Theoretical and numerical comparisons of GMRES and WZ-GMRES
type: Item
instance of: scholarly article
title: Theoretical and numerical comparisons of GMRES and WZ-GMRES
zbMATH Open document ID: 1071.65037
DOI: 10.1016/S0898-1221(04)90127-7
author: Guizhi Chen
author: Zhongxiao Jia
publication date: 2005-03-08T00:00:00Z
review text: The authors study the numerical stability of the WZ-GMRES method proposed by \textit{H. F. Walker} and \textit{L. Zhou} [Numer. Linear Algebra Appl. 1, No. 6, 571--581 (1994; Zbl 0838.65030)] and compare the stability of the WZ-GMRES method with that of the GMRES method for solving systems of linear equations \(Ax = b\) with a non-symmetric matrix \(A\). The main result of the paper is that the mathematically equivalent GMRES and WZ-GMRES methods are not numerically equivalent when the methods are near convergence. Some relationships between the condition numbers of the triangular matrices \(R_{G}\) and \(R_{WZ}\) and the matrix \(A\) are derived. Here, \(R_{G}\) and \(R_{WZ}\) are the matrices of the systems of equations which one has to solve within the GMRES and the WZ-GMRES iteration, respectively. It is shown that generally the matrix \(R_{WZ}\) is worse conditioned than the matrix \(R_{G}\) and that the matrix \(R_{WZ}\) is ill-conditioned when the method is near convergence. This makes the WZ-GMRES method numerically unstable. Therefore, it is more preferable to use the original GMRES implementation. Additionally, an upper bound of the residual norm is established. Numerical experiments confirm the theoretical results.
reviewed by: Michael Jung
Mathematics Subject Classification ID: 65F10
zbMATH DE Number: 2142392
zbMATH Keywords: non-symmetric matrix
zbMATH Keywords: iterative methods
zbMATH Keywords: GMRES method
zbMATH Keywords: Krylov subspace method
zbMATH Keywords: Arnoldi process
zbMATH Keywords: numerical experiments
zbMATH Keywords: stability
zbMATH Keywords: finite precision
zbMATH Keywords: convergence
zbMATH Keywords: condition numbers
describes a project that uses: Harwell-Boeing sparse matrix collection
MaRDI profile type: Publication
cites work: GMRES: A Generalized Minimal Residual Algorithm for Solving Nonsymmetric Linear Systems
cites work: The convergence of Krylov subspace methods for large unsymmetric linear systems
cites work: Computable Convergence Bounds for GMRES
cites work: A Theoretical Comparison of the Arnoldi and GMRES Algorithms
cites work: GMRES/CR and Arnoldi/Lanczos as Matrix Approximation Problems
cites work: A simpler GMRES
cites work: Numerical stability of GMRES
cites work: Implementation of the GMRES Method Using Householder Transformations
cites work: Estimating the Attainable Accuracy of Recursively Computed Residual Methods
cites work: Least squares residuals and minimal residual methods
cites work: Q5689624
cites work: Roundoff error analysis of algorithms based on Krylov subspace methods
cites work: Sparse matrix test problems
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published in: Computers & Mathematics with Applications
description: scientific article; zbMATH DE number 2142392