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Page 5752 of Mathematical Reviews Vol. , Issue 94j [page]

1994 Mathematical Reviews  
Manfred Sommer (Eichstatt) 94j:41008 41A15 Wu, Dong Bing (PRC-BJ; Beijing) Dual bases of a Bernstein polynomial basis on simplices. (English summary) Comput. Aided Geom.  ...  The author constructs dual bases for the space spanned by the classical Bernstein basis polynomials of degree n on a simplex in R”+!.  ... 

Lattices and Algorithms for Bivariate Bernstein, Lagrange, Newton, and Other Related Polynomial Bases Based on Duality betweenL-Bases andB-Bases

Suresh Kumar Lodha, Ron Goldman
1998 Journal of Approximation Theory  
We describe, in particular, a new change of basis algorithm from a bivariate Lagrange L-basis to a bivariate Bernstein basis with computational complexity O(n 3 ). Academic Press  ...  It is well known that the Bernstein and multinomial (or Taylor) bases are special cases of both L-bases and B-bases.  ...  ACKNOWLEDGMENTS We would like to thank Phil Barry of the University of Minnesota for discussing some of the topics presented here, and for helping us to improve our presentation.  ... 
doi:10.1006/jath.1997.3162 fatcat:bo4prnb6hjbq3nk3l4mjielfgq

Numerical Solutions of the Nonlinear Fractional-Order Brusselator System by Bernstein Polynomials

Hasib Khan, Hossein Jafari, Rahmat Ali Khan, Haleh Tajadodi, Sarah Jane Johnston
2014 The Scientific World Journal  
We use operational matrices of fractional integration and multiplication of Bernstein polynomials, which turns the nonlinear fractional-order Brusselator system to a system of algebraic equations.  ...  In this paper we propose the Bernstein polynomials to achieve the numerical solutions of nonlinear fractional-order chaotic system known by fractional-order Brusselator system.  ...  Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper.  ... 
doi:10.1155/2014/257484 pmid:25485293 pmcid:PMC4251784 fatcat:ei53selkpfbdhobnkclwnmlotm

New Operational Matrices of Seventh Degree Orthonormal Bernstein Polynomials

Baghdad Science Journal
2015 Baghdad Science Journal  
Based on analyzing the properties of Bernstein polynomials, the extended orthonormal Bernstein polynomials, defined on the interval [0, 1] for n=7 is achieved.  ...  Also the result of the proposed method is compared with true answers to show the convergence and advantages of the new method.  ...  Theorem: [9] The first derivatives of nth degree generalized Bernstein basis polynomials can be written as a linear combination of the generalized Bernstein basis polynomials of degree n ( )  ... 
doi:10.21123/bsj.12.4.846-853 fatcat:aumki4xb25dedouj55h4ovyw3y

Lyapunov Function Synthesis - Infeasibility and Farkas' Lemma * *This work is supported by the Danish Council for Independent Research under grant number DFF - 4005-00452 in the project CodeMe

Tobias Leth, Christoffer Sloth, Rafał Wisniewski, Sriram Sankaranarayanan
2017 IFAC-PapersOnLine  
In this paper we prove the convergence of an algorithm synthesising continuous piecewise-polynomial Lyapunov functions for polynomial vector fields defined on simplices.  ...  Abstract: In this paper we prove the convergence of an algorithm synthesising continuous piecewise-polynomial Lyapunov functions for polynomial vector fields defined on simplices.  ...  Let a polynomial p of degree d be defined in the Bernstein basis on a simplexσ, and let b(p, d,σ) ≥ 0.  ... 
doi:10.1016/j.ifacol.2017.08.339 fatcat:25l34y2zmnedzizy2x3yceioku

Pyramid Algorithms for Bernstein--Bézier Finite Elements of High, Nonuniform Order in Any Dimension

Mark Ainsworth
2014 SIAM Journal on Scientific Computing  
Pyramid algorithms replace an operation on single high order polynomial by a recursive sequence of self-similar affine combinations, and are ubiquitous in CAGD for computations involving high order curves  ...  A new, non-uniform order, variant of the de Casteljau algorithm is developed that is applicable to the variable polynomial order case but incurs no additional complexity compared with the original algorithm  ...  How are the Bernstein polynomials defined on a face F related to the Bernstein polynomials defined on the original simplex T ?  ... 
doi:10.1137/130914048 fatcat:pyrgyigq6zdyhnq55j7axldjxy

Differential-recurrence properties of dual Bernstein polynomials [article]

Filip Chudy, Paweł Woźny
2018 arXiv   pre-print
New differential-recurrence properties of dual Bernstein polynomials are given which follow from relations between dual Bernstein and orthogonal Hahn and Jacobi polynomials.  ...  Using these results, a fourth-order differential equation satisfied by dual Bernstein polynomials has been constructed.  ...  Bernstein basis polynomials B n i are given by B n i (x) := n i x i (1 − x) n−i (i = 0, 1, . . . , n; n ∈ N). (2.8) One can easily check that polynomials B n 0 , B n 1 , . . . , B n n form a basis of the  ... 
arXiv:1803.01735v2 fatcat:j3mswjyicjeebceizqjgrkn4be

Page 1467 of Mathematical Reviews Vol. , Issue 91C [page]

1991 Mathematical Reviews  
The authors introduce a B-net representation of polynomials on simplices and splines over triangulations.  ...  Again the author takes the B-net (Bernstein-Bézier-net) approach in his construction of a local basis for S.  ... 

The apolar bilinear form in geometric modeling

Gert Vegter
1999 Mathematics of Computation  
These results are applied to study so-called lineal polynomial bases and their dual bases. The B-patch basis is introduced in [6] as the dual of a special lineal basis.  ...  A similar binary form on the space of univariate polynomials of a fixed degree has been studied by Goldman [12] .  ...  Acknowledgments I wish to thank Lyle Ramshaw for his interest in this paper, and the anonymous referee for suggesting a useful addition to the introductory section.  ... 
doi:10.1090/s0025-5718-99-01144-8 fatcat:j4wpj3hhwzftldljbzt6k4ht6m

Legendre–Bernstein basis transformations

Rida T. Farouki
2000 Journal of Computational and Applied Mathematics  
We derive the transformation matrices that map the Bernstein and Legendre forms of a degree-n polynomial on [0; 1] into each other, and examine the stability of this linear map.  ...  The Bernstein form of a polynomial o ers valuable insight into its geometrical behavior, and has thus won widespread acceptance as the basis for BÃ ezier curves and surfaces.  ...  One approach [18] to direct least-squares approximation by polynomials in Bernstein form relies on construction of the basis d n 0 (u); : : : ; d n n (u) that is "dual" to the Bernstein basis J uttler  ... 
doi:10.1016/s0377-0427(00)00376-9 fatcat:5n2xq6wvjnf67d5qldourigrly

Bézier form of dual bivariate Bernstein polynomials [article]

Stanisław Lewanowicz, Paweł Keller, Paweł Woźny
2016 arXiv   pre-print
Dual Bernstein polynomials of one or two variables have proved to be very useful in obtaining Bézier form of the L^2-solution of the problem of best polynomial approximation of Bézier curve or surface.  ...  In this connection, the Bézier coefficients of dual Bernstein polynomials are to be evaluated at a reasonable cost.  ...  Remark that the striking simplicity of Algorithm 2.1 was obtained thanks to using properties of dual bivariate Bernstein polynomials, investigated in Appendix A.  ... 
arXiv:1510.08246v4 fatcat:vrnj6yy2brdmlh7wfme6rjtosy

Connections between two-variable Bernstein and Jacobi polynomials on the triangle

Stanisław Lewanowicz, Paweł Woźny
2006 Journal of Computational and Applied Mathematics  
Dual two-variable Bernstein polynomials are introduced. Explicit formula in terms of two-variable Jacobi polynomials and a recurrence relation are given.  ...  Connection coefficients between the two-variable Bernstein and Jacobi polynomial families on the triangle are given explicitly as evaluations of two-variable Hahn polynomials.  ...  Appendix A.  ... 
doi:10.1016/ fatcat:ftdj5htq5bcbvei5qfvr6dhmay

Efficient merging of multiple segments of Bézier curves [article]

Paweł Woźny, Przemysław Gospodarczyk, Stanisław Lewanowicz
2015 arXiv   pre-print
We present a novel method which is based on the idea of using constrained dual Bernstein polynomial basis (P. Wo\'zny, S. Lewanowicz, Comput. Aided Geom.  ...  Thanks to using fast schemes of evaluation of certain connections involving Bernstein and dual Bernstein polynomials, the complexity of our algorithm is significantly less than complexity of other merging  ...  We present a novel method which is based on the idea of using constrained dual Bernstein polynomial basis [12] to compute the control points r i .  ... 
arXiv:1409.2671v3 fatcat:4ogl3oklmzcglhqvbeuyikqmwy

Fast simplicial quadrature-based finite element operators using Bernstein polynomials

Robert C. Kirby, Kieu Tri Thinh
2011 Numerische Mathematik  
We derive low-complexity matrix-free finite element algorithms for simplicial Bernstein polynomials on simplices.  ...  Our techniques, based on a sparse representation of differentiation and special block structure in the matrices evaluating B-form polynomials at warped Gauss points, apply to variable coefficient problems  ...  The complete set of Bernstein polynoimals, {B n α } |α|=n , form a basis of polynomials of complete degree n on S d . They form a nonnegative partition of unity on S d .  ... 
doi:10.1007/s00211-011-0431-y fatcat:uiub7xmxsnawxmioy3c2dhjacq

Bernstein dual-Petrov–Galerkin method: application to 2D time fractional diffusion equation

M. Jani, S. Javadi, E. Babolian, D. Bhatta
2017 Computational and Applied Mathematics  
A spectral discretization is applied by introducing suitable combinations of dual Bernstein polynomials as the test functions and the Bernstein polynomials as the trial ones.  ...  We derive the exact sparse operational matrix of differentiation for the dual Bernstein basis which provides a matrix based approach for the spatial discretization.  ...  So it is of interest to explore some new aspects of this basis in order to facilitate the numerical methods for differential equations that are based on Bernstein polynomials and to present a method for  ... 
doi:10.1007/s40314-017-0455-8 fatcat:6n5drszugnh4zen3rdobhwqxqu
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