Newton's Approximation Algorithm

A Newton's Approximation Algorithm is an Root-Finding Algorithm that is an Iterative Approximation Algorithm that ...

• AKA: Newton–Raphson Method.
• Context:
  • It can be applied by a Newton's Approximation System (to solve a Newton's Approximation Task).
  • It performs an Iteration Step based on a rearrangement of the Taylor Series.
• Example(s):
  • a Quasi Newton's Method.
  • a Truncated Newton's Method.
  • …
• Counter-Example(s):
  • Halley's Method.
• See: Numerical Analysis, Root of a Function, Real Number, Derivative, Numerical Optimization Algorithm, Householder's Method, Non-Linear Equation, Function Fitting 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.  :[math]\displaystyle{ x : f(x) = 0 \,. }[/math]

    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  :[math]\displaystyle{ x_{1} = x_0 - \frac{f(x_0)}{f'(x_0)} \,. }[/math]

    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  :[math]\displaystyle{ x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \, }[/math]

    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.

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