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Approximation of the set of solutions

Let consider the parameters space i.e. a $n$ dimensional space where each of the dimension corresponds to one of the $n$ parameters. A point in this space corresponds to a unique value for all the parameters and therefore to a specific polynomial. In the parameters space there are possibly a set ${\cal S}$ of regions such that for any point in the region(s) the corresponding polynomial has all its root within the given interval. The purpose of the following procedure is to determine an approximation of ${\cal S}$. This approximation ${\cal A}$ will be constituted of a set of $n$ dimensional boxes which are guaranteed to be included in ${\cal S}$ and that will be written in a file. During the calculation the boxes whose width is lower than a given threshold $\epsilon$ and for which the algorithm has been unable to determine if they are fully enclosed in ${\cal S}$ will be neglected. A possible index for measuring the quality of the approximation ${\cal A}$ is the ratio $V/V_n$ between the total volume $V$ of the boxes written into the file over the total volume $V_n$ of the boxes that have been neglected as the volume of ${\cal S}$ is lower or equal to $V+V_n$.

The procedure is:

 
int ALIAS_Min_Max_EigenValues_Area(int Degree,int Nb_Parameter,
          int Has_Interval,
          INTERVAL_VECTOR (* TheCoeff)(INTERVAL_VECTOR &), 
          INTERVAL_VECTOR (* TheCoeffCentered)(INTERVAL_VECTOR &,double), 
          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 Nb_Points,int Use_Solve,int rand,int Strong,int Iteration,
          double Accuracy_Variable,double Accuracy,double AccuracyM,double AccuracyB,
          double *Volume_Result,double *Volume_Neglected,double Seuil,
          char *FileName,int Has_Input,char *File_Input,
          int (* Solve_Poly)(double *, int *,double *),int RealRoot,
          INTERVAL_VECTOR (* Evaluate_Complex)(int,int,INTERVAL_VECTOR &), 
          int (* Simp_Proc)(INTERVAL_VECTOR &))
where the arguments are similar to the one of the previous procedure except for:

This procedure returns the number of boxes written in the result file or a negative number if the calculation has failed. The possible negative return code are:


next up previous contents
Next: Largest square enclosed in Up: Possible parameters values for Previous: Possible parameters values for   Contents
Jean-Pierre Merlet 2012-12-20