MathematicaÒ programming: an advanced introduction Leonid

Shifrin

Part I: The core language

Version 1.01

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Mathematica programming: an advanced introduction Leonid Shifrin Copyright © Leonid Shifrin, 2008

This work is licensed under the Creative Commons Attribution-Noncommercial-Share Alike 3.0 United States License. To view a copy of this license, visit http://creativecommons.org/licenses/by-nc-sa/3.0/us/ or send a letter to Creative Commons, 171 Second Street, Suite 300, San Francisco, California, 94105, USA.

MathematicaTM is a registered trademark of Wolfram Research Inc. Other symbols, where used, respective owners.

Leonid Shifrin

Digitally signed by Leonid Shifrin DN: cn=Leonid Shifrin, o=Brunel University, ou, [email protected] ming-intro.org, c=GB Date: 2009.02.04 11:30:22 -08'00'

are registered trademarks of their

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To my parents

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Contents Preface.......................................................................................18 I. Introduction..............................................................................26 Ÿ 1.1

First principle: everything is an expression................................................................26

Ÿ 1.1.1

Atoms and the built-in AtomQ predicate......................................................................26

Ÿ 1.1.2

Mathematica normal (composite) expressions............................................................26

Ÿ 1.1.3

Literal equivalents of built-in functions, and FullForm command................................26

Ÿ 1.1.4.

All normal expressions are trees - TreeForm command............................................ .27

Ÿ 1.1.5.

Heads of expressions and the Head command...........................................................27

Ÿ 1.1.6

Accessing individual parts of expressions through indexing ......................................28

Ÿ 1.1.7

Levels of expressions and the Level command...........................................................28

Ÿ 1.2

Second principle: pattern-matching and rule substitution...........................................30

Ÿ 1.2.1

Rewrite Rules..............................................................................................................30

Ÿ 1.2.2

An example of a simple pattern-defined function........................................................30

Ÿ 1.2.3

Functions are really rules : DownValues command....................................................31

Ÿ 1.2.4

Example of a function based on a restricted pattern...................................................31

Ÿ 1.2.5

A bit about evaluation..................................................................................................31

Ÿ 1.2.6

Patterns allow for multiple definitions of the same function.........................................31

Ÿ 1.2.7

Non - commutativity of rules substitution.....................................................................32

Ÿ 1.2.8

Automatic rule reordering.............................................................................................32

Ÿ 1.3

Third principle: expression evaluation........................................................................33

Ÿ Summary...............................................................................................................................33

II. Elementary operations..............................................................34 Ÿ 2.1

Introduction........................