.. _potrs_batch-buffer-strided-version:

potrs_batch (Buffer Strided Version)
====================================

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


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

The routine solves for ``X``\ :sub:`i` the system of linear equations
``A``\ :sub:`i`\ \*\ ``X``\ :sub:`i` = ``B``\ :sub:`i` with a
symmetric positive-definite or, for complex data, Hermitian
positive-definite matrices ``A``\ :sub:`i`, given the Cholesky
factorization of ``A``\ :sub:`i`, ``i ϵ{1...batch_size}`` :


- ``A``\ :sub:`i` = ``U``\ :sub:`i`\ :sup:`T`\ \*\ ``U``\ :sub:`i` for real data, ``A``\ :sub:`i` = ``U``\ :sub:`i`\ :sup:`H`\ \*\ ``U``\ :sub:`i` for complex data if uplo=\ ``mkl::uplo::upper``

- ``A``\ :sub:`i` = ``L``\ :sub:`i`\ \*\ ``L``\ :sub:`i`\ :sup:`T` for real data, ``A``\ :sub:`i` = ``L``\ :sub:`i`\ \*\ ``L``\ :sub:`i`\ :sup:`H` for complex data if uplo=\ ``mkl::uplo::lower``


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


Before calling this routine, matrices ``A``\ :sub:`i` should be
factorized by a call to :ref:`potrf_batch-buffer-strided-version`.


API
***


Syntax
------

.. code-block:: cpp

   namespace oneapi::mkl::lapack {
     void potrs_batch(cl::sycl::queue &queue,
     mkl::uplo uplo,
     std::int64_t n,
     std::int64_t nrhs,
     cl::sycl::buffer<T> &a,
     std::int64_t lda,
     std::int64_t stride_a,
     cl::sycl::buffer<T> &b,
     std::int64_t ldb,
     std::int64_t stride_b,
     std::int64_t batch_size,
     cl::sycl::buffer<T> &scratchpad,
     std::int64_t scratchpad_size)
   }

Function 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``\ :sub:`i`
   of ``A``\ :sub:`i` is stored, where ``A``\ :sub:`i` =
   ``U``\ :sub:`i`\ :sup:`T`\ \*\ ``U``\ :sub:`i` for real data,
   ``A``\ :sub:`i` = ``U``\ :sub:`i`\ :sup:`H`\ \*\ ``U``\ :sub:`i`
   for complex data.


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


n
   The order of the matrices ``A``\ :sub:`i` (``0 ≤ n``).


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


a
   Array containing the batch of factorizations of the matrices
   ``A``\ :sub:`i`, as returned by :ref:`potrf_batch-buffer-strided-version`.


lda
   The leading dimension of ``A``\ :sub:`i`.


stride_a
   The stride between the beginnings of matrices inside the batch
   array ``a``.


b
   The array containing the batch of matrices ``B``\ :sub:`i` whose
   columns are the right-hand sides for the systems of equations.


ldb
   The leading dimensions of ``B``\ :sub:`i`.


stride_b
   The stride between the beginnings of matrices ``B``\ :sub:`i`
   inside the batch array ``b``.


batch_size
   Specifies the number of problems in a batch.


scratchpad
   Scratchpad memory to be used by 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 then the value returned by
   stride version of :ref:`potrs_batch_scratchpad_size-strided-version`
   function.


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

b
   The batch array b is overwritten by the solution matrix
   ``X``\ :sub:`i`.


Exceptions
----------


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

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

   * - Exception
     - Description

   * -     ``mkl::lapack::batch_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 = -n``, the ``n``-th parameter had an illegal   value.

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

           If ``info`` is zero, then the diagonal   element of some of ``U``\ :sub:`i` is zero, and the solve could not   be completed. The indexes of such matrices in the batch can be   obtained with the ids() method of the exception object. You can   obtain the indexes of the first zero diagonal elements in these   ``U``\ :sub:`i` matrices using the infos() method of the exception   object.