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Computing the least fixed point of positive polynomial systems

By Javier Esparza, Stefan Kiefer and Michael Luttenberger

Abstract

We consider equation systems of the form X1 = f1(X1,...,Xn),..., Xn = fn(X1,...,Xn), where f1,...,fn are polynomials with positive real coefficients. In vector form we denote such an equation system by X = f(X) andcallf a system of positive polynomials (SPP). Equation systems of this kind appear naturally in the analysis of stochastic models like stochastic context-free grammars (with numerous applications to natural language processing and computational biology), probabilistic programs with procedures, web-surfing models with back buttons, and branching processes. The least nonnegative solution μf of an SPP equation X = f(X) is of central interest for these models. Etessami and Yannakakis [J. ACM, 56 (2009), pp. 1–66] have suggested a particular version of Newton’s method to approximate μf. We extend a result of Etessami and Yannakakis and show that Newton’s method starting at 0 always converges to μf. We obtain lower bounds on the convergence speed of the method. For so-called strongly connected SPPs we prove the existence of a threshold kf ∈ N such that for every i ≥ 0the(kf + i)th iteration of Newton’s method has at least i valid bits of μf. The proof yields an explicit bound for kf depending only on syntactic parameters of f. We further show that for arbitrary SPP equations, Newton’s method still converges linearly: there exists a threshold kf and an αf> 0 such that for every i ≥ 0the(kf + αf · i)th iteration of Newton’s method has at least i valid bits of μf. The proof yields an explicit bound for αf;the bound is exponential in the number of equations in X = f(X), but we also show that it is essentially optimal. The proof does not yield any bound for kf, but only proves its existence. Constructing a bound for kf is still an open problem. Finally, we also provide a geometric interpretation of Newton’s method for SPPs

Year: 2010
OAI identifier: oai:CiteSeerX.psu:10.1.1.181.1010
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