A non-linear structure-preserving matrix method for the computation of the coefficients of an approximate greatest common divisor of two Bernstein polynomials

Highlights

• An approximate greatest common divisor of two Bernstein polynomials is computed.

• An approximate polynomial factorisation and the Sylvester matrix are used.

• The results from the two methods are very similar.

Abstract

This paper describes a non-linear structure-preserving matrix method for the computation of the coefficients of an approximate greatest common divisor ($\mathrm{AGCD}$) of degree $t$ of two Bernstein polynomials $f\left(y\right)$ and $g\left(y\right)$. This method is applied to a modified form $S_t(f,g)Q_t$ of the $t$th subresultant matrix $S_t(f,g)$ of the Sylvester resultant matrix $S(f,g)$ of $f\left(y\right)$ and $g\left(y\right)$, where $Q_t$ is a diagonal matrix of combinatorial terms. This modified subresultant matrix has significant computational advantages with respect to the standard subresultant matrix $S_t(f,g)$, and it yields better results for $\mathrm{AGCD}$ computations. It is shown that $f\left(y\right)$ and $g\left(y\right)$ must be processed by three operations before $S_t(f,g)Q_t$ is formed, and the consequence of these operations is the introduction of two parameters, $\alpha$ and $\theta$, such that the entries of $S_t(f,g)Q_t$ are non-linear functions of $\alpha ,\theta$ and the coefficients of $f\left(y\right)$ and $g\left(y\right)$. The values of $\alpha$ and $\theta$ are optimised, and it is shown that these optimal values allow an $\mathrm{AGCD}$ that has a small error, and a structured low rank approximation of $S(f,g)$, to be computed.