README.md

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