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When it is preferable to use Bernstein polynomials to approximate a continuous function instead of using the only following preliminary Numerical Analysis methods: "Lagrange Polynomials", "Simple finite differences operators".

The question is about compairing these method.

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Why is BERNSTEIN capitalized? Is this in reference to a particular software package? –  Thomas Andrews Dec 5 '12 at 15:01
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One aspect of my question was nearly to your question, I want to see is there any predominance to use this method instead of those were mentioned even at very especial case? BERNSTEIN Polynomials are nice themselves and have lots of properties, but are they better to use for example in computer program or other situations too? –  AmirHosein SadeghiManesh Dec 5 '12 at 15:08
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4 Answers 4

Bernstein polynomials and Lagrange polynomials both span the same spaces. So in terms of the possible functions one can represent, using one or the other makes no difference. However, if you are thinking of using these as basis functions in either a finite element method or an interpolation problem, the spectral properties of the linear operator you create will depend on the polynomials you choose as a basis. This can cause differences in the convergence of iterative solvers. However in the absence of linear algebra error, you will get the same answer using either basis.

Comparing this to finite difference operators is a different story. Using polynomials will give you error approximations on a continuous norm. I am not as well versed in finite differences, but my understanding is that you will only get an error estimate at the locations that you choose to discretize. What happens in between these points is not as clear.

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I use Bernstein polynomials in a collocation method to solve boundary value problems for ODEs and PDEs. They are quite interesting.

Convergence was exponential for some linear BVPs, but little slower compared to Chebyshev collocation, Legendre Galerkin, and Tau.

Here's the figure comparing convergence rates with some Chebyshev spectral methods. The example problem is linear BVP:

$ \frac{d^2u}{dx^2}-4\frac{du}{dx}+4u = e^x +C \;,\; x \in [-1,1]$

with homogenous Dirichlet BCs, and C is a constant $C=-4e/(1+e)^2$.

enter image description here

I also uploaded this figure to figshare.

If you want, you may check out the code I'm writing:

http://code.google.com/p/bernstein-poly/

And here's the arxiv paper I wrote about solving elliptic BVPs on a square using Bernstein polynomial collocation.

Last year they celebrated a centennial of Bernstein polynomials - one more interesting fact.

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About the centennial, see Rida T. Farouki, The Bernstein polynomial basis: A centennial retrospective, Computer Aided Geometric Design, Volume 29, Issue 6, August 2012, Pages 379-419, DOI: 10.1016/j.cagd.2012.03.001. –  lhf Jan 23 '13 at 23:28
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Also interesting: Reliable computing - Special Issue on the Use of Bernstein Polynomials in Reliable Computing: A Centennial Anniversary interval.louisiana.edu/reliable-computing-journal/… –  Johntra Volta Jan 24 '13 at 8:05
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There is some theory for the use of Bernstein polynomials in a collocation method. When you go to multiple spans (elements) you need to use at least $C^1$ B-splines. See ISOGEOMETRIC COLLOCATION METHODS F. AURICCHIO, L. BEIRÃO DA VEIGA, T. J. R. HUGHES, A. REALI, and G. SANGALLI, Mathematical Models and Methods in Applied Sciences 2010 20:11, 2075-2107 –  Nathan Collier Jan 27 '13 at 6:24
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The paper below shows that representing polynomials in Bernstein form leads to numerically stable algorithms in many cases:

R.T. Farouki, V.T. Rajan, On the numerical condition of polynomials in Bernstein form, Computer Aided Geometric Design, Volume 4, Issue 3, November 1987, Pages 191-216, DOI: 10.1016/0167-8396(87)90012-4

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The control points of a Bézier curve are close to the curve, but not necessarily on the curve. This is exactly the same situation as for the approximation by Bernstein polynomials, and in fact the Bernstein polynomials are the basis for the Bézier curve. You could use a high order Bézier curve to draw a smooth line through a curve given by noisy points, also nobody would do this due to the high computational effort. In fact, high order polynomial interpolation is only rarely used for exactly that reason, only Chebyshev interpolation is occasionally an exception from that rule.

But if we are only talking about low order polynomial interpolation, then the intuitive specification of a Bézier curve via control points is a clear advantage over other methods. However, in this respect NURBS are even better, but at least a Bézier curve is a special case of a NURBS, and Bernstein polynomials are also an important ingredient for NURBS.

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