Mathematical background

Let a system of $n$ equations in $n$ unknowns:

$$F=\{ F_i(x_1,\ldots,x_n)=0, i\in [1,n]\}$$

We will denote by $F^{(k)}$ the matrix of the derivatives of order $k$ of $F$ with respect to the variable. Let us consider a point $X_0$ and define $\gamma$ as

$$\gamma = Sup (\frac{\vert\vert(F^{(1)}(X_0))^{-1}F^{(k)}(X_0)\vert\vert^{1/k-1}}{k!})~~k \ge 2$$

$$\beta =\vert\vert (F^{(1)}(X_0))^{-1}F(X_0)\vert\vert$$

and $\alpha = \beta \gamma$. If $\alpha$ is strictly lower than $3-2\sqrt{2}$, then $F$ has a single root in a ball centered at $X_0$ with radius $(1+\alpha-\sqrt{\alpha^2-6*\alpha+1})/(4*\gamma)$ and the Newton scheme with initial guess $X_0$ will converge to the solution.

The most difficult part for using this theorem is to determine $\gamma$. For algebraic equations it is easy to determine a value $k_1$, that we will call the order of Rouche theorem, such that $F^{(k_1)}=0$ and consequently $\gamma$ may be obtained by computing

$$S_k=\frac{\vert\vert(F^{(1)}(X_0))^{-1}F^{(k)}(X_0)\vert\vert^{1/k-1}}{k!}$$

for all $k$ in $[2,k_1-1]$ and taking $\gamma$ as the Sup of all $S_k$.

For non algebraic finding $\gamma$ requires an analysis of the system.

Rouche theorem may be more efficient than Moore or Kantorovitvh theorems. For example when combined with a polynomial deflation (see section 5.9.6) it allows one to solve Wilkinson polynomial of order up to 18 with the C++ arithmetic on a PC, while stand solving procedure fails for order 13.