Monotone systems of polynomial equations (MSPEs) are systems of fixed-point
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 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 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 μf of f(X). Using this result we
show that kf 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/w2h new
bits of the solution, where w and h are the width and height of the DAG of
strongly connected components.Comment: version 2 deposited February 29, after the end of the STACS
conference. Two minor mistakes correcte