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 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 a so-called modified Newton method, which is a generalization of the gradient-projection method, by searching the solution iteratively. We demonstrate that our algorithm calculates approximate GCD with perturbations as small as those calculated by a method based on the structured total least norm (STLN) method, while our method runs significantly faster than theirs by approximately up to 30 times, compared with their implementation. We also show that our algorithm properly handles some ill-conditioned problems with GCD containing small or large leading coefficient.

GPGCD, an Iterative Method for Calculating Approximate GCD, for Multiple Univariate Polynomials

“…Based on our previous research ( [17], [21]), we have extended our GPGCD method for more than two input polynomials with the real coefficients. We have shown that, at least theoretically, our algorithm properly calculates an approximate GCD under certain conditions for multiple polynomial inputs.…”

A Quadratically Convergent Algorithm for Structured Low-Rank Approximation

“…In Table 1, we compare the steps' sizes of NewtonSLRA with those of GPGCD, a state-of-the art algorithm dedicated to the computation of approximate GCDs [42]. The experimental results give evidence of the practical quadratic convergence of NewtonSLRA, as predicted by Theorem 4.1.…”

“…In some articles, the goal is to find a pair (f * , g * ) which minimizes the distance (f − f * , g − g * ) (see e.g. [42] and references therein). In particular, [11] yields a certified quadratically convergent algorithm in the particular case d = 1 (i.e.…”

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

“…Remark 3. In this experiment, we have compared our implementation designed 3 Our previous test results ( [26,Section 5.2]) have shown that there were test cases (input polynomials with the real coefficients) in which the GPGCD method was not able to calculate an approximate GCD with sufficiently small magnitude of perturbations. After thorough investigation, we have found that such input polynomials accidentally have an approximate GCD of degree exceeding d. for problems of two univariate polynomials against the implementation of the STLN-based method designed for multivariate multi-polynomial problems with additional linear coefficient constraints.…”