matrix.h File Reference

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

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

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

Functions
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	adjoint (const Mtype &arg)
	Return adjoint 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 , 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<typename Ntype , typename T , int n, int m, std::enable_if_t< hila::is_arithmetic_or_extended< T >::value, int > = 0>
Matrix< n, m, Ntype >	hila::cast_to (const Matrix< n, m, T > &mat)
	Change basic number type of Matrix/Vector.
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 , 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 >
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 >
Matrix_t< n, m, T, MT >	chsexp (const Matrix_t< n, m, T, MT > &mat)
	Calculate exp of a square matrix.
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	dagger (const Mtype &arg)
	Return dagger of Matrix.
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>
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 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_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 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<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()*mmat*exp(mat) and trace(exp(mat).dagger*mmat*dexp(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()*mmat*exp(mat) and trace(exp(mat).dagger*mmat*dexp(mat)) from output of chexpk.
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 mmat*exp(mat) and trace(mmat*dexp(mat))
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<typename Mt , std::enable_if_t< Mt::is_matrix(), int > = 0>
auto	norm (const Mt &rhs)
	Returns vector norm of Matrix.
template<typename Mt , std::enable_if_t< Mt::is_matrix() &&Mt::rows()==Mt::columns(), int > = 0>
Mt	operator* (const Mt &a, const Mt &b)
	Multiplication operator Square Matrix * Square Matrix.
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+ (Mtype1 a, const Mtype2 &b)
	Real micro-optimization Matrix + Matrix - no extra creation of variable and copy.
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 Mtype , std::enable_if_t< Mtype::is_matrix(), int > = 0>
auto	real (const Mtype &arg)
	Return real of Matrix or Vector.
template<typename Mt , std::enable_if_t< Mt::is_matrix(), int > = 0>
auto	squarenorm (const Mt &rhs)
	Returns square norm of Matrix.
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<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	transpose (const Mtype &arg)
	Return transposed Matrix or Vector.

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);

Template Parameters

Mtype

Matrix type

Parameters

arg	Object to compute absolute value of

Returns

auto Mtype absolute value Matrix or Vector

Definition at line 2018 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 1982 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 3135 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 3102 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 3403 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 3215 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 3851 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 4123 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);

Template Parameters

Mtype

Matrix type

Parameters

arg	Object to be conjugated

Returns

auto Mtype conjugated matrix

Definition at line 1962 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 1998 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);

Template Parameters

Mtype

Matrix type

Parameters

arg	Object to compute imag part of

Returns

auto Mtype imag part of arg

Definition at line 2079 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)

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 2586 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 2604 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 2664 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 2690 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 2633 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 2714 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 2744 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 2770 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()*mmat*exp(mat) gets stored
domat	matrix to which trace(exp(mat).dagger()*mmat*dexp(mat)) gets stored

Returns

void

Definition at line 3629 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()*mmat*exp(mat) gets stored
domat	matrix to which trace(exp(mat).dagger()*mmat*dexp(mat)) gets stored

Returns

void

Definition at line 4046 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 3170 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 2795 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 2825 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 2851 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 3047 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);

Template Parameters

Mtype

Matrix type

Parameters

arg	Object to compute real part of

Returns

auto Mtype real part of arg

Definition at line 2059 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);

Template Parameters

Mt	Matrix type

Parameters

rhs	Matrix to compute square norm of

Returns

auto

Definition at line 3033 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 2039 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 1942 of file matrix.h.