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How to make Simpler GMRES and GCR more Stable
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SYSNO ASEP 0310698 Druh ASEP J - Článek v odborném periodiku Zařazení RIV J - Článek v odborném periodiku Poddruh J Článek ve WOS Název How to make Simpler GMRES and GCR more Stable Překlad názvu Jak stabilizovat metody Simpler GMRES and GCR? Tvůrce(i) Jiránek, P. (CZ)
Rozložník, Miroslav (UIVT-O) SAI, RID, ORCID
Gutknecht, M. H. (CH)Zdroj.dok. SIAM Journal on Matrix Analysis and Applications. - : SIAM Society for Industrial and Applied Mathematics - ISSN 0895-4798
Roč. 30, č. 4 (2008), s. 1483-1499Poč.str. 17 s. Jazyk dok. eng - angličtina Země vyd. US - Spojené státy americké Klíč. slova large-scale nonsymmetric linear systems ; Krylov subspace methods ; minimum residual methods ; numerical stability ; rounding errors Vědní obor RIV BA - Obecná matematika CEP 1M0554 GA MŠMT - Ministerstvo školství, mládeže a tělovýchovy IAA100300802 GA AV ČR - Akademie věd IAA1030405 GA AV ČR - Akademie věd CEZ AV0Z10300504 - UIVT-O (2005-2011) UT WOS 000263103700013 EID SCOPUS 70449371819 DOI 10.1137/070707373 Anotace 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. Pracoviště Ústav informatiky Kontakt Tereza Šírová, sirova@cs.cas.cz, Tel.: 266 053 800 Rok sběru 2010
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