Some modifications and extensions of Popovski’s and Laguerre’s families for solving systems of nonlinear equations
DOI:
https://doi.org/10.5564/mmj.v27i25.4013Keywords:
Multipoint iterative methods, Order of convergence, Nonlinear systems, Popovski’s and Laguerre’s methodsAbstract
Two modifications of Popovski’s and Laguerre’s families of methods are developed, which free of second derivatives. The improved modifications have fourth-order of convergence. Moreover, we propose the extensions of modifications to solve nonlinear systems. The convergence order of two and three-step iterations equal to four, six and seven respectively. The numerical results confirm the order of convergence. In addition, we investigate the basin of attraction of the methods and its dependence on the convergence behavior. The comparison is made based on the performance on examples of nonlinear problems and the CPU time.
Downloads
36
References
1. Argyros, I.K., Kansal, M., Kanwar, V., & Bajaj, S. (2017) ”Higher-order derivative-free families of Chebyshev-Halley type methods with or without memory for solving nonlinear equations”. Applied Mathematics and Computation, 315, 224–245. https://doi.org/10.1016/j.amc.2017.07.051
2. Behl, R. (2020) ”An efficient family of Chebyshev–Halley’s methods for sys tem of nonlinear equations”. Journal of Mathematical Chemistry, 58, 868–885. https://doi.org/10.1016/j.amc.2017.07.051
3. Chun, Changbum, & Neta, B. (2015) ”Basins of attraction for several third order methods to f ind multiple roots of nonlinear equations”. Applied Mathematics and Computation, 268, 129-137. https://doi.org/10.1016/j.amc.2015.06.068
4. Chun, Changbum, & Neta, B. (2013) ”On a family of Laguerre methods to find multi ple roots of nonlinear equations”. Applied Mathematics and Computation 219, 10987–11004. https://doi.org/10.1016/j.amc.2013.05.002
5. Cuyt, A., & Cruyssen, P. (1983) ”Abstract Pad´e-approximants for the solution of a sys tem of nonlinear equations”. Computers & Mathematics with Applications, 9, 617–624. https://doi.org/10.1016/0898-1221(83)90119-0
6. Dehghan, M., & Shirilord, A. (2022) ”Three-step iterative methods for numerical solu tion of systems of nonlinear equations”. Engineering with Computers, 38, 1015–1028. https://doi.org/10.1007/s00366-020-01072-1
7. Narang, M., Bhatia, S., & Kanwar, V. (2013) ”New two parameter Chebyshev–Halley like family of fourth and sixth-order methods for systems of nonlinear equations”. Applied Mathematics and Computation, 275, 394–403. https://doi.org/10.1016/j.amc.2015.11.063
8. Neta, B. (2022) ”Basins of attraction for family of Popovski’s methods and their extension to multiple roots”. Journal of Numerical Analysis and Approximation Theory, 51, 88–102. https://doi.org/10.33993/jnaat511-1248
9. Neta, B. (2008) ”On Popovski’s method for nonlinear equations”. Applied Mathematics and Computation, 201, 710–715. https://doi.org/10.1016/j.amc.2008.01.012
10. Kansal, M., Kanwar, V., & Bhatia, S. (2019) ”New modifications of Hansen–Patrick’s family with optimal fourth and eighth orders of convergence”. Applied Mathematics and Computation, 269, 507–519. https://doi.org/10.1016/j.amc.2015.07.101
11. Kansal, M., Kanwar, V., & Bhatia, S. (2016) ”Efficient derivative-free variants of Hansen-Patrick’s family with memory for solving nonlinear equations”. Numerical Algorithms, 73, 1017-–1036. https://doi.org/10.1007/s11075-016-0127-6
12. Grau-S´anchez, M., & Noguera, M., Guti´ errez, J. M. (2010) ”On some computational orders of con vergence”. Applied Mathematics Letters, 23, 472–478. https://doi.org/10.1016/j.aml.2009.12.006
13. Popovski, D. B. (1979) ”A Family of one-point iteration formulae for finding roots”. International Journal of Computer Mathematics, 8, 85–88. https://doi.org/10.1080/00207168008803193
14. Kou, J., & Li, Y. (2007) ”Modified Chebyshev’s method free from second deriva tive for nonlinear equations”. Applied Mathematics and Computation, 187, 1027-–1032. https://doi.org/10.1016/j.amc.2006.09.021
15. Kou, J., Li, Y., & Wang, X. (2006) ”Modified Halley’s method free from second derivative”. Applied Mathematics and Computation, 183, 704-–708. https://doi.org/10.1016/j.amc.2006.05.097
16. Sharma, J. R., Guha, R. K., & Sharma, R. (2013) ”An efficient fourth-order weighted Newton method for systems of nonlinear equations”. Numerical Algorithms, 62, 307–323. https://doi.org/10.1007/s11075-012-9585-7
17. Singh, H., & Sharma, J.R. (2023) ”Simple yet highly efficient numerical techniques for systems of nonlinear equations”. Computational and Applied Mathematics, 42, 22. https://doi.org/10.1007/s40314-022-02159-9
18. Schmidt, J. W. (1981) ”On the R-Order of Coupled Sequences”. Computing, 26, 333—342. https://doi.org/10.1007/BF02237952
19. Zhanlav, T., & Chuluunbaatar, O. (2022) ”New developments of Newton-type iterations for solving nonlinear problems”. Ulaanbaatar, Moscow.
20. Zhanlav, T. & Otgondorj, Kh. (2021) ”On the optimal choice of parameters in two-point iterative methods for solving nonlinear equations”. Computational Mathematics and Mathematical Physics, 61, 29–42. https://doi.org/10.1134/S0965542520120180
21. Zhanlav, T., Chun, Changbum, Otgondorj, Kh. & Ulziibayar, V. (2020) ”High-order iterations for systems of nonlinear equations”. International Journal of Computer Mathematics, 97, 1704–1724. https://doi.org/10.1080/00207160.2019.1652739
22. Zhanlav, T., & Khongorzul, D.(2013) ”Semilocal convergence with R-order three theorems for Chebyshev method its applications”. A. Chinchuluun et al (eds), Optimization, Simulation and Control, Springer optimization and its applications, 76, 331–345. https://doi.org/10.1007/978-1 4614-5131-021
23. Zhanlav, T., Chuluunbaatar, O., & Ulziibayar, V. (2017) ”The necessary and sufficient conditions for two and three-point iterative method of Newton’s type iterations”. Computational Mathematics and Mathematical Physics, 57, 1093–1102. https://doi.org/10.1134/S0965542517070120
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2024 Tugal Zhanlav, Khuder Otgondorj
This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.
Copyright on any research article in the Mongolian Mathematical Journal is retained by the author(s).
The authors grant the Mongolian Mathematical Journal a license to publish the article and identify itself as the original publisher.
Articles in the Mongolian Mathematical Journal are Open Access articles published under a Creative Commons Attribution NonCommercial (CC-BY-NC)
This license permits use, distribution and reproduction in any medium, provided the original work is properly cited, does not allow for commercial use of the original work.
