mikera.matrixx.decompose.impl.qr

Class HouseholderQR

java.lang.Object

mikera.matrixx.decompose.impl.qr.HouseholderQR

All Implemented Interfaces:

QRDecomposition

public class HouseholderQR
extends Object
implements QRDecomposition

This variation of QR decomposition uses reflections to compute the Q matrix. Each reflection uses a householder operations, hence its name. To provide a meaningful solution the original matrix must have full rank. This is intended for processing of small to medium matrices.

Both Q and R are stored in the same m by n matrix. Q is not stored directly, instead the u from Q_k=(I-γ*u*u^T) is stored. Decomposition requires about 2n*m²-2m²/3 flops.

See the QR reflections algorithm described in:
David S. Watkins, "Fundamentals of Matrix Computations" 2nd Edition, 2002

For the most part this is a straight forward implementation. To improve performance on large matrices a column is written to an array and the order of some of the loops has been changed. This will degrade performance noticeably on small matrices. Since it is unlikely that the QR decomposition would be a bottle neck when small matrices are involved only one implementation is provided.

Author:

Peter Abeles

Field Summary

Fields

Modifier and Type	Field and Description
protected double[]	dataQR
protected boolean	error
protected double	gamma
protected double[]	gammas
protected int	minLength
protected int	numCols
protected int	numRows
protected Matrix	QR Where the Q and R matrices are stored.
protected double	tau
protected double[]	u
protected double[]	v

Constructor Summary

Constructors

Constructor and Description
HouseholderQR(boolean compact)

Method Summary

Modifier and Type	Method and Description
protected AMatrix	computeQ() Computes the Q matrix from the information stored in the QR matrix.
protected AMatrix	computeR() Returns an upper triangular matrix which is the R in the QR decomposition.
QRResult	decompose(AMatrix A) In order to decompose the matrix 'A' it must have full rank.
double[]	getGammas()
AMatrix	getQ()
AMatrix	getQR() Returns a single matrix which contains the combined values of Q and R.
AMatrix	getR()
protected void	householder(int j) Computes the householder vector "u" for the first column of submatrix j.
protected void	updateA(int w) Takes the results from the householder computation and updates the 'A' matrix. A = (I - γ*u*uT)A

Methods inherited from class java.lang.Object

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

Field Detail

QR

protected Matrix QR

Where the Q and R matrices are stored. R is stored in the upper triangular portion and Q on the lower bit. Lower columns are where u is stored. Q_k = (I - gamma_k*u_k*u_k^T).

u

protected double[] u

v

protected double[] v

numCols

protected int numCols

numRows

protected int numRows

minLength

protected int minLength

dataQR

protected double[] dataQR

gammas

protected double[] gammas

gamma

protected double gamma

tau

protected double tau

error

protected boolean error

Constructor Detail

HouseholderQR

public HouseholderQR(boolean compact)

Method Detail

getQR

public AMatrix getQR()

Returns a single matrix which contains the combined values of Q and R. This is possible since Q is symmetric and R is upper triangular.

Returns:

The combined Q R matrix.

getQ

public AMatrix getQ()

Returns:

The Q matrix from the decomposition.

getR

public AMatrix getR()

Returns:

The R matrix from the decomposition.

computeQ

protected AMatrix computeQ()

Computes the Q matrix from the information stored in the QR matrix. This operation requires about 4(m²n-mn²+n³/3) flops.

computeR

protected AMatrix computeR()

Returns an upper triangular matrix which is the R in the QR decomposition.

decompose

public QRResult decompose(AMatrix A)

In order to decompose the matrix 'A' it must have full rank. 'A' is a 'm' by 'n' matrix. It requires about 2n*m²-2m²/3 flops.

The matrix provided here can be of different dimension than the one specified in the constructor. It just has to be smaller than or equal to it.

Specified by:

decompose in interface QRDecomposition

householder

protected void householder(int j)

Computes the householder vector "u" for the first column of submatrix j. Note this is a specialised householder for this problem. There is some protection against overflow and underflow.

Q = I - γuu^T

This function finds the values of 'u' and 'γ'.

Parameters:

j - Which submatrix to work off of.

updateA

protected void updateA(int w)

Takes the results from the householder computation and updates the 'A' matrix.

A = (I - γ*u*u^T)A

Parameters:

w - The submatrix.

getGammas

public double[] getGammas()