The computation of the degree of the greatest common divisor of three Bernstein basis polynomials

Abstract

This paper considers the application of the Sylvester resultant matrix to the computation of the degree of the greatest common divisor (GCD) of three Bernstein basis polynomials f(y),g(y) and h(y). It is shown that the governing equations can be written in two forms, which lead to different Sylvester matrices. The first form requires that the polynomials be considered in pairs, but different pairs of polynomials may yield different computational answers, for example, the solution of the computations GCD(f,g) and GCD(g,h) may differ from the solution of the computations GCD(f,g) and GCD(f,h), depending on f(y),g(y) and h(y). This problem does not arise when the second form is considered, which requires that the three polynomials be considered simultaneously. Complications arise in both forms because of the combinatorial terms in the Bernstein basis functions, which cause the entries of the matrices to span several orders of magnitude, even if the coefficients of the polynomials are of the same order of magnitude. It is shown that the adverse effects of this wide range of magnitudes can be mitigated by postmultiplying both forms of the Sylvester matrix by a diagonal matrix of combinatorial terms and preprocessing f(y),g(y) and h(y) by three operations. Results of GCD computations from the two forms of the Sylvester matrix when f(y),g(y) and h(y) are perturbed by noise, and with the omission and inclusion of the preprocessing operations, are shown.