Optimal Fourth-Order Iterative Methods for Solving Nonlinear Equations

I.A. Al-Subaihi

doi:https://doi.org/10.14445/22315373/IJMTT-V13P503

International Journal of Mathematics Trends and Technology

Research Article | Open Access | Download PDF

Volume 13 | Number 1 | Year 2014 | Article Id. IJMTT-V13P503 | DOI : https://doi.org/10.14445/22315373/IJMTT-V13P503

Optimal Fourth-Order Iterative Methods for Solving Nonlinear Equations

I.A. Al-Subaihi

Abstract

This paper presents a new optimal fourth-order iterative methods for solving nonlinear equations f(x)=0. The proposed iterative methods are obtained by composing the fourth-order iterative family method proposed by King 1973 with the ellipse method described by Gupta et al.(1998), which is always defined even if the first directive of the function at a point is zero. The convergence analysis of the proposed iterative methods was performed. Several numerical examples are also considered to illustrate the efficiency and the accuracy of the proposed iterative methods.

Keywords

Nonlinear Equations, Optimal Iterative Methods, Order of Convergence, Newton-Type Method, Ellipse Method

References

1. S. Abbasbandy, Improving Newton Raphson method for nonlinear equations by modified Adomian decomposition method. Appl. Math. Comput. 145, pp. 887–893, 2003.
2. I.K. Argyros, D. Chen, Q. Qian. The Jarratt method in Banach space setting. J. Comput. Appl. Math. 51, pp. 103-106, 1994.
3. K.E. Atkinson. A Survey of Numerical Methods for Solving Nonlinear Integral. Journal of Integral Equations and Applications, 4, 1, pp. 15-46, 1992.
4. W. Bi, H. Ren, Q. Wu. A new family of eight-order iterative methods for solving nonlinear equations. Appl. Math. Comput. 214,pp. 236i–245, 2009.
5. R.L. Burden, J.D. Faires. Numerical Analysis, Ninth Edition, Brooks/Cole, Richard Stratton, 2011.
6. C.Chun. Some fourth-order iterative methods for solving nonlinear equations. Appl. Math. Comute. 195, pp. 454-459, 2008.
7. C. Chun, Yoon Mee Ham. Some fourth-order modification of Newton’s method. Appl. Math. Comput. 197, pp. 654-658, 2008.
8. C. Chun. Some Variants of King’s Fourth-Order Family of Methods for Nonlinear Equations. Applied Mathe- matics and Computation. 190, 1, pp. 57- 62, 2007.
9. C. Chun. A family of composite fourth-order iterative methods for solving nonlinear equations. Appl. Math. Comute. 187, pp. 951–956, 2007.
10. K. C. Gupta, V. Kanwar, Sanjeev Kumar. A Family of Ellipse Methods for Solving Nonlinear Equations. International Journal of Mathematical Education in Science and Technology 40, 4, pp. 571-575, 2009.
11. L.W. Johnson, D.R. Scholz. On Steffensen’s Method. SIAM Journal on Numerical Analysis. 5, 2, 296–302, 1968.
12. V. Kanwar, S. K..Tomar. Exponentially Fitted Variants of Newton’s Method with Quadratic and Cubic Convergence. International Journal of Computer Mathematics 86, 9, pp. 1603-1611, 2009.
13. V. Kanwar, S.K., Tomar. Modified Families of Newton, Halley and Chebyshev Methods. Applied Mathe- matics and Computation 192, 1, pp. 20-26, 2007.
14. Richard F. King. A Family of Fourth Order Methods for Nonlinear Equations. SIAM J. NUMER. ANAL. 10, 5, pp. 876-879, 1973.
15. J. Kou, Y. Li, X. Wan. A family of fourth-order methods for solving nonlinear equations. Appl. Math. Comput. 188, pp. 1031-1036, 2007.
16. J. Kou, Y. Li, X. Wang. A composite fourth-order iterative method for solving nonlinear equations. Appl. Math. Comput. 184,pp. 471-475, 2007.
17. Sanjeev Kumar, Vinay Kanwar, Sukhjit Singh. Modified Efficient Families of Two and Three-Step Predictor-Corrector Iterative Methods for Solving Nonlinear Equations. Applied Mathematics, 1, pp. 153-158, 2010.
18. H.T. Kung, J.F. Traub. Optimal order of one-point and multi-point iteration. J. ACM 21, pp. 643-651, 1974.
19. S. Li, J. Tan, J. Xie, Y. Dong. A New Fourth-order Convergent Iterative Method Based on Thiele’s Continued Fraction for Solving Equations. Journal of Information & Computational Science. 8, 1, pp. 139–145, 2011.
20. T. Lotfi. A new optimal method of fourth-order convergence for solving nonlinear equations. Int. J. Industrial Mathematics. 6, 2, pp. 121-124, 2014.
21. N.A. Mir, K. Ayub, A. Rafiq. A Third-Order Convergent Iterative Method for Solving Non-Linear Equations. International Journal of Computer Mathematics 87, 4, pp. 849-854, 2010.
22. A.M. Ostrowski. Solution of Equations in Euclidean and Banach Space. 3rd Edition, Academic Press, New York, 1973.
23. Miodrag S. Petkovic'. Beny Neta, , Ljiljana D. Petkovic', Jovana Džunic. Multipoint methods for solving nonlinear equations: A survey. Applied Mathematics and Computation. 226, pp. 635-660, 2014.
24. S. Miodrag Petkovic'. On Optimal Multipoint Methods for Solving Nonlinear Equations. Novi Sad J. Math. Vol. 39, 1, pp. 123-130, 2009.
25. Miodrag S. Petkovic', Ljiljana D. Petkovic'. Families of Optimal Multipont Methods for Solving Nonlinear Equations: A Survey. Appl. Anal. Discrete Math. 4, pp. 1–22, 2010.
26. S. Weerakoon, G.I. Fernando,. A variant of Newton’s method with accelerated third-order convergence. Appl. Math. Lett. 17, 8, pp. 87-93, 2000.
27. Janak Raj Sharma, Rajni Sharma. A new family of modifed Ostrowski’s methods with accelerated eighth order convergence. Numer Algor. 54, pp. 445–458, 2010.
28. Hani I. Siyyam, , I.A. Al-Subaihi. A New Class of Optimal Fourth-Order Iterative Methods for Solving Nonlinear Equations. European Journal of Scientific Research. 109, 2, pp. 224-231, 2013.
29. R. Thukral, M. S. Petkovi´c. Family of three-point methods of optimal order for solving nonlinear equations. J. Comput. Appl. Math. 233, pp. 2278–2284, 2010.
30. J.F. Traub. Iterative Methods for the Solution of Equations. Prentice Hall, New Jersey, 1964.

Citation :

I.A. Al-Subaihi, "Optimal Fourth-Order Iterative Methods for Solving Nonlinear Equations," International Journal of Mathematics Trends and Technology (IJMTT), vol. 13, no. 1, pp. 13-18, 2014. Crossref, https://doi.org/10.14445/22315373/IJMTT-V13P503