Newton's Basin
From Math Images
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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 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 | + | 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 the x-value where y = 0 and the graph of an equation crosses the horizontal x-axis [[Image:Root.gif|200px]]</span> (or solutions) of equations. |
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[[Image:NewtonRoot_Animation.gif|right]] | [[Image:NewtonRoot_Animation.gif|right]] | ||
Newton's Method in calculus is a procedure to find roots of polynomials, using an estimated value as a starting point. Newton devised an iterated method (animated to the right) with the following steps: | Newton's Method in calculus is a procedure to find roots of polynomials, using an estimated value as a starting point. Newton devised an iterated method (animated to the right) with the following steps: | ||
- | :#Estimate a starting value (<math>x_o</math>) on the graph near to the root | + | :#Estimate a starting x-value (<math>x_o</math>) on the graph near to the root |
- | :#Find the tangent line at that starting value | + | :#Find the tangent line at that starting x-value |
:#Find the root of the tangent line | :#Find the root of the tangent line | ||
- | :#Using the tangent's root as new starting value (<math>x_{n}, x_{n+1},...</math>), iterate the method to find a better estimate | + | :#Using the tangent's root as new starting x-value (<math>x_{n}, x_{n+1},...</math>), iterate the method to find a better estimate |
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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|210px]] |
- | <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{0 - y_n}{x_n - x_{n+1}}</math> |
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+ | <math>f'(x_n) = \frac{f(x_n)}{x_n - x_{n+1}}</math> | ||
<math>x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}</math> | <math>x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}</math> | ||
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- | Each root has a set of complex numbers (or pixels)that converge to the root (algebraically, this set would include all of the <math>z_0</math> values referenced above). This set of coordinates is called the root's ''basin of attraction'', where the name of this fractal comes from. | + | Each root has a set of complex numbers (or pixels)that converge to the root (algebraically, this set would include all of the <math>z_0</math> values referenced above). This set of coordinates is called the root's '''basin of attraction''', where the name of this fractal comes from. |
Revision as of 14:26, 9 July 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 (or solutions) of equations.
Each pixel in a Newton's Basin corresponds to a unique coordinate, or point. 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 into the complex plane.
Newton's Method
Newton's Basin
To produce an interesting fractal, the Newton Method needs to be extended to the complex plane. Newton's Basin is created using a complex polynomial , with real and/or complex roots. In addition, each root in a Newton's Basin fractal is usually given a distinctive color. Thus, the fractal on the right is generated by a polynomial with a total of five roots colored magenta, yellow, red, green, and blue.
Every pixel in the image represents a complex number. Each complex number is applied to the equation and iterated continually with the output of the previous iteration becoming the input of the next iteration. This iteration is done by using the same equations discussed in the previous Newton's method section, where x is now a complex number z, y is now a complex number p, and is the nth estimate:
Coloring
An Example
Self-Similarity
Teaching Materials
- There are currently no teaching materials for this page. Add teaching materials.
References
Wikipedia, Newton fractal page and Newton's Method page
Simon Tatham, Fractals derived from Newton-Raphson iteration
David E. Joyce, Newton Basins
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