Structured backward error and condition number for linear systems of the type \(A^* Ax = b\)
type: Item
instance of: scholarly article
title: Structured backward error and condition number for linear systems of the type \(A^* Ax = b\)
zbMATH Open document ID: 0958.65054
DOI: 10.1023/A:1022366318322
author: V. Toumazou
author: Valérie Frayssé
author: Serge Gratton
published in: BIT
publication date: 2001-03-28T00:00:00Z
review text: This paper presents a summary of a thorough study of the structured backward error analysis and the condition number of systems \[ A^*Ax=b,\tag{1} \] where \(A\) is an \(m\times n\) matrix of rank \(n\). It is important to point out that the system (1) is not the system of normal equations resulting from a least-squares problem. The paper derives a formulation of the structured condition number of (1.1) when the matrix \(A\) is the object of normwise perturbations. Then the corresponding backward error analysis is obtained, whose result can be viewed as the solution of a minimization problem over a sphere. The proofs of these results, whose mere enunciation is quite involved, can be found in technical reports by the same authors. A major feature in the paper is the section devoted to reporting results on numerical experiments performed with MATLAB on test matrices taken from Nick Higham's collection. The experiments show that the structured analysis proposed succeeds in predicting errors faithfully.
reviewed by: Juan Pedro Milaszewicz
Mathematics Subject Classification ID: 65F35
Mathematics Subject Classification ID: 65G50
zbMATH DE Number: 1471139
zbMATH Keywords: structured perturbations
zbMATH Keywords: condition number
zbMATH Keywords: backward error analysis
zbMATH Keywords: numerical experiments
describes a project that uses: AAFAC
describes a project that uses: Matlab
MaRDI profile type: Publication
full work available at URL: a:1022366318322
OpenAlex ID: W150509771
description: scientific article; zbMATH DE number 1471139