Structured low rank approximations of the sylvester resultant matrix for approximate GCDS of Bernstein basis polynomials

Winkler, Joab R. ; Allan, John D.

Electronic transactions on numerical analysis, Tome 31 (2008), pp. 141-155

Résumé

A structured low rank approximation of the Sylvester resultant matrix $S(f, g)$ of the Bernstein basis polynomials f = f (y) and $g = g(y)$, for the determination of their approximate greatest common divisors (GCDs), is computed using the method of structured total least norm. Since the GCD of f (y) and $g(y)$ is equal to the GCD of f (y) and $\alpha g(y)$, where $\alpha $is an arbitrary non-zero constant, it is more appropriate to consider a structured low rank approximation $S( \tilde f , \tilde g)$ of $S(f, \alpha g)$, where the polynomials $\tilde f = \tilde f$ (y) and $\tilde g = \tilde g(y)$ approximate the polynomials f (y) and $\alpha g(y)$, respectively. Different values of $\alpha $yield different structured low rank approximations $S( \tilde f , \tilde g)$, and therefore different approximate GCDs. It is shown that the inclusion of $\alpha $allows to obtain considerably improved approximations, as measured by the decrease of the singular values $\sigma i$ of $S( \tilde f , \tilde g)$, with respect to the approximation obtained when the default value $\alpha = 1$ is used. An example that illustrates the theory is presented and future work is discussed.

Détail

Classification : 15A12, 65F35
Keywords: Bernstein polynomials, structured low rank approximation, sylvester resultant matrix