I'm confused...
What does \(O(n)\) mean, do you mean \(O(z^n)\)? Or do you mean the taylor coefficients grow like \(O(n)\), but then the \(O(constant)\) wouldn't make sense. I'm confused. Could you clarify?
Because:
\[
f^a(f^b(z)) - f^{a+b}(z) = 0\\
\]
And it equals this across all the taylor coefficients. So that:
\[
f^a(f^b(z)) = \sum_{j=0}^\infty c_j z^j\\
\]
\[
f^{a+b}(z) = \sum_{j=0}^\infty c_j z^j\\
\]
They're the same series...
Could you clarify what you mean?
EDIT:
Oh I just realized you meant digit precision! (At least I think you did)
I can usually cap out at about 200 digits using the beta method on the Shell-Thron region, it can vary a bit. Outside the Shell-thron region, the beta method isn't holomorphic (just C^\infty), so the taylor series doesn't converge, but I can still get about 200 digits accuracy for this functional equation. Using the standard Schroder method I can get desired accuracy, but it eats the hell out of my ram.
Using pari-gp basically lets you set precision, and if you program carefully it'll usually agree to the precision minus some odd digits. So If you want 300 digit precision, set about 350, and you should get the Taylor series converging to about 300. But then you should also have series precision really large...
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.
What does \(O(n)\) mean, do you mean \(O(z^n)\)? Or do you mean the taylor coefficients grow like \(O(n)\), but then the \(O(constant)\) wouldn't make sense. I'm confused. Could you clarify?
Because:
\[
f^a(f^b(z)) - f^{a+b}(z) = 0\\
\]
And it equals this across all the taylor coefficients. So that:
\[
f^a(f^b(z)) = \sum_{j=0}^\infty c_j z^j\\
\]
\[
f^{a+b}(z) = \sum_{j=0}^\infty c_j z^j\\
\]
They're the same series...
Could you clarify what you mean?
EDIT:
Oh I just realized you meant digit precision! (At least I think you did)
I can usually cap out at about 200 digits using the beta method on the Shell-Thron region, it can vary a bit. Outside the Shell-thron region, the beta method isn't holomorphic (just C^\infty), so the taylor series doesn't converge, but I can still get about 200 digits accuracy for this functional equation. Using the standard Schroder method I can get desired accuracy, but it eats the hell out of my ram.
Using pari-gp basically lets you set precision, and if you program carefully it'll usually agree to the precision minus some odd digits. So If you want 300 digit precision, set about 350, and you should get the Taylor series converging to about 300. But then you should also have series precision really large...
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.