Improved Interval Bounds for Ranges of Functions

Abstract : Evaluation of a real function f on an interval X using interval arithmetic yields an interval extension F(X) containing the range R(f;X) of f on X. Unfortunately, F(X) is sometimes excessively wider than R(f;X). Evaluation of f epsilon C1 by interval differentiation arithmetic gives F(X) and the extension F'(X) of f' on X. If F' (X) 0 (or F'(X) 0), then f is monotone on X = (a,b), and R(f;X) = (f(a), f(b)) (or R(f;X) = (f(b), f(a))), giving improved bounds for R(f;X). If 0 epsilon int(F'(X)), then X is divided into subintervals on which f is either guaranteed to be monotone or has possible extremal points. A Pascal-SC program for this simple algorithm is given, and numerical results are presented. As a byproduct of the computation, possible extremal points of f are isolated. Keywords: Range of functions; Extremal points; Automatic differentiation; Interval computation; Optimization; Pascal-SC.