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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 $C^1$. Let ${\bf X}$ be a range vector for $\{x_1,\ldots,x_n\}$, $y$ a point inside ${\bf X}$ and $Y$ an arbitrary nonsingular real matrix. Let define $K$ as:

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

and let $r_0$ be the norm of the matrix $I-YF^\prime({\bf X})$. If

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

then there is a unique solution [16] of $F$ in ${\bf X}$. This unique solution can be found using Krawczyk solving method (see section 2.10).

Jean-Pierre Merlet 2012-12-20