Let \(f= (f_i): D\subseteq\mathbb{R}^n\to \mathbb{R}^m\) be a twice continuously differentiable function, let \(Y\) be a given box in \(\mathbb{R}^m\) and let \(X_0\) be a given box in \(D\). It is known that a parameter \(\overline x\in \mathbb{R}^n\) satisfies \(\overline x\in L:= f^{-1}(Y)\cap X_0\). Unless further information on \(\overline x\in \mathbb{R}^n\) is available the best possible inclusion \(X\) of \(\overline x\in \mathbb{R}^n\) by a box is the smallest box containing \(L\).
It is the aim of the paper to compute boxes \(X^{(i)}\), \(X^{(o)}\) such that \(X^{(i)}\subseteq X\subseteq X^{(o)}\) holds, i.e., \(X^{(i)}\), \(X^{(o)}\) are inner and outer enclosures of \(X\), respectively. Choosing \(x_0\in X_0\) the method being presented is based on computed approximations of \(f_j(x_0)\) and of grad \(f_j(x_0)\) and on some enclosure involving Hessians. Polyhedra \(p^{(i)}\), \(p^{(0)}\) are defined which satisfy \(p^{(i)}\cap X_0\subseteq L\subseteq p^{(o)}\cap X_0\). Then \(X^{(i)}\), \(X^{(o)}\) can be obtained as good inner and outer approximations, respectively, of the interval hull of these intersections.
A typical practical case from geodesy is discussed and illustrated by two numerical examples. Additional examples conclude the paper.