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+--- Thread: Comparing the Known Tetration Solutions (/showthread.php?tid=29)
RE: Comparing the Known Tetration Solutions - Gottfried - 08/29/2007
jaydfox Wrote:Using gp, I computed Andrew's solution for base e, using a 50x50 matrix. (Side question: Andrew, how much faster are other libraries at solving these large matrices?)Jay,
could you provide me your gp-code for comparision? You may put it here or send me an email.
Also I would like to check, whether your last computations can be smoothed by accelerating convergence using Euler-summation of terms. But I've only Pari/GP and maxima(the latter with extremely little experience) and I don't know how complex your computations were and how much work it would be to get it implemented.
What do you think?
Gottfried
RE: Comparing the Known Tetration Solutions - bo198214 - 08/29/2007
@Jay
Wow, this is indeed interesting.
I mean theoretically it is clear that \( h(x):=f^{-1}(g(x))-x \) must have period 1 for two (strictly increasing continuous) solutions \( f \) and \( g \) of \( \phi(x+1)=F(\phi(x)) \).
Because \( h(x+1)=f^{-1}(g(x+1))-(x+1)=f^{-1}(F(g(x))-(x+1)=f^{-1}(g(x))+1-(x+1)=h(x) \).
But to see it as graphs is another thing
.And I would think that it is in no way a sinus wave. However the periodic analytic functions can built by Fourier series.
RE: Comparing the Known Tetration Solutions - andydude - 08/29/2007
Several things. I have never used pari/gp but I have 3 implementations of my super-log code and Carleman-matrix stuff in languages other than Maple/Mathematica:
- C with GMP (super-logarithm)
- Perl with BigInt (super-logarithm)
- Maxima (for Carleman-matrix)
However, I'm not sure I know where the first two are, but I just found my maxima code, so I can post that soon.
On another note, I like these graphs, although I still don't know how to use Jay's method.
I can understand the oscillation, since the further you get away from integers the "less defined" tetration is... so it makes sense to me.
I would like to post some graphs of Daniel's and my extensions, and I think I'll stick with the (critical) interval -1<x<0 since its the most well-behaved, and use the close-to-zero plots for \( {}^{x}b - (x+1) \) and \( slog_b(x) - (x-1) \) so that any oscillation is immediately obvious. My next post should have them, but I need more time to make the graphs.
Andrew Robbins
PS. That was a beautiful derivation Henryk
RE: Comparing the Known Tetration Solutions - jaydfox - 08/29/2007
Gottfried Wrote:jaydfox Wrote:Using gp, I computed Andrew's solution for base e, using a 50x50 matrix. (Side question: Andrew, how much faster are other libraries at solving these large matrices?)Jay,
could you provide me your gp-code for comparision? You may put it here or send me an email.
Also I would like to check, whether your last computations can be smoothed by accelerating convergence using Euler-summation of terms. But I've only Pari/GP and maxima(the latter with extremely little experience) and I don't know how complex your computations were and how much work it would be to get it implemented.
What do you think?
Gottfried
I posted my code here:
http://math.eretrandre.org/tetrationforum/showthread.php?tid=33&pid=213#pid213
I gave up on PARI/gp, the solve_right() function in SAGE (using maxima) was about 50-100 times faster. The matsolve function in gp must not be a low-level function, and I didn't bother looking for one once I got it working with maxima.