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<h1 id="homework-4">Homework #4</h1>
<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 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>
</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 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="./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">
<li>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 when calculating the displacements of the 4 masses? Include the analysis and results in your README.</li>
</ol>
<ol style="list-style-type: lower-alpha">
<li><p>K1=K2=K3=K4=1000 N/m</p></li>
<li><p>K1=K3=K4=1000 N/m, K2=1000e12 N/m</p></li>
<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>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="./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>
<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>
<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 approaches 0, how many eigenvalues would there be?</p>
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	<h1 id="homework-4">Homework #4</h1>
	<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 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>
	</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 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="./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">
	<li>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 when calculating the displacements of the 4 masses? Include the analysis and results in your README.</li>
	</ol>
	<ol style="list-style-type: lower-alpha">
	<li><p>K1=K2=K3=K4=1000 N/m</p></li>
	<li><p>K1=K3=K4=1000 N/m, K2=1000e12 N/m</p></li>
	<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>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="./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>
	<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>
	<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 approaches 0, how many eigenvalues would there be?</p>
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