Classes
struct	Inverse

Functions
template<int dim, typename Number >
void	tridiagonalize (const ::SymmetricTensor< 2, dim, Number > &A, ::Tensor< 2, dim, Number > &Q, std::array< Number, dim > &d, std::array< Number, dim-1 > &e)

template<int dim, typename Number >
std::array< std::pair< Number, Tensor< 1, dim, Number > >, dim >	ql_implicit_shifts (const ::SymmetricTensor< 2, dim, Number > &A)

template<int dim, typename Number >
std::array< std::pair< Number, Tensor< 1, dim, Number > >, dim >	jacobi (::SymmetricTensor< 2, dim, Number > A)

template<typename Number >
std::array< std::pair< Number, Tensor< 1, 2, Number > >, 2 >	hybrid (const ::SymmetricTensor< 2, 2, Number > &A)

template<typename Number >
std::array< std::pair< Number, Tensor< 1, 3, Number > >, 3 >	hybrid (const ::SymmetricTensor< 2, 3, Number > &A)

Detailed Description

A namespace for functions and classes that are internal to how the SymmetricTensor class (and its associate functions) works.

Function Documentation

◆ tridiagonalize()

template<int dim, typename Number >

void internal::SymmetricTensorImplementation::tridiagonalize	(	const ::SymmetricTensor< 2, dim, Number > &	A,
		::Tensor< 2, dim, Number > &	Q,
		std::array< Number, dim > &	d,
		std::array< Number, dim-1 > &	e
	)

Tridiagonalize a rank-2 symmetric tensor using the Householder method. The specialized algorithm implemented here is given in

@Article{Kopp2008,
  title        = {Efficient numerical diagonalization of hermitian 3x3 matrices},
  author       = {Kopp, J.},
  journal      = {International Journal of Modern Physics C},
  year         = {2008},
  volume       = {19},
  number       = {3},
  pages        = {523--548},
  doi          = {10.1142/S0129183108012303},
  eprinttype   = {arXiv},
  eprint       = {physics/0610206v3},
  eprintclass  = {physics.comp-ph},
  url          = {https://www.mpi-hd.mpg.de/personalhomes/globes/3x3/index.html}
}

and is based off of the generic algorithm presented in section 11.3.2 of

@Book{Press2007,
  title        = {Numerical recipes 3rd edition: The art of scientific computing},
  author       = {Press, W. H.},
  journal      = {Cambridge university press},
  year         = {2007}
}

Parameters

[in]	A	This tensor to be tridiagonalized
[out]	Q	The orthogonal matrix effecting the transformation
[out]	d	The diagonal elements of the tridiagonal matrix
[out]	e	The off-diagonal elements of the tridiagonal matrix

Author: Joachim Kopp, Jean-Paul Pelteret, 2017

◆ ql_implicit_shifts()

template<int dim, typename Number >

std::array<std::pair<Number, Tensor<1,dim,Number> >,dim> internal::SymmetricTensorImplementation::ql_implicit_shifts ( const ::SymmetricTensor< 2, dim, Number > & A )

Compute the eigenvalues and eigenvectors of a real-valued rank-2 symmetric tensor using the QL algorithm with implicit shifts. The specialized algorithm implemented here is given in

@Article{Kopp2008,
  title        = {Efficient numerical diagonalization of hermitian 3x3 matrices},
  author       = {Kopp, J.},
  journal      = {International Journal of Modern Physics C},
  year         = {2008},
  volume       = {19},
  number       = {3},
  pages        = {523--548},
  doi          = {10.1142/S0129183108012303},
  eprinttype   = {arXiv},
  eprint       = {physics/0610206v3},
  eprintclass  = {physics.comp-ph},
  url          = {https://www.mpi-hd.mpg.de/personalhomes/globes/3x3/index.html}
}

and is based off of the generic algorithm presented in section 11.4.3 of

@Book{Press2007,
  title        = {Numerical recipes 3rd edition: The art of scientific computing},
  author       = {Press, W. H.},
  journal      = {Cambridge university press},
  year         = {2007}
}

Parameters

[in] A The tensor of which the eigenvectors and eigenvalues are to be computed.

Returns: An array containing the eigenvectors and the associated eigenvalues. The array is not sorted in any particular order.

Author: Joachim Kopp, Jean-Paul Pelteret, 2017

◆ jacobi()

template<int dim, typename Number >

std::array<std::pair<Number, Tensor<1,dim,Number> >,dim> internal::SymmetricTensorImplementation::jacobi ( ::SymmetricTensor< 2, dim, Number > A )

Compute the eigenvalues and eigenvectors of a real-valued rank-2 symmetric tensor using the Jacobi algorithm. The specialized algorithm implemented here is given in

@Article{Kopp2008,
  title        = {Efficient numerical diagonalization of hermitian 3x3 matrices},
  author       = {Kopp, J.},
  journal      = {International Journal of Modern Physics C},
  year         = {2008},
  volume       = {19},
  number       = {3},
  pages        = {523--548},
  doi          = {10.1142/S0129183108012303},
  eprinttype   = {arXiv},
  eprint       = {physics/0610206v3},
  eprintclass  = {physics.comp-ph},
  url          = {https://www.mpi-hd.mpg.de/personalhomes/globes/3x3/index.html}
}

and is based off of the generic algorithm presented in section 11.4.3 of

@Book{Press2007,
  title        = {Numerical recipes 3rd edition: The art of scientific computing},
  author       = {Press, W. H.},
  journal      = {Cambridge university press},
  year         = {2007}
}

Parameters

[in] A The tensor of which the eigenvectors and eigenvalues are to be computed.

Returns: An array containing the eigenvectors and the associated eigenvalues. The array is not sorted in any particular order.

Author: Joachim Kopp, Jean-Paul Pelteret, 2017

◆ hybrid() [1/2]

template<typename Number >

std::array<std::pair<Number, Tensor<1,2,Number> >,2> internal::SymmetricTensorImplementation::hybrid ( const ::SymmetricTensor< 2, 2, Number > & A )

Compute the eigenvalues and eigenvectors of a real-valued rank-2 symmetric 2x2 tensor using the characteristic equation to compute eigenvalues and an analytical approach based on the cross-product for the eigenvectors. If the computations are deemed too inaccurate then the method falls back to ql_implicit_shifts.

Parameters

[in] A The tensor of which the eigenvectors and eigenvalues are to be computed.

Returns: An array containing the eigenvectors and the associated eigenvalues. The array is not sorted in any particular order.

Author: Joachim Kopp, Jean-Paul Pelteret, 2017

◆ hybrid() [2/2]

template<typename Number >

std::array<std::pair<Number, Tensor<1,3,Number> >,3> internal::SymmetricTensorImplementation::hybrid ( const ::SymmetricTensor< 2, 3, Number > & A )

Compute the eigenvalues and eigenvectors of a real-valued rank-2 symmetric 3x3 tensor using the characteristic equation to compute eigenvalues and an analytical approach based on the cross-product for the eigenvectors. If the computations are deemed too inaccurate then the method falls back to ql_implicit_shifts. The specialized algorithm implemented here is given in

@Article{Kopp2008,
  title        = {Efficient numerical diagonalization of hermitian 3x3 matrices},
  author       = {Kopp, J.},
  journal      = {International Journal of Modern Physics C},
  year         = {2008},
  volume       = {19},
  number       = {3},
  pages        = {523--548},
  doi          = {10.1142/S0129183108012303},
  eprinttype   = {arXiv},
  eprint       = {physics/0610206v3},
  eprintclass  = {physics.comp-ph},
  url          = {https://www.mpi-hd.mpg.de/personalhomes/globes/3x3/index.html}
}

Parameters

[in] A The tensor of which the eigenvectors and eigenvalues are to be computed.

Returns: An array containing the eigenvectors and the associated eigenvalues. The array is not sorted in any particular order.

Author: Joachim Kopp, Jean-Paul Pelteret, 2017

Classes

Functions

Detailed Description

Function Documentation

◆ tridiagonalize()

◆ ql_implicit_shifts()

◆ jacobi()

◆ hybrid() [1/2]

◆ hybrid() [2/2]