A new criterion for the H-matrix property
type: Item
instance of: scholarly article
title: A new criterion for the H-matrix property
zbMATH Open document ID: 1022.65048
DOI: 10.1016/S0377-0427(02)00666-0
author: Keiko Ojiro
author: Hiroshi Niki
author: Masataka Usui
published in: Journal of Computational and Applied Mathematics
publication date: 2003-03-11T00:00:00Z
review text: The comparison matrix \({\mathcal M}(A)=(\alpha_{ij}) \in {\mathbb R}^{n,n}\) of a matrix \(A=(a_{ij}) \in {\mathbb R}^{n,n}\) is defined by \(\alpha_{ii}=|a_{ii}|\), \(\alpha_{ij}=-|a_{ij}|\) if \(i \neq j\). A matrix is called an H-matrix if the eigenvalues of its comparison matrix have positive real parts. It is known that \(A\) is an H-matrix if and only if \(A\) is generalized strictly diagonally dominant, i.e. there exists a diagonal matrix \(D\) with positive entries such that \(AD\) is strictly diagonally dominant. The latter property is important in the proofs of convergence theorems for certain iterative methods to solve linear systems with \(A\), notably Gauss-Seidel, Jacobi and successive overrelaxation. The present authors construct an iterative algorithm that decides in a finite number of steps if a given matrix (irreducible and with at least one dominant diagonal element) is an H-matrix. The cost per iteration step is only \(O(n)\) where earlier methods had \(O(n^2)\). The authors prove this result and provide several numerical examples for small values of \(n\).
Mathematics Subject Classification ID: 65F30
Mathematics Subject Classification ID: 65F10
zbMATH DE Number: 1879844
zbMATH Keywords: diagonally dominant matrix
zbMATH Keywords: comparison matrix
zbMATH Keywords: Gauss-Seidel method
zbMATH Keywords: Jacobi method
zbMATH Keywords: H-matrix
zbMATH Keywords: iterative methods
zbMATH Keywords: convergence
zbMATH Keywords: algorithm
zbMATH Keywords: numerical examples
zbMATH Keywords: successive overrelaxation
reviewed by: Willy J. F. Govaerts
MaRDI profile type: Publication
cites work: Iterative Solution Methods
cites work: Q4326384
cites work: Criteria for generalized diagonally dominant matrices and \(M\)-matrices
cites work: An extension of the criteria for generalized diagonally dominant matrices
cites work: Q4040931
cites work: An iterative test for \(H\)-matrix
cites work: Über die Determinanten mit überwiegender Hauptdiagonale
cites work: Q5342712
cites work: On recurring theorems on diagonal dominance
description: scientific article; zbMATH DE number 1879844