On 16.03.2014 17:52, Avichal Dayal wrote:
I'm little surpised. Can you provide an example of this
from the Mathematica?
I don't have the software. But I have seen examples in
the references.
Wolfram gives both the truncated series with the order
term and using generating functions as shown below:
In[1] := FormalSeries(exp(x), x)
Out[1] := Summation(x**k/k!, (k, 0, oo))
In[2] := FormalSeries(cos(x), x)
Out[2] := Summation((-1)**k * x**(2*k) / (2*k)!, (k, 0, oo))
The nearest similar thing in the last version of Mathematica 9.0 is
SeriesCoefficient:
https://reference.wolfram.com/mathematica/ref/SeriesCoefficient.html
And the old package of Wolfram Koepf (not Steven) for Mathematica 2.0
http://arxiv.org/abs/math/9404219
I don't think it's a good idea. We should have instead a
basic container type (cf. tuple or list) for infinite sequence
Thanks for the suggestion. We will already have the
formula produced by the algorithm or given by the user.
Then we can use stream to calculate and store the infinte
sequence. But is it encouraged to use "fn" library?
Is it possible strategically to combine this model with
multy
variable cases development in [1] with usage of polys
(and some special
fast polynomials bases)? In this case there is no special
structures as
Sequences, Generating functions and so on.
I'll drop the idea of sequence and use Stream class in
"fn" as @skirpichev pointed out.
In this case take into account
http://www.sagemath.org/doc/reference/sage/combinat/species/series.html
Can Stream (or wrapper) be used for a few variables?
Other than that I only need to store the formula used to
generate the sequence.
I can store that as an expression or as a tuple of (coeff and exponent)?
It depends of use-cases.
For example [1] used some polynomial bases.
And not principal before undefined strategic.
Firstly we must divergent tasks:
a) Formal Power Series (it is one thing: generating functions, possibly
infinity number of coefficients, which defien FPS)
b) Series Expansion, then Transeries Expansion.
Used for for the expansion, limits
And extend all/one of them for multivariables.
And, possibly, combine (as FindGenerationFunctions (that is formula)
from a few terms, that is the reverse task of expansions)
In my opinion we must concentrate only for the one tasks a) or b), but
taking into account both.
Links:
[0]
https://github.com/sympy/sympy/wiki/GSoC-2014-Application-Avichal-Dayal-Series-Expansion
[1] https://github.com/sympy/sympy/blob/master/sympy/polys/ring_series.py
[2]
https://github.com/sympy/sympy/wiki/UD-Sequences-and-formal-power-series-prototype
[3] http://www.goodok.ru/sympy/series/series-sympy-example.pdf
[4] http://arxiv.org/abs/0801.4877 "Transseries for beginners" G. A. Edgar
[5] http://www.sagemath.org/doc/reference/sage/combinat/species/series.html
[6] http://arxiv.org/abs/math/9404219
--
Alexey Gudchenko
I'll improve the FormalSeries class description and try to add
multivariable case soon. Any other suggestions?
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