To the question of efficiency of iterative methods

Abstract

In this paper we prove that, among all one-point iterative processes without memory of order p , the most efficient processes are of order p = 3 . Moreover, the computational efficiency of one-point iterative processes without memory decreases to 1 as p increases, i.e., the efficiency index of higher order of convergence methods is low. We find the upper and lower bounds of the Ostrowski–Traub index of computational efficiency in a wider class of iterative methods with unit informational efficiency.