A CSR formatted sparse matrix. More...

Public Types
using	policy = dg::get_execution_policy<Vector<Value>>
	Execution policy of the matrix.

using	index_type = Index
	The index type (on GPU must be either `int` or `long`)

using	value_type = Value
	The value type (on GPU must be either `float` or `double`)

template<class OtherMatrix >
using	enable_if_serial = std::enable_if_t<std::is_same_v<typename OtherMatrix::policy, SerialTag>, OtherMatrix>
	A static guard to allow certain members only on host (for serial execution)

Public Member Functions
	SparseMatrix ()=default
	Empty matrix.

	SparseMatrix (size_t num_rows, size_t num_cols, const Vector< Index > &row_offsets, const Vector< Index > &column_indices, const Vector< Value > &values)
	Directly construct from given vectors.

template<class I , class V , template< class > class Vec>
	SparseMatrix (const SparseMatrix< I, V, Vec > &src)
	Copy construct from differen type.

void	setFromCoo (size_t num_rows, size_t num_cols, const Vector< Index > &row_indices, const Vector< Index > &column_indices, const Vector< Value > &values, bool sort=false)
	Set csr values from coo formatted sparse matrix.

void	set (size_t num_rows, size_t num_cols, const Vector< Index > &row_offsets, const Vector< Index > column_indices, const Vector< Value > &values, bool sort=false)
	Set csr values directly.

void	sort_indices ()
	Sort by row and column.

size_t	total_num_rows () const
	Alias for `num_rows`.

size_t	total_num_cols () const
	Alias for `num_cols`.

size_t	num_rows () const
	Number of rows in matrix.

size_t	num_cols () const
	Number of columns in matrix.

size_t	num_vals () const
	Alias for `num_nnz`.

size_t	num_nnz () const
	Number of nonzero elements in matrix.

size_t	num_entries () const
	Alias for `num_nnz`.

const Vector< Index > &	row_offsets () const
	Read row_offsets vector.

const Vector< Index > &	column_indices () const
	Read column indices vector.

const Vector< Value > &	values () const
	Read values vector.

Vector< Value > &	values ()
	Change values vector.

SparseMatrix	transpose () const
	Transposition.

template<class OtherMatrix = SparseMatrix>
std::enable_if_t< dg::has_policy_v< OtherMatrix, SerialTag >, bool >	operator!= (const SparseMatrix &rhs) const
	Two Matrices are considered equal if elements and sparsity pattern are equal.

template<class OtherMatrix = SparseMatrix>
std::enable_if_t< dg::has_policy_v< OtherMatrix, SerialTag >, bool >	operator== (const SparseMatrix &rhs) const
	Two Matrices are considered equal if elements and sparsity pattern are equal.

SparseMatrix	operator- () const
	Negate.

template<class OtherMatrix = SparseMatrix>
enable_if_serial< OtherMatrix > &	operator+= (const SparseMatrix &op)
	Add.

template<class OtherMatrix = SparseMatrix>
enable_if_serial< OtherMatrix > &	operator-= (const SparseMatrix &op)
	subtract

SparseMatrix &	operator*= (const Value &value)
	scalar multiply

Friends
template<class I , class V , template< class > class Vec>
class	SparseMatrix
	Enable copy constructor from foreign types.

template<class OtherMatrix = SparseMatrix>
enable_if_serial< OtherMatrix >	operator+ (const SparseMatrix &lhs, const SparseMatrix &rhs)
	add

template<class OtherMatrix = SparseMatrix>
enable_if_serial< OtherMatrix >	operator- (const SparseMatrix &lhs, const SparseMatrix &rhs)
	subtract

template<class OtherMatrix = SparseMatrix>
enable_if_serial< OtherMatrix >	operator* (const Value &value, const SparseMatrix &rhs)
	scalar multiplication

template<class OtherMatrix = SparseMatrix>
enable_if_serial< OtherMatrix >	operator* (const SparseMatrix &lhs, const Value &value)
	scalar multiplication

template<class OtherMatrix = SparseMatrix>
enable_if_serial< OtherMatrix >	operator* (const SparseMatrix &lhs, const SparseMatrix &rhs)
	matrix-matrix multiplication \( C = A*B\)

template<class ContainerType , class = std::enable_if_t < dg::has_policy_v<ContainerType, policy> and dg::is_vector_v<ContainerType> >>
ContainerType	operator* (const SparseMatrix &S, const ContainerType &x)
	matrix-vector multiplication \( y = S x\)

template<class Ostream , class OtherMatrix = SparseMatrix>
std::enable_if_t< dg::has_policy_v< OtherMatrix, SerialTag >, Ostream > &	operator<< (Ostream &os, const SparseMatrix &mat)
	puts a matrix linewise in output stream

Detailed Description

template<class Index = int, class Value = double, template< class > class Vector = thrust::host_vector>
struct dg::SparseMatrix< Index, Value, Vector >

A CSR formatted sparse matrix.

This class was designed to replace our dependency on cusp::csr_matrix. On the host arithmetic operators like + and * are overloaded allowing for expressive code to concisely assemble the matrix:

// 1 0 0 2
// 2 0 5 0
// 0 4 0 0
// 0 1 1 0
unsigned num_rows = 4, num_cols = 4;
std::vector<int> rows = {0,2,4,5,7}, cols = {0,3,0,2,1,1,2};
std::vector<double> vals = {1,2,2,5,4,1,1};
dg::SparseMatrix<int,double,std::vector> A ( num_rows, num_cols, rows, cols, vals);
//
// 2 0 0 0
// 0 3 3 5
// 0 0 4 0
// 0 0 0 5
// 1 0 0 2
num_rows = 5, num_cols = 4;
rows = {0,1,4,5,6,8};
cols = {0,1,2,3,2,3,0,3};
vals = {2,3,3,5,4,5,1,2};
dg::SparseMatrix<int,double,std::vector> B ( num_rows, num_cols, rows, cols, vals);
//
// 3 4 0 0
// 10 16.5 13.5 8.5
// 0 20 2 4
// 10 0 0 2.5
// 5.5 2 0 1
auto C = B*A.transpose()+0.5*B;
//
CHECK( C.num_rows() == 5);
CHECK( C.num_cols() == 4);
CHECK( C.row_offsets() == std::vector<int>{ 0,2,6,9,11,14});
CHECK( C.column_indices() == std::vector<int>{ 0,1,0,1,2,3,1,2,3,0,3,0,1,3});
CHECK( C.values() == std::vector<double>{ 3, 4, 10, 16.5, 13.5, 8.5, 20, 2, 4, 10, 2.5, 5.5, 2, 1});
//
// Matrix-vector multiplication can be done on device
dg::SparseMatrix<int,double,thrust::device_vector> dC = C;
std::vector<double>  v = { 2,4,6,8}, w = {1,2,3,4,5};
thrust::device_vector<double> dv( v.begin(), v.end()), dw( w.begin(), w.end());
// dw = dC * dv + 0.5 dw
dg::blas2::gemv( 1., dC, dv, 0.5, dw);
//
thrust::copy( dw.begin(), dw.end(), w.begin());
CHECK( w == std::vector{22.5,236.,125.5,42.,29.5});

On the device like OpenMP or GPU the only currently allowed operations are transpose and the matrix-vector multiplications through dg::blas2::symv (and its aliases dg::blas2::gemv and dg::apply). We use the cusparse library to dispatch matrix-vector multiplication on the GPU.

See also: The CSR format is nicely explained at the cusparse documentation We use the zero-base index

Template Parameters

Index	The index type (on GPU must be either `int` or `long`)
Value	The value type (on GPU must be either `float` or `double`)
Vector	The vector class to store indices and values (typically `thrust::host_vector` or `thrust::device_vector`)

See also: IHMatrix, IDMatrix

Member Typedef Documentation

◆ enable_if_serial

template<class Index = int, class Value = double, template< class > class Vector = thrust::host_vector>

template<class OtherMatrix >

using dg::SparseMatrix< Index, Value, Vector >::enable_if_serial = std::enable_if_t<std::is_same_v<typename OtherMatrix::policy, SerialTag>, OtherMatrix>

A static guard to allow certain members only on host (for serial execution)

◆ index_type

template<class Index = int, class Value = double, template< class > class Vector = thrust::host_vector>

using dg::SparseMatrix< Index, Value, Vector >::index_type = Index

The index type (on GPU must be either int or long)

◆ policy

template<class Index = int, class Value = double, template< class > class Vector = thrust::host_vector>

using dg::SparseMatrix< Index, Value, Vector >::policy = dg::get_execution_policy<Vector<Value>>

Execution policy of the matrix.

◆ value_type

template<class Index = int, class Value = double, template< class > class Vector = thrust::host_vector>

using dg::SparseMatrix< Index, Value, Vector >::value_type = Value

The value type (on GPU must be either float or double)

Constructor & Destructor Documentation

◆ SparseMatrix() [1/3]

template<class Index = int, class Value = double, template< class > class Vector = thrust::host_vector>

dg::SparseMatrix< Index, Value, Vector >::SparseMatrix ( )

default

Empty matrix.

Set values using either setFromCoo or set members

◆ SparseMatrix() [2/3]

template<class Index = int, class Value = double, template< class > class Vector = thrust::host_vector>

dg::SparseMatrix< Index, Value, Vector >::SparseMatrix	(	size_t	num_rows,
		size_t	num_cols,
		const Vector< Index > &	row_offsets,
		const Vector< Index > &	column_indices,
		const Vector< Value > &	values )

inline

Directly construct from given vectors.

// 1 0 0 2 3
// 2 0 5 0 0
// 0 4 0 0 1
unsigned num_rows = 3, num_cols = 5;
std::vector<int> rows = {0,3,5,7}, cols = {0,3,4,0,2,1,4};
std::vector<double> vals = {1,2,3,2,5,4,1};
dg::SparseMatrix<int,double,std::vector> A ( num_rows, num_cols, rows, cols, vals);
//
CHECK( A.num_rows() == num_rows);
CHECK( A.num_cols() == num_cols);
CHECK( A.num_vals() == vals.size());
CHECK( A.row_offsets()    == rows);
CHECK( A.column_indices() == cols);
CHECK( A.values()         == vals);

Parameters

num_rows	Number of rows in the matrix
num_cols	Number of columns in the matrix
row_offsets	Vector of size `num_rows+1` representing the starting position of each row in the column_indices and values arrays. `row_offsets.back()` equals the number of non-zero elements in the matrix
column_indices	Vector of size `row_offsets.back()` containing column indices. The column indices may be unsorted within each row and should be unique. If duplicates exist symv may not be correct (from cusparse).
values	Vector of size `row_offsets.back()` containing nonzero values

◆ SparseMatrix() [3/3]

template<class Index = int, class Value = double, template< class > class Vector = thrust::host_vector>

template<class I , class V , template< class > class Vec>

dg::SparseMatrix< Index, Value, Vector >::SparseMatrix ( const SparseMatrix< I, V, Vec > & src )

inline

Copy construct from differen type.

Typically used to transfer a matrix from host to device

// 1 0 0 2 3
// 2 0 5 0 0
// 0 4 0 0 1
unsigned num_rows = 3, num_cols = 5;
// unsorted!!
std::vector<int> rows = {0,3,5,7}, cols = {0,4,3,0,2,1,4};
std::vector<double> vals = {1,3,2,2,5,4,1};
dg::SparseMatrix<int,double,thrust::host_vector> A ( num_rows, num_cols, rows, cols, vals);
dg::SparseMatrix<int,double, thrust::device_vector> dA( A);

Parameters

src	Copy the matrix

Member Function Documentation

◆ column_indices()

template<class Index = int, class Value = double, template< class > class Vector = thrust::host_vector>

const Vector< Index > & dg::SparseMatrix< Index, Value, Vector >::column_indices ( ) const

inline

Read column indices vector.

Returns: column indices

◆ num_cols()

template<class Index = int, class Value = double, template< class > class Vector = thrust::host_vector>

size_t dg::SparseMatrix< Index, Value, Vector >::num_cols ( ) const

inline

Number of columns in matrix.

◆ num_entries()

template<class Index = int, class Value = double, template< class > class Vector = thrust::host_vector>

size_t dg::SparseMatrix< Index, Value, Vector >::num_entries ( ) const

inline

Alias for num_nnz.

◆ num_nnz()

template<class Index = int, class Value = double, template< class > class Vector = thrust::host_vector>

size_t dg::SparseMatrix< Index, Value, Vector >::num_nnz ( ) const

inline

Number of nonzero elements in matrix.

◆ num_rows()

template<class Index = int, class Value = double, template< class > class Vector = thrust::host_vector>

size_t dg::SparseMatrix< Index, Value, Vector >::num_rows ( ) const

inline

Number of rows in matrix.

◆ num_vals()

template<class Index = int, class Value = double, template< class > class Vector = thrust::host_vector>

size_t dg::SparseMatrix< Index, Value, Vector >::num_vals ( ) const

inline

Alias for num_nnz.

◆ operator!=()

template<class Index = int, class Value = double, template< class > class Vector = thrust::host_vector>

template<class OtherMatrix = SparseMatrix>

std::enable_if_t< dg::has_policy_v< OtherMatrix, SerialTag >, bool > dg::SparseMatrix< Index, Value, Vector >::operator!= ( const SparseMatrix< Index, Value, Vector > & rhs ) const

inline

Two Matrices are considered equal if elements and sparsity pattern are equal.

Parameters

rhs	Matrix to be compared to this

Returns: true if rhs does not equal this

◆ operator*=()

template<class Index = int, class Value = double, template< class > class Vector = thrust::host_vector>

SparseMatrix & dg::SparseMatrix< Index, Value, Vector >::operator*= ( const Value & value )

inline

scalar multiply

Parameters

value

Returns

◆ operator+=()

template<class Index = int, class Value = double, template< class > class Vector = thrust::host_vector>

template<class OtherMatrix = SparseMatrix>

enable_if_serial< OtherMatrix > & dg::SparseMatrix< Index, Value, Vector >::operator+= ( const SparseMatrix< Index, Value, Vector > & op )

inline

Add.

Parameters

op

Returns

◆ operator-()

template<class Index = int, class Value = double, template< class > class Vector = thrust::host_vector>

SparseMatrix dg::SparseMatrix< Index, Value, Vector >::operator- ( ) const

inline

Negate.

Returns

◆ operator-=()

template<class Index = int, class Value = double, template< class > class Vector = thrust::host_vector>

template<class OtherMatrix = SparseMatrix>

enable_if_serial< OtherMatrix > & dg::SparseMatrix< Index, Value, Vector >::operator-= ( const SparseMatrix< Index, Value, Vector > & op )

inline

subtract

Parameters

op

Returns

◆ operator==()

template<class Index = int, class Value = double, template< class > class Vector = thrust::host_vector>

template<class OtherMatrix = SparseMatrix>

std::enable_if_t< dg::has_policy_v< OtherMatrix, SerialTag >, bool > dg::SparseMatrix< Index, Value, Vector >::operator== ( const SparseMatrix< Index, Value, Vector > & rhs ) const

inline

Two Matrices are considered equal if elements and sparsity pattern are equal.

Parameters

rhs	Matrix to be compared to this

Returns: true if rhs equals this

◆ row_offsets()

template<class Index = int, class Value = double, template< class > class Vector = thrust::host_vector>

const Vector< Index > & dg::SparseMatrix< Index, Value, Vector >::row_offsets ( ) const

inline

Read row_offsets vector.

Returns: row_offsets

◆ set()

template<class Index = int, class Value = double, template< class > class Vector = thrust::host_vector>

void dg::SparseMatrix< Index, Value, Vector >::set	(	size_t	num_rows,
		size_t	num_cols,
		const Vector< Index > &	row_offsets,
		const Vector< Index >	column_indices,
		const Vector< Value > &	values,
		bool	sort = false )

inline

Set csr values directly.

// 0 2 0 0 4
// 0 1 2 0 0
// 0 0 0 0 0
num_rows = 3, num_cols = 5;
rows = {0,2,4,4}, cols = {1,4,1,2};
vals = {2,4,1,2};
dg::SparseMatrix<int,double,std::vector> D;
D.set( num_rows, num_cols, rows, cols, vals);

Parameters

num_rows	Number of rows in the matrix
num_cols	Number of columns in the matrix
row_offsets	Vector of size `num_rows+1` representing the starting position of each row in the column_indices and values arrays. `row_offsets.back()` equals the number of non-zero elements in the matrix
column_indices	Vector of size `row_offsets.back()` containing column indices. The column indices may be unsorted within each row and should be unique. If duplicates exist symv may not be correct (from cusparse).
values	Vector of size `row_offsets.back()` containing nonzero values
sort	If `true` the given vectors are sorted by column_indices in each row. It is allowed to have unsorted column indices in each row so sorting is not strictly necessary. It may (or may not) have an effect on performance.

◆ setFromCoo()

template<class Index = int, class Value = double, template< class > class Vector = thrust::host_vector>

void dg::SparseMatrix< Index, Value, Vector >::setFromCoo	(	size_t	num_rows,
		size_t	num_cols,
		const Vector< Index > &	row_indices,
		const Vector< Index > &	column_indices,
		const Vector< Value > &	values,
		bool	sort = false )

inline

Set csr values from coo formatted sparse matrix.

// 1 0 0 2 3
// 2 0 5 0 0
// 0 4 0 0 1
unsigned num_rows = 3, num_cols = 5;
std::vector<int> rows = {2,2,0,0,0,1,1}, cols = {4,1,4,3,0,2,0};
std::vector<double> vals = {1,4, 3,2,1,5,2};
dg::SparseMatrix<int,double,std::vector> A;
A.setFromCoo( num_rows, num_cols, rows, cols, vals, true);
CHECK( A.row_offsets() == std::vector{0,3,5,7});
CHECK( A.column_indices() == std::vector{0,3,4,0,2,1,4});
CHECK( A.values() == std::vector<double>{1,2,3,2,5,4,1});

Note: May be unsorted, in which case the sort parameter should be set to true; if duplicates exist symv may not be correct (from cusparse)

Parameters

num_rows	Number of rows in the matrix
num_cols	Number of columns in the matrix
row_indices	Contains row indices of the corresponding elements in the values vector
column_indices	Vector of size `row_indices.size()()` containing column indices.
values	Vector of size `row_indices.size()` containing nonzero values
sort	If `true` the given vectors are sorted by row and column before converted to the csr format. Per default we assume that the values are sorted by row (not necessarily by column). In this case there is no need to sort the indices, which may be somewhat expensive. If the row indices are unsorted however, `sort` must be set to `true` otherwise the coo2csr conversion will fail.

◆ sort_indices()

template<class Index = int, class Value = double, template< class > class Vector = thrust::host_vector>

void dg::SparseMatrix< Index, Value, Vector >::sort_indices ( )

inline

Sort by row and column.

Sort is somewhat expensive and it is allowed to have unsorted column indices (at least in cusparse). Sorting may or may not have an influence on performance. Never sort a stencil for dg::blas2::stencil.

◆ total_num_cols()

template<class Index = int, class Value = double, template< class > class Vector = thrust::host_vector>

size_t dg::SparseMatrix< Index, Value, Vector >::total_num_cols ( ) const

inline

Alias for num_cols.

◆ total_num_rows()

template<class Index = int, class Value = double, template< class > class Vector = thrust::host_vector>

size_t dg::SparseMatrix< Index, Value, Vector >::total_num_rows ( ) const

inline

Alias for num_rows.

◆ transpose()

template<class Index = int, class Value = double, template< class > class Vector = thrust::host_vector>

SparseMatrix dg::SparseMatrix< Index, Value, Vector >::transpose ( ) const

inline

Transposition.

The transpose is sorted even if the original is not

Returns: A newly generated SparseMatrix containing the transpose.

◆ values() [1/2]

template<class Index = int, class Value = double, template< class > class Vector = thrust::host_vector>

Vector< Value > & dg::SparseMatrix< Index, Value, Vector >::values ( )

inline

Change values vector.

Note: The reason why values can be changed directly while row_offsets and column_indices cannot, is that changing the values does not influence the performance cache, while changing the sparsity pattern through row_offsets and column_indices does

Returns: Values array reference

◆ values() [2/2]

template<class Index = int, class Value = double, template< class > class Vector = thrust::host_vector>

const Vector< Value > & dg::SparseMatrix< Index, Value, Vector >::values ( ) const

inline

Read values vector.

Returns: values

Friends And Related Symbol Documentation

◆ operator* [1/4]

template<class Index = int, class Value = double, template< class > class Vector = thrust::host_vector>

template<class OtherMatrix = SparseMatrix>

enable_if_serial< OtherMatrix > operator*	(	const SparseMatrix< Index, Value, Vector > &	lhs,
		const SparseMatrix< Index, Value, Vector > &	rhs )

friend

matrix-matrix multiplication \( C = A*B\)

Parameters

lhs
rhs

Returns: multiplication (sorted even if lhs and rhs are not)

◆ operator* [2/4]

template<class Index = int, class Value = double, template< class > class Vector = thrust::host_vector>

template<class OtherMatrix = SparseMatrix>

enable_if_serial< OtherMatrix > operator*	(	const SparseMatrix< Index, Value, Vector > &	lhs,
		const Value &	value )

friend

scalar multiplication

Simply multiply values array by given value

Parameters

lhs
value

Returns

◆ operator* [3/4]

template<class Index = int, class Value = double, template< class > class Vector = thrust::host_vector>

template<class ContainerType , class = std::enable_if_t < dg::has_policy_v<ContainerType, policy> and dg::is_vector_v<ContainerType> >>

ContainerType operator*	(	const SparseMatrix< Index, Value, Vector > &	S,
		const ContainerType &	x )

friend

matrix-vector multiplication \( y = S x\)

Works if ContainerType has the same execution policy as the SparseMatrix

auto mat = dg::SquareMatrix<double>({0,1,2, 3,4,5, 6,7,8});
std::vector<double> vec = {1,2,3};
std::vector<double> res = mat*vec;
CHECK( res == std::vector<double>{8, 26, 44});

Parameters

S	Matrix
x	Vector

Returns: Vector

◆ operator* [4/4]

template<class Index = int, class Value = double, template< class > class Vector = thrust::host_vector>

template<class OtherMatrix = SparseMatrix>

enable_if_serial< OtherMatrix > operator*	(	const Value &	value,
		const SparseMatrix< Index, Value, Vector > &	rhs )

friend

scalar multiplication

Simply multiply values array by given value

Parameters

value
rhs

Returns

◆ operator+

template<class Index = int, class Value = double, template< class > class Vector = thrust::host_vector>

template<class OtherMatrix = SparseMatrix>

enable_if_serial< OtherMatrix > operator+	(	const SparseMatrix< Index, Value, Vector > &	lhs,
		const SparseMatrix< Index, Value, Vector > &	rhs )

friend

add

Parameters

lhs
rhs

Returns: addition (sorted even if lhs and rhs are not)

◆ operator-

template<class Index = int, class Value = double, template< class > class Vector = thrust::host_vector>

template<class OtherMatrix = SparseMatrix>

enable_if_serial< OtherMatrix > operator-	(	const SparseMatrix< Index, Value, Vector > &	lhs,
		const SparseMatrix< Index, Value, Vector > &	rhs )

friend

subtract

Parameters

lhs
rhs

Returns: subtraction (sorted even if lhs and rhs are not)

◆ operator<<

template<class Index = int, class Value = double, template< class > class Vector = thrust::host_vector>

template<class Ostream , class OtherMatrix = SparseMatrix>

std::enable_if_t< dg::has_policy_v< OtherMatrix, SerialTag >, Ostream > & operator<<	(	Ostream &	os,
		const SparseMatrix< Index, Value, Vector > &	mat )

friend

puts a matrix linewise in output stream

Template Parameters

Ostream The stream e.g. std::cout

Parameters

os	the outstream
mat	the matrix to output

Returns: the outstream

◆ SparseMatrix

template<class Index = int, class Value = double, template< class > class Vector = thrust::host_vector>

template<class I , class V , template< class > class Vec>

friend class SparseMatrix

friend

Enable copy constructor from foreign types.

The documentation for this struct was generated from the following file:

backend/sparsematrix.h

Public Types

Public Member Functions

Friends

Detailed Description

Member Typedef Documentation

◆ enable_if_serial

◆ index_type

◆ policy

◆ value_type

Constructor & Destructor Documentation

◆ SparseMatrix() [1/3]

◆ SparseMatrix() [2/3]

◆ SparseMatrix() [3/3]

Member Function Documentation

◆ column_indices()

◆ num_cols()

◆ num_entries()

◆ num_nnz()

◆ num_rows()

◆ num_vals()

◆ operator!=()

◆ operator*=()

◆ operator+=()

◆ operator-()

◆ operator-=()

◆ operator==()

◆ row_offsets()

◆ set()

◆ setFromCoo()

◆ sort_indices()

◆ total_num_cols()

◆ total_num_rows()

◆ transpose()

◆ values() [1/2]

◆ values() [2/2]

Friends And Related Symbol Documentation

◆ operator* [1/4]

◆ operator* [2/4]

◆ operator* [3/4]

◆ operator* [4/4]

◆ operator+

◆ operator-

◆ operator<<

◆ SparseMatrix