AbstractThis note addresses the input and output of intervals in the sense of intervalarithmetic and interval constraints. The most obvious, and so far most widely usednotation, for intervals has drawbacks that we remedy with a new notation that wepropose to call factored notation. It is more compact and allows one to ﬁnd a goodtrade-oﬀ between interval width and ease of reading. We describe how such a trade-oﬀ can be based on the information yield (in the sense of information theory) of thelast decimal shown. 1 Introduction Once upon a time, it was a matter of professional ethics among computers never to writea meaningless decimal. Since then computers have become machines and thereby lost anyform of ethics, professional or otherwise. The human computers of yore were helped intheir ethical behaviour by the fact that it took eﬀort to write spurious decimals. Now thesituation is reversed: the lazy way is to use the default precision of the I O library function.As a result it is common to see ﬁfteen decimals, all but three of which are meaningless.Of course interval arithmetic is not guilty of such negligence. After all, the very raisond’ˆetre of the subject is to be explicit about the precision of computed results. Yet, eveninterval arithmetic is plagued by phoney decimals, albeit in a more subtle way. Just as con-ventional computation often needs more care in the presentation of computational results,the most obvious interval notation with default precision needs improvement.As a bounded interval has two bounds, say, l and u, the most straightforward notationis something like [l,u]. Written like this, it may not be immediately obvious what is wrongwith writing it that way. But when confronted with a real-life consequence
Hansen @cite_3 , @cite_5 , and Kearfott @cite_1 opt for the straightforward @math notation. Hansen mostly presents bounds with few digits, but for instance on page 178 we find @math demonstrating the problems addressed here.
@cite_5 @cite_1 @cite_3
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From the Publisher: This book offers a general discussion on arithmetic and computational reliability, analytical mathematics and verification techniques, algorithms, and (most importantly) actual C++ implementations. In each chapter, examples, exercises, and numerical results demonstrate the application of the routines presented. The book introduces many computational verification techniques. It is not assumed that the reader has any prior formal knowledge of numerical verfication or any familiarity with interval analysis. The necessary concepts are introduced. 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. Employing a closed set-theoretic foundation for interval computations, Global Optimization Using Interval Analysis simplifies algorithm construction and increases generality of interval arithmetic. This Second Edition contains an up-to-date discussion of interval methods for solving systems of nonlinear equations and global optimization problems. It expands and improves various aspects of its forerunner and features significant new discussions, such as those on the use of consistency methods to enhance algorithm performance. Provided algorithms are guaranteed to find and bound all solutions to these problems despite bounded errors in data, in approximations, and from use of rounded arithmetic.