#17601: Meta-Ticket: Asymptotic Expressions in Sage
-------------------------------------------------+-------------------------
Reporter: behackl | Owner:
Type: enhancement | Status: new
Priority: major | Milestone: sage-6.6
Component: symbolics | Resolution:
Keywords: asymptotics | Merged in:
Authors: Benjamin Hackl, Clemens | Reviewers:
Heuberger, Daniel Krenn | Work issues:
Report Upstream: N/A | Commit:
Branch: | Stopgaps:
Dependencies: #17600, #17693, #17715, |
#17716, #18182, #18222, #18223, #18586, |
#18587 |
-------------------------------------------------+-------------------------
Changes (by dkrenn):
* dependencies: #17600, #17693, #17715, #17716, #18182, #18222, #18223 =>
#17600, #17693, #17715, #17716, #18182, #18222, #18223, #18586, #18587
Old description:
> We intend to implement asymptotic expressions in Sage. We would like to
> do computations with simple expressions such as
>
> n^2^ + n^3/2^ + O(n^1/2^),
>
> but also with expressions such as
>
> 2^n^ * n + O(n*log(n))
>
> or even multivariate expressions such as
>
> 3*k/n + O(k^2^ / n^2^) with |k| <= n^(1/2)^.
>
> Of course, O(n) - O(n) = O(n) must hold.
>
> Eventually, specified O-constants shall also be supported.
>
> The current plan is to implement the following classes (plus derivatives
> for more concrete situations). For simplicity, the corresponding parents
> are not listed here.
>
> !AsymptoticGrowthElement::
> hold _one_ term, e.g. n^2^ or k/n or n*log(n). This can compare,
> multiply etc.,
> but has **no** coefficient. Here, only the order of magnitude shall be
> managed.
>
> !AsymptoticTerm::
>
> holds one !AsymptoticGrowthElement, plus information on the coefficient
> or that it is an O-term etc.
>
> !AsymptoticExpression::
>
> represents the sum of several !AsymptoticTerms.
>
> The idea is to override !AsymptoticGrowthElement to obtain specific
> behaviour (as mentioned in our wishlist) because it seems unlikely to be
> able to handle everything in one class.
> For starters, there will be an !GrowthGroupPowerElement.
>
> !AsymptoticTerm is expected to be more general; it might be necessary to
> override it for the case of specified O-constants.
>
> !AsymptoticExpression, however, can be general enough to deal with all
> cases; here, the sum, the product, the exponential function, etc. are
> implemented in a generic setting.
>
> Related Tickets:
> #17600 (!AsmyptoticGrowthElement): elements which handle the
> asymptotic growth.
> #17715 (!AsymptoticTerm): "building blocks" for asymptotic
> expressions, growth + additional information (OTerm, !ExactTerm, ...).
> #17716 (!AsymptoticExpression): sum of multiple asymptotic terms.
> #17693 (!MutablePoset): data structure for storing asymptotic terms
> within an asymptotic expression.
>
> Other Dependencies:
> #18182: pushout construction and finding common parents for/including
> cartesian products
> #18222: provide <=, <, >=, > for poset elements by the category
> (depends on #10130)
> #18223: new categories for cartesian products with orders
New description:
We intend to implement asymptotic expressions in Sage. We would like to do
computations with simple expressions such as
n^2^ + n^3/2^ + O(n^1/2^),
but also with expressions such as
2^n^ * n + O(n*log(n))
or even multivariate expressions such as
3*k/n + O(k^2^ / n^2^) with |k| <= n^(1/2)^.
Of course, O(n) - O(n) = O(n) must hold.
Eventually, specified O-constants shall also be supported.
The current plan is to implement the following classes (plus derivatives
for more concrete situations). For simplicity, the corresponding parents
are not listed here.
!AsymptoticGrowthElement::
hold _one_ term, e.g. n^2^ or k/n or n*log(n). This can compare,
multiply etc.,
but has **no** coefficient. Here, only the order of magnitude shall be
managed.
!AsymptoticTerm::
holds one !AsymptoticGrowthElement, plus information on the coefficient
or that it is an O-term etc.
!AsymptoticExpression::
represents the sum of several !AsymptoticTerms.
The idea is to override !AsymptoticGrowthElement to obtain specific
behaviour (as mentioned in our wishlist) because it seems unlikely to be
able to handle everything in one class.
For starters, there will be an !GrowthGroupPowerElement.
!AsymptoticTerm is expected to be more general; it might be necessary to
override it for the case of specified O-constants.
!AsymptoticExpression, however, can be general enough to deal with all
cases; here, the sum, the product, the exponential function, etc. are
implemented in a generic setting.
Related Tickets:
#17600 (!AsmyptoticGrowthElement): elements which handle the
asymptotic growth.
#17715 (!AsymptoticTerm): "building blocks" for asymptotic
expressions, growth + additional information (OTerm, !ExactTerm, ...).
#17716 (!AsymptoticExpression): sum of multiple asymptotic terms.
#17693 (!MutablePoset): data structure for storing asymptotic terms
within an asymptotic expression.
#18587: cartesian products of growth groups
Other Dependencies:
#18182: pushout construction and finding common parents for/including
cartesian products
#18222: provide <=, <, >=, > for poset elements by the category
(depends on #10130)
#18223: cartesian products with orders
#18586: passing on parameters and extra_category
--
--
Ticket URL: <http://trac.sagemath.org/ticket/17601#comment:17>
Sage <http://www.sagemath.org>
Sage: Creating a Viable Open Source Alternative to Magma, Maple, Mathematica,
and MATLAB
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