org.apache.commons.math3.linear

## Class QRDecomposition

• Direct Known Subclasses:
RRQRDecomposition

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:

Since:
1.2 (changed to concrete class in 3.0)
MathWorld, Wikipedia
• ### 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
• ### 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

• #### 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, 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.

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