{
  "id": "1009.1843",
  "version": "v2",
  "published": "2010-09-09T18:31:59.000Z",
  "updated": "2011-08-29T22:24:31.000Z",
  "title": "A small probabilistic universal set of starting points for finding roots of complex polynomials by Newton's method",
  "authors": [
    "Béla Bollobás",
    "Malte Lackmann",
    "Dierk Schleicher"
  ],
  "comment": "17 pages, 6 figures. Minor update and slight improvements upon referee recommendations",
  "categories": [
    "math.DS",
    "math.NA"
  ],
  "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.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2011-08-29T22:24:31.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37F10",
      "49M15"
    ],
    "keywords": [
      "small probabilistic universal set",
      "newtons method",
      "starting points",
      "complex polynomials",
      "finding roots"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1009.1843B"
    }
  }
}