Newton's Method

Newton's Method is a method of solving an equation for which there is no convenient closed form solution. The first derivative of the equation has to be knownin order to use Newton's Method.

Assume the equation is of the form:
 * $y=f(x)$

And the value of x is wanted for a known y.

Then an iterative improvement on an initial guess is of the form
 * $\displaystyle x_2=x_1-\frac{f(x_1)-y}{f^\prime(x_1)}$

Where $f^\prime(x)$ is the first derivative evaluated at $x$.

Proof
The function $f(x)$ can be expanded in Taylor's Expansion:
 * $\displaystyle f(x_2)=f(x_1)+f^\prime(x_1)(x_2-x_1)+\frac 1 2 f^{\prime\prime}(x_1)(x_2-x_1)^2+\cdots$

As $x_2$ gets closer to $x_1$, this series can be truncated to:
 * $\displaystyle f(x_2)=f(x_1)+f^\prime(x_1)(x_2-x_1)$

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

Let $\epsilon$ be the difference from the new estimate to the solution:
 * $x_\infty = x_2 + \epsilon$

Then the function expanded around the new estimate is:
 * $y=f(x_2)+f^\prime(x_2)\epsilon$

Solving for $x_2$ produces:
 * $\displaystyle x_2=x_1-\frac{f(x_1)-y}{f^\prime(x_1)}-\epsilon\frac{f^\prime(x_2)}{f^\prime(x_1)}$

For $\epsilon$ small enough the final term can be neglected:
 * $\displaystyle x_2=x_1-\frac{f(x_1)-y}{f^\prime(x_1)}$