Reference documentation for deal.II version Git 3f1f337db3 20211023 13:19:02 0600

#include <deal.II/lac/householder.h>
Public Types  
using  size_type = types::global_dof_index 
Public Member Functions  
Householder ()=default  
template<typename number2 >  
Householder (const FullMatrix< number2 > &A)  
template<typename number2 >  
void  initialize (const FullMatrix< number2 > &A) 
template<typename number2 >  
double  least_squares (Vector< number2 > &dst, const Vector< number2 > &src) const 
template<typename number2 >  
double  least_squares (BlockVector< number2 > &dst, const BlockVector< number2 > &src) const 
template<class VectorType >  
void  vmult (VectorType &dst, const VectorType &src) const 
template<class VectorType >  
void  Tvmult (VectorType &dst, const VectorType &src) const 
Private Attributes  
std::vector< number >  diagonal 
FullMatrix< double >  storage 
QRdecomposition of a full matrix.
This class computes the QRdecomposition of given matrix by the Householder algorithm. Then, the function least_squares() can be used to compute the vector \(x\) minimizing \(\Axb\\) for a given vector \(b\). The QR decomposition of \(A\) is useful for this purpose because the minimizer is given by the equation \(x=(A^TA)^{1}A^Tb=(R^TQ^TQR)^{1}R^TQ^Tb\) which is easy to compute because \(Q\) is an orthogonal matrix, and consequently \(Q^TQ=I\). Thus, \(x=(R^TR)^{1}R^TQ^Tb=R^{1}R^{T}R^TQ^Tb=R^{1}Q^Tb\). Furthermore, \(R\) is triangular, so applying \(R^{1}\) to a vector only involves a backward or forward solve.
The class does not in fact store the \(Q\) and \(R\) factors explicitly as matrices. It does store \(R\), but the \(Q\) factor is stored as the product of Householder reflections of the form \(Q_i = Iv_i v_i^T\) where the vectors \(v_i\) are so that they can be stored in the lowertriangular part of an underlying matrix object, whereas \(R\) is stored in the upper triangular part.
The \(v_i\) vectors and the \(R\) matrix now are in conflict because they both want to use the diagonal entry of the matrix, but we can only store one in these positions, of course. Consequently, the entries \((v_i)_i\) are stored separately in the diagonal
member variable.
<float> and <double>
; others can be generated in application programs (see the section on Template instantiations in the manual). Definition at line 79 of file householder.h.
using Householder< number >::size_type = types::global_dof_index 
Declare type of container size type.
Definition at line 85 of file householder.h.

default 
Create an empty object.
Householder< number >::Householder  (  const FullMatrix< number2 > &  A  ) 
Create an object holding the QRdecomposition of the matrix \(A\).
void Householder< number >::initialize  (  const FullMatrix< number2 > &  A  ) 
Compute the QRdecomposition of the given matrix \(A\).
This overwrites any previously computed QR decomposition.
double Householder< number >::least_squares  (  Vector< number2 > &  dst, 
const Vector< number2 > &  src  
)  const 
Solve the leastsquares problem for the right hand side src
. The returned scalar value is the Euclidean norm of the approximation error.
dst
contains the solution of the least squares problem on return.src
contains the right hand side b of the least squares problem. It will be changed during the algorithm and is unusable on return. double Householder< number >::least_squares  (  BlockVector< number2 > &  dst, 
const BlockVector< number2 > &  src  
)  const 
This function does the same as the previous one, but for BlockVectors.
void Householder< number >::vmult  (  VectorType &  dst, 
const VectorType &  src  
)  const 
A wrapper to least_squares(), implementing the standard MatrixType interface.
void Householder< number >::Tvmult  (  VectorType &  dst, 
const VectorType &  src  
)  const 
A wrapper to least_squares() that implements multiplication with the transpose matrix.

private 
Storage for the diagonal elements of the orthogonal transformation. See the class documentation for more information.
Definition at line 151 of file householder.h.

private 
Storage that is internally used for the Householder transformation.
Definition at line 156 of file householder.h.