matrix function like iteration without power series expansion

[bo198214]

Awhile back, I did some test with parabolic iteration, and convinced myself that Woon's formula and Jabotinsky's formula produce exactly the same results for parabolic iteration (for symbolic coefficients). So I could write up something about this if needed.

The problematic thing about parabolic iteration is convergence. You usually only have an asymptotic development at the fixed point. I.e. the series is not developable at the fixed point however in every neighborhood and all the coefficients converge (towards the fixed point in a certain sector) to that of the asymptotic development.

I also see that I made mistake in my first post about the convergence of that series. It is not true that the series always converges if the function has a finite attracting fixed point.