# A two-point eighth-order method based on the weight function for solving nonlinear equations

## Keywords:

Method with memory, Accelerator parameter, Weight function, Newton’s interpolatory polynomial, Order of convergence, nonlinear equations in R, iterative method## Abstract

In this work, we have designed a family of with-memory methods with eighth-order convergence. We have used the weight function technique. The proposed methods have three parameters. Three self-accelerating parameters are calculated in each iterative step employing only information from the current and all previous iteration. Numerical experiments are carried out to demonstrate the convergence and the e?ciency of our iterative method.

### Downloads

## References

I.K. Argyros,Approximating solutions of equations using Newton’s method with a mod-ified Newton’s method iterate as a starting point, J. Numer. Anal. Approx. Theory,36(2)(2007), pp. 123–138.

P. Bassiri, P. Bakhtiari, S. Abbasbandy,A new iterative with memory class forsolving nonlinear equations, Int. J. Ind. Math.8(3) (2016) , pp. 225–229.

R. Behl, S.S. Motsa,Higher order two-point efficient family of Hal ley type methodsfor simple roots, Intl. J. Comput. Meth.13(2016) 4, pp. 1–16.

N. Choubey, B. Panday, J.P. Jaiswal,Several two-point with memory iterative meth-ods for solving nonlinear equations, Afr. Mat. (2018), pp. 1–15.

C. Chun,Some fourth-order iterative methods for solving nonlinear equationss, Appl.Math. Comput.,195(2008), pp. 454–459.

A. Cordero, T. Lotfi, K. Mahdiani, J.R. Torregrosa,Two optimal general classesof iterative methods with eighth-order, Acta Appl. Math.,134(1) (2014), pp. 61–74.

A. Cordero, T. Lotfi, J.R. Torregrosa, P. Assari, K. Mahdiani,Some new bi-accelarator two-point methods for solving nonlinear equations, Comput. Appl. Math.35(2016) pp. 251–267.

E. Catinas,A survey on the convergence orders and the computational convergenceorders of sequences, Appl. Math. Comput.343(2019), pp. 1–20.

T. Eftekhari,An efficient class of multipoint root-solvers with and without memory fornonlinear equations, Acta. Math. Viet. 2015 (2015) pp. 1–13.

Y. H. Geum, Y. I. Kim,A biparametric family optimal ly convergent sixteenth-ordermultipoint methods with their fourth-step weighting function as a sum of a rational and ageneric two-variable function, J. Comput. Appl. Math.235(2011), pp. 3178–3188.

J. P. Jaiswal,Two efficient bi-parametric derivative free with memory methods forfinding simple roots of nonlinear equations, J. Adv. Appl. Math,1(2016) 4, pp. 203–210.

H.T. Kung, J.F. Traub,Optimal order of one-point and multipoint iteration, J. Assoc.Comput. Mach.21(1974) 4, pp. 643–651.

T. Lotfi, K. Mahdiani, P. Bakhtiari, F. Soleymani,Constructing two-step itera-tive methods with and without memory, Comput. Math. Math. Phys,55(2) (2015), pp. 183–193.

T. Lotfi, F. Soleymani, S. Sharifi, S. Shateyi, F.K. Haghani,Multipoint iterativemethods for finding al l the simple zeros in an interval, J. Appl. Math. 2014, pp. (1–14)

T. Lotfi, P. Assari,A new two-step class of methods with memory for solving non-linear equations with high efficiency index, Int. J. Math. Model. Comput.4(3) (2014),pp. 277–288.

T. Lotfi, F. Soleymani, M. Ghorbanzadeh, P. Assari,On the cunstruction of sometri-parametric iterative methods with memory, Numer. Algor.,70(4) (2015), pp. 835–845

T. Lotfi, F. Soleymani, Z. Noori, A. Kilicman, F. Khaksar Haghani,Efficientiterative methods with and without memory possessing high efficiency indices, Disc. Dyna.Natu. Soci. 2014 (2014), pp. 1–9.

A.K. Maheshwari,A fourth order iterative method for solving nonlinear equations,Appl. Math. Comput.211(2009, pp. 383–391.

M. Mohamadi Zadeh,T. Lotfi, M. Amirfakhrian,Developing two efficient adaptiveNewton-type methods with memory, Math. Meth. Appl. Sci. 2018 (2018), pp. 1–9.

B. Neta,A new family of higher order methods for solving equations, Int. J. Comput.Math.14(1983), pp. 191–195.

J.M. Ortega, W.G. Rheinboldt,Iterative Solutions of Nonlinear Equations in Sev-eral Variables, Academic Press, New York, 1970.

A.M. Ostrowski,Solution of Equations and Systems of Equations, Academic Press,New York, 1960.

M.S. Petkovic, B. Neta, L.D. Petkovic, J. Dzunic,Multipoint Methods for SolvingNonlinear Equations, Elsevier, Amsterdam, 2013.

M.S. Petkovic, S. Ilic, J. Dzunic,Derivative free two-point methods with and withoutmemory for solving nonlinear equations, Appl. Math. Comput.,217(2010), pp. 1887–1895.

S. Sharifi, M. Salimi, S. Siegmund, T. Lotfi,A new class of optimal four-pointmethods with convergence order 16 for solving nonlinear equations, Math. Comput. Simu.2015 (2015), pp. 1–26.

F. Soleymani,On a bi-parametric classes of optimal eight-order derivative-free meth-ods, Int. J. Pure Appl. Math. bf 72 (1) (2011), pp. 27–37.

F. Soleymani, T. Lotfi, E. Tavakoli, F. Khaksar Haghani,Several iterative meth-ods with memory using self-accelerators, Appl. Math. Comput.254(2015), pp. 452–458.

F. Soleymani,Some optimal iterative methods and their with memory variants, J. Egypt.Math. Soc. 2013 (2013) 1-9.

V. Torkashvand, T. Lotfi, M.A. Fariborzi Araghi,A new family of adaptivemethods with memory for solving nonlinear equations, Math. Sci.13(2019), pp. 1–20.

V. Torkashvand, M. Kazemi,On an efficient family with memory with high order ofconvergence for solving nonlinear equations, Int. J. Ind. Math.12(2) (2020), pp. 209–224.

J.F. Traub,Iterative Methods for the Solution of Equations, Prentice Hall, New York,USA, 1964.

X. Wang,A new accelerating technique applied to a variant of Cordero-Torregrosamethod, J. Comput. Appl. Math.330(2018), pp. 695–709.

X. Wang, T. Zhang,High-order Newton-type iterative methods with memory for solv-ing nonlinear equations, Math. Commun.19(2014), pp. 91–109.

T. Zhanlav, O. Chuluunbaatar, V. Ulziibayar,Accelerating the convergence ofNewton-type iterations, J. Numer. Anal. Approx. Theory,46(2) (2017), pp. 162–180

## Downloads

## Published

## How to Cite

*J. Numer. Anal. Approx. Theory*,

*50*(1), 73–93. Retrieved from https://ictp.acad.ro/jnaat/journal/article/view/1230

## Issue

## Section

## License

**Open Access. **This article is distributed under the terms of the Creative Commons Attribution 4.0 International License, which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.