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I found this Program without series approximation but was wondering if the radius has to have arbitrary precision as well. As is the radius would be zero when zooming in and reaching doubles precision limit, or not?

Hi,

https://en.wikibooks.org/wiki/Fractals/Iterations_in_the_complex_plane/Mandelbrot_set#Perturbation_theory
 http://code.mathr.co.uk/mandelbrot-perturbator/
http://mathr.co.uk/blog/2014-03-31_perturbation_glitches.html

HTH

Adam

you are right it is not easy; there are lots of aspects both to the algorithms and their implementation, and it all has to be perfect or it will blow up and not work right.  furthermore not all aspects to it necessarily even have the best possible solution worked out yet, nor have current techniques necessarily been set in stone or fully agreed upon even now, primarily regarding how to deal with precision loss (both in the perturbed iterations and in the computation of the series approximation) and also regarding how to approach implementing a stop condition for the iteration of the series coefficients.

i think a concise yet comprehensive writeup designed to actually teach and convey real understanding of the whole thing from the ground up (particularly for those who are not already a seasoned math/fractal/programming expert) is yet to appear, and any requests for such a thing will likely fall on deaf ears and disappear into the vacuum of exclusivity.  "i trudged through the 50000 pages of extraneous and poorly organized forum posts, and spent a bunch of time figuring a bunch of stuff out the hard way, i'll be damned if i see it made easy for you" is the general sentiment you are most likely to encounter.  of course that is just humans generally, not necessarily specific to mandelbrot perturbation  

honestly what i did when i first started out was to trudge through the endless wasteland of forum posts in an attempt to sift for gold and copy-paste what looked to be the most relevant bits of actual information into a word processor and then use that to study from and refer to whilst attempting to understand it and implement it    i dont think im the only one with such a tale to tell either, though i am one of those not-an-expert-in-any-of-the-things people... maybe everyone here but me is an expert.

well, it is simply an area of ongoing research.  from time to time people are still coming up with interesting ideas for better ways to do things, like knighty's application of interval arithmetic to create a proper stop condition for the iteration of the series coefficients, and claude's novel approach to glitch handling.  one could certainly write up a nice article/tutorial/book/whatever if they were so inclined, it is just a matter of doing it.


as i said, up to 10¹⁶ or so people are doing the standard mandelbrot iterations with hardware floating point, as there is nothing to gain from series approximation or perturbation at those shallow depths.  in fact, the overhead is likely to be slower in comparison.  the perturbation stuff is what is employed to break the 10¹⁶ or so barrier.  with regard to your implementation, you indeed need to use arbitrary precision for the location and for the reference iteration, as this is limited by the precision of the mantissa as opposed to the exponent.

Here is some C++ code that tries to be simple without lots of extra optimisations (but the trouble is the whole thing is quite complicated, so I guess it's still pretty hard to get to grips with):

https://code.mathr.co.uk/fractal-bits/blob/HEAD:/mandelbrot-series-approximation/m.cpp

It could do with lots more comments, to be sure.