Box-Bisection for Solving Second-Degree Systems and the Problem of Clustering

Abstract

Box-bisection is a method for solving nonlinear systems. Space is subdivided into boxes of smallerand smaller diameter, and each subbox is tested for the existence of solutions by a test that eithereliminates it from further consideration or marks it for subdivision. Simple bisection uses a test forthe exclusion of subboxes, but no test that guarantees the existence of a unique solution in a subbox. Using this simple bisection, we show that the passed boxes tend to cluster in geometrical configurations whose number is stable under subdivision. This implies for many problems that the workrequired to do simple bisection may be prohibitive. However, improvements may be possible bygrouping clusters and dynamically redefining the box proportions. The restriction to second-degreesystems is sufficient to display this behavior. © 1987, ACM. All rights reserved.