The Universe of Discourse

Fixed points and attractors, part 3

Last week I wrote about a super-lightweight variation on Newton's method, in which one takes this function: $$f_n : \frac ab \mapsto \frac{a+nb}{a+b}$$

or equivalently

$$f_n : x \mapsto \frac{x+n}{x+1}$$

Iterating !!f_n!! for a suitable initial value (say, !!1!!) converges to !!\sqrt n!!:

$$ \begin{array}{rr} x & f_3(x) \\ \hline 1.0 & 2.0 \\ 2.0 & 1.667 \\ 1.667 & 1.75 \\ 1.75 & 1.727 \\ 1.727 & 1.733 \\ 1.733 & 1.732 \\ 1.732 & 1.732 \end{array} $$

Later I remembered that a few months back I wrote a couple of articles about a more general method that includes this as a special case:

• How to calculate the square root of 2
• More about fixed points and attractors

The general idea was:

Suppose we were to pick a function !!f!! that had !!\sqrt 2!! as a fixed point. Then !!\sqrt 2!! might be an attractor, in which case iterating !!f!! will get us increasingly accurate approximations to !!\sqrt 2!!.

We can see that !!\sqrt n!! is a fixed point of !!f_n!!:

$$ \begin{align} f_n(\sqrt n) & = \frac{\sqrt n + n}{\sqrt n + 1} \\ & = \frac{\sqrt n(1 + \sqrt n)}{1 + \sqrt n} \\ & = \sqrt n \end{align} $$

And in fact, it is an attracting fixed point, because if !!x = \sqrt n + \epsilon!! then

$$\begin{align} f_n(\sqrt n + \epsilon) & = \frac{\sqrt n + \epsilon + n}{\sqrt n + \epsilon + 1} \\ & = \frac{(\sqrt n + \sqrt n\epsilon + n) - (\sqrt n -1)\epsilon}{\sqrt n + \epsilon + 1} \\ & = \sqrt n - \frac{(\sqrt n -1)\epsilon}{\sqrt n + \epsilon + 1} \end{align}$$

Disregarding the !!\epsilon!! in the denominator we obtain $$f_n(\sqrt n + \epsilon) \approx \sqrt n - \frac{\sqrt n - 1}{\sqrt n + 1} \epsilon $$

The error term !!-\frac{\sqrt n - 1}{\sqrt n + 1} \epsilon!! is strictly smaller than the original error !!\epsilon!!, because !!0 < \frac{x-1}{x+1} < 1!! whenever !!x>1!!. This shows that the fixed point !!\sqrt n!! is attractive.

In the previous articles I considered several different simple functions that had fixed points at !!\sqrt n!!, but I didn't think to consider this unusally simple one. I said at the time:

I had meant to write about Möbius transformations, but that will have to wait until next week, I think.

but I never did get around to the Möbius transformations, and I have long since forgotten what I planned to say. !!f_n!! is an example of a Möbius transformation, and I wonder if my idea was to systematically find all the Möbius transformations that have !!\sqrt n!! as a fixed point, and see what they look like. It is probably possible to automate the analysis of whether the fixed point is attractive, and if not to apply one of the transformations from the previous article to make it attractive.

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