Mathematical background

This section is freely inspired from the book [5]. An interval number is a real, closed interval $(\underline{x},\overline{x})$. Arithmetic rules exist for interval numbers. For example let two interval numbers $X=(\underline{x},\overline{x})$, $Y=(\underline{y},\overline{y})$, then:

$$\begin{eqnarray*} &&X+Y= [\underline{x}+\underline{y},\overline{x}+\overline{y}]\\ &&X-Y= [\underline{x}-\overline{y},\overline{x}-\underline{y}] \end{eqnarray*}$$

An interval function is an interval-valued function of one or more interval arguments. An interval function $F$ is said to be inclusion monotonic if $X_i \subset Y_i$ for $i$ in $[1,n]$ implies:

$$F(X_1,\ldots,X_n) \subset F(Y_1,\ldots,Y_n)$$

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 $x$ will be denoted by $(\underline{x},\overline{x})$. The width or diameter of an interval $(\underline{x},\overline{x})$ is the positive difference $\overline{x}-\underline{x}$. The mid-point of an interval is defined as $(\overline{x}+\underline{x})/2$.

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.