Three-point methods with and without memory for solving nonlinear equations. (English) Zbl 1248.65049

A new family of three-point derivative free methods for solving nonlinear equations is presented. It is proved that the order of convergence of the basic family without memory is eight requiring four function-evaluations, which means that this family is optimal in the sense of the Kung-Traub conjecture. Further accelerations of convergence speed are attained by suitable variation of a free parameter in each iterative step. This self-accelerating parameter is calculated using information from the current and previous iteration so that the presented methods may be regarded as the methods with memory. The self-correcting parameter is calculated applying the secant-type method in three different ways and Newton’s interpolatory polynomial of the second degree. The increase of convergence order is attained without any additional function calculations, providing a very high computational efficiency of the proposed methods with memory. Another advantage is a convenient fact that these methods do not use derivatives. Numerical examples and the comparison with existing three-point methods are included to confirm theoretical results and high computational efficiency.


65H05 Numerical computation of solutions to single equations
65Y20 Complexity and performance of numerical algorithms
Full Text: DOI


[1] Alefeld, G.; Herzberger, J., Introduction to interval computation, (1983), Academic Press New York
[2] Bi, W.; Wu, Q.; Ren, H., A new family of eight-order iterative methods for solving nonlinear equations, Appl. math. comput., 214, 236-245, (2009) · Zbl 1173.65030
[3] Bi, W.; Ren, H.; Wu, Q., Three-step iterative methods with eight-order convergence for solving nonlinear equations, J. comput. appl. math., 225, 105-112, (2009) · Zbl 1161.65039
[4] Cordero, A.; Hueso, J.L.; Martínez, E.; Torregrosa, J.R., New modifications of potra – pták’s method with optimal fourth and eighth orders of convergence, J. comput. appl. math., 234, 2969-2976, (2010) · Zbl 1191.65048
[5] Džunić, J.; Petković, M.S.; Petković, L.D., A family of optimal three-point methods for solving nonlinear equations using two parametric functions, Appl. math. comput., 217, 7612-7619, (2011) · Zbl 1216.65056
[6] Geum, Y.H.; Kim, Y.I., A multi-parameter family of three-step eighth-order iterative methods locating a simple root, Appl. math. comput., 215, 3375-3382, (2010) · Zbl 1183.65049
[7] Geum, Y.H.; Kim, Y.I., A uniparametric family of three-step eighth-order multipoint iterative methods for simple roots, Appl. math. lett., 24, 929-935, (2011) · Zbl 1215.65092
[8] Kung, H.T.; Traub, J.F., Optimal order of one-point and multipoint iteration, J. ACM, 21, 643-651, (1974) · Zbl 0289.65023
[9] Neta, B.; Petković, M.S., Construction of optimal order nonlinear solvers using inverse interpolation, Appl. math. comput., 217, 2448-2455, (2010) · Zbl 1202.65062
[10] Ortega, J.M.; Rheiboldt, W.C., Iterative solution of nonlinear equations in several variables, (1970), Academic Press New York · Zbl 0241.65046
[11] Petković, M.S., Remarks on “on a general class of multipoint root-finding methods of high computational efficiency”, SIAM J. numer. math., 49, 1317-1319, (2011) · Zbl 1231.65087
[12] Petković, M.S.; Ilić, S.; Džunić, J., Derivative free two-point methods with and without memory for solving nonlinear equations, Appl. math. comput., 217, 1887-1895, (2010) · Zbl 1200.65034
[13] Petković, M.S.; Džunić, J.; Petković, L.D., A family of two-point methods with memory for solving nonlinear equations, Appl. anal. discrete math., 5, 298-317, (2011) · Zbl 1265.65097
[14] Sharma, J.R.; Sharma, R., A new family of modified ostrowskis methods with accelerated eighth order convergence, Numer. algorithms, 54, 445-458, (2010) · Zbl 1195.65067
[15] Thukral, R.; Petković, M.S., Family of three-point methods of optimal order for solving nonlinear equations, J. comput. appl. math., 233, 2278-2284, (2010) · Zbl 1180.65058
[16] Traub, J.F., Iterative methods for the solution of equations, (1964), Prentice-Hall Englewood Cliffs, New Jersey · Zbl 0121.11204
[17] Wang, X.; Liu, L., New eighth-order iterative methods for solving nonlinear equations, J. comput. appl. math., 234, 1611-1620, (2010) · Zbl 1190.65081
[18] Wang, X.; Liu, L., Modified ostrowski’s method with eighth-order convergence and high efficiency index, Appl. math. lett., 23, 549-554, (2010) · Zbl 1191.65050
[19] Zheng, Q.; Li, J.; Huang, F., Optimal Steffensen-type families for solving nonlinear equations, Appl. math. comp., 217, 9592-9597, (2011) · Zbl 1227.65044
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.