Some optimal iterative methods and their with memory variants. (English) Zbl 1315.65047

The paper concerns some optimal iterative methods for solving nonlinear equations. Based on some existing works, the author constructs a new method, which needs only 4 function evaluations per step and achieves the optimal 8th-order accuracy. This result is also proved rigorously. Furthermore, by carefully choosing the free parameter in the method, the author obtains a new method with memory and obtains a much better convergence rate, namely, order 10. Numerical results are presented to confirm the theoretical results.


65H05 Numerical computation of solutions to single equations


Full Text: DOI


[1] Sauer, T., Numerical Analysis (2012), Pearson: Pearson Boston
[2] Steffensen, J. F., Remarks on iteration, Skand. Aktuarietidskr, 16, 64-72 (1933) · Zbl 0007.02601
[3] Yun, B. I.; Petkovic, M. S., Iterative methods based on the Signum function approach for solving nonlinear equations, Numer. Algor., 52, 649-662 (2009) · Zbl 1178.65046
[6] Neta, B.; Petkovic, M. S., Construction of optimal order nonlinear solvers using inverse interpolation, Appl. Math. Comput., 217, 2445-2448 (2010) · Zbl 1202.65062
[8] Kung, H. T.; Traub, J. F., Optimal order of one-point and multipoint iteration, J. ACM, 21, 643-651 (1974) · Zbl 0289.65023
[9] Zheng, Q.; Zhao, P.; Li, Z.; Ma, W., Variants of Steffensen-secant method and applications, Appl. Math. Comput., 216, 3486-3496 (2010) · Zbl 1200.65036
[10] Liu, Z.; Zheng, Q.; Zhao, P., A variant of Steffensen’s method of fourth-order convergence and its applications, Appl. Math. Comput., 216, 1978-1983 (2010) · Zbl 1208.65064
[11] Cordero, A.; Hueso, J. L.; Martinez, E.; Torregrosa, J. R., Steffensen type methods for solving nonlinear equations, J. Comput. Appl. Math., 236, 3058-3064 (2012) · Zbl 1237.65049
[12] Li, X.; Mu, C.; Ma, J.; Hou, L., Fifth-order iterative method for finding multiple roots of nonlinear equations, Numer. Algor., 57, 389-398 (2011) · Zbl 1222.65047
[13] Schroder, E., Uber unendlich viele algorithmen zur Auflösung der Gleichungen, Math. Ann., 2, 317-365 (1870)
[14] Traub, J. F., Iterative Methods for the Solution of Equations (1964), Prentice Hall: Prentice Hall New York · Zbl 0121.11204
[16] Soleymani, F.; Khattri, S. K.; Karimi Vanani, S., Two new classes of optimal Jarratt-type fourth-order methods, Appl. Math. Lett., 25, 847-853 (2012) · Zbl 1239.65030
[17] Soleymani, F.; Karimi Vanani, S., Optimal Steffensen-type methods with eighth order of convergence, Comput. Math. Appl., 62, 4619-4626 (2011) · Zbl 1236.65056
[18] Wagon, S., Mathematica in Action (2010), Springer
[19] Trott, M., The Mathematica GuideBook for Numerics (2006), Springer · Zbl 1101.65001
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.