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Numerical algorithm for Fourier continuum sum tetration theory
The problem is the initial guess. I mentioned in the main posts:

Quote:It seems that with a large number of nodes, the convergence with this initial guess only works for small bases like base 2 -- not sure what's going on there, but that can be used as an initial guess for a wide variety of other bases.

I wasn't specific enough -- the base has to be real small, even e seems too large, at this initial guess. Not sure why this is. Try a loop or two on base 2 first, then use the result of that as the initial guess to run base e. On my system, a single loop at base 2, followed by six loops at base e, using the parameters you give, yields tet(pi) ~ 3.716e10, which is closer to the expected value of 3.71498e10. You'll probably need more than 81 nodes and 32*I period to get more accuracy than that.

Yes, the Kneser process seems to yield more precision more quickly, but is restricted only to real bases (where the conjugate symmetry holds).

ADDENDUM: Actually, it does work in your case, but with more nodes than what you used, it won't. You just need to iterate it a lot more times -- try 30 or so.

Messages In This Thread
RE: Numerical algorithm for Fourier continuum sum tetration theory - by mike3 - 09/17/2010, 10:10 AM

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