Convergence Thresholds of Newton's Method for Monotone Polynomial Equations

Abstract. Monotone systems of polynomial equations (MSPEs) are systems of fixedpoint equations X1 = f1(X1, . . . , Xn), . . . , Xn = fn(X1, . . . , Xn) where each fi is a polynomial with positive real coefficients. The question of computing the least non-negative solution of a given MSPE X = f(X) arises naturally in the analysis of stochastic models such as stochastic context-free grammars, probabilistic pushdown automata, and backbutton 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 kf for strongly connected MSPEs, such that after kf 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 kf as a function of the minimal component of the least fixed-point

• Type: Journal
• Year: 2008
• Where: CORR
• Authors: Javier Esparza, Stefan Kiefer, Michael Luttenberger
• Keywords: CORR 2008, Education, Positive Real Coefficients, Probabilistic Pushdown Automata, Strongly Connected