Julia Basics¶

Here is some italic text. Even equations $\int_0^1 \frac{1}{1+x^2} dx$.

I can use Julia as a calculator:

In :
3 + 4
Out:
7
In :
3 * 4
Out:
12
In :
3^4
Out:
81
In :
sin(3)
Out:
0.1411200080598672
In :
1 / (1 + sin(7))
Out:
0.6035051827052901

Variables¶

In :
x = 17
Out:
17
In :
y = sin(x)
Out:
-0.9613974918795568
In :
α = 3.74 # Unicode variable names — type it by "\alpha<tab>"
Out:
3.74
In :
α̂₂ = 3
Out:
3

A complex number $3 + 5i$:

In :
z = 3 + 5im
Out:
3 + 5im
In :
z^3
Out:
-198 + 10im
In :
exp(z) # compute eᶻ
Out:
5.697507299833739 - 19.26050892528742im
In :
sin(z)
Out:
10.472508533940392 - 73.46062169567367im

In :
?sin
search: sin sinh sind sinc sinpi sincos sincosd asin using isinf asinh asind

Out:
sin(x)

Compute sine of x, where x is in radians.

sin(A::AbstractMatrix)

Compute the matrix sine of a square matrix A.

If A is symmetric or Hermitian, its eigendecomposition (eigen) is used to compute the sine. Otherwise, the sine is determined by calling exp.

Examples¶

jldoctest
julia> sin(fill(1.0, (2,2)))
2×2 Array{Float64,2}:
0.454649  0.454649
0.454649  0.454649
In :
?α̂₂
"α̂₂" can be typed by \alpha<tab>\hat<tab>\_2<tab>

search: α̂

Out:

No documentation found.

α̂₂ is of type Int64.

Summary¶

primitive type Int64 <: Signed

Supertype Hierarchy¶

Int64 <: Signed <: Integer <: Real <: Number <: Any

Vectors¶

In :
x = [1, 17, 32, 15]  # elements separated by commas
Out:
4-element Array{Int64,1}:
1
17
32
15

Array{Int64,1} is a 1-dimensional array (a "vector") of 64-bit integers (Int64).

In :
y = [15, 2, 6, -9]
Out:
4-element Array{Int64,1}:
15
2
6
-9
In :
x + y
Out:
4-element Array{Int64,1}:
16
19
38
6
In :
x * y # not allowed: vector * vector is not a linear-algebra operation
MethodError: no method matching *(::Array{Int64,1}, ::Array{Int64,1})
Closest candidates are:
*(::Any, ::Any, !Matched::Any, !Matched::Any...) at operators.jl:538
*(!Matched::LinearAlgebra.Adjoint{var"#s826",var"#s8261"} where var"#s8261"<:(AbstractArray{T,1} where T) where var"#s826"<:Number, ::AbstractArray{var"#s825",1} where var"#s825"<:Number) at /Users/julia/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.5/LinearAlgebra/src/adjtrans.jl:283
*(!Matched::LinearAlgebra.Transpose{T,var"#s826"} where var"#s826"<:(AbstractArray{T,1} where T), ::AbstractArray{T,1}) where T<:Real at /Users/julia/buildbot/worker/package_macos64/build/usr/share/julia/stdlib/v1.5/LinearAlgebra/src/adjtrans.jl:284
...

Stacktrace:
 top-level scope at In:1
 execute_code(::String, ::String) at /Users/stevenj/.julia/dev/IJulia/src/execute_request.jl:27
 execute_request(::ZMQ.Socket, ::IJulia.Msg) at /Users/stevenj/.julia/dev/IJulia/src/execute_request.jl:86
 #invokelatest#1 at ./essentials.jl:710 [inlined]
 invokelatest at ./essentials.jl:709 [inlined]
 eventloop(::ZMQ.Socket) at /Users/stevenj/.julia/dev/IJulia/src/eventloop.jl:8
In :
transpose(x)
Out:
1×4 LinearAlgebra.Transpose{Int64,Array{Int64,1}}:
1  17  32  15
In :
transpose(x) * y # allowed = a dot product
Out:
106
In :
x .* y # elementwise product
Out:
4-element Array{Int64,1}:
15
34
192
-135
In :
float(y) .^ x
Out:
4-element Array{Float64,1}:
15.0
131072.0
7.958661109946401e24
-2.05891132094649e14
In :
float(y)
Out:
4-element Array{Float64,1}:
15.0
2.0
6.0
-9.0
In :
z = [3.7, 4.2, 6.1]
Out:
3-element Array{Float64,1}:
3.7
4.2
6.1
In :
z .^ [0,1,2]
Out:
3-element Array{Float64,1}:
1.0
4.2
37.209999999999994
In :
z .^ transpose([0,1,2]) # a Vandermonde matrix
Out:
3×3 Array{Float64,2}:
1.0  3.7  13.69
1.0  4.2  17.64
1.0  6.1  37.21
In :
z .^ [0 1 2]
Out:
3×3 Array{Float64,2}:
1.0  3.7  13.69
1.0  4.2  17.64
1.0  6.1  37.21
In :
[0 1 2] # another row vector
Out:
1×3 Array{Int64,2}:
0  1  2
In :
@show x
x # second element
x = [1, 17, 32, 15]
Out:
17
In :
x[2:3] # elements 2 to 3 — "slicing"
Out:
2-element Array{Int64,1}:
17
32
In :
x[2:4] # elements 2,3,4
Out:
3-element Array{Int64,1}:
17
32
15
In :
2:4
Out:
2:4
In :
0:4 # analogous to range(5) in Python
Out:
0:4
In :
collect(0:4)
Out:
5-element Array{Int64,1}:
0
1
2
3
4
In :
y .^ 3 # elementwise cube
Out:
4-element Array{Int64,1}:
3375
8
216
-729
In :
sin.(y) # elementwise sin
Out:
4-element Array{Float64,1}:
0.6502878401571168
0.9092974268256817
-0.27941549819892586
-0.4121184852417566
In :
rand(7) # 7 random numbers in [0,1)
Out:
7-element Array{Float64,1}:
0.9895925568529049
0.43720897273586745
0.18692127626397315
0.5248400288925346
0.7725784322036084
0.2755926598616645
0.046487055457531845
In :
randn(7) # 7 normally-distributed random numbers (mean 0, std. dev. 1)
Out:
7-element Array{Float64,1}:
-0.7766970200438933
0.8375893322814147
-1.0161992755239175
-1.4656970471375195
-0.2380323626857652
0.3324373560431813
-0.2545786106508578
In :
collect(0:2:7) # 0 to 7 in steps of 2
Out:
4-element Array{Int64,1}:
0
2
4
6
In :
collect(0:0.1:7) # 0 to 7 in steps of 0.1
Out:
71-element Array{Float64,1}:
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
⋮
5.9
6.0
6.1
6.2
6.3
6.4
6.5
6.6
6.7
6.8
6.9
7.0
In :
r = range(0, 2π, length=50) # 50 numbers from 0 to 2π, equally spaced
Out:
0.0:0.1282282715750936:6.283185307179586
In :
collect(r)
Out:
50-element Array{Float64,1}:
0.0
0.1282282715750936
0.2564565431501872
0.38468481472528077
0.5129130863003744
0.641141357875468
0.7693696294505615
0.8975979010256552
1.0258261726007487
1.1540544441758425
1.282282715750936
1.4105109873260295
1.538739258901123
⋮
4.872674319853557
5.00090259142865
5.129130863003744
5.257359134578837
5.385587406153931
5.513815677729025
5.642043949304118
5.770272220879212
5.898500492454305
6.026728764029399
6.154957035604492
6.283185307179586
In :
r = range(0, 2π, length=10^8) # not computed explicitly
Out:
0.0:6.28318537001144e-8:6.283185307179586
In :
collect(r[1:10])
Out:
10-element Array{Float64,1}:
0.0
6.28318537001144e-8
1.256637074002288e-7
1.884955611003432e-7
2.513274148004576e-7
3.14159268500572e-7
3.769911222006864e-7
4.398229759008008e-7
5.026548296009151e-7
5.654866833010296e-7

Matrices¶

In :
A = [1 3 7
4 7 2
0 1 1]
Out:
3×3 Array{Int64,2}:
1  3  7
4  7  2
0  1  1

Notice that the type is Array{Int64,2} — the 2 here is for a 2d array, a "matrix"

In :
b = [3, 2, 1] # a vector
Out:
3-element Array{Int64,1}:
3
2
1
In :
A * b # matrix-vector product
Out:
3-element Array{Int64,1}:
16
28
3
In :
inv(A)
Out:
3×3 Array{Float64,2}:
0.238095   0.190476  -2.04762
-0.190476   0.047619   1.2381
0.190476  -0.047619  -0.238095

Let's solve a linear system: $$Ax = b$$ for $x = A^{-1} b$:

In :
x = inv(A) * b
Out:
3-element Array{Float64,1}:
-0.9523809523809521
0.761904761904762
0.23809523809523808
In :
A * x
Out:
3-element Array{Float64,1}:
3.0000000000000004
2.0000000000000018
1.0
In :
A * x - b # not zero due to roundoff errors — more about this in 18.330 or 18.335 in spring
Out:
3-element Array{Float64,1}:
4.440892098500626e-16
1.7763568394002505e-15
0.0

Alternatively:

In :
x = A \ b # effectively equivalent to inv(A) * b, but faster and better
Out:
3-element Array{Float64,1}:
-0.9523809523809528
0.7619047619047621
0.23809523809523808
In :
A * x - b # slightly different (smaller in this case)
Out:
3-element Array{Float64,1}:
0.0
0.0
2.220446049250313e-16
In :
sin.(A) # elementwise sine
Out:
3×3 Array{Float64,2}:
0.841471  0.14112   0.656987
-0.756802  0.656987  0.909297
0.0       0.841471  0.841471
In :
sin(A) # is this meaningful????
Out:
3×3 Array{Float64,2}:
-0.888487  -0.296349   6.91956
0.962251   0.264022  -3.59102
-0.581735   0.240563   0.565852

In 18.06 (or maybe 18.S096), you'll eventually learn that $\sin(A)$ is perfectly meaningful for a square matrix. It's defined by the Taylor series: $$\sin(A) = A - \frac{A^3}{3!} + \frac{A^5}{5!} - \frac{A^7}{7!} + \cdots$$

Functions¶

In :
function f(x)
return 3x^3 - 5x^2 + sin(x) - x
end
Out:
f (generic function with 1 method)
In :
f(3)
Out:
34.141120008059865
In :
f(A)
Out:
3×3 Array{Float64,2}:
344.112    669.704  418.92
688.962   1421.26   956.409
87.4183   172.241  125.566
In :
f.(x) # applied elementwise
Out:
3-element Array{Float64,1}:
-6.989077393122971
-1.6472437070737587
-0.24519753391390525
In :
f.(A)
Out:
3×3 Array{Float64,2}:
-2.15853   33.1411   777.657
107.243    777.657      2.9093
0.0       -2.15853   -2.15853
In :
g(x) = 4x^3 - 5x^2 + 6x + 2 # one-line function definition
Out:
g (generic function with 1 method)
In :
g.(A)
Out:
3×3 Array{Int64,2}:
7    83  1171
202  1171    26
2     7     7

More Linear Algebra¶

Most linear-algebra routines are in the LinearAlgebra standard library:

In :
using LinearAlgebra # similar to "from LinearAlgebra import *" in Python
In :
eigvals(A) # eigenvalues
Out:
3-element Array{Complex{Float64},1}:
-0.07094370057923445 - 1.5139635518670342im
-0.07094370057923445 + 1.5139635518670342im
9.141887401158474 + 0.0im
In :
F = eigen(A) # eigenvectors and eigenvalues
Out:
Eigen{Complex{Float64},Complex{Float64},Array{Complex{Float64},2},Array{Complex{Float64},1}}
values:
3-element Array{Complex{Float64},1}:
-0.07094370057923445 - 1.5139635518670342im
-0.07094370057923445 + 1.5139635518670342im
9.141887401158474 + 0.0im
vectors:
3×3 Array{Complex{Float64},2}:
0.834342-0.0im        0.834342+0.0im        0.42575+0.0im
-0.454852+0.168903im  -0.454852-0.168903im  0.898092+0.0im
0.0672889-0.252839im  0.0672889+0.252839im  0.110305+0.0im
In :
F.values # the eigenvalues
Out:
3-element Array{Complex{Float64},1}:
-0.07094370057923445 - 1.5139635518670342im
-0.07094370057923445 + 1.5139635518670342im
9.141887401158474 + 0.0im
In :
F.vectors # the eigenvectors
Out:
3×3 Array{Complex{Float64},2}:
0.834342-0.0im        0.834342+0.0im        0.42575+0.0im
-0.454852+0.168903im  -0.454852-0.168903im  0.898092+0.0im
0.0672889-0.252839im  0.0672889+0.252839im  0.110305+0.0im
In :
qr(A) # QR factorization = Gram-Schmidt orthogonalization
Out:
LinearAlgebra.QRCompactWY{Float64,Array{Float64,2}}
Q factor:
3×3 LinearAlgebra.QRCompactWYQ{Float64,Array{Float64,2}}:
-0.242536   0.748481  -0.617213
-0.970143  -0.18712    0.154303
0.0        0.636209   0.771517
R factor:
3×3 Array{Float64,2}:
-4.12311  -7.5186   -3.63803
0.0       1.57181   5.50134
0.0       0.0      -3.24037
In :
lu(A) # Gaussian elimination = LU factorization
Out:
LU{Float64,Array{Float64,2}}
L factor:
3×3 Array{Float64,2}:
1.0   0.0  0.0
0.25  1.0  0.0
0.0   0.8  1.0
U factor:
3×3 Array{Float64,2}:
4.0  7.0    2.0
0.0  1.25   6.5
0.0  0.0   -4.2
In :
svd(A)
Out:
SVD{Float64,Float64,Array{Float64,2}}
U factor:
3×3 Array{Float64,2}:
-0.657006   0.742232   -0.132041
-0.742831  -0.667249   -0.0545987
-0.128629   0.0622125   0.989739
singular values:
3-element Array{Float64,1}:
10.249509228237182
4.9777646539733755
0.4116061587661843
Vt factor:
3×3 Array{Float64,2}:
-0.354     -0.712177  -0.606208
-0.387074  -0.478494   0.788173
-0.851386   0.513661  -0.106278

A little plotting¶

You must load a plotting library, similar to Python.

There are lots to choose from. If you are used to Python, you've probably used Matplotlib, and you can call Matplotlib in Julia via PyPlot:

In :
using PyPlot
In :
plot(x, "bo-") Out:
1-element Array{PyCall.PyObject,1}:
PyObject <matplotlib.lines.Line2D object at 0x7ffd3f6b1fd0>
In :
x = range(0, 2π, length=1000)
y = sin.(3x + 4cos.(2x))
plot(x / 2π, y, "r-")
plot(x / 2π, y.^2, "b-")
plot(x / 2π, 1 ./ sqrt.(1 .+ y.^2), "b-")
xlabel("x / 2π")
ylabel("our funny function")
title("our second plot")
legend([L"y", L"y^2", L"\frac{1}{\sqrt{1+y^2}}"]) Out:
PyObject <matplotlib.legend.Legend object at 0x7ffd472bf790>
In :
?plot
search: plot plot3D plotfile plot_date plot_trisurf plot_surface plot_wireframe

Out:
Plot y versus x as lines and/or markers.

Call signatures::

plot([x], y, [fmt], *, data=None, **kwargs)
plot([x], y, [fmt], [x2], y2, [fmt2], ..., **kwargs)

The coordinates of the points or line nodes are given by *x*, *y*.

The optional parameter *fmt* is a convenient way for defining basic
formatting like color, marker and linestyle. It's a shortcut string
notation described in the *Notes* section below.

>>> plot(x, y)        # plot x and y using default line style and color
>>> plot(x, y, 'bo')  # plot x and y using blue circle markers
>>> plot(y)           # plot y using x as index array 0..N-1
>>> plot(y, 'r+')     # ditto, but with red plusses

You can use .Line2D properties as keyword arguments for more
control on the appearance. Line properties and *fmt* can be mixed.
The following two calls yield identical results:

>>> plot(x, y, 'go--', linewidth=2, markersize=12)
>>> plot(x, y, color='green', marker='o', linestyle='dashed',
...      linewidth=2, markersize=12)

When conflicting with *fmt*, keyword arguments take precedence.

**Plotting labelled data**

There's a convenient way for plotting objects with labelled data (i.e.
data that can be accessed by index obj['y']). Instead of giving
the data in *x* and *y*, you can provide the object in the *data*
parameter and just give the labels for *x* and *y*::

>>> plot('xlabel', 'ylabel', data=obj)

All indexable objects are supported. This could e.g. be a dict, a
pandas.DataFame or a structured numpy array.

**Plotting multiple sets of data**

There are various ways to plot multiple sets of data.

- The most straight forward way is just to call plot multiple times.
Example:

>>> plot(x1, y1, 'bo')
>>> plot(x2, y2, 'go')

- Alternatively, if your data is already a 2d array, you can pass it
directly to *x*, *y*. A separate data set will be drawn for every
column.

Example: an array a where the first column represents the *x*
values and the other columns are the *y* columns::

>>> plot(a, a[1:])

- The third way is to specify multiple sets of *[x]*, *y*, *[fmt]*
groups::

>>> plot(x1, y1, 'g^', x2, y2, 'g-')

In this case, any additional keyword argument applies to all
datasets. Also this syntax cannot be combined with the *data*
parameter.

By default, each line is assigned a different style specified by a
'style cycle'. The *fmt* and line property parameters are only
necessary if you want explicit deviations from these defaults.
Alternatively, you can also change the style cycle using the
'axes.prop_cycle' rcParam.

Parameters
----------
x, y : array-like or scalar
The horizontal / vertical coordinates of the data points.
*x* values are optional and default to range(len(y)).

Commonly, these parameters are 1D arrays.

They can also be scalars, or two-dimensional (in that case, the
columns represent separate data sets).

These arguments cannot be passed as keywords.

fmt : str, optional
A format string, e.g. 'ro' for red circles. See the *Notes*
section for a full description of the format strings.

Format strings are just an abbreviation for quickly setting
basic line properties. All of these and more can also be
controlled by keyword arguments.

This argument cannot be passed as keyword.

data : indexable object, optional
An object with labelled data. If given, provide the label names to
plot in *x* and *y*.

.. note::
Technically there's a slight ambiguity in calls where the
second label is a valid *fmt*. plot('n', 'o', data=obj)
could be plt(x, y) or plt(y, fmt). In such cases,
the former interpretation is chosen, but a warning is issued.
You may suppress the warning by adding an empty format string
plot('n', 'o', '', data=obj).

Other Parameters
----------------
scalex, scaley : bool, optional, default: True
These parameters determined if the view limits are adapted to
the data limits. The values are passed on to autoscale_view.

**kwargs : .Line2D properties, optional
*kwargs* are used to specify properties like a line label (for
auto legends), linewidth, antialiasing, marker face color.
Example::

>>> plot([1,2,3], [1,2,3], 'go-', label='line 1', linewidth=2)
>>> plot([1,2,3], [1,4,9], 'rs',  label='line 2')

If you make multiple lines with one plot command, the kwargs
apply to all those lines.

Here is a list of available .Line2D properties:

agg_filter: a filter function, which takes a (m, n, 3) float array and a dpi value, and returns a (m, n, 3) array
alpha: float
animated: bool
antialiased or aa: bool
clip_box: .Bbox
clip_on: bool
clip_path: [(~matplotlib.path.Path, .Transform) | .Patch | None]
color or c: color
contains: callable
dash_capstyle: {'butt', 'round', 'projecting'}
dash_joinstyle: {'miter', 'round', 'bevel'}
dashes: sequence of floats (on/off ink in points) or (None, None)
drawstyle or ds: {'default', 'steps', 'steps-pre', 'steps-mid', 'steps-post'}, default: 'default'
figure: .Figure
fillstyle: {'full', 'left', 'right', 'bottom', 'top', 'none'}
gid: str
in_layout: bool
label: object
linestyle or ls: {'-', '--', '-.', ':', '', (offset, on-off-seq), ...}
linewidth or lw: float
marker: marker style
markeredgecolor or mec: color
markeredgewidth or mew: float
markerfacecolor or mfc: color
markerfacecoloralt or mfcalt: color
markersize or ms: float
markevery: None or int or (int, int) or slice or List[int] or float or (float, float)
path_effects: .AbstractPathEffect
picker: float or callable[[Artist, Event], Tuple[bool, dict]]
rasterized: bool or None
sketch_params: (scale: float, length: float, randomness: float)
snap: bool or None
solid_capstyle: {'butt', 'round', 'projecting'}
solid_joinstyle: {'miter', 'round', 'bevel'}
transform: matplotlib.transforms.Transform
url: str
visible: bool
xdata: 1D array
ydata: 1D array
zorder: float

Returns
-------
lines
A list of .Line2D objects representing the plotted data.

--------
scatter : XY scatter plot with markers of varying size and/or color (
sometimes also called bubble chart).

Notes
-----
**Format Strings**

A format string consists of a part for color, marker and line::

fmt = '[marker][line][color]'

Each of them is optional. If not provided, the value from the style
cycle is used. Exception: If line is given, but no marker,
the data will be a line without markers.

Other combinations such as [color][marker][line] are also
supported, but note that their parsing may be ambiguous.

**Markers**

=============    ===============================
character        description
=============    ===============================
'.'          point marker
','          pixel marker
'o'          circle marker
'v'          triangle_down marker
'^'          triangle_up marker
'<'          triangle_left marker
'>'          triangle_right marker
'1'          tri_down marker
'2'          tri_up marker
'3'          tri_left marker
'4'          tri_right marker
's'          square marker
'p'          pentagon marker
'*'          star marker
'h'          hexagon1 marker
'H'          hexagon2 marker
'+'          plus marker
'x'          x marker
'D'          diamond marker
'd'          thin_diamond marker
'|'          vline marker
'_'          hline marker
=============    ===============================

**Line Styles**

=============    ===============================
character        description
=============    ===============================
'-'          solid line style
'--'         dashed line style
'-.'         dash-dot line style
':'          dotted line style
=============    ===============================

Example format strings::

'b'    # blue markers with default shape
'or'   # red circles
'-g'   # green solid line
'--'   # dashed line with default color
'^k:'  # black triangle_up markers connected by a dotted line

**Colors**

The supported color abbreviations are the single letter codes

=============    ===============================
character        color
=============    ===============================
'b'          blue
'g'          green
'r'          red
'c'          cyan
'm'          magenta
'y'          yellow
'k'          black
'w'          white
=============    ===============================

and the 'CN' colors that index into the default property cycle.

If the color is the only part of the format string, you can
additionally use any  matplotlib.colors spec, e.g. full names
('green') or hex strings ('#008000').

Multiple dispatch¶

In an OOP language like Python or C++, given an object o, you would type o.method(x,y) to call a method, and it would choose the correct method based on the runtime ("dynamic") type of o (for a "virtual" method).

In Julia, the analogous code would call method(o, x, y)o is just one of the arguments. But Julia looks at the runtime ("dynamic") types of all the arguments to decide on the method to call.

In this sense it is a strict superset of multiple dispatch. It is not equivalent to C++ overloading, because that is static overloading. There was a nice talk about the differences between OOP, static overloading, and dynamic multiple dispatch at JuliaCon 2019 that I strongly encourage you to go watch.

In :
foo(x, y) = x + y
Out:
foo (generic function with 1 method)
In :
foo(3, 4)
Out:
7
In :
foo(A, 4)
MethodError: no method matching +(::Array{Int64,2}, ::Int64)
Closest candidates are:
+(::Any, ::Any, !Matched::Any, !Matched::Any...) at operators.jl:538
+(!Matched::Missing, ::Number) at missing.jl:115
+(!Matched::BigFloat, ::Union{Int16, Int32, Int64, Int8}) at mpfr.jl:378
...

Stacktrace:
 foo(::Array{Int64,2}, ::Int64) at ./In:1
 top-level scope at In:1
 execute_code(::String, ::String) at /Users/stevenj/.julia/dev/IJulia/src/execute_request.jl:27
 execute_request(::ZMQ.Socket, ::IJulia.Msg) at /Users/stevenj/.julia/dev/IJulia/src/execute_request.jl:86
 #invokelatest#1 at ./essentials.jl:710 [inlined]
 invokelatest at ./essentials.jl:709 [inlined]
 eventloop(::ZMQ.Socket) at /Users/stevenj/.julia/dev/IJulia/src/eventloop.jl:8
In :
foo(A::Matrix, y::Number) = A + y*I
Out:
foo (generic function with 2 methods)
In :
foo(A, 100)
Out:
3×3 Array{Int64,2}:
101    3    7
4  107    2
0    1  101
In :
foo(A, A)
Out:
3×3 Array{Int64,2}:
2   6  14
8  14   4
0   2   2
In [ ]: