Arrays
Constructors and Types
Core.AbstractArray
— TypeAbstractArray{T,N}
Supertype for N
-dimensional arrays (or array-like types) with elements of type T
. Array
and other types are subtypes of this. See the manual section on the AbstractArray
interface.
See also: AbstractVector
, AbstractMatrix
, eltype
, ndims
.
Base.AbstractVector
— TypeAbstractVector{T}
Supertype for one-dimensional arrays (or array-like types) with elements of type T
. Alias for AbstractArray{T,1}
.
Base.AbstractMatrix
— TypeAbstractMatrix{T}
Supertype for two-dimensional arrays (or array-like types) with elements of type T
. Alias for AbstractArray{T,2}
.
Base.AbstractVecOrMat
— TypeAbstractVecOrMat{T}
Union type of AbstractVector{T}
and AbstractMatrix{T}
.
Core.Array
— TypeArray{T,N} <: AbstractArray{T,N}
N
-dimensional dense array with elements of type T
.
Core.Array
— MethodArray{T}(undef, dims)
Array{T,N}(undef, dims)
Construct an uninitialized N
-dimensional Array
containing elements of type T
. N
can either be supplied explicitly, as in Array{T,N}(undef, dims)
, or be determined by the length or number of dims
. dims
may be a tuple or a series of integer arguments corresponding to the lengths in each dimension. If the rank N
is supplied explicitly, then it must match the length or number of dims
. Here undef
is the UndefInitializer
.
Examples
julia> A = Array{Float64, 2}(undef, 2, 3) # N given explicitly
2×3 Matrix{Float64}:
6.90198e-310 6.90198e-310 6.90198e-310
6.90198e-310 6.90198e-310 0.0
julia> B = Array{Float64}(undef, 4) # N determined by the input
4-element Vector{Float64}:
2.360075077e-314
NaN
2.2671131793e-314
2.299821756e-314
julia> similar(B, 2, 4, 1) # use typeof(B), and the given size
2×4×1 Array{Float64, 3}:
[:, :, 1] =
2.26703e-314 2.26708e-314 0.0 2.80997e-314
0.0 2.26703e-314 2.26708e-314 0.0
Core.Array
— MethodArray{T}(nothing, dims)
Array{T,N}(nothing, dims)
Construct an N
-dimensional Array
containing elements of type T
, initialized with nothing
entries. Element type T
must be able to hold these values, i.e. Nothing <: T
.
Examples
julia> Array{Union{Nothing, String}}(nothing, 2)
2-element Vector{Union{Nothing, String}}:
nothing
nothing
julia> Array{Union{Nothing, Int}}(nothing, 2, 3)
2×3 Matrix{Union{Nothing, Int64}}:
nothing nothing nothing
nothing nothing nothing
Core.Array
— MethodArray{T}(missing, dims)
Array{T,N}(missing, dims)
Construct an N
-dimensional Array
containing elements of type T
, initialized with missing
entries. Element type T
must be able to hold these values, i.e. Missing <: T
.
Examples
julia> Array{Union{Missing, String}}(missing, 2)
2-element Vector{Union{Missing, String}}:
missing
missing
julia> Array{Union{Missing, Int}}(missing, 2, 3)
2×3 Matrix{Union{Missing, Int64}}:
missing missing missing
missing missing missing
Core.UndefInitializer
— TypeUndefInitializer
Singleton type used in array initialization, indicating the array-constructor-caller would like an uninitialized array. See also undef
, an alias for UndefInitializer()
.
Examples
julia> Array{Float64, 1}(UndefInitializer(), 3)
3-element Array{Float64, 1}:
2.2752528595e-314
2.202942107e-314
2.275252907e-314
Core.undef
— Constantundef
Alias for UndefInitializer()
, which constructs an instance of the singleton type UndefInitializer
, used in array initialization to indicate the array-constructor-caller would like an uninitialized array.
Examples
julia> Array{Float64, 1}(undef, 3)
3-element Vector{Float64}:
2.2752528595e-314
2.202942107e-314
2.275252907e-314
Base.Vector
— TypeVector{T} <: AbstractVector{T}
One-dimensional dense array with elements of type T
, often used to represent a mathematical vector. Alias for Array{T,1}
.
Base.Vector
— MethodVector{T}(undef, n)
Construct an uninitialized Vector{T}
of length n
.
Examples
julia> Vector{Float64}(undef, 3)
3-element Array{Float64, 1}:
6.90966e-310
6.90966e-310
6.90966e-310
Base.Vector
— MethodVector{T}(nothing, m)
Construct a Vector{T}
of length m
, initialized with nothing
entries. Element type T
must be able to hold these values, i.e. Nothing <: T
.
Examples
julia> Vector{Union{Nothing, String}}(nothing, 2)
2-element Vector{Union{Nothing, String}}:
nothing
nothing
Base.Vector
— MethodVector{T}(missing, m)
Construct a Vector{T}
of length m
, initialized with missing
entries. Element type T
must be able to hold these values, i.e. Missing <: T
.
Examples
julia> Vector{Union{Missing, String}}(missing, 2)
2-element Vector{Union{Missing, String}}:
missing
missing
Base.Matrix
— TypeMatrix{T} <: AbstractMatrix{T}
Two-dimensional dense array with elements of type T
, often used to represent a mathematical matrix. Alias for Array{T,2}
.
See also fill
, zeros
, undef
and similar
for creating matrices.
Base.Matrix
— MethodMatrix{T}(undef, m, n)
Construct an uninitialized Matrix{T}
of size m
×n
.
Examples
julia> Matrix{Float64}(undef, 2, 3)
2×3 Array{Float64, 2}:
2.36365e-314 2.28473e-314 5.0e-324
2.26704e-314 2.26711e-314 NaN
julia> similar(ans, Int32, 2, 2)
2×2 Matrix{Int32}:
490537216 1277177453
1 1936748399
Base.Matrix
— MethodMatrix{T}(nothing, m, n)
Construct a Matrix{T}
of size m
×n
, initialized with nothing
entries. Element type T
must be able to hold these values, i.e. Nothing <: T
.
Examples
julia> Matrix{Union{Nothing, String}}(nothing, 2, 3)
2×3 Matrix{Union{Nothing, String}}:
nothing nothing nothing
nothing nothing nothing
Base.Matrix
— MethodMatrix{T}(missing, m, n)
Construct a Matrix{T}
of size m
×n
, initialized with missing
entries. Element type T
must be able to hold these values, i.e. Missing <: T
.
Examples
julia> Matrix{Union{Missing, String}}(missing, 2, 3)
2×3 Matrix{Union{Missing, String}}:
missing missing missing
missing missing missing
Base.VecOrMat
— TypeVecOrMat{T}
Union type of Vector{T}
and Matrix{T}
which allows functions to accept either a Matrix or a Vector.
Examples
julia> Vector{Float64} <: VecOrMat{Float64}
true
julia> Matrix{Float64} <: VecOrMat{Float64}
true
julia> Array{Float64, 3} <: VecOrMat{Float64}
false
Core.DenseArray
— TypeDenseArray{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
— TypeDenseVector{T}
One-dimensional DenseArray
with elements of type T
. Alias for DenseArray{T,1}
.
Base.DenseMatrix
— TypeDenseMatrix{T}
Two-dimensional DenseArray
with elements of type T
. Alias for DenseArray{T,2}
.
Base.DenseVecOrMat
— TypeDenseVecOrMat{T}
Union type of DenseVector{T}
and DenseMatrix{T}
.
Base.StridedArray
— TypeStridedArray{T, N}
A hard-coded Union
of common array types that follow the strided array interface, with elements of type T
and N
dimensions.
If A
is a StridedArray
, then its elements are stored in memory with offsets, which may vary between dimensions but are constant within a dimension. For example, A
could have stride 2 in dimension 1, and stride 3 in dimension 2. Incrementing A
along dimension d
jumps in memory by [strides(A, d)
] slots. Strided arrays are particularly important and useful because they can sometimes be passed directly as pointers to foreign language libraries like BLAS.
Base.StridedVector
— TypeStridedVector{T}
One dimensional StridedArray
with elements of type T
.
Base.StridedMatrix
— TypeStridedMatrix{T}
Two dimensional StridedArray
with elements of type T
.
Base.StridedVecOrMat
— TypeStridedVecOrMat{T}
Union type of StridedVector
and StridedMatrix
with elements of type T
.
Base.getindex
— Methodgetindex(type[, elements...])
Construct a 1-d array of the specified type. This is usually called with the syntax Type[]
. Element values can be specified using Type[a,b,c,...]
.
Examples
julia> Int8[1, 2, 3]
3-element Vector{Int8}:
1
2
3
julia> getindex(Int8, 1, 2, 3)
3-element Vector{Int8}:
1
2
3
Base.zeros
— Functionzeros([T=Float64,] dims::Tuple)
zeros([T=Float64,] dims...)
Create an Array
, with element type T
, of all zeros with size specified by dims
. See also fill
, ones
, zero
.
Examples
julia> zeros(1)
1-element Vector{Float64}:
0.0
julia> zeros(Int8, 2, 3)
2×3 Matrix{Int8}:
0 0 0
0 0 0
Base.ones
— Functionones([T=Float64,] dims::Tuple)
ones([T=Float64,] dims...)
Create an Array
, with element type T
, of all ones with size specified by dims
. See also fill
, zeros
.
Examples
julia> ones(1,2)
1×2 Matrix{Float64}:
1.0 1.0
julia> ones(ComplexF64, 2, 3)
2×3 Matrix{ComplexF64}:
1.0+0.0im 1.0+0.0im 1.0+0.0im
1.0+0.0im 1.0+0.0im 1.0+0.0im
Base.BitArray
— TypeBitArray{N} <: AbstractArray{Bool, N}
Space-efficient N
-dimensional boolean array, using just one bit for each boolean value.
BitArray
s pack up to 64 values into every 8 bytes, resulting in an 8x space efficiency over Array{Bool, N}
and allowing some operations to work on 64 values at once.
By default, Julia returns BitArrays
from broadcasting operations that generate boolean elements (including dotted-comparisons like .==
) as well as from the functions trues
and falses
.
Due to its packed storage format, concurrent access to the elements of a BitArray
where at least one of them is a write is not thread safe.
Base.BitArray
— MethodBitArray(undef, dims::Integer...)
BitArray{N}(undef, dims::NTuple{N,Int})
Construct an undef BitArray
with the given dimensions. Behaves identically to the Array
constructor. See undef
.
Examples
julia> BitArray(undef, 2, 2)
2×2 BitMatrix:
0 0
0 0
julia> BitArray(undef, (3, 1))
3×1 BitMatrix:
0
0
0
Base.BitArray
— MethodBitArray(itr)
Construct a BitArray
generated by the given iterable object. The shape is inferred from the itr
object.
Examples
julia> BitArray([1 0; 0 1])
2×2 BitMatrix:
1 0
0 1
julia> BitArray(x+y == 3 for x = 1:2, y = 1:3)
2×3 BitMatrix:
0 1 0
1 0 0
julia> BitArray(x+y == 3 for x = 1:2 for y = 1:3)
6-element BitVector:
0
1
0
1
0
0
Base.trues
— Functiontrues(dims)
Create a BitArray
with all values set to true
.
Examples
julia> trues(2,3)
2×3 BitMatrix:
1 1 1
1 1 1
Base.falses
— Functionfalses(dims)
Create a BitArray
with all values set to false
.
Examples
julia> falses(2,3)
2×3 BitMatrix:
0 0 0
0 0 0
Base.fill
— Functionfill(x, dims::Tuple)
fill(x, dims...)
Create an array filled with the value x
. For example, fill(1.0, (5,5))
returns a 5×5 array of floats, with each element initialized to 1.0
.
dims
may be specified as either a tuple or a sequence of arguments. For example, the common idiom fill(x)
creates a zero-dimensional array containing the single value x
.
See also: fill!
, zeros
, ones
, similar
.
Examples
julia> fill(1.0, (2,3))
2×3 Matrix{Float64}:
1.0 1.0 1.0
1.0 1.0 1.0
julia> fill(42)
0-dimensional Array{Int64, 0}:
42
If x
is an object reference, all elements will refer to the same object:
julia> A = fill(zeros(2), 2);
julia> A[1][1] = 42; # modifies both A[1][1] and A[2][1]
julia> A
2-element Vector{Vector{Float64}}:
[42.0, 0.0]
[42.0, 0.0]
Base.fill!
— Functionfill!(A, x)
Fill array A
with the value x
. If x
is an object reference, all elements will refer to the same object. fill!(A, Foo())
will return A
filled with the result of evaluating Foo()
once.
Examples
julia> A = zeros(2,3)
2×3 Matrix{Float64}:
0.0 0.0 0.0
0.0 0.0 0.0
julia> fill!(A, 2.)
2×3 Matrix{Float64}:
2.0 2.0 2.0
2.0 2.0 2.0
julia> a = [1, 1, 1]; A = fill!(Vector{Vector{Int}}(undef, 3), a); a[1] = 2; A
3-element Vector{Vector{Int64}}:
[2, 1, 1]
[2, 1, 1]
[2, 1, 1]
julia> x = 0; f() = (global x += 1; x); fill!(Vector{Int}(undef, 3), f())
3-element Vector{Int64}:
1
1
1
Base.empty
— Functionempty(x::Tuple)
Returns an empty tuple, ()
.
empty(v::AbstractVector, [eltype])
Create an empty vector similar to v
, optionally changing the eltype
.
See also: empty!
, isempty
, isassigned
.
Examples
julia> empty([1.0, 2.0, 3.0])
Float64[]
julia> empty([1.0, 2.0, 3.0], String)
String[]
empty(a::AbstractDict, [index_type=keytype(a)], [value_type=valtype(a)])
Create an empty AbstractDict
container which can accept indices of type index_type
and values of type value_type
. The second and third arguments are optional and default to the input's keytype
and valtype
, respectively. (If only one of the two types is specified, it is assumed to be the value_type
, and the index_type
we default to keytype(a)
).
Custom AbstractDict
subtypes may choose which specific dictionary type is best suited to return for the given index and value types, by specializing on the three-argument signature. The default is to return an empty Dict
.
Base.similar
— Functionsimilar(array, [element_type=eltype(array)], [dims=size(array)])
Create an uninitialized mutable array with the given element type and size, based upon the given source array. The second and third arguments are both optional, defaulting to the given array's eltype
and size
. The dimensions may be specified either as a single tuple argument or as a series of integer arguments.
Custom AbstractArray subtypes may choose which specific array type is best-suited to return for the given element type and dimensionality. If they do not specialize this method, the default is an Array{element_type}(undef, dims...)
.
For example, similar(1:10, 1, 4)
returns an uninitialized Array{Int,2}
since ranges are neither mutable nor support 2 dimensions:
julia> similar(1:10, 1, 4)
1×4 Array{Int64,2}:
4419743872 4374413872 4419743888 0
Conversely, similar(trues(10,10), 2)
returns an uninitialized BitVector
with two elements since BitArray
s are both mutable and can support 1-dimensional arrays:
julia> similar(trues(10,10), 2)
2-element BitVector:
0
0
Since BitArray
s can only store elements of type Bool
, however, if you request a different element type it will create a regular Array
instead:
julia> similar(falses(10), Float64, 2, 4)
2×4 Array{Float64,2}:
2.18425e-314 2.18425e-314 2.18425e-314 2.18425e-314
2.18425e-314 2.18425e-314 2.18425e-314 2.18425e-314
See also: undef
, isassigned
.
similar(storagetype, axes)
Create an uninitialized mutable array analogous to that specified by storagetype
, but with axes
specified by the last argument.
Examples:
similar(Array{Int}, axes(A))
creates an array that "acts like" an Array{Int}
(and might indeed be backed by one), but which is indexed identically to A
. If A
has conventional indexing, this will be identical to Array{Int}(undef, size(A))
, but if A
has unconventional indexing then the indices of the result will match A
.
similar(BitArray, (axes(A, 2),))
would create a 1-dimensional logical array whose indices match those of the columns of A
.
Basic functions
Base.ndims
— Functionndims(A::AbstractArray) -> Integer
Return the number of dimensions of A
.
Examples
julia> A = fill(1, (3,4,5));
julia> ndims(A)
3
Base.size
— Functionsize(A::AbstractArray, [dim])
Return a tuple containing the dimensions of A
. Optionally you can specify a dimension to just get the length of that dimension.
Note that size
may not be defined for arrays with non-standard indices, in which case axes
may be useful. See the manual chapter on arrays with custom indices.
See also: length
, ndims
, eachindex
, sizeof
.
Examples
julia> A = fill(1, (2,3,4));
julia> size(A)
(2, 3, 4)
julia> size(A, 2)
3
Base.axes
— Methodaxes(A)
Return the tuple of valid indices for array A
.
See also: size
, keys
, eachindex
.
Examples
julia> A = fill(1, (5,6,7));
julia> axes(A)
(Base.OneTo(5), Base.OneTo(6), Base.OneTo(7))
Base.axes
— Methodaxes(A, d)
Return the valid range of indices for array A
along dimension d
.
See also size
, and the manual chapter on arrays with custom indices.
Examples
julia> A = fill(1, (5,6,7));
julia> axes(A, 2)
Base.OneTo(6)
Usage note
Each of the indices has to be an AbstractUnitRange{<:Integer}
, but at the same time can be a type that uses custom indices. So, for example, if you need a subset, use generalized indexing constructs like begin
/end
or firstindex
/lastindex
:
ix = axes(v, 1)
ix[2:end] # will work for eg Vector, but may fail in general
ix[(begin+1):end] # works for generalized indexes
Base.length
— Methodlength(A::AbstractArray)
Return the number of elements in the array, defaults to prod(size(A))
.
Examples
julia> length([1, 2, 3, 4])
4
julia> length([1 2; 3 4])
4
Base.keys
— Methodkeys(a::AbstractArray)
Return an efficient array describing all valid indices for a
arranged in the shape of a
itself.
They keys of 1-dimensional arrays (vectors) are integers, whereas all other N-dimensional arrays use CartesianIndex
to describe their locations. Often the special array types LinearIndices
and CartesianIndices
are used to efficiently represent these arrays of integers and CartesianIndex
es, respectively.
Note that the keys
of an array might not be the most efficient index type; for maximum performance use eachindex
instead.
Base.eachindex
— Functioneachindex(A...)
Create an iterable object for visiting each index of an AbstractArray
A
in an efficient manner. For array types that have opted into fast linear indexing (like Array
), this is simply the range 1:length(A)
. For other array types, return a specialized Cartesian range to efficiently index into the array with indices specified for every dimension. For other iterables, including strings and dictionaries, return an iterator object supporting arbitrary index types (e.g. unevenly spaced or non-integer indices).
If you supply more than one AbstractArray
argument, eachindex
will create an iterable object that is fast for all arguments (a UnitRange
if all inputs have fast linear indexing, a CartesianIndices
otherwise). If the arrays have different sizes and/or dimensionalities, a DimensionMismatch exception will be thrown.
Examples
julia> A = [1 2; 3 4];
julia> for i in eachindex(A) # linear indexing
println(i)
end
1
2
3
4
julia> for i in eachindex(view(A, 1:2, 1:1)) # Cartesian indexing
println(i)
end
CartesianIndex(1, 1)
CartesianIndex(2, 1)
Base.IndexStyle
— TypeIndexStyle(A)
IndexStyle(typeof(A))
IndexStyle
specifies the "native indexing style" for array A
. When you define a new AbstractArray
type, you can choose to implement either linear indexing (with IndexLinear
) or cartesian indexing. If you decide to only implement linear indexing, then you must set this trait for your array type:
Base.IndexStyle(::Type{<:MyArray}) = IndexLinear()
The default is IndexCartesian()
.
Julia's internal indexing machinery will automatically (and invisibly) recompute all indexing operations into the preferred style. This allows users to access elements of your array using any indexing style, even when explicit methods have not been provided.
If you define both styles of indexing for your AbstractArray
, this trait can be used to select the most performant indexing style. Some methods check this trait on their inputs, and dispatch to different algorithms depending on the most efficient access pattern. In particular, eachindex
creates an iterator whose type depends on the setting of this trait.
Base.IndexLinear
— TypeIndexLinear()
Subtype of IndexStyle
used to describe arrays which are optimally indexed by one linear index.
A linear indexing style uses one integer index to describe the position in the array (even if it's a multidimensional array) and column-major ordering is used to efficiently access the elements. This means that requesting eachindex
from an array that is IndexLinear
will return a simple one-dimensional range, even if it is multidimensional.
A custom array that reports its IndexStyle
as IndexLinear
only needs to implement indexing (and indexed assignment) with a single Int
index; all other indexing expressions — including multidimensional accesses — will be recomputed to the linear index. For example, if A
were a 2×3
custom matrix with linear indexing, and we referenced A[1, 3]
, this would be recomputed to the equivalent linear index and call A[5]
since 2*1 + 3 = 5
.
See also IndexCartesian
.
Base.IndexCartesian
— TypeIndexCartesian()
Subtype of IndexStyle
used to describe arrays which are optimally indexed by a Cartesian index. This is the default for new custom AbstractArray
subtypes.
A Cartesian indexing style uses multiple integer indices to describe the position in a multidimensional array, with exactly one index per dimension. This means that requesting eachindex
from an array that is IndexCartesian
will return a range of CartesianIndices
.
A N
-dimensional custom array that reports its IndexStyle
as IndexCartesian
needs to implement indexing (and indexed assignment) with exactly N
Int
indices; all other indexing expressions — including linear indexing — will be recomputed to the equivalent Cartesian location. For example, if A
were a 2×3
custom matrix with cartesian indexing, and we referenced A[5]
, this would be recomputed to the equivalent Cartesian index and call A[1, 3]
since 5 = 2*1 + 3
.
It is significantly more expensive to compute Cartesian indices from a linear index than it is to go the other way. The former operation requires division — a very costly operation — whereas the latter only uses multiplication and addition and is essentially free. This asymmetry means it is far more costly to use linear indexing with an IndexCartesian
array than it is to use Cartesian indexing with an IndexLinear
array.
See also IndexLinear
.
Base.conj!
— Functionconj!(A)
Transform an array to its complex conjugate in-place.
See also conj
.
Examples
julia> A = [1+im 2-im; 2+2im 3+im]
2×2 Matrix{Complex{Int64}}:
1+1im 2-1im
2+2im 3+1im
julia> conj!(A);
julia> A
2×2 Matrix{Complex{Int64}}:
1-1im 2+1im
2-2im 3-1im
Base.stride
— Functionstride(A, k::Integer)
Return the distance in memory (in number of elements) between adjacent elements in dimension k
.
See also: strides
.
Examples
julia> A = fill(1, (3,4,5));
julia> stride(A,2)
3
julia> stride(A,3)
12
Base.strides
— Functionstrides(A)
Return a tuple of the memory strides in each dimension.
See also: stride
.
Examples
julia> A = fill(1, (3,4,5));
julia> strides(A)
(1, 3, 12)
Broadcast and vectorization
See also the dot syntax for vectorizing functions; for example, f.(args...)
implicitly calls broadcast(f, args...)
. Rather than relying on "vectorized" methods of functions like sin
to operate on arrays, you should use sin.(a)
to vectorize via broadcast
.
Base.Broadcast.broadcast
— Functionbroadcast(f, As...)
Broadcast the function f
over the arrays, tuples, collections, Ref
s and/or scalars As
.
Broadcasting applies the function f
over the elements of the container arguments and the scalars themselves in As
. Singleton and missing dimensions are expanded to match the extents of the other arguments by virtually repeating the value. By default, only a limited number of types are considered scalars, including Number
s, String
s, Symbol
s, Type
s, Function
s and some common singletons like missing
and nothing
. All other arguments are iterated over or indexed into elementwise.
The resulting container type is established by the following rules:
- If all the arguments are scalars or zero-dimensional arrays, it returns an unwrapped scalar.
- If at least one argument is a tuple and all others are scalars or zero-dimensional arrays, it returns a tuple.
- All other combinations of arguments default to returning an
Array
, but custom container types can define their own implementation and promotion-like rules to customize the result when they appear as arguments.
A special syntax exists for broadcasting: f.(args...)
is equivalent to broadcast(f, args...)
, and nested f.(g.(args...))
calls are fused into a single broadcast loop.
Examples
julia> A = [1, 2, 3, 4, 5]
5-element Vector{Int64}:
1
2
3
4
5
julia> B = [1 2; 3 4; 5 6; 7 8; 9 10]
5×2 Matrix{Int64}:
1 2
3 4
5 6
7 8
9 10
julia> broadcast(+, A, B)
5×2 Matrix{Int64}:
2 3
5 6
8 9
11 12
14 15
julia> parse.(Int, ["1", "2"])
2-element Vector{Int64}:
1
2
julia> abs.((1, -2))
(1, 2)
julia> broadcast(+, 1.0, (0, -2.0))
(1.0, -1.0)
julia> (+).([[0,2], [1,3]], Ref{Vector{Int}}([1,-1]))
2-element Vector{Vector{Int64}}:
[1, 1]
[2, 2]
julia> string.(("one","two","three","four"), ": ", 1:4)
4-element Vector{String}:
"one: 1"
"two: 2"
"three: 3"
"four: 4"
Base.Broadcast.broadcast!
— Functionbroadcast!(f, dest, As...)
Like broadcast
, but store the result of broadcast(f, As...)
in the dest
array. Note that dest
is only used to store the result, and does not supply arguments to f
unless it is also listed in the As
, as in broadcast!(f, A, A, B)
to perform A[:] = broadcast(f, A, B)
.
Examples
julia> A = [1.0; 0.0]; B = [0.0; 0.0];
julia> broadcast!(+, B, A, (0, -2.0));
julia> B
2-element Vector{Float64}:
1.0
-2.0
julia> A
2-element Vector{Float64}:
1.0
0.0
julia> broadcast!(+, A, A, (0, -2.0));
julia> A
2-element Vector{Float64}:
1.0
-2.0
Base.Broadcast.@__dot__
— Macro@. expr
Convert every function call or operator in expr
into a "dot call" (e.g. convert f(x)
to f.(x)
), and convert every assignment in expr
to a "dot assignment" (e.g. convert +=
to .+=
).
If you want to avoid adding dots for selected function calls in expr
, splice those function calls in with $
. For example, @. sqrt(abs($sort(x)))
is equivalent to sqrt.(abs.(sort(x)))
(no dot for sort
).
(@.
is equivalent to a call to @__dot__
.)
Examples
julia> x = 1.0:3.0; y = similar(x);
julia> @. y = x + 3 * sin(x)
3-element Vector{Float64}:
3.5244129544236893
4.727892280477045
3.4233600241796016
For specializing broadcast on custom types, see
Base.Broadcast.BroadcastStyle
— TypeBroadcastStyle
is an abstract type and trait-function used to determine behavior of objects under broadcasting. BroadcastStyle(typeof(x))
returns the style associated with x
. To customize the broadcasting behavior of a type, one can declare a style by defining a type/method pair
struct MyContainerStyle <: BroadcastStyle end
Base.BroadcastStyle(::Type{<:MyContainer}) = MyContainerStyle()
One then writes method(s) (at least similar
) operating on Broadcasted{MyContainerStyle}
. There are also several pre-defined subtypes of BroadcastStyle
that you may be able to leverage; see the Interfaces chapter for more information.
Base.Broadcast.AbstractArrayStyle
— TypeBroadcast.AbstractArrayStyle{N} <: BroadcastStyle
is the abstract supertype for any style associated with an AbstractArray
type. The N
parameter is the dimensionality, which can be handy for AbstractArray types that only support specific dimensionalities:
struct SparseMatrixStyle <: Broadcast.AbstractArrayStyle{2} end
Base.BroadcastStyle(::Type{<:SparseMatrixCSC}) = SparseMatrixStyle()
For AbstractArray
types that support arbitrary dimensionality, N
can be set to Any
:
struct MyArrayStyle <: Broadcast.AbstractArrayStyle{Any} end
Base.BroadcastStyle(::Type{<:MyArray}) = MyArrayStyle()
In cases where you want to be able to mix multiple AbstractArrayStyle
s and keep track of dimensionality, your style needs to support a Val
constructor:
struct MyArrayStyleDim{N} <: Broadcast.AbstractArrayStyle{N} end
(::Type{<:MyArrayStyleDim})(::Val{N}) where N = MyArrayStyleDim{N}()
Note that if two or more AbstractArrayStyle
subtypes conflict, broadcasting machinery will fall back to producing Array
s. If this is undesirable, you may need to define binary BroadcastStyle
rules to control the output type.
See also Broadcast.DefaultArrayStyle
.
Base.Broadcast.ArrayStyle
— TypeBroadcast.ArrayStyle{MyArrayType}()
is a BroadcastStyle
indicating that an object behaves as an array for broadcasting. It presents a simple way to construct Broadcast.AbstractArrayStyle
s for specific AbstractArray
container types. Broadcast styles created this way lose track of dimensionality; if keeping track is important for your type, you should create your own custom Broadcast.AbstractArrayStyle
.
Base.Broadcast.DefaultArrayStyle
— TypeBroadcast.DefaultArrayStyle{N}()
is a BroadcastStyle
indicating that an object behaves as an N
-dimensional array for broadcasting. Specifically, DefaultArrayStyle
is used for any AbstractArray
type that hasn't defined a specialized style, and in the absence of overrides from other broadcast
arguments the resulting output type is Array
. When there are multiple inputs to broadcast
, DefaultArrayStyle
"loses" to any other Broadcast.ArrayStyle
.
Base.Broadcast.broadcastable
— FunctionBroadcast.broadcastable(x)
Return either x
or an object like x
such that it supports axes
, indexing, and its type supports ndims
.
If x
supports iteration, the returned value should have the same axes
and indexing behaviors as collect(x)
.
If x
is not an AbstractArray
but it supports axes
, indexing, and its type supports ndims
, then broadcastable(::typeof(x))
may be implemented to just return itself. Further, if x
defines its own BroadcastStyle
, then it must define its broadcastable
method to return itself for the custom style to have any effect.
Examples
julia> Broadcast.broadcastable([1,2,3]) # like `identity` since arrays already support axes and indexing
3-element Vector{Int64}:
1
2
3
julia> Broadcast.broadcastable(Int) # Types don't support axes, indexing, or iteration but are commonly used as scalars
Base.RefValue{Type{Int64}}(Int64)
julia> Broadcast.broadcastable("hello") # Strings break convention of matching iteration and act like a scalar instead
Base.RefValue{String}("hello")
Base.Broadcast.combine_axes
— Functioncombine_axes(As...) -> Tuple
Determine the result axes for broadcasting across all values in As
.
julia> Broadcast.combine_axes([1], [1 2; 3 4; 5 6])
(Base.OneTo(3), Base.OneTo(2))
julia> Broadcast.combine_axes(1, 1, 1)
()
Base.Broadcast.combine_styles
— Functioncombine_styles(cs...) -> BroadcastStyle
Decides which BroadcastStyle
to use for any number of value arguments. Uses BroadcastStyle
to get the style for each argument, and uses result_style
to combine styles.
Examples
julia> Broadcast.combine_styles([1], [1 2; 3 4])
Base.Broadcast.DefaultArrayStyle{2}()
Base.Broadcast.result_style
— Functionresult_style(s1::BroadcastStyle[, s2::BroadcastStyle]) -> BroadcastStyle
Takes one or two BroadcastStyle
s and combines them using BroadcastStyle
to determine a common BroadcastStyle
.
Examples
julia> Broadcast.result_style(Broadcast.DefaultArrayStyle{0}(), Broadcast.DefaultArrayStyle{3}())
Base.Broadcast.DefaultArrayStyle{3}()
julia> Broadcast.result_style(Broadcast.Unknown(), Broadcast.DefaultArrayStyle{1}())
Base.Broadcast.DefaultArrayStyle{1}()
Indexing and assignment
Base.getindex
— Methodgetindex(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
Base.setindex!
— Methodsetindex!(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
Base.copyto!
— Methodcopyto!(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.
Base.copy!
— Functioncopy!(dst, src) -> dst
In-place copy
of src
into dst
, discarding any pre-existing elements in dst
. If dst
and src
are of the same type, dst == src
should hold after the call. If dst
and src
are multidimensional arrays, they must have equal axes
.
See also copyto!
.
This method requires at least Julia 1.1. In Julia 1.0 this method is available from the Future
standard library as Future.copy!
.
Base.isassigned
— Functionisassigned(array, i) -> Bool
Test whether the given array has a value associated with index i
. Return false
if the index is out of bounds, or has an undefined reference.
Examples
julia> isassigned(rand(3, 3), 5)
true
julia> isassigned(rand(3, 3), 3 * 3 + 1)
false
julia> mutable struct Foo end
julia> v = similar(rand(3), Foo)
3-element Vector{Foo}:
#undef
#undef
#undef
julia> isassigned(v, 1)
false
Base.Colon
— TypeColon()
Colons (:) are used to signify indexing entire objects or dimensions at once.
Very few operations are defined on Colons directly; instead they are converted by to_indices
to an internal vector type (Base.Slice
) to represent the collection of indices they span before being used.
The singleton instance of Colon
is also a function used to construct ranges; see :
.
Base.IteratorsMD.CartesianIndex
— TypeCartesianIndex(i, j, k...) -> I
CartesianIndex((i, j, k...)) -> I
Create a multidimensional index I
, which can be used for indexing a multidimensional array A
. In particular, A[I]
is equivalent to A[i,j,k...]
. One can freely mix integer and CartesianIndex
indices; for example, A[Ipre, i, Ipost]
(where Ipre
and Ipost
are CartesianIndex
indices and i
is an Int
) can be a useful expression when writing algorithms that work along a single dimension of an array of arbitrary dimensionality.
A CartesianIndex
is sometimes produced by eachindex
, and always when iterating with an explicit CartesianIndices
.
Examples
julia> A = reshape(Vector(1:16), (2, 2, 2, 2))
2×2×2×2 Array{Int64, 4}:
[:, :, 1, 1] =
1 3
2 4
[:, :, 2, 1] =
5 7
6 8
[:, :, 1, 2] =
9 11
10 12
[:, :, 2, 2] =
13 15
14 16
julia> A[CartesianIndex((1, 1, 1, 1))]
1
julia> A[CartesianIndex((1, 1, 1, 2))]
9
julia> A[CartesianIndex((1, 1, 2, 1))]
5
Base.IteratorsMD.CartesianIndices
— TypeCartesianIndices(sz::Dims) -> R
CartesianIndices((istart:[istep:]istop, jstart:[jstep:]jstop, ...)) -> R
Define a region R
spanning a multidimensional rectangular range of integer indices. These are most commonly encountered in the context of iteration, where for I in R ... end
will return CartesianIndex
indices I
equivalent to the nested loops
for j = jstart:jstep:jstop
for i = istart:istep:istop
...
end
end
Consequently these can be useful for writing algorithms that work in arbitrary dimensions.
CartesianIndices(A::AbstractArray) -> R
As a convenience, constructing a CartesianIndices
from an array makes a range of its indices.
The step range method CartesianIndices((istart:istep:istop, jstart:[jstep:]jstop, ...))
requires at least Julia 1.6.
Examples
julia> foreach(println, CartesianIndices((2, 2, 2)))
CartesianIndex(1, 1, 1)
CartesianIndex(2, 1, 1)
CartesianIndex(1, 2, 1)
CartesianIndex(2, 2, 1)
CartesianIndex(1, 1, 2)
CartesianIndex(2, 1, 2)
CartesianIndex(1, 2, 2)
CartesianIndex(2, 2, 2)
julia> CartesianIndices(fill(1, (2,3)))
2×3 CartesianIndices{2, Tuple{Base.OneTo{Int64}, Base.OneTo{Int64}}}:
CartesianIndex(1, 1) CartesianIndex(1, 2) CartesianIndex(1, 3)
CartesianIndex(2, 1) CartesianIndex(2, 2) CartesianIndex(2, 3)
Conversion between linear and cartesian indices
Linear index to cartesian index conversion exploits the fact that a CartesianIndices
is an AbstractArray
and can be indexed linearly:
julia> cartesian = CartesianIndices((1:3, 1:2))
3×2 CartesianIndices{2, Tuple{UnitRange{Int64}, UnitRange{Int64}}}:
CartesianIndex(1, 1) CartesianIndex(1, 2)
CartesianIndex(2, 1) CartesianIndex(2, 2)
CartesianIndex(3, 1) CartesianIndex(3, 2)
julia> cartesian[4]
CartesianIndex(1, 2)
julia> cartesian = CartesianIndices((1:2:5, 1:2))
3×2 CartesianIndices{2, Tuple{StepRange{Int64, Int64}, UnitRange{Int64}}}:
CartesianIndex(1, 1) CartesianIndex(1, 2)
CartesianIndex(3, 1) CartesianIndex(3, 2)
CartesianIndex(5, 1) CartesianIndex(5, 2)
julia> cartesian[2, 2]
CartesianIndex(3, 2)
Broadcasting
CartesianIndices
support broadcasting arithmetic (+ and -) with a CartesianIndex
.
Broadcasting of CartesianIndices requires at least Julia 1.1.
julia> CIs = CartesianIndices((2:3, 5:6))
2×2 CartesianIndices{2, Tuple{UnitRange{Int64}, UnitRange{Int64}}}:
CartesianIndex(2, 5) CartesianIndex(2, 6)
CartesianIndex(3, 5) CartesianIndex(3, 6)
julia> CI = CartesianIndex(3, 4)
CartesianIndex(3, 4)
julia> CIs .+ CI
2×2 CartesianIndices{2, Tuple{UnitRange{Int64}, UnitRange{Int64}}}:
CartesianIndex(5, 9) CartesianIndex(5, 10)
CartesianIndex(6, 9) CartesianIndex(6, 10)
For cartesian to linear index conversion, see LinearIndices
.
Base.Dims
— TypeDims{N}
An NTuple
of N
Int
s used to represent the dimensions of an AbstractArray
.
Base.LinearIndices
— TypeLinearIndices(A::AbstractArray)
Return a LinearIndices
array with the same shape and axes
as A
, holding the linear index of each entry in A
. Indexing this array with cartesian indices allows mapping them to linear indices.
For arrays with conventional indexing (indices start at 1), or any multidimensional array, linear indices range from 1 to length(A)
. However, for AbstractVector
s linear indices are axes(A, 1)
, and therefore do not start at 1 for vectors with unconventional indexing.
Calling this function is the "safe" way to write algorithms that exploit linear indexing.
Examples
julia> A = fill(1, (5,6,7));
julia> b = LinearIndices(A);
julia> extrema(b)
(1, 210)
LinearIndices(inds::CartesianIndices) -> R
LinearIndices(sz::Dims) -> R
LinearIndices((istart:istop, jstart:jstop, ...)) -> R
Return a LinearIndices
array with the specified shape or axes
.
Example
The main purpose of this constructor is intuitive conversion from cartesian to linear indexing:
julia> linear = LinearIndices((1:3, 1:2))
3×2 LinearIndices{2, Tuple{UnitRange{Int64}, UnitRange{Int64}}}:
1 4
2 5
3 6
julia> linear[1,2]
4
Base.to_indices
— Functionto_indices(A, I::Tuple)
Convert the tuple I
to a tuple of indices for use in indexing into array A
.
The returned tuple must only contain either Int
s or AbstractArray
s of scalar indices that are supported by array A
. It will error upon encountering a novel index type that it does not know how to process.
For simple index types, it defers to the unexported Base.to_index(A, i)
to process each index i
. While this internal function is not intended to be called directly, Base.to_index
may be extended by custom array or index types to provide custom indexing behaviors.
More complicated index types may require more context about the dimension into which they index. To support those cases, to_indices(A, I)
calls to_indices(A, axes(A), I)
, which then recursively walks through both the given tuple of indices and the dimensional indices of A
in tandem. As such, not all index types are guaranteed to propagate to Base.to_index
.
Base.checkbounds
— Functioncheckbounds(Bool, A, I...)
Return true
if the specified indices I
are in bounds for the given array A
. Subtypes of AbstractArray
should specialize this method if they need to provide custom bounds checking behaviors; however, in many cases one can rely on A
's indices and checkindex
.
See also checkindex
.
Examples
julia> A = rand(3, 3);
julia> checkbounds(Bool, A, 2)
true
julia> checkbounds(Bool, A, 3, 4)
false
julia> checkbounds(Bool, A, 1:3)
true
julia> checkbounds(Bool, A, 1:3, 2:4)
false
checkbounds(A, I...)
Throw an error if the specified indices I
are not in bounds for the given array A
.
Base.checkindex
— Functioncheckindex(Bool, inds::AbstractUnitRange, index)
Return true
if the given index
is within the bounds of inds
. Custom types that would like to behave as indices for all arrays can extend this method in order to provide a specialized bounds checking implementation.
See also checkbounds
.
Examples
julia> checkindex(Bool, 1:20, 8)
true
julia> checkindex(Bool, 1:20, 21)
false
Base.elsize
— Functionelsize(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.
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
— Functionview(A, inds...)
Like getindex
, but returns a lightweight array that lazily references (or is effectively a view into) the parent array A
at the given index or indices inds
instead of eagerly extracting elements or constructing a copied subset. Calling getindex
or setindex!
on the returned value (often a SubArray
) computes the indices to access or modify the parent array on the fly. The behavior is undefined if the shape of the parent array is changed after view
is called because there is no bound check for the parent array; e.g., it may cause a segmentation fault.
Some immutable parent arrays (like ranges) may choose to simply recompute a new array in some circumstances instead of returning a SubArray
if doing so is efficient and provides compatible semantics.
In Julia 1.6 or later, view
can be called on an AbstractString
, returning a SubString
.
Examples
julia> A = [1 2; 3 4]
2×2 Matrix{Int64}:
1 2
3 4
julia> b = view(A, :, 1)
2-element view(::Matrix{Int64}, :, 1) with eltype Int64:
1
3
julia> fill!(b, 0)
2-element view(::Matrix{Int64}, :, 1) with eltype Int64:
0
0
julia> A # Note A has changed even though we modified b
2×2 Matrix{Int64}:
0 2
0 4
julia> view(2:5, 2:3) # returns a range as type is immutable
3:4
Base.@view
— Macro@view A[inds...]
Transform the indexing expression A[inds...]
into the equivalent view
call.
This can only be applied directly to a single indexing expression and is particularly helpful for expressions that include the special begin
or end
indexing syntaxes like A[begin, 2:end-1]
(as those are not supported by the normal view
function).
Note that @view
cannot be used as the target of a regular assignment (e.g., @view(A[1, 2:end]) = ...
), nor would the un-decorated indexed assignment (A[1, 2:end] = ...
) or broadcasted indexed assignment (A[1, 2:end] .= ...
) make a copy. It can be useful, however, for updating broadcasted assignments like @view(A[1, 2:end]) .+= 1
because this is a simple syntax for @view(A[1, 2:end]) .= @view(A[1, 2:end]) + 1
, and the indexing expression on the right-hand side would otherwise make a copy without the @view
.
See also @views
to switch an entire block of code to use views for non-scalar indexing.
Using begin
in an indexing expression to refer to the first index requires at least Julia 1.5.
Examples
julia> A = [1 2; 3 4]
2×2 Matrix{Int64}:
1 2
3 4
julia> b = @view A[:, 1]
2-element view(::Matrix{Int64}, :, 1) with eltype Int64:
1
3
julia> fill!(b, 0)
2-element view(::Matrix{Int64}, :, 1) with eltype Int64:
0
0
julia> A
2×2 Matrix{Int64}:
0 2
0 4
Base.@views
— Macro@views expression
Convert every array-slicing operation in the given expression (which may be a begin
/end
block, loop, function, etc.) to return a view. Scalar indices, non-array types, and explicit getindex
calls (as opposed to array[...]
) are unaffected.
The @views
macro only affects array[...]
expressions that appear explicitly in the given expression
, not array slicing that occurs in functions called by that code.
Using begin
in an indexing expression to refer to the first index 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
Base.parent
— Functionparent(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
Base.parentindices
— Functionparentindices(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)))
Base.selectdim
— Functionselectdim(A, d::Integer, i)
Return a view of all the data of A
where the index for dimension d
equals i
.
Equivalent to view(A,:,:,...,i,:,:,...)
where i
is in position d
.
See also: eachslice
.
Examples
julia> A = [1 2 3 4; 5 6 7 8]
2×4 Matrix{Int64}:
1 2 3 4
5 6 7 8
julia> selectdim(A, 2, 3)
2-element view(::Matrix{Int64}, :, 3) with eltype Int64:
3
7
julia> selectdim(A, 2, 3:4)
2×2 view(::Matrix{Int64}, :, 3:4) with eltype Int64:
3 4
7 8
Base.reinterpret
— Functionreinterpret(type, A)
Change the type-interpretation of a block of memory. For arrays, this constructs a view of the array with the same binary data as the given array, but with the specified element type. For example, reinterpret(Float32, UInt32(7))
interprets the 4 bytes corresponding to UInt32(7)
as a Float32
.
Examples
julia> reinterpret(Float32, UInt32(7))
1.0f-44
julia> reinterpret(Float32, UInt32[1 2 3 4 5])
1×5 reinterpret(Float32, ::Matrix{UInt32}):
1.0f-45 3.0f-45 4.0f-45 6.0f-45 7.0f-45
reinterpret(reshape, T, A::AbstractArray{S}) -> B
Change the type-interpretation of A
while consuming or adding a "channel dimension."
If sizeof(T) = n*sizeof(S)
for n>1
, A
's first dimension must be of size n
and B
lacks A
's first dimension. Conversely, if sizeof(S) = n*sizeof(T)
for n>1
, B
gets a new first dimension of size n
. The dimensionality is unchanged if sizeof(T) == sizeof(S)
.
This method requires at least Julia 1.6.
Examples
julia> A = [1 2; 3 4]
2×2 Matrix{Int64}:
1 2
3 4
julia> reinterpret(reshape, Complex{Int}, A) # the result is a vector
2-element reinterpret(reshape, Complex{Int64}, ::Matrix{Int64}) with eltype Complex{Int64}:
1 + 3im
2 + 4im
julia> a = [(1,2,3), (4,5,6)]
2-element Vector{Tuple{Int64, Int64, Int64}}:
(1, 2, 3)
(4, 5, 6)
julia> reinterpret(reshape, Int, a) # the result is a matrix
3×2 reinterpret(reshape, Int64, ::Vector{Tuple{Int64, Int64, Int64}}) with eltype Int64:
1 4
2 5
3 6
Base.reshape
— Functionreshape(A, dims...) -> AbstractArray
reshape(A, dims) -> AbstractArray
Return an array with the same data as A
, but with different dimension sizes or number of dimensions. The two arrays share the same underlying data, so that the result is mutable if and only if A
is mutable, and setting elements of one alters the values of the other.
The new dimensions may be specified either as a list of arguments or as a shape tuple. At most one dimension may be specified with a :
, in which case its length is computed such that its product with all the specified dimensions is equal to the length of the original array A
. The total number of elements must not change.
Examples
julia> A = Vector(1:16)
16-element Vector{Int64}:
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
julia> reshape(A, (4, 4))
4×4 Matrix{Int64}:
1 5 9 13
2 6 10 14
3 7 11 15
4 8 12 16
julia> reshape(A, 2, :)
2×8 Matrix{Int64}:
1 3 5 7 9 11 13 15
2 4 6 8 10 12 14 16
julia> reshape(1:6, 2, 3)
2×3 reshape(::UnitRange{Int64}, 2, 3) with eltype Int64:
1 3 5
2 4 6
Base.dropdims
— Functiondropdims(A; dims)
Return an array with the same data as A
, but with the dimensions specified by dims
removed. size(A,d)
must equal 1 for every d
in dims
, and repeated dimensions or numbers outside 1:ndims(A)
are forbidden.
The result shares the same underlying data as A
, such that the result is mutable if and only if A
is mutable, and setting elements of one alters the values of the other.
Examples
julia> a = reshape(Vector(1:4),(2,2,1,1))
2×2×1×1 Array{Int64, 4}:
[:, :, 1, 1] =
1 3
2 4
julia> b = dropdims(a; dims=3)
2×2×1 Array{Int64, 3}:
[:, :, 1] =
1 3
2 4
julia> b[1,1,1] = 5; a
2×2×1×1 Array{Int64, 4}:
[:, :, 1, 1] =
5 3
2 4
Base.vec
— Functionvec(a::AbstractArray) -> AbstractVector
Reshape the array a
as a one-dimensional column vector. Return a
if it is already an AbstractVector
. The resulting array shares the same underlying data as a
, so it will only be mutable if a
is mutable, in which case modifying one will also modify the other.
Examples
julia> a = [1 2 3; 4 5 6]
2×3 Matrix{Int64}:
1 2 3
4 5 6
julia> vec(a)
6-element Vector{Int64}:
1
4
2
5
3
6
julia> vec(1:3)
1:3
Base.SubArray
— TypeSubArray{T,N,P,I,L} <: AbstractArray{T,N}
N
-dimensional view into a parent array (of type P
) with an element type T
, restricted by a tuple of indices (of type I
). L
is true for types that support fast linear indexing, and false
otherwise.
Construct SubArray
s using the view
function.
Concatenation and permutation
Base.cat
— Functioncat(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
Base.vcat
— Functionvcat(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
Base.hcat
— Functionhcat(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
Base.hvcat
— Functionhvcat(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.
Base.hvncat
— Functionhvncat(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.
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
Base.vect
— Functionvect(X...)
Create a Vector
with element type computed from the promote_typeof
of the argument, containing the argument list.
Examples
julia> a = Base.vect(UInt8(1), 2.5, 1//2)
3-element Vector{Float64}:
1.0
2.5
0.5
Base.circshift
— Functioncircshift(A, shifts)
Circularly shift, i.e. rotate, the data in an array. The second argument is a tuple or vector giving the amount to shift in each dimension, or an integer to shift only in the first dimension.
See also: circshift!
, circcopy!
, bitrotate
, <<
.
Examples
julia> b = reshape(Vector(1:16), (4,4))
4×4 Matrix{Int64}:
1 5 9 13
2 6 10 14
3 7 11 15
4 8 12 16
julia> circshift(b, (0,2))
4×4 Matrix{Int64}:
9 13 1 5
10 14 2 6
11 15 3 7
12 16 4 8
julia> circshift(b, (-1,0))
4×4 Matrix{Int64}:
2 6 10 14
3 7 11 15
4 8 12 16
1 5 9 13
julia> a = BitArray([true, true, false, false, true])
5-element BitVector:
1
1
0
0
1
julia> circshift(a, 1)
5-element BitVector:
1
1
1
0
0
julia> circshift(a, -1)
5-element BitVector:
1
0
0
1
1
Base.circshift!
— Functioncircshift!(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!
— Functioncirccopy!(dest, src)
Copy src
to dest
, indexing each dimension modulo its length. src
and dest
must have the same size, but can be offset in their indices; any offset results in a (circular) wraparound. If the arrays have overlapping indices, then on the domain of the overlap dest
agrees with src
.
See also: circshift
.
Examples
julia> src = reshape(Vector(1:16), (4,4))
4×4 Array{Int64,2}:
1 5 9 13
2 6 10 14
3 7 11 15
4 8 12 16
julia> dest = OffsetArray{Int}(undef, (0:3,2:5))
julia> circcopy!(dest, src)
OffsetArrays.OffsetArray{Int64,2,Array{Int64,2}} with indices 0:3×2:5:
8 12 16 4
5 9 13 1
6 10 14 2
7 11 15 3
julia> dest[1:3,2:4] == src[1:3,2:4]
true
Base.findall
— Methodfindall(A)
Return a vector I
of the true
indices or keys of A
. If there are no such elements of A
, return an empty array. To search for other kinds of values, pass a predicate as the first argument.
Indices or keys are of the same type as those returned by keys(A)
and pairs(A)
.
See also: findfirst
, searchsorted
.
Examples
julia> A = [true, false, false, true]
4-element Vector{Bool}:
1
0
0
1
julia> findall(A)
2-element Vector{Int64}:
1
4
julia> A = [true false; false true]
2×2 Matrix{Bool}:
1 0
0 1
julia> findall(A)
2-element Vector{CartesianIndex{2}}:
CartesianIndex(1, 1)
CartesianIndex(2, 2)
julia> findall(falses(3))
Int64[]
Base.findall
— Methodfindall(f::Function, A)
Return a vector I
of the indices or keys of A
where f(A[I])
returns true
. If there are no such elements of A
, return an empty array.
Indices or keys are of the same type as those returned by keys(A)
and pairs(A)
.
Examples
julia> x = [1, 3, 4]
3-element Vector{Int64}:
1
3
4
julia> findall(isodd, x)
2-element Vector{Int64}:
1
2
julia> A = [1 2 0; 3 4 0]
2×3 Matrix{Int64}:
1 2 0
3 4 0
julia> findall(isodd, A)
2-element Vector{CartesianIndex{2}}:
CartesianIndex(1, 1)
CartesianIndex(2, 1)
julia> findall(!iszero, A)
4-element Vector{CartesianIndex{2}}:
CartesianIndex(1, 1)
CartesianIndex(2, 1)
CartesianIndex(1, 2)
CartesianIndex(2, 2)
julia> d = Dict(:A => 10, :B => -1, :C => 0)
Dict{Symbol, Int64} with 3 entries:
:A => 10
:B => -1
:C => 0
julia> findall(x -> x >= 0, d)
2-element Vector{Symbol}:
:A
:C
Base.findfirst
— Methodfindfirst(A)
Return the index or key of the first true
value in A
. Return nothing
if no such value is found. To search for other kinds of values, pass a predicate as the first argument.
Indices or keys are of the same type as those returned by keys(A)
and pairs(A)
.
See also: findall
, findnext
, findlast
, searchsortedfirst
.
Examples
julia> A = [false, false, true, false]
4-element Vector{Bool}:
0
0
1
0
julia> findfirst(A)
3
julia> findfirst(falses(3)) # returns nothing, but not printed in the REPL
julia> A = [false false; true false]
2×2 Matrix{Bool}:
0 0
1 0
julia> findfirst(A)
CartesianIndex(2, 1)
Base.findfirst
— Methodfindfirst(predicate::Function, A)
Return the index or key of the first element of A
for which predicate
returns true
. Return nothing
if there is no such element.
Indices or keys are of the same type as those returned by keys(A)
and pairs(A)
.
Examples
julia> A = [1, 4, 2, 2]
4-element Vector{Int64}:
1
4
2
2
julia> findfirst(iseven, A)
2
julia> findfirst(x -> x>10, A) # returns nothing, but not printed in the REPL
julia> findfirst(isequal(4), A)
2
julia> A = [1 4; 2 2]
2×2 Matrix{Int64}:
1 4
2 2
julia> findfirst(iseven, A)
CartesianIndex(2, 1)
Base.findlast
— Methodfindlast(A)
Return the index or key of the last true
value in A
. Return nothing
if there is no true
value in A
.
Indices or keys are of the same type as those returned by keys(A)
and pairs(A)
.
See also: findfirst
, findprev
, findall
.
Examples
julia> A = [true, false, true, false]
4-element Vector{Bool}:
1
0
1
0
julia> findlast(A)
3
julia> A = falses(2,2);
julia> findlast(A) # returns nothing, but not printed in the REPL
julia> A = [true false; true false]
2×2 Matrix{Bool}:
1 0
1 0
julia> findlast(A)
CartesianIndex(2, 1)
Base.findlast
— Methodfindlast(predicate::Function, A)
Return the index or key of the last element of A
for which predicate
returns true
. Return nothing
if there is no such element.
Indices or keys are of the same type as those returned by keys(A)
and pairs(A)
.
Examples
julia> A = [1, 2, 3, 4]
4-element Vector{Int64}:
1
2
3
4
julia> findlast(isodd, A)
3
julia> findlast(x -> x > 5, A) # returns nothing, but not printed in the REPL
julia> A = [1 2; 3 4]
2×2 Matrix{Int64}:
1 2
3 4
julia> findlast(isodd, A)
CartesianIndex(2, 1)
Base.findnext
— Methodfindnext(A, i)
Find the next index after or including i
of a true
element of A
, or nothing
if not found.
Indices are of the same type as those returned by keys(A)
and pairs(A)
.
Examples
julia> A = [false, false, true, false]
4-element Vector{Bool}:
0
0
1
0
julia> findnext(A, 1)
3
julia> findnext(A, 4) # returns nothing, but not printed in the REPL
julia> A = [false false; true false]
2×2 Matrix{Bool}:
0 0
1 0
julia> findnext(A, CartesianIndex(1, 1))
CartesianIndex(2, 1)
Base.findnext
— Methodfindnext(predicate::Function, A, i)
Find the next index after or including i
of an element of A
for which predicate
returns true
, or nothing
if not found.
Indices are of the same type as those returned by keys(A)
and pairs(A)
.
Examples
julia> A = [1, 4, 2, 2];
julia> findnext(isodd, A, 1)
1
julia> findnext(isodd, A, 2) # returns nothing, but not printed in the REPL
julia> A = [1 4; 2 2];
julia> findnext(isodd, A, CartesianIndex(1, 1))
CartesianIndex(1, 1)
Base.findprev
— Methodfindprev(A, i)
Find the previous index before or including i
of a true
element of A
, or nothing
if not found.
Indices are of the same type as those returned by keys(A)
and pairs(A)
.
See also: findnext
, findfirst
, findall
.
Examples
julia> A = [false, false, true, true]
4-element Vector{Bool}:
0
0
1
1
julia> findprev(A, 3)
3
julia> findprev(A, 1) # returns nothing, but not printed in the REPL
julia> A = [false false; true true]
2×2 Matrix{Bool}:
0 0
1 1
julia> findprev(A, CartesianIndex(2, 1))
CartesianIndex(2, 1)
Base.findprev
— Methodfindprev(predicate::Function, A, i)
Find the previous index before or including i
of an element of A
for which predicate
returns true
, or nothing
if not found.
Indices are of the same type as those returned by keys(A)
and pairs(A)
.
Examples
julia> A = [4, 6, 1, 2]
4-element Vector{Int64}:
4
6
1
2
julia> findprev(isodd, A, 1) # returns nothing, but not printed in the REPL
julia> findprev(isodd, A, 3)
3
julia> A = [4 6; 1 2]
2×2 Matrix{Int64}:
4 6
1 2
julia> findprev(isodd, A, CartesianIndex(1, 2))
CartesianIndex(2, 1)
Base.permutedims
— Functionpermutedims(A::AbstractArray, perm)
Permute the dimensions of array A
. perm
is a vector or a tuple of length ndims(A)
specifying the permutation.
See also permutedims!
, PermutedDimsArray
, transpose
, invperm
.
Examples
julia> A = reshape(Vector(1:8), (2,2,2))
2×2×2 Array{Int64, 3}:
[:, :, 1] =
1 3
2 4
[:, :, 2] =
5 7
6 8
julia> permutedims(A, (3, 2, 1))
2×2×2 Array{Int64, 3}:
[:, :, 1] =
1 3
5 7
[:, :, 2] =
2 4
6 8
julia> B = randn(5, 7, 11, 13);
julia> perm = [4,1,3,2];
julia> size(permutedims(B, perm))
(13, 5, 11, 7)
julia> size(B)[perm] == ans
true
permutedims(m::AbstractMatrix)
Permute the dimensions of the matrix m
, by flipping the elements across the diagonal of the matrix. Differs from LinearAlgebra
's transpose
in that the operation is not recursive.
Examples
julia> a = [1 2; 3 4];
julia> b = [5 6; 7 8];
julia> c = [9 10; 11 12];
julia> d = [13 14; 15 16];
julia> X = [[a] [b]; [c] [d]]
2×2 Matrix{Matrix{Int64}}:
[1 2; 3 4] [5 6; 7 8]
[9 10; 11 12] [13 14; 15 16]
julia> permutedims(X)
2×2 Matrix{Matrix{Int64}}:
[1 2; 3 4] [9 10; 11 12]
[5 6; 7 8] [13 14; 15 16]
julia> transpose(X)
2×2 transpose(::Matrix{Matrix{Int64}}) with eltype Transpose{Int64, Matrix{Int64}}:
[1 3; 2 4] [9 11; 10 12]
[5 7; 6 8] [13 15; 14 16]
permutedims(v::AbstractVector)
Reshape vector v
into a 1 × length(v)
row matrix. Differs from LinearAlgebra
's transpose
in that the operation is not recursive.
Examples
julia> permutedims([1, 2, 3, 4])
1×4 Matrix{Int64}:
1 2 3 4
julia> V = [[[1 2; 3 4]]; [[5 6; 7 8]]]
2-element Vector{Matrix{Int64}}:
[1 2; 3 4]
[5 6; 7 8]
julia> permutedims(V)
1×2 Matrix{Matrix{Int64}}:
[1 2; 3 4] [5 6; 7 8]
julia> transpose(V)
1×2 transpose(::Vector{Matrix{Int64}}) with eltype Transpose{Int64, Matrix{Int64}}:
[1 3; 2 4] [5 7; 6 8]
Base.permutedims!
— Functionpermutedims!(dest, src, perm)
Permute the dimensions of array src
and store the result in the array dest
. perm
is a vector specifying a permutation of length ndims(src)
. The preallocated array dest
should have size(dest) == size(src)[perm]
and is completely overwritten. No in-place permutation is supported and unexpected results will happen if src
and dest
have overlapping memory regions.
See also permutedims
.
Base.PermutedDimsArrays.PermutedDimsArray
— TypePermutedDimsArray(A, perm) -> B
Given an AbstractArray A
, create a view B
such that the dimensions appear to be permuted. Similar to permutedims
, except that no copying occurs (B
shares storage with A
).
See also permutedims
, invperm
.
Examples
julia> A = rand(3,5,4);
julia> B = PermutedDimsArray(A, (3,1,2));
julia> size(B)
(4, 3, 5)
julia> B[3,1,2] == A[1,2,3]
true
Base.promote_shape
— Functionpromote_shape(s1, s2)
Check two array shapes for compatibility, allowing trailing singleton dimensions, and return whichever shape has more dimensions.
Examples
julia> a = fill(1, (3,4,1,1,1));
julia> b = fill(1, (3,4));
julia> promote_shape(a,b)
(Base.OneTo(3), Base.OneTo(4), Base.OneTo(1), Base.OneTo(1), Base.OneTo(1))
julia> promote_shape((2,3,1,4), (2, 3, 1, 4, 1))
(2, 3, 1, 4, 1)
Array functions
Base.accumulate
— Functionaccumulate(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
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
Base.accumulate!
— Functionaccumulate!(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
Base.cumprod
— Functioncumprod(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
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
.
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
Base.cumprod!
— Functioncumprod!(B, A; dims::Integer)
Cumulative product of A
along the dimension dims
, storing the result in B
. See also cumprod
.
cumprod!(y::AbstractVector, x::AbstractVector)
Cumulative product of a vector x
, storing the result in y
. See also cumprod
.
Base.cumsum
— Functioncumsum(A; dims::Integer)
Cumulative sum along the dimension dims
. See also cumsum!
to use a preallocated output array, both for performance and to control the precision of the output (e.g. to avoid overflow).
Examples
julia> a = [1 2 3; 4 5 6]
2×3 Matrix{Int64}:
1 2 3
4 5 6
julia> cumsum(a, dims=1)
2×3 Matrix{Int64}:
1 2 3
5 7 9
julia> cumsum(a, dims=2)
2×3 Matrix{Int64}:
1 3 6
4 9 15
The return array's eltype
is Int
for signed integers of less than system word size and UInt
for unsigned integers of less than system word size. To preserve eltype
of arrays with small signed or unsigned integer accumulate(+, A)
should be used.
julia> cumsum(Int8[100, 28])
2-element Vector{Int64}:
100
128
julia> accumulate(+,Int8[100, 28])
2-element Vector{Int8}:
100
-128
In the former case, the integers are widened to system word size and therefore the result is Int64[100, 128]
. In the latter case, no such widening happens and integer overflow results in Int8[100, -128]
.
cumsum(itr)
Cumulative sum 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).
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
Base.cumsum!
— Functioncumsum!(B, A; dims::Integer)
Cumulative sum of A
along the dimension dims
, storing the result in B
. See also cumsum
.
Base.diff
— Functiondiff(A::AbstractVector)
diff(A::AbstractArray; dims::Integer)
Finite difference operator on a vector or a multidimensional array A
. In the latter case the dimension to operate on needs to be specified with the dims
keyword argument.
diff
for arrays with dimension higher than 2 requires at least Julia 1.1.
Examples
julia> a = [2 4; 6 16]
2×2 Matrix{Int64}:
2 4
6 16
julia> diff(a, dims=2)
2×1 Matrix{Int64}:
2
10
julia> diff(vec(a))
3-element Vector{Int64}:
4
-2
12
Base.repeat
— Functionrepeat(A::AbstractArray, counts::Integer...)
Construct an array by repeating array A
a given number of times in each dimension, specified by counts
.
See also: fill
, Iterators.repeated
, Iterators.cycle
.
Examples
julia> repeat([1, 2, 3], 2)
6-element Vector{Int64}:
1
2
3
1
2
3
julia> repeat([1, 2, 3], 2, 3)
6×3 Matrix{Int64}:
1 1 1
2 2 2
3 3 3
1 1 1
2 2 2
3 3 3
repeat(A::AbstractArray; inner=ntuple(Returns(1), ndims(A)), outer=ntuple(Returns(1), ndims(A)))
Construct an array by repeating the entries of A
. The i-th element of inner
specifies the number of times that the individual entries of the i-th dimension of A
should be repeated. The i-th element of outer
specifies the number of times that a slice along the i-th dimension of A
should be repeated. If inner
or outer
are omitted, no repetition is performed.
Examples
julia> repeat(1:2, inner=2)
4-element Vector{Int64}:
1
1
2
2
julia> repeat(1:2, outer=2)
4-element Vector{Int64}:
1
2
1
2
julia> repeat([1 2; 3 4], inner=(2, 1), outer=(1, 3))
4×6 Matrix{Int64}:
1 2 1 2 1 2
1 2 1 2 1 2
3 4 3 4 3 4
3 4 3 4 3 4
repeat(s::AbstractString, r::Integer)
Repeat a string r
times. This can be written as s^r
.
See also ^
.
Examples
julia> repeat("ha", 3)
"hahaha"
repeat(c::AbstractChar, r::Integer) -> String
Repeat a character r
times. This can equivalently be accomplished by calling c^r
.
Examples
julia> repeat('A', 3)
"AAA"
Base.rot180
— Functionrot180(A)
Rotate matrix A
180 degrees.
Examples
julia> a = [1 2; 3 4]
2×2 Matrix{Int64}:
1 2
3 4
julia> rot180(a)
2×2 Matrix{Int64}:
4 3
2 1
rot180(A, k)
Rotate matrix A
180 degrees an integer k
number of times. If k
is even, this is equivalent to a copy
.
Examples
julia> a = [1 2; 3 4]
2×2 Matrix{Int64}:
1 2
3 4
julia> rot180(a,1)
2×2 Matrix{Int64}:
4 3
2 1
julia> rot180(a,2)
2×2 Matrix{Int64}:
1 2
3 4
Base.rotl90
— Functionrotl90(A)
Rotate matrix A
left 90 degrees.
Examples
julia> a = [1 2; 3 4]
2×2 Matrix{Int64}:
1 2
3 4
julia> rotl90(a)
2×2 Matrix{Int64}:
2 4
1 3
rotl90(A, k)
Left-rotate matrix A
90 degrees counterclockwise an integer k
number of times. If k
is a multiple of four (including zero), this is equivalent to a copy
.
Examples
julia> a = [1 2; 3 4]
2×2 Matrix{Int64}:
1 2
3 4
julia> rotl90(a,1)
2×2 Matrix{Int64}:
2 4
1 3
julia> rotl90(a,2)
2×2 Matrix{Int64}:
4 3
2 1
julia> rotl90(a,3)
2×2 Matrix{Int64}:
3 1
4 2
julia> rotl90(a,4)
2×2 Matrix{Int64}:
1 2
3 4
Base.rotr90
— Functionrotr90(A)
Rotate matrix A
right 90 degrees.
Examples
julia> a = [1 2; 3 4]
2×2 Matrix{Int64}:
1 2
3 4
julia> rotr90(a)
2×2 Matrix{Int64}:
3 1
4 2
rotr90(A, k)
Right-rotate matrix A
90 degrees clockwise an integer k
number of times. If k
is a multiple of four (including zero), this is equivalent to a copy
.
Examples
julia> a = [1 2; 3 4]
2×2 Matrix{Int64}:
1 2
3 4
julia> rotr90(a,1)
2×2 Matrix{Int64}:
3 1
4 2
julia> rotr90(a,2)
2×2 Matrix{Int64}:
4 3
2 1
julia> rotr90(a,3)
2×2 Matrix{Int64}:
2 4
1 3
julia> rotr90(a,4)
2×2 Matrix{Int64}:
1 2
3 4
Base.mapslices
— Functionmapslices(f, A; dims)
Transform the given dimensions of array A
using function f
. f
is called on each slice of A
of the form A[...,:,...,:,...]
. dims
is an integer vector specifying where the colons go in this expression. The results are concatenated along the remaining dimensions. For example, if dims
is [1,2]
and A
is 4-dimensional, f
is called on A[:,:,i,j]
for all i
and j
.
Examples
julia> a = reshape(Vector(1:16),(2,2,2,2))
2×2×2×2 Array{Int64, 4}:
[:, :, 1, 1] =
1 3
2 4
[:, :, 2, 1] =
5 7
6 8
[:, :, 1, 2] =
9 11
10 12
[:, :, 2, 2] =
13 15
14 16
julia> mapslices(sum, a, dims = [1,2])
1×1×2×2 Array{Int64, 4}:
[:, :, 1, 1] =
10
[:, :, 2, 1] =
26
[:, :, 1, 2] =
42
[:, :, 2, 2] =
58
Base.eachrow
— Functioneachrow(A::AbstractVecOrMat)
Create a generator that iterates over the first dimension of vector or matrix A
, returning the rows as AbstractVector
views.
See also eachcol
, eachslice
, mapslices
.
This function requires at least Julia 1.1.
Example
julia> a = [1 2; 3 4]
2×2 Matrix{Int64}:
1 2
3 4
julia> first(eachrow(a))
2-element view(::Matrix{Int64}, 1, :) with eltype Int64:
1
2
julia> collect(eachrow(a))
2-element Vector{SubArray{Int64, 1, Matrix{Int64}, Tuple{Int64, Base.Slice{Base.OneTo{Int64}}}, true}}:
[1, 2]
[3, 4]
Base.eachcol
— Functioneachcol(A::AbstractVecOrMat)
Create a generator that iterates over the second dimension of matrix A
, returning the columns as AbstractVector
views.
See also eachrow
and eachslice
.
This function requires at least Julia 1.1.
Example
julia> a = [1 2; 3 4]
2×2 Matrix{Int64}:
1 2
3 4
julia> first(eachcol(a))
2-element view(::Matrix{Int64}, :, 1) with eltype Int64:
1
3
julia> collect(eachcol(a))
2-element Vector{SubArray{Int64, 1, Matrix{Int64}, Tuple{Base.Slice{Base.OneTo{Int64}}, Int64}, true}}:
[1, 3]
[2, 4]
Base.eachslice
— Functioneachslice(A::AbstractArray; dims)
Create a generator that iterates over dimensions dims
of A
, returning views that select all the data from the other dimensions in A
.
Only a single dimension in dims
is currently supported. Equivalent to (view(A,:,:,...,i,:,: ...)) for i in axes(A, dims))
, where i
is in position dims
.
See also eachrow
, eachcol
, mapslices
, and selectdim
.
This function requires at least Julia 1.1.
Example
julia> M = [1 2 3; 4 5 6; 7 8 9]
3×3 Matrix{Int64}:
1 2 3
4 5 6
7 8 9
julia> first(eachslice(M, dims=1))
3-element view(::Matrix{Int64}, 1, :) with eltype Int64:
1
2
3
julia> collect(eachslice(M, dims=2))
3-element Vector{SubArray{Int64, 1, Matrix{Int64}, Tuple{Base.Slice{Base.OneTo{Int64}}, Int64}, true}}:
[1, 4, 7]
[2, 5, 8]
[3, 6, 9]
Combinatorics
Base.invperm
— Functioninvperm(v)
Return the inverse permutation of v
. If B = A[v]
, then A == B[invperm(v)]
.
See also sortperm
, invpermute!
, isperm
, permutedims
.
Examples
julia> p = (2, 3, 1);
julia> invperm(p)
(3, 1, 2)
julia> v = [2; 4; 3; 1];
julia> invperm(v)
4-element Vector{Int64}:
4
1
3
2
julia> A = ['a','b','c','d'];
julia> B = A[v]
4-element Vector{Char}:
'b': ASCII/Unicode U+0062 (category Ll: Letter, lowercase)
'd': ASCII/Unicode U+0064 (category Ll: Letter, lowercase)
'c': ASCII/Unicode U+0063 (category Ll: Letter, lowercase)
'a': ASCII/Unicode U+0061 (category Ll: Letter, lowercase)
julia> B[invperm(v)]
4-element Vector{Char}:
'a': ASCII/Unicode U+0061 (category Ll: Letter, lowercase)
'b': ASCII/Unicode U+0062 (category Ll: Letter, lowercase)
'c': ASCII/Unicode U+0063 (category Ll: Letter, lowercase)
'd': ASCII/Unicode U+0064 (category Ll: Letter, lowercase)
Base.isperm
— Functionisperm(v) -> Bool
Return true
if v
is a valid permutation.
Examples
julia> isperm([1; 2])
true
julia> isperm([1; 3])
false
Base.permute!
— Methodpermute!(v, p)
Permute vector v
in-place, according to permutation p
. No checking is done to verify that p
is a permutation.
To return a new permutation, use v[p]
. Note that this is generally faster than permute!(v,p)
for large vectors.
See also invpermute!
.
Examples
julia> A = [1, 1, 3, 4];
julia> perm = [2, 4, 3, 1];
julia> permute!(A, perm);
julia> A
4-element Vector{Int64}:
1
4
3
1
Base.invpermute!
— Functioninvpermute!(v, p)
Like permute!
, but the inverse of the given permutation is applied.
Examples
julia> A = [1, 1, 3, 4];
julia> perm = [2, 4, 3, 1];
julia> invpermute!(A, perm);
julia> A
4-element Vector{Int64}:
4
1
3
1
Base.reverse
— Methodreverse(A; dims=:)
Reverse A
along dimension dims
, which can be an integer (a single dimension), a tuple of integers (a tuple of dimensions) or :
(reverse along all the dimensions, the default). See also reverse!
for in-place reversal.
Examples
julia> b = Int64[1 2; 3 4]
2×2 Matrix{Int64}:
1 2
3 4
julia> reverse(b, dims=2)
2×2 Matrix{Int64}:
2 1
4 3
julia> reverse(b)
2×2 Matrix{Int64}:
4 3
2 1
Prior to Julia 1.6, only single-integer dims
are supported in reverse
.
Base.reverseind
— Functionreverseind(v, i)
Given an index i
in reverse(v)
, return the corresponding index in v
so that v[reverseind(v,i)] == reverse(v)[i]
. (This can be nontrivial in cases where v
contains non-ASCII characters.)
Examples
julia> s = "Julia🚀"
"Julia🚀"
julia> r = reverse(s)
"🚀ailuJ"
julia> for i in eachindex(s)
print(r[reverseind(r, i)])
end
Julia🚀
Base.reverse!
— Functionreverse!(v [, start=1 [, stop=length(v) ]]) -> v
In-place version of reverse
.
Examples
julia> A = Vector(1:5)
5-element Vector{Int64}:
1
2
3
4
5
julia> reverse!(A);
julia> A
5-element Vector{Int64}:
5
4
3
2
1
reverse!(A; dims=:)
Like reverse
, but operates in-place in A
.
Multidimensional reverse!
requires Julia 1.6.