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

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 log² d; if all the roots of the polynomial are real, it can be 1.30 d.

Authors and Affiliations

1. Department of Mathematics, Malott Hall, Cornell University, Ithaca, NY 14853-4201, USA (e-mail: jhh8@cornell.edu), USA

   John Hubbard

2. Centre de Mathématiques et Informatique, Université de Provence, 39 rue F. Joliot-Curie, F-13453 Marseille cedex 13, France, France

   John Hubbard

3. Mathematisches Institut, Ludwig-Maximilians-Universität, Theresienstrasse 39, D-80333 München, Germany (e-mail: dierk@rz.mathematik.uni-muenchen.de), Germany

   Dierk Schleicher

4. Institute for Mathematical Sciences, State University of New York, Stony Brook, NY 11794-3660, USA (e-mail: scott@math.sunysb.edu), USA

   Scott Sutherland

Oblatum 24-II-2000 & 14-II-2001¶Published online: 20 July 2001

Reprints and permissions