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Two optimal eighth-order derivative-free classes of iterative methods. (English) Zbl 1253.65100
Summary: Optimization problems defined by (objective) functions for which derivatives are unavailable or available at an expensive cost are emerging in computational science. Due to this, the main aim of this paper is to attain as high as possible of local convergence order by using fixed number of (functional) evaluations to find efficient solvers for one-variable nonlinear equations, while the procedure to achieve this goal is totally free from derivative. To this end, we consider the fourth-order uniparametric family of Kung and Traub to suggest and demonstrate two classes of three-step derivative-free methods using only four pieces of information per full iteration to reach the optimal order eight and the optimal efficiency index 1.682. Moreover, a large number of numerical tests are considered to confirm the applicability and efficiency of the produced methods from the new classes.

65K05Mathematical programming (numerical methods)
90C29Multi-objective programming; goal programming
Full Text: DOI
[1] A. Iliev and N. Kyurkchiev, Nontrivial Methods in Numerical Analysis (Selected Topics in Numerical Analysis), Lambert Academy, 2010.
[2] B. H. Dayton, T.-Y. Li, and Z. Zeng, “Multiple zeros of nonlinear systems,” Mathematics of Computation, vol. 80, no. 276, pp. 2143-2168, 2011. · Zbl 1242.65102 · doi:10.1090/S0025-5718-2011-02462-2
[3] F. Soleymani, M. Sharifi, and B. S. Mousavi, “An improvement of Ostrowski’s and King’s techniques with optimal convergence order eight,” Journal of Optimization Theory and Applications, vol. 153, no. 1, pp. 225-236, 2012. · Zbl 1237.90229 · doi:10.1007/s10957-011-9929-9
[4] 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
[5] J. F. Steffensen, “Remarks on iteration,” Scandinavian Aktuarietidskr, vol. 16, pp. 64-72, 1933. · Zbl 0007.02601
[6] F. Soleymani, “Two classes of iterative schemes for approximating simple roots,” Journal of Applied Sciences, vol. 11, no. 19, pp. 3442-3446, 2011. · doi:10.3923/jas.2011.3442.3446
[7] M. Sharifi, D. K. R. Babajee, and F. Soleymani, “Finding the solution of nonlinear equations by a class of optimal methods,” Computers & Mathematics with Applications, vol. 63, no. 4, pp. 764-774, 2012. · Zbl 1247.65066 · doi:10.1016/j.camwa.2011.11.040
[8] F. Soleymani and S. Karimi Vanani, “Optimal Steffensen-type methods with eighth order of convergence,” Computers & Mathematics with Applications, vol. 62, no. 12, pp. 4619-4626, 2011. · Zbl 1236.65056 · doi:10.1016/j.camwa.2011.10.047
[9] F. Soleymani, S. K. Khattri, and S. Karimi Vanani, “Two new classes of optimal Jarratt-type fourth-order methods,” Applied Mathematics Letters, vol. 25, no. 5, pp. 847-853, 2012. · Zbl 1239.65030 · doi:10.1016/j.aml.2011.10.030
[10] Q. Zheng, J. Li, and F. Huang, “An optimal Steffensen-type family for solving nonlinear equations,” Applied Mathematics and Computation, vol. 217, no. 23, pp. 9592-9597, 2011. · Zbl 1227.65044 · doi:10.1016/j.amc.2011.04.035
[11] S. K. Khattri and I. K. Argyros, “Sixth order derivative free family of iterative methods,” Applied Mathematics and Computation, vol. 217, no. 12, pp. 5500-5507, 2011. · Zbl 1229.65080 · doi:10.1016/j.amc.2010.12.021
[12] A. Cordero, J. L. Hueso, E. Martínez, and J. R. Torregrosa, “A family of derivative-free methods with high order of convergence and its application to nonsmooth equations,” Abstract and Applied Analysis, vol. 2012, Article ID Article number836901, 15 pages, 2012. · Zbl 1246.65079 · doi:10.1155/2012/836901
[13] G. Alefeld, “Verified numerical computation for nonlinear equations,” Japan Journal of Industrial and Applied Mathematics, vol. 26, no. 2-3, pp. 297-315, 2009. · Zbl 1186.65056 · doi:10.1007/BF03186536 · http://www.projecteuclid.org/euclid.jjiam/1265033783
[14] M. Trott, The Mathematica Guidebook for Numerics, Springer, New York, NY, USA, 2006. · Zbl 1101.65001
[15] http://www.mathematica.stackexchange.com/questions/5663/about-multi-root-search-inmathematica-for-transcendental-equations?lq=1.