Rouche

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 $k$-th derivatives of the equations with respect to the variable, that will be denoted $F^(k)$ and to be able to find $k_0$ such that there is a $k_j$ in $[2,k_0]$

$$\frac{\vert\vert(F^{(1)}(X_0))^{-1}F^{(k)}(X_0)\vert\vert^{1... ...F^{(1)}(X_0))^{-1}F^{(k_j)}(X_0)\vert\vert^{1/(k_0-1)}}{k_j!}$$

for all $X_0$ in the search space and all possible $k\ge 2$. The value $k_0$ will be called the order of the Rouche method. Two different cases will be considered here:

• algebraic equations
• non-algebraic equations

For algebraic equations there will be clearly a $k_1$ such that $F^{(k)}=0$ for all $k\ge k_1$. Hence we may use $k1-1$ as order for the Rouche method.

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

A good example is

 
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.