HLIBpro  2.1
Classes | Functions
Algebra

Classes

struct  tri_eval_option_t
 determines characteristics of triangular system More...
 
struct  fac_options_t
 options for matrix factorisations More...
 
class  TLU
 Computes LU factorisation $ A = LU $. More...
 
class  TLDU
 Computes LDU factorisation $ A = LDU $. More...
 
class  TLL
 computes Cholesky factorisation $ A = LL^T $ or $ A=LL^H $ More...
 
class  TLDL
 computes LDL factorisation $ A = LDL^T $ or $ A = LDL^H $ More...
 
struct  inv_options_t
 options for matrix inversion More...
 
struct  solve_option_t
 determines characteristics of triangular system 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 tri_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 tri_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 tri_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 tri_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...
 
TVector * inverse_diag (TMatrix *A, const TTruncAcc &acc, const inv_options_t opts=inv_options_t())
 Compute diagonal of $ 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.

TLinearOperator * factorise (TMatrix *A, const TTruncAcc &acc, const fac_options_t &options=fac_options_t())
 compute factorisation of A More...
 
TLinearOperator * 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...
 
size_t multiply_steps (const matop_t op_A, const TMatrix *A, const matop_t op_B, const TMatrix *B, const TMatrix *C)
 return number of steps for computation of C ≔ C + op(A)·op(B) for progress meter initialisation
 
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)
 

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

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
alphascaling factor for A
Aupdate matrix for C
betascaling factor for C
Cmatrix to update
accaccuracy of summation
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)
void HLIB::eval ( const TMatrix *  A,
TVector *  v,
const matop_t  op,
const tri_eval_option_t &  options 
)
  • L and U are both stored in A
  • on entry: v = x
  • on exit : v = y
void HLIB::eval_diag ( const TMatrix *  A,
TVector *  v,
const matop_t  op,
const tri_eval_option_t &  options 
)
  • on entry: v = x
  • on exit : v = y
void HLIB::eval_lower ( const TMatrix *  A,
TVector *  v,
const matop_t  op,
const tri_eval_option_t &  options 
)
  • on entry: v = x
  • on exit : v = y
void HLIB::eval_upper ( const TMatrix *  A,
TVector *  v,
const matop_t  op,
const tri_eval_option_t &  options 
)
  • on entry: v = x
  • on exit : v = y
TLinearOperator* 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.

TLinearOperator* 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).

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.

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.

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
psprocessor set defining processors involved in computation
alphascaling factor for multiplication
Amatrix to multiply with
xvector to multiply with
betascaling factor of updated vector
yvector to update with multiplication result
opdefines transformation of matrix, e.g. transposed, adjoint (
See Also
matop_t)
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
alphascaling factor for update
op_Atransformation of matrix A, e.g. transposed, adjoint (
See Also
matop_t)
Parameters
Aarbitrary matrix
op_Dtransformation of matrix D
Ddiagonal matrix
xsource vector
betascaling factor for destination
ydestination vector
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
alphascaling factor of product
op_Amatrix modifier for A
Afirst matrix factor
op_Bmatrix modifier for B
Bsecond matrix factor
betascaling factor for C
Cmatrix to update
accaccuracy of multiplication
progressoptional progress bar
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)

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