An analogue of the results of Saad and Stewart for harmonic Ritz vectors (Q596175)

The harmonic Ritz method approximates eigenvalues \(\lambda\) of a large matrix \(A\) near a point \(\tau\), and the corresponding eigenvectors \(x\), with respect to a subspace \(\mathcal{K}\), by computing harmonic Ritz pairs \((\theta ,w)\), where \(w\in \mathcal{K}\) and \(Aw-\theta w\) is orthogonal to \((A-\tau I)\mathcal{K}\). This paper establishes bounds for \(\sin \angle (x,w)\) analogous to those for the classical Ritz vectors given by Theorem 4.6 of \textit{Y. Saad} [Numerical methods for large eigenvalue problems (1992; Zbl 0991.65039)] for Hermitian \(A\) and by \textit{G. W. Stewart} [Linear Algebra Appl. 327, 115--119 (2001; Zbl 0982.15023)] for non-Hermitian \(A\).