On 12.03.2014 00:12, Avichal Dayal wrote:
The paper you mention works for functions containing only bounded
functions like:- (sin(x**2) - sin(x))/(3 + cos(E*x**2))
And I think that's what we need. Heuristics handle cases like
sin(x)/x but not when all are bounded. This can be a useful
addition.
I've partly understood the algorithm and I plan to add it to my
proposal. Can you give some suggestions regarding my proposal? I plan
to do formal power series, asymptotic series and hopefully limits.
About strategic (for all)
Here is the link:-
https://github.com/sympy/sympy/wiki/GSoC-2014-Application-Avichal-Dayal-Series-Expansion
Formal Power Series:
Other CAS are capable of returning an infinite series in terms of a
generating function or a sequence. I feel we should have this.
Currently series() returns something like 1 + x + x**2/2 + O(x**3)
while Mathematica and others are also capable of returning something
like summation(x**n/factorial(n), (n, 0, oo)).
1. 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.
2. What is more "symbolical": description of series: via some structure,
or only as the coeffitients (symbolical thou) of some restricted number
of terms? What is common for them?
Asymptotic series:
Most of the routines are implemented in gruntz.py. However still work
is to be done to implement it. Series is extensively used by gruntz
and asymptotic expansion is not implemented in SymPy. This would
improve gruntz greatly.
3. Is it transseries [4]?
4. If it is transseries, what is better (for what aims) to implement
firstly: Formal Transseries (without consideration of limits), or series
expansion ( restricted number of terms).
5. Is it possible or impossible to combine transseries with [1] or with
lazy Series structure [2]?
And may be outside the context of your Application
6. Is it possible to implement multi variable cases so, to keep in mind
abstract FPS, functions and Sequences (or, obviously, something instead
of abstract Sequences but rather tensors) as in [3]?
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
--
Alexey Gudchenko
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