Abstract generalized bisection and a cost bound

Abstract

The purpose of this paper is to study desirable properties of binary search algorithms for isolating all solutions to nonlinear systems of equations F ( X ) = 0 F(X) = 0 within a given compact domain D ∈ R n {\mathbf {D}} \in {{\mathbf {R}}^n} . We devise a general framework such that any algorithm fitting into the general framework will always isolate all solutions Z ∈ D Z \in {\mathbf {D}} such that F ( Z ) = 0 F(Z) = 0 ; this framework contains a new idea for handling the occurrence of roots on boundaries. We then present and prove a bound on the total amount of computation which is valid for any algorithm in the class. Finally, we define a specific prototypical algorithm valid for F satisfying certain natural smoothness properties; we show that it satisfies the hypotheses for the general framework. This algorithm is based on "bisection" of generalized rectangles, the Kantorovich theorem, and second-order Taylor type models for F. It is meant to provide further guidelines for the development of effective heuristics, etc., for actual implementations.