## 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 Numerical computation of solutions to single equations
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### 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 [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 [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 [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 [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 [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 [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 [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
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