complex iteration (complex "height")

[Gottfried]

I was only interested whether our results match. But one couldnt see from the graph what \( \sqrt{2}[4]I \) is, so I would be glad if you could compare this with my result of
\( \sqrt{2}[4]I=1.210309011+.5058275611*I \)

Yepp, that's exactly the value that I've got for height h=I (up to 7'th digit)
In my excel-file it is real=1.2103090 imag = 0.50582757


[update] with higher precision I recomputed the value and got in the 92..96 partial-sums the following approximations:

Code:

´
  1.210309025559961+0.5058275713618201*I
  1.210309025559961+0.5058275713618201*I
  1.210309025559961+0.5058275713618201*I
  1.210309025559961+0.5058275713618201*I
  1.210309025559961+0.5058275713618201*I
---------------------------------------------
  1.210309011      + .5058275611          *I  <--- yours

So this differs from the 7'th digit; anyway - I just computed this with the function
U_t (x) = t^x - 1
T_b(x) = b^x

U_t°h(x) and T_b°h(x) as their iterates of general height h

and the shift
T_b°h(x) = (U_t°h(x/t-1) +1)*t

and the height-iteration by applying powers to the diagonal of the diagonalized operator for U_t(x) with 96 terms, using b=sqrt(2),t=2,x=1

If your values have full accuracy, then there must be a methodical difference, which I would like to find; I'll check today with other fixpoint-shifts.

Gottfried