Properties of the Companion Matrix Resultant for Bernstein Polynomials

The computational implementation in a floating point environment of the companion matrix resultant is considered and it is shown that the numerical condition of the resultant matrix is strongly dependent on the basis in which the polynomials are expressed. In particular, a companion matrix of a Bernstein polynomial is derived and this is used to construct a resultant matrix for two Bernstein polynomials. A measure of the numerical condition of a resultant matrix is developed and then used to compare the stability of the resultant matrices of the same polynomials that are expressed in different bases. It is shown that it is desirable to express the polynomials in the Bernstein basis, but since the power basis is the natural choice in many applications, a transformation of the resultant matrix between these bases is required. It is shown that this transformation of the resultant matrix between the bases cannot be achieved by performing a basis transformation of each polynomial. Rather, the equation that defines the transformation of the companion matrix resultant between the bases is derived by considering the eigenvectors of the companion matrix of a polynomial in each basis. The numerical condition of this equation is considered and it is shown that it is ill-conditioned, even for polynomials of low degree.