zbMATH — the first resource for mathematics

High-order nonlinear solver for multiple roots. (English) Zbl 1142.65044
Summary: A method of order four for finding multiple zeros of nonlinear functions is developed. The method is based on P. Jarratt’s fifth-order method (for simple roots) [Comput. J. 8, 398–400 (1966; Zbl 0141.13404)] and it requires one evaluation of the function and three evaluations of the derivative. The informational efficiency of the method is the same as previously developed schemes of lower order. For the special case of double root, we found a family of fourth-order methods requiring one less derivative. Thus this family is more efficient than all others. All these methods require the knowledge of the multiplicity.

MSC:
 65H05 Numerical computation of solutions to single equations
Zbl 0141.13404
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  Neta, B., Numerical methods for the solution of equations, (1983), Net-A-Sof California · 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 family of multipoint iterative functions for finding multiple roots of equations, Int. J. comput. math., 21, 363-367, (1987) · Zbl 0656.65050  Halley, E., A new, exact and easy method of finding the roots of equations generally and that without any previous reduction, Phil. trans. R. soc. London, 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  Jarratt, P., Multipoint iterative methods for solving certain equations, Comput. J., 8, 398-400, (1966) · Zbl 0141.13404  Redfern, D., The Maple handbook, (1994), Springer-Verlag New York · Zbl 0820.68002
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.