Monographs on Numerical Analysis. Oxford: Clarendon Press. xiii, 322 p. £ 37.50 (1993).

As the author points out, the special structure of Sturm-Liouville eigenvalue problems allows for numerical methods which give an order of magnitude better performance than do the general-purpose codes.
The present text contains a selective survey on those methods in the literature that the author found most useful for automatic computation, with a strong emphasis on built-in error estimation and control.
The author’s interest is mainly in the construction of fast and robust algorithms for singular eigenvalue problems. The text is not intended as a theoretical development from the ground up. But it presents enough theoretical background to understand the various options and possible difficulties.
The book is divided into four main parts: Chapters 1-2 are introductory theory, referred to in the subsequent chapters, Chapters 3-6 are about the basic methods as simple finite difference and variational methods; shooting methods using a standard code for ordinary differential equations; and `Pruess methods’ of piecewise constant approximation. Then there is a jump in the level of theoretical sophistication in Chapters 7- 8, which cover theory and numerics of singular Sturm-Liouville problems. The problem is set in the context of operators in Hilbert space, to cover the Weyl-Kodaira classification and the theory of limit points and (oscillatory or non-oscillatory) limit circles. In this part, there is more emphasis on proofs. These chapters clarify why simple methods sometimes fail, and how other algorithms successfully handle difficult singular problems.
The final part (Chapters 9-12) covers a great variety of material, for instance computing eigenfunctions, multiparameter and vector problems. An appendix contains a list of 60 test problems of various type, and a list of available Sturm-Liouville software.
The book is most clearly written and is highly recommended for algorithms to handle singular and non-singular Sturm-Liouville problems.