### A new iterative linearization approach for solving nonlinear equations systems

Gizem Temelcan, Mustafa Sivri, Inci Albayrak

#### Abstract

Nonlinear equations arise frequently while modeling chemistry, physics, economy and engineering problems. In this paper, a new iterative approach for finding a solution of a nonlinear equations system (NLES) is presented by applying a linearization technique. The proposed approach is based on computational method that converts NLES into a linear equations system by using Taylor series expansion at the chosen arbitrary nonnegative initial point. Using the obtained solution of the linear equations system, a linear programming (LP) problem is constructed by considering the equations as constraints and minimizing the objective function constructed as the summation of balancing variables. At the end of the presented algorithm, the exact solution of the NLES is obtained. The performance of the proposed approach has been demonstrated by considering different numerical examples from literature.

#### Keywords

Nonlinear Equations System; Linear Programming Problem; Taylor Series

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#### References

Grapsa, T.N. and Vrahatis, M.N. (2003). Dimension reducing methods for systems of nonlinear equations and unconstrained optimization: A review. Recent Advances in Mechanics and Related Fields, 215-225.

Frontini, M. and Sormani, E. (2004). Third-order methods from quadrature formulae for solving systems of nonlinear equations. Applied Mathematics and Computation, 149(3), 771-782.

Babolian, E., Biazar, J. and Vahidi, A.R. (2004). Solution of a system of nonlinear equations by Adomian decomposition method. Applied Mathematics and Computation, 150(3), 847-854.

Nie, P. (2004). A null space method for solving system of equations. Applied Mathematics and Computation, 149(1), 215-226.

Nie, P. (2006). An SQP approach with line search for a system of nonlinear equations. Mathematical and Computer Modelling, 43(3-4), 368-373.

Jafari, H. and Daftardar-Gejji, V. (2006). Revised Adomian decomposition method for solving a system of nonlinear equations. Applied Mathematics and Computation, 175(1), 1-7.

Darvishi, M.T. and Barati, A. (2007). A third-order Newton-type method to solve systems of nonlinear equations. Applied Mathematics and Computation, 187(2), 630-635.

Golbabai, A. and Javidi, M. (2007). A new family of iterative methods for solving system of nonlinear algebric equations. Applied Mathematics and Computation, 190(2), 1717-1722.

Biazar, J. and Ghanbary, B. (2008). A new approach for solving systems of nonlinear equations. International Mathematical Forum, 3(38), 1885-1889.

Grosan, C. and Abraham, A. (2008). A new approach for solving nonlinear equations systems. IEEE Transactions on Systems, Man, and Cybernetics-Part A: Systems and Humans, 38(3), 698-714.

Hosseini, M.M. and Kafash, B. (2010). An efficient algorithm for solving system of non- linear equations. Applied Mathematical Sciences, 4(3), 119-131.

Gu, C. and Zhu, D. (2012). A filter algorithm for nonlinear systems of equalities and inequalities. Applied Mathematics and Computation, 218(20), 10289-10298.

Vahidi, A.R., Javadi, S. and Khorasani, S.M. (2012). Solving system of nonlinear equations by restarted Adomains method. Applied Mathematical Sciences, 6(11), 509-516.

Sharma, J.R. and Gupta, P. (2013). On some efficient techniques for solving systems of nonlinear equations. Advances in Numerical Analysis, 2013.

Wang, H. and Pu, D. (2013). A nonmonotone filter trust region method for the system of nonlinear equations. Applied Mathematical Modelling, 37(1-2), 498-506.

Zhang, W. (2013). Methods for solving non-linear systems of equations (Technical report). Department of Mathematics, University of Washington, Seattle, WA, USA.

Dhamacharoen, A. (2014). An efficient hybrid method for solving systems of nonlinear equations. Journal of Computational and Applied Mathematics, 263, 59-68.

Izadian, J., Abrishami, R. and Jalili, M. (2014). A new approach for solving nonlinear system of equations using Newton method and HAM. Iranian Journal of Numerical Analysis and Optimization, 4(2), 57-72.

Narang, M., Bhatia, S. and Kanwar, V. (2016). 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.

Saheya, B., Chen, G., Sui, Y. and Wu, C. (2016) A new Newton-like method for solving nonlinear equations. SpringerPlus, 5(1), 1269.

Wang, X. and Fan, X. (2016). Two efficient derivative-free iterative methods for solving nonlinear systems. Algorithms, 9(1), 14.

Xiao, X.Y. and Yin, H.W. (2016). Increasing the order of convergence for iterative methods to solve nonlinear systems. Calcolo, 53(3), 285-300.

Balaji, S., Venkataraman, V., Sastry, D. and Raghul, M. (2017). Solution of system of non-linear equations using integrated RADM and ADM. International Journal of Pure and Applied Mathematics, 117(3), 367-373.

Madhu, K., Babajee, D.K.R. and Jayaraman, J. (2017). An improvement to double step Newton method and its multi-step version for solving system of nonlinear equations and its applications. Numerical Algorithms, 74(2), 593-607.

Sharma, J.R. and Arora, H. (2017). Improved Newton-like methods for solving systems of nonlinear equations. SeMA Journal, 74(2), 147-163.

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.

Dai, J., Wu, G., Wu, Y. and Zhu, G. (2008). Helicopter trim research based on hybrid genetic algorithm. 7th World Congress on Intelligent Control and Automation, 2007-2011.

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

Remani, C. (2012). Numerical methods for solving systems of nonlinear equations (Technical report). Lakehead University,

DOI: http://dx.doi.org/10.11121/ijocta.01.2020.00684

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