New estimates for Ritz vectors

Source abstract

The following estimate for the Rayleigh–Ritz method is proved: $$| λ ~ − λ | | ( u ~ , u ) | ≤ ‖ A u ~ − λ ~ u ~ ‖ sin ⁡ ∠ { u ; U ~ } , ‖ u ‖ = 1. | \tilde \lambda - \lambda | |( \tilde u , u )| \le { \| A \tilde u - \tilde \lambda \tilde u \| } \sin \angle \{ u ; \tilde U \}, \| u \| =1.$$ Here A A is a bounded self-adjoint operator in a real Hilbert/euclidian space, { λ , u } \{ \lambda , u \} one of its eigenpairs, U ~ \tilde U a trial subspace for the Rayleigh–Ritz method, and { λ ~ , u ~ } \{ \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 | ( u ~ , u ) | ≤ C ϵ 2 , |( \tilde u , u )| \le C \epsilon ^2, if an eigenvector u u is close to the trial subspace with accuracy ϵ \epsilon and a Ritz vector u ~ \tilde u is an ϵ \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.