zbMATH — the first resource for mathematics

New modifications of Potra-Pták’s method with optimal fourth and eighth orders of convergence. (English) Zbl 1191.65048
Summary: We present two new iterative methods for solving nonlinear equations by using suitable Taylor and divided difference approximations. Both methods are obtained by modifying Potra-Pták’s method trying to get optimal order. We prove that the new methods reach orders of convergence four and eight with three and four functional evaluations, respectively. So, Kung and Traub’s conjecture [H. T. Kung and J. F. Traub, J. Assoc. Comput. Mach. 21, 643–651 (1974; Zbl 0289.65023)], that establishes for an iterative method based on \(n\) evaluations an optimal order \(p=2^{n - 1}\) is fulfilled, getting the highest efficiency indices for orders \(p=4\) and \(p=8\), which are 1.587 and 1.682.
We also perform different numerical tests that confirm the theoretical results and allow us to compare these methods with Potra-Pták’s method from which they have been derived, and with other recently published eighth-order methods.

65H05 Numerical computation of solutions to single equations
Full Text: DOI
[1] Ostrowski, A.M., Solutions of equations and systems of equations, (1966), Academic Press New York, London · Zbl 0222.65070
[2] 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
[3] Jarrat, P., Some fourth order multipoint iterative methods for solving equations, Mathematical computation, 20, 434-437, (1966) · Zbl 0229.65049
[4] King, R., A family of fourth order methods for nonlinear equations, SIAM journal on numerical analysis, 10, 876-879, (1973) · Zbl 0266.65040
[5] Basu, D., From third to fourth order variant of newton’s method for simple roots, Applied mathematics and computation, 202, 2, 886-892, (2008) · Zbl 1147.65037
[6] Maheshwari, A.K., A fourth order iterative method for solving nonlinear equations, Applied mathematics and computation, 211, 2, 383-391, (2009) · Zbl 1162.65346
[7] Li, X.; Mu, C.; Ma, J.; Wang, C., Sixteenth-order method for nonlinear equations, Journal of computational and applied mathematics, 215, 10, 3754-3758, (2010) · Zbl 1205.65171
[8] 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
[9] J.R. Sharma, R. Sharma, A new family of modified Ostrowski’s methods with accelerated eighth order convergence, Numerical Algoritms, in press (doi:10.1007/s11075-009.9345-5). · Zbl 1195.65067
[10] Bi, W.; Wu, Q.; Ren, H., A new family of eighth-order iterative methods for solving nonlinear equations, Applied mathematics and computation, 214, 1, 236-245, (2009) · Zbl 1173.65030
[11] Thukral, R.; Petkovic, M.S., A family of three-point methods of optimal order for solving nonlinear equations, Journal of computational and applied mathematics, 233, 9, 2278-2284, (2010) · Zbl 1180.65058
[12] 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
[13] Potra, F.A.; Pták, V., Nondiscrete introduction and iterative processes, () · Zbl 0549.41001
[14] Traub, J.F., Iterative methods for the solution of equations, (1982), Chelsea Publishing Company New York · Zbl 0472.65040
[15] Danby, J.M.A.; Burkardt, T.M., The solution of kepler’s equation, I, Celestial mechanics, 31, 95-107, (1983) · Zbl 0572.70014
[16] Weerakoon, S.; Fernando, T.G.I., A variant of newton’s method with accelerated third-order convergence, Applied mathematics letters, 13, 8, 87-93, (2000) · Zbl 0973.65037
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.