Programming in Mathematica (3rd Edition)

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. This book is no replacement for the Mathematica manual "The Mathematica Book" [40]. I do not expect that you have read everything in the Mathematica book, 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 tum 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 Mathematica book. All this information is also available in electronic form and can be accessed through the Help Browser of Mathematica. Title Page Preface Contents About This Book Chapter Overview About the Examples Notation and Terminology The Programming in Mathematica Web Site Teaching Mathematica Programming 1 Introduction 1.1 From Calculations to Programs 1.2 Basic Ingredients of a Package 1.3 A Second Function in the Package 1.4 Options 1.5 Defaults for Positional Arguments 1.6 Parameter Type Checking 2 Packages 2.1 Contexts 2.2 Packages That Use Other Packages 2.3 Protection of Symbols in a Package 2.4 Package Framework and Documentation 2.5 Loading Packages 2.6 Large Projects 3 Defaults and Options 3.1 Default Values 3.2 Options for Your Functions 3.3 Setting Options of Several Commands 4 Functional and Procedural Programming 4.1 Procedures and Local Variables 4.2 Loops 4.3 Structured Iteration 4.4 Iterated Function Application 4.5 Map and Apply 4.6 Application: The Platonic Solids 4.7 Operations on Lists and Matrices 5 Evaluation 5.1 Evaluation of the Body of a Rule 5.2 Pure Functions 5.3 Nonstandard Evaluation 5.4 Nonlocal Flow of Control 5.5 Definitions 5.6 Advanced Topic: Scopes of Names 6 Transformation Rules 6.1 Simplification Rules and Normal Forms 6.2 Application: Trigonometric Simplifications 6.3 Globally Defined Rules 6.4 Pattern Matching for Rules 6.5 Traversing Expressions 7 Numerical Computations 7.1 Numbers 7.2 Numerical Evaluation 7.3 Numeric Quantities 7.4 Application: Differential Equations 8 Interaction with Built-in Rules 8.1 Modifying the Main Evaluation Loop 8.2 User-Defined Rules Take Precedence 8.3 Modifying System Function 8.4 Advanced Topic: A New Mathematical Function 9 Input and Output 9.1 Input and Output Formatting 9.2 Input from Files and Programs 9.3 Running Mathematica Unattended 9.4 Session Logging 9.5 Advanced Topic: Typesetting Mathematics 10 Graphics Programming 10.1 Graphics Packages 10.2 Animated Graphics. 10.3 The Chapter Pictures 11 Notebooks 11.1 Packages and Notebooks 11.2 The Structure of Notebooks 11.3 Frontend Programming 12 Application: Iterated Function Systems 12.1 Affine Maps 12.2 Iterated Function Systems 12.3 Examples of Invariant Sets 12.4 Documentation: Help Notebooks and Manuals Appendix A Exercises A.1 Programming Exercises A.2 Solutions Appendix B Bibliography B.1 Background Information and Further Reading B.2 References Index Programs Subjects and Names This revised and expanded edition of the standard reference on programming in Mathematica addresses all the new features in the latest Version 3 of the software. The support for developing larger applications has been improved, and the book now discusses the software engineering issues related to writing and using larger programs in Mathematica. As before, Roman Maeder, one of the original authors of the Mathematica system, explains how to take advantage of its powerful built-in programming language. Roman E. Maeder. Includes Bibliographical References (p. 347-351) And Index.