In Julia, sparse matrices are stored in the . Julia sparse matrices have the type SparseMatrixCSC{Tv,Ti}, where Tv is the type of the stored values, and Ti is the integer type for storing column pointers and row indices. The internal representation of SparseMatrixCSC is as follows:

    The compressed sparse column storage makes it easy and quick to access the elements in the column of a sparse matrix, whereas accessing the sparse matrix by rows is considerably slower. Operations such as insertion of previously unstored entries one at a time in the CSC structure tend to be slow. This is because all elements of the sparse matrix that are beyond the point of insertion have to be moved one place over.

    All operations on sparse matrices are carefully implemented to exploit the CSC data structure for performance, and to avoid expensive operations.

    If you have data in CSC format from a different application or library, and wish to import it in Julia, make sure that you use 1-based indexing. The row indices in every column need to be sorted. If your SparseMatrixCSC object contains unsorted row indices, one quick way to sort them is by doing a double transpose.

    In some applications, it is convenient to store explicit zero values in a SparseMatrixCSC. These are accepted by functions in Base (but there is no guarantee that they will be preserved in mutating operations). Such explicitly stored zeros are treated as structural nonzeros by many routines. The function returns the number of elements explicitly stored in the sparse data structure, including non-structural zeros. In order to count the exact number of numerical nonzeros, use count(!iszero, x), which inspects every stored element of a sparse matrix. , and the in-place dropzeros!, can be used to remove stored zeros from the sparse matrix.

    1. julia> A = sparse([1, 1, 2, 3], [1, 3, 2, 3], [0, 1, 2, 0])
    2. 3×3 SparseMatrixCSC{Int64, Int64} with 4 stored entries:
    3.  0  ⋅  1
    4.  ⋅  2  ⋅
    5.  ⋅  ⋅  0
    7. julia> dropzeros(A)
    8. 3×3 SparseMatrixCSC{Int64, Int64} with 2 stored entries:
    9.  ⋅  ⋅  1
    10.  ⋅  2  ⋅
    11.  ⋅  ⋅  ⋅

    Sparse vectors are stored in a close analog to compressed sparse column format for sparse matrices. In Julia, sparse vectors have the type SparseVector{Tv,Ti} where Tv is the type of the stored values and Ti the integer type for the indices. The internal representation is as follows:

    1. struct SparseVector{Tv,Ti<:Integer} <: AbstractSparseVector{Tv,Ti}
    2.     n::Int              # Length of the sparse vector
    3.     nzind::Vector{Ti}   # Indices of stored values
    4.     nzval::Vector{Tv}   # Stored values, typically nonzeros
    5. end

    As for , the SparseVector type can also contain explicitly stored zeros. (See Sparse Matrix Storage.).

    The simplest way to create a sparse array is to use a function equivalent to the zeros function that Julia provides for working with dense arrays. To produce a sparse array instead, you can use the same name with an sp prefix:

    1. julia> spzeros(3)
    2. 3-element SparseVector{Float64, Int64} with 0 stored entries

    The function is often a handy way to construct sparse arrays. For example, to construct a sparse matrix we can input a vector I of row indices, a vector J of column indices, and a vector V of stored values (this is also known as the COO (coordinate) format). sparse(I,J,V) then constructs a sparse matrix such that S[I[k], J[k]] = V[k]. The equivalent sparse vector constructor is , which takes the (row) index vector I and the vector V with the stored values and constructs a sparse vector R such that R[I[k]] = V[k].

    1. julia> I = [1, 4, 3, 5]; J = [4, 7, 18, 9]; V = [1, 2, -5, 3];
    3. julia> S = sparse(I,J,V)
    4. 5×18 SparseMatrixCSC{Int64, Int64} with 4 stored entries:
    5. ⠀⠈⠀⡀⠀⠀⠀⠀⠠
    6. ⠀⠀⠀⠀⠁⠀⠀⠀⠀
    8. julia> R = sparsevec(I,V)
    9. 5-element SparseVector{Int64, Int64} with 4 stored entries:
    10.   [1]  =  1
    11.   [3]  =  -5
    12.   [4]  =  2
    13.   [5]  =  3

    The inverse of the sparse and functions is findnz, which retrieves the inputs used to create the sparse array. returns the cartesian indices of non-zero entries in x (including stored entries equal to zero).

    1. julia> findnz(S)
    2. ([1, 4, 5, 3], [4, 7, 9, 18], [1, 2, 3, -5])
    4. julia> findall(!iszero, S)
    5. 4-element Vector{CartesianIndex{2}}:
    6.  CartesianIndex(1, 4)
    7.  CartesianIndex(4, 7)
    8.  CartesianIndex(5, 9)
    9.  CartesianIndex(3, 18)
    11. julia> findnz(R)
    12. ([1, 3, 4, 5], [1, -5, 2, 3])
    14. julia> findall(!iszero, R)
    15. 4-element Vector{Int64}:
    16.  1
    17.  3
    18.  4
    19.  5

    Another way to create a sparse array is to convert a dense array into a sparse array using the sparse function:

    1. julia> sparse(Matrix(1.0I, 5, 5))
    2. 5×5 SparseMatrixCSC{Float64, Int64} with 5 stored entries:
    3.  1.0   ⋅    ⋅    ⋅    ⋅
    4.   ⋅   1.0   ⋅    ⋅    ⋅
    5.   ⋅    ⋅   1.0   ⋅    ⋅
    6.   ⋅    ⋅    ⋅   1.0   ⋅
    7.   ⋅    ⋅    ⋅    ⋅   1.0
    9. julia> sparse([1.0, 0.0, 1.0])
    10. 3-element SparseVector{Float64, Int64} with 2 stored entries:
    11.   [1]  =  1.0
    12.   [3]  =  1.0

    You can go in the other direction using the constructor. The issparse function can be used to query if a matrix is sparse.

    1. julia> issparse(spzeros(5))
    2. true

    Arithmetic operations on sparse matrices also work as they do on dense matrices. Indexing of, assignment into, and concatenation of sparse matrices work in the same way as dense matrices. Indexing operations, especially assignment, are expensive, when carried out one element at a time. In many cases it may be better to convert the sparse matrix into (I,J,V) format using findnz, manipulate the values or the structure in the dense vectors (I,J,V), and then reconstruct the sparse matrix.

    The following table gives a correspondence between built-in methods on sparse matrices and their corresponding methods on dense matrix types. In general, methods that generate sparse matrices differ from their dense counterparts in that the resulting matrix follows the same sparsity pattern as a given sparse matrix S, or that the resulting sparse matrix has density d, i.e. each matrix element has a probability d of being non-zero.

    Details can be found in the Sparse Vectors and Matrices section of the standard library reference.

    Sparse Arrays

    — Type

    1. AbstractSparseArray{Tv,Ti,N}

    Supertype for N-dimensional sparse arrays (or array-like types) with elements of type Tv and index type Ti. SparseMatrixCSC, and SuiteSparse.CHOLMOD.Sparse are subtypes of this.

    SparseArrays.AbstractSparseVector — Type

    1. AbstractSparseVector{Tv,Ti}

    Supertype for one-dimensional sparse arrays (or array-like types) with elements of type Tv and index type Ti. Alias for AbstractSparseArray{Tv,Ti,1}.

    — Type

    1. AbstractSparseMatrix{Tv,Ti}

    Supertype for two-dimensional sparse arrays (or array-like types) with elements of type Tv and index type Ti. Alias for AbstractSparseArray{Tv,Ti,2}.

    SparseArrays.SparseVector — Type

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

    Vector type for storing sparse vectors.

    — Type

    1. SparseMatrixCSC{Tv,Ti<:Integer} <: AbstractSparseMatrixCSC{Tv,Ti}

    Matrix type for storing sparse matrices in the Compressed Sparse Column format. The standard way of constructing SparseMatrixCSC is through the function. See also spzeros, and sprand.

    — Function

    1. sparse(A)

    Convert an AbstractMatrix A into a sparse matrix.

    Examples

    1. julia> A = Matrix(1.0I, 3, 3)
    2. 3×3 Matrix{Float64}:
    3.  1.0  0.0  0.0
    4.  0.0  1.0  0.0
    5.  0.0  0.0  1.0
    7. julia> sparse(A)
    8. 3×3 SparseMatrixCSC{Float64, Int64} with 3 stored entries:
    9.  1.0   ⋅    ⋅
    10.   ⋅   1.0   ⋅
    11.   ⋅    ⋅   1.0
    1. 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 SparseArrays.sparse!.

    Examples

    1. julia> Is = [1; 2; 3];
    3. julia> Js = [1; 2; 3];
    5. julia> Vs = [1; 2; 3];
    7. julia> sparse(Is, Js, Vs)
    8. 3×3 SparseMatrixCSC{Int64, Int64} with 3 stored entries:
    9.  1  ⋅  ⋅
    10.  ⋅  2  ⋅
    11.  ⋅  ⋅  3

    — Function

    1. 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 |.

    Examples

    1. julia> II = [1, 3, 3, 5]; V = [0.1, 0.2, 0.3, 0.2];
    2. julia> sparsevec(II, V)
    3. 5-element SparseVector{Float64, Int64} with 3 stored entries:
    4.   [1]  =  0.1
    5.   [3]  =  0.5
    6.   [5]  =  0.2
    8. julia> sparsevec(II, V, 8, -)
    9. 8-element SparseVector{Float64, Int64} with 3 stored entries:
    10.   [1]  =  0.1
    11.   [3]  =  -0.1
    12.   [5]  =  0.2
    14. julia> sparsevec([1, 3, 1, 2, 2], [true, true, false, false, false])
    15. 3-element SparseVector{Bool, Int64} with 3 stored entries:
    16.   [1]  =  1
    17.   [2]  =  0
    18.   [3]  =  1
    1. 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.

    Examples

    1. julia> sparsevec(Dict(1 => 3, 2 => 2))
    2. 2-element SparseVector{Int64, Int64} with 2 stored entries:
    3.   [1]  =  3
    4.   [2]  =  2

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

    1. julia> sparsevec([1.0, 2.0, 0.0, 0.0, 3.0, 0.0])
    2. 6-element SparseVector{Float64, Int64} with 3 stored entries:
    3.   [1]  =  1.0
    4.   [2]  =  2.0
    5.   [5]  =  3.0

    SparseArrays.issparse — Function

    1. issparse(S)

    Returns true if S is sparse, and false otherwise.

    Examples

    1. julia> sv = sparsevec([1, 4], [2.3, 2.2], 10)
    2. 10-element SparseVector{Float64, Int64} with 2 stored entries:
    3.   [1 ]  =  2.3
    4.   [4 ]  =  2.2
    6. julia> issparse(sv)
    7. true
    9. julia> issparse(Array(sv))
    10. false

    — Function

    1. nnz(A)

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

    Examples

    1. julia> A = sparse(2I, 3, 3)
    2. 3×3 SparseMatrixCSC{Int64, Int64} with 3 stored entries:
    3.  2  ⋅  ⋅
    4.  ⋅  2  ⋅
    5.  ⋅  ⋅  2
    7. julia> nnz(A)
    8. 3

    SparseArrays.findnz — Function

    1. findnz(A::SparseMatrixCSC)

    Return a tuple (I, J, V) where I and J are the row and column indices of the stored (“structurally non-zero”) values in sparse matrix A, and V is a vector of the values.

    Examples

    1. julia> A = sparse([1 2 0; 0 0 3; 0 4 0])
    2. 3×3 SparseMatrixCSC{Int64, Int64} with 4 stored entries:
    3.  1  2  ⋅
    4.  ⋅  ⋅  3
    5.  ⋅  4  ⋅
    7. julia> findnz(A)
    8. ([1, 1, 3, 2], [1, 2, 2, 3], [1, 2, 4, 3])

    — Function

    1. 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

    1. julia> spzeros(3, 3)
    2. 3×3 SparseMatrixCSC{Float64, Int64} with 0 stored entries:
    3.   ⋅    ⋅    ⋅
    4.   ⋅    ⋅    ⋅
    5.   ⋅    ⋅    ⋅
    7. julia> spzeros(Float32, 4)
    8. 4-element SparseVector{Float32, Int64} with 0 stored entries

    — Function

    1. spdiagm(kv::Pair{<:Integer,<:AbstractVector}...)
    2. spdiagm(m::Integer, n::Integer, kv::Pair{<:Integer,<:AbstractVector}...)

    Construct a sparse diagonal matrix from Pairs of vectors and diagonals. Each vector kv.second will be placed on the kv.first diagonal. By default, the matrix is square and its size is inferred from kv, but a non-square size m×n (padded with zeros as needed) can be specified by passing m,n as the first arguments.

    Examples

    1. julia> spdiagm(-1 => [1,2,3,4], 1 => [4,3,2,1])
    2. 5×5 SparseMatrixCSC{Int64, Int64} with 8 stored entries:
    3.  ⋅  4  ⋅  ⋅  ⋅
    4.  1  ⋅  3  ⋅  ⋅
    5.  ⋅  2  ⋅  2  ⋅
    6.  ⋅  ⋅  3  ⋅  1
    7.  ⋅  ⋅  ⋅  4  ⋅
    1. spdiagm(v::AbstractVector)
    2. spdiagm(m::Integer, n::Integer, v::AbstractVector)

    Construct a sparse matrix with elements of the vector as diagonal elements. By default (no given m and n), the matrix is square and its size is given by length(v), but a non-square size m×n can be specified by passing m and n as the first arguments.

    Julia 1.6

    These functions require at least Julia 1.6.

    Examples

    1. julia> spdiagm([1,2,3])
    2. 3×3 SparseMatrixCSC{Int64, Int64} with 3 stored entries:
    3.  1  ⋅  ⋅
    4.  ⋅  2  ⋅
    5.  ⋅  ⋅  3
    7. julia> spdiagm(sparse([1,0,3]))
    8. 3×3 SparseMatrixCSC{Int64, Int64} with 2 stored entries:
    9.  1  ⋅  ⋅
    10.  ⋅  ⋅  ⋅
    11.  ⋅  ⋅  3

    SparseArrays.sparse_hcat — Function

    1. sparse_hcat(A...)

    Concatenate along dimension 2. Return a SparseMatrixCSC object.

    Julia 1.8

    This method was added in Julia 1.8. It mimicks previous concatenation behavior, where the concatenation with specialized “sparse” matrix types from LinearAlgebra.jl automatically yielded sparse output even in the absence of any SparseArray argument.

    — Function

    1. sparse_vcat(A...)

    Concatenate along dimension 1. Return a SparseMatrixCSC object.

    Julia 1.8

    This method was added in Julia 1.8. It mimicks previous concatenation behavior, where the concatenation with specialized “sparse” matrix types from LinearAlgebra.jl automatically yielded sparse output even in the absence of any SparseArray argument.

    SparseArrays.sparse_hvcat — Function

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

    Sparse 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 1.8

    This method was added in Julia 1.8. It mimicks previous concatenation behavior, where the concatenation with specialized “sparse” matrix types from LinearAlgebra.jl automatically yielded sparse output even in the absence of any SparseArray argument.

    — Function

    1. blockdiag(A...)

    Concatenate matrices block-diagonally. Currently only implemented for sparse matrices.

    Examples

    1. julia> blockdiag(sparse(2I, 3, 3), sparse(4I, 2, 2))
    2. 5×5 SparseMatrixCSC{Int64, Int64} with 5 stored entries:
    3.  2  ⋅  ⋅  ⋅  ⋅
    4.  ⋅  2  ⋅  ⋅  ⋅
    5.  ⋅  ⋅  2  ⋅  ⋅
    6.  ⋅  ⋅  ⋅  4  ⋅
    7.  ⋅  ⋅  ⋅  ⋅  4

    SparseArrays.sprand — Function

    1. 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 .

    Examples

    1. julia> sprand(Bool, 2, 2, 0.5)
    2. 2×2 SparseMatrixCSC{Bool, Int64} with 2 stored entries:
    3.  ⋅  ⋅
    5. julia> sprand(Float64, 3, 0.75)
    6. 3-element SparseVector{Float64, Int64} with 2 stored entries:
    7.   [1]  =  0.795547
    8.   [2]  =  0.49425

    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.

    Julia 1.1

    Specifying the output element type Type requires at least Julia 1.1.

    Examples

    1. julia> sprandn(2, 2, 0.75)
    3.  -1.20577     ⋅
    4.   0.311817  -0.234641

    — Function

    1. 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 .

    Examples

    1. julia> A = sparse(2I, 3, 3)
    2. 3×3 SparseMatrixCSC{Int64, Int64} with 3 stored entries:
    3.  2  ⋅  ⋅
    4.  ⋅  2  ⋅
    5.  ⋅  ⋅  2
    7. julia> nonzeros(A)
    8. 3-element Vector{Int64}:
    9.  2
    10.  2
    11.  2

    SparseArrays.rowvals — Function

    1. rowvals(A::AbstractSparseMatrixCSC)

    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 and nzrange.

    Examples

    1. julia> A = sparse(2I, 3, 3)
    2. 3×3 SparseMatrixCSC{Int64, Int64} with 3 stored entries:
    3.  2  ⋅  ⋅
    4.  ⋅  2  ⋅
    5.  ⋅  ⋅  2
    7. julia> rowvals(A)
    8. 3-element Vector{Int64}:
    9.  1
    10.  2
    11.  3

    — Function

    1. nzrange(A::AbstractSparseMatrixCSC, col::Integer)

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

    1. A = sparse(I,J,V)
    2. rows = rowvals(A)
    3. vals = nonzeros(A)
    4. m, n = size(A)
    5. for j = 1:n
    6.    for i in nzrange(A, j)
    7.       row = rows[i]
    8.       val = vals[i]
    9.       # perform sparse wizardry...
    10.    end
    11. end
    1. nzrange(x::SparseVectorUnion, col)

    Give the range of indices to the structural nonzero values of a sparse vector. The column index col is ignored (assumed to be 1).

    SparseArrays.droptol! — Function

    1. droptol!(A::AbstractSparseMatrixCSC, tol)

    Removes stored values from A whose absolute value is less than or equal to tol.

    1. droptol!(x::SparseVector, tol)

    Removes stored values from x whose absolute value is less than or equal to tol.

    — Function

    1. dropzeros!(A::AbstractSparseMatrixCSC;)

    Removes stored numerical zeros from A.

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

    1. dropzeros!(x::SparseVector)

    Removes stored numerical zeros from x.

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

    SparseArrays.dropzeros — Function

    1. dropzeros(A::AbstractSparseMatrixCSC;)

    Generates a copy of A and removes stored numerical zeros from that copy.

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

    Examples

    1. julia> A = sparse([1, 2, 3], [1, 2, 3], [1.0, 0.0, 1.0])
    2. 3×3 SparseMatrixCSC{Float64, Int64} with 3 stored entries:
    3.  1.0   ⋅    ⋅
    4.   ⋅   0.0   ⋅
    5.   ⋅    ⋅   1.0
    7. julia> dropzeros(A)
    8. 3×3 SparseMatrixCSC{Float64, Int64} with 2 stored entries:
    9.  1.0   ⋅    ⋅
    10.   ⋅    ⋅    ⋅
    11.   ⋅    ⋅   1.0
    1. dropzeros(x::SparseVector)

    Generates a copy of x and removes numerical zeros from that copy.

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

    Examples

    1. julia> A = sparsevec([1, 2, 3], [1.0, 0.0, 1.0])
    2. 3-element SparseVector{Float64, Int64} with 3 stored entries:
    3.   [1]  =  1.0
    4.   [2]  =  0.0
    5.   [3]  =  1.0
    7. julia> dropzeros(A)
    8. 3-element SparseVector{Float64, Int64} with 2 stored entries:
    9.   [1]  =  1.0
    10.   [3]  =  1.0

    — Function

    1. permute(A::AbstractSparseMatrixCSC{Tv,Ti}, p::AbstractVector{<:Integer},
    2.         q::AbstractVector{<:Integer}) where {Tv,Ti}

    Bilaterally permute A, returning PAQ (A[p,q]). Column-permutation q‘s length must match A‘s column count (length(q) == size(A, 2)). Row-permutation p‘s length must match A‘s row count (length(p) == size(A, 1)).

    For expert drivers and additional information, see permute!.

    Examples

    1. julia> A = spdiagm(0 => [1, 2, 3, 4], 1 => [5, 6, 7])
    2. 4×4 SparseMatrixCSC{Int64, Int64} with 7 stored entries:
    3.  1  5  ⋅  ⋅
    4.  ⋅  2  6  ⋅
    5.  ⋅  ⋅  3  7
    6.  ⋅  ⋅  ⋅  4
    8. julia> permute(A, [4, 3, 2, 1], [1, 2, 3, 4])
    9. 4×4 SparseMatrixCSC{Int64, Int64} with 7 stored entries:
    10.  ⋅  ⋅  ⋅  4
    11.  ⋅  ⋅  3  7
    12.  ⋅  2  6  ⋅
    13.  1  5  ⋅  ⋅
    15. julia> permute(A, [1, 2, 3, 4], [4, 3, 2, 1])
    16. 4×4 SparseMatrixCSC{Int64, Int64} with 7 stored entries:
    17.  ⋅  ⋅  5  1
    18.  ⋅  6  2  ⋅
    19.  7  3  ⋅  ⋅
    20.  4  ⋅  ⋅  ⋅

    — Method

    1. permute!(X::AbstractSparseMatrixCSC{Tv,Ti}, A::AbstractSparseMatrixCSC{Tv,Ti},
    2.          p::AbstractVector{<:Integer}, q::AbstractVector{<:Integer},
    3.          [C::AbstractSparseMatrixCSC{Tv,Ti}]) where {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:

    1. permute!(A::AbstractSparseMatrixCSC{Tv,Ti}, p::AbstractVector{<:Integer},
    2.          q::AbstractVector{<:Integer}[, C::AbstractSparseMatrixCSC{Tv,Ti},

    X‘s dimensions must match those of A (size(X, 1) == size(A, 1) and size(X, 2) == size(A, 2)), and X must have enough storage to accommodate all allocated entries in A (length(rowvals(X)) >= nnz(A) and length(nonzeros(X)) >= nnz(A)). Column-permutation q‘s length must match A‘s column count (length(q) == size(A, 2)). Row-permutation p‘s length must match A‘s row count (length(p) == size(A, 1)).

    C‘s dimensions must match those of transpose(A) (size(C, 1) == size(A, 2) and size(C, 2) == size(A, 1)), and C must have enough storage to accommodate all allocated entries in A (length(rowvals(C)) >= nnz(A) and length(nonzeros(C)) >= 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!.