Acta Applicandae Mathematica

, Volume 13, Issue 1, pp 81–121

The continuous, desingularized Newton method for meromorphic functions

  • H. Th. Jongen
  • P. Jonker
  • F. Twilt
Article

DOI: 10.1007/BF00047503

Cite this article as:
Jongen, H.T., Jonker, P. & Twilt, F. Acta Appl Math (1988) 13: 81. doi:10.1007/BF00047503

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.

AMS subject classifications (1980)

05C0505C3030D3030D3534C3558F09

Key words

Newton methodmeromorphic functiondynamical systemphase-portraitplane (sphere) graphasymptotic value

Copyright information

© Kluwer Academic Publishers 1988

Authors and Affiliations

  • H. Th. Jongen
    • 1
  • P. Jonker
    • 1
  • F. Twilt
    • 1
  1. 1.Department of Applied MathematicsTwente University of TechnologyEnschedeThe Netherlands