Condition number

The condition number of a polynomial may be defined either as the ratio lowest root over largest root or as the ratio ${\rm Min}(\vert x_i\vert)$ over ${\rm Max}(\vert x_i\vert)$ where $x_i$ are the roots of the polynomial. In the later case the condition number has a value between 0 and 1. The minimal and maximal values of the condition number of a parametric polynomial in both form may be calculated using the procedure:

 
MinMax_Condition_Number_Gradient(Func,Vars,Init,Type,Absolute,Rand,Min,Pm,Max,PM)

where Absolute has to be set to 0 if looking for the ratio minimal root over maximal root or 1 if looking for the ratio in absolute value. The parametric polynomial should appear as last argument of Func. Note that is this last element is a matrix then the considered polynomial will be the characteristic polynomial of the matrix. This procedure may hence be used to compute the minimal and maximal value of the condition number of a matrix. Note here that the derivatives of the polynomial and of its coefficients must be available.