.. _getrs:

getrs
=====

Solves a system of linear equations with an LU-factored square
coefficient matrix, with multiple right-hand sides. This routine belongs
to the ``oneapi::mkl::lapack`` namespace.


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

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

The routine solves for ``X`` the following systems of linear
equations:


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

   * - Equation
     - Condition

   * -     \ ``A*X = B``\
     -     if ``trans``\ =\ ``mkl::transpose::nontrans``\
   * -     \ ``AT*X = B``\
     -     if ``trans``\ =\ ``mkl::transpose::trans``\
   * -     \ ``AH*X = B``\
     -     if ``trans``\ =\ ``mkl::transpose::conjtrans``\




Before calling this routine, you must call :ref:`getrf` to compute the ``LU`` factorization of ``A``.


API
***


Syntax
------

.. code-block:: cpp

   namespace oneapi::mkl::lapack {
     void getrs(cl::sycl::queue &queue,
     mkl::transpose trans,
     std::int64_t n,
     std::int64_t nrhs,
     cl::sycl::buffer<T> &a,
     std::int64_t lda,
     std::int64_t *ipiv,
     cl::sycl::buffer<T> &b,
     std::int64_t ldb,
     cl::sycl::buffer<T> &scratchpad,
     std::int64_t   scratchpad_size)
   }

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

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

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


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

queue
   Device queue where calculations will be performed.


trans
   Indicates the form of the equations:


   If ``trans=mkl::transpose::nontrans``, then ``A*X = B`` is solved
   for ``X``.


   If ``trans=mkl::transpose::trans``, then ``AT*X = B`` is solved
   for ``X``.


   If ``trans=mkl::transpose::conjtrans``, then ``AH*X = B`` is
   solved for ``X``.


n
   The order of the matrix ``A`` and the number of rows in matrix
   ``B(0≤n)``.


nrhs
   The number of right-hand sides (``0≤nrhs``).


a
   Buffer holding the array containing the factorization of the
   matrix ``A``, as returned by
   :ref:`getrf`. The
   second dimension of a must be at least ``max(1, n)``.


lda
   The leading dimension of a.


ipiv
   Array, size at least ``max(1, n)``. The ipiv array, as returned by
   :ref:`getrf`.


b
   The array b contains the matrix ``B`` whose columns are the
   right-hand sides for the systems of equations. The second
   dimension of b must be at least ``max(1,nrhs)``.


ldb
   The leading dimension of b.


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:`getrs_scratchpad_size`
   function.


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

b
   The buffer b is overwritten by the solution matrix ``X``.


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 = i``, the ``i``-th diagonal element   of ``U`` is zero, and the solve could not be   completed.

           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.


Known Limitations
-----------------

For GPU support, errors are reported through the info parameter, but computation does not halt for an algorithmic error.