where the elements are functions of the unknowns . The above equation describes a family of linear system and the

A classical scheme for finding the enclosure is to use an interval
variant of the *Gauss elimination* scheme [18].

When the unknowns lie in given ranges we may compute an interval
evaluation of and an interval evaluation
of .
The Gauss elimination scheme may be written as [18]

(7.2) | |||

(7.3) |

The enclosure of the variable can then be obtained from by

(7.4) |

A drawback of this scheme is that the family of linear systems that will be obtained for all instances of is usually a subset of the family of linear systems defined by , , as we do not take into account the dependency of the elements of . Furthermore the calculation in the scheme involves products, sums and ratio of elements of and their direct interval evaluation again do not take into account the dependency between the elements. Hence a direct application of the Gauss scheme will usually lead to an overestimation of the enclosure.

A possible method to reduce this overestimation is to consider the
system

where is an arbitrary matrix. The above system has the same solutions than the system (7.1) but for some matrix the enclosure of the above interval system may be included in the enclosure obtained by the Gauss elimination scheme on the initial system: this is called a

But even with a possible good the dependency between the elements of are not taken into account and hence the overestimation of the enclosure may be large.

A first possible way to reduce this overestimation is to improve the
interval evaluation of , by using the
derivatives of their elements with respect to
and the procedure described in
section 2.4.2.3. Note that the procedures necessary to compute
the elements of , and their derivatives may
be obtained by using the `MakeF, MakeJ` procedures of
ALIAS-maple.

But to improve the efficiency of the procedure it must be noticed that at iteration an interval evaluation of the derivatives of with respect to may be deduced for the derivatives of the elements computed at iteration . As we have the derivatives of the elements at iteration 0 we may then deduce the derivatives of the elements at any iteration and use these derivatives to improve the interval evaluation of these elements (see sections 2.4.1.1 and 2.4.2.3).