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Comparing the Known Tetration Solutions
#12
Unfortunately, my method and Andrew's would appear to give different results. The differences are fairly small, at least close to the origin. Obviously, even a tiny discrepancy gets magnified tremendously after a few exponentiations. The data points you provided weren't precise enough for me to calculate a usable fifth derivative, which is where the differences between our methods really seem to show up. And to be honest, Andrew's fourth derivative looks a little better than mine (aesthetically, if that makes sense). I'd like to see graphs of the fifth derivative of Andrew's version.

For comparison, here is my graph, along with the first five derivatives. Blue, light-blue, and sea green are the 0th, 2nd, and 4th derivatives, while red, orange, and yellow-orange are the 1st, 3rd, and 5th derivatives, respectively.

   

To be honest, I'm less than satisfied with the results. It's not numerical inaccuracies: my numbers are "precise" to about 100 decimal digits. By precise I mean higher numerical precision in the calculations won't change the results if we only look at the first 100 digits. Accuracy might be another matter altogether.

Anyway, could someone generate a dataset for Andrew's numbers, sufficient to extract the 5th derivative? More data points, and at least 13-15 digits of precision?

And until I figure out how to output the results to a text file, here are a few data points of interest:

0.5, 1.64515080754212070699721
e, 2058.05985438912517767154
pi, 36940638694.936515311638
~ Jay Daniel Fox
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Messages In This Thread
RE: Comparing the Known Tetration Solutions - by jaydfox - 08/20/2007, 09:59 AM
RE: computing the iterated exp(x)-1 - by andydude - 08/17/2007, 11:20 PM
RE: computing the iterated exp(x)-1 - by jaydfox - 08/17/2007, 11:38 PM
RE: computing the iterated exp(x)-1 - by bo198214 - 08/17/2007, 11:45 PM
RE: computing the iterated exp(x)-1 - by jaydfox - 08/18/2007, 12:19 AM
RE: computing the iterated exp(x)-1 - by bo198214 - 08/18/2007, 08:19 AM
RE: computing the iterated exp(x)-1 - by andydude - 08/18/2007, 09:35 AM
RE: computing the iterated exp(x)-1 - by bo198214 - 08/18/2007, 11:59 AM
RE: computing the iterated exp(x)-1 - by jaydfox - 08/18/2007, 03:49 PM
RE: computing the iterated exp(x)-1 - by jaydfox - 08/19/2007, 12:50 AM

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