The `Rouche` procedure has to be used for a system of equations
and will create a simplification procedure that may return 11, i.e. it
provide a ball that include a single solution of the system, that may
be found using a Newton iteration.
A key-point for using this procedure is to consider the matrix
constituted of the -th derivatives of the equations with respect to
the variable, that will be denoted and to be able to find
such that there is a in

for all in the search space and all possible . The value will be called the

- algebraic equations
- non-algebraic equations

As soon as there are non algebraic terms in at least one equation, then we cannot determine in advance the order of the Rouche method without an in-depth analysis of the system.

The syntax of this procedure is:

Rouche(Func,nfunc,Order,Vars,ProcName)where

`Func`: the list of equations`nfunc`: the number of equations in`Func``Order`: the order of the Rouche method. For algebraic equation you may specify "p" for this order as the procedure will determine automatically the correct order. Otherwise it is your responsibility to provide a numerical value for the order`Vars`: a list of variable name`ProcName`: the name of the simplification procedure which will be written in the file`ProcName.C`

EQ:=[x^2+y^2+z^2-9,(x-1)^2+(y-1)^2+z^2-9,(x-1)^2+(y+1)^2+z^2-16]]: VAR:=[x,y,z]: Rouche(EQ,"p",VAR,"rouche");

This procedure will be called only if the width of the box is lower
than `ALIAS/maxnewton` and uses at most `ALIAS/newton_iteration` of the Newton scheme to compute an
approximation of the root such that the residues of the system are
lower than `ALIAS/fepsilon`. The generated C++ program will also
use the epsilon inflation method to enlarge as much as possible the
ball that includes the root.

The Rouche procedure may be quite powerful to find a ball that
includes a single root of the system (even more powerful than the
Kantorovitch scheme). For example by using `Rouche` in combination
with `DeflationUP` (see section 8.1.2.1) we have been
able to solve the Wilkinson polynomial of order 19 (see
section 12.3), while the general
procedure failed certifying roots starting at order 13.