Convergence of asynchronous iteration with arbitrary splitting form
type: Item
instance of: scholarly article
title: Convergence of asynchronous iteration with arbitrary splitting form
zbMATH Open document ID: 0661.65035
DOI: 10.1016/0024-3795(89)90290-5
author: Lei Li
published in: Linear Algebra and its Applications
publication date: 1989-01-01T00:00:00Z
review text: Let \(x_ i=\sum^{n}_{j=1}a_{ij}x_ j+b_ i\), \(i=1,2,...,n\) be a linear system and \(\{p_{ik}\}^{\infty}_{k=0}\), \(\{q_{ik}\}^{\infty}_{k=0}\), \(i=1,2,...,n\) integer sequences such that \(0=q_{i0}\leq p_{i1}<q_{i1}\leq p_{i2}<... \). The following iterative method defined for integer \(t\geq 0\) and \(i=1,2,...,n\) by \(x_{it}=x^ 0_ i\quad if\quad t=0,\quad x_{it}=\sum^{n}_{j=1}a_{ij}x_{j,p_{ik}}+b_ i\) if \(t=q_{ik}\), \(k=1,2,...\), and \(x_{it}=x_{i,q_{ik}}\) if \(q_{ik}<t<q_{i,k+1}\), \(k=0,1,2,..\). is an asynchronous iterative method. The convergence of \(x_{it}\) to \(x_ i\) for \(t\to \infty\) is guaranteed if \(\rho (| A|)<1\), where \(A=(a_{ij})\), \(| A| =(| a_{ij}|)\) and \(\rho\) (\(\cdot)\) denotes the spectral radius. This result is proved using a new method and is then generalized to an iterative method of the same form obtained from splitting the matrix A in such a way that \(A=A_ 1+A_ 2=(a_{ij}^{(1)}+(a_{ij}^{(2)})\) with \(a_{ij}^{(1)}a_{ij}^{(2)}=0\) for every \(i,j=1,2,...,n\). Further extension of the result to the case of iteration with relaxation parameter is also proved.
reviewed by: P.Polcar
Mathematics Subject Classification ID: 65F10
zbMATH DE Number: 4080758
zbMATH Keywords: matrix splitting
zbMATH Keywords: asynchronous iterative method
zbMATH Keywords: convergence
zbMATH Keywords: relaxation
MaRDI profile type: Publication
full work available at URL: 0024-3795(89)90290-5
OpenAlex ID: W2048743071
cites work: Chaotic relaxation
cites work: Asynchronous Iterative Methods for Multiprocessors
cites work: Some convergence results for asynchronous algorithms
cites work: Q5342712
description: scientific article; zbMATH DE number 4080758