Bernstein Bases are Optimal , but , sometimes , Lagrange Bases are Better

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.