Computing the Least Fixed Point of Positive Polynomial Systems

• Authors: Javier Esparza;Stefan Kiefer;Michael Luttenberger
• Venue: SIAM Journal on Computing
• Year: 2010

We consider equation systems of the form $X_1=f_1(X_1,\dots,X_n)$, $\dots$, $X_n = f_n(X_1,\dots,X_n)$, where $f_1,\dots,f_n$ are polynomials with positive real coefficients. In vector form we denote such an equation system by ${\bf X}={\bf f}({\bf X})$ and call ${\bf f}$ 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 $\mu{\bf f}$ of an SPP equation ${\bf X}={\bf f}({\bf X})$ 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 $\mu{\bf f}$. We extend a result of Etessami and Yannakakis and show that Newton's method starting at ${\bf 0}$ always converges to $\mu{\bf f}$. We obtain lower bounds on the convergence speed of the method. For so-called strongly connected SPPs we prove the existence of a threshold $k_{{\bf f}}\in\mathbb{N}$ such that for every $i\geq0$ the $(k_{{\bf f}}+i)$th iteration of Newton's method has at least $i$ valid bits of $\mu{\bf f}$. The proof yields an explicit bound for $k_{{\bf f}}$ depending only on syntactic parameters of ${\bf f}$. We further show that for arbitrary SPP equations, Newton's method still converges linearly: there exists a threshold $k_{{\bf f}}$ and an $\alpha_{{\bf f}}0$ such that for every $i\geq0$ the $(k_{{\bf f}}+\alpha_{{\bf f}}\cdot i)$th iteration of Newton's method has at least $i$ valid bits of $\mu{\bf f}$. The proof yields an explicit bound for $\alpha_{{\bf f}}$; the bound is exponential in the number of equations in ${\bf X}={\bf f}({\bf X})$, but we also show that it is essentially optimal. The proof does not yield any bound for $k_{{\bf f}}$, but only proves its existence. Constructing a bound for $k_{{\bf f}}$ is still an open problem. Finally, we also provide a geometric interpretation of Newton's method for SPPs.