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05/28/2019, 02:53 PM
(This post was last modified: 05/28/2019, 02:55 PM by Ember Edison.)
I was reading the article[1] ,it use 4 Method to Evaluate exp_b^z and b,z is complex.
but i was not find the Inversefunction in the article.
So,Can we get the holomorphic superroot and superlogarithm function and all parameter is complex?
[1]https://link.springer.com/article/10.1007/s1044401896157
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05/29/2019, 03:49 PM
(This post was last modified: 05/29/2019, 05:05 PM by sheldonison.)
(05/28/2019, 02:53 PM)Ember Edison Wrote: ... So,Can we get the holomorphic superroot and superlogarithm function and all parameter is complex? https://math.eretrandre.org/tetrationfor...p?tid=1017
fatou.gp implements Kneser's superlogarithm or inverse of tetration for complex bases and complex heights.
There is a proof of uniqueness of Kneser's slog if the slog between the two primary fixed points is defined, unique and one to one mapping in a region bounded by a sickle between the fixed points. One side of the sickle is a curve connecting the two fixed points, and the other side is the exponent of the curve. The algorithm for fatou.gp is a bit complicated, but it compute's Kneser's slog for a very wide range of real and complex bases.
I have reviewed other papers by the author, but not this one. The results from the other paper match fatou.gp exactly (within error limits), so I'm pretty sure that my program would also be the inverse slog function matching the author's results for complex base tetration as well. There is only one valid extension of Kneser's solution to complex tetration bases.
 Sheldon
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05/30/2019, 09:16 AM
(This post was last modified: 05/30/2019, 09:28 AM by Ember Edison.)
(05/29/2019, 03:49 PM)sheldonison Wrote: (05/28/2019, 02:53 PM)Ember Edison Wrote: ... So,Can we get the holomorphic superroot and superlogarithm function and all parameter is complex? https://math.eretrandre.org/tetrationfor...p?tid=1017
fatou.gp implements Kneser's superlogarithm or inverse of tetration for complex bases and complex heights.
There is a proof of uniqueness of Kneser's slog if the slog between the two primary fixed points is defined, unique and one to one mapping in a region bounded by a sickle between the fixed points. One side of the sickle is a curve connecting the two fixed points, and the other side is the exponent of the curve. The algorithm for fatou.gp is a bit complicated, but it compute's Kneser's slog for a very wide range of real and complex bases.
I have reviewed other papers by the author, but not this one. The results from the other paper match fatou.gp exactly (within error limits), so I'm pretty sure that my program would also be the inverse slog function matching the author's results for complex base tetration as well. There is only one valid extension of Kneser's solution to complex tetration bases. He use Cross Track method to evaluate tetration in real bases, larger than e^(1/e), Your's method to evaluate bases e^(1/e), SwordTrack to evaluate Shell Thron boundary, Double Dagger Track to Evaluate other complex bases.
I am reading your superlogarithm code,it look like can evaluate all complex bases. Thank you for your work.
So now what conclusion can get with complex superroot?
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05/30/2019, 11:33 PM
(This post was last modified: 06/01/2019, 05:40 PM by sheldonison.)
(05/30/2019, 09:16 AM)Ember Edison Wrote: (05/29/2019, 03:49 PM)sheldonison Wrote: (05/28/2019, 02:53 PM)Ember Edison Wrote: ... So,Can we get the holomorphic superroot and superlogarithm function and all parameter is complex? https://math.eretrandre.org/tetrationfor...p?tid=1017
fatou.gp implements Kneser's superlogarithm or inverse of tetration for complex bases and complex heights.
There is a proof of uniqueness of Kneser's slog if the slog between the two primary fixed points is defined, unique and one to one mapping in a region bounded by a sickle between the fixed points. One side of the sickle is a curve connecting the two fixed points, and the other side is the exponent of the curve. The algorithm for fatou.gp is a bit complicated, but it compute's Kneser's slog for a very wide range of real and complex bases.
I have reviewed other papers by the author, but not this one. The results from the other paper match fatou.gp exactly (within error limits), so I'm pretty sure that my program would also be the inverse slog function matching the author's results for complex base tetration as well. There is only one valid extension of Kneser's solution to complex tetration bases. He use Cross Track method to evaluate tetration in real bases, larger than e^(1/e), Your's method to evaluate bases e^(1/e), SwordTrack to evaluate Shell Thron boundary, Double Dagger Track to Evaluate other complex bases.
I am reading your superlogarithm code,it look like can evaluate all complex bases. Thank you for your work.
So now what conclusion can get with complex superroot?
I don't have access to a university right now, so I haven't downloaded Paulson's latest paper; I have downloaded two of his other papers. I would guess that it would be nice to show formally that one of these algorithms rigorously converge to Kneser. That's a complicated problem; whenever I think I'm close to being able to rigorously prove fatou.gp actually converges, I get sidetracked.
Also, fatou.gp struggles with bases on the Shell Thron boundary since one of the fixed points is neutral and the Schroder function may not converge so there is no theta mapping so my algorithm converges poorly for bases on the Shell Thron boundary. For bases on either side of the boundary, inside or outside the Shell region, my algorithm still works good, but it slows down quite a bit
sexpinit(2+1.1*I) works good
sexpinit(2+1.2*I) works good
sexpinit(2+1.15*I) initialization slow; takes 40 seconds; normal precision 34 decimal digits
sexpinit(2+1.16*I) base too close to the Shell Thron boundary; no upper theta mapping; precision 14 decimal digits
I haven't worked with superroots, and I don't have a super root algorithm for real or complex bases. The function is analytic in the teration base b, and instead one may use other values like n=0.5i; or n=4; or n=2.5+0.25i or any other value of interest. Then is the superroot for the value n in question and is also analytic.
 Sheldon
Posts: 62
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06/01/2019, 03:59 PM
(This post was last modified: 06/03/2019, 12:57 PM by Ember Edison.)
(05/30/2019, 11:33 PM)sheldonison Wrote: (05/30/2019, 09:16 AM)Ember Edison Wrote: (05/29/2019, 03:49 PM)sheldonison Wrote: (05/28/2019, 02:53 PM)Ember Edison Wrote: ... So,Can we get the holomorphic superroot and superlogarithm function and all parameter is complex? https://math.eretrandre.org/tetrationfor...p?tid=1017
fatou.gp implements Kneser's superlogarithm or inverse of tetration for complex bases and complex heights.
There is a proof of uniqueness of Kneser's slog if the slog between the two primary fixed points is defined, unique and one to one mapping in a region bounded by a sickle between the fixed points. One side of the sickle is a curve connecting the two fixed points, and the other side is the exponent of the curve. The algorithm for fatou.gp is a bit complicated, but it compute's Kneser's slog for a very wide range of real and complex bases.
I have reviewed other papers by the author, but not this one. The results from the other paper match fatou.gp exactly (within error limits), so I'm pretty sure that my program would also be the inverse slog function matching the author's results for complex base tetration as well. There is only one valid extension of Kneser's solution to complex tetration bases. He use Cross Track method to evaluate tetration in real bases, larger than e^(1/e), Your's method to evaluate bases e^(1/e), SwordTrack to evaluate Shell Thron boundary, Double Dagger Track to Evaluate other complex bases.
I am reading your superlogarithm code,it look like can evaluate all complex bases. Thank you for your work.
So now what conclusion can get with complex superroot?
I don't have access to a university right now, so I haven't downloaded Paulson's latest paper; I have downloaded two of his other papers. I would guess that it would be nice to show formally that one of these algorithms rigorously converge to Kneser. That's a complicated problem; whenever I think I'm close to being able to rigorously prove fatou.gp actually converges, I get sidetracked.
Also, fatou.gp struggles with bases on the Shell Thron region since there is no upper theta mapping so my algorithm converges poorly for bases on the Shell Thron region itself. For bases near the Shell region, it still works good, but it slows down quite a bit
sexpinit(2+1.1*I) works good
sexpinit(2+1.2*I) works good
sexpinit(2+1.15*I) initialization slow; takes 40 seconds; normal precision 34 decimal digits
sexpinit(2+1.16*I) base too close to the Shell Thron region; no theta mapping; precision 14 decimal digits
I haven't worked with superroots, and I don't have a super root algorithm for real or complex bases. The function is analytic in the teration base b, and instead one may use other values like n=0.5i; or n=4; or n=2.5+0.25i or any other value of interest. Then is the superroot for the value n in question and is also analytic.
You can see the Wolfram (Mathematica) code and javascript code about his article in here.
http://myweb.astate.edu/wpaulsen/tetration.html
But the javascript code can only evaluate 8 different base, and the Wolfram code look like a full of bullshit.I can't find the OffSet[] algorithm in double dagger track method. In the SwordTrack Method Wolfram code say he can Evaluate Shell Thron, But psi2[] will be fucking infinite loop with Sheldon base  and javascript code can work in Sheldon base,is fucking carzy! And ready work for new base is very slowly (900s ~ 500s).
The most fucking thing is, the Wolfram Kernel sometime will crash or Memory leak when Abs@Tetrate[b,z] > 10^10^8 or Abs@Tetrate[b,z] < 10^10^8.The shit Kernel destroy my plot function when i want to use your plot style to test the code. If you want to debug the bullshit, you would be best to use Wolfram 12 and be careful the memory leak.DO NOT USE Wolfram 11/10!
https://drive.google.com/drive/folders/1...HxaG1x087A
Code: <<"code.CrossTrack.nb"
(* Load method.
Use Cross in real bases, larger than e^(1/e),
SwordTrack in Shell Thron boundary,
Double Dagger Track in other bases. *)
Boot[E];(* set base *)
(* CrossTrackPrecision[60] *)(* reset a better Precision. Automatic = nn *)
Main[1+I] (* Tetrate[b,z] === Boot[b];Main[z] *)
Ps: How to get the photo like you in pari/gp? I need some plot code.
https://math.eretrandre.org/tetrationforum/showthread.php?tid=1017
ok, i see MakeGraph(). But why i use write() nothing in my test file? ok, i use the wrong slash.
Ps2:How far are we get a perfect code to evaluate superroots if we get the perfect tetration code?
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06/02/2019, 07:51 AM
(This post was last modified: 06/02/2019, 08:53 AM by sheldonison.)
(06/01/2019, 03:59 PM)Ember Edison Wrote: Ps: How to get the photo like you in pari/gp? I need some plot code.
https://math.eretrandre.org/tetrationforum/showthread.php?tid=1017
ok, i see MakeGraph().But why i use write() nothing in my test file?
Ps2:How far are we get a perfect code to evaluate superroots if we get the perfect tetration code?
Thanks for the links. I had seen Paulson's javascript code before, when I exchanged emails with the author last May.
I'm assuming you got MakeGraph to generate plots; This MakeGraph example takes 20minutes???
sexpinit(exp(1));
fmode=3; /* f(z) is used by MakeGraph; fmode=3 sets f(z) to safesexp(z) */
MakeGraph(1080,540,3,3,9,3,"sexp_e.ppm",1); /* from 3 to +9 at real axis, 3 to +3 imaginary */
write ("foo.txt","hello");
parigp is really slow writing one pixel at a time; attached is a faster version of MakeGraph; I also have faster sexp inversion code, but the faster sexp code has bugs with some complex bases, so I didn't include it.
MakeGraphSpeedup.gp (Size: 3 KB / Downloads: 347)
I think one needs to start by understanding the inverse of superroot before one can understand superroot since you need to take the inverse of sexp_b(n) for some particular value of n for which the superroot is interested in, so I would think you want to understand how sexp_b(1+0.5*I) behaves for example, for the variable b. What if n is between 0 and 1? Is sexp_b(0.5) bounded as base(b) changes? I have no idea. So obviously, I don't understand the inverse of the function, much less have a perfect code ... My suggestion is to start with Newton's method and iterate on finding approximations for the inverse, but Newton's method relies on a guess being close enough to the correct answer to converge. For some particular value of n, I think I might know how to numerically compute the Taylor series for the function given some starting point since I had done something like that earlier to show complex tetration is analytic in the base.
 Sheldon
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06/08/2019, 05:22 AM
(This post was last modified: 06/08/2019, 02:14 PM by Ember Edison.)
(06/02/2019, 07:51 AM)sheldonison Wrote: (06/01/2019, 03:59 PM)Ember Edison Wrote: Ps: How to get the photo like you in pari/gp? I need some plot code.
https://math.eretrandre.org/tetrationforum/showthread.php?tid=1017
ok, i see MakeGraph().But why i use write() nothing in my test file?
Ps2:How far are we get a perfect code to evaluate superroots if we get the perfect tetration code?
Thanks for the links. I had seen Paulson's javascript code before, when I exchanged emails with the author last May.
I'm assuming you got MakeGraph to generate plots; This MakeGraph example takes 20minutes???
sexpinit(exp(1));
fmode=3; /* f(z) is used by MakeGraph; fmode=3 sets f(z) to safesexp(z) */
MakeGraph(1080,540,3,3,9,3,"sexp_e.ppm",1); /* from 3 to +9 at real axis, 3 to +3 imaginary */
write ("foo.txt","hello");
parigp is really slow writing one pixel at a time; attached is a faster version of MakeGraph; I also have faster sexp inversion code, but the faster sexp code has bugs with some complex bases, so I didn't include it.
I think one needs to start by understanding the inverse of superroot before one can understand superroot since you need to take the inverse of sexp_b(n) for some particular value of n for which the superroot is interested in, so I would think you want to understand how sexp_b(1+0.5*I) behaves for example, for the variable b. What if n is between 0 and 1? Is sexp_b(0.5) bounded as base(b) changes? I have no idea. So obviously, I don't understand the inverse of the function, much less have a perfect code ... My suggestion is to start with Newton's method and iterate on finding approximations for the inverse, but Newton's method relies on a guess being close enough to the correct answer to converge. For some particular value of n, I think I might know how to numerically compute the Taylor series for the function given some starting point since I had done something like that earlier to show complex tetration is analytic in the base. This sound like research the tetration can not get key helpful for superroot.
This is too regrettable, the all inverse function for tetration just has superroot was nonComplex.
Ps:the Tetration Wiki not have any superroot information. It's too carzy.
Ps2:If i just need 6 decimal digits for precision, can you speed up the code? You know, the human's eye can not get too high precision for color.
Ps3:How to recode the sexp and slog to catch Underflow and Overflow? Yon know, underflow will become overflow in pari/gp.
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06/09/2019, 12:24 AM
(This post was last modified: 06/09/2019, 02:29 PM by sheldonison.)
(06/08/2019, 05:22 AM)Ember Edison Wrote: This sound like research the tetration can not get key helpful for superroot.
This is too regrettable, the all inverse function for tetration just has superroot was nonComplex.
Ps:the Tetration Wiki not have any superroot information. It's too carzy.
Ps2:If i just need 6 decimal digits for precision, can you speed up the code? You know, the human's eye can not get too high precision for color.
Ps3:How to recode the sexp and slog to catch Underflow and Overflow? Yon know, underflow will become overflow in pari/gp.
There aren't that many people in the world who even know how to write a complex base tetration program ... For complex bases, fatou.gp probably initializes 100x faster than everything else written so it makes sense that there isn't much out there on superroots for non integer values. Moreover, I consider it a technical tourdeforce that fatou.gp converges over as much of the upper complex plane as it does, which is also over a larger region than anything else written. Anyway, I spent a fair amount of time playing with the superroot of height n=1.5. Most of the limits of where I can't get accurate values for the superroot involve tetration to bases close to zero. So these are the same limitations we have in using fatou.gp for less well behaved bases. Yet the superroot is clearly an analytic function with a well behaved Taylor series in its well behaved regions.
The nearest singularity is at base eta=exp(1/e) where there is a surprisingly quiet branch so that extrending Kneser to real bases<eta are no longer real valued at the real axis, but the imaginary compoenent of sexp starts out quiet. So, I'm interested in in solving the problem f(z)=Tet_z(1.5); superroot_1.5(z)=f^{1}(z). Since iterated exponentials always misbehave, we can equivalently sample
f(z)=z^z^Tet_z(0.5); superroot_1.5(z)=f^{1}(z)
I know there is a singularity at eta. I start by sampling a circle big enough to just touch the negative real axis so I could start understanding how to solve the superroot_1.5(z). But I also need to avoid the singularity at eta so the left edge of the circle is at 1.5 which is a little bigger than eta. Tet_z(1.5) is analytic and one to one in this entire circle, although Tet_z(0.5) is better behaved in terms of Taylor series approximations.
This shows the mapping of the circle above.
And this shows a blowup of negative real axis where the complex logarithmic branch of Tet_z(1.5) starts to take on negative values.
So I think I can get the superroot_1.5 anywhere in the complex plane except for a region pretty close to the origin, by using the routines I posted below. I can also generate the Taylor series for the superroot, by taking the inverse of the z^^h. This example takes about 34 minutes to sample 120 Tetration bases, to get 240 sample points around a circle centered at 4, radius=2, for Tet_1.5. This is a smaller more well behaved circle than the one above, and the sroot(z) is accurate to >30 decimal digits where the Taylor series is well behaved.
superroot.gp (Size: 3.89 KB / Downloads: 345)
Code: \r fatou.gp
\r superroot.gp
gen_sroot(4,2,1.5);
helps();
zb= sroot(5)); /* base~=2.674... via taylor series of gm */
sexpinit(zb);sexp(1.5) /* sexp(1.5) almost exactly 5; accurate to >32 digits */
zb = sroot(6+5*I);
sexpinit(zb);sexp(6+5*I); /* another example */
What if you want fast results and results over more of the complex plane?
gen_sroot(15,13.5,0.5); /* initialize over a fairly big circle ... */
zb=invzzztth(10) /* approx 2.232942146388 + 4.126815378582*I */
sexpinit(zb); sexp(1.5) /*10.00002382582  1.225114519020 E5*I */
zb=newtonsroot(10,zb) /* more exact value */
sexp(1.5) /* 10 */
This approximation is camera accurate if Re(z)<=9.2 or Re(z)>=1.7, or Im(z)>2.8i, as can be seen in the region of poor behavior shown below.
newtonsroot(z,invzzztth(z)) /* For accurate results inside poorly behaved region with Im(z)>0 */
This approach should also work with other heights with real(height) between 1 and 2; as long as imag(z) is small. This is the only other height I've tried so far.
/* setup approximation for superroot(1.5+0.5*I)(z); 34 minutes */
gen_sroot(15,13.5,0.5+0.5*I); /* initialize over a fairly big circle */
zb=invzzztth(8*I); /* 4.239026742708 + 0.01949705866750*I */
sexpinit(zb); /* 0.0004576177155245 + 8.000948294760*I */
sexp(1.5+0.5*i);
zb=newtonsroot(8*I,zb);
fyi; the gen_sroot only supports centering samples on the real axis. One could imagine extensions of invzzztth to try to get better approximations over more of the poorly behaved region... It would be easy to implement the inverse of z^z^z^(ztth(z)) to handle superroot with Re(height) between 2 and 3 etc.
 Sheldon
Posts: 62
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06/09/2019, 04:03 PM
(This post was last modified: 06/10/2019, 12:39 AM by Ember Edison.)
(06/09/2019, 12:24 AM)sheldonison Wrote: (06/08/2019, 05:22 AM)Ember Edison Wrote: This sound like research the tetration can not get key helpful for superroot.
This is too regrettable, the all inverse function for tetration just has superroot was nonComplex.
Ps:the Tetration Wiki not have any superroot information. It's too carzy.
Ps2:If i just need 6 decimal digits for precision, can you speed up the code? You know, the human's eye can not get too high precision for color.
Ps3:How to recode the sexp and slog to catch Underflow and Overflow? Yon know, underflow will become overflow in pari/gp.
There aren't that many people in the world who even know how to write a complex base tetration program ... For complex bases, fatou.gp probably initializes 100x faster than everything else written so it makes sense that there isn't much out there on superroots for non integer values. Moreover, I consider it a technical tourdeforce that fatou.gp converges over as much of the upper complex plane as it does, which is also over a larger region than anything else written. Anyway, I spent a fair amount of time playing with the superroot of height n=1.5. Most of the limits of where I can't get accurate values for the superroot involve tetration to bases close to zero. So these are the same limitations we have in using fatou.gp for less well behaved bases. Yet the superroot is clearly an analytic function with a well behaved Taylor series in its well behaved regions.
The nearest singularity is at base eta=exp(1/e) where there is a surprisingly quiet branch so that extrending Kneser to real bases<eta are no longer real valued at the real axis, but the imaginary compoenent of sexp starts out quiet. So, I'm interested in in solving the problem f(z)=Tet_z(1.5); superroot_1.5(z)=f^{1}(z). Since iterated exponentials always misbehave, we can equivalently sample
f(z)=z^z^Tet_z(0.5); superroot_1.5(z)=f^{1}(z)
I know there is a singularity at eta. I start by sampling a circle big enough to just touch the negative real axis so I could start understanding how to solve the superroot_1.5(z). But I also need to avoid the singularity at eta so the left edge of the circle is at 1.5 which is a little bigger than eta. Tet_z(1.5) is analytic and one to one in this entire circle, although Tet_z(0.5) is better behaved in terms of Taylor series approximations.
This shows the mapping of the circle above.
And this shows a blowup of negative real axis where the complex logarithmic branch of Tet_z(1.5) starts to take on negative values.
So I think I can get the superroot_1.5 anywhere in the complex plane except for a region pretty close to the origin, by using the routines I posted below. I can also generate the Taylor series for the superroot, by taking the inverse of the z^^h. This example takes about 34 minutes to sample 120 Tetration bases, to get 240 sample points around a circle centered at 4, radius=2, for Tet_1.5. This is a smaller more well behaved circle than the one above, and the sroot(z) is accurate to >30 decimal digits where the Taylor series is well behaved.
Code: \r fatou.gp
\r superroot.gp
gen_sroot(4,2,1.5);
helps();
zb= sroot(5)); /* base~=2.674... via taylor series of gm */
sexpinit(zb);sexp(1.5) /* sexp(1.5) almost exactly 5; accurate to >32 digits */
zb = sroot(6+5*I);
sexpinit(zb);sexp(6+5*I); /* another example */
What if you want fast results and results over more of the complex plane?
gen_sroot(15,13.5,0.5); /* initialize over a fairly big circle ... */
zb=invzzztth(10) /* approx 2.232942146388 + 4.126815378582*I */
sexpinit(zb); sexp(1.5) /*10.00002382582  1.225114519020 E5*I */
zb=newtonsroot(10,zb) /* more exact value */
sexp(1.5) /* 10 */
This approximation is camera accurate if Re(z)<=9.2 or Re(z)>=1.7, or Im(z)>2.8i, as can be seen in the region of poor behavior shown below.
newtonsroot(z,invzzztth(z)) /* For accurate results inside poorly behaved region with Im(z)>0 */
This approach should also work with other heights with real(height) between 1 and 2; as long as imag(z) is small. This is the only other height I've tried so far.
/* setup approximation for superroot(1.5+0.5*I)(z); 34 minutes */
gen_sroot(15,13.5,0.5+0.5*I); /* initialize over a fairly big circle */
zb=invzzztth(8*I); /* 4.239026742708 + 0.01949705866750*I */
sexpinit(zb); /* 0.0004576177155245 + 8.000948294760*I */
sexp(1.5+0.5*i);
zb=newtonsroot(8*I,zb);
fyi; the gen_sroot only supports centering samples on the real axis. One could imagine extensions of invzzztth to try to get better approximations over more of the poorly behaved region... It would be easy to implement the inverse of z^z^z^(ztth(z)) to handle superroot with Re(height) between 2 and 3 etc.
Thank you for your work.It's very helpful.
by the way...I can't use sexpinit() set the base with 0.1 <= x <= 0.62, 0.88 <= x <= 1.042, how can i do something to fix it?
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06/09/2019, 04:31 PM
(This post was last modified: 06/10/2019, 03:33 AM by sheldonison.)
(06/09/2019, 04:03 PM)Ember Edison Wrote: Thank you for your work.It's very helpful.
by the way...I can't set the base with 0.1 <= x <= 0.62, 0.88 <= x <= 1.042, how can i do something to fix it?
Instead of b, try b+1E25*I so that its not on the real axis; that might help a little. But if using fatou.gp for bases close to zero, the sexpinit breaks since the real part of the pseudo period of the fixed points gets too small so the theta mapping falls apart. base=0 is a singularity. For real bases>1 and less than 1.042, the problems are different as the imaginary period gets smaller and smaller. base=1 is also a singularity. Its probably best to accept these as the unavoidable computation limits. So then the superroot problem mostly has the same computation limits as extending Kneser to complex bases and bases<eta.
edit: Its been awhile since I've looked closely at the singularity for circling around the base=1. sexp(0.5) is complex valued for bases<1, but the branch is more significant than for 1<b<eta where both fixed points are still real valued. It is because the secondary fixed point is no longer real valued. for sexpinit(0.7) the two fixed points are L1~=0.762 and L2~=7.3505 +11.632i. Kneser's solution requires us to weave together the two Schroder/Abel functions from the two fixed points. This is done with a 1periodic theta mapping, z+theta(z). This this allows the complex conjugate pair of fixed points for real bases>eta to generate a real valued tetration function. Perhaps I will post more later.
 Sheldon
