Abstract
For any (nonconstant) meromorphic function, we present a real analytic dynamical system, which may be interpreted as an infinitesimal version of Newton's method for finding its zeros. A fairly complete description of the local and global features of the phase portrait of such a system is obtained (especially, if the function behaves not too bizarre at infinity). Moreover, in the case of rational functions, structural stability aspects are studied. For a generic class of rational functions, we give a complete graph-theoretical characterization, resp. classification, of these systems. Finally, we present some results on the asymptotic behaviour of meromorphic functions.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Andronov A. A., Leontovich E. L., Gordon I. I., and Maier A. G.:Theory of bifurcations of dynamical systems on a plane, Wiley, New York, 1973.
Braess D.: Ueber die Einzugbereiche der Nullstellen von Polynomen beim Newton-Verfahren,Numer. Math. 29 (1977), 123–132.
Branin, F. H.: A widely convergent method for finding multiple solutions of simultaneous non-linear equations,I.B.M. J. Res. Develop. (1972), 504–522.
Deimling K.:Nicht-linearen Gleichungen und Abildungsgrade, Hochschultext, Springer-Verlag, Berlin, 1974.
Garcia C. B. and Gould F. J.: Relations beteween several path-following algorithms and local and global Newton methods, SIAMRev.,22, (1980) 263–274.
Giblin P. J.:Graphs, Surfaces and Homology, Wiley, New York, 1977.
Gomulka J.: Remarks on Branin's method for solving non-linear equations, in L. C. W. Dixon and G. P. Szegö (eds.),Towards Global Optimization Academic Press, New York, 1976.
Guillemin V. and Pollack A.:Differential Topology, Prentice Hall, Englewood Cliffs, 1974.
Harary F., Prins G., and Tutte W. T.: The number of plane trees,Indag. Math. 26 (1964), 319–329.
Harary F.:Graph Theory, Addison-Wesley, Reading, Mass., 1969.
Harary F. and Robinson R. W.: The number of achiral trees,J. Reine Angew. Math. 278 (1975), 322–335.
Hartman P.:Ordinary Differential Equations, John Wiley, New York, 1964.
Hirsch M. W. and Smale S.:Differential Equations, Dynamical Systems and Linear Algebra, Academic Press, New York, 1970.
Hirsch M. W. and Smale S.: Algorithms for solvingf(x)=0,Comm. Pure Appl. Math. 32 (1979), 281–312.
Holland A. S. B.:Introduction to the Theory of Entire Functions, Academic Press, New York, 1973.
Hurewicz W.:Lectures on Ordinary Differential Equations, MIT Press, Cambridge, Mass., 1970.
Hurley M. and Martin C.: Newton's algorithm and chaotic dynamical systems,SIAM J. Math. Anal. 15 (1984), 238–252.
Jongen H. Th., Jonker P., and Twilt F.: On Newton-flows in optimization,Methods of Operations Research 31 (1979), 345–359.
Jongen H. Th., Jonker P., and Twilt F.: The continuous Newton method for meromorphic functions, in R. Martini, (ed.),Geometric Approaches to Differential Equations, Lect. Notes in Math., Vol. 810, Springer-Verlag, Berlin 1980, pp. 181–239.
Jongen, H. Th., Jonker, P., and Twilt, F.: Some reflections on the continuous Newton-method for rational functions, in L. Collatz, G. Meinardus and W. Wetterling, (eds.),Konstruktive Methoden der finiten nichtlinearen Optimierung Int. Series Num. math., Vol. 55, 1980, pp. 131–147.
Jongen H. Th., Jonker P., and Twilt F.: On the classification of plane-graphs represepting structurally stable rational Newton flows, Memorandum Nr 645, University of Twente, Enschede, The Netherlands (1987).
Lefschetz, S.:Differential Equations: Geometric Theory, Interscience Publ.
Markushevich A. I.:Theory of Functions of a Complex Variable, Vol. II, Prentice Hall, Englewood Cliffs, 1965.
Markushevich A. I.:Theory of Functions of a Complex Variable, Vol. III, Prentice Hall, Englewood Cliffs, 1965.
Milnor, J. W.:Topology from a Differential Viewpoint, The University Press of Virginia, 1965.
Nevanlinna R.:Eindeutige Analytische Funktionen, Springer-Verlag, Berlin, 1957.
Peitgen, H. O., Saupe, D., and Haeseler, F. V.: Newton's methods and Julia sets, Report No. 104, Forschungsschwerpunkt Dynamische Systeme, Universität Bremen (1983).
Peixoto, M. C. and Peixoto, M. M.: Structural stability in the plane with enlarged boundary conditions,An. Acad. Bras. de Ciêns, (1959), 135–160.
Peixoto M. M.: Structural stability on two-dimensional manifolds,Topology 1 (1962), 101–120.
Peixoto M. M.: On the classification of flows on 2-manifolds, in M. M. Peixoto, (ed.),Dynamical Systems. Academic Press, New York, 1973, pp. 389–419.
Smale S.: A convergent process of price adjustment and global Newton-methods,J. Math. Economics 3 (1976), 107–120.
Smale, S.: The fundamental theorem of algebra and complexity theory,Bull. Am. Math. Soc., (1981), 1–36.
Twilt, F.: Newton-flows for meromorphic functions, Ph.D. Thesis, Twente University of Technology, The Netherlands, 1981.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Jongen, H.T., Jonker, P. & Twilt, F. The continuous, desingularized Newton method for meromorphic functions. Acta Appl Math 13, 81–121 (1988). https://doi.org/10.1007/BF00047503
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00047503