Abstract
In this manuscript, we present a new class of highly efficient two-parameter optimal iterative methods for solving nonlinear systems that generalizes Ostrowski’s method, King’s Family, Chun’s method, and KLAM Family in multidimensional context. This class is an extension to systems of the Ermakov’s Hyperfamily. The fourth order of convergence of the members of the class is demonstrated, thus obtaining optimal schemes for solving nonlinear systems. The high efficiency of the elements of the class is studied, compared with other known methods of the same order or even higher, and some numerical proofs are presented. We also analyze its robustness.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Data availability
Not applicable.
Code availability
Not applicable.
References
Sharma, J.R., Kumar, S.: A class of accurate Newton-Jarratt-like methods with applications to nonlinear models. Computational and Applied Mathematics 41(1), 46 (2022)
Behl, R., Bhalla, S., Magreñán, Á.A., Kumar, S.: An efficient high order iterative scheme for large nonlinear systems with dynamics. Journal of Computational and Applied Mathematics 404, 113249 (2022)
Solaiman, O.S., Hashim, I.: An iterative scheme of arbitrary odd order and its basins of attraction for nonlinear systems. Comput. Mater. Contin. Computers 66, 1427–1444 (2021)
Yaseen, S., Zafar, F., Chicharro, F.I.: A seventh order family of Jarratt type iterative method for electrical power systems. Fractal and Fractional 7(4), 317 (2023)
Amat, S., Busquier, S.: Advances in iterative methods for nonlinear equations. SEMA SIMAI Springer Series, Switzerland (2016)
Ermakov, V.V., Kalitkin, N.N.: The optimal step and regularization for Newton’s method. USSR Computational Mathematics and Mathematical Physics 21(2), 235–242 (1981)
Ortega, J.M., Rheinboldt, W.G.: Iterative solutions of nonlinear equations in several variables. Society for Industrial and Applied Mathematics (2000)
Kung, H.T., Traub, J.F.: Optimal order of one-point and multi-point iteration. Journal of the ACM (JACM) 21(4), 643–651 (1974)
Arroyo, V., Cordero, A., Torregrosa, J.R.: Approximation of artificial satellites’ preliminary orbits: the efficiency challenge. Mathematical and Computer Modelling 54(161–271), 1802–1807 (2011)
Jarratt, P.: Some fourth order multipoint iterative methods for solving equations. Mathematics of computation 20(95), 434–437 (1966)
Khirallah, M.Q., Hafiz, M.A.: Solving system of non-linear equations using family of Jarratt methods. Int. J. Differ. Equ. Appl. 12(2), 69–83 (2013)
Budzko, D.A., Cordero, A., Torregrosa, J.R.: A new family of iterative methods widening areas of convergence. Appl. Math. Comput. 252, 405–417 (2015)
Cordero, A., Rojas-Hiciano, R.V., Torregrosa, J.R., Vassileva, M.P.: Fractal complexity of a new biparametric family of fourth optimal order based on the Ermakov-Kalitkin scheme. Fractal and Fractional. Accepted
Ostrowski, A.M.: Solutions of equations and systems of equations. Academic Press, (1966)
King, R.: A family of fourth order methods for nonlinear equations. SIAM Journal on Numerical Analysis 10(5), 876–879 (1973)
Chun, C., Lee, M.Y., Neta, B., Dzunic, J.: On optimal fourth-order iterative methods free from second derivative and their dynamics. Appl. Math. Comput. 218(11), 6427–6438 (2012)
Singh, H., Sharma, J. R., Kumar, S.: A simple yet efficient two-step fifth-order weighted-Newton method for nonlinear models. Numerical Algorithms 1-23 (2022)
Traub, J.F.: Iterative methods for the solution of equation. Chelsea Publishing Company (1982)
Cordero, A., Hueso, J.L., Martínez, E., Torregrosa, J.R.: Increasing the convergence order of an iterative method for nonlinear systems. Appl. Math. Lett. 25, 2369–2374 (2012)
Cordero, A., Hueso, J.L., Torregrosa, J.R.: Variants of Newton’s method using fifth-order quadrature formulas. Applied Mathematics and Computation 190, 686–698 (2012)
Grau-Sánchez, M., Noguera, M., Gutiérrez, J.: On some computational orders of convergence. Applied Mathematics Letters 23, 472–478 (2010)
Grau-Sánchez, M., Noguera, M., Gutiérrez, J.: On new computational local orders of convergence. Applied Mathematics Letters 25, 2023–2030 (2012)
Burden, R.L., Faires, J.D.: Numerical analysis. Boundary-value problems for ordinary differential equations, 641-685, Belmount: Thomson Brooks/Cole (2005)
Acknowledgements
The authors would thank to the anonymous reviewer for his/her suggestions and comments that have improved the final version of this manuscript.
Author information
Authors and Affiliations
Contributions
Conceptualization, A.C.; methodology, R.V.R.; software, M.P.V.; validation, J.R.T.; formal analysis, A.C.; investigation, R.V.R.; writing—original draft preparation, R.V.R. and M.P.V.; writing—review and editing, A.C. and J.R.T.; supervision, J.R.T. All authors have read and agreed to the published version of the manuscript.
Corresponding author
Ethics declarations
Ethics approval
Not applicable.
Conflict of interest
The authors declare no competing interests.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Cordero, A., Rojas-Hiciano, R.V., Torregrosa, J.R. et al. A highly efficient class of optimal fourth-order methods for solving nonlinear systems. Numer Algor 95, 1879–1904 (2024). https://doi.org/10.1007/s11075-023-01631-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11075-023-01631-9