Cauchy Integral Experiment

[Kouznetsov]

Well, I say "improved" because when it does converge it will converge for very high number of nodes.

After some tens of iterations, do your curves become smooth?
You may have a problem related with the different weight of nodes in the Simpson,
the even nodes are twice "heavier" than the odd ones. The updating of each third node could boost the convergence, but the code becomes cumbersom. With the even amount of nodes I did not have such a problem, the weight of even nodes is similar to that of hte odd ones (except the tips, but there the weight of each node is small, so, it does no matter) and I could do it in a single loop.

..At 811 nodes with A = 24 it converges exquisitely with the residual after 32 normalized followed by 6 unnormalized iterations having a magnitude smaller than 10^-9.

Congratulations! I see, improving the precision, you go by the "extensive" way: you increase the number of nodes instead of to change for some high-order approximation;
each 3 addifional significant digits cost you an order of magnitude in the CPU time.
If you plan to get a precision better than that with complex<double> (15 digits),
try to finish your calculus in this century.
Another note: do you still evaluate the residual in few single points?
Could you plot the residual F(z)-exp(F(z-1)) in the complex z-plane?