08/15/2007, 08:54 AM
Since I don't have Mathematica, could you by chance post graphs of the root test to 150 or 200 terms for half-iterates (1/2 for starters, -1/2, etc., if not too much trouble)? I'm curious to see if it continues what appears to be a linear climb. I've downloaded Paritty and am hoping to learn how to use it over the next few days, but it'll be a while before I'm up and running.
By the way, the non-infinite radius of convergence for negative half-iterates is to be *expected*, because those involve partial logarithms, and those will have a radius of convergence (as can be seen by the iterate -1).
Also, looking at the graph for the 1/2 iterate, it would seem that the first 50 terms should behave convergently for z values less than 0.6 or so. Considering that I plan to use z values less than 0.01, this is way more than sufficient for several hundred digits of accuracy, especially if I go out to 100 terms or so.
By the way, the non-infinite radius of convergence for negative half-iterates is to be *expected*, because those involve partial logarithms, and those will have a radius of convergence (as can be seen by the iterate -1).
Also, looking at the graph for the 1/2 iterate, it would seem that the first 50 terms should behave convergently for z values less than 0.6 or so. Considering that I plan to use z values less than 0.01, this is way more than sufficient for several hundred digits of accuracy, especially if I go out to 100 terms or so.
~ Jay Daniel Fox