# Generalizations of harmonic and refined Rayleigh-Ritz

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Electronic Transactions on Numerical Analysis. Volume 20, pp. 235-252, 2005. Copyright 2005, Kent State University. ISSN 1068-9613. ETNA Kent State University etna@mcs.kent.edu GENERALIZATIONS OF HARMONIC AND REFINED RAYLEIGH–RITZ � MICHIEL E. HOCHSTENBACH � Abstract. We investigate several generalizations of the harmonic and re�ned 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 ma- trix, Rayleigh Ritz, harmonic Rayleigh Ritz, re�ned 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 norm �� � � �� is small. Since for an eigenvector we have �� ��� �� ��� �� �� � �� ��� �� �� � ���(1.1) we see that these eigenvectors correspond to eigenvalues � that have a small � �� �� � � . An example is the situation where we are interested in (interior) eigenpairs �� ��� � of � of which the eigenvalue � close to a target ff . Then a natural choice is �� �fi � � fiffifl ff �(1.2) indeed, if � is an eigenvector with a small � fl ff"! � �� , where ! is the identity matrix, its corresponding eigenvalue must be close to ff . Subspace methods are often used for th

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