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

Inheritance diagram for TensorProductMatrixSymmetricSum< dim, Number, size >:

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)

Public Member Functions inherited from TensorProductMatrixSymmetricSumBase< dim, Number, size >
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

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

Additional Inherited Members
Protected Member Functions inherited from TensorProductMatrixSymmetricSumBase< dim, Number, size >
	TensorProductMatrixSymmetricSumBase ()=default

Protected Attributes inherited from TensorProductMatrixSymmetricSumBase< dim, Number, size >
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

Detailed Description

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

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 for the size 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<float> and VectorizedArray<double> exists. This is necessary to perform LAPACK calculations for each vectorization lane, i.e. for the supported float and double numbers.
size	Compile-time array lengths. By default at -1, which means that the run-time information stored in the matrices passed to the reinit() function is used.

Author: Martin Kronbichler and Julius Witte, 2017

Definition at line 225 of file tensor_product_matrix.h.