## org.apache.commons.math3.linear Class QRDecomposition

java.lang.Object
org.apache.commons.math3.linear.QRDecomposition

public class QRDecomposition
extends 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 (QTQ = 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• the solve and isFullRank methods have been replaced by a getSolver method and the equivalent methods provided by the returned DecompositionSolver.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)
Version:
\$Id: QRDecomposition.java 1416643 2012-12-03 19:37:14Z tn \$
MathWorld, Wikipedia

Constructor Summary
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
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.

Methods inherited from class java.lang.Object
clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

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.
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

### 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, QT

### 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.

Returns:
a solver