Numerical Examples

We consider the complex function $f:\mathbb {C}\to\mathbb {C}$ with $f(x)=2x^3-3x^2+x-5$. This function can be imagined if we plot only the absolute value $\vert f(x)\vert$ whereby we consider $x\in\mathbb {C}$ as a two dimensional value, e.g. we plot the function $g:\mathbb {R}^2\to\mathbb {R}$ with $g(x_1,x_2):=\vert f(x_1+\ix_2)\vert$. The graph of this function is shown in figure 10.

Figure 10: Absolute value of the complex function $f(x)=2x^3-3x^2+x-5$

We can now use the Newton iteration method to obtain one of the three roots of this function. It depends on the start value $x_0$ which root we will obtain. Figure 11 and 12 show which root is obtained depending on the initial value $x_0\in\mathbb {C}$. This is done by assigning each start point $x_0\in\mathbb {C}$ the colour which hints to the root to which the newton method will converge. The result is a fractal.

Figure 11: Newton iteration with the complex function $f(x)=2x^3-3x^2+x-5$

Figure 12: Newton iteration (zoomed)

More colourful pictures are created if we assign each start point $x_0\in\mathbb {C}$ the number of iterations needed to get a sufficient numerical approximation of the root:

Figure 13: Newton iteration, iteration depth

Figure 14: Newton iteration, iteration depth (zoomed)