TNT::Linear_Algebra::QR< Real > Class Template Reference

#include <tnt_linalg.h>

Public Member Functions
	for (k=0;k< n;k++)
int	isFullRank () const
Matrix< Real >	getHouseholder (void) const
Matrix< Real >	getR () const
Matrix< Real >	getQ () const
Vector< Real >	solve (const Vector< Real > &b) const
Matrix< Real >	solve (const Matrix< Real > &B) const

Public Attributes
	m = A.num_rows()
	n = A.num_cols()
	Rdiag = Vector<Real>(n)
int	i =0
int	j =0
int	k =0

Detailed Description

template<class Real>
class TNT::Linear_Algebra::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).

Member Function Documentation

getHouseholder()

template<class Real >

Matrix< Real > TNT::Linear_Algebra::QR< Real >::getHouseholder ( void ) const

inline

Retreive the Householder vectors from QR factorization

Returns

lower trapezoidal matrix whose columns define the reflections

getQ()

template<class Real >

Matrix< Real > TNT::Linear_Algebra::QR< Real >::getQ ( ) const

inline

Returns

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

getR()

template<class Real >

Matrix< Real > TNT::Linear_Algebra::QR< Real >::getR ( ) const

inline

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

Returns

R

isFullRank()

template<class Real >

int TNT::Linear_Algebra::QR< Real >::isFullRank ( ) const

inline

Flag to denote the matrix is of full rank.

Returns

1 if matrix is full rank, 0 otherwise.

solve() [1/2]

template<class Real >

Matrix< Real > TNT::Linear_Algebra::QR< Real >::solve ( const Matrix< 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 (Real(0.0)) array.

solve() [2/2]

template<class Real >

Vector< Real > TNT::Linear_Algebra::QR< Real >::solve ( const Vector< Real > & b ) const

inline

Least squares solution of A*x = b

Parameters

b	right hand side (m-length vector).

Returns

x n-length 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.

Member Data Documentation

m

template<class Real >

TNT::Linear_Algebra::QR< Real >::m = A.num_rows()

Create a QR factorization object for A.

Parameters

A	rectangular (m>=n) matrix.

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

• tnt_linalg.h