Possible parameters values for a given range for the roots

It is possible to determine an approximation of the region of the parameters space (a $n$ dimensional space where each of the dimension corresponds to one of the $n$ parameters) that contain all the possible values of the parameters such that the corresponding polynomial has all its roots in a given range. This approximation will be constituted of a set of $n$ dimensional boxes written in a file. The procedure to be used is:

 
MinMax_Polynom_Area(Func,Vars,Init,Strong,Solve,Rand,Points,Out,Lim,Type)

where

• Out: a string which is the name of the file in which the result will be written
• Lim: during the calculation all the boxes that have a width lower than this value will be neglected
• Type: a string that may be "Real" (if only the real root or the polynomial are considered), "RealPart" (for the real part of the root), "AllReal" (for polynomial having only real root), "OneReal" (for polynomial having at least one real root)

Strong is a list of two elements: the first element may be 1 or 2. If it is 2 the derivative of the coefficients of the polynomial will be used. This algorithm uses a secondary bisection process and the second element of Strong gives the maximum number of boxes that can be used for the secondary algorithm.

The range for the real roots of the polynomial is defined as:

 
[`ALIAS/opt_sol_min`,`ALIAS/opt_sol_max`]

It is possible to improve incrementally the quality of the approximation. During the first run the flag `ALIAS/ND` will be set to 1 and the neglected boxes will be written in the file `ALIAS/ND_file`. During the next run Lim has to decreased and the flag `ALIAS/ND` has to be set to 2. This has the effect that the initial box considered by the algorithm will not be Init but the set of boxes stored in `ALIAS/ND_file`. In this run the neglected boxes will still be stored in the file, thereby allowing another run of the algorithm with a lower Lim.

The largest square enclosed in the parameters space such that the polynomials defined by parameter values within the square have all their real roots within a given range can be computed using:

 
 MinMax_Square_Polynom_Gradient(Func,Grad,Vars,Range,Init,Rand,Points,Center,Edge)

where Center will be the center of the largest square while Edge will be the lengths of the edge of the square. If you have imperative constraint you may use the variable `ALIAS/Imperatif` as a list with 0 for non imperative constraints and 1 for imperative constraints. Note that this algorithm uses a main algorithm for getting the largest square and a secondary algorithm for calculating the minimal and maximal root of the polynomial in a given square. You may specify two different bisection mode, one for the main algorithm (`ALIAS/single_bisectiong`) and one for for the secondary algorithm (`ALIAS/single_bisection). Similarly the maximal number of boxes used by the main algorithm may be specified by `ALIAS/maxboxg` while the number of boxes used by the secondary algorithm is specified by `ALIAS/maxbox`