DocTestSetup = :(using SparseArrays, LinearAlgebra)
Julia has support for sparse vectors and sparse matrices
in the SparseArrays stdlib module. Sparse arrays are arrays that contain enough zeros
that storing them in a special data structure leads to savings in space and execution time,
compared to dense arrays.
In Julia, sparse matrices are stored in the Compressed Sparse Column (CSC) format.
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:
struct SparseMatrixCSC{Tv,Ti<:Integer} <: AbstractSparseMatrixCSC{Tv,Ti}
m::Int # Number of rows
n::Int # Number of columns
colptr::Vector{Ti} # Column j is in colptr[j]:(colptr[j+1]-1)
rowval::Vector{Ti} # Row indices of stored values
nzval::Vector{Tv} # Stored values, typically nonzeros
endThe 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 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 count(!iszero, x), which inspects every stored element of a sparse
matrix. dropzeros, and the in-place dropzeros!, can be used to
remove stored zeros from the sparse matrix.
julia> A = sparse([1, 1, 2, 3], [1, 3, 2, 3], [0, 1, 2, 0])
3×3 SparseMatrixCSC{Int64, Int64} with 4 stored entries:
0 ⋅ 1
⋅ 2 ⋅
⋅ ⋅ 0
julia> dropzeros(A)
3×3 SparseMatrixCSC{Int64, Int64} with 2 stored entries:
⋅ ⋅ 1
⋅ 2 ⋅
⋅ ⋅ ⋅
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:
struct SparseVector{Tv,Ti<:Integer} <: AbstractSparseVector{Tv,Ti}
n::Int # Length of the sparse vector
nzind::Vector{Ti} # Indices of stored values
nzval::Vector{Tv} # Stored values, typically nonzeros
endAs for SparseMatrixCSC, the SparseVector type can also contain explicitly
stored zeros. (See [Sparse Matrix Storage](@ref man-csc).).
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:
julia> spzeros(3)
3-element SparseVector{Float64, Int64} with 0 stored entries
The sparse 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 sparsevec, 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].
julia> I = [1, 4, 3, 5]; J = [4, 7, 18, 9]; V = [1, 2, -5, 3];
julia> S = sparse(I,J,V)
5×18 SparseMatrixCSC{Int64, Int64} with 4 stored entries:
⠀⠈⠀⡀⠀⠀⠀⠀⠠
⠀⠀⠀⠀⠁⠀⠀⠀⠀
julia> R = sparsevec(I,V)
5-element SparseVector{Int64, Int64} with 4 stored entries:
[1] = 1
[3] = -5
[4] = 2
[5] = 3
The inverse of the sparse and sparsevec functions is
findnz, 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).
julia> findnz(S)
([1, 4, 5, 3], [4, 7, 9, 18], [1, 2, 3, -5])
julia> findall(!iszero, S)
4-element Vector{CartesianIndex{2}}:
CartesianIndex(1, 4)
CartesianIndex(4, 7)
CartesianIndex(5, 9)
CartesianIndex(3, 18)
julia> findnz(R)
([1, 3, 4, 5], [1, -5, 2, 3])
julia> findall(!iszero, R)
4-element Vector{Int64}:
1
3
4
5
Another way to create a sparse array is to convert a dense array into a sparse array using
the sparse function:
julia> sparse(Matrix(1.0I, 5, 5))
5×5 SparseMatrixCSC{Float64, Int64} with 5 stored entries:
1.0 ⋅ ⋅ ⋅ ⋅
⋅ 1.0 ⋅ ⋅ ⋅
⋅ ⋅ 1.0 ⋅ ⋅
⋅ ⋅ ⋅ 1.0 ⋅
⋅ ⋅ ⋅ ⋅ 1.0
julia> sparse([1.0, 0.0, 1.0])
3-element SparseVector{Float64, Int64} with 2 stored entries:
[1] = 1.0
[3] = 1.0
You can go in the other direction using the Array constructor. The issparse
function can be used to query if a matrix is sparse.
julia> issparse(spzeros(5))
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](@ref stdlib-sparse-arrays) section of the standard library reference.
| Sparse | Dense | Description |
|---|---|---|
spzeros(m,n) |
zeros(m,n) |
Creates a m-by-n matrix of zeros. (spzeros(m,n) is empty.) |
sparse(I,n,n) |
Matrix(I,n,n) |
Creates a n-by-n identity matrix. |
sparse(A) |
Array(S) |
Interconverts between dense and sparse formats. |
sprand(m,n,d) |
rand(m,n) |
Creates a m-by-n random matrix (of density d) with iid non-zero elements distributed uniformly on the half-open interval [0, 1). |
sprandn(m,n,d) |
randn(m,n) |
Creates a m-by-n random matrix (of density d) with iid non-zero elements distributed according to the standard normal (Gaussian) distribution. |
sprandn(rng,m,n,d) |
randn(rng,m,n) |
Creates a m-by-n random matrix (of density d) with iid non-zero elements generated with the rng random number generator |
SparseArrays.AbstractSparseArray
SparseArrays.AbstractSparseVector
SparseArrays.AbstractSparseMatrix
SparseArrays.SparseVector
SparseArrays.SparseMatrixCSC
SparseArrays.sparse
SparseArrays.sparsevec
SparseArrays.issparse
SparseArrays.nnz
SparseArrays.findnz
SparseArrays.spzeros
SparseArrays.spdiagm
SparseArrays.blockdiag
SparseArrays.sprand
SparseArrays.sprandn
SparseArrays.nonzeros
SparseArrays.rowvals
SparseArrays.nzrange
SparseArrays.droptol!
SparseArrays.dropzeros!
SparseArrays.dropzeros
SparseArrays.permute
permute!{Tv, Ti, Tp <: Integer, Tq <: Integer}(::SparseMatrixCSC{Tv,Ti}, ::SparseMatrixCSC{Tv,Ti}, ::AbstractArray{Tp,1}, ::AbstractArray{Tq,1})
DocTestSetup = nothing