.. ******************************************************************************
.. * Copyright 2020 Intel Corporation
.. *
.. * Licensed under the Apache License, Version 2.0 (the "License");
.. * you may not use this file except in compliance with the License.
.. * You may obtain a copy of the License at
.. *
.. *     http://www.apache.org/licenses/LICENSE-2.0
.. *
.. * Unless required by applicable law or agreed to in writing, software
.. * distributed under the License is distributed on an "AS IS" BASIS,
.. * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
.. * See the License for the specific language governing permissions and
.. * limitations under the License.
.. *******************************************************************************/

Pivoted QR Decomposition
========================

Given the matrix :math:`X` of size :math:`n \times p`, the problem is to compute the QR decomposition
with column pivoting :math:`XP = 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`
- :math:`P` is a permutation matrix of size :math:`n \times n`

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

.. math::

    XP = 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`.

Batch Processing
****************

Algorithm Input
---------------

Pivoted QR decomposition accepts the input described below.
Pass the ``Input ID`` as a parameter to the methods that provide input for your algorithm.
For more details, see :ref:`algorithms`.

.. tabularcolumns::  |\Y{0.2}|\Y{0.8}|

.. list-table:: Algorithm Input for Pivoted QR Decomposition (Batch Processing)
   :widths: 10 60
   :header-rows: 1

   * - Input ID
     - Input
   * - ``data``
     - Pointer to the numeric table that represents the :math:`n \times p` matrix :math:`X` to be factorized.
       The input can be an object of any class derived from ``NumericTable``.

Algorithm Parameters
--------------------

Pivoted QR decomposition has the following parameters:

.. tabularcolumns::  |\Y{0.15}|\Y{0.15}|\Y{0.7}|

.. list-table:: Algorithm Parameters for Pivoted QR Decomposition (Batch Processing)
   :header-rows: 1
   :widths: 10 10 60
   :align: left
   :class: longtable

   * - Parameter
     - Default Value
     - Description
   * - ``algorithmFPType``
     - ``float``
     - The floating-point type that the algorithm uses for intermediate computations. Can be ``float`` or ``double``.
   * - ``method``
     - ``defaultDense``
     - Performance-oriented computation method, the only method supported by the algorithm.
   * - ``permutedColumns``
     - Not applicable
     - Pointer to the numeric table with the :math:`1 \times p` matrix with the information for the permutation:

       - If the :math:`i`-th element is zero, the :math:`i`-th column of the input matrix is a free column
         and may be permuted with any other free column during the computation.

       - If the :math:`i`-th element is non-zero, the :math:`i`-th column of the input matrix is moved
         to the beginning of XP before the computation and remains in its place during the computation.

       .. note::
           By default, this parameter is an object of the ``HomogenNumericTable`` class, filled by zeros.
           However, you can define this parameter as an object of any class derived from ``NumericTable``
           except the ``PackedSymmetricMatrix`` class, ``CSRNumericTable`` class, and ``PackedTriangularMatrix`` class
           with the ``lowerPackedTriangularMatrix`` layout.

Algorithm Output
----------------

Pivoted QR decomposition calculates the results described below.
Pass the ``Result ID`` as a parameter to the methods that access the results of your algorithm.
For more details, see :ref:`algorithms`.

.. tabularcolumns::  |\Y{0.2}|\Y{0.8}|

.. list-table:: Algorithm Output for Pivoted QR Decomposition (Batch Processing)
   :widths: 10 60
   :header-rows: 1
   :class: longtable

   * - Result ID
     - Result
   * - ``matrixQ``
     - Pointer to the numeric table with the :math:`n \times p` matrix :math:`Q_1`.

       .. note::
           By default, this result is an object of the ``HomogenNumericTable`` class, but you can define the result as an object of any class
           derived from ``NumericTable`` except ``PackedSymmetricMatrix``, ``PackedTriangularMatrix``, and ``CSRNumericTable``.

   * - ``matrixR``
     - Pointer to the numeric table with the :math:`p \times p` upper triangular matrix :math:`R_1`.

       .. note::
           By default, this result is an object of the ``HomogenNumericTable`` class, but you can define the result as an object of any class
           derived from ``NumericTable`` except the ``PackedSymmetricMatrix`` class, ``CSRNumericTable`` class,
           and ``PackedTriangularMatrix`` class with the ``lowerPackedTriangularMatrix`` layout.

   * - ``permutationMatrix``
     - Pointer to the numeric table with the :math:`1 \times p` matrix such that
       :math:`\text{permutationMatrix}(i) = k` if the column :math:`k` of the full matrix :math:`X` is permuted into the position :math:`i` in :math:`XP`.

       .. note::
           By default, this result is an object of the ``HomogenNumericTable`` class, but you can define the result as an object of any class
           derived from ``NumericTable`` except the ``PackedSymmetricMatrix`` class, ``CSRNumericTable`` class,
           and ``PackedTriangularMatrix`` class with the ``lowerPackedTriangularMatrix`` layout.

Examples
********

.. tabs::

  .. tab:: C++ (CPU)

    Batch Processing:

    - :cpp_example:`pivoted_qr_dense_batch.cpp <pivoted_qr/pivoted_qr_dense_batch.cpp>`


  .. tab:: Python*

    Batch Processing:

    - :daal4py_example:`pivoted_qr.py`