Polynomial root-finding with matrix eigen-solving

Abstract

Numerical matrix methods are increasingly popular for polynomial root-finding. This approach usually amounts to the application of the QR algorithm to the highly structured Frobenius companion matrix of the input polynomial. The structure, however, is routinely destroyed already in the first iteration steps. To accelerate this approach, we exploit the matrix structure of the Frobenius and generalized companion matrices, employ various known and novel techniques for eigen-solving and polynomial root-finding, and in addition to the Frobenius input allow other highly structured generalized companion matrices. Employing polynomial root-finders for eigen-solving is a harder task because of the potential numerical stability problems, but we found some new promising directions, particularly for sparse and/or structured input matrices.