A Homotopy Method for Finding All Solutions of a Multiparameter Eigenvalue Problem

Abstract

Multiparameter eigenvalue problems arise naturally in a variety of applications, especially in certain boundary value problems when the technique of separation of variables is applied. The homotopy method has been successfully used to solve the classical eigenvalue problems, the generalized eigenvalue problems, the right definite two-parameter eigenvalue problems and the weakly elliptic two-parameter eigenvalue problems. In this paper, we propose applying homotopy methods to solve the general multiparameter eigenvalue problems. Comparing with the existing homotopy methods for multiparameter eigenvalue problems, a distinct feature of our proposed method is that it is suitable for multiparameter eigenvalue problems without any limitations on the special properties of the coefficient matrices. Comparing with the method of transforming the multiparameter eigenvalue problem into the simultaneous eigenvalue problem, our proposed method is more efficient for coefficient matrices of large order, and has some advantages on the storage requirement and the implementation of the parallel algorithm. The results of our numerical experiments on illustrative problems show that our proposed method works well.