# Tetration Forum

Full Version: naming: natural -> intuitive
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I want to suggest to rename what until now we called "natural slog" to "intuitive slog".
I think this is necessary, as "natural" leads to two misinterpretations:
1. Natural numbers: If one extends the term to "natural sexp" and to "natural iteration" then it can be misunderstood to be the superexponential on natural numbers or the iteration for natural numbers
2. Base e: Many people associate base e with the word "natural" like ln = logarithmus naturali
So we can then make use of intuitive slog, intuitive sexp and intuitive iteration without fear of misinterpretation.
bo198214 Wrote:I think this is necessary, as "natural" leads to two misinterpretations

True, not to mention the possibility of confusion with natural transformation which could be used in relation to the Carleman matrix, because the Carleman matrix can be viewed as a functor.
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-$e^{1/e}$ 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 $x=x_0$ where $x_0$ is a fixed point, is called regular iteration.
• Analytic iteration of f(x) about $x=x_0$ where $\mathbf{J}(\mathbf{B}[f(x + x_0)] - I)\mathbf{K}$ is invertible (which requires at the very least that $x_0$ 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
andydude Wrote:I think that both regular iteration and natural/intuitive iteration can be viewed as different techniques for applying matrix iteration in general.

Haha, that would introduce more ambiguity!
Matrix iteration would mean matrix power method for me!
Which is applicable to fixed points and non-fixed points.
If it is applied to fixed points then it is regular iteration.

But intuitive iteration ($\operatorname{slog}^{-1}(t+\operatorname{slog}(x))$) is imho not equal to matrix power iteration, at least not a priori.

Quote:like parabolic and hyperbolic fixed points

Where also it is not quite clear what "hyperbolic" means ($|f'(x_0)|>1$). Is there also an elliptic ($0<|f'(x_0)|<1$)?

Quote:Analytic iteration of f(x) about $x=x_0$ where $\mathbf{J}(\mathbf{B}[f(x + x_0)] - I)\mathbf{K}$ is invertible (which requires at the very least that $x_0$ is not a fixed point), is called natural/intuitive iteration.

No, analytic iteration means just that the iterates are analytic. It can not presumed uniquely to be intuitive iteration until equality is shown.