#include <jama_qr.h>

Public Member Functions
	QR (const TNT::Array2D< Real > &A)

int	isFullRank () const

TNT::Array2D< Real >	getHouseholder (void) const

TNT::Array2D< Real >	getR () const

TNT::Array2D< Real >	getQ () const

TNT::Array1D< Real >	solve (const TNT::Array1D< Real > &b) const

TNT::Array2D< Real >	solve (const TNT::Array2D< Real > &B) const

Private Attributes
TNT::Array2D< Real >	QR_

int	m

int	n

TNT::Array1D< Real >	Rdiag

Detailed Description

template<class Real>
class JAMA::QR< Real >

Classical QR Decompisition: for an m-by-n matrix A with m >= n, the QR decomposition is an m-by-n orthogonal matrix Q and an n-by-n upper triangular matrix R so that A = Q*R.

The QR decompostion always exists, even if the matrix does not have full rank, so the constructor will never fail. The primary use of the QR decomposition is in the least squares solution of nonsquare systems of simultaneous linear equations. This will fail if isFullRank() returns 0 (false).

The Q and R factors can be retrived via the getQ() and getR() methods. Furthermore, a solve() method is provided to find the least squares solution of Ax=b using the QR factors.

(Adapted from JAMA, a Java Matrix Library, developed by jointly by the Mathworks and NIST; see http://math.nist.gov/javanumerics/jama).

Constructor & Destructor Documentation

◆ QR()

QR ( const TNT::Array2D< Real > & A )

inline

Create a QR factorization object for A.

Parameters

A	rectangular (m>=n) matrix.

References Array2D< T >::copy(), Array2D< T >::dim1(), Array2D< T >::dim2(), TNT::hypot(), QR< Real >::m, QR< Real >::n, QR< Real >::QR_, and QR< Real >::Rdiag.

Member Function Documentation

◆ getHouseholder()

TNT::Array2D<Real> getHouseholder ( void ) const

inline

Retreive the Householder vectors from QR factorization

Returns: lower trapezoidal matrix whose columns define the reflections

References QR< Real >::m, QR< Real >::n, and QR< Real >::QR_.

◆ getQ()

TNT::Array2D<Real> getQ ( ) const

inline

    Generate and return the (economy-sized) orthogonal factor

Parameters

Q	the (ecnomy-sized) orthogonal factor (Q*R=A).

References QR< Real >::m, QR< Real >::n, and QR< Real >::QR_.

◆ getR()

TNT::Array2D<Real> getR ( ) const

inline

Return the upper triangular factor, R, of the QR factorization

Returns: R

References QR< Real >::n, QR< Real >::QR_, and QR< Real >::Rdiag.

◆ isFullRank()

int isFullRank ( ) const

inline

Flag to denote the matrix is of full rank.

Returns: 1 if matrix is full rank, 0 otherwise.

References QR< Real >::n, and QR< Real >::Rdiag.

Referenced by QR< Real >::solve().

◆ solve() [1/2]

TNT::Array1D<Real> solve ( const TNT::Array1D< Real > & b ) const

inline

Least squares solution of A*x = b

Parameters

B	m-length array (vector).

Returns: x n-length array (vector) that minimizes the two norm of Q*R*X-B. If B is non-conformant, or if QR.isFullRank() is false, the routine returns a null (0-length) vector.

References Array1D< T >::copy(), Array1D< T >::dim1(), QR< Real >::isFullRank(), QR< Real >::m, QR< Real >::n, QR< Real >::QR_, and QR< Real >::Rdiag.

◆ solve() [2/2]

TNT::Array2D<Real> solve ( const TNT::Array2D< Real > & B ) const

inline

Least squares solution of A*X = B

Parameters

B	m x k Array (must conform).

Returns: X n x k Array that minimizes the two norm of Q*R*X-B. If B is non-conformant, or if QR.isFullRank() is false, the routine returns a null (0x0) array.

References Array2D< T >::copy(), Array2D< T >::dim1(), Array2D< T >::dim2(), QR< Real >::isFullRank(), QR< Real >::m, QR< Real >::n, QR< Real >::QR_, and QR< Real >::Rdiag.