A Jacobi--Davidson Type Method for a Right Definite Two-Parameter Eigenvalue Problem

Abstract

We present a new numerical iterative method for computing selected eigenpairs of a right de nite two-parameter eigenvalue problem. The method works even without good initial approximations and is able to tackle large problems that are too expensive for existing methods. The new method is similar to the Jacobi{ Davidson method for the eigenvalue problem. In each s t e p w e rst compute Ritz pairs of a small projected right de nite two-parameter eigenvalue problem and then expand the search spaces using approximate solutions of appropriate correction equations. We p r e s e n t t wo alternatives for the correction equations, introduce a selection technique that makes it possible to compute more than one eigenpair, and give some numerical results.

Computing several eigenvalues of nonlinear eigenvalue problems by selection

“…While criterion (4.1) has turned out to perform satisfactorily in the numerical tests in [16,12], it may be unnecessarily strict: if one eigenvalue has been detected with right and left eigenvector x 1 ⊗ x 2 and y 1 ⊗ y 2 for which the right-hand side of (4.1) is small, the selection procedure may reject many or all candidate Ritz pairs. Therefore, instead of (4.1), we propose the new modified criterion (cf.…”

“…In [16] the special but important right-definite case has been treated, where all matrices A i , B i , and C i are Hermitian, and ∆ 0 is positive definite. In this situation, the right and left eigenvectors coincide, and therefore eigenvectors x 1 ⊗ x 2 and x 1 ⊗ x 2 corresponding to different eigenvalues are ∆ 0 -orthogonal: (x 1 ⊗ x 2 ) * ∆ 0 ( x 1 ⊗ x 2 ) = 0.…”

The Jacobi–Davidson method

“…We mention that Jacobi-Davidson type methods have also been developed for eigenvalue problems of more general type: the generalized eigenvalue problem [58,4,75], constrained eigenvalue problem [75], polynomial eigenvalue problem [58,4,43,75,32,31], nonlinear eigenvalue problem [8,81], singular value problem [24,25], generalized singular value problem [27], and multiparameter eigenvalue problem [29,28]. These papers show that JD type methods may be quite attractive for "more complicated" eigenproblems, also without preconditioning.…”

Structured Eigenvalue Problems

“…The result of the separation is a multiparameter system of ordinary differential equations. The multiparameter eigenproblems can be considered as particularly structured generalized eigenvalue problems [3]; see [67,68,104] for algorithms and perturbation results based on this connection.…”