Resultant matrices and the computation of the degree of an approximate greatest common divisor of two inexact Bernstein basis polynomials

Abstract

The computation of the degree d of an approximate greatest common divisor of two Bernstein basis polynomials f(y) and g(y) that are noisy forms of, respectively, the exact polynomials fˆ(y) and gˆ(y) that have a non-constant common divisor is considered using the singular value decomposition of their Sylvester S(f,g) and Bézout B(f,g) resultant matrices. It is shown that the best estimate of d is obtained when S(f,g) is postmultiplied by a diagonal matrix Q that is derived from the vectors that lie in the null space of S(f,g), where the correct value of d is defined as the degree of the greatest common divisor of the exact polynomials fˆ(y) and gˆ(y). The computed value of d is improved further by preprocessing f(y) and g(y), and examples of the computation of d using S(f,g), S(f,g)Q and B(f,g) are presented.