This paper considers the computation of the degree t of an approximate greatestudcommon divisor d(y) of two Bernstein polynomials f(y) and g(y), which areudof degrees m and n respectively. The value of t is computed from the QRuddecomposition of the Sylvester resultant matrix S(f, g) and its subresultantudmatrices Sk(f, g), k = 2, . . . , min(m, n), where S1(f, g) = S(f, g). It is shownudthat the computation of t is significantly more complicated than its equivalentudfor two power basis polynomials because (a) Sk(f, g) can be written in severaludforms that differ in the complexity of the computation of their entries, (b)uddifferent forms of Sk(f, g) may yield different values of t, and (c) the binomialudterms in the entries of Sk(f, g) may cause the ratio of its entry of maximumudmagnitude to its entry of minimum magnitude to be large, which may lead toudnumerical problems. It is shown that the QR decomposition and singular valueuddecomposition (SVD) of the Sylvester matrix and its subresultant matrices yieldudbetter results than the SVD of the B´ezout matrix, and that f(y) and g(y)udmust be processed before computations are performed on these resultant andudsubresultant matrices in order to obtain good results.
展开▼
