Ramon E. Moore

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.

Interval analysis and fuzzy set theory

An overview of interval analysis, its development, and its relationship to fuzzy set theory is given. Possible areas of further fruitful research are highlighted.

SAFE STARTING REGIONS FOR ITERATIVE METHODS

A search procedure based on interval computation is given for finding safe starting regions in n dimensions for iterative methods for solving systems of nonlinear equations. The procedure can search an arbitrary n-dimensional rectangle for a safe starting region for a quadratically convergent iterative method. The procedure is more powerful than continuation methods.

Inclusion functions and global optimization 11

This paper discusses algorithms of Moore, Skelboe, Ichida, Fujii and Hansen for solving the global unconstrained optimization problem. These algorithms have been tried on computers, but a thorough theoretical discussion of their convergence properties has been missing. The discussion was started in part I of this paper (Mathematical Programming 33 (1985) 300–317) where the convergence to the global minimum was studied. The present paper is concerned with the different behaviours of these algorithms when they are used for the determination of global minimum points. The solution sets of the algorithms can be a subset of the set of global minimum points,G, a superset ofG, or exactlyG. The algorithms are applicable to a very general class of functions: functions which are continuous, and have suitable inclusion functions. The number of global minimum points can be infinite.

Global optimization to prescribed accuracy

Abstract Programmable methods are outlined for rigorous global optimization to prescribed accuracy with references to enough detail to enable the implementation of the techniques on any computer. Some directions for future research are indicated.

On computing the range of values

A simple algorithm is given for computing the range of values of a differentiable function over ann-dimensional rectangle.ZusammenfassungGegeben ist ein einfaches Verfahren zur Berechnung des Wertebereichs einer differenzierbaren Funktion auf einemn-dimensionalen Quader.

On computing the range of a rational function of n variables over a bounded region

We seek an efficient method for computing the range of values of a function ofn variables over a bounded domain.ZusammenfassungWir suchen ein wirksames Verfahren zur Berechnung des Wertebereichs einer Funktion auf einemn-dimensionalen Quader.

A Computational Test for Convergence of Iterative Methods for Nonlinear Systems

A simple computational test for existence of a solution to a nonlinear system of equations and convergence of iterative methods is given for n-cubes. The test is eventually satisfied by any convergent Newton-type sequence.

Parameter sets for bounded-error data

A simple approach to the development of user-friendly software for the bounded-error approach to parameter estimation is offered. It enables the computer determination of the boundaries of the set of all parameters satisfying the bounded-error constraints, whether the problem is linear or nonlinear, and whether the parameter set is connected or disconnected.

Interval Methods for Nonlinear Systems

Interval methods provide computational tests for the existence or non-existence of a Solution to a given system of nonlinear equations in an n-dimensional rectangle. Tests are also provided for the convergence of certain iterative methods within suitable regions (safe starting regions). Using bisection procedures, we can search an arbitrary n-dimensional rectangle for a safe starting region. Various bisection rules are discussed.