The Convergence of Harmonic Ritz Values, Harmonic Ritz Vectors and Refined Harmonic Ritz Vectors

This paper concerns the harmonic projection methods for computing an approximation to an eigenpair (λ, ) of a large matrix A. Given a subspace that contains an approximation to , a harmonic method returns an approximation to (λ, ). Three convergence results are established as the deviation ε of from approaches zero. First, the harmonic Ritz value converges to λ if certain Rayleigh quotient is uniformly nonsingular. Second, converges to if the Rayleigh quotient is uniformly nonsingular and remains well separated from the other harmonic Ritz values. Third, better error bounds for the convergence of are derived when converges. We explain, in some detail, why the harmonic projection methods may fail to find the desired eigenvalue λ or in other words they may miss λ if it is very close to the given target point . We deduce from a known result that under the assumption that the Rayleigh quotient is uniformly nonsingular certain refined harmonic Ritz vectors or more generally refined eigenvector approximations introduced by the author converge