×

zbMATH — the first resource for mathematics

A class of Steffensen type methods with optimal order of convergence. (English) Zbl 1216.65055
Summary: A family of Steffensen type methods of fourth-order convergence for solving nonlinear smooth equations is suggested. In the proposed methods, a linear combination of divided differences is used to get a better approximation to the derivative of the given function. Each derivative-free member of the family requires only three evaluations of the given function per iteration. Therefore, this class of methods has efficiency index equal to 1.587. H. T. Kung and J. F. Traub [J. Assoc. Comput. Mach. 21, 643–651 (1974; Zbl 0289.65023)] conjectured that the order of convergence of any multipoint method without memory cannot exceed the bound \(2^{d-1}\), where \(d\) is the number of functional evaluations per step. The new class of methods agrees with this conjecture for the case \(d=3\). Numerical examples are presented to show the performance of the presented methods, on smooth and nonsmooth equations, and to compare them with others.

MSC:
65H05 Numerical computation of solutions to single equations
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Ortega, J.M.; Rheinboldt, W.G., Iterative solutions of nonlinear equations in several variables, (1970), Academic Press New York · Zbl 0241.65046
[2] Jain, P., Steffensen type methods for solving nonlinear equations, Applied mathematics and computation, 194, 527-533, (2007) · Zbl 1193.65063
[3] Dehghan, M.; Hajarian, M., An some derivative free quadratic and cubic convergence iterative formulas for solving nonlinear equations, Journal of computational and applied mathematics, 29, 19-30, (2010) · Zbl 1189.65091
[4] Ren, H.; Wu, Q.; Bi, W., A class of two-step Steffensen type methods with fourth-order convergence, Applied mathematics and computation, 209, 206-210, (2009) · Zbl 1166.65338
[5] Zheng, Q.; Wang, J.; Zhao, P.; Zhang, L., A Steffensen-like method and its higher-order variants, Applied mathematics and computation, 214, 10-16, (2009) · Zbl 1179.65052
[6] Feng, X.; He, Y., High order oterative methods without derivatives for solving nonlinear equations, Applied mathematics and computation, 186, 1617-1623, (2007) · Zbl 1119.65036
[7] Ostrowski, A.M., Solutions of equations and systems of equations, (1966), Academic Press New York-London · Zbl 0222.65070
[8] 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
[9] Amat, S.; Busquier, S., On a steffensen’s type method and its behavior for semismooth equations, Applied mathematics and computation, 177, 819-823, (2006) · Zbl 1096.65047
[10] Potra, F.A.; Qi, L.; Sun, D., Secant methods for semismooth equations, Numerical mathematics, 80, 305-324, (1998) · Zbl 0914.65051
[11] Cordero, A.; Torregrosa, J.R., Variants of newton’s method using fifth-order quadrature formulas, Applied mathematics and computation, 190, 686-698, (2007) · Zbl 1122.65350
[12] Amat, S.; Busquier, S., On a higher order secant methods, Applied mathematics and computation, 141, 321-329, (2003) · Zbl 1035.65057
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.