Solution of Systems of Polynomial Equations by Using Bernstein Expansion

Abstract

Systems of polynomial equations appear in a great variety of applications, e.g., in geometrie interseetion eomputations (Hu et al. 1996), ehemieal equilibrium problems, combustion, and kinematies, to name only a few. Examples ean be found in the monograph Morgan (1987). Following Sherbrooke and Patrikalakis (1993), most of the methods for the solution of sueh a system ean be dassitied as techniques based on elimination theory, continuation, and subdivision. Elimination theory-based methods for eonstructing Gröbner bases rely on symbolic manipulations, making those methods seem somewhat unsuitable for larger problems. This class and also the second of the methods based on continuation frequently give us more information than we need since they determine all complex solutions of the system, whereas in applications often only the solutions in a given area of interest - typically a box - are sought. In the last category we collect all methods which apply a domain-splitting approach: Starting with the box of interest, such an algorithrn sequentially splits it into subboxes, eliminating infeasible boxes by using bounds for the range of the polynomials under consideration over each of them, and ending up with a union of boxes that contains all solutions to the system which lie within the given box. Methods utilising this approach indude interval computation techniques as well as methods which apply the expansion of a multivariate polynomial into Bernstein polynomials