Numerical Computation of a Polynomial GCD and Extensions

In the rst part of this paper, we deene approximate polynomial gcds (greatest common divisors) and extended gcds provided that approximations to the zeros of the input polynomials are available. We relate our novel deenition to the older and weaker ones, based on perturbation of the coeecients of the input polynomials, we demonstrate some deeciency of the latter deenitions (which our deenition avoids), and we propose new eeective sequential and parallel (RNC and NC) algorithms for computing approximate gcds and extended gcds. Our stronger results are obtained with no increase of the asymptotic bounds on the computational cost. This is partly due to application of our recent nearly optimal algorithms for approximating polynomial zeros. In the second part of our paper, working under the older and more customary deenition of approximate gcds, we modify and develop an alternative approach, which was previously based on the computation of the Singular Value Decomposition (SVD) of the associated Sylvester (resultant) matrix. We observe that only a small part of the SVD computation is needed in our case, and we also yield further simpliication by using the techniques of Padd approximation and computations with Hankel and Bezout matrices. Finally, in the last part of our paper, we show an extension of the numerical computation of the gcd to the problem of computing numerical rank of a Hankel matrix, which is a bottleneck of Padd and Berlekamp-Massey computations, having important applications to coding and transmission of information. RRsumm : Nous dddnissons les pgcd (plus grand commn diviseur) approchhs et ggnnralisss de polynnmes en fonction des valeurs approchhes des racines des polynnmes en entrre. Nous relions cette nouvelle dddnition aux plus anciennes et plus faibles en termes de perturbation des coeecients des polynnmes en entrre ; nous montrons des lacunes des dddnitions ddjj existantes (lacunes qu''vite notre mmthode) ; puis nous proposons de nouveaux algorithmes eeec-tifs ssquentiels et paralllles (RNC et NC) pour le calcul des pgcd approchhs et ggnnralisss. Nos meilleurs rrsultats n'ammliorent pas les bornes asympto-tiques du coot des calculs. Ceci est partiellement dd une application de notre rrcent algorithme presque optimal pour l'approximation des racines d'un po-lynnme. Nous tudions aussi et ddveloppons une autre approche inspirre des techniques d'approximation de Padd et de calcul de matrices de Hanhel et de Bezout. Nous montrons ensuite une nouvelle extension du probllme de calcul nummrique de rang de matrice de Hankel, goulet d''tranglement des calculs de …