Minimal and maximal real roots of a parametric polynomial

The basic procedures for computing the minimal and maximal roots of a parametric polynomial are:

 
MinMax_Polynom(Func,Vars,Init,Type,Solve,Rand,Points,Min,Pm,Max,PM)
MinMax_Polynom_Gradient(Func,Vars,Init,Type,Solve,Rand,Points,Min,Pm,Max,PM)

where

• Func: the list of constraints between the parameters and polynomial unknown. The polynomial must be the last element of this list. If this last element is a matrix then the considered polynomial is the characteristic polynomial of the matrix (but this is not the more efficient procedure for this case, see next section). If this last argument is Fast_CharPoly(S),Medium_CharPoly(S) or Slow_CharPoly(S) where S is a matrix the polynomial is the characteristic polynomial of the matrix but this polynomial will not be computed by Maple. Instead the procedure Coeff_CharPoly will be used to compute the coefficients of the polynomial: this may be interesting for relatively large matrix.
• Vars:a list giving the name of the unknown starting with the name of the polynomial unknown
• Init: a list of ranges for the unknown starting with the range for the polynomial unknown (this element should be set to a large interval and is updated automatically) followed by the ranges for the parameters
• Type: 0 to find only the minimal root, 1 to find only the maximal root and 2 to find both.
• Solve: a flag that should usually be set to 3, see the on-line help
• Rand, Points: the equivalent of the rand and Nb_Points arguments of the C++ procedure
  ALIAS_Min_Max_EigenValues, see the ALIAS-C++ manual
• Min, Max: minimum and maximum root
• Pm,PM: values of the parameters at which the minimum and maximum have been obtained

The difference between these two procedures is that the second one uses the derivatives of the polynomial with respect to the variables together with the derivative of the coefficients of the polynomial.

For these procedures the flag `ALIAS/stop_opt_sol`, `ALIAS/opt_sol_max` will play the same role than Stop, Seuil[0] and Seuil[1] of the C++ procedure ALIAS_Min_Max_EigenValues, see the ALIAS-C++ reference manual.

In the special case where the polynomial is the characteristic polynomial it may be interesting to use the specific procedures:

 
MinMax_Char_Poly(Func,Vars,Init,Type,Solve,Rand,Points,Min,Pm,Max,PM)
MinMax_Char_Poly_Gradient(Func,Vars,Init,Type,Solve,Rand,Points,Min,Pm,Max,PM)

which has the same argument that the previous procedures except that the last argument of Func must be a matrix.