Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Banded Matrices and Discrete Sturm-Liouville Eigenvalue Problems | Hindawi Publishing Corporation Advances in Difference Equations Volume 2009 Article ID 362627 18 pages doi 2009 362627 Research Article Banded Matrices and Discrete Sturm-Liouville Eigenvalue Problems Werner Kratz Institute of Applied Analysis University of Ulm 89069 Ulm Germany Correspondence should be addressed to Werner Kratz Received 31 August 2009 Accepted 19 November 2009 Recommended by Ondrej Dosly 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. 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. 1. Introduction In 1 it was shown that every discrete Sturm-Liouville eigenvalue problem where hwk Wk 1 - Wk n L y k A fo i .- J Ằyk 1 for0 k N SL 0 with Dirichlet boundary conditions y1-n y0 yN 2-n yN 1 0 is equivalent with an algebraic eigenvalue problem 2 for a symmetric banded N 1 - n X N 1 - n -matrix with bandwidth 2n 1 where N and n are fixed integers with 1 n N see 1 Theorem 1 Remark 1 i . Note that SL is irrelevant for N - n 1 k N in the case of Dirichlet boundary .
