# 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 $y$.

Then an iterative improvement on an initial guess 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 $f$ with respect to $x$ evaluated at $x_1$.

## Proof

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

- $\map f {x_2} = \map f {x_1} + \map {f'} {x_1} \paren {x_2 - x_1} + \dfrac 1 2 \map {f''} {x_1} \paren {x_2 - x_1}^2 + \dotsb$

As $x_2$ gets closer to $x_1$, this series can be truncated to:

- $\map f {x_2} = \map f {x_1} + \map {f'} {x_1} \paren {x_2 - x_1}$

Let $x_\infty$ be the exact solution where:

- $\map f {x_\infty} = y$

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:

- $y = \map f {x_2} + \map {f'} {x_2} \epsilon$

Solving for $x_2$ produces:

- $x_2 = x_1 - \dfrac {\map f {x_1} - y} {\map {f'} {x_1} } - \epsilon \dfrac {\map {f'} {x_2} } {\map {f'} {x_1} }$

For $\epsilon$ small enough the final term can be neglected:

- $x_2 = x_1 - \dfrac {\map f {x_1} - y} {\map {f'} {x_1} }$

$\blacksquare$

## Example

### Newton's Method: $x^3 - 2 x - 5 = 0$

- $x^3 - 2 x - 5 = 0$

can be found by using Newton's Method.

Its approximate value is:

- $2 \cdotp 09455 \, 1$

## Source of Name

This entry was named for Isaac Newton.

## Sources

- 2014: Christopher Clapham and James Nicholson:
*The Concise Oxford Dictionary of Mathematics*(5th ed.) ... (previous) ... (next): Entry:**Newton's method**