Maximal Stationary Iterative Methods for the Solution of Operator Equations

Abstract

We study stationary iterative methods of maximal order for calculating zeros of operator equations. These methods use the values of the operator and its first s Frechet derivatives at n previous iteration points. We introduce a sufficient condition for an iterative method to have maximal order in a certain class of admissible methods. We prove the maximality of the interpolatory method $I_{n,s} $ in the scalar case (see Traub [11, p. 60 and ff.]). For the m-dimensional case, $2 \leqq m \leqq + \infty $, we prove that interpolatory iteration is maximal for $n = 0$ in the class of iterations using values of the first s derivatives at n previous points.