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Convergence of GMRES for Tridiagonal Toeplitz Matrices
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SYSNO ASEP 0103272 Document Type J - Journal Article R&D Document Type Journal Article Subsidiary J Článek ve WOS Title Convergence of GMRES for Tridiagonal Toeplitz Matrices Title Konvergence metody GMRES pro třídiagonální Toeplitzovské matice Author(s) Liesen, J. (DE)
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. 26, č. 1 (2004), s. 233-251Number of pages 19 s. Language eng - English Country US - United States Keywords Krylov subspace methods ; GMRES ; minimal residual methods ; convergence analysis ; Jordan blocks ; Toeplitz matrices Subject RIV BA - General Mathematics R&D Projects GA201/02/0595 GA ČR - Czech Science Foundation (CSF) CEZ AV0Z1030915 - UIVT-O UT WOS 000225642900011 EID SCOPUS 14544308440 DOI https://doi.org/10.1137/S0895479803424967 Annotation We analyze the residuals of GMRES, when the method is applied to tridiagonal Toeplitz matrices. We first derive formulas for the residuals as well as their norms when GMRES is applied to scaled Jordan blocks. This problem has been studied previously by Ipsen and Eiermann and Ernst, but we formulate and prove our results in a different way. Intuitively, when a scaled Jordan block is extended to a tridiagonal Toeplitz matrix by a superdiagonal of small modulus (compared to the modulus of the subdiagonal), the GMRES residual norms for both matrices and the same initial residual should be close to each other. We confirm and quantify this intuitive statement. We also demonstrate principal difficulties of any GMRES convergence analysis which is based on eigenvector expansion of the initial residual when the eigenvector matrix is ill-conditioned. Workplace Institute of Computer Science Contact Tereza Šírová, sirova@cs.cas.cz, Tel.: 266 053 800 Year of Publishing 2005
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