02/22/2009, 10:20 PM

Hi -

as I mentioned in the earlier thread

http://math.eretrandre.org/tetrationforu...hp?tid=186

the idea of series of powertowers, which define fractional heights, excited me much. After the translation of the Binomial-formula into my matrix-concept there was no progress in this; but after some fiddling last days, it seems, that I found the key to translate and thus to apply at least the core concepts in the "matrix-method" also to such powertower-series.

I'll call them from now more generally "iteration-series" because they also occured in some iteration-exercises with iterable functions other than tetration.

I defined in my matrix-concept the vandermonde-vector V(x) as infinite vector of powers of x and got this way a very handy way to formulate functions of powerseries, compositions and iterations, as the forum-fellows know already (at least my opinion in this regard :cool.

While

and have called this "Vandermonde-vector" I define now

as "iteration-vector" where the related function f(x) shall be defined in the near context.

If it is needed to start at a different h_0 then I extend the definition to

so a change of h_0 to, say 1/2 changes the stepwidth to 1/2 (and the second element of the vector has the value of f°(1/2)(x))

--------------------------------------------

I cannot yet post the full procedere of my computations of last days; but the first main result is that, similarly to the nice binomial-formula (Newton/Woon/Henryk...:-) ) I can show that the computation of fractional heigths can be done based on IT-vectors/-series the same way as fractional powers using logarithms, let me call it here: the function-logarithm- or stirling-formula.

Instead of fractional binomials, as in the binomial-formula, I can use the Stirling-numbers (which define the logarithmic- and exponential-series) on IT-series/-vectors, as one would do this with powerseries/Vandermonde-vectors.

I do not yet recognize, whether this gives an improvement of convergence compared to the binomial-formula for fractional iterates, I'll see tomorrow.

But unfortunately, it lacks the same problem as the binomial-formula, that if we want to use an IT-(iteration)-series, then the divergence-problem, if it exists, gets us very early, and with little hope to have a faster approximation-/summation-method.

It's late, I'll continue this tomorrow.

Gottfried

as I mentioned in the earlier thread

http://math.eretrandre.org/tetrationforu...hp?tid=186

the idea of series of powertowers, which define fractional heights, excited me much. After the translation of the Binomial-formula into my matrix-concept there was no progress in this; but after some fiddling last days, it seems, that I found the key to translate and thus to apply at least the core concepts in the "matrix-method" also to such powertower-series.

I'll call them from now more generally "iteration-series" because they also occured in some iteration-exercises with iterable functions other than tetration.

I defined in my matrix-concept the vandermonde-vector V(x) as infinite vector of powers of x and got this way a very handy way to formulate functions of powerseries, compositions and iterations, as the forum-fellows know already (at least my opinion in this regard :cool.

While

and have called this "Vandermonde-vector" I define now

as "iteration-vector" where the related function f(x) shall be defined in the near context.

If it is needed to start at a different h_0 then I extend the definition to

so a change of h_0 to, say 1/2 changes the stepwidth to 1/2 (and the second element of the vector has the value of f°(1/2)(x))

--------------------------------------------

I cannot yet post the full procedere of my computations of last days; but the first main result is that, similarly to the nice binomial-formula (Newton/Woon/Henryk...:-) ) I can show that the computation of fractional heigths can be done based on IT-vectors/-series the same way as fractional powers using logarithms, let me call it here: the function-logarithm- or stirling-formula.

Instead of fractional binomials, as in the binomial-formula, I can use the Stirling-numbers (which define the logarithmic- and exponential-series) on IT-series/-vectors, as one would do this with powerseries/Vandermonde-vectors.

I do not yet recognize, whether this gives an improvement of convergence compared to the binomial-formula for fractional iterates, I'll see tomorrow.

But unfortunately, it lacks the same problem as the binomial-formula, that if we want to use an IT-(iteration)-series, then the divergence-problem, if it exists, gets us very early, and with little hope to have a faster approximation-/summation-method.

It's late, I'll continue this tomorrow.

Gottfried

Gottfried Helms, Kassel