Abstract
The research reflected 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 approximation of the solution of an equation in a linear normed space, together with the inverse of the Fréchet differential on this solution. An important place is given in the paper to the notion of convergence speed 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), sequence which finds its convergence speed order ameliorated. We will analyze certain special cases and, in the same time, we will determine optimal methods from the point of view of the convergence speed order.
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