×

zbMATH — the first resource for mathematics

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
Citations:
Zbl 0814.65045
Software:
Maple
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Ostrowski, A.M., Solutions of equations and system of equations, (1960), Academic Press New York · Zbl 0115.11201
[2] Traub, J.F., Iterative methods for the solution of equations, (1964), Prentice Hall New Jersey · Zbl 0121.11204
[3] B. Neta, Numerical Methods for the Solution of Equations, Net-A-Sof, California, 1983. · Zbl 0514.65029
[4] Rall, L.B., Convergence of the Newton process to multiple solutions, Numer. math., 9, 23-37, (1966) · Zbl 0163.38702
[5] Schröder, E., Über unendlich viele algorithmen zur auflösung der gleichungen, Math. ann., 2, 317-365, (1870)
[6] Hansen, E.; Patrick, M., A family of root finding methods, Numer. math., 27, 257-269, (1977) · Zbl 0361.65041
[7] 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
[8] 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
[9] 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
[10] 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
[11] B. Neta, Extension of Murakami’s high order nonlinear solver to multiple roots, Int. J. Comput. Math., submitted for publication. · Zbl 1192.65052
[12] Werner, W., Iterationsverfahren höherer ordnung zur Lösung nicht linearer gleichungen, Z. angew. math. mech., 61, T322-T324, (1981) · Zbl 0494.65024
[13] 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)
[14] King, R.F., A family of fourth order methods for nonlinear equations, SIAM J. numer. anal., 10, 876-879, (1973) · Zbl 0266.65040
[15] Jarratt, P., Some fourth order multipoint methods for solving equations, Math. comp., 20, 434-437, (1966) · Zbl 0229.65049
[16] Osada, N., An optimal multiple root-finding method of order three, J. comput. appl. math., 51, 131-133, (1994) · Zbl 0814.65045
[17] Jarratt, P., Multipoint iterative methods for solving certain equations, Comput. J., 8, 398-400, (1966) · Zbl 0141.13404
[18] Murakami, T., Some fifth order multipoint iterative formulae for solving equations, J. inform. process., 1, 138-139, (1978) · Zbl 0394.65015
[19] Candela, V.F.; Marquina, A., Recurrence relations for rational cubic methods II: the Chebyshev method, Computing, 45, 355-367, (1990) · Zbl 0714.65061
[20] 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
[21] Popovski, D.B., A family of one point iteration formulae for finding roots, Int. J. comput. math., 8, 85-88, (1980) · Zbl 0423.65028
[22] Redfern, D., The Maple handbook, (1994), Springer-Verlag New York · Zbl 0820.68002
[23] 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
[24] Neta, B., On popovski’s method for nonlinear equations, Appl. math. comp., 201, 1-2, 710-715, (2008) · Zbl 1155.65337
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.