Numerical and Algebraic Properties of Bernstein Basis Resultant Matrices

Abstract

Algebraic properties of the power and Bernstein forms of the companion, Sylvester and Bézout resultant matrices are compared and it is shown that some properties of the power basis form of these matrices are not shared by their Bernstein basis equivalents because of the combinatorial factors in the Bernstein basis functions. Several condition numbers of a resultant matrix are considered and it is shown that the most refined measure is NP-hard, and that a simpler, sub-optimal measure is easily computed. The transformation of the companion and Bézout resultant matrices between the power and Bernstein bases is considered numerically and algebraically. In particular, it is shown that these transformations are ill—conditioned, even for polynomials of low degree, and that the matrices that occur in these basis transformation equations share some properties.