On the convergence of harmonic Ritz vectors and harmonic Ritz values
For researchers using harmonic projection methods to compute eigenpairs of large non-Hermitian matrices, this work relaxes a restrictive theoretical assumption, making convergence guarantees more broadly applicable.
The paper removes the uniform nonsingularity assumption required in prior work for convergence of harmonic Ritz vectors and values in harmonic Rayleigh-Ritz projection methods for non-Hermitian matrices, deriving new convergence bounds under a uniform separation condition.
We are interested in computing a simple eigenpair $(λ,{\bf x})$ 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 $τ$, we focus on the convergence of the harmonic Ritz vector $\widetilde{\bf x}$ and harmonic Ritz value $\widetildeλ$. 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 $τ$ is not close to $λ$. 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 $τ$ is satisfied with the so-called {\it uniform separation condition}, the harmonic Ritz value converges, and the harmonic Ritz vector converges as $\frac{1}{λ-τ}$ is well separated from other Ritz values of $(A-τI)^{-1}$ in the orthogonal complement of $(A-τI)\widetilde{\bf x}$ with respect to $(A-τI)\mathcal{K}$.