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 >
	Defines a reference countable memory block. More...

class	NullMemBlock< T >
	special version of a memory block for NULL pointers More...

class	MemBlockRef< T >
	Pointer to handle delete operations for memory blocks like refptr. More...

class	Range
	defines an indexset [ first, last ] with stepwidth stride More...

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

Vector Algebra
template<typename T >
void	fill (const T f, Vector< T > &x)
	fill vector with constant

template<typename T >
void	conj (Vector< T > &x)
	conjugate entries in vector

template<typename T >
void	scale (const T f, Vector< T > &x)
	scale vector by constant

template<typename T >
void	copy (const Vector< T > &x, Vector< T > &y)
	copy x into y

template<typename T >
void	swap (Vector< T > &x, Vector< T > &y)
	exchange x and y

template<typename T >
idx_t	max_idx (const Vector< T > &x)
	determine index with maximal absolute value in x

template<typename T >
idx_t	min_idx (const Vector< T > &x)
	determine index with minimax absolute value in x

template<typename T >
void	add (const T alpha, const Vector< T > &x, Vector< T > &y)
	compute y ≔ y + α·x

template<typename T >
T	dot (const Vector< T > &x, const Vector< T > &y)
	compute <x,y> = x^H · y

template<typename T >
T	dotu (const Vector< T > &x, const Vector< T > &y)
	compute <x,y> without conjugating x, e.g. x^T · y

template<typename T >
num_traits< T >::real_t	norm2 (const Vector< T > &x)
	compute ∥x∥₂

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

template<typename T >
T	stable_dotu (const Vector< T > &x, const Vector< T > &y)
	compute dot product x · y numerically stable More...

template<typename T >
T	stable_sum (const Vector< T > &x)
	compute sum of elements in x numerically stable More...

Basic Matrix Algebra
template<typename T >
void	fill (const T f, Matrix< T > &M)
	set M to f entrywise

template<typename T >
void	conj (Matrix< T > &M)
	conjugate entries in vector

template<typename T >
void	scale (const T f, Matrix< T > &M)
	compute M ≔ f · M

template<typename T >
void	copy (const Matrix< T > &A, Matrix< T > &B)
	copy A to B

template<typename T_value , typename T_derived >
void	copy (const MatrixBase< T_derived > &A, Matrix< T_value > &B)
	copy A to B

template<typename T >
void	transpose (Matrix< T > &A)
	transpose matrix A: A → A^T ASSUMPTION: A is square matrix

template<typename T >
void	conj_transpose (Matrix< T > &A)
	conjugate transpose matrix A: A → A^H ASSUMPTION: A is square matrix

template<typename T >
void	max_idx (const Matrix< T > &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 T >
void	add (const T f, const Matrix< T > &A, Matrix< T > &B)
	compute B = B + f A

template<typename T >
void	add_r1 (const T alpha, const Vector< T > &x, const Vector< T > &y, Matrix< T > &A)
	compute A ≔ A + α·x·y^H

template<typename T_value , typename T_derived >
void	mulvec (const T_value alpha, const MatrixBase< T_derived > &A, const Vector< T_value > &x, const T_value beta, Vector< T_value > &y)
	compute y ≔ β·y + α·A·x

template<typename T_value , typename T_derived >
void	mulvec_tri (const tri_type_t shape, const diag_type_t diag, const MatrixBase< T_derived > &A, Vector< T_value > &x)
	compute x ≔ M·x, where M is upper or lower triangular with unit or non-unit diagonal

template<typename T >
void	solve (Matrix< T > &A, Vector< T > &b)
	solve A·x = b with known A and b; x overwrites b (A is overwritten upon exit!)

template<typename T_value , typename T_derived_A , typename T_derived_B >
void	prod (const T_value alpha, const MatrixBase< T_derived_A > &A, const MatrixBase< T_derived_B > &B, const T_value beta, Matrix< T_value > &C)
	compute C ≔ β·C + α·A·B

template<typename T_derived_A , typename T_derived_B >
void	hadamard_prod (const MatrixBase< T_derived_A > &A, MatrixBase< T_derived_B > &B)
	compute B ≔ A⊙B, e.g. Hadamard product

template<typename T_value , typename T_derived >
void	prod_tri (const eval_side_t side, const tri_type_t uplo, const diag_type_t diag, const T_value alpha, const MatrixBase< T_derived > &A, Matrix< T_value > &B)
	compute B ≔ α·A·B or B ≔ α·B·A

template<typename T >
void	prod_diag (Matrix< T > &M, const Vector< typename num_traits< T >::real_t > &D, const idx_t k)
	multiply k columns of M with diagonal matrix D, e.g. compute M ≔ M·D

template<typename T >
num_traits< T >::real_t	norm2 (const Matrix< T > &M)
	return spectral norm of M

template<typename T >
num_traits< T >::real_t	normF (const Matrix< T > &M)
	return Frobenius norm of M

template<typename T >
num_traits< T >::real_t	normF (const Matrix< T > &A, const Matrix< T > &B)
	compute Frobenius norm of A-B

template<typename T >
num_traits< T >::real_t	cond (const Matrix< T > &M)
	return condition of M More...

template<typename T >
void	make_symmetric (Matrix< T > &A)
	make given matrix symmetric, e.g. copy lower left part to upper right part

template<typename T >
void	make_hermitian (Matrix< T > &A)
	make given matrix hermitian, e.g. copy conjugated lower left part to upper right part and make diagonal real

template<typename T_value , typename T_derived >
void	add (const T_value f, const MatrixBase< T_derived > &A, Matrix< T_value > &B)

Advanced Matrix Algebra
template<typename T >
void	solve (Matrix< T > &A, Matrix< T > &B)
	solve A·X = B with known A and B; X overwrites B

template<typename T_value , typename T_derived >
void	solve_tri (const tri_type_t uplo, const diag_type_t diag, const MatrixBase< T_derived > &A, Vector< T_value > &b)
	solve A·x = b with known A and b; x overwrites b

template<typename T_value , typename T_derived >
void	solve_tri (const eval_side_t side, const tri_type_t uplo, const diag_type_t diag, const T_value alpha, const MatrixBase< T_derived > &A, Matrix< T_value > &B)
	solve A·X = α·B or X·A· = α·B with known A and B; X overwrites B

template<typename T >
void	invert (Matrix< T > &A)
	invert matrix A; A will be overwritten with A^-1 upon exit

template<typename T >
void	invert (Matrix< T > &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 T >
void	lu (Matrix< T > &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 T >
void	llt (Matrix< T > &A)
	compute L·L^T factorisation of given symmetric n×n matrix A

template<typename T >
void	llh (Matrix< T > &A)
	compute L·L^H factorisation of given hermitian n×n matrix A

template<typename T >
void	ldlt (Matrix< T > &A)
	compute L·D·L^T factorisation of given symmetric n×n matrix A

template<typename T >
void	ldlh (Matrix< T > &A)
	compute L·D·L^H factorisation of given hermitian n×n matrix A

template<typename T >
void	qr (Matrix< T > &A, Matrix< 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

template<typename T >
void	eigen (Matrix< T > &M, Vector< T > &eig_val, Matrix< T > &eig_vec)
	compute eigenvalues and eigenvectors of matrix M

template<typename T >
void	eigen (Matrix< T > &M, const Range &eig_range, Vector< T > &eig_val, Matrix< T > &eig_vec)
	compute selected (by eig_range) eigenvalues and eigenvectors of the symmetric matrix M More...

template<typename T >
void	eigen (Vector< T > &diag, Vector< T > &subdiag, Vector< T > &eig_val, Matrix< 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 T >
void	svd (Matrix< T > &A, Vector< typename num_traits< T >::real_t > &S, Matrix< T > &V)
	compute SVD decomposition 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 T >
void	svd (Matrix< T > &A, Vector< typename num_traits< T >::real_t > &S, const bool left=true)
	compute SVD decomposition 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 T >
void	sv (Matrix< T > &A, Vector< typename num_traits< T >::real_t > &S)
	compute SVD decomposition 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 T >
void	sv (Matrix< T > &A, Matrix< T > &B, Vector< typename num_traits< T >::real_t > &S)
	compute SVD decomposition 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 T >
size_t	approx (Matrix< T > &M, const TTruncAcc &acc, Matrix< T > &A, Matrix< 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 T >
size_t	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

template<typename T >
void	factorise_ortho (Matrix< T > &A, Matrix< 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...

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

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

template<typename T_matrix >
MatrixView< T_matrix >	mat_view (const matop_t op, const T_matrix &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

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

Parameters

M	matrix to conjugate transpose

num_traits< T >::real_t HLIB::BLAS::cond ( const Matrix< T > & M)

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

void HLIB::BLAS::eigen	(	Matrix< T > &	M,
		const Range &	eig_range,
		Vector< T > &	eig_val,
		Matrix< T > &	eig_vec
	)

the lower half of M is accessed

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

A is overwritten with Q upon exit

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

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

Parameters

op	matop_t value
M	matrix to create view for

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.

T HLIB::BLAS::stable_dotu	(	const Vector< T > &	x,
		const Vector< T > &	y
	)

Parameters

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

T HLIB::BLAS::stable_sum ( const Vector< T > & x)

Parameters

x	vector holding coefficients to sum up

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

Parameters

M	matrix to transpose

Classes

Vector Algebra

Basic Matrix Algebra

Advanced Matrix Algebra

Matrix Modifiers

Detailed Description

Function Documentation