Newton Fractal - Generalization of Newton Fractals

Generalization of Newton Fractals

A generalization of Newton's iteration is

where is any complex number. The special choice corresponds to the Newton fractal. The fixed points of this map are stable when lies inside the disk of radius 1 centered at 1. When 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 is a polynomial of degree, then the sequence is bounded provided that is inside a disk of radius centered at .

More generally, Newton's fractal is a special case of a Julia set.

• Newton fractal for three degree-3 roots, coloured by number of iterations required

• Newton fractal for three degree-3 roots, coloured by root reached

• Newton fractal for . Points in the red basins do not reach a root.

• Newton fractal for a 7th order polynomial, colored by root reached and shaded by rate of convergence.

• Newton fractal for

• Newton fractal for, coloured by root reached, shaded by number of iterations required.

• Newton fractal for, coloured by root reached, shaded by number of iterations required

• Another Newton fractal for

• Generalized Newton fractal for, The colour was chosen based on the argument after 40 iterations.

• Generalized Newton fractal for,

• Generalized Newton fractal for,

• Generalized Newton fractal for,