Computing a matrix symmetrizer exactly using modified multiple modulus residue arithmetic
type: Item
instance of: scholarly article
title: Computing a matrix symmetrizer exactly using modified multiple modulus residue arithmetic
zbMATH Open document ID: 0635.65046
DOI: 10.1016/0377-0427(88)90385-8
author: V. Ch. Venkaiah
author: Syamal K. Sen
published in: Journal of Computational and Applied Mathematics
publication date: 1988-01-01T00:00:00Z
review text: A multiple-modulus residue arithmetic is introduced and applied to computing an error-free matrix symmetrizer (i.e., a symmetric nonsingular X satisfying \(XA=A\) tX). In particular, when \(A=(a_{ij})\) is a lower Hessenberg matrix, with nonzero codiagonal elements, then the method recovers the following simple and elegant recursive algorithm to compute a symmetrizer, suggested by Datta (unpublished): Let x 1,x 2,...,x n be the rows of a symmetrizer X to be computed. Step 1: Choose \(x\quad n=(\alpha,0,...,0),\) where \(\alpha\neq 0\) is arbitrary; Step 2 Compute \(x^{n-1},x^{n-2},...,x\) 1 recursively from \(x\quad i=(1/a_{i,i+1})\times (x^{i+1}A-a_{i+1,i+1}x^{i+1}-a_{i+2,\quad i+1}x^{i+2}-...-a_{n,i+1}x\quad n),\) \(i=n-1,...,1\); Step 3: Print x 1,x 2,...,x n and stop.
Mathematics Subject Classification ID: 65F30
Mathematics Subject Classification ID: 65F15
Mathematics Subject Classification ID: 15A24
zbMATH DE Number: 4034997
zbMATH Keywords: Euclid's algorithm
zbMATH Keywords: Gauss elimination
zbMATH Keywords: similar matrices
zbMATH Keywords: non-symmetric eigenvalue problem
zbMATH Keywords: floating-point modular arithmetic
zbMATH Keywords: multiple-modulus residue arithmetic
zbMATH Keywords: error-free matrix symmetrizer
zbMATH Keywords: recursive algorithm
reviewed by: Sanjo Zlobec
MaRDI profile type: Publication
full work available at URL: 0377-0427(88)90385-8
OpenAlex ID: W2134760331
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description: scientific article; zbMATH DE number 4034997