de.lmu.ifi.dbs.elki.math.linearalgebra
Class QRDecomposition

java.lang.Object
  de.lmu.ifi.dbs.elki.math.linearalgebra.QRDecomposition

All Implemented Interfaces:

Serializable

public class QRDecomposition
extends Object
implements Serializable

QR Decomposition.

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 false.

See Also:

Serialized Form

Field Summary

private int	m Row and column dimensions.
private int	n Row and column dimensions.
private double[][]	QR Array for internal storage of decomposition.
private double[]	Rdiag Array for internal storage of diagonal of R.

Constructor Summary

QRDecomposition(Matrix A)
QR Decomposition, computed by Householder reflections.

Method Summary

Matrix	getH() Return the Householder vectors
Matrix	getQ() Generate and return the (economy-sized) orthogonal factor
Matrix	getR() Return the upper triangular factor
boolean	isFullRank() Is the matrix full rank?
Matrix	solve(Matrix B) Least squares solution of A*X = B

Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Field Detail

QR

private double[][] QR

Array for internal storage of decomposition.

m

private int m

Row and column dimensions.

n

private int n

Row and column dimensions.

Rdiag

private double[] Rdiag

Array for internal storage of diagonal of R.

Constructor Detail

QRDecomposition

public QRDecomposition(Matrix A)

QR Decomposition, computed by Householder reflections.

Parameters:

A - Rectangular matrix

Method Detail

isFullRank

public boolean isFullRank()

Is the matrix full rank?

Returns:

true if R, and hence A, has full rank.

getH

public Matrix getH()

Return the Householder vectors

Returns:

Lower trapezoidal matrix whose columns define the reflections

getR

public Matrix getR()

Return the upper triangular factor

Returns:

R

getQ

public Matrix getQ()

Generate and return the (economy-sized) orthogonal factor

Returns:

Q

solve

public Matrix solve(Matrix B)

Least squares solution of A*X = B

Parameters:

B - A Matrix with as many rows as A and any number of columns.

Returns:

X that minimizes the two norm of Q*R*X-B.

Throws:

IllegalArgumentException - Matrix row dimensions must agree.

RuntimeException - Matrix is rank deficient.