Fun with TI-83: billions upon billions of cosines

Okay, maybe not billions. But by taking cosines repeatedly, one can find the solution of the equation {\cos x = x} with high precision in under a minute.

Step 1: Enter any number, for example 0, and press Enter.
Step 2: Enter cos(Ans) and press Enter

Step 2
Step 2

Step 3: Keep pushing Enter. (Unfortunately, press-and-hold-to-repeat does not work on TI-83). This will repeatedly execute the command cos(Ans).

Step 3
Step 3

After a few iterations, the numbers begin to settle down:

Convergence
Convergence

and eventually stabilize at 0.7390851332

Limit
Limit

Explanation: the graph of cosine meets the line {y = x} at one point: this is a unique fixed point of the function {f(x)=\cos x}.

Fixed point of f(x)=cos x
Fixed point of f(x)=cos x

Since the derivative {f'(x)=-\sin x} at the fixed point is less than 1 in absolute value, the fixed point is attracting.

Now try the same with the equation {10 \cos x =x}.

Beginning
Beginning

This time, the numbers flat out refuse to converge:

Chaos!
Chaos!

Explanation: the graph of {f(x)=10\cos x} meets the line {y = x} at seven point: thus, this function has seven fixed points.

Seven fixed points
Seven fixed points

And it so happens that {|f'(x)|>1} at each of those fixed points. This makes them repelling. The sequence has nowhere to converge, because every candidate for the limit pushes it away. All that’s left to it is to jump chaotically around the interval {[-10,10]}. Here are the first 1024 terms, plotted with OpenOffice:

There is some system in this chaos
There is a pattern in this chaos…

Clearly, the distribution of the sequence is not uniform. I divided the interval {[-10,10]} into subintervals of length {0.05} and counted the number of terms falling into each.

Distribution of terms
Distribution of terms

What is going on here? Stay tuned.

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