The Convergence of Harmonic Ritz Vectors and Harmonic Ritz Values, Revisited

Abstract

We are interested in computing a simple eigenpair $(\lambda,{\bf x})$ of a large non-Hermitian matrix $A$ by using a general harmonic Rayleigh--Ritz projection method. Given a search subspace $\mathcal{K}$ and a target point $\tau$ that is not an eigenvalue of $A$, we focus on the convergence of the harmonic Ritz vector $\widetilde{\bf x}$ and the harmonic Ritz value $\widetilde{\lambda}$. In [Z. Jia, Math. Comp., 74 (2004), pp. 1441--1456], Jia showed that for the convergence of the harmonic Ritz vector and the harmonic Ritz value, it is essential to assume a certain Rayleigh quotient matrix is uniformly nonsingular as $\angle({\bf x},\mathcal{K})\rightarrow 0$. However, this assumption cannot be guaranteed both in theory and in practice, and the Rayleigh quotient matrix can be singular or near singular even if $\tau$ is not close to $\lambda$. In this paper, we abolish this constraint and derive new bounds for the convergence of harmonic Rayleigh--Ritz projection methods. We show that as the distance be...