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interval analysis methods(区间分析).pdf

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    Contents
    Preface
    Chapter 1 Introduction
    1.1 Enclosing a Solution
    1.2 Bounding Roundoff Error
    1.3 Number Pair Extensions
    Chapter 2 The Interval Number System
    2.1 Basic Terms and Concepts
    2.2 Order Relations for Intervals
    2.3 Operations of Interval Arithmetic
    2.4 Interval Vectors and Matrices
    2.5 Some Historical References
    Chapter 3 First Applications of Interval Arithmetic
    3.1 Examples
    3.2 Outwardly Rounded Interval Arithmetic
    3.3 INTLAB
    3.4 Other Systems and Considerations
    Chapter 4 Further Properties of Interval Arithmetic
    4.1 Algebraic Properties
    4.2 Symmetric Intervals
    4.3 Inclusion Isotonicity of Interval Arithmetic
    Chapter 5 Introduction to Interval Functions
    5.1 Set Images and United Extension
    5.2 Elementary Functions of Interval Arguments
    5.3 Interval-Valued Extensions of Real Functions
    5.4 The Fundamental Theorem and Its Applications
    5.5 Remarks on Numerical Computation
    Chapter 6 Sequences of Intervals and Interval Functions
    6.1 A Metric for the Set of Intervals
    6.2 Refinement
    6.3 Finite Convergence and Stopping Criteria
    6.4 More Efficient Refinements
    6.5 Summary
    Chapter 7 Interval Matrices
    7.1 Definitions
    7.2 Interval Matrices and Dependency
    7.3 INTLAB Support for Matrix Operations
    7.4 Systems of Linear Equations
    7.5 Linear Systems with Inexact Data
    7.6 More on Gaussian Elimination
    7.7 Sparse Linear Systems Within INTLAB
    7.8 Final Notes
    Chapter 8 Interval Newton Methods
    8.1 Newton’s Method in One Dimension
    8.2 The Krawczyk Method
    8.3 Safe Starting Intervals
    8.4 Multivariate Interval Newton Methods
    8.5 Concluding Remarks
    Chapter 9 Integration of Interval Functions
    9.1 Definition and Properties of the Integral
    9.2 Integration of Polynomials
    9.3 Polynomial Enclosure and Automatic Differentiation
    9.4 Computing Enclosures for Integrals
    9.5 Further Remarks on Interval Integration
    9.6 Software and Further References
    Chapter 10 Integral and Differential Equations
    10.1 Integral Equations
    10.2 ODEs and Initial Value Problems
    10.3 ODEs and Boundary Value Problems
    10.4 Partial Differential Equations
    Chapter 11 Applications
    11.1 Computer-Assisted Proofs
    11.2 Global Optimization and Constraint Satisfaction
    11.3 Structural Engineering Applications
    11.4 Computer Graphics
    11.5 Computation of Physical Constants
    11.6 Other Applications
    11.7 For Further Study
    Appendix A Sets and Functions
    Appendix B Formulary
    Appendix C Hints for Selected Exercises
    Appendix D Internet Resources
    Appendix E INTLAB Commands and Functions
    Index
    References
    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, info@mathworks.com, 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 i interval 2008/11/18 page v i Contents Preface Introduction 1.1 1.2 1.3 Enclosing a Solution . . . . . Bounding Roundoff Error . . . . . Number Pair Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix 1 1 3 5 1 2 3 4 5 6 The Interval Number System 2.1 2.2 2.3 2.4 2.5 Basic Terms and Concepts . . . . . Order Relations for Intervals Operations of Interval Arithmetic . . . . . Interval Vectors and Matrices . . . . . Some Historical References . . . . . . . . . . . . . . 7 7 9 . . . . . . . . . 10 . 14 . . 16 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . First Applications of Interval Arithmetic 3.1 . . . . . . . . 3.2 3.3 3.4 Examples . Outwardly Rounded Interval Arithmetic INTLAB . Other Systems and Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 . . . . 19 . . . . . 22 . . . . . . . . 22 . 28 . . . . . Further Properties of Interval Arithmetic 4.1 4.2 4.3 Algebraic Properties . . . . . Symmetric Intervals . . . . . Inclusion Isotonicity of Interval Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 . . 31 . . 33 . . 34 Introduction to Interval Functions 5.1 5.2 5.3 5.4 5.5 Set Images and United Extension . . . . . . . . . . . Elementary Functions of Interval Arguments . . . . . Interval-Valued Extensions of Real Functions The Fundamental Theorem and Its Applications Remarks on Numerical Computation . . . . . . . . . 37 . . . . . . . . . 37 . . . . . . . . . 38 . . 42 . 45 . . . . . 49 . . . . . . . . . . . . . . . . . . . . . . . . . . Interval Sequences 6.1 6.2 A Metric for the Set of Intervals . . . . . Refinement . . . . . . . . . . . . . . . . . . . . . . . . 51 . . . . . . . . . . 51 . . . 53 . . . . . . . . . . . v i i i i
    i i vi 7 8 9 10 11 i interval 2008/11/18 page vi i Contents 6.3 6.4 6.5 Finite Convergence and Stopping Criteria . . . . More Efficient Refinements . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 . . 64 . . . . 83 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Interval Matrices 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 Definitions . Interval Matrices and Dependency . . . . . INTLAB Support for Matrix Operations Systems of Linear Equations Linear Systems with Inexact Data . . . . . More on Gaussian Elimination . . . . . Sparse Linear Systems Within INTLAB . . . . . Final Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 . . . 85 . . . . . . . . 86 . . . . . 87 . . . . 88 . . . . . . . . . . . . . . 92 . . . . . . . . . . 100 . . . . . . . . . . . 101 . . . . . . . . . . . 103 . . . . . . . . . . . . Interval Newton Methods 8.1 8.2 8.3 8.4 8.5 Newton’s Method in One Dimension . . . . . The Krawczyk Method . . . . . Safe Starting Intervals . . . . . Multivariate Interval Newton Methods . . . . . Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 . . . . . . . 105 . . . . 116 . 121 . . . . . . 123 . . . . . . . . 127 . . . . Integration of Interval Functions 9.1 9.2 9.3 9.4 9.5 9.6 Definition and Properties of the Integral Integration of Polynomials Polynomial Enclosure, Automatic Differentiation . . . . . Computing Enclosures for Integrals . . . . . . . . . . . . . Further Remarks on Interval Integration . . . . . Software and Further References . . . . . 129 . . . . . 129 . . . . . 133 . . . . . . 135 . . . . . . 141 . . . . . 145 . . . . . . . . . 147 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Integral and Differential Equations Integral Equations . . . . . 10.1 ODEs and Initial Value Problems . . . . . 10.2 ODEs and Boundary Value Problems . . . . . 10.3 10.4 Partial Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 . . . 149 . . . . . . . . . 151 . . . . . . . 156 . 156 . . . . . . . . . . . . . . . . Applications 11.1 11.2 . . . . . . . . . . . . . . . . Computer-Assisted Proofs . . . . . Global Optimization and Constraint Satisfaction . . . . 11.2.1 11.2.2 11.2.3 11.2.4 11.2.5 11.2.6 11.2.7 11.2.8 A Prototypical Algorithm . . . . . . . . . . . Parameter Estimation . . . . . Robotics Applications . . . . . . . . . . . Chemical Engineering Applications . . . . Water Distribution Network Design . . . . Pitfalls and Clarifications Additional Centers of Study . . . . . Summary of Links for Further Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 . . . 157 . . . . . . . . 159 . . . . . . 159 . . . . . . . . 161 . . . . . . . . 162 . . . . . . . . 163 . . . . . . . . 164 . . . . . . 164 . . . . . 167 . . . . . . 168 i i i i
    i i Contents 11.3 11.4 11.5 11.6 11.7 i interval 2008/11/18 page vii i vii Structural Engineering Applications . . . . Computer Graphics Computation of Physical Constants . . . . . Other Applications . . . . . For Further Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 . . 169 . . . . . . . . 169 . . . 170 . . . 170 . . . . . . . . A B C D E Sets and Functions Formulary Hints for Selected Exercises Internet Resources INTLAB Commands and Functions References Index 171 177 185 195 197 201 219 i i i i
    i i i interval 2008/11/18 page viii i i i i i
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