Re: Bernstein polynomails
- To: mathgroup at smc.vnet.net
- Subject: [mg45131] Re: Bernstein polynomails
- From: Paul Abbott <paul at physics.uwa.edu.au>
- Date: Wed, 17 Dec 2003 07:54:45 -0500 (EST)
- Organization: The University of Western Australia
- References: <brk4u6$sr1$1@smc.vnet.net>
- Sender: owner-wri-mathgroup at wolfram.com
In article <brk4u6$sr1$1 at smc.vnet.net>,
"fiddlehead" <_mietek@//no-spam//wp.pl> wrote:
> I need solution or same help:
>
> en = max|f(x) -Bn(x)|
> [0,1]
>
> Bn(x)= \sum_j=0^n [((n!)/(k!)(n-k)!) f(j\n) x^j (1-x)^n-j]
This should read, in Mathematica notation,
BernsteinExpansion[n_,f_]:=
Sum[Binomial[n, j] x^j (1 - x)^(n - j) f[j/n], {j, 0, n}]
> f(x)=|x|
Since x is in [0,1], f(x) = x.
You can compute the Bernstein polynomial expansion for order n as
follows:
f[x_] = x;
FullSimplify[BernsteinExpansion[n, f], -1 <= x <= 1]
and, not suprisingly, the result is always just x. Hence the maximum
error for any n > 0 is 0.
Also see http://mathworld.wolfram.com/BernsteinPolynomial.html
[There are some typos on this page. The envelope f(x) in equation (15)
incorrectly refers to the variable t and includes n. The following
definition is not, IMHO, of the Bernstein polynomials but of the
Bernstein polynomial expansion for order n. It contains f which is _not_
the same f used in equation (15).]
Cheers,
Paul
--
Paul Abbott Phone: +61 8 9380 2734
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