Applying the iteration methods to simpler functions

[bo198214]

You may also have a look at my operators-article... I discussed the three operations in connection with the matrix-method in few lines at the second last page. See here:http://math.eretrandre.org/tetrationforu...hp?aid=124

hmm. Don't know, seems, there is something broken with the pdf-attachments. Here is an external
link

The link works well. You write in this article:

The eigensystem of P is degenerate; but it has an exceptional simple matrix-logarithm, by which then a general power can be easily computed when just multiplied with the h-parameter.

There is even a general method that works with every matrix via the Jordan normal form. However it suffices to use a more relaxed form
where the blocks of the Jordan normal form consist of upper triangular matrices with the eigenvalue \( \lambda \) on the diagonal (instead having the eigenvalue on the diagonal and 1 on the diagonal above them, see wikipedia). We can take the power of such a block \( B \) by the formula \( B^t = \lambda^t(\frac{1}{\lambda}B-I+I)^t =\lambda^t \sum_{k=0}^\infty \left(t\\k\right)(\frac{1}{\lambda}B-I)^k \). Which however involves no limits for the entries. And the \( t \)th power of the whole matrix, which consists of several such blocks on the diagonal, is simply the \( t \)th power of each block on the diagonal.

The matrix in the considered case above consisted of just one such block with eigenvalue \( \lambda=1 \).

Do you consider parabolic/hyperbolic iteration (and Daniel Geisler's methods) to be part of regular iteration?

Yes I consider both as regular iteration (though if directly handled they require different treatment) but both coincide with the matrix operator method if developed at the fixed point.