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Convergence Thresholds of Newton's Method for Monotone Polynomial Equations

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Abstract

Monotone systems of polynomial equations (MSPEs) are systems of fixed-point equations $X_1 = f_1(X_1, ldots, X_n),$ $ldots, X_n = f_n(X_1, ldots, X_n)$ where each $f_i$ is a polynomial with positive real coefficients. The question of computing the least non-negative solution of a given MSPE $vec X = vec f(vec X)$ arises naturally in the analysis of stochastic models such as stochastic context-free grammars, probabilistic pushdown automata, and back-button processes. Etessami and Yannakakis have recently adapted Newton's iterative method to MSPEs. In a previous paper we have proved the existence of a threshold $k_{vec f}$ for strongly connected MSPEs, such that after $k_{vec f}$ iterations of Newton's method each new iteration computes at least 1 new bit of the solution. However, the proof was purely existential. In this paper we give an upper bound for $k_{vec f}$ as a function of the minimal component of the least fixed-point $muvec f$ of $vec f(vec X)$. Using this result we show that $k_{vec f}$ is at most single exponential resp. linear for strongly connected MSPEs derived from probabilistic pushdown automata resp. from back-button processes. Further, we prove the existence of a threshold for arbitrary MSPEs after which each new iteration computes at least $1/w2^h$ new bits of the solution, where $w$ and $h$ are the width and height of the DAG of strongly connected components.

BibTeX - Entry

@InProceedings{esparza_et_al:LIPIcs:2008:1351,
  author =	{Javier Esparza and Stefan Kiefer and Michael Luttenberger},
  title =	{{Convergence Thresholds of Newton's Method for Monotone Polynomial Equations}},
  booktitle =	{25th International Symposium on Theoretical Aspects of Computer Science},
  pages =	{289--300},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-939897-06-4},
  ISSN =	{1868-8969},
  year =	{2008},
  volume =	{1},
  editor =	{Susanne Albers and Pascal Weil},
  publisher =	{Schloss Dagstuhl--Leibniz-Zentrum fuer Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{http://drops.dagstuhl.de/opus/volltexte/2008/1351},
  URN =		{urn:nbn:de:0030-drops-13516},
  doi =		{http://dx.doi.org/10.4230/LIPIcs.STACS.2008.1351},
  annote =	{Keywords: Newton's Method, Fixed-Point Equations, Formal Verification of Software, Probabilistic Pushdown Systems}
}

Keywords: Newton's Method, Fixed-Point Equations, Formal Verification of Software, Probabilistic Pushdown Systems
Seminar: 25th International Symposium on Theoretical Aspects of Computer Science
Issue date: 2008
Date of publication: 2008


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