Efficient computation of high index SturmLiouville eigenvalues for problems in physics
Abstract
Finding the eigenvalues of a SturmLiouville problem can be a computationally challenging task, especially when a large set of eigenvalues is computed, or just when particularly large eigenvalues are sought. This is a consequence of the highly oscillatory behavior of the solutions corresponding to high eigenvalues, which forces a naive integrator to take increasingly smaller steps. We will discuss the most used approaches to the numerical solution of the SturmLiouville problem: finite differences and variational methods, both leading to a matrix eigenvalue problem; shooting methods using an initialvalue solver; and coefficient approximation methods. Special attention will be paid to techniques that yield uniform approximation over the whole eigenvalue spectrum and that allow large steps even for high eigenvalues.
 Publication:

Computer Physics Communications
 Pub Date:
 February 2009
 DOI:
 10.1016/j.cpc.2008.10.001
 arXiv:
 arXiv:0804.2605
 Bibcode:
 2009CoPhC.180..241L
 Keywords:

 02.30.Hq;
 02.60.Lj;
 02.60.Jh;
 03.65.w;
 Ordinary differential equations;
 Ordinary and partial differential equations;
 boundary value problems;
 Numerical differentiation and integration;
 Quantum mechanics;
 Mathematics  Numerical Analysis;
 Mathematical Physics;
 34B24;
 34L16
 EPrint:
 doi:10.1016/j.cpc.2008.10.001