Handy Polynomial Fitting with Bernstein Polynomials
11-11-2018, 05:12 AM
Post: #3
 Thomas Klemm Senior Member Posts: 1,688 Joined: Dec 2013
RE: Handy Polynomial Fitting with Bernstein Polynomials
(11-11-2018 03:10 AM)Valentin Albillo Wrote:  What values does produce your Bernstein fit for, say, order 2 ?
What are the coefficients of the Bernstein Polynomial fit ?

The Bernstein Polynomial reproduces the function for $$f(x)=1$$ and $$f(x)=x$$:

$$\begin{matrix} B_n(1;x)=1 \\ B_n(x;x)=x \end{matrix}$$

However for $$f(x)=x^2$$ we get:

$$B_n(x^2;x) = x^2 + \frac{1}{n}x(1−x)$$

Thus for $$n=10$$ it's off by $$\frac{1}{40}$$ at $$x=\frac{1}{2}$$.

While it can be shown that $$\lim _{n\to \infty }{B_{n}(f)}=f$$ uniformly on the interval [0, 1] the convergence is rather slow.

Cheers
Thomas
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 Messages In This Thread Handy Polynomial Fitting with Bernstein Polynomials - Namir - 11-10-2018, 10:22 PM RE: Handy Polynomial Fitting with Bernstein Polynomials - Valentin Albillo - 11-11-2018, 03:10 AM RE: Handy Polynomial Fitting with Bernstein Polynomials - Namir - 11-11-2018, 06:00 AM RE: Handy Polynomial Fitting with Bernstein Polynomials - Thomas Klemm - 11-11-2018 05:12 AM RE: Handy Polynomial Fitting with Bernstein Polynomials - Thomas Klemm - 11-11-2018, 01:39 PM RE: Handy Polynomial Fitting with Bernstein Polynomials - Namir - 11-11-2018, 01:51 PM RE: Handy Polynomial Fitting with Bernstein Polynomials - Thomas Klemm - 11-12-2018, 05:39 AM RE: Handy Polynomial Fitting with Bernstein Polynomials - Thomas Okken - 11-12-2018, 01:21 PM RE: Handy Polynomial Fitting with Bernstein Polynomials - Thomas Klemm - 11-12-2018, 02:09 PM