This notebook is a preview demo of IJulia: a Julia-language backend combined with the IPython interactive environment. This combination allows you to interact with the Julia language using IPython's powerful graphical notebook, which combines code, formatted text, math, and multimedia in a single document.
Basic mathematical expressions work like you expect:
1 + sin(3)
You can define variables, write loops, and execute arbitrary multiline code blocks. Here is an example of an alternating harmonic series $\sum_{n=1}^\infty \frac{(-1)^n}{n}$ from a Julia tutorial by Homer Reid:
s = 0.0
for n = 1:2:10000
s += 1/n - 1/(n+1)
end
s # an expression on the last line (if it doesn't end with ";") is printed as "Out"
Previous outputs can be referred to via Out[n], following the IPython, for example Out[2] for the result above. You can also use the shorthand _2, or _ for the previous result, as in IPython. Like in Matlab, ans can also be used to refer to the previous result, even if it was not printed (when the command ended with ;).
For example, the harmonic series above should be converging (slowly) to $\ln 2$, and we can check this:
Out[2] - log(2)
Like Matlab or Scipy + Numpy, Julia has lots of mathematical functions and linear algebra built in. For example, we can define a $500\times500$ random matrix $R$ and form the positive-definite matrix $R^* R$:
R = rand(500,500)
R' * R
(Notice that, by default, only a portion of a large matrix is shown. You didn't really want to read $500^2 = 250,000$ numbers, did you?)
Standard output from Julia is also captured and sent to the IJulia notebook as you'd expect:
println("Hello world!\n")
println(STDERR, "Börk börk börk, some unicode output on stderr...\n")
IJulia even captures output from external C libraries (and notice how easy it is using Julia's ccall intrinsic):
ccall(:printf, Cint, (Ptr{Uint8},), "Hello from C!!\n");
We can define functions, of course, and use them in later input cells:
f(x) = x + 1
println(f(3))
f([1,1,2,3,5,8])
Notice that the input above both printed an scalar to STDOUT and also returned a vector, the latter using Julia's ability to write polymorphic functions and built-in array operations.
On the other hand adding a string to a number is not defined (there is no + method defined for those types, although we could easily add one), and attempting to do so will throw an exception:
f("Hello?")
Julia can easily and transparently call external Python code using a package called PyCall, and to illustrate that capability we will show some examples calling SciPy and Matplotlib from Julia.
For example, we can use the Newton solver in scipy.optimize to solve a transcendental equation $\cos(x) - x = 0$ given a Julia function:
using PyCall
@pyimport scipy.optimize as so
so.newton(x -> cos(x) - x, 1)
We can use the same @pyimport syntax to import Matplotlib (specifically, the matplotlib.pyplot module), but to integrate Matplotlib's graphics with the IJulia display requires a little more work. To simplify this, we've created a PyPlot module for Julia:
using PyPlot
x = linspace(0,2*pi,1000)
y = sin(3*x + 4*cos(2*x))
plot(x, y, color="red", linewidth=2.0, linestyle="--")
ylabel("the y axis")
xlabel("the x axis")
title("a sinusoidally-modulated sinusoid")
Notice that, by default, the plots are displayed inline (just as for the %pylab inline "magic" in IPython). This kind of multimedia display can be enabled for any Julia object, as explained in the next section.
Like most programming languages, Julia has a built-in print(x) function for outputting an object x as text, and you can override the resulting text representation of a user-defined type by overloading Julia's show function. The next version of Julia, however, will extend this to a more general mechanism to display arbitrary multimedia representations of objects, as defined by standard MIME types. More specifically, the Julia multimedia I/O API provides:
display(x) function requests the richest available multimedia display of a Julia object x (with a text/plain fallback).writemime allows one to indicate arbitrary multimedia representations (keyed by standard MIME types) of user-defined types.Display type. IJulia provides one such backend which, thanks to the IPython notebook, is capable of displaying HTML, LaTeX, SVG, PNG, and JPEG media formats.The last two points are critical, because they separate multimedia export (which is defined by functions associated with the originating Julia data) from multimedia display (defined by backends which know nothing about the source of the data).
Precisely these mechanism were used to create the inline PyPlot plots above. To start with, the simplest thing is to provide the MIME type of the data when you call display, which allows you to pass "raw" data in the corresponding format:
display("text/html", """Hello <b>world</b> in <font color="red">HTML</font>!""")
However, it will be more common to attach this information to types, so that they display correctly automatically. For example, let's define a simple HTML type in Julia that contains a string and automatically displays as HTML (given an HTML-capable backend such as IJulia):
type HTML
s::String
end
import Base.writemime
writemime(io::IO, ::@MIME("text/html"), x::HTML) = print(io, x.s)
Here, writemime is just a function that writes x in the corresponding format (text/html) to the I/O stream io. The @MIME is a bit of magic to allow Julia's multiple dispatch to automatically select the correct writemime function for a given MIME type (here "text/html") and object type (here HTML). We also needed an import statement in order to add new methods to an existing function from another module.
This writemime definition is all that we need to make any object of type HTML display automatically as HTML text in IJulia:
x = HTML("<ul> <li> Hello from a bulleted list! </ul>")
display(x)
println(x)
Once this functionality becomes available in a Julia release, we expect that many Julia modules will provide rich representations of their objects for display in IJulia, and moreover that other backends will appear. Not only can other backends (such as Tim Holy's ImageView package) provide more full-featured display of images etcetera than IJulia's inline graphics, but they can also add support for displaying MIME types not handled by the IPython notebook (such as video or audio).