fractional iteration/another progress Gottfried Ultimate Fellow Posts: 889 Threads: 130 Joined: Aug 2007 07/24/2008, 02:25 PM Hi - I am currently reviewing the problem of divergence of powerseries (for the fractional iteration of Ut°h(x), but real bases). First I produced some plots with few different bases to get a better overview over the general characterisitcs of the divergence of the powerseries. Today I found a start to overcome that nasty type of divergence: using a Stirling-transform (as I call it at the moment) of the powerseries I get nicely convergent series even for the fractional case (just had only time to check this for one (divergent) base b=exp(2)). This is a big progress for my implementation of diagonalization! Look at http://go.helms-net.de/math/tetdocs/Coef...Height.htm Gottfried Gottfried Helms, Kassel Gottfried Ultimate Fellow Posts: 889 Threads: 130 Joined: Aug 2007 07/24/2008, 10:47 PM Gottfried Wrote:Look at http://go.helms-net.de/math/tetdocs/Coef...Height.htm Just updated the text. The said "Stirling-transform" seems to stretch the radius of convergence for x from zero - perhaps to infinity, at least (and surprising) for bases b(="t") > 2 . It seems to be not useful for classical bases like b=exp(1) and for the range below 2 (I'll have to investigate that more) Gottfried Gottfried Helms, Kassel Gottfried Ultimate Fellow Posts: 889 Threads: 130 Joined: Aug 2007 07/26/2008, 02:13 PM Gottfried Wrote:Gottfried Wrote:Look at http://go.helms-net.de/math/tetdocs/Coef...Height.htm Just updated the text. The said "Stirling-transform" seems to stretch the radius of convergence for x from zero - perhaps to infinity, at least (and surprising) for bases b(="t") > 2 . It seems to be not useful for classical bases like b=exp(1) and for the range below 2 (I'll have to investigate that more) GottfriedAnother update. Some more explixite example and improving of plots. Hellgate - in summer... Gottfried Gottfried Helms, Kassel « Next Oldest | Next Newest »

 Possibly Related Threads… Thread Author Replies Views Last Post Calculating the fractional iterates of $$f(z) = e^z-1$$ to reasonable accuracy JmsNxn 6 594 11/19/2022, 04:26 AM Last Post: JmsNxn fractional iteration with complex bases/a bit of progress Gottfried 1 7,048 07/21/2008, 10:58 PM Last Post: Gottfried

Users browsing this thread: 1 Guest(s)