Generalizations of harmonic and refined Rayleigh-Ritz
Michiel E. Hochstenbach · TU/e Research Portal · 2005
Abstract. We investigate several generalizations of the harmonic and refined Rayleigh–Ritz method. These may be practical when one is interested in eigenvalues close to one of two targets (for instance, when the eigenproblem has Hamiltonian structure such that eigenvalues come in pairs or quadruples), or in rightmost eigenvalues close to (for instance) the imaginary axis. Our goal is to develop new methods to extract promising approximate eigenpairs from a search space, for instance one generated by the Arnoldi or Jacobi–Davidson method. We give theoretical as well as numerical results of the methods, and recommendations for their use. AMS subject classifications. 65F15, 65F50 Key words. Rational harmonic Rayleigh–Ritz, rightmost eigenvalue, structured eigenproblem, Hamiltonian matrix, Rayleigh–Ritz, harmonic Rayleigh–Ritz, refined Rayleigh–Ritz, subspace method, subspace extraction, Jacobi– Davidson 1. Introduction. Let ¢ be a (real or complex) large £¥¤¦ £ sparse matrix. Suppose that is a polynomial, and that we are looking for one or more normalized eigenvectors ¨ such that the Euclidean © §� � ¢��