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QR Decomposition without Pivoting
=================================

Given the matrix :math:`X` of size :math:`n \times p`, the problem is to compute the QR decomposition
:math:`X = QR`, where

- :math:`Q` is an orthogonal matrix of size :math:`n \times n`
- :math:`R` is a rectangular upper triangular matrix of size :math:`n \times p`

The library requires :math:`n > p`. In this case:

.. math::

    X = QR = [Q_1, Q_2] \cdot \begin{bmatrix} R_1 \\ 0 \end{bmatrix} = Q_1 R_1

where the matrix :math:`Q_1` has the size :math:`n \times p` and :math:`R_1` has the size :math:`p \times p`.

Computation
***********

The following computation modes are available:

.. toctree::
   :maxdepth: 1

   without-pivoting/computation-batch-online.rst
   without-pivoting/computation-distributed.rst

Examples
********

.. tabs::

  .. tab:: C++ (CPU)

    Batch Processing:

    - :cpp_example:`qr_dense_batch.cpp <qr/qr_dense_batch.cpp>`

    Online Processing:

    - :cpp_example:`qr_dense_online.cpp <qr/qr_dense_online.cpp>`

    Distributed Processing:

    - :cpp_example:`qr_dense_distr.cpp <qr/qr_dense_distr.cpp>`

  .. tab:: Java*

    .. note:: There is no support for Java on GPU.

    Batch Processing:

    - :java_example:`QRDenseBatch.java <qr/QRDenseBatch.java>`

    Online Processing:

    - :java_example:`QRDenseOnline.java <qr/QRDenseOnline.java>`

    Distributed Processing:

    - :java_example:`QRDenseDistr.java <qr/QRDenseDistr.java>`

  .. tab:: Python*

    Batch Processing:

    - :daal4py_example:`qr_batch.py`

    Online Processing:

    - :daal4py_example:`qr_streaming.py`

    Distributed Processing:

    - :daal4py_example:`qr_spmd.py`