Classes
class	Matrix< T_value >
	Standard dense matrix in basic linear algebra, i.e. BLAS/LAPACK. More...

class	TransposeView< T_matrix >
	Provide transposed view of a matrix. More...

class	AdjoinView< T_matrix >
	Provide adjoint view, e.g. conjugate transposed of a given matrix. More...

class	MatrixView< T_matrix >
	Provide generic view to a matrix, e.g. transposed or adjoint. More...

class	MatrixBase< T_derived >
	defines basic interface for matrices More...

class	MemBlock< T_value >
	Defines a reference countable memory block. More...

class	Range
	indexset with modified ctors More...

class	Vector< T_value >
	Standard vector in basic linear algebra, i.e. BLAS/LAPACK. More...

Functions
matop_t	conjugate (const matop_t op)
	return conjugate of given matrix operation More...

matop_t	transposed (const matop_t op)
	return transposed of given matrix operation More...

matop_t	adjoint (const matop_t op)
	return adjoint of given matrix operation More...

Vector Algebra
template<typename T1 , typename T2 >
enable_if< is_vector< T2 >::value &&is_same_type< T1, typename T2::value_t >::value >::result	fill (const T1 f, T2 &x)
	fill vector with constant

template<typename T1 >
enable_if< is_vector< T1 >::value >::result	conj (T1 &x)
	conjugate entries in vector

template<typename T1 , typename T2 >
enable_if< is_vector< T2 >::value &&is_same_type< T1, typename T2::value_t >::value >::result	scale (const T1 f, T2 &x)
	scale vector by constant

template<typename T1 , typename T2 >
enable_if< is_vector< T1 >::value &&is_vector< T2 >::value &&is_same_type< typename T1::value_t, typename T2::value_t >::value >::result	copy (const T1 &x, T2 &y)
	copy x into y

template<typename T1 , typename T2 >
enable_if< is_vector< T1 >::value &&is_vector< T2 >::value &&is_same_type< typename T1::value_t, typename T2::value_t >::value >::result	swap (T1 &x, T2 &y)
	exchange x and y

template<typename T1 >
enable_if_res< is_vector< T1 >::value, idx_t >::result	max_idx (const T1 &x)
	determine index with maximal absolute value in x

template<typename T1 >
enable_if_res< is_vector< T1 >::value, idx_t >::result	min_idx (const T1 &x)
	determine index with minimax absolute value in x

template<typename T1 , typename T2 , typename T3 >
enable_if< is_vector< T2 >::value &&is_vector< T3 >::value &&is_same_type< T1, typename T2::value_t >::value &&is_same_type< T1, typename T3::value_t >::value >::result	add (const T1 alpha, const T2 &x, T3 &y)
	compute y ≔ y + α·x

template<typename T1 , typename T2 >
enable_if_res< is_vector< T1 >::value &&is_vector< T2 >::value &&is_same_type< typename T1::value_t, typename T2::value_t >::value, typename T1::value_t >::result	dot (const T1 &x, const T2 &y)
	compute <x,y> = x^H · y

template<typename T1 , typename T2 >
enable_if_res< is_vector< T1 >::value &&is_vector< T2 >::value &&is_same_type< typename T1::value_t, typename T2::value_t >::value, typename T1::value_t >::result	dotu (const T1 &x, const T2 &y)
	compute <x,y> without conjugating x, e.g. x^T · y

template<typename T1 >
enable_if_res< is_vector< T1 >::value, typename real_type< typename T1::value_t >::type_t >::result	norm2 (const T1 &x)
	compute ∥x∥₂ More...

template<typename T1 >
bool	abs_lt (const T1 a1, const T1 a2)

template<typename T1 , typename T2 >
enable_if_res< is_vector< T1 >::value &&is_vector< T2 >::value &&is_same_type< typename T1::value_t, typename T2::value_t >::value, typename T1::value_t >::result	stable_dotu (const T1 &x, const T2 &y)
	compute dot product x · y numerically stable More...

template<typename T1 >
enable_if_res< is_vector< T1 >::value, typename T1::value_t >::result	stable_sum (const T1 &x)
	compute sum of elements in x numerically stable More...

Basic Matrix Algebra
template<typename T1 >
void	fill_rand (Matrix< T1 > &M)
	set M to f entrywise

template<typename T1 >
enable_if< is_matrix< T1 >::value >::result	transpose (T1 &A)
	transpose matrix A: A → A^T ASSUMPTION: A is square matrix

template<typename T1 >
enable_if< is_matrix< T1 >::value >::result	conj_transpose (T1 &A)
	conjugate transpose matrix A: A → A^H ASSUMPTION: A is square matrix

template<typename T1 >
enable_if< is_matrix< T1 >::value >::result	max_idx (const T1 &M, idx_t &row, idx_t &col)
	determine index (i,j) with maximal absolute value in M and return in row and col

template<typename T1 , typename T2 , typename T3 , typename T4 >
enable_if< is_vector< T2 >::value &&is_vector< T3 >::value &&is_matrix< T4 >::value &&is_same_type< T1, typename T2::value_t >::value &&is_same_type< T1, typename T3::value_t >::value &&is_same_type< T1, typename T4::value_t >::value >::result	add_r1 (const T1 alpha, const T2 &x, const T3 &y, T4 &A)
	compute A ≔ A + α·x·y^H

template<typename T1 , typename T2 , typename T3 , typename T4 >
enable_if< is_vector< T2 >::value &&is_vector< T3 >::value &&is_matrix< T4 >::value &&is_same_type< T1, typename T2::value_t >::value &&is_same_type< T1, typename T3::value_t >::value &&is_same_type< T1, typename T4::value_t >::value >::result	add_r1u (const T1 alpha, const T2 &x, const T3 &y, T4 &A)
	compute A ≔ A + α·x·y^T

template<typename T1 , typename T2 , typename T3 , typename T4 >
enable_if< is_matrix< T2 >::value &&is_vector< T3 >::value &&is_vector< T4 >::value &&is_same_type< T1, typename T2::value_t >::value &&is_same_type< T1, typename T3::value_t >::value &&is_same_type< T1, typename T4::value_t >::value >::result	mulvec (const T1 alpha, const T2 &A, const T3 &x, const T1 beta, T4 &y)
	compute y ≔ β·y + α·A·x

template<typename T1 , typename T2 , typename T3 >
enable_if_res< is_matrix< T2 >::value &&is_vector< T3 >::value &&is_same_type< T1, typename T2::value_t >::value &&is_same_type< T1, typename T3::value_t >::value, Vector< typename T2::value_t > >::result	mulvec (const T1 alpha, const T2 &A, const T3 &x)
	compute y ≔ α·A·x

template<typename T1 , typename T2 >
enable_if< is_matrix< T1 >::value &&is_vector< T2 >::value &&is_same_type< typename T1::value_t, typename T2::value_t >::value >::result	mulvec_tri (const tri_type_t shape, const diag_type_t diag, const T1 &A, T2 &x)
	compute x ≔ M·x, where M is upper or lower triangular with unit or non-unit diagonal

template<typename T1 , typename T2 >
enable_if< is_matrix< T1 >::value &&is_vector< T2 >::value &&is_same_type< typename T1::value_t, typename T2::value_t >::value >::result	solve (T1 &A, T2 &b)
	solve A·x = b with known A and b; x overwrites b (A is overwritten upon exit!)

template<typename T1 , typename T2 , typename T3 , typename T4 >
enable_if< is_matrix< T2 >::value &&is_matrix< T3 >::value &&is_matrix< T4 >::value &&is_same_type< T1, typename T2::value_t >::value &&is_same_type< T1, typename T3::value_t >::value &&is_same_type< T1, typename T4::value_t >::value >::result	prod (const T1 alpha, const T2 &A, const T3 &B, const T1 beta, T4 &C)
	compute C ≔ β·C + α·A·B

template<typename T1 , typename T2 , typename T3 >
enable_if_res< is_matrix< T2 >::value &&is_matrix< T3 >::value &&is_same_type< T1, typename T2::value_t >::value &&is_same_type< T1, typename T3::value_t >::value, Matrix< typename T2::value_t > >::result	prod (const T1 alpha, const T2 &A, const T3 &B)
	compute C ≔ α·A·B

template<typename T1 , typename T2 >
enable_if< is_matrix< T1 >::value &&is_matrix< T2 >::value &&is_same_type< typename T1::value_t, typename T2::value_t >::value >::result	hadamard_prod (const T1 &A, T2 &B)
	compute B ≔ A⊙B, e.g. Hadamard product

template<typename T1 , typename T2 , typename T3 >
enable_if< is_matrix< T2 >::value &&is_matrix< T3 >::value &&is_same_type< T1, typename T2::value_t >::value &&is_same_type< T1, typename T3::value_t >::value >::result	prod_tri (const eval_side_t side, const tri_type_t uplo, const diag_type_t diag, const T1 alpha, const T2 &A, T3 &B)
	compute B ≔ α·A·B or B ≔ α·B·A with triangular matrix A

template<typename T1 , typename T2 >
enable_if< is_matrix< T1 >::value &&is_vector< T2 >::value &&is_same_type< typename real_type< typename T1::value_t >::type_t, typename real_type< typename T2::value_t >::type_t >::value >::result	prod_diag (T1 &M, const T2 &D, const idx_t k)
	multiply k columns of M with diagonal matrix D, e.g. compute M ≔ M·D

template<typename T1 >
enable_if_res< is_matrix< T1 >::value, typename real_type< typename T1::value_t >::type_t >::result	normF (const T1 &M)
	return Frobenius norm of M

template<typename T1 , typename T2 >
enable_if_res< is_matrix< T1 >::value &&is_matrix< T2 >::value &&is_same_type< typename T1::value_t, typename T2::value_t >::value, typename real_type< typename T1::value_t >::type_t >::result	diff_normF (const T1 &A, const T2 &B)
	compute Frobenius norm of A-B

template<typename T1 , typename T2 >
enable_if_res< is_matrix< T1 >::value &&is_matrix< T2 >::value &&is_same_type< typename T1::value_t, typename T2::value_t >::value, typename real_type< typename T1::value_t >::type_t >::result	lr_normF (const T1 &A, const T2 &B)
	compute Frobenius norm of M=A-B^H

template<typename T1 >
enable_if_res< is_matrix< T1 >::value, typename real_type< typename T1::value_t >::type_t >::result	cond (const T1 &M)
	return condition of M More...

template<typename T1 >
enable_if< is_matrix< T1 >::value >::result	make_symmetric (T1 &A)
	make given matrix symmetric, e.g. copy lower left part to upper right part

template<typename T1 >
enable_if< is_matrix< T1 >::value >::result	make_hermitian (T1 &A)
	make given matrix hermitian, e.g. copy conjugated lower left part to upper right part and make diagonal real

Advanced Matrix Algebra
template<typename T1 , typename T2 >
enable_if< is_matrix< T1 >::value &&is_vector< T2 >::value &&is_same_type< typename T1::value_t, typename T2::value_t >::value >::result	solve_tri (const tri_type_t uplo, const diag_type_t diag, const T1 &A, T2 &b)
	solve A·x = b with known A and b; x overwrites b

template<typename T1 , typename T2 , typename T3 >
enable_if< is_matrix< T2 >::value &&is_matrix< T3 >::value &&is_same_type< T1, typename T2::value_t >::value &&is_same_type< T1, typename T3::value_t >::value >::result	solve_tri (const eval_side_t side, const tri_type_t uplo, const diag_type_t diag, const T1 alpha, const T2 &A, T3 &B)
	solve A·X = α·B or X·A· = α·B with known A and B; X overwrites B

template<typename T1 >
enable_if< is_matrix< T1 >::value >::result	invert (T1 &A)
	invert matrix A; A will be overwritten with A^-1 upon exit

template<typename T1 >
enable_if< is_matrix< T1 >::value >::result	invert (T1 &A, const tri_type_t tri_type, const diag_type_t diag_type)
	invert lower or upper triangular matrix A with unit or non-unit diagonal; A will be overwritten with A^-1 upon exit

template<typename T >
void	pseudo_invert (Matrix< T > &A, const TTruncAcc &acc)
	compute pseudo inverse of matrix A with precision acc More...

template<typename T1 >
enable_if< is_matrix< T1 >::value >::result	lu (T1 &A)
	compute LU factorisation of the n×m matrix A with n×min(n,m) unit diagonal lower triangular matrix L and min(n,m)xm upper triangular matrix U; A will be overwritten with L and U upon exit

template<typename T1 >
enable_if< is_matrix< T1 >::value >::result	llt (T1 &A)
	compute L·L^T factorisation of given symmetric n×n matrix A

template<typename T1 >
enable_if< is_matrix< T1 >::value >::result	llh (T1 &A)
	compute L·L^H factorisation of given hermitian n×n matrix A

template<typename T1 >
enable_if< is_matrix< T1 >::value >::result	ldlt (T1 &A)
	compute L·D·L^T factorisation of given symmetric n×n matrix A

template<typename T1 >
enable_if< is_matrix< T1 >::value >::result	ldlh (T1 &A)
	compute L·D·L^H factorisation of given hermitian n×n matrix A

template<typename T1 >
enable_if< is_matrix< T1 >::value >::result	qr (T1 &A, Matrix< typename T1::value_t > &R)
	Compute QR factorisation of the matrix A. More...

template<typename T1 >
enable_if< is_matrix< T1 >::value >::result	qrp (T1 &A, Matrix< typename T1::value_t > &R, std::vector< blas_int_t > &P)
	Compute QR factorisation with column pivoting of the matrix A. More...

template<typename T1 >
enable_if< is_matrix< T1 >::value >::result	eigen (T1 &M, Vector< typename T1::value_t > &eig_val, Matrix< typename T1::value_t > &eig_vec)
	compute eigenvalues and eigenvectors of matrix M

template<typename T1 >
enable_if< is_matrix< T1 >::value >::result	eigen (T1 &M, const Range &eig_range, Vector< typename T1::value_t > &eig_val, Matrix< typename T1::value_t > &eig_vec)
	compute selected (by eig_range) eigenvalues and eigenvectors of the symmetric matrix M More...

template<typename T1 , typename T2 >
enable_if< is_vector< T1 >::value &&is_vector< T2 >::value &&is_same_type< typename T1::value_t, typename T2::value_t >::value >::result	eigen (T1 &diag, T2 &subdiag, Vector< typename T1::value_t > &eig_val, Matrix< typename T1::value_t > &eig_vec)
	compute eigenvalues and eigenvectors of the symmetric, tridiagonal matrix defines by diagonal coefficients in diag and off-diagonal coefficients subdiag

template<typename T1 >
enable_if< is_matrix< T1 >::value >::result	svd (T1 &A, Vector< typename real_type< typename T1::value_t >::type_t > &S, Matrix< typename T1::value_t > &V)
	compute SVD decomposition \( A = U·S·V^H \) of the nxm matrix A with n×min(n,m) matrix U, min(n,m)×min(n,m) matrix S (diagonal) and m×min(n,m) matrix V; A will be overwritten with U upon exit

template<typename T1 >
enable_if< is_matrix< T1 >::value >::result	svd (T1 &A, Vector< typename real_type< typename T1::value_t >::type_t > &S, const bool left=true)
	compute SVD decomposition \( A = U·S·V^H \) of the nxm matrix A but return only the left/right singular vectors and the singular values S ∈ ℝ^min(n,m); upon exit, A will be contain the corresponding sing. vectors

template<typename T1 >
enable_if< is_matrix< T1 >::value >::result	sv (T1 &A, Vector< typename real_type< typename T1::value_t >::type_t > &S)
	compute SVD decomposition \( A = U·S·V^H \) of the nxm matrix A but return only the singular values S ∈ ℝ^min(n,m); A will be overwritten with U upon exit

template<typename T1 , typename T2 >
enable_if< is_matrix< T1 >::value &&is_matrix< T2 >::value &&is_same_type< typename T1::value_t, typename T2::value_t >::value >::result	sv (T1 &A, T2 &B, Vector< typename real_type< typename T1::value_t >::type_t > &S)
	compute SVD decomposition \( M = A·B^H = U·S·V^H \) of the nxm low-rank matrix M but return only the singular values S ∈ ℝ^min(n,m); A and B will be overwritten upon exit

template<typename T1 >
enable_if_res< is_matrix< T1 >::value, size_t >::result	approx (T1 &M, const TTruncAcc &acc, Matrix< typename T1::value_t > &A, Matrix< typename T1::value_t > &B)
	approximate given dense matrix M by low rank matrix according to accuracy acc. The low rank matrix will be stored in A and B

template<typename T1 >
enable_if_res< is_matrix< T1 >::value, size_t >::result	approx_svd (T1 &M, const TTruncAcc &acc, Matrix< typename T1::value_t > &A, Matrix< typename T1::value_t > &B)
	approximate given dense matrix M by low rank matrix according to accuracy acc using SVD. The low rank matrix will be stored in A and B

template<typename T1 >
enable_if_res< is_matrix< T1 >::value, size_t >::result	approx_rrqr (T1 &M, const TTruncAcc &acc, Matrix< typename T1::value_t > &A, Matrix< typename T1::value_t > &B)
	approximate given dense matrix M by low rank matrix according to accuracy acc using RRQR. The low rank matrix will be stored in A and B.

template<typename T1 >
enable_if_res< is_matrix< T1 >::value, size_t >::result	approx_rand (T1 &M, const TTruncAcc &acc, Matrix< typename T1::value_t > &A, Matrix< typename T1::value_t > &B, const uint power_steps=CFG::BLAS::power_steps, const size_t oversampling=CFG::BLAS::oversampling)
	approximate given dense matrix M by low rank matrix according to accuracy acc using randomized SVD. The low rank matrix will be stored in A and B.

template<typename T >
size_t	truncate (Matrix< T > &A, Matrix< T > &B, const TTruncAcc &acc)
	truncate A · B^H based on accuracy acc. More...

template<typename T >
size_t	truncate_svd (Matrix< T > &A, Matrix< T > &B, const TTruncAcc &acc)
	truncate A · B^H based on accuracy acc using SVD.

template<typename T >
size_t	truncate_rrqr (Matrix< T > &A, Matrix< T > &B, const TTruncAcc &acc)
	truncate A · B^H based on accuracy acc using RRQR.

template<typename T >
size_t	truncate_rand (Matrix< T > &A, Matrix< T > &B, const TTruncAcc &acc)
	truncate A · B^H based on accuracy acc using randomized SVD.

template<typename T1 >
enable_if< is_matrix< T1 >::value >::result	factorise_ortho (T1 &A, Matrix< typename T1::value_t > &R)
	construct factorisation A = Q·R of A, with orthonormal Q More...

template<typename T >
void	factorise_ortho (Matrix< T > &A, Matrix< T > &R, const TTruncAcc &acc)
	construct approximate factorisation A = Q·R of A, with orthonormal Q More...

void	print_statistics ()
	print statistics for Algebra functions

void	reset_statistics ()
	reset statistics for Algebra functions

Matrix Modifiers
Classes for matrix modifiers, e.g. transposed and adjoint view.
template<typename T >
TransposeView< T >	transposed (const T &M)
	return transposed view object for matrix More...

template<typename T >
AdjoinView< T >	adjoint (const T &M)
	return adjoint view object for matrix More...

template<typename T >
MatrixView< T >	mat_view (const matop_t op, const T &M)
	convert matop_t into view object More...

Detailed Description

This modules provides most low level algebra functions, e.g. vector dot products, matrix multiplication, factorisation and singular value decomposition. See also BLAS/LAPACK Interface for an introduction.

To include all BLAS algebra functions and classes add

#include <hlib-blas.hh>

to your source files.

Function Documentation

◆ adjoint() [1/2]

matop_t HLIB::BLAS::adjoint ( const matop_t op )

inline

Parameters

op	matrix op. to be adjoint

◆ adjoint() [2/2]

AdjoinView< T > HLIB::BLAS::adjoint ( const T & M )

Parameters

M	matrix to conjugate transpose

◆ cond()

enable_if_res< is_matrix< T1 >::value, typename real_type< typename T1::value_t >::type_t >::result HLIB::BLAS::cond ( const T1 & M )

if M ≡ 0 or size(M) = 0, 0 is returned

◆ conjugate()

matop_t HLIB::BLAS::conjugate ( const matop_t op )

inline

Parameters

op	matrix op. to be conjugated

◆ eigen()

enable_if< is_matrix< T1 >::value >::result HLIB::BLAS::eigen	(	T1 &	M,
		const Range &	eig_range,
		Vector< typename T1::value_t > &	eig_val,
		Matrix< typename T1::value_t > &	eig_vec
	)

the lower half of M is accessed

◆ factorise_ortho() [1/2]

enable_if< is_matrix< T1 >::value >::result HLIB::BLAS::factorise_ortho	(	T1 &	A,
		Matrix< typename T1::value_t > &	R
	)

A is overwritten with Q upon exit

◆ factorise_ortho() [2/2]

void HLIB::BLAS::factorise_ortho	(	Matrix< T > &	A,
		Matrix< T > &	R,
		const TTruncAcc &	acc
	)

approximation quality is definded by acc
A is overwritten with Q upon exit

◆ mat_view()

MatrixView< T > HLIB::BLAS::mat_view	(	const matop_t	op,
		const T &	M
	)

Parameters

op	matop_t value
M	matrix to create view for

◆ norm2()

enable_if_res< is_vector< T1 >::value, typename real_type< typename T1::value_t >::type_t >::result HLIB::BLAS::norm2 ( const T1 & x )

return spectral norm of M

◆ pseudo_invert()

void HLIB::BLAS::pseudo_invert	(	Matrix< T > &	A,
		const TTruncAcc &	acc
	)

Compute pseudo inverse B of matrix A up to precision acc, e.g. \(\|A-B\|\le \epsilon\) with \(\epsilon\) defined by acc.

◆ qr()

enable_if< is_matrix< T1 >::value >::result HLIB::BLAS::qr	(	T1 &	A,
		Matrix< typename T1::value_t > &	R
	)

Compute QR factorisation of the n×m matrix A with n×m matrix Q and mxm matrix R (n >= m); A will be overwritten with Q upon exit

◆ qrp()

enable_if< is_matrix< T1 >::value >::result HLIB::BLAS::qrp	(	T1 &	A,
		Matrix< typename T1::value_t > &	R,
		std::vector< blas_int_t > &	P
	)

Compute QR factorisation with column pivoting of the n×m matrix A, i.e., \( AP = QR \), with n×m matrix Q and mxm matrix R (n >= m). P_i = j means, column i of AP was column j of A; A will be overwritten with Q upon exit

◆ stable_dotu()

enable_if_res< is_vector< T1 >::value && is_vector< T2 >::value && is_same_type< typename T1::value_t, typename T2::value_t >::value, typename T1::value_t >::result HLIB::BLAS::stable_dotu	(	const T1 &	x,
		const T2 &	y
	)

Parameters

x	first argument of dot product
y	second argument of dot product

◆ stable_sum()

enable_if_res< is_vector< T1 >::value, typename T1::value_t >::result HLIB::BLAS::stable_sum ( const T1 & x )

Parameters

x	vector holding coefficients to sum up

◆ transposed() [1/2]

matop_t HLIB::BLAS::transposed ( const matop_t op )

inline

Parameters

op	matrix op. to be transposed

◆ transposed() [2/2]

TransposeView< T > HLIB::BLAS::transposed ( const T & M )

Parameters

M	matrix to transpose

◆ truncate()

size_t HLIB::BLAS::truncate	(	Matrix< T > &	A,
		Matrix< T > &	B,
		const TTruncAcc &	acc
	)

Truncate given A · B^H low rank matrix matrix (A being n×k and B being m×k) with respect to given accuracy acc; store truncated matrix in A(:,0:k-1) and B(:,0:k-1) where k is the returned new rank after truncation

Classes

Functions

Vector Algebra

Basic Matrix Algebra

Advanced Matrix Algebra

Matrix Modifiers

Detailed Description

Function Documentation

◆ adjoint() [1/2]

◆ adjoint() [2/2]

◆ cond()

◆ conjugate()

◆ eigen()

◆ factorise_ortho() [1/2]

◆ factorise_ortho() [2/2]

◆ mat_view()

◆ norm2()

◆ pseudo_invert()

◆ qr()

◆ qrp()

◆ stable_dotu()

◆ stable_sum()

◆ transposed() [1/2]

◆ transposed() [2/2]

◆ truncate()