At the same time, I kind of like "natural", but I also dislike ambiguity... so perhaps if we talked about this a year ago, I would instantly change my terminology, but now that we have used the term extensively, I would have to think about it for a long while before changing terminology.
I think this problem is quite common. Especially with the 3-argument Ackermann function / hyperoperations which are commonly equated. I think some of this is alleviated with the use of author names, for example, if we were to call it
Walker iteration instead. But this is also ambiguous, since Walker discussed 2 methods: a recursive method and a matrix method.
I think that both regular iteration and natural/intuitive iteration can be viewed as different techniques for applying
matrix iteration in general. Even though the two seem to work well for base-
tetration, I see the two as mutually exclusive (like parabolic and hyperbolic fixed points). So I personally make the distinction as follows:
- Analytic iteration of f(x) about
where 
is a fixed point, is called regular iteration.
- Analytic iteration of f(x) about where
] - I)\mathbf{K})
is invertible (which requires at the very least that is not a fixed point), is called natural/intuitive iteration.
Since the points at which each technique works are mutually exclusive, they can be viewed as special techniques for analytic iteration. That is, if anyone can prove they are equivalent when the domains of the continuous iterates overlap!
Andrew Robbins
