MathProgrammingIntro.pdf

Mathematica Ò programming: an

advanced introduction

Leonid Shifrin

Part I: The core language

Version 1.01

To my parents

Contents

Preface.......................................................................................18

I. Introduction..............................................................................26

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Summary...............................................................................................................................33

II. Elementary operations..............................................................34

2.1 Introduction..................................................................................................................34

2.2 Symbols and variables............................................................................................. ..34

2.2.1 Legitimate symbol names............................................................................................. 34

2.2.2 Getting information about symbols.................................................................................35

2.2.3 "Proper" variables and OwnValues.................................................................................36

2.2.4 Indexed variables and DownValues................................................................................36

2.2.5 Composite variables and SubValues..............................................................................37

2.2.6 Clearing variables...........................................................................................................37

2.2.7 Does a variable have a value? ValueQ.......................................................................39

2.2.8 Assignments attach rules to Symbol-s...........................................................................39

2.2.9 Summary........................................................................................................................40

2.3 Dynamic data typing.......................................................................................................40

2.4 Assignments...................................................................................................................41

2.4.1 Immediate and delayed assignments: Set and SetDelayed...........................................41

2.4.2 The difference between Set and SetDelayed : an example...........................................41

2.4.3 Chain assignments.........................................................................................................42

2.4.4 Don’ t use SetDelayed with more than two arguments...................................................43

2.4.5 Set and SetDelayed : when which one is used...............................................................44

2.5 Equality checks............................................................................................................ ..44

2.5.1 Equal...............................................................................................................................44

2.5.2 Beware: Equal may return "unevaluated".......................................................................44

2.5.3 Equal is used by built-in solvers to build equations..........................................................45

2.5.4 Assigning the result of Equal to a variable, and more on evaluation..............................45

2.5.5 The role of side effects...................................................................................................46

2.5.6 A digression: standard and non-standard evaluation........................................................46

2.5.7 Pass by reference semantics - possible to imitate..........................................................47

2.5.8 SameQ............................................................................................................................47

2.5.9 TrueQ..............................................................................................................................48

2.6 Logical operators..........................................................................................................49

2.7 Conditionals...................................................................................................................50

2.7.1 The If operator................................................................................................................50

2.7.2 If may return "unevaluated"............................................................................................50

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