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.

source

Base.AbstractVector — Type

AbstractVector{T}

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

source

Base.AbstractMatrix — Type

AbstractMatrix{T}

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

source

Base.AbstractVecOrMat — Type

AbstractVecOrMat{T}

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

source

Core.Array — Type

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

N-dimensional dense array with elements of type T.

source

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

source

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

source

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

source

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

source

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.

Basic functions

Base.ndims — Function

ndims(A::AbstractArray) -> Integer

Return the number of dimensions of A.

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, 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

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

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

julia> abs.((1, -2))
(1, 2)

julia> broadcast(+, 1.0, (0, -2.0))
(1.0, -1.0)

julia> (+).([[0,2], [1,3]], Ref{Vector{Int}}([1,-1]))
2-element Vector{Vector{Int64}}:
 [1, 1]
 [2, 2]

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

source

Base.Broadcast.broadcast! — Function

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

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

Examples

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

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

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

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

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

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

source

Base.Broadcast.@__dot__ — Macro

@. expr

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

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

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

Examples

julia> x = 1.0:3.0; y = similar(x);

julia> @. y = x + 3 * sin(x)
3-element Vector{Float64}:
 3.5244129544236893
 4.727892280477045
 3.4233600241796016

source

For specializing broadcast on custom types, see

Base.Broadcast.BroadcastStyle — Type

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

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

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

source

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 AbstractArrayStyles and keep track of dimensionality, your style needs to support a Val constructor:

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

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

Indexing and assignment

Base.getindex — Method

getindex(A, inds...)

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

Examples

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

julia> getindex(A, 1)
1

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

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

source

Base.setindex! — Method

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

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

Examples

julia> A = zeros(2,2);

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

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

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

source

Base.copyto! — Method

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

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

source

Base.copy! — Function

copy!(dst, src) -> dst

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

Views (SubArrays and other view types)

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

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

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

Base.view — Function

view(A, inds...)

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

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

Julia 1.6

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

Examples

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

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

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

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

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

source

Base.@view — Macro

@view A[inds...]

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

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

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

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

Julia 1.5

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

Examples

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

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

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

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

source

Base.@views — Macro

@views expression

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

Note

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

Julia 1.5

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

Examples

julia> A = zeros(3, 3);

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

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

source

Base.parent — Function

parent(A)

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

Examples

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

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

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

source

Base.parentindices — Function

parentindices(A)

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

Examples

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

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

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

source

Base.selectdim — Function

selectdim(A, d::Integer, i)

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

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

Concatenation and permutation

Base.cat — Function

cat(A...; dims)

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

See also hcat, vcat, hvcat, repeat.

Examples

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

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

source

Base.vcat — Function

vcat(A...)

Concatenate along dimension 1.

Examples

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

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

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

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

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

source

Base.hcat — Function

hcat(A...)

Concatenate along dimension 2.

Examples

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

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

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

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

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

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

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

source

Base.hvcat — Function

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

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

Examples

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

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

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

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

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

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

source

Base.hvncat — Function

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

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

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

Examples

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

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

[:, :, 2] =
 4  5  6

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

[:, :, 2] =
 3
 4

[:, :, 3] =
 5
 6

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

[:, :, 2] =
 3  4

[:, :, 3] =
 5  6

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

[:, :, 2] =
 4  5  6

Examples for construction of the arguments:

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

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

source

Base.vect — Function

vect(X...)

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

Examples

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

source

Base.circshift — Function

circshift(A, shifts)

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

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

Examples

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

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

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

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

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

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

source

Base.circshift! — Function

circshift!(dest, src, shifts)

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

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

Array functions

Base.accumulate — Function

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

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

Julia 1.5

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

Examples

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

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

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

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

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

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

source

Base.accumulate! — Function

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

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

Examples

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

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

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

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

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

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

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

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

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

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

source

Base.cumprod — Function

cumprod(A; dims::Integer)

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

Examples

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

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

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

source

cumprod(itr)

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

See also cumprod!, accumulate, cumsum.

Julia 1.5

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

Examples

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

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

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

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

source

Base.cumprod! — Function

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

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

source

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

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

source

Base.cumsum — Function

cumsum(A; dims::Integer)

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

Examples

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

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

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

Note

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

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

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

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

source

cumsum(itr)

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

Julia 1.5

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

Examples

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

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

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

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

source

Base.cumsum! — Function

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

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

source

Base.diff — Function

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

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

Julia 1.1

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

Examples

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

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

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

source

Base.repeat — Function

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

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

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.

Combinatorics

Base.invperm — Function

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.isperm — Function

isperm(v) -> Bool

Return true if v is a valid permutation.

Examples

julia> isperm([1; 2])
true

julia> isperm([1; 3])
false

source

Base.permute! — Method

permute!(v, p)

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

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