08/16/2007, 05:24 AM
Nice, I'm definitely learning about this uniqueness/convergence/analiciy thing. I guess the series expansions of a flow based on fixed-points does not need to converge for the flow to be analytic elsewhere. If this is true, and an analytic iterate can be found by some other means, then can we perhaps derive an accuracy function?
An accuracy function would be a function E(n) associated with the flow of a particular function f(x) such that:
This would allow us to use the series to compute the flow of a function instead of whatever other methods we might use to compute the more "accurate" version. We could use this even when the series for the flow does not converge (which appears likely for many functions). Since it is stated in terms of a function of t instead of a polynomial of t, this method should work for hyperbolic iteration series as well as parabolic iteration series.
Andrew Robbins
An accuracy function would be a function E(n) associated with the flow of a particular function f(x) such that:
\( E(n) > \left| f^{[t]}(x) - \sum_{k=1}^{n}f_k(t) x^k \right| \)
for all |t|<1, |x|<1 or something.This would allow us to use the series to compute the flow of a function instead of whatever other methods we might use to compute the more "accurate" version. We could use this even when the series for the flow does not converge (which appears likely for many functions). Since it is stated in terms of a function of t instead of a polynomial of t, this method should work for hyperbolic iteration series as well as parabolic iteration series.
Andrew Robbins