Julia/julia_express.pdf.txt

The Julia Express

Bogumił Kami ´nski

December 31, 2022

Contents

1 Introduction

2 Getting around

3 Basic literals and types

4 Special literals and types

4.1 Tuples and NamedTuples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.2 Arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.3 Composite types

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4.4 Dictionaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 Strings

6 Programming constructs

7 Variable scoping

8 Modules

9 Operators

10 Essential general usage functions

11 Reading and writing data

12 Random numbers

13 Statistics and machine learning

14 Macros

15 Plotting

16 Working with tabular data

17 The Joy of Julia

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The Julia Express

1 Introduction

2

The purpose of this document is to introduce programmers to the Julia programming by example. This is a simplified
exposition of the language.1
It is simplest to execute these examples by copy-pasting to the Julia REPL (https://docs.julialang.org/en/latest/
stdlib/REPL/) or copying them to a file and next running them using include function. The difference is that copy-
paste approach will echo output of each instruction to the terminal.

If some package is missing on your system switch to the package manager mode by pressing ] in the Julia REPL, and
then write add [package name] to require installing it.

Over the years of using Julia I have learned that for each project you have it is best to have a separate project environ-
ment that keeps track of its dependencies. You can check out this blog post https://bkamins.github.io/julialang/
2020/05/18/project-workflow.html to read more about it.

This is an introductory document. Important topics that a person learning the Julia should be aware of, that are not
covered are:

1) parametric types;
2) parallel and distributed processing;
3) advanced I/O operations;
4) advanced package management;
5) interaction with system shell; see run;
6) exception handling; see try;
7) creation of coroutines;
8) integration with C, Fortran, Python and R.

You can read about them in the latest Julia documentation at http://julia.readthedocs.org/en/latest/manual/.

The Julia Express was tested using the following 64-bit Julia version (you can check your version by calling running
versioninfo() in your Julia session):

Julia Version 1.8.4

Commit 00177ebc4f (2022-12-23 21:32 UTC)

Platform Info:

OS: Windows (x86_64-w64-mingw32)
CPU: 12 x Intel(R) Core(TM) i7-10850H CPU @ 2.70GHz
WORD_SIZE: 64
LIBM: libopenlibm

LLVM: libLLVM-13.0.1 (ORCJIT, skylake)

Threads: 1 on 12 virtual cores

If you prefer to read this document in a different format than PDF then the simplest approach to do it is to clone the
project repository https://github.com/bkamins/The-Julia-Express to your local folder and use Pandoc to perform an
appropriate conversion. For instance running the command

pandoc -i julia_express.tex -f latex -t html5 -s -o julia_express.html

will produce you a HTML output.

All suggestions how this guide can be improved are welcomed. Please contact me at [email protected].

2 Getting around

Running julia invokes an interactive (REPL) mode. In this mode some useful commands are:

1) ^D (exits Julia);
2) ^C (interrupts computations);
3) ? (enters help mode);
4) ; (enters system shell mode);
5) ] (enters package manager mode);
6) Ctrl-l clears screen;
7) putting ; after the expression will disable showing its value in REPL (not needed in scripts).

Examples of some essential functions in the Julia REPL (they can be also invoked in scripts):

@edit max(1,2)

# show the definition of max function when invoked with arguments 1 and 2

varinfo()

# list of global variables and their types

1The rocket ship clip is free for download at http://www.clipartlord.com/free-cartoon-rocketship-clip-art-2/.

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3

cd("D:/")

pwd()

# change working directory to D:/ (on Windows you can use /)

# get current working directory

include("file.jl") # execute source file

exit(1)

# exit Julia with code 1 (exit code 0 is used by default)

clipboard([1,2])

# copy data to system clipboard

clipboard()

# load data from system clipboard as a string

You can execute a Julia script from OS shell by running julia script.jl.

Try saving the following example script to a file and run it (more examples of all the constructs used are given in
following sections):

"""Sieve of Eratosthenes function docstring"""

function es(n::Int) # accepts one integer argument

isprime = trues(n) # n-element vector of true-s

isprime[1] = false # 1 is not a prime

for i in 2:isqrt(n) # loop integers less or equal than sqrt(n)

if isprime[i] # conditional evaluation

for j in i^2:i:n # sequence with step i

isprime[j] = false

end

end

end

return filter(x -> isprime[x], 1:n) # filter using an anonymous function

end

println(es(100))

# print all primes less or equal than 100

@time length(es(10^6))

# check function execution time and memory usage

3 Basic literals and types

Basic scalar literals are the following:

1::Int

# 64 bit integer on 64 bit Julia, no overflow warnings

1.0::Float64 # 64 bit float, defines NaN, -Inf, Inf

true::Bool

# boolean, allows "true" and "false"

'c'::Char

# character, allows Unicode

"s"::String # strings, allows Unicode, see also Strings below

The syntax x::Type is a literal x with type Type assertion. In practice the type assertion is not needed. Here we use it
only to show the type of each kind of a literal. All basic types listed above are immutable.

Type assertions for variables are made in the same way and they can be useful to catch bugs in your code.

An important feature of integers in Julia is that by default they are 64 bit on 64 bit Julia and 32 bit on 32 bit Julia. This
means that 1::Int32 assertion will fail on 64-bit Julia.

Notably Int is a constant whose value is either Int64 or Int32 depending on version (the same with unsigned integer
UInt).

There is no automatic type conversion, unless some function explicitly performs it. This is especially important in
function calls. The simplest, and preferred, way to perform the construction of a value x to type T by writing T(x), for
example:

Int64('a')

# character to integer

Int64(2.0)

# float to integer

Int64(1.3)

# inexact error

Int64("a")

# error no conversion possible

Float64(1)

# integer to float

Bool(1)

Bool(0)

Bool(2)

# constructs to boolean true

# constructs to boolean false

# construction error

Char(89)

# integer to char

string(true)

# cast Bool to string (works with other types, note small caps)

string(1, true) # string can take more than one argument and concatenate them

zero(10.0)

# zero of type of 10.0

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Number

Complex{T<:Real}

Real

Irrational{sym}

Rational{T<:Integer}

Integer

AbstractFloat

Float16

Float32

Float64

Bool

Signed

Unsigned

BigFloat

BigInt

Int8

Int16

Int32

Int64

UInt8

UInt16

UInt32

UInt64

Int128

UInt128

Figure 1: Hierarchy of numeric types

one(Int64)

# one of type Int64

Conversion can be done using convert(Type, x) (typically convert will not perform a copy if x already has type Type;
for this reason it is typically preferable to perform construction, as it was explained above, rather than conversion):

convert(Int64, 1.0) # convert float to integer

Julia is throwing an inexact conversion error, if it cannot perform it exactly.

convert(Int64, 1.3) # convert float to integer -> throws inexact conversion error

Use floor(Int64, 1.3), ceil(Int64, 1.3) or round(Int64, 1.3) to perform rounding.

Parsing strings can be done using parse(Type, str):

parse(Int64, "1") # parse "1" string as Int64

Automatic promotion of many arguments to common type (if any exists) can be achieved using promote (typically you
will not need to perform such promotion yourself):

promote(true, BigInt(1) // 3, 1.0) # tuple (see Tuples) of BigFloats, true promoted to 1.0

promote("a", 1)

# error, promotion to a common type is not possible

Many operations (arithmetic, assignment) are defined in a way that performs automatic type promotion (so this is a
way to work around no automatic type conversion rule in Julia).

One can verify type of a value in the following way:

typeof("abc")

# String returned which is a AbstractString subtype

isa("abc", AbstractString) # true

isa(1, Float64)

# false, integer is not a float

isa(1.0, Float64)

# true

1.0 isa Number

# an alternative syntax; true, Number is abstract type

supertype(Int64)

# supertype of Int64

subtypes(Real)

Int <: Real

# subtypes of abstract type Real

# true, <: checks if type is subtype of other type

It is possible to perform calculations using arbitrary precision arithmetic, complex and rational numbers:

BigInt(10) ^ 1000

# big integer

BigFloat(10) ^ 1000 # big float, see documentation how to change default precision

big(1.5)

1 + 1im

# value of big type chosen appropriately, in this case BigFloat

# a complex number

123 // 456

# rational numbers are created using // operator

Type hierarchy of all standard numeric types is given in Figure 1.

An important feature is that Bool is considered to be Integer. This is useful for performing arithmetic operations, but
sometimes can be surprising.

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4 Special literals and types

Any

# all objects are of this type

Union{} # subtype of all types, no object can have this type

Nothing # type indicating nothing (absence of a value), a subtype of Any

nothing # only instance of Nothing

Missing # type indicating missing value (a value exists but is unknown), a subtype of Any

missing # only instance of Missing

Additionally #undef indicates an incompletely initialized object element (see documentation for details).

4.1 Tuples and NamedTuples

Tuples are immutable sequences indexed from 1:

()

(1,)

# an empty tuple

# a one element tuple

("a", 1)

# a two element tuple

('a', false)::Tuple{Char, Bool} # tuple type assertion

x = (1, 2, 3)

x[1]

x[1:2]

x[4]

x[1] = 1

a, b = x

# 1 (element)

# (1, 2) (tuple)

# bounds error

# error - a tuple is not mutable

# tuple unpacking a == 1, b == 2

Additionally you can add names to tuple entries (via named tuples):

NamedTuple() # an empty named tuple

(a=1,)

# a one element named tuple

(x="a", y=1) # a two element named tuple

x = (p=1, q=2, r=3)

x.p

# access to element p of a tuple

typeof(x)

# NamedTuple{(:p, :q, :r),Tuple{Int64,Int64,Int64}}, field names are part of type

a, b, c = x

# NamedTuple also supports unpacking; here we define a, b, and c variables

(; a, b, c)

# a convenience syntax to create a named tuple with a, b, c fields from variables

NamedTuple can be thought of as an anonymous struct — see composite types below (so they behave in a different way
than tuples when testing for subtyping). This is an advanced topic that we do not cover in this introduction, see the
Julia manual for the details https://docs.julialang.org/en/latest/manual/types/).

4.2 Arrays

Arrays are mutable and passed by reference.

Useful array creation functions are the following:

Array{Char}(undef, 2, 3, 4) # uninitialized 2x3x4 array of Chars

Array{Int64}(undef, 0, 0)

# degenerate 0x0 array of Int64

zeros(5)

ones(5)

# vector of Float64 zeros

# vector of Float64 ones

ones(Int64, 2, 1)

# 2x1 array of Int64 ones

trues(3), falses(3) # a tuple of a vector of trues and a vector of falses

Matrix(I, 3, 3)

# 3x3 Bool identity matrix, requires to run first: using LinearAlgebra

x = range(0, stop=1, length=11) # an iterator having 11 equally spaced elements

collect(x)

# converts an iterator to vector

1:10

1:2:10

# iterable from 1 to 10

# iterable from 1 to 9 with 2 skip

reshape(1:12, 3, 4) # a 3x4 matrix like object filled columnwise with values from 1 to 12

fill("a", 2, 2)

# a 2x2 array filled with "a"

repeat(rand(2,2), 3, 2) # a 2x2 random matrix repeated 3x2 times

x = [1, 2]

# a two element vector

resize!(x, 5)

# resize x in place to hold 5 values (filled with garbage)

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[1]

[1;;]

# a vector with one element (not a scalar)

# a matrix with one element (not a scalar)

[x * y for x in 1:2, y in 1:3] # a comprehension generating 2x3 array
Float64[x^2 for x in 1:4] # casting comprehension result element type to Float64

[1 2]

[1 2]'

# 1x2 matrix (hcat function)

# 2x1 Adjoint matrix (reuses memory)

permutedims([1 2])

# 2x1 matrix (permuted dimensions, new memory)

[1, 2]

[1; 2]

# vector (concatenation)

# vector (vcat function)

[1 2 3; 1 2 3]

# 2x3 matrix (hvcat function)

[1; 2] == [1 2]'

# false, different array dimensions

hcat(1:2)==[1 2]'

# true, dimensions match

[(1, 2)]

# 1-element vector

collect((1, 2))

# 2-element vector by tuple unpacking

[[1 2] 3]

[[1; 2]; 3]

# concatenate along rows (hcat)

# concatenate along columns (vcat)

tuple([1,2,3])

# a 1-element tuple containing a vector

Tuple([1,2,3])

# a 3-element tuple unpacking a vector

Vectors (1D arrays) are treated as column vectors.

Most of the functionality for working with matrices are in LinearAlgebra module. Additionally Julia offers sparse and
distributed matrices (see the documentation for details).

Commonly needed array utility functions:

a = [x * y for x in 1:2, y in 1, z in 1:3] # 2x3 array of Int64; a singleton dimension is dropped
a = [x * y for x in 1:2, y in 1:1, z in 1:3] # 2x1x3 array of Int64; a singleton dimension is not dropped
ndims(a)

# number of dimensions in a

eltype(a)

length(a)

size(a)

axes(a)

# type of elements in a

# number of elements in a

# a tuple containing dimension sizes of a

# a tuple of ranges specifying array axes

eachindex(a)

# each index to an array a

CartesianIndices(a) # a lazy iterator over Cartesian indices into a

LinearIndices(a)

# a lazy iterator over linear indices into a

vec(a)

# cast an array to vector (single dimension); reuses memory

dropdims(a, dims=2) # remove the 2nd dimension as it has length 1

sum(a, dims=3)

# calculate sums for 3rd dimensions, similarly: mean, std,

# prod, minimum, maximum, any, all;

# using Statistics is required for statistical functions

count(>(0), a)

# count number of times a predicate is true, similar: all, any

# note that we create an anonymous function with >(0) here

Access functions:

a = 0:0.01:1

# range with step 0.01

a[1]

a[begin]

a[end]

# get scalar 0.0

# get scalar 0.0 (first position)

# get scalar 1.0 (last position)

a[begin:2:end]

# every second element from range

view(a, 1:2:101)

# a view into a (a subarray of a)

a[[1, 3, 6]]

lastindex(a)

# 1st, 3rd and 6th element of a, Array{Float64,1}

# last index of the collection a; similarly firstindex

Observe the treatment of singleton dimensions:

a = reshape(1:12, 3, 4)

a[:, 1:2]

# 3x2 matrix

a[:, 1]

a[1, :]

# 3 element vector

# 4 element vector

a[1:1, :]

# 1x4 matrix

a[:, :, 1, 1]

# works 3x4 matrix

a[:, :, :, [true]] # works 3x4x1x1 matrix

a[1, 1, [false]]

# works 0-element Array{Int64,1}

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The Julia Express

Array assignment:

x = collect(reshape(1:8, 2, 4))

x[:,2:3] = [1 2]

# error; size mismatch

x[:,2:3] .= [1 2]

# OK, broadcasting with .

x[:,2:3] = repeat([1 2], 2) # OK

x[:,2:3] .= 3

# OK, need to use broadcast with .

Arrays are assigned and passed by reference. Therefore copying is provided:

x = Array{Any}(undef, 2)

x[1] = ones(2)

x[2] = trues(3)

a = x

b = copy(x)

# shallow copy

c = deepcopy(x) # deep copy

x[1] = "Bang"

x[2][1] = false

a

b

c

# identical as x

# only x[2][1] changed from the original x

# contents of the original x

Array types syntax examples:

[1 2]::Array{Int64, 2}

# 2 dimensional array of Int64

[true; false]::Vector{Bool} # vector of Bool

[1 2; 3 4]::Matrix{Int64}

# matrix of Int64

Numbers are treated as 0-dimensional containers:

x = 10

# an integer

x[]

# returns 10

x[1, 1]

# also returns 10, as trailing 1-s are ignored by Julia

size(x)

# an empty tuple

x = [10] # a one element array can be also indexed with []

x[]

# gets you 10, this will only work for arrays with exactly 1 element

only(x)

# returns the one and only element of x, otherwise errors

Ref is a special 0-dimensional container that can store any value:

x = Ref("a") # an 0-dimensional containers storing "a" string

x[]

# returns "a"

4.3 Composite types

You can define and access composite types. Here is an example of a mutable composite type:

mutable struct Point

x::Int64

y::Float64

meta

end

p = Point(0, 0.0, "Origin")

p.x

p.meta = 2

p.x = 1.5

p.z = 1

# access field

# change field value

# error, wrong data type

# error - no such field

fieldnames(Point) # get names of type fields

Similarly you can define some type to be immutable by removing mutable keyword (named tuples are anonymous
immutable structs).

There are also union types (see documentation of Type Unions in the Julia manual for details).

Finally you can define that your type is a subtype of an abstract type to properly position it in the type hierarchy, or
even define your own abstract types (see documentation of Abstract Types in the Julia manual for details).

The Julia Express

4.4 Dictionaries

8

Associative collections (key-value dictionaries):

x = Dict{Float64, Int64}()

# an empty dictionary mapping floats to integers

y = Dict("a"=>1, "b"=>2)

# a filled dictionary

y["a"]

y["c"]

y["c"] = 3

haskey(y, "b")

keys(y), values(y)

delete!(y, "b")

# element retrieval

# error

# added element

# check if y contains key "b"

# tuple of collections returning keys and values in y

# delete a key from a collection, see also: pop!

get(y, "c", "default")

# return y["c"] or "default" if not haskey(y,"c")

Julia also supports operations on sets, created similarly with Set constructor (please refer to the documentation for
details).

5 Strings

String operations:

"Hi " * "there!"
"Ho " ^ 3

# string concatenation

# repeat string

string("a = ", 123.3) # create using print function

repr(123.3)

# fetch value of show function to a string

occursin("CD", "ABCD") # check if the second string contains the first

"\"\n\t\$"

x = 123

# C-like escaping in strings, new \$ escape

"$x + 3 = $(x+3)"

# unescaped $ is used for interpolation

"\$199"

# to get a $ symbol you must escape it

raw"D:\path"

# a raw string literal; useful for paths under Windows

s = "abc"

chop(s)

# a string of type String

# remove last character from s, returns a SubString

Both String and SubString are subtypes of AbstractString. The SubString type is used to avoid copying of strings.
Usually, when writing your own code, it is best to assume that the user will pass an arbitrary AbstractString.

PCRE regular expressions handling:

r = r"A|B"

# create new regexp

occursin(r, "CD")

# false, no match found

m = match(r, "ACBD") # find first regexp match, see the documentation for details

There is a vast number of string functions — please refer to the documentation.

Warning! Note that you can index-into a string, e.g. "abc"[1] will return you a character 'a'. However, in general
Julia encodes standard strings using UTF-8 and indexing is based on bytes not characters, so correct string indexing
requires you to understand how UTF-8 encoding works. See the documentation for details.

6 Programming constructs

The simplest way to bind a value to a new variable is by an assignment:

x = 1.0

x = 1

# x is bound to Float64 value

# now x is bound to value Int32 on 32 bit machine and Int64 on 64 bit machine

Expressions can be compound using ; or begin end block:

x = (a = 1; 2 * a) # after: x = 2; a = 1
y = begin

b = 3

3 * b

end

# after: y = 9; b = 3

There are standard programming constructs:

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if false

# if clause requires Bool test

z = 1

elseif 1 == 2

z = 2

else

a = 3

end

# after this a = 3 and z is undefined

1==2 ? "A" : "B" # standard ternary operator

i = 1

while true

global i += 1

if i > 10

break

end

end

for x in 1:10

# x in collection, can also use = here instead of in

if 3 < x < 6

continue # skip one iteration

end

println(x)

end

# x is defined in the inner scope of the loop

You can define your own functions:

f(x, y = 10) = x + y

# one line definition of a new function f with y defaulting to 10

function f(x, y=10)

# the same as above but allowing multiple expressions

# last expression result returned

x + y

end

f(3, 2)

f(3)

(x -> x^2)(3)

() -> 0

# in the body of the function

# a simple call, 5 returned

# 13 returned

# an anonymous function with a call example

# an anonymous function with no arguments

h(x...) = sum(x)/length(x) - mean(x) # a vararg function; x is a tuple; call first: using Statistics

h(1, 2, 3)

x = (2, 3)

f(x)

f(x...)

# the result is 0

# a tuple

# error - we try to add 10 to (2,3)

# OK - tuple unpacking

s(x; a = 1, b = 1) = x * a / b # function with keyword arguments a and b
s(3, b = 2)
q(f::Function, x) = 2 * f(x)
q(x -> 2x, 10)

# call with a keyword argument

# a function can be passed around; here we require that f is a Function
# 40 returned, no need to use * in 2x (means 2*x)
# creation of an anonymous function by do construct, useful eg. in IO

q(10) do x
2 * x

end

m = reshape(1:12, 3, 4)

map(x -> x ^ 2, m)

# 3x4 array returned with transformed data

filter(x -> bitstring(x)[end] == '0', 1:12) # a fancy way to choose even integers from the range

==(1)

# returns a function that tests for equality to 1

findall(==(1), 1:10)

# find indices of all elements equal to 1, similar: findfirst, findlast

As a convention functions with name ending with ! change their arguments in-place. See for example resize! in this
document.

Default function arguments are evaluated left to right:

y = 10

f1(x=y) = x; f1()

# 10

f2(x=y,y=1) = x; f2() # 10

f3(y=1,x=y) = x; f3() # 1

f4(;x=y) = x; f4()

# 10

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f5(;x=y,y=1) = x; f5() # 10

f6(;y=1,x=y) = x; f6() # 1

There is an important part of Julia terminology is that a function can have multiple methods. Each method specifies a
behavior of a function for a given set of argument types. This behavior is called multiple dispatch and works only for
positional arguments. Here are some short examples. More details are given in Methods section of the Julia manual.

g(x, y) = println("all accepted") # method for g function accepting any type of x and y

function g(x::Int, y::Int)

# method called when both x and y are Int

y, x

end

g(x::Int, y::Bool) = x * y
g(1.0, 1)

g(1, 1)

g(1, true)

methods(g)

# this will be called when x is Int and y is Bool

# the first definition is invoked

# the second definition is invoked

# the third definition is invoked

# list all methods defined for g

t(; x::Int64 = 2) = x

# a single keyword argument

t()

# 2 returned

t(; x::Bool = true) = x

# no multiple dispatch for keyword arguments; function overwritten

t()

# true; old function was overwritten

7 Variable scoping

The following constructs introduce a new variable scope: function, while, for, try/catch, let, struct, mutable struct.

Additionally you can define variables as:

• global: use variable from a global scope of the current module;

• local: define a new variable in a current scope (useful to avoid variable name conflicts, or to make sure variable

is present in outer scope; see examples below);

• const: signal to the compiler that variable type is constant (global only).

Special cases:

t

# error, a variable t does not exist

f() = global t = 1

f()

# after the call t is defined globally

function f1(n)

x = 0

for i = 1:n

x = i

end

x

end

f1(10)

function f2(n)

x = 0

for i = 1:n

local x

x = i

end

x

end

f2(10)

function f3(n)

for i = 1:n

h = i

end

# 10; inside the loop we use the outer local variable

# 0; inside loop we use new local variable

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h

end

f3(10)

function f4(n)

local h

for i = 1:n

h = i

end

h

end

f4(10)

const x = 2

# error; h not defined in outer scope

# 10; h is defined in outer scope

x = 3 # warning, value changed; but you should never do this as it might break compiled code

x = 3.0 # error, wrong type

function f()

x::Int = 1 # this is a way to specify a required type of a variable

x = 2.5 # error will be thrown when f() is called as x has to have type Int

end

Global constants speed up code execution as the compiler knows their type.

Loops and comprehensions rebind variables on each iteration, so they are safe to use then creating closures in itera-
tion:

Fs = Array{Any}(undef, 2)

for i in 1:2

Fs[i] = () -> i

end

Fs[1](), Fs[2]() # (1, 2)

Note that for, while, try, and struct use a so called soft local scope. Simplifying a bit this means that if they are used in
a top level (global) scope they overwrite existing global variables if you are in an interactive mode (REPL, notebook):

julia> x = 5

5

julia> for i in 1:10

x = i

end

julia> x

10

However, the same code passed in an non-interactive session prints a warning and does not overwrite a global vari-
able:

| Warning: Assignment to

~$ julia -e "x=5; for i in 1:10 x = i end; println(x)"
‘
‘
x
‘
‘
x

by the same name exists:
‘
‘
local x

to suppress this warning or

‘
‘
global x

in soft scope is ambiguous because a global variable

will be treated as a new local. Disambiguate by using

to assign to the existing global variable.

| @ none:1

5

8 Modules

Modules encapsulate code and each module has its own global name space (module name of Julia REPL is Main).

module M # module name

export x # what module exposes for the world

x = 1

The Julia Express

12

y = 2 # hidden variable

end

varinfo(M) # list exported variables

x

M.y

# not found in global scope

# direct variable access possible

# import all exported variables

# also load standard packages this way, but without . prefix

using .M

#import variable y to global scope (even if not exported)

import .M.y

Rebinding values of variables defined in other modules is not allowed. Here is a short typical example that often
surprises people:

sin(1) # works

sin = 1 # fails in module Main you cannot rebind a value defined in module Base

cos = 1 # works, as cos was not called yet so it was not imported from Base into Main

cos # gives 1

cos(1) # fails - cos is bound to 1 in the module Main

Base.cos(1) # works

9 Operators

Julia follows standard operators with the following quirks:

true || false

# binary or operator (singletons only), || and && use short-circuit evaluation

[1 2] .& [2 1]

# bitwise and operator (vectorized by .)

[true, false] .&& [true false] # logical and operator (vectorized by .); the syntax creates truth table

1 < 2 < 3

# chaining conditions is OK (singletons only without .)

[1 2] .< [2 1]

# for vectorized operators need to add '.' in front

x = [1 2 3]

2x + 2(x .+ 1)

# multiplication can be omitted between a literal and a variable or a left parenthesis

y = [1, 2, 3]

x + y # an error - dimensions do not match

x .+ y # a 3x3 matrix, dimension broadcasting

x + y' # a 1x3 matrix

x * y # array multiplication, a 1-element vector (not scalar)
x .* y # element-wise multiplication, a 3x3 array

x == [1 2 3] # true, object looks the same

x === [1 2 3] # false, objects not identical

z = reshape(1:9, 3, 3)

z + x # error - dimensions do not match

z .+ x # x broadcasted vertically

z .+ y # y broadcasted horizontally

# an explicit broadcast of singleton dimensions

# function + is called for each array element

broadcast(+, [1 2], [1; 2])

# broadcasting using . operator

using Random

length([randstring(10) for i in 1:5]) # 5 - length of an array

length.([randstring(10) for i in 1:5]) # 5-element array of 10s - lengths of strings

Function broadcasting examples:

t(x::Float64, y::Float64 = 1.0) = x * y
t(1.0, 2.0)

# OK

The Julia Express

13

t([1.0 2.0])

t.([1.0 2.0])

t([1.0 2.0], 2.0)

t.([1.0 2.0], 2.0)

t.(2.0, [1.0 2.0])

# error

# OK

# error

# OK

# OK

t.([1.0 2.0], [1.0 2.0]) # OK

t.([1.0, 2.0], [1.0 2.0]) # OK

10 Essential general usage functions

show(collect(1:100)) # show text representation of an object

eps()

# distance from 1.0 to next representable Float64

nextfloat(2.0)

# next float representable, similarly provided prevfloat

isequal(NaN, NaN) # true

NaN == NaN

NaN === NaN

# false

# true

isequal(1, 1.0)

# true

1 == 1.0

1 === 1.0

0.0 == -0.0

0.0 === -0.0

# true

# false

# true

# false

isfinite(Inf)

# false, similarly provided: isinf, isnan

fld(-5, 3), mod(-5, 3) # (-2, 1), division towards minus infinity

div(-5, 3), rem(-5, 3) # (-1, -2), division towards zero

findall(x -> mod(x, 2) == 0, 1:8) # find indices for which function returns true

x = [1 2]; identity(x) === x # true, identity function

@info "Info"

# print information, similarly @warn and @error (see Logging module)

ntuple(x->2x, 3) # create tuple by calling x->2x with values 1, 2 and 3

@isdefined x

# if variable x is defined

y = Array{Any}(undef,2); isassigned(y, 3) # is 3 in array is assigned (not out of bounds or #undef)

fieldtype(typeof(1:2),:start) # get type of the field in composite type (passed as symbol)

fieldnames(typeof(1:2)) # get field names of a type

zip(1:3, 1:3) |> collect # convert iterables to iterable tuple and pass it to collect

enumerate("abc") # create iterator of tuples (index, collection element)

collect(enumerate("abc")) # and materialize it

isempty("abc")

# check if a collection is empty; strings are treated as collections of characters

'b' in "abc"

# check if element is in a collection

indexin(collect("abc"), collect("abrakadabra")) # [1, 2, nothing] ('c' not found), needs arrays

findall(in("abrakadabra"), "abc") # [1, 2] ('c' was not found)

unique("abrakadabra") # return unique elements

issubset("abc", "abcd") # check if every element in the first collection is in the second

argmax("abrakadabra") # an index of maximal element (3 - 'r' in this case)

findmax("abrakadabra") # tuple: a maximal element and its index

filter(x->mod(x,2)==0, 1:10) # retain elements of a collection that meet predicate

dump(1:2:5)

# show all user-visible structure of an object

sort(rand(10))

# sort 10 uniform random values, sort! for in-place operation

11 Reading and writing data

For I/O details refer documentation. There are numerous packages providing this functionality. Basic operations from
DelimitedFiles module:

• readdlm: read from file

• writedlm: write to a file

Warning! Trailing spaces are not discarded if delim=' ' in file reading.

The Julia Express

14

12 Random numbers

Basic random numbers:

Random.seed!(1) # set random number generator seed to 1; needs calling first: using Random

rand()

# generate random number from U[0,1)

rand(3, 4)

# generate 3x4 matrix of random numbers from U[0,1]

rand(2:5, 10)

# generate vector of 10 random integer numbers in range form 2 to 5

randn(10)

# generate vector of 10 random numbers from standard normal distribution

Advanced randomness form Distributions.jl package (version 0.25.79):

using Distributions # load package

sample(1:10, 10)

# single bootstrap sample from set 1-10

b = Beta(0.4, 0.8) # Beta distribution with parameters 0.4 and 0.8

# see documentation for supported distributions

mean(b)

# expected value of distribution b

rand(b, 100)

# 100 independent random samples from distribution b

# see documentation for other supported statistics

13 Statistics and machine learning

Visit http://juliastats.github.io/ for the details (in particular R-like data frames).

There is a core language construct Missing that allows to represent missing value.

missing # Missing value

ismissing(missing) # true

coalesce(missing, 1, 2) # return first non-missing value, or missing if all are missing

Here are some most basic statistical functions shipped with Julia:

using Statistics # module defining the statistical functions

mean([1, 2, 3]) # mean

var([1, 2, 3]) # variance

std([1, 2, 3]) # standard deviation

quantile(1:100, 0.05) # quantiles

median(1:100) # median

cov(1:5, 5:-1:1) # covariance

cor(1:5, 5:-1:1) # Pearson's correlation

14 Macros

You can define macros (see documentation for details). Useful standard macros.

Assertions:

@assert 1 == 2 "ERROR"

# 2 macro arguments; error raised

using Test

# load Test package

@test 1 == 2
@test_throws DomainError sqrt(-1) # passed, sqrt(-1) is not possible

# similar to assert; error

Benchmarking:

@time [x for x in 1:10^6];

# print time and memory

@timed [x for x in 1:10^6];

# return value, time and memory

@elapsed [x for x in 1:10^6]

# return time

@allocated [x for x in 1:10^6] # return memory

Use BenchmarkTools.jl package (version 1.3.2) for a more powerful benchmarking functionality.

The Julia Express

15 Plotting

15

There are several plotting packages for Julia like Plots.jl (which is an umbrella packages for several plotting backends).
Here we show how to use it (version 1.38.0):

using Plots

using Random

Random.seed!(1) # make the plot reproducible

x, y = 1:100, randn(100)

plot(x, y) # line plot

scatter(x, y) # scatter plot

histogram(y) # histogram

bar(y) # barplot

An example reproducing https://matplotlib.org/1.2.1/examples/pylab_examples/histogram_demo.html in which we
also use StatsPlots.jl package (version 0.15.4):

using Distributions

using Plots

using Random

using StatsPlots

Random.seed!(1234)

mu, sigma = 100, 15

x = mu .+ sigma * randn(10000)

histogram(x,

title="Histogram of IQ: \\mu=100, \\sigma=15",

label=nothing,

xlabel="Smarts",

ylabel="Probability",

color="green",

normalize=true)

plot!(Normal(mu, sigma),

color="red",

label=nothing,

linestyle=:dash)

savefig("hist.pdf")

producing:

16 Working with tabular data

There are multiple packages supporting tabular data for the Julia language.

Here we will show how DataFrames.jl (version 1.4.4) and CSV.jl (version 0.10.8) packages can be used.

Loading a CSV file:

16

The Julia Express

using DataFrames

using CSV

path = joinpath(dirname(pathof(DataFrames)), "../docs/src/assets/iris.csv")

df = CSV.read(path, DataFrame);

first(df, 5) # print first 5 rows of a data frame; use the last function for last rows

produces the following output:

5x5 DataFrame

Row | SepalLength SepalWidth PetalLength PetalWidth Species

| Float64

Float64

Float64

Float64

String

---------------------------------------------------------------------

1 |

2 |

3 |

4 |

5 |

5.1

4.9

4.7

4.6

5.0

3.5

3.0

3.2

3.1

3.6

1.4

1.4

1.3

1.5

1.4

0.2 Iris-setosa

0.2 Iris-setosa

0.2 Iris-setosa

0.2 Iris-setosa

0.2 Iris-setosa

Here is a limited list of most useful operations on a DataFrame we have just read in:

DataFrame(a=1:10, b=rand(10)) # manual creation of a DataFrame from a list of columns

describe(df) # get a summary information about a data frame

df.Species # get a column Species from a data frame without copying

df[!, :Species] # the same as above

df[:, :Species] # the same as above, but perform a copy

df[1, 5] # get a value from row 1 and column 5 (Species again) from a DataFrame

df[1:2, 1:2] # subset a data frame getting two first rows and two first columns

Matrix(df[:, 1:4]) # convert columns 1 to 4 to a matrix

names(df) # get column names in the data frame as strings

nrow(df), ncol(df) # number of rows and columns in a data frame

sort(df, :SepalWidth) # return a new data frame that is sorted by SepalWidth column

filter(:SepalWidth => >(3), df) # return a new data frame with only rows that meet the predicate

push!(df, (1, 2, 3, 4, "Some species")) # add a new row at the end of the data frame

df.key = axes(df, 1) # add a new variable named key to a data frame

# calculate sum of SepalLength column by Species column and store it in x column

combine(groupby(df, :Species), :SepalLength => sum)

# transform df to long format taking SepalLength as values and key and Species as id variable

df2 = stack(df, :SepalLength, [:key, :Species])

unstack(df2, [:key, :variable], :Species, :value) # a reverse operation - wide to long format

17 The Joy of Julia

The Julia language is so flexible that it allows you to transform

this

to this

You can fully appreciate it (actually it is an animation of revolving hearts) when you run the code yourself. You can
find it here https://github.com/maxbennedich/code-golf/tree/cea06287689868f2342959f9c12f0b629a1d0cf4/hearts.

Please make sure that you run the code on a fast terminal with font that supports Unicode.