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Mathematical background

Let a system $F(X)=0$ with $X=\{x_1,\ldots,x_n\}$. Let us consider a ball ${\cal B}=\{[\underline{x_1},\overline{x_1}],\ldots,[\underline{x_n},\overline{x_n}]$ for $X$ and define

&&[X]_j^+ = \{X \in {\cal B}~{\rm such~that}~x_j=\overline{x_j...
..._j^- = \{X \in {\cal B}~{\rm such~that}~x_j=\underline{x_j}\}\\

for $j$ in $[1,n]$. If

F_j(X)\ge 0 {\rm and}~ F_j(Y)\le 0~~\forall~X\in [X]_j^+,
\forall~Y\in [X]_j^-, j=1,\ldots,n

or if

F_j(X)\le 0 {\rm and}~ F_j(Y)\ge 0~~\forall~X\in [X]_j^+,
\forall~Y\in [X]_j^-, j=1,\ldots,n

then $F$ has at least one zero in ${\cal B}$ [15].

Jean-Pierre Merlet 2012-12-20