Harmonic and refined Rayleigh-Ritz for the polynomial eigenvalue problem

After reviewing the harmonic Rayleigh–Ritz approach for the standard and generalized eigenvalue problem, we discuss different extraction processes for subspace methods for the polynomial eigenvalue problem. We generalize the harmonic and refined Rayleigh–Ritz approach, which are new approaches to extract promising approximate eigenpairs from a search space. We give theoretical as well as numerical results of the methods. In addition, we take a new look at the Jacobi–Davidson method for polynomial eigenvalue problems. AMS subject classifications. 65F15, 65F50.