#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 | 2af8888cad7403af8b396a406713d23dab6a0150
Report Upstream: N/A | Stopgaps:
Branch: public/asy/trunk |
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, #19510, #19521, #19528, |
#19532, #19540, #19576, #19577, |
#19580 |
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Comment (by dkrenn):
Replying to [comment:83 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.
We've planned to do so, once asymptotic expansions at `0` are implemented
(currently everything is only at `oo`).
> 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)
We are aware of Puiseux series.
The restrictions imposed by Puiseux series are that they cannot handle
non-rational exponents like `sqrt(2)` or symbolic constants. However, we'd
like to be able to handle these.
> 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).
In addition to the relation between `n` and `log(n)`, our asymptotic
expansions will be able to handle dependencies between the variables as
well, e.g. `x <= sqrt(y)`.
> 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.
As mentioned above, once we have expansions at `0`, we have a
!MultivariatePuiseuxSeriesRing as a special case; renaming it now is not
expedient.
--
Ticket URL: <http://trac.sagemath.org/ticket/17601#comment:101>
Sage <http://www.sagemath.org>
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