Cauchy Integral Experiment

[Kouznetsov]

Yes, it turns smooth if the number of nodes meets the requirement I mentioned, otherwise it fails as the posted graphs show.

1. Can you explicitly formulate the requirement?
What value of A and how many nodes should one use in order to get 3 decimal digits, to get 9 decimal digits, to get 12 decimal digits, etc?

2. Will you print the table of values of your approximation of tetration along the imaginary axis?

It's interesting you mention about the weight, though I do not update nodes during the Simpson procedure, and why does it seems that odd node numbers that converge alternate with those that don't?

This may be due to the jumping distribution of weights in the nodes of the Simpson formula.
The Simpsons in the cartoon are better than in the precise evaluation of the tetrational.

..are still a few issues to work out, then I'll switch to a more sophisticated method (cubic, maybe the Gauss-Legendre even).

You already use the cubic one.
You may take the algorithm for the Gauss-Legendre at
http://en.citizendium.org/wiki/GauLegExample/code
The routine for the evaluation takes only 11 lines at very beginning of the code;
it is easy to translate to any language.

That's what I'm after here, trying to get out the bugs so they don't bite me later

I do not understand how do you treat the conjugation, some bug may be there.
if you do not use f(z^*)=f(z)^* ,
then the simultaneous update of F(z) and f(z^*) does not have any sense.

3. In order to catch bugs, calculate the residual at some sufficiently dense mesh and plot it. The mesh is really good tool to catch bugs.