ABSTRACT In this paper, seven self-accelerating iterative
methods with memory are derived from an optimal two-step Steffensen-type method
without memory for solving nonlinear equations, their orders of convergence are
proved to be increased, numerical examples are demonstrat-ed demonstrated to verify the theoretical results, and applications for solving systems of
nonlinear equations and BVPs of nonlinear ODEs are illustrated.
Cite this paper
nullZheng, Q. , Guo, X. and Huang, F. (2012) Self-accelerating two-step Steffensen-type methods with memory and their applications on the solution of nonlinear BVPs. Open Journal of Applied Sciences, 2, 70-73. doi: 10.4236/ojapps.2012.24B017.
 J.M. Ortega, W.G. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, 1970.
 H.T. Kung, J.F. Traub, Optimal order of one-point and multipoint iteration, J. Assoc. Comput. Math. 21 (1974) 634-651.
 J.F. Traub, Iterative Methods for the Solution of Equations, Prentice-Hall, Englewood Cliffs, New Jersey, 1964.
 V. Alarc?n, S. Amat, S. Busquier, D. J. L?pez, A Stef-fensen’s type method in Banach spaces with applications on boundary-value problems, J. Comput. Appl. Math. 216 (2008) 243-250.
 W. Bi, H. Ren, Q. Wu, A class of two-step Steffensen type methods with fourth-order convergence, Appl. Math. Comput. 209 (2009) 206-210.
 F. Soleymani, S.K. Vanani, Optimal Steffensen-type meth-ods with eighth order of convergence, Comput. Math. Appl. 62 (2011) 4619-4626.
 Q. Zheng, J. Li, F. Huang, An optimal Steffensen-type family for solving nonlinear equations, Appl. Math. Comput. 217 (2011) 9592-9597.
 Q. Zheng, J. Wang, P. Zhao, L. Zhang, A Steffensen-like method and its higher-order variants, Appl. Math. Comput. 214 (2009) 10-16.
 Q. Zheng, P. Zhao, L. Zhang, W. Ma, Variants of Steffensen-secant method and applications, Appl. Math. Comput. 216 (2010) 3486-3496.
 M.S. Petkovi?, S. Ili?, J. D?uni?, Derivative free two-point methods with and without memory for solving nonlinear equations, Appl. Math. Comput. 217 (2010) 1887-1895.