On the development and extensions of some classes of optimal three-point iterations for solving nonlinear equations

Tugal Zhanlav; Otgondorj Khuder

doi:10.33993/jnaat502-1238

Authors

Tugal Zhanlav Institute of Mathematics and Digital Technology, Mongolian Academy of Sciences, Mongolia
Otgondorj Khuder School of Applied Sciences, Mongolian University of Science and Technology, Mongolia

DOI:

https://doi.org/10.33993/jnaat502-1238

Keywords:

nonlinear equations in R, order of convergence, multipoint methods

Abstract views: 194

Abstract

We develop a new families of optimal eight--order methods for solving nonlinear equations. We also extend some classes of optimal methods for any suitable choice of iteration parameter. Such development and extension was made using sufficient convergence conditions given in [14]. Numerical examples are considered to check the convergence order of new families and extensions of some well-known methods.

Downloads

Download data is not yet available.

References

M.S. Petkovic, B. Neta, L.D. Petković, J. ǅunić, Multipoint methods for solving nonlinear equations. Elsevier (2013).

X. Wang, L. Liu, Modified Ostrowski’s method with eighth-order convergence and high efficiency index, Appl. Math. Lett. 23 (2010) 549–554, https://doi.org/10.1016/j.aml.2010.01.009

R. Thukral, M.S. Petković, Family of three-point methods of optimal order for solving nonlinear equations, J. Comput. Appl. Math. 233 (2010) 2278–2284, http://doi.org/10.1016/j.cam.2009.10.012

W. Bi, H. Ren, Q. Wu, Three-step iterative methods with eight-order convergence for solving nonlinear equations, J. Comput. Appl. Math. 225 (2009) 105–112, https://doi.org/10.1016/j.cam.2008.07.004

T. Lotfi, S. Sharifi, M. Salimi, S. Siegmund, New class of three-point methods with optimal convergence order eight and its dynamics, Numer. Algor. 68 (2015) 261–288, https://doi.org/10.1007/s11075-014-9843-y

A. Cordero, M. Fardi, M. Ghasemi, J.R. Torregrosa, Accelerated iterative methods for finding solutions of nonlinear equations and their dynamical behavior, Calcolo 51 (2014) 17–30, http://doi.org/10.1007/s10092-012-0073-1

J.R. Sharma, R. Sharma, A new family of modified Ostrowski’s method with accelerated eighth–order convergence, Numer. Algorithms, 54 (2010) 445–458, https://doi.org/10.1007/s11075-009-9345-5

C. Chun, M.Y. Lee, A new optimal eighth-order family of iterative methods for the solution of nonlinear equations, Appl. Math. Comput. 223 (2013), 506–519, https://doi.org/10.1016/j.amc.2013.08.033

W. Bi, Q. Wu, H. Ren, A new family of eighth–order iterative methods for solving nonlinear equations, Appl. Math. Comput. 214 (2009) 236–245, https://doi.org/10.1016/j.amc.2009.03.077

J. R. Sharma, H. Arora, A new family of optimal eighth order methods with dynamics for nonlinear equations. Appl. Math. Comput. 273 (2016) 924–933, https://doi.org/10.1016/j.amc.2015.10.049

Y.H. Geum, Y.I. Kim, A uniparamtric family of three-step eighth-order multipoint iterative methods for simple roots. Appl. Math. Lett. 24 (2011) 929–935, https://doi.org/10.1016/j.aml.2011.01.002

J. R. Sharma, H. Arora, An efficient family of weighted-Newton methods with optimal eighth order convergence, Appl. Math. Lett. 29 (2014) 1–6, https://doi.org/10.1016/j.aml.2013.10.002

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

T. Zhanlav, V. Ulziibayar, O. Chuluunbaatar, The necessary and sufficient conditions for two and three-point iterative method of Newton’s type, Comput. Math. Math. Phys. 57: 1093–1102. http://doi.org/10.1134/S0965542517070120

T. Zhanlav, O. Chuluunbaatar, V. Ulziibayar, Generating function method for constructing new iterations, Appl. Math. Comput. 315 (2017) 414–423, https://doi.org/10.1016/j.amc.2017.07.078

T. Zhanlav, V. Ulziibayar, Modified King’s methods with optimal eighth-order of convergence and high efficiency index, Amer. J. Comput. Appl. Math.6 (2016) 177–181, http://doi.org/10.5923/j.ajcam.20160605.01

T. Zhanlav, Kh. Otgondorj, A new family of optimal eighth-order methods for solving nonlinear equations, Amer. J. Comput. Appl. Math. 8 (2018) 15–19., http://doi.org/10.5923/j.ajcam.20180801.02

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

R. Sharma, A. Bahl, Optimal eighth order convergent iteration scheme based on Lagrange interpolation, Acta Mathematicae Applicatae Sinica, English Series 33 (2017) 1093–1102, https://doi.org/10.1007/s10255-017-0722-x

R. Thukral, Eighth-order iterative Methods without derivatives for solving nonlinear equations, ISRN. Appl. Math. Vol (2011), Article ID 693787, 12 pages. https://doi.org/10.5402/2011/693787

S.K. Khattri, R.P. Agarwal, Derivative-free optimal iterative methods, Comput. Methods. Appl. Math. 10 (2010) 368–375, http://doi.org/10.2478/cmam-2010-0022