The solutions of many problems in various areas of computational mathematics require that one is able to approximate integrals in an efficient and reliable way. Thus, there is a high demand for practically useful algorithms for numerical integration, and the mathematics community has produced a vast number of such algorithms in recent decades. For the user with a concrete integration problem at hand, this means he has to find an algorithm that fits his specific needs. Traditionally, one would look for a suitable algorithm in the classic book by Davis and Rabinowitz [1], which gives an excellent overview of many different types of integration problems and the corresponding algorithms. However, it is now more than 20 years old, and does not reflect today’s state-of-the-art algorithms.

It seems as if the authors of this book intended to create a successor to this classic reference. They have been quite successful. The book is a good guide for the practitioner who has a concrete integration problem and needs to find a suitable algorithm. It covers not only the classical types of problems, but also more recent developments like algorithms for integrals with various types of singularities. In addition, numerical methods for integral transforms (Fourier, inverse Laplace, and wavelet) and algorithms for the numerical solution of Fredholm integral equations are discussed. In each case, the theoretical background is introduced and discussed briefly, without going into too much detail. The authors manage to keep a very good balance in this respect, giving enough background information to the user to allow him to make a good choice, and still writing the text concisely enough to keep it readable for nonexperts in numerical integration.

One of the very attractive features of the book is its very large list of concrete quadrature formulas, both in printed and electronic form (as formulas or actual source code in various programming languages, such as C, FORTRAN, MATLAB, and Mathematica). (The electronic version is provided on the accompanying CD.) It should be noted, however, that the authors have strictly restricted their efforts to one-dimensional integrals.