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Newtonian Graph
Newton's method for finding roots of a complex polynomial
entails iterating the function![z-[f(z)/f^'(z)]](/image.pl?url=http%3a%2f%2fmathworld.wolfram.com%2fimages%2fequations%2fNewtonianGraph%2fInline2.svg&f=jpg&w=240)
, which can be viewed as applying theEuler backward method with step size unity to the so-calledNewtonian vector field
. The rescaled and desingularizedvector field
then has sinks at roots of
and has saddle points at roots of
that are not also roots of
. The union of the closures of the unstable manifolds of the saddles of
defines a directed graph whose vertices are the roots of
and of
, and whose edges are the unstable curves oriented by the flow direction. This graph, along with the labelling of each vertex
with the multiplicity
of
as a root of
, is defined to be the Newtonian graph of
(Smale 1985, Shubet al.1988, Kozen and Stefánsson 1997).
See also
Newton's Method,NewtonianVector Field,Vector FieldExplore with Wolfram|Alpha
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References
Airapetyan, R. "Continuous Newton Method and Its Modification."Appl. Anal.73, 463-484, 1999.Airapetyan, R.; Ramm, A. G.; and Smirnova, A. "Continuous Analog of the Gauss-Newton Method."Math. Models Methods Appl. Sci.9, 463-474, 1999.Diener, I. "Trajectory Methods in Global Optimization." InHandbook of Global Optimization, 2 (Ed. R. Horst and P. M. Pardalos). Dordrecht, Netherlands: Kluwer, pp. 649-668, 1995.Jongen, H. T.; Jonker, P.; and Twilt, F. "The Continuous Newton-Method for Meromorphic Functions." InGeometrical Approaches to Differential Equations (Proc. Fourth Scheveningen Conf., Scheveningen, 1979) (Ed. R. Martini). Berlin: Springer-Verlag, pp. 181-239, 1980.Jongen, H. T.; Jonker, P.; and Twilt, F. "The Continuous, Desingularized Newton Method for Meromorphic Functions."Acta Appl. Math.13, 81-121, 1988.Kozen, D. and Stefánsson, K. "Computing the Newtonian Graph."J. Symb. Comput.24, 125-136, 1997.Shub, M.; Tischler, D.; Williams, R. F. "The Newtonian Graph of a Complex Polynomial."SIAM J. Math. Anal.19, 246-256, 1988.Smale, S. "On the Efficiency of Algorithms of Analysis."Bull. Amer. Math. Soc.13, 87-121, 1985.Referenced on Wolfram|Alpha
Newtonian GraphCite this as:
Weisstein, Eric W. "Newtonian Graph."FromMathWorld--A Wolfram Resource.https://mathworld.wolfram.com/NewtonianGraph.html