Generalized companion matrix for approximate GCD

TL;DR

This paper introduces a novel method for approximate GCD computation of two polynomials, leveraging the structure of a generalized companion matrix to achieve efficient quadratic-time algorithms.

Contribution

It presents a new approach using the properties of a generalized companion matrix to efficiently compute approximate GCDs with known and perturbed polynomial coefficients.

Findings

The method exploits the null space of the multiplication matrix for GCD information.

The multiplication matrix has a displacement structure enabling fast algorithms.

The approach achieves quadratic complexity in polynomial degrees.

Abstract

We study a variant of the univariate approximate GCD problem, where the coefficients of one polynomial f(x)are known exactly, whereas the coefficients of the second polynomial g(x)may be perturbed. Our approach relies on the properties of the matrix which describes the operator of multiplication by gin the quotient ring C[x]=(f). In particular, the structure of the null space of the multiplication matrix contains all the essential information about GCD(f; g). Moreover, the multiplication matrix exhibits a displacement structure that allows us to design a fast algorithm for approximate GCD computation with quadratic complexity w.r.t. polynomial degrees.