#9706: Symbolic Chebyshev polynomials
-------------------------------------+-------------------------------------
Reporter: maldun | Owner: burcin, maldun
Type: enhancement | Status: needs_review
Priority: major | Milestone: sage-5.13
Component: symbolics | Resolution:
Keywords: orthogonal | Merged in:
polynomials, symbolics | Reviewers: Burcin Erocal, Travis
Authors: Stefan Reiterer, | Scrimshaw, Stefan Reiterer, Jeroen
Travis Scrimshaw | Demeyer
Report Upstream: N/A | Work issues:
Branch: | Commit:
Dependencies: #864, #9640, | Stopgaps:
#10018, #11868, #15422 |
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Comment (by maldun):
Replying to [comment:111 jdemeyer]:
> Replying to [comment:110 maldun]:
> > Is it really a good idea to replace *args and **kwds in the
OrthogonalPolynomial class?
> >
> > Since not all ortho polys only have 2 arguments, e.g. Gegenbauer
polynomials [http://en.wikipedia.org/wiki/Gegenbauer_polynomials] which
have an additional parameter alpha
> > It makes perfect sense for the chebyshev polys though
> Ok, it's always difficult to design an interface for something you don't
have an implementation for, but I made the change such that the Gegenbauer
polynomials should (in theory) work.
Maybe you should have a short look ino this patch:
[http://trac.sagemath.org/attachment/ticket/9706/trac_9706_orthogonal_polys.patch]
it contains already prototypes of the most ortho polys.
I have some remarks concerning your patch:
{{{
sage: chebyshev_T(n,Mod(0,8))
1/2*(-1)^(1/2*n)*((-1)^n + 1)
}}}
but this makes no sense since 1/2 is not defined in Mod(8). This was the
reason for the 0 in CC check at this point.
You evaluate numerical expressions for n in NN with recursion. this is
favorable for chebyshev polynomials, but not for all ortho polys you can
evaluate numeric values in O(log n). You have already problems with the
legendre polynomials, since the coefficients depend on n, and the
recursion is not stable. Thus other evaluation methods should be used.
Thats the reason why _evalf_ with mpmath should come first.
In case of chebyshev I catched this with call of the recursion.
The _old_maxima_ method is used for some oddballs, where the only useful
implementation is in maxima, and for some special cases. So removing is
probably not a good idea.
The reason why negative values are checked in special values, is that in
later versions, or for other polys non integral negative values can be
treated analogously, as in the case of their algebraic counterpart. E.g
negative legendre polynomials would return associated legendre functions,
but have no algebraic sense.
Another reason to allow the special values check for non symbolic input,
is that e.g. (-1)^100000000 is evaluated faster, than applying the
recursion, or other special points.
I use this feature sometimes to evaluate certain polynomials at the end
points of an intervall e.g. [-1,1]
--
Ticket URL: <http://trac.sagemath.org/ticket/9706#comment:112>
Sage <http://www.sagemath.org>
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