Harmonic Rayleigh-ritz for the Multiparameter Eigenvalue Problem
نویسنده
چکیده
Harmonic extraction methods for the multiparameter eigenvalue problem willbe presented. These techniques are generalizations of their counterparts forthe standard and generalized eigenvalue problem. The methods aim to ap-proximate interior eigenpairs, generally more accurately than the standardextraction does. The process can be combined with any subspace expansionapproach, for instance a Jacobi-Davidson type technique, to form a subspacemethod for multiparameter eigenproblems of high dimension.We will focus on the two-parameter eigenvalue problemA1x1 = B1x1 + C1x1,A2x2 = B2x2 + C2x2,for given ni × ni (real or complex) matrices Ai, Bi, Ci for i = 1, 2; we areinterested in eigenpairs ((λ1, λ2), x1 ⊗ x2) where x1 and x2 have unit norm. References[1] M. E. Hochstenbach and B. Plestenjak, Harmonic Rayleigh-Ritz for the multi-parameter eigenvalue problem, CASA Report 06-35, TU Eindhoven, October 2006.
منابع مشابه
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 numeri...
متن کاملVariations on Harmonic Rayleigh–ritz for Standard and Generalized Eigenproblems
We present several variations on the harmonic Rayleigh–Ritz method. First, we introduce a relative harmonic approach for the standard, generalized, and polynomial eigenproblem. Second, a harmonic extraction method is studied for rightmost eigenvalues of generalized eigenvalue problems. Third, we propose harmonic extraction methods for large eigenvalues of generalized and polynomial eigenproblem...
متن کاملHarmonic Ritz Values and Their Reciprocals
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...
متن کاملVariations of Ritz and Lehmann Bounds
Eigenvalue estimates that are optimal in some sense have selfevident appeal and leave estimators with a sense of virtue and economy. So, it is natural that ongoing searches for effective strategies for difficult tasks such as estimating matrix eigenvalues that are situated well into the interior of the spectrum revisit from time to time methods that are known to yield optimal bounds. This artic...
متن کاملThe 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...
متن کامل