Arrays

Constructors and Types

Core.ArrayType
Array{T,N} <: AbstractArray{T,N}

N-dimensional dense array with elements of type T.

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Core.ArrayMethod
Array{T}(undef, dims)
Array{T,N}(undef, dims)

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

Examples

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

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

julia> similar(B, 2, 4, 1) # use typeof(B), and the given size
2×4×1 Array{Float64, 3}:
[:, :, 1] =
2.26703e-314  2.26708e-314  0.0           2.80997e-314
0.0           2.26703e-314  2.26708e-314  0.0
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Core.ArrayMethod
Array{T}(nothing, dims)
Array{T,N}(nothing, dims)

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

Examples

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

julia> Array{Union{Nothing, Int}}(nothing, 2, 3)
2×3 Matrix{Union{Nothing, Int64}}:
nothing  nothing  nothing
nothing  nothing  nothing
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Core.ArrayMethod
Array{T}(missing, dims)
Array{T,N}(missing, dims)

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

Examples

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

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

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

Examples

julia> Array{Float64, 1}(UndefInitializer(), 3)
3-element Array{Float64, 1}:
2.2752528595e-314
2.202942107e-314
2.275252907e-314
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Core.undefConstant
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
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Base.VectorType
Vector{T} <: AbstractVector{T}

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

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Base.VectorMethod
Vector{T}(undef, n)

Construct an uninitialized Vector{T} of length n.

Examples

julia> Vector{Float64}(undef, 3)
3-element Array{Float64, 1}:
6.90966e-310
6.90966e-310
6.90966e-310
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Base.VectorMethod
Vector{T}(nothing, m)

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

Examples

julia> Vector{Union{Nothing, String}}(nothing, 2)
2-element Vector{Union{Nothing, String}}:
nothing
nothing
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Base.VectorMethod
Vector{T}(missing, m)

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

Examples

julia> Vector{Union{Missing, String}}(missing, 2)
2-element Vector{Union{Missing, String}}:
missing
missing
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Base.MatrixMethod
Matrix{T}(undef, m, n)

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

Examples

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

julia> similar(ans, Int32, 2, 2)
2×2 Matrix{Int32}:
490537216  1277177453
1  1936748399
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Base.MatrixMethod
Matrix{T}(nothing, m, n)

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

Examples

julia> Matrix{Union{Nothing, String}}(nothing, 2, 3)
2×3 Matrix{Union{Nothing, String}}:
nothing  nothing  nothing
nothing  nothing  nothing
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Base.MatrixMethod
Matrix{T}(missing, m, n)

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

Examples

julia> Matrix{Union{Missing, String}}(missing, 2, 3)
2×3 Matrix{Union{Missing, String}}:
missing  missing  missing
missing  missing  missing
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Base.VecOrMatType
VecOrMat{T}

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

Examples

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

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

julia> Array{Float64, 3} <: VecOrMat{Float64}
false
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Core.DenseArrayType
DenseArray{T, N} <: AbstractArray{T,N}

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

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Base.StridedArrayType
StridedArray{T, N}

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

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

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Base.getindexMethod
getindex(type[, elements...])

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

Examples

julia> Int8[1, 2, 3]
3-element Vector{Int8}:
1
2
3

julia> getindex(Int8, 1, 2, 3)
3-element Vector{Int8}:
1
2
3
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Base.zerosFunction
zeros([T=Float64,] dims::Tuple)
zeros([T=Float64,] dims...)

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

Examples

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

julia> zeros(Int8, 2, 3)
2×3 Matrix{Int8}:
0  0  0
0  0  0
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Base.onesFunction
ones([T=Float64,] dims::Tuple)
ones([T=Float64,] dims...)

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

Examples

julia> ones(1,2)
1×2 Matrix{Float64}:
1.0  1.0

julia> ones(ComplexF64, 2, 3)
2×3 Matrix{ComplexF64}:
1.0+0.0im  1.0+0.0im  1.0+0.0im
1.0+0.0im  1.0+0.0im  1.0+0.0im
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Base.BitArrayType
BitArray{N} <: AbstractArray{Bool, N}

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

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

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

Note

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

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

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

Examples

julia> BitArray(undef, 2, 2)
2×2 BitMatrix:
0  0
0  0

julia> BitArray(undef, (3, 1))
3×1 BitMatrix:
0
0
0
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Base.BitArrayMethod
BitArray(itr)

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

Examples

julia> BitArray([1 0; 0 1])
2×2 BitMatrix:
1  0
0  1

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

julia> BitArray(x+y == 3 for x = 1:2 for y = 1:3)
6-element BitVector:
0
1
0
1
0
0
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Base.truesFunction
trues(dims)

Create a BitArray with all values set to true.

Examples

julia> trues(2,3)
2×3 BitMatrix:
1  1  1
1  1  1
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Base.falsesFunction
falses(dims)

Create a BitArray with all values set to false.

Examples

julia> falses(2,3)
2×3 BitMatrix:
0  0  0
0  0  0
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Base.fillFunction
fill(x, dims::Tuple)
fill(x, dims...)

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

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

Examples

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

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

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

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

julia> A = 42; # modifies both A and A

julia> A
2-element Vector{Vector{Float64}}:
[42.0, 0.0]
[42.0, 0.0]
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Base.fill!Function
fill!(A, x)

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

Examples

julia> A = zeros(2,3)
2×3 Matrix{Float64}:
0.0  0.0  0.0
0.0  0.0  0.0

julia> fill!(A, 2.)
2×3 Matrix{Float64}:
2.0  2.0  2.0
2.0  2.0  2.0

julia> a = [1, 1, 1]; A = fill!(Vector{Vector{Int}}(undef, 3), a); a = 2; A
3-element Vector{Vector{Int64}}:
[2, 1, 1]
[2, 1, 1]
[2, 1, 1]

julia> x = 0; f() = (global x += 1; x); fill!(Vector{Int}(undef, 3), f())
3-element Vector{Int64}:
1
1
1
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Base.emptyFunction
empty(x::Tuple)

Returns an empty tuple, ().

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empty(v::AbstractVector, [eltype])

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

Examples

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

julia> empty([1.0, 2.0, 3.0], String)
String[]
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empty(a::AbstractDict, [index_type=keytype(a)], [value_type=valtype(a)])

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

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

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

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

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

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

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

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

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

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

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

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similar(storagetype, axes)

Create an uninitialized mutable array analogous to that specified by storagetype, but with axes specified by the last argument.

Examples:

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

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

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

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

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Basic functions

Base.ndimsFunction
ndims(A::AbstractArray) -> Integer

Return the number of dimensions of A.

Examples

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

julia> ndims(A)
3
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Base.sizeFunction
size(A::AbstractArray, [dim])

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

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

Examples

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

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

julia> size(A, 2)
3
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Base.axesMethod
axes(A)

Return the tuple of valid indices for array A.

Examples

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

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

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

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

Examples

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

julia> axes(A, 2)
Base.OneTo(6)

Usage note

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

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

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

Examples

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

julia> length([1 2; 3 4])
4
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Base.keysMethod
keys(a::AbstractArray)

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

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

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

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Base.eachindexFunction
eachindex(A...)

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

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

Examples

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

julia> for i in eachindex(A) # linear indexing
println(i)
end
1
2
3
4

julia> for i in eachindex(view(A, 1:2, 1:1)) # Cartesian indexing
println(i)
end
CartesianIndex(1, 1)
CartesianIndex(2, 1)
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Base.IndexStyleType
IndexStyle(A)
IndexStyle(typeof(A))

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

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

The default is IndexCartesian().

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

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

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Base.IndexLinearType
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 since 2*1 + 3 = 5.

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Base.IndexCartesianType
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, this would be recomputed to the equivalent Cartesian index and call A[1, 3] since 5 = 2*1 + 3.

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

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Base.conj!Function
conj!(A)

Transform an array to its complex conjugate in-place.

Examples

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

julia> conj!(A);

julia> A
2×2 Matrix{Complex{Int64}}:
1-1im  2+1im
2-2im  3-1im
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Base.strideFunction
stride(A, k::Integer)

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

Examples

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

julia> stride(A,2)
3

julia> stride(A,3)
12
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Base.stridesFunction
strides(A)

Return a tuple of the memory strides in each dimension.

Examples

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

julia> strides(A)
(1, 3, 12)
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See also the dot syntax for vectorizing functions; for example, f.(args...) implicitly calls broadcast(f, args...). Rather than relying on "vectorized" methods of functions like sin to operate on arrays, you should use sin.(a) to vectorize via broadcast.

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

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

The resulting container type is established by the following rules:

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

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

Examples

julia> A = [1, 2, 3, 4, 5]
5-element 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

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)

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

Examples

julia> A = [1.0; 0.0]; B = [0.0; 0.0];

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

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

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

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

julia> A
2-element Vector{Float64}:
1.0
-2.0
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@. 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
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For specializing broadcast on custom types, see

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

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

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

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

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

(::Type{<:MyArrayStyleDim})(::Val{N}) where N = MyArrayStyleDim{N}()

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

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

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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)

Base.RefValue{String}("hello")
source
combine_axes(As...) -> Tuple

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

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

()
source

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 2; 3 4])
source

Examples

source

Indexing and assignment

Base.getindexMethod
getindex(A, inds...)

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

Examples

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

julia> getindex(A, 1)
1

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

julia> getindex(A, 2:4)
3-element Vector{Int64}:
3
2
4
source
Base.setindex!Method
setindex!(A, X, inds...)
A[inds...] = X

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

Examples

julia> A = zeros(2,2);

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

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

julia> A
2×2 Matrix{Float64}:
10.0  30.0
20.0  40.0
source
Base.copyto!Method
copyto!(dest, Rdest::CartesianIndices, src, Rsrc::CartesianIndices) -> dest

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

source
Base.copy!Function
copy!(dst, src) -> dst

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

Julia 1.1

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

source
Base.isassignedFunction
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
source
Base.ColonType
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 :.

source
Base.IteratorsMD.CartesianIndexType
CartesianIndex(i, j, k...)   -> I
CartesianIndex((i, j, k...)) -> I

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

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

Examples

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

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

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

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

julia> A[CartesianIndex((1, 1, 1, 1))]
1

julia> A[CartesianIndex((1, 1, 1, 2))]
9

julia> A[CartesianIndex((1, 1, 2, 1))]
5
source
Base.IteratorsMD.CartesianIndicesType
CartesianIndices(sz::Dims) -> R
CartesianIndices((istart:[istep:]istop, jstart:[jstep:]jstop, ...)) -> R

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

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

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

CartesianIndices(A::AbstractArray) -> R

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

Julia 1.6

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

Examples

julia> foreach(println, CartesianIndices((2, 2, 2)))
CartesianIndex(1, 1, 1)
CartesianIndex(2, 1, 1)
CartesianIndex(1, 2, 1)
CartesianIndex(2, 2, 1)
CartesianIndex(1, 1, 2)
CartesianIndex(2, 1, 2)
CartesianIndex(1, 2, 2)
CartesianIndex(2, 2, 2)

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

Conversion between linear and cartesian indices

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

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

julia> cartesian
CartesianIndex(1, 2)

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

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

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

Julia 1.1

Broadcasting of CartesianIndices requires at least Julia 1.1.

julia> CIs = CartesianIndices((2:3, 5:6))
2×2 CartesianIndices{2, Tuple{UnitRange{Int64}, UnitRange{Int64}}}:
CartesianIndex(2, 5)  CartesianIndex(2, 6)
CartesianIndex(3, 5)  CartesianIndex(3, 6)

julia> CI = CartesianIndex(3, 4)
CartesianIndex(3, 4)

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

For cartesian to linear index conversion, see LinearIndices.

source
Base.LinearIndicesType
LinearIndices(A::AbstractArray)

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

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

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

Examples

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

julia> b = LinearIndices(A);

julia> extrema(b)
(1, 210)
LinearIndices(inds::CartesianIndices) -> R
LinearIndices(sz::Dims) -> R
LinearIndices((istart:istop, jstart:jstop, ...)) -> R

Return a LinearIndices array with the specified shape or axes.

Example

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

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

julia> linear[1,2]
4
source
Base.to_indicesFunction
to_indices(A, I::Tuple)

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

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

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

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

source
Base.checkboundsFunction
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.

Examples

julia> A = rand(3, 3);

julia> checkbounds(Bool, A, 2)
true

julia> checkbounds(Bool, A, 3, 4)
false

julia> checkbounds(Bool, A, 1:3)
true

julia> checkbounds(Bool, A, 1:3, 2:4)
false
source
checkbounds(A, I...)

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

source
Base.checkindexFunction
checkindex(Bool, inds::AbstractUnitRange, index)

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

Examples

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

julia> checkindex(Bool, 1:20, 21)
false
source
Base.elsizeFunction
elsize(type)

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

source

Views (SubArrays and other view types)

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

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

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.viewFunction
view(A, inds...)

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

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

Julia 1.6

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

Examples

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

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

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

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

julia> view(2:5, 2:3) # returns a range as type is immutable
3:4
source
Base.@viewMacro
@view A[inds...]

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

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

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

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

Julia 1.5

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

Examples

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

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

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

julia> A
2×2 Matrix{Int64}:
0  2
0  4
source
Base.@viewsMacro
@views expression

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

Note

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

Julia 1.5

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

Examples

julia> A = zeros(3, 3);

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

julia> A
3×3 Matrix{Float64}:
1.0  1.0  1.0
2.0  2.0  2.0
3.0  3.0  3.0
source
Base.parentFunction
parent(A)

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

Examples

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

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

julia> parent(V)
2×2 Matrix{Int64}:
1  2
3  4
source
Base.parentindicesFunction
parentindices(A)

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

Examples

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

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

julia> parentindices(V)
(1, Base.Slice(Base.OneTo(2)))
source
Base.selectdimFunction
selectdim(A, d::Integer, i)

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

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

Examples

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

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

julia> selectdim(A, 2, 3:4)
2×2 view(::Matrix{Int64}, :, 3:4) with eltype Int64:
3  4
7  8
source
Base.reinterpretFunction
reinterpret(type, A)

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

Examples

julia> reinterpret(Float32, UInt32(7))
1.0f-44

julia> reinterpret(Float32, UInt32[1 2 3 4 5])
1×5 reinterpret(Float32, ::Matrix{UInt32}):
1.0f-45  3.0f-45  4.0f-45  6.0f-45  7.0f-45
source
reinterpret(reshape, T, A::AbstractArray{S}) -> B

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

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

Julia 1.6

This method requires at least Julia 1.6.

Examples

julia> A = [1 2; 3 4]
2×2 Matrix{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
source
Base.reshapeFunction
reshape(A, dims...) -> AbstractArray
reshape(A, dims) -> AbstractArray

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

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

Examples

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

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

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

julia> reshape(1:6, 2, 3)
2×3 reshape(::UnitRange{Int64}, 2, 3) with eltype Int64:
1  3  5
2  4  6
source
Base.dropdimsFunction
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
source
Base.vecFunction
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

source
Base.SubArrayType
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 SubArrays using the view function.

source

Concatenation and permutation

Base.catFunction
cat(A...; dims)

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

Examples

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

julia> cat(true, trues(2,2), trues(4)', dims=(1,2))
4×7 Matrix{Bool}:
1  0  0  0  0  0  0
0  1  1  0  0  0  0
0  1  1  0  0  0  0
0  0  0  1  1  1  1
source
Base.vcatFunction
vcat(A...)

Concatenate along dimension 1.

Examples

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

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

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

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

julia> vcat(c...)
2×3 Matrix{Int64}:
1  2  3
4  5  6
source
Base.hcatFunction
hcat(A...)

Concatenate along dimension 2.

Examples

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

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

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

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

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

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

julia> hcat(x, [1; 2; 3])
3×1 Matrix{Any}:
1
2
3
source
Base.hvcatFunction
hvcat(rows::Tuple{Vararg{Int}}, values...)

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

Examples

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

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

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

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

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

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

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

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

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

Examples

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

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

[:, :, 2] =
4  5  6

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

[:, :, 2] =
3
4

[:, :, 3] =
5
6

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

[:, :, 2] =
3  4

[:, :, 3] =
5  6

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

[:, :, 2] =
4  5  6

Examples for construction of the arguments:

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

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

source
Base.vectFunction
vect(X...)

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

Examples

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

Examples

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

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

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

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

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

julia> circshift(a, -1)
5-element BitVector:
1
0
0
1
1
source
Base.circshift!Function
circshift!(dest, src, shifts)

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

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

source
Base.circcopy!Function
circcopy!(dest, src)

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

Examples

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

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

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

julia> dest[1:3,2:4] == src[1:3,2:4]
true
source
Base.findallMethod
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).

Examples

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

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

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

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

julia> findall(falses(3))
Int64[]
source
Base.findallMethod
findall(f::Function, A)

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

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

Examples

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

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

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

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

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

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

Examples

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

julia> findfirst(A)
3

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

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

julia> findfirst(A)
CartesianIndex(2, 1)
source
Base.findfirstMethod
findfirst(predicate::Function, A)

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

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

Examples

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

julia> findfirst(iseven, A)
2

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

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

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

julia> findfirst(iseven, A)
CartesianIndex(2, 1)
source
Base.findlastMethod
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).

Examples

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

julia> findlast(A)
3

julia> A = falses(2,2);

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

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

julia> findlast(A)
CartesianIndex(2, 1)
source
Base.findlastMethod
findlast(predicate::Function, A)

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

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

Examples

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

julia> findlast(isodd, A)
3

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

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

julia> findlast(isodd, A)
CartesianIndex(2, 1)
source
Base.findnextMethod
findnext(A, i)

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

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

Examples

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

julia> findnext(A, 1)
3

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

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

julia> findnext(A, CartesianIndex(1, 1))
CartesianIndex(2, 1)
source
Base.findnextMethod
findnext(predicate::Function, A, i)

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

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

Examples

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

julia> findnext(isodd, A, 1)
1

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

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

julia> findnext(isodd, A, CartesianIndex(1, 1))
CartesianIndex(1, 1)
source
Base.findprevMethod
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).

Examples

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

julia> findprev(A, 3)
3

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

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

julia> findprev(A, CartesianIndex(2, 1))
CartesianIndex(2, 1)
source
Base.findprevMethod
findprev(predicate::Function, A, i)

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

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

Examples

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

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

julia> findprev(isodd, A, 3)
3

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

julia> findprev(isodd, A, CartesianIndex(1, 2))
CartesianIndex(2, 1)
source
Base.permutedimsFunction
permutedims(A::AbstractArray, perm)

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

Examples

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

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

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

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

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

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

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

julia> size(B)[perm] == ans
true
source
permutedims(m::AbstractMatrix)

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

Examples

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

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

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

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

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

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

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

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

Examples

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

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

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

julia> transpose(V)
1×2 transpose(::Vector{Matrix{Int64}}) with eltype Transpose{Int64, Matrix{Int64}}:
[1 3; 2 4]  [5 7; 6 8]
source
Base.permutedims!Function
permutedims!(dest, src, perm)

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

source
Base.PermutedDimsArrays.PermutedDimsArrayType
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).

Examples

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

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

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

julia> B[3,1,2] == A[1,2,3]
true
source
Base.promote_shapeFunction
promote_shape(s1, s2)

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

Examples

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

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

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

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

Array functions

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

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

Julia 1.5

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

Examples

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

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

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

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

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

julia> accumulate(+, fill(1, 3, 3), dims=2)
3×3 Matrix{Int64}:
1  2  3
1  2  3
1  2  3
source
Base.accumulate!Function
accumulate!(op, B, A; [dims], [init])

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

Examples

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

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

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

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

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

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

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

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

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

julia> B
2×2 Matrix{Int64}:
1  -1
3  -1
source
Base.cumprodFunction
cumprod(A; dims::Integer)

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

Examples

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

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

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

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

Julia 1.5

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

Examples

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

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

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

julia> cumprod(x^2 for x in 1:3)
3-element Vector{Int64}:
1
4
36
source
Base.cumprod!Function
cumprod!(B, A; dims::Integer)

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

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

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

source
Base.cumsumFunction
cumsum(A; dims::Integer)

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

Examples

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

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

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

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

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

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

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

source
cumsum(itr)

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

Julia 1.5

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

Examples

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

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

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

julia> cumsum(x^2 for x in 1:3)
3-element Vector{Int64}:
1
5
14
source
Base.cumsum!Function
cumsum!(B, A; dims::Integer)

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

source
Base.diffFunction
diff(A::AbstractVector)
diff(A::AbstractArray; dims::Integer)

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

Julia 1.1

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

Examples

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

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

julia> diff(vec(a))
3-element Vector{Int64}:
4
-2
12
source
Base.repeatFunction
repeat(A::AbstractArray, counts::Integer...)

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

Examples

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

julia> repeat([1, 2, 3], 2, 3)
6×3 Matrix{Int64}:
1  1  1
2  2  2
3  3  3
1  1  1
2  2  2
3  3  3
source
repeat(A::AbstractArray; inner=ntuple(Returns(1), ndims(A)), outer=ntuple(Returns(1), ndims(A)))

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

Examples

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

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

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

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

Examples

julia> repeat("ha", 3)
"hahaha"
source
repeat(c::AbstractChar, r::Integer) -> String

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

Examples

julia> repeat('A', 3)
"AAA"
source
Base.rot180Function
rot180(A)

Rotate matrix A 180 degrees.

Examples

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

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

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

Examples

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

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

julia> rot180(a,2)
2×2 Matrix{Int64}:
1  2
3  4
source
Base.rotl90Function
rotl90(A)

Rotate matrix A left 90 degrees.

Examples

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

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

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

Examples

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

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

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

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

julia> rotl90(a,4)
2×2 Matrix{Int64}:
1  2
3  4
source
Base.rotr90Function
rotr90(A)

Rotate matrix A right 90 degrees.

Examples

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

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

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

Examples

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

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

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

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

julia> rotr90(a,4)
2×2 Matrix{Int64}:
1  2
3  4
source
Base.mapslicesFunction
mapslices(f, A; dims)

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

Examples

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

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

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

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

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

[:, :, 2, 1] =
26

[:, :, 1, 2] =
42

[:, :, 2, 2] =
58
source
Base.eachrowFunction
eachrow(A::AbstractVecOrMat)

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

Julia 1.1

This function requires at least Julia 1.1.

Example

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

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

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

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

Julia 1.1

This function requires at least Julia 1.1.

Example

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

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

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

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

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

Julia 1.1

This function requires at least Julia 1.1.

Example

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

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

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

Combinatorics

Base.invpermFunction
invperm(v)

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

Examples

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

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

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

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

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

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

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

Return true if v is a valid permutation.

Examples

julia> isperm([1; 2])
true

julia> isperm([1; 3])
false
source
Base.permute!Method
permute!(v, p)

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

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

Examples

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

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

julia> permute!(A, perm);

julia> A
4-element Vector{Int64}:
1
4
3
1
source
Base.invpermute!Function
invpermute!(v, p)

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

Examples

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

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

julia> invpermute!(A, perm);

julia> A
4-element Vector{Int64}:
4
1
3
1
source
Base.reverseMethod
reverse(A; dims=:)

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

Examples

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

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

julia> reverse(b)
2×2 Matrix{Int64}:
4  3
2  1
Julia 1.6

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

source
Base.reverseindFunction
reverseind(v, i)

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

Examples

julia> s = "Julia🚀"
"Julia🚀"

julia> r = reverse(s)
"🚀ailuJ"

julia> for i in eachindex(s)
print(r[reverseind(r, i)])
end
Julia🚀
source
Base.reverse!Function
reverse!(v [, start=1 [, stop=length(v) ]]) -> v

In-place version of reverse.

Examples

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

julia> reverse!(A);

julia> A
5-element Vector{Int64}:
5
4
3
2
1
source
reverse!(A; dims=:)

Like reverse, but operates in-place in A.

Julia 1.6

Multidimensional reverse! requires Julia 1.6.

source