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:
 * $y = f \left({x}\right)$

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

Then an iterative improvement on an initial guess is of the form
 * $x_2 = x_1 - \dfrac {f \left({x_1}\right) -y} {f' \left({x_1}\right)}$

where $f' \left({x}\right)$ is the derivative of $f$ evaluated at $x$.

Proof
The function $f \left({x}\right)$ can be expanded using Taylor's Theorem:


 * $f \left({x_2}\right) = f \left({x_1}\right) + f' \left({x_1}\right) \left({x_2 - x_1}\right) + \dfrac 1 2 f'' \left({x_1}\right) \left({x_2 - x_1}\right)^2 + \dotsb$

As $x_2$ gets closer to $x_1$, this series can be truncated to:
 * $f \left({x_2}\right) = f \left({x_1}\right) + f' \left({x_1}\right) \left({x_2 - x_1}\right)$

Let $x_\infty$ be the exact solution where:
 * $f \left({x_\infty}\right) = y$

Let $\epsilon \in \R_{>0}$ be the difference from the new estimate to the solution:
 * $x_\infty = x_2 + \epsilon$

Then the function expanded around the new estimate is:
 * $y = f \left({x_2}\right) + f' \left({x_2}\right) \epsilon$

Solving for $x_2$ produces:
 * $x_2 = x_1 - \dfrac {f \left({x_1}\right) - y} {f' \left({x_1}\right)} - \epsilon \dfrac {f' \left({x_2}\right)} {f' \left({x_1}\right)}$

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