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A class of three-step derivative-free root solvers with optimal convergence order. (English) Zbl 1235.65047
Summary: A class of three-step eighth-order root solvers is constructed in this study. Our aim is fulfilled by using an interpolatory rational function in the third step of a three-step cycle. Each method of the class reaches the optimal efficiency index according to the Kung-Traub conjecture concerning multipoint iterative methods without memory. Moreover, the class is free from derivative calculation per full iteration, which is important in engineering problems. One method of the class is established analytically. To test the derived methods from the class, we apply them to a lot of nonlinear scalar equations. Numerical examples suggest that the novel class of derivative-free methods is better than the existing methods of the same type in the literature.

65H05Single nonlinear equations (numerical methods)
Full Text: DOI
[1] A. Iliev and N. Kyurkchiev, Nontrivial Methods in Numerical Analysis: Selected Topics in Numerical Analysis, LAP LAMBERT Academic Publishing, 2010. · Zbl 1217.65090
[2] M. A. Hernández and N. Romero, “On the efficiency index of one-point iterative processes,” Numerical Algorithms, vol. 46, no. 1, pp. 35-44, 2007. · Zbl 1132.65043 · doi:10.1007/s11075-007-9125-z
[3] F. Soleymani and M. Sharifi, “On a general efficient class of four-step root-finding methods,” International Journal of Mathematics and Computers in Simulation, vol. 5, pp. 181-189, 2011.
[4] F. Soleymani, “New optimal iterative methods in solving nonlinear equations,” International Journal of Pure and Applied Mathematics, vol. 72, no. 2, pp. 195-202, 2011. · Zbl 1246.65081
[5] F. Soleymani, S. Karimi Vanani, and A. Afghani, “A general three-step class of optimal iterations for nonlinear equations,” Mathematical Problems in Engineering, vol. 2011, Article ID 469512, 10 pages, 2011. · Zbl 1235.74002 · doi:10.1155/2011/469512
[6] P. Sargolzaei and F. Soleymani, “Accurate fourteenth-order methods for solving nonlinear equations,” Numerical Algorithms, vol. 58, no. 4, pp. 513-527, 2011. · Zbl 1242.65100 · doi:10.1007/s11075-011-9467-4
[7] F. Soleymani, “A novel and precise sixth-order method for solving nonlinear equations,” International Journal of Mathematical Models and Methods in Applied Sciences, vol. 5, pp. 730-737, 2011.
[8] F. Soleymani, “On a bi-parametric class of optimal eighth-order derivative-free methods,” International Journal of Pure and Applied Mathematics, vol. 72, pp. 27-37, 2011. · Zbl 1248.65050
[9] 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
[10] Y. Peng, H. Feng, Q. Li, and X. Zhang, “A fourth-order derivative-free algorithm for nonlinear equations,” Journal of Computational and Applied Mathematics, vol. 235, no. 8, pp. 2551-2559, 2011. · Zbl 1229.65083 · doi:10.1016/j.cam.2010.11.007
[11] H. Ren, Q. Wu, and W. Bi, “A class of two-step Steffensen type methods with fourth-order convergence,” Applied Mathematics and Computation, vol. 209, no. 2, pp. 206-210, 2009. · Zbl 1166.65338 · doi:10.1016/j.amc.2008.12.039
[12] 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
[13] 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
[14] R. Thukral, “Eighth-order iterative methods without derivatives for solving nonlinear equations,” ISRN Applied Mathematics, vol. 2011, Article ID 693787, 12 pages, 2011. · doi:10.5402/2011/693787
[15] F. Soleymani and V. Hosseinabadi, “New third- and sixth-order derivative-free techniques for nonlinear equations,” Journal of Mathematics Research, vol. 3, pp. 107-112, 2011. · Zbl 1221.65119 · doi:10.5539/jmr.v3n2p107 · http://ccsenet.org/journal/index.php/jmr/article/view/8251
[16] F. Soleymani and S. Karimi Vanani, “Optimal Steffensen-type methods with eighth order of convergence,” Computers and Mathematics with Applications, vol. 62, no. 12, pp. 4619-4626, 2011. · Zbl 1236.65056 · doi:10.1016/j.camwa.2011.10.047
[17] 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
[18] F. Soleymani, “Regarding the accuracy of optimal eighth-order methods,” Mathematical and Computer Modelling, vol. 53, no. 5-6, pp. 1351-1357, 2011. · Zbl 1217.65089 · doi:10.1016/j.mcm.2010.12.032
[19] 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
[20] F. Soleymani, “Revisit of Jarratt method for solving nonlinear equations,” Numerical Algorithms, vol. 57, no. 3, pp. 377-388, 2011. · Zbl 1222.65048 · doi:10.1007/s11075-010-9433-6
[21] D. K. R. Babajee, Analysis of higher order variants of Newton’s method and their applications to differential and integral equations and in ocean acidification, Ph.D. thesis, University of Mauritius, 2010.
[22] F. Soleymani, “On a novel optimal quartically class of methods,” Far East Journal of Mathematical Sciences (FJMS), vol. 58, no. 2, pp. 199-206, 2011. · Zbl 1252.41015
[23] F. Soleymani, S. Karimi Vanani, M. Khan, and M. Sharifi, “Some modifications of King’s family with optimal eighth order of convergence,” Mathematical and Computer Modelling, vol. 55, no. 3-4, pp. 1373-1380, 2012. · Zbl 1255.65097 · doi:10.1016/j.mcm.2011.10.016
[24] F. Soleymani and B. S. Mousavi, “A novel computational technique for finding simple roots of nonlinear equations,” International Journal of Mathematical Analysis, vol. 5, pp. 1813-1819, 2011. · Zbl 1251.65072
[25] F. Soleymani and M. Sharifi, “On a class of fifteenth-order iterative formulas for simple roots,” International Electronic Journal of Pure and Applied Mathematics, vol. 3, pp. 245-252, 2011.
[26] L. D. Petković, M. S. Petković, and J. D\vzunić, “A class of three-point root-solvers of optimal order of convergence,” Applied Mathematics and Computation, vol. 216, no. 2, pp. 671-676, 2010. · Zbl 1188.65068 · doi:10.1016/j.amc.2010.01.123
[27] X. Wang and L. Liu, “New eighth-order iterative methods for solving nonlinear equations,” Journal of Computational and Applied Mathematics, vol. 234, no. 5, pp. 1611-1620, 2010. · Zbl 1190.65081 · doi:10.1016/j.cam.2010.03.002
[28] B. I. Yun, “A non-iterative method for solving non-linear equations,” Applied Mathematics and Computation, vol. 198, no. 2, pp. 691-699, 2008. · Zbl 1138.65035 · doi:10.1016/j.amc.2007.09.006
[29] J. F. Traub, Iterative Methods for the Solution of Equations, Chelsea Publishing Company, New York, NY, USA, 1982. · Zbl 0472.65040