APL with a Mathematical Accent

Clifford A. Reiter and William R. Jones, Brooks-Cole Publishing Co., Pacific Grove (1990) (now distributed by CRC Press)
ISBN 0-534-12864-5

CHAPTER 1 First Steps With APL 1

1.1    Interacting with APL 1
1.2    Order of Execution 2
1.3    Negative Sign, Minus Sign 2
1.4    Assignment and Variable Names 3
1.5    Example: Heron's Formula 4
1.6    The Equals Function 4
1.7    Examples: Right Triangle Test, Conditional Incrementing 4
1.8    Dyadic, Monadic Functions 4
1.9    Functions with Vector Arguments 5
1.10   Examples: Some Uses of Vector Arithmetic 6
1.11   Shape, Reduction Operator 7
1.12   Examples: Average, Dot Product 7
1.13   More on the Reduction Operator 8
1.14   Index Generator 8
1.15   Examples: A Partial Sum, A Sequence for e 9
1.16   Number Forms and Print Precision 9
1.17   Workspaces, System Commands 9
1.18   Exiting APL 10
Exercises 10

CHAPTER 2 Fundamentals of Arrays 12

2.1    Example: Polynomial Evaluation 12
2.2    Elementwise Arithmetic 13
2.3    Reshape: Defining Arravs 14
2.4   Reduction 14
2.5   Outer Product 15
2.6   Example: Evaluation of Polynomials at Several Points 15
2.7   Catenation 16
2.8   Catenation of Matrices 16
2.9   Example: Forming a Blockwise Diagonal Matrix 17
2.10  Catenation of Matrices with Vectors 18
2.11  Indexing 18
2.12  Example: Elementary Row Operations on Matrices 19
2.13  Shape 20
2.14  Empty Arrays 22 Exercises 22

CHAPTER 3 Elementary Function Definition 26

3.1   Function Definition 26
3.2   Example: A Function for Polynomial Evaluation 28
3.3   Example: Approximating Areas Using Right-Hand Endpoints 28
3.4   Random Integers, Random Reals      30
3.5   Correcting a Function Line 30
3.6   Commands for Editing Functions     31
3.7   Suspended Functions; Interrupts   32
3.8   System Commands 33
3.9   Branching: Simple Loops 34
3.10  Example: Newton's Method 35
3.11  Statement Separation, Diamond 36
3.12  Explicit Output 37
3.13  Example: Fibonacci Numbers 38
Exercises 38

CHAPTER 4 Matrix Algebra 41

4.1   Matrix Product 41
4.2   Example: Markov Processes 42
4.3   Solving Linear Systems, Matrix Divide 43
4.4   Example: Some Solutions of Linear Systems 44
4.5   Outer Product 45
4.6   Example: Polynomial Interpolation 46
4.7    Example: Polynomial Evaluation Revisited 46
4.8    Least-Squares Solutions of Linear Systems 47
4.9    Example: Least-Squares Polynomial Curve Fitting 48
4.10   Matrix Inverse 49
4.11   Matrix Transpose 49
4.12   Example: A Formula Using Transpose 50
4.13   Example: Gram-Schmidt Orthogonalization 50
4.14   Example: Testing for Orthonormality 50
4.15   The Power Method 51
Exercises 53

CHAPTER 5 Data Comparison and Logical Functions 57

5.1    Relational Functions: Comparatives 57
5.2    Example: Averages on Selected Scores 58
5.3    Logical Functions 59
5.4    Example: Frequency Distribution 60
5.5    Generalized Inner Product: Vectors 60
5.6    Example: Weighted Average 61
5.7    Generalized Inner Product: General Arrays 61
5.8    Example: Paths in Graphs 63
5.9    Floor, Ceiling 64
5.10   Example: Rounding to a Decimal Position 65
5.11   Membership Function 65
5.12   Comparison Tolerance 66
Exercises 67

CHAPTER 6 Simulation and More Mathematical Functions 69

6.1    Roll, Deal, and Random Link 69
6.2    Example: Matching Partners 70
6.3    Example: Simulating A Classical Probability Experiment 71
6.4    Example: Polya's Um Scheme 72
6.5    ]Factorial and Binomial 74
6.6    Example: Binomial Distribution 74
6.7    Example: Polynomial Translation 75
6.8    Trigonometric Functions 77
6.9    Example: Monte Carlo Integration 78
Exercises 79

CHAPTER 7 Statistics and Graphics 81

7.1    Mean and Standard Deviation 81
7.2    Maximum and Minimum 82
7.3    Example: Range 82
7.4    Grade Up 83
7.5    Example: Median 83
7.6    Character Arrays 84
7.7    Example: Frequencies 84
7.8    Example: Histograms 85
7.9    Replicate on Vectors 86
7.10   Example: Separating Data 87
7.11   Scan 88
7.12   Example: Cumulative Histogram 88
7.13   Plotting X-Y Points 89
7.14   Example:  Least-Squares Line Fitting 90
7.15   Sampling 93
7.16   Example: Sampling from an Exponential Population 93
7.17   Example: Sampling from a Binomial Distribution 94
7.18   Example: Sampling from a Normal Population 94
Exercises 95

CHAPTER 8 More Array Manipuladon 99

8.1    Take and Drop on Vectors 99
8.2    Example: Polynomial Addition 100
8.3    Take and Drop on Matrices 100
8.4    Ravel 101
8.5    Replicate, Compression, and Expansion 102
8.6    Example: Constructing Data with a Given Distribution 103
8.7    Reverse 104
8.8    Rotate 104
8.9    Example: Polynomial Multiplication 105
8.10   Dyadic Transpose 106
8.11   Transpose on Higher Dimensional Arrays 106
8.12   Laminate 108
8.13   Example: Constructing Banded Matrices 109
Exercises 109

CHAPTER 9 Sorting and Coding 113

9.1    Grade Up, Grade Down 113
9.2    Grade Up, Grade Down on Matrices 114
9.3    Example: Manipulation of Grade Point Average Statistics 115
9.4    Index Relative to a Vector 115
9.5    Example: Alphabetization 116
9.6    Example: Removing Duplicates and Frequency Revisited 116
9.7    Index Origin 118
9.8    Example: Polynomial Evaluation in Both Index Origins 119
9.9    Residue: Modular Reduction 119
9.10   Example: Finite Field Tables 120
9.11   Example: Alphabetization with Mixed-Case Letters 120
9.12   Represent 121
9.13   Base Value 122
9.14   Example: Polynomial Evaluation, Finis 122
9.15   Represent and Base Value on Arrays 123
9.16   Atomic Vector 124
9.17   Example: Encoding a String into Binary 124
9.18   Example: Hamming Code for Error Correction 125
9.19   Example: Hamming Code on Natural Text 127
Exercises 131

CHAPTER 10 More Function Definition 134

10.1   Functions for Programming Uses 134
10.2   Example: Descriptive Statistics 136
10.3   Branching 136
10.4   Line Labels 138
10.5   Examples: Aggregating Preferential Ballots 139
10.6   Multioption Branches 141
10.7   Example: The Bisection Method 141
10.8   Recursive Functions 142
10.9   Example: Generating Permutations 143
10.10  Example: Adaptive Integration 144
10.11  Quad: Input and Output 145
10.12  Quote-Quad: Input and Output 146
10.13  Example: Entering a Matrix Name-List 148
10.14  Example: More Adaptive Integration 149
10.15  Format 150
10.16  Example: Formatting an Interest Table 151
10.17  Execute 152
10.18  Example: Entering a Matrix Name-List, Revisited 153
10.19  Example: Matrix to a Power-of-2 Power 153
10.20  Functions as Arguments of Functions 154
Exercises 155

APPENDIX A Help, Error Messages, and Debugging 159

A. I   Surprises for New Users 159
A.2    Error Messages 160
A.3    Suspended Functions 165
A.4    Example: Debugging with Quad Output 167
A.5    Setting Traces and Stops 169

APPENDIX B Workspace Environment 172

B. I   Workspace Contents: Saving and Loading 172
B.2    Workspace Management 174
B.3    System Commands 175
B.4    System Variables 176
B.5    System Reports and System Functions 177

APPENDIX C Keyboards 180

APPENDIX D Answers to Selected Exercises 182


Bibliography 196
References 197
Index 198

APL with a Mathematical Accent

Table of Contents