Comparison theorems for splittings of matrices
type: Item
instance of: scholarly article
title: Comparison theorems for splittings of matrices
zbMATH Open document ID: 1012.65028
author: Yongzhong Song
published in: Numerische Mathematik
publication date: 2002-12-01T00:00:00Z
review text: Let \(A\in \mathbb{C}^{n\times n}\) and \(x,b\in \mathbb{C}^n\) with \(x\) unknown. In order to solve a linear system of the form \(Ax=b\) with iterative methods, the author splits the matrix \(A\) into \(A=M-N\) (or \(A=M_{k}-N_{k}\), \(k=1,2,\dots ,m\) ``parallel multisplitting method'') with \(M\) nonsingular (\(M_{k}\) nonsingular) and then uses iterative formulas for solving the system. The transition matrices used in these iterative formula are functions of the matrices \(M,N\) (or \(M_{k},N_{k}\)). This paper suggests two different nonnegative splitting methods (nonnegative parallel multisplitting methods) and then, based on the nonnegative matrix theory, compares the asymptotic rates of convergence of its iteration matrices. The above comparisons are also applied for the special case where \(A\) is a Hermitian positive (semi)definite matrix.
Mathematics Subject Classification ID: 65F10
Mathematics Subject Classification ID: 65Y05
Mathematics Subject Classification ID: 15A06
zbMATH DE Number: 1837367
zbMATH Keywords: comparison of methods
zbMATH Keywords: parallel computation
zbMATH Keywords: iterative methods
zbMATH Keywords: multisplitting method
zbMATH Keywords: nonnegative splitting methods
zbMATH Keywords: convergence
zbMATH Keywords: Hermitian positive semidefinite matrix
reviewed by: Nicholas Karampetakis
MaRDI profile type: Publication
full work available at URL: s002110100333
OpenAlex ID: W2061592236
DOI: 10.1007/S002110100333
description: scientific article; zbMATH DE number 1837367