org.apache.commons.math4.linear

## Class RRQRDecomposition

• public class RRQRDecomposition
extends QRDecomposition
Calculates the rank-revealing QR-decomposition of a matrix, with column pivoting.

The rank-revealing QR-decomposition of a matrix A consists of three matrices Q, R and P such that AP=QR. Q is orthogonal (QTQ = I), and R is upper triangular. If A is m×n, Q is m×m and R is m×n and P is n×n.

QR decomposition with column pivoting produces a rank-revealing QR decomposition and the getRank(double) method may be used to return the rank of the input matrix A.

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:
3.2
MathWorld, Wikipedia
• ### Constructor Summary

Constructors
Constructor and Description
RRQRDecomposition(RealMatrix matrix)
Calculates the QR-decomposition of the given matrix.
RRQRDecomposition(RealMatrix matrix, double threshold)
Calculates the QR-decomposition of the given matrix.
• ### Method Summary

All Methods
Modifier and Type Method and Description
protected void decompose(double[][] qrt)
Decompose matrix.
RealMatrix getP()
Returns the pivot matrix, P, used in the QR Decomposition of matrix A such that AP = QR.
int getRank(double dropThreshold)
Return the effective numerical matrix rank.
DecompositionSolver getSolver()
Get a solver for finding the A × X = B solution in least square sense.
protected void performHouseholderReflection(int minor, double[][] qrt)
Perform Householder reflection for a minor A(minor, minor) of A.
• ### Methods inherited from class org.apache.commons.math4.linear.QRDecomposition

getH, getQ, getQT, getR
• ### Methods inherited from class java.lang.Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait
• ### Constructor Detail

• #### RRQRDecomposition

public RRQRDecomposition(RealMatrix matrix)
Calculates the QR-decomposition of the given matrix. The singularity threshold defaults to zero.
Parameters:
matrix - The matrix to decompose.
RRQRDecomposition(RealMatrix, double)
• #### RRQRDecomposition

public RRQRDecomposition(RealMatrix matrix,
double threshold)
Calculates the QR-decomposition of the given matrix.
Parameters:
matrix - The matrix to decompose.
threshold - Singularity threshold.
RRQRDecomposition(RealMatrix)
• ### Method Detail

• #### decompose

protected void decompose(double[][] qrt)
Decompose matrix.
Overrides:
decompose in class QRDecomposition
Parameters:
qrt - transposed matrix
• #### performHouseholderReflection

protected void performHouseholderReflection(int minor,
double[][] qrt)
Perform Householder reflection for a minor A(minor, minor) of A.
Overrides:
performHouseholderReflection in class QRDecomposition
Parameters:
minor - minor index
qrt - transposed matrix
• #### getP

public RealMatrix getP()
Returns the pivot matrix, P, used in the QR Decomposition of matrix A such that AP = QR. If no pivoting is used in this decomposition then P is equal to the identity matrix.
Returns:
a permutation matrix.
• #### getRank

public int getRank(double dropThreshold)
Return the effective numerical matrix rank.

The effective numerical rank is the number of non-negligible singular values.

This implementation looks at Frobenius norms of the sequence of bottom right submatrices. When a large fall in norm is seen, the rank is returned. The drop is computed as:

   (thisNorm/lastNorm) * rNorm < dropThreshold


where thisNorm is the Frobenius norm of the current submatrix, lastNorm is the Frobenius norm of the previous submatrix, rNorm is is the Frobenius norm of the complete matrix

Parameters:
dropThreshold - threshold triggering rank computation
Returns:
effective numerical matrix rank
• #### 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.

Overrides:
getSolver in class QRDecomposition
Returns:
a solver