**2.1 Interval arithmetic**

Since its formalization in 1962 by R. Moore (Moore, 1962), Interval Arithmetic (IA) has been widely used to bound uncertainties in complex systems (Moore, 1966). The main advantage of traditional IA is that it is able to obtain the range of all the possible results of a given function. On the other hand, it suffers from three different types of problems (Neumaier, 2002): the dependency problem, the cancellation problem, and the wrapping effect.

The dependency problem expresses that IA computations overestimate the output range of a given function whenever it depends on one or more of its variables through two or more different paths. The cancellation problem occurs when the width of the intervals is not canceled in the inverse functions. In particular, this situation occurs in the subtraction operations (i.e., given the non-empty interval *I1 – I1* z *0*), what can be seen as a particular case of the dependency problem, but its effect is clearly identified. The wrapping effect occurs because the intervals are not able to accurately represent regions of space whose boundaries are not parallel to the coordinate axes.

These overestimations are propagated in the computations and make the results inaccurate, and even useless in some cases. For this reason, the Overestimation Factor (*OF*) (Makino & Berz, 2003; Neumaier, 2002) has been defined as

$$\text{OF} = \text{(Estimated Range - Exact Range)} \;/\; \text{(Exact Range)}.\tag{1}$$

to quantify the accuracy of the results. Another interesting definition used to evaluate the performance of these methods is the Approximation Order (Makino & Berz, 2003; Neumaier, 2002), defined as the minimum order of the monomial *C*H *<sup>S</sup>* (where *C* is constant, and H [0,1]) that contains the difference between the bounds of the interval function and the target function in the range of interest.
