## Google Summer of Code 2015: Conclusion

##### By behackl, 22nd August 2015

The “Google Summer of Code 2015” program has ended yesterday, on the 21. of August at 19.00 UTC. This blog entry shall provide a short wrap-up of our GSoC project.

The aim of our project was to implement a basic framework that enables us to do computations with asymptotic expressions in SageMath — and I am very happy to say that we very much succeeded to do so. An overview of all our developments can be found at meta ticket #17601.

Although we did not really follow the timeline suggested in my original proposal (mainly because the implementation of the Asymptotic Ring took way longer than originally anticipated) we managed to implement the majority of ideas from my proposal — with the most important part being that our current prototype is stable. In particular, this means that we do not expect to make major design changes at this point. Every detail of our design is well-discussed and can be explained.

Of course, our “Asymptotic Expressions” project is far from finished, and we will continue to extend the functionality of our framework. For example, although working with exponential and logarithmic terms is currently possible, it is not very convenient because the $\log$, $\exp$, and power functions are not fully implemented. Furthermore, it would be interesting to investigate the performance-gain obtained by cythonizing the central parts of this framework (e.g. parts of the MutablePoset…) — and so on…

To conclude, I want to thank Daniel Krenn for his hard work and helpful advice als my mentor, as well as the SageMath community for giving me the opportunity to work on this project within the Google Summer of Code program! 🙂

## Asymptotic Expressions: Current Developments

##### By behackl, 17th August 2015

Since my last blog entry on the status of our implementation of Asymptotic Expressions in SageMath quite a lot of improvements have happened. Essentially, all the pieces required in order to have a basic working implementation of multivariate asymptotics are there. The remaining tasks within my Google Summer of Code project are:

• Polish the documentation of our minimal prototype, which consists of #17716 and the respective dependencies. Afterwards, we will set this to needs_review.
• Open a ticket for the multivariate asymptotic ring and put together everything that we have written so far there.

In this blog post I want to give some more examples of what can be done with our implementation right now and what we would like to be able to handle in the future.

### Status Quo

After I wrote my last blog entry, we introduced a central idea/interface to our project: short notations. By using the short notation factory for growth groups (introduced in #18930) it becomes very simple to construct the desired growth group. Essentially, monomial growth groups (cf. #17600), i.e. groups that contain elements of the form variable^power (for a fixed variable and powers from some base ring, e.g. the Integer Ring or even the Rational Field) are represented by variable^base , where the base ring is also specified via its shortened name. The short notation factory then enables us to do the following:

Naturally, this interface carries over to the generation of asymptotic rings: instead of the (slightly dubious) "monomial" keyword advertised in my last blog entry, we can now actually construct the growth group by specifying the respective growth group via its short representation:

Recently, we also implemented another type of growth group: exponential growth groups (see #19028). These groups represent elements of the form base^variable where the base is from some multiplicative group. For example, we could do the following:

Note: unfortunately, we did not implement a function that allows taking some element from some growth group (e.g. x from a monomial growth group) as the variable in an exponential growth group yet. Implementing some way to “change” between growth groups by taking the log or the exponential function is one of our next steps.

We also made this short notation a central interface for working with cartesian products. This is implemented in #18587. For example, this allows to construct growth groups containing elements like $2^x \sqrt[5]{x^2} \log(x)^2$:

Simple parsing from the symbolic ring (and from strings) is implemented. Like I have written above, operations like 2^G(x) or log(G(x)) are one of the next steps on our roadmap.

### Further Steps

Of course, having an easy way to generate growth groups (and thus also asymptotic rings) is nice — however, it would be even better if the process of finding the correct parent would be even more automated. Unfortunately, this requires some non-trivial effort regarding the pushout construction — which will certainly not happen within the GSoC project.

As soon as we have an efficient way to “switch” between factors of a growth group (e.g. by taking the logarithm or the exponential function), this has to be carried over up to the asymptotic ring. Operations like

where the output could also be 2^(x^2) * O(x^g) , where $g$ is determined by series_precision() .

Division of asymptotic expressions can be realized with just about the same idea, for example:

$\frac{1}{x^2 + O(x)} = \frac{1}{x^2} \frac{1}{1 + O(1/x)} = x^{-2} + O(x^{-3}),$

and so on. If an infinite series occurs, it will have to be cut using an $O$-Term, most likely somehow depending on series_precision() as well.

Ultimately, we would also like to incorporate, for example, Stirling’s approximation of the factorial such that we could do something like

which then can be used to obtain asymptotic expansions of binomial coefficients like $\binom{2n}{n}$:

As you can see, there is still a lot of work within our “Asymptotic Expressions” project — nevertheless, with the minimal working prototype and the ability to create cartesian products of growth groups, the fundament for all of this is already implemented! 😉

## Computing with Asymptotic Expressions

##### By behackl, 16th July 2015

It has been quite some time since my last update on the progress of my Google Summer of Code project, which has two reasons. On the one hand, I have been busy because of the end of the semester, as well as because of the finalization of my Master’s thesis — and on the other hand, it is not very interesting to write a post on discussing and implementing rather technical details. Nevertheless, Daniel Krenn and myself have been quite busy in order to bring asymptotic expressions to SageMath. Fortunately, these efforts are starting to become quite fruitful.

In this post I want to discuss our current implementation roadmap (i.e. not only for the remaining Summer of Code, but also for the time afterwards), and give some examples for what we are currently able to do.

An overview of the entire roadmap can be found at here (trac #17601). Recall that the overall goal of this project is to bring asymptotic expressions like $2^n + n^2 \log n + O(n)$ to Sage. Our implementation (which aims to be as general and expandable as possible) tackles this problem with a three-layer approach:

• GrowthGroups and GrowthElements (trac #17600). These elements and parents manage the growth (and just the growth!) of a summand in an asymptotic expression like above. The simplest cases are monomial and logarithmic growth groups. For example, their elements are given by $n^r$ and $\log(n)^r$ where the exponent $r$ is from some ordered ring like $\mathbb{Z}$ or $\mathbb{Q}$. Both cases (monomial and logarithmic growth groups) can be handled in the current implementation — however, growth elements like $n^2 \log n$ are intended to live in the cartesian product of a monomial and a logarithmic growth group (in the same variable). Parts of this infrastructure are already prepared (see trac #18587).
• AsymptoticTerms and TermMonoids (trac #17715). While GrowthElements only represent the growth, AsymptoticTerms have more information: basically, they represent a summand in an asymptotic expression. There are different classes for each type of asymptotic term (e.g. ExactTerm and OTerm — with more to come). Additionally to a growth element, some types of asymptotic terms (like exact terms) also possess a coefficient.
• AsymptoticExpression and AsymptoticRing (trac #17716). This is what we are currently working on, and we do have a running prototype! 🙂 The version that can be found on trac is only missing some doctests and a bit of documentation. Asymptotic expressions are the central objects within this project, and essentially they are sums of several asymptotic terms. In the background, we use a special data structure (“mutable posets“, trac #17693) in order to model the (partial) order induced by the various growth elements belonging to an asymptotic expression. This allows to perform critical operations like absorption (when an $O$-term absorbs “weaker” terms) efficiently and in a simple way.

The resulting minimal prototype can, in some sense, be compared to Sage’s PowerSeriesRing: however, we also allow non-integer exponents, and extending this prototype to work with multivariate expressions should not be too hard now, as the necessary infrastructure is there.

Following the finalization of the minimal prototype, there are several improvements to be made. Here are some examples:

• Besides addition and multiplication, we also want to divide asymptotic expressions, and higher-order operations like exponentiation and taking the logarithm would be interesting as well.
• Also, conversion from, for example, the symbolic ring is important when it comes to usability of our tools. We will implement and enhance this conversion gradually.

### Examples

An asymptotic ring (over a monomial growth group with coefficients and exponents from the rational field) can be created with

Note that we marked the code as experimental, meaning that you will see some warnings regarding the stability of the code. Now, as we have an asymptotic ring, we can do some calculations. For example, take $(2\sqrt{x} + O(1))^{15}$:

We can also have a look at the underlying structure:

As you might have noticed, the “O”-constructor that is used for the PowerSeriesRing and related structures, can also be used here. In particular, $O(\mathit{expr})$ acts exactly as expected:

Of course, the usual rules for computing with asymptotic expressions hold:

So far, so good. Our next step is making the multivariate growth groups usable for the AsymptoticRing and then improving the overall user interface of the ring.

## Asymptotic Expressions: Motivation

##### By behackl, 29th May 2015

$\def\R{\mathbb{R}}$So, as Google Summer of Code started on Monday, May 25th it is time to give a proper motivation for the project I have proposed. The working title of my project is (Multivariate) Asymptotic Expressions, and its overall goal is to bring asymptotic expressions to SageMath.

### What are Asymptotic Expressions?

A motivating answer for this question comes from the theory of Taylor series. Assume that we have a sufficiently nice (in this case meaning smooth) function $f : \R \to \R$ that we want to approximate in a neighborhood of some point $x_0 \in \R$. Taylor’s theorem allows us to write $f(x) = T_n(x) + R_n(x)$ where

$T_n(x) = \sum_{j=0}^n \frac{f^{(j)}(x_0)}{j!}\cdot (x-x_0)^j = f(x_0) + f'(x_0)\cdot (x-x_0) + \cdots + \frac{f^{(n)}(x_0)}{n!}\cdot (x-x_0)^n,$

and $R_n(x) = \frac{f^{(n+1)}(\xi)}{(n+1)!} \cdot (x-x_0)^{n+1}$, where $\xi$ lies in a neighborhood of $x_0$. Note that for $x\to x_0$, $R_n(x)$ “behaves like” $(x-x_0)^{n+1}$. In particular, we can certainly find a constant $C > 0$ such that $|R_n(x)| \leq C\cdot |x-x_0|^{n+1}$, or, in other words: for $x\to x_0$ the growth of the function $R_n(x)$ is bounded from above by the growth of $(x-x_0)^{n+1}$.

The idea of bounding the growth of a function by the growth of another function when the argument approaches some number (or $\infty$) is the central idea behind the big O notationFor function $f, g : \R \to \R$ we write $f(x) = O(g(x))$ for $x\to x_0$ if there is a constant $C > 0$ such that $|f(x)| \leq C\cdot |g(x)|$ for all $x$ in some neighborhood of $x_0$.

A case that is particularly important is the case of $x_0 = \infty$, that is if we want to compare and/or characterize the behavior of some function for $x\to\infty$, which is also called the functions asymptotic behavior. For example, consider the functions $\log x$, $x^3$ and $e^x$. All of them are growing unbounded for $x\to\infty$ — however, their asymptotic behavior differs. This can be seen by considering pairwise quotients of these functions: $\frac{x^3}{e^x} \to 0$ for $x\to\infty$, and therefore the asymptotic growth of $x^3$ can be bounded above by the growth of $e^x$, meaning $x^3 = O(e^x)$ for $x\to\infty$.

The analysis of a functions asymptotic behavior is important for many applications, for example when determining time and space complexity of algorithms in computer science, or for describing the growth of classes of combinatorial objects: take, for example, binary strings of length $2n$ that contain equally many zeros and ones. If $s_n$ denotes the number of such strings, then we have

$s_n = \binom{2n}{n} = \frac{4^n}{\sqrt{n\pi}} \left(1 + O\left(\frac{1}{n}\right)\right) \quad\text{ for } n\to\infty.$

Expressions like these are asymptotic expressions. When we consider asymptotic expressions in only one variable, everything works out nicely as a total order is induced. But as soon as multiple variables are involved, we don’t have a total order any more. Consider, for example, $x^2 y$ and $xy^2$ when $x$ and $y$ approach $\infty$. These two elements cannot be compared to each other, which complicates computing with these expressions as they may contain multiple “irreducible” O-terms.

The following univariate and multivariate examples shall demonstrate how computing with such expressions looks like (all variables are assumed to go to $\infty$):

$x + O(x) = O(x),\quad x^2 \cdot (x + O(1)) = x^3 + O(x^2),\quad O(x^2) \cdot O(x^3) = O(x^5),$

$x y + O(x^2 y) = O(x^2y),\quad (y \log y + O(y)) (x^2 y + O(4^x \sqrt{x})) = x^2 y^2 \log y + O(x^2 y^2) + O(4^x \sqrt{x} y \log y).$

Our plan is to provide an implementation based on which computations with these and more complicated expressions are possible.

### Planned Structure

There are four core concepts of our implementation.

• Asymptotic Growth Groups: These are multiplicative groups that contain growth elements like $x^2$, $\log x$, $2^x \cdot x \cdot \log x$. For starters, only univariate power growth groups will be implemented.
• Asymptotic Term Monoids: These monoids contain asymptotic terms — in essence, these are summands of asymptotic terms. Apart from exact term monoids (growth elements with a coefficient), we will also implement O-term monoids as well as a term monoid for a deviation of O-terms. Asymptotic terms have (in addition to their group operation, multiplication) absorption as an additional operation: for example, O-terms are able to absorb all asymptotically “smaller” elements.
• Mutable Poset: As we have mentioned above, due to the fact that multivariate asymptotic expressions do not have a total order with respect to their growth, we need a partially ordered set (“Poset”) that deals with this structure such that operations like absorbing terms can be performed efficiently. The mutable poset is the central data structure that asymptotic expressions are built upon.
• Asymptotic Ring: This is our top-level structure which is also supposed to be the main interaction object for users. The asymptotic ring contains the asymptotic expressions, i.e. intelligently managed sums of asymptotic terms. All common operations shall be possible here. Furthermore, the interface should be intelligent enough such that admissible expressions from the symbolic ring can be directly converted into elements of the asymptotic ring.

Obviously, this “planned structure” is rather superficial. However, this is only to supplement the motivation for my project with some ideas on the implementation. I’ll go a lot more into the details of what I am currently implementing in the next few blog posts!

## Google Summer of Code — Countdown

##### By behackl, 23rd May 2015

Today I received the welcome package for attending this year’s “Google Summer of Code”! Actually, it’s pretty cool; the following things were included:

• a blue notebook with a monochromatic GSoC 15 logo (in dark blue) printed on it
• a sticker with a colored GSoC 15 logo
• a pen that is both a blue ballpoint pen as well as a mechanical pencil (0.5)

Here is a photo of all this stuff:

The work on our project (multivariate) Asymptotic Expressions (in cooperation with Daniel Krenn and Clemens Heuberger) begins (or rather continues) on Monday, the 25th of May. Over the course of next week (probably in a $\varepsilon$-neighborhood of Monday) I will blog about the status quo, as well as about the motivation for the project. 🙂