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Main Title: The worst-case GMRES for normal matrices
Author(s): Liesen, Jörg
Tichý, Petr
Type: Research Paper
Abstract: We study the convergence of GMRES for linear algebraic systems with normal matrices. In particular, we explore the standard bound based on a min-max approximation problem on the discrete set of the matrix eigenvalues. We completely characterize the worst-case GMRES-related quantities in the next-to-last iteration step and evaluate the standard bound in terms of explicit polynomials involving the matrix eigenvalues. For a general iteration step, we develop a computable lower and upper bound on the standard bound. Our bounds allow to study the worst-case GMRES residual norm in dependence of the eigenvalue distribution. For hermitian matrices the lower bound is equal to the worst-case residual norm. In addition, numerical experiments show that the lower bound is generally very tight, and support our conjecture that it is to within a constant factor of the actual worst-case residual norm. Since the worst-case residual norm in each step is to within a factor of the square root of the matrix size to what is considered an ``average'' residual norm, our results are of relevance beyond the worst case.
Subject(s): GMRES
evaluation of convergence
ideal GMRES
normal matrices
min-max problem
Issue Date: 15-Sep-2003
Date Available: 17-Dec-2021
Language Code: en
DDC Class: 510 Mathematik
MSC 2000: 65F10 Iterative methods for linear systems
65F15 Eigenvalues, eigenvectors
65F20 Overdetermined systems, pseudoinverses
15A06 Linear equations
15A09 Matrix inversion, generalized inverses
15A18 Eigenvalues, singular values, and eigenvectors
Series: Preprint-Reihe des Instituts für Mathematik, Technische Universität Berlin
Series Number: 2003, 27
ISSN: 2197-8085
TU Affiliation(s): Fak. 2 Mathematik und Naturwissenschaften » Inst. Mathematik
Appears in Collections:Technische Universität Berlin » Publications

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