On a Newton–Steffensen type method

In this paper we study the convergence of a Newton–Steffensen type method for solving nonlinear equations in RR, 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 qq-convergence order of the iterations is 3. The efficiency index of the method is 33, and is larger than I2=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.