Advances in Difference Equations
Volume 2009 (2009), Article ID 362627, 18 pages
Research Article

Banded Matrices and Discrete Sturm-Liouville Eigenvalue Problems

Institute of Applied Analysis, University of Ulm, 89069 Ulm, Germany

Received 31 August 2009; Accepted 19 November 2009

Academic Editor: Ondřej Dosly

Copyright © 2009 Werner Kratz. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


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.