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5d. Initializing Vectors

Martin Alfaro
PhD in Economics

Introduction

We continue introducing preliminary concepts for mutations. The previous section distinguished between the use of = for assignments and mutations. Now, we'll deal with approaches to creating vectors.

Our presentation starts by outlining the process of initializing vectors, where memory is reserved without assigning initial values. We'll then discuss how to create vectors filled with predefined values such as zeros or ones. Finally, we show how to concatenate multiple vectors into new ones.

Initializing Vectors

Creating an array involves two steps: reserving memory for holding its content and assigning initial values to its elements. However, if you don't intend to populate the array with values right away, it's more efficient to only initialize the array. This means reserving memory space, but without setting any initial values.

Technically, initializing an array entails creating an array filled with undef values. These values represent arbitrary content in memory at the moment of allocation. Importantly, while undef displays concrete numbers when you output the array's content, they're meaningless and vary every time you initialize a new array.

There are two methods for creating vectors with undef values. The first one requires you to explicitly specify the type and length of the array, which is accomplished via Vector{<elements' type>}(undef, <length>). The second approach is based on the function similar(x), which creates a vector with the same type and dimensions as an existing vector x.

x_length = 3

x = Vector{Int64}(undef, x_length)  # `x` can hold `Int64` values, and is initialized with 3 undefined elements
Output in REPL
julia>
x
3-element Vector{Int64}:
 140724480121488
   2497084710592
   2497285012816
y = [3,4,5]

x = similar(y)                      # `x` has the same type as `y`, which is Vector{Int64}(undef, 3)
Output in REPL
julia>
x
3-element Vector{Int64}:
 2497063587648
 2497063587664
 2497355825296

The example demonstrates that undef values don't follow any particular pattern. Moreover, these values vary in each execution, as they reflect any content held in RAM at the moment of allocation. In fact, a more descriptive way to call undef values would be uninitialized values.

Creating Vectors with Given Values

In the following, we present several approaches to creating arrays filled with predefined values.

Vectors With Range

If our goal is to generate a sequence of values, we can employ the function collect(<range>). Recall that the syntax for defining ranges is <start>: <steps>: <stop>, where <steps> establishes the gap between elements.

some_range = 2:5

x          = collect(some_range)
Output in REPL
julia>
x
4-element Vector{Int64}:
 2
 3
 4
 5

Notice that when a range is created, <steps> implicitly dictates the number of elements to be generated. Alternatively, you could specify the number of elements to be stored, letting <steps> be implicitly determined. This is achieved by the function range, whose syntax is range(<start>, <end>, <number of elements>). [note] Note that range represents a convenient syntax for <start> : 1 / <number of elements> : <end>.

The following code snippet demonstrates the use of range, by generating five evenly spaced elements between 0 and 1.

x = range(0, 1, 5)
Output in REPL
julia>
x
0.0:0.25:1.0
x = range(start=0, stop=1, length=5)
Output in REPL
julia>
x
0.0:0.25:1.0
x = range(start=0, length=5, stop=1)    # any order for keyword arguments
Output in REPL
julia>
x
0.0:0.25:1.0

Vectors With Specific Values

We can also create vectors of some given length filled with the same repeated value. In particular, the functions zeros and ones respectively create vectors with zeros and ones. By default, these functions define Float64 elements, although you can specify a different type in the first argument of the function.

length_vector = 3

x             = zeros(length_vector)
Output in REPL
julia>
x
3-element Vector{Float64}:
 0.0
 0.0
 0.0
length_vector = 3

x             = zeros(Int, length_vector)
Output in REPL
julia>
x
3-element Vector{Int64}:
 0
 0
 0
length_vector = 3

x             = ones(length_vector)
Output in REPL
julia>
x
3-element Vector{Float64}:
 1.0
 1.0
 1.0
length_vector = 3

x             = ones(Int, length_vector)
Output in REPL
julia>
x
3-element Vector{Int64}:
 1
 1
 1

For creating Boolean vectors, Julia provides two convenient functions called trues and falses.

length_vector = 3

x             = trues(length_vector)
Output in REPL
julia>
x
3-element BitVector:
 1
 1
 1
length_vector = 3

x             = falses(length_vector)
Output in REPL
julia>
x
3-element BitVector:
 0
 0
 0

Vectors Filled With a Repeated Object

To define vectors comprising elements different from zeros or ones, Julia provides the fill function. Unlike the previous functions, this accepts any arbitrary scalar to be repeated.

length_vector    = 3
filling_object   = 1

x                = fill(filling_object, length_vector)
Output in REPL
julia>
x
3-element Vector{Int64}:
 1
 1
 1
length_vector    = 3
filling_object   = [1,2]

x                = fill(filling_object, length_vector)
Output in REPL
julia>
x
3-element Vector{Vector{Int64}}:
 [1, 2]
 [1, 2]
 [1, 2]
length_vector    = 3
filling_object   = [1]

x                = fill(filling_object, length_vector)
Output in REPL
julia>
x
3-element Vector{Vector{Int64}}:
 [1]
 [1]
 [1]

Concatenating Vectors

Finally, we can create a vector z that merges all the elements of two vectors x and y. One simple approach for doing this is through z = [x ; y]. While this method is suitable for concatenating a few vectors, it becomes impractical with a large number of vectors, and directly infeasible when the number of vectors to concatenate is unknown.

For these scenarios, we can instead employ the function vcat, which merges all its arguments into one vector. By use of the splat operator ...., the function can also be applied with a single argument that comprises a list of vectors. [note] Recall that the operator ... splits a collection into multiple arguments. This enables the use of a vector or tuple to denote multiple function arguments. For further details, see here under the subsection "Splatting".

x = [3,4,5]
y = [6,7,8]


z = vcat(x,y)
Output in REPL
julia>
x
6-element Vector{Int64}:
 3
 4
 â‹®
 7
 8
x = [3,4,5]
y = [6,7,8]

A = [x, y]
z = vcat(A...)
Output in REPL
julia>
x
6-element Vector{Int64}:
 3
 4
 â‹®
 7
 8

Closely related to vector concatenation is the repeat function, which defines a vector containing the same object multiple times. Importantly, unlike fill, repeat requires an array as its input, throwing an error if a scalar is passed in particular.

nr_repetitions   = 3
vector_to_repeat = [1,2]

x                = repeat(vector_to_repeat, nr_repetitions)
Output in REPL
julia>
x
6-element Vector{Int64}:
 1
 2
 â‹®
 1
 2
nr_repetitions   = 3
vector_to_repeat = [1]

x                = repeat(vector_to_repeat, nr_repetitions)
Output in REPL
julia>
x
3-element Vector{Int64}:
 1
 1
 1
nr_repetitions   = 3
vector_to_repeat = 1

x                = repeat(vector_to_repeat, nr_repetitions)
Output in REPL
ERROR: MethodError: no method matching repeat(::Int64, ::Int64)

Adding, Removing, and Replacing Elements (OPTIONAL)

Warning!
This subsection requires knowledge of a few concepts we haven't discussed yet. As such, it's marked as optional.

One such concept is that of in-place functions, identified by the symbol ! appended to the function's name. The symbol is simply a convention chosen by developers to indicate that the function modifies at least one of its arguments. A detailed discussion of in-place functions will be provided later.

Another concept introduced is that of pairs, which will be throughly examined in a future sectiontoo. For the purposes of this subsection, it's sufficient to know that pairs are written in the form a => b. In our applications, a will represent a given value and b denotes its corresponding replacement value.

Next, we demonstrate how to add, remove, and replace elements within a vector. Below, we begin by presenting methods for adding a single element.

x                 = [3,4,5]
element_to_insert = 0


push!(x, element_to_insert)                 # add 0 at the end - faster
Output in REPL
julia>
x
4-element Vector{Int64}:
 3
 4
 5
 0
x                 = [3,4,5]
element_to_insert = 0


pushfirst!(x, element_to_insert)            # add 0 at the beginning - slower
Output in REPL
julia>
x
4-element Vector{Int64}:
 0
 3
 4
 5
x                 = [3,4,5]
element_to_insert = 0
at_index          = 2

insert!(x, at_index, element_to_insert)     # add 0 at index 2
Output in REPL
julia>
x
4-element Vector{Int64}:
 3
 0
 4
 5
x                 = [3,4,5]
vector_to_insert  = [6,7]


append!(x, vector_to_insert)                # add 6 and 7 at the end
Output in REPL
julia>
x
5-element Vector{Int64}:
 3
 4
 5
 6
 7

The function push! is particularly helpful to collect outputs in a vector. Since it doesn't require prior knowledge of how many elements will be stored, the vector can grow dynamically as new results are added. Notice that adding elements at the end with push! is generally faster than inserting them at the beginning with pushfirst!.

Analogous functions exist to remove elements, as shown below.

x                  = [5,6,7]


pop!(x)                            # delete last element
Output in REPL
julia>
x
2-element Vector{Int64}:
 5
 6
x                  = [5,6,7]


popfirst!(x)                       # delete first element
Output in REPL
julia>
x
2-element Vector{Int64}:
 6
 7
x                  = [5,6,7]
index_of_removal   = 2

deleteat!(x, index_of_removal)      # delete element at index 2
Output in REPL
julia>
x
2-element Vector{Int64}:
 5
 7
x                  = [5,6,7]
indices_of_removal = [1,3]

deleteat!(x, indices_of_removal)    # delete elements at indices 1 and 3
Output in REPL
julia>
x
1-element Vector{Int64}:
 6

By analogy with the behavior of deleteat!, it's also possible to specify which elements should be retained.

x               = [5,6,7]
index_to_keep   = 2

keepat!(x, index_to_keep)
Output in REPL
julia>
x
1-element Vector{Int64}:
 6
x               = [5,6,7]
indices_to_keep = [2,3]

keepat!(x, index_to_keep)
Output in REPL
julia>
x
1-element Vector{Int64}:
 6

Finally, specific values can be replaced with new ones. This can be done by either creating a new copy using replace or by updating the original vector with replace!.

Both functions make use of pairs a => b, where a denotes a given value and b specifies its replacement. Note that these functions perform substitutions based on values, not on indices.

x = [3,3,5]

replace!(x, 3 => 0)              # in-place (it updates x)
Output in REPL
julia>
x
3-element Vector{Int64}:
 0
 0
 5
x = [3,3,5]

replace!(x, 3 => 0, 5 => 1)      # in-place (it updates x)
Output in REPL
julia>
x
3-element Vector{Int64}:
 0
 0
 1
x = [3,3,5]

y = replace(x, 3 => 0)           # new copy
Output in REPL
julia>
y
3-element Vector{Int64}:
 0
 0
 5
x = [3,3,5]

y = replace(x, 3 => 0, 5 => 1)   # new copy
Output in REPL
julia>
y
3-element Vector{Int64}:
 0
 0
 1