Arrays

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

    Base.AbstractVector — Type

    1. AbstractVector{T}

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

    source

    — Type

    1. AbstractMatrix{T}

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

    Base.AbstractVecOrMat — Type

    1. AbstractVecOrMat{T}

    Union type of and AbstractMatrix{T}.

    Core.Array — Type

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

    N-dimensional dense array with elements of type T.

    Core.Array — Method

    1. Array{T}(undef, dims)
    2. Array{T,N}(undef, dims)

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

    Examples

    1. julia> A = Array{Float64, 2}(undef, 2, 3) # N given explicitly
    2. 2×3 Array{Float64, 2}:
    3.  6.90198e-310  6.90198e-310  6.90198e-310
    4.  6.90198e-310  6.90198e-310  0.0
    6. julia> B = Array{Float64}(undef, 2) # N determined by the input
    7. 2-element Array{Float64, 1}:
    8.  1.87103e-320
    9.  0.0

    Core.Array — Method

    1. Array{T}(nothing, dims)
    2. Array{T,N}(nothing, dims)

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

    Examples

    1. julia> Array{Union{Nothing, String}}(nothing, 2)
    2. 2-element Vector{Union{Nothing, String}}:
    3.  nothing
    4.  nothing
    6. julia> Array{Union{Nothing, Int}}(nothing, 2, 3)
    7. 2×3 Matrix{Union{Nothing, Int64}}:
    8.  nothing  nothing  nothing
    9.  nothing  nothing  nothing

    Core.Array — Method

    1. Array{T}(missing, dims)
    2. Array{T,N}(missing, dims)

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

    Examples

    1. julia> Array{Union{Missing, String}}(missing, 2)
    2. 2-element Vector{Union{Missing, String}}:
    3.  missing
    4.  missing
    6. julia> Array{Union{Missing, Int}}(missing, 2, 3)
    7. 2×3 Matrix{Union{Missing, Int64}}:
    8.  missing  missing  missing
    9.  missing  missing  missing

    Core.UndefInitializer — Type

    1. UndefInitializer

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

    Examples

    1. julia> Array{Float64, 1}(UndefInitializer(), 3)
    2. 3-element Array{Float64, 1}:
    3.  2.2752528595e-314
    4.  2.202942107e-314
    5.  2.275252907e-314

    source

    — Constant

    1. 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

    1. julia> Array{Float64, 1}(undef, 3)
    2. 3-element Array{Float64, 1}:
    3.  2.2752528595e-314
    4.  2.202942107e-314
    5.  2.275252907e-314

    Base.Vector — Type

    1. Vector{T} <: AbstractVector{T}

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

    source

    — Method

    1. Vector{T}(undef, n)

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

    Examples

    1. julia> Vector{Float64}(undef, 3)
    2. 3-element Array{Float64, 1}:
    3.  6.90966e-310
    4.  6.90966e-310
    5.  6.90966e-310

    source

    — Method

    1. Vector{T}(nothing, m)

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

    Examples

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

    source

    — Method

    1. Vector{T}(missing, m)

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

    Examples

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

    source

    — Type

    1. Matrix{T} <: AbstractMatrix{T}

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

    Base.Matrix — Method

    1. Matrix{T}(undef, m, n)

    Construct an uninitialized of size m×n. See undef.

    Examples

    1. julia> Matrix{Float64}(undef, 2, 3)
    2. 2×3 Array{Float64, 2}:
    3.  6.93517e-310  6.93517e-310  6.93517e-310
    4.  6.93517e-310  6.93517e-310  1.29396e-320

    Base.Matrix — Method

    1. Matrix{T}(nothing, m, n)

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

    Examples

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

    Base.Matrix — Method

    1. Matrix{T}(missing, m, n)

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

    Examples

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

    Base.VecOrMat — Type

    1. VecOrMat{T}

    Union type of and Matrix{T}.

    Core.DenseArray — Type

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

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

    Base.DenseVector — Type

    1. DenseVector{T}

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

    source

    — Type

    1. DenseMatrix{T}

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

    Base.DenseVecOrMat — Type

    1. DenseVecOrMat{T}

    Union type of and DenseMatrix{T}.

    Base.StridedArray — Type

    1. StridedArray{T, N}

    A hard-coded 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.

    Base.StridedVector — Type

    1. StridedVector{T}

    One dimensional with elements of type T.

    source

    — Type

    1. StridedMatrix{T}

    Two dimensional StridedArray with elements of type T.

    Base.StridedVecOrMat — Type

    1. StridedVecOrMat{T}

    Union type of and StridedMatrix with elements of type T.

    Base.getindex — Method

    1. 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

    1. julia> Int8[1, 2, 3]
    2. 3-element Vector{Int8}:
    3.  1
    4.  2
    5.  3
    7. julia> getindex(Int8, 1, 2, 3)
    8. 3-element Vector{Int8}:
    9.  1
    10.  2
    11.  3

    Base.zeros — Function

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

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

    Examples

    1. julia> zeros(1)
    2. 1-element Vector{Float64}:
    3.  0.0
    5. julia> zeros(Int8, 2, 3)
    6. 2×3 Matrix{Int8}:
    7.  0  0  0
    8.  0  0  0

    Base.ones — Function

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

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

    Examples

    1. julia> ones(1,2)
    2. 1×2 Matrix{Float64}:
    3.  1.0  1.0
    5. julia> ones(ComplexF64, 2, 3)
    6. 2×3 Matrix{ComplexF64}:
    7.  1.0+0.0im  1.0+0.0im  1.0+0.0im
    8.  1.0+0.0im  1.0+0.0im  1.0+0.0im

    Base.BitArray — Type

    1. 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 operations that generate boolean elements (including dotted-comparisons like .==) as well as from the functions trues and .

    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.

    source

    — Method

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

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

    Examples

    1. julia> BitArray(undef, 2, 2)
    2. 2×2 BitMatrix:
    3.  0  0
    4.  0  0
    6. julia> BitArray(undef, (3, 1))
    7. 3×1 BitMatrix:
    8.  0
    9.  0
    10.  0

    Base.BitArray — Method

    1. BitArray(itr)

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

    Examples

    1. julia> BitArray([1 0; 0 1])
    2. 2×2 BitMatrix:
    3.  1  0
    4.  0  1
    6. julia> BitArray(x+y == 3 for x = 1:2, y = 1:3)
    7. 2×3 BitMatrix:
    8.  0  1  0
    9.  1  0  0
    11. julia> BitArray(x+y == 3 for x = 1:2 for y = 1:3)
    12. 6-element BitVector:
    13.  0
    14.  1
    15.  0
    16.  1
    17.  0
    18.  0

    source

    — Function

    1. trues(dims)

    Create a BitArray with all values set to true.

    Examples

    1. julia> trues(2,3)
    2. 2×3 BitMatrix:
    3.  1  1  1
    4.  1  1  1

    source

    — Function

    1. falses(dims)

    Create a BitArray with all values set to false.

    Examples

    1. julia> falses(2,3)
    2. 2×3 BitMatrix:
    3.  0  0  0
    4.  0  0  0

    source

    — Function

    1. fill(x, dims::Tuple)
    2. 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

    1. julia> fill(1.0, (2,3))
    2. 2×3 Matrix{Float64}:
    3.  1.0  1.0  1.0
    4.  1.0  1.0  1.0
    6. julia> fill(42)
    7. 0-dimensional Array{Int64, 0}:
    8. 42

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

    1. julia> A = fill(zeros(2), 2);
    3. julia> A[1][1] = 42; # modifies both A[1][1] and A[2][1]
    5. julia> A
    6. 2-element Vector{Vector{Float64}}:
    7.  [42.0, 0.0]
    8.  [42.0, 0.0]

    source

    — Function

    1. 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

    1. julia> A = zeros(2,3)
    2. 2×3 Matrix{Float64}:
    3.  0.0  0.0  0.0
    4.  0.0  0.0  0.0
    6. julia> fill!(A, 2.)
    7. 2×3 Matrix{Float64}:
    8.  2.0  2.0  2.0
    9.  2.0  2.0  2.0
    11. julia> a = [1, 1, 1]; A = fill!(Vector{Vector{Int}}(undef, 3), a); a[1] = 2; A
    12. 3-element Vector{Vector{Int64}}:
    13.  [2, 1, 1]
    14.  [2, 1, 1]
    15.  [2, 1, 1]
    17. julia> x = 0; f() = (global x += 1; x); fill!(Vector{Int}(undef, 3), f())
    18. 3-element Vector{Int64}:
    19.  1
    20.  1
    21.  1

    source

    — Function

    1. 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:

    1. julia> similar(1:10, 1, 4)
    2. 1×4 Array{Int64,2}:
    3.  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:

    1. julia> similar(trues(10,10), 2)
    2. 2-element BitVector:
    3.  0
    4.  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:

    1. julia> similar(falses(10), Float64, 2, 4)
    2. 2×4 Array{Float64,2}:
    3.  2.18425e-314  2.18425e-314  2.18425e-314  2.18425e-314
    4.  2.18425e-314  2.18425e-314  2.18425e-314  2.18425e-314

    1. similar(storagetype, axes)

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

    Examples:

    1. 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.

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

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

    source

    Basic functions

    — Function

    1. ndims(A::AbstractArray) -> Integer

    Return the number of dimensions of A.

    Examples

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

    source

    — Function

    1. 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 .

    Examples

    1. julia> A = fill(1, (2,3,4));
    3. julia> size(A)
    4. (2, 3, 4)
    6. julia> size(A, 2)
    7. 3

    source

    — Method

    1. axes(A)

    Return the tuple of valid indices for array A.

    Examples

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

    source

    — Method

    1. axes(A, d)

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

    See also size, and the manual chapter on .

    Examples

    1. julia> A = fill(1, (5,6,7));
    3. julia> axes(A, 2)
    4. 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/:

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

    source

    — Method

    1. length(A::AbstractArray)

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

    Examples

    1. julia> length([1, 2, 3, 4])
    2. 4
    4. julia> length([1 2; 3 4])
    5. 4

    source

    — Function

    1. 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 otherwise). If the arrays have different sizes and/or dimensionalities, a DimensionMismatch exception will be thrown.

    Examples

    1. julia> A = [1 2; 3 4];
    3. julia> for i in eachindex(A) # linear indexing
    4.            println(i)
    5.        end
    6. 1
    7. 2
    8. 3
    9. 4
    11. julia> for i in eachindex(view(A, 1:2, 1:1)) # Cartesian indexing
    12.            println(i)
    13.        end
    14. CartesianIndex(1, 1)
    15. CartesianIndex(2, 1)

    source

    — Type

    1. IndexStyle(A)
    2. 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 ) or cartesian indexing. If you decide to only implement linear indexing, then you must set this trait for your array type:

    1. 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, creates an iterator whose type depends on the setting of this trait.

    source

    — Type

    1. 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 from an array that is IndexLinear will return a simple one-dimensional range, even if it is multidimensional.

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

    See also IndexCartesian.

    Base.IndexCartesian — Type

    1. IndexCartesian()

    Subtype of 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 from an array that is IndexCartesian will return a range of CartesianIndices.

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

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

    See also .

    source

    — Function

    1. conj!(A)

    Transform an array to its complex conjugate in-place.

    See also conj.

    Examples

    1. julia> A = [1+im 2-im; 2+2im 3+im]
    2. 2×2 Matrix{Complex{Int64}}:
    3.  1+1im  2-1im
    4.  2+2im  3+1im
    6. julia> conj!(A);
    8. julia> A
    9. 2×2 Matrix{Complex{Int64}}:
    10.  1-1im  2+1im
    11.  2-2im  3-1im

    Base.stride — Function

    1. stride(A, k::Integer)

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

    Examples

    1. julia> A = fill(1, (3,4,5));
    3. julia> stride(A,2)
    4. 3
    6. julia> stride(A,3)
    7. 12

    Base.strides — Function

    1. strides(A)

    Return a tuple of the memory strides in each dimension.

    Examples

    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.

    — Function

    1. 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 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.

    1. julia> A = [1, 2, 3, 4, 5]
    2. 5-element Vector{Int64}:
    3.  1
    4.  2
    5.  3
    6.  4
    7.  5
    9. julia> B = [1 2; 3 4; 5 6; 7 8; 9 10]
    10. 5×2 Matrix{Int64}:
    11.  1   2
    12.  3   4
    13.  5   6
    14.  7   8
    15.  9  10
    17. julia> broadcast(+, A, B)
    18. 5×2 Matrix{Int64}:
    19.   2   3
    20.   5   6
    21.   8   9
    22.  11  12
    23.  14  15
    25. julia> parse.(Int, ["1", "2"])
    26. 2-element Vector{Int64}:
    27.  1
    28.  2
    30. julia> abs.((1, -2))
    31. (1, 2)
    33. julia> broadcast(+, 1.0, (0, -2.0))
    34. (1.0, -1.0)
    36. julia> (+).([[0,2], [1,3]], Ref{Vector{Int}}([1,-1]))
    37. 2-element Vector{Vector{Int64}}:
    38.  [1, 1]
    39.  [2, 2]
    41. julia> string.(("one","two","three","four"), ": ", 1:4)
    42. 4-element Vector{String}:
    43.  "one: 1"
    44.  "two: 2"
    45.  "three: 3"
    46.  "four: 4"

    Base.Broadcast.broadcast! — Function

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

    Like , 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

    1. julia> A = [1.0; 0.0]; B = [0.0; 0.0];
    3. julia> broadcast!(+, B, A, (0, -2.0));
    5. julia> B
    6. 2-element Vector{Float64}:
    7.   1.0
    8.  -2.0
    10. julia> A
    11. 2-element Vector{Float64}:
    12.  1.0
    13.  0.0
    15. julia> broadcast!(+, A, A, (0, -2.0));
    17. julia> A
    18. 2-element Vector{Float64}:
    19.   1.0
    20.  -2.0

    source

    — Macro

    1. @. 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

    1. julia> x = 1.0:3.0; y = similar(x);
    3. julia> @. y = x + 3 * sin(x)
    4. 3-element Vector{Float64}:
    5.  3.5244129544236893
    6.  4.727892280477045
    7.  3.4233600241796016

    source

    For specializing broadcast on custom types, see

    — 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

    1. struct MyContainerStyle <: BroadcastStyle end
    2. 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 for more information.

    source

    — 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:

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

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

    1. struct MyArrayStyle <: Broadcast.AbstractArrayStyle{Any} end
    2. 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:

    1. struct MyArrayStyleDim{N} <: Broadcast.AbstractArrayStyle{N} end
    2. (::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 rules to control the output type.

    See also Broadcast.DefaultArrayStyle.

    Base.Broadcast.ArrayStyle — Type

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

    source

    — Type

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

    source

    — Function

    1. Broadcast.broadcastable(x)

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

    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 , then it must define its broadcastable method to return itself for the custom style to have any effect.

    Examples

    1. julia> Broadcast.broadcastable([1,2,3]) # like `identity` since arrays already support axes and indexing
    2. 3-element Vector{Int64}:
    3.  1
    4.  2
    5.  3
    7. julia> Broadcast.broadcastable(Int) # Types don't support axes, indexing, or iteration but are commonly used as scalars
    8. Base.RefValue{Type{Int64}}(Int64)
    10. julia> Broadcast.broadcastable("hello") # Strings break convention of matching iteration and act like a scalar instead
    11. Base.RefValue{String}("hello")

    source

    — Function

    1. combine_axes(As...) -> Tuple

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

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

    source

    — Function

    1. combine_styles(cs...) -> BroadcastStyle

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

    Examples

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

    source

    — Function

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

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

    Examples

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

    Base.getindex — Method

    1. getindex(A, inds...)

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

    Examples

    1. julia> A = [1 2; 3 4]
    2. 2×2 Matrix{Int64}:
    3.  1  2
    4.  3  4
    6. julia> getindex(A, 1)
    7. 1
    9. julia> getindex(A, [2, 1])
    10. 2-element Vector{Int64}:
    11.  3
    12.  1
    14. julia> getindex(A, 2:4)
    15. 3-element Vector{Int64}:
    16.  3
    17.  2
    18.  4

    source

    — Method

    1. setindex!(A, X, inds...)
    2. 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

    1. julia> A = zeros(2,2);
    3. julia> setindex!(A, [10, 20], [1, 2]);
    5. julia> A[[3, 4]] = [30, 40];
    7. julia> A
    8. 2×2 Matrix{Float64}:
    9.  10.0  30.0
    10.  20.0  40.0

    source

    — Method

    1. 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

    — Function

    1. 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

    1. julia> isassigned(rand(3, 3), 5)
    2. true
    4. julia> isassigned(rand(3, 3), 3 * 3 + 1)
    5. false
    7. julia> mutable struct Foo end
    9. julia> v = similar(rand(3), Foo)
    10. 3-element Vector{Foo}:
    11.  #undef
    12.  #undef
    13.  #undef
    15. julia> isassigned(v, 1)
    16. false

    source

    — Type

    1. 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

    — Type

    1. CartesianIndex(i, j, k...)   -> I
    2. 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 .

    Examples

    1. julia> A = reshape(Vector(1:16), (2, 2, 2, 2))
    2. 2×2×2×2 Array{Int64, 4}:
    3. [:, :, 1, 1] =
    4.  1  3
    5.  2  4
    7. [:, :, 2, 1] =
    8.  5  7
    9.  6  8
    11. [:, :, 1, 2] =
    12.   9  11
    13.  10  12
    15. [:, :, 2, 2] =
    16.  13  15
    17.  14  16
    19. julia> A[CartesianIndex((1, 1, 1, 1))]
    20. 1
    22. julia> A[CartesianIndex((1, 1, 1, 2))]
    23. 9
    25. julia> A[CartesianIndex((1, 1, 2, 1))]
    26. 5

    source

    — Type

    1. CartesianIndices(sz::Dims) -> R
    2. 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 will return CartesianIndex indices I equivalent to the nested loops

    1. for j = jstart:jstep:jstop
    2.     for i = istart:istep:istop
    3.         ...
    4.     end
    5. end

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

    1. 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

    1. julia> foreach(println, CartesianIndices((2, 2, 2)))
    2. CartesianIndex(1, 1, 1)
    3. CartesianIndex(2, 1, 1)
    4. CartesianIndex(1, 2, 1)
    5. CartesianIndex(2, 2, 1)
    6. CartesianIndex(1, 1, 2)
    7. CartesianIndex(2, 1, 2)
    8. CartesianIndex(1, 2, 2)
    9. CartesianIndex(2, 2, 2)
    11. julia> CartesianIndices(fill(1, (2,3)))
    12. 2×3 CartesianIndices{2, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}}}:
    13.  CartesianIndex(1, 1)  CartesianIndex(1, 2)  CartesianIndex(1, 3)
    14.  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:

    1. julia> cartesian = CartesianIndices((1:3, 1:2))
    2. 3×2 CartesianIndices{2, Tuple{UnitRange{Int64}, UnitRange{Int64}}}:
    3.  CartesianIndex(1, 1)  CartesianIndex(1, 2)
    4.  CartesianIndex(2, 1)  CartesianIndex(2, 2)
    5.  CartesianIndex(3, 1)  CartesianIndex(3, 2)
    7. julia> cartesian[4]
    8. CartesianIndex(1, 2)
    10. julia> cartesian = CartesianIndices((1:2:5, 1:2))
    11. 3×2 CartesianIndices{2, Tuple{StepRange{Int64, Int64}, UnitRange{Int64}}}:
    12.  CartesianIndex(1, 1)  CartesianIndex(1, 2)
    13.  CartesianIndex(3, 1)  CartesianIndex(3, 2)
    14.  CartesianIndex(5, 1)  CartesianIndex(5, 2)
    16. julia> cartesian[2, 2]
    17. CartesianIndex(3, 2)

    Broadcasting

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

    Julia 1.1

    Broadcasting of CartesianIndices requires at least Julia 1.1.

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

    For cartesian to linear index conversion, see .

    source

    — Type

    1. Dims{N}

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

    Base.LinearIndices — Type

    1. LinearIndices(A::AbstractArray)

    Return a LinearIndices array with the same shape and 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

    1. julia> A = fill(1, (5,6,7));
    3. julia> b = LinearIndices(A);
    5. julia> extrema(b)
    6. (1, 210)
    1. LinearIndices(inds::CartesianIndices) -> R
    2. LinearIndices(sz::Dims) -> R
    3. 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:

    1. 3×2 LinearIndices{2, Tuple{UnitRange{Int64}, UnitRange{Int64}}}:
    2.  1  4
    3.  2  5
    4.  3  6
    6. julia> linear[1,2]
    7. 4

    Base.to_indices — Function

    1. 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.

    Base.checkbounds — Function

    1. 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 .

    See also checkindex.

    Examples

    1. julia> A = rand(3, 3);
    3. julia> checkbounds(Bool, A, 2)
    4. true
    6. julia> checkbounds(Bool, A, 3, 4)
    7. false
    9. julia> checkbounds(Bool, A, 1:3)
    10. true
    12. julia> checkbounds(Bool, A, 1:3, 2:4)
    13. false

    1. checkbounds(A, I...)

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

    source

    — Function

    1. 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

    1. julia> checkindex(Bool, 1:20, 8)
    2. true
    4. julia> checkindex(Bool, 1:20, 21)
    5. false

    source

    — Function

    1. 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

    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 .

    Base.view — Function

    1. view(A, inds...)

    Like , 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 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

    1. julia> A = [1 2; 3 4]
    2. 2×2 Matrix{Int64}:
    3.  1  2
    4.  3  4
    6. julia> b = view(A, :, 1)
    7. 2-element view(::Matrix{Int64}, :, 1) with eltype Int64:
    8.  1
    9.  3
    11. julia> fill!(b, 0)
    12. 2-element view(::Matrix{Int64}, :, 1) with eltype Int64:
    13.  0
    14.  0
    16. julia> A # Note A has changed even though we modified b
    17. 2×2 Matrix{Int64}:
    18.  0  2
    19.  0  4
    21. julia> view(2:5, 2:3) # returns a range as type is immutable
    22. 3:4

    Base.@view — Macro

    1. @view A[inds...]

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

    Julia 1.5

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

    Examples

    1. julia> A = [1 2; 3 4]
    2. 2×2 Matrix{Int64}:
    3.  1  2
    4.  3  4
    6. julia> b = @view A[:, 1]
    7. 2-element view(::Matrix{Int64}, :, 1) with eltype Int64:
    8.  1
    9.  3
    11. julia> fill!(b, 0)
    12. 2-element view(::Matrix{Int64}, :, 1) with eltype Int64:
    13.  0
    14.  0
    16. julia> A
    17. 2×2 Matrix{Int64}:
    18.  0  2
    19.  0  4

    source

    — Macro

    1. @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

    1. julia> A = zeros(3, 3);
    3. julia> @views for row in 1:3
    4.            b = A[row, :]
    5.            b[:] .= row
    6.        end
    8. julia> A
    9. 3×3 Matrix{Float64}:
    10.  1.0  1.0  1.0
    11.  2.0  2.0  2.0
    12.  3.0  3.0  3.0

    Base.parent — Function

    1. 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.

    Examples

    1. julia> A = [1 2; 3 4]
    2. 2×2 Matrix{Int64}:
    3.  1  2
    4.  3  4
    6. julia> V = view(A, 1:2, :)
    7. 2×2 view(::Matrix{Int64}, 1:2, :) with eltype Int64:
    8.  1  2
    9.  3  4
    11. julia> parent(V)
    12. 2×2 Matrix{Int64}:
    13.  1  2
    14.  3  4

    Base.parentindices — Function

    1. parentindices(A)

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

    Examples

    1. julia> A = [1 2; 3 4];
    3. julia> V = view(A, 1, :)
    4. 2-element view(::Matrix{Int64}, 1, :) with eltype Int64:
    5.  1
    6.  2
    8. julia> parentindices(V)
    9. (1, Base.Slice(Base.OneTo(2)))

    source

    — Function

    1. 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

    1. julia> A = [1 2 3 4; 5 6 7 8]
    2. 2×4 Matrix{Int64}:
    3.  1  2  3  4
    4.  5  6  7  8
    6. julia> selectdim(A, 2, 3)
    7. 2-element view(::Matrix{Int64}, :, 3) with eltype Int64:
    8.  3
    9.  7

    source

    — Function

    1. 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

    1. julia> reinterpret(Float32, UInt32(7))
    2. 1.0f-44
    4. julia> reinterpret(Float32, UInt32[1 2 3 4 5])
    5. 1×5 reinterpret(Float32, ::Matrix{UInt32}):
    6.  1.0f-45  3.0f-45  4.0f-45  6.0f-45  7.0f-45

    1. 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

    1. julia> A = [1 2; 3 4]
    2. 2×2 Matrix{Int64}:
    3.  1  2
    4.  3  4
    6. julia> reinterpret(reshape, Complex{Int}, A)    # the result is a vector
    7. 2-element reinterpret(reshape, Complex{Int64}, ::Matrix{Int64}) with eltype Complex{Int64}:
    8.  1 + 3im
    9.  2 + 4im
    11. julia> a = [(1,2,3), (4,5,6)]
    12. 2-element Vector{Tuple{Int64, Int64, Int64}}:
    13.  (1, 2, 3)
    14.  (4, 5, 6)
    16. julia> reinterpret(reshape, Int, a)             # the result is a matrix
    17. 3×2 reinterpret(reshape, Int64, ::Vector{Tuple{Int64, Int64, Int64}}) with eltype Int64:
    18.  1  4
    19.  2  5
    20.  3  6

    source

    — Function

    1. reshape(A, dims...) -> AbstractArray
    2. 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

    1. julia> A = Vector(1:16)
    2. 16-element Vector{Int64}:
    3.   1
    4.   2
    5.   3
    6.   4
    7.   5
    8.   6
    9.   7
    10.   8
    11.   9
    12.  10
    13.  11
    14.  12
    15.  13
    16.  14
    17.  15
    18.  16
    20. julia> reshape(A, (4, 4))
    21. 4×4 Matrix{Int64}:
    22.  1  5   9  13
    23.  2  6  10  14
    24.  3  7  11  15
    25.  4  8  12  16
    27. julia> reshape(A, 2, :)
    28. 2×8 Matrix{Int64}:
    29.  1  3  5  7   9  11  13  15
    30.  2  4  6  8  10  12  14  16
    32. julia> reshape(1:6, 2, 3)
    33. 2×3 reshape(::UnitRange{Int64}, 2, 3) with eltype Int64:
    34.  1  3  5
    35.  2  4  6

    source

    — Function

    1. dropdims(A; dims)

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

    Examples

    1. julia> a = reshape(Vector(1:4),(2,2,1,1))
    2. 2×2×1×1 Array{Int64, 4}:
    3. [:, :, 1, 1] =
    4.  1  3
    5.  2  4
    7. julia> dropdims(a; dims=3)
    8. 2×2×1 Array{Int64, 3}:
    9. [:, :, 1] =
    10.  1  3
    11.  2  4

    source

    — Function

    1. 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

    1. julia> a = [1 2 3; 4 5 6]
    2. 2×3 Matrix{Int64}:
    3.  1  2  3
    4.  4  5  6
    6. julia> vec(a)
    7. 6-element Vector{Int64}:
    8.  1
    9.  4
    10.  2
    11.  5
    12.  3
    13.  6
    15. julia> vec(1:3)
    16. 1:3

    See also reshape.

    Base.SubArray — Type

    1. 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 function.

    source

    Concatenation and permutation

    — Function

    1. cat(A...; dims=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.

    source

    — Function

    1. vcat(A...)

    Concatenate along dimension 1.

    Examples

    1. julia> a = [1 2 3 4 5]
    2. 1×5 Matrix{Int64}:
    3.  1  2  3  4  5
    5. julia> b = [6 7 8 9 10; 11 12 13 14 15]
    6. 2×5 Matrix{Int64}:
    7.   6   7   8   9  10
    8.  11  12  13  14  15
    10. julia> vcat(a,b)
    11. 3×5 Matrix{Int64}:
    12.   1   2   3   4   5
    13.   6   7   8   9  10
    14.  11  12  13  14  15
    16. julia> c = ([1 2 3], [4 5 6])
    17. ([1 2 3], [4 5 6])
    19. julia> vcat(c...)
    20. 2×3 Matrix{Int64}:
    21.  1  2  3
    22.  4  5  6

    source

    — Function

    1. hcat(A...)

    Concatenate along dimension 2.

    Examples

    1. julia> a = [1; 2; 3; 4; 5]
    2. 5-element Vector{Int64}:
    3.  1
    4.  2
    5.  3
    6.  4
    7.  5
    9. julia> b = [6 7; 8 9; 10 11; 12 13; 14 15]
    10. 5×2 Matrix{Int64}:
    11.   6   7
    12.   8   9
    13.  10  11
    14.  12  13
    15.  14  15
    17. julia> hcat(a,b)
    18. 5×3 Matrix{Int64}:
    19.  1   6   7
    20.  2   8   9
    21.  3  10  11
    22.  4  12  13
    23.  5  14  15
    25. julia> c = ([1; 2; 3], [4; 5; 6])
    26. ([1, 2, 3], [4, 5, 6])
    28. julia> hcat(c...)
    29. 3×2 Matrix{Int64}:
    30.  1  4
    31.  2  5
    32.  3  6
    34. julia> x = Matrix(undef, 3, 0)  # x = [] would have created an Array{Any, 1}, but need an Array{Any, 2}
    35. 3×0 Matrix{Any}
    37. julia> hcat(x, [1; 2; 3])
    38. 3×1 Matrix{Any}:
    39.  1
    40.  2
    41.  3

    source

    — Function

    1. 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

    1. julia> a, b, c, d, e, f = 1, 2, 3, 4, 5, 6
    2. (1, 2, 3, 4, 5, 6)
    4. julia> [a b c; d e f]
    5. 2×3 Matrix{Int64}:
    6.  1  2  3
    7.  4  5  6
    9. julia> hvcat((3,3), a,b,c,d,e,f)
    10. 2×3 Matrix{Int64}:
    11.  1  2  3
    12.  4  5  6
    14. julia> [a b;c d; e f]
    15. 3×2 Matrix{Int64}:
    16.  1  2
    17.  3  4
    18.  5  6
    20. julia> hvcat((2,2,2), a,b,c,d,e,f)
    21. 3×2 Matrix{Int64}:
    22.  1  2
    23.  3  4
    24.  5  6

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

    source

    — Function

    1. vect(X...)

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

    Examples

    1. julia> a = Base.vect(UInt8(1), 2.5, 1//2)
    2. 3-element Vector{Float64}:
    3.  1.0
    4.  2.5
    5.  0.5

    Base.circshift — Function

    1. 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

    1. julia> b = reshape(Vector(1:16), (4,4))
    2. 4×4 Matrix{Int64}:
    3.  1  5   9  13
    4.  2  6  10  14
    5.  3  7  11  15
    6.  4  8  12  16
    8. julia> circshift(b, (0,2))
    9. 4×4 Matrix{Int64}:
    10.   9  13  1  5
    11.  10  14  2  6
    12.  11  15  3  7
    13.  12  16  4  8
    15. julia> circshift(b, (-1,0))
    16. 4×4 Matrix{Int64}:
    17.  2  6  10  14
    18.  3  7  11  15
    19.  4  8  12  16
    20.  1  5   9  13
    22. julia> a = BitArray([true, true, false, false, true])
    23. 5-element BitVector:
    24.  1
    25.  1
    26.  0
    27.  0
    28.  1
    30. julia> circshift(a, 1)
    31. 5-element BitVector:
    32.  1
    33.  1
    34.  1
    35.  0
    36.  0
    38. julia> circshift(a, -1)
    39. 5-element BitVector:
    40.  1
    41.  0
    42.  0
    43.  1
    44.  1

    See also .

    source

    — Function

    1. circshift!(dest, src, shifts)

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

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

    See also circshift.

    Base.circcopy! — Function

    1. 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

    1. julia> src = reshape(Vector(1:16), (4,4))
    2. 4×4 Array{Int64,2}:
    3.  1  5   9  13
    4.  2  6  10  14
    5.  3  7  11  15
    6.  4  8  12  16
    8. julia> dest = OffsetArray{Int}(undef, (0:3,2:5))
    10. julia> circcopy!(dest, src)
    11. OffsetArrays.OffsetArray{Int64,2,Array{Int64,2}} with indices 0:3×2:5:
    12.  8  12  16  4
    13.  5   9  13  1
    14.  6  10  14  2
    15.  7  11  15  3
    17. julia> dest[1:3,2:4] == src[1:3,2:4]
    18. true

    — Method

    1. 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 .

    Examples

    1. julia> A = [true, false, false, true]
    2. 4-element Vector{Bool}:
    3.  1
    4.  0
    5.  0
    6.  1
    8. julia> findall(A)
    9. 2-element Vector{Int64}:
    10.  1
    11.  4
    13. julia> A = [true false; false true]
    14. 2×2 Matrix{Bool}:
    15.  1  0
    16.  0  1
    18. julia> findall(A)
    19. 2-element Vector{CartesianIndex{2}}:
    20.  CartesianIndex(1, 1)
    21.  CartesianIndex(2, 2)
    23. julia> findall(falses(3))
    24. Int64[]

    source

    — Method

    1. 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 .

    Examples

    source

    — Method

    1. 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 .

    Examples

    1. julia> A = [false, false, true, false]
    2. 4-element Vector{Bool}:
    3.  0
    4.  0
    5.  1
    6.  0
    8. julia> findfirst(A)
    9. 3
    11. julia> findfirst(falses(3)) # returns nothing, but not printed in the REPL
    13. julia> A = [false false; true false]
    14. 2×2 Matrix{Bool}:
    15.  0  0
    16.  1  0
    18. julia> findfirst(A)
    19. CartesianIndex(2, 1)

    source

    — Method

    1. 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 .

    Examples

    1. julia> A = [1, 4, 2, 2]
    2. 4-element Vector{Int64}:
    3.  1
    4.  4
    5.  2
    6.  2
    8. julia> findfirst(iseven, A)
    9. 2
    11. julia> findfirst(x -> x>10, A) # returns nothing, but not printed in the REPL
    13. julia> findfirst(isequal(4), A)
    14. 2
    16. julia> A = [1 4; 2 2]
    17. 2×2 Matrix{Int64}:
    18.  1  4
    19.  2  2
    21. julia> findfirst(iseven, A)
    22. CartesianIndex(2, 1)

    source

    — Method

    1. 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 .

    Examples

    1. julia> A = [true, false, true, false]
    2. 4-element Vector{Bool}:
    3.  1
    4.  0
    5.  1
    6.  0
    8. julia> findlast(A)
    9. 3
    11. julia> A = falses(2,2);
    13. julia> findlast(A) # returns nothing, but not printed in the REPL
    15. julia> A = [true false; true false]
    16. 2×2 Matrix{Bool}:
    17.  1  0
    18.  1  0
    20. julia> findlast(A)
    21. CartesianIndex(2, 1)

    source

    — Method

    1. 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 .

    Examples

    1. julia> A = [1, 2, 3, 4]
    2. 4-element Vector{Int64}:
    3.  1
    4.  2
    5.  3
    6.  4
    8. julia> findlast(isodd, A)
    9. 3
    11. julia> findlast(x -> x > 5, A) # returns nothing, but not printed in the REPL
    13. julia> A = [1 2; 3 4]
    14. 2×2 Matrix{Int64}:
    15.  1  2
    16.  3  4
    18. julia> findlast(isodd, A)
    19. CartesianIndex(2, 1)

    source

    — Method

    1. 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 .

    Examples

    1. julia> A = [false, false, true, false]
    2. 4-element Vector{Bool}:
    3.  0
    4.  0
    5.  1
    6.  0
    8. julia> findnext(A, 1)
    9. 3
    11. julia> findnext(A, 4) # returns nothing, but not printed in the REPL
    13. julia> A = [false false; true false]
    14. 2×2 Matrix{Bool}:
    15.  0  0
    16.  1  0
    18. julia> findnext(A, CartesianIndex(1, 1))
    19. CartesianIndex(2, 1)

    source

    — Method

    1. findnext(predicate::Function, A, i)

    Find the next index after or including 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 .

    Examples

    1. julia> A = [1, 4, 2, 2];
    3. julia> findnext(isodd, A, 1)
    4. 1
    6. julia> findnext(isodd, A, 2) # returns nothing, but not printed in the REPL
    8. julia> A = [1 4; 2 2];
    10. julia> findnext(isodd, A, CartesianIndex(1, 1))
    11. CartesianIndex(1, 1)

    source

    — Method

    1. 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 .

    Examples

    1. julia> A = [false, false, true, true]
    2. 4-element Vector{Bool}:
    3.  0
    4.  0
    5.  1
    6.  1
    8. julia> findprev(A, 3)
    9. 3
    11. julia> findprev(A, 1) # returns nothing, but not printed in the REPL
    13. julia> A = [false false; true true]
    14. 2×2 Matrix{Bool}:
    15.  0  0
    16.  1  1
    18. julia> findprev(A, CartesianIndex(2, 1))
    19. CartesianIndex(2, 1)

    source

    — Method

    1. 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 .

    Examples

    1. julia> A = [4, 6, 1, 2]
    2. 4-element Vector{Int64}:
    3.  4
    4.  6
    5.  1
    6.  2
    8. julia> findprev(isodd, A, 1) # returns nothing, but not printed in the REPL
    10. julia> findprev(isodd, A, 3)
    11. 3
    13. julia> A = [4 6; 1 2]
    14. 2×2 Matrix{Int64}:
    15.  1  2
    17. julia> findprev(isodd, A, CartesianIndex(1, 2))
    18. CartesianIndex(2, 1)

    source

    — Function

    1. permutedims(A::AbstractArray, perm)

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

    See also: PermutedDimsArray.

    Examples

    1. julia> A = reshape(Vector(1:8), (2,2,2))
    2. 2×2×2 Array{Int64, 3}:
    3. [:, :, 1] =
    4.  1  3
    5.  2  4
    7. [:, :, 2] =
    8.  5  7
    9.  6  8
    11. julia> permutedims(A, [3, 2, 1])
    12. 2×2×2 Array{Int64, 3}:
    13. [:, :, 1] =
    14.  1  3
    15.  5  7
    17. [:, :, 2] =
    18.  2  4
    19.  6  8

    1. 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

    1. julia> a = [1 2; 3 4];
    3. julia> b = [5 6; 7 8];
    5. julia> c = [9 10; 11 12];
    7. julia> d = [13 14; 15 16];
    9. julia> X = [[a] [b]; [c] [d]]
    10. 2×2 Matrix{Matrix{Int64}}:
    11.  [1 2; 3 4]     [5 6; 7 8]
    12.  [9 10; 11 12]  [13 14; 15 16]
    14. julia> permutedims(X)
    15. 2×2 Matrix{Matrix{Int64}}:
    16.  [1 2; 3 4]  [9 10; 11 12]
    17.  [5 6; 7 8]  [13 14; 15 16]
    19. julia> transpose(X)
    20. 2×2 transpose(::Matrix{Matrix{Int64}}) with eltype Transpose{Int64, Matrix{Int64}}:
    21.  [1 3; 2 4]  [9 11; 10 12]
    22.  [5 7; 6 8]  [13 15; 14 16]

    1. 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

    1. julia> permutedims([1, 2, 3, 4])
    2. 1×4 Matrix{Int64}:
    3.  1  2  3  4
    5. julia> V = [[[1 2; 3 4]]; [[5 6; 7 8]]]
    6. 2-element Vector{Matrix{Int64}}:
    7.  [1 2; 3 4]
    8.  [5 6; 7 8]
    10. julia> permutedims(V)
    11. 1×2 Matrix{Matrix{Int64}}:
    12.  [1 2; 3 4]  [5 6; 7 8]
    14. julia> transpose(V)
    15. 1×2 transpose(::Vector{Matrix{Int64}}) with eltype Transpose{Int64, Matrix{Int64}}:
    16.  [1 3; 2 4]  [5 7; 6 8]

    Base.permutedims! — Function

    1. permutedims!(dest, src, perm)

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

    See also .

    source

    — Type

    1. PermutedDimsArray(A, perm) -> B

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

    See also: permutedims.

    Examples

    1. julia> A = rand(3,5,4);
    3. julia> B = PermutedDimsArray(A, (3,1,2));
    5. julia> size(B)
    6. (4, 3, 5)
    8. julia> B[3,1,2] == A[1,2,3]
    9. true

    Base.promote_shape — Function

    1. promote_shape(s1, s2)

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

    Examples

    1. julia> a = fill(1, (3,4,1,1,1));
    3. julia> b = fill(1, (3,4));
    5. julia> promote_shape(a,b)
    6. (Base.OneTo(3), Base.OneTo(4), Base.OneTo(1), Base.OneTo(1), Base.OneTo(1))
    8. julia> promote_shape((2,3,1,4), (2, 3, 1, 4, 1))
    9. (2, 3, 1, 4, 1)

    Base.accumulate — Function

    1. 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 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,

    Julia 1.5

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

    Examples

    1. julia> accumulate(+, [1,2,3])
    2. 3-element Vector{Int64}:
    3.  1
    4.  3
    5.  6
    7. julia> accumulate(*, [1,2,3])
    8. 3-element Vector{Int64}:
    9.  1
    10.  2
    11.  6
    13. julia> accumulate(+, [1,2,3]; init=100)
    14. 3-element Vector{Int64}:
    15.  101
    16.  103
    17.  106
    19. julia> accumulate(min, [1,2,-1]; init=0)
    20. 3-element Vector{Int64}:
    21.   0
    22.   0
    23.  -1
    25. julia> accumulate(+, fill(1, 3, 3), dims=1)
    26. 3×3 Matrix{Int64}:
    27.  1  1  1
    28.  2  2  2
    29.  3  3  3
    31. julia> accumulate(+, fill(1, 3, 3), dims=2)
    32. 3×3 Matrix{Int64}:
    33.  1  2  3
    34.  1  2  3
    35.  1  2  3

    source

    — Function

    1. 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

    1. julia> x = [1, 0, 2, 0, 3];
    3. julia> y = [0, 0, 0, 0, 0];
    5. julia> accumulate!(+, y, x);
    7. julia> y
    8. 5-element Vector{Int64}:
    9.  1
    10.  1
    11.  3
    12.  3
    13.  6
    15. julia> A = [1 2; 3 4];
    17. julia> B = [0 0; 0 0];
    19. julia> accumulate!(-, B, A, dims=1);
    21. julia> B
    22. 2×2 Matrix{Int64}:
    23.   1   2
    24.  -2  -2
    26. julia> accumulate!(-, B, A, dims=2);
    28. julia> B
    29. 2×2 Matrix{Int64}:
    30.  1  -1
    31.  3  -1

    Base.cumprod — Function

    1. cumprod(A; dims::Integer)

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

    Examples

    1. julia> a = [1 2 3; 4 5 6]
    2. 2×3 Matrix{Int64}:
    3.  1  2  3
    4.  4  5  6
    6. julia> cumprod(a, dims=1)
    7. 2×3 Matrix{Int64}:
    8.  1   2   3
    9.  4  10  18
    11. julia> cumprod(a, dims=2)
    12. 2×3 Matrix{Int64}:
    13.  1   2    6
    14.  4  20  120

    source

    1. cumprod(itr)

    Cumulative product of an iterator. See also 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

    1. julia> cumprod(fill(1//2, 3))
    2. 3-element Vector{Rational{Int64}}:
    3.  1//2
    4.  1//4
    5.  1//8
    7. julia> cumprod([fill(1//3, 2, 2) for i in 1:3])
    8. 3-element Vector{Matrix{Rational{Int64}}}:
    9.  [1//3 1//3; 1//3 1//3]
    10.  [2//9 2//9; 2//9 2//9]
    11.  [4//27 4//27; 4//27 4//27]
    13. julia> cumprod((1, 2, 1))
    14. (1, 2, 2)
    16. julia> cumprod(x^2 for x in 1:3)
    17. 3-element Vector{Int64}:
    18.   1
    19.   4
    20.  36

    source

    — Function

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

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

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

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

    Base.cumsum — Function

    1. cumsum(A; dims::Integer)

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

    Examples

    1. julia> a = [1 2 3; 4 5 6]
    2. 2×3 Matrix{Int64}:
    3.  1  2  3
    4.  4  5  6
    6. julia> cumsum(a, dims=1)
    7. 2×3 Matrix{Int64}:
    8.  1  2  3
    9.  5  7  9
    11. julia> cumsum(a, dims=2)
    12. 2×3 Matrix{Int64}:
    13.  1  3   6
    14.  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.

    1. julia> cumsum(Int8[100, 28])
    2. 2-element Vector{Int64}:
    3.  100
    4.  128
    6. julia> accumulate(+,Int8[100, 28])
    7. 2-element Vector{Int8}:
    8.   100
    9.  -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

    1. cumsum(itr)

    Cumulative sum an iterator. See also 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

    1. julia> cumsum([1, 1, 1])
    2. 3-element Vector{Int64}:
    3.  1
    4.  2
    5.  3
    7. julia> cumsum([fill(1, 2) for i in 1:3])
    8. 3-element Vector{Vector{Int64}}:
    9.  [1, 1]
    10.  [2, 2]
    11.  [3, 3]
    13. julia> cumsum((1, 1, 1))
    14. (1, 2, 3)
    16. julia> cumsum(x^2 for x in 1:3)
    17. 3-element Vector{Int64}:
    18.   1
    19.   5
    20.  14

    source

    — Function

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

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

    Base.diff — Function

    1. diff(A::AbstractVector)
    2. 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

    1. julia> a = [2 4; 6 16]
    2. 2×2 Matrix{Int64}:
    3.  2   4
    4.  6  16
    6. julia> diff(a, dims=2)
    7. 2×1 Matrix{Int64}:
    8.   2
    9.  10
    11. julia> diff(vec(a))
    12. 3-element Vector{Int64}:
    13.   4
    14.  -2
    15.  12

    Base.repeat — Function

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

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

    Examples

    1. julia> repeat([1, 2, 3], 2)
    2. 6-element Vector{Int64}:
    3.  1
    4.  2
    5.  3
    6.  1
    7.  2
    8.  3
    10. julia> repeat([1, 2, 3], 2, 3)
    11. 6×3 Matrix{Int64}:
    12.  1  1  1
    13.  2  2  2
    14.  3  3  3
    15.  1  1  1
    16.  2  2  2
    17.  3  3  3

    1. repeat(A::AbstractArray; inner=ntuple(x->1, ndims(A)), outer=ntuple(x->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

    1. julia> repeat(1:2, inner=2)
    2. 4-element Vector{Int64}:
    3.  1
    4.  1
    5.  2
    6.  2
    8. julia> repeat(1:2, outer=2)
    9. 4-element Vector{Int64}:
    10.  1
    11.  2
    12.  1
    13.  2
    15. julia> repeat([1 2; 3 4], inner=(2, 1), outer=(1, 3))
    16. 4×6 Matrix{Int64}:
    17.  1  2  1  2  1  2
    18.  1  2  1  2  1  2
    19.  3  4  3  4  3  4
    20.  3  4  3  4  3  4

    source

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

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

    See also:

    Examples

    1. julia> repeat("ha", 3)
    2. "hahaha"

    source

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

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

    Examples

    1. julia> repeat('A', 3)
    2. "AAA"

    source

    — Function

    1. rot180(A)

    Rotate matrix A 180 degrees.

    Examples

    1. julia> a = [1 2; 3 4]
    2. 2×2 Matrix{Int64}:
    3.  1  2
    4.  3  4
    6. julia> rot180(a)
    7. 2×2 Matrix{Int64}:
    8.  4  3
    9.  2  1

    source

    1. rot180(A, k)

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

    Examples

    1. julia> a = [1 2; 3 4]
    2. 2×2 Matrix{Int64}:
    3.  1  2
    4.  3  4
    6. julia> rot180(a,1)
    7. 2×2 Matrix{Int64}:
    8.  4  3
    9.  2  1
    11. julia> rot180(a,2)
    12. 2×2 Matrix{Int64}:
    13.  1  2
    14.  3  4

    Base.rotl90 — Function

    1. rotl90(A)

    Rotate matrix A left 90 degrees.

    Examples

    1. julia> a = [1 2; 3 4]
    2. 2×2 Matrix{Int64}:
    3.  1  2
    4.  3  4
    6. julia> rotl90(a)
    7. 2×2 Matrix{Int64}:
    8.  2  4
    9.  1  3

    1. 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

    1. julia> a = [1 2; 3 4]
    2. 2×2 Matrix{Int64}:
    3.  1  2
    4.  3  4
    6. julia> rotl90(a,1)
    7. 2×2 Matrix{Int64}:
    8.  2  4
    9.  1  3
    11. julia> rotl90(a,2)
    12. 2×2 Matrix{Int64}:
    13.  4  3
    14.  2  1
    16. julia> rotl90(a,3)
    17. 2×2 Matrix{Int64}:
    18.  3  1
    19.  4  2
    21. julia> rotl90(a,4)
    22. 2×2 Matrix{Int64}:
    23.  1  2
    24.  3  4

    source

    — Function

    1. rotr90(A)

    Rotate matrix A right 90 degrees.

    Examples

    1. julia> a = [1 2; 3 4]
    2. 2×2 Matrix{Int64}:
    3.  1  2
    4.  3  4
    6. julia> rotr90(a)
    7. 2×2 Matrix{Int64}:
    8.  3  1
    9.  4  2

    source

    1. 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

    1. julia> a = [1 2; 3 4]
    2. 2×2 Matrix{Int64}:
    3.  1  2
    4.  3  4
    6. julia> rotr90(a,1)
    7. 2×2 Matrix{Int64}:
    8.  3  1
    9.  4  2
    11. julia> rotr90(a,2)
    12. 2×2 Matrix{Int64}:
    13.  4  3
    14.  2  1
    16. julia> rotr90(a,3)
    17. 2×2 Matrix{Int64}:
    18.  2  4
    19.  1  3
    21. julia> rotr90(a,4)
    22. 2×2 Matrix{Int64}:
    23.  1  2
    24.  3  4

    Base.mapslices — Function

    1. 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

    1. julia> a = reshape(Vector(1:16),(2,2,2,2))
    2. 2×2×2×2 Array{Int64, 4}:
    3. [:, :, 1, 1] =
    4.  1  3
    5.  2  4
    7. [:, :, 2, 1] =
    8.  5  7
    9.  6  8
    11. [:, :, 1, 2] =
    12.   9  11
    13.  10  12
    15. [:, :, 2, 2] =
    16.  13  15
    17.  14  16
    19. julia> mapslices(sum, a, dims = [1,2])
    20. 1×1×2×2 Array{Int64, 4}:
    21. [:, :, 1, 1] =
    22.  10
    24. [:, :, 2, 1] =
    25.  26
    27. [:, :, 1, 2] =
    28.  42
    30. [:, :, 2, 2] =
    31.  58

    Base.eachrow — Function

    1. eachrow(A::AbstractVecOrMat)

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

    See also and eachslice.

    Julia 1.1

    This function requires at least Julia 1.1.

    Example

    1. julia> a = [1 2; 3 4]
    2. 2×2 Matrix{Int64}:
    3.  1  2
    4.  3  4
    6. julia> first(eachrow(a))
    7. 2-element view(::Matrix{Int64}, 1, :) with eltype Int64:
    8.  1
    9.  2
    11. julia> collect(eachrow(a))
    12. 2-element Vector{SubArray{Int64, 1, Matrix{Int64}, Tuple{Int64, Base.Slice{Base.OneTo{Int64}}}, true}}:
    13.  [1, 2]
    14.  [3, 4]

    Base.eachcol — Function

    1. eachcol(A::AbstractVecOrMat)

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

    See also and eachslice.

    Julia 1.1

    This function requires at least Julia 1.1.

    Example

    1. julia> a = [1 2; 3 4]
    2. 2×2 Matrix{Int64}:
    3.  1  2
    4.  3  4
    6. julia> first(eachcol(a))
    7. 2-element view(::Matrix{Int64}, :, 1) with eltype Int64:
    8.  1
    9.  3
    11. julia> collect(eachcol(a))
    12. 2-element Vector{SubArray{Int64, 1, Matrix{Int64}, Tuple{Base.Slice{Base.OneTo{Int64}}, Int64}, true}}:
    13.  [1, 3]
    14.  [2, 4]

    Base.eachslice — Function

    1. eachslice(A::AbstractArray; dims)

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

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

    See also , eachcol, and .

    Julia 1.1

    This function requires at least Julia 1.1.

    Example

    1. julia> M = [1 2 3; 4 5 6; 7 8 9]
    2. 3×3 Matrix{Int64}:
    3.  1  2  3
    4.  4  5  6
    5.  7  8  9
    7. julia> first(eachslice(M, dims=1))
    8. 3-element view(::Matrix{Int64}, 1, :) with eltype Int64:
    9.  1
    10.  2
    11.  3
    13. julia> collect(eachslice(M, dims=2))
    14. 3-element Vector{SubArray{Int64, 1, Matrix{Int64}, Tuple{Base.Slice{Base.OneTo{Int64}}, Int64}, true}}:
    15.  [1, 4, 7]
    16.  [2, 5, 8]
    17.  [3, 6, 9]

    source

    Combinatorics

    — Function

    1. invperm(v)

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

    Examples

    1. julia> v = [2; 4; 3; 1];
    3. julia> invperm(v)
    4. 4-element Vector{Int64}:
    5.  4
    6.  1
    7.  3
    8.  2
    10. julia> A = ['a','b','c','d'];
    12. julia> B = A[v]
    13. 4-element Vector{Char}:
    14.  'b': ASCII/Unicode U+0062 (category Ll: Letter, lowercase)
    15.  'd': ASCII/Unicode U+0064 (category Ll: Letter, lowercase)
    16.  'c': ASCII/Unicode U+0063 (category Ll: Letter, lowercase)
    17.  'a': ASCII/Unicode U+0061 (category Ll: Letter, lowercase)
    19. julia> B[invperm(v)]
    20. 4-element Vector{Char}:
    21.  'a': ASCII/Unicode U+0061 (category Ll: Letter, lowercase)
    22.  'b': ASCII/Unicode U+0062 (category Ll: Letter, lowercase)
    23.  'c': ASCII/Unicode U+0063 (category Ll: Letter, lowercase)
    24.  'd': ASCII/Unicode U+0064 (category Ll: Letter, lowercase)

    source

    — Function

    1. isperm(v) -> Bool

    Return true if v is a valid permutation.

    Examples

    1. julia> isperm([1; 2])
    2. true
    4. julia> isperm([1; 3])
    5. false

    source

    — Method

    1. permute!(v, p)

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

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

    See also invpermute!.

    Examples

    1. julia> A = [1, 1, 3, 4];
    3. julia> perm = [2, 4, 3, 1];
    5. julia> permute!(A, perm);
    7. julia> A
    8. 4-element Vector{Int64}:
    9.  1
    10.  4
    11.  3
    12.  1

    Base.invpermute! — Function

    1. invpermute!(v, p)

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

    Examples

    1. julia> A = [1, 1, 3, 4];
    3. julia> perm = [2, 4, 3, 1];
    5. julia> invpermute!(A, perm);
    7. julia> A
    8. 4-element Vector{Int64}:
    9.  4
    10.  1
    11.  3
    12.  1

    source

    — Method

    1. 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

    1. julia> b = Int64[1 2; 3 4]
    2. 2×2 Matrix{Int64}:
    3.  1  2
    4.  3  4
    6. julia> reverse(b, dims=2)
    7. 2×2 Matrix{Int64}:
    8.  2  1
    9.  4  3
    11. julia> reverse(b)
    12. 2×2 Matrix{Int64}:
    13.  4  3
    14.  2  1

    Julia 1.6

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

    Base.reverseind — Function

    1. reverseind(v, i)

    Given an index i in , 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

    1. julia> r = reverse("Julia")
    2. "ailuJ"
    4. julia> for i in 1:length(r)
    5.            print(r[reverseind("Julia", i)])
    6.        end
    7. Julia

    source

    — Function

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

    In-place version of reverse.

    Examples

    1. reverse!(A; dims=:)

    Like reverse, but operates in-place in A.

    Julia 1.6

    Multidimensional reverse! requires Julia 1.6.