Inventiones mathematicae

, Volume 146, Issue 1, pp 1–33

How to find all roots of complex polynomials by Newton’s method

  • John Hubbard
  • Dierk Schleicher
  • Scott Sutherland

DOI: 10.1007/s002220100149

Cite this article as:
Hubbard, J., Schleicher, D. & Sutherland, S. Invent. math. (2001) 146: 1. doi:10.1007/s002220100149


We investigate Newton’s method to find roots of polynomials of fixed degree d, appropriately normalized: we construct a finite set of points such that, for every root of every such polynomial, at least one of these points will converge to this root under Newton’s map. The cardinality of such a set can be as small as 1.11 d log2d; if all the roots of the polynomial are real, it can be 1.30 d.

Copyright information

© Springer-Verlag Berlin Heidelberg 2001

Authors and Affiliations

  • John Hubbard
    • 1
    • 2
  • Dierk Schleicher
    • 3
  • Scott Sutherland
    • 4
  1. 1.Department of Mathematics, Malott Hall, Cornell University, Ithaca, NY 14853-4201, USA (e-mail:
  2. 2.Centre de Mathématiques et Informatique, Université de Provence, 39 rue F. Joliot-Curie, F-13453 Marseille cedex 13, FranceFrance
  3. 3.Mathematisches Institut, Ludwig-Maximilians-Universität, Theresienstrasse 39, D-80333 München, Germany (e-mail:
  4. 4.Institute for Mathematical Sciences, State University of New York, Stony Brook, NY 11794-3660, USA (e-mail: