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Produktbild: From Mathematics to Generic Programming
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From Mathematics to Generic Programming

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Beschreibung

Produktdetails

Einband

Taschenbuch

Erscheinungsdatum

07.11.2014

Verlag

Pearson Education Limited

Seitenzahl

320

Maße (L/B/H)

22,8/15,2/1,8 cm

Gewicht

408 g

Auflage

1

Sprache

Englisch

ISBN

978-0-321-94204-3

Beschreibung

Produktdetails

Einband

Taschenbuch

Erscheinungsdatum

07.11.2014

Verlag

Pearson Education Limited

Seitenzahl

320

Maße (L/B/H)

22,8/15,2/1,8 cm

Gewicht

408 g

Auflage

1

Sprache

Englisch

ISBN

978-0-321-94204-3

Herstelleradresse

Financial Times Prent.
St.-Martin-Straße 82
81541 München
DE

Email: GPSR Kontakt

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  • Produktbild: From Mathematics to Generic Programming

  • Acknowledgments ix

    About the Authors xi

    Authors’ Note xiii

    Chapter 1: What This Book Is About 1

    1.1 Programming and Mathematics 2

    1.2 A Historical Perspective 2

    1.3 Prerequisites 3

    1.4 Roadmap 4

    Chapter 2: The First Algorithm 7

    2.1 Egyptian Multiplication 8

    2.2 Improving the Algorithm 11

    2.3 Thoughts on the Chapter 15

    Chapter 3: Ancient Greek Number Theory 17

    3.1 Geometric Properties of Integers 17

    3.2 Sifting Primes 20

    3.3 Implementing and Optimizing the Code 23

    3.4 Perfect Numbers 28

    3.5 The Pythagorean Program 32

    3.6 A Fatal Flaw in the Program 34

    3.7 Thoughts on the Chapter 38

    Chapter 4: Euclid’s Algorithm 41

    4.1 Athens and Alexandria 41

    4.2 Euclid’s Greatest Common Measure Algorithm 45

    4.3 A Millennium without Mathematics 50

    4.4 The Strange History of Zero 51

    4.5 Remainder and Quotient Algorithms 53

    4.6 Sharing the Code 57

    4.7 Validating the Algorithm 59

    4.8 Thoughts on the Chapter 61

    Chapter 5: The Emergence of Modern Number Theory 63

    5.1 Mersenne Primes and Fermat Primes 63

    5.2 Fermat’s Little Theorem 69

    5.3 Cancellation 72

    5.4 Proving Fermat’s Little Theorem 77

    5.5 Euler’s Theorem 79

    5.6 Applying Modular Arithmetic 83

    5.7 Thoughts on the Chapter 84

    Chapter 6: Abstraction in Mathematics 85

    6.1 Groups 85

    6.2 Monoids and Semigroups 89

    6.3 Some Theorems about Groups 92

    6.4 Subgroups and Cyclic Groups 95

    6.5 Lagrange’s Theorem 97

    6.6 Theories and Models 102

    6.7 Examples of Categorical and Non-categorical Theories 104

    6.8 Thoughts on the Chapter 107

    Chapter 7: Deriving a Generic Algorithm 111

    7.1 Untangling Algorithm Requirements 111

    7.2 Requirements on A 113

    7.3 Requirements on N 116

    7.4 New Requirements 118

    7.5 Turning Multiply into Power 119

    7.6 Generalizing the Operation 121

    7.7 Computing Fibonacci Numbers 124

    7.8 Thoughts on the Chapter 127

    Chapter 8: More Algebraic Structures 129

    8.1 Stevin, Polynomials, and GCD 129

    8.2 Göttingen and German Mathematics 135

    8.3 Noether and the Birth of Abstract Algebra 140

    8.4 Rings 142

    8.5 Matrix Multiplication and Semirings 145

    8.6 Application: Social Networks and Shortest Paths 147

    8.7 Euclidean Domains 150

    8.8 Fields and Other Algebraic Structures 151

    8.9 Thoughts on the Chapter 152

    Chapter 9: Organizing Mathematical Knowledge 155

    9.1 Proofs 155

    9.2 The First Theorem 159

    9.3 Euclid and the Axiomatic Method 161

    9.4 Alternatives to Euclidean Geometry 164

    9.5 Hilbert’s Formalist Approach 167

    9.6 Peano and His Axioms 169

    9.7 Building Arithmetic 173

    9.8 Thoughts on the Chapter 176

    Chapter 10: Fundamental Programming Concepts 177

    10.1 Aristotle and Abstraction 177

    10.2 Values and Types 180

    10.3 Concepts 181

    10.4 Iterators 184

    10.5 Iterator Categories, Operations, and Traits 185

    10.6 Ranges 188

    10.7 Linear Search 190

    10.8 Binary Search 191

    10.9 Thoughts on the Chapter 196

    Chapter 11: Permutation Algorithms 197

    11.1 Permutations and Transpositions 197

    11.2 Swapping Ranges 201

    11.3 Rotation 204

    11.4 Using Cycles 207

    11.5 Reverse 212

    11.6 Space Complexity 215

    11.7 Memory-Adaptive Algorithms 216

    11.8 Thoughts on the Chapter 217

    Chapter 12: Extensions of GCD 219

    12.1 Hardware Constraints and a More Efficient Algorithm 219

    12.2 Generalizing Stein’s Algorithm 222

    12.3 Bézout’s Identity 225

    12.4 Extended GCD 229

    12.5 Applications of GCD 234

    12.6 Thoughts on the Chapter 234

    Chapter 13: A Real-World Application 237

    13.1 Cryptology 237

    13.2 Primality Testing 240

    13.3 The Miller-Rabin Test 243

    13.4 The RSA Algorithm: How and Why It Works 245

    13.5 Thoughts on the Chapter 248

    Chapter 14: Conclusions 249

    Further Reading 251

    Appendix A: Notation 257

    Appendix B: Common Proof Techniques 261

    B.1 Proof by Contradiction 261

    B.2 Proof by Induction 262

    B.3 The Pigeonhole Principle 263

    Appendix C: C++ for Non-C++ Programmers 265

    C.1 Template Functions 265

    C.2 Concepts 266

    C.3 Declaration Syntax and Typed Constants 267

    C.4 Function Objects 268

    C.5 Preconditions, Postconditions, and Assertions 269

    C.6 STL Algorithms and Data Structures 269

    C.7 Iterators and Ranges 270

    C.8 Type Aliases and Type Functions with using in C++11 272

    C.9 Initializer Lists in C++11 272

    C.10 Lambda Functions in C++11 272

    C.11 A Note about inline 273

    Bibliography 275

    Index 281