Computation of Approximate Polynomial GCDs and an Extension

Abstract

Computation of approximate polynomial greatest common divisors (GCDs) is important both theoretically and due to its applications to control linear systems, network theory, and computer-aided design. We study two approaches to the solution so far omitted by the researchers, despite intensive recent work in this area. Correlation to numerical Padé approximation enabled us to improve computations for both problems (GCDs and Padé). Reduction to the approximation of polynomial zeros enabled us to obtain a new insight into the GCD problem and to devise effective solution algorithms. In particular, unlike the known algorithms, we estimate the degree of approximate GCDs at a low computational cost, and this enables us to obtain certified correct solution for a large class of input polynomials. We also restate the problem in terms of the norm of the perturbation of the zeros (rather than the coefficients) of the input polynomials, which leads us to the fast certified solution for any pair of input polynomials via the computation of their roots and the maximum matchings or connected components in the associated bipartite graph.