**Newton's method** is a method for approximating the value of the roots of a function that cannot be solved for algebraically. Given the function f(x) and an estimate value for the root x_{0}, the first approximation is

The second is

and in general

The more times this process is repeated, the better the approximation will be.

## Example

Suppose we are given the function

We will start with the approximation x_{0} = -0.5. The first approximation will be

The second will be

Plugging this into the original equation, we get

The more approximations we make, the closer to zero the function will become.