Opened 6 years ago

# Meta-Ticket: Asymptotic Expressions in Sage — at Version 16

Reported by: Owned by: behackl major sage-7.4 asymptotic expansions asymptotics, gsoc15 dkrenn, cheuberg, ncohen, vdelecroix, malb, mmezzarobba, rws, kalvotom Benjamin Hackl, Clemens Heuberger, Daniel Krenn N/A #17600, #17693, #17715, #17716, #18182, #18222, #18223

We intend to implement asymptotic expressions in Sage. We would like to do computations with simple expressions such as

n2 + n3/2 + O(n1/2),

but also with expressions such as

2n * n + O(n*log(n))

or even multivariate expressions such as

3*k/n + O(k2 / n2) 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. n2 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

### comment:1 Changed 6 years ago by behackl

• Dependencies set to 17600

### comment:2 Changed 6 years ago by behackl

• Dependencies changed from 17600 to #17600

### comment:3 follow-up: ↓ 4 Changed 6 years ago by tscrim

#10519 might be of interest.

### comment:4 in reply to: ↑ 3 Changed 6 years ago by dkrenn

#10519 might be of interest.

Thanks---I'm involved in both tickets ;)

At the moment both are independent, but when the asymptotic expressions are created, one can use them in the calculations (or at least as a possible output format) in #10519.

### comment:5 Changed 6 years ago by ncohen

• Cc ncohen added

### comment:6 Changed 6 years ago by dkrenn

• Dependencies changed from #17600 to #17600, #17693
• Description modified (diff)

### comment:7 Changed 6 years ago by vdelecroix

• Cc vdelecroix added

### comment:8 Changed 6 years ago by behackl

• Dependencies changed from #17600, #17693 to #17600, #17693, #17715, #17716
• Description modified (diff)

### comment:9 Changed 6 years ago by malb

• Cc malb added

### comment:10 Changed 6 years ago by mmezzarobba

• Cc mmezzarobba added

### comment:11 Changed 6 years ago by fredrik.johansson

Are "asymptotic expressions" equivalent to "transseries" (http://arxiv.org/abs/0801.4877, http://www.texmacs.org/joris/ln/ln-abs.html)? Or are they more general, less general, or partially overlapping in scope?

### comment:12 Changed 6 years ago by rws

• Cc rws added
• Milestone changed from sage-6.5 to sage-6.6

### comment:13 follow-up: ↓ 14 Changed 6 years ago by vdelecroix

Hi,

Whatever you propose, I would say that the most important thing to do is to consider the integration into Sage. In other words:

• how it will be used from Sage
• how it does interact with the Symbolic ring, polynomials, fraction fields, power series and any objects where asymptotic makes sens

I do not see any of this in the ticket description. And it is definitely important to think of it before starting the implementation.

I only see a list of classes, parents and elements whose goal is basically to mimic the symbolic ring by adding some big Oh. I do not see the point of creating so much classes to handle asymptotic terms. Please, motivate and explain your choices.

Vincent

### comment:14 in reply to: ↑ 13 Changed 6 years ago by cheuberg

I only see a list of classes, parents and elements whose goal is basically to mimic the symbolic ring by adding some big Oh.

I rather think of it as a version of the `PowerSeriesRing` with additional features (non-integer exponents, several (not completely independent) variables).

### comment:15 Changed 6 years ago by kalvotom

• Cc kalvotom added

### comment:16 Changed 6 years ago by dkrenn

• Dependencies changed from #17600, #17693, #17715, #17716 to #17600, #17693, #17715, #17716, #18182, #18222, #18223
• Description modified (diff)
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