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This paper concerns the harmonic shift-invert residual Arnoldi (HSIRA) andJacobi--Davidson (HJD) methods as well as their refined variants RHSIRA andRHJD for the interior eigenvalue problem. Each method needs to solve an innerlinear system to expand the subspace successively. When the linear systems aresolved only approximately, we are led to the inexact methods. We prove that theinexact HSIRA, RHSIRA, HJD and RHJD methods mimic their exact counterparts wellwhen the inner linear systems are solved with only low or modest accuracy. Weshow that (i) the exact HSIRA and HJD expand subspaces better than the exactSIRA and JD and (ii) the exact RHSIRA and RHJD expand subspaces better than theexact HSIRA and HJD. Based on the theory, we design stopping criteria for innersolves. To be practical, we present restarted HSIRA, HJD, RHSIRA and RHJDalgorithms. Numerical results demonstrate that these algorithms are much moreefficient than the restarted standard SIRA and JD algorithms and furthermorethe refined harmonic algorithms outperform the harmonic ones verysubstantially.

Harmonic and Refined Harmonic Shift-Invert Residual Arnoldi and Jacobi--Davidson Methods for Interior Eigenvalue Problems