Givens rotations, Cholesky, symmetric eigendecomposition, SVD, polar decompositions and condition numbers. More...

Functions
template<typename T , Index N>
KOKKOS_INLINE_FUNCTION void	minitensor::givens_left (T const &c, T const &s, Index i, Index k, Tensor< T, N > &A)

template<typename T , Index N>
KOKKOS_INLINE_FUNCTION void	minitensor::givens_right (T const &c, T const &s, Index i, Index k, Tensor< T, N > &A)

template<typename T , Index N>
std::pair< Vector< T, N >, Tensor< T, N > >	minitensor::sort_permutation (Vector< T, N > const &u)

template<typename T , Index N>
std::tuple< Tensor< T, N >, Tensor< T, N >, Tensor< T, N > >	minitensor::svd (Tensor< T, N > const &A)

template<typename T , Index N>
KOKKOS_INLINE_FUNCTION Tensor< T, N >	minitensor::polar_rotation (Tensor< T, N > const &A)

template<typename T , Index N>
std::pair< Tensor< T, N >, Tensor< T, N > >	minitensor::polar_left (Tensor< T, N > const &A)

template<typename T , Index N>
std::pair< Tensor< T, N >, Tensor< T, N > >	minitensor::polar_right (Tensor< T, N > const &A)

template<typename T , Index N>
std::pair< Tensor< T, N >, Tensor< T, N > >	minitensor::polar_left_eig (Tensor< T, N > const &A)

template<typename T , Index N>
std::pair< Tensor< T, N >, Tensor< T, N > >	minitensor::polar_right_eig (Tensor< T, N > const &A)

template<typename T , Index N>
std::tuple< Tensor< T, N >, Tensor< T, N >, Tensor< T, N > >	minitensor::polar_left_logV (Tensor< T, N > const &F)

template<typename T , Index N>
std::tuple< Tensor< T, N >, Tensor< T, N >, Tensor< T, N > >	minitensor::polar_left_logV_eig (Tensor< T, N > const &F)

template<typename T , Index N>
std::tuple< Tensor< T, N >, Tensor< T, N >, Tensor< T, N > >	minitensor::polar_left_logV_lame (Tensor< T, N > const &F)

template<typename T >
std::pair< T, T >	minitensor::schur_sym (const T f, const T g, const T h)

template<typename T >
std::pair< T, T >	minitensor::givens (T const &a, T const &b)

template<typename T , Index N>
std::pair< Tensor< T, N >, Tensor< T, N > >	minitensor::eig_sym (Tensor< T, N > const &A)

template<typename T , Index N>
std::pair< Tensor< T, N >, Tensor< T, N > >	minitensor::eig_spd (Tensor< T, N > const &A)

template<typename T , Index N>
std::pair< Tensor< T, N >, Tensor< T, N > >	minitensor::eig_spd_cos (Tensor< T, N > const &A)

template<typename T , Index N>
std::pair< Tensor< T, N >, bool >	minitensor::cholesky (Tensor< T, N > const &A)

template<typename T , Index N>
T	minitensor::cond (Tensor< T, N > const &A)

template<typename T , Index N>
T	minitensor::inv_cond (Tensor< T, N > const &A)

Detailed Description

Givens rotations, Cholesky, symmetric eigendecomposition, SVD, polar decompositions and condition numbers.

Function Documentation

◆ givens_left()

template<typename T , Index N>

KOKKOS_INLINE_FUNCTION void minitensor::givens_left	(	T const &	c,
		T const &	s,
		Index	i,
		Index	k,
		Tensor< T, N > &	A
	)

Apply Givens-Jacobi rotation on the left in place.

Parameters

c	cosine defining the rotation G in form [c, s; -s, c]
s	sine defining the rotation G in form [c, s; -s, c]
i	row index where the rotation is applied
k	column index where the rotation is applied
A	tensor to rotate

Definition at line 261 of file MiniTensor_Factorizations.h.

◆ givens_right()

template<typename T , Index N>

KOKKOS_INLINE_FUNCTION void minitensor::givens_right	(	T const &	c,
		T const &	s,
		Index	i,
		Index	k,
		Tensor< T, N > &	A
	)

Apply Givens-Jacobi rotation on the right in place.

Parameters

c	cosine defining the rotation G in form [c, s; -s, c]
s	sine defining the rotation G in form [c, s; -s, c]
i	row index where the rotation is applied
k	column index where the rotation is applied
A	tensor to rotate

Definition at line 281 of file MiniTensor_Factorizations.h.

◆ sort_permutation()

template<typename T , Index N>

std::pair< Vector< T, N >, Tensor< T, N > > minitensor::sort_permutation ( Vector< T, N > const & u )

Sort and index. Useful for ordering singular values and eigenvalues and corresponding vectors in the respective decompositions.

Parameters

u	vector to sort

Returns: v P sorted vector, permutation matrix such that v = P^T u

Definition at line 227 of file MiniTensor_Factorizations.h.

◆ svd()

template<typename T , Index N>

std::tuple< Tensor< T, N >, Tensor< T, N >, Tensor< T, N > > minitensor::svd ( Tensor< T, N > const & A )

Singular value decomposition (SVD)

Returns: \( A = USV^T\)

Definition at line 546 of file MiniTensor_Factorizations.h.

◆ polar_rotation()

template<typename T , Index N>

KOKKOS_INLINE_FUNCTION Tensor< T, N > minitensor::polar_rotation ( Tensor< T, N > const & A )

Project to O(N) (Orthogonal Group) using a Newton-type algorithm. See Higham's Functions of Matrices p210 [2008]

Parameters

A	tensor (often a deformation-gradient-like tensor)

Returns: \( R = \arg min_Q \|A - Q\|\) This algorithm projects a given tensor in GL(N) to O(N). The rotation/reflection obtained through this projection is the orthogonal component of the real polar decomposition

Definition at line 606 of file MiniTensor_Factorizations.h.

◆ polar_left()

template<typename T , Index N>

std::pair< Tensor< T, N >, Tensor< T, N > > minitensor::polar_left ( Tensor< T, N > const & A )

Left polar decomposition

Parameters

A	tensor (often a deformation-gradient-like tensor)

Returns: \( VR = A \) with \( R \in SO(N) \) and \( V \in SPD(N) \)

Definition at line 688 of file MiniTensor_Factorizations.h.

◆ polar_right()

template<typename T , Index N>

std::pair< Tensor< T, N >, Tensor< T, N > > minitensor::polar_right ( Tensor< T, N > const & A )

Right polar decomposition

Parameters

A	tensor (often a deformation-gradient-like tensor)

Returns: \( RU = A \) with \( R \in SO(N) \) and \( U \in SPD(N) \)

Definition at line 704 of file MiniTensor_Factorizations.h.

◆ polar_left_eig()

template<typename T , Index N>

std::pair< Tensor< T, N >, Tensor< T, N > > minitensor::polar_left_eig ( Tensor< T, N > const & A )

Left polar decomposition computed with eigenvalue decomposition

Parameters

A	tensor (often a deformation-gradient-like tensor)

Returns: \( VR = A \) with \( R \in SO(N) \) and \( V \in SPD(N) \)

Definition at line 720 of file MiniTensor_Factorizations.h.

◆ polar_right_eig()

template<typename T , Index N>

std::pair< Tensor< T, N >, Tensor< T, N > > minitensor::polar_right_eig ( Tensor< T, N > const & A )

R^3 right polar decomposition

Parameters

A	tensor (often a deformation-gradient-like tensor)

Returns: \( RU = F \) with \( R \in SO(N) \) and \( U \in SPD(N) \)

Definition at line 780 of file MiniTensor_Factorizations.h.

◆ polar_left_logV()

template<typename T , Index N>

std::tuple< Tensor< T, N >, Tensor< T, N >, Tensor< T, N > > minitensor::polar_left_logV ( Tensor< T, N > const & F )

Left polar decomposition with matrix logarithm for V

Parameters

F	tensor (often a deformation-gradient-like tensor)

Returns: \( VR = F \) with \( R \in SO(N) \) and V SPD, and log V

Definition at line 845 of file MiniTensor_Factorizations.h.

◆ polar_left_logV_eig()

template<typename T , Index N>

std::tuple< Tensor< T, N >, Tensor< T, N >, Tensor< T, N > > minitensor::polar_left_logV_eig ( Tensor< T, N > const & F )

R^N left polar decomposition with eigenvalue decomposition, returning also \( \log V \).

Definition at line 879 of file MiniTensor_Factorizations.h.

◆ polar_left_logV_lame()

template<typename T , Index N>

std::tuple< Tensor< T, N >, Tensor< T, N >, Tensor< T, N > > minitensor::polar_left_logV_lame ( Tensor< T, N > const & F )

Left polar decomposition with matrix logarithm for V using eig_spd_cos

Parameters

F	tensor (often a deformation-gradient-like tensor)

Returns: \( VR = F \) with \( R \in SO(N) \) and V SPD, and log V

Definition at line 922 of file MiniTensor_Factorizations.h.

◆ schur_sym()

template<typename T >

std::pair< T, T > minitensor::schur_sym	(	const T	f,
		const T	g,
		const T	h
	)

Symmetric Schur algorithm for R^2.

Parameters

f	component of \( A = [f, g; g, h] \in S(2) \)
g	component of \( A = [f, g; g, h] \in S(2) \)
h	component of \( A = [f, g; g, h] \in S(2) \)

Returns: \( c, s \) such that \( [c, -s; s, c] \) diagonalizes A

Definition at line 970 of file MiniTensor_Factorizations.h.

◆ givens()

template<typename T >

std::pair< T, T > minitensor::givens	(	T const &	a,
		T const &	b
	)

Givens rotation. [c, -s; s, c] [a; b] = [r; 0]

Returns: c and s

Definition at line 994 of file MiniTensor_Factorizations.h.

◆ eig_sym()

template<typename T , Index N>

std::pair< Tensor< T, N >, Tensor< T, N > > minitensor::eig_sym ( Tensor< T, N > const & A )

Eigenvalue decomposition for symmetric 2nd-order tensor

Returns: V eigenvectors, D eigenvalues in diagonal Matlab-style

Definition at line 1211 of file MiniTensor_Factorizations.h.

◆ eig_spd()

template<typename T , Index N>

std::pair< Tensor< T, N >, Tensor< T, N > > minitensor::eig_spd ( Tensor< T, N > const & A )

Eigenvalue decomposition for SPD 2nd-order tensor

Returns: V eigenvectors, D eigenvalues in diagonal Matlab-style

Definition at line 1239 of file MiniTensor_Factorizations.h.

◆ eig_spd_cos()

template<typename T , Index N>

std::pair< Tensor< T, N >, Tensor< T, N > > minitensor::eig_spd_cos ( Tensor< T, N > const & A )

Eigenvalue decomposition for SPD 2nd-order tensor

Returns: V eigenvectors, D eigenvalues in diagonal Matlab-style This algorithm comes from the journal article Scherzinger and Dohrmann, CMAME 197 (2008) 4007-4015

Definition at line 1249 of file MiniTensor_Factorizations.h.

◆ cholesky()

template<typename T , Index N>

std::pair< Tensor< T, N >, bool > minitensor::cholesky ( Tensor< T, N > const & A )

Cholesky decomposition, rank-1 update algorithm (Matrix Computations 3rd ed., Golub & Van Loan, p145)

Parameters

A	assumed symmetric tensor

Returns: G Cholesky factor A = GG^T and completed (bool) algorithm ran to completion

Definition at line 1503 of file MiniTensor_Factorizations.h.

◆ cond()

template<typename T , Index N>

T minitensor::cond ( Tensor< T, N > const & A )

Condition number: ratio of largest to smalest singular values.

Definition at line 195 of file MiniTensor_Factorizations.h.

◆ inv_cond()

template<typename T , Index N>

T minitensor::inv_cond ( Tensor< T, N > const & A )

Reciprocal condition number: ratio of smallest to largest singular values.

Definition at line 210 of file MiniTensor_Factorizations.h.