# Arrays

## Constructors and Types

`Core.AbstractArray` — Type

    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.

See also: `AbstractVector`, `AbstractMatrix`, `eltype`, `ndims`.

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`Base.AbstractVector` — Type

    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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`Base.AbstractMatrix` — Type

    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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`Base.AbstractVecOrMat` — Type

    AbstractVecOrMat{T}

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

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`Core.Array` — Type

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

`N`-dimensional dense array with elements of type `T`.

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`Core.Array` — Method

    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`. Here `undef` is the `UndefInitializer`.

Examples

    julia> A = Array{Float64, 2}(undef, 2, 3) # N given explicitly
    2×3 Matrix{Float64}:
     6.90198e-310  6.90198e-310  6.90198e-310
     6.90198e-310  6.90198e-310  0.0

    julia> B = Array{Float64}(undef, 4) # N determined by the input
    4-element Vector{Float64}:
       2.360075077e-314
     NaN
       2.2671131793e-314
       2.299821756e-314

    julia> similar(B, 2, 4, 1) # use typeof(B), and the given size
    2×4×1 Array{Float64, 3}:
    [:, :, 1] =
     2.26703e-314  2.26708e-314  0.0           2.80997e-314
     0.0           2.26703e-314  2.26708e-314  0.0

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`Core.Array` — Method

    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 Vector{Union{Nothing, String}}:
     nothing
     nothing

    julia> Array{Union{Nothing, Int}}(nothing, 2, 3)
    2×3 Matrix{Union{Nothing, Int64}}:
     nothing  nothing  nothing
     nothing  nothing  nothing

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`Core.Array` — Method

    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 Vector{Union{Missing, String}}:
     missing
     missing

    julia> Array{Union{Missing, Int}}(missing, 2, 3)
    2×3 Matrix{Union{Missing, Int64}}:
     missing  missing  missing
     missing  missing  missing

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`Core.UndefInitializer` — Type

    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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`Core.undef` — Constant

    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.

See also: `missing`, `similar`.

Examples

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

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`Base.Vector` — Type

    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}`.

See also `empty`, `similar` and `zero` for creating vectors.

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`Base.Vector` — Method

    Vector{T}(undef, n)

Construct an uninitialized `Vector{T}` of length `n`.

Examples

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

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`Base.Vector` — Method

    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 Vector{Union{Nothing, String}}:
     nothing
     nothing

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`Base.Vector` — Method

    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 Vector{Union{Missing, String}}:
     missing
     missing

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`Base.Matrix` — Type

    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}`.

See also `fill`, `zeros`, `undef` and `similar` for creating matrices.

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`Base.Matrix` — Method

    Matrix{T}(undef, m, n)

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

Examples

    julia> Matrix{Float64}(undef, 2, 3)
    2×3 Array{Float64, 2}:
     2.36365e-314  2.28473e-314    5.0e-324
     2.26704e-314  2.26711e-314  NaN

    julia> similar(ans, Int32, 2, 2)
    2×2 Matrix{Int32}:
     490537216  1277177453
             1  1936748399

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`Base.Matrix` — Method

    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 Matrix{Union{Nothing, String}}:
     nothing  nothing  nothing
     nothing  nothing  nothing

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`Base.Matrix` — Method

    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 Matrix{Union{Missing, String}}:
     missing  missing  missing
     missing  missing  missing

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`Base.VecOrMat` — Type

    VecOrMat{T}

Union type of `Vector{T}` and `Matrix{T}` which allows functions to accept either a Matrix or a Vector.

Examples

    julia> Vector{Float64} <: VecOrMat{Float64}
    true

    julia> Matrix{Float64} <: VecOrMat{Float64}
    true

    julia> Array{Float64, 3} <: VecOrMat{Float64}
    false

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`Core.DenseArray` — Type

    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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`Base.DenseVector` — Type

    DenseVector{T}

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

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`Base.DenseMatrix` — Type

    DenseMatrix{T}

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

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`Base.DenseVecOrMat` — Type

    DenseVecOrMat{T}

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

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`Base.StridedArray` — Type

    StridedArray{T, N}

A hard-coded `Union` of common array types that follow the strided array interface, with elements of type `T` and `N` dimensions.

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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`Base.StridedVector` — Type

    StridedVector{T}

One dimensional `StridedArray` with elements of type `T`.

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`Base.StridedMatrix` — Type

    StridedMatrix{T}

Two dimensional `StridedArray` with elements of type `T`.

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`Base.StridedVecOrMat` — Type

    StridedVecOrMat{T}

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

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`Base.getindex` — Method

    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 Vector{Int8}:
     1
     2
     3

    julia> getindex(Int8, 1, 2, 3)
    3-element Vector{Int8}:
     1
     2
     3

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`Base.zeros` — Function

    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`, `zero`.

Examples

    julia> zeros(1)
    1-element Vector{Float64}:
     0.0

    julia> zeros(Int8, 2, 3)
    2×3 Matrix{Int8}:
     0  0  0
     0  0  0

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`Base.ones` — Function

    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 Matrix{Float64}:
     1.0  1.0

    julia> ones(ComplexF64, 2, 3)
    2×3 Matrix{ComplexF64}:
     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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`Base.BitArray` — Type

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

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

`BitArray`s 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`.

Note

Due to its packed storage format, concurrent access to the elements of a `BitArray` where at least one of them is a write is not thread safe.

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`Base.BitArray` — Method

    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 BitMatrix:
     0  0
     0  0

    julia> BitArray(undef, (3, 1))
    3×1 BitMatrix:
     0
     0
     0

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`Base.BitArray` — Method

    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 BitMatrix:
     1  0
     0  1

    julia> BitArray(x+y == 3 for x = 1:2, y = 1:3)
    2×3 BitMatrix:
     0  1  0
     1  0  0

    julia> BitArray(x+y == 3 for x = 1:2 for y = 1:3)
    6-element BitVector:
     0
     1
     0
     1
     0
     0

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`Base.trues` — Function

    trues(dims)

Create a `BitArray` with all values set to `true`.

Examples

    julia> trues(2,3)
    2×3 BitMatrix:
     1  1  1
     1  1  1

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`Base.falses` — Function

    falses(dims)

Create a `BitArray` with all values set to `false`.

Examples

    julia> falses(2,3)
    2×3 BitMatrix:
     0  0  0
     0  0  0

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`Base.fill` — Function

    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`.

See also: `fill!`, `zeros`, `ones`, `similar`.

Examples

    julia> fill(1.0, (2,3))
    2×3 Matrix{Float64}:
     1.0  1.0  1.0
     1.0  1.0  1.0

    julia> fill(42)
    0-dimensional Array{Int64, 0}:
    42

If `x` is an object reference, all elements will refer to the same object:

    julia> A = fill(zeros(2), 2);

    julia> A[1][1] = 42; # modifies both A[1][1] and A[2][1]

    julia> A
    2-element Vector{Vector{Float64}}:
     [42.0, 0.0]
     [42.0, 0.0]

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`Base.fill!` — Function

    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 Matrix{Float64}:
     0.0  0.0  0.0
     0.0  0.0  0.0

    julia> fill!(A, 2.)
    2×3 Matrix{Float64}:
     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 Vector{Vector{Int64}}:
     [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 Vector{Int64}:
     1
     1
     1

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`Base.empty` — Function

    empty(x::Tuple)

Returns an empty tuple, `()`.

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    empty(v::AbstractVector, [eltype])

Create an empty vector similar to `v`, optionally changing the `eltype`.

See also: `empty!`, `isempty`, `isassigned`.

Examples

    julia> empty([1.0, 2.0, 3.0])
    Float64[]

    julia> empty([1.0, 2.0, 3.0], String)
    String[]

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    empty(a::AbstractDict, [index_type=keytype(a)], [value_type=valtype(a)])

Create an empty `AbstractDict` container which can accept indices of type `index_type` and values of type `value_type`. The second and third arguments are optional and default to the input's `keytype` and `valtype`, respectively. (If only one of the two types is specified, it is assumed to be the `value_type`, and the `index_type` we default to `keytype(a)`).

Custom `AbstractDict` subtypes may choose which specific dictionary type is best suited to return for the given index and value types, by specializing on the three-argument signature. The default is to return an empty `Dict`.

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`Base.similar` — Function

    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 `BitArray`s are both mutable and can support 1-dimensional arrays:

    julia> similar(trues(10,10), 2)
    2-element BitVector:
     0
     0

Since `BitArray`s 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

See also: `undef`, `isassigned`.

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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.

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

`Base.ndims` — Function

    ndims(A::AbstractArray) -> Integer

Return the number of dimensions of `A`.

See also: `size`, `axes`.

Examples

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

    julia> ndims(A)
    3

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`Base.size` — Function

    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.

See also: `length`, `ndims`, `eachindex`, `sizeof`.

Examples

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

    julia> size(A)
    (2, 3, 4)

    julia> size(A, 2)
    3

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`Base.axes` — Method

    axes(A)

Return the tuple of valid indices for array `A`.

See also: `size`, `keys`, `eachindex`.

Examples

    julia> A = fill(1, (5,6,7));

    julia> axes(A)
    (Base.OneTo(5), Base.OneTo(6), Base.OneTo(7))

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`Base.axes` — Method

    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)

Usage note

Each of the indices has to be an `AbstractUnitRange{<:Integer}`, but at the same time can be a type that uses custom indices. So, for example, if you need a subset, use generalized indexing constructs like `begin`/`end` or `firstindex`/`lastindex`:

    ix = axes(v, 1)
    ix[2:end]          # will work for eg Vector, but may fail in general
    ix[(begin+1):end]  # works for generalized indexes

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`Base.length` — Method

    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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`Base.keys` — Method

    keys(a::AbstractArray)

Return an efficient array describing all valid indices for `a` arranged in the shape of `a` itself.

They keys of 1-dimensional arrays (vectors) are integers, whereas all other N-dimensional arrays use `CartesianIndex` to describe their locations. Often the special array types `LinearIndices` and `CartesianIndices` are used to efficiently represent these arrays of integers and `CartesianIndex`es, respectively.

Note that the `keys` of an array might not be the most efficient index type; for maximum performance use `eachindex` instead.

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`Base.eachindex` — Function

    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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`Base.IndexStyle` — Type

    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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`Base.IndexLinear` — Type

    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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`Base.IndexCartesian` — Type

    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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`Base.conj!` — Function

    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 Matrix{Complex{Int64}}:
     1+1im  2-1im
     2+2im  3+1im

    julia> conj!(A);

    julia> A
    2×2 Matrix{Complex{Int64}}:
     1-1im  2+1im
     2-2im  3-1im

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`Base.stride` — Function

    stride(A, k::Integer)

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

See also: `strides`.

Examples

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

    julia> stride(A,2)
    3

    julia> stride(A,3)
    12

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`Base.strides` — Function

    strides(A)

Return a tuple of the memory strides in each dimension.

See also: `stride`.

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`.

`Base.Broadcast.broadcast` — Function

    broadcast(f, As...)

Broadcast the function `f` over the arrays, tuples, collections, `Ref`s 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 `Number`s, `String`s, `Symbol`s, `Type`s, `Function`s 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 Vector{Int64}:
     1
     2
     3
     4
     5

    julia> B = [1 2; 3 4; 5 6; 7 8; 9 10]
    5×2 Matrix{Int64}:
     1   2
     3   4
     5   6
     7   8
     9  10

    julia> broadcast(+, A, B)
    5×2 Matrix{Int64}:
      2   3
      5   6
      8   9
     11  12
     14  15

    julia> parse.(Int, ["1", "2"])
    2-element Vector{Int64}:
     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 Vector{Vector{Int64}}:
     [1, 1]
     [2, 2]

    julia> string.(("one","two","three","four"), ": ", 1:4)
    4-element Vector{String}:
     "one: 1"
     "two: 2"
     "three: 3"
     "four: 4"

source
`Base.Broadcast.broadcast!` — Function

    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 Vector{Float64}:
      1.0
     -2.0

    julia> A
    2-element Vector{Float64}:
     1.0
     0.0

    julia> broadcast!(+, A, A, (0, -2.0));

    julia> A
    2-element Vector{Float64}:
      1.0
     -2.0

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`Base.Broadcast.@__dot__` — Macro

    @. 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 Vector{Float64}:
     3.5244129544236893
     4.727892280477045
     3.4233600241796016

source

For specializing broadcast on custom types, see

`Base.Broadcast.BroadcastStyle` — Type

`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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`Base.Broadcast.AbstractArrayStyle` — Type

`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 `AbstractArrayStyle`s 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 `Array`s. 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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`Base.Broadcast.ArrayStyle` — Type

`Broadcast.ArrayStyle{MyArrayType}()` is a `BroadcastStyle` indicating that an object behaves as an array for broadcasting. It presents a simple way to construct `Broadcast.AbstractArrayStyle`s 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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`Base.Broadcast.DefaultArrayStyle` — Type

`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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`Base.Broadcast.broadcastable` — Function

    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 Vector{Int64}:
     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")

source
`Base.Broadcast.combine_axes` — Function

    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)
    ()

source
`Base.Broadcast.combine_styles` — Function

    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}()

source
`Base.Broadcast.result_style` — Function

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

Takes one or two `BroadcastStyle`s 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

`Base.getindex` — Method

    getindex(A, inds...)

Return a subset of array `A` as specified by `inds`, where each `ind` may be, for example, 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 Matrix{Int64}:
     1  2
     3  4

    julia> getindex(A, 1)
    1

    julia> getindex(A, [2, 1])
    2-element Vector{Int64}:
     3
     1

    julia> getindex(A, 2:4)
    3-element Vector{Int64}:
     3
     2
     4

source
`Base.setindex!` — Method

    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...); X)`.

Examples

    julia> A = zeros(2,2);

    julia> setindex!(A, [10, 20], [1, 2]);

    julia> A[[3, 4]] = [30, 40];

    julia> A
    2×2 Matrix{Float64}:
     10.0  30.0
     20.0  40.0

source
`Base.copyto!` — Method

    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.

source
`Base.copy!` — Function

    copy!(dst, src) -> dst

In-place `copy` of `src` into `dst`, discarding any pre-existing elements in `dst`. If `dst` and `src` are of the same type, `dst == src` should hold after the call. If `dst` and `src` are multidimensional arrays, they must have equal `axes`.

See also `copyto!`.

Julia 1.1

This method requires at least Julia 1.1. In Julia 1.0 this method is available from the `Future` standard library as `Future.copy!`.

source
`Base.isassigned` — Function

    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 Vector{Foo}:
     #undef
     #undef
     #undef

    julia> isassigned(v, 1)
    false

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`Base.Colon` — Type

    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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`Base.IteratorsMD.CartesianIndex` — Type

    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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`Base.IteratorsMD.CartesianIndices` — Type

    CartesianIndices(sz::Dims) -> R
    CartesianIndices((istart:[istep:]istop, jstart:[jstep:]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:jstep:jstop
        for i = istart:istep: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.

Julia 1.6

The step range method `CartesianIndices((istart:istep:istop, jstart:[jstep:]jstop, ...))` requires at least Julia 1.6.

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)

    julia> cartesian = CartesianIndices((1:2:5, 1:2))
    3×2 CartesianIndices{2, Tuple{StepRange{Int64, Int64}, UnitRange{Int64}}}:
     CartesianIndex(1, 1)  CartesianIndex(1, 2)
     CartesianIndex(3, 1)  CartesianIndex(3, 2)
     CartesianIndex(5, 1)  CartesianIndex(5, 2)

    julia> cartesian[2, 2]
    CartesianIndex(3, 2)

Broadcasting

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

Julia 1.1

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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`Base.Dims` — Type

    Dims{N}

An `NTuple` of `N` `Int`s used to represent the dimensions of an `AbstractArray`.

source
`Base.LinearIndices` — Type

    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 `AbstractVector`s 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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`Base.to_indices` — Function

    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 `Int`s or `AbstractArray`s 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`.

source
`Base.checkbounds` — Function

    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

source

    checkbounds(A, I...)

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

source
`Base.checkindex` — Function

    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.

See also `checkbounds`.

Examples

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

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

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`Base.elsize` — Function

    elsize(type)

Compute the memory stride in bytes between consecutive elements of `eltype` stored inside the given `type`, if the array elements are stored densely with a uniform linear stride.

source

## Views (SubArrays and other view types)

A “view” is a data structure that acts like an array (it is a subtype of `AbstractArray`), but the underlying data is actually part of another array.

For example, if `x` is an array and `v = @view x[1:10]`, then `v` acts like a 10-element array, but its data is actually accessing the first 10 elements of `x`. Writing to a view, e.g. `v[3] = 2`, writes directly to the underlying array `x` (in this case modifying `x[3]`).

Slicing operations like `x[1:10]` create a copy by default in Julia. `@view x[1:10]` changes it to make a view. The `@views` macro can be used on a whole block of code (e.g. `@views function foo() .... end` or `@views begin ... end`) to change all the slicing operations in that block to use views. Sometimes making a copy of the data is faster and sometimes using a view is faster, as described in the performance tips.

`Base.view` — Function

    view(A, inds...)

Like `getindex`, but returns a lightweight array that lazily references (or is effectively a view into) the parent array `A` at the given index or indices `inds` instead of eagerly extracting elements or constructing a copied subset. Calling `getindex` or `setindex!` on the returned value (often a `SubArray`) computes the indices to access or modify the parent array on the fly. The behavior is undefined if the shape of the parent array is changed after `view` is called because there is no bound check for the parent array; e.g., it may cause a segmentation fault.

Some immutable parent arrays (like ranges) may choose to simply recompute a new array in some circumstances instead of returning a `SubArray` if doing so is efficient and provides compatible semantics.

Julia 1.6

In Julia 1.6 or later, `view` can be called on an `AbstractString`, returning a `SubString`.

Examples

    julia> A = [1 2; 3 4]
    2×2 Matrix{Int64}:
     1  2
     3  4

    julia> b = view(A, :, 1)
    2-element view(::Matrix{Int64}, :, 1) with eltype Int64:
     1
     3

    julia> fill!(b, 0)
    2-element view(::Matrix{Int64}, :, 1) with eltype Int64:
     0
     0

    julia> A # Note A has changed even though we modified b
    2×2 Matrix{Int64}:
     0  2
     0  4

    julia> view(2:5, 2:3) # returns a range as type is immutable
    3:4

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`Base.@view` — Macro

    @view A[inds...]

Transform the indexing expression `A[inds...]` into the equivalent `view` call.

This can only be applied directly to a single indexing expression and is particularly helpful for expressions that include the special `begin` or `end` indexing syntaxes like `A[begin, 2:end-1]` (as those are not supported by the normal `view` function).

Note that `@view` cannot be used as the target of a regular assignment (e.g., `@view(A[1, 2:end]) = ...`), nor would the un-decorated indexed assignment (`A[1, 2:end] = ...`) or broadcasted indexed assignment (`A[1, 2:end] .= ...`) make a copy. It can be useful, however, for updating broadcasted assignments like `@view(A[1, 2:end]) .+= 1` because this is a simple syntax for `@view(A[1, 2:end]) .= @view(A[1, 2:end]) + 1`, and the indexing expression on the right-hand side would otherwise make a copy without the `@view`.

See also `@views` to switch an entire block of code to use views for non-scalar indexing.

Julia 1.5

Using `begin` in an indexing expression to refer to the first index requires at least Julia 1.5.

Examples

    julia> A = [1 2; 3 4]
    2×2 Matrix{Int64}:
     1  2
     3  4

    julia> b = @view A[:, 1]
    2-element view(::Matrix{Int64}, :, 1) with eltype Int64:
     1
     3

    julia> fill!(b, 0)
    2-element view(::Matrix{Int64}, :, 1) with eltype Int64:
     0
     0

    julia> A
    2×2 Matrix{Int64}:
     0  2
     0  4

source
`Base.@views` — Macro

    @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.

Note

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.

Julia 1.5

Using `begin` in an indexing expression to refer to the first index requires at least Julia 1.5.

Examples

    julia> A = zeros(3, 3);

    julia> @views for row in 1:3
               b = A[row, :]
               b[:] .= row
           end

    julia> A
    3×3 Matrix{Float64}:
     1.0  1.0  1.0
     2.0  2.0  2.0
     3.0  3.0  3.0

source
`Base.parent` — Function

    parent(A)

Return the underlying "parent array”. This parent array of objects of types `SubArray`, `ReshapedArray` or `LinearAlgebra.Transpose` is what was passed as an argument to `view`, `reshape`, `transpose`, etc. during object creation. If the input is not a wrapped object, return the input itself. If the input is wrapped multiple times, only the outermost wrapper will be removed.

Examples

    julia> A = [1 2; 3 4]
    2×2 Matrix{Int64}:
     1  2
     3  4

    julia> V = view(A, 1:2, :)
    2×2 view(::Matrix{Int64}, 1:2, :) with eltype Int64:
     1  2
     3  4

    julia> parent(V)
    2×2 Matrix{Int64}:
     1  2
     3  4

source
`Base.parentindices` — Function

    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(::Matrix{Int64}, 1, :) with eltype Int64:
     1
     2

    julia> parentindices(V)
    (1, Base.Slice(Base.OneTo(2)))

source
`Base.selectdim` — Function

    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`.

See also: `eachslice`.

Examples

    julia> A = [1 2 3 4; 5 6 7 8]
    2×4 Matrix{Int64}:
     1  2  3  4
     5  6  7  8

    julia> selectdim(A, 2, 3)
    2-element view(::Matrix{Int64}, :, 3) with eltype Int64:
     3
     7

    julia> selectdim(A, 2, 3:4)
    2×2 view(::Matrix{Int64}, :, 3:4) with eltype Int64:
     3  4
     7  8

source
`Base.reinterpret` — Function

    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, ::Matrix{UInt32}):
     1.0f-45  3.0f-45  4.0f-45  6.0f-45  7.0f-45

source

    reinterpret(reshape, T, A::AbstractArray{S}) -> B

Change the type-interpretation of `A` while consuming or adding a "channel dimension."

If `sizeof(T) = n*sizeof(S)` for `n>1`, `A`'s first dimension must be of size `n` and `B` lacks `A`'s first dimension. Conversely, if `sizeof(S) = n*sizeof(T)` for `n>1`, `B` gets a new first dimension of size `n`. The dimensionality is unchanged if `sizeof(T) == sizeof(S)`.

Julia 1.6

This method requires at least Julia 1.6.

Examples

    julia> A = [1 2; 3 4]
    2×2 Matrix{Int32}:
     1  2
     3  4

    julia> reinterpret(reshape, Complex{Int}, A)    # the result is a vector
    2-element reinterpret(reshape, Complex{Int32}, ::Matrix{Int32}) with eltype Complex{Int32}:
     1 + 3im
     2 + 4im

    julia> a = [(1,2,3), (4,5,6)]
    2-element Vector{Tuple{Int32, Int32, Int32}}:
     (1, 2, 3)
     (4, 5, 6)

    julia> reinterpret(reshape, Int, a)             # the result is a matrix
    3×2 reinterpret(reshape, Int32, ::Vector{Tuple{Int32, Int32, Int32}}) with eltype Int32:
     1  4
     2  5
     3  6

source
`Base.reshape` — Function

    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 Vector{Int64}:
      1
      2
      3
      4
      5
      6
      7
      8
      9
     10
     11
     12
     13
     14
     15
     16

    julia> reshape(A, (4, 4))
    4×4 Matrix{Int64}:
     1  5   9  13
     2  6  10  14
     3  7  11  15
     4  8  12  16

    julia> reshape(A, 2, :)
    2×8 Matrix{Int64}:
     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

source
`Base.dropdims` — Function

    dropdims(A; dims)

Return an array with the same data as `A`, but with the dimensions specified by `dims` removed. `size(A,d)` must equal 1 for every `d` in `dims`, and repeated dimensions or numbers outside `1:ndims(A)` are forbidden.

The result shares the same underlying data as `A`, such that the result is mutable if and only if `A` is mutable, and setting elements of one alters the values of the other.

See also: `reshape`, `vec`.

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

    julia> b[1,1,1] = 5; a
    2×2×1×1 Array{Int64, 4}:
    [:, :, 1, 1] =
     5  3
     2  4

source
`Base.vec` — Function

    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 Matrix{Int64}:
     1  2  3
     4  5  6

    julia> vec(a)
    6-element Vector{Int64}:
     1
     4
     2
     5
     3
     6

    julia> vec(1:3)
    1:3

See also `reshape`, `dropdims`.

source
`Base.SubArray` — Type

    SubArray{T,N,P,I,L} <: AbstractArray{T,N}

`N`-dimensional view into a parent array (of type `P`) with an element type `T`, restricted by a tuple of indices (of type `I`). `L` is true for types that support fast linear indexing, and `false` otherwise.

Construct `SubArray`s using the `view` function.

source

## Concatenation and permutation

`Base.cat` — Function

    cat(A...; dims)

Concatenate the input arrays along the specified dimensions in the iterable `dims`. For dimensions not in `dims`, all input arrays should have the same size, which will also be the size of the output array along that dimension. For dimensions in `dims`, the size of the output array is the sum of the sizes of the input arrays along that dimension. If `dims` is a single number, the different arrays are tightly stacked along that dimension. If `dims` is an iterable containing several dimensions, this allows one to construct block diagonal matrices and their higher-dimensional analogues by simultaneously increasing several dimensions for every new input array and putting zero blocks elsewhere. For example, `cat(matrices...; dims=(1,2))` builds a block diagonal matrix, i.e. a block matrix with `matrices[1]`, `matrices[2]`, ... as diagonal blocks and matching zero blocks away from the diagonal.

See also `hcat`, `vcat`, `hvcat`, `repeat`.

Examples

    julia> cat([1 2; 3 4], [pi, pi], fill(10, 2,3,1); dims=2)
    2×6×1 Array{Float64, 3}:
    [:, :, 1] =
     1.0  2.0  3.14159  10.0  10.0  10.0
     3.0  4.0  3.14159  10.0  10.0  10.0

    julia> cat(true, trues(2,2), trues(4)', dims=(1,2))
    4×7 Matrix{Bool}:
     1  0  0  0  0  0  0
     0  1  1  0  0  0  0
     0  1  1  0  0  0  0
     0  0  0  1  1  1  1

source
`Base.vcat` — Function

    vcat(A...)

Concatenate along dimension 1.

Examples

    julia> a = [1 2 3 4 5]
    1×5 Matrix{Int64}:
     1  2  3  4  5

    julia> b = [6 7 8 9 10; 11 12 13 14 15]
    2×5 Matrix{Int64}:
      6   7   8   9  10
     11  12  13  14  15

    julia> vcat(a,b)
    3×5 Matrix{Int64}:
      1   2   3   4   5
      6   7   8   9  10
     11  12  13  14  15

    julia> c = ([1 2 3], [4 5 6])
    ([1 2 3], [4 5 6])

    julia> vcat(c...)
    2×3 Matrix{Int64}:
     1  2  3
     4  5  6

source
`Base.hcat` — Function

    hcat(A...)

Concatenate along dimension 2.

Examples

    julia> a = [1; 2; 3; 4; 5]
    5-element Vector{Int64}:
     1
     2
     3
     4
     5

    julia> b = [6 7; 8 9; 10 11; 12 13; 14 15]
    5×2 Matrix{Int64}:
      6   7
      8   9
     10  11
     12  13
     14  15

    julia> hcat(a,b)
    5×3 Matrix{Int64}:
     1   6   7
     2   8   9
     3  10  11
     4  12  13
     5  14  15

    julia> c = ([1; 2; 3], [4; 5; 6])
    ([1, 2, 3], [4, 5, 6])

    julia> hcat(c...)
    3×2 Matrix{Int64}:
     1  4
     2  5
     3  6

    julia> x = Matrix(undef, 3, 0)  # x = [] would have created an Array{Any, 1}, but need an Array{Any, 2}
    3×0 Matrix{Any}

    julia> hcat(x, [1; 2; 3])
    3×1 Matrix{Any}:
     1
     2
     3

source
`Base.hvcat` — Function

    hvcat(rows::Tuple{Vararg{Int}}, values...)

Horizontal and vertical concatenation in one call. This function is called for block matrix syntax. The first argument specifies the number of arguments to concatenate in each block row.

Examples

    julia> a, b, c, d, e, f = 1, 2, 3, 4, 5, 6
    (1, 2, 3, 4, 5, 6)

    julia> [a b c; d e f]
    2×3 Matrix{Int64}:
     1  2  3
     4  5  6

    julia> hvcat((3,3), a,b,c,d,e,f)
    2×3 Matrix{Int64}:
     1  2  3
     4  5  6

    julia> [a b; c d; e f]
    3×2 Matrix{Int64}:
     1  2
     3  4
     5  6

    julia> hvcat((2,2,2), a,b,c,d,e,f)
    3×2 Matrix{Int64}:
     1  2
     3  4
     5  6

If the first argument is a single integer `n`, then all block rows are assumed to have `n` block columns.

source
`Base.hvncat` — Function

    hvncat(dim::Int, row_first, values...)
    hvncat(dims::Tuple{Vararg{Int}}, row_first, values...)
    hvncat(shape::Tuple{Vararg{Tuple}}, row_first, values...)

Horizontal, vertical, and n-dimensional concatenation of many `values` in one call.

This function is called for block matrix syntax. The first argument either specifies the shape of the concatenation, similar to `hvcat`, as a tuple of tuples, or the dimensions that specify the key number of elements along each axis, and is used to determine the output dimensions. The `dims` form is more performant, and is used by default when the concatenation operation has the same number of elements along each axis (e.g., [a b; c d;;; e f ; g h]). The `shape` form is used when the number of elements along each axis is unbalanced (e.g., [a b ; c]). Unbalanced syntax needs additional validation overhead. The `dim` form is an optimization for concatenation along just one dimension. `row_first` indicates how `values` are ordered. The meaning of the first and second elements of `shape` are also swapped based on `row_first`.

Examples

    julia> a, b, c, d, e, f = 1, 2, 3, 4, 5, 6
    (1, 2, 3, 4, 5, 6)

    julia> [a b c;;; d e f]
    1×3×2 Array{Int64, 3}:
    [:, :, 1] =
     1  2  3

    [:, :, 2] =
     4  5  6

    julia> hvncat((2,1,3), false, a,b,c,d,e,f)
    2×1×3 Array{Int64, 3}:
    [:, :, 1] =
     1
     2

    [:, :, 2] =
     3
     4

    [:, :, 3] =
     5
     6

    julia> [a b;;; c d;;; e f]
    1×2×3 Array{Int64, 3}:
    [:, :, 1] =
     1  2

    [:, :, 2] =
     3  4

    [:, :, 3] =
     5  6

    julia> hvncat(((3, 3), (3, 3), (6,)), true, a, b, c, d, e, f)
    1×3×2 Array{Int64, 3}:
    [:, :, 1] =
     1  2  3

    [:, :, 2] =
     4  5  6

Examples for construction of the arguments:

[a b c ; d e f ;;; g h i ; j k l ;;; m n o ; p q r ;;; s t u ; v w x] => dims = (2, 3, 4)

[a b ; c ;;; d ;;;;] ___ _ _ 2 1 1 = elements in each row (2, 1, 1) _______ _ 3 1 = elements in each column (3, 1) _____________ 4 = elements in each 3d slice (4,) _____________ 4 = elements in each 4d slice (4,) => shape = ((2, 1, 1), (3, 1), (4,), (4,)) with `rowfirst` = true

source
`Base.vect` — Function

    vect(X...)

Create a `Vector` with element type computed from the `promote_typeof` of the argument, containing the argument list.

Examples

    julia> a = Base.vect(UInt8(1), 2.5, 1//2)
    3-element Vector{Float64}:
     1.0
     2.5
     0.5

source
`Base.circshift` — Function

    circshift(A, shifts)

Circularly shift, i.e. rotate, the data in an array. The second argument is a tuple or vector giving the amount to shift in each dimension, or an integer to shift only in the first dimension.

See also: `circshift!`, `circcopy!`, `bitrotate`, `<<`.

Examples

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

    julia> circshift(b, (0,2))
    4×4 Matrix{Int64}:
      9  13  1  5
     10  14  2  6
     11  15  3  7
     12  16  4  8

    julia> circshift(b, (-1,0))
    4×4 Matrix{Int64}:
     2  6  10  14
     3  7  11  15
     4  8  12  16
     1  5   9  13

    julia> a = BitArray([true, true, false, false, true])
    5-element BitVector:
     1
     1
     0
     0
     1

    julia> circshift(a, 1)
    5-element BitVector:
     1
     1
     1
     0
     0

    julia> circshift(a, -1)
    5-element BitVector:
     1
     0
     0
     1
     1

source
`Base.circshift!` — Function

    circshift!(dest, src, shifts)

Circularly shift, i.e. rotate, the data in `src`, storing the result in `dest`. `shifts` specifies the amount to shift in each dimension.

The `dest` array must be distinct from the `src` array (they cannot alias each other).

See also `circshift`.

source
`Base.circcopy!` — Function

    circcopy!(dest, src)

Copy `src` to `dest`, indexing each dimension modulo its length. `src` and `dest` must have the same size, but can be offset in their indices; any offset results in a (circular) wraparound. If the arrays have overlapping indices, then on the domain of the overlap `dest` agrees with `src`.

See also: `circshift`.

Examples

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

    julia> dest = OffsetArray{Int}(undef, (0:3,2:5))

    julia> circcopy!(dest, src)
    OffsetArrays.OffsetArray{Int64,2,Array{Int64,2}} with indices 0:3×2:5:
     8  12  16  4
     5   9  13  1
     6  10  14  2
     7  11  15  3

    julia> dest[1:3,2:4] == src[1:3,2:4]
    true

source
`Base.findall` — Method

    findall(A)

Return a vector `I` of the `true` indices or keys of `A`. If there are no such elements of `A`, return an empty array. To search for other kinds of values, pass a predicate as the first argument.

Indices or keys are of the same type as those returned by `keys(A)` and `pairs(A)`.

See also: `findfirst`, `searchsorted`.

Examples

    julia> A = [true, false, false, true]
    4-element Vector{Bool}:
     1
     0
     0
     1

    julia> findall(A)
    2-element Vector{Int64}:
     1
     4

    julia> A = [true false; false true]
    2×2 Matrix{Bool}:
     1  0
     0  1

    julia> findall(A)
    2-element Vector{CartesianIndex{2}}:
     CartesianIndex(1, 1)
     CartesianIndex(2, 2)

    julia> findall(falses(3))
    Int64[]

source
`Base.findall` — Method

    findall(f::Function, A)

Return a vector `I` of the indices or keys of `A` where `f(A[I])` returns `true`. If there are no such elements of `A`, return an empty array.

Indices or keys are of the same type as those returned by `keys(A)` and `pairs(A)`.

Examples

    julia> x = [1, 3, 4]
    3-element Vector{Int64}:
     1
     3
     4

    julia> findall(isodd, x)
    2-element Vector{Int64}:
     1
     2

    julia> A = [1 2 0; 3 4 0]
    2×3 Matrix{Int64}:
     1  2  0
     3  4  0
    julia> findall(isodd, A)
    2-element Vector{CartesianIndex{2}}:
     CartesianIndex(1, 1)
     CartesianIndex(2, 1)

    julia> findall(!iszero, A)
    4-element Vector{CartesianIndex{2}}:
     CartesianIndex(1, 1)
     CartesianIndex(2, 1)
     CartesianIndex(1, 2)
     CartesianIndex(2, 2)

    julia> d = Dict(:A => 10, :B => -1, :C => 0)
    Dict{Symbol, Int64} with 3 entries:
      :A => 10
      :B => -1
      :C => 0

    julia> findall(x -> x >= 0, d)
    2-element Vector{Symbol}:
     :A
     :C

source
`Base.findfirst` — Method

    findfirst(A)

Return the index or key of the first `true` value in `A`. Return `nothing` if no such value is found. To search for other kinds of values, pass a predicate as the first argument.

Indices or keys are of the same type as those returned by `keys(A)` and `pairs(A)`.

See also: `findall`, `findnext`, `findlast`, `searchsortedfirst`.

Examples

    julia> A = [false, false, true, false]
    4-element Vector{Bool}:
     0
     0
     1
     0

    julia> findfirst(A)
    3

    julia> findfirst(falses(3)) # returns nothing, but not printed in the REPL

    julia> A = [false false; true false]
    2×2 Matrix{Bool}:
     0  0
     1  0

    julia> findfirst(A)
    CartesianIndex(2, 1)

source
`Base.findfirst` — Method

    findfirst(predicate::Function, A)

Return the index or key of the first element of `A` for which `predicate` returns `true`. Return `nothing` if there is no such element.

Indices or keys are of the same type as those returned by `keys(A)` and `pairs(A)`.

Examples

    julia> A = [1, 4, 2, 2]
    4-element Vector{Int64}:
     1
     4
     2
     2

    julia> findfirst(iseven, A)
    2

    julia> findfirst(x -> x>10, A) # returns nothing, but not printed in the REPL

    julia> findfirst(isequal(4), A)
    2

    julia> A = [1 4; 2 2]
    2×2 Matrix{Int64}:
     1  4
     2  2

    julia> findfirst(iseven, A)
    CartesianIndex(2, 1)

source
`Base.findlast` — Method

    findlast(A)

Return the index or key of the last `true` value in `A`. Return `nothing` if there is no `true` value in `A`.

Indices or keys are of the same type as those returned by `keys(A)` and `pairs(A)`.

See also: `findfirst`, `findprev`, `findall`.

Examples

    julia> A = [true, false, true, false]
    4-element Vector{Bool}:
     1
     0
     1
     0

    julia> findlast(A)
    3

    julia> A = falses(2,2);

    julia> findlast(A) # returns nothing, but not printed in the REPL

    julia> A = [true false; true false]
    2×2 Matrix{Bool}:
     1  0
     1  0

    julia> findlast(A)
    CartesianIndex(2, 1)

source
`Base.findlast` — Method

    findlast(predicate::Function, A)

Return the index or key of the last element of `A` for which `predicate` returns `true`. Return `nothing` if there is no such element.

Indices or keys are of the same type as those returned by `keys(A)` and `pairs(A)`.

Examples

    julia> A = [1, 2, 3, 4]
    4-element Vector{Int64}:
     1
     2
     3
     4

    julia> findlast(isodd, A)
    3

    julia> findlast(x -> x > 5, A) # returns nothing, but not printed in the REPL

    julia> A = [1 2; 3 4]
    2×2 Matrix{Int64}:
     1  2
     3  4

    julia> findlast(isodd, A)
    CartesianIndex(2, 1)

source
`Base.findnext` — Method

    findnext(A, i)

Find the next index after or including `i` of a `true` element of `A`, or `nothing` if not found.

Indices are of the same type as those returned by `keys(A)` and `pairs(A)`.

Examples

    julia> A = [false, false, true, false]
    4-element Vector{Bool}:
     0
     0
     1
     0

    julia> findnext(A, 1)
    3

    julia> findnext(A, 4) # returns nothing, but not printed in the REPL

    julia> A = [false false; true false]
    2×2 Matrix{Bool}:
     0  0
     1  0

    julia> findnext(A, CartesianIndex(1, 1))
    CartesianIndex(2, 1)

source
`Base.findnext` — Method

    findnext(predicate::Function, A, i)

Find the next index after or including `i` of an element of `A` for which `predicate` returns `true`, or `nothing` if not found.

Indices are of the same type as those returned by `keys(A)` and `pairs(A)`.

Examples

    julia> A = [1, 4, 2, 2];

    julia> findnext(isodd, A, 1)
    1

    julia> findnext(isodd, A, 2) # returns nothing, but not printed in the REPL

    julia> A = [1 4; 2 2];

    julia> findnext(isodd, A, CartesianIndex(1, 1))
    CartesianIndex(1, 1)

source
`Base.findprev` — Method

    findprev(A, i)

Find the previous index before or including `i` of a `true` element of `A`, or `nothing` if not found.

Indices are of the same type as those returned by `keys(A)` and `pairs(A)`.

See also: `findnext`, `findfirst`, `findall`.

Examples

    julia> A = [false, false, true, true]
    4-element Vector{Bool}:
     0
     0
     1
     1

    julia> findprev(A, 3)
    3

    julia> findprev(A, 1) # returns nothing, but not printed in the REPL

    julia> A = [false false; true true]
    2×2 Matrix{Bool}:
     0  0
     1  1

    julia> findprev(A, CartesianIndex(2, 1))
    CartesianIndex(2, 1)

source
`Base.findprev` — Method

    findprev(predicate::Function, A, i)

Find the previous index before or including `i` of an element of `A` for which `predicate` returns `true`, or `nothing` if not found.

Indices are of the same type as those returned by `keys(A)` and `pairs(A)`.

Examples

    julia> A = [4, 6, 1, 2]
    4-element Vector{Int64}:
     4
     6
     1
     2

    julia> findprev(isodd, A, 1) # returns nothing, but not printed in the REPL

    julia> findprev(isodd, A, 3)
    3

    julia> A = [4 6; 1 2]
    2×2 Matrix{Int64}:
     4  6
     1  2

    julia> findprev(isodd, A, CartesianIndex(1, 2))
    CartesianIndex(2, 1)

source
`Base.permutedims` — Function

    permutedims(A::AbstractArray, perm)

Permute the dimensions of array `A`. `perm` is a vector or a tuple of length `ndims(A)` specifying the permutation.

See also `permutedims!`, `PermutedDimsArray`, `transpose`, `invperm`.

Examples

    julia> A = reshape(Vector(1:8), (2,2,2))
    2×2×2 Array{Int64, 3}:
    [:, :, 1] =
     1  3
     2  4

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

    julia> permutedims(A, (3, 2, 1))
    2×2×2 Array{Int64, 3}:
    [:, :, 1] =
     1  3
     5  7

    [:, :, 2] =
     2  4
     6  8

    julia> B = randn(5, 7, 11, 13);

    julia> perm = [4,1,3,2];

    julia> size(permutedims(B, perm))
    (13, 5, 11, 7)

    julia> size(B)[perm] == ans
    true

source

    permutedims(m::AbstractMatrix)

Permute the dimensions of the matrix `m`, by flipping the elements across the diagonal of the matrix. Differs from `LinearAlgebra`'s `transpose` in that the operation is not recursive.

Examples

    julia> a = [1 2; 3 4];

    julia> b = [5 6; 7 8];

    julia> c = [9 10; 11 12];

    julia> d = [13 14; 15 16];

    julia> X = [[a] [b]; [c] [d]]
    2×2 Matrix{Matrix{Int64}}:
     [1 2; 3 4]     [5 6; 7 8]
     [9 10; 11 12]  [13 14; 15 16]

    julia> permutedims(X)
    2×2 Matrix{Matrix{Int64}}:
     [1 2; 3 4]  [9 10; 11 12]
     [5 6; 7 8]  [13 14; 15 16]

    julia> transpose(X)
    2×2 transpose(::Matrix{Matrix{Int64}}) with eltype Transpose{Int64, Matrix{Int64}}:
     [1 3; 2 4]  [9 11; 10 12]
     [5 7; 6 8]  [13 15; 14 16]

source

    permutedims(v::AbstractVector)

Reshape vector `v` into a `1 × length(v)` row matrix. Differs from `LinearAlgebra`'s `transpose` in that the operation is not recursive.

Examples

    julia> permutedims([1, 2, 3, 4])
    1×4 Matrix{Int64}:
     1  2  3  4

    julia> V = [[[1 2; 3 4]]; [[5 6; 7 8]]]
    2-element Vector{Matrix{Int64}}:
     [1 2; 3 4]
     [5 6; 7 8]

    julia> permutedims(V)
    1×2 Matrix{Matrix{Int64}}:
     [1 2; 3 4]  [5 6; 7 8]

    julia> transpose(V)
    1×2 transpose(::Vector{Matrix{Int64}}) with eltype Transpose{Int64, Matrix{Int64}}:
     [1 3; 2 4]  [5 7; 6 8]

source
`Base.permutedims!` — Function

    permutedims!(dest, src, perm)

Permute the dimensions of array `src` and store the result in the array `dest`. `perm` is a vector specifying a permutation of length `ndims(src)`. The preallocated array `dest` should have `size(dest) == size(src)[perm]` and is completely overwritten. No in-place permutation is supported and unexpected results will happen if `src` and `dest` have overlapping memory regions.

See also `permutedims`.

source
`Base.PermutedDimsArrays.PermutedDimsArray` — Type

    PermutedDimsArray(A, perm) -> B

Given an AbstractArray `A`, create a view `B` such that the dimensions appear to be permuted. Similar to `permutedims`, except that no copying occurs (`B` shares storage with `A`).

See also `permutedims`, `invperm`.

Examples

    julia> A = rand(3,5,4);

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

    julia> size(B)
    (4, 3, 5)

    julia> B[3,1,2] == A[1,2,3]
    true

source
`Base.promote_shape` — Function

    promote_shape(s1, s2)

Check two array shapes for compatibility, allowing trailing singleton dimensions, and return whichever shape has more dimensions.

Examples

    julia> a = fill(1, (3,4,1,1,1));

    julia> b = fill(1, (3,4));

    julia> promote_shape(a,b)
    (Base.OneTo(3), Base.OneTo(4), Base.OneTo(1), Base.OneTo(1), Base.OneTo(1))

    julia> promote_shape((2,3,1,4), (2, 3, 1, 4, 1))
    (2, 3, 1, 4, 1)

source

## Array functions

`Base.accumulate` — Function

    accumulate(op, A; dims::Integer, [init])

Cumulative operation `op` along the dimension `dims` of `A` (providing `dims` is optional for vectors). An initial value `init` may optionally be provided by a keyword argument. See also `accumulate!` to use a preallocated output array, both for performance and to control the precision of the output (e.g. to avoid overflow). For common operations there are specialized variants of `accumulate`, see: `cumsum`, `cumprod`

Julia 1.5

`accumulate` on a non-array iterator requires at least Julia 1.5.

Examples

    julia> accumulate(+, [1,2,3])
    3-element Vector{Int64}:
     1
     3
     6

    julia> accumulate(*, [1,2,3])
    3-element Vector{Int64}:
     1
     2
     6

    julia> accumulate(+, [1,2,3]; init=100)
    3-element Vector{Int64}:
     101
     103
     106

    julia> accumulate(min, [1,2,-1]; init=0)
    3-element Vector{Int64}:
      0
      0
     -1

    julia> accumulate(+, fill(1, 3, 3), dims=1)
    3×3 Matrix{Int64}:
     1  1  1
     2  2  2
     3  3  3

    julia> accumulate(+, fill(1, 3, 3), dims=2)
    3×3 Matrix{Int64}:
     1  2  3
     1  2  3
     1  2  3

source
`Base.accumulate!` — Function

    accumulate!(op, B, A; [dims], [init])

Cumulative operation `op` on `A` along the dimension `dims`, storing the result in `B`. Providing `dims` is optional for vectors. If the keyword argument `init` is given, its value is used to instantiate the accumulation. See also `accumulate`.

Examples

    julia> x = [1, 0, 2, 0, 3];

    julia> y = [0, 0, 0, 0, 0];

    julia> accumulate!(+, y, x);

    julia> y
    5-element Vector{Int64}:
     1
     1
     3
     3
     6

    julia> A = [1 2; 3 4];

    julia> B = [0 0; 0 0];

    julia> accumulate!(-, B, A, dims=1);

    julia> B
    2×2 Matrix{Int64}:
      1   2
     -2  -2

    julia> accumulate!(-, B, A, dims=2);

    julia> B
    2×2 Matrix{Int64}:
     1  -1
     3  -1

source
`Base.cumprod` — Function

    cumprod(A; dims::Integer)

Cumulative product along the dimension `dim`. See also `cumprod!` to use a preallocated output array, both for performance and to control the precision of the output (e.g. to avoid overflow).

Examples

    julia> a = [1 2 3; 4 5 6]
    2×3 Matrix{Int64}:
     1  2  3
     4  5  6

    julia> cumprod(a, dims=1)
    2×3 Matrix{Int64}:
     1   2   3
     4  10  18

    julia> cumprod(a, dims=2)
    2×3 Matrix{Int64}:
     1   2    6
     4  20  120

source

    cumprod(itr)

Cumulative product of an iterator. See also `cumprod!` to use a preallocated output array, both for performance and to control the precision of the output (e.g. to avoid overflow).

See also `cumprod!`, `accumulate`, `cumsum`.

Julia 1.5

`cumprod` on a non-array iterator requires at least Julia 1.5.

Examples

    julia> cumprod(fill(1//2, 3))
    3-element Vector{Rational{Int64}}:
     1//2
     1//4
     1//8

    julia> cumprod([fill(1//3, 2, 2) for i in 1:3])
    3-element Vector{Matrix{Rational{Int64}}}:
     [1//3 1//3; 1//3 1//3]
     [2//9 2//9; 2//9 2//9]
     [4//27 4//27; 4//27 4//27]

    julia> cumprod((1, 2, 1))
    (1, 2, 2)

    julia> cumprod(x^2 for x in 1:3)
    3-element Vector{Int64}:
      1
      4
     36

source
`Base.cumprod!` — Function

    cumprod!(B, A; dims::Integer)

Cumulative product of `A` along the dimension `dims`, storing the result in `B`. See also `cumprod`.

source

    cumprod!(y::AbstractVector, x::AbstractVector)

Cumulative product of a vector `x`, storing the result in `y`. See also `cumprod`.

source
`Base.cumsum` — Function

    cumsum(A; dims::Integer)

Cumulative sum along the dimension `dims`. See also `cumsum!` to use a preallocated output array, both for performance and to control the precision of the output (e.g. to avoid overflow).

Examples

    julia> a = [1 2 3; 4 5 6]
    2×3 Matrix{Int64}:
     1  2  3
     4  5  6

    julia> cumsum(a, dims=1)
    2×3 Matrix{Int64}:
     1  2  3
     5  7  9

    julia> cumsum(a, dims=2)
    2×3 Matrix{Int64}:
     1  3   6
     4  9  15

Note

The return array's `eltype` is `Int` for signed integers of less than system word size and `UInt` for unsigned integers of less than system word size. To preserve `eltype` of arrays with small signed or unsigned integer `accumulate(+, A)` should be used.

    julia> cumsum(Int8[100, 28])
    2-element Vector{Int64}:
     100
     128

    julia> accumulate(+,Int8[100, 28])
    2-element Vector{Int8}:
      100
     -128

In the former case, the integers are widened to system word size and therefore the result is `Int64[100, 128]`. In the latter case, no such widening happens and integer overflow results in `Int8[100, -128]`.

source

    cumsum(itr)

Cumulative sum an iterator. See also `cumsum!` to use a preallocated output array, both for performance and to control the precision of the output (e.g. to avoid overflow).

Julia 1.5

`cumsum` on a non-array iterator requires at least Julia 1.5.

Examples

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

    julia> cumsum([fill(1, 2) for i in 1:3])
    3-element Vector{Vector{Int64}}:
     [1, 1]
     [2, 2]
     [3, 3]

    julia> cumsum((1, 1, 1))
    (1, 2, 3)

    julia> cumsum(x^2 for x in 1:3)
    3-element Vector{Int64}:
      1
      5
     14

source
`Base.cumsum!` — Function

    cumsum!(B, A; dims::Integer)

Cumulative sum of `A` along the dimension `dims`, storing the result in `B`. See also `cumsum`.

source
`Base.diff` — Function

    diff(A::AbstractVector)
    diff(A::AbstractArray; dims::Integer)

Finite difference operator on a vector or a multidimensional array `A`. In the latter case the dimension to operate on needs to be specified with the `dims` keyword argument.

Julia 1.1

`diff` for arrays with dimension higher than 2 requires at least Julia 1.1.

Examples

    julia> a = [2 4; 6 16]
    2×2 Matrix{Int64}:
     2   4
     6  16

    julia> diff(a, dims=2)
    2×1 Matrix{Int64}:
      2
     10

    julia> diff(vec(a))
    3-element Vector{Int64}:
      4
     -2
     12

source
`Base.repeat` — Function

    repeat(A::AbstractArray, counts::Integer...)

Construct an array by repeating array `A` a given number of times in each dimension, specified by `counts`.

See also: `fill`, `Iterators.repeated`, `Iterators.cycle`.

Examples

    julia> repeat([1, 2, 3], 2)
    6-element Vector{Int64}:
     1
     2
     3
     1
     2
     3

    julia> repeat([1, 2, 3], 2, 3)
    6×3 Matrix{Int64}:
     1  1  1
     2  2  2
     3  3  3
     1  1  1
     2  2  2
     3  3  3

source

    repeat(A::AbstractArray; inner=ntuple(Returns(1), ndims(A)), outer=ntuple(Returns(1), ndims(A)))

Construct an array by repeating the entries of `A`. The i-th element of `inner` specifies the number of times that the individual entries of the i-th dimension of `A` should be repeated. The i-th element of `outer` specifies the number of times that a slice along the i-th dimension of `A` should be repeated. If `inner` or `outer` are omitted, no repetition is performed.

Examples

    julia> repeat(1:2, inner=2)
    4-element Vector{Int64}:
     1
     1
     2
     2

    julia> repeat(1:2, outer=2)
    4-element Vector{Int64}:
     1
     2
     1
     2

    julia> repeat([1 2; 3 4], inner=(2, 1), outer=(1, 3))
    4×6 Matrix{Int64}:
     1  2  1  2  1  2
     1  2  1  2  1  2
     3  4  3  4  3  4
     3  4  3  4  3  4

source

    repeat(s::AbstractString, r::Integer)

Repeat a string `r` times. This can be written as `s^r`.

See also `^`.

Examples

    julia> repeat("ha", 3)
    "hahaha"

source

    repeat(c::AbstractChar, r::Integer) -> String

Repeat a character `r` times. This can equivalently be accomplished by calling `c^r`.

Examples

    julia> repeat('A', 3)
    "AAA"

source
`Base.rot180` — Function

    rot180(A)

Rotate matrix `A` 180 degrees.

Examples

    julia> a = [1 2; 3 4]
    2×2 Matrix{Int64}:
     1  2
     3  4

    julia> rot180(a)
    2×2 Matrix{Int64}:
     4  3
     2  1

source

    rot180(A, k)

Rotate matrix `A` 180 degrees an integer `k` number of times. If `k` is even, this is equivalent to a `copy`.

Examples

    julia> a = [1 2; 3 4]
    2×2 Matrix{Int64}:
     1  2
     3  4

    julia> rot180(a,1)
    2×2 Matrix{Int64}:
     4  3
     2  1

    julia> rot180(a,2)
    2×2 Matrix{Int64}:
     1  2
     3  4

source
`Base.rotl90` — Function

    rotl90(A)

Rotate matrix `A` left 90 degrees.

Examples

    julia> a = [1 2; 3 4]
    2×2 Matrix{Int64}:
     1  2
     3  4

    julia> rotl90(a)
    2×2 Matrix{Int64}:
     2  4
     1  3

source

    rotl90(A, k)

Left-rotate matrix `A` 90 degrees counterclockwise an integer `k` number of times. If `k` is a multiple of four (including zero), this is equivalent to a `copy`.

Examples

    julia> a = [1 2; 3 4]
    2×2 Matrix{Int64}:
     1  2
     3  4

    julia> rotl90(a,1)
    2×2 Matrix{Int64}:
     2  4
     1  3

    julia> rotl90(a,2)
    2×2 Matrix{Int64}:
     4  3
     2  1

    julia> rotl90(a,3)
    2×2 Matrix{Int64}:
     3  1
     4  2

    julia> rotl90(a,4)
    2×2 Matrix{Int64}:
     1  2
     3  4

source
`Base.rotr90` — Function

    rotr90(A)

Rotate matrix `A` right 90 degrees.

Examples

    julia> a = [1 2; 3 4]
    2×2 Matrix{Int64}:
     1  2
     3  4

    julia> rotr90(a)
    2×2 Matrix{Int64}:
     3  1
     4  2

source

    rotr90(A, k)

Right-rotate matrix `A` 90 degrees clockwise an integer `k` number of times. If `k` is a multiple of four (including zero), this is equivalent to a `copy`.

Examples

    julia> a = [1 2; 3 4]
    2×2 Matrix{Int64}:
     1  2
     3  4

    julia> rotr90(a,1)
    2×2 Matrix{Int64}:
     3  1
     4  2

    julia> rotr90(a,2)
    2×2 Matrix{Int64}:
     4  3
     2  1

    julia> rotr90(a,3)
    2×2 Matrix{Int64}:
     2  4
     1  3

    julia> rotr90(a,4)
    2×2 Matrix{Int64}:
     1  2
     3  4

source
`Base.mapslices` — Function

    mapslices(f, A; dims)

Transform the given dimensions of array `A` using function `f`. `f` is called on each slice of `A` of the form `A[...,:,...,:,...]`. `dims` is an integer vector specifying where the colons go in this expression. The results are concatenated along the remaining dimensions. For example, if `dims` is `[1,2]` and `A` is 4-dimensional, `f` is called on `A[:,:,i,j]` for all `i` and `j`.

See also `eachcol`, `eachslice`.

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> mapslices(sum, a, dims = [1,2])
    1×1×2×2 Array{Int64, 4}:
    [:, :, 1, 1] =
     10

    [:, :, 2, 1] =
     26

    [:, :, 1, 2] =
     42

    [:, :, 2, 2] =
     58

source
`Base.eachrow` — Function

    eachrow(A::AbstractVecOrMat)

Create a generator that iterates over the first dimension of vector or matrix `A`, returning the rows as `AbstractVector` views.

See also `eachcol`, `eachslice`, `mapslices`.

Julia 1.1

This function requires at least Julia 1.1.

Example

    julia> a = [1 2; 3 4]
    2×2 Matrix{Int64}:
     1  2
     3  4

    julia> first(eachrow(a))
    2-element view(::Matrix{Int64}, 1, :) with eltype Int64:
     1
     2

    julia> collect(eachrow(a))
    2-element Vector{SubArray{Int64, 1, Matrix{Int64}, Tuple{Int64, Base.Slice{Base.OneTo{Int64}}}, true}}:
     [1, 2]
     [3, 4]

source
`Base.eachcol` — Function

    eachcol(A::AbstractVecOrMat)

Create a generator that iterates over the second dimension of matrix `A`, returning the columns as `AbstractVector` views.

See also `eachrow` and `eachslice`.

Julia 1.1

This function requires at least Julia 1.1.

Example

    julia> a = [1 2; 3 4]
    2×2 Matrix{Int64}:
     1  2
     3  4

    julia> first(eachcol(a))
    2-element view(::Matrix{Int64}, :, 1) with eltype Int64:
     1
     3

    julia> collect(eachcol(a))
    2-element Vector{SubArray{Int64, 1, Matrix{Int64}, Tuple{Base.Slice{Base.OneTo{Int64}}, Int64}, true}}:
     [1, 3]
     [2, 4]

source
`Base.eachslice` — Function

    eachslice(A::AbstractArray; dims)

Create a generator that iterates over dimensions `dims` of `A`, returning views that select all the data from the other dimensions in `A`.

Only a single dimension in `dims` is currently supported. Equivalent to `(view(A,:,:,...,i,:,: ...)) for i in axes(A, dims))`, where `i` is in position `dims`.

See also `eachrow`, `eachcol`, `mapslices`, and `selectdim`.

Julia 1.1

This function requires at least Julia 1.1.

Example

    julia> M = [1 2 3; 4 5 6; 7 8 9]
    3×3 Matrix{Int64}:
     1  2  3
     4  5  6
     7  8  9

    julia> first(eachslice(M, dims=1))
    3-element view(::Matrix{Int64}, 1, :) with eltype Int64:
     1
     2
     3

    julia> collect(eachslice(M, dims=2))
    3-element Vector{SubArray{Int64, 1, Matrix{Int64}, Tuple{Base.Slice{Base.OneTo{Int64}}, Int64}, true}}:
     [1, 4, 7]
     [2, 5, 8]
     [3, 6, 9]

source

## Combinatorics

`Base.invperm` — Function

    invperm(v)

Return the inverse permutation of `v`. If `B = A[v]`, then `A == B[invperm(v)]`.

See also `sortperm`, `invpermute!`, `isperm`, `permutedims`.

Examples

    julia> p = (2, 3, 1);

    julia> invperm(p)
    (3, 1, 2)

    julia> v = [2; 4; 3; 1];

    julia> invperm(v)
    4-element Vector{Int64}:
     4
     1
     3
     2

    julia> A = ['a','b','c','d'];

    julia> B = A[v]
    4-element Vector{Char}:
     'b': ASCII/Unicode U+0062 (category Ll: Letter, lowercase)
     'd': ASCII/Unicode U+0064 (category Ll: Letter, lowercase)
     'c': ASCII/Unicode U+0063 (category Ll: Letter, lowercase)
     'a': ASCII/Unicode U+0061 (category Ll: Letter, lowercase)

    julia> B[invperm(v)]
    4-element Vector{Char}:
     'a': ASCII/Unicode U+0061 (category Ll: Letter, lowercase)
     'b': ASCII/Unicode U+0062 (category Ll: Letter, lowercase)
     'c': ASCII/Unicode U+0063 (category Ll: Letter, lowercase)
     'd': ASCII/Unicode U+0064 (category Ll: Letter, lowercase)

source
`Base.isperm` — Function

    isperm(v) -> Bool

Return `true` if `v` is a valid permutation.

Examples

    julia> isperm([1; 2])
    true

    julia> isperm([1; 3])
    false

source
`Base.permute!` — Method

    permute!(v, p)

Permute vector `v` in-place, according to permutation `p`. No checking is done to verify that `p` is a permutation.

To return a new permutation, use `v[p]`. Note that this is generally faster than `permute!(v,p)` for large vectors.

See also `invpermute!`.

Examples

    julia> A = [1, 1, 3, 4];

    julia> perm = [2, 4, 3, 1];

    julia> permute!(A, perm);

    julia> A
    4-element Vector{Int64}:
     1
     4
     3
     1

source
`Base.invpermute!` — Function

    invpermute!(v, p)

Like `permute!`, but the inverse of the given permutation is applied.

Examples

    julia> A = [1, 1, 3, 4];

    julia> perm = [2, 4, 3, 1];

    julia> invpermute!(A, perm);

    julia> A
    4-element Vector{Int64}:
     4
     1
     3
     1

source
`Base.reverse` — Method

    reverse(A; dims=:)

Reverse `A` along dimension `dims`, which can be an integer (a single dimension), a tuple of integers (a tuple of dimensions) or `:` (reverse along all the dimensions, the default). See also `reverse!` for in-place reversal.

Examples

    julia> b = Int64[1 2; 3 4]
    2×2 Matrix{Int64}:
     1  2
     3  4

    julia> reverse(b, dims=2)
    2×2 Matrix{Int64}:
     2  1
     4  3

    julia> reverse(b)
    2×2 Matrix{Int64}:
     4  3
     2  1

Julia 1.6

Prior to Julia 1.6, only single-integer `dims` are supported in `reverse`.

source
`Base.reverseind` — Function

    reverseind(v, i)

Given an index `i` in `reverse(v)`, return the corresponding index in `v` so that `v[reverseind(v,i)] == reverse(v)[i]`. (This can be nontrivial in cases where `v` contains non-ASCII characters.)

Examples

    julia> s = "Julia🚀"
    "Julia🚀"

    julia> r = reverse(s)
    "🚀ailuJ"

    julia> for i in eachindex(s)
               print(r[reverseind(r, i)])
           end
    Julia🚀

source
`Base.reverse!` — Function

    reverse!(v [, start=1 [, stop=length(v) ]]) -> v

In-place version of `reverse`.

Examples

    julia> A = Vector(1:5)
    5-element Vector{Int64}:
     1
     2
     3
     4
     5

    julia> reverse!(A);

    julia> A
    5-element Vector{Int64}:
     5
     4
     3
     2
     1

source

    reverse!(A; dims=:)

Like `reverse`, but operates in-place in `A`.

Julia 1.6

Multidimensional `reverse!` requires Julia 1.6.

source

Theme

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