## The Nonlinear Workbook

### by Willi-Hans Steeb.

### World Scientific 2011, 5th edition.

Beginning with chaos and dynamic systems, from one-dimensional maps and fractals to ordinary differential equations, the usual topics are presented succinctly in the first half of the book. Short chapters on chaos control and synchronization are also included. Then, the second half deals with topics that should be more familiar to computer scientists, such as neural networks, genetic algorithms, optimization, wavelets and fuzzy logic. This is a huge span of topics that cannot be covered in depth in a single volume. According to Steeb, most of these disparate fields are interrelated. If so, there is a valid motivation for presenting them in the same book and highlighting their relations. However, many interesting and often difficult fields are only presented in glimpses. The chapter on wavelets is a case in point; that material is not used in the rest of the book so the chapter might as well have been left out.

To be clear, this is not a book for beginners. It serves better as a complement to other literature, and to some extent it offers a different point of view than many other sources.

The book claims to balance a theoretical exposition with practical computer code. Several short stand-alone programs written in C++, Java and Symbolic C++ (the latter being a library developed by the author) form the backbone of the text. There is even a short routine in assembler. The treatment of chaotic systems is different from many other comparable texts in that exact numerical algorithms are used to study the iterations of maps. The SymbolicC++ library is used for this, but unfortunately this code is neither included nor explained in the book despite the many programs relying upon it.

The solution of ODEs is for a large part done with the Lie series technique, which is not very accessibly explained. Another frequently used technique is the Runge-Kutta-Fehlberg method, which makes use of a set of coefficients that are copied and pasted into each code snippet where they are needed. Needless to say, this makes for a lot of redundancy, which could easily have been avoided by placing this part of the code in an include file. In Chapter 11, where the integration techniques for ODEs should presumable be explained, the same mysterious coefficients appear again without any hint as to how they are derived (there is just a reference to the literature). This avoidance of explanations stands in stark contrast to the style of another, excellent source on scientific computation — the Numerical Recipes by Press

*et al*. Although the Nonlinear Workbook is not at all on a par with Numerical Recipes regarding stylistic issues, clarity of presentation, and general usefulness, it does contain much material that cannot be found there (at least not in the third printed edition).

Already in its fifth edition, one would think that the Nonlinear Workbook has all kinds of editorial flaws sorted out by now. Unfortunately, this is not the case. The text suffers from a lack of efforts to organize the material. Skimming through the text, there appears to be no single illustration; in fact there are less than a dozen across its more than six hundred pages, but they are not very visually striking. Most books about chaos theory and matters cognate will include a number of elegant pictures of attractors or fractals; this one does not, except for the one on its cover. Certainly most readers will have already seen an assortment of representative fractals and attractors and do not need them printed in yet another book; better yet, using the code in the book one should be able to generate and experiment with them for oneself. However, a few explanatory diagrams would often make ideas in the text more accessible to the reader. Or how about trying to explain a PoincarĂ© section in words and formulae, with no illustrations? That is surely a recipe for making simple things look hard.

On the whole, the book suffers from editorial neglect. Code examples are often given without any indentation, making nested routines hard to read. The copy-and-paste manner of coding does not make things any better. The prose is awkward in many places, such as the following:

If we want to compare two hidden Markov models then we need a measure for the distance between two hidden Markov models.There is very little cross-referencing inside the book and the equations are almost never numbered. On the positive side, one never needs to turn the pages to find some equation that was introduced earlier.

The level of exposition is also a bit uneven. Some sections assume rather much mathematical background, whereas others are quite accessible. Presumably someone who already is at ease with Lie series, exterior products and what not will have little need for the guidance provided by the code examples. Some programs are rather trivial and probably pose no difficulty to a first semester student of computer science. A few programs are far from trivial, however, and the lack of explanatory comments in the code as well as in the main text make them hard to understand. There are almost as little exercises as there are illustrations. Why should a text book have exercises at all, though? The intelligent reader will find his or her own problems to solve, spurred by puzzling remarks in the text or unproven propositions.

To round up, this workbook provides glimpses into many fascinating topics, albeit presented in a less than ideal way. The text is too tersely written to serve as an introduction to any of the many fields that it covers, but may be valuable to someone who has studied the theory before.

See for yourself, sample chapters are available for free.

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