com.numericalmethod.suanshu.matrix.doubles.factorization.qr
Class QR

java.lang.Object
  com.numericalmethod.suanshu.matrix.doubles.factorization.qr.QR

All Implemented Interfaces:

QRDecomposition

public class QR
extends java.lang.Object
implements QRDecomposition

QR decomposition of a matrix decomposes an m x n matrix A so that A = Q * R.

• Q is an m x n orthogonal matrix;
• R is a n x n upper triangular matrix.

Alternatively, we can have A = sqQ * tallR, where

• sqQ is a square m x m orthogonal matrix;
• tallR is a m x n matrix.

See Also:

Wikipedia: QR decomposition

Constructor Summary

QR(Matrix A)
Run the QR decomposition on a matrix.

QR(Matrix A, double epsilon)
Run the QR decomposition on a matrix.

Method Summary

PermutationMatrix	P() Get P, the pivoting matrix in the QR decomposition.
Matrix	Q() Get the orthogonal Q matrix in the QR decomposition, A = QR.
UpperTriangularMatrix	R() Get the upper triangular matrix R in the QR decomposition, A = QR.
int	rank() Get the numerical rank of A as computed by the QR decomposition.
Matrix	squareQ() Get the square Q matrix.
Matrix	tallR() Get the tall R matrix.

Methods inherited from class java.lang.Object

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

Constructor Detail

QR

public QR(Matrix A,
          double epsilon)

Run the QR decomposition on a matrix.

Parameters:

A - a matrix

epsilon - a precision parameter: when a number |x| ≤ ε, it is considered 0

QR

public QR(Matrix A)

Run the QR decomposition on a matrix.

Parameters:

A - a matrix

Method Detail

P

public PermutationMatrix P()

Description copied from interface: QRDecomposition

Get P, the pivoting matrix in the QR decomposition.

Specified by:

P in interface QRDecomposition

Returns:

P

Q

public Matrix Q()

Description copied from interface: QRDecomposition

Get the orthogonal Q matrix in the QR decomposition, A = QR. The dimension of Q is m x n, the same as A, the matrix to be orthogonalized.

Specified by:

Q in interface QRDecomposition

Returns:

Q

R

public UpperTriangularMatrix R()

Description copied from interface: QRDecomposition

Get the upper triangular matrix R in the QR decomposition, A = QR. The dimension of R is n x n, a square matrix.

Specified by:

R in interface QRDecomposition

Returns:

R

rank

public int rank()

Description copied from interface: QRDecomposition

Get the numerical rank of A as computed by the QR decomposition. Numerical determination of rank requires a criterion to decide when a value should be treated as zero, hence a precision parameter. This is a practical choice which depends on both the matrix and the application. For instance, for a matrix with a big first eigenvector, we should accordingly decrease the precision to compute the rank.

Specified by:

rank in interface QRDecomposition

Returns:

the rank of A

squareQ

public Matrix squareQ()

Description copied from interface: QRDecomposition

Get the square Q matrix. This is an arbitrary orthogonal completion of the Q matrix in the QR decomposition. The dimension is m x m (square). We have A = sqQ * tallR.

Specified by:

squareQ in interface QRDecomposition

Returns:

the square Q matrix

tallR

public Matrix tallR()

Description copied from interface: QRDecomposition

Get the tall R matrix. This is completed by binding zero rows beneath the square upper triangular matrix R in the QR decomposition. The dimension is m x n. It may not be square. We have A = sqQ * tallR.

Specified by:

tallR in interface QRDecomposition

Returns:

the tall R matrix