Mandelbrot set Newton basins
Newton's root finding method has a few applications when exploring the Mandelbrot set. You can use it to find the center of a component given a reasonable location estimate, or find a particular point on the boundary of a component given its center, and it is also used when tracing external rays from infinity inwards towards the boundary.
Tracing external rays is a slow process, needing many steps with many iterations of Newton's method for each step. When tracing rays to a particular component, it would be desireable to switch to Newton's method for center finding as soon as possible. A rough heuristic (ie, I haven't proved that it works everywhere) might be to trace a few rays in parallel, and check the atom domain at the ray end points, switching when all ray end points are in an atom domain of the target period using the average of the endpoints as initial estimate.
The images show fractal basins of convergence for Newton's method for a particular period, with the atom domains of the target period highlighted, overlayed with the boundary of the Mandelbrot set. It seems that the atom domain is wholy within its own Newton basin, and also significantly larger than the corresponding component.