Abstract
Solving nonlinear equations by using iterative methods is discussed in this paper. An optimally convergent class of efficient three-point three-step methods without memory is suggested. Analytical proof for the class of methods is given to show the eighth-order convergence and also reveal its consistency with the conjecture of Kung and Traub. The beauty in the proposed methods from the class can be seen because of the optimization in important effecting factors, i.e. optimality order, lesser number of functional evaluations; as well as in viewpoint of efficiency index. The accuracy of some iterative methods from the proposed derivative-involved scheme is illustrated by solving numerical test problems and comparing with the available methods in the literatures.
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Cordero A., Hueso J.L., Martinez 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)
Durazzi C., Ruggiero V.: A Newton inexact interiorpoint method for large-scale nonlinear optimization problems. Annali dell’Universita’ di Ferrara 79, 333–357 (2003)
Ezzati R., Saleki F.: On the construction of new iterative methods with fourth-order convergence by combining previous methods. Int. Math. Forum 6, 1319–1326 (2011)
Iliev, A., Kyurkchiev, N.: Nontrivial Methods in Numerical Analysis: Selected Topics in Numerical Analysis. LAP Lambert Academic Publishing (2010)
Kung H.T., Traub J.F.: Optimal order of one-point and multipoint iteration. J. Assoc. Comput. Mach. 21, 643–651 (1974)
Kou J., Wang X., Sun S.: Some new root-finding methods with eighth-order convergence. Bull. Math. Soc. Sci. Math. Roumanie 53, 133–143 (2010)
Liu L., Wang X.: Eighth-order methods with high efficiency index for solving nonlinear equations. Appl. Math. comput. 215, 3449–3454 (2010)
Mir N.A., Rafiq N., Akram S.: An efficient three-step iterative method for non-linear equations. Int. J. Math. Anal. 3, 1989–1996 (2009)
Neta B., Petkovic M.S.: Construction of optimal order nonlinear solvers using inverse interpolation. Appl. Math. Comput. 217, 2448–2455 (2010)
Sargolzaei P., Soleymani F.: Accurate fourteenth-order methods for solving nonlinear equations. Numer. Algorithms 58, 513–527 (2011)
Sharma J.R., Sharma R.: A new family of modified Ostrowski’s methods with accelerated eighth order convergence. Numer. Algorithms 54, 445–458 (2010)
Soleymani F., Karimi Vanani S.: Optimal Steffensen-type methods with eighth order of convergence. Comput. Math. Appl. 62, 4619–4626 (2011)
Siyyam H.I.: An iterative method with fifth-order convergence for nonlinear equations. Appl. Math. Sci. 3, 2041–2053 (2009)
Soleymani F., Soleimani F.: Novel computational derivative-free methods for simple roots. Fixed Point Theory 13, 247–258 (2012)
Soleymani F.: On a bi-parametric class of optimal eighth-order derivative-free methods. Int. J. Pure Appl. Math. 72, 27–37 (2011)
Soleymani, F., Karimi Vanani, S., Jamali Paghaleh, M.: A class of three-step derivative-free root-solvers with optimal convergence order. J. Appl. Math., vol. 2012, Article ID 568740. doi:10.1155/2012/568740
Soleymani F., Mousavi B.S.: On novel classes of iterative methods for solving nonlinear equations. Comput. Math. Math. Physics 52, 203–210 (2012)
Sharifi M., Babajee D.K.R., Soleymani F.: Finding the solution of nonlinear equations by a class of optimal methods. Comput. Math. Appl. 63, 764–774 (2012)
Soleymani F., Sharifi M., Somayeh Mousavi B.: An improvement of Ostrowski’s and King’s techniques with optimal convergence order eight. J. Optim. Theory Appl. 153, 225–236 (2012)
Soleymani F.: Optimal fourth-order iterative methods free from derivative. Miskolc Math. Notes 12, 255–264 (2011)
Soleymani, F., Karimi Vanani, S., Afghani, A.: A general three-step class of optimal iterations for nonlinear equations. Math. Prob. Eng., vol. 2011, Article ID 469512. doi:10.1155/2011/469512
Soleymani F., Sharma R., Li X., Tohidi E.: An optimized derivative-free form of the Potra-Ptak method. Math. Comput. Model. 56, 97–104 (2012)
Traub J.F.: Iterative Methods for the Solution of Equations. Chelsea, New York (1976)
Wang X., Liu L.: New eighth-order iterative methods for solving nonlinear equations. J. Comput. Appl. Math. 234, 1611–1620 (2010)
Weerakoon S., Fernando T.G.I.: A variant of Newton’s method with accelerated third-order convergence. Appl. Math. Lett. 13, 87–93 (2000)
Ye Q., Xu X.: A class of Newton-like methods with cubic convergence for nonlinear equations. Fixed Point Theory 11, 161–168 (2010)
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Soleymani, F., Karimi Vanani, S., Siyyam, H.I. et al. Numerical solution of nonlinear equations by an optimal eighth-order class of iterative methods. Ann Univ Ferrara 59, 159–171 (2013). https://doi.org/10.1007/s11565-012-0165-5
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DOI: https://doi.org/10.1007/s11565-012-0165-5