[UFO] Attracting Fixpoints or attracting line?

[tommy1729]

Second - range of convergence is an important topic. If we use the powerseries for iteration and a point is too far from the fixpoint, we get divergence. So we have the option to use another fixpoint.
Say, base=sqrt(2). Then iterating x0=5 using the schröder-function to implement fractional iteration gives divergence, when the series is developed around the fixpoint 2. But that series developed around the other fixpoint 4 gives nicely converging sums.
Now the fixpoints are separated; if we can use some arbitrary value from the "fixed-line" for the construction of the powerseries, say the complex coordinate nearest to x (x=5 in the example), then -perhaps- we can improve convergence - don't know yet.

The whole idea is just an "UFO" - unidentified floating observation ;-) ; let's see, what we can do with it...

Gottfried

i think the range of convergence problem can be solved by using formal power series and then using mittag-leffler expansion for the ' actual numerical computation '.

regards

tommy1729