11/02/2007, 08:30 PM

jaydfox Wrote:And getting back to the line of inquiry that started this thread, all of this begs another question: which of the two fixed points for base sqrt(2) gives the "correct" solution? Or is it something else, perhaps roughly between the two? Or is there no definitively "correct" solution, just a collection of solutions which satisfy basic properties?

If I was asked, we need the lower fixed point, because only this is reachable by .

However the real dilemma is as I already pointed out, that every other (analytic) iteration than the regular iteration at the lower and the regular iteration at the upper fixed point must be singular at both fixed points. And for each analytic iteration one of the fixed points is a singularity.

In this line I would be really interested in the difference of Andrew's slog with the regular Abel function at the lower fixed point.

The formula for the latter, derived from here, is:

, where is the lower fixed point of .

Can you make a comparison of with Andrew's computed by your super sophisticated algorithm and post a graph of the difference in the range say [-1,1.9] somewhere?!

If the difference turns into a smooth curve starting from some precision then we know they are different, if the result is at any precision rather a random curve this would favour the equality of both solutions.