Fractals

For complex functions $f:\mathbb{C}\to\mathbb{C}$, however, Newton's method can be directly applied to find their zeros. For many complex functions, the boundary of the set (also known as the basin of attraction) of all starting values that cause the method to converge to a particular zero is a fractal^6.13

For example, the function $f(x)=x^5-1$, $x\in \mathbb{C}$, has five roots, equally spaced around the unit circle in the complex plane. If $x_0$ is a starting point which converges to the root at $x=1$, color $x_0$ yellow. Repeat this using four other colors (blue, red, green, purple) for the other four roots of $f$. The resulting image is in Figure 6.14.

Figure 6.14: Basins of attraction for $x^5 - 1 = 0$; darker means more iterations to converge.