Stopping Criteria for Rational Matrix Functions of Hermitian and Symmetric Matrices
Abstract
Building upon earlier work by Golub, Meurant, Strakoš, and Tichý, we derive new a posteriori error bounds for Krylov subspace approximations to $f(A)b$, the action of a function f of a real symmetric or complex Hermitian matrix A on a vector b. To this purpose we assume that a rational function in partial fraction expansion form is used to approximate f, and the Krylov subspace approximations are obtained as linear combinations of Galerkin approximations to the individual terms in the partial fraction expansion. Our error estimates come at very low computational cost. In certain important special situations they can be shown to actually be lower bounds of the error. Our numerical results include experiments with the matrix exponential, as used in exponential integrators, and with the matrix sign function, as used in lattice quantum chromodynamics simulations, and demonstrate the accuracy of the estimates. The use of our error estimates within acceleration procedures is also discussed.
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Information & Authors
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Published In
SIAM Journal on Scientific Computing
Pages: 1387 - 1412
DOI: 10.1137/070684598
ISSN (online): 1095-7197
Copyright
Copyright © 2008 Society for Industrial and Applied Mathematics.
History
Submitted: 7 March 2007
Accepted: 10 September 2007
Published online: 28 March 2008
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