) operates within an OpenMP* execution environment, you can\ncontrol the number of threads for Intel® oneAPI Math Kernel Library (oneMKL) using OpenMP* runtime library\nroutines and environment variables (see the OpenMP* specification for details). Additionally Intel® oneAPI\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2490\n\n\nMath Kernel Library (oneMKL) providesoptionalthreading control functions and environment variables that\nenable you to specify the number of threads for Intel® oneAPI Math Kernel Library (oneMKL) and to control\ndynamic adjustment of the number of threadsindependentlyof the OpenMP* settings. The settings made with\nthe Intel® oneAPI Math Kernel Library (oneMKL) threading control functions and environment variables do not\naffect OpenMP* settings but take precedence over them.\nIf functions are used, Intel® oneAPI Math Kernel Library (oneMKL) environment variables may control Intel®\noneAPI Math Kernel Library (oneMKL) threading. For details of those environment variables, see the Intel®\noneAPI Math Kernel Library (oneMKL) Developer Guide.\nYou can specify the number of threads for Intel® oneAPI Math Kernel Library (oneMKL) function domains with\nthe mkl_set_num_threads or mkl_domain_set_num_threads function. While mkl_set_num_threads\nspecifies the number of threads for the entire Intel® oneAPI Math Kernel Library (oneMKL),\nmkl_domain_set_num_threads does it for a specific function domain. The following table lists the function\ndomains that support independent threading control. The table also provides named constants to pass to\nthreading control functions as a parameter that specifies the function domain.\noneMKL Function Domains\nFunction Domain\nNamed Constant\nBasic Linear Algebra Subroutines (BLAS)\nMKL_DOMAIN_BLAS\nFast Fourier Transform (FFT) functions, except Cluster FFT functions\nMKL_DOMAIN_FFT\nVector Math (VM) functions\nMKL_DOMAIN_VML\nParallel Direct Solver (PARDISO) functions\nMKL_DOMAIN_PARDISO\nAll Intel® oneAPI Math Kernel Library (oneMKL) functions except the\nfunctions from the domains where the number of threads is set\nexplicitly.\nMKL_DOMAIN_ALL\nWarning\nDo not increase the number of OpenMP threads used for cluster_sparse_solver between the first call\nand the factorization or solution phase. Because the minimum amount of memory required for out-of-\ncore execution depends on the number of OpenMP threads, increasing it after the initial call can cause\nincorrect results.\nBoth mkl_set_num_threads and mkl_domain_set_num_threads functions set the number of threads for\nall subsequent calls to Intel® oneAPI Math Kernel Library (oneMKL) from all applications threads. Use\nthemkl_set_num_threads_local function to specify different numbers of threads for Intel® oneAPI Math Kernel\nLibrary (oneMKL) on different execution threads of your application. The thread-local settings take\nprecedence over the global settings. However, the thread-local settings may have undesirable side effects\n(see the description of themkl_set_num_threads_local function for details).\nBy default, Intel® oneAPI Math Kernel Library (oneMKL) canadjust the specified number of threads\ndynamically. For example, Intel® oneAPI Math Kernel Library (oneMKL) may use fewer threads if the size of\nthe computation is not big enough or not create parallel regions when running within an OpenMP* parallel\nregion. Although Intel® oneAPI Math Kernel Library (oneMKL) may actually use a different number of threads\nfrom the number specified, the library does not create parallel regions with more threads than specified. If\ndynamic adjustment of the number of threads is disabled, Intel® oneAPI Math Kernel Library (oneMKL)\nattempts to use the specified number of threads in internal parallel regions (for more information, see the\nIntel® oneAPI Math Kernel Library (oneMKL) Developer Guide). Use the mkl_set_dynamic function to control\ndynamic adjustment of the number of threads.\nmkl_set_num_threads\nSpecifies the number of OpenMP* threads to use.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2491\n\n\nSyntax\nvoid mkl_set_num_threads( int nt );\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nnt\nint\nnt > 0 - The number of threads suggested by the user.\nnt≤ 0 - Invalid value, which is ignored.\nDescription\nThis function enables you to specify how many OpenMP threads Intel® oneAPI Math Kernel Library (oneMKL)\nshould use for internal parallel regions. If this number is not set (default), Intel® oneAPI Math Kernel Library\n(oneMKL) functions use the default number of threads for the OpenMP run-time library. The specified number\nof threads applies:\n•\nTo all Intel® oneAPI Math Kernel Library (oneMKL) functions except the functions from the domains where\nthe number of threads is set withmkl_domain_set_num_threads\n•\nTo all execution threads except the threads where the number of threads is set with \nmkl_set_num_threads_local\nThe number specified is a hint, and Intel® oneAPI Math Kernel Library (oneMKL) may actually use a smaller\nnumber.\nNOTE\nThis function takes precedence over the MKL_NUM_THREADS environment variable.\nExample\n#include \"mkl.h\"\n…\nmkl_set_num_threads(4);\nmy_compute_using_mkl();    // Intel MKL uses up to 4 OpenMP threads\nmkl_domain_set_num_threads\nSpecifies the number of OpenMP* threads for a\nparticular function domain.\nSyntax\nint mkl_domain_set_num_threads (int nt, int domain);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nnt\nint\nnt > 0 - The number of threads suggested by the user.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2492\n\n\nName\nType\nDescription\nnt = 0 - The default number of threads for the OpenMP\nrun-time library.\nnt < 0 - Invalid value, which is ignored.\ndomain\nint\nThe named constant that defines the targeted domain.\nDescription\nThis function specifies how many OpenMP threads a particular function domain of Intel® oneAPI Math Kernel\nLibrary (oneMKL) should use. If this number is not set (default) or if it is set to zero in a call to this function,\nIntel® oneAPI Math Kernel Library (oneMKL) uses the default number of threads for the OpenMP run-time\nlibrary. The number of threads specified applies to the specified function domain on all execution threads\nexcept the threads where the number of threads is set withmkl_set_num_threads_local. For a list of\nsupported values of the domain argument, see Table \"Intel MKL Function Domains\".\nThe number of threads specified is only a hint, and Intel® oneAPI Math Kernel Library (oneMKL) may actually\nuse a smaller number.\nNOTE\nThis function takes precedence over the MKL_DOMAIN_NUM_THREADS environment variable.\nReturn Values\nName\nType\nDescription\nierr\nint\n1 - Indicates no error, execution is successful.\n0 - Indicates a failure, possibly because of invalid input\nparameters.\nExample\n#include \"mkl.h\"\n…\nmkl_domain_set_num_threads(4, MKL_DOMAIN_BLAS);\nmy_compute_using_mkl_blas();    // Intel MKL BLAS functions use up to 4 threads\nmy_compute_using_mkl_dft();     // Intel MKL FFT functions use the default number of threads\nmkl_set_num_threads_local\nSpecifies the number of OpenMP* threads for all Intel®\noneAPI Math Kernel Library (oneMKL) functions on the\ncurrent execution thread.\nSyntax\nint mkl_set_num_threads_local( int nt );\nInclude Files\n•\nmkl.h\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2493\n\n\nInput Parameters\nName\nType\nDescription\nnt\nint\nnt> 0 - The number of threads for Intel® oneAPI Math\nKernel Library (oneMKL) functions to use on the current\nexecution thread.\nnt = 0 - A request to reset the thread-local number of\nthreads and use the global number.\nDescription\nThis function sets the number of OpenMP threads that Intel® oneAPI Math Kernel Library (oneMKL) functions\nshould request for parallel computation. The number of threads is thread-local, which means that it only\naffects the current execution thread of the application. If the thread-local number is not set or if this number\nis set to zero in a call to this function, Intel® oneAPI Math Kernel Library (oneMKL) functions use the global\nnumber of threads. You can set the global number of threads using themkl_set_num_threads or \nmkl_domain_set_num_threads function.\nThe thread-local number of threads takes precedence over the global number: if the thread-local number is\nnon-zero, changes to the global number of threads have no effect on the current thread.\nCaution\nIf your application is threaded with OpenMP* andparallelization of Intel® oneAPI Math Kernel\nLibrary (oneMKL) is based on nested OpenMP parallelism,different OpenMP parallel regions\nreuse OpenMP threads. Therefore a thread-local setting in one OpenMP parallel region may\ncontinue to affect not only the master thread after the parallel region ends, but also\nsubsequent parallel regions. To avoid performance implications of this side effect, reset the\nthread-local number of threads before leaving the OpenMP parallel region (see Examples for\nhow to do it).\nReturn Values\nName\nType\nDescription\nsave_nt\nint\nThe value of the thread-local number of threads that was\nused before this function call. Zero means that the global\nnumber of threads was used.\nExamples\nThis example shows how to avoid the side effect of a thread-local number of threads by reverting to the\nglobal setting:\n#include \"omp.h\"\n#include \"mkl.h\"\n…\nmkl_set_num_threads(16);\nmy_compute_using_mkl();        // Intel MKL functions use up to 16 threads\n#pragma omp parallel num_threads(2)\n{\n  if (0 == omp_get_thread_num())\n    mkl_set_num_threads_local(4);\n  else\n    mkl_set_num_threads_local(12);\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2494\n\n\n  my_compute_using_mkl();        // Intel MKL functions use up to 4 threads on thread 0\n//   and up to 12 threads on thread 1\n}\nmy_compute_using_mkl();        // Intel MKL functions use up to 4 threads (!)\nmkl_set_num_threads_local( 0 );    // make master thread use global setting\nmy_compute_using_mkl();        // Intel MKL functions use up to 16 threads\nThis example shows how to avoid the side effect of a thread-local number of threads by saving and restoring\nthe existing setting:\n#include \"mkl.h\"\nvoid my_compute( int nt )\n{\n    int save = mkl_set_num_threads_local( nt ); // save the Intel® oneAPI Math Kernel Library \n(oneMKL) number of threads\n    my_compute_using_mkl();    // Intel MKL functions use up to nt threads on this thread\n    mkl_set_num_threads_local( save ); // restore the Intel® oneAPI Math Kernel Library (oneMKL) \nnumber of threads\n}\nmkl_set_dynamic\nEnables Intel® oneAPI Math Kernel Library (oneMKL)\nto dynamically change the number of OpenMP*\nthreads.\nSyntax\nvoid mkl_set_dynamic (int flag);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nflag\nint\nflag = 0 - Requests disabling dynamic adjustment of the\nnumber of threads.\nflag≠ 0 - Requests enabling dynamic adjustment of the\nnumber of threads.\nDescription\nThis function indicates whether Intel® oneAPI Math Kernel Library (oneMKL) can dynamically change the\nnumber of OpenMP threads or should avoid doing this. The setting applies to all Intel® oneAPI Math Kernel\nLibrary (oneMKL) functions on all execution threads. This function takes precedence over theMKL_DYNAMIC\nenvironment variable.\nDynamic adjustment of the number of threads is enabled by default. Specifically, Intel® oneAPI Math Kernel\nLibrary (oneMKL) may use fewer threads in parallel regions than the number returned by\nthemkl_get_max_threadsfunction. Disabling dynamic adjustment of the number of threads does not ensure\nthat Intel® oneAPI Math Kernel Library (oneMKL) actually uses the specified number of threads, although the\nlibrary attempts to use that number.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2495\n\n\nTip\nIf you call Intel® oneAPI Math Kernel Library (oneMKL) from within an OpenMP parallel\nregion and want to create internal parallel regions, either disable dynamic adjustment of the\nnumber of threads or set the thread-local number of threads\n(seemkl_set_num_threads_local for how to do it).\nExample\n#include \"mkl.h\"\n…\nmkl_set_num_threads( 8 );\n#pragma omp parallel\n{\n    my_compute_with_mkl();    // Intel MKL uses 1 thread, being called from OpenMP parallel \nregion\n    mkl_set_dynamic( 0 );    // disable adjustment of the number of threads\n    my_compute_with_mkl();    // Intel MKL uses 8 threads\n}\nmkl_get_max_threads\nGets the number of OpenMP* threads targeted for\nparallelism.\nSyntax\nint mkl_get_max_threads (void);\nInclude Files\n•\nmkl.h\nDescription\nThis function returns the number of OpenMP threads available for Intel® oneAPI Math Kernel Library\n(oneMKL) to use in internal parallel regions.\nReturn Values\nName\nType\nDescription\nnt\nint\nThe maximum number of threads for Intel® oneAPI Math\nKernel Library (oneMKL) functions to use in internal parallel\nregions.\nExample\n#include \"mkl.h\"\n…\nif (1 == mkl_get_max_threads()) puts(\"Intel MKL does not employ threading\");\nSee Also\nmkl_set_dynamic\nmkl_get_dynamic\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2496\n\n\nmkl_domain_get_max_threads\nGets the number of OpenMP* threads targeted for\nparallelism for a particular function domain.\nSyntax\nint mkl_domain_get_max_threads (int domain);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\ndomain\nint\nThe named constant that defines the targeted domain.\nDescription\nComputational functions of the Intel® oneAPI Math Kernel Library (oneMKL) function domain defined by\nthedomain parameter use the value returned by this function as a limit of the number of OpenMP threads\nthey should request for parallel computations. The mkl_domain_get_max_threads function returns the\nthread-local number of threads or, if that value is zero or not set, the global number of threads. To determine\nthis number, the function inspects the environment settings and return values of the function calls below in\nthe order they are listed until it finds a non-zero value:\n•\nA call to mkl_set_num_threads_local\n•\nThe last of the calls to mkl_set_num_threads or mkl_domain_set_num_threads( …, MKL_DOMAIN_ALL)\n•\nA call to mkl_domain_set_num_threads( …, domain)\n•\nThe MKL_DOMAIN_NUM_THREADS environment variable with the MKL_DOMAIN_ALL tag\n•\nThe MKL_DOMAIN_NUM_THREADS environment variable (with the specific domain tag)\n•\nThe MKL_NUM_THREADS environment variable\n•\nA call to omp_set_num_threads\n•\nThe OMP_NUM_THREADS environment variable\nActual number of threads used by the Intel® oneAPI Math Kernel Library (oneMKL) computational functions\nmay vary depending on the problem size and on whether dynamic adjustment of the number of threads is\nenabled (see the description ofmkl_set_dynamic). For a list of supported values of the domain argument, see \nTable \"Intel MKL Function Domains\".\nReturn Values\nName\nType\nDescription\nnt\nint\nThe maximum number of threads for Intel® oneAPI Math\nKernel Library (oneMKL) functions from a given domain to\nuse in internal parallel regions.\nIf an invalid value of domain is supplied, the function\nreturns the number of threads for MKL_DOMAIN_ALL\nExample\n#include \"mkl.h\"\n…\nif (1 < mkl_domain_get_max_threads(MKL_DOMAIN_BLAS)) \nputs(\"Intel MKL BLAS functions employ threading\");\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2497\n\n\nmkl_get_dynamic\nDetermines whether Intel® oneAPI Math Kernel Library\n(oneMKL) is enabled to dynamically change the\nnumber of OpenMP* threads.\nSyntax\nint mkl_get_dynamic(void);\nInclude Files\n•\nmkl.h\nDescription\nThis function returns the status of dynamic adjustment of the number of OpenMP* threads. To determine this\nstatus, the function inspects the return value of the following function call and if it is undefined, inspects the\nenvironment setting below:\n•\nA call to mkl_set_dynamic\n•\nThe MKL_DYNAMIC environment variable\nNOTE\nDynamic adjustment of the number of threads is enabled by default.\nThe dynamic adjustment works as follows. Suppose that the mkl_get_max_threads function returns the\nnumber of threads equal to N. If dynamic adjustment is enabled, Intel® oneAPI Math Kernel Library (oneMKL)\nmay request up toNthreads, depending on the size of the problem. If dynamic adjustment is disabled, Intel®\noneAPI Math Kernel Library (oneMKL) requests exactlyN threads for internal parallel regions (provided it uses\na threaded algorithm with at least Ncomputations that can be done in parallel). However, the OpenMP* run-\ntime library may be configured to supply fewer threads than Intel® oneAPI Math Kernel Library (oneMKL)\nrequests, depending on the OpenMP* setting of dynamic adjustment.\nReturn Values\nName\nType\nDescription\nret\nint\n0 - Dynamic adjustment of the number of threads is\ndisabled.\n1 - Dynamic adjustment of the number of threads is\nenabled.\nExample\n#include \"mkl.h\"\n…\nint nt = mkl_get_max_threads();\nif (1 == mkl_get_dynamic()) \nprintf(\"Intel MKL may use less than %i threads for a large problem\", nt);\nelse\nprintf(\"Intel MKL should use %i threads for a large problem\", nt);\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2498\n\n\nmkl_set_num_stripes\nSpecifies the number of partitions along the leading\ndimension of the output matrix for parallel ?gemm\nfunctions.\nSyntax\nvoid mkl_set_num_stripes( int ns );\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nns\nint\nns > 0 - Specifies the number of partitions to use.\nns= 0 - Instructs Intel® oneAPI Math Kernel Library\n(oneMKL) to use the default partitioning algorithm.\nns < 0 - Invalid value; ignored.\nDescription\nThis function enables you to specify the number of stripes, or partitions along the leading dimension of the\noutput matrix, for parallel ?gemmfunctions. If this number is not set (default) or if it is set to zero, Intel®\noneAPI Math Kernel Library (oneMKL)?gemm functions use the default partitioning algorithm. The specified\nnumber of partitions only applies to ?gemm functions.\nThe number specified is a hint, and Intel® oneAPI Math Kernel Library (oneMKL) may actually use a smaller\nnumber.\nNOTE\nThis function takes precedence over the MKL_NUM_STRIPES environment variable.\nExample\n#include \"mkl.h\"\n…\nmkl_set_num_stripes(4);\ndgemm(...);     // Intel MKL uses up to 4 stripes for dgemm\nSee Also\nmkl_get_num_stripes\nmkl_get_num_stripes\nGets the number of partitions along the leading\ndimension of the output matrix for parallel ?gemm\nfunctions.\nSyntax\nint mkl_get_num_stripes(void);\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2499\n\n\nInclude Files\n•\nmkl.h\nDescription\nThis function returns the number of stripes, that is, partitions along the leading dimension of the output\nmatrix, for parallel ?gemm functions. The number of partitions only applies to ?gemm functions.\nThe number returned is a hint, and Intel® oneAPI Math Kernel Library (oneMKL) may actually use a smaller\nnumber.\nReturn Values\nName\nType\nDescription\nns\nint\nThe number of stripes for Intel® oneAPI Math Kernel Library\n(oneMKL)?gemm functions to use.\nExample\n#include \"mkl.h\"\n…\nint ns = mkl_get_num_stripes();\nif (ns > 0) printf(\"Intel MKL uses %d number of stripes\\n\", ns);\nSee Also\nmkl_set_num_stripes\nError Handling\nError Handling for Linear Algebra Routines\nxerbla\nError handling function called by BLAS, LAPACK,\nVector Math, and Vector Statistics functions.\nSyntax\nvoid xerbla( const char * srname, const int* info, const int len );\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nsrname\nconst char*\nThe name of the routine that called xerbla\ninfo\nconst int*\nThe position of the invalid parameter in the parameter list\nof the calling function or an error code\nlen\nconst int\nLength of the source string\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2500\n\n\nDescription\nThe xerbla function is an error handler for Intel® oneAPI Math Kernel Library (oneMKL) BLAS, LAPACK,\nVector Math, and Vector Statistics functions. These functions call xerbla if an issue is encountered on entry\nor during the function execution.\nxerbla operates as follows:\n1.\nPrints a message that depends on the value of the info parameter as explained in the following table.\nNOTE\nA specific message can differ from the listed messages in numeric values and/or function names.\n2.\nReturns to the calling application.\nError Messages Printed by xerbla\nValue of info\nError Message\n1001\nIntel MKL ERROR: Incompatible optional parameters on entry to\nDGEMM.\n1000 or 1089\nIntel MKL INTERNAL ERROR: Insufficient workspace available in\nfunction CGELSD.\n< 0\nIntel MKL INTERNAL ERROR: Condition 1 detected in function DLASD8.\nOther\nThe position of the invalid parameter in the parameter list of the\ncalling function.\nNote that xerbla is an internal function. You can change or disable printing of an error message by providing\nyour own xerbla function. The following examples illustrate usage of xerbla.\nExample\nvoid xerbla(char* srname, int* info, int len){\n// srname - name of the function that called xerbla\n// info - position of the invalid parameter in the parameter list\n// len - length of the name in bytes\nprintf(\"\\nXERBLA is called :%s: %d\\n\",srname,*info);\n}\nSee Also\nmkl_set_xerbla\npxerbla\nError handling routine called by ScaLAPACK routines.\nSyntax\nvoid pxerbla (MKL_INT* ictxt, char* srname, MKL_INT* info, MKL_INT srname_len);\nInclude Files\n•\nmkl_scalapack.h\nInput Parameters\nictxt\n(local)\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2501\n\n\nMKL_INT*\nThe BLACS context handle, indicating the global context of the operation.\nThe context itself is global.\nsrname\n(global)\nchar*\nThe name of the routine that called pxerbla.\ninfo\n(global)\nMKL_INT*\nThe position of the invalid parameter in the parameter list of the calling\nroutine.\nsrname_len\n(global)\nMKL_INT\nThe length of the calling routine name.\nDescription\nThis routine is an error handler for the ScaLAPACK routines. It is called if an input parameter has an invalid\nvalue. A message is printed and program execution continues. For ScaLAPACK driver and computational\nroutines, a RETURN statement is issued following the call to pxerbla.\nControl returns to the higher-level calling routine, and you can determine how the program should proceed.\nHowever, in the specialized low-level ScaLAPACK routines (auxiliary routines that are Level 2 equivalents of\ncomputational routines), the call to pxerbla() is immediately followed by a call to BLACS_ABORT() to\nterminate program execution since recovery from an error at this level in the computation is not possible.\nIt is always good practice to check for a non-zero value of info on return from a ScaLAPACK routine.\nInstallers may consider modifying this routine in order to call system-specific exception-handling facilities.\nLAPACKE_xerbla\nError handling function called by the C interface to\nLAPACK functions.\nSyntax\nvoid LAPACKE_xerbla( const char * name, lapack_int info );\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nname\nconst char*\nThe name of the routine that called LAPACKE_xerbla\ninfo\nlapack_int\nThe position of the invalid parameter in the parameter list\nof the calling function or an error code\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2502\n\n\nDescription\nThe LAPACKE_xerblafunction is an error handler for Intel® oneAPI Math Kernel Library (oneMKL) LAPACKE\nfunctions (the C interface to LAPACK functionality). If a LAPACKE function encounters an issue on entry or\nduring the function execution, it callsLAPACKE_xerbla to print an error message and return an error code.\nNOTE\nThe LAPACKE_xerbla routine does not replace the xerbla routine. For instance, if an issue\noccurs when a LAPACK function is called by a LAPACKE function, the LAPACK function calls\nxerbla.\nError Messages Printed by LAPACKE_xerbla\nValue of info\nExample Error Message\nLAPACK_WORK_MEMORY_ERROR\nNot enough memory to allocate work array\nin LAPACKE_dgees\nLAPACK_TRANSPOSE_MEMORY_ERROR\nNot enough memory to transpose matrix in\nLAPACKE_dgetrf_work\n< 0\nWrong parameter 1 in LAPACKE_dgetrf\nNOTE\nLAPACKE_xerbla is an internal function. You can change or disable printing of an error\nmessage by providing your own LAPACKE_xerblafunction. Intel® oneAPI Math Kernel Library\n(oneMKL) does not provide functionality for dynamic replacement ofLAPACKE_xerbla.\nSee Also\nxerbla\nHandling Fatal Errors\nA fatal error is a circumstance under which Intel® oneAPI Math Kernel Library (oneMKL) cannot continue the\ncomputation. For example, a fatal error occurs when Intel® oneAPI Math Kernel Library (oneMKL) cannot load\na dynamic library or confronts an unsupported CPU type. In case of a fatal error, the default Intel® oneAPI\nMath Kernel Library (oneMKL) behavior is to print an explanatory message to the console and call an internal\nfunction that terminates the application with a call to the systemexit()function. Intel® oneAPI Math Kernel\nLibrary (oneMKL) enables you to override this behavior by setting a custom handler of fatal errors. The\ncustom error handler can be configured to throw a C++ exception, set a global variable indicating the failure,\nor otherwise handle cannot-continue situations. It is not necessary for the custom error handler to call the\nsystemexit()function. Once execution of the error handler completes, a call to Intel® oneAPI Math Kernel\nLibrary (oneMKL) returns to the calling program without performing any computations and leaves no memory\nallocated by Intel® oneAPI Math Kernel Library (oneMKL) and no thread synchronization pending on return.\nTo specify a custom fatal error handler, call the mkl_set_exit_handler function.\nmkl_set_exit_handler\nSets the custom handler of fatal errors.\nSyntax\nint mkl_set_exit_handler (MKLExitHandler myexit);\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2503\n\n\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nPrototype\nDescription\nmyexit\nvoid (*myexit)(int why);\nThe error handler to set.\nDescription\nThis function sets the custom handler of fatal errors. If the input parameter is NULL, the system exit()\nfunction is set.\nThe following example shows how to use a custom handler of fatal errors in your C++ application:\n#include \"mkl.h\"\nvoid my_exit(int why){\n    throw my_exception();\n}\nint ComputationFunction()\n{\n    mkl_set_exit_handler( my_exit ); \n    try {\n        compute_using_mkl();\n    }\n    catch (const my_exception& e) {\n        handle_exception();\n    }\n}\nCharacter Equality Testing\nlsame\nTests two characters for equality regardless of the\ncase.\nSyntax\nint lsame( const char* ca, const char* cb, int lca, int lcb );\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nca, cb\nconst char*\nPointers to the single characters to be compared\nlca, lcb\nint\nLengths of the input character strings, equal to one.\nDescription\nThis logical function checks whether two characters are equal regardless of the case.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2504\n\n\nReturn Values\nName\nType\nDescription\nval\nint\nResult of the comparison:\n•\na non-zero value if ca is the same letter as cb, maybe\nexcept for the case.\n•\nzero if ca and cb are different letters for whatever cases.\nlsamen\nTests two character strings for equality regardless of\nthe case.\nSyntax\nMKL_INTlsamen( const MKL_INT* n, const char* ca, const char* cb );\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nn\nconst MKL_INT*\nPointer to the number of characters in ca and cb to be\ncompared.\nca, cb\nconst char*\nCharacter strings of length at least n to be compared. Only\nthe first n characters of each string will be accessed.\nDescription\nThis logical function tests whether the first n letters of one string are the same as the first n letters of the\nother string, regardless of the case.\nReturn Values\nName\nType\nDescription\nval\nMKL_INT\nResult of the comparison:\n•\na non-zero value if the first n letters in ca and cb\ncharacter strings are equal, maybe except for the case,\nor if the length of character string ca or cb is less than n.\n•\nzero if the first n letters in ca and cb character strings\nare different for whatever cases.\nTiming\nsecond/dsecnd\nReturns elapsed time in seconds. Use to estimate real\ntime between two calls to this function.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2505\n\n\nSyntax\nfloat second( void );\ndouble dsecnd( void );\nInclude Files\n•\nmkl.h\nDescription\nThe second/dsecnd function returns time in seconds to be used to estimate real time between two calls to\nthe function. The difference between these functions is in the precision of the floating-point type of the\nresult: while second returns the single-precision type, dsecnd returns the double-precision type.\nUse these functions to measure durations. To do this, call each of these functions twice. For example, to\nmeasure performance of a routine, call the appropriate function directly before a call to the routine to be\nmeasured, and then after the call of the routine. The difference between the returned values shows real time\nspent in the routine.\nInitializations may take some time when the second/dsecnd function runs for the first time. To eliminate the\neffect of this extra time on your measurements, make the first call to second/dsecnd in advance.\nDo not use second to measure short time intervals because the single-precision format is not capable of\nholding sufficient timer precision.\nReturn Values\nName\nType\nDescription\nval\nfloat for second\ndouble for dsecnd\nElapsed real time in seconds\nmkl_get_cpu_clocks\nReturns elapsed CPU clocks.\nSyntax\nvoid mkl_get_cpu_clocks (unsigned MKL_INT64 *clocks);\nInclude Files\n•\nmkl.h\nOutput Parameters\nName\nType\nDescription\nclocks\nunsigned MKL_INT64\nElapsed CPU clocks\nDescription\nThe mkl_get_cpu_clocks function returns the elapsed CPU clocks.\nThis may be useful when timing short intervals with high resolution. The mkl_get_cpu_clocks function is\nalso applied in pairs like second/dsecnd. Note that out-of-order code execution on IA-32 or Intel® 64\narchitecture processors may disturb the exact elapsed CPU clocks value a little bit, which may be important\nwhile measuring extremely short time intervals.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2506\n\n\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nmkl_get_cpu_frequency\nReturns the current CPU frequency value in GHz.\nSyntax\ndoublemkl_get_cpu_frequency(void);\nInclude Files\n•\nmkl.h\nDescription\nThe function mkl_get_cpu_frequency returns the current CPU frequency in GHz.\nNOTE\nThe returned value may vary from run to run if power management or Intel® Turbo Boost\nTechnology is enabled.\nReturn Values\nName\nType\nDescription\nfreq\ndouble\nCurrent CPU frequency value in GHz\nmkl_get_max_cpu_frequency\nReturns the maximum CPU frequency value in GHz.\nSyntax\ndouble mkl_get_max_cpu_frequency(void);\nInclude Files\n•\nmkl.h\nDescription\nThe function mkl_get_max_cpu_frequency returns the maximum CPU frequency in GHz.\nReturn Values\nName\nType\nDescription\nfreq\ndouble\nMaximum CPU frequency value in GHz\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2507\n\n\nmkl_get_clocks_frequency\nReturns the frequency value in GHz based on\nconstant-rate Time Stamp Counter.\nSyntax\ndouble mkl_get_clocks_frequency (void);\nInclude Files\n•\nmkl.h\nDescription\nThe function mkl_get_clocks_frequency returns the CPU frequency value (in GHz) based on constant-rate\nTime Stamp Counter (TSC). Use of the constant-rate TSC ensures that each clock tick is constant even if the\nCPU frequency changes. Therefore, the returned frequency is constant.\nNOTE\nObtaining the frequency may take some time when mkl_get_clocks_frequency is called for\nthe first time. The same holds for functions second/dsecnd, which call\nmkl_get_clocks_frequency.\nReturn Values\nName\nType\nDescription\nfreq\ndouble\nFrequency value in GHz\nSee Also\nsecond/dsecnd\nMemory Management\nThis section describes the Intel® oneAPI Math Kernel Library (oneMKL) memory functions. See the Intel®\noneAPI Math Kernel Library (oneMKL) Developer Guide for more memory usage information.\nmkl_free_buffers\nFrees unused memory allocated by Intel® oneAPI Math\nKernel Library (oneMKL) on the Host, including both\nCPU- and GPU-related buffers.\nSyntax\nvoid mkl_free_buffers (void);\nmkl.h\nDescription\nTo improve performance of Intel® oneAPI Math Kernel Library (oneMKL) on CPU, the Memory Allocator uses\nper-thread memory pools where buffers may be collected for fast reuse. Intel® oneAPI Math Kernel Library\n(oneMKL) also allocates temporary buffers on the host memory to improve performance of GPU kernels. The\nmkl_free_buffers function frees both types of memory.\nSee theIntel® oneAPI Math Kernel Library (oneMKL) Developer Guide for details.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2508\n\n\nYou should call mkl_free_buffers after the last call to Intel® oneAPI Math Kernel Library (oneMKL)\nfunctions. In large applications, if you suspect that the memory may get insufficient, you may call this\nfunction earlier, but anticipate a drop in performance that may occur due to reallocation of buffers for\nsubsequent calls to Intel® oneAPI Math Kernel Library (oneMKL) functions.\nNOTE\nmkl_free_buffers is triggered automatically during oneMKL unloading when it is possible as\npart of mkl_finalize; however, in the case of statically linked oneMKL or oneMKL using GPU\non Windows, the user must call mkl_free_buffers or mkl_finalize manually in order to\nclean up oneMKL internal buffers when oneMKL is no longer needed.\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nUsage of mkl_free_buffers with FFT Functions\nDFTI_DESCRIPTOR_HANDLE hand1; \nDFTI_DESCRIPTOR_HANDLE hand2; \nvoid mkl_free_buffers(void);\n. . . . . .\n/* Using Intel MKL FFT */\nStatus = DftiCreateDescriptor(&hand1, DFTI_SINGLE, DFTI_COMPLEX, dim, m1); \nStatus = DftiCommitDescriptor(hand1);\nStatus = DftiComputeForward(hand1, s_array1);\n. . . . . .\nStatus = DftiCreateDescriptor(&hand2, DFTI_SINGLE, DFTI_COMPLEX, dim, m2); \nStatus = DftiCommitDescriptor(hand2);\n. . . . . .\nStatus = DftiFreeDescriptor(&hand1);\n. . . . . .\nStatus = DftiComputeBackward(hand2, s_array2)); \nStatus = DftiFreeDescriptor(&hand2);\n/* Here you finish using Intel MKL FFT */\n/* Memory leak will be triggered by any memory control tool */\n/* Use mkl_free_buffers() to avoid memory leaking */\nmkl_free_buffers();\nmkl_thread_free_buffers\nFrees unused memory allocated by the Intel® oneAPI\nMath Kernel Library (oneMKL) Memory Allocator in the\ncurrent thread.\nSyntax\nvoid mkl_thread_free_buffers (void);\nInclude Files\n•\nmkl.h\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2509\n\n\nDescription\nTo improve performance of Intel® oneAPI Math Kernel Library (oneMKL), the Memory Allocator uses per-\nthread memory pools where buffers may be collected for fast reuse. Themkl_thread_free_buffers\nfunction frees unused memory allocated by the Memory Allocator in the current thread only.\nYou should call mkl_thread_free_buffersafter the last call to Intel® oneAPI Math Kernel Library (oneMKL)\nfunctions in the current thread. In large applications, if you suspect that the memory may get insufficient,\nyou may call this function earlier, but anticipate a drop in performance that may occur due to reallocation of\nbuffers for subsequent calls to Intel® oneAPI Math Kernel Library (oneMKL) functions.\nSee Also\nmkl_free_buffers\nmkl_disable_fast_mm\nTurns off the Intel® oneAPI Math Kernel Library\n(oneMKL) Memory Allocator for Intel® oneAPI Math\nKernel Library (oneMKL) functions to directly use the\nsystemmalloc/free functions.\nSyntax\nint mkl_disable_fast_mm (void);\nInclude Files\n•\nmkl.h\nDescription\nThe mkl_disable_fast_mmfunction turns the Intel® oneAPI Math Kernel Library (oneMKL) Memory Allocator\noff for Intel® oneAPI Math Kernel Library (oneMKL) functions to directly use the systemmalloc/\nfreefunctions. Intel® oneAPI Math Kernel Library (oneMKL) Memory Allocator uses per-thread memory pools\nwhere buffers may be collected for fast reuse. The Memory Allocator is turned on by default for better\nperformance. To turn it off, you can use themkl_disable_fast_mm function or the MKL_DISABLE_FAST_MM\nenvironment variable (See the Intel® oneAPI Math Kernel Library (oneMKL) Developer Guide for details.) Call\nmkl_disable_fast_mmbefore calling any Intel® oneAPI Math Kernel Library (oneMKL) functions that require\nallocation of memory buffers.\nNOTE\nTurning the Memory Allocator off negatively impacts performance of some Intel® oneAPI\nMath Kernel Library (oneMKL) routines, especially for small problem sizes.\nReturn Values\nName\nType\nDescription\nmm\nint\n1 - The Memory Allocator is successfully turned\noff.\n0 - Turning the Memory Allocator off failed.\nmkl_mem_stat\nReports the status of the Intel® oneAPI Math Kernel\nLibrary (oneMKL) Memory Allocator.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2510\n\n\nSyntax\nMKL_INT64 mkl_mem_stat (int* AllocatedBuffers);\nInclude Files\n•\nmkl.h\nOutput Parameters\nName\nType\nDescription\nAllocatedBuffers\nint\nThe number of buffers allocated by Intel® oneAPI\nMath Kernel Library (oneMKL).\nDescription\nThe function returns the number of buffers allocated by Intel® oneAPI Math Kernel Library (oneMKL) and the\namount of memory in these buffers. Intel® oneAPI Math Kernel Library (oneMKL) can allocate the memory\nbuffers internally or in a call tomkl_malloc/mkl_calloc. If no buffers are allocated at the moment, the\nmkl_mem_stat function returns 0. Call mkl_mem_statto check the Intel® oneAPI Math Kernel Library\n(oneMKL) memory status.\nNOTE\nIf you free all the memory allocated in calls to mkl_malloc or mkl_calloc and then call \nmkl_free_buffers, a subsequent call to mkl_mem_stat normally returns 0.\nReturn Values\nName\nType\nDescription\nAllocatedBytes\nMKL_INT64\nThe amount of allocated memory (in bytes).\nSee Also\nUsage Example for the Memory Functions\nmkl_peak_mem_usage\nReports the peak memory allocated by the Intel®\noneAPI Math Kernel Library (oneMKL) Memory\nAllocator.\nSyntax\nMKL_INT64 mkl_peak_mem_usage (intmode);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nmode\nint\nRequested mode of the function's operation.\nPossible values:\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2511\n\n\nName\nType\nDescription\n•\nMKL_PEAK_MEM_ENABLE - start gathering the\npeak memory data\n•\nMKL_PEAK_MEM_DISABLE - stop gathering the\npeak memory data\n•\nMKL_PEAK_MEM - return the peak memory\n•\nMKL_PEAK_MEM_RESET - return the peak\nmemory and reset the counter to start\ngathering the peak memory data from scratch\nDescription\nThe mkl_peak_mem_usagefunction reports the peak memory allocated by the Intel® oneAPI Math Kernel\nLibrary (oneMKL) Memory Allocator.\nGathering the peak memory data is turned off by default. If you need to know the peak memory, explicitly\nturn the data gathering mode on by calling the function with the MKL_PEAK_MEM_ENABLE value of the\nparameter. Use the MKL_PEAK_MEM and MKL_PEAK_MEM_RESET values only when the data gathering mode is\nturned on. Otherwise the function returns -1. The data gathering mode leads to performance degradation, so\nwhen the mode is turned on, you can turn it off by calling the function with the MKL_PEAK_MEM_DISABLE\nvalue of the parameter.\nNOTE\n•\nIf Intel® oneAPI Math Kernel Library (oneMKL) is running in a threaded mode,\nthemkl_peak_mem_usage function may return different amounts of memory from run to run.\n•\nThe function reports the peak memory for the entire application, not just for the calling thread.\nReturn Values\nName\nType\nDescription\nAllocatedBytes\nMKL_INT64\nThe peak memory allocated by the Memory\nAllocator (in bytes) or -1 in case of errors.\nSee Also\nUsage Example for the Memory Functions\nmkl_malloc\nAllocates an aligned memory buffer.\nSyntax\nvoid* mkl_malloc (size_t alloc_size, int alignment);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nalloc_size\nsize_t\nSize of the buffer to be allocated.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2512\n\n\nName\nType\nDescription\nalignment\nint\nAlignment of the buffer.\nDescription\nThe function allocates an alloc_size-byte buffer aligned on the alignment-byte boundary.\nIf alignment is not a power of 2, the 64-byte alignment is used.\nReturn Values\nName\nType\nDescription\na_ptr\nvoid*\nPointer to the allocated buffer if alloc_size≥ 1,\nNULL if alloc_size < 1.\nSee Also\nmkl_free\nUsage Example for the Memory Functions\nmkl_calloc\nAllocates and initializes an aligned memory buffer.\nSyntax\nvoid* mkl_calloc (size_t num, size_t size, int alignment);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nnum\nsize_t\nThe number of elements in the buffer to be\nallocated.\nsize\nsize_t\nThe size of the element.\nalignment\nint\nAlignment of the buffer.\nDescription\nThe function allocates a num*size-byte buffer, aligned on the alignment-byte boundary, and initializes the\nbuffer with zeros.\nIf alignment is not a power of 2, the 64-byte alignment is used.\nReturn Values\nName\nType\nDescription\na_ptr\nvoid*\nPointer to the allocated buffer if size≥ 1,\nNULL if size < 1.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2513\n\n\nSee Also\nmkl_malloc\nmkl_realloc\nmkl_free\nUsage Example for the Memory Functions\nmkl_realloc\nChanges the size of memory buffer allocated by\nmkl_malloc/mkl_calloc.\nSyntax\nvoid* mkl_realloc (void *ptr, size_t size);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nptr\nvoid*\nPointer to the memory buffer allocated by the\nmkl_malloc or mkl_calloc function or a NULL\npointer.\nsize\nsize_t\nNew size of the buffer.\nDescription\nThe function changes the size of the memory buffer allocated by the mkl_malloc or mkl_calloc function to\nsize bytes. The first bytes of the returned buffer up to the minimum of the old and new sizes keep the\ncontent of the input buffer. The returned memory buffer can have a different location than the input one. If\nptr is NULL, the function works as mkl_malloc.\nReturn Values\nName\nType\nDescription\na_ptr\nvoid*\n•\nPointer to the re-allocated buffer if re-\nallocation is successful.\n•\nNULL if re-allocation is unsuccessful.\nSee Also\nmkl_malloc\nmkl_calloc\nmkl_free\nUsage Example for the Memory Functions\nmkl_free\nFrees the aligned memory buffer allocated by\nmkl_malloc/mkl_calloc.\nSyntax\nvoid mkl_free (void *a_ptr);\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2514\n\n\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\na_ptr\nvoid*\nPointer to the buffer to be freed.\nDescription\nThe function frees the buffer pointed by a_ptr and allocated by the mkl_malloc() or mkl_calloc()\nfunction and does nothing if a_ptr is NULL.\nSee Also\nmkl_malloc\nmkl_calloc\nUsage Example for the Memory Functions\nmkl_set_memory_limit\nOn Linux, sets the limit of memory that Intel® oneAPI\nMath Kernel Library (oneMKL) can allocate for a\nspecified type of memory.\nSyntax\nint mkl_set_memory_limit (int mem_type, size_t limit);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nmem_type\nint\nType of memory to limit. Possible values:\nMKL_MEM_MCDRAM - Multi-Channel Dynamic Random Access\nMemory (MCDRAM).\nlimit\nsize_t\nMemory limit in megabytes.\nDescription\nThis function sets the limit for the amount of memory that Intel® oneAPI Math Kernel Library (oneMKL) can\nallocate for the specified memory type. The limit bounds both internal allocations (inside Intel® oneAPI Math\nKernel Library (oneMKL) computation routines) and external allocations (in a call tomkl_malloc,\nmkl_calloc, or mkl_realloc). By default no limit is set for memory allocation.\nCall mkl_set_memory_limitat most once, prior to calling any other Intel® oneAPI Math Kernel Library\n(oneMKL) function in your application except mkl_set_interface_layer and mkl_set_threading_layer.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2515\n\n\nNOTE\n•\nAllocation in MCDRAM requires libmemkind and libjemalloc dynamic libraries which are a part of\nIntel® Manycore Platform Software Package (Intel® MPSP) for Linux*.\n•\nThe mkl_set_memory_limit function takes precedence over the MKL_FAST_MEMORY_LIMIT\nenvironment variable.\nReturn Values\nType\nDescription\nint\nStatus of the function completion:\n•\n1 - the limit is set\n•\n0 - the limit is not set\nSee Also\nmkl_malloc\nmkl_calloc\nmkl_realloc\nUsage Example for the Memory Functions\nUsage Example for the Memory Functions\n#include <stdio.h>\n#include <mkl.h>\nint main(void) {\n  double *a, *b, *c;\n  int n, i;\n  double alpha, beta;\n  MKL_INT64 AllocatedBytes;\n  int N_AllocatedBuffers;\n \n  alpha = 1.1; beta = -1.2;\n  n = 1000;\n  mkl_peak_mem_usage(MKL_PEAK_MEM_ENABLE);\n  a = (double*)mkl_malloc(n*n*sizeof(double),64);\n  b = (double*)mkl_malloc(n*n*sizeof(double),64);\n  c = (double*)mkl_calloc(n*n,sizeof(double),64);\n  for (i=0;i<(n*n);i++) {\n     a[i] = (double)(i+1);\n     b[i] = (double)(-i-1);\n  }\n \n  dgemm(\"N\",\"N\",&n,&n,&n,&alpha,a,&n,b,&n,&beta,c,&n);\n  AllocatedBytes = mkl_mem_stat(&N_AllocatedBuffers);\n  printf(\"\\nDGEMM uses %d bytes in %d buffers\",AllocatedBytes,N_AllocatedBuffers);\n \n  mkl_free_buffers();\n  mkl_free(a);\n  mkl_free(b);\n  mkl_free(c);\n \n  AllocatedBytes = mkl_mem_stat(&N_AllocatedBuffers);\n  if (AllocatedBytes > 0) {\n      printf(\"\\nMKL memory leak!\");\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2516\n\n\n      printf(\"\\nAfter mkl_free_buffers there are %d bytes in %d buffers\",\n         AllocatedBytes,N_AllocatedBuffers);\n  }\n  printf(\"\\nPeak memory allocated by Intel MKL memory allocator %d bytes. Start to count new \nmemory peak\",\n         mkl_peak_mem_usage(MKL_PEAK_MEM_RESET));\n  a = (double*)mkl_malloc(n*n*sizeof(double),64);\n  a = (double*)mkl_realloc(a,2*n*n*sizeof(double));\n  mkl_free(a);\n  printf(\"\\nPeak memory allocated by Intel MKL memory allocator after reset of peak memory \ncounter %d bytes\\n\",\n         mkl_peak_mem_usage(MKL_PEAK_MEM));\n \n  return 0;\n}\nSingle Dynamic Library Control\nIntel® oneAPI Math Kernel Library (oneMKL) provides the Single Dynamic Library (SDL), which enables\nsetting the interface and threading layer for Intel® oneAPI Math Kernel Library (oneMKL) at run time.\nSeeIntel® oneAPI Math Kernel Library (oneMKL) Developer Guide for details of SDL and layered model\nconcept. This section describes the functions supporting SDL.\nmkl_set_interface_layer\nSets the interface layer for Intel® oneAPI Math Kernel\nLibrary (oneMKL) at run time. Use with the Single\nDynamic Library.\nSyntax\nint mkl_set_interface_layer (int required_interface);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nrequired_interface\nint\nDetermines the interface layer. Possible values depend on the\nsystem architecture. Some of the values are only available on\nLinux* OS:\n•\nIntel® 64 architecture:\nMKL_INTERFACE_LP64 for the Intel LP64 interface.\nMKL_INTERFACE_ILP64 for the Intel ILP64 interface.\nMKL_INTERFACE_LP64+MKL_INTERFACE_GNU for the GNU*\nLP64 interface on Linux OS.\nMKL_INTERFACE_ILP64+MKL_INTERFACE_GNU for the GNU\nILP64 interface on Linux OS.\n•\nIA-32 architecture:\nMKL_INTERFACE_LP64 for the Intel interface on Linux OS.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2517\n\n\nName\nType\nDescription\nMKL_INTERFACE_LP64+MKL_INTERFACE_GNU or\nMKL_INTERFACE_GNU for the GNU interface on Linux OS.\nDescription\nIf you are using the Single Dynamic Library (SDL), the mkl_set_interface_layerfunction sets the\nspecified interface layer for Intel® oneAPI Math Kernel Library (oneMKL) at run time.\nCall this function prior to calling any other Intel® oneAPI Math Kernel Library (oneMKL) function in your\napplication exceptmkl_set_threading_layer. You can call mkl_set_interface_layer and\nmkl_set_threading_layer in any order.\nThe mkl_set_interface_layer function takes precedence over the MKL_INTERFACE_LAYER environment\nvariable.\nSee Intel® oneAPI Math Kernel Library (oneMKL) Developer Guide for the layered model concept and usage\ndetails of the SDL.\nReturn Values\nType\nDescription\nint\n•\nCurrent interface layer if it is set in a call to\nmkl_set_interface_layer or specified by environment variables or\ndefaults.\nPossible values are specified in Input Parameters.\n•\n-1, if the layer was not specified prior to the call and the input\nparameter is incorrect.\nmkl_set_threading_layer\nSets the threading layer for Intel® oneAPI Math Kernel\nLibrary (oneMKL) at run time. Use with the Single\nDynamic Library (SDL).\nSyntax\nint mkl_set_threading_layer (int required_threading);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nrequired_threading\nint\nDetermines the threading layer. Possible values:\nMKL_THREADING_INTEL for Intel threading.\nMKL_THREADING_SEQUENTIALfor the sequential mode of Intel®\noneAPI Math Kernel Library (oneMKL).\nMKL_THREADING_TBB for threading with the Intel® Threading\nBuilding Blocks.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2518\n\n\nName\nType\nDescription\nMKL_THREADING_PGI for PGI threading on Windows* or Linux*\noperating system only. Do not use this value with the SDL for\nIntel® Many Integrated Core (Intel® MIC) Architecture.\nNOTE PGI* support is deprecated and will be removed in the\noneMKL 2025.0 release.\nMKL_THREADING_GNU for GNU threading on Linux* operating\nsystem only. Do not use this value with the SDL for Intel MIC\nArchitecture.\nDescription\nIf you are using the Single Dynamic Library (SDL), the mkl_set_threading_layerfunction sets the\nspecified threading layer for Intel® oneAPI Math Kernel Library (oneMKL) at run time.\nCall this function prior to calling any other Intel® oneAPI Math Kernel Library (oneMKL) function in your\napplication except mkl_set_interface_layer.\nYou can call mkl_set_threading_layer and mkl_set_interface_layer in any order.\nThe mkl_set_threading_layer function takes precedence over the MKL_THREADING_LAYER environment\nvariable.\nSee Intel® oneAPI Math Kernel Library (oneMKL) Developer Guide for the layered model concept and usage\ndetails of the SDL.\nReturn Values\nType\nDescription\nint\n•\nCurrent threading layer if it is set in a call to\nmkl_set_threading_layer or specified by environment variables or\ndefaults. Possible values are specified in Input Parameters.\n•\n-1, if the layer was not specified prior to the call and the input\nparameter is incorrect.\nmkl_set_xerbla\nReplaces the error handling routine. Use with the\nSingle Dynamic Library .\nSyntax\nXerblaEntry mkl_set_xerbla (XerblaEntry new_xerbla_ptr);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nnew_xerbla_ptr\nXerblaEntry\nPointer to the error handling routine to be used.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2519\n\n\nDescription\nThe mkl_set_xerblafunction replaces the error handling routine that is called by Intel® oneAPI Math Kernel\nLibrary (oneMKL) functions with the routine specified by the parameter.\nSee Intel® oneAPI Math Kernel Library (oneMKL) Developer Guide for details about SDL.\nReturn Values\nThe function returns the pointer to the replaced error handling routine.\nSee Also\nxerbla\nmkl_set_progress\nReplaces the progress information routine.\nSyntax\nProgressEntry mkl_set_progress (ProgressEntry new_progress_ptr);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nnew_progress_ptr\nProgressEntry\nPointer to the progress information routine to be used.\nDescription\nThe mkl_set_progress function replaces the currently used progress information routine with the routine\nspecified by the parameter.\nUsually a user-supplied mkl_progress function redefines the default mkl_progress function automatically.\nHowever, you must call mkl_set_progress to replace the default mkl_progress on Windows* in any of the\nfollowing cases:\n•\nYou are using the Single Dynamic Library (SDL) mkl_rt.lib.\n•\nYou link dynamically with ScaLAPACK.\nNOTE\nIn a future release, a user-supplied mkl_progress function will not redefine the default\nmkl_progress function automatically. You will need to use the mkl_set_progress function to\nspecify any overrides.\nSee the Intel® oneAPI Math Kernel Library (oneMKL) Developer Guide for details of SDL.\nReturn Values\nThe function returns the pointer to the replaced progress information routine.\nSee Also\nmkl_progress\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2520\n\n\nmkl_set_pardiso_pivot\nReplaces the routine handling Intel® oneAPI Math\nKernel Library (oneMKL) PARDISO pivots with a user-\ndefined routine. Use with the Single Dynamic Library\n(SDL).\nSyntax\nPardisopivotEntry mkl_set_pardiso_pivot (PardisopivotEntry new_pardiso_pivot_ptr);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nnew_pardiso_pivot_pt\nr\nPardisopivo\ntEntry\nPointer to the pivot setting routine to be used.\nDescription\nIf you are using the Single Dynamic Library (SDL), the mkl_set_pardiso_pivotfunction replaces the pivot\nsetting routine that is called by Intel® oneAPI Math Kernel Library (oneMKL) functions with the routine\nspecified by the parameter.\nSee Intel® oneAPI Math Kernel Library (oneMKL) Developer Guide for usage details of the SDL.\nReturn Values\nType\nDescription\nPardisopivotEntry\nPointer to the replaced pivot setting routine.\nSee Also\nmkl_pardiso_pivot\nConditional Numerical Reproducibility Control\nThe CNR mode of Intel® oneAPI Math Kernel Library (oneMKL) ensures bitwise reproducible results from run\nto run of Intel® oneAPI Math Kernel Library (oneMKL) functions on a fixed number of threads for a specific\nIntel instruction set architecture (ISA) under the following conditions:\n•\nCalls to Intel® oneAPI Math Kernel Library (oneMKL) occur in a single executable\n•\nThe number of computational threads used by the library does not change in the run\nIntel® oneAPI Math Kernel Library (oneMKL) offers both functions and environment variables to support\nconditional numerical reproducibility. See theIntel® oneAPI Math Kernel Library (oneMKL) Developer Guide for\nmore information on bitwise reproducible results of computations and for details about the environment\nvariables.\nThe support functions enable you to configure the CNR mode and also provide information on the current and\noptimal CNR branch on your system. Usage Examples for CNR Support Functions illustrate usage of these\nfunctions.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2521\n\n\nImportant\nCall the functions that define the behavior of CNR before any of the math library functions that they\ncontrol.\nIntel® oneAPI Math Kernel Library (oneMKL) provides named constants for use as input and output\nparameters of the functions instead of integer values. SeeNamed Constants for CNR Control for a list of the\nnamed constants.\nAlthough you can configure the CNR mode using either the support functions or the environment variables,\nthe functions offer more flexible configuration and control than the environment variables. Settings specified\nby the functions take precedence over the settings specified by the environment variables.\nUse Intel® oneAPI Math Kernel Library (oneMKL) in the CNR mode only in case a need for bitwise reproducible\nresults is critical. Otherwise, run Intel® oneAPI Math Kernel Library (oneMKL) as usual to avoid performance\ndegradation.\nWhile you can supply unaligned input and output data to Intel® oneAPI Math Kernel Library (oneMKL)\nfunctions running in the CNR mode, use of aligned data is recommended. Refer toReproducibility Conditions\nfor more details.\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nmkl_cbwr_set\nConfigures the CNR mode of Intel® oneAPI Math\nKernel Library (oneMKL). The mkl_cbwr_set function\nmust be called only once, before any other Intel®\noneAPI Math Kernel Library (oneMKL) functions.\nSyntax\nint mkl_cbwr_set (int setting);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nsetting\nint\nCNR branch to set. See Named Constants for CNR Control\nfor a list of named constants that specify the settings.\nDescription\nThe mkl_cbwr_set function configures the CNR mode. (Specifically, it sets the CNR branch and then turns on\nthe CNR mode.)\nThe mkl_cbwr_set function must be called only once, before any other Intel® oneAPI Math Kernel Library\n(oneMKL) functions.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2522\n\n\nNOTE\nSettings specified by the mkl_cbwr_set function take precedence over the settings specified\nby the MKL_CBWR environment variable.\nReturn Values\nName\nType\nDescription\nstatus\nint\nThe status of the function completion:\n•\nMKL_CBWR_SUCCESS - the function completed\nsuccessfully.\n•\nMKL_CBWR_ERR_INVALID_INPUT - an invalid setting is\nrequested.\n•\nMKL_CBWR_ERR_UNSUPPORTED_BRANCH - the input value\nof the branch does not match the instruction set\narchitecture (ISA) of your system. See Named Constants\nfor CNR Control for more details.\n•\nMKL_CBWR_ERR_MODE_CHANGE_FAILURE - the\nmkl_cbwr_setfunction requested to change the current\nCNR branch after a call to some Intel® oneAPI Math\nKernel Library (oneMKL) function other than a CNR\nfunction.\nSee Also\nUsage Examples for CNR Support Functions\nmkl_cbwr_get\nReturns the current CNR settings.\nSyntax\nint mkl_cbwr_get (int option);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\noption\nint\nSpecifies the CNR settings requested. Named constants\ndefine possible values of option:\n•\nMKL_CBWR_BRANCH - returns the current CNR branch\nonly.\n•\nMKL_CBWR_ALL - returns all CNR settings including strict\nCNR setting.\nDescription\nThe mkl_cbwr_get function returns the requested CNR settings. The function returns\nMKL_CBWR_ERR_INVALID_INPUT if an invalid option is specified.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2523\n\n\nNOTE\nTo enable CNR mode, use the mkl_cbwr_set function or environment variables. For more\ndetails, see the Intel® oneAPI Math Kernel Library (oneMKL) Developer Guide.\nReturn Values\nName\nType\nDescription\nsetting\nint\nRequested CNR settings. See Named Constants for CNR\nControl for a list of named constants that specify the\nsettings.\nIf the value of the option parameter is not permitted,\ncontains the MKL_CBWR_ERR_INVALID_INPUT error code.\nSee Also\nUsage Examples for CNR Support Functions\nmkl_cbwr_set\nmkl_cbwr_get_auto_branch\nAutomatically detects the CNR code branch for your\nplatform.\nSyntax\nint mkl_cbwr_get_auto_branch (void);\nInclude Files\n•\nmkl.h\nDescription\nThe mkl_cbwr_get_auto_branch function uses a run-time CPU check to return a CNR branch that is\noptimized for the processor where the program is currently running.\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nReturn Values\nName\nType\nDescription\nsetting\nint\nAutomatically detected CNR branch. May be any specific\nbranch listed in Named Constants for CNR Control.\nSee Also\nUsage Examples for CNR Support Functions\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2524\n\n\nNamed Constants for CNR Control\nUse the conditional numerical reproducibility (CNR) functionality in Intel® oneAPI Math Kernel Library\n(oneMKL) to obtain reproducible results from MKL routines. When enabling CNR, you choose a specific code\nbranch of Intel® oneAPI Math Kernel Library (oneMKL) that corresponds to the instruction set architecture\n(ISA) that you target. Use these named constants to specify the code branch and other CNR options.\nNamed Constant\nValue\nDescription\nCNR Branches\nMKL_CBWR_OFF\n0\nDisable CNR mode\nMKL_CBWR_BRANCH_OFF\n1\nCNR mode is disabled\nMKL_CBWR_AUTO\n2\nChoose branch automatically. CNR mode uses the\nstandard ISA-based dispatching model while\nensuring fixed cache sizes, deterministic reductions,\nand static scheduling\nMKL_CBWR_COMPATIBLE\n3\nIntel® Streaming SIMD Extensions 2 (Intel® SSE2)\nwithout rcpps/rsqrtps instructions\nMKL_CBWR_SSE2\n4\nIntel SSE2\nMKL_CBWR_SSE3\n5\nDEPRECATED. Intel® Streaming SIMD Extensions 3\n(Intel® SSE3). This setting is kept for backward\ncompatibility and is equivalent to MKL_CBWR_SSE2.\nMKL_CBWR_SSSE3\n6\nSupplemental Streaming SIMD Extensions 3\n(SSSE3)\nMKL_CBWR_SSE4_1\n7\nIntel® Streaming SIMD Extensions 4-1 (SSE4-1)\nMKL_CBWR_SSE4_2\n8\nIntel® Streaming SIMD Extensions 4-2 (SSE4-2)\nMKL_CBWR_AVX\n9\nIntel® Advanced Vector Extensions (Intel® AVX)\nMKL_CBWR_AVX2\n10\nIntel® Advanced Vector Extensions 2 (Intel® AVX2)\nMKL_CBWR_AVX512_MIC\n11\nDEPRECATED. Intel® Advanced Vector Extensions\n512 (Intel® AVX-512) on Intel® Xeon Phi™\nprocessors. This setting is kept for backward\ncompatibility and is equivalent to MKL_CBWR_AVX2.\nMKL_CBWR_AVX512\n12\nIntel AVX-512 on Intel® Xeon® processors\nMKL_CBWR_AVX512_MIC_E1\n13\nDEPRECATED. Intel® Advanced Vector Extensions\n512 (Intel® AVX-512) for Intel® Many Integrated\nCore Architecture (Intel® MIC Architecture) with\nsupport of AVX512_4FMAPS and AVX512_4VNNIW\ninstruction groups enabled processors. This setting\nis kept for backward compatibility and is equivalent\nto MKL_CBWR_AVX2.\nMKL_CBWR_AVX512_E1\n14\nIntel® Advanced Vector Extensions 512 (Intel®\nAVX-512) with support of Vector Neural Network\nInstructions enabled processors\nCNR Flags\nMKL_CBWR_STRICT\n65536 or\n0x10000\nStrict CNR mode enabled. See Reproducibility\nConditions for more information.\nWhen specifying the CNR branch with the named constants, be aware of the following:\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2525\n\n\n•\nReproducible results are provided under Reproducibility Conditions.\n•\nSettings other than MKL_CBWR_AUTO or MKL_CBWR_COMPATIBLE are available only for Intel processors.\n•\nIntel and Intel compatible CPUs have a few instructions, such as approximation instructions rcpps/rsqrtps,\nthat may return different results. Setting the branch to MKL_CBWR_COMPATIBLEensures that Intel® oneAPI\nMath Kernel Library (oneMKL) does not use these instructions and forces a single Intel SSE2-only code\npath to be executed.\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nSee Also\nUsage Examples for CNR Support Functions\nReproducibility Conditions\nTo get reproducible results from run to run, ensure that the number of threads is fixed and constant.\nSpecifically:\n•\nIf you are running your program with OpenMP* parallelization on different processors, explicitly specify\nthe number of threads.\n•\nTo ensure that your application has deterministic behavior with OpenMP* parallelization and does not\nadjust the number of threads dynamically at run time, set MKL_DYNAMIC and OMP_DYNAMIC to FALSE. This\nis especially needed if you are running your program on different systems.\n•\nIf you are running your program with the Intel® Threading Building Blocks parallelization, numerical\nreproducibility is not guaranteed.\nOpenMP* Offload\nStarting in version 2024.1, numerical reproducibility is supported for using OpenMP* offload to execute BLAS\nlevel-3 routines and batched extensions on the GPU. CNR will be enabled for GPU whenever any CNR code\nbranch is enabled (that is, in the case of a setting other than MKL_CBWR_OFF or MKL_CBWR_BRANCH_OFF).\nFor more information on CNR support for GPU, see the oneMKL Developer Guide.\nStrict CNR Mode\nIn strict CNR mode, oneAPI Math Kernel Library provides bitwise reproducible results for a limited set of\nfunctions and code branches even when the number of threads changes. These routines and branches\nsupport strict CNR mode (64-bit libraries only):\n•\n?gemm, ?symm, ?hemm, ?trsm, and their CBLAS equivalents (cblas_?gemm, cblas_?symm, cblas_?hemm,\nand cblas_?trsm.\n•\nIntel® Advanced Vector Extensions 2 (Intel® AVX2) or Intel® Advanced Vector Extensions 512 (Intel®\nAVX-512).\nWhen using other routines or CNR branches,oneAPI Math Kernel Library operates in standard (non-strict)\nCNR mode, subject to the restrictions described above. Enabling strict CNR mode can reduce performance.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2526\n\n\nNOTE\n•\nAs usual, you should align your data, even in CNR mode, to obtain the best possible performance.\nWhile CNR mode also fully supports unaligned input and output data, the use of it might reduce the\nperformance of some oneAPI Math Kernel Library functions on earlier Intel processors. To ensure\nproper alignment of arrays, allocate memory for them using mkl_malloc/mkl_calloc.\n•\nConditional Numerical Reproducibility does not ensure that bitwise-identical NaN values are\ngenerated when the input data contains NaN values.\n•\nIf dynamic memory allocation fails on one run but succeeds on another run, you may fail to get\nreproducible results between these two runs.\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nSee Also\nmkl_malloc\nmkl_calloc\nUsage Examples for CNR Support Functions\nThe following examples illustrate usage of support functions for conditional numerical reproducibility.\nSetting Automatically Detected CNR Branch\n#include <mkl.h>\nint main(void) {\n   int my_cbwr_branch;\n   /* Find the available MKL_CBWR_BRANCH automatically */\n   my_cbwr_branch = mkl_cbwr_get_auto_branch();\n   /* User code without Intel MKL calls */\n   /* Piece of the code where CNR of Intel MKL is needed */\n   /* The performance of Intel MKL functions might be reduced for CNR mode */\n   if (mkl_cbwr_set(my_cbwr_branch)!=MKL_CBWR_SUCCESS) {\n      printf(\"Error in setting MKL_CBWR_BRANCH! Aborting…\\n\");\n      return;\n   }\n   /* CNR calls to Intel MKL + any other code */\n}\nUse of the mkl_cbwr_get Function\n#include <mkl.h>\nint main(void) {\n   int my_cbwr_branch;\n   /* Piece of the code where CNR of Intel MKL is analyzed */\n   my_cbwr_branch = mkl_cbwr_get(MKL_CBWR_BRANCH);\n   switch (my_cbwr_branch) {\n      case MKL_CBWR_AUTO:\n              /* actions in case of automatic mode */\n              break;\n      case MKL_CBWR_SSSE3:\n              /* actions for SSSE3 code */\n              break;\n         default:\n              /* all other cases */\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2527\n\n\n   }\n   /* User code */\n}\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nMiscellaneous\nmkl_progress\nProvides progress information.\nSyntax\nint mkl_progress (int* thread_process, int* step, char* stage, int lstage);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nthread_pr\nocess\nconst int*\nIndicates the number of thread or process the progress\nroutine is called from:\n•\nThe thread number for non-cluster components linked\nwith OpenMP threading layer\n•\nZero for non-cluster components linked with sequential\nthreading layer\n•\nThe process number (MPI rank) for cluster components\nstep\nconst int*\nPointer to the linear progress indicator that shows the\namount of work done. Increases from 0 to the linear size of\nthe problem during the computation.\nstage\nconst char*\nMessage indicating the name of the routine or the name of\nthe computation stage the progress routine is called from.\nlstage\nint\nThe length of a stage string excluding the trailing NULL\ncharacter.\nDescription\nThe mkl_progress function is intended to track progress of a lengthy computation and/or interrupt the\ncomputation. By default this routine does nothing but the user application can redefine it to obtain the\ncomputation progress information. You can set it to perform certain operations during the routine\ncomputation, for instance, to print a progress indicator. A non-zero return value may be supplied by the\nredefined function to break the computation.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2528\n\n\nNOTE\nThe user-defined mkl_progress function must be thread-safe.\nSome Intel® oneAPI Math Kernel Library (oneMKL) functions from LAPACK, ScaLAPACK, DSS/PARDISO, and\nParallel Direct Sparse Solver for Clusters regularly call themkl_progress function during the computation.\nRefer to the description of a specific function from those domains to see whether the function supports this\nfeature or not.\nIf a LAPACK function returns info=-1002, the function was interrupted by mkl_progress. Because\nScaLAPACK does not support interruption of the computation, Intel® oneAPI Math Kernel Library (oneMKL)\nignores any value returned bymkl_progress.\nWhile a user-supplied mkl_progress function usually redefines the default mkl_progress function\nautomatically, some configurations require calling the mkl_set_progress function to replace the default\nmkl_progress function. Call mkl_set_progress to replace the default mkl_progress on Windows* in any\nof the following cases:\n•\nYou are using the Single Dynamic Library (SDL) mkl_rt.lib.\n•\nYou link dynamically with ScaLAPACK.\nWarning\nThe mkl_progress function supports OpenMP*/TBB threading and sequential execution for\nspecific routines.\nReturn Values\nName\nType\nDescription\nstopflag\nint\nThe stopping flag. A non-zero flag forces the routine to be\ninterrupted. The zero flag is the default return value.\nExample\nThe following example prints the progress information to the standard output device:\n#include <stdio.h>\n#include <string.h> \n#define BUFLEN 16 \nint mkl_progress( int* thread_process, int* step, char* stage, int lstage )\n{\n  char buf[BUFLEN];\n  if( lstage >= BUFLEN ) lstage = BUFLEN-1;\n  strncpy( buf, stage, lstage );\n  buf[lstage] = '\\0';\n  printf( \"In thread %i, at stage %s, steps passed %i\\n\", *thread_process, buf, *step );\n  return 0; \n}\nmkl_enable_instructions\nEnables dispatching for new Intel® architectures or\nrestricts the set of Intel® instruction sets available for\ndispatching. The mkl_enable_instructions function\nmust be called only once, before any other Intel®\noneAPI Math Kernel Library (oneMKL) functions.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2529\n\n\nSyntax\nint mkl_enable_instructions (int isa);\nInput Parameters\nName\nType\nDescription\nisa\nint\nThe latest Intel® instruction-set architecture (ISA) for Intel®\noneAPI Math Kernel Library (oneMKL) to dispatch.\nMKL_ENABLE_AVX512\nIntel® Advanced Vector\nExtensions 512 (Intel®\nAVX-512)\nMKL_ENABLE_AVX512_E1\nIntel® Advanced Vector\nExtensions 512 (Intel®\nAVX-512) with support for Intel®\nDeep Learning Boost (Intel® DL\nBoost).\nMKL_ENABLE_AVX512_E2\nIntel® Advanced Vector\nExtensions 512 (Intel®\nAVX-512) with support for Intel®\nDeep Learning Boost (Intel® DL\nBoost), EVEX-encoded AES, and\nCarry-Less Multiplication\nQuadword instructions\nMKL_ENABLE_AVX512_E3\nIntel® Advanced Vector\nExtensions 512 (Intel®\nAVX-512) with support for Intel®\nDeep Learning Boost (Intel® DL\nBoost) and bfloat16\nMKL_ENABLE_AVX512_E4\nIntel® Advanced Vector\nExtensions 512 (Intel®\nAVX-512) with support for INT8,\nBF16, FP16 (limited)\ninstructions, and Intel®\nAdvanced Matrix Extensions\n(Intel® AMX) with INT8 and\nBF16\nMKL_ENABLE_AVX512_E5\nIntel® Advanced Vector\nExtensions 512 (Intel®\nAVX-512) with support for INT8,\nBF16, FP16 (limited)\ninstructions, and Intel®\nAdvanced Matrix Extensions\n(Intel® AMX) with INT8, BF16,\nand FP16\nNOTE Not dispatched by\ndefault.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2530\n\n\nName\nType\nDescription\nMKL_ENABLE_AVX2\nIntel® Advanced Vector\nExtensions 2 (Intel® AVX2)\nMKL_ENABLE_AVX2_E1\nIntel® Advanced Vector\nExtensions 2 (Intel® AVX2) with\nsupport for Intel® Deep Learning\nBoost (Intel® DL Boost)\nMKL_ENABLE_SSE4_2\nIntel® Streaming SIMD\nExtensions 4.2 (Intel® SSE4.2)\nDescription\nIntel® oneAPI Math Kernel Library (oneMKL) does run-time processor dispatching to identify appropriate\ninternal code paths to traverse for Intel® oneAPI Math Kernel Library (oneMKL) functions called by the\napplication. The mkl_enable_instructions function controls the behavior of the dispatcher to do either of\nthe following:\n•\nEnable dispatching for new Intel architectures.\nIntel® oneAPI Math Kernel Library (oneMKL) does not dispatch instruction sets that do not have silicon\navailable at time of the product launch. Callmkl_enable_instructions to enable dispatching the code\npath for such an ISA in a simulator environment or on hardware that supports this ISA.\n•\nRestrict the set of Intel instruction sets available for dispatching.\nCall mkl_enable_instructions to restrict dispatching to code paths for earlier ISA. For example, if the\nhardware supports Intel AVX, a call to mkl_enable_instructions with the MKL_ENABLE_SSE4_2\nparameter forces the dispatcher to use the Intel SSE4-2 code path.\nIf the system does not support the instruction set specified by the isa parameter or if the system is based\non a non-Intel architecture, mkl_enable_instructions does nothing and returns zero.\nSettings specified by the mkl_enable_instructions function set an upper limit to settings specified by the \nmkl_cbwr_set function.\nYou can use the MKL_ENABLE_INSTRUCTIONS environment variable instead of calling\nmkl_enable_instructions (for more details, see the Intel® oneAPI Math Kernel Library (oneMKL)\nDeveloper Guide); however, the settings specified by the function take precedence over the settings specified\nby the environment variable.\nReturn Values\nName\nType\nDescription\nirc\nint\nFunction completion status:\n1 - Intel® oneAPI Math Kernel Library (oneMKL) dispatches\nthe code path for the specified ISA by default.\n0 - The request is rejected. Usually this occurs if\nmkl_enable_instructions was called:\n•\nAfter another Intel® oneAPI Math Kernel Library\n(oneMKL) function\n•\nOn a non-Intel architecture\n•\nWith an incompatible ISA specified\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2531\n\n\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nmkl_set_env_mode\nSets up the mode that ignores environment settings\nspecific to Intel® oneAPI Math Kernel Library\n(oneMKL).\nSyntax\nint mkl_set_env_mode(int mode);\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nmode\nint\nSpecifies what mode to set. For details, see Description.\nPossible values:\n•\n0 - Do nothing.\nUse this value to query the current environment mode.\n•\n1 - Make Intel® oneAPI Math Kernel Library (oneMKL)\nignore environment settings specific to the library.\nDescription\nIn the default environment mode, Intel® oneAPI Math Kernel Library (oneMKL) can control its behavior using\nenvironment variables for threading, memory management, Conditional Numerical Reproducibility, automatic\noffload, and so on. Themkl_set_env_mode function sets up the environment mode that ignores all settings\nspecified by Intel® oneAPI Math Kernel Library (oneMKL) environment variables\nexceptMIC_LD_LIBRARY_PATH and MKLROOT.\nReturn Values\nName\nType\nDescription\ncurrent_m\node\nint\nEnvironment mode that was used before the function call:\n•\n0 - Default\n•\n1 - Ignore environment settings specific to Intel® oneAPI\nMath Kernel Library (oneMKL).\nmkl_verbose\nEnables or disables Intel® oneAPI Math Kernel Library\n(oneMKL) Verbose mode.\nSyntax\nint mkl_verbose (int enable);\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2532\n\n\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nenable\nint\nDesired state of the Intel® oneAPI Math Kernel Library\n(oneMKL) Verbose mode. Indicates whether printing Intel®\noneAPI Math Kernel Library (oneMKL) function call\ninformation should be turned on or off. Possible values:\n•\n0 – disable the Verbose mode\n•\n1 – enable the Verbose mode (GPU application: enable\nthe Verbose mode without timing)\n•\n2 – enable the Verbose mode (GPU application: enable\nthe Verbose mode with synchronous timing)\nDescription\nThis function enables or disables the Intel® oneAPI Math Kernel Library (oneMKL) Verbose mode, in which\ncomputational functions print call description information. For details of the Verbose mode, see theIntel®\noneAPI Math Kernel Library (oneMKL) Developer Guide, available in the Intel® Software Documentation\nLibrary.\nNOTE\nThe setting for the Verbose mode specified by the mkl_verbose function takes precedence\nover the setting specified by the MKL_VERBOSE environment variable.\nReturn Values\nName\nType\nDescription\nstatus\nint\n•\nIf the requested operation completed successfully,\ncontains previous state of the verbose mode:\n•\n0 – Verbose mode was disabled\n•\n1 – Verbose mode was enabled (GPU application:\nVerbose mode was enabled without timing)\n•\n2 – Verbose mode was enabled (GPU application:\nVerbose mode was enabled with synchronous timing)\n•\nIf the function failed to complete the operation because\nof an incorrect input parameter, equals –1.\nSee Also\nIntel Software Documentation Library\nmkl_verbose_output_file\nWrite output in Intel® oneAPI Math Kernel Library\n(oneMKL) Verbose mode to a file.\nSyntax\nint mkl_verbose_output_file (const char*filename);\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2533\n\n\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nfilename\nchar\nName of file. Specify the complete path of the output file.\nDescription\nThis function writes the output in Verbose mode to the file specified in the path.\nIf the write operation is successful, the function returns 0.\nIf the file does not exist or cannot be opened, the write operation is unsuccessful. The function returns 1 and\ndefaults to mkl_verbose behavior by printing to stdout.\nNOTE\nYou can alternatively use MKL_VERBOSE_OUTPUT_FILE environment variable instead of\ncalling the mkl_verbose_output_file function. If you want to use the environment variable\noption, you must set it to the complete path of the output file.\nImportant The setting for the verbose output file specified by the mkl_verbose_output_file\nfunction takes precedence over the setting specified by the MKL_VERBOSE_OUTPUT_FILE\nenvironment variable.\nFor more information on the Verbose mode, see the Intel® oneAPI Math Kernel Library (oneMKL)\nDeveloper Guide, available in the Intel® Software Documentation Library.\nReturn Values\nName\nType\nDescription\nstatus\nint\n•\n0 indicates that the write operation was successful.\n•\n1 indicates that the write operation was unsuccessful.\nSee Also\nIntel Software Documentation Library\nmkl_set_mpi\nSets the implementation of the message-passing\ninterface to be used by Intel® oneAPI Math Kernel\nLibrary (oneMKL).\nSyntax\nint mkl_set_mpi (int vendor, const char *custom_library_name);\nInclude Files\n•\nmkl.h\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2534\n\n\nInput Parameters\nName\nType\nDescription\nvendor\nint\nSpecifies the implementation of the message-passing\ninterface (MPI) to use:\nPossible values:\n•\nMKL_BLACS_CUSTOM - a custom MPI library. Requires a\nprebuilt custom MPI BLACS library.\n•\nMKL_BLACS_MSMPI - Microsoft MPI library.\n•\nMKL_BLACS_INTELMPI - Intel® MPI library.\n•\nMKL_BLACS_MPICH - MPICH MPI library.\ncustom_li\nbrary_nam\nevendor\nconst char *\nThe filename (without a directory name) of the custom\nBLACS dynamic library to use. This library must be located\nin the directory with your application executable or with\nIntel® oneAPI Math Kernel Library (oneMKL) dynamic\nlibraries. Can beNULL or an empty string.\nDescription\nCall this function to set the MPI implementation to be used by Intel® oneAPI Math Kernel Library (oneMKL) on\nWindows* OS when dynamic Intel® oneAPI Math Kernel Library (oneMKL) libraries are used. For all other\nconfigurations, the function returns an error indicating that you cannot set the MPI implementation. You can\nspecify your own prebuilt dynamic BLACS library for a custom MPI by settingvendor to MKL_BLACS_CUSTOM\nand optionally passing the name of the custom BLACS dynamic library. If the custom_library_path\nparameter is NULLor an empty string, Intel® oneAPI Math Kernel Library (oneMKL) uses the default platform-\nspecific library name:mkl_blacs_custom_lp64.dll or mkl_blacs_custom_ilp64.dll, depending on\nwhether the BLACS interface linked against your application is LP64 or ILP64.\nReturn Values\nName\nType\nDescription\nstatus\nint\nThe return status:\n•\n0 - The function completed successfully.\n•\n-1 - The vendor parameter is invalid.\n•\n-2 - The custom_library_name parameter is invalid.\n•\n-3 - The MPI library cannot be set at this point.\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nmkl_finalize\nTerminates Intel® oneAPI Math Kernel Library\n(oneMKL) execution environment and frees resources\nallocated by the library.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2535\n\n\nSyntax\nvoid mkl_finalize(void);\nInclude Files\n•\nmkl.h\nDescription\nThis function frees resources allocated by Intel® oneAPI Math Kernel Library (oneMKL). Once this function is\ncalled, the application can no longer call Intel® oneAPI Math Kernel Library (oneMKL) functions other\nthanmkl_finalize.\nIn particular, the mkl_finalizefunction enables you to free resources when a third-party shared library is\nstatically linked to Intel® oneAPI Math Kernel Library (oneMKL). To avoid resource leaks that may happen\nwhen a shared library is loaded and unloaded multiple times, callmkl_finalize each time the library is\nunloaded. The recommended method to do this depends on the operating system:\n•\nOn Linux* or macOS*, place the call into a shared library destructor.\n•\nOn Windows*, call mkl_finalize from the DLL_PROCESS_DETACH handler of DllMain.\nNOTE\nIntel® oneAPI Math Kernel Library (oneMKL) shared libraries automatically perform\nfinalization when they are unloaded. If an application is statically linked to Intel® oneAPI\nMath Kernel Library (oneMKL), the operating system frees all resources allocated by Intel®\noneAPI Math Kernel Library (oneMKL) during termination of the process associated with the\napplication.\nBLACS Routines\nIntel® oneAPI Math Kernel Libraryimplements FORTRAN 77 routines from the BLACS (Basic Linear Algebra\nCommunication Subprograms) package. These routines are used to support a linear algebra oriented\nmessage passing interface that may be implemented efficiently and uniformly across a large range of\ndistributed memory platforms.\nThe BLACS routines make linear algebra applications both easier to program and more portable. For this\npurpose, they are used in Intel® oneAPI Math Kernel Library (oneMKL) intended for the Linux* and Windows*\nOSs as the communication layer of ScaLAPACK and Cluster FFT.\nOn computers, a linear algebra matrix is represented by a two dimensional array (2D array), and therefore\nthe BLACS operate on 2D arrays. See description of the basic matrix shapes in a special topic.\nThe BLACS routines implemented in Intel® oneAPI Math Kernel Library (oneMKL) are of four categories:\n•\nCombines\n•\nPoint to Point Communication\n•\nBroadcast\n•\nSupport.\nThe Combines take data distributed over processes and combine the data to produce a result. The Point to\nPoint routines are intended for point-to-point communication and Broadcast routines send data possessed by\none process to all processes within a scope.\nThe Support routines perform distinct tasks that can be used for initialization, destruction, information, and\nmiscellaneous tasks.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2536\n\n\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nMatrix Shapes\nThe BLACS routines recognize the two most common classes of matrices for dense linear algebra. The first of\nthese classes consists of general rectangular matrices, which in machine storage are 2D arrays consisting of\nm rows and n columns, with a leading dimension, lda, that determines the distance between successive\ncolumns in memory.\nThe general rectangular matrices take the following parameters as input when determining what array to\noperate on:\nm\n(input) INTEGER. The number of matrix rows to be operated on.\nn\n(input) INTEGER. The number of matrix columns to be operated on.\na\n(input/output) TYPE (depends on routine), array of dimension (lda,n).\nA pointer to the beginning of the (sub)array to be sent.\nlda\n(input) INTEGER. The distance between two elements in matrix row.\nThe second class of matrices recognized by the BLACS are trapezoidal matrices (triangular matrices are a\nsub-class of trapezoidal). Trapezoidal arrays are defined by m, n, and lda, as above, but they have two\nadditional parameters as well. These parameters are:\nuplo\n(input) CHARACTER*1 . Indicates whether the matrix is upper or lower\ntrapezoidal, as discussed below.\ndiag\n(input) CHARACTER*1 . Indicates whether the diagonal of the matrix is unit\ndiagonal (will not be operated on) or otherwise (will be operated on).\nThe shape of the trapezoidal arrays is determined by these parameters as follows:\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2537\n\n\n__border__top\nTrapezoidal Arrays Shapes\nThe packing of arrays, if required, so that they may be sent efficiently is hidden, allowing the user to\nconcentrate on the logical matrix, rather than on how the data is organized in the system memory.\nRepeatability and Coherence\nFloating point computations are not exact on almost all modern architectures. This lack of precision is\nparticularly problematic in parallel operations. Since floating point computations are inexact, algorithms are\nclassified according to whether they are repeatable and to what degree they guarantee coherence.\n•\nRepeatable: a routine is repeatable if it is guaranteed to give the same answer if called multiple times\nwith the same parallel configuration and input.\n•\nCoherent: a routine is coherent if all processes selected to receive the answer get identical results.\nNOTE\nRepeatability and coherence do not effect correctness. A routine may be both incoherent and non-\nrepeatable, and still give correct output. But inaccuracies in floating point calculations may cause the\nroutine to return differing values, all of which are equally valid.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2538\n\n\nRepeatability\nBecause the precision of floating point arithmetic is limited, it is not truly associative: (a + b) + c might\nnot be the same as a + (b + c). The lack of exact arithmetic can cause problems whenever the possibility\nfor reordering of floating point calculations exists. This problem becomes prevalent in parallel computing due\nto race conditions in message passing. For example, consider a routine which sums numbers stored on\ndifferent processes. Assume this routine runs on four processes, with the numbers to be added being the\nprocess numbers themselves. Therefore, process 0 has the value 0:0, process 1 has the value 1:0, and son\non.\nOne algorithm for the computation of this result is to have all processes send their process numbers to\nprocess 0; process 0 adds them up, and sends the result back to all processes. So, process 0 would add a\nnumber to 0:0 in the first step. If receiving the process numbers is ordered so that process 0 always receives\nthe message from process 1 first, then 2, and finally 3, this results in a repeatable algorithm, which\nevaluates the expression ((0:0+ 1:0) + 2:0) + 3:0.\nHowever, to get the best parallel performance, it is better not to require a particular ordering, and just have\nprocess 0 add the first available number to its value and continue to do so until all numbers have been added\nin. Using this method, a race condition occurs, because the order of the operation is determined by the order\nin which process 0 receives the messages, which can be effected by any number of things. This\nimplementation is not repeatable, because the answer can vary between invocations, even if the input is the\nsame. For instance, one run might produce the sequence ((0:0+1:0)+2:0)+3:0, while a subsequent run\ncould produce ((0:0 + 2:0) + 1:0) + 3:0. Both of these results are correct summations of the given\nnumbers, but because of floating point roundoff, they might be different.\nCoherence\nA routine produces coherent output if all processes are guaranteed to produce the exact same results.\nObviously, almost no algorithm involving communication is coherent if communication can change the values\nbeing communicated. Therefore, if the parallel system being studied cannot guarantee that communication\nbetween processes preserves values, no routine is guaranteed to produce coherent results.\nIf communication is assumed to be coherent, there are still various levels of coherent algorithms. Some\nalgorithms guarantee coherence only if floating point operations are done in the exact same order on every\nnode. This is homogeneous coherence: the result will be coherent if the parallel machine is homogeneous in\nits handling of floating point operations.\nA stronger assertion of coherence is heterogeneous coherence, which does not require all processes to have\nthe same handling of floating point operations.\nIn general, a routine that is homogeneous coherent performs computations redundantly on all nodes, so that\nall processes get the same answer only if all processes perform arithmetic in the exact same way, whereas a\nroutine which is heterogeneous coherent is usually constrained to having one process calculate the final\nresult, and broadcast it to all other processes.\nExample of Incoherence\nAn incoherent algorithm is one which does not guarantee that all processes get the same result even on a\nhomogeneous system with coherent communication. The previous example of summing the process numbers\ndemonstrates this kind of behavior. One way to perform such a sum is to have every process broadcast its\nnumber to all other processes. Each process then adds these numbers, starting with its own. The calculations\nperformed by each process receives would then be:\n•\nProcess 0 : ((0:0+ 1:0) + 2:0) + 3:0\n•\nProcess 1 : ((1:0+ 2:0) + 3:0) + 0:0\n•\nProcess 2 : ((2:0+ 3:0) + 0:0) + 1:0\n•\nProcess 3 : ((3:0+ 0:0) + 1:0) + 0:0\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2539\n\n\nAll of these results are equally valid, and since all the results might be different from each other, this\nalgorithm is incoherent. Notice, however, that this algorithm is repeatable: each process will get the same\nresult if the algorithm is called again on the same data.\nExample of Homogeneous Coherence\nAnother way to perform this summation is for all processes to send their data to all other processes, and to\nensure the result is not incoherent, enforce the ordering so that the calculation each node performs is\n((0:0+ 1:0) + 2:0) + 3:0. This answer is the same for all processes only if all processes do the floating\npoint arithmetic in the same way. Otherwise, each process may make different floating point errors during\nthe addition, leading to incoherence of the output. Notice that since there is a specific ordering to the\naddition, this algorithm is repeatable.\nExample of Heterogeneous Coherence\nIn the final example, all processes send the result to process 0, which adds the numbers and broadcasts the\nresult to the rest of the processes. Since one process does all the computation, it can perform the operations\nin any order and it will give coherent results as long as communication is itself coherent. If a particular order\nis not forced on the the addition, the algorithm will not be repeatable. If a particular order is forced, it will be\nrepeatable.\nSummary\nRepeatability and coherence are separate issues which may occur in parallel computations. These concepts\nmay be summarized as:\n•\nRepeatability: The routine will yield the exact same result if it run multiple times on an identical problem.\nEach process may get a different result than the others (i.e., repeatability does not imply coherence), but\nthat value will not change if the routine is invoked multiple times.\n•\nHomogeneous coherence: All processes selected to possess the result will receive the exact same answer\nif:\n•\nCommunication does not change the value of the communicated data.\n•\nAll processes perform floating point arithmetic exactly the same.\n•\nHeterogeneous coherence: All processes will receive the exact same answer if communication does not\nchange the value of the communicated data.\nIn general, lack of the associative property for floating point calculations may cause both incoherence and\nnon-repeatability. Algorithms that rely on redundant computations are at best homogeneous coherent, and\nalgorithms in which one process broadcasts the result are heterogeneous coherent. Repeatability does not\nimply coherence, nor does coherence imply repeatability.\nSince these issues do not effect the correctness of the answer, they can usually be ignored. However, in very\nspecific situations, these issues may become very important. A stopping criteria should not be based on\nincoherent results, for instance. Also, a user creating and debugging a parallel program may wish to enforce\nrepeatability so the exact same program sequence occurs on every run.\nIn the BLACS, coherence and repeatability apply only in the context of the combine operations. As mentioned\nabove, it is possible to have communication which is incoherent (for instance, two machines which store\nfloating point numbers differently may easily produce incoherent communication, since a number stored on\nmachine A may not have a representation on machine B). However, the BLACS cannot control this issue.\nCommunication is assumed to be coherent, which for communication implies that it is also repeatable.\nFor combine operations, the BLACS allow you to set flags indicating that you would like combines to be\nrepeatable and/or heterogeneous coherent (see blacs_get and blacs_set for details on setting these\nflags).\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2540\n\n\nIf the BLACS are instructed to guarantee heterogeneous coherency, the BLACS restrict the topologies which\ncan be used so that one process calculates the final result of the combine, and if necessary, broadcasts the\nanswer to all other processes.\nIf the BLACS are instructed to guarantee repeatability, orderings will be enforced in the topologies which are\nselected. This may result in loss of performance which can range from negligible to serious depending on the\napplication.\nA couple of additional notes are in order. Incoherence and nonrepeatability can arise as a result of floating\npoint errors, as discussed previously. This might lead you to suspect that integer calculations are always\nrepeatable and coherent, since they involve exact arithmetic. This is true if overflow is ignored. With overflow\ntaken into consideration, even integer calculations can display incoherence and non-repeatability. Therefore,\nif the repeatability or coherence flags are set, the BLACS treats integer combines the same as floating point\ncombines in enforcing repeatability and coherence guards.\nBy their nature, maximization and minimization should always be repeatable. In the complex precisions,\nhowever, the real and imaginary parts must be combined in order to obtain a magnitude value used to do the\ncomparison (this is typically |r| + |i| or sqr(r2 + i2)). This allows for the possibility of heterogeneous\nincoherence. The BLACS therefore restrict which topologies are used for maximization and minimization in\nthe complex routines when the heterogeneous coherence flag is set.\nBLACS Combine Operations\nThis topic describes BLACS routines that combine the data to produce a result.\nIn a combine operation, each participating process contributes data that is combined with other processes’\ndata to produce a result. This result can be given to a particular process (called the destination process), or\nto all participating processes. If the result is given to only one process, the operation is referred to as a\nleave-on-one combine, and if the result is given to all participating processes the operation is referenced as\na leave-on-all combine.\nAt present, three kinds of combines are supported. They are:\n•\nelement-wise summation\n•\nelement-wise absolute value maximization\n•\nelement-wise absolute value minimization\nof general rectangular arrays.\nNote that a combine operation combines data between processes. By definition, a combine performed across\na scope of only one process does not change the input data. This is why the operations (max/min/sum) are\nspecified as element-wise. Element-wise indicates that each element of the input array will be combined\nwith the corresponding element from all other processes’ arrays to produce the result. Thus, a 4 x 2 array of\ninputs produces a 4 x 2 answer array.\nWhen the max/min comparison is being performed, absolute value is used. For example, -5 and 5 are\nequivalent. However, the returned value is unchanged; that is, it is not the absolute value, but is a signed\nvalue instead. Therefore, if you performed a BLACS absolute value maximum combine on the numbers -5, 3,\n1, 8 the result would be -8.\nThe initial symbol ? in the routine names below masks the data type:\ni\ninteger\ns\nsingle precision real\nd\ndouble precision real\nc\nsingle precision complex\nz\ndouble precision complex.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2541\n\n\nBLACS Combines\nRoutine name\nResults of operation\ngamx2d\nEntries of result matrix will have the value of the greatest absolute\nvalue found in that position.\ngamn2d\nEntries of result matrix will have the value of the smallest absolute\nvalue found in that position.\ngsum2d\nEntries of result matrix will have the summation of that position.\n?gamx2d\nPerforms element-wise absolute value maximization.\nSyntax\ncall igamx2d( icontxt, scope, top, m, n, a, lda, ra, ca, rcflag, rdest, cdest )\ncall sgamx2d( icontxt, scope, top, m, n, a, lda, ra, ca, rcflag, rdest, cdest )\ncall dgamx2d( icontxt, scope, top, m, n, a, lda, ra, ca, rcflag, rdest, cdest )\ncall cgamx2d( icontxt, scope, top, m, n, a, lda, ra, ca, rcflag, rdest, cdest )\ncall zgamx2d( icontxt, scope, top, m, n, a, lda, ra, ca, rcflag, rdest, cdest )\nInput Parameters\nicontxt\nINTEGER. Integer handle that indicates the context.\nscope\nCHARACTER*1. Indicates what scope the combine should proceed on.\nLimited to ROW, COLUMN, or ALL.\ntop\nCHARACTER*1. Communication pattern to use during the combine\noperation.\nm\nINTEGER. The number of matrix rows to be combined.\nn\nINTEGER. The number of matrix columns to be combined.\na\nTYPE array (lda, n). Matrix to be compared with to produce the\nmaximum.\nlda\nINTEGER. The leading dimension of the matrix A, that is, the distance\nbetween two successive elements in a matrix row.\nrcflag\nINTEGER.\nIf rcflag = -1, the arrays ra and ca are not referenced and need not\nexist. Otherwise, rcflag indicates the leading dimension of these\narrays, and so must be ≥ m.\nrdest\nINTEGER.\nThe process row coordinate of the process that should receive the\nresult. If rdest or cdest = -1, all processes within the indicated\nscope receive the answer.\ncdest\nINTEGER.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2542\n\n\nThe process column coordinate of the process that should receive the\nresult. If rdest or cdest = -1, all processes within the indicated\nscope receive the answer.\nOutput Parameters\na\nTYPE array (lda, n). Contains the result if this process is selected to\nreceive the answer, or intermediate results if the process is not\nselected to receive the result.\nra\nINTEGER array (rcflag, n).\nIf rcflag = -1, this array will not be referenced, and need not exist.\nOtherwise, it is an integer array (of size at least rcflag x n)\nindicating the row index of the process that provided the maximum. If\nthe calling process is not selected to receive the result, this array will\ncontain intermediate (useless) results.\nca\nINTEGER array (rcflag, n).\nIf rcflag = -1, this array will not be referenced, and need not exist.\nOtherwise, it is an integer array (of size at least rcflag x n)\nindicating the row index of the process that provided the maximum. If\nthe calling process is not selected to receive the result, this array will\ncontain intermediate (useless) results.\nDescription\nThis routine performs element-wise absolute value maximization, that is, each element of matrix A is\ncompared with the corresponding element of the other process's matrices. Note that the value of A is\nreturned, but the absolute value is used to determine the maximum (the 1-norm is used for complex\nnumbers). Combines may be globally-blocking, so they must be programmed as if no process returns until all\nhave called the routine.\nSee Also\nExamples of BLACS Routines Usage\n?gamn2d\nPerforms element-wise absolute value minimization.\nSyntax\ncall igamn2d( icontxt, scope, top, m, n, a, lda, ra, ca, rcflag, rdest, cdest )\ncall sgamn2d( icontxt, scope, top, m, n, a, lda, ra, ca, rcflag, rdest, cdest )\ncall dgamn2d( icontxt, scope, top, m, n, a, lda, ra, ca, rcflag, rdest, cdest )\ncall cgamn2d( icontxt, scope, top, m, n, a, lda, ra, ca, rcflag, rdest, cdest )\ncall zgamn2d( icontxt, scope, top, m, n, a, lda, ra, ca, rcflag, rdest, cdest )\nInput Parameters\nicontxt\nINTEGER. Integer handle that indicates the context.\nscope\nCHARACTER*1. Indicates what scope the combine should proceed on.\nLimited to ROW, COLUMN, or ALL.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2543\n\n\ntop\nCHARACTER*1. Communication pattern to use during the combine\noperation.\nm\nINTEGER. The number of matrix rows to be combined.\nn\nINTEGER. The number of matrix columns to be combined.\na\nTYPE array (lda, n). Matrix to be compared with to produce the\nminimum.\nlda\nINTEGER. The leading dimension of the matrix A, that is, the distance\nbetween two successive elements in a matrix row.\nrcflag\nINTEGER.\nIf rcflag = -1, the arrays ra and ca are not referenced and need not\nexist. Otherwise, rcflag indicates the leading dimension of these\narrays, and so must be ≥ m.\nrdest\nINTEGER.\nThe process row coordinate of the process that should receive the\nresult. If rdest or cdest = -1, all processes within the indicated\nscope receive the answer.\ncdest\nINTEGER.\nThe process column coordinate of the process that should receive the\nresult. If rdest or cdest = -1, all processes within the indicated\nscope receive the answer.\nOutput Parameters\na\nTYPE array (lda, n). Contains the result if this process is selected to\nreceive the answer, or intermediate results if the process is not\nselected to receive the result.\nra\nINTEGER array (rcflag, n).\nIf rcflag = -1, this array will not be referenced, and need not exist.\nOtherwise, it is an integer array (of size at least rcflag x n)\nindicating the row index of the process that provided the minimum. If\nthe calling process is not selected to receive the result, this array will\ncontain intermediate (useless) results.\nca\nINTEGER array (rcflag, n).\nIf rcflag = -1, this array will not be referenced, and need not exist.\nOtherwise, it is an integer array (of size at least rcflag x n)\nindicating the row index of the process that provided the minimum. If\nthe calling process is not selected to receive the result, this array will\ncontain intermediate (useless) results.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2544\n\n\nDescription\nThis routine performs element-wise absolute value minimization, that is, each element of matrix A is\ncompared with the corresponding element of the other process's matrices. Note that the value of A is\nreturned, but the absolute value is used to determine the minimum (the 1-norm is used for complex\nnumbers). Combines may be globally-blocking, so they must be programmed as if no process returns until all\nhave called the routine.\nSee Also\nExamples of BLACS Routines Usage\n?gsum2d\nPerforms element-wise summation.\nSyntax\ncall igsum2d( icontxt, scope, top, m, n, a, lda, rdest, cdest )\ncall sgsum2d( icontxt, scope, top, m, n, a, lda, rdest, cdest )\ncall dgsum2d( icontxt, scope, top, m, n, a, lda, rdest, cdest )\ncall cgsum2d( icontxt, scope, top, m, n, a, lda, rdest, cdest )\ncall zgsum2d( icontxt, scope, top, m, n, a, lda, rdest, cdest )\nInput Parameters\nicontxt\nINTEGER. Integer handle that indicates the context.\nscope\nCHARACTER*1. Indicates what scope the combine should proceed on.\nLimited to ROW, COLUMN, or ALL.\ntop\nCHARACTER*1. Communication pattern to use during the combine\noperation.\nm\nINTEGER. The number of matrix rows to be combined.\nn\nINTEGER. The number of matrix columns to be combined.\na\nTYPE array (lda, n). Matrix to be added to produce the sum.\nlda\nINTEGER. The leading dimension of the matrix A, that is, the distance\nbetween two successive elements in a matrix row.\nrdest\nINTEGER.\nThe process row coordinate of the process that should receive the\nresult. If rdest or cdest = -1, all processes within the indicated\nscope receive the answer.\ncdest\nINTEGER.\nThe process column coordinate of the process that should receive the\nresult. If rdest or cdest = -1, all processes within the indicated\nscope receive the answer.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2545\n\n\nOutput Parameters\na\nTYPE array (lda, n). Contains the result if this process is selected to\nreceive the answer, or intermediate results if the process is not\nselected to receive the result.\nDescription\nThis routine performs element-wise summation, that is, each element of matrix A is summed with the\ncorresponding element of the other process's matrices. Combines may be globally-blocking, so they must be\nprogrammed as if no process returns until all have called the routine.\nSee Also\nExamples of BLACS Routines Usage\nBLACS Point To Point Communication\nThis topic describes BLACS routines for point to point communication.\nPoint to point communication requires two complementary operations. The send operation produces a\nmessage that is then consumed by the receive operation. These operations have various resources\nassociated with them. The main such resource is the buffer that holds the data to be sent or serves as the\narea where the incoming data is to be received. The level of blocking indicates what correlation the return\nfrom a send/receive operation has with the availability of these resources and with the status of message.\nNon-blocking\nThe return from the send or receive operations does not imply that the resources may be reused, that the\nmessage has been sent/received or that the complementary operation has been called. Return means only\nthat the send/receive has been started, and will be completed at some later date. Polling is required to\ndetermine when the operation has finished.\nIn non-blocking message passing, the concept of communication/computation overlap (abbreviated C/C\noverlap) is important. If a system possesses C/C overlap, independent computation can occur at the same\ntime as communication. That means a nonblocking operation can be posted, and unrelated work can be done\nwhile the message is sent/received in parallel. If C/C overlap is not present, after returning from the routine\ncall, computation will be interrupted at some later date when the message is actually sent or received.\nLocally-blocking\nReturn from the send or receive operations indicates that the resources may be reused. However, since this\nonly depends on local information, it is unknown whether the complementary operation has been called.\nThere are no locally-blocking receives: the send must be completed before the receive buffer is available for\nre-use.\nIf a receive has not been posted at the time a locally-blocking send is issued, buffering will be required to\navoid losing the message. Buffering can be done on the sending process, the receiving process, or not done\nat all, losing the message.\nGlobally-blocking\nReturn from a globally-blocking procedure indicates that the operation resources may be reused, and that\ncomplement of the operation has at least been posted. Since the receive has been posted, there is no\nbuffering required for globally-blocking sends: the message is always sent directly into the user's receive\nbuffer.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2546\n\n\nAlmost all processors support non-blocking communication, as well as some other level of blocking sends.\nWhat level of blocking the send possesses varies between platforms. For instance, the Intel® processors\nsupport locally-blocking sends, with buffering done on the receiving process. This is a very important\ndistinction, because codes written assuming locally-blocking sends will hang on platforms with globally-\nblocking sends. Below is a simple example of how this can occur:\nIAM = MY_PROCESS_ID()\n IF (IAM .EQ. 0) THEN\n   SEND TO PROCESS 1\n   RECV FROM PROCESS 1\nELSE IF (IAM .EQ. 1) THEN\n   SEND TO PROCESS 0\n   RECV FROM PROCESS 0\nEND IF\nIf the send is globally-blocking, process 0 enters the send, and waits for process 1 to start its receive before\ncontinuing. In the meantime, process 1 starts to send to 0, and waits for 0 to receive before continuing. Both\nprocesses are now waiting on each other, and the program will never continue.\nThe solution for this case is obvious. One of the processes simply reverses the order of its communication\ncalls and the hang is avoided. However, when the communication is not just between two processes, but\nrather involves a hierarchy of processes, determining how to avoid this kind of difficulty can become\nproblematic.\nFor this reason, it was decided the BLACS would support locally-blocking sends. On systems natively\nsupporting globally-blocking sends, non-blocking sends coupled with buffering is used to simulate locally-\nblocking sends. The BLACS support globally-blocking receives.\nIn addition, the BLACS specify that point to point messages between two given processes will be strictly\nordered. If process 0 sends three messages (label them A, B, and C) to process 1, process 1 must receive A\nbefore it can receive B, and message C can be received only after both A and B. The main reason for this\nrestriction is that it allows for the computation of message identifiers.\nNote, however, that messages from different processes are not ordered. If processes 0, . . ., 3 send\nmessages A, . . ., D to process 4, process 4 may receive these messages in any order that is convenient.\nConvention\nThe convention used in the communication routine names follows the template ?xxyy2d, where the letter in\nthe ? position indicates the data type being sent, xx is replaced to indicate the shape of the matrix, and the\nyy positions are used to indicate the type of communication to perform:\ni\ninteger\ns\nsingle precision real\nd\ndouble precision real\nc\nsingle precision complex\nz\ndouble precision complex\nge\nThe data to be communicated is stored in a general rectangular matrix.\ntr\nThe data to be communicated is stored in a trapezoidal matrix.\nsd\nSend. One process sends to another.\nrv\nReceive. One process receives from another.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2547\n\n\nBLACS Point To Point Communication\nRoutine name\nOperation performed\ngesd2d\ntrsd2d\nTake the indicated matrix and send it to the destination process.\ngerv2d\ntrrv2d\nReceive a message from the process into the matrix.\nAs a simple example, the pseudo code given above is rewritten below in terms of the BLACS. It is further\nspecifed that the data being exchanged is the double precision vector X, which is 5 elements long.\nCALL GRIDINFO(NPROW, NPCOL, MYPROW, MYPCOL)\nIF (MYPROW.EQ.0 .AND. MYPCOL.EQ.0) THEN\n   CALL DGESD2D(5, 1, X, 5, 1, 0)\n   CALL DGERV2D(5, 1, X, 5, 1, 0)\nELSE IF (MYPROW.EQ.1 .AND. MYPCOL.EQ.0) THEN\n   CALL DGESD2D(5, 1, X, 5, 0, 0)\n   CALL DGERV2D(5, 1, X, 5, 0, 0)\nEND IF\n?gesd2d\nTakes a general rectangular matrix and sends it to the\ndestination process.\nSyntax\ncall igesd2d( icontxt, m, n, a, lda, rdest, cdest )\ncall sgesd2d( icontxt, m, n, a, lda, rdest, cdest )\ncall dgesd2d( icontxt, m, n, a, lda, rdest, cdest )\ncall cgesd2d( icontxt, m, n, a, lda, rdest, cdest )\ncall zgesd2d( icontxt, m, n, a, lda, rdest, cdest )\nInput Parameters\nicontxt\nINTEGER. Integer handle that indicates the context.\nm, n, a, lda\nDescribe the matrix to be sent. See Matrix Shapes for details.\nrdest\nINTEGER.\nThe process row coordinate of the process to send the message to.\ncdest\nINTEGER.\nThe process column coordinate of the process to send the message to.\nDescription\nThis routine takes the indicated general rectangular matrix and sends it to the destination process located at\n{RDEST, CDEST} in the process grid. Return from the routine indicates that the buffer (the matrix A) may be\nreused. The routine is locally-blocking, that is, it will return even if the corresponding receive is not posted.\nSee Also\nExamples of BLACS Routines Usage\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2548\n\n\n?trsd2d\nTakes a trapezoidal matrix and sends it to the\ndestination process.\nSyntax\ncall itrsd2d( icontxt, uplo, diag, m, n, a, lda, rdest, cdest )\ncall strsd2d( icontxt, uplo, diag, m, n, a, lda, rdest, cdest )\ncall dtrsd2d( icontxt, uplo, diag, m, n, a, lda, rdest, cdest )\ncall ctrsd2d( icontxt, uplo, diag, m, n, a, lda, rdest, cdest )\ncall ztrsd2d( icontxt, uplo, diag, m, n, a, lda, rdest, cdest )\nInput Parameters\nicontxt\nINTEGER. Integer handle that indicates the context.\nuplo, diag, m,\nn, a, lda\nDescribe the matrix to be sent. See Matrix Shapes for details.\nrdest\nINTEGER.\nThe process row coordinate of the process to send the message to.\ncdest\nINTEGER.\nThe process column coordinate of the process to send the message to.\nDescription\nThis routine takes the indicated trapezoidal matrix and sends it to the destination process located at {RDEST,\nCDEST} in the process grid. Return from the routine indicates that the buffer (the matrix A) may be reused.\nThe routine is locally-blocking, that is, it will return even if the corresponding receive is not posted.\n?gerv2d\nReceives a message from the process into the general\nrectangular matrix.\nSyntax\ncall igerv2d( icontxt, m, n, a, lda, rsrc, csrc )\ncall sgerv2d( icontxt, m, n, a, lda, rsrc, csrc )\ncall dgerv2d( icontxt, m, n, a, lda, rsrc, csrc )\ncall cgerv2d( icontxt, m, n, a, lda, rsrc, csrc )\ncall zgerv2d( icontxt, m, n, a, lda, rsrc, csrc )\nInput Parameters\nicontxt\nINTEGER. Integer handle that indicates the context.\nm, n, lda\nDescribe the matrix to be sent. See Matrix Shapes for details.\nrsrc\nINTEGER.\nThe process row coordinate of the source of the message.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2549\n\n\ncsrc\nINTEGER.\nThe process column coordinate of the source of the message.\nOutput Parameters\na\nAn array of dimension (lda,n) to receive the incoming message into.\nDescription\nThis routine receives a message from process {RSRC, CSRC} into the general rectangular matrix A. This\nroutine is globally-blocking, that is, return from the routine indicates that the message has been received\ninto A.\nSee Also\nExamples of BLACS Routines Usage\n?trrv2d\nReceives a message from the process into the\ntrapezoidal matrix.\nSyntax\ncall itrrv2d( icontxt, uplo, diag, m, n, a, lda, rsrc, csrc )\ncall strrv2d( icontxt, uplo, diag, m, n, a, lda, rsrc, csrc )\ncall dtrrv2d( icontxt, uplo, diag, m, n, a, lda, rsrc, csrc )\ncall ctrrv2d( icontxt, uplo, diag, m, n, a, lda, rsrc, csrc )\ncall ztrrv2d( icontxt, uplo, diag, m, n, a, lda, rsrc, csrc )\nInput Parameters\nicontxt\nINTEGER. Integer handle that indicates the context.\nuplo, diag, m, n, lda\nDescribe the matrix to be sent. See Matrix Shapes for details.\nrsrc\nINTEGER.\nThe process row coordinate of the source of the message.\ncsrc\nINTEGER.\nThe process column coordinate of the source of the message.\nOutput Parameters\na\nAn array of dimension (lda,n) to receive the incoming message into.\nDescription\nThis routine receives a message from process {RSRC, CSRC} into the trapezoidal matrix A. This routine is\nglobally-blocking, that is, return from the routine indicates that the message has been received into A.\nBLACS Broadcast Routines\nThis topic describes BLACS broadcast routines.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2550\n\n\nA broadcast sends data possessed by one process to all processes within a scope. Broadcast, much like point\nto point communication, has two complementary operations. The process that owns the data to be broadcast\nissues a broadcast/send. All processes within the same scope must then issue the complementary\nbroadcast/receive.\nThe BLACS define that both broadcast/send and broadcast/receive are globally-blocking. Broadcasts/\nreceives cannot be locally-blocking since they must post a receive. Note that receives cannot be locally-\nblocking. When a given process can leave, a broadcast/receive operation is topology dependent, so, to avoid\na hang as topology is varied, the broadcast/receive must be treated as if no process can leave until all\nprocesses have called the operation.\nBroadcast/sends could be defined to be locally-blocking. Since no information is being received, as long as\nlocally-blocking point to point sends are used, the broadcast/send will be locally blocking. However, defining\none process within a scope to be locally-blocking while all other processes are globally-blocking adds little to\nthe programmability of the code. On the other hand, leaving the option open to have globally-blocking\nbroadcast/sends may allow for optimization on some platforms.\nThe fact that broadcasts are defined as globally-blocking has several important implications. The first is that\nscoped operations (broadcasts or combines) must be strictly ordered, that is, all processes within a scope\nmust agree on the order of calls to separate scoped operations. This constraint falls in line with that already\nin place for the computation of message IDs, and is present in point to point communication as well.\nA less obvious result is that scoped operations with SCOPE = 'ALL' must be ordered with respect to any\nother scoped operation. This means that if there are two broadcasts to be done, one along a column, and one\ninvolving the entire process grid, all processes within the process column issuing the column broadcast must\nagree on which broadcast will be performed first.\nThe convention used in the communication routine names follows the template ?xxyy2d, where the letter in\nthe ? position indicates the data type being sent, xx is replaced to indicate the shape of the matrix, and the\nyy positions are used to indicate the type of communication to perform:\ni\ninteger\ns\nsingle precision real\nd\ndouble precision real\nc\nsingle precision complex\nz\ndouble precision complex\nge\nThe data to be communicated is stored in a general rectangular matrix.\ntr\nThe data to be communicated is stored in a trapezoidal matrix.\nbs\nBroadcast/send. A process begins the broadcast of data within a scope.\nbr\nBroadcast/receive A process receives and participates in the broadcast of data\nwithin a scope.\nBLACS Broadcast Routines\nRoutine name\nOperation performed\ngebs2d\ntrbs2d\nStart a broadcast along a scope.\ngebr2d\ntrbr2d\nReceive and participate in a broadcast along a scope.\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2551\n\n\nProduct and Performance Information\nNotice revision #20201201\n?gebs2d\nStarts a broadcast along a scope for a general\nrectangular matrix.\nSyntax\ncall igebs2d( icontxt, scope, top, m, n, a, lda )\ncall sgebs2d( icontxt, scope, top, m, n, a, lda )\ncall dgebs2d( icontxt, scope, top, m, n, a, lda )\ncall cgebs2d( icontxt, scope, top, m, n, a, lda )\ncall zgebs2d( icontxt, scope, top, m, n, a, lda )\nInput Parameters\nicontxt\nINTEGER. Integer handle that indicates the context.\nscope\nCHARACTER*1. Indicates what scope the broadcast should proceed on.\nLimited to 'Row', 'Column', or 'All'.\ntop\nCHARACTER*1. Indicates the communication pattern to use for the\nbroadcast.\nm, n, a, lda\nDescribe the matrix to be sent. See Matrix Shapes for details.\nDescription\nThis routine starts a broadcast along a scope. All other processes within the scope must call broadcast/\nreceive for the broadcast to proceed. At the end of a broadcast, all processes within the scope will possess\nthe data in the general rectangular matrix A.\nBroadcasts may be globally-blocking. This means no process is guaranteed to return from a broadcast until\nall processes in the scope have called the appropriate routine (broadcast/send or broadcast/receive).\nSee Also\nExamples of BLACS Routines Usage\n?trbs2d\nStarts a broadcast along a scope for a trapezoidal\nmatrix.\nSyntax\ncall itrbs2d( icontxt, scope, top, uplo, diag, m, n, a, lda )\ncall strbs2d( icontxt, scope, top, uplo, diag, m, n, a, lda )\ncall dtrbs2d( icontxt, scope, top, uplo, diag, m, n, a, lda )\ncall ctrbs2d( icontxt, scope, top, uplo, diag, m, n, a, lda )\ncall ztrbs2d( icontxt, scope, top, uplo, diag, m, n, a, lda )\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2552\n\n\nInput Parameters\nicontxt\nINTEGER. Integer handle that indicates the context.\nscope\nCHARACTER*1. Indicates what scope the broadcast should proceed on.\nLimited to 'Row', 'Column', or 'All'.\ntop\nCHARACTER*1. Indicates the communication pattern to use for the\nbroadcast.\nuplo, diag, m,\nn, a, lda\nDescribe the matrix to be sent. See Matrix Shapes for details.\nDescription\nThis routine starts a broadcast along a scope. All other processes within the scope must call broadcast/\nreceive for the broadcast to proceed. At the end of a broadcast, all processes within the scope will possess\nthe data in the trapezoidal matrix A.\nBroadcasts may be globally-blocking. This means no process is guaranteed to return from a broadcast until\nall processes in the scope have called the appropriate routine (broadcast/send or broadcast/receive).\n?gebr2d\nReceives and participates in a broadcast along a scope\nfor a general rectangular matrix.\nSyntax\ncall igebr2d( icontxt, scope, top, m, n, a, lda, rsrc, csrc )\ncall sgebr2d( icontxt, scope, top, m, n, a, lda, rsrc, csrc )\ncall dgebr2d( icontxt, scope, top, m, n, a, lda, rsrc, csrc )\ncall cgebr2d( icontxt, scope, top, m, n, a, lda, rsrc, csrc )\ncall zgebr2d( icontxt, scope, top, m, n, a, lda, rsrc, csrc )\nInput Parameters\nicontxt\nINTEGER. Integer handle that indicates the context.\nscope\nCHARACTER*1. Indicates what scope the broadcast should proceed on.\nLimited to 'Row', 'Column', or 'All'.\ntop\nCHARACTER*1. Indicates the communication pattern to use for the\nbroadcast.\nm, n, lda\nDescribe the matrix to be sent. See Matrix Shapes for details.\nrsrc\nINTEGER.\nThe process row coordinate of the process that called broadcast/send.\ncsrc\nINTEGER.\nThe process column coordinate of the process that called broadcast/\nsend.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2553\n\n\nOutput Parameters\na\nAn array of dimension (lda,n) to receive the incoming message into.\nDescription\nThis routine receives and participates in a broadcast along a scope. At the end of a broadcast, all processes\nwithin the scope will possess the data in the general rectangular matrix A. Broadcasts may be globally-\nblocking. This means no process is guaranteed to return from a broadcast until all processes in the scope\nhave called the appropriate routine (broadcast/send or broadcast/receive).\nSee Also\nExamples of BLACS Routines Usage\n?trbr2d\nReceives and participates in a broadcast along a scope\nfor a trapezoidal matrix.\nSyntax\ncall itrbr2d( icontxt, scope, top, uplo, diag, m, n, a, lda, rsrc, csrc )\ncall strbr2d( icontxt, scope, top, uplo, diag, m, n, a, lda, rsrc, csrc )\ncall dtrbr2d( icontxt, scope, top, uplo, diag, m, n, a, lda, rsrc, csrc )\ncall ctrbr2d( icontxt, scope, top, uplo, diag, m, n, a, lda, rsrc, csrc )\ncall ztrbr2d( icontxt, scope, top, uplo, diag, m, n, a, lda, rsrc, csrc )\nInput Parameters\nicontxt\nINTEGER. Integer handle that indicates the context.\nscope\nCHARACTER*1. Indicates what scope the broadcast should proceed on.\nLimited to 'Row', 'Column', or 'All'.\ntop\nCHARACTER*1. Indicates the communication pattern to use for the\nbroadcast.\nuplo, diag, m, n, lda\nDescribe the matrix to be sent. See Matrix Shapes for details.\nrsrc\nINTEGER.\nThe process row coordinate of the process that called broadcast/send.\ncsrc\nINTEGER.\nThe process column coordinate of the process that called broadcast/\nsend.\nOutput Parameters\na\nAn array of dimension (lda,n) to receive the incoming message into.\nDescription\nThis routine receives and participates in a broadcast along a scope. At the end of a broadcast, all processes\nwithin the scope will possess the data in the trapezoidal matrix A. Broadcasts may be globally-blocking. This\nmeans no process is guaranteed to return from a broadcast until all processes in the scope have called the\nappropriate routine (broadcast/send or broadcast/receive).\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2554\n\n\nBLACS Support Routines\nThe support routines perform distinct tasks that can be used for:\nInitialization\nDestruction\nInformation Purposes\nMiscellaneous Tasks.\nInitialization Routines\nThis topic describes BLACS routines that deal with grid/context creation, and processing before the grid/\ncontext has been defined.\nBLACS Initialization Routines\nRoutine name\nOperation performed\nblacs_pinfo\nReturns the number of processes available for use.\nblacs_setup\nAllocates virtual machine and spawns processes.\nblacs_get\nGets values that BLACS use for internal defaults.\nblacs_set\nSets values that BLACS use for internal defaults.\nblacs_gridinit\nAssigns available processes into BLACS process grid.\nblacs_gridmap\nMaps available processes into BLACS process grid.\nblacs_pinfo\nReturns the number of processes available for use.\nSyntax\ncall blacs_pinfo( mypnum, nprocs )\nOutput Parameters\nmypnum\nINTEGER. An integer between 0 and (nprocs - 1) that uniquely\nidentifies each process.\nnprocs\nINTEGER.The number of processes available for BLACS use.\nDescription\nThis routine is used when some initial system information is required before the BLACS are set up. On all\nplatforms except PVM, nprocs is the actual number of processes available for use, that is, nprows * npcols\n<= nprocs. In PVM, the virtual machine may not have been set up before this call, and therefore no parallel\nmachine exists. In this case, nprocs is returned as less than one. If a process has been spawned via the\nkeyboard, it receives mypnum of 0, and all other processes get mypnum of -1. As a result, the user can\ndistinguish between processes. Only after the virtual machine has been set up via a call to BLACS_SETUP,\nthis routine returns the correct values for mypnum and nprocs.\nSee Also\nExamples of BLACS Routines Usage\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2555\n\n\nblacs_setup\nAllocates virtual machine and spawns processes.\nSyntax\ncall blacs_setup( mypnum, nprocs )\nInput Parameters\nnprocs\nINTEGER. On the process spawned from the keyboard rather than\nfrom pvmspawn, this parameter indicates the number of processes to\ncreate when building the virtual machine.\nOutput Parameters\nmypnum\nINTEGER. An integer between 0 and (nprocs - 1) that uniquely\nidentifies each process.\nnprocs\nINTEGER. For all processes other than spawned from the keyboard,\nthis parameter means the number of processes available for BLACS\nuse.\nDescription\nThis routine only accomplishes meaningful work in the PVM BLACS. On all other platforms, it is functionally\nequivalent to blacs_pinfo. The BLACS assume a static system, that is, the given number of processes does\nnot change. PVM supplies a dynamic system, allowing processes to be added to the system on the fly.\nblacs_setup is used to allocate the virtual machine and spawn off processes. It reads in a file called\nblacs_setup.dat, in which the first line must be the name of your executable. The second line is optional,\nbut if it exists, it should be a PVM spawn flag. Legal values at this time are 0 (PvmTaskDefault), 4\n(PvmTaskDebug), 8 (PvmTaskTrace), and 12 (PvmTaskDebug + PvmTaskTrace). The primary reason for this\nline is to allow the user to easily turn on and off PVM debugging. Additional lines, if any, specify what\nmachines should be added to the current configuration before spawning nprocs-1 processes to the machines\nin a round robin fashion.\nnprocs is input on the process which has no PVM parent (that is, mypnum=0), and both parameters are\noutput for all processes. So, on PVM systems, the call to blacs_pinfo informs you that the virtual machine\nhas not been set up, and a call to blacs_setup then sets up the machine and returns the real values for\nmypnum and nprocs.\nNote that if the file blacs_setup.dat does not exist, the BLACS prompt the user for the executable name,\nand processes are spawned to the current PVM configuration.\nSee Also\nExamples of BLACS Routines Usage\nblacs_get\nGets values that BLACS use for internal defaults.\nSyntax\ncall blacs_get( icontxt, what, val )\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2556\n\n\nInput Parameters\nicontxt\nINTEGER. On values of what that are tied to a particular context, this\nparameter is the integer handle indicating the context. Otherwise,\nignored.\nwhat\nINTEGER. Indicates what BLACS internal(s) should be returned in val.\nPresent options are:\n•\nwhat = 0 : Handle indicating default system context.\n•\nwhat = 1 : The BLACS message ID range.\n•\nwhat = 2 : The BLACS debug level the library was compiled with.\n•\nwhat = 10 : Handle indicating the system context used to define\nthe BLACS context whose handle is icontxt.\n•\nwhat = 11 : Number of rings multiring broadcast topology is\npresently using.\n•\nwhat = 12 : Number of branches general tree broadcast topology\nis presently using.\n•\nwhat = 13 : Number of rings multiring combine topology is\npresently using.\n•\nwhat = 14 : Number of branches general tree combine topology is\npresently using.\n•\nwhat = 15 : Whether topologies are forced to be repeatable or not.\nA non-zero return value indicates that topologies are being forced\nto be repeatable. See Repeatability and Coherence for more\ninformation about repeatability.\n•\nwhat = 16 : Whether topologies are forced to be heterogenous\ncoherent or not. A non-zero return value indicates that topologies\nare being forced to be heterogenous coherent. See Repeatability\nand Coherence for more information about coherence.\nOutput Parameters\nval\nINTEGER. The value of the BLACS internal.\nDescription\nThis routine gets the values that the BLACS are using for internal defaults. Some values are tied to a BLACS\ncontext, and some are more general. The most common use is in retrieving a default system context for\ninput into blacs_gridinit or blacs_gridmap.\nSome systems, such as MPI*, supply their own version of context. For those users who mix system code with\nBLACS code, a BLACS context should be formed in reference to a system context. Thus, the grid creation\nroutines take a system context as input. If you wish to have strictly portable code, you may use blacs_get\nto retrieve a default system context that will include all available processes. This value is not tied to a BLACS\ncontext, so the parameter icontxt is unused.\nblacs_get returns information on three quantities that are tied to an individual BLACS context, which is\npassed in as icontxt. The information that may be retrieved is:\n•\nThe handle of the system context upon which this BLACS context was defined\n•\nThe number of rings for TOP = 'M' (multiring broadcast/combine)\n•\nThe number of branches for TOP = 'T' (general tree broadcast/general tree gather).\n•\nWhether topologies are being forced to be repeatable or heterogenous coherent.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2557\n\n\nSee Also\nExamples of BLACS Routines Usage\nblacs_set\nSets values that BLACS use for internal defaults.\nSyntax\ncall blacs_set( icontxt, what, val )\nInput Parameters\nicontxt\nINTEGER. For values of what that are tied to a particular context, this\nparameter is the integer handle indicating the context. Otherwise,\nignored.\nwhat\nINTEGER. Indicates what BLACS internal(s) should be set. Present\nvalues are:\n•\n1 = Set the BLACS message ID range\n•\n11 = Number of rings for multiring broadcast topology to use\n•\n12 = Number of branches for general tree broadcast topology to\nuse\n•\n13 = Number of rings for multiring combine topology to use\n•\n14 = Number of branches for general tree combine topology to use\n•\n15 = Force topologies to be repeatable or not\n•\n16 = Force topologies to be heterogenous coherent or not\nval\nINTEGER. Array of dimension (*). Indicates the value(s) the internals\nshould be set to. The specific meanings depend on what values, as\ndiscussed in Description below.\nDescription\nThis routine sets the BLACS internal defaults depending on what values:\nwhat = 1\nSetting the BLACS message ID range.\nIf you wish to mix the BLACS with other message-passing packages, restrict the\nBLACS to a certain message ID range not to be used by the non-BLACS routines.\nThe message ID range must be set before the first call to blacs_gridinit or \nblacs_gridmap. Subsequent calls will have no effect. Because the message ID\nrange is not tied to a particular context, the parameter icontxt is ignored, and\nval is defined as:\nVAL (input) INTEGER array of dimension (2)\n    VAL(1) : The smallest message ID (also called message type or message tag)\nthe BLACS should use.\n    VAL(2) : The largest message ID (also called message type or message tag)\nthe BLACS should use.\nwhat = 11\nSet number of rings for TOP = 'M' (multiring broadcast).This quantity is tied to\na context, so icontxt is used, and val is defined as:\nVAL (input) INTEGER array of dimension (1)\n    VAL(1) : The number of rings for multiring topology to use.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2558\n\n\nwhat = 12\nSet number of branches for TOP = 'T' (general tree broadcast). This quantity is\ntied to a context, so icontxt is used, and val is defined as:\nVAL (input) INTEGER array of dimension (1)\n    VAL(1) : The number of branches for general tree topology to use.\nwhat = 13\nSet number of rings for TOP = 'M' (multiring combine).This quantity is tied to a\ncontext, so icontxt is used, and val is defined as:\nVAL (input) INTEGER array of dimension (1)\n    VAL(1) : The number of rings for multiring topology to use.\nwhat = 14\nSet number of branches for TOP = 'T' (general tree gather). This quantity is\ntied to a context, so icontxt is used, and val is defined as:\nVAL (input) INTEGER array of dimension (1)\n    VAL(1) : The number of branches for general tree topology to use.\nwhat = 15\nForce topologies to be repeatable or not (see Repeatability and Coherence for\nmore information about repeatability).\nVAL (input) INTEGER array of dimension (1)\nVAL(1) = 0 (default)\nTopologies are not required to be repeatable.\nVAL(1) ≠ 0\nAll used topologies are required to be repeatable,\nwhich might degrade performance.\nwhat = 16\nForce topologies to be heterogenous coherent or not (see Repeatability and\nCoherence for more information about coherence).\nVAL (input) INTEGER array of dimension (1)\nVAL(1) = 0 (default)\nTopologies are not required to be heterogenous\ncoherent.\nVAL(1) ≠ 0\nAll used topologies are required to be heterogenous\ncoherent, which might degrade performance.\nblacs_gridinit\nAssigns available processes into BLACS process grid.\nSyntax\ncall blacs_gridinit( icontxt, layout, nprow, npcol )\nInput Parameters\nicontxt\nINTEGER. Integer handle indicating the system context to be used in\ncreating the BLACS context. Call blacs_get to obtain a default\nsystem context.\nlayout\nCHARACTER*1. Indicates how to map processes to BLACS grid. Options\nare:\n•\n'R' : Use row-major natural ordering\n•\n'C' : Use column-major natural ordering\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2559\n\n\n•\nELSE : Use row-major natural ordering\nnprow\nINTEGER. Indicates how many process rows the process grid should\ncontain.\nnpcol\nINTEGER. Indicates how many process columns the process grid\nshould contain.\nOutput Parameters\nicontxt\nINTEGER. Integer handle to the created BLACS context.\nDescription\nAll BLACS codes must call this routine, or its sister routine blacs_gridmap. These routines take the available\nprocesses, and assign, or map, them into a BLACS process grid. In other words, they establish how the\nBLACS coordinate system maps into the native machine process numbering system. Each BLACS grid is\ncontained in a context, so that it does not interfere with distributed operations that occur within other grids/\ncontexts. These grid creation routines may be called repeatedly to define additional contexts/grids.\nThe creation of a grid requires input from all processes that are defined to be in this grid. Processes\nbelonging to more than one grid have to agree on which grid formation will be serviced first, much like the\nglobally blocking sum or broadcast.\nThese grid creation routines set up various internals for the BLACS, and one of them must be called before\nany calls are made to the non-initialization BLACS.\nNote that these routines map already existing processes to a grid: the processes are not created dynamically.\nOn most parallel machines, the processes are \"created\" when you run your executable. When using the PVM\nBLACS, if the virtual machine has not been set up yet, the routine blacs_setup should be used to create the\nvirtual machine.\nThis routine creates a simple nprow x npcol process grid. This process grid uses the first nprow * npcol\nprocesses, and assigns them to the grid in a row- or column-major natural ordering. If these process-to-grid\nmappings are unacceptable, call blacs_gridmap.\nSee Also\nExamples of BLACS Routines Usage\nblacs_get\nblacs_gridmap\nblacs_setup\nblacs_gridmap\nMaps available processes into BLACS process grid.\nSyntax\ncall blacs_gridmap( icontxt, usermap, ldumap, nprow, npcol )\nInput Parameters\nicontxt\nINTEGER. Integer handle indicating the system context to be used in\ncreating the BLACS context. Call blacs_get to obtain a default\nsystem context.\nusermap\nINTEGER. Array, dimension (ldumap, npcol), indicating the process-\nto-grid mapping.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2560\n\n\nldumap\nINTEGER. Leading dimension of the 2D array usermap. ldumap ≥\nnprow.\nnprow\nINTEGER. Indicates how many process rows the process grid should\ncontain.\nnpcol\nINTEGER. Indicates how many process columns the process grid\nshould contain.\nOutput Parameters\nicontxt\nINTEGER. Integer handle to the created BLACS context.\nDescription\nAll BLACS codes must call this routine, or its sister routine blacs_gridinit. These routines take the\navailable processes, and assign, or map, them into a BLACS process grid. In other words, they establish how\nthe BLACS coordinate system maps into the native machine process numbering system. Each BLACS grid is\ncontained in a context, so that it does not interfere with distributed operations that occur within other grids/\ncontexts. These grid creation routines may be called repeatedly to define additional contexts/grids.\nThe creation of a grid requires input from all processes that are defined to be in this grid. Processes\nbelonging to more than one grid have to agree on which grid formation will be serviced first, much like the\nglobally blocking sum or broadcast.\nThese grid creation routines set up various internals for the BLACS, and one of them must be called before\nany calls are made to the non-initialization BLACS.\nNote that these routines map already existing processes to a grid: the processes are not created dynamically.\nOn most parallel machines, the processes are actual processors (hardware), and they are \"created\" when you\nrun your executable. When using the PVM BLACS, if the virtual machine has not been set up yet, the routine\nblacs_setup should be used to create the virtual machine.\nThis routine allows the user to map processes to the process grid in an arbitrary manner. usermap(i,j)\nholds the process number of the process to be placed in {i, j} of the process grid. On most distributed\nsystems, this process number is a machine defined number between 0 ... nprow-1. For PVM, these node\nnumbers are the PVM TIDS (Task IDs). The blacs_gridmap routine is intended for an experienced user. The\nblacs_gridinit routine is much simpler. blacs_gridinit simply performs a gridmap where the first\nnprow * npcol processes are mapped into the current grid in a row-major natural ordering. If you are an\nexperienced user, blacs_gridmap allows you to take advantage of your system's actual layout. That is, you\ncan map nodes that are physically connected to be neighbors in the BLACS grid, etc. The blacs_gridmap\nroutine also opens the way for multigridding: you can separate your nodes into arbitrary grids, join them\ntogether at some later date, and then re-split them into new grids. blacs_gridmap also provides the ability\nto make arbitrary grids or subgrids (for example, a \"nearest neighbor\" grid), which can greatly facilitate\noperations among processes that do not fall on a row or column of the main process grid.\nSee Also\nExamples of BLACS Routines Usage\nblacs_get\nblacs_gridinit\nblacs_setup\nDestruction Routines\nThis topic describes BLACS routines that destroy grids, abort processes, and free resources.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2561\n\n\nBLACS Destruction Routines\nRoutine name\nOperation performed\nblacs_freebuff\nFrees BLACS buffer.\nblacs_gridexit\nFrees a BLACS context.\nblacs_abort\nAborts all processes.\nblacs_exit\nFrees all BLACS contexts and releases all allocated memory.\nblacs_freebuff\nFrees BLACS buffer.\nSyntax\ncall blacs_freebuff( icontxt, wait )\nInput Parameters\nicontxt\nINTEGER. Integer handle that indicates the BLACS context.\nwait\nINTEGER. Parameter indicating whether to wait for non-blocking\noperations or not. If equals 0, the operations should not be waited for;\nfree only unused buffers. Otherwise, wait in order to free all buffers.\nDescription\nThis routine releases the BLACS buffer.\nThe BLACS have at least one internal buffer that is used for packing messages. The number of internal\nbuffers depends on what platform you are running the BLACS on. On systems where memory is tight,\nkeeping this buffer or buffers may become expensive. Call freebuff to release the buffer. However, the next\ncall of a communication routine that requires packing reallocates the buffer.\nThe wait parameter determines whether the BLACS should wait for any non-blocking operations to be\ncompleted or not. If wait = 0, the BLACS free any buffers that can be freed without waiting. If wait is not 0,\nthe BLACS free all internal buffers, even if non-blocking operations must be completed first.\nblacs_gridexit\nFrees a BLACS context.\nSyntax\ncall blacs_gridexit( icontxt )\nInput Parameters\nicontxt\nINTEGER. Integer handle that indicates the BLACS context to be freed.\nDescription\nThis routine frees a BLACS context.\nRelease the resources when contexts are no longer needed. After freeing a context, the context no longer\nexists, and its handle may be re-used if new contexts are defined.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2562\n\n\nblacs_abort\nAborts all processes.\nSyntax\ncall blacs_abort( icontxt, errornum )\nInput Parameters\nicontxt\nINTEGER. Integer handle that indicates the BLACS context to be\naborted.\nerrornum\nINTEGER. User-defined integer error number.\nDescription\nThis routine aborts all the BLACS processes, not only those confined to a particular context.\nUse blacs_abort to abort all the processes in case of a serious error. Note that both parameters are input,\nbut the routine uses them only in printing out the error message. The context handle passed in is not\nrequired to be a valid context handle.\nblacs_exit\nFrees all BLACS contexts and releases all allocated\nmemory.\nSyntax\ncall blacs_exit( continue )\nInput Parameters\ncontinue\nINTEGER. Flag indicating whether message passing continues after the\nBLACS are done. If continue is non-zero, the user is assumed to\ncontinue using the machine after completing the BLACS. Otherwise,\nno message passing is assumed after calling this routine.\nDescription\nThis routine frees all BLACS contexts and releases all allocated memory.\nThis routine should be called when a process has finished all use of the BLACS. The continue parameter\nindicates whether the user will be using the underlying communication platform after the BLACS are finished.\nThis information is most important for the PVM BLACS. If continue is set to 0, then pvm_exit is called;\notherwise, it is not called. Setting continue not equal to 0 indicates that explicit PVM send/recvs will be\ncalled after the BLACS routines are used. Make sure your code calls pvm_exit. PVM users should either call\nblacs_exit or explicitly call pvm_exit to avoid PVM problems.\nSee Also\nExamples of BLACS Routines Usage\nInformational Routines\nThis topic describes BLACS routines that return information involving the process grid.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2563\n\n\nBLACS Informational Routines\nRoutine name\nOperation performed\nblacs_gridinfo\nReturns information on the current grid.\nblacs_pnum\nReturns the system process number of the process in the process grid.\nblacs_pcoord\nReturns the row and column coordinates in the process grid.\nblacs_gridinfo\nReturns information on the current grid.\nSyntax\ncall blacs_gridinfo( icontxt, nprow, npcol, myprow, mypcol )\nInput Parameters\nicontxt\nINTEGER. Integer handle that indicates the context.\nOutput Parameters\nnprow\nINTEGER. Number of process rows in the current process grid.\nnpcol\nINTEGER. Number of process columns in the current process grid.\nmyprow\nINTEGER. Row coordinate of the calling process in the process grid.\nmypcol\nINTEGER. Column coordinate of the calling process in the process grid.\nDescription\nThis routine returns information on the current grid. If the context handle does not point at a valid context,\nall quantities are returned as -1.\nSee Also\nExamples of BLACS Routines Usage\nblacs_pnum\nReturns the system process number of the process in\nthe process grid.\nSyntax\ncall blacs_pnum( icontxt, prow, pcol )\nInput Parameters\nicontxt\nINTEGER. Integer handle that indicates the context.\nprow\nINTEGER. Row coordinate of the process the system process number\nof which is to be determined.\npcol\nINTEGER. Column coordinate of the process the system process\nnumber of which is to be determined.\nDescription\nThis function returns the system process number of the process at {PROW, PCOL} in the process grid.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2564\n\n\nSee Also\nExamples of BLACS Routines Usage\nblacs_pcoord\nReturns the row and column coordinates in the\nprocess grid.\nSyntax\ncall blacs_pcoord( icontxt, pnum, prow, pcol )\nInput Parameters\nicontxt\nINTEGER. Integer handle that indicates the context.\npnum\nINTEGER. Process number the coordinates of which are to be\ndetermined. This parameter stand for the process number of the\nunderlying machine, that is, it is a tid for PVM.\nOutput Parameters\nprow\nINTEGER. Row coordinates of the pnum process in the BLACS grid.\npcol\nINTEGER. Column coordinates of the pnum process in the BLACS grid.\nDescription\nGiven the system process number, this function returns the row and column coordinates in the BLACS\nprocess grid.\nSee Also\nExamples of BLACS Routines Usage\nMiscellaneous Routines\nThis topic describes blacs_barrier routine.\nBLACS Informational Routines\nRoutine name\nOperation performed\nblacs_barrier\nHolds up execution of all processes within the indicated scope until\nthey have all called the routine.\nblacs_barrier\nHolds up execution of all processes within the\nindicated scope.\nSyntax\ncall blacs_barrier( icontxt, scope )\nInput Parameters\nicontxt\nINTEGER. Integer handle that indicates the context.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2565\n\n\nscope\nCHARACTER*1. Parameter that indicates whether a process row\n(scope='R'), column ('C'), or entire grid ('A') will participate in the\nbarrier.\nDescription\nThis routine holds up execution of all processes within the indicated scope until they have all called the\nroutine.\nExamples of BLACS Routines Usage\nData Fitting Functions\nData Fitting functions in Intel® oneAPI Math Kernel Library (oneMKL) provide spline-based interpolation\ncapabilities that you can use to approximate functions, function derivatives or integrals, and perform cell\nsearch operations.\nThe Data Fitting component is task based. The task is a data structure or descriptor that holds the\nparameters related to a specific Data Fitting operation. You can modify the task parameters using the task\nediting functionality of the library.\nFor definition of the implemented operations, see Mathematical Conventions.\nData Fitting routines use the following workflow to process a task:\n1.\nCreate a task or multiple tasks.\n2.\nModify the task parameters.\n3.\nPerform a Data Fitting computation.\n4.\nDestroy the task or tasks.\nAll Data Fitting functions fall into the following categories:\nTask Creation and Initialization Routines - routines that create a new Data Fitting task descriptor and initialize\nthe most common parameters, such as partition of the interpolation interval, values of the vector-valued\nfunction, and the parameters describing their structure.\nTask Configuration Routines - routines that set, modify, or query parameters in an existing Data Fitting task.\nComputational Routines - routines that perform Data Fitting computations, such as construction of a spline,\ninterpolation, computation of derivatives and integrals, and search.\nTask Destructors - routines that delete Data Fitting task descriptors and deallocate resources.\nYou can access the Data Fitting routines through the Fortran and C89/C99 language interfaces. You can also\nuse the C89 interface with more recent versions of C/C++, or the Fortran 90 interface with programs written\nin Fortran 95.\nThe ${MKL}/includedirectory of the Intel® oneAPI Math Kernel Library (oneMKL) contains the following Data\nFitting header files:\n•\nmkl_df.h\nYou can find examples that demonstrate usage of Data Fitting routines in the ${MKL}/examples/\ndatafittingc directory .\nData Fitting Function Naming Conventions\nThe interface of the Data Fitting functions, types, and constants are case-sensitive and can be in lowercase,\nuppercase, and mixed case.\nThe names of all routines have the following structure:\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2566\n\n\ndf[datatype]<base_name>\nwhere\n•\ndfis a prefix indicating that the routine belongs to the Data Fitting component of Intel® oneAPI Math\nKernel Library (oneMKL).\n•\n[datatype] field specifies the type of the input and/or output data and can be s (for the single precision\nreal type), d (for the double precision real type), or i (for the integer type). This field is omitted in the\nnames of the routines that are not data type dependent.\n•\n<base_name> field specifies the functionality the routine performs. For example, this field can be\nNewTask1D, Interpolate1D, or DeleteTask\nData Fitting Function Data Types\nThe Data Fitting component provides routines for processing single and double precision real data types. The\nresults of cell search operations are returned as a generic integer data type.\nAll Data Fitting routines use the following data type:\nType\nData Object\nDFTaskPtr\nPointer to a task\nNOTE\nThe actual size of the generic integer type is platform-dependent. Before compiling your application,\nyou need to set an appropriate byte size for integers. For details, see section Using the ILP64 Interface\nvs. LP64 Interface of the Intel® oneAPI Math Kernel Library (oneMKL) Developer Guide.\nMathematical Conventions for Data Fitting Functions\nThis section explains the notation used for Data Fitting function descriptions. Spline notations are based on\nthe terminology and definitions of [deBoor2001]. The Subbotin quadratic spline definition follows the\nconventions of [StechSub76]. The quasi-uniform partition definition is based on [Schumaker2007].\nMathematical Notation in the Data Fitting Component\nConcept\nMathematical Notation\nPartition of interpolation interval [a, b] , where\n•\nxi denotes breakpoints.\n•\n[xi, xi+1) denotes a sub-interval (cell) of size\nΔi=xi+1-xi .\n{xi}i=1,...,n, where a = x1 < x2<... <xn = b\nQuasi-uniform partition of interpolation interval [a,\nb]\nPartition {xi}i=1,...,n which meets the constraint with\na constant C defined as\n1 ≤M/ m≤C,\nwhere\n•\nM = maxi=1,...,n-1 (Δi)\n•\nm = mini=1,...,n-1 (Δi)\n•\nΔi = xi+1 - xi\nVector-valued function of dimension p being fit\nƒ(x) = (ƒ1(x),..., ƒp(x))\nPiecewise polynomial (PP) function ƒ of order k+1\nƒ(x) ≔ Pi (x), if x ∈ [ xi, xi+1), i = 1,..., n-1\nwhere\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2567\n\n\nConcept\nMathematical Notation\n•\n{xi}i= 1,..., n is a strictly increasing sequence of\nbreakpoints.\n•\nPi(x) = ci,0 + ci,1(x - xi) + ... + ci,k(x - xi)k is a\npolynomial of degree k (order k+1) over the\ninterval x ∈ [ xi, xi+1).\nFunction p agrees with function ƒ at the points\n{xi}i=1,...,n .\nFor every point ζ in sequence {xi}i=1,...,n that occurs\nm times, the equality p(i-1)(ζ) = ƒ(i-1)(ζ) holds for all\ni = 1,...,m, where p(i)(t) is the derivative of the i-th\norder.\nThe k-th divided difference of function ƒ at points\nxi,..., xi + k. This difference is the leading coefficient\nof the polynomial of order k+1 that agrees with ƒ at\nxi,..., xi + k.\n[ xi,..., xi + k] ƒ\nIn particular,\n•\n[x1]ƒ = ƒ(x1)\n•\n[ x1, x2] ƒ = (ƒ(x1) - ƒ(x2)) / (x1 - x2)\nA k-order derivative of interpolant ƒ(x) at\ninterpolation site \n.\nInterpolants to the Function ƒ at x1,..., xn and Boundary Conditions\nConcept\nMathematical Notation\nLinear interpolant\nPi(x) = c1, i + c2, i(x - xi),\nwhere\n•\nx ∈ [ xi, xi+1)\n•\nc1, i = ƒ(xi)\n•\nc2, i = [xi, xi+1 ]ƒ\n•\ni = 1,..., n-1\nPiecewise parabolic interpolant\nPi(x) = c1, i + c2, i(x - xi) + c3, i(x - xi)2, x ∈ [ xi, xi+1)\nCoefficients c1, i, c2, i, and c3, i depend on the conditions:\n•\nPi(xi) = ƒ(xi)\n•\nPi(xi+1) = ƒ(xi+1)\n•\nPi((xi+1 + xi) / 2) = vi+1\nwhere parameter vi+1 depends on the interpolant being\ncontinuously differentiable:\nPi-1(1)(xi) = Pi(1)(xi)\nPiecewise parabolic Subbotin\ninterpolant\nP(x) = Pi(x) = c1,i+c2,i(x-xi)+c3,i(x-xi)2+d3,i((x-ti)+)2,\nwhere\n•\nx ∈ [ ti, ti+1)\n•\n{ti}i=1,...,n+1 is a sequence of knots such that\n•\nt1 = x1, tn+1 = xn\n•\nti ∈ (xi-1, xi), i = 2,..., n\n•\nCoefficients c1,i, c2,i, c3,i, and d3,i depend on the following\nconditions:\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2568\n\n\nConcept\nMathematical Notation\n•\nPi(xi) = ƒ(xi), Pi(xi+1) = ƒ(xi+1)\n•\nP(x) is a continuously differentiable polynomial of the second\ndegree on [ ti, ti+1), i = 1,..., n.\nPiecewise cubic Hermite interpolant\nPi(x) = c1,i + c2,i(x - xi) + c3,i(x - xi)2 + c4,i(x - xi)3,\nwhere\n•\nx ∈ [ xi, xi+1)\n•\nc1,i = ƒ(xi)\n•\nc2,i = si\n•\nc3,i = ([xi, xi+1]ƒ - si ) / (Δxi) - c4,i(Δxi)\n•\nc4,i = (si + si+1 - 2[xi, xi+1]ƒ) / (Δxi)2\n•\ni = 1,..., n-1\n•\nsi = ƒ(1)(xi)\nPiecewise cubic Bessel interpolant\nPi(x) = c1,i + c2,i(x - xi) + c3,i(x - xi)2 + c4,i(x - xi)3,\nwhere\n•\nx ∈ [ xi, xi+1)\n•\nc1,i = ƒ(xi)\n•\nc2,i = si\n•\nc3,i = ([xi, xi+1]ƒ - si ) / (Δxi) - c4,i(Δxi)\n•\nc4,i = (si + si+1 - 2[xi, xi+1]ƒ) / (Δxi)2\n•\ni = 1,..., n-1\n•\nsi = (Δxi[xi-1, xi]ƒ + Δxi-1[xi, xi+1]ƒ) / (Δxi + Δxi+1)\nPiecewise cubic Akima interpolant\nPi(x) = c1,i + c2,i(x - xi) + c3,i(x - xi)2 + c4,i(x - xi)3,\nwhere\n•\nx ∈ [ xi, xi+1)\n•\nc1,i = ƒ(xi)\n•\nc2,i = si\n•\nc3,i = ([xi, xi+1]ƒ - si ) / (Δxi) - c4,i(Δxi)\n•\nc4,i = (si + si+1 - 2[xi, xi+1]ƒ) / (Δxi)2\n•\ni = 1,..., n-1\n•\nsi = (wi+1[xi-1, xi]ƒ + wi-1[xƒi, xi+1]ƒ) / (wi+1 + wi-1),\nwhere\nwi = |[xi, xi+1]ƒ - [xi-1, xi]ƒ|\nPiecewise natural cubic interpolant\nPi(x) = c1,i + c2,i(x - xi) + c3,i(x - xi)2 + c4,i(x - xi)3,\nwhere\n•\nx ∈ [ xi, xi+1)\n•\nc1,i = ƒ(xi)\n•\nc2,i = si\n•\nc3,i = ([xi, xi+1]ƒ - si ) / (Δxi) - c4,i(Δxi)\n•\nc4,i = (si + si+1 - 2[xi, xi+1]ƒ) / (Δxi)2\n•\ni = 1,..., n-1\n•\nParameter si depends on the condition that the interpolant is\ntwice continuously differentiable: Pi-1(2)(xi) = Pi(2)(xi).\nNot-a-knot boundary condition.\nParameters s1 and sn provide P1 = P2 and Pn-1 = Pn, so that the\nfirst and the last interior breakpoints are inactive.\nFree-end boundary condition.\nƒ\"(x1) = ƒ\"(xn) = 0\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2569\n\n\nConcept\nMathematical Notation\nLook-up interpolator for discrete set\nof points (x1, y1),..., (xn, yn) .\nStep-wise constant continuous right\ninterpolator.\nStep-wise constant continuous left\ninterpolator.\nData Fitting Usage Model\nConsider an algorithm that uses the Data Fitting functions. Typically, such algorithms consist of four steps or\nstages:\n1.\nCreate a task. You can call the Data Fitting function several times to create multiple tasks.\nstatus = dfdNewTask1D( &task, nx, x, xhint, ny, y, yhint );\n2.\nModify the task parameters.\nstatus = dfdEditPPSpline1D( task, s_order, c_type, bc_type, bc, ic_type, ic,\nscoeff, scoeffhint );\n3.\nPerform Data Fitting spline-based computations. You may reiterate steps 2-3 as needed.\nstatus = dfdInterpolate1D(task, estimate, method, nsite, site, sitehint, ndorder,\ndorder, datahint, r, rhint, cell );\n4.\nDestroy the task or tasks.\nstatus = dfDeleteTask( &task );\nSee Also\nData Fitting Usage Examples\nData Fitting Usage Examples\nThe examples below illustrate several operations that you can perform with Data Fitting routines.\nYou can get source code for similar examples in the .\\examples\\datafittingcsubdirectory of the Intel®\noneAPI Math Kernel Library (oneMKL) installation directory.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2570\n\n\nThe following example demonstrates the construction of a linear spline using Data Fitting routines. The spline\napproximates a scalar function defined on non-uniform partition. The coefficients of the spline are returned\nas a one-dimensional array:\nExample of Linear Spline Construction\n#include \"mkl.h\"\n#define N 500                     /* Size of partition, number of breakpoints */\n#define SPLINE_ORDER DF_PP_LINEAR /* Linear spline to construct */\nint main()\n{\n    int status;          /* Status of a Data Fitting operation */\n    DFTaskPtr task;      /* Data Fitting operations are task based */\n    /* Parameters describing the partition */\n    MKL_INT nx;          /* The size of partition x */\n    double x[N];         /* Partition x */\n    MKL_INT xhint;       /* Additional information about the structure of breakpoints */\n    /* Parameters describing the function */\n    MKL_INT ny;          /* Function dimension */\n    double y[N];         /* Function values at the breakpoints */\n    MKL_INT yhint;       /* Additional information about the function */\n    \n    /* Parameters describing the spline */\n    MKL_INT  s_order;    /* Spline order */\n    MKL_INT  s_type;     /* Spline type */\n    MKL_INT  ic_type;    /* Type of internal conditions */\n    double* ic;         /* Array of internal conditions */\n    MKL_INT  bc_type;    /* Type of boundary conditions */\n    double* bc;         /* Array of boundary conditions */\n    double scoeff[(N-1)* SPLINE_ORDER];   /* Array of spline coefficients */\n    MKL_INT scoeffhint;            /* Additional information about the coefficients */\n    /* Initialize the partition */\n    nx = N;\n    /* Set values of partition x */\n    ...\n    xhint = DF_NO_HINT;    /* No additional information about the function is provided.\n                              By default, the partition is non-uniform. */\n    /* Initialize the function */\n     ny = 1;               /* The function is scalar. */\n   \n    /* Set function values */\n    ...\n    yhint = DF_NO_HINT;    /* No additional information about the function is provided. */\n    /* Create a Data Fitting task */\n    status = dfdNewTask1D( &task, nx, x, xhint, ny, y, yhint );\n    /* Check the Data Fitting operation status */\n    ...\n    /* Initialize spline parameters */\n    s_order = DF_PP_LINEAR;    /* Spline is of the second order. */ \n    s_type = DF_PP_DEFAULT;    /* Spline is of the default type. */ \nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2571\n\n\n    /* Define internal conditions for linear spline construction (none in this example) */\n    ic_type = DF_NO_IC; \n    ic = NULL;\n    /* Define boundary conditions for linear spline construction (none in this example) */\n    bc_type = DF_NO_BC; \n    bc = NULL;\n    scoeffhint = DF_NO_HINT;    /* No additional information about the spline. */ \n    /* Set spline parameters  in the Data Fitting task */\n    status = dfdEditPPSpline1D( task, s_order, s_type, bc_type, bc, ic_type,\n                                ic, scoeff, scoeffhint );\n    \n    /* Check the Data Fitting operation status */\n    ...\n    /* Use a standard computation method to construct a linear spline: */\n    /* Pi(x) = ci,0+ci,1(x-xi), i=0,..., N-2 */\n    /* The library packs spline coefficients to array scoeff. */ \n    /* scoeff[2*i+0]=ci,0 and scoeff[2*i+1]=ci,1, i=0,..., N-2 */\n    status = dfdConstruct1D( task, DF_PP_SPLINE, DF_METHOD_STD );\n    \n    /* Check the Data Fitting operation status */\n    ...\n    /* Process spline coefficients */\n    ...\n    /* Deallocate Data Fitting task resources */\n    status = dfDeleteTask( &task ) ;\n    /* Check the Data Fitting operation status */\n    ...\n    return 0 ;\n}\nThe following example demonstrates cubic spline-based interpolation using Data Fitting routines. In this\nexample, a scalar function defined on non-uniform partition is approximated by Bessel cubic spline using not-\na-knot boundary conditions. Once the spline is constructed, you can use the spline to compute spline values\nat the given sites. Computation results are packed by the Data Fitting routine in row-major format.\nExample of Cubic Spline-Based Interpolation\n#include \"mkl.h\"\n#define NX 100                     /* Size of partition, number of breakpoints */\n#define NSITE 1000                 /* Number of interpolation sites */\n#define SPLINE_ORDER DF_PP_CUBIC   /* A cubic spline to construct */\nint main()\n{\n    int status;          /* Status of a Data Fitting operation */\n    DFTaskPtr task;      /* Data Fitting operations are task based */\n    /* Parameters describing the partition */\n    MKL_INT nx;          /* The size of partition x */\n    double x[NX];         /* Partition x */\n    MKL_INT xhint;       /* Additional information about the structure of breakpoints */\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2572\n\n\n    /* Parameters describing the function */\n    MKL_INT ny;          /* Function dimension */\n    double y[NX];         /* Function values at the breakpoints */\n    MKL_INT yhint;       /* Additional information about the function */\n    \n    /* Parameters describing the spline */\n    MKL_INT  s_order;    /* Spline order */\n    MKL_INT  s_type;     /* Spline type */\n    MKL_INT  ic_type;    /* Type of internal conditions */\n    double* ic;         /* Array of internal conditions */\n    MKL_INT  bc_type;    /* Type of boundary conditions */\n    double* bc;         /* Array of boundary conditions */\n    double scoeff[(NX-1)* SPLINE_ORDER];   /* Array of spline coefficients */\n    MKL_INT scoeffhint;            /* Additional information about the coefficients */\n    /* Parameters describing interpolation computations */\n    MKL_INT nsite;        /* Number of interpolation sites */\n    double site[NSITE];   /* Array of interpolation sites */\n    MKL_INT sitehint;     /* Additional information about the structure of \n                             interpolation sites */\n    MKL_INT ndorder, dorder;    /* Parameters defining the type of interpolation */\n    double* datahint;   /* Additional information on partition and interpolation sites */\n    double r[NSITE];    /* Array of interpolation results */\n    MKL_INT rhint;     /* Additional information on the structure of the results */\n    MKL_INT* cell;      /* Array of cell indices */\n    /* Initialize the partition */\n    nx = NX;\n    /* Set values of partition x */\n    ...\n    xhint = DF_NON_UNIFORM_PARTITION;  /* The partition is non-uniform. */\n    /* Initialize the function */\n     ny = 1;               /* The function is scalar. */\n     \n    /* Set function values */\n    ...\n    yhint = DF_NO_HINT;    /* No additional information about the function is provided. */\n    /* Create a Data Fitting task */\n    status = dfdNewTask1D( &task, nx, x, xhint, ny, y, yhint );\n    /* Check the Data Fitting operation status */\n    ...\n    /* Initialize spline parameters */\n    s_order = DF_PP_CUBIC;     /* Spline is of the fourth order (cubic spline). */ \n    s_type = DF_PP_BESSEL;     /* Spline is of the Bessel cubic type. */ \n    /* Define internal conditions for cubic spline construction (none in this example) */\n    ic_type = DF_NO_IC; \n    ic = NULL;\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2573\n\n\n    /* Use not-a-knot boundary conditions. In this case, the is first and the last \n     interior breakpoints are inactive, no additional values are provided. */\n    bc_type = DF_BC_NOT_A_KNOT; \n    bc = NULL;\n    scoeffhint = DF_NO_HINT;    /* No additional information about the spline. */ \n    /* Set spline parameters  in the Data Fitting task */\n    status = dfdEditPPSpline1D( task, s_order, s_type, bc_type, bc, ic_type,\n                                ic, scoeff, scoeffhint );\n    \n    /* Check the Data Fitting operation status */\n    ...\n    /* Use a standard method to construct a cubic Bessel spline: */\n    /* Pi(x) = ci,0 + ci,1(x - xi) + ci,2(x - xi)2 + ci,3(x - xi)3, */\n    /* The library packs spline coefficients to array scoeff: */\n    /* scoeff[4*i+0] = ci,0, scoef[4*i+1] = ci,1,         */   \n    /* scoeff[4*i+2] = ci,2, scoef[4*i+1] = ci,3,         */   \n    /* i=0,...,N-2  */\n    status = dfdConstruct1D( task, DF_PP_SPLINE, DF_METHOD_STD );\n    /* Check the Data Fitting operation status */\n    ...\n    /* Initialize interpolation parameters */\n    nsite = NSITE;\n    /* Set site values */\n    ...\n    sitehint = DF_NON_UNIFORM_PARTITION; /* Partition of sites is non-uniform */\n    /* Request to compute spline values */\n    ndorder = 1;\n    dorder = 1;\n    datahint = DF_NO_APRIORI_INFO;  /* No additional information about breakpoints or\n                                       sites is provided. */\n    rhint = DF_MATRIX_STORAGE_ROWS; /* The library packs interpolation results \n                                       in row-major format. */\n    cell = NULL;                    /* Cell indices are not required. */\n    /* Solve interpolation problem using the default method: compute the spline values\n       at the points site(i), i=0,..., nsite-1 and place the results to array r */ \n    status = dfdInterpolate1D( task, DF_INTERP, DF_METHOD_PP, nsite, site,\n    sitehint, ndorder, &dorder, datahint, r, rhint, cell );\n \n    /* Check Data Fitting operation status */ \n     ...\n    /* De-allocate Data Fitting task resources */\n     status = dfDeleteTask( &task );\n    /* Check Data Fitting operation status */ \n     ...\n    return 0;\n}\nThe following example demonstrates how to compute indices of cells containing given sites. This example\nuses uniform partition presented with two boundary points. The sites are in the ascending order.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2574\n\n\nExample of Cell Search\n#include \"mkl.h\"\n#define NX 100                     /* Size of partition, number of breakpoints */\n#define NSITE 1000                 /* Number of interpolation sites */\nint main()\n{    \n    int status;          /* Status of a Data Fitting operation */\n    DFTaskPtr task;      /* Data Fitting operations are task based */\n    /* Parameters describing the partition */\n    MKL_INT nx;          /* The size of partition x */\n    float x[2];         /* Partition x is uniform and holds endpoints \n                            of interpolation interval [a, b] */\n    MKL_INT xhint;       /* Additional information about the structure of breakpoints */\n    /* Parameters describing the function */\n    MKL_INT ny;          /* Function dimension */\n    float   *y;          /* Function values at the breakpoints */\n    MKL_INT yhint;       /* Additional information about the function */\n    \n    /* Parameters describing cell search */\n    MKL_INT nsite;       /* Number of interpolation sites */\n    float  site[NSITE]; /* Array of interpolation sites  */\n    MKL_INT sitehint;    /* Additional information about the structure of sites */\n    float* datahint;     /* Additional information on partition and interpolation sites */\n    MKL_INT cell[NSITE]; /* Array for cell indices */ \n    /* Initialize a uniform partition */    \n    nx = NX;\n    /* Set values of partition x: for uniform partition,           */\n    /* provide end-points of the interpolation interval [-1.0,1.0] */\n    x[0] = -1.0f; x[1] = 1.0f;\n    xhint = DF_UNIFORM_PARTITION; /* Partition is uniform */\n    /* Initialize function parameters */ \n    /* In cell search, function values are not necessary and are set to zero/NULL values */\n    ny    = 0;\n    y     = NULL; \n    yhint = DF_NO_HINT;\n    /* Create a Data Fitting task */\n    status = dfsNewTask1D( &task, nx, x, xhint, ny, y, yhint );\n    /* Check Data Fitting operation status */ \n    ...\n    /* Initialize interpolation (cell search) parameters */\n    nsite = NSITE;\n   /* Set sites in the ascending order */\n    ...\n    sitehint = DF_SORTED_DATA;      /* Sites are provided in the ascending order. */\n    datahint = DF_NO_APRIORI_INFO;  /* No additional information \nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2575\n\n\n                                       about breakpoints/sites is provided.*/\n    /* Use a standard method to compute indices of the cells that contain \n       interpolation sites. The library places the index of the cell containing     \n       site(i) to the cell(i), i=0,...,nsite-1 */\n    status = dfsSearchCells1D( task, DF_METHOD_STD, nsite, site, sitehint,\n                            datahint, cell );\n    /* Check Data Fitting operation status */ \n     ...\n    /* Process cell indices */   \n     ...  \n    /* Deallocate Data Fitting task resources */\n    status = dfDeleteTask( &task );\n    /* Check Data Fitting operation status */ \n     ...\n    return 0;\n}\nData Fitting Function Task Status and Error Reporting\nThe Data Fitting routines report a task status through integer values. Negative status values indicate errors,\nwhile positive values indicate warnings. An error can be caused by invalid parameter values or a memory\nallocation failure.\nThe status codes have symbolic names predefined in the header file as macros via the #define statements.\nIf no error occurred, the function returns the DF_STATUS_OK code defined as zero:\n#define DF_STATUS_OK 0\nIn case of an error, the function returns a non-zero error code that specifies the origin of the failure. Header\nfiles define the following status codes:\nStatus Codes in the Data Fitting Component\nStatus Code\nDescription\nCommon Status Codes\nDF_STATUS_OK\nOperation completed successfully.\nDF_ERROR_NULL_TASK\nData Fitting task is a NULL pointer.\nDF_ERROR_MEM_FAILURE\nMemory allocation failure.\nDF_ERROR_METHOD_NOT_SUPPORTED\nRequested method is not supported.\nDF_ERROR_COMP_TYPE_NOT_SUPPORTED\nRequested computation type is not supported.\nDF_ERROR_NULL_PTR\nPointer to parameter is null.\nData Fitting Task Creation and Initialization, and Generic Editing Operations\nDF_ERROR_BAD_NX\nInvalid number of breakpoints.\nDF_ERROR_BAD_X\nArray of breakpoints is invalid.\nDF_ERROR_BAD_X_HINT\nInvalid hint describing the structure of the partition.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2576\n\n\nStatus Code\nDescription\nDF_ERROR_BAD_NY\nInvalid dimension of vector-valued function y.\nDF_ERROR_BAD_Y\nArray of function values is invalid.\nDF_ERROR_BAD_Y_HINT\nInvalid flag describing the structure of function y\nData Fitting Task-Specific Editing Operations\nDF_ERROR_BAD_SPLINE_ORDER\nInvalid spline order.\nDF_ERROR_BAD_SPLINE_TYPE\nInvalid spline type.\nDF_ERROR_BAD_IC_TYPE\nType of internal conditions used for spline\nconstruction is invalid.\nDF_ERROR_BAD_IC\nArray of internal conditions for spline construction is\nnot defined.\nDF_ERROR_BAD_BC_TYPE\nType of boundary conditions used in spline\nconstruction is invalid.\nDF_ERROR_BAD_BC\nArray of boundary conditions for spline construction\nis not defined.\nDF_ERROR_BAD_PP_COEFF\nArray of piecewise polynomial spline coefficients is\nnot defined.\nDF_ERROR_BAD_PP_COEFF_HINT\nInvalid flag describing the structure of the\npiecewise polynomial spline coefficients.\nDF_ERROR_BAD_PERIODIC_VAL\nFunction values at the endpoints of the interpolation\ninterval are not equal as required in periodic\nboundary conditions.\nDF_ERROR_BAD_DATA_ATTR\nInvalid attribute of the pointer to be set or modified\nin Data Fitting task descriptor with the df?\nEditIdxPtr task editor.\nDF_ERROR_BAD_DATA_IDX\nIndex of the pointer to be set or modified in the\nData Fitting task descriptor with the df?\nEditIdxPtr task editor is out of the pre-defined\nrange.\nData Fitting Computation Operations\nDF_ERROR_BAD_NSITE\nInvalid number of interpolation sites.\nDF_ERROR_BAD_SITE\nArray of interpolation sites is not defined.\nDF_ERROR_BAD_SITE_HINT\nInvalid flag describing the structure of interpolation\nsites.\nDF_ERROR_BAD_NDORDER\nInvalid size of the array defining derivative orders\nto be computed at interpolation sites.\nDF_ERROR_BAD_DORDER\nArray defining derivative orders to be computed at\ninterpolation sites is not defined.\nDF_ERROR_BAD_DATA_HINT\nInvalid flag providing additional information about\npartition or interpolation sites.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2577\n\n\nStatus Code\nDescription\nDF_ERROR_BAD_INTERP\nArray of spline-based interpolation results is not\ndefined.\nDF_ERROR_BAD_INTERP_HINT\nInvalid flag defining the structure of spline-based\ninterpolation results.\nDF_ERROR_BAD_CELL_IDX\nArray of indices of partition cells containing\ninterpolation sites is not defined.\nDF_ERROR_BAD_NLIM\nInvalid size of arrays containing integration limits.\nDF_ERROR_BAD_LLIM\nArray of the left-side integration limits is not\ndefined.\nDF_ERROR_BAD_RLIM\nArray of the right-side integration limits is not\ndefined.\nDF_ERROR_BAD_INTEGR\nArray of spline-based integration results is not\ndefined.\nDF_ERROR_BAD_INTEGR_HINT\nInvalid flag providing the structure of the array of\nspline-based integration results.\nDF_ERROR_BAD_LOOKUP_INTERP_SITE\nBad site provided for interpolation with look-up\ninterpolator.\nNOTE\nThe routine that estimates piecewise polynomial cubic spline coefficients can return internal error\ncodes related to the specifics of the implementation. Such error codes indicate invalid input data or\nother issues unrelated to Data Fitting routines.\nData Fitting Task Creation and Initialization Routines\nTask creation and initialization routines are functions used to create a new task descriptor and initialize its\nparameters. The Data Fitting component provides the df?NewTask1D routine that creates and initializes a\nnew task descriptor for a one-dimensional Data Fitting task.\ndf?NewTask1D\nCreates and initializes a new task descriptor for a one-\ndimensional Data Fitting task.\nSyntax\nstatus = dfsNewTask1D(&task, nx, x, xhint, ny, y, yhint)\nstatus = dfdNewTask1D(&task, nx, x, xhint, ny, y, yhint)\nInclude Files\n•\nmkl.h\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2578\n\n\nInput Parameters\nName\nType\nDescription\nnx\nconst MKL_INT\nNumber of breakpoints representing partition of\ninterpolation interval [a, b].\nx\nconst float* for\ndfsNewTask1D\nconst double* for\ndfdNewTask1D\nOne-dimensional array containing the strictly sorted\nbreakpoints from interpolation interval [a, b]. The structure\nof the array is defined by parameter xhint:\n•\nIf partition is non-uniform or quasi-uniform, the array\nshould contain nx strictly ordered values.\n•\nIf partition is uniform, the array should contain two\nentries that represent endpoints of interpolation interval\n[a, b].\nCaution\nThe array must be strictly sorted. If it is unordered, the\nresults of data fitting routines are not correct.\nxhint\nconst MKL_INT\nA flag describing the structure of partition x. For the list of\npossible values of xhint, see table \"Hint Values for\nPartition x\". If you set the flag to the DF_NO_HINT value,\nthe library interprets the partition as non-uniform.\nny\nconst MKL_INT\nDimension of vector-valued function y.\ny\nconst float* for dfsNewTask\nconst double* for dfdNewTask\nVector-valued function y, array of size nx*ny.\nThe storage format of function values in the array is defined\nby the value of flag yhint.\nyhint\nconst MKL_INT\nA flag describing the structure of array y. Valid hint values\nare listed in table \"Hint Values for Vector-Valued Function\ny\". If you set the flag to the DF_NO_HINT value, the library\nassumes that all ny coordinates of the vector-valued\nfunction y are provided and stored in row-major format.\nOutput Parameters\nName\nType\nDescription\ntask\nDFTaskPtr\nDescriptor of the task.\nstatus\nint\nStatus of the routine:\n•\nDF_STATUS_OK if the task is created successfully.\n•\nNon-zero error code if the task creation failed. See \"Task\nStatus and Error Reporting\" for error code definitions.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2579\n\n\nDescription\nThe df?NewTask1D routine creates and initializes a new Data Fitting task descriptor with user-specified\nparameters for a one-dimensional Data Fitting task. The x and nx parameters representing the partition of\ninterpolation interval [a, b] are mandatory. If you provide invalid values for these parameters, such as a\nNULL pointer x or the number of breakpoints smaller than two, the routine does not create the Data Fitting\ntask and returns an error code.\nIf you provide a vector-valued function y, make sure that the function dimension ny and the array of function\nvalues y are both valid. If any of these parameters are invalid, the routine does not create the Data Fitting\ntask and returns an error code.\nIf you store coordinates of the vector-valued function y in non-contiguous memory locations, you can set the\nyhint flag to DF_1ST_COORDINATE, and pass only the first coordinate of the function into the task creation\nroutine. After successful creation of the Data Fitting task, you can pass the remaining coordinates using the\ndf?EditIdxPtr task editor.\nIf the routine fails to create the task descriptor, it returns a NULL task pointer.\nThe routine supports the following hint values for partition x:\nHint Values for Partition x\nValue\nDescription\nDF_NON_UNIFORM_PARTITION\nPartition is non-uniform.\nDF_QUASI_UNIFORM_PARTITION\nPartition is quasi-uniform.\nDF_UNIFORM_PARTITION\nPartition is uniform.\nDF_NO_HINT\nNo hint is provided. By default, partition is interpreted as non-\nuniform.\nThe routine supports the following hint values for the vector-valued function:\nHint Values for Vector-Valued Function y\nValue\nDescription\nDF_MATRIX_STORAGE_ROWS\nData is stored in row-major format according to C conventions.\nDF_MATRIX_STORAGE_COLS\nData is stored in column-major format according to Fortran\nconventions.\nDF_1ST_COORDINATE\nThe first coordinate of vector-valued data is provided.\nDF_NO_HINT\nNo hint is provided. By default, the coordinates of vector-valued\nfunction y are provided and stored in row-major format.\nNOTE\nYou must preserve the arrays x (breakpoints) and y (vector-valued functions) through the\nentire workflow of the Data Fitting computations for a task, as the task stores the addresses\nof the arrays for spline-based computations.\nTask Configuration Routines\nIn order to configure tasks, you can use task editors and task query routines.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2580\n\n\nTask editors initialize or change the predefined Data Fitting task parameters. You can use task editors to\ninitialize or modify pointers to arrays or parameter values.\nTask editors can be task-specific or generic. Task-specific editors can modify more than one parameter\nrelated to a specific task. Generic editors modify a single parameter at a time.\nThe Data Fitting component of Intel® oneAPI Math Kernel Library (oneMKL) provides the following task\neditors:\nData Fitting Task Editors\nEditor\nDescription\nType\ndf?\nEditPPSpline1D\nChanges parameters of the piecewise polynomial\nspline.\nTask-specific\ndf?EditPtr\nChanges a pointer in the task descriptor.\nGeneric\ndfiEditVal\nChanges a value in the task descriptor.\nGeneric\ndf?EditIdxPtr\nChanges a coordinate of data represented in\nmatrix format, such as a vector-valued function or\nspline coefficients.\nGeneric\nTask query routines are used to read the predefined Data Fitting task parameters. You can use task query\nroutines to read the values of pointers or parameters.\nTask query routines are generic (not task-specific), allowing you to read a single parameter at a time.\nThe Data Fitting component of the Intel® oneAPI Math Kernel Library (oneMKL) provides the following task\nquery routines:\nData Fitting Task Query Routines\nEditor\nDescription\nType\ndf?QueryPtr\nQueries a pointer in the task descriptor.\nGeneric\ndfiQueryVal\nQueries a value in the task descriptor.\nGeneric\ndf?QueryIdxPtr\nQueries a coordinate of data represented in matrix\nformat, such as a vector-valued function or spline\ncoefficients.\nGeneric\ndf?EditPPSpline1D\nModifies parameters representing a spline in a Data\nFitting task descriptor.\nSyntax\nstatus = dfsEditPPSpline1D(task, s_order, s_type, bc_type, bc, ic_type, ic, scoeff,\nscoeffhint)\nstatus = dfdEditPPSpline1D(task, s_order, s_type, bc_type, bc, ic_type, ic, scoeff,\nscoeffhint)\nInclude Files\n•\nmkl.h\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2581\n\n\nInput Parameters\nName\nType\nDescription\ntask\nDFTaskPtr\nDescriptor of the task.\ns_order\nconst MKL_INT\nSpline order. The parameter takes one of the values\ndescribed in table \"Spline Orders Supported by Data Fitting\nFunctions\".\ns_type\nconst MKL_INT\nSpline type. The parameter takes one of the values\ndescribed in table \"Spline Types Supported by Data Fitting\nFunctions\".\nbc_type\nconst MKL_INT\nType of boundary conditions. The parameter takes one of\nthe values described in table \"Boundary Conditions\nSupported by Data Fitting Functions\".\n \n \n \nbc\nconst float* for\ndfsEditPPSpline1D\nconst double* for\ndfdEditPPSpline1D\nPointer to boundary conditions. The size of the array is\ndefined by the value of parameter bc_type:\n•\nIf you set free-end or not-a-knot boundary conditions,\npass the NULL pointer to this parameter.\n•\nIf you combine boundary conditions at the endpoints of\nthe interpolation interval, pass an array of two elements.\n•\nIf you set a boundary condition for the default quadratic\nspline or a periodic condition for Hermite or the default\ncubic spline, pass an array of one element.\nic_type\nconst MKL_INT\nType of internal conditions. The parameter takes one of the\nvalues described in table \"Internal Conditions Supported by\nData Fitting Functions\".\nic\nconst float* for\ndfsEditPPSpline1D\nconst double* for\ndfdEditPPSpline1D\nA non-NULL pointer to the array of internal conditions. The\nsize of the array is defined by the value of parameter\nic_type:\n•\nIf you set first derivatives or second\nderivatives internal conditions\n(ic_type=DF_IC_1ST_DER or\nic_type=DF_IC_2ND_DER), pass an array of n-1\nderivative values at the internal points of the\ninterpolation interval.\n•\nIf you set the knot values internal condition for\nSubbotin spline (ic_type=DF_IC_Q_KNOT) and the knot\npartition is non-uniform, pass an array of n+1 elements.\n•\nIf you set the knot values internal condition for\nSubbotin spline (ic_type=DF_IC_Q_KNOT) and the knot\npartition is uniform, pass an array of four elements.\nscoeff\nconst float* for\ndfsEditPPSpline1D\nconst double* for\ndfdEditPPSpline1D\nSpline coefficients. An array of size ny*s_order*(nx-1).\nThe storage format of the coefficients in the array is defined\nby the value of flag scoeffhint.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2582\n\n\nName\nType\nDescription\nscoeffhint\nconst MKL_INT\nA flag describing the structure of the array of spline\ncoefficients. For valid hint values, see table \"Hint Values for\nSpline Coefficients\". The library stores the coefficients in\nrow-major format. The default value is DF_NO_HINT.\nOutput Parameters\nName\nType\nDescription\nstatus\nint\nStatus of the routine:\n•\nDF_STATUS_OK if the routine execution completed\nsuccessfully.\n•\nNon-zero error code if the routine execution failed. See \n\"Task Status and Error Reporting\" for error code\ndefinitions.\nDescription\nThe editor modifies parameters that describe the order, type, boundary conditions, internal conditions, and\ncoefficients of a spline. The spline order definition is provided in the \"Mathematical Conventions\" section. You\ncan set the spline order to any value supported by Data Fitting functions. The table below lists the available\nvalues:\nSpline Orders Supported by the Data Fitting Functions\nOrder\nDescription\nDF_PP_STD\nArtificial value. Use this value for look-up and step-\nwise constant interpolants only.\nDF_PP_LINEAR\nPiecewise polynomial spline of the second order\n(linear spline).\nDF_PP_QUADRATIC\nPiecewise polynomial spline of the third order\n(quadratic spline).\nDF_PP_CUBIC\nPiecewise polynomial spline of the fourth order\n(cubic spline).\nTo perform computations with a spline not supported by Data Fitting routines, set the parameter defining the\nspline order and pass the spline coefficients to the library in the supported format. For format description,\nsee figure \"Row-major Coefficient Storage Format\".\nThe table below lists the supported spline types:\nSpline Types Supported by Data Fitting Functions\nType\nDescription\nDF_PP_DEFAULT\nThe default spline type. You can use this type with\nlinear, quadratic, or user-defined splines.\nDF_PP_SUBBOTIN\nQuadratic splines based on Subbotin algorithm,\n[StechSub76].\nDF_PP_NATURAL\nNatural cubic spline.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2583\n\n\nType\nDescription\nDF_PP_HERMITE\nHermite cubic spline.\nDF_PP_BESSEL\nBessel cubic spline.\nDF_PP_AKIMA\nAkima cubic spline.\nDF_LOOKUP_INTERPOLANT\nLook-up interpolant.\nDF_CR_STEPWISE_CONST_INTERPOLANT\nContinuous right step-wise constant interpolant.\nDF_CL_STEPWISE_CONST_INTERPOLANT\nContinuous left step-wise constant interpolant.\nIf you perform computations with look-up or step-wise constant interpolants, set the spline order to the\nDF_PP_STD value.\nConstruction of specific splines may require boundary or internal conditions. To compute coefficients of such\nsplines, you should pass boundary or internal conditions to the library by specifying the type of the conditions\nand providing the necessary values. For splines that do not require additional conditions, such as linear\nsplines, set condition types to DF_NO_BC and DF_NO_IC, and pass NULL pointers to the conditions. The table\nbelow defines the supported boundary conditions:\nBoundary Conditions Supported by Data Fitting Functions\nBoundary Condition\nDescription\nSpline\nDF_NO_BC\nNo boundary conditions provided.\nAll\nDF_BC_NOT_A_KNOT\nNot-a-knot boundary conditions.\nAkima, Bessel, Hermite, natural\ncubic\nDF_BC_FREE_END\nFree-end boundary conditions.\nAkima, Bessel, Hermite, natural\ncubic, quadratic Subbotin\nDF_BC_1ST_LEFT_DER\nThe first derivative at the left\nendpoint.\nAkima, Bessel, Hermite, natural\ncubic, quadratic Subbotin\nDF_BC_1ST_RIGHT_DER\nThe first derivative at the right\nendpoint.\nAkima, Bessel, Hermite, natural\ncubic, quadratic Subbotin\nDF_BC_2ST_LEFT_DER\nThe second derivative at the left\nendpoint.\nAkima, Bessel, Hermite, natural\ncubic, quadratic Subbotin\nDF_BC_2ND_RIGHT_DER\nThe second derivative at the right\nendpoint.\nAkima, Bessel, Hermite, natural\ncubic, quadratic Subbotin\nDF_BC_PERIODIC\nPeriodic boundary conditions.\nLinear, all cubic splines\nDF_BC_Q_VAL\nFunction value at point\n(x0 + x1)/2\nDefault quadratic\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2584\n\n\nNOTE\nTo construct a natural cubic spline, pass these settings to the editor:\n•\nDF_PP_CUBIC as the spline order,\n•\nDF_PP_NATURAL as the spline type, and\n•\nDF_BC_FREE_END as the boundary condition.\nTo construct a cubic spline with other boundary conditions, pass these settings to the editor:\n•\nDF_PP_CUBIC as the spline order,\n•\nDF_PP_NATURAL as the spline type, and\n•\nthe required type of boundary condition.\nFor Akima, Hermite, Bessel, and default cubic splines use the corresponding type defined in Table\nSpline Types Supported by Data Fitting Functions.\nYou can combine the values of boundary conditions with a bitwise OR operation. This permits you to pass\ncombinations of first and second derivatives at the endpoints of the interpolation interval into the library. To\npass a first derivative at the left endpoint and a second derivative at the right endpoint, set the boundary\nconditions to DF_BC_1ST_LEFT_DER OR DF_BC_2ND_RIGHT_DER.\nYou should pass the combined boundary conditions as an array of two elements. The first entry of the array\ncontains the value of the boundary condition for the left endpoint of the interpolation interval, and the second\nentry - for the right endpoint. Pass other boundary conditions as arrays of one element.\nFor the conditions defined as a combination of valid values, the library applies the following rules to identify\nthe boundary condition type:\n•\nIf not required for spline construction, the value of boundary conditions is ignored.\n•\nNot-a-knot condition has the highest priority. If set, other boundary conditions are ignored.\n•\nFree-end condition has the second priority after the not-a-knot condition. If set, other boundary\nconditions are ignored.\n•\nPeriodic boundary condition has the next priority after the free-end condition.\n•\nThe first derivative has higher priority than the second derivative at the right and left endpoints.\nIf you set the periodic boundary condition, make sure that function values at the endpoints of the\ninterpolation interval are identical. Otherwise, the library returns an error code. The table below specifies the\nvalues to be provided for each type of spline if the periodic boundary condition is set.\nBoundary Requirements for Periodic Conditions\nSpline Type\nPeriodic Boundary Condition\nSupport\nBoundary Value\nLinear\nYes\nNot required\nDefault quadratic\nNo\nSubbotin quadratic\nNo\nNatural cubic\nYes\nNot required\nBessel\nYes\nNot required\nAkima\nYes\nNot required\nHermite cubic\nYes\nFirst derivative\nDefault cubic\nYes\nSecond derivative\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2585\n\n\nInternal conditions supported in the Data Fitting domain that you can use for the ic_type parameter are the\nfollowing:\nInternal Conditions Supported by Data Fitting Functions\nInternal Condition\nDescription\nSpline\nDF_NO_IC\nNo internal conditions provided.\nDF_IC_1ST_DER\nArray of first derivatives of size\nn-2, where n is the number of\nbreakpoints. Derivatives are\napplicable to each coordinate of\nthe vector-valued function.\nHermite cubic\nDF_IC_2ND_DER\nArray of second derivatives of\nsize n-2, where n is the number\nof breakpoints. Derivatives are\napplicable to each coordinate of\nthe vector-valued function.\nDefault cubic\nDF_IC_Q_KNOT\nKnot array of size n+1, where n\nis the number of breakpoints.\nSubbotin quadratic\nTo construct a Subbotin quadratic spline, you have three options to get the array of knots in the library:\n•\nIf you do not provide the knots, the library uses the default values of knots t = {ti}, i = 0, ..., n according\nto the rule:\nt0 = x0, tn = xn-1, ti = (xi + xi-1)/2, i = 1, ..., n - 1.\n•\nIf you provide the knots in an array of size n + 1, the knots form a non-uniform partition. Make sure that\nthe knot values you provide meet the following conditions:\nt0 = x0, tn = xn-1, ti ∈ (xi-1, xi), i = 1,..., n - 1.\n•\nIf you provide the knots in an array of size 4, the knots form a uniform partition\nt0 = x0, t1 = l, t2 = r, t3 = xn - 1, where l ∈ (x0, x1) and r ∈ (xn - 2, xn - 1).\nIn this case, you need to set the value of the ic_type parameter holding the type of internal conditions\nto DF_IC_Q_KNOT OR DF_UNIFORM_PARTITION.\nNOTE\nSince the partition is uniform, perform an OR operation with the DF_UNIFORM_PARTITION partition hint\nvalue described in Table Hint Values for Partition x.\nFor computations based on look-up and step-wise constant interpolants, you can avoid calling the df?\nEditPPSpline1D editor and directly call one of the routines for spline-based computation of spline values,\nderivatives, or integrals. For example, you can call the df?Construct1D routine to construct the required\nspline with the given attributes, such as order or type.\nThe memory location of the spline coefficients is defined by the scoeff parameter. Make sure that the size of\nthe array is sufficient to hold ny*s_order * (nx-1) values.\nThe df?EditPPSpline1D routine supports the following hint values for spline coefficients:\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2586\n\n\nHint Values for Spline Coefficients\nOrder\nDescription\nDF_1ST_COORDINATE\nThe first coordinate of vector-valued data is\nprovided.\nDF_NO_HINT\nNo hint is provided. By default, all sets of spline\ncoefficients are stored in row-major format.\nThe coefficients for all coordinates of the vector-valued function are packed in memory one by one in\nsuccessive order, from function y1 to function yny.\nWithin each coordinate, the library stores the coefficients as an array, in row-major format:\nc1, 0, c1, 1, ..., c1, k, c2, 0, c2, 1, ..., c2, k, ..., cn-1, 0, cn-1, 1, ..., cn-1, k\nMapping of the coefficients to storage in the scoeff array is described below, where ci,j is the jth coefficient\nof the function\n.\nSee Mathematical Conventions for more details on nomenclature and interpolants.\nRow-major Coefficient Storage Format\nIf you store splines corresponding to different coordinates of the vector-valued function at non-contiguous\nmemory locations, do the following:\n1.\nSet the scoeffhint flag to DF_1ST_COORDINATE and provide the spline for the first coordinate.\n2.\nPass the spline coefficients for the remaining coordinates into the Data Fitting task using the df?\nEditIdxPtr task editor.\nUsing the df?EditPPSpline1D task editor, you can provide to the Data Fitting task an already constructed\nspline that you want to use in computations. To ensure correct interpretation of the memory content, you\nshould set the following parameters:\n•\nSpline order and type, if appropriate. If the spline is not supported by the library, set the s_type\nparameter to DF_PP_DEFAULT.\n•\nPointer to the array of spline coefficients in row-major format.\n•\nThe scoeffhint parameter describing the structure of the array:\n•\nSet the scoeffhint flag to the DF_1ST_COORDINATE value to pass spline coefficients stored at\ndifferent memory locations. In this case, you can set the parameters that describe boundary and\ninternal conditions to zero.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2587\n\n\n•\nUse the default value DF_NO_HINT for all other cases.\nBefore passing an already constructed spline into the library, you should call the dfiEditVal task editor to\nprovide the dimension of the spline DF_NY. See table \"Parameters Supported by the dfiEditVal Task Editor\"\nfor details.\nAfter you provide the spline to the Data Fitting task, you can run computations that use this spline.\nNOTE\nYou must preserve the arrays bc (boundary conditions), ic (internal conditions), and scoeff\n(spline coefficients) through the entire workflow of the Data Fitting computations for a task,\nas the task stores the addresses of the arrays for spline-based computations.\ndf?EditPtr\nModifies a pointer to an array held in a Data Fitting\ntask descriptor.\nSyntax\nstatus = dfsEditPtr(task, ptr_attr, ptr)\nstatus = dfdEditPtr(task, ptr_attr, ptr)\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\ntask\nDFTaskPtr\nDescriptor of the task.\nptr_attr\nconst MKL_INT\nThe parameter to change. For details, see the Pointer\nAttribute column in table \"Pointers Supported by the df?\nEditPtr Task Editor\".\nptr\nconst float* for dfsEditPtr\nconst double* for dfdEditPtr\nNew pointer. For details, see the Purpose column in table \n\"Pointers Supported by the df?EditPtr Task Editor\".\nOutput Parameters\nName\nType\nDescription\nstatus\nint\nStatus of the routine:\n•\nDF_STATUS_OK if the routine execution completed\nsuccessfully.\n•\nNon-zero error code otherwise. See \"Task Status and\nError Reporting\" for error code definitions.\nDescription\nThe df?EditPtr editor replaces the pointer of type ptr_attr stored in a Data Fitting task descriptor with a\nnew pointer ptr. The table below describes types of pointers supported by the editor:\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2588\n\n\nPointers Supported by the df?EditPtr Task Editor\nPointer Attribute\nPurpose\nDF_X\nPartition x of the interpolation interval, an array of strictly sorted\nbreakpoints.\nCaution\nThe array must be strictly sorted. If it is unordered, the results of data\nfitting routines are not correct.\nDF_Y\nVector-valued function y\nDF_IC\nInternal conditions for spline construction. For details, see table \n\"Internal Conditions Supported by Data Fitting Functions\".\nDF_BC\nBoundary conditions for spline construction. For details, see table \n\"Boundary Conditions Supported by Data Fitting Functions\".\nDF_PP_SCOEFF\nSpline coefficients\nYou can use df?EditPtr to modify different types of pointers including pointers to the vector-valued\nfunction and spline coefficients stored in contiguous memory. Use the df?EditIdxPtr editor if you need to\nmodify pointers to coordinates of the vector-valued function or spline coefficients stored at non-contiguous\nmemory locations.\nIf you modify a partition of the interpolation interval, then you should call the dfiEditVal task editor with\nthe corresponding value of DF_XHINT, even if the structure of the partition remains the same.\nIf you pass a NULL pointer to the df?EditPtr task editor, the task remains unchanged and the routine\nreturns an error code. For the predefined error codes, please see \"Task Status and Error Reporting\".\nNOTE\nYou must preserve the arrays x (breakpoints), y (vector-valued functions), bc (boundary\nconditions), ic (internal conditions), and scoeff (spline coefficients) through the entire\nworkflow of the Data Fitting computations which use those arrays, as the task stores the\naddresses of the arrays for spline-based computations.\ndfiEditVal\nModifies a parameter value in a Data Fitting task\ndescriptor.\nSyntax\nstatus = dfiEditVal(task, val_attr, val)\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\ntask\nDFTaskPtr\nDescriptor of the task.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2589\n\n\nName\nType\nDescription\nval_attr\nconst MKL_INT\nThe parameter to change. See table \"Parameters Supported\nby the dfiEditVal Task Editor\".\nval\nconst MKL_INT\nA new parameter value. See table \"Parameters Supported\nby the dfiEditVal Task Editor\".\nOutput Parameters\nName\nType\nDescription\nstatus\nint\nStatus of the routine:\n•\nDF_STATUS_OK if the routine execution completed\nsuccessfully.\n•\nNon-zero error code otherwise. See \"Task Status and\nError Reporting\" for error code definitions.\nDescription\nThe dfiEditVal task editor replaces the parameter of type val_attr stored in a Data Fitting task descriptor\nwith a new value val. The table below describes valid types of parameter val_attr supported by the editor:\nParameters Supported by the dfiEditVal Task Editor\nParameter Attribute\nPurpose\nDF_NX\nNumber of breakpoints\nDF_XHINT\nA flag describing the structure of partition. See table \"Hint Values\nfor Partition x\" for the list of available values.\nDF_NY\nDimension of the vector-valued function\nDF_YHINT\nA flag describing the structure of the vector-valued function. See\ntable \"Hint Values for Vector Function y\" for the list of available\nvalues.\nDF_SPLINE_ORDER\nSpline order. See table \"Spline Orders Supported by Data Fitting\nFunctions\" for the list of available values.\nDF_SPLINE_TYPE\nSpline type. See table \"Spline Types Supported by Data Fitting\nFunctions\" for the list of available values.\nDF_BC_TYPE\nType of boundary conditions used in spline construction. See table \n\"Boundary Conditions Supported by Data Fitting Functions\" for the\nlist of available values.\nDF_IC_TYPE\nType of internal conditions used in spline construction. See table \n\"Internal Conditions Supported by Data Fitting Functions\" for the\nlist of available values.\nDF_PP_COEFF_HINT\nA flag describing the structure of spline coefficients. See table \"Hint\nValues for Spline Coefficients\" for the list of available values.\nDF_CHECK_FLAG\nA flag which controls checking of Data Fitting parameters. See\ntable \"Possible Values for the DF_CHECK_FLAG Parameter\" for the\nlist of available values.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2590\n\n\nIf you pass a zero value for the parameter describing the size of the arrays that hold coefficients for a\npartition, a vector-valued function, or a spline, the parameter held in the Data fitting task remains\nunchanged and the routine returns an error code. For the predefined error codes, see \"Task Status and Error\nReporting\".\nPossible Values for the DF_CHECK_FLAG Parameter\nValue\nDescription\nDF_ENABLE_CHECK_FLAG\nChecks the correctness of parameters of Data\nFitting computational routines (default mode).\nDF_DISABLE_CHECK_FLAG\nDisables checking of the correctness of parameters\nof Data Fitting computational routines.\nUse DF_CHECK_FLAG for val_attr in order to control validation of parameters of Data Fitting computational\nroutines such as df?Construct1D, df?Interpolate1D/df?InterpolateEx1D, and df?\nSearchCells1D/df?SearchCellsEx1D, which can perform better with a small number of interpolation sites\nor integration limits (fewer than one dozen). The default mode, with checking of parameters enabled, should\nbe used as you develop a Data Fitting-based application. After you complete development you can disable\nparameter checking in order to improve the performance of your application.\nIf you modify the parameter describing dimensions of the arrays that hold the vector-valued function or\nspline coefficients in contiguous memory, you should call the df?EditPtr task editor with the corresponding\npointers to the vector-valued function or spline coefficients even when this pointer remains unchanged. Call\nthe df?EditIdxPtr editor if those arrays are stored in non-contiguous memory locations.\nYou must call the dfiEditVal task editor to edit the structure of the partition DF_XHINT every time you\nmodify a partition using df?EditPtr, even if the structure of the partition remains the same.\ndf?EditIdxPtr\nModifies a pointer to the memory representing a\ncoordinate of the data stored in matrix format.\nSyntax\nstatus = dfsEditIdxPtr(task, ptr_attr, idx, ptr)\nstatus = dfdEditIdxPtr(task, ptr_attr, idx, ptr)\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\ntask\nDFTaskPtr\nDescriptor of the task.\nptr_attr\nconst MKL_INT\nType of the data to be modified. The parameter takes one\nof the values described in \"Data Attributes Supported by\nthe df?EditIdxPtr Task Editor\".\nidx\nconst MKL_INT\nIndex of the coordinate whose pointer is to be modified.\nptr\nconst float* for\ndfsEditIdxPtr\nPointer to the data that holds values of coordinate idx. For\ndetails, see table \"Data Attributes Supported by the df?\nEditIdxPtr Task Editor\".\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2591\n\n\nName\nType\nDescription\nconst double* for\ndfdEditIdxPtr\nOutput Parameters\nName\nType\nDescription\nstatus\nint\nStatus of the routine:\n•\nDF_STATUS_OK if the routine execution completed\nsuccessfully.\n•\nNon-zero error code otherwise. See \"Task Status and\nError Reporting\" for error code definitions.\nDescription\nThe routine modifies a pointer to the array that holds the idx coordinate of vector-valued function y or the\npointer to the array of spline coefficients corresponding to the given coordinate.\nYou can use the editor if you need to pass into a Data Fitting task or modify the pointer to coordinates of the\nvector-valued function or spline coefficients held at non-contiguous memory locations. Do not use the editor\nfor coordinates at contiguous memory locations in row-major format.\nBefore calling this editor, make sure that you have created and initialized the task using a task creation\nfunction or a relevant editor such as the generic or specific df?EditPPSpline1D editor.\nData Attributes Supported by the df?EditIdxPtr Task Editor\nData Attribute\nDescription\nDF_Y\nVector-valued function y\nDF_PP_SCOEFF\nPiecewise polynomial spline coefficients\nWhen using df?EditIdxPtr, you might receive an error code in the following cases:\n•\nYou passed an unsupported parameter value into the editor.\n•\nThe value of the index exceeds the predefined value that equals the dimension ny of the vector-valued\nfunction.\n•\nYou pass a NULL pointer to the editor. In this case, the task remains unchanged.\n•\nYou pass a pointer to the idx coordinate of the vector-valued function you provided to contiguous\nmemory in column-major format.\nThe code example below demonstrates how to use the editor for providing values of a vector-valued function\nstored in two non-contiguous arrays:\n#define NX 1000  /* number of break points       */\n#define NY 2     /* dimension of vector-valued function */\nint main()\n{\n    DFTaskPtr task;\n    double x[NX];\n    double y1[NX], y2[NX]; /* vector-valued function is stored as two arrays */\n    /* Provide first coordinate of two-dimensional function y into creation routine */\n    status = dfdNewTask1D( &task, NX, x, DF_NON_UNIFORM_PARTITION, NY, y1,\n                           DF_1ST_COORDINATE );\n    /* Provide second coordiante of two–dimensional function */\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2592\n\n\n    status = dfdEditIdxPtr(task, DF_Y, 1, y2 );\n    ...\n}\ndf?QueryPtr\nReads a pointer to an array held in a Data Fitting task\ndescriptor.\nSyntax\nstatus = dfsQueryPtr(task, ptr_attr, ptr)\nstatus = dfdQueryPtr(task, ptr_attr, ptr)\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\ntask\nDFTaskPtr\nDescriptor of the task.\nptr_attr\nconst MKL_INT\nThe parameter to query. The query routine supports pointer\nattributes described in the table \"Pointers Supported by the\ndf?EditPtr Task Editor\". For details, see the Pointer\nAttribute column in the table.\nOutput Parameters\nName\nType\nDescription\nptr\nfloat** for dfsQueryPtr\ndouble** for dfdQueryPtr\nPointer to array returned by the query routine. For details,\nsee the Purpose column in table \"Pointers Supported by the\ndf?EditPtr Task Editor\".\nstatus\nint\nStatus of the routine:\n•\nDF_STATUS_OK if the routine execution completed\nsuccessfully.\n•\nNon-zero error code otherwise. See \"Task Status and\nError Reporting\" for error code definitions.\nDescription\nThe df?QueryPtr routine returns the pointer of type ptr_attr stored in a Data Fitting task descriptor as\nparameter ptr. Attributes of the pointers supported by the query function are identical to those supported by\nthe editor df?EditPtr editor in the table \"Pointers Supported by the df?EditPtr Task Editor\".\nYou can use df?QueryPtr to read different types of pointers including pointers to the vector-valued function\nand spline coefficients stored in contiguous memory.\ndfiQueryVal\nReads a parameter value in a Data Fitting task\ndescriptor.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2593\n\n\nSyntax\nstatus = dfiQueryVal(task, val_attr, val)\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\ntask\nDFTaskPtr\nDescriptor of the task.\nval_attr\nconst MKL_INT\nThe parameter to query. The query function supports the\nparameter attributes described in \"Parameters Supported\nby the dfiEditVal Task Editor\".\nOutput Parameters\nName\nType\nDescription\nval\nMKL_INT\nThe parameter value returned by the query function. See\ntable \"Parameters Supported by the dfiEditVal Task\nEditor\".\nstatus\nint\nStatus of the routine:\n•\nDF_STATUS_OK if the routine execution completed\nsuccessfully.\n•\nNon-zero error code otherwise. See \"Task Status and\nError Reporting\" for error code definitions.\nDescription\nThe dfiQueryVal routine returns a parameter of type val_attr stored in a Data Fitting task descriptor as\nparameter val. The query function supports the parameter attributes described in \"Parameters Supported by\nthe dfiEditVal Task Editor\".\ndf?QueryIdxPtr\nReads a pointer to the memory representing a\ncoordinate of the data stored in matrix format.\nSyntax\nstatus = dfsQueryIdxPtr(task, ptr_attr, idx, ptr)\nstatus = dfdQueryIdxPtr(task, ptr_attr, idx, ptr)\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\ntask\nDFTaskPtr\nDescriptor of the task.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2594\n\n\nName\nType\nDescription\nptr_attr\nconst MKL_INT\nPointer attribute to query. The parameter takes one of the\nattributes described in \"Data Attributes Supported by the\ndf?EditIdxPtr Task Editor\".\nidx\nconst MKL_INT\nIndex of the coordinate of the pointer to query.\nOutput Parameters\nName\nType\nDescription\nptr\nfloat* for dfsQueryIdxPtr\ndouble* for dfdQueryIdxPtr\nPointer to the data that holds values of coordinate idx\nreturned. For details, see table \"Data Attributes Supported\nby the df?EditIdxPtr Task Editor\".\nstatus\nint\nStatus of the routine:\n•\nDF_STATUS_OK if the routine execution completed\nsuccessfully.\n•\nNon-zero error code otherwise. See \"Task Status and\nError Reporting\" for error code definitions.\nDescription\nThe routine returns a pointer to the array that holds the idx coordinate of vector-valued function y or the\npointer to the array of spline coefficients corresponding to the given coordinate.\nYou can use the query routine if you need the pointer to coordinates of the vector-valued function or spline\ncoefficients held at non-contiguous memory locations or at a contiguous memory location in row-major\nformat (the default storage format for spline coefficients).\nBefore calling this query routine, make sure that you have created and initialized the task using a task\ncreation function or a relevant editor such as the generic or specific df?EditPPSpline1D editor.\nWhen using df?QueryIdxPtr, you might receive an error code in the following cases:\n•\nYou passed an unsupported parameter value into the editor.\n•\nThe value of the index exceeds the predefined value that equals the dimension ny of the vector-valued\nfunction.\n•\nYou request the pointer to the idx coordinate of the vector-valued function you provided to contiguous\nmemory in column-major format.\nData Fitting Computational Routines\nData Fitting computational routines are functions used to perform spline-based computations, such as:\n•\nspline construction\n•\ncomputation of values, derivatives, and integrals of the predefined order\n•\ncell search\nOnce you create a Data Fitting task and initialize the required parameters, you can call computational\nroutines as many times as necessary.\nThe table below lists the available computational routines:\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2595\n\n\nData Fitting Computational Routines\nRoutine\nDescription\ndf?Construct1D\nConstructs a spline for a one-dimensional Data\nFitting task.\ndf?Interpolate1D\nComputes spline values and derivatives.\ndf?InterpolateEx1D\nComputes spline values and derivatives by calling\nuser-provided interpolants.\ndf?Integrate1D\nComputes spline-based integrals.\ndf?IntegrateEx1D\nComputes spline-based integrals by calling user-\nprovided integrators.\ndf?SearchCells1D\nFinds indices of cells containing interpolation sites.\ndf?SearchCellsEx1D\nFinds indices of cells containing interpolation sites\nby calling user-provided cell searchers.\nIf a Data Fitting computation completes successfully, the computational routines return the DF_STATUS_OK\ncode. If an error occurs, the routines return an error code specifying the origin of the failure. Some possible\nerrors are the following:\n•\nThe task pointer is NULL.\n•\nMemory allocation failed.\n•\nThe computation failed for another reason.\nFor the list of available status codes, see \"Task Status and Error Reporting\".\nNOTE\nData Fitting computational routines do not control errors for floating-point conditions, such as\noverflow, gradual underflow, or operations with Not a Number (NaN) values.\ndf?Construct1D\nSyntax\nConstructs a spline of the given type.\nstatus = dfsConstruct1D(task, s_format, method)\nstatus = dfdConstruct1D(task, s_format, method)\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\ntask\nDFTaskPtr\nDescriptor of the task.\ns_format\nconst MKL_INT\nSpline format. The supported value is DF_PP_SPLINE.\nmethod\nconst MKL_INT\nConstruction method. The supported value is\nDF_METHOD_STD.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2596\n\n\nOutput Parameters\nName\nType\nDescription\nstatus\nint\nStatus of the routine:\n•\nDF_STATUS_OK if the routine execution completed\nsuccessfully.\n•\nNon-zero error code if the routine execution failed. See \"Task\nStatus and Error Reporting\" for error code definitions.\nDescription\nBefore calling df?Construct1D, you need to create and initialize the task, and set the parameters\nrepresenting the spline. Then you can call the df?Construct1D routine to construct the spline. The format of\nthe spline is defined by parameter s_format. The method for spline construction is defined by parameter\nmethod. Upon successful construction, the spline coefficients are available in the user-provided memory\nlocation in the format you set through the Data Fitting editor. For the available storage formats, see table \n\"Hint Values for Spline Coefficients\".\ndf?Interpolate1D/df?InterpolateEx1D\nRuns data fitting computations.\nSyntax\nstatus = dfsInterpolate1D(task, type, method, nsite, site, sitehint, ndorder, dorder,\ndatahint, r, rhint, cell)\nstatus = dfdInterpolate1D(task, type, method, nsite, site, sitehint, ndorder, dorder,\ndatahint, r, rhint, cell)\nstatus = dfsInterpolateEx1D(task, type, method, nsite, site, sitehint, ndorder, dorder,\ndatahint, r, rhint, cell, le_cb, le_params, re_cb, re_params, i_cb, i_params, search_cb,\nsearch_params)\nstatus = dfdInterpolateEx1D(task, type, method, nsite, site, sitehint, ndorder, dorder,\ndatahint, r, rhint, cell, le_cb, le_params, re_cb, re_params, i_cb, i_params, search_cb,\nsearch_params)\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\ntask\nDFTaskPtr\nDescriptor of the task.\ntype\nconst MKL_INT\nType of spline-based computations. The parameter takes\none or more values combined with an OR operation. For the\nlist of possible values, see table \"Computation Types\nSupported by the df?Interpolate1D/ df?Interpolate1D\nRoutines\".\nmethod\nconst MKL_INT\nComputation method. The supported value is\nDF_METHOD_PP.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2597\n\n\nName\nType\nDescription\nnsite\nconst MKL_INT\nNumber of interpolation sites.\nsite\nconst float* for\ndfsInterpolate1D/\ndfsInterpolateEx1D\nconst double* for\ndfdInterpolate1D/\ndfdInterpolateEx1D\nArray of interpolation sites of size nsite. The structure of\nthe array is defined by the sitehint parameter:\n•\nIf sites form a non-uniform partition, the array should\ncontain nsite values.\n•\nIf sites form a uniform partition, the array should\ncontain two entries that represent the left and the right\ninterpolation sites. The first entry of the array contains\nthe left-most interpolation point. The second entry of the\narray contains the right-most interpolation point.\nsitehint\nconst MKL_INT\nA flag describing the structure of the interpolation sites. For\nthe list of possible values of sitehint, see table \"Hint\nValues for Interpolation Sites\". If you set the flag to\nDF_NO_HINT, the library interprets the site-defined partition\nas non-uniform.\nndorder\nconst MKL_INT\nMaximal derivative order increased by one to be computed\nat interpolation sites.\ndorder\nconst MKL_INT*\nArray of size ndorder that defines the order of the\nderivatives to be computed at the interpolation sites. If all\nthe elements in dorder are zero, the library computes the\nspline values only. If you do not need interpolation\ncomputations, set ndorder to zero and pass a NULL pointer\nto dorder.\ndatahint\nconst float* for\ndfsInterpolate1D/\ndfsInterpolateEx1D\nconst double* for\ndfdInterpolate1D/\ndfdInterpolateEx1D\nArray that contains additional information about the\nstructure of partition x and interpolation sites. This data\nhelps to speed up the computation. If you provide a NULL\npointer, the routine uses the default settings for\ncomputations. For details on the datahint array, see table \n\"Structure of the datahint Array\".\nr\nfloat* for\ndfsInterpolate1D/\ndfsInterpolateEx1D\ndouble* for\ndfdInterpolate1D/\ndfdInterpolateEx1D\nArray for results. If you do not need spline-based\ninterpolation, set this pointer to NULL.\nrhint\nconst MKL_INT\nA flag describing the structure of the results. For the list of\npossible values of rhint, see table \"Hint Values for the\nrhint Parameter\". If you set the flag to DF_NO_HINT, the\nlibrary stores the result in row-major format.\ncell\nMKL_INT*\nArray of cell indices in partition x that contain the\ninterpolation sites. Provide this parameter as input if type\nis DF_INTERP_USER_CELL. If you do not need cell indices,\nset this parameter to NULL.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2598\n\n\nName\nType\nDescription\nle_cb\nconstdfsInterpCallBack\nfor dfsInterpolateEx1D\nconstdfdInterpCallBack\nfor dfdInterpolateEx1D\nUser-defined callback function for extrapolation at the sites\nto the left of the interpolation interval.\nSet to NULL if you are not supplying a callback function.\nle_params\nconst void*\nPointer to additional user-defined parameters passed by the\nlibrary to the le_cb function.\nSet to NULL if there are no additional parameters or if you\nare not supplying a callback function.\nre_cb\nconstdfsInterpCallBack\nfor dfsInterpolateEx1D\nconstdfdInterpCallBack\nfor dfdInterpolateEx1D\nUser-defined callback function for extrapolation at the sites\nto the right of the interpolation interval.\nSet to NULL if you are not supplying a callback function.\nre_params\nconst void*\nPointer to additional user-defined parameters passed by the\nlibrary to the re_cb function.\nSet to NULL if there are no additional parameters or if you\nare not supplying a callback function.\ni_cb\nconstdfsInterpCallBack\nfor dfsInterpolateEx1D\nconstdfdInterpCallBack\nfor dfdInterpolateEx1D\nUser-defined callback function for interpolation within the\ninterpolation interval.\nSet to NULL if you are not supplying a callback function.\ni_params\nconst void*\nPointer to additional user-defined parameters passed by the\nlibrary to the i_cb function.\nSet to NULL if there are no additional parameters or if you\nare not supplying a callback function.\nsearch_cb\nconstdfsSearchCellsCal\nlBack for\ndfsInterpolateEx1D\nconstdfdSearchCellsCal\nlBack for\ndfdInterpolateEx1D\nUser-defined callback function for computing indices of cells\nthat can contain interpolation sites.\nSet to NULL if you are not supplying a callback function.\nsearch_params\nconst void*\nPointer to additional user-defined parameters passed by the\nlibrary to the search_cb function.\nSet to NULL if there are no additional parameters or if you\nare not supplying a callback function.\nOutput Parameters\nName\nType\nDescription\nstatus\nint\nStatus of the routine:\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2599\n\n\nName\nType\nDescription\n•\nDF_STATUS_OK if the routine execution completed\nsuccessfully.\n•\nNon-zero error code if the routine execution failed. See \n\"Task Status and Error Reporting\" for error code\ndefinitions.\nr\nContains results of computations at the interpolation sites.\nCaution\nThe df?Interpolate1D/df?InterpolateEx1D routines\ndo not support in-place computations. You must provide\nnon-aliasing memory locations for the site input\nparameter and the r output parameter.\ncell\nArray of cell indices in partition x that contain the\ninterpolation sites, which is computed if type is DF_CELL.\nDescription\nThe df?Interpolate1D/df?InterpolateEx1D routine performs spline-based computations with user-\ndefined settings. The routine supports two types of computations for interpolation sites provided in array\nsite:\nComputation Types Supported by the df?Interpolate1D/df?InterpolateEx1D Routines\nType\nDescription\nDF_INTERP\nCompute derivatives of predefined order. The\nderivative of the zero order is the spline value.\nDF_INTERP_USER_CELL\nCompute derivatives of predefined order given\nuser-provided cell indices. The derivative of the\nzero order is the spline value.\nFor this type of the computations you should\nprovide a valid cell array, which holds the indices\nof cells in the site array containing relevant\ninterpolation sites.\nDF_CELL\nCompute indices of cells in partition x that contain\nthe sites.\nIf the indices of cells which contain interpolation types are available before the call to df?Interpolate1D/\ndf?InterpolateEx1D, you can improve performance by using the DF_INTERP_USER_CELL computation\ntype.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2600\n\n\nNOTE\nIf you pass any combination of DF_INTERP, DF_INTERP_USER_CELL, and DF_CELL computation\ntypes to the routine, the library uses the DF_INTERP_USER_CELL computation mode.\nIf you specify DF_INTERP_USER_CELL computation mode and a user-defined callback function for\ncomputing cell indices to df?InterpolateEx1D, the library uses the DF_INTERP_USER_CELL\ncomputation mode, and the call-back function is not called.\nIf the sites do not belong to interpolation interval [a, b] , the library uses:\n•\npolynomial P0 of the spline constructed on interval [x0, x1] for computations at the sites to the left of a.\n•\npolynomial Pn-2 of the spline constructed on interval [xn-2, xn-1] for computations at the sites to the right\nof b.\nInterpolation sites support the following hints:\nHint Values for Interpolation Sites\nValue\nDescription\nDF_NON_UNIFORM_PARTITION\nPartition is non-uniform.\nDF_UNIFORM_PARTITION\nPartition is uniform.\nDF_SORTED_DATA\nInterpolation sites are sorted in the ascending order and define\na non-uniform partition.\nDF_NO_HINT\nNo hint is provided. By default, the partition defined by\ninterpolation sites is interpreted as non-uniform.\nNOTE\nIf you pass a sorted array of interpolation sites to the Intel® oneAPI Math Kernel Library\n(oneMKL), set thesitehint parameter to the DF_SORTED_DATA value. The library uses this\ninformation when choosing the search algorithm and ignores any other data hints about the\nstructure of the interpolation sites.\nData Fitting computation routines can use the following hints to speed up the computation:\n•\nDF_UNIFORM_PARTITION describes the structure of breakpoints and the interpolation sites.\n•\nDF_QUASI_UNIFORM_PARTITION describes the structure of breakpoints.\nPass the above hints to the library when appropriate.\nFor spline-based interpolation, you should set the derivatives whose values are required for the computation.\nYou can provide the derivatives by setting the dorder array of size ndorder as follows:\ndorder i = 1, if derivative of the i-th order is required   \n0, otherwise\ni = 0, ..., ndorder −1\nOrders of derivatives id(d = 0, 1, .., nder - 1), corresponding to non-zero derivatives to be calculated, form\nthe array {id} of length nder≤ndorder.\nThe storage format for the interpolation results is specified using the rhint parameter values. For each\nstorage format, Table Hint Values for the rhint Parameter describes how to get the result R(j, s, id) from\narray r, for function index j(0 ≤j≤yn - 1), site number s(0 ≤s≤nsite - 1), and derivative index id(0 ≤d≤nder -\n1), where yn is the number of functions, nsite is the number of sites, and nder is the total number of non-\nzero derivatives for interpolation. The array r can be either a one-dimensional array of size ny*nder*nsite\nor a three-dimensional array with the dimensions described in the table.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2601\n\n\nHint Values for the rhint Parameter\nValue\nLocation of R(j, s, id), One-\ndimensional Array Storage\nLocation of R(j, s, id), Three-\ndimensional Array Storage\nDF_MATRIX_STORAGE_FU\nNCS_SITES_DERS\n(DF_MATRIX_STORAGE_R\nOWS)\nr[d + nder*(s + nsite*j)]\nr[j][s][d]\nr declared as r[ny][nsite]\n[nder].\nDF_MATRIX_STORAGE_FU\nNCS_DERS_SITES\n(DF_MATRIX_STORAGE_C\nOLS)\nr[s + nsite*(d + nder*j)]\nr[j][d][s]\nr declared as r[ny][nder]\n[nsite].\nDF_MATRIX_STORAGE_SI\nTES_FUNCS_DERS\nr[d + nder*(j + ny*s)]\nr[s][j][d]\nr declared as r[nsite][ny]\n[nder].\nDF_MATRIX_STORAGE_SI\nTES_DERS_FUNCS\nr[j + ny*(d + nder*s)]\nr[s][d][j]\nr declared as r[nsite][nder]\n[ny].\nDF_NO_HINT\nNo hint is provided. By default, the results are stored as in rhint =\nDF_MATRIX_STORAGE_FUNCS_SITES_DERS.\nThe following figures show the structure of the storage formats. Each shows sequential memory layout line\nby line, left to right.\n•\nStorage in r for rhint = DF_MATRIX_STORAGE_FUNCS_SITES_DERS (DF_MATRIX_STORAGE_ROWS):\nR 0, 0, i0\nR 0, 0, i1\n…\nR 0, 0, inder −1\nR 0, 1, i0\nR 0, 1, i1\n…\nR 0, 1, inder −1\n…\n…\n…\n…\nR 0, nsite −1, i0 R 0, nsite −1, i1 … R 0, nsite −1, inder −1\nR 1, 0, i0\nR 1, 0, i1\n…\nR 1, 0, inder −1\nR 1, 1, i0\nR 1, 1, i1\n…\nR 1, 1, inder −1\n…\n…\n…\n…\nR 1, nsite −1, i0 R 1, nsite −1, i1 … R 1, nsite −1, inder −1\n…\n…\n…\n…\n•\nStorage in r for rhint = DF_MATRIX_STORAGE_FUNCS_DERS_SITES (DF_MATRIX_STORAGE_COLS):\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2602\n\n\nR 0, 0, i0\nR 0, 1, i0\n…\nR 0, nsite −1, i0\nR 0, 0, i1\nR 0, 1, i1\n…\nR 0, nsite −1, i1\n…\n…\n…\n…\nR 0, 0, inder −1 R 0, 1, inder −1 … R 0, nsite −1, inder −1\nR 1, 0, i0\nR 1, 1, i0\n…\nR 1, nsite −1, i0\nR 1, 0, i1\nR 1, 1, i1\n…\nR 1, nsite −1, i1\n…\n…\n…\n…\nR 1, 0, inder −1 R 1, 1, inder −1 … R 1, nsite −1, inder −1\n…\n…\n…\n…\n•\nStorage in r for rhint = DF_MATRIX_STORAGE_SITES_FUNCS_DERS:\nR 0, 0, i0\nR 0, 0, i1\n…\nR 0, 0, inder −1\nR 1, 0, i0\nR 1, 0, i1\n…\nR 1, 0, inder −1\n…\n…\n…\n…\nR ny −1, 0, i0 R ny −1, 0, i1 … R ny −1, 0, inder −1\nR 0, 1, i0\nR 0, 1, i1\n…\nR 0, 1, inder −1\nR 1, 1, i0\nR 1, 1, i1\n…\nR 1, 1, inder −1\n…\n…\n…\n…\nR ny −1, 1, i0 R ny −1, 1, i1 … R ny −1, 1, inder −1\n…\n…\n…\n…\n•\nStorage in r for rhint = DF_MATRIX_STORAGE_SITES_DERS_FUNCS:\nR 0, 0, i0\nR 1, 0, i0\n…\nR ny −1, 0, i0\nR 0, 0, i1\nR 1, 0, i1\n…\nR ny −1, 0, i1\n…\n…\n…\n…\nR 0, 0, inder −1 R 1, 0, inder −1 … R ny −1, 0, inder −1\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2603\n\n\nR 0, 1, i0\nR 1, 1, i0\n…\nR ny −1, 1, i0\nR 0, 1, i1\nR 1, 1, i1\n…\nR ny −1, 1, i1\n…\n…\n…\n…\nR 0, 1, inder −1 R 1, 1, inder −1 … R ny −1, 1, inder −1\n…\n…\n…\n…\nTo speed up Data Fitting computations, use the datahint parameter that provides additional information\nabout the structure of the partition and interpolation sites. This data represents a floating-point or a double\narray with the following structure:\nStructure of the datahint Array\nElement Number\nDescription\n0\nTask dimension\n1\nType of additional information\n2\nReserved field\n3\nThe total number q of elements containing additional information.\n4\nElement (1)\n...\n...\nq+3\nElement (q)\nData Fitting computation functions support the following types of additional information for datahint[1]:\nTypes of Additional Information\nType\nElement Number\nParameter\nDF_NO_APRIORI_INFO\n0\nNo parameters are provided.\nInformation about the data\nstructure is absent.\nDF_APRIORI_MOST_LIKELY_CELL\n1\nIndex of the cell that is likely to\ncontain interpolation sites.\nTo compute indices of the cells that contain interpolation sites, provide the pointer to the array of size nsite\nfor the results. The library supports the following scheme of cell indexing for the given partition{xi},\ni=1,...,nx:\ncell[j] = i, if site[j] ∈[xi, xi+1), i = 0,..., nx - 2,\ncell[j] = nx - 1, if site[j] ∈[xnx - 1, xnx],\ncell[j] = nx, if site[j] ∈(xnx, xnx + 1],\nwhere\n•\nx0 = -∞\n•\nxnx+1 = +∞\n•\nj = 0,..., nsite-1\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2604\n\n\nTo perform interpolation computations with spline types unsupported in the Data Fitting component, use the\nextended version of the routine df?InterpolateEx1D. With this routine, you can provide user-defined\ncallback functions for computations within, to the left of, or to the right of interpolaton interval [a, b]. The\ncallback functions compute indices of the cells that contain the specified interpolation sites or can serve as an\napproximation for computing the exact indices of such cells.\nIf you do not pass any function for computations at the sites outside the interval [a, b], the routine uses the\ndefault settings.\nSee Also\nMathematical Conventions for Data Fitting Functions\ndf?InterpCallBack\ndf?SearchCellsCallBack\ndf?Integrate1D/df?IntegrateEx1D\nComputes a spline-based integral.\nSyntax\nstatus = dfsIntegrate1D(task, method, nlim, llim, llimhint, rlim, rlimhint, ldatahint,\nrdatahint, r, rhint)\nstatus = dfdIntegrate1D(task, method, nlim, llim, llimhint, rlim, rlimhint, ldatahint,\nrdatahint, r, rhint)\nstatus = dfsIntegrateEx1D(task, method, nlim, llim, llimhint, rlim, rlimhint, ldatahint,\nrdatahint, r, rhint, le_cb, le_params, re_cb, re_params, i_cb, i_params, search_cb,\nsearch_params)\nstatus = dfdIntegrateEx1D(task, method, nlim, llim, llimhint, rlim, rlimhint, ldatahint,\nrdatahint, r, rhint, le_cb, le_params, re_cb, re_params, i_cb, i_params, search_cb,\nsearch_params)\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\ntask\nDFTaskPtr\nDescriptor of the task.\nmethod\nconst MKL_INT\nIntegration method. The supported value is DF_METHOD_PP.\nnlim\nconst MKL_INT\nNumber of pairs of integration limits.\nllim\nconst float* for\ndfsIntegrate1D/\ndfsIntegrateEx1D\nconst double* for\ndfdIntegrate1D/\ndfdIntegrateEx1D\nArray of size nlim that defines the left-side integration\nlimits.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2605\n\n\nName\nType\nDescription\nllimhint\nconst MKL_INT\nA flag describing the structure of the left-side integration\nlimits llim. For the list of possible values of llimhint, see\ntable \"Hint Values for Integration Limits\". If you set the flag\nto the DF_NO_HINT value, the library assumes that the left-\nside integration limits define a non-uniform partition.\nrlim\nconst float* for\ndfsIntegrate1D/\ndfsIntegrateEx1D\nconst double* for\ndfdIntegrate1D/\ndfdIntegrateEx1D\nArray of size nlim that defines the right-side integration\nlimits.\nrlimhint\nconst MKL_INT\nA flag describing the structure of the right-side integration\nlimits rlim. For the list of possible values of rlimhint, see\ntable \"Hint Values for Integration Limits\". If you set the flag\nto the DF_NO_HINT value, the library assumes that the\nright-side integration limits define a non-uniform partition.\nldatahint\nconst float* for\ndfsIntegrate1D/\ndfsIntegrateEx1D\nconst double* for\ndfdIntegrate1D/\ndfdIntegrateEx1D\nArray that contains additional information about the\nstructure of partition x and left-side integration limits. For\ndetails on the ldatahint array, see table \"Structure of the\ndatahint Array\" in the description of the df?\nInterpolate1D function.\nrdatahint\nconst float* for\ndfsIntegrate1D/\ndfsIntegrateEx1D\nconst double* for\ndfdIntegrate1D/\ndfdIntegrateEx1D\nArray that contains additional information about the\nstructure of partition x and right-side integration limits. For\ndetails on the rdatahint array, see table \"Structure of the\ndatahint Array\" in the description of the df?\nInterpolate1D function.\nrhint\nconst MKL_INT\nA flag describing the structure of the results. For the list of\npossible values of rhint, see table \"Hint Values for\nIntegration Results\". If you set the flag to the DF_NO_HINT\nvalue, the library stores the results in row-major format.\n \n \n \nle_cb\nconstdfsIntegrCallBack\nfor dfsIntegrateEx1D\nconstdfdIntegrCallBack\nfor dfdIntegrateEx1D\nUser-defined callback function for integration on interval\n[ llim[i], min(rlim[i], a)) for llim[i] < a .\nSet to NULL if you are not supplying a callback function.\nle_params\nconst void*\nPointer to additional user-defined parameters passed by the\nlibrary to the le_cb function.\nSet to NULL if there are no additional parameters or if you\nare not supplying a callback function.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2606\n\n\nName\nType\nDescription\nre_cb\nconstdfsInterpCallBack\nfor dfsIntegrateEx1D\nconstdfdInterpCallBack\nfor dfdIntegrateEx1D\nUser-defined callback function for integration on interval\n[max(llim[i], b), rlim[i]) for rlim[i]≥b.\nSet to NULL if you are not supplying a callback function.\nre_params\nconst void*\nPointer to additional user-defined parameters passed by the\nlibrary to the re_cb function.\nSet to NULL if there are no additional parameters or if you\nare not supplying a callback function.\ni_cb\nconstdfsIntegrCallBack\nfor dfsIntegrateEx1D\nconstdfdIntegrCallBack\nfor dfdIntegrateEx1D\nUser-defined callback function for integration on interval\n[max(a, llim[i], ), min(rlim[i], b)).\nSet to NULL if you are not supplying a callback function.\ni_params\nconst void*\nPointer to additional user-defined parameters passed by the\nlibrary to the i_cb function.\nSet to NULL if there are no additional parameters or if you\nare not supplying a callback function.\nsearch_cb\nconstdfsSearchCellsCal\nlBack for\ndfsIntegrateEx1D\nconstdfdSearchCellsCal\nlBack for\ndfdIntegrateEx1D\nUser-defined callback function for computing indices of cells\nthat can contain interpolation sites.\nSet to NULL if you are not supplying a callback function.\nsearch_params\nconst void*\nPointer to additional user-defined parameters passed by the\nlibrary to the search_cb function.\nSet to NULL if there are no additional parameters or if you\nare not supplying a callback function.\nOutput Parameters\nName\nType\nDescription\nstatus\nint\nStatus of the routine:\n•\nDF_STATUS_OK if the routine execution completed\nsuccessfully.\n•\nNon-zero error code if the routine execution failed. See \n\"Task Status and Error Reporting\" for error code\ndefinitions.\nr\nfloat* for dfsIntegrate1D/\ndfsIntegrateEx1D\ndouble* for dfdIntegrate1D/\ndfdIntegrateEx1D\nArray of integration results. The size of the array should\nbe sufficient to hold nlim*ny values, where ny is the\ndimension of the vector-valued function. The integration\nresults are packed according to the settings in rhint.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2607\n\n\nName\nType\nDescription\nCaution\nThe df?Integrate1D/df?IntegrateEx1D routines do\nnot support in-place computations. You must provide\nnon-aliasing memory locations for the llim and rlim\ninput parameters and the r output parameter.\nDescription\nThe df?Integrate1D/df?IntegrateEx1D routine computes spline-based integral on user-defined intervals\n,\nwhere rli = rlim[i], lli = llim[i], and i = 0, ..., ny - 1.\nIf rlim[i] < llim[i], the routine returns\nThe routine supports the following hint values for integration results:\nHint Values for Integration Results\nValue\nDescription\nDF_MATRIX_STORAGE_ROWS\nData is stored in row-major format according to C conventions.\nDF_MATRIX_STORAGE_COLS\nData is stored in column-major format according to Fortran\nconventions.\nDF_NO_HINT\nNo hint is provided. By default, the coordinates of vector-valued\nfunction y are provided and stored in row-major format.\nA common structure of the storage formats for the integration results is as follows:\n•\nRow-major format\nI(0, 0)\n...\nI(0, nlim - 1)\n...\n...\n...\nI (ny - 1, 0)\n...\nI(ny - 1, nlim - 1)\n•\nColumn-major format\nI(0, 0)\n...\nI (ny - 1, 0)\n...\n...\n...\nI(0, nlim - 1)\n...\nI(ny - 1, nlim - 1)\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2608\n\n\nUsing the llimhint and rlimhint parameters, you can provide the following hint values for integration\nlimits:\nHint Values for Integration Limits\nValue\nDescription\nDF_SORTED_DATA\nIntegration limits are sorted in the ascending order and define a\nnon-uniform partition.\nDF_NON_UNIFORM_PARTITION\nPartition defined by integration limits is non-uniform.\nDF_UNIFORM_PARTITION\nPartition defined by integration limits is uniform.\nDF_NO_HINT\nNo hint is provided. By default, partition defined by integration\nlimits is interpreted as non-uniform.\nTo compute integration with splines unsupported in the Data Fitting component, use the extended version of\nthe routine df?IntegrateEx1D. With this routine, you can provide user-defined callback functions that\ncompute:\n•\nintegrals within, to the left of, or to the right of the interpolation interval [a, b]\n•\nindices of cells that contain the provided integration limits or can serve as an approximation for computing\nthe exact indices of such cells\nIf you do not pass callback functions, the routine uses the default settings.\nSee Also\nMathematical Conventions for Data Fitting Functions\ndf?Interpolate1D/df?InterpolateEx1D\ndf?IntegrCallBack\ndf?SearchCellsCallBack\ndf?SearchCells1D/df?SearchCellsEx1D\nSearches sub-intervals containing interpolation sites.\nSyntax\nstatus = dfsSearchCells1D(task, method, nsite, site, sitehint, datahint, cell)\nstatus = dfdSearchCells1D(task, method, nsite, site, sitehint, datahint, cell)\nstatus = dfsSearchCellsEx1D(task, method, nsite, site, sitehint, datahint, cell,\nsearch_cb, search_params)\nstatus = dfdSearchCellsEx1D(task, method, nsite, site, sitehint, datahint, cell,\nsearch_cb, search_params)\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\ntask\nDFTaskPtr\nDescriptor of the task.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2609\n\n\nName\nType\nDescription\nmethod\nconst MKL_INT\nSearch method. The supported value is DF_METHOD_STD.\nnsite\nconst MKL_INT*\nNumber of interpolation sites.\nsite\nconst float* for\ndfsSearchCells1D/\ndfsSearchCellsEx1D\nconst double* for\ndfdSearchCells1D/\ndfdSearchCellsEx1D\nArray of interpolation sites of size nsite. The structure of\nthe array is defined by the sitehint parameter:\n•\nIf the sites form a non-uniform partition, the array\nshould contain nsite values.\n•\nIf the sites form a uniform partition, the array should\ncontain two entries that represent the left-most and the\nright-most interpolation sites. The first entry of the array\ncontains the left-most interpolation point. The second\nentry of the array contains the right-most interpolation\npoint.\nsitehint\nconst MKL_INT\nA flag describing the structure of the interpolation sites. For\nthe list of possible values of sitehint, see table \"Hint\nValues for Interpolation Sites\". If you set the flag to\nDF_NO_HINT, the library interprets the site-defined partition\nas non-uniform.\ndatahint\nconst float* for\ndfsSearchCells1D/\ndfsSearchCellsEx1D\nconst double* for\ndfdSearchCells1D/\ndfdSearchCellsEx1D\nArray that contains additional information about the\nstructure of the partition and interpolation sites. This data\nhelps to speed up the computation. If you provide a NULL\npointer, the routine uses the default settings for\ncomputations. For details on the datahint array, see table \n\"Structure of the datahint Array\".\nsearch_cb\nconstdfsSearchCellsCallBac\nk for dfsSearchCellsEx1D\nconstdfdSearchCellsCallBac\nk for dfdSearchCellsEx1D\nUser-defined callback function for computing indices of cells\nthat can contain interpolation sites.\nSet to NULL if you are not supplying a callback function.\nsearch_pa\nrams\nconst void*\nPointer to additional user-defined parameters passed by the\nlibrary to the search_cb function.\nSet to NULL if there are no additional parameters or if you\nare not supplying a callback function.\nOutput Parameters\nName\nType\nDescription\nstatus\nint\nStatus of the routine:\n•\nDF_STATUS_OK if the routine execution completed\nsuccessfully.\n•\nNon-zero error code if the routine execution failed. See \n\"Task Status and Error Reporting\" for error code\ndefinitions.\ncell\nMKL_INT*\nArray of cell indices in the partition that contain the\ninterpolation sites.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2610\n\n\nDescription\nThe df?SearchCells1D/df?SearchCellsEx1D routines return array cell of indices of sub-intervals (cells)\nin the partition that contain interpolation sites available in array site. For details on the cell indexing\nscheme, see the description of the df?Interpolate1D/df?InterpolateEx1D computation routines.\nUse the datahint parameter to provide additional information about the structure of the partition and/or\ninterpolation sites. The definition of the datahint parameter is availalbe in the description of the df?\nInterpolate1D/df?InterpolateEx1D computation routines.\nFor description of the user-defined callback for computation of cell indices, see df?SearchCellsCallBack.\nSee Also\nMathematical Conventions for Data Fitting Functions\ndf?Interpolate1D/df?InterpolateEx1D\ndf?SearchCellsCallBack\ndf?InterpCallBack\nA callback function for user-defined interpolation to be\npassed into df?InterpolateEx1D.\nSyntax\nstatus = dfsInterpCallBack(n, cell, site, r, user_params, library_params)\nstatus = dfdInterpCallBack(n, cell, site, r, user_params, library_params)\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nn\nlong long*\nNumber of interpolation sites.\ncell\nlong long*\nArray of size n containing indices of the cells to which the\ninterpolation sites in array site belong.\nsite\nfloat* for dfsInterpCallBack\ndouble* for\ndfdInterpCallBack\nArray of interpolation sites of size n.\nuser_para\nms\nvoid*\nPointer to user-defined parameters of the callback function.\nlibrary_p\narams\ndfInterpCallBackLibraryPar\nams*\nPointer to library-defined parameters of the callback\nfunction.\n \n \n \n \n \n \nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2611\n\n\nOutput Parameters\nName\nType\nDescription\nstatus\nint\nThe status returned by the callback function:\n•\nZero indicates successful completion of the callback\noperation.\n•\nA negative value indicates an error.\n•\nA positive value indicates a warning.\nSee \"Task Status and Error Reporting\" for error code\ndefinitions.\nr\nfloat* for dfsInterpCallBack\ndouble* for\ndfdInterpCallBack\nArray of the computed interpolation results packed in row-\nmajor format.\nDescription\nWhen passed into the df?InterpolateEx1D routine, this function performs user-defined interpolation\noperation.\nThe library_params parameter allows the library to provide extra parameters. Currently no parameters are\nprovided.\nSee Also\ndf?interpolate1d/df?interpolateex1d\ndf?searchcellscallback\ndf?IntegrCallBack\nA callback function that you can pass into df?\nIntegrateEx1D to define integration computations.\nSyntax\nstatus = dfsIntegrCallBack(n, lcell, llim, rcell, rlim, r, user_params, library_params)\nstatus = dfdIntegrCallBack(n, lcell, llim, rcell, rlim, r, user_params, library_params)\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nn\nlong long*\nNumber of pairs of integration limits.\nlcell\nlong long*\nArray of size n with indices of the cells that contain the left-\nside integration limits in array llim.\nllim\nfloat* for dfsIntegrCallBack\ndouble* for\ndfdIntegrCallBack\nArray of size n that holds the left-side integration limits.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2612\n\n\nName\nType\nDescription\nrcell\nlong long*\nArray of size n with indices of the cells that contain the\nright-side integration limits in array rlim.\nrlim\nfloat* for dfsIntegrCallBack\ndouble* for\ndfdIntegrCallBack\nArray of size n that holds the right-side integration limits.\nuser_para\nms\nvoid*\nPointer to user-defined parameters of the callback function.\n \n \n \n \n \n \nlibrary_p\narams\ndfIntegrCallBackLibraryPar\nams*\nPointer to library-defined parameters of the callback\nfunction.\nOutput Parameters\nName\nType\nDescription\nstatus\nint\nThe status returned by the callback function:\n•\nZero indicates successful completion of the callback\noperation.\n•\nA negative value indicates an error.\n•\nA positive value indicates a warning.\nSee \"Task Status and Error Reporting\" for error code\ndefinitions.\nr\nfloat* for dfsIntegrCallBack\ndouble* for\ndfdIntegrCallBack\nArray of integration results. For packing the results in row-\nmajor format, follow the instructions described in df?\nInterpolate1D/df?InterpolateEx1D.\nDescription\nWhen passed into the df?IntegrateEx1D routine, this function defines integration computations. If at least\none of the integration limits is outside the interpolation interval [a, b], the library decomposes the integration\ninto sub-intervals that belong to the extrapolation range to the left of a, the extrapolation range to the right\nof b, and the interpolation interval [a, b], as follows:\n•\nIf the left integration limit is to the left of the interpolation interval (llim< a), the df?IntegrateEx1D\nroutine passes llim as the left integration limit and min(rlim, a) as the right integration limit to the\nuser-defined callback function.\n•\nIf the right integration limit is to the right of the interpolation interval (rlim> b), the df?IntegrateEx1D\nroutine passes max(llim, b) as the left integration limit and rlim as the right integration limit to the\nuser-defined callback function.\n•\nIf the left and the right integration limits belong to the interpolation interval, the df?IntegrateEx1D\nroutine passes them to the user-defined callback function unchanged.\nThe value of the integral is the sum of integral values obtained on the sub-intervals.\nThe library_params parameter allows the library to provide extra parameters. Currently no parameters are\nprovided.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2613\n\n\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nSee Also\ndf?Integrate1D/df?IntegrateEx1D\ndf?IntegrCallBack\ndf?SearchCellsCallBack\ndf?SearchCellsCallBack\nA callback function for user-defined search to be\npassed into df?InterpolateEx1D,\ndf?IntegrateEx1D, or df?SearchCellsEx1D.\nSyntax\nstatus = dfsSearchCellsCallBack(n, site, cell, flag, user_params, library_params)\nstatus = dfdSearchCellsCallBack(n, site, cell, flag, user_params, library_params)\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\nn\nlong long*\nNumber of interpolation sites or integration limits.\nsite\nfloat* for\ndfsSearchCellsCallBack\ndouble* for\ndfdSearchCellsCallBack\nArray, size n, of interpolation sites or integration limits.\nflag\nint*\nArray of size n, with values set as follows:\n•\nIf the cell with index cell[i] contains site[i], set\nflag[i] to 1.\n•\nOtherwise, set flag[i] to zero. In this case, the library\ninterprets the index as an approximation and computes\nthe index of the cell containing site[i] by using the\nprovided index as a starting point for the search.\nuser_para\nms\nvoid*\nPointer to user-defined parameters of the callback function.\n \n \n \n \n \n \n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2614\n\n\nName\nType\nDescription\nlibrary_p\narams\ndfSearchCallBackLibraryPar\nams*\nPointer to library-defined parameters of the callback\nfunction.\nOutput Parameters\nName\nType\nDescription\nstatus\nint\nThe status returned by the callback function:\n•\nZero indicates successful completion of the callback\noperation.\n•\nA negative value indicates an error.\n•\nThe DF_STATUS_EXACT_RESULT status indicates that cell\nindices returned by the callback function are exact. In\nthis case, you do not need to initialize entries of the\nflag array.\n•\nA positive value indicates a warning.\nSee \"Task Status and Error Reporting\" for error code\ndefinitions.\ncell\nlong long*\nArray of size n that returns indices of the cells computed by\nthe callback function.\nDescription\nWhen passed into the df?InterpolateEx1D, df?IntegrateEx1D, or df?SearchCellsEx1D routine, this\nfunction performs a user-defined search.\nThe library_params parameter allows the library to provide extra parameters. The df?InterpolateEx1D,\nand df?SearchCellsEx1D routines do not provide extra parameters and set library_params to NULL. The\ndf?IntegrateEx1D routines use this parameter to specify which type of integration limits, left or right, are\nprovided for the callback. To do this the library declares the dfSearchCallBackLibraryParams structure. It\ncurrently contains one field, limit_type_flag, of type int. The field is set by the library to one of two\npossible values: DF_INTEGR_SEARCH_CB_LLIM_FLAG if the left integration limits are provided, or\nDF_INTEGR_SEARCH_CB_RLIM_FLAG if the right integration limits are provided.\nSee Also\ndf?Interpolate1D/df?InterpolateEx1D\ndf?InterpCallBack\nData Fitting Task Destructors\nTask destructors are routines used to delete task descriptors and deallocate the corresponding memory\nresources. The Data Fitting task destructor dfDeleteTask destroys a Data Fitting task and frees the\nmemory.\ndfDeleteTask\nDestroys a Data Fitting task object and frees the\nmemory.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2615\n\n\nSyntax\nstatus = dfDeleteTask(&task)\nInclude Files\n•\nmkl.h\nInput Parameters\nName\nType\nDescription\ntask\nDFTaskPtr\nDescriptor of the task to destroy.\nOutput Parameters\nName\nType\nDescription\nstatus\nint\nStatus of the routine:\n•\nDF_STATUS_OK if the task is deleted successfully.\n•\nNon-zero error code if the operation failed. See \"Task\nStatus and Error Reporting\" for error code definitions.\nDescription\nGiven a pointer to a task descriptor, this routine deletes the Data Fitting task descriptor and frees the\nmemory allocated for the structure. If the task is deleted successfully, the routine sets the task pointer to\nNULL. Otherwise, the routine returns an error code.\nAppendix A: Linear Solvers Basics\nMany applications in science and engineering require the solution of a system of linear equations. This\nproblem is usually expressed mathematically by the matrix-vector equation, Ax = b, where A is an m-by-n\nmatrix, x is the n element column vector and b is the m element column vector. The matrix A is usually\nreferred to as the coefficient matrix, and the vectors x and b are referred to as the solution vector and the\nright-hand side, respectively.\nBasic concepts related to solving linear systems with sparse matrices are described in Sparse Linear Systems\nand various storage schemes for sparse matrices are described in Sparse Matrix Storage Formats.\nSparse Linear Systems\nIn many real-life applications, most of the elements in A are zero. Such a matrix is referred to as sparse.\nConversely, matrices with very few zero elements are called dense. For sparse matrices, computing the\nsolution to the equation Ax = b can be made much more efficient with respect to both storage and\ncomputation time, if the sparsity of the matrix can be exploited. The more an algorithm can exploit the\nsparsity without sacrificing the correctness, the better the algorithm.\nGenerally speaking, computer software that finds solutions to systems of linear equations is called a solver. A\nsolver designed to work specifically on sparse systems of equations is called a sparse solver. Solvers are\nusually classified into two groups - direct and iterative.\nIterative Solvers start with an initial approximation to a solution and attempt to estimate the difference\nbetween the approximation and the true result. Based on the difference, an iterative solver calculates a new\napproximation that is closer to the true result than the initial approximation. This process is repeated until\nthe difference between the approximation and the true result is sufficiently small. The main drawback to\niterative solvers is that the rate of convergence depends greatly on the values in the matrix A. Consequently,\nit is not possible to predict how long it will take for an iterative solver to produce a solution. In fact, for ill-\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2616\n\n\nconditioned matrices, the iterative process will not converge to a solution at all. However, for well-conditioned\nmatrices it is possible for iterative solvers to converge to a solution very quickly. Consequently, if an\napplication involves well-conditioned matrices iterative solvers can be very efficient.\nDirect Solvers, on the other hand, factor the matrix A into the product of two triangular matrices and then\nperform a forward and backward triangular solve.\nThis approach makes the time required to solve a systems of linear equations relatively predictable, based on\nthe size of the matrix. In fact, for sparse matrices, the solution time can be predicted based on the number\nof non-zero elements in the array A.\nMatrix Fundamentals\nA matrix is a rectangular array of either real or complex numbers. A matrix is denoted by a capital letter; its\nelements are denoted by the same lower case letter with row/column subscripts. Thus, the value of the\nelement in row i and column j in matrix A is denoted by a(i,j). For example, a 3 by 4 matrix A, is written\nas follows:\nNote that with the above notation, we assume the standard Fortran programming language convention of\nstarting array indices at 1 rather than the C programming language convention of starting them at 0.\nA matrix in which all of the elements are real numbers is called a real matrix. A matrix that contains at least\none complex number is called a complex matrix. A real or complex matrix A with the property that a(i,j) =\na(j,i), is called a symmetric matrix. A complex matrix A with the property that a(i,j) = conj(a(j,i)), is called a\nHermitian matrix. Note that programs that manipulate symmetric and Hermitian matrices need only store\nhalf of the matrix values, since the values of the non-stored elements can be quickly reconstructed from the\nstored values.\nA matrix that has the same number of rows as it has columns is referred to as a square matrix. The elements\nin a square matrix that have same row index and column index are called the diagonal elements of the\nmatrix, or simply the diagonal of the matrix.\nThe transpose of a matrix A is the matrix obtained by “flipping” the elements of the array about its diagonal.\nThat is, we exchange the elements a(i,j) and a(j,i). For a complex matrix, if we both flip the elements\nabout the diagonal and then take the complex conjugate of the element, the resulting matrix is called the\nHermitian transpose or conjugate transpose of the original matrix. The transpose and Hermitian transpose of\na matrix A are denoted by AT and AH respectively.\nA column vector, or simply a vector, is a n × 1 matrix, and a row vector is a 1 × n matrix. A real or complex\nmatrix A is said to be positive definite if the vector-matrix product xTAx is greater than zero for all non-zero\nvectors x. A matrix that is not positive definite is referred to as indefinite.\nAn upper (or lower) triangular matrix, is a square matrix in which all elements below (or above) the diagonal\nare zero. A unit triangular matrix is an upper or lower triangular matrix with all 1's along the diagonal.\nA matrix P is called a permutation matrix if, for any matrix A, the result of the matrix product PA is identical\nto A except for interchanging the rows of A. For a square matrix, it can be shown that if PA is a permutation\nof the rows of A, then APT is the same permutation of the columns of A. Additionally, it can be shown that the\ninverse of P is PT.\nIn order to save space, a permutation matrix is usually stored as a linear array, called a permutation vector,\nrather than as an array. Specifically, if the permutation matrix maps the i-th row of a matrix to the j-th row,\nthen the i-th element of the permutation vector is j.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2617\n\n\nA matrix with non-zero elements only on the diagonal is called a diagonal matrix. As is the case with a\npermutation matrix, it is usually stored as a vector of values, rather than as a matrix.\nDirect Method\nFor solvers that use the direct method, the basic technique employed in finding the solution of the system Ax\n= b is to first factor A into triangular matrices. That is, find a lower triangular matrix L and an upper\ntriangular matrix U, such that A = LU. Having obtained such a factorization (usually referred to as an LU\ndecomposition or LU factorization), the solution to the original problem can be rewritten as follows.\nAx = b\nLUx = b\nL(Ux) = b\nThis leads to the following two-step process for finding the solution to the original system of equations:\n1.\nSolve the systems of equations Ly = b.\n2.\nSolve the system Ux = y.\nSolving the systems Ly = b and Ux = y is referred to as a forward solve and a backward solve, respectively.\nIf a symmetric matrix A is also positive definite, it can be shown that A can be factored as LLT where L is a\nlower triangular matrix. Similarly, a Hermitian matrix, A, that is positive definite can be factored as A = LLH.\nFor both symmetric and Hermitian matrices, a factorization of this form is called a Cholesky factorization.\nIn a Cholesky factorization, the matrix U in an LU decomposition is either LT or LH. Consequently, a solver can\nincrease its efficiency by only storing L, and one-half of A, and not computing U. Therefore, users who can\nexpress their application as the solution of a system of positive definite equations will gain a significant\nperformance improvement over using a general representation.\nFor matrices that are symmetric (or Hermitian) but not positive definite, there are still some significant\nefficiencies to be had. It can be shown that if A is symmetric but not positive definite, then A can be factored\nas A = LDLT, where D is a diagonal matrix and L is a lower unit triangular matrix. Similarly, if A is Hermitian,\nit can be factored as A = LDLH. In either case, we again only need to store L, D, and half of A and we need\nnot compute U. However, the backward solve phases must be amended to solving LTx = D-1y rather than\nLTx = y.\nFill-In and Reordering of Sparse Matrices\nTwo important concepts associated with the solution of sparse systems of equations are fill-in and reordering.\nThe following example illustrates these concepts.\nConsider the system of linear equation Ax = b, where A is a symmetric positive definite sparse matrix, and A\nand b are defined by the following:\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2618\n\n\nA star (*) is used to represent zeros and to emphasize the sparsity of A. The Cholesky factorization of A is: A\n= LLT, where L is the following:\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2619\n\n\nNotice that even though the matrix A is relatively sparse, the lower triangular matrix L has no zeros below\nthe diagonal. If we computed L and then used it for the forward and backward solve phase, we would do as\nmuch computation as if A had been dense.\nThe situation of L having non-zeros in places where A has zeros is referred to as fill-in. Computationally, it\nwould be more efficient if a solver could exploit the non-zero structure of A in such a way as to reduce the\nfill-in when computing L. By doing this, the solver would only need to compute the non-zero entries in L.\nToward this end, consider permuting the rows and columns of A. As described in Matrix Fundamentals, the\npermutations of the rows of A can be represented as a permutation matrix, P. The result of permuting the\nrows is the product of P and A. Suppose, in the above example, we swap the first and fifth row of A, then\nswap the first and fifth columns of A, and call the resulting matrix B. Mathematically, we can express the\nprocess of permuting the rows and columns of A to get B as B = PAPT. After permuting the rows and\ncolumns of A, we see that B is given by the following:\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2620\n\n\nSince B is obtained from A by simply switching rows and columns, the numbers of non-zero entries in A and\nB are the same. However, when we find the Cholesky factorization, B = LLT, we see the following:\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2621\n\n\nThe fill-in associated with B is much smaller than the fill-in associated with A. Consequently, the storage and\ncomputation time needed to factor B is much smaller than to factor A. Based on this, we see that an efficient\nsparse solver needs to find permutation P of the matrix A, which minimizes the fill-in for factoring B = PAPT,\nand then use the factorization of B to solve the original system of equations.\nAlthough the above example is based on a symmetric positive definite matrix and a Cholesky decomposition,\nthe same approach works for a general LU decomposition. Specifically, let P be a permutation matrix, B =\nPAPT and suppose that B can be factored as B = LU. Then\nAx = b\n   PA(P-1P)x = Pb\n   PA(PTP)x = Pb\n   (PAPT)(Px) = Pb\n   B(Px) = Pb\n   LU(Px) = Pb\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2622\n\n\nIt follows that if we obtain an LU factorization for B, we can solve the original system of equations by a three\nstep process:\n1.\nSolve Ly = Pb.\n2.\nSolve Uz = y.\n3.\nSet x = PTz.\nIf we apply this three-step process to the current example, we first need to perform the forward solve of the\nsystems of equation Ly = Pb:\nThis gives:\nThe second step is to perform the backward solve, Uz = y. Or, in this case, since a Cholesky factorization is\nused, LTz = y.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2623\n\n\nThis gives\nThe third and final step is to set x = PTz. This gives\nSparse Matrix Storage Formats\nIt is more efficient to store only the non-zero elements of a sparse matrix. There are a number of common\nstorage formats used for sparse matrices, but most of them employ the same basic technique. That is, store\nall non-zero elements of the matrix into a linear array and provide auxiliary arrays to describe the locations\nof the non-zero elements in the original matrix.\nStorage Formats for the Direct Sparse Solvers\nStoring the non-zero elements of a sparse matrix into a linear array is done by walking down each column\n(column-major format) or across each row (row-major format) in order, and writing the non-zero elements to\na linear array in the order they appear in the walk.\n•\nDSS Symmetric Matrix Storage\n•\nDSS Nonsymmetric Matrix Storage\n•\nDSS Structurally Symmetric Matrix Storage\n•\nDSS Distributed Symmetric Matrix Storage\nSparse Matrix Storage Formats for Sparse BLAS Levels 2 and Level 3\nThese sections describe in detail the sparse matrix storage formats supported in the current version of the\nIntel® oneAPI Math Kernel Library (oneMKL) Sparse BLAS Level 2 and Level 3.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2624\n\n\n•\nSparse BLAS CSR Matrix Storage\n•\nSparse BLAS CSC Matrix Storage\n•\nSparse BLAS Coordinate Matrix Storage\n•\nSparse BLAS Diagonal Matrix Storage\n•\nSparse BLAS Skyline Matrix Storage\n•\nSparse BLAS BSR Matrix Storage\nDSS Symmetric Matrix Storage\nFor symmetric matrices, it is necessary to store only the upper triangular half of the matrix (upper triangular\nformat) or the lower triangular half of the matrix (lower triangular format).\nThe Intel® oneAPI Math Kernel Library (oneMKL) direct sparse solvers use a row-major upper triangular\nstorage format: the matrix is compressed row-by-row and for symmetric matrices only non-zero elements in\nthe upper triangular half of the matrix are stored.\nThe Intel® oneAPI Math Kernel Library (oneMKL) sparse matrix storage format for direct sparse solvers is\nspecified by three arrays:values, columns, and rowIndex. The following table describes the arrays in terms of\nthe values, row, and column positions of the non-zero elements in a sparse matrix.\nvalues\nA real or complex array that contains the non-zero elements of a sparse matrix.\nThe non-zero elements are mapped into the values array using the row-major\nupper triangular storage mapping described above.\ncolumns\nElement i of the integer array columns is the number of the column that contains\nthe i-th element in the values array.\nrowIndex\nElement j of the integer array rowIndex gives the index of the element in the\nvalues array that is first non-zero element in a row j.\nThe length of the values and columns arrays is equal to the number of non-zero elements in the matrix.\nAs the rowIndex array gives the location of the first non-zero element within a row, and the non-zero\nelements are stored consecutively, the number of non-zero elements in the i-th row is equal to the difference\nof rowIndex[i] and rowIndex[i+1].\nTo have this relationship hold for the last row of the matrix, an additional entry (dummy entry) is added to\nthe end of rowIndex. Its value is equal to the number of non-zero elements plus one. This makes the total\nlength of the rowIndex array one larger than the number of rows in the matrix.\nNOTE\nThe Intel® oneAPI Math Kernel Library (oneMKL) sparse storage scheme for the direct sparse solvers\nsupports both one-based indexing and zero-based indexing.\nConsider the symmetric matrix A:\nA =\n1\n−1 * −3\n*\n−1\n5\n*\n*\n*\n*\n*\n4\n6\n4\n−3\n*\n6\n7\n*\n*\n*\n4\n*\n−5\nOnly elements from the upper triangle are stored. The actual arrays for the matrix A are as follows:\nStorage Arrays for a Symmetric Matrix\none-based indexing\nvalues\n=\n(1\n-1\n-3\n5\n4\n6\n4\n7\n-5)\ncolumns\n=\n(1\n2\n4\n2\n3\n4\n5\n4\n5)\nrowIndex\n=\n(1\n4\n5\n8\n9\n10)\n \n \n \nzero-based indexing\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2625\n\n\nvalues\n=\n(1\n-1\n-3\n5\n4\n6\n4\n7\n-5)\ncolumns\n=\n(0\n1\n3\n1\n2\n3\n4\n3\n4)\nrowIndex\n=\n(0\n3\n4\n7\n8\n9)\n \n \n \nStorage Format Restrictions\nThe storage format for the sparse solver must conform to two important restrictions:\n•\nthe non-zero values in a given row must be placed into the values array in the order in which they occur\nin the row (from left to right);\n•\nno diagonal element can be omitted from the values array for any symmetric or structurally symmetric\nmatrix.\nThe second restriction implies that if symmetric or structurally symmetric matrices have zero diagonal\nelements, then they must be explicitly represented in the values array.\nDSS Nonsymmetric Matrix Storage\nFor a non-symmetric or non-Hermitian matrix, all non-zero elements need to be stored. Consider the non-\nsymmetric matrix B:\n1\n−1 * −3\n*\n−2\n5\n*\n*\n*\n*\n*\n4\n6\n4\n−4\n*\n2\n7\n*\n*\n8\n*\n*\n−5\nThe matrix B has 13 non-zero elements, and all of them are stored as follows:\nStorage Arrays for a Non-Symmetric Matrix\none-based\nindexing\nvalues\n=\n(1\n-1\n-3\n-2\n5\n4\n6\n4\n-4\n2\n7\n8\n-5)\ncolumns\n=\n(1\n2\n4\n1\n2\n3\n4\n5\n1\n3\n4\n2\n5)\nrowIndex\n=\n(1\n4\n6\n9\n12\n14)\n \n \n \n \n \n \n \nzero-based\nindexing\nvalues\n=\n(1\n-1\n-3\n-2\n5\n4\n6\n4\n-4\n2\n7\n8\n-5)\ncolumns\n=\n(0\n1\n3\n0\n1\n2\n3\n4\n0\n2\n3\n1\n4)\nrowIndex\n=\n(0\n3\n5\n8\n11\n13)\n \n \n \n \n \n \n \nStorage Format Restrictions\nThe storage format for the sparse solver must conform to two important restrictions:\n•\nthe non-zero values in a given row must be placed into the values array in the order in which they occur\nin the row (from left to right);\n•\nno diagonal element can be omitted from the values array for any symmetric or structurally symmetric\nmatrix.\nThe second restriction implies that if symmetric or structurally symmetric matrices have zero diagonal\nelements, then they must be explicitly represented in the values array.\nDSS Structurally Symmetric Matrix Storage\nDirect sparse solvers can also solve symmetrically structured systems of equations. A symmetrically\nstructured system of equations is one where the pattern of non-zero elements is symmetric. That is, a matrix\nhas a symmetric structure if aj,i is not zero if and only if ai, j is not zero. From the point of view of the solver\nsoftware, a \"non-zero\" element of a matrix is any element stored in the values array, even if its value is\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2626\n\n\nequal to 0. In that sense, any non-symmetric matrix can be turned into a symmetrically structured matrix by\ncarefully adding zeros to the values array. For example, the above matrix B can be turned into a\nsymmetrically structured matrix by adding two non-zero entries:\nB =\n1\n−1 * 3\n*\n−2\n5\n* *\n0\n*\n*\n4 6\n4\n−4\n*\n2 7\n*\n*\n8\n0 * −5\nThe matrix B can be considered to be symmetrically structured with 15 non-zero elements and represented\nas:\nStorage Arrays for a Symmetrically Structured Matrix\none-based\nindexing\nvalues\n=\n(1\n-1\n-3\n-2\n5\n0\n4\n6\n4\n-4\n2\n7\n8\n0\n-5)\ncolumns\n=\n(1\n2\n4\n1\n2\n5\n3\n4\n5\n1\n3\n4\n2\n3\n5)\nrowIndex\n=\n(1\n4\n7\n10\n13\n16)\n \n \n \n \n \n \n \n \n \nzero-based\nindexing\nvalues\n=\n(1\n-1\n-3\n-2\n5\n0\n4\n6\n4\n-4\n2\n7\n8\n0\n-5)\ncolumns\n=\n(0\n1\n3\n0\n1\n4\n2\n3\n4\n0\n2\n3\n1\n2\n4)\nrowIndex\n=\n(0\n3\n6\n9\n12\n15)\nStorage Format Restrictions\nThe storage format for the sparse solver must conform to two important restrictions:\n•\nthe non-zero values in a given row must be placed into the values array in the order in which they occur\nin the row (from left to right);\n•\nno diagonal element can be omitted from the values array for any symmetric or structurally symmetric\nmatrix.\nThe second restriction implies that if symmetric or structurally symmetric matrices have zero diagonal\nelements, then they must be explicitly represented in the values array.\nDSS Distributed Symmetric Matrix Storage\nThe distributed assembled matrix input format can be used by the Parallel Direct Sparse Solver for Clusters\nInterface.\nIn this format, the symmetric input matrix A is divided into sequential row subsets, or domains. Each domain\nbelongs to an MPI process. Neighboring domains can overlap. For such intersection between two domains,\nthe element values of the full matrix can be obtained by summing the respective elements of both domains.\nAs in the centralized format, the distributed format uses three arrays to describe the input data, but the\nvalues, columns, and rowIndex arrays on each processor only describe the domain belonging to that\nparticular processor and not the entire matrix.\nFor example, consider a symmetric matrix A:\nA =\n6\n−1 * −3 *\n−1\n5\n*\n*\n*\n*\n*\n11\n5\n4\n−3\n*\n5\n10 *\n*\n*\n4\n*\n5\nThis array could be distributed between two domains corresponding to two MPI processes, with the first\ncontaining rows 1 through 3, and the second containing rows 3 through 5.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2627\n\n\nNOTE\nFor the symmetric input matrix, it is not necessary to store the values from the lower triangle.\nADomain1 =\n6\n−1 * −3 *\n−1\n5\n*\n*\n*\n*\n*\n3\n*\n2\nDistributed Storage Arrays for a Symmetric Matrix, Domain 1\none-based indexing\nvalues\n=\n(6\n-1\n-3\n5\n3\n2)\ncolumns\n=\n(1\n2\n4\n2\n3\n5)\nrowIndex\n=\n(1\n4\n5\n7)\nzero-based indexing\nvalues\n=\n(6\n-1\n-3\n5\n3\n2)\ncolumns\n=\n(0\n1\n3\n1\n2\n4)\nrowIndex\n=\n(0\n3\n4\n6)\nADomain2 =\n*\n* 8 5 2\n−3 * 5 10 *\n*\n* 4 * 5\nDistributed Storage Arrays for a Symmetric Matrix, Domain 2\none-based indexing\nvalues\n=\n(8\n5\n2\n10\n5)\ncolumns\n=\n(3\n4\n5\n4\n5)\nrowIndex\n=\n(1\n4\n5\n6)\nzero-based indexing\nvalues\n=\n(8\n5\n2\n10\n5)\ncolumns\n=\n(2\n3\n4\n3\n4)\nrowIndex\n=\n(0\n3\n4\n5)\nThe third row of matrix A is common between domain 1 and domain 2. The values of row 3 of matrix A are\nthe sums of the respective elements of row 3 of matrix ADomain1 and row 1 of matrix ADomain2.\nStorage Format Restrictions\nThe storage format for the sparse solver must conform to two important restrictions:\n•\nthe non-zero values in a given row must be placed into the values array in the order in which they occur\nin the row (from left to right);\n•\nno diagonal element can be omitted from the values array for any symmetric or structurally symmetric\nmatrix.\nThe second restriction implies that if symmetric or structurally symmetric matrices have zero diagonal\nelements, then they must be explicitly represented in the values array.\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nSparse BLAS CSR Matrix Storage Format\nThe Intel® oneAPI Math Kernel Library (oneMKL) Sparse BLAS compressed sparse row (CSR) format is\nspecified by four arrays:\n•\nvalues\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2628\n\n\n•\ncolumns\n•\npointerB\n•\npointerE\nIn addition, each sparse matrix has an associated variable ,indexing, which specifies if the matrix indices\nare 0-based (indexing=0) or 1-based (indexing=1). These are descriptions of the arrays in terms of the\nvalues, row, and column positions of the non-zero elements in a sparse matrix A.\nvalues\nA real or complex array that contains the non-zero elements of A. Values of the\nnon-zero elements of A are mapped into the values array using the row-major\nstorage mapping described above.\ncolumns\nElement i of the integer array columns is the number of the column in A that\ncontains the i-th value in the values array.\npointerB\nElement j of this integer array gives the index of the element in the values array\nthat is first non-zero element in a row j of A. Note that this index is equal to\npointerB[j]-indexing .\npointerE\nAn integer array that contains row indices, such that pointerE[j]-1-indexing\nis the index of the element in the values array that is last non-zero element in a\nrow j of A.\nThe length of the values and columns arrays is equal to the number of non-zero elements in A.The length of\nthe pointerB and pointerE arrays is equal to the number of rows in A.\nNOTE\nNote that the Intel® oneAPI Math Kernel Library (oneMKL) Sparse BLAS routines support the CSR\nformat both with one-based indexing and zero-based indexing.\nYou can represent the matrix B\nB =\n1\n−1 * −3\n*\n−2\n5\n*\n*\n*\n*\n*\n4\n6\n4\n−4\n*\n2\n7\n*\n*\n8\n*\n*\n−5\nin the CSR format as:\nStorage Arrays for a Matrix in CSR Format\none-based indexing\nvalues\n=\n(1\n-1\n-3\n-2\n5\n4\n6\n4\n-4\n2\n7\n8\n-5)\ncolumns\n=\n(1\n2\n4\n1\n2\n3\n4\n5\n1\n3\n4\n2\n5)\npointerB\n=\n(1\n4\n6\n9\n12)\n \n \n \n \n \n \n \n \npointerE\n=\n(4\n6\n9\n12\n14)\n \n \n \n \n \n \n \n \nzero-based indexing\nvalues\n=\n(1\n-1\n-3\n-2\n5\n4\n6\n4\n-4\n2\n7\n8\n-5)\ncolumns\n=\n(0\n1\n3\n0\n1\n2\n3\n4\n0\n2\n3\n1\n4)\npointerB\n=\n(0\n3\n5\n8\n11)\n \n \n \n \n \n \n \n \npointerE\n=\n(3\n5\n8\n11\n13)\n \n \n \n \n \n \n \n \nAdditionally, you can define submatrices with different pointerB and pointerE arrays that share the same\nvalues and columns arrays of a CSR matrix. For example, you can represent the lower right 3x3 submatrix of\nB as:\nStorage Arrays for a Matrix in CSR Format\none-based indexing\nsubpointerB\n=\n(6\n10\n13)\n \n \n \n \n \n \n \n \nsubpointerE\n=\n(9\n12\n14)\n \n \n \n \n \n \n \n \nzero-based indexing\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2629\n\n\nsubpointerB\n=\n(5\n9\n12)\n \n \n \n \n \n \n \n \nsubpointerE\n=\n(8\n11\n13)\n \n \n \n \n \n \n \n \nNOTE The CSR matrix must have a monotonically increasing row index. That is, pointerB[i] ≤\npointerB[j] and pointerE[i] ≤ pointerE[j] for all indices i <j.\nThis storage format is used in the NIST Sparse BLAS library [Rem05].\nThree Array Variation of CSR Format\nThe storage format accepted for the direct sparse solvers is a variation of the CSR format. It also is used in\nthe Intel® oneAPI Math Kernel Library (oneMKL) Sparse BLAS Level 2 both with one-based indexing and zero-\nbased indexing. The above matrixB can be represented in this format (referred to as the 3-array variation of\nthe CSR format or CSR3) as:\nStorage Arrays for a Matrix in CSR Format (3-Array Variation)\none-based indexing\nvalues\n=\n(1\n-1\n-3\n-2\n5\n4\n6\n4\n-4\n2\n7\n8\n-5)\ncolumns\n=\n(1\n2\n4\n1\n2\n3\n4\n5\n1\n3\n4\n2\n5)\nrowIndex\n=\n(1\n4\n6\n9\n12\n14)\n \n \n \n \n \n \n \nzero-based\nindexing\nvalues\n=\n(1\n-1\n-3\n-2\n5\n4\n6\n4\n-4\n2\n7\n8\n-5)\ncolumns\n=\n(0\n1\n3\n0\n1\n2\n3\n4\n0\n2\n3\n1\n4)\nrowIndex\n=\n(0\n3\n5\n8\n11\n13)\n \n \n \n \n \n \n \nThe 3-array variation of the CSR format has a restriction: all non-zero elements are stored continuously, that\nis the set of non-zero elements in the row J goes just after the set of non-zero elements in the row J-1.\nThere are no such restrictions in the general (NIST) CSR format. This may be useful, for example, if there is\na need to operate with different submatrices of the matrix at the same time. In this case, it is enough to\ndefine the arrays pointerB and pointerE for each needed submatrix so that all these arrays are pointers to the\nsame array values.\nBy definition, the array rowIndex from the Table \"Storage Arrays for a Non-Symmetric Example Matrix\" is\nrelated to the arrays pointerB and pointerE from the Table \"Storage Arrays for an Example Matrix in CSR\nFormat\", and you can see that\npointerB[i] = rowIndex[i] for i=0, ..4;\n                pointerE[i] = rowIndex[i+1] for i=0, ..4.\nThis enables calling a routine that has values, columns, pointerB and pointerE as input parameters for a\nsparse matrix stored in the format accepted for the direct sparse solvers. For example, a routine with the\ninterface:\n   void name_routine(.... ,  double *values, MKL_INT *columns, MKL_INT *pointerB, MKL_INT \n*pointerE, ...)\ncan be called with parameters values, columns, rowIndex as follows:\n   name_routine(.... ,  values, columns, rowIndex, rowIndex+1, ...).\nSparse BLAS CSC Matrix Storage Format\nThe compressed sparse column format (CSC) is similar to the CSR format, but the columns are used instead\nthe rows. In other words, the CSC format is identical to the CSR format for the transposed matrix. The CSR\nformat is specified by four arrays: values, columns, pointerB, and pointerE. The following table describes the\narrays in terms of the values, row, and column positions of the non-zero elements in a sparse matrix A.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2630\n\n\nvalues\nA real or complex array that contains the non-zero elements of A. Values of the\nnon-zero elements of A are mapped into the values array using the column-\nmajor storage mapping.\nrows\nElement i of the integer array rows is the number of the row in A that contains\nthe i-th value in the values array.\npointerB\nElement j of this integer array gives the index of the element in the values array\nthat is first non-zero element in a column j of A. Note that this index is equal to\npointerB[j]-indexing for Inspector-executor Sparse BLAS CSC arrays.\npointerE\nAn integer array that contains column indices, such that pointerE[j]-\nindexing is the index of the element in the values array that is last non-zero\nelement in a column j of A.\nThe length of the values and columns arrays is equal to the number of non-zero elements in A. The length of\nthe pointerB and pointerE arrays is equal to the number of columns in A.\nNOTE\nNote that the Intel® oneAPI Math Kernel Library (oneMKL) Sparse BLAS routines support the CSC\nformat both with one-based indexing and zero-based indexing.\nFor example, consider matrix B:\nB =\n1\n−1 * −3\n*\n−2\n5\n*\n*\n*\n*\n*\n4\n6\n4\n−4\n*\n2\n7\n*\n*\n8\n*\n*\n−5\nIt can be represented in the CSC format as:\nStorage Arrays for a Matrix in CSC Format\none-based indexing\nvalues\n=\n(1\n-2\n-4\n-1\n5\n8\n4\n2\n-3\n6\n7\n4\n-5)\nrows\n=\n(1\n2\n4\n1\n2\n5\n3\n4\n1\n3\n4\n3\n5)\npointerB\n=\n(1\n4\n7\n9\n12)\n \n \n \n \n \n \n \n \npointerE\n=\n(4\n7\n9\n12\n14)\n \n \n \n \n \n \n \n \nzero-based indexing\nvalues\n=\n(1\n-2\n-4\n-1\n5\n8\n4\n2\n-3\n6\n7\n4\n-5)\nrows\n=\n(0\n1\n3\n0\n1\n4\n2\n3\n0\n2\n3\n2\n4)\npointerB\n=\n(0\n3\n6\n8\n11)\n \n \n \n \n \n \n \n \npointerE\n=\n(3\n6\n8\n11\n13)\n \n \n \n \n \n \n \n \nSparse BLAS Coordinate Matrix Storage Format\nThe coordinate format is the most flexible and simplest format for the sparse matrix representation. Only\nnon-zero elements are stored, and the coordinates of each non-zero element are given explicitly. Many\ncommercial libraries support the matrix-vector multiplication for the sparse matrices in the coordinate\nformat.\nThe Intel® oneAPI Math Kernel Library (oneMKL) coordinate format is specified by three arrays:values, rows,\nand column, and a parameter nnz which is number of non-zero elements in A. All three arrays have\ndimension nnz. The following table describes the arrays in terms of the values, row, and column positions of\nthe non-zero elements in a sparse matrix A.\nvalues\nA real or complex array that contains the non-zero elements of A in any order.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2631\n\n\nrows\nElement i of the integer array rows is the number of the row in A that contains\nthe i-th value in the values array.\ncolumns\nElement i of the integer array columns is the number of the column in A that\ncontains the i-th value in the values array.\nNOTE\nNote that the Intel® oneAPI Math Kernel Library (oneMKL) Sparse BLAS routines support the\ncoordinate format both with one-based indexing and zero-based indexing.\nFor example, the sparse matrix C\nC =\n1\n−1 −3 0\n0\n−2\n5\n0\n0\n0\n0\n0\n4\n6\n4\n−4\n0\n2\n7\n0\n0\n8\n0\n0 −5\ncan be represented in the coordinate format as follows:\nStorage Arrays for an Example Matrix in case of the coordinate format\none-based indexing\nvalues\n=\n(1\n-1\n-3\n-2\n5\n4\n6\n4\n-4\n2\n7\n8\n-5)\nrows\n=\n(1\n1\n1\n2\n2\n3\n3\n3\n4\n4\n4\n5\n5)\ncolumns\n=\n(1\n2\n3\n1\n2\n3\n4\n5\n1\n3\n4\n2\n5)\nzero-based\nindexing\nvalues\n=\n(1\n-1\n-3\n-2\n5\n4\n6\n4\n-4\n2\n7\n8\n-5)\nrows\n=\n(0\n0\n0\n1\n1\n2\n2\n2\n3\n3\n3\n4\n4)\ncolumns\n=\n(0\n1\n2\n0\n1\n2\n3\n4\n0\n2\n3\n1\n4)\nSparse BLAS Diagonal Matrix Storage Format\nIf the sparse matrix has diagonals containing only zero elements, then the diagonal storage format can be\nused to reduce the amount of information needed to locate the non-zero elements. This storage format is\nparticularly useful in many applications where the matrix arises from a finite element or finite difference\ndiscretization. The Intel® oneAPI Math Kernel Library (oneMKL) diagonal storage format is specified by two\narrays:values and distance, and two parameters: ndiag, which is the number of non-empty diagonals, and\nlval, which is the declared leading dimension in the calling (sub)programs. The following table describes the\narrays values and distance:\nvalues\nA real or complex two-dimensional array is dimensioned as lval by ndiag. Each\ncolumn of it contains the non-zero elements of certain diagonal of A. The key\npoint of the storage is that each element in values retains the row number of the\noriginal matrix. To achieve this diagonals in the lower triangular part of the\nmatrix are padded from the top, and those in the upper triangular part are\npadded from the bottom. Note that the value of distance[i] is the number of\nelements to be padded for diagonal i.\ndistance\nAn integer array with dimension ndiag. Element i of the array distance is the\ndistance between i-diagonal and the main diagonal. The distance is positive if the\ndiagonal is above the main diagonal, and negative if the diagonal is below the\nmain diagonal. The main diagonal has a distance equal to zero.\nThe above matrix C can be represented in the diagonal storage format as follows:\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2632\n\n\ndistance = (−3 −1 0 1 2)\nvalues\n=\n*\n*\n1\n−1 −3\n*\n−2\n5\n0\n0\n*\n0\n4\n6\n4\n−4\n2\n7\n0\n*\n8\n0\n−5\n*\n*\nwhere the asterisks denote padded elements.\nWhen storing symmetric, Hermitian, or skew-symmetric matrices, it is necessary to store only the upper or\nthe lower triangular part of the matrix.\nFor the Intel® oneAPI Math Kernel Library (oneMKL) triangular solver routines elements of the arraydistance\nmust be sorted in increasing order. In all other cases the diagonals and distances can be stored in arbitrary\norder.\nSparse BLAS Skyline Matrix Storage Format\nThe skyline storage format is important for the direct sparse solvers, and it is well suited for Cholesky or LU\ndecomposition when no pivoting is required.\nThe skyline storage format accepted in Intel® oneAPI Math Kernel Library (oneMKL) can store only triangular\nmatrix or triangular part of a matrix. This format is specified by two arrays:values and pointers. The\nfollowing table describes these arrays:\nvalues\nA scalar array. For a lower triangular matrix it contains the set of elements from\neach row of the matrix starting from the first non-zero element to and including\nthe diagonal element. For an upper triangular matrix it contains the set of\nelements from each column of the matrix starting with the first non-zero element\ndown to and including the diagonal element. Encountered zero elements are\nincluded in the sets.\npointers\nAn integer array with dimension (m+1), where m is the number of rows for lower\ntriangle (columns for the upper triangle). pointers]i] - pointers[0]+1 gives\nthe index of element in values that is first non-zero element in row (column) i.\nThe value of pointers[m] is set to nnz+pointers[0], where nnz is the number\nof elements in the array values.\nFor example, consider the matrix C:\nC =\n1\n−1 −3 0\n0\n−2\n5\n0\n0\n0\n0\n0\n4\n6\n4\n−4\n0\n2\n7\n0\n0\n8\n0\n0 −5\nThe low triangle of the matrix C given above can be stored as follows:\nvalues  =  [ 1  -2   5   4  -4   0   2   7   8   0   0   -5 ]\n                pointers = [ 0   1   3   4   8   12 ]\nand the upper triangle of this matrix C can be stored as follows:\nvalues   = [ 1  -1   5  -3   0   4   6   7  4   0   -5 ]\n                pointers = [ 0   1   3   6   8   11 ]\nThis storage format is supported by the NIST Sparse BLAS library [Rem05].\nNote that the Intel® oneAPI Math Kernel Library (oneMKL) Sparse BLAS routines operating with the skyline\nstorage format do not support general matrices.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2633\n\n\nSparse BLAS BSR Matrix Storage Format\nThe Intel® oneAPI Math Kernel Library (oneMKL) block compressed sparse row (BSR) format for sparse\nmatrices is specified by four arrays:values, columns, pointerB, and pointerE. The following table describes\nthese arrays.\nvalues\nA real array that contains the elements of the non-zero blocks of a sparse matrix.\nThe elements are stored block-by-block in row-major order. A non-zero block is\nthe block that contains at least one non-zero element. All elements of non-zero\nblocks are stored, even if some of them are equal to zero. Within each non-zero\nblock elements are stored in column-major order in the case of one-based\nindexing, and in row-major order in the case of the zero-based indexing.\ncolumns\nElement i of the integer array columns is the number of the column in the block\nmatrix that contains the i-th non-zero block.\npointerB\nElement j of this integer array gives the index of the element in the columns\narray that is first non-zero block in a row j of the block matrix.\npointerE\nElement j of this integer array gives the index of the element in the columns\narray that contains the last non-zero block in a row j of the block matrix plus 1.\nThe length of the values array is equal to the number of all elements in the non-zero blocks, the length of the\ncolumns array is equal to the number of non-zero blocks. The length of the pointerB and pointerE arrays is\nequal to the number of block rows in the block matrix.\nNOTE\nNote that the Intel® oneAPI Math Kernel Library (oneMKL) Sparse BLAS routines support BSR format\nboth with one-based indexing and zero-based indexing.\nFor example, consider the sparse matrix D\nD =\n1 0 6 7 * *\n2 1 8 2 * *\n* * 1 4 * *\n* * 5 1 * *\n* * 4 3 7 2\n* * 0 0 0 0\nIf the size of the block equals 2, then the sparse matrix D can be represented as a 3x3 block matrix E with\nthe following structure:\nE =\nL M *\n* N *\n* P Q\nwhere\nL =\n1 0\n2 1\n, M =\n6 7\n8 2\n, N =\n1 4\n5 1 , P =\n4 3\n0 0\n, Q =\n7 2\n0 0\nThe matrix D can be represented in the BSR format as follows:\none-based indexing\nvalues  =  (1 2 0 1 6 8 7 2 1 5 4 1 4 0 3 0 7 0 2 0)\n                columns  = (1   2   2   2   3)\n                pointerB = (1   3   4)\n                pointerE = (3   4   6)\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2634\n\n\nzero-based indexing\nvalues  =  [1 0 2 1 6 7 8 2 1 4 5 1 4 3 0 0 7 2 0 0]\n                columns  = [0   1   1   1   2]\n                pointerB = [0   2   3]\n                pointerE = [2   3   5]\nThis storage format is supported by the NIST Sparse BLAS library [Rem05].\nThree Array Variation of BSR Format\nIntel® oneAPI Math Kernel Library (oneMKL) supports the variation of the BSR format that is specified by\nthree arrays:values, columns, and rowIndex. The following table describes these arrays.\nvalues\nA real array that contains the elements of the non-zero blocks of a sparse matrix.\nThe elements are stored block by block in row-major order. A non-zero block is\nthe block that contains at least one non-zero element. All elements of non-zero\nblocks are stored, even if some of them is equal to zero. Within each non-zero\nblock the elements are stored in column major order in the case of the one-\nbased indexing, and in row major order in the case of the zero-based indexing.\ncolumns\nElement i of the integer array columns is the number of the column in the block\nmatrix that contains the i-th non-zero block.\nrowIndex\nElement j of this integer array gives the index of the element in the columns\narray that is first non-zero block in a row j of the block matrix.\nThe length of the values array is equal to the number of all elements in the non-zero blocks, the length of the\ncolumns array is equal to the number of non-zero blocks.\nAs the rowIndex array gives the location of the first non-zero block within a row, and the non-zero blocks are\nstored consecutively, the number of non-zero blocks in the i-th row is equal to the difference of rowIndex[i]\nand rowIndex[i+1].\nTo retain this relationship for the last row of the block matrix, an additional entry (dummy entry) is added to\nthe end of rowIndex with value equal to the number of non-zero blocks plus one. This makes the total length\nof the rowIndex array one larger than the number of rows of the block matrix.\nThe above matrix D can be represented in this 3-array variation of the BSR format as follows:\none-based indexing\nvalues  =  [1 2 0 1 6 8 7 2 1 5 4 2 4 0 3 0 7 0 2 0]\n                columns  = [1   2   2   2   3]\n                rowIndex = [1   3   4   6]\nzero-based indexing\nvalues  =  [1 0 2 1 6 7 8 2 1 4 5 1 4 3 0 0 7 2 0 0]\n                columns  = [0   1   1   1   2]\n                rowIndex = [0   2   3 5]\nWhen storing symmetric matrices, it is necessary to store only the upper or the lower triangular part of the\nmatrix.\nFor example, consider the symmetric sparse matrix F:\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2635\n\n\nF =\n1 0 6 7 * *\n2 1 8 2 * *\n6 8 1 4 * *\n7 2 5 2 * *\n* * * * 7 2\n* * * * 0 0\nIf the size of the block equals 2, then the sparse matrix F can be represented as a 3x3 block matrix G with\nthe following structure:\nG =\nL M *\nM′ N *\n*\n* Q\nwhere\nL = 1 0\n2 1 , M = 6 7\n8 2 , M′ = 6 8\n7 2 , N = 1 4\n5 2 , and Q = 7 2\n0 0\nThe symmetric matrix F can be represented in this 3-array variation of the BSR format (storing only the\nupper triangular part) as follows:\none-based indexing\nvalues  =  [1 2 0 1 6 8 7 2 1 5 4 2 7 0 2 0]\n                columns  = [1   2   2   3]\n                rowIndex = [1   3   4 5]\nzero-based indexing\nvalues  =  [1 0 2 1 6 7 8 2 1 4 5 2 7 2 0 0]\n                columns  = [0   1   1   2]\n                rowIndex = [0   2   3 4]\nVariable BSR Format\nA variation of BSR3 is variable block compressed sparse row format. For a trust level t, 0 ≤t≤ 100, rows\nsimilar up to t percent are placed in one supernode.\nAppendix B: Routine and Function Arguments\nThe major arguments in the BLAS routines are vector and matrix, whereas VM functions work on vector\narguments only. The sections that follow discuss each of these arguments and provide examples.\nVector Arguments in BLAS\nVector arguments are passed in one-dimensional arrays. The array dimension (length) and vector increment\nare passed as integer variables. The length determines the number of elements in the vector. The increment\n(also called stride) determines the spacing between vector elements and the order of the elements in the\narray in which the vector is passed.\nA vector of length n and increment incx is passed in a one-dimensional array x whose values are defined as\nx[0], x[|incx|], ..., x[(n-1)* |incx|]\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2636\n\n\nIf incx is positive, then the elements in array x are stored in increasing order. If incx is negative, the\nelements in array x are stored in decreasing order with the first element defined as x[(n-1)* |incx|]. If\nincx is zero, then all elements of the vector have the same value, x[0]. The size of the one-dimensional\narray that stores the vector must always be at least\nidimx = 1 + (n-1)* |incx |\nExample. One-dimensional Real Array\nLet x[0:6] be the one-dimensional real array\nx = [1.0, 3.0, 5.0, 7.0, 9.0, 11.0, 13.0].\nIf incx =2 and n = 3, then the vector argument with elements in order from first to last is [1.0, 5.0,\n9.0].\nIf incx = -2 and n = 4, then the vector elements in order from first to last is [13.0, 9.0, 5.0, 1.0].\nIf incx = 0 and n = 4, then the vector elements in order from first to last is [1.0, 1.0, 1.0, 1.0].\nOne-dimensional substructures of a matrix, such as the rows, columns, and diagonals, can be passed as \nvector arguments with the starting address and increment specified.\nStorage of the m-by-n matrix can be based on either column-major ordering where the increment between\nelements in the same column is 1, the increment between elements in the same row is m, and the increment\nbetween elements on the same diagonal is m + 1; or row-major ordering where the increment between\nelements in the same row is 1, the increment between elements in the same column is n, and the increment\nbetween elements on the same diagonal is n + 1.\nExample. Two-dimensional Real Matrix\nLet a be a real 5 x 4 matrix declared as .\nTo scale the third column of a by 2.0, use the BLAS routine sscal with the following calling sequence:\ncblas_sscal (5, 2.0, a[2], 4)\nTo scale the second row, use the statement:\ncblas_sscal (4, 2.0, a[4], 1)\nTo scale the main diagonal of a by 2.0, use the statement:\ncblas_sscal (4, 2.0, a[0], 5)\nNOTE\nThe default vector argument is assumed to be 1.\nVector Arguments in Vector Math\nVector arguments of classic VM mathematical functions are passed in one-dimensional arrays with unit vector\nincrement. It means that a vector of length n is passed contiguously in an array a whose values are defined\nas\na[0], a[1], ..., a[n-1].\nStrided VM mathematical functions allow using positive increments for all input and output vector arguments.\nTo accommodate for arrays with other increments, or more complicated indexing, VM contains auxiliary Pack/\nUnpack functions that gather the array elements into a contiguous vector and then scatter them after the\ncomputation is complete.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2637\n\n\nGenerally, if the vector elements are stored in a one-dimensional array a as\na[m0], a[m1], ..., a[mn-1]\nand need to be regrouped into an array y as\ny[k0], y[k1], ..., y[kn-1],.\nVM Pack/Unpack functions can use one of the following indexing methods:\nPositive Increment Indexing\nkj = incy * j, mj = inca * j, j = 0 ,..., n-1.\nConstraint: incy > 0 and inca > 0.\nFor example, setting incy = 1 specifies gathering array elements into a contiguous vector.\nThis method is similar to that used in BLAS, with the exception that negative and zero increments are not\npermitted.\nIndex Vector Indexing\n.\nkj = iy[j], mj = ia[j], j = 0 ,..., n-1.\nwhere ia and iy are arrays of length n that contain index vectors for the input and output arrays a and y,\nrespectively.\nMask Vector Indexing\nIndices kj , mj are such that:\n.\nmy[kj] ≠ 0, ma[mj] ≠ 0 , j = 0,..., n-1.\nwhere ma and my are arrays that contain mask vectors for the input and output arrays a and y, respectively.\nVector Mathematical Functions\nMatrix Arguments\nMatrix arguments of the Intel® oneAPI Math Kernel Library routines can be stored in arrays, using the\nfollowing storage schemes:\n•\nconventional full storage\n•\npacked storage for Hermitian, symmetric, or triangular matrices\n•\nband storage for band matrices\n•\nrectangular full packed storage for symmetric, Hermitian, or triangular matrices as compact as the Packed\nstorage while maintaining efficiency by using Level 3 BLAS/LAPACK kernels.\nFull storage is the simplest scheme. . A matrix A is stored in a one-dimensional array a, with the matrix\nelement aij stored in the array element a[i - 1 + (j - 1)*lda], where lda is the leading dimension of\narray a.\nIf a matrix is triangular (upper or lower, as specified by the argument uplo), only the elements of the\nrelevant triangle are stored; the remaining elements of the array need not be set.\nRoutines that handle symmetric or Hermitian matrices allow for either the upper or lower triangle of the\nmatrix to be stored in the corresponding elements of the array:\nif uplo ='U',\naij is stored as described for i ≤ j, other elements of a need not be set.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2638\n\n\nif uplo ='L',\naij is stored as described for j ≤ i, other elements of a need not be set.\nPacked storage allows you to store symmetric, Hermitian, or triangular matrices more compactly: the\nrelevant triangle (again, as specified by the argument uplo) is packed by columns in a one-dimensional array\nap:\nif uplo ='U', aij is stored in ap[i - 1 +j(j - 1)/2] for i ≤ j\nif uplo ='L', aij is stored in ap[i - 1 + (2*n - j)*(j - 1)/2] for j ≤ i.\nIn descriptions of LAPACK routines, arrays with packed matrices have names ending in p.\nBand storage is as follows: an m-by-n band matrix with kl non-zero sub-diagonals and ku non-zero super-\ndiagonals is stored compactly in an array ab with (kl+ku + 1)*n elements. Thus,\naij is stored in ab(ku+1+i-j,j) for max(1,j-ku) ≤ i ≤ min(n,j+kl).\nUse the band storage scheme only when kl and ku are much less than the matrix size n. Although the\nroutines work correctly for all values of kl and ku, using the band storage is inefficient if your matrices are\nnot really banded.\nThe band storage scheme is illustrated by the following example, when\nm = n = 6, kl = 2, ku = 1\nArray elements marked * are not used by the routines:\nWhen a general band matrix is supplied for LU factorization, space must be allowed to store kl additional\nsuper-diagonals generated by fill-in as a result of row interchanges. This means that the matrix is stored\naccording to the above scheme, but with kl + ku super-diagonals. Thus,\naij is stored in ab(kl+ku+1+i-j,j) for max(1,j-ku) ≤ i ≤ min(n,j+kl).\nThe band storage scheme for LU factorization is illustrated by the following example, whenm = n = 6, kl =\n2, ku = 1:\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2639\n\n\nArray elements marked * are not used by the routines; elements marked + need not be set on entry, but are\nrequired by the LU factorization routines to store the results. The input array will be overwritten on exit by\nthe details of the LU factorization as follows:\nwhere uij are the elements of the upper triangular matrix U, and mij are the multipliers used during\nfactorization.\nTriangular band matrices are stored in the same format, with either kl= 0 if upper triangular, or ku = 0 if\nlower triangular. For symmetric or Hermitian band matrices with k sub-diagonals or super-diagonals, you\nneed to store only the upper or lower triangle, as specified by the argument uplo:\nif uplo ='U', aij is stored in ab(k+1+i-j,j) for max(1,j-k) ≤ i ≤ j\nif uplo ='L', aij is stored in ab(1+i-j,j) for j ≤ i ≤ min(n,j+k).\nIn descriptions of LAPACK routines, arrays that hold matrices in band storage have names ending in b.\nIn Fortran, column-major ordering of storage is assumed. This means that elements of the same column\noccupy successive storage locations.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2640\n\n\nThree quantities are usually associated with a two-dimensional array argument: its leading dimension, which\nspecifies the number of storage locations between elements in the same row, its number of rows, and its \nnumber of columns. For a matrix in full storage, the leading dimension of the array must be at least as large\nas the number of rows in the matrix.\nA character transposition parameter is often passed to indicate whether the matrix argument is to be used in\nnormal or transposed form or, for a complex matrix, if the conjugate transpose of the matrix is to be used.\nThe values of the transposition parameter for these three cases are the following:\n'N' or 'n'\nnormal (no conjugation, no transposition)\n'T' or 't'\ntranspose\n'C' or 'c'\nconjugate transpose.\nExample. Two-Dimensional Complex Array\nSuppose A (1:5, 1:4) is the complex two-dimensional array presented by matrix\nLet transa be the transposition parameter, m be the number of rows, n be the number of columns, and lda be\nthe leading dimension. Then if\ntransa = 'N', m = 4, n = 2, and lda = 5, the matrix argument would be\nIf transa = 'T', m = 4, n = 2, and lda =5, the matrix argument would be\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2641\n\n\nIf transa = 'C', m = 4, n = 2, and lda =5, the matrix argument would be\nNote that care should be taken when using a leading dimension value which is different from the number of\nrows specified in the declaration of the two-dimensional array. For example, suppose the array A above is\ndeclared as a complex 5-by-4 matrix.\nThen if transa = 'N', m = 3, n = 4, and lda = 4, the matrix argument will be\nRectangular Full Packed storage allows you to store symmetric, Hermitian, or triangular matrices as\ncompact as the Packed storage while maintaining efficiency by using Level 3 BLAS/LAPACK kernels. To store\nan n-by-n triangle (and suppose for simplicity that n is even), you partition the triangle into three parts: two\nn/2-by-n/2 triangles and an n/2-by-n/2 square, then pack this as an n-by-n/2 rectangle (or n/2-by-n\nrectangle), by transposing (or transpose-conjugating) one of the triangles and packing it next to the other\ntriangle. Since the two triangles are stored in full storage, you can use existing efficient routines on them.\nThere are eight cases of RFP storage representation: when n is even or odd, the packed matrix is transposed\nor not, the triangular matrix is lower or upper. See below for all the eight storage schemes illustrated:\nn is odd, A is lower triangular\nn is even, A is lower triangular\nn is odd, A is upper triangular\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2642\n\n\nn is even, A is upper triangular\nIntel® oneAPI Math Kernel Library (oneMKL) provides a number of routines such as?hfrk, ?sfrk performing\nBLAS operations working directly on RFP matrices, as well as some conversion routines, for instance, ?tpttf\ngoes from the standard packed format to RFP and ?trttf goes from the full format to RFP.\nPlease refer to the Netlib site for more information.\nNote that in the descriptions of LAPACK routines, arrays with RFP matrices have names ending in fp.\nAppendix C: FFTW Interface to Intel® Math Kernel Library\nIntel® oneAPI Math Kernel Library (oneMKL) offers FFTW2 and FFTW3 interfaces to Intel® oneAPI Math Kernel\nLibrary (oneMKL) Fast Fourier Transform and Trigonometric Transform functionality. The purpose of these\ninterfaces is to enable applications using FFTW (www.fftw.org) to gain performance with Intel® oneAPI Math\nKernel Library (oneMKL) without changing the program source code.\nBoth FFTW2 and FFTW3 interfaces are provided in open source as FFTW wrappers to Intel® oneAPI Math\nKernel Library (oneMKL). For ease of use, FFTW3 interface is also integrated in Intel® oneAPI Math Kernel\nLibrary (oneMKL).\nFFTW Notational Conventions\nThis appendix typically employs path notations for Windows* OS.\nFFTW2 Interface to Intel® oneAPI Math Kernel Library\nThis section describes a collection of C and Fortran wrappers providing FFTW 2.x interface to Intel® oneAPI\nMath Kernel Library (oneMKL). The wrappers translate calls to FFTW 2.x functions into the calls of the Intel®\noneAPI Math Kernel Library (oneMKL) Fast Fourier Transform interface (FFT interface).\nNote that Intel® oneAPI Math Kernel Library (oneMKL) FFT interface operates on both single- and double-\nprecision floating-point data types.\nBecause of differences between FFTW and Intel® oneAPI Math Kernel Library (oneMKL) FFT functionalities,\nthere are restrictions on using wrappers instead of the FFTW functions. Some FFTW functions have empty\nwrappers. However, many typical FFTs can be computed using these wrappers.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2643\n\n\nRefer to Fourier Transform Functions, for better understanding the effects from the use of the wrappers.\nWrappers Reference\nThe section provides a brief reference for the FFTW 2.x C interface. For details please refer to the original\nFFTW 2.x documentation available at www.fftw.org.\nEach FFTW function has its own wrapper. Some of them, which are not expressly listed in this section, are\nempty and do nothing, but they are provided to avoid link errors and satisfy the function calls.\nSee Also\nLimitations of the FFTW2 Interface to Intel® oneAPI Math Kernel Library (oneMKL)\nOne-dimensional Complex-to-complex FFTs\nThe following functions compute a one-dimensional complex-to-complex Fast Fourier transform.\nfftw_plan fftw_create_plan(int n, fftw_direction dir, int flags);\nfftw_plan fftw_create_plan_specific(int n, fftw_direction dir, int flags, fftw_complex\n*in, int istride, fftw_complex *out, int ostride);\nvoid fftw(fftw_plan plan, int howmany, fftw_complex *in, int istride, int idist,\nfftw_complex *out, int ostride, int odist);\nvoid fftw_one(fftw_plan plan, fftw_complex *in , fftw_complex *out);\nvoid fftw_destroy_plan(fftw_plan plan);\nMulti-dimensional Complex-to-complex FFTs\nThe following functions compute a multi-dimensional complex-to-complex Fast Fourier transform.\nfftwnd_plan fftwnd_create_plan(int rank, const int *n, fftw_direction dir, int flags);\nfftwnd_plan fftw2d_create_plan(int nx, int ny, fftw_direction dir, int flags);\nfftwnd_plan fftw3d_create_plan(int nx, int ny, int nz, fftw_direction dir, int flags);\nfftwnd_plan fftwnd_create_plan_specific(int rank, const int *n, fftw_direction dir, int\nflags, fftw_complex *in, int istride, fftw_complex *out, int ostride);\nfftwnd_plan fftw2d_create_plan_specific(int nx, int ny, fftw_direction dir, int flags,\nfftw_complex *in, int istride, fftw_complex *out, int ostride);\nfftwnd_plan fftw3d_create_plan_specific(int nx, int ny, int nz, fftw_direction dir, int\nflags, fftw_complex *in, int istride, fftw_complex *out, int ostride);\nvoid fftwnd(fftwnd_plan plan, int howmany, fftw_complex *in, int istride, int idist,\nfftw_complex *out, int ostride, int odist);\nvoid fftwnd_one(fftwnd_plan plan, fftw_complex *in, fftw_complex *out);\nvoid fftwnd_destroy_plan(fftwnd_plan plan);\nOne-dimensional Real-to-half-complex/Half-complex-to-real FFTs\nHalf-complex representation of a conjugate-even symmetric vector of size N in a real array of the same size\nN consists of N/2+1 real parts of the elements of the vector followed by non-zero imaginary parts in the\nreverse order. Because the Intel® oneAPI Math Kernel Library (oneMKL) FFT interface does not currently\nsupport this representation, all wrappers of this kind are empty and do nothing.\nNevertheless, you can perform one-dimensional real-to-complex and complex-to-real transforms using\nrfftwnd functions with rank=1.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2644\n\n\nSee Also\nMulti-dimensional Real-to-complex/complex-to-real FFTs\nMulti-dimensional Real-to-complex/Complex-to-real FFTs\nThe following functions compute multi-dimensional real-to-complex and complex-to-real Fast Fourier\ntransforms.\nrfftwnd_plan rfftwnd_create_plan(int rank, const int *n, fftw_direction dir, int\nflags);\nrfftwnd_plan rfftw2d_create_plan(int nx, int ny, fftw_direction dir, int flags);\nrfftwnd_plan rfftw3d_create_plan(int nx, int ny, int nz, fftw_direction dir, int\nflags);\nrfftwnd_plan rfftwnd_create_plan_specific(int rank, const int *n, fftw_direction dir,\nint flags, fftw_real *in, int istride, fftw_real *out, int ostride);\nrfftwnd_plan rfftw2d_create_plan_specific(int nx, int ny, fftw_direction dir, int\nflags, fftw_real *in, int istride, fftw_real *out, int ostride);\nrfftwnd_plan rfftw3d_create_plan_specific(int nx, int ny, int nz, fftw_direction dir,\nint flags, fftw_real *in, int istride, fftw_real *out, int ostride);\nvoid rfftwnd_real_to_complex(rfftwnd_plan plan, int howmany, fftw_real *in, int\nistride, int idist, fftw_complex *out, int ostride, int odist);\nvoid rfftwnd_complex_to_real(rfftwnd_plan plan, int howmany, fftw_complex *in, int\nistride, int idist, fftw_real *out, int ostride, int odist);\nvoid rfftwnd_one_real_to_complex(rfftwnd_plan plan, fftw_real *in, fftw_complex *out);\nvoid rfftwnd_one_complex_to_real(rfftwnd_plan plan, fftw_complex *in, fftw_real *out);\nvoid rfftwnd_destroy_plan(rfftwnd_plan plan);\nMulti-threaded FFTW\nThis section discusses multi-threaded FFTW wrappers only. MPI FFTW wrappers, available only with Intel®\noneAPI Math Kernel Library (oneMKL) for the Linux* and Windows* operating systems, are described in a\nseparate section.\nUnlike the original FFTW interface, every computational function in the FFTW2 interface to Intel® oneAPI Math\nKernel Library (oneMKL) provides multithreaded computation by default, with the maximum number of\nthreads permitted in FFT functions (see \"Techniques to Set the Number of Threads\" in Intel® oneAPI Math\nKernel Library (oneMKL) Developer Guide). To limit the number of threads, call the threaded FFTW\ncomputational functions:\nvoid fftw_threads(int nthreads, fftw_plan plan, int howmany, fftw_complex *in, int\nistride, int idist, fftw_complex *out, int ostride, int odist);\nvoid fftw_threads_one(int nthreads, rfftwnd_plan plan, fftw_complex *in, fftw_complex\n*out);\n...\nvoid rfftwnd_threads_real_to_complex( int nthreads, rfftwnd_plan plan, int howmany,\nfftw_real *in, int istride, int idist, fftw_complex *out, int ostride, int odist);\nCompared to its non-threaded counterpart, every threaded computational function has threads_ as the\nsecond part of its name and additional first parameter nthreads. Set the nthreads parameter to the thread\nlimit to ensure that the computation requires at most that number of threads.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2645\n\n\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nFFTW Support Functions\nThe FFTW wrappers provide memory allocation functions to be used with FFTW:\nvoid* fftw_malloc(size_t n);\nvoid fftw_free(void* x);\nThe fftw_malloc wrapper aligns the memory on a 16-byte boundary.\nIf fftw_malloc fails to allocate memory, it aborts the application. To override this behavior, set a global\nvariable fftw_malloc_hook and optionally the complementary variable fftw_free_hook:\nvoid *(*fftw_malloc_hook) (size_t n);\nvoid (*fftw_free_hook) (void *p);\nThe wrappers use the function fftw_die to abort the application in cases when a caller cannot be informed\nof an error otherwise (for example, in computational functions that return void). To override this behavior,\nset a global variable fftw_die_hook:\nvoid (*fftw_die_hook)(const char *error_string);\nvoid fftw_die(const char *s);\nLimitations of the FFTW2 Interface to Intel® oneAPI Math Kernel Library (oneMKL)\nThe FFTW2 wrappers implement the functionality of only those FFTW functions that Intel® oneAPI Math\nKernel Library (oneMKL) can reasonably support. Other functions are provided as no-operation functions,\nwhose only purpose is to satisfy link-time symbol resolution. Specifically, no-operation functions include:\n•\nReal-to-half-complex and respective backward transforms\n•\nPrint plan functions\n•\nFunctions for importing/exporting/forgetting wisdom\n•\nMost of the FFTW functions not covered by the original FFTW2 documentation\nBecause the Intel® oneAPI Math Kernel Library (oneMKL) implementation of FFTW2 wrappers does not use\nplan and plan node structures declared in fftw.h, the behavior of an application that relies on the internals\nof the plan structures defined in that header file is undefined.\nFFTW2 wrappers define plan as a set of attributes, such as strides, used to commit the Intel® oneAPI Math\nKernel Library (oneMKL) FFT descriptor structure. If an FFTW2 computational function is called with attributes\ndifferent from those recorded in the plan, the function attempts to adjust the attributes of the plan and\nrecommit the descriptor. So, repeated calls of a computational function with the same plan but different\nstrides, distances, and other parameters may be performance inefficient.\nPlan creation functions disregard most planner flags passed through the flags parameter. These functions\ntake into account only the following values of flags:\n•\nFFTW_IN_PLACE\nIf this value of flags is supplied, the plan is marked so that computational functions using that plan\nignore the parameters related to output (out, ostride, and odist). Unlike the original FFTW interface,\nthe wrappers never use the out parameter as a scratch space for in-place transforms.\n•\nFFTW_THREADSAFE\nIf this value of flags is supplied, the plan is marked read-only. An attempt to change attributes of a\nread-only plan aborts the application.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2646\n\n\nFFTW wrappers are generally not thread safe. Therefore, do not use the same plan in parallel user threads\nsimultaneously.\nInstalling FFTW2 Interface Wrappers\nWrappers are delivered as source code, which you must compile to build the wrapper library. Then you can\nsubstitute the wrapper and Intel® oneAPI Math Kernel Library (oneMKL) libraries for the FFTW library. The\nsource code for the wrappers, makefiles, and files with lists of wrappers are located in the .\\interfaces\n\\fftw2xcsubdirectory in the Intel® oneAPI Math Kernel Library (oneMKL) directory.\nCreating the Wrapper Library\nTwo header files are used to compile the C wrapper library: fftw2_mkl.h and fftw.h. The fftw2_mkl.h file\nis located in the .\\interfaces\\fftw2xc\\wrappers subdirectory in the Intel® oneAPI Math Kernel Library\n(oneMKL) directory.\nThe file fftw.h, used to compile libraries and located in the .\\include\\fftwsubdirectory in the Intel®\noneAPI Math Kernel Library (oneMKL) directory, slightly differs from the original FFTW (www.fftw.org) header\nfilefftw.h.\nThe source code for the wrappers, makefiles, and files with lists of functions are located in\nthe .\\interfaces\\fftw2xc subdirectory in the Intel® oneAPI Math Kernel Library (oneMKL) directory.\nA wrapper library contains wrappers for complex and real transforms in a serial and multi-threaded mode for\ndouble- or single-precision floating-point data types. A makefile parameter manages the data type.\nParameters of a makefile also specify the platform (required), compiler, and data precision. The makefile\ncomment heading provides the exact description of these parameters.\nTo build the library, run the make command on Linux* OS and macOS* or the nmake command on Windows*\nOS with appropriate parameters.\nFor example, on Linux OS the command\nmake libintel64\nbuilds a double-precision wrapper library for Intel® 64 architecture based applications using the Intel® oneAPI\nDPC++/C++ Compiler or the Intel® Fortran Compiler (Compilers and data precision are chosen by default.)\nEach makefile creates the library in the directory with Intel® oneAPI Math Kernel Library (oneMKL) libraries\ncorresponding to the platform used. For example,./lib/ia32 (on Linux OS and macOS) or .\\lib\\ia32 (on\nWindows* OS).\nIn the names of a wrapper library, the suffix corresponds to the compiler used and the letter preceding the\nunderscore is \"c\" for the C programming language.\nFor example,\nfftw2xc_intel.lib (on Windows OS); libfftw2xc_intel.a (on Linux OS and macOS);\nfftw2xc_ms.lib (on Windows OS); libfftw2xc_gnu.a (on Linux OS and macOS).\nApplication Assembling\nUse the necessary original FFTW (www.fftw.org) header files without any modifications. Use the created\nwrapper library and the Intel® oneAPI Math Kernel Library (oneMKL) library instead of the FFTW library.\nRunning FFTW2 Interface Wrapper Examples\nIntel® oneAPI Math Kernel Library (oneMKL) provides examples to demonstrate how to use the MPI FFTW\nwrapper library. The source code for the examples, makefiles used to run them, and files with lists of\nexamples are located in the .\\examples\\fftw2xc subdirectory in the Intel® oneAPI Math Kernel Library\n(oneMKL) directory . To build examples, several additional files are needed: fftw.h, fftw_threads.h,\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2647\n\n\nrfftw.h, and rfftw_threads.h. These files are distributed with permission from FFTW and are available\nin .\\include\\fftw. The original files can also be found in FFTW 2.1.5 at http://www.fftw.org/\ndownload.html.\nAn example makefile uses the function parameter in addition to the parameters of the corresponding\nwrapper library makefile (see Creating a Wrapper Library). The makefile comment heading provides the exact\ndescription of these parameters.\nAn example makefile normally invokes examples. However, if the appropriate wrapper library is not yet\ncreated, the makefile first builds the library the same way as the wrapper library makefile does and then\nproceeds to examples.\nIf the parameter function=<example_name> is defined, only the specified example runs. Otherwise, all\nexamples from the appropriate subdirectory run. The subdirectory .\\_results is created, and the results\nare stored there in the <example_name>.res files.\nProduct and Performance Information\nPerformance varies by use, configuration and other factors. Learn more at www.Intel.com/\nPerformanceIndex.\nNotice revision #20201201\nMPI FFTW2 Wrappers\nMPI FFTW wrappers for FFTW 2 are available only with Intel® oneAPI Math Kernel Library (oneMKL) for the\nLinux* and Windows* operating systems.\nMPI FFTW Wrappers Reference\nThe section provides a reference for MPI FFTW C interface.\nComplex MPI FFTW\nComplex One-dimensional MPI FFTW Transforms\nfftw_mpi_plan fftw_mpi_create_plan(MPI_Comm comm, int n, fftw_direction dir, int\nflags);\nvoid fftw_mpi(fftw_mpi_plan p, int n_fields, fftw_complex *local_data, fftw_complex\n*work);\nvoid fftw_mpi_local_sizes(fftw_mpi_plan p, int *local_n, int *local_start, int\n*local_n_after_transform, int *local_start_after_transform, int *total_local_size);\nvoid fftw_mpi_destroy_plan(fftw_mpi_plan plan);\nArgument restrictions:\n•\nSupported values of flags are FFTW_ESTIMATE, FFTW_MEASURE, FFTW_SCRAMBLED_INPUT and\nFFTW_SCRAMBLED_OUTPUT. The same algorithm corresponds to all these values of the flags parameter. If\nany other flags value is supplied, the wrapper library reports an error 'CDFT error in wrapper: unknown\nflags'.\n•\nThe only supported value of n_fields is 1.\nComplex Multi-dimensional MPI FFTW Transforms\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2648\n\n\nfftwnd_mpi_plan fftw2d_mpi_create_plan(MPI_Comm comm, int nx, int ny, fftw_direction\ndir, int flags);\nfftwnd_mpi_plan fftw3d_mpi_create_plan(MPI_Comm comm, int nx, int ny, int nz,\nfftw_direction dir, int flags);\nfftwnd_mpi_plan fftwnd_mpi_create_plan(MPI_Comm comm, int dim, int *n, fftw_direction\ndir, int flags);\nvoid fftwnd_mpi(fftwnd_mpi_plan p, int n_fields, fftw_complex *local_data, fftw_complex\n*work, fftwnd_mpi_output_order output_order);\nvoid fftwnd_mpi_local_sizes(fftwnd_mpi_plan p, int *local_nx, int *local_x_start, int\n*local_ny_after_transpose, int *local_y_start_after_transpose, int *total_local_size);\nvoid fftwnd_mpi_destroy_plan(fftwnd_mpi_plan plan);\nArgument restrictions:\n•\nSupported values of flags are FFTW_ESTIMATE and FFTW_MEASURE. If any other value of flags is\nsupplied, the wrapper library reports an error 'CDFT error in wrapper: unknown flags'.\n•\nThe only supported value of n_fields is 1.\nReal MPI FFTW\nReal-to-Complex MPI FFTW Transforms\nrfftwnd_mpi_plan rfftw2d_mpi_create_plan(MPI_Comm comm, int nx, int ny, fftw_direction\ndir, int flags);\nrfftwnd_mpi_plan rfftw3d_mpi_create_plan(MPI_Comm comm, int nx, int ny, int nz,\nfftw_direction dir, int flags);\nrfftwnd_mpi_plan rfftwnd_mpi_create_plan(MPI_Comm comm, int dim, int *n, fftw_direction\ndir, int flags);\nvoid rfftwnd_mpi(rfftwnd_mpi_plan p, int n_fields, fftw_real *local_data, fftw_real\n*work, fftwnd_mpi_output_order output_order);\nvoid rfftwnd_mpi_local_sizes(rfftwnd_mpi_plan p, int *local_nx, int *local_x_start, int\n*local_ny_after_transpose, int *local_y_start_after_transpose, int *total_local_size);\nvoid rfftwnd_mpi_destroy_plan(rfftwnd_mpi_plan plan);\nArgument restrictions:\n•\nSupported values of flags are FFTW_ESTIMATE and FFTW_MEASURE. If any other value of flags is\nsupplied, the wrapper library reports an error 'CDFT error in wrapper: unknown flags'.\n•\nThe only supported value of n_fields is 1.\nNOTE\n•\nFunction rfftwnd_mpi_create_plan can be used for both one-dimensional and multi-dimensional\ntransforms.\n•\nBoth values of the output_order parameter are supported: FFTW_NORMAL_ORDER and\nFFTW_TRANSPOSED_ORDER.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2649\n\n\nCreating MPI FFTW2 Wrapper Library\nThe source code for the wrappers, makefiles, and files with lists of wrappers are located in\nthe .\\interfaces\\fftw2x_cdft subdirectory in the Intel® oneAPI Math Kernel Library (oneMKL) directory.\nA wrapper library contains C wrappers for Complex One-dimensional MPI FFTW Transforms and Complex\nMulti-dimensional MPI FFTW Transforms. The library also contains empty C wrappers for Real Multi-\ndimensional MPI FFTW Transforms. For details, see MPI FFTW Wrappers Reference.\nParameters of a makefile specify the platform (required), compiler, and data precision. Specifying the\nplatform is required. The makefile comment heading provides the exact description of these parameters.\nTo build the library, run the make command on Linux* OS and macOS* or the nmake command on Windows*\nOS with appropriate parameters.\nFor example, on Linux OS the command\nmake libintel64\nbuilds a double-precision wrapper library for Intel® 64 architecture based applications using Intel MPI and the\nIntel® oneAPI DPC++/C++ Compiler (compilers and data precision are chosen by default.).\nA makefile creates the wrapper library in the directory with the Intel® oneAPI Math Kernel Library (oneMKL)\nlibraries corresponding to the used platform. For example,./lib/ia32 (on Linux OS) or .\\lib\\ia32 (on\nWindows* OS).\nIn the wrapper library names, the suffix corresponds to the used data precision. For example,\nfftw2x_cdft_SINGLE.lib on Windows OS;\nlibfftw2x_cdft_DOUBLE.a on Linux OS.\nApplication Assembling with MPI FFTW Wrapper Library\nUse the necessary original FFTW (www.fftw.org) header files without any modifications. Use the created MPI\nFFTW wrapper library and the Intel® oneAPI Math Kernel Library (oneMKL) library instead of the FFTW library.\nRunning MPI FFTW2 Wrapper Examples\nThere are some examples that demonstrate how to use the MPI FFTW wrapper library for FFTW2. The source\nC code for the examples, makefiles used to run them, and files with lists of examples are located in\nthe .\\examples\\fftw2x_cdft subdirectory in the Intel® oneAPI Math Kernel Library (oneMKL) directory. To\nbuild examples, one additional file, fftw_mpi.h, is needed. This file is distributed with permission from FFTW\nand is available in .\\include\\fftw. The original file can also be found in FFTW 2.1.5 at http://\nwww.fftw.org/download.html.\nParameters for the example makefiles are described in the makefile comment headings and are similar to the\nparameters of the wrapper library makefiles (see Creating MPI FFTW Wrapper Library).\nThe table below lists examples available in the .\\examples\\fftw2x_cdft\\source subdirectory.\nExamples of MPI FFTW Wrappers\nSource file for the example\nDescription\nwrappers_c1d.c\nOne-dimensional Complex MPI FFTW transform,\nusing plan = fftw_mpi_create_plan(...)\nwrappers_c2d.c\nTwo-dimensional Complex MPI FFTW transform,\nusing plan = fftw2d_mpi_create_plan(...)\nwrappers_c3d.c\nThree-dimensional Complex MPI FFTW transform,\nusing plan = fftw3d_mpi_create_plan(...)\nwrappers_c4d.c\nFour-dimensional Complex MPI FFTW transform,\nusing plan = fftwnd_mpi_create_plan(...)\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2650\n\n\nSource file for the example\nDescription\nwrappers_r1d.c\nOne-dimensional Real MPI FFTW transform, using\nplan = rfftw_mpi_create_plan(...)\nwrappers_r2d.c\nTwo-dimensional Real MPI FFTW transform, using\nplan = rfftw2d_mpi_create_plan(...)\nwrappers_r3d.c\nThree-dimensional Real MPI FFTW transform, using\nplan = rfftw3d_mpi_create_plan(...)\nwrappers_r4d.c\nFour-dimensional Real MPI FFTW transform, using\nplan = rfftwnd_mpi_create_plan(...)\nFFTW3 Interface to Intel® oneAPI Math Kernel Library\nThis section describes a collection of FFTW3 wrappers to Intel® oneAPI Math Kernel Library (oneMKL). The\nwrappers translate calls of FFTW3 functions to the calls of the Intel® oneAPI Math Kernel Library (oneMKL)\nFourier transform (FFT) or Trigonometric Transform (TT) functions. The purpose of FFTW3 wrappers is to\nenable developers whose programs currently use the FFTW3 library to gain performance with the Intel®\noneAPI Math Kernel Library (oneMKL) Fourier transforms without changing the program source code.\nThe FFTW3 wrappers provide a limited functionality compared to the original FFTW 3.x library, because of\ndifferences between FFTW and Intel® oneAPI Math Kernel Library (oneMKL) FFT and TT functionality. This\nsection describes limitations of the FFTW3 wrappers and hints for their usage. Nevertheless, many typical FFT\ntasks can be performed using the FFTW3 wrappers to Intel® oneAPI Math Kernel Library (oneMKL).\nThe FFTW3 wrappers are integrated in Intel® oneAPI Math Kernel Library (oneMKL). The only change required\nto use Intel® oneAPI Math Kernel Library (oneMKL) through the FFTW3 wrappers is to link your application\nusing FFTW3 against Intel® oneAPI Math Kernel Library (oneMKL).\nA reference implementation of the FFTW3 wrappers is also provided in open source. You can find it in the\ninterfaces directory of the Intel® oneAPI Math Kernel Library (oneMKL) distribution. You can use the\nreference implementation to create your own wrapper library (see Building Your Own Wrapper Library)\nSee also these resources:\nIntel® oneAPI Math Kernel Library\n(oneMKL) Release Notes\nfor the version of the FFTW3 library supported by the wrappers.\nwww.fftw.org\nfor a description of the FFTW interface.\nFourier Transform Functions\nfor a description of the Intel® oneAPI Math Kernel Library (oneMKL)\nFFT interface.\nTrigonometric Transform Routines\nfor a description of Intel® oneAPI Math Kernel Library (oneMKL) TT\ninterface.\nUsing FFTW3 Wrappers\nThe FFTW3 wrappers are a set of functions and data structures depending on one another. The wrappers are\nnot designed to provide the interface on a function-per-function basis. Some FFTW3 wrapper functions are\nempty and do nothing, but they are present to avoid link errors and satisfy function calls.\nThis document does not list the declarations of the functions that the FFTW3 wrappers provide (you can find\nthe declarations in the fftw3.h header file). Instead, this section comments on particular limitations of the\nwrappers and provides usage hints:. These are some known limitations of FFTW3 wrappers and their usage\nin Intel® oneAPI Math Kernel Library (oneMKL).\n•\nThe FFTW3 wrappers do not support long double precision because Intel® oneAPI Math Kernel Library\n(oneMKL) FFT functions operate only on single- and double-precision floating-point data types (float and\ndouble, respectively). Therefore the functions with prefix fftwl_, supporting the long double data\ntype, are not provided.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2651\n\n\n•\nThe wrappers provide equivalent implementation for double- and single-precision functions (those with\nprefixes fftw_ and fftwf_, respectively). So, all these comments equally apply to the double- and\nsingle-precision functions and will refer to functions with prefix fftw_, that is, double-precision functions,\nfor brevity.\n•\nThe FFTW3 interface that the wrappers provide is defined in the fftw3.h header file. This file is borrowed\nfrom the FFTW3.x package and distributed within Intel® oneAPI Math Kernel Library (oneMKL) with\npermission. Additionally, the fftw3_mkl.h header file defines supporting structures and supplementary\nconstants and macros.\n•\nActual functionality of the plan creation wrappers is implemented in guru64 set of functions. Basic\ninterface, advanced interface, and guru interface plan creation functions call the guru64 interface\nfunctions. So, all types of the FFTW3 plan creation interface in the wrappers are functional.\n•\nPlan creation functions may return a NULL plan, indicating that the functionality is not supported. So,\nplease carefully check the result returned by plan creation functions in your application. In particular, the\nfollowing problems return a NULL plan:\n–\nc2r and r2c problems with a split storage of complex data.\n–\nr2r problems with kind values FFTW_R2HC, FFTW_HC2R, and FFTW_DHT. The only supported r2r kinds\nare even/odd DFTs (sine/cosine transforms).\n–\nMultidimensional r2r transforms.\n–\nTransforms of multidimensional vectors. That is, the only supported values for parameter\nhowmany_rank in guru and guru64 plan creation functions are 0 and 1.\n–\nMultidimensional transforms with rank > MKL_MAXRANK.\n•\nThe MKL_RODFT00 value of the kind parameter is introduced by the FFTW3 wrappers. For better\nperformance, you are strongly encouraged to use this value rather than FFTW_RODFT00. To use this kind\nvalue, provide an extra first element equal to 0.0 for the input/output vectors. Consider the following\nexample:\nplan1 = fftw_plan_r2r_1d(n, in1, out1, FFTW_RODFT00, FFTW_ESTIMATE);\nplan2 = fftw_plan_r2r_1d(n, in2, out2, MKL_RODFT00, FFTW_ESTIMATE);\n \nBoth plans perform the same transform, except that the in2/out2 arrays have one extra zero element at\nlocation 0. For example, if n=3, in1={x,y,z} and out1={u,v,w}, then in2={0,x,y,z} and\nout2={0,u,v,w}.\n•\nThe flags parameter in plan creation functions is always ignored. The same algorithm is used regardless\nof the value of this parameter. In particular, flags values FFTW_ESTIMATE, FFTW_MEASURE, etc. have no\neffect.\n•\nFor multithreaded plans, use normal sequence of calls to the fftw_init_threads() and\nfftw_plan_with_nthreads() functions (refer to FFTW documentation).\n•\nMemory allocation function fftw_malloc returns memory aligned at a 16-byte boundary. You must free\nthe memory with fftw_free.\n•\nThe FFTW3 wrappers to Intel® oneAPI Math Kernel Library (oneMKL) use the 32-bit int type in both LP64\nand ILP64 interfaces of Intel® oneAPI Math Kernel Library (oneMKL). Use guru64 FFTW3 interfaces for 64-\nbit sizes.\n•\nThe wrappers typically indicate a problem by returning a NULL plan. In a few cases, the wrappers may\nreport a descriptive message of the problem detected. By default the reporting is turned off. To turn it on,\nset variable fftw3_mkl.verbose to a non-zero value, for example:\n#include \"fftw3.h\"\n#include \"fftw3_mkl.h\"\nfftw3_mkl.verbose = 0;\nplan = fftw_plan_r2r(...);\n \n•\nThe following functions are empty:\n–\nFor saving, loading, and printing plans\n–\nFor saving and loading wisdom\n–\nFor estimating arithmetic cost of the transforms.\n•\nDo not use macro FFTW_DLL with the FFTW3 wrappers to Intel® oneAPI Math Kernel Library (oneMKL).\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2652\n\n\n•\nDo not use negative stride values. Though FFTW3 wrappers support negative strides in the part of\nadvanced and guru FFTW interface, the underlying implementation does not.\n•\nDo not set a FFTW2 wrapper library before a FFTW3 wrapper library or Intel® oneAPI Math Kernel Library\n(oneMKL) in your link line application. All libraries define \"fftw_destroy_plan\" symbol and linkage in\nincorrect order results into expected errors.\nBuilding Your Own FFTW3 Interface Wrapper Library\nThe FFTW3 wrappers to Intel® oneAPI Math Kernel Library (oneMKL) are delivered both integrated in Intel®\noneAPI Math Kernel Library (oneMKL) and as source code, which can be compiled to build a standalone\nwrapper library with exactly the same functionality. Normally you do not need to build the wrappers yourself.\nThe source code for the wrappers, makefiles, and files with lists of functions are located in\nthe .\\interfaces\\fftw3xc subdirectory in the Intel® oneAPI Math Kernel Library (oneMKL) directory.\nTo build the wrappers,\n1.\nChange the current directory to the wrapper directory\n2.\nRun the make command on Linux* OS and macOS* or the nmake command on Windows* OS with a\nrequired target and optionally several parameters.\nThe target libia32 or libintel64 defines the platform architecture, and the other parameters specify the\ncompiler, size of the default integer type, and placement of the resulting wrapper library. You can find a\ndetailed and up-to-date description of the parameters in the makefile.\nIn the following example, the make command is used to build the FFTW3 C wrappers to Intel® oneAPI Math\nKernel Library (oneMKL) for use from the GNU gcc* compiler on Linux OS based on Intel® 64 architecture:\ncd interfaces/fftw3xc\nmake libintel64 compiler=gnu INSTALL_DIR=/my/path\n \nThis command builds the wrapper library and places the result, named libfftw3xc_gnu.a, into the /my/\npath directory. The name of the resulting library is composed of the name of the compiler used and may be\nchanged by an optional parameter INSTALL_LIBNAME.\nBuilding an Application With FFTW3 Interface Wrappers\nNormally, the only change needed to build your application with FFTW3 wrappers replacing original FFTW\nlibrary is to add Intel® oneAPI Math Kernel Library (oneMKL) at the link stage (see section\"Linking Your\nApplication with Intel® oneAPI Math Kernel Library\" in the Intel® oneAPI Math Kernel Library (oneMKL)\nDeveloper Guide).\nIf you recompile your application, add subdirectory include\\fftw to the search path for header files to\navoid FFTW3 version conflicts.\nSometimes, you may have to modify your application according to the following recommendations:\n•\nThe application requires\n#include \"fftw3.h\" ,\nwhich it probably already includes.\n•\nThe application does not require\n#include \"mkl_dfti.h\" .\n•\nThe application does not require\n#include \"fftw3_mkl.h\" .\nIt is required only in case you want to use the MKL_RODFT00 constant.\n•\nIf the application does not check whether a NULL plan is returned by plan creation functions, this check\nmust be added, because the FFTW3 to Intel® oneAPI Math Kernel Library (oneMKL) wrappers do not\nprovide 100% of FFTW3 functionality.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2653\n\n\nRunning FFTW3 Interface Wrapper Examples\nThere are some examples that demonstrate how to use the wrapper library. The source code for the\nexamples, makefiles used to run them, and files with lists of examples are located in the .\\examples\n\\fftw3xc subdirectory in the Intel® oneAPI Math Kernel Library (oneMKL) directory.\nParameters of the example makefiles are similar to the parameters of the wrapper library makefiles. Example\nmakefiles normally build and invoke the examples. If the parameter function=<example_name> is defined,\nthen only the specified example will run. Otherwise, all examples will be executed. Results of running the\nexamples are saved in subdirectory .\\_results in files with extension .res.\nFor detailed information about options for the example makefile, refer to the makefile.\nMPI FFTW3 Wrappers\nThis section describes a collection of MPI FFTW wrappers to Intel® oneAPI Math Kernel Library (oneMKL).\nMPI FFTW wrappers are available only with Intel® oneAPI Math Kernel Library (oneMKL) for the Linux* and\nWindows* operating systems.\nThese wrappers translate calls of MPI FFTW functions to the calls of the Intel® oneAPI Math Kernel Library\n(oneMKL) cluster Fourier transform (CFFT) functions. The purpose of the wrappers is to enable users of MPI\nFFTW functions improve performance of the applications without changing the program source code.\nAlthough the MPI FFTW wrappers provide less functionality than the original FFTW3 because of differences\nbetween MPI FFTW and Intel® oneAPI Math Kernel Library (oneMKL) CFFT, the wrappers cover many typical\nCFFT use cases.\nThe MPI FFTW wrappers are provided as source code. To use the wrappers, you need to build your own\nwrapper library (see Building Your Own Wrapper Library).\nSee also these resources:\nIntel® oneAPI Math Kernel Library\n(oneMKL) Release Notes\nfor the version of the FFTW3 library supported by the wrappers.\nwww.fftw.org\nfor a description of the MPI FFTW interface.\nCluster FFT Functions\nfor a description of the Intel® oneAPI Math Kernel Library (oneMKL)\nCFFT interface.\nBuilding Your Own Wrapper Library\nThe MPI FFTW wrappers for FFTW3 are delivered as source code, which can be compiled to build a wrapper\nlibrary.\nThe source code for the wrappers, makefiles, and files with lists of functions are located in\nsubdirectory .\\interfaces\\fftw3x_cdft in the Intel® oneAPI Math Kernel Library (oneMKL) directory.\nTo build the wrappers,\n1.\nChange the current directory to the wrapper directory\n2.\nRun the make command on Linux* OS or the nmake command on Windows* OS with a required target\nand optionally several parameters.\nThe target libia32 or libintel64 defines the platform architecture, and the other parameters specify the\ncompiler, size of the default INTEGER type, as well as the name and placement of the resulting wrapper\nlibrary. You can find a detailed and up-to-date description of the parameters in the makefile.\nIn the following example, the make command is used to build the MPI FFTW wrappers to Intel® oneAPI Math\nKernel Library (oneMKL) for use from the GNU C compiler on Linux OS based on Intel® 64 architecture:\ncd interfaces/fftw3x_cdft\nmake libintel64 compiler=gnu mpi=openmpi INSTALL_DIR=/my/path\n \n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2654\n\n\nThis command builds the wrapper library using the GNU gcc compiler so that the final executable can use\nOpen MPI, and places the result, named libfftw3x_cdft_DOUBLE.a, into directory /my/path.\nBuilding an Application\nNormally, the only change needed to build your application with MPI FFTW wrappers replacing original FFTW3\nlibrary is to add Intel® oneAPI Math Kernel Library (oneMKL) and the wrapper library at the link stage (see\nsection \"Linking Your Application with Intel® oneAPI Math Kernel Library\" in the Intel® oneAPI Math Kernel\nLibrary (oneMKL) Developer Guide).\nWhen you are recompiling your application, add subdirectory include\\fftw to the search path for header\nfiles to avoid FFTW3 version conflicts.\nRunning Examples\nThere are some examples that demonstrate how to use the MPI FFTW wrapper library for FFTW3. The source\ncode for the examples, makefiles used to run them, and files with lists of examples are located in\nthe .\\examples\\fftw3x_cdft subdirectory in the Intel® oneAPI Math Kernel Library (oneMKL) directory.\nParameters of the example makefiles are similar to the parameters of the wrapper library makefiles. Example\nmakefiles normally build and invoke the examples. Results of running the examples are saved in\nsubdirectory .\\_results in files with extension .res.\nFor detailed information about options for the example makefile, refer to the makefile.\nSee Also\nBuilding Your Own Wrapper Library \nAppendix D: Code Examples\nThis appendix presents code examples of using some Intel® oneAPI Math Kernel Library (oneMKL) routines\nand functions.\nPlease refer to respective sections in the document for detailed descriptions of function parameters and\noperation.\nBLAS Code Examples\nExample. Using BLAS Level 1 Function\nThe following example illustrates a call to the BLAS Level 1 function sdot. This function performs a vector-\nvector operation of computing a scalar product of two single-precision real vectors x and y.\nParameters\n \nn\nSpecifies the number of elements in vectors x and y.\nincx\nSpecifies the increment for the elements of x.\nincy\nSpecifies the increment for the elements of y.\n#include <stdio.h>\n#include <stdlib.h>\n#include \"mkl_example.h\"\nint main()\n{\n      MKL_INT  n, incx, incy, i;\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2655\n\n\n      float   *x, *y;\n      float    res;\n      MKL_INT  len_x, len_y;\n      n = 5;\n      incx = 2;\n      incy = 1;\n      len_x = 1+(n-1)*abs(incx);\n      len_y = 1+(n-1)*abs(incy);\n      x    = (float *)calloc( len_x, sizeof( float ) );\n      y    = (float *)calloc( len_y, sizeof( float ) );\n      if( x == NULL || y == NULL ) {\n          printf( \"\\n Can't allocate memory for arrays\\n\");\n          return 1;\n      }\n      for (i = 0; i < n; i++) {\n          x[i*abs(incx)] = 2.0;\n          y[i*abs(incy)] = 1.0;\n      }\n      res = cblas_sdot(n, x, incx, y, incy);\n      printf(\"\\n       SDOT = %7.3f\", res);\n      free(x);\n      free(y);\n      return 0;\n}\nAs a result of this program execution, the following line is printed:\nSDOT = 10.000\nExample. Using BLAS Level 1 Routine\nThe following example illustrates a call to the BLAS Level 1 routine scopy. This routine performs a vector-\nvector operation of copying a single-precision real vector x to a vector y.\nParameters\n \nn\nSpecifies the number of elements in vectors x and y.\nincx\nSpecifies the increment for the elements of x.\nincy\nSpecifies the increment for the elements of y.\n#include <stdio.h>\n#include <stdlib.h>\n#include \"mkl_example.h\"\nint main()\n{\n      MKL_INT  n, incx, incy, i;\n      float   *x, *y;\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2656\n\n\n      MKL_INT  len_x, len_y;\n      n = 3;\n      incx = 3;\n      incy = 1;\n      len_x = 10;\n      len_y = 10;\n      x    = (float *)calloc( len_x, sizeof( float ) );\n      y    = (float *)calloc( len_y, sizeof( float ) );\n      if( x == NULL || y == NULL ) {\n          printf( \"\\n Can't allocate memory for arrays\\n\");\n          return 1;\n      }\n      for (i = 0; i < 10; i++) {\n          x[i] = i + 1;\n      }\n      cblas_scopy(n, x, incx, y, incy);\n/*       Print output data                                     */\n      printf(\"\\n\\n     OUTPUT DATA\");\n      PrintVectorS(FULLPRINT, n, y, incy, \"Y\");\n      free(x);\n      free(y);\n      return 0;\n}\nAs a result of this program execution, the following line is printed:\nY = 1.00000 4.00000 7.00000\nExample. Using BLAS Level 2 Routine\nThe following example illustrates a call to the BLAS Level 2 routine sger. This routine performs a matrix-\nvector operation\na :=  alpha*x*y' + a.\nParameters\n \nalpha\nSpecifies a scalar alpha.\nx\nm-element vector.\ny\nn-element vector.\na\nm-by-n matrix.\n#include <stdio.h>\n#include <stdlib.h>\n#include \"mkl_example.h\"\nint main()\n{\n      MKL_INT         m, n, lda, incx, incy, i, j;\n      MKL_INT         rmaxa, cmaxa;\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2657\n\n\n      float           alpha;\n      float          *a, *x, *y;\n      CBLAS_LAYOUT    layout;\n      MKL_INT         len_x, len_y;\n      m = 2;\n      n = 3;\n      lda = 5;\n      incx = 2;\n      incy = 1;\n      alpha = 0.5;\n                        layout = CblasRowMajor;\n      len_x = 10;\n      len_y = 10;\n      rmaxa = m + 1;\n      cmaxa = n;\n      a = (float *)calloc( rmaxa*cmaxa, sizeof(float) );\n      x = (float *)calloc( len_x, sizeof(float) );\n      y = (float *)calloc( len_y, sizeof(float) );\n      if( a == NULL || x == NULL || y == NULL ) {\n          printf( \"\\n Can't allocate memory for arrays\\n\");\n          return 1;\n      }\n      if( layout == CblasRowMajor )\n         lda=cmaxa;\n      else\n         lda=rmaxa;\n      for (i = 0; i < 10; i++) {\n          x[i] = 1.0;\n          y[i] = 1.0;\n      }\n      \n      for (i = 0; i < m; i++) {\n          for (j = 0; j < n; j++) {\n              a[i + j*lda] = j + 1;\n          }\n      }\n      cblas_sger(layout, m, n, alpha, x, incx, y, incy, a, lda);\n      PrintArrayS(&layout, FULLPRINT, GENERAL_MATRIX, &m, &n, a, &lda, \"A\");\n      free(a);\n      free(x);\n      free(y);\n      return 0;\n}\nAs a result of this program execution, matrix a is printed as follows:\nMatrix A:\n1.50000 2.50000 3.50000\n1.50000 2.50000 3.50000\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2658\n\n\nExample. Using BLAS Level 3 Routine\nThe following example illustrates a call to the BLAS Level 3 routine ssymm. This routine performs a matrix-\nmatrix operation\nc :=  alpha*a*b' + beta*c.\nParameters\n \nalpha\nSpecifies a scalar alpha.\nbeta\nSpecifies a scalar beta.\na\nSymmetric matrix\nb\nm-by-n matrix\nc\nm-by-n matrix\n#include <stdio.h>\n#include <stdlib.h>\n#include \"mkl_example.h\"\nint main(int argc, char *argv[])\n{\n      MKL_INT         m, n, i, j;\n      MKL_INT         lda, ldb, ldc;\n      MKL_INT         rmaxa, cmaxa, rmaxb, cmaxb, rmaxc, cmaxc;\n      float           alpha, beta;\n      float          *a, *b, *c;\n      CBLAS_LAYOUT    layout;\n      CBLAS_SIDE      side;\n      CBLAS_UPLO      uplo;\n      MKL_INT         ma, na, typeA;\n      uplo = 'u';\n      side = 'l';\n      layout = CblasRowMajor;\n      m = 3;\n      n = 2;\n      lda = 3;\n      ldb = 3;\n      ldc = 3;\n      alpha = 0.5;\n      beta = 2.0;\n      if( side == CblasLeft ) {\n          rmaxa = m + 1;\n          cmaxa = m;\n          ma    = m;\n          na    = m;\n      } else {\n          rmaxa = n + 1;\n          cmaxa = n;\n          ma    = n;\n          na    = n;\n      }\n      rmaxb = m + 1;\n      cmaxb = n;\n      rmaxc = m + 1;\n      cmaxc = n;\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2659\n\n\n      a = (float *)calloc( rmaxa*cmaxa, sizeof(float) );\n      b = (float *)calloc( rmaxb*cmaxb, sizeof(float) );\n      c = (float *)calloc( rmaxc*cmaxc, sizeof(float) );\n      if ( a == NULL || b == NULL || c == NULL ) {\n           printf(\"\\n Can't allocate memory arrays\");\n           return 1;\n      }\n      if( layout == CblasRowMajor ) {\n         lda=cmaxa;\n         ldb=cmaxb;\n         ldc=cmaxc;\n      } else {\n         lda=rmaxa;\n         ldb=rmaxb;\n         ldc=rmaxc;\n      }\n      if (uplo == CblasUpper) typeA = UPPER_MATRIX;\n      else                    typeA = LOWER_MATRIX;\n      for (i = 0; i < m; i++) {\n         for (j = 0; j < m; j++) {\n            a[i + j*lda] = 1.0;\n         }\n      }\n      for (i = 0; i < m; i++) {\n         for (j = 0; j < n; j++) {\n            c[i + j*ldc] = 1.0;\n            b[i + j*ldb] = 2.0;\n         }\n      }\n      cblas_ssymm(layout, side, uplo, m, n, alpha, a, lda,\n                  b, ldb, beta, c, ldc);\n      printf(\"\\n\\n     OUTPUT DATA\");\n      PrintArrayS(&layout, FULLPRINT, GENERAL_MATRIX, &m, &n, c, &ldc, \"C\");\n      free(a);\n      free(b);\n      free(c);\n      return 0;\n}\nAs a result of this program execution, matrix c is printed as follows:\nMatrix C:\n5.00000 5.00000\n5.00000 5.00000\n5.00000 5.00000\nThe following example illustrates a call from a C program to the Fortran version of the complex BLAS Level 1\nfunction zdotc(). This function computes the dot product of two double-precision complex vectors.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2660\n\n\nExample. Calling a Complex BLAS Level 1 Function from C\nIn this example, the complex dot product is returned in the structure c.\n#include <cstdio>\n#include \"mkl_blas.h\"\n#define N 5\nvoid main()\n{\n  int n, inca = 1, incb = 1, i;\n  MKL_Complex16 a[N], b[N], c;\n  void zdotc();\n  n = N;\n  for( i = 0; i < n; i++ ){\n    a[i].real = (double)i; a[i].imag = (double)i * 2.0;\n    b[i].real = (double)(n - i); b[i].imag = (double)i * 2.0;\n  }\n  zdotc( &c, &n, a, &inca, b, &incb );\n  printf( \"The complex dot product is: ( %6.2f, %6.2f )\\n\", c.real, c.imag );\n}\nNOTE\nInstead of calling BLAS directly from C programs, you might wish to use the C interface to the Basic\nLinear Algebra Subprograms (CBLAS) implemented in Intel® oneAPI Math Kernel Library (oneMKL).\nSeeC Interface Conventions for more information.\nFourier Transform Functions Code Examples\nThis section presents code examples for functions described in the “FFT Functions” and “Cluster FFT\nFunctions” subsections in the “Fourier Transform Functions” section. The examples are grouped in\nsubsections\n•\nExamples for FFT Functions, including Examples of Using Multi-Threading for FFT Computation\n•\nExamples for Cluster FFT Functions\n•\nAuxiliary data transformations.\nFFT Code Examples\nThis section presents examples of using the FFT interface functions described in \"Fourier Transform\nFunctions\".\nHere are the examples of two one-dimensional computations. These examples use the default settings for all\nof the configuration parameters, which are specified in \"Configuration Settings\".\nOne-dimensional In-place FFT\n/* C example, float _Complex is defined in C9X */\n#include \"mkl_dfti.h\"\nfloat _Complex c2c_data[32];\nfloat r2c_data[34];\nDFTI_DESCRIPTOR_HANDLE my_desc1_handle = NULL;\nDFTI_DESCRIPTOR_HANDLE my_desc2_handle = NULL;\nMKL_LONG status;\n/* ...put values into c2c_data[i] 0<=i<=31 */\n/* ...put values into r2c_data[i] 0<=i<=31 */\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2661\n\n\nstatus = DftiCreateDescriptor(&my_desc1_handle, DFTI_SINGLE,\n                              DFTI_COMPLEX, 1, 32);\nstatus = DftiCommitDescriptor(my_desc1_handle);\nstatus = DftiComputeForward(my_desc1_handle, c2c_data);\nstatus = DftiFreeDescriptor(&my_desc1_handle);\n/* result is c2c_data[i] 0<=i<=31 */\nstatus = DftiCreateDescriptor(&my_desc2_handle, DFTI_SINGLE,\n                              DFTI_REAL, 1, 32);\nstatus = DftiCommitDescriptor(my_desc2_handle);\nstatus = DftiComputeForward(my_desc2_handle, r2c_data);\nstatus = DftiFreeDescriptor(&my_desc2_handle);\n/* result is the complex value r2c_data[i] 0<=i<=31 */\n/* and is stored in CCS format*/\n \n \nOne-dimensional Out-of-place FFT\n/* C example, float _Complex is defined in C9X */\n#include \"mkl_dfti.h\"\nfloat _Complex c2c_input[32];\nfloat _Complex c2c_output[32];\nfloat r2c_input[32];\nfloat r2c_output[34];\nDFTI_DESCRIPTOR_HANDLE my_desc1_handle = NULL;\nDFTI_DESCRIPTOR_HANDLE my_desc2_handle = NULL;\nMKL_LONG status;\n/* ...put values into c2c_input[i] 0<=i<=31 */\n/* ...put values into r2c_input[i] 0<=i<=31 */\nstatus = DftiCreateDescriptor(&my_desc1_handle, DFTI_SINGLE,\n                              DFTI_COMPLEX, 1, 32);\nstatus = DftiSetValue(my_desc1_handle, DFTI_PLACEMENT, DFTI_NOT_INPLACE);\nstatus = DftiCommitDescriptor(my_desc1_handle);\nstatus = DftiComputeForward(my_desc1_handle, c2c_input, c2c_output);\nstatus = DftiFreeDescriptor(&my_desc1_handle);\n/* result is c2c_output[i] 0<=i<=31 */\nstatus = DftiCreateDescriptor(&my_desc2_handle, DFTI_SINGLE,\n                              DFTI_REAL, 1, 32);\nStatus = DftiSetValue(my_desc1_handle, DFTI_PLACEMENT, DFTI_NOT_INPLACE);\nstatus = DftiCommitDescriptor(my_desc2_handle);\nstatus = DftiComputeForward(my_desc2_handle, r2c_input, r2c_output);\nstatus = DftiFreeDescriptor(&my_desc2_handle);\n/* result is the complex r2c_data[i] 0<=i<=31   and is stored in CCS format*/\n \n \nTwo-dimensional FFT\n/* C99 example */\n#include \"mkl_dfti.h\"\n/* complex data in, complex data out */\nfloat _Complex c2c_data[32][100];\n/* real data in, complex data out */\nfloat r2c_data[34][102];\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2662\n\n\nDFTI_DESCRIPTOR_HANDLE my_desc1_handle = NULL;\nDFTI_DESCRIPTOR_HANDLE my_desc2_handle = NULL;\nMKL_LONG status;MKL_LONG dim_sizes[2] = {32, 100};\n/* ...put values into c2c_data[i][j] 0<=i<=31, 0<=j<=99 */\n/* ...put values into r2c_data[i][j] 0<=i<=31, 0<=j<=99 */\nstatus = DftiCreateDescriptor(&my_desc1_handle, DFTI_SINGLE,\n                              DFTI_COMPLEX, 2, dim_sizes);\nstatus = DftiCommitDescriptor(my_desc1_handle);\nstatus = DftiComputeForward(my_desc1_handle, c2c_data);\nstatus = DftiFreeDescriptor(&my_desc1_handle);\n/* result is the complex value c2c_data[i][j], 0<=i<=31, 0<=j<=99 */\nstatus = DftiCreateDescriptor(&my_desc2_handle, DFTI_SINGLE,\n                              DFTI_REAL, 2, dim_sizes);\nstatus = DftiCommitDescriptor(my_desc2_handle);\nstatus = DftiComputeForward(my_desc2_handle, r2c_data);\nstatus = DftiFreeDescriptor(&my_desc2_handle);\n/* result is the complex r2c_data[i][j] 0<=i<=31, 0<=j<=99\n   and is stored in CCS format*/\n \n \nThe following example demonstrates how you can change the default configuration settings by using the\nDftiSetValue function.\nFor instance, to preserve the input data after the FFT computation, the configuration of DFTI_PLACEMENT\nshould be changed to \"not in place\" from the default choice of \"in place.\"\nThe code below illustrates how this can be done:\nChanging Default Settings\n/* C99 example */\n#include \"mkl_dfti.h\"\nfloat  _Complex x_in[32], x_out[32];\nDFTI_DESCRIPTOR_HANDLE my_desc_handle = NULL;\nMKL_LONG status;\n/* ...put values into x_in[i] 0<=i<=31 */\nstatus = DftiCreateDescriptor(&my_desc_handle, DFTI_SINGLE,\n                              DFTI_COMPLEX, 1, 32);\nstatus = DftiSetValue(my_desc_handle, DFTI_PLACEMENT, DFTI_NOT_INPLACE);\nstatus = DftiCommitDescriptor(my_desc_handle);\nstatus = DftiComputeForward(my_desc_handle, x_in, x_out);\nstatus = DftiFreeDescriptor(&my_desc_handle);\n/* result is the complex value x_out[i], 0<=i<=31 */\n \n \nUsing Status Checking Functions\nThis example illustrates the use of status checking functions described in \"Fourier Transform Functions\".\n/* C99 example */\n#include \"mkl_dfti.h\"\nDFTI_DESCRIPTOR_HANDLE desc = NULL;\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2663\n\n\nMKL_LONG status;\n/* ...descriptor creation and other code */\nstatus = DftiCommitDescriptor(desc);\nif (status && !DftiErrorClass(status, DFTI_NO_ERROR))\n{\n   printf('Error: %s\\n', DftiErrorMessage(status));\n}\n \nComputing 2D FFT by One-Dimensional Transforms\nBelow is an example where a 20-by-40 two-dimensional FFT is computed explicitly using one-dimensional\ntransforms.\n/* C */\n#include \"mkl_dfti.h\"\nfloat _Complex data[20][40];\nMKL_LONG status;\nDFTI_DESCRIPTOR_HANDLE desc_handle_dim1 = NULL;\nDFTI_DESCRIPTOR_HANDLE desc_handle_dim2 = NULL;\n/* ...put values into data[i][j] 0<=i<=19, 0<=j<=39 */\nstatus = DftiCreateDescriptor(&desc_handle_dim1, DFTI_SINGLE,\n                              DFTI_COMPLEX, 1, 20);\nstatus = DftiCreateDescriptor(&desc_handle_dim2, DFTI_SINGLE,\n                              DFTI_COMPLEX, 1, 40);\n/* perform 40 one-dimensional transforms along 1st dimension */\n/* note that the 1st dimension data are not unit-stride */\nMKL_LONG stride[2] = {0, 40};\nstatus = DftiSetValue(desc_handle_dim1, DFTI_NUMBER_OF_TRANSFORMS, 40);\nstatus = DftiSetValue(desc_handle_dim1, DFTI_INPUT_DISTANCE, 1);\nstatus = DftiSetValue(desc_handle_dim1, DFTI_OUTPUT_DISTANCE, 1);\nstatus = DftiSetValue(desc_handle_dim1, DFTI_INPUT_STRIDES, stride);\nstatus = DftiSetValue(desc_handle_dim1, DFTI_OUTPUT_STRIDES, stride);\nstatus = DftiCommitDescriptor(desc_handle_dim1);\nstatus = DftiComputeForward(desc_handle_dim1, data);\n/* perform 20 one-dimensional transforms along 2nd dimension */\n/* note that the 2nd dimension is unit stride */\nstatus = DftiSetValue(desc_handle_dim2, DFTI_NUMBER_OF_TRANSFORMS, 20);\nstatus = DftiSetValue(desc_handle_dim2, DFTI_INPUT_DISTANCE, 40);\nstatus = DftiSetValue(desc_handle_dim2, DFTI_OUTPUT_DISTANCE, 40);\nstatus = DftiCommitDescriptor(desc_handle_dim2);\nstatus = DftiComputeForward(desc_handle_dim2, data);\nstatus = DftiFreeDescriptor(&desc_handle_dim1);\nstatus = DftiFreeDescriptor(&desc_handle_dim2); \n \n \n \nThe following code illustrates real multi-dimensional transforms with CCE format storage of conjugate-even\ncomplex matrix. Example \"Three-Dimensional REAL FFT (C Interface)\" is three-dimensional out-of-place\ntransform in C interface.\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2664\n\n\nThree-Dimensional REAL FFT\n \n/* C99 example */\n#include \"mkl_dfti.h\"\nfloat input[32][100][19];\nfloat _Complex output[32][100][10];  /* 10 = 19/2 + 1 */\nDFTI_DESCRIPTOR_HANDLE my_desc_handle = NULL;\nMKL_LONG status;\nMKL_LONG dim_sizes[3] = {32, 100, 19};\nMKL_LONG strides_out[4] = {0, 100*10*1, 10*1, 1};\n//...put values into input[i][j][k] 0<=i<=31; 0<=j<=99, 0<=k<=9\nstatus = DftiCreateDescriptor(&my_desc_handle,\n                              DFTI_SINGLE, DFTI_REAL, 3, dim_sizes);\nstatus = DftiSetValue(my_desc_handle,\n                      DFTI_CONJUGATE_EVEN_STORAGE, DFTI_COMPLEX_COMPLEX);\nstatus = DftiSetValue(my_desc_handle, DFTI_PLACEMENT, DFTI_NOT_INPLACE);\nstatus = DftiSetValue(my_desc_handle, DFTI_OUTPUT_STRIDES, strides_out);\nstatus = DftiCommitDescriptor(my_desc_handle);\nstatus = DftiComputeForward(my_desc_handle, input, output);\nstatus = DftiFreeDescriptor(&my_desc_handle);\n/* result is the complex output[i][j][k] 0<=i<=31; 0<=j<=99, 0<=k<=9\n   and is stored in CCE format. */\n \nExamples of Using OpenMP* Threading for FFT Computation\nThe following sample program shows how to employ internal OpenMP* threading in Intel® oneAPI Math\nKernel Library (oneMKL) for FFT computation.\nTo specify the number of threads inside Intel® oneAPI Math Kernel Library (oneMKL), use the following\nsettings:\nset MKL_NUM_THREADS = 1 for one-threaded mode;\nset MKL_NUM_THREADS = 4 for multi-threaded mode.\nUsing oneMKL Internal Threading Mode (C Example)\n \n/* C99 example */\n#include \"mkl_dfti.h\"\nfloat data[200][100];\nDFTI_DESCRIPTOR_HANDLE fft = NULL;\nMKL_LONG dim_sizes[2] = {200, 100};\n/* ...put values into data[i][j] 0<=i<=199, 0<=j<=99 */\nDftiCreateDescriptor(&fft, DFTI_SINGLE, DFTI_REAL, 2, dim_sizes);\nDftiCommitDescriptor(fft);\nDftiComputeForward(fft, data);\nDftiFreeDescriptor(&fft);\n \nThe following Example “Using Parallel Mode with Multiple Descriptors Initialized in a Parallel Region” and \nExample “Using Parallel Mode with Multiple Descriptors Initialized in One Thread” illustrate a parallel\ncustomer program with each descriptor instance used only in a single thread.\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2665\n\n\nSpecify the number of OpenMP threads for Example “Using Parallel Mode with Multiple Descriptors Initialized\nin a Parallel Region” like this:\nset MKL_NUM_THREADS = 1 for Intel® oneAPI Math Kernel Library (oneMKL) to work in the single-threaded\nmode (recommended);\nset OMP_NUM_THREADS = 4 for the customer program to work in the multi-threaded mode.\nUsing Parallel Mode with Multiple Descriptors Initialized in a Parallel Region\nNote that in this example, the program can be transformed to become single-threaded at the customer level\nbut using parallel mode within Intel® oneAPI Math Kernel Library (oneMKL). To achieve this, you must set the\nparameter DFTI_NUMBER_OF_TRANSFORMS = 4 and to set the corresponding parameter\nDFTI_INPUT_DISTANCE = 5000.\n/* C99 example */\n#include \"mkl_dfti.h\"\n#include <omp.h>\n#define ARRAY_LEN(a) sizeof(a)/sizeof(a[0])\n// 4 OMP threads, each does 2D FFT 50x100 points\nMKL_Complex8 data[4][50][100];\nint nth = ARRAY_LEN(data);\nMKL_LONG dim_sizes[2] = {\n    ARRAY_LEN(data[0]),\n    ARRAY_LEN(data[0][0])\n};  /* {50, 100} */\nint th;\n/* ...put values into data[i][j][k] 0<=i<=3, 0<=j<=49, 0<=k<=99 */\n// assume data is initialized and do 2D FFTs\n#pragma omp parallel for shared(dim_sizes, data)\nfor (th = 0; th < nth; ++th)\n{\n    DFTI_DESCRIPTOR_HANDLE myFFT = NULL;\n    DftiCreateDescriptor(&myFFT, DFTI_SINGLE, DFTI_COMPLEX, 2, dim_sizes);\n    DftiCommitDescriptor(myFFT);\n    DftiComputeForward(myFFT, data[th]);\n    DftiFreeDescriptor(&myFFT);\n}\nSpecify the number of OpenMP threads for Example “Using Parallel Mode with Multiple Descriptors Initialized\nin One Thread” like this:\nset MKL_NUM_THREADS = 1 for Intel® oneAPI Math Kernel Library (oneMKL) to work in the single-threaded\nmode (obligatory);\nset OMP_NUM_THREADS = 4 for the customer program to work in the multi-threaded mode.\nUsing Parallel Mode with Multiple Descriptors Initialized in One Thread\n/* C99 example */\n#include \"mkl_dfti.h\"\n#include <omp.h>#\ndefine ARRAY_LEN(a) sizeof(a)/sizeof(a[0])\n// 4 OMP threads, each does 2D FFT 50x100 points\nMKL_Complex8 data[4][50][100];\nint nth = ARRAY_LEN(data);\nMKL_LONG dim_sizes[2] = {\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2666\n\n\n    ARRAY_LEN(data[0]),\n    ARRAY_LEN(data[0][0])\n};  /* {50, 100} */\nDFTI_DESCRIPTOR_HANDLE FFT[ARRAY_LEN(data)];\nint th;\n/* ...put values into data[i][j][k] 0<=i<=3, 0<=j<=49, 0<=k<=99 */\nfor (th = 0; th < nth; ++th)\n    DftiCreateDescriptor(&FFT[th], DFTI_SINGLE, DFTI_COMPLEX, 2, dim_sizes);\nfor (th = 0; th < nth; ++th)\n    DftiCommitDescriptor(FFT[th]);\n// assume data is initialized and do 2D FFTs\n#pragma omp parallel for shared(FFT, data)\nfor (th = 0; th < nth; ++th)\n    DftiComputeForward(FFT[th], data[th]);\nfor (th = 0; th < nth; ++th)\n    DftiFreeDescriptor(&FFT[th]);\nThe following Example “Using Parallel Mode with a Common Descriptor” illustrates a parallel customer\nprogram with a common descriptor used in several threads.\nUsing Parallel Mode with a Common Descriptor\n#include \"mkl_dfti.h\"\n#include <omp.h>\n#define ARRAY_LEN(a) sizeof(a)/sizeof(a[0])\n// 4 OMP threads, each does 2D FFT 50x100 points\nMKL_Complex8 data[4][50][100];\nint nth = ARRAY_LEN(data);\nMKL_LONG len[2] = {ARRAY_LEN(data[0]), ARRAY_LEN(data[0][0])};\nDFTI_DESCRIPTOR_HANDLE FFT;\nint th;\n/* ...put values into data[i][j][k] 0<=i<=3, 0<=j<=49, 0<=k<=99 */\nDftiCreateDescriptor(&FFT, DFTI_SINGLE, DFTI_COMPLEX, 2, len);\nDftiCommitDescriptor(FFT);\n// assume data is initialized and do 2D FFTs\n#pragma omp parallel for shared(FFT, data)\nfor (th = 0; th < nth; ++th)\n    DftiComputeForward(FFT, data[th]);\nDftiFreeDescriptor(&FFT);\nExamples for Cluster FFT Functions\nThe following C example computes a 2-dimensional out-of-place FFT using the cluster FFT interface:\n2D Out-of-place Cluster FFT Computation\n/* C99 example */\n#include \"mpi.h\"\n#include \"mkl_cdft.h\"\nDFTI_DESCRIPTOR_DM_HANDLE desc = NULL;\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2667\n\n\nMKL_LONG v, i, j, n, s;\nComplex *in, *out;\nMKL_LONG dim_sizes[2] = {nx, ny};\nMPI_Init(...);\n/* Create descriptor for 2D FFT */\nDftiCreateDescriptorDM(MPI_COMM_WORLD,\n                       &desc, DFTI_DOUBLE, DFTI_COMPLEX, 2, dim_sizes);\n/* Ask necessary length of in and out arrays and allocate memory */\nDftiGetValueDM(desc,CDFT_LOCAL_SIZE,&v);\nin = (Complex*) malloc(v*sizeof(Complex));\nout = (Complex*) malloc(v*sizeof(Complex));\n/* Fill local array with initial data. Current process performs n rows,\n   0 row of in corresponds to s row of virtual global array */\nDftiGetValueDM(desc, CDFT_LOCAL_NX, &n);\nDftiGetValueDM(desc, CDFT_LOCAL_X_START, &s);\n/* Virtual global array globalIN is defined by function f as\n   globalIN[i*ny+j]=f(i,j) */\nfor(i = 0; i < n; ++i)\n   for(j = 0; j < ny; ++j) in[i*ny+j] = f(i+s,j);\n/* Set that we want out-of-place transform (default is DFTI_INPLACE) */\nDftiSetValueDM(desc, DFTI_PLACEMENT, DFTI_NOT_INPLACE);\n/* Commit descriptor, calculate FFT, free descriptor */\nDftiCommitDescriptorDM(desc);\nDftiComputeForwardDM(desc, in, out);\n/* Virtual global array globalOUT is defined by function g as\n   globalOUT[i*ny+j]=g(i,j)   Now out contains result of FFT. out[i*ny+j]=g(i+s,j) */\nDftiFreeDescriptorDM(&desc);\nfree(in);\nfree(out);\nMPI_Finalize();\n \n1D In-place Cluster FFT Computations\nThe C example below illustrates one-dimensional in-place cluster FFT computations effected with a user-\ndefined workspace:\n/* C99 example */\n#include \"mpi.h\"\n#include \"mkl_cdft.h\"\nDFTI_DESCRIPTOR_DM_HANDLE desc = NULL;\nMKL_LONG N, v, i, n_out, s_out;\nComplex *in, *work;\nMPI_Init(...);\n/* Create descriptor for 1D FFT */\nDftiCreateDescriptorDM(MPI_COMM_WORLD, &desc, DFTI_DOUBLE, DFTI_COMPLEX, 1, N);\n/* Ask necessary length of array and workspace and allocate memory */\nDftiGetValueDM(desc,CDFT_LOCAL_SIZE,&v);\nin = (Complex*) malloc(v*sizeof(Complex));\nwork = (Complex*) malloc(v*sizeof(Complex));\n/* Fill local array with initial data. Local array has n elements,\n   0 element of in corresponds to s element of virtual global array */\nDftiGetValueDM(desc, CDFT_LOCAL_NX, &n);\nDftiGetValueDM(desc, CDFT_LOCAL_X_START, &s);\n/* Set work array as a workspace */\n  1   Developer Reference for Intel® oneAPI Math Kernel Library for C\n2668\n\n\nDftiSetValueDM(desc, CDFT_WORKSPACE, work);\n/* Virtual global array globalIN is defined by function f as globalIN[i]=f(i) */\nfor(i = 0; i < n; ++i) in[i] = f(i+s);\n/* Commit descriptor, calculate FFT, free descriptor */\nDftiCommitDescriptorDM(desc);\nDftiComputeForwardDM(desc,in);\nDftiGetValueDM(desc, CDFT_LOCAL_OUT_NX, &n_out);\nDftiGetValueDM(desc, CDFT_LOCAL_OUT_X_START, &s_out);\n/* Virtual global array globalOUT is defined by function g as globalOUT[i]=g(i)\n   Now in contains result of FFT. Local array has n_out elements,\n   0 element of in corresponds to s_out element of virtual global array.\n   in[i]==g(i+s_out) */\nDftiFreeDescriptorDM(&desc);\nfree(in);\nfree(work);\nMPI_Finalize();\n \n \nAuxiliary Data Transformations\nThis section presents C examples for conversion from the Cartesian to polar representation of complex data\nand vice versa.\nConversion from Cartesian to polar representation of complex data\n// Cartesian->polar conversion of complex data\n// Cartesian representation: z = re + I*im\n// Polar representation: z = r * exp( I*phi )\n#include <mkl_vml.h>\n \nvoid \nvariant1_Cartesian2Polar(int n,const double *re,const double *im,\n                         double *r,double *phi)\n{\n    vdHypot(n,re,im,r);         // compute radii r[]\n    vdAtan2(n,im,re,phi);       // compute phases phi[]\n}\n \nvoid \nvariant2_Cartesian2Polar(int n,const MKL_Complex16 *z,double *r,double *phi,\n                         double *temp_re,double *temp_im)\n{\n    vzAbs(n,z,r);               // compute radii r[]\n    vdPackI(n, (double*)z + 0, 2, temp_re);\n    vdPackI(n, (double*)z + 1, 2, temp_im);\n    vdAtan2(n,temp_im,temp_re,phi); // compute phases phi[]\n}\n \nConversion from polar to Cartesian representation of complex data\n// Polar->Cartesian conversion of complex data.\n// Polar representation: z = r * exp( I*phi )\n// Cartesian representation: z = re + I*im\n#include <mkl_vml.h>\n \nvoid\nDeveloper Reference for Intel® oneAPI Math Kernel Library - C  1  \n2669\n\n\nvariant1_Polar2Cartesian(int n,const double *r,const double *phi,\n                         double *re,double *im)\n{\n    vdSinCos(n,phi,im,re);      // compute direction, i.e. z[]/abs(z[])\n    vdMul(n,r,re,re);           // scale real part\n    vdMul(n,r,im,im);           // scale imaginary part\n}\n \nvoid\nvariant2_Polar2Cartesian(int n,const double *r,const double *phi,\n                         MKL_Complex16 *z,\n                         double *temp_re,double *temp_im)\n{\n    vdSinCos(n,phi,temp_im,temp_re); // compute direction, i.e. z[]/abs(z[])\n    vdMul(n,r,temp_im,temp_im); // scale imaginary part\n    vdMul(n,r,temp_re,temp_re); // scale real part\n    vdUnpackI(n,temp_re,(double*)z + 0, 2); // fill in result.re\n    vdUnpackI(n,temp_im,(double*)z + 1, 2); // fill in result.im\n}","difficulty":"hard","domain":"Long In-context Learning","length":"long","question":"According to the documentation, when using the pardiso function in onemkl pardiso to solve a sparse linear system, for which of the following inputs, the solution must be solved without iterative refinement?","sub_domain":"User guide QA"}

Source: https://huggingface.co/datasets/zai-org/LongBench-v2

initial import

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