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New third order nonlinear solvers for multiple roots. (English) Zbl 1151.65041
Summary: Two third order methods for finding multiple zeros of nonlinear functions are developed. One method is based on Chebyshev’s third order scheme (for simple roots) and the other is a family based on a variant of Chebyshev’s which does not require the second derivative. Two other more efficient methods of lower order are also given. These last two methods are variants of Chebyshev’s and N. Osada’s schemes [J. Comput. Appl. Math. 51, No. 1, 131–133 (1994; Zbl 0814.65045)]. The informational efficiency of the methods is discussed. All these methods require the knowledge of the multiplicity.

MSC:
 65H05 Numerical computation of solutions to single equations
Zbl 0814.65045
Maple
Full Text:
References:
  Ostrowski, A.M., Solutions of equations and system of equations, (1960), Academic Press New York · Zbl 0115.11201  Traub, J.F., Iterative methods for the solution of equations, (1964), Prentice Hall New Jersey · Zbl 0121.11204  B. Neta, Numerical Methods for the Solution of Equations, Net-A-Sof, California, 1983. · Zbl 0514.65029  Rall, L.B., Convergence of the Newton process to multiple solutions, Numer. math., 9, 23-37, (1966) · Zbl 0163.38702  Schröder, E., Über unendlich viele algorithmen zur auflösung der gleichungen, Math. ann., 2, 317-365, (1870)  Hansen, E.; Patrick, M., A family of root finding methods, Numer. math., 27, 257-269, (1977) · Zbl 0361.65041  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  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  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  B. Neta, A.N. Johnson, High order nonlinear solver for multiple roots, Comput. Math. Appl., doi:10.1016/j.camwa.2007.09.001. · Zbl 1142.65044  B. Neta, Extension of Murakami’s high order nonlinear solver to multiple roots, Int. J. Comput. Math., submitted for publication. · Zbl 1192.65052  Werner, W., Iterationsverfahren höherer ordnung zur Lösung nicht linearer gleichungen, Z. angew. math. mech., 61, T322-T324, (1981) · Zbl 0494.65024  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. lond., 18, 136-148, (1694)  King, R.F., A family of fourth order methods for nonlinear equations, SIAM J. numer. anal., 10, 876-879, (1973) · Zbl 0266.65040  Jarratt, P., Some fourth order multipoint methods for solving equations, Math. comp., 20, 434-437, (1966) · Zbl 0229.65049  Osada, N., An optimal multiple root-finding method of order three, J. comput. appl. math., 51, 131-133, (1994) · Zbl 0814.65045  Jarratt, P., Multipoint iterative methods for solving certain equations, Comput. J., 8, 398-400, (1966) · Zbl 0141.13404  Murakami, T., Some fifth order multipoint iterative formulae for solving equations, J. inform. process., 1, 138-139, (1978) · Zbl 0394.65015  Candela, V.F.; Marquina, A., Recurrence relations for rational cubic methods II: the Chebyshev method, Computing, 45, 355-367, (1990) · Zbl 0714.65061  Hofsommer, D.J., Note on the computation of the zeros of functions satisfying a second order differential equation, Math. table other aids comp., 12, 58-60, (1958) · Zbl 0083.11901  Popovski, D.B., A family of one point iteration formulae for finding roots, Int. J. comput. math., 8, 85-88, (1980) · Zbl 0423.65028  Redfern, D., The Maple handbook, (1994), Springer-Verlag New York · Zbl 0820.68002  Kou, J.; Li, Y., Modified chebyshev’s method free from second derivative for nonlinear equations, Appl. math. comp., 187, 1027-1032, (2007) · Zbl 1116.65055  Neta, B., On popovski’s method for nonlinear equations, Appl. math. comp., 201, 1-2, 710-715, (2008) · Zbl 1155.65337
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