Interval expansion method for nonlinear equation in several variables (Q1026277)

The authors present a new algorithm for verifying and enclosing zeros of a nonlinear continuously differentiable function \(f: D\subseteq\mathbb{R}^n\to \mathbb{R}\). To this end they use interval arithmetic and two new operators: The first one is called optimum center form and is defined by means of the mean value form \(F_M(X, c)= f(c)+ F'(X)(X- c)\) as \[ F_0(X,\overline c,\underline c)= [\text{inf\,}F_M(X,\underline c),\,\sup F_M(X,\overline c)]. \] Here, \(X= [\underline X,\overline X]= ([\underline X_i,\overline X_i])\) is an interval vector in \(D\), and \(\underline c= (\underline c_i)\), \(\overline c= (\overline c_i)\) are real vectors with \[ \underline c_i= \begin{cases} \overline X_i,\quad & \overline l_i\leq 0,\\ \underline X_i,\quad & \underline l_i\geq 0,\\ {\overline l_i\underline X_i-\underline l_i\overline X_i\over \overline l_i-\underline l_i},\quad &\text{otherwise},\end{cases}\quad \overline c_i= \begin{cases} \underline i,\quad & \overline l_i\leq 0,\\ \overline X_i,\quad & \underline l_i\geq 0,\\ {\underline l_i\underline X_i-\overline l_i\overline X_i\over \underline l_i-\overline l_i},\quad & \text{otherwise},\end{cases} \] for \(i= 1,2,\dots, n\), where \(F'(X)= [\underline l_i,\overline l_i])\) is an interval extension of \((\text{grad\,}f(x))^T\) on \(X\). The second operator is defined as \[ {\mathcal K}(X,\overline c,\underline c)= X-\Gamma\cdot(F_N(X)\cap F_0(X,\overline c,\underline c)), \] where \(F_N(X)= f(X)\) is the natural interval extension of \(f\) on \(X\) and \(\Gamma\) is a vector in \(\mathbb{R}^n\) with non-zero components. Existence and nonexistence results for zeros \(x^*\in X\) of \(f\) are proved. Numerical examples illustrate the theory.