Sixteenth-Order Iterative Method for Solving Nonlinear Equations

ผู้วิจัย

Chalermwut Comemuang , Pairat Janngam

บทคัดย่อ

In this paper, we suggest and analyze some new sixteen-order iterative methods by using Householder’s method free from second derivative for solving nonlinear equations. Here we use a new and different technique for implementation of sixteen-order derivative of the function. The efficiency index equals 16^(1/6) ≈ 1.587. Numerical examples of the new methods are compared with other methods by exhibiting the effectiveness of the method presented in this paper.

บรรณานุกรม

[1] D. K. R. Babajee, R. Thukral, On a 4-Point Sixteenth-Order King Family of Iterative Methods for Solving Nonlinear Equations, International Journal of Mathematics and Mathematical Sciences, (2012), 13 pages. [2] Y. H. Geum, Y. I. Kim, A biparametric family of optimally convergent sixteenth-order multipoint methods with their fourth-step weighting function as a sum of a rational and a generic two-variable function, Journal of Computational and Applied Mathematics, 235, no. 10, (2011), 3178–3188. [3] Y. H. Geum, Y. I. Kim, A family of optimal sixteenth-order multipoint methods with a linear fraction plus a trivariate polynomial as the fourthstep weighting function, Comp. Math. Appl., 61, (2011), 3278–3287. [4] A. S. Householder, The Numerical Treatment of a Single Nonlinear Equation, McGraw-Hill, New York, 1970. [5] S. Huang, A. Rafiq, M. R. Shahzad, F. Ali, New higher order iterative methods for solving nonlinear equations, Hacettepe Journal of Mathematics and Statistics, 47, no. 1, (2018), 77–91. [6] B. Kalantary, Polynomial Root-Finding and Polynomiography, World Sci. Publishing Co., Hackensack, 2009. [7] S. K. Khattri, I. K. Argyros, Sixteenth Order Iterative Methods without Restraint on Derivatives, Applied Mathematical Sciences, 6, (2012), 6477–6486. [8] B. Kongied, Two new eighth and twelfth order iterative methods for solving nonlinear equations, International Journal of Mathematics and Computer Science, 16, no. 1, (2021), 333–344. [9] X. Li, C. Mu, J. Ma, C. Wang, Sixteenth-order method for nonlinear equations, Appl. Math. Comput., 215, (2010), 3754–3758. [10] M. S. K. Mylapalli, R. K. Palli, V. B. K. Vatti, An Iterative Method with Twelfth Order Coonvergence for Solving Non-linear Equations, Advances and Applications in Mathematical Sciences 20, no. 8, (2021), 1633–1643. [11] M. Rafiullah, D. Jabeen, New Eighth and Sixteenth Order Iterative Methods to Solve Nonlinear Equations, Int. J. Appl. Comput. Math., 3, (2017), 2467–2476. [12] O. S. Solaiman, I. Hashim, Efficacy of Optimal Methods for Nonlinear Equations with Chemical Engineering Applications, Mathematical Problems in Engineering, (2019), 1–11. [13] R. Thukral, New Sixteenth-Order Derivative-Free Methods for Solving Nonlinear Equations, American Journal of Computational and Applied Mathematics, 2, no. 3, (2012), 112–118. [14] J. F. Traub, Iterative Methods for the Solution of Equations, Chelsea Publishing Company, New York, 1977. [15] M. Turkyilmazoglu, A simple algorithm for high order Newton iteration formulae and some new variants, Hacettepe Journal of Mathematics and Statistics 49, no. 1, (2020), 425–438. [16] M. Z. Ullah, A. S. Al-Fhaid, F. Ahmad, Four-Point Optimal SixteenthOrder Iterative Method for Solving Nonlinear Equations, Journal of Applied Mathematics, (2013), 5 pages. [17] F. Zafar, N. Hussain, Z. Fatimah, K. Kharal, Optimal Sixteenth Order Convergent Method Based on Quasi-Hermite Interpolation for Computing Roots, The Scientific World Journal, (2014), 18 pages.