Julia, like most technical computing languages, provides a firstclass 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 multidimensional 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 (compare, 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.
Function  Description 

eltype(A) 
the type of the elements contained in A

length(A) 
the number of elements in A

ndims(A) 
the number of dimensions of A

size(A) 
a tuple containing the dimensions of A

size(A,n) 
the size of A along dimension n

axes(A) 
a tuple containing the valid indices of A

axes(A,n) 
a range expressing the valid indices along dimension n

eachindex(A) 
an efficient iterator for visiting each position in A

stride(A,k) 
the stride (linear index distance between adjacent elements) along dimension k

strides(A) 
a tuple of the strides in each dimension 
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
.
Function  Description 

Array{T}(undef, dims...) 
an uninitialized dense Array

zeros(T, dims...) 
an Array of all zeros 
ones(T, dims...) 
an Array of all ones 
trues(dims...) 
a BitArray with all values true

falses(dims...) 
a BitArray with all values false

reshape(A, dims...) 
an array containing the same data as A , but with different dimensions 
copy(A) 
copy A

deepcopy(A) 
copy A , recursively copying its elements 
similar(A, T, dims...) 
an uninitialized array of the same type as A (dense, sparse, etc.), but with the specified element type and dimensions. The second and third arguments are both optional, defaulting to the element type and dimensions of A if omitted. 
reinterpret(T, A) 
an array with the same binary data as A , but with element type T

rand(T, dims...) 
an Array with random, iid [1] and uniformly distributed values in the halfopen interval $[0, 1)$

randn(T, dims...) 
an Array with random, iid and standard normally distributed values 
Matrix{T}(I, m, n) 
m byn identity matrix 
range(start, stop=stop, length=n) 
range of n linearly spaced elements from start to stop

fill!(A, x) 
fill the array A with the value x

fill(x, dims...) 
an Array filled with the value x

iid, independently and identically distributed.
The syntax [A, B, C, ...]
constructs a 1d array (i.e., a 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, 3) 2×3 Array{Int8,2}: 0 0 0 0 0 0 julia> zeros(Int8, (2, 3)) 2×3 Array{Int8,2}: 0 0 0 0 0 0 julia> zeros((2, 3)) 2×3 Array{Float64,2}: 0.0 0.0 0.0 0.0 0.0 0.0
Here, (2, 3)
is a Tuple
.
Arrays can be constructed and also concatenated using the following functions:
Function  Description 

cat(A...; dims=k) 
concatenate input arrays along dimension(s) k

vcat(A...) 
shorthand for cat(A...; dims=1)

hcat(A...) 
shorthand for cat(A...; dims=2)

Scalar values passed to these functions are treated as 1element arrays. For example,
julia> vcat([1, 2], 3) 3element 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:
Expression  Calls 

[A; B; C; ...] 
vcat 
[A B C ...] 
hcat 
[A B; C D; ...] 
hvcat 
hvcat
concatenates in both dimension 1 (with semicolons) and dimension 2 (with spaces). Consider these examples of this syntax:
julia> [[1; 2]; [3, 4]] 4element 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
An array with a specific element type can be constructed using the syntax T[A, B, C, ...]
. This will construct a 1d 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 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:(n1)
, or explicit arrays of values like [1.2, 3.4, 5.7]
. The result is an Nd 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 1d grid. :
julia> x = rand(8) 8element Array{Float64,1}: 0.843025 0.869052 0.365105 0.699456 0.977653 0.994953 0.41084 0.809411 julia> [ 0.25*x[i1] + 0.5*x[i] + 0.25*x[i+1] for i=2:length(x)1 ] 6element 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[i1] + 0.5*x[i] + 0.25*x[i+1] for i=2:length(x)1 ]
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 commaseparated 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. Just like inner functions used elsewhere 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; see 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] 6element Array{Tuple{Int64,Int64},1}: (1, 1) (2, 1) (2, 2) (3, 1) (3, 2) (3, 3)
In such cases, the result is always 1d.
Generated values can be filtered using the if
keyword:
julia> [(i,j) for i=1:3 for j=1:i if i+j == 4] 2element Array{Tuple{Int64,Int64},1}: (2, 2) (3, 1)
The general syntax for indexing into an ndimensional 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 I_k
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 twodimensional 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, if J
is an array of indices, then the result of A[2, J, 3]
is an array with size size(J)
. Its j
th element is populated by A[2, J[j], 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:end1] 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
The general syntax for assigning values in an ndimensional 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 all indices I_k
are integers, then the value in location I_1, I_2, ..., I_n
of A
is overwritten with the value of X
, convert
ing to the eltype
of A
if necessary.
If any index I_k
selects more than one location, then the right hand side X
must be an array with the same shape as the result of indexing A[I_1, I_2, ..., I_n]
or a vector with the same number of elements. 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]
, converting if necessary. The elementwise assignment operator .=
may be used to broadcast X
across the selected locations:
A[I_1, I_2, ..., I_n] .= X
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
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
:
CartesianIndex{N}
s, which behave like an N
tuple of integers spanning multiple dimensions (see below for more details)[]
, which select no elementsa:c
or a:b:c
, which select contiguous or strided subsections from a
to c
(inclusive)AbstractArray
CartesianIndex{N}
(see below for more details)to_indices
. By default this includes:Colon()
(:
), which represents all indices within an entire dimension or across the entire arraytrue
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]] 3element Array{Int64,1}: 3 9 15 julia> A[[1 4; 3 8]] 2×2 Array{Int64,2}: 1 7 5 15 julia> A[[]] 0element Array{Int64,1} julia> A[1:2:5] 3element Array{Int64,1}: 1 5 9 julia> A[2, :] 3element Array{Int64,1}: 3 9 15 julia> A[:, 3] 3element Array{Int64,1}: 13 15 17
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 CartesianIndex
es, however, this can produce 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)]] 4element 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] 4element 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
CartesianIndex
and arrays of CartesianIndex
are not compatible with the end
keyword to represent the last index of a dimension. Do not use end
in indexing expressions that may contain either CartesianIndex
or arrays thereof.
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}: 1 0 0 0 1 0 0 0 0 0 0 0 1 1 0 1 julia> x[mask] 5element Array{Int64,1}: 1 2 4 8 16
The ordinary way to index into an N
dimensional array is to use exactly N
indices; each index selects the position(s) in its particular dimension. For example, in the threedimensional array A = rand(4, 3, 2)
, A[2, 3, 1]
will select the number in the second row of the third column in the first "page" of the array. This is often referred to as cartesian indexing.
When exactly one index i
is provided, that index no longer represents a location in a particular dimension of the array. Instead, it selects the i
th element using the columnmajor iteration order that linearly spans the entire array. This is known as linear indexing. It essentially treats the array as though it had been reshaped into a onedimensional vector with vec
.
julia> A = [2 6; 4 7; 3 1] 3×2 Array{Int64,2}: 2 6 4 7 3 1 julia> A[5] 7 julia> vec(A)[5] 7
A linear index into the array A
can be converted to a CartesianIndex
for cartesian indexing with CartesianIndices(A)[i]
(see CartesianIndices
), and a set of N
cartesian indices can be converted to a linear index with LinearIndices(A)[i_1, i_2, ..., i_N]
(see LinearIndices
).
julia> CartesianIndices(A)[5] CartesianIndex(2, 2) julia> LinearIndices(A)[2, 2] 5
It's important to note that there's a very large assymmetry in the performance of these conversions. Converting a linear index to a set of cartesian indices requires dividing and taking the remainder, whereas going the other way is just multiplies and adds. In modern processors, integer division can be 1050 times slower than multiplication. While some arrays — like Array
itself — are implemented using a linear chunk of memory and directly use a linear index in their implementations, other arrays — like Diagonal
— need the full set of cartesian indices to do their lookup (see IndexStyle
to introspect which is which). As such, when iterating over an entire array, it's much better to iterate over eachindex(A)
instead of 1:length(A)
. Not only will the former be much faster in cases where A
is IndexCartesian
, but it will also support OffsetArrays, too.
In addition to linear indexing, an N
dimensional array may be indexed with fewer or more than N
indices in certain situations.
Indices may be omitted if the trailing dimensions that are not indexed into are all length one. In other words, trailing indices can be omitted only if there is only one possible value that those omitted indices could be for an inbounds indexing expression. For example, a fourdimensional array with size (3, 4, 2, 1)
may be indexed with only three indices as the dimension that gets skipped (the fourth dimension) has length one. Note that linear indexing takes precedence over this rule.
julia> A = reshape(1:24, 3, 4, 2, 1) 3×4×2×1 reshape(::UnitRange{Int64}, 3, 4, 2, 1) with eltype Int64: [:, :, 1, 1] = 1 4 7 10 2 5 8 11 3 6 9 12 [:, :, 2, 1] = 13 16 19 22 14 17 20 23 15 18 21 24 julia> A[1, 3, 2] # Omits the fourth dimension (length 1) 19 julia> A[1, 3] # Attempts to omit dimensions 3 & 4 (lengths 2 and 1) ERROR: BoundsError: attempt to access 3×4×2×1 reshape(::UnitRange{Int64}, 3, 4, 2, 1) with eltype Int64 at index [1, 3] julia> A[19] # Linear indexing 19
When omitting all indices with A[]
, this semantic provides a simple idiom to retrieve the only element in an array and simultaneously ensure that there was only one element.
Similarly, more than N
indices may be provided if all the indices beyond the dimensionality of the array are 1
(or more generally are the first and only element of axes(A, d)
where d
is that particular dimension number). This allows vectors to be indexed like onecolumn matrices, for example:
julia> A = [8,6,7] 3element Array{Int64,1}: 8 6 7 julia> A[2,1] 6
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.
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.
The following operators are supported for arrays:

, +

, +
, *
, /
, \
, ^
==
, !=
, ≈
(isapprox
), ≉
To enable convenient vectorization of mathematical and other operations, Julia provides the dot syntax f.(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 broadcast
s 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)
.
It is sometimes useful to perform elementbyelement 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 provides 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). In fact, 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]) 2element 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"]) 3element Array{String,1}: "1. First" "2. Second" "3. Third"
The base array type in Julia is the abstract type AbstractArray{T,N}
. It is parameterized by the number of dimensions N
and the element type T
. AbstractVector
and AbstractMatrix
are aliases for the 1d and 2d 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 arraylike, 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 outofplace 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 columnmajor order (see additional notes in Performance Tips). The Array
type is a specific instance of DenseArray
; Vector
and Matrix
are aliases for the 1d and 2d cases. Very few operations are implemented specifically for Array
beyond those that are required for all AbstractArray
s; 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.
BitArray
s are spaceefficient "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 k
th 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 appropriate 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
© 2009–2019 Jeff Bezanson, Stefan Karpinski, Viral B. Shah, and other contributors
Licensed under the MIT License.
https://docs.julialang.org/en/v1.2.0/manual/arrays/