Why Monotonicity in Interval Computations ?

Monotonicity of functions has been successfully used in many problems of interval computations. However, in the context of interval computations, monotonicity seems somewhat ad hoc. In this paper, we show that monotonicity can be reformulated in interval terms and is, therefore, a natural condition for interval mathematics. One of the main problems of interval computations it to compute the range solution of this problem often requires long computations. It is known that these computations can be made faster if the function f(x 1 ; :::; x n) is strictly monotonic (strictly increasing or strictly decreasing) in some of the variables. From a practical viewpoint, this is a great idea that has been successfully applied to many real-life problems (see, e. But from the theoretical viewpoint, the idea of monotonicity sounds too ad hoc, unrelated to intervals. In this short paper, we make monotonicity more theoretically acceptable by showing that monotonicity can be easily reformulated in interval terms. two conditions are equivalent to each other: 1) f is either strictly increasing or strictly decreasing; 2) for every two intervals a and b, if a b then f u (a) f u (b).