Well, I didn't get the regular tetration for complex fixpoints properly working until now, but because you show that picture I'll give it a new try. Surprise, surprise... I got the regular tetration for base exp(1) working, also for fractional heights... simply when I applied my newer routines to the old stuff...
What I did was to use the integer-height tetrate z1 = itet(z0,-10) of height -10 (where z0 was taken from x0 in my initial post); computed via the schroeder-function and its inverse the fractional values in the interval of h=0..2 and lifted the results by integer-height tetration of +10.
With this I got the graph below.
Now: in my computation I had no situation where I had to change the branch for logarithm. What's going on here? I understood your msg such, that you had to change the branch in the near of h=1.49 ? Or did you mean, one *had to* change the branch *if* one wanted to proceed on the real axis alone (at h=1.49** )?
Besides, in my regular tetration version now the real point is at height 1.498+eps - does this conform with your value?
Gottfried
[Update2]: Well, just it occurs to me, that at h1~1.43 and h1~1.93 ~ h1+0.5 the curve crosses itself. After some binary search I find, that the height-difference is not exactly 0.5, so this does not indicate a "half-iterate-fixpoint", but something possibly with some irrational period?
The point is in the height-intervals h1 in 1.4371584...1.4372544 and h2 in 1.94060196...1.94062644 .
The point itself is in the area
[ -30.6552101206 + 12.4460532739*I, -30.6707389330 + 12.4324857727*I ;
-30.6634224893 + 12.4493356383*I, -30.6641828863 + 12.4300037481*I]
[/update2]
Appendix: The first picture is the overview using z0=2.38551673096 + 1.30321372969*I for h=0..2 in steps of delta-h=0.01
The second picture shows the detail at h=1.49..1.5 where the curve crosses the real axis. By binary-search I get h_r=1.49800142290, z_r = -39.4162061931 + 0*î
Gottfried
What I did was to use the integer-height tetrate z1 = itet(z0,-10) of height -10 (where z0 was taken from x0 in my initial post); computed via the schroeder-function and its inverse the fractional values in the interval of h=0..2 and lifted the results by integer-height tetration of +10.
With this I got the graph below.
Now: in my computation I had no situation where I had to change the branch for logarithm. What's going on here? I understood your msg such, that you had to change the branch in the near of h=1.49 ? Or did you mean, one *had to* change the branch *if* one wanted to proceed on the real axis alone (at h=1.49** )?
Besides, in my regular tetration version now the real point is at height 1.498+eps - does this conform with your value?
Gottfried
[Update2]: Well, just it occurs to me, that at h1~1.43 and h1~1.93 ~ h1+0.5 the curve crosses itself. After some binary search I find, that the height-difference is not exactly 0.5, so this does not indicate a "half-iterate-fixpoint", but something possibly with some irrational period?
The point is in the height-intervals h1 in 1.4371584...1.4372544 and h2 in 1.94060196...1.94062644 .
The point itself is in the area
[ -30.6552101206 + 12.4460532739*I, -30.6707389330 + 12.4324857727*I ;
-30.6634224893 + 12.4493356383*I, -30.6641828863 + 12.4300037481*I]
[/update2]
Appendix: The first picture is the overview using z0=2.38551673096 + 1.30321372969*I for h=0..2 in steps of delta-h=0.01
The second picture shows the detail at h=1.49..1.5 where the curve crosses the real axis. By binary-search I get h_r=1.49800142290, z_r = -39.4162061931 + 0*î
Gottfried
Gottfried Helms, Kassel
