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Properties of Worst-Case GMRES
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SYSNO ASEP 0421797 Document Type J - Journal Article R&D Document Type Journal Article Subsidiary J Článek ve WOS Title Properties of Worst-Case GMRES Author(s) Faber, V. (US)
Liesen, J. (DE)
Tichý, Petr (UIVT-O) SAI, RID, ORCIDSource Title SIAM Journal on Matrix Analysis and Applications. - : SIAM Society for Industrial and Applied Mathematics - ISSN 0895-4798
Roč. 34, č. 4 (2013), s. 1500-1519Number of pages 20 s. Language eng - English Country US - United States Keywords GMRES method ; worst-case convergence ; ideal GMRES ; matrix approximation problems ; minmax Subject RIV BA - General Mathematics R&D Projects GA13-06684S GA ČR - Czech Science Foundation (CSF) Institutional support UIVT-O - RVO:67985807 UT WOS 000328902900004 EID SCOPUS 84892418917 DOI 10.1137/13091066X Annotation In the convergence analysis of the GMRES method for a given matrix A, one quantity of interest is the largest possible residual norm that can be attained, at a given iteration step k, over all unit norm initial vectors. This quantity is called the worst-case GMRES residual norm for A and k. We show that the worst case behavior of GMRES for the matrices A and A transposed is the same, and we analyze properties of initial vectors for which the worst-case residual norm is attained. In particular, we prove that such vectors satisfy a certain "cross equality". We show that the worst-case GMRES polynomial may not be uniquely determined, and we consider the relation between the worst-case and the ideal GMRES approximations, giving new examples in which the inequality between the two quantities is strict at all iteration steps k greater than 3. Workplace Institute of Computer Science Contact Tereza Šírová, sirova@cs.cas.cz, Tel.: 266 053 800 Year of Publishing 2014
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