Abstract
In this paper, we present a new tri-parametric derivative-free family of Hansen-Patrick type methods for solving nonlinear equations numerically. The proposed family requires only three functional evaluations to achieve optimal fourth order of convergence. In addition, acceleration of convergence speed is attained by suitable variation of free parameters in each iterative step. The self-accelerating parameters are estimated from the current and previous iteration. These self-accelerating parameters are calculated using Newton’s interpolation polynomials of third and fourth degrees. Consequently, the R-order of convergence is increased from 4 to 7, without any additional functional evaluation. Furthermore, the most striking feature of this contribution is that the proposed schemes can also determine the complex zeros without having to start from a complex initial guess as would be necessary with other methods. Numerical experiments and the comparison of the existing robust methods are included to confirm the theoretical results and high computational efficiency.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Traub, J.F.: Iterative Methods for the Solution of Equations. Prentice-Hall, Englewood Cliffs (1964)
Weerakon, S., Fernando, T.G.I.: A variant of Newton’s method with accelerated third-order convergence. Appl. Math. Lett. 13, 87–93 (2000)
Amat, S., Busquier, S., Gutiérrez, J.M.: Geometric constructions of iterative functions to solve nonlinear equations. J. Comput. Appl. Math. 157, 197–205 (2003)
Hansen, E., Patrick, M.: A family of root finding methods. Numer. Math. 27, 257–269 (1977)
Kung, H.T., Traub, J.F.: Optimal order of one-point and multi-point iteration. J. Assoc. Comput. Math. 21, 643–651 (1974)
Ostrowski, A.M.: Solutions of Equations and System of Equations. Academic Press, New York (1960)
King, R.F.: A family of fourth order methods for nonlinear equations. SIAM J. Numer. Anal. 10, 876–879 (1973)
Jarratt, P.: Some efficient fourth-order multipoint methods for solving equations. BIT 9, 119–124 (1969)
Sharma, J.R., Guha, R.K., Sharma, R.: Some variants of Hansen-Patrick method with third and fourth order convergence. Appl. Math. Comput. 214, 171–177 (2009)
Kansal, M., Kanwar, V., Bhatia, S.: New modifications of Hansen-Patrick’s family with optimal fourth and eighth orders of convergence. Appl. Math. Comput. 269, 507–519 (2015)
Ortega, J.M., Rheinboldt, W.C.: Iterative Solutions of Nonlinear Equations in Several Variables. Academic Press, New York (1970)
Petković, M.S., Dz̆unić, J., Petković, L.D.: A family of two-point methods with memory for solving nonlinear equations. Appl. Anal. Discrete Math. 5, 298–317 (2011)
Sharifi, S., Siegmund, S., Salimi, M.: Solving nonlinear equations by a derivative-free form of the King’s family with memory. Calcolo. doi:10.1007/s10092-015-0144-1
Sharma, J.R., Guha, R.K., Gupta, P.: Some efficient derivative free methods with memory for solving nonlinear equations. Appl. Math. Comput. 219, 699–707 (2012)
Dz̆unić, J., Petković, M.S., Petković, L.D.: Three-point methods with and without memory for solving nonlinear equations. Appl. Math. Comput. 218, 4917–4927 (2012)
Cordero, A., Fardi, M., Ghasemi, M., Torregrosa, J.R.: Accelerated iterative methods for ?nding solutions of nonlinear equations and their dynamical behavior. Calcolo 51, 17–30 (2014)
Cordero, A., Torregrosa, J.R., Vassileva, M.P.: Three-step iterative methods with optimal eighth-order convergence. J. Comput. Appl. Math. 235, 3189–3194 (2011)
Andreu, C., Cambil, N., Cordero, A., Torregrosa, J.R.: A class of optimal eighth-order derivative free methods for solving the Danchick-Guass problem. Appl. Math. Comput. 232, 237–246 (2014)
Dzunic, J., Petkovic, M.S.: On generalized multipoint root-solvers with memory. J. Comput. Appl. Math. 236, 2909–2920 (2012)
Hazrat, R.: Mathematica: A Problem-Centered Approach. Springer, New York (2010)
Zheng, Q., Li, J., Huang, F.: An optimal Steffensen-type family for solving nonlinear equations. Appl. Math. Comput. 217, 9592–9597 (2011)
Soleymani, F., Sharma, R., Li, X., Tohidi, E.: An optimized derivative-free form of the Potra-Pták method. Math. Comput. Modell. 56, 97–104 (2012)
Cordero, A., Lotfi, T., Bakhtiari, P., Torregrosa, J.R.: An efficient two-parametric family with memory for nonlinear equations. Numer. Algoritm. 68, 323–335 (2015)
Dzunic, J.: On efficient two-parameter methods for solving nonlinear equations. Numer. Algoritm. 63(3), 549–569 (2013)
Jay, I.O.: A note on Q-order of convergence. BIT Numer. Math. 41, 422–429 (2011)
Danby, J.M.A., Burkardt, T.M.: The solution of Kepler’s equation. I. Celest. Mech. 31, 95–107 (1983)
Otolorin, O.: A new Newton-like iterative method for roots of analytic functions. Int. J. Math. Ed. Sci. Tech. 36, 539–572 (2005)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Kansal, M., Kanwar, V. & Bhatia, S. Efficient derivative-free variants of Hansen-Patrick’s family with memory for solving nonlinear equations. Numer Algor 73, 1017–1036 (2016). https://doi.org/10.1007/s11075-016-0127-6
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11075-016-0127-6
Keywords
- Multipoint iterative methods
- Derivative-free methods
- Methods with memory
- R-order of convergence
- Computational efficiency