An application of the "polynomial method/Diagonalization" (matrix-size 64x64) which was in the instance of the article shown with base \( b=4 \) and in that article likely asymptotic to the Kneser method.
Here I provide a picture with focus on complex iteration from one real starting point \( z_0 = 1 \) with base \( b=1.3 \). (I've not yet checked against Sheldon's Kneser-implementation ).
The picture shows roughly circles: along the circumferences the iteration-height is purely imaginary; one revolving means \( 2 \pi i / v \) where v is the log of the log of the fixpoint \( t \approx 1.48 \)
It is still surprising to me that we can proceed from one real point below the fixpoint to some other real point above the fixpoint - which means to avoid/surpass the infinite height-iteration: just by using imaginary heights...
Gottfried
Here I added two more (hopefully instructive) views;
Here the base-point is z0=0
and here is the iteration to one more negative height, where I had to leave out the infinitely distant point z0 (-> - infinity)
Now I've got my matrices for base b=1.44, near eta. What a mess!
I don't have any idea - there is no obvious divergence in the power series with 64 terms. I also took z0=b^b ~ 1.69 as initial point; the curves originating from z0=1 were even more messed up.
This is the picture base b=1.44 z0=1+0î - I've no explanation so far for the messed curves.
<hr>
P.s.: A 3-D picture with colors indicating height, and "isobares"-grid and the fixpoint shown as peak of infinite height were nicer but I do not know how to draw one(which software). If someone else likes to play with this I can provide the coefficients of the diagonalization matrices in Pari/GP-convention: the computation of that matrices is extremely costly (it needed 6000 secs to be computed and more than 3000 digits decimal precision;I chose then 4000 digits) so it might be interesting to get the ready-made numbers by download instead by a new computation.
Here I provide a picture with focus on complex iteration from one real starting point \( z_0 = 1 \) with base \( b=1.3 \). (I've not yet checked against Sheldon's Kneser-implementation ).
The picture shows roughly circles: along the circumferences the iteration-height is purely imaginary; one revolving means \( 2 \pi i / v \) where v is the log of the log of the fixpoint \( t \approx 1.48 \)
It is still surprising to me that we can proceed from one real point below the fixpoint to some other real point above the fixpoint - which means to avoid/surpass the infinite height-iteration: just by using imaginary heights...
Gottfried
Here I added two more (hopefully instructive) views;
Here the base-point is z0=0
and here is the iteration to one more negative height, where I had to leave out the infinitely distant point z0 (-> - infinity)
Now I've got my matrices for base b=1.44, near eta. What a mess!
I don't have any idea - there is no obvious divergence in the power series with 64 terms. I also took z0=b^b ~ 1.69 as initial point; the curves originating from z0=1 were even more messed up.
This is the picture base b=1.44 z0=1+0î - I've no explanation so far for the messed curves.
<hr>
P.s.: A 3-D picture with colors indicating height, and "isobares"-grid and the fixpoint shown as peak of infinite height were nicer but I do not know how to draw one(which software). If someone else likes to play with this I can provide the coefficients of the diagonalization matrices in Pari/GP-convention: the computation of that matrices is extremely costly (it needed 6000 secs to be computed and more than 3000 digits decimal precision;I chose then 4000 digits) so it might be interesting to get the ready-made numbers by download instead by a new computation.
Gottfried Helms, Kassel
