Well, Henryk,

that opended eyes, thanks. With the following parametrization I got the difference.

Starting with the general

x' = x/e-d, x"=(x+d)*e, e*d=fixpoint

and fixing e=1

e= 1 , d0=t0=fixpoint_0, d1=t1=fixpoint_1

and development of the matrices accordingly I got differences, too. So thanks for the hints!

Now I'd like to understand, what is the source of the difference of selection e=1 vs d=1. The most obvious difference was in the given case

f(x) = x^2 + x - 1/16 = x + (x - 1/4) (x + 1/4) = x + (x-t0)*(x-t1)

that I kept the eigenvalues in both developments equal. Maybe my transformation introduces somehow a triviality - but note, that in my tetration-examples I also used d=1, e0=t0 versus e1=t1 for the definition of the triangular matrices, just this selection, that you say is "illegal"

I tried with a general definition of a quadratic function (fixpoints p,q)

f(x) = x + (x - p) (x - q) = x^2 + (1 - (p+q))x + pq

g(z) = a*z^2 + b*z

z = x/e-d

f(x) = (g(z) + d)*e

By comparing coefficients at powers of x in f(x) and g(z) I get conditions for a and b in g(z):

a = e

b = 1 - (p+q) + 2de , b-1 = 2de - (p+q)

and then for d and e the condition occurs

(de - p) (de - q) = 0

where either e or d is free.

It seems then, if b equals for each selection of fixpoints, such that either de=p or de=q and the resulting b's are equal, then the resulting powerseries seem to be compatible and give equal results (I've to check that with more examples)

But again: maybe this introduces some triviality... On the other hand: it is intriguing, that using d=1 , e=fixpoint is the appropriate selection to convert the tetration-matrices into triangular matrices for decremented iterated exponentiation.

Hmmm.

Gottfried

that opended eyes, thanks. With the following parametrization I got the difference.

Starting with the general

x' = x/e-d, x"=(x+d)*e, e*d=fixpoint

and fixing e=1

e= 1 , d0=t0=fixpoint_0, d1=t1=fixpoint_1

and development of the matrices accordingly I got differences, too. So thanks for the hints!

Now I'd like to understand, what is the source of the difference of selection e=1 vs d=1. The most obvious difference was in the given case

f(x) = x^2 + x - 1/16 = x + (x - 1/4) (x + 1/4) = x + (x-t0)*(x-t1)

that I kept the eigenvalues in both developments equal. Maybe my transformation introduces somehow a triviality - but note, that in my tetration-examples I also used d=1, e0=t0 versus e1=t1 for the definition of the triangular matrices, just this selection, that you say is "illegal"

I tried with a general definition of a quadratic function (fixpoints p,q)

f(x) = x + (x - p) (x - q) = x^2 + (1 - (p+q))x + pq

g(z) = a*z^2 + b*z

z = x/e-d

f(x) = (g(z) + d)*e

By comparing coefficients at powers of x in f(x) and g(z) I get conditions for a and b in g(z):

a = e

b = 1 - (p+q) + 2de , b-1 = 2de - (p+q)

and then for d and e the condition occurs

(de - p) (de - q) = 0

where either e or d is free.

It seems then, if b equals for each selection of fixpoints, such that either de=p or de=q and the resulting b's are equal, then the resulting powerseries seem to be compatible and give equal results (I've to check that with more examples)

But again: maybe this introduces some triviality... On the other hand: it is intriguing, that using d=1 , e=fixpoint is the appropriate selection to convert the tetration-matrices into triangular matrices for decremented iterated exponentiation.

Hmmm.

Gottfried

Gottfried Helms, Kassel