.. _hegvd-usm-version:

hegvd (USM Version)
===================

Computes all the eigenvalues, and optionally, the eigenvectors of a
complex generalized Hermitian positive-definite eigenproblem using a
divide and conquer method. This routine belongs to the
``oneapi::mkl::lapack`` namespace.


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

The routine computes all the eigenvalues, and optionally, the
eigenvectors of a complex generalized Hermitian positive-definite
eigenproblem, of the form


``A*x = λ*B*x, A*B*x = λ*x``, or ``B*A*x = λ*x``.


Here ``A`` and ``B`` are assumed to be Hermitian and ``B`` is also
positive definite.


It uses a divide and conquer algorithm.


API
***


Syntax
------

.. code-block:: cpp

   namespace oneapi::mkl::lapack {
     cl::sycl::event hegvd(cl::sycl::queue &queue,
     std::int64_t itype,
     mkl::job jobz,
     mkl::uplo uplo,
     std::int64_t n,
     T *a,
     std::int64_t lda,
     T *b,
     std::int64_t ldb,
     T *w,
     T *scratchpad,
     std::int64_t scratchpad_size,
     const std::vector<cl::sycl::event> &events = {})
   }

``hegvd`` (USM version) supports the following precision and
devices.

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

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


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

queue
   Device queue where calculations will be performed.


itype
   Must be 1 or 2 or 3. Specifies the problem type to be solved:


   if itype\ ``= 1``, the problem type is ``A*x = lambda*B*x;``


   if itype\ ``= 2``, the problem type is ``A*B*x = lambda*x;``


   if itype\ ``= 3``, the problem type is ``B*A*x = lambda*x``.


jobz
   Must be ``job::novec`` or ``job::vec``.


   If ``jobz = job::novec``, then only eigenvalues are computed.


   If ``jobz = job::vec``, then eigenvalues and eigenvectors are
   computed.


uplo
   Must be ``uplo::upper`` or ``uplo::lower``.


   If ``uplo = uplo::upper``, a and b store the upper triangular part
   of ``A`` and ``B``.


   If ``uplo = uplo::lower``, a and b stores the lower triangular
   part of ``A`` and ``B``.


n
   The order of the matrices ``A`` and ``B`` (``0≤n``).


a
   Pointer to the array of size ``a(lda,*)`` containing the upper or
   lower triangle of the Hermitian matrix ``A``, as specified by
   uplo.


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


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


b
   Pointer to the array of size ``b(ldb,*)`` containing the upper or
   lower triangle of the Hermitian matrix ``B``, as specified by
   uplo.


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


ldb
   The leading dimension of b; at least ``max(1,n)``.


scratchpad
   Pointer to 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:`hegvd_scratchpad_size`
   function.


events
   List of events to wait for before starting computation. Defaults
   to empty list.


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

a
   On exit, if ``jobz = job::vec``, then if ``info = 0``, a contains
   the matrix ``Z`` of eigenvectors. The eigenvectors are normalized
   as follows:


   if itype\ ``= 1`` or ``2``, ``Z``\ :sup:`H`\ ``*B*Z = I``;


   if itype\ ``= 3``, ``Z``\ :sup:`H`\ \*\ ``inv(B)*Z = I``;


   If ``jobz = job::novec``, then on exit the upper triangle (if
   ``uplo = uplo::upper``) or the lower triangle (if
   ``uplo = uplo::lower``) of ``A``, including the diagonal, is
   destroyed.


b
   On exit, if ``info≤n``, the part of b containing the matrix is
   overwritten by the triangular factor ``U`` or ``L`` from the
   Cholesky factorization ``B = U``\ :sup:`H`\ ``*U``\ or ``B`` =
   ``L``\ \*\ ``L``\ :sup:`H`.


w
   Pointer to the array of size at least n. If ``info = 0``, contains
   the eigenvalues of the matrix ``A`` in ascending order. See also
   info.


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.      For ``info ≤ n``:

           If   ``info = i``, and ``jobz = job::novec``, then the algorithm failed to   converge; ``i`` indicates the number of off-diagonal elements of an   intermediate tridiagonal form which did not converge to   zero.

           If ``info = i``, and ``jobz = job:vec``, then   the algorithm failed to compute an eigenvalue while working on the   submatrix lying in rows and columns ``info/(n+1)`` through   ``mod(info,n+1)``.       For ``info > n``:

           If ``info = n + i``, for ``1 ≤ i ≤ n``, then the leading minor of   order ``i`` of ``B`` is not positive-definite. The factorization of   ``B`` could not be completed and no eigenvalues or eigenvectors were   computed.

           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
-------------

Output event to wait on to ensure computation is complete.