Package org.apache.commons.math4.linear

Class QRDecomposition

java.lang.Object
org.apache.commons.math4.linear.QRDecomposition
Direct Known Subclasses:
RRQRDecomposition
public class QRDecomposition
extends java.lang.Object
Calculates the QR-decomposition of a matrix.

The QR-decomposition of a matrix A consists of two matrices Q and R that satisfy: A = QR, Q is orthogonal (QTQ = I), and R is upper triangular. If A is m×n, Q is m×m and R m×n.

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:

  • a getQT method has been added,
  • the solve and isFullRank methods have been replaced by a getSolver method and the equivalent methods provided by the returned DecompositionSolver.
Since:
1.2 (changed to concrete class in 3.0)
See Also:
MathWorld, Wikipedia

  • Constructor Summary

    Constructors 
    Constructor Description
    QRDecomposition​(RealMatrix matrix)
    Calculates the QR-decomposition of the given matrix.
    QRDecomposition​(RealMatrix matrix, double threshold)
    Calculates the QR-decomposition of the given matrix.

  • Method Summary

    Modifier and Type Method Description
    protected void decompose​(double[][] matrix)
    Decompose matrix.
    RealMatrix getH()
    Returns the Householder reflector vectors.
    RealMatrix getQ()
    Returns the matrix Q of the decomposition.
    RealMatrix getQT()
    Returns the transpose of the matrix Q of the decomposition.
    RealMatrix getR()
    Returns the matrix R of the decomposition.
    DecompositionSolver getSolver()
    Get a solver for finding the A × X = B solution in least square sense.
    protected void performHouseholderReflection​(int minor, double[][] matrix)
    Perform Householder reflection for a minor A(minor, minor) of A.

    Methods inherited from class java.lang.Object

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

  • Constructor Details

    • QRDecomposition

      public QRDecomposition​(RealMatrix matrix)
      Calculates the QR-decomposition of the given matrix. The singularity threshold defaults to zero.
      Parameters:
      matrix - The matrix to decompose.
      See Also:
      QRDecomposition(RealMatrix,double)

    • QRDecomposition

      public QRDecomposition​(RealMatrix matrix, double threshold)
      Calculates the QR-decomposition of the given matrix.
      Parameters:
      matrix - The matrix to decompose.
      threshold - Singularity threshold. The matrix will be considered singular if the absolute value of any of the diagonal elements of the "R" matrix is smaller than the threshold.

  • Method Details

    • decompose

      protected void decompose​(double[][] matrix)
      Decompose matrix.
      Parameters:
      matrix - transposed matrix
      Since:
      3.2

    • performHouseholderReflection

      protected void performHouseholderReflection​(int minor, double[][] matrix)
      Perform Householder reflection for a minor A(minor, minor) of A.
      Parameters:
      minor - minor index
      matrix - transposed matrix
      Since:
      3.2

    • getR

      public RealMatrix getR()
      Returns the matrix R of the decomposition.

      R is an upper-triangular matrix

      Returns:
      the R matrix

    • getQ

      public RealMatrix getQ()
      Returns the matrix Q of the decomposition.

      Q is an orthogonal matrix

      Returns:
      the Q matrix

    • getQT

      public RealMatrix getQT()
      Returns the transpose of the matrix Q of the decomposition.

      Q is an orthogonal matrix

      Returns:
      the transpose of the Q matrix, QT

    • getH

      public RealMatrix getH()
      Returns the Householder reflector vectors.

      H is a lower trapezoidal matrix whose columns represent each successive Householder reflector vector. This matrix is used to compute Q.

      Returns:
      a matrix containing the Householder reflector vectors

    • 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.

      Returns:
      a solver