Convergence proof of the Harmonic Ritz pairs of iterative projection methods with restart strategies for symmetric eigenvalue problems

We consider numerical methods for computing eigenvalues located in the interior part of the spectrum of a large symmetric matrix. For such difficult eigenvalue problems, an effective solution is to use the Harmonic Ritz pairs in projection methods because the error bounds on the Harmonic Ritz pairs are well studied. In this paper, we prove global convergence of the iterative projection methods with the Harmonic Ritz pairs in an abstract form, where the standard restart strategy is employed. To this end, we reformulate the existing convergence proof of the Ritz pairs to be successfully applied to the Harmonic Ritz pairs with the inexact linear system solvers. Our main theorem obtained by the above convergence analysis shows important features concerning the global convergence of the Harmonic Ritz pairs.