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Reference documentation for deal.II version 9.6.0
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#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<typename VectorType > | |
void | vmult (VectorType &dst, const VectorType &src) const |
template<typename VectorType > | |
void | Tvmult (VectorType &dst, const VectorType &src) const |
Private Attributes | |
std::vector< number > | diagonal |
FullMatrix< double > | storage |
QR-decomposition of a full matrix.
This class computes the QR-decomposition of given matrix by the Householder algorithm. Then, the function least_squares() can be used to compute the vector \(x\) minimizing \(\|Ax-b\|\) 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 \(Q_i = I-v_i v_i^T\) where the vectors \(v_i\) are so that they can be stored in the lower-triangular 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.
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default |
Create an empty object.
Householder< number >::Householder | ( | const FullMatrix< number2 > & | A | ) |
Create an object holding the QR-decomposition of the matrix \(A\).
void Householder< number >::initialize | ( | const FullMatrix< number2 > & | A | ) |
Compute the QR-decomposition 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 least-squares 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.
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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.
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private |
Storage that is internally used for the Householder transformation.
Definition at line 156 of file householder.h.