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Optimal Steffensen-type methods with eighth order of convergence. (English) Zbl 1236.65056
Summary: We propose two classes of three-step without memory iterations based on the well known second-order method of Steffensen. Per computing step, the methods from the developed classes reach the order of convergence eight using only four evaluations, while they are totally free from derivative evaluation. Hence, they agree with the optimality conjecture of Kung-Traub for providing multi-point iterations without memory. As things develop, numerical examples are employed to support the underlying theory developed for the contributed classes of optimal Steffensen-type eighth-order methods.

65H05Single nonlinear equations (numerical methods)
Full Text: DOI
[1] Sargolzaei, P.; Soleymani, F.: Accurate fourteenth-order methods for solving nonlinear equations, Numerical algorithms (2011) · Zbl 1242.65100
[2] Soleymani, F.; Sharifi, M.: On a general efficient class of four-step root-finding methods, International journal of mathematics and computers in simulation 5, 181-189 (2011)
[3] Iliev, A.; Kyurkchiev, N.: Nontrivial methods in numerical analysis, Selected topics in numerical analysis (2010)
[4] Wang, X.; Liu, L.: New eighth-order iterative methods for solving nonlinear equations, Journal of computational and applied mathematics 234, 1611-1620 (2010) · Zbl 1190.65081 · doi:10.1016/j.cam.2010.03.002
[5] Kung, H. T.; Traub, J. F.: Optimal order of one-point and multipoint iteration, Journal of ACM 21, 643-651 (1974) · Zbl 0289.65023 · doi:10.1145/321850.321860
[6] Soleymani, F.; Sharifi, M.; Mousavi, B. S.: An improvement of Ostrowski’s and King’s techniques with optimal convergence order eight, Journal of optimization theory and applications (2011) · Zbl 1237.90229
[7] F. Soleymani, S. Karimi Vanani, A. Afghani, A general three-step class of optimal iterations for nonlinear equations, Mathematical Problems in Engineering, 10. doi:10.1155/2011/469512. Article ID 469512. · Zbl 1235.74002
[8] Soleymani, F.; Vanani, S. Karimi; Khan, M.; Sharifi, M.: Some modifications of King’s family with optimal eighth order of convergence, Mathematical and computer modelling (2011) · Zbl 1255.65097
[9] Steffensen, J. F.: Remarks on iteration, Skand aktuar tidsr 16, 64-72 (1933) · Zbl 0007.02601
[10] Zheng, Q.; Li, J.; Huang, F.: An optimal Steffensen-type family for solving nonlinear equations, Applied mathematics and computation 217, 9592-9597 (2011) · Zbl 1227.65044 · doi:10.1016/j.amc.2011.04.035
[11] Soleymani, F.: On a bi-parametric class of optimal eighth-order derivative-free methods, International journal of pure and applied mathematics 72, 27-37 (2011) · Zbl 1248.65050
[12] Yun, B. I.: A non-iterative method for solving non-linear equations, Applied mathematics and computation 198, 691-699 (2008) · Zbl 1138.65035 · doi:10.1016/j.amc.2007.09.006