## 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.## References

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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