# Finding the root of a nested function with small values

I have reduced an error in my program to this line of code:

FindRoot[Nest[# (1 - #) k &, 1/2, 2^4] - 1/2, {k, 3.5}]


It works for $2^1, 2^2, 2^3, 2^4,$ but then for $2^n, n\ge5,$ it stops. Nothing happens. Mathematica is "thinking" forever. FindRoot is not doing a single iteration.

Why does this happen?

I have some ideas of my own, none of which have gotten me anywhere:

1. The values are very small, is it a precision problem?

2. Maybe I'm not using Nest properly?

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You do realize that you trying to find the root for polynomial of order 2^2^n? So for n=5 it's 4294967296 order! – swish Apr 30 '13 at 21:53
To speed things up a bit I suggest to Expand your expression and Chop the small parts at every nesting: Nest[Chop[Expand[# (1 - #) k]] &, 0.5, 2^4] - 0.5 – swish Apr 30 '13 at 21:57

Make it a purely numeric function so that FindRoot cannot do anything fancy with the symbolic form. This can be done as below.
ff[k_?NumberQ] := Nest[# (1 - #) k &, 1/2, 2^4]
(* Out[22]= {k -> 3.49856169933} *)
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