Convergence Thresholds of Newton's Method for Monotone Polynomial Equations

Javier Esparza, Stefan Kiefer & Michael Luttenberger
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...