Consistency in the composition of iterations

[Daniel]

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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? Consider \( f(0)=0 \), then the second derivative of \( f^m(z) \) is always is true regardless of whether Schoeder or Abel is used.

For \( f^m(z) \) we have \( D^2 f^m(z)= \(\sum_{k=0}^{m-1}f'(0)^{2m-k-2}\)f''(0) \).