The Forth programming language is not usually encountered within the context of scientific or engineering computation, although most users of personal computers or workstations have unwittingly experienced it in one form or another. Known as "one of the best-kept secrets in computing", it lurks unseen in automatic bank teller machines, computer games, industrial control devices and robots. Some scientists and engineers have gained familiarity with Forth because it is fast, compact, and easily debugged; and because it simplifies interfacing microprocessors with machines and laboratory equipment. It was indeed originally developed for just this purpose.

Scientific computation deserves a language that is easy to program, debug and maintain and that embodies the desirable features of modern structured languages without sacrificing the best parts of FORTRAN. The ideal language would simplify linking with machine code and with compiled subroutines from other languages. It would execute quickly while using memory parsimoniously for clearly, the more memory devoted to program, the less is available for data. Finally, it would be extensible both to new data structures and to new operations, so it could grow and evolve without having to be redesigned.

Despite its small size and simple structure, Forth has all but one of these characteristics: In its pristine form Forth lacks any resemblance to FORTRAN either in structure or functionality. Fortunately, Forth is an easily extensible language. In this book Julian Noble takes the reader through the process of providing a FORTRAN based mixed arithmetic function library (real, complex, single- and double precision), complete with examples and descriptions directed to the programmer.

While the book is intended primarily for programmers with some familiarity with Forth, a brief review of Forth is given, so that non-Forth programmers will be able to understand the program fragments. Unfortunately it was written some two to three years before the new ANS Forth standard was developed, with the consequence that both the text and and some of the program examples, appear a bit dated. This introduction chapter not intended as a tutorial and should not be treated as one. A Forth compiler is provided on an accompanying disk, along with the programming examples, including many that are only discussed in outline in the book.

The book investigates the requirements of a floating point library by considering its use in a number of standard functions, including three random number routines. The library is built from basic building blocks, providing the reader with a unique insight into the operation of such a library. The use of dedicated co-processors are discussed, in a very readable and high-level manner, such that the 80x86 co-processor library provided is not only very usable, but could be converted for use with other co-processors (such as the MC6881, NS32081 or the PowerPC) with ease.

Scientific work requires the use of a number of complex data structures. These are developed in the book from the same basic building blocks as the floating point library, with the objective of providing ease of use à la FORTRAN, without making the operations so "smart" they slow up the machine or become inflexible. The details relating to implementation methods are discussed in a very clear manner. The use of these data structures and the use of floating point techniques are then demonstrated with a number of worked examples, including: summing infinite series; solving transcendental equations; solving differential equations.

The next chapter is devoted to the analysis of complex numbers, their implementation and use, and was found to be a very clear description of the complex number, intended for the programmer. The floating point library is developed further to incorporate complex numbers. Again, a number of examples are given, including: numerical quadrature, contour integration, fast Fourier transform (FFT) . This latter is particularly well explained for the non-mathamatician (viz programmers). The flexibility of the Forth compiler allows the generation of tables of random variates at compile time, permitting fast Monte-Carlo calculations that use arbitrary distribution functions. This is fairly original and shows the power of splitting the operation of a complex calculation between compile-time and run-time. This could be duplicated in C/C++ but it would almost certainly not be as simple, or as fast.

Due to its prominence, linear algebra is covered in intricate detail, and again, the discussion is based around examples. The approach is down to earth, helps the programmer to see why the scientist would use these functions, and how one would go about their implementation. In the discussion dealing with linear algebra, the solving of linear equations, simultaneous linear equations, and matrix inversion are examined. The optimised program described will solve 350 simultaneous linear equations (single precision) in about 16 minutes on a old PC (an original IBM PC with 8087 co-processor), and about 2 minutes on the reviewer's office machine (a 486DX4-100).

Finally a FORmula TRANslator is described, that translates FORTRAN arithmetic assignment statements into the corresponding Forth. This introduces some advanced programming techniques, such as finite state machines and recursive-descent parsing, in the development of the translator. Here the translator provides a focus for the programming examples, making the descriptions relevant for the programmer as well as giving the scientist an understanding of the requirements of programmers. The translator works by translating the FORTRAN expression into Forth at compile time, and not interfering with (and thus slowing down) the run-time system.

It would have been more useful for the FORmula TRANslator to be given up front towards the start of the book, thus putting it all in context. A problem with this would be the use of formula to explain the algorithms, putring many programmers off. As it stands, however, the algorithms have to be justified by a number of detailed examples. As all of these examples are taken from the scientific programming domain, its easy for a programmer to see why one would need such an algorithm and a good idea as to how it would be implemented. This is true for most, if not all, of the examples.

The examples used in the book are presented at a fairly low level, and uses some of the more esoteric features of the Forth language. However, you do not need to know Forth to get the most from them. The code is presented in a highly readable and usable manner. Indeed the Forth Interest Group (FIG) has adopted this style for its "scientific library" project, which aims to bring together many of the algorithms that simply could not be covered by a single book. In this regard it provides a wonderful collection of tools and shows how one should develop good software. Worked algorithms are given for the majority of examples in the book. When these are not derived into code as part of the discussion, the code is provided on the accompanying disk.

For those who provide interfaces between the microprocessor and industrial control systems, robots, lab equipment, etc, the book provides a good introduction to the field of numerical analysis, and advanced programming techniques. It puts the algorithm into perspective by showing how the Scientist/Engineer would use it, and investigates some of the trade-offs necessary for implementation. For those who are trying to improve their interface (either making it smaller, faster, or providing more functionality) it presents an alternative that many would otherwise overlook.

Although this book is undoubtedly aimed at programmers, the scientist/engineer will be introduced to the restrictions, constraints and trade-offs programmers are required to work with on a regular basis, giving an insight into those strange decisions programmers seem to make. For those wanting more control over their equipment, or wanting to reduce the size of their system, this might be one way forward, that they would probably have dismissed, if reviewed at all.