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.

It decomposes a 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

  This decomposition can be used to

  • solve a linear least square problem
  • compute the rank

  See Also:

  Wikipedia: QR decomposition

  Nested Class Summary

  static class

  QR.Method the methods available to do the QR decomposition

  Field Summary

  double

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

  Constructor Summary

  QR(Matrix A)
  Construct an instance of the QR decomposition.

  QR(Matrix A, QR.Method method, double epsilon)
  Construct an instance of the QR decomposition.

  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.

  Methods inherited from class java.lang.Object

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

  Field Detail

  epsilon

  public final double epsilon

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

  Constructor Detail

  QR

  public QR(Matrix A,
            QR.Method method,
            double epsilon)

  Construct an instance of the QR decomposition. The user specifies one of the algorithm in QR.Method.

  Parameters:

  A - a matrix

  method - a QR algorithm implementation

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

  QR

  public QR(Matrix A)

  Construct an instance of the QR decomposition.

  Parameters:

  A - a matrix

  Method Detail

  Q

  public Matrix Q()

  Description copied from interface: QRDecomposition

  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.

  Specified by:

  Q in interface QRDecomposition

  Returns:

  a copy of the Q matrix in the QR decomposition

  R

  public UpperTriangularMatrix R()

  Description copied from interface: QRDecomposition

  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.

  Specified by:

  R in interface QRDecomposition

  Returns:

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

  P

  public PermutationMatrix P()

  Description copied from interface: QRDecomposition

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

  Specified by:

  P in interface QRDecomposition

  Returns:

  a copy of the P pivoting matrix in the QR decomposition

  rank

  public int rank()

  Description copied from interface: QRDecomposition

  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.

  Specified by:

  rank in interface QRDecomposition

  Returns:

  the rank of A

  squareQ

  public Matrix squareQ()

  Description copied from interface: QRDecomposition

  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
   

  Specified by:

  squareQ in interface QRDecomposition

  Returns:

  a copy of the square Q matrix

  tallR

  public Matrix tallR()

  Description copied from interface: QRDecomposition

  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
   

  Specified by:

  tallR in interface QRDecomposition

  Returns:

  a copy of the tall R matrix