Applicable Mathematics Series. Boston-London-Melbourne: Pitman Advanced Publishing Program. XIII, 310 p. £ 29.95 (1984).
This welcome and exhaustive survey of the use of recurrence relations in numerical computation brings together for the first time in one volume the wide range of computational techniques for both the linear and nonlinear cases. Furthermore it is written with the authority of one who has himself made major contributions in this field. Roughly three quarters of the book is devoted to linear algorithms, beginning with the classical algorithm for first and second order homogeneous equations applied by J. C. P. Miller in 1952 to the compilation of tables of the Bessel function $I\sb n(x)$, though apparently mentioned by Lord Rayleigh as early as 1910. The whole family of related methods for both the homogeneous and nonhomogeneous cases, owing much to the work of such other authors as Olver, Clenshaw and Gautschi, are discussed in detail. Indeed, as the author remarks, ”a lexicon of techniques has been developed that can be used to do almost everything but peel apples, as Norbert Wiener once said of the Fourier transform”.
Also surveyed are the applications of these recurrence techniques to computing particular special functions in applied mathematics, and series solutions of ordinary differential equations. The latter part of the book is concerned with the more challenging nonlinear case, where the current research effort is rightly described by the author as frenzied. The motivation for such work comes from its application to the study of discrete time dynamical systems, arising in such varied disciplines as the theory of turbulence, population growth, and biosystems. Given the rapid pace of development, this text cannot give such a coherent and complete account as it does in the linear case, but the author has certainly given an excellent foundation for further study of the continuing achievements in the nonlinear field. The provision of illuminating examples is on a generous scale, and the book also contains three useful appendices concerned with the general and asymptotic theory of linear difference equations, and recursion formulas for hypergeometric functions. The inclusion also of a very comprehensive bibliography makes this presentation of computational techniques useful to both numerial analysts and those who wish to apply recurrence relations in their own problem areas.