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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 illregion for fatou.gp. In fact, the illregion for slog is little bigger the sexp.
But, There's look like other problem out of the illregion.This gif is slog for base = Pi*(1)^(x/30), 0<=x<=59.
You can see the program be pathological close to branch cut.
This plot is slog for Sheldon base. It look like wellbehaved.
This plot is slog for base = eta. It look like wellbehaved too.
base = eta. It look like pathological.
base = sqrt(2). It look like wellbehaved.
base = sqrt(2).
base = (1)^(1/100). It's the closest base to 1. Is look like wellbehaved? No, the program crash when Im(z)>
base = 0.8.
base = 0.105. It's the closest base to 0.
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07/13/2019, 01:10 PM
(This post was last modified: 07/13/2019, 09:27 PM by sheldonison.
Edit Reason: Typo
)
(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 illregion for fatou.gp. In fact, the illregion 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 1cyclic 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 1cyclic 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 1cyclic 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 1cyclic 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, parigp 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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(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 illregion for fatou.gp. In fact, the illregion 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 1cyclic 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 1cyclic 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 1cyclic 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 1cyclic 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, parigp 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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07/31/2019, 01:40 PM
(This post was last modified: 08/02/2019, 05:46 PM by sheldonison.)
I made a lot more progress than I thought I would. This post concerns fatou.gp boundary conditions and how it works. First, I'm going to put out an experimental version of fatou.gp, along with some routines that may be helpful to Ember. I'm going to try to answer some of the questions that Ember has.
 How do I get better results for bases close to zero?
examples, 0.2*I, 0.2, 0.1*I, 0.1, 0.15
 Why there are bases for which it is so difficult to get convergence? real bases<0.15, bases exp(e), 0.1
 This same method works to get results for larger bases, like b=30000+5000i
Eventually, this routine will move to the main branch once I have more time to make sure it works just as well.
fatou_experiment.gp (Size: 74.73 KB / Downloads: 89)
Besides the sexpinit(B) routine that works mostly automatically, I have two other matrix based routines for initializing more difficult problems, with lots of parameters that effect speed, memory usage, and accuracy.
 matrix_ir(B,lctr,ltht,myctr,myir)
lctr is the number of sample points around a circle centered between the two fixed points
The radius is myctr*(ll2)/2. myctr is a real number less than 1.
The closer to 1 myctr is, the worst the convergence, but sometimes the routine works on difficult bases.
myir is a second multiplier, also less than 1, that defines the inner circle where all the sample points are.
Again, the closer to 1, the slower the convergence is, but the better it works on difficult bases.
ltht is the number of sample points in the theta mapping, which is indirectly included in the main matrix. This should be a smaller number, usually half or one quarter of lctr. The "matrix" version of the basee solution is "matrix_ir(exp(1),244,19,4/5,7/10)" which gives almost exactly the same approximation as sexpinit(exp(1))
 matrix_r(B,lctr);
similar without a theta mapping, so convergence improves more slowly. Originally I thought if this routine doesn't converge because he periodicity is too close to 2 for example, than there would be no hope using fatou.gp but it turns out there are some bases where the theta mapping in matrix_ir allows convergence and yet this routine, matrix_r, cannot converge. For example
matrix_ir(0.15,72,14,14/15,45/46); /* converges */
matrix_r(0.15,256); /* cannot converge no matter how many points there are */
 future work items
improve sexp/invabel routine, which has some difficulties for these bases
plot Henryk Trapmann's uniqueness sickles for some of these bases
why do bases 0.2, 0.3, 0.4 require an imaginary offset to converge? I*1E30 is sufficient for convergence.
Here is a picture. Compare this to the base "e" version https://math.eretrandre.org/tetrationfor...22#pid9022
This pictures comes from an excel spreadsheet, by pasting in the circchart.csv file.
In this picture, with the theta mapping from l2 represented in green, and the theta mapping from l represented in purple. I am sampling 72 points around the middle circle with brown points paired with b^z, and yellow points paired with log_b(z). green point are generated from the theta mapping for the fixed point of l2, and pink points are generated from the fixed point of l. The function is centered between (l+l2)/2, and uses Jay's slog representation to accelerate convergence. The radius of the middle circle is (14/15)*radius of the outer circle, so the fixed points singularity doesn't prevent convergence, and there is some exponential convergence of approximately (14/15)^n bits plus the small accelerated convergence term. All of the points in the middle circle were generated from points inside the smaller circle whose radius is 21/23*radius of the outer circle. Compare this to base(e), where the middle circle is 0.8x the radius of the bigger circle and the inner circle is 0.56x the radius of the bigger circle, which gives much cleaner convergence properties.
Several interesting things about base=0.15
 the two fixed points are l=0.436708722498860; l2=0.467424941870507 + 2.38182350871689i
the fixed points are "upside down" from where you might expect them, but if you follow a path from tetration base(e) to base(0.15) going counter clockwise from above, avoiding the singularities at 1 and eta, then that is what you get.
 The fixed point of l is weakly attracting with a Schroder function period of 1.99285186874753  0.119353117783046i
To me, it is surprising that we can get a workable theta mapping solution with the period so close to 2!
and the real part of the period is <2! Nearby is exp(e), which has a period of exactly 2 can't converge.
 This base doesn't have great convergence at only 5.5 decimal digits with 72 sample points.
 Convergence is exponential and with 400 sample points, convergence improves to 1617 decimal digits. Visually, the problem is the poorly behaved Schroder function ...
 This base cannot converge at all without a theta mapping since the sample points are outside of a circle!
 The sexp(z) invabel(z) functions don't work well enough to do a MakeGraph this base; but I can plot sexp(2 ... 2.5)
This matrix_ir approach also works for other difficult to converge bases like Ember's recent example of ~32000+5000i. Tetration bases 0.1i, 0.2i, 0.3i all converge with this algorithm, as do 0.1 and 0.2 etc.
The higher precision settings I tried were matrix_ir(B,400,90,14/15,45/46) which seems to provide about 2021 decimal digits of accuracy for these examples, requires about 64meg of memory, and initializes in 2025 seconds.
Tetration base 0.1 doesn't work with these settings though perhaps with myctr and myir both closer to 1, it might work. Bases (0.2,0.3,0.4) require a small I*1E30 imaginary offset to converge; I'm not sure why. exp(e)~=0.06599 can't converge since the fixed point has a period of exactly 2 and the upper fixed point doesn't have a Schrocder function, so there isn't a theta mapping; a different algorithm would be required.
I will post more when I have more time.
 Sheldon
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08/02/2019, 02:09 PM
(This post was last modified: 08/02/2019, 05:40 PM by Ember Edison.)
(07/31/2019, 01:40 PM)sheldonison Wrote: I made a lot more progress than I thought I would. This post concerns fatou.gp boundary conditions and how it works. First, I'm going to put out an experimental version of fatou.gp, along with some routines that may be helpful to Ember. I'm going to try to answer some of the questions that Ember has.
 How do I get better results for bases close to zero?
examples, 0.2*I, 0.2, 0.1*I, 0.1, 0.15
 Why there are bases for which it is so difficult to get convergence? real bases<0.15, bases exp(e), 0.1
 This same method works to get results for larger bases, like b=30000+5000i
Eventually, this routine will move to the main branch once I have more time to make sure it works just as well.
…
Congratulations, you have completed the key work.
All my calculations are testing superroot.gp, I will test it when I have finish the current work.
(New Edit)
So, What happened between fatou_experiment.gp and fatou.gp when bases is no theta mapping?
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08/02/2019, 09:34 PM
(This post was last modified: 08/04/2019, 06:32 AM by sheldonison.)
(08/02/2019, 02:09 PM)Ember Edison Wrote: Congratulations, you have completed the key work.
All my calculations are testing superroot.gp, I will test it when I have finish the current work.
(New Edit)
So, What happened between fatou_experiment.gp and fatou.gp when bases is no theta mapping?
Thanks for testing, Ember. The bases with no theta mapping are the bases very close the the Shell Thron region, whose period's have imaginary part of zero. sexpinit(B) hasn't changed in fatou_experiment.gp. In fatou_experiment, there is a function bfromp(period). Let's consider two tetration bases, one near the Shell Thron boundary, and one practically on top of the of the Shell Thron boundary.
Code: \r fatou_experiment.gp
zb4=1.989+1.193*I; /* period is 4.0000.0003i; too close to Shell Thron boundary; no theta mapping */
zb4_005i=1.995+1.195*I; /* period is 4.000+0.005i; has a theta mapping; but its slow */
limitp=20; /* limit precision so it doesn't run toooo long */
sexpinit(zb4_005i); /* initializes after 25 iterations, 22 seconds, 20 decimal digits */
matrix_ir(zb4_005i,132,82,8/10,15/16); /* using settings from sexpinit(zb_005i) but faster; 3 seconds */
sexpinit(zb4); /* no theta mapping; 14 decimal digits after 70 iterations 40 seconds, 888 samples */
matrix_r(zb4,888); /* no theta mapping, 13.6 decimal digits; 2min 35 seconds */
So that's a Period=4 example. For period=4, we get some convergence without a theta mapping by taking advantage of Jay's accelerated slog representation, which I use both with and without a theta mapping to improve convergence.
Near the origin is a much more difficult problem. The period from the upper fixed point for B=exp(e) is exactly 2. The accelerated representation doesn't help as much as the Period gets closer to 2, and, as can be seen in the plot for B=0.015 above, the Schroeder function for some of the nearby bases is really badly behaved so it takes a lot of theta points, plus the sample radius has to be closer to 1 to get convergence. The posted results used matrix_ir(0.15,400,90,14/15,45/46). What settings to use for matrix_ir is a matter for experimentation. Relaxing the sample radius, and increasing the number of theta samples seems to give good results too; matrix_ir(0.15,248,160,8/9,45/46). However, matrix_r(0.15,n) without a theta mapping doesn't not converge, as is shown by the sample chart showing the points need to pair up with points outside of the sampling circle. The matrix becomes unstable, and the results have negative precision.
[edit1] The theta mapping for these bases greater than exp(e) seems poorly behaved and requires a larger number of coefficients to converge than one might expect. It would be nice to better understand the theta mapping and its radius of convergence for base=0.15, and its nearest singularity. For B=0.1+I*E30; I needed 250 terms in the theta mapping! matrix_ir(B,400, 250,14/15,45/46) to get a bit over 16 decimal digits of precision as compared with 90 terms for B=0.15 matrix_ir(0.15,400, 90,14/15,45/46). Either way, where is the nearest singularity in the theta function? I also don't know why I needed the small imaginary offset for B=0.1; otherwise the Schroeder function to Abel function conversion gets confused, but I don't know why.
[edit2] The Schroeder function for neutral points does not converge if the period is a rational number. So a period=4 Schroeder function doesn't exist since it doesn't converge. But if the period is a well behaved irrational number (with imaginary part=0) then the Schroeder function does converge. For example, a period=pi Schroeder function does converge. See
https://en.wikipedia.org/wiki/Brjuno_number, and http://www.scholarpedia.org/article/Sieg...egel_disks. The Kneser Tetration and its inverse are analytic even if the period is rational. The experiment is to look at all of the Taylor series coefficients of the slog and see how they vary as the base changes in the complex plane. There is an analytic function for each of these Taylor series coefficients. So from the bases around the rational base, one can construct an arbitrarily accurate series even though the fatou.gp algorithm isn't ideal for computation of such neutral rational period bases.
 Sheldon
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08/04/2019, 12:55 PM
(This post was last modified: 08/04/2019, 01:18 PM by Ember Edison.)
(08/02/2019, 09:34 PM)sheldonison Wrote: ……
[edit1] The theta mapping for these bases greater than exp(e) seems poorly behaved and requires a larger number of coefficients to converge than one might expect. It would be nice to better understand the theta mapping and its radius of convergence for base=0.15, and its nearest singularity. For B=0.1+I*E30; I needed 250 terms in the theta mapping! matrix_ir(B,400,250,14/15,45/46) to get a bit over 16 decimal digits of precision as compared with 90 terms for B=0.15 matrix_ir(0.15,400,90,14/15,45/46). Either way, where is the nearest singularity in the theta function? I also don't know why I needed the small imaginary offset for B=0.1; otherwise the Schroeder function to Abel function conversion gets confused, but I don't know why.
[edit2] The Schroeder function for neutral points does not converge if the period is a rational number. So a period=4 Schroeder function doesn't exist since it doesn't converge. But if the period is a well behaved irrational number (with imaginary part=0) then the Schroeder function does converge. For example, a period=pi Schroeder function does converge. See
https://en.wikipedia.org/wiki/Brjuno_number, and http://www.scholarpedia.org/article/Sieg...egel_disks. The Kneser Tetration and its inverse are analytic even if the period is rational. The experiment is to look at all of the Taylor series coefficients of the slog and see how they vary as the base changes in the complex plane. There is an analytic function for each of these Taylor series coefficients. So from the bases around the rational base, one can construct an arbitrarily accurate series even though the fatou.gp algorithm isn't ideal for computation of such neutral rational period bases. The last edit it's sounds strange. So Will the Kneser Tetration and its inverse can exist if Schroeder function doesn't exist?
The base = E^E is a singularity?
And Can you get more digital evidence to "proof" add small imaginary will not add side effects?
Ps: As long as the fatou.gp be faster, I can accept more memory usage. 2GB is good, but I can accept >10GB if you can really speed up (but I don't think you can lot).
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08/04/2019, 01:56 PM
(This post was last modified: 08/04/2019, 07:02 PM by sheldonison.)
(08/04/2019, 12:55 PM)Ember Edison Wrote: (08/02/2019, 09:34 PM)sheldonison Wrote: [edit2] The Schroeder function for neutral points does not converge if the period is a rational number. So a period=4 Schroeder function doesn't exist since it doesn't converge. But if the period is a well behaved irrational number (with imaginary part=0) then the Schroeder function does converge. For example, a period=pi Schroeder function does converge. See
https://en.wikipedia.org/wiki/Brjuno_number, and http://www.scholarpedia.org/article/Sieg...egel_disks. The Kneser Tetration and its inverse are analytic even if the period is rational. The experiment is to look at all of the Taylor series coefficients of the slog and see how they vary as the base changes in the complex plane. There is an analytic function for each of these Taylor series coefficients. So from the bases around the rational base, one can construct an arbitrarily accurate series even though the fatou.gp algorithm isn't ideal for computation of such neutral rational period bases. The last edit it's sounds strange. So Will the Kneser Tetration and its inverse can exist if Schroeder function doesn't exist?
The base = E^E is a singularity? B~=1.98933207608102 + 1.19328219946665*I;
B=bfromp(4); /* using fatou_experiment.gp */
l~=0.540302305868140 + 0.841470984807897*I;
There is no Schroeder function from this neutral fixed point for base B. even though Kneser Tetration is analytic for base B. For this base, the multiplier at the fixed point is , which is a period 4 multiplier since
I know how to use the Riemann mapping theorem showing that Kneser exists for real bases at the real axis. For complex bases, they start with a kernal around the base with a multipier of 1, where the fixed point bifurcates. For tetration that is eta=exp(1/e). And then one can use the Measurable Riemann mapping theorem to show the Abel function exists. But this is a pretty recent result in complex dynamics and I don't understand the details in spite of many attempts to read the paper by Lei Tan and Shishikura, reference here:
https://math.eretrandre.org/hyperops_wik...oordinates
Just like base B above is not a singularity for Kneser, base exp(e) with is not a singularity for Kneser either. I have no algorithm for calculating results for base(exp(e)), or to even to get approximate results. I am unaware of anyone else who can do calculations for this base either. The proofs using both the Riemann mapping theorem and the measurable Riemann mapping are proofs of existence that don't give a usable algorithm for calculations.
Adrienne Douady conjectured the merged Abel function can be analytically extended until you hit the neutral boundary a second time (the Shell Thron boundary for tetration). I posted a tetcomplex.gp program, which doesn't work anymore with current parigp, but there are some interesting results at this link: https://math.eretrandre.org/tetrationfor...hp?tid=729
 Sheldon
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(08/04/2019, 01:56 PM)sheldonison Wrote: B~=1.98933207608102 + 1.19328219946665*I;
B=bfromp(4); /* using fatou_experiment.gp */
l~=0.540302305868140 + 0.841470984807897*I;
There is no Schroeder function from this neutral fixed point for base B. even though Kneser Tetration is analytic for base B. For this base, the multiplier at the fixed point is , which is a period 4 multiplier since
I know how to use the Riemann mapping theorem showing that Kneser exists for real bases at the real axis. For complex bases, they start with a kernal around the base with a multipier of 1, where the fixed point bifurcates. For tetration that is eta=exp(1/e). And then one can use the Measurable Riemann mapping theorem to show the Abel function exists. But this is a pretty recent result in complex dynamics and I don't understand the details in spite of many attempts to read the paper by Lei Tan and Shishikura, reference here:
https://math.eretrandre.org/hyperops_wik...oordinates
Just like B, above is not a singularity for Kneser, base exp(e) with is not a singularity for Kneser either. I have no algorithm for calculating results for base(exp(e)), or to even to get approximate results. I am unaware of anyone else who can do calculations for this base either. The proofs using both the Riemann mapping theorem and the measurable Riemann mapping are proofs of existence that don't give a usable algorithm for calculations.
Adrienne Douady conjectured the merged Abel function can be analytically extended until you hit the neutral boundary a second time (the Shell Thron boundary for tetration). I posted a tetcomplex.gp program, which doesn't work anymore with current parigp, but there are some interesting results at this link: https://math.eretrandre.org/tetrationfor...hp?tid=729
Thank you.
by the why, how can I get the photo like this? https://math.eretrandre.org/tetrationfor...p?aid=1311
i need some code.
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08/06/2019, 04:17 PM
(This post was last modified: 08/06/2019, 05:03 PM by sheldonison.)
(08/04/2019, 07:06 PM)Ember Edison Wrote: Thank you.
by the why, how can I get the photo like this? https://math.eretrandre.org/tetrationfor...p?aid=1311
i need some code.
Its a combination of an excel spreadsheet; uniquecharts.xls download and the circchart function I recently added to fatou_experiment
fatou_experiment.gp (Size: 74.91 KB / Downloads: 82)
(also other cleanup and updates). I think I'm ready to update the main fatou.gp link. Also, I understand the singularities in the theta mapping for bases like B=0.1, and why so many sample points are needed for theta mapping (more later). Also, I understand why there is a difficult programming problem where the slog(B=0.1) converges beautifully within a circle connecting the fixed points, but even with the theta mapping, there are points outside the circle where the computation of the Schroder function leads to the slog being misbehaved even though the slog is accurate everywhere else. Obviously, this effects the sexp as well, since it is the inverse. More later, time is very limited right now.
B=0.1+I*E30; matrix_ir(B,300,300,14/15,24/25);
plot of slog between fixed points; l=0.399012978260252; l2=0.301845266020991+1.98089260122376*I;
The two misbehaving slivers are when the algorithm for determining which branch of the logarithm of Psi is broken; but this is a very difficult programming problem and my solution works except for these Tetration bases on the real axis near the origin.
 Sheldon
