Block LU factors of generalized companion matrix pencils
Theoretical Computer Science
New progress in real and complex polynomial root-finding
Computers & Mathematics with Applications
Efficient polynomial root-refiners: A survey and new record efficiency estimates
Computers & Mathematics with Applications
Journal of Computational and Applied Mathematics
Computer Aided Geometric Design
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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.