On the stability of the incomplete LU-factorizations and characterizations of \(H\)-matrices
type: Item
instance of: scholarly article
title: On the stability of the incomplete LU-factorizations and characterizations of \(H\)-matrices
zbMATH Open document ID: 0822.65018
author: V. Pereyra
published in: Numerische Mathematik
publication date: 1995-10-15T00:00:00Z
review text: An \(n\times n\) real matrix \(A= [a_{i,j}]\) is an M-matrix if \(a_{i,i}\geq 0\), if \(a_{i,j}\leq 0\) for all \(i\neq j\), and if \(A^{- 1}\geq 0\). The comparison matrix \({\mathcal M}(A)= [\alpha_{i, j}]\) of an arbitrary \(n\times n\) complex matrix \(A= [a_{i, j}]\) is defined by \(\alpha_{i, j}:= | a_{i, j}|\) if \(i= j\) and \(\alpha_{ij}:= -| a_{i, j}|\) if \(i\neq j\). An \(n\times n\) complex matrix \(A\) is an H-matrix iff \({\mathcal M}(A)\) is an M-matrix. The computational stability of incomplete LU-factorizations of H-matrices is investigated. The main results: The incomplete LU-factorization of an H-matrix \(A\) is at least as stable as the incomplete LU-factorization of its comparison matrix \({\mathcal M}(A)\). The incomplete LU-factorization of an H-matrix \(A\) is at least as stable as the complete LU-factorization of its comparison matrix \({\mathcal M}(A)\) without pivoting. If \(A\) is a Hermitian H-matrix and the Cholesky method is applied then an incomplete \(\text{LL}^*\)-factorization for \(A\) is at least as stable as the incomplete factorization of \({\mathcal M}(A)\). If \(A\) is a Hermitian H- matrix and \(A\) is symmetric then the incomplete \(\text{LL}^*\)- factorization of \(A\) is at least as stable as Cholesky's method applied to \({\mathcal M}(A)\). Some new characteristics of H-matrices in connection with their incomplete LU-factorizations are given.
Mathematics Subject Classification ID: 65F10
Mathematics Subject Classification ID: 65F05
zbMATH DE Number: 739414
zbMATH Keywords: M-matrix
zbMATH Keywords: comparison matrix
zbMATH Keywords: H-matrix
zbMATH Keywords: computational stability
zbMATH Keywords: incomplete LU-factorizations
zbMATH Keywords: Cholesky method
MaRDI profile type: Publication
full work available at URL: s002110050095
OpenAlex ID: W1968421132
Wikidata QID: Q114231043
DOI: 10.1007/S002110050095
description: scientific article; zbMATH DE number 739414