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<h1 id="homework-2">Homework #2</h1>
<h2 id="due-10617-by-1159pm">due 10/6/17 by 11:59pm</h2>
<p><strong>1.</strong> Create a new github repository called ‘02_roots_and_optimization’.</p>
<ol style="list-style-type: lower-alpha">
<li><p>Add rcc02007 and zhs15101 as collaborators.</p></li>
<li><p>submit the clone repository URL to: <a href="https://goo.gl/forms/svFKpfiCfLO9Zvfz1" class="uri">https://goo.gl/forms/svFKpfiCfLO9Zvfz1</a></p></li>
</ol>
<p><strong>2.</strong> You’re installing a powerline in a residential neighborhood. The lowest point on the cable is 30 m above the ground, but 30 m away is a tree that is 35 m tall. Another engineer informs you that this is a catenary cable problem with the following solution</p>
<div class="figure">
<img src="./equations/eq1.png" alt="eq. 1" />
<p class="caption">eq. 1</p>
</div>
<p><span class="math inline">\(y(x)=\frac{T}{w}\cosh\left(\frac{w}{T}x\right)+y_{0}-\frac{T}{w}\)</span>.</p>
<p>where y(x) is the height of the cable at a distance, x, from the lowest point, <span class="math inline">\(y_{0}\)</span>, T is the tension in the cable, and w is the weight per unit length of the cable. Your supervisor wants to know which numerical solver to use when they have to install these powerlines in similar places.</p>
<ol style="list-style-type: lower-alpha">
<li><p>Use the three solvers <code>falsepos.m</code>, <code>bisect.m</code>, and <code>mod_secant.m</code> to solve for the tension neededi, T, to reach y(30 m)=35 m, with w=10 N/m, and <span class="math inline">\(y_{0}\)</span>=30 m.</p></li>
<li><p>Compare the number of iterations that each function needed to reach an accuracy of 0.00001%. Include a table in your README.md with:</p>
<pre><code>| solver | initial guess(es) | ea | number of iterations|
| --- | --- | --- | --- |
|falsepos   |  |  |  |
|mod_secant |  |  |  |
|bisect     |  |  |  |</code></pre></li>
<li><p>Add a figure to your README that plots the final shape of the powerline (<img src="./equations/eq2.png" alt="eq2" />) from x=-10 to 50 m.</p></li>
</ol>
<p><strong>3.</strong> The Newton-Raphson method and the modified secant method do not always converge to a solution. One simple example is the function f(x) = (x-1)*exp(-(x-1)^2). The root is at 1, but using the numerical solvers, <code>newtraph.m</code> and <code>mod_secant.m</code>, there are certain initial guesses that do not converge.</p>
<ol style="list-style-type: lower-alpha">
<li><p>Calculate the first 5 iterations for the Newton-Raphson method with an initial guess of x_i=3 for f(x)=(x-1)*exp(-(x-1)^2).</p></li>
<li><p>Add the results to a table in the <code>README.md</code> with:</p>
<pre><code>### divergence of Newton-Raphson method

| iteration | x_i | approx error |
| --- | --- | --- |
| 0 | 3 | n/a |
| 1 |   |     |
| 2 |   |     |
| 3 |   |     |
| 4 |   |     |
| 5 |   |     |</code></pre></li>
<li><p>Repeat steps a-b for an initial guess of 1.2. (But change the heading from ‘divergence’ to ‘convergence’)</p></li>
</ol>
<div class="figure">
<img src="../08_optimization/Auchain_model.png" alt="Model of Gold chain, from molecular dynamics simulation" />
<p class="caption">Model of Gold chain, from molecular dynamics simulation</p>
</div>
<p><strong>4.</strong> Determine the nonlinear spring constants of a single-atom gold chain. You can assume the gold atoms are aligned in a one dimensional network and the potential energy is described by the Lennard-Jones potential as such</p>
<div class="figure">
<img src="./equations/eq3.png" alt="eq3" />
<p class="caption">eq3</p>
</div>
<p><span class="math inline">\(E_{LJ}(x)=4\epsilon  \left(\left(\frac{\sigma}{x}\right)^{12}-\left(\frac{\sigma}{x}\right)^{6}\right)\)</span>.</p>
<p>Where x is the distance between atoms in nm, <span class="math inline">\(\epsilon\)</span>=2.71E-4 aJ, and <span class="math inline">\(\sigma\)</span>=0.2934 nm. The energy term that must be minimized is</p>
<div class="figure">
<img src="./equations/eq4.png" alt="eq4" />
<p class="caption">eq4</p>
</div>
<p><span class="math inline">\(E_{total}(\Delta x)=E_{LJ}(x_{0}+\Delta x)-F\Delta x\)</span>.</p>
<p>Where <span class="math inline">\(x_{0}\)</span> is the distance between atoms with no force applied and <img src="./equations/deltax.png" alt="dx" /> is the amount each gold atom has moved under a given force, F.</p>
<ol style="list-style-type: lower-alpha">
<li><p>Determine <img src="./equations/x0.png" alt="x0" /> when F=0 nN using the golden ratio and parabolic methods. <em>Show your script and output in your README and include your functions</em></p></li>
<li><p>Solve for <img src="./equations/deltax.png" alt="dx" /> is the amount each gold atom has mov for F=0 to 0.0022 nN with 30 steps. *Use the golden ratio solver or the matlab/octave <code>fminsearch</code></p></li>
<li><p>create a sum of squares error function <code>sse_of_parabola.m</code> that calculates the sum of squares error between a function <span class="math inline">\(F(x)=K_{1}\Delta x+1/2K_{2}\Delta x^{2}\)</span> and the Forces used in part B for each <img src="./equations/deltax.png" alt="dx" />.</p></li>
<li><p>Use the <code>fminsearch</code> matlab/octave function to determine <img src="./equations/k1k2.png" alt="k1k2" />.</p></li>
<li><p>Plot the force vs calculated <img src="./equations/deltax.png" alt="dx" /> and the best-fit parabola using <img src="./equations/k1k2.png" alt="k1k2" /> in part d.</p></li>
</ol>
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	<h1 id="homework-2">Homework #2</h1>
	<h2 id="due-10617-by-1159pm">due 10/6/17 by 11:59pm</h2>
	<p><strong>1.</strong> Create a new github repository called ‘02_roots_and_optimization’.</p>
	<ol style="list-style-type: lower-alpha">
	<li><p>Add rcc02007 and zhs15101 as collaborators.</p></li>
	<li><p>submit the clone repository URL to: <a href="https://goo.gl/forms/svFKpfiCfLO9Zvfz1" class="uri">https://goo.gl/forms/svFKpfiCfLO9Zvfz1</a></p></li>
	</ol>
	<p><strong>2.</strong> You’re installing a powerline in a residential neighborhood. The lowest point on the cable is 30 m above the ground, but 30 m away is a tree that is 35 m tall. Another engineer informs you that this is a catenary cable problem with the following solution</p>
	<div class="figure">
	<img src="./equations/eq1.png" alt="eq. 1" />
	<p class="caption">eq. 1</p>
	</div>
	<p><span class="math inline">\(y(x)=\frac{T}{w}\cosh\left(\frac{w}{T}x\right)+y_{0}-\frac{T}{w}\)</span>.</p>
	<p>where y(x) is the height of the cable at a distance, x, from the lowest point, <span class="math inline">\(y_{0}\)</span>, T is the tension in the cable, and w is the weight per unit length of the cable. Your supervisor wants to know which numerical solver to use when they have to install these powerlines in similar places.</p>
	<ol style="list-style-type: lower-alpha">
	<li><p>Use the three solvers <code>falsepos.m</code>, <code>bisect.m</code>, and <code>mod_secant.m</code> to solve for the tension neededi, T, to reach y(30 m)=35 m, with w=10 N/m, and <span class="math inline">\(y_{0}\)</span>=30 m.</p></li>
	<li><p>Compare the number of iterations that each function needed to reach an accuracy of 0.00001%. Include a table in your README.md with:</p>
	<pre><code>\| solver \| initial guess(es) \| ea \| number of iterations\|
	\| --- \| --- \| --- \| --- \|
	\|falsepos \| \| \| \|
	\|mod_secant \| \| \| \|
	\|bisect \| \| \| \|</code></pre></li>
	<li><p>Add a figure to your README that plots the final shape of the powerline (<img src="./equations/eq2.png" alt="eq2" />) from x=-10 to 50 m.</p></li>
	</ol>
	<p><strong>3.</strong> The Newton-Raphson method and the modified secant method do not always converge to a solution. One simple example is the function f(x) = (x-1)*exp(-(x-1)^2). The root is at 1, but using the numerical solvers, <code>newtraph.m</code> and <code>mod_secant.m</code>, there are certain initial guesses that do not converge.</p>
	<ol style="list-style-type: lower-alpha">
	<li><p>Calculate the first 5 iterations for the Newton-Raphson method with an initial guess of x_i=3 for f(x)=(x-1)*exp(-(x-1)^2).</p></li>
	<li><p>Add the results to a table in the <code>README.md</code> with:</p>
	<pre><code>### divergence of Newton-Raphson method

	\| iteration \| x_i \| approx error \|
	\| --- \| --- \| --- \|
	\| 0 \| 3 \| n/a \|
	\| 1 \| \| \|
	\| 2 \| \| \|
	\| 3 \| \| \|
	\| 4 \| \| \|
	\| 5 \| \| \|</code></pre></li>
	<li><p>Repeat steps a-b for an initial guess of 1.2. (But change the heading from ‘divergence’ to ‘convergence’)</p></li>
	</ol>
	<div class="figure">
	<img src="../08_optimization/Auchain_model.png" alt="Model of Gold chain, from molecular dynamics simulation" />
	<p class="caption">Model of Gold chain, from molecular dynamics simulation</p>
	</div>
	<p><strong>4.</strong> Determine the nonlinear spring constants of a single-atom gold chain. You can assume the gold atoms are aligned in a one dimensional network and the potential energy is described by the Lennard-Jones potential as such</p>
	<div class="figure">
	<img src="./equations/eq3.png" alt="eq3" />
	<p class="caption">eq3</p>
	</div>
	<p><span class="math inline">\(E_{LJ}(x)=4\epsilon \left(\left(\frac{\sigma}{x}\right)^{12}-\left(\frac{\sigma}{x}\right)^{6}\right)\)</span>.</p>
	<p>Where x is the distance between atoms in nm, <span class="math inline">\(\epsilon\)</span>=2.71E-4 aJ, and <span class="math inline">\(\sigma\)</span>=0.2934 nm. The energy term that must be minimized is</p>
	<div class="figure">
	<img src="./equations/eq4.png" alt="eq4" />
	<p class="caption">eq4</p>
	</div>
	<p><span class="math inline">\(E_{total}(\Delta x)=E_{LJ}(x_{0}+\Delta x)-F\Delta x\)</span>.</p>
	<p>Where <span class="math inline">\(x_{0}\)</span> is the distance between atoms with no force applied and <img src="./equations/deltax.png" alt="dx" /> is the amount each gold atom has moved under a given force, F.</p>
	<ol style="list-style-type: lower-alpha">
	<li><p>Determine <img src="./equations/x0.png" alt="x0" /> when F=0 nN using the golden ratio and parabolic methods. <em>Show your script and output in your README and include your functions</em></p></li>
	<li><p>Solve for <img src="./equations/deltax.png" alt="dx" /> is the amount each gold atom has mov for F=0 to 0.0022 nN with 30 steps. *Use the golden ratio solver or the matlab/octave <code>fminsearch</code></p></li>
	<li><p>create a sum of squares error function <code>sse_of_parabola.m</code> that calculates the sum of squares error between a function <span class="math inline">\(F(x)=K_{1}\Delta x+1/2K_{2}\Delta x^{2}\)</span> and the Forces used in part B for each <img src="./equations/deltax.png" alt="dx" />.</p></li>
	<li><p>Use the <code>fminsearch</code> matlab/octave function to determine <img src="./equations/k1k2.png" alt="k1k2" />.</p></li>
	<li><p>Plot the force vs calculated <img src="./equations/deltax.png" alt="dx" /> and the best-fit parabola using <img src="./equations/k1k2.png" alt="k1k2" /> in part d.</p></li>
	</ol>
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