AM  Vol.4 No.12 , December 2013
New Ninth Order J-Halley Method for Solving Nonlinear Equations
ABSTRACT

In the paper [1], authors have suggested and analyzed a predictor-corrector Halley method for solving nonlinear equations. In this paper, we modified this method by using the finite difference scheme, which had a quantic convergence. We have compared this modified Halley method with some other iterative methods of ninth order, which shows that this new proposed method is a robust one. Some examples are given to illustrate the efficiency and the performance of this new method.


Cite this paper
Ahmad, F. , Hussain, S. , Hussain, S. and Rafiq, A. (2013) New Ninth Order J-Halley Method for Solving Nonlinear Equations. Applied Mathematics, 4, 1709-1713. doi: 10.4236/am.2013.412233.
References
[1]   K. I. Noor and M. Aslam Noor, “Predictor-Corrector Halley Method for Nonlinear Equations,” Applied Mathematics and Computation, Vol. 188, No. 2, 2007, pp. 15871591.
http://dx.doi.org/10.1016/j.amc.2006.11.023

[2]   J. Kou and Y. T. Li, “The Improvements of ChebyshevHalley Methods with Fifth-Order Convergence,” Applied Mathematics and Computation, Vol. 188, No. 1, 2007, pp. 143-147.
http://dx.doi.org/10.1016/j.amc.2006.09.097

[3]   J. Kou, Y. T. Li and X. H. Wang, “A Family of FifthOrder Iterations Composed of Newton and Third-Order Methods,” Applied Mathematics and Computation, Vol. 186, No. 2, 2007, pp. 1258-1262.
http://dx.doi.org/10.1016/j.amc.2006.07.150

[4]   J. Kou and Y. Li, “An Improvement of the Jarrat Method,” Applied Mathematics and Computation, Vol. 189, No. 2, 2007, pp. 1816-1821.
http://dx.doi.org/10.1016/j.amc.2006.12.062

[5]   Z. Y. Hu, L. Guocai and L. Tian, “An Iterative Method with Ninth-Order Convergence for Solving Nonlinear Equations,” International Journal of Contemporary Mathematical Sciences, Vol. 6, No. 1, 2011, pp. 17-23.

[6]   M. A. Noor and K. I. Noor, “Fifth-Order Iterative Methods for Solving Nonlinear Equations,” Applied Mathematics and Computation, Vol. 188, No. 1, 2007, pp. 406410.
http://dx.doi.org/10.1016/j.amc.2006.10.007

[7]   S. Amat, S. Busquier and J. M. Gutierrez, “Geometric Construction of Iterative Functions to Solve Nonlinear Equations,” Journal of Computational and Applied Mathematics, Vol. 157, No. 1, 2003, pp. 197-205.
http://dx.doi.org/10.1016/S0377-0427(03)00420-5

[8]   I. K. Argyros, D. Chen and Q. Qian, “The Jarratt Method in Banach Space Setting,” Journal of Computational and Applied Mathematics, Vol. 51, No. 1, 1994, pp. 103-106.
http://dx.doi.org/10.1016/0377-0427(94)90093-0

[9]   J. A. Ezquerro and M. A. Hernandez, “A Uniparametric Halley-Type Iteration with Free Second Derivative,” International Journal of Pure and Applied Mathematics, Vol. 6, No. 1, 2003, pp. 103-114.

[10]   J. A. Ezquerro and M. A. Hernandez, “On Halley-Type Iterations with Free Second Derivative,” Journal of Computational and Applied Mathematics, Vol. 170, No. 2, 2004, pp. 455-459.
http://dx.doi.org/10.1016/j.cam.2004.02.020

[11]   E. Halley, “A New Exact and Easy Method for Finding the Roots of Equations Generally and without any Previous Reduction,” Philosophical Transactions of the Royal Society of London, Vol. 18, 1964, pp. 136-147.
http://dx.doi.org/10.1098/rstl.1694.0029

[12]   A. Melman, “Geometry and Convergence of Halley’s Method,” SIAM Review, Vol. 39, No. 4, 1997, pp. 728735. http://dx.doi.org/10.1137/S0036144595301140

[13]   M. A. Noor, “Numerical Analysis and Optimization,” Lecture Notes, Mathematics Department, COMSATS Institute of Information Technology, Islamabad, 2006.

[14]   J. F. Traub, “Iterative Methods for Solution of Equations,” Prentice-Hall, Englewood, 1964.

 
 
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