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On Newton-type methods for multiple roots with cubic convergence. (English) Zbl 1168.65024
Summary: We introduce two families of Newton-type methods for multiple roots with cubic convergence. A further Newton-type method for multiple roots with cubic convergence is presented that is related to quadrature. We also provide numerical tests that show that these new methods are competitive to other known methods for multiple roots.

65H05 Numerical computation of solutions to single equations
Full Text: DOI
[1] 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
[2] Hansen, E.; Patrick, M., A family of root finding methods, Numer. math., 27, 257-269, (1977) · Zbl 0361.65041
[3] Kravanja, P.; Haegemans, A., A modification of newton’s method for analytic mappings having multiple zeros, Computing, 62, 129-145, (1999) · Zbl 0933.65049
[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] Schröder, E., Über unendlich viele algorithmen zur auflösung der gleichungen, Math. ann., 2, 317-365, (1870)
[6] Traub, J.F., Iterative methods for the solution of equations, (1982), Chelsea Publishing Company New York · Zbl 0472.65040
[7] Ostrowski, A.M., Solution of equations in Banach spaces, (1973), Academic Press New York · Zbl 0304.65002
[8] Rump, Siegfried M., Ten methods to bound multiple roots of polynomials, J. comput. appl. math., 156, 403-432, (2003) · Zbl 1030.65046
[9] Osada, N., An optimal multiple root-finding method of order three, J. comput. appl. math., 51, 131-133, (1994) · Zbl 0814.65045
[10] Osada, N., Asymptotic error constants of cubically convergent zero finding methods, J. comput. appl. math., 196, 347-357, (2006) · Zbl 1128.65031
[11] Osada, N., A one parameter family of locally quartically convergent zero-finding methods, J. comput. appl. math., 205, 116-128, (2007) · Zbl 1120.65063
[12] Osada, N., Chebyshev – halley methods for analytic functions, J. comput. appl. math., 216, 585-599, (2008) · Zbl 1146.65043
[13] 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
[14] Halley, E., Methodus nova accurata & facilis inveniendi radices aequationum quarumcumque genereliter, sine praevia reductionei, Philos. trans. roy. soc. London, 18, 136-148, (1694)
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