Homework #2
due 10/6/17 by 11:59pm
1. Create a new github repository called '02_roots_and_optimization'.
a. Add rcc02007 and zhs15101 as collaborators.
b. submit the clone repository URL to: https://goo.gl/forms/svFKpfiCfLO9Zvfz1
2. You're installing a powerline in a residential neighborhood. The lowest point on the cable is 30 m above the ground, but 30 m away is a tree that is 35 m tall. Another engineer informs you that this is a catenary cable problem with the following solution
where y(x) is the height of the cable at a distance, x, from the lowest point,
a. Use the three solvers falsepos.m
, bisect.m
, and mod_secant.m
to solve for the tension neededi, T, to reach y(30 m)=35 m, with w=10 N/m, and
b. 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 | | | |
|mod_secant | | | |
|bisect | | | |
```
c. Add a figure to your README that plots the final shape of the powerline
(
3. 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-1)*exp(-(x-1)^2). The root is at 1, 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=3 for f(x)=(x-1)*exp(-(x-1)^2).
b. Add the results to a table in the README.md
with:
```
### divergence of Newton-Raphson method
| iteration | x_i | approx error |
| --- | --- | --- |
| 0 | 3 | n/a |
| 1 | | |
| 2 | | |
| 3 | | |
| 4 | | |
| 5 | | |
```
c. Repeat steps a-b for an initial guess of 1.2. (But change the heading from 'divergence' to 'convergence')
4. Determine the nonlinear spring constants of a single-atom gold chain. You can assume the gold atoms are aligned in a one dimensional network and the potential energy is described by the Lennard-Jones potential as such
Where x is the distance between atoms in nm,
Where
a. Determine
b. Solve for fminsearch
c. create a sum of squares error function sse_of_parabola.m
that calculates the sum of
squares error between a function
d. Use the fminsearch
matlab/octave function to determine
e. Plot the force vs calculated