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
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.