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A note on backward errors for structured linear systems
Author(s) -
Sun JiGuang
Publication year - 2005
Publication title -
numerical linear algebra with applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.02
H-Index - 53
eISSN - 1099-1506
pISSN - 1070-5325
DOI - 10.1002/nla.422
Subject(s) - mathematics , toeplitz matrix , measure (data warehouse) , vandermonde matrix , linear system , system of linear equations , karush–kuhn–tucker conditions , mathematical analysis , pure mathematics , mathematical optimization , computer science , eigenvalues and eigenvectors , physics , quantum mechanics , database
Abstract Let 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 thatand 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 thatCan 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.