Sixth-Order Two-Point Efficient Family of Super-Halley Type Methods

The main focus of this manuscript is to provide a
highly efficient two-point sixth-order family of super-Halley type
methods that do not require any second-order derivative evaluation
for obtaining simple roots of nonlinear equations, numerically. Each
member of the proposed family requires two evaluations of the given
function and two evaluations of the first-order derivative per iteration.
By using Mathematica-9 with its high precision compatibility, a
variety of concrete numerical experiments and relevant results are
extensively treated to confirm t he t heoretical d evelopment. From
their basins of attraction, it has been observed that the proposed
methods have better stability and robustness as compared to the other
sixth-order methods available in the literature.

Authors:

• Ramandeep Behl
• S. S. Motsa

Keywords:

• Basins of attraction
• nonlinear equations
• simple roots
• Super-Halley.

References:

[1] J. F. Traub, Iterative methods for the solution of equations, Prentice-Hall,
Englewood Cliffs, 1964.
[2] M. S. Petkovi´c, B. Neta, L. D. Petkovi´c, and J. D˘zuni´c, Multipoint
methods for solving nonlinear equations, Academic Press, 2012.
[3] J. R. Sharma, R. K. Guha, A family of modified Ostrowski methods with
accelerated sixth-order convergence, Appl. Math. Comput. 190 (2007),
111–115.
[4] C. Chun, Some improvements of Jarratt’s method with sixth-order
convergence, Appl. Math. Comput. 190 (2007), 1432–1437.
[5] X. Wang, L. Liu, Two new families of sixth-order methods for solving
non-linear equations, Appl. Math. Comput. 213 (2009), 73–78.
[6] Changbum Chun, YoonMee Ham, Some sixth-order variants of Ostrowski
root-finding methods, Appl. Math. Comput. 193 (2007), 389–394
[7] Xiuhua Wang, Jisheng Kou, Yitian Li, A variant of Jarratt method with
sixth-order convergence, Appl. Math. Comput. 204 (2008) 14–19.
[8] S.K. Parhi, D. K. Gupta, A sixth order method for nonlinear equations,
Appl. Math. Comput. 203 (2008), 50–55.
[9] V. Kanwar, R. Behl, Kapil K. Sharma, Simply constructed family of a
Ostrowski’s method with optimal order of convergence, Comput. Math.
Appl. 62 (2011), 4021–4027.
[10] R. Behl, V. Kanwar, K. K. Sharma, Optimal equi-scaled families of
Jarratt’s method, Int. J. Comput. Math. 90 (2013), 408–422.
[11] Y. H. Geum, Y. I. Kim, B. Neta, On developing a higher-order family of
double-Newton methods with a bivariate weighting function Appl. Math. Comput. 254 (2015), 277–290.
[12] B. Neta, M. Scot, C. Chun, Basins of attraction for several methods
to find simple roots of nonlinear equations, Appl. Math. Comput. 218
(2012), 10548–10556. 2012.
[13] R. Behl and S. S. Motsa, Geometric Construction of Eighth-Order
Optimal Families of Ostrowskis Method, Sci. Wor. J. Article ID 614612.