The Gauss-Seidel iterative method with the preconditioning matrix \((I+S+S_m)\)
type: Item
instance of: scholarly article
title: The Gauss-Seidel iterative method with the preconditioning matrix \((I+S+S_m)\)
zbMATH Open document ID: 1058.65042
DOI: 10.1007/BF03167430
author: Munenori Morimoto
author: Kyouji Harada
author: Michio Sakakihara
author: Hideo Sawami
published in: Japan Journal of Industrial and Applied Mathematics
publication date: 2004-09-27T00:00:00Z
review text: An improved preconditioner is discussed for the Gauss-Seidel method applied to a linear system \(Ax=b\) where \(A\) is a matrix with ones on the diagonal and all other elements non-positive. \textit{A. D.~Gunawardena, S. K.~Jain}, and \textit{L.~Snyder} [Linear Algebra Appl. 154--156, 123--143 (1991; Zbl 0731.65016)] have proposed a preconditioner of the form \(I+S\) with \(S\) the zero matrix, except for the elements \((i,i+1)\) which are replaced by \(a_{ij}\), the corresponding element from \(A\). Here it is proposed to add a correction term and use \(I+S+S_m\) where \(S_m\) is the zero matrix, except for the elements \((i,j)\) which is \(-a_{ij}\) where \(a_{ij}\) is the first element in row \(i\) of \(A\) satisfying \(| a_{ij}| =\max_{i+1<k}| a_{ik}| \). The different precontitioners \(I\) (no preconditioning), \(I+S\) (Gunawardena et al. [loc. cit.]) and \(I+S+S_m\) are compared and it is shown, in theory and by numerical examples, that the the latter gives indeed a slight improvement over \(I+S\) (the spectral radius is slightly smaller and execution time is shorter).
reviewed by: Adhemar Bultheel
Mathematics Subject Classification ID: 65F10
Mathematics Subject Classification ID: 65F35
zbMATH DE Number: 2103572
zbMATH Keywords: Gauss-Seidel method
zbMATH Keywords: preconditioning
zbMATH Keywords: numerical examples
zbMATH Keywords: linear system
MaRDI profile type: Publication
cites work: Some comparison theorems for weak nonnegative splittings of bounded operators
cites work: Modified iterative methods for consistent linear systems
cites work: Improving the modified Gauss-Seidel method for \(Z\)-matrices
cites work: Comparison theorems for weak splittings of bounded operators
cites work: Q4947421
full work available at URL: https://doi.org/10.1007/bf03167430
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description: scientific article; zbMATH DE number 2103572