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

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Inventiones mathematicae Aims and scope

Abstract.

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.

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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

Authors
  1. John Hubbard
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  2. Dierk Schleicher
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  3. Scott Sutherland
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Oblatum 24-II-2000 & 14-II-2001¶Published online: 20 July 2001

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Hubbard, J., Schleicher, D. & Sutherland, S. How to find all roots of complex polynomials by Newton’s method. Invent. math. 146, 1–33 (2001). https://doi.org/10.1007/s002220100149

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  • DOI: https://doi.org/10.1007/s002220100149

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