Numerical Solution of Sturm-Liouville Problems

Sturm Liouville Problems (SLPs) were an applied mathematics tool in the 19th century (Fourier series and separation-of-variables), a driving force of pure mathematics in the early 20th century (Hilbert space operator theory), and of vital interest to physicists when Schrodinger's equations came along. The variety of interesting SL-related computations reflects this triple background. Numerical methods date from the 1920s: in quantum physics literature, often for one type of problem and of limited accuracy; in numerical literature, accurate and efficient on a class of (usually regular) problem but hard to automate. General ODE boundary value software solves SLPs reliably but inefficiently. It is worth developing special methods to cope with the variety of behaviour singular SLPs display. The book is intended for the scientist/engineer who wants simple methods for simple SLPs but needs to know their limitations, the algorithms that overcome these and the software that embodies these algorithms. It is also for the numerical analyst who wants a reference on good SLP methods, their theory, implementation and performance. The basic mathematical theory as it relates to algorithms is covered in some detail. There are numerous problems.Table of ContentIntroductory background; 1. Elementary theory of the classical SLP; 2. Simple matrix methods; 3. Variational methods; 4. Shooting and the scaled Prufer method; 5. Pruess methods; 6. Singular SLPs: theory; 7. Singular SLPs: numerical treatments; 8. Computing and manipulating eigenfunctions; 9. The computation of resonances; 10. Further topics; 11. Conclusions; A: Eigenvalues of Paine problems; B: List of test problems; C: Available Sturm-Liouville software