Reference documentation for deal.II version Git c1e8229cd1 2021-05-06 12:37:45 -0600
LAPACKFullMatrix< number > Class Template Reference

#include <deal.II/base/array_view.h>

Inheritance diagram for LAPACKFullMatrix< number >:
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## Public Types

using size_type = std::make_unsigned< types::blas_int >::type

using value_type = typename AlignedVector< number >::value_type

using reference = typename AlignedVector< number >::reference

using const_reference = typename AlignedVector< number >::const_reference

using const_iterator = MatrixTableIterators::Iterator< TransposeTable< number >, true, MatrixTableIterators::Storage::column_major >

using iterator = MatrixTableIterators::Iterator< TransposeTable< number >, false, MatrixTableIterators::Storage::column_major >

## Public Member Functions

LAPACKFullMatrix (const size_type size=0)

LAPACKFullMatrix (const size_type rows, const size_type cols)

LAPACKFullMatrix (const LAPACKFullMatrix &)

LAPACKFullMatrix< number > & operator= (const LAPACKFullMatrix< number > &)

template<typename number2 >
LAPACKFullMatrix< number > & operator= (const FullMatrix< number2 > &)

template<typename number2 >
LAPACKFullMatrix< number > & operator= (const SparseMatrix< number2 > &)

LAPACKFullMatrix< number > & operator= (const number d)

LAPACKFullMatrix< number > & operator*= (const number factor)

LAPACKFullMatrix< number > & operator/= (const number factor)

void set (const size_type i, const size_type j, const number value)

void add (const number a, const LAPACKFullMatrix< number > &B)

void rank1_update (const number a, const Vector< number > &v)

void apply_givens_rotation (const std::array< number, 3 > &csr, const size_type i, const size_type k, const bool left=true)

template<typename MatrixType >
void copy_from (const MatrixType &)

void reinit (const size_type size)

void grow_or_shrink (const size_type size)

void remove_row_and_column (const size_type row, const size_type col)

void reinit (const size_type rows, const size_type cols)

void set_property (const LAPACKSupport::Property property)

size_type m () const

size_type n () const

template<typename MatrixType >
void fill (const MatrixType &src, const size_type dst_offset_i=0, const size_type dst_offset_j=0, const size_type src_offset_i=0, const size_type src_offset_j=0, const number factor=1., const bool transpose=false)

template<typename number2 >
void vmult (Vector< number2 > &w, const Vector< number2 > &v, const bool adding=false) const

void vmult (Vector< number > &w, const Vector< number > &v, const bool adding=false) const

template<typename number2 >
void vmult_add (Vector< number2 > &w, const Vector< number2 > &v) const

void vmult_add (Vector< number > &w, const Vector< number > &v) const

template<typename number2 >
void Tvmult (Vector< number2 > &w, const Vector< number2 > &v, const bool adding=false) const

void Tvmult (Vector< number > &w, const Vector< number > &v, const bool adding=false) const

template<typename number2 >
void Tvmult_add (Vector< number2 > &w, const Vector< number2 > &v) const

void Tvmult_add (Vector< number > &w, const Vector< number > &v) const

void mmult (LAPACKFullMatrix< number > &C, const LAPACKFullMatrix< number > &B, const bool adding=false) const

void mmult (FullMatrix< number > &C, const LAPACKFullMatrix< number > &B, const bool adding=false) const

void Tmmult (LAPACKFullMatrix< number > &C, const LAPACKFullMatrix< number > &B, const bool adding=false) const

void Tmmult (FullMatrix< number > &C, const LAPACKFullMatrix< number > &B, const bool adding=false) const

void Tmmult (LAPACKFullMatrix< number > &C, const LAPACKFullMatrix< number > &B, const Vector< number > &V, const bool adding=false) const

void mTmult (LAPACKFullMatrix< number > &C, const LAPACKFullMatrix< number > &B, const bool adding=false) const

void mTmult (FullMatrix< number > &C, const LAPACKFullMatrix< number > &B, const bool adding=false) const

void TmTmult (LAPACKFullMatrix< number > &C, const LAPACKFullMatrix< number > &B, const bool adding=false) const

void TmTmult (FullMatrix< number > &C, const LAPACKFullMatrix< number > &B, const bool adding=false) const

void transpose (LAPACKFullMatrix< number > &B) const

void scale_rows (const Vector< number > &V)

void compute_lu_factorization ()

void compute_cholesky_factorization ()

number reciprocal_condition_number (const number l1_norm) const

number reciprocal_condition_number () const

number determinant () const

number l1_norm () const

number linfty_norm () const

number frobenius_norm () const

number trace () const

void invert ()

void solve (Vector< number > &v, const bool transposed=false) const

void solve (LAPACKFullMatrix< number > &B, const bool transposed=false) const

void compute_eigenvalues (const bool right_eigenvectors=false, const bool left_eigenvectors=false)

void compute_eigenvalues_symmetric (const number lower_bound, const number upper_bound, const number abs_accuracy, Vector< number > &eigenvalues, FullMatrix< number > &eigenvectors)

void compute_generalized_eigenvalues_symmetric (LAPACKFullMatrix< number > &B, const number lower_bound, const number upper_bound, const number abs_accuracy, Vector< number > &eigenvalues, std::vector< Vector< number >> &eigenvectors, const types::blas_int itype=1)

void compute_generalized_eigenvalues_symmetric (LAPACKFullMatrix< number > &B, std::vector< Vector< number >> &eigenvectors, const types::blas_int itype=1)

void compute_svd ()

void compute_inverse_svd (const double threshold=0.)

void compute_inverse_svd_with_kernel (const unsigned int kernel_size)

std::complex< number > eigenvalue (const size_type i) const

number singular_value (const size_type i) const

const LAPACKFullMatrix< number > & get_svd_u () const

const LAPACKFullMatrix< number > & get_svd_vt () const

void print_formatted (std::ostream &out, const unsigned int precision=3, const bool scientific=true, const unsigned int width=0, const char *zero_string=" ", const double denominator=1., const double threshold=0.) const

void reinit (const size_type size1, const size_type size2, const bool omit_default_initialization=false)

void reinit (const TableIndices< N > &new_size, 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)

AlignedVector< number >::reference operator() (const TableIndices< N > &indices)

AlignedVector< number >::const_reference operator() (const TableIndices< N > &indices) const

size_type n_rows () const

size_type n_cols () const

iterator begin ()

const_iterator begin () const

iterator end ()

const_iterator end () const

bool operator== (const TableBase< N, number > &T2) const

void reset_values ()

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 number &value)

void replicate_across_communicator (const MPI_Comm &communicator, const unsigned int root_process)

void swap (TableBase< N, number > &v)

std::size_t memory_consumption () const

void serialize (Archive &ar, const unsigned int version)

Subscriptor functionality

Classes derived from Subscriptor provide a facility to subscribe to this object. This is mostly used by the SmartPointer class.

void subscribe (std::atomic< bool > *const validity, const std::string &identifier="") const

void unsubscribe (std::atomic< bool > *const validity, const std::string &identifier="") const

unsigned int n_subscriptions () const

template<typename StreamType >
void list_subscribers (StreamType &stream) const

void list_subscribers () const

## Static Public Member Functions

static ::ExceptionBaseExcInUse (int arg1, std::string arg2, std::string arg3)

static ::ExceptionBaseExcNoSubscriber (std::string arg1, std::string arg2)

## Protected Member Functions

reference el (const size_type i, const size_type j)

const_reference el (const size_type i, const size_type j) const

AlignedVector< number >::reference el (const TableIndices< N > &indices)

AlignedVector< number >::const_reference el (const TableIndices< N > &indices) const

size_type position (const TableIndices< N > &indices) const

## Protected Attributes

AlignedVector< number > values

TableIndices< N > table_size

## Private Member Functions

number norm (const char type) const

## Private Attributes

LAPACKSupport::State state

LAPACKSupport::Property property

std::vector< number > work

std::vector< types::blas_intiwork

std::vector< types::blas_intipiv

std::vector< number > inv_work

std::vector< typename numbers::NumberTraits< number >::real_type > wr

std::vector< number > wi

std::vector< number > vl

std::vector< number > vr

std::unique_ptr< LAPACKFullMatrix< number > > svd_u

std::unique_ptr< LAPACKFullMatrix< number > > svd_vt

std::mutex mutex

## Detailed Description

### template<typename number> class LAPACKFullMatrix< number >

A variant of FullMatrix using LAPACK functions wherever possible. In order to do this, the matrix is stored in transposed order. The element access functions hide this fact by reverting the transposition.

Note
In order to perform LAPACK functions, the class contains a lot of auxiliary data in the private section. The names of these data vectors are usually the names chosen for the arguments in the LAPACK documentation.

Definition at line 36 of file array_view.h.

## ◆ size_type

template<typename number>
 using LAPACKFullMatrix< number >::size_type = std::make_unsigned::type

Declare type for container size.

Definition at line 65 of file lapack_full_matrix.h.

## ◆ value_type

 using TransposeTable< number >::value_type = typename AlignedVector::value_type
inherited

Typedef for the values in the table.

Definition at line 1964 of file table.h.

## ◆ reference

 using TransposeTable< number >::reference = typename AlignedVector::reference
inherited

Typedef for the references in the table.

Definition at line 1969 of file table.h.

## ◆ const_reference

 using TransposeTable< number >::const_reference = typename AlignedVector::const_reference
inherited

Typedef for the constant references in the table.

Definition at line 1974 of file table.h.

## ◆ const_iterator

 using TransposeTable< number >::const_iterator = MatrixTableIterators::Iterator, true, MatrixTableIterators::Storage::column_major>
inherited

Typedef for a constant iterator that traverses the table in column-major order.

Definition at line 1983 of file table.h.

## ◆ iterator

 using TransposeTable< number >::iterator = MatrixTableIterators::Iterator, false, MatrixTableIterators::Storage::column_major>
inherited

Typedef for an iterator that traverses the table in column-major order.

Definition at line 1991 of file table.h.

## ◆ LAPACKFullMatrix() [1/3]

template<typename number >
 LAPACKFullMatrix< number >::LAPACKFullMatrix ( const size_type size = 0 )
explicit

Constructor. Initialize the matrix as a square matrix with dimension size.

In order to avoid the implicit conversion of integers and other types to a matrix, this constructor is declared explicit.

By default, no memory is allocated.

Definition at line 244 of file lapack_full_matrix.cc.

## ◆ LAPACKFullMatrix() [2/3]

template<typename number >
 LAPACKFullMatrix< number >::LAPACKFullMatrix ( const size_type rows, const size_type cols )

Constructor. Initialize the matrix as a rectangular matrix $$\rm{rows} \times \rm{cols}$$.

Definition at line 252 of file lapack_full_matrix.cc.

## ◆ LAPACKFullMatrix() [3/3]

template<typename number >
 LAPACKFullMatrix< number >::LAPACKFullMatrix ( const LAPACKFullMatrix< number > & M )

Copy constructor. This constructor does a deep copy of the matrix. Therefore, it poses a possible efficiency problem, if for example, function arguments are passed by value rather than by reference. Unfortunately, we can't mark this copy constructor explicit, since that prevents the use of this class in containers, such as std::vector. The responsibility to check performance of programs must therefore remain with the user of this class.

Definition at line 260 of file lapack_full_matrix.cc.

## ◆ operator=() [1/4]

template<typename number>
 LAPACKFullMatrix< number > & LAPACKFullMatrix< number >::operator= ( const LAPACKFullMatrix< number > & M )

Assignment operator.

Definition at line 269 of file lapack_full_matrix.cc.

## ◆ operator=() [2/4]

template<typename number >
template<typename number2 >
 LAPACKFullMatrix< number > & LAPACKFullMatrix< number >::operator= ( const FullMatrix< number2 > & M )

Assignment operator from a regular FullMatrix.

Note
Since LAPACK expects matrices in transposed order, this transposition is included here.

Definition at line 376 of file lapack_full_matrix.cc.

## ◆ operator=() [3/4]

template<typename number >
template<typename number2 >
 LAPACKFullMatrix< number > & LAPACKFullMatrix< number >::operator= ( const SparseMatrix< number2 > & M )

Assignment operator from a regular SparseMatrix.

Note
Since LAPACK expects matrices in transposed order, this transposition is included here.

Definition at line 393 of file lapack_full_matrix.cc.

## ◆ operator=() [4/4]

template<typename number>
 LAPACKFullMatrix< number > & LAPACKFullMatrix< number >::operator= ( const number d )

This operator assigns a scalar to a matrix. To avoid confusion with constructors, zero (when cast to the number type) is the only value allowed for d.

Definition at line 409 of file lapack_full_matrix.cc.

## ◆ operator*=()

template<typename number>
 LAPACKFullMatrix< number > & LAPACKFullMatrix< number >::operator*= ( const number factor )

This operator multiplies all entries by a fixed factor.

Definition at line 424 of file lapack_full_matrix.cc.

## ◆ operator/=()

template<typename number>
 LAPACKFullMatrix< number > & LAPACKFullMatrix< number >::operator/= ( const number factor )

This operator divides all entries by a fixed factor.

Definition at line 452 of file lapack_full_matrix.cc.

## ◆ set()

template<typename number>
 void LAPACKFullMatrix< number >::set ( const size_type i, const size_type j, const number value )
inline

Set a particular entry of the matrix to a value. Thus, calling A.set(1,2,3.141); is entirely equivalent to the operation A(1,2) = 3.141;. This function exists for compatibility with the various sparse matrix objects.

Parameters
 i The row index of the element to be set. j The column index of the element to be set. value The value to be written into the element.

Definition at line 1011 of file lapack_full_matrix.h.

template<typename number>
 void LAPACKFullMatrix< number >::add ( const number a, const LAPACKFullMatrix< number > & B )

Simple addition of a scaled matrix, i.e. $$\mathbf A \mathrel{+}= a \, \mathbf B$$.

Definition at line 483 of file lapack_full_matrix.cc.

## ◆ rank1_update()

template<typename number>
 void LAPACKFullMatrix< number >::rank1_update ( const number a, const Vector< number > & v )

Perform a rank-1 update of a symmetric matrix $$\mathbf A \leftarrow \mathbf A + a \, \mathbf v \mathbf v^T$$.

This function also works for Cholesky factorization. In that case, updating ( $$a>0$$) is performed via Givens rotations, whereas downdating ( $$a<0$$) via hyperbolic rotations. Note that the latter case might lead to a negative definite matrix in which case the error will be thrown (because Cholesky factorizations are only valid for symmetric and positive definite matrices).

Definition at line 610 of file lapack_full_matrix.cc.

## ◆ apply_givens_rotation()

template<typename number>
 void LAPACKFullMatrix< number >::apply_givens_rotation ( const std::array< number, 3 > & csr, const size_type i, const size_type k, const bool left = true )

Apply Givens rotation csr (a triplet of cosine, sine and radius, see Utilities::LinearAlgebra::givens_rotation() for the definition of the rotation matrix $$\mathbf G$$) to this matrix in the plane spanned by the i'th and k'th unit vectors. If left is true, the rotation is applied from left $$\mathbf A \leftarrow \mathbf G \mathbf A$$ and only rows i and k are affected. Otherwise, transpose of the rotation matrix is applied from right $$\mathbf A \leftarrow \mathbf A \mathbf G^T$$ and only columns i and k are affected.

Definition at line 305 of file lapack_full_matrix.cc.

## ◆ copy_from()

template<typename number >
template<typename MatrixType >
 void LAPACKFullMatrix< number >::copy_from ( const MatrixType & M )
inline

Assignment from different matrix classes, performing the usual conversion to the transposed format expected by LAPACK. This assignment operator uses iterators of the typename MatrixType. Therefore, sparse matrices are possible sources.

Definition at line 1036 of file lapack_full_matrix.h.

## ◆ reinit() [1/4]

template<typename number >
 void LAPACKFullMatrix< number >::reinit ( const size_type size )

Regenerate the current matrix by one that has the same properties as if it were created by the constructor of this class with the same argument list as this present function.

Definition at line 281 of file lapack_full_matrix.cc.

## ◆ grow_or_shrink()

template<typename number >
 void LAPACKFullMatrix< number >::grow_or_shrink ( const size_type size )

Same as above but will preserve the values of matrix upon resizing. The original values of the matrix are kept on increasing the size

$\mathbf A \rightarrow \left( \begin{array}{cc} \mathbf A & \mathbf 0 \\ \mathbf 0 & \mathbf 0 \end{array} \right)$

Whereas if the new size is smaller, the matrix will contain the upper left block of the original one

$\left( \begin{array}{cc} \mathbf A_{11} & \mathbf A_{12} \\ \mathbf A_{21} & \mathbf A_{22} \end{array} \right) \rightarrow \mathbf A_{11}$

Definition at line 291 of file lapack_full_matrix.cc.

## ◆ remove_row_and_column()

template<typename number >
 void LAPACKFullMatrix< number >::remove_row_and_column ( const size_type row, const size_type col )

Remove row row and column col from the matrix.

$\left( \begin{array}{ccc} \mathbf A_{11} & \mathbf a_{12} & \mathbf A_{13} \\ \mathbf a_{21}^T & a_{22} & \mathbf a_{23}^T \\ \mathbf A_{31} & \mathbf a_{32} & \mathbf A_{33} \end{array} \right) \rightarrow \left( \begin{array}{cc} \mathbf A_{11} & \mathbf A_{13} \\ \mathbf A_{31} & \mathbf A_{33} \end{array} \right)$

Definition at line 339 of file lapack_full_matrix.cc.

## ◆ reinit() [2/4]

template<typename number >
 void LAPACKFullMatrix< number >::reinit ( const size_type rows, const size_type cols )

Regenerate the current matrix by one that has the same properties as if it were created by the constructor of this class with the same argument list as this present function.

Definition at line 366 of file lapack_full_matrix.cc.

## ◆ set_property()

template<typename number >
 void LAPACKFullMatrix< number >::set_property ( const LAPACKSupport::Property property )

Assign property to this matrix.

Definition at line 1388 of file lapack_full_matrix.cc.

## ◆ m()

template<typename number >
 LAPACKFullMatrix< number >::size_type LAPACKFullMatrix< number >::m ( ) const
inline

Return the dimension of the codomain (or range) space.

Note
The matrix is of dimension $$m \times n$$.

Definition at line 1021 of file lapack_full_matrix.h.

## ◆ n()

template<typename number >
 LAPACKFullMatrix< number >::size_type LAPACKFullMatrix< number >::n ( ) const
inline

Return the dimension of the domain space.

Note
The matrix is of dimension $$m \times n$$.

Definition at line 1028 of file lapack_full_matrix.h.

## ◆ fill() [1/3]

template<typename number>
template<typename MatrixType >
 void LAPACKFullMatrix< number >::fill ( const MatrixType & src, const size_type dst_offset_i = 0, const size_type dst_offset_j = 0, const size_type src_offset_i = 0, const size_type src_offset_j = 0, const number factor = 1., const bool transpose = false )
inline

Fill rectangular block.

A rectangular block of the matrix src is copied into this. The upper left corner of the block being copied is (src_offset_i,src_offset_j). The upper left corner of the copied block is (dst_offset_i,dst_offset_j). The size of the rectangular block being copied is the maximum size possible, determined either by the size of this or src.

The final two arguments allow to enter a multiple of the source or its transpose.

Definition at line 1060 of file lapack_full_matrix.h.

## ◆ vmult() [1/2]

template<typename number >
template<typename number2 >
 void LAPACKFullMatrix< number >::vmult ( Vector< number2 > & w, const Vector< number2 > & v, const bool adding = false ) const

Matrix-vector-multiplication.

Depending on previous transformations recorded in state, the result of this function is one of

The optional parameter adding determines, whether the result is stored in the vector $$\mathbf w = \mathbf A \cdot \mathbf v$$ or added to it $$\mathbf w \mathrel{+}= \mathbf A \cdot \mathbf v$$.

Note
Source and destination must not be the same vector.
The template with number2 only exists for compile-time compatibility with FullMatrix. Only the case number2 = number is implemented due to limitations in the underlying LAPACK interface. All other variants throw an error upon invocation.

Definition at line 1094 of file lapack_full_matrix.h.

## ◆ vmult() [2/2]

template<typename number>
 void LAPACKFullMatrix< number >::vmult ( Vector< number > & w, const Vector< number > & v, const bool adding = false ) const

Specialization of above function for compatible Vector::value_type.

Definition at line 650 of file lapack_full_matrix.cc.

## ◆ vmult_add() [1/2]

template<typename number >
template<typename number2 >
 void LAPACKFullMatrix< number >::vmult_add ( Vector< number2 > & w, const Vector< number2 > & v ) const

Adding Matrix-vector-multiplication $$\mathbf w \mathrel{+}= \mathbf A \cdot \mathbf v$$.

See the documentation of vmult() for details on the implementation.

Definition at line 1107 of file lapack_full_matrix.h.

## ◆ vmult_add() [2/2]

template<typename number>
 void LAPACKFullMatrix< number >::vmult_add ( Vector< number > & w, const Vector< number > & v ) const

Specialization of above function for compatible Vector::value_type.

Definition at line 923 of file lapack_full_matrix.cc.

## ◆ Tvmult() [1/2]

template<typename number >
template<typename number2 >
 void LAPACKFullMatrix< number >::Tvmult ( Vector< number2 > & w, const Vector< number2 > & v, const bool adding = false ) const

Transpose matrix-vector-multiplication.

The optional parameter adding determines, whether the result is stored in the vector $$\mathbf w = \mathbf A^T \cdot \mathbf v$$ or added to it $$\mathbf w \mathrel{+}= \mathbf A^T \cdot \mathbf v$$.

See the documentation of vmult() for details on the implementation.

Definition at line 1119 of file lapack_full_matrix.h.

## ◆ Tvmult() [2/2]

template<typename number>
 void LAPACKFullMatrix< number >::Tvmult ( Vector< number > & w, const Vector< number > & v, const bool adding = false ) const

Specialization of above function for compatible Vector::value_type.

Definition at line 785 of file lapack_full_matrix.cc.

## ◆ Tvmult_add() [1/2]

template<typename number >
template<typename number2 >
 void LAPACKFullMatrix< number >::Tvmult_add ( Vector< number2 > & w, const Vector< number2 > & v ) const

Adding transpose matrix-vector-multiplication $$\mathbf w \mathrel{+}= \mathbf A^T \cdot \mathbf v$$.

See the documentation of vmult() for details on the implementation.

Definition at line 1132 of file lapack_full_matrix.h.

## ◆ Tvmult_add() [2/2]

template<typename number>
 void LAPACKFullMatrix< number >::Tvmult_add ( Vector< number > & w, const Vector< number > & v ) const

Specialization of above function for compatible Vector::value_type.

Definition at line 932 of file lapack_full_matrix.cc.

## ◆ mmult() [1/2]

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

Matrix-matrix-multiplication.

The optional parameter adding determines, whether the result is stored in the matrix $$\mathbf C = \mathbf A \cdot \mathbf B$$ or added to it $$\mathbf C \mathrel{+}= \mathbf A \cdot \mathbf B$$.

Note
It is assumed that A and B have compatible sizes and that C already has the right size.

This function uses the BLAS function Xgemm.

Definition at line 941 of file lapack_full_matrix.cc.

## ◆ mmult() [2/2]

template<typename number>
 void LAPACKFullMatrix< number >::mmult ( FullMatrix< number > & C, const LAPACKFullMatrix< number > & B, const bool adding = false ) const

Same as before, but stores the result in a FullMatrix, not in a LAPACKFullMatrix.

Definition at line 975 of file lapack_full_matrix.cc.

## ◆ Tmmult() [1/3]

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

Matrix-matrix-multiplication using transpose of this.

The optional parameter adding determines, whether the result is stored in the matrix $$\mathbf C = \mathbf A^T \cdot \mathbf B$$ or added to it $$\mathbf C \mathrel{+}= \mathbf A^T \cdot \mathbf B$$.

Note
It is assumed that A and B have compatible sizes and that C already has the right size.
This function uses the BLAS function Xgemm.

Definition at line 1110 of file lapack_full_matrix.cc.

## ◆ Tmmult() [2/3]

template<typename number>
 void LAPACKFullMatrix< number >::Tmmult ( FullMatrix< number > & C, const LAPACKFullMatrix< number > & B, const bool adding = false ) const

Same as before, but stores the result in a FullMatrix, not in a LAPACKFullMatrix.

Definition at line 1167 of file lapack_full_matrix.cc.

## ◆ Tmmult() [3/3]

template<typename number>
 void LAPACKFullMatrix< number >::Tmmult ( LAPACKFullMatrix< number > & C, const LAPACKFullMatrix< number > & B, const Vector< number > & V, const bool adding = false ) const

Matrix-matrix-multiplication using transpose of this and a diagonal vector V.

If the adding=false then the result is stored in the matrix $$\mathbf C = \mathbf A^T \cdot \rm{diag}(\mathbf V) \cdot \mathbf B$$ otherwise it is added $$\mathbf C \mathrel{+}= \mathbf A^T \cdot \rm{diag}(\mathbf V) \cdot \mathbf B$$.

Note
It is assumed that A, B and V have compatible sizes and that C already has the right size.
This function is not provided by LAPACK. The function first forms $$\rm{diag}(\mathbf V) \cdot \mathbf B$$ product and then uses Xgemm function.

Definition at line 1011 of file lapack_full_matrix.cc.

## ◆ mTmult() [1/2]

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

Matrix-matrix-multiplication using transpose of B.

The optional parameter adding determines, whether the result is stored in the matrix $$\mathbf C = \mathbf A \cdot \mathbf B^T$$ or added to it $$\mathbf C \mathrel{+}= \mathbf A \cdot \mathbf B^T$$.

Note
It is assumed that A and B have compatible sizes and that C already has the right size.
This function uses the BLAS function Xgemm.

Definition at line 1202 of file lapack_full_matrix.cc.

## ◆ mTmult() [2/2]

template<typename number>
 void LAPACKFullMatrix< number >::mTmult ( FullMatrix< number > & C, const LAPACKFullMatrix< number > & B, const bool adding = false ) const

Same as before, but stores the result in a FullMatrix, not in a LAPACKFullMatrix.

Definition at line 1260 of file lapack_full_matrix.cc.

## ◆ TmTmult() [1/2]

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

Matrix-matrix-multiplication using transpose of this and B.

The optional parameter adding determines, whether the result is stored in the matrix $$\mathbf C = \mathbf A^T \cdot \mathbf B^T$$ or added to it $$\mathbf C \mathrel{+}= \mathbf A^T \cdot \mathbf B^T$$.

Note
It is assumed that A and B have compatible sizes and that C already has the right size.
This function uses the BLAS function Xgemm.

Definition at line 1295 of file lapack_full_matrix.cc.

## ◆ TmTmult() [2/2]

template<typename number>
 void LAPACKFullMatrix< number >::TmTmult ( FullMatrix< number > & C, const LAPACKFullMatrix< number > & B, const bool adding = false ) const

Same as before, but stores the result in a FullMatrix, not in a LAPACKFullMatrix.

Definition at line 1329 of file lapack_full_matrix.cc.

## ◆ transpose()

template<typename number>
 void LAPACKFullMatrix< number >::transpose ( LAPACKFullMatrix< number > & B ) const

Performs out-place transposition. Matrix B should be appropriately sized.

Note
for complex number types, conjugate transpose will be performed.
If deal.II is configured with Intel-MKL, mkl_?omatcopy will be used, otherwise transposition is done element by element.

Definition at line 1073 of file lapack_full_matrix.cc.

## ◆ scale_rows()

template<typename number>
 void LAPACKFullMatrix< number >::scale_rows ( const Vector< number > & V )

Scale rows of this matrix by V . This is equivalent to premultiplication with a diagonal matrix $$\mathbf A\leftarrow {\rm diag}(\mathbf V)\mathbf A$$.

Definition at line 1093 of file lapack_full_matrix.cc.

## ◆ compute_lu_factorization()

template<typename number >
 void LAPACKFullMatrix< number >::compute_lu_factorization ( )

Compute the LU factorization of the matrix using LAPACK function Xgetrf.

Definition at line 1364 of file lapack_full_matrix.cc.

## ◆ compute_cholesky_factorization()

template<typename number >
 void LAPACKFullMatrix< number >::compute_cholesky_factorization ( )

Compute the Cholesky factorization of the matrix using LAPACK function Xpotrf.

Note
The factorization is stored in the lower-triangular part of the matrix.

Definition at line 1478 of file lapack_full_matrix.cc.

## ◆ reciprocal_condition_number() [1/2]

template<typename number>
 number LAPACKFullMatrix< number >::reciprocal_condition_number ( const number l1_norm ) const

Estimate the reciprocal of the condition number $$1/k(\mathbf A)$$ in $$L_1$$ norm ( $$1/(||\mathbf A||_1 \, ||\mathbf A^{-1}||_1)$$) of a symmetric positive definite matrix using Cholesky factorization. This function can only be called if the matrix is already factorized.

Note
The condition number $$k(\mathbf A)$$ can be used to estimate the numerical error related to the matrix inversion or the solution of the system of linear algebraic equations as error = std::numeric_limits<Number>::epsilon * k. Alternatively one can get the number of accurate digits std::floor(std::log10(k)).
The function computes reciprocal of the condition number to avoid possible overflow if the matrix is nearly singular.
Parameters
 [in] l1_norm Is the $$L_1$$ norm of the matrix before calling Cholesky factorization. It can be obtained by calling l1_norm().

Definition at line 1505 of file lapack_full_matrix.cc.

## ◆ reciprocal_condition_number() [2/2]

template<typename number>
 number LAPACKFullMatrix< number >::reciprocal_condition_number ( ) const

Estimate the reciprocal of the condition number $$1/k(\mathbf A)$$ in $$L_1$$ norm for triangular matrices. The matrix has to have the LAPACKSupport::Property set to either LAPACKSupport::Property::upper_triangular or LAPACKSupport::Property::lower_triangular, see set_property().

Definition at line 1538 of file lapack_full_matrix.cc.

## ◆ determinant()

template<typename number >
 number LAPACKFullMatrix< number >::determinant ( ) const

Compute the determinant of a matrix. As it requires the LU factorization of the matrix, this function can only be called after compute_lu_factorization() has been called.

Definition at line 1850 of file lapack_full_matrix.cc.

## ◆ l1_norm()

template<typename number >
 number LAPACKFullMatrix< number >::l1_norm ( ) const

Compute $$L_1$$ norm.

Definition at line 1397 of file lapack_full_matrix.cc.

## ◆ linfty_norm()

template<typename number >
 number LAPACKFullMatrix< number >::linfty_norm ( ) const

Compute $$L_\infty$$ norm.

Definition at line 1407 of file lapack_full_matrix.cc.

## ◆ frobenius_norm()

template<typename number >
 number LAPACKFullMatrix< number >::frobenius_norm ( ) const

Compute Frobenius norm

Definition at line 1417 of file lapack_full_matrix.cc.

## ◆ trace()

template<typename number >
 number LAPACKFullMatrix< number >::trace ( ) const

Compute trace of the matrix, i.e. the sum of the diagonal values. Obviously, the matrix needs to be quadratic for this function.

Definition at line 1460 of file lapack_full_matrix.cc.

## ◆ invert()

template<typename number >
 void LAPACKFullMatrix< number >::invert ( )

Invert the matrix by first computing an LU/Cholesky factorization with the LAPACK function Xgetrf/Xpotrf and then building the actual inverse using Xgetri/Xpotri.

Definition at line 1696 of file lapack_full_matrix.cc.

## ◆ solve() [1/2]

template<typename number>
 void LAPACKFullMatrix< number >::solve ( Vector< number > & v, const bool transposed = false ) const

Solve the linear system with right hand side v and put the solution back to v. The matrix should be either triangular or LU/Cholesky factorization should be previously computed.

The flag transposed indicates whether the solution of the transposed system is to be performed.

Definition at line 1739 of file lapack_full_matrix.cc.

## ◆ solve() [2/2]

template<typename number>
 void LAPACKFullMatrix< number >::solve ( LAPACKFullMatrix< number > & B, const bool transposed = false ) const

Same as above but for multiple right hand sides (as many as there are columns in the matrix B).

Definition at line 1782 of file lapack_full_matrix.cc.

## ◆ compute_eigenvalues()

template<typename number >
 void LAPACKFullMatrix< number >::compute_eigenvalues ( const bool right_eigenvectors = false, const bool left_eigenvectors = false )

Compute eigenvalues of the matrix. After this routine has been called, eigenvalues can be retrieved using the eigenvalue() function. The matrix itself will be LAPACKSupport::unusable after this operation.

The optional arguments allow to compute left and right eigenvectors as well.

Note that the function does not return the computed eigenvalues right away since that involves copying data around between the output arrays of the LAPACK functions and any return array. This is often unnecessary since one may not be interested in all eigenvalues at once, but for example only the extreme ones. In that case, it is cheaper to just have this function compute the eigenvalues and have a separate function that returns whatever eigenvalue is requested.

Note
Calls the LAPACK function Xgeev.

Definition at line 1881 of file lapack_full_matrix.cc.

## ◆ compute_eigenvalues_symmetric()

template<typename number>
 void LAPACKFullMatrix< number >::compute_eigenvalues_symmetric ( const number lower_bound, const number upper_bound, const number abs_accuracy, Vector< number > & eigenvalues, FullMatrix< number > & eigenvectors )

Compute eigenvalues and eigenvectors of a real symmetric matrix. Only eigenvalues in the interval $$(\rm{lower\_bound}, \rm{upper\_bound}]$$ are computed with the absolute tolerance $$\rm abs\_accuracy$$. An approximate eigenvalue is accepted as converged when it is determined to lie in an interval $$[a,b]$$ of width less than or equal to $$\rm{abs\_accuracy} + eps * \rm{max}(|a|,|b|)$$, where $$eps$$ is the machine precision. If $$\rm{abs\_accuracy}$$ is less than or equal to zero, then $$eps\,|\mathbf{T}|_1$$ will be used in its place, where $$|\mathbf{T}|_1$$ is the 1-norm of the tridiagonal matrix obtained by reducing $$\mathbf A$$ to tridiagonal form. Eigenvalues will be computed most accurately when $$\rm{abs\_accuracy}$$ is set to twice the underflow threshold, not zero. After this routine has been called, all eigenvalues in $$(\rm{lower\_bound}, \rm{upper\_bound}]$$ will be stored in eigenvalues and the corresponding eigenvectors will be stored in the columns of eigenvectors, whose dimension is set accordingly.

Note
Calls the LAPACK function Xsyevx.

Definition at line 1961 of file lapack_full_matrix.cc.

## ◆ compute_generalized_eigenvalues_symmetric() [1/2]

template<typename number>
 void LAPACKFullMatrix< number >::compute_generalized_eigenvalues_symmetric ( LAPACKFullMatrix< number > & B, const number lower_bound, const number upper_bound, const number abs_accuracy, Vector< number > & eigenvalues, std::vector< Vector< number >> & eigenvectors, const types::blas_int itype = 1 )

Compute generalized eigenvalues and eigenvectors of a real generalized symmetric eigenproblem of the form

• itype = 1: $$\mathbf A \cdot \mathbf x=\lambda \mathbf B \cdot \mathbf x$$
• itype = 2: $$\mathbf A \cdot \mathbf B \cdot \mathbf x=\lambda \mathbf x$$
• itype = 3: $$\mathbf B \cdot \mathbf A \cdot \mathbf x=\lambda \mathbf x$$

where $$\mathbf A$$ is this matrix. $$\mathbf A$$ and $$\mathbf B$$ are assumed to be symmetric, and $$\mathbf B$$ has to be positive definite. Only eigenvalues in the interval $$(\rm{lower\_bound}, \rm{upper\_bound}]$$ are computed with the absolute tolerance $$\rm{abs\_accuracy}$$. An approximate eigenvalue is accepted as converged when it is determined to lie in an interval $$[a,b]$$ of width less than or equal to $$\rm{abs\_accuracy} + eps * \rm{max}( |a|,|b| )$$, where $$eps$$ is the machine precision. If $$\rm{abs\_accuracy}$$ is less than or equal to zero, then $$eps \, |\mathbf{T}|_1$$ will be used in its place, where $$|\mathbf{T}|_1$$ is the 1-norm of the tridiagonal matrix obtained by reducing $$\mathbf A$$ to tridiagonal form. Eigenvalues will be computed most accurately when $$\rm{abs\_accuracy}$$ is set to twice the underflow threshold, not zero. After this routine has been called, all eigenvalues in $$(\rm{lower\_bound}, \rm{upper\_bound}]$$ will be stored in eigenvalues and the corresponding eigenvectors will be stored in eigenvectors, whose dimension is set accordingly.

Note
Calls the LAPACK function Xsygvx.

Definition at line 2072 of file lapack_full_matrix.cc.

## ◆ compute_generalized_eigenvalues_symmetric() [2/2]

template<typename number>
 void LAPACKFullMatrix< number >::compute_generalized_eigenvalues_symmetric ( LAPACKFullMatrix< number > & B, std::vector< Vector< number >> & eigenvectors, const types::blas_int itype = 1 )

Same as the other compute_generalized_eigenvalues_symmetric function except that all eigenvalues are computed and the tolerance is set automatically. Note that this function does not return the computed eigenvalues right away since that involves copying data around between the output arrays of the LAPACK functions and any return array. This is often unnecessary since one may not be interested in all eigenvalues at once, but for example only the extreme ones. In that case, it is cheaper to just have this function compute the eigenvalues and have a separate function that returns whatever eigenvalue is requested. Eigenvalues can be retrieved using the eigenvalue() function. The number of computed eigenvectors is equal to eigenvectors.size()

Note
Calls the LAPACK function Xsygv.

Definition at line 2197 of file lapack_full_matrix.cc.

## ◆ compute_svd()

template<typename number >
 void LAPACKFullMatrix< number >::compute_svd ( )

Compute the singular value decomposition of the matrix using LAPACK function Xgesdd.

Requires that the state is LAPACKSupport::matrix, fills the data members wr, svd_u, and svd_vt, and leaves the object in the state LAPACKSupport::svd.

The singular value decomposition factorizes the provided matrix (A) into three parts: U, sigma, and the transpose of V (V^T), such that A = U sigma V^T. Sigma is a MxN matrix which contains the singular values of A on the diagonal while all the other elements are zero. U is a MxM orthogonal matrix containing the left singular vectors corresponding to the singular values of A. V is a NxN orthonal matrix containing the right singular vectors corresponding the singular values of A.

Note that the variable svd_vt contains the tranpose of V and can be accessed by get_svd_vt(), while U is accessed with get_svd_u().

Definition at line 1576 of file lapack_full_matrix.cc.

## ◆ compute_inverse_svd()

template<typename number >
 void LAPACKFullMatrix< number >::compute_inverse_svd ( const double threshold = 0. )

Compute the inverse of the matrix by singular value decomposition.

Requires that state is either LAPACKSupport::matrix or LAPACKSupport::svd. In the first case, this function calls compute_svd(). After this function, the object will have the state LAPACKSupport::inverse_svd.

For a singular value decomposition, the inverse is simply computed by replacing all singular values by their reciprocal values. If the matrix does not have maximal rank, singular values 0 are not touched, thus computing the minimal norm right inverse of the matrix.

The parameter threshold determines, when a singular value should be considered zero. It is the ratio of the smallest to the largest nonzero singular value $$s_{max}$$. Thus, the inverses of all singular values less than $$s_{max}/\rm{threshold}$$ will be set to zero.

Definition at line 1652 of file lapack_full_matrix.cc.

## ◆ compute_inverse_svd_with_kernel()

template<typename number >
 void LAPACKFullMatrix< number >::compute_inverse_svd_with_kernel ( const unsigned int kernel_size )

Same as above but provide the size of the kernel instead of a threshold, i.e. the kernel_size smallest eigenvalues.

Definition at line 1675 of file lapack_full_matrix.cc.

## ◆ eigenvalue()

template<typename number >
 std::complex< number > LAPACKFullMatrix< number >::eigenvalue ( const size_type i ) const
inline

Retrieve eigenvalue after compute_eigenvalues() was called.

Definition at line 1144 of file lapack_full_matrix.h.

## ◆ singular_value()

template<typename number >
 number LAPACKFullMatrix< number >::singular_value ( const size_type i ) const
inline

Retrieve singular values after compute_svd() or compute_inverse_svd() was called.

Definition at line 1160 of file lapack_full_matrix.h.

## ◆ get_svd_u()

template<typename number >
 const LAPACKFullMatrix< number > & LAPACKFullMatrix< number >::get_svd_u ( ) const
inline

Retrieve the matrix svd_u after compute_svd() or compute_inverse_svd() was called.

Definition at line 1172 of file lapack_full_matrix.h.

## ◆ get_svd_vt()

template<typename number >
 const LAPACKFullMatrix< number > & LAPACKFullMatrix< number >::get_svd_vt ( ) const
inline

Retrieve the matrix svd_vt after compute_svd() or compute_inverse_svd() was called.

Definition at line 1183 of file lapack_full_matrix.h.

## ◆ print_formatted()

template<typename number >
 void LAPACKFullMatrix< number >::print_formatted ( std::ostream & out, const unsigned int precision = 3, const bool scientific = true, const unsigned int width = 0, const char * zero_string = " ", const double denominator = 1., const double threshold = 0. ) const

Print the matrix and allow formatting of entries.

The parameters allow for a flexible setting of the output format:

Parameters
 out This specifies the stream to write to. precision denotes the number of trailing digits. scientific is used to determine the number format, where scientific = false means fixed point notation. width denotes the with of each column. A zero entry for width makes the function compute a width, but it may be changed to a positive value, if output is crude. zero_string specifies a string printed for zero entries. denominator Multiply the whole matrix by this common denominator to get nicer numbers. threshold all entries with absolute value smaller than this are considered zero.
Note
The entries stored resemble a matrix only if the state is either LAPACKSupport::matrix or LAPACK::inverse_matrix. Otherwise, calling this function is not allowed.

Definition at line 2289 of file lapack_full_matrix.cc.

## ◆ norm()

template<typename number >
 number LAPACKFullMatrix< number >::norm ( const char type ) const
private

Internal function to compute various norms.

Definition at line 1427 of file lapack_full_matrix.cc.

## ◆ reinit() [3/4]

 void TransposeTable< number >::reinit ( const size_type size1, const size_type size2, const bool omit_default_initialization = false )
inherited

Reinitialize the object. This function is mostly here for compatibility with the earlier vector2d class. Passes down to the base class by converting the arguments to the data type requested by the base class.

## ◆ reinit() [4/4]

 void TableBase< N, number >::reinit ( const TableIndices< N > & new_size, const bool omit_default_initialization = false )
inherited

Set the dimensions of this object to the sizes given in the first argument, and allocate the required memory for table entries to accommodate these sizes. If omit_default_initialization is set to false, all elements of the table are set to a default constructed object for the element type. Otherwise the memory is left in an uninitialized or otherwise undefined state.

## ◆ operator()() [1/4]

 const_reference TransposeTable< number >::operator() ( const size_type i, const size_type j ) const
inherited

Direct access to one element of the table by specifying all indices at the same time. Range checks are performed.

This version of the function only allows read access.

## ◆ operator()() [2/4]

 reference TransposeTable< number >::operator() ( const size_type i, const size_type j )
inherited

Direct access to one element of the table by specifying all indices at the same time. Range checks are performed.

This version of the function allows read-write access.

## ◆ operator()() [3/4]

 AlignedVector::reference TableBase< N, number >::operator() ( const TableIndices< N > & indices )
inherited

Return a read-write reference to the indicated element.

## ◆ operator()() [4/4]

 AlignedVector::const_reference TableBase< N, number >::operator() ( const TableIndices< N > & indices ) const
inherited

Return the value of the indicated element as a read-only reference.

We return the requested value as a constant reference rather than by value since this object may hold data types that may be large, and we don't know here whether copying is expensive or not.

## ◆ n_rows()

 size_type TransposeTable< number >::n_rows ( ) const
inherited

Number of rows. This function really makes only sense since we have a two-dimensional object here.

## ◆ n_cols()

 size_type TransposeTable< number >::n_cols ( ) const
inherited

Number of columns. This function really makes only sense since we have a two-dimensional object here.

## ◆ begin() [1/2]

 iterator TransposeTable< number >::begin ( )
inherited

Return an iterator pointing to the first entry.

## ◆ begin() [2/2]

 const_iterator TransposeTable< number >::begin ( ) const
inherited

Return a constant iterator pointing to the first entry.

## ◆ end() [1/2]

 iterator TransposeTable< number >::end ( )
inherited

Return an iterator pointing to one past the last entry.

## ◆ end() [2/2]

 const_iterator TransposeTable< number >::end ( ) const
inherited

Return a constant iterator pointing to one past the last entry.

## ◆ el() [1/4]

 reference TransposeTable< number >::el ( const size_type i, const size_type j )
protectedinherited

Return a read-write reference to the element (i,j).

This function does no bounds checking and is only to be used internally and in functions already checked.

These functions are mainly here for compatibility with a former implementation of these table classes for 2d arrays, then called vector2d.

## ◆ el() [2/4]

 const_reference TransposeTable< number >::el ( const size_type i, const size_type j ) const
protectedinherited

Return the value of the element (i,j) as a read-only reference.

This function does no bounds checking and is only to be used internally and in functions already checked.

We return the requested value as a constant reference rather than by value since this object may hold data types that may be large, and we don't know here whether copying is expensive or not.

These functions are mainly here for compatibility with a former implementation of these table classes for 2d arrays, then called vector2d.

## ◆ el() [3/4]

 AlignedVector::reference TableBase< N, number >::el ( const TableIndices< N > & indices )
protectedinherited

Return a read-write reference to the indicated element.

This function does no bounds checking and is only to be used internally and in functions already checked.

## ◆ el() [4/4]

 AlignedVector::const_reference TableBase< N, number >::el ( const TableIndices< N > & indices ) const
protectedinherited

Return the value of the indicated element as a read-only reference.

This function does no bounds checking and is only to be used internally and in functions already checked.

We return the requested value as a constant reference rather than by value since this object may hold data types that may be large, and we don't know here whether copying is expensive or not.

## ◆ operator==()

 bool TableBase< N, number >::operator== ( const TableBase< N, number > & T2 ) const
inherited

Test for equality of two tables.

## ◆ reset_values()

 void TableBase< N, number >::reset_values ( )
inherited

Set all entries to their default value (i.e. copy them over with default constructed objects). Do not change the size of the table, though.

## ◆ size() [1/2]

 size_type TableBase< N, number >::size ( const unsigned int i ) const
inherited

Size of the table in direction i.

## ◆ size() [2/2]

 const TableIndices& TableBase< N, number >::size ( ) const
inherited

Return the sizes of this object in each direction.

## ◆ n_elements()

 size_type TableBase< N, number >::n_elements ( ) const
inherited

Return the number of elements stored in this object, which is the product of the extensions in each dimension.

## ◆ empty()

 bool TableBase< N, number >::empty ( ) const
inherited

Return whether the object is empty, i.e. one of the directions is zero. This is equivalent to n_elements()==0.

## ◆ fill() [2/3]

 void TableBase< N, number >::fill ( InputIterator entries, const bool C_style_indexing = true )
inherited

Fill this table (which is assumed to already have the correct size) from a source given by dereferencing the given forward iterator (which could, for example, be a pointer to the first element of an array, or an inserting std::istream_iterator). The second argument denotes whether the elements pointed to are arranged in a way that corresponds to the last index running fastest or slowest. The default is to use C-style indexing where the last index runs fastest (as opposed to Fortran-style where the first index runs fastest when traversing multidimensional arrays. For example, if you try to fill an object of type Table<2,T>, then calling this function with the default value for the second argument will result in the equivalent of doing

for (unsigned int i=0; i<t.size(0); ++i)
for (unsigned int j=0; j<t.size(1); ++j)
t[i][j] = *entries++;

On the other hand, if the second argument to this function is false, then this would result in code of the following form:

for (unsigned int j=0; j<t.size(1); ++j)
for (unsigned int i=0; i<t.size(0); ++i)
t[i][j] = *entries++;

Note the switched order in which we fill the table elements by traversing the given set of iterators.

Parameters
 entries An iterator to a set of elements from which to initialize this table. It is assumed that iterator can be incremented and dereferenced a sufficient number of times to fill this table. C_style_indexing If true, run over elements of the table with the last index changing fastest as we dereference subsequent elements of the input range. If false, change the first index fastest.

## ◆ fill() [3/3]

 void TableBase< N, number >::fill ( const number & value )
inherited

Fill all table entries with the same value.

## ◆ replicate_across_communicator()

 void TableBase< N, number >::replicate_across_communicator ( const MPI_Comm & communicator, const unsigned int root_process )
inherited

This function replicates the state found on the process indicated by root_process across all processes of the MPI communicator. The current state found on any of the processes other than root_process is lost in this process. One can imagine this operation to act like a call to Utilities::MPI::broadcast() from the root process to all other processes, though in practice the function may try to move the data into shared memory regions on each of the machines that host MPI processes and let all MPI processes on this machine then access this shared memory region instead of keeping their own copy. See the general documentation of this class for a code example.

The intent of this function is to quickly exchange large arrays from one process to others, rather than having to compute or create it on all processes. This is specifically the case for data loaded from disk – say, large data tables – that are more easily dealt with by reading once and then distributing across all processes in an MPI universe, than letting each process read the data from disk itself. Specifically, the use of shared memory regions allows for replicating the data only once per multicore machine in the MPI universe, rather than replicating data once for each MPI process. This results in large memory savings if the data is large on today's machines that can easily house several dozen MPI processes per shared memory space.

This function does not imply a model of keeping data on different processes in sync, as parallel::distributed::Vector and other vector classes do where there exists a notion of certain elements of the vector owned by each process and possibly ghost elements that are mirrored from its owning process to other processes. Rather, the elements of the current object are simply copied to the other processes, and it is useful to think of this operation as creating a set of const AlignedVector objects on all processes that should not be changed any more after the replication operation, as this is the only way to ensure that the vectors remain the same on all processes. This is particularly true because of the use of shared memory regions where any modification of a vector element on one MPI process may also result in a modification of elements visible on other processes, assuming they are located within one shared memory node.

Note
The use of shared memory between MPI processes requires that the detected MPI installation supports the necessary operations. This is the case for MPI 3.0 and higher.
This function is not cheap. It needs to create sub-communicators of the provided communicator object, which is generally an expensive operation. Likewise, the generation of shared memory spaces is not a cheap operation. As a consequence, this function primarily makes sense when the goal is to share large read-only data tables among processes; examples are data tables that are loaded at start-up time and then used over the course of the run time of the program. In such cases, the start-up cost of running this function can be amortized over time, and the potential memory savings from not having to store the table on each process may be substantial on machines with large core counts on which many MPI processes run on the same machine.
This function only makes sense if the data type T is "self-contained", i.e., all of its information is stored in its member variables, and if none of the member variables are pointers to other parts of the memory. This is because if a type T does have pointers to other parts of memory, then moving T into a shared memory space does not result in the other processes having access to data that the object points to with its member variable pointers: These continue to live only on one process, and are typically in memory areas not accessible to the other processes. As a consequence, the usual use case for this function is to share arrays of simple objects such as doubles or ints.
After calling this function, objects on different MPI processes share a common state. That means that certain operations become "collective", i.e., they must be called on all participating processors at the same time. In particular, you can no longer call resize(), reserve(), or clear() on one MPI process – you have to do so on all processes at the same time, because they have to communicate for these operations. If you do not do so, you will likely get a deadlock that may be difficult to debug. By extension, this rule of only collectively resizing extends to this function itself: You can not call it twice in a row because that implies that first all but the root_process throw away their data, which is not a collective operation. Generally, these restrictions on what can and can not be done hint at the correctness of the comments above: You should treat an AlignedVector on which the current function has been called as const, on which no further operations can be performed until the destructor is called.

## ◆ swap()

 void TableBase< N, number >::swap ( TableBase< N, number > & v )
inherited

Swap the contents of this table and the other table v. One could do this operation with a temporary variable and copying over the data elements, but this function is significantly more efficient since it only swaps the pointers to the data of the two vectors and therefore does not need to allocate temporary storage and move data around.

This function is analogous to the swap function of all C++ standard containers. Also, there is a global function swap(u,v) that simply calls u.swap(v), again in analogy to standard functions.

## ◆ memory_consumption()

 std::size_t TableBase< N, number >::memory_consumption ( ) const
inherited

Determine an estimate for the memory consumption (in bytes) of this object.

## ◆ serialize()

 void TableBase< N, number >::serialize ( Archive & ar, const unsigned int version )
inherited

Write or read the data of this object to or from a stream for the purpose of serialization using the BOOST serialization library.

## ◆ position()

 size_type TableBase< N, number >::position ( const TableIndices< N > & indices ) const
protectedinherited

Return the position of the indicated element within the array of elements stored one after the other. This function does no index checking.

## ◆ subscribe()

 void Subscriptor::subscribe ( std::atomic< bool > *const validity, const std::string & identifier = "" ) const
inherited

Subscribes a user of the object by storing the pointer validity. The subscriber may be identified by text supplied as identifier.

Definition at line 136 of file subscriptor.cc.

## ◆ unsubscribe()

 void Subscriptor::unsubscribe ( std::atomic< bool > *const validity, const std::string & identifier = "" ) const
inherited

Unsubscribes a user from the object.

Note
The identifier and the validity pointer must be the same as the one supplied to subscribe().

Definition at line 156 of file subscriptor.cc.

## ◆ n_subscriptions()

 unsigned int Subscriptor::n_subscriptions ( ) const
inlineinherited

Return the present number of subscriptions to this object. This allows to use this class for reference counted lifetime determination where the last one to unsubscribe also deletes the object.

Definition at line 301 of file subscriptor.h.

## ◆ list_subscribers() [1/2]

template<typename StreamType >
 void Subscriptor::list_subscribers ( StreamType & stream ) const
inlineinherited

List the subscribers to the input stream.

Definition at line 318 of file subscriptor.h.

## ◆ list_subscribers() [2/2]

 void Subscriptor::list_subscribers ( ) const
inherited

List the subscribers to deallog.

Definition at line 204 of file subscriptor.cc.

## ◆ state

template<typename number>
 LAPACKSupport::State LAPACKFullMatrix< number >::state
private

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

Definition at line 907 of file lapack_full_matrix.h.

## ◆ property

template<typename number>
 LAPACKSupport::Property LAPACKFullMatrix< number >::property
private

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

Definition at line 913 of file lapack_full_matrix.h.

## ◆ work

template<typename number>
 std::vector LAPACKFullMatrix< number >::work
mutableprivate

The working array used for some LAPACK functions.

Definition at line 918 of file lapack_full_matrix.h.

## ◆ iwork

template<typename number>
 std::vector LAPACKFullMatrix< number >::iwork
mutableprivate

Integer working array used for some LAPACK functions.

Definition at line 923 of file lapack_full_matrix.h.

## ◆ ipiv

template<typename number>
 std::vector LAPACKFullMatrix< number >::ipiv
private

The vector storing the permutations applied for pivoting in the LU- factorization.

Also used as the scratch array IWORK for LAPACK functions needing it.

Definition at line 931 of file lapack_full_matrix.h.

## ◆ inv_work

template<typename number>
 std::vector LAPACKFullMatrix< number >::inv_work
private

Workspace for calculating the inverse matrix from an LU factorization.

Definition at line 936 of file lapack_full_matrix.h.

## ◆ wr

template<typename number>
 std::vector::real_type> LAPACKFullMatrix< number >::wr
private

Real parts of eigenvalues or the singular values. Filled by compute_eigenvalues() or compute_svd().

Definition at line 942 of file lapack_full_matrix.h.

## ◆ wi

template<typename number>
 std::vector LAPACKFullMatrix< number >::wi
private

Imaginary parts of eigenvalues, or, in the complex scalar case, the eigenvalues themselves. Filled by compute_eigenvalues.

Definition at line 948 of file lapack_full_matrix.h.

## ◆ vl

template<typename number>
 std::vector LAPACKFullMatrix< number >::vl
private

Space where left eigenvectors can be stored.

Definition at line 953 of file lapack_full_matrix.h.

## ◆ vr

template<typename number>
 std::vector LAPACKFullMatrix< number >::vr
private

Space where right eigenvectors can be stored.

Definition at line 958 of file lapack_full_matrix.h.

## ◆ svd_u

template<typename number>
 std::unique_ptr > LAPACKFullMatrix< number >::svd_u
private

The matrix $$\mathbf U$$ in the singular value decomposition $$\mathbf U \cdot \mathbf S \cdot \mathbf V^T$$.

Definition at line 964 of file lapack_full_matrix.h.

## ◆ svd_vt

template<typename number>
 std::unique_ptr > LAPACKFullMatrix< number >::svd_vt
private

The matrix $$\mathbf V^T$$ in the singular value decomposition $$\mathbf U \cdot \mathbf S \cdot \mathbf V^T$$.

Definition at line 970 of file lapack_full_matrix.h.

## ◆ mutex

template<typename number>
 std::mutex LAPACKFullMatrix< number >::mutex
mutableprivate

Definition at line 975 of file lapack_full_matrix.h.

## ◆ values

 AlignedVector TableBase< N, number >::values
protectedinherited

Component-array.

Definition at line 787 of file table.h.

## ◆ table_size

 TableIndices TableBase< N, number >::table_size
protectedinherited

Size in each direction of the table.

Definition at line 792 of file table.h.

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