  This paper concerns the harmonic shift-invert residual Arnoldi (HSIRA) and
Jacobi--Davidson (HJD) methods as well as their refined variants RHSIRA and
RHJD for the interior eigenvalue problem. Each method needs to solve an inner
linear system to expand the subspace successively. When the linear systems are
solved only approximately, we are led to the inexact methods. We prove that the
inexact HSIRA, RHSIRA, HJD and RHJD methods mimic their exact counterparts well
when the inner linear systems are solved with only low or modest accuracy. We
show that (i) the exact HSIRA and HJD expand subspaces better than the exact
SIRA and JD and (ii) the exact RHSIRA and RHJD expand subspaces better than the
exact HSIRA and HJD. Based on the theory, we design stopping criteria for inner
solves. To be practical, we present restarted HSIRA, HJD, RHSIRA and RHJD
algorithms. Numerical results demonstrate that these algorithms are much more
efficient than the restarted standard SIRA and JD algorithms and furthermore
the refined harmonic algorithms outperform the harmonic ones very
substantially.
