Homework 3ΒΆ

Submit via the Canvas page by 11:00pm on November 5, 2015. Homework can be submitted up to 24 hours late with a 10% reduction in points possible. If special circumstances warrant turning in assignments late, please make arrangements in advance.

  • Create a GitHub account for yourself if you don’t already have one, see https://github.com.
  • Watch the Intro to Git video from 26 October 2015 if you are not already familiar with git.
  • See the Canvas Homework 3 assignment page for a link that will create a private repository for you named am570-username based on your GitHub username. See Using Git and GitHub for information on how to clone this repository.
  • Watch the Intro to GitHub video from 26 October 2015, even if you are familiar with GitHub, since it contains information about the class repositories and walks you through the next steps:
    • Clone the repository am570-username
    • cd into the directory this creates.
    • Edit README.md so the title is correct for your username and so that it your full name appears in the file.
    • Use git add and git commit to commit this change.
    • Use git push to push this change to GitHub.
    • Check the GitHub webpage to see that it appears as desired.
    • Create a subdirectory hw3
    • Copy the files cheb.m and p11.m from SMM into hw3
    • git add and git commit these files (original version)
    • Run p11.m in Matlab and print -djpeg p11.jpg to create a jpeg file.
    • Add and commit this file and git push to GitHub.
    • Note that you can view the image on GitHub.
    • Modify p11.m so that the upper figure is for N=30 rather than N=10.
    • Create a new version of p11.jpg.
    • Add and commit these changes and git push to GitHub.
    • Note that on GitHub you can view the changes introduced in a commit by clicking on the commit number.

For the remaining problems you are encouraged to work in your git repository and commit often as you go along to get in the habit of using version control.

You may submit the homework as usual through the Canvas submission link.

If you want to use your GitHub repository instead, please submit a file to Canvas by the due date that contains your GitHub username and the commit number for the version you want me to grade (the 40-digit hexadecimal string). This can just be a text file or a pdf file.

  • ATAP #5.1, 5.2.
  • ATAP #5.7. In addition to what is asked for, also:
    • Produce a log-log plot of the max-norm error in \(p_n(x)\) vs. \(n\) for \(n=1,~2,~\ldots,~25\) and on the same plot, the error in the polynomial interpolating at the Chebyshev points.
    • Do this for \(f(x) = |x|\), and then also try your code on \(f(x) = \exp(-|x|^3)\) and \(f(x) = e^x\). Comment on what you observe in each case.
  • ATAP #6.5. Note that the binomial coefficient formula for “\(k\) choose \(n\)” given by \(k(k-1)...(k-n+1)/n!\) can be used also when \(k=1/2\). Be careful about showing convergence of your series approximation. You can use the ratio test, but may need to rescale \(x\) to get a series that coverges everywhere in \([-1,1]\), using e.g. \(|x| = 2|x/2|\).
  • ATAP #7.1, 8.1, 8.2, 8.3.