Simple and efficient fifth order solvers for systems of nonlinear problems
Abstract
In this study, two multi-step iterative techniques of fifth order convergence are explored to solve nonlinear equations. The techniques are designed with the prime objective of keeping the computational cost as low as possible. To claim this objective, the efficiency indices are determined and compared with the efficiencies of the existing techniques of same order. The outcome of comparison analysis is remarkable from the view of high computational efficiency of new methods. Performance and stability are illustrated by executing the numerical tests on some nonlinear problems of diverse nature. The entire analysis significantly favors the new techniques compared to their existing counterparts, especially for the case of large dimensional systems.
Keyword : nonlinear systems, iterative techniques, convergence order, computational complexity
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
A. Cordero, J.L. Hueso, E. Martínez and J.R. Torregrosa. Efficient high-order methods based on golden ratio for nonlinear systems. Applied Mathematics and Computation, 217(9):4548–4556, 2011. https://doi.org/10.1016/j.amc.2010.11.006
A. Cordero, J.L. Hueso, E. Martínez and J.R. Torregrosa. Increasing the convergence order of an iterative method for nonlinear systems. Applied Mathematics Letters, 25(12):2369–2374, 2012. https://doi.org/10.1016/j.aml.2012.07.005
Z. Liu, Q. Zheng and C.E. Huang. Third-and fifth-order Newton–Gauss methods for solving nonlinear equations with n variables. Applied Mathematics and Computation, 290:250–257, 2016. https://doi.org/10.1016/j.amc.2016.06.010
J.M. Ortega and W.C. Rheinboldt. Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, New York, 1970.
A.M. Ostrowski. Solution of Equation and Systems of Equations. Academic Press, New York, 1960.
F.A. Potra and V. Pták. On a class of modified Newton processes. Numerical Functional Analysis and Optimization, 2(1):107–120, 1980. https://doi.org/10.1080/01630568008816049
J.R. Sharma and P. Gupta. An efficient fifth order method for solving systems of nonlinear equations. Computers & Mathematics with Applications, 67(3):591– 601, 2014. https://doi.org/10.1016/j.camwa.2013.12.004
R. Sihwail, O.S. Solaiman, K. Omar, K.A.Z. Ariffin, M. Alswaitti and I. Hashim. A hybrid approach for solving systems of nonlinear equations using Harris Hawks optimization and Newton’s method. IEEE Access, 9:95791–95807, 2021. https://doi.org/10.1109/ACCESS.2021.3094471
O.S. Solaiman and I. Hashim. An iterative scheme of arbitrary odd order and its basins of attraction for nonlinear systems. Computers, Materials & Continua, 66(2):1427–1444, 2021. https://doi.org/10.32604/cmc.2020.012610
J.F. Traub. Iterative Methods for the Solution of Equations. Chelsea Publishing Company, New York, 1982.
S. Wolfram. The Mathematica Book (5th edition). Wolfram Media, USA, 2003.
X.Y. Xiao and H.W. Yin. A new class of methods with higher order of convergence for solving systems of nonlinear equations. Applied Mathematics and Computation, 264:300–309, 2015. https://doi.org/10.1016/j.amc.2015.04.094
X.Y. Xiao and H.W. Yin. Increasing the order of convergence for iterative methods to solve nonlinear systems. Calcolo, 53(3):285–300, 2016. https://doi.org/10.1007/s10092-015-0149-9
Z. Xu and T. Jieqing. The fifth order of three-step iterative methods for solving systems of nonlinear equations. Mathematica Numerica Sinica, 35(3):297–304, 2013.
T. Zhanlav, C. Chun, K. Otgondorj and V. Ulziibayar. High-order iterations for systems of nonlinear equations. International Journal of Computer Mathematics, 97(8):1704–1724, 2020. https://doi.org/10.1080/00207160.2019.1652739
T. Zhanlav and K. Otgondorj. Higher order Jarratt-like iterations for solving systems of nonlinear equations. Applied Mathematics and Computation, 395:125849, 2021. https://doi.org/10.1016/j.amc.2020.125849