elpa2_compute.F90

!    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 Naturwissenschaftrn,
!      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".


#include "config-f90.h"

module ELPA2_compute

! Version 1.1.2, 2011-02-21

  use elpa_utilities
  USE ELPA1_compute
  use elpa1, only : elpa_print_times, time_evp_back, time_evp_fwd, time_evp_solve
  use elpa2_utilities
  use elpa_pdgeqrf

  implicit none

  PRIVATE ! By default, all routines contained are private

  public :: bandred_real
  public :: tridiag_band_real
  public :: trans_ev_tridi_to_band_real
  public :: trans_ev_band_to_full_real

  public :: bandred_complex
  public :: tridiag_band_complex
  public :: trans_ev_tridi_to_band_complex
  public :: trans_ev_band_to_full_complex

  public :: band_band_real
  public :: divide_band

  integer, public :: which_qr_decomposition = 1     ! defines, which QR-decomposition algorithm will be used
                                                    ! 0 for unblocked
                                                    ! 1 for blocked (maxrank: nblk)
  include 'mpif.h'

  contains

    subroutine bandred_real(na, a, lda, nblk, nbw, matrixCols, numBlocks, mpi_comm_rows, mpi_comm_cols, &
                            tmat, wantDebug, useGPU, success, useQR)

  !-------------------------------------------------------------------------------
  !  bandred_real: Reduces a distributed symmetric matrix to band form
  !
  !  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

#ifdef HAVE_DETAILED_TIMINGS
      use timings
#endif
#ifdef WITH_OPENMP
      use omp_lib
#endif
      implicit none

      integer             :: na, lda, nblk, nbw, matrixCols, numBlocks, mpi_comm_rows, mpi_comm_cols
      real*8              :: a(lda,matrixCols), tmat(nbw,nbw,numBlocks)

      real*8              :: eps

      logical, intent(in) :: useGPU

      integer             :: my_prow, my_pcol, np_rows, np_cols, mpierr
      integer             :: l_cols, l_rows
      integer             :: i, j, lcs, lce, lrs, lre, lc, lr, cur_pcol, n_cols, nrow
      integer             :: istep, ncol, lch, lcx, nlc, mynlc
      integer             :: tile_size, l_rows_tile, l_cols_tile

      real*8              :: vnorm2, xf, aux1(nbw), aux2(nbw), vrl, tau, vav(nbw,nbw)

      real*8, allocatable :: tmpCUDA(:),  vmrCUDA(:),  umcCUDA(:)
      real*8, allocatable :: tmpCPU(:,:), vmrCPU(:,:), umcCPU(:,:)
      real*8, allocatable :: vr(:)

      ! needed for blocked QR decomposition
      integer                  :: PQRPARAM(11), work_size
      real*8                   :: dwork_size(1)
      real*8, allocatable      :: work_blocked(:), tauvector(:), blockheuristic(:)

      integer(kind=C_intptr_T) :: a_dev, vmr_dev, umc_dev, tmat_dev, vav_dev
      integer, external        :: numroc
      integer                  :: ierr
      integer                  :: cur_l_rows, cur_l_cols, vmr_size, umc_size
      integer(kind=c_size_t)   :: lc_start, lc_end
      integer                  :: lr_end
      integer                  :: na_rows, na_cols


      logical             :: successCUDA
      integer             :: istat
      character(200)      :: errorMessage
      logical, intent(in) :: wantDebug
      logical, intent(out):: success

      logical, intent(in) :: useQR

      integer :: mystart, myend, m_way, n_way, work_per_thread, m_id, n_id, n_threads, ii, pp, transformChunkSize

#ifdef HAVE_DETAILED_TIMINGS
      call timer%start("bandred_real")
#endif
      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)
      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
            write(error_unit,*) 'ELPA2_bandred_real: ERROR: nbw=',nbw,', nblk=',nblk
            write(error_unit,*) 'ELPA2_bandred_real: ELPA2 works only for nbw==n*nblk'
          endif
          success = .false.
          return
        endif
      endif

      if (useGPU) then
        na_rows = numroc(na, nblk, my_prow, 0, np_rows)
        na_cols = numroc(na, nblk, my_pcol, 0, np_cols)
      endif

      ! 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

      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,    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

          call qr_pdgeqrf_2dcomm(a, lda, vmrCPU, max(l_rows,1), tauvector, tmat(1,1,1), nbw, dwork_size(1), -1, na, &
                                nbw, nblk, nblk, na, na, 1, 0, PQRPARAM, mpi_comm_rows, mpi_comm_cols, blockheuristic)
          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

          work_blocked = 0.0d0
          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

      if (useGPU) then
        ! Here we convert the regular host array into a pinned host array
        successCUDA = cuda_malloc(a_dev, lda*na_cols*size_of_real_datatype)
        if (.not.(successCUDA)) then
          print *,"bandred_real: error in cudaMalloc"
          stop
        endif

        successCUDA = cuda_malloc(tmat_dev, nbw*nbw*size_of_real_datatype)
        if (.not.(successCUDA)) then
          print *,"bandred_real: error in cudaMalloc"
          stop
        endif

        successCUDA = cuda_malloc(vav_dev, nbw*nbw*size_of_real_datatype)
        if (.not.(successCUDA)) then
          print *,"bandred_real: error in cudaMalloc"
          stop
        endif

        cur_l_rows = 0
        cur_l_cols = 0

        successCUDA = cuda_memcpy(a_dev, loc(a(1,1)), (lda)*(na_cols)*size_of_real_datatype,cudaMemcpyHostToDevice)
        if (.not.(successCUDA)) then
          print *,"bandred_real: error in cudaMemcpy"
          stop
        endif
      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)

        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
                print *,"bandred_real: error when deallocating vr "//errorMessage
                stop
              endif
            endif
            allocate(vr(l_rows + 1), stat=istat, errmsg=errorMessage)
            if (istat .ne. 0) then
              print *,"bandred_real: error when allocating vr "//errorMessage
              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
                print *,"bandred_real: error when allocating vmrCUDA "//errorMessage
                stop
              endif

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

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

            successCUDA = cuda_malloc(vmr_dev, vmr_size*size_of_real_datatype)
            if (.not.(successCUDA)) then
              print *,"bandred_real: error in cudaMalloc"
              stop
            endif

          endif

          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
                print *,"bandred_real: error when deallocating umcCUDA "//errorMessage
                stop
              endif

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

            endif

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

            successCUDA = cuda_malloc(umc_dev, umc_size*size_of_real_datatype)
            if (.not.(successCUDA)) then
              print *,"bandred_real: error in cudaMalloc"
              stop
            endif

          endif
        else ! GPU not used
          ! Allocate vmr and umc to their exact sizes so that they can be used in bcasts and reduces

          allocate(vmrCPU(max(l_rows,1),2*n_cols), stat=istat, errmsg=errorMessage)
          if (istat .ne. 0) then
            print *,"bandred_real: error when allocating vmrCPU "//errorMessage
            stop
          endif

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

          allocate(vr(l_rows+1), stat=istat, errmsg=errorMessage)
          if (istat .ne. 0) then
            print *,"bandred_real: error when allocating vr "//errorMessage
            stop
          endif
        endif ! use GPU

        if (useGPU) then
          vmrCUDA(1 : cur_l_rows * n_cols) = 0.
        else
          vmrCPU(1:l_rows,1:n_cols) = 0.
        endif

        vr(:) = 0
        tmat(:,:,istep) = 0

        if (useGPU) then
          umcCUDA(1 : umc_size) = 0.

          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)

          if(lc_start .le. 0) lc_start = 1

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

          if(my_pcol == cur_pcol) then

            successCUDA = cuda_memcpy2d(loc(a(1, lc_start)), lda*size_of_real_datatype,         &
                                       (a_dev + ((lc_start-1) * lda*size_of_real_datatype)),    &
                                       lda*size_of_real_datatype, lr_end*size_of_real_datatype, &
                                       (lc_end - lc_start+1), cudaMemcpyDeviceToHost)
            if (.not.(successCUDA)) then
              print *,"bandred_real: error in cudaMemcpy2d"
              stop
            endif

          endif
        endif ! useGPU

        ! Reduce current block to lower triangular form

        if (useQR) then
          if (which_qr_decomposition == 1) then
            call qr_pdgeqrf_2dcomm(a, lda, vmrCPU, max(l_rows,1), tauvector(1), &
                                  tmat(1,1,istep), nbw, work_blocked,       &
                                  work_size, na, n_cols, nblk, nblk,        &
                                  istep*nbw+n_cols-nbw, istep*nbw+n_cols, 1,&
                                  0, PQRPARAM, mpi_comm_rows, mpi_comm_cols,&
                                  blockheuristic)
          endif
       else !useQR

         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))
               aux1(2) = 0.
             endif

             call mpi_allreduce(aux1,aux2,2,MPI_REAL8,MPI_SUM,mpi_comm_rows,mpierr)

             vnorm2 = aux2(1)
             vrl    = aux2(2)

             ! Householder transformation

             call hh_transform_real(vrl, vnorm2, xf, tau)

             ! 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
               vr(lr) = 1.
             else
               a(1:lr,lch) = vr(1:lr)
             endif

           endif

           ! Broadcast Householder vector and tau along columns

           vr(lr+1) = tau
           call MPI_Bcast(vr,lr+1,MPI_REAL8,cur_pcol,mpi_comm_cols,mpierr)

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

           tau = vr(lr+1)
           tmat(lc,lc,istep) = tau ! Store tau in diagonal of tmat

           ! Transform remaining columns in current block with Householder vector
           ! Local dot product

           aux1 = 0

#ifdef WITH_OPENMP
           !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
           if (mynlc>0) call mpi_allreduce(aux1,aux2,mynlc,MPI_REAL8,MPI_SUM,mpi_comm_rows,mpierr)
           !$omp end single
           !$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
                          a(ii+pp,lcx) = a(ii+pp,lcx) - tau*aux2(mynlc)*vr(ii+pp)
                  enddo
               enddo
             endif
           enddo
           !$omp end parallel
#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
           if (nlc>0) call mpi_allreduce(aux1,aux2,nlc,MPI_REAL8,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) - tau*aux2(nlc)*vr(1:lr)
             endif
           enddo
#endif /* WITH_OPENMP */
         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
             successCUDA = cuda_memcpy2d((a_dev+((lc_start-1)*lda*size_of_real_datatype)),          &
                                          lda*size_of_real_datatype, loc(a(1, lc_start)),           &
                                          lda*size_of_real_datatype,  lr_end*size_of_real_datatype, &
                                          (lc_end - lc_start+1),cudaMemcpyHostToDevice)
             if (.not.(successCUDA)) then
               print *,"bandred_real: error in cudaMemcpy2d"
               stop
             endif

           endif
         endif

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

         vav = 0

         if (useGPU) then
           if (l_rows>0) &
             call dsyrk('U','T',n_cols,l_rows,1.d0,vmrCUDA,cur_l_rows,0.d0,vav,ubound(vav,dim=1))
         else
           if (l_rows>0) &
             call dsyrk('U','T',n_cols,l_rows,1.d0,vmrCPU,ubound(vmrCPU,dim=1),0.d0,vav,ubound(vav,dim=1))

         endif

         call symm_matrix_allreduce(n_cols,vav, nbw, nbw,mpi_comm_rows)

         ! Calculate triangular matrix T for block Householder Transformation

         do lc=n_cols,1,-1
           tau = tmat(lc,lc,istep)
           if (lc<n_cols) then
             call dtrmv('U','T','N',n_cols-lc,tmat(lc+1,lc+1,istep),ubound(tmat,dim=1),vav(lc+1,lc),1)
             tmat(lc,lc+1:n_cols,istep) = -tau * vav(lc+1:n_cols,lc)
           endif
         enddo
       endif

       ! Transpose vmr -> vmc (stored in umc, second half)

       if (useGPU) then
         call elpa_transpose_vectors_real  (vmrCUDA, cur_l_rows, mpi_comm_rows, &
                                            umcCUDA(cur_l_cols * n_cols + 1), cur_l_cols, mpi_comm_cols, &
                                            1, istep*nbw, n_cols, nblk)
       else
         call elpa_transpose_vectors_real  (vmrCPU, ubound(vmrCPU,dim=1), mpi_comm_rows, &
                                            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

       ! 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
         umcCUDA(1 : l_cols * n_cols) = 0.d0
         vmrCUDA(cur_l_rows * n_cols + 1 : cur_l_rows * n_cols * 2) = 0

         if (l_cols>0 .and. l_rows>0) then
           successCUDA = cuda_memcpy(vmr_dev, loc(vmrCUDA(1)), vmr_size*size_of_real_datatype,cudaMemcpyHostToDevice)
           if (.not.(successCUDA)) then
             print *,"bandred_real: error in cudaMemcpy"
             stop
           endif

           successCUDA = cuda_memcpy(umc_dev, loc(umcCUDA(1)), umc_size*size_of_real_datatype,cudaMemcpyHostToDevice)
           if (.not.(successCUDA)) then
             print *,"bandred_real: error in cudaMemcpy"
             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

             lre = min(l_rows,(i+1)*l_rows_tile)

             call cublas_dgemm('T','N',lce-lcs+1,n_cols,lre, &
                               1.d0, (a_dev + ((lcs-1)*lda*size_of_real_datatype)), lda, vmr_dev,cur_l_rows, &
                               1.d0, (umc_dev+ (lcs-1)*size_of_real_datatype), cur_l_cols)

             if(i==0) cycle
             lre = min(l_rows,i*l_rows_tile)

             call cublas_dgemm('N','N',lre,n_cols,lce-lcs+1,&
                               1.d0, (a_dev+ ((lcs-1)*lda*size_of_real_datatype)),lda,                  &
                               (umc_dev+(cur_l_cols * n_cols+lcs-1)*size_of_real_datatype), cur_l_cols, &
                               1.d0, (vmr_dev+(cur_l_rows * n_cols)*size_of_real_datatype), cur_l_rows)
           enddo

           successCUDA = cuda_memcpy(loc(vmrCUDA(1)), vmr_dev,vmr_size*size_of_real_datatype,cudaMemcpyDeviceToHost)
           if (.not.(successCUDA)) then
             print *,"bandred_real: error in cudaMemcpy"
             stop
           endif

           successCUDA = cuda_memcpy(loc(umcCUDA(1)), umc_dev, umc_size*size_of_real_datatype,cudaMemcpyDeviceToHost)
           if (.not.(successCUDA)) then
             print *,"bandred_real: error in cudaMemcpy"
             stop
           endif

         endif ! l_cols>0 .and. l_rows>0

       else ! do not useGPU version
         !Code for Algorithm 4

         n_way = 1
#ifdef WITH_OPENMP
         n_way = omp_get_max_threads()
#endif
         !umc(1:l_cols,1:n_cols) = 0.d0
         !vmr(1:l_rows,n_cols+1:2*n_cols) = 0
#ifdef WITH_OPENMP
         !$omp parallel private( i,lcs,lce,lrs,lre)
#endif
         if (n_way > 1) then
           !$omp do
           do i=1,min(l_cols_tile, l_cols)
             umcCPU(i,1:n_cols) = 0.d0
           enddo
           !$omp do
           do i=1,l_rows
             vmrCPU(i,n_cols+1:2*n_cols) = 0.d0
           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

             !$omp do schedule(static,1)
             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
                 call DGEMM('N','N', lre-lrs+1, n_cols, l_cols-lcs+1,          &
                            1.d0, a(lrs,lcs), ubound(a,dim=1),                 &
                                  umcCPU(lcs,n_cols+1), ubound(umcCPU,dim=1),  &
                            0.d0, vmrCPU(lrs,n_cols+1), ubound(vmrCPU,dim=1))
               endif

               ! C1 += A10' B0
               if ( lce > lcs .and. i > 0 ) then
                 call DGEMM('T','N', lce-lcs+1, n_cols, lrs-1,           &
                            1.d0, a(1,lcs),   ubound(a,dim=1),           &
                                  vmrCPU(1,1),   ubound(vmrCPU,dim=1),   &
                            0.d0, umcCPU(lcs,1), ubound(umcCPU,dim=1))
               endif
             enddo
           endif ! l_cols>0 .and. l_rows>0
         else ! n_way > 1
           umcCPU(1:l_cols,1:n_cols) = 0.d0
           vmrCPU(1:l_rows,n_cols+1:2*n_cols) = 0
           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

               lre = min(l_rows,(i+1)*l_rows_tile)
               call DGEMM('T','N',lce-lcs+1,n_cols,lre,1.d0,a(1,lcs),ubound(a,dim=1), &
                            vmrCPU,ubound(vmrCPU,dim=1),1.d0,umcCPU(lcs,1),ubound(umcCPU,dim=1))

               if (i==0) cycle
                 lre = min(l_rows,i*l_rows_tile)
                 call DGEMM('N','N',lre,n_cols,lce-lcs+1,1.d0,a(1,lcs),lda, &
                            umcCPU(lcs,n_cols+1),ubound(umcCPU,dim=1),1.d0,vmrCPU(1,n_cols+1),ubound(vmrCPU,dim=1))
             enddo
           endif
         endif ! n_way > 1
#ifdef WITH_OPENMP
        !$omp end parallel
#endif
       endif ! do not useGPU version

       ! 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
           call elpa_reduce_add_vectors_real  (vmrCUDA(cur_l_rows * n_cols + 1),cur_l_rows,mpi_comm_rows, &
                                               umcCUDA, cur_l_cols, mpi_comm_cols, &
                                               istep*nbw, n_cols, nblk)
         endif

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

           call mpi_allreduce(umcCUDA,tmpCUDA,l_cols*n_cols,MPI_REAL8,MPI_SUM,mpi_comm_rows,ierr)
           umcCUDA(1 : l_cols * n_cols) = tmpCUDA(1 : l_cols * n_cols)

           if (allocated(tmpCUDA)) then
             deallocate(tmpCUDA, stat=istat, errmsg=errorMessage)
             if (istat .ne. 0) then
               print *,"bandred_real: error when deallocating tmpCUDA "//errorMessage
               stop
             endif
           endif
         endif ! l_cols

         ! U = U * Tmat**T
         successCUDA = cuda_memcpy(umc_dev, loc(umcCUDA(1)), umc_size*size_of_real_datatype, cudaMemcpyHostToDevice)
         if (.not.(successCUDA)) then
           print *,"bandred_real: error in cudaMemcpy"
           stop
         endif

         successCUDA = cuda_memcpy(tmat_dev,loc(tmat(1,1,istep)),nbw*nbw*size_of_real_datatype,cudaMemcpyHostToDevice)
         if (.not.(successCUDA)) then
           print *,"bandred_real: error in cudaMemcpy"
           stop
         endif

         call cublas_dtrmm('Right','Upper','Trans','Nonunit',l_cols,n_cols, &
                           1.d0, tmat_dev,nbw,umc_dev,cur_l_cols)

         ! VAV = Tmat * V**T * A * V * Tmat**T = (U*Tmat**T)**T * V * Tmat**T

         successCUDA = cuda_memcpy(vav_dev,loc(vav(1,1)), nbw*nbw*size_of_real_datatype,cudaMemcpyHostToDevice)
         if (.not.(successCUDA)) then
           print *,"bandred_real: error in cudaMemcpy"
           stop
         endif

         call cublas_dgemm('T','N',n_cols,n_cols,l_cols, &
                           1.d0, umc_dev,cur_l_cols,(umc_dev+(cur_l_cols * n_cols )*size_of_real_datatype),cur_l_cols, &
                           0.d0, vav_dev,nbw)

         call cublas_dtrmm('Right','Upper','Trans','Nonunit',n_cols,n_cols, &
                           1.d0, tmat_dev,nbw, vav_dev, nbw)


         successCUDA = cuda_memcpy(loc(vav(1,1)), vav_dev, nbw*nbw*size_of_real_datatype, cudaMemcpyDeviceToHost)
         if (.not.(successCUDA)) then
           print *,"bandred_real: error in cudaMemcpy"
           stop
         endif

         call symm_matrix_allreduce(n_cols,vav, nbw,nbw,mpi_comm_cols)

         successCUDA = cuda_memcpy(vav_dev, loc(vav(1,1)), nbw*nbw*size_of_real_datatype,cudaMemcpyHostToDevice)
         if (.not.(successCUDA)) then
           print *,"bandred_real: error in cudaMemcpy"
           stop
         endif

         ! U = U - 0.5 * V * VAV
         call cublas_dgemm('N','N',l_cols,n_cols,n_cols,&
                           -0.5d0, (umc_dev+(cur_l_cols * n_cols )*size_of_real_datatype),cur_l_cols, vav_dev,nbw,&
                           1.0d0, umc_dev,cur_l_cols)

         successCUDA = cuda_memcpy(loc(umcCUDA(1)), umc_dev, umc_size*size_of_real_datatype, cudaMemcpyDeviceToHost)
         if (.not.(successCUDA)) then
           print *,"bandred_real: error in cudaMemcpy"
           stop
         endif

         ! Transpose umc -> umr (stored in vmr, second half)

         call elpa_transpose_vectors_real  (umcCUDA, cur_l_cols, mpi_comm_cols, &
                                            vmrCUDA(cur_l_rows * n_cols + 1), cur_l_rows, mpi_comm_rows, &
                                            1, istep*nbw, n_cols, nblk)
         successCUDA = cuda_memcpy(vmr_dev, loc(vmrCUDA(1)), vmr_size*size_of_real_datatype, cudaMemcpyHostToDevice)
         if (.not.(successCUDA)) then
           print *,"bandred_real: error in cudaMemcpy"
           stop
         endif

         successCUDA = cuda_memcpy(umc_dev, loc(umcCUDA(1)), umc_size*size_of_real_datatype, cudaMemcpyHostToDevice)
         if (.not.(successCUDA)) then
           print *,"bandred_real: error in cudaMemcpy"
           stop
         endif

         ! A = A - V*U**T - U*V**T
         do i=0,(istep*nbw-1)/tile_size
           lcs = i*l_cols_tile+1
           lce = min(l_cols,(i+1)*l_cols_tile)
           lre = min(l_rows,(i+1)*l_rows_tile)
           if (lce<lcs .or. lre<1) cycle

           call cublas_dgemm('N', 'T', lre, lce-lcs+1, 2*n_cols, -1.d0, &
                             vmr_dev,cur_l_rows,(umc_dev +(lcs-1)*size_of_real_datatype),cur_l_cols, &
                             1.d0,(a_dev+(lcs-1)*lda*size_of_real_datatype),lda)
         enddo
       else ! do not useGPU
         ! Or if we used the Algorithm 4
         if (tile_size < istep*nbw .or. n_way > 1) then
         call elpa_reduce_add_vectors_real  (vmrCPU(1,n_cols+1),ubound(vmrCPU,dim=1),mpi_comm_rows, &
                                             umcCPU, ubound(umcCPU,dim=1), mpi_comm_cols, &
                                             istep*nbw, n_cols, nblk)
         endif

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

           call mpi_allreduce(umcCPU,tmpCPU,l_cols*n_cols,MPI_REAL8,MPI_SUM,mpi_comm_rows,mpierr)
           umcCPU(1:l_cols,1:n_cols) = tmpCPU(1:l_cols,1:n_cols)

           deallocate(tmpCPU, stat=istat, errmsg=errorMessage)
           if (istat .ne. 0) then
             print *,"bandred_real: error when deallocating tmpCPU "//errorMessage
             stop
           endif
         endif

         ! U = U * Tmat**T

         call dtrmm('Right','Upper','Trans','Nonunit',l_cols,n_cols,1.d0,tmat(1,1,istep),ubound(tmat,dim=1), &
                    umcCPU,ubound(umcCPU,dim=1))

         ! VAV = Tmat * V**T * A * V * Tmat**T = (U*Tmat**T)**T * V * Tmat**T

         call dgemm('T','N',n_cols,n_cols,l_cols,1.d0,umcCPU,ubound(umcCPU,dim=1),umcCPU(1,n_cols+1), &
                    ubound(umcCPU,dim=1),0.d0,vav,ubound(vav,dim=1))

         call dtrmm('Right','Upper','Trans','Nonunit',n_cols,n_cols,1.d0,tmat(1,1,istep),    &
                    ubound(tmat,dim=1),vav,ubound(vav,dim=1))

         call symm_matrix_allreduce(n_cols,vav, nbw, nbw ,mpi_comm_cols)

         ! U = U - 0.5 * V * VAV
         call dgemm('N','N',l_cols,n_cols,n_cols,-0.5d0,umcCPU(1,n_cols+1),ubound(umcCPU,dim=1),vav, &
                     ubound(vav,dim=1),1.d0,umcCPU,ubound(umcCPU,dim=1))

         ! Transpose umc -> umr (stored in vmr, second half)

         call elpa_transpose_vectors_real(umcCPU, ubound(umcCPU,dim=1), mpi_comm_cols, &
                                         vmrCPU(1,n_cols+1), ubound(vmrCPU,dim=1), mpi_comm_rows, &
                                         1, istep*nbw, n_cols, nblk)

         ! A = A - V*U**T - U*V**T
#ifdef WITH_OPENMP
         !$omp parallel private( ii, i, lcs, lce, lre, n_way, m_way, m_id, n_id, work_per_thread, mystart, myend  )
         n_threads = omp_get_num_threads()
         if (mod(n_threads, 2) == 0) then
           n_way = 2
         else
           n_way = 1
         endif

         m_way = n_threads / n_way

         m_id = mod(omp_get_thread_num(),  m_way)
         n_id = omp_get_thread_num() / m_way

         do ii=n_id*tile_size,(istep*nbw-1),tile_size*n_way
           i = ii / tile_size
           lcs = i*l_cols_tile+1
           lce = min(l_cols,(i+1)*l_cols_tile)
           lre = min(l_rows,(i+1)*l_rows_tile)
           if (lce<lcs .or. lre<1) cycle

           !Figure out this thread's range
           work_per_thread = lre / m_way
           if (work_per_thread * m_way < lre) work_per_thread = work_per_thread + 1