Well, I have constructed a function to find the values that GFR is looking for. The function is defined such that: \( y''(f(b)) = 0 \) where \( y(x) = {}^{x}b \). I have attached a plot of this function for the first few approximations (n=5..9) of tetraiton using the natural/inverse-slog tetration method. It looks as though this function is very well-behaved for \( b > e \), but is quite slow to converge for lower bases. So my guess is that GFR's conjecture that f(2)=0 is probably not true.
Two bases for which this function seems to converge quickly are \( (e, \pi) \), so I have included numerical data for these. I have been meaning to implement my own version of Jay's accelerated natural slog, but until I do that I can only use very small approximations.
@GFR
Finding such a differential equation would be amazing! But maybe we should model it after the differential equation of exponentiation:
Andrew Robbins
Two bases for which this function seems to converge quickly are \( (e, \pi) \), so I have included numerical data for these. I have been meaning to implement my own version of Jay's accelerated natural slog, but until I do that I can only use very small approximations.
\(
\begin{tabular}{c|ccc}
n & f_n(2) & f_n(e) & f_n(\pi) \\
\hline
5 & -0.2351 & -0.5129 & -0.6231 \\
6 & -0.1167 & -0.5037 & -0.6276 \\
7 & -0.0387 & -0.5012 & -0.6282 \\
8 & -0.0056 & -0.5108 & -0.6337 \\
9 & +0.0154 & -0.5170 & -0.6357
\end{tabular}
\)
\begin{tabular}{c|ccc}
n & f_n(2) & f_n(e) & f_n(\pi) \\
\hline
5 & -0.2351 & -0.5129 & -0.6231 \\
6 & -0.1167 & -0.5037 & -0.6276 \\
7 & -0.0387 & -0.5012 & -0.6282 \\
8 & -0.0056 & -0.5108 & -0.6337 \\
9 & +0.0154 & -0.5170 & -0.6357
\end{tabular}
\)
@GFR
Finding such a differential equation would be amazing! But maybe we should model it after the differential equation of exponentiation:
\( \frac{d}{dt}(x^y) = x^{y-1}\left(y \frac{dx}{dt} + x \ln(x) \frac{dy}{dt}\right) \)
although I think your idea has a better chance of working.Andrew Robbins
