NDLI: Residual and Backward Error Bounds in Minimum Residual Krylov Subspace Methods

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Paige, Christopher C. Strakos, Zdenvek
Copyright Year	2002
Abstract	Minimum residual norm iterative methods for solving linear systems Ax=b can be viewed as, and are often implemented as, sequences of least squares problems involving Krylov subspaces of increasing dimensions. The minimum residual method (MINRES) [C. Paige and M. Saunders, SIAM J. Numer. Anal., 12 (1975), pp. 617--629] and generalized minimum residual method (GMRES) [Y. Saad and M. Schultz, SIAM J. Sci. Statist. Comput., 7 (1986), pp. 856--869] represent typical examples. In [C. Paige and Z. Strakos, Bounds for the least squares distance using scaled total least squares, Numer. Math., to appear] revealing upper and lower bounds on the residual norm of any linear least squares (LS) problem were derived in terms of the total least squares (TLS) correction of the corresponding scaled TLS problem. In this paper theoretical results of [C. Paige and Z. Strakos, Bounds for the least squares distance using scaled total least squares, Numer. Math., to appear] are extended to the GMRES context. The bounds that are developed are important in theory, but they also have fundamental practical implications for the finite precision behavior of the modified Gram--Schmidt implementation of GMRES, and perhaps for other minimum norm methods.
Starting Page	1898
Ending Page	1923
Page Count	26
File Format	PDF
ISSN	10648275
DOI	10.1137/S1064827500381239
e-ISSN	10957197
Issue Number	6
Volume Number	23
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2006-07-25
Access Restriction	Subscribed
Subject Keyword	Krylov subspace methods total least squares Arnoldi method Sparse matrices eigenproblem Inequalities involving eigenvalues and eigenvectors Orthogonalization Iterative methods for linear systems least squares modified Gram--Schmidt iterative solution singular values large sparse matrices generalized minimum residual method Overdetermined systems, pseudoinverses linear equations
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Computational Mathematics

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