Classes
struct	fac_options_t
	options for matrix factorisations More...

class	TLU
	Computes LU factorisation . More...

class	TLDU
	Computes LDU factorisation . More...

class	TLL
	computes Cholesky factorisation or More...

class	TLDL
	computes LDL factorisation or More...

struct	inv_options_t
	options for matrix inversion More...

struct	solve_option_t
	options for how to solve with given matrix More...

struct	eval_option_t
	options for how to evaluate given matrix More...

class	TLowRankApx
	base class for all low rank approximation techniques More...

class	TZeroLRApx
	Approximate all low-rank blocks by zero, e.g. for nearfield only. More...

class	TSVDLRApx< T >
	Uses exact SVD to compute low rank approximation (WARNING: O(n³) complexity) More...

class	TACA< T >
	Defines interface for all ACA algorithms and implements classical ACA. More...

class	TACAPlus< T >
	Implements ACA+, which corrects some of the deficits of the original ACA algorithm. More...

class	TACAFull< T >
	ACA with full pivot search (complexity: O(n²)) More...

class	THCA< T >
	uses hybrid cross approximation (HCA) for computing low rank approximation More...

class	TPermHCAGeneratorFn< T_val >
	base class for HCA generator functions using row/column permutations More...

Functions
void	eval_diag (const TMatrix A, TVector v, const matop_t op, const eval_option_t &options)
	evaluate D·x=y with (block) diagonal D More...

void	eval (const TMatrix A, TVector v, const matop_t op, const eval_option_t &options)
	evaluate L·U·x=y with lower triangular L and upper triangular U More...

void	eval_lower (const TMatrix A, TVector v, const matop_t op, const eval_option_t &options)
	evaluate A·x=y with lower triangular A More...

void	eval_upper (const TMatrix A, TVector v, const matop_t op, const eval_option_t &options)
	evaluate A·x = y with upper triangular A More...

void	invert (TMatrix *A, const TTruncAcc &acc, const inv_options_t &opts=inv_options_t())
	compute $A^{-1}$ More...

void	invert_ll (TMatrix *A, const TTruncAcc &acc, const inv_options_t &opts=inv_options_t())
	compute $A^{-1}$ More...

void	invert_ll (TMatrix L, TMatrix X, const TTruncAcc &acc, const inv_options_t &opts=inv_options_t())
	compute $X = L^{-1}$ More...

void	invert_ur (TMatrix *A, const TTruncAcc &acc, const inv_options_t &opts=inv_options_t())
	compute $A^{-1}$ More...

void	invert_diag (TMatrix *A, const TTruncAcc &acc, const inv_options_t &opts=inv_options_t())
	compute $A^{-1}$ More...

TVector *	inverse_diag (TMatrix *A, const TTruncAcc &acc, const inv_options_t &opts=inv_options_t())
	Compute diagonal of $A^{-1}$ . More...

void	gauss_elim (TMatrix A, TMatrix C, const TTruncAcc &acc, const inv_options_t &opts=inv_options_t())
	Compute $C = A^{-1}$ via Gaussian elimination. More...

void	invert (TDenseMatrix *A)
	compute $A^{-1}$ More...

std::unique_ptr< TDenseMatrix >	inverse (const TDenseMatrix *A)
	compute and return $A^{-1}$ More...

void	multiply_diag (const real alpha, const matop_t op_A, const TMatrix A, const matop_t op_D, const TMatrix D, const matop_t op_B, const TMatrix B, const real beta, TMatrix C, const TTruncAcc &acc, TProgressBar *progress=NULL)
	compute C ≔ β·C + α·op(A)·op(D)·op(B) where D is a block diagonal matrix

size_t	multiply_diag_steps (const matop_t op_A, const TMatrix A, const matop_t op_D, const TMatrix D, const matop_t op_B, const TMatrix B, const TMatrix C)
	return number of steps for computing C ≔ C + op(A)·op(D)·op(B)

void	mul_diag_left (const TScalarVector &v, TMatrix *A)
	compute B = diag(v)·A and overwrite A

void	mul_diag_right (TMatrix *A, const TScalarVector &v)
	compute B = A·diag(v) and overwrite A

Fourier Transformation
Functions for the forward and backward Fourier transformation of vectors.
void	fft (TVector *v)
	perform FFT for vector v (inplace)

void	ifft (TVector *v)
	perform inverse FFT for vector v (inplace)

Matrix Addition
Functions for matrix addition
template<typename T_value >
void	add (const T_value alpha, const TMatrix A, const T_value beta, TMatrix C, const TTruncAcc &acc)
	C ≔ α·A + β·C. More...

template<typename T_value >
void	add_identity (TMatrix *A, const T_value lambda)
	compute A ≔ A + λ·I

Matrix Factorisation
Functions related to matrix factorisation, e.g. LU or Cholesky factorisation.
std::unique_ptr< TFacMatrix >	factorise (TMatrix *A, const TTruncAcc &acc, const fac_options_t &options=fac_options_t())
	compute factorisation of A More...

std::unique_ptr< TFacInvMatrix >	factorise_inv (TMatrix *A, const TTruncAcc &acc, const fac_options_t &options=fac_options_t())
	compute factorisation of A and return inverse operator More...

void	lu (TMatrix *A, const TTruncAcc &acc, const fac_options_t &options=fac_options_t())
	compute LU factorisation using TLU

void	ldu (TMatrix *A, const TTruncAcc &acc, const fac_options_t &options=fac_options_t())
	compute LDU factorisation using TLDU

void	ll (TMatrix *A, const TTruncAcc &acc, const fac_options_t &options=fac_options_t())
	compute LL^H factorisation using TLL

void	ldl (TMatrix *A, const TTruncAcc &acc, const fac_options_t &options=fac_options_t())
	compute LDL^H factorisation using TLDL

Matrix Multiplication
Functions for matrix multiplication.
template<typename T_value >
void	multiply (const T_value alpha, const matop_t op_A, const TMatrix A, const matop_t op_B, const TMatrix B, const T_value beta, TMatrix C, const TTruncAcc &acc, TProgressBar progress=NULL)
	compute C ≔ β·C + α·op(A)·op(B) More...

template<typename T_value >
void	multiply (const T_value alpha, const TMatrix A, const TMatrix B, const T_value beta, TMatrix C, const TTruncAcc &acc, TProgressBar progress=NULL)
	compute C = β·C + α·A·B More...

void	multiply_ll_left (const real alpha, const TMatrix L, TMatrix A, const TTruncAcc &acc, const eval_option_t &opts)
	compute A ≔ α·L·A with lower left tridiagonal L

void	multiply_ll_right (const real alpha, TMatrix A, const TMatrix L, const TTruncAcc &acc, const eval_option_t &opts)
	compute A ≔ α·A·L with lower left tridiagonal L

void	multiply_ll_right (const real alpha, const TMatrix A, const TMatrix B, TMatrix *C, const TTruncAcc &acc, const eval_option_t &opts)
	compute C ≔ C + α·A·B with lower left tridiagonal B

void	multiply_ur_left (const real alpha, const TMatrix U, TMatrix A, const TTruncAcc &acc, const eval_option_t &opts)
	compute A ≔ α·U·A with upper right tridiagonal U

void	multiply_ur_right (const real alpha, TMatrix A, const TMatrix U, const TTruncAcc &acc, const eval_option_t &opts)
	compute A ≔ α·A·U with upper right tridiagonal U

void	multiply_ur_ll (const TMatrix U, const TMatrix L, TMatrix *A, const TTruncAcc &acc)
	Compute A ≔ A + U·L. More...

void	multiply_llt_d_ll (const TMatrix L, const TMatrix D, TMatrix *C, const matop_t op_L, const TTruncAcc &acc)
	Compute C ≔ C + L^T·D·L. More...

void	multiply_llt_d_left (const real alpha, const TMatrix L, const TMatrix D, TMatrix *B, const matop_t op_L, const TTruncAcc &acc)
	Compute B ≔ op(L)·D·B inplace, overwriting B. More...

void	multiply_llt_d_left (const real alpha, const TMatrix L, const TMatrix D, const TMatrix B, TMatrix C, const matop_t op_L, const TTruncAcc &acc)
	Compute C ≔ C + op(L)·D·B. More...

template<typename T_value >
void	multiply (const T_value alpha, const TMatrixView &A, const TMatrix B, const T_value beta, TMatrix C, const TTruncAcc &acc, TProgressBar *progress=NULL)

template<typename T_value >
void	multiply (const T_value alpha, const TMatrix A, const TMatrixView &B, const T_value beta, TMatrix C, const TTruncAcc &acc, TProgressBar *progress=NULL)

template<typename T_value >
void	multiply (const T_value alpha, const TMatrixView &A, const TMatrixView &B, const T_value beta, TMatrix C, const TTruncAcc &acc, TProgressBar progress=NULL)

template<typename T_value >
std::unique_ptr< TMatrix >	multiply (const T_value alpha, const matop_t op_A, const TRkMatrix A, const matop_t op_B, const TMatrix B)

template<typename T_value >
std::unique_ptr< TMatrix >	multiply (const T_value alpha, const matop_t op_A, const TMatrix A, const matop_t op_B, const TRkMatrix B)

template<typename T_value >
std::unique_ptr< TMatrix >	multiply (const T_value alpha, const matop_t op_A, const TDenseMatrix A, const matop_t op_B, const TMatrix B)

template<typename T_value >
std::unique_ptr< TMatrix >	multiply (const T_value alpha, const matop_t op_A, const TMatrix A, const matop_t op_B, const TDenseMatrix B)

size_t	multiply_steps (const matop_t op_A, const TMatrix A, const matop_t op_B, const TMatrix B, const TMatrix *C)

size_t	multiply_ll_left_steps (const TMatrix A, const TMatrix B)

size_t	multiply_ll_right_steps (const TMatrix A, const TMatrix B)

size_t	multiply_ur_left_steps (const TMatrix A, const TMatrix B)

size_t	multiply_ur_right_steps (const TMatrix A, const TMatrix B)

size_t	multiply_ur_ll_steps (const TMatrix *A)

size_t	multiply_llt_d_ll_steps (const TMatrix *L)

size_t	multiply_llt_d_left_steps (const TMatrix L, const TMatrix B)

Matrix-Vector Multiplication
Functions for (parallel) matrix-vector multiplication
void	mul_vec (const TProcSet &ps, const real alpha, const TMatrix A, const TVector x, const real beta, TVector *y, const matop_t op)
	compute y ≔ α·A·x + β·y on (possibly) distributed matrix A on all processors in ps More...

void	cmul_vec (const TProcSet &ps, const complex alpha, const TMatrix A, const TVector x, const complex beta, TVector *y, const matop_t op)
	same as mul_vec but with complex valued scalars ( More...

void	mul_vec_diag (const real alpha, const matop_t op_A, const TMatrix A, const matop_t op_D, const TMatrix D, const TVector x, const real beta, TVector y)
	compute y ≔ α·A·D·x + β·y with diagonal matrix D More...

Detailed Description

This module provides most higher level algebra functions, e.g. matrix multiplication, inversion and factorisation. See also Basic Matrix Algebra and Matrix Factorisation for an introduction into ℋ-arithmetic.

To include all algebra functions and classes add

#include <hlib-alg.hh>

to your source files.

Function Documentation

◆ add()

void HLIB::add	(	const T_value	alpha,
		const TMatrix *	A,
		const T_value	beta,
		TMatrix *	C,
		const TTruncAcc &	acc
	)

The function computes the sum $C := \alpha A + \beta C$ with a predefined accuracy acc. Thread parallel execution is supported.

Parameters

alpha	scaling factor for A
A	update matrix for C
beta	scaling factor for C
C	matrix to update
acc	accuracy of summation

◆ cmul_vec()

void HLIB::cmul_vec	(	const TProcSet &	ps,
		const complex	alpha,
		const TMatrix *	A,
		const TVector *	x,
		const complex	beta,
		TVector *	y,
		const matop_t	op
	)

See also: mul_vec)

◆ eval()

void HLIB::eval	(	const TMatrix *	A,
		TVector *	v,
		const matop_t	op,
		const eval_option_t &	options
	)

L and U are both stored in A
on entry: v = x
on exit : v = y

◆ eval_diag()

void HLIB::eval_diag	(	const TMatrix *	A,
		TVector *	v,
		const matop_t	op,
		const eval_option_t &	options
	)

on entry: v = x
on exit : v = y

◆ eval_lower()

void HLIB::eval_lower	(	const TMatrix *	A,
		TVector *	v,
		const matop_t	op,
		const eval_option_t &	options
	)

on entry: v = x
on exit : v = y

◆ eval_upper()

void HLIB::eval_upper	(	const TMatrix *	A,
		TVector *	v,
		const matop_t	op,
		const eval_option_t &	options
	)

on entry: v = x
on exit : v = y

◆ factorise()

std::unique_ptr< TFacMatrix > HLIB::factorise	(	TMatrix *	A,
		const TTruncAcc &	acc,
		const fac_options_t &	options = `fac_options_t()`
	)

Compute triangular factorisation of A while choosing appropriate factorisation method depending on the format of A, e.g. if unsymmetric, symmetric or hermitian.

The return value is an operator object representing the factorised form and suitable for evaluation, e.g. matrix-vector multiplication (see TFacMatrix). A will be overwritten with the actual factorisation data.

◆ factorise_inv()

std::unique_ptr< TFacInvMatrix > HLIB::factorise_inv	(	TMatrix *	A,
		const TTruncAcc &	acc,
		const fac_options_t &	options = `fac_options_t()`
	)

Compute triangular factorisation of A as in factorise but instead of an operator for evaluation of the factorised A, an operator for evaluation of the inverse of A is returned (see TFacInvMatrix).

◆ gauss_elim()

void HLIB::gauss_elim	(	TMatrix *	A,
		TMatrix *	C,
		const TTruncAcc &	acc,
		const inv_options_t &	opts = `inv_options_t()`
	)

A is modified during computation.

◆ inverse()

std::unique_ptr< TDenseMatrix > HLIB::inverse ( const TDenseMatrix * A )

Compute inverse of $ A $ (dense matrix version) but do not modify $ A $ .

◆ inverse_diag()

TVector* HLIB::inverse_diag	(	TMatrix *	A,
		const TTruncAcc &	acc,
		const inv_options_t &	opts = `inv_options_t()`
	)

Compute only the diagonal of A and return the resulting vector. A is modified during computation.

◆ invert() [1/2]

void HLIB::invert	(	TMatrix *	A,
		const TTruncAcc &	acc,
		const inv_options_t &	opts = `inv_options_t()`
	)

Compute inverse of $ A $ with block-wise accuracy acc. A is overwritten by $A^{-1}$ during inversion.

◆ invert() [2/2]

void HLIB::invert ( TDenseMatrix * A )

Compute inverse of $ A $ (dense matrix version). A is overwritten by $A^{-1}$ during inversion.

◆ invert_diag()

void HLIB::invert_diag	(	TMatrix *	A,
		const TTruncAcc &	acc,
		const inv_options_t &	opts = `inv_options_t()`
	)

Compute inverse of diagonal matrix $ A $ with block-wise accuracy acc. A is overwritten by $A^{-1}$ during inversion.

◆ invert_ll() [1/2]

void HLIB::invert_ll	(	TMatrix *	A,
		const TTruncAcc &	acc,
		const inv_options_t &	opts = `inv_options_t()`
	)

Compute inverse of lower-left triangular matrix $ A $ with block-wise accuracy acc. A is overwritten by $A^{-1}$ during inversion.

◆ invert_ll() [2/2]

void HLIB::invert_ll	(	TMatrix *	L,
		TMatrix *	X,
		const TTruncAcc &	acc,
		const inv_options_t &	opts = `inv_options_t()`
	)

Compute inverse of lower-left triangular matrix $ L $ with block-wise accuracy acc. L may be modified during inversion.

◆ invert_ur()

void HLIB::invert_ur	(	TMatrix *	A,
		const TTruncAcc &	acc,
		const inv_options_t &	opts = `inv_options_t()`
	)

Compute inverse of upper-right triangular matrix $ A $ with block-wise accuracy acc. A is overwritten by $A^{-1}$ during inversion.

◆ mul_vec()

void HLIB::mul_vec	(	const TProcSet &	ps,
		const real	alpha,
		const TMatrix *	A,
		const TVector *	x,
		const real	beta,
		TVector *	y,
		const matop_t	op
	)

Parameters

ps	processor set defining processors involved in computation
alpha	scaling factor for multiplication
A	matrix to multiply with
x	vector to multiply with
beta	scaling factor of updated vector
y	vector to update with multiplication result
op	defines transformation of matrix, e.g. transposed, adjoint (

See also: matop_t)

◆ mul_vec_diag()

void HLIB::mul_vec_diag	(	const real	alpha,
		const matop_t	op_A,
		const TMatrix *	A,
		const matop_t	op_D,
		const TMatrix *	D,
		const TVector *	x,
		const real	beta,
		TVector *	y
	)

Parameters

alpha	scaling factor for update
op_A	transformation of matrix A, e.g. transposed, adjoint (

See also: matop_t)

Parameters

A	arbitrary matrix
op_D	transformation of matrix D
D	diagonal matrix
x	source vector
beta	scaling factor for destination
y	destination vector

◆ multiply() [1/2]

void HLIB::multiply	(	const T_value	alpha,
		const matop_t	op_A,
		const TMatrix *	A,
		const matop_t	op_B,
		const TMatrix *	B,
		const T_value	beta,
		TMatrix *	C,
		const TTruncAcc &	acc,
		TProgressBar *	progress = `NULL`
	)

The function computes to matrix product $C := \beta C + \alpha \tilde A \tilde B$ where $\tilde A$ and $\tilde B$ may be the non-modified, transposed or adjoint matrices $A$ and $B$ respectively.

The result of the multiplication is written to $C$ , whereby the block structure of C is not changed, e.g. the resulting block structure of the product is defined by C.

Thread-parallel execution is supported by the matrix multiplication.

This function is available in various versions without corresponding parameters, e.g. without op_A, op_B.

Parameters

alpha	scaling factor of product
op_A	matrix modifier for A
A	first matrix factor
op_B	matrix modifier for B
B	second matrix factor
beta	scaling factor for C
C	matrix to update
acc	accuracy of multiplication
progress	optional progress bar

◆ multiply() [2/2]

void HLIB::multiply	(	const T_value	alpha,
		const TMatrix *	A,
		const TMatrix *	B,
		const T_value	beta,
		TMatrix *	C,
		const TTruncAcc &	acc,
		TProgressBar *	progress = `NULL`
	)

inline

matrices are provided as matrix view (or in combination with matrix)

◆ multiply_llt_d_left() [1/2]

void HLIB::multiply_llt_d_left	(	const real	alpha,
		const TMatrix *	L,
		const TMatrix *	D,
		TMatrix *	B,
		const matop_t	op_L,
		const TTruncAcc &	acc
	)

Compute B ≔ op(L)·D·B with lower left tridiagonal L and diagonal D.

◆ multiply_llt_d_left() [2/2]

void HLIB::multiply_llt_d_left	(	const real	alpha,
		const TMatrix *	L,
		const TMatrix *	D,
		const TMatrix *	B,
		TMatrix *	C,
		const matop_t	op_L,
		const TTruncAcc &	acc
	)

Compute C ≔ C + op(L)·D·B with lower left tridiagonal L, diagonal D and general B

◆ multiply_llt_d_ll()

void HLIB::multiply_llt_d_ll	(	const TMatrix *	L,
		const TMatrix *	D,
		TMatrix *	C,
		const matop_t	op_L,
		const TTruncAcc &	acc
	)

Compute C ≔ C + L^T·D·L with lower left tridiagonal L and diagonal D.

◆ multiply_ur_ll()

void HLIB::multiply_ur_ll	(	const TMatrix *	U,
		const TMatrix *	L,
		TMatrix *	A,
		const TTruncAcc &	acc
	)

Compute A ≔ A + U·L with lower left tridiagonal L and upper right tridiagonal U.

Classes

Functions

Fourier Transformation

Matrix Addition

Matrix Factorisation

Matrix Multiplication

Matrix-Vector Multiplication

Detailed Description

Function Documentation

◆ add()

◆ cmul_vec()

◆ eval()

◆ eval_diag()

◆ eval_lower()

◆ eval_upper()

◆ factorise()

◆ factorise_inv()

◆ gauss_elim()

◆ inverse()

◆ inverse_diag()

◆ invert() [1/2]

◆ invert() [2/2]

◆ invert_diag()

◆ invert_ll() [1/2]

◆ invert_ll() [2/2]

◆ invert_ur()

◆ mul_vec()

◆ mul_vec_diag()

◆ multiply() [1/2]

◆ multiply() [2/2]

◆ multiply_llt_d_left() [1/2]

◆ multiply_llt_d_left() [2/2]

◆ multiply_llt_d_ll()

◆ multiply_ur_ll()