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Index

! , Factorial, 16

!=, Unequal, 13

#,Slot, 84

$Context, 268

$ContextPath, 268

$IterationLimit, 143

$MachinePrecision, 194

$Recursion1imit, 170

&, Function, 84

', Derivative, 30

->,Rule, 110

I, Divide, 7

I., ReplaceAll, 110

II., ReplaceRepeated, 112

I;, Condition, 104

:=, SetDelayed, 94

:>, RuleDelayed, 111

;, CompoundExpression, 8, 16, 80

<,Less, 13

<<, Get, 26, 44, 266

<=, LessEqual, 13

=,Set, 94

==, Equal, 13

>,Greater, 13

>=, GreaterEqual, 13

?, (information escape), 6, 36

%,Out, 2

&&,And, 14

·,Power, 7

··,(number base), 200

_, Blank, 46, 101

__ , BlankSequence, 103

___ , BlankNullSequence, 103

I, Alternatives, 105

II, Or, 14

1/f music, 263

Aborting calculations, 6

Abs, 196

acce!Newton,217

Accuracy, 198,208

definition of, 206

internal representation, 198

Action menu, 7

addPairs, 133

addTriples, 134

Algorithms, nondeterministic, 242

Alternatives

and pattern matching, 105

Alternatives, (I), 105

Ampersand (&),Function, 84

And, (&&), 14

angle, 245

Anonymous functions, 83

Function, 83

and Module, 86

evaluation of, 84

example of, 90

example of predicate, 104

286 INDEX

functional form, 83

multiple variables, 84

naming, 84

nested, 85

precedence, 104

shorthand notation, 85

standard input form, 84

Anti-derivative Integrate, 31

Append, 59

Apply, 63

used to change the head of expres-

sion, 98

applyChar, 119-121

ArcCos, 28

ArcSin, 28

ArcTan,245

areEltsEven, 77, 84

Arg,196

Arguments to expressions, 97

Arithmetic

different data types, 197

exact, 206

extended precision, 198

inexact, 206

machine, 194

modular, 261

unexpected results, 199

Array, 55

Arrays, see Matrices

ASCII character codes, 67

table of, 67, 154

AspectRatio, 39, 223

Atoms, 98

internal representation, 99

numbers, 99

strings, 99

structure of, 99

symbols, 98

Attributes, 18

of Plus, 209

changing of, 19

Flat, 18

HoldAll, 186

HoldFirst, 186

Listable, 63

Oneidentity, 19

Orderless, 18

Protected, 19

Augmented matrix, 147

Auto-correlation, of sound, 258

Automatically loading packages, 37

Auxiliary functions, 80

Axes, 222

Bach, J.S., 262

balanced, 15 3

Balanced trees, 153

BarChart, 202

BaseConvert, 272

BaseForm, 199

Bases

numbers in various, 200

of real numbers, 200

Begin, 267

BeginPackage,271

multiple arguments to, 272

Bernoulli numbers, BernoulliB, 16

BernoulliB, 16

Binary numbers, 201

Binary shifts, 201

Binary trees, 150

definition of, 151 Binomial, 32

Binomial coefficients, Binomial, 32

Binomial theorem

Isaac Newton, 32

bisect, 180

Bisection method

for root finding, 179

Blank(_), 46

Blanks, 101

double blanks, ( __ ), 103

single blank, (_), 101

triple blank, ( ___ ), 103

Body of function, 76

Boolean operators, 14

Brandenburg Concerto, 262

Brownian movement

sounds of, 260

Brownian music, 260

Built-in graphics

internal representation, 219

options for, 239

structure of, 233

C major scale, 259

C programming language, xv Calculus'VectorAnalysis', 266 canFollow, 117 Canonical order, 18

of lists, 58

carddeck, 78 Caret, Power("), 7

Casti, John, 262

catNodes, 153

Cells,21

groups of, 21 in notebooks, 21

Character codes, 67

variable length, 155 Characters, 66

Characters, 67

Chemical elements

abbreviations for, 36

atomic weights of, 36

ChemicalElements.m, 35 electron configuration of, 36

heat of vaporization of, 36 Chi-square test, 205

chooseWithReplacement, 79

Circle, 225

Circles

represented parametrically, 39 Clear, 76, 77 Clearing values, Clear, 76 Closed path, 242

Collatz sequence conjectures concerning, 166 definition of, 131

Color wheel

creation of graphic, 232

ColorFunction, 241

Command completion, 7

Command line interface, 5, 25

starting Mathematica, 25

Comments, 9

Complement, 61

Complex, 197

Complex numbers, 196

absolute value of, Abs, 196

arithmetic with, 196

conjugate of, Conjugate, 196

Index 287

geometric representation, 197

imaginary part, lm, 196 pattern matching, 197

phase angle, Arg, 196 real part, Re, 196

Component assignment

in lists, 181

Compound functions, 76, 80

and global rule base, 96

creation of, 80

order of evaluation, 80, 81

syntax, 80 compress, 185

Compression of data Huffman encoding, 154

Computer systems

variation between, 5

Computers in technical education, xiii

use of in research, xiii

Condition,/;, 104

Condition numbers, 218

Conditional functions, 118

If, 118 nesting of, 124

Switch, 120

Which,120

within clauses, 125 Conditional pattern matching, 104

using predicates, 104 con~uniber, 218 Conjugate, 196

Conjunction And, (&&), 14

Constants

computing with, 195

mathematical, 28

representation of, 195

Contexts, 265, 267

and package names, 270

conventions for, 268

current, $Context, 267

End,269

Global', 267

naming conventions, 267

nesting of, 269

path of, $ContextPath, 268

288 INDEX

private, BeginPackage, 271

removing definitions, Remove, 2 76

starting new, Begin, 267

System', 267

Contour plots

simple examples of, 41

ContourGraphics, 234

ContourPlot, 41, 234

convert, 201

convex, 248

Convex hull, 248

Convex polygons, 248

Cos, 28

Count, 89

CrossProduct, 266

Cuboid, 228

Curl, 266

D, 30

Damping Factor

argument to FindRoot, 217

Dashing, 223, 231

Data

plotting with ListPlot, 44

representation of, 44

visualizing with graphics, 38

Data compression

Huffman encoding, 154

runEncode, 135

Data types, 10

DataManipulation, 202

ddx, 141

deal, 79, 81, 82, 86, 133

Debugging, 167

printing variables, 168

using Trace, 15, 167

using TracePrint, 15, 167

DeclarePackage, 37, 276

Declaring values, 96

decode,158

Default values of iterators, 52, 175

Delayed evaluation, 95

Delete, 57, 78

Denominator, 193

Density plots, DensityPlot, 42

DensityGraphics, 233

DensityPlot, 42

Dependent system of equations, 148

Derivatives

Derivative, 30

notation for, 173

numerical, 211

programming symbolically, 140

Deterministic algorithms, 242

Diagonal

of matrix, 149

Differential equations

symbolic solution of, DSolve, 31

Differentiation

Digits

symbolic, D, 30

conversion to numbers, 199

of integers, IntegerDigits, 199

of real numbers, Re~igits, 200

Dimensions, 54

Disjunction Or (Ill. 14

Disk, 232

distance, 79

Divide (1). 7

divide 1 OOBy, 121

Divisors, 49

Do, 174

compared to Nest, 175

compared to Table, 174

Dodecahedron,228

Dot, 64

Dot product, using Inner, 65

Double blank, 103

Double integrals, Integrate, 31

doubleUptoZero, 135

drawSepTree, 251

Drop, 57

DSolve, 31

Dynamic programming, 108, 162

rules in global rule base, 164

E,28

E, Euler's constant, 195

Ears, and hearing, 253

Efficiency of pattern matching, 124

eliminPlace, 188

elimxl, 147

Empty list, 51

enclose, 248

encodeChar, 157

encodeString, 15 7, 161

Encoding

Huffman, 154 End, 269

EndPackage, 271

Entering input, 5, 22

Equal(==), 13

Equal-tempered scale, 259

Equations

matrix, LinearSolve, 145

solution of, 171

solution of, Solve, 29

solving by bisection, 179

systems of, 145

Eratosthenes

sieve of, 182

Error messages

debugging, 169

Errors

common, 169

due to roundoff, 208

infinite recursion, 170

omitting arguments, 169

sampling for graphics, 236

syntax, 169

Euclidean plane classifying points in, 122

Euler's constant, EulerGamma, 28, 48

Euler's totient function, EulerPhi, 32

EulerGamma, 28, 48

EulerPhi, 32

Evaluation

delayed, using SetDelayed, 95

delayed, RuleDelayed, 111

example of steps in, 107

functions, 77

holding, Hold, 109

immediate, using Set, 95

in definitions, 116

of anonymous functions, 84

of expressions, 21, 106

order of in compound functions,

80, 81

precedence, 104

priority of, 107

procedure for, 93

Index 289

replacement rules, 111

term rewriting, 106

Evaluation of expressions, 15

EvenQ, 13 Exact numbers, 192

Exclusive or

Xor, 15

Expand, 29

trigonometric expressions, 29

Exponentiation

notation for, 7

Exponentiation ("), 7

Expressions, 97

arguments to, 97

atoms, 98

changing the head of, Apply, 98

composed of atoms, 97

evaluation of, 15, 106

extracting parts of, Part, 97

head of, 97

internal form, 97

internal form of, 10

manipulating with built-in functions,

98

parts of, 97

priority of evaluation, 107

types of, 10

Extended precision integers, 192 Extended precision numbers

internal representation, 207

Factor, 1, 11

Factorial (!), 16

Factoring

of polynomials, Factor, 1

False, 13

Fibonacci numbers, 129

computed iteratively, 175

cost of computation, 162

definition of, 12 9

dynamically defined, 162

Fibonacci, Leonardo, 129

FindRoot, 30

findRoot, 177

First, 57

Fit, 46

Fixed precision real numbers, 193

290 INDEX

Flat, 18

Flatten, 58, 73

options to, 73

Flattening lists, Flatten, 58

Floating point numbers, 193

variable precision, 195

FM synthesis, 258

Fold, 65, 166

FoldList, 166

FontForm, 224

Formatting conventions, 120

Fortran programming language, xv

Fourier, 263

Fourier transform, 263

FreeQ, 144

Frequencies, 202

frequencies, 91

Frequency modulation synthesis, 258

FromCharacterCode, 68

FullForm, 10, 191

Function, 83

Function(&), 84

Function arguments, 76

Function Browser, 23-25, 36

Function calls, 70

Function names, 76

Functions, 69

anonymous, 83

arguments to, 76

attributes of, 18

automatically loading, DeclarePack-age,37

auxiliary, 80

body of, 76

checking for arguments to, 119

compound, 76, 80

conditional, 118

converting nested function calls to

programs, 77

defined parametrically, 39

defined with several rules, 115

evaluation of, 77

function calls, 70

higher order, 61

information about, 6, 36

multi-clause, 116

names, 76

nested anonymous, 85

nested call, 70

notation for, 7

one-liners, 87

pure, 83

recursion and higher-order, 164

repeated application of, Fold, 65

repeated application of, Nest, 64

simple examples of, 32

syntax of user-defined, 76

threading onto elements, 63

user defined, 46, 76

visualizing with graphics, 38

Gardner, Martin, 259

Gaussian elimination, 145

conditions for failure, 212

failure of, 148

interchanging rows, 148

iterative version, 18 7 numerical considerations, 211

recursively defined, 146

zero pivot, 148

Get,(<<), 26, 44, 266

Global' context, 267

Global rule base, 95

built-in functions, 95

compound functions, 96

dynamic programming effects, 164

examining values in, 163

getting information about, 95

new rule replaces older, 96

preventing addition of rules, 96

querying, 95

user-defined functions, 95

Golden mean, GoldenRatio, 39

GoldenRatio, 39

Graphics, 219

built-in functions, 232

coloring of, ColorFunction, 241

displaying with Show, 222

internal representation of Plot, 219

labelling with Text, 223

logarithmic plots, LogLogListPlot,

45

parametric, ParametricPlot, 39

primitives, 219

rendering of, 40

sampling errors, 236

smoothing, PlotPoints, 41

smoothing of, 237

structure of, 233

three-dimensional, Plot3D, 40

two dimensional, Plot, 38

using fonts with, FontForm, 224

Graphics, 222

Graphics directives, 221

scoping of, 221, 229

table of, 232

Graphics options, 222, 229

listing of, Options, 239

overriding default values, 240

Graphics output, 233

Graphics primitives

table of, 226

three-dimensional, 227

two-dimensional, 219

Graphics programming, 242

Graphics3D, 227, 234

Graphics'MultipleListPlot', 266

GrayLevel, 232

Greater(>), 13

GreaterEqual (>=), 13

Hamming distance, 109 Harmonic numbers, 46

Hayes, Allan, 201

Head,10,191

Head of expressions, 97

Heat production, 44

Height

of trees, 152

height, 152, 154

Help, 6

using the Function Browser, 23

Hexahedron, Cuboid, 228

Higher-order functions, 61

Hilbert matrices, 145, 215

Hold, 109

HoldAll, 186

HoldFirst, 179, 186

Hue, 232

Huffman, David, 155

Huffman encoding, 154

Index 291

efficiency considerations, 156

Huffman trees

construction of, 158

encoding of, 155

representation of, 159

I, v=t. 28, 196

Icon

Mathematica, 5

Icosahedron, 228

If, 118

and higher-order functions, 121

evaluation in, 118

nested, 118

Ill-conditioned matrices, 215, 218

Illinois lottery, 91

Illinois lotto

Pick3, 79

Im, 196

Image compression

runEncode, 135

incrementNumbers, 121

Infinite precision, 192

Infix notation, 10

Information

on functions, 6, 36

init.m, 37, 276

Initialization file, init.m, 3 7 Inner, 65

generalization of dot product, 65

Input

entering, 22

Input forms

standard, 10

Insert, 59

Integer-valued expression, 181

IntegerDigits, 199

IntegerQ, 13

Integers, 191

digits of, IntegerDigits, 199

extended precision, 192

machine size, 192

size limitations of, 191

Integrate, 30

Integration

definite, 1, 30

indefinite, 1, 30

292 INDEX

numerical, Nlntegrate, 31

symbolic, Integrate, 30

Interface

command line, 5, 25

graphical, 5, 20

Notebook, 5, 20

textual, 5

Interior nodes

of trees, 150

interleave, 134

interleave2, 137

interleaveS, 139

Intermediate value theorem, 180

Intersection, 61

Irrational numbers

sound of, 256

Iteration

Do loops, 174

error in assignment, 178, 186

Newton's method, 171

termination, 176, 179

While loops, 176

Iterators

Join, 60

default values of, 52, 53

dependent, 54 in Do loop, 174

nested lists with multiple, 53

omitting from expression, 53

order of multiple, 53

Josephus problem, 87

Josephus, Flavius, 87

Kleiber, Max, 45

Klein bottle, ParametricPlot3D, 43

kSublists, 139

Labelled pattern, 94

Labels

of trees, 150

landMineWalk, 166

Last, 57

Lattice

square, 60

Leaf

of a tree, 150

Least-squares fits, Fit, 46

Lehtonen, Ari, 43

Length, 54

length, 132

Less(<), 13

LessEqual (<=), 13

Line, 221, 244

Linear congruential method, 204

Linear fits, Fit, 46

Linear systems

magnification of error in, 212

solution of, 211

solving with scaled pivoting, 218

LinearSolve, 145

LISP programming language, xv

Lissajous curves, ParametricPlot, 40

List, 51, 62

creation of, 52

creation of, Table, 52

elements, 51

empty, 51

location of elements, 51

notation for, 51

Listability, 62, 63

Listable, 62, 63

ListDensityPlot, 233

listLevel, 153

ListPlay, 256

ListPlot, 44

Lists, 51

adding elements to, Append, 59

adding elements to, Insert, 59

adding elements to, Prepend, 59

applying functions to, Map, 61

as sets, 61

as vectors or matrices, 55

changing the Head of, Apply, 63

choosing elements that pass test,

Select, 58

combining into one, Join, 60

component assignment, 181

creation of, Table, 52

creation of, Array, 55

creation of, Range, 52

displaying with TableForm, 54

elements, 51

empty, 51

extracting consecutive elements, 57 extracting elements from back, Take,

56 extracting elements from front, Take,

56 extracting elements, Part, 56

forming outer products, 62

location of elements, Position, 51,

56

measuring size, Length, 54

measuring with Dimensions, 54

modifying elements of, 181, 182

non-rectangular, 54 notation for, 8, 51

of different length, 137, 166

pairing elements of, MapThread,

62 plotting, ListPlot, 44

rearranging, Partition, 58

removing braces, Flatten, 58

removing elements from, Delete,

57 removing elements from, Drop, 57

repeated application of functions, FoldList, 65

repeated application of functions,

NestList, 64 replacing elements of, 59 representing data with, 44

reversing order of, 58 rotating elements of, RotateLeft, 58 rotating elements of, RotateRight,

58 sorting, Sort, 58

the complement of, Complement,

61

the intersection of, Intersection, 61

threading functions onto, 63

transposing elements, Transpose,

59 union of,Union, 60

used to represent trees, 151

working with multiple, 60

Loading packages, 35, 266 Local rewrite rule, 110

Local variables

Module, 178

Index 293

use of, 178 Localizing names, Module, 81

Log attributes of, 63

Log, 28

Logarithms base of natural, E, 28

bases of, 28

Logarithms, Log, 28

Logical operators, 14

table of, 15

LogLogListPlot, 45

Loops Do, 174

While, 176

lottol, 91

lotto2, 92

Lower triangular matrix, 149

LU decomposition, 149, 190

LUdecompl, 150 LUdecomp2, 150

Machine integer, 192 Machine integers

addition of, 192

Machine numbers

display of, 194

Machine precision arithmetic, 194

MachinePrecisionQ, 194 Macintosh

starting Mathematica, 5 Maeder, Roman, 93 makeChange, 92 MakeSyrnbol,266

Mandelbrot, BenOit, 258

Map, 61

map, 164

map2, 167

MapThread,62, 108

difference from Thread, 109

MatchQ, 101

math file, 5

Mathematica features of, xiv

Mathematica Conference

programming contest, 136

294 INDEX

Mathematica icon, 5

Mathematical constants, 28

MathSource, xvi

Matrices, 180

as lists, 55

augmented, 147

condition number of, 218

diagonal of, 149

Hilbert, 145, 215

ill-conditioned, 215

lower triangular, 149

LU-decomposition, 149

modifying arrays, 185

multiplication, Dot, 64

nonsingular, 213

norms of, 217

passing arrays to functions, 185

rotating as list, 64

upper triangular, 149

matrixNorm, 218

MaxBend, 235

maxima, 77, 85, 137, 139

maxPairs, 134

Messages

usage, 273

Metabolic heat rate

Kleiber's constant, 45

Middle square method, 205

minlnTree, 151, 154

Mobius band, ParametricPlot3D, 43

Mod, 32

Modular arithmetic, 261

Module, 81, 96, 119, 178

and anonymous functions, 86

syntax of, 81

Modulo, Mod, 32

Morse code, 155

multAllPairs, 134

Multi-clause definition, 116

Multiple precision arithmetic, 195

Multiplication

notation for, 7

Multipliers

in Gaussian elimination, 211

multPairwi.se, 133

Musical scales, 259

N, 27, 28

Names

precision in, 28, 206

localizing, Module, 81

removing definitions, Clear, 77

value of, 74

Names, 276

syntax of, 7 4

Needs, 266

Nest, 64, 78

compared to Do, 175

nest, 165

Nested functions, 70

anonymous, 85

converting to programs, 77

Nested lists, 53

modifying elements of, 182

NestList, 166

newton,210

Newton's method, 171, 209

acceleration of, 217

example of failure, 171

Finc!Root, 30

increased precision, 210

no symbolic derivative, 210

zero derivative, 210

Newton, Isaac, 32

NeXT

starting Mathematica, 5

Nintegrate, 31

Nodes

of trees, 150

Noise

sound of, 256

Non-rectangular lists, 54

Nonsingular matrix, 213

Norm

of a matrix, 217

Normal, 241

Notation

division, 7

exponentiation, 7

multiplication, 7

Notebook Front End, 20

Notebook interface, 5, 20

starting Mathematica, 20

NP-complete problems, 242

NSum, 47

Null, 174

Number base, ", 200

Numbers

Bernoulli, BernoulliB, 16

as atoms, 99

bases of, 199

complex, 196

display of, 194

exact, 192

fibonacci, 129

floating point, 193

harmonic, 46

heads of, 99

integer, 191

irrational, 256

perfect, 49, 82

prime, 182

random, 34

random, Random, 202

rational, 191, 256

real, 193

super-perfect, 83

triangular, 33

Numerator, 193

Numerical calculations

simple examples, 27

Numerical derivative, 211

Numerical summation, NSum, 47

Octahedron, 228

OddQ,13

Offspring

on trees, 150

One-dimensional random walk, 55

One-liners, 87

examples of, 88, 90

Oneldentity, 19

Operators

Boolean, 14

logical, 14, 15

relational, 13

Options

graphics, 239

Options, 239

Or (11),14

Order of operations, 7

Index 295

Orderless, 18

Out(%), 2

Outer, 62

Outer products, Outer, 62

Packages, 35, 265

and Module's, 265

automatic loading of, 276

automatically loading of, 37

Calculus'VectorAnalysis', 266

ChemicalElements.m, 35

customizing Mathematica with, 35

DataManipulation, 202

example of, 272

folder, 35

Graphics, 202

Graphics'MultipleListPlot', 266

hierarchy of, 270

listing names, Names, 276

loading, 35

name localizing, 265

names and contexts, 270

naming conventions, 272

Polyhedra.m, 228

private contexts, 271

reading in,(<<), 26, 44, 266

reading in, (Needs), 266

usage messages, 273 Parametric functions, 39

Parametric surfaces

Klein bottle, ParametricPlot3D, 43

Mobius band, ParametricPlot3D, 43

ParametricPlot, 39

ParametricPlot3D, 43, 234

Part, 56

shorthand notation, 56

Partial pivoting, 213

Partition, 58

Partitioning lists, Partition, 59

Pascal, Blaise, 33

Pascal's Triangle, 33

path, 244

Paths, closed, 242

Pattern matching, 100

alternatives, 105

and conditions, 104

blanks, 101

296 INDEX

complex numbers, 197

conditional, 104

efficiency considerations, 124

preferred format, 117

rational numbers, 193

sequences, 103

sequences of arguments, 103-104

triple blank, 103

using MatchQ, 101

using single blank, 101

with sequences, 116

Pattern variable, 94

Patterns, 100

definition, 101

labelled, 94

labelling of, 101

variable, 94

perfect, 82

Perfect numbers, 49, 82

conjectures, 50

Perfect shuffle, 73

perfect3, 83

perfect4, 83

perfectK, 83

perfecto, 83

Periodicity, of numbers, 256

Permutations, 60

permute, 180

Phase angle

of complex number, Arg, 196

Pi, 28, 195

pick, 79

Pick3

Illinois lotto, 79

Pivoting

on zero element, 213

placeTree, 250, 251

Platonic solids, 228

Play, 234, 254

Plot, 38, 219

Plot3D, 234, 236

PlotDivision, 235

PlotPoints, 41, 236

Plots

default values for options, 235

displaying options of, Options, 239

evaluation of functions in, 234

fonts in, FontForm, 224

internal algorithms, 234

internal representation, 219

of data, ListDensityPlot, 233

plotting surfaces, Plot3D, 40

points to sample in, PlotPoints, 236

shape of, AspectRatio, 39, 223

simple examples of, 38

singularities, 38

smoothing of, 237

structure of built-in, 233

subdivision level in, PlotDivision,

235

text in, Text, 223

three-dimensional, Graphics3D, 227

three-dimensional, Plot3D, 40, 236

tick marks in, Ticks, 223

view point for, ViewPoint, 239

Plus, 10, 63

plus4mod20, 166

Pi NP, 242

Point, 221

pointLoc, 122

PointSize, 221

Polygons

convex,248

convex hull, 248

Polynomials

factoring of, Factor, 1

Position, 56

Postfix notation, 11

Pound symbol, (#), 84

Power("), 7

Power spectra

in music, 263

Precedence

and anonymous functions, 104

of operations, 7

ofrelational operators, 14

Precision

definition of, 206

extended integer, 192

in multiplication, 198

infinite, 192

internal representation, 198

machine, $MachinePrecision, 194

real numbers, 193

setting of, SetPrecision, 207

significant digits, 192

Predicates

and conditional pattern matching,

104

example of, 49

example of anonymous, 104

testing expressions using, 13

prefixMatch, 138

Prepend, 59

Prime numbers

algorithms for finding, 182

PrimeQ, 13

primes, 183

primes2, 184

Print, 175, 177

example of, 184

used to debug, 168

printlndented, 152

Printing

of messages, 175, 177

Printing variables

as debugging tool, 168

printTree, 152, 154

Programming

as a tool, 2

debugging, 167

dynamic, 108, 162 efficiency considerations, 92, 125

errors in, 169

graphical problems, 242

other languages, 3

principles of, 3

simple examples of, 49

using prototypes, 73

Programming languages

C,xv

Fortran,xv

LISP, XV

Programs

acce!Newton, 217

addPairs, 133

add'l'riples, 134

applyChar, 119-121

areEltsEven, 77, 84

balanced, 153

BaseConvert, 272

Index 297

bisect, 180

bowlingScore, 2

canFollow, 117

carddeck,78

catNodes,153

chooseWithReplacement, 79

compress, 185

con(Uqunruber, 218

convert, 201

convex,248

ddx, 141

deal,79,81,82,86, 133

decode,158

distance, 79

dividelOOBy, 121

doubleUptoZero, 135

drawSepTree, 251

eliminPlace, 188

elimxl, 147

enclose, 248

encodeChar, 157

encodeString, 157, 161

findRoot, 177

FreeQ, 144

frequencies, 91

height, 152, 154

incrementNunrubers, 121

interleave, 134 interleave2, 137

interleave3, 139

kSublists, 139

landMineWalk, 166

lengthlength, 132

listLevel, 15 3

lottol, 91

lotto2, 92

LUdecompl, 150

LUdecomp2, 150

makeChange, 92

map, 164

map2, 167

matrixNorm, 218

maxhna,77,85,137,139

maxPairs, 134

ntirUnTree, 151, 154

multAllPairs, 134

multPairwise, 133

298 INDEX

nest, 165

newton,210

angle, 245

path, 244

perfect, 82

perfect3, 83

perfect4, 83

perfectK, 83

perfecto, 83

permute, 180

pick, 79

placeTree, 250, 251

plus4mod20, 166

pocketChange, 90

pointLoc, 122

prefixMatch, 138

primes, 183

primes2, 184

printindented, 152

priniTree, 152, 154

randomWalk3D, 227

removeLeadingSpaces, 134

removeRand, 78, 85

removeRepetitions, 122

remove~Spaces, 118 removeWhiteSpace, 115, 116, 118

repeat, 165

repeatCount, 165

reverse, 185

reverselnPlace, 18 7

rollEm, 204

rotateRight, 185

rotateRows, 185

rotateRowsByS, 185

runDecode,139

runEncode, 135, 136, 138

secant, 211

setOfDistances, 86

showRandomWalk3D, 227, 232

sieve, 185

simpleClosedPath, 246

soln, 189

solve, 147

solve2, 148

solve!, 187, 188

solveLower, 149

solvePP, 213, 218

solveUpper, 149

splitl, 91

split2, 91

sqrElement, 186

subtractEl, 211

sumElts, 132

sumEveryOtherElt, 134

sumNodes, 152

sumOddElts, 134

sumsOfCubes, 201

superperfect, 83

survivor, 88, 202

swap, 187

transpose, 185

vowelQ, 117

Protected, 19

Prototyping programs, 73

Pure functions, 83

Quadratic congruential method, 205

Quit, 5, 26

Quitting Mathematica, 5

Random,34, 78,202

arguments of, 195 Random number algorithms

linear congruential method, 204

middle square method, 205

quadratic congruential method, 205

Random numbers, 34, 202

distribution of, 202

integers, 34

testing hypothesis with, 203

testing randomness, 205

Random sampling, 165

with replacement, 79

Random walks

one-dimensional, 55, 66

plots of three dimensional, 227

two-dimensional, 60, 66, 92, 166

randomWalk3D, 227

Range,49,52

Rational, 193

Rational numbers, 191

arithmetic, 193

pattern matching, 193

sound of, 256

Re,196

Reading packages, Get(<<), 26, 44, 266

Real numbers, 193

arithmetic, 194

digits of, RealDigits, 200

extended precision, 195

fixed precision, 193

multiple precision, 195

variable precision, 195

RealDigits, 200

Recursion

and symbolic computations, 140

on lists, 132

Recursive functions

and trees, 151

base cases, 131

errors in definition, 132, 141, 143

symbolic computations with, 140

tracing output of, 130

writing, 134

Relational operators, 13

precedence of, 14

table of, 14

ReleaseHold, 109

Remove, 276

removeLeadingSpaces, 134

removeRand, 78, 85

removeRepetitions, 122 removeTwoSpaces, 118

removeWhiteSpace, 115, 116, 118

Removing a value, Clear, 76

repeat, 165

repeatCount, 165

ReplaceAll, (/.), 110

ReplacePart, 59

ReplaceRepeated, (II.), 112

Replacing elements of lists, ReplacePart,

59

Representation of data, 44

Rest, 57,88

Reverse, 58, 61

reverse, 185

reverselnPlace, 187

Rewrite rules, 94

and dynamic programming, 108

built-in functions, 94

compound functions, 96

Index 299

creating, Set, 94

creating, SetDelayed, 94

difference between Set and SetDe-

layed, 94

global rule base, 95

local, 110

localizing, Module, 96

names and, 96

new rule replaces older, 96

order of, 162

priority of, 107

simple examples, 94

steps in term rewriting, 107

transformation rules, 110

user-defined, 94

rollEm, 204

Root finding

Roots

accuracy of, 209

bisection method, 179

Newton's method, 171

Secant method, 211

of trees, 150

RotateLeft, 58, 88

RotateRight, 58

rotateRight, 185

rotateRows, 185

rotateRowsByS, 185 Roundoff errors, 208

Rule(->), 110

RuleDelayed (:>), 111

Rules

evaluation in, 116

priority of evaluation, 116

runDecode, 139

runEncode, 135, 136, 138

SampleRate, 255

Sawtooth waves

sound of, 258

Scaled pivoting, 218

Scaling noises, 258

Sec,28

secant, 211

Secant method, 211

SeedRandom, 35

Select, 49, 58

300 INDEX

Semicolon, (;). 8, 16, 80

Separation tree, 251

Sequences, 103

and pattern matching, 103, 116

and triple blanks, 103

Collatz, 131

testing randomness of, 205

Series, 241

Set(=), 94

difference from SetDelayed, 94

notation, 94

used to make value declarations,

96

SetAttributes, 19

SetCoordinates, 266

SetDelayed (:=), 94

setOfDistances, 86

SetPrecision, 207

Short, 220

Show, 222

showRandomWalk3D, 227, 232

sieve, 185

Sieve of Eratosthenes, 182

Significant digits

and Accuracy, 198

and Precision, 192

Simple closed paths, 242

simpleClosedPath, 246

Sin, 28

soln, 189

Solution of equations

numerical, FindRoot, 30

symbolic, Solve, 29

Solve, 29

solve, 147

solve2, 148

solve!, 187, 188

solveLower, 149

solvePP, 213, 218

solveUpper, 149

Solving equations

bisection method, 179

Solving equations, Solve, 29

Sort, 58, 61, 245

Sound

1/f music, 263

auto-correlation, 258

brownian music, 260

noise, 256

of functions, Play, 254

periodic, 254

physics of, 253

sampling rates, 255

sawtooth waves, 258

scaling noises, 258

speeding execution of, 259

square waves, 257

using to find periodicity, 256

white music, 258

Sound,234

splitl, 91

split2, 91

sqrElement, 186

Sqrt, 28

Square brackets, 46

Square lattice, 60

Square roots, Sqrt, 28

Square waves

sound of, 257

Standard input form, 10

Starting Mathematica, 5

command line interface, 25

Notebook interface, 20

Statistical analysis

random sampling, 165

StringDrop, 67, 79

Stringlnsert, 67, 79

StringJoin, 6 7

StringLength, 66

StringPosition, 67

StringReplace, 67

StringReverse, 66

Strings

and trees, 151

as atoms, 99

character codes, 67

converting from characters,

StringJoin, 67

converting to characters, Charac­

ters, 67

determining position in, StringPo­

sition, 67

finding length of, StringLength, 66

heads of, 99

inserting character in, Stringlnsert,

67

internal form, 66

manipulating, 66

removing characters from, String­

Drop, 67

replacing characters in, StringRe­

place, 67

reversal of, StringReverse, 66

StringTake, 66

subtractE 1, 211

Sum, 33, 203

sumElts, 132

sumEveryOtherElt, 134

sumNodes, 152

sumOddE!ts, 134

Sums, Sum, 33

sumsOfCubes, 201

Super-perfect numbers, 83

superperfect, 83

Suppressing output

with semicolon, 16

SurfaceGraphics, 234

survivor, 88, 202

swap, 187

Switch, 120

evaluation of, 121

Symbolic calculations simple examples of, 27

Symbolic computations

and recursion, 140

derivatives, 140

Symbolic sums, SymbolicSum, 48

SymbolicSum, 48

Symbols

as atoms, 98

computation with, 195

heads of, 99

System', 267

Systems of equations

dependent, 148

order of, 148

solution of, 145

solved recursively, 146

Table, 33, 52

compared to Do, 174

Index 301

nested lists with, 53

TableForm, 33, 54

Take, 56,57

Tan,28

Taylor polynomials

series of, Series, 241

Templates, 7

Term rewriting, 106

dynamic programming, 108

priority of, 107

steps of, 107

Tetrahedron, 228

Text, 223

Textual interface, 5

Thread, 108

difference from MapThread, 109

Threading functions onto lists, 63

Ticks, 223

Timing

factors affecting, 16, 17

Timing, 16

ToCharacterCode, 68, 79

Totient function, EulerPhi, 32

Trace, 15, 70, 71, 108, 167

TraceOriginal, 108

TracePrint, 15, 88, 167

second argument of, 168

Tracing specific patterns, 168

Transformation rules, 110

applying repeatedly,

ReplaceRepeated, 112

example of, 136

local rewrite rule, 110

multiple rules, 112

order of, 162

using labelled patterns, 111

using symbols, 111

Transpose, 59, 73

transpose, 185

Transposing lists, Transpose, 59

Traveling salesman problems, 242

Trees

of strings, 151

balanced, 153

binary, 150, 151

children of, 150

display of, 152

302 INDEX

drawing, 249 Which, 120

finding minimum string, minln'I'ree, While, 176

151 White music, 258

height of, 152 White noise, 258

Huffman encoding, 155 Windows

interior node, 150 starting Mathematica, 5

labels of, 150 Word size, 192

leaf of, 150

nodes of, 150

representation of, 151

root of, 150

weight of node, 159

Triangular numbers, 33

Trig, 29

False, 13

True, 13

Tungsten, W, 36

Unequal, 144

Unequal(!=), 13

Union, 60

UnsameQ, 144

Upper triangular matrix, 149

usage messages, 273

User defined functions, 46, 76

syntax, 76

Value declaration, 96

Values

nicknames, 75

of names, 74

removing, Clear, 76

result of evaluation, 75

Variable length codes, 155

Variables

in anonymous functions, 84

local, 178

Vectors, 180

as lists, 55

ViewPoint, 239

von Neumann, John, 205

vowelQ, 117

Voxel, Cuboid, 228

Waves

sound of sawtooth, 258

sound of square, 257

Xor, 15

Zizza, Frank, 136