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Bernstein–Bezoutian matrices and curve implicitization

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. ...

##
###
On de Casteljau-type algorithms for rational Bézier curves

2015
*
Journal of Computational and Applied Mathematics
*

different de Casteljau-type

doi:10.1016/j.cam.2015.01.037
fatcat:vua5llxe7zgx7okdlqxk76srsu
*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. ...##
###
Low-Complexity Finite Element Algorithms for the de Rham Complex on Simplices

2014
*
SIAM Journal on Scientific Computing
*

We combine recently-developed finite element

doi:10.1137/130927693
fatcat:w4ob6vrppraajkii36pgxk5gsa
*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*. ...##
###
A fast and accurate algorithm for solving Bernstein-Vandermonde linear sytem
[article]

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. ...

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

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. ...

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

2015
*
arXiv
*
pre-print

We also show

arXiv:1504.03990v1
fatcat:ylnrlv67tnbzbbvef6wrxtqzkm
*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. ...##
###
Bernstein–Bézier Finite Elements of Arbitrary Order and Optimal Assembly Procedures

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. ...

##
###
A unified approach to evaluation algorithms for multivariate polynomials

1997
*
Mathematics of Computation
*

We present

doi:10.1090/s0025-5718-97-00862-4
fatcat:37audtlqq5a4zkbodg6lmeyg64
*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. ...##
###
An iterative approximate method of solving boundary value problems using dual Bernstein polynomials
[article]

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. ...

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

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*). ...

##
###
Accurate computations with totally positive Bernstein-Vandermonde matrices

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. ...

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

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. ...

##
###
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
*

We describe, in particular,

doi:10.1006/jath.1997.3162
fatcat:bo4prnb6hjbq3nk3l4mjielfgq
*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. ...##
###
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

2012
*
Computer Aided Geometric Design
*

This survey provides

doi:10.1016/j.cagd.2012.03.001
fatcat:eiqucogpb5gh3lh5gnoj7p32cq
*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. ...
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