elpa2_compute.F90 215 KB
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!    This file is part of ELPA.
!
!    The ELPA library was originally created by the ELPA consortium,
!    consisting of the following organizations:
!
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!    - Max Planck Computing and Data Facility (MPCDF), fomerly known as
!      Rechenzentrum Garching der Max-Planck-Gesellschaft (RZG),
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!    - 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:
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!    http://elpa.mpcdf.mpg.de/
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!
!    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
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  use elpa_mpi
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  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)
  contains

    subroutine bandred_real(na, a, lda, nblk, nbw, matrixCols, numBlocks, mpi_comm_rows, mpi_comm_cols, &
                            tmat, wantDebug, 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
    !
    !-------------------------------------------------------------------------------
#ifdef HAVE_DETAILED_TIMINGS
      use timings
#endif
#ifdef WITH_OPENMP
      use omp_lib
#endif
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      use precision
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      implicit none

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      integer(kind=ik)           :: na, lda, nblk, nbw, matrixCols, numBlocks, mpi_comm_rows, mpi_comm_cols
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#ifdef DESPERATELY_WANT_ASSUMED_SIZE
      real(kind=rk)              :: a(lda,*), tmat(nbw,nbw,*)
#else
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      real(kind=rk)              :: a(lda,matrixCols), tmat(nbw,nbw,numBlocks)
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#endif
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      integer(kind=ik)           :: my_prow, my_pcol, np_rows, np_cols, mpierr
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      integer(kind=ik)           :: l_cols, l_rows, vmrCols
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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, mynlc
      integer(kind=ik)           :: tile_size, l_rows_tile, l_cols_tile
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      real(kind=rk)              :: vnorm2, xf, aux1(nbw), aux2(nbw), vrl, tau, vav(nbw,nbw)
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      real(kind=rk), allocatable :: tmp(:,:), vr(:), vmr(:,:), umc(:,:)
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      ! needed for blocked QR decomposition
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      integer(kind=ik)           :: PQRPARAM(11), work_size
      real(kind=rk)              :: dwork_size(1)
      real(kind=rk), allocatable :: work_blocked(:), tauvector(:), blockheuristic(:)
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      logical, intent(in)        :: wantDebug
      logical, intent(out)       :: success
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      logical, intent(in)        :: useQR
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      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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#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

      ! 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 (which_qr_decomposition == 1) then
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          call qr_pqrparam_init(pqrparam(1:11),    nblk,'M',0,   nblk,'M',0,   nblk,'M',1,'s')
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          allocate(tauvector(na))
          allocate(blockheuristic(nblk))
          l_rows = local_index(na, my_prow, np_rows, nblk, -1)
          allocate(vmr(max(l_rows,1),na))

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          vmrCols = na
#ifdef DESPERATELY_WANT_ASSUMED_SIZE_QR
          call qr_pdgeqrf_2dcomm(a, lda, matrixCols, vmr, max(l_rows,1), vmrCols, tauvector(1), na, tmat(1,1,1), &
                                 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
          call qr_pdgeqrf_2dcomm(a(1:lda,1:matrixCols), matrixCols, lda, vmr(1:max(l_rows,1),1:vmrCols), max(l_rows,1), &
                                 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
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          work_size = dwork_size(1)
          allocate(work_blocked(work_size))

          work_blocked = 0.0d0
          deallocate(vmr)
        endif
      endif

      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)

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

        allocate(vmr(max(l_rows,1),2*n_cols))
        allocate(umc(max(l_cols,1),2*n_cols))

        allocate(vr(l_rows+1))

        vmr(1:l_rows,1:n_cols) = 0.
        vr(:) = 0
        tmat(:,:,istep) = 0

        ! Reduce current block to lower triangular form

        if (useQR) then
          if (which_qr_decomposition == 1) then
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            vmrCols = 2*n_cols
#ifdef DESPERATELY_WANT_ASSUMED_SIZE_QR
            call qr_pdgeqrf_2dcomm(a, lda, matrixCols, vmr, max(l_rows,1), vmrCols, tauvector(1), &
                                   na, tmat(1,1,istep), nbw, nbw, work_blocked, work_size,        &
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                                     work_size, na, n_cols, nblk, nblk,        &
                                     istep*nbw+n_cols-nbw, istep*nbw+n_cols, 1,&
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                                     0, PQRPARAM(1:11), mpi_comm_rows, mpi_comm_cols,&
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                                     blockheuristic)
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#else
            call qr_pdgeqrf_2dcomm(a(1:lda,1:matrixCols), lda, matrixCols, vmr(1:max(l_rows,1),1:vmrCols) ,   &
                                    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
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          endif
        else

          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
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#ifdef WITH_MPI
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              call mpi_allreduce(aux1,aux2,2,MPI_REAL8,MPI_SUM,mpi_comm_rows,mpierr)
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#else
              aux2 = aux1
#endif
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              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
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#ifdef WITH_MPI
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            call MPI_Bcast(vr,lr+1,MPI_REAL8,cur_pcol,mpi_comm_cols,mpierr)
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#endif
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            vmr(1:lr,lc) = vr(1:lr)
            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
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#ifdef WITH_MPI
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            if (mynlc>0) call mpi_allreduce(aux1,aux2,mynlc,MPI_REAL8,MPI_SUM,mpi_comm_rows,mpierr)
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#else
            if (mynlc>0) aux2 = aux1
#endif
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            !$omp end single
            !$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
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#else /* WITH_OPENMP */
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            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
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#ifdef WITH_MPI
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            if (nlc>0) call mpi_allreduce(aux1,aux2,nlc,MPI_REAL8,MPI_SUM,mpi_comm_rows,mpierr)
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#else
            if (nlc>0) aux2=aux1
#endif
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            ! 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
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#endif /* WITH_OPENMP */

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          enddo

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

          vav = 0
          if (l_rows>0) &
              call dsyrk('U','T',n_cols,l_rows,1.d0,vmr,ubound(vmr,dim=1),0.d0,vav,ubound(vav,dim=1))
          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)

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

        ! 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
        !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)
            umc(i,1:n_cols) = 0.d0
          enddo
          !$omp do
          do i=1,l_rows
            vmr(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),           &
                               umc(lcs,n_cols+1), ubound(umc,dim=1),  &
                         0.d0, vmr(lrs,n_cols+1), ubound(vmr,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),           &
                               vmr(1,1),   ubound(vmr,dim=1),         &
                         0.d0, umc(lcs,1), ubound(umc,dim=1))
              endif
            enddo
          endif
        else
          umc(1:l_cols,1:n_cols) = 0.d0
          vmr(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), &
                           vmr,ubound(vmr,dim=1),1.d0,umc(lcs,1),ubound(umc,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, &
                           umc(lcs,n_cols+1),ubound(umc,dim=1),1.d0,vmr(1,n_cols+1),ubound(vmr,dim=1))
            enddo
          endif
        endif
#ifdef WITH_OPENMP
        !$omp end parallel
#endif
        ! 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
        ! Or if we used the Algorithm 4
        if (tile_size < istep*nbw .or. n_way > 1) then
        call elpa_reduce_add_vectors_real  (vmr(1,n_cols+1),ubound(vmr,dim=1),mpi_comm_rows, &
                                            umc, ubound(umc,dim=1), mpi_comm_cols, &
                                            istep*nbw, n_cols, nblk)
        endif
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#ifdef WITH_MPI
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        if (l_cols>0) then
          allocate(tmp(l_cols,n_cols))
          call mpi_allreduce(umc,tmp,l_cols*n_cols,MPI_REAL8,MPI_SUM,mpi_comm_rows,mpierr)
          umc(1:l_cols,1:n_cols) = tmp(1:l_cols,1:n_cols)
          deallocate(tmp)
        endif
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#endif
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        ! U = U * Tmat**T

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

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

        call elpa_transpose_vectors_real  (umc, ubound(umc,dim=1), mpi_comm_cols, &
                                       vmr(1,n_cols+1), ubound(vmr,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
          mystart = m_id * work_per_thread + 1
          myend   = mystart + work_per_thread - 1
          if ( myend > lre ) myend = lre
          if ( myend-mystart+1 < 1) cycle

          call dgemm('N','T',myend-mystart+1, lce-lcs+1, 2*n_cols, -1.d0, &
                      vmr(mystart, 1), ubound(vmr,1), umc(lcs,1), ubound(umc,1), &
                      1.d0,a(mystart,lcs),ubound(a,1))
        enddo
        !$omp end parallel

#else /* WITH_OPENMP */
        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 dgemm('N','T',lre,lce-lcs+1,2*n_cols,-1.d0, &
                      vmr,ubound(vmr,dim=1),umc(lcs,1),ubound(umc,dim=1), &
                      1.d0,a(1,lcs),lda)
        enddo
#endif /* WITH_OPENMP */
        deallocate(vmr, umc, vr)

      enddo

      if (useQR) then
        if (which_qr_decomposition == 1) then
          deallocate(work_blocked)
          deallocate(tauvector)
        endif
      endif

#ifdef HAVE_DETAILED_TIMINGS
      call timer%stop("bandred_real")
#endif
    end subroutine bandred_real

    subroutine symm_matrix_allreduce(n,a,lda,ldb,comm)

    !-------------------------------------------------------------------------------
    !  symm_matrix_allreduce: Does an mpi_allreduce for a symmetric matrix A.
    !  On entry, only the upper half of A needs to be set
    !  On exit, the complete matrix is set
    !-------------------------------------------------------------------------------
#ifdef HAVE_DETAILED_TIMINGS
      use timings
#endif
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      use precision
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      implicit none
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      integer(kind=ik)  :: n, lda, ldb, comm
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#ifdef DESPERATELY_WANT_ASSUMED_SIZE
      real(kind=rk)     :: a(lda,*)
#else
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      real(kind=rk)     :: a(lda,ldb)
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#endif
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      integer(kind=ik)  :: i, nc, mpierr
      real(kind=rk)     :: h1(n*n), h2(n*n)
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#ifdef HAVE_DETAILED_TIMINGS
      call timer%start("symm_matrix_allreduce")
#endif

      nc = 0
      do i=1,n
        h1(nc+1:nc+i) = a(1:i,i)
        nc = nc+i
      enddo
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#ifdef WITH_MPI
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      call mpi_allreduce(h1,h2,nc,MPI_REAL8,MPI_SUM,comm,mpierr)
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#else
      h2=h1
#endif
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      nc = 0
      do i=1,n
        a(1:i,i) = h2(nc+1:nc+i)
        a(i,1:i-1) = a(1:i-1,i)
        nc = nc+i
      enddo

#ifdef HAVE_DETAILED_TIMINGS
      call timer%stop("symm_matrix_allreduce")
#endif

    end subroutine symm_matrix_allreduce

    subroutine trans_ev_band_to_full_real(na, nqc, nblk, nbw, a, lda, tmat, q, ldq, matrixCols, numBlocks, mpi_comm_rows, &
                                      mpi_comm_cols, useQR)
    !-------------------------------------------------------------------------------
    !  trans_ev_band_to_full_real:
    !  Transforms the eigenvectors of a band matrix back to the eigenvectors of the original matrix
    !
    !  Parameters
    !
    !  na          Order of matrix a, number of rows of matrix q
    !
    !  nqc         Number of columns of matrix q
    !
    !  nblk        blocksize of cyclic distribution, must be the same in both directions!
    !
    !  nbw         semi bandwith
    !
    !  a(lda,matrixCols)    Matrix containing the Householder vectors (i.e. matrix a after bandred_real)
    !              Distribution is like in Scalapack.
    !
    !  lda         Leading dimension of a
    !  matrixCols  local columns of matrix a and q
    !
    !  tmat(nbw,nbw,numBlocks) Factors returned by bandred_real
    !
    !  q           On input: Eigenvectors of band matrix
    !              On output: Transformed eigenvectors
    !              Distribution is like in Scalapack.
    !
    !  ldq         Leading dimension of q
    !
    !  mpi_comm_rows
    !  mpi_comm_cols
    !              MPI-Communicators for rows/columns
    !
    !-------------------------------------------------------------------------------
#ifdef HAVE_DETAILED_TIMINGS
      use timings
#endif
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      use precision
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      implicit none

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      integer(kind=ik)            :: na, nqc, lda, ldq, nblk, nbw, matrixCols, numBlocks, mpi_comm_rows, mpi_comm_cols
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#ifdef DESPERATELY_WANT_ASSUMED_SIZE
      real(kind=rk)               :: a(lda,*), q(ldq,*), tmat(nbw,nbw,*)
#else
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      real(kind=rk)               :: a(lda,matrixCols), q(ldq,matrixCols), tmat(nbw, nbw, numBlocks)
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#endif
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      integer(kind=ik)            :: my_prow, my_pcol, np_rows, np_cols, mpierr
      integer(kind=ik)            :: max_blocks_row, max_blocks_col, max_local_rows, &
                                     max_local_cols
      integer(kind=ik)            :: l_cols, l_rows, l_colh, n_cols
      integer(kind=ik)            :: istep, lc, ncol, nrow, nb, ns
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      real(kind=rk), allocatable  :: tmp1(:), tmp2(:), hvb(:), hvm(:,:)
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      integer(kind=ik)            :: i
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      real(kind=rk), allocatable  :: tmat_complete(:,:), t_tmp(:,:), t_tmp2(:,:)
      integer(kind=ik)            :: cwy_blocking, t_blocking, t_cols, t_rows
      logical, intent(in)         :: useQR
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#ifdef HAVE_DETAILED_TIMINGS
      call timer%start("trans_ev_band_to_full_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)
      max_blocks_row = ((na -1)/nblk)/np_rows + 1  ! Rows of A
      max_blocks_col = ((nqc-1)/nblk)/np_cols + 1  ! Columns of q!

      max_local_rows = max_blocks_row*nblk
      max_local_cols = max_blocks_col*nblk

      ! t_blocking was formerly 2; 3 is a better choice
      t_blocking = 3 ! number of matrices T (tmat) which are aggregated into a new (larger) T matrix (tmat_complete) and applied at once

      ! we only use the t_blocking if we could call it fully, this is might be better but needs to benchmarked.
!     if ( na >= ((t_blocking+1)*nbw) ) then
      cwy_blocking = t_blocking * nbw

      allocate(tmp1(max_local_cols*cwy_blocking))
      allocate(tmp2(max_local_cols*cwy_blocking))
      allocate(hvb(max_local_rows*cwy_blocking))
      allocate(hvm(max_local_rows,cwy_blocking))
      allocate(tmat_complete(cwy_blocking,cwy_blocking))
      allocate(t_tmp(cwy_blocking,nbw))
      allocate(t_tmp2(cwy_blocking,nbw))
!      else
!        allocate(tmp1(max_local_cols*nbw))
!        allocate(tmp2(max_local_cols*nbw))
!        allocate(hvb(max_local_rows*nbw))
!        allocate(hvm(max_local_rows,nbw))
!      endif

      hvm = 0   ! Must be set to 0 !!!
      hvb = 0   ! Safety only

      l_cols = local_index(nqc, my_pcol, np_cols, nblk, -1) ! Local columns of q

!     if ( na >= ((t_blocking+1)*nbw) ) then

      do istep=1,((na-1)/nbw-1)/t_blocking + 1
        ! This the call when using  na >= ((t_blocking+1)*nbw)
        !      n_cols = MIN(na,istep*cwy_blocking+nbw) - (istep-1)*cwy_blocking - nbw ! Number of columns in current step
        ! As an alternative we add some special case handling if na < cwy_blocking
        IF (na < cwy_blocking) THEN
          n_cols = MAX(0, na-nbw)
          IF ( n_cols .eq. 0 ) THEN
            EXIT
          END IF
        ELSE
          n_cols = MIN(na,istep*cwy_blocking+nbw) - (istep-1)*cwy_blocking - nbw ! Number of columns in current step
        END IF

        ! Broadcast all Householder vectors for current step compressed in hvb

        nb = 0
        ns = 0

        do lc = 1, n_cols
          ncol = (istep-1)*cwy_blocking + nbw + lc ! absolute column number of householder vector
          nrow = ncol - nbw ! absolute number of pivot row

          l_rows = local_index(nrow-1, my_prow, np_rows, nblk, -1) ! row length for bcast
          l_colh = local_index(ncol  , my_pcol, np_cols, nblk, -1) ! HV local column number

          if (my_pcol==pcol(ncol, nblk, np_cols)) hvb(nb+1:nb+l_rows) = a(1:l_rows,l_colh)

          nb = nb+l_rows

          if (lc==n_cols .or. mod(ncol,nblk)==0) then
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#ifdef WITH_MPI
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            call MPI_Bcast(hvb(ns+1),nb-ns,MPI_REAL8,pcol(ncol, nblk, np_cols),mpi_comm_cols,mpierr)
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#endif
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            ns = nb
          endif
        enddo

        ! Expand compressed Householder vectors into matrix hvm

        nb = 0
        do lc = 1, n_cols
          nrow = (istep-1)*cwy_blocking + lc ! absolute number of pivot row
          l_rows = local_index(nrow-1, my_prow, np_rows, nblk, -1) ! row length for bcast

          hvm(1:l_rows,lc) = hvb(nb+1:nb+l_rows)
          if (my_prow==prow(nrow, nblk, np_rows)) hvm(l_rows+1,lc) = 1.

          nb = nb+l_rows
        enddo

        l_rows = local_index(MIN(na,(istep+1)*cwy_blocking), my_prow, np_rows, nblk, -1)

        ! compute tmat2 out of tmat(:,:,)
        tmat_complete = 0
        do i = 1, t_blocking
          t_cols = MIN(nbw, n_cols - (i-1)*nbw)
          if (t_cols <= 0) exit
          t_rows = (i - 1) * nbw
          tmat_complete(t_rows+1:t_rows+t_cols,t_rows+1:t_rows+t_cols) = tmat(1:t_cols,1:t_cols,(istep-1)*t_blocking + i)
          if (i > 1) then
            call dgemm('T', 'N', t_rows, t_cols, l_rows, 1.d0, hvm(1,1), max_local_rows, hvm(1,(i-1)*nbw+1), &
                      max_local_rows, 0.d0, t_tmp, cwy_blocking)
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#ifdef WITH_MPI
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            call mpi_allreduce(t_tmp,t_tmp2,cwy_blocking*nbw,MPI_REAL8,MPI_SUM,mpi_comm_rows,mpierr)
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#else
            t_tmp2 = t_tmp
#endif
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            call dtrmm('L','U','N','N',t_rows,t_cols,1.0d0,tmat_complete,cwy_blocking,t_tmp2,cwy_blocking)
            call dtrmm('R','U','N','N',t_rows,t_cols,-1.0d0,tmat_complete(t_rows+1,t_rows+1),cwy_blocking,t_tmp2,cwy_blocking)
            tmat_complete(1:t_rows,t_rows+1:t_rows+t_cols) = t_tmp2(1:t_rows,1:t_cols)
          endif
        enddo

        ! Q = Q - V * T**T * V**T * Q

        if (l_rows>0) then
          call dgemm('T','N',n_cols,l_cols,l_rows,1.d0,hvm,ubound(hvm,dim=1), &
                     q,ldq,0.d0,tmp1,n_cols)
        else
          tmp1(1:l_cols*n_cols) = 0
        endif
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        call mpi_allreduce(tmp1,tmp2,n_cols*l_cols,MPI_REAL8,MPI_SUM,mpi_comm_rows,mpierr)
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#else
        tmp2=tmp1
#endif
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        if (l_rows>0) then
          call dtrmm('L','U','T','N',n_cols,l_cols,1.0d0,tmat_complete,cwy_blocking,tmp2,n_cols)
          call dgemm('N','N',l_rows,l_cols,n_cols,-1.d0,hvm,ubound(hvm,dim=1), tmp2,n_cols,1.d0,q,ldq)
        endif
      enddo

!   else 
!
!     do istep=1,(na-1)/nbw
!
!       n_cols = MIN(na,(istep+1)*nbw) - istep*nbw ! Number of columns in current step
!
!       ! Broadcast all Householder vectors for current step compressed in hvb
!
!       nb = 0
!       ns = 0
!
!       do lc = 1, n_cols
!         ncol = istep*nbw + lc ! absolute column number of householder vector
!         nrow = ncol - nbw ! absolute number of pivot row
!
!         l_rows = local_index(nrow-1, my_prow, np_rows, nblk, -1) ! row length for bcast
!         l_colh = local_index(ncol  , my_pcol, np_cols, nblk, -1) ! HV local column number
!
!         if (my_pcol==pcol(ncol, nblk, np_cols)) hvb(nb+1:nb+l_rows) = a(1:l_rows,l_colh)
!
!         nb = nb+l_rows
!
!         if (lc==n_cols .or. mod(ncol,nblk)==0) then
!           call MPI_Bcast(hvb(ns+1),nb-ns,MPI_REAL8,pcol(ncol, nblk, np_cols),mpi_comm_cols,mpierr)
!           ns = nb
!         endif
!       enddo
!
!       ! Expand compressed Householder vectors into matrix hvm
!
!       nb = 0
!       do lc = 1, n_cols
!         nrow = (istep-1)*nbw+lc ! absolute number of pivot row
!         l_rows = local_index(nrow-1, my_prow, np_rows, nblk, -1) ! row length for bcast
!
!         hvm(1:l_rows,lc) = hvb(nb+1:nb+l_rows)
!         if (my_prow==prow(nrow, nblk, np_rows)) hvm(l_rows+1,lc) = 1.
!
!         nb = nb+l_rows
!       enddo
!
!       l_rows = local_index(MIN(na,(istep+1)*nbw), my_prow, np_rows, nblk, -1)
!
!       ! Q = Q - V * T**T * V**T * Q
!
!       if (l_rows>0) then
!         call dgemm('T','N',n_cols,l_cols,l_rows,1.d0,hvm,ubound(hvm,dim=1), &
!                    q,ldq,0.d0,tmp1,n_cols)
!       else
!         tmp1(1:l_cols*n_cols) = 0
!       endif
!
!       call mpi_allreduce(tmp1,tmp2,n_cols*l_cols,MPI_REAL8,MPI_SUM,mpi_comm_rows,mpierr)
!
!       if (l_rows>0) then
!         call dtrmm('L','U','T','N',n_cols,l_cols,1.0d0,tmat(1,1,istep),ubound(tmat,dim=1),tmp2,n_cols)
!         call dgemm('N','N',l_rows,l_cols,n_cols,-1.d0,hvm,ubound(hvm,dim=1), &
!                    tmp2,n_cols,1.d0,q,ldq)
!       endif
!     enddo
!   endif

      deallocate(tmp1, tmp2, hvb, hvm)
!   if ( na >= ((t_blocking+1)*nbw) ) then
      deallocate(tmat_complete, t_tmp, t_tmp2)
!   endif

#ifdef HAVE_DETAILED_TIMINGS
      call timer%stop("trans_ev_band_to_full_real")
#endif
    end subroutine trans_ev_band_to_full_real

    subroutine tridiag_band_real(na, nb, nblk, a, lda, d, e, matrixCols, hh_trans_real, &
                                 mpi_comm_rows, mpi_comm_cols, mpi_comm)

    !-------------------------------------------------------------------------------
    ! tridiag_band_real:
    ! Reduces a real symmetric band matrix to tridiagonal form
    !
    !  na          Order of matrix a
    !
    !  nb          Semi bandwith
    !
    !  nblk        blocksize of cyclic distribution, must be the same in both directions!
    !
    !  a(lda,matrixCols)    Distributed system matrix reduced to banded form in the upper diagonal
    !
    !  lda         Leading dimension of a
    !  matrixCols  local columns of matrix a
    !
    !  d(na)       Diagonal of tridiagonal matrix, set only on PE 0 (output)
    !
    !  e(na)       Subdiagonal of tridiagonal matrix, set only on PE 0 (output)
    !
    !  mpi_comm_rows
    !  mpi_comm_cols
    !              MPI-Communicators for rows/columns
    !  mpi_comm
    !              MPI-Communicator for the total processor set
    !-------------------------------------------------------------------------------
#ifdef HAVE_DETAILED_TIMINGS
      use timings
#endif
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      use precision
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      implicit none

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      integer(kind=ik), intent(in)  ::  na, nb, nblk, lda, matrixCols, mpi_comm_rows, mpi_comm_cols, mpi_comm
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#ifdef DESPERATELY_WANT_ASSUMED_SIZE
      real(kind=rk), intent(in)     :: a(lda,*)
#else
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      real(kind=rk), intent(in)     :: a(lda,matrixCols)
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#endif
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      real(kind=rk), intent(out)    :: d(na), e(na) ! set only on PE 0
      real(kind=rk), intent(out), &
          allocatable               :: hh_trans_real(:,:)
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      real(kind=rk)                 :: vnorm2, hv(nb), tau, x, h(nb), ab_s(1+nb), hv_s(nb), hv_new(nb), tau_new, hf
      real(kind=rk)                 :: hd(nb), hs(nb)
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      integer(kind=ik)              :: i, j, n, nc, nr, ns, ne, istep, iblk, nblocks_total, nblocks, nt
      integer(kind=ik)              :: my_pe, n_pes, mpierr
      integer(kind=ik)              :: my_prow, np_rows, my_pcol, np_cols
      integer(kind=ik)              :: ireq_ab, ireq_hv
      integer(kind=ik)              :: na_s, nx, num_hh_vecs, num_chunks, local_size, max_blk_size, n_off
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#ifdef WITH_OPENMP
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      integer(kind=ik)              :: max_threads, my_thread, my_block_s, my_block_e, iter
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#ifdef WITH_MPI
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      integer(kind=ik)              :: mpi_status(MPI_STATUS_SIZE)
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#endif
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      integer(kind=ik), allocatable :: mpi_statuses(:,:), global_id_tmp(:,:)
      integer(kind=ik), allocatable :: omp_block_limits(:)
      real(kind=rk), allocatable    :: hv_t(:,:), tau_t(:)
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#endif
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      integer(kind=ik), allocatable :: ireq_hhr(:), ireq_hhs(:), global_id(:,:), hh_cnt(:), hh_dst(:)
      integer(kind=ik), allocatable :: limits(:), snd_limits(:,:)
      integer(kind=ik), allocatable :: block_limits(:)
      real(kind=rk), allocatable    :: ab(:,:), hh_gath(:,:,:), hh_send(:,:,:)
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#ifdef WITH_OPENMP
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      integer(kind=ik)              :: omp_get_max_threads
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#endif

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#ifndef WITH_MPI
      integer(kind=ik)             :: startAddr
#endif

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#ifdef HAVE_DETAILED_TIMINGS
      call timer%start("tridiag_band_real")
#endif
      call mpi_comm_rank(mpi_comm,my_pe,mpierr)
      call mpi_comm_size(mpi_comm,n_pes,mpierr)

      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)
      ! Get global_id mapping 2D procssor coordinates to global id

      allocate(global_id(0:np_rows-1,0:np_cols-1))
      global_id(:,:) = 0
      global_id(my_prow, my_pcol) = my_pe
#ifdef WITH_OPENMP
      allocate(global_id_tmp(0:np_rows-1,0:np_cols-1))
#endif

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#ifdef WITH_MPI

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#ifndef WITH_OPENMP
      call mpi_allreduce(mpi_in_place, global_id, np_rows*np_cols, mpi_integer, mpi_sum, mpi_comm, mpierr)
#else
      global_id_tmp(:,:) = global_id(:,:)
      call mpi_allreduce(global_id_tmp, global_id, np_rows*np_cols, mpi_integer, mpi_sum, mpi_comm, mpierr)
      deallocate(global_id_tmp)
#endif

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#endif /* WITH_MPI */
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      ! Total number of blocks in the band:

      nblocks_total = (na-1)/nb + 1

      ! Set work distribution

      allocate(block_limits(0:n_pes))
      call divide_band(nblocks_total, n_pes, block_limits)

      ! nblocks: the number of blocks for my task
      nblocks = block_limits(my_pe+1) - block_limits(my_pe)

      ! allocate the part of the band matrix which is needed by this PE
      ! The size is 1 block larger than needed to avoid extensive shifts
      allocate(ab(2*nb,(nblocks+1)*nb))
      ab = 0 ! needed for lower half, the extra block should also be set to 0 for safety

      ! n_off: Offset of ab within band
      n_off = block_limits(my_pe)*nb

      ! Redistribute band in a to ab
      call redist_band_real(a, lda, na, nblk, nb, matrixCols, mpi_comm_rows, mpi_comm_cols, mpi_comm, ab)

      ! Calculate the workload for each sweep in the back transformation
      ! and the space requirements to hold the HH vectors

      allocate(limits(0:np_rows))
      call determine_workload(na, nb, np_rows, limits)
      max_blk_size = maxval(limits(1:np_rows) - limits(0:np_rows-1))

      num_hh_vecs = 0
      num_chunks  = 0
      nx = na
      do n = 1, nblocks_total
        call determine_workload(nx, nb, np_rows, limits)
        local_size = limits(my_prow+1) - limits(my_prow)
        ! add to number of householder vectors
        ! please note: for nx==1 the one and only HH vector is 0 and is neither calculated nor send below!
        if (mod(n-1,np_cols) == my_pcol .and. local_size>0 .and. nx>1) then
          num_hh_vecs = num_hh_vecs + local_size
          num_chunks  = num_chunks+1
        endif
        nx = nx - nb
      enddo

      ! Allocate space for HH vectors

      allocate(hh_trans_real(nb,num_hh_vecs))

      ! Allocate and init MPI requests

      allocate(ireq_hhr(num_chunks)) ! Recv requests
      allocate(ireq_hhs(nblocks))    ! Send requests

      num_hh_vecs = 0
      num_chunks  = 0
      nx = na
      nt = 0
      do n = 1, nblocks_total
        call determine_workload(nx, nb, np_rows, limits)
        local_size = limits(my_prow+1) - limits(my_prow)
        if (mod(n-1,np_cols) == my_pcol .and. local_size>0 .and. nx>1) then
          num_chunks  = num_chunks+1
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#ifdef WITH_MPI

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          call mpi_irecv(hh_trans_real(1,num_hh_vecs+1), nb*local_size, mpi_real8, nt, &
                           10+n-block_limits(nt), mpi_comm, ireq_hhr(num_chunks), mpierr)
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#else
          ! carefull non-block recv data copy must be done at wait or send
          ! hh_trans_real(1:nb*local_size,num_hh_vecs+1) = hh_send(1:nb*hh_cnt(iblk),1,iblk)
#endif
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          num_hh_vecs = num_hh_vecs + local_size
        endif
        nx = nx - nb
        if (n == block_limits(nt+1)) then
          nt = nt + 1
        endif
      enddo
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#ifdef WITH_MPI
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      ireq_hhs(:) = MPI_REQUEST_NULL
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#endif
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      ! Buffers for gathering/sending the HH vectors

      allocate(hh_gath(nb,max_blk_size,nblocks)) ! gathers HH vectors
      allocate(hh_send(nb,max_blk_size,nblocks)) ! send buffer for HH vectors
      hh_gath(:,:,:) = 0
      hh_send(:,:,:) = 0

      ! Some counters

      allocate(hh_cnt(nblocks))
      allocate(hh_dst(nblocks))

      hh_cnt(:) = 1 ! The first transfomation vector is always 0 and not calculated at all
      hh_dst(:) = 0 ! PE number for receive
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#ifdef WITH_MPI
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      ireq_ab = MPI_REQUEST_NULL
      ireq_hv = MPI_REQUEST_NULL
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#endif
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      ! Limits for sending

      allocate(snd_limits(0:np_rows,nblocks))

      do iblk=1,nblocks
        call determine_workload(na-(iblk+block_limits(my_pe)-1)*nb, nb, np_rows, snd_limits(:,iblk))
      enddo

#ifdef WITH_OPENMP
      ! OpenMP work distribution:

      max_threads = 1
      max_threads = omp_get_max_threads()

      ! For OpenMP we need at least 2 blocks for every thread
      max_threads = MIN(max_threads, nblocks/2)
      if (max_threads==0) max_threads = 1

      allocate(omp_block_limits(0:max_threads))

      ! Get the OpenMP block limits
      call divide_band(nblocks, max_threads, omp_block_limits)

      allocate(hv_t(nb,max_threads), tau_t(max_threads))
      hv_t = 0
      tau_t = 0
#endif

      ! ---------------------------------------------------------------------------
      ! Start of calculations

      na_s = block_limits(my_pe)*nb + 1

      if (my_pe>0 .and. na_s<=na) then
        ! send first column to previous PE
        ! Only the PE owning the diagonal does that (sending 1 element of the subdiagonal block also)
        ab_s(1:nb+1) = ab(1:nb+1,na_s-n_off)
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        call mpi_isend(ab_s,nb+1,mpi_real8,my_pe-1,1,mpi_comm,ireq_ab,mpierr)
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#endif
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      endif

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#ifndef WITH_MPI
          startAddr   = ubound(hh_trans_real,dim=2)
#endif

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#ifdef WITH_OPENMP
      do istep=1,na-1-block_limits(my_pe)*nb
#else
      do istep=1,na-1
#endif

        if (my_pe==0) then
          n = MIN(na-na_s,nb) ! number of rows to be reduced
          hv(:) = 0
          tau = 0
          ! The last step (istep=na-1) is only needed for sending the last HH vectors.
          ! We don't want the sign of the last element flipped (analogous to the other sweeps)
          if (istep < na-1) then
            ! Transform first column of remaining matrix
            vnorm2 = sum(ab(3:n+1,na_s-n_off)**2)
            call hh_transform_real(ab(2,na_s-n_off),vnorm2,hf,tau)
            hv(1) = 1
            hv(2:n) = ab(3:n+1,na_s-n_off)*hf
          endif
          d(istep) = ab(1,na_s-n_off)
          e(istep) = ab(2,na_s-n_off)
          if (istep == na-1) then
            d(na) = ab(1,na_s+1-n_off)
            e(na) = 0
          endif
        else
          if (na>na_s) then
            ! Receive Householder vector from previous task, from PE owning subdiagonal
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#ifdef WITH_MPI

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#ifdef WITH_OPENMP
            call mpi_recv(hv,nb,mpi_real8,my_pe-1,2,mpi_comm,MPI_STATUS,mpierr)
#else
            call mpi_recv(hv,nb,mpi_real8,my_pe-1,2,mpi_comm,MPI_STATUS_IGNORE,mpierr)
#endif
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#else /* WITH_MPI */

#ifdef WITH_OPENMP
            hv(1:nb) = hv_s(1:nb)
#else
            hv(1:nb) = hv_s(1:nb)
#endif

#endif /* WITH_MPI */
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            tau = hv(1)
            hv(1) = 1.
          endif
        endif

        na_s = na_s+1
        if (na_s-n_off > nb) then
          ab(:,1:nblocks*nb) = ab(:,nb+1:(nblocks+1)*nb)
          ab(:,nblocks*nb+1:(nblocks+1)*nb) = 0
          n_off = n_off + nb
        endif


#ifdef WITH_OPENMP
        if (max_threads > 1) then

          ! Codepath for OpenMP

          ! Please note that in this case it is absolutely necessary to have at least 2 blocks per thread!
          ! Every thread is one reduction cycle behind its predecessor and thus starts one step later.
          ! This simulates the behaviour of the MPI tasks which also work after each other.
          ! The code would be considerably easier, if the MPI communication would be made within
          ! the parallel region - this is avoided here since this would require
          ! MPI_Init_thread(MPI_THREAD_MULTIPLE) at the start of the program.

          hv_t(:,1) = hv
          tau_t(1) = tau

          do iter = 1, 2

            ! iter=1 : work on first block
            ! iter=2 : work on remaining blocks
            ! This is done in 2 iterations so that we have a barrier in between:
            ! After the first iteration, it is guaranteed that the last row of the last block
            ! is completed by the next thread.
            ! After the first iteration it is also the place to exchange the last row
            ! with MPI calls
#ifdef HAVE_DETAILED_TIMINGS
            call timer%start("OpenMP parallel")
#endif

!$omp parallel do private(my_thread, my_block_s, my_block_e, iblk, ns, ne, hv, tau, &
!$omp&                    nc, nr, hs, hd, vnorm2, hf, x, h, i), schedule(static,1), num_threads(max_threads)
            do my_thread = 1, max_threads

              if (iter == 1) then
                my_block_s = omp_block_limits(my_thread-1) + 1
                my_block_e = my_block_s
              else
                my_block_s = omp_block_limits(my_thread-1) + 2
                my_block_e = omp_block_limits(my_thread)
              endif

              do iblk = my_block_s, my_block_e

                ns = na_s + (iblk-1)*nb - n_off - my_thread + 1 ! first column in block
                ne = ns+nb-1                    ! last column in block

                if (istep<my_thread .or. ns+n_off>na) exit

                hv = hv_t(:,my_thread)
                tau = tau_t(my_thread)

                ! Store Householder vector for back transformation

                hh_cnt(iblk) = hh_cnt(iblk) + 1

                hh_gath(1   ,hh_cnt(iblk),iblk) = tau
                hh_gath(2:nb,hh_cnt(iblk),iblk) = hv(2:nb)

                nc = MIN(na-ns-n_off+1,nb) ! number of columns in diagonal block
                nr = MIN(na-nb-ns-n_off+1,nb) ! rows in subdiagonal block (may be < 0!!!)
                                          ! Note that nr>=0 implies that diagonal block is full (nc==nb)!

                ! Transform diagonal block

                call DSYMV('L',nc,tau,ab(1,ns),2*nb-1,hv,1,0.d0,hd,1)

                x = dot_product(hv(1:nc),hd(1:nc))*tau
                hd(1:nc) = hd(1:nc) - 0.5*x*hv(1:nc)

                call DSYR2('L',nc,-1.d0,hd,1,hv,1,ab(1,ns),2*nb-1)

                hv_t(:,my_thread) = 0
                tau_t(my_thread)  = 0

                if (nr<=0) cycle ! No subdiagonal block present any more

                ! Transform subdiagonal block

                call DGEMV('N',nr,nb,tau,ab(nb+1,ns),2*nb-1,hv,1,0.d0,hs,1)

                if (nr>1) then

                  ! complete (old) Householder transformation for first column

                  ab(nb+1:nb+nr,ns) = ab(nb+1:nb+nr,ns) - hs(1:nr) ! Note: hv(1) == 1

                  ! calculate new Householder transformation for first column
                  ! (stored in hv_t(:,my_thread) and tau_t(my_thread))

                  vnorm2 = sum(ab(nb+2:nb+nr,ns)**2)
                  call hh_transform_real(ab(nb+1,ns),vnorm2,hf,tau_t(my_thread))
                  hv_t(1   ,my_thread) = 1.
                  hv_t(2:nr,my_thread) = ab(nb+2:nb+nr,ns)*hf
                  ab(nb+2:,ns) = 0

                  ! update subdiagonal block for old and new Householder transformation
                  ! This way we can use a nonsymmetric rank 2 update which is (hopefully) faster

                  call DGEMV('T',nr,nb-1,tau_t(my_thread),ab(nb,ns+1),2*nb-1,hv_t(1,my_thread),1,0.d0,h(2),1)
                  x = dot_product(hs(1:nr),hv_t(1:nr,my_thread))*tau_t(my_thread)
                  h(2:nb) = h(2:nb) - x*hv(2:nb)
                  ! Unfortunately there is no BLAS routine like DSYR2 for a nonsymmetric rank 2 update ("DGER2")
                  do i=2,nb
                    ab(2+nb-i:1+nb+nr-i,i+ns-1) = ab(2+nb-i:1+nb+nr-i,i+ns-1) - hv_t(1:nr,my_thread)*h(i) - hs(1:nr)*hv(i)
                  enddo

                else

                  ! No new Householder transformation for nr=1, just complete the old one
                  ab(nb+1,ns) = ab(nb+1,ns) - hs(1) ! Note: hv(1) == 1
                  do i=2,nb
                    ab(2+nb-i,i+ns-1) = ab(2+nb-i,i+ns-1) - hs(1)*hv(i)
                  enddo
                  ! For safety: there is one remaining dummy transformation (but tau is 0 anyways)
                  hv_t(1,my_thread) = 1.

                endif

              enddo

            enddo ! my_thread
!$omp end parallel do
#ifdef HAVE_DETAILED_TIMINGS
            call timer%stop("OpenMP parallel")
#endif

            if (iter==1) then
              ! We are at the end of the first block

              ! Send our first column to previous PE
              if (my_pe>0 .and. na_s <= na) then
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#ifdef WITH_MPI
1392
                call mpi_wait(ireq_ab,mpi_status,mpierr)
1393
#endif
1394
                ab_s(1:nb+1) = ab(1:nb+1,na_s-n_off)
1395
#ifdef WITH_MPI
1396
                call mpi_isend(ab_s,nb+1,mpi_real8,my_pe-1,1,mpi_comm,ireq_ab,mpierr)
1397
#endif
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1401
              endif

              ! Request last column from next PE
              ne = na_s + nblocks*nb - (max_threads-1) - 1
1402
#ifdef WITH_MPI
1403
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1405
              if (istep>=max_threads .and. ne <= na) then
                call mpi_recv(ab(1,ne-n_off),nb+1,mpi_real8,my_pe+1,1,mpi_comm,mpi_status,mpierr)
              endif
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#else
              if (istep>=max_threads .and. ne <= na) then
                ab(1:nb+1,ne-n_off) = ab_s(1:nb+1)
              endif
#endif
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            else
              ! We are at the end of all blocks

              ! Send last HH vector and TAU to next PE if it has been calculated above
              ne = na_s + nblocks*nb - (max_threads-1) - 1
              if (istep>=max_threads .and. ne < na) then
1417
#ifdef WITH_MPI
1418
                call mpi_wait(ireq_hv,mpi_status,mpierr)
1419
#endif
1420
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                hv_s(1) = tau_t(max_threads)
                hv_s(2:) = hv_t(2:,max_threads)
1422
#ifdef WITH_MPI
1423
                call mpi_isend(hv_s,nb,mpi_real8,my_pe+1,2,mpi_comm,ireq_hv,mpierr)
1424
#endif
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              endif

              ! "Send" HH vector and TAU to next OpenMP thread
              do my_thread = max_threads, 2, -1
                hv_t(:,my_thread) = hv_t(:,my_thread-1)
                tau_t(my_thread)  = tau_t(my_thread-1)
              enddo

            endif
          enddo ! iter

        else

          ! Codepath for 1 thread without OpenMP

          ! The following code is structured in a way to keep waiting times for
          ! other PEs at a minimum, especially if there is only one block.
          ! For this reason, it requests the last column as late as possible
          ! and sends the Householder vector and the first column as early
          ! as possible.

#endif /* WITH_OPENMP */

          do iblk=1,nblocks
            ns = na_s + (iblk-1)*nb - n_off ! first column in block
            ne = ns+nb-1                    ! last column in block

            if (ns+n_off>na) exit

            ! Store Householder vector for back transformation

            hh_cnt(iblk) = hh_cnt(iblk) + 1

            hh_gath(1   ,hh_cnt(iblk),iblk) = tau
            hh_gath(2:nb,hh_cnt(iblk),iblk) = hv(2:nb)

#ifndef WITH_OPENMP
            if (hh_cnt(iblk) == snd_limits(hh_dst(iblk)+1,iblk)-snd_limits(hh_dst(iblk),iblk)) then
              ! Wait for last transfer to finish
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#ifdef WITH_MPI
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              call mpi_wait(ireq_hhs(iblk), MPI_STATUS_IGNORE, mpierr)
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#endif
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              ! Copy vectors into send buffer
              hh_send(:,1:hh_cnt(iblk),iblk) = hh_gath(:,1:hh_cnt(iblk),iblk)
              ! Send to destination
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#ifdef WITH_MPI
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              call mpi_isend(hh_send(1,1,iblk), nb*hh_cnt(iblk), mpi_real8, &
                           global_id(hh_dst(iblk),mod(iblk+block_limits(my_pe)-1,np_cols)), &
                           10+iblk, mpi_comm, ireq_hhs(iblk), mpierr)
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#else
             startAddr = startAddr - hh_cnt(iblk)
             hh_trans_real(1:nb,startAddr+1:startAddr+hh_cnt(iblk)) = hh_send(1:nb,1:hh_cnt(iblk),iblk)
#endif /* WITH_MPI */

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            ! Reset counter and increase destination row
              hh_cnt(iblk) = 0
              hh_dst(iblk) = hh_dst(iblk)+1
            endif

            ! The following code is structured in a way to keep waiting times for
            ! other PEs at a minimum, especially if there is only one block.
            ! For this reason, it requests the last column as late as possible
            ! and sends the Householder vector and the first column as early
            ! as possible.
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#endif /* WITH_OPENMP */
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            nc = MIN(na-ns-n_off+1,nb) ! number of columns in diagonal block
            nr = MIN(na-nb-ns-n_off+1,nb) ! rows in subdiagonal block (may be < 0!!!)
                                          ! Note that nr>=0 implies that diagonal block is full (nc==nb)!

            ! Multiply diagonal block and subdiagonal block with Householder vector

            if (iblk==nblocks .and. nc==nb) then

              ! We need the last column from the next PE.
              ! First do the matrix multiplications without last column ...

              ! Diagonal block, the contribution of the last element is added below!
              ab(1,ne) = 0
              call DSYMV('L',nc,tau,ab(1,ns),2*nb-1,hv,1,0.d0,hd,1)

              ! Subdiagonal block
              if (nr>0) call DGEMV('N',nr,nb-1,tau,ab(nb+1,ns),2*nb-1,hv,1,0.d0,hs,1)

              ! ... then request last column ...
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#ifdef WITH_MPI

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#ifdef WITH_OPENMP
              call mpi_recv(ab(1,ne),nb+1,mpi_real8,my_pe+1,1,mpi_comm,MPI_STATUS,mpierr)
#else
              call mpi_recv(ab(1,ne),nb+1,mpi_real8,my_pe+1,1,mpi_comm,MPI_STATUS_IGNORE,mpierr)
#endif

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#else /* WITH_MPI */

#ifdef WITH_OPENMP
              ab(1:nb+1,ne) = ab_s(1:nb+1)
#else
              ab(1:nb+1,ne) = ab_s(1:nb+1)
#endif

#endif /* WITH_MPI */
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              ! ... and complete the result
              hs(1:nr) = hs(1:nr) + ab(2:nr+1,ne)*tau*hv(nb)
              hd(nb) = hd(nb) + ab(1,ne)*hv(nb)*tau

            else

              ! Normal matrix multiply
              call DSYMV('L',nc,tau,ab(1,ns),2*nb-1,hv,1,0.d0,hd,1)
              if (nr>0) call DGEMV('N',nr,nb,tau,ab(nb+1,ns),2*nb-1,hv,1,0.d0,hs,1)

            endif

            ! Calculate first column of subdiagonal block and calculate new
            ! Householder transformation for this column

            hv_new(:) = 0 ! Needed, last rows must be 0 for nr < nb
            tau_new = 0

            if (nr>0) then

              ! complete (old) Householder transformation for first column

              ab(nb+1:nb+nr,ns) = ab(nb+1:nb+nr,ns) - hs(1:nr) ! Note: hv(1) == 1

              ! calculate new Householder transformation ...
              if (nr>1) then
                vnorm2 = sum(ab(nb+2:nb+nr,ns)**2)
                call hh_transform_real(ab(nb+1,ns),vnorm2,hf,tau_new)
                hv_new(1) = 1.
                hv_new(2:nr) = ab(nb+2:nb+nr,ns)*hf
                ab(nb+2:,ns) = 0
              endif

              ! ... and send it away immediatly if this is the last block

              if (iblk==nblocks) then
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#ifdef WITH_MPI

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#ifdef WITH_OPENMP
                call mpi_wait(ireq_hv,MPI_STATUS,mpierr)
#else
                call mpi_wait(ireq_hv,MPI_STATUS_IGNORE,mpierr)
#endif
1569
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#endif /* WITH_MPI */
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                hv_s(1) = tau_new
                hv_s(2:) = hv_new(2:)
1573
#ifdef WITH_MPI
1574
                call mpi_isend(hv_s,nb,mpi_real8,my_pe+1,2,mpi_comm,ireq_hv,mpierr)
1575
#endif
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              endif

            endif

            ! Transform diagonal block
            x = dot_product(hv(1:nc),hd(1:nc))*tau
            hd(1:nc) = hd(1:nc) - 0.5*x*hv(1:nc)

            if (my_pe>0 .and. iblk==1) then

              ! The first column of the diagonal block has to be send to the previous PE
              ! Calculate first column only ...

              ab(1:nc,ns) = ab(1:nc,ns) - hd(1:nc)*hv(1) - hv(1:nc)*hd(1)

              ! ... send it away ...
1592
#ifdef WITH_MPI
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#ifdef WITH_OPENMP
              call mpi_wait(ireq_ab,MPI_STATUS,mpierr)
#else
              call mpi_wait(ireq_ab,MPI_STATUS_IGNORE,mpierr)
#endif
1599
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#endif /* WITH_MPI */
1601
              ab_s(1:nb+1) = ab(1:nb+1,ns)
1602
#ifdef WITH_MPI
1603
              call mpi_isend(ab_s,nb+1,mpi_real8,my_pe-1,1,mpi_comm,ireq_ab,mpierr)
1604
#endif
1605
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              ! ... and calculate remaining columns with rank-2 update
              if (nc>1) call DSYR2('L',nc-1,-1.d0,hd(2),1,hv(2),1,ab(1,ns+1),2*nb-1)
            else
              ! No need to  send, just a rank-2 update
              call DSYR2('L',nc,-1.d0,hd,1,hv,1,ab(1,ns),2*nb-1)
            endif

            ! Do the remaining double Householder transformation on the subdiagonal block cols 2 ... nb

            if (nr>0) then
              if (nr>1) then
                call DGEMV('T',nr,nb-1,tau_new,ab(nb,ns+1),2*nb-1,hv_new,1,0.d0,h(2),1)
                x = dot_product(hs(1:nr),hv_new(1:nr))*tau_new
                h(2:nb) = h(2:nb) - x*hv(2:nb)
                ! Unfortunately there is no BLAS routine like DSYR2 for a nonsymmetric rank 2 update
                do i=2,nb
                  ab(2+nb-i:1+nb+nr-i,i+ns-1) = ab(2+nb-i:1+nb+nr-i,i+ns-1) - hv_new(1:nr)*h(i) - hs(1:nr)*hv(i)
                enddo
              else
                ! No double Householder transformation for nr=1, just complete the row
                do i=2,nb
                  ab(2+nb-i,i+ns-1) = ab(2+nb-i,i+ns-1) - hs(1)*hv(i)
                enddo