Newtonian Graph

Newton's method for finding roots of a complex polynomial  entails iterating the function , which can be viewed as applying the Euler backward method with step size unity to the so-called Newtonian vector field . The rescaled and desingularized vector 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, Shub et al. 1988, Kozen and Stefánsson 1997).

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." In Handbook 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." In Geometrical 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.