A refined harmonic Rayleigh-Ritz procedure and an explicitly restarted refined harmonic Arnoldi algorithm

The work is fourfold. First, a refined harmonic Rayleigh-Ritz procedure is proposed, some relationships are established between the refined harmonic Ritz vector and the harmonic Ritz vector, an a priori error bound is derived for the refined harmonic Ritz vector, and some properties are established on Rayleigh quotients and residual norms. Second, a resulting refined harmonic Arnoldi method is discussed, and how to compute the refined harmonic Ritz vectors cheaply and accurately is considered. Third, an explicitly restarted refined harmonic Arnoldi algorithm is developed over an augmented Krylov subspace. Finally, numerical examples are reported that compare the new algorithm with the implicitly restarted harmonic Arnoldi algorithm (IRHA) and the implicitly restarted refined harmonic Arnoldi algorithm (IRRHA). Numerical results confirm efficiency of the new algorithm.