Improved Ostrowski-Like Methods Based on Cubic Curve Interpolation
ABSTRACT
In this paper, we derive two higher order multipoint methods for solving nonlinear equations. The methodology is based on Ostrowski’s method and further developed by using cubic interpolation process. The adaptation of this strategy increases the order of Ostrowski’s method from four to eight and its efficiency index from 1.587 to 1.682. The methods are compared with closest competitors in a series of numerical examples. Moreover, theoretical order of convergence is verified on the examples.
In this paper, we derive two higher order multipoint methods for solving nonlinear equations. The methodology is based on Ostrowski’s method and further developed by using cubic interpolation process. The adaptation of this strategy increases the order of Ostrowski’s method from four to eight and its efficiency index from 1.587 to 1.682. The methods are compared with closest competitors in a series of numerical examples. Moreover, theoretical order of convergence is verified on the examples.
KEYWORDS
Nonlinear Equations, Ostrowski’s Method, Root-Finding, Order of Convergence, Cubic Interpolation
Nonlinear Equations, Ostrowski’s Method, Root-Finding, Order of Convergence, Cubic Interpolation
Cite this paper
nullJ. Sharma, R. Guha and R. Sharma, "Improved Ostrowski-Like Methods Based on Cubic Curve Interpolation," Applied Mathematics, Vol. 2 No. 7, 2011, pp. 816-823. doi: 10.4236/am.2011.27109.
nullJ. Sharma, R. Guha and R. Sharma, "Improved Ostrowski-Like Methods Based on Cubic Curve Interpolation," Applied Mathematics, Vol. 2 No. 7, 2011, pp. 816-823. doi: 10.4236/am.2011.27109.
References
[1] J. F. Traub, “Iterative Methods for the Solution of Equations,” Prentice Hall, Englewood Cliffs, 1964. [2] H. T. Kung and J. F. Traub, “Optimal Order of One-Point and Multipoint Iteration,” Journal of the Association for Computing Machinery, Vol. 21, No. 4, 1974, pp. 643-651. doi: 10.1145/321850.321860 [3] A. M. Ostrowski, “Solutions of Equations and System of Equations,” Academic Press, New York, 1960. [4] M. Grau and J. L. Díaz-Barrero, “An Improvement to Ostrowski Root-Finding Method,” Applied Mathematics and Computation, Vol. 173, No. 1, 2006, pp. 450-456. doi:10.1016/j.amc.2005.04.043 [5] J. R. Sharma and R. K. Guha, “A Family of Modified Ostrowski Methods with Accelerated Sixth Order Convergence,” Applied Mathematics and Computation, Vol. 190, No. 1, 2007, pp. 111-115. doi:10.1016/j.amc.2007.01.009 [6] G. M. Phillips and P. J. Taylor, “Theory and Applications of Numerical Analysis,” Academic Press, New York, 1996. [7] S. Weerakoon and T. G. I Fernando, “A Variant of Newton’s Method with Accelerated Third-Order Convergence,” Applied Mathematics Letters, Vol. 13, No. 8, 2000, pp. 87-93. doi:10.1016/S0893-9659(00)00100-2
[1] J. F. Traub, “Iterative Methods for the Solution of Equations,” Prentice Hall, Englewood Cliffs, 1964. [2] H. T. Kung and J. F. Traub, “Optimal Order of One-Point and Multipoint Iteration,” Journal of the Association for Computing Machinery, Vol. 21, No. 4, 1974, pp. 643-651. doi: 10.1145/321850.321860 [3] A. M. Ostrowski, “Solutions of Equations and System of Equations,” Academic Press, New York, 1960. [4] M. Grau and J. L. Díaz-Barrero, “An Improvement to Ostrowski Root-Finding Method,” Applied Mathematics and Computation, Vol. 173, No. 1, 2006, pp. 450-456. doi:10.1016/j.amc.2005.04.043 [5] J. R. Sharma and R. K. Guha, “A Family of Modified Ostrowski Methods with Accelerated Sixth Order Convergence,” Applied Mathematics and Computation, Vol. 190, No. 1, 2007, pp. 111-115. doi:10.1016/j.amc.2007.01.009 [6] G. M. Phillips and P. J. Taylor, “Theory and Applications of Numerical Analysis,” Academic Press, New York, 1996. [7] S. Weerakoon and T. G. I Fernando, “A Variant of Newton’s Method with Accelerated Third-Order Convergence,” Applied Mathematics Letters, Vol. 13, No. 8, 2000, pp. 87-93. doi:10.1016/S0893-9659(00)00100-2