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How to make Simpler GMRES and GCR more Stable
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SYSNO ASEP 0310698 Document Type J - Journal Article R&D Document Type Journal Article Subsidiary J Článek ve WOS Title How to make Simpler GMRES and GCR more Stable Title Jak stabilizovat metody Simpler GMRES and GCR? Author(s) Jiránek, P. (CZ)
Rozložník, Miroslav (UIVT-O) SAI, RID, ORCID
Gutknecht, M. H. (CH)Source Title SIAM Journal on Matrix Analysis and Applications. - : SIAM Society for Industrial and Applied Mathematics - ISSN 0895-4798
Roč. 30, č. 4 (2008), s. 1483-1499Number of pages 17 s. Language eng - English Country US - United States Keywords large-scale nonsymmetric linear systems ; Krylov subspace methods ; minimum residual methods ; numerical stability ; rounding errors Subject RIV BA - General Mathematics R&D Projects 1M0554 GA MŠMT - Ministry of Education, Youth and Sports (MEYS) IAA100300802 GA AV ČR - Academy of Sciences of the Czech Republic (AV ČR) IAA1030405 GA AV ČR - Academy of Sciences of the Czech Republic (AV ČR) CEZ AV0Z10300504 - UIVT-O (2005-2011) UT WOS 000263103700013 EID SCOPUS 70449371819 DOI 10.1137/070707373 Annotation In this paper we analyze the numerical behavior of several minimum residual methods, which are mathematically equivalent to the GMRES method. Two main approaches are compared: the one that computes the approximate solution in terms of a Krylov space basis from an upper triangular linear system for the coordinates, and the one where the approximate solutions are updated with a simple recursion formula. We show that a different choice of the basis can significantly influence the numerical behavior of the resulting implementation. While Simpler GMRES and ORTHODIR are less stable due to the ill-conditioning of the basis used, the residual basis is well-conditioned as long as we have a reasonable residual norm decrease. These results lead to a new implementation, which is conditionally backward stable, and they explain the experimentally observed fact that the GCR method delivers very accurate approximate solutions when it converges fast enough without stagnation. Workplace Institute of Computer Science Contact Tereza Šírová, sirova@cs.cas.cz, Tel.: 266 053 800 Year of Publishing 2010
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