internal::SymmetricTensorImplementation Namespace Reference

Classes
struct	Inverse
struct	SortEigenValuesVectors

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

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

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

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

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

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