.. _trtrs:

trtrs
=====

Solves a system of linear equations with a triangular 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 with a triangular matrix ``A``, with multiple right-hand
sides stored in ``B``:

.. list-table::
   :header-rows: 0

   * -     A*X = B
     -
     -     if ``transa`` =\ ``transpose::nontrans``,
   * -     \ ``AT*X = B``\
     -
     -     if ``transa`` =\ ``transpose::trans``,
   * -     A\ :sup:`H`\ ``*X`` = B
     -
     -     if ``transa`` =\ ``transpose::conjtrans`` (for complex    matrices only).


API
***


Syntax
------

.. code-block:: cpp

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

``trtrs`` 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.


uplo
   Indicates whether ``A`` is upper or lower triangular:


   If uplo = ``uplo::upper``, then ``A`` is upper triangular.


   If uplo = ``uplo::lower``, then ``A`` is lower triangular.


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


   If transa = ``transpose::trans``, then ``A``\ :sup:`T`\ \*\ ``X``
   = ``B`` is solved for ``X``.


   If transa = ``transpose::conjtrans``, then
   ``A``\ :sup:`H`\ \*\ ``X`` = ``B`` is solved for ``X``.


diag
   If diag = ``diag::nonunit``, then ``A`` is not a unit triangular
   matrix.


   If diag = ``diag::unit``, then ``A`` is unit triangular: diagonal
   elements of ``A`` are assumed to be 1 and not referenced in the
   array a.


n
   The order of ``A``; the number of rows in ``B``; n\ ``≥ 0``.


nrhs
   The number of right-hand sides; nrhs\ ``≥ 0``.


a
   Array containing the matrix ``A``.


   The second dimension of a must be at least ``max(1,n)``.


lda
   The leading dimension of ``a``; lda\ ``≥ max(1, n)``.


b
   Array containing the matrix ``B`` whose columns are the right-hand
   sides for the systems of equations.


   The second dimension of b at least ``max(1,nrhs)``.


ldb
   The leading dimension of b; ldb\ ``≥ max(1, n)``.


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:`trtrs_scratchpad_size`
   function.


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


b
   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`` 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.


Return Values
-------------

info
   Buffer containing error information.

   If ``info`` = 0, the execution is successful.

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


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

GPU support for this function does not include error reporting through the ``info`` parameter.