Application to Newton-Raphson method

Consider the problem of finding the roots of an equation

$$f(x) = 0$$

If an approximation to the root $x_n$ is already available then the next approximation

$$x_{n+1}= x_n + \Delta x_n$$

is calculated by requiring that

$$f(x_n + \Delta x_n) = 0$$

Using Taylors formula and assuming that $f$ is differentiable we can approximate the left hand side,

$$f(x_n) + \Delta x_n f^\prime(x_n) = 0 \Longrightarrow \ \Delta x_n = - \frac{f(x_n)}{ f^\prime(x_n) }$$

so that the new approximation is given by

$$x_{n+1} = x_n - \frac{f(x_n)}{f^\prime(x_n) }$$

This defines the iterative scheme for Newton-Raphson method and the solution if it exists, is the fixed point of

$$F(x) = x - \frac{ f(x) }{ f^\prime(x) }$$

If we assume that

$$\vert F^\prime(x) \vert \le \alpha < 1$$ (11)

in some neighbourhood of the solution then

$$F(y) - F(x) = \int^y_x F^\prime(z) \textrm{d}z \Longrightarrow \vert F(y) - F(x)\vert \le \alpha \vert y-x\vert$$

Thus condition (11) guarantees that $F$ is contractive and the Newton iterations will converge.