Soleymani, Fazlollah; Sharifi, Mahdi; Somayeh Mousavi, Bibi An improvement of Ostrowski’s and King’s techniques with optimal convergence order eight. (English) Zbl 1237.90229 J. Optim. Theory Appl. 153, No. 1, 225-236 (2012). Summary: 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 H. T. Kung and J. F. Traub [J. Assoc. Comput. Mach. 21, 643–651 (1974; Zbl 0289.65023)] 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. Cited in 26 Documents MSC: 90C30 Nonlinear programming Keywords:simple root; three-step iterative methods; derivative-involved methods; optimal convergence rate; weight function Citations:Zbl 0289.65023 PDFBibTeX XMLCite \textit{F. Soleymani} et al., J. Optim. Theory Appl. 153, No. 1, 225--236 (2012; Zbl 1237.90229) Full Text: DOI References: [1] Sargolzaei, P., Soleymani, F.: Accurate fourteenth-order methods for solving nonlinear equations. Numer. Algorithms (2011). doi: 10.1007/s11075-011-9467-4 · Zbl 1242.65100 [2] Iliev, A., Kyurkchiev, N.: Nontrivial Methods in Numerical Analysis: Selected Topics in Numerical Analysis. LAP Lambert Academic, Saarbrücker (2006) [3] King, R.F.: Family of fourth-order methods for nonlinear equations. SIAM J. Numer. Anal. 10, 876–879 (1973) · Zbl 0266.65040 · doi:10.1137/0710072 [4] Kung, H.T., Traub, J.F.: Optimal order of one-point and multipoint iteration. J. ACM 643–651 (1974) · Zbl 0289.65023 [5] 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) · Zbl 1161.65039 · doi:10.1016/j.cam.2008.07.004 [6] 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) · Zbl 1215.65091 · doi:10.1016/j.cam.2011.01.004 [7] 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) · Zbl 1299.65092 · doi:10.2298/AADM110228012H [8] 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) · Zbl 1277.65030 · doi:10.1007/s10957-011-9796-4 [9] 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) · Zbl 1118.65034 · doi:10.1007/s10957-006-9154-0 [10] Soleymani, F., Sharifi, M.: On a general efficient class of four-step root-finding methods. Int. J. Math. Comput. Simul. 5, 181–189 (2011) This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.