Systematic use of Newton

It may be interesting to systematically use the Newton scheme in a solving procedure in order to quickly determine the solutions of a system of equations.

For that purpose we may use the TryNewton procedure whose purpose is to run a few iterations of the Newton scheme for a given box. The syntax of this procedure is:

 
int TryNewton(int DimensionEq,int DimVar,
              INTERVAL_VECTOR (* TheIntervalFunction)(int,int,INTERVAL_VECTOR &),
              INTERVAL_MATRIX (* Gradient)(int, int, INTERVAL_VECTOR &), 
              INTERVAL_MATRIX (* Hessian)(int, int, INTERVAL_VECTOR &), 
              double Accuracy,
              int MaxIter,
              INTERVAL_VECTOR &Input,
              INTERVAL_VECTOR &Domain,
              INTERVAL_VECTOR &UnicityBox)

where

• DimensionEq: number of equations
• DimVar: number of variables
• TheIntervalFunction: a procedure in MakeF format for computing an interval evaluation of the equations
• Gradient: a procedure that compute the jacobian in MakeJ format
• Hessian: a procedure in MakeH format that computes the Hessian of the system
• Domain: the domain in which we are looking for solutions of the system
• Input: a sub-box of Domain

The mid-point of Input is used as initial guess of the Newton scheme. The parameters Accuracy is used in the Newton scheme to determine if Newton has converged i.e. if the residues are lower than Accuracy. A maximum of MaxIter iterations are performed.

If the Newton scheme converges, the presence of a single solution in the neighborhood of the approximated solution is checked by using the Kantorovitch theorem (see section 3.1.2). If this check is positive, then a ball that includes this single solution is determined and returned in UnicityBox. If the flag ALIAS_Epsilon_Inflation is set to 1, then the inflation scheme is used to try to enlarge this unicity box.

This procedure returns 11 if an unicity box has been determined, 0 otherwise. Note that this procedure is already embedded in HessianSolve.