Arrays

# Arrays

## Constructors and Types

AbstractArray{T, N}

Abstract array supertype which arrays inherit from.

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Core.ArrayType.
Array{T}(dims)
Array{T,N}(dims)

Construct an uninitialized N-dimensional dense array with element type T, where N is determined from 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 as in Array{T,N}(dims), then it must match the length or number of dims.

Example

julia> A = Array{Float64, 2}(2, 2);

julia> ndims(A)
2

julia> eltype(A)
Float64
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Base.getindexMethod.
getindex(type[, elements...])

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

Example

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

julia> getindex(Int8, 1, 2, 3)
3-element Array{Int8,1}:
1
2
3
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Base.zerosFunction.
zeros([A::AbstractArray,] [T=eltype(A)::Type,] [dims=size(A)::Tuple])

Create an array of all zeros with the same layout as A, element type T and size dims. The A argument can be skipped, which behaves like Array{Float64,0}() was passed. For convenience dims may also be passed in variadic form.

Examples

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

julia> zeros(Int8, 2, 3)
2×3 Array{Int8,2}:
0  0  0
0  0  0

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

julia> zeros(A)
2×2 Array{Int64,2}:
0  0
0  0

julia> zeros(A, Float64)
2×2 Array{Float64,2}:
0.0  0.0
0.0  0.0

julia> zeros(A, Bool, (3,))
3-element Array{Bool,1}:
false
false
false

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Base.onesFunction.
ones([A::AbstractArray,] [T=eltype(A)::Type,] [dims=size(A)::Tuple])

Create an array of all ones with the same layout as A, element type T and size dims. The A argument can be skipped, which behaves like Array{Float64,0}() was passed. For convenience dims may also be passed in variadic form.

Examples

julia> ones(Complex128, 2, 3)
2×3 Array{Complex{Float64},2}:
1.0+0.0im  1.0+0.0im  1.0+0.0im
1.0+0.0im  1.0+0.0im  1.0+0.0im

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

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

julia> ones(A)
2×2 Array{Int64,2}:
1  1
1  1

julia> ones(A, Float64)
2×2 Array{Float64,2}:
1.0  1.0
1.0  1.0

julia> ones(A, Bool, (3,))
3-element Array{Bool,1}:
true
true
true

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

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

julia> BitArray(2, 2)
2×2 BitArray{2}:
false  false
false  true

julia> BitArray((3, 1))
3×1 BitArray{2}:
false
true
false
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BitArray(itr)

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

julia> BitArray([1 0; 0 1])
2×2 BitArray{2}:
true  false
false   true

julia> BitArray(x+y == 3 for x = 1:2, y = 1:3)
2×3 BitArray{2}:
false   true  false
true  false  false

julia> BitArray(x+y == 3 for x = 1:2 for y = 1:3)
6-element BitArray{1}:
false
true
false
true
false
false
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Base.truesFunction.
trues(dims)

Create a BitArray with all values set to true.

julia> trues(2,3)
2×3 BitArray{2}:
true  true  true
true  true  true
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trues(A)

Create a BitArray with all values set to true of the same shape as A.

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

julia> trues(A)
2×2 BitArray{2}:
true  true
true  true
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Base.falsesFunction.
falses(dims)

Create a BitArray with all values set to false.

julia> falses(2,3)
2×3 BitArray{2}:
false  false  false
false  false  false
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falses(A)

Create a BitArray with all values set to false of the same shape as A.

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

julia> falses(A)
2×2 BitArray{2}:
false  false
false  false
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Base.fillFunction.
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.

julia> fill(1.0, (5,5))
5×5 Array{Float64,2}:
1.0  1.0  1.0  1.0  1.0
1.0  1.0  1.0  1.0  1.0
1.0  1.0  1.0  1.0  1.0
1.0  1.0  1.0  1.0  1.0
1.0  1.0  1.0  1.0  1.0

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

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Base.fill!Function.
fill!(A, x)

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

Examples

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

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

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

julia> x = 0; f() = (global x += 1; x); fill!(Vector{Int}(3), f())
3-element Array{Int64,1}:
1
1
1
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Base.similarMethod.
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}(dims...).

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

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

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

julia> similar(trues(10,10), 2)
2-element BitArray{1}:
false
false

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

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

Create an uninitialized mutable array analogous to that specified by storagetype, but with indices specified by the last argument. storagetype might be a type or a function.

Examples:

similar(Array{Int}, indices(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}(size(A)), but if A has unconventional indexing then the indices of the result will match A.

similar(BitArray, (indices(A, 2),))

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

similar(dims->zeros(Int, dims), indices(A))

would create an array of Int, initialized to zero, matching the indices of A.

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Base.eyeFunction.
eye([T::Type=Float64,] m::Integer, n::Integer)

m-by-n identity matrix. The default element type is Float64.

Examples

julia> eye(3, 4)
3×4 Array{Float64,2}:
1.0  0.0  0.0  0.0
0.0  1.0  0.0  0.0
0.0  0.0  1.0  0.0

julia> eye(2, 2)
2×2 Array{Float64,2}:
1.0  0.0
0.0  1.0

julia> eye(Int, 2, 2)
2×2 Array{Int64,2}:
1  0
0  1
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eye(m, n)

m-by-n identity matrix.

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eye([T::Type=Float64,] n::Integer)

n-by-n identity matrix. The default element type is Float64.

Examples

julia> eye(Int, 2)
2×2 Array{Int64,2}:
1  0
0  1

julia> eye(2)
2×2 Array{Float64,2}:
1.0  0.0
0.0  1.0
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eye(A)

Constructs an identity matrix of the same dimensions and type as A.

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

julia> eye(A)
3×3 Array{Int64,2}:
1  0  0
0  1  0
0  0  1

Note the difference from ones.

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Base.linspaceFunction.
linspace(start, stop, n=50)

Construct a range of n linearly spaced elements from start to stop.

julia> linspace(1.3,2.9,9)
1.3:0.2:2.9
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Base.logspaceFunction.
logspace(start::Real, stop::Real, n::Integer=50)

Construct a vector of n logarithmically spaced numbers from 10^start to 10^stop.

julia> logspace(1.,10.,5)
5-element Array{Float64,1}:
10.0
1778.28
3.16228e5
5.62341e7
1.0e10
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randsubseq(A, p) -> Vector

Return a vector consisting of a random subsequence of the given array A, where each element of A is included (in order) with independent probability p. (Complexity is linear in p*length(A), so this function is efficient even if p is small and A is large.) Technically, this process is known as "Bernoulli sampling" of A.

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randsubseq!(S, A, p)

Like randsubseq, but the results are stored in S (which is resized as needed).

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## Basic functions

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

Returns the number of dimensions of A.

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

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

Returns a tuple containing the dimensions of A. Optionally you can specify the dimension(s) you want the length of, and get the length of that dimension, or a tuple of the lengths of dimensions you asked for.

julia> A = ones(2,3,4);

julia> size(A, 2)
3

julia> size(A,3,2)
(4, 3)
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Base.indicesMethod.
indices(A)

Returns the tuple of valid indices for array A.

julia> A = ones(5,6,7);

julia> indices(A)
(Base.OneTo(5), Base.OneTo(6), Base.OneTo(7))
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Base.indicesMethod.
indices(A, d)

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

julia> A = ones(5,6,7);

julia> indices(A,2)
Base.OneTo(6)
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Base.lengthMethod.
length(A::AbstractArray) -> Integer

Returns the number of elements in A.

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

julia> length(A)
60
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Base.eachindexFunction.
eachindex(A...)

Creates 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, this returns a specialized Cartesian range to efficiently index into the array with indices specified for every dimension. For other iterables, including strings and dictionaries, this returns an iterator object supporting arbitrary index types (e.g. unevenly spaced or non-integer indices).

Example for a sparse 2-d array:

julia> A = sparse([1, 1, 2], [1, 3, 1], [1, 2, -5])
2×3 SparseMatrixCSC{Int64,Int64} with 3 stored entries:
[1, 1]  =  1
[2, 1]  =  -5
[1, 3]  =  2

julia> for iter in eachindex(A)
@show iter.I[1], iter.I[2]
@show A[iter]
end
(iter.I[1], iter.I[2]) = (1, 1)
A[iter] = 1
(iter.I[1], iter.I[2]) = (2, 1)
A[iter] = -5
(iter.I[1], iter.I[2]) = (1, 2)
A[iter] = 0
(iter.I[1], iter.I[2]) = (2, 2)
A[iter] = 0
(iter.I[1], iter.I[2]) = (1, 3)
A[iter] = 2
(iter.I[1], iter.I[2]) = (2, 3)
A[iter] = 0

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 CartesianRange otherwise). If the arrays have different sizes and/or dimensionalities, eachindex returns an iterable that spans the largest range along each dimension.

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Base.linearindicesFunction.
linearindices(A)

Returns a UnitRange specifying the valid range of indices for A[i] where i is an Int. For arrays with conventional indexing (indices start at 1), or any multidimensional array, this is 1:length(A); however, for one-dimensional arrays with unconventional indices, this is indices(A, 1).

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

julia> A = ones(5,6,7);

julia> b = linearindices(A);

julia> extrema(b)
(1, 210)
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Base.IndexStyleType.
IndexStyle(A)
IndexStyle(typeof(A))

IndexStyle specifies the "native indexing style" for array A. When you define a new AbstractArray type, you can choose to implement either linear indexing or cartesian indexing. If you decide to 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) convert all indexing operations into the preferred style using sub2ind or ind2sub. This allows users to access elements of your array using any indexing style, even when explicit methods have not been provided.

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

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Base.countnzFunction.
countnz(A) -> Integer

Counts the number of nonzero values in array A (dense or sparse). Note that this is not a constant-time operation. For sparse matrices, one should usually use nnz, which returns the number of stored values.

julia> A = [1 2 4; 0 0 1; 1 1 0]
3×3 Array{Int64,2}:
1  2  4
0  0  1
1  1  0

julia> countnz(A)
6
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Base.conj!Function.
conj!(A)

Transform an array to its complex conjugate in-place.

Example

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

julia> conj!(A);

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

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

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

julia> stride(A,2)
3

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

Returns a tuple of the memory strides in each dimension.

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

julia> strides(A)
(1, 3, 12)
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Base.ind2subFunction.
ind2sub(a, index) -> subscripts

Returns a tuple of subscripts into array a corresponding to the linear index index.

julia> A = ones(5,6,7);

julia> ind2sub(A,35)
(5, 1, 2)

julia> ind2sub(A,70)
(5, 2, 3)
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ind2sub(dims, index) -> subscripts

Returns a tuple of subscripts into an array with dimensions dims, corresponding to the linear index index.

Example:

i, j, ... = ind2sub(size(A), indmax(A))

provides the indices of the maximum element.

julia> ind2sub((3,4),2)
(2, 1)

julia> ind2sub((3,4),3)
(3, 1)

julia> ind2sub((3,4),4)
(1, 2)
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Base.sub2indFunction.
sub2ind(dims, i, j, k...) -> index

The inverse of ind2sub, returns the linear index corresponding to the provided subscripts.

julia> sub2ind((5,6,7),1,2,3)
66

julia> sub2ind((5,6,7),1,6,3)
86
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LinAlg.checksquare(A)

Check that a matrix is square, then return its common dimension. For multiple arguments, return a vector.

Example

julia> A = ones(4,4); B = zeros(5,5);

julia> LinAlg.checksquare(A, B)
2-element Array{Int64,1}:
4
5
source

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.

Broadcasts the arrays, tuples, Refs, nullables, and/or scalars As to a container of the appropriate type and dimensions. In this context, anything that is not a subtype of AbstractArray, Ref (except for Ptrs), Tuple, or Nullable is considered a scalar. The resulting container is established by the following rules:

• If all the arguments are scalars, it returns a scalar.

• If the arguments are tuples and zero or more scalars, it returns a tuple.

• If the arguments contain at least one array or Ref, it returns an array (expanding singleton dimensions), and treats Refs as 0-dimensional arrays, and tuples as 1-dimensional arrays.

The following additional rule applies to Nullable arguments: If there is at least one Nullable, and all the arguments are scalars or Nullable, it returns a Nullable treating Nullables as "containers".

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.

julia> A = [1, 2, 3, 4, 5]
5-element Array{Int64,1}:
1
2
3
4
5

julia> B = [1 2; 3 4; 5 6; 7 8; 9 10]
5×2 Array{Int64,2}:
1   2
3   4
5   6
7   8
9  10

5×2 Array{Int64,2}:
2   3
5   6
8   9
11  12
14  15

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

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

(1.0, -1.0)

julia> broadcast(+, 1.0, (0, -2.0), Ref(1))
2-element Array{Float64,1}:
2.0
0.0

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

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

julia> Nullable("X") .* "Y"
Nullable{String}("XY")

Nullable{Float64}(0.5)

julia> (1 + im) ./ Nullable{Int}()
Nullable{Complex{Float64}}()
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Like broadcast, but store the result of broadcast(f, As...) in the dest array. Note that dest is only used to store the result, and does not supply arguments to f unless it is also listed in the As, as in broadcast!(f, A, A, B) to perform A[:] = broadcast(f, A, B).

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@. expr

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

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

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

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

julia> @. y = x + 3 * sin(x)
3-element Array{Float64,1}:
3.52441
4.72789
3.42336
source

Broadcasts the inds arrays to a common size like broadcast and returns an array of the results A[ks...], where ks goes over the positions in the broadcast result A.

julia> A = [1, 2, 3, 4, 5]
5-element Array{Int64,1}:
1
2
3
4
5

julia> B = [1 2; 3 4; 5 6; 7 8; 9 10]
5×2 Array{Int64,2}:
1   2
3   4
5   6
7   8
9  10

5×2 Array{Int64,2}:
2   3
5   6
8   9
11  12
14  15

3-element Array{Int64,1}:
2
5
15
source

Broadcasts the X and inds arrays to a common size and stores the value from each position in X at the indices in A given by the same positions in inds.

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## Indexing and assignment

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

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

Examples

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

julia> getindex(A, 1)
1

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

julia> getindex(A, 2:4)
3-element Array{Int64,1}:
3
2
4
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Base.setindex!Method.
setindex!(A, X, inds...)

Store values from array X within some subset of A as specified by inds.

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Base.copy!Method.
copy!(dest, Rdest::CartesianRange, src, Rsrc::CartesianRange) -> 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.

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Base.isassignedFunction.
isassigned(array, i) -> Bool

Tests whether the given array has a value associated with index i. Returns false if the index is out of bounds, or has an undefined reference.

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 Array{Foo,1}:
#undef
#undef
#undef

julia> isassigned(v, 1)
false
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Base.ColonType.
Colon()

Colons (:) are used to signify indexing entire objects or dimensions at once.

Very few operations are defined on Colons directly; instead they are converted by to_indices to an internal vector type (Base.Slice) to represent the collection of indices they span before being used.

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

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

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

Example

julia> A = reshape(collect(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
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CartesianRange(Istart::CartesianIndex, Istop::CartesianIndex) -> R
CartesianRange(sz::Dims) -> R
CartesianRange(istart:istop, jstart: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:jstop
for i = istart:istop
...
end
end

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

julia> foreach(println, CartesianRange((2, 2, 2)))
CartesianIndex{3}((1, 1, 1))
CartesianIndex{3}((2, 1, 1))
CartesianIndex{3}((1, 2, 1))
CartesianIndex{3}((2, 2, 1))
CartesianIndex{3}((1, 1, 2))
CartesianIndex{3}((2, 1, 2))
CartesianIndex{3}((1, 2, 2))
CartesianIndex{3}((2, 2, 2))
source
Base.to_indicesFunction.
to_indices(A, I::Tuple)

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

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

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

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

source
Base.checkboundsFunction.
checkbounds(Bool, A, I...)

Return true if the specified indices I are in bounds for the given array A. Subtypes of AbstractArray should specialize this method if they need to provide custom bounds checking behaviors; however, in many cases one can rely on A's indices and checkindex.

julia> A = rand(3, 3);

julia> checkbounds(Bool, A, 2)
true

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

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

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

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

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

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

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

julia> checkindex(Bool,1:20,21)
false
source

## Views (SubArrays and other view types)

Base.viewFunction.
view(A, inds...)

Like getindex, but returns a view into the parent array A with the given indices instead of making a copy. Calling getindex or setindex! on the returned SubArray computes the indices to the parent array on the fly without checking bounds.

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

julia> b = view(A, :, 1)
2-element SubArray{Int64,1,Array{Int64,2},Tuple{Base.Slice{Base.OneTo{Int64}},Int64},true}:
1
3

julia> fill!(b, 0)
2-element SubArray{Int64,1,Array{Int64,2},Tuple{Base.Slice{Base.OneTo{Int64}},Int64},true}:
0
0

julia> A # Note A has changed even though we modified b
2×2 Array{Int64,2}:
0  2
0  4
source
Base.@viewMacro.
@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 @views to switch an entire block of code to use views for slicing.

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

julia> b = @view A[:, 1]
2-element SubArray{Int64,1,Array{Int64,2},Tuple{Base.Slice{Base.OneTo{Int64}},Int64},true}:
1
3

julia> fill!(b, 0)
2-element SubArray{Int64,1,Array{Int64,2},Tuple{Base.Slice{Base.OneTo{Int64}},Int64},true}:
0
0

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

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

Note that 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.

source
Base.parentFunction.
parent(A)

Returns the "parent array" of an array view type (e.g., SubArray), or the array itself if it is not a view.

source
Base.parentindexesFunction.
parentindexes(A)

From an array view A, returns the corresponding indexes in the parent.

source
Base.slicedimFunction.
slicedim(A, d::Integer, i)

Return all the data of A where the index for dimension d equals i. Equivalent to A[:,:,...,i,:,:,...] where i is in position d.

Example

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

julia> slicedim(A,2,3)
2-element Array{Int64,1}:
3
7
source
Base.reinterpretFunction.
reinterpret(type, A)

Change the type-interpretation of a block of memory. For arrays, this constructs an 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.

Warning

It is not allowed to reinterpret an array to an element type with a larger alignment then the alignment of the array. For a normal Array, this is the alignment of its element type. For a reinterpreted array, this is the alignment of the Array it was reinterpreted from. For example, reinterpret(UInt32, UInt8[0, 0, 0, 0]) is not allowed but reinterpret(UInt32, reinterpret(UInt8, Float32[1.0])) is allowed.

Examples

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

julia> reinterpret(Float32, UInt32[1 2 3 4 5])
1×5 Array{Float32,2}:
1.4013f-45  2.8026f-45  4.2039f-45  5.60519f-45  7.00649f-45
source
Base.reshapeFunction.
reshape(A, dims...) -> R
reshape(A, dims) -> R

Return an array R 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 setting elements of R alters the values of A and vice versa.

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.

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

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

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

Example

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

julia> squeeze(a,3)
2×2×1 Array{Int64,3}:
[:, :, 1] =
1  3
2  4
source
Base.vecFunction.
vec(a::AbstractArray) -> Vector

Reshape the array a as a one-dimensional column vector. The resulting array shares the same underlying data as a, so modifying one will also modify the other.

Example

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

julia> vec(a)
6-element Array{Int64,1}:
1
4
2
5
3
6

source

## Concatenation and permutation

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

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([1,2], matrices...) 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
Base.vcatFunction.
vcat(A...)

Concatenate along dimension 1.

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

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

julia> vcat(a,b)
3×5 Array{Int64,2}:
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 Array{Int64,2}:
1  2  3
4  5  6
source
Base.hcatFunction.
hcat(A...)

Concatenate along dimension 2.

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

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

julia> hcat(a,b)
5×3 Array{Int64,2}:
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 Array{Int64,2}:
1  4
2  5
3  6
source
Base.hvcatFunction.
hvcat(rows::Tuple{Vararg{Int}}, values...)

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

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 Array{Int64,2}:
1  2  3
4  5  6

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

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

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

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

source
Base.flipdimFunction.
flipdim(A, d::Integer)

Reverse A in dimension d.

Example

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

julia> flipdim(b,2)
2×2 Array{Int64,2}:
2  1
4  3
source
Base.circshiftFunction.
circshift(A, shifts)

Circularly shift the data in an array. The second argument is a vector giving the amount to shift in each dimension.

Example

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

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

source
Base.circshift!Function.
circshift!(dest, src, shifts)

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

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

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

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

Example

julia> src = reshape(collect(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}((0:3,2:5))

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

julia> dest[1:3,2:4] == src[1:3,2:4]
true
source
Base.containsMethod.
contains(fun, itr, x) -> Bool

Returns true if there is at least one element y in itr such that fun(y,x) is true.

julia> vec = [10, 100, 200]
3-element Array{Int64,1}:
10
100
200

julia> contains(==, vec, 200)
true

julia> contains(==, vec, 300)
false

julia> contains(>, vec, 100)
true

julia> contains(>, vec, 200)
false
source
Base.findMethod.
find(A)

Return a vector of the linear indexes of the non-zeros in A (determined by A[i]!=0). A common use of this is to convert a boolean array to an array of indexes of the true elements. If there are no non-zero elements of A, find returns an empty array.

Examples

julia> A = [true false; false true]
2×2 Array{Bool,2}:
true  false
false   true

julia> find(A)
2-element Array{Int64,1}:
1
4

julia> find(zeros(3))
0-element Array{Int64,1}
source
Base.findMethod.
find(f::Function, A)

Return a vector I of the linear indexes of A where f(A[I]) returns true. If there are no such elements of A, find returns an empty array.

Examples

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

julia> find(isodd,A)
2-element Array{Int64,1}:
1
2

julia> find(isodd, [2, 4])
0-element Array{Int64,1}
source
Base.findnFunction.
findn(A)

Return a vector of indexes for each dimension giving the locations of the non-zeros in A (determined by A[i]!=0). If there are no non-zero elements of A, findn returns a 2-tuple of empty arrays.

Examples

julia> A = [1 2 0; 0 0 3; 0 4 0]
3×3 Array{Int64,2}:
1  2  0
0  0  3
0  4  0

julia> findn(A)
([1, 1, 3, 2], [1, 2, 2, 3])

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

julia> findn(A)
(Int64[], Int64[])
source
Base.findnzFunction.
findnz(A)

Return a tuple (I, J, V) where I and J are the row and column indexes of the non-zero values in matrix A, and V is a vector of the non-zero values.

Example

julia> A = [1 2 0; 0 0 3; 0 4 0]
3×3 Array{Int64,2}:
1  2  0
0  0  3
0  4  0

julia> findnz(A)
([1, 1, 3, 2], [1, 2, 2, 3], [1, 2, 4, 3])
source
Base.findfirstMethod.
findfirst(A)

Return the linear index of the first non-zero value in A (determined by A[i]!=0). Returns 0 if no such value is found.

Examples

julia> A = [0 0; 1 0]
2×2 Array{Int64,2}:
0  0
1  0

julia> findfirst(A)
2

julia> findfirst(zeros(3))
0
source
Base.findfirstMethod.
findfirst(A, v)

Return the linear index of the first element equal to v in A. Returns 0 if v is not found.

Examples

julia> A = [4 6; 2 2]
2×2 Array{Int64,2}:
4  6
2  2

julia> findfirst(A,2)
2

julia> findfirst(A,3)
0
source
Base.findfirstMethod.
findfirst(predicate::Function, A)

Return the linear index of the first element of A for which predicate returns true. Returns 0 if there is no such element.

Examples

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

julia> findfirst(iseven, A)
2

julia> findfirst(x -> x>10, A)
0
source
Base.findlastMethod.
findlast(A)

Return the linear index of the last non-zero value in A (determined by A[i]!=0). Returns 0 if there is no non-zero value in A.

Examples

julia> A = [1 0; 1 0]
2×2 Array{Int64,2}:
1  0
1  0

julia> findlast(A)
2

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

julia> findlast(A)
0
source
Base.findlastMethod.
findlast(A, v)

Return the linear index of the last element equal to v in A. Returns 0 if there is no element of A equal to v.

Examples

julia> A = [1 2; 2 1]
2×2 Array{Int64,2}:
1  2
2  1

julia> findlast(A,1)
4

julia> findlast(A,2)
3

julia> findlast(A,3)
0
source
Base.findlastMethod.
findlast(predicate::Function, A)

Return the linear index of the last element of A for which predicate returns true. Returns 0 if there is no such element.

Examples

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

julia> findlast(isodd, A)
2

julia> findlast(x -> x > 5, A)
0
source
Base.findnextMethod.
findnext(A, i::Integer)

Find the next linear index >= i of a non-zero element of A, or 0 if not found.

Examples

julia> A = [0 0; 1 0]
2×2 Array{Int64,2}:
0  0
1  0

julia> findnext(A,1)
2

julia> findnext(A,3)
0
source
Base.findnextMethod.
findnext(predicate::Function, A, i::Integer)

Find the next linear index >= i of an element of A for which predicate returns true, or 0 if not found.

Examples

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

julia> findnext(isodd, A, 1)
1

julia> findnext(isodd, A, 2)
0
source
Base.findnextMethod.
findnext(A, v, i::Integer)

Find the next linear index >= i of an element of A equal to v (using ==), or 0 if not found.

Examples

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

julia> findnext(A,4,4)
0

julia> findnext(A,4,3)
3
source
Base.findprevMethod.
findprev(A, i::Integer)

Find the previous linear index <= i of a non-zero element of A, or 0 if not found.

Examples

julia> A = [0 0; 1 2]
2×2 Array{Int64,2}:
0  0
1  2

julia> findprev(A,2)
2

julia> findprev(A,1)
0
source
Base.findprevMethod.
findprev(predicate::Function, A, i::Integer)

Find the previous linear index <= i of an element of A for which predicate returns true, or 0 if not found.

Examples

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

julia> findprev(isodd, A, 1)
0

julia> findprev(isodd, A, 3)
2
source
Base.findprevMethod.
findprev(A, v, i::Integer)

Find the previous linear index <= i of an element of A equal to v (using ==), or 0 if not found.

Examples

julia> A = [0 0; 1 2]
2×2 Array{Int64,2}:
0  0
1  2

julia> findprev(A, 1, 4)
2

julia> findprev(A, 1, 1)
0
source
Base.permutedimsFunction.
permutedims(A, perm)

Permute the dimensions of array A. perm is a vector specifying a permutation of length ndims(A). This is a generalization of transpose for multi-dimensional arrays. Transpose is equivalent to permutedims(A, [2,1]).

Example

julia> A = reshape(collect(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
source
Base.permutedims!Function.
permutedims!(dest, src, perm)

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

source
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).

Example

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

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

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

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

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

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

julia> b = ones(3,4);

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

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

## Array functions

Base.accumulateMethod.
accumulate(op, A, dim=1)

Cumulative operation op along a dimension dim (defaults to 1). See also accumulate! to use a preallocated output array, both for performance and to control the precision of the output (e.g. to avoid overflow). For common operations there are specialized variants of accumulate, see: cumsum, cumprod

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

julia> accumulate(*, [1,2,3])
3-element Array{Int64,1}:
1
2
6
source
accumulate(op, v0, A)

Like accumulate, but using a starting element v0. The first entry of the result will be op(v0, first(A)). For example:

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

julia> accumulate(min, 0, [1,2,-1])
3-element Array{Int64,1}:
0
0
-1
source
Base.accumulate!Function.
accumulate!(op, B, A, dim=1)

Cumulative operation op on A along a dimension, storing the result in B. The dimension defaults to 1. See also accumulate.

source
Base.cumprodFunction.
cumprod(A, dim=1)

Cumulative product along a dimension dim (defaults to 1). See also cumprod! to use a preallocated output array, both for performance and to control the precision of the output (e.g. to avoid overflow).

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

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

julia> cumprod(a,2)
2×3 Array{Int64,2}:
1   2    6
4  20  120
source
Base.cumprod!Function.
cumprod!(B, A, dim::Integer=1)

Cumulative product of A along a dimension, storing the result in B. The dimension defaults to 1. See also cumprod.

source
Base.cumsumFunction.
cumsum(A, dim=1)

Cumulative sum along a dimension dim (defaults to 1). See also cumsum! to use a preallocated output array, both for performance and to control the precision of the output (e.g. to avoid overflow).

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

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

julia> cumsum(a,2)
2×3 Array{Int64,2}:
1  3   6
4  9  15
source
Base.cumsum!Function.
cumsum!(B, A, dim::Integer=1)

Cumulative sum of A along a dimension, storing the result in B. The dimension defaults to 1. See also cumsum.

source
Base.cumsum_kbnFunction.
cumsum_kbn(A, [dim::Integer=1])

Cumulative sum along a dimension, using the Kahan-Babuska-Neumaier compensated summation algorithm for additional accuracy. The dimension defaults to 1.

source
Base.LinAlg.diffFunction.
diff(A, [dim::Integer=1])

Finite difference operator of matrix or vector A. If A is a matrix, compute the finite difference over a dimension dim (default 1).

Example

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

julia> diff(a,2)
2×1 Array{Int64,2}:
2
10
source

Compute differences along vector F, using h as the spacing between points. The default spacing is one.

Example

julia> a = [2,4,6,8];

4-element Array{Float64,1}:
2.0
2.0
2.0
2.0
source
Base.rot180Function.
rot180(A)

Rotate matrix A 180 degrees.

Example

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

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

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

Examples

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

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

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

Rotate matrix A left 90 degrees.

Example

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

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

Rotate matrix A left 90 degrees an integer k number of times. If k is zero or a multiple of four, this is equivalent to a copy.

Examples

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

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

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

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

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

Rotate matrix A right 90 degrees.

Example

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

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

Rotate matrix A right 90 degrees an integer k number of times. If k is zero or a multiple of four, this is equivalent to a copy.

Examples

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

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

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

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

julia> rotr90(a,4)
2×2 Array{Int64,2}:
1  2
3  4
source
Base.reducedimFunction.
reducedim(f, A, region[, v0])

Reduce 2-argument function f along dimensions of A. region is a vector specifying the dimensions to reduce, and v0 is the initial value to use in the reductions. For +, *, max and min the v0 argument is optional.

The associativity of the reduction is implementation-dependent; if you need a particular associativity, e.g. left-to-right, you should write your own loop. See documentation for reduce.

Examples

julia> a = reshape(collect(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> reducedim(max, a, 2)
4×1 Array{Int64,2}:
13
14
15
16

julia> reducedim(max, a, 1)
1×4 Array{Int64,2}:
4  8  12  16
source
Base.mapreducedimFunction.
mapreducedim(f, op, A, region[, v0])

Evaluates to the same as reducedim(op, map(f, A), region, f(v0)), but is generally faster because the intermediate array is avoided.

Examples

julia> a = reshape(collect(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> mapreducedim(isodd, *, a, 1)
1×4 Array{Bool,2}:
false  false  false  false

julia> mapreducedim(isodd, |, a, 1, true)
1×4 Array{Bool,2}:
true  true  true  true
source
Base.mapslicesFunction.
mapslices(f, A, dims)

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

Examples

julia> a = reshape(collect(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, [1,2])
1×1×2×2 Array{Int64,4}:
[:, :, 1, 1] =
10

[:, :, 2, 1] =
26

[:, :, 1, 2] =
42

[:, :, 2, 2] =
58
source
Base.sum_kbnFunction.
sum_kbn(A)

Returns the sum of all elements of A, using the Kahan-Babuska-Neumaier compensated summation algorithm for additional accuracy.

source

## Combinatorics

Base.Random.randpermFunction.
randperm([rng=GLOBAL_RNG,] n::Integer)

Construct a random permutation of length n. The optional rng argument specifies a random number generator (see Random Numbers). To randomly permute a arbitrary vector, see shuffle or shuffle!.

Example

julia> rng = MersenneTwister(1234);

julia> randperm(rng, 4)
4-element Array{Int64,1}:
2
1
4
3
source
Base.invpermFunction.
invperm(v)

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

Example

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

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

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

julia> B = A[v]
4-element Array{Char,1}:
'b'
'd'
'c'
'a'

julia> B[invperm(v)]
4-element Array{Char,1}:
'a'
'b'
'c'
'd'
source
Base.ispermFunction.
isperm(v) -> Bool

Returns true if v is a valid permutation.

Examples

julia> isperm([1; 2])
true

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

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

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

Example

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

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

julia> permute!(A, perm);

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

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

Example

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

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

julia> ipermute!(A, perm);

julia> A
4-element Array{Int64,1}:
4
1
3
1
source
Base.Random.randcycleFunction.
randcycle([rng=GLOBAL_RNG,] n::Integer)

Construct a random cyclic permutation of length n. The optional rng argument specifies a random number generator, see Random Numbers.

Example

julia> rng = MersenneTwister(1234);

julia> randcycle(rng, 6)
6-element Array{Int64,1}:
3
5
4
6
1
2
source
Base.Random.shuffleFunction.
shuffle([rng=GLOBAL_RNG,] v)

Return a randomly permuted copy of v. The optional rng argument specifies a random number generator (see Random Numbers). To permute v in-place, see shuffle!. To obtain randomly permuted indices, see randperm.

Example

julia> rng = MersenneTwister(1234);

julia> shuffle(rng, collect(1:10))
10-element Array{Int64,1}:
6
1
10
2
3
9
5
7
4
8
source
Base.Random.shuffle!Function.
shuffle!([rng=GLOBAL_RNG,] v)

In-place version of shuffle: randomly permute the array v in-place, optionally supplying the random-number generator rng.

Example

julia> rng = MersenneTwister(1234);

julia> shuffle!(rng, collect(1:16))
16-element Array{Int64,1}:
2
15
5
14
1
9
10
6
11
3
16
7
4
12
8
13
source
Base.reverseFunction.
reverse(v [, start=1 [, stop=length(v) ]] )

Return a copy of v reversed from start to stop.

Examples

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

julia> reverse(A)
5-element Array{Int64,1}:
5
4
3
2
1

julia> reverse(A, 1, 4)
5-element Array{Int64,1}:
4
3
2
1
5

julia> reverse(A, 3, 5)
5-element Array{Int64,1}:
1
2
5
4
3
source
Base.reverseindFunction.
reverseind(v, i)

Given an index i in reverse(v), return the corresponding index in v so that v[reverseind(v,i)] == reverse(v)[i]. (This can be nontrivial in the case where v is a Unicode string.)

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

In-place version of reverse.

source

## BitArrays

BitArrays are space-efficient "packed" boolean arrays, which store one bit per boolean value. They can be used similarly to Array{Bool} arrays (which store one byte per boolean value), and can be converted to/from the latter via Array(bitarray) and BitArray(array), respectively.

Base.flipbits!Function.
flipbits!(B::BitArray{N}) -> BitArray{N}

Performs a bitwise not operation on B. See ~.

Example

julia> A = trues(2,2)
2×2 BitArray{2}:
true  true
true  true

julia> flipbits!(A)
2×2 BitArray{2}:
false  false
false  false
source
Base.rol!Function.
rol!(dest::BitVector, src::BitVector, i::Integer) -> BitVector

Performs a left rotation operation on src and puts the result into dest. i controls how far to rotate the bits.

source
rol!(B::BitVector, i::Integer) -> BitVector

Performs a left rotation operation in-place on B. i controls how far to rotate the bits.

source
Base.rolFunction.
rol(B::BitVector, i::Integer) -> BitVector

Performs a left rotation operation, returning a new BitVector. i controls how far to rotate the bits. See also rol!.

Examples

julia> A = BitArray([true, true, false, false, true])
5-element BitArray{1}:
true
true
false
false
true

julia> rol(A,1)
5-element BitArray{1}:
true
false
false
true
true

julia> rol(A,2)
5-element BitArray{1}:
false
false
true
true
true

julia> rol(A,5)
5-element BitArray{1}:
true
true
false
false
true
source
Base.ror!Function.
ror!(dest::BitVector, src::BitVector, i::Integer) -> BitVector

Performs a right rotation operation on src and puts the result into dest. i controls how far to rotate the bits.

source
ror!(B::BitVector, i::Integer) -> BitVector

Performs a right rotation operation in-place on B. i controls how far to rotate the bits.

source
Base.rorFunction.
ror(B::BitVector, i::Integer) -> BitVector

Performs a right rotation operation on B, returning a new BitVector. i controls how far to rotate the bits. See also ror!.

Examples

julia> A = BitArray([true, true, false, false, true])
5-element BitArray{1}:
true
true
false
false
true

julia> ror(A,1)
5-element BitArray{1}:
true
true
true
false
false

julia> ror(A,2)
5-element BitArray{1}:
false
true
true
true
false

julia> ror(A,5)
5-element BitArray{1}:
true
true
false
false
true
source

## Sparse Vectors and Matrices

Sparse vectors and matrices largely support the same set of operations as their dense counterparts. The following functions are specific to sparse arrays.

SparseVector{Tv,Ti<:Integer} <: AbstractSparseVector{Tv,Ti}

Vector type for storing sparse vectors.

source
SparseMatrixCSC{Tv,Ti<:Integer} <: AbstractSparseMatrix{Tv,Ti}

Matrix type for storing sparse matrices in the Compressed Sparse Column format.

source
sparse(A)

Convert an AbstractMatrix A into a sparse matrix.

Example

julia> A = eye(3)
3×3 Array{Float64,2}:
1.0  0.0  0.0
0.0  1.0  0.0
0.0  0.0  1.0

julia> sparse(A)
3×3 SparseMatrixCSC{Float64,Int64} with 3 stored entries:
[1, 1]  =  1.0
[2, 2]  =  1.0
[3, 3]  =  1.0
source
sparse(I, J, V,[ m, n, combine])

Create a sparse matrix S of dimensions m x n such that S[I[k], J[k]] = V[k]. The combine function is used to combine duplicates. If m and n are not specified, they are set to maximum(I) and maximum(J) respectively. If the combine function is not supplied, combine defaults to + unless the elements of V are Booleans in which case combine defaults to |. All elements of I must satisfy 1 <= I[k] <= m, and all elements of J must satisfy 1 <= J[k] <= n. Numerical zeros in (I, J, V) are retained as structural nonzeros; to drop numerical zeros, use dropzeros!.

For additional documentation and an expert driver, see Base.SparseArrays.sparse!.

Example

julia> Is = [1; 2; 3];

julia> Js = [1; 2; 3];

julia> Vs = [1; 2; 3];

julia> sparse(Is, Js, Vs)
3×3 SparseMatrixCSC{Int64,Int64} with 3 stored entries:
[1, 1]  =  1
[2, 2]  =  2
[3, 3]  =  3
source
sparsevec(I, V, [m, combine])

Create a sparse vector S of length m such that S[I[k]] = V[k]. Duplicates are combined using the combine function, which defaults to + if no combine argument is provided, unless the elements of V are Booleans in which case combine defaults to |.

julia> II = [1, 3, 3, 5]; V = [0.1, 0.2, 0.3, 0.2];

julia> sparsevec(II, V)
5-element SparseVector{Float64,Int64} with 3 stored entries:
[1]  =  0.1
[3]  =  0.5
[5]  =  0.2

julia> sparsevec(II, V, 8, -)
8-element SparseVector{Float64,Int64} with 3 stored entries:
[1]  =  0.1
[3]  =  -0.1
[5]  =  0.2

julia> sparsevec([1, 3, 1, 2, 2], [true, true, false, false, false])
3-element SparseVector{Bool,Int64} with 3 stored entries:
[1]  =  true
[2]  =  false
[3]  =  true
source
sparsevec(d::Dict, [m])

Create a sparse vector of length m where the nonzero indices are keys from the dictionary, and the nonzero values are the values from the dictionary.

julia> sparsevec(Dict(1 => 3, 2 => 2))
2-element SparseVector{Int64,Int64} with 2 stored entries:
[1]  =  3
[2]  =  2
source
sparsevec(A)

Convert a vector A into a sparse vector of length m.

Example

julia> sparsevec([1.0, 2.0, 0.0, 0.0, 3.0, 0.0])
6-element SparseVector{Float64,Int64} with 3 stored entries:
[1]  =  1.0
[2]  =  2.0
[5]  =  3.0
source
issparse(S)

Returns true if S is sparse, and false otherwise.

source
Base.fullFunction.
full(S)

Convert a sparse matrix or vector S into a dense matrix or vector.

Example

julia> A = speye(3)
3×3 SparseMatrixCSC{Float64,Int64} with 3 stored entries:
[1, 1]  =  1.0
[2, 2]  =  1.0
[3, 3]  =  1.0

julia> full(A)
3×3 Array{Float64,2}:
1.0  0.0  0.0
0.0  1.0  0.0
0.0  0.0  1.0
source
Base.SparseArrays.nnzFunction.
nnz(A)

Returns the number of stored (filled) elements in a sparse array.

Example

julia> A = speye(3)
3×3 SparseMatrixCSC{Float64,Int64} with 3 stored entries:
[1, 1]  =  1.0
[2, 2]  =  1.0
[3, 3]  =  1.0

julia> nnz(A)
3
source
spzeros([type,]m[,n])

Create a sparse vector of length m or sparse matrix of size m x n. This sparse array will not contain any nonzero values. No storage will be allocated for nonzero values during construction. The type defaults to Float64 if not specified.

Examples

julia> spzeros(3, 3)
3×3 SparseMatrixCSC{Float64,Int64} with 0 stored entries

julia> spzeros(Float32, 4)
4-element SparseVector{Float32,Int64} with 0 stored entries
source
spones(S)

Create a sparse array with the same structure as that of S, but with every nonzero element having the value 1.0.

Example

julia> A = sparse([1,2,3,4],[2,4,3,1],[5.,4.,3.,2.])
4×4 SparseMatrixCSC{Float64,Int64} with 4 stored entries:
[4, 1]  =  2.0
[1, 2]  =  5.0
[3, 3]  =  3.0
[2, 4]  =  4.0

julia> spones(A)
4×4 SparseMatrixCSC{Float64,Int64} with 4 stored entries:
[4, 1]  =  1.0
[1, 2]  =  1.0
[3, 3]  =  1.0
[2, 4]  =  1.0

Note the difference from speye.

source
speye([type,]m[,n])

Create a sparse identity matrix of size m x m. When n is supplied, create a sparse identity matrix of size m x n. The type defaults to Float64 if not specified.

sparse(I, m, n) is equivalent to speye(Int, m, n), and sparse(α*I, m, n) can be used to efficiently create a sparse multiple α of the identity matrix.

source
speye(S)

Create a sparse identity matrix with the same size as S.

Example

julia> A = sparse([1,2,3,4],[2,4,3,1],[5.,4.,3.,2.])
4×4 SparseMatrixCSC{Float64,Int64} with 4 stored entries:
[4, 1]  =  2.0
[1, 2]  =  5.0
[3, 3]  =  3.0
[2, 4]  =  4.0

julia> speye(A)
4×4 SparseMatrixCSC{Float64,Int64} with 4 stored entries:
[1, 1]  =  1.0
[2, 2]  =  1.0
[3, 3]  =  1.0
[4, 4]  =  1.0

Note the difference from spones.

source
speye([type,]m[,n])

Create a sparse identity matrix of size m x m. When n is supplied, create a sparse identity matrix of size m x n. The type defaults to Float64 if not specified.

sparse(I, m, n) is equivalent to speye(Int, m, n), and sparse(α*I, m, n) can be used to efficiently create a sparse multiple α of the identity matrix.

source
spdiagm(B, d[, m, n])

Construct a sparse diagonal matrix. B is a tuple of vectors containing the diagonals and d is a tuple containing the positions of the diagonals. In the case the input contains only one diagonal, B can be a vector (instead of a tuple) and d can be the diagonal position (instead of a tuple), defaulting to 0 (diagonal). Optionally, m and n specify the size of the resulting sparse matrix.

Example

julia> spdiagm(([1,2,3,4],[4,3,2,1]),(-1,1))
5×5 SparseMatrixCSC{Int64,Int64} with 8 stored entries:
[2, 1]  =  1
[1, 2]  =  4
[3, 2]  =  2
[2, 3]  =  3
[4, 3]  =  3
[3, 4]  =  2
[5, 4]  =  4
[4, 5]  =  1
source
sprand([rng],[type],m,[n],p::AbstractFloat,[rfn])

Create a random length m sparse vector or m by n sparse matrix, in which the probability of any element being nonzero is independently given by p (and hence the mean density of nonzeros is also exactly p). Nonzero values are sampled from the distribution specified by rfn and have the type type. The uniform distribution is used in case rfn is not specified. The optional rng argument specifies a random number generator, see Random Numbers.

Example

julia> rng = MersenneTwister(1234);

julia> sprand(rng, Bool, 2, 2, 0.5)
2×2 SparseMatrixCSC{Bool,Int64} with 2 stored entries:
[1, 1]  =  true
[2, 1]  =  true

julia> sprand(rng, Float64, 3, 0.75)
3-element SparseVector{Float64,Int64} with 1 stored entry:
[3]  =  0.298614
source
sprandn([rng], m[,n],p::AbstractFloat)

Create a random sparse vector of length m or sparse matrix of size m by n with the specified (independent) probability p of any entry being nonzero, where nonzero values are sampled from the normal distribution. The optional rng argument specifies a random number generator, see Random Numbers.

Example

julia> rng = MersenneTwister(1234);

julia> sprandn(rng, 2, 2, 0.75)
2×2 SparseMatrixCSC{Float64,Int64} with 3 stored entries:
[1, 1]  =  0.532813
[2, 1]  =  -0.271735
[2, 2]  =  0.502334
source
nonzeros(A)

Return a vector of the structural nonzero values in sparse array A. This includes zeros that are explicitly stored in the sparse array. The returned vector points directly to the internal nonzero storage of A, and any modifications to the returned vector will mutate A as well. See rowvals and nzrange.

Example

julia> A = speye(3)
3×3 SparseMatrixCSC{Float64,Int64} with 3 stored entries:
[1, 1]  =  1.0
[2, 2]  =  1.0
[3, 3]  =  1.0

julia> nonzeros(A)
3-element Array{Float64,1}:
1.0
1.0
1.0
source
rowvals(A::SparseMatrixCSC)

Return a vector of the row indices of A. Any modifications to the returned vector will mutate A as well. Providing access to how the row indices are stored internally can be useful in conjunction with iterating over structural nonzero values. See also nonzeros and nzrange.

Example

julia> A = speye(3)
3×3 SparseMatrixCSC{Float64,Int64} with 3 stored entries:
[1, 1]  =  1.0
[2, 2]  =  1.0
[3, 3]  =  1.0

julia> rowvals(A)
3-element Array{Int64,1}:
1
2
3
source
nzrange(A::SparseMatrixCSC, col::Integer)

Return the range of indices to the structural nonzero values of a sparse matrix column. In conjunction with nonzeros and rowvals, this allows for convenient iterating over a sparse matrix :

A = sparse(I,J,V)
rows = rowvals(A)
vals = nonzeros(A)
m, n = size(A)
for i = 1:n
for j in nzrange(A, i)
row = rows[j]
val = vals[j]
# perform sparse wizardry...
end
end
source
dropzeros!(A::SparseMatrixCSC, trim::Bool = true)

Removes stored numerical zeros from A, optionally trimming resulting excess space from A.rowval and A.nzval when trim is true.

For an out-of-place version, see dropzeros. For algorithmic information, see fkeep!.

source
dropzeros(A::SparseMatrixCSC, trim::Bool = true)

Generates a copy of A and removes stored numerical zeros from that copy, optionally trimming excess space from the result's rowval and nzval arrays when trim is true.

For an in-place version and algorithmic information, see dropzeros!.

Example

julia> A = sparse([1, 2, 3], [1, 2, 3], [1.0, 0.0, 1.0])
3×3 SparseMatrixCSC{Float64,Int64} with 3 stored entries:
[1, 1]  =  1.0
[2, 2]  =  0.0
[3, 3]  =  1.0

julia> dropzeros(A)
3×3 SparseMatrixCSC{Float64,Int64} with 2 stored entries:
[1, 1]  =  1.0
[3, 3]  =  1.0
source
dropzeros!(x::SparseVector, trim::Bool = true)

Removes stored numerical zeros from x, optionally trimming resulting excess space from x.nzind and x.nzval when trim is true.

For an out-of-place version, see dropzeros. For algorithmic information, see fkeep!.

source
dropzeros(x::SparseVector, trim::Bool = true)

Generates a copy of x and removes numerical zeros from that copy, optionally trimming excess space from the result's nzind and nzval arrays when trim is true.

For an in-place version and algorithmic information, see dropzeros!.

Example

julia> A = sparsevec([1, 2, 3], [1.0, 0.0, 1.0])
3-element SparseVector{Float64,Int64} with 3 stored entries:
[1]  =  1.0
[2]  =  0.0
[3]  =  1.0

julia> dropzeros(A)
3-element SparseVector{Float64,Int64} with 2 stored entries:
[1]  =  1.0
[3]  =  1.0
source
permute{Tv,Ti}(A::SparseMatrixCSC{Tv,Ti}, p::AbstractVector{<:Integer},
q::AbstractVector{<:Integer})

Bilaterally permute A, returning PAQ (A[p,q]). Column-permutation q's length must match A's column count (length(q) == A.n). Row-permutation p's length must match A's row count (length(p) == A.m).

For expert drivers and additional information, see permute!.

Example

julia> A = spdiagm([1, 2, 3, 4], 0, 4, 4) + spdiagm([5, 6, 7], 1, 4, 4)
4×4 SparseMatrixCSC{Int64,Int64} with 7 stored entries:
[1, 1]  =  1
[1, 2]  =  5
[2, 2]  =  2
[2, 3]  =  6
[3, 3]  =  3
[3, 4]  =  7
[4, 4]  =  4

julia> permute(A, [4, 3, 2, 1], [1, 2, 3, 4])
4×4 SparseMatrixCSC{Int64,Int64} with 7 stored entries:
[4, 1]  =  1
[3, 2]  =  2
[4, 2]  =  5
[2, 3]  =  3
[3, 3]  =  6
[1, 4]  =  4
[2, 4]  =  7

julia> permute(A, [1, 2, 3, 4], [4, 3, 2, 1])
4×4 SparseMatrixCSC{Int64,Int64} with 7 stored entries:
[3, 1]  =  7
[4, 1]  =  4
[2, 2]  =  6
[3, 2]  =  3
[1, 3]  =  5
[2, 3]  =  2
[1, 4]  =  1
source
Base.permute!Method.
permute!{Tv,Ti}(X::SparseMatrixCSC{Tv,Ti}, A::SparseMatrixCSC{Tv,Ti},
p::AbstractVector{<:Integer}, q::AbstractVector{<:Integer}[, C::SparseMatrixCSC{Tv,Ti}])

Bilaterally permute A, storing result PAQ (A[p,q]) in X. Stores intermediate result (AQ)^T (transpose(A[:,q])) in optional argument C if present. Requires that none of X, A, and, if present, C alias each other; to store result PAQ back into A, use the following method lacking X:

permute!{Tv,Ti}(A::SparseMatrixCSC{Tv,Ti}, p::AbstractVector{<:Integer},
q::AbstractVector{<:Integer}[, C::SparseMatrixCSC{Tv,Ti}[, workcolptr::Vector{Ti}]])

X's dimensions must match those of A (X.m == A.m and X.n == A.n), and X must have enough storage to accommodate all allocated entries in A (length(X.rowval) >= nnz(A) and length(X.nzval) >= nnz(A)). Column-permutation q's length must match A's column count (length(q) == A.n). Row-permutation p's length must match A's row count (length(p) == A.m).

C's dimensions must match those of transpose(A) (C.m == A.n and C.n == A.m), and C must have enough storage to accommodate all allocated entries in A (length(C.rowval) >= nnz(A) and length(C.nzval) >= nnz(A)).

For additional (algorithmic) information, and for versions of these methods that forgo argument checking, see (unexported) parent methods unchecked_noalias_permute! and unchecked_aliasing_permute!.