#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, gsoc15 | 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, #18930, #19017, #19028, #19047, |
#19048, #19068, #19073, #19079 |
-------------------------------------------------+-------------------------
Description changed by dkrenn:
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 and we want to perform various
> arithmetic operations with these asymptotic expressions. Eventually,
> specified O-constants shall also be supported.
>
> See #17716 for more examples.
> -------
>
> **Roadmap**:
>
> * Implementing a minimal working example
> * #17600 (!AsymptoticGrowthElement): elements which handle the
> asymptotic growth. Such an element holds, e.g. n^2^ or k/n or n*log(n).
> This can compare, multiply etc., but has **no** coefficient; the order of
> magnitude is managed here. Concretely for this ticket:
> !MonomialGrowthElement, implementation for powers.
> * #18930: Factory for user-friendly generation of growth groups
> * #17715 (!AsymptoticTerm): a summand for asymptotic expressions.
> They contain the growth and additional information on the type of the
> summand. For starters, there will be big-Oh terms (e.g. `O(n)` and exact
> terms (e.g. `3*n^2`).
> * #17693 (!MutablePoset): data structure for storing asymptotic terms
> within an asymptotic expression.
> * #17716 (!AsymptoticRing and !AsymptoticExpression): sum of
> asymptotic terms.
>
> * Extending the functionality of the !AsymptoticRing and
> !AsymptoticExpression
> * #19017: Easy access to the `O`-constructor in `big_oh.py`.
> * #19068: Implement Division for asymptotic Expressions.
> * Implement higher-order operations like `exp` and `log` for
> asymptotic expressions.
> * Improve the user interface: extend the conversion from the symbolic
> ring such that more than just monomials can be converted.
> * Implement comparison for asymptotic expressions.
> * Improve the performance of computations in the !AsymptoticRing.
> * #19048: `AsymptoticRing.an_element()`
> * #19047: `QQ.some_elements()`
> * #19073: categorial constructions, pushout and coercions (extended)
> for asymptotic ring and growth groups
> * #18182: pushout construction and finding common parents
> for/including cartesian products
>
> * Extending the functionality of growth groups
> * #19028: More growth group implementations: exponential growth
> groups.
> * #18587: cartesian products for growth groups (allowing the
> construction of more complicated univariate as well as multivariate
> asymptotic expressions)
> * #18223: cartesian products with orders
> * #18586: passing on parameters and extra_category for cartesian
> products
> * implement dependencies like |k| <= n^1/2^ for different growth
> group variables.
>
> * Further plans
> * growth groups with asymptotic at a non-infinity point
> * Implementation of more types of asymptotic terms (little-oh terms,
> omega-terms, variations of big-Oh terms ...)
>
> * Additional Dependencies:
> * #18222: provide <=, <, >=, > for poset elements by the category
> (depends on #10130)
> * #19079: ConstructionFunctor: remove `__str__`
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 and we want to perform various
arithmetic operations with these asymptotic expressions. Eventually,
specified O-constants shall also be supported.
See #17716 for more examples.
-------
**Roadmap**:
* Implementing a minimal working example
* #17600 (!AsymptoticGrowthElement): elements which handle the
asymptotic growth. Such an element holds, e.g. n^2^ or k/n or n*log(n).
This can compare, multiply etc., but has **no** coefficient; the order of
magnitude is managed here. Concretely for this ticket:
!MonomialGrowthElement, implementation for powers.
* #18930: Factory for user-friendly generation of growth groups
* #17715 (!AsymptoticTerm): a summand for asymptotic expressions. They
contain the growth and additional information on the type of the summand.
For starters, there will be big-Oh terms (e.g. `O(n)` and exact terms
(e.g. `3*n^2`).
* #17693 (!MutablePoset): data structure for storing asymptotic terms
within an asymptotic expression.
* #17716 (!AsymptoticRing and !AsymptoticExpression): sum of
asymptotic terms.
* Extending the functionality of the !AsymptoticRing and
!AsymptoticExpression
* #19017: Easy access to the `O`-constructor in `big_oh.py`.
* #19068: Implement Division for asymptotic Expressions.
* Implement higher-order operations like `exp` and `log` for
asymptotic expressions.
* Improve the user interface: extend the conversion from the symbolic
ring such that more than just monomials can be converted.
* Implement comparison for asymptotic expressions.
* Improve the performance of computations in the !AsymptoticRing.
* #19048: `AsymptoticRing.an_element()`
* #19047: `QQ.some_elements()`
* #19073: categorial constructions, pushout and coercions (extended)
for asymptotic ring and growth groups
* #18182: pushout construction and finding common parents
for/including cartesian products
* #19079: !ConstructionFunctor: remove `__str__`
* Extending the functionality of growth groups
* #19028: More growth group implementations: exponential growth
groups.
* #18587: cartesian products for growth groups (allowing the
construction of more complicated univariate as well as multivariate
asymptotic expressions)
* #18223: cartesian products with orders
* #18586: passing on parameters and extra_category for cartesian
products
* implement dependencies like |k| <= n^1/2^ for different growth
group variables.
* Further plans
* growth groups with asymptotic at a non-infinity point
* Implementation of more types of asymptotic terms (little-oh terms,
omega-terms, variations of big-Oh terms ...)
* Additional Dependencies:
* #18222: provide <=, <, >=, > for poset elements by the category
(depends on #10130)
--
--
Ticket URL: <http://trac.sagemath.org/ticket/17601#comment:28>
Sage <http://www.sagemath.org>
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and MATLAB
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