Specificity of the procedure

Note that there is no need to use the HullConsistency simplification procedure when using SolveDistance as it is already embedded in the C++ method. The consistency method in the algorithm updates the value of the variables and starts again the update if the change in at least one interval exceed the threshold `ALIAS/seuil2B` (which has the default value 0.1).

As the SolveDistance algorithm uses systematically a Newton scheme with as estimate of the solution the center of the box, it may be interesting to switch the current box with the largest one in the list of boxes to process in order to find new solutions to the system very quickly. Indeed if the Newton scheme appears to converge toward an approximate solution $S$ we will use a special version of the Kantorovitch test to determine a box $B$ centered at $S$ that contains only one solution. This box will be further enlarged by using a specific version of the Neumaier test (this is called an inflation of the box). Then we will determine if $B$ has an intersection with each box in the list of boxes to process and if this is the case we will modify the list of boxes so that it has only boxes that are the complementary of $B$: this process will speed up the algorithm.

Switching the current box with the largest one is done by setting the flag `ALIAS/permute` to the number of bisection after which the boxes will be permuted. The default value for this flag is 1000 and if it is set to 0 no permutation will be done.

Note that if you have a system of distance equations with one parameter, you may fix the value of this parameter to a given value $a$ and then follow the solutions when the parameter changes in the range [$a, b$] by using a specific continuation procedure (see section 7.3)