On Stein-Rosenberg type theorems for nonnegative and Perron-Frobenius splittings
type: Item
instance of: scholarly article
title: On Stein-Rosenberg type theorems for nonnegative and Perron-Frobenius splittings
zbMATH Open document ID: 1153.65033
author: Dimitrios Noutsos
published in: Linear Algebra and its Applications
publication date: 2008-10-06T00:00:00Z
review text: The author discusses the iterative solution of linear systems by extending and generalizing the Stein-Rosenberg theorem. \textit{P. Stein} and \textit{R. L. Rosenberg} stated and proved an important theorem which compares the spectral radius of the Jacobi and Gauss-Seidel iterative methods [see J. Lond. Math. Soc. 23, 111--118 (1948; Zbl 0036.36501)]. Since 1948 the theorem of Stein and Rosenberg was generalized in many different ways. In this paper Stein-Rosenberg type theorems for a class of nonnegative splittings of the matrix of the linear system are discussed. Also, the class of Perron-Frobenius splittings are considered. Two types of Stein-Rosenberg theorems for each class of splittings are stated and proved. A comparison of these theorems to characterize which theorem is stronger than the other is presented. Characteristic examples to confirm the theoretical results are presented.
reviewed by: Iulian Coroian
Mathematics Subject Classification ID: 65F10
Mathematics Subject Classification ID: 15A42
zbMATH DE Number: 5349165
zbMATH Keywords: iterative methods
zbMATH Keywords: linear systems
zbMATH Keywords: Stein-Rosenberg theorem
zbMATH Keywords: nonnegative splittings
zbMATH Keywords: Perron-Frobenius theory
zbMATH Keywords: numerical examples
zbMATH Keywords: spectral radius
zbMATH Keywords: Gauss-Seidel iterative methods
MaRDI profile type: Publication
full work available at URL: j.laa.2008.05.033
OpenAlex ID: W2000327419
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DOI: 10.1016/J.LAA.2008.05.033
description: scientific article; zbMATH DE number 5349165