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Loops

While if conditions are evaluated only once, loops are assessed multiple times.

for and while loops

As in other languages, Julia supports two basic constructs for repeated evaluation: while andfor loops. Loops are useful to repeat the same computation multiple times with different values. A typical example is performing operations on array elements.

The while loop evaluates the condition, and as long it remains true, it evaluates the body of the while loop. If the condition is false, the while loop is terminated. If the condition is false before the first iteration, the whole while loop is skipped.

julia> i = 1
1

julia> while i <= 5
           @show i
           i += 1
       end
i = 1
i = 2
i = 3
i = 4
i = 5

The @show macro in this example prints the results of an expression. It can also be used to print multiple variables at once.

julia> a, b, c = 1, "hello", :world;

julia> @show (a, b, c);
(a, b, c) = (1, "hello", :world)

The for loops are created similarly to Matlab. The following example iterates over all integers from 1 to 5, and in each iteration, we use the @show macro to print the current value of the variable i.

julia> for i in 1:5
           @show i
       end
i = 1
i = 2
i = 3
i = 4
i = 5
An alternative notation for for loops:

There are two alternative notations for the for loop. It is possible to use the = or ∈ symbol instead of the in keyword.

julia> for i = 1:5
           @show i
       end
i = 1
i = 2
i = 3
i = 4
i = 5

However, it is better to use the in keyword to improve code readability. Regardless of which notation is used, it is essential to be consistent and use the same notation in all for loops.

In Julia (similarly to Python), it is possible to loop not only over ranges but over any iterable object such as arrays or tuples. This is advantageous because it allows getting elements of iterable objects directly without using indices.

julia> persons = ["Alice", "Bob", "Carla", "Daniel"];

julia> for name in persons
           println("Hi, my name is $name.")
       end
Hi, my name is Alice.
Hi, my name is Bob.
Hi, my name is Carla.
Hi, my name is Daniel.

It is also possible to iterate over other data structures such as dictionaries. In such a case, we get a tuple of the key and the corresponding value in each iteration.

julia> persons = Dict("Alice" => 10, "Bob" => 23, "Carla" => 14, "Daniel" => 34);

julia> for (name, age) in persons
           println("Hi, my name is $name and I am $age old.")
       end
Hi, my name is Carla and I am 14 old.
Hi, my name is Alice and I am 10 old.
Hi, my name is Daniel and I am 34 old.
Hi, my name is Bob and I am 23 old.
Exercise:

Use for or while loop to print all integers between 1 and 100 which can be divided by both 3 and 7.

Hint: use the mod function.

Solution:

First, we need to check if a given integer is divisible by both 3 and 7. This can be performed using the mod function in combination with the if-else statement as follows:

julia> i = 21
21

julia> if mod(i, 3) == 0 && mod(i, 7) == 0
           println("$(i) is divisible by 3 and 7")
       end
21 is divisible by 3 and 7

or using the short-circuit evaluation

julia> i = 21
21

julia> mod(i, 3) == mod(i, 7) == 0 && println("$(i) is divisible by 3 and 7")
21 is divisible by 3 and 7

When we know how to check the conditions, it is easy to write a for loop to iterate over integers from 1 to 100.

julia> for i in 1:100
           mod(i, 3) == mod(i, 7) == 0 && @show i
       end
i = 21
i = 42
i = 63
i = 84

A while loop can be created in a similar way

julia> i = 0;

julia> while i <= 100
           i += 1
           mod(i, 3) == mod(i, 7) == 0 && @show i
       end
i = 21
i = 42
i = 63
i = 84

The for loop should be used here because the range is known before-hand and unlike the while loop, it does not require to initialize i.

break and continue

Sometimes it is useful to stop the loop when some condition is satisfied. This is done by the break keyword. In the following example, the loop iterates over the range from 1 to 10 and breaks when i == 4, i.e., only the first three numbers are printed.

julia> for i in 1:10
           i == 4 && break
           @show i
       end
i = 1
i = 2
i = 3

Another useful feature is to skip elements using the continue keyword. The following code prints all even numbers from 1 to 10.

julia> for i in 1:10
           mod(i, 2) == 0 || continue
           @show i
       end
i = 2
i = 4
i = 6
i = 8
i = 10

The code after the continue keyword is not evaluated.

Exercise:

Rewrite the code from the exercise above. Use a combination of the while loop and the keyword continue to print all integers between 1 and 100 divisible by both 3 and 7. In the declaration of the while loop use the true value instead of a condition. Use the break keyword and a proper condition to terminate the loop.

Solution:

The true value creates an infinite loop, i.e., it is necessary to end the loop with the break keyword. Because the variable i represents an integer and we want to iterate over integers between 1 and 100, the correct termination condition is i > 100.

julia> i = 0;

julia> while true
           i += 1
           i > 100 && break
           mod(i, 3) == mod(i, 7) == 0 || continue
           @show i
       end
i = 21
i = 42
i = 63
i = 84

We used the short-circuit evaluation to break the loop. To check that the integer is divisible, we use the same condition as before. However, we must use || instead of && because we want to use the continue keyword.

Nested loops

In Julia, nested loops can be created in a similar way as in other languages.

julia> for i in 1:3
           for j in i:3
               @show (i, j)
           end
       end
(i, j) = (1, 1)
(i, j) = (1, 2)
(i, j) = (1, 3)
(i, j) = (2, 2)
(i, j) = (2, 3)
(i, j) = (3, 3)

The range of the inner loop depends on the variable i from the outer loop. This style of writing nested loops is typical in other languages. Julia allows for an additional shorter syntax:

julia> for i in 1:3, j in i:3
           @show (i, j)
       end
(i, j) = (1, 1)
(i, j) = (1, 2)
(i, j) = (1, 3)
(i, j) = (2, 2)
(i, j) = (2, 3)
(i, j) = (3, 3)

Even though the output is the same, this syntax is not equivalent to the previous one. The main difference is when using the break keyword. If we use the traditional syntax and the break keyword inside the inner loop, it breaks only the inner loop. On the other hand, if we use the shorter syntax, the break keyword breaks both loops.

julia> for i in 1:3
           for j in i:10
               j > 3 && break
               @show (i, j)
           end
       end
(i, j) = (1, 1)
(i, j) = (1, 2)
(i, j) = (1, 3)
(i, j) = (2, 2)
(i, j) = (2, 3)
(i, j) = (3, 3)

julia> for i in 1:3, j in i:10
           j > 3 && break
           @show (i, j)
       end
(i, j) = (1, 1)
(i, j) = (1, 2)
(i, j) = (1, 3)

There are other limitations of the shorter syntax, such as the impossibility to perform any operation outside the inner loop. Nevertheless, it is a useful syntax in many cases.

Exercise:

Use nested loops to create a matrix with elements given by the formula

\[A_{i, j} = \frac{1}{2}\exp\left\{\frac{1}{2} (x_{i}^2 - y_{j}^2) \right\} \quad i \in \{1, 2, 3\}, \quad j \in \{1, 2, 3, 4\},\]

where $x \in \{0.4, 2.3, 4.6\}$ and $y \in \{1.4, -3.1, 2.4, 5.2\}$.

Bonus: try to create the same matrix in a more effective way.

Solution:

First, we have to define vectors x and y, and an empty array of the proper size and element type to use in nested loops.

x = [0.4, 2.3, 4.6]
y = [1.4, -3.1, 2.4, 5.2]
A = zeros(Float64, length(x), length(y))

The element type specification can be omitted since the default value type is Float64. Now we have to use proper indices to fill A. In this case, we use the indices 1:length(x) for x and 1:length(y) for y.

julia> for i in 1:length(x), j in 1:length(y)
           A[i, j] = exp((x[i]^2 - y[j]^2)/2)/2
       end

julia> A
3×4 Matrix{Float64}:
    0.203285    0.00443536     0.030405  7.27867e-7
    2.64284     0.0576626      0.395285  9.46275e-6
 7382.39      161.072       1104.17      0.0264329

There are more efficient ways to create this array. The one way is to use broadcasting.

julia> y_row = y'
1×4 adjoint(::Vector{Float64}) with eltype Float64:
 1.4  -3.1  2.4  5.2

julia> A = @. exp((x^2 - y_row^2)/2)/2
3×4 Matrix{Float64}:
    0.203285    0.00443536     0.030405  7.27867e-7
    2.64284     0.0576626      0.395285  9.46275e-6
 7382.39      161.072       1104.17      0.0264329

We use the @ . macro to perform all operations elementwise. Since x is a column vector and y_row is a row vector, x - y_row uses broadcasting to create a matrix.

The third way to create this matrix is to use list comprehension. Due to its importance, we dedicate a whole section to it.

List comprehension

As we mentioned in the last exercise, one way to create an array with prescribed elements is to use list comprehension. Comprehensions provide a general and powerful way to construct arrays, and the syntax is similar to the set construction notation from mathematics.

A = [f(x, y, ...) for x in X, y in Y, ...]

The previous example reads: The function f will be evaluated for each combination of elements of iterable objects X, Y, etc. The result will be an n-dimensional array of size (length(X), length(Y), ...). Returning to the previous exercise, we can create the required array as follows:

julia> X = [0.4, 2.3, 4.6];

julia> Y = [1.4, -3.1, 2.4, 5.2];

julia> A = [exp((x^2 - y^2)/2)/2 for x in X, y in Y]
3×4 Matrix{Float64}:
    0.203285    0.00443536     0.030405  7.27867e-7
    2.64284     0.0576626      0.395285  9.46275e-6
 7382.39      161.072       1104.17      0.0264329

The resulting array type depends on the types of the computed elements. A type can be prepended to the comprehension to control the type explicitly.

julia> A = Float32[exp((x^2 - y^2)/2)/2 for x in X, y in Y]
3×4 Matrix{Float32}:
    0.203285    0.00443536     0.030405  7.27867f-7
    2.64284     0.0576626      0.395285  9.46275f-6
 7382.39      161.072       1104.17      0.0264329

A handy feature is the possibility to filter values when creating list comprehensions by the if keyword. In such a case, the result will always be a vector. In the next example, we create a vector of tuples (x, y, x + y), where x + y < 5.

julia> [(x, y, x + y)  for x in 1:10, y in 1:10 if x + y < 5]
6-element Vector{Tuple{Int64, Int64, Int64}}:
 (1, 1, 2)
 (2, 1, 3)
 (3, 1, 4)
 (1, 2, 3)
 (2, 2, 4)
 (1, 3, 4)
Exercise:

Use the list comprehension to create a vector of all integers from 1 to 100 divisible by 3 and 7 simultaneously. What is the sum of all these integers?

Solution:

We can use list comprehension with the same condition that we used in the exercise in the first section.

julia> v = [i for i in 1:100 if mod(i, 3) == mod(i, 7) == 0]
4-element Vector{Int64}:
 21
 42
 63
 84

Then we can use the sum function to get their sum.

julia> sum(v)
210

Generator expressions

List comprehensions can also be written without the enclosing square brackets. This produces an object known as a generator. When iterating over a generator, the values are generated on demand instead of pre-allocating an array. For example, the following expression sums a series without allocating the full array in memory.

julia> gen = (1/n^2 for n in 1:1000);

julia> sum(gen)
1.6439345666815615

julia> sum(1/n^2 for n in 1:1000)
1.6439345666815615

It is possible to write nested list comprehensions and generators. The syntax is similar to writing nested loops.

julia> [(i,j) for i in 1:3, j in 1:2]
3×2 Matrix{Tuple{Int64, Int64}}:
 (1, 1)  (1, 2)
 (2, 1)  (2, 2)
 (3, 1)  (3, 2)

julia> gen = ((i,j) for i in 1:3, j in 1:2);

julia> collect(gen)
3×2 Matrix{Tuple{Int64, Int64}}:
 (1, 1)  (1, 2)
 (2, 1)  (2, 2)
 (3, 1)  (3, 2)

Iterables may refer to outer loop variables. However, in such a case, it is necessary to use the for keyword before each iterable statement, and the result will be a vector.

julia> gen = ((i,j) for i in 1:3 for j in 1:i);

julia> collect(gen)
6-element Vector{Tuple{Int64, Int64}}:
 (1, 1)
 (2, 1)
 (2, 2)
 (3, 1)
 (3, 2)
 (3, 3)

Generated values can also be filtered using the if keyword. Similarly to list comprehensions, the result in such a case is a vector.

julia> gen = ((i,j) for i in 1:3 for j in 1:i if i+j == 4);

julia> collect(gen)
2-element Vector{Tuple{Int64, Int64}}:
 (2, 2)
 (3, 1)
Exercise:

Use a generator to sum the square of all integers from 1 to 100, which are divisible by 3 and 7 simultaneously.

Solution:

There are two ways how to solve this exercise. The first one creates a generator and then uses the sum function.

julia> gen = (i^2 for i in 1:100 if mod(i, 3) == mod(i, 7) == 0);

julia> typeof(gen)
Base.Generator{Base.Iterators.Filter{var"#2#4",UnitRange{Int64}},var"#1#3"}

julia> sum(gen)
13230

It is worth noting that gen is a Generator object and not an array. The second way uses the shorter syntax that allows us to write a generator inside the sum function.

julia> sum(i^2 for i in 1:100 if mod(i, 3) == mod(i, 7) == 0)
13230

Iterators

Many structures are iterable in Julia. However, it is not sufficient to iterate only over elements of a structure in many cases. Consider the situation when we have the following array, and we want to iterate over all its elements and print all indices and the corresponding values.

julia> A = [2.3 4.5; 6.7 7.1]
2×2 Matrix{Float64}:
 2.3  4.5
 6.7  7.1

julia> for i in 1:length(A)
           println("i = $(i) and A[i] = $(A[i])")
       end
i = 1 and A[i] = 2.3
i = 2 and A[i] = 6.7
i = 3 and A[i] = 4.5
i = 4 and A[i] = 7.1

There is an even more straightforward way. We can do the same using the enumerate function that returns an iterator (an iterable object that can be iterated in for loops). It produces couples of the form (i, x[i]).

julia> for (i, val) in enumerate(A)
           println("i = $(i) and A[i] = $(val)")
       end
i = 1 and A[i] = 2.3
i = 2 and A[i] = 6.7
i = 3 and A[i] = 4.5
i = 4 and A[i] = 7.1

Other beneficial functions return iterators. For example, the eachcol function returns an iterator that iterates over matrix columns.

julia> for col in eachcol(A)
           println("col = $(col)")
       end
col = [2.3, 6.7]
col = [4.5, 7.1]

Similarly, eachrow returns an iterator that iterates over matrix rows. Another convenient function is the zip function, which zips together multiple iterable objects and iterates over them simultaneously.

julia> for (i, j, k) in zip([1, 4, 2, 5], 2:12, (:a, :b, :c))
           @show (i, j, k)
       end
(i, j, k) = (1, 2, :a)
(i, j, k) = (4, 3, :b)
(i, j, k) = (2, 4, :c)

In this case, the iterable objects were of different lengths. The iterator returned by the zip function will have the same length as the shortest of its inputs. It is also possible to combine these handy functions.

julia> for (i, vals) in enumerate(zip([1, 4, 2, 5], 2:12, (:a, :b, :c)))
           @show (i, vals)
       end
(i, vals) = (1, (1, 2, :a))
(i, vals) = (2, (4, 3, :b))
(i, vals) = (3, (2, 4, :c))
Exercise:

Create a matrix with elements given by the following formula

\[A_{i, j} = \frac{1}{2}\exp\left\{\frac{1}{2} (x_{i}^2 - y_{j}^2) \right\} \quad i \in \{1, 2, 3\}, \; j \in \{1, 2, 3, 4\}\]

where $x \in \{0.4, 2.3, 4.6\}$, $y \in \{1.4, -3.1, 2.4, 5.2\}$. Compute the sum of all elements in each row and print the following message:

Sum of all elements in a row i is i_sum

where i represents row's number and i_sum the sum of all elements in this row. Do the same for each column and print the following message:

Sum of all elements in a column i is i_sum

Hint: use iterators eachcol and eachrow.

Solution:

First, we have to generate the matrix A. It can be done using list comprehension as follows:

X = [0.4, 2.3, 4.6]
Y = [1.4, -3.1, 2.4, 5.2]
A = [exp((x^2 - y^2)/2)/2 for x in X, y in Y]

To compute the sum of each row and print the appropriate message, we use the combination of enumerate and eachrow functions.

julia> for (i, row) in enumerate(eachrow(A))
           println("Sum of all elements in a row $(i) is $(sum(row))")
       end
Sum of all elements in a row 1 is 0.2381259460051036
Sum of all elements in a row 2 is 3.0957940729669864
Sum of all elements in a row 3 is 8647.66342895583

Similarly, to compute the sum of each column and print the appropriate message, we use the combination of enumerate and eachcol functions.

julia> for (i, row) in enumerate(eachcol(A))
           println("Sum of all elements in a column $(i) is $(sum(row))")
       end
Sum of all elements in a column 1 is 7385.236904243371
Sum of all elements in a column 2 is 161.13431527671185
Sum of all elements in a column 3 is 1104.5996863997295
Sum of all elements in a column 4 is 0.026443054989612996