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

Interface QRDecomposition

All Superinterfaces:

java.io.Serializable

All Known Implementing Classes:

GramSchmidt, HouseholderReflection, QR

public interface QRDecomposition
extends java.io.Serializable

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

Method Summary

Modifier and Type	Method and Description
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.

Method Detail

P

PermutationMatrix P()

Get P, the pivoting matrix in the QR decomposition.

Returns:

P

Q

Matrix Q()

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.

Returns:

Q

R

UpperTriangularMatrix R()

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

Returns:

R

rank

int rank()

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.

Returns:

the rank of A

squareQ

Matrix squareQ()

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.

Returns:

the square Q matrix

tallR

Matrix tallR()

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.

Returns:

the tall R matrix