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| <!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> | ||
| </head> | ||
| <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></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 | ||
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||
| | 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> |
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| # Homework #3 | ||
| ## due 2/15/17 by 11:59pm | ||
|
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||
| 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: | ||
|
|
||
| `` | ||
|
|
||
| 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) |
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