from formulas to pari implementations

[Xorter]

Hmm, I don't really get it what you are after.      
For one: perhaps my very introductiory essay ContinuousfunctionalIteration  is enough to explain the method which leads to the Carleman-method and Schroeder-function (where I did not know in my first writing that already known/used terms and concepts and thus had rediscovered them on my own), and for second, this is all done for examples  with a not too large set of procedures (mostly matrix-functions) in Pari/GP.

I read your paper, it is great, understandable. Thank you for sharing.
In fact, this year in january I have (re)invented an iteration method which is actually the functional form of 2.1.5. b) but I discarded the idea because I misunderstood it. It was the following:
(y^ox')(x) = oexp(x' × olog(y(x)))
Substitute the nullelement to x and x to x', then you give the superfunction of y(x), where
oexp(y) = lim (x + y/h)^oh = sum from k=0 to infinity y^ok / k!
olog(y) = lim h×(y^o(1/h) - x) = sum from k=1 to infinity (-1)^(k+1) y^ok / (k+1)

And I now know that [ oexp(y) ] = exp [y] and [ olog(y) ] = log [y], so I was right. But I thought that:
oexp^ot(x' × olog^ot (y)) is the t-th superfunction of y and t could be real or even complex. It was a mistake, and this is why I had beleived I was wrong until now, despite I could calculate some superfunctions by hand.

Lets try your methods with trigonometric function in no pari/gp but Mathematica because I could not write the code of matrix logarithm in pari/gp.
Here are some codes and plots: definitions, sucos (eigen decomposition), susin (matrix log method) and some wrong results.

It seems almost correct but it is not. Some idea to fix it?