Discrete and Continuous Models and Applied Computational ScienceDiscrete and Continuous Models and Applied Computational ScienceDiscrete and Continuous Models and Applied Computational Science2658-46702658-7149Peoples' Friendship University of Russia named after Patrice Lumumba (RUDN University)4750410.22363/2658-4670-2025-33-4-389-403HZYRKNModeling and SimulationМатематическое моделированиеResearch ArticleOptimal eight-order three-step iterative methods for solving systems of nonlinear equationsОптимальные трёхшаговые итерационные методы восьмого порядка для решения систем нелинейных уравненийhttps://orcid.org/0000-0003-0743-558724484328800ZhanlavTugalЖанлавТ.
Russian Federation

Full member of Mongolian Academy of Sciences, professor, doctor of sciences in physics and mathematics, Honorary Doctor of JINR

tzhanlav@yahoo.comhttps://orcid.org/0000-0003-1635-797157209734799OtgondorjKhuderОтгондоржХ.
Russian Federation

Associate Professor of Department of Mathematics at School of Applied Sciences

otgondorj@gmail.comInstitute of Mathematics and Digital Technology, Mongolian Academy of SciencesИнститут математики и цифровой технологии, Монгольская академия наукMongolian University of Science and TechnologyМонгольский государственный университет науки и технологии07122025334VOL 33, No4 (2025)ТОМ 33, №4 (2025)38940306122025Copyright ©; 2025, Zhanlav T., Otgondorj K.Copyright ©; 2025, Жанлав Т., Отгондорж Х.2025Zhanlav T., Otgondorj K.Жанлав Т., Отгондорж Х.https://creativecommons.org/licenses/by-nc/4.0https://journals.rudn.ru/miph/article/view/47504

In this paper, we for the first time propose the extension of optimal eighth-order methods to multidimensional case. It is shown that these extensions maintained the optimality properties of the original methods. The computational efficiency of the proposed methods is compared with that of known methods. Numerical experiments are included to confirm the theoretical results and to demonstrate the efficiency of the methods.

В данной статье мы впервые предлагаем расширение оптимальных методов восьмого порядка на многомерный случай. Показано, что эти расширения сохранили свойства оптимальности исходных методов. Вычислительная эффективность предлагаемых методов сравнивается с известными методами. Проводится сравнение с другими методами. Для подтверждения теоретических результатов и эффективности методов включены численные эксперименты.

newton-type methodssystems of nonlinear equationsconvergence orderoptimality and extension of methodsefficiency indexметоды ньютоновского типасистемы нелинейных уравненийпорядок сходимостиоптимальность и расширение методовиндекс эффективностиBi, W., Ren, H. & Wu, Q. Three-step iterative methods with eighth-order convergence for solving nonlinear equations. Journal of Computational and Applied Mathematics 225, 105-112. doi:10.1016/j.cam.2008.07.004 (2009).Changbum, C. & Neta, B.Comparison of several families of optimal eighth order methods. Applied Mathematics and Computation 274, 762-773. doi:10.1016/j.amc.2015.10.092 (2016).Changbum, C. & Lee, M. Y. A new optimal eighth-order family of iterative methods for the solution of nonlinear equations. Applied Mathematics and Computation 223, 506-519. doi:doi:.1016/j.amc.2013.08.033 (2013).Cordero, A., Rojas-Hiciano, R. V., Torregrosa, J. R. & Vassileva, M. P. A highly efficient class of optimal fourth-order methods for solving nonlinear systems. Numerical Algorithms 95, 1879- 1904. doi:10.1007/s11075-023-01631-9 (2024).Dehghan, M. & Shirilord, A. Three-step iterative methods for numerical solution of systems of nonlinear equations. Engineering with Computers 38, 1015-1028. doi:10.1007/s00366-020-01072-1 (2020).Sharma, J. R. & Sharma, R. A new family of modified Ostrowski’s methods with accelerated eighth order convergence. Numerical Algorithms 54, 445-458. doi:10.1007/s11075-009-9345-5 (2010).Singh, H., Sharma, J. R. & Kumar, S. A simple yet efficient two-step fifth-order weighted-Newton method for nonlinear models. Numerical Algorithms 93, 203-225. doi:10.1007/s11075-022-01412-w (2023).Zhanlav, T., Chuluunbaatar, O. & Ulziibayar, V. Necessary and sufficient conditions for two and three-point iterative method of Newton’s type iterations.Computational Mathematics and Mathematical Physics 57, 1090-1100. doi:10.1134/S0965542517070120 (2017).Zhanlav, T. & Chuluunbaatar, O. New development of Newton-type iterations for solving nonlinear problems 281 pp. doi:10.1007/978-3-031-63361-4 (Switzerland, Springer Nature, 2024).Zhanlav, T. & Otgondorj, K. On the development and extensions of some classes of optimal threepoint iterations for solving nonlinear equations. Journal of Numerical Analysis and Approximation Theory 50, 180-193. doi:10.33993/jnaat502-1238180--193 (2021).Zhanlav, T. & Otgondorj, K. High efficient iterative methods with scalar parameter coefficients for systems of nonlinear equations. Journal of Mathematical Sciences 279, 866-875. doi:10.1007/s10958-024-07066-4 (2024).Zhanlav, T. & Otgondorj, K. Development and adaptation of higher-order iterative methods in 𝑅𝑛 with specific rules. Discrete and Continuous Models and Applied Computational Science 32, 425- 444. doi:10.22363/2658-4670-2024-32-4-425-444 (2024).Wang, X. & Liu, L. Modified Ostrowski’s method with eighth-order convergence and high efficiency index. Applied Mathematics Letters 23, 549-554. doi:10.1016/j.aml.2010.01.009 (2010).Cordero, A., Torregrosa, J. R. & Triguero-Navarro, P. First optimal vectorial eighth-order iterative scheme for solving non-linear systems. Applied Mathematics and Computation 498, 129401. doi:10.1016/j.amc.2025.129401 (2025).Sharma, J. & Arora, H. Improved Newton-like methods for solving systems of nonlinear equations. SeMA Journal 74, 147-163. doi:10.1007/s40324-016-0085-x (2017).Zhanlav, T. & Otgondorj, K. Higher order Jarratt-like iterations for solving systems of nonlinear equations. Applied Mathematics and Computation 395, 125849. doi:doi:10.1016/j.amc.2020.125849 (2021).