A Six-order Variant of Newton’s Method for Solving Nonlinear Equations
Department of Mathematics, Faculty of Science
Banaras Hindu University
Varanasi, India
e-mail: manoj07777@gmail.com
Received:
Received: 30 October 2009; revised: 05 December 2009; accepted: 10 December 2009; published online: 30 December 2009
DOI: 10.12921/cmst.2009.15.02.185-193
OAI: oai:lib.psnc.pl:674
Abstract:
A new variant of Newton’s method based on contra harmonic mean has been developed and its convergence properties have been discussed. The order of convergence of the proposed method is six. Starting with a suitably chosen x0, the method generates a sequence of iterates converging to the root. The convergence analysis is provided to establish its sixth order of convergence. In terms of computational cost, it requires evaluations of only two functions and two first order derivatives per iteration. This implies that efficiency index of our method is 1.5651. The proposed method is comparable with the methods of Parhi, and Gupta [15] and that of Kou and Li [8]. It does not require the evaluation of the second order derivative of the given function as required in the family of Chebyshev-Halley type methods. The efficiency of the method is tested on a number of numerical examples. It is observed that our method takes lesser number of iterations than Newton’s method and the other third order variants of Newton´s method. In comparison with the sixth order methods, it behaves either similarly or better for the examples considered.
Key words:
efficiency index, function evaluations, iteration function, Newton's method, order of convergence
References:
[1] C. Chun, Some improvements of Jarratt’s method with sixth-order convergence. Appl. Math. Comput. 190, 1432- 1437 (2007).
[2] J. Chen, Some new iterative methods with three-order convergence. Appl. Math. Comput. 181, 1519-1522 (2006).
[3] M. Frontini and E. Sormani, Some variant of Newton’s method with third-order convergence. Appl. Math. Comput. 140, 419-426 (2003).
[4] M. Grau and M. Noguera, A variant of Cauchy’s method with accelerated fifth-order convergence. Appl. Math. Lett. 17, 509-517 (2004).
[5] M.K. Jain, S.R.K. Iyengar and R.K. Jain, Numerical Methods for Scientific and Engineering Computation. New York, NY: Halsted Press, 1985.
[6] J. Kou and X. Wang, Sixth-order variants of Chebyshev-Halley methods for solving non-linear equations. Appl. Math. Comput. 190, 1839-1843 (2007).
[7] J. Kou, On Chebyshev-Halley methods with sixth-order convergence for solving non-linear equations. Appl. Math. Comput. 190, 126-131 (2007).
[8] J. Kou and Y. Li, An improvement of Jarratt method. Appl. Math. Comput. 189, 1816-1821 (2007).
[9] J. Kou, Y. Li, and X. Wang, Some modifications of Newton’s method with fifth order convergence. Journal of Computational and Applied Mathematics 209, 146- 152 (2009).
[10] T. Lukic and N.M. Ralevie, Geometric mean Newton’s method for simple and multiple roots. Applied Mathematics Letters 21, 30-36 (2008).
[11] A.K. Maheshwari, A fourth-order iterative method for solving non-linear equations. Applied Mathematics and Computation 211, 383-391 (2009).
[12] A.M. Ostrowski, Solution of Equations and Systems of Equations. Academic Press Inc., 1966.
[13] J.M. Ortega and W.C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, New York, 1970.
[14] A.Y. Ozban, Some new variants of Newton’s method. Applied Mathematics Letters 17, 677-682 (2004).
[15] S.K. Parhi and D.K. Gupta, A sixth order method for nonlinear equations. Applied Mathematics and Computation 203, 50-55 (2008).
[16] J.R. Sharma and R.K. Guha, A family of modified Ostrowski methods with accelerated sixth order convergence. Appl. Math. Comput. 190, 111-115 (2007).
[17] J.F. Traub, Iterative Methods for the Solution of Equations. Prentice Hall, Clifford, NJ, 1964.
[18] S. Weerakoon and T.G.I. Fernando, A variant of Newton’s method with accelerated third-order convergence. Appl. Math. Lett. 13, 87-93 (2000).
A new variant of Newton’s method based on contra harmonic mean has been developed and its convergence properties have been discussed. The order of convergence of the proposed method is six. Starting with a suitably chosen x0, the method generates a sequence of iterates converging to the root. The convergence analysis is provided to establish its sixth order of convergence. In terms of computational cost, it requires evaluations of only two functions and two first order derivatives per iteration. This implies that efficiency index of our method is 1.5651. The proposed method is comparable with the methods of Parhi, and Gupta [15] and that of Kou and Li [8]. It does not require the evaluation of the second order derivative of the given function as required in the family of Chebyshev-Halley type methods. The efficiency of the method is tested on a number of numerical examples. It is observed that our method takes lesser number of iterations than Newton’s method and the other third order variants of Newton´s method. In comparison with the sixth order methods, it behaves either similarly or better for the examples considered.
Key words:
efficiency index, function evaluations, iteration function, Newton's method, order of convergence
References:
[1] C. Chun, Some improvements of Jarratt’s method with sixth-order convergence. Appl. Math. Comput. 190, 1432- 1437 (2007).
[2] J. Chen, Some new iterative methods with three-order convergence. Appl. Math. Comput. 181, 1519-1522 (2006).
[3] M. Frontini and E. Sormani, Some variant of Newton’s method with third-order convergence. Appl. Math. Comput. 140, 419-426 (2003).
[4] M. Grau and M. Noguera, A variant of Cauchy’s method with accelerated fifth-order convergence. Appl. Math. Lett. 17, 509-517 (2004).
[5] M.K. Jain, S.R.K. Iyengar and R.K. Jain, Numerical Methods for Scientific and Engineering Computation. New York, NY: Halsted Press, 1985.
[6] J. Kou and X. Wang, Sixth-order variants of Chebyshev-Halley methods for solving non-linear equations. Appl. Math. Comput. 190, 1839-1843 (2007).
[7] J. Kou, On Chebyshev-Halley methods with sixth-order convergence for solving non-linear equations. Appl. Math. Comput. 190, 126-131 (2007).
[8] J. Kou and Y. Li, An improvement of Jarratt method. Appl. Math. Comput. 189, 1816-1821 (2007).
[9] J. Kou, Y. Li, and X. Wang, Some modifications of Newton’s method with fifth order convergence. Journal of Computational and Applied Mathematics 209, 146- 152 (2009).
[10] T. Lukic and N.M. Ralevie, Geometric mean Newton’s method for simple and multiple roots. Applied Mathematics Letters 21, 30-36 (2008).
[11] A.K. Maheshwari, A fourth-order iterative method for solving non-linear equations. Applied Mathematics and Computation 211, 383-391 (2009).
[12] A.M. Ostrowski, Solution of Equations and Systems of Equations. Academic Press Inc., 1966.
[13] J.M. Ortega and W.C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, New York, 1970.
[14] A.Y. Ozban, Some new variants of Newton’s method. Applied Mathematics Letters 17, 677-682 (2004).
[15] S.K. Parhi and D.K. Gupta, A sixth order method for nonlinear equations. Applied Mathematics and Computation 203, 50-55 (2008).
[16] J.R. Sharma and R.K. Guha, A family of modified Ostrowski methods with accelerated sixth order convergence. Appl. Math. Comput. 190, 111-115 (2007).
[17] J.F. Traub, Iterative Methods for the Solution of Equations. Prentice Hall, Clifford, NJ, 1964.
[18] S. Weerakoon and T.G.I. Fernando, A variant of Newton’s method with accelerated third-order convergence. Appl. Math. Lett. 13, 87-93 (2000).
