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Zbl 0795.65053
Pryce, John D.
Numerical solution of Sturm-Liouville problems. (English)
[B] Monographs on Numerical Analysis. Oxford: Clarendon Press. xiii, 322 p. \sterling 37.50 (1993). ISBN 0-19-853415-9/hbk

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.\par 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.\par 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.\par 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.\par 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.\par The book is most clearly written and is highly recommended for algorithms to handle singular and non-singular Sturm-Liouville problems.
[W.Velte (Würzburg)]

MSC 2000:: *65L15 Eigenvalue problems for ODE (numerical methods)
65-02 Research monographs (numerical analysis)
65L12 Finite difference methods for ODE
65L60 Finite numerical methods for ODE
65F15 Eigenvalues (numerical linear algebra)
34B24 Sturm-Liouville theory
34L15 Estimation of eigenvalues for OD operators

Keywords: monograph; Pruess methods; Sturm-Liouville eigenvalue problems; performance; error estimation; fast and robust algorithms; singular eigenvalue problems; finite difference and variational methods; Hilbert space; Weyl-Kodaira classification; limit points; limit circles; eigenfunctions; multiparameter and vector problems; test problems; Sturm- Liouville software

Cited in: Zbl 0941.34013 Zbl 0814.65086

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