trying out homotopy methods (construct an ordinary differential equation to interpolate from root at iteration k to two roots at iteration k+1) to solve f_c^n(0) = z for z non-zero; where f_c(w) = w^2+c and f^n denotes n-fold composition. a paper I found describes it for z zero, wasn't too much algebra work to adapt to my use case.

image shows paths between roots to iteration 12 or so, for z = i.

hoping eventually to use it to accelerate zooming into nebulabrot aka buddhabrot, nothing to show for that yet though. not finished debugging the root finder yet.

the idea is to find little balls around the roots that when iterated will hit the viewing area around z (ideally ball contains some boundary of mandelbrot set => some parts of the ball are near (pre)periodic points and will hit the viewing area many times).

will use derivatives to work out how big each ball should be, and distance estimate to check if ball contains boundary.

the tough part is that balls may overlap.

this approach is deterministic. afaik most buddhabrot zoomers use stochastic methods like metropolis-hastings.

reference:
https://ir.lib.uwo.ca/etd/4028/ https://uwo.scholaris.ca/server/api/core/bitstreams/2cf58b26-c757-48e1-833c-bda6c70a1e18/content
"A comparison of solution methods for Mandelbrot-like polynomials"
Eunice Y. S. Chan, MSc thesis, The University of Western Ontario

#maths #homotopy #RootFinding #fractals

`We show that the Weierstrass method, like the well known #Newton method, is not generally convergent: there are open sets of #polynomials p of every degree d≥3 such that the dynamics of the Weierstrass method applied to p exhibits attracting periodic orbits.`

https://arxiv.org/abs/2004.04777

#rootFinding #optimizationTheory #optimization #computation #computing