.. Created by Adam Cunnningham on Fri Jun 3 2016.

**Newton in the Complex Plane**
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Report Description
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The report consists of the following exercises.

**Exercise 1.**

Use Newton's method to find which root different points in a rectangular portion of the complex plane converge to for the equation :math:`f(z) = z^3 - 1`.

**Exercise 2.**

Color the points in this region according to which root they converge to using red, green, and blue (or some other colors of your choice).

**Exercise 3.**

Make an additional zoomed in picture of some interesting region.

**Exercise 4.**

Make a picture encoding the number of iterations needed to converge in the brightness of the color.

Hints and Suggestions for the Report
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Generating the Image
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- Use ``meshgrid`` to generate the array of points in the complex plane, then iterate using Newton's method on the whole array.
- Find the root that each point converges to by checking if each point ends up closer to the root than some tolerance.
- Set each component of the color separately, or use NumPy indexing with boolean arrays to set the entire color according to the root.
- Increase the number of points used, since the image shown on the website is of low resolution.
- Increase the number of iterations enough to remove any black areas from the image. These correspond to points that have not converged to any of the three roots.
- Images can be displayed using ``imshow``.
- Images can be saved using ``imsave``. They can also be saved directly from the IPython notebook, or from the window that the image is being displayed in.

Making a Zoomed in Picture of Some Interesting Region
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

- The displayed region of the complex plane can be changed by modifying the arguments to ``linspace`` in the website code.
- How you define "interesting" is up to you.
- Within limits. For example, a picture of a solid red square does not count as interesting in this context.

Encoding the Number of Iterations
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

- The "number of iterations" refers to how many steps Newton's method requires for a given point in the complex plane to converge to a given root.
- The current code on the website can already detect if a point has converged to a given root. This is only run once though, after Newton's method has finished.
- Testing for convergence after *each* iteration of Newton's method can be used as the basis for finding the number of iterations needed.
- The number of iterations needed will be an integer less than or equal to the maximum number of iterations. To be encoded as a brightness, this will need to be scaled somehow to a float in the interval [0, 1] and applied to the appropriate color components.