Inhalt des Dokuments
Preprint 27-2003
The worst-case GMRES for normal matrices
Author(s) :
Jörg Liesen
,
Petr Tichy
Preprint series of the Institute of Mathematics, Technische Universität Berlin
Preprint 27-2003
MSC 2000
- 65F10 Iterative methods for linear systems
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65F15 Eigenvalues, eigenvectors
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65F20 Overdetermined systems, pseudoinverses
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15A06 Linear equations
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15A09 Matrix inversion, generalized inverses
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15A18 Eigenvalues, singular values, and eigenvectors
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.
Keywords :
GMRES, evaluation of convergence, ideal GMRES, normal matrices, min-max problem