.. _gerqf:

gerqf
=====

Computes the RQ factorization of a general m-by-n matrix. This routine
belongs to the ``oneapi::mkl::lapack`` namespace.


.. contents::
    :local:
    :depth: 1

Description
***********

The routine forms the ``RQ`` factorization of a general
``m``-by-``n`` matrix ``A`` No pivoting is performed.


The routine does not form the matrix ``Q`` explicitly. Instead, ``Q``
is represented as a product of min(``m``, ``n``) elementary
reflectors. Routines are provided to work with ``Q`` in this
representation.


.. note::
   This routine supports the Progress Routine feature.


API
***


Syntax
------

.. code-block:: cpp

    namespace oneapi::mkl::lapack {
      void gerqf(cl::sycl::queue &queue,
      std::int64_t m,
      std::int64_t n,
      cl::sycl::buffer<T> &a,
      std::int64_t lda,
      cl::sycl::buffer<T> &tau,
      cl::sycl::buffer<T> &scratchpad,
      std::int64_t scratchpad_size)
   }

``gerqf`` supports the following precisions and devices:


.. list-table::
   :header-rows: 1

   * -  T
     -  Devices supported
   * -  ``float``
     -  Host and CPU
   * -  ``double``
     -  Host and CPU
   * -  ``std::complex<float>``
     -  Host and CPU
   * -  ``std::complex<double>``
     -  Host and CPU


Input Parameters
----------------

queue
   Device queue where calculations will be performed.


m
   The number of rows in the matrix ``A`` (``0≤m``).


n
   The number of columns in the matrix ``A`` (``0≤n``).


a
   Buffer holding input matrix ``A``. The second dimension of ``a``
   must be at least ``max(1, n)``.


lda
   The leading dimension of ``a``, at least ``max(1, m)``.


scratchpad
   Buffer holding scratchpad memory to be used by the routine for
   storing intermediate results.


scratchpad_size
   Size of scratchpad memory as a number of floating point elements
   of type ``T``. Size should not be less than the value returned by
   the :ref:`gerqf_scratchpad_size` function.


Output Parameters
-----------------

a
   Overwritten by the factorization data as follows:


   if ``m ≤ n``, the upper triangle of the subarray
   ``a(1:m, n-m+1:n )`` contains the ``m``-by-``m`` upper triangular
   matrix ``R``; if ``m ≥ n``, the elements on and above the
   (``m``-``n``)-th subdiagonal contain the ``m``-by-``n`` upper
   trapezoidal matrix ``R``


   In both cases, the remaining elements, with the array\ ``tau``,
   represent the orthogonal/unitary matrix ``Q`` as a product of
   ``min(m,n)`` elementary reflectors.


tau
   Array, size at least ``min(m,n)``.


   Contains scalars that define elementary reflectors for the matrix
   ``Q`` in its decomposition in a product of elementary reflectors.


Exceptions
----------

.. tabularcolumns:: |\Y{0.3}|\Y{0.7}|

.. list-table::
   :header-rows: 1

   * - Exception
     - Description

   * -     ``mkl::lapack::exception``
     -     This exception is thrown when problems occur during    calculations. You can obtain the info code of the problem using the   info() method of the exception object:

           If   ``info = -i``, the ``i``-th parameter had an illegal   value.

           If ``info`` is equal to the value passed as   scratchpad size, and detail() returns non zero, then the passed   scratchpad has an insufficient size, and the required size should not   be less than the value returned by the detail() method of the   exception object.