org.apache.commons.math3.linear

## Class RRQRDecomposition

- java.lang.Object
- - org.apache.commons.math3.linear.QRDecomposition
  - - org.apache.commons.math3.linear.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  - the `solve` and `isFullRank` methods have been replaced by a `getSolver` method and the equivalent methods provided by the returned `DecompositionSolver`.
  Since:
    3.2
  See Also:
    MathWorld, Wikipedia

- - ### Constructor Summary

    Constructors
    Constructor and Description
    `RRQRDecomposition(RealMatrix matrix)` Calculates the QR-decomposition of the given matrix.
    `RRQRDecomposition(RealMatrix matrix, double threshold)` Calculates the QR-decomposition of the given matrix.

  - ### Method Summary

    Methods
    Modifier and Type | Method and Description
    --- | ---
    `protected void` | `decompose(double[][] qrt)` Decompose matrix.
    `RealMatrix` | `getP()` Returns the pivot matrix, P, used in the QR Decomposition of matrix A such that AP = QR.
    `int` | `getRank(double dropThreshold)` Return the effective numerical matrix rank.
    `DecompositionSolver` | `getSolver()` Get a solver for finding the A × X = B solution in least square sense.
    `protected void` | `performHouseholderReflection(int minor, double[][] qrt)` Perform Householder reflection for a minor A(minor, minor) of A.

    - ### Methods inherited from class org.apache.commons.math3.linear.QRDecomposition

      `getH, getQ, getQT, getR`

    - ### Methods inherited from class java.lang.Object

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

- - ### Constructor Detail

    - #### RRQRDecomposition

          public RRQRDecomposition(RealMatrix matrix)

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

    - #### RRQRDecomposition

          public RRQRDecomposition(RealMatrix matrix,
                           double threshold)

      Calculates the QR-decomposition of the given matrix.
      Parameters:
        `matrix` - The matrix to decompose.
        `threshold` - Singularity threshold.
      See Also:
        `RRQRDecomposition(RealMatrix)`

  - ### Method Detail

    - #### decompose

          protected void decompose(double[][] qrt)

      Decompose matrix.
      Overrides:
        `decompose` in class `QRDecomposition`
      Parameters:
        `qrt` - transposed matrix

    - #### performHouseholderReflection

          protected void performHouseholderReflection(int minor,
                                          double[][] qrt)

      Perform Householder reflection for a minor A(minor, minor) of A.
      Overrides:
        `performHouseholderReflection` in class `QRDecomposition`
      Parameters:
        `minor` - minor index
        `qrt` - transposed matrix

    - #### 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.
      Returns:
        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

      Parameters:
        `dropThreshold` - threshold triggering rank computation
      Returns:
        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.

      Overrides:
        `getSolver` in class `QRDecomposition`
      Returns:
        a solver

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