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Ill-Conditioned Inclusions

By A. S. Lewis

Abstract

A square system of linear equations is \ill-conditioned" when the norm of the corresponding inverse matrix is large. This norm bounds the size of the solution, and measures how close the system is to being inconsistent: it is thus of fundamental computational signicance. We generalize this idea from linear equations to inclusions governed by closed convex processes, and hence to \conic linear systems". 1 Research partially supported by the Natural Sciences and Engineering Research Council of Canada. The author thanks the University of Montpellier II for their hospitality while this work was completed. 1 1 Introduction Given an invertible n n real matrix F and a vector b in R n , consider the linear system Fx = b; x 2 R n : It is a simple consequence of the Eckart-Young theorem (see [7], for example) that the smallest operator norm of a matrix G making the perturbed matrix F +G singular is just kF 1 k 1 . This quantity is a fundamental measure of the \conditioning..

Year: 2000
OAI identifier: oai:CiteSeerX.psu:10.1.1.32.1139
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