# A couple of questions about implementing perturbation for the Mandelbrot set

I followed this thread on perturbation of the Mandelbrot set iterations: Perturbation of Mandelbrot set fractal

I was wondering what accuracy these different variables need to be calculated to (high accuracy or like normal floating point) i.e.

1. the original values of the reference point iterations, X_n in the link
2. the A, B and C coefficients
3. the approximated (perturbed) point

What do you do if the perturbed point requires more iterations than the reference point iterations? Do you always have to pick a reference point such that it has more iterations (before escaping) than the points you want to approximate?

Thanks

• if you know $p,q$ to say 10 digits, but the first 7 digits are the same, you only need 3 digits to represent the difference between p and q – Claude Sep 8 '20 at 1:44
• But what when $\Delta c\sim 10^{-1200}$ for a really deep plot, this offset is not representable in double FP. One could some scaling parameter $\rho$ to set $\Delta c=\rho s$ and $\Delta z_n=\rho w_n$. How then to arrange to compute $\Delta z_{n+1}$ or here $w_{n+1}=2z_nw_n+\rho w_n^2+s$. In the reduction to double precision the second term would fall away. Is that admissible or is that caught and compensated in some way? But then one would have to reconnect the scales to compute $|z_n+\rho w_n|>2$. To get non-trivial results from this, one would have to adapt $rho$ dynamically. – Lutz Lehmann Sep 12 '20 at 7:38