Newton's Basin
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The results of this method lead to very close estimates to the root of the polynomial. Newton's Method can also be expressed algebraically as follows, where <math>x_n</math> is the nth estimate: | The results of this method lead to very close estimates to the root of the polynomial. Newton's Method can also be expressed algebraically as follows, where <math>x_n</math> is the nth estimate: | ||
- | [[Image:NewtonsMethod.gif|left| | + | [[Image:NewtonsMethod.gif|left|150px]] |
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:<math>f'(x_n) = \frac{\mathrm{\Delta y}}{\mathrm{\Delta x}} = \frac{f(x_n)}{x_n - x_{n+1}}</math> | :<math>f'(x_n) = \frac{\mathrm{\Delta y}}{\mathrm{\Delta x}} = \frac{f(x_n)}{x_n - x_{n+1}}</math> | ||
Revision as of 13:33, 10 June 2009
- Newton's Basin is a visual representation of Newton's Method, which is a procedure for estimating the root of a function.
Newton's Basin |
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Contents |
Basic Description
This image is one of many examples of Newton's Basin or Newton's Fractal. Newton's Basin is based on a calculus technique called Newton's Method, a procedure Newton developed to estimate roots of equations.
The colors in a Newton's Basin usually correspond to each individual root of the equation, and can be used to infer where each root is located. Each color region reflects the set of points, which, after undergoing iteration with the equation describing the fractal, will eventually get closer and closer to the value of the root associated with that color.
The animation emphasizes the roots in a Newton's Basin, whose equation clearly has three roots. The image featured at the top of this page is also a Newton's Basin with three roots.
A More Mathematical Explanation
- Note: understanding of this explanation requires: *Calculus
The image at the top of this page is a visual representation of Newton's Method in calculus expanded [...]
The image at the top of this page is a visual representation of Newton's Method in calculus expanded into the complex plane.
Newton's Method
Newton's Basin
Teaching Materials
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Related Links
Additional Resources
- http://www.chiark.greenend.org.uk/~sgtatham/newton/ for further mathematical explanation
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