#17601: Meta ticket: Asymptotic Expansions in SageMath
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Reporter: behackl | Owner:
Type: enhancement | Status: new
Priority: major | Milestone: sage-6.10
Component: asymptotic | Resolution:
expansions | Merged in:
Keywords: asymptotics, | Reviewers:
gsoc15 | Work issues:
Authors: Benjamin Hackl, | Commit:
Daniel Krenn | a848139a35e95bfd67d9964fa7412743942de4a4
Report Upstream: N/A | Stopgaps:
Branch: |
u/dkrenn/asy/prototype |
Dependencies: #17600, #17693, |
#17715, #17716, #18182, #18222, |
#18223, #18586, #18587, #18930, |
#19017, #19028, #19047, #19048, |
#19068, #19073, #19079, #19083, |
#19088, #19094, #19110, #19259, |
#19269, #19300, #19305, #19306, |
#19316, #19319, #19399, #19400, |
#19411, #19412, #19420, #19421, |
#19423, #19424, #19425, #19426, |
#19429, #19431, #19436, #19437, |
#19504 |
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Comment (by nbruin):
Replying to [comment:14 cheuberg]:
> I rather think of it as a version of the `PowerSeriesRing` with
additional features (non-integer exponents, several (not completely
independent) variables).
Yes, and you could get a lot of leverage out of making that link more
prominent. In fact, the appropriate concept would be "Puiseux series",
which are Laurent series (with negative exponents allowed) in fractional
powers of your variables.
For asymptotic expansions you have x+O(x^(1/2)) = O(x^(1/2)), which is
consistent with Puiseux series in t=1/x.
The usual implementation for Puiseux series is as
[d,N,a[N],a[N+1],a[N+2],...,finite number of terms]
meaning
sum_{i=N..} a_i* x^(i/d)
For arithmetic you just first bring series in common denominator "d" and
then do power series arithmetic.
For multivariate series, the appropriate behaviour is caught by "local
term orders". SingularLib might offer some useful things already.
Note that a series in n and log(n) can be treated as a bivariate series,
with an appropriate term order on the variables signifying "n" and
"log(n)", for asymptotic series probably again modelling these with X=1/n
and Y= 1/log(n).
So I think this ticket can be realized by implementing "multivariate
puiseux series", which would be useful in a lot of settings. It would be
better if (the underlying ring) would also be called "multivariate puiseux
series ring" because that would improve discoverability.
Searching the literature for these terms will probably also make it easier
to find relevant algorithms.
--
Ticket URL: <http://trac.sagemath.org/ticket/17601#comment:83>
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