GPGCD: An iterative method for calculating approximate GCD of univariate polynomials

Abstract

We present an iterative algorithm for calculating approximate greatest common divisor (GCD) of univariate polynomials with the real or the complex coefficients. For a given pair of polynomials and a degree, our algorithm finds a pair of polynomials which has a GCD of the given degree and whose coefficients are perturbed from those in the original inputs, making the perturbations as small as possible, along with the GCD. The problem of approximate GCD is transfered to a constrained minimization problem, then solved with the so-called modified Newton method, which is a generalization of the gradient-projection method, by searching the solution iteratively. We demonstrate that, in some test cases, our algorithm calculates approximate GCD with perturbations as small as those calculated by a method based on the structured total least norm (STLN) method and the UVGCD method, while our method runs significantly faster than theirs by approximately up to 30 or 10 times, respectively, compared with their implementation. We also show that our algorithm properly handles some ill-conditioned polynomials which have a GCD with small or large leading coefficient.

The Computation of Multiple Roots of a Bernstein Basis Polynomial

“…3. The clear gap between the subresultant matrices that are, and are not, rank deficient in Figure 4 does not require a threshold for its determination, even though the upper bound of the relative error in the coefficients of f (y) and g(y) is a random variable that spans two orders of magnitude, as shown in (19) and (20). This is a measure of the efficacy of D −1 k T k (f k ,g k )Q k for the calculation of the degree of the GCD of f (y) and g(y).…”

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

“…It is considered a classical Sylvester matrix in the case of two polynomials [32], a generalized Sylvester matrix for more than two polynomials [13,14,24]. More algorithms for the AGCD computation in the same framework are the structured total least norm (STLN) approach [13,16], the gradient projection method [26], and a recent Structured Low Rank Approximation algorithm [25] based on a Newton-like iteration.…”

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.…”