Numerical methods for computing the greatest common divisor of univariate polynomials using floating point arithmetic

The Numerical Greatest Common Divisor of Univariate Polynomials

This paper presents a regularization theory for numerical computation of polynomial greatest common divisors and a convergence analysis, along with a detailed description of a blackbox-type algorithm. The root of the ill-posedness in conventional GCD computation is identified by its geometry where polynomials form differentiable manifolds entangled in a stratification structure. With a proper r...

Numerical and Symbolical Methods Computing the Greatest Common Divisor of Several Polynomials

The computation of the Greatest Common Divisor (GCD) of a set of polynomials is an important issue in computational mathematics and it is linked to Control Theory very strong. In this paper we present different matrix-based methods, which are developed for the efficient computation of the GCD of several polynomials. Some of these methods are naturally developed for dealing with numerical inaccu...

Barnett's Theorems About the Greatest Common Divisor of Several Univariate Polynomials Through Bezout-like Matrices

This article provides a new presentation of Barnett’s theorems giving the degree (resp. coefficients) of the greatest common divisor of several univariate polynomials with coefficients in an integral domain by means of the rank (resp. linear dependencies of the columns) of several Bezout-like matrices. This new presentation uses Bezout or hybrid Bezout matrices instead of polynomials evaluated ...

Irreducibility and Greatest Common Divisor Algorithms for Sparse Polynomials

Complexity of Algorithms for Computing Greatest Common Divisors of Parametric Univariate Polynomials

This paper presents a comparison between the complexity bounds of different algorithms for computing greatest common divisor of a finite set of parametric univariate polynomials. Each algorithm decomposes the parameters space into a finite number of constructible sets such that a greatest common divisor of the parametric univariate polynomials is given uniformly in each constructible set. The f...