edu.northwestern.at.utils.math.matrix
Class QRDecomposition

java.lang.Object
  edu.northwestern.at.utils.math.matrix.QRDecomposition

public class QRDecomposition
extends java.lang.Object

QR Decomposition of a matrix using Householder reflections.

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.

This is the JAMA implementation modified for use with our Matrix class.

Constructor Summary

QRDecomposition(Matrix matrix)
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

Constructor Detail

QRDecomposition

public QRDecomposition(Matrix matrix)

QR Decomposition computed by Householder reflections.

Parameters:

matrix - Rectangular matrix to decompose.

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 Householder reflections.

getR

public Matrix getR()

Return the upper triangular factor.

Returns:

The upper triangular factor R.

getQ

public Matrix getQ()

Generate and return the (economy-sized) orthogonal factor.

Returns:

The orthogonal factor 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:

java.lang.IllegalArgumentException - Matrix row dimensions must agree.

java.lang.RuntimeException - Matrix is rank deficient.