# On a Newton-Steffensen type method

## Abstract

In this paper we study the convergence of a Newton-Steffensen type method for solving nonlinear equations in R, introduced by Sharma [J.R. Sharma, A composite third order Newton–Steffensen method for solving nonlinear equations, Appl. Math. Comput. 169 (2005), 242–246]. Under simplified assumptions regarding the smoothness of the nonlinear function, we show that the q-convergence order of the iterations is 3. The efficiency index of the method is $$\sqrt[3]{3}$$ and is larger than $$I_2=\sqrt{2}$$, which corresponds to the Newton method or the Steffensen method. Moreover, we show that if the nonlinear function maintains the same monotony and convexity on an interval containing the solution, and the initial approximation satisfies the Fourier condition, then the iterations converge monotonically to the solution. We also obtain a posteriori formulas for controlling the errors. The numerical examples confirm the theoretical results.

## Authors

Ion Păvăloiu
(Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy)

Emil Cătinaş
(Tiberiu Popoviciu Institute of Numerical Analysis, Romanian Academy)

## Keywords

Nonlinear equations in R; Newton method; Steffensen type methods; inverse interpolatory polynomials; divided differences.

## Paper coordinates

I. Păvăloiu, E. Cătinaş, On a Newton-Steffensen type method, Appl. Math. Lett., 26 (2013) no. 6, pp. 659-663
doi: 10.1016/j.aml.2013.01.003

## PDF

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Elsevier

0893-9659