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
\( V(x) = colvector_{r=0}^{\infty} \lbrace x^r \rbrace \)
and have called this "Vandermonde-vector" I define now
\( IT(x) = colvector_{h=0}^{\infty} \lbrace f^{\circ h}(x) \rbrace \)
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
\( IT(x,h_0) = colvector_{h=0}^{\infty} \lbrace f^{\circ h_0 *h}(x) \rbrace \)
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
\( V(x) = colvector_{r=0}^{\infty} \lbrace x^r \rbrace \)
and have called this "Vandermonde-vector" I define now
\( IT(x) = colvector_{h=0}^{\infty} \lbrace f^{\circ h}(x) \rbrace \)
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
\( IT(x,h_0) = colvector_{h=0}^{\infty} \lbrace f^{\circ h_0 *h}(x) \rbrace \)
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
