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Mathematics of Computation

Published by the American Mathematical Society since 1960 (published as Mathematical Tables and other Aids to Computation 1943-1959), Mathematics of Computation is devoted to research articles of the highest quality in computational mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

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New estimates for Ritz vectors
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by Andrew V. Knyazev PDF
Math. Comp. 66 (1997), 985-995 Request permission

Abstract:

The following estimate for the Rayleigh–Ritz method is proved: \[ | \tilde \lambda - \lambda | |( \tilde u , u )| \le { \| A \tilde u - \tilde \lambda \tilde u \| } \sin \angle \{ u ; \tilde U \}, \| u \| =1. \] 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
  • Email: knyazev@na-net.ornl.gov
  • 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.
  • © Copyright 1997 American Mathematical Society
  • Journal: Math. Comp. 66 (1997), 985-995
  • MSC (1991): Primary 65F35
  • DOI: https://doi.org/10.1090/S0025-5718-97-00855-7
  • MathSciNet review: 1415802