new fatou.gp program

[sheldonison]

Hey, Sheldonison, why fatou.gp only shows 15 digits?

Is only for aesthetic reasons? I mean, can the other digits be used as valid data?

A 512x512 matrix gives results accurate to about 15-16 decimal digits for base(e), \( z\mapsto \exp(z) \). The fatou.gp pari-program uses theta mapping to iteratively solve p(z), to dramatically improve convergence.

^^oh. I see...

Also, yes, the formatting is entirely cosmetic, and has no impact on the computation precision. The computation precision is "\p 28" by default, 28 decimal digits, which normally gives 23-25 decimal digits precision for fatou.gp. But there are bases near the Shell Thron boundary where there's no theta mapping. Without a theta mapping, you only get 15-16 decimal digits of precision in fatou.gp

For bases with imag(b)<0, you could write your own code, as follows. Someday I may support bases<eta with imag(b)<0 where the rotation angle is >Pi, which would be similar to tetcomplex, but it turns out its a harder problem, so I can't get the sexp(z) to work for those upside down bases. So I disabled it. But that's probably not what you're interested in anyway, you just want bases with imag(b)<0, hopefully for bases greater than eta. So here's some quick code which I haven't tested which will probably does exactly what you want. Also, there's a newer version of fatou.gp in your thread that may be of interest to you.

Code:

mysexpinit(b) = {
if (imag(b)<0,
  sexpinit(conj(b));
  conjugatemode=1;
,
  sexpinit(b)
  conjugatemode=0;
);
}
mysexp(z) = {
  if (conjugatemode,
    return(conj(sexp(conj(z))));
  ,
    return(sexp(z));
  );
}
myslog(z) = {
  if (conjugatemode,
    return(conj(slog(conj(z))));
  ,
    return(slog(z));
  );
}