A New Family of Nonlinear Fifth-Order Solvers for Finding Simple Roots

Author(s)
Behzad Ghanbari

ABSTRACT

In this paper, we present a new family of iterative methods for solving nonlinear equations. It is proved that the order of convergence of this family is five. Two functions and two derivative evaluations should be computed per iteration. To demonstrate convergence properties of the proposed family of methods, some numerical examples are given. Further numerical comparisons are made with several other existing fifth-order methods.

In this paper, we present a new family of iterative methods for solving nonlinear equations. It is proved that the order of convergence of this family is five. Two functions and two derivative evaluations should be computed per iteration. To demonstrate convergence properties of the proposed family of methods, some numerical examples are given. Further numerical comparisons are made with several other existing fifth-order methods.

Cite this paper

B. Ghanbari, "A New Family of Nonlinear Fifth-Order Solvers for Finding Simple Roots,"*Applied Mathematics*, Vol. 3 No. 6, 2012, pp. 577-580. doi: 10.4236/am.2012.36088.

B. Ghanbari, "A New Family of Nonlinear Fifth-Order Solvers for Finding Simple Roots,"

References

[1] J. F. Traub, “Iterative Methods for the Solution of Equations,” Prentice Hall, Upper Saddle River, 1964.

[2] C. Chun, “Some Variants of King’s Fourth-Order Family of Methods for Nonlinear Equations,” Applied Mathematics and Computation, Vol. 190, No. 1, 2007, pp. 5762. doi:10.1016/j.amc.2007.01.006

[3] A. M. Ostrowski, “Solution of Equations in Euclidean and Banach Space,” Academic Press, New York, 1973.

[4] B. Ghanbari, “A New General Fourth-Order Family of Methods for Findingsimple Roots of Nonlinear Equations,” Journal of King Saud University—Science, Vol. 23, No. 1, 2011, pp. 395-398.

[5] L. Fang, L. Sun and G. He, “An Efficient Newton-Type Method with Fifth-Order Convergence for Solving Nonlinear Equations,” Computational Applied Mathematics, Vol. 27, No. 3, 2008, pp. 269-274.

[6] Y. M. Ham and C. Chun, “A Fifth-Order Iterative Method for Solving Nonlinear Equations,” Applied Mathematics and Computation, Vol. 194, No. 1, 2007, pp. 287-290. doi:10.1016/j.amc.2007.04.005

[7] L. Fang and G. He, “Some Modifications of Newton’s Method with Higher-Order Convergence for Solving Nonlinear Equations,” Journal of Computational and Applied Mathematics, Vol. 228, No. 1, 2009, pp. 296-303. doi:10.1016/j.cam.2008.09.023

[8] J. Biazar and B. Ghanbari, “Some Higher-Order Families of Methods for Finding Simple Roots of Nonlinear Equations,” General Mathematics Notes, Vol. 7, No. 1, 2011, pp. 46-51.

[9] W. Gautschi, “Numerical Analysis: An Introduction,” Birkh?user, Boston, 1997.

[10] M. Grau and J. L. Diaz-Barrero, “An Improvement of the Euler-Chebyshev Iterative Method,” Journal of Mathematical Analysis and Applications, Vol. 315, No. 1, 2006, pp. 1-7. doi:10.1016/j.jmaa.2005.09.086

[11] Y. M. Ham, C. Chun and S. Lee, “Some Higher-Order Modifications of Newton’s Method for Solving Nonlinear Equations,” Journal of Computational and Applied Mathematics, Vol. 222, No. 2, 2008, pp. 477-486. doi:10.1016/j.cam.2007.11.018

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

[1] J. F. Traub, “Iterative Methods for the Solution of Equations,” Prentice Hall, Upper Saddle River, 1964.

[2] C. Chun, “Some Variants of King’s Fourth-Order Family of Methods for Nonlinear Equations,” Applied Mathematics and Computation, Vol. 190, No. 1, 2007, pp. 5762. doi:10.1016/j.amc.2007.01.006

[3] A. M. Ostrowski, “Solution of Equations in Euclidean and Banach Space,” Academic Press, New York, 1973.

[4] B. Ghanbari, “A New General Fourth-Order Family of Methods for Findingsimple Roots of Nonlinear Equations,” Journal of King Saud University—Science, Vol. 23, No. 1, 2011, pp. 395-398.

[5] L. Fang, L. Sun and G. He, “An Efficient Newton-Type Method with Fifth-Order Convergence for Solving Nonlinear Equations,” Computational Applied Mathematics, Vol. 27, No. 3, 2008, pp. 269-274.

[6] Y. M. Ham and C. Chun, “A Fifth-Order Iterative Method for Solving Nonlinear Equations,” Applied Mathematics and Computation, Vol. 194, No. 1, 2007, pp. 287-290. doi:10.1016/j.amc.2007.04.005

[7] L. Fang and G. He, “Some Modifications of Newton’s Method with Higher-Order Convergence for Solving Nonlinear Equations,” Journal of Computational and Applied Mathematics, Vol. 228, No. 1, 2009, pp. 296-303. doi:10.1016/j.cam.2008.09.023

[8] J. Biazar and B. Ghanbari, “Some Higher-Order Families of Methods for Finding Simple Roots of Nonlinear Equations,” General Mathematics Notes, Vol. 7, No. 1, 2011, pp. 46-51.

[9] W. Gautschi, “Numerical Analysis: An Introduction,” Birkh?user, Boston, 1997.

[10] M. Grau and J. L. Diaz-Barrero, “An Improvement of the Euler-Chebyshev Iterative Method,” Journal of Mathematical Analysis and Applications, Vol. 315, No. 1, 2006, pp. 1-7. doi:10.1016/j.jmaa.2005.09.086

[11] Y. M. Ham, C. Chun and S. Lee, “Some Higher-Order Modifications of Newton’s Method for Solving Nonlinear Equations,” Journal of Computational and Applied Mathematics, Vol. 222, No. 2, 2008, pp. 477-486. doi:10.1016/j.cam.2007.11.018

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