A new shift strategy for the implicitly restarted refined harmonic Lanczos method

The implicitly restarted refined harmonic Lanczos method is widely used for computing the interior eigenpairs of large real symmetric matrices. In each restart, it takes the wanted refined harmonic Ritz pairs to be the approximate eigenpairs, and takes the refined harmonic shifts to implicitly restart the algorithm. In this paper, we modify this method by replacing the refined harmonic shifts by new shifts. Based on a special direct sum decomposition of the projected subspace, we build a new subspace in each restart and then take the harmonic Ritz values onto it to be the new shifts. The new shifts can be obtained from a small generalized eigenvalue problem and the computational cost is negligible. Numerical experiments show that the new shifts are generally better than the refined harmonic shifts.