Class 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    • the solve and isFullRank methods have been replaced by a getSolver method and the equivalent methods provided by the returned DecompositionSolver.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
    • Nested Class Summary

      Nested Classes 
      Modifier and Type Class Description
      private static class  QRDecomposition.Solver
      Specialized solver.
    • 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

      All Methods Instance Methods Concrete Methods 
      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
    • Field Detail

      • 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 Detail

      • 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.
    • Method Detail

      • 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