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

TL;DR

This paper extends the GPGCD iterative method to compute approximate GCDs for multiple univariate polynomials, minimizing coefficient perturbations through a constrained optimization approach.

Contribution

The paper introduces an extension of the GPGCD method to handle multiple polynomials, enabling approximate GCD computation with minimal coefficient perturbations.

Findings

Successfully extended GPGCD to multiple polynomials.

Achieved minimal perturbations in coefficients.

Demonstrated effectiveness through numerical experiments.

Abstract

We present an extension of our GPGCD method, an iterative method for calculating approximate greatest common divisor (GCD) of univariate polynomials, to multiple polynomial inputs. 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. In our GPGCD method, the problem of approximate GCD is transferred 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. In this paper, we extend our method to accept more than two polynomials with the real coefficients as an input.