Programming in Mathematica

Mathematica was officially announced in June of 1988. Since then it has found many uses in very diverse fields. While it is useful to do a few calculations interactively, its real strength lies in the programming language it offers. Writing programs, one extends Mathematica with specialized new functions in one's own field of interest.

Mathematica's programming language is unlike any you have encountered before. The language's manual explains all its features and gives some very basic examples of their use. For writing good programs, however, this is not enough: there is clearly a need for a book explaining all the many features in context and giving more extensive examples of their use. The first edition of this book filled this gap for version 1.2 of Mathematica. Like the second edition, this third edition was prompted by a major new version of Mathematica, Version 3.0.

I have been in contact with many early users of Mathematica and I have also given students at the University of Illinois and the Swiss Federal Institute of Technology an opportunity to learn about it. I have seen many programs written in Mathematica, good ones and bad ones. The bad ones invariably solve a particular problem in an unnecessarily complicated way, perhaps because the programmers were unaware of the elegant constructs available in Mathematica. Let me point out these constructs and show you how they can be used to write concise, elegant, and efficient programs.

The best way to teach these Mathematica-specific programming methods is to look at examples of complete programs that solve some nontrivial problem. Even if the example chosen does not lie in the particular field of application that you are interested in, you will be able to use similar ideas for your own programs. Many of the examples presented here deal with graphics. Graphics applications are especially suited for learning programming, because it is easy to see whether your code is correct simply by looking at the picture it produces. Advanced graphics applications are also sufficiently challenging to require advanced programming methods. Other examples come from symbolic computation, geometry, dynamics, and numerical mathematics. Along the way, we also develop many short pieces of code that can be used as parts of larger programs or that help to customize Mathematica to your particular needs. Many of these found their way into the standard Mathematica distribution and are now available to all users.

Because there exists an excellent manual, this book does not explain everything from scratch, but assumes some familiarity with Mathematica. I assume also that you have access to a computer on which to try out the examples. In this book I want to tell you how you can use Mathematica for more than just typing in single commands. If you are using Mathematica for your work, teaching, or solving homework assignments, sooner or later you will encounter more advanced problems requiring many commands to solve, or you will be faced with doing the same calculation steps over and over again with different input. This is the point at which you want to start writing programs. Mathematica includes a rich and powerful programming language. Unlike the usual languages such as BASIC and C, it is not restricted to a small number of data types, but allows you to perform all its symbolic computations. The Mathematica manual can only hint at the possibilities. It explains all the features but does not show you which ones to use for a particular problem or how to fit things together into larger programs.

Through design and tradition, each programming language has developed a certain preferred style of good programming. It is possible to solve the same problem in many different ways, but there is usually some idea about what is a good or a bad program. In this book, I want to present examples of what the designers of

Mathematica think is good programming style and show you why this is so. Even if you write your programs strictly for personal use, you will benefit from following a good style. For developing programs for others to use, adhering to this style is indispensable.

The programming examples in this book serve two purposes. First, they help explain concepts and show how things fit together to make up complete programs. Second, they are designed to be more than mere toy programs and should prove useful in their own right. In developing an example we always use the same method. We start out with a few commands or definitions that could be entered directly into Mathematica. We then extract the parts of the computation that are the same regardless of the input and define some functions or procedures that automate these steps. Then we apply standard techniques to these functions to make them into a package, adding documentation and certain programming constructs that make such a package easier to use. The goal is to write a program that would be useful not only to its author, who knows how it works, but also to other people. Finally, we might add a few more functions to the package or look at alternatives to what we did so far. In Chapter 1 these steps are described in full detail. Later on, when we concentrate on other aspects, we assume that you are familiar with these basic concepts and we shall not mention all the steps in detail. All the programs developed in this book are either part of the standard Mathematica distribution from Wolfram Research, or they are available free in electronic form.

Version 3.0 brings major new features. The programming language itself has not changed all that much, but many problems that required elaborate workarounds and many inconsistencies have been remedied. As a consequence, it is now possible to present the material on packages in a more logical fashion. The support for developing larger applications has been improved, and this edition discusses the software engineering issues of writing and using larger programs in Mathematica. The treatment of exact numeric quantities is another area of improvement, and the way numerical code should be written has changed. All programs have been revised to take advantage of the many new built-in functions. I added more material about functional and structural programming. These techniques are fundamental to writing good Mathematica code, but they are not available in most other programming languages and therefore need an expanded treatment.

The most important addition to Version 3.0 is, of course, the new frontend and the typesetting capabilities. The fact that notebooks and typeset formulae are represented as ordinary Mathematica expressions means that they can be manipulated easily with programs. This capability leads to some fascinating applications, and the new material on the frontend and typesetting teaches you how to develop such applications.

A complete, larger application (iterated function systems), more exercises, and an updated bibliography complete this expanded and revised edition.

This book is no replacement for the Mathematica manual The Mathematica Book. I do not expect that you have read everything in the manual, but you should have some basic experience with Mathematica before reading this book. Single commands are usually used without detailed explanation. You can use the index in The Mathematica Book to look up a description of a command that you did not know about. We also give references to places in The Mathematica Book where you can find explanations of concepts that are particularly relevant to a topic in this book. You should always turn to The Mathematica Book for explanations of features that are assumed known here, but that you have not used yet. The place to look for an explanation of all variants, defaults, or options for a particular command is the Reference Guide in the back of the book. All this information is also available in electronic form and can be accessed through the Help Browser of Mathematica.

All explanations about how Mathematica works are based on Version 3.0. The first edition of this book was about Version 1.2. Many things have changed in the new version. I added a few sections called "Changes from Earlier Editions"for the benefit of readers familiar with earlier versions of Mathematica or the first and second edition of this book.

I am grateful to many people who have contributed to this book. My thanks go first to the other developers of Mathematica. The language was shaped through countless discussions and many heated arguments. Trying to explain to each other why we think a certain feature should be done in a certain way has deepened our understanding of the matters involved and has given the language its overall consistency, despite the fact that it contains hundreds of commands and unifies many diverse programming paradigms.

Helpful ideas for this book came from Jim Feagin, Theodore Gray, Dan Grayson, Jerry Keiper, Silvio Levy, Troels Petersen, Will Self, Bruce Smith, Ilan Vardi, Ferrell Wheeler, and Stephen Wolfram. Some of the examples I used were inspired by Henry Cejtin, John Gray, Lee Rubel, William Thurston, Ilan Vardi, Jnrg Waldvogel, and Dave With