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Main Title: Convergence of GMRES for tridiagonal Toeplitz matrices
Author(s): Liesen, Jörg
Strakoš, Zdeněk
Type: Article
Language Code: en
Abstract: We analyze the residuals of GMRES [Y. Saad and M. H. Schultz, SIAM J. Sci. Statist. Comput., 7 (1986), pp. 856--859], when the method is applied totridiagonal 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 [BIT, 40 (2000), pp. 524--535] and Eiermann and Ernst [Private communication, 2002], but we formulate and prove our results in a different way. We then extend the (lower) bidiagonal Jordan blocks to tridiagonal Toeplitz matrices and study extensions of our bidiagonal analysis to the tridiagonal case. 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. Such analyses are complicated by a cancellation of possibly huge components due to close eigenvectors, which can prevent achieving well-justified conclusions.
Issue Date: 2006
Date Available: 20-Dec-2017
DDC Class: 518 Numerische Analysis
512 Algebra
Subject(s): Krylov subspace methods
minimal residual methods
convergence analysis
Jordan blocks
Toeplitz matrices
Journal Title: SIAM Journal on Matrix Analysis and Applications
Publisher: Society for Industrial and Applied Mathematics
Publisher Place: Philadelphia, Pa
Volume: 26
Issue: 1
Publisher DOI: 10.1137/S0895479803424967
Page Start: 233
Page End: 251
EISSN: 1095-7162
ISSN: 0895-4798
Appears in Collections:FG Numerische Lineare Algebra » Publications

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