Interval Arithmetic

Abstract

The fundamental idea behind the interval arithmetic (IA) is that the values of a variable can be expressed as ranging over a certain interval. If one computes a number A as an approximation to some unknown number X such that ǀX − Aǀ ≤ B, where B is a precise bound on the overall error in A, we will know for sure that X lies in the interval [A − B, A + B], no matter how A and B are computed. The idea behind IA was to investigate computations with intervals, instead of simple numbers. In fact, when we use a computer to make real number computations, we are limited to a finite set of floating-point numbers imposed by the hardware. In these circumstances, there are two main options for approximating a real number. One is to use a simple floating point approximation of the number and to propagate the error of this approximation whenever the number is used in a calculation. The other is to bind the number in an interval (whose ends may also be floating point values) within which the number is guaranteed to lie. In the latter case, any calculation that uses the number can just as well use its interval approximation instead. This chapter deals with computations involving two floating-point numbers as intervals—the subject covered by interval arithmetic. Approximations carried out with a single floating-point number are studied in the next chapter.