A method for the improvement of the convergence rates of stationary iterative schemes for the solution of complex linear systems
type: Item
instance of: scholarly article
title: A method for the improvement of the convergence rates of stationary iterative schemes for the solution of complex linear systems
zbMATH Open document ID: 0624.65026
DOI: 10.1016/0377-0427(86)90234-7
published in: Journal of Computational and Applied Mathematics
publication date: 1986-01-01T00:00:00Z
review text: This paper presents some theorems and lemmas which may prove useful in improving the convergence rates of stationary iterative schemes for complex linear systems. The coefficient matrix of the system \(Ax=b\) is written in the form (k-part splitting) \(A=A_ 0-A_ 1-...-A_ k\) leading to the stationary iterative scheme (1) \(A_ 0x_{n+k}=A_ 1x_{n+k-1}+...+A_ kx_ n+b\). \textit{J. de Pillis} and \textit{M. Neumann} [IMA J. Numer. Anal. 1, 65-79 (1981; Zbl 0476.65017)] have shown how the asymptotic behaviour of (1) may be obtained from that of the equivalent first-order scheme \(y_{n+1}=Wy_ n+C\) with approximate choice of W. The present authors use a modification of the power method to derive a polynomial whose roots, are, asymptotically, the m eigenvalues of W greatest in modulus. A discussion of implementation of this result is given and an ad hoc `dynamic process' to approximate the solution of the equations from current estimates of the eigenvalues which makes substantial savings on the number of iterations for slowly convergent problems is described. The paper concludes with some numerical applications based on a Dirichlet boundary value problem.
Mathematics Subject Classification ID: 65F10
Mathematics Subject Classification ID: 65F15
Mathematics Subject Classification ID: 65N22
zbMATH DE Number: 4013863
zbMATH Keywords: numerical examples
zbMATH Keywords: complex linear systems
zbMATH Keywords: convergence acceleration
zbMATH Keywords: convergence rates
zbMATH Keywords: k-part splitting
zbMATH Keywords: stationary iterative scheme
zbMATH Keywords: power method
author: G. Avdelas
author: Alexandros Leontitsis
MaRDI profile type: Publication
full work available at URL: 0377-0427(86)90234-7
OpenAlex ID: W2078230582
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description: scientific article; zbMATH DE number 4013863