Proceedings of the 2019 on International Symposium on Symbolic and Algebraic Computation | 2019
Effective Subdivision Algorithm for Isolating Zeros of Real Systems of Equations, with Complexity Analysis
Abstract
We describe a new algorithm Miranda for isolating the simple zeros of a function \\boldsymbolf :\\mathbbR ^n\\to\\mathbbR ^n within a box B_0\\subseteq\\mathbbR ^n. The function \\boldsymbolf 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 that is sufficient to certify an implementation with any standard BigFloat number type. The main predicate is the Moore-Kioustelidis (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.