A sufficient condition for non-overestimation in interval arithmetic

Abstract

The range of an arithmetic expression on an interval can be estimated by evaluating the expression using interval arithmetic. This gives always a superset of the range, which is usually proper. However, there are cases where no overestimation error occurs. The subject of this paper is to study such cases systematically and to give a sufficient condition when interval evaluation yields the exact range. Testing the condition is inexpensive: Given the intermediate values which are obtained during the evaluation of the expression, only some comparisons and bookkeeping but no additional arithmetic operations need to be performed. We show that for certain classes of expressions there is no overestimation if the input interval has a sufficiently large distance to zero. Formulae for the minimal distance are derived for polynomials in Horner form and in factorized form.