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 (λ,x) be an eigenpair of the matrix A of order n and let (µ,u) be a Ritz pair of A with respect to a subspace K. Saad has derived a simple priori error bound for sin ∠(x, u) in terms of sin ∠(x, K) for A Hermitian. Similar to Saad's result, Stewart has got an equally simple inequality for A non-Hermitian. In this paper, let (θ, w) be a harmonic Ritz pair from a subspace K, a similar priori error bound for sin ∠(x, w) is established in terms of sin ∠(x, K).