On the convergence of harmonic Ritz vectors and harmonic Ritz values

We are interested in computing a simple eigenpair $(\lambda,{\bf x})$ of a large non-Hermitian matrix $A$, by a general harmonic Rayleigh-Ritz projection method. Given a search subspace $\mathcal{K}$ and a target point $\tau$, we focus on the convergence of the harmonic Ritz vector $\widetilde{\bf x}$ and harmonic Ritz value $\widetilde{\lambda}$. In [{Z. Jia}, {\em The convergence of harmonic Ritz values, harmonic Ritz vectors, and refined harmonic Ritz vectors}, Math. Comput., 74 (2004), pp. 1441--1456.], Jia showed that for the convergence of harmonic Ritz vector and harmonic Ritz value, it is essential to assume certain Rayleigh quotient matrix being {\it uniformly nonsingular} as $\angle({\bf x},\mathcal{K})\rightarrow 0$. However, this assumption can not be guaranteed theoretically for a general matrix $A$, 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 between ${\bf x}$ and $\mathcal{K}$ tends to zero and $\tau$ is satisfied with the so-called {\it uniform separation condition}, the harmonic Ritz value converges, and the harmonic Ritz vector converges as $\frac{1}{\lambda-\tau}$ is well separated from other Ritz values of $(A-\tau I)^{-1}$ in the orthogonal complement of $(A-\tau I)\widetilde{\bf x}$ with respect to $(A-\tau I)\mathcal{K}$.