在 Julia 中，稀疏矩阵是按照压缩稀疏列（CSC）格式存储的。Julia 稀疏矩阵具有 类型，其中 Tv 是存储值的类型，Ti 是存储列指针和行索引的整型类型。SparseMatrixCSC 的内部表示如下所示：

    压缩稀疏列存储格式使得访问稀疏矩阵的列元素非常简单快速，而访问稀疏矩阵的行会非常缓慢。在 CSC 稀疏矩阵中执行类似插入新元素的操作也会非常慢。这是由于在稀疏矩阵中插入新元素时，在插入点之后的所有元素都要向后移动一位。

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

    如果你有来自不同应用或库的 CSC 格式数据，并且想要将它导入 Julia，确保使用基于 1 的索引。每个列中的行索引都要是有序的。如果你的 SparseMatrixCSC 对象包含无序的行索引，一个快速将它们排序的方法是做一次二重转置。

    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 nnz 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 , which inspects every stored element of a sparse matrix. dropzeros, and the in-place , 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

    对于 ， SparseVector 类型也能包含显示存储的，零值。（见 稀疏矩阵存储。）

    创建一个稀疏矩阵的最简单的方法是使用一个与 Julia 提供的用来处理稠密矩阵的 等价的函数。要产生一个稀疏矩阵，你可以用同样的名字加上 sp 前缀：

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

    sparse 函数通常是一个构建稀疏矩阵的便捷方法。例如，要构建一个稀疏矩阵，我们可以输入一个列索引向量 I，一个行索引向量 J，一个储存值的向量 V（这也叫作 ）。 然后 sparse(I,J,V) 创建一个满足 S[I[k], J[k]] = V[k] 的稀疏矩阵。等价的稀疏向量构建函数是 sparsevec，它接受（行）索引向量 I 和储存值的向量 V 并创建一个满足 R[I[k]] = V[k] 的向量 R。

    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 and sparsevec functions is , which retrieves the inputs used to create the sparse array. findall(!iszero, x) 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

    另一个创建稀疏数组的方法是使用 函数将一个稠密数组转化为稀疏数组：

    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 Array constructor. The 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 section of the standard library reference.

    SparseArrays.AbstractSparseArray — 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. , SparseVector and SuiteSparse.CHOLMOD.Sparse are subtypes of this.

    — 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}.

    SparseArrays.AbstractSparseMatrix — 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}.

    — Type

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

    Vector type for storing sparse vectors.

    SparseArrays.SparseMatrixCSC — Type

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

    Matrix type for storing sparse matrices in the format. The standard way of constructing SparseMatrixCSC is through the sparse function. See also , spdiagm and .

    SparseArrays.sparse — 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 .

    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  ⋅

    SparseArrays.sparsevec — Function

    1. sparsevec(I, V, [m, combine])

    Create a sparse vector S of length 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 |.

    1. julia> II = [1, 3, 3, 5]; V = [0.1, 0.2, 0.3, 0.2];
    3. julia> sparsevec(II, V)
    4. 5-element SparseVector{Float64, Int64} with 3 stored entries:
    5.   [1]  =  0.1
    6.   [3]  =  0.5
    7.   [5]  =  0.2
    9. julia> sparsevec(II, V, 8, -)
    10. 8-element SparseVector{Float64, Int64} with 3 stored entries:
    11.   [1]  =  0.1
    12.   [3]  =  -0.1
    13.   [5]  =  0.2
    15. julia> sparsevec([1, 3, 1, 2, 2], [true, true, false, false, false])
    16. 3-element SparseVector{Bool, Int64} with 3 stored entries:
    17.   [1]  =  1
    18.   [2]  =  0
    19.   [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. sparsevec(A)

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

    Examples

    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

    — 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

    SparseArrays.nnz — 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

    — 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])

    SparseArrays.spzeros — 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 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

    SparseArrays.spdiagm — 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

    — 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

    SparseArrays.sprandn — Function

    1. sprandn([rng][,Type],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 .

    Julia 1.1

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

    SparseArrays.nonzeros — Function

    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 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> nonzeros(A)
    8. 3-element Vector{Int64}:
    9.  2
    10.  2
    11.  2

    — 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 nonzeros 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> rowvals(A)
    8. 3-element Vector{Int64}:
    9.  1
    10.  2
    11.  3

    SparseArrays.nzrange — 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 and rowvals, 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).

    — 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.

    SparseArrays.dropzeros! — Function

    1. dropzeros!(A::AbstractSparseMatrixCSC;)

    Removes stored numerical zeros from A.

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

    1. dropzeros!(x::SparseVector)

    Removes stored numerical zeros from x.

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

    — 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 dropzeros!.

    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 .

    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

    SparseArrays.permute — 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 .

    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  ⋅  ⋅  ⋅

    Base.permute! — 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!.