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Is bugs or features for fatou.gp super-logarithm?
#1
I have been a big pressure test for fatou.gp in one month.
In the post(https://math.eretrandre.org/tetrationfor...p?pid=8952) I say is ill-region for fatou.gp. In fact, the ill-region for slog is little bigger the sexp.

But, There's look like other problem out of the ill-region.This gif is slog for base = Pi*(-1)^(x/30), 0<=x<=59.[Image: Pathological_phenomenon_for_fatou_super-...9%3DPi.gif]

You can see the program be pathological close to branch cut.
This plot is slog for Sheldon base. It look like well-behaved.
   
This plot is slog for base = eta. It look like well-behaved too.
[Image: Complex_super-logarithm%2C_base_%3D_eta.png]
base = -eta. It look like pathological.
   
base = sqrt(2). It look like well-behaved.
[Image: Complex_Kneser%27s_super-logarithm%2C_ba...282%29.png]
base = -sqrt(2). 
   
base = (-1)^(1/100). It's the closest base to 1. Is look like well-behaved? No, the program crash when Im(z)>
   
base = 0.8. 
[Image: Complex_Kneser%27s_super-logarithm%2C_base_%3D_0.8.png]
base = -0.105. It's the closest base to 0. 
   
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#2
(07/10/2019, 12:46 PM)Ember Edison Wrote: I have been a big pressure test for fatou.gp in one month.
In the post(https://math.eretrandre.org/tetrationfor...p?pid=8952) I say is ill-region for fatou.gp. In fact, the ill-region for slog is little bigger the sexp.
....
base = -eta. It look like pathological.
First of all cool pictures!   Thanks for posting.  base = -exp(1/e) ~= -1.445 is a good example to look at.  The slog abel function Taylor series is sandwiched between the two fixed points
0.25032 + 0.31754*I, -0.95550 - 0.099101*I, centered in between them
It is well defined and accurate along the sickle as required.  There is also two 1-cyclic theta mappings, one for each fixed point's Schroder function so that it matches the abel function series.
But the logic doesn't always know whether to take the log_b or exp_b, to get back to the well behaved region before evaluating the slog.  Also, like most (all?) Kneser slog's, the slog has a Periodic region as well. 
   

zoomed up view of well behaved region, showing the two fixed points and the cutpoints.
   

The fatou.gp code is trying to generate the slog cutpoints roughly along a line extending away from both fixed points, using the following two equations, but for these complicated bases, the algorithm of weather to take the log or the exp doesn't work.  
slog(z)=slog(exp(z))-1
slog(z)=slog(log(z))+1
Then it tries to get into the well behaved region, and uses either the slog taylor series (which takes advantage of Jay D Fox's accelerated representation), or uses a 1-cyclic theta mapping of the Schroder/Abel function from one of the two fixed points.

Working with hundered of bases makes cool pictures (thanks btw), but makes it difficult to debug.  Unfortunately, I simply do not have the time to properly debug this base; but I might come back to it.  If interested, I could take this base, along with a more well behaved base, and show the sample points around a circle that defines the slog well behaved region between the two fixed points, and how the program combines the two 1-cyclic theta mappings.  

fatou.gp works ith a different mathematically identical "congruent" iteration equation.  Instead of iterating

I generate the Abel function for iterating the following.  The Abel function, stitches together the two Schroeder function solutions for both fixed points with a pair of 1-cyclic mappings.  The Abel function I generate is centered exactly between the two fixed points to get a well defined Trapmann uniqueness sickle between the two fixed points.


Then I translate back to the slog/sexp by using the linear transformation between y and z.  

Actually, I use k+1 so k=0 corresponds to base eta=exp(1/e).  Then I found a straightforward series to help find the two fixed primary fixed points based on +/-sqrt(k+1), which is required for these complicated bases.  This is simpler than using the lambertW function to generate the fixed points, pari-gp doesn't have a good implementation of lambertW anyway.  I can post more if interested and when I have time.
- Sheldon
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#3
(07/13/2019, 01:10 PM)sheldonison Wrote:
(07/10/2019, 12:46 PM)Ember Edison Wrote: I have been a big pressure test for fatou.gp in one month.
In the post(https://math.eretrandre.org/tetrationfor...p?pid=8952) I say is ill-region for fatou.gp. In fact, the ill-region for slog is little bigger the sexp.
....
base = -eta. It look like pathological.
First of all cool pictures!   Thanks for posting.  base = -exp(1/e) ~= -1.445 is a good example to look at.  The slog abel function Taylor series is sandwiched between the two fixed points
0.25032 + 0.31754*I, -0.95550 - 0.099101*I, centered in between them
It is well defined and accurate along the sickle as required.  There is also two 1-cyclic theta mappings, one for each fixed point's Schroder function so that it matches the abel function series.
But the logic doesn't always know whether to take the log_b or exp_b, to get back to the well behaved region before evaluating the slog.  Also, like most (all?) Kneser slog's, the slog has a Periodic region as well. 


zoomed up view of well behaved region, showing the two fixed points and the cutpoints.


The fatou.gp code is trying to generate the slog cutpoints roughly along a line extending away from both fixed points, using the following two equations, but for these complicated bases, the algorithm of weather to take the log or the exp doesn't work.  
slog(z)=slog(exp(z))-1
slog(z)=slog(log(z))+1
Then it tries to get into the well behaved region, and uses either the slog taylor series (which takes advantage of Jay D Fox's accelerated representation), or uses a 1-cyclic theta mapping of the Schroder/Abel function from one of the two fixed points.

Working with hundered of bases makes cool pictures (thanks btw), but makes it difficult to debug.  Unfortunately, I simply do not have the time to properly debug this base; but I might come back to it.  If interested, I could take this base, along with a more well behaved base, and show the sample points around a circle that defines the slog well behaved region between the two fixed points, and how the program combines the two 1-cyclic theta mappings.  

fatou.gp works ith a different mathematically identical "congruent" iteration equation.  Instead of iterating

I generate the Abel function for iterating the following.  The Abel function, stitches together the two Schroeder function solutions for both fixed points with a pair of 1-cyclic mappings.  The Abel function I generate is centered exactly between the two fixed points to get a well defined Trapmann uniqueness sickle between the two fixed points.


Then I translate back to the slog/sexp from the abel function and invabel function using the sexp_invabel function or invabel_sexp fucntion.

Actually, I use k+1 so k=0 corresponds to base eta=exp(1/e).  Then I found a straightforward series to help find the two fixed primary fixed points based on +/-sqrt(k+1), which is required for these complicated bases.  This is simpler than using the lambertW function to generate the fixed points, pari-gp doesn't have a good implementation of lambertW anyway.  I can post more if interested and when I have time.

Maybe you need a Development schedule? I wish the arg(base) close to imaginary axis can be reduce first…
How serious is this problem when I use sexp? I want to test some big abs(base) in the next month.
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