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Published: December 14,2021

The solution of a system of nonlinear equations is presumably one of the most common but difficult features in numerical analysis in the sense of different aspirations for instance; high accuracy, minimum computation time, a small number of iterations along with less computational cost. In this study, four distinct iterative methods are presented for solving a system of nonlinear equations such as Broyden’s method (BM), Optimal fourth-order method (OFOM), Optimal sixth order method (OSOM), and Homotopy continuation method (HCM). Detail formulations are explained along with the solution procedure. In addition, the rate of convergence and computational complexities are also explained within the formulations. Furthermore, to demonstrate and compare the efficiency of these methods, we solve two real-world practical nonlinear models. However, the results are presented numerically in a tabular form and the approximate results are compared with the exact and approximate solutions of other existing iterative methods. After analyzing the results, we conclude which method is better in what aspects.

[1] Abbasbandy, S. (2003). Improving Newton-Raphson method for nonlinear equations by modified Adomian decomposition method. Applied Mathematics and Computation, 145(2-3), 887-893.

[2] Chun, C. and Ham, Y. (2007). A one-parameter fourth-order family of iterative methods for nonlinear equations. Applied Mathematics and Computation, 189(1), 610614.

[3] Daftardar-Gejji, V. and Jafari, H. (2006). An iterative method for solving nonlinear functional equations. Journal of Mathematical Analysis and Applications, 316(2), 753-763.

[4] He, J. H. (2003). A new iteration method for solving algebraic equations. Applied Mathematics and Computation, 135(1), 81-84.

[5] Aslam Noor, M. and Inayat Noor, K. (2006). Three-step iterative methods for nonlinear equations. Applied Mathematics and Computation, 183(1), 322-327.

[6] Noor, M. A., and Noor, K. I. (2006). Some iterative schemes for nonlinear equations. Applied Mathematics and Computation, 183(2), 774-779.

[7] Noor, M. A., Noor, K. I., and Waseem, M. (2010). Fourth-order iterative methods for solving nonlinear equations. International Journal of Applied Mathematics and Engineering Sciences, 4(1), 43-52.

[8] Ortega, J. M. and Rheinboldt, W. C. (2000). Iterative solution of nonlinear equations in several variables. Society for Industrial and Applied Mathematics.

[9] Broyden, C. G. (1965). A class of methods for solving nonlinear simultaneous equations. Mathematics of computation, 19(92), 577-593.

[10] Behl, R., Íñigo Sarría, Ruben Gonzalez Crespo, and Ángel Alberto Magreñán. (2019). Highly efficient family of iterative methods for solving nonlinear models. Journal of Computational and Applied Mathematics, 346, 110-132.

[11] Sharma, R. and Bahl, A. (2015). An optimal fourth order iterative method for solving nonlinear equations and its dynamics. Journal of Complex Analysis, 2015(8), 1-9.

[12] Burden, R. L. and Faires, J. D. (2005). Numerical analysis. 8th ed. Thomson Brooks/Cole.

[13] Hirsch, M. J., Pardalos, P. M., and Resende, M. G. (2009). Solving systems of nonlinear equations with continuous GRASP. Nonlinear Analysis: Real World Applications, 10(4), 2000-2006.

[14] Henderson, N., Sacco, W. F., and Platt, G. M. (2010). Finding more than one root of nonlinear equations via a pola-rization technique: An application to double retrograde vaporization. Chemical Engineering Research and Design, 88(5-6), 551-561.

[15] Floudas, C. A. (1999). Recent advances in global optimization for process synthesis, design and control: enclosure of all solutions. Computers and Chemical Engineering, 23, S963-S973.

[16] Pourjafari, E. and Mojallali, H. (2012). Solving nonlinear equations systems with a new approach based on invasive weed optimization algorithm and clustering. Swarm and Evolutionary Computation, 4, 33-43.

[17] Luo, Y. Z., Tang, G. J., and Zhou, L. N. (2008). Hybrid approach for solving systems of nonlinear equations using chaos optimization and quasi-Newton method. Applied Soft Computing, 8(2), 1068-1073.

[18] Ali, H., Kamrujjaman, M., and Shirin, A. (2021). Numerical solution of a fractional-order Bagley–Torvik equation by quadratic finite element method. Journal of Applied Mathematics and Computing, 66(1), 351-367.

[19] Akter, S. I., Mahmud, M. S., Kamrujjaman, M., and Ali, H. (2020). Global Spectral Collocation Method with Fourier Transform to Solve Differential Equations. GANIT: Journal of Bangladesh Mathematical Society, 40(1), 28-42.

[20] Ali, H. and Islam, M. S. (2017). Generalized Galerkin finite element formulation for the numerical solutions of second order nonlinear boundary value problems. GANIT: Journal of Bangladesh Mathematical Society, 37, 147-159.

[21] Ali, H., Kamrujjaman, M., and Islam, M. S. (2020). Numerical Computation of Fitzhugh-Nagumo Equation: A Novel Galerkin Finite Element Approach. International Journal of Mathematical Research, 9(1), 20-27.

[22] Ali, H., and Kamrujjaman, M. Numerical solutions of nonlinear parabolic equations with Robin condition: Galerkin approach.TWMS J. App. and Eng. Math. In press.

[23] Lima, S. A., Kamrujjaman, M., and Islam, M. S. (2021). Numerical solution of convection-diffusion-reaction equations by a finite element method with error correlation. AIP Advances, 11(8), 085225.

[24] Temelcan, G., Sivri, M., and Albayrak, I. (2020). A new iterative linearization approach for solving nonlinear equa-tions systems. An International Journal of Optimization and Control: Theories & Applications, 10(1), 47-54.

[25] Dehghan, M. and Shirilord, A. (2020). Three-step iterative methods for numerical solution of systems of nonlinear equations. Engineering with Computers, 1-14.

[26] Sayevand, K., Erfanifar, R., and Esmaeili, H. (2020). On Computational Efficiency and Dynamical Analysis for a Class of Novel Multi-step Iterative Schemes. International Journal of Applied and Computational Mathematics, 6(6), 1-23.

[27] Srivastava, H. M., Iqbal, J., Arif, M., Khan, A., Gasimov, Y. S., and Chinram, R. (2021). A new application of Gauss quadrature method for solving systems of nonlinear equations. Symmetry, 13(3), 432.

**Efficient Family of Iterative Methods for Solving Nonlinear Simultaneous Equations: A Comparative Study**

**How to cite this paper:** Hazrat Ali, Trishna Datta, Md. Kamrujjaman. (2021) Efficient Family of Iterative Methods for Solving Nonlinear Simultaneous Equations: A Comparative Study. *Journal of Applied Mathematics and Computation*, **5**(**4**), 331-337.

DOI: http://dx.doi.org/10.26855/jamc.2021.12.011

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