V2EX cholesky decomposition

Cholesky Decomposition

释义 Definition

乔列斯基分解:一种矩阵分解方法,适用于对称正定矩阵 \(A\),将其分解为
\[ A = LL^{\mathsf T} \] 其中 \(L\) 是下三角矩阵(或也常写为 \(A = R^{\mathsf T}R\),\(R\) 为上三角矩阵)。它在数值计算中常用于更高效、稳定地求解线性方程组、进行高斯过程/协方差计算等。

发音 Pronunciation (IPA)

/tolski dikmpzn/

例句 Examples

Cholesky decomposition is fast for solving systems with symmetric positive-definite matrices.
乔列斯基分解在求解对称正定矩阵的线性方程组时速度很快。

In Gaussian process regression, we often use Cholesky decomposition of the covariance matrix to compute the log-likelihood efficiently and stably.
在高斯过程回归中,我们常对协方差矩阵做乔列斯基分解,以高效且稳定地计算对数似然。

词源 Etymology

Cholesky”来自法国数学家与工程师 André-Louis Cholesky(安德烈-路易乔列斯基) 的姓氏;“decomposition”意为“分解”。该方法最早与数值线性代数中的实际计算需求相关,后来成为处理对称正定矩阵的经典工具。(在很多教材中也称 Cholesky factorization“乔列斯基分解/因子分解”。)

相关词 Related Words

文学与经典著作中的用例 Literary Works

  • Matrix Computations(Golub & Van Loan)数值线性代数经典教材,系统讲解乔列斯基分解及其数值性质
  • Numerical Recipes(Press et al.)介绍在实际计算中用乔列斯基分解求解方程组与处理协方差矩阵
  • Convex Optimization(Boyd & Vandenberghe)在优化问题的数值实现与二次型/正定矩阵背景中出现
  • Numerical Linear Algebra(Trefethen & Bau)以算法视角讲解分解方法,包含乔列斯基分解
  • Accuracy and Stability of Numerical Algorithms(Higham)讨论包括乔列斯基在内的数值稳定性与误差分析
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