Generalized companion matrix for approximate GCD

Paola Boito (1) , Olivier Ruatta (1)

1 DMI

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 $g$in the quotient ring $\mathbb{C}[x] / (f)$. 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.

Mots clés: Approximate greatest common divisor; Approximate geometry; Symbolic-numeric computation; Structured matrices

Domaines: Géométrie algébrique [math.AG]; Calcul formel [cs.SC]

Paola Boito, Olivier Ruatta. Generalized companion matrix for approximate GCD. 2011. ⟨hal-00564448⟩