On the numerical condition of polynomials in Bernstein form

Abstract

The Bernstein-Bézier curve and surface forms have enjoyed considerable popularity in computer aided design applications, due to their elegant geometric properties and the simple recursive algorithms available for processing them. In this paper we describe a different and largely unexplored favorable aspect of Bernstein-Bézier methods: their inherent numerical stability under the influence of imprecise computer arithmetic or perturbed input data. We show that the condition numbers of the simple real roots of a polynomial on the unit interval are always smaller in the Bernstein basis than in the power basis. These arguments are then generalized to accommodate multiple roots and roots on an arbitrary interval. We further show that the standard Bernstein polynomial subdivision and degree elevation procedures induce a strictly monotonic decrease in root condition numbers. Finally, we prove that among a large family of polynomial bases characterized by certain simple properties, the Bernstein basis exhibits optimal root conditioning. Some well-known examples are employed to illustrate these results, and their practical implications for floating-point implementation of curve and surface intersection algorithms in a geometric modeling system are discussed. Important issues requiring further attention in the processing of polynomials in Bernstein form are also briefly indicated.