Growth Factors of Orthogonal Matrices and Local Behavior of Gaussian Elimination with Partial and Complete Pivoting

IF 1.5 2区数学 Q2 MATHEMATICS, APPLIED

SIAM Journal on Matrix Analysis and Applications Pub Date : 2024-08-22 DOI:10.1137/23m1597733

John Peca-Medlin

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引用次数: 0

Abstract

SIAM Journal on Matrix Analysis and Applications, Volume 45, Issue 3, Page 1599-1620, September 2024.
Abstract. Gaussian elimination (GE) is the most used dense linear solver. Error analysis of GE with selected pivoting strategies on well-conditioned systems can focus on studying the behavior of growth factors. Although exponential growth is possible with GE with partial pivoting (GEPP), growth tends to stay much smaller in practice. Support for this behavior was provided recently by Huang and Tikhomirov’s average-case analysis of GEPP, which showed GEPP growth factors for Gaussian matrices stay at most polynomial with very high probability. GE with complete pivoting (GECP) has also seen a lot of recent interest, with improvements to both lower and upper bounds on worst-case GECP growth provided by Bisain, Edelman, and Urschel in 2023. We are interested in studying how GEPP and GECP behave on the same linear systems as well as studying large growth on particular subclasses of matrices, including orthogonal matrices. Moreover, as a means to better address the question of why large growth is rarely encountered, we further study matrices with a large difference in growth between using GEPP and GECP, and we explore how the smaller growth strategy dominates behavior in a small neighborhood of the initial matrix.

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正交矩阵的增长因子以及部分和完全透视高斯消元的局部行为

SIAM 矩阵分析与应用期刊》，第 45 卷第 3 期，第 1599-1620 页，2024 年 9 月。摘要高斯消元（GE）是最常用的密集线性求解器。利用选定的枢轴策略对条件良好的系统进行高斯消元误差分析，可以重点研究增长因子的行为。尽管使用部分支点策略（GEPP）的几何级数增长是可能的，但在实践中增长往往要小得多。最近，Huang 和 Tikhomirov 对 GEPP 的平均情况分析为这一行为提供了支持，该分析表明高斯矩阵的 GEPP 增长因子以极高的概率最多保持多项式。具有完全支点的通用计算（GECP）最近也受到了广泛关注，比萨恩、埃德尔曼和乌尔谢尔在 2023 年对最坏情况下 GECP 增长的下限和上限进行了改进。我们有兴趣研究 GEPP 和 GECP 在相同线性系统上的表现，以及研究特定子类矩阵（包括正交矩阵）上的大增长。此外，为了更好地解决为什么很少出现大增长的问题，我们进一步研究了使用 GEPP 和 GECP 时增长差异较大的矩阵，并探讨了较小增长策略如何在初始矩阵的小邻域内主导行为。

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来源期刊

SIAM Journal on Matrix Analysis and Applications 数学-应用数学

CiteScore

2.90

自引率

6.70%

发文量

审稿时长

6-12 weeks

期刊介绍： The SIAM Journal on Matrix Analysis and Applications contains research articles in matrix analysis and its applications and papers of interest to the numerical linear algebra community. Applications include such areas as signal processing, systems and control theory, statistics, Markov chains, and mathematical biology. Also contains papers that are of a theoretical nature but have a possible impact on applications.