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`

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`.