02/19/2009, 12:10 AM
(This post was last modified: 02/19/2009, 05:37 PM by sheldonison.)

I did a little more work, enough to convince me that using the taylor series posted by Kouznetsov in this_post gives different results different results from what I'm getting. I think that means converting between tetration bases, even for very large numbers, is never going to give an exact constant value, but "wobbles" a little bit. The values I derived for base "e" below, are pretty smooth even out to the 4th derivative, which can be estimated by using linear difference equations. The 3rd order polynomial curve fit I'm using for base 1.485 should be accurate to more than 3 significant digits, I would guess it would be accurate to 5 or more digits.

converting from base 1.485 to base "e"

x my_value correct_value error term

-1.00 0.00000 0.00000 +0.00000

-0.95 0.05383 0.05382 +0.00001

-0.90 0.10624 0.10626 -0.00002

-0.85 0.15747 0.15754 -0.00008

-0.80 0.20770 0.20786 -0.00016

-0.75 0.25713 0.25739 -0.00026

-0.70 0.30593 0.30630 -0.00037

-0.65 0.35426 0.35473 -0.00047

-0.60 0.40226 0.40283 -0.00056

-0.55 0.45009 0.45073 -0.00064

-0.50 0.49788 0.49856 -0.00069

-0.45 0.54575 0.54645 -0.00070

-0.40 0.59383 0.59451 -0.00068

-0.35 0.64223 0.64286 -0.00063

-0.30 0.69108 0.69163 -0.00055

-0.25 0.74049 0.74094 -0.00045

-0.20 0.79056 0.79090 -0.00034

-0.15 0.84143 0.84166 -0.00023

-0.10 0.89321 0.89333 -0.00013

-0.05 0.94602 0.94606 -0.00005

+0.00 1.00000 1.00000 +0.00000

I also tried going backwards, from the published correct tetration values for base e, to base 1.485. The wobble going in the "backward" direction was much much more pronounced, and even the first derivative curve for base 1.485 already showed defects in it when graphed.

converting from base 1.485 to base "e"

x my_value correct_value error term

-1.00 0.00000 0.00000 +0.00000

-0.95 0.05383 0.05382 +0.00001

-0.90 0.10624 0.10626 -0.00002

-0.85 0.15747 0.15754 -0.00008

-0.80 0.20770 0.20786 -0.00016

-0.75 0.25713 0.25739 -0.00026

-0.70 0.30593 0.30630 -0.00037

-0.65 0.35426 0.35473 -0.00047

-0.60 0.40226 0.40283 -0.00056

-0.55 0.45009 0.45073 -0.00064

-0.50 0.49788 0.49856 -0.00069

-0.45 0.54575 0.54645 -0.00070

-0.40 0.59383 0.59451 -0.00068

-0.35 0.64223 0.64286 -0.00063

-0.30 0.69108 0.69163 -0.00055

-0.25 0.74049 0.74094 -0.00045

-0.20 0.79056 0.79090 -0.00034

-0.15 0.84143 0.84166 -0.00023

-0.10 0.89321 0.89333 -0.00013

-0.05 0.94602 0.94606 -0.00005

+0.00 1.00000 1.00000 +0.00000

I also tried going backwards, from the published correct tetration values for base e, to base 1.485. The wobble going in the "backward" direction was much much more pronounced, and even the first derivative curve for base 1.485 already showed defects in it when graphed.