Detailed Description

Linear algebra routines.

Functions
void	CASM::get_Hermitian (Eigen::MatrixXcd &original_mat, Eigen::MatrixXcd &hermitian_mat, Eigen::MatrixXcd &antihermitian_mat)

bool	CASM::is_Hermitian (Eigen::MatrixXcd &mat)

std::pair< Eigen::MatrixXi, Eigen::MatrixXi >	CASM::hermite_normal_form (const Eigen::MatrixXi &M)
	Return the hermite normal form, M == H*V. More...

double	CASM::angle (const Eigen::Ref< const Eigen::Vector3d > &a, const Eigen::Ref< const Eigen::Vector3d > &b)
	Get angle, in radians, between two vectors on range [0,pi]. More...

double	CASM::signed_angle (const Eigen::Ref< const Eigen::Vector3d > &a, const Eigen::Ref< const Eigen::Vector3d > &b, const Eigen::Ref< const Eigen::Vector3d > &pos_ref)
	signed angle, in radians, between -pi and pi that describe separation in direction of two vectors More...

Eigen::MatrixXd	CASM::pretty (const Eigen::MatrixXd &M, double tol)
	Round entries that are within tol of being integer to that integer value. More...

template<typename Derived >
Derived::Scalar	CASM::triple_product (const Derived &vec0, const Derived &vec1, const Derived &vec2)

template<typename Derived >
bool	CASM::is_integer (const Eigen::MatrixBase< Derived > &M, double tol)
	Check if Eigen::Matrix is integer. More...

template<typename Derived >
bool	CASM::is_unimodular (const Eigen::MatrixBase< Derived > &M, double tol)
	Check if Eigen::Matrix is unimodular. More...

template<typename Derived >
bool	CASM::is_diagonal (const Eigen::MatrixBase< Derived > &M, double tol=TOL)
	Check if Eigen::Matrix is diagonal. More...

template<typename Derived >
Eigen::CwiseUnaryOp< decltype(Local::round_l< typename Derived::Scalar >), const Derived >	CASM::round (const Eigen::MatrixBase< Derived > &val)
	Round Eigen::MatrixXd. More...

template<typename Derived >
Eigen::CwiseUnaryOp< decltype(Local::iround_l< typename Derived::Scalar >), const Derived >	CASM::iround (const Eigen::MatrixBase< Derived > &val)
	Round Eigen::MatrixXd to Eigen::MatrixXi. More...

template<typename Derived >
Eigen::CwiseUnaryOp< decltype(Local::lround_l< typename Derived::Scalar >), const Derived >	CASM::lround (const Eigen::MatrixBase< Derived > &val)
	Round Eigen::MatrixXd to Eigen::MatrixXl. More...

template<typename Derived >
Derived::Scalar	CASM::matrix_minor (const Eigen::MatrixBase< Derived > &M, int row, int col)
	Return the minor of integer Matrix M element row, col. More...

template<typename Derived >
Eigen::Matrix< typename Derived::Scalar, Derived::RowsAtCompileTime, Derived::ColsAtCompileTime >	CASM::cofactor (const Eigen::MatrixBase< Derived > &M)
	Return cofactor matrix. More...

template<typename Derived >
Eigen::Matrix< typename Derived::Scalar, Derived::RowsAtCompileTime, Derived::ColsAtCompileTime >	CASM::adjugate (const Eigen::MatrixBase< Derived > &M)
	Return adjugate matrix. More...

template<typename Derived >
Eigen::Matrix< typename Derived::Scalar, Derived::RowsAtCompileTime, Derived::ColsAtCompileTime >	CASM::inverse (const Eigen::MatrixBase< Derived > &M)
	Return the integer inverse matrix of an invertible integer matrix. More...

template<typename DerivedIn , typename DerivedOut >
void	CASM::smith_normal_form (const Eigen::MatrixBase< DerivedIn > &M, Eigen::MatrixBase< DerivedOut > &U, Eigen::MatrixBase< DerivedOut > &S, Eigen::MatrixBase< DerivedOut > &V)
	Return the smith normal form, M == USV. More...

Eigen::Matrix3d	CASM::polar_decomposition (Eigen::Matrix3d const &mat)

std::vector< Eigen::Matrix3i >	CASM::_unimodular_matrices (bool positive, bool negative, int range=1)

const std::vector< Eigen::Matrix3i > &	CASM::positive_unimodular_matrices ()

const std::vector< Eigen::Matrix3i > &	CASM::negative_unimodular_matrices ()

template<int range = 1>
const std::vector< Eigen::Matrix3i > &	CASM::unimodular_matrices ()

Function Documentation

◆ _unimodular_matrices()

std::vector< Eigen::Matrix3i > CASM::_unimodular_matrices	(	bool	positive,
		bool	negative,
		int	range = `1`
	)

Definition at line 646 of file CASM_Eigen_math.cc.

◆ adjugate()

template<typename Derived >

Eigen::Matrix<typename Derived::Scalar, Derived::RowsAtCompileTime, Derived::ColsAtCompileTime> CASM::adjugate ( const Eigen::MatrixBase< Derived > & M )

Return adjugate matrix.

Returns: adjugate matrix

Parameters

M matrix

The adjugate matrix is the transpose of the cofactor matrix. The adjugate matrix is related to the inverse of a matrix through

M.inverse() == adjugate(M)/M.determinant()

CASM::adjugate

Eigen::Matrix< typename Derived::Scalar, Derived::RowsAtCompileTime, Derived::ColsAtCompileTime > adjugate(const Eigen::MatrixBase< Derived > &M)

Return adjugate matrix.

Definition: CASM_Eigen_math.hh:320

Definition at line 320 of file CASM_Eigen_math.hh.

◆ angle()

double CASM::angle	(	const Eigen::Ref< const Eigen::Vector3d > &	a,
		const Eigen::Ref< const Eigen::Vector3d > &	b
	)

Get angle, in radians, between two vectors on range [0,pi].

Definition at line 611 of file CASM_Eigen_math.cc.

◆ cofactor()

template<typename Derived >

Eigen::Matrix<typename Derived::Scalar, Derived::RowsAtCompileTime, Derived::ColsAtCompileTime> CASM::cofactor ( const Eigen::MatrixBase< Derived > & M )

Return cofactor matrix.

Returns: cofactor matrix

Parameters

M matrix

The cofactor matrix is

C(i,j) = pow(-1,i+j)*matrix_minor(M,i,j)

CASM::matrix_minor

Derived::Scalar matrix_minor(const Eigen::MatrixBase< Derived > &M, int row, int col)

Return the minor of integer Matrix M element row, col.

Definition: CASM_Eigen_math.hh:250

Definition at line 292 of file CASM_Eigen_math.hh.

◆ get_Hermitian()

void CASM::get_Hermitian	(	Eigen::MatrixXcd &	original_mat,
		Eigen::MatrixXcd &	hermitian_mat,
		Eigen::MatrixXcd &	antihermitian_mat
	)

Given a matrix, original_mat, this function calculates both the Hermitian and anti-Hermitian parts.

Parameters

[in]	original_mat	The matrix we want to get the Hermitian and anti-Hermitian parts of
[in,out]	hermitian_mat	The Hermitian matrix
[in,out]	antihermitian_mat	The anti-Hermitian matrix

Definition at line 486 of file CASM_Eigen_math.cc.

◆ hermite_normal_form()

std::pair< Eigen::MatrixXi, Eigen::MatrixXi > CASM::hermite_normal_form ( const Eigen::MatrixXi & M )

Return the hermite normal form, M == H*V.

Parameters

M	a square integer matrix

Exceptions

std::runtime_error if M is not full rank

Returns: std::pair<H,V>

H set to upper triangular matrix H with:

H(i,i) > 0
0 <= H(r,i) < H(i,i) for r > i
H(r,i) == 0 for r < i V set to unimodular matrix V
V.determinant() = +/- 1

Definition at line 524 of file CASM_Eigen_math.cc.

◆ inverse()

template<typename Derived >

Eigen::Matrix<typename Derived::Scalar, Derived::RowsAtCompileTime, Derived::ColsAtCompileTime> CASM::inverse ( const Eigen::MatrixBase< Derived > & M )

Return the integer inverse matrix of an invertible integer matrix.

Returns: a 3x3 integer matrix inverse

Parameters

M	an 3x3 invertible integer matrix

Definition at line 359 of file CASM_Eigen_math.hh.

◆ iround()

template<typename Derived >

Eigen::CwiseUnaryOp<decltype(Local::iround_l<typename Derived::Scalar>), const Derived> CASM::iround ( const Eigen::MatrixBase< Derived > & val )

Round Eigen::MatrixXd to Eigen::MatrixXi.

Returns: an Eigen:MatrixXi

Parameters

M	Eigen::MatrixXd to be rounded to integer

For each coefficient, sets

Mint(i,j) = boost::math::iround(Mdouble(i,

j))

CASM::iround

Eigen::CwiseUnaryOp< decltype(Local::iround_l< typename Derived::Scalar >), const Derived > iround(const Eigen::MatrixBase< Derived > &val)

Round Eigen::MatrixXd to Eigen::MatrixXi.

Definition: CASM_Eigen_math.hh:218

Definition at line 218 of file CASM_Eigen_math.hh.

◆ is_diagonal()

template<typename Derived >

bool CASM::is_diagonal	(	const Eigen::MatrixBase< Derived > &	M,
		double	tol = `TOL`
	)

Check if Eigen::Matrix is diagonal.

Definition at line 186 of file CASM_Eigen_math.hh.

◆ is_Hermitian()

bool CASM::is_Hermitian ( Eigen::MatrixXcd & mat )

Returns true if the specified matrix is Hermitian.

Checks if the matrix is the same as the conjugate

Definition at line 503 of file CASM_Eigen_math.cc.

◆ is_integer()

template<typename Derived >

bool CASM::is_integer	(	const Eigen::MatrixBase< Derived > &	M,
		double	tol
	)

Check if Eigen::Matrix is integer.

Definition at line 166 of file CASM_Eigen_math.hh.

◆ is_unimodular()

template<typename Derived >

bool CASM::is_unimodular	(	const Eigen::MatrixBase< Derived > &	M,
		double	tol
	)

Check if Eigen::Matrix is unimodular.

Definition at line 178 of file CASM_Eigen_math.hh.

◆ lround()

template<typename Derived >

Eigen::CwiseUnaryOp<decltype(Local::lround_l<typename Derived::Scalar>), const Derived> CASM::lround ( const Eigen::MatrixBase< Derived > & val )

Round Eigen::MatrixXd to Eigen::MatrixXl.

Returns: an Eigen:MatrixXl

Parameters

M	Eigen::MatrixXd to be rounded to integer

For each coefficient, sets

Mint(i,j) = std::lround(Mdouble(i, j))

CASM::lround

Eigen::CwiseUnaryOp< decltype(Local::lround_l< typename Derived::Scalar >), const Derived > lround(const Eigen::MatrixBase< Derived > &val)

Round Eigen::MatrixXd to Eigen::MatrixXl.

Definition: CASM_Eigen_math.hh:234

Definition at line 234 of file CASM_Eigen_math.hh.

◆ matrix_minor()

template<typename Derived >

Derived::Scalar CASM::matrix_minor	(	const Eigen::MatrixBase< Derived > &	M,
		int	row,
		int	col
	)

Return the minor of integer Matrix M element row, col.

Returns: the minor of element row, col

Parameters

M	matrix
row,col	element row and column

The minor of element row, col of a matrix is the determinant of the submatrix of M which does not include any elements from row 'row' or column 'col'

Definition at line 250 of file CASM_Eigen_math.hh.

◆ negative_unimodular_matrices()

const std::vector< Eigen::Matrix3i > & CASM::negative_unimodular_matrices ( )

Definition at line 680 of file CASM_Eigen_math.cc.

◆ polar_decomposition()

Eigen::Matrix3d CASM::polar_decomposition ( Eigen::Matrix3d const & mat )

Definition at line 641 of file CASM_Eigen_math.cc.

◆ positive_unimodular_matrices()

const std::vector< Eigen::Matrix3i > & CASM::positive_unimodular_matrices ( )

Definition at line 674 of file CASM_Eigen_math.cc.

◆ pretty()

Eigen::MatrixXd CASM::pretty	(	const Eigen::MatrixXd &	M,
		double	tol
	)

Round entries that are within tol of being integer to that integer value.

Definition at line 629 of file CASM_Eigen_math.cc.

◆ round()

template<typename Derived >

Eigen::CwiseUnaryOp<decltype(Local::round_l<typename Derived::Scalar>), const Derived> CASM::round ( const Eigen::MatrixBase< Derived > & val )

Round Eigen::MatrixXd.

Returns: an Eigen:MatrixXd

Parameters

M	Eigen::MatrixXd to be rounded

For each coefficient, sets

M(i,j) = boost::math::round(Mdouble(i, j))

CASM::round

Eigen::CwiseUnaryOp< decltype(Local::round_l< typename Derived::Scalar >), const Derived > round(const Eigen::MatrixBase< Derived > &val)

Round Eigen::MatrixXd.

Definition: CASM_Eigen_math.hh:202

Definition at line 202 of file CASM_Eigen_math.hh.

◆ signed_angle()

double CASM::signed_angle	(	const Eigen::Ref< const Eigen::Vector3d > &	a,
		const Eigen::Ref< const Eigen::Vector3d > &	b,
		const Eigen::Ref< const Eigen::Vector3d > &	pos_ref
	)

signed angle, in radians, between -pi and pi that describe separation in direction of two vectors

return signed angle, in radians, between -pi and pi that describe separation in direction of two vectors

Definition at line 618 of file CASM_Eigen_math.cc.

◆ smith_normal_form()

template<typename DerivedIn , typename DerivedOut >

void CASM::smith_normal_form	(	const Eigen::MatrixBase< DerivedIn > &	M,
		Eigen::MatrixBase< DerivedOut > &	U,
		Eigen::MatrixBase< DerivedOut > &	S,
		Eigen::MatrixBase< DerivedOut > &	V
	)

Return the smith normal form, M == U*S*V.

Parameters

	M	3x3 integer matrix
[out]	U	set to U: is integer, U.determinant() == 1
[out]	S	set to S: is diagonal, integer, and nonnegative, S(i,i) divides S(i+1,i+1) for all i
[out]	V	set to V: is integer, V.determinant() == 1

Adapted from Matlab implementation written by John Gilbert (gilbe.nosp@m.rt@p.nosp@m.arc.x.nosp@m.erox.nosp@m..com):

http://www.mathworks.com/matlabcentral/newsreader/view_thread/13728

Definition at line 376 of file CASM_Eigen_math.hh.

◆ triple_product()

template<typename Derived >

Derived::Scalar CASM::triple_product	(	const Derived &	vec0,
		const Derived &	vec1,
		const Derived &	vec2
	)

Definition at line 158 of file CASM_Eigen_math.hh.

◆ unimodular_matrices()

template<int range = 1>

const std::vector<Eigen::Matrix3i>& CASM::unimodular_matrices ( )

Definition at line 528 of file CASM_Eigen_math.hh.

Detailed Description

Functions

Function Documentation

◆ _unimodular_matrices()

◆ adjugate()

◆ angle()

◆ cofactor()

◆ get_Hermitian()

◆ hermite_normal_form()

◆ inverse()

◆ iround()

◆ is_diagonal()

◆ is_Hermitian()

◆ is_integer()

◆ is_unimodular()

◆ lround()

◆ matrix_minor()

◆ negative_unimodular_matrices()

◆ polar_decomposition()

◆ positive_unimodular_matrices()

◆ pretty()

◆ round()

◆ signed_angle()

◆ smith_normal_form()

◆ triple_product()

◆ unimodular_matrices()