A unified approach to the construction of higher-order derivative-free iterative methods for solving systems of nonlinear equations
DOI:
https://doi.org/10.5564/pmas.v64i02.3649Keywords:
Nonlinear systems, Higher order methods, Derivative-free methods, Order of convergenceAbstract
In this article, we introduce a unified approach to constructing a higher-order derivative-free scheme based on the approximations of F'(zk)-1. A family of order p=6,7 derivative-free method is proposed and compared to some well-known methods. The necessary and sufficient condition for p-th order of convergence are given in terms of parameter matrices τ(k) and α(k) . Some good choices of and are offered. Numerical experiments were carried out to confirm the theoretical results.
Downloads
679
References
F. Ahmad, F. Soleymani, F. K. Haghani, S. Serra-Capizzano, Higher order derivative-free iterative methods with and without memory for systems of nonlinear equations, Appl. Math. Comput., 314 (2017) 199-211. https://doi.org/10.1016/j.amc.2017.07.012
A. Amiri, A. Cordero, M. T. Darvishi, J. R. Torregrosa, A fast algorithm to solve systems of nonlinear equations, J. Comput. App. Math., 354 (2019) 242–258. https://doi.org/10.1016/j.cam.2018.03.048
M. Grau-Sánchez, M. Noguera , S. Amat, On the approximation of derivatives using divided difference operators preserving the local convergence order of iterative methods, J. Comput. App. Math., 237 (2013) 363–372. https://doi.org/10.1016/j.cam.2012.06.005
S. Bhalla, S. Kumar, I. K. Argyros, R. Behl, A family of higher order derivative free methods for nonlinear systems with local convergence analysis, Comp. Appl. Math., 37 (2018) 5807–5828. https://doi.org/10.1007/s40314-018-0663-x
Z. Liu, Q. Zheng, P. Zhao, A variant of Steffensen’s method of fourth-order convergence and its applications, Appl. Math. Comput., 216 (2010) 1978–1983. https://doi.org/10.1016/j.amc.2010.03.028
M. Narang, S. Bhatia, V. Kanwar, New efficient derivative free family of seventh-order methods for solving systems of nonlinear equations, Numer. Algor., 76 (2017) 283–307. https://doi.org/10.1007/s11075-016-0254-0
M. Narang, S. Bhatia, A. S. Alshomrani, V. Kanwar, General efficient class of Steffensen type methods with memory for solving systems of nonlinear equations, J. Comput. App. Math., 352 (2019) 23–39. https://doi.org/10.1016/j.cam.2018.10.048
M. S. Petković, J. R. Sharma, On some efficient derivative-free iterative methods with memory for solving systems of nonlinear equations, Numer. Algor., 71 (2016) 457–474. https://doi.org/10.1007/s11075-015-0003-9
J. R. Sharma, H. Arora, M. S. Petković, An efficient derivative free family of fourth order methods for solving systems of nonlinear equations, Appl. Math. Comput., 235 (2014) 383–393. https://doi.org/10.1016/j.amc.2014.02.103
J. R. Sharma and H. Arora, Efficient derivative-free numerical methods for solving systems of nonlinear equations, Comput. Appl. Math., 35 (2016) 269–284. https://doi.org/10.1007/s40314-014-0193-0
J. R. Sharma and H. Arora, An efficient derivative free iterative method for solving systems of nonlinear equations, Appl. Anal. Disc. Math., 7 (2013) 390–403. https://doi.org/10.2298/AADM130725016S
J. R. Sharma, H. Arora, A novel derivative free algorithm with seventh order convergence for solving systems of nonlinear equations, Numer. Algor., 67 (2014) 917–933. https://doi.org/10.1007/s11075-014-9832-1
J. R. Sharma, H. Arora, Efficient higher order derivative-free multipoint methods with and without memory for systems of nonlinear equations, Int. J. Comput. Math., 95 (2018) 920–938. https://doi.org/10.1080/00207160.2017.1298747
J. R. Sharma, H. Arora, A simple yet efficient derivative free family of seventh order methods for systems of nonlinear equations, SeMA, 73 (2016) 59–75. https://doi.org/10.1007/s40324-015-0055-8
X. Wang, X. Fan, Two Efficient Derivative-Free Iterative Methods for Solving Nonlinear Systems, Algorithms 2016, 9, 14; https://doi.org/10.3390/a9010014
X. Wang, T. Zhang, A family of Steffensen type methods with seventh-order convergence, Numer. Algor., 62 (2013) 429–464. https://doi.org/10.1007/s11075-012-9597-3
X. Wang, T. Zhang, W. Qian, M. Teng, Seventh-order derivative-free iterative method for solving nonlinear systems, Numer. Algor., 70 (2015) 545–558. https://doi.org/10.1007/s11075-015-9960-2
A. R. Amiri, A. Cordero, M. T. Darvishi, J. R. Torregrosa, Preserving the order of convergence: Low-complexity Jacobian-free iterative schemes for solving nonlinear systems, J. Comput. Appl. Math., 337 (2018) 87–9.https://doi.org/10.1016/j.cam.2018.01.004
J. M. Ortega, W. C. Rheinbolt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, 1970.
J. F. Traub, Iterative methods for the solution of equations, Prentice-Hall, New Jersey, 1964.
F. A. Potra, V. Ptak, Nondiscrete induction and iterative processes, Pitman Publishing, Boston, 1984.
T. Zhanlav, Changbum Chun, Kh. Otgondorj, V. Ulziibayar, High-order iterations for systems of nonlinear equations, Int. J. Comput. Math., 97(8) (2020) 1704-1724. https://doi.org/10.1080/00207160.2019.1652739.
T. Zhanlav, Kh. Otgondorj, L. Saruul, R. Mijiddorj, Optimal choice of parameters in higher-order derivative-free iterative methods for systems of nonlinear equations, Springer Proceedings in Mathematics and Statistics, 434 (2023), 165–185. https://doi.org/10.1007/978-3-031-41229-5_13-0945.
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2024 Tugal Zhanlav, Khuder Otgondorj, Renchin-Ochir Mijiddorj, Lkhagvadash Saruul
This work is licensed under a Creative Commons Attribution 4.0 International License.
Copyright on any research article in the Proceedings of the Mongolian Academy of Sciences is retained by the author(s).
The authors grant the Proceedings of the Mongolian Academy of Sciences a license to publish the article and identify itself as the original publisher.
Articles in the Proceedings of the Mongolian Academy of Sciences are Open Access articles published under a Creative Commons Attribution 4.0 International License CC BY.
This license permits use, distribution and reproduction in any medium, provided the original work is properly cited.
