Abstract
In this paper, we first establish a new class of three-point methods based on the two-point optimal method of Ostrowski. Analysis of convergence shows that any method of our class arrives at eighth order of convergence by using three evaluations of the function and one evaluation of the first derivative per iteration. Thus, this order agrees with the conjecture of Kung and Traub (J. ACM 643–651, 1974) for constructing multipoint optimal iterations without memory. We second present another optimal eighth-order class based on the King’s fourth-order family and the first attained class. To support the underlying theory developed in this work, we examine some methods of the proposed classes by comparison with some of the existing optimal eighth-order methods in literature. Numerical experience suggests that the new classes would be valuable alternatives for solving nonlinear equations.
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Sargolzaei, P., Soleymani, F.: Accurate fourteenth-order methods for solving nonlinear equations. Numer. Algorithms (2011). doi:10.1007/s11075-011-9467-4
Iliev, A., Kyurkchiev, N.: Nontrivial Methods in Numerical Analysis: Selected Topics in Numerical Analysis. LAP Lambert Academic, Saarbrücker (2006)
King, R.F.: Family of fourth-order methods for nonlinear equations. SIAM J. Numer. Anal. 10, 876–879 (1973)
Kung, H.T., Traub, J.F.: Optimal order of one-point and multipoint iteration. J. ACM 643–651 (1974)
Bi, W., Ren, H., Wu, Q.: Three-step iterative methods with eighth-order convergence for solving nonlinear equations. J. Comput. Appl. Math. 225, 105–112 (2009)
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)
Heydari, M., Hosseini, S.M., Loghmani, G.B.: On two new families of iterative methods for solving nonlinear equations with optimal order. Appl. Anal. Discrete Math. 5, 93–109 (2011)
Chen, M., Chang, T.-S.: On the higher-order method for the solution of a nonlinear scalar equation. J. Optim. Theory Appl. 149, 647–664 (2011)
Germani, A., Manes, C., Palumbo, P.: Higher-Order method for the solution of a nonlinear scalar equation. J. Optim. Theory Appl. 131, 347–364 (2006)
Soleymani, F., Sharifi, M.: On a general efficient class of four-step root-finding methods. Int. J. Math. Comput. Simul. 5, 181–189 (2011)
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Communicated by Ilio Galligani.
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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). https://doi.org/10.1007/s10957-011-9929-9
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DOI: https://doi.org/10.1007/s10957-011-9929-9