02/29/2012, 10:28 PM
(This post was last modified: 03/12/2012, 03:32 PM by sheldonison.)

This is the link to tetcomplex.gp pari gp code, which generates tetration for arbitrary bases in the complex plane, using the merged fixed point solution. You'll need to download par-gp.. Then load tetcomplex.gp and init(b), for a complex base, and then sexp(z) gives the resulting tetration function at tet(z). tet(0)=1.

tetcomplex.gp (Size: 59.81 KB / Downloads: 304) updated March 11th, improved convergence near eta, and works with arbitrary precision settings. Previous updates improved convergence near Shell Thron boundary, and for real bases<eta, and as imag(z) gets bigger The program is now working reasonably well, but is not fully debugged. The program generalizes the ideas Mike and I talked about, and Mike implemented, for generating merged fixed point tetration solutions in the complex plane. Mike has been advocating these merged fixed point solutions as the natural extension of tetration at the real axis, for bases>eta, as well as Dimitrii Kouznetsov. This program goes beyond what Mike did, in that it works for arbitrary bases, inside and outside the Shell Thron region. For bases not too close to one and not too close to the Shell Thron boundary, this program will generate both primary fixed points, using this algorithm, and figures out if they're both repelling, or if one of them is attracting. Then it will generate both Schroder functions and their inverses for both fixed points, and use those to generate superfunctions and superfunction inverses for both fixed points. Then the program iterates, generating two theta(z) mappings, one for the upper fixed point superfunction, and one for the low fixed point superfunction. Between these, is an sexp(z) taylor series. For imaginary(z)>0.175, , and for imaginary(z)<-0.175, . For bases at the real axis > eta, this is mathematically identical to my Kneser tetration algorithm.

The program is fast. The program does have bugs, although it is already working surprisingly well. Also, if the base is too close to the Shell Thron boundary, the program gets really slow, and eventually stops converging. The program comes with a fast complex plane graphing algorithm, MakeGraph. So the default precision is \p 28, which allows very fast accurate results to 12 decimal digits, for making graphs. You can increase the prcecision to \p 67, and results should work for bases between 1.1 and 3. For some bases, higher precision works, up to \p 134, probably depending on the accuracy of the theta, and the superfunction and the initialization for that base. This routine lacks many of the complicated optimizations, that allow kneser.gp to work for a very wide range of real bases. The program will aid in the study of tetration in the complex plane. Below, are a few graphing examples. The program works best with imag(base)>0, and gives novel interesting solutions for some bases with imag(b)<0, where the base is inside the shell thron boundary.

- Sheldon

base=2+0.56i, which is just inside the Shell Thron boundary. This pair of solutions (this one and the one below), seems to indicate that a merged tetration solution on the Shell Thron boundary itself works!

base=2+0.51i, which is just outside the Shell Thron boundary, with both fixed points repelling. This is a natural extension of tetration for real bases>eta.

base=1.45+0.3i, which is tetration where the two superfunctions are roughly at 45 degree angles, and -135 degree angles, from the sexp(z) at z=0.

base=1.45-0.3i. This result totally blew me away, because the upper fixed point, for imag(z)>0, for is 1.2125955-0.6939i, which has imag(z)<0, and the lower fixed point is 1.6459+3.82111i, which has imag(z)>0. Also, the superfunction gets more chaotic as imag(z) decreases for the lower fixed point, which is also interesting. The alternative solution, rotating in the other direction around eta, is the complex conjugate solution of the solution above, which makes much more sense, with the fixed points in the "correct" locations. I wonder if this solution can be extended, as we continue rotating further around eta?

- Sheldon

tetcomplex.gp (Size: 59.81 KB / Downloads: 304) updated March 11th, improved convergence near eta, and works with arbitrary precision settings. Previous updates improved convergence near Shell Thron boundary, and for real bases<eta, and as imag(z) gets bigger The program is now working reasonably well, but is not fully debugged. The program generalizes the ideas Mike and I talked about, and Mike implemented, for generating merged fixed point tetration solutions in the complex plane. Mike has been advocating these merged fixed point solutions as the natural extension of tetration at the real axis, for bases>eta, as well as Dimitrii Kouznetsov. This program goes beyond what Mike did, in that it works for arbitrary bases, inside and outside the Shell Thron region. For bases not too close to one and not too close to the Shell Thron boundary, this program will generate both primary fixed points, using this algorithm, and figures out if they're both repelling, or if one of them is attracting. Then it will generate both Schroder functions and their inverses for both fixed points, and use those to generate superfunctions and superfunction inverses for both fixed points. Then the program iterates, generating two theta(z) mappings, one for the upper fixed point superfunction, and one for the low fixed point superfunction. Between these, is an sexp(z) taylor series. For imaginary(z)>0.175, , and for imaginary(z)<-0.175, . For bases at the real axis > eta, this is mathematically identical to my Kneser tetration algorithm.

The program is fast. The program does have bugs, although it is already working surprisingly well. Also, if the base is too close to the Shell Thron boundary, the program gets really slow, and eventually stops converging. The program comes with a fast complex plane graphing algorithm, MakeGraph. So the default precision is \p 28, which allows very fast accurate results to 12 decimal digits, for making graphs. You can increase the prcecision to \p 67, and results should work for bases between 1.1 and 3. For some bases, higher precision works, up to \p 134, probably depending on the accuracy of the theta, and the superfunction and the initialization for that base. This routine lacks many of the complicated optimizations, that allow kneser.gp to work for a very wide range of real bases. The program will aid in the study of tetration in the complex plane. Below, are a few graphing examples. The program works best with imag(base)>0, and gives novel interesting solutions for some bases with imag(b)<0, where the base is inside the shell thron boundary.

- Sheldon

base=2+0.56i, which is just inside the Shell Thron boundary. This pair of solutions (this one and the one below), seems to indicate that a merged tetration solution on the Shell Thron boundary itself works!

base=2+0.51i, which is just outside the Shell Thron boundary, with both fixed points repelling. This is a natural extension of tetration for real bases>eta.

base=1.45+0.3i, which is tetration where the two superfunctions are roughly at 45 degree angles, and -135 degree angles, from the sexp(z) at z=0.

base=1.45-0.3i. This result totally blew me away, because the upper fixed point, for imag(z)>0, for is 1.2125955-0.6939i, which has imag(z)<0, and the lower fixed point is 1.6459+3.82111i, which has imag(z)>0. Also, the superfunction gets more chaotic as imag(z) decreases for the lower fixed point, which is also interesting. The alternative solution, rotating in the other direction around eta, is the complex conjugate solution of the solution above, which makes much more sense, with the fixed points in the "correct" locations. I wonder if this solution can be extended, as we continue rotating further around eta?

- Sheldon