We prove a Saad's type bound for harmonic Ritz vectors of a Hermitian matrix. The new bound reveals a dependence of the harmonic Rayleigh--Ritz procedure on the condition number of a shifted problem operator. Several practical implications are discussed. In particular, the bound motivates incorporation of preconditioning into the harmonic Rayleigh--Ritz scheme.
In particular, if @math is Hermitian, the result of @cite_4 states that eq:chen.jia (x, v) 1 + ^2 |B^ -1 |^2 sep ( ,G)^2 (x, K), where @math , @math , and @math denotes an orthogonal projector onto @math . The quantity @math describes separation of the targeted eigenvalue @math from the harmonic Ritz values different from the one associated with @math , which are the eigenvalues of @math ; see @cite_4 for the precise definition of @math .
@cite_4
1975625541
Let A be a real mn matrix with mn. It is well known (cf. [4]) that @math (1) where @math The matrix U consists of n orthonormalized eigenvectors associated with the n largest eigenvalues of AA T , and the matrix V consists of the orthonormalized eigenvectors of A T A. The diagonal elements of  are the non-negative square roots of the eigenvalues of A T A; they are called singular values. We shall assume that @math Thus if rank(A)=r,  r+1 =  r+2== n = 0. The decomposition (1) is called the singular value decomposition (SVD).