The Extended Real Interval System - CiteSeerX

The Extended Real Interval System G. William Walster April 29, 1998

Abstract

Three extended real interval systems are de ned and distinguished by their implementation complexity and result sharpness. The three systems are closed with respect to interval arithmetic and the enclosure of functions and relations, notwithstanding domain restrictions or the presence of singularities.

1 Overview Section 2 introduces the problem of de ning closed interval systems. In Section 3, real and extended points and intervals are de ned. In Section 4, the empty and entire intervals are used to close the extended interval system. Section ?? shows how incorrect conclusions have been reached about the result of certain interval arithmetic operator-operand combinations. The author is grateful to Professor Arnold Neumaier for originally raising this issue. Section 9 describes how to legitimately use IEEE oating-point arithmetic to obtain the sharp results described in Section ??. Section 6 generalizes extended interval arithmetic to de ne interval enclosures of functions, with the objective of removing all restrictions on their domains. Sections 7 and 8, contain brief remarks concerning, respectively: the relationship between extended intervals and the fundamental theorem of interval arithmetic and its generalization in [9]; and, extended dependent interval variables and interval enclosures. Section 10 de nes three dierent mappings of extended intervals onto binary compatible IEEE representations. Two of these mappings support the sharp results introduced in Section ??. Pseudo code for implementation algorithms are included.

1

2 Introduction The real and extended real number systems are not closed under arithmetic operations. This causes diculties for compiler developers, because exceptional provision must be made for unde ned outcomes. Extended real interval systems can be closed because intervals can be used to represent sets of values. This paper is con ned to real numbers and intervals. Therefore real numbers, variables, and intervals are often referred to as simply numbers, variables, and intervals, respectively.

3 De nitions The following de nitions are generally accepted in real point and interval analysis.

3.1 Points

The following point (as opposed to interval) de nitions are paraphrased from [3]:

De nition 1 A variable is a symbol used to represent an unspeci ed number

in a set of numbers. A variable is a \place holder" or a \blank" for an element of a set of numbers. Any member of the set is a value of the variable. The set, itself is the range of the variable. If the set has only one member, the variable is a constant.

The distinction between variables and constants is important, particularly in the case of interval constants and variables, see Section 8 and [9].

De nition 2 The extended real number system contains the set of real numbers (denoted by 0.

Figure 8: Z = [0:4; 0:6]  Y

Figure 9: Z = [1:4; 1:6]  Y 



Finally, combining the above results, the limit set 00