Sharpness in Interval Computations

Abstract

Let f(x) be a rational function and let F(X) be an interval extension of f(x). When we evaluate F(X) using interval arithmetic, we obtain an interval which bounds the range of f(x) for all x in the interval X. In some cases, the lower or upper bound (or both) may be sharp. We show that we can determine whether an endpoint is sharp or not merely by keeping track of which endpoints of X are used in each step of the evaluation of F(X). We show that in certain cases, this procedure can prove that f(x) is monotonic in the interval X.