Indexing, slicing, and all that · Scientific computing

In the previous module, we have introduced the notion of Arrays, and experimented with the shape of vectors and matrices. In this module, we will continue our exploration of these objects, and see how we can modify and access the information they store.

In order to facilitate our work, we will create a simple matrix, which will be full of ones:

A = ones(Float64, 4, 7)

4×7 Matrix{Float64}:
 1.0  1.0  1.0  1.0  1.0  1.0  1.0
 1.0  1.0  1.0  1.0  1.0  1.0  1.0
 1.0  1.0  1.0  1.0  1.0  1.0  1.0
 1.0  1.0  1.0  1.0  1.0  1.0  1.0

The ones and zeros functions are extremely useful to initialize objects of a given size and type, and we strongly recommend you check out their documentation.

One fairly important question is, in what order are these elements stored in the matrix? We can start looking at the first indexing approach, also known as linear indexing:

LinearIndices(A)

4×7 LinearIndices{2, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}}}:
 1  5   9  13  17  21  25
 2  6  10  14  18  22  26
 3  7  11  15  19  23  27
 4  8  12  16  20  24  28

The first element is at the top-left of the matrix, elements are stored alongside columns, and the final element is at the bottom-right.

This may seem a little awkward as we think of matrices as having two dimensions, but it is perfectly appropriate to ask for “the ninth value in A”:

A[9]

1.0

It’s one. Of course it’s one, because we generated a matrix that is filled with ones. So let’s change this value:

A[9] = 9.0

9.0

What is going one behind the scenes is in fact a call to two different methods. We can get values out of a structure with getindex:

getindex(A, 9)

9.0

We can write values in a structure with setindex!:

setindex!(A, 2.0, 9)

4×7 Matrix{Float64}:
 1.0  1.0  2.0  1.0  1.0  1.0  1.0
 1.0  1.0  1.0  1.0  1.0  1.0  1.0
 1.0  1.0  1.0  1.0  1.0  1.0  1.0
 1.0  1.0  1.0  1.0  1.0  1.0  1.0

The setindex! function ends with an exclamation mark to let you know that it will mutate its first argument. This is not a feature of the language (adding ! to a function name does not change anything), but a very strongly adhered to social contract. Much later, we will see how we use this design pattern in practice.

There is a second way to access the coordinates of an array:

collect(CartesianIndices(A))

4×7 Matrix{CartesianIndex{2}}:
 CartesianIndex(1, 1)  CartesianIndex(1, 2)  CartesianIndex(1, 3)  CartesianIndex(1, 4)  CartesianIndex(1, 5)  CartesianIndex(1, 6)  CartesianIndex(1, 7)
 CartesianIndex(2, 1)  CartesianIndex(2, 2)  CartesianIndex(2, 3)  CartesianIndex(2, 4)  CartesianIndex(2, 5)  CartesianIndex(2, 6)  CartesianIndex(2, 7)
 CartesianIndex(3, 1)  CartesianIndex(3, 2)  CartesianIndex(3, 3)  CartesianIndex(3, 4)  CartesianIndex(3, 5)  CartesianIndex(3, 6)  CartesianIndex(3, 7)
 CartesianIndex(4, 1)  CartesianIndex(4, 2)  CartesianIndex(4, 3)  CartesianIndex(4, 4)  CartesianIndex(4, 5)  CartesianIndex(4, 6)  CartesianIndex(4, 7)

This is a type of indexing we are more familiar with, as each entry is specified by a (row,column) pair of values. Notice that this is a matrix with the same shape as the A matrix, so we can check what the coordinates of the ninth position are:

CartesianIndices(A)[9]

CartesianIndex(1, 3)

Line 1, row 3 – what can we do with a CartesianIndex? Well, we can get information out of matrix:

A[CartesianIndex(1, 3)]

2.0

Wait a minute, you say, this sounds very complicated. Why can’t I write A[1,3]?

A[1, 3]

2.0

You can. You can also use it to modify a value!

A[1, 3] = 4.0
A

4×7 Matrix{Float64}:
 1.0  1.0  4.0  1.0  1.0  1.0  1.0
 1.0  1.0  1.0  1.0  1.0  1.0  1.0
 1.0  1.0  1.0  1.0  1.0  1.0  1.0
 1.0  1.0  1.0  1.0  1.0  1.0  1.0

These different ways to access/write information in an array all have their uses. For now, it is enough that you remember that it is possible to access information using the position you want by increasing dimension (rows, then columns, then …). But at some point, we will have to leverage some unique functions of Cartesian indexing, so you can put a mental pin in this.

Let’s now come up with a slightly more interesting matrix:

B = reshape(1:30, (5, 6))

5×6 reshape(::UnitRange{Int64}, 5, 6) with eltype Int64:
 1   6  11  16  21  26
 2   7  12  17  22  27
 3   8  13  18  23  28
 4   9  14  19  24  29
 5  10  15  20  25  30

We can extract the third row of this matrix with:

B[3, :]

6-element Vector{Int64}:
  3
  8
 13
 18
 23
 28

We can get the fourth column with:

B[:, 4]

5-element Vector{Int64}:
 16
 17
 18
 19
 20

We can even get the entire matrix with:

B[:, :]

5×6 Matrix{Int64}:
 1   6  11  16  21  26
 2   7  12  17  22  27
 3   8  13  18  23  28
 4   9  14  19  24  29
 5  10  15  20  25  30

This seems useless, but not really. What we have just introduced is a mechanism called slicing, in which we give a range of values that we want. It just so happens that : in this context is a shortcut for begin:end.

B[begin:end, 3]

5-element Vector{Int64}:
 11
 12
 13
 14
 15

As our B[:,:] example shows, we can actually use two ranges:

B[1:2, 1:2]

2×2 Matrix{Int64}:
 1  6
 2  7

We can also use ranges that are defined relative to the start or the end of the array alongside this dimension:

B[(begin + 1):(end - 1), begin:(end - 2)]

3×4 Matrix{Int64}:
 2  7  12  17
 3  8  13  18
 4  9  14  19

This is a very interesting way to access elements, and it also works with vectors:

u = [1, 2, 3, 4, 5, 6]
u[(begin + 2):(end - 2)]

2-element Vector{Int64}:
 3
 4

In fact, it works with arrays of any dimensions, as you need to specificy one range for each dimension. You can try with an array X = reshape(Array(1:27), (3,3,3)), to get X[1:2,:,2:3].

A final piece of information to know before moving forward is that we can use slices to rapidly change a lot of values in an array. Let’s imagine a stochastic block matrix where we have two blocks of 5×5 with probability values, and the rest of the matrix is 0:

SBM = round.(rand(Float64, 10, 10); digits = 1);

We can slice our way through this matrix to replace the parts that we want to set to 0:

SBM[6:end, begin:5] .= 0.0;
SBM[begin:5, 6:end] .= 0.0;

We use .= to replace multiple values at once. This is a specific bit of Julia notation that we will take a deep dive into in a later module. For now, keep in mind that .= will replace more than one thing.

SBM

10×10 Matrix{Float64}:
 0.1  0.5  0.8  0.9  0.7  0.0  0.0  0.0  0.0  0.0
 0.4  0.9  0.1  0.7  0.4  0.0  0.0  0.0  0.0  0.0
 0.8  1.0  0.4  0.8  0.5  0.0  0.0  0.0  0.0  0.0
 0.7  0.7  0.1  0.5  0.7  0.0  0.0  0.0  0.0  0.0
 0.1  0.9  0.0  0.6  0.4  0.0  0.0  0.0  0.0  0.0
 0.0  0.0  0.0  0.0  0.0  0.7  0.8  0.3  0.2  0.9
 0.0  0.0  0.0  0.0  0.0  0.1  0.7  0.5  0.3  0.8
 0.0  0.0  0.0  0.0  0.0  0.1  0.3  0.8  0.2  0.4
 0.0  0.0  0.0  0.0  0.0  0.6  0.1  0.1  0.4  0.5
 0.0  0.0  0.0  0.0  0.0  0.6  0.0  0.8  0.9  0.1

At the end of this module, we have covered the ways we can index positions in an array, and how we can use slices to access multiple rows/columns at once, and if need be over-write them. This is a strong foundation to start building more ambitious code, as we will start to do in the next section!