Solving Polynomials with Small Leading Coefficients
Guðbjörn F. Jónsson, Stephen A. Vavasis · SIAM Journal on Matrix Analysis and Applications · 2004
We explore the computation of roots of polynomials via eigenvalue problems. In particular, we look at the case when the leading coefficient is relatively very small. We argue that the companion matrix algorithm (used, for instance, by the Matlab {\tt 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 Bézier polynomials and present computational experiments.