Reference documentation for deal.II version Git b85d307 2018-08-20 17:45:16 -0600
Public Types | Public Member Functions | Private Member Functions | Private Attributes | List of all members
ScaLAPACKMatrix< NumberType > Class Template Reference

#include <deal.II/lac/scalapack.h>

Inheritance diagram for ScaLAPACKMatrix< NumberType >:
[legend]

Public Types

using size_type = unsigned int
 

Public Member Functions

 ScaLAPACKMatrix (const size_type n_rows, const size_type n_columns, const std::shared_ptr< const Utilities::MPI::ProcessGrid > &process_grid, const size_type row_block_size=32, const size_type column_block_size=32, const LAPACKSupport::Property property=LAPACKSupport::Property::general)
 
 ScaLAPACKMatrix (const size_type size, const std::shared_ptr< const Utilities::MPI::ProcessGrid > process_grid, const size_type block_size=32, const LAPACKSupport::Property property=LAPACKSupport::Property::symmetric)
 
 ~ScaLAPACKMatrix () override=default
 
void reinit (const size_type n_rows, const size_type n_columns, const std::shared_ptr< const Utilities::MPI::ProcessGrid > &process_grid, const size_type row_block_size=32, const size_type column_block_size=32, const LAPACKSupport::Property property=LAPACKSupport::Property::general)
 
void reinit (const size_type size, const std::shared_ptr< const Utilities::MPI::ProcessGrid > process_grid, const size_type block_size=32, const LAPACKSupport::Property property=LAPACKSupport::Property::symmetric)
 
void set_property (const LAPACKSupport::Property property)
 
LAPACKSupport::Property get_property () const
 
LAPACKSupport::State get_state () const
 
ScaLAPACKMatrix< NumberType > & operator= (const FullMatrix< NumberType > &)
 
void copy_from (const LAPACKFullMatrix< NumberType > &matrix, const unsigned int rank)
 
void copy_to (FullMatrix< NumberType > &matrix) const
 
void copy_to (LAPACKFullMatrix< NumberType > &matrix, const unsigned int rank) const
 
void copy_to (ScaLAPACKMatrix< NumberType > &dest) const
 
void copy_to (ScaLAPACKMatrix< NumberType > &B, const std::pair< unsigned int, unsigned int > &offset_A, const std::pair< unsigned int, unsigned int > &offset_B, const std::pair< unsigned int, unsigned int > &submatrix_size) const
 
void copy_transposed (const ScaLAPACKMatrix< NumberType > &B)
 
void add (const ScaLAPACKMatrix< NumberType > &B, const NumberType a=0., const NumberType b=1., const bool transpose_B=false)
 
void add (const NumberType b, const ScaLAPACKMatrix< NumberType > &B)
 
void Tadd (const NumberType b, const ScaLAPACKMatrix< NumberType > &B)
 
void mult (const NumberType b, const ScaLAPACKMatrix< NumberType > &B, const NumberType c, ScaLAPACKMatrix< NumberType > &C, const bool transpose_A=false, const bool transpose_B=false) const
 
void mmult (ScaLAPACKMatrix< NumberType > &C, const ScaLAPACKMatrix< NumberType > &B, const bool adding=false) const
 
void Tmmult (ScaLAPACKMatrix< NumberType > &C, const ScaLAPACKMatrix< NumberType > &B, const bool adding=false) const
 
void mTmult (ScaLAPACKMatrix< NumberType > &C, const ScaLAPACKMatrix< NumberType > &B, const bool adding=false) const
 
void TmTmult (ScaLAPACKMatrix< NumberType > &C, const ScaLAPACKMatrix< NumberType > &B, const bool adding=false) const
 
void save (const char *filename, const std::pair< unsigned int, unsigned int > &chunk_size=std::make_pair(numbers::invalid_unsigned_int, numbers::invalid_unsigned_int)) const
 
void load (const char *filename)
 
void compute_cholesky_factorization ()
 
void compute_lu_factorization ()
 
void invert ()
 
std::vector< NumberType > eigenpairs_symmetric_by_index (const std::pair< unsigned int, unsigned int > &index_limits, const bool compute_eigenvectors)
 
std::vector< NumberType > eigenpairs_symmetric_by_value (const std::pair< NumberType, NumberType > &value_limits, const bool compute_eigenvectors)
 
std::vector< NumberType > eigenpairs_symmetric_by_index_MRRR (const std::pair< unsigned int, unsigned int > &index_limits, const bool compute_eigenvectors)
 
std::vector< NumberType > eigenpairs_symmetric_by_value_MRRR (const std::pair< NumberType, NumberType > &value_limits, const bool compute_eigenvectors)
 
std::vector< NumberType > compute_SVD (ScaLAPACKMatrix< NumberType > *U=nullptr, ScaLAPACKMatrix< NumberType > *VT=nullptr)
 
void least_squares (ScaLAPACKMatrix< NumberType > &B, const bool transpose=false)
 
unsigned int pseudoinverse (const NumberType ratio)
 
NumberType reciprocal_condition_number (const NumberType a_norm) const
 
NumberType l1_norm () const
 
NumberType linfty_norm () const
 
NumberType frobenius_norm () const
 
size_type m () const
 
size_type n () const
 
unsigned int local_m () const
 
unsigned int local_n () const
 
unsigned int global_row (const unsigned int loc_row) const
 
unsigned int global_column (const unsigned int loc_column) const
 
NumberType local_el (const unsigned int loc_row, const unsigned int loc_column) const
 
NumberType & local_el (const unsigned int loc_row, const unsigned int loc_column)
 
template<class InputVector >
void scale_columns (const InputVector &factors)
 
template<class InputVector >
void scale_rows (const InputVector &factors)
 

Private Member Functions

NumberType norm_symmetric (const char type) const
 
NumberType norm_general (const char type) const
 
std::vector< NumberType > eigenpairs_symmetric (const bool compute_eigenvectors, const std::pair< unsigned int, unsigned int > &index_limits=std::make_pair(numbers::invalid_unsigned_int, numbers::invalid_unsigned_int), const std::pair< NumberType, NumberType > &value_limits=std::make_pair(std::numeric_limits< NumberType >::quiet_NaN(), std::numeric_limits< NumberType >::quiet_NaN()))
 
std::vector< NumberType > eigenpairs_symmetric_MRRR (const bool compute_eigenvectors, const std::pair< unsigned int, unsigned int > &index_limits=std::make_pair(numbers::invalid_unsigned_int, numbers::invalid_unsigned_int), const std::pair< NumberType, NumberType > &value_limits=std::make_pair(std::numeric_limits< NumberType >::quiet_NaN(), std::numeric_limits< NumberType >::quiet_NaN()))
 

Private Attributes

LAPACKSupport::State state
 
LAPACKSupport::Property property
 
std::shared_ptr< const Utilities::MPI::ProcessGridgrid
 
int n_rows
 
int n_columns
 
int row_block_size
 
int column_block_size
 
int n_local_rows
 
int n_local_columns
 
int descriptor [9]
 
std::vector< NumberType > work
 
std::vector< int > iwork
 
std::vector< int > ipiv
 
const char uplo
 
const int first_process_row
 
const int first_process_column
 
const int submatrix_row
 
const int submatrix_column
 
Threads::Mutex mutex
 

Additional Inherited Members

- Protected Types inherited from TransposeTable< NumberType >
using size_type = typename TableBase< 2, NumberType >::size_type
 
using value_type = typename AlignedVector< NumberType >::value_type
 
using reference = typename AlignedVector< NumberType >::reference
 
using const_reference = typename AlignedVector< NumberType >::const_reference
 
using const_iterator = TransposeTableIterators::Iterator< NumberType, true >
 
using iterator = TransposeTableIterators::Iterator< NumberType, false >
 
- Protected Types inherited from TableBase< 2, NumberType >
using size_type = typename AlignedVector< NumberType >::size_type
 
- Protected Member Functions inherited from TransposeTable< NumberType >
reference el (const size_type i, const size_type j)
 
const_reference el (const size_type i, const size_type j) const
 
 TransposeTable ()=default
 
 TransposeTable (const size_type size1, const size_type size2)
 
void reinit (const size_type size1, const size_type size2, const bool omit_default_initialization=false)
 
const_reference operator() (const size_type i, const size_type j) const
 
reference operator() (const size_type i, const size_type j)
 
size_type n_rows () const
 
size_type n_cols () const
 
iterator begin ()
 
const_iterator begin () const
 
iterator end ()
 
const_iterator end () const
 
- Protected Member Functions inherited from TableBase< 2, NumberType >
size_type position (const TableIndices< N > &indices) const
 
AlignedVector< NumberType >::reference el (const TableIndices< N > &indices)
 
AlignedVector< NumberType >::const_reference el (const TableIndices< N > &indices) const
 
 TableBase ()=default
 
 TableBase (const TableIndices< N > &sizes)
 
 TableBase (const TableIndices< N > &sizes, InputIterator entries, const bool C_style_indexing=true)
 
 TableBase (const TableBase< N, NumberType > &src)
 
 TableBase (const TableBase< N, T2 > &src)
 
 TableBase (TableBase< N, NumberType > &&src) noexcept
 
 ~TableBase () override=default
 
TableBase< N, NumberType > & operator= (const TableBase< N, NumberType > &src)
 
TableBase< N, NumberType > & operator= (const TableBase< N, T2 > &src)
 
TableBase< N, NumberType > & operator= (TableBase< N, NumberType > &&src) noexcept
 
bool operator== (const TableBase< N, NumberType > &T2) const
 
void reset_values ()
 
void reinit (const TableIndices< N > &new_size, const bool omit_default_initialization=false)
 
size_type size (const unsigned int i) const
 
const TableIndices< N > & size () const
 
size_type n_elements () const
 
bool empty () const
 
void fill (InputIterator entries, const bool C_style_indexing=true)
 
void fill (const NumberType &value)
 
AlignedVector< NumberType >::reference operator() (const TableIndices< N > &indices)
 
AlignedVector< NumberType >::const_reference operator() (const TableIndices< N > &indices) const
 
void swap (TableBase< N, NumberType > &v)
 
std::size_t memory_consumption () const
 
void serialize (Archive &ar, const unsigned int version)
 
- Protected Member Functions inherited from Subscriptor
 Subscriptor ()
 
 Subscriptor (const Subscriptor &)
 
 Subscriptor (Subscriptor &&) noexcept
 
virtual ~Subscriptor ()
 
Subscriptoroperator= (const Subscriptor &)
 
Subscriptoroperator= (Subscriptor &&) noexcept
 
void subscribe (const char *identifier=nullptr) const
 
void unsubscribe (const char *identifier=nullptr) const
 
unsigned int n_subscriptions () const
 
template<typename StreamType >
void list_subscribers (StreamType &stream) const
 
void list_subscribers () const
 
template<class Archive >
void serialize (Archive &ar, const unsigned int version)
 
- Static Protected Member Functions inherited from Subscriptor
static::ExceptionBase & ExcInUse (int arg1, std::string arg2, std::string arg3)
 
static::ExceptionBase & ExcNoSubscriber (std::string arg1, std::string arg2)
 
- Protected Attributes inherited from TableBase< 2, NumberType >
AlignedVector< NumberType > values
 
TableIndices< N > table_size
 

Detailed Description

template<typename NumberType>
class ScaLAPACKMatrix< NumberType >

A wrapper class around ScaLAPACK parallel dense linear algebra.

ScaLAPACK assumes that matrices are distributed according to the block-cyclic decomposition scheme. An \(M\) by \(N\) matrix is first decomposed into \(MB\) by \(NB\) blocks which are then uniformly distributed across the 2D process grid \(p*q \le Np\), where \(p,q\) are grid dimensions and \(Np\) is the total number of processes.

For example, a global real symmetric matrix of size \(9\times 9\) is stored in upper storage mode with block sizes 4 × 4:

0 1 2
┌ ┐
| -6.0 0.0 0.0 0.0 | 0.0 -2.0 -2.0 0.0 | -2.0 |
| . -6.0 -2.0 0.0 | -2.0 -4.0 0.0 -4.0 | -2.0 |
0 | . . -6.0 -2.0 | -2.0 0.0 2.0 0.0 | 6.0 |
| . . . -6.0 | 2.0 0.0 2.0 0.0 | 2.0 |
| ---------------------|-----------------------|------- |
| . . . . | -8.0 -4.0 0.0 -2.0 | 0.0 |
| . . . . | . -6.0 0.0 -4.0 | -6.0 |
1 | . . . . | . . -4.0 0.0 | 0.0 |
| . . . . | . . . -4.0 | -4.0 |
| ---------------------|-----------------------|------- |
2 | . . . . | . . . . | -16.0 |
└ ┘

may be distributed using the 2x2 process grid:

| 0 2 | 1
-----| ------- |-----
0 | P00 | P01
2 | |
-----| ------- |-----
1 | P10 | P11

with the following local arrays:

p,q | 0 | 1
-----|----------------------------|----------------------
| -6.0 0.0 0.0 0.0 -2.0 | 0.0 -2.0 -2.0 0.0
| . -6.0 -2.0 0.0 -2.0 | -2.0 -4.0 0.0 -4.0
0 | . . -6.0 -2.0 6.0 | -2.0 0.0 2.0 0.0
| . . . -6.0 2.0 | 2.0 0.0 2.0 0.0
| . . . . -16.0 | . . . .
-----|----------------------------|----------------------
| . . . . 0.0 | -8.0 -4.0 0.0 -2.0
| . . . . -6.0 | . -6.0 0.0 -4.0
1 | . . . . 0.0 | . . -4.0 0.0
| . . . . -4.0 | . . . -4.0

Note how processes \((0,0)\) and \((1,0)\) of the process grid store an extra column to represent the last column of the original matrix that did not fit the decomposition into \(4\times 4\) sub-blocks.

The choice of the block size is a compromise between a sufficiently large size for efficient local/serial BLAS, but one that is also small enough to achieve good parallel load balance.

Below we show a strong scaling example of ScaLAPACKMatrix::invert() on up to 5 nodes each composed of two Intel Xeon 2660v2 IvyBridge sockets 2.20GHz, 10 cores/socket. Calculations are performed on square processor grids 1x1, 2x2, 3x3, 4x4, 5x5, 6x6, 7x7, 8x8, 9x9, 10x10.

scalapack_invert.png
Author
Denis Davydov, Benjamin Brands, 2017

Definition at line 31 of file process_grid.h.

Member Typedef Documentation

template<typename NumberType>
using ScaLAPACKMatrix< NumberType >::size_type = unsigned int

Declare the type for container size.

Definition at line 114 of file scalapack.h.

Constructor & Destructor Documentation

template<typename NumberType >
ScaLAPACKMatrix< NumberType >::ScaLAPACKMatrix ( const size_type  n_rows,
const size_type  n_columns,
const std::shared_ptr< const Utilities::MPI::ProcessGrid > &  process_grid,
const size_type  row_block_size = 32,
const size_type  column_block_size = 32,
const LAPACKSupport::Property  property = LAPACKSupport::Property::general 
)

Constructor for a rectangular matrix with n_rows and n_cols and distributed using the grid process_grid.

Definition at line 79 of file scalapack.cc.

template<typename NumberType >
ScaLAPACKMatrix< NumberType >::ScaLAPACKMatrix ( const size_type  size,
const std::shared_ptr< const Utilities::MPI::ProcessGrid process_grid,
const size_type  block_size = 32,
const LAPACKSupport::Property  property = LAPACKSupport::Property::symmetric 
)

Constructor for a square matrix of size size, and distributed using the process grid in process_grid.

Definition at line 104 of file scalapack.cc.

template<typename NumberType>
ScaLAPACKMatrix< NumberType >::~ScaLAPACKMatrix ( )
overridedefault

Destructor

Member Function Documentation

template<typename NumberType >
void ScaLAPACKMatrix< NumberType >::reinit ( const size_type  n_rows,
const size_type  n_columns,
const std::shared_ptr< const Utilities::MPI::ProcessGrid > &  process_grid,
const size_type  row_block_size = 32,
const size_type  column_block_size = 32,
const LAPACKSupport::Property  property = LAPACKSupport::Property::general 
)

Initialize the rectangular matrix with n_rows and n_cols and distributed using the grid process_grid.

Definition at line 121 of file scalapack.cc.

template<typename NumberType >
void ScaLAPACKMatrix< NumberType >::reinit ( const size_type  size,
const std::shared_ptr< const Utilities::MPI::ProcessGrid process_grid,
const size_type  block_size = 32,
const LAPACKSupport::Property  property = LAPACKSupport::Property::symmetric 
)

Initialize the square matrix of size size and distributed using the grid process_grid.

Definition at line 196 of file scalapack.cc.

template<typename NumberType >
void ScaLAPACKMatrix< NumberType >::set_property ( const LAPACKSupport::Property  property)

Assign property to this matrix.

Definition at line 209 of file scalapack.cc.

template<typename NumberType >
LAPACKSupport::Property ScaLAPACKMatrix< NumberType >::get_property ( ) const

Return current property of this matrix

Definition at line 219 of file scalapack.cc.

template<typename NumberType >
LAPACKSupport::State ScaLAPACKMatrix< NumberType >::get_state ( ) const

Return current state of this matrix

Definition at line 228 of file scalapack.cc.

template<typename NumberType >
ScaLAPACKMatrix< NumberType > & ScaLAPACKMatrix< NumberType >::operator= ( const FullMatrix< NumberType > &  matrix)

Assignment operator from a regular FullMatrix.

Note
This function should only be used for relatively small matrix dimensions. It is primarily intended for debugging purposes.

Definition at line 237 of file scalapack.cc.

template<typename NumberType >
void ScaLAPACKMatrix< NumberType >::copy_from ( const LAPACKFullMatrix< NumberType > &  matrix,
const unsigned int  rank 
)

Copies the content of the locally owned matrix to the distributed matrix. The distributed matrix and matrix on process rank must have matching dimensions.

For all processes except the process with rank rank the serial matrix is not referenced. The user has to ensure that all processes call this with identical rank. The rank refers to a process of the MPI communicator used to create the process grid of the distributed matrix.

Definition at line 268 of file scalapack.cc.

template<typename NumberType >
void ScaLAPACKMatrix< NumberType >::copy_to ( FullMatrix< NumberType > &  matrix) const

Copy the contents of the distributed matrix into matrix.

Note
This function should only be used for relatively small matrix dimensions. It is primarily intended for debugging purposes.

Definition at line 566 of file scalapack.cc.

template<typename NumberType >
void ScaLAPACKMatrix< NumberType >::copy_to ( LAPACKFullMatrix< NumberType > &  matrix,
const unsigned int  rank 
) const

Copies the content of the distributed matrix into the locally replicated matrix on the process with rank . For all processes except rank matrix is not referenced. The distributed matrix and matrix on the process rank must have matching dimensions.

The user has to ensure that all processes call this with identical rank. The rank refers to a process of the MPI communicator used to create the process grid of the distributed matrix.

Definition at line 435 of file scalapack.cc.

template<typename NumberType >
void ScaLAPACKMatrix< NumberType >::copy_to ( ScaLAPACKMatrix< NumberType > &  dest) const

Copy the contents of the distributed matrix into a differently distributed matrix dest. The function also works for matrices with different process grids or block-cyclic distributions.

Definition at line 758 of file scalapack.cc.

template<typename NumberType >
void ScaLAPACKMatrix< NumberType >::copy_to ( ScaLAPACKMatrix< NumberType > &  B,
const std::pair< unsigned int, unsigned int > &  offset_A,
const std::pair< unsigned int, unsigned int > &  offset_B,
const std::pair< unsigned int, unsigned int > &  submatrix_size 
) const

Copy a submatrix (subset) of the distributed matrix A to a submatrix of the distributed matrix B.

  • The global row and column index of the first element of the submatrix A is provided by offset_A with row index=offset_A.first and column index=offset_A.second.
  • The global row and column index of the first element of the submatrix B is provided by offset_B with row index=offset_B.first and column index=offset_B.second.
  • The dimension of the submatrix to be copied is given by submatrix_size with number of rows=submatrix_size.first and number of columns=submatrix_size.second.

If it is necessary to copy complete matrices with an identical block-cyclic distribution, use ScaLAPACKMatrix<NumberType>::copy_to(ScaLAPACKMatrix<NumberType> &dest) with only one argument to avoid communication.

The underlying process grids of the matrices A and B must have been built with the same MPI communicator.

Definition at line 623 of file scalapack.cc.

template<typename NumberType >
void ScaLAPACKMatrix< NumberType >::copy_transposed ( const ScaLAPACKMatrix< NumberType > &  B)

Transposing assignment: \(\mathbf{A} = \mathbf{B}^T\)

The matrices \(\mathbf{A}\) and \(\mathbf{B}\) must have the same process grid.

The following alignment conditions have to be fulfilled: \(MB_A=NB_B\) and \(NB_A=MB_B\).

Definition at line 887 of file scalapack.cc.

template<typename NumberType >
void ScaLAPACKMatrix< NumberType >::add ( const ScaLAPACKMatrix< NumberType > &  B,
const NumberType  a = 0.,
const NumberType  b = 1.,
const bool  transpose_B = false 
)

The operations based on the input parameter transpose_B and the alignment conditions are summarized in the following table:

transpose_B Block Sizes Operation
false \(MB_A=MB_B\)
\(NB_A=NB_B\)
\(\mathbf{A} = a \mathbf{A} + b \mathbf{B}\)
true \(MB_A=NB_B\)
\(NB_A=MB_B\)
\(\mathbf{A} = a \mathbf{A} + b \mathbf{B}^T\)

The matrices \(\mathbf{A}\) and \(\mathbf{B}\) must have the same process grid.

Definition at line 897 of file scalapack.cc.

template<typename NumberType >
void ScaLAPACKMatrix< NumberType >::add ( const NumberType  b,
const ScaLAPACKMatrix< NumberType > &  B 
)

Matrix-addition: \(\mathbf{A} = \mathbf{A} + b\, \mathbf{B}\)

The matrices \(\mathbf{A}\) and \(\mathbf{B}\) must have the same process grid.

The following alignment conditions have to be fulfilled: \(MB_A=MB_B\) and \(NB_A=NB_B\).

Definition at line 952 of file scalapack.cc.

template<typename NumberType >
void ScaLAPACKMatrix< NumberType >::Tadd ( const NumberType  b,
const ScaLAPACKMatrix< NumberType > &  B 
)

Matrix-addition: \(\mathbf{A} = \mathbf{A} + b\, \mathbf{B}^T\)

The matrices \(\mathbf{A}\) and \(\mathbf{B}\) must have the same process grid.

The following alignment conditions have to be fulfilled: \(MB_A=NB_B\) and \(NB_A=MB_B\).

Definition at line 962 of file scalapack.cc.

template<typename NumberType >
void ScaLAPACKMatrix< NumberType >::mult ( const NumberType  b,
const ScaLAPACKMatrix< NumberType > &  B,
const NumberType  c,
ScaLAPACKMatrix< NumberType > &  C,
const bool  transpose_A = false,
const bool  transpose_B = false 
) const

Matrix-matrix-multiplication:

The operations based on the input parameters and the alignment conditions are summarized in the following table:

transpose_A transpose_B Block Sizes Operation
false false \(MB_A=MB_C\)
\(NB_A=MB_B\)
\(NB_B=NB_C\)
\(\mathbf{C} = b \mathbf{A} \cdot \mathbf{B} + c \mathbf{C}\)
false true \(MB_A=MB_C\)
\(NB_A=NB_B\)
\(MB_B=NB_C\)
\(\mathbf{C} = b \mathbf{A} \cdot \mathbf{B}^T + c \mathbf{C}\)
true false \(MB_A=MB_B\)
\(NB_A=MB_C\)
\(NB_B=NB_C\)
\(\mathbf{C} = b \mathbf{A}^T \cdot \mathbf{B} + c \mathbf{C}\)
true true \(MB_A=NB_B\)
\(NB_A=MB_C\)
\(MB_B=NB_C\)
\(\mathbf{C} = b \mathbf{A}^T \cdot \mathbf{B}^T + c \mathbf{C}\)

It is assumed that \(\mathbf{A}\) and \(\mathbf{B}\) have compatible sizes and that \(\mathbf{C}\) already has the right size.

The matrices \(\mathbf{A}\), \(\mathbf{B}\) and \(\mathbf{C}\) must have the same process grid.

Definition at line 972 of file scalapack.cc.

template<typename NumberType >
void ScaLAPACKMatrix< NumberType >::mmult ( ScaLAPACKMatrix< NumberType > &  C,
const ScaLAPACKMatrix< NumberType > &  B,
const bool  adding = false 
) const

Matrix-matrix-multiplication.

The optional parameter adding determines whether the result is stored in \(\mathbf{C}\) or added to \(\mathbf{C}\).

if (adding) \(\mathbf{C} = \mathbf{C} + \mathbf{A} \cdot \mathbf{B}\)

else \(\mathbf{C} = \mathbf{A} \cdot \mathbf{B}\)

It is assumed that \(\mathbf{A}\) and \(\mathbf{B}\) have compatible sizes and that \(\mathbf{C}\) already has the right size.

The following alignment conditions have to be fulfilled: \(MB_A=MB_C\), \(NB_A=MB_B\) and \(NB_B=NB_C\).

Definition at line 1089 of file scalapack.cc.

template<typename NumberType >
void ScaLAPACKMatrix< NumberType >::Tmmult ( ScaLAPACKMatrix< NumberType > &  C,
const ScaLAPACKMatrix< NumberType > &  B,
const bool  adding = false 
) const

Matrix-matrix-multiplication using transpose of \(\mathbf{A}\).

The optional parameter adding determines whether the result is stored in \(\mathbf{C}\) or added to \(\mathbf{C}\).

if (adding) \(\mathbf{C} = \mathbf{C} + \mathbf{A}^T \cdot \mathbf{B}\)

else \(\mathbf{C} = \mathbf{A}^T \cdot \mathbf{B}\)

It is assumed that \(\mathbf{A}\) and \(\mathbf{B}\) have compatible sizes and that \(\mathbf{C}\) already has the right size.

The following alignment conditions have to be fulfilled: \(MB_A=MB_B\), \(NB_A=MB_C\) and \(NB_B=NB_C\).

Definition at line 1103 of file scalapack.cc.

template<typename NumberType >
void ScaLAPACKMatrix< NumberType >::mTmult ( ScaLAPACKMatrix< NumberType > &  C,
const ScaLAPACKMatrix< NumberType > &  B,
const bool  adding = false 
) const

Matrix-matrix-multiplication using the transpose of \(\mathbf{B}\).

The optional parameter adding determines whether the result is stored in \(\mathbf{C}\) or added to \(\mathbf{C}\).

if (adding) \(\mathbf{C} = \mathbf{C} + \mathbf{A} \cdot \mathbf{B}^T\)

else \(\mathbf{C} = \mathbf{A} \cdot \mathbf{B}^T\)

It is assumed that \(\mathbf{A}\) and \(\mathbf{B}\) have compatible sizes and that \(\mathbf{C}\) already has the right size.

The following alignment conditions have to be fulfilled: \(MB_A=MB_C\), \(NB_A=NB_B\) and \(MB_B=NB_C\).

Definition at line 1117 of file scalapack.cc.

template<typename NumberType >
void ScaLAPACKMatrix< NumberType >::TmTmult ( ScaLAPACKMatrix< NumberType > &  C,
const ScaLAPACKMatrix< NumberType > &  B,
const bool  adding = false 
) const

Matrix-matrix-multiplication using transpose of \(\mathbf{A}\) and \(\mathbf{B}\).

The optional parameter adding determines whether the result is stored in \(\mathbf{C}\) or added to \(\mathbf{C}\).

if (adding) \(\mathbf{C} = \mathbf{C} + \mathbf{A}^T \cdot \mathbf{B}^T\)

else \(\mathbf{C} = \mathbf{A}^T \cdot \mathbf{B}^T\)

It is assumed that \(\mathbf{A}\) and \(\mathbf{B}\) have compatible sizes and that \(\mathbf{C}\) already has the right size.

The following alignment conditions have to be fulfilled: \(MB_A=NB_B\), \(NB_A=MB_C\) and \(MB_B=NB_C\).

Definition at line 1131 of file scalapack.cc.

template<typename NumberType >
void ScaLAPACKMatrix< NumberType >::save ( const char *  filename,
const std::pair< unsigned int, unsigned int > &  chunk_size = std::make_pair(numbers::invalid_unsigned_int,                        numbers::invalid_unsigned_int) 
) const

Stores the distributed matrix in filename using HDF5.

In case that deal.II was built without HDF5 a call to this function will cause an exception to be thrown.

If HDF5 was built with MPI, parallel I/O is used to save the matrix. Otherwise, just one process will do the output. This means that internally the distributed matrix is copied to one process, which does the output. Therefore, the matrix has to fit into the memory of one process.

To tweak the I/O performance, especially for parallel I/O, the user may define the optional parameter chunk_size. All MPI processes need to call the function with the same value. The matrix is written in chunks to the file, therefore the properties of the system define the optimal chunk size. Internally, HDF5 splits the matrix into chunk_size.first x chunk_size.second sized blocks, with chunk_size.first being the number of rows of a chunk and chunk_size.second the number of columns.

Definition at line 2518 of file scalapack.cc.

template<typename NumberType >
void ScaLAPACKMatrix< NumberType >::load ( const char *  filename)

Loads the distributed matrix from file filename using HDF5. In case that deal.II was built without HDF5 a call to this function will cause an exception to be thrown.

The matrix must have the same dimensions as the matrix stored in the file.

If HDF5 was build with MPI, parallel I/O is used to load the matrix. Otherwise, just one process will load the matrix from storage and distribute the content to the other processes subsequently.

Definition at line 2935 of file scalapack.cc.

template<typename NumberType >
void ScaLAPACKMatrix< NumberType >::compute_cholesky_factorization ( )

Compute the Cholesky factorization of the matrix using ScaLAPACK function pXpotrf. The result of the factorization is stored in this object.

Definition at line 1145 of file scalapack.cc.

template<typename NumberType >
void ScaLAPACKMatrix< NumberType >::compute_lu_factorization ( )

Compute the LU factorization of the matrix using ScaLAPACK function pXgetrf and partial pivoting with row interchanges. The result of the factorization is stored in this object.

Definition at line 1178 of file scalapack.cc.

template<typename NumberType >
void ScaLAPACKMatrix< NumberType >::invert ( )

Invert the matrix by first computing a Cholesky for symmetric matrices or a LU factorization for general matrices and then building the actual inverse using pXpotri or pXgetri. If the matrix is triangular, the LU factorization step is skipped, and pXtrtri is used directly.

If a Cholesky or LU factorization has been applied previously, pXpotri or pXgetri are called directly.

The inverse is stored in this object.

Definition at line 1219 of file scalapack.cc.

template<typename NumberType >
std::vector< NumberType > ScaLAPACKMatrix< NumberType >::eigenpairs_symmetric_by_index ( const std::pair< unsigned int, unsigned int > &  index_limits,
const bool  compute_eigenvectors 
)

Computing selected eigenvalues and, optionally, the eigenvectors of the real symmetric matrix \(\mathbf{A} \in \mathbb{R}^{M \times M}\).

The eigenvalues/eigenvectors are selected by prescribing a range of indices index_limits.

If successful, the computed eigenvalues are arranged in ascending order. The eigenvectors are stored in the columns of the matrix, thereby overwriting the original content of the matrix.

If all eigenvalues/eigenvectors have to be computed, pass the closed interval \( \left[ 0, M-1 \right] \) in index_limits.

Pass the closed interval \( \left[ M-r, M-1 \right] \) if the \(r\) largest eigenvalues/eigenvectors are desired.

Definition at line 1335 of file scalapack.cc.

template<typename NumberType >
std::vector< NumberType > ScaLAPACKMatrix< NumberType >::eigenpairs_symmetric_by_value ( const std::pair< NumberType, NumberType > &  value_limits,
const bool  compute_eigenvectors 
)

Computing selected eigenvalues and, optionally, the eigenvectors. The eigenvalues/eigenvectors are selected by prescribing a range of values value_limits for the eigenvalues.

If successful, the computed eigenvalues are arranged in ascending order. The eigenvectors are stored in the columns of the matrix, thereby overwriting the original content of the matrix.

Definition at line 1360 of file scalapack.cc.

template<typename NumberType >
std::vector< NumberType > ScaLAPACKMatrix< NumberType >::eigenpairs_symmetric_by_index_MRRR ( const std::pair< unsigned int, unsigned int > &  index_limits,
const bool  compute_eigenvectors 
)

Computing selected eigenvalues and, optionally, the eigenvectors of the real symmetric matrix \(\mathbf{A} \in \mathbb{R}^{M \times M}\) using the MRRR algorithm.

The eigenvalues/eigenvectors are selected by prescribing a range of indices index_limits.

If successful, the computed eigenvalues are arranged in ascending order. The eigenvectors are stored in the columns of the matrix, thereby overwriting the original content of the matrix.

If all eigenvalues/eigenvectors have to be computed, pass the closed interval \( \left[ 0, M-1 \right] \) in index_limits.

Pass the closed interval \( \left[ M-r, M-1 \right] \) if the \(r\) largest eigenvalues/eigenvectors are desired.

Definition at line 1670 of file scalapack.cc.

template<typename NumberType >
std::vector< NumberType > ScaLAPACKMatrix< NumberType >::eigenpairs_symmetric_by_value_MRRR ( const std::pair< NumberType, NumberType > &  value_limits,
const bool  compute_eigenvectors 
)

Computing selected eigenvalues and, optionally, the eigenvectors of the real symmetric matrix \(\mathbf{A} \in \mathbb{R}^{M \times M}\) using the MRRR algorithm. The eigenvalues/eigenvectors are selected by prescribing a range of values value_limits for the eigenvalues.

If successful, the computed eigenvalues are arranged in ascending order. The eigenvectors are stored in the columns of the matrix, thereby overwriting the original content of the matrix.

Definition at line 1693 of file scalapack.cc.

template<typename NumberType >
std::vector< NumberType > ScaLAPACKMatrix< NumberType >::compute_SVD ( ScaLAPACKMatrix< NumberType > *  U = nullptr,
ScaLAPACKMatrix< NumberType > *  VT = nullptr 
)

Computing the singular value decomposition (SVD) of a matrix \(\mathbf{A} \in \mathbb{R}^{M \times N}\), optionally computing the left and/or right singular vectors. The SVD is written as \(\mathbf{A} = \mathbf{U} \cdot \mathbf{\Sigma} \cdot \mathbf{V}^T\) with \(\mathbf{\Sigma} \in \mathbb{R}^{M \times N}\) as a diagonal matrix, \(\mathbf{U} \in \mathbb{R}^{M \times M}\) and \(\mathbf{V} \in \mathbb{R}^{M \times M}\) as orthogonal matrices. The diagonal elements of \(\mathbf{\Sigma}\) are the singular values of \(A\) and the columns of \(\mathbf{U}\) and \(\mathbf{V}\) are the corresponding left and right singular vectors, respectively. The singular values are returned in decreasing order and only the first \(\min(M,N)\) columns of \(\mathbf{U}\) and rows of \(\mathbf{V}^T\) are computed.

Upon return the content of the matrix is unusable. The matrix \(\mathbf{A}\) must have identical block cyclic distribution for the rows and column.

If left singular vectors are required matrices \(\mathbf{A}\) and \(\mathbf{U}\) have to be constructed with the same process grid and block cyclic distribution. If right singular vectors are required matrices \(\mathbf{A}\) and \(\mathbf{V}^T\) have to be constructed with the same process grid and block cyclic distribution.

To avoid computing the left and/or right singular vectors the function accepts nullptr for U and/or VT.

Definition at line 1921 of file scalapack.cc.

template<typename NumberType >
void ScaLAPACKMatrix< NumberType >::least_squares ( ScaLAPACKMatrix< NumberType > &  B,
const bool  transpose = false 
)

Solving overdetermined or underdetermined real linear systems involving matrix \(\mathbf{A} \in \mathbb{R}^{M \times N}\), or its transpose \(\mathbf{A}^T\), using a QR or LQ factorization of \(\mathbf{A}\) for \(N_{\rm RHS}\) RHS vectors in the columns of matrix \(\mathbf{B}\)

It is assumed that \(\mathbf{A}\) has full rank: \(\rm{rank}(\mathbf{A}) = \min(M,N)\).

The following options are supported:

  1. If(!transpose) and \(M \geq N\): least squares solution of overdetermined system \(\min \Vert \mathbf{B} - \mathbf{A}\cdot \mathbf{X}\Vert\).
    Upon exit the rows \(0\) to \(N-1\) of \(\mathbf{B}\) contain the least square solution vectors. The residual sum of squares for each column is given by the sum of squares of elements \(N\) to \(M-1\) in that column.
  2. If(!transpose) and \(M < N\): find minimum norm solutions of underdetermined systems \(\mathbf{A} \cdot \mathbf{X} = \mathbf{B}\).
    Upon exit the columns of \(\mathbf{B}\) contain the minimum norm solution vectors.
  3. If(transpose) and \(M \geq N\): find minimum norm solutions of underdetermined system \( \mathbf{A}^\top \cdot \mathbf{X} = \mathbf{B}\).
    Upon exit the columns of \(\mathbf{B}\) contain the minimum norm solution vectors.
  4. If(transpose) and \(M < N\): least squares solution of overdetermined system \(\min \Vert \mathbf{B} - \mathbf{A}^\top \cdot \mathbf{X}\Vert\).
    Upon exit the rows \(0\) to \(M-1\) contain the least square solution vectors. The residual sum of squares for each column is given by the sum of squares of elements \(M\) to \(N-1\) in that column.

If(!tranpose) then \(\mathbf{B} \in \mathbb{R}^{M \times N_{\rm RHS}}\), otherwise \(\mathbf{B} \in \mathbb{R}^{N \times N_{\rm RHS}}\). The matrices \(\mathbf{A}\) and \(\mathbf{B}\) must have an identical block cyclic distribution for rows and columns.

Definition at line 2041 of file scalapack.cc.

template<typename NumberType >
unsigned int ScaLAPACKMatrix< NumberType >::pseudoinverse ( const NumberType  ratio)

Compute the pseudoinverse \(\mathbf{A}^+ \in \mathbb{R}^{N \times M}\) (Moore-Penrose inverse) of a real matrix \(\mathbf{A} \in \mathbb{R}^{M \times N}\) using the singular value decomposition \(\mathbf{A} = \mathbf{U} \cdot \mathbf{\Sigma} \cdot \mathbf{V}^T\).

Unlike the inverse, the pseudoinverse \(\mathbf{A}^+ = \mathbf{V} \cdot \mathbf{\Sigma}^+ \cdot \mathbf{U}^T\) exists for both rectangular as well as singular matrices \(\mathbf{A}\).

For a rectangular \(\mathbf{\Sigma}\) the pseudoinverse is computed by taking the reciprocal of each non-zero element on the diagonal, leaving the zeros in place, and then transposing \(\mathbf{\Sigma}\). For the numerical computation only the singular values \(\sigma_i > \sigma_{\text{max}} \, \text{ratio}\) are taken into account. Upon successful exit, the function returns the number of singular values fulfilling that condition. That value can be interpreted as the rank of \(\mathbf{A}\).

Upon return this object contains the pseudoinverse \(\mathbf{A}^+ \in \mathbb{R}^{N \times M}\).

The following alignment conditions have to be fulfilled: \(MB_A = NB_A\).

Definition at line 2133 of file scalapack.cc.

template<typename NumberType >
NumberType ScaLAPACKMatrix< NumberType >::reciprocal_condition_number ( const NumberType  a_norm) const

Estimate the condition number of a SPD matrix in the \(l_1\)-norm. The matrix has to be in the Cholesky state (see compute_cholesky_factorization()). The reciprocal of the condition number is returned in order to avoid the possibility of overflow when the condition number is very large.

a_norm must contain the \(l_1\)-norm of the matrix prior to calling Cholesky factorization (see l1_norm()).

Note
An alternative is to compute the inverse of the matrix explicitly and manually construct \(k_1 = ||\mathbf{A}||_1 \, ||\mathbf{A}^{-1}||_1\).

Definition at line 2220 of file scalapack.cc.

template<typename NumberType >
NumberType ScaLAPACKMatrix< NumberType >::l1_norm ( ) const

Compute the \(l_1\)-norm of the matrix.

Definition at line 2282 of file scalapack.cc.

template<typename NumberType >
NumberType ScaLAPACKMatrix< NumberType >::linfty_norm ( ) const

Compute the \(l_{\infty}\) norm of the matrix.

Definition at line 2296 of file scalapack.cc.

template<typename NumberType >
NumberType ScaLAPACKMatrix< NumberType >::frobenius_norm ( ) const

Compute the Frobenius norm of the matrix.

Definition at line 2310 of file scalapack.cc.

template<typename NumberType>
size_type ScaLAPACKMatrix< NumberType >::m ( ) const

Number of rows of the \(M \times N\) matrix.

template<typename NumberType>
size_type ScaLAPACKMatrix< NumberType >::n ( ) const

Number of columns of the \(M \times N\) matrix.

template<typename NumberType>
unsigned int ScaLAPACKMatrix< NumberType >::local_m ( ) const

Number of local rows on this MPI processes.

template<typename NumberType>
unsigned int ScaLAPACKMatrix< NumberType >::local_n ( ) const

Number of local columns on this MPI process.

template<typename NumberType >
unsigned int ScaLAPACKMatrix< NumberType >::global_row ( const unsigned int  loc_row) const

Return the global row number for the given local row loc_row .

Definition at line 400 of file scalapack.cc.

template<typename NumberType >
unsigned int ScaLAPACKMatrix< NumberType >::global_column ( const unsigned int  loc_column) const

Return the global column number for the given local column loc_column.

Definition at line 417 of file scalapack.cc.

template<typename NumberType>
NumberType ScaLAPACKMatrix< NumberType >::local_el ( const unsigned int  loc_row,
const unsigned int  loc_column 
) const

Read access to local element.

template<typename NumberType>
NumberType& ScaLAPACKMatrix< NumberType >::local_el ( const unsigned int  loc_row,
const unsigned int  loc_column 
)

Write access to local element.

template<typename NumberType >
template<class InputVector >
void ScaLAPACKMatrix< NumberType >::scale_columns ( const InputVector &  factors)

Scale the columns of the distributed matrix by the scalars provided in the array factors.

The array factors must have as many entries as the matrix columns.

Copies of factors have to be available on all processes of the underlying MPI communicator.

Note
The fundamental prerequisite for the InputVector is that it must be possible to create an ArrayView from it; this is satisfied by the std::vector and Vector classes.

Definition at line 3383 of file scalapack.cc.

template<typename NumberType >
template<class InputVector >
void ScaLAPACKMatrix< NumberType >::scale_rows ( const InputVector &  factors)

Scale the rows of the distributed matrix by the scalars provided in the array factors.

The array factors must have as many entries as the matrix rows.

Copies of factors have to be available on all processes of the underlying MPI communicator.

Note
The fundamental prerequisite for the InputVector is that it must be possible to create an ArrayView from it; this is satisfied by the std::vector and Vector classes.

Definition at line 3394 of file scalapack.cc.

template<typename NumberType >
NumberType ScaLAPACKMatrix< NumberType >::norm_symmetric ( const char  type) const
private

Calculate the norm of a distributed symmetric dense matrix using ScaLAPACK's internal function.

Definition at line 2383 of file scalapack.cc.

template<typename NumberType >
NumberType ScaLAPACKMatrix< NumberType >::norm_general ( const char  type) const
private

Calculate the norm of a distributed dense matrix using ScaLAPACK's internal function.

Definition at line 2324 of file scalapack.cc.

template<typename NumberType >
std::vector< NumberType > ScaLAPACKMatrix< NumberType >::eigenpairs_symmetric ( const bool  compute_eigenvectors,
const std::pair< unsigned int, unsigned int > &  index_limits = std::make_pair(numbers::invalid_unsigned_int,                     numbers::invalid_unsigned_int),
const std::pair< NumberType, NumberType > &  value_limits = std::make_pair(std::numeric_limits<NumberType>::quiet_NaN(),                     std::numeric_limits<NumberType>::quiet_NaN()) 
)
private

Computing selected eigenvalues and, optionally, the eigenvectors. The eigenvalues/eigenvectors are selected by either prescribing a range of indices index_limits or a range of values value_limits for the eigenvalues. The function will throw an exception if both ranges are prescribed (meaning that both ranges differ from the default value) as this ambiguity is prohibited. If successful, the computed eigenvalues are arranged in ascending order. The eigenvectors are stored in the columns of the matrix, thereby overwriting the original content of the matrix.

Definition at line 1380 of file scalapack.cc.

template<typename NumberType >
std::vector< NumberType > ScaLAPACKMatrix< NumberType >::eigenpairs_symmetric_MRRR ( const bool  compute_eigenvectors,
const std::pair< unsigned int, unsigned int > &  index_limits = std::make_pair(numbers::invalid_unsigned_int,                     numbers::invalid_unsigned_int),
const std::pair< NumberType, NumberType > &  value_limits = std::make_pair(std::numeric_limits<NumberType>::quiet_NaN(),                     std::numeric_limits<NumberType>::quiet_NaN()) 
)
private

Computing selected eigenvalues and, optionally, the eigenvectors of the real symmetric matrix \(\mathbf{A} \in \mathbb{R}^{M \times M}\) using the MRRR algorithm. The eigenvalues/eigenvectors are selected by either prescribing a range of indices index_limits or a range of values value_limits for the eigenvalues. The function will throw an exception if both ranges are prescribed (meaning that both ranges differ from the default value) as this ambiguity is prohibited.

By calling this function the original content of the matrix will be overwritten. If requested, the eigenvectors are stored in the columns of the matrix. Also in the case that just the eigenvalues are required, the content of the matrix will be overwritten.

If successful, the computed eigenvalues are arranged in ascending order.

Note
Due to a bug in Netlib-ScaLAPACK, either all or no eigenvectors can be computed. Therefore, the input index_limits has to be set accordingly. Using Intel-MKL this restriction is not required.

Definition at line 1711 of file scalapack.cc.

Member Data Documentation

template<typename NumberType>
LAPACKSupport::State ScaLAPACKMatrix< NumberType >::state
private

Since ScaLAPACK operations notoriously change the meaning of the matrix entries, we record the current state after the last operation here.

Definition at line 885 of file scalapack.h.

template<typename NumberType>
LAPACKSupport::Property ScaLAPACKMatrix< NumberType >::property
private

Additional property of the matrix which may help to select more efficient ScaLAPACK functions.

Definition at line 891 of file scalapack.h.

template<typename NumberType>
std::shared_ptr<const Utilities::MPI::ProcessGrid> ScaLAPACKMatrix< NumberType >::grid
private

A shared pointer to a Utilities::MPI::ProcessGrid object which contains a BLACS context and a MPI communicator, as well as other necessary data structures.

Definition at line 898 of file scalapack.h.

template<typename NumberType>
int ScaLAPACKMatrix< NumberType >::n_rows
private

Number of rows in the matrix.

Definition at line 903 of file scalapack.h.

template<typename NumberType>
int ScaLAPACKMatrix< NumberType >::n_columns
private

Number of columns in the matrix.

Definition at line 908 of file scalapack.h.

template<typename NumberType>
int ScaLAPACKMatrix< NumberType >::row_block_size
private

Row block size.

Definition at line 913 of file scalapack.h.

template<typename NumberType>
int ScaLAPACKMatrix< NumberType >::column_block_size
private

Column block size.

Definition at line 918 of file scalapack.h.

template<typename NumberType>
int ScaLAPACKMatrix< NumberType >::n_local_rows
private

Number of rows in the matrix owned by the current process.

Definition at line 923 of file scalapack.h.

template<typename NumberType>
int ScaLAPACKMatrix< NumberType >::n_local_columns
private

Number of columns in the matrix owned by the current process.

Definition at line 928 of file scalapack.h.

template<typename NumberType>
int ScaLAPACKMatrix< NumberType >::descriptor[9]
private

ScaLAPACK description vector.

Definition at line 933 of file scalapack.h.

template<typename NumberType>
std::vector<NumberType> ScaLAPACKMatrix< NumberType >::work
mutableprivate

Workspace array.

Definition at line 938 of file scalapack.h.

template<typename NumberType>
std::vector<int> ScaLAPACKMatrix< NumberType >::iwork
mutableprivate

Integer workspace array.

Definition at line 943 of file scalapack.h.

template<typename NumberType>
std::vector<int> ScaLAPACKMatrix< NumberType >::ipiv
private

Integer array holding pivoting information required by ScaLAPACK's matrix factorization routines.

Definition at line 949 of file scalapack.h.

template<typename NumberType>
const char ScaLAPACKMatrix< NumberType >::uplo
private

A character to define where elements are stored in case ScaLAPACK operations support this.

Definition at line 955 of file scalapack.h.

template<typename NumberType>
const int ScaLAPACKMatrix< NumberType >::first_process_row
private

The process row of the process grid over which the first row of the global matrix is distributed.

Definition at line 961 of file scalapack.h.

template<typename NumberType>
const int ScaLAPACKMatrix< NumberType >::first_process_column
private

The process column of the process grid over which the first column of the global matrix is distributed.

Definition at line 967 of file scalapack.h.

template<typename NumberType>
const int ScaLAPACKMatrix< NumberType >::submatrix_row
private

Global row index that determines where to start a submatrix. Currently this equals unity, as we don't use submatrices.

Definition at line 973 of file scalapack.h.

template<typename NumberType>
const int ScaLAPACKMatrix< NumberType >::submatrix_column
private

Global column index that determines where to start a submatrix. Currently this equals unity, as we don't use submatrices.

Definition at line 979 of file scalapack.h.

template<typename NumberType>
Threads::Mutex ScaLAPACKMatrix< NumberType >::mutex
mutableprivate

Thread mutex.

Definition at line 984 of file scalapack.h.


The documentation for this class was generated from the following files: