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

All Known Implementing Classes:

GramSchmidt, HouseholderReflection, QR

public interface QRDecomposition

All QR algorithms implement this interface.

QR decomposition decomposes a m x n matrix A so that

 A = Q %*% R
 

where

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

  Method Summary

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

  Method Detail

  P

  PermutationMatrix P()

  Get a copy of P, the pivoting matrix in the QR decomposition.

  Returns:

  a copy of the P pivoting matrix in the QR decomposition

  Q

  Matrix Q()

  Get a copy of the orthogonal Q matrix in the QR decomposition.

   A = QR
   

  Dimension of Q is nrows x ncols, same as A, the matrix to orthogonalize.

  Returns:

  a copy of the Q matrix in the QR decomposition

  R

  UpperTriangularMatrix R()

  Get a copy of the upper triangular matrix R in the QR decomposition.

   A = QR
   

  Dimension of R is ncols x ncols, a square matrix.

  Returns:

  a copy of the upper triangular matrix R in the QR decomposition

  rank

  int rank()

  Get the numerical rank of the matrix A as computed by the QR decomposition.

  Numerical determination of rank requires a criterion to decide when a value should be treated as zero.

  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.

  You may need to change the precision parameter to accurately compute the rank. See the test cases for example.

  Returns:

  the rank of A

  squareQ

  Matrix squareQ()

  Get a copy of the square Q matrix. This is an arbitrary orthogonal completion of the Q matrix in the QR decomposition. Dimension is nrows x nrows (square).

   A = square_Q %*% tall_R
   

  Returns:

  a copy of the square Q matrix

  tallR

  Matrix tallR()

  Get a copy of the tall R matrix. This is completed by binding zero rows beneath the square upper triangular matrix R in the QR decomposition. Dimension is nrows x ncols. Note that this may no longer be square.

   A = square_Q %*% tall_R
   

  Returns:

  a copy of the tall R matrix