## A third-order modification of Newton’s method for multiple roots.(English)Zbl 1162.65342

Summary: We present a new third-order modification of Newton’s method for multiple roots, which is based on existing third-order multiple root-finding methods. Numerical examples show that the new method is competitive to other methods for multiple roots.

### MSC:

 65H05 Numerical computation of solutions to single equations

Maple
Full Text:

### References:

 [1] Schröder, E., Über unendlich viele algorithmen zur auflösung der gleichungen, Math. ann., 2, 317-365, (1870) [2] Traub, J.F., Iterative methods for the solution of equations, (1964), Prentice Hall New Jersey · Zbl 0121.11204 [3] Hansen, E.; Patrick, M., A family of root finding methods, Numer. math., 27, 257-269, (1977) · Zbl 0361.65041 [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 basic theorem of constructing an iterative formula of the higher order for computing multiple roots of an equation, Math. numer. sinica, 11, 445-450, (1982) · Zbl 0511.65030 [6] 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 [7] Neta, B.; Johnson, A.N., High order nonlinear solver for multiple roots, Comput. math. appl., 55, 2012-2017, (2008) · Zbl 1142.65044 [8] B. Neta, Extension of Murakami’s High order nonlinear solver to multiple roots, Int. J. Comput. Math., in press, doi:10.1080/00207160802272263. · Zbl 1192.65052 [9] Werner, W., Iterationsverfahren höherer ordnung zur Lösung nicht linearer gleichungen, Z. angew. math. mech., 61, T322-T324, (1981) · Zbl 0494.65024 [10] B. Neta, Numerical Methods for the Solution of Equations, Net-A-Sof, California, 1983. · Zbl 0514.65029 [11] Halley, E., A new, exact and easy method of finding the roots of equations generally and that without any previous reduction, Phil. trans. roy. soc. London, 18, 136-148, (1694) [12] King, R.F., A family of fourth order methods for nonlinear equations, SIAM J. numer. anal., 10, 876-879, (1973) · Zbl 0266.65040 [13] Jarratt, P., Some fourth order multipoint methods for solving equations, Math. comput., 20, 434-437, (1966) · Zbl 0229.65049 [14] Osada, N., An optimal multiple root-finding method of order three, J. comput. appl. math., 51, 131-133, (1994) · Zbl 0814.65045 [15] Jarratt, P., Multipoint iterative methods for solving certain equations, Comput. J., 8, 398-400, (1966) · Zbl 0141.13404 [16] Murakami, T., Some fifth order multipoint iterative formulae for solving equations, J. inform. process., 1, 138-139, (1978) · Zbl 0394.65015 [17] Chun, C., A simply constructed third-order modifications of newton’s method, J. comput. appl. math., (2007) [18] Redfern, D., The Maple handbook, (1994), Springer-Verlag New York · Zbl 0820.68002 [19] Neta, B., New third order nonlinear solvers for multiple roots, Appl. math. comput., 202, 162-170, (2008) · Zbl 1151.65041
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