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Families of third and fourth order methods for multiple roots of nonlinear equations. (English) Zbl 1273.65064
Summary: This paper presents two families of higher-order iterative methods for solving multiple roots of nonlinear equations. One is of order three and the other is of order four. The presented iterative families all require two evaluations of the function and one evaluation of its first derivative, thus the latter is of optimal order. The third-order family contains several iterative methods known already. And, different from the optimal fourth-order methods for multiple roots known already, the presented fourth-order family use the modified Newton’s method as its first step. Local convergence analyses and some special cases of the presented families are given. We also carry out some numerical examples to show their performance.

##### MSC:
 65H05 Numerical computation of solutions to single equations
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##### References:
 [1] Schröder, E., Über unendlich viele algorithmen zur auflösung der gleichungen, Math. Ann., 2, 317-365, (1870) [2] Hansen, E.; Patrick, M., A family of root finding methods, Numer. Math., 27, 257-269, (1977) · Zbl 0361.65041 [3] Dong, C., A basic theorem of constructing an iterative formula of the higher order for computing multiple roots of an equation, Math. Numer. Sin., 11, 445-450, (1982) · Zbl 0511.65030 [4] Victory, H. D.; Neta, B., A higher order method for multiple zeros of nonlinear functions, Int. J. Comput. Math., 12, 329-335, (1983) · Zbl 0499.65026 [5] Dong, C., A family of multipoint iterative functions for finding multiple roots of equations, Int. J. Comput. Math., 21, 363-367, (1987) · Zbl 0656.65050 [6] Osada, N., An optimal multiple root-finding method of order three, J. Comput. Appl. Math., 51, 131-133, (1994) · Zbl 0814.65045 [7] Neta, B.; Johnson, A. N., High-order nonlinear solver for multiple roots, Comput. Math. Appl., 55, 2012-2017, (2008) · Zbl 1142.65044 [8] Chun, C.; Bae, H.; Neta, B., New families of nonlinear third-order solvers for finding multiple roots, Comput. Math. Appl., 57, 1574-1582, (2009) · Zbl 1186.65060 [9] Homeier, H. H.H., On Newton-type methods for multiple roots with cubic convergence, J. Comput. Appl. Math., 231, 249-254, (2009) · Zbl 1168.65024 [10] Sharma, J. R.; Sharma, R., Modified jarratt method for computing multiple roots, Appl. Math. Comput., 217, 878-881, (2010) · Zbl 1203.65084 [11] Li, S.; Cheng, L.; Neta, B., Some fourth-order nonlinear solvers with closed formulae for multiple roots, Comput. Math. Appl., 59, 126-135, (2010) · Zbl 1189.65093 [12] Zhou, X.; Chen, X.; Song, Y., Constructing higher-order methods for obtaining the multiple roots of nonlinear equations, J. Comput. Appl. Math., 235, 4199-4206, (2011) · Zbl 1219.65048 [13] Kung, H. T.; Traub, J. F., Optimal order of one-point and multipoint iteration, J. Assoc. Comput. Mach., 21, 643-651, (1974) · Zbl 0289.65023 [14] Neta, B.; Petković, M. S., Construction of optimal order nonlinear solvers using inverse interpolation, Appl. Math. Comput., 217, 2448-2455, (2010) · Zbl 1202.65062 [15] Petković, M. S., On a general class of multipoint root-finding methods of high computational efficiency, SIAM J. Numer. Math., 47, 4402-4414, (2010) · Zbl 1209.65053 [16] 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 [17] Džunić, J.; Petković, M. S., On generalized multipoint root-solvers with memory, J. Comput. Appl. Math., 236, 2909-2920, (2012) · Zbl 1343.65051 [18] King, R., A family of fourth-order methods for nonlinear equations, SIAM J. Numer. Anal., 10, 876-879, (1973) · Zbl 0266.65040 [19] Chun, C., Some fourth-order iterative methods for solving nonlinear equations, Appl. Math. Comput., 195, 454-459, (2008) · Zbl 1173.65031 [20] Chun, C.; Ham, Y. M., Some fourth-order modifications of newton’s method, Appl. Math. Comput., 197, 654-658, (2008) · Zbl 1137.65028 [21] Sharma, J. R., A composite third order Newton-Steffensen method for solving nonlinear equations, Appl. Math. Comput., 169, 242-246, (2005) · Zbl 1084.65054 [22] Neta, B., New third order nonlinear solvers for multiple roots, Appl. Math. Comput., 202, 162-170, (2008) · Zbl 1151.65041
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