The SolveSimplex and SolveSimplexGradient procedures

This procedure allows to solve systems that have algebraic terms in at least at two equations. Each equation is transformed as the sum of a linear part and a non linear part. The linear part is substituted into a new variable that is submitted to a linear constraints. Then the simplex method is used to improve the range for the unknowns. The difference between the SolveSimplex and SolveSimplexGradient procedure is that in the second case the derivatives of the non linear parts of the equations are used to improve their interval evaluation and hence the linear constraints used by the simplex.

The following variables plays a role in these procedures:

• `ALIAS/diam_simplex`: the simplex will not be applied if the width of the current box is lower than this value (default value:0.1)
• `ALIAS/full_simplex`: if the value of this variable is $n$ the simplex will be applied only for searching the minimum and maximum of the first $n-1$ variables when they are ordered by decreasing order of width. This is probably the most important parameter to set: a compromise has to be found between the gain of considering all variables and the computation time of the simplex
• `ALIAS/init_simplex_linear`: if some of the coefficients of the linear part of the expressions have an identical value you may assign this value to this variable so that a faster assignation of the coefficients will be performed (available only for the parallel version of these procedures). For example if a majority of the linear coefficients are 2 setting `ALIAS/init_simplex_linear` to 2 implies that the C++ procedure that calculate the linear coefficients will just compute the coefficients that are not equal to 2
• `ALIAS/min_diam_simplex`: the simplex will not be applied if the width of the current box is larger than this value (default value: 1.e7)
• `ALIAS/min_improve_simplex`: if the width of one range of the box has been improved by a value larger than this variable, then the simplex method will be repeated (default value: 0.1)
• `ALIAS/no_simplex`: if this parameter is set to 1 the simplex will not be used. This may be interesting for the parallel implementation where the computation involved by the simplex may be to computer intensive for the master
• `ALIAS/simplex_expanded`: for the simplex method using the gradient setting this flag to 1 indicates that the expression will be expanded with respect to the lower bound of each variable (i.e. the unknown $x$ with lower bound $\underline{x}$ will be written as $x$=$X$+$\underline{x}$ and the simplex method will use the variable $X$)