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

Abstract

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

The Computation of Multiple Roots of a Bernstein Basis Polynomial

“…, m, (1) because the entries in the matrices that arise in the computations may span several orders of magnitude. Also, the Sylvester matrix and its subresultant matrices, which are used for the GCD computations, for two power basis polynomials are formed by the concatenation of two Toeplitz matrices, but these matrices for two Bernstein basis polynomials do not have this structure [3,4,24], from which it follows that algorithms that require the Sylvester matrix and its subresultant matrices are more complicated for Bernstein basis polynomials.…”

“…because D −1 k and Q k , which are defined in (10) and 14, respectively, are nonsingular. It is shown in [3,4] that the best form of the Sylvester matrix and its subresultant matrices is D −1 k T k (f, g)Q k because it yields significantly better results than the other forms in (15) for the degree and coefficients of the GCD of f (y) and g(y). This result follows from consideration of the combinatorial terms, which are of the forms…”

The Sylvester and Bézout Resultant Matrices for Blind Image Deconvolution

“…2, thereby yielding the polynomialsf (w) and α 0ḡ (w). The calculation of the coefficients oft(w), the GCD off (w) and α 0ḡ (w), is addressed for Bernstein basis polynomials in [9], and the same method can be used for power basis polynomials. In particular, Theorem 3 shows that…”

“…, has unit rank loss and (9) shows that the coprime polynomialsv d (w) andū d (w) lie in the null space of S d (f , α 0ḡ ). There is therefore one equation that defines the linear dependence of the columns of S d (f , α 0ḡ ), and if the pth column of S d (f , α 0ḡ ) is one of these linearly dependent columns and it is denoted by b, then the homogeneous equation (9) can be written as…”

“…whereĒ d has the same structure asĀ d , andē has the same structure asb, from which it follows that a structurepreserving matrix method is used to compute its solution, that is, the matrixĒ d and the vectorsē andx [25]. The solution is under-determined but it is shown in [9] that the addition of a constraint allows a unique solution to be computed. The approximation (14) and the exact equation (15) can be interpreted in terms of polynomial computations.…”