Arrays

Constructors and Types

AbstractArray{T,N}

Supertype for N-dimensional arrays (or array-like types) with elements of type T. Array and other types are subtypes of this. See the manual section on the AbstractArray interface.

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AbstractVector{T}

Supertype for one-dimensional arrays (or array-like types) with elements of type T. Alias for AbstractArray{T,1}.

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AbstractMatrix{T}

Supertype for two-dimensional arrays (or array-like types) with elements of type T. Alias for AbstractArray{T,2}.

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AbstractVecOrMat{T}

Union type of AbstractVector{T} and AbstractMatrix{T}.

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Array{T,N} <: AbstractArray{T,N}

N-dimensional dense array with elements of type T.

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Array{T}(undef, dims)
Array{T,N}(undef, dims)

Construct an uninitialized N-dimensional Array containing elements of type T. N can either be supplied explicitly, as in Array{T,N}(undef, dims), or be determined by the length or number of dims. dims may be a tuple or a series of integer arguments corresponding to the lengths in each dimension. If the rank N is supplied explicitly, then it must match the length or number of dims. See undef.

Examples

julia> A = Array{Float64,2}(undef, 2, 3) # N given explicitly
2×3 Array{Float64,2}:
 6.90198e-310  6.90198e-310  6.90198e-310
 6.90198e-310  6.90198e-310  0.0
julia> B = Array{Float64}(undef, 2) # N determined by the input
2-element Array{Float64,1}:
 1.87103e-320
 0.0

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Array{T}(nothing, dims)
Array{T,N}(nothing, dims)

Construct an N-dimensional Array containing elements of type T, initialized with nothing entries. Element type T must be able to hold these values, i.e. Nothing <: T.

Examples

julia> Array{Union{Nothing, String}}(nothing, 2)
2-element Array{Union{Nothing, String},1}:
 nothing
 nothing
julia> Array{Union{Nothing, Int}}(nothing, 2, 3)
2×3 Array{Union{Nothing, Int64},2}:
 nothing  nothing  nothing
 nothing  nothing  nothing

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Array{T}(missing, dims)
Array{T,N}(missing, dims)

Construct an N-dimensional Array containing elements of type T, initialized with missing entries. Element type T must be able to hold these values, i.e. Missing <: T.

Examples

julia> Array{Union{Missing, String}}(missing, 2)
2-element Array{Union{Missing, String},1}:
 missing
 missing
julia> Array{Union{Missing, Int}}(missing, 2, 3)
2×3 Array{Union{Missing, Int64},2}:
 missing  missing  missing
 missing  missing  missing

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UndefInitializer

Singleton type used in array initialization, indicating the array-constructor-caller would like an uninitialized array. See also undef, an alias for UndefInitializer().

Examples

julia> Array{Float64,1}(UndefInitializer(), 3)
3-element Array{Float64,1}:
 2.2752528595e-314
 2.202942107e-314
 2.275252907e-314

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undef

Alias for UndefInitializer(), which constructs an instance of the singleton type UndefInitializer, used in array initialization to indicate the array-constructor-caller would like an uninitialized array.

Examples

julia> Array{Float64,1}(undef, 3)
3-element Array{Float64,1}:
 2.2752528595e-314
 2.202942107e-314
 2.275252907e-314

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Vector{T} <: AbstractVector{T}

One-dimensional dense array with elements of type T, often used to represent a mathematical vector. Alias for Array{T,1}.

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Vector{T}(undef, n)

Construct an uninitialized Vector{T} of length n. See undef.

Examples

julia> Vector{Float64}(undef, 3)
3-element Array{Float64,1}:
 6.90966e-310
 6.90966e-310
 6.90966e-310

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Vector{T}(nothing, m)

Construct a Vector{T} of length m, initialized with nothing entries. Element type T must be able to hold these values, i.e. Nothing <: T.

Examples

julia> Vector{Union{Nothing, String}}(nothing, 2)
2-element Array{Union{Nothing, String},1}:
 nothing
 nothing

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Vector{T}(missing, m)

Construct a Vector{T} of length m, initialized with missing entries. Element type T must be able to hold these values, i.e. Missing <: T.

Examples

julia> Vector{Union{Missing, String}}(missing, 2)
2-element Array{Union{Missing, String},1}:
 missing
 missing

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Matrix{T} <: AbstractMatrix{T}

Two-dimensional dense array with elements of type T, often used to represent a mathematical matrix. Alias for Array{T,2}.

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Matrix{T}(undef, m, n)

Construct an uninitialized Matrix{T} of size m×n. See undef.

Examples

julia> Matrix{Float64}(undef, 2, 3)
2×3 Array{Float64,2}:
 6.93517e-310  6.93517e-310  6.93517e-310
 6.93517e-310  6.93517e-310  1.29396e-320

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Matrix{T}(nothing, m, n)

Construct a Matrix{T} of size m×n, initialized with nothing entries. Element type T must be able to hold these values, i.e. Nothing <: T.

Examples

julia> Matrix{Union{Nothing, String}}(nothing, 2, 3)
2×3 Array{Union{Nothing, String},2}:
 nothing  nothing  nothing
 nothing  nothing  nothing

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Matrix{T}(missing, m, n)

Construct a Matrix{T} of size m×n, initialized with missing entries. Element type T must be able to hold these values, i.e. Missing <: T.

Examples

julia> Matrix{Union{Missing, String}}(missing, 2, 3)
2×3 Array{Union{Missing, String},2}:
 missing  missing  missing
 missing  missing  missing

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VecOrMat{T}

Union type of Vector{T} and Matrix{T}.

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DenseArray{T, N} <: AbstractArray{T,N}

N-dimensional dense array with elements of type T. The elements of a dense array are stored contiguously in memory.

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DenseVector{T}

One-dimensional DenseArray with elements of type T. Alias for DenseArray{T,1}.

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DenseMatrix{T}

Two-dimensional DenseArray with elements of type T. Alias for DenseArray{T,2}.

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DenseVecOrMat{T}

Union type of DenseVector{T} and DenseMatrix{T}.

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StridedArray{T, N}

An N dimensional strided array with elements of type T. These arrays follow the strided array interface. If A is a StridedArray, then its elements are stored in memory with offsets, which may vary between dimensions but are constant within a dimension. For example, A could have stride 2 in dimension 1, and stride 3 in dimension 2. Incrementing A along dimension d jumps in memory by [strides(A, d)] slots. Strided arrays are particularly important and useful because they can sometimes be passed directly as pointers to foreign language libraries like BLAS.

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StridedVector{T}

One dimensional StridedArray with elements of type T.

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StridedMatrix{T}

Two dimensional StridedArray with elements of type T.

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StridedVecOrMat{T}

Union type of StridedVector and StridedMatrix with elements of type T.

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getindex(type[, elements...])

Construct a 1-d array of the specified type. This is usually called with the syntax Type[]. Element values can be specified using Type[a,b,c,...].

Examples

julia> Int8[1, 2, 3]
3-element Array{Int8,1}:
 1
 2
 3
julia> getindex(Int8, 1, 2, 3)
3-element Array{Int8,1}:
 1
 2
 3

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zeros([T=Float64,] dims::Tuple)
zeros([T=Float64,] dims...)

Create an Array, with element type T, of all zeros with size specified by dims. See also fill, ones.

Examples

julia> zeros(1)
1-element Array{Float64,1}:
 0.0
julia> zeros(Int8, 2, 3)
2×3 Array{Int8,2}:
 0  0  0
 0  0  0

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ones([T=Float64,] dims::Tuple)
ones([T=Float64,] dims...)

Create an Array, with element type T, of all ones with size specified by dims. See also: fill, zeros.

Examples

julia> ones(1,2)
1×2 Array{Float64,2}:
 1.0  1.0
julia> ones(ComplexF64, 2, 3)
2×3 Array{Complex{Float64},2}:
 1.0+0.0im  1.0+0.0im  1.0+0.0im
 1.0+0.0im  1.0+0.0im  1.0+0.0im

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BitArray{N} <: AbstractArray{Bool, N}

Space-efficient N-dimensional boolean array, using just one bit for each boolean value.

BitArrays pack up to 64 values into every 8 bytes, resulting in an 8x space efficiency over Array{Bool, N} and allowing some operations to work on 64 values at once.

By default, Julia returns BitArrays from broadcasting operations that generate boolean elements (including dotted-comparisons like .==) as well as from the functions trues and falses.

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BitArray(undef, dims::Integer...)
BitArray{N}(undef, dims::NTuple{N,Int})

Construct an undef BitArray with the given dimensions. Behaves identically to the Array constructor. See undef.

Examples

julia> BitArray(undef, 2, 2)
2×2 BitArray{2}:
 0  0
 0  0
julia> BitArray(undef, (3, 1))
3×1 BitArray{2}:
 0
 0
 0

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BitArray(itr)

Construct a BitArray generated by the given iterable object. The shape is inferred from the itr object.

Examples

julia> BitArray([1 0; 0 1])
2×2 BitArray{2}:
 1  0
 0  1
julia> BitArray(x+y == 3 for x = 1:2, y = 1:3)
2×3 BitArray{2}:
 0  1  0
 1  0  0
julia> BitArray(x+y == 3 for x = 1:2 for y = 1:3)
6-element BitArray{1}:
 0
 1
 0
 1
 0
 0

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trues(dims)

Create a BitArray with all values set to true.

Examples

julia> trues(2,3)
2×3 BitArray{2}:
 1  1  1
 1  1  1

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falses(dims)

Create a BitArray with all values set to false.

Examples

julia> falses(2,3)
2×3 BitArray{2}:
 0  0  0
 0  0  0

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fill(x, dims::Tuple)
fill(x, dims...)

Create an array filled with the value x. For example, fill(1.0, (5,5)) returns a 5×5 array of floats, with each element initialized to 1.0.

dims may be specified as either a tuple or a sequence of arguments. For example, the common idiom fill(x) creates a zero-dimensional array containing the single value x.

Examples

julia> fill(1.0, (5,5))
5×5 Array{Float64,2}:
 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
julia> fill(0.5, 1, 2)
1×2 Array{Float64,2}:
 0.5  0.5
julia> fill(42)
0-dimensional Array{Int64,0}:
42

If x is an object reference, all elements will refer to the same object. fill(Foo(), dims) will return an array filled with the result of evaluating Foo() once.

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fill!(A, x)

Fill array A with the value x. If x is an object reference, all elements will refer to the same object. fill!(A, Foo()) will return A filled with the result of evaluating Foo() once.

Examples

julia> A = zeros(2,3)
2×3 Array{Float64,2}:
 0.0  0.0  0.0
 0.0  0.0  0.0
julia> fill!(A, 2.)
2×3 Array{Float64,2}:
 2.0  2.0  2.0
 2.0  2.0  2.0
julia> a = [1, 1, 1]; A = fill!(Vector{Vector{Int}}(undef, 3), a); a[1] = 2; A
3-element Array{Array{Int64,1},1}:
 [2, 1, 1]
 [2, 1, 1]
 [2, 1, 1]
julia> x = 0; f() = (global x += 1; x); fill!(Vector{Int}(undef, 3), f())
3-element Array{Int64,1}:
 1
 1
 1

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similar(array, [element_type=eltype(array)], [dims=size(array)])

Create an uninitialized mutable array with the given element type and size, based upon the given source array. The second and third arguments are both optional, defaulting to the given array’s eltype and size. The dimensions may be specified either as a single tuple argument or as a series of integer arguments.

Custom AbstractArray subtypes may choose which specific array type is best-suited to return for the given element type and dimensionality. If they do not specialize this method, the default is an Array{element_type}(undef, dims...).

For example, similar(1:10, 1, 4) returns an uninitialized Array{Int,2} since ranges are neither mutable nor support 2 dimensions:

julia> similar(1:10, 1, 4)
1×4 Array{Int64,2}:
 4419743872  4374413872  4419743888  0

Conversely, similar(trues(10,10), 2) returns an uninitialized BitVector with two elements since BitArrays are both mutable and can support 1-dimensional arrays:

julia> similar(trues(10,10), 2)
2-element BitArray{1}:
 0
 0

Since BitArrays can only store elements of type Bool, however, if you request a different element type it will create a regular Array instead:

julia> similar(falses(10), Float64, 2, 4)
2×4 Array{Float64,2}:
 2.18425e-314  2.18425e-314  2.18425e-314  2.18425e-314
 2.18425e-314  2.18425e-314  2.18425e-314  2.18425e-314

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similar(storagetype, axes)

Create an uninitialized mutable array analogous to that specified by storagetype, but with axes specified by the last argument. storagetype might be a type or a function.

Examples:

similar(Array{Int}, axes(A))

creates an array that “acts like” an Array{Int} (and might indeed be backed by one), but which is indexed identically to A. If A has conventional indexing, this will be identical to Array{Int}(undef, size(A)), but if A has unconventional indexing then the indices of the result will match A.

similar(BitArray, (axes(A, 2),))

would create a 1-dimensional logical array whose indices match those of the columns of A.

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Basic functions

ndims(A::AbstractArray) -> Integer

Return the number of dimensions of A.

Examples

julia> A = fill(1, (3,4,5));
julia> ndims(A)
3

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size(A::AbstractArray, [dim])

Return a tuple containing the dimensions of A. Optionally you can specify a dimension to just get the length of that dimension.

Note that size may not be defined for arrays with non-standard indices, in which case axes may be useful. See the manual chapter on arrays with custom indices.

Examples

julia> A = fill(1, (2,3,4));
julia> size(A)
(2, 3, 4)
julia> size(A, 2)
3

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axes(A)

Return the tuple of valid indices for array A.

Examples

julia> A = fill(1, (5,6,7));
julia> axes(A)
(Base.OneTo(5), Base.OneTo(6), Base.OneTo(7))

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axes(A, d)

Return the valid range of indices for array A along dimension d.

See also size, and the manual chapter on arrays with custom indices.

Examples

julia> A = fill(1, (5,6,7));
julia> axes(A, 2)
Base.OneTo(6)

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length(A::AbstractArray)

Return the number of elements in the array, defaults to prod(size(A)).

Examples

julia> length([1, 2, 3, 4])
4
julia> length([1 2; 3 4])
4

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eachindex(A...)

Create an iterable object for visiting each index of an AbstractArray A in an efficient manner. For array types that have opted into fast linear indexing (like Array), this is simply the range 1:length(A). For other array types, return a specialized Cartesian range to efficiently index into the array with indices specified for every dimension. For other iterables, including strings and dictionaries, return an iterator object supporting arbitrary index types (e.g. unevenly spaced or non-integer indices).

If you supply more than one AbstractArray argument, eachindex will create an iterable object that is fast for all arguments (a UnitRange if all inputs have fast linear indexing, a CartesianIndices otherwise). If the arrays have different sizes and/or dimensionalities, a DimensionMismatch exception will be thrown.

Examples

julia> A = [1 2; 3 4];
julia> for i in eachindex(A) # linear indexing
           println(i)
       end
1
2
3
4
julia> for i in eachindex(view(A, 1:2, 1:1)) # Cartesian indexing
           println(i)
       end
CartesianIndex(1, 1)
CartesianIndex(2, 1)

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IndexStyle(A)
IndexStyle(typeof(A))

IndexStyle specifies the “native indexing style” for array A. When you define a new AbstractArray type, you can choose to implement either linear indexing (with IndexLinear) or cartesian indexing. If you decide to only implement linear indexing, then you must set this trait for your array type:

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

The default is IndexCartesian().

Julia’s internal indexing machinery will automatically (and invisibly) recompute all indexing operations into the preferred style. This allows users to access elements of your array using any indexing style, even when explicit methods have not been provided.

If you define both styles of indexing for your AbstractArray, this trait can be used to select the most performant indexing style. Some methods check this trait on their inputs, and dispatch to different algorithms depending on the most efficient access pattern. In particular, eachindex creates an iterator whose type depends on the setting of this trait.

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IndexLinear()

Subtype of IndexStyle used to describe arrays which are optimally indexed by one linear index.

A linear indexing style uses one integer index to describe the position in the array (even if it’s a multidimensional array) and column-major ordering is used to efficiently access the elements. This means that requesting eachindex from an array that is IndexLinear will return a simple one-dimensional range, even if it is multidimensional.

A custom array that reports its IndexStyle as IndexLinear only needs to implement indexing (and indexed assignment) with a single Int index; all other indexing expressions — including multidimensional accesses — will be recomputed to the linear index. For example, if A were a 2×3 custom matrix with linear indexing, and we referenced A[1, 3], this would be recomputed to the equivalent linear index and call A[5] since 2*1 + 3 = 5.

See also IndexCartesian.

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IndexCartesian()

Subtype of IndexStyle used to describe arrays which are optimally indexed by a Cartesian index. This is the default for new custom AbstractArray subtypes.

A Cartesian indexing style uses multiple integer indices to describe the position in a multidimensional array, with exactly one index per dimension. This means that requesting eachindex from an array that is IndexCartesian will return a range of CartesianIndices.

A N-dimensional custom array that reports its IndexStyle as IndexCartesian needs to implement indexing (and indexed assignment) with exactly N Int indices; all other indexing expressions — including linear indexing — will be recomputed to the equivalent Cartesian location. For example, if A were a 2×3 custom matrix with cartesian indexing, and we referenced A[5], this would be recomputed to the equivalent Cartesian index and call A[1, 3] since 5 = 2*1 + 3.

It is significantly more expensive to compute Cartesian indices from a linear index than it is to go the other way. The former operation requires division — a very costly operation — whereas the latter only uses multiplication and addition and is essentially free. This asymmetry means it is far more costly to use linear indexing with an IndexCartesian array than it is to use Cartesian indexing with an IndexLinear array.

See also IndexLinear.

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conj!(A)

Transform an array to its complex conjugate in-place.

See also conj.

Examples

julia> A = [1+im 2-im; 2+2im 3+im]
2×2 Array{Complex{Int64},2}:
 1+1im  2-1im
 2+2im  3+1im
julia> conj!(A);
julia> A
2×2 Array{Complex{Int64},2}:
 1-1im  2+1im
 2-2im  3-1im

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stride(A, k::Integer)

Return the distance in memory (in number of elements) between adjacent elements in dimension k.

Examples

julia> A = fill(1, (3,4,5));
julia> stride(A,2)
3
julia> stride(A,3)
12

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strides(A)

Return a tuple of the memory strides in each dimension.

Examples

julia> A = fill(1, (3,4,5));
julia> strides(A)
(1, 3, 12)

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Broadcast and vectorization

See also the dot syntax for vectorizing functions; for example, f.(args...) implicitly calls broadcast(f, args...). Rather than relying on “vectorized” methods of functions like sin to operate on arrays, you should use sin.(a) to vectorize via broadcast.

broadcast(f, As...)

Broadcast the function f over the arrays, tuples, collections, Refs and/or scalars As.

Broadcasting applies the function f over the elements of the container arguments and the scalars themselves in As. Singleton and missing dimensions are expanded to match the extents of the other arguments by virtually repeating the value. By default, only a limited number of types are considered scalars, including Numbers, Strings, Symbols, Types, Functions and some common singletons like missing and nothing. All other arguments are iterated over or indexed into elementwise.

The resulting container type is established by the following rules:

• If all the arguments are scalars or zero-dimensional arrays, it returns an unwrapped scalar.
• If at least one argument is a tuple and all others are scalars or zero-dimensional arrays, it returns a tuple.
• All other combinations of arguments default to returning an Array, but custom container types can define their own implementation and promotion-like rules to customize the result when they appear as arguments.

A special syntax exists for broadcasting: f.(args...) is equivalent to broadcast(f, args...), and nested f.(g.(args...)) calls are fused into a single broadcast loop.

Examples

julia> A = [1, 2, 3, 4, 5]
5-element Array{Int64,1}:
 1
 2
 3
 4
 5
julia> B = [1 2; 3 4; 5 6; 7 8; 9 10]
5×2 Array{Int64,2}:
 1   2
 3   4
 5   6
 7   8
 9  10
julia> broadcast(+, A, B)
5×2 Array{Int64,2}:
  2   3
  5   6
  8   9
 11  12
 14  15
julia> parse.(Int, ["1", "2"])
2-element Array{Int64,1}:
 1
 2
julia> abs.((1, -2))
(1, 2)
julia> broadcast(+, 1.0, (0, -2.0))
(1.0, -1.0)
julia> (+).([[0,2], [1,3]], Ref{Vector{Int}}([1,-1]))
2-element Array{Array{Int64,1},1}:
 [1, 1]
 [2, 2]
julia> string.(("one","two","three","four"), ": ", 1:4)
4-element Array{String,1}:
 "one: 1"
 "two: 2"
 "three: 3"
 "four: 4"

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broadcast!(f, dest, As...)

Like broadcast, but store the result of broadcast(f, As...) in the dest array. Note that dest is only used to store the result, and does not supply arguments to f unless it is also listed in the As, as in broadcast!(f, A, A, B) to perform A[:] = broadcast(f, A, B).

Examples

julia> A = [1.0; 0.0]; B = [0.0; 0.0];
julia> broadcast!(+, B, A, (0, -2.0));
julia> B
2-element Array{Float64,1}:
  1.0
 -2.0
julia> A
2-element Array{Float64,1}:
 1.0
 0.0
julia> broadcast!(+, A, A, (0, -2.0));
julia> A
2-element Array{Float64,1}:
  1.0
 -2.0

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@. expr

Convert every function call or operator in expr into a “dot call” (e.g. convert f(x) to f.(x)), and convert every assignment in expr to a “dot assignment” (e.g. convert += to .+=).

If you want to avoid adding dots for selected function calls in expr, splice those function calls in with $. For example, @. sqrt(abs($sort(x))) is equivalent to sqrt.(abs.(sort(x))) (no dot for sort).

(@. is equivalent to a call to @__dot__.)

Examples

julia> x = 1.0:3.0; y = similar(x);
julia> @. y = x + 3 * sin(x)
3-element Array{Float64,1}:
 3.5244129544236893
 4.727892280477045
 3.4233600241796016

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For specializing broadcast on custom types, see

BroadcastStyle is an abstract type and trait-function used to determine behavior of objects under broadcasting. BroadcastStyle(typeof(x)) returns the style associated with x. To customize the broadcasting behavior of a type, one can declare a style by defining a type/method pair

struct MyContainerStyle <: BroadcastStyle end
Base.BroadcastStyle(::Type{<:MyContainer}) = MyContainerStyle()

One then writes method(s) (at least similar) operating on Broadcasted{MyContainerStyle}. There are also several pre-defined subtypes of BroadcastStyle that you may be able to leverage; see the Interfaces chapter for more information.

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Broadcast.AbstractArrayStyle{N} <: BroadcastStyle is the abstract supertype for any style associated with an AbstractArray type. The N parameter is the dimensionality, which can be handy for AbstractArray types that only support specific dimensionalities:

struct SparseMatrixStyle <: Broadcast.AbstractArrayStyle{2} end
Base.BroadcastStyle(::Type{<:SparseMatrixCSC}) = SparseMatrixStyle()

For AbstractArray types that support arbitrary dimensionality, N can be set to Any:

struct MyArrayStyle <: Broadcast.AbstractArrayStyle{Any} end
Base.BroadcastStyle(::Type{<:MyArray}) = MyArrayStyle()

In cases where you want to be able to mix multiple AbstractArrayStyles and keep track of dimensionality, your style needs to support a Val constructor:

struct MyArrayStyleDim{N} <: Broadcast.AbstractArrayStyle{N} end
(::Type{<:MyArrayStyleDim})(::Val{N}) where N = MyArrayStyleDim{N}()

Note that if two or more AbstractArrayStyle subtypes conflict, broadcasting machinery will fall back to producing Arrays. If this is undesirable, you may need to define binary BroadcastStyle rules to control the output type.

See also Broadcast.DefaultArrayStyle.

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Broadcast.ArrayStyle{MyArrayType}() is a BroadcastStyle indicating that an object behaves as an array for broadcasting. It presents a simple way to construct Broadcast.AbstractArrayStyles for specific AbstractArray container types. Broadcast styles created this way lose track of dimensionality; if keeping track is important for your type, you should create your own custom Broadcast.AbstractArrayStyle.

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Broadcast.DefaultArrayStyle{N}() is a BroadcastStyle indicating that an object behaves as an N-dimensional array for broadcasting. Specifically, DefaultArrayStyle is used for any AbstractArray type that hasn’t defined a specialized style, and in the absence of overrides from other broadcast arguments the resulting output type is Array. When there are multiple inputs to broadcast, DefaultArrayStyle “loses” to any other Broadcast.ArrayStyle.

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Broadcast.broadcastable(x)

Return either x or an object like x such that it supports axes, indexing, and its type supports ndims.

If x supports iteration, the returned value should have the same axes and indexing behaviors as collect(x).

If x is not an AbstractArray but it supports axes, indexing, and its type supports ndims, then broadcastable(::typeof(x)) may be implemented to just return itself. Further, if x defines its own BroadcastStyle, then it must define its broadcastable method to return itself for the custom style to have any effect.

Examples

julia> Broadcast.broadcastable([1,2,3]) # like `identity` since arrays already support axes and indexing
3-element Array{Int64,1}:
 1
 2
 3
julia> Broadcast.broadcastable(Int) # Types don't support axes, indexing, or iteration but are commonly used as scalars
Base.RefValue{Type{Int64}}(Int64)
julia> Broadcast.broadcastable("hello") # Strings break convention of matching iteration and act like a scalar instead
Base.RefValue{String}("hello")

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combine_axes(As...) -> Tuple

Determine the result axes for broadcasting across all values in As.

julia> Broadcast.combine_axes([1], [1 2; 3 4; 5 6])
(Base.OneTo(3), Base.OneTo(2))
julia> Broadcast.combine_axes(1, 1, 1)
()

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combine_styles(cs...) -> BroadcastStyle

Decides which BroadcastStyle to use for any number of value arguments. Uses BroadcastStyle to get the style for each argument, and uses result_style to combine styles.

Examples

julia> Broadcast.combine_styles([1], [1 2; 3 4])
Base.Broadcast.DefaultArrayStyle{2}()

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result_style(s1::BroadcastStyle[, s2::BroadcastStyle]) -> BroadcastStyle

Takes one or two BroadcastStyles and combines them using BroadcastStyle to determine a common BroadcastStyle.

Examples

julia> Broadcast.result_style(Broadcast.DefaultArrayStyle{0}(), Broadcast.DefaultArrayStyle{3}())
Base.Broadcast.DefaultArrayStyle{3}()
julia> Broadcast.result_style(Broadcast.Unknown(), Broadcast.DefaultArrayStyle{1}())
Base.Broadcast.DefaultArrayStyle{1}()

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Indexing and assignment

getindex(A, inds...)

Return a subset of array A as specified by inds, where each ind may be an Int, an AbstractRange, or a Vector. See the manual section on array indexing for details.

Examples

julia> A = [1 2; 3 4]
2×2 Array{Int64,2}:
 1  2
 3  4
julia> getindex(A, 1)
1
julia> getindex(A, [2, 1])
2-element Array{Int64,1}:
 3
 1
julia> getindex(A, 2:4)
3-element Array{Int64,1}:
 3
 2
 4

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setindex!(A, X, inds...)
A[inds...] = X

Store values from array X within some subset of A as specified by inds. The syntax A[inds...] = X is equivalent to setindex!(A, X, inds...).

Examples

julia> A = zeros(2,2);
julia> setindex!(A, [10, 20], [1, 2]);
julia> A[[3, 4]] = [30, 40];
julia> A
2×2 Array{Float64,2}:
 10.0  30.0
 20.0  40.0

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copyto!(dest, Rdest::CartesianIndices, src, Rsrc::CartesianIndices) -> dest

Copy the block of src in the range of Rsrc to the block of dest in the range of Rdest. The sizes of the two regions must match.

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isassigned(array, i) -> Bool

Test whether the given array has a value associated with index i. Return false if the index is out of bounds, or has an undefined reference.

Examples

julia> isassigned(rand(3, 3), 5)
true
julia> isassigned(rand(3, 3), 3 * 3 + 1)
false
julia> mutable struct Foo end
julia> v = similar(rand(3), Foo)
3-element Array{Foo,1}:
 #undef
 #undef
 #undef
julia> isassigned(v, 1)
false

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Colon()

Colons (:) are used to signify indexing entire objects or dimensions at once.

Very few operations are defined on Colons directly; instead they are converted by to_indices to an internal vector type (Base.Slice) to represent the collection of indices they span before being used.

The singleton instance of Colon is also a function used to construct ranges; see :.

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CartesianIndex(i, j, k...)   -> I
CartesianIndex((i, j, k...)) -> I

Create a multidimensional index I, which can be used for indexing a multidimensional array A. In particular, A[I] is equivalent to A[i,j,k...]. One can freely mix integer and CartesianIndex indices; for example, A[Ipre, i, Ipost] (where Ipre and Ipost are CartesianIndex indices and i is an Int) can be a useful expression when writing algorithms that work along a single dimension of an array of arbitrary dimensionality.

A CartesianIndex is sometimes produced by eachindex, and always when iterating with an explicit CartesianIndices.

Examples

julia> A = reshape(Vector(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[CartesianIndex((1, 1, 1, 1))]
1
julia> A[CartesianIndex((1, 1, 1, 2))]
9
julia> A[CartesianIndex((1, 1, 2, 1))]
5

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CartesianIndices(sz::Dims) -> R
CartesianIndices((istart:istop, jstart:jstop, ...)) -> R

Define a region R spanning a multidimensional rectangular range of integer indices. These are most commonly encountered in the context of iteration, where for I in R ... end will return CartesianIndex indices I equivalent to the nested loops

for j = jstart:jstop
    for i = istart:istop
        ...
    end
end

Consequently these can be useful for writing algorithms that work in arbitrary dimensions.

CartesianIndices(A::AbstractArray) -> R

As a convenience, constructing a CartesianIndices from an array makes a range of its indices.

Examples

julia> foreach(println, CartesianIndices((2, 2, 2)))
CartesianIndex(1, 1, 1)
CartesianIndex(2, 1, 1)
CartesianIndex(1, 2, 1)
CartesianIndex(2, 2, 1)
CartesianIndex(1, 1, 2)
CartesianIndex(2, 1, 2)
CartesianIndex(1, 2, 2)
CartesianIndex(2, 2, 2)
julia> CartesianIndices(fill(1, (2,3)))
2×3 CartesianIndices{2,Tuple{Base.OneTo{Int64},Base.OneTo{Int64}}}:
 CartesianIndex(1, 1)  CartesianIndex(1, 2)  CartesianIndex(1, 3)
 CartesianIndex(2, 1)  CartesianIndex(2, 2)  CartesianIndex(2, 3)

Conversion between linear and cartesian indices

Linear index to cartesian index conversion exploits the fact that a CartesianIndices is an AbstractArray and can be indexed linearly:

julia> cartesian = CartesianIndices((1:3, 1:2))
3×2 CartesianIndices{2,Tuple{UnitRange{Int64},UnitRange{Int64}}}:
 CartesianIndex(1, 1)  CartesianIndex(1, 2)
 CartesianIndex(2, 1)  CartesianIndex(2, 2)
 CartesianIndex(3, 1)  CartesianIndex(3, 2)
julia> cartesian[4]
CartesianIndex(1, 2)

Broadcasting

CartesianIndices support broadcasting arithmetic (+ and -) with a CartesianIndex.

Broadcasting of CartesianIndices requires at least Julia 1.1.

julia> CIs = CartesianIndices((2:3, 5:6))
2×2 CartesianIndices{2,Tuple{UnitRange{Int64},UnitRange{Int64}}}:
 CartesianIndex(2, 5)  CartesianIndex(2, 6)
 CartesianIndex(3, 5)  CartesianIndex(3, 6)
julia> CI = CartesianIndex(3, 4)
CartesianIndex(3, 4)
julia> CIs .+ CI
2×2 CartesianIndices{2,Tuple{UnitRange{Int64},UnitRange{Int64}}}:
 CartesianIndex(5, 9)  CartesianIndex(5, 10)
 CartesianIndex(6, 9)  CartesianIndex(6, 10)

For cartesian to linear index conversion, see LinearIndices.

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Dims{N}

An NTuple of N Ints used to represent the dimensions of an AbstractArray.

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LinearIndices(A::AbstractArray)

Return a LinearIndices array with the same shape and axes as A, holding the linear index of each entry in A. Indexing this array with cartesian indices allows mapping them to linear indices.

For arrays with conventional indexing (indices start at 1), or any multidimensional array, linear indices range from 1 to length(A). However, for AbstractVectors linear indices are axes(A, 1), and therefore do not start at 1 for vectors with unconventional indexing.

Calling this function is the “safe” way to write algorithms that exploit linear indexing.

Examples

julia> A = fill(1, (5,6,7));
julia> b = LinearIndices(A);
julia> extrema(b)
(1, 210)

LinearIndices(inds::CartesianIndices) -> R
LinearIndices(sz::Dims) -> R
LinearIndices((istart:istop, jstart:jstop, ...)) -> R

Return a LinearIndices array with the specified shape or axes.

Example

The main purpose of this constructor is intuitive conversion from cartesian to linear indexing:

julia> linear = LinearIndices((1:3, 1:2))
3×2 LinearIndices{2,Tuple{UnitRange{Int64},UnitRange{Int64}}}:
 1  4
 2  5
 3  6
julia> linear[1,2]
4

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to_indices(A, I::Tuple)

Convert the tuple I to a tuple of indices for use in indexing into array A.

The returned tuple must only contain either Ints or AbstractArrays of scalar indices that are supported by array A. It will error upon encountering a novel index type that it does not know how to process.

For simple index types, it defers to the unexported Base.to_index(A, i) to process each index i. While this internal function is not intended to be called directly, Base.to_index may be extended by custom array or index types to provide custom indexing behaviors.

More complicated index types may require more context about the dimension into which they index. To support those cases, to_indices(A, I) calls to_indices(A, axes(A), I), which then recursively walks through both the given tuple of indices and the dimensional indices of A in tandem. As such, not all index types are guaranteed to propagate to Base.to_index.

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checkbounds(Bool, A, I...)

Return true if the specified indices I are in bounds for the given array A. Subtypes of AbstractArray should specialize this method if they need to provide custom bounds checking behaviors; however, in many cases one can rely on A‘s indices and checkindex.

See also checkindex.

Examples

julia> A = rand(3, 3);
julia> checkbounds(Bool, A, 2)
true
julia> checkbounds(Bool, A, 3, 4)
false
julia> checkbounds(Bool, A, 1:3)
true
julia> checkbounds(Bool, A, 1:3, 2:4)
false

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checkbounds(A, I...)

Throw an error if the specified indices I are not in bounds for the given array A.

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checkindex(Bool, inds::AbstractUnitRange, index)

Return true if the given index is within the bounds of inds. Custom types that would like to behave as indices for all arrays can extend this method in order to provide a specialized bounds checking implementation.

Examples

julia> checkindex(Bool, 1:20, 8)
true
julia> checkindex(Bool, 1:20, 21)
false

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Views (SubArrays and other view types)

view(A, inds...)

Like getindex, but returns a view into the parent array A with the given indices instead of making a copy. Calling getindex or setindex! on the returned SubArray computes the indices to the parent array on the fly without checking bounds.

Examples

julia> A = [1 2; 3 4]
2×2 Array{Int64,2}:
 1  2
 3  4
julia> b = view(A, :, 1)
2-element view(::Array{Int64,2}, :, 1) with eltype Int64:
 1
 3
julia> fill!(b, 0)
2-element view(::Array{Int64,2}, :, 1) with eltype Int64:
 0
 0
julia> A # Note A has changed even though we modified b
2×2 Array{Int64,2}:
 0  2
 0  4

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@view A[inds...]

Creates a SubArray from an indexing expression. This can only be applied directly to a reference expression (e.g. @view A[1,2:end]), and should not be used as the target of an assignment (e.g. @view(A[1,2:end]) = ...). See also @views to switch an entire block of code to use views for slicing.

Examples

julia> A = [1 2; 3 4]
2×2 Array{Int64,2}:
 1  2
 3  4
julia> b = @view A[:, 1]
2-element view(::Array{Int64,2}, :, 1) with eltype Int64:
 1
 3
julia> fill!(b, 0)
2-element view(::Array{Int64,2}, :, 1) with eltype Int64:
 0
 0
julia> A
2×2 Array{Int64,2}:
 0  2
 0  4

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@views expression

Convert every array-slicing operation in the given expression (which may be a begin/end block, loop, function, etc.) to return a view. Scalar indices, non-array types, and explicit getindex calls (as opposed to array[...]) are unaffected.

The @views macro only affects array[...] expressions that appear explicitly in the given expression, not array slicing that occurs in functions called by that code.

Examples

julia> A = zeros(3, 3);
julia> @views for row in 1:3
           b = A[row, :]
           b[:] .= row
       end
julia> A
3×3 Array{Float64,2}:
 1.0  1.0  1.0
 2.0  2.0  2.0
 3.0  3.0  3.0

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parent(A)

Returns the “parent array” of an array view type (e.g., SubArray), or the array itself if it is not a view.

Examples

julia> A = [1 2; 3 4]
2×2 Array{Int64,2}:
 1  2
 3  4
julia> V = view(A, 1:2, :)
2×2 view(::Array{Int64,2}, 1:2, :) with eltype Int64:
 1  2
 3  4
julia> parent(V)
2×2 Array{Int64,2}:
 1  2
 3  4

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parentindices(A)

Return the indices in the parent which correspond to the array view A.

Examples

julia> A = [1 2; 3 4];
julia> V = view(A, 1, :)
2-element view(::Array{Int64,2}, 1, :) with eltype Int64:
 1
 2
julia> parentindices(V)
(1, Base.Slice(Base.OneTo(2)))

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selectdim(A, d::Integer, i)

Return a view of all the data of A where the index for dimension d equals i.

Equivalent to view(A,:,:,...,i,:,:,...) where i is in position d.

Examples

julia> A = [1 2 3 4; 5 6 7 8]
2×4 Array{Int64,2}:
 1  2  3  4
 5  6  7  8
julia> selectdim(A, 2, 3)
2-element view(::Array{Int64,2}, :, 3) with eltype Int64:
 3
 7

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reinterpret(type, A)

Change the type-interpretation of a block of memory. For arrays, this constructs a view of the array with the same binary data as the given array, but with the specified element type. For example, reinterpret(Float32, UInt32(7)) interprets the 4 bytes corresponding to UInt32(7) as a Float32.

Examples

julia> reinterpret(Float32, UInt32(7))
1.0f-44
julia> reinterpret(Float32, UInt32[1 2 3 4 5])
1×5 reinterpret(Float32, ::Array{UInt32,2}):
 1.0f-45  3.0f-45  4.0f-45  6.0f-45  7.0f-45

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reshape(A, dims...) -> AbstractArray
reshape(A, dims) -> AbstractArray

Return an array with the same data as A, but with different dimension sizes or number of dimensions. The two arrays share the same underlying data, so that the result is mutable if and only if A is mutable, and setting elements of one alters the values of the other.

The new dimensions may be specified either as a list of arguments or as a shape tuple. At most one dimension may be specified with a :, in which case its length is computed such that its product with all the specified dimensions is equal to the length of the original array A. The total number of elements must not change.

Examples

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

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dropdims(A; dims)

Remove the dimensions specified by dims from array A. Elements of dims must be unique and within the range 1:ndims(A). size(A,i) must equal 1 for all i in dims.

Examples

julia> a = reshape(Vector(1:4),(2,2,1,1))
2×2×1×1 Array{Int64,4}:
[:, :, 1, 1] =
 1  3
 2  4
julia> dropdims(a; dims=3)
2×2×1 Array{Int64,3}:
[:, :, 1] =
 1  3
 2  4

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vec(a::AbstractArray) -> AbstractVector

Reshape the array a as a one-dimensional column vector. Return a if it is already an AbstractVector. The resulting array shares the same underlying data as a, so it will only be mutable if a is mutable, in which case modifying one will also modify the other.

Examples

julia> a = [1 2 3; 4 5 6]
2×3 Array{Int64,2}:
 1  2  3
 4  5  6
julia> vec(a)
6-element Array{Int64,1}:
 1
 4
 2
 5
 3
 6
julia> vec(1:3)
1:3