[SciPy-User] orthonormal polynomials (cont.)

[Charles R Harris]

On Tue, Jun 14, 2011 at 12:10 PM, nicky van foreest <vanforeest at gmail.com>wrote:

> Hi,
>
> Without understanding the details... I recall from numerical recipes
> in C that Gram Schmidt is a very risky recipe. I don't know whether
> this advice also pertains to fitting polynomials, however,
>
> Nicky
>
> On 14 June 2011 18:58,  <josef.pktd at gmail.com> wrote:
> > (I'm continuing the story with orthogonal polynomial density
> > estimation, and found a nice new paper
> > http://www.informaworld.com/smpp/content~db=all~content=a933669464 )
> >
> > Last time I managed to get orthonormal polynomials out of scipy with
> > weight 1, and it worked well for density estimation.
> >
> > Now, I would like to construct my own orthonormal polynomials for
> > arbitrary weights. (The weights represent a base density around which
> > we make the polynomial expansion).
> >
> > The reference refers to Gram-Schmidt or Emmerson recurrence.
> >
> > Is there a reasonably easy way to get the polynomial coefficients for
> > this with numscipython?
> >
>

What do you mean by 'polynomial'? If you want the values of a set of
polynomials orthonormal on a given set of points, you want the 'q' in a qr
factorization of a (row) weighted Vandermonde matrix.  However, I would
suggest using a weighted chebvander instead for numerical stability.

You can also solve for the three term recursion (Emerson?), but that is more
work.

Chuck
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