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.
KEYWORDS
Newton’s Method, Steffensen’s Method, Derivative Free, Super-Cubic Convergence, Nonlinear Equation
Newton’s Method, Steffensen’s Method, Derivative Free, Super-Cubic Convergence, Nonlinear Equation
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.
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.
References
[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. http://dx.doi.org/10.1145/321850.321860 [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.
http://dx.doi.org/10.1016/j.amc.2009.03.053 [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. http://dx.doi.org/10.1016/j.amc.2010.04.058 [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.
http://dx.doi.org/10.1016/j.amc.2010.06.043 [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.
http://dx.doi.org/10.1016/j.cam.2007.05.008
[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. http://dx.doi.org/10.1145/321850.321860 [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.
http://dx.doi.org/10.1016/j.amc.2009.03.053 [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. http://dx.doi.org/10.1016/j.amc.2010.04.058 [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.
http://dx.doi.org/10.1016/j.amc.2010.06.043 [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.
http://dx.doi.org/10.1016/j.cam.2007.05.008