A nice series for b^^h , base sqrt(2), by diagonalization

[Gottfried]

For me matrix power method means exactly what I wrote. Take the truncations \( B_N \) of an infinite matrix \( B \), apply the matrix power \( (B_N)^h \) and take the limit
\( B^h := \lim_{N\to\infty} (B_N)^h \).

The difference for the matrix-power may be "small"; but it may be significant when applied to the matrix-inverse: "take-the-truncated,invert,use-as-approximation" may give then different results from "conclude-the-exact-entries-for-infinite-case,truncate,use-as-approximation". It is even more significant, if we discuss more complex entities like the set of eigenvalues.
So these different views of things should be still explicite, and it would be good to keep identifying nomenclature. I tended to give the finite-matrix-based the attribute "polynomially", but this might be not the best choice...

Now one can apply this method at different development points, i.e. the original function \( f(x)=b^x \) is conjugated to the development point \( p \):

Surely. No dissent here.

And I really honored this method (applied to a *non-fixed point* like 0) because it can do where regular iteration fails: to be able to compute real iterates for \( b>e^{1/e} \). This also puzzles me that despite you insinst on regular iteration for \( b>e^{1/e} \).

:-) As it comes to honor... Well, that's not my problem. I surely should have come to my full description-text about my way of thinking in a new pdf-file - I've some first chapters, but it's very complex and I stuck several times soon. I'll be "honoring" that method, too, so we have also no dissent here.
I've just left this field and am digging at the other one for the gold. I think if a definitive description for the matrices in the infinite case can be given, (based on the hypothese about eigenvalues) this would be very good and if then also a method for the actual computation were found - this would please me much more than the approximating of 7^^Pi using ad-hoc-eigensystems of finite-size-matrices. Maybe the latter will even be the only way to get to practical values; but then: well, there'll be many people, programs which could do that, very fine, why should I bother, it's not my job/profession/money to calculate values?

But if you focus too much on only matrices and nothing else, such interesting relations like the convergence radius of the iterates is just out of scope for you. Because it is derived by the interelation of the limit formulas for regular iteration (which is power series free) and power series formulas for regular iteration.

Here you made a point. However, not in the sense of missing the aspect of convergence-radius; in the contrary: I think I need the matrix-layout for the infinite case to have even better conditions for convergence considerations. And since in important cases we'll miss convergence anyway we can check for summability methods to overstep the range of converge, but in a well-founded manner.

But as I learned in some discussions in sci.math in the last monthes it is fruitful to discuss iteration also in terms of the functions themselves - even some very nice and surprising closed forms were discussed which I never could have found with the formal powerseries/matrix-approach. Here opened a much interesting field and I'm actually fiddling with that on a casual manner (you've notices my casual "iteration exercises" also here in the forum). I think I'll go into this much more if I have the feeling, that my questions/ideas with the infinite matrices are solved (or shown to be unsolvable) and I can close that case. Just currently I've applied the (infinite) matrix-concept to Andrew's slog with a nice achievement of insight... :-) So there's still something in it. (I'll need it also for the discussion of iteration series, I think)