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New families of nonlinear third-order solvers for finding multiple roots. (English) Zbl 1186.65060
Summary: We present two new families of iterative methods for multiple roots of nonlinear equations. One of the families require one-function and two-derivative evaluation per step, and the other family requires two-function and one-derivative evaluation. It is shown that both are third-order convergent for multiple roots. Numerical examples suggest that each family member can be competitive to other third-order methods and Newton’s method for multiple roots. In fact the second family is even better than the first.

65H05Single nonlinear equations (numerical methods)
Full Text: DOI
[1] Schröder, E.: Über unendlich viele algorithmen zur auflösung der gleichungen, Math. ann. 2, 317-365 (1870) · Zbl 02.0042.02
[2] Halley, E.: A new, exact and easy method of finding the roots of equations generally and without any previous reduction, Phil. trans. R. soc. London 18, 136-148 (1694)
[3] Hansen, E.; Patrick, M.: A family of root finding methods, Numer. math. 27, 257-269 (1977) · Zbl 0361.65041 · doi:10.1007/BF01396176
[4] Osada, N.: An optimal multiple root-finding method of order three, J. comput. Appl. math. 51, 131-133 (1994) · Zbl 0814.65045 · doi:10.1016/0377-0427(94)00044-1
[5] Dong, C.: A family of multipoint iterative functions for finding multiple roots, Int. J. Comput. math. 21, 363-367 (1987) · Zbl 0656.65050 · doi:10.1080/00207168708803576
[6] 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 · doi:10.1080/00207168208803346
[7] Frontini, M.; Sormani, E.: Modified Newton’s method with third-order convergence and multiple roots, J. comput. Appl. math. 156, 345-354 (2003) · Zbl 1030.65044 · doi:10.1016/S0377-0427(02)00920-2
[8] Kou, J.; Li, Y.; Wang, X.: A composite fourth-order iterative method for solving non-linear equations, Appl. math. Comput. 184, 471-475 (2007) · Zbl 1114.65045 · doi:10.1016/j.amc.2006.05.181
[9] B. Neta, Extension of Murakami’s High order nonlinear solver to multiple roots, Int. J. Comput. Math. (in press) · Zbl 1192.65052 · doi:10.1080/00207160802272263
[10] Neta, B.: New third order nonlinear solvers for multiple roots, Appl. math. Comput. 202, 162-170 (2008) · Zbl 1151.65041 · doi:10.1016/j.amc.2008.01.031
[11] C. Chun, B. Neta, A third-order modification of Newton’s method for multiple roots, Appl. Math. Comput. AMC-S-08-01123 (submitted for publication) · Zbl 1162.65342
[12] Neta, B.; Jhonson, A. N.: High-order nonlinear solver for multiple roots, Comput. math. Appl. 55, 2012-2017 (2008) · Zbl 1142.65044 · doi:10.1016/j.camwa.2007.09.001
[13] Neta, B.: Numerical methods for the solution of equations, (1983) · Zbl 0514.65029
[14] Hansen, E.; Patrick, M.: A family of root finding methods, Numer. math. 27, 257-269 (1977) · Zbl 0361.65041 · doi:10.1007/BF01396176
[15] Traub, J. F.: Iterative methods for the solution of equations, (1977) · Zbl 0383.68041