sdp::QRDecomposition< T > Class Template Reference

Provides the QR decomposition of a matrix with entries of type T. More...

#include <qrdec.h>

Public Member Functions
	QRDecomposition (const std::vector< T > &A, int m, int n, double sing_eps=1e-10)
	Initializes an instance by computing the QR decomposition. More...
int	rank () const
	Returns the rank of the matrix, as determined by the QR decomposition. More...
int	perm (int j) const
	Returns the original index of the j-th column after the pivoting. More...
const T &	scaling_factor () const
	Returns the scaling factor used. More...
const T &	R (int i, int j) const
	Returns entry (i, j) of matrix \(R\) of the decomposition.
void	Q_mul (std::vector< T > &x) const
	Multiplies a vector x on the left by the matrix \(Q\).
void	Q_transpose_mul (std::vector< T > &x) const
	Multiplies a vector x on the left by the matrix \(Q^{\sf T}\).

Detailed Description

template<class T>
class sdp::QRDecomposition< T >

Provides the QR decomposition of a matrix with entries of type T.

The decomposition is computed using Householder reflectors with column pivoting. The algorithm is column-oriented, and the input matrix should be provided as a vector in column-major order.

Scaling is performed to avoid overflow, the scaling factor should be taken into account when using the decomposition.

Constructor & Destructor Documentation

template<class T >

sdp::QRDecomposition< T >::QRDecomposition ( const std::vector< T > &  A, int  m, int  n, double  sing_eps = 1e-10  )

inline

Initializes an instance by computing the QR decomposition.

Parameters

A	is the input matrix as a vector in column-major order.
m	number of rows in A.
n	number of columns in A.
sing_eps	is used during the process to decide the rank of A; during the decomposition, a column whose squared norm is less than sing_eps is considered to be zero.

Exceptions

ValueError

if there are less rows than columns.

Member Function Documentation

template<class T >

int sdp::QRDecomposition< T >::perm ( int  j) const

inline

Returns the original index of the j-th column after the pivoting.

The decomposition is implemented with column pivoting; we have \(AP = QR\), where \(P\) is a permutation matrix that permutes the columns of \(A\). So we have a decomposition for \(AP\), which is a permuted version of \(A\). The permutation is given as follows: the j-th column of \(AP\) is the column of \(A\) with index QR.perm(j).

template<class T >

int sdp::QRDecomposition< T >::rank ( ) const

inline

Returns the rank of the matrix, as determined by the QR decomposition.

If the rank is \(r\), then

\[ R = \begin{pmatrix} R_{11}&R_{12}\\ 0&0 \end{pmatrix} \]

where \(R_{11}\) is an \(r \times r\) matrix.

template<class T >

const T& sdp::QRDecomposition< T >::scaling_factor ( ) const

inline

Returns the scaling factor used.

The decomposition is not computed for the given matrix \(A\), but for \(s^{-1} A\), where \(s\) is the scaling factor used.

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The documentation for this class was generated from the following file:

• include/sdpsl/qrdec.h