10/15/2007, 08:36 PM

This problem is something I could easily see occupying us for a while, as far as trying to really, deeply understand it all. While I wasn't surprised, I suppose I should phrase it more as I wasn't "shocked". For bases between 1 and eta, I would have the least hoped for the solutions to the upper and lower fixed points to be consistent with each other. But given:

1) My change of base formula does not give identical results with "good old fashioned" continuous iteration from fixed points, and

2) Andrew's slog, at least for base e, appears to be loosely based on continuous iteration only from the primary fixed points (and hence the other fixed points would almost certainly give different results),

it does not come as a total shock that continuous iteration from two different real fixed points of a base between 1 and eta gives two different results.

I'm quite frustrated by this, because it adds to the problem of defining a unique solution. Yes, we already knew that we could take "the" solution and distort it by applying a cyclic, infinitely differentiable transform to the input. But it would at least have been nice if there were one solution that stood out as the obviously correct solution for a given base. So far, the only base I've seen this be true for is base eta, with parabolic iteration, which has essentially been solved for years if not decades already.

Andrew's slog does appear to yield the very nice property of "total monotonicity" for base e, but given how much accuracy was needed to show differences for the upper and lower fixed points of base sqrt(2), I'm tempted to go back with my newer, far more accurate power series for the slog and recalculate the first few hundred derivatives, to make sure the property still appears to hold.

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?

1) My change of base formula does not give identical results with "good old fashioned" continuous iteration from fixed points, and

2) Andrew's slog, at least for base e, appears to be loosely based on continuous iteration only from the primary fixed points (and hence the other fixed points would almost certainly give different results),

it does not come as a total shock that continuous iteration from two different real fixed points of a base between 1 and eta gives two different results.

I'm quite frustrated by this, because it adds to the problem of defining a unique solution. Yes, we already knew that we could take "the" solution and distort it by applying a cyclic, infinitely differentiable transform to the input. But it would at least have been nice if there were one solution that stood out as the obviously correct solution for a given base. So far, the only base I've seen this be true for is base eta, with parabolic iteration, which has essentially been solved for years if not decades already.

Andrew's slog does appear to yield the very nice property of "total monotonicity" for base e, but given how much accuracy was needed to show differences for the upper and lower fixed points of base sqrt(2), I'm tempted to go back with my newer, far more accurate power series for the slog and recalculate the first few hundred derivatives, to make sure the property still appears to hold.

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?

~ Jay Daniel Fox