JAMP  Vol.2 No.7 , June 2014
Steffensen-Type Method of Super Third-Order Convergence for Solving Nonlinear Equations
Abstract: 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.
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.