This paper highlights a formal connection between two families of widely used matrix factorization algorithms in numerical linear algebra. One family consists of the Jacobi eigenvalue algorithm and its variants for computing the Hermitian eigendecomposition and singular value decomposition. The other consists of Gaussian elimination and the Gram-Schmidt procedure with various pivoting rules for computing the Cholesky decomposition and QR decomposition respectively. Both families are cast as special cases of a more general class of factorization algorithms. We provide a randomized pivoting rule that applies to this general class (which differs substantially from the usual pivoting rules for Gaussian elimination / Gram-Schmidt) which admits a unified analysis of the entire class of algorithms. The result is the same linear rate of convergence for each algorithm, irrespective of which factorization it computes. One important consequence of this randomized pivoting rule is a provable, effective bound on the numerical stability of the Jacobi eigenvalue algorithm, which addresses a longstanding open problem of Demmel and Veseli\'c `92.

翻译：本文揭示了数值线性代数中两个广泛使用的矩阵分解算法族之间的形式化联系。一族包括用于计算埃尔米特特征分解和奇异值分解的雅可比特征值算法及其变体。另一族包括分别用于计算楚列斯基分解和QR分解的高斯消元法和具有各种主元选取规则的格拉姆-施密特过程。这两族算法均可视为更广义的分解算法类的特例。我们提出了一种适用于该广义算法类的随机主元选取规则（该规则与高斯消元/格拉姆-施密特的常规主元选取规则有本质区别），使得能够对这一整类算法进行统一分析。分析结果表明，无论算法计算何种分解，均具有相同的线性收敛速率。该随机主元选取规则的一个重要推论是：为雅可比特征值算法的数值稳定性提供了可证明的有效界，这解决了Demmel和Veselić于1992年提出的长期悬而未决的公开问题。