Multi-dimensional Arrays

Julia, like most technical computing languages, provides a first-class array implementation. Most technical computing languages pay a lot of attention to their array implementation at the expense of other containers. Julia does not treat arrays in any special way. The array library is implemented almost completely in Julia itself, and derives its performance from the compiler, just like any other code written in Julia. As such, it's also possible to define custom array types by inheriting from AbstractArray. See the manual section on the AbstractArray interface for more details on implementing a custom array type.

An array is a collection of objects stored in a multi-dimensional grid. In the most general case, an array may contain objects of type Any. For most computational purposes, arrays should contain objects of a more specific type, such as Float64 or Int32.

In general, unlike many other technical computing languages, Julia does not expect programs to be written in a vectorized style for performance. Julia's compiler uses type inference and generates optimized code for scalar array indexing, allowing programs to be written in a style that is convenient and readable, without sacrificing performance, and using less memory at times.

In Julia, all arguments to functions are passed by sharing (i.e. by pointers). Some technical computing languages pass arrays by value, and while this prevents accidental modification by callees of a value in the caller, it makes avoiding unwanted copying of arrays difficult. By convention, a function name ending with a ! indicates that it will mutate or destroy the value of one or more of its arguments (see, for example, sort and sort!. Callees must make explicit copies to ensure that they don't modify inputs that they don't intend to change. Many non- mutating functions are implemented by calling a function of the same name with an added ! at the end on an explicit copy of the input, and returning that copy.

Basic Functions

Construction and Initialization

Many functions for constructing and initializing arrays are provided. In the following list of such functions, calls with a dims... argument can either take a single tuple of dimension sizes or a series of dimension sizes passed as a variable number of arguments. Most of these functions also accept a first input T, which is the element type of the array. If the type T is omitted it will default to Float64.

The syntax [A, B, C, ...] constructs a 1-d array (vector) of its arguments. If all arguments have a common promotion type then they get converted to that type using convert.

To see the various ways we can pass dimensions to these constructors, consider the following examples:

julia> zeros(Int8, 2, 2)
2×2 Array{Int8,2}:
 0  0
 0  0

julia> zeros(Int8, (2, 2))
2×2 Array{Int8,2}:
 0  0
 0  0

julia> zeros((2, 2))
2×2 Array{Float64,2}:
 0.0  0.0
 0.0  0.0

Here, (2, 2) is a Tuple.

Concatenation

Arrays can be constructed and also concatenated using the following functions:

Scalar values passed to these functions are treated as 1-element arrays. For example,

julia> vcat([1, 2], 3)
3-element Array{Int64,1}:
 1
 2
 3

julia> hcat([1 2], 3)
1×3 Array{Int64,2}:
 1  2  3

The concatenation functions are used so often that they have special syntax:

hvcat concatenates in both dimension 1 (with semicolons) and dimension 2 (with spaces). Consider these examples of this syntax:

julia> [[1; 2]; [3, 4]]
4-element Array{Int64,1}:
 1
 2
 3
 4

julia> [[1 2] [3 4]]
1×4 Array{Int64,2}:
 1  2  3  4

julia> [[1 2]; [3 4]]
2×2 Array{Int64,2}:
 1  2
 3  4

Typed array initializers

An array with a specific element type can be constructed using the syntax T[A, B, C, ...]. This will construct a 1-d array with element type T, initialized to contain elements A, B, C, etc. For example, Any[x, y, z] constructs a heterogeneous array that can contain any values.

Concatenation syntax can similarly be prefixed with a type to specify the element type of the result.

julia> [[1 2] [3 4]]
1×4 Array{Int64,2}:
 1  2  3  4

julia> Int8[[1 2] [3 4]]
1×4 Array{Int8,2}:
 1  2  3  4

Comprehensions

Comprehensions provide a general and powerful way to construct arrays. Comprehension syntax is similar to set construction notation in mathematics:

A = [ F(x,y,...) for x=rx, y=ry, ... ]

The meaning of this form is that F(x,y,...) is evaluated with the variables x, y, etc. taking on each value in their given list of values. Values can be specified as any iterable object, but will commonly be ranges like 1:n or 2:(n-1), or explicit arrays of values like [1.2, 3.4, 5.7]. The result is an N-d dense array with dimensions that are the concatenation of the dimensions of the variable ranges rx, ry, etc. and each F(x,y,...) evaluation returns a scalar.

The following example computes a weighted average of the current element and its left and right neighbor along a 1-d grid. :

julia> x = rand(8)
8-element Array{Float64,1}:
 0.843025
 0.869052
 0.365105
 0.699456
 0.977653
 0.994953
 0.41084
 0.809411

julia> [ 0.25*x[i-1] + 0.5*x[i] + 0.25*x[i+1] for i=2:length(x)-1 ]
6-element Array{Float64,1}:
 0.736559
 0.57468
 0.685417
 0.912429
 0.8446
 0.656511

The resulting array type depends on the types of the computed elements. In order to control the type explicitly, a type can be prepended to the comprehension. For example, we could have requested the result in single precision by writing:

Float32[ 0.25*x[i-1] + 0.5*x[i] + 0.25*x[i+1] for i=2:length(x)-1 ]

Generator Expressions

Comprehensions can also be written without the enclosing square brackets, producing an object known as a generator. This object can be iterated to produce values on demand, instead of allocating an array and storing them in advance (see Iteration). For example, the following expression sums a series without allocating memory:

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

When writing a generator expression with multiple dimensions inside an argument list, parentheses are needed to separate the generator from subsequent arguments:

julia> map(tuple, 1/(i+j) for i=1:2, j=1:2, [1:4;])
ERROR: syntax: invalid iteration specification

All comma-separated expressions after for are interpreted as ranges. Adding parentheses lets us add a third argument to map:

julia> map(tuple, (1/(i+j) for i=1:2, j=1:2), [1 3; 2 4])
2×2 Array{Tuple{Float64,Int64},2}:
 (0.5, 1)       (0.333333, 3)
 (0.333333, 2)  (0.25, 4)

Generators are implemented via inner functions. As in other cases of inner functions in the language, variables from the enclosing scope can be "captured" in the inner function. For example, sum(p[i] - q[i] for i=1:n) captures the three variables p, q and n from the enclosing scope. Captured variables can present performance challenges described in performance tips.

Ranges in generators and comprehensions can depend on previous ranges by writing multiple for keywords:

julia> [(i,j) for i=1:3 for j=1:i]
6-element Array{Tuple{Int64,Int64},1}:
 (1, 1)
 (2, 1)
 (2, 2)
 (3, 1)
 (3, 2)
 (3, 3)

In such cases, the result is always 1-d.

Generated values can be filtered using the if keyword:

julia> [(i,j) for i=1:3 for j=1:i if i+j == 4]
2-element Array{Tuple{Int64,Int64},1}:
 (2, 2)
 (3, 1)

Indexing

The general syntax for indexing into an n-dimensional array A is:

X = A[I_1, I_2, ..., I_n]

where each I_k may be a scalar integer, an array of integers, or any other supported index. This includes Colon (:) to select all indices within the entire dimension, ranges of the form a:c or a:b:c to select contiguous or strided subsections, and arrays of booleans to select elements at their true indices.

If all the indices are scalars, then the result X is a single element from the array A. Otherwise, X is an array with the same number of dimensions as the sum of the dimensionalities of all the indices.

If all indices are vectors, for example, then the shape of X would be (length(I_1), length(I_2), ..., length(I_n)), with location (i_1, i_2, ..., i_n) of X containing the value A[I_1[i_1], I_2[i_2], ..., I_n[i_n]].

Example:

julia> A = reshape(collect(1:16), (2, 2, 2, 2))
2×2×2×2 Array{Int64,4}:
[:, :, 1, 1] =
 1  3
 2  4

[:, :, 2, 1] =
 5  7
 6  8

[:, :, 1, 2] =
  9  11
 10  12

[:, :, 2, 2] =
 13  15
 14  16

julia> A[1, 2, 1, 1] # all scalar indices
3

julia> A[[1, 2], [1], [1, 2], [1]] # all vector indices
2×1×2×1 Array{Int64,4}:
[:, :, 1, 1] =
 1
 2

[:, :, 2, 1] =
 5
 6

julia> A[[1, 2], [1], [1, 2], 1] # a mix of index types
2×1×2 Array{Int64,3}:
[:, :, 1] =
 1
 2

[:, :, 2] =
 5
 6

Note how the size of the resulting array is different in the last two cases.

If I_1 is changed to a two-dimensional matrix, then X becomes an n+1-dimensional array of shape (size(I_1, 1), size(I_1, 2), length(I_2), ..., length(I_n)). The matrix adds a dimension.

Example:

julia> A = reshape(collect(1:16), (2, 2, 2, 2));

julia> A[[1 2; 1 2]]
2×2 Array{Int64,2}:
 1  2
 1  2

julia> A[[1 2; 1 2], 1, 2, 1]
2×2 Array{Int64,2}:
 5  6
 5  6

The location (i_1, i_2, i_3, ..., i_{n+1}) contains the value at A[I_1[i_1, i_2], I_2[i_3], ..., I_n[i_{n+1}]]. All dimensions indexed with scalars are dropped. For example, the result of A[2, I, 3] is an array with size size(I). Its ith element is populated by A[2, I[i], 3].

As a special part of this syntax, the end keyword may be used to represent the last index of each dimension within the indexing brackets, as determined by the size of the innermost array being indexed. Indexing syntax without the end keyword is equivalent to a call to getindex:

X = getindex(A, I_1, I_2, ..., I_n)

Example:

julia> x = reshape(1:16, 4, 4)
4×4 reshape(::UnitRange{Int64}, 4, 4) with eltype Int64:
 1  5   9  13
 2  6  10  14
 3  7  11  15
 4  8  12  16

julia> x[2:3, 2:end-1]
2×2 Array{Int64,2}:
 6  10
 7  11

julia> x[1, [2 3; 4 1]]
2×2 Array{Int64,2}:
  5  9
 13  1

Empty ranges of the form n:n-1 are sometimes used to indicate the inter-index location between n-1 and n. For example, the searchsorted function uses this convention to indicate the insertion point of a value not found in a sorted array:

julia> a = [1,2,5,6,7];

julia> searchsorted(a, 3)
3:2

Assignment

The general syntax for assigning values in an n-dimensional array A is:

A[I_1, I_2, ..., I_n] = X

where each I_k may be a scalar integer, an array of integers, or any other supported index. This includes Colon (:) to select all indices within the entire dimension, ranges of the form a:c or a:b:c to select contiguous or strided subsections, and arrays of booleans to select elements at their true indices.

If X is an array, it must have the same number of elements as the product of the lengths of the indices: prod(length(I_1), length(I_2), ..., length(I_n)). The value in location I_1[i_1], I_2[i_2], ..., I_n[i_n] of A is overwritten with the value X[i_1, i_2, ..., i_n]. If X is not an array, its value is written to all referenced locations of A.

Just as in Indexing, the end keyword may be used to represent the last index of each dimension within the indexing brackets, as determined by the size of the array being assigned into. Indexed assignment syntax without the end keyword is equivalent to a call to setindex!:

setindex!(A, X, I_1, I_2, ..., I_n)

Example:

julia> x = collect(reshape(1:9, 3, 3))
3×3 Array{Int64,2}:
 1  4  7
 2  5  8
 3  6  9

julia> x[3, 3] = -9;

julia> x[1:2, 1:2] = [-1 -4; -2 -5];

julia> x
3×3 Array{Int64,2}:
 -1  -4   7
 -2  -5   8
  3   6  -9

Supported index types

In the expression A[I_1, I_2, ..., I_n], each I_k may be a scalar index, an array of scalar indices, or an object that represents an array of scalar indices and can be converted to such by to_indices:

1. A scalar index. By default this includes:
   • Non-boolean integers
   • CartesianIndex{N}s, which behave like an N-tuple of integers spanning multiple dimensions (see below for more details)
2. An array of scalar indices. This includes:
   • Vectors and multidimensional arrays of integers
   • Empty arrays like [], which select no elements
   • Ranges like a:c or a:b:c, which select contiguous or strided subsections from a to c (inclusive)
   • Any custom array of scalar indices that is a subtype of AbstractArray
   • Arrays of CartesianIndex{N} (see below for more details)
3. An object that represents an array of scalar indices and can be converted to such by to_indices. By default this includes:
   • Colon() (:), which represents all indices within an entire dimension or across the entire array
   • Arrays of booleans, which select elements at their true indices (see below for more details)

Some examples:

julia> A = reshape(collect(1:2:18), (3, 3))
3×3 Array{Int64,2}:
 1   7  13
 3   9  15
 5  11  17

julia> A[4]
7

julia> A[[2, 5, 8]]
3-element Array{Int64,1}:
  3
  9
 15

julia> A[[1 4; 3 8]]
2×2 Array{Int64,2}:
 1   7
 5  15

julia> A[[]]
0-element Array{Int64,1}

julia> A[1:2:5]
3-element Array{Int64,1}:
 1
 5
 9

julia> A[2, :]
3-element Array{Int64,1}:
  3
  9
 15

julia> A[:, 3]
3-element Array{Int64,1}:
 13
 15
 17

Cartesian indices

The special CartesianIndex{N} object represents a scalar index that behaves like an N-tuple of integers spanning multiple dimensions. For example:

julia> A = reshape(1:32, 4, 4, 2);

julia> A[3, 2, 1]
7

julia> A[CartesianIndex(3, 2, 1)] == A[3, 2, 1] == 7
true

Considered alone, this may seem relatively trivial; CartesianIndex simply gathers multiple integers together into one object that represents a single multidimensional index. When combined with other indexing forms and iterators that yield CartesianIndexes, however, this can lead directly to very elegant and efficient code. See Iteration below, and for some more advanced examples, see this blog post on multidimensional algorithms and iteration.

Arrays of CartesianIndex{N} are also supported. They represent a collection of scalar indices that each span N dimensions, enabling a form of indexing that is sometimes referred to as pointwise indexing. For example, it enables accessing the diagonal elements from the first "page" of A from above:

julia> page = A[:,:,1]
4×4 Array{Int64,2}:
 1  5   9  13
 2  6  10  14
 3  7  11  15
 4  8  12  16

julia> page[[CartesianIndex(1,1),
             CartesianIndex(2,2),
             CartesianIndex(3,3),
             CartesianIndex(4,4)]]
4-element Array{Int64,1}:
  1
  6
 11
 16

This can be expressed much more simply with dot broadcasting and by combining it with a normal integer index (instead of extracting the first page from A as a separate step). It can even be combined with a : to extract both diagonals from the two pages at the same time:

julia> A[CartesianIndex.(axes(A, 1), axes(A, 2)), 1]
4-element Array{Int64,1}:
  1
  6
 11
 16

julia> A[CartesianIndex.(axes(A, 1), axes(A, 2)), :]
4×2 Array{Int64,2}:
  1  17
  6  22
 11  27
 16  32

Logical indexing

Often referred to as logical indexing or indexing with a logical mask, indexing by a boolean array selects elements at the indices where its values are true. Indexing by a boolean vector B is effectively the same as indexing by the vector of integers that is returned by findall(B). Similarly, indexing by a N-dimensional boolean array is effectively the same as indexing by the vector of CartesianIndex{N}s where its values are true. A logical index must be a vector of the same length as the dimension it indexes into, or it must be the only index provided and match the size and dimensionality of the array it indexes into. It is generally more efficient to use boolean arrays as indices directly instead of first calling findall.

julia> x = reshape(1:16, 4, 4)
4×4 reshape(::UnitRange{Int64}, 4, 4) with eltype Int64:
 1  5   9  13
 2  6  10  14
 3  7  11  15
 4  8  12  16

julia> x[[false, true, true, false], :]
2×4 Array{Int64,2}:
 2  6  10  14
 3  7  11  15

julia> mask = map(ispow2, x)
4×4 Array{Bool,2}:
  true  false  false  false
  true  false  false  false
 false  false  false  false
  true   true  false   true

julia> x[mask]
5-element Array{Int64,1}:
  1
  2
  4
  8
 16

Iteration

The recommended ways to iterate over a whole array are

for a in A
    # Do something with the element a
end

for i in eachindex(A)
    # Do something with i and/or A[i]
end

The first construct is used when you need the value, but not index, of each element. In the second construct, i will be an Int if A is an array type with fast linear indexing; otherwise, it will be a CartesianIndex:

julia> A = rand(4,3);

julia> B = view(A, 1:3, 2:3);

julia> for i in eachindex(B)
           @show i
       end
i = CartesianIndex(1, 1)
i = CartesianIndex(2, 1)
i = CartesianIndex(3, 1)
i = CartesianIndex(1, 2)
i = CartesianIndex(2, 2)
i = CartesianIndex(3, 2)

In contrast with for i = 1:length(A), iterating with eachindex provides an efficient way to iterate over any array type.

Array traits

If you write a custom AbstractArray type, you can specify that it has fast linear indexing using

Base.IndexStyle(::Type{<:MyArray}) = IndexLinear()

This setting will cause eachindex iteration over a MyArray to use integers. If you don't specify this trait, the default value IndexCartesian() is used.

Array and Vectorized Operators and Functions

The following operators are supported for arrays:

1. Unary arithmetic – -, +
2. Binary arithmetic – -, +, *, /, \, ^
3. Comparison – ==, !=, ≈ (isapprox), ≉

Most of the binary arithmetic operators listed above also operate elementwise when one argument is scalar: -, +, and * when either argument is scalar, and / and \ when the denominator is scalar. For example, [1, 2] + 3 == [4, 5] and [6, 4] / 2 == [3, 2].

Additionally, to enable convenient vectorization of mathematical and other operations, Julia provides the dot syntaxf.(args...), e.g. sin.(x) or min.(x,y), for elementwise operations over arrays or mixtures of arrays and scalars (a Broadcasting operation); these have the additional advantage of "fusing" into a single loop when combined with other dot calls, e.g. sin.(cos.(x)).

Also, every binary operator supports a dot version that can be applied to arrays (and combinations of arrays and scalars) in such fused broadcasting operations, e.g. z .== sin.(x .* y).

Note that comparisons such as == operate on whole arrays, giving a single boolean answer. Use dot operators like .== for elementwise comparisons. (For comparison operations like <, only the elementwise .< version is applicable to arrays.)

Also notice the difference between max.(a,b), which broadcasts max elementwise over a and b, and maximum(a), which finds the largest value within a. The same relationship holds for min.(a,b) and minimum(a).

Broadcasting

It is sometimes useful to perform element-by-element binary operations on arrays of different sizes, such as adding a vector to each column of a matrix. An inefficient way to do this would be to replicate the vector to the size of the matrix:

julia> a = rand(2,1); A = rand(2,3);

julia> repeat(a,1,3)+A
2×3 Array{Float64,2}:
 1.20813  1.82068  1.25387
 1.56851  1.86401  1.67846

This is wasteful when dimensions get large, so Julia offers broadcast, which expands singleton dimensions in array arguments to match the corresponding dimension in the other array without using extra memory, and applies the given function elementwise:

julia> broadcast(+, a, A)
2×3 Array{Float64,2}:
 1.20813  1.82068  1.25387
 1.56851  1.86401  1.67846

julia> b = rand(1,2)
1×2 Array{Float64,2}:
 0.867535  0.00457906

julia> broadcast(+, a, b)
2×2 Array{Float64,2}:
 1.71056  0.847604
 1.73659  0.873631

Dotted operators such as .+ and .* are equivalent to broadcast calls (except that they fuse, as described below). There is also a broadcast! function to specify an explicit destination (which can also be accessed in a fusing fashion by .= assignment). Moreover, f.(args...) is equivalent to broadcast(f, args...), providing a convenient syntax to broadcast any function (dot syntax). Nested "dot calls" f.(...) (including calls to .+ etcetera) automatically fuse into a single broadcast call.

Additionally, broadcast is not limited to arrays (see the function documentation), it also handles tuples and treats any argument that is not an array, tuple or Ref (except for Ptr) as a "scalar".

julia> convert.(Float32, [1, 2])
2-element Array{Float32,1}:
 1.0
 2.0

julia> ceil.((UInt8,), [1.2 3.4; 5.6 6.7])
2×2 Array{UInt8,2}:
 0x02  0x04
 0x06  0x07

julia> string.(1:3, ". ", ["First", "Second", "Third"])
3-element Array{String,1}:
 "1. First"
 "2. Second"
 "3. Third"

Implementation

The base array type in Julia is the abstract type AbstractArray{T,N}. It is parametrized by the number of dimensions N and the element type T. AbstractVector and AbstractMatrix are aliases for the 1-d and 2-d cases. Operations on AbstractArray objects are defined using higher level operators and functions, in a way that is independent of the underlying storage. These operations generally work correctly as a fallback for any specific array implementation.

The AbstractArray type includes anything vaguely array-like, and implementations of it might be quite different from conventional arrays. For example, elements might be computed on request rather than stored. However, any concrete AbstractArray{T,N} type should generally implement at least size(A) (returning an Int tuple), getindex(A,i) and getindex(A,i1,...,iN); mutable arrays should also implement setindex!. It is recommended that these operations have nearly constant time complexity, or technically Õ(1) complexity, as otherwise some array functions may be unexpectedly slow. Concrete types should also typically provide a similar(A,T=eltype(A),dims=size(A)) method, which is used to allocate a similar array for copy and other out-of-place operations. No matter how an AbstractArray{T,N} is represented internally, T is the type of object returned by integer indexing (A[1, ..., 1], when A is not empty) and N should be the length of the tuple returned by size. For more details on defining custom AbstractArray implementations, see the array interface guide in the interfaces chapter.

DenseArray is an abstract subtype of AbstractArray intended to include all arrays where elements are stored contiguously in column-major order (see additional notes in Performance Tips). The Array type is a specific instance of DenseArrayVector and Matrix are aliases for the 1-d and 2-d cases. Very few operations are implemented specifically for Array beyond those that are required for all AbstractArrayss; much of the array library is implemented in a generic manner that allows all custom arrays to behave similarly.

SubArray is a specialization of AbstractArray that performs indexing by sharing memory with the original array rather than by copying it. A SubArray is created with the view function, which is called the same way as getindex (with an array and a series of index arguments). The result of view looks the same as the result of getindex, except the data is left in place. view stores the input index vectors in a SubArray object, which can later be used to index the original array indirectly. By putting the @views macro in front of an expression or block of code, any array[...] slice in that expression will be converted to create a SubArray view instead.

BitArrays are space-efficient "packed" boolean arrays, which store one bit per boolean value. They can be used similarly to Array{Bool} arrays (which store one byte per boolean value), and can be converted to/from the latter via Array(bitarray) and BitArray(array), respectively.

A "strided" array is stored in memory with elements laid out in regular offsets such that an instance with a supported isbits element type can be passed to external C and Fortran functions that expect this memory layout. Strided arrays must define a strides(A) method that returns a tuple of "strides" for each dimension; a provided stride(A,k) method accesses the kth element within this tuple. Increasing the index of dimension k by 1 should increase the index i of getindex(A,i) by stride(A,k). If a pointer conversion method Base.unsafe_convert(Ptr{T}, A) is provided, the memory layout must correspond in the same way to these strides. DenseArray is a very specific example of a strided array where the elements are arranged contiguously, thus it provides its subtypes with the approporiate definition of strides. More concrete examples can be found within the interface guide for strided arrays. StridedVector and StridedMatrix are convenient aliases for many of the builtin array types that are considered strided arrays, allowing them to dispatch to select specialized implementations that call highly tuned and optimized BLAS and LAPACK functions using just the pointer and strides.

The following example computes the QR decomposition of a small section of a larger array, without creating any temporaries, and by calling the appropriate LAPACK function with the right leading dimension size and stride parameters.

julia> a = rand(10, 10)
10×10 Array{Float64,2}:
 0.517515  0.0348206  0.749042   0.0979679  …  0.75984     0.950481   0.579513
 0.901092  0.873479   0.134533   0.0697848     0.0586695   0.193254   0.726898
 0.976808  0.0901881  0.208332   0.920358      0.288535    0.705941   0.337137
 0.657127  0.0317896  0.772837   0.534457      0.0966037   0.700694   0.675999
 0.471777  0.144969   0.0718405  0.0827916     0.527233    0.173132   0.694304
 0.160872  0.455168   0.489254   0.827851   …  0.62226     0.0995456  0.946522
 0.291857  0.769492   0.68043    0.629461      0.727558    0.910796   0.834837
 0.775774  0.700731   0.700177   0.0126213     0.00822304  0.327502   0.955181
 0.9715    0.64354    0.848441   0.241474      0.591611    0.792573   0.194357
 0.646596  0.575456   0.0995212  0.038517      0.709233    0.477657   0.0507231

julia> b = view(a, 2:2:8,2:2:4)
4×2 view(::Array{Float64,2}, 2:2:8, 2:2:4) with eltype Float64:
 0.873479   0.0697848
 0.0317896  0.534457
 0.455168   0.827851
 0.700731   0.0126213

julia> (q, r) = qr(b);

julia> q
4×4 LinearAlgebra.QRCompactWYQ{Float64,Array{Float64,2}}:
 -0.722358    0.227524  -0.247784    -0.604181
 -0.0262896  -0.575919  -0.804227     0.144377
 -0.376419   -0.75072    0.540177    -0.0541979
 -0.579497    0.230151  -0.00552346   0.781782

julia> r
2×2 Array{Float64,2}:
 -1.20921  -0.383393
  0.0      -0.910506