Ill-conditionedness need not be componentwise near to ill-posedness for least squares problems
type: Item
instance of: scholarly article
title: Ill-conditionedness need not be componentwise near to ill-posedness for least squares problems
zbMATH Open document ID: 0970.65039
DOI: 10.1023/A:1022377410087
published in: BIT
publication date: 2001-10-24T00:00:00Z
review text: The condition number of a problem measures the sensitivity of the answer to small changes in the input, where ``small'' refers to some distance measure. A problem is called ill-conditioned if the condition number is large, and it is called ill-posed if the condition number is infinity. It is known that for many problems the (normwise) distance to the nearest ill-posed problem is proportional to the reciprocal of the condition number. Recently it has been shown that for linear systems and matrix inversion this is also true for componentwise distances. In this paper the author shows that this is no longer true for least squares problems and other problems involving rectangular matrices. The author identifies problems which are arbitrarily ill-conditioned (in a componentwise sense) whereas any componetwise relative perturbation less than 1 cannot produce an ill-posed problem. Bounds are given using additional information on the matrix.
Mathematics Subject Classification ID: 65F22
Mathematics Subject Classification ID: 65F20
zbMATH DE Number: 1275247
zbMATH Keywords: componentwise distance
zbMATH Keywords: condition number
zbMATH Keywords: pseudoinverse
zbMATH Keywords: undetermined linear systems
zbMATH Keywords: ill-posed problem
zbMATH Keywords: least squares problems
author: Siegfried M. Rump
reviewed by: G. Dimitriu
describes a project that uses: Maple
MaRDI profile type: Publication
full work available at URL: a:1022377410087
OpenAlex ID: W1539082189
description: scientific article; zbMATH DE number 1275247