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Bernstein–Bezoutian matrices and curve implicitization

Ana Marco, José-Javier Martínez
2007 Theoretical Computer Science  
A new application of Bernstein-Bezoutian matrices, a type of resultant matrices constructed when the polynomials are given in the Bernstein basis, is presented.  ...  appearing in the rational parametric equations of the curve are expressed in the Bernstein basis, avoiding the basis conversion from the Bernstein to the power basis.  ...  Acknowledgements This research has been partially supported by Spanish Research Grant MTM2006-03388 from the Spanish Ministerio de Educación y Ciencia.  ... 
doi:10.1016/j.tcs.2007.02.008 fatcat:vszjphdksjcbvazgch3gpj7qaq

On de Casteljau-type algorithms for rational Bézier curves

Zbyněk Šír, Bert Jüttler
2015 Journal of Computational and Applied Mathematics  
different de Casteljau-type algorithms, depending on the ordering of the elementary factors of the polynomial.  ...  We consider the space of rational functions of degree n with a common denominator.  ...  Figure 5 : 5 Three different de Casteljau-type algorithms for value t = 2/3 for a rational curve with two complex conjugate linear factors in the denominator.  ... 
doi:10.1016/j.cam.2015.01.037 fatcat:vua5llxe7zgx7okdlqxk76srsu

Low-Complexity Finite Element Algorithms for the de Rham Complex on Simplices

Robert C. Kirby
2014 SIAM Journal on Scientific Computing  
We combine recently-developed finite element algorithms based on Bernstein polynomials [1, 14] with the explicit basis construction of the finite element exterior calculus [5] to give a family of algorithms  ...  In particular, we take the explicit basis construction [5] provided by Arnold, Falk, and Winther for the de Rham complex [4] .  ...  If the linear combinations are short, then the same order of complexity holds for these algorithms.  ... 
doi:10.1137/130927693 fatcat:w4ob6vrppraajkii36pgxk5gsa

A fast and accurate algorithm for solving Bernstein-Vandermonde linear sytem [article]

A. Marco, J. J. Martinez
2006 arXiv   pre-print
A fast and accurate algorithm for solving a Bernstein-Vandermonde linear system is presented.  ...  The use of explicit expressions for the determinants involved in the process serves to make the algorithm both fast and accurate.  ...  We are grateful to the referees for their suggestion of extending the numerical experiments.  ... 
arXiv:math/0605577v2 fatcat:xb7b3rkj6nhovnjdja4tepmso4

A fast and accurate algorithm for solving Bernstein–Vandermonde linear systems

Ana Marco, José-Javier Martı´nez
2007 Linear Algebra and its Applications  
A fast and accurate algorithm for solving a Bernstein-Vandermonde linear system is presented.  ...  The use of explicit expressions for the determinants involved in the process serves to make the algorithm both fast and accurate.  ...  We are grateful to the referees for their suggestion of extending the numerical experiments.  ... 
doi:10.1016/j.laa.2006.11.020 fatcat:jhkmqnfunrbzznbcx22aaws27m

Efficient discontinuous Galerkin finite element methods via Bernstein polynomials [article]

Robert C. Kirby
2015 arXiv   pre-print
We also show a fast algorithm for solving linear systems involving the element mass matrix to preserve the overall complexity of the DG method.  ...  We consider the discontinuous Galerkin method for hyperbolic conservation laws, with some particular attention to the linear acoustic equation, using Bernstein polynomials as local bases.  ...  Bernstein polynomials admit optimal-complexity algorithms for discontinuous Galerkin methods for conservation laws.  ... 
arXiv:1504.03990v1 fatcat:ylnrlv67tnbzbbvef6wrxtqzkm

Bernstein–Bézier Finite Elements of Arbitrary Order and Optimal Assembly Procedures

Mark Ainsworth, Gaelle Andriamaro, Oleg Davydov
2011 SIAM Journal on Scientific Computing  
The elements are based on Bernstein polynomials and are the first to achieve optimal complexity for the standard finite element spaces on simplicial elements. 1991 Mathematics Subject Classification. 65N30  ...  The algorithms (i) take account of numerical quadrature; (ii) are applicable to non-linear problems; and, (iii) do not rely on pre-computed arrays containing values of one-dimensional basis functions at  ...  Thus, the algorithms attain optimal complexity for the assembly of the element matrix.  ... 
doi:10.1137/11082539x fatcat:65k3khmm75d3nm7urzaj7gr3me

A unified approach to evaluation algorithms for multivariate polynomials

Suresh K. Lodha, Ron Goldman
1997 Mathematics of Computation  
We present a unified framework for most of the known and a few new evaluation algorithms for multivariate polynomials expressed in a wide variety of bases including the Bernstein-Bézier, multinomial (or  ...  The second class of algorithms, based on certain change of basis algorithms between L-bases, include a nested multiplication algorithm with computational complexity O(n s ), a divided difference algorithm  ...  Acknowledgments We wish to thank the anonymous referee whose comments helped us to improve the presentation of this work.  ... 
doi:10.1090/s0025-5718-97-00862-4 fatcat:37audtlqq5a4zkbodg6lmeyg64

An iterative approximate method of solving boundary value problems using dual Bernstein polynomials [article]

Przemysław Gospodarczyk, Paweł Woźny
2017 arXiv   pre-print
The idea is to compute approximate polynomial solutions in the Bernstein form using least squares approximation combined with some properties of dual Bernstein polynomials which guarantee high efficiency  ...  The method can deal with both linear and nonlinear differential equations. Moreover, not only second order differential equations can be solved but also higher order differential equations.  ...  Notice that our idea could be generalized to work for any pair of dual bases, not only Bernstein and dual Bernstein bases. However, the algorithm would be much more complicated.  ... 
arXiv:1709.02162v1 fatcat:vvhmrctrpjauxfsoofllxzo3aa

A new bivariate basis representation for Bézier-based triangular patches with quadratic complexity

Natasha Dejdumrong
2011 Computers and Mathematics with Applications  
A new class of bivariate bases for the triangular surface construction, based on quadratic and cubic bivariate Bernstein polynomials, is proposed, by extending a model for the univariate basis with linear  ...  In addition, a recursive algorithm for calculating a point on this triangular surface is recursively defined in the same manner as in the well known de Casteljau algorithm.  ...  The reason that we chose this univariate polynomial basis is that it possesses an efficient recursive algorithm with (linear complexity).  ... 
doi:10.1016/j.camwa.2010.09.051 fatcat:doe3ji3omnduvjcqrghqgsyveq

Accurate computations with totally positive Bernstein-Vandermonde matrices

Ana Marco, Jose Javier Martinez
2013 The Electronic Journal of Linear Algebra  
Bernstein-Vandermonde matrices are a generalization of Vandermonde matrices arising when considering the Bernstein basis instead of the monomial basis for the space of the algebraic polynomials of degree  ...  The accurate solution of some of the main problems in numerical linear algebra (linear system solving, eigenvalue computation, singular value computation and the least squares problem) for a totally positive  ...  The authors are grateful to one anonymous referee and to the associate editor for suggesting useful ideas to extend and improve the initial version of the paper.  ... 
doi:10.13001/1081-3810.1658 fatcat:kyi5cylwkjfh5njzne67vg53ku

Accurate numerical linear algebra with Bernstein-Vandermonde matrices [article]

Ana Marco, Jose-Javier Martinez
2008 arXiv   pre-print
Bernstein-Vandermonde matrices are a generalization of Vandermonde matrices arising when considering for the space of the algebraic polynomials of degree less than or equal to n the Bernstein basis, a  ...  The accurate solution of some of the main problems in numerical linear algebra (linear system solving, eigenvalue computation, singular value computation and the least squares problem) for a totally positive  ...  This research has been partially supported by Spanish Research Grant MTM2006-03388 from the Spanish Ministerio de Educación y Ciencia.  ... 
arXiv:0812.3115v1 fatcat:l2zmrnrdxravzjnbvi76pmlx4e

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  ...  Duality can also be used to develop change of basis algorithms with computational complexity O(n 3 ) between any two L-bases andÂor 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

Page 2786 of Mathematical Reviews Vol. , Issue 93e [page]

1993 Mathematical Reviews  
Under certain mild restrictions on these linear functions, the polynomials Cj(t) form a basis for the space of polynomials of degree n.  ...  It should be noted that these basis functions are in fact complex- valued partitions of unity if we view R? as the complex plane.  ... 

The Bernstein polynomial basis: A centennial retrospective

Rida T. Farouki
2012 Computer Aided Geometric Design  
This survey provides a brief historical perspective on the evolution of the Bernstein polynomial basis, and a synopsis of the current state of associated algorithms and applications.  ...  One hundred years after the introduction of the Bernstein polynomial basis, we survey the historical development and current state of theory, algorithms, and applications associated with this remarkable  ...  Acknowledgements The idea for this survey originated among discussions between Ron Goldman, Hartmut Prautzsch, and others at a meeting in Schloss Dagstuhl in May 2011.  ... 
doi:10.1016/j.cagd.2012.03.001 fatcat:eiqucogpb5gh3lh5gnoj7p32cq
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