Iterationsverfahren höherer Ordnung in Banach-Räumen

Abstract

The Newton process for operator equations in say a linear normed complete space converges under certain hypothesis about the Fréchet-derivatives of the operator with at least the order two. There are different ways to improve this Newton process. For instance you obtain a process of order three if you add a correction element containing the second Fréchet-derivative of the operator [1]. In the following note we will generalize this idea. In a recursive manner — by adding higher derivatives — we will construct iterative processes of any orderk (k > 1). A general theorem due toCollatz provides us error estimates for this processes. Last we will illustrate the processes by several examples.