Trefethen & Bau & MATLAB & Julia, Lecture 3: Norms

Here are the MATLAB and julia notebooks.

The big issue this time around was graphics. This topic dramatically illustrates the advantages on both sides of the commercial/open source fence. On the MATLAB side, it’s perfectly clear what you should do. There are many options that have been well constructed, and it’s all under a relatively consistent umbrella. There are things to learn and options to choose, but it’s clear what functions you will be using to make, say, a scatter plot, and a lot of similarity across commands.

Julia graphics are another story. At this writing, there are two options recommended on Julia’s official page about plotting packages: PyPlot and Gadfly. It doesn’t take much exploration to decide that the former is favored by MATLAB veterans and the latter, by R devotees. Confusingly, the general download page for Julia mentions a third package called Plots that is supposed to integrate all of the backends. It’s still early days for Julia, and I’m sure much remains in flux.

Moreover, because you can (quite easily) import and run Python code in Julia, in principle you have access to all Python plotting packages. One of the big players is matplotlib, which is more or less what Julia’s PyPlot is supposed to provide. But there are also Bokeh, plotly, and pyqtgraph—for all I know, many more besides. All of these can make gorgeous graphics, often highly interactive and even hosted in the cloud. The relative merits are not at all clear.

Here we run into the paradox of choice: having many options, even good ones, can provoke anxiety rather than satisfaction. Which package do I invest time in learning? MATLAB limits choice but provides a sort of editorial, almost paternal, reassurance.

My personal goal is to learn Julia from the standpoint of a MATLAB user, so PyPlot it is. All in all, the transition isn’t bad, though there are some twists.

In the last few years I’ve been more often turning to automatic function plotting in MATLAB, using fplot, ezsurf, and ezcontour. If PyPlot supports those, I have yet to find out about them. So it’s back to the world of evaluating functions on tensor product grids.  A MATLAB veteran turns to meshgrid, but Julia supports broadcasting across singleton dimensions. For example:

using PyPlot
x = linspace(-1,1,90);
y = x';
contour(x[:],y[:],sqrt(x.^2 .+ y.^2))</pre>

Because x has a column shape while y has a row shape, the .+ operator broadcasts each along the “missing” dimension. It’s a clever shortcut once you know it. It works just as well for contours of the vector 1-norm, but for the max norm I had to broadcast manually:

contour(x[:],y[:],broadcast(max,abs(x),abs(y)))

It’s not clear to me why that broadcast should not happen automatically, given that max is a dedicated elementwise operator.

There’s more Julia subtlety hiding in this notebook, but those issues will wait for another time.

Professor of Mathematical Sciences

My research interests are in scientific computation, mathematical software, and applications of mathematics in the life sciences.