Imposing symmetry in augmented linear systems |
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Authors: | M Szularz |
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Institution: | School of Computing and Information Engineering, University of Ulster, , Coleraine BT52 1SA, Northern Ireland |
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Abstract: | This paper discusses the methods of imposing symmetry in the augmented system formulation (ASF) for least‐squares (LS) problems. A particular emphasis is on upper Hessenberg problems, where the challenge lies in leaving all zero‐by‐definition elements of the LS matrix unperturbed. Analytical solutions for optimal perturbation matrices are given, including upper Hessenberg matrices. Finally, the upper Hessenberg LS problems represented by unsymmetric ASF that indicate a normwise backward stability of the problem (which is not the case in general) are identified. It is observed that such problems normally arise from Arnoldi factorization (for example, in the generalized minimal residual (GMRES) algorithm). The problem is illustrated with a number of practical (arising in the GMRES algorithm) and some ‘purpose‐built’ examples. Copyright © 2014 John Wiley & Sons, Ltd. |
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Keywords: | GMRES algorithm least‐squares problem Hessenberg matrix Arnoldi method augmented system formulation generalized Kieł basiń ski– Schwetlick lemma sensitivity of QR factorization quadratic programming with box constraints |
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