[SciPy-dev] Difference between polynomial.trimcoef and trimseq
David Goldsmith
d.l.goldsmith@gmail....
Sun Jan 24 16:54:05 CST 2010
On Sun, Jan 24, 2010 at 2:50 PM, Charles R Harris <charlesr.harris@gmail.com
> wrote:
>
> On Sun, Jan 24, 2010 at 3:04 PM, David Goldsmith <d.l.goldsmith@gmail.com>wrote:
>
>> On Sun, Jan 24, 2010 at 1:10 PM, Charles R Harris <
>> charlesr.harris@gmail.com> wrote:
>>
>>>
>>> On Sun, Jan 24, 2010 at 2:09 AM, David Goldsmith <
>>> d.l.goldsmith@gmail.com> wrote:
>>>
>>>> from
>>>> Charles R Harris <charlesr.harris@gmail.com>
>>>>
>>>> On Sat, Jan 23, 2010 at 11:08 PM, David Goldsmith <
>>>> d.l.goldsmith@gmail.com> wrote:
>>>>
>>>>> Do you think a typical user would ever use both? (Or is this an
>>>>> efficiency that most can live w/out? I'm just curious how much we should
>>>>> "explain ourselves" in their docstrings.)
>>>>>
>>>>> > Hard to say ;) I wrote the docstrings for the helper funtions
>>>>
>>>> And in this case the "helper function" is the trimseq, correct?
>>>>
>>>> Yes, and pretty much the rest of the functions in polyutils, but trimseq
>>> is sort of lower level than the others.
>>>
>>
>> OK, thanks for the clarification.
>>
>>
>>> > mostly for my own use and think of those helper functions as private.
>>>> They are in the standard import just in case anyone wants
>>>> > to do their own stuff.
>>>>
>>>>
>>>>> PS: If I were to use chebyshev as my "template," what would you say is
>>>>> the next most useful/algorithmically-studied polynomial basis to implement?
>>>>
>>>> > The power/Chebyshev series have the special property that it is easy
>>>> to multiply/divide them, so the template needs to lose a
>>>> > few features to be useful for functions where that is far more
>>>> difficult.
>>>>
>>>> Yeah, that's what I meant by "algorithmically-studied": AFAYK,
>>>> numericists haven't derived/discovered nearly as efficient "tricks" for
>>>> operating on the other orthos/classes as they have for the standard and
>>>> Chebyshev bases? BTW: on the subject of "numerical tricks," are there such
>>>> for trigonometric polynomials?
>>>
>>>
>>> Trigonometric polynomials could pretty much follow the Chebyshev pattern,
>>> they are essentially the z-series. The trick is to decide how to represent
>>> the coefficients. The complex exponential form is easy to work with but not
>>> so easy to enter as data, the sin/cos version is easier in that respect but
>>> effectively requires two sets of coefficients.
>>>
>>
>> Sounds like the ideal sit. would be to implement both w/ go between
>> functions.
>>
>>
>
> Yeah, that could be done. The template approach is a bit of a stunt for
> just the two polynomial types, but maybe it will justify itself if the
> trigonometric polynomials are added. Hmm....
>
>
>> The main virtue of such a trigonometric series relative to using an fft is
>>> that the sample/interpolation points can be more general.
>>>
>>
>> That, and pedagogical purposes (trigonometric poly's are still taught in
>> various contexts, aren't they? Plus instructors might like to have both
>> implemented to illustrate the relationships and relative advantages. You
>> can see where I'm coming from: part of what I consider to be my charge is to
>> assure some suitability of NumPy/SciPy as an instructional tool, not just a
>> research/professional tool.)
>>
>>
>>> The drawback is that the fft is much faster for large degree.
>>>
>>
>> Of course, thus the name. ;-)
>>
>>
>>> Polynomials by value would be a valuable addition. But I'm thinking the
>>> framework should specific to that problem and not try to be more general.
>>> It's a tradeoff between simplicity and generality and I incline towards
>>> simplicity here along with numerical speed.
>>>
>>
>> Is this a case where that relationship doesn't lend itself to
>> class/subclass? (E.g., because the implementation details are vastly
>> different to achieve the speed gains of the specialized?)
>>
>>
> Looking back on the discussion, I think we were shooting for too much
> generality in a single implementation. An more productive approach might be
> to treat each area -- graded polynomials, lagrange, Bernstein, and B-splines
> -- in a way most appropriate to each. The experience gained in such an
> approach could help us if at some point a generalized framework looks
> desirable.
>
Gotchya, sounds reasonable.
DG
>
> Chuck
>
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