{"kind":"task","effective_mode":"full","benchmark":{"kind":"benchmark","effective_mode":"full","slug":"scicode","formal_name":"SciCode","introduction":"科学研究の問題をコードで解く能力を評価するベンチマークです。親問題を複数の小問題に分けており、今回のdev取得では15親問題と50小問題の関係を保持します。\n\nSciCode evaluates the ability to solve scientific research problems through code. Problems are decomposed into subproblems; this dev import preserves the relationships between 15 parent problems and 50 subproblems.","introduction_ja":"","introduction_en":"","category":"Category not supplied","task_count":null,"acquisition_status":"Acquisition status not supplied","official_url":"https://huggingface.co/datasets/SciCode1/SciCode","indexing_mode":"noindex"},"task_id":"7d62c663-5b81-5b67-b93f-5e3fdc928fc5","task_key":"dev--4","task_revision_id":"2","upstream_id":"4","short_description":"IncomChol","config":"","split":"dev","body":"{\"problem_background_main\":\"Background:\\nAn incomplete Cholesky factorization provides a sparse approximation of the Cholesky factorization for a symmetric positive definite matrix. This factorization is commonly employed as a preconditioner for iterative algorithms such as the conjugate gradient method.\\n\\nIn the Cholesky factorization of a positive definite matrix $A$, we have $A = LL*$, where $L$ is a lower triangular matrix. The incomplete Cholesky factorization yields a sparse lower triangular matrix $K$ that closely approximates $L$. The corresponding preconditioner is $KK*$.\\n\\nA popular approach to find the matrix $K$ is to adapt the algorithm for the exact Cholesky decomposition, ensuring that $K$ retains the same sparsity pattern as $A$ (any zero entry in $A$ leads to a zero entry in $K$). This method produces an incomplete Cholesky factorization that is as sparse as matrix $A$.\\n\\nFor $i$ from $1$ to $N$ :\\n$$\\nL_{i i}=\\\\left(a_{i i}-\\\\sum_{k=1}^{i-1} L_{i k}^2\\\\right)^{\\\\frac{1}{2}}\\n$$\\n\\nFor $j$ from $i+1$ to $N$ :\\n$$\\nL_{j i}=\\\\frac{1}{L_{i i}}\\\\left(a_{j i}-\\\\sum_{k=1}^{i-1} L_{i k} L_{j k}\\\\right)\\n$$\",\"problem_description_main\":\"Create a function to compute the incomplete Cholesky factorization of an input matrix.\",\"problem_io\":\"\\\"\\\"\\\"\\nInputs:\\nA : Matrix, 2d array M * M\\n\\nOutputs:\\nA : Matrix, 2d array M * M\\n\\\"\\\"\\\"\",\"problem_name\":\"IncomChol\",\"required_dependencies\":\"import numpy as np\"}","display_format":"scicode","language":"","answer_status":"published","assets":[],"source_url":"https://huggingface.co/datasets/SciCode1/SciCode","history":"initial import","indexing_mode":"noindex","subproblems":[],"grids":[]}