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A geometric theory for preconditioned inverse iteration applied to a subspace

Author: Klaus Neymeyr
Journal: Math. Comp. 71 (2002), 197-216
MSC (2000): Primary 65N30, 65N25; Secondary 65F10, 65F15
Published electronically: September 17, 2001
MathSciNet review: 1862995
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Abstract: The aim of this paper is to provide a convergence analysis for a preconditioned subspace iteration, which is designated to determine a modest number of the smallest eigenvalues and its corresponding invariant subspace of eigenvectors of a large, symmetric positive definite matrix. The algorithm is built upon a subspace implementation of preconditioned inverse iteration, i.e., the well-known inverse iteration procedure, where the associated system of linear equations is solved approximately by using a preconditioner. This step is followed by a Rayleigh-Ritz projection so that preconditioned inverse iteration is always applied to the Ritz vectors of the actual subspace of approximate eigenvectors. The given theory provides sharp convergence estimates for the Ritz values and is mainly built on arguments exploiting the geometry underlying preconditioned inverse iteration.

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Additional Information

Klaus Neymeyr
Affiliation: Mathematisches Institut der Universität Tübingen, Auf der Morgenstelle 10, 72076 Tübingen, Germany

Keywords: Symmetric eigenvalue problem, subspace iteration, preconditioning, multigrid, inverse iteration
Received by editor(s): July 27, 1999
Published electronically: September 17, 2001
Article copyright: © Copyright 2001 American Mathematical Society