The stuff in this directory involves computing the continued fraction expansion of the Feigenbaum constants delta and alpha. I compute the simple continued fraction expansion of each, e.g.

delta = a1 + 1/(a2 + 1/(a3 + ....

The program makes a plot of the the sequence of coefficients a1, a2, a3, ... I use the mpfr library to compute the expansions to high order.

I make a plot of a vs. order. If the continued fraction was periodic I would observe periodicity in the plot. However, I see none.

The python program uses the value of delta from Andrea Molteni's website to get the value of alpha and delta:

http://converge.to/feigenbaum/alpha.txt

http://converge.to/feigenbaum/delta.txt

I also compute the Khinchin constant as described in https://en.wikipedia.org/wiki/Khinchin%27s_constant and make a plot of the computed constant vs. number of convergents computed. I find the expansions of both alpha and delta converge to the Khinchin constant.
Both plots are shown below.

To test the program I have a facility to create high-precision representations of the square root of numbers, for example sqrt2, sqrt3, etc. Square roots yeild simple continued fraction expansions which repeat, and the repeating digits are known and tabulated (e.g. Wolfram MathWorld). Therefore, I can use the known coefficients from the continued fraction expansions of these quantities to test my program.

Stuart Brorson February 2021