QZ algorithm with two‐sided generalized Rayleigh quotient shifts

IF 1.8 3区数学 Q1 MATHEMATICS

Numerical Linear Algebra with Applications Pub Date : 2022-11-10 DOI:10.1002/nla.2475

X. Chen, Hongguo Xu

引用次数: 0

Abstract

We generalize the recently proposed two‐sided Rayleigh quotient single‐shift and the two‐sided Grassmann–Rayleigh quotient double‐shift used in the QR algorithm and apply the generalized versions to the QZ algorithm. With such shift strategies the QZ algorithm normally has a cubic local convergence rate. Our main focus is on the modified shift strategies and their corresponding truncated versions. Numerical examples are provided to demonstrate the convergence properties and the efficiency of the QZ algorithm equipped with the proposed shifts. For the truncated versions, local convergence analysis is not provided. Numerical examples show they outperform the modified shifts and the standard Rayleigh quotient single‐shift and Francis double‐shift.

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具有双侧广义瑞利商位移的QZ算法

我们推广了最近提出的QR算法中使用的双侧瑞利商单频移和双侧Grassmann–Rayleigh商双频移，并将广义版本应用于QZ算法。对于这样的移位策略，QZ算法通常具有三次局部收敛速度。我们的主要关注点是修改后的转换策略及其相应的截断版本。数值算例证明了带有所提出的移位的QZ算法的收敛性和效率。对于截断的版本，不提供局部收敛性分析。数值算例表明，它们优于修正位移和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.