Consistency in the composition of iterations

[JmsNxn]

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EDIT2:

Oh so it means Daniel was referring to series precision of the Taylor series.

I'll answer again.

This is a bit more difficult using the beta method, it'll start to lag about 100 terms, but I can usually get about \(O(z^{100})\) and that's usually what I shoot for to double check everything. And to do that you have to set the series precision usually larger (some terms start turning to junk). With Schroder, it's again, desired accuracy. So technically you could get it to \(O(z^{1000})\) but be prepared to eat like 10 gbs of ram, lol.

I usually set my series precision to get at like 50-100, depending on what I'm running, then double check that it's continuing to work at something ridiculous like 200-300. But that's the final stages because it's super slow and eatttsss ram like the cookie monster.

My precision and accuracy are guaranteed to be prefect because I only use symbolic and rational numbers in my calculations. Are you using floating point?

No, I am not using floating point. I'm using pari, which is symbolic, but you'll still get some garbage which comes out in the 100th digit or so, it's just a consequence of any calculator. 

For example, if I take the Schroder function:

\[
\Psi(z)\\
\]

I can set arbitrary precision, and arbitrary series precision (how many terms to the Taylor series), but it will create some garbage data. This is also because I try to optimize the speed, I could technically make it absolutely 100% lossless, but who has time to watch that compile. Then, you have to series invert this, which is not lossless, despite how good Pari is, this is a polynomial inversion, so it can lose a bit of data. Then when you actually plug in the iteration:

\[
f^{\circ s}(z) = \Psi^{-1}(\lambda^s \Psi(z))\\
\]

You do lose a good amount of data, especially for large \(s\) (negative or positive depending on if \(|\lambda|\) is greater/less than 1) and \(z\), but you can recover it by keeping \(z \approx L\), and iterating the functional equation. But still, you will lose data. I could program my code to make sure it's 100% no errors, but, again, it'd be too slow and impractical.