Majorizing functions and two-point Newton-type methods
The authors are concerned with the problem of approximating a locally unique solution \(x^{*}\) of a nonlinear equation \(F(x)=0\) where \(F\) is a twice Fréchet differentiable operator defined on a convex domain \({\mathcal D}\) of the Banach space \({\mathcal X}\) with values in the Banach space \({\mathcal Y}\). The approximating sequences \(\{x_{n}\},\{y_{n}\}\) \((n\geq 0) \) are given by \[ \begin{aligned} z_{n}&= \tfrac12(x_n+y_n), \quad n\geq 0,\\ x_{n+1}&= x_{n}- F'(z_n)^{-1} F(x_{n}), \\ y_{n+1}&= x_{n+1}- F'(z_n)^{-1} F(x_{n+1}). \end{aligned} \] Denote \(m=\frac{2}{\beta +\sqrt{\beta^{2}+2\alpha}}\), \(\alpha>0\), \(\beta>0\), and \(f= \frac{\alpha}{6}t^{3}+\frac{\beta}{2}t^{2}-t+ \gamma\); \(\gamma>0\). It is known that for \(f(m)\leq 0\) is necessary and sufficient that \(f\) have positive roots \(t^{*},t^{**}\). It will be assumed that \(y_{0}\in \overline{U}(x_{0},m)\subseteq \mathcal{D}\). The main convergence result is given by the theorem: The approximating sequences are defined for \(x_{n},y_{n}\in U(x_{0},t^{*}) \) and converge to the unique solution \(x^{*}\in \overline{U}(x_{0},m)\) with \(\|x_{n}-x^{*} \|\leq t^{*}-t_{0}\). Here \(U(x_{0},m)\) is an open ball with radius \(r\) and center \(x_{0}\).
scientific article; zbMATH DE number 5710754