Validated Numerics

Warwick Tucker is professor of mathematics and principal investigator for the Computer-Aided Proofs in Analysis (CAPA) Group at Uppsala University in Sweden. He has been honored with several awards, including the European Mathematical Society's Prize for Distinguished Contributions in Mathematics, the R. E. Moore Prize for Applications of Interval Analysis, and the Swedish Mathematical Society's Wallenberg Prize.

"Beyond obvious practical value, this material offers students an excellent opportunity to revisit and rethink some crucial, fundamental college mathematics."--Choice "[T]his little book is a very important supplement to existing books on validated numerics. It is a must for researchers working in this field."--G. Alefeld, Mathematical Reviews "The book contains a lot of exercises, various small programs written in MATLAB code, and four sections with numerous problems provided for experimenting on a computer. It is written at an elementary level corresponding to its aims. But it is also a pleasure for specialists to leaf through the book."--Gunter Mayer, Zentralblatt MATH "This book is an essential resource for those entering this fast-developing field, and it is also the ideal textbook for graduate students and advanced undergraduates needing an accessible introduction to the subject."--World Book Industry

• 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