This paper considers the computation of the degree t of an approximate greatest common divisor d(y) of two Bernstein polynomials f (y) and g(y), which are of degrees m and n respectively. The value of t is computed from the QR decomposition of the Sylvester resultant matrix S(f, g) and its subresultant matrices S k (f, g), k = 2,. .. , min(m, n), where S 1 (f, g) = S(f, g). It is shown that the computation of t is significantly more complicated than its equivalent for two power basis polynomials because (a) S k (f, g) can be written in several forms that differ in the complexity of the computation of their entries, (b) different forms of S k (f, g) may yield different values of t, and (c) the binomial terms in the entries of S k (f, g) may cause the ratio of its entry of maximum magnitude to its entry of minimum magnitude to be large, which may lead to numerical problems. It is shown that the QR decomposition and singular value decomposition (SVD) of the Sylvester matrix and its subresultant matrices yield better results than the SVD of the Bézout matrix, and that f (y) and g(y) must be processed before computations are performed on these resultant and subresultant matrices in order to obtain good results.
This paper describes a non-linear structure-preserving matrix method for the computation of the coefficients of an approximate greatest common divisor (AGCD) of degree t of two Bernstein polynomials f (y) and g(y). This method is applied to a modified form S t (f, g)Q t of the tth subresultant matrix S t (f, g) of the Sylvester resultant matrix S(f, g) of f (y) and g(y), 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 AGCD computations. It is shown that f (y) and g(y) 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, α and θ, such that the entries of S t (f, g)Q t are non-linear functions of α, θ and the coefficients of f (y) and g(y). The values of α and θ are optimised, and it is shown that these optimal values allow an AGCD that has a small error, and a structured low rank approximation of S(f, g), to be computed.
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