Fast algorithms for applying finite element mass and stiffness operators to the B-form of polynomials over d-dimensional simplices are derived. These rely on special properties of the Bernstein basis and lead to stiffness matrix algorithms with the same asymptotic complexity as tensor-product techniques in rectangular domains. First, special structure leading to fast application of mass matrice...

A-patches[BCX94a] are implicit surfaces in Bernstein-Bezier(BB) form that are smooth and single-sheeted. In this paper, we present algorithms to utilize the extra degrees of freedom of these patches for local shape co·ntrol. A ray shooting scheme is also given to rapidly generate polygonal approximations of A-patches for graphic display. A distributed implementation of this scheme gives near "r...

Multi-degree Tchebycheffian splines are with pieces drawn from extended (complete) Tchebycheff spaces, which may differ interval to interval, and possibly of different dimensions. These a natural extension multi-degree polynomial splines. Under quite mild assumptions, they can be represented in terms so-called B-spline (MDTB-spline) basis; such basis possesses all the characterizing properties ...

Recently, a new class of surface-divergence free radial basis function interpolants has been developed for surfaces in R3. In this paper, several approximation results for this class of interpolants will be derived in the case of the sphere, S2. In particular, Sobolev-type error estimates are obtained, as well as optimal stability estimates for the associated interpolation matrices. In addition...

We derive the stability inequality ‖C‖ 6 γ ‖∑i cibi‖ for the B-splines bi from the formula for knot insertion. The key observation is that knot removal increases the norm of the B-spline coefficients C = {ci}i∈Z at most by a constant factor, which is independent of the knot sequence. As a consequence, stability for splines follows from the stability of the Bernstein basis.  2000 Elsevier Scien...

Sets of desirable gambles constitute a quite general type of uncertainty model with an interesting geometrical interpretation. We study inĄnite exchangeability assessments for them, and give a counterpart of de FinettiŠs inĄnite representation theorem. We show how the inĄnite representation in terms of frequency vectors is tied up with multivariate Bernstein (basis) polynomials. We also lay bar...

S d := {x = (x1, . . . , xd) ∈ R : 0 6 x1, . . . , xd 6 1, 0 6 x1 + · · ·+ xd 6 1} denote the standard simplex in R. We denote by ∂S the boundary of S. We will also use barycentric coordinates on the simplex which we denote by the boldface symbol x = (x0, x1, . . . , xd), x0 := 1−x1−· · ·−xd. We will use standard multiindex notation such as x := x0 0 x α1 1 · · ·xd d and α n := (α0 n , α1 n , ·...

‎In this paper‎, ‎we present a numerical method based on Bernstein polynomials to solve optimal control systems with constant and pantograph delays‎. ‎Constant or pantograph delays may appear in state-control or both‎. ‎We derive delay operational matrix and pantograph operational matrix for Bernstein polynomials then‎, ‎these are utilized to reduce the solution of optimal control with constant...

Bernstein coe cients provide a discrete approximation of the behavior of a polynomial inside an interval. This can be used for example to isolate real roots of polynomials. We prove a criterion for the existence of a single root in an interval and the correctness of the de Casteljau algorithm to compute e ciently Bernstein coe cients. Key-words: Polynomials, de Casteljau, Bernstein polynomials,...

We propose an algorithm based on Newton’s method and subdivision for finding all zeros of a polynomial system in a bounded region of the plane. This algorithm can be used to find the intersections between a line and a surface, which has applications in graphics and computer-aided geometric design. The algorithm can operate on polynomials represented in any basis that satisfies a few conditions....