On the acceleration of the convergence of certain iterative proceedings (I)

Abstract

The research elaborated in this paper has its origin in the study of the convergence of the sequences generated through the use of certain methods derived from the well-known Newton-Kantorovich method for the simultaneous approximation of the solution of an equation in a linear normed space, and of the inverse of the Fréchet differential at this solution. An important place is given in the paper to the notion of convergence order of an approximant sequence of the solution of an equation. Considering given an approximant sequence which verifies certain conditions expressed through the inequalities (25), we will build another approximant sequence through the relations (22), with ameliorated convergence order. We will analyse certain special cases and, in the same time, we will determine optimal methods from the point of view of the convergence order.