Validated Numerics: A Short Introduction to Rigorous Computations

by Warwick Tucker

Overview

A comprehensive, self-contained primer on validated numerics

This textbook provides a comprehensive introduction to the theory and practice of validated numerics, an emerging new field that combines the strengths of scientific computing and pure mathematics. In numerous fields ranging from pharmaceutics and engineering to weather prediction and robotics, fast and precise computations are essential. Based on the theory of set-valued analysis, a new suite of numerical methods is developed, producing efficient and reliable solvers for numerous problems in nonlinear analysis. Validated numerics yields rigorous computations that can find all possible solutions to a problem while taking into account all possible sources of error—fast, and with guaranteed accuracy.

Validated Numerics offers a self-contained primer on the subject, guiding readers from the basics to more advanced concepts and techniques. 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. Validated Numerics features many examples, exercises, and computer labs using MATLAB/C++, as well as detailed appendixes and an extensive bibliography for further reading.

• Provides a comprehensive, self-contained introduction to validated numerics
• Requires no advanced mathematics or programming skills
• Features many examples, exercises, and computer labs
• Includes code snippets that illustrate implementation
• Suitable as a textbook for graduate students and advanced undergraduates

Product Details

ISBN-13:	9780691247656
Publisher:	Princeton University Press
Publication date:	06/06/2023
Pages:	152
Product dimensions:	6.12(w) x 9.25(h) x (d)

About the Author

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.

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

What People are Saying About This

From the Publisher

"Validated Numerics contains introductory material on interval arithmetic and rigorous computations that is easily accessible to students with little background in mathematics and computer programming. I am not aware of any other book like it. The exercises and computer labs make it ideal for the classroom, and the references offer a good starting point for readers trying to gain deeper knowledge in this area."—Zbigniew Galias, AGH University of Science and Technology, Kraków

"A significant contribution, particularly since there are not many texts in this area. Validated Numerics will be read by those interested in interval arithmetic, numerical analysis, and ways to make computer simulations more robust and less susceptible to errors. It is well written and well organized."—A. J. Meir, Auburn University

Editorial Reviews

"Beyond obvious practical value, this material offers students an excellent opportunity to revisit and rethink some crucial, fundamental college mathematics."

Choice

"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

"[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

From the Publisher

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.

Zentralblatt MATH - Gunter Mayer

[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.

Mathematical Reviews - G. Alefeld

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 Database

[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