Introduction To Interval Analysis

This unique book provides an introduction to a subject whose use has steadily increased over the past 40 years. An update of Ramon Moore s previous books on the topic, it provides broad coverage of the subject as well as the historical perspective of one of the originators of modern interval analysis. The authors provide a hands-on introduction to INTLAB, a high-quality, comprehensive MATLAB® toolbox for interval computations, making this the first interval analysis book that does with INTLAB what general numerical analysis texts do with MATLAB. Readers will find the following features of interest: elementary motivating examples and notes that help maximize the reader s chance of success in applying the techniques; exercises and hands-on MATLAB-based examples woven into the text; INTLAB-based examples and explanations integrated into the text, along with a comprehensive set of exercises and solutions, and an appendix with INTLAB commands; an extensive bibliography and appendices that will continue to be valuable resources once the reader is familiar with the subject; and a Web page with links to computational tools and other resources of interest. Audience: Introduction to Interval Analysis will be valuable to engineers and scientists interested in scientific computation, especially in reliability, effects of roundoff error, and automatic verification of results. The introductory material is particularly important for experts in global optimization and constraint solution algorithms. This book is suitable for introducing the subject to students in these areas. Contents: Preface; Chapter 1: Introduction; Chapter 2: The Interval Number System; Chapter 3: First Applications of Interval Arithmetic; Chapter 4: Further Properties of Interval Arithmetic; Chapter 5: Introduction to Interval Functions; Chapter 6: Interval Sequences; Chapter 7: Interval Matrices; Chapter 8: Interval Newton Methods; Chapter 9: Integration of Interval Functions; Chapter 10: Integral and Differential Equations; Chapter 11: Applications; Appendix A: Sets and Functions; Appendix B: Formulary; Appendix C: Hints for Selected Exercises; Appendix D: Internet Resources; Appendix E: INTLAB Commands and Functions; References; Index. Contents......Page 5 Preface......Page 9 1.1 Enclosing a Solution......Page 13 1.2 Bounding Roundoff Error......Page 15 1.3 Number Pair Extensions......Page 17 2.1 Basic Terms and Concepts......Page 19 2.2 Order Relations for Intervals......Page 21 2.3 Operations of Interval Arithmetic......Page 22 2.4 Interval Vectors and Matrices......Page 26 2.5 Some Historical References......Page 28 3.1 Examples......Page 31 3.3 INTLAB......Page 34 3.4 Other Systems and Considerations......Page 40 4.1 Algebraic Properties......Page 43 4.2 Symmetric Intervals......Page 45 4.3 Inclusion Isotonicity of Interval Arithmetic......Page 46 5.1 Set Images and United Extension......Page 49 5.2 Elementary Functions of Interval Arguments......Page 50 5.3 Interval-Valued Extensions of Real Functions......Page 54 5.4 The Fundamental Theorem and Its Applications......Page 57 5.5 Remarks on Numerical Computation......Page 61 6.1 A Metric for the Set of Intervals......Page 63 6.2 Refinement......Page 65 6.3 Finite Convergence and Stopping Criteria......Page 69 6.4 More Efficient Refinements......Page 76 6.5 Summary......Page 95 7.1 Definitions......Page 97 7.2 Interval Matrices and Dependency......Page 98 7.3 INTLAB Support for Matrix Operations......Page 99 7.4 Systems of Linear Equations......Page 100 7.5 Linear Systems with Inexact Data......Page 104 7.6 More on Gaussian Elimination......Page 112 7.7 Sparse Linear Systems Within INTLAB......Page 113 7.8 Final Notes......Page 115 8.1 Newton’s Method in One Dimension......Page 117 8.2 The Krawczyk Method......Page 128 8.3 Safe Starting Intervals......Page 133 8.4 Multivariate Interval Newton Methods......Page 135 8.5 Concluding Remarks......Page 139 9.1 Definition and Properties of the Integral......Page 141 9.2 Integration of Polynomials......Page 145 9.3 Polynomial Enclosure and Automatic Differentiation......Page 147 9.4 Computing Enclosures for Integrals......Page 153 9.5 Further Remarks on Interval Integration......Page 157 9.6 Software and Further References......Page 159 10.1 Integral Equations......Page 161 10.2 ODEs and Initial Value Problems......Page 163 10.4 Partial Differential Equations......Page 168 11.1 Computer-Assisted Proofs......Page 169 11.2 Global Optimization and Constraint Satisfaction......Page 171 11.3 Structural Engineering Applications......Page 180 11.5 Computation of Physical Constants......Page 181 11.7 For Further Study......Page 182 Appendix A Sets and Functions......Page 183 Appendix B Formulary......Page 189 Appendix C Hints for Selected Exercises......Page 197 Appendix D Internet Resources......Page 207 Appendix E INTLAB Commands and Functions......Page 209 Index......Page 231 References......Page 213 This book is a unique introduction to interval analysis, which has grown steadily in popularity over the past 40 years. An update of Ramon Moore's previous books on the topic, it gives broad coverage of the subject as well as the historical perspective of one of the originators of modern interval analysis. The authors provide a hands-on introduction to INTLAB, a high-quality, comprehensive MATLAB toolbox for interval computations, making this the first interval analysis book that does with INTLAB what general numerical analysis texts do with MATLAB. The book combines a gentle introduction and simple motivating examples for those not previously familiar with the subject, with reference material (as a bibliography and appendices) that will continue to be valuable once the subject is familiar. An Update On The Author's Previous Books, This Introduction To Interval Analysis Provides An Introduction To Intlab, A High-quality, Comprehensive Matlab Toolbox For Interval Computations, Making This The First Interval Analysis Book That Does With Intlab What General Numerical Analysis Texts Do With Matlab.