An optimal three-point eighth-order iterative method without memory for solving nonlinear equations with its dynamics
- 212 Downloads
We present a three-point iterative method without memory for solving nonlinear equations in one variable. The proposed method provides convergence order eight with four function evaluations per iteration. Hence, it possesses a very high computational efficiency and supports Kung–Traub’s conjecture. The construction, the convergence analysis, and the numerical implementation of the method will be presented. Using several test problems, the proposed method will be compared with existing methods of convergence order eight concerning accuracy and basins of attraction. Furthermore, some measures are used to judge methods with respect to their performance in finding the basins of attraction.
KeywordsOptimal multi-point iterative methods Simple root Order of convergence Kung–Traub’s conjecture Basins of attraction
Mathematics Subject Classification65H05 37F10
The research of the fourth author is supported by Grant MTM2015-65888-C4-4-P (MINECO/FEDER).
- 1.Amat, S., Busquier, S., Magreñán, Á.A.: Reducing chaos and bifurcations in Newton-type methods. Abstr. Appl. Anal. 2013, Art. ID 726701 (2013)Google Scholar
- 9.Ferrara, M., Sharifi, S., Salimi, M.: Computing multiple zeros by using a parameter in Newton–Secant method. SeMA J. (2016). doi: 10.1007/s40324-016-0074-0
- 22.Salimi, M., Lotfi, T., Sharifi, S., Siegmund, S.: Optimal Newton–Secant like methods without memory for solving nonlinear equations with its dynamics. Int. J. Comput. Math. (2016). doi: 10.1080/00207160.2016.1227800
- 27.Stewart, B.D.: Attractor Basins of Various Root-Finding Methods. M.S. thesis, Naval Postgraduate School, Monterey, CA (2001)Google Scholar