A small probabilistic universal set of starting points for finding roots of complex polynomials by Newton's method

Béla Bollobás; Malte Lackmann; Dierk Schleicher

A small probabilistic universal set of starting points for finding roots of complex polynomials by Newton's method

Dynamical Systems 2011-08-31 v2 Numerical Analysis

Authors: Béla Bollobás , Malte Lackmann , Dierk Schleicher

Abstract

We specify a small set, consisting of $O(d(\log\log d)^2)$ points, that intersects the basins under Newton's method of \emph{all} roots of \emph{all} (suitably normalized) complex polynomials of fixed degrees $d$ , with arbitrarily high probability. This set is an efficient and universal \emph{probabilistic} set of starting points to find all roots of polynomials of degree $d$ using Newton's method; the best known \emph{deterministic} set of starting points consists of $\lceil 1.1d(\log d)^2\rceil$ points.

Keywords

Newton's method complex polynomials root finding probabilistic starting points basins of attraction universal set

Cite

@article{arxiv.1009.1843,
  title  = {A small probabilistic universal set of starting points for finding roots of complex polynomials by Newton's method},
  author = {Béla Bollobás and Malte Lackmann and Dierk Schleicher},
  journal= {arXiv preprint arXiv:1009.1843},
  year   = {2011}
}

Comments

17 pages, 6 figures. Minor update and slight improvements upon referee recommendations

A small probabilistic universal set of starting points for finding roots of complex polynomials by Newton's method

Abstract

Keywords

Cite

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