added optimization lecture

08_optimization/goldmin.m

@@ -0,0 +1,36 @@
+function [x,fx,ea,iter]=goldmin(f,xl,xu,es,maxit,varargin)
+% goldmin: minimization golden section search
+%   [x,fx,ea,iter]=goldmin(f,xl,xu,es,maxit,p1,p2,...):
+%     uses golden section search to find the minimum of f
+% input:
+%   f = name of function
+%   xl, xu = lower and upper guesses
+%   es = desired relative error (default = 0.0001%)
+%   maxit = maximum allowable iterations (default = 50)
+%   p1,p2,... = additional parameters used by f
+% output:
+%   x = location of minimum
+%   fx = minimum function value
+%   ea = approximate relative error (%)
+%   iter = number of iterations
+if nargin<3,error('at least 3 input arguments required'),end
+if nargin<4|isempty(es), es=0.0001;end
+if nargin<5|isempty(maxit), maxit=50;end
+phi=(1+sqrt(5))/2;
+iter=0;
+while(1)
+  d = (phi-1)*(xu - xl);
+  x1 = xl + d;
+  x2 = xu - d;
+  if f(x1,varargin{:}) < f(x2,varargin{:})
+    xopt = x1;
+    xl = x2;
+  else
+    xopt = x2;
+    xu = x1;
+  end
+  iter=iter+1;
+  if xopt~=0, ea = (2 - phi) * abs((xu - xl) / xopt) * 100;end
+  if ea <= es | iter >= maxit,break,end
+end
+x=xopt;fx=f(xopt,varargin{:});

08_optimization/lennard_jones.m

@@ -0,0 +1,4 @@
+function E_LJ =lennard_jones(x,sigma,epsilon)
+  E_LJ = 4*epsilon*((sigma./x).^12-(sigma./x).^6);
+end
+

HW2/README.md

@@ -0,0 +1,99 @@
+# 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](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
+
+  $y(x)=\frac{T}{w}\cosh\left(\frac{w}{T}x\right)+y_{0}-\frac{T}{w}$.
+
+  where y(x) is the height of the cable at a distance, x, from the lowest point, $y_{0}$,
+  T is the tension in the cable, and w is the weight per unit length of the cable. Your
+  supervisor wants to know which numerical solver to use when they have to install these
+  powerlines in similar places. 
+
+  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 $y_{0}$=30 m. 
+
+  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
+  ($y(x)~vs.~x$) from x=-10 to 50 m. 
+
+**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')
+
+![Model of Gold chain, from molecular dynamics simulation](../08_optimization/Auchain_model.png)
+
+**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
+
+  $E_{LJ}(x)=4\epsilon
+  \left(\left(\frac{\sigma}{x}\right)^{12}-\left(\frac{\sigma}{x}\right)^{6}\right)$.
+
+  Where x is the distance between atoms in nm, $\epsilon$=2.71E-4 aJ, and $\sigma$=0.2934
+  nm. The energy term that must be minimized is 
+
+  $E_{total}(\Delta x)=E_{LJ}(x_{0}+\Delta x)-F\Delta x$.
+
+  Where $x_{0}$ is the distance between atoms with no force applied and $\Delta x$ is the
+  amount each gold atom has moved under a given force, F.
+
+  a. Determine $x_{0}$ when F=0 nN using the golden ratio and parabolic methods. *Show
+  your script and output in your README and include your functions*
+
+  b. Solve for $\Delta x$ for F=0 to 0.0022 nN with 30 steps. *Use the golden ratio
+  solver or the matlab/octave `fminsearch`
+
+  c. create a sum of squares error function `sse_of_parabola.m` that calculates the sum of
+  squares error between a function $F(x)=K_{1}\Delta x+1/2K_{2}\Delta x^{2}$ and the
+  Forces used in part B for each $\Delta x$. 
+
+  d. Use the `fminsearch` matlab/octave function to determine $K_{1}$ and $K_{2}$. 
+
+  e. Plot the force vs calculated $\Delta x$ and the best-fit parabola using $K_{1}$ and
+  $K_{2}$ in part d. 
+