# Newton's Approximation Algorithm

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## References

### 2014

• (Wikipedia, 2014) ⇒ http://en.wikipedia.org/wiki/Newton's_method Retrieved:2014-6-28.
• In numerical analysis, Newton's method (also known as the 'Newton–Raphson method), named after Isaac Newton and Joseph Raphson, is a method for finding successively better approximations to the roots (or zeroes) of a real-valued function.  :$\displaystyle{ x : f(x) = 0 \,. }$

The Newton–Raphson method in one variable is implemented as follows:

Given a function ƒ defined over the reals x, and its derivative ƒTemplate:Void', we begin with a first guess x0 for a root of the function f. Provided the function satisfies all the assumptions made in the derivation of the formula, a better approximation x1 is  :$\displaystyle{ x_{1} = x_0 - \frac{f(x_0)}{f'(x_0)} \,. }$

Geometrically, (x1, 0) is the intersection with the x-axis of the tangent to the graph of f at (x0, f (x0)).

The process is repeated as  :$\displaystyle{ x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \, }$

until a sufficiently accurate value is reached.

This algorithm is first in the class of Householder's methods, succeeded by Halley's method. The method can also be extended to complex functions and to systems of equations.