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Reviewer: Andrew Donald Booth

The merits of this book are shown by the fact that the first edition [1], published in 1986, was reprinted several times. The scientific and engineering communities have been awaiting the appearance of an update that would reflect both the progress of the subject in nearly seven years and the requests for additional material that the authors received, as testified by a monthly column in Computers in Physics . The new book exceeds, if possible, the excellence of its predecessor: it is about 50 percent longer and has been thoroughly updated. It contains 20 chapters as compared to 17 in the original, and 994 pages as compared to 735. The bibliographical material has been considerably extended and updated. Perhaps the most important change has been in the nature of C itself. The current standard for the language as implemented in most compilers has the requirement for prototype definitions, and the absence of these in the program libraries supplied with the first edition was a constant source of annoyance to less skillful users of the software. This absence has now been remedied both in the text and in the associated program disk. What has not been done is a conversion to the now prevalent C++ standard. The authors admit this fact, and perhaps avoidance of the complexities of object-oriented programming is justified; however, the use of // comments and cin>> and cout << is so simple, and so valuable in the elimination of the obscure I/O errors generated by forgetting the “&” directive for some input variables, that it would have been well worth the small extra effort needed to include them. To list the chapter titles of so well known a book would be pointless. For new users, it is sufficient to say that practically every aspect of numerical analysis is covered, broadly speaking: computational technique and errors; random numbers; linear algebra; solution of nonlinear equations; evaluation of integrals and the calculation of the most important functions of mathematical physics; Fourier analysis; differential and integral equations; and data coding. Each topic is illustrated by readable explanations and one or more C functions to effect the calculation. The authors point out, and I agree, that most of the material in such expensive packages as the NAG and IMSL libraries and in LINPACK is to be found in the book. I will now turn to some of the new things I have had the opportunity to try. The chapter on random number generation and applications to Monte Carlo integration has been greatly expanded. The most interesting feature here is the discussion of the Sobol method, which enables one- and two-dimensional pseudorandom number sequences to be generated in such a way that space is uniformly filled to a packing density that can be used as a parameter for terminating a process at a specified accuracy. This method is shown to allow Monte Carlo integration in which the errors are of order 1/N instead of the usual 1/N 1/2 . Second, the book contains an update on simulated annealing and two new programs that combine the annealing technique with the Nelder-Mead process for minimum location. This new device for avoiding descent into relative minima has shown promise in such diverse areas as the traveling salesperson problem and the analysis of ocean acoustic signals in the presence of multipath contamination. A third important addition is a section on periodogram analysis without direct use of the fast Fourier transform. The Lomb method is described and two programs are provided; one is slow and simple, and the other is fast and complex. Important features of the Lomb method are that it does not require that data sets have a length 2 N or that the items are equally spaced. An additional plus is that, under suitable conditions, aliasing is avoided and no Gibbs phenomenon occurs. A fourth addition is a discussion of wavelet analysis, a new technique whose applications range from image compression to the solution of integral equations. The emphasis is on the Daubechies transform, and only the real form is discussed. The material on the Lamarie transform is skeletal. As a method of data compression, the wavelet transform should be better known; in suitable cases, it can lead to compression ratios of better than 10:1. Other additions are in the areas of partial differential equations, integral equations, and data coding. The last covers Gray, Huffman, and RLL methods and has useful programs for the engineer. Finally, a chapter on infinite precision arithmetic includes a program to calculate p to several thousand places and a suggestion that the reader try it on the Ramanujan identity for 1/ p . This chapter contains the only really unsatisfactory pieces of code in the collection. For example, the program for p , when run on an IBM 80486 (66MHz) machine, requires 105 seconds for 100 digits. A C program that will do the same thing in about .05 second is freely available. Experience over the past seven years has shown me that the material produced by the authors is reliable and near to optimal. The necessarily superficial tests on the new material show that it lives up to its predecessors. It is worth mentioning that all the programs are available on a floppy disk at minimal cost. Unfortunately, the new version is much less convenient to use than the old disk because the data are compressed and cannot be accessed directly from most text editors. This minor cavil has nothing to do with the book as such, however. This monumental and classic work is beautifully produced and of literary as well as mathematical quality. It is an essential component of any serious scientific or engineering library.