fixed point formula

[sheldonison]

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Another benefit of a DFT (compared to, say, Lagrange or Newton interpolation), is that it uses considerably less memory, and it's quite a bit faster if you use an FFT (Fast Fourier Transform). I'll see what I can put together over the next few days.
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Okay, here's a somewhat stable version of my DFT/FFT library, a slightly updated version of the fixed point finder, and a piece of code that calculates the fixed point Taylor series using a DFT (i.e., a "Cauchy integral").

I was able to calculate 512 terms to 96 digits of precision, in just a couple seconds. It took my other code (closed-form calculation of terms, using reversion of a power series) over 2 minutes to get the same result. I then calculated 1600 terms, again in only a few seconds. It took nearly two days using my other code.

Needless to say, in this specific scenario, the DFT method is extremely fast. And just as accurate.
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Note the performance of the FFT. I can use thousands of samples and still get a result in a few seconds. Heck, I can do tens of thousands of samples and still get a result in a reasonable timeframe. I did 8! (40320) samples in 11 seconds at 67 digits of precision, or 30 seconds at 144 digits.

I missed the posts in the computation section! This looks like great fun. I've never written an FFT, but I would like to. Its just that a generic DFT works pretty good within reasonable radius of convergence scenarios.

Back to the original topic, generating the fixed points from the \( \ln(\ln(b))+1 \). The \( k=\ln(\ln(b))+1 \) form is also very interesting because it means the tetration itreration function switches from iterating \( z \mapsto b^z \) to iterating \( z \mapsto \exp(z)-1+k \). Here, \( \text{sexp}_k(z) \mapsto \text{sexp}_b(z)\cdot \ln(b)+\ln(\ln(b)) \)

Another comment. Sometime after this post, I did succeed in getting a complex base tetration algorithm to work; its posted online here http://math.eretrandre.org/tetrationforu...hp?tid=729. But it doesn't converge for bases near the Shell Thron boundary. Its also dog slow for bases close to \( \eta=\exp(1/e) \). So I'm thinking about a tetration algorithm that iterates generating the slog, centered exactly between the two fixed points, using the "k" form ... and using the very same fixed point Taylor series as from above; well actually the \( L_k \mapsto \ln(\ln(L_b))+1 \). The algorithm I have in mind should have convergences of \( n^{-r} \), where I think r should be the real(pseudo period+1), so a 500 Taylor series terms for base e would be accurate to 15 decimal digits or so. Smaller values of k (base closer to eta) have larger real periods.... An FFT would help because it allow for thousands of terms instead of hundreds of terms.... So I may eventually need your FFT.