Mathematica Cookbook

827 Pages·2010·11.16 MB·English

by  Sal Mangano

Mathematica Cookbook
Sal Mangano
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Mathematica Cookbook
by Sal Mangano
Copyright © 2010 Salvatore Mangano. All rights reserved.
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ISBN: 978-0-596-52099-1
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To Wanda, Leonardo and Salvatore:
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Table of Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1. Numerics ..................................................................................................11
1.1 Controlling Precision and Accuracy 4
1.2 Mixing Different Numerical Types 9
1.3 Representing Numbers in Other Bases 12
1.4 Extracting the Digits of a Number 13
1.5 Working with Intervals 16
1.6 Converting Between Numerical Types 18
1.7 Displaying Numbers in Alternate Forms 20
2. Functional Programming ..........................................................................23
2.0 Introduction 23
2.1 Mapping Functions with More Than One Argument 32
2.2 Holding Arbitrary Arguments 36
2.3 Creating Functions That Automatically Map Over Lists 37
2.4 Mapping Multiple Functions in a Single Pass 38
2.5 Keeping Track of the Index of Each Item As You Map 41
2.6 Mapping a Function over a Moving Sublist 43
2.7 Using Prefix and Postfix Notation to Produce More
Readable Code 48
2.8 Defining Indexed Functions 51
2.9 Understanding the Use of Fold As an Alternative to Recursion 53
2.10 Incremental Construction of Lists 57
2.11 Computing Through Repeated Function Application 59
2.12 Building a Function Through Iteration 62
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2.13 Exploiting Function Composition and Inverse Functions 63
2.14 Implementing Closures 66
2.15 Currying in Mathematica 73
2.16 Creating Functions with Default Values 77
2.17 Creating Functions That Accept Options 79
3. Data Structures .......................................................................................85
3.0 Introduction 85
3.1 Ensuring the Most Efficient Representation of Numerical Lists 95
3.2 Sorting Lists 97
3.3 Determining Order Without Sorting 100
3.4 Extracting the Diagonals of a Matrix 102
3.5 Constructing Matrices of Specific Structure 103
3.6 Constructing Permutation and Shift Matrices 105
3.7 Manipulating Rows and Columns of Matrices 110
3.8 Using Sparse Arrays to Conserve Memory 112
3.9 Manipulating Deeply Nested Lists Using Functions
with Level Specifications 114
3.10 Implementing Bit Vectors and Using Format to Customize
Their Presentation 119
3.11 Implementing Trees and Traversals Using Lists 121
3.12 Implementing Ordered Associative Lookup Using
a Red-Black Tree 125
3.13 Exploiting Mathematica’s Built-In Associative Lookup 130
3.14 Constructing Graphs Using the Combinatorica` Package 134
3.15 Using Graph Algorithms to Extract Information from Graphs 140
4. Patterns and Rule-Based Programming . ...............................................145
4.0 Introduction 145
4.1 Collecting Items That Match (or Don’t Match) a Pattern 151
4.2 Excluding Items That Match (or Don’t Match) a Pattern 153
4.3 Counting Items That Match a Pattern 155
4.4 Replacing Parts of an Expression 157
4.5 Finding the Longest (or Shortest) Match for a Pattern 159
4.6 Implementing Algorithms in Terms of Rules 161
4.7 Debugging Infinite Loops When Using ReplaceRepeated 165
4.8 Preventing Evaluation Until Replace Is Complete 168
4.9 Manipulating Patterns with Patterns 169
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4.10 Optimizing Rules 170
4.11 Using Patterns As a Query Language 171
4.12 Semantic Pattern Matching 177
4.13 Unification Pattern Matching 178
5. String and Text Processing . ....................................................................181
5.0 Introduction 181
5.1 Comparing Strings 187
5.2 Removing and Replacing Characters from Strings 188
5.3 Extracting Characters and Substrings 192
5.4 Duplicating a String 196
5.5 Matching and Searching Text 198
5.6 Tokenizing Text 201
5.7 Working with Natural Language Dictionaries 202
5.8 Importing XML 209
5.9 Transforming XML Using Patterns and Rules 213
5.10 Transforming XML Using Recursive Functions (à la XSLT) 218
5.11 Writing Parsers and Grammars in Mathematica 227
6. Two-Dimensional Graphics and Plots ......................................................237
6.0 Introduction 237
6.1 Plotting Functions in Cartesian Coordinates 238
6.2 Plotting in Polar Coordinates 247
6.3 Creating Plots Parametrically 249
6.4 Plotting Data 252
6.5 Mixing Two or More Graphs into a Single Graph 255
6.6 Displaying Multiple Graphs in a Grid 258
6.7 Creating Plots with Legends 260
6.8 Displaying 2D Geometric Shapes 263
6.9 Annotating Graphics with Text 269
6.10 Creating Custom Arrows 270
7. Three-Dimensional Plots and Graphics . .................................................275
7.0 Introduction 275
7.1 Plotting Functions of Two Variables in Cartesian Coordinates 276
7.2 Plotting Functions in Spherical Coordinates 283
7.3 Plotting Surfaces in Cylindrical Coordinates 285
7.4 Plotting 3D Surfaces Parametrically 290
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7.5 Creating 3D Contour Plots 292
7.6 Combining 2D Contours with 3D Plots 295
7.7 Constraining Plots to Specified Regions 296
7.8 Plotting Data in 3D 298
7.9 Plotting 3D Regions Where a Predicate Is Satisfied 301
7.10 Displaying 3D Geometrical Shapes 302
7.11 Constructing Wireframe Models from Mesh 306
7.12 Controlling Viewing Geometry 309
7.13 Controlling Lighting and Surface Properties 313
7.14 Transforming 3D Graphics 317
7.15 Exploring Polyhedra 320
7.16 Importing 3D Graphics from CAD and Other 3D Software 326
8. Image Processing . ................................................................................329
8.0 Introduction 329
8.1 Extracting Image Information 332
8.2 Converting Images from RGB Color Space to HSV Color Space 335
8.3 Enhancing Images Using Histogram Equalization 341
8.4 Correcting Images Using Histogram Specification 347
8.5 Sharpening Images Using Laplacian Transforms 351
8.6 Sharpening and Smoothing with Fourier Transforms 356
8.7 Detecting Edges in Images 361
8.8 Image Recognition Using Eigenvectors (Eigenimages) 365
9. Audio and Music Processing . ..................................................................373
9.0 Introduction 373
9.1 Creating Musical Notes 374
9.2 Creating a Scale or a Melody 375
9.3 Adding Rhythm to a Melody 376
9.4 Controlling the Volume 377
9.5 Creating Chords 378
9.6 Playing a Chord Progression 379
9.7 Writing Music with Traditional Chord Notation 380
9.8 Creating Percussion Grooves 384
9.9 Creating More Complex Percussion Grooves 386
9.10 Exporting MIDI files 389
9.11 Playing Functions As Sound 390
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