R. Baker Kearfott

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.

Rigorous Global Search: Continuous Problems

List of Figures. List of Tables. Preface. 1. Preliminaries. 2. Software Environments. 3. On Preconditioning. 4. Verified Solution of Nonlinear Systems. 5. Optimization. 6. Non-Differentiable Problems. 7. Use of Intermediate Quantities in the Expression Values. References. Index.

Applications of interval computations

Preface. 1. Applications of Interval Computations: An Introduction R.B. Kearfott, V. Kreinovich. 2. A Review of Techniques in the Verified Solution of Constrained Global Optimization Problems R.B. Kearfott. 3. The Shape of the Symmetric Solution Set G. Alefeld, et al. 4. Linear Interval Equations: Computing Enclosures with Bounded Relative Overestimation is NP-Hard J. Rohn. 5. Quality Improvement Via Optimization of Tolerance Intervals During the Design Stage S. Hadjihassan, et al. 6. Applications of Interval Computations to Regional Economic Input-Output Models M.E. Jerrell. 7. Interval Arithmetic in Quantum Mechanics C.L. Fefferman, L.A. Seco. 8. Interval Computations on the Spreadsheet E. Hyvonen, S. De Pascale. 9. Solving Optimization Problems with Help of the UniCalc Solver A.L. Semenov. 10. Automatically Verified Arithmetic on Probability Distributions and Intervals D. Berleant. 11. Nested Intervals and Sets: Concepts, Relations to Fuzzy Sets, and Applications H.T. Nguyen, V. Kreinovich. 12. Fuzzy Interval Inference Utilizing the Checklist Paradigm and BK-Relational Products L.J. Kohout, W. Bandler. 13. Computing Uncertainty in Interval Based Sets L.M. Rocha, et al. 14. Software and Hardware Techniques for Accurate, Self-Validating Arithmetic M.J. Schulte, E.E. Swartzlander, Jr. 15. Stimulating Hardware and Software Support for Interval Arithmetic G.W. Walster. Index.

Algorithm 681: INTBIS, a portable interval Newton/bisection package

We present a portable software package for finding all real roots of a system of nonlinear equations within a region defined by bounds on the variables. Where practical, the package should find all roots with mathematical certainty. Though based on interval Newton methods, it is self-contained. It allows various control and output options and does not require programming if the equations are polynomials; it is structured for further algorithmic research. Its practicality does not depend in a simple way on the dimension of the system or on the degree of nonlinearity.

Some tests of generalized bisection

This paper addresses the task of reliably finding approximations to all solutions to a system of nonlinear equations within a region defined by bounds on each of the individual coordinates. Various forms of generalized bisection were proposed some time ago for this task. This paper systematically compares such generalized bisection algorithms to themselves, to continuation methods, and to hybrid steepest descent/quasi-Newton methods. A specific algorithm containing novel “expansion” and “exclusion” steps is fully described, and the effectiveness of these steps is evaluated. A test problem consisting of a small, high-degree polynomial system that is appropriate for generalized bisection, but very difficult for continuation methods, is presented. This problem forms part of a set of 17 test problems from published literature on the methods being compared; this test set is fully described here.

The cluster problem in multivariate global optimization

We consider branch and bound methods for enclosing all unconstrained global minimizers of a nonconvex nonlinear twice-continuously differentiable objective function. In particular, we consider bounds obtained with interval arithmetic, with the “midpoint test,” but no acceleration procedures. Unless the lower bound is exact, the algorithm without acceleration procedures in general gives an undesirable cluster of boxes around each minimizer. In a previous paper, we analyzed this problem for univariate objective functions. In this paper, we generalize that analysis to multi-dimensional objective functions. As in the univariate case, the results show that the problem is highly related to the behavior of the objective function near the global minimizers and to the order of the corresponding interval extension.

Decomposition of arithmetic expressions to improve the behavior of interval iteration for nonlinear systems

Interval iteration can be used, in conjunction with other techniques, for rigorously bounding all solutions to a nonlinear system of equations within a given region, or for verifying approximate solutions. However, because of overestimation which occurs when the interval Jacobian matrix is accumulated and applied, straightforward linearization of the original nonlinear system sometimes leads to nonconvergent iteration.In this paper, we examine interval iterations based on an expanded system obtained from the intermediate quantities in the original system. In this system, there is no overestimation in entries of the interval Jacobi matrix, and nonlinearities can be taken into account to obtain sharp bounds. We present an example in detail, algorithms, and detailed experimental results obtained from applying our algorithms to the example.ZusammenfassungIntervalliterationen können in Verbindung mit anderen Verfahren verwendet werden, um alle Lösungen eines nichlinearen Gleichungsystems in einem gegebenen Gebiet mit Sicherheit abzuschätzen, und auch um Approximationen der Lösungen solcher Systeme zu verifizieren. Die Abschätzungen in den Verfahren sind jedoch manchmal nicht hinreichend genau, da Überschätzungen in der Berechnung und in dem Gebrauch der Invervall-Jacobi Matrix auftreten.In der vorliegenden Arbeit werden Intervalliterationen auf einem erweiterten Gleichungssystem behandelt. In diesem System gibt es keine Überschätzungen der Einzelkomponenten der Intervall-Jacobi Matrix, und für die Nichtlinearitären können Abschätzungen angegeben werden. Anhand eines Beispiels wird die Wirkungsweise der behandelten Algorithmen demonstriert.

Algorithm 763: INTERVAL_ARITHMETIC: a Fortran 90 module for an interval data type

Interval arithmetic is useful in automatically verified computations, that is, in computations in which the algorithm itself rigorously proves that the answer must lie within certain bounds. In addition to rigor, interval arithmetic also provides a simple and sometimes sharp method of bounding ranges of functions for global optimization and other tasks. Convenient use of interval arithmetic requires an interval data type in the programming language. Although various packages supply such a data type, previous ones are machine specific, obsolete, and unsupported, for languages other than Fortran, or commercial. The Fortran 90 module INTERVAL_ARITHMETIC provides a portable interval data type in Fortran 90. This data type is based on two double-precision real Fortran storage units. Module INTERVAL_ARTHMETIC uses the Fortran 77 library INTLIB (ACM TOMS Algorithm 737) as a supporting library. The module has been employed extensively in the authors own research.

A Fortran 90 environment for research and prototyping of enclosure algorithms for nonlinear equations and global optimization

An environment for general research into and prototyping of algorithms for reliable constrained and unconstrained global nonlinear optimization and reliable enclosure of all roots of nonlinear systems of equations, with or without inequality constraints, is being developed. This environment should be portable, easy to learn, use, and maintain, and sufficiently fast for some production work. The motivation, design principles, uses, and capabilities for this environment are outlined. The environment includes an interval data type, a symbolic form of automatic differentiation to obtain an internal representation for functions, a special technique to allow conditional branches with operator overloading and interval computations, and generic routines to give interval and noninterval function and derivative information. Some of these generic routines use a special version of the backward mode of automatic differentiation. The package also includes dynamic data structures for exhaustive search algorithms.

An interval step control for continuation methods

The authors present a step control for continuation methods that is deterministic in the sense that (i) it computationally but rigorously verifies that the corrector iteration will converge to a point on the same curve as the previous point (i.e., the predictor/corrector iteration will never jump across paths), and (ii) each predictor step is as large as possible, subject to verification that the curve is unique with the given interval extension. The authors present performance data and comparisons with an approximate step control method (PITCON version 6.1). A comparison of plots obtained from both step controls reveals that an approximate step control will behave erratically in situations where the interval step control leads to orderly progression along the curve. This is true even if the maximum allowable stepsize for the approximate method is set to be smaller than many of the steps actually taken by the interval algorithm. Limitations of interval step controls are also discussed.