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.

- Note: this is a
*preview*, because it relies on pre-release bleeding-edge versions of Julia, IPython, and several Julia packages, as explained on the IJulia github page, and functionality is evolving rapidly. We hope to have a more polished release soon.

Basic mathematical expressions work like you expect:

In [1]:

```
1 + sin(3)
```

Out[1]:

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:

In [2]:

```
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"
```

Out[2]:

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:

In [3]:

```
Out[2] - log(2)
```

Out[3]:

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 $R^* R$:

In [4]:

```
R = rand(500,500)
R' * R
```

Out[4]:

(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:

In [5]:

```
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):

In [5]:

```
ccall(:printf, Cint, (Ptr{Uint8},), "Hello from C!!\n");
```

We can define functions, of course, and use them in later input cells:

In [6]:

```
f(x) = x + 1
```

Out[6]:

In [7]:

```
println(f(3))
f([1,1,2,3,5,8])
```

Out[7]:

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:

In [8]:

```
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:

In [9]:

```
using PyCall
@pyimport scipy.optimize as so
so.newton(x -> cos(x) - x, 1)
```

Out[9]:

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:

In [10]:

```
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")
```

Out[10]:

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:

- A
`display(x)`

function requests the richest available multimedia display of a Julia object x (with a`text/plain`

fallback). - Overloading
`writemime`

allows one to indicate arbitrary multimedia representations (keyed by standard MIME types) of user-defined types. - Multimedia-capable display backends may be registered by subclassing a generic
`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:

In [11]:

```
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):

In [12]:

```
type HTML
s::String
end
import Base.writemime
writemime(io::IO, ::@MIME("text/html"), x::HTML) = print(io, x.s)
```

Out[12]:

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:

In [13]:

```
x = HTML("<ul> <li> Hello from a bulleted list! </ul>")
```

Out[13]:

In [14]:

```
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).