A lower bound for range enclosure in interval arithmetic

Including the range of a rational function over an interval is an important problem in numerical computation. A direct interval arithmetic evaluation of a formula for the function yields in general a superset with an error linear in the width of the interval. Special formulas like the centered forms yield a better approximation with a quadratic error. Alefeld posed the question whether in general there exists a formula whose interval arithmetic evaluation gives an approximation of better than quadratic order. In this paper we show that the answer to this question is negative if in the interval arithmetic evaluation of a formula only the basic four interval operations +; ; ; = are used.