The Method

The Newton iteration method is a numerical method to approximate a root of a function. The idea is that on a given start point $x_0\in\mathbb {R}^n$ the function is approximated by a linear function and the root of this linear function is determined. In the next iteration this point is used in the hope that this is a better approximation to the root of our function. For a function $f:\mathbb {R}\to\mathbb {R}$ this idea is illustrated in figure 9.

Figure 9: Idea of the Newton method

The iteration method is quickly derived. The linear approximation of a differentiable function $f:\mathbb {R}^n\to\mathbb {R}^n$ in a point $x_0\in\mathbb {R}^n$ is given by

$$f(x_0)+\big({\bf D}f(x_0)\big)(x-x_0)$$

which is zero at $x_1$ if ${\bf D}f(x_0)$ is non singular and

$$x_1:=x_0-\big({\bf D}f(x_0)\big)^{-1}f(x_0)$$

In a one dimensional case this iteration method simplifies to

$$x_{i+1}=x_{i}-\frac{f(x_0)}{f'(x_0)}$$

This idea, of course, can be applied to a complex function $f:\mathbb {C}\to\mathbb {C}$.