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Steffensen type methods for solving nonlinear equations. (English) Zbl 1237.65049
Summary: By approximating the derivatives in the well known fourth-order Ostrowski’s method and in a sixth-order improved Ostrowski’s method by central-difference quotients, we obtain new modifications of these methods free from derivatives. We prove the important fact that the methods obtained preserve their convergence orders $4$ and $6$, respectively, without calculating any derivatives. Finally, numerical tests confirm the theoretical results and allow us to compare these variants with the corresponding methods that make use of derivatives and with the classical Newton’s method.

65H05Single nonlinear equations (numerical methods)
Full Text: DOI
[1] Zhu, X.: Modified Chebyshev--halley methods free from second derivative, Applied mathematics and computation 203, 824-827 (2008) · Zbl 1157.65373 · doi:10.1016/j.amc.2008.05.092
[2] Chun, C.: Some second derivative free variants of Chebyshev--halley methods, Applied mathematics and computation 191, 410-414 (2007) · Zbl 1193.65054 · doi:10.1016/j.amc.2007.02.105
[3] Chun, C.; Ham, Y.: Some second-derivative-free variants of super-halley method with fourth-order convergence, Applied mathematics and computation 195, 537-541 (2008) · Zbl 1132.65041 · doi:10.1016/j.amc.2007.05.003
[4] Jain, P.: Steffensen type methods for solving nonlinear equations, Applied mathematics and computation 194, 527-533 (2007) · Zbl 1193.65063 · doi:10.1016/j.amc.2007.04.087
[5] Feng, X.; He, Y.: High order iterative methods without derivatives for solving nonlinear equations, Applied mathematics and computation 186, 1617-1623 (2007) · Zbl 1119.65036 · doi:10.1016/j.amc.2006.08.070
[6] Weerakoon, S.; Fernando, T. G. I.: A variant of Newton’s methods with accelerated third-order convergence, Applied mathematics letters 13, 87-93 (2000) · Zbl 0973.65037 · doi:10.1016/S0893-9659(00)00100-2
[7] 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 · doi:10.1016/j.amc.2009.03.053
[8] 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 · doi:10.1016/j.amc.2005.11.032
[9] Alarcón, V.; Amat, S.; Busquier, S.; López, D. J.: A Steffensen’s type method in Banach spaces with applications on boundary-value problems, Journal of computational and applied mathematics 216, 243-250 (2008) · Zbl 1139.65040 · doi:10.1016/j.cam.2007.05.008
[10] Ostrowski, A. M.: Solutions of equations and systems of equations, (1960) · Zbl 0115.11201
[11] 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 · doi:10.1590/S1807-03022010000100002
[12] Grau, M.; Díaz-Barrero, J. L.: An improvement to Ostrowski root-finding method, Applied mathematics and computation 173, 450-456 (2006) · Zbl 1090.65053 · doi:10.1016/j.amc.2005.04.043
[13] A. Cordero, J.L. Hueso, E. Martínez, J.R. Torregrosa, A modified Newton--Jarratt’s composition, Numerical Algorithm. doi:10.1007/s11075-009-9359-z. · Zbl 1251.65074
[14] 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 · doi:10.1016/j.amc.2007.01.062