Number of the records: 1
Modified Gram-Schmidt (MGS), Least Squares, and Backward Stability of MGS-GMRES
- 1.
SYSNO ASEP 0405455 Document Type J - Journal Article R&D Document Type Journal Article Subsidiary J Článek ve WOS Title Modified Gram-Schmidt (MGS), Least Squares, and Backward Stability of MGS-GMRES Title Modifikovaný Gram-Schmidtův algoritmus, úloha nejmenších čtverců a zpětná stabilita metody GMRES Author(s) Paige, C. C. (CA)
Rozložník, Miroslav (UIVT-O) SAI, RID, ORCID
Strakoš, Zdeněk (UIVT-O) SAI, RID, ORCIDSource Title SIAM Journal on Matrix Analysis and Applications. - : SIAM Society for Industrial and Applied Mathematics - ISSN 0895-4798
Roč. 28, č. 1 (2006), s. 264-284Number of pages 21 s. Language eng - English Country US - United States Keywords rounding error analysis ; modified Gram-Schmidt ; QR factorization ; loss of orthogonality ; least squares ; singular values ; backward stability ; linear equations ; condition numbers ; large sparse matrices ; iterative solution ; Krylov subspace methods ; Arnoldi method ; generalized minimum residual method Subject RIV BA - General Mathematics R&D Projects 1ET400300415 GA AV ČR - Academy of Sciences of the Czech Republic (AV ČR) CEZ AV0Z10300504 - UIVT-O (2005-2011) UT WOS 000237145900016 EID SCOPUS 33748334951 DOI 10.1137/050630416 Annotation The generalized minimum residual method (GMRES) [Y. Saad and M. Schultz,SIAM J. Sci. Statist. Comput., 7 (1986), pp. 856-869] for solving linear systems Ax=b is implemented as a sequence of least squares problems involving Krylov subspaces of increasing dimensions. The most usual implementation is modified Gram-Schmidt GMRES (MGS-GMRES). Here we show that MGS-GMRES is backward stable. The result depends on a more general result on the backward stability of a variant of the MGS algorithm applied to solving a linear least squares problem, and uses other new results on MGS and its loss of orthogonality, together with an important but neglected condition number, and a relation between residual norms and certain singular values. Workplace Institute of Computer Science Contact Tereza Šírová, sirova@cs.cas.cz, Tel.: 266 053 800 Year of Publishing 2007
Number of the records: 1