Detailed Description

Linear algebra routines.

Namespaces
	CASM::normal_form_impl

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

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

void	CASM::poly_fit (Eigen::VectorXcd &xvec, Eigen::VectorXcd &yvec, Eigen::VectorXcd &coeffs, int degree)

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_prod (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(std::ptr_fun(boost::math::iround < 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(std::ptr_fun(boost::math::lround < 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...

std::vector< Eigen::Matrix3i >	CASM::_unimodular_matrices (bool positive, bool negative)

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

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

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

Function Documentation

std::vector< Eigen::Matrix3i > CASM::_unimodular_matrices	(	bool	positive,
		bool	negative
	)

Definition at line 182 of file LinearAlgebra.cc.

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()

Definition at line 184 of file LinearAlgebra.hh.

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 154 of file LinearAlgebra.cc.

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)

Definition at line 156 of file LinearAlgebra.hh.

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 21 of file LinearAlgebra.cc.

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 61 of file LinearAlgebra.cc.

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 222 of file LinearAlgebra.hh.

template<typename Derived >

Eigen::CwiseUnaryOp< decltype(std::ptr_fun(boost::math::iround<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))

Definition at line 93 of file LinearAlgebra.hh.

template<typename Derived >

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

Check if Eigen::Matrix is diagonal.

Definition at line 79 of file LinearAlgebra.hh.

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 37 of file LinearAlgebra.cc.

template<typename Derived >

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

Check if Eigen::Matrix is integer.

Definition at line 61 of file LinearAlgebra.hh.

template<typename Derived >

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

Check if Eigen::Matrix is unimodular.

Definition at line 73 of file LinearAlgebra.hh.

template<typename Derived >

Eigen::CwiseUnaryOp< decltype(std::ptr_fun(boost::math::lround<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))

Definition at line 107 of file LinearAlgebra.hh.

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 122 of file LinearAlgebra.hh.

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

Definition at line 212 of file LinearAlgebra.cc.

void CASM::poly_fit	(	Eigen::VectorXcd &	xvec,
		Eigen::VectorXcd &	yvec,
		Eigen::VectorXcd &	coeffs,
		int	degree
	)

Definition at line 308 of file CASM_math.cc.

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

Definition at line 207 of file LinearAlgebra.cc.

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 170 of file LinearAlgebra.cc.

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 159 of file LinearAlgebra.cc.

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 237 of file LinearAlgebra.hh.

template<typename Derived >

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

Definition at line 53 of file LinearAlgebra.hh.

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

Definition at line 217 of file LinearAlgebra.cc.

Detailed Description

Namespaces

Functions

Function Documentation