Newton fractal

When there are no attractive cycles (of order greater than 1), it divides the complex plane into regions Gk, each of which is associated with a root ζk of the polynomial, k = 1, …, deg(p).

It is relevant to numerical analysis because it shows that (outside the region of quadratic convergence) the Newton method can be very sensitive to its choice of start point.

, one finds the index k(m,n) of the corresponding root ζk(m,n) and uses this to fill an M × N raster grid by assigning to each point (m,n) a color fk(m,n).

When a is outside this disk, the fixed points are locally unstable, however the map still exhibits a fractal structure in the sense of Julia set.

If p is a polynomial of degree d, then the sequence zn is bounded provided that a is inside a disk of radius d centered at d. More generally, Newton's fractal is a special case of a Julia set.