A parallel iterative system solver
type: Item
instance of: scholarly article
title: A parallel iterative system solver
zbMATH Open document ID: 0575.65026
DOI: 10.1016/0024-3795(85)90085-0
author: Nikolaos M. Missirlis
published in: Linear Algebra and its Applications
publication date: 1985-01-01T00:00:00Z
review text: The author examines an iterative method for solution of the linear system \(Au=b\), where the matrix A has real eigenvalues. Writing \(A=D(I-B)\), where D is diagonal and I the identity, the iterative method is given by \[ u^{(n+1)}=u^{(n)}+\tau (\sum^{l-1}_{k=0}\omega_ kB^ k)D^{-1}(b-Au^{(n)}), \] where l,\(\tau\), and \(\{\omega_ k\}_ 0^{l-1}\) are predetermined, fixed parameters. This method is appropriate for pipelined or parallel (SIMD or MIMD) computers. With the parameters \(\omega_ k=1\), \(k=0...(l-1)\), the author generalizes known results for \(l=1,2\) to compute the optimal value of \(\tau\) and the resulting convergence rate in terms of the extreme eigenvalues of B. Based on this analysis, the best choice for l is given as \(l=2\). For the case when \(l=2\), when the matrix B has precisely two distinct real eigenvalues, and when \(\omega_ k=\omega\), \(k=0...(l-1)\), the author computes the optimal value of \(\omega\) and the resulting convergence rate. The optimal value turns out to be \(\omega =1\).
Mathematics Subject Classification ID: 65F10
zbMATH DE Number: 3919647
zbMATH Keywords: parallel algorithms
zbMATH Keywords: convergence rate
reviewed by: Myron Sussman
MaRDI profile type: Publication
full work available at URL: 0024-3795(85)90085-0
OpenAlex ID: W2001001100
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description: scientific article; zbMATH DE number 3919647