Class RRQRDecomposition

  • public class RRQRDecomposition
    extends QRDecomposition
    Calculates the rank-revealing QR-decomposition of a matrix, with column pivoting.

    The rank-revealing QR-decomposition of a matrix A consists of three matrices Q, R and P such that AP=QR. Q is orthogonal (QTQ = I), and R is upper triangular. If A is m×n, Q is m×m and R is m×n and P is n×n.

    QR decomposition with column pivoting produces a rank-revealing QR decomposition and the getRank(double) method may be used to return the rank of the input matrix A.

    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    • the solve and isFullRank methods have been replaced by a getSolver method and the equivalent methods provided by the returned DecompositionSolver.
    See Also:
    MathWorld, Wikipedia
    • Method Detail

      • getP

        public RealMatrix getP()
        Returns the pivot matrix, P, used in the QR Decomposition of matrix A such that AP = QR. If no pivoting is used in this decomposition then P is equal to the identity matrix.
        a permutation matrix.
      • getRank

        public int getRank​(double dropThreshold)
        Return the effective numerical matrix rank.

        The effective numerical rank is the number of non-negligible singular values.

        This implementation looks at Frobenius norms of the sequence of bottom right submatrices. When a large fall in norm is seen, the rank is returned. The drop is computed as:

           (thisNorm/lastNorm) * rNorm < dropThreshold

        where thisNorm is the Frobenius norm of the current submatrix, lastNorm is the Frobenius norm of the previous submatrix, rNorm is is the Frobenius norm of the complete matrix

        dropThreshold - threshold triggering rank computation
        effective numerical matrix rank
      • 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.

        getSolver in class QRDecomposition
        a solver