## Archive for January, 2017

### Using git and github for course assignments

Tuesday, January 10th, 2017

Here’s my tested plan for using git and github to manage student assignments for my upcoming course.

## Initial setup as the instructor:

We’re going to create a student repo and an instructor fork for each assignment. The motivation for separating assignments is to ensure that the diffs that students are submitting for their solutions are tidy. In my case, there’s not much shared code between assignments

Create public* repo that students will see: test-gp-hw0-student.

If you try to fork this on Github, it will tell you that you already have a fork. So instead trick github into forking it via the github importer, import test-gp-hw0-student as a private test-gp-hw0-instructor. Now you have two repos. We’ll set up a branch on test-gp-hw0-instructor to track test-gp-hw0-student for easy merging:

INSTRUCTOR_REPO=test-gp-hw0-instructor
STUDENT_REPO=test-gp-hw0-student
git clone https://github.com/alecjacobson/$INSTRUCTOR_REPO.git cd$INSTRUCTOR_REPO
git remote add student https://github.com/alecjacobson/$STUDENT_REPO.git git fetch student git checkout -b student-tracker --track student/master git checkout master cd ..  To test this out, let’s create a file visible to everyone (students and instructors) in the student repo: git clone https://github.com/alecjacobson/$STUDENT_REPO.git
cd $STUDENT_REPO echo "visible to everyone" >> hello-everyone.md git add hello-everyone.md git commit -m "add file visible to everyone" hello-everyone.md git push cd ../  Now, let’s merge change into the instructor repo cd$INSTRUCTOR_REPO
git checkout master
git pull student master
cat hello-everyone.md
cd ..


Just to be thorough, let’s try the reverse: creating a file just visible to instructors:

cd stewartdent
git remote add instructor https://stewartdent@github.com/alecjacobson/$INSTRUCTOR_REPO.git git fetch instructor  You should see an error: remote: Repository not found. fatal: repository 'https://stewartdent@github.com/alecjacobson/test-gp-hw0-instructor.git/' not found  Now, let’s act as the student to submit the homework # cd stewartdent echo "stewartdent was here" >> assignment-01.md git add assignment-01.md git commit -m "Stewart Dent's Assignment 01 submission" assignment-01.md git push cd ..  The student has pushed their submission onto their own repo. They can continue to work on it using git in all its glory. When they’re down they “hand in” their homework via a pull request onto the public test-gp-hw0-student repo. The student does this by visiting the Github page for their own fork of test-gp-hw0-student and clicking “New Pull Request”. ## Grading as the instructor The instructor can now grade this. I’m imagining that the test-gp-hw0-instructor repo has a solution that can be compared (in some way or another) to the students work. To pull the students work into the instructor copy (but never merge): cd$INSTRUCTOR_REPO
git checkout -b stewartdent-master master
git pull https://github.com/stewartdent/test-gp-hw0-student.git master
cat empty-form.md
git checkout master
git branch -D stewartdent-master


Then the instructor can close the pull request on test-gp-hw0-student.

## Other perks

Questions or discussions related to the homework can be conducted on the “issues” page of test-gp-hw0-student

* The fact that this repo is public should immediately indicate that student’s work submitted via pull request will also be public and importantly will be visible to other students. Unlike the most common form of cheating, this means that Student A can copy the work of Student B without the explicit consent of Student B. Student B’s only possible defense against this would be to submit their work just before the deadline. As far as I know, I can’t hide pull requests from the public (that would be a nice solution) and I prefer not to use any method that relies on student’s having to create a private fork (since they cost money; although, other class require very over-priced textbooks, so a 6-month github subscription is not so crazy, but it does go against the open-source spirit; there’s also the Big Tobacco style get’em while their young https://education.github.com/pack).

### How does matlab’s hypot function work?

Saturday, January 7th, 2017

The hypot function in matlab purports to be more numerically stable at computing the hypotenuse of a (right-)triangle in 2D. The example the help hypot gives is:

a = 3*[1e300 1e-300];
b = 4*[1e300 1e-300];
c1 = sqrt(a.^2 + b.^2)
c2 = hypot(a,b)


where you see as output:

c1 =

Inf     0


and

c2 =

5e+300       5e-300


this is a compiled built-in function so you can’t just open hypot to find out what its doing. It might just be pre-dividing by the maximum absolute value. There’s probably a better reference for this, but I found it in: “Vector length and normalization difficulties” by Mike Day.

Continuing this example:

m = max(abs([a;b]))
c3 = m.*sqrt((a./m).^2 + (b./m).^2)


produces

c3 =

5e+300       5e-300


While matlab’s hypot only accepts two inputs, pre-dividing by the maximum obviously extends to any dimension.

### Triangle area in nD

Saturday, January 7th, 2017

My typical go-to formula for the area of a triangle with vertices in nD is Heron’s formula. Actually this is a formula for the area of a triangle given its side lengths. Presumably it’s easy to compute side lengths in any dimension: e.g., a = sqrt((B-C).(B-C)).

Kahan showed that the vanilla Heron’s formula is numerically poor if your (valid) triangle side lengths are given as floating point numbers. He improved this formula by sorting the side-lengths and carefully rearranging the terms in the formula to reduce roundoff.

However, if you’re vertices don’t actually live in R^n but rather F^n where F is the fixed precision floating-point grid, then computing side lengths will in general incur some roundoff error. And it may so happen (and it does so happen) that this roundoff error invalidates the triangle side lengths. Here, invalid means that the side lengths don’t obey the triangle inequality. This might happen for nearly degenerate (zero area) triangles: one (floating point) side length ends up longer than the sum of the others. Kahan provides an assertion to check for this, but there’s no proposed remedy that I could find. One could just declare that these edge-lengths must of come from a zero-area triangle. But returning zero might let the user happily work with very nonsensical triangle side lengths in other contexts not coming from an embedded triangle. You could have two versions: one for “known embeddings” (with assertions off and returning 0) and one for “intrinsic/metric only” (with assertions on and returning NaN). Or you could try to fudge in an epsilon a nd hope you choose it above the roundoff error threshold but below the tolerance for perceived “nonsense”. These are messy solutions.

An open question (open in the personal sense of “I don’t know the answer”) is what is the most robust way to compute triangle area in nD with floating point vertex positions. A possible answer might be: compute the darn floating point edge lengths, use Kahan’s algorithm and replace NaNs with zeros. That seems unlikely to me because (as far as I can tell) there are 4 sqrt calls, major sources of floating point error.

Re-reading Shewchuk’s robustness notes, I saw his formula for triangle area for points A,B,C in 3D:

Area3D(A,B,C) = sqrt( Area2D(Axy,Bxy,Cxy)^2 + Area2D(Ayz,Byz,Cyz)^2 + Area2D(Azx,Bzx,Czx)^2 ) / 2

where Area2D(A,B,C) is computed as the determinant of [[A;B;C] 1]. This formula reduces the problem of computing 3D triangle area into computing 2D triangle area on the “shadows” (projections) of the triangle on each axis-aligned plane. This lead me to think of a natural generalization to nD:

AreaND(A,B,C) = sqrt( ∑_{i<j} ( Area2D(Aij,Bij,Cij)^2 ) / 2

This formula computes the area of the shadow on all (N choose 2) axis-aligned planes. Since the sqrt receives a sum of squares as in argument there’s no risk of getting a NaN. There’s a clear drawback that this formula is O(N^2) vs Heron’s/Kahan’s O(N).

Friday, January 6th, 2017

Like clockwork, my matlab updates have gotten out of sync with Xcode updates. It seems like fixing this SDK error always requires a different hack each year. This year I got the error:

No supported compiler or SDK was found. For options, visit
http://www.mathworks.com/support/compilers/current_release/.


To fix it, I replaced all occurrences of 10.9 with 10.11 in /Applications/MATLAB_R2017a.app/bin/maci64/mexopts/clang{++,}_maci64.xml

ld: warning: object file was built for newer OSX version (10.11) than being linked (10.9)