Matrix_t< n, m, T, Mtype > Class Template Reference

The main \( n \times m \) matrix type template Matrix_t. This is a base class type for "useful" types which are derived from this. More...

#include <matrix.h>

Public Member Functions
auto	abs () const
	Returns absolute value of Matrix.
template<typename Rtype = typename std::conditional<n == m, Mtype, Matrix<m, n, T>>::type>
Rtype	adjoint () const
	Adjoint of matrix.
const Array< n, m, T > &	asArray () const
	Cast Matrix to Array.
template<int mm = m, std::enable_if_t< mm==1, int > = 0>
const DiagonalMatrix< n, T > &	asDiagonalMatrix () const
	Cast Vector to DiagonalMatrix.
Vector< n, T >	column (int c) const
	Returns column vector as value at index c.
Mtype	conj () const
	Returns complex conjugate of Matrix.
template<typename Rtype = typename std::conditional<n == m, Mtype, Matrix<m, n, T>>::type>
Rtype	dagger () const
	Hermitian conjugate of matrix.
T	det () const
	determinant function - if matrix size is < 5, use Laplace, otherwise LU
T	det_laplace () const
	Determinant of matrix, using Laplace method.
T	det_lu () const
	following calculate the determinant of a square matrix det() is the generic interface, using laplace for small matrices and LU for large
DiagonalMatrix< n, T >	diagonal ()
	Return diagonal of square matrix.
template<int p, int q, typename S , typename R = hila::type_mul<T, S>>
R	dot (const Matrix< p, q, S > &rhs) const
	Dot product.
template<int q = n, int p = m, std::enable_if_t<(q==1||p==1), int > = 0>
T	e (const int i) const
	Standard array indexing operation for vectors.
T	e (const int i, const int j) const
	Standard array indexing operation for matrices.
template<typename Et , typename Mt , typename MT >
int	eigen_hermitean (DiagonalMatrix< n, Et > &eigenvalues, Matrix_t< n, n, Mt, MT > &eigenvectors, enum hila::sort sorted=hila::sort::unsorted) const
	Calculate eigenvalues and -vectors of an hermitean (or symmetric) matrix.
eigen_result< Mtype >	eigen_hermitean (enum hila::sort sorted=hila::sort::unsorted) const
	eigenvalues and -vectors of hermitean/symmetric matrix, alternative interface
template<typename S , std::enable_if_t< hila::is_assignable< T &, S >::value, int > = 0>
const auto &	fill (const S rhs)
	Matrix fill.
Mtype &	gaussian_random (base_type width=1.0)
	Fills Matrix with gaussian random elements from gaussian distribution.
Matrix< n, m, hila::arithmetic_type< T > >	imag () const
	Returns imaginary part of Matrix or Vector.
	Matrix_t ()=default
	Define default constructors to ensure std::is_trivial.
template<typename S = T, std::enable_if_t< hila::is_arithmetic< S >::value, int > = 0>
T	max () const
	Find max of Matrix only for arithmetic types.
template<typename S = T, std::enable_if_t< hila::is_arithmetic< S >::value, int > = 0>
T	min () const
	Find min of Matrix only for arithmetic types.
template<int p, int q, typename S , typename MT >
hila::type_mul< T, S >	mul_trace (const Matrix_t< p, q, S, MT > &rm) const
	Multiply with given matrix and compute trace of result.
template<typename R , typename Mt >
void	mult_by_2x2_left (int p, int q, const Matrix_t< 2, 2, R, Mt > &M)
	Multiply \( n \times m \)-matrix from the left by \( n \times m \) matrix defined by \( 2 \times 2 \) sub matrix.
template<typename R , typename Mt >
void	mult_by_2x2_right (int p, int q, const Matrix_t< 2, 2, R, Mt > &M)
	Multiply \( n \times m \)-matrix from the right by \( n \times m \) matrix defined by \( 2 \times 2 \) sub matrix.
template<typename S = T, std::enable_if_t< hila::is_floating_point< hila::arithmetic_type< S > >::value, int > = 0>
hila::arithmetic_type< T >	norm () const
	Calculate vector norm - sqrt of squarenorm.
template<typename S >
bool	operator!= (const Matrix< n, m, S > &rhs) const
	Boolean operator != to check if matrices are exactly different.
template<int m, int n, typename T1 , typename T2 , typename Rt = hila::ntype_op<T1, T2>>
Rt	operator* (const Matrix< 1, m, T1 > &A, const Matrix< n, 1, T2 > &B)
	Multiplication operator RowVector * Vector.
template<typename Mt , typename S , std::enable_if_t<(Mt::is_matrix() &&hila::is_complex_or_arithmetic< S >::value), int > = 0, typename Rt = hila::mat_scalar_type<Mt, S>>
Rt	operator* (const Mt &mat, const S &rhs)
	Multiplication operator Matrix * scalar.
template<typename Mt1 , typename Mt2 , std::enable_if_t< Mt1::is_matrix() &&Mt2::is_matrix() &&!std::is_same< Mt1, Mt2 >::value, int > = 0, typename R = hila::type_mul<hila::number_type<Mt1>, hila::number_type<Mt2>>, int n = Mt1::rows(), int m = Mt2::columns()>
Matrix< n, m, R >	operator* (const Mt1 &a, const Mt2 &b)
	Multiplication operator.
template<typename Mt , typename S , std::enable_if_t<(Mt::is_matrix() &&hila::is_complex_or_arithmetic< S >::value), int > = 0, typename Rt = hila::mat_scalar_type<Mt, S>>
Rt	operator* (const S &rhs, const Mt &mat)
	Multiplication operator Scalar * Matrix.
template<typename S , std::enable_if_t< hila::is_assignable< T &, hila::type_mul< T, S > >::value, int > = 0>
auto &	operator*= (const DiagonalMatrix< m, S > &rhs) &
	Multiply assign operator for DiagonalMatrix to Matrix.
template<int p, typename S , typename MT , std::enable_if_t< hila::is_assignable< T &, hila::type_mul< T, S > >::value, int > = 0>
auto &	operator*= (const Matrix_t< m, p, S, MT > &rhs) &
	Multiply assign scalar or matrix.
template<typename S , std::enable_if_t< hila::is_assignable< T &, hila::type_mul< T, S > >::value, int > = 0>
auto &	operator*= (const S rhs) &
	Multiply assign operator scalar to Matrix.
const auto &	operator+ () const
	Addition operator.
template<typename Mtype , typename S , std::enable_if_t< Mtype::is_matrix() &&hila::is_complex_or_arithmetic< S >::value, int > = 0, typename Rtype = hila::mat_scalar_type<Mtype, S>>
Rtype	operator+ (const Mtype &a, const S &b)
	Addition operator Matrix + scalar.
template<typename Mtype1 , typename Mtype2 , std::enable_if_t< Mtype1::is_matrix() &&Mtype2::is_matrix(), int > = 0, typename Rtype = hila::mat_x_mat_type<Mtype1, Mtype2>, std::enable_if_t<!std::is_same< Mtype1, Rtype >::value, int > = 0>
Rtype	operator+ (const Mtype1 &a, const Mtype2 &b)
	Addition operator Matrix + Matrix.
template<typename Mtype , typename S , std::enable_if_t< Mtype::is_matrix() &&hila::is_complex_or_arithmetic< S >::value, int > = 0, typename Rtype = hila::mat_scalar_type<Mtype, S>>
Rtype	operator+ (const S &b, const Mtype &a)
	Addition operator scalar + Matrix.
template<typename S , std::enable_if_t< hila::is_assignable< T &, hila::type_plus< T, S > >::value, int > = 0>
auto &	operator+= (const DiagonalMatrix< n, S > &rhs) &
	Addition assign operator for DiagonalMatrix to Matrix.
template<typename S , typename MT , std::enable_if_t< hila::is_assignable< T &, S >::value, int > = 0>
auto &	operator+= (const Matrix_t< n, m, S, MT > &rhs) &
	Add assign operator Matrix to Matrix.
template<typename S , std::enable_if_t< hila::is_assignable< T &, hila::type_plus< T, S > >::value, int > = 0>
auto &	operator+= (const S &rhs) &
	Add assign operator scalar to Matrix.
Mtype	operator- () const
	Unary operator.
template<typename Mtype , typename S , std::enable_if_t< Mtype::is_matrix() &&hila::is_complex_or_arithmetic< S >::value, int > = 0, typename Rtype = hila::mat_scalar_type<Mtype, S>>
Rtype	operator- (const Mtype &a, const S &b)
	Subtraction operator Matrix - scalar.
template<typename Mtype1 , typename Mtype2 , std::enable_if_t< Mtype1::is_matrix() &&Mtype2::is_matrix(), int > = 0, typename Rtype = hila::mat_x_mat_type<Mtype1, Mtype2>>
Rtype	operator- (const Mtype1 &a, const Mtype2 &b)
	Subtraction operator Matrix - Matrix.
template<typename Mtype , typename S , std::enable_if_t< Mtype::is_matrix() &&hila::is_complex_or_arithmetic< S >::value, int > = 0, typename Rtype = hila::mat_scalar_type<Mtype, S>>
Rtype	operator- (const S &b, const Mtype &a)
	Subtraction operator Scalar - Matrix.
template<typename S , std::enable_if_t< hila::is_assignable< T &, hila::type_plus< T, S > >::value, int > = 0>
auto &	operator-= (const DiagonalMatrix< n, S > &rhs) &
	Subtract assign operator for DiagonalMatrix to Matrix.
template<typename S , typename MT , std::enable_if_t< hila::is_assignable< T &, S >::value, int > = 0>
auto &	operator-= (const Matrix_t< n, m, S, MT > &rhs) &
	Subtract assign operator Matrix to MAtrix.
template<typename S , std::enable_if_t< hila::is_assignable< T &, hila::type_minus< T, S > >::value, int > = 0>
auto &	operator-= (const S rhs) &
	Subtract assign operator scalar to Matrix.
template<typename Mt , typename S , std::enable_if_t<(Mt::is_matrix() &&hila::is_complex_or_arithmetic< S >::value), int > = 0, typename Rt = hila::mat_scalar_type<Mt, S>>
Rt	operator/ (const Mt &mat, const S &rhs)
	Division operator.
template<typename S , std::enable_if_t< hila::is_assignable< T &, hila::type_div< T, S > >::value, int > = 0>
auto &	operator/= (const DiagonalMatrix< m, S > &rhs) &
	Divide assign operator for DiagonalMatrix to Matrix.
template<typename S , std::enable_if_t< hila::is_assignable< T &, hila::type_div< T, S > >::value, int > = 0>
auto &	operator/= (const S rhs) &
	Divide assign oprator scalar with matrix.
template<typename S , std::enable_if_t< hila::is_assignable< T &, S >::value, int > = 0>
auto &	operator= (const DiagonalMatrix< n, S > &rhs) &
	Assignment from diagonal matrix.
template<typename S , typename MT , std::enable_if_t< hila::is_assignable< T &, S >::value, int > = 0>
auto &	operator= (const Matrix_t< n, m, S, MT > &rhs) &
	Copy matrix assignment.
template<typename S , std::enable_if_t< hila::is_assignable< T &, S >::value &&n==m, int > = 0>
auto &	operator= (const S rhs) &
	Assignment from scalar.
Mtype &	operator= (const std::nullptr_t &z)
	Zero assignment.
template<typename S , std::enable_if_t< hila::is_assignable< T &, S >::value, int > = 0>
auto &	operator= (std::initializer_list< S > rhs) &
	Initializer list assignment.
template<typename S , int n2, int m2>
bool	operator== (const Matrix< n2, m2, S > &rhs) const
	Boolean operator == to determine if two matrices are exactly the same.
template<int q = n, int p = m, std::enable_if_t<(q==1||p==1), int > = 0>
T	operator[] (const int i) const
	Indexing operation [] defined only for vectors.
template<int p, int q, typename S , typename R = hila::type_mul<T, S>>
Matrix< n, p, R >	outer_product (const Matrix< p, q, S > &rhs) const
	Outer product.
template<int N>
Mtype	permute (const Vector< N, int > &permutation) const
	Permute elements of vector.
Mtype	permute_columns (const Vector< m, int > &permutation) const
	permute columns of Matrix
Mtype	permute_rows (const Vector< n, int > &permutation) const
	permute rows of Matrix
Mtype &	random ()
	dot with matrix - matrix
Matrix< n, m, hila::arithmetic_type< T > >	real () const
	Returns real part of Matrix or Vector.
RowVector< m, T >	row (int r) const
	Return reference to row in a matrix.
template<typename S , std::enable_if_t< hila::is_assignable< T &, S >::value, int > = 0>
void	set_column (int c, const Vector< n, S > &v)
	get column of a matrix
template<typename S , std::enable_if_t< hila::is_assignable< T &, S >::value, int > = 0>
void	set_diagonal (const Vector< n, S > &v)
	Set diagonal of square matrix to Vector which is passed to the method.
template<typename S , std::enable_if_t< hila::is_assignable< T &, S >::value, int > = 0>
void	set_row (int r, const RowVector< m, S > &v)
	Set row of Matrix with RowVector if types are assignable.
Mtype	sort (hila::sort order=hila::sort::ascending) const
	Sort method for Vector.
template<int N>
Mtype	sort (Vector< N, int > &permutation, hila::sort order=hila::sort::ascending) const
	Sort method for Vector which returns permutation order.
hila::arithmetic_type< T >	squarenorm () const
	Calculate square norm - sum of squared elements.
std::string	str (int prec=8, char separator=' ') const
svd_result< Mtype >	svd (enum hila::sort sorted=hila::sort::unsorted) const
	svd and svd_pivot - alternative interface
template<typename Et , typename Mt , typename MT >
int	svd (Matrix_t< n, n, Mt, MT > &_U, DiagonalMatrix< n, Et > &_D, Matrix_t< n, n, Mt, MT > &_V, enum hila::sort sorted=hila::sort::unsorted) const
	Singular value decomposition: divide matrix A = U S V*, where U,V unitary and S diagonal matrix of real singular values. Unpivoted Jacobi rotations.
template<typename Et , typename Mt , typename MT >
int	svd_pivot (Matrix_t< n, n, Mt, MT > &_U, DiagonalMatrix< n, Et > &_D, Matrix_t< n, n, Mt, MT > &_V, enum hila::sort sorted=hila::sort::unsorted) const
	Singular value decomposition: divide matrix A = U S V*, where U,V unitary and S diagonal matrix of real singular values. Fully pivoted Jacobi rotations.
T	trace () const
	Computes Trace for Matrix.
template<int mm = m, typename Rtype = typename std::conditional<n == m, Matrix_t, Matrix<m, n, T>>::type, std::enable_if_t<(mm !=1 &&n !=1), int > = 0>
Rtype	transpose () const
	Transpose of matrix.
template<int mm = m, std::enable_if_t< mm==1, int > = 0>
const RowVector< n, T > &	transpose () const
	Transpose of vector.
template<int nn = n, std::enable_if_t< nn==1 &&m !=1, int > = 0>
const Vector< m, T > &	transpose () const
	Transpose of RowVector.

Static Public Member Functions
static constexpr int	columns ()
	Returns column length.
static constexpr bool	is_square ()
	Returns true if matrix is a square matrix.
static constexpr bool	is_vector ()
	Returns true if Matrix is a vector.
static constexpr int	rows ()
	Define constant methods rows(), columns() - may be useful in template code.
template<int q = n, int p = m, std::enable_if_t< q==1, int > = 0>
static constexpr int	size ()
	Returns size of Vector or square Matrix.

Public Attributes
T	c [n *m]
	The data as a one dimensional array.

Detailed Description

template<const int n, const int m, typename T, typename Mtype>
class Matrix_t< n, m, T, Mtype >

The main \( n \times m \) matrix type template Matrix_t. This is a base class type for "useful" types which are derived from this.

Uses curiously recurring template pattern (CRTP), where the last template parameter is the template itself

Example: the Matrix type below is defined as

template <int n, int m, typename T>
class Matrix : public Matrix_t<n, m, T, Matrix<n, m, T>> { .. }

This pattern is used because stupid c++ makes it complicated to write generic code, in this case derived functions to return derived type

Template Parameters

n	Number of rows
m	Number of columns
T	Matrix element type
Mtype	Specific "Matrix" type for CRTP

Definition at line 106 of file matrix.h.

Member Function Documentation

abs()

template<const int n, const int m, typename T , typename Mtype >

auto Matrix_t< n, m, T, Mtype >::abs ( ) const

inline

Returns absolute value of Matrix.

For Matrix<n,m,Complex<T>> case type is changed to Matrix<n,m,T> as expected since absolute value of a complex number is real

Template Parameters

M

Returns

Mtype

Definition at line 1123 of file matrix.h.

adjoint()

template<const int n, const int m, typename T , typename Mtype >

template<typename Rtype = typename std::conditional<n == m, Mtype, Matrix<m, n, T>>::type>

Rtype Matrix_t< n, m, T, Mtype >::adjoint ( ) const

inline

Adjoint of matrix.

Alias to dagger

Template Parameters

std::conditional<n,Mtype,Matrix<m,n,T>>::type

Returns

Rtype

Definition at line 1111 of file matrix.h.

asArray()

template<const int n, const int m, typename T , typename Mtype >

const Array< n, m, T > & Matrix_t< n, m, T, Mtype >::asArray ( ) const

inline

Cast Matrix to Array.

used for array operations

Returns

Array<n, m, T>&

Definition at line 476 of file matrix.h.

asDiagonalMatrix()

template<const int n, const int m, typename T , typename Mtype >

template<int mm = m, std::enable_if_t< mm==1, int > = 0>

const DiagonalMatrix< n, T > & Matrix_t< n, m, T, Mtype >::asDiagonalMatrix ( ) const

inline

Cast Vector to DiagonalMatrix.

Returns

DiagonalMatrix<n,T>

Definition at line 491 of file matrix.h.

column()

template<const int n, const int m, typename T , typename Mtype >

Vector< n, T > Matrix_t< n, m, T, Mtype >::column ( int  c ) const

inline

Returns column vector as value at index c.

Matrix<n,n,T> M;
M.random();
Vector<n,T> V = M.column(i);

Parameters

c	index of column vector to be returned

Returns

const Vector<n, T>

Definition at line 391 of file matrix.h.

columns()

template<const int n, const int m, typename T , typename Mtype >

static constexpr int Matrix_t< n, m, T, Mtype >::columns ( )

inlinestaticconstexpr

Returns column length.

Returns

constexpr int

Definition at line 234 of file matrix.h.

conj()

template<const int n, const int m, typename T , typename Mtype >

Mtype Matrix_t< n, m, T, Mtype >::conj ( ) const

inline

Returns complex conjugate of Matrix.

Returns

const Mtype

Definition at line 1078 of file matrix.h.

dagger()

template<const int n, const int m, typename T , typename Mtype >

template<typename Rtype = typename std::conditional<n == m, Mtype, Matrix<m, n, T>>::type>

Rtype Matrix_t< n, m, T, Mtype >::dagger ( ) const

inline

Hermitian conjugate of matrix.

for square matrix return type is same, for non square it is Matrix<m,n,MyType>

Template Parameters

std::conditional<n,Mtype,Matrix<m,n,T>>::type

Returns

const Rtype

Definition at line 1094 of file matrix.h.

det_laplace()

template<int n, int m, typename T , typename Mtype >

T Matrix_t< n, m, T, Mtype >::det_laplace

Determinant of matrix, using Laplace method.

Defined only for square matrices

For perfomance overloads exist for \( 1 \times 1 \), \( 2 \times 2 \) and \( 3 \times 3 \) matrices.

Parameters

mat	matrix to compute determinant for

Returns

T result determinant

Definition at line 637 of file matrix_linalg.h.

det_lu()

template<int n, int m, typename T , typename Mtype >

T Matrix_t< n, m, T, Mtype >::det_lu

following calculate the determinant of a square matrix det() is the generic interface, using laplace for small matrices and LU for large

Matrix determinant with LU decomposition.

Algorithm: numerical Recipes, 2nd ed. p. 47 ff Works for Real and Complex matrices Defined only for square matrices

Returns

Complex<radix> Determinant

Definition at line 677 of file matrix_linalg.h.

diagonal()

template<const int n, const int m, typename T , typename Mtype >

DiagonalMatrix< n, T > Matrix_t< n, m, T, Mtype >::diagonal ( )

inline

Return diagonal of square matrix.

If called for non square matrix the program will throw an error.

Matrix<n,n,T> M = 1;
DiagonalMatrix<n,T> D = M.diagonal();

Returns

Vector<n, T> returned vector.

Definition at line 439 of file matrix.h.

dot()

template<const int n, const int m, typename T , typename Mtype >

template<int p, int q, typename S , typename R = hila::type_mul<T, S>>

R Matrix_t< n, m, T, Mtype >::dot ( const Matrix< p, q, S > &  rhs ) const

inline

Dot product.

Only works between two Vector objects

Vector<m,MyType> V,W;
.
.
.
V.dot(W); // valid operation

RowVector<m,MyType> V,W;
.
.
.
V.dot(W); // not valid operation

Template Parameters

p	Row length for rhs
q	Column length for rhs
S	Type for rhs
R	Gives resulting type of lhs and rhs multiplication

Parameters

rhs	Vector to compute dot product with

Returns

R Value of dot product

Definition at line 1314 of file matrix.h.

e() [1/2]

template<const int n, const int m, typename T , typename Mtype >

template<int q = n, int p = m, std::enable_if_t<(q==1||p==1), int > = 0>

T Matrix_t< n, m, T, Mtype >::e ( const int  i ) const

inline

Standard array indexing operation for vectors.

Accessing singular elements is insufficient, but Vector elements are often quite small.

 Vector<n,MyType> V;
MyType a = V.e(i) \\ i <= n

Template Parameters

q	Number of rows
p	Number of columns

Parameters

i

Index

Returns

T

Definition at line 305 of file matrix.h.

e() [2/2]

template<const int n, const int m, typename T , typename Mtype >

T Matrix_t< n, m, T, Mtype >::e ( const int  i, const int  j  ) const

inline

Standard array indexing operation for matrices.

Accessing singular elements is insufficient, but Matrix elements are often quite small.

Exammple for matrix:

Matrix<n,m,MyType> M;
MyType a = M.e(i,j); \\ i <= n, j <= m

Parameters

i	Row index
j	Column index

Returns

T matrix element type

Definition at line 279 of file matrix.h.

eigen_hermitean() [1/2]

template<int n, int m, typename T , typename Mtype >

template<typename Et , typename Mt , typename MT >

int Matrix_t< n, m, T, Mtype >::eigen_hermitean	(	DiagonalMatrix< n, Et > &	E,
		Matrix_t< n, n, Mt, MT > &	U,
		enum hila::sort	sorted = hila::sort::unsorted
	)		const

Calculate eigenvalues and -vectors of an hermitean (or symmetric) matrix.

Calculate eigenvalues and vectors of an hermitean or real symmetric matrix.

Returns the number of Jacobi iterations, or -1 if did not converge. Arguments will contain eigenvalues and eigenvectors in columns of the "eigenvectors" matrix. Computation is done in double precision despite the input matrix types.

Parameters

eigenvaluevec	Vector of computed eigenvalue
eigenvectors	Eigenvectors as columns of \( n \times n \) Matrix

Algorithm uses fully pivoted Jacobi rotations.

Two interfaces:

H.eigen_hermitean(E, U, [optional: sort]); E: output is DiagnoalMatrix containing real eigenvalues U: nxn unitary matrix, columns are normalized eigenvectors

auto res = H.eigen_hermitean([optional: sort]); This saves the trouble of defining E and U (see below)

Thus, H = U E U^* and H U = U E

Both interfaces allow optional sorting according to eigenvalues: hila::sort::unsorted [default] hila::sort::ascending / descending

Example:

SquareMatrix<n,Complex<double>> M;
M = ... // make unitary
DiagonalMatrix<n,double> eigenvalues;
SquareMatrix<n,Complex<double>> eigenvectors;
 
int rotations = M.eigen_hermitean(eigenvalues,eigenvectors,hila::sort::ascending);

Note

The matrix has to be hermitean/symmetric

Parameters

E	diagonal matrix of real eigenvalues
U	Unitary nxn matrix of eigenvectors
sorted	sorting of eigenvalues (default:unsorted)

Returns

int number of jacobi rotations

Definition at line 200 of file matrix_linalg.h.

eigen_hermitean() [2/2]

template<int n, int m, typename T , typename Mtype >

eigen_result< Mtype > Matrix_t< n, m, T, Mtype >::eigen_hermitean

(

enum hila::sort

sorted = hila::sort::unsorted

)

const

eigenvalues and -vectors of hermitean/symmetric matrix, alternative interface

result = eigen_hermitean(hila::sort [optional]) returns struct eigen_result<Mtype>, with fields eigenvalues: DiagonalMatrix of eigenvalues eigenvectors: nxn unitary matrix of eigenvectors

Example:

Field<SquareMatrix<n,double>> M, sn;
M[ALL] = .. // make symmetric
 
onsites(ALL) {
    auto R = M[X].svd();
 
    // Use EV decomposition to evaluate function: sin(M) = U sin(eigenvals) U^T
    sn[X] = R.eigenvectors * sin(R.eigenvalues) * R.eigenvectors.dagger();
}

This interface saves the trouble of defining the eigenvalue and -vector variables.

Definition at line 306 of file matrix_linalg.h.

fill()

template<const int n, const int m, typename T , typename Mtype >

template<typename S , std::enable_if_t< hila::is_assignable< T &, S >::value, int > = 0>

const auto & Matrix_t< n, m, T, Mtype >::fill ( const S  rhs )

inline

Matrix fill.

Fills the matrix with element if it is assignable to matrix type T

Works as follows:

Matrix<n,m,MyType> M;
M.fill(2) \\ Matrix is filled with 2

Template Parameters

S	Scalar type to of rhs

Parameters

rhs	Element to fill matrix with

Returns

const Mtype&

Definition at line 1022 of file matrix.h.

gaussian_random()

template<const int n, const int m, typename T , typename Mtype >

Mtype & Matrix_t< n, m, T, Mtype >::gaussian_random ( base_type  width = 1.0 )

inline

Fills Matrix with gaussian random elements from gaussian distribution.

Matrix<n,m,T> M;
M.gaussian_random();

Parameters

width

Returns

Mtype&

Definition at line 1415 of file matrix.h.

imag()

template<const int n, const int m, typename T , typename Mtype >

Matrix< n, m, hila::arithmetic_type< T > > Matrix_t< n, m, T, Mtype >::imag ( ) const

inline

Returns imaginary part of Matrix or Vector.

Returns

Matrix<n, m, hila::arithmetic_type<T>>

Definition at line 1156 of file matrix.h.

is_square()

template<const int n, const int m, typename T , typename Mtype >

static constexpr bool Matrix_t< n, m, T, Mtype >::is_square ( )

inlinestaticconstexpr

Returns true if matrix is a square matrix.

Returns

true

false

Definition at line 142 of file matrix.h.

is_vector()

template<const int n, const int m, typename T , typename Mtype >

static constexpr bool Matrix_t< n, m, T, Mtype >::is_vector ( )

inlinestaticconstexpr

Returns true if Matrix is a vector.

Returns

true

false

Definition at line 132 of file matrix.h.

mul_trace()

template<const int n, const int m, typename T , typename Mtype >

template<int p, int q, typename S , typename MT >

hila::type_mul< T, S > Matrix_t< n, m, T, Mtype >::mul_trace ( const Matrix_t< p, q, S, MT > &  rm ) const

inline

Multiply with given matrix and compute trace of result.

Slightly cheaper operation than

(M*N).trace() 

Template Parameters

p
q
S
MT

Parameters

rm

Returns

hila::type_mul<T, S>

Definition at line 1192 of file matrix.h.

mult_by_2x2_left()

template<const int n, const int m, typename T , typename Mtype >

template<typename R , typename Mt >

void Matrix_t< n, m, T, Mtype >::mult_by_2x2_left ( int  p, int  q, const Matrix_t< 2, 2, R, Mt > &  M  )

inline

Multiply \( n \times m \)-matrix from the left by \( n \times m \) matrix defined by \( 2 \times 2 \) sub matrix.

The multiplication is defined as follows, let \(M\) as the \( 2 \times 2 \) input matrix and \(B\) be (this) matrix, being the matrix stored in the object this method is called for. Let \(A = I\) be a \( n \times m \) unit matrix. We then set the values of A to be:

\begin{align} A_{p,p} = M_{0,0}, \hspace{5px} A_{p,q} = M_{0,1}, \hspace{5px} A_{q,p} = M_{1,0}, \hspace{5px} A_{q,q} = M_{1,1}. \end{align}

Then the resulting matrix will be:

\begin{align} B = A \cdot B \end{align}

Template Parameters

R	Element type of M
Mt	Matrix type of M

Parameters

p	First row and column
q	Second row and column
M	\( 2 \times 2\) Matrix to multiply with

Definition at line 1595 of file matrix.h.

mult_by_2x2_right()

template<const int n, const int m, typename T , typename Mtype >

template<typename R , typename Mt >

void Matrix_t< n, m, T, Mtype >::mult_by_2x2_right ( int  p, int  q, const Matrix_t< 2, 2, R, Mt > &  M  )

inline

Multiply \( n \times m \)-matrix from the right by \( n \times m \) matrix defined by \( 2 \times 2 \) sub matrix.

See Matrix::mult_by_2x2_left, only difference being that the multiplication is from the right.

Template Parameters

R	Element type of M
Mt	Matrix type of M

Parameters

p	First row and column
q	Second row and column
M	\( 2 \times 2\) Matrix to multiply with

Definition at line 1624 of file matrix.h.

norm()

template<const int n, const int m, typename T , typename Mtype >

template<typename S = T, std::enable_if_t< hila::is_floating_point< hila::arithmetic_type< S > >::value, int > = 0>

hila::arithmetic_type< T > Matrix_t< n, m, T, Mtype >::norm ( ) const

inline

Calculate vector norm - sqrt of squarenorm.

Template Parameters

S

Returns

hila::arithmetic_type<T>

Definition at line 1225 of file matrix.h.

operator!=()

template<const int n, const int m, typename T , typename Mtype >

template<typename S >

bool Matrix_t< n, m, T, Mtype >::operator!= ( const Matrix< n, m, S > &  rhs ) const

inline

Boolean operator != to check if matrices are exactly different.

if matrices are exactly the same then this will return false

Template Parameters

S	Type for MAtrix which is being compared to

Parameters

rhs	right hand side Matrix which we are comparing

Returns

true

false

Definition at line 593 of file matrix.h.

operator*() [1/4]

template<int m, int n, typename T1 , typename T2 , typename Rt = hila::ntype_op<T1, T2>>

Rt operator*	(	const Matrix< 1, m, T1 > &	A,
		const Matrix< n, 1, T2 > &	B
	)

Multiplication operator RowVector * Vector.

Defined as standard dot product between vectors as long as vectors are of same length

RowVector<n,MyType> V;
Vector<n,MyType> W;
//
// Fill V and W
//
auto S = V*W; // Results in MyType scalar

Template Parameters

m	Dimensions of RowVector
n	Dimension of Vector
T1	Element type of RowVector
T2	Element type of Vector
Rt	Return type of T1 \( \cdot \) T2 product

Parameters

A	RowVector to multiply
B	Vector to multiply

Returns

Rt

Definition at line 2468 of file matrix.h.

operator*() [2/4]

template<typename Mt , typename S , std::enable_if_t<(Mt::is_matrix() &&hila::is_complex_or_arithmetic< S >::value), int > = 0, typename Rt = hila::mat_scalar_type<Mt, S>>

Rt operator* ( const Mt &  mat, const S &  rhs  )

inline

Multiplication operator Matrix * scalar.

Multiplication operator between Matrix and Scalar are defined in the usual way (element wise)

Matrix<n,n,MyType> M;
M.fill(1);
auto S = M*2; // Resulting Matrix is uniformly 2

Template Parameters

Mt	Matrix type of mat
s	Scalar type rhs

Parameters

mat	Matrix to multiply
rhs	Scalar to multiply

Returns

Rt

Definition at line 2503 of file matrix.h.

operator*() [3/4]

template<typename Mt1 , typename Mt2 , std::enable_if_t< Mt1::is_matrix() &&Mt2::is_matrix() &&!std::is_same< Mt1, Mt2 >::value, int > = 0, typename R = hila::type_mul<hila::number_type<Mt1>, hila::number_type<Mt2>>, int n = Mt1::rows(), int m = Mt2::columns()>

Matrix< n, m, R > operator* ( const Mt1 &  a, const Mt2 &  b  )

inline

Multiplication operator.

Multiplication type depends on the original types of the multiplied matrices. Defined for the following operations.

Matrix * Matrix

Standard matrix multiplication where LHS columns must match RHS rows

Matrix<2, 3, double> M;
Matrix<3, 2, double> N;
M.random();
N.random();
auto S = N * M; // Results in a 3 x 3 Matrix since N.rows() = 3 and M.columns = 3

Vector * RowVector is same as outer product which is equivalent to a matrix multiplication

auto S = W*V // Result in n x n Matrix of type MyType

Template Parameters

Mt1	Matrix type for a
Mt2	Matrix type for b
n	Number of rows
m	Number of columns

Parameters

a	Left Matrix or Vector
b	Right Matrix or RowVector

Returns

Matrix<n, m, R>

Definition at line 2406 of file matrix.h.

operator*() [4/4]

template<typename Mt , typename S , std::enable_if_t<(Mt::is_matrix() &&hila::is_complex_or_arithmetic< S >::value), int > = 0, typename Rt = hila::mat_scalar_type<Mt, S>>

Rt operator* ( const S &  rhs, const Mt &  mat  )

inline

Multiplication operator Scalar * Matrix.

Multiplication operator between Matrix and Scalar are defined in the usual way (element wise)

Matrix<n,n,MyType> M;
M.fill(1);
auto S = 2*M; // Resulting Matrix is uniformly 2

Template Parameters

Mt	Matrix type of mat
s	Scalar type rhs

Parameters

mat	Matrix to multiply
rhs	Scalar to multiply

Returns

Rt

Definition at line 2532 of file matrix.h.

operator*=() [1/3]

template<const int n, const int m, typename T , typename Mtype >

template<typename S , std::enable_if_t< hila::is_assignable< T &, hila::type_mul< T, S > >::value, int > = 0>

auto & Matrix_t< n, m, T, Mtype >::operator*= ( const DiagonalMatrix< m, S > &  rhs ) &

inline

Multiply assign operator for DiagonalMatrix to Matrix.

Simply defined as matrix multiplication, but since rhs is guaranteed to be diagonal the method is optimized to skip most of the elements.

Template Parameters

S	Element type of rhs

Parameters

rhs	DiagonalMatrix to multiply

Returns

Mtype&

Definition at line 975 of file matrix.h.

operator*=() [2/3]

template<const int n, const int m, typename T , typename Mtype >

template<int p, typename S , typename MT , std::enable_if_t< hila::is_assignable< T &, hila::type_mul< T, S > >::value, int > = 0>

auto & Matrix_t< n, m, T, Mtype >::operator*= ( const Matrix_t< m, p, S, MT > &  rhs ) &

inline

Multiply assign scalar or matrix.

Multiplication works as defined for matrix multiplication

Matrix multiply assign only defined for square matrices, since the matrix dimensions would change otherwise.

Matrix<n,m,MyType> M,N;
.
. Fill matrices M and N
.
M *= N; \\ M = M*N

Parameters

rhs	Matrix to multiply with

Returns

template <int p, typename S, typename MT, std::enable_if_t<hila::is_assignable<T &, hila::type_mul<T, S>>::value, int> = 0>&

Definition at line 845 of file matrix.h.

operator*=() [3/3]

template<const int n, const int m, typename T , typename Mtype >

template<typename S , std::enable_if_t< hila::is_assignable< T &, hila::type_mul< T, S > >::value, int > = 0>

auto & Matrix_t< n, m, T, Mtype >::operator*= ( const S  rhs ) &

inline

Multiply assign operator scalar to Matrix.

Multiply Matrix uniformly with scalar

Matrix<n,m,MyType> M;
.
. Fill whole matrix with 1
.
M *= 2 ; \\ M is filled with 2

Template Parameters

S	Scalar type of rhs

Parameters

rhs	Scalar to multiply

Returns

Mtype&

Definition at line 897 of file matrix.h.

operator+() [1/4]

template<const int n, const int m, typename T , typename Mtype >

const auto & operator+ ( ) const

inline

Addition operator.

Identity operation

M == +M;

Template Parameters

Mtype1	Matrix type for a
Mtype2	Matrix type for b

Parameters

a	Left matrix
b	Right matrix

Returns

Rtype

Definition at line 556 of file matrix.h.

operator+() [2/4]

template<typename Mtype , typename S , std::enable_if_t< Mtype::is_matrix() &&hila::is_complex_or_arithmetic< S >::value, int > = 0, typename Rtype = hila::mat_scalar_type<Mtype, S>>

Rtype operator+ ( const Mtype &  a, const S &  b  )

inline

Addition operator Matrix + scalar.

Addition operator between matrix and scalar is defined in the usual way, where the scalar is added to the diagonal elements.

• \( M + 2 = M + 2\cdot\mathbb{1} \)

Matrix<n,m,MyType> M,S;
M.fill(0);
S = M + 1; // Resulting matrix is identity matrix

Template Parameters

Mtype	Matrix type of b
S	Type of scalar

Parameters

a	Matrix to add to
b	Scalar to add

Returns

Rtype

Definition at line 2216 of file matrix.h.

operator+() [3/4]

template<typename Mtype1 , typename Mtype2 , std::enable_if_t< Mtype1::is_matrix() &&Mtype2::is_matrix(), int > = 0, typename Rtype = hila::mat_x_mat_type<Mtype1, Mtype2>, std::enable_if_t<!std::is_same< Mtype1, Rtype >::value, int > = 0>

Rtype operator+ ( const Mtype1 &  a, const Mtype2 &  b  )

inline

Addition operator Matrix + Matrix.

Addition operator between matrices is defined in the usual way (element wise).

NOTE: Matrices must share same dimensionality.

Matrix<n,m,MyType> M, N, S;
M.fill(1);
N.fill(1);
S = M + N; // Resulting matrix is uniformly 2

Template Parameters

Mtype1	Matrix type of a
Mtype2	Matrix type of b

Parameters

a	Matrix to add
b	Matrix to add

Returns

Rtype Return matrix of compatible type between Mtype1 and Mtype2

Definition at line 2111 of file matrix.h.

operator+() [4/4]

template<typename Mtype , typename S , std::enable_if_t< Mtype::is_matrix() &&hila::is_complex_or_arithmetic< S >::value, int > = 0, typename Rtype = hila::mat_scalar_type<Mtype, S>>

Rtype operator+ ( const S &  b, const Mtype &  a  )

inline

Addition operator scalar + Matrix.

Addition operator between Scalar and Matrix is defined in the usual way, where the scalar is treated as diagonal matrix which is then added to. \( 2 + M = 2\cdot\mathbb{1} + M \)

Matrix<n,m,MyType> M,R;
M = 0; // M = 0*I
R = 1 + M; // Resulting matrix is identity matrix

Template Parameters

Mtype	Matrix type of a
S	scalar type of b

Parameters

b	Matrix to add
a	Scalar to add to

Returns

Rtype

Definition at line 2253 of file matrix.h.

operator+=() [1/3]

template<const int n, const int m, typename T , typename Mtype >

template<typename S , std::enable_if_t< hila::is_assignable< T &, hila::type_plus< T, S > >::value, int > = 0>

auto & Matrix_t< n, m, T, Mtype >::operator+= ( const DiagonalMatrix< n, S > &  rhs ) &

inline

Addition assign operator for DiagonalMatrix to Matrix.

Works as long as Matrix to assign to is square

Template Parameters

S	Element type of rhs

Parameters

rhs	DiagonalMatrix to add

Returns

Mtype&

Definition at line 939 of file matrix.h.

operator+=() [2/3]

template<const int n, const int m, typename T , typename Mtype >

template<typename S , typename MT , std::enable_if_t< hila::is_assignable< T &, S >::value, int > = 0>

auto & Matrix_t< n, m, T, Mtype >::operator+= ( const Matrix_t< n, m, S, MT > &  rhs ) &

inline

Add assign operator Matrix to Matrix.

Matrix<n,m,MyType> M,N;
M = 1;
N = 1;
M += N; \\M = 2*I

Template Parameters

S	Element type of rhs
MT	Matrix type of rhs

Parameters

rhs	Matrix to add

Returns

Mtype&

Definition at line 740 of file matrix.h.

operator+=() [3/3]

template<const int n, const int m, typename T , typename Mtype >

template<typename S , std::enable_if_t< hila::is_assignable< T &, hila::type_plus< T, S > >::value, int > = 0>

auto & Matrix_t< n, m, T, Mtype >::operator+= ( const S &  rhs ) &

inline

Add assign operator scalar to Matrix.

Adds scalar \( a \) to square matrix as \( M + a\cdot\mathbb{1} \)

Matrix<n,m,MyType> M = 1;
M += 1 ; \\ M = 2*I

Template Parameters

S	Element type of scalar rhs

Parameters

rhs	Sclar to add

Returns

Mtype&

Definition at line 789 of file matrix.h.

operator-() [1/4]

template<const int n, const int m, typename T , typename Mtype >

Mtype Matrix_t< n, m, T, Mtype >::operator- ( ) const

inline

Unary operator.

Negation operation

M == -M;

Definition at line 531 of file matrix.h.

operator-() [2/4]

template<typename Mtype , typename S , std::enable_if_t< Mtype::is_matrix() &&hila::is_complex_or_arithmetic< S >::value, int > = 0, typename Rtype = hila::mat_scalar_type<Mtype, S>>

Rtype operator- ( const Mtype &  a, const S &  b  )

inline

Subtraction operator Matrix - scalar.

Subtraction operator between matrix and scalar is defined in the usual way, where the scalar is subtracted from the diagonal elements.

\( M - 2 = M - 2\cdot\mathbb{1} \)

Matrix<n,m,MyType> M,R;
M = 2; // M = 2*I
R = M - 1; // Resulting matrix is identity matrix

Template Parameters

Mtype	Matrix type of a
S	Scalar type of b

Parameters

a	Matrix to subtract from
b	Scalar to subtract

Returns

Rtype

Definition at line 2285 of file matrix.h.

operator-() [3/4]

template<typename Mtype1 , typename Mtype2 , std::enable_if_t< Mtype1::is_matrix() &&Mtype2::is_matrix(), int > = 0, typename Rtype = hila::mat_x_mat_type<Mtype1, Mtype2>>

Rtype operator- ( const Mtype1 &  a, const Mtype2 &  b  )

inline

Subtraction operator Matrix - Matrix.

Subtraction operator between matrices is defined in the usual way (element wise).

NOTE: Matrices must share same dimensionality.

Matrix<n,m,MyType> M, N, S;
M.fill(1);
N.fill(1);
S = M - N; // Resulting matrix is uniformly 0

Template Parameters

Mtype1	Matrix type of a
Mtype2	Matrix type of b

Parameters

a	Matrix to subtract from
b	Matrix to subtract

Returns

Rtype Return matrix of compatible type between Mtype1 and Mtype2

Definition at line 2178 of file matrix.h.

operator-() [4/4]

template<typename Mtype , typename S , std::enable_if_t< Mtype::is_matrix() &&hila::is_complex_or_arithmetic< S >::value, int > = 0, typename Rtype = hila::mat_scalar_type<Mtype, S>>

Rtype operator- ( const S &  b, const Mtype &  a  )

inline

Subtraction operator Scalar - Matrix.

Subtraction operator between Scalar and Matrix is defined in the usual way, where the scalar is treated as diagonal matrix which is then subtracted from.

\( 2 - M = 2\cdot\mathbb{1} - M \)

Matrix<n,m,MyType> M,R;
M = 1; // M = 1*I
R = 2 - M; // Resulting matrix is identity matrix

Template Parameters

Mtype	Matrix type a
S	Scalar type of b

Parameters

b	Scalar to subtract from
a	Matrix to subtract

Returns

Rtype

Definition at line 2323 of file matrix.h.

operator-=() [1/3]

template<const int n, const int m, typename T , typename Mtype >

template<typename S , std::enable_if_t< hila::is_assignable< T &, hila::type_plus< T, S > >::value, int > = 0>

auto & Matrix_t< n, m, T, Mtype >::operator-= ( const DiagonalMatrix< n, S > &  rhs ) &

inline

Subtract assign operator for DiagonalMatrix to Matrix.

Works as long as Matrix to assign to is square

Template Parameters

S	Element type of rhs

Parameters

rhs	DiagonalMatrix to add

Returns

Mtype&

Definition at line 956 of file matrix.h.

operator-=() [2/3]

template<const int n, const int m, typename T , typename Mtype >

template<typename S , typename MT , std::enable_if_t< hila::is_assignable< T &, S >::value, int > = 0>

auto & Matrix_t< n, m, T, Mtype >::operator-= ( const Matrix_t< n, m, S, MT > &  rhs ) &

inline

Subtract assign operator Matrix to MAtrix.

Matrix<n,m,MyType> M,N;
M = 3;
N = 1;
M -= N; \\M = 2*I

Parameters

rhs	Matrix to subtract with

Returns

Mtype&

Definition at line 764 of file matrix.h.

operator-=() [3/3]

template<const int n, const int m, typename T , typename Mtype >

template<typename S , std::enable_if_t< hila::is_assignable< T &, hila::type_minus< T, S > >::value, int > = 0>

auto & Matrix_t< n, m, T, Mtype >::operator-= ( const S  rhs ) &

inline

Subtract assign operator scalar to Matrix.

Subtract scalar \( a \) to square matrix as \( M - a\cdot\mathbb{1} \)

Matrix<n,m,MyType> M = 3;
M -= 1 ; \\ M = 2*I

Template Parameters

S	Element type of scalar rhs

Parameters

rhs	scalar to add

Returns

Mtype&

Definition at line 815 of file matrix.h.

operator/()

template<typename Mt , typename S , std::enable_if_t<(Mt::is_matrix() &&hila::is_complex_or_arithmetic< S >::value), int > = 0, typename Rt = hila::mat_scalar_type<Mt, S>>

Rt operator/ ( const Mt &  mat, const S &  rhs  )

inline

Division operator.

Defined for following operations

Matrix / Scalar:

Division operator between Matrix and Scalar are defined in the usual way (element wise)

Matrix<n,m,MyType> M;
M.fill(2);
auto S = M/2; // Resulting matrix is uniformly 1

Template Parameters

Mt	Matrix type
S	Scalar type

Parameters

mat	Matrix to divide scalar with
rhs	Scalar to divide with

Returns

Rt Resulting Matrix

Definition at line 2562 of file matrix.h.

operator/=() [1/2]

template<const int n, const int m, typename T , typename Mtype >

template<typename S , std::enable_if_t< hila::is_assignable< T &, hila::type_div< T, S > >::value, int > = 0>

auto & Matrix_t< n, m, T, Mtype >::operator/= ( const DiagonalMatrix< m, S > &  rhs ) &

inline

Divide assign operator for DiagonalMatrix to Matrix.

Well defined since rhs is guaranteed to be Diagonal.

Let M be the Matrix which we divide and D be the DiagonalMatrix which we divide with then the operation is defined as \( M \cdot D^{-1} \).

Template Parameters

S	Element type of rhs

Parameters

rhs	DiagonalMatrix to divide

Returns

Mtype&

Definition at line 996 of file matrix.h.

operator/=() [2/2]

template<const int n, const int m, typename T , typename Mtype >

template<typename S , std::enable_if_t< hila::is_assignable< T &, hila::type_div< T, S > >::value, int > = 0>

auto & Matrix_t< n, m, T, Mtype >::operator/= ( const S  rhs ) &

inline

Divide assign oprator scalar with matrix.

Divide works as defined for scalar matrix division.

Matrix<n,m,MyType> M;
.
. Fill whole matrix with 2
.
M /= 2 ; \\ M is filled with 1

Template Parameters

S	Scalar type of rhs

Parameters

rhs	Scalar to divide with

Returns

Definition at line 923 of file matrix.h.

operator=() [1/5]

template<const int n, const int m, typename T , typename Mtype >

template<typename S , std::enable_if_t< hila::is_assignable< T &, S >::value, int > = 0>

auto & Matrix_t< n, m, T, Mtype >::operator= ( const DiagonalMatrix< n, S > &  rhs ) &

inline

Assignment from diagonal matrix.

Template Parameters

S	Element type of Diagonal matrix

Parameters

rhs	Diagonal matrix to assign

Returns

Mtype&

Definition at line 681 of file matrix.h.

operator=() [2/5]

template<const int n, const int m, typename T , typename Mtype >

template<typename S , typename MT , std::enable_if_t< hila::is_assignable< T &, S >::value, int > = 0>

auto & Matrix_t< n, m, T, Mtype >::operator= ( const Matrix_t< n, m, S, MT > &  rhs ) &

inline

Copy matrix assignment.

Matrix<n,m,MyType> M_0;
.
. M_0 has values assigned to it
.
Matrix<n,m,MyType> M; \\ undefined matrix
M = M_0; \\ Assignment from M_0

Parameters

rhs	Matrix to assign

Returns

Definition at line 614 of file matrix.h.

operator=() [3/5]

template<const int n, const int m, typename T , typename Mtype >

template<typename S , std::enable_if_t< hila::is_assignable< T &, S >::value &&n==m, int > = 0>

auto & Matrix_t< n, m, T, Mtype >::operator= ( const S  rhs ) &

inline

Assignment from scalar.

Assigns the scalar to the diagonal elements as \( M = I\cdot a\)

MyType a = hila::random();
Matrix<n,m,MyType> M;
M = a; M = I*a

Template Parameters

S	Scalar type to assign

Parameters

rhs	Scalar to assign

Returns

Mtype&

Definition at line 658 of file matrix.h.

operator=() [4/5]

template<const int n, const int m, typename T , typename Mtype >

Mtype & Matrix_t< n, m, T, Mtype >::operator= ( const std::nullptr_t &  z )

inline

Zero assignment.

Matrix<n,m,MyType> M;
M = 0; Zero matrix;

Parameters

z

0

Returns

Mtype&

Definition at line 634 of file matrix.h.

operator=() [5/5]

template<const int n, const int m, typename T , typename Mtype >

template<typename S , std::enable_if_t< hila::is_assignable< T &, S >::value, int > = 0>

auto & Matrix_t< n, m, T, Mtype >::operator= ( std::initializer_list< S >  rhs ) &

inline

Initializer list assignment.

Matrix<2,2,int> M ;
M = {1, 0
     0, 1};

Template Parameters

S	Element type of initializer list

Parameters

rhs	Initializer list to assign

Returns

Mtype&

Definition at line 708 of file matrix.h.

operator==()

template<const int n, const int m, typename T , typename Mtype >

template<typename S , int n2, int m2>

bool Matrix_t< n, m, T, Mtype >::operator== ( const Matrix< n2, m2, S > &  rhs ) const

inline

Boolean operator == to determine if two matrices are exactly the same.

Tolerance for equivalence is zero, meaning that the matrices must be exactly the same.

Template Parameters

S	Type for Matrix which is being compared to

Parameters

rhs	right hand side Matrix which we are comparing

Returns

true

false

Definition at line 571 of file matrix.h.

operator[]()

template<const int n, const int m, typename T , typename Mtype >

template<int q = n, int p = m, std::enable_if_t<(q==1||p==1), int > = 0>

T Matrix_t< n, m, T, Mtype >::operator[] ( const int  i ) const

inline

Indexing operation [] defined only for vectors.

Example:

Vector<n,MyType> V;
MyType a = V[i] \\ i <= n

Template Parameters

q	row size n
p	column size m

Parameters

i	row or vector index depending on which is being indexed

Returns

T

Definition at line 332 of file matrix.h.

outer_product()

template<const int n, const int m, typename T , typename Mtype >

template<int p, int q, typename S , typename R = hila::type_mul<T, S>>

Matrix< n, p, R > Matrix_t< n, m, T, Mtype >::outer_product ( const Matrix< p, q, S > &  rhs ) const

inline

Outer product.

Only works between two Vector objects

Vector<n,MyType> V,W;
.
.
.
V.outer_product(W); \\ Valid operation

RowVector<m,MyType> V,W;
.
.
.
V.outer_product(W); // not valid operation

Template Parameters

p	Row length for rhs
q	Column length for rhs
S	Element type for rhs
R	Type between lhs and rhs multiplication

Parameters

rhs	Vector to compute outer product with

Returns

Matrix<n, p, R>

Definition at line 1357 of file matrix.h.

permute()

template<const int n, const int m, typename T , typename Mtype >

template<int N>

Mtype Matrix_t< n, m, T, Mtype >::permute ( const Vector< N, int > &  permutation ) const

inline

Permute elements of vector.

Reordering is done based off of permutation vector

Parameters

permutation

Vector of integers to permute rows

Returns

Mtype

Definition at line 1478 of file matrix.h.

permute_columns()

template<const int n, const int m, typename T , typename Mtype >

Mtype Matrix_t< n, m, T, Mtype >::permute_columns ( const Vector< m, int > &  permutation ) const

inline

permute columns of Matrix

Reordering is done based off of permutation vector

Parameters

permutation

Vector of integers to permute columns

Returns

Mtype

Definition at line 1448 of file matrix.h.

permute_rows()

template<const int n, const int m, typename T , typename Mtype >

Mtype Matrix_t< n, m, T, Mtype >::permute_rows ( const Vector< n, int > &  permutation ) const

inline

permute rows of Matrix

Reordering is done based off of permutation vector

Parameters

permutation

Vector of integers to permute rows

Returns

Mtype

Definition at line 1462 of file matrix.h.

random()

template<const int n, const int m, typename T , typename Mtype >

Mtype & Matrix_t< n, m, T, Mtype >::random ( )

inline

dot with matrix - matrix

Generate random elements

Fills Matrix with random elements from uniform distribution

Matrix<n,m,T> M;
M.random();

Returns

Mtype&

Definition at line 1393 of file matrix.h.

real()

template<const int n, const int m, typename T , typename Mtype >

Matrix< n, m, hila::arithmetic_type< T > > Matrix_t< n, m, T, Mtype >::real ( ) const

inline

Returns real part of Matrix or Vector.

Returns

Matrix<n, m, hila::arithmetic_type<T>>

Definition at line 1143 of file matrix.h.

row()

template<const int n, const int m, typename T , typename Mtype >

RowVector< m, T > Matrix_t< n, m, T, Mtype >::row ( int  r ) const

inline

Return reference to row in a matrix.

Since the Matrix data is ordered in a row major order accessing a row returns a reference to the row.

Matrix<n,n,T> M;
M.random();
RowVector<n,T> V = M.row(i);

Parameters

r	index of row to be referenced

Returns

const RowVector<m, T>&

Definition at line 355 of file matrix.h.

rows()

template<const int n, const int m, typename T , typename Mtype >

static constexpr int Matrix_t< n, m, T, Mtype >::rows ( )

inlinestaticconstexpr

Define constant methods rows(), columns() - may be useful in template code.

Returns row length

Returns

constexpr int

Definition at line 226 of file matrix.h.

set_column()

template<const int n, const int m, typename T , typename Mtype >

template<typename S , std::enable_if_t< hila::is_assignable< T &, S >::value, int > = 0>

void Matrix_t< n, m, T, Mtype >::set_column ( int  c, const Vector< n, S > &  v  )

inline

get column of a matrix

Set column of Matrix with Vector if types are assignable

Vector<n,T> V;
V.random();
Matrix<n,n,T> M;
M.set_column(i,V);

Template Parameters

S	Vector type

Parameters

c	Index of column to be set
v	Vector to be set

Definition at line 424 of file matrix.h.

set_diagonal()

template<const int n, const int m, typename T , typename Mtype >

template<typename S , std::enable_if_t< hila::is_assignable< T &, S >::value, int > = 0>

void Matrix_t< n, m, T, Mtype >::set_diagonal ( const Vector< n, S > &  v )

inline

Set diagonal of square matrix to Vector which is passed to the method.

If called for non square matrix the program will throw an error.

Example:

SquareMatrix<n,MyType> S = 0; \\ Zero matrix
Vector<n,MyType> V = 1; \\ Vector assigned to 1 at all elements
S.set_diagonal(V); \\ Results in Identity matrix of size n

Template Parameters

S	type vector to assign values to

Parameters

v	Vector to assign to diagonal

Definition at line 463 of file matrix.h.

set_row()

template<const int n, const int m, typename T , typename Mtype >

template<typename S , std::enable_if_t< hila::is_assignable< T &, S >::value, int > = 0>

void Matrix_t< n, m, T, Mtype >::set_row ( int  r, const RowVector< m, S > &  v  )

inline

Set row of Matrix with RowVector if types are assignable.

RowVector<n,T> V;
V.random();
Matrix<n,n,T> M;
M.set_row(i,V);

Template Parameters

S	RowVector type

Parameters

r	Index of row to be set
v	RowVector to be set

Definition at line 376 of file matrix.h.

size()

template<const int n, const int m, typename T , typename Mtype >

template<int q = n, int p = m, std::enable_if_t< q==1, int > = 0>

static constexpr int Matrix_t< n, m, T, Mtype >::size ( )

inlinestaticconstexpr

Returns size of Vector or square Matrix.

Template Parameters

q	row size n
p	column size m

Returns

constexpr int

Definition at line 248 of file matrix.h.

sort() [1/2]

template<const int n, const int m, typename T , typename Mtype >

Mtype Matrix_t< n, m, T, Mtype >::sort ( hila::sort  order = hila::sort::ascending ) const

inline

Sort method for Vector.

Order can be past as argument as either ascending (default) or descending

Ascending

Vector<n,MyType> V;
V.random();
V.sort(); // V is sorted in ascending order

Descending

V.sort(hila::sort::descending);

Parameters

order

Order to sort in

Returns

Mtype

Definition at line 1567 of file matrix.h.

sort() [2/2]

template<const int n, const int m, typename T , typename Mtype >

template<int N>

Mtype Matrix_t< n, m, T, Mtype >::sort ( Vector< N, int > &  permutation, hila::sort  order = hila::sort::ascending  ) const

inline

Sort method for Vector which returns permutation order.

Order can be past as argument as either ascending (default) or descending

Ascending

Vector<n,MyType> V;
Vector<n,int> perm;
V.random();
V.sort(perm);
V.permute(perm);

Descending

Vector<n,MyType> V;
Vector<n,int> perm;
V.random();
V.sort(perm,hila::sort::descending);
V.permute(perm);

Template Parameters

N

Parameters

permutation
order

Returns

Mtype

Definition at line 1519 of file matrix.h.

squarenorm()

template<const int n, const int m, typename T , typename Mtype >

hila::arithmetic_type< T > Matrix_t< n, m, T, Mtype >::squarenorm ( ) const

inline

Calculate square norm - sum of squared elements.

Returns

hila::arithmetic_type<T>

Definition at line 1209 of file matrix.h.

str()

template<const int n, const int m, typename T , typename Mtype >

std::string Matrix_t< n, m, T, Mtype >::str ( int  prec = 8, char  separator = ' '  ) const

inline

Convert to string for printing

Definition at line 1709 of file matrix.h.

svd() [1/2]

template<int n, int m, typename T , typename Mtype >

svd_result< Mtype > Matrix_t< n, m, T, Mtype >::svd

(

enum hila::sort

sorted = hila::sort::unsorted

)

const

svd and svd_pivot - alternative interface

svd(hila::sort [optional]) returns struct svd_result<Mtype>, with fields U: nxn unitary matrix singularvalues: DiagonalMatrix of singular values (real, > 0) V: nxn unitary matrix

auto res = M.svd(); now res.U : nxn unitary res.singularvalues : DiagonalMatrix of singular values res.V : nxn unitary

result satifies M = res.U res.singularvalues res.V.dagger()

Definition at line 577 of file matrix_linalg.h.

svd() [2/2]

template<int n, int m, typename T , typename Mtype >

template<typename Et , typename Mt , typename MT >

int Matrix_t< n, m, T, Mtype >::svd	(	Matrix_t< n, n, Mt, MT > &	_U,
		DiagonalMatrix< n, Et > &	_S,
		Matrix_t< n, n, Mt, MT > &	_V,
		enum hila::sort	sorted = hila::sort::unsorted
	)		const

Singular value decomposition: divide matrix A = U S V*, where U,V unitary and S diagonal matrix of real singular values. Unpivoted Jacobi rotations.

Unpivoted rotation is generally faster than pivoted (esp. on gpu), but a bit less accurate

Use: M.svd(U, S, V, [sorted]);

Result satisfies M = U S V.dagger()

The algorithm works by one-sided Jacobi rotations.

• U = V = 1
• Loop
  • for i=0..(n-2); j=(i+1)..(n-1)
    • calculate B_ii, B_ij, B_jj, (B_ij = (A^* A)_ij)
    • If B_ij > 0
      • diagonalize J^* [ B_ii B_ij ] J = diag(D_ii, D_jj) [ B_ji B_jj ]
      • A <- A J , Columns i and j change
      • V <- V J
  • endfor
  • if nothing changed, break Loop
• Endloop Now A = U S, A^* A = S* S = S^2
• S_ii = A_ii / |A_i| (norm of column i)
• U = A / S

Definition at line 469 of file matrix_linalg.h.

svd_pivot()

template<int n, int m, typename T , typename Mtype >

template<typename Et , typename Mt , typename MT >

int Matrix_t< n, m, T, Mtype >::svd_pivot	(	Matrix_t< n, n, Mt, MT > &	_U,
		DiagonalMatrix< n, Et > &	_S,
		Matrix_t< n, n, Mt, MT > &	_V,
		enum hila::sort	sorted = hila::sort::unsorted
	)		const

Singular value decomposition: divide matrix A = U S V*, where U,V unitary and S diagonal matrix of real singular values. Fully pivoted Jacobi rotations.

Use: M.svd_pivot(U, S, V, [sorted]);

The algorithm works by one-sided Jacobi rotations.

• U = V = 1
• Compute Gram matrix B = A^* A. B is Hermitean, with B_ii >= 0.
• Loop:
  • Find largest B_ij, j > i - full pivoting if |B_ij|^2 <= tol * B_ii B_jj we're done
  • find J to diagonalize J^* [ B_ii B_ij ] J = diag(D_ii, D_jj) [ B_ji B_jj ]
  • A <- A J , Columns i and j change
  • V <- V J
  • Recompute B_ik and B_jk, k /= i,j (B_ii = D_ii, B_jj = D_jj, B_ij = 0) (n^2)
• Endloop Now A = U S, A^* A = S* S = S^2
• S_ii = A_ii / |A_i| (norm of column i)
• U = A / S

Parameters

U

Definition at line 344 of file matrix_linalg.h.

trace()

template<const int n, const int m, typename T , typename Mtype >

T Matrix_t< n, m, T, Mtype >::trace ( ) const

inline

Computes Trace for Matrix.

Not define for non square matrices. Static assert will stop code from compiling if executed for non square matrices.

Returns

T

Definition at line 1171 of file matrix.h.

transpose() [1/3]

template<const int n, const int m, typename T , typename Mtype >

template<int mm = m, typename Rtype = typename std::conditional<n == m, Matrix_t, Matrix<m, n, T>>::type, std::enable_if_t<(mm !=1 &&n !=1), int > = 0>

Rtype Matrix_t< n, m, T, Mtype >::transpose ( ) const

inline

Transpose of matrix.

Return type for square matrix is same input, for non square return type is Matrix<n,m,MyType>

Template Parameters

mm
std::conditional<n,Mtype,Matrix<m,n,T>>::type

Returns

const Rtype

Definition at line 1039 of file matrix.h.

transpose() [2/3]

template<const int n, const int m, typename T , typename Mtype >

template<int mm = m, std::enable_if_t< mm==1, int > = 0>

const RowVector< n, T > & Matrix_t< n, m, T, Mtype >::transpose ( ) const

inline

Transpose of vector.

Returns reference

Template Parameters

mm

Returns

const RowVector<n, T>&

Definition at line 1056 of file matrix.h.

transpose() [3/3]

template<const int n, const int m, typename T , typename Mtype >

template<int nn = n, std::enable_if_t< nn==1 &&m !=1, int > = 0>

const Vector< m, T > & Matrix_t< n, m, T, Mtype >::transpose ( ) const

inline

Transpose of RowVector.

Returns reference

Template Parameters

mm

Returns

const RowVector<n, T>&

Definition at line 1068 of file matrix.h.

---

The documentation for this class was generated from the following files:

• matrix.h
• matrix_linalg.h