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:
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[[Image:NewtonsMethod.gif|left|200px]]
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[[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

Image:inprogress.png
Newton's Basin
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
Fields: Fractals and Calculus
Created By: Ashley T.

Contents

Basic Description

Animation Emphasizing Roots
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 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 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:

  1. Estimate a starting value (x_o) on the graph near to the root
  2. Find the tangent line at that starting value
  3. Find the root of the tangent line
  4. Using the tangent's root as new starting value (x_{n}, x_{n+1},...), iterate the method to find a better 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 x_n is the nth estimate:

f'(x_n) = \frac{\mathrm{\Delta y}}{\mathrm{\Delta x}} = \frac{f(x_n)}{x_n - x_{n+1}}
x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}


Newton's Basin

Creating Newton's Basin

Newton Basin with 5 Roots
Newton Basin with 5 Roots

To produce an interesting fractal, the Newton Method needs to be extended to the complex plane and to imaginary numbers. 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 left has a total of five roots colored magenta, yellow, red, green, and blue.

Every pixel in the image is assigned a complex number coordinate. The coordinates are applied to the equation and iterated continually with the output of the previous iteration becoming the input of the next iteration. If the iterations lead the x-values of the coordinates to converge towards a particular root, the pixel is colored accordingly. If the iterations lead to a loop and not a root, then the pixel is usually colored black because the x-values do not converge.

Each root has a set of initial (or pixel) coordinates x_0 that converge to the root. This set of coordinates that are complex number values is called the root's basin of attraction- where the name of this fractal comes from. In addition, some images including shading in each basin. The shading is determined by the number of iterations (aka. escape time) it takes each pixel to converge to a particular root, and it allows us to see the location of the root more clearly.



Solutions
Solutions p(z) = z^3 - 2z + 2

For example, the image below, as well as the featured image, was created from the equation p(z) = z^3 - 2z + 2. Since this equation is a 3rd degree complex polynomial, it has three roots, two of which are complex: z = -1.7693, 0.8846 + 0.5897i, and 0.8846 - 0.5897i. The resulting map of these solutions are to the right, and you can see that the Newton's Basin created from this complex polynomial has three roots (yellow, blue, and green) that correspond to the solution map.

Newton Basin with 3 Roots



Self-Similarity

As with all other fractals, Newton's Basin exhibits self-similarity. The video below is an interactive representation of the continual self-similarity displayed by a Newton's Basin with a root degree of 5 (similar to the fractal shown in the previous section). Towards the end of the video, you will notice that the pixels are no longer adequate to continue magnifying the image...however, the fractal still goes on.




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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