The Computation of Multiple Roots of a Bernstein Basis Polynomial

2008

In this article we want to determinate a recursive formula for Bernstein polynomials associated to the functions ep(x) = xp, p ∈ N, and an expresion for the central moments of the Bernstein polinomyals. 2000 Mathematics Subject Classification. 41A10; 41A63.

2004

We introduce polynomials B n i (x; ω|q), depending on two parameters q and ω, which generalize classical Bernstein polynomials, discrete Bernstein polynomials defined by Sablonnière, as well as q-Bernstein polynomials introduced by Phillips. Basic properties of the new polynomials are given. Also, formulas relating B n i (x; ω|q), big q-Jacobi and q-Hahn (or dual q-Hahn) polynomials are presented.

Journal of Computational and Applied Mathematics, 2019

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.

We construct multiple representations relative to different bases of the generalized Tschebyscheff polynomials of second kind. Also, we provide an explicit closed from of The generalized Polynomials of degree r less than or equal n in terms of the Bernstein basis of fixed degree n. In addition, we create the change-of-basis matrices between the generalized Tschebyscheff of the second kind polynomial basis and Bernstein polynomial basis

arXiv (Cornell University), 2015

We construct multiple representations relative to different bases of the generalized Tschebyscheff polynomials of second kind. Also, we provide an explicit closed from of The generalized Polynomials of degree r less than or equal n in terms of the Bernstein basis of fixed degree n. In addition, we create the change-of-basis matrices between the generalized Tschebyscheff of the second kind polynomial basis and Bernstein polynomial basis.

In this paper, the problem of computing zeros of a general degree bivariate Bernstein polynomial is considered. An efficient and robust algorithm is presented that takes into full account particular properties of the function considered. The algorithm works for rectangular as well as triangular domains. The outlined procedure can also be applied for the computation of the intersection of a Bézier patch and a plane as well as in the determination of an algebraic curve restricted to a compact domain. In particular, singular points of the algebraic curve are reliably detected.

International Journal of Mathematics, 2016

This paper provides an explicit closed form of generalized Jacobi–Koornwinder’s polynomials of degree [Formula: see text] in terms of the Bernstein basis of fixed degree [Formula: see text] Moreover, explicit forms of generalized Jacobi–Koornwinder’s type and Bernstein polynomials bases transformations are considered.

Journal of Computational and Applied Mathematics, 2008

The aim of this paper is to transform a polynomial expressed as a weighted sum of discrete orthogonal polynomials on Gauss-Lobatto nodes into Bernstein form and vice versa. Explicit formulas and recursion expressions are derived. Moreover, an efficient algorithm for the transformation from Gauss-Lobatto to Bernstein is proposed. Finally, in order to show the robustness of the proposed algorithm, experimental results are reported.

Journal of Approximation Theory, 2006

The generalized Bernstein basis in the space n of polynomials of degree at most n, being an extension of the q-Bernstein basis introduced by Philips [Bernstein polynomials based on the q-integers, Ann. Numer. Math. 4 (1997) 511-518], is given by the formula [S. Lewanowicz, P. Woźny, Generalized Bernstein polynomials, BIT Numer. Math. 44 (2004) 63-78] B n i (x; | q) := 1 (; q) n n i q x i (x −1 ; q) i (x; q) n−i (i = 0, 1,. .. , n). We give explicitly the dual basis functions D n k (x; a, b, | q) for the polynomials B n i (x; | q), in terms of big q-Jacobi polynomials P k (x; a, b, /q; q), a and b being parameters; the connection coefficients are evaluations of the q-Hahn polynomials. An inverse formula-relating big q-Jacobi, dual generalized Bernstein, and dual q-Hahn polynomials-is also given. Further, an alternative formula is given, representing the dual polynomial D n j (0 j n) as a linear combination of min(j, n − j) + 1 big q-Jacobi polynomials with shifted parameters and argument. Finally, we give a recurrence relation satisfied by D n k , as well as an identity which may be seen as an analogue of the extended Marsden's identity [R.N. Goldman, Dual polynomial bases, J. Approx. Theory 79 (1994) 311-346].

Journal of Computational and Applied Mathematics, 2021

This paper considers an approximate factorisation of three bivariate Bernstein basis polynomials that are defined in a triangular domain. This problem is important for the computation of the intersection points and curves of surfaces in computer-aided design systems, and it reduces to the determination of an approximate greatest common divisor (AGCD) d(y) of the polynomials. The Sylvester matrix and its subresultant matrices of these three polynomials are formed and it is shown that there are four forms of these matrices. The most difficult part of the computation is the determination of the degree of d(y) because it reduces to the determination of the rank loss of these matrices. This computation is made harder by the presence of trinomial terms in the Bernstein basis functions because they cause the entries of the matrices to span many orders of magnitude. The adverse numerical effects of this wide range of magnitudes of the entries of the four forms of the Sylvester matrix and its subresultant matrices are mitigated by processing the polynomials before these matrices are formed. It is shown that significantly improved results are obtained if the polynomials are processed before computations are performed on their Sylvester matrices and subresultant matrices.