Numerical stability of GMRES
type: Item
instance of: scholarly article
title: Numerical stability of GMRES
zbMATH Open document ID: 0837.65040
DOI: 10.1007/BF01732607
author: Zdeněk Strakoš
author: Jitka Drkošova
author: Anne Greenbaum
author: Miroslav Rozložník
published in: BIT
publication date: 1996-05-21T00:00:00Z
review text: The generalized minimal residual (GMRES) method is one of the most popular methods for solving systems of linear equations with nonsymmetric coefficient matrices. The authors study the numerical stability of GMRES when the computation of approximations is based on constructing an orthonormal basis of Krylov subspaces (Arnoldi basis) and after that the transformed least squares problem is solved. It is shown that if the Arnoldi basis is computed via Householder orthogonalization and the transformed least squares problem is solved using Givens rotations, then the computed GMRES approximation has a guaranteed backward error of size at worst \(O(N^{{5\over 2}}\varepsilon)\), where \(N\) is the size of the coefficient matrix and \(\varepsilon\) is the machine precision. Reading the paper, one can find the description of the Arnoldi recurrence for the quantities actually computed in finite precision arithmetic and an analysis of the relation between the true and Arnoldi residuals in the presence of rounding errors.
reviewed by: Krassimir Georgiev
Mathematics Subject Classification ID: 65F20
Mathematics Subject Classification ID: 65F10
zbMATH DE Number: 827429
zbMATH Keywords: nonsymmetric linear system
zbMATH Keywords: generalized minimal residual method
zbMATH Keywords: iterative methods
zbMATH Keywords: Arnoldi basis
zbMATH Keywords: numerical stability
zbMATH Keywords: GMRES
zbMATH Keywords: Krylov subspaces
zbMATH Keywords: least squares problem
zbMATH Keywords: Householder orthogonalization
zbMATH Keywords: Givens rotations
zbMATH Keywords: backward error
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01732607
OpenAlex ID: W2037785139
description: scientific article; zbMATH DE number 827429