.. _gesvd-usm-version:

gesvd (USM Version)
===================

Computes the singular value decomposition of a general rectangular
matrix. This routine belongs to the ``oneapi::mkl::lapack`` namespace.


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

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

The routine computes the singular value decomposition (SVD) of a
real/complex ``m``-by-``n`` matrix ``A``, optionally computing the
left and/or right singular vectors. The SVD is written as:

- ``A = U*Σ*VT`` for real routines

- ``A = U*Σ*VH`` for complex routines


where Σ is an ``m``-by-``n`` diagonal matrix, ``U`` is an
``m``-by-``m`` orthogonal/unitary matrix, and ``V`` is an
``n``-by-``n`` orthogonal/unitary matrix. The diagonal elements of Σ
are the singular values of ``A``; they are real and non-negative, and
are returned in descending order. The first ``min(m, n)`` columns of
``U`` and ``V`` are the left and right singular vectors of ``A``.


API
***


Syntax
------

.. code-block:: cpp

   namespace oneapi::mkl::lapack {
     cl::sycl::event gesvd(cl::sycl::queue &queue,
     mkl::jobsvd jobu,
     mkl::jobsvd jobvt,
     std::int64_t m,
     std::int64_t n,
     T *a,
     std::int64_t lda,
     RealT *s,
     T *u,
     std::int64_t ldu,
     T *vt,
     std::int64_t ldvt,
     T *scratchpad,
     std::int64_t scratchpad_size,
     const std::vector<cl::sycl::event> &events = {})
   }

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

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

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


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

queue
   Device queue where calculations will be performed.


jobu
   Must be ``jobsvd::vectors``, ``job::somevec``,
   ``jobsvd::vectorsina``, or ``job::novec``. Specifies options for
   computing all or part of the matrix ``U``.


   If ``jobu = jobsvd::vectors``, all ``m`` columns of ``U`` are
   returned in the array u;


   if ``jobu = job::somevec``, the first ``min(m, n)`` columns of
   ``U`` (the left singular vectors) are returned in the array u;


   if ``jobu = jobsvd::vectorsina``, the first ``min(m, n)`` columns
   of ``U`` (the left singular vectors) are overwritten on the array
   a;


   if ``jobu = job::novec``, no columns of ``U`` (no left singular
   vectors) are computed.


jobvt
   Must be ``jobsvd::vectors, job::somevec``, ``jobsvd::vectorsina``,
   or ``job::novec``. Specifies options for computing all or part of
   the matrix *V\ T/V\ H*.


   If ``jobvt = jobsvd::vectors``, all n columns of *V\ T/V\ H* are
   returned in the array vt;


   if ``jobvt = job::somevec``, the first ``min(m, n)`` columns of
   *V\ T/V\ H* (the left singular vectors) are returned in the array
   vt;


   if ``jobvt = jobsvd::vectorsina``, the first ``min(m, n)`` columns
   of *V\ T/V\ H* (the left singular vectors) are overwritten on the
   array a;


   if ``jobvt = job::novec``, no columns of *V\ T/V\ H* (no left
   singular vectors) are computed.


   jobvt and jobu cannot both be ``jobsvd::vectorsina``.


m
   The number of rows in the matrix ``A`` (``0≤m``).


n
   The number of columns in the matrix ``A`` (``0≤n``).


a
   Pointer to array a, size ``(lda,*)``. The second dimension of
   ``a`` must be at least ``max(1, m)``.


lda
   The leading dimension of a.


ldu
   The leading dimension of u.


ldvt
   The leading dimension of vt.


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:`gesvd_scratchpad_size` function.


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


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

a
   On exit,


   If ``jobu = jobsvd::vectorsina``, a is overwritten with the first
   ``min(m,n)`` columns of ``U`` (the left singular vectors stored
   columnwise);


   If ``jobvt = jobsvd::vectorsina``, a is overwritten with the first
   ``min(m, n)`` rows of ``V``\ :sup:`T`/``V``\ :sup:`H` (the right
   singular vectors stored rowwise);


   If ``jobu ≠ jobsvd::vectorsina`` and
   ``jobvt ≠ jobsvd::vectorsina``, the contents of a are destroyed.


s
   Array containing the singular values, size at least
   ``max(1, min(m,n))``. Contains the singular values of ``A`` sorted
   so that ``s(i) ≥ s(i+1)``.


u
   Array containing ``U``; the second dimension of u must be at least
   ``max(1, m)`` if ``jobu = jobsvd::vectors``, and at least
   ``max(1, min(m, n))`` if ``jobu = job::somevec``.


   If ``jobu = jobsvd::vectors``, u contains the m-by-m
   orthogonal/unitary matrix ``U``.


   If ``jobu = job::somevec``, u contains the first ``min(m, n)``
   columns of ``U`` (the left singular vectors stored column-wise).


   If ``jobu = job::novec`` or ``jobsvd::vectorsina``, u is not
   referenced.


vt
   Array containing ``V``\ :sup:`T`; the second dimension of vt must
   be at least ``max(1, n)``.


   If ``jobvt = jobsvd::vectors``, vt contains the n-by-n
   orthogonal/unitary matrix ``V``\ :sup:`T`/``V``\ :sup:`H`.


   If ``jobvt = job::somevec``, vt contains the first ``min(m, n)``
   rows of ``V``\ :sup:`T`/``V``\ :sup:`H` (the right singular
   vectors stored row-wise).


   If ``jobvt = job::novec`` or ``jobsvd::vectorsina``, vt is not
   referenced.


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   get_info() method of the exception object:

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

           If ``info = i``, then if bdsqr did not   converge, ``i`` specifies how many superdiagonals of the intermediate   bidiagonal form ``B`` did not converge to zero, and   ``scratchpad(2:min(m,n))`` contains the unconverged superdiagonal   elements of an upper bidiagonal matrix ``B`` whose diagonal is in   ``s`` (not necessarily sorted). ``B`` satisfies ``A = U*B*VT``, so it   has the same singular values as ``A``, and singular vectors related   by ``U`` and ``V``\ :sup:`T`.

           If ``info`` is equal to   the value passed as scratchpad size, and get_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   get_detail() method of the exception object.


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

Output event to wait on to ensure computation is complete.