#include <deal.II/base/array_view.h>
#include <deal.II/base/tensor.h>
#include <deal.II/lac/exceptions.h>
#include <deal.II/lac/lapack_templates.h>
#include <array>

Functions
template<int dim, typename Number>
Tensor< 2, dim, Number >	project_onto_orthogonal_tensors (const Tensor< 2, dim, Number > &A)
template Tensor< 2, 1, float >	project_onto_orthogonal_tensors (const Tensor< 2, 1, float > &)
template Tensor< 2, 2, float >	project_onto_orthogonal_tensors (const Tensor< 2, 2, float > &)
template Tensor< 2, 3, float >	project_onto_orthogonal_tensors (const Tensor< 2, 3, float > &)
template Tensor< 2, 1, double >	project_onto_orthogonal_tensors (const Tensor< 2, 1, double > &)
template Tensor< 2, 2, double >	project_onto_orthogonal_tensors (const Tensor< 2, 2, double > &)
template Tensor< 2, 3, double >	project_onto_orthogonal_tensors (const Tensor< 2, 3, double > &)

Function Documentation

◆ project_onto_orthogonal_tensors() [1/7]

template<int dim, typename Number>

Tensor< 2, dim, Number > project_onto_orthogonal_tensors ( const Tensor< 2, dim, Number > & A )

Return the nearest orthogonal matrix $\hat {\mathbf A}=\mathbf U \mathbf{V}^T$ by combining the products of the singular value decomposition (SVD) ${\mathbf A}=\mathbf U \mathbf S \mathbf V^T$ for a given input ${\mathbf A}$ , effectively replacing $\mathbf S$ with the identity matrix.

This is a (nonlinear) projection operation since when applied twice, we have $\hat{\hat{\mathbf A}}=\hat{\mathbf A}$ as is easy to see. (That is because the SVD of $\hat {\mathbf A}$ is simply $\mathbf U \mathbf I \mathbf{V}^T$ .) Furthermore, $\hat {\mathbf A}$ is really an orthogonal matrix because orthogonal matrices have to satisfy ${\hat {\mathbf A}}^T \hat {\mathbf A}={\mathbf I}$ , which here implies that

$\begin{align*} {\hat {\mathbf A}}^T \hat {\mathbf A} &= \left(\mathbf U \mathbf{V}^T\right)^T\left(\mathbf U \mathbf{V}^T\right) \\ &= \mathbf V \mathbf{U}^T \mathbf U \mathbf{V}^T \\ &= \mathbf V \left(\mathbf{U}^T \mathbf U\right) \mathbf{V}^T \\ &= \mathbf V \mathbf I \mathbf{V}^T \\ &= \mathbf V \mathbf{V}^T \\ &= \mathbf I \end{align*}$

due to the fact that the $\mathbf U$ and $\mathbf V$ factors that come out of the SVD are themselves orthogonal matrices.

Parameters

A	The tensor for which to find the closest orthogonal tensor.

Template Parameters

Number The type used to store the entries of the tensor. Must be either float or double.

Precondition: In order to use this function, this program must be linked with the LAPACK library.; A must not be singular. This is because, conceptually, the problem to be solved here is trying to find a matrix $\hat{\mathbf A}$ that minimizes some kind of distance from $\mathbf A$ while satisfying the quadratic constraint ${\hat {\mathbf A}}^T \hat {\mathbf A}={\mathbf I}$ . This is not so dissimilar to the kind of problem where one wants to find a vector $\hat{\mathbf x}\in{\mathbb R}^n$ that minimizes the quadratic objective function $\|\hat {\mathbf x} - \mathbf x\|^2$ for a given $\mathbf x$ subject to the constraint $\|\mathbf x\|^2=1$ – in other words, we are seeking the point $\hat{\mathbf x}$ on the unit sphere that is closest to $\mathbf x$ . This problem has a solution for all $\mathbf x$ except if $\mathbf x=0$ . The corresponding condition for the problem we are considering here is that $\mathbf A$ must not have a zero eigenvalue.

Definition at line 81 of file tensor.cc.

◆ project_onto_orthogonal_tensors() [2/7]

template Tensor< 2, 1, float > project_onto_orthogonal_tensors ( const Tensor< 2, 1, float > & )

◆ project_onto_orthogonal_tensors() [3/7]

template Tensor< 2, 2, float > project_onto_orthogonal_tensors ( const Tensor< 2, 2, float > & )

◆ project_onto_orthogonal_tensors() [4/7]

template Tensor< 2, 3, float > project_onto_orthogonal_tensors ( const Tensor< 2, 3, float > & )

◆ project_onto_orthogonal_tensors() [5/7]

template Tensor< 2, 1, double > project_onto_orthogonal_tensors ( const Tensor< 2, 1, double > & )

◆ project_onto_orthogonal_tensors() [6/7]

template Tensor< 2, 2, double > project_onto_orthogonal_tensors ( const Tensor< 2, 2, double > & )

◆ project_onto_orthogonal_tensors() [7/7]

template Tensor< 2, 3, double > project_onto_orthogonal_tensors ( const Tensor< 2, 3, double > & )

Functions