A note on backward errors for structured linear systems

Abstract

AbstractLet x̃ be a computed solution to a linear system Ax=b with , where is a proper subclass of matrices in . A structured backward error (SBE) of x̃ is defined by a measure of the minimal perturbations and such that equation image and that the SBE can be used to distinguish the structured backward stability of the computed solution x̃. For simplicity, we may define a partial SBE of x̃ by a measure of the minimal perturbation such that equation image Can one use the partial SBE to distinguish the structured backward stability of x̃? In this note we show that the partial SBE may be much larger than the SBE for certain structured linear systems such as symmetric Toeplitz systems, KKT systems, and dual Vandermonde systems. Besides, certain backward errors for linear least squares are discussed. Copyright © 2004 John Wiley & Sons, Ltd.