An

A fundamental theorem is that any rational interval function evaluated with a fixed sequence of operations involving only addition, subtraction, multiplication and division is inclusion monotonic. This means in practice that the interval evaluation of a function gives bounds (very often overestimated) for the value of the function: for any specific values of the unknowns within their range the value of the function for these values will be included in the interval evaluation of the function. A very interesting point is that the above statement will be true even taking into account numerical errors. For example the number 1/3, which has no exact representation in a computer, will be represented by an interval (whose bounds are the highest floating point number less than 1/3 and the smallest the lowest floating point number greater than 1/3) in such way that the multiplication of this interval by 3 will include the value 1. A straightforward consequence is that if the interval evaluation of a function does not include 0, then there is no root of the function within the ranges for the unknowns.

In all the following sections an interval for the variable will be
denoted by
. The *width* or *diameter* of an interval
is the
positive difference
.
The *mid-point* of an interval is defined as
.

A *box*
is a set of intervals. The width of a box is the largest width of the
intervals in the set and the *center* of the box is the vector
constituted with the mid-point of all the intervals in the set.