Matrix factorization of multivariate Bernstein polynomials

Abstract

Ordinary univariate Bernstein polynomials can be represented in matrix form using factor matrices. In this paper we present the definition and basic properties of such factor matrices extended from the univariate case to the general case of arbitrary number of variables by using barycentric coordinates in the hyper-simplices of respective dimension. The main results in the paper are related to the design of an iterative algorithm for fast convex computation of multivariate Bernstein polynomials based on sparse-matrix factorization. In the process of derivation of this algorithm, we investigate some properties of the factorization, including symmetry, commutativity and differentiability of the factor matrices, and address the relevance of this factorization to the de Casteljau algorithm for evaluating curves and surfaces on B´ezier form. A set of representative examples is provided, including a geometric interpretation of the de Casteljau algorithm, and representation by factor matrices of multivariate surfaces and their derivatives in B´ezier form. Another new result is the observation that inverting the order of steps of a part of the new factorization algorithm provides a new, matrix-based, algebraic representation of a multivariate generalization of a special case of the de Boor-Cox computational algorithm.