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

摘要

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 W that contains an approximation to x, the harmonic projection method returns an approximation (mu + tau, (x) over tilde) to (lambda, x). Three convergence results are established as the deviation epsilon of x from 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 (x) over tilde 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 (x) over tilde 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 t. To this end, we propose to compute the Rayleigh quotient rho of A with respect to (x) over tilde and take it as a new approximate eigenvalue. rho is shown to converge to lambda once (x) over tilde 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.