An Introduction to Programming with Mathematica®

Accompanying the book, as with all TELOS sponsored publications, is an electronic component. In this case it is a DOS-Diskette produced by one of the coauthors, Paul Wellin. This diskette consists of Mathematica notebooks and packages which contain the codes for all examples and exercises in the book, as well as additional materials intended to extend many ideas covered in the text. It is of great value to teachers, students, and others using this book to learn how to effectively program with Mathematica .

Introduction to Programming with Mathematica is designed to teach Mathematica programming to scientists, engineers, mathematicians, and computer scientists so that they can fully utilize Mathematica for their work in research or education. No prior familiarity with Mathematica or programming is assumed. The text can be used either for individual study by students and professionals, or in a Mathematica-related university course.

1 Preliminaries.- 1.1 Introduction.- 1.2 Using Mathematica.- 1.2.1 Getting Into and Out of Mathematica.- 1.2.2 Getting Help.- 1.2.3 The Syntax of Inputs.- 1.2.4 Errors.- 1.3 The Mathematica Language.- 1.3.1 Internal Forms of Expressions.- 1.3.2 Predicates and Boolean Operations.- 1.3.3 Evaluation of Expressions.- 1.3.4 Attributes.- 1.4 The Mathematica Interface.- 1.4.1 The Notebook Front End.- 1.4.2 The Command Line Interface.- 2 A Brief Overview of Mathematica.- 2.1 Numerical and Symbolic Computations.- 2.2 Functions.- 2.2.1 Functions of Number Theory.- 2.2.2 Functions of Linear Algebra.- 2.2.3 Random Number Generators.- 2.2.4 Packages.- 2.3 Graphics.- 2.3.1 Two-Dimensional Plots.- 2.3.2 Parametric Plots.- 2.3.3 Three-Dimensional Plots.- 2.4 Representation of Data.- 2.5 Programming.- 2.5.1 Example-Harmonic Numbers.- 2.5.2 Example-Perfect Numbers.- 3 List Manipulation.- 3.1 Introduction.- 3.2 Creating and Measuring Lists.- 3.2.1 List Construction.- 3.2.2 Dimensions of Lists.- 3.3 Working With the Elements of a List.- 3.3.1 Positions in a List.- 3.3.2 Extracting Elements and Rearranging Lists.- 3.4 Working with Several Lists.- 3.5 Higher-Order Functions.- 3.6 Applying Functions to Lists Repeatedly.- 3.7 Strings and Characters.- 4 Functions.- 4.1 Introduction.- 4.2 Programs as Functions.- 4.2.1 Nesting Function Calls.- 4.2.2 Value Names.- 4.3 User-Defined Functions.- 4.4 Auxiliary Functions.- 4.4.1 Compound Functions.- 4.4.2 Localizing Names.- 4.5 Anonymous Functions.- 4.6 One-Liners.- 4.6.1 The Josephus Problem.- 4.6.2 Pocket Change.- 5 Evaluation of Expressions.- 5.1 Introduction.- 5.2 Creating Rewrite Rules.- 5.2.1 The Global Rule Base.- 5.3 Expressions.- 5.3.1 Atoms.- 5.4 Patterns.- 5.4.1 Blanks.- 5.4.2 Expression Pattern-Matching.- 5.4.3 Sequence Pattern-Matching.- 5.4.4 Conditional Pattern-Matching.- 5.4.5 Alternatives.- 5.5 Term Rewriting.- 5.6 Transformation Rules.- 6 Conditional Function Definitions.- 6.1 Introduction.- 6.2 Conditional Functions.- 6.3 Example-Classifying Points.- 7 Recursion.- 7.1 Fibonacci Numbers.- 7.2 List Functions.- 7.3 Thinking Recursively.- 7.4 Recursion and Symbolic Computations.- 7.5 Gaussian Elimination.- 7.6 Trees.- 7.6.1 Binary Trees.- 7.6.2 Huffman Encoding.- 7.7 Dynamic Programming.- 7.8 Higher-Order Functions and Recursion.- 7.9 Debugging.- 7.9.1 Tracing Evaluation.- 7.9.2 Printing Variables.- 7.9.3 Common Errors.- 8 Iteration.- 8.1 Newtons Method.- 8.1.1 Do Loops.- 8.1.2 While Loops.- 8.2 Vectors and Matrices.- 8.2.1 List Component Assignment.- 8.2.2 Finding prime numbers.- 8.3 Passing Arrays to Functions.- 8.4 Gaussian Elimination Revisited.- 9 Numerics.- 9.1 Types of Numbers.- 9.1.1 Integers and Rationals.- 9.1.2 Real Numbers.- 9.1.3 Complex Numbers.- 9.1.4 Computing with Different Number Types.- 9.1.5 Digits and Number Bases.- 9.2 Random Numbers.- 9.3 Precision and Accuracy.- 9.3.1 Roundoff Errors.- 9.4 Numerical Computations.- 9.4.1 Newton's Method Revisited.- 9.4.2 Gaussian Elimination Revisited Again.- 10 Graphics Programming.- 10.1 Graphics Primitives.- 10.1.1 Two-Dimensional Graphics Primitives.- 10.1.2 Three-Dimensional Graphics Primitives.- 10.2 Graphics Directives and Options.- 10.3 Built-in Graphics Functions.- 10.3.1 The Structure of Built-in Graphics.- 10.3.2 Graphics Anomalies.- 10.3.3 Options for Built-in Graphics Functions.- 10.4 Graphics Programming.- 10.4.1 Simple Closed Paths.- 10.4.2 Drawing Trees.- 10.5 Sound.- 10.5.1 The Sound of Mathematics.- 10.5.2 White Music, Brownian Music, and Fractal Noise.- 11 Applications.- 11.1 The Random Walk.- 11.1.1 Introduction.- 11.1.2 The One-Dimensional Random Walk.- 11.1.3 The Two-Dimensional Lattice Walk.- 11.1.4 Visualizing The Two-Dimensional Lattice Walk.- 11.1.5 Numerical Analysis Of The Two-Dimensional Lattice Walk.- 11.2 The Game of Life.- 11.3 Implementing Languages.- 12 Contexts and Packages.- 12.1 Introduction.- 12.2 Using Packages.- 12.2.1 Loading Packages.- 12.2.2 Finding Out What s in a Package.- 12.3 Contexts.- 12.4 Packages.- 12.5 Avoiding Name Collisions.- 12.6 The BaseConvert Package.- 12.7 The RandomWalks Package.- 12.7.1 Adding Options and Defaults.- 12.7.2 The RandomWalks Package.- References.