diff --git a/HW3/README.md b/HW3/README.md
index 494e0d5..45214a4 100644
--- a/HW3/README.md
+++ b/HW3/README.md
@@ -11,8 +11,8 @@
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`, `incsearch.m`, `newtraph.m` and `mod_secant.m`
- to solve for the angle needed to reach h=1.72 m, with an initial speed of 1.5 m/s.
+ 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
@@ -29,7 +29,7 @@
| solver | initial guess(es) | ea | number of iterations|
| --- | --- | --- | --- |
|falsepos | | | |
- |incsearch | | | |
+ |bisect | | | |
|newtraph | | | |
|mod_secant | | | |
```
@@ -49,7 +49,7 @@ using the numerical solvers, `newtraph.m` and `mod_secant.m`, there are certain
guesses that do not converge.
a. Calculate the first 5 iterations for the Newton-Raphson method with an initial
- guess of x_i=2.
+ guess of x_i=2 for f(x)=x*exp(-x^2).
b. Add the results to a table in the `README.md` with:
@@ -70,5 +70,4 @@ guesses that do not converge.
'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)
+copy and paste the "clone URL" into the following Google Form [Homework 3](https://goo.gl/forms/UJBGwp0fQcSxImkq2)
diff --git a/README.md b/README.md
index 8059993..73ca14a 100644
--- a/README.md
+++ b/README.md
@@ -74,8 +74,8 @@ general, I will not post homework solutions.
|3|1/31||Consistent Coding habits|
| |2/2|5|Root Finding|
|4|2/7|6|Root Finding con’d|
-| |2/9|7|Optimization|
-|5|2/14||Intro to Linear Algebra|
+| |2/9|7| **Snow Day**|
+|5|2/14|| Optimization |
| |2/16|8|Linear Algebra|
|6|2/21|9|Linear systems: Gauss elimination|
| |2/23|10|Linear Systems: LU factorization|
diff --git a/lecture_07/.lecture_07.md.swp b/lecture_07/.lecture_07.md.swp
new file mode 100644
index 0000000..0998c86
Binary files /dev/null and b/lecture_07/.lecture_07.md.swp differ
diff --git a/lecture_08/.ipynb_checkpoints/lecture_08-checkpoint.ipynb b/lecture_08/.ipynb_checkpoints/lecture_08-checkpoint.ipynb
new file mode 100644
index 0000000..2fd6442
--- /dev/null
+++ b/lecture_08/.ipynb_checkpoints/lecture_08-checkpoint.ipynb
@@ -0,0 +1,6 @@
+{
+ "cells": [],
+ "metadata": {},
+ "nbformat": 4,
+ "nbformat_minor": 2
+}
diff --git a/lecture_08/Auchain_model.gif b/lecture_08/Auchain_model.gif
new file mode 100644
index 0000000..2f8c78c
Binary files /dev/null and b/lecture_08/Auchain_model.gif differ
diff --git a/lecture_08/Auchain_model.png b/lecture_08/Auchain_model.png
new file mode 100644
index 0000000..dce9cd7
Binary files /dev/null and b/lecture_08/Auchain_model.png differ
diff --git a/lecture_08/au_chain.jpg b/lecture_08/au_chain.jpg
new file mode 100644
index 0000000..492fee8
Binary files /dev/null and b/lecture_08/au_chain.jpg differ
diff --git a/lecture_08/goldenratio.png b/lecture_08/goldenratio.png
new file mode 100644
index 0000000..9493c8c
Binary files /dev/null and b/lecture_08/goldenratio.png differ
diff --git a/lecture_08/goldmin.m b/lecture_08/goldmin.m
new file mode 100644
index 0000000..fb4ce0b
--- /dev/null
+++ b/lecture_08/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{:});
diff --git a/lecture_08/lecture_08.ipynb b/lecture_08/lecture_08.ipynb
new file mode 100644
index 0000000..32b973c
--- /dev/null
+++ b/lecture_08/lecture_08.ipynb
@@ -0,0 +1,2762 @@
+{
+ "cells": [
+ {
+ "cell_type": "code",
+ "execution_count": 6,
+ "metadata": {
+ "collapsed": true
+ },
+ "outputs": [],
+ "source": [
+ "function h = parabola(x)\n",
+ " y=1;\n",
+ " h=sum(x.^2+y.^2-1);\n",
+ "end\n",
+ " "
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 9,
+ "metadata": {
+ "collapsed": false
+ },
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "ans = 0\r\n"
+ ]
+ }
+ ],
+ "source": [
+ "fzero(@(x) parabola(x),1)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 10,
+ "metadata": {
+ "collapsed": false
+ },
+ "outputs": [],
+ "source": [
+ "%plot --format svg"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "# Optimization\n",
+ "\n",
+ "Many problems involve finding a minimum or maximum based on given constraints. Engineers and scientists typically use energy balance equations to find the conditions of minimum energy, but this value may never go to 0 and the actual value of energy may not be of interest. "
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "The Lennard-Jones potential is commonly used to model interatomic bonding. \n",
+ "\n",
+ "$E_{LJ}(x)=4\\epsilon \\left(\\left(\\frac{\\sigma}{x}\\right)^{12}-\\left(\\frac{\\sigma}{x}\\right)^{6}\\right)$\n",
+ "\n",
+ "Considering a 1-D gold chain, we can calculate the bond length, $x_{b}$, with no force applied to the chain and even for tension, F. This will allow us to calculate the nonlinear spring constant of a 1-D gold chain. \n",
+ "\n",
+ "\n",
+ "\n",
+ "Computational Tools to Study and Predict the Long-Term Stability of Nanowires.\n",
+ "By Martin E. Zoloff Michoff, Patricio Vélez, Sergio A. Dassie and Ezequiel P. M. Leiva \n",
+ "\n",
+ "\n",
+ "\n",
+ "[Single atom gold chain mechanics](http://www.uam.es/personal_pdi/ciencias/agrait/)\n",
+ "\n",
+ "### First, let's find the minimum energy $\\min(E_{LJ}(x))$\n",
+ "\n",
+ "## Brute force"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 11,
+ "metadata": {
+ "collapsed": false
+ },
+ "outputs": [
+ {
+ "data": {
+ "image/svg+xml": [
+ ""
+ ],
+ "text/plain": [
+ ""
+ ]
+ },
+ "metadata": {},
+ "output_type": "display_data"
+ }
+ ],
+ "source": [
+ "setdefaults\n",
+ "epsilon = 0.039; % kcal/mol\n",
+ "sigma = 2.934; % Angstrom\n",
+ "x=linspace(2.8,6,200); % bond length in Angstrom\n",
+ "\n",
+ "Ex = lennard_jones(x,sigma,epsilon);\n",
+ "\n",
+ "[Emin,imin]=min(Ex);\n",
+ "\n",
+ "plot(x,Ex,x(imin),Emin,'o')"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 12,
+ "metadata": {
+ "collapsed": false
+ },
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "ans = 3.2824\n",
+ "ans = 3.2985\n",
+ "ans = 3.3146\n"
+ ]
+ }
+ ],
+ "source": [
+ "x(imin-1)\n",
+ "x(imin)\n",
+ "x(imin+1)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Golden Search Algorithm\n",
+ "\n",
+ "We can't just look for a sign change for the problem (unless we can take a derivative) so we need a new approach to determine whether we have a maximum between the two bounds.\n",
+ "\n",
+ "Rather than using the midpoint of initial bounds, here the problem is more difficult. We need to compare the values of 4 function evaluations. The golden search uses the golden ratio to determine two interior points. \n",
+ "\n",
+ "\n",
+ "\n",
+ "Start with bounds of 2.5 and 6 Angstrom. "
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 13,
+ "metadata": {
+ "collapsed": false
+ },
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "current_min = -0.019959\r\n"
+ ]
+ },
+ {
+ "data": {
+ "image/svg+xml": [
+ ""
+ ],
+ "text/plain": [
+ ""
+ ]
+ },
+ "metadata": {},
+ "output_type": "display_data"
+ }
+ ],
+ "source": [
+ "% define Au atomic potential\n",
+ "epsilon = 0.039; % kcal/mol\n",
+ "sigma = 2.934; % Angstrom\n",
+ "Au_x= @(x) lennard_jones(x,sigma,epsilon);\n",
+ "\n",
+ "% calculate golden ratio\n",
+ "phi = 1/2+sqrt(5)/2;\n",
+ "% set initial limits\n",
+ "x_l=2.8; \n",
+ "x_u=6; \n",
+ "\n",
+ "% Iteration #1\n",
+ "d=(phi-1)*(x_u-x_l);\n",
+ "\n",
+ "x1=x_l+d; % define point 1\n",
+ "x2=x_u-d; % define point 2\n",
+ "\n",
+ "\n",
+ "% evaluate Au_x(x1) and Au_x(x2)\n",
+ "\n",
+ "f1=Au_x(x1);\n",
+ "f2=Au_x(x2);\n",
+ "plot(x,Au_x(x),x_l,Au_x(x_l),'ro',x2,f2,'rs',x1,f1,'gs',x_u,Au_x(x_u),'go')\n",
+ "hold on;\n",
+ "\n",
+ "if f2\n",
+ "\n",
+ "Gnuplot\n",
+ "Produced by GNUPLOT 5.0 patchlevel 3 \n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "\t\n",
+ "\t\n",
+ "\t\n",
+ "\t\n",
+ "\t\n",
+ "\t\n",
+ "\t\n",
+ "\t\n",
+ "\n",
+ "\n",
+ "\n",
+ "\t\n",
+ "\t\t\n",
+ "\t\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "\t\t\n",
+ "\t\t-0.04\n",
+ "\t\n",
+ "\n",
+ "\n",
+ "\t\t\n",
+ "\t\t-0.02\n",
+ "\t\n",
+ "\n",
+ "\n",
+ "\t\t\n",
+ "\t\t0\n",
+ "\t\n",
+ "\n",
+ "\n",
+ "\t\t\n",
+ "\t\t0.02\n",
+ "\t\n",
+ "\n",
+ "\n",
+ "\t\t\n",
+ "\t\t0.04\n",
+ "\t\n",
+ "\n",
+ "\n",
+ "\t\t\n",
+ "\t\t0.06\n",
+ "\t\n",
+ "\n",
+ "\n",
+ "\t\t\n",
+ "\t\t0.08\n",
+ "\t\n",
+ "\n",
+ "\n",
+ "\t\t\n",
+ "\t\t2.5\n",
+ "\t\n",
+ "\n",
+ "\n",
+ "\t\t\n",
+ "\t\t3\n",
+ "\t\n",
+ "\n",
+ "\n",
+ "\t\t\n",
+ "\t\t3.5\n",
+ "\t\n",
+ "\n",
+ "\n",
+ "\t\t\n",
+ "\t\t4\n",
+ "\t\n",
+ "\n",
+ "\n",
+ "\t\t\n",
+ "\t\t4.5\n",
+ "\t\n",
+ "\n",
+ "\n",
+ "\t\t\n",
+ "\t\t5\n",
+ "\t\n",
+ "\n",
+ "\n",
+ "\t\t\n",
+ "\t\t5.5\n",
+ "\t\n",
+ "\n",
+ "\n",
+ "\t\t\n",
+ "\t\t6\n",
+ "\t\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "\t\n",
+ "\n",
+ "\n",
+ "\tgnuplot_plot_1a\n",
+ "\n",
+ "\n",
+ "\n",
+ "\t\n",
+ "\t\n",
+ "\tgnuplot_plot_2a\n",
+ "\n",
+ "\t \n",
+ "\t\n",
+ "\n",
+ "\t\n",
+ "\tgnuplot_plot_3a\n",
+ "\n",
+ "\t \n",
+ "\t\n",
+ "\n",
+ "\t\n",
+ "\tgnuplot_plot_4a\n",
+ "\n",
+ "\t \n",
+ "\t\n",
+ "\n",
+ "\t\n",
+ "\tgnuplot_plot_5a\n",
+ "\n",
+ "\t \n",
+ "\t\n",
+ "\n",
+ "\t\n",
+ "\tgnuplot_plot_6a\n",
+ "\n",
+ "\t\n",
+ "\t\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ ""
+ ],
+ "text/plain": [
+ ""
+ ]
+ },
+ "metadata": {},
+ "output_type": "display_data"
+ }
+ ],
+ "source": [
+ "% Iteration #2\n",
+ "d=(phi-1)*(x_u-x_l);\n",
+ "\n",
+ "x1=x_l+d; % define point 1\n",
+ "x2=x_u-d; % define point 2\n",
+ "\n",
+ "% evaluate Au_x(x1) and Au_x(x2)\n",
+ "\n",
+ "f1=Au_x(x1);\n",
+ "f2=Au_x(x2);\n",
+ "plot(x,Au_x(x),x_l,Au_x(x_l),'ro',x2,f2,'rs',x1,f1,'gs',x_u,Au_x(x_u),'go')\n",
+ "hold on;\n",
+ "\n",
+ "if f2\n",
+ "\n",
+ "Gnuplot\n",
+ "Produced by GNUPLOT 5.0 patchlevel 3 \n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "\t\n",
+ "\t\n",
+ "\t\n",
+ "\t\n",
+ "\t\n",
+ "\t\n",
+ "\t\n",
+ "\t\n",
+ "\n",
+ "\n",
+ "\n",
+ "\t\n",
+ "\t\t\n",
+ "\t\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "\t\t\n",
+ "\t\t-0.04\n",
+ "\t\n",
+ "\n",
+ "\n",
+ "\t\t\n",
+ "\t\t-0.02\n",
+ "\t\n",
+ "\n",
+ "\n",
+ "\t\t\n",
+ "\t\t0\n",
+ "\t\n",
+ "\n",
+ "\n",
+ "\t\t\n",
+ "\t\t0.02\n",
+ "\t\n",
+ "\n",
+ "\n",
+ "\t\t\n",
+ "\t\t0.04\n",
+ "\t\n",
+ "\n",
+ "\n",
+ "\t\t\n",
+ "\t\t0.06\n",
+ "\t\n",
+ "\n",
+ "\n",
+ "\t\t\n",
+ "\t\t0.08\n",
+ "\t\n",
+ "\n",
+ "\n",
+ "\t\t\n",
+ "\t\t2.5\n",
+ "\t\n",
+ "\n",
+ "\n",
+ "\t\t\n",
+ "\t\t3\n",
+ "\t\n",
+ "\n",
+ "\n",
+ "\t\t\n",
+ "\t\t3.5\n",
+ "\t\n",
+ "\n",
+ "\n",
+ "\t\t\n",
+ "\t\t4\n",
+ "\t\n",
+ "\n",
+ "\n",
+ "\t\t\n",
+ "\t\t4.5\n",
+ "\t\n",
+ "\n",
+ "\n",
+ "\t\t\n",
+ "\t\t5\n",
+ "\t\n",
+ "\n",
+ "\n",
+ "\t\t\n",
+ "\t\t5.5\n",
+ "\t\n",
+ "\n",
+ "\n",
+ "\t\t\n",
+ "\t\t6\n",
+ "\t\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "\t\n",
+ "\n",
+ "\n",
+ "\tgnuplot_plot_1a\n",
+ "\n",
+ "\n",
+ "\n",
+ "\t\n",
+ "\t\n",
+ "\tgnuplot_plot_2a\n",
+ "\n",
+ "\t \n",
+ "\t\n",
+ "\n",
+ "\t\n",
+ "\tgnuplot_plot_3a\n",
+ "\n",
+ "\t \n",
+ "\t\n",
+ "\n",
+ "\t\n",
+ "\tgnuplot_plot_4a\n",
+ "\n",
+ "\t \n",
+ "\t\n",
+ "\n",
+ "\t\n",
+ "\tgnuplot_plot_5a\n",
+ "\n",
+ "\t \n",
+ "\t\n",
+ "\n",
+ "\t\n",
+ "\tgnuplot_plot_6a\n",
+ "\n",
+ "\t\n",
+ "\t\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ ""
+ ],
+ "text/plain": [
+ ""
+ ]
+ },
+ "metadata": {},
+ "output_type": "display_data"
+ }
+ ],
+ "source": [
+ "% Iteration #3\n",
+ "d=(phi-1)*(x_u-x_l);\n",
+ "\n",
+ "x1=x_l+d; % define point 1\n",
+ "x2=x_u-d; % define point 2\n",
+ "\n",
+ "% evaluate Au_x(x1) and Au_x(x2)\n",
+ "\n",
+ "f1=Au_x(x1);\n",
+ "f2=Au_x(x2);\n",
+ "plot(x,Au_x(x),x_l,Au_x(x_l),'ro',x2,f2,'rs',x1,f1,'gs',x_u,Au_x(x_u),'go')\n",
+ "hold on;\n",
+ "\n",
+ "if f2\n",
+ "\n",
+ "Gnuplot\n",
+ "Produced by GNUPLOT 5.0 patchlevel 3 \n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "\t\n",
+ "\t\n",
+ "\t\n",
+ "\t\n",
+ "\t\n",
+ "\t\n",
+ "\t\n",
+ "\t\n",
+ "\n",
+ "\n",
+ "\n",
+ "\t\n",
+ "\t\t\n",
+ "\t\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "\t\t\n",
+ "\t\t-0.04\n",
+ "\t\n",
+ "\n",
+ "\n",
+ "\t\t\n",
+ "\t\t-0.02\n",
+ "\t\n",
+ "\n",
+ "\n",
+ "\t\t\n",
+ "\t\t0\n",
+ "\t\n",
+ "\n",
+ "\n",
+ "\t\t\n",
+ "\t\t0.02\n",
+ "\t\n",
+ "\n",
+ "\n",
+ "\t\t\n",
+ "\t\t0.04\n",
+ "\t\n",
+ "\n",
+ "\n",
+ "\t\t\n",
+ "\t\t0.06\n",
+ "\t\n",
+ "\n",
+ "\n",
+ "\t\t\n",
+ "\t\t0.08\n",
+ "\t\n",
+ "\n",
+ "\n",
+ "\t\t\n",
+ "\t\t2.5\n",
+ "\t\n",
+ "\n",
+ "\n",
+ "\t\t\n",
+ "\t\t3\n",
+ "\t\n",
+ "\n",
+ "\n",
+ "\t\t\n",
+ "\t\t3.5\n",
+ "\t\n",
+ "\n",
+ "\n",
+ "\t\t\n",
+ "\t\t4\n",
+ "\t\n",
+ "\n",
+ "\n",
+ "\t\t\n",
+ "\t\t4.5\n",
+ "\t\n",
+ "\n",
+ "\n",
+ "\t\t\n",
+ "\t\t5\n",
+ "\t\n",
+ "\n",
+ "\n",
+ "\t\t\n",
+ "\t\t5.5\n",
+ "\t\n",
+ "\n",
+ "\n",
+ "\t\t\n",
+ "\t\t6\n",
+ "\t\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "\t\n",
+ "\n",
+ "\n",
+ "\tgnuplot_plot_1a\n",
+ "\n",
+ "\n",
+ "\n",
+ "\t\n",
+ "\t\n",
+ "\tgnuplot_plot_2a\n",
+ "\n",
+ "\t \n",
+ "\t\n",
+ "\n",
+ "\t\n",
+ "\tgnuplot_plot_3a\n",
+ "\n",
+ "\t \n",
+ "\t\n",
+ "\n",
+ "\t\n",
+ "\tgnuplot_plot_4a\n",
+ "\n",
+ "\t \n",
+ "\t\n",
+ "\n",
+ "\t\n",
+ "\tgnuplot_plot_5a\n",
+ "\n",
+ "\t \n",
+ "\t\n",
+ "\n",
+ "\t\n",
+ "\tgnuplot_plot_6a\n",
+ "\n",
+ "\t\n",
+ "\t\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ ""
+ ],
+ "text/plain": [
+ ""
+ ]
+ },
+ "metadata": {},
+ "output_type": "display_data"
+ }
+ ],
+ "source": [
+ "% Iteration #3\n",
+ "d=(phi-1)*(x_u-x_l);\n",
+ "\n",
+ "x1=x_l+d; % define point 1\n",
+ "x2=x_u-d; % define point 2\n",
+ "\n",
+ "% evaluate Au_x(x1) and Au_x(x2)\n",
+ "\n",
+ "f1=Au_x(x1);\n",
+ "f2=Au_x(x2);\n",
+ "plot(x,Au_x(x),x_l,Au_x(x_l),'ro',x2,f2,'rs',x1,f1,'gs',x_u,Au_x(x_u),'go')\n",
+ "hold on;\n",
+ "\n",
+ "if f2\n",
+ "\n",
+ "Gnuplot\n",
+ "Produced by GNUPLOT 5.0 patchlevel 3 \n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "\t\n",
+ "\t\n",
+ "\t\n",
+ "\t\n",
+ "\t\n",
+ "\t\n",
+ "\t\n",
+ "\t\n",
+ "\n",
+ "\n",
+ "\n",
+ "\t\n",
+ "\t\t\n",
+ "\t\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "\t\t\n",
+ "\t\t-0.06\n",
+ "\t\n",
+ "\n",
+ "\n",
+ "\t\t\n",
+ "\t\t-0.04\n",
+ "\t\n",
+ "\n",
+ "\n",
+ "\t\t\n",
+ "\t\t-0.02\n",
+ "\t\n",
+ "\n",
+ "\n",
+ "\t\t\n",
+ "\t\t0\n",
+ "\t\n",
+ "\n",
+ "\n",
+ "\t\t\n",
+ "\t\t0.02\n",
+ "\t\n",
+ "\n",
+ "\n",
+ "\t\t\n",
+ "\t\t0.04\n",
+ "\t\n",
+ "\n",
+ "\n",
+ "\t\t\n",
+ "\t\t0.06\n",
+ "\t\n",
+ "\n",
+ "\n",
+ "\t\t\n",
+ "\t\t0.08\n",
+ "\t\n",
+ "\n",
+ "\n",
+ "\t\t\n",
+ "\t\t2.5\n",
+ "\t\n",
+ "\n",
+ "\n",
+ "\t\t\n",
+ "\t\t3\n",
+ "\t\n",
+ "\n",
+ "\n",
+ "\t\t\n",
+ "\t\t3.5\n",
+ "\t\n",
+ "\n",
+ "\n",
+ "\t\t\n",
+ "\t\t4\n",
+ "\t\n",
+ "\n",
+ "\n",
+ "\t\t\n",
+ "\t\t4.5\n",
+ "\t\n",
+ "\n",
+ "\n",
+ "\t\t\n",
+ "\t\t5\n",
+ "\t\n",
+ "\n",
+ "\n",
+ "\t\t\n",
+ "\t\t5.5\n",
+ "\t\n",
+ "\n",
+ "\n",
+ "\t\t\n",
+ "\t\t6\n",
+ "\t\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "\t\n",
+ "\n",
+ "\n",
+ "\tgnuplot_plot_1a\n",
+ "\n",
+ "\n",
+ "\n",
+ "\t\n",
+ "\t\n",
+ "\tgnuplot_plot_2a\n",
+ "\n",
+ "\t\n",
+ "\t\n",
+ "\tgnuplot_plot_3a\n",
+ "\n",
+ "\t \n",
+ "\t\n",
+ "\t\n",
+ "\t\n",
+ "\tgnuplot_plot_4a\n",
+ "\n",
+ "\t \n",
+ "\t\n",
+ "\n",
+ "\t\n",
+ "\tgnuplot_plot_5a\n",
+ "\n",
+ "\t\n",
+ "\t\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ "\n",
+ ""
+ ],
+ "text/plain": [
+ ""
+ ]
+ },
+ "metadata": {},
+ "output_type": "display_data"
+ }
+ ],
+ "source": [
+ "% define Au atomic potential\n",
+ "epsilon = 0.039; % kcal/mol\n",
+ "sigma = 2.934; % Angstrom\n",
+ "Au_x= @(x) lennard_jones(x,sigma,epsilon);\n",
+ "\n",
+ "% set initial limits\n",
+ "x_l=2.8; \n",
+ "x_u=3.5; \n",
+ "\n",
+ "% Iteration #1\n",
+ "x1=x_l;\n",
+ "x2=mean([x_l,x_u]);\n",
+ "x3=x_u;\n",
+ "\n",
+ "% evaluate Au_x(x1), Au_x(x2) and Au_x(x3)\n",
+ " \n",
+ "f1=Au_x(x1);\n",
+ "f2=Au_x(x2);\n",
+ "f3=Au_x(x3);\n",
+ "p = polyfit([x1,x2,x3],[f1,f2,f3],2);\n",
+ "x_fit = linspace(x1,x3,20);\n",
+ "y_fit = polyval(p,x_fit);\n",
+ "\n",
+ "plot(x,Au_x(x),x_fit,y_fit,[x1,x2,x3],[f1,f2,f3],'o')\n",
+ "hold on\n",
+ "if f2x2\n",
+ " plot(x4,f4,'*',[x1,x2],[f1,f2])\n",
+ " x1=x2;\n",
+ " f1=f2;\n",
+ " else\n",
+ " plot(x4,f4,'*',[x3,x2],[f3,f2])\n",
+ " x3=x2;\n",
+ " f3=f2;\n",
+ " end\n",
+ " x2=x4; f2=f4;\n",
+ "else\n",
+ " error('no minimum in bracket')\n",
+ "end\n",
+ "hold off"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 24,
+ "metadata": {
+ "collapsed": false
+ },
+ "outputs": [
+ {
+ "data": {
+ "image/svg+xml": [
+ ""
+ ],
+ "text/plain": [
+ ""
+ ]
+ },
+ "metadata": {},
+ "output_type": "display_data"
+ }
+ ],
+ "source": [
+ "p = polyfit([x1,x2,x3],[f1,f2,f3],2);\n",
+ "x_fit = linspace(x1,x3,20);\n",
+ "y_fit = polyval(p,x_fit);\n",
+ "\n",
+ "plot(x,Au_x(x),x_fit,y_fit,[x1,x2,x3],[f1,f2,f3],'o')\n",
+ "hold on\n",
+ "if f2x2\n",
+ " plot(x4,f4,'*',[x1,x2],[f1,f2])\n",
+ " x1=x2;\n",
+ " f1=f2;\n",
+ " else\n",
+ " plot(x4,f4,'*',[x3,x2],[f3,f2])\n",
+ " x3=x2;\n",
+ " f3=f2;\n",
+ " end\n",
+ " x2=x4; f2=f4;\n",
+ "else\n",
+ " error('no minimum in bracket')\n",
+ "end\n",
+ "hold off\n"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 25,
+ "metadata": {
+ "collapsed": false
+ },
+ "outputs": [
+ {
+ "data": {
+ "image/svg+xml": [
+ ""
+ ],
+ "text/plain": [
+ ""
+ ]
+ },
+ "metadata": {},
+ "output_type": "display_data"
+ }
+ ],
+ "source": [
+ "p = polyfit([x1,x2,x3],[f1,f2,f3],2);\n",
+ "x_fit = linspace(x1,x3,20);\n",
+ "y_fit = polyval(p,x_fit);\n",
+ "\n",
+ "plot(x,Au_x(x),x_fit,y_fit,[x1,x2,x3],[f1,f2,f3],'o')\n",
+ "hold on\n",
+ "if f2x2\n",
+ " plot(x4,f4,'*',[x1,x2],[f1,f2])\n",
+ " x1=x2;\n",
+ " f1=f2;\n",
+ " else\n",
+ " plot(x4,f4,'*',[x3,x2],[f3,f2])\n",
+ " x3=x2;\n",
+ " f3=f2;\n",
+ " end\n",
+ " x2=x4; f2=f4;\n",
+ "else\n",
+ " error('no minimum in bracket')\n",
+ "end\n",
+ "hold off\n"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "Parabolic interpolation does not converge in many scenarios even though it it a bracketing method. Instead, functions like `fminbnd` in Matlab and Octave use a combination of the two (Golden Ratio and Parabolic)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Using the solutions to minimization for the nonlinear spring constant\n",
+ "\n",
+ "Now, we have two routines to find minimums of a univariate function (Golden Ratio and Parabolic). Let's use these to solve for the minimum energy given a range of applied forces to the single atom gold chain\n",
+ "\n",
+ "$E_{total}(\\Delta x) = E_{LJ}(x_{min}+\\Delta x) - F \\cdot \\Delta x$\n",
+ "\n",
+ "$1 aJ = 10^{-18} J = 1~nN* 1~nm = 10^{-9}N * 10^{-9} m$"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 26,
+ "metadata": {
+ "collapsed": false
+ },
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "warning: Matlab-style short-circuit operation performed for operator |\n",
+ "warning: called from\n",
+ " goldmin at line 17 column 1\n",
+ "warning: Matlab-style short-circuit operation performed for operator |\n",
+ "warning: Matlab-style short-circuit operation performed for operator |\n",
+ "xmin = 0.32933\n",
+ "Emin = -2.7096e-04\n"
+ ]
+ },
+ {
+ "data": {
+ "image/svg+xml": [
+ ""
+ ],
+ "text/plain": [
+ ""
+ ]
+ },
+ "metadata": {},
+ "output_type": "display_data"
+ }
+ ],
+ "source": [
+ "epsilon = 0.039; % kcal/mol\n",
+ "epsilon = epsilon*6.9477e-21; % J/atom\n",
+ "epsilon = epsilon*1e18; % aJ/J\n",
+ "% final units for epsilon are aJ\n",
+ "\n",
+ "sigma = 2.934; % Angstrom\n",
+ "sigma = sigma*0.10; % nm/Angstrom\n",
+ "x=linspace(2.8,6,200)*0.10; % bond length in um\n",
+ "\n",
+ "Ex = lennard_jones(x,sigma,epsilon);\n",
+ "\n",
+ "%[Emin,imin]=min(Ex);\n",
+ "\n",
+ "[xmin,Emin] = goldmin(@(x) lennard_jones(x,sigma,epsilon),0.28,0.6)\n",
+ "\n",
+ "plot(x,Ex,xmin,Emin,'o')\n",
+ "ylabel('Lennard Jones Potential (aJ/atom)')\n",
+ "xlabel('bond length (nm)')\n",
+ "\n",
+ "Etotal = @(dx,F) lennard_jones(xmin+dx,sigma,epsilon)-F.*dx;\n"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 31,
+ "metadata": {
+ "collapsed": false
+ },
+ "outputs": [
+ {
+ "data": {
+ "image/svg+xml": [
+ ""
+ ],
+ "text/plain": [
+ ""
+ ]
+ },
+ "metadata": {},
+ "output_type": "display_data"
+ }
+ ],
+ "source": [
+ "N=50;\n",
+ "warning('off')\n",
+ "dx = zeros(1,N); % [in nm]\n",
+ "F_applied=linspace(0,0.0022,N); % [in nN]\n",
+ "for i=1:N\n",
+ " optmin=goldmin(@(dx) Etotal(dx,F_applied(i)),-0.001,0.035);\n",
+ " dx(i)=optmin;\n",
+ "end\n",
+ "\n",
+ "plot(dx,F_applied)\n",
+ "xlabel('dx (nm)')\n",
+ "ylabel('Force (nN)')"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "For this function, it is possible to take a derivative and compare the analytical result:\n",
+ "\n",
+ "dE/dx = 0 = d(E_LJ)/dx - F\n",
+ "\n",
+ "F= d(E_LJ)/dx"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 32,
+ "metadata": {
+ "collapsed": false
+ },
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "F =\n",
+ "\n",
+ "@(dx) 4 * epsilon * 6 * (sigma ^ 6 ./ (xmin + dx) .^ 7 - 2 * sigma ^ 12 ./ (xmin + dx) .^ 13)\n",
+ "\n"
+ ]
+ },
+ {
+ "data": {
+ "image/svg+xml": [
+ ""
+ ],
+ "text/plain": [
+ ""
+ ]
+ },
+ "metadata": {},
+ "output_type": "display_data"
+ }
+ ],
+ "source": [
+ "dx_full=linspace(0,0.06,50);\n",
+ "F= @(dx) 4*epsilon*6*(sigma^6./(xmin+dx).^7-2*sigma^12./(xmin+dx).^13)\n",
+ "plot(dx_full,F(dx_full),dx,F_applied)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {
+ "collapsed": true
+ },
+ "source": [
+ "## Curve-fitting\n",
+ "Another example is minimizing error in your approximation of a function. If you have data (now we have Force-displacement data) we can fit this to a function, such as:\n",
+ "\n",
+ "$F(x) = K_{1}\\Delta x + \\frac{1}{2} K_{2}(\\Delta x)^{2}$\n"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 33,
+ "metadata": {
+ "collapsed": false
+ },
+ "outputs": [],
+ "source": [
+ "function SSE = sse_of_parabola(K,xdata,ydata)\n",
+ " % calculate the sum of squares error for a parabola given a function, func, and xdata and ydata\n",
+ " % output is SSE=sum of squares error\n",
+ " K1=K(1);\n",
+ " K2=K(2);\n",
+ " y_function = K1*xdata+1/2*K2*xdata.^2;\n",
+ " SSE = sum((ydata-y_function).^2);\n",
+ "end\n",
+ " "
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 34,
+ "metadata": {
+ "collapsed": false
+ },
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "\n",
+ "The nonlinear spring constants are K1=0.16 nN/nm and K2=-5.98 nN/nm^2\n",
+ "The mininum sum of squares error = 7.35e-08\n"
+ ]
+ }
+ ],
+ "source": [
+ "[K,SSE_min]=fminsearch(@(K) sse_of_parabola(K,dx,F_applied),[1,1]);\n",
+ "fprintf('\\nThe nonlinear spring constants are K1=%1.2f nN/nm and K2=%1.2f nN/nm^2\\n',K)\n",
+ "fprintf('The mininum sum of squares error = %1.2e',SSE_min)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 35,
+ "metadata": {
+ "collapsed": false
+ },
+ "outputs": [
+ {
+ "data": {
+ "image/svg+xml": [
+ ""
+ ],
+ "text/plain": [
+ ""
+ ]
+ },
+ "metadata": {},
+ "output_type": "display_data"
+ }
+ ],
+ "source": [
+ "plot(dx,F_applied,'o',dx,K(1)*dx+1/2*K(2)*dx.^2)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": null,
+ "metadata": {
+ "collapsed": true
+ },
+ "outputs": [],
+ "source": []
+ }
+ ],
+ "metadata": {
+ "kernelspec": {
+ "display_name": "Octave",
+ "language": "octave",
+ "name": "octave"
+ },
+ "language_info": {
+ "file_extension": ".m",
+ "help_links": [
+ {
+ "text": "MetaKernel Magics",
+ "url": "https://github.com/calysto/metakernel/blob/master/metakernel/magics/README.md"
+ }
+ ],
+ "mimetype": "text/x-octave",
+ "name": "octave",
+ "version": "0.19.14"
+ }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 2
+}
diff --git a/lecture_08/lecture_08.md b/lecture_08/lecture_08.md
new file mode 100644
index 0000000..79edc0a
--- /dev/null
+++ b/lecture_08/lecture_08.md
@@ -0,0 +1,626 @@
+
+
+```octave
+%plot --format svg
+```
+
+# Optimization
+
+Many problems involve finding a minimum or maximum based on given constraints. Engineers and scientists typically use energy balance equations to find the conditions of minimum energy, but this value may never go to 0 and the actual value of energy may not be of interest.
+
+The Lennard-Jones potential is commonly used to model interatomic bonding.
+
+$E_{LJ}(x)=4\epsilon \left(\left(\frac{\sigma}{x}\right)^{12}-\left(\frac{\sigma}{x}\right)^{6}\right)$
+
+Considering a 1-D gold chain, we can calculate the bond length, $x_{b}$, with no force applied to the chain and even for tension, F. This will allow us to calculate the nonlinear spring constant of a 1-D gold chain.
+
+
+
+Computational Tools to Study and Predict the Long-Term Stability of Nanowires.
+By Martin E. Zoloff Michoff, Patricio Vélez, Sergio A. Dassie and Ezequiel P. M. Leiva
+
+
+
+[Single atom gold chain mechanics](http://www.uam.es/personal_pdi/ciencias/agrait/)
+
+### First, let's find the minimum energy $\min(E_{LJ}(x))$
+
+## Brute force
+
+
+```octave
+setdefaults
+epsilon = 0.039; % kcal/mol
+sigma = 2.934; % Angstrom
+x=linspace(2.8,6,200); % bond length in Angstrom
+
+Ex = lennard_jones(x,sigma,epsilon);
+
+[Emin,imin]=min(Ex);
+
+plot(x,Ex,x(imin),Emin,'o')
+```
+
+
+
+
+
+
+```octave
+x(imin-1)
+x(imin)
+x(imin+1)
+```
+
+ ans = 3.2824
+ ans = 3.2985
+ ans = 3.3146
+
+
+## Golden Search Algorithm
+
+We can't just look for a sign change for the problem (unless we can take a derivative) so we need a new approach to determine whether we have a maximum between the two bounds.
+
+Rather than using the midpoint of initial bounds, here the problem is more difficult. We need to compare the values of 4 function evaluations. The golden search uses the golden ratio to determine two interior points.
+
+
+
+Start with bounds of 2.5 and 6 Angstrom.
+
+
+```octave
+% define Au atomic potential
+epsilon = 0.039; % kcal/mol
+sigma = 2.934; % Angstrom
+Au_x= @(x) lennard_jones(x,sigma,epsilon);
+
+% calculate golden ratio
+phi = 1/2+sqrt(5)/2;
+% set initial limits
+x_l=2.8;
+x_u=6;
+
+% Iteration #1
+d=(phi-1)*(x_u-x_l);
+
+x1=x_l+d; % define point 1
+x2=x_u-d; % define point 2
+
+
+% evaluate Au_x(x1) and Au_x(x2)
+
+f1=Au_x(x1);
+f2=Au_x(x2);
+plot(x,Au_x(x),x_l,Au_x(x_l),'ro',x2,f2,'rs',x1,f1,'gs',x_u,Au_x(x_u),'go')
+hold on;
+
+if f2x2
+ plot(x4,f4,'*',[x1,x2],[f1,f2])
+ x1=x2;
+ f1=f2;
+ else
+ plot(x4,f4,'*',[x3,x2],[f3,f2])
+ x3=x2;
+ f3=f2;
+ end
+ x2=x4; f2=f4;
+else
+ error('no minimum in bracket')
+end
+hold off
+```
+
+
+
+
+
+
+```octave
+p = polyfit([x1,x2,x3],[f1,f2,f3],2);
+x_fit = linspace(x1,x3,20);
+y_fit = polyval(p,x_fit);
+
+plot(x,Au_x(x),x_fit,y_fit,[x1,x2,x3],[f1,f2,f3],'o')
+hold on
+if f2x2
+ plot(x4,f4,'*',[x1,x2],[f1,f2])
+ x1=x2;
+ f1=f2;
+ else
+ plot(x4,f4,'*',[x3,x2],[f3,f2])
+ x3=x2;
+ f3=f2;
+ end
+ x2=x4; f2=f4;
+else
+ error('no minimum in bracket')
+end
+hold off
+
+```
+
+
+
+
+
+
+```octave
+p = polyfit([x1,x2,x3],[f1,f2,f3],2);
+x_fit = linspace(x1,x3,20);
+y_fit = polyval(p,x_fit);
+
+plot(x,Au_x(x),x_fit,y_fit,[x1,x2,x3],[f1,f2,f3],'o')
+hold on
+if f2x2
+ plot(x4,f4,'*',[x1,x2],[f1,f2])
+ x1=x2;
+ f1=f2;
+ else
+ plot(x4,f4,'*',[x3,x2],[f3,f2])
+ x3=x2;
+ f3=f2;
+ end
+ x2=x4; f2=f4;
+else
+ error('no minimum in bracket')
+end
+hold off
+
+```
+
+
+
+
+
+Parabolic interpolation does not converge in many scenarios even though it it a bracketing method. Instead, functions like `fminbnd` in Matlab and Octave use a combination of the two (Golden Ratio and Parabolic)
+
+## Using the solutions to minimization for the nonlinear spring constant
+
+Now, we have two routines to find minimums of a univariate function (Golden Ratio and Parabolic). Let's use these to solve for the minimum energy given a range of applied forces to the single atom gold chain
+
+$E_{total}(\Delta x) = E_{LJ}(x_{min}+\Delta x) - F \cdot \Delta x$
+
+
+```octave
+epsilon = 0.039; % kcal/mol
+epsilon = epsilon*6.9477e-21; % J/atom
+epsilon = epsilon*1e18; % aJ/J
+% final units for epsilon are aJ
+
+sigma = 2.934; % Angstrom
+sigma = sigma*0.10; % nm/Angstrom
+x=linspace(2.8,6,200)*0.10; % bond length in um
+
+Ex = lennard_jones(x,sigma,epsilon);
+
+%[Emin,imin]=min(Ex);
+
+[xmin,Emin] = fminbnd(@(x) lennard_jones(x,sigma,epsilon),0.28,0.6)
+
+plot(x,Ex,xmin,Emin,'o')
+ylabel('Lennard Jones Potential (aJ/atom)')
+xlabel('bond length (nm)')
+
+Etotal = @(dx,F) lennard_jones(xmin+dx,sigma,epsilon)-F.*dx;
+
+```
+
+ xmin = 0.32933
+ Emin = -2.7096e-04
+
+
+
+
+
+
+
+```octave
+N=50;
+dx = zeros(1,N); % [in nm]
+F_applied=linspace(0,0.0022,N); % [in nN]
+for i=1:N
+ optmin=goldmin(@(dx) Etotal(dx,F_applied(i)),-0.001,0.035);
+ dx(i)=optmin;
+end
+
+plot(dx,F_applied)
+xlabel('dx (nm)')
+ylabel('Force (nN)')
+```
+
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: called from
+ goldmin at line 17 column 1
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+ warning: Matlab-style short-circuit operation performed for operator |
+
+
+
+
+
+
+For this function, it is possible to take a derivative and compare the analytical result:
+
+
+```octave
+dx_full=linspace(0,0.06,50);
+F= @(dx) 4*epsilon*6*(sigma^6./(xmin+dx).^7-2*sigma^12./(xmin+dx).^13)
+plot(dx_full,F(dx_full),dx,F_applied)
+```
+
+ F =
+
+ @(dx) 4 * epsilon * 6 * (sigma ^ 6 ./ (xmin + dx) .^ 7 - 2 * sigma ^ 12 ./ (xmin + dx) .^ 13)
+
+
+
+
+
+
+
+## Curve-fitting
+Another example is minimizing error in your approximation of a function. If you have data (now we have Force-displacement data) we can fit this to a function, such as:
+
+$F(x) = K_{1}\Delta x + \frac{1}{2} K_{2}(\Delta x)^{2}$
+
+
+
+```octave
+function SSE = sse_of_parabola(K,xdata,ydata)
+ % calculate the sum of squares error for a parabola given a function, func, and xdata and ydata
+ % output is SSE=sum of squares error
+ K1=K(1);
+ K2=K(2);
+ y_function = K1*xdata+1/2*K2*xdata.^2;
+ SSE = sum((ydata-y_function).^2);
+end
+
+```
+
+
+```octave
+[K,SSE_min]=fminsearch(@(K) sse_of_parabola(K,dx,F_applied),[1,1]);
+fprintf('\nThe nonlinear spring constants are K1=%1.2f nN/nm and K2=%1.2f nN/nm^2\n',K)
+fprintf('The mininum sum of squares error = %1.2e',SSE_min)
+```
+
+
+ The nonlinear spring constants are K1=0.16 nN/nm and K2=-5.98 nN/nm^2
+ The mininum sum of squares error = 7.35e-08
+
+
+
+```octave
+plot(dx,F_applied,'o',dx,K(1)*dx+1/2*K(2)*dx.^2)
+```
+
+
+
+
diff --git a/lecture_08/lecture_08.pdf b/lecture_08/lecture_08.pdf
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+
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+
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+
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+
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+
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+
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+
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+
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diff --git a/lecture_08/lennard_jones.m b/lecture_08/lennard_jones.m
new file mode 100644
index 0000000..d18a6ad
--- /dev/null
+++ b/lecture_08/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
+
diff --git a/lecture_08/octave-workspace b/lecture_08/octave-workspace
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--- /dev/null
+++ b/lecture_09/.ipynb_checkpoints/lecture_09-checkpoint.ipynb
@@ -0,0 +1,6 @@
+{
+ "cells": [],
+ "metadata": {},
+ "nbformat": 4,
+ "nbformat_minor": 2
+}
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+
+
+
+
diff --git a/lecture_09/lecture_09.ipynb b/lecture_09/lecture_09.ipynb
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+++ b/lecture_09/lecture_09.ipynb
@@ -0,0 +1,1151 @@
+{
+ "cells": [
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "# Linear Algebra (Review/Introduction)\n",
+ "\n",
+ "Representation of linear equations:\n",
+ "\n",
+ "1. $5x_{1}+3x_{2} =1$\n",
+ "\n",
+ "2. $x_{1}+2x_{2}+3x_{3} =2$\n",
+ "\n",
+ "3. $x_{1}+x_{2}+x_{3} =3$\n",
+ "\n",
+ "in matrix form:\n",
+ "\n",
+ "$\\left[ \\begin{array}{ccc}\n",
+ "5 & 3 & 0 \\\\\n",
+ "1 & 2 & 3 \\\\\n",
+ "1 & 1 & 1 \\end{array} \\right]\n",
+ "\\left[\\begin{array}{c} \n",
+ "x_{1} \\\\ \n",
+ "x_{2} \\\\\n",
+ "x_{3}\\end{array}\\right]=\\left[\\begin{array}{c} \n",
+ "1 \\\\\n",
+ "2 \\\\\n",
+ "3\\end{array}\\right]$\n",
+ "\n",
+ "$Ax=b$"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "### Vectors \n",
+ "\n",
+ "column vector x (length of 3):\n",
+ "\n",
+ "$\\left[\\begin{array}{c} \n",
+ "x_{1} \\\\ \n",
+ "x_{2} \\\\\n",
+ "x_{3}\\end{array}\\right]$\n",
+ "\n",
+ "row vector y (length of 3):\n",
+ "\n",
+ "$\\left[\\begin{array}{ccc} \n",
+ "y_{1}~ \n",
+ "y_{2}~ \n",
+ "y_{3}\\end{array}\\right]$\n",
+ "\n",
+ "vector of length N:\n",
+ "\n",
+ "$\\left[\\begin{array}{c} \n",
+ "x_{1} \\\\ \n",
+ "x_{2} \\\\\n",
+ "\\vdots \\\\\n",
+ "x_{N}\\end{array}\\right]$\n",
+ "\n",
+ "The $i^{th}$ element of x is $x_{i}$"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 1,
+ "metadata": {
+ "collapsed": false
+ },
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "x =\n",
+ "\n",
+ " 1 2 3 4 5 6 7 8 9 10\n",
+ "\n"
+ ]
+ }
+ ],
+ "source": [
+ "x=[1:10]"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 2,
+ "metadata": {
+ "collapsed": false
+ },
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "ans =\n",
+ "\n",
+ " 1\n",
+ " 2\n",
+ " 3\n",
+ " 4\n",
+ " 5\n",
+ " 6\n",
+ " 7\n",
+ " 8\n",
+ " 9\n",
+ " 10\n",
+ "\n"
+ ]
+ }
+ ],
+ "source": [
+ "x'"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "### Matrices\n",
+ "\n",
+ "Matrix A is 3x3:\n",
+ "\n",
+ "$A=\\left[ \\begin{array}{ccc}\n",
+ "5 & 3 & 0 \\\\\n",
+ "1 & 2 & 3 \\\\\n",
+ "1 & 1 & 1 \\end{array} \\right]$\n",
+ "\n",
+ "elements in the matrix are denoted $A_{row~column}$, $A_{23}=3$\n",
+ "\n",
+ "In general, matrix, B, can be any size, $M \\times N$ (M-rows and N-columns):\n",
+ "\n",
+ "$B=\\left[ \\begin{array}{cccc}\n",
+ "B_{11} & B_{12} &...& B_{1N} \\\\\n",
+ "B_{21} & B_{22} &...& B_{2N} \\\\\n",
+ "\\vdots & \\vdots &\\ddots& \\vdots \\\\\n",
+ "B_{M1} & B_{M2} &...& B_{MN}\\end{array} \\right]$"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 3,
+ "metadata": {
+ "collapsed": false
+ },
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "A =\n",
+ "\n",
+ " 5 3 0\n",
+ " 1 2 3\n",
+ " 1 1 1\n",
+ "\n"
+ ]
+ }
+ ],
+ "source": [
+ "A=[5,3,0;1,2,3;1,1,1]"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Multiplication\n",
+ "\n",
+ "A column vector is a $1\\times N$ matrix and a row vector is a $M\\times 1$ matrix\n",
+ "\n",
+ "**Multiplying matrices is not commutative**\n",
+ "\n",
+ "$A B \\neq B A$\n",
+ "\n",
+ "Inner dimensions must agree, output is outer dimensions. \n",
+ "\n",
+ "A is $M1 \\times N1$ and B is $M2 \\times N2$\n",
+ "\n",
+ "C=AB\n",
+ "\n",
+ "Therefore N1=M2 and C is $M1 \\times N2$\n",
+ "\n",
+ "If $C'=BA$, then N2=M1 and C is $M2 \\times N1$\n",
+ "\n",
+ "\n",
+ "e.g. \n",
+ "$A=\\left[ \\begin{array}{cc}\n",
+ "5 & 3 & 0 \\\\\n",
+ "1 & 2 & 3 \\end{array} \\right]$\n",
+ "\n",
+ "$B=\\left[ \\begin{array}{cccc}\n",
+ "1 & 2 & 3 & 4 \\\\\n",
+ "5 & 6 & 7 & 8 \\\\\n",
+ "9 & 10 & 11 & 12 \\end{array} \\right]$\n",
+ "\n",
+ "C=AB\n",
+ "\n",
+ "$[2\\times 4] = [2 \\times 3][3 \\times 4]$\n",
+ "\n",
+ "The rule for multiplying matrices, A and B, is\n",
+ "\n",
+ "$C_{ij} = \\sum_{k=1}^{n}A_{ik}B_{kj}$\n",
+ "\n",
+ "In the previous example, \n",
+ "\n",
+ "$C_{11} = A_{11}B_{11}+A_{12}B_{21}+A_{13}B_{31} = 5*1+3*5+0*9=20$\n",
+ "\n",
+ "\n",
+ "Multiplication is associative:\n",
+ "\n",
+ "$(AB)C = A(BC)$\n",
+ "\n",
+ "and distributive:\n",
+ "\n",
+ "$A(B+C)=AB+AC$"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 4,
+ "metadata": {
+ "collapsed": false
+ },
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "A =\n",
+ "\n",
+ " 5 3 0\n",
+ " 1 2 3\n",
+ "\n",
+ "B =\n",
+ "\n",
+ " 1 2 3 4\n",
+ " 5 6 7 8\n",
+ " 9 10 11 12\n",
+ "\n"
+ ]
+ }
+ ],
+ "source": [
+ "A=[5,3,0;1,2,3] \n",
+ "B=[1,2,3,4;5,6,7,8;9,10,11,12]"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 7,
+ "metadata": {
+ "collapsed": false
+ },
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "C =\n",
+ "\n",
+ " 0 0 0 0\n",
+ " 0 0 0 0\n",
+ "\n",
+ "C =\n",
+ "\n",
+ " 20 28 36 44\n",
+ " 38 44 50 56\n",
+ "\n"
+ ]
+ }
+ ],
+ "source": [
+ "C=zeros(2,4)\n",
+ "C(1,1)=A(1,1)*B(1,1)+A(1,2)*B(2,1)+A(1,3)*B(3,1);\n",
+ "C(1,2)=A(1,1)*B(1,2)+A(1,2)*B(2,2)+A(1,3)*B(3,2);\n",
+ "\n",
+ "C=A*B"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 8,
+ "metadata": {
+ "collapsed": false
+ },
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "error: operator *: nonconformant arguments (op1 is 3x4, op2 is 2x3)\r\n"
+ ]
+ }
+ ],
+ "source": [
+ "Cp=B*A"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Representation of a problem with Matrices and Vectors\n",
+ "\n",
+ "If you have a set of known output, $y_{1},~y_{2},~...y_{N}$ and a set of equations that\n",
+ "relate unknown inputs, $x_{1},~x_{2},~...x_{N}$, then these can be written in a vector\n",
+ "matrix format as:\n",
+ "\n",
+ "$y=Ax=\\left[\\begin{array}{c} \n",
+ "y_{1} \\\\ \n",
+ "y_{2} \\\\\n",
+ "\\vdots \\\\\n",
+ "y_{N}\\end{array}\\right]\n",
+ "=\n",
+ "A\\left[\\begin{array}{c} \n",
+ "x_{1} \\\\ \n",
+ "x_{2} \\\\\n",
+ "\\vdots \\\\\n",
+ "x_{N}\\end{array}\\right]$\n",
+ "\n",
+ "$A=\n",
+ "\\left[\\begin{array}{cccc} \n",
+ "| & | & & | \\\\ \n",
+ "a_{1} & a_{2} & ... & a_{N} \\\\ \n",
+ "| & | & & | \\end{array}\\right]$\n",
+ "\n",
+ "or \n",
+ "\n",
+ "$y = a_{1}x_{1} + a_{2}x_{2} +...+a_{N}x_{N}$\n",
+ "\n",
+ "where each $a_{i}$ is a column vector and $x_{i}$ is a scalar taken from the $i^{th}$\n",
+ "element of x.\n",
+ "\n",
+ "Consider the following problem, 4 masses are connected in series to 4 springs with K=10 N/m. What are the final positions of the masses? \n",
+ "\n",
+ "\n",
+ "\n",
+ "The masses haves the following amounts, 1, 2, 3, and 4 kg for masses 1-4. Using a FBD for each mass:\n",
+ "\n",
+ "$m_{1}g+k(x_{2}-x_{1})-kx_{1}=0$\n",
+ "\n",
+ "$m_{2}g+k(x_{3}-x_{2})-k(x_{2}-x_{1})=0$\n",
+ "\n",
+ "$m_{3}g+k(x_{4}-x_{3})-k(x_{3}-x_{2})=0$\n",
+ "\n",
+ "$m_{4}g-k(x_{4}-x_{3})=0$\n",
+ "\n",
+ "in matrix form:\n",
+ "\n",
+ "$\\left[ \\begin{array}{cccc}\n",
+ "2k & -k & 0 & 0 \\\\\n",
+ "-k & 2k & -k & 0 \\\\\n",
+ "0 & -k & 2k & -k \\\\\n",
+ "0 & 0 & -k & k \\end{array} \\right]\n",
+ "\\left[ \\begin{array}{c}\n",
+ "x_{1} \\\\\n",
+ "x_{2} \\\\\n",
+ "x_{3} \\\\\n",
+ "x_{4} \\end{array} \\right]=\n",
+ "\\left[ \\begin{array}{c}\n",
+ "m_{1}g \\\\\n",
+ "m_{2}g \\\\\n",
+ "m_{3}g \\\\\n",
+ "m_{4}g \\end{array} \\right]$"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 11,
+ "metadata": {
+ "collapsed": false
+ },
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "K =\n",
+ "\n",
+ " 20 -10 0 0\n",
+ " -10 20 -10 0\n",
+ " 0 -10 20 -10\n",
+ " 0 0 -10 10\n",
+ "\n",
+ "y =\n",
+ "\n",
+ " 9.8100\n",
+ " 19.6200\n",
+ " 29.4300\n",
+ " 39.2400\n",
+ "\n",
+ "x =\n",
+ "\n",
+ " 9.8100\n",
+ " 18.6390\n",
+ " 25.5060\n",
+ " 29.4300\n",
+ "\n"
+ ]
+ }
+ ],
+ "source": [
+ "k=10; % N/m\n",
+ "m1=1; % kg\n",
+ "m2=2;\n",
+ "m3=3;\n",
+ "m4=4;\n",
+ "g=9.81; % m/s^2\n",
+ "K=[2*k -k 0 0; -k 2*k -k 0; 0 -k 2*k -k; 0 0 -k k]\n",
+ "y=[m1*g;m2*g;m3*g;m4*g]\n",
+ "\n",
+ "x=K\\y"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Matrix Operations\n",
+ "\n",
+ "Identity matrix `eye(M,N)` **Assume M=N unless specfied**\n",
+ "\n",
+ "$I_{ij} =1$ if $i=j$ \n",
+ "\n",
+ "$I_{ij} =0$ if $i\\neq j$ \n",
+ "\n",
+ "AI=A=IA\n",
+ "\n",
+ "Diagonal matrix, D, is similar to the identity matrix, but each diagonal element can be any\n",
+ "number\n",
+ "\n",
+ "$D_{ij} =d_{i}$ if $i=j$ \n",
+ "\n",
+ "$D_{ij} =0$ if $i\\neq j$ "
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "### Transpose\n",
+ "\n",
+ "The transpose of a matrix changes the rows -> columns and columns-> rows\n",
+ "\n",
+ "$(A^{T})_{ij} = A_{ji}$\n",
+ "\n",
+ "for diagonal matrices, $D^{T}=D$\n",
+ "\n",
+ "in general, the transpose has the following qualities:\n",
+ " \n",
+ "1. $(A^{T})^{T}=A$\n",
+ "\n",
+ "2. $(AB)^{T} = B^{T}A^{T}$\n",
+ "\n",
+ "3. $(A+B)^{T} = A^{T} +B^{T}$ \n",
+ "\n",
+ "All matrices have a symmetric and antisymmetric part:\n",
+ "\n",
+ "$A= 1/2(A+A^{T}) +1/2(A-A^{T})$\n",
+ "\n",
+ "If $A=A^{T}$, then A is symmetric, if $A=-A^{T}$, then A is antisymmetric."
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 18,
+ "metadata": {
+ "collapsed": false
+ },
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "ans =\n",
+ "\n",
+ " 20 38\n",
+ " 28 44\n",
+ " 36 50\n",
+ " 44 56\n",
+ "\n",
+ "ans =\n",
+ "\n",
+ " 20 38\n",
+ " 28 44\n",
+ " 36 50\n",
+ " 44 56\n",
+ "\n",
+ "error: operator *: nonconformant arguments (op1 is 3x2, op2 is 4x3)\n"
+ ]
+ }
+ ],
+ "source": [
+ "(A*B)'\n",
+ "\n",
+ "B'*A' % if this wasnt true, then inner dimensions wouldn;t match\n",
+ "\n",
+ "A'*B' == (A*B)'"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "## Square matrix operations\n",
+ "\n",
+ "### Trace\n",
+ "\n",
+ "The trace of a square matrix is the sum of the diagonal elements\n",
+ "\n",
+ "$tr~A=\\sum_{i=1}^{N}A_{ii}$\n",
+ "\n",
+ "The trace is invariant, meaning that if you change the basis through a rotation, then the\n",
+ "trace remains the same. \n",
+ "\n",
+ "It also has the following properties:\n",
+ "\n",
+ "1. $tr~A=tr~A^{T}$\n",
+ "\n",
+ "2. $tr~A+tr~B=tr(A+B)$\n",
+ "\n",
+ "3. $tr(kA)=k tr~A$\n",
+ "\n",
+ "4. $tr(AB)=tr(BA)$"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 19,
+ "metadata": {
+ "collapsed": false
+ },
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "id_m =\n",
+ "\n",
+ "Diagonal Matrix\n",
+ "\n",
+ " 1 0 0\n",
+ " 0 1 0\n",
+ " 0 0 1\n",
+ "\n",
+ "ans = 3\n"
+ ]
+ }
+ ],
+ "source": [
+ "id_m=eye(3)\n",
+ "trace(id_m)"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "### Inverse\n",
+ "\n",
+ "The inverse of a square matrix, $A^{-1}$ is defined such that\n",
+ "\n",
+ "$A^{-1}A=I=AA^{-1}$\n",
+ "\n",
+ "Not all square matrices have an inverse, they can be *singular* or *non-invertible*\n",
+ "\n",
+ "The inverse has the following properties:\n",
+ "\n",
+ "1. $(A^{-1})^{-1}=A$\n",
+ "\n",
+ "2. $(AB)^{-1}=B^{-1}A^{-1}$\n",
+ "\n",
+ "3. $(A^{-1})^{T}=(A^{T})^{-1}$"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 20,
+ "metadata": {
+ "collapsed": false
+ },
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "A =\n",
+ "\n",
+ " 0.5762106 0.3533174 0.7172134\n",
+ " 0.7061664 0.4863733 0.9423436\n",
+ " 0.4255961 0.0016613 0.3561407\n",
+ "\n"
+ ]
+ }
+ ],
+ "source": [
+ "A=rand(3,3)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 22,
+ "metadata": {
+ "collapsed": false
+ },
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "Ainv =\n",
+ "\n",
+ " 41.5613 -30.1783 -3.8467\n",
+ " 36.2130 -24.2201 -8.8415\n",
+ " -49.8356 36.1767 7.4460\n",
+ "\n"
+ ]
+ }
+ ],
+ "source": [
+ "Ainv=inv(A)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 26,
+ "metadata": {
+ "collapsed": false
+ },
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "B =\n",
+ "\n",
+ " 0.524529 0.470856 0.708116\n",
+ " 0.084491 0.730986 0.528292\n",
+ " 0.303545 0.782522 0.389282\n",
+ "\n"
+ ]
+ }
+ ],
+ "source": [
+ "B=rand(3,3)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 28,
+ "metadata": {
+ "collapsed": false
+ },
+ "outputs": [
+ {
+ "name": "stdout",
+ "output_type": "stream",
+ "text": [
+ "ans =\n",
+ "\n",
+ " -182.185 125.738 40.598\n",
+ " -133.512 97.116 17.079\n",
+ " 282.422 -200.333 -46.861\n",
+ "\n",
+ "ans =\n",
+ "\n",
+ " -182.185 125.738 40.598\n",
+ " -133.512 97.116 17.079\n",
+ " 282.422 -200.333 -46.861\n",
+ "\n",
+ "ans =\n",
+ "\n",
+ " 41.5613 36.2130 -49.8356\n",
+ " -30.1783 -24.2201 36.1767\n",
+ " -3.8467 -8.8415 7.4460\n",
+ "\n",
+ "ans =\n",
+ "\n",
+ " 41.5613 36.2130 -49.8356\n",
+ " -30.1783 -24.2201 36.1767\n",
+ " -3.8467 -8.8415 7.4460\n",
+ "\n"
+ ]
+ }
+ ],
+ "source": [
+ "inv(A*B)\n",
+ "inv(B)*inv(A)\n",
+ "\n",
+ "inv(A')\n",
+ "\n",
+ "inv(A)'"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "metadata": {},
+ "source": [
+ "### Orthogonal Matrices\n",
+ "\n",
+ "Vectors are *orthogonal* if $x^{T}$ y=0, and a vector is *normalized* if $||x||_{2}=1$. A\n",
+ "square matrix is *orthogonal* if all its column vectors are orthogonal to each other and\n",
+ "normalized. The column vectors are then called *orthonormal* and the following results\n",
+ "\n",
+ "$U^{T}U=I=UU^{T}$\n",
+ "\n",
+ "and \n",
+ "\n",
+ "$||Ux||_{2}=||x||_{2}$\n",
+ "\n",
+ "### Determinant\n",
+ "\n",
+ "The **determinant** of a matrix has 3 properties\n",
+ "\n",
+ "1. The determinant of the identity matrix is one, $|I|=1$\n",
+ "\n",
+ "2. If you multiply a single row by scalar $t$ then the determinant is $t|A|$:\n",
+ "\n",
+ "$t|A|=\\left[ \\begin{array}{cccc}\n",
+ "tA_{11} & tA_{12} &...& tA_{1N} \\\\\n",
+ "A_{21} & A_{22} &...& A_{2N} \\\\\n",
+ "\\vdots & \\vdots &\\ddots& \\vdots \\\\\n",
+ "A_{M1} & A_{M2} &...& A_{MN}\\end{array} \\right]$\n",
+ "\n",
+ "3. If you switch 2 rows, the determinant changes sign:\n",
+ "\n",
+ "\n",
+ "$-|A|=\\left[ \\begin{array}{cccc}\n",
+ "A_{21} & A_{22} &...& A_{2N} \\\\\n",
+ "A_{11} & A_{12} &...& A_{1N} \\\\\n",
+ "\\vdots & \\vdots &\\ddots& \\vdots \\\\\n",
+ "A_{M1} & A_{M2} &...& A_{MN}\\end{array} \\right]$\n",
+ "\n",
+ "4. inverse of the determinant is the determinant of the inverse:\n",
+ "\n",
+ "$|A^{-1}|=\\frac{1}{|A|}=|A|^{-1}$\n",
+ "\n",
+ "For a $2\\times2$ matrix, \n",
+ "\n",
+ "$|A|=\\left|\\left[ \\begin{array}{cc}\n",
+ "A_{11} & A_{12} \\\\\n",
+ "A_{21} & A_{22} \\\\\n",
+ "\\end{array} \\right]\\right| = A_{11}A_{22}-A_{21}A_{12}$\n",
+ "\n",
+ "For a $3\\times3$ matrix,\n",
+ "\n",
+ "$|A|=\\left|\\left[ \\begin{array}{ccc}\n",
+ "A_{11} & A_{12} & A_{13} \\\\\n",
+ "A_{21} & A_{22} & A_{23} \\\\\n",
+ "A_{31} & A_{32} & A_{33}\\end{array} \\right]\\right|=$\n",
+ "\n",
+ "$A_{11}A_{22}A_{33}+A_{12}A_{23}A_{31}+A_{13}A_{21}A_{32}\n",
+ "-A_{31}A_{22}A_{13}-A_{32}A_{23}A_{11}-A_{33}A_{21}A_{12}$\n",
+ "\n",
+ "For larger matrices, the determinant is \n",
+ "\n",
+ "$|A|=\n",
+ "\\frac{1}{n!}\n",
+ "\\sum_{i_{r}}^{N}\n",
+ "\\sum_{j_{r}}^{N}\n",
+ "\\epsilon_{i_{1}...i_{N}}\n",
+ "\\epsilon_{j_{1}...j_{N}}\n",
+ "A_{i_{1}j_{1}}\n",
+ "A_{i_{2}j_{2}}\n",
+ "...\n",
+ "A_{i_{N}j_{N}}$\n",
+ "\n",
+ "Where the Levi-Cevita symbol is $\\epsilon_{i_{1}i_{2}...i_{N}} = 1$ if it is an even\n",
+ "permutation and $\\epsilon_{i_{1}i_{2}...i_{N}} = -1$ if it is an odd permutation and\n",
+ "$\\epsilon_{i_{1}i_{2}...i_{N}} = 0$ if $i_{n}=i_{m}$ e.g. $i_{1}=i_{7}$\n",
+ "\n",
+ "\n",
+ "\n",
+ "So a $4\\times4$ matrix determinant,\n",
+ "\n",
+ "$|A|=\\left|\\left[ \\begin{array}{cccc}\n",
+ "A_{11} & A_{12} & A_{13} & A_{14} \\\\\n",
+ "A_{21} & A_{22} & A_{23} & A_{24} \\\\\n",
+ "A_{31} & A_{32} & A_{33} & A_{34} \\\\\n",
+ "A_{41} & A_{42} & A_{43} & A_{44} \\end{array} \\right]\\right|$\n",
+ "\n",
+ "$=\\epsilon_{1234}\\epsilon_{1234}A_{11}A_{22}A_{33}A_{44}+\n",
+ "\\epsilon_{2341}\\epsilon_{1234}A_{21}A_{32}A_{43}A_{14}+\n",
+ "\\epsilon_{3412}\\epsilon_{1234}A_{31}A_{42}A_{13}A_{24}+\n",
+ "\\epsilon_{4123}\\epsilon_{1234}A_{41}A_{12}A_{23}A_{34}+...\n",
+ "\\epsilon_{4321}\\epsilon_{4321}A_{44}A_{33}A_{22}A_{11}$\n",
+ "\n",
+ "### Special Case for determinants\n",
+ "\n",
+ "The determinant of a diagonal matrix $|D|=D_{11}D_{22}D_{33}...D_{NN}$. \n",
+ "\n",
+ "Similarly, if a matrix is upper triangular (so all values of $A_{ij}=0$ when $j
+Babel <3.9q> and hyphenation patterns for 81 language(s) loaded.
+! You can't use `macro parameter character #' in vertical mode.
+l.2 #
+ Linear Algebra (Review/Introduction)
+?
+! Interruption.
+
+
+l.2 #
+ Linear Algebra (Review/Introduction)
+? xx
+
+Here is how much of TeX's memory you used:
+ 6 strings out of 493030
+ 147 string characters out of 6136260
+ 53067 words of memory out of 5000000
+ 3641 multiletter control sequences out of 15000+600000
+ 3640 words of font info for 14 fonts, out of 8000000 for 9000
+ 1141 hyphenation exceptions out of 8191
+ 5i,0n,1p,57b,8s stack positions out of 5000i,500n,10000p,200000b,80000s
+No pages of output.
diff --git a/lecture_09/lecture_09.md b/lecture_09/lecture_09.md
new file mode 100644
index 0000000..a1d53d1
--- /dev/null
+++ b/lecture_09/lecture_09.md
@@ -0,0 +1,802 @@
+
+# Linear Algebra (Review/Introduction)
+
+Representation of linear equations:
+
+1. $5x_{1}+3x_{2} =1$
+
+2. $x_{1}+2x_{2}+3x_{3} =2$
+
+3. $x_{1}+x_{2}+x_{3} =3$
+
+in matrix form:
+
+$\left[ \begin{array}{ccc}
+5 & 3 & 0 \\
+1 & 2 & 3 \\
+1 & 1 & 1 \end{array} \right]
+\left[\begin{array}{c}
+x_{1} \\
+x_{2} \\
+x_{3}\end{array}\right]=\left[\begin{array}{c}
+1 \\
+2 \\
+3\end{array}\right]$
+
+$Ax=b$
+
+### Vectors
+
+column vector x (length of 3):
+
+$\left[\begin{array}{c}
+x_{1} \\
+x_{2} \\
+x_{3}\end{array}\right]$
+
+row vector y (length of 3):
+
+$\left[\begin{array}{ccc}
+y_{1}~
+y_{2}~
+y_{3}\end{array}\right]$
+
+vector of length N:
+
+$\left[\begin{array}{c}
+x_{1} \\
+x_{2} \\
+\vdots \\
+x_{N}\end{array}\right]$
+
+The $i^{th}$ element of x is $x_{i}$
+
+
+```octave
+x=[1:10]
+```
+
+ x =
+
+ 1 2 3 4 5 6 7 8 9 10
+
+
+
+
+```octave
+x'
+```
+
+ ans =
+
+ 1
+ 2
+ 3
+ 4
+ 5
+ 6
+ 7
+ 8
+ 9
+ 10
+
+
+
+### Matrices
+
+Matrix A is 3x3:
+
+$A=\left[ \begin{array}{ccc}
+5 & 3 & 0 \\
+1 & 2 & 3 \\
+1 & 1 & 1 \end{array} \right]$
+
+elements in the matrix are denoted $A_{row~column}$, $A_{23}=3$
+
+In general, matrix, B, can be any size, $M \times N$ (M-rows and N-columns):
+
+$B=\left[ \begin{array}{cccc}
+B_{11} & B_{12} &...& B_{1N} \\
+B_{21} & B_{22} &...& B_{2N} \\
+\vdots & \vdots &\ddots& \vdots \\
+B_{M1} & B_{M2} &...& B_{MN}\end{array} \right]$
+
+
+```octave
+A=[5,3,0;1,2,3;1,1,1]
+```
+
+ A =
+
+ 5 3 0
+ 1 2 3
+ 1 1 1
+
+
+
+## Multiplication
+
+A column vector is a $1\times N$ matrix and a row vector is a $M\times 1$ matrix
+
+**Multiplying matrices is not commutative**
+
+$A B \neq B A$
+
+Inner dimensions must agree, output is outer dimensions.
+
+A is $M1 \times N1$ and B is $M2 \times N2$
+
+C=AB
+
+Therefore N1=M2 and C is $M1 \times N2$
+
+If $C'=BA$, then N2=M1 and C is $M2 \times N1$
+
+
+e.g.
+$A=\left[ \begin{array}{cc}
+5 & 3 & 0 \\
+1 & 2 & 3 \end{array} \right]$
+
+$B=\left[ \begin{array}{cccc}
+1 & 2 & 3 & 4 \\
+5 & 6 & 7 & 8 \\
+9 & 10 & 11 & 12 \end{array} \right]$
+
+C=AB
+
+$[2\times 4] = [2 \times 3][3 \times 4]$
+
+The rule for multiplying matrices, A and B, is
+
+$C_{ij} = \sum_{k=1}^{n}A_{ik}B_{kj}$
+
+In the previous example,
+
+$C_{11} = A_{11}B_{11}+A_{12}B_{21}+A_{13}B_{31} = 5*1+3*5+0*9=20$
+
+
+Multiplication is associative:
+
+$(AB)C = A(BC)$
+
+and distributive:
+
+$A(B+C)=AB+AC$
+
+
+```octave
+A=[5,3,0;1,2,3]
+B=[1,2,3,4;5,6,7,8;9,10,11,12]
+```
+
+ A =
+
+ 5 3 0
+ 1 2 3
+
+ B =
+
+ 1 2 3 4
+ 5 6 7 8
+ 9 10 11 12
+
+
+
+
+```octave
+C=zeros(2,4)
+C(1,1)=A(1,1)*B(1,1)+A(1,2)*B(2,1)+A(1,3)*B(3,1);
+C(1,2)=A(1,1)*B(1,2)+A(1,2)*B(2,2)+A(1,3)*B(3,2);
+
+C=A*B
+```
+
+ C =
+
+ 0 0 0 0
+ 0 0 0 0
+
+ C =
+
+ 20 28 36 44
+ 38 44 50 56
+
+
+
+
+```octave
+Cp=B*A
+```
+
+ error: operator *: nonconformant arguments (op1 is 3x4, op2 is 2x3)
+
+
+## Representation of a problem with Matrices and Vectors
+
+If you have a set of known output, $y_{1},~y_{2},~...y_{N}$ and a set of equations that
+relate unknown inputs, $x_{1},~x_{2},~...x_{N}$, then these can be written in a vector
+matrix format as:
+
+$y=Ax=\left[\begin{array}{c}
+y_{1} \\
+y_{2} \\
+\vdots \\
+y_{N}\end{array}\right]
+=
+A\left[\begin{array}{c}
+x_{1} \\
+x_{2} \\
+\vdots \\
+x_{N}\end{array}\right]$
+
+$A=
+\left[\begin{array}{cccc}
+| & | & & | \\
+a_{1} & a_{2} & ... & a_{N} \\
+| & | & & | \end{array}\right]$
+
+or
+
+$y = a_{1}x_{1} + a_{2}x_{2} +...+a_{N}x_{N}$
+
+where each $a_{i}$ is a column vector and $x_{i}$ is a scalar taken from the $i^{th}$
+element of x.
+
+Consider the following problem, 4 masses are connected in series to 4 springs with K=10 N/m. What are the final positions of the masses?
+
+
+
+The masses haves the following amounts, 1, 2, 3, and 4 kg for masses 1-4. Using a FBD for each mass:
+
+$m_{1}g+k(x_{2}-x_{1})-kx_{1}=0$
+
+$m_{2}g+k(x_{3}-x_{2})-k(x_{2}-x_{1})=0$
+
+$m_{3}g+k(x_{4}-x_{3})-k(x_{3}-x_{2})=0$
+
+$m_{4}g-k(x_{4}-x_{3})=0$
+
+in matrix form:
+
+$\left[ \begin{array}{cccc}
+2k & -k & 0 & 0 \\
+-k & 2k & -k & 0 \\
+0 & -k & 2k & -k \\
+0 & 0 & -k & k \end{array} \right]
+\left[ \begin{array}{c}
+x_{1} \\
+x_{2} \\
+x_{3} \\
+x_{4} \end{array} \right]=
+\left[ \begin{array}{c}
+m_{1}g \\
+m_{2}g \\
+m_{3}g \\
+m_{4}g \end{array} \right]$
+
+
+```octave
+k=10; % N/m
+m1=1; % kg
+m2=2;
+m3=3;
+m4=4;
+g=9.81; % m/s^2
+K=[2*k -k 0 0; -k 2*k -k 0; 0 -k 2*k -k; 0 0 -k k]
+y=[m1*g;m2*g;m3*g;m4*g]
+
+x=K\y
+```
+
+ K =
+
+ 20 -10 0 0
+ -10 20 -10 0
+ 0 -10 20 -10
+ 0 0 -10 10
+
+ y =
+
+ 9.8100
+ 19.6200
+ 29.4300
+ 39.2400
+
+ x =
+
+ 9.8100
+ 18.6390
+ 25.5060
+ 29.4300
+
+
+
+## Matrix Operations
+
+Identity matrix `eye(M,N)` **Assume M=N unless specfied**
+
+$I_{ij} =1$ if $i=j$
+
+$I_{ij} =0$ if $i\neq j$
+
+AI=A=IA
+
+Diagonal matrix, D, is similar to the identity matrix, but each diagonal element can be any
+number
+
+$D_{ij} =d_{i}$ if $i=j$
+
+$D_{ij} =0$ if $i\neq j$
+
+### Transpose
+
+The transpose of a matrix changes the rows -> columns and columns-> rows
+
+$(A^{T})_{ij} = A_{ji}$
+
+for diagonal matrices, $D^{T}=D$
+
+in general, the transpose has the following qualities:
+
+1. $(A^{T})^{T}=A$
+
+2. $(AB)^{T} = B^{T}A^{T}$
+
+3. $(A+B)^{T} = A^{T} +B^{T}$
+
+All matrices have a symmetric and antisymmetric part:
+
+$A= 1/2(A+A^{T}) +1/2(A-A^{T})$
+
+If $A=A^{T}$, then A is symmetric, if $A=-A^{T}$, then A is antisymmetric.
+
+
+```octave
+(A*B)'
+
+B'*A' % if this wasnt true, then inner dimensions wouldn;t match
+
+A'*B' == (A*B)'
+```
+
+ ans =
+
+ 20 38
+ 28 44
+ 36 50
+ 44 56
+
+ ans =
+
+ 20 38
+ 28 44
+ 36 50
+ 44 56
+
+ error: operator *: nonconformant arguments (op1 is 3x2, op2 is 4x3)
+
+
+## Square matrix operations
+
+### Trace
+
+The trace of a square matrix is the sum of the diagonal elements
+
+$tr~A=\sum_{i=1}^{N}A_{ii}$
+
+The trace is invariant, meaning that if you change the basis through a rotation, then the
+trace remains the same.
+
+It also has the following properties:
+
+1. $tr~A=tr~A^{T}$
+
+2. $tr~A+tr~B=tr(A+B)$
+
+3. $tr(kA)=k tr~A$
+
+4. $tr(AB)=tr(BA)$
+
+
+```octave
+id_m=eye(3)
+trace(id_m)
+```
+
+ id_m =
+
+ Diagonal Matrix
+
+ 1 0 0
+ 0 1 0
+ 0 0 1
+
+ ans = 3
+
+
+### Inverse
+
+The inverse of a square matrix, $A^{-1}$ is defined such that
+
+$A^{-1}A=I=AA^{-1}$
+
+Not all square matrices have an inverse, they can be *singular* or *non-invertible*
+
+The inverse has the following properties:
+
+1. $(A^{-1})^{-1}=A$
+
+2. $(AB)^{-1}=B^{-1}A^{-1}$
+
+3. $(A^{-1})^{T}=(A^{T})^{-1}$
+
+
+```octave
+A=rand(3,3)
+```
+
+ A =
+
+ 0.5762106 0.3533174 0.7172134
+ 0.7061664 0.4863733 0.9423436
+ 0.4255961 0.0016613 0.3561407
+
+
+
+
+```octave
+Ainv=inv(A)
+```
+
+ Ainv =
+
+ 41.5613 -30.1783 -3.8467
+ 36.2130 -24.2201 -8.8415
+ -49.8356 36.1767 7.4460
+
+
+
+
+```octave
+B=rand(3,3)
+```
+
+ B =
+
+ 0.524529 0.470856 0.708116
+ 0.084491 0.730986 0.528292
+ 0.303545 0.782522 0.389282
+
+
+
+
+```octave
+inv(A*B)
+inv(B)*inv(A)
+
+inv(A')
+
+inv(A)'
+```
+
+ ans =
+
+ -182.185 125.738 40.598
+ -133.512 97.116 17.079
+ 282.422 -200.333 -46.861
+
+ ans =
+
+ -182.185 125.738 40.598
+ -133.512 97.116 17.079
+ 282.422 -200.333 -46.861
+
+ ans =
+
+ 41.5613 36.2130 -49.8356
+ -30.1783 -24.2201 36.1767
+ -3.8467 -8.8415 7.4460
+
+ ans =
+
+ 41.5613 36.2130 -49.8356
+ -30.1783 -24.2201 36.1767
+ -3.8467 -8.8415 7.4460
+
+
+
+### Orthogonal Matrices
+
+Vectors are *orthogonal* if $x^{T}$ y=0, and a vector is *normalized* if $||x||_{2}=1$. A
+square matrix is *orthogonal* if all its column vectors are orthogonal to each other and
+normalized. The column vectors are then called *orthonormal* and the following results
+
+$U^{T}U=I=UU^{T}$
+
+and
+
+$||Ux||_{2}=||x||_{2}$
+
+### Determinant
+
+The **determinant** of a matrix has 3 properties
+
+1. The determinant of the identity matrix is one, $|I|=1$
+
+2. If you multiply a single row by scalar $t$ then the determinant is $t|A|$:
+
+$t|A|=\left[ \begin{array}{cccc}
+tA_{11} & tA_{12} &...& tA_{1N} \\
+A_{21} & A_{22} &...& A_{2N} \\
+\vdots & \vdots &\ddots& \vdots \\
+A_{M1} & A_{M2} &...& A_{MN}\end{array} \right]$
+
+3. If you switch 2 rows, the determinant changes sign:
+
+
+$-|A|=\left[ \begin{array}{cccc}
+A_{21} & A_{22} &...& A_{2N} \\
+A_{11} & A_{12} &...& A_{1N} \\
+\vdots & \vdots &\ddots& \vdots \\
+A_{M1} & A_{M2} &...& A_{MN}\end{array} \right]$
+
+4. inverse of the determinant is the determinant of the inverse:
+
+$|A^{-1}|=\frac{1}{|A|}=|A|^{-1}$
+
+For a $2\times2$ matrix,
+
+$|A|=\left|\left[ \begin{array}{cc}
+A_{11} & A_{12} \\
+A_{21} & A_{22} \\
+\end{array} \right]\right| = A_{11}A_{22}-A_{21}A_{12}$
+
+For a $3\times3$ matrix,
+
+$|A|=\left|\left[ \begin{array}{ccc}
+A_{11} & A_{12} & A_{13} \\
+A_{21} & A_{22} & A_{23} \\
+A_{31} & A_{32} & A_{33}\end{array} \right]\right|=$
+
+$A_{11}A_{22}A_{33}+A_{12}A_{23}A_{31}+A_{13}A_{21}A_{32}
+-A_{31}A_{22}A_{13}-A_{32}A_{23}A_{11}-A_{33}A_{21}A_{12}$
+
+For larger matrices, the determinant is
+
+$|A|=
+\frac{1}{n!}
+\sum_{i_{r}}^{N}
+\sum_{j_{r}}^{N}
+\epsilon_{i_{1}...i_{N}}
+\epsilon_{j_{1}...j_{N}}
+A_{i_{1}j_{1}}
+A_{i_{2}j_{2}}
+...
+A_{i_{N}j_{N}}$
+
+Where the Levi-Cevita symbol is $\epsilon_{i_{1}i_{2}...i_{N}} = 1$ if it is an even
+permutation and $\epsilon_{i_{1}i_{2}...i_{N}} = -1$ if it is an odd permutation and
+$\epsilon_{i_{1}i_{2}...i_{N}} = 0$ if $i_{n}=i_{m}$ e.g. $i_{1}=i_{7}$
+
+
+
+So a $4\times4$ matrix determinant,
+
+$|A|=\left|\left[ \begin{array}{cccc}
+A_{11} & A_{12} & A_{13} & A_{14} \\
+A_{21} & A_{22} & A_{23} & A_{24} \\
+A_{31} & A_{32} & A_{33} & A_{34} \\
+A_{41} & A_{42} & A_{43} & A_{44} \end{array} \right]\right|$
+
+$=\epsilon_{1234}\epsilon_{1234}A_{11}A_{22}A_{33}A_{44}+
+\epsilon_{2341}\epsilon_{1234}A_{21}A_{32}A_{43}A_{14}+
+\epsilon_{3412}\epsilon_{1234}A_{31}A_{42}A_{13}A_{24}+
+\epsilon_{4123}\epsilon_{1234}A_{41}A_{12}A_{23}A_{34}+...
+\epsilon_{4321}\epsilon_{4321}A_{44}A_{33}A_{22}A_{11}$
+
+### Special Case for determinants
+
+The determinant of a diagonal matrix $|D|=D_{11}D_{22}D_{33}...D_{NN}$.
+
+Similarly, if a matrix is upper triangular (so all values of $A_{ij}=0$ when $j
+
+
+