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New estimates for Ritz vectors

Authors:	Andrew V Knyazev

Institution:	Department of Mathematics, University of Colorado at Denver, Denver, Colorado 80217

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 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 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.

Keywords:	Eigenvalue problem Rayleigh--Ritz method approximation error estimate

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