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Iteration Asymptotics

17 Jun 2021 - Tags: sage , featured

I really like recurrences, and the kind of asymptotic analysis that shows up in combinatorics and computer science. I think I’m drawn to it because it melds something I enjoy (combinatorics and computer science) with something I historically struggle with (analysis).

My usual tool for handling reccurences (particularly for getting asymptotic information about their solutions) is generating functions. They slaughter linear recurrences (which nowadays I just solve with sage), but through functional equations, lagrange inversion, and complex analysis, they form an extremely sophisticated theory1. Plus, I would be lying if I didn’t say I was drawn to them because of how cool they are conceptually. I’m a sucker for applying one branch of math to another, so the combination of complex analysis and enumerative combinatorics is irresistable.

Unfortunately, you can’t solve all asymptotics problems with generating functions, and it’s good to have some other tools around as well. Today we’ll be working with the following question:

If $f$ is some function, what are the asymptotics of

\[x_{n+1} = f(x_n)\]

where we allow $x_0$ and $n$ to vary?

If $f$ is continuous and $x_n \to r$, it’s clear that $r$ must be a fixed point of $f$. If moreover, $f’(r)$ exists, and $|f’(r)| \lt 1$, then anything which starts out near $r$ will get pulled into $r$. Also, we might as well assume $r = 0$, since we can replace $f$ by $f(x+r) - r$ without loss of generality.

These observations tell us we should restrict attention to those systems where $f(x) = a_1 x + a_2 x^2 + \ldots$ is analytic at $0$ with $f(0) = 0$ and $|a_1| \lt 1$. Indeed, we’ll focus on exactly this case for the rest of the post2.

As a case study, let’s take a simple problem I found on mse the other day. I can’t actually find the question I saw anymore, or else I’d link it. It looks like it’s been asked a few times now, but none of the options are recent enough to be the one I saw. Oh well.

Define $x_0 = 2$, $x_{n+1} = \frac{x_n + 3}{3 x_n + 1}$. What is $\lim_{n \to \infty} x_n$?

The original problem is fairly routine, and we can solve it using cobweb diagrams. The asymptotics are more interesting, though. It turns out we can read the asymptotics right off of $f$, which is super cool! I guess I hadn’t seen any examples because people who are in the know feel like it’s too obvious to talk about, but that makes it the perfect topic for a blog post!

Notice $f(x) = \frac{x+3}{3x+1}$ has a fixed point at $1$, so we’ll need to translate it to the origin. We’ll replace $f$ by $g(x) = f(x+1) - 1 = \frac{-2x}{3x+4}$, remembering to replace $x_0 = 2$ by $y_0 = x_0 - 1 = 1$ as well.

Notice $g(x) = - \frac{1}{2} x + \frac{3}{8}x^2 - O(x^3)$ is analytic at $0$ with $| - \frac{1}{2} | \lt 1$.

Let’s get started!

Simple Asymptotics

We’ll be following de Bruijn’s Asymptotic Methods in Analysis in both this section and the next. In the interest of showcasing how to actually use these tools, I’m going to gloss over a lot of details. You can find everything made precise in chapter $8$.

First, if $x \approx 0$, then $f(x) \approx f’(0) \cdot x$. Since we are assuming $|f’(0)| \lt 1$, if $x_n \approx 0$, then $x_{n+1} \approx f’(0) x_n \approx 0$ too. An easy induction then shows that

\[x_n \approx (f'(0))^n x_0.\]

But now we have our asymptotics! If we formalize all of the $\approx$ signs above, we find

For each $|f’(0)| \lt b \lt 1$, there is a radius $\delta$ so that as long as $x_0 \in (-\delta, \delta)$ we’re guaranteed

\(|x_n| \lt b^n |x_0|\)

Since $x_n \to 0$, we’re guaranteed that eventually our $x_n$s will be inside however tight a radius we want! Since big-oh notation ignores the first finitely many terms anyways, this tells us

Life Pro Tip:

If $x_n \to r$ (a fixed point of $f$) with $\lvert f’(r) \rvert \lt 1$, then $x_n \to r$ exponentially quickly. More precisely, for any $\epsilon$ you want3

\(x_n = r \pm O((\lvert a_1 \rvert + \epsilon)^n)\)

What about our concrete example? We know $f$ has $1$ as a fixed point, and $f’(1) = - \frac{1}{2}$. Then for $x_0 = 2$, we get (by choosing $\epsilon = 0.00001$) that $x_n = 1 \pm O ( 0.50001^n )$. Which is fast enough convergence for most practical purposes.

Asymptotic Expansion

But what if you’re a real masochist, and the exponential convergence above isn’t enough for you? Well don’t worry, becaue de Bruijn has more to say.

If $x_0$ is fixed, then $x_n a_1^{-n}$ converges to a limit, which we’ll call $\omega(x_0)$. But since $x_{n+1} = f(x_n)$, we get an important restriction on $\omega$ by considering $x_1$ as the start of its own iteration4:

\[\omega(f(x)) = a_1 \omega(x) \quad \quad \quad \quad (\star)\]

In the proof that $\omega$ exists (which you can find in de Bruijn’s book), we also show that $\omega$ is analytic whenever $f$ is! Then using lagrange inversion, we can find an analytic $\Omega$ so that $\Omega(\omega(x)) = x$.

But now we can get a great approximation for $x_n$! By repeatedly applying $(\star)$ we find

\[\omega(x_n) = a_1^n \omega(x_0)\]

which we can then invert to find

\[x_n = \Omega(a_1^n \omega(x_0)).\]

If we use the first, say, $5$ terms of the expansion of $\Omega$, this will give us accuracy up to $\tilde{O}(a_1^{5n})$. There are also some lower order terms which come from how much of $\omega$ we use, but I’m sweeping under the $\tilde{O}$.

How do we actually do this in practice, though? The answer is “with sage”!

This code will take a function $f$ like we’ve been discussing, and will recursively compute the first $N$ coefficients of $\omega$. It turns out the $j$th coefficient of $\omega$ depends only on the first $j-1$ coefficients plus equation $(\star)$. Then it will lagrange invert to get the first $N$ terms of $\Omega$, and it will use these to compute an asymptotic expansion for $x_n$ in terms of $n$ and $x_0$ (which it’s writing as $x$).

Since I had to (rather brutally) convert back and forth between symbolic and ordinary power series, sage wasn’t able to keep track of the error term for us. Thankfully, it’s pretty easy to see that the error term is always $O(a_1^n x_0^N)$, so I just manually wrote it in.

As a nice exercise, you might try to modify the above code to work with functions with a fixed point at $r \neq 0$. You can do this either by taylor expanding at $r$ directly, or by translating $r$ to $0$, then using this code, then translating back.

Be careful, though! We get much more numerical precision near $0$, so if you do things near $r$ you might want to work with arbitrary precision reals.

So, in our last moments together, let’s finish up that concrete example in far more detail than anyone ever wanted. The default function that I put in the code above is our function translated to $0$. If you look at the first $10$ terms of the sequence (that is, set $N=5$) and work with $x_0 - 1 = 1$ (since we translated everything left by $1$) we find

\[x_n - 1 \approx \frac{5}{8} \left ( \frac{-1}{2} \right )^{n} + \frac{3}{8} \left ( \frac{-1}{2} \right )^{2n} - \frac{1}{8} \left ( \frac{-1}{2} \right )^{3n} + \frac{1}{8} \left ( \frac{-1}{2} \right )^{4n}\]

For $n = 10$, say, we would expect

\[x_n \approx 1.0006107\]

the actual answer is

\[x_n = 1.0006512\]

which, seeing as $x_0 = 2$ is pretty far away from $1$ (the limit), $5$ is a pretty small number of terms to use, and $10$ really isn’t that many iterations, is good enough for me.

Of course, you should look at the statistics in the output of the code above to see how close we get for $n=100$, or any other number you like ^_^.

  1. I know the basics, but there’s some real black magic people are able to do by considering what type of singularities your function has. This seems to be outlined in Flajolet and Sedgewick’s Analytic Combinatorics, but every time I’ve tried to read that book I’ve gotten quite lost quite quickly. I want to find some other references at some point, ideally at a slower pace, but if I never do I’ll just have to write up a post about it once I finally muster up the energy to really understand that book. 

  2. It turns out you can sometimes say things if $|a_1| = 1$. But convergence is slow (if you get it at all) and the entire discussion is a bit more delicate. You should see de Bruijn’s Asymptotic Methods in Analysis (chapter $8$) for more details. 

  3. Notice these techniques can’t remove the $\epsilon$. For instance, $n C^n = O((C+\epsilon)^n)$ for each $\epsilon$, but is not $O(C^n)$. 

  4. Which is apparently called Schröder’s equation