[Numpy-discussion] Generating special polynomials (Chebyshev, Hermite etc.)
Kumar Appaiah
a.kumar@alumni.iitm.ac...
Fri Jun 14 23:29:11 CDT 2013
On Fri, Jun 14, 2013 at 08:07:57PM -0600, Charles R Harris wrote:
> I am trying to find out a way by which I can easily generate the n-th
> order "special" polynomial, where "special" could refer to Hermite,
> Chebyshev etc. Numpy 1.7 introduces several methods for such
> polynomials, but I couldn't find a convenience function that gives me
> a polynomial directly based on degree. For instance, I'd like:
>
> hermite(3) to result in array([ �0., -12., � 0., � 8.])
> hermite(6) to result in array([-120., � �0., �720., � �0., -480., � �0.,
> � 64.])
> and so on.
>
> Generally that is a bad idea, polynomials tend to be numerically unstable
> and you lose all the virtue of the Hermite basis. However, you can do
This may be true, but I am not concerned with the numerical
instability in my current application. Nevertheless, I understand it
if you don't want to introduce such a function.
> In [1]: from numpy.polynomial import Polynomial, Hermite
>
> In [2]: p = Hermite.basis(5)
>
> In [3]: p.convert(kind=Polynomial)
> Out[3]: Polynomial([�� 0.,� 120.,��� 0., -160.,��� 0.,�� 32.], [-1.,� 1.],
> [-1.,� 1.])
>
> In [4]: Polynomial.cast(p)
> Out[4]: Polynomial([�� 0.,� 120.,��� 0., -160.,��� 0.,�� 32.], [-1.,� 1.],
> [-1.,� 1.])
>
> In [5]: from numpy.polynomial import Chebyshev
>
> In [6]: Chebyshev.cast(p)
> Out[6]: Chebyshev([� 0.,� 20.,�� 0., -30.,�� 0.,�� 2.], [-1.,� 1.], [-1.,�
> 1.])
>
> Hmm, it should be possible to make the constructor take polynomials of
> different kinds since they all derive from PolyBase and can be detected.
> That could replace the cast method in a nice way.
I now see that the polynomial structure is intended to be "rich", as
opposed to the naïve function that I proposed. In the least, though,
the documentation could reflect the example you gave me. I could send
a patch that adds an example for each of the polynomial types in the
documentation, much like yours, if that would be useful.
Thanks for the clarification.
Kumar
--
Kumar Appaiah
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