Definition of Matrix types. More...

#include <type_traits>
#include <sstream>
#include "plumbing/defs.h"
#include "datatypes/cmplx.h"
#include "datatypes/array.h"
#include "datatypes/diagonal_matrix.h"
#include "datatypes/matrix_linalg.h"

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Classes
struct	svd_result< M >
	type to store the return combo of svd: {U, D, V} where U and V are nxn unitary / orthogonal, and D is real diagonal singular value matrices. More...

struct	eigen_result< M >
	type to store the return value of eigen_hermitean(): {E, U} where E is nxn DiagonalMatrix containing eigenvalues and U nxn unitary matrix, with eigenvector columns More...

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. More...

class	Matrix< n, m, T >
	\( n \times m \) Matrix class which defines matrix operations. More...

Namespaces
namespace	hila
	Implement hila::swap for gauge fields.

Typedefs
template<int n, typename T >
using	Vector = Matrix< n, 1, T >
	Vector is defined as 1-column Matrix.

template<int n, typename T >
using	RowVector = Matrix< 1, n, T >
	RowVector is a 1-row Matrix.

template<int n, typename T >
using	SquareMatrix = Matrix< n, n, T >
	Square matrix is defined as alias with special case of Matrix<n,n,T>

Functions
template<typename Mtype , std::enable_if_t< Mtype::is_matrix(), int > = 0>
auto	transpose (const Mtype &arg)
	Return transposed Matrix or Vector.

template<typename Mtype , std::enable_if_t< Mtype::is_matrix(), int > = 0>
auto	conj (const Mtype &arg)
	Return conjugate Matrix or Vector.

template<typename Mtype , std::enable_if_t< Mtype::is_matrix(), int > = 0>
auto	adjoint (const Mtype &arg)
	Return adjoint Matrix.

template<typename Mtype , std::enable_if_t< Mtype::is_matrix(), int > = 0>
auto	dagger (const Mtype &arg)
	Return dagger of Matrix.

template<typename Mtype , std::enable_if_t< Mtype::is_matrix(), int > = 0>
auto	abs (const Mtype &arg)
	Return absolute value Matrix or Vector.

template<typename Mtype , std::enable_if_t< Mtype::is_matrix(), int > = 0>
auto	trace (const Mtype &arg)
	Return trace of square Matrix.

template<typename Mtype , std::enable_if_t< Mtype::is_matrix(), int > = 0>
auto	real (const Mtype &arg)
	Return real of Matrix or Vector.

template<typename Mtype , std::enable_if_t< Mtype::is_matrix(), int > = 0>
auto	imag (const Mtype &arg)
	Return imaginary of Matrix or Vector.

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.

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.

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 Mt &mat, const S &rhs)
	Division operator.

template<typename Mtype1 , typename Mtype2 , std::enable_if_t< Mtype1::is_matrix() &&Mtype2::is_matrix(), int > = 0>
auto	mul_trace (const Mtype1 &a, const Mtype2 &b)
	Returns trace of product of two matrices.

template<typename Mt1 , typename Mt2 , typename Mt3 , std::enable_if_t< Mt1::is_matrix() &&Mt2::is_matrix() &&Mt3::is_matrix(), int > = 0>
void	mult (const Mt1 &a, const Mt2 &b, Mt3 &res)
	compute product of two matrices and write result to existing matrix

template<typename Mt1 , typename Mt2 , typename Mt3 , std::enable_if_t< Mt1::is_matrix() &&Mt2::is_matrix() &&Mt3::is_matrix(), int > = 0>
void	mult_aa (const Mt1 &a, const Mt2 &b, Mt3 &res)
	compute hermitian conjugate of product of two matrices and write result to existing matrix

template<typename Mt1 , typename S , typename Mt2 , std::enable_if_t<(Mt1::is_matrix() &&Mt2::is_matrix() &&hila::is_complex_or_arithmetic< S >::value), int > = 0>
void	mult (const Mt1 &a, const S &b, Mt2 &res)
	compute product of a matrix and a scalar and write result to existing matrix

template<typename S , typename Mt1 , typename Mt2 , std::enable_if_t<(Mt1::is_matrix() &&Mt2::is_matrix() &&hila::is_complex_or_arithmetic< S >::value), int > = 0>
void	mult (const S &a, const Mt1 &b, Mt2 &res)
	compute product of a scalar and a matrix and write result to existing matrix

template<typename Mt1 , typename Mt2 , typename Mt3 , std::enable_if_t< Mt1::is_matrix() &&Mt2::is_matrix() &&Mt3::is_matrix(), int > = 0>
void	mult_add (const Mt1 &a, const Mt2 &b, Mt3 &res)
	compute product of two matrices and add result to existing matrix

template<typename Mt1 , typename S , typename Mt2 , std::enable_if_t<(Mt1::is_matrix() &&Mt2::is_matrix() &&hila::is_complex_or_arithmetic< S >::value), int > = 0>
void	mult_add (const Mt1 &a, const S &b, Mt2 &res)
	compute product of a matrix and a scalar and add result to existing matrix

template<typename S , typename Mt1 , typename Mt2 , std::enable_if_t<(Mt1::is_matrix() &&Mt2::is_matrix() &&hila::is_complex_or_arithmetic< S >::value), int > = 0>
void	mult_add (const S &a, const Mt1 &b, Mt2 &res)
	compute product of a scalar and a matrix and add result to existing matrix

template<typename Mt1 , typename Mt2 , typename Mt3 , std::enable_if_t< Mt1::is_matrix() &&Mt2::is_matrix() &&Mt3::is_matrix(), int > = 0>
void	mult_sub (const Mt1 &a, const Mt2 &b, Mt3 &res)
	compute product of two matrices and subtract result from existing matrix

template<typename Mt1 , typename S , typename Mt2 , std::enable_if_t<(Mt1::is_matrix() &&Mt2::is_matrix() &&hila::is_complex_or_arithmetic< S >::value), int > = 0>
void	mult_sub (const Mt1 &a, const S &b, Mt2 &res)
	compute product of a matrix and a scalar and subtract result from existing matrix

template<typename S , typename Mt1 , typename Mt2 , std::enable_if_t<(Mt1::is_matrix() &&Mt2::is_matrix() &&hila::is_complex_or_arithmetic< S >::value), int > = 0>
void	mult_sub (const S &a, const Mt1 &b, Mt2 &res)
	compute product of a scalar and a matrix and subtract result from existing matrix

template<int n, int m, typename T , typename MT >
std::ostream &	operator<< (std::ostream &strm, const Matrix_t< n, m, T, MT > &A)
	Stream operator.

template<int n, int m, typename T , typename MT >
std::string	hila::to_string (const Matrix_t< n, m, T, MT > &A, int prec=8, char separator=' ')
	Converts Matrix_t object to string.

template<int n, int m, typename T , typename MT >
std::string	hila::prettyprint (const Matrix_t< n, m, T, MT > &A, int prec=8)
	Formats Matrix_t object in a human readable way.

template<typename Mt , std::enable_if_t< Mt::is_matrix(), int > = 0>
auto	squarenorm (const Mt &rhs)
	Returns square norm of Matrix.

template<typename Mt , std::enable_if_t< Mt::is_matrix(), int > = 0>
auto	norm (const Mt &rhs)
	Returns vector norm of Matrix.

template<int n, int m, typename T , typename MT >
Matrix_t< n, m, T, MT >	exp (const Matrix_t< n, m, T, MT > &mat, const int order=20)
	Calculate exp of a square matrix.

template<int n, int m, typename T , typename MT , typename atype = hila::arithmetic_type<T>>
Matrix_t< n, m, T, MT >	altexp (const Matrix_t< n, m, T, MT > &mat, int &niter)
	Calculate exp of a square matrix.

template<int n, int m, typename T , typename MT , typename atype = hila::arithmetic_type<T>>
Matrix_t< n, m, T, MT >	altexp (const Matrix_t< n, m, T, MT > &mat)
	Calculate exp of a square matrix.

template<int n, int m, typename T , typename MT >
void	mult_exp (const Matrix_t< n, m, T, MT > &mat, const Matrix_t< n, m, T, MT > &mmat, Matrix_t< n, m, T, MT > &r, Matrix_t< n, m, T, MT > &dr, const int order=20)
	Calculate mmatexp(mat) and trace(mmatdexp(mat))

template<int n, int m, typename T , typename MT , typename Mt , std::enable_if_t< Mt::is_matrix() &&Mt::is_square() &&Mt::rows()==n, int > = 0>
int	chexp (const Matrix_t< n, m, T, MT > &mat, Matrix_t< n, m, T, MT > &omat, Mt(&pl)[n])
	Calculate exp of a square matrix.

template<int n, int m, typename T , typename MT , typename Mt , std::enable_if_t< Mt::is_matrix() &&Mt::is_square() &&Mt::rows()==n, int > = 0>
void	chexp (const Matrix_t< n, m, T, MT > &mat, Matrix_t< n, m, T, MT > &omat, Mt(&domat)[n][m])
	Calculate exp and dexp of a square matrix.

template<int n, int m, typename T , typename MT >
void	mult_chexp (const Matrix_t< n, m, T, MT > &mat, const Matrix_t< n, m, T, MT > &mmat, Matrix_t< n, m, T, MT > &omat, Matrix_t< n, m, T, MT > &domat)
	Calculate exp(mat).dagger()mmatexp(mat) and trace(exp(mat).daggermmatdexp(mat))

template<int n, int m, typename T , typename MT >
void	chexpk (const Matrix_t< n, m, T, MT > &mat, Matrix_t< n, m, T, MT > &omat, Matrix_t< n, m, T, MT > &kmats)
	Calculate exp(mat) and the decomposition k_{i,j} of dexp in terms bilinears of powers of mat.

template<int n, int m, typename T , typename MT >
void	mult_chexpk_fast (const Matrix_t< n, m, T, MT > &mat, const Matrix_t< n, m, T, MT > &texp, const Matrix_t< n, m, T, MT > &kmats, const Matrix_t< n, m, T, MT > &mmat, Matrix_t< n, m, T, MT > &omat, Matrix_t< n, m, T, MT > &domat)
	Calculate exp(mat).dagger()mmatexp(mat) and trace(exp(mat).daggermmatdexp(mat)) from output of chexpk.

template<int n, int m, typename T , typename MT >
Matrix_t< n, m, T, MT >	chsexp (const Matrix_t< n, m, T, MT > &mat)
	Calculate exp of a square matrix.

Detailed Description

Definition of Matrix types.

This file contains base matrix type Matrix_t which defines all general matrix type operations Matrix types are Matrix, Vector, RowVector, SquareMatrix of which Matrix is defined as a class and the rest are special cases of the Matrix class.

Definition in file matrix.h.

Function Documentation

◆ abs()

template<typename Mtype , std::enable_if_t< Mtype::is_matrix(), int > = 0>

auto abs ( const Mtype & arg )

inline

Return absolute value Matrix or Vector.

Wrapper around Matrix::abs

Can be called as:

Matrix<n,m,MyType> M,M_abs;

M_abs = abs(M);

Matrix

Matrix class which defines matrix operations.

Definition matrix.h:1620

abs

T abs(const Complex< T > &a)

Return absolute value of Complex number.

Definition cmplx.h:1187

Template Parameters

Mtype Matrix type

Parameters

arg	Object to compute absolute value of

Returns: auto Mtype absolute value Matrix or Vector

Definition at line 1833 of file matrix.h.

◆ adjoint()

template<typename Mtype , std::enable_if_t< Mtype::is_matrix(), int > = 0>

auto adjoint ( const Mtype & arg )

inline

Return adjoint Matrix.

Wrapper around Matrix::adjoint

Can be called as:

Matrix<n,m,MyType> M,M_adjoint;

M_conjugate = adjoint(M);

Template Parameters

Mtype Matrix type

Parameters

arg	Object to compute adjoint of

Returns: auto Mtype adjoint matrix

Definition at line 1797 of file matrix.h.

◆ altexp() [1/2]

template<int n, int m, typename T , typename MT , typename atype = hila::arithmetic_type<T>>

Matrix_t< n, m, T, MT > altexp ( const Matrix_t< n, m, T, MT > & mat )

inline

Calculate exp of a square matrix.

Adds hihger order terms till matrix norm converges within floating point accuracy

Template Parameters

n	Number of rowsMa
T	Matrix element type
MT	Matrix type

Parameters

mat	Matrix to compute exponential for

Returns: Matrix_t<n, m, T, MT>

Definition at line 2853 of file matrix.h.

◆ altexp() [2/2]

template<int n, int m, typename T , typename MT , typename atype = hila::arithmetic_type<T>>

Matrix_t< n, m, T, MT > altexp	(	const Matrix_t< n, m, T, MT > &	mat,
		int &	niter
	)

inline

Calculate exp of a square matrix.

Adds higher order terms till matrix norm converges within floating point accuracy and returns required number of iterations

Template Parameters

n	Number of rowsMa
T	Matrix element type
MT	Matrix type

Parameters

mat	Matrix to compute exponential for
niter	(output) number of iteration performed till converges was reached

Returns: Matrix_t<n, m, T, MT>

Definition at line 2820 of file matrix.h.

◆ chexp() [1/2]

template<int n, int m, typename T , typename MT , typename Mt , std::enable_if_t< Mt::is_matrix() &&Mt::is_square() &&Mt::rows()==n, int > = 0>

void chexp	(	const Matrix_t< n, m, T, MT > &	mat,
		Matrix_t< n, m, T, MT > &	omat,
		Mt(&)	domat[n][m]
	)

inline

Calculate exp and dexp of a square matrix.

Computation is done using iterative Cayley-Hamilton (cf. from arXiv:2404.07704)

Template Parameters

n	Number of rowsMa
T	Matrix element type
MT	Matrix type

Parameters

mat	Matrix to compute exponential for
omat	Matrix to which exponential of mat gets stored
domat	matrix of Matrices to which the derivatives of the exponential with respect to the components of mat gets stored

Returns: void

Definition at line 3093 of file matrix.h.

◆ chexp() [2/2]

template<int n, int m, typename T , typename MT , typename Mt , std::enable_if_t< Mt::is_matrix() &&Mt::is_square() &&Mt::rows()==n, int > = 0>

int chexp	(	const Matrix_t< n, m, T, MT > &	mat,
		Matrix_t< n, m, T, MT > &	omat,
		Mt(&)	pl[n]
	)

inline

Calculate exp of a square matrix.

Computation is done using iterative Cayley-Hamilton (cf. from arXiv:2404.07704)

Template Parameters

n	Number of rowsMa
T	Matrix element type
MT	Matrix type

Parameters

mat	Matrix to compute exponential for
omat	Matrix to which exponential of mat gets stored (optional)
pl	array of n+1 temporary nxn Matrices (optional)

Returns: void (if omat is provided) or Matrix_t<n,m,T,MT>

Definition at line 2933 of file matrix.h.

◆ chexpk()

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

void chexpk	(	const Matrix_t< n, m, T, MT > &	mat,
		Matrix_t< n, m, T, MT > &	omat,
		Matrix_t< n, m, T, MT > &	kmats
	)

inline

Calculate exp(mat) and the decomposition k_{i,j} of dexp in terms bilinears of powers of mat.

Computation is done using iterative Cayley-Hamilton (cf. from arXiv:2404.07704) exp(mat)^a_b = \sum_{i=0}^{n-1} r_{i} (mat^i)^a_b dexp(X)^a_b/dX^c_d|_X=mat = \sum_{i,j=0}^{n-1} k_{i,j} (mat^i)^d_b (mat^j)^a_c

Template Parameters

n	Number of rows of Matrix
m	Number of columns of Matrix
T	Matrix element type
MT	Matrix type

Parameters

mat	Matrix of which to compute exponential
omat	Matrix to which exponential of mat gets stored
kmats	Matrix to which decomposition coefficients of dexp/dX\|X=mat in terms of bilinears of powers of mat get stroed

Returns: void

Definition at line 3453 of file matrix.h.

◆ chsexp()

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

Matrix_t< n, m, T, MT > chsexp ( const Matrix_t< n, m, T, MT > & mat )

inline

Calculate exp of a square matrix.

Computation is done using iterative Cayley-Hamilton (cf. from arXiv:2404.07704) with minimal temporary storage

Template Parameters

n	Number of rowsMa
T	Matrix element type
MT	Matrix type

Parameters

mat	Matrix to compute exponential for

Returns: Matrix_t<n, m, T, MT>

Definition at line 3672 of file matrix.h.

◆ conj()

template<typename Mtype , std::enable_if_t< Mtype::is_matrix(), int > = 0>

auto conj ( const Mtype & arg )

inline

Return conjugate Matrix or Vector.

Wrapper around Matrix::conj

Can be called as:

Matrix<n,m,MyType> M,M_conjugate;

M_conjugate = conj(M);

conj

Array< n, m, T > conj(const Array< n, m, T > &arg)

Return conjugate Array.

Definition array.h:674

Template Parameters

Mtype Matrix type

Parameters

arg	Object to be conjugated

Returns: auto Mtype conjugated matrix

Definition at line 1777 of file matrix.h.

◆ dagger()

template<typename Mtype , std::enable_if_t< Mtype::is_matrix(), int > = 0>

auto dagger ( const Mtype & arg )

inline

Return dagger of Matrix.

Wrapper around Matrix::adjoint

Same as adjoint

Template Parameters

Mtype Matrix type

Parameters

arg	Object to compute dagger of

Returns: auto Mtype dagger matrix

Definition at line 1813 of file matrix.h.

◆ exp()

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

Matrix_t< n, m, T, MT > exp	(	const Matrix_t< n, m, T, MT > &	mat,
		const int	order = `20`
	)

inline

Calculate exp of a square matrix.

Computes up to certain order given as argument

Evaluation is done as:

\begin{align} \exp(H) &= 1 + H + \frac{H^2}{2!} + \frac{H^2}{2!} + \frac{H^3}{3!} \\ &= 1 + H\cdot(1 + (\frac{H}{2})\cdot (1 + (\frac{H}{3})\cdot(...))) \end{align}

Done backwards in order to reduce accumulation of errors

Template Parameters

n	Number of rowsMa
T	Matrix element type
MT	Matrix type

Parameters

mat	Matrix to compute exponential for
order	order to compute exponential to

Returns: Matrix_t<n, m, T, MT>

Definition at line 2786 of file matrix.h.

◆ imag()

template<typename Mtype , std::enable_if_t< Mtype::is_matrix(), int > = 0>

auto imag ( const Mtype & arg )

inline

Return imaginary of Matrix or Vector.

Wrapper around Matrix::imag

Can be called as:

Matrix<n,m,MyType> M,M_imag;

M_imag = imag(M);

imag

Array< n, m, hila::arithmetic_type< T > > imag(const Array< n, m, T > &arg)

Return imaginary part of Array.

Definition array.h:702

Template Parameters

Mtype Matrix type

Parameters

arg	Object to compute imag part of

Returns: auto Mtype imag part of arg

Definition at line 1894 of file matrix.h.

◆ mul_trace()

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

auto mul_trace	(	const Mtype1 &	a,
		const Mtype2 &	b
	)

inline

Returns trace of product of two matrices.

Wrapper around Matrix::mul_trace in the form

mul_trace(a,b) // -> a.mul_trace(b)

mul_trace

auto mul_trace(const Mtype1 &a, const Mtype2 &b)

Returns trace of product of two matrices.

Definition matrix.h:2309

Template Parameters

Mtype1	Matrix type for a
Mtype2	Matrix type for b

Parameters

a	Left Matrix
b	Right Matrix

Returns: auto Resulting trace

Definition at line 2309 of file matrix.h.

◆ mult() [1/3]

template<typename Mt1 , typename Mt2 , typename Mt3 , std::enable_if_t< Mt1::is_matrix() &&Mt2::is_matrix() &&Mt3::is_matrix(), int > = 0>

void mult	(	const Mt1 &	a,
		const Mt2 &	b,
		Mt3 &	res
	)

inline

compute product of two matrices and write result to existing matrix

Template Parameters

Mt1,Mt2,Mt3 Matrix types

Parameters

a	Left Matrix of type Mt1
b	Right Matrix of type Mt2
c	Matrix of type Mt3 to which result gets written

Returns: void

Definition at line 2327 of file matrix.h.

◆ mult() [2/3]

template<typename Mt1 , typename S , typename Mt2 , std::enable_if_t<(Mt1::is_matrix() &&Mt2::is_matrix() &&hila::is_complex_or_arithmetic< S >::value), int > = 0>

void mult	(	const Mt1 &	a,
		const S &	b,
		Mt2 &	res
	)

inline

compute product of a matrix and a scalar and write result to existing matrix

Template Parameters

Mt1	Matrix type1, S Scalar type, Mt2 Matrix type2

Parameters

a	Left Matrix of type Mt1
b	Right Scalar of type S
c	Matrix of type Mt2 to which result gets written

Returns: void

Definition at line 2387 of file matrix.h.

◆ mult() [3/3]

template<typename S , typename Mt1 , typename Mt2 , std::enable_if_t<(Mt1::is_matrix() &&Mt2::is_matrix() &&hila::is_complex_or_arithmetic< S >::value), int > = 0>

void mult	(	const S &	a,
		const Mt1 &	b,
		Mt2 &	res
	)

inline

compute product of a scalar and a matrix and write result to existing matrix

Template Parameters

S	Scalar type, Mt1 Matrix type1, Mt2 Matrix type2

Parameters

a	Left Scalar of type S
b	Right Matrix of type Mt1
c	Matrix of type Mt2 to which result gets written

Returns: void

Definition at line 2413 of file matrix.h.

◆ mult_aa()

template<typename Mt1 , typename Mt2 , typename Mt3 , std::enable_if_t< Mt1::is_matrix() &&Mt2::is_matrix() &&Mt3::is_matrix(), int > = 0>

void mult_aa	(	const Mt1 &	a,
		const Mt2 &	b,
		Mt3 &	res
	)

inline

compute hermitian conjugate of product of two matrices and write result to existing matrix

Template Parameters

Mt1,Mt2,Mt3 Matrix types

Parameters

a	Left Matrix of type Mt1
b	Right Matrix of type Mt2
c	Matrix of type Mt3 to which result gets written

Returns: void

Definition at line 2356 of file matrix.h.

◆ mult_add() [1/3]

template<typename Mt1 , typename Mt2 , typename Mt3 , std::enable_if_t< Mt1::is_matrix() &&Mt2::is_matrix() &&Mt3::is_matrix(), int > = 0>

void mult_add	(	const Mt1 &	a,
		const Mt2 &	b,
		Mt3 &	res
	)

inline

compute product of two matrices and add result to existing matrix

Template Parameters

Mt1,Mt2,Mt3 Matrix types

Parameters

a	Left Matrix of type Mt1
b	Right Matrix of type Mt2
c	Matrix of type Mt3 to which result gets written

Returns: void

Definition at line 2437 of file matrix.h.

◆ mult_add() [2/3]

template<typename Mt1 , typename S , typename Mt2 , std::enable_if_t<(Mt1::is_matrix() &&Mt2::is_matrix() &&hila::is_complex_or_arithmetic< S >::value), int > = 0>

void mult_add	(	const Mt1 &	a,
		const S &	b,
		Mt2 &	res
	)

inline

compute product of a matrix and a scalar and add result to existing matrix

Template Parameters

Mt1	Matrix type1, S Scalar type, Mt2 Matrix type2

Parameters

a	Left Matrix of type Mt1
b	Right Scalar of type S
c	Matrix of type Mt2 to which result gets written

Returns: void

Definition at line 2467 of file matrix.h.

◆ mult_add() [3/3]

template<typename S , typename Mt1 , typename Mt2 , std::enable_if_t<(Mt1::is_matrix() &&Mt2::is_matrix() &&hila::is_complex_or_arithmetic< S >::value), int > = 0>

void mult_add	(	const S &	a,
		const Mt1 &	b,
		Mt2 &	res
	)

inline

compute product of a scalar and a matrix and add result to existing matrix

Template Parameters

S	Scalar type, Mt1 Matrix type1, Mt2 Matrix type2

Parameters

a	Left Scalar of type S
b	Right Matrix of type Mt1
c	Matrix of type Mt2 to which result gets written

Returns: void

Definition at line 2493 of file matrix.h.

◆ mult_chexp()

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

void mult_chexp	(	const Matrix_t< n, m, T, MT > &	mat,
		const Matrix_t< n, m, T, MT > &	mmat,
		Matrix_t< n, m, T, MT > &	omat,
		Matrix_t< n, m, T, MT > &	domat
	)

inline

Calculate exp(mat).dagger()*mmat*exp(mat) and trace(exp(mat).dagger*mmat*dexp(mat))

exp and dexp computation is done using iterative Cayley-Hamilton (arXiv:2404.07704) Calculate omat[i][j] = (exp(mat).dagger() * mmat * exp(mat))[i][j] and domat[i][j] = trace(exp(mat).dagger() * mmat * dexp(mat)/dmat[j][i]) for given matrices mat and mmat, using iterative Cayley-Hamilton method (arXiv:2404.07704) for computing exp(mat) and dexp(mat)/dmat[][]

Template Parameters

n	Number of rowsMa
T	Matrix element type
MT	Matrix type

Parameters

mat	Matrix to compute exponential for
mmat	Matrix to multiply with
omat	Matrix to which exp(mat).dagger()mmatexp(mat) gets stored
domat	matrix to which trace(exp(mat).dagger()mmatdexp(mat)) gets stored

Returns: void

Definition at line 3275 of file matrix.h.

◆ mult_chexpk_fast()

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

void mult_chexpk_fast	(	const Matrix_t< n, m, T, MT > &	mat,
		const Matrix_t< n, m, T, MT > &	texp,
		const Matrix_t< n, m, T, MT > &	kmats,
		const Matrix_t< n, m, T, MT > &	mmat,
		Matrix_t< n, m, T, MT > &	omat,
		Matrix_t< n, m, T, MT > &	domat
	)

inline

Calculate exp(mat).dagger()*mmat*exp(mat) and trace(exp(mat).dagger*mmat*dexp(mat)) from output of chexpk.

exp is given and dexp computation is done using decompisition matrix kmats_{i,j}, as computed by chexpk using iterative Cayley-Hamilton (arXiv:2404.07704) Calculate omat[i][j] = (exp(mat).dagger() * mmat * exp(mat))[i][j] and domat[i][j] = trace(exp(mat).dagger() * mmat * dexp(mat)/dmat[j][i]) for given matrices mat, exp(mat), kmats, and mmat

Template Parameters

n	Number of rowsMa
T	Matrix element type
MT	Matrix type

Parameters

mat	Matrix to compute exponential for
texp	Matrix containing exp(mat)
kmats	Matrix containing decomposition coefficients k_{i,j} of dexp(X)^a_b/dX^c_d = k_{i,j} (X^i)^d_b (X^j)^a_c
mmat	Matrix to multiply with
omat	Matrix to which exp(mat).dagger()mmatexp(mat) gets stored
domat	matrix to which trace(exp(mat).dagger()mmatdexp(mat)) gets stored

Returns: void

Definition at line 3604 of file matrix.h.

◆ mult_exp()

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

void mult_exp	(	const Matrix_t< n, m, T, MT > &	mat,
		const Matrix_t< n, m, T, MT > &	mmat,
		Matrix_t< n, m, T, MT > &	r,
		Matrix_t< n, m, T, MT > &	dr,
		const int	order = `20`
	)

inline

Calculate mmat*exp(mat) and trace(mmat*dexp(mat))

exp and dexp computation is done using factorized, truncated Taylor series method

Template Parameters

n	Number of rowsMa
T	Matrix element type
MT	Matrix type

Parameters

mat	Matrix to compute exponential for
mmat	Matrix to multiply with
r	Matrix to which mmat*exp(mat) gets stored
dr	matrix to which trace(mmat*dexp(mat)) gets stored
order	= 20 integer input parameter giving the Taylor series truncation order

Returns: void

Definition at line 2888 of file matrix.h.

◆ mult_sub() [1/3]

template<typename Mt1 , typename Mt2 , typename Mt3 , std::enable_if_t< Mt1::is_matrix() &&Mt2::is_matrix() &&Mt3::is_matrix(), int > = 0>

void mult_sub	(	const Mt1 &	a,
		const Mt2 &	b,
		Mt3 &	res
	)

inline

compute product of two matrices and subtract result from existing matrix

Template Parameters

Mt1,Mt2,Mt3 Matrix types

Parameters

a	Left Matrix of type Mt1
b	Right Matrix of type Mt2
c	Matrix of type Mt3 to which result gets written

Returns: void

Definition at line 2518 of file matrix.h.

◆ mult_sub() [2/3]

template<typename Mt1 , typename S , typename Mt2 , std::enable_if_t<(Mt1::is_matrix() &&Mt2::is_matrix() &&hila::is_complex_or_arithmetic< S >::value), int > = 0>

void mult_sub	(	const Mt1 &	a,
		const S &	b,
		Mt2 &	res
	)

inline

compute product of a matrix and a scalar and subtract result from existing matrix

Template Parameters

Mt1	Matrix type1, S Scalar type, Mt2 Matrix type2

Parameters

a	Left Matrix of type Mt1
b	Right Scalar of type S
c	Matrix of type Mt2 to which result gets written

Returns: void

Definition at line 2548 of file matrix.h.

◆ mult_sub() [3/3]

template<typename S , typename Mt1 , typename Mt2 , std::enable_if_t<(Mt1::is_matrix() &&Mt2::is_matrix() &&hila::is_complex_or_arithmetic< S >::value), int > = 0>

void mult_sub	(	const S &	a,
		const Mt1 &	b,
		Mt2 &	res
	)

inline

compute product of a scalar and a matrix and subtract result from existing matrix

Template Parameters

S	Scalar type, Mt1 Matrix type1, Mt2 Matrix type2

Parameters

a	Left Scalar of type S
b	Right Matrix of type Mt1
c	Matrix of type Mt2 to which result gets written

Returns: void

Definition at line 2574 of file matrix.h.

◆ norm()

template<typename Mt , std::enable_if_t< Mt::is_matrix(), int > = 0>

auto norm ( const Mt & rhs )

inline

Returns vector norm of Matrix.

Wrapper around Matrix::norm - sqrt of sum of squared elements

Template Parameters

Mt	Matrix type

Parameters

rhs	Matrix to compute norm of

Returns: auto

Definition at line 2743 of file matrix.h.

◆ operator*()

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

RowVector * Vector / Vector * RowVector:

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

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

Matrix * Scalar / 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 = M*2; // Resulting Matrix is uniformly 2

Template Parameters

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

Parameters

a	Left Matrix, Vector or Scalar
b	Right Matrix, Vector or Scalar

Returns: Matrix<n, m, R>

Definition at line 2187 of file matrix.h.

◆ operator+()

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.

Defined for the following operations

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

Scalar + Matrix / Matrix + Scalar:

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

NOTE: Exact definition exist in overloaded functions that can be viewed in source code.

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

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

Template Parameters

Mtype1	Matrix type for a
Mtype2	Matrix type for b

Parameters

a	Left matrix or scalar
b	Right matrix or scalar

Returns: Rtype Return matrix of compatible type between Mtype1 and Mtype2

Definition at line 1946 of file matrix.h.

◆ operator-()

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.

Defined for the following operations

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

Scalar - Matrix / Matrix - Scalar:

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

NOTE: Exact definition exist in overloaded functions that can be viewed in source code.

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

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

Template Parameters

Mtype1	Matrix type for a
Mtype2	Matrix type for b

Parameters

a	Left matrix or scalar
b	Right matrix or scalar

Returns: Rtype Return matrix of compatible type between Mtype1 and Mtype2

Definition at line 2022 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 2285 of file matrix.h.

◆ operator<<()

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

std::ostream & operator<<	(	std::ostream &	strm,
		const Matrix_t< n, m, T, MT > &	A
	)

Stream operator.

Naive Stream operator for formatting Matrix for printing. Simply puts elements one after the other in row major order

Template Parameters

n
m
T
MT

Parameters

strm
A

Returns: std::ostream&

Definition at line 2605 of file matrix.h.

◆ real()

template<typename Mtype , std::enable_if_t< Mtype::is_matrix(), int > = 0>

auto real ( const Mtype & arg )

inline

Return real of Matrix or Vector.

Wrapper around Matrix::real

Can be called as:

Matrix<n,m,MyType> M,M_real;

M_real = real(M);

real

Array< n, m, hila::arithmetic_type< T > > real(const Array< n, m, T > &arg)

Return real part of Array.

Definition array.h:688

Template Parameters

Mtype Matrix type

Parameters

arg	Object to compute real part of

Returns: auto Mtype real part of arg

Definition at line 1874 of file matrix.h.

◆ squarenorm()

template<typename Mt , std::enable_if_t< Mt::is_matrix(), int > = 0>

auto squarenorm ( const Mt & rhs )

inline

Returns square norm of Matrix.

Wrapper around Matrix::squarenorm - sum of squared elements

Can be called as:

Matrix<n,m,MyType> M;

auto a = squarenorm(M);

squarenorm

hila::arithmetic_type< T > squarenorm(const Array< n, m, T > &rhs)

Return square norm of Array.

Definition array.h:1018

Template Parameters

Mt	Matrix type

Parameters

rhs	Matrix to compute square norm of

Returns: auto

Definition at line 2729 of file matrix.h.

◆ trace()

template<typename Mtype , std::enable_if_t< Mtype::is_matrix(), int > = 0>

auto trace ( const Mtype & arg )

inline

Return trace of square Matrix.

Wrapper around Matrix::trace

Can be called as:

Matrix<n,m,MyType> M;
MyType T;
T = trace(M);

Template Parameters

Mtype Matrix type

Parameters

arg	Object to compute trace of

Returns: auto Trace of Matrix

Definition at line 1854 of file matrix.h.

◆ transpose()

template<typename Mtype , std::enable_if_t< Mtype::is_matrix(), int > = 0>

auto transpose ( const Mtype & arg )

inline

Return transposed Matrix or Vector.

Wrapper around Matrix::transpose

Can be called as:

Matrix<n,m,MyType> M,M_transposed;

M_transposed = transpose(M);

Template Parameters

Mtype Matrix type

Parameters

arg	Object to be transposed

Returns: auto Mtype transposed matrix

Definition at line 1757 of file matrix.h.

Classes

Namespaces

Typedefs

Functions

Detailed Description

Function Documentation

◆ abs()

◆ adjoint()

◆ altexp() [1/2]

◆ altexp() [2/2]

◆ chexp() [1/2]

◆ chexp() [2/2]

◆ chexpk()

◆ chsexp()

◆ conj()

◆ dagger()

◆ exp()

◆ imag()

◆ mul_trace()

◆ mult() [1/3]

◆ mult() [2/3]

◆ mult() [3/3]

◆ mult_aa()

◆ mult_add() [1/3]

◆ mult_add() [2/3]

◆ mult_add() [3/3]

◆ mult_chexp()

◆ mult_chexpk_fast()

◆ mult_exp()

◆ mult_sub() [1/3]

◆ mult_sub() [2/3]

◆ mult_sub() [3/3]

◆ norm()

◆ operator*()

◆ operator+()

◆ operator-()

◆ operator/()

◆ operator<<()

◆ real()

◆ squarenorm()

◆ trace()

◆ transpose()