The GPGCD Algorithm with the B\'ezout Matrix for Multiple Univariate Polynomials

TL;DR

This paper introduces a modified GPGCD algorithm utilizing the Bézout matrix to efficiently compute approximate GCDs of multiple univariate polynomials, improving efficiency while maintaining accuracy.

Contribution

The paper presents a novel modification of the GPGCD algorithm that replaces the Sylvester matrix with the Bézout matrix for better efficiency in multiple polynomial GCD computation.

Findings

The proposed algorithm is more efficient than the original GPGCD.

It maintains similar accuracy levels in most cases.

Experimental results support the improved efficiency.

Abstract

We propose a modification of the GPGCD algorithm, which has been presented in our previous research, for calculating approximate greatest common divisor (GCD) of more than 2 univariate polynomials with real coefficients and a given degree. In transferring the approximate GCD problem to a constrained minimization problem, different from the original GPGCD algorithm for multiple polynomials which uses the Sylvester subresultant matrix, the proposed algorithm uses the B\'ezout matrix. Experiments show that the proposed algorithm is more efficient than the original GPGCD algorithm for multiple polynomials with maintaining almost the same accuracy for most of the cases.