Share This Article:

Steffensen-Type Method of Super Third-Order Convergence for Solving Nonlinear Equations

Abstract Full-Text HTML XML Download Download as PDF (Size:295KB) PP. 581-586
DOI: 10.4236/jamp.2014.27064    3,774 Downloads   4,739 Views   Citations


In this paper, a one-step Steffensen-type method with super-cubic convergence for solving nonlinear equations is suggested. The convergence order 3.383 is proved theoretically and demonstrated numerically. This super-cubic convergence is obtained by self-accelerating second-order Steffensen’s method twice with memory, but without any new function evaluations. The proposed method is very efficient and convenient, since it is still a derivative-free two-point method. Its theoretical results and high computational efficiency is confirmed by Numerical examples.

Conflicts of Interest

The authors declare no conflicts of interest.

Cite this paper

Liu, Z. and Zhang, H. (2014) Steffensen-Type Method of Super Third-Order Convergence for Solving Nonlinear Equations. Journal of Applied Mathematics and Physics, 2, 581-586. doi: 10.4236/jamp.2014.27064.


[1] Ortega, J.M. and Rheinboldt, W.G. (1970) Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, New York.
[2] Kung, H.T. and Traub, J.F. (1974) Optimal Order of One-Point and Multipoint Iteration. Journal of the ACM, 21, 634-651.
[3] Traub, J.F. (1964) Iterative Methods for the Solution of Equations. Prentice-Hall, Englewood Cliffs.
[4] Zheng, Q., Wang, J., Zhao, P. and Zhang, L. (2009) A Steffensen-Like Method and Its Higher-Order Variants. Applied Mathematics and Computation, 214, 10-16.
[5] Zheng, Q., Zhao, P., Zhang, L. and Ma, W. (2010) Variants of Steffensen-Secant Method and Applications. Applied Mathematics and Computation, 216, 3486-3496.
[6] Petkovic, M.S., Ilic, S. and Dzunic, J. (2010) Derivative Free Two-Point Methods with and without Memory for Solving Nonlinear Equations. Applied Mathematics and Computation, 217, 1887-1895.
[7] Dzunic, J. and Petkovic, M.S. (2012) A Cubically Convergent Steffensen-Like Method for Solving Nonlinear Equations. Applied Mathematics Letters, 25, 1881-1886.
[8] Alarcón, V., Amat, S., Busquier, S. and López, D.J. (2008) A Steffensen’s Type Method in Banach Spaces with Applications on Boundary-Value Problems. Journal of Computational and Applied Mathematics, 216, 243-250.

comments powered by Disqus

Copyright © 2019 by authors and Scientific Research Publishing Inc.

Creative Commons License

This work and the related PDF file are licensed under a Creative Commons Attribution 4.0 International License.