Class QRDecomposition

java.lang.Object
org.apache.commons.math3.linear.QRDecomposition
Direct Known Subclasses:
RRQRDecomposition

public class QRDecomposition extends 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  • the solve and isFullRank methods have been replaced by a getSolver method and the equivalent methods provided by the returned DecompositionSolver.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:
  • Field Details

    • qrt

      private double[][] qrt
      A packed TRANSPOSED representation of the QR decomposition.

      The elements BELOW the diagonal are the elements of the UPPER triangular matrix R, and the rows ABOVE the diagonal are the Householder reflector vectors from which an explicit form of Q can be recomputed if desired.

    • rDiag

      private double[] rDiag
      The diagonal elements of R.
    • cachedQ

      private RealMatrix cachedQ
      Cached value of Q.
    • cachedQT

      private RealMatrix cachedQT
      Cached value of QT.
    • cachedR

      private RealMatrix cachedR
      Cached value of R.
    • cachedH

      private RealMatrix cachedH
      Cached value of H.
    • threshold

      private final double threshold
      Singularity threshold.
  • 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

      public QRDecomposition(RealMatrix matrix, double threshold)
      Calculates the QR-decomposition of the given matrix.
      Parameters:
      matrix - The matrix to decompose.
      threshold - Singularity 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