# 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 $f$ with respect to $x$ 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:

\(\ds \map f {x_1} + \map {f'} {x_1} \paren {x - x_1} + \epsilon\) | \(=\) | \(\ds y\) | ||||||||||||

\(\ds \map {f'} {x_1} \paren {x - x_1}\) | \(=\) | \(\ds y - \map f {x_1} - \epsilon\) | ||||||||||||

\(\ds x\) | \(=\) | \(\ds x_1 + \dfrac {y - \map f {x_1} - \epsilon} {\map {f'} {x_1} }\) | ||||||||||||

\(\ds \) | \(=\) | \(\ds x_1 - \dfrac {\map f {x_1} - y + \epsilon} {\map {f'} {x_1} }\) |

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

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- 2014: Christopher Clapham and James Nicholson:
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