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HW3/README.md

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+# Homework #3
+## due 2/15/17 by 11:59pm
+
+
+1. Create a new github repository called 'roots_and_optimization'. 
+
+    a. Add rcc02007 and pez16103 as collaborators.
+
+    b. Clone the repository to your computer.
+
+    c. Copy your `projectile.m` function into the 'roots_and_optimization' folder.
+    *Disable the plotting routine for the solvers*
+
+    d. Use the four solvers `falsepos.m`, `bisect.m`, `newtraph.m` and `mod_secant.m`
+    to solve for the angle needed to reach h=1.72 m, with an initial speed of 15 m/s. 
+
+    e. The `newtraph.m` function needs a derivative, calculate the derivative of your
+    function with respect to theta, `dprojectile_dtheta.m`. This function should
+    output d(h)/d(theta). 
+
+
+    f. In your `README.md` file, document the following under the heading `#
+    Homework #3`:
+
+        i. Compare the number of iterations that each function needed to reach an
+        accuracy of 0.00001%. Include a table in your README.md with:
+
+        ```
+        | solver | initial guess(es) | ea | number of iterations|
+        | --- | --- | --- | --- |
+        |falsepos   |  |  |  |
+        |bisect     |  |  |  |
+        |newtraph   |  |  |  |
+        |mod_secant |  |  |  |
+        ```
+
+        ii. Compare the convergence of the 4 methods. Plot the approximate error vs the
+        number of iterations that the solver has calculated. Save the plot as
+        `convergence.png` and display the plot in your `README.md` with:
+
+        `![Plot of convergence for four numerical solvers.](convergence.png)`
+
+        iii. In the `README.md` provide a description of the files used to create the
+        table and the convergence plot. 
+
+2. The Newton-Raphson method and the modified secant method do not always converge to a
+solution. One simple example is the function f(x) = x*exp(-x^2). The root is at 0, but
+using the numerical solvers, `newtraph.m` and `mod_secant.m`, there are certain initial
+guesses that do not converge. 
+
+    a. Calculate the first 5 iterations for the Newton-Raphson method with an initial
+    guess of x_i=2 for f(x)=x*exp(-x^2).
+
+    b. Add the results to a table in the `README.md` with:
+
+    ```
+    ### divergence of Newton-Raphson method
+
+    | iteration | x_i | approx error |
+    | --- | --- | --- |
+    | 0 | 2 | n/a |
+    | 1 |   |     |
+    | 2 |   |     |
+    | 3 |   |     |
+    | 4 |   |     |
+    | 5 |   |     |
+    ```
+
+    c. Repeat steps a-b for an initial guess of 0.2. (But change the heading from
+    'divergence' to 'convergence')
+
+3. Commit your changes to your repository. Sync your local repository with github. Then
+copy and paste the "clone URL" into the following Google Form [Homework 3](https://goo.gl/forms/UJBGwp0fQcSxImkq2)