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+---
+title: sparse
+---
+
+# The `stdlib_sparse` module
+
+[TOC]
+
+## Introduction
+
+The `stdlib_sparse` module provides derived types for standard sparse matrix data structures. It also provides math kernels such as sparse matrix-vector product and conversion between matrix types.
+
+## Sparse matrix derived types
+
+<!-- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -->
+### The `sparse_type` abstract derived type
+#### Status
+
+Experimental
+
+#### Description
+The parent `sparse_type` is as an abstract derived type holding the basic common meta data needed to define a sparse matrix, as well as shared APIs. All sparse matrix flavors are extended from the `sparse_type`.
+
+```Fortran
+type, public, abstract :: sparse_type
+integer :: nrows !! number of rows
+integer :: ncols !! number of columns
+integer :: nnz !! number of non-zero values
+integer :: storage !! assumed storage symmetry
+end type
+```
+
+The storage integer label should be assigned from the module's internal enumerator containing the following three enums:
+
+```Fortran
+enum, bind(C)
+enumerator :: sparse_full !! Full Sparse matrix (no symmetry considerations)
+enumerator :: sparse_lower !! Symmetric Sparse matrix with triangular inferior storage
+enumerator :: sparse_upper !! Symmetric Sparse matrix with triangular supperior storage
+end enum
+```
+In the following, all sparse kinds will be presented in two main flavors: a data-less type `<matrix>_type` useful for topological graph operations. And real/complex valued types `<matrix>_<kind>_type` containing the `data` buffer for the matrix values. The following rectangular matrix will be used to showcase how each sparse matrix holds the data internally:
+
+$$ M = \begin{bmatrix}
+9 & 0 & 0 & 0 & -3 \\
+4 & 7 & 0 & 0 & 0 \\
+0 & 8 & -1 & 8 & 0 \\
+4 & 0 & 5 & 6 & 0 \\
+\end{bmatrix} $$
+<!-- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -->
+### `COO`: The COOrdinates compressed sparse format
+#### Status
+
+Experimental
+
+#### Description
+The `COO`, triplet or `ijv` format defines all non-zero elements of the matrix by explicitly allocating the `i,j` index and the value of the matrix. While some implementations use separate `row` and `col` arrays for the index, here we use a 2D array in order to promote fast memory acces to `ij`.
+
+```Fortran
+type(COO_sp_type) :: COO
+call COO%malloc(4,5,10)
+COO%data(:) = real([9,-3,4,7,8,-1,8,4,5,6])
+COO%index(1:2,1) = [1,1]
+COO%index(1:2,2) = [1,5]
+COO%index(1:2,3) = [2,1]
+COO%index(1:2,4) = [2,2]
+COO%index(1:2,5) = [3,2]
+COO%index(1:2,6) = [3,3]
+COO%index(1:2,7) = [3,4]
+COO%index(1:2,8) = [4,1]
+COO%index(1:2,9) = [4,3]
+COO%index(1:2,10) = [4,4]
+```
+<!-- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -->
+### `CSR`: The Compressed Sparse Row or Yale format
+#### Status
+
+Experimental
+
+#### Description
+The Compressed Sparse Row or Yale format `CSR` stores the matrix structure by compressing the row indices with a counter pointer `rowptr` enabling to know the first and last non-zero column index `col` of the given row.
+
+```Fortran
+type(CSR_sp_type) :: CSR
+call CSR%malloc(4,5,10)
+CSR%data(:) = real([9,-3,4,7,8,-1,8,4,5,6])
+CSR%col(:) = [1,5,1,2,2,3,4,1,3,4]
+CSR%rowptr(:) = [1,3,5,8,11]
+```
+<!-- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -->
+### `CSC`: The Compressed Sparse Column format
+#### Status
+
+Experimental
+
+#### Description
+The Compressed Sparse Colum `CSC` is similar to the `CSR` format but values are accesed first by column, thus an index counter is given by `colptr` which enables to know the first and last non-zero row index of a given colum.
+
+```Fortran
+type(CSC_sp_type) :: CSC
+call CSC%malloc(4,5,10)
+CSC%data(:) = real([9,4,4,7,8,-1,5,8,6,-3])
+CSC%row(:) = [1,2,4,2,3,3,4,3,4,1]
+CSC%colptr(:) = [1,4,6,8,10,11]
+```
+<!-- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -->
+### `ELLPACK`: ELL-pack storage format
+#### Status
+
+Experimental
+
+#### Description
+The `ELL` format stores data in a dense matrix of $nrows \times K$ in column major order. By imposing a constant number of elements per row $K$, this format will incur in additional zeros being stored, but it enables efficient vectorization as memory acces is carried out by constant sized strides.
+
+```Fortran
+type(ELL_sp_type) :: ELL
+call ELL%malloc(num_rows=4,num_cols=5,num_nz_row=3)
+ELL%data(1,1:3) = real([9,-3,0])
+ELL%data(2,1:3) = real([4,7,0])
+ELL%data(3,1:3) = real([8,-1,8])
+ELL%data(4,1:3) = real([4,5,6])
+
+ELL%index(1,1:3) = [1,5,0]
+ELL%index(2,1:3) = [1,2,0]
+ELL%index(3,1:3) = [2,3,4]
+ELL%index(4,1:3) = [1,3,4]
+```
+<!-- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -->
+### `SELL-C`: The Sliced ELLPACK with Constant blocks format
+#### Status
+
+Experimental
+
+#### Description
+The Sliced ELLPACK format `SELLC` is a variation of the `ELLPACK` format. This modification reduces the storage size compared to the `ELLPACK` format but maintaining its efficient data access scheme. It can be seen as an intermediate format between `CSR` and `ELLPACK`. For more details read [the reference](https://arxiv.org/pdf/1307.6209v1)
+
+<!-- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -->
+## `add`- sparse matrix data accessors
+
+### Status
+
+Experimental
+
+### Description
+Type-bound procedures to enable adding data in a sparse matrix.
+
+### Syntax
+
+`call matrix%add(i,j,v)` or
+`call matrix%add(i(:),j(:),v(:,:))`
+
+### Arguments
+
+`i`: Shall be an integer value or rank-1 array. It is an `intent(in)` argument.
+
+`j`: Shall be an integer value or rank-1 array. It is an `intent(in)` argument.
+
+`v`: Shall be a `real` or `complex` value or rank-2 array. The type shall be in accordance to the declared sparse matrix object. It is an `intent(in)` argument.
+
+## `at`- sparse matrix data accessors
+
+### Status
+
+Experimental
+
+### Description
+Type-bound procedures to enable requesting data from a sparse matrix.
+
+### Syntax
+
+`v = matrix%at(i,j)`
+
+### Arguments
+
+`i` : Shall be an integer value. It is an `intent(in)` argument.
+
+`j` : Shall be an integer value. It is an `intent(in)` argument.
+
+`v` : Shall be a `real` or `complex` value in accordance to the declared sparse matrix object. If the `ij` tuple is within the sparse pattern, `v` contains the value in the data buffer. If the `ij` tuple is outside the sparse pattern, `v` is equal `0`. If the `ij` tuple is outside the matrix pattern `(nrows,ncols)`, `v` is `NaN`.
+
+## Example
+```fortran
+{!example/linalg/example_sparse_data_accessors.f90!}
+```
+
+<!-- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -->
+## `spmv` - Sparse Matrix-Vector product
+
+### Status
+
+Experimental
+
+### Description
+
+Provide sparse matrix-vector product kernels for the current supported sparse matrix types.
+
+$$y=\alpha*op(M)*x+\beta*y$$
+
+### Syntax
+
+`call ` [[stdlib_sparse_spmv(module):spmv(interface)]] `(matrix,vec_x,vec_y [,alpha,beta,op])`
+
+### Arguments
+
+`matrix`: Shall be a `real` or `complex` sparse type matrix. It is an `intent(in)` argument.
+
+`vec_x`: Shall be a rank-1 or rank-2 array of `real` or `complex` type array. It is an `intent(in)` argument.
+
+`vec_y`: Shall be a rank-1 or rank-2 array of `real` or `complex` type array. . It is an `intent(inout)` argument.
+
+`alpha`, `optional` : Shall be a scalar value of the same type as `vec_x`. Default value `alpha=1`. It is an `intent(in)` argument.
+
+`beta`, `optional` : Shall be a scalar value of the same type as `vec_x`. Default value `beta=0`. It is an `intent(in)` argument.
+
+`op`, `optional`: In-place operator identifier. Shall be a `character(1)` argument. It can have any of the following values: `N`: no transpose, `T`: transpose, `H`: hermitian or complex transpose. These values are provided as constants by the `stdlib_sparse` module: `sparse_op_none`, `sparse_op_transpose`, `sparse_op_hermitian`
+
+<!-- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -- -->
+## Sparse matrix to matrix conversions
+
+### Status
+
+Experimental
+
+### Description
+
+This module provides facility functions for converting between storage formats.
+
+### Syntax
+
+`call ` [[stdlib_sparse_conversion(module):coo2ordered(interface)]] `(coo[,sort_data])`
+
+### Arguments
+
+`COO` : Shall be any `COO` type. The same object will be returned with the arrays reallocated to the correct size after removing duplicates. It is an `intent(inout)` argument.
+
+`sort_data`, `optional` : Shall be a `logical` argument to determine whether data in the COO graph should be sorted while sorting the index array, default `.false.`. It is an `intent(in)` argument.
+
+### Syntax
+
+`call ` [[stdlib_sparse_conversion(module):from_ijv(interface)]] `(sparse,row,col[,data,nrows,ncols,num_nz_rows,chunk])`
+
+### Arguments
+
+`sparse` : Shall be a `COO`, `CSR`, `ELL` or `SELLC` type. The graph object will be returned with a canonical shape after sorting and removing duplicates from the `(row,col,data)` triplet. If the graph is `COO_type` no data buffer is allowed. It is an `intent(inout)` argument.
+
+`row` : rows index array. It is an `intent(in)` argument.
+
+`col` : columns index array. It is an `intent(in)` argument.
+
+`data`, `optional`: `real` or `complex` data array. It is an `intent(in)` argument.
+
+`nrows`, `optional`: number of rows, if not given it will be computed from the `row` array. It is an `intent(in)` argument.
+
+`ncols`, `optional`: number of columns, if not given it will be computed from the `col` array. It is an `intent(in)` argument.
+
+`num_nz_rows`, `optional`: number of non zeros per row, only valid in the case of an `ELL` matrix, by default it will computed from the largest row. It is an `intent(in)` argument.
+
+`chunk`, `optional`: chunk size, only valid in the case of a `SELLC` matrix, by default it will be taken from the `SELLC` default attribute chunk size. It is an `intent(in)` argument.
+
+## Example
+```fortran
+{!example/linalg/example_sparse_from_ijv.f90!}
+```
+### Syntax
+
+`call ` [[stdlib_sparse_conversion(module):diag(interface)]] `(matrix,diagonal)`
+
+### Arguments
+
+`matrix` : Shall be a `dense`, `COO`, `CSR` or `ELL` type. It is an `intent(in)` argument.
+
+`diagonal` : A rank-1 array of the same type as the `matrix`. It is an `intent(inout)` and `allocatable` argument.
+
+#### Note
+If the `diagonal` array has not been previously allocated, the `diag` subroutine will allocate it using the `nrows` of the `matrix`.
+
+### Syntax
+
+`call ` [[stdlib_sparse_conversion(module):dense2coo(interface)]] `(dense,coo)`
+
+### Arguments
+
+`dense` : Shall be a rank-2 array of `real` or `complex` type. It is an `intent(in)` argument.
+
+`coo` : Shall be a `COO` type of `real` or `complex` type. It is an `intent(out)` argument.
+
+### Syntax
+
+`call ` [[stdlib_sparse_conversion(module):coo2dense(interface)]] `(coo,dense)`
+
+### Arguments
+
+`coo` : Shall be a `COO` type of `real` or `complex` type. It is an `intent(in)` argument.
+
+`dense` : Shall be a rank-2 array of `real` or `complex` type. It is an `intent(out)` argument.
+
+### Syntax
+
+`call ` [[stdlib_sparse_conversion(module):coo2csr(interface)]] `(coo,csr)`
+
+### Arguments
+
+`coo` : Shall be a `COO` type of `real` or `complex` type. It is an `intent(in)` argument.
+
+`csr` : Shall be a `CSR` type of `real` or `complex` type. It is an `intent(out)` argument.
+
+### Syntax
+
+`call ` [[stdlib_sparse_conversion(module):coo2csc(interface)]] `(coo,csc)`
+
+### Arguments
+
+`coo` : Shall be a `COO` type of `real` or `complex` type. It is an `intent(in)` argument.
+
+`csc` : Shall be a `CSC` type of `real` or `complex` type. It is an `intent(out)` argument.
+
+### Syntax
+
+`call ` [[stdlib_sparse_conversion(module):csr2coo(interface)]] `(csr,coo)`
+
+### Arguments
+
+`csr` : Shall be a `CSR` type of `real` or `complex` type. It is an `intent(in)` argument.
+
+`coo` : Shall be a `COO` type of `real` or `complex` type. It is an `intent(out)` argument.
+
+### Syntax
+
+`call ` [[stdlib_sparse_conversion(module):csr2sellc(interface)]] `(csr,sellc[,chunk])`
+
+### Arguments
+
+`csr` : Shall be a `CSR` type of `real` or `complex` type. It is an `intent(in)` argument.
+
+`sellc` : Shall be a `SELLC` type of `real` or `complex` type. It is an `intent(out)` argument.
+
+`chunk`, `optional`: chunk size for the Sliced ELLPACK format. It is an `intent(in)` argument.
+
+### Syntax
+
+`call ` [[stdlib_sparse_conversion(module):csr2sellc(interface)]] `(csr,ell[,num_nz_rows])`
+
+### Arguments
+
+`csr` : Shall be a `CSR` type of `real` or `complex` type. It is an `intent(in)` argument.
+
+`ell` : Shall be a `ELL` type of `real` or `complex` type. It is an `intent(out)` argument.
+
+`num_nz_rows`, `optional`: number of non zeros per row. If not give, it will correspond to the size of the longest row in the `CSR` matrix. It is an `intent(in)` argument.
+
+### Syntax
+
+`call ` [[stdlib_sparse_conversion(module):csc2coo(interface)]] `(csc,coo)`
+
+### Arguments
+
+`csc` : Shall be a `CSC` type of `real` or `complex` type. It is an `intent(in)` argument.
+
+`coo` : Shall be a `COO` type of `real` or `complex` type. It is an `intent(out)` argument.
+
+## Example
+```fortran
+{!example/linalg/example_sparse_spmv.f90!}
+```