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Main Title: Properties of worst-case GMRES
Author(s): Faber, Vance
Liesen, Jörg
Tichý, Petr
Type: Article
Language Code: en
Abstract: 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 $A^T$ 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 $k\geq 3$. Finally, we give a complete characterization of how the values of the approximation problems change in the context of worst-case and ideal GMRES for a real matrix, when one considers complex (rather than real) polynomials and initial vectors.
Issue Date: 2013
Date Available: 14-Dec-2017
DDC Class: 518 Numerische Analysis
Subject(s): GMRES method
worst-case convergence
ideal GMRES
matrix approximation problems
Journal Title: SIAM Journal on Matrix Analysis and Applications
Publisher: Society for Industrial and Applied Mathematics
Publisher Place: Philadelphia, Pa.
Volume: 34
Issue: 4
Publisher DOI: 10.1137/13091066X
Page Start: 1500
Page End: 1519
EISSN: 1095-7162
ISSN: 0895-4798
Appears in Collections:FG Numerische Lineare Algebra » Publications

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