On extension to "other" iteration roots

[Leo.W]

It looks like my improved/simplified version of your "P method" yields the same result as your "P method" and can also handle the case \(b<\eta_-\), except for the condition \(\text{tet}_b(0)=1\) because it is a repelling fixed point and if you start to iterate you can not use 1.

It's pretty nice to have a simplification, I see that you probably use 1-order approx about superfunctions T and its inverse, so you get a simplification, great idea
And it shall be very easily to show it's a special case of "P method"(this name so wierd lol)

The difference is it's not easy to determine which function to use as a theta mapping here, also exactly "P method" uses a kinda theta-mapping as well, it's
\(T_merged\approx{T(z+\theta(z))}=\lim{L+s^{z+\theta(z)}}=\lim{L+\theta_1(z)s^z}\) where 2 thetas are 1-periodic.
I'm not tryna act stubborn or pretentious, but simplification is one good thing, while a detailed construction can be helpful, P can be accurately determine what value a merged function gains at specific points, while by simple theta mapping it's hard to find a right theta mapping
and simplified version is indeed awesome



I personally defined the method as a merging tech for 2 different superfunction but has the same limiting value at some point, only with a naive and open and undefinite initial guess. This is why I say yours is a special case as:
1. an initial guess
2. merging or theta-mapping

Leave this aside, I have to point out that the second superfunction you're making, is in fact a merged superfunction of \(log_b(z)\) and the hidden 2 different superfunction (the limit is ~0.2528 meets at negative infinity. Besides your function will never meets tet(0)=1, so still a special P method.
And I insist that P method or your simplified version won't work for such base, if you want to get a tet_b, let f(z)=b^b^z, you have to take a decreasing superfunction of f(z) which will pass 1, but it also has a logarithmic singularity at -2, and another rising superfunction that goes from 0 at -1, merging these 2 together will get u the desired tetration \(T(z)\)

However, because you must wanna include continuity for tet_b, when you merge these 2, your initial guess \(T_0(z)\) must go across the fixed point \(L\) of b^z between [0,1], which must be repelling (easy to check), let's say the crossing point \(T_0©=L, c\in[0,1]\), we'll write an asymp that \(T_0(z)=L+s_0(z-c)+O(z-c)^2\), lets denote \(T_n(z)\) as the nth approximation. 
Because the branch cutting issue you can only use \(T_n(z)=\log_b(T_{n-1}(z+1))\) to approximate the final T, you must  notuse \(T_n(z)=exp_b(T_{n-1}(z-1))\)
doesnt matter
Denote \(c_n\) as the solution to \(T_n(z)=L\) in [0,1], \(s_n\) as the derivative of T_n at it. I'll show you a specific example for b=0.06 by pics

1. if you use \(T_n(z)=\log_b(T_{n-1}(z+1))\), we focus on the asym at c_n, we must obtain that \(T_n(z)=L+s_n^{-n}(z-c_n)+O(z-c_n)^2\), you can verify that \(s_n\to0\), and because this time T gets 3 limiting value at inf, L and 2 other fixed point of \(f(z)=b^{b^z}\), T_n would behave constant around a big neighborhood of c_n, but a sudden leap at integers. Thus for very large n, the T will converge at a function which has discontinuity at integer and remain L constantly otherwise.

and pic1 that shows such manner: 
blue: initial guess T_0 with values at integer correct
yellow: after 40 iterations, T_40 tends to be constant around c_n
green: after 80 iterations, T_80 more "constant" around c_n



2. if you use \(T_n(z)=b^{T{n-1}(z-1)}\),(even ignored the branch cuts) we still focus on the asym at c_n, this time we'll have \(T_n(z)=L+s_n^{n}(z-c_n)+O(z-c_n)^2\\), and \(s_n^n\to\infty\), thus as n getting very big, T will remain constant at integers and have infinitely many jumps at c_n, c_n+1, c_n+2, and so on, still not working

no pic at [-1,10] this time because branch cuts makes T_n complex in [-1,n-1]
pic2 showing how \(T_n(z)=b^{T{n-1}(z-1)}\) works:
blue: initial guess T_0 in [80,90]
yellow: after 40 iterations, seems T_40 begins to jump at the c_n or solutions of \(T_{40}(z)=L_{base=0.06}\approx0.36158\)
green: after 80 iterations, T_80 stays constant at 2 exotic fixed point of b^b^z, and jumps at c_n


If u got a great idea on such realvalued tet plz call me up