Analytical results of Schröder's Equation According to the Wiki page on Schröder's equation, $\Psi(f(x)) = s\Psi(x)$ can be solved for $\Psi(x)$ analytically if there is a fixed point $a$ such that $0 < f'(a) < 1$.
That implies that there is an algorithm to put in an arbitrary $f(x)$ and $a$ and get an answer. What is this algorithm? I can't find Schröder's paper (which is in German anyway), and the Koenig paper linked there is in French. I can find similar questions, such as this one, but none dealing with a straightforward approach given a function with a fixed point satisfying the condition above. Some questions mention, "Use Schröder's method." That's my problem: what is Schröder's method, and how did he solve for the examples here?
 A: I don't know whether this is interesting for you, but when I started to look at functional iteration I did this with focus on powerseries and their evolution under iteration. Naturally this leads to a notation in matrices & matrix-powers, I learned later that such matrices are called "Carleman matrix" and are well known in the math-community. A function, having a power series, can be associated by such a Carleman matrix (simply containing the coefficients of the power series and their powers), and functional iteration then simply by powers of that Carleman matrices.
Fractional iteration means then fractional powers of that matrices and this leads to the concept of diagonalization, such that if $\mathcal C$ is the Carleman-matrix for a function $f(x)$, say $f(x) = 2^x - 1 = \exp(\ln(2) \cdot x)-1$ and by diagonalization $\mathcal C = \mathcal M \cdot \mathcal D \cdot \mathcal M^{-1}$ with $\mathcal D$ diagonal, then $\mathcal M$ is the Carlemanmatrix of the Schroederfunction, and $\mathcal M^{-1} $ the Carlemanmatrix of the inverse Schroeder-function.
This must all be a bit more explained, but after the basic matrix-notation for the powerseries operations it becomes much simple and intuitive.
I've made a couple of essays on this problem:

*

*introduction this explains in length & broadth the matrix-notation for the powerseries operations with some examples, when I wrote this I even didn't know that the diagonalization would represent the Schroeder-mechanism, so although chap 4.3 shows a relevant example to explain this, the term "Schroeder function" is not in there at all


*technical ecplanation of diagonalization of triangular Carlemanmatrix for the function $f=b^x-1$ where $0 \lt \ln(b) \lt \exp(1)$ and thus $f()$ is suitable for application of the Schroederfunction. There are also explicite decompositions of the coefficients of the Schroederfunction which you would find nowhere else.
If this is already helpful enough (or otherwise completely uninteresting) I'd like to leave this answer as it is for the moment to avoid needless typing-typing-typing  re-explanations. If there is something more to the point, and some more explanations were helpful, please note about it in a comment.
Late addition: in an answer to a question at MathOverflow (Jun 2017) I explained the Schroeder function using Matrix-methods ("Carleman matrix") down to small details (and bountied by 300 pts), see MO-answer
A: One of the "explicit" procedures is the following. For simplicity, let us assume $a=0$ is the fixed point - this is not a strong restriction since we can always shift the fixed point to $0$. Thus, $f(0)=0$ is the fixed point and $0<s=f'(0)<1$, and plus some assumptions on the analyticity of $f$, then
$$
 \Psi(x)=\lim_{n\to\infty}s^{-n}\underbrace{f\circ f\circ...\circ f}_\text{n}(x).
$$
It satisfies the Schröder functional equation
$$
 \Psi(f(x))=s\Psi(x),\ \ \Psi(0)=0,\ \ \ \Psi'(0)=1.
$$
Substituting unknown power series
$$
 \Psi(z)=z+\psi_2z^2+\psi_3z^3+...
$$
along with the known power series
$$
 f(z)=sz+f_2z^2+f_3z^3+...
$$
into the Schröder functional equation, one may found
$$
 \psi_2=\frac{f_2}{s-s^2},\ \ \ \psi_3=\frac{f_3+2s\psi_2f_2}{s-s^3},\ \ \ ...
$$
step by step. It should be noted that even if $f$ is a polynomial, the solution $\Psi$ usually can not be expressed in terms of elementary functions. Very roughly speaking, this is because the domain of definition of $\Psi$ is a component of the filled Julia set for $f$, which has a complex fractal structure. There are no elementary functions or their finite combinations that have such a complex domain of the definition. Of course, for some specific $f$, the solution $\Psi$ can be written explicitly. This is a preliminary situation with the Schröder-type functional equations. I am sorry if I was inaccurate in some places.
