Newton's Method

Proof Technique
Newton's method is a method of solving an equation expressed as a real function for which there may be no convenient closed form solution.

The derivative of the function has to be known in order to use Newton's method.

Let the equation to be solved be of the form:
 * $y = \map f x$

Let the value of $x$ be required for a given $y$.

Let $x_1$ be an initial guess.

Then an iterative improvement on $x_1$ is of the form:
 * $x_2 = x_1 - \dfrac {\map f {x_1} - y} {\map {f'} {x_1} }$

where $\map {f'} {x_1}$ is the derivative of $f$ $x$ evaluated at $x_1$.

Proof
The function $\map f x$ can be expanded about $x_1$ using Taylor's Theorem:


 * $\map f x = \map f {x_1} + \map {f'} {x_1} \paren {x - x_1} + \dfrac 1 2 \map {f''} {x_1} \paren {x - x_1}^2 + \dotsb$

This series can be rewritten with an error term as follows:
 * $\map f x = \map f {x_1} + \map {f'} {x_1} \paren {x - x_1} + \epsilon$

This gives:

For $\epsilon$ small enough the final term can be neglected:
 * $x_2 = x_1 - \dfrac {\map f {x_1} - y} {\map {f'} {x_1} }$