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 Can we get the holomorphic super-root and super-logarithm function? Ember Edison Fellow Posts: 67 Threads: 7 Joined: May 2019 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 super-root and super-logarithm function and all parameter is complex? [1]https://link.springer.com/article/10.1007/s10444-018-9615-7 sheldonison Long Time Fellow Posts: 683 Threads: 24 Joined: Oct 2008 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 super-root and super-logarithm function and all parameter is complex?https://math.eretrandre.org/tetrationfor...p?tid=1017 fatou.gp implements Kneser's super-logarithm 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 Ember Edison Fellow Posts: 67 Threads: 7 Joined: May 2019 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 super-root and super-logarithm function and all parameter is complex?https://math.eretrandre.org/tetrationfor...p?tid=1017 fatou.gp implements Kneser's super-logarithm 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), Sword-Track to evaluate Shell Thron boundary, Double Dagger Track to Evaluate other complex bases. I am reading your super-logarithm code,it look like can evaluate all complex bases. Thank you for your work. So now what conclusion can get with complex super-root? sheldonison Long Time Fellow Posts: 683 Threads: 24 Joined: Oct 2008 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 super-root and super-logarithm function and all parameter is complex?https://math.eretrandre.org/tetrationfor...p?tid=1017 fatou.gp implements Kneser's super-logarithm 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), Sword-Track to evaluate Shell Thron boundary, Double Dagger Track to Evaluate other complex bases. I am reading your super-logarithm code,it look like can evaluate all complex bases. Thank you for your work. So now what conclusion can get with complex super-root? 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 super-roots, and I don't have a super root algorithm for real or complex bases.  The function $f(b) = \text{Tet}_b(0.5i)\;\;$ 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 $f^{-1}(z)$ is the super-root for the value n in question and is also analytic. - Sheldon Ember Edison Fellow Posts: 67 Threads: 7 Joined: May 2019 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 super-root and super-logarithm function and all parameter is complex?https://math.eretrandre.org/tetrationfor...p?tid=1017 fatou.gp implements Kneser's super-logarithm 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), Sword-Track to evaluate Shell Thron boundary, Double Dagger Track to Evaluate other complex bases. I am reading your super-logarithm code,it look like can evaluate all complex bases. Thank you for your work. So now what conclusion can get with complex super-root? 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 super-roots, and I don't have a super root algorithm for real or complex bases.  The function $f(b) = \text{Tet}_b(0.5i)\;\;$ 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 $f^{-1}(z)$ is the super-root 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 Sword-Track 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),  Sword-Track 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 super-roots if we get the perfect tetration code? sheldonison Long Time Fellow Posts: 683 Threads: 24 Joined: Oct 2008 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 super-roots 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"); pari-gp 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: 388)   I think one needs to start by understanding the inverse of super-root before one can understand super-root since you need to take the inverse of sexp_b(n) for some particular value of n for which the super-root 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 Ember Edison Fellow Posts: 67 Threads: 7 Joined: May 2019 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 super-roots 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"); pari-gp 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 super-root before one can understand super-root since you need to take the inverse of sexp_b(n) for some particular value of n for which the super-root 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 super-root. This is too regrettable, the all inverse function for tetration just has super-root was non-Complex. Ps:the Tetration Wiki not have any super-root 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. sheldonison Long Time Fellow Posts: 683 Threads: 24 Joined: Oct 2008 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 super-root. This is too regrettable, the all inverse function for tetration just has super-root was non-Complex. Ps:the Tetration Wiki not have any super-root 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 tour-de-force 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 bases30 decimal digits where the Taylor series is well behaved.   superroot.gp (Size: 3.89 KB / Downloads: 387) 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 E-5*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); 3-4 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 Ember Edison Fellow Posts: 67 Threads: 7 Joined: May 2019 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 super-root. This is too regrettable, the all inverse function for tetration just has super-root was non-Complex. Ps:the Tetration Wiki not have any super-root 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 tour-de-force 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 bases30 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 E-5*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); 3-4 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? sheldonison Long Time Fellow Posts: 683 Threads: 24 Joined: Oct 2008 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+1E-25*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 baseseta to generate a real valued tetration function.  Perhaps I will post more later. - Sheldon « Next Oldest | Next Newest »

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