Exact SOR convergence regions for a general class of \(p\)-cyclic matrices
type: Item
instance of: scholarly article
title: Exact SOR convergence regions for a general class of \(p\)-cyclic matrices
zbMATH Open document ID: 0839.65035
DOI: 10.1007/BF01739821
published in: BIT
publication date: 1996-06-16T00:00:00Z
review text: To solve the system \(Ax= b\) by successive overrelaxation (SOR), the iteration is \(x^{m+ 1}= {\mathcal L} x^m+ c\) with \({\mathcal L}= (I- \omega L)^{- 1}[(1- \omega)I+ \omega U]\), \(c= \omega(I- \omega L)^{- 1} D^{- 1} b\), with \(\omega\) the relaxation parameter. Here \(A\) is a \(p\times p\) block matrix and \(A= L+ D+ U\) is the usual decomposition in lower, diagonal and upper block parts. It is assumed that the Jacobi matrix \(B= I- D^{- 1} A\) is weakly cyclic, i.e., that it is generated from a block diagonal matrix by a cyclic permutation which can not be split in subcycles. Convergence of the SOR method depends on the spectrum \(\sigma({\mathcal L})\) of \(\mathcal L\) being inside or outside a disk centered at the origin. Since there is a relation between the spectra \(\sigma(B)\) and \(\sigma({\mathcal L})\), there is a largest region in the complex plane containing \(\sigma(B)\) for which the method converges. This region is described in this paper. Also an answer is given to the question for what pairs \((\rho(B),\omega)\) (\(\rho(B)\) is the spectral radius of \(B\)) the method does converge. To describe these regions, extensive use is made of hypocycloid curves in the complex plane. The usefulness of these curves follows naturally from the relation between \(\sigma(B)\) and \(\sigma({\mathcal L})\). Especially the case \(p= 5\) is studied in more detail and explicit formulas are given. The paper generalizes many exemplary results that were previously obtained by several authors.
reviewed by: Adhemar Bultheel
Mathematics Subject Classification ID: 65F10
zbMATH DE Number: 844663
zbMATH Keywords: convergence regions
zbMATH Keywords: iterative methods
zbMATH Keywords: SOR method
zbMATH Keywords: \(p\)-cyclic block matrices
zbMATH Keywords: successive overrelaxation
zbMATH Keywords: relaxation parameter
zbMATH Keywords: Jacobi matrix
zbMATH Keywords: spectral radius
zbMATH Keywords: hypocycloid curves
author: A. Hadjidimos
author: Dimitrios Noutsos
author: M. Tzoumas
MaRDI profile type: Publication
cites work: Q4326384
cites work: On the solution of singular linear systems of algebraic equations by semiiterative methods
cites work: Optimal successive overrelaxation iterative methods for \(p\)-cyclic matrices
cites work: Best Cyclic Repartitioning for Optimal Successive Overrelaxation Convergence
cites work: On different classes of monoparametric stationary iterative methods for the solution of linear systems
cites work: On the convergence of monoparametrick-step iterative euler methods for the solution of linear systems
cites work: On the equivalence of the k-step iterative Euler methods and successive overrelaxation (SOR) methods for k-cyclic matrices
cites work: On the optimum relaxation factor associated with \(p\)-cyclic matrices
cites work: On the Optimization of the Classical Iterative Schemes for the Solution of Complex Singular Linear Systems
cites work: On the convergence domains of the \(p\)-cyclic SOR
cites work: Q4281297
cites work: Optimal \(p\)-cyclic SOR
cites work: Block cyclic SOR for Markov chains with \(p\)-cyclic infinitesimal generator
cites work: On complex successive overrelaxation
cites work: A note on the SSOR and USSOR iterative methods applied to p-cyclic matrices
cites work: Convergence of a direct-iterative method for large-scale least-squares problems
cites work: Generalised consistent ordering and the optimum successive overrelaxation factor
cites work: The analysis of k-step iterative methods for linear systems from summability theory
cites work: Convergence of block iterative methods applied to sparse least-squares problems
cites work: Optimal stretched parameters for the SOR iterative method
cites work: An Operator Relation of the Ussor and the Jacobi Iteration Matrices of a p-Cyclic Matrix
cites work: Optimality relationships for p-cyclic SOR
cites work: Q4054161
cites work: \(p\)-cyclic matrices: A generalization of the Young-Frankel successive overrelaxation scheme
cites work: Characterization of matrix variate normal distributions
cites work: On generalizations of the theory of consistent orderings for successive overrelaxation methods
cites work: Over- and underrelaxation for linear systems with weakly cyclic Jacobi matrices of index p
cites work: Iterative Methods for Solving Partial Difference Equations of Elliptic Type
cites work: Q5639278
Recommended article: Q4359037
Recommended article: Precise domains of convergence for the block SSOR method associated with p-cyclic matrices
Recommended article: On the convergence domains of the \(p\)-cyclic SOR
Recommended article: On the exact \(p\)-cyclic SSOR convergence domains
Recommended article: Convergence domains of the SSOR method for a class of generalized consistently ordered matrices
Recommended article: Q5453454
Recommended article: Q3610290
Recommended article: Semiconvergence of block SOR method for singular linear systems with \(p\)-cyclic matrices
Recommended article: The optimal parameter of SOR-\(k\) method for \(p\)-cyclic matrices
Recommended article: The SOR-kmethod for linear systems withp-cyclic matrices
description: scientific article; zbMATH DE number 844663