Newton-Raphson Method (Uni-Variate)

To solve an algebraic equation $f(x)=0$, select a random initial guess $x_0$ and follow the iteration:

$$x \Leftarrow x-\frac{f(x)}{f'(x)}$$

This Newton-Raphson formula can be derived below. The equation of the tangent of $f(x)$ at $x=x_0$ is

$$f'(x_0) = \frac{f(x_1)-f(x_0)}{x_1-x_0}$$

If we let $f(x_1)=0$, i.e., $x_1$ is the zero crossing of the tangent, we get

$$x_1=x_0-\frac{f(x_0)}{f'(x_0)}$$

which is closer to the desired solution than $x_0$. Repeating the process we will get $x_2, x_3, \cdots,$ which approach the actual solution.