regular vs intuitive (formerly: natural)

[bo198214]

I computed \( \delta:=\text{rsexp}_{\sqrt{2}}(\text{nslog}_{\sqrt{2}}(0.3))-0.3 \) with a precision of rsexp of 20 digits and varying matrix sizes for nslog from 10 to 100. The following graphs show \( \delta \) in dependence on the matrix size (x-Axis).

   
   

This looks quite as if regular and natural method are equal. Yippee!

Guys, I think we are on a great trail of recognizing natural, regular and diagonal tetration being equal.

[GFR]

Congratulations !!!

[bo198214]

Congratulations !!!

There is not yet much to congratulate, everything is in the state of conjectures and proofs are reluctant to reveal so far.

[GFR]

Hey, Henryk!

I just wished to sincerely congratulate you for the good possibility of acceptable coincident results of theese different strategies. The problems that we are trying to solve are really difficult.

I wonder why they are so difficult.

GFR

[bo198214]

I computed \( \delta:=\text{rsexp}_{\sqrt{2}}(\text{nslog}_{\sqrt{2}}(0.3))-0.3 \) with a precision of rsexp of 20 digits and varying matrix sizes for nslog from 10 to 100. The following graphs show \( \delta \) in dependence on the matrix size (x-Axis).
...
This looks quite as if regular and natural method are equal. Yippee!

And here we see again how numerics can be misleading.
Actually I increased the matrix size to 300.
And plotted the difference \( \text{rslog}-\text{islog} \), the result is the expected dilated oscillation for two different Abel functions:
   

In other words we can be quite sure that regular and intuitive tetration is different.

[mike3]

I'm curious: What programs and methods are you using to compute this? Esp. how do you compute the solution in reasonable time?

Also, can you try to expand, say, the \( \exp_{\sqrt{2}}^{1/2}(z) \) generated by this method about the fixed point 2? If it's not the same as the regular iteration then it should fail to be holomorphic there, right? (as that's the thing that characterizes the regular iteration -- that it's differentiable at the fixed point, no?) If not, then what happens if you try comparing coefficients to those obtained from the regular iteration and looking for the difference?

[bo198214]

I'm curious: What programs and methods are you using to compute this? Esp. how do you compute the solution in reasonable time?

You mean the intuitive Abel function via the Abel matrix? Well I use some self written algorithm in Sage. It takes some time to compute the coefficients for a 300 matrix. I get 100x100 in less than a minute. But regardless how long it takes you only need to compute it once (even if it takes 5 days ), save it, and later you can use it anytime. If you need code, just tell me.

But J.D. Fox development much more sophisticated algorithms, giving better precision. I will have a look at it later and improve my own algorithms accordingly.

Also, can you try to expand, say, the \( \exp_{\sqrt{2}}^{1/2}(z) \) generated by this method about the fixed point 2? If it's not the same as the regular iteration then it should fail to be holomorphic there, right? (as that's the thing that characterizes the regular iteration -- that it's differentiable at the fixed point, no?)

yes. I will have a go at it (when I implemented Fox's improvements).

[Gottfried]

I'm curious: What programs and methods are you using to compute this? Esp. how do you compute the solution in reasonable time?

You mean the intuitive Abel function via the Abel matrix? Well I use some self written algorithm in Sage. It takes some time to compute the coefficients for a 300 matrix. I get 100x100 in less than a minute. But regardless how long it takes you only need to compute it once (even if it takes 5 days ), save it, and later you can use it anytime. If you need code, just tell me.

But J.D. Fox development much more sophisticated algorithms, giving better precision. I will have a look at it later and improve my own algorithms accordingly.

Also you could look at my computation at
http://math.eretrandre.org/tetrationforu...27#pid2827

Gottfried