[SciPy-User] orthonormal polynomials (cont.)

[Charles R Harris]

On Tue, Jun 14, 2011 at 3:01 PM, <josef.pktd at gmail.com> wrote:

> On Tue, Jun 14, 2011 at 4:40 PM, Charles R Harris
> <charlesr.harris at gmail.com> wrote:
> >
> >
> > 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,
>
> I read some warnings about the stability of Gram Schmidt, but the idea
> is that if I can choose the right weight function, then we should need
> only a few polynomials. So, I guess in that case numerical stability
> wouldn't be very relevant.
>
> >>
> >> 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.
>
> Following your suggestion last time to use QR, I had figured out how
> to get the orthonormal basis for a given set of points.
> Now, I would like to get the functional version (not just for a given
> set of points), that is an orthonormal polynomial basis like Hermite,
> Legendre, Laguerre and Jacobi, only for any kind of weight function,
> where the weight function is chosen depending on the data.
>
>
But in what basis? The columns of the inverse of 'R' in QR will give you the
orthonormal polynomials as series in whatever basis you used for the columns
of the pseudo-Vandermonde matrix.

Chuck
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