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.


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

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


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

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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



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[1] M.A. Ostrowski, Solution of Equations in Euclidian and Banach Spaces Academic Press, New York and London (1973)

[2] M. Frontini, Hermite interpolation and a new iterative method for the computation of the roots of non-linear equations, Calcolo, 40 (2003), pp. 109-119 View Record in Scopus

[3] I.K. Argyros, A new convergence theorem for the Steffensen method in Banach space and applications, Rev. Anal. Numer. Theor. Approx., 29 (2) (2000), pp. 119-127

[4] S. Amat, J. Blanda, S. Busquier, A Steffensen type method with modified functions, Riv. Mat. Univ. Parma, 7 (2007), pp. 125-133

[5] E. Cătinaș E., On some Steffensen-type iterative methods for a class of nonlinear equations, Rev. Anal. Numer. Theor. Approx., 24 (1-2) (1995), pp. 37-43

[6] A. Cordero, J.R. Torregrosa, A class of Steffensen type methods with optimal order of convergence, Appl. Math. Comput., 217 (2011), pp. 7653-7659

[7] Hongmin Ren, Qingbiao Wu, Weihong Bi, A class of two-step Steffensen type methods with fourth-order convergence, Appl. Math. Comput., 209 (2009), pp. 206-210

[8] P. Jain, Steffensen type method for solving non-linear equations, Appl. Math. Comput., 194 (2007), pp. 527-533

[9] I. Păvăloiu I., Approximation of the root of equations by Aitken–Steffensen-type monotonic sequences, Calcolo, 32 (1995), pp. 69-82

[10] I. Păvăloiu, E. Cătinaș, On a Steffensen type method, Proceedings of SYNASC 2007, 9th International Symposium on Symbolic and Numeric Algoritms for Scientific Computing, Timișoara, Romania, September 26-29, IEEE Computer Society (2007), pp. 369-375

[11] J.R. Sharma, A composite third order Newton–Steffensen method for solving nonlinear equations, Appl. Math. Comput., 169 (2005), pp. 242-246

[12] Quan Zheng, Peng Zhao, Li Zhang, Wenchao Ma, Variants of Steffensen-secant method and applications, Appl. Math. Comput., 216 (12) (2010), pp. 3486-3496

[13] J.F. Traub, Iterative Method for Solution of Equations Prentice-Hall, Englewood Cliffs, New Jersey (1964)

[14] Shaohua Yu, Xiubin Xu, Jianqiu Li, Younghui Ling, Convergence behavior for Newton-Steffensen’s method under Lipschitz condition of second derivative Taiwanese, J. Math., 15 (6) (2011), pp. 2577-2600

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