The calculation of the degree of an approximate greatest common divisor of two polynomials

The calculation of the degree of an approximate greatest common divisor (AGCD) of two inexact polynomials f(y)f(y) and g(y)g(y) is a non-trivial computation because it reduces to the estimation of the rank loss of a resultant matrix R(f,g)R(f,g). This computation is usually performed by placing a threshold on the small singular values of R(f,g)R(f,g), but this method suffers from disadvantages because the numerical rank of R(f,g)R(f,g) may not be defined, or the noise level imposed on the coefficients of f(y)f(y) and g(y)g(y) may not be known, or it may only be known approximately. This paper addresses this problem by considering two methods for estimating the degree of an AGCD of f(y)f(y) and g(y)g(y), such that knowledge of the noise level is not required. The first method involves the calculation of the smallest angle between two subspaces that are apparent from the structure of the Sylvester resultant matrix S(f,g)S(f,g), and the second method uses the theory of subresultant matrices, which are derived from S(f,g)S(f,g) by the deletion of some of its rows and columns. The two methods are compared computationally on non-trivial polynomials.