Newton's Basin
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[[Image:NewtonBasin_Animate.gif|thumb|left|250px|Animation Emphasizing Roots]] | [[Image:NewtonBasin_Animate.gif|thumb|left|250px|Animation Emphasizing Roots]] | ||
- | This image is one of many examples of Newton's Basin or Newton's Fractal. Newton's Basin is based on a calculus | + | 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 <balloon title="load:myContent">roots</balloon><span id="myContent" style="display:none">A root is located where y = 0 and the graph of an equation crosses the horizontal x-axis [[Image:Root.gif|200px]]</span> 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 | + | 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 | + | 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. |
- | |ImageDesc=The | + | |ImageDesc=The image at the top of this page is a visual representation of Newton's Method in calculus expanded into the [[Complex Numbers|complex plane]]. |
===Newton's Method=== | ===Newton's Method=== | ||
{{hide|1= | {{hide|1= | ||
- | Newton's Method | + | Newton's Method in calculus is a procedure to find roots of polynomials, using an estimated value as a starting point. |
[[Image:NewtonRoot_Animation.gif|right]] | [[Image:NewtonRoot_Animation.gif|right]] | ||
Newton devised an iterated method (animated to the right) with the following steps: | Newton devised an iterated method (animated to the right) with the following steps: | ||
- | :#Estimate a starting | + | :#Estimate a starting value (<math>x_o</math>) on the graph near to the root |
- | :#Find the tangent line at that starting | + | :#Find the tangent line at that starting value |
:#Find the root of the tangent line | :#Find the root of the tangent line | ||
- | :#Using the root as | + | :#Using the tangent's root as new starting value (<math>x_{n+1}</math>), iterate the method to find a better estimate |
- | The results of this method lead to very close estimates to the | + | 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: |
:<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 13:09, 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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