Homework #3
due 2/15/17 by 11:59pm

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 solversd. Use the four solvers
falsepos.m
,bisect.m
,newtraph.m
andmod_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.

The NewtonRaphson 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
andmod_secant.m
, there are certain initial guesses that do not converge.a. Calculate the first 5 iterations for the NewtonRaphson 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 NewtonRaphson method  iteration  x_i  approx error          0  2  n/a   1     2     3     4     5   
c. Repeat steps ab for an initial guess of 0.2. (But change the heading from 'divergence' to 'convergence')

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