Newton's Method

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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:

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


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

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

$\blacksquare$


Graphical Illustration

Newton's method is generally used to find the zeros of a polynomial.

The main idea is to draw a tangent line to a function at a given value (e.g. $x_1$) and the subsequent value (e.g. $x_2$) is where the tangent line intersects the $x$-axis.


Newton Method.jpg


Example

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

The real root of the cubic:

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

can be found by using Newton's Method.

Its approximate value is:

$2 \cdotp 09455 \, 1$


Also see


Source of Name

This entry was named for Isaac Newton.


Sources


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