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Condition number

The condition number of a polynomial may be defined either as the ratio lowest root over largest root or as the ratio over where 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 &),
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.    Next: Kharitonov polynomials Up: Parametric polynomials and eigenvalues Previous: Largest square enclosed in   Contents
Jean-Pierre Merlet 2012-12-20