Mathematical background

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

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

each $F_i$ being at least $C^2$. Let ${\bf x_0}$ be a point and $U=\{ {\bf x} / \vert\vert{\bf x} -{\bf x_0}\vert\vert\le 2B_0\}$, the norm being $\vert\vert A\vert\vert= {\rm Max}_i \sum_j \vert a_{ij}\vert$. Assume that ${\bf x_0}$ is such that:

1. the Jacobian matrix of the system has an inverse $\Gamma_0$ at ${\bf x_0}$ such that $\vert\vert\Gamma_0\vert\vert \le A_0$
2. $\vert\vert\Gamma_0 F({\bf x_0})\vert\vert \le 2B_0$
3. $\sum_{k=1}^{n}\vert\frac{\partial^2F_i({\bf x}}{\partial x_j\partial x_k}\vert \le C$ for $i,j=1,\ldots,n$ and ${\bf x} \in U$
4. the constants $A_0, B_0, C$ satisfy $2nA_0B_0C \le 1$

Then there is an unique solution of $F=0$ in $U$ and Newton method used with ${\bf x_0}$ as estimate of the solution will converge toward this solution [3]. An interesting use of Kantorovitch theorem can be found in section 2.5.