Solving Polynomials with Small Leading Coefficients

We explore the computation of roots of polynomials via eigenvalue problems. In particular, we look at the case when the leading coeecient is relatively very small. We argue that the companion matrix algorithm (used, for instance, by the Matlab roots function) is inaccurate in this case. The accuracy problem is addressed by using matrix pencils instead. This improvement can be predicted from the backward error bound of Edelman and Murakami (for companion matrices) versus the bound of Van Dooren and Dewilde (for pencils). We then show how to extend the accurate algorithm to B ezier polynomials and present computational experiments.