regular slog

[Gottfried]

Now there is the the so called principal Schroeder function \( \sigma_f \) of a function \( f \) with fixed point 0 with slope \( s:=f'(0) \), \( 0<s<1 \) given by:

\( \sigma_f(x) = \lim_{n\to\infty} \frac{f^{\circ n}(x)}{s^n} \)

This function particularly yields the regular iteration at 0, via \( f^{\circ t}(x)=\sigma^{-1}(s^t\sigma(x)) \).

Sometimes a thing needs a whole life to be recognized...

In the matrix-method I dealt with the eigen-decomposition of the (triangular) dxp_t() -Bellmatrix U_t to satisfy the relation

\( \hspace{24} U_t = W * D * W^{-1 } \)

While the recursion to compute W and W^-1 efficiently is easy and is working well, I did not have a deeper idea about the structure of the columns in W. Now I found, it just agrees with the above formula:

\( \hspace{24} W = \lim_{h\to\infty} {U_t}^h * diag({U_t}^h)^{-1} \)

which is exactly the above formula; we even can write this, if we refer to the second column of U_t^h as F°h, the second column of W as S, and s = F[1] while F°h[1] = F[1]^h =s^h , then we have

\( \hspace{24} S = \lim_{h\to\infty} \frac {F^{\circ h}}{s^h} \)

Something *very* stupid ... <sigh>
But, well, now also this detail is explained for me.

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<Hmmm I don't know why the forum software merges my two replies (to two previous posts of Henryk) into one So here is the second post>

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This is the diagonal matrix:
\( M=\begin{pmatrix}
c &0 & 0 &\dots &0\\
0 & c^2 & 0&\dots& 0\\
&&\vdots&\\
0 & &\dots& 0 & c^n\\
\end{pmatrix} \)
I think Gottfried calls this the Vandermonde matrix.

Not exactly. I call the Vandermonde-*matrix* VZ (and ZV=VZ~) the *collection* of consecutive vandermonde V(x)-vectors

´ VZ = [V(0), V(1) , V(2), V(3), ...] \\ Vandermondematrix

Your M is just c*dV( c) in my notation: the vandermondevector V( c) used as diagonalmatrix (and since the first entry is not c^0 I noted the additional factor c)

Gottfried