pentation and hexation

[sheldonison]

Now onto Pentation.  The latest version of fatou.gp also has built in support for pentation for real bases, but lets start with the integer values for Pentation.

Hi, Do you have any idea make pentinit() support complex base? The real base>eta is Kneser's solution in sexpinit(), like base<E^(-E) and complex base.


Take a step back, base>eta is not All real field.

pentation, as implemented, relies on the Schroeder/Konig Abel function from the lower fixed point of tetration where tet(z)=z.  This is somewhere between tet(-2)=-infinity and tet(-1)=0.  Its a simpler solution than Kneser's solution; also less elegent and less well behaved.

Unfortunately, I'm too busy to write any code right now, and complex base pentation would seem to be a very large effort project.  The existing pentation code starts with an estimate of -1.97 and finds the fixed point iterating z=slog(z), and generates the Pentation solution, assuming it finds a repelling fixed point.  I think for complex bases it crashes when it tries to estimate the accuracy of the resulting Pentation solution, so it might not be too difficult for you to experiment with it.
  n=0;while (z1<1,z1=sexp(z1);n++);  

Code:

  z1 = subst(pentz,x,pentr);
  z2 = subst(pentz,x,pentr/lambdat);
  n=0;while (z1<1,z1=sexp(z1);n++);  
  n++;while (n>0, z2=sexp(z2);n--;);
  errz = -log(abs(z1-z2))/log(10);
...