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Introduction to INTERVAL ANALYSIS

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Introduction to INTERVAL ANALYSIS Introduction to INTERVAL ANALYSIS Ramon E. Moore Worthington, Ohio R. Baker Kearfott University of Louisiana at Lafayette Lafayette, Louisiana Michael J. Cloud Lawrence Technological University Southfield, Michigan Society for Industrial and Applied Mathematics Philadelphia Copyright © 2009 by the Society for Industrial and Applied Mathematics 10 9 8 7 6 5 4 3 2 1 All rights reserved. Printed in the United States of America. No part of this book may be reproduced, stored, or transmitted in any manner without the written permission of the publisher. For information, write to the Society for Industrial and Applied Mathematics, 3600 Market Street, 6th Floor, Philadelphia, PA, 19104-2688 USA. Trademarked names may be used in this book without the inclusion of a trademark symbol. These names are used in an editorial context only; no infringement of trademark is intended. COSY INFINITY is copyrighted by the Board of Trustees of Michigan State University. GlobSol is covered by the Boost Software License Version 1.0, August 17th, 2003. Permission is hereby granted, free of charge, to any person or organization obtaining a copy of the software and accompanying documentation covered by this license (the “Software”) to use, reproduce, display, distribute, execute, and transmit the Software, and to prepare derivative works of the software, and to permit third-parties to whom the Software is furnished to do so, all subject to the following: The copyright notices in the Software and this entire statement, including he above license grant, this restriction and the following disclaimer, must be included in all copies of the Software, in whole or in part, and all derivative works of the Software, unless such copies or derivative works are solely in the form of machine-executable object code generated by a source language processor. THE SOFTWARE IS PROVIDED “AS IS”, WITHOUT WARRANTY OF ANY KIND, EXPRESS OR IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY, FITNESS FOR A PARTICULAR PURPOSE, TITLE AND NON-INFRINGEMENT. IN NO EVENT SHALL THE COPYRIGHT HOLDERS OR ANYONE DISTRIBUTING THE SOFTWARE BE LIABLE FOR ANY DAMAGES OR OTHER LIABILITY, WHETHER IN CONTRACT, TORT OR OTHERWISE, ARISING FROM, OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE SOFTWARE. INTLAB is copyrighted © 1998-2008 by Siegfried M. Rump @ TUHH, Institute for Reliable Computing. Linux is a registered trademark of Linus Torvalds. Mac OS is a trademark of Apple Computer, Inc., registered in the United States and other countries. Introduction to Interval Analysis is an independent publication and has not been authorized, sponsored, or otherwise approved by Apple Computer, Inc. Maple is a registered trademark of Waterloo Maple, Inc. Mathematica is a registered trademark of Wolfram Research, Inc. MATLAB is a registered trademark of The MathWorks, Inc. For MATLAB product information, please contact The MathWorks, Inc., 3 Apple Hill Drive, Natick, MA 01760-2098 USA, 508-647-7000, Fax: 508-647-7001, [email protected], www.mathworks.com. Windows is a registered trademark of Microsoft Corporation in the United States and/or other countries. Library of Congress Cataloging-in-Publication Data Moore, Ramon E. Introduction to interval analysis / Ramon E. Moore, R. Baker Kearfott, Michael J. Cloud. p. cm. Includes bibliographical references and index. ISBN 978-0-898716-69-6 1. Interval analysis (Mathematics) I. Kearfott, R. Baker. II. Cloud, Michael J. III. Title. QA297.75.M656 2009 511’.42—dc22 2008042348 is a registered trademark. i i interval 2008/11/18 page v i i Contents Preface ix 1 Introduction 1 1.1 Enclosing a Solution ........................... 1 1.2 Bounding Roundoff Error ........................ 3 1.3 Number Pair Extensions ......................... 5 2 The Interval Number System 7 2.1 Basic Terms and Concepts ........................ 7 2.2 Order Relations for Intervals ...................... 9 2.3 Operations of Interval Arithmetic .................... 10 2.4 Interval Vectors and Matrices ...................... 14 2.5 Some Historical References ....................... 16 3 First Applications of Interval Arithmetic 19 3.1 Examples ................................. 19 3.2 Outwardly Rounded Interval Arithmetic ................ 22 3.3 INTLAB ................................. 22 3.4 Other Systems and Considerations ................... 28 4 Further Properties of Interval Arithmetic 31 4.1 Algebraic Properties ........................... 31 4.2 Symmetric Intervals ........................... 33 4.3 Inclusion Isotonicity of Interval Arithmetic ............... 34 5 Introduction to Interval Functions 37 5.1 Set Images and United Extension .................... 37 5.2 Elementary Functions of Interval Arguments .............. 38 5.3 Interval-Valued Extensions of Real Functions ............. 42 5.4 The Fundamental Theorem and Its Applications ............ 45 5.5 Remarks on Numerical Computation .................. 49 6 Interval Sequences 51 6.1 A Metric for the Set of Intervals ..................... 51 6.2 Refinement ................................ 53 v i i i i i i interval 2008/11/18 page vi i i vi Contents 6.3 Finite Convergence and Stopping Criteria ................ 57 6.4 More Efficient Refinements ....................... 64 6.5 Summary ................................. 83 7 Interval Matrices 85 7.1 Definitions ................................ 85 7.2 Interval Matrices and Dependency ................... 86 7.3 INTLAB Support for Matrix Operations ................ 87 7.4 Systems of Linear Equations ...................... 88 7.5 Linear Systems with Inexact Data .................... 92 7.6 More on Gaussian Elimination .....................100 7.7 Sparse Linear Systems Within INTLAB ................101 7.8 Final Notes ................................103 8 Interval Newton Methods 105 8.1 Newton’s Method in One Dimension ..................105 8.2 The Krawczyk Method .........................116 8.3 Safe Starting Intervals ..........................121 8.4 Multivariate Interval Newton Methods .................123 8.5 Concluding Remarks ..........................127 9 Integration of Interval Functions 129 9.1 Definition and Properties of the Integral ................129 9.2 Integration of Polynomials .......................133 9.3 Polynomial Enclosure, Automatic Differentiation ...........135 9.4 Computing Enclosures for Integrals ...................141 9.5 Further Remarks on Interval Integration ................145 9.6 Software and Further References ....................147 10 Integral and Differential Equations 149 10.1 Integral Equations ............................149 10.2 ODEs and Initial Value Problems ....................151 10.3 ODEs and Boundary Value Problems ..................156 10.4 Partial Differential Equations ......................156 11 Applications 157 11.1 Computer-Assisted Proofs ........................157 11.2 Global Optimization and Constraint Satisfaction ............159 11.2.1 A Prototypical Algorithm .................159 11.2.2 Parameter Estimation ...................161 11.2.3 Robotics Applications ...................162 11.2.4 Chemical Engineering Applications ............163 11.2.5 Water Distribution Network Design ............164 11.2.6 Pitfalls and Clarifications .................164 11.2.7 Additional Centers of Study ................167 11.2.8 Summary of Links for Further Study ...........168 i i i i i i interval 2008/11/18 page vii i i Contents vii 11.3 Structural Engineering Applications ...................168 11.4 Computer Graphics ...........................169 11.5 Computation of Physical Constants ...................169 11.6 Other Applications ............................170 11.7 For Further Study ............................170 A Sets and Functions 171 B Formulary 177 C Hints for Selected Exercises 185 D Internet Resources 195 E INTLAB Commands and Functions 197 References 201 Index 219 i i i i i i interval 2008/11/18 page viii i i i i i i i i interval 2008/11/18 page ix i i Preface This book is intended primarily for those not yet familiar with methods for computing with intervals of real numbers and what can be done with these methods. Using a pair [a,b] of computer numbers to represent an interval of real numbers a ≤ x ≤ b, we define an arithmetic for intervals and interval valued extensions of functions commonly used in computing. In this way, an interval [a,b] has a dual nature. It is a new kind of number pair, and it represents a set [a,b]={x : a ≤ x ≤ b}. We combine set operations on intervals with interval function evaluations to get algorithms for computing enclosures of sets of solutions to computational problems. A procedure known as outward rounding guarantees that these enclosures are rigorous, despite the roundoff errors that are inherent in finite machine arithmetic. With interval computation we can program a computer to find intervals that contain—with absolute certainty—the exact answers to various mathematical problems. In effect, interval analysis allows us to compute with sets on the real line. Interval vectors give us sets in higher-dimensional spaces. Using multinomials with interval coefficients, we can compute with sets in function spaces. In applications, interval analysis provides rigorous enclosures of solutions to model equations. In this way we can at least know for sure what a mathematical model tells us, and, from that, we might determine whether it adequately represents reality. Without rigorous bounds on computational errors, a comparison of numerical results with physical measurements
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