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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.  ...  In the p = 1 and ∞ norms, the condition number of the Legendre-Bernstein transformation matrix grows at a signiÿcantly slower rate with n than in the well-studied power-Bernstein case, and at a dramatically  ...  Acknowledgements This work was supported in part by the National Science Foundation under grant CCR-9902669.  ... 
doi:10.1016/s0377-0427(00)00376-9 fatcat:5n2xq6wvjnf67d5qldourigrly

Algebraic manipulation in the Bernstein form made simple via convolutions

J. Sánchez-Reyes
2003 Computer-Aided Design  
Examples of applications in geometry processing are provided, such as conversions between the triangular and tensor-product Bézier forms. q  ...  Traditional methods for algebraic manipulation of polynomials in Bernstein form try to obtain an explicit formula for each coefficient of the result of a given procedure, such us multiplication, arbitrarily  ...  Acknowledgements This work is supported by the Spanish Ministerio de Ciencia y Tecnología, under research grant DPI2000-0676.  ... 
doi:10.1016/s0010-4485(03)00021-6 fatcat:mavvmqdux5cyfgn4srzitstdhm

A companion matrix resultant for Bernstein polynomials

Joab R. Winkler
2003 Linear Algebra and its Applications  
It is shown that M differs from its equivalent form for a power basis polynomial because an upper triangular Hankel matrix does not define a similarity transformation between M and M T .  ...  A closed form expression for a companion matrix M of a Bernstein polynomial is obtained, and this is used to derive an expression for a resultant matrix of two Bernstein polynomials.  ...  substitution (2) is used to transform the polynomials r(x) and s(x) to a power basis form, and then using the resultant matrix that is based on a companion matrix of a polynomial in this basis.  ... 
doi:10.1016/s0024-3795(02)00486-x fatcat:aqjyw5zxpzab7n6ntkkjto4ami

On the optimal stability of the Bernstein basis

R. T. Farouki, T. N. T. Goodman
1996 Mathematics of Computation  
These ideas are illustrated by comparing the stability properties of the power, Bernstein, and generalized Ball bases.  ...  of arbitrary polynomials on that interval.  ...  The enhanced stability of the Bernstein form, as compared to the power form, can be attributed to two simple facts: (i) the power and Bernstein bases are both nonnegative on [ 0, 1 ]; and (ii) the latter  ... 
doi:10.1090/s0025-5718-96-00759-4 fatcat:767j4x3djbgo5gpjopexmy3c6e

Application of Chebyshev II–Bernstein basis transformations to degree reduction of Bézier curves

Lizheng Lu, Guozhao Wang
2008 Journal of Computational and Applied Mathematics  
A polynomial curve on [0, 1] can be expressed in terms of Bernstein polynomials and Chebyshev polynomials of the second kind.  ...  We derive the transformation matrices that map the Bernstein and Chebyshev coefficients into each other, and examine the stability of this linear map.  ...  This work is supported by the Natural Science Foundation of China (No. 60773179) and Foundation of State Key Basic Research 973 Development Programming Item of China (No. G2004CB318000).  ... 
doi:10.1016/j.cam.2007.10.032 fatcat:galkoegkrzeybbbrc3lhhp4baa

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.  ...  With the desire to exploit the power of computers for geometric design applications, however, the Bernstein form began to enjoy widespread use as a versatile means of intuitively constructing and manipulating  ...  The paper has been greatly improved by the efforts of the anonymous referees, through their diligent reading of and perceptive comments on an initial draft.  ... 
doi:10.1016/j.cagd.2012.03.001 fatcat:eiqucogpb5gh3lh5gnoj7p32cq

Generalized Shifted Chebyshev Koornwinder's Type Polynomials: Basis Transformations

Mohammad AlQudah, Maalee AlMheidat
2018 Symmetry  
On the other hand, polynomials can be characterized in several ways using different bases, where every form of basis has its advantages and power.  ...  In addition, explicit formulas of conversion matrices between generalized shifted Chebyshev Koornwinder's type polynomials and Bernstein polynomial bases were given.  ...  Acknowledgments: The authors would like to thank the anonymous referees for their comments that helped us to improve this article. Conflicts of Interest: The authors declare no conflict of interest.  ... 
doi:10.3390/sym10120692 fatcat:bszgu6e3gbeofpbwrs74xkvb5a

Robust normalizing flows using Bernstein-type polynomials [article]

Sameera Ramasinghe, Kasun Fernando, Salman Khan, Nick Barnes
2021 arXiv   pre-print
To this end, we utilize Bernstein-type polynomials inspired by the optimal stability of the Bernstein basis.  ...  We conduct a thorough theoretical analysis and empirically demonstrate the efficacy of the proposed technique using experiments on both real-world and synthetic datasets.  ...  On [a, b] the Bernstein polynomials and the power monomials, {1, x, x 2 , . . . , x n }, are non-negative bases.  ... 
arXiv:2102.03509v3 fatcat:izxciik32fhtxnlnmtcbjcuhcu

Linear relaxations of polynomial positivity for polynomial Lyapunov function synthesis

Mohamed Amin Ben Sassi, Sriram Sankaranarayanan, Xin Chen, Erika Ábrahám
2015 IMA Journal of Mathematical Control and Information  
Next, we present a series of increasingly powerful LP relaxations based on expressing the given polynomial in its Bernstein form, as a linear combination of Bernstein polynomials.  ...  We first compare two classes of relaxations for encoding polynomial positivity: relaxations by sum-of-squares (SOS) programs, against relaxations based on Handelman representations and Bernstein polynomials  ...  Let D represent the matrix form of the Lie derivatives on the monomial basis m, T represent the transformation of the monomials from R x to [0, 1] n , and finally B represent the transformation to Bernstein  ... 
doi:10.1093/imamci/dnv003 fatcat:p65om6rrbnewpg5pexcu4nm63i

Transformation of Chebyshev–Bernstein Polynomial Basis

Abedallah Rababah
2003 Computational Methods in Applied Mathematics  
In paper [4] , transformation matrices mapping the Legendre and Bernstein forms of a polynomial of degree n into each other are derived and examined.  ...  Chebyshev polynomials with the geometrical insight of the Bernstein form.  ...  Acknowledgement: The author wishes to thank the referee for useful comments.  ... 
doi:10.2478/cmam-2003-0038 fatcat:es5yniykqbeznmremzzku6vbxy

Wavelet-based Multiresolution Isosurface Rendering [article]

Markus Steinberger, Markus Grabner
2010 International Workshop on Volume Graphics, Proceedings of the  
Both wavelet reconstruction and root finding are implemented in CUDA to utilize the high computational performance of Nvidia's hardware and to obtain high quality results.  ...  The underlying trivariate function is represented as a spline wavelet hierarchy, which allows for adaptive (view-dependent) selection of the desired level-of-detail by superimposing appropriately weighted  ...  Acknowledgements We would like to thank Johan Seland (SINTEF ICT, Norway) for providing the implementation of their method [RS08] .  ... 
doi:10.2312/vg/vg10/013-020 fatcat:hlyh6zlxhjaf5ejbljqrx5tmyi

Linear Relaxations of Polynomial Positivity for Polynomial Lyapunov Function Synthesis [article]

Mohamed Amin Ben Sassi and Sriram Sankaranarayanan and Xin Chen and Erika Abraham
2014 arXiv   pre-print
Next, we present a progression of increasingly more powerful LP relaxations based on expressing the given polynomial in its Bernstein form, as a linear combination of Bernstein polynomials.  ...  The well-known bounds on Bernstein polynomials over the unit box help us formulate increasingly precise LP relaxations that help us establish the positive definiteness of a polynomial over a bounded domain  ...  Let D represent the matrix form of the Lie derivatives on the monomial basis m, T represent the transformation of the monomials from R x to [0, 1] n , and finally B represent the transformation to Bernstein  ... 
arXiv:1407.2952v2 fatcat:43rnag74lrb6zovft6uai4buym

On Stability of Parametrized Families of Polynomials and Matrices

Handan Akyar, Taner Büyükköroğlu, Vakıf Dzhafarov
2010 Abstract and Applied Analysis  
Based on the Bernstein expansion of a multivariable polynomial and extremal properties of a multilinear function, fast algorithms are suggested.  ...  The Schur and Hurwitz stability problems for a parametric polynomial family as well as the Schur stability problem for a compact set of real matrix family are considered.  ...  Acknowledgment The authors would like to thank the reviewers for their helpful suggestions to the improvement of this paper.  ... 
doi:10.1155/2010/687951 fatcat:nkf3chjyqfgh7lzoualuwcs4sa

Fixed Order Controller for Schur Stability

Taner Büyükköroğlu
2016 Mathematical and Computational Applications  
For this purpose, the distance function between the Schur stability region and the affine controller subset is investigated.  ...  If the characteristic polynomial of a discrete-time system has all its roots in the open unit disc of the complex plane, the system is called Schur stable.  ...  Acknowledgments: The author thanks the referees for many valuable comments. Conflicts of Interest: The author declares no conflict of interest.  ... 
doi:10.3390/mca21020025 fatcat:j7g5cefsfrh2nbh3i3s2kfemfy

Page 6783 of Mathematical Reviews Vol. , Issue 91M [page]

1991 Mathematical Reviews  
{For the entire collection see MR 91¢:65003.} 91m:65042 65D17 68U07 Farouki, R. T. (1-IBM) On the stability of transformations between power and Bernstein polynomial forms. Comput. Aided Geom.  ...  One of the characteristics of this criterion is that the functions P;(u) and P,(u) (Q;(v) and Q>(v)) are polynomials of one variable.  ... 
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