{
  "id": "1202.2475",
  "version": "v2",
  "published": "2012-02-11T21:02:36.000Z",
  "updated": "2014-08-25T16:24:31.000Z",
  "title": "On the speed of convergence of Newton's method for complex polynomials",
  "authors": [
    "Todor Bilarev",
    "Magnus Aspenberg",
    "Dierk Schleicher"
  ],
  "comment": "13 pages, 1 figure, to appear in AMS Mathematics of Computation",
  "categories": [
    "math.DS"
  ],
  "abstract": "We investigate Newton's method for complex polynomials of arbitrary degree $d$, normalized so that all their roots are in the unit disk. For each degree $d$, we give an explicit set $\\mathcal{S}_d$ of $3.33d\\log^2 d(1 + o(1))$ points with the following universal property: for every normalized polynomial of degree $d$ there are $d$ starting points in $\\mathcal{S}_d$ whose Newton iterations find all the roots with a low number of iterations: if the roots are uniformly and independently distributed, we show that with probability at least $1-2/d$ the number of iterations for these $d$ starting points to reach all roots with precision $\\varepsilon$ is $O(d^2\\log^4 d + d\\log|\\log \\varepsilon|)$. This is an improvement of an earlier result in \\cite{Schleicher}, where the number of iterations is shown to be $O(d^4\\log^2 d + d^3\\log^2d|\\log \\varepsilon|)$ in the worst case (allowing multiple roots) and $O(d^3\\log^2 d(\\log d + \\log \\delta) + d\\log|\\log \\varepsilon|)$ for well-separated (so-called $\\delta$-separated) roots. Our result is almost optimal for this kind of starting points in the sense that the number of iterations can never be smaller than $O(d^2)$ for fixed $\\varepsilon$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-02-11T21:02:36.000Z",
      "abstract": "We investigate Newton's method for complex polynomials of arbitrary degree $d$, normalized so that all their roots are in the unit disk. For each degree $d$, we give an explicit set $\\mathcal{S}_d$ of $3.33d\\log^2 d(1 + o(1))$ points with the following universal property: for every normalized polynomial of degree $d$ there are $d$ starting points in $\\mathcal{S}_d$ whose Newton iterations find all the roots. If the roots are uniformly and independently distributed, we show that the number of iterations for these $d$ starting points to reach all roots with precision $\\varepsilon$ is $O(d^2\\log^4 d + d\\log|\\log \\varepsilon|)$ (with probability $p_d$ tending to 1 as $d\\to\\infty$). This is an improvement of an earlier result in \\cite{D}, where the number of iterations is shown to be $O(d^4\\log^2 d + d^3\\log^2d|\\log \\varepsilon|)$ in the worst case (allowing multiple roots) and $O(d^3\\log^2 d(\\log d + \\log \\delta) + d\\log|\\log \\varepsilon|)$ for well-separated (so-called $\\delta$-separated) roots. Our result is almost optimal for this kind of starting points in the sense that the number of iterations can never be smaller than $O(d^2)$ for fixed $\\eps$.",
      "comment": "11 pages, 1 figure",
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2014-08-25T16:24:31.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "newtons method",
      "complex polynomials",
      "starting points",
      "convergence",
      "universal property"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 13,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1202.2475B"
    }
  }
}