Blending Set and Interval Arithmetic for Maximal Reliability

Abstract

In both [3] and [8], the authors review the implementation of the basic operations in interval arithmetic, and in particular discuss the different approaches given in the literature for interval division when the divisor interval contains zero. In these papers, and in the references therein, the basic operations are defined for real or extended real interval operands.

Division by an interval containing zero is a special case of an interval function for which the input arguments contain points outside the domain of the underlying point function. A number of approaches exist in the literature, [7], [12], to remove restrictions on the domain of interval functions and hence obtain a closed, exception-free interval system.

In this paper, we present an alternative approach to remove restrictions on the domain of interval functions and to guarantee the inclusion property in all situations, even when some input intervals contain points that lie outside the domain of the underlying point function. To achieve this, we allow for the (efficient) set-based representation of non-real results. The computed intervals are sharp, yet contain more information and the resulting interval system is closed and exception-free. We also show how the presented ideas can be implemented in an interval arithmetic library. The performance overhead is negligible compared to the fact that the implementation using the new approach offers 100% reliability in return.

The structure of the paper is as follows. We set off with a motivating example in Sect. 1. In Sect. 2, we review various approaches to interval division and then introduce vset-division of real intervals, based on the newly introduced concept of value set or vset. In Sect. 3, we give a formal definition of real vset-intervals and arithmetic on these intervals. We prove a number of essential properties and point out the likenesses and differences with other approaches. Finally, in Sect. 4, we discuss the implementation of vset-interval arithmetic in a floating-point context.