# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
A family of derivative-free methods with high order of convergence and its application to nonsmooth equations. (English) Zbl 1246.65079
Summary: A family of derivative-free methods of seventh-order convergence for solving nonlinear equations is suggested. In the proposed methods, several linear combinations of divided differences are used in order to get a good estimation of the derivative of the given function at the different steps of the iteration. The efficiency indices of the members of this family are equal to $1.6266$. Also, numerical examples are used to show the performance of the presented methods, on smooth and nonsmooth equations, and to compare with other derivative-free methods, including some optimal fourth-order ones, in the sense of Kung-Traub’s conjecture.

##### MSC:
 65H05 Single nonlinear equations (numerical methods)
Full Text:
##### References:
 [1] J. M. Ortega and W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables, Academic Press, New York, NY, USA, 1970. · Zbl 0241.65046 [2] A. M. Ostrowski, Solution of Equations and Systems of Equations, Pure and Applied Mathematics, Vol. 9, Academic Press, New York, NY, USA, 1966. · Zbl 0222.65070 [3] H. T. Kung and J. F. Traub, “Optimal order of one-point and multipoint iteration,” Journal of the Association for Computing Machinery, vol. 21, pp. 643-651, 1974. · Zbl 0289.65023 · doi:10.1145/321850.321860 [4] Z. Liu, Q. Zheng, and P. Zhao, “A variant of Steffensen’s method of fourth-order convergence and its applications,” Applied Mathematics and Computation, vol. 216, no. 7, pp. 1978-1983, 2010. · Zbl 1208.65064 · doi:10.1016/j.amc.2010.03.028 [5] M. Dehghan and M. Hajarian, “Some derivative free quadratic and cubic convergence iterative formulas for solving nonlinear equations,” Computational & Applied Mathematics, vol. 29, no. 1, pp. 19-30, 2010. · Zbl 1189.65091 · doi:10.1590/S1807-03022010000100002 [6] A. Cordero and J. R. Torregrosa, “A class of Steffensen type methods with optimal order of convergence,” Applied Mathematics and Computation, vol. 217, no. 19, pp. 7653-7659, 2011. · Zbl 1216.65055 · doi:10.1016/j.amc.2011.02.067 [7] Y. Hu and L. Fang, “A seventh-order convergence Newton-type method for solving nonlinear equations,” in 2nd International Conference on Computational Intelligence and Natural Computing, 2010. [8] M. A. Noor, W. A. Khan, K. I. Noor, and E. Al-said, “High-order iterative method free from second derivative for solving nonlinear equations,” International Journal of the Physical Science, vol. 6, no. 8, pp. 1887-1893, 2011. [9] F. Soleymani and S. K. Khattri, “Finding simple roots by seventh and eighthorder derivative-free methods,” International Journal of Mathematical Models and Methods in Applied Sciences, vol. 1, no. 6, pp. 45-52, 2012. [10] S. Amat and S. Busquier, “On a Steffensen’s type method and its behavior for semismooth equations,” Applied Mathematics and Computation, vol. 177, no. 2, pp. 819-823, 2006. · Zbl 1096.65047 · doi:10.1016/j.amc.2005.11.032 [11] A. Cordero, J. L. Hueso, E. Martínez, and J. R. Torregrosa, “A modified Newton-Jarratt’s composition,” Numerical Algorithms, vol. 55, no. 1, pp. 87-99, 2010. · Zbl 1251.65074 · doi:10.1007/s11075-009-9359-z [12] A. Cordero and J. R. Torregrosa, “Variants of Newton’s method using fifth-order quadrature formulas,” Applied Mathematics and Computation, vol. 190, no. 1, pp. 686-698, 2007. · Zbl 1122.65350 · doi:10.1016/j.amc.2007.01.062 [13] S. Amat and S. Busquier, “On a higher order secant method,” Applied Mathematics and Computation, vol. 141, no. 2-3, pp. 321-329, 2003. · Zbl 1035.65057 · doi:10.1016/S0096-3003(02)00257-6