The convergence of harmonic Ritz values, harmonic Ritz vectors and refined harmonic Ritz vectors
نویسندگان
چکیده
منابع مشابه
The convergence of harmonic Ritz values, harmonic Ritz vectors and refined harmonic Ritz vectors
This paper concerns a harmonic projection method for computing an approximation to an eigenpair (λ, x) of a large matrix A. Given a target point τ and a subspace W that contains an approximation to x, the harmonic projection method returns an approximation (μ + τ, x̃) to (λ, x). Three convergence results are established as the deviation of x from W approaches zero. First, the harmonic Ritz value...
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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...
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FOM and GMRES are Krylov subspace iterative methods for solving nonsymmetric linear systems. The Ritz values are approximate eigenvalues, which can be computed cheaply within these algorithms. In this paper, we generalise the concept of Harmonic Ritz values, introduced by Paige et al. for symmetric matrices, to nonsymmetric matrices. We show that the zeroes of the residual polynomials of FOM an...
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One application of harmonic Ritz values is to approximate, with a projection method, the interior eigenvalues of a matrix A while avoiding the explicit use of the inverse A. In this context, harmonic Ritz values are commonly derived from a Petrov-Galerkin condition for the residual of a vector from the test space. In this paper, we investigate harmonic Ritz values from a slightly different pers...
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Eigenvalue iterative methods, such as Arnoldi and Jacobi-Davidson, are typically used with restarting. This has signiicant performance shortcomings, since important components of the invariant subspace may be discarded. One way of saving more information at restart is the idea of \thick" restarting which keeps more Ritz vectors than needed. Our previously proposed dynamic thick restarting choos...
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ژورنال
عنوان ژورنال: Mathematics of Computation
سال: 2004
ISSN: 0025-5718
DOI: 10.1090/s0025-5718-04-01684-9