QR algorithm with two‐sided Rayleigh quotient shifts

IF 1.8 3区数学 Q1 MATHEMATICS

Numerical Linear Algebra with Applications 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

Numerical Linear Algebra with Applications 数学-数学

CiteScore

3.40

自引率

2.30%

发文量

审稿时长

12 months

期刊介绍： Manuscripts submitted to Numerical Linear Algebra with Applications should include large-scale broad-interest applications in which challenging computational results are integral to the approach investigated and analysed. Manuscripts that, in the Editor’s view, do not satisfy these conditions will not be accepted for review. Numerical Linear Algebra with Applications receives submissions in areas that address developing, analysing and applying linear algebra algorithms for solving problems arising in multilinear (tensor) algebra, in statistics, such as Markov Chains, as well as in deterministic and stochastic modelling of large-scale networks, algorithm development, performance analysis or related computational aspects. Topics covered include: Standard and Generalized Conjugate Gradients, Multigrid and Other Iterative Methods; Preconditioning Methods; Direct Solution Methods; Numerical Methods for Eigenproblems; Newton-like Methods for Nonlinear Equations; Parallel and Vectorizable Algorithms in Numerical Linear Algebra; Application of Methods of Numerical Linear Algebra in Science, Engineering and Economics.