The convergence of harmonic Ritz values, harmonic Ritz vectors, and refined harmonic Ritz vectors

This paper concerns a harmonic projection method for computing an approximation to an eigenpair $(\lambda, x)$ of a large matrix $A$. Given a target point $\tau$ and a subspace $\mathcal{W}$ that contains an approximation to $x$, the harmonic projection method returns an approximation $(\mu+\tau, \tilde x)$ to $(\lambda,x)$. Three convergence results are established as the deviation $\epsilon$ of $x$ from $\mathcal{W}$ approaches zero. First, the harmonic Ritz value $\mu+\tau$ converges to $\lambda$ if a certain Rayleigh quotient matrix is uniformly nonsingular. Second, the harmonic Ritz vector $\tilde x$ converges to $x$ if the Rayleigh quotient matrix is uniformly nonsingular and $\mu+\tau$ remains well separated from the other harmonic Ritz values. Third, better error bounds for the convergence of $\mu+\tau$ are derived when $\tilde x$ converges. However, we show that the harmonic projection method can fail to find the desired eigenvalue $\lambda$--in other words, the method can miss $\lambda$ if it is very close to $\tau$. To this end, we propose to compute the Rayleigh quotient $\rho$ of $A$ with respect to $\tilde x$ and take it as a new approximate eigenvalue. $\rho$ is shown to converge to $\lambda$ once $\tilde x$ tends to $x$, no matter how $\tau$ is close to $\lambda$. Finally, we show that if the Rayleigh quotient matrix is uniformly nonsingular, then the refined harmonic Ritz vector, or more generally the refined eigenvector approximation introduced by the author, converges. We construct examples to illustrate our theory.