added HW3

rfa10002 · rfa10002 · Feb 14, 2017 · Jan 31, 2017 · Jan 31, 2017 · Feb 2, 2017
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+<body>
+<h1 id="homework-3">Homework #3</h1>
+<h2 id="due-21517-by-1159pm">due 2/15/17 by 11:59pm</h2>
+<ol style="list-style-type: decimal">
+<li><p>Create a new github repository called ‘roots_and_optimization’.</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>
+<li><p>Copy your <code>projectile.m</code> function into the ‘roots_and_optimization’ folder. <em>Disable the plotting routine for the solvers</em></p></li>
+<li><p>Use the four solvers <code>falsepos.m</code>, <code>incsearch.m</code>, <code>newtraph.m</code> and <code>mod_secant.m</code> to solve for the angle needed to reach h=1.72, with an initial speed of 1.5 m/s.</p></li>
+<li><p>The <code>newtraph.m</code> function needs a derivative, calculate the derivative of your function with respect to theta, <code>dprojectile_dtheta.m</code>. This function should output d(h)/d(theta).</p></li>
+<li><p>In your <code>README.md</code> file, you will document the following under the heading <code># Homework #3</code>:</p>
+<ol style="list-style-type: lower-roman">
+<li>Compare the number of iterations that each function needed to reach an accuracy of 0.00001%. Include a table in your README.md with:</li>
+</ol>
+<pre><code>| solver | initial guess(es) | ea | number of iterations|
+| --- | --- | --- | --- |
+|falsepos   |  |  |  |
+|incsearch  |  |  |  |
+|newtraph   |  |  |  |
+|mod_secant |  |  |  |</code></pre>
+<ol start="2" style="list-style-type: lower-roman">
+<li>Compare the convergence of the 4 methods. Plot the approximate error vs the number of iterations that the solver has calculated. Save the plot as <code>convergence.png</code> and display the plot in your <code>README.md</code> with:</li>
+</ol>
+<p><code>![Plot of convergence for four numerical solvers.](convergence.png)</code></p>
+<ol start="3" style="list-style-type: lower-roman">
+<li>In the <code>README.md</code> provide a description of the files used to create the table and the convergence plot.</li>
+</ol></li>
+</ol></li>
+<li><p>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*exp(-x^2). The root is at 0, 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=2.</p></li>
+<li><p>Add the results to a table in the <code>README.md</code> with:</p></li>
+</ol>
+<pre><code>### divergence of Newton-Raphson method
+
+| iteration | x_i | approx error |
+| --- | --- | --- |
+| 0 | 2 | n/a |
+| 1 |   |     |
+| 2 |   |     |
+| 3 |   |     |
+| 4 |   |     |
+| 5 |   |     |</code></pre>
+<ol start="3" style="list-style-type: lower-alpha">
+<li>Repeat steps a-b for an initial guess of 0.2. (But change the heading from ‘divergence’ to ‘convergence’)</li>
+</ol></li>
+<li><p>Commit your changes to your repository. Sync your local repository with github. Then copy and paste the “clone URL” into the following Google Form <a href="https://goo.gl/forms/UJBGwp0fQcSxImkq2">Homework #3</a></p></li>
+</ol>
+</body>
+</html>
diff --git a/HW3/README.md b/HW3/README.md
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+# Homework #3
+## due 2/15/17 by 11:59pm
+
+
+1. Create a new github repository called 'roots_and_optimization'. 
+
+    a. Add rcc02007 and pez16103 as collaborators.
+
+    b. Clone the repository to your computer.
+
+    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. 
+
+    e. The `newtraph.m` function needs a derivative, calculate the derivative of your
+    function with respect to theta, `dprojectile_dtheta.m`. This function should
+    output d(h)/d(theta). 
+
+
+    f. In your `README.md` file, document the following under the heading `#
+    Homework #3`:
+
+        i. Compare the number of iterations that each function needed to reach an
+        accuracy of 0.00001%. Include a table in your README.md with:
+
+        ```
+        | solver | initial guess(es) | ea | number of iterations|
+        | --- | --- | --- | --- |
+        |falsepos   |  |  |  |
+        |incsearch  |  |  |  |
+        |newtraph   |  |  |  |
+        |mod_secant |  |  |  |
+        ```
+
+        ii. Compare the convergence of the 4 methods. Plot the approximate error vs the
+        number of iterations that the solver has calculated. Save the plot as
+        `convergence.png` and display the plot in your `README.md` with:
+
+        `![Plot of convergence for four numerical solvers.](convergence.png)`
+
+        iii. In the `README.md` provide a description of the files used to create the
+        table and the convergence plot. 
+
+2. 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*exp(-x^2). The root is at 0, but
+using the numerical solvers, `newtraph.m` and `mod_secant.m`, there are certain initial
+guesses that do not converge. 
+
+    a. Calculate the first 5 iterations for the Newton-Raphson method with an initial
+    guess of x_i=2. 
+
+    b. Add the results to a table in the `README.md` with:
+
+    ```
+    ### divergence of Newton-Raphson method
+
+    | iteration | x_i | approx error |
+    | --- | --- | --- |
+    | 0 | 2 | n/a |
+    | 1 |   |     |
+    | 2 |   |     |
+    | 3 |   |     |
+    | 4 |   |     |
+    | 5 |   |     |
+    ```
+
+    c. Repeat steps a-b for an initial guess of 0.2. (But change the heading from
+    '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)
diff --git a/lecture_07/.newtraph.m.swp b/lecture_07/.newtraph.m.swp