Comparison  of some optimal derivative-free three-point iterations

Thugal Zhanlav; Khuder Otgondorj

doi:10.33993/jnaat491-1179

Authors

Thugal Zhanlav Mongolian University of Science and Technology, Mongolia
Khuder Otgondorj School of Applied Sciences, Mongolian University of Science and Technology, Mongolia

DOI:

https://doi.org/10.33993/jnaat491-1179

Keywords:

nonlinear equations in R, derivative-free methods, optimal three-point iterative methods

Abstract views: 393

Abstract

We show that the well-known Khattri et al. methods and Zheng et al. methods are identical. In passing, we propose a suitable calculation formula for Khattri et al. methods. We also show that the families of eighth-order derivative-free methods obtained in [8] include some existing methods, among them the above-mentioned ones as particular cases. We also give the sufficient convergence condition of these families. Numerical examples and comparison with some existing methods were made. In addition, the dynamical behavior of methods of these families is analyzed.

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References

I.K. Argyros, M. Kansal, V. Kanwar, S. Bajaj, Higher-order derivative–free families of Chebyshev–Hal ley type methods with or without memory for solving nonlinear equations, Appl. Math. Comput., 315 (2017), pp. 224- 245, https://doi.org/10.1016/j.amc.2017.07.051 DOI: https://doi.org/10.1016/j.amc.2017.07.051

R. Behl, D. Gonzalez, P. Maroju, S.S. Motsa, An optimal and efficient general eighth-order derivative–free scheme for simple roots, J. Comput. Appl. Math., 330 (2018), pp. 666-675. https://doi.org/10.1016/j.cam.2017.07.036 DOI: https://doi.org/10.1016/j.cam.2017.07.036

A. Cordero, J.L. Hueso E. Martinez, J.R. Torregrosa, A new technique to obtain derivative–free optimal iterative methods for solving nonlinear equations, J. Comput. Appl. Math., 252 (2013), pp. 95-102. https://doi.org/10.1016/j.cam.2012.03.030 DOI: https://doi.org/10.1016/j.cam.2012.03.030

C. Chun, B. Neta, Comparative study of eighth-order methods for finding simple roots of nonlinear equations, Numer. Algor., 74 (2017), pp. 1169-1201. https://doi.org/10.1007/s11075-016-0191-y DOI: https://doi.org/10.1007/s11075-016-0191-y

S.K. Khattri,T. Steihaug, Algorithm for forming derivative–free optimal methods, Numer. Algor., 65 (2014), pp. 809-824 https://doi.org/10.1007/s11075-013-9715-x DOI: https://doi.org/10.1007/s11075-013-9715-x

T. Lotfi, F. Soleymani, M. Ghorbanzadeh, P. Assari, On the construction of some triparametric iterative methods with memory, Numer. Algor., 70 ( 2015), pp. 835-845. https://doi.org/10.1007/s11075-015-9976-7 DOI: https://doi.org/10.1007/s11075-015-9976-7

S. Sharifi, S. Siegmund, M. Salimi, Solving nonlinear equations by a derivative–free form of the King’s family with memory, Calcolo, 53 (2016), pp. 201-215. https://doi.org/10.1007/s10092-015-0144-1 DOI: https://doi.org/10.1007/s10092-015-0144-1

M. Petkovic, B. Neta, L. Petkovic J. Dzunic, Multipoint Methods for Solving Nonlinear Equations, Elsevier, 2013.

J.R. Sharma, R.K. Guha, P. Gupta, Some efficient derivative free methods with memory for solving nonlinear equations, Appl. Math. Comput., 219 (2012), pp. 699-707. https://doi.org/10.1016/j.amc.2012.06.062 DOI: https://doi.org/10.1016/j.amc.2012.06.062

F. Soleymani, S. Shateyi, Two optimal eighth-order derivative–free classes of iterative methods, Abstr. Appl. Anal., 2012, ID 318165, pp. 1-14. http://doi.org/10.1155/2012/318165 DOI: https://doi.org/10.1155/2012/318165

F. Soleymani, S.K. Vanani, Optimal Steffensen–type methods with eighth order of convergence, Comput. Math. Appl., 62 (2011), pp. 4619-4626. https://doi.org/10.1016/j.camwa.2011.10.047 DOI: https://doi.org/10.1016/j.camwa.2011.10.047

R. Thukral, Eighth–Order iterative Methods without derivatives for solving nonlinear equations, ISRN. Appl. Math., 2011, ID 693787, pp. 1–12. https://doi.org/10.5402/2011/693787. DOI: https://doi.org/10.5402/2011/693787

T. Zhanlav, O. Chuluunbaatar, Kh. Otgondorj, A derivative–free families of optimal two–and three–point iterative methods for solving nonlinear equations, Comput. Math. Math. Phys., 59 (2019), pp. 920–936. DOI: https://doi.org/10.1134/S0965542519060149

Q. Zheng, J. Li, F. Huang, An optimal Steffensen–type family for solving nonlinear equations, Appl. Math. Comput., 217 (2011), pp. 9592–9597. https://doi.org/10.1016/j.amc.2011.04.035 DOI: https://doi.org/10.1016/j.amc.2011.04.035

T. Zhanlav, O. Chuluunbaatar, G. Ankhbayar, Relationship between inexact Newton method and the continuous analogy of Newton’s method, J. Numer. Anal. Approx. Theory, 40 (2011) no. 2, pp. 182-189, https://ictp.acad.ro/jnaat/journal/article/view/2011-vol40-no2-art8

R.W. Hamming, Numerical methods for scientist and engineers, McGraw–Hill, New–York, 1962.

H.T. Kung, J.F. Traub, Optimal order of one–point and multi–point iteration, J. Assoc. Comput. Math., 21(1974), pp. 643-651. https://doi.org/10.1145/321850.321860 DOI: https://doi.org/10.1145/321850.321860

H. Veiseh, T. Lotfi, T. Allahviranloo,A study on the local convergence and dynamics of the two-step and derivative-free Kung–Traub’s method, Comp. Appl. Math., 37 (2018), pp. 2428–2444. https://doi.org/10.1007/s40314-017-0458-5 DOI: https://doi.org/10.1007/s40314-017-0458-5

E. Catinas, A survey on the high convergence orders and computational convergence orders of sequences, Appl. Math. Comput., 343 (2019), pp. 1-20. https://doi.org/10.1016/j.amc.2018.08.006 DOI: https://doi.org/10.1016/j.amc.2018.08.006

F. A. Potra, Nondiscrete induction and iterative processes, Pitman, London, 1984.

J.M. Ortega, W.C. Rheinboldt, Iterative Solutions of Nonlinear Equations in Several Variables, Academic Press, New York, 1970.