An ODE-based method for computing the approximate greatest common divisor of polynomials

Abstract

Computing the greatest common divisor of a set of polynomials is a problem which plays an important role in different fields, such as linear system, control and network theory. In practice, the polynomials are obtained through measurements and computations, so that their coefficients are inexact. This poses the problem of computing an approximate common factor. We propose an improvement and a generalization of the method recently proposed in [12], which restates the problem as a (structured) distance to singularity of the Sylvester matrix. We generalize the algorithm in order to work with more than 2 polynomials and to compute an Approximate GCD (Greatest Common Divisor) of degree k ≥ 1; moreover we show that the algorithm becomes faster by replacing the eigenvalues by the singular values.

Computing common factors of matrix polynomials with applications in system and control theory

“…The computation of approximate common factors has been extensively studied in the case of scalar polynomials (e.g. [5], [6], [7], [8], [9], [10], [11], [12], [13], [14] and some references therein) and it is still an active research topic. However, the case of matrix polynomials has not been considered in the scientific literature.…”

“…This is a straightforward generalization of the approximate GCD computation for scalar polynomials ( [5], [15], [12], [9], [10]). As in the case of scalar polynomials, matrix polynomials having a common divisor define a variety of the Grassman type [16].…”

“…The approximate greatest common divisor problem is a non-convex optimization problem. To the best of our knowledge, there is no algorithm in the scientific literature for approximate GCD computation for matrix polynomials, so we propose a generalization of the algorithm in [5] (based on a local optimization approach) from scalar to matrix polynomials. We describe in the appendix how the algorithm works, and we use it later on a numerical example.…”

“…In the following, we propose a numerical example in order to show the benefits of our approach (the behavioral setting) and the performances of our algorithm (the one we briefly illustate in the Appendix; it generalizes the local optimization method proposed in [5], based on integration of a system of ODEs describing the gradient system associated with a suitable functional). Problem 4 could be stated in the same way it was developed for Single-Input Single-Output systems; the novelty consists in the algorithm we have for its numerical solution (there is no algorithm in the scientific literature for computing approximate common factors of matrix polynomials, to the best of our knowledge).…”

“…the computation of approximate left common factors for two matrix polynomials. It generalizes the algorithm presented in [5] from scalar to matrix polynomials.…”