×

An optimized derivative-free form of the Potra-Pták method. (English) Zbl 1255.65096

Summary: We discuss iterative methods for solving univariate nonlinear equations. First of all, we construct a family of methods with optimal convergence rate 4 based upon the Potra-Pták scheme and provide its error equation theoretically. Second, by using this derivative-involved family, a novel derivative-free family of two-step iterations without memory is derived. This derivative-free family agrees with the Kung-Traub conjecture [H. T. Kung and J. F. Traub, J. Assoc. Comput. Mach. 21, 643–651 (1974; Zbl 0289.65023)] for building optimal multi-point iterations without memory, since it is proven that each derivative-free method of the family reaches the convergence rate 4 requiring only three function evaluations per full iteration. Finally, numerical test problems are also provided to confirm the theoretical results.

MSC:

65H05 Numerical computation of solutions to single equations

Citations:

Zbl 0289.65023
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Traub, J. F., Iterative Methods for the Solution of Equations (1982), Chelsea Publishing Company: Chelsea Publishing Company New York · Zbl 0472.65040
[2] Kung, H. T.; Traub, J. F., Optimal order of one-point and multi-point iteration, Journal of the ACM, 21, 643-651 (1974) · Zbl 0289.65023
[3] Yun, B. I., Solving nonlinear equations by a new derivative free iterative method, Applied Mathematics and Computation, 217, 5768-5773 (2011) · Zbl 1229.65084
[4] Soleymani, F., Regarding the accuracy of optimal eighth-order methods, Mathematical and Computer Modelling, 53, 1351-1357 (2011) · Zbl 1217.65089
[5] Potra, F. A.; Pták, V., Nondiscrete introduction and iterative processes, (Research Notes in Mathematics, vol 103 (1984), Pitman: Pitman Boston) · Zbl 0549.41001
[6] D.K.R. Babajee, Analysis of higher order variants of Newton’s method and their applications to differential and integral equations and in ocean acidification, Ph.D. Thesis, University of Mauritius, December 2010.; D.K.R. Babajee, Analysis of higher order variants of Newton’s method and their applications to differential and integral equations and in ocean acidification, Ph.D. Thesis, University of Mauritius, December 2010.
[7] Petkovic, L. D.; Petkovic, M. S., A note on some recent methods for solving nonlinear equations, Applied Mathematics and Computation, 185, 368-374 (2007) · Zbl 1121.65321
[8] 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)
[9] Soleymani, F.; Karimi Vanani, S.; Khan, M.; Sharifi, M., Some modifications of King’s family with optimal eighth order of convergence, Mathematical and Computer Modelling (2011)
[10] Soleymani, F.; Karimi Vanani, S., Optimal Steffensen-type methods with eighth order of convergence, Computers and Mathematics with Applications, 62, 4619-4626 (2011) · Zbl 1236.65056
[11] Soleymani, F.; Khattri, S. K.; Karimi Vanani, S., Two new classes of optimal Jarratt-type fourth-order methods, Applied Mathematics Letters (2011) · Zbl 1239.65030
[12] Iliev, A.; Kyurkchiev, N., Nontrivial Methods in Numerical Analysis (Selected Topics in Numerical Analysis) (2010), Lambert Acad. Publishing
[13] Soleymani, F.; Mousavi, B. S., A novel computational technique for finding simple roots of nonlinear equations, International Journal of Mathematical Analysis, 5, 1813-1819 (2011) · Zbl 1251.65072
[14] Soleymani, F.; Sharifi, M., On a class of fifteenth-order iterative formulas for simple roots, International Electronic Journal of Pure and Applied Mathematics, 3, 245-252 (2011) · Zbl 1413.65187
[15] Sharifi, M.; Babajee, D. K.R.; Soleymani, F., Finding the solution of nonlinear equations by a class of optimal methods, Computers and Mathematics with Applications (2011)
[16] Cordero, A.; Hueso, J. L.; Martínez, E.; Torregrosa, J. R., New modifications of Potra-Pták’s method with optimal fourth and eighth orders of convergence, Journal of Computational and Applied Mathematics, 234, 2969-2976 (2010) · Zbl 1191.65048
[17] Liu, Z.; Zheng, Q.; Zhao, P., A variant of Steffensen’s method of fourth-order convergence and its applications, Applied Mathematics and Computation, 216, 1978-1983 (2010) · Zbl 1208.65064
[18] Ren, H.; Wu, Q.; Bi, W., A class of two-step Steffensen type methods with fourth-order convergence, Applied Mathematics and Computation, 209, 206-210 (2009) · Zbl 1166.65338
[19] Petkovic, M. S.; Ilic, S.; Dzunic, J., Derivative free two-point methods with and without memory for solving nonlinear equations, Applied Mathematics and Computation, 217, 1887-1895 (2010) · Zbl 1200.65034
[20] Peng, Y.; Feng, H.; Li, Q.; Zhang, X., A fourth-order derivative-free algorithm for nonlinear equations, Journal of Computational and Applied Mathematics (2010)
[21] Zheng, Q.; Zhao, P.; Huang, F., A family of fourth-order Steffensen-type methods with the applications on solving nonlinear ODEs, Applied Mathematics and Computation (2011) · Zbl 1223.65034
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.