.. _potrs:

potrs
=====


Solves a system of linear equations with a Cholesky-factored symmetric
(Hermitian) positive-definite coefficient matrix. This routine belongs
to the ``oneapi::mkl::lapack`` namespace.


.. contents::
    :local:
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Description
***********

The routine solves for ``X`` the system of linear equations
``A*X = B`` with a symmetric positive-definite or, for complex data,
Hermitian positive-definite matrix ``A``, given the Cholesky
factorization of ``A``:


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

   * -  ``A = UT*U`` for real data, ``A = UH*U`` for complex data
     -  if ``uplo=mkl::uplo::upper``
   * -  ``A = L*LT`` for real data, ``A = L*LH`` for complex data
     -  if ``uplo=mkl::uplo::lower``

where ``L`` is a lower triangular matrix and ``U`` is upper
triangular. The system is solved with multiple right-hand sides
stored in the columns of the matrix ``B``.


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


API
***


Syntax
------

.. code-block:: cpp

   namespace oneapi::mkl::lapack {
     void potrs(cl::sycl::queue &queue, mkl::uplo uplo,
     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)
   }

``potrs`` 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 how the input matrix has been factored:


   If uplo=\ ``mkl::uplo::upper``, the upper triangle ``U`` of ``A``
   is stored, where ``A`` = ``U``\ :sup:`T`\ \*\ ``U`` for real data,
   ``A`` = ``U``\ :sup:`H`\ \*\ ``U`` for complex data.


   If uplo=\ ``mkl::uplo::lower``, the upper triangle ``L`` of ``A``
   is stored, where ``A`` = ``L``\ \*\ ``L``\ :sup:`T` for real data,
   ``A`` = ``L``\ \*\ ``L``\ :sup:`H` for complex data.


n
   The order of matrix ``A`` (0≤n).


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


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


lda
   The leading dimension of a.


b
   Buffer holding the data of 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:`potrs_scratchpad_size`
   function.


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

b
   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 the Cholesky factor 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
-----------------

- GPU support is for only real precisions.

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