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Bernstein–Bezoutian matrices and curve implicitization
2007
Theoretical Computer Science
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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
2015
Journal of Computational and Applied Mathematics
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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
2014
SIAM Journal on Scientific Computing
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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]
2006
arXiv
pre-print
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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
2007
Linear Algebra and its Applications
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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]
2015
arXiv
pre-print
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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
2011
SIAM Journal on Scientific Computing
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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
1997
Mathematics of Computation
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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]
2017
arXiv
pre-print
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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
2011
Computers and Mathematics with Applications
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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
2013
The Electronic Journal of Linear Algebra
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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]
2008
arXiv
pre-print
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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
1998
Journal of Approximation Theory
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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
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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
2012
Computer Aided Geometric Design
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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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