Is an interval the right result of arithmetic operations on intervals? (original) (raw)

Is the conventional interval arithmetic correct?

Interval arithmetic as part of interval mathematics and Granular Computing is unusually im-portant for development of science and engineering in connection with necessity of taking into account uncertainty and approximativeness of data occurring in almost all calculations. Interval arithmetic also conditions development of Artificial Intelligence and especially of automatic think-ing, Computing with Words, grey systems, fuzzy arithmetic and probabilistic arithmetic. However, the mostly used conventional Moore-arithmetic has evident weak-points. These weak-points are well known, but nonetheless it is further on frequently used. The paper presents basic operations of RDM-arithmetic that does not possess faults of Moore-arithmetic. The RDM-arithmetic is based on multi-dimensional approach, the Moore-arithmetic on one-dimensional approach to interval calculations. The paper also presents a testing method, which allows for clear checking whether results of any interval arithmetic are cor...

Theories of Interval Arithmetic: Mathematical Foundations and Applications

Scientists are, all the time, in a struggle with uncertainty which is always a threat to a trustworthy scientific knowledge. A very simple and natural idea, to defeat uncertainty, is that of enclosing uncertain measured values in real closed intervals. On the basis of this idea, interval arithmetic is constructed. The idea of calculating with intervals is not completely new in mathematics: the concept has been known since Archimedes, who used guaranteed lower and upper bounds to compute his constant Pi. Interval arithmetic is now a broad field in which rigorous mathematics is associated with scientific computing. This connection makes it possible to solve uncertainty problems that cannot be efficiently solved by floating-point arithmetic. Today, application areas of interval methods include electrical engineering, control theory, remote sensing, experimental and computational physics, chaotic systems, celestial mechanics, signal processing, computer graphics, robotics, and computer-assisted proofs. The purpose of this book is to be a concise but informative introduction to the theories of interval arithmetic as well as to some of their computational and scientific applications.

Interval arithmetic: From principles to implementation

Journal of the ACM, 2001

We start with a mathematical definition of a real interval as a closed, connected set of reals. Interval arithmetic operations (addition, subtraction, multiplication, and division) are likewise defined mathematically and we provide algorithms for computing these operations assuming exact real arithmetic. Next, we define interval arithmetic operations on intervals with IEEE 754 floating point endpoints to be sound and optimal approximations of the real interval operations and we show that the IEEE standard's specification of operations involving the signed infinities, signed zeros, and the exact/inexact flag are such as to make a correct and optimal implementation more efficient. From the resulting theorems, we derive data that are sufficiently detailed to convert directly to a program for efficiently implementing the interval operations. Finally, we extend these results to the case of general intervals, which are defined as connected sets of reals that are not necessarily closed.

Interval Mathematics as a Potential Weapon against Uncertainty

Mathematics of Uncertainty Modeling in the Analysis of Engineering and Science Problems, 2014

This chapter is devoted to introducing the theories of interval algebra to people who are interested in applying the interval methods to uncertainty analysis in science and engineering. In view of this purpose, we shall introduce the key concepts of the algebraic theories of intervals that form the foundations of the interval techniques as they are now practised, provide a historical and epistemological background of interval mathematics and uncertainty in science and technology, and finally describe some typical applications that clarify the need for interval computations to cope with uncertainty in a wide variety of scientific disciplines. Keywords. Interval mathematics, Uncertainty, Quantitative Knowledge, Reliability, Complex interval arithmetic, Machine interval arithmetic, Interval automatic differentiation, Computer graphics, Ray tracing, Interval root isolation. Recommended Citation: Hend Dawood. Interval Mathematics as a Potential Weapon against Uncertainty. In S. Chakraverty, editor, Mathematics of Uncertainty Modeling in the Analysis of Engineering and Science Problems. chapter 1, pages 1-38. IGI Global, Hershey, PA, 2014. ISBN 978-1-4666-4991-0. The final publication is available at IGI Global via http://dx.doi.org/10.4018/978-1-4666-4991-0.ch001

New Operations on Fuzzy Numbers and Intervals

This paper introduces new operations on fuzzy numbers and intervals. These operations allow keeping the shape of a membership function intact and constructing complex linguistic terms corresponding to such linguistic hedges as "very" and "more or less". The article contains mathematical equations which allow us to determine the characteristic points of operation results for particular types of membership functions without integral evaluation.

Power and beauty of interval methods

Interval calculus is a relatively new branch of mathematics. Initially understood as a set of tools to assess the quality of numerical calculations (rigorous control of rounding errors), it became a discipline in its own rights today. Interval methods are usefull whenever we have to deal with uncertainties, which can be rigorously bounded. Fuzzy sets, rough sets and probability calculus can perform similar tasks, yet only the interval methods are able to (dis)prove, with mathematical rigor, the (non)existence of desired solution(s). Known are several problems, not presented here, which cannot be effectively solved by any other means. This paper presents basic notions and main ideas of interval calculus and two examples of useful algorithms.