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Topic: Bernstein polynomial

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	Bernstein polynomial - Wikipedia, the free encyclopedia
		For the Bernstein polynomial in D-module theory, see Bernstein-Sato polynomial.
		The n + 1 Bernstein basis polynomials of degree n are defined as
		Bernstein polynomials thus afford one way to prove the Stone-Weierstrass approximation theorem that every real-valued continuous function on a real interval [a,b] can be uniformly approximated by polynomial functions over R.
	en.wikipedia.org /wiki/Bernstein_polynomial (443 words)

	polyApprox.html
		This latter class of polynomials are simplier for their construction uses neither the techniques of differentiation nor of integration.
		This is the Bernstein polynomial approximation for g.
		Bernstein Polynomials will converge uniformly to the function being approximated, if the function is continuous.
	www.adeptscience.co.uk /maplearticles/f50.html (1490 words)

	Bézier Curves in Bernstein Form
		Bernstein polynomials: The Bernstein polynomials of degree n are defined by
		Generalized Bernstein polynomials: Recall that every point p in the interior of a triangle with vertices U, V, and W can be expressed in terms of the barycentric coordinates (u,v,w) which are determined by p(u,v,w) = uU + vV + wW, subject to the constraints u+v+w=1; u,v,w > 0.
		Since the generalized Bernstein polynomials are linearly independent, they form a basis for an (n+1)(n+2)/2 dimensional linear subspace of the space of polynomials of degree (n, n).
	www.math.hmc.edu /~gu/math142/mellon/Application_to_CAGD/Bezier_curves/Bernstein_Form.html (395 words)

	List Of Articles (Site not responding. Last check: 2007-10-07)
		Here we have used Bernstein polynomial method to fit QCD predictions for the moments of $g_1^p$ structure function, to suitably the constructed appropriate average quantities of the E143 and SMC experimental data.
		Bernstein polynomial averages are used to compare parameterized valon distribution to the $xg_1^p$ of the E143 and SMC data, and direct fits for unknown parameters, over the range $3\leq Q^2\leq 10$ GeV$^2$, were performed.
		Bernstein polynomial averages are used to get unknown parameters of polarized valon distributions by fitting to available experimental data.
	theory.ipm.ac.ir /papers/atashbar.html (970 words)

	[No title]
		The n-variate polynomial p of degree EMBED Equation.3 can again be represented in the form (1), the Bernstein polynomials are given by (2), and the relations (3) — (6) generalize to the multivariate case.
		Bernstein expansion is applicable to the solution of this problem if we use algebraic stability criteria: In the case of Hurwitz stability, we have to test the so-called Hurwitz determinant for positivity over Q, see, e.g., Sect.
		By extending bounds known from the literature on the positive zeros of a polynomial with constant coefficients to the case in which the coefficients of the polynomial are depending on parameters, cf.
	www-home.fh-konstanz.de /~garloff/BeExAppl.doc (4139 words)

	Geisow's Algorithm
		It represents the polynomial in the Bernstein basis { choose(N,i)(1-x)^{N-i}x^i }, because the coefficients of the polynomial in that basis have the property that if ALL Bernstein coefficients have the same sign (pos or neg), then the polynomial does not have any zeros on the interval [0,1].
		Check coeffs of 2D Bernstein polynomial if they all have the same sign, poly has no zeros ==> DONE else { look at first partial derivatives (as Bernstein polys) if they have coeffs of same sign poly is monotone, therefore find zero and draw soln.
		For example, if the polynomial is singular at the origin, then the algorithm will "locate" the singularity by virtue of the fact that the singularity was on one of the grid points for the algorithm.
	www.geom.uiuc.edu /~streed/pisces/docs/algorithms/geisow.html (697 words)

	[No title]
		Bernstein Polynomials First we introduce the Bernstein polynomials which are the foundation for building Bezier curves.
		In contrast, in the case of the polynomial splines and B-splines, the coefficients are generally different from sub-interval to sub-interval.
		Prove that the Bernstein polynomial EMBED Equation.3 is linear in regard to function y.
	www.esm.psu.edu /courses/emch407/njs/notes02/ch6_3.doc (3071 words)

	PCO: Mathematic methods
		Since Ehrhart polynomials represent the exact number of integer points contained within a parameterized polytope or a union of parameterized polytopes, the approach we would like to explore considers any polynomial as the Ehrhart polynomial of a parameterized polytope or a union of parameterized polytopes.
		Bernstein polynomials are particular polynomials that form a basis for the space of polynomials.
		Due to the Bernstein convex hull property, the value of the polynomial is then bounded by the values of the minimum and maximum Bernstein coefficients.
	icps.u-strasbg.fr /pco/polynomial.htm (1673 words)

	MSc Projects 1999-2000 (Site not responding. Last check: 2007-10-07)
		The Bernstein basis is used extensively in computer graphics and computer aided geometric design because of its elegant geometric properties and numerical stability.
		The determination of the roots of a Bernstein polynomial is frequently required, for example, in intersection problems.
		This project is concerned with the development of a suite of MATLAB programs for the computation of the roots of a Bernstein polynomial.
	www.dcs.shef.ac.uk /~joab/Teaching/MScProjects/proj03.html (433 words)

	Atlas: Convergence of q-Bernstein polynomials by Sofiya Ostrovska
		In 1912 S.N. Bernstein found the proof of the Weierstrass Approximation Theorem based on the Law of Large Numbers for a sequence of Bernoulli trials.
		It was proved that convergence properties of q-Bernstein polynomials for q =/= 1 differ essentially from those in the classical case.
		The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cakq-17.
	atlas-conferences.com /cgi-bin/abstract/cakq-17 (271 words)

	Bernstein polynomial (Site not responding. Last check: 2007-10-07)
		The polynomial occurred as result of the work of Sergei Natanovich Bernstein, an Ukrainian mathematician (1880-1968).
		Bernstein introduced the curve in 1911, using it for a constructive proof for the Stone-Weierstrass approximation theorem.
		His initial idea was to construct a basic curve as the intersection of two elliptic cylinders, later he moved to polynomials.
	www.2dcurves.com /polynomial/polynomialb.html (210 words)

	ECS EPrints Service - Generalised neurofuzzy network modelling algorithms using Bezier Bernstein polynomial functions ...
		This paper is generalised in that it copes with n-dimensional inputs by utilising an additive decomposition construction to overcome the curse of dimensionality associated with high n..
		This new construction algorithm also introduces univariate Bezier Bernstein polynomial functions for the completeness of the generalised procedure.
		This new modelling network is based on additive decomposition approach together with two separate basis function formation approaches for both univariate and bivariate Bezier Bernstein polynomial functions used in model construction.
	eprints.ecs.soton.ac.uk /669 (208 words)

	[No title] (Site not responding. Last check: 2007-10-07)
		function yval = bp_approx (n, a, b, ydata, xval) %% BP_APPROX evaluates the Bernstein polynomial for F(X) on [A,B].
		However, for a fixed interval % [A,B], if we let N increase, the Bernstein polynomial converges % uniformly to F everywhere in [A,B], provided only that F is continuous.
		% % Parameters: % % Input, integer N, the degree of the Bernstein polynomial to be used.
	www.csit.fsu.edu /~burkardt/m_src/spline/bp_approx.m (277 words)

	Jürgen Garloff´s Homepage
		A fundamental property of the Bernstein expansion is its convex hull property which states that the graph of p over a box is contained in the convex hull of the control points associated with its Bernstein coefficients.
		We have developed two algorithms to check a polynomial family with polynomial parameter dependency for robust stability for a parameter box Q. Both algorithms rely on the expansion of a multivariate polynomial into Bernstein polynomials.
		The first one tests the Hurwitz determinant associated with the polynomial family for strict sign over Q. The second algorithm is designed for larger control problems and avoids the blowing up of the problem caused by using the Hurwitz determinant.
	www-home.fh-konstanz.de /~garloff (2136 words)

	Vrr: bernstein.h File Reference (Site not responding. Last check: 2007-10-07)
		Converts polynomial in bernstein form to power form.
		Function evaluates a real polynomial in bernstein form at given parameter t.
		Calculates first derivation of a polynomial in bernstein form.
	atrey.karlin.mff.cuni.cz /projekty/vrr/docs/html/bernstein_8h.html (336 words)

	Weierstrass Approximation Theorem. Bernstein's Polynomials. (Site not responding. Last check: 2007-10-07)
		of polynomials uniformly approaching f on the [0,1].
		It is called n-th Bernstein polynomial for the function f.
		Factorization of a polynomial, which defines values of sine function (angles n*pi/17).
	www.geocities.com /literka/mathcountry/approximation.htm (330 words)

	Final-year Geometric Modelling Course
		The Bernstein basis has the nice property that it is very numerically stable, so rounding errors in the computer with the high-degree terms will not be a problem, but the curve may take a while to compute.
		The fact that the whole high-degree curve is a single polynomial function also means that it has a very high degree of continuity - we can differentiate it lots of times without the derivatives becoming discontinuous.
		There are `more' implicit polynomials than parametric ones (in fact, there are an infinite number of both, of course, but there isn't a one-to-one mapping between the two sets).
	www.bath.ac.uk /~ensab/G_mod/FYGM/c5.htm (2388 words)

	Nonparametric Estimation of Multivariate Distributions with Given Marginals
		From the nonparametric Bernstein copula, the nonparametric Bernstein copula density is derived.
		It is shown that the nonparametric Bernstein copula density is closely related to the histogram estimator, but has the smoothing properties of kernel estimators.
		The optimal order of polynomial under the L2 norm is shown to be closely related to the inverse of the optimal smoothing factor for common nonparametric estimator.
	ideas.repec.org /p/cam/camdae/0320.html (301 words)

	Title page for ETD 583
		Multi-objective particle swarm optimization of a modified Bernstein polynomial for curved phased array synthesis using Bézier curves, surfaces, and volumes
		This dissertation develops the multi-objective particle swarm optimization of a modified Bernstein polynomial for curved phased array synthesis.
		Chapter 2 also introduces the novel modified Bernstein polynomial, which provides a new way to specify a variety of smooth, realizable, and unimodal array amplitude distributions using just five parameters.
	etda.libraries.psu.edu /theses/approved/WithheldIndex/ETD-583 (387 words)

	Undergraduate Lectures in Mathematics
		Show that if p(x) is a polynomial of degree n and if r is a root of the polynomial, then there exists a polynomial q(x) of degree at most n-1 with p(x) = (x-r)q(x).
		Use the intermediate value theorem to show that a polynomial of odd degree has a real root.
		Find the quadratic polynomial which is the best approximation to the function f(x) = 1/x at the points x=1,2,3,4.
	www.ms.uky.edu /~larry/lecture_notes.html (853 words)

	Bezier Curve Notes
		The Bezier polynomial is related to the Bernstein polynomial.
		Thus the Bezier curve is said to have a Bernstein basis.
		N is the degree of the polynomial and i is the particular vertex in the ordered set (from 0 to N).
	www.mtsu.edu /~csjudy/4250/BezierCurveNotes.htm (234 words)

	Publications
		Winkler, J.R. and Ragozin, D.L., "A class of Bernstein polynomials that satisfy Descartes' rule of signs exactly", The Mathematics of Surfaces IX : Proceedings of the Ninth IMA Conference on The Mathematics of Surfaces, The University of Cambridge, 4-6 September 2000.
		Winkler, J.R., "The numerical condition of roots of polynomials in Bernstein form", The Mathematics of Surfaces VIII, Proceedings of the 8th IMA Conference on The Mathematics of Surfaces, ed.
		Winkler, J.R., "Polynomial basis conversion made stable by truncated singular value decomposition", Applied Mathematical Modelling, 21, September 1997, 557-568.
	www.dcs.shef.ac.uk /~joab/publications.html (912 words)

	polynomial
		As said, here the polynomial in x, each term is an entire power of x, and the function is also called an entire (rational) function.
		The highest power in x is the degree (or: order) of the polynomial.
		A broken function is a function that is the quotient of two polynomials.Together with the polynomials do they form the group of rational functions, also named the rational polynomial functions.
	www.2dcurves.com /polynomial/polynomial.html (178 words)

	CS184 Lecture 20 summary
		The old basis was just the power basis, where the basis polynomials are powers of u.
		Also notice that only one of the Bernstein polynomials is non-zero at u = 0 (namely B
		Notice also that all the Bernstein polynomials are non-negative in the range u = 0,...,1.
	www.cs.berkeley.edu /~jfc/cs184f98/lec20/lec20.html (574 words)

	PlanetMath: Bernstein polynomial (Site not responding. Last check: 2007-10-07)
		Bernstein polynomials are used extensively in interpolation theory and in computer graphics.
		Gerald Farin, Curves and Surfaces for CAGD, A Practical Guide, 5th edition, Academic Press, 2002.
		This is version 3 of Bernstein polynomial, born on 2004-03-14, modified 2004-04-30.
	planetmath.org /encyclopedia/BernsteinPolynomial.html (59 words)

	[No title]
		These functions behave (within certain limits) like ordinary polynomials but their number of terms is uncountably infinite.
		These 'infinitely long' polynomials (they are infinitely much longer than any power series) provide a tool whose flexibility is unmatched in 'ordinary' analysis.
		In this paper each distribution with compact support is identified to a sequence of polynomials.
	cage.rug.ac.be /~ci/onderz.htm (480 words)

	Pisces: Geisow's Algorithm
		This algorithm (also known as the "2D Bernstein Method") traces the level set of a POLYNOMIAL function of two variables.
		It represents the polynomial in the Bernstein basis
		Sometimes symmetries in the polynomial being studied give us a misleading view of how the algorithm behaves.
	www.geom.uiuc.edu /~fjw/pisces/docs/panels/algorithm/geisow.html (684 words)

	OpenGL 1.1 Reference: glMap2
		Evaluators provide a way to use polynomial or rational polynomial mapping to produce vertices, normals, texture coordinates, and colors.
		All polynomial or rational polynomial splines of any degree (up to the maximum degree supported by the GL implementation) can be described using evaluators.
		Evaluators define surfaces based on bivariate Bernstein polynomials.
	www.talisman.org /opengl-1.1/Reference/glMap2.html (867 words)

	DC MetaData pour: On the Brieskorn $(a,b)$-module of an isolated hypersurface singularity (Site not responding. Last check: 2007-10-07)
		Résumé: We show in this note that for a germ \ $g$ \ of holomorphic function with an isolated singularity at the origin of \ $\mathbb{C}^n$ \ there is a pole for the meromorphic extension of the distribution
		at \ $- n - \alpha$ when \ $ \alpha$ \ is the smallest root in its class modulo \ $\mathbb{Z}$ \ of the reduce Bernstein-Sato polynomial of \ $g$.
		This duality gives also the relation between the "dual" Bernstein-Sato polynomial and the usual one, which is the key of the proof of the theorem.
	www.iecn.u-nancy.fr /Preprint/publis/2005_Daniel._Barlet.Fri_Nov_18_11_22_16_CET_2005 (205 words)

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