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.
Generalized eigenvalue problems involving a singular pencil are very challenging to solve, both with respect to accuracy and efficiency. The existing package Guptri is very elegant but may be time-demanding, even for small and medium-sized matrices. We propose a simple method to compute the eigenvalues of singular pencils, based on one perturbation of the original problem of a certain specific rank. For many problems, the method is both fast and robust. This approach may be seen as a welcome alternative to staircase methods.
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.
In various applications, for instance in the detection of a Hopf bifurcation or in solving separable boundary value problems using the two-parameter eigenvalue problem, one has to solve a generalized eigenvalue problem of the form (B 1 ⊗ A 2 − A 1 ⊗ B 2)z = µ(B 1 ⊗ C 2 − C 1 ⊗ B 2)z, where matrices are 2 × 2 operator determinants. We present efficient methods that can be used to compute a small subset of the eigenvalues. For full matrices of moderate size we propose either the standard implicitly restarted Arnoldi or Krylov-Schur iteration with shift-and-invert transformation, performed efficiently by solving a Sylvester equation. For large problems, it is more efficient to use subspace iteration based on low-rank approximations of the solution of the Sylvester equation combined with a Krylov-Schur method for the projected problems.
scite is a Brooklyn-based organization that helps researchers better discover and understand research articles through Smart Citations–citations that display the context of the citation and describe whether the article provides supporting or contrasting evidence. scite is used by students and researchers from around the world and is funded in part by the National Science Foundation and the National Institute on Drug Abuse of the National Institutes of Health.