# 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`

.

`Base.AbstractVector`

— Type`AbstractVector{T}`

Supertype for one-dimensional arrays (or array-like types) with elements of type `T`

. Alias for `AbstractArray{T,1}`

.

`Base.AbstractMatrix`

— Type`AbstractMatrix{T}`

Supertype for two-dimensional arrays (or array-like types) with elements of type `T`

. Alias for `AbstractArray{T,2}`

.

`Base.AbstractVecOrMat`

— Type`AbstractVecOrMat{T}`

Union type of `AbstractVector{T}`

and `AbstractMatrix{T}`

.

`Core.Array`

— Type`Array{T,N} <: AbstractArray{T,N}`

`N`

-dimensional dense array with elements of type `T`

.

`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
```

`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
```

`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
```

`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
```

`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.

**Examples**

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

`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}`

.

`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
```

`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
```

`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
```

`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.

`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
```

`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
```

`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
```

`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
```

`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.

`Base.DenseVector`

— Type`DenseVector{T}`

One-dimensional `DenseArray`

with elements of type `T`

. Alias for `DenseArray{T,1}`

.

`Base.DenseMatrix`

— Type`DenseMatrix{T}`

Two-dimensional `DenseArray`

with elements of type `T`

. Alias for `DenseArray{T,2}`

.

`Base.DenseVecOrMat`

— Type`DenseVecOrMat{T}`

Union type of `DenseVector{T}`

and `DenseMatrix{T}`

.

`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 [`stride(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.

`Base.StridedVector`

— Type`StridedVector{T}`

One dimensional `StridedArray`

with elements of type `T`

.

`Base.StridedMatrix`

— Type`StridedMatrix{T}`

Two dimensional `StridedArray`

with elements of type `T`

.

`Base.StridedVecOrMat`

— Type`StridedVecOrMat{T}`

Union type of `StridedVector`

and `StridedMatrix`

with elements of type `T`

.

`Core.Memory`

— Type`Memory{T} == GenericMemory{:not_atomic, T, Core.CPU}`

One-dimensional dense array with elements of type `T`

.

This type requires Julia 1.11 or later.

`Core.MemoryRef`

— Type`MemoryRef(memory)`

Construct a MemoryRef from a memory object. This does not fail, but the resulting memory may point out-of-bounds if the memory is empty.

```
MemoryRef(::Memory, index::Integer)
MemoryRef(::MemoryRef, index::Integer)
```

Construct a MemoryRef from a memory object and an offset index (1-based) which can also be negative. This always returns an inbounds object, and will throw an error if that is not possible (because the index would result in a shift out-of-bounds of the underlying memory).

`Base.Slices`

— Type`Slices{P,SM,AX,S,N} <: AbstractSlices{S,N}`

An `AbstractArray`

of slices into a parent array over specified dimension(s), returning views that select all the data from the other dimension(s).

These should typically be constructed by `eachslice`

, `eachcol`

or `eachrow`

.

`parent(s::Slices)`

will return the parent array.

`Base.RowSlices`

— Type`RowSlices{M,AX,S}`

A special case of `Slices`

that is a vector of row slices of a matrix, as constructed by `eachrow`

.

`parent`

can be used to get the underlying matrix.

`Base.ColumnSlices`

— Type`ColumnSlices{M,AX,S}`

A special case of `Slices`

that is a vector of column slices of a matrix, as constructed by `eachcol`

.

`parent`

can be used to get the underlying matrix.

`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
```

`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
```

`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
```

`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`

.

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.

`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
```

`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
```

`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
```

`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
```

`Base.fill`

— Function```
fill(value, dims::Tuple)
fill(value, dims...)
```

Create an array of size `dims`

with every location set to `value`

.

For example, `fill(1.0, (5,5))`

returns a 5×5 array of floats, with `1.0`

in every location of the array.

The dimension lengths `dims`

may be specified as either a tuple or a sequence of arguments. An `N`

-length tuple or `N`

arguments following the `value`

specify an `N`

-dimensional array. Thus, a common idiom for creating a zero-dimensional array with its only location set to `x`

is `fill(x)`

.

Every location of the returned array is set to (and is thus `===`

to) the `value`

that was passed; this means that if the `value`

is itself modified, all elements of the `fill`

ed array will reflect that modification because they're *still* that very `value`

. This is of no concern with `fill(1.0, (5,5))`

as the `value`

`1.0`

is immutable and cannot itself be modified, but can be unexpected with mutable values like — most commonly — arrays. For example, `fill([], 3)`

places *the very same* empty array in all three locations of the returned vector:

```
julia> v = fill([], 3)
3-element Vector{Vector{Any}}:
[]
[]
[]
julia> v[1] === v[2] === v[3]
true
julia> value = v[1]
Any[]
julia> push!(value, 867_5309)
1-element Vector{Any}:
8675309
julia> v
3-element Vector{Vector{Any}}:
[8675309]
[8675309]
[8675309]
```

To create an array of many independent inner arrays, use a comprehension instead. This creates a new and distinct array on each iteration of the loop:

```
julia> v2 = [[] for _ in 1:3]
3-element Vector{Vector{Any}}:
[]
[]
[]
julia> v2[1] === v2[2] === v2[3]
false
julia> push!(v2[1], 8675309)
1-element Vector{Any}:
8675309
julia> v2
3-element Vector{Vector{Any}}:
[8675309]
[]
[]
```

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
julia> A = fill(zeros(2), 2) # sets both elements to the same [0.0, 0.0] vector
2-element Vector{Vector{Float64}}:
[0.0, 0.0]
[0.0, 0.0]
julia> A[1][1] = 42; # modifies the filled value to be [42.0, 0.0]
julia> A # both A[1] and A[2] are the very same vector
2-element Vector{Vector{Float64}}:
[42.0, 0.0]
[42.0, 0.0]
```

`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
```

`Base.empty`

— Function`empty(x::Tuple)`

Return an empty tuple, `()`

.

`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[]
```

`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`

.

`Base.similar`

— Function`similar(A::AbstractSparseMatrixCSC{Tv,Ti}, [::Type{TvNew}, ::Type{TiNew}, m::Integer, n::Integer]) where {Tv,Ti}`

Create an uninitialized mutable array with the given element type, index type, and size, based upon the given source `SparseMatrixCSC`

. The new sparse matrix maintains the structure of the original sparse matrix, except in the case where dimensions of the output matrix are different from the output.

The output matrix has zeros in the same locations as the input, but uninitialized values for the nonzero locations.

`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 Matrix{Int64}:
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 Matrix{Float64}:
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`

.

`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`

.

## Basic functions

`Base.ndims`

— Function`ndims(A::AbstractArray) -> Integer`

Return the number of dimensions of `A`

.

**Examples**

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

`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
```

`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))
```

`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)
julia> axes(A, 4) == 1:1 # all dimensions d > ndims(A) have size 1
true
```

**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
```

`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
```

`Base.keys`

— Method`keys(a::AbstractArray)`

Return an efficient array describing all valid indices for `a`

arranged in the shape of `a`

itself.

The 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.

**Examples**

```
julia> keys([4, 5, 6])
3-element LinearIndices{1, Tuple{Base.OneTo{Int64}}}:
1
2
3
julia> keys([4 5; 6 7])
CartesianIndices((2, 2))
```

`Base.eachindex`

— Function```
eachindex(A...)
eachindex(::IndexStyle, A::AbstractArray...)
```

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)`

if they use 1-based indexing. For array types that have not opted into fast linear indexing, a specialized Cartesian range is typically returned to efficiently index into the array with indices specified for every dimension.

In general `eachindex`

accepts arbitrary iterables, including strings and dictionaries, and returns an iterator object supporting arbitrary index types (e.g. unevenly spaced or non-integer indices).

If `A`

is `AbstractArray`

it is possible to explicitly specify the style of the indices that should be returned by `eachindex`

by passing a value having `IndexStyle`

type as its first argument (typically `IndexLinear()`

if linear indices are required or `IndexCartesian()`

if Cartesian range is wanted).

If you supply more than one `AbstractArray`

argument, `eachindex`

will create an iterable object that is fast for all arguments (typically 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.

See also `pairs`

`(A)`

to iterate over indices and values together, and `axes`

`(A, 2)`

for valid indices along one dimension.

**Examples**

```
julia> A = [10 20; 30 40];
julia> for i in eachindex(A) # linear indexing
println("A[", i, "] == ", A[i])
end
A[1] == 10
A[2] == 30
A[3] == 20
A[4] == 40
julia> for i in eachindex(view(A, 1:2, 1:1)) # Cartesian indexing
println(i)
end
CartesianIndex(1, 1)
CartesianIndex(2, 1)
```

`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.

`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 `1 + 2*(3 - 1) = 5`

.

See also `IndexCartesian`

.

`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 = 1 + 2*(3 - 1)`

.

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`

.

`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
```

`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
```

`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)
```

## 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"
```

`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
```

`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
```

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.

`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`

.

`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`

.

`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`

.

`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")
```

`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)
()
```

`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}()
```

`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}()
```

## Indexing and assignment

`Base.getindex`

— Method`getindex(A, inds...)`

Return a subset of array `A`

as selected by the indices `inds`

.

Each index may be any supported index type, such as an `Integer`

, `CartesianIndex`

, range, or array of supported indices. A : may be used to select all elements along a specific dimension, and a boolean array (e.g. an `Array{Bool}`

or a `BitArray`

) may be used to filter for elements where the corresponding index is `true`

.

When `inds`

selects multiple elements, this function returns a newly allocated array. To index multiple elements without making a copy, use `view`

instead.

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
julia> getindex(A, 2, 1)
3
julia> getindex(A, CartesianIndex(2, 1))
3
julia> getindex(A, :, 2)
2-element Vector{Int64}:
2
4
julia> getindex(A, 2, :)
2-element Vector{Int64}:
3
4
julia> getindex(A, A .> 2)
2-element Vector{Int64}:
3
4
```

`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)`

.

Behavior can be unexpected when any mutated argument shares memory with any other argument.

**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
```

`Base.nextind`

— Function`nextind(A, i)`

Return the index after `i`

in `A`

. The returned index is often equivalent to `i

- 1
`for an integer`

i`. This function can be useful for generic code.

The returned index might be out of bounds. Consider using `checkbounds`

.

See also: `prevind`

.

**Examples**

```
julia> x = [1 2; 3 4]
2×2 Matrix{Int64}:
1 2
3 4
julia> nextind(x, 1) # valid result
2
julia> nextind(x, 4) # invalid result
5
julia> nextind(x, CartesianIndex(1, 1)) # valid result
CartesianIndex(2, 1)
julia> nextind(x, CartesianIndex(2, 2)) # invalid result
CartesianIndex(1, 3)
```

`Base.prevind`

— Function`prevind(A, i)`

Return the index before `i`

in `A`

. The returned index is often equivalent to `i

- 1
`for an integer`

i`. This function can be useful for generic code.

The returned index might be out of bounds. Consider using `checkbounds`

.

See also: `nextind`

.

**Examples**

```
julia> x = [1 2; 3 4]
2×2 Matrix{Int64}:
1 2
3 4
julia> prevind(x, 4) # valid result
3
julia> prevind(x, 1) # invalid result
0
julia> prevind(x, CartesianIndex(2, 2)) # valid result
CartesianIndex(1, 2)
julia> prevind(x, CartesianIndex(1, 1)) # invalid result
CartesianIndex(2, 0)
```

`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.

**Examples**

```
julia> A = zeros(5, 5);
julia> B = [1 2; 3 4];
julia> Ainds = CartesianIndices((2:3, 2:3));
julia> Binds = CartesianIndices(B);
julia> copyto!(A, Ainds, B, Binds)
5×5 Matrix{Float64}:
0.0 0.0 0.0 0.0 0.0
0.0 1.0 2.0 0.0 0.0
0.0 3.0 4.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0
0.0 0.0 0.0 0.0 0.0
```

`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`

.

Behavior can be unexpected when any mutated argument shares memory with any other argument.

See also `copyto!`

.

This method requires at least Julia 1.1. In Julia 1.0 this method is available from the `Future`

standard library as `Future.copy!`

.

`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
```

`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 `:`

.

`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`

.

An `I::CartesianIndex`

is treated as a "scalar" (not a container) for `broadcast`

. In order to iterate over the components of a `CartesianIndex`

, convert it to a tuple with `Tuple(I)`

.

**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
```

Using a `CartesianIndex`

as a "scalar" for `broadcast`

requires Julia 1.10; in previous releases, use `Ref(I)`

.

`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.

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)))
CartesianIndices((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))
CartesianIndices((1:3, 1:2))
julia> cartesian[4]
CartesianIndex(1, 2)
julia> cartesian = CartesianIndices((1:2:5, 1:2))
CartesianIndices((1:2:5, 1:2))
julia> cartesian[2, 2]
CartesianIndex(3, 2)
```

**Broadcasting**

`CartesianIndices`

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

.

Broadcasting of CartesianIndices requires at least Julia 1.1.

```
julia> CIs = CartesianIndices((2:3, 5:6))
CartesianIndices((2:3, 5:6))
julia> CI = CartesianIndex(3, 4)
CartesianIndex(3, 4)
julia> CIs .+ CI
CartesianIndices((5:6, 9:10))
```

For cartesian to linear index conversion, see `LinearIndices`

.

`Base.Dims`

— Type`Dims{N}`

An `NTuple`

of `N`

`Int`

s used to represent the dimensions of an `AbstractArray`

.

`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`

.

**Examples**

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
```

`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`

.

**Examples**

```
julia> A = zeros(1,2,3,4);
julia> to_indices(A, (1,1,2,2))
(1, 1, 2, 2)
julia> to_indices(A, (1,1,2,20)) # no bounds checking
(1, 1, 2, 20)
julia> to_indices(A, (CartesianIndex((1,)), 2, CartesianIndex((3,4)))) # exotic index
(1, 2, 3, 4)
julia> to_indices(A, ([1,1], 1:2, 3, 4))
([1, 1], 1:2, 3, 4)
julia> to_indices(A, (1,2)) # no shape checking
(1, 2)
```

`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
```

`checkbounds(A, I...)`

Throw an error if the specified indices `I`

are not in bounds for the given array `A`

.

`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
```

`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.

**Examples**

```
julia> Base.elsize(rand(Float32, 10))
4
```

## 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(m::GenericMemory{M, T}, inds::Union{UnitRange, OneTo})`

Create a vector `v::Vector{T}`

backed by the specified indices of `m`

. It is only safe to resize `v`

if `m`

is subseqently not used.

`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.

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
```

`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.

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
```

`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.

Similarly, `@views`

converts string slices into `SubString`

views.

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.

Using `begin`

in an indexing expression to refer to the first index was implemented in Julia 1.4, but was only supported by `@views`

starting in Julia 1.5.

**Examples**

```
julia> A = zeros(3, 3);
julia> @views for row in 1:3
b = A[row, :] # b is a view, not a copy
b .= row # assign every element to the row index
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
```

`Base.parent`

— Function`parent(A)`

Return the underlying parent object of the view. This parent of objects of types `SubArray`

, `SubString`

, `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
```

`Base.parentindices`

— Function`parentindices(A)`

Return the indices in the `parent`

which correspond to the 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)))
```

`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
```

`Base.reinterpret`

— Function`reinterpret(::Type{Out}, x::In)`

Change the type-interpretation of the binary data in the isbits value `x`

to that of the isbits type `Out`

. The size (ignoring padding) of `Out`

has to be the same as that of the type of `x`

. For example, `reinterpret(Float32, UInt32(7))`

interprets the 4 bytes corresponding to `UInt32(7)`

as a `Float32`

. Note that `reinterpret(In, reinterpret(Out, x)) === x`

```
julia> reinterpret(Float32, UInt32(7))
1.0f-44
julia> reinterpret(NTuple{2, UInt8}, 0x1234)
(0x34, 0x12)
julia> reinterpret(UInt16, (0x34, 0x12))
0x1234
julia> reinterpret(Tuple{UInt16, UInt8}, (0x01, 0x0203))
(0x0301, 0x02)
```

The treatment of padding differs from reinterpret(::DataType, ::AbstractArray).

Use caution if some combinations of bits in `Out`

are not considered valid and would otherwise be prevented by the type's constructors and methods. Unexpected behavior may result without additional validation.

`reinterpret(T::DataType, A::AbstractArray)`

Construct a view of the array with the same binary data as the given array, but with `T`

as element type.

This function also works on "lazy" array whose elements are not computed until they are explicitly retrieved. For instance, `reinterpret`

on the range `1:6`

works similarly as on the dense vector `collect(1:6)`

:

```
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
julia> reinterpret(Complex{Int}, 1:6)
3-element reinterpret(Complex{Int64}, ::UnitRange{Int64}):
1 + 2im
3 + 4im
5 + 6im
```

If the location of padding bits does not line up between `T`

and `eltype(A)`

, the resulting array will be read-only or write-only, to prevent invalid bits from being written to or read from, respectively.

```
julia> a = reinterpret(Tuple{UInt8, UInt32}, UInt32[1, 2])
1-element reinterpret(Tuple{UInt8, UInt32}, ::Vector{UInt32}):
(0x01, 0x00000002)
julia> a[1] = 3
ERROR: Padding of type Tuple{UInt8, UInt32} is not compatible with type UInt32.
julia> b = reinterpret(UInt32, Tuple{UInt8, UInt32}[(0x01, 0x00000002)]); # showing will error
julia> b[1]
ERROR: Padding of type UInt32 is not compatible with type Tuple{UInt8, UInt32}.
```

`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)`

.

This method requires at least Julia 1.6.

**Examples**

```
julia> A = [1 2; 3 4]
2×2 Matrix{Int64}:
1 2
3 4
julia> reinterpret(reshape, Complex{Int}, A) # the result is a vector
2-element reinterpret(reshape, Complex{Int64}, ::Matrix{Int64}) with eltype Complex{Int64}:
1 + 3im
2 + 4im
julia> a = [(1,2,3), (4,5,6)]
2-element Vector{Tuple{Int64, Int64, Int64}}:
(1, 2, 3)
(4, 5, 6)
julia> reinterpret(reshape, Int, a) # the result is a matrix
3×2 reinterpret(reshape, Int64, ::Vector{Tuple{Int64, Int64, Int64}}) with eltype Int64:
1 4
2 5
3 6
```

`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
```

`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.

**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
```

`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
```

`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.

## Concatenation and permutation

`Base.cat`

— Function`cat(A...; dims)`

Concatenate the input arrays along the dimensions specified in `dims`

.

Along a dimension `d in dims`

, the size of the output array is `sum(size(a,d) for a in A)`

. Along other dimensions, all input arrays should have the same size, which will also be the size of the output array along those dimensions.

If `dims`

is a single number, the different arrays are tightly packed along that dimension. If `dims`

is an iterable containing several dimensions, the positions along these dimensions are increased simultaneously for each input array, filling with zero elsewhere. This allows one to construct block-diagonal matrices as `cat(matrices...; dims=(1,2))`

, and their higher-dimensional analogues.

The special case `dims=1`

is `vcat`

, and `dims=2`

is `hcat`

. See also `hvcat`

, `hvncat`

, `stack`

, `repeat`

.

The keyword also accepts `Val(dims)`

.

For multiple dimensions `dims = Val(::Tuple)`

was added in Julia 1.8.

**Examples**

Concatenate two arrays in different dimensions:

```
julia> a = [1 2 3]
1×3 Matrix{Int64}:
1 2 3
julia> b = [4 5 6]
1×3 Matrix{Int64}:
4 5 6
julia> cat(a, b; dims=1)
2×3 Matrix{Int64}:
1 2 3
4 5 6
julia> cat(a, b; dims=2)
1×6 Matrix{Int64}:
1 2 3 4 5 6
julia> cat(a, b; dims=(1, 2))
2×6 Matrix{Int64}:
1 2 3 0 0 0
0 0 0 4 5 6
```

**Extended Help**

Concatenate 3D arrays:

```
julia> a = ones(2, 2, 3);
julia> b = ones(2, 2, 4);
julia> c = cat(a, b; dims=3);
julia> size(c) == (2, 2, 7)
true
```

Concatenate arrays of different sizes:

```
julia> cat([1 2; 3 4], [pi, pi], fill(10, 2,3,1); dims=2) # same as hcat
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
```

Construct a block diagonal matrix:

```
julia> cat(true, trues(2,2), trues(4)', dims=(1,2)) # block-diagonal
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
```

```
julia> cat(1, [2], [3;;]; dims=Val(2))
1×3 Matrix{Int64}:
1 2 3
```

`cat`

does not join two strings, you may want to use `*`

.

```
julia> a = "aaa";
julia> b = "bbb";
julia> cat(a, b; dims=1)
2-element Vector{String}:
"aaa"
"bbb"
julia> cat(a, b; dims=2)
1×2 Matrix{String}:
"aaa" "bbb"
julia> a * b
"aaabbb"
```

`Base.vcat`

— Function`vcat(A...)`

Concatenate arrays or numbers vertically. Equivalent to `cat`

`(A...; dims=1)`

, and to the syntax `[a; b; c]`

.

To concatenate a large vector of arrays, `reduce(vcat, A)`

calls an efficient method when `A isa AbstractVector{<:AbstractVecOrMat}`

, rather than working pairwise.

See also `hcat`

, `Iterators.flatten`

, `stack`

.

**Examples**

```
julia> v = vcat([1,2], [3,4])
4-element Vector{Int64}:
1
2
3
4
julia> v == vcat(1, 2, [3,4]) # accepts numbers
true
julia> v == [1; 2; [3,4]] # syntax for the same operation
true
julia> summary(ComplexF64[1; 2; [3,4]]) # syntax for supplying the element type
"4-element Vector{ComplexF64}"
julia> vcat(range(1, 2, length=3)) # collects lazy ranges
3-element Vector{Float64}:
1.0
1.5
2.0
julia> two = ([10, 20, 30]', Float64[4 5 6; 7 8 9]) # row vector and a matrix
([10 20 30], [4.0 5.0 6.0; 7.0 8.0 9.0])
julia> vcat(two...)
3×3 Matrix{Float64}:
10.0 20.0 30.0
4.0 5.0 6.0
7.0 8.0 9.0
julia> vs = [[1, 2], [3, 4], [5, 6]];
julia> reduce(vcat, vs) # more efficient than vcat(vs...)
6-element Vector{Int64}:
1
2
3
4
5
6
julia> ans == collect(Iterators.flatten(vs))
true
```

`Base.hcat`

— Function`hcat(A...)`

Concatenate arrays or numbers horizontally. Equivalent to `cat`

`(A...; dims=2)`

, and to the syntax `[a b c]`

or `[a;; b;; c]`

.

For a large vector of arrays, `reduce(hcat, A)`

calls an efficient method when `A isa AbstractVector{<:AbstractVecOrMat}`

. For a vector of vectors, this can also be written `stack`

`(A)`

.

**Examples**

```
julia> hcat([1,2], [3,4], [5,6])
2×3 Matrix{Int64}:
1 3 5
2 4 6
julia> hcat(1, 2, [30 40], [5, 6, 7]') # accepts numbers
1×7 Matrix{Int64}:
1 2 30 40 5 6 7
julia> ans == [1 2 [30 40] [5, 6, 7]'] # syntax for the same operation
true
julia> Float32[1 2 [30 40] [5, 6, 7]'] # syntax for supplying the eltype
1×7 Matrix{Float32}:
1.0 2.0 30.0 40.0 5.0 6.0 7.0
julia> ms = [zeros(2,2), [1 2; 3 4], [50 60; 70 80]];
julia> reduce(hcat, ms) # more efficient than hcat(ms...)
2×6 Matrix{Float64}:
0.0 0.0 1.0 2.0 50.0 60.0
0.0 0.0 3.0 4.0 70.0 80.0
julia> stack(ms) |> summary # disagrees on a vector of matrices
"2×2×3 Array{Float64, 3}"
julia> hcat(Int[], Int[], Int[]) # empty vectors, each of size (0,)
0×3 Matrix{Int64}
julia> hcat([1.1, 9.9], Matrix(undef, 2, 0)) # hcat with empty 2×0 Matrix
2×1 Matrix{Any}:
1.1
9.9
```

`Base.hvcat`

— Function`hvcat(blocks_per_row::Union{Tuple{Vararg{Int}}, 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. If the first argument is a single integer `n`

, then all block rows are assumed to have `n`

block columns.

**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
julia> hvcat((2,2,2), a,b,c,d,e,f) == hvcat(2, a,b,c,d,e,f)
true
```

`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 `row_first` = true
```

`Base.stack`

— Function`stack(iter; [dims])`

Combine a collection of arrays (or other iterable objects) of equal size into one larger array, by arranging them along one or more new dimensions.

By default the axes of the elements are placed first, giving `size(result) = (size(first(iter))..., size(iter)...)`

. This has the same order of elements as `Iterators.flatten`

`(iter)`

.

With keyword `dims::Integer`

, instead the `i`

th element of `iter`

becomes the slice `selectdim`

`(result, dims, i)`

, so that `size(result, dims) == length(iter)`

. In this case `stack`

reverses the action of `eachslice`

with the same `dims`

.

The various `cat`

functions also combine arrays. However, these all extend the arrays' existing (possibly trivial) dimensions, rather than placing the arrays along new dimensions. They also accept arrays as separate arguments, rather than a single collection.

This function requires at least Julia 1.9.

**Examples**

```
julia> vecs = (1:2, [30, 40], Float32[500, 600]);
julia> mat = stack(vecs)
2×3 Matrix{Float32}:
1.0 30.0 500.0
2.0 40.0 600.0
julia> mat == hcat(vecs...) == reduce(hcat, collect(vecs))
true
julia> vec(mat) == vcat(vecs...) == reduce(vcat, collect(vecs))
true
julia> stack(zip(1:4, 10:99)) # accepts any iterators of iterators
2×4 Matrix{Int64}:
1 2 3 4
10 11 12 13
julia> vec(ans) == collect(Iterators.flatten(zip(1:4, 10:99)))
true
julia> stack(vecs; dims=1) # unlike any cat function, 1st axis of vecs[1] is 2nd axis of result
3×2 Matrix{Float32}:
1.0 2.0
30.0 40.0
500.0 600.0
julia> x = rand(3,4);
julia> x == stack(eachcol(x)) == stack(eachrow(x), dims=1) # inverse of eachslice
true
```

Higher-dimensional examples:

```
julia> A = rand(5, 7, 11);
julia> E = eachslice(A, dims=2); # a vector of matrices
julia> (element = size(first(E)), container = size(E))
(element = (5, 11), container = (7,))
julia> stack(E) |> size
(5, 11, 7)
julia> stack(E) == stack(E; dims=3) == cat(E...; dims=3)
true
julia> A == stack(E; dims=2)
true
julia> M = (fill(10i+j, 2, 3) for i in 1:5, j in 1:7);
julia> (element = size(first(M)), container = size(M))
(element = (2, 3), container = (5, 7))
julia> stack(M) |> size # keeps all dimensions
(2, 3, 5, 7)
julia> stack(M; dims=1) |> size # vec(container) along dims=1
(35, 2, 3)
julia> hvcat(5, M...) |> size # hvcat puts matrices next to each other
(14, 15)
```

`stack(f, args...; [dims])`

Apply a function to each element of a collection, and `stack`

the result. Or to several collections, `zip`

ped together.

The function should return arrays (or tuples, or other iterators) all of the same size. These become slices of the result, each separated along `dims`

(if given) or by default along the last dimensions.

**Examples**

```
julia> stack(c -> (c, c-32), "julia")
2×5 Matrix{Char}:
'j' 'u' 'l' 'i' 'a'
'J' 'U' 'L' 'I' 'A'
julia> stack(eachrow([1 2 3; 4 5 6]), (10, 100); dims=1) do row, n
vcat(row, row .* n, row ./ n)
end
2×9 Matrix{Float64}:
1.0 2.0 3.0 10.0 20.0 30.0 0.1 0.2 0.3
4.0 5.0 6.0 400.0 500.0 600.0 0.04 0.05 0.06
```

`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
```

`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
```

`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.

Behavior can be unexpected when any mutated argument shares memory with any other argument.

See also `circshift`

.

`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`

.

Behavior can be unexpected when any mutated argument shares memory with any other argument.

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
```

`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[]
```

`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
```

`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)
```

`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)
```

`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)
```

`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)
```

`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)
```

`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. This works for Arrays, Strings, and most other collections that support `getindex`

, `keys(A)`

, and `nextind`

.

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)
julia> findnext(isspace, "a b c", 3)
4
```

`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)
```

`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. This works for Arrays, Strings, and most other collections that support `getindex`

, `keys(A)`

, and `nextind`

.

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)
julia> findprev(isspace, "a b c", 3)
2
```

`Base.permutedims`

— Function```
permutedims(A::AbstractArray, perm)
permutedims(A::AbstractMatrix)
```

Permute the dimensions (axes) of array `A`

. `perm`

is a tuple or vector of `ndims(A)`

integers specifying the permutation.

If `A`

is a 2d array (`AbstractMatrix`

), then `perm`

defaults to `(2,1)`

, swapping the two axes of `A`

(the rows and columns of the matrix). This differs from `transpose`

in that the operation is not recursive, which is especially useful for arrays of non-numeric values (where the recursive `transpose`

would throw an error) and/or 2d arrays that do not represent linear operators.

For 1d arrays, see `permutedims(v::AbstractVector)`

, which returns a 1-row “matrix”.

See also `permutedims!`

, `PermutedDimsArray`

, `transpose`

, `invperm`

.

**Examples**

**2d arrays:**

Unlike `transpose`

, `permutedims`

can be used to swap rows and columns of 2d arrays of arbitrary non-numeric elements, such as strings:

```
julia> A = ["a" "b" "c"
"d" "e" "f"]
2×3 Matrix{String}:
"a" "b" "c"
"d" "e" "f"
julia> permutedims(A)
3×2 Matrix{String}:
"a" "d"
"b" "e"
"c" "f"
```

And `permutedims`

produces results that differ from `transpose`

for matrices whose elements are themselves numeric matrices:

```
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]
```

**Multi-dimensional arrays**

```
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> perm = (3, 1, 2); # put the last dimension first
julia> B = permutedims(A, perm)
2×2×2 Array{Int64, 3}:
[:, :, 1] =
1 2
5 6
[:, :, 2] =
3 4
7 8
julia> A == permutedims(B, invperm(perm)) # the inverse permutation
true
```

For each dimension `i`

of `B = permutedims(A, perm)`

, its corresponding dimension of `A`

will be `perm[i]`

. This means the equality `size(B, i) == size(A, perm[i])`

holds.

```
julia> A = randn(5, 7, 11, 13);
julia> perm = [4, 1, 3, 2];
julia> B = permutedims(A, perm);
julia> size(B)
(13, 5, 11, 7)
julia> size(A)[perm] == ans
true
```

`permutedims(v::AbstractVector)`

Reshape vector `v`

into a `1 × length(v)`

row matrix. Differs from `transpose`

in that the operation is not recursive, which is especially useful for arrays of non-numeric values (where the recursive `transpose`

might throw an error).

**Examples**

Unlike `transpose`

, `permutedims`

can be used on vectors of arbitrary non-numeric elements, such as strings:

```
julia> permutedims(["a", "b", "c"])
1×3 Matrix{String}:
"a" "b" "c"
```

For vectors of numbers, `permutedims(v)`

works much like `transpose(v)`

except that the return type differs (it uses `reshape`

rather than a `LinearAlgebra.Transpose`

view, though both share memory with the original array `v`

):

```
julia> v = [1, 2, 3, 4]
4-element Vector{Int64}:
1
2
3
4
julia> p = permutedims(v)
1×4 Matrix{Int64}:
1 2 3 4
julia> r = transpose(v)
1×4 transpose(::Vector{Int64}) with eltype Int64:
1 2 3 4
julia> p == r
true
julia> typeof(r)
Transpose{Int64, Vector{Int64}}
julia> p[1] = 5; r[2] = 6; # mutating p or r also changes v
julia> v # shares memory with both p and r
4-element Vector{Int64}:
5
6
3
4
```

However, `permutedims`

produces results that differ from `transpose`

for vectors whose elements are themselves numeric matrices:

```
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]
```

`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`

.

`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
```

`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)
```

## 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`

. For a lazy version, see `Iterators.accumulate`

.

`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(min, (1, -2, 3, -4, 5), init=0)
(0, -2, -2, -4, -4)
julia> accumulate(/, (2, 4, Inf), init=100)
(50.0, 12.5, 0.0)
julia> accumulate(=>, i^2 for i in 1:3)
3-element Vector{Any}:
1
1 => 4
(1 => 4) => 9
julia> accumulate(+, fill(1, 3, 4))
3×4 Matrix{Int64}:
1 4 7 10
2 5 8 11
3 6 9 12
julia> accumulate(+, fill(1, 2, 5), dims=2, init=100.0)
2×5 Matrix{Float64}:
101.0 102.0 103.0 104.0 105.0
101.0 102.0 103.0 104.0 105.0
```

`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.

Behavior can be unexpected when any mutated argument shares memory with any other argument.

See also `accumulate`

, `cumsum!`

, `cumprod!`

.

**Examples**

```
julia> x = [1, 0, 2, 0, 3];
julia> y = rand(5);
julia> accumulate!(+, y, x);
julia> y
5-element Vector{Float64}:
1.0
1.0
3.0
3.0
6.0
julia> A = [1 2 3; 4 5 6];
julia> B = similar(A);
julia> accumulate!(-, B, A, dims=1)
2×3 Matrix{Int64}:
1 2 3
-3 -3 -3
julia> accumulate!(*, B, A, dims=2, init=10)
2×3 Matrix{Int64}:
10 20 60
40 200 1200
```

`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 = Int8[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
```

`cumprod(itr)`

Cumulative product of an iterator.

See also `cumprod!`

, `accumulate`

, `cumsum`

.

`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((1, 2, 1, 3, 1))
(1, 2, 2, 6, 6)
julia> cumprod("julia")
5-element Vector{String}:
"j"
"ju"
"jul"
"juli"
"julia"
```

`Base.cumprod!`

— Function`cumprod!(B, A; dims::Integer)`

Cumulative product of `A`

along the dimension `dims`

, storing the result in `B`

. See also `cumprod`

.

Behavior can be unexpected when any mutated argument shares memory with any other argument.

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

Cumulative product of a vector `x`

, storing the result in `y`

. See also `cumprod`

.

Behavior can be unexpected when any mutated argument shares memory with any other argument.

`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
```

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]`

.

`cumsum(itr)`

Cumulative sum of an iterator.

See also `accumulate`

to apply functions other than `+`

.

`cumsum`

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

**Examples**

```
julia> cumsum(1:3)
3-element Vector{Int64}:
1
3
6
julia> cumsum((true, false, true, false, true))
(1, 1, 2, 2, 3)
julia> cumsum(fill(1, 2) for i in 1:3)
3-element Vector{Vector{Int64}}:
[1, 1]
[2, 2]
[3, 3]
```

`Base.cumsum!`

— Function`cumsum!(B, A; dims::Integer)`

Cumulative sum of `A`

along the dimension `dims`

, storing the result in `B`

. See also `cumsum`

.

Behavior can be unexpected when any mutated argument shares memory with any other argument.

`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.

`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
```

`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
```

`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
```

`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"
```

`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"
```

`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
```

`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
```

`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
```

`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
```

`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
```

`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
```

`Base.mapslices`

— Function`mapslices(f, A; dims)`

Transform the given dimensions of array `A`

by applying a function `f`

on each slice of the form `A[..., :, ..., :, ...]`

, with a colon at each `d`

in `dims`

. The results are concatenated along the remaining dimensions.

For example, if `dims = [1,2]`

and `A`

is 4-dimensional, then `f`

is called on `x = A[:,:,i,j]`

for all `i`

and `j`

, and `f(x)`

becomes `R[:,:,i,j]`

in the result `R`

.

See also `eachcol`

or `eachslice`

, used with `map`

or `stack`

.

**Examples**

```
julia> A = reshape(1:30,(2,5,3))
2×5×3 reshape(::UnitRange{Int64}, 2, 5, 3) with eltype Int64:
[:, :, 1] =
1 3 5 7 9
2 4 6 8 10
[:, :, 2] =
11 13 15 17 19
12 14 16 18 20
[:, :, 3] =
21 23 25 27 29
22 24 26 28 30
julia> f(x::Matrix) = fill(x[1,1], 1,4); # returns a 1×4 matrix
julia> B = mapslices(f, A, dims=(1,2))
1×4×3 Array{Int64, 3}:
[:, :, 1] =
1 1 1 1
[:, :, 2] =
11 11 11 11
[:, :, 3] =
21 21 21 21
julia> f2(x::AbstractMatrix) = fill(x[1,1], 1,4);
julia> B == stack(f2, eachslice(A, dims=3))
true
julia> g(x) = x[begin] // x[end-1]; # returns a number
julia> mapslices(g, A, dims=[1,3])
1×5×1 Array{Rational{Int64}, 3}:
[:, :, 1] =
1//21 3//23 1//5 7//27 9//29
julia> map(g, eachslice(A, dims=2))
5-element Vector{Rational{Int64}}:
1//21
3//23
1//5
7//27
9//29
julia> mapslices(sum, A; dims=(1,3)) == sum(A; dims=(1,3))
true
```

Notice that in `eachslice(A; dims=2)`

, the specified dimension is the one *without* a colon in the slice. This is `view(A,:,i,:)`

, whereas `mapslices(f, A; dims=(1,3))`

uses `A[:,i,:]`

. The function `f`

may mutate values in the slice without affecting `A`

.

`Base.eachrow`

— Function`eachrow(A::AbstractVecOrMat) <: AbstractVector`

Create a `RowSlices`

object that is a vector of rows of matrix or vector `A`

. Row slices are returned as `AbstractVector`

views of `A`

.

For the inverse, see `stack`

`(rows; dims=1)`

.

See also `eachcol`

, `eachslice`

and `mapslices`

.

This function requires at least Julia 1.1.

Prior to Julia 1.9, this returned an iterator.

**Examples**

```
julia> a = [1 2; 3 4]
2×2 Matrix{Int64}:
1 2
3 4
julia> s = eachrow(a)
2-element RowSlices{Matrix{Int64}, Tuple{Base.OneTo{Int64}}, SubArray{Int64, 1, Matrix{Int64}, Tuple{Int64, Base.Slice{Base.OneTo{Int64}}}, true}}:
[1, 2]
[3, 4]
julia> s[1]
2-element view(::Matrix{Int64}, 1, :) with eltype Int64:
1
2
```

`Base.eachcol`

— Function`eachcol(A::AbstractVecOrMat) <: AbstractVector`

Create a `ColumnSlices`

object that is a vector of columns of matrix or vector `A`

. Column slices are returned as `AbstractVector`

views of `A`

.

For the inverse, see `stack`

`(cols)`

or `reduce(`

`hcat`

`, cols)`

.

See also `eachrow`

, `eachslice`

and `mapslices`

.

This function requires at least Julia 1.1.

Prior to Julia 1.9, this returned an iterator.

**Examples**

```
julia> a = [1 2; 3 4]
2×2 Matrix{Int64}:
1 2
3 4
julia> s = eachcol(a)
2-element ColumnSlices{Matrix{Int64}, Tuple{Base.OneTo{Int64}}, SubArray{Int64, 1, Matrix{Int64}, Tuple{Base.Slice{Base.OneTo{Int64}}, Int64}, true}}:
[1, 3]
[2, 4]
julia> s[1]
2-element view(::Matrix{Int64}, :, 1) with eltype Int64:
1
3
```

`Base.eachslice`

— Function`eachslice(A::AbstractArray; dims, drop=true)`

Create a `Slices`

object that is an array of slices over dimensions `dims`

of `A`

, returning views that select all the data from the other dimensions in `A`

. `dims`

can either be an integer or a tuple of integers.

If `drop = true`

(the default), the outer `Slices`

will drop the inner dimensions, and the ordering of the dimensions will match those in `dims`

. If `drop = false`

, then the `Slices`

will have the same dimensionality as the underlying array, with inner dimensions having size 1.

See `stack`

`(slices; dims)`

for the inverse of `eachslice(A; dims::Integer)`

.

See also `eachrow`

, `eachcol`

, `mapslices`

and `selectdim`

.

This function requires at least Julia 1.1.

Prior to Julia 1.9, this returned an iterator, and only a single dimension `dims`

was supported.

**Examples**

```
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> s = eachslice(m, dims=1)
3-element RowSlices{Matrix{Int64}, Tuple{Base.OneTo{Int64}}, SubArray{Int64, 1, Matrix{Int64}, Tuple{Int64, Base.Slice{Base.OneTo{Int64}}}, true}}:
[1, 2, 3]
[4, 5, 6]
[7, 8, 9]
julia> s[1]
3-element view(::Matrix{Int64}, 1, :) with eltype Int64:
1
2
3
julia> eachslice(m, dims=1, drop=false)
3×1 Slices{Matrix{Int64}, Tuple{Int64, Colon}, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}}, SubArray{Int64, 1, Matrix{Int64}, Tuple{Int64, Base.Slice{Base.OneTo{Int64}}}, true}, 2}:
[1, 2, 3]
[4, 5, 6]
[7, 8, 9]
```

## 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)
```

`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
```

`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]`

. This is generally faster than `permute!(v, p)`

; it is even faster to write into a pre-allocated output array with `u .= @view v[p]`

. (Even though `permute!`

overwrites `v`

in-place, it internally requires some allocation to keep track of which elements have been moved.)

Behavior can be unexpected when any mutated argument shares memory with any other argument.

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
```

`Base.invpermute!`

— Function`invpermute!(v, p)`

Like `permute!`

, but the inverse of the given permutation is applied.

Note that if you have a pre-allocated output array (e.g. `u = similar(v)`

), it is quicker to instead employ `u[p] = v`

. (`invpermute!`

internally allocates a copy of the data.)

Behavior can be unexpected when any mutated argument shares memory with any other argument.

**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
```

`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
```

Prior to Julia 1.6, only single-integer `dims`

are supported in `reverse`

.

`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🚀
```

`Base.reverse!`

— Function`reverse!(v [, start=firstindex(v) [, stop=lastindex(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
```

`reverse!(A; dims=:)`

Like `reverse`

, but operates in-place in `A`

.

Multidimensional `reverse!`

requires Julia 1.6.