Comparison theorems for weak splittings in respect to a proper cone of nonsingular matrices
type: Item
instance of: scholarly article
title: Comparison theorems for weak splittings in respect to a proper cone of nonsingular matrices
zbMATH Open document ID: 0948.65030
DOI: 10.1016/S0024-3795(99)00158-5
author: Joan-Josep Climent
author: Carmen Perea
published in: Linear Algebra and its Applications
publication date: 2000-03-21T00:00:00Z
review text: Let \(K\) be a proper cone in \(\mathbb{R}^n\) and call the real \(n\times n\) matrix \(A\) \(K\)-nonnegative if \(AK\subseteq K\) writing \(A\geq 0\) in this case. A splitting \(A= M- N\) (\(M\) nonsingular) is called weak nonnegative if \(M^{-1}\geq 0\) an \(M^{-1}N\geq 0\) or if \(M^{-1}\geq 0\) and \(NM^{-1}\geq 0\). Denoting by \(\rho(A)\) the spectral radius of \(A\) the authors present conditions on two weak nonnegative splittings \(A= M_1- N_1= M_2- N_2\) such that the comparison result \[ \rho(M^{-1}_1 N_1)< \rho(M^{-1}_2 N_2)< 1\tag{1} \] holds with the first `\(<\)' being replaced by `\(\leq\)' in a part of the cases. As a main result they show that some of their conditions can be retrieved if (1) holds together with appropriate additional assumptions.
Mathematics Subject Classification ID: 65F10
Mathematics Subject Classification ID: 15B48
Mathematics Subject Classification ID: 15A42
zbMATH DE Number: 1419875
zbMATH Keywords: comparison conditions
zbMATH Keywords: iterative method
zbMATH Keywords: \(K\)-nonnegative matrix
zbMATH Keywords: \(K\)-irreducible matrix
zbMATH Keywords: proper cone
zbMATH Keywords: weak nonnegative splittings
reviewed by: Günter Mayer
MaRDI profile type: Publication
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Wikidata QID: Q127662430
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description: scientific article; zbMATH DE number 1419875