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

TL;DR

This paper introduces GPGCD, an iterative algorithm that efficiently computes approximate GCDs of univariate polynomials by minimizing coefficient perturbations, outperforming existing methods in speed while maintaining accuracy.

Contribution

The paper presents a novel iterative method for approximate GCD computation that is faster and handles ill-conditioned polynomials effectively.

Findings

Achieves perturbation levels comparable to existing methods.

Runs up to 30 times faster than structured total least norm method.

Effectively handles ill-conditioned polynomials with small or large GCD coefficients.

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.