On Fri, Nov 29, 2013 at 11:07 AM, Pablo Puente <[email protected]> wrote:
> I definitely agree that Series infrastructure is important and SymPy should
> have it.
>
> Maybe the base class can be just a pure interface (in Java terms, pure
> virtual functions in C++), not requiring actual storage sparse or dense. So
> that coefficients can be retrieved through a function. Then in Sparse or
> Dense subclass coefficients can be cached or pre-calculated in the
> appropriate manner.
I think you are right. And we can implement this in parallel to what
we have already in sympy.
>
> Other nice to have series also supported in the new structure would be
> Fourier trigonometric series:
> - SUM( an*sin(n*x)+bn*cos (n*x) )
> - SUM( cn*exp(i*n*x) )
Interesting --- so you would subclass Series as FourierSeries,
and let FourierSeries return a Fourier series of the given function?
So internally it would
calculate the cn coefficients (or an, bn) and internally it understands
that the series is not given using the x, x^2, x^3 basis, but using
exp(i*n*x) basis.
The Series would be abstract enough so that subclasses can choose
both how coefficients are calculated as well as how the series is represented,
i.e. linear combination of {x, x^2, x^3, ...} or {exp(i*x), exp(2*i*x), ...}.
So DenseSeries, SparseSeries and FourierSeries would be on the same level.
> And thinking even more general any projection on a Hilbert space with a
> given base of functions (of which particular cases are Fourier series,
> Legendre polynomial series, etc.).
Good idea.
Ondrej
>
> Pablo.
>
> On Thursday, November 28, 2013 5:39:30 AM UTC+1, Ondřej Čertík wrote:
>>
>> On Tue, Nov 26, 2013 at 5:38 AM, mario <[email protected]> wrote:
>> >
>> > In PR 609 and PR 2630 power series are implemented with dictionaries,
>> > that
>> > is in a sparse representation.
>> > The sparse representation is efficient when there are many variables. PR
>> > 2630 uses ``ring()``.
>> >
>> > For power series in one variable the dense representation is usually
>> > faster.
>> > In `compose_add` in PR 2630 series in one variable are used, so using
>> > the
>> > dense representation
>> > would be faster, but in a negligible way, since `compose_add` takes
>> > roughly
>> > 5% of the time in `minpoly`.
>> >
>> > In PR 609 there is lpoly.py, which deals with power series, and
>> > ltaylor.py.
>> > lpoly.py can now be implemented using ``ring()``; this is partially done
>> > in
>> > ring_series.py in PR 2630,
>> > where there are only the functions used in the computed sum. It seems to
>> > me
>> > that ring_series.py fits
>> > well in SymPy; if one wants to compute efficiently power series,
>> > one can use ``ring`` with ring_series.py functions (once all the series
>> > functions in lpoly.py are ported).
>> >
>> > To do a thorough treatment of power series, one should also implement
>> > the dense representation; consider the case of Laurent series;
>> > add functions to manipulate series at a higher level, possibly using a
>> > Series class with an Order function.
>> >
>> > Mateusz Paprocki might want to comment on this.
>> >
>> > PowerSeriesRing in Sage could be an example of higher level
>> > representation.
>>
>> Thanks a lot for the feedback. So I think we should try to merge your
>> code, or improve it, so that it can be merged.
>>
>>
>> > ltaylor.py uses lpoly.py to compute the taylor series of SymPy
>> > functions;
>> > if the function is not smooth enough, the computation is passed to
>> > series.series.
>> > So if the function is smooth enough, the taylor series is computed fast
>> > (more or less the speed of `taylor` in Sage); otherwise it can be
>> > slightly slower than series.series.
>> > I do not know if it is a good approach; anyway, since it uses power
>> > series,
>> > for the moment I will leave it aside.
>>
>> Btw, just today I discovered a real life example where series() is really
>> slow:
>>
>> https://github.com/sympy/sympy/issues/2635
>>
>> Ondrej
>
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