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Orthogonal Polynomials

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  Orthogonal Polynomials Mate <mmatica@personal.ro> - 2011-07-08 10:41 -0700
    Re: Orthogonal Polynomials Axel Vogt <&noreply@axelvogt.de> - 2011-07-09 16:35 +0200

#205 — Orthogonal Polynomials

Hello to all,

I'd like to share here a recent experience in using
Maple in a project requesting the manipulation of some
orthogonal polynomials. It was a rather frustrating one
and I mention it hoping to be useful for other users and also for
Mapelsoft's stuff (if they read this forum) in order to do
something about this.

The OthogonalSeries package is not new in Maple, so
it is supposed to be a reliable one. It is one of the few packages
I did not use until now, so I started to read the documentation,
and I was very disappointed.

I tried to find first for which orthogonal polynomial the package
works.
The help pages contain a few examples and mention an internal
database. But the list of these polynomial is absent.
The database for the accepted polynomials is not easily accessible
(it is a local procedure in the module).
(let me say that using the library browser was not possible because
the scroll bar and the search do not work well - due to our beloved
slow buggy Java with which the Maple users are blessed.)

Anyway I discovered that the Hahn polynomials are implemented.
Trying to see the exact syntax (parameters' order etc) and
searching for "Hahn", the name appears only in the dictionary
and is related to some of Hahn's theorems (e.g. Hahn-Banach) which
have nothing to do with Maple. So, the user has to guess the order
of the parameters and apply the trial & error method; or he/she may
use the GetInfo procedure and apply some "reverse engineering".
   I wonder if this package was really used; I am almost sure that
most of its capabilities are unknown.
   The syntax for the differential operators is not explained; it is
actually the same as in the LinearOperators package but no link is
provided.
Operators such as deltastruct or nablastruct (nabla-structural
operator)
are supposed to be well known.

I have also found that some orthogonal functions are not well
integrated into the system.
For example, the obvious:

int(ChebyshevT(40,x)^2*(1-x^2)^(-1/2),x=-1..1);

cannot be computed to Pi/2 (at least in a reasonable time); it works
however if the polynomial is expanded, or if its degree is e.g. 10.

Finally the package was useful, but I would be interested if other
users had similar experiences and if there is a better approach
in such cases.

Mate

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#206

On 08.07.2011 19:41, Mate wrote:
> Hello to all,
>
> I'd like to share here a recent experience in using
> Maple in a project requesting the manipulation of some
> orthogonal polynomials. It was a rather frustrating one
> and I mention it hoping to be useful for other users and also for
> Mapelsoft's stuff (if they read this forum) in order to do
> something about this.
>
> The OthogonalSeries package is not new in Maple, so

... (snipped to shorten for my reply)

> I have also found that some orthogonal functions are not well
> integrated into the system.
> For example, the obvious:
>
> int(ChebyshevT(40,x)^2*(1-x^2)^(-1/2),x=-1..1);
>
> cannot be computed to Pi/2 (at least in a reasonable time); it works
> however if the polynomial is expanded, or if its degree is e.g. 10.
>
> Finally the package was useful, but I would be interested if other
> users had similar experiences and if there is a better approach
> in such cases.
>
> Mate

I only used it occasionally and then only for computing
a base change.

For the above integral I could imagine, that Maple uses some
general representation for ChebyshevT (hypergeom 2F1 ?) and
can not find a solution, even for the specific situation (but
after 'expand' becomes aware of it).

For that I used Cheb:= (n,x) -> cos(n*arccos(x)) and then
   Int(ChebyshevT(40,x)^2*(1-x^2)^(-1/2),x=-1..1);
   subs(ChebyshevT = Cheb, %);
   value(%);
is quite fast.

However for orthogonality Cheb(m,x), Cheb(n,x) some care is
needed, M15 seems to ignore the special case m=n.

There is a package by Sergey Moiseev at Maple's Application
Center, which may be of interest for you.

Axel

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