updated HW4

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diff --git a/HW4/README.html b/HW4/README.html
@@ -10,29 +10,34 @@
 </head>
 <body>
 <h1 id="homework-4">Homework #4</h1>
-<h2 id="due-32817-by-1159pm">due 3/28/17 by 11:59pm</h2>
+<h2 id="final-commit-due-11217-by-1159pm">final commit due 11/2/17 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 ‘04_linear_algebra’.</p>
 <ol style="list-style-type: lower-alpha">
-<li><p>Add rcc02007 and pez16103 as collaborators.</p></li>
+<li><p>Add rcc02007 and zhs15101 as collaborators.</p></li>
 <li><p>Clone the repository to your computer.</p></li>
+<li><p>Complete this before 10/26 by midnight</p></li>
 </ol></li>
-<li><p>Create the 4x4 and 5x5 Hilbert matrix as H:</p></li>
+</ol>
+<h3 id="p2-due-1026">P2 Due 10/26</h3>
+<ol start="2" style="list-style-type: decimal">
+<li>Create the 4x4 and 5x5 <a href="https://en.wikipedia.org/wiki/Hilbert_matrix">Hilbert matrix</a> as H. Include the following results in your README before 10/26 by midnight:</li>
 </ol>
 <ol style="list-style-type: lower-alpha">
 <li><p>What are the 2-norm, frobenius-norm, 0-norm and infinity-norm of the 4x4 and 5x5 Hilbert matrices?</p></li>
 <li><p>What are the 2-norm, frobenius-norm, 0-norm and infinity-norm of the 4x4 and 5x5 inverse Hilbert matrices?</p></li>
 <li><p>What are the condition numbers for the 2-norm, frobenius-norm, 0-norm and infinity-norm of the 4x4 and 5x5 Hilbert matrices?</p></li>
 </ol>
+<h3 id="p3-4-due-1030">P3-4 Due 10/30</h3>
 <ol start="3" style="list-style-type: decimal">
-<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>Create a Cholesky factorization function called <code>chol_tridiag.m</code> that takes 2 vectors as inputs and calculates the Cholseky factorization of a tridiagonal matrix. The output should be 2 vectors, the diagonal and the off-diagonal vector of the Cholesky matrix.</p>
+<p><code>[d,u]=chol_tridiag(e,f);</code></p></li>
+<li><p>Use the output from <code>chol_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 [d,u]=lu_tridiag(e,f). <em>Note: do not use the backslash solver <code>\</code>, create an algebraic solution</em></p>
+<p><code>x=solve_tridiag(d,u,b);</code></p></li>
 </ol>
 <div class="figure">
-<img src="mass_springs.png" alt="Spring-mass system for problem 5" />
+<img src="./figures/mass_springs.png" alt="Spring-mass system for problem 5" />
 <p class="caption">Spring-mass system for problem 5</p>
 </div>
 <ol start="5" style="list-style-type: decimal">
@@ -44,28 +49,36 @@
 <li><p>K1=K3=K4=1000 N/m, K2=1000e-12 N/m</p></li>
 </ol>
 <ol start="6" style="list-style-type: decimal">
-<li>Use <code>lu_tridiag.m</code> and <code>solve_tridiag.m</code> to solve for the displacements of hanging masses 1-4 shown above, if all masses are 1 kg.</li>
+<li>Use <code>chol_tridiag.m</code> and <code>solve_tridiag.m</code> to solve for the displacements of hanging masses 1-4 shown above in 5a-c, if all masses are 1 kg.</li>
 </ol>
 <div class="figure">
-<img src="spring_mass.png" alt="Spring-mass system for analysis" />
+<img src="./figures/spring_mass.png" alt="Spring-mass system for analysis" />
 <p class="caption">Spring-mass system for analysis</p>
 </div>
 <ol start="7" 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. Create a function, <code>mass_spring_vibrate.m</code> that outputs the vibration modes and natural frequencies based upon the parameters, k1, k2, k3, and k4.</p></li>
 <li><p>The curvature of a slender column subject to an axial load P 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>
+<div class="figure">
+<img src="./equations/d2ydx2.png" />
+
+</div>
+<p>where <span class="math inline">\(p^{2} = \frac{P}{EI}\)</span> <img src="./equations/p2.png" /></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>
+<div class="figure">
+<img src="./equations/delta2y.png" />
+
+</div>
+<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> <img src="./equations/solve.png" /> 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 loads in the beam based upon the different shapes? How many eigenvalues are there?</p>
 <pre><code>| # of segments | largest load (N) | smallest load (N) | # of eigenvalues |
 | --- | --- | --- | --- |
 | 5 | ... | ... | ... |
 | 6 | ... | ... | ... |
 | 10 | ... | ... | ... |</code></pre>
-<p>If the segment length (<span class="math inline">\(\Delta x\)</span>) approaches 0, how many eigenvalues would there be?</p>
+<p>If the segment length approaches 0, how many eigenvalues would there be?</p>
 </body>
 </html>
diff --git a/HW4/README.md b/HW4/README.md
@@ -1,18 +1,21 @@
 # Homework #4
-## due 11/2/17 by 11:59pm
+## final commit due 11/2/17 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*
 
+### P1-2 Due 10/26
+
 1. Create a new github repository called '04_linear_algebra'. 
 
     a. Add rcc02007 and zhs15101 as collaborators.
 
     b. Clone the repository to your computer.
 
-    c. Complete this before 10/26 by midnight
+    c. Submit clone repo link to
+    [https://goo.gl/forms/gFNxhNM4qJJKj8hE3](https://goo.gl/forms/gFNxhNM4qJJKj8hE3)
 
-2. Create the  4x4 and 5x5 Hilbert matrix as H. Include the following results in your
+2. Create the  4x4 and 5x5 [Hilbert matrix](https://en.wikipedia.org/wiki/Hilbert_matrix) as H. Include the following results in your
 README before 10/26 by midnight:
 
   a. What are the 2-norm, frobenius-norm, 0-norm and infinity-norm of the 4x4 and 5x5
@@ -24,21 +27,22 @@ README before 10/26 by midnight:
   c. What are the condition numbers for the 2-norm, frobenius-norm, 0-norm and
   infinity-norm of the 4x4 and 5x5 Hilbert matrices?
 
-3. 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. 
+### P3-4 Due 10/30
+
+3. Create a Cholesky factorization function called `chol_tridiag.m` that takes 2 vectors
+as inputs and calculates the Cholseky factorization of a tridiagonal matrix. The output
+should be 2 vectors, the diagonal and the off-diagonal vector of the Cholesky matrix. 
 
-    ```[ud,uo,lo]=lu_tridiag(e,f,g);```
+    ```[d,u]=chol_tridiag(e,f);```
 
-4. Use the output from `lu_tridiag.m` to create a forward substitution and
+4. Use the output from `chol_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
+Ax=b given the vectors from the output of [d,u]=lu_tridiag(e,f). *Note: do not use
 the backslash solver `\`, create an algebraic solution*
 
-    ```x=solve_tridiag(ud,uo,lo,b);```
+    ```x=solve_tridiag(d,u,b);```
 
-![Spring-mass system for problem 5](mass_springs.png)
+![Spring-mass system for problem 5](./figures/mass_springs.png)
 
 5. Create the stiffness matrix for the 4-mass system shown above
 for cases a-c. Calculate the condition of the stiffness matrices. What is the expected error
@@ -50,10 +54,10 @@ when calculating the displacements of the 4 masses? Include the analysis and res
 
   c. K1=K3=K4=1000 N/m, K2=1000e-12 N/m
 
-6. Use `lu_tridiag.m` and `solve_tridiag.m` to solve for the displacements of hanging
-  masses 1-4 shown above, if all masses are 1 kg.
+6. Use `chol_tridiag.m` and `solve_tridiag.m` to solve for the displacements of hanging
+  masses 1-4 shown above in 5a-c, if all masses are 1 kg.
 
-![Spring-mass system for analysis](spring_mass.png)
+![Spring-mass system for analysis](./figures/spring_mass.png)
 
 7. 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

diff --git a/HW4/chol_tridiag.m b/HW4/chol_tridiag.m
@@ -0,0 +1,11 @@
+function [d,u]=lu_tridiag(e,f);
+  % chol_tridiag calculates the components for Cholesky factorization of a symmetric
+  % positive definite tridiagonal matrix
+  % given its off-diagonal vector, e  
+  % and diagonal vector f
+  % the output is 
+  % the diagonal of the Upper matrix, d
+  % the off-diagonal of the Upper matrix, u
+
+end
+
diff --git a/HW4/lu_tridiag.m b/HW4/lu_tridiag.m
diff --git a/HW4/solve_tridiag.m b/HW4/solve_tridiag.m
@@ -1,11 +1,9 @@
 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 diagonal of the Cholesky matrix, d
+  % the off-diagonal of the Upper matrix, u
   % the vector b
-  % note: the diagonal of the Lower matrix is all ones
 
 end