I just realized that it is quite easy in maple to compute the matrix power via diagonalization (the function is called "MatrixPower" and you can put float values as exponents), I just compare it with the natural tetration.

To get the dsexp (diagonalization super exponential) I compute the Carlemann matrix of then just take the -th matrix power via "MatrixPower" and get the value of row 1 and column 0, which is then so the diagonalization tetration is .

For the comparison I compute which is always a periodic function with period 1.

And this is the resulting for matrix size of dsexp and nslog being 9 and precision 90 digits:

I think even in this low precision its recognizable that they are not equal. However I am currently preparing a plot in doubled precision which though takes some time, so I will add the graph later to this post.

edit: and here it is now:

hm, the amplitude decreased a lot, so I am again unsure ...

To get the dsexp (diagonalization super exponential) I compute the Carlemann matrix of then just take the -th matrix power via "MatrixPower" and get the value of row 1 and column 0, which is then so the diagonalization tetration is .

For the comparison I compute which is always a periodic function with period 1.

And this is the resulting for matrix size of dsexp and nslog being 9 and precision 90 digits:

I think even in this low precision its recognizable that they are not equal. However I am currently preparing a plot in doubled precision which though takes some time, so I will add the graph later to this post.

edit: and here it is now:

hm, the amplitude decreased a lot, so I am again unsure ...