Class QRDecomposition

  • Direct Known Subclasses:

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    • the solve and isFullRank methods have been replaced by a getSolver method and the equivalent methods provided by the returned DecompositionSolver.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

      • QRDecomposition

        public QRDecomposition​(RealMatrix matrix,
                               double threshold)
        Calculates the QR-decomposition of the given matrix.
        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 Detail

      • decompose

        protected void decompose​(double[][] matrix)
        Decompose matrix.
        matrix - transposed matrix
      • performHouseholderReflection

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

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

        R is an upper-triangular matrix

        the R matrix
      • getQ

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

        Q is an orthogonal matrix

        the Q matrix
      • getQT

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

        Q is an orthogonal matrix

        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.

        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.

        a solver