Local Convergence Analysis of an Efficient Fourth Order Weighted-Newton Method under Weak Conditions

Ioannis K. Argyros; Santhosh George

doi:10.2478/awutm-2018-0002

.blurhash-client-img { display: none !important; }

Local Convergence Analysis of an Efficient Fourth Order Weighted-Newton Method under Weak Conditions

Annals of West University of Timisoara - Mathematics and Computer Science

Volume 56 (2018): Issue 1 (July 2018)

By: Ioannis K. Argyros and Santhosh George

Open Access

|Dec 2018

Abstract

Local convergence analysis of a fourth order method considered by Sharma et. al in [19] for solving systems of nonlinear equations. Using conditions on derivatives upto the order five, they proved that the method is of order four. In this study using conditions only on the first derivative, we prove the convergence of the method in [19]. This way we extended the applicability of the method. Numerical example which do not satisfy earlier conditions but satisfy our conditions are presented in this study.

References

[1] S. Amat, S. Busquier, and M. Negra, Adaptive approximation of nonlinear operators, Numer. Funct. Anal. Optim.25, (2004), 397–40510.1081/NFA-200042628
Search in Google Scholar Back to article
[2] I.K. Argyros, Computational theory of iterative methods, Series: Studies in Computational Mathematics, 15, Editors: C.K. Chui and L. Wuytack, Elsevier Publ. Co. New York, U.S.A, (2007)
Search in Google Scholar Back to article
[3] I.K. Argyros and H. Ren, Improved local analysis for certain class of iterative methods with cubic convergence, Numerical Algorithms, 59, (2012), 505–52110.1007/s11075-011-9501-6
Search in Google Scholar Back to article
[4] I. K. Argyros, Yeol Je Cho, and S. George, Local convergence for some third-order iterative methods under weak conditions, J. Korean Math. Soc.53 (4), (2016), 781–79310.4134/JKMS.j150244
Search in Google Scholar Back to article
[5] I. K. Argyros and S. George, Ball convergence of a sixth order iterative method with one parameter for solving equations under weak conditions, ISSN 0008-0624, Calcolo, DOI 10.1007/s10092-015-0163-y
Search in Google Scholar Back to article
[6] I.K. Argyros and A. A. Magreñán, Local convergence analysis of proximal Gauss-Newton method for penalized nonlinear least square problems, Appl. Math. Comput. V.241, 15, (2014), 401–40810.1016/j.amc.2014.04.087
Search in Google Scholar Back to article
[7] I.K. Argyros and F. Szidarovszky, The theory and Applications of Iterative methods, CRC Press, Boca Raton Florida, USA, (1993)
Search in Google Scholar Back to article
[8] A. Cordero, J. Hueso, E. Martinez, and J.R. Torregrosa, A modified Newton-Jarratt’s composition, Numer. Algor.55, (2010), 87–9910.1007/s11075-009-9359-z
Search in Google Scholar Back to article
[9] A. Cordero and J.R. Torregrosa, Variants of Newton’s method for functions of several variables, Appl.Math. Comput.183, (2006), 199–20810.1016/j.amc.2006.05.062
Search in Google Scholar Back to article
[10] A. Cordero and J.R. Torregrosa, Variants of Newton’s method using fifth order quadrature formulas, Appl.Math. Comput.190, (2007), 686-69810.1016/j.amc.2007.01.062
Search in Google Scholar Back to article
[11] J.M. Gutiérrez and M.A. Hernández, Newton’s method under weak Kantorovich conditions, IMA J. Numer. Anal.20, (2000), 521–53210.1093/imanum/20.4.521
Search in Google Scholar Back to article
[12] G.M Grau-Sanchez, A. Grau, and M. Noguera, On the computational efficiency index and some iterative methods for solving systems of non-linear equations, J. Comput. Appl Math.236, (2011), 1259–126610.1016/j.cam.2011.08.008
Search in Google Scholar Back to article
[13] H.H. Homeier, A modified Newton method with cubic convergence, the multivariable case, J.Comput.Appl.Math.169, (2004), 161–16910.1016/j.cam.2003.12.041
Search in Google Scholar Back to article
[14] H.H. Homeier, On Newton type methods with cubic convergence, J.Comput.Appl.Math.176, (2005), 425–43210.1016/j.cam.2004.07.027
Search in Google Scholar Back to article
[15] J.S Kou, Y.T. Li, and X.H. Wang, A modification of Newton method with fifth-order convergence, J. Comput. Appl. Math.209, (2007), 146–15210.1016/j.cam.2006.10.072
Search in Google Scholar Back to article
[16] M.A. Noor and M. Waseem, Some iterative methods for solving a system of nonlinear equations, Computers and Mathematics with Applications, 57, (2009), 101–10610.1016/j.camwa.2008.10.067
Search in Google Scholar Back to article
[17] A.N. Romero, J.A. Ezquerro, and M.A. Hernandez, Approximacion de soluciones de algunas equacuaciones integrals de Hammerstein mediante metodos iterativos tipo. Newton, XXI Congresode ecuaciones diferenciales y aplicaciones Universidad de Castilla-La Mancha, (2009)
Search in Google Scholar Back to article
[18] W.C. Rheinboldt, An adaptive continuation process for solving systems of nonlinear equations, Mathematical models and numerical methods (A.N.Tikhonov et al. eds.) pub.3, (1977), 129–14210.4064/-3-1-129-142
Search in Google Scholar Back to article
[19] J.R. Sharma, R.K. Guha, and R. Sharma, An efficient fourth order weighted-Newton method for systems of nonlinear equations, Numer. Algor.62, (2013), 307–32310.1007/s11075-012-9585-7
Search in Google Scholar Back to article
[20] J.R. Sharma and P.K Gupta, An efficient fifth order method for solving systems of nonlinear equations, Comput. Math. Appl.67, (2014), 591–60110.1016/j.camwa.2013.12.004
Search in Google Scholar Back to article
[21] F.A. Shah and M.A Noor, Some numerical methods for solving nonlinear equations by using decomposition technique, Appl. Math. Comput.251, (2015), 378–38610.1016/j.amc.2014.11.065
Search in Google Scholar Back to article
[22] J.F. Traub, Iterative methods for the solution of equations, AMS Chelsea Publishing, (1982)
Search in Google Scholar Back to article

Articles in this issue

DOI: https://doi.org/10.2478/awutm-2018-0002 | Journal eISSN: 1841-3307 | Journal ISSN: 1841-3293

Journal RSS Feed

Language: English

Page range: 23 - 34

Submitted on: Sep 20, 2016

Accepted on: Nov 4, 2017

Published on: Dec 7, 2018

Published by: West University of Timisoara

In partnership with: Paradigm Publishing Services

Publication frequency: Volume open

Keywords:

local convergence,

weighted Newton method,

Fréchet derivative

Related subjects:

Mathematics,

General mathematics

© 2018 Ioannis K. Argyros, Santhosh George, published by West University of Timisoara
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Volume 56 (2018): Issue 1 (July 2018)