org.apache.commons.math3.linear

## 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  - the `solve` and `isFullRank` methods have been replaced by a `getSolver` method and the equivalent methods provided by the returned `DecompositionSolver`.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:
    MathWorld, Wikipedia

- - ### Constructor Summary

    Constructors
    Constructor and 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

    Methods
    Modifier and Type | Method and 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`

- - ### 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

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