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

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 St(f,g)Qt of the tth subresultant matrix St(f,g) of the Sylvester resultant matrix S(f,g) of f(y) and g(y), where Qt is a diagonal matrix of combinatorial terms. This modified subresultant matrix has significant computational advantages with respect to the standard subresultant matrix St(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 St(f,g)Qt is formed, and the consequence of these operations is the introduction of two parameters, α and θ, such that the entries of St(f,g)Qt 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.