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:

 
int ALIAS_Min_Max_CN(int Degree,  
     int Nb_Parameter,INTERVAL_VECTOR (* TheCoeff)(INTERVAL_VECTOR &), 
     int Nb_Constraints,INTEGER_VECTOR &Type_Eq,
     int (* TheMatrix)(INTERVAL_VECTOR &, INTERVAL_MATRIX &), 
     int Has_Matrix,
     INTERVAL_VECTOR (* IntervalFunction)(int,int,INTERVAL_VECTOR &), 
     int Has_Gradient,
     INTERVAL_MATRIX (* Gradient)(int, int,INTERVAL_VECTOR &), 
     INTERVAL & TheDomain,INTERVAL_VECTOR & TheDomain_Parameter, int Type,
     int Nb_Points,int Absolute,
     int rand,int Iteration,
     double Accuracy_Variable,double Accuracy,double AccuracyM,
     INTERVAL &Lowest,INTERVAL &Highest,
     INTERVAL_MATRIX &Place,int  Stop, double *Seuil,
     int (* Solve_Poly)(double *, int *,double *), 
     int (* Simp_Proc)(INTERVAL_VECTOR &))

where the arguments are identical than for the previous procedure except for:

• Absolute: 0 if looking for the ratio minimal root over maximal root, 1 if looking for the ratio in absolute value
• AccuracyM: the accuracy with which the ratio will be computed
• Lowest, Highest: minimal and maximal value of the condition number

This procedure allows for the calculation of the minimal and maximal condition number of a matrix. Various bisection methods are available through the use of the integer Single_Bisection:

• 1: mode 1 of section 2.4.1.3
• 2: mode 6 of section 2.4.1.3 if the gradient is not available
• 3,4: mode 1 of section 2.4.1.3
• 5: mode 5 of section 2.4.1.3

A specific procedure is used when the gradient is available. If the interval for the root includes 0 we bisect it. Otherwise we use the smear function to decide which other variable should be bisected.