In the old forum, there was an exciting challenge 10 years ago (when I was only computing Lyapunov diagrams, so I missed it) about one million roots in the Mandelbrot set (

http://www.fractalforums.com/theory/the-mandelbrot-polynomial-roots-challenge/) and a few years back there was a similar article by

*Schleicher et al.* Newton's method in practice: Finding all roots of polynomials of degree one million efficiently, 2017.

I have yet to read those articles completely and in depth, but I thought, I try it with my interval arithmetics based subdivision implementation to see how far/fast I can get there - and what types of optimizations need to be implemented to reach a million (if at all possible).

Below are the first levels to detect the hyperbolic centers of period 7 and 1 using the z-constant term of the 7-fold composition f[7] of f(z)=z^2+c, i.e. g[7] for g[n]=g[n-1]^2+c, g[0]=0 with degree 64 (so quite a long distance to a million to go

).

Upper row shows the development of some levels. The gray region shrinks in an interesting fashion from outwards in streaks and at early levels follow roughly the Mset shape. It looks a bit like a crowded game of life.

At deepest level L16 so far it found 40 definite root regions (lower left, red rectangles around black pixels; probably 40 roots then from the image, but see details' last item). Some regions are yet to be judged (lower middle, gray circumferenced gray regions) which interestingly cover a lot of the components sitting on the real axis (including the main cardioid).

Computationally the number of boxes per level increases exponentially as expected (lower right) - about 2^1.8 fold at start, so not quite the maximal 2^2-fold increase due to subdivision into 4 squares - but at some point the complexity drops rapidly and box numbers go down, so interestingly no plateau there. It seems to be a "climb-the-hill-and-then-roll-down-fast" type of complexity. There are only 248 boxes currently left in total at end of L16, so the present 64 root regions might be in reach soon.

Has anyone attempted this (with or w/o IA) recently? Probably one can get to more than a million roots with nowadays computers?

**Technical details**- the image shows the complex 2-square. It is for visualization and not intended to provide a reliable filter (but region coordinates are available).
- regions fully in the positive imaginary half are not analyzed due to symmetry.
- I use my custom-made fixed-point complex interval type with arbitrary (at compile time) precision of about 270 decimal digits for now with simulated outward rounding when needed.
- subdivision uses the interval Newton operator as described in forum link.
- the polynomial g[n] is implemented in iterative form, the inverse Jacobian is computed using elementary relations between complex and partial derivatives.
- computation was performed on-disk in a single-thread manner and took 4 hrs in total.
- as a plausibility check, the midpoint (double precision calculated) of a definite root region was analyzed by the TSA cell mapping. In 2*13/40, a cycle of length 7 could be verified at level 18 (no other cycles were found).
- root regions have not yet been checked whether they are pairwise disjoint (they could touch if a root lies on a rationally complex border of two adjacent regions).

Linkback: https://fractalforums.org/programming/11/the-mandelbrot-set-root-challenge-10-years-later/3952/