I think I've resolved my brain-scattering concerning the ordering in my previous post now. It was just confusion, when I tried to get access to the understanding of the sequence of increasing heights "from the scratch" by composing appropriate and "near-appropriate" functions.
Anyway, what is still surprising, is, that the \( \sigma \)-function seems to have an exponential characteristic. I'd expected something near a logarithmic characteristic instead.
This is \( \sigma(x) \) or "p_b(x)" (as denoted in the previous post) for b~1.345 (u=0.5, t=exp(u), b=exp(u/t)=t^(1/t))
Strange...
Gottfried
[update]: Plot updated ,code updated for better readability [/update]
Anyway, what is still surprising, is, that the \( \sigma \)-function seems to have an exponential characteristic. I'd expected something near a logarithmic characteristic instead.
This is \( \sigma(x) \) or "p_b(x)" (as denoted in the previous post) for b~1.345 (u=0.5, t=exp(u), b=exp(u/t)=t^(1/t))
Strange...
Gottfried
[update]: Plot updated ,code updated for better readability [/update]
Code:
n=96 \\ set matrix-size; 96 terms for powerseries
fmt(200,12) \\ set internal float-precision to 200 dec digits
\\ set base-parameter b
[u = 0.5,t=exp(u),b=exp(u/t),bl = u/t] \\ b = t^(1/t)
\\ the carleman-matrix Bb is not needed, instead we use the analytically
\\ determined Bell-matrix UtA of the decomposition Bb = (P^(-t))~ * UtA * (P^t)~
\\ This decomposition is only correct for the untruncated case, so we have to determine
\\ UtA analytically
\\ create Bell-matrix for regular iteration b^x using fixpoint t
Ut = dV(u)*fS2F; \\ analytically; from factorially scaled Stirlingnumbers 2'nd kind
\\ Ut performs x -> t^x - 1
\\ uses fixpoint-shift " x'=x/t-1 "
UtA = dV(1/t)* Ut * dV(t); \\ modifies the fixpoint-shift to " x'=x-t "
\\ so UtA performs x -> b^(x+t) - t
\\ diagonalization of Bell-matrix, returns parameters and X,dV(u),X^-1
UtAKenn = APT_Init2EW(UtA);
X = UtAKenn[2]; \\ the eigenvector-matrix of UtA
\\ note, that the eigenvector-matrix W of the Carleman-matrix Bb is W= P~^t * X
\\ Schroeder-function-plot
evec=ESum(2.5)[n,]; \\ create coefficients-vector for Eulersummation order 2.5
\\ this was checked against smaller orders, differences occur 3.2<x<5 (1e-9 <diff<1e6)
sigma(x,x0=t) = evec*dV(x-x0)*X[,2] \\ define Schroeder function (perform implicitely fixpoint-shift using X=UtAKenn[2])
\\ uses also Euler-summation to extend range for x
list = matrix(n,2); \\ n=96
for(k=0,n-1,k1=1.0*k/10-3;list[1+k,]=[k1,sigma(k1)])
%box >plot list \\ from box "plot" transfer coordinates to Excel "Schroederfunction.xls" in csv-format
Gottfried Helms, Kassel