The problem is the initial guess. I mentioned in the main posts:
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 \( b > e^{1/e} \) (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.
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 \( b > e^{1/e} \) (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.