New Modification of Third-Order Iterative Method with Optimal Fourth-Order Convergence for Solving Nonlinear Equations

Wartono ¹ , Rahmawati ² , R. Agustin³

Section:Research Paper, Product Type: Isroset-Journal
Vol.6 , Issue.1 , pp.155-161, Feb-2019

CrossRef-DOI:   https://doi.org/10.26438/ijsrmss/v6i1.155161

Online published on Feb 28, 2019

Copyright © Wartono, Rahmawati, R. Agustin . This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
 

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Abstract :
Behl’s method is a third-order iterative method involving three functional evaluations for solving nonlinear equation. In this paper, we presented a new iterative method free second derivative with three real parameters by using second-order Taylor expansion. In order to avoid the second derivative, it is approximated by using equality of Chun-Kim’s method and Newton-Steffensen’s method. The result of study shows that the proposed method converges quartically and requires three evaluation of functions per iteration with efficiency index equal to 1.587401. Numerical simulation is presented to examine the performance of the proposed method by using several real test functions. The final results show that the proposed method is more efficient and perform better than some other kind of methods

Key-Words / Index Term :
Behl’s method, Chun-Kim’s method, Third-order iterative method, Newton-Steffensen’s method, Numerical simulation

References :
[1] A. C. Chapra, R. P. Canale, “Numerical Methods for Engineering with Programming and Software Application”, McGraw-Hill Publication, Singapore, pp. 194 – 209, 1998.
[2] K. C. Gupta, V. Kanwar, S. Kumar, “A Family of Ellips Methods for Solving Non-Linear Equations”, International Journal of Mathematical Educations in Science and Technology, pp. 571 – 575. 2009.
[3] S. Amat, S. Busquier, “On An Improvement of the Steffensen Method”, IMHOTEP, Vol. 4, No. 1, pp. 41 – 48, 2003.
[4] S. Amat, S. Busquier, J. M. Gutierrez, “Geometric Construction of Iterative Functions to Solve Nonlinear Equations”, Journal of Computational and Applied Mathematics, Vol. 157, pp. 197-205, 2003.
[5] S. Amat, S. Busquier, J. M. Gutierrez, M. A. Hernandez, “On Global Convergence of Chebyshev’s Iterative Method”, Vol. 230, pp. 17 – 21, 2008.
[6] R. Behl, V. Kanwar, K. K. Sharma “Another Simple Way of Deriving Several Iterative Functions to Solve Nonlinear Equatiaons”. Journal of Applied Mathematics Vol. 2012, pp. 1- 22, 2012.
[7] D. Jiang, D. Han,”Some One-Parameter Families of Third-Order Methods for Solving Nonlinear Equations”, Applied Mathematics and Computations, Vol. 195, pp. 392 – 396, 2008.
[8] V. Kanwar, S. K. Tomar, “Exponentially Fitted Variants of Newton’s Method with Quadratic and Cubic Convergence”, International Journal of Computer Mathematics, Vol. 86, No. 9, pp. 1603 – 1611, 2009.
[9] A. Melman, “Geometry and Convergence fo Euler’s and Halley’s Methods”, SIAM Review, Vol. 39, No. 4, pp. 726 – 735, 1997.
[10] A. Rafiq, A. Javeria, “New Iterative Methods for Solving Nonlinear Equations by Using Modified Homotopy Pertubation Method”, Acta Unversitatis Apulensis, Vol. 18, pp. 129 – 137, 2009.
[11] J. F. Traub, “Iterative Methods for the Solution of Equations”, Prentice-Hall, Inc, Englewood Cliffs, N.Y, pp. 1 – 36, 1964.
[12] C. Chun, Y. Kim, “Several New Third-Order Iterative Methods for Solving Nonlinear Equations”. Acta Applied Mathematics, Vol. 109, pp. 1053 – 1063. 2010.
[13] S. Abbasbandy, “Modified Homotopy Perturbation Method for Nonlinear Equations and Comparison with Adomian Decomposition Method”, Applied Mathematics and Computation. Vol. 172, pp. 431–438, 2006.
[14] C. Chun, “Iterative Methods Improving Newton’s Method by the Decomposition Method”, Computer and Mathematics with Applications, Vol. 50, pp. 1559 – 1568, 2005.
[15] C. Chun, “A One-Parameter Family of Third-Order Methods to solve nonlinear Equations”, Applied Mathematics and Computation. Vol. 189, pp. 126 – 130, 2007.
[16] M. A. Noor, “Iterative Methods for Nonlinear Equations Using Homotopy Pertubation Technique”, Applied Mathematics & Information Sciences, Vol 4, No. 2, pp. 227 – 235, 2010.
[17] S. Abbasbandy, “Improving Newton-Raphson Method for Solving Nonlinear Equation by Modified Adomian Decomposition Method,”Applied Mathematics and Computation. Vol. 145, pp. 887-893, 2003.
[18] M. Frontini, E. Sormani, “Some Variant of Newton’s Method with Third-Order Convergence”, Applied Mathematics and Computation, Vol. 140, pp. 419 – 426, 2003.
[19] A. Y. Ozban, “Some New Variants of Newton’s Method”, Applied Mathematics Letters, Vol. 17, pp. 677 – 682, 2004.
[20] J. R. Sharma, “A Composite Third-Order Newton-Steffensen Method for Solving Nonlinear Equations”. Applied Mathematics and Computation, Vol. 169, pp. 242 - 246. 2005.
[21] S. Weerakoon, T. G. I. Fernando, “A Variant of Newton’s Method with Accelerated Third-Order Convergence”, Applied Mathematics Letters, Vol. 13, pp. 87 – 93, 2000.
[22] H. T. Kung, J. F. Traub, “Optimal Order of One-point and Multipoint Iteration”, Journal of the Association for Computing Machinery, Vol 21, No. 4, 643 – 651, 1974.
[23] M. Kansal, V. Kanwar, S. Bhatia, “Optimized Mean Based Second Derivative-Free Families of Chebyshev-Halley Type Methods”, Numerical Analysis and Applications, Vol. 8, No.2, pp. 129 – 140, 2016.
[24] Wartono, M. Soleh, I. Suryani, Muhafzan, “Chebysev-Halley’s Method without Second Derivative of Eight-Order Convergence”. Global Journal of Pure and Applied Mathematics. Vol. 12, No. 4, pp. 2987-2997, 2016.
[25] Wartono, M. Soleh, I. Suryani, Zulakmal, Muhafzan, “A New Variant of Chebyshev-Halley’s Method without Second Derivative with Convergence of Optimal Order”. Asian Journal of Scientific Research, Vol. 11, No. 3, pp. 409 - 414, 2018.