Optimal Scaling of Companion Pencils for the QZ-Algorithm∗†

Computing roots of a monic polynomial may be done by computing the eigenvalues of the corresponding companion matrix using for instance the well-known QR-algorithm. We know this algorithm to be backward stable since it computes exact eigenvalues of a slightly modified matrix. But it may yield very poor backward errors in the coefficients of the polynomial. In this paper we investigate what can be done to improve these errors, using a geometric approach. We will see that preconditioning the companion matrix using some carefully chosen similarity may achieve this goal. In particular, we will give a geometric interpretation of what balancing the companion matrix does. We then naturally extend these results for the nonmonic polynomial case where the algorithm we deal with is now the QZ-algorithm acting on companion pencils instead of companion matrices. The article is divided into two parts: in the first one we examine the monic scalar polynomial case and in the second one the general non-monic scalar case. In each part, we begin by explaining the problem in terms of error analysis, and then we look at the problem from a geometric point of view.