Reference documentation for deal.II version Git 67353a5f2d 2021-01-26 18:33:38 +0100
TensorProductMatrixSymmetricSum< dim, Number, n_rows_1d > Class Template Reference

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

Inheritance diagram for TensorProductMatrixSymmetricSum< dim, Number, n_rows_1d >:
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## Public Types

using value_type = Number

## Public Member Functions

TensorProductMatrixSymmetricSum ()=default

TensorProductMatrixSymmetricSum (const std::array< Table< 2, Number >, dim > &mass_matrix, const std::array< Table< 2, Number >, dim > &derivative_matrix)

TensorProductMatrixSymmetricSum (const std::array< FullMatrix< Number >, dim > &mass_matrix, const std::array< FullMatrix< Number >, dim > &derivative_matrix)

TensorProductMatrixSymmetricSum (const Table< 2, Number > &mass_matrix, const Table< 2, Number > &derivative_matrix)

void reinit (const std::array< Table< 2, Number >, dim > &mass_matrix, const std::array< Table< 2, Number >, dim > &derivative_matrix)

void reinit (const std::array< FullMatrix< Number >, dim > &mass_matrix, const std::array< FullMatrix< Number >, dim > &derivative_matrix)

void reinit (const Table< 2, Number > &mass_matrix, const Table< 2, Number > &derivative_matrix)

unsigned int m () const

unsigned int n () const

void vmult (const ArrayView< Number > &dst, const ArrayView< const Number > &src) const

void apply_inverse (const ArrayView< Number > &dst, const ArrayView< const Number > &src) const

## Static Public Attributes

static constexpr int n_rows_1d_static = n_rows_1d

## Protected Attributes

std::array< Table< 2, Number >, dim > mass_matrix

std::array< Table< 2, Number >, dim > derivative_matrix

std::array< AlignedVector< Number >, dim > eigenvalues

std::array< Table< 2, Number >, dim > eigenvectors

## Private Member Functions

template<typename MatrixArray >
void reinit_impl (MatrixArray &&mass_matrix, MatrixArray &&derivative_matrix)

## Detailed Description

### template<int dim, typename Number, int n_rows_1d = -1> class TensorProductMatrixSymmetricSum< dim, Number, n_rows_1d >

This is a special matrix class defined as the tensor product (or Kronecker product) of 1D matrices of the type

\begin{align*} L &= A_1 \otimes M_0 + M_1 \otimes A_0 \end{align*}

in 2D and

\begin{align*} L &= A_2 \otimes M_1 \otimes M_0 + M_2 \otimes A_1 \otimes M_0 + M_2 \otimes M_1 \otimes A_0 \end{align*}

in 3D. The typical application setting is a discretization of the Laplacian $$L$$ on a Cartesian (axis-aligned) geometry, where it can be exactly represented by the Kronecker or tensor product of a 1D mass matrix $$M$$ and a 1D Laplace matrix $$A$$ in each tensor direction (due to symmetry $$M$$ and $$A$$ are the same in each dimension). The dimension of the resulting class is the product of the one-dimensional matrices.

This class implements two basic operations, namely the usual multiplication by a vector and the inverse. For both operations, fast tensorial techniques can be applied that implement the operator evaluation in $$\text{size}(M)^{d+1}$$ arithmetic operations, considerably less than $$\text{size}(M)^{2d}$$ for the naive forward transformation and $$\text{size}(M)^{3d}$$ for setting up the inverse of $$L$$.

Interestingly, the exact inverse of the matrix $$L$$ can be found through tensor products due to an article by R. E. Lynch, J. R. Rice, D. H. Thomas, Direct solution of partial difference equations by tensor product methods, Numerische Mathematik 6, 185-199 from 1964,

\begin{align*} L^{-1} &= S_1 \otimes S_0 (\Lambda_1 \otimes I + I \otimes \Lambda_0)^{-1} S_1^\mathrm T \otimes S_0^\mathrm T, \end{align*}

where $$S_d$$ is the matrix of eigenvectors to the generalized eigenvalue problem in the given tensor direction $$d$$:

\begin{align*} A_d s &= \lambda M_d s, d = 0, \quad \ldots,\mathrm{dim}, \end{align*}

and $$\Lambda_d$$ is the diagonal matrix representing the generalized eigenvalues $$\lambda$$. Note that the vectors $$s$$ are such that they simultaneously diagonalize $$A_d$$ and $$M_d$$, i.e. $$S_d^{\mathrm T} A_d S_d = \Lambda_d$$ and $$S_d^{\mathrm T} M_d S_d = I$$. This method of matrix inversion is called fast diagonalization method.

This class requires LAPACK support.

Note that this class allows for two modes of usage. The first is a use case with run time constants for the matrix dimensions that is achieved by setting the optional template parameter n_rows_1d to -1. The second mode of usage that is faster allows to set the template parameter as a compile time constant, giving significantly faster code in particular for small sizes of the matrix.

Template Parameters
 dim Dimension of the problem. Currently, 1D, 2D, and 3D codes are implemented. Number Arithmetic type of the underlying array elements. Note that the underlying LAPACK implementation supports only float and double numbers, so only these two types are currently supported by the generic class. Nevertheless, a template specialization for the vectorized types VectorizedArray and VectorizedArray exists. This is necessary to perform LAPACK calculations for each vectorization lane, i.e. for the supported float and double numbers. n_rows_1d Compile-time number of rows of 1D matrices (only valid if the number of rows and columns coincide for each dimension). By default at -1, which means that the number of rows is determined at run-time by means of the matrices passed to the reinit() function.

Definition at line 238 of file tensor_product_matrix.h.

## ◆ value_type

template<int dim, typename Number, int n_rows_1d = -1>
 using TensorProductMatrixSymmetricSumBase< dim, Number, n_rows_1d >::value_type = Number
inherited

Type of matrix entries. This alias is analogous to value_type in the standard library containers.

Definition at line 81 of file tensor_product_matrix.h.

## ◆ TensorProductMatrixSymmetricSum() [1/4]

template<int dim, typename Number , int n_rows_1d = -1>
 TensorProductMatrixSymmetricSum< dim, Number, n_rows_1d >::TensorProductMatrixSymmetricSum ( )
default

Default constructor.

## ◆ TensorProductMatrixSymmetricSum() [2/4]

template<int dim, typename Number , int n_rows_1d = -1>
 TensorProductMatrixSymmetricSum< dim, Number, n_rows_1d >::TensorProductMatrixSymmetricSum ( const std::array< Table< 2, Number >, dim > & mass_matrix, const std::array< Table< 2, Number >, dim > & derivative_matrix )

Constructor that is equivalent to the empty constructor and immediately calling reinit(const std::array<Table<2,Number>, dim>&,const std::array<Table<2,Number>, dim>&).

## ◆ TensorProductMatrixSymmetricSum() [3/4]

template<int dim, typename Number , int n_rows_1d = -1>
 TensorProductMatrixSymmetricSum< dim, Number, n_rows_1d >::TensorProductMatrixSymmetricSum ( const std::array< FullMatrix< Number >, dim > & mass_matrix, const std::array< FullMatrix< Number >, dim > & derivative_matrix )

Constructor that is equivalent to the empty constructor and immediately calling reinit(const std::array<FullMatrix<Number>,dim>&,const std::array<FullMatrix<Number>,dim>&).

## ◆ TensorProductMatrixSymmetricSum() [4/4]

template<int dim, typename Number , int n_rows_1d = -1>
 TensorProductMatrixSymmetricSum< dim, Number, n_rows_1d >::TensorProductMatrixSymmetricSum ( const Table< 2, Number > & mass_matrix, const Table< 2, Number > & derivative_matrix )

Constructor that is equivalent to the empty constructor and immediately calling reinit(const Table<2,Number>&,const Table<2,Number>&).

## ◆ reinit() [1/3]

template<int dim, typename Number , int n_rows_1d = -1>
 void TensorProductMatrixSymmetricSum< dim, Number, n_rows_1d >::reinit ( const std::array< Table< 2, Number >, dim > & mass_matrix, const std::array< Table< 2, Number >, dim > & derivative_matrix )

Initializes the tensor product matrix by copying the arrays of 1D mass matrices mass_matrix and 1D derivative matrices derivative_matrix into its base class counterparts, respectively, and by assembling the regarding generalized eigenvalues and eigenvectors in TensorProductMatrixSymmetricSumBase::eigenvalues and TensorProductMatrixSymmetricSumBase::eigenvectors, respectively. Note that the current implementation requires each $$M_{d}$$ to be symmetric and positive definite and every $$A_{d}$$ to be symmetric and invertible but not necessarily positive definite.

## ◆ reinit() [2/3]

template<int dim, typename Number , int n_rows_1d = -1>
 void TensorProductMatrixSymmetricSum< dim, Number, n_rows_1d >::reinit ( const std::array< FullMatrix< Number >, dim > & mass_matrix, const std::array< FullMatrix< Number >, dim > & derivative_matrix )

This function is equivalent to the previous reinit() except that the 1D matrices in mass_matrix and derivative_matrix are passed in terms of a FullMatrix, respectively.

## ◆ reinit() [3/3]

template<int dim, typename Number , int n_rows_1d = -1>
 void TensorProductMatrixSymmetricSum< dim, Number, n_rows_1d >::reinit ( const Table< 2, Number > & mass_matrix, const Table< 2, Number > & derivative_matrix )

This function is equivalent to the first reinit() except that we consider the same 1D mass matrix mass_matrix and the same 1D derivative matrix derivative_matrix for each tensor direction.

## ◆ reinit_impl()

template<int dim, typename Number , int n_rows_1d = -1>
template<typename MatrixArray >
 void TensorProductMatrixSymmetricSum< dim, Number, n_rows_1d >::reinit_impl ( MatrixArray && mass_matrix, MatrixArray && derivative_matrix )
private

A generic implementation of all reinit() functions based on perfect forwarding, that allows to pass lvalue as well as rvalue arguments.

Template Parameters
 MatrixArray Has to be convertible to the underlying type of TensorProductMatrixSymmetricSumBase::mass_matrix and TensorProductMatrixSymmetricSumBase::derivative_matrix.

## ◆ m()

template<int dim, typename Number, int n_rows_1d = -1>
 unsigned int TensorProductMatrixSymmetricSumBase< dim, Number, n_rows_1d >::m ( ) const
inherited

Return the number of rows of the tensor product matrix resulting from the Kronecker product of 1D matrices, which is described in the main documentation of TensorProductMatrixSymmetricSum.

## ◆ n()

template<int dim, typename Number, int n_rows_1d = -1>
 unsigned int TensorProductMatrixSymmetricSumBase< dim, Number, n_rows_1d >::n ( ) const
inherited

Return the number of columns of the tensor product matrix resulting from the Kronecker product of 1D matrices, which is described in the main documentation of TensorProductMatrixSymmetricSum.

## ◆ vmult()

template<int dim, typename Number, int n_rows_1d = -1>
 void TensorProductMatrixSymmetricSumBase< dim, Number, n_rows_1d >::vmult ( const ArrayView< Number > & dst, const ArrayView< const Number > & src ) const
inherited

Implements a matrix-vector product with the underlying matrix as described in the main documentation of TensorProductMatrixSymmetricSum. This function is operating on ArrayView to allow checks of array bounds with respect to dst and src.

## ◆ apply_inverse()

template<int dim, typename Number, int n_rows_1d = -1>
 void TensorProductMatrixSymmetricSumBase< dim, Number, n_rows_1d >::apply_inverse ( const ArrayView< Number > & dst, const ArrayView< const Number > & src ) const
inherited

Implements a matrix-vector product with the underlying matrix as described in the main documentation of TensorProductMatrixSymmetricSum. This function is operating on ArrayView to allow checks of array bounds with respect to dst and src.

## ◆ n_rows_1d_static

template<int dim, typename Number, int n_rows_1d = -1>
 constexpr int TensorProductMatrixSymmetricSumBase< dim, Number, n_rows_1d >::n_rows_1d_static = n_rows_1d
staticinherited

The static number of rows of the 1D matrices. For more details, see the description of the template parameter n_rows_1d.

Definition at line 87 of file tensor_product_matrix.h.

## ◆ mass_matrix

template<int dim, typename Number, int n_rows_1d = -1>
 std::array, dim> TensorProductMatrixSymmetricSumBase< dim, Number, n_rows_1d >::mass_matrix
protectedinherited

An array containing a mass matrix for each tensor direction.

Definition at line 133 of file tensor_product_matrix.h.

## ◆ derivative_matrix

template<int dim, typename Number, int n_rows_1d = -1>
 std::array, dim> TensorProductMatrixSymmetricSumBase< dim, Number, n_rows_1d >::derivative_matrix
protectedinherited

An array containing a derivative matrix for each tensor direction.

Definition at line 138 of file tensor_product_matrix.h.

## ◆ eigenvalues

template<int dim, typename Number, int n_rows_1d = -1>
 std::array, dim> TensorProductMatrixSymmetricSumBase< dim, Number, n_rows_1d >::eigenvalues
protectedinherited

An array storing the generalized eigenvalues for each tensor direction.

Definition at line 144 of file tensor_product_matrix.h.

## ◆ eigenvectors

template<int dim, typename Number, int n_rows_1d = -1>
 std::array, dim> TensorProductMatrixSymmetricSumBase< dim, Number, n_rows_1d >::eigenvectors
protectedinherited

An array storing the generalized eigenvectors for each tensor direction.

Definition at line 150 of file tensor_product_matrix.h.

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