论文信息 - New estimates for Ritz vectors

New estimates for Ritz vectors

The following estimate for the Rayleigh-Ritz method is proved: |λ-λ||(u, u)| ≤:|Au - λu|| sin <{u; U}, ||u|| = 1. Here A is a bounded self-adjoint operator in a real Hilbert/euclidian space, {λ, u} one of its eigenpairs, U a trial subspace for the Rayleigh-Ritz method, and {λ, 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 |(u, u)| ≤ C ∈ 2 , if an eigenvector u is close to the trial subspace with accuracy ∈ and a Ritz vector u is an ∈ 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.

Andrew Knyazev | A. Knyazev

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