Hey, so I did run some protocols on the cusp of the Shell-Thron region. It does get a little whacky. But on the main strip it works very well. You should note here, that the glitchy areas are actually a plethora of branch cuts and mixing problem between the lower/upper planes. I will definitely have to figure out how to code this better, it looks a little off. But still the branch cuts are measure zero under \(\mathbb{R}^2\), but definitely something is going on here I'll have to examine.
Here is, \(b = \exp(-0.1-e+0.1i)\) for \(\lambda = 1\):
Here is, \(b = \exp(0.3-e+i)\) for \(\lambda = 1\):
Here is, \(b = exp(0.1-e+0.5i)\) for \(\lambda = 1\):
The trouble is probably forcing a real period on a tetration that has a naturally complex period. I'll run some more tests, and see if I can modify some of the parameters to work around these TV static pictures. It could also be related to the fact these graphs were made with very low precision, so it could be a precision error.
Regards, James
EDIT:
So definitely the TV static in the above pictures is low precision error. I'm rerunning the graphs with higher precision/higher series precision, and the TV static is totally disappearing, forming an actual detail.
My suggestion would then be, when dealing with the left cusp of the Shell-Thron region, make sure you use enough precision/series precision.
EDIT2:
So it took wayyyy longer to run but here is: \(b = exp(0.1-e+0.5i)\) and \(\lambda = 1\) . I did about 350 iterations/series precision and 50 digit precision. We get something way nicer. Way slower to compile, I made a smaller graph for that reason but it works far better:
So all the TV static can be fixed by higher precision/series precision. But the low-precision graphs won't work in the cusp of the area you asked me to test. You have to run at least mid level precision to work in this area. Be prepared for really long wait times though. This is definitely a case of it works; but it's slow as fuck.
Regards, james
All the above graphs are done over \(0 \le \Re(s) \le 6\) and \(|\Im(s)| \le 3\); recalling that we have an exact period of \(2\pi i \approx 6i\), we've basically graphed the entire behaviour of the function.
Here is, \(b = \exp(-0.1-e+0.1i)\) for \(\lambda = 1\):
Here is, \(b = \exp(0.3-e+i)\) for \(\lambda = 1\):
Here is, \(b = exp(0.1-e+0.5i)\) for \(\lambda = 1\):
The trouble is probably forcing a real period on a tetration that has a naturally complex period. I'll run some more tests, and see if I can modify some of the parameters to work around these TV static pictures. It could also be related to the fact these graphs were made with very low precision, so it could be a precision error.
Regards, James
EDIT:
So definitely the TV static in the above pictures is low precision error. I'm rerunning the graphs with higher precision/higher series precision, and the TV static is totally disappearing, forming an actual detail.
My suggestion would then be, when dealing with the left cusp of the Shell-Thron region, make sure you use enough precision/series precision.
EDIT2:
So it took wayyyy longer to run but here is: \(b = exp(0.1-e+0.5i)\) and \(\lambda = 1\) . I did about 350 iterations/series precision and 50 digit precision. We get something way nicer. Way slower to compile, I made a smaller graph for that reason but it works far better:
So all the TV static can be fixed by higher precision/series precision. But the low-precision graphs won't work in the cusp of the area you asked me to test. You have to run at least mid level precision to work in this area. Be prepared for really long wait times though. This is definitely a case of it works; but it's slow as fuck.
Regards, james
All the above graphs are done over \(0 \le \Re(s) \le 6\) and \(|\Im(s)| \le 3\); recalling that we have an exact period of \(2\pi i \approx 6i\), we've basically graphed the entire behaviour of the function.
