Ganesh Swami

A couple of months back, Prof. Gilbert Strang had come to SFU as part of the distinguished speakers series (as I had written earlier.) After the talk, I got a chance to chat with the guru. I owe a lot of my understanding of linear algebra to his books and his kick-ass animations (check out these eigen-analysis demos,) and so I asked him about the single most important topic in linear algebra. Without hesitation, he immediately responded: “the SVD!”

The Singular Value Decomposition is the swiss-knife of linear algebra. Every matrix Y can be factored into three matrices: U, S, and V as

U and V are orthogonal matrices and S is a diagonal matrix. Some uses of this factorization of the matrix Y: calculating 2-norms, Frobenius norms, ranks, null spaces and ranges, (pseudo)inverses and determinants, and by extension solving systems of equations (exact, over- and under- determined), eigenvalues and eigenvectors, and approximations.

Almost every problem in engineering becomes an optimization problem (as in reducing the error under some norm) and the method of least squares makes extensive use of the SVD.

The stage has been set for the next few posts of mine.

I’ve recently become a heavy user of the numerical capabilities of Python. I’ve written about my experiments before, but now I’m writing production quality code with numpy and matplotlib.

The above is an actual plot that I created for some Hall measurements I was doing. I was supposed for find functional relationships between temperature and majority charge carriers, which in my case were electrons because of the n-type doping. The simple case was a least squares fit: scipy.optimize.leastsq to the rescue. The more complicated part was solving a non-linear equation for roots and then doing a least squares fit. The root-finding module in scientific python provides lots of options. At this point, I can confidently say that this environment has more features than Octave.

Just today, I wanted to use the Fourier method on a differential equation (plug: the advantages of which are here) and numerical python with fft, fftshift and fftfreq are exact substitutes for their Matlab equivalents. You can also put actual LaTeX equations on plots, which is a major plus.

That is all.

I strongly think that unit tests are absolutely necessary to keep the overall quality of ones code high. I’ve written about Unit Tests before, so I’ll continue on that trend.

Most high performance numerical codes use some variant of Fortran or just raw C (which is my staple language.) In that case, as Micheal Feathers rightly points out, unit tests can “collide” with C.

I’ll want to expand on his solution of link seams.

GNU/Linux (and other Unix-like operating systems I think), provide a mechanism of loading shared object files at runtime. Using dlopen(3) and dlsym(3) you can re-create much of the reflection functionality that the Java community enjoys. So by loading your unit test dynamically at runtime, and having a predetermined symbol that behaves as the entry point, you can nicely wrap your module/function with a unit test. Interface this with ctest from the cmake project, and you are set. If you want to have a private function whose symbol isn’t visible on the outside, make it static. The nm utility will give you a list of public symbols in an object.

I’m pretty sure this is supported across the board. I remember doing this under DOS using DJGPP (remember that?) so this isn’t exactly novel.