Abstract. Experimental observations of rootfinding by generalized companion matrix pencils expressed in the Lagrange basis show that the method can sometimes be numerically stable, and indeed sometimes be much more stable than rootfinding of polynomials expressed in even the Bernstein basis. This paper details some of those experiments and provides a theoretical justification for this. We prove that a new condition number, defined for points on a set containing the interpolation points, is never larger than the rootfinding condition number for the Bernstein polynomial; and computation shows that sometimes it can be much smaller. This result may be of interest for those who wish to find the zeros of polynomials given simply by values. 1

A closed form expression for a companion matrix M of a Bernstein polynomial is obtained, and this is used to derive an expression for a resultant matrix of two Bernstein polynomials. It is shown that M diers from its equivalent form for a power basis polynomial because an upper triangular Hankel matrix does not de ne a similarity transformation between M and M . A measure of the numerical condition of a resultant matrix, for polynomials in an arbitrary basis, is reviewed and this is used to compare the stability of two resultant matrices. In particular, computational tests are performed and it is shown that the resultant matrix of two Bernstein polynomials is numerically better conditioned than the resultant matrix that is obtained when a simple parameter substitution is used to transform the polynomials to the power basis.

Abstract. Experimental observations of univariate rootfinding by generalized companion matrix pencils expressed in the Lagrange basis show that the method can sometimes be numerically stable. It has recently been proved that a new condition number, defined for points on a set containing the interpolation points, is never larger than the rootfinding condition number for the Bernstein polynomial (which is itself optimal in a certain sense); and computation shows that sometimes it can be much smaller. These results also hold for the matrix polynomial case, when we are not looking for polynomial roots but rather for eigenvalues where the matrix polynomial is singular. This current paper explores the influence of the geometry of the interpolation nodes on the conditioning of the rootfinding and eigenvalue problems.

The computational implementation in a oating 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 dierent 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 de nes 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.