Two new three-step classes of optimal iterative methods to approximate simple roots of nonlinear equations, satisfying the Kung-Traub’s conjecture, are designed. The development of the methods and their convergence analysis are provided joint with a generalization of both processes. In order to check the goodness of the theoretical results, some concrete methods are extracted and numerical and dynamically compared with some known methods.
This is a preview of subscription content, log in to check access.
Buy single article
Instant unlimited access to the full article PDF.
Price includes VAT for USA
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
This is the net price. Taxes to be calculated in checkout.
Higham, N.J.: Funstions of Matrices: Theory and Computation. SIAM, Philadelphia (2008)
Chun, C., Kim, Y.: Several new third-order iterative methods for solving nonlinear equations. Acta Appl. Math. 109(3), 1053–1063 (2010)
Cordero, A., Torregrosa, J.R.: Variants of Newton’s method using fifth-order quadrature formulas. Appl. Math. Comput. 190, 686–698 (2007)
Cordero, A., Hueso, J.L., Martínez, E., Torregrosa, J.R.: A family of iterative methods with sixth and seventh order convergence for nonlinear equations. Math. Comput. Model. 52, 1490–1496 (2010)
Weerakoon, S., Fernando, T.G.I.: A variant of Newton’s method with accelerated third-order convergence. Appl. Math. Lett. 13(8), 87–93 (2000)
Wang, H., Liu, H.: Note on a cubically convergent Newton-type method under weak conditions. Acta Appl. Math. 110(2), 725–735 (2010)
Ostrowski, A.M.: Solution of Equations and Systems of Equations. Prentice-Hall, Englewood Cliffs (1964)
Kung, H.T., Traub, J.F.: Optimal order of one-point and multi-point iteration. J. Assoc. Comput. Math. 21, 643–651 (1974)
Petković, M.S., Neta, B., Petković, L.D., Dz̆nić, J.: Multipoint Methods for Solving Nonlinear Equations. Elsevier, Amsterdam (2013)
Petković, M.S., Petković, L.D.: Families of optimal multipoint methods for solving polynomial equations. Appl. Anal. Discrete Math. 4, 1–22 (2010)
Soleymani, F.: Two novel classes of two-step optimal methods for all the zeros in an interval. Afr. Math. (2012). doi:10.1007/s13370-012-0112-8
Džunić, J., Petković, M.S., Petković, L.D.: A family of optimal three-point methods for solving nonlinear equations using two parametric functions. Appl. Math. Comput. 217(19), 7612–7619 (2011)
Thukral, R., Petković, M.S.: A family of three-point methods of optimal order for solving nonlinear equation. J. Comput. Appl. Math. 233(9), 2278–2284 (2010)
Obrechkoff, N.: Sur la solution numeriue des equations. God. Sofij. Univ. 56(1), 73–83 (1963)
Jarratt, P.: Some fourth order multipoint iterative methods for solving equations. Math. Comput. 20, 434–437 (1966)
Petković, M.S.: Multipoint methods for solving nonlinear equations: a survey. Appl. Math. Comput. 226, 635–660 (2014)
Džunić, J., Petković, M.S.: A family of three-point methods of Ostrowski’s type for solving nonlinear equations. J. Appl. Math. 2012, 425867 (2012)
Soleymani, F., Vanani, S.K., Afghani, A.: A general three-step class of optimal iterations for nonlinear equations. Math. Probl. Eng. 2011, 469512 (2011). 10 pp.
Geum, Y.H., Kim, Y.I.: A uniparametric family of three-step eighth-order multipoint iterative methods for simple roots. Appl. Math. Lett. 24, 929–935 (2011)
Geum, Y.H., Kim, Y.I.: A biparametric family of eighth-order methods with their third-step weighting function decomposed into a one-variable linear fraction and a two-variable generic function. Comput. Math. Appl. 61, 708–714 (2011)
Jay, I.O.: A note on Q-order of convergence. BIT Numer. Math. 41, 422–429 (2001)
Chicharro, F., Cordero, A., Torregrosa, J.R.: Drawing dynamical and parameters planes of iterative families and methods. Sci. World J. 2013, 780153 (2013). 11 pp.
The authors would like to render thanks to the reviewers for their comments and suggestions that have improved the readability of the paper.
This research was supported by Ministerio de Ciencia y Tecnología MTM2011-28636-C02-02.
About this article
Cite this article
Cordero, A., Lotfi, T., Mahdiani, K. et al. Two Optimal General Classes of Iterative Methods with Eighth-Order. Acta Appl Math 134, 61–74 (2014) doi:10.1007/s10440-014-9869-0
- Multipoint iterative method
- Nonlinear equation
- Optimal order
- Kung-Traub’s conjecture
- Kung-Traub’s method
Mathematics Subject Classification