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New estimates for Ritz vectors

Author: Andrew V. Knyazev
Journal: Math. Comp. 66 (1997), 985-995
MSC (1991): Primary 65F35
MathSciNet review: 1415802
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Abstract: The following estimate for the Rayleigh-Ritz method is proved:

\begin{displaymath}| \tilde \lambda - \lambda | |( \tilde u , u )| \le { \| A \tilde u - \tilde \lambda \tilde u \| } \sin \angle \{ u ; \tilde U \}, \ \| u \| =1. \end{displaymath}

Here $A$ is a bounded self-adjoint operator in a real Hilbert/euclidian space, $\{ \lambda , u \}$ one of its eigenpairs, $\tilde U$ a trial subspace for the Rayleigh-Ritz method, and $\{ \tilde \lambda , \tilde u \}$ a Ritz pair. This inequality makes it possible to analyze the fine structure of the error of the Rayleigh-Ritz method, in particular, it shows that $ |( \tilde u , u )| \le C \epsilon ^2, $ if an eigenvector $u$ is close to the trial subspace with accuracy $\epsilon $ and a Ritz vector $\tilde u$ is an $\epsilon $ approximation to another eigenvector, with a different eigenvalue. Generalizations of the estimate to the cases of eigenspaces and invariant subspaces are suggested, and estimates of approximation of eigenspaces and invariant subspaces are proved.

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Additional Information

Andrew V. Knyazev
Affiliation: Department of Mathematics, University of Colorado at Denver, Denver, Colorado 80217

Keywords: Eigenvalue problem, Rayleigh--Ritz method, approximation, error estimate
Received by editor(s): May 10, 1995
Received by editor(s) in revised form: September 5, 1995, and June 3, 1996
Additional Notes: This research was supported by the National Science Foundation under grant NSF-CCR-9204255 and was performed while the author was visiting the Courant Institute.
Article copyright: © Copyright 1997 American Mathematical Society