The transformation of the companion matrix resultant between the power and Bernstein polynomial bases Original Research Article

Many problems in applied mathematics require the computation of the resultant of two polynomials, and it is nearly always assumed that the polynomials are expressed in the power basis. Recent work has shown that significantly improved numerical answers are obtained if the polynomials are expressed in the Bernstein basis, and thus the transformation of a resultant matrix between these bases is required. In this paper, this transformation is considered for one type of resultant, the companion matrix resultant. It is shown that this change of basis of the resultant matrix is defined by a similarity transformation, and that this transformation is ill-conditioned, even for matrices of low order. It is concluded that the companion matrix resultant should be constructed and computed in the Bernstein basis, such that the power basis is not used.