Improving convergence of Andrew's slog

[bo198214]

Does anybody know what JayD meant here with "coefficients for the two known singularities", and how he arranges them and their conjugates in the B vector?

Add the power series for the two known singularities (i.e., the singularities assumed to be closest to the origin, and hence whose terms would limit the radius of convergence): \( \log_c(z-c) \) and \( \log_{\bar{c}}(z-\bar{c}) \)

The power series for \( \log_c(z-c) \) is \( a_0 + \sum_{k=1}^{\infty} \frac{-x^k}{k\,c^{k+1}} \)

Ya in the mean time I looked into your code and guessed something like that.

So this is based on your observation that the slog behaves like log (which is like rslog) at the fixed points (which you uttered somewhere if my memory is right)?! And based on your assumption that the slog should be constructed somehow from both fixed points (though not in the simple way i proposed by just adding the two regular slogs at both fixed points, which anyway is not analytic at all points on the real axis).

Now that I know what you are doing, I am a bit skeptical that it indeed converges toward the same solution that the original system would. Do you have some justification that it is not biased (towards your dream solution) by choosing this initial guess of the coefficients, I mean other than numerical justification?

Dont get me wrong I think that your accelerated solution is the best in the sense that it satisfies the uniqueness criterion in my (in the meantime published) article [1] (which I found out later matches some known criteria in holomorphic dynamics, see this thread), particularly that it is equal to Kneser's solution (which I think is in turn equal to Kouznetsov's solution); however for me there are just different approaches and I would like to know whether they are equal or not and what are the benefits and drawbacks of the methods.

In this sense, I also would like to see a numerical comparison of the regular slog at base sqrt(2) with the islog for that base. I would think that your way of acceleration also works with that base, could you make a picture of acclerated islog vs rslog? (I surely dont want to steal your time, and I anyway surprised to see a reply here by you; on the other hand you would be most efficient with it, as you considered both methods already in detail)

[1] H. Trappmann, D. Kouznetsov. Uniqueness of Holomorphic Abel Functions at a Complex Fixed Point Pair. Aequationes Mathematicae, 2010.