Mathematical background

Let ${\cal F}$ be a system of $n$ equations in the $n$ unknowns ${\bf x}$. Let ${\bf X}$ be a range vector for ${\bf x}$ and $y_0=Mid({\bf X})$. Let $r_0$ be the norm of the matrix $I-YF^\prime({\bf X})$. Let the following iterative scheme for $k\ge 1$:

$$\begin{eqnarray*} &&y_k=Mid({\bf X}_k)\\ &&Y_k=\left\{ \begin{array}{l} (Mid(F... ...} \right.\\ &&r_k=\vert\vert I-Y_kF^\prime({\bf X}_k)\vert\vert \end{eqnarray*}$$

Let define $K$ as:

$$K{\bf X}=y-YF(y)+\{I-YF^\prime({\bf X})\}({\bf X}-y)$$

If

$$K({\bf X}_0) \subseteq {\bf X}_0~~~{\rm and}~~r_0<1$$

then the previous iterative scheme will converge to the unique solution of $F$ in ${\bf X}$ [8]. The procedure described in section 3.1.1 enable to verify if the scheme will be convergent.