Effective Subdivision Algorithm for Isolating Zeros of Real Systems of Equations, with Complexity Analysis

Abstract

We describe a new algorithm \texttt{Miranda} for isolating the simple zeros of a function $\boldsymbol{f}:{\mathbb R}^n\to{\mathbb R}^n$ within a box $B_0\subseteq {\mathbb R}^n$. The function $\boldsymbol{f}$ and its partial derivatives must have interval forms, but need not be polynomial. Our subdivision-based algorithm is "effective" in the sense that our algorithmic description also specifies the numerical precision hat is sufficient to certify an implementation with any standard BigFloat number type. The main predicate is the Moore-Kioustelides (MK) test, based on Miranda's Theorem (1940). Although the MK test is well-known, this paper appears to be the first synthesis of this test into a complete root isolation algorithm. We provide a complexity analysis of our algorithm based on intrinsic geometric parameters of the system. Our algorithm and complexity analysis are developed using 3 levels of description (Abstract, Interval, Effective). This methodology provides a systematic pathway for achieving effective subdivision algorithms in general.