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Three-step iterative methods with optimal eighth-order convergence. (English) Zbl 1215.65091
Authors’ abstract: Based on Ostrowski’s method, a new family of eighth-order methods for solving nonlinear equations is derived. In terms of computational cost, each iteration of these methods requires three evaluations of the function and one evaluation of its first derivative, so that their efficiency indices are 1.682, which is optimal according to Kung and Traub’s conjecture. Numerical comparisons are made to show the performance of the new family.

65H05Single nonlinear equations (numerical methods)
Full Text: DOI
[1] Bi, W.; Ren, H.; Wu, Q.: Three-step iterative methods with eighth-order convergence for solving nonlinear equations, Journal of computational and applied mathematics 255, 105-112 (2009) · Zbl 1161.65039 · doi:10.1016/j.cam.2008.07.004
[2] Grau, M.; Díaz-Barrero, J. L.: An improvement to Ostrowski root-finding method, Applied mathematics and computation 173, 450-456 (2006) · Zbl 1090.65053 · doi:10.1016/j.amc.2005.04.043
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[4] Kou, J.; Wang, X.: Some improvements of Ostrowski’s method, Applied mathematics letters 23, 92-96 (2010) · Zbl 1191.65049 · doi:10.1016/j.aml.2009.08.010
[5] Liu, L.; Wang, X.: Eighth-order methods with high efficiency index for solving nonlinear equations, Applied mathematics and computation 215, 3449-3454 (2010) · Zbl 1183.65051 · doi:10.1016/j.amc.2009.10.040
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[7] Cordero, A.; Hueso, J. L.; Martínez, E.; Torregrosa, J. R.: New modifications of potra-pták’s method with optimal fourth and eighth order of convergence, Journal of computational and applied mathematics 234, 2969-2976 (2010) · Zbl 1191.65048 · doi:10.1016/j.cam.2010.04.009
[8] Ostrowski, A. M.: Solutions of equations and systems of equations, (1966) · Zbl 0222.65070
[9] Kung, H. T.; Traub, J. F.: Optimal order of one-point and multi-point iteration, Applied mathematics and computation 21, 643-651 (1974) · Zbl 0289.65023 · doi:10.1145/321850.321860
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[11] Traub, J. F.: Iterative methods for the solution of equations, (1982) · Zbl 0472.65040
[12] Chun, C.; Ham, Y.: Some sixth-order variants of Ostrowski root-finding methods, Applied mathematics and computation 193, 389-394 (2007) · Zbl 1193.65055 · doi:10.1016/j.amc.2007.03.074
[13] Weerakoon, S.; Fernando, T. G. I.: A variant of Newton’s method with accelerated third-order convergence, Applied mathematics letters 13, No. 8, 87-93 (2000) · Zbl 0973.65037 · doi:10.1016/S0893-9659(00)00100-2