Backward error and condition of structured linear systems

Higham, Desmond J. and Higham, Nicholas J. (1992) Backward error and condition of structured linear systems. SIAM Journal On Matrix Analysis And Applications, 13 (1). pp. 162-175. ISSN 1095-7162

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Abstract

Existing definitions of backward error and condition number for linear systems do not cater to structure in the coefficient matrix, except possibly for sparsity. The definitions are extended so that when the coefficient matrix has structure the perturbed matrix has this structure too. It is shown that when the structure comprises linear dependence on a set of parameters, the structured componentwise backward error is given by the solution of minimal $\infty $ -norm to an underdetermined linear system; an explicit expression for the condition number in this linear case is also obtained. Applications to symmetric matrices, Toeplitz matrices and the least squares problem are discussed and illustrated through numerical examples.

Item Type:	Article
Uncontrolled Keywords:	componentwise backward error, condition number, underdetermined system, symmetric matrix, Toeplitz matrix, least squares problem, augmented system
Subjects:	MSC 2010, the AMS's Mathematics Subject Classification > 15 Linear and multilinear algebra; matrix theory MSC 2010, the AMS's Mathematics Subject Classification > 65 Numerical analysis
Depositing User:	Ms Lucy van Russelt
Date Deposited:	03 Jul 2006
Last Modified:	20 Oct 2017 14:12
URI:	https://eprints.maths.manchester.ac.uk/id/eprint/347

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