A Jacobi--Davidson Type Method for the Two-Parameter Eigenvalue Problem

Abstract

We present a new numerical method for computing selected eigenvalues and eigenvectors of the two-parameter eigenvalue problem. The method does not require good initial approximations and is able to tackle large problems that are too expensive for methods that compute all eigenvalues. The new method uses a two-sided approach and is a generalization of the Jacobi-Davidson type method for right definite two-parameter eigenvalue problems [M. E. Hochstenbach and B. Plestenjak, SIAM J. Matrix Anal. Appl., 24 (2002), pp. 392-410]. Here we consider the much wider class of nonsingular problems. In each step we first compute Petrov triples of a small projected two-parameter eigenvalue problem and then expand the left and right search spaces using approximate solutions to appropriate correction equations. Using a selection technique, it is possible to compute more than one eigenpair. Some numerical examples are presented.

Spectral collocation solutions to multiparameter Mathieu’s system

“…For the numerical solution we exploit a Jacobi-Davidson method as developed in [16,19,20]. In this method the eigenvectors x and y are sought in search spaces U and V, respectively.…”

“…If m is small, we can apply the existing numerical methods for the generalized eigenvalue problem to solve the coupled pair (7). An algorithm of this kind, which is based on the QZ algorithm, is presented in [16].…”

“…In case when we are interested in more than just one eigenpair and we do not have any initial approximations, a method of choice is a Jacobi-Davidson type method [16]. The state-of-the-art, which uses harmonic Ritz values [19], can be used to compute a small number of eigenvalues of (6), which are closest to a given target (λ T , µ T ) ∈ C 2 .…”

Computing the common zeros of two bivariate functions via Bézout resultants

“…This approach has advantages because the two-parameter eigenvalue problem can be solved with the QZ algorithm [24], or other techniques [24]. However, the construction of a determinantal expression and hence, the matrices A i , B i , C i for i = 1, 2 currently requires the solution of a multivariate polynomial system [36].…”

“…This example is of small degree, with the functions f and g being approximated by polynomial interpolants of degrees (m p , n p , m q , n q ) = (20,20,24,30):…”