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Validated Numerics:
A Short Introduction to Rigorous Computations
Warwick Tucker

Book Description | Reviews
Introduction [in PDF format] | Chapter 1 [in PDF format]

TABLE OF CONTENTS:

Preface ix
Introduction xi
What Is Validated Numerics? xi
The Scope and Aim of This Book xi
Further Reading xii
Acknowledgments xii

Chapter 1. Computer Arithmetic 1
1.1 Positional Systems 1
1.2 Floating Point Numbers 2
1.3 Rounding 5
1.4 Floating Point Arithmetic 12
1.5 The IEEE Standard 14
1.6 Examples of Floating Point Computations 19
1.7 Computer Lab I 23

Chapter 2. Interval Arithmetic 24
2.1 Real Intervals 24
2.2 Real Interval Arithmetic 27
2.3 Extended Interval Arithmetic 30
2.4 Floating Point Interval Arithmetic 37

Chapter 3. Interval Analysis 46
3.1 Interval Functions 46
3.2 Centered Forms 55
3.3 Monotonicity 58
3.4 Computer Lab II 59

Chapter 4. Automatic Differentiation 60
4.1 First-Order Derivatives 60
4.2 Higher-Order Derivatives 64
4.3 Higher-Order Enclosures 71
4.4 Computer Lab III 72

Chapter 5. Interval Analysis in Action 73
5.1 Zero-Finding Methods 73
5.2 Optimization 87
5.3 Quadrature 94
5.4 Computer Lab IV 105

Chapter 6. Ordinary Differential Equations 106
6.1 A Gentle Mathematical Introduction 106
6.2 Simple Enclosure Methods 107
6.3 High-Order Methods 111
6.4 Rigorous High-Order Examples 112

Appendix A. Mathematical Foundations 118
A.1 The Rational Numbers 118
A.2 What Is a Real Number? 120
A.3 Completeness 122
A.4 Fixed-Point Theorems 124

Appendix B. Program Codes 126
B.1 IEEE Constants 126
B.2 Changing Rounding Modes 127
B.3 A Sample Code in C++ 128
Bibliography 131
Index 137

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File created: 7/11/2014

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