matrix function like iteration without power series expansion

[bo198214]

Using a matrix-expression this would be

t°h(x) = W^-1 sum k=0..inf sum j=0..k (-1)^j * binomial(k,j) *diag(1,u^j,u^2j,...) * W

sum j=0..k (-1)^j * binomial(k,j) *dV(u^j) = diag(u^j-1) *PPow(
sum k=0..inf

what?

"not regularly" Euler-summable

Yes, I see. Its somehow similar to your tetra series \( x-f(x)+f(f(x))-f(f(f(x)))+\dots \) its only summable via the matrix method for \( f \) not having an attracting (finite) fixed point, i.e. in our case \( b>e^{1/e} \).

However the aim was to not use the matrix method for computation.
This is faster and does not require the function to be developable into a powerseries. For regular iteration we have both: limit formulas and powerseries coefficient formulas. And now we have also both for the matrix function method (if it can still be called that way for the limit formula).

The limit formula yields exactly the same iterates as the matrix function method if it converges. With this formula its perhaps easier to verify that the matrix function method just gives the regular iteration at the attracting fixed point, though I didnt try to prove this yet.