QR algorithm with two‐sided Rayleigh quotient shifts

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2023-01-02 DOI:10.1002/nla.2487

X. Chen, Hongguo Xu

引用次数: 1

Abstract

We introduce the two‐sided Rayleigh quotient shift to the QR algorithm for non‐Hermitian matrices to achieve a cubic local convergence rate. For the singly shifted case, the two‐sided Rayleigh quotient iteration is incorporated into the QR iteration. A modified version of the method and its truncated version are developed to improve the efficiency. Based on the observation that the Francis double‐shift QR iteration is related to a 2D Grassmann–Rayleigh quotient iteration, A doubly shifted QR algorithm with the two‐sided 2D Grassmann–Rayleigh quotient double‐shift is proposed. A modified version of the method and its truncated version are also developed. Numerical examples are presented to show the convergence behavior of the proposed algorithms. Numerical examples also show that the truncated versions of the modified methods outperform their counterparts including the standard Rayleigh quotient single‐shift and the Francis double‐shift.

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二维瑞利商移位QR算法

我们在非厄米矩阵的QR算法中引入了双侧瑞利商移位，以达到三次局部收敛速率。对于单位移情况，将双面瑞利商迭代纳入QR迭代中。为了提高效率，提出了该方法的改进版本及其截断版本。在观察到Francis双移QR迭代与二维Grassmann-Rayleigh商迭代相关的基础上，提出了一种二维Grassmann-Rayleigh商双移的双移QR算法。还开发了该方法的修改版本及其截断版本。数值算例表明了所提算法的收敛性。数值算例还表明，改进方法的截断版本优于标准瑞利商单位移和弗朗西斯双位移。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

期刊介绍： Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.