# Sparse Arrays

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.

External packages which implement different sparse storage types, multidimensional sparse arrays, and more can be found in Noteworthy External Sparse Packages

## Compressed Sparse Column (CSC) Sparse Matrix Storage

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
end
```

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, and if they are not, the matrix will display incorrectly. If your `SparseMatrixCSC`

object contains unsorted row indices, one quick way to sort them is by doing a double transpose. Since the transpose operation is lazy, make a copy to materialize each 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 Vector Storage

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
end
```

As for `SparseMatrixCSC`

, the `SparseVector`

type can also contain explicitly stored zeros. (See Sparse Matrix Storage.).

## Sparse Vector and Matrix Constructors

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 (including stored entries equal to zero). `findall(!iszero, x)`

returns the Cartesian indices of non-zero entries in `x`

(not 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
```

## Sparse matrix operations

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.

## Correspondence of dense and sparse methods

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 | 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 API

`SparseArrays.AbstractSparseArray`

— Type`AbstractSparseArray{Tv,Ti,N}`

Supertype for `N`

-dimensional sparse arrays (or array-like types) with elements of type `Tv`

and index type `Ti`

. `SparseMatrixCSC`

, `SparseVector`

and `SuiteSparse.CHOLMOD.Sparse`

are subtypes of this.

`SparseArrays.AbstractSparseVector`

— Type`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`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`SparseVector{Tv,Ti<:Integer} <: AbstractSparseVector{Tv,Ti}`

Vector type for storing sparse vectors. Can be created by passing the length of the vector, a *sorted* vector of non-zero indices, and a vector of non-zero values.

For instance, the vector `[5, 6, 0, 7]`

can be represented as

`SparseVector(4, [1, 2, 4], [5, 6, 7])`

This indicates that the element at index 1 is 5, at index 2 is 6, at index 3 is `zero(Int)`

, and at index 4 is 7.

It may be more convenient to create sparse vectors directly from dense vectors using `sparse`

as

`sparse([5, 6, 0, 7])`

yields the same sparse vector.

`SparseArrays.SparseMatrixCSC`

— Type`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 `sparse`

function. See also `spzeros`

, `spdiagm`

and `sprand`

.

`SparseArrays.sparse`

— Function`sparse(A)`

Convert an AbstractMatrix `A`

into a sparse matrix.

**Examples**

```
julia> A = Matrix(1.0I, 3, 3)
3×3 Matrix{Float64}:
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:
1.0 ⋅ ⋅
⋅ 1.0 ⋅
⋅ ⋅ 1.0
```

`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**

```
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 ⋅ ⋅
⋅ 2 ⋅
⋅ ⋅ 3
```

`SparseArrays.sparse!`

— Function```
sparse!(I::AbstractVector{Ti}, J::AbstractVector{Ti}, V::AbstractVector{Tv},
m::Integer, n::Integer, combine, klasttouch::Vector{Ti},
csrrowptr::Vector{Ti}, csrcolval::Vector{Ti}, csrnzval::Vector{Tv},
[csccolptr::Vector{Ti}], [cscrowval::Vector{Ti}, cscnzval::Vector{Tv}] ) where {Tv,Ti<:Integer}
```

Parent of and expert driver for `sparse`

; see `sparse`

for basic usage. This method allows the user to provide preallocated storage for `sparse`

's intermediate objects and result as described below. This capability enables more efficient successive construction of `SparseMatrixCSC`

s from coordinate representations, and also enables extraction of an unsorted-column representation of the result's transpose at no additional cost.

This method consists of three major steps: (1) Counting-sort the provided coordinate representation into an unsorted-row CSR form including repeated entries. (2) Sweep through the CSR form, simultaneously calculating the desired CSC form's column-pointer array, detecting repeated entries, and repacking the CSR form with repeated entries combined; this stage yields an unsorted-row CSR form with no repeated entries. (3) Counting-sort the preceding CSR form into a fully-sorted CSC form with no repeated entries.

Input arrays `csrrowptr`

, `csrcolval`

, and `csrnzval`

constitute storage for the intermediate CSR forms and require `length(csrrowptr) >= m + 1`

, `length(csrcolval) >= length(I)`

, and `length(csrnzval >= length(I))`

. Input array `klasttouch`

, workspace for the second stage, requires `length(klasttouch) >= n`

. Optional input arrays `csccolptr`

, `cscrowval`

, and `cscnzval`

constitute storage for the returned CSC form `S`

. If necessary, these are resized automatically to satisfy `length(csccolptr) = n + 1`

, `length(cscrowval) = nnz(S)`

and `length(cscnzval) = nnz(S)`

; hence, if `nnz(S)`

is unknown at the outset, passing in empty vectors of the appropriate type (`Vector{Ti}()`

and `Vector{Tv}()`

respectively) suffices, or calling the `sparse!`

method neglecting `cscrowval`

and `cscnzval`

.

On return, `csrrowptr`

, `csrcolval`

, and `csrnzval`

contain an unsorted-column representation of the result's transpose.

You may reuse the input arrays' storage (`I`

, `J`

, `V`

) for the output arrays (`csccolptr`

, `cscrowval`

, `cscnzval`

). For example, you may call `sparse!(I, J, V, csrrowptr, csrcolval, csrnzval, I, J, V)`

. Note that they will be resized to satisfy the conditions above.

For the sake of efficiency, this method performs no argument checking beyond `1 <= I[k] <= m`

and `1 <= J[k] <= n`

. Use with care. Testing with `--check-bounds=yes`

is wise.

This method runs in `O(m, n, length(I))`

time. The HALFPERM algorithm described in F. Gustavson, "Two fast algorithms for sparse matrices: multiplication and permuted transposition," ACM TOMS 4(3), 250-269 (1978) inspired this method's use of a pair of counting sorts.

`SparseArrays.sparse!(I, J, V, [m, n, combine]) -> SparseMatrixCSC`

Variant of `sparse!`

that re-uses the input vectors (`I`

, `J`

, `V`

) for the final matrix storage. After construction the input vectors will alias the matrix buffers; `S.colptr === I`

, `S.rowval === J`

, and `S.nzval === V`

holds, and they will be `resize!`

d as necessary.

Note that some work buffers will still be allocated. Specifically, this method is a convenience wrapper around `sparse!(I, J, V, m, n, combine, klasttouch, csrrowptr, csrcolval, csrnzval, csccolptr, cscrowval, cscnzval)`

where this method allocates `klasttouch`

, `csrrowptr`

, `csrcolval`

, and `csrnzval`

of appropriate size, but reuses `I`

, `J`

, and `V`

for `csccolptr`

, `cscrowval`

, and `cscnzval`

.

Arguments `m`

, `n`

, and `combine`

defaults to `maximum(I)`

, `maximum(J)`

, and `+`

, respectively.

This method requires Julia version 1.10 or later.

`SparseArrays.sparsevec`

— Function`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**

```
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] = 1
[2] = 0
[3] = 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**

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

`sparsevec(A)`

Convert a vector `A`

into a sparse vector of length `m`

.

**Examples**

```
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
```

`Base.similar`

— Method`similar(A::AbstractSparseMatrixCSC{Tv,Ti}, [::Type{TvNew}, ::Type{TiNew}, m::Integer, n::Integer]) where {Tv,Ti}`

Create an uninitialized mutable array with the given element type, index type, and size, based upon the given source `SparseMatrixCSC`

. The new sparse matrix maintains the structure of the original sparse matrix, except in the case where dimensions of the output matrix are different from the output.

The output matrix has zeros in the same locations as the input, but uninitialized values for the nonzero locations.

`SparseArrays.issparse`

— Function`issparse(S)`

Returns `true`

if `S`

is sparse, and `false`

otherwise.

**Examples**

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

`SparseArrays.nnz`

— Function`nnz(A)`

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

**Examples**

```
julia> A = sparse(2I, 3, 3)
3×3 SparseMatrixCSC{Int64, Int64} with 3 stored entries:
2 ⋅ ⋅
⋅ 2 ⋅
⋅ ⋅ 2
julia> nnz(A)
3
```

`SparseArrays.findnz`

— Function`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**

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

`SparseArrays.spzeros`

— Function`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
```

`spzeros([type], I::AbstractVector, J::AbstractVector, [m, n])`

Create a sparse matrix `S`

of dimensions `m x n`

with structural zeros at `S[I[k], J[k]]`

.

This method can be used to construct the sparsity pattern of the matrix, and is more efficient than using e.g. `sparse(I, J, zeros(length(I)))`

.

For additional documentation and an expert driver, see `SparseArrays.spzeros!`

.

This methods requires Julia version 1.10 or later.

`SparseArrays.spzeros!`

— Function```
spzeros!(::Type{Tv}, I::AbstractVector{Ti}, J::AbstractVector{Ti}, m::Integer, n::Integer,
klasttouch::Vector{Ti}, csrrowptr::Vector{Ti}, csrcolval::Vector{Ti},
[csccolptr::Vector{Ti}], [cscrowval::Vector{Ti}, cscnzval::Vector{Tv}]) where {Tv,Ti<:Integer}
```

Parent of and expert driver for `spzeros(I, J)`

allowing user to provide preallocated storage for intermediate objects. This method is to `spzeros`

what `SparseArrays.sparse!`

is to `sparse`

. See documentation for `SparseArrays.sparse!`

for details and required buffer lengths.

This methods requires Julia version 1.10 or later.

`SparseArrays.spzeros!(::Type{Tv}, I, J, [m, n]) -> SparseMatrixCSC{Tv}`

Variant of `spzeros!`

that re-uses the input vectors `I`

and `J`

for the final matrix storage. After construction the input vectors will alias the matrix buffers; `S.colptr === I`

and `S.rowval === J`

holds, and they will be `resize!`

d as necessary.

Note that some work buffers will still be allocated. Specifically, this method is a convenience wrapper around `spzeros!(Tv, I, J, m, n, klasttouch, csrrowptr, csrcolval, csccolptr, cscrowval)`

where this method allocates `klasttouch`

, `csrrowptr`

, and `csrcolval`

of appropriate size, but reuses `I`

and `J`

for `csccolptr`

and `cscrowval`

.

Arguments `m`

and `n`

defaults to `maximum(I)`

and `maximum(J)`

.

This method requires Julia version 1.10 or later.

`SparseArrays.spdiagm`

— Function```
spdiagm(kv::Pair{<:Integer,<:AbstractVector}...)
spdiagm(m::Integer, n::Integer, kv::Pair{<:Integer,<:AbstractVector}...)
```

Construct a sparse diagonal matrix from `Pair`

s 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**

```
julia> spdiagm(-1 => [1,2,3,4], 1 => [4,3,2,1])
5×5 SparseMatrixCSC{Int64, Int64} with 8 stored entries:
⋅ 4 ⋅ ⋅ ⋅
1 ⋅ 3 ⋅ ⋅
⋅ 2 ⋅ 2 ⋅
⋅ ⋅ 3 ⋅ 1
⋅ ⋅ ⋅ 4 ⋅
```

```
spdiagm(v::AbstractVector)
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.

These functions require at least Julia 1.6.

**Examples**

```
julia> spdiagm([1,2,3])
3×3 SparseMatrixCSC{Int64, Int64} with 3 stored entries:
1 ⋅ ⋅
⋅ 2 ⋅
⋅ ⋅ 3
julia> spdiagm(sparse([1,0,3]))
3×3 SparseMatrixCSC{Int64, Int64} with 2 stored entries:
1 ⋅ ⋅
⋅ ⋅ ⋅
⋅ ⋅ 3
```

`SparseArrays.sparse_hcat`

— Function`sparse_hcat(A...)`

Concatenate along dimension 2. Return a SparseMatrixCSC object.

This method was added in Julia 1.8. It mimics 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_vcat`

— Function`sparse_vcat(A...)`

Concatenate along dimension 1. Return a SparseMatrixCSC object.

This method was added in Julia 1.8. It mimics 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`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.

This method was added in Julia 1.8. It mimics 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.blockdiag`

— Function`blockdiag(A...)`

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

**Examples**

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

`SparseArrays.sprand`

— Function```
sprand([rng],[T::Type],m,[n],p::AbstractFloat)
sprand([rng],m,[n],p::AbstractFloat,[rfn=rand])
```

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`

). The optional `rng`

argument specifies a random number generator, see Random Numbers. The optional `T`

argument specifies the element type, which defaults to `Float64`

.

By default, nonzero values are sampled from a uniform distribution using the `rand`

function, i.e. by `rand(T)`

, or `rand(rng, T)`

if `rng`

is supplied; for the default `T=Float64`

, this corresponds to nonzero values sampled uniformly in `[0,1)`

.

You can sample nonzero values from a different distribution by passing a custom `rfn`

function instead of `rand`

. This should be a function `rfn(k)`

that returns an array of `k`

random numbers sampled from the desired distribution; alternatively, if `rng`

is supplied, it should instead be a function `rfn(rng, k)`

.

**Examples**

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

`SparseArrays.sprandn`

— Function`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 Random Numbers.

Specifying the output element type `Type`

requires at least Julia 1.1.

**Examples**

```
julia> sprandn(2, 2, 0.75)
2×2 SparseMatrixCSC{Float64, Int64} with 3 stored entries:
-1.20577 ⋅
0.311817 -0.234641
```

`SparseArrays.nonzeros`

— Function`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`

.

**Examples**

```
julia> A = sparse(2I, 3, 3)
3×3 SparseMatrixCSC{Int64, Int64} with 3 stored entries:
2 ⋅ ⋅
⋅ 2 ⋅
⋅ ⋅ 2
julia> nonzeros(A)
3-element Vector{Int64}:
2
2
2
```

`SparseArrays.rowvals`

— Function`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 `nzrange`

.

**Examples**

```
julia> A = sparse(2I, 3, 3)
3×3 SparseMatrixCSC{Int64, Int64} with 3 stored entries:
2 ⋅ ⋅
⋅ 2 ⋅
⋅ ⋅ 2
julia> rowvals(A)
3-element Vector{Int64}:
1
2
3
```

`SparseArrays.nzrange`

— Function`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 `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 j = 1:n
for i in nzrange(A, j)
row = rows[i]
val = vals[i]
# perform sparse wizardry...
end
end
```

Adding or removing nonzero elements to the matrix may invalidate the `nzrange`

, one should not mutate the matrix while iterating.

`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`droptol!(A::AbstractSparseMatrixCSC, tol)`

Removes stored values from `A`

whose absolute value is less than or equal to `tol`

.

`droptol!(x::AbstractCompressedVector, tol)`

Removes stored values from `x`

whose absolute value is less than or equal to `tol`

.

`SparseArrays.dropzeros!`

— Function`dropzeros!(x::AbstractCompressedVector)`

Removes stored numerical zeros from `x`

.

For an out-of-place version, see `dropzeros`

. For algorithmic information, see `fkeep!`

.

`SparseArrays.dropzeros`

— Function`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**

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

`dropzeros(x::AbstractCompressedVector)`

Generates a copy of `x`

and removes numerical zeros from that copy.

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

.

**Examples**

```
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
```

`SparseArrays.permute`

— Function```
permute(A::AbstractSparseMatrixCSC{Tv,Ti}, p::AbstractVector{<:Integer},
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**

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

`Base.permute!`

— Method```
permute!(X::AbstractSparseMatrixCSC{Tv,Ti}, A::AbstractSparseMatrixCSC{Tv,Ti},
p::AbstractVector{<:Integer}, q::AbstractVector{<:Integer},
[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`

:

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

.

See also `permute`

.

`SparseArrays.halfperm!`

— Function```
halfperm!(X::AbstractSparseMatrixCSC{Tv,Ti}, A::AbstractSparseMatrixCSC{TvA,Ti},
q::AbstractVector{<:Integer}, f::Function = identity) where {Tv,TvA,Ti}
```

Column-permute and transpose `A`

, simultaneously applying `f`

to each entry of `A`

, storing the result `(f(A)Q)^T`

(`map(f, transpose(A[:,q]))`

) in `X`

.

Element type `Tv`

of `X`

must match `f(::TvA)`

, where `TvA`

is the element type of `A`

. `X`

's dimensions must match those of `transpose(A)`

(`size(X, 1) == size(A, 2)`

and `size(X, 2) == size(A, 1)`

), 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)`

).

This method is the parent of several methods performing transposition and permutation operations on `SparseMatrixCSC`

s. As this method performs no argument checking, prefer the safer child methods (`[c]transpose[!]`

, `permute[!]`

) to direct use.

This method implements the `HALFPERM`

algorithm described in F. Gustavson, "Two fast algorithms for sparse matrices: multiplication and permuted transposition," ACM TOMS 4(3), 250-269 (1978). The algorithm runs in `O(size(A, 1), size(A, 2), nnz(A))`

time and requires no space beyond that passed in.

`SparseArrays.ftranspose!`

— Function`ftranspose!(X::AbstractSparseMatrixCSC{Tv,Ti}, A::AbstractSparseMatrixCSC{Tv,Ti}, f::Function) where {Tv,Ti}`

Transpose `A`

and store it in `X`

while applying the function `f`

to the non-zero elements. Does not remove the zeros created by `f`

. `size(X)`

must be equal to `size(transpose(A))`

. No additional memory is allocated other than resizing the rowval and nzval of `X`

, if needed.

See `halfperm!`

# Noteworthy External Sparse Packages

Several other Julia packages provide sparse matrix implementations that should be mentioned:

SuiteSparseGraphBLAS.jl is a wrapper over the fast, multithreaded SuiteSparse:GraphBLAS C library. On CPU this is typically the fastest option, often significantly outperforming MKLSparse.

CUDA.jl exposes the CUSPARSE library for GPU sparse matrix operations.

SparseMatricesCSR.jl provides a Julia native implementation of the Compressed Sparse Rows (CSR) format.

MKLSparse.jl accelerates SparseArrays sparse-dense matrix operations using Intel's MKL library.

SparseArrayKit.jl available for multidimensional sparse arrays.

LuxurySparse.jl provides static sparse array formats, as well as a coordinate format.

ExtendableSparse.jl enables fast insertion into sparse matrices using a lazy approach to new stored indices.

Finch.jl supports extensive multidimensional sparse array formats and operations through a mini tensor language and compiler, all in native Julia. Support for COO, CSF, CSR, CSC and more, as well as operations like broadcast, reduce, etc. and custom operations.