On the convergence of harmonic Ritz vectors and harmonic Ritz values

TL;DR

This paper investigates the convergence behavior of harmonic Ritz vectors and values when computing eigenpairs of large non-Hermitian matrices, removing previous restrictive assumptions and establishing new convergence bounds under certain conditions.

Contribution

It introduces new convergence bounds for harmonic Ritz methods that do not require the Rayleigh quotient matrix to be nonsingular, broadening applicability.

Findings

Harmonic Ritz values converge under the uniform separation condition.

Harmonic Ritz vectors converge as the subspace approaches the eigenvector.

Convergence is characterized by the separation of Ritz values of shifted matrices.

Abstract

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 $(A-\tau I)\mathcal{K}$.