Iteration exercises: f(x)=x^2 - 0.5 ; Fixpoint-irritation...

[sheldonison]

...
1) The Carleman-matrix is always based on the power series of a function f(x) and more specifically of a function g(x+t_0)-t_0 where t_0 is the attracting fixpoint for the function f(x). For that option the Carleman-matrix-based and the serial summation approach evaluate to the same value.             

2) But for the other direction of the iteration series, with iterates of the inverse function f^[-1] () we need the Carleman matrix developed at that fixpoint t_1 which is attracting for f^[-1](x) ...

So with the correct adapation of the required two Carleman-matrices and their Neumann-series we reproduce correctly the iteration-series in question in both directions.         

Gottfried

Is there a connection between the Carlemann-matrix and the Schröder's equation, \( \Psi(z)\;\;\;\Psi(f(z))=\lambda\cdot\Psi(z) \)?  Here lambda is the derivative at the fixed point; \( \lambda=f'(t_0) \), and then the iterated function g(x+1)= f(g(x)) can be generated from the inverse Schröder's equation:  \( g(z)=t_0+\Psi^{-1}(\lambda^z) \)

Does the solution to the Carlemann Matrix give you the power series for \( \Psi^{-1} \)?
I would like a Matrix solution for the Schröder's equation.  I have a pari-gp program for the formal power series for both \( \Psi,\;\;\Psi^{-1} \), iterating using Pari-gp's polynomials, but a Matrix solution would be easier to port over to a more accessible programming language and I thought maybe your Carlemann solution might be what I'm looking for