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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) |