org.apache.commons.math3.linear

Class QRDecomposition

java.lang.Object

org.apache.commons.math3.linear.QRDecomposition

Direct Known Subclasses:

RRQRDecomposition

public class QRDecomposition
extends java.lang.Object

Calculates the QR-decomposition of a matrix.

The QR-decomposition of a matrix A consists of two matrices Q and R that satisfy: A = QR, Q is orthogonal (Q^TQ = I), and R is upper triangular. If A is m×n, Q is m×m and R m×n.

This class compute the decomposition using Householder reflectors.

For efficiency purposes, the decomposition in packed form is transposed. This allows inner loop to iterate inside rows, which is much more cache-efficient in Java.

This class is based on the class with similar name from the JAMA library, with the following changes:

• a getQT method has been added,
• the solve and isFullRank methods have been replaced by a getSolver method and the equivalent methods provided by the returned DecompositionSolver.

Since:

1.2 (changed to concrete class in 3.0)

See Also:

MathWorld, Wikipedia

Nested Class Summary

Nested Classes

Modifier and Type	Class and Description
private static class	QRDecomposition.Solver Specialized solver.

Field Summary

Fields

Modifier and Type	Field and Description
private RealMatrix	cachedH Cached value of H.
private RealMatrix	cachedQ Cached value of Q.
private RealMatrix	cachedQT Cached value of QT.
private RealMatrix	cachedR Cached value of R.
private double[][]	qrt A packed TRANSPOSED representation of the QR decomposition.
private double[]	rDiag The diagonal elements of R.
private double	threshold Singularity threshold.

Constructor Summary

Constructors

Constructor and Description
QRDecomposition(RealMatrix matrix) Calculates the QR-decomposition of the given matrix.
QRDecomposition(RealMatrix matrix, double threshold) Calculates the QR-decomposition of the given matrix.

Method Summary

Methods

Modifier and Type	Method and Description
protected void	decompose(double[][] matrix) Decompose matrix.
RealMatrix	getH() Returns the Householder reflector vectors.
RealMatrix	getQ() Returns the matrix Q of the decomposition.
RealMatrix	getQT() Returns the transpose of the matrix Q of the decomposition.
RealMatrix	getR() Returns the matrix R of the decomposition.
DecompositionSolver	getSolver() Get a solver for finding the A × X = B solution in least square sense.
protected void	performHouseholderReflection(int minor, double[][] matrix) Perform Householder reflection for a minor A(minor, minor) of A.

Methods inherited from class java.lang.Object

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

Field Detail

qrt

private double[][] qrt

A packed TRANSPOSED representation of the QR decomposition.

The elements BELOW the diagonal are the elements of the UPPER triangular matrix R, and the rows ABOVE the diagonal are the Householder reflector vectors from which an explicit form of Q can be recomputed if desired.

rDiag

private double[] rDiag

The diagonal elements of R.

cachedQ

private RealMatrix cachedQ

Cached value of Q.

cachedQT

private RealMatrix cachedQT

Cached value of QT.

cachedR

private RealMatrix cachedR

Cached value of R.

cachedH

private RealMatrix cachedH

Cached value of H.

threshold

private final double threshold

Singularity threshold.

Constructor Detail

QRDecomposition

public QRDecomposition(RealMatrix matrix)

Calculates the QR-decomposition of the given matrix. The singularity threshold defaults to zero.

Parameters:

matrix - The matrix to decompose.

See Also:

QRDecomposition(RealMatrix,double)

QRDecomposition

public QRDecomposition(RealMatrix matrix,
               double threshold)

Calculates the QR-decomposition of the given matrix.

Parameters:

matrix - The matrix to decompose.

threshold - Singularity threshold.

Method Detail

decompose

protected void decompose(double[][] matrix)

Decompose matrix.

Parameters:

matrix - transposed matrix

Since:

3.2

performHouseholderReflection

protected void performHouseholderReflection(int minor,
                                double[][] matrix)

Perform Householder reflection for a minor A(minor, minor) of A.

Parameters:

minor - minor index

matrix - transposed matrix

Since:

3.2

getR

public RealMatrix getR()

Returns the matrix R of the decomposition.

R is an upper-triangular matrix

Returns:

the R matrix

getQ

public RealMatrix getQ()

Returns the matrix Q of the decomposition.

Q is an orthogonal matrix

Returns:

the Q matrix

getQT

public RealMatrix getQT()

Returns the transpose of the matrix Q of the decomposition.

Q is an orthogonal matrix

Returns:

the transpose of the Q matrix, Q^T

getH

public RealMatrix getH()

Returns the Householder reflector vectors.

H is a lower trapezoidal matrix whose columns represent each successive Householder reflector vector. This matrix is used to compute Q.

Returns:

a matrix containing the Householder reflector vectors

getSolver

public DecompositionSolver getSolver()

Get a solver for finding the A × X = B solution in least square sense.

Least Square sense means a solver can be computed for an overdetermined system, (i.e. a system with more equations than unknowns, which corresponds to a tall A matrix with more rows than columns). In any case, if the matrix is singular within the tolerance set at construction, an error will be triggered when the solve method will be called.

Returns:

a solver