线性系统数值解的反射法

IF 10.8 1区数学 Q1 MATHEMATICS, APPLIED

SIAM Review Pub Date : 2023-11-07 DOI:10.1137/22m1470463

Margherita Guida, Carlo Sbordone

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引用次数: 0

摘要

SIAM评论，第65卷第4期，第1137-1151页，2023年11月。我们提出了线性系统数值解的Cimmino反射算法，该算法从$\mathbb｛R｝^n$中的任意点开始，该点相对于系统的超平面得到反射。随后的点集合的质心成为下一次迭代的起点。我们提供了每一步收敛的误差估计。还设计了一个概率论证来改进这种优雅的几何算法。本课程向学生展示了线性代数如何不仅与代数、几何和数值分析，而且与概率论和方法进行富有成效的互动。

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The Reflection Method for the Numerical Solution of Linear Systems

SIAM Review, Volume 65, Issue 4, Page 1137-1151, November 2023.
We present Cimmino's reflection algorithm for the numerical solution of linear systems, which starts with an arbitrary point in $\mathbb{R}^n$ that gets reflected with respect to the system's hyperplanes. The centroid of the ensuing collection of points becomes the starting point of the next iteration. We provide error estimates for the convergence at each step. A probabilistic argument is also devised to improve this elegant geometrical algorithm. This subject is an opportunity to show students how linear algebra can interact fruitfully not only with algebra, geometry, and numerical analysis, but also with probability theory and methods.

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来源期刊

SIAM Review 数学-应用数学

CiteScore

16.90

自引率

0.00%

发文量

期刊介绍： Survey and Review feature papers that provide an integrative and current viewpoint on important topics in applied or computational mathematics and scientific computing. These papers aim to offer a comprehensive perspective on the subject matter. Research Spotlights publish concise research papers in applied and computational mathematics that are of interest to a wide range of readers in SIAM Review. The papers in this section present innovative ideas that are clearly explained and motivated. They stand out from regular publications in specific SIAM journals due to their accessibility and potential for widespread and long-lasting influence.