Banded Matrices and Discrete Sturm-Liouville Eigenvalue Problems

We consider eigenvalue problems for self-adjoint Sturm-Liouville difference equations of any even order. It is well known that such problems with Dirichlet boundary conditions can be transformed into an algebraic eigenvalue problem for a banded, real-symmetric matrix, and vice versa. In this article it is shown that such a transform exists for general separated, self-adjoint boundary conditions also. But the main result is an explicit procedure algorithm for the numerical computation of this banded, real-symmetric matrix. This construction can be used for numerical purposes, since in the recent paper by Kratz and Tentler 2008 there is given a stable and superfast algorithm to compute the eigenvalues of banded, real-symmetric matrices. Hence, the Sturm-Liouville problems considered here may now be treated by this algorithm.