The ultimate sanity check

[Daniel]

Oh, Daniel, I really don't know where to start - there are so many implicit assumptions in your question.

Start with my first impression: Somehow it's asking like: "I want to sanity check an algorithm that I found (after long research and encountering a lot of interesting mathematical side topic) that can compute the length of a third side of a right triangle." People would answer "well there is something that is called theorem of Pythagoras, why not use that?!" 

If you only would compare numerical results of this theorem with your own algorithm, you would miss a lot! First a numerical "equality " is no proof. (You could easily be fooled for example by the half-iterate developed at 2 and 4 of sqrt(2)^x, which coincide by many digits, but in the end are essentially different functions - this was already several times said on the forum). 
Second an algorithm is not a proof. Theorem of Pythagoras contains a proof that an algorithm will always yield the right answer.
Similar issue with the Schroeder iteration: There we have a proof of convergence of different ways of computing. We have a proof of the resulting function being analytic, etc etc
You see how much richer that is than just having numerical coincidence? Of course as a sanity check on should do some numerical comparisons.

And then there is another category like the Kneser construction: which gives proof of existence, or the Leau-Fatou construction, which proofs existence and uniqueness. But both have no easily accessible means of numerical computation. (So the only sanity check is that many mathematicians looked at it and confirmed, typically by peer-reviewed publication or teaching it at university)

With Tetration we are unfortunately mostly (except regular iteration) in the situation of having some numerical methods with no proof of convergence, with no proof of holomorphy (in case of convergence) and with no clue about identity (which algorithms converge to the same function).

So what I want to say: it is essential to understand the well-known theory (e.g. Schröder-Iteration, Kneser construction, Leau-Fatou construction) if only to have a common base of understanding each other. Otherwise we praise ourselves in our snail house, never having seen the world in comparison.

Wow, that was a disappointing response Bo. I ask a simple question and I get a bunch of random criticisms of me and my work. Good criticism is wonderful. Uninformed criticism sucks. If you want to know what mathematics you are criticizing, see Bell Polynomials of Iterated Functions. Yiannis supports my work and my paper. I submitted the paper to the Annals of Mathematics. They only publish one out of twenty papers they receive. Their one unresolved issue was that I didn't base my work on germs.

Now back to the original simple question I asked.

My ultimate sanity test is to prove symbolically that using the Taylor's series for \( f^n(z) \) that \( f^{a+b}(z)-f^{a}(f^{b}(z))=\mathcal{O}(z^k) \)

So, no talk about algorithms or numerical simulations. This identity can be used to determine if different techniques are equivalent. Also, is is always good to have more than one proof for a problem. Wasn't it quadratic reciprocity that Gauss discovered six different proofs for? Given the comments made, I assume that none of the techniques represented by current involved members of this forum can satisfy my sanity test. The exception is I believe Gottfried's matrix methods are consistent with my work. If Wolfram fixes me up with another copy of Mathematica I will contact Paulsen and ask him for copies of his Mathematica notebooks.

Edit: I just wrote Dr. Paulsen, so maybe I can arrange to get copies of the notebooks.