eib12003 · eib12003 · Mar 22, 2017 · Mar 9, 2017 · Mar 21, 2017 · Mar 21, 2017
diff --git a/HW4/README.md b/HW4/README.md
@@ -1,4 +1,4 @@
-# Homework #3
+# Homework #4
 ## due 3/1/17 by 11:59pm
 
 

diff --git a/HW5/README.html b/HW5/README.html
@@ -0,0 +1,53 @@
+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-transitional.dtd">
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head>
+  <meta http-equiv="Content-Type" content="text/html; charset=utf-8" />
+  <meta http-equiv="Content-Style-Type" content="text/css" />
+  <meta name="generator" content="pandoc" />
+  <title></title>
+  <style type="text/css">code{white-space: pre;}</style>
+  <script src="https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML" type="text/javascript"></script>
+</head>
+<body>
+<h1 id="homework-5">Homework #5</h1>
+<h2 id="due-32817-by-1159pm">due 3/28/17 by 11:59pm</h2>
+<p><em>Include all work as either an m-file script, m-file function, or example code included with ``` and document your code in the README.md file</em></p>
+<ol style="list-style-type: decimal">
+<li><p>Create a new github repository called ‘linear_algebra’.</p>
+<ol style="list-style-type: lower-alpha">
+<li><p>Add rcc02007 and pez16103 as collaborators.</p></li>
+<li><p>Clone the repository to your computer.</p></li>
+</ol></li>
+<li><p>Create an LU-decomposition function called <code>lu_tridiag.m</code> that takes 3 vectors as inputs and calculates the LU-decomposition of a tridiagonal matrix. The output should be 3 vectors, the diagonal of the Upper matrix, and the two off-diagonal vectors of the Lower and Upper matrices.</p>
+<p><code>[ud,uo,lo]=lu_tridiag(e,f,g);</code></p></li>
+<li><p>Use the output from <code>lu_tridiag.m</code> to create a forward substitution and back-substitution function called <code>solve_tridiag.m</code> that provides the solution of Ax=b given the vectors from the output of [ud,uo,lo]=lu_tridiag(e,f,g). <em>Note: do not use the backslash solver <code>\</code>, create an algebraic solution</em></p>
+<p><code>x=solve_tridiag(ud,uo,lo,b);</code></p></li>
+<li><p>Test your function on the matrices A3, A4, …, A10 generated with <code>test_arrays.m</code> Solving for <code>b=ones(N,1);</code> where N is the size of A. In your <code>README.md</code> file, compare the norm of the error between your result and the result of AN.</p></li>
+</ol>
+<pre><code>| size of A | norm(error) |
+|-----------|-------------|
+|  3 | ... |
+| 4 | ... |</code></pre>
+<div class="figure">
+<img src="spring_mass.png" alt="Spring-mass system for analysis" />
+<p class="caption">Spring-mass system for analysis</p>
+</div>
+<ol start="5" style="list-style-type: decimal">
+<li><p>In the system shown above, determine the three differential equations for the position of masses 1, 2, and 3. Solve for the vibrational modes of the spring-mass system if k1=10 N/m, k2=k3=20 N/m, and k4=10 N/m. The masses are m1=1 kg, m2=2 kg and m3=4 kg. Determine the eigenvalues and natural frequencies.</p></li>
+<li><p>The curvature of a slender column subject to an axial load P (Fig. P13.10) can be modeled by</p></li>
+</ol>
+<p><span class="math inline">\(\frac{d^{2}y}{dx^{2}} + p^{2} y = 0\)</span></p>
+<p>where <span class="math inline">\(p^{2} = \frac{P}{EI}\)</span></p>
+<p>where E = the modulus of elasticity, and I = the moment of inertia of the cross section about its neutral axis.</p>
+<p>This model can be converted into an eigenvalue problem by substituting a centered finite-difference approximation for the second derivative to give <span class="math inline">\(\frac{y_{i+1} − 2y_{i} + y_{i−1} }{\Delta x^{2}}+ p^{2} y_{i}\)</span></p>
+<p>where i = a node located at a position along the rod’s interior, and <span class="math inline">\(\Delta x\)</span> = the spacing between nodes. This equation can be expressed as <span class="math inline">\(y_{i−1} − (2 − \Delta x^{2} p^{2} )y_{i} + y_{i+1} = 0\)</span> Writing this equation for a series of interior nodes along the axis of the column yields a homogeneous system of equations. (See 13.10 for 4 interior-node example)</p>
+<p>Determine the eigenvalues for a 5-segment (4-interior nodes), 6-segment (5-interior nodes), and 10-segment (9-interior nodes). Using the modulus and moment of inertia of a pole for pole-vaulting ( <a href="http://people.bath.ac.uk/taf21/sports_whole.htm" class="uri">http://people.bath.ac.uk/taf21/sports_whole.htm</a>) E=76E9 Pa, I=4E-8 m^4, and L= 5m.</p>
+<p>Include a table in the <code>README.md</code> that shows the following results: What are the largest and smallest eigenvalues for the beam? How many eigenvalues are there?</p>
+<pre><code>| # of segments | largest | smallest | # of eigenvalues |
+| --- | --- | --- | --- |
+| 5 | ... | ... | ... |
+| 6 | ... | ... | ... |
+| 10 | ... | ... | ... |</code></pre>
+<p>If the segment length approaches 0, how many eigenvalues would there be?</p>
+</body>
+</html>
diff --git a/HW5/README.md b/HW5/README.md
@@ -0,0 +1,83 @@
+# Homework #5
+## due 3/28/17 by 11:59pm
+
+*Include all work as either an m-file script, m-file function, or example code included
+with \`\`\` and document your code in the README.md file*
+
+1. Create a new github repository called 'linear_algebra'. 
+
+    a. Add rcc02007 and pez16103 as collaborators.
+
+    b. Clone the repository to your computer.
+
+2. Create an LU-decomposition function called `lu_tridiag.m` that takes 3 vectors as inputs
+and calculates the LU-decomposition of a tridiagonal matrix. The output should be 3
+vectors, the diagonal of the Upper matrix, and the two off-diagonal vectors of the Lower
+and Upper matrices. 
+
+    ```[ud,uo,lo]=lu_tridiag(e,f,g);```
+
+3. Use the output from `lu_tridiag.m` to create a forward substitution and
+back-substitution function called `solve_tridiag.m` that provides the solution of
+Ax=b given the vectors from the output of [ud,uo,lo]=lu_tridiag(e,f,g). *Note: do not use
+the backslash solver `\`, create an algebraic solution*
+
+    ```x=solve_tridiag(ud,uo,lo,b);```
+
+4. Test your function on the matrices A3, A4, ..., A10 generated with `test_arrays.m`
+Solving for `b=ones(N,1);` where N is the size of A.  In your `README.md` file, compare
+the norm of the error between your result and the result of AN\b. 
+
+```
+| size of A | norm(error) |
+|-----------|-------------|
+|  3 | ... |
+| 4 | ... |
+```
+
+![Spring-mass system for analysis](spring_mass.png)
+
+5. In the system shown above, determine the three differential equations for the position
+of masses 1, 2, and 3. Solve for the vibrational modes of the spring-mass system if k1=10
+N/m, k2=k3=20 N/m, and k4=10 N/m. The masses are m1=1 kg, m2=2 kg and m3=4 kg. Determine
+the eigenvalues and natural frequencies. 
+
+6. The curvature of a slender column subject to an axial load P (Fig. P13.10) can be
+modeled by 
+
+$\frac{d^{2}y}{dx^{2}} + p^{2} y = 0$
+
+where $p^{2} = \frac{P}{EI}$
+
+where E = the modulus of elasticity, and I = the moment of inertia of the cross section
+about its neutral axis.  
+
+This model can be converted into an eigenvalue problem by
+substituting a centered finite-difference approximation for the second derivative to give
+$\frac{y_{i+1} -2y_{i} + y_{i-1} }{\Delta x^{2}}+ p^{2} y_{i}$ 
+
+where i = a node located at a position along the rod’s interior, and $\Delta x$ = the
+spacing between nodes. This equation can be expressed as $y_{i-1} - (2 - \Delta x^{2}
+p^{2} )y_{i}  y_{i+1} = 0$ Writing this equation for a series of interior nodes along the
+axis of the column yields a homogeneous system of equations. (See 13.10 for 4
+interior-node example)
+
+Determine the eigenvalues for a 5-segment (4-interior nodes), 6-segment (5-interior
+nodes), and 10-segment (9-interior nodes). Using the modulus and moment of inertia of a
+pole for pole-vaulting (
+[http://people.bath.ac.uk/taf21/sports_whole.htm](http://people.bath.ac.uk/taf21/sports_whole.htm))
+E=76E9 Pa, I=4E-8 m^4, and L= 5m. 
+
+Include a table in the `README.md` that shows the following results:
+What are the largest and smallest eigenvalues for the beam? How many eigenvalues are
+there? 
+
+```
+| # of segments | largest | smallest | # of eigenvalues |
+| --- | --- | --- | --- |
+| 5 | ... | ... | ... |
+| 6 | ... | ... | ... |
+| 10 | ... | ... | ... |
+```
+
+If the segment length ($\Delta x$) approaches 0, how many eigenvalues would there be?
diff --git a/HW5/README.pdf b/HW5/README.pdf
diff --git a/HW5/lu_tridiag.m b/HW5/lu_tridiag.m
@@ -0,0 +1,12 @@
+function [ud,uo,lo]=lu_tridiag(e,f,g);```
+  % lu_tridiag calculates the components for LU-decomposition of a tridiagonal matrix
+  % given its off-diagonal vectors, e and g 
+  % and diagonal vector f
+  % the output is 
+  % the diagonal of the Upper matrix, ud
+  % the off-diagonal of the Upper matrix, uo
+  % and the off-diagonal of the Lower matrix, lo
+  % note: the diagonal of the Lower matrix is all ones
+
+end
+
diff --git a/HW5/solve_tridiag.m b/HW5/solve_tridiag.m
@@ -0,0 +1,11 @@
+function x=solve_tridiag(ud,uo,lo,b);```
+  % solve_tridiag solves Ax=b  for x
+  % given 
+  % the diagonal of the Upper matrix, ud
+  % the off-diagonal of the Upper matrix, uo
+  % the off-diagonal of the Lower matrix, lo
+  % the vector b
+  % note: the diagonal of the Lower matrix is all ones
+
+end
+
diff --git a/HW5/spring_mass.png b/HW5/spring_mass.png