elpa2_bandred_template.X90 71.6 KB
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#if 0
!    This file is part of ELPA.
!
!    The ELPA library was originally created by the ELPA consortium,
!    consisting of the following organizations:
!
!    - Max Planck Computing and Data Facility (MPCDF), fomerly known as
!      Rechenzentrum Garching der Max-Planck-Gesellschaft (RZG),
!    - Bergische Universität Wuppertal, Lehrstuhl für angewandte
!      Informatik,
!    - Technische Universität München, Lehrstuhl für Informatik mit
!      Schwerpunkt Wissenschaftliches Rechnen ,
!    - Fritz-Haber-Institut, Berlin, Abt. Theorie,
!    - Max-Plack-Institut für Mathematik in den Naturwissenschaften,
!      Leipzig, Abt. Komplexe Strukutren in Biologie und Kognition,
!      and
!    - IBM Deutschland GmbH
!
!    This particular source code file contains additions, changes and
!    enhancements authored by Intel Corporation which is not part of
!    the ELPA consortium.
!
!    More information can be found here:
!    http://elpa.mpcdf.mpg.de/
!
!    ELPA is free software: you can redistribute it and/or modify
!    it under the terms of the version 3 of the license of the
!    GNU Lesser General Public License as published by the Free
!    Software Foundation.
!
!    ELPA is distributed in the hope that it will be useful,
!    but WITHOUT ANY WARRANTY; without even the implied warranty of
!    MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
!    GNU Lesser General Public License for more details.
!
!    You should have received a copy of the GNU Lesser General Public License
!    along with ELPA.  If not, see <http://www.gnu.org/licenses/>
!
!    ELPA reflects a substantial effort on the part of the original
!    ELPA consortium, and we ask you to respect the spirit of the
!    license that we chose: i.e., please contribute any changes you
!    may have back to the original ELPA library distribution, and keep
!    any derivatives of ELPA under the same license that we chose for
!    the original distribution, the GNU Lesser General Public License.
!
!
! ELPA1 -- Faster replacements for ScaLAPACK symmetric eigenvalue routines
!
! Copyright of the original code rests with the authors inside the ELPA
! consortium. The copyright of any additional modifications shall rest
! with their original authors, but shall adhere to the licensing terms
! distributed along with the original code in the file "COPYING".



! ELPA2 -- 2-stage solver for ELPA
!
! Copyright of the original code rests with the authors inside the ELPA
! consortium. The copyright of any additional modifications shall rest
! with their original authors, but shall adhere to the licensing terms
! distributed along with the original code in the file "COPYING".
#endif
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    subroutine bandred_&
    &MATH_DATATYPE&
    &_&
    &PRECISION &
    (na, a, &
#if REALCASE == 1
     a_dev, &
#endif
     lda, nblk, nbw, matrixCols, numBlocks, mpi_comm_rows, mpi_comm_cols, tmat, &
#if REALCASE == 1
     tmat_dev, &
#endif
     wantDebug, useGPU, success &
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#if REALCASE == 1
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     , useQR)
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#endif
#if COMPLEXCASE == 1
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     )
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#endif
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  !-------------------------------------------------------------------------------
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  !  bandred_real/complex: Reduces a distributed symmetric matrix to band form
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  !
  !  Parameters
  !
  !  na          Order of matrix
  !
  !  a(lda,matrixCols)    Distributed matrix which should be reduced.
  !              Distribution is like in Scalapack.
  !              Opposed to Scalapack, a(:,:) must be set completely (upper and lower half)
  !              a(:,:) is overwritten on exit with the band and the Householder vectors
  !              in the upper half.
  !
  !  lda         Leading dimension of a
  !  matrixCols  local columns of matrix a
  !
  !  nblk        blocksize of cyclic distribution, must be the same in both directions!
  !
  !  nbw         semi bandwith of output matrix
  !
  !  mpi_comm_rows
  !  mpi_comm_cols
  !              MPI-Communicators for rows/columns
  !
  !  tmat(nbw,nbw,numBlocks)    where numBlocks = (na-1)/nbw + 1
  !              Factors for the Householder vectors (returned), needed for back transformation
  !
  !-------------------------------------------------------------------------------

      use cuda_functions
      use iso_c_binding
      use elpa1_compute
#ifdef HAVE_DETAILED_TIMINGS
      use timings
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#else
      use timings_dummy
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#endif
#ifdef WITH_OPENMP
      use omp_lib
#endif
      use precision
      implicit none

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      integer(kind=ik)                            :: na, lda, nblk, nbw, matrixCols, numBlocks, mpi_comm_rows, mpi_comm_cols

#if REALCASE == 1
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#ifdef USE_ASSUMED_SIZE
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      real(kind=REAL_DATATYPE)                    :: a(lda,*), tmat(nbw,nbw,*)
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#else
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      real(kind=REAL_DATATYPE)                    :: a(lda,matrixCols), tmat(nbw,nbw,numBlocks)
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#endif
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#endif
#if COMPLEXCASE == 1
#ifdef USE_ASSUMED_SIZE
      complex(kind=COMPLEX_DATATYPE)              :: a(lda,*), tmat(nbw,nbw,*)
#else
      complex(kind=COMPLEX_DATATYPE)              :: a(lda,matrixCols), tmat(nbw,nbw,numBlocks)
#endif
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#endif /* COMPLEXCASE */

#if REALCASE == 1
#ifdef DOUBLE_PRECISION_REAL
      real(kind=REAL_DATATYPE), parameter         :: ZERO = 0.0_rk8, ONE = 1.0_rk8

#else
      real(kind=REAL_DATATYPE), parameter         :: ZERO = 0.0_rk4, ONE = 1.0_rk4
#endif
#endif

#if COMPLEXCASE == 1
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#ifdef DOUBLE_PRECISION_COMPLEX
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      complex(kind=COMPLEX_DATATYPE), parameter   :: ZERO = (0.0_rk8, 0.0_rk8), ONE = (1.0_rk8, 0.0_rk8)
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#else
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      complex(kind=COMPLEX_DATATYPE), parameter   :: ZERO = (0.0_rk4, 0.0_rk4), ONE = (1.0_rk4, 0.0_rk4)
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#endif
#endif /* COMPLEXCASE == 1 */
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#if REALCASE == 1
      real(kind=REAL_DATATYPE)                    :: eps
#endif
      logical, intent(in)                         :: useGPU
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      integer(kind=ik)                            :: my_prow, my_pcol, np_rows, np_cols, mpierr
      integer(kind=ik)                            :: l_cols, l_rows
#if REALCASE == 1
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      integer(kind=ik)                            :: vmrCols
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#endif
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      integer(kind=ik)                            :: mynlc
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      integer(kind=ik)                            :: i, j, lcs, lce, lrs, lre, lc, lr, cur_pcol, n_cols, nrow
      integer(kind=ik)                            :: istep, ncol, lch, lcx, nlc
      integer(kind=ik)                            :: tile_size, l_rows_tile, l_cols_tile
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      real(kind=REAL_DATATYPE)                    :: vnorm2
#if REALCASE == 1
      real(kind=REAL_DATATYPE)                    :: xf, aux1(nbw), aux2(nbw), vrl, tau, vav(nbw,nbw)
#endif
#if COMPLEXCASE == 1
      complex(kind=COMPLEX_DATATYPE)              :: xf, aux1(nbw), aux2(nbw), vrl, tau, vav(nbw,nbw)
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#endif
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#if COMPLEXCASE == 1
      complex(kind=COMPLEX_DATATYPE), allocatable :: tmpCUDA(:,:), vmrCUDA(:,:), umcCUDA(:,:) ! note the different dimension in real case
      complex(kind=COMPLEX_DATATYPE), allocatable :: tmpCPU(:,:), vmrCPU(:,:), umcCPU(:,:)
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      complex(kind=COMPLEX_DATATYPE), allocatable :: vr(:)
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#endif
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#if REALCASE == 1
      real(kind=REAL_DATATYPE), allocatable       :: tmpCUDA(:),  vmrCUDA(:),  umcCUDA(:)
      real(kind=REAL_DATATYPE), allocatable       :: tmpCPU(:,:), vmrCPU(:,:), umcCPU(:,:)
      real(kind=REAL_DATATYPE), allocatable       :: vr(:)
#endif

#if REALCASE == 1
      ! needed for blocked QR decomposition
      integer(kind=ik)                            :: PQRPARAM(11), work_size
      real(kind=REAL_DATATYPE)                    :: dwork_size(1)
      real(kind=REAL_DATATYPE), allocatable       :: work_blocked(:), tauvector(:), blockheuristic(:)
#endif
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      ! a_dev is passed from bandred_real to trans_ev_band
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      integer(kind=C_intptr_T)                    :: a_dev, vmr_dev, umc_dev, tmat_dev, vav_dev
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#ifdef WITH_MPI
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      integer(kind=ik), external                  :: numroc
#endif
      integer(kind=ik)                            :: ierr
      integer(kind=ik)                            :: cur_l_rows, cur_l_cols, vmr_size, umc_size
      integer(kind=c_size_t)                      :: lc_start, lc_end
#if COMPLEXCASE == 1
      integer(kind=c_size_t)                      :: lce_1, lcs_1, lre_1
#endif
      integer(kind=ik)                            :: lr_end
      integer(kind=ik)                            :: na_cols
#if COMPLEXCASE == 1
      integer(kind=ik)                            :: na_rows
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#endif

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      logical, intent(in)                         :: wantDebug
      logical, intent(out)                        :: success
      logical                                     :: successCUDA
      integer(kind=ik)                            :: istat
      character(200)                              :: errorMessage
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#if REALCASE == 1
      logical, intent(in)                         :: useQR
#endif
      integer(kind=ik)                            :: mystart, myend, m_way, n_way, work_per_thread, m_id, n_id, n_threads, &
                                                    ii, pp, transformChunkSize
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      call timer%start("bandred_&
      &MATH_DATATYPE&
      &" // &
      &PRECISION_SUFFIX &
      )
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      call timer%start("mpi_communication")

      call mpi_comm_rank(mpi_comm_rows,my_prow,mpierr)
      call mpi_comm_size(mpi_comm_rows,np_rows,mpierr)
      call mpi_comm_rank(mpi_comm_cols,my_pcol,mpierr)
      call mpi_comm_size(mpi_comm_cols,np_cols,mpierr)
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      call timer%stop("mpi_communication")
      success = .true.


      ! Semibandwith nbw must be a multiple of blocksize nblk
      if (mod(nbw,nblk)/=0) then
        if (my_prow==0 .and. my_pcol==0) then
          if (wantDebug) then
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            write(error_unit,*) 'ELPA2_bandred_&
	    &MATH_DATATYPE&
	    &: ERROR: nbw=',nbw,', nblk=',nblk
            write(error_unit,*) 'ELPA2_bandred_&
	    &MATH_DATATYPE&
	    &: ELPA2 works only for nbw==n*nblk'
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          endif
          success = .false.
          return
        endif
      endif

! na_rows in used nowhere; only na_cols
      if (useGPU) then
#ifdef WITH_MPI
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#if COMPLEXCASE == 1
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        na_rows = numroc(na, nblk, my_prow, 0, np_rows)
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#endif
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        na_cols = numroc(na, nblk, my_pcol, 0, np_cols)
#else
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#if COMPLEXCASE == 1
         na_rows = na
#endif
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        na_cols = na
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#endif /* WITH_MPI */
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        ! Here we convert the regular host array into a pinned host array
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        successCUDA = cuda_malloc(a_dev, lda*na_cols*  &
#if REALCASE == 1
	                          size_of_PRECISION_real)
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#endif
#if COMPLEXCASE == 1
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                                  size_of_PRECISION_complex)
#endif
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        if (.not.(successCUDA)) then
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          print *,"bandred_&
	  &MATH_DATATYPE&
	  &: error in cudaMalloc"
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          stop
        endif

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        successCUDA = cuda_malloc(tmat_dev, nbw*nbw*   &
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#if REALCASE == 1
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	                          size_of_PRECISION_real)
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#endif
#if COMPLEXCASE == 1
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                                  size_of_PRECISION_complex)
#endif
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        if (.not.(successCUDA)) then
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          print *,"bandred_&
	  &MATH_DATATYPE&
	  &: error in cudaMalloc"
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          stop
        endif

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        successCUDA = cuda_malloc(vav_dev, nbw*nbw*   &
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#if REALCASE == 1
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	                          size_of_PRECISION_real)
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#endif
#if COMPLEXCASE == 1
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                                  size_of_PRECISION_complex)
#endif
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        if (.not.(successCUDA)) then
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          print *,"bandred_&
	  &MATH_DATATYPE&
	  &: error in cudaMalloc"
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          stop
        endif
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      endif ! useGPU

      ! Matrix is split into tiles; work is done only for tiles on the diagonal or above

      tile_size = nblk*least_common_multiple(np_rows,np_cols) ! minimum global tile size
      tile_size = ((128*max(np_rows,np_cols)-1)/tile_size+1)*tile_size ! make local tiles at least 128 wide

      l_rows_tile = tile_size/np_rows ! local rows of a tile
      l_cols_tile = tile_size/np_cols ! local cols of a tile

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#if REALCASE == 1
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      if (useQR) then

        if (useGPU) then
          print *,"qr decomposition at the moment not supported with GPU"
          stop
        endif

        if (which_qr_decomposition == 1) then
          call qr_pqrparam_init(pqrparam(1:11),    nblk,'M',0,   nblk,'M',0,   nblk,'M',1,'s')
          allocate(tauvector(na), stat=istat, errmsg=errorMessage)
          if (istat .ne. 0) then
            print *,"bandred_real: error when allocating tauvector "//errorMessage
            stop
          endif

          allocate(blockheuristic(nblk), stat=istat, errmsg=errorMessage)
          if (istat .ne. 0) then
            print *,"bandred_real: error when allocating blockheuristic "//errorMessage
            stop
          endif

          l_rows = local_index(na, my_prow, np_rows, nblk, -1)
          allocate(vmrCPU(max(l_rows,1),na), stat=istat, errmsg=errorMessage)
          if (istat .ne. 0) then
            print *,"bandred_real: error when allocating vmrCPU "//errorMessage
            stop
          endif

          vmrCols = na

#ifdef USE_ASSUMED_SIZE_QR
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          call qr_pdgeqrf_2dcomm_&
	  &PRECISION&
	  &(a, lda, matrixCols, vmrCPU, max(l_rows,1), vmrCols, tauvector(1), na, tmat(1,1,1), &
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                                 nbw, nbw, dwork_size, 1, -1, na, nbw, nblk, nblk, na, na, 1, 0, PQRPARAM(1:11), &
                                 mpi_comm_rows, mpi_comm_cols, blockheuristic)

#else
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          call qr_pdgeqrf_2dcomm_&
	  &PRECISION&
	  &(a(1:lda,1:matrixCols), matrixCols, lda, vmrCPU(1:max(l_rows,1),1:vmrCols), max(l_rows,1), &
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                                 vmrCols, tauvector(1:na), na, tmat(1:nbw,1:nbw,1), nbw, &
                                 nbw, dwork_size(1:1), 1, -1, na, nbw, nblk, nblk, na, na, 1, 0, PQRPARAM(1:11), &
                                 mpi_comm_rows, mpi_comm_cols, blockheuristic)
#endif

          work_size = dwork_size(1)
          allocate(work_blocked(work_size), stat=istat, errmsg=errorMessage)
          if (istat .ne. 0) then
            print *,"bandred_real: error when allocating work_blocked "//errorMessage
            stop
          endif
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          work_blocked = CONST_0_0
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          deallocate(vmrCPU, stat=istat, errmsg=errorMessage)
          if (istat .ne. 0) then
            print *,"bandred_real: error when deallocating vmrCPU "//errorMessage
            stop
          endif

        endif ! which_qr_decomposition

      endif ! useQr
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#endif /* REALCASE */
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      if (useGPU) then

        cur_l_rows = 0
        cur_l_cols = 0
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        successCUDA = cuda_memcpy(a_dev, loc(a(1,1)), (lda)*(na_cols)*   &
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#if REALCASE == 1
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	                          size_of_PRECISION_real,    &
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#endif
#if COMPLEXCASE == 1
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                                  size_of_PRECISION_complex, &
#endif
				  cudaMemcpyHostToDevice)
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        if (.not.(successCUDA)) then
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          print *,"bandred_&
	  &MATH_DATATYPE&
	  &: error in cudaMemcpy"
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          stop
        endif
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      endif ! useGPU


      do istep = (na-1)/nbw, 1, -1

        n_cols = MIN(na,(istep+1)*nbw) - istep*nbw ! Number of columns in current step

        ! Number of local columns/rows of remaining matrix
        l_cols = local_index(istep*nbw, my_pcol, np_cols, nblk, -1)
        l_rows = local_index(istep*nbw, my_prow, np_rows, nblk, -1)

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        ! Allocate vmr and umc to their exact sizes so that they can be used in bcasts and reduces

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        if (useGPU) then
          cur_l_rows = max(l_rows, 1)
          cur_l_cols = max(l_cols, 1)

          vmr_size = cur_l_rows * 2 * n_cols
          umc_size = cur_l_cols * 2 * n_cols

          ! Allocate vmr and umc only if the inew size exceeds their current capacity
          ! Added for FORTRAN CALLS
          if ((.not. allocated(vr)) .or. (l_rows + 1 .gt. ubound(vr, dim=1))) then
            if (allocated(vr)) then
              deallocate(vr, stat=istat, errmsg=errorMessage)
              if (istat .ne. 0) then
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                print *,"bandred_&
		&MATH_DATATYPE&
		&: error when deallocating vr "//errorMessage
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                stop
              endif
            endif
            allocate(vr(l_rows + 1), stat=istat, errmsg=errorMessage)
            if (istat .ne. 0) then
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              print *,"bandred_&
	      &MATH_DATATYPE&
	      &: error when allocating vr "//errorMessage
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              stop
            endif

          endif

          if ((.not. allocated(vmrCUDA)) .or. (vmr_size .gt. ubound(vmrCUDA, dim=1))) then
            if (allocated(vmrCUDA)) then
              deallocate(vmrCUDA, stat=istat, errmsg=errorMessage)
              if (istat .ne. 0) then
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                print *,"bandred_&
		&MATH_DATATYPE&
		&: error when allocating vmrCUDA "//errorMessage
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                stop
              endif

              successCUDA = cuda_free(vmr_dev)
              if (.not.(successCUDA)) then
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                print *,"bandred_&
		&MATH_DATATYPE&: error in cuda_free"
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                stop
              endif
            endif

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#if REALCASE == 1
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            allocate(vmrCUDA(vmr_size), stat=istat, errmsg=errorMessage)
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#endif
#if COMPLEXCASE == 1
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            allocate(vmrCUDA(max(l_rows,1),2*n_cols), stat=istat, errmsg=errorMessage)
#endif
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            if (istat .ne. 0) then
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              print *,"bandred_&
	      &MATH_DATATYPE&
	      &: error when allocating vmrCUDA "//errorMessage
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              stop
            endif
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            successCUDA = cuda_malloc(vmr_dev, vmr_size*    &
#if REALCASE == 1
	                              size_of_PRECISION_real)
#endif
#if COMPLEXCASE == 1
                                      size_of_PRECISION_complex)
#endif
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            if (.not.(successCUDA)) then
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              print *,"bandred_&
	      &MATH_DATATYPE&
	      &: error in cudaMalloc: vmr_dev"
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              stop
            endif

          endif
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          if ((.not. allocated(umcCUDA)) .or. (umc_size .gt. ubound(umcCUDA, dim=1))) then
            if (allocated(umcCUDA)) then
              deallocate(umcCUDA, stat=istat, errmsg=errorMessage)
              if (istat .ne. 0) then
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                print *,"bandred_&
		&MATH_DATATYPE&
		&: error when deallocating umcCUDA "//errorMessage
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                stop
              endif

              successCUDA = cuda_free(umc_dev)
              if (.not.(successCUDA)) then
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                 print *,"bandred_&
		 &MATH_DATATYPE&
		 &: error in cudaFree umc_dev"
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                 stop
              endif

            endif
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#if REALCASE == 1
            allocate(umcCUDA(umc_size), stat=istat, errmsg=errorMessage)
#endif
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#if COMPLEXCASE == 1
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            allocate(umcCUDA(max(l_cols,1),2*n_cols), stat=istat, errmsg=errorMessage)
#endif
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            if (istat .ne. 0) then
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              print *,"bandred_&
	      &MATH_DATATYPE&
	      &: error when deallocating umcCUDA "//errorMessage
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              stop
            endif

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            successCUDA = cuda_malloc(umc_dev, umc_size*     &
#if REALCASE == 1
	                              size_of_PRECISION_real)
#endif
#if COMPLEXCASE == 1
                                      size_of_PRECISION_complex)
#endif
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            if (.not.(successCUDA)) then
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              print *,"bandred_&
	      &MATH_DATATYPE&
	      &: error in cudaMalloc umc_dev"
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              stop
            endif
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          endif
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        else ! GPU not used

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          ! unify the the name vmr and vmrCPU, as well as vmrGPU
          ! the same for umcCPU and umcGPU
          ! Allocate vmr and umcCPU to their exact sizes so that they can be used in bcasts and reduces
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          allocate(vmrCPU(max(l_rows,1),2*n_cols), stat=istat, errmsg=errorMessage)
          if (istat .ne. 0) then
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            print *,"bandred_&
	    &MATH_DATATYPE&
	    &: error when allocating vmrCPU "//errorMessage
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            stop
          endif

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          allocate(umcCPU(max(l_cols,1),2*n_cols), stat=istat, errmsg=errorMessage)
          if (istat .ne. 0) then
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            print *,"bandred_&
	    &MATH_DATATYPE&
	    &: error when allocating umcCPU "//errorMessage
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            stop
          endif
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          allocate(vr(l_rows+1), stat=istat, errmsg=errorMessage)
          if (istat .ne. 0) then
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            print *,"bandred_&
	    &MATH_DATATYPE&
	    &: error when allocating vr "//errorMessage
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            stop
          endif

        endif ! use GPU

        if (useGPU) then
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#if REALCASE == 1
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          vmrCUDA(1 : cur_l_rows * n_cols) = CONST_0_0
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#endif
#if COMPLEXCASE == 1
          vmrCUDA(1:l_rows,1:n_cols) = CONST_COMPLEX_0_0
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#endif
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        else
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#if REALCASE == 1
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          vmrCPU(1:l_rows,1:n_cols) = CONST_0_0
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#endif
#if COMPLEXCASE == 1
          vmrCPU(1:l_rows,1:n_cols) = CONST_COMPLEX_0_0
#endif
        endif ! useGPU

#if REALCASE == 1
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        vr(:) = CONST_0_0
        tmat(:,:,istep) = CONST_0_0
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#endif
#if COMPLEXCASE == 1
        vr(:) = CONST_COMPLEX_0_0
        tmat(:,:,istep) = CONST_COMPLEX_0_0
#endif
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        if (useGPU) then
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#if REALCASE == 1
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          umcCUDA(1 : umc_size) = CONST_0_0
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#endif
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          lc_start = local_index(istep*nbw+1, my_pcol, np_cols, nblk, -1)
          lc_end   = local_index(istep*nbw+n_cols, my_pcol, np_cols, nblk, -1)
          lr_end   = local_index((istep-1)*nbw + n_cols, my_prow, np_rows, nblk, -1)

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          if (lc_start .le. 0) lc_start = 1
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          ! Here we assume that the processor grid and the block grid are aligned
          cur_pcol = pcol(istep*nbw+1, nblk, np_cols)

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          if (my_pcol == cur_pcol) then
            successCUDA = cuda_memcpy2d(loc(a(1, lc_start)), &
#if REALCASE == 1
                                            lda*size_of_PRECISION_real,         &
#endif
#if COMPLEXCASE == 1
                                        int(lda*size_of_PRECISION_complex,kind=c_size_t), &
#endif
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#if REALCASE == 1
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                                        (a_dev + ((lc_start-1) * lda*size_of_PRECISION_real)),    &
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#endif
#if COMPLEXCASE == 1
                                        (a_dev + int( ( (lc_start-1) * lda*size_of_PRECISION_complex),kind=c_size_t )),      &
#endif
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#if REALCASE == 1
                                        lda*size_of_PRECISION_real, lr_end*size_of_PRECISION_real, &
#endif
#if COMPLEXCASE == 1
                                         int(lda*size_of_PRECISION_complex,kind=c_size_t),              &
                                         int(lr_end*size_of_PRECISION_complex,kind=c_size_t),           &
#endif
#if REALCASE == 1
                                         (lc_end - lc_start+1), cudaMemcpyDeviceToHost)
#endif
#if COMPLEXCASE == 1
                                         int((lc_end - lc_start+1),kind=c_size_t),int(cudaMemcpyDeviceToHost,kind=c_int))
#endif

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            if (.not.(successCUDA)) then
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              print *,"bandred_&
	      &MATH_DATATYPE&
	      &: error in cudaMemcpy2d"
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              stop
            endif

          endif
        endif ! useGPU

        ! Reduce current block to lower triangular form
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#if REALCASE == 1
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        if (useQR) then
          if (which_qr_decomposition == 1) then
            vmrCols = 2*n_cols
#ifdef USE_ASSUMED_SIZE_QR
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            call qr_pdgeqrf_2dcomm_&
	    &PRECISION&
	    &(a, lda, matrixCols, vmrCPU, max(l_rows,1), vmrCols, tauvector(1), &
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                                   na, tmat(1,1,istep), nbw, nbw, work_blocked, work_size,        &
                                     work_size, na, n_cols, nblk, nblk,        &
                                     istep*nbw+n_cols-nbw, istep*nbw+n_cols, 1,&
                                     0, PQRPARAM(1:11), mpi_comm_rows, mpi_comm_cols,&
                                     blockheuristic)

#else
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            call qr_pdgeqrf_2dcomm_&
	    &PRECISION&
	    &(a(1:lda,1:matrixCols), lda, matrixCols, vmrCPU(1:max(l_rows,1),1:vmrCols) ,   &
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                                    max(l_rows,1), vmrCols, tauvector(1:na), na, &
                                     tmat(1:nbw,1:nbw,istep), nbw, nbw, work_blocked(1:work_size), work_size, &
                                     work_size, na, n_cols, nblk, nblk,        &
                                     istep*nbw+n_cols-nbw, istep*nbw+n_cols, 1,&
                                     0, PQRPARAM(1:11), mpi_comm_rows, mpi_comm_cols,&
                                     blockheuristic)
#endif
          endif

       else !useQR
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#endif /* REALCASE == 1 */
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         do lc = n_cols, 1, -1

           ncol = istep*nbw + lc ! absolute column number of householder vector
           nrow = ncol - nbw ! Absolute number of pivot row

           lr  = local_index(nrow, my_prow, np_rows, nblk, -1) ! current row length
           lch = local_index(ncol, my_pcol, np_cols, nblk, -1) ! HV local column number

           tau = 0

           if (nrow == 1) exit ! Nothing to do

           cur_pcol = pcol(ncol, nblk, np_cols) ! Processor column owning current block

           if (my_pcol==cur_pcol) then

             ! Get vector to be transformed; distribute last element and norm of
             ! remaining elements to all procs in current column

             vr(1:lr) = a(1:lr,lch) ! vector to be transformed

             if (my_prow==prow(nrow, nblk, np_rows)) then
               aux1(1) = dot_product(vr(1:lr-1),vr(1:lr-1))
               aux1(2) = vr(lr)
             else
               aux1(1) = dot_product(vr(1:lr),vr(1:lr))
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#if REALCASE == 1
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               aux1(2) = CONST_0_0
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#endif
#if COMPLEXCASE == 1
               aux1(2) = CONST_COMPLEX_0_0
#endif
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             endif

#ifdef WITH_MPI
             call timer%start("mpi_communication")
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             call mpi_allreduce(aux1, aux2, 2, &
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#if REALCASE == 1
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                                MPI_REAL_PRECISION, &
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#endif
#if COMPLEXCASE == 1
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                                MPI_COMPLEX_PRECISION, &
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#endif
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                                MPI_SUM, mpi_comm_rows, mpierr)
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             call timer%stop("mpi_communication")

#else /* WITH_MPI */
              aux2 = aux1 ! this should be optimized
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#endif
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             vnorm2 = aux2(1)
             vrl    = aux2(2)

             ! Householder transformation
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#if REALCASE == 1
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	     call hh_transform_real_&
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#endif
#if COMPLEXCASE == 1
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	     call hh_transform_complex_&
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#endif
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             &PRECISION &
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                              (vrl, vnorm2, xf, tau)
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             ! Scale vr and store Householder vector for back transformation

             vr(1:lr) = vr(1:lr) * xf
             if (my_prow==prow(nrow, nblk, np_rows)) then
               a(1:lr-1,lch) = vr(1:lr-1)
               a(lr,lch) = vrl
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#if REALCASE == 1
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               vr(lr) = CONST_1_0
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#endif
#if COMPLEXCASE == 1
               vr(lr) = CONST_COMPLEX_1_0
#endif
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             else
               a(1:lr,lch) = vr(1:lr)
             endif

           endif

           ! Broadcast Householder vector and tau along columns

           vr(lr+1) = tau
#ifdef WITH_MPI
           call timer%start("mpi_communication")
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	   call MPI_Bcast(vr, lr+1, &
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#if REALCASE == 1
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                          MPI_REAL_PRECISION, &
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#endif
#if COMPLEXCASE == 1
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                          MPI_COMPLEX_PRECISION, &
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#endif
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                          cur_pcol, mpi_comm_cols, mpierr)
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           call timer%stop("mpi_communication")

#endif /* WITH_MPI */
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           if (useGPU) then
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             vmrCUDA(cur_l_rows * (lc - 1) + 1 : cur_l_rows * (lc - 1) + lr) = vr(1:lr)
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#endif
#if COMPLEXCASE == 1
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             vmrCUDA(1:lr,lc) = vr(1:lr)
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#endif
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           else
             vmrCPU(1:lr,lc) = vr(1:lr)
           endif
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           tau = vr(lr+1)

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#if REALCASE == 1
           tmat(lc,lc,istep) = tau ! Store tau in diagonal of tmat
#endif
#if COMPLEXCASE == 1
           tmat(lc,lc,istep) = conjg(tau) ! Store tau in diagonal of tmat
#endif
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           ! Transform remaining columns in current block with Householder vector
           ! Local dot product

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#if REALCASE == 1
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           aux1 = 0
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#endif
#if COMPLEXCASE == 1
          aux1 = CONST_COMPLEX_0_0
#endif
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#ifdef WITH_OPENMP
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#if 0
 ! original complex implementation without openmp. check performance
            nlc = 0 ! number of local columns
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           do j=1,lc-1
             lcx = local_index(istep*nbw+j, my_pcol, np_cols, nblk, 0)
             if (lcx>0) then
               nlc = nlc+1
               aux1(nlc) = dot_product(vr(1:lr),a(1:lr,lcx))
             endif
           enddo

           ! Get global dot products
#ifdef WITH_MPI
           call timer%start("mpi_communication")
           if (nlc>0) call mpi_allreduce(aux1, aux2, nlc, MPI_COMPLEX_PRECISION, MPI_SUM, mpi_comm_rows, mpierr)

           ! Transform

           nlc = 0
           do j=1,lc-1
             lcx = local_index(istep*nbw+j, my_pcol, np_cols, nblk, 0)
             if (lcx>0) then
               nlc = nlc+1
               a(1:lr,lcx) = a(1:lr,lcx) - conjg(tau)*aux2(nlc)*vr(1:lr)
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             endif
           enddo

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           call timer%stop("mpi_communication")

#else /* WITH_MPI */
!          if (nlc>0) aux2=aux1

           ! Transform

           nlc = 0
           do j=1,lc-1
             lcx = local_index(istep*nbw+j, my_pcol, np_cols, nblk, 0)
             if (lcx>0) then
               nlc = nlc+1
               a(1:lr,lcx) = a(1:lr,lcx) - conjg(tau)*aux1(nlc)*vr(1:lr)
             endif
           enddo

#endif /* WITH_MPI */
!
!           ! Transform
!
!           nlc = 0
!           do j=1,lc-1
!             lcx = local_index(istep*nbw+j, my_pcol, np_cols, nblk, 0)
!             if (lcx>0) then
!               nlc = nlc+1
!               a(1:lr,lcx) = a(1:lr,lcx) - conjg(tau)*aux2(nlc)*vr(1:lr)
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!             endif
!           enddo
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#endif /* if 0 */
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           !Open up one omp region to avoid paying openmp overhead.
           !This does not help performance due to the addition of two openmp barriers around the MPI call,
           !But in the future this may be beneficial if these barriers are replaced with a faster implementation

           !$omp parallel private(mynlc, j, lcx, ii, pp ) shared(aux1)
           mynlc = 0 ! number of local columns

           !This loop does not have independent iterations,
           !'mynlc' is incremented each iteration, and it is difficult to remove this dependency
           !Thus each thread executes every iteration of the loop, except it only does the work if it 'owns' that iteration
           !That is, a thread only executes the work associated with an iteration if its thread id is congruent to
           !the iteration number modulo the number of threads
           do j=1,lc-1
             lcx = local_index(istep*nbw+j, my_pcol, np_cols, nblk, 0)
             if (lcx>0 ) then
               mynlc = mynlc+1
               if ( mod((j-1), omp_get_num_threads()) .eq. omp_get_thread_num() ) then
                   if (lr>0) aux1(mynlc) = dot_product(vr(1:lr),a(1:lr,lcx))
               endif
             endif
           enddo

           ! Get global dot products

           !$omp barrier
           !$omp single
#ifdef WITH_MPI
           call timer%start("mpi_communication")
           if (mynlc>0) call mpi_allreduce(aux1, aux2, mynlc,     &
#if REALCASE == 1
	                                   MPI_REAL_PRECISION,    &
#endif
#if COMPLEXCASE == 1
                                           MPI_COMPLEX_PRECISION, &
#endif
        			           MPI_SUM, mpi_comm_rows, mpierr)
           call timer%stop("mpi_communication")
#else /* WITH_MPI */
           if (mynlc>0) aux2 = aux1
#endif /* WITH_MPI */
           !$omp end single
           !$omp barrier

           ! Transform
           transformChunkSize=32
           mynlc = 0
           do j=1,lc-1
             lcx = local_index(istep*nbw+j, my_pcol, np_cols, nblk, 0)
             if (lcx>0) then
               mynlc = mynlc+1
               !This loop could be parallelized with an openmp pragma with static scheduling and chunk size 32
               !However, for some reason this is slower than doing it manually, so it is parallelized as below.
               do ii=omp_get_thread_num()*transformChunkSize,lr,omp_get_num_threads()*transformChunkSize
                  do pp = 1,transformChunkSize
                      if (pp + ii > lr) exit
#if REALCASE == 1
                          a(ii+pp,lcx) = a(ii+pp,lcx) - tau*aux2(mynlc)*vr(ii+pp)
#endif
#if COMPLEXCASE == 1
                          a(ii+pp,lcx) = a(ii+pp,lcx) - conjg(tau)*aux2(mynlc)*vr(ii+pp)
#endif
                  enddo
               enddo
             endif
           enddo
           !$omp end parallel
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#else /* WITH_OPENMP */

           nlc = 0 ! number of local columns
           do j=1,lc-1
             lcx = local_index(istep*nbw+j, my_pcol, np_cols, nblk, 0)
             if (lcx>0) then
               nlc = nlc+1
               if (lr>0) aux1(nlc) = dot_product(vr(1:lr),a(1:lr,lcx))
             endif
           enddo

           ! Get global dot products
#ifdef WITH_MPI
           call timer%start("mpi_communication")
           if (nlc>0) call mpi_allreduce(aux1, aux2, nlc,      &
#if REALCASE == 1
	                                 MPI_REAL_PRECISION,   &
#endif
#if COMPLEXCASE == 1
                                         MPI_COMPLEX_PRECISION,&
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#endif
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					 MPI_SUM, mpi_comm_rows, mpierr)
           call timer%stop("mpi_communication")
#else /* WITH_MPI */
           if (nlc>0) aux2=aux1
#endif /* WITH_MPI */
           ! Transform

           nlc = 0
           do j=1,lc-1
             lcx = local_index(istep*nbw+j, my_pcol, np_cols, nblk, 0)
             if (lcx>0) then
               nlc = nlc+1
#if REALCASE == 1
               a(1:lr,lcx) = a(1:lr,lcx) - tau*aux2(nlc)*vr(1:lr)
#endif
#if COMPLEXCASE == 1
               a(1:lr,lcx) = a(1:lr,lcx) - conjg(tau)*aux2(nlc)*vr(1:lr)
#endif
             endif
           enddo
#endif /* WITH_OPENMP */
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         enddo ! lc

         if (useGPU) then
           ! store column tiles back to GPU
           cur_pcol = pcol(istep*nbw+1, nblk, np_cols)
           if (my_pcol == cur_pcol) then
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             successCUDA = cuda_memcpy2d((a_dev+        &
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#if REALCASE == 1
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                                         ((lc_start-1)*lda*size_of_PRECISION_real)),          &
#endif
#if COMPLEXCASE == 1
                                         int(((lc_start-1)*lda*size_of_PRECISION_complex),kind=c_size_t)),    &
#endif
#if REALCASE == 1
                                         lda*size_of_PRECISION_real, loc(a(1, lc_start)),           &
#endif
#if COMPLEXCASE == 1
                                         int(lda*size_of_PRECISION_complex,kind=c_size_t), loc(a(1,lc_start)), &
#endif
#if REALCASE == 1
                                         lda*size_of_PRECISION_real,  lr_end*size_of_PRECISION_real, &
#endif
#if COMPLEXCASE == 1
                                         int(lda*size_of_PRECISION_complex,kind=c_size_t),           &
                                         int(lr_end*size_of_PRECISION_complex,kind=c_size_t),        &
#endif
#if REALCASE == 1
                                         (lc_end - lc_start+1),cudaMemcpyHostToDevice)
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#endif
#if COMPLEXCASE == 1
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                                         int((lc_end - lc_start+1),kind=c_size_t), &
                                         int(cudaMemcpyHostToDevice,kind=c_int))
#endif
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             if (.not.(successCUDA)) then
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               print *, "bandred_&
	       &MATH_DATATYPE&
	       &: cuda memcpy a_dev  failed ", istat
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               stop
             endif
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           endif
         endif

         ! Calculate scalar products of stored Householder vectors.
         ! This can be done in different ways, we use dsyrk

         vav = 0
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	 call timer%start("blas")
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         if (useGPU) then
           if (l_rows>0) &
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#if REALCASE == 1
             call PRECISION_SYRK('U', 'T',            &
#endif
#if COMPLEXCASE == 1
             call PRECISION_HERK('U', 'C',            &
#endif
	                         n_cols, l_rows, ONE, &
#if REALCASE == 1
				 vmrCUDA, cur_l_rows, &
#endif
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#if COMPLEXCASE == 1
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                                 vmrCUDA, ubound(vmrCUDA,dim=1), &
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#endif
				 ZERO, vav, ubound(vav,dim=1))
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         else ! useGPU
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           if (l_rows>0) &
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#if REALCASE == 1
             call PRECISION_SYRK('U', 'T',           &
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#endif
#if COMPLEXCASE == 1
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             call PRECISION_HERK('U', 'C',           &
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#endif
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	                         n_cols, l_rows, ONE, vmrCPU, ubound(vmrCPU,dim=1), ZERO, vav, ubound(vav,dim=1))
         endif
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	 call timer%stop("blas")
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#if REALCASE == 1
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	 call symm_matrix_allreduce_&
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#endif
#if COMPLEXCASE == 1
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         call herm_matrix_allreduce_&
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#endif
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         &PRECISION &
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                         (n_cols,vav, nbw, nbw,mpi_comm_rows)
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         ! Calculate triangular matrix T for block Householder Transformation
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	 call timer%start("blas")
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         do lc=n_cols,1,-1
           tau = tmat(lc,lc,istep)
           if (lc<n_cols) then
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#if REALCASE == 1
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             call PRECISION_TRMV('U', 'T', 'N',          &
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#endif
#if COMPLEXCASE == 1
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             call PRECISION_TRMV('U', 'C', 'N',          &
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#endif
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                                 n_cols-lc, tmat(lc+1,lc+1,istep), ubound(tmat,dim=1), vav(lc+1,lc), 1)
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#if REALCASE == 1
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             tmat(lc,lc+1:n_cols,istep) = -tau * vav(lc+1:n_cols,lc)
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#endif
#if COMPLEXCASE == 1
             tmat(lc,lc+1:n_cols,istep) = -tau * conjg(vav(lc+1:n_cols,lc))
#endif
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           endif
         enddo
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 	 call timer%stop("blas")
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#if REALCASE == 1
       endif !useQR
#endif
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       ! Transpose vmr -> vmc (stored in umc, second half)
       if (useGPU) then
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         call elpa_transpose_vectors_&
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              &MATH_DATATYPE&
              &_&
              &PRECISION &
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#if REALCASE == 1
	                                   (vmrCUDA, cur_l_rows,  &
#endif
#if COMPLEXCASE == 1
                                           (vmrCUDA, ubound(vmrCUDA,dim=1), &
#endif
					    mpi_comm_rows, &
#if REALCASE == 1
                                            umcCUDA(cur_l_cols * n_cols + 1), cur_l_cols, &
#endif
#if COMPLEXCASE == 1
                                            umcCUDA(1,n_cols+1), ubound(umcCUDA,dim=1),     &
#endif
					    mpi_comm_cols, 1, istep*nbw, n_cols, nblk)
       else ! useGPU
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         call elpa_transpose_vectors_&
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              &MATH_DATATYPE&
              &_&
              &PRECISION &
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                                           (vmrCPU, ubound(vmrCPU,dim=1), mpi_comm_rows, &
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                                            umcCPU(1,n_cols+1), ubound(umcCPU,dim=1), mpi_comm_cols, &
                                            1, istep*nbw, n_cols, nblk)
       endif

       ! Calculate umc = A**T * vmr
       ! Note that the distributed A has to be transposed
       ! Opposed to direct tridiagonalization there is no need to use the cache locality
       ! of the tiles, so we can use strips of the matrix
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       ! here the GPU version and CPU version diverged substantially, due to the newest
       ! optimizations due to Intel. The GPU version has to be re-written
       if (useGPU) then
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#if REALCASE == 1
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         umcCUDA(1 : l_cols * n_cols) = CONST_0_0
         vmrCUDA(cur_l_rows * n_cols + 1 : cur_l_rows * n_cols * 2) = CONST_0_0
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#endif
#if COMPLEXCASE == 1
         umcCUDA(1:l_cols,1:n_cols) = CONST_COMPLEX_0_0
         vmrCUDA(1:l_rows,n_cols+1:2*n_cols) = CONST_COMPLEX_0_0
#endif
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         if (l_cols>0 .and. l_rows>0) then
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#if REALCASE == 1
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           successCUDA = cuda_memcpy(vmr_dev, loc(vmrCUDA(1)), vmr_size*size_of_PRECISION_real,cudaMemcpyHostToDevice)
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#endif
#if COMPLEXCASE == 1
           successCUDA = cuda_memcpy(vmr_dev, loc(vmrCUDA(1,1)),vmr_size*size_of_PRECISION_complex,cudaMemcpyHostToDevice)
#endif
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           if (.not.(successCUDA)) then
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             print *,"bandred_&
	     &MATH_DATATYPE&
	     &: error in cudaMemcpy"
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             stop
           endif
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#if REALCASE == 1
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           successCUDA = cuda_memcpy(umc_dev, loc(umcCUDA(1)), umc_size*size_of_PRECISION_real,cudaMemcpyHostToDevice)
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#endif
#if COMPLEXCASE == 1
           successCUDA = cuda_memcpy(umc_dev, loc(umcCUDA(1,1)),umc_size*size_of_PRECISION_complex,cudaMemcpyHostToDevice)
#endif
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           if (.not.(successCUDA)) then
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             print *,"bandred_&
	     &MATH_DATATYPE&
	     &: error in cudaMemcpy"
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             stop
           endif

           do i=0,(istep*nbw-1)/tile_size

             lcs = i*l_cols_tile+1
             lce = min(l_cols,(i+1)*l_cols_tile)
             if (lce<lcs) cycle
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             call timer%start("cublas")
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             lre = min(l_rows,(i+1)*l_rows_tile)
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#if REALCASE == 1
             call cublas_PRECISION_GEMM('T', 'N',               &
#endif
#if COMPLEXCASE == 1
             call cublas_PRECISION_GEMM('C', 'N',               &
#endif
	                                lce-lcs+1, n_cols, lre, &
                                        ONE, (a_dev + ((lcs-1)*lda* &
#if REALCASE == 1
					size_of_PRECISION_real)),   &
#endif
#if COMPLEXCASE == 1
                                        size_of_PRECISION_complex)),&
#endif
					lda, vmr_dev,cur_l_rows, ONE, &
                                        (umc_dev+ (lcs-1)*            &
#if REALCASE == 1
					size_of_PRECISION_real),      &
#endif
#if COMPLEXCASE == 1
                                        size_of_PRECISION_complex),   &
#endif
					cur_l_cols)

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             if(i==0) cycle
             lre = min(l_rows,i*l_rows_tile)
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             call cublas_PRECISION_GEMM('N', 'N', lre,n_cols, lce-lcs+1, ONE, &
                                        (a_dev+ ((lcs-1)*lda*                 &
#if REALCASE == 1
					size_of_PRECISION_real)),             &
#endif
#if COMPLEXCASE == 1
                                        size_of_PRECISION_complex)),          &
#endif
					lda, (umc_dev+(cur_l_cols * n_cols+lcs-1)* &
#if REALCASE == 1
					size_of_PRECISION_real),              &
#endif
#if COMPLEXCASE == 1
                                        size_of_PRECISION_complex),           &
#endif
					cur_l_cols, ONE, (vmr_dev+(cur_l_rows * n_cols)* &
#if REALCASE == 1
					size_of_PRECISION_real),              &
#endif
#if COMPLEXCASE == 1
                                        size_of_PRECISION_complex),           &
#endif
					cur_l_rows)
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             call timer%stop("cublas")
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           enddo
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#if REALCASE == 1
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           successCUDA = cuda_memcpy(loc(vmrCUDA(1)), vmr_dev,vmr_size*size_of_PRECISION_real,cudaMemcpyDeviceToHost)
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#endif
#if COMPLEXCASE == 1
           successCUDA = cuda_memcpy(loc(vmrCUDA(1,1)),vmr_dev,vmr_size*size_of_PRECISION_complex,cudaMemcpyDeviceToHost)
#endif
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           if (.not.(successCUDA)) then
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             print *,"bandred_&
	     &MATH_DATATYPE&
	     &: error in cudaMemcpy"
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             stop
           endif
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#if REALCASE == 1
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           successCUDA = cuda_memcpy(loc(umcCUDA(1)), umc_dev, umc_size*size_of_PRECISION_real,cudaMemcpyDeviceToHost)
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#endif
#if COMPLEXCASE == 1
           successCUDA = cuda_memcpy(loc(umcCUDA(1,1)), umc_dev,umc_size*size_of_PRECISION_complex,cudaMemcpyDeviceToHost)
#endif
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           if (.not.(successCUDA)) then
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             print *,"bandred_&
	     &MATH_DATATYPE&
	     &: error in cudaMemcpy"
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             stop
           endif

         endif ! l_cols>0 .and. l_rows>0

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       else ! do not useGPU version
#if 0
! original complex implemetation check for performance
          umcCPU(1:l_cols,1:n_cols) = CONST_COMPLEX_0_0
          vmrCPU(1:l_rows,n_cols+1:2*n_cols) = CONST_COMPLEX_0_0

        if (l_cols>0 .and. l_rows>0) then
          do i=0,(istep*nbw-1)/tile_size
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            lcs = i*l_cols_tile+1
            lce = min(l_cols,(i+1)*l_cols_tile)
            if (lce<lcs) cycle
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            lre = min(l_rows,(i+1)*l_rows_tile)

              call timer%start("blas")
              call PRECISION_GEMM('C', 'N', lce-lcs+1, n_cols, lre, ONE, a(1,lcs), ubound(a,dim=1), &
                         vmrCPU, ubound(vmrCPU,dim=1), ONE, umcCPU(lcs,1), ubound(umcCPU,dim=1))
              call timer%stop("blas")

            if (i==0) cycle
            lre = min(l_rows,i*l_rows_tile)
              call timer%start("blas")
              call PRECISION_GEMM('N', 'N', lre, n_cols, lce-lcs+1, ONE, a(1,lcs), lda, &
                         umcCPU(lcs,n_cols+1), ubound(umcCPU,dim=1), ONE, vmrCPU(1,n_cols+1), ubound(vmrCPU,dim=1))
              call timer%stop("blas")
          enddo

        endif ! (l_cols>0 .and. l_rows>0)
#endif /* if 0 */
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         !Code for Algorithm 4

         n_way = 1
#ifdef WITH_OPENMP
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#if REALCASE == 1
         ! check whether this can also be done in complex case
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         n_way = omp_get_max_threads()
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#endif
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#endif
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         !umcCPU(1:l_cols,1:n_cols) = 0.d0
         !vmrCPU(1:l_rows,n_cols+1:2*n_cols) = 0
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#ifdef WITH_OPENMP
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#if REALCASE == 1
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         !$omp parallel private( i,lcs,lce,lrs,lre)
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#endif
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#endif
         if (n_way > 1) then
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#ifdef WITH_OPENMP
#if REALCASE == 1
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           !$omp do
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#endif
#endif
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           do i=1,min(l_cols_tile, l_cols)
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#if REALCASE == 1
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             umcCPU(i,1:n_cols) = CONST_0_0
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#endif
#if COMPLEXCASE == 1
             umcCPU(i,1:n_cols) = CONST_COMPLEX_0_0
#endif
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           enddo
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#ifdef WITH_OPENMP
#if REALCASE == 1
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           !$omp do
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#endif
#endif
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           do i=1,l_rows
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#if REALCASE == 1
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             vmrCPU(i,n_cols+1:2*n_cols) = CONST_0_0
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#endif
#if COMPLEXCASE == 1
             vmrCPU(i,n_cols+1:2*n_cols) = CONST_COMPLEX_0_0
#endif
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           enddo
           if (l_cols>0 .and. l_rows>0) then

             !SYMM variant 4
             !Partitioned Matrix Expression:
             ! Ct = Atl Bt + Atr Bb
             ! Cb = Atr' Bt + Abl Bb
             !
             !Loop invariant:
             ! Ct = Atl Bt + Atr Bb
             !
             !Update:
             ! C1 = A10'B0 + A11B1 + A21 B2
             !
             !This algorithm chosen because in this algoirhtm, the loop around the dgemm calls
             !is easily parallelized, and regardless of choise of algorithm,
             !the startup cost for parallelizing the dgemms inside the loop is too great
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#ifdef WITH_OPENMP
#if REALCASE == 1
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             !$omp do schedule(static,1)
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#endif
#endif
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             do i=0,(istep*nbw-1)/tile_size
               lcs = i*l_cols_tile+1                   ! local column start
               lce = min(l_cols, (i+1)*l_cols_tile)    ! local column end

               lrs = i*l_rows_tile+1                   ! local row start
               lre = min(l_rows, (i+1)*l_rows_tile)    ! local row end

               !C1 += [A11 A12] [B1
               !                 B2]
               if ( lre > lrs .and. l_cols > lcs ) then
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	         call timer%start("blas")
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                 call PRECISION_GEMM('N', 'N', lre-lrs+1, n_cols, l_cols-lcs+1,          &
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                                     ONE, a(lrs,lcs), ubound(a,dim=1),                 &
                                     umcCPU(lcs,n_cols+1), ubound(umcCPU,dim=1),  &
                                     ZERO, vmrCPU(lrs,n_cols+1), ubound(vmrCPU,dim=1))
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	         call timer%stop("blas")
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               endif

               ! C1 += A10' B0
               if ( lce > lcs .and. i > 0 ) then
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	       	 call timer%start("blas")
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#if REALCASE == 1
                 call PRECISION_GEMM('T', 'N',     &
#endif
#if COMPLEXCASE == 1
                 call PRECISION_GEMM('T', 'N',     &
#endif
		                    lce-lcs+1, n_cols, lrs-1,              &
                                    ONE, a(1,lcs),   ubound(a,dim=1),      &
                                    vmrCPU(1,1),   ubound(vmrCPU,dim=1),   &
                                    ZERO, umcCPU(lcs,1), ubound(umcCPU,dim=1))
                 call timer%stop("blas")
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               endif
             enddo
           endif ! l_cols>0 .and. l_rows>0
         else ! n_way > 1
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#if REALCASE == 1
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           umcCPU(1:l_cols,1:n_cols) = CONST_0_0
           vmrCPU(1:l_rows,n_cols+1:2*n_cols) = CONST_0_0
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#endif
#if COMPLEXCASE == 1
           umcCPU(1:l_cols,1:n_cols) = CONST_COMPLEX_0_0
	   vmrCPU(1:l_rows,n_cols+1:2*n_cols) = CONST_COMPLEX_0_0

#endif
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           if (l_cols>0 .and. l_rows>0) then
             do i=0,(istep*nbw-1)/tile_size

               lcs = i*l_cols_tile+1
               lce = min(l_cols,(i+1)*l_cols_tile)
               if (lce<lcs) cycle
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	       call timer%start("blas")
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               lre = min(l_rows,(i+1)*l_rows_tile)
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#if REALCASE == 1
               call PRECISION_GEMM('T', 'N',          &
#endif
#if COMPLEXCASE == 1
               call PRECISION_GEMM('C', 'N',          &
#endif
	                           lce-lcs+1, n_cols, lre, ONE, a(1,lcs), ubound(a,dim=1), &
                                   vmrCPU, ubound(vmrCPU,dim=1), ONE, umcCPU(lcs,1), ubound(umcCPU,dim=1))
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	       call timer%stop("blas")
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               if (i==0) cycle
                 lre = min(l_rows,i*l_rows_tile)
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	       	 call timer%start("blas")
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                 call PRECISION_GEMM('N', 'N', lre, n_cols, lce-lcs+1, ONE, a(1,lcs), lda, &
                                     umcCPU(lcs,n_cols+1), ubound(umcCPU,dim=1), ONE,      &
				     vmrCPU(1,n_cols+1), ubound(vmrCPU,dim=1))
                 call timer%stop("blas")
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             enddo
           endif
         endif ! n_way > 1
#ifdef WITH_OPENMP
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#if REALCASE == 1
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        !$omp end parallel
#endif
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#endif
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       endif ! do not useGPU version

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       ! Sum up all ur(:) parts along rows and add them to the uc(:) parts
       ! on the processors containing the diagonal
       ! This is only necessary if ur has been calculated, i.e. if the
       ! global tile size is smaller than the global remaining matrix

       if (useGPU) then
         ! here the GPU version and CPU version divereged due to the same reasons as above

         if (tile_size < istep*nbw) then
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           call elpa_reduce_add_vectors_&
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                &MATH_DATATYPE&
                &_&
                &PRECISION &
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#if REALCASE == 1
                                               (vmrCUDA(cur_l_rows * n_cols + 1),cur_l_rows,  &
#endif
#if COMPLEXCASE == 1
                                               (vmrCUDA(1,n_cols+1),ubound(vmrCUDA,dim=1),    &
#endif
					       mpi_comm_rows, umcCUDA,                        &
#if REALCASE == 1
					       cur_l_cols,                                    &
#endif
#if COMPLEXCASE == 1
                                               ubound(umcCUDA,dim=1),                         &
#endif
					       mpi_comm_cols, istep*nbw, n_cols, nblk)
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         endif

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         if (l_cols>0) then
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#ifdef WITH_MPI
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#if REALCASE == 1
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           allocate(tmpCUDA(l_cols * n_cols), stat=istat, errmsg=errorMessage)
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#endif
#if COMPLEXCASE == 1
           allocate(tmpCUDA(l_cols,n_cols), stat=istat, errmsg=errorMessage)
#endif
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