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This paper concerns the harmonic projection methods for computing an approximation to an eigenpair (λ, x ) of a large matrix A . Given a target point τ and a subspace that contains an approximation to x , a harmonic projection method returns an approximation to (λ, x ). Three convergence results are established as the deviation ε of x from approaches zero. First, the harmonic Ritz value μ + τ converges to λ if certain Rayleigh quotient is uniformly nonsingular. Second, the harmonic Ritz vector converges to x if the Rayleigh quotient is uniformly nonsingular and μ+τ remains well separated from the other harmonic Ritz values. Third, better error bounds for the convergence of μ+τ are derived when converges. However, we show that the harmonic projection method can fail to find the desired eigenvalue λ or in other words they can miss λ if it is very close to τ. We then propose to compute the Rayleigh quotient ρ of A with respect to and take it as a new approximate eigenvalue, which is shown to converge to λ no matter how τ is close to λ whenever tends to x . Finally, we deduce from a known result that under the assumption that the Rayleigh quotient is uniformly nonsingular certain refined harmonic Ritz vector or more generally refined eigenvector approximation introduced by the author converge. We construct examples to illustrate some of our theory. Computational Mathematics Numerical Analysis Applied_mathematics/0303025 Supplementary material Preprint version 1

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