elpa2.F90 170 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:
!
!    - 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
!
!
!    More information can be found here:
!    http://elpa.rzg.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

! Version 1.1.2, 2011-02-21

  USE ELPA1

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#ifdef HAVE_ISO_FORTRAN_ENV
  use iso_fortran_env, only : error_unit
#endif
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  implicit none

  PRIVATE ! By default, all routines contained are private

  ! The following routines are public:

  public :: solve_evp_real_2stage
  public :: solve_evp_complex_2stage

  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

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#ifndef HAVE_ISO_FORTRAN_ENV
  integer, parameter :: error_unit = 6
#endif

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!-------------------------------------------------------------------------------

  ! The following array contains the Householder vectors of the
  ! transformation band -> tridiagonal.
  ! It is allocated and set in tridiag_band_real and used in
  ! trans_ev_tridi_to_band_real.
  ! It must be deallocated by the user after trans_ev_tridi_to_band_real!

  real*8, allocatable :: hh_trans_real(:,:)
  complex*16, allocatable :: hh_trans_complex(:,:)

!-------------------------------------------------------------------------------

  include 'mpif.h'


!******
contains

subroutine solve_evp_real_2stage(na, nev, a, lda, ev, q, ldq, nblk, mpi_comm_rows, mpi_comm_cols, mpi_comm_all)

!-------------------------------------------------------------------------------
!  solve_evp_real_2stage: Solves the real eigenvalue problem with a 2 stage approach
!
!  Parameters
!
!  na          Order of matrix a
!
!  nev         Number of eigenvalues needed
!
!  a(lda,*)    Distributed matrix for which eigenvalues are to be computed.
!              Distribution is like in Scalapack.
!              The full matrix must be set (not only one half like in scalapack).
!              Destroyed on exit (upper and lower half).
!
!  lda         Leading dimension of a
!
!  ev(na)      On output: eigenvalues of a, every processor gets the complete set
!
!  q(ldq,*)    On output: Eigenvectors of a
!              Distribution is like in Scalapack.
!              Must be always dimensioned to the full size (corresponding to (na,na))
!              even if only a part of the eigenvalues is needed.
!
!  ldq         Leading dimension of q
!
!  nblk        blocksize of cyclic distribution, must be the same in both directions!
!
!  mpi_comm_rows
!  mpi_comm_cols
!              MPI-Communicators for rows/columns
!  mpi_comm_all
!              MPI-Communicator for the total processor set
!
!-------------------------------------------------------------------------------

   implicit none

   integer, intent(in) :: na, nev, lda, ldq, nblk, mpi_comm_rows, mpi_comm_cols, mpi_comm_all
   real*8, intent(inout) :: a(lda,*), ev(na), q(ldq,*)

   integer my_pe, n_pes, my_prow, my_pcol, np_rows, np_cols, mpierr
   integer nbw, num_blocks
   real*8, allocatable :: tmat(:,:,:), e(:)
   real*8 ttt0, ttt1, ttts

   call mpi_comm_rank(mpi_comm_all,my_pe,mpierr)
   call mpi_comm_size(mpi_comm_all,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)

   ! Choose bandwidth, must be a multiple of nblk, set to a value >= 32

   nbw = (31/nblk+1)*nblk

   num_blocks = (na-1)/nbw + 1

   allocate(tmat(nbw,nbw,num_blocks))

   ! Reduction full -> band

   ttt0 = MPI_Wtime()
   ttts = ttt0
   call bandred_real(na, a, lda, nblk, nbw, mpi_comm_rows, mpi_comm_cols, tmat)
   ttt1 = MPI_Wtime()
   if(my_prow==0 .and. my_pcol==0 .and. elpa_print_times) &
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      write(error_unit,*) 'Time bandred_real               :',ttt1-ttt0
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   ! Reduction band -> tridiagonal

   allocate(e(na))

   ttt0 = MPI_Wtime()
   call tridiag_band_real(na, nbw, nblk, a, lda, ev, e, mpi_comm_rows, mpi_comm_cols, mpi_comm_all)
   ttt1 = MPI_Wtime()
   if(my_prow==0 .and. my_pcol==0 .and. elpa_print_times) &
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      write(error_unit,*) 'Time tridiag_band_real          :',ttt1-ttt0
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   call mpi_bcast(ev,na,MPI_REAL8,0,mpi_comm_all,mpierr)
   call mpi_bcast(e,na,MPI_REAL8,0,mpi_comm_all,mpierr)

   ttt1 = MPI_Wtime()
   time_evp_fwd = ttt1-ttts

   ! Solve tridiagonal system

   ttt0 = MPI_Wtime()
   call solve_tridi(na, nev, ev, e, q, ldq, nblk, mpi_comm_rows, mpi_comm_cols)
   ttt1 = MPI_Wtime()
   if(my_prow==0 .and. my_pcol==0 .and. elpa_print_times) &
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      write(error_unit,*) 'Time solve_tridi                :',ttt1-ttt0
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   time_evp_solve = ttt1-ttt0
   ttts = ttt1

   deallocate(e)

   ! Backtransform stage 1

   ttt0 = MPI_Wtime()
   call trans_ev_tridi_to_band_real(na, nev, nblk, nbw, q, ldq, mpi_comm_rows, mpi_comm_cols)
   ttt1 = MPI_Wtime()
   if(my_prow==0 .and. my_pcol==0 .and. elpa_print_times) &
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      write(error_unit,*) 'Time trans_ev_tridi_to_band_real:',ttt1-ttt0
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   ! We can now deallocate the stored householder vectors
   deallocate(hh_trans_real)

   ! Backtransform stage 2

   ttt0 = MPI_Wtime()
   call trans_ev_band_to_full_real(na, nev, nblk, nbw, a, lda, tmat, q, ldq, mpi_comm_rows, mpi_comm_cols)
   ttt1 = MPI_Wtime()
   if(my_prow==0 .and. my_pcol==0 .and. elpa_print_times) &
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      write(error_unit,*) 'Time trans_ev_band_to_full_real :',ttt1-ttt0
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   time_evp_back = ttt1-ttts

   deallocate(tmat)

1  format(a,f10.3)

end subroutine solve_evp_real_2stage

!-------------------------------------------------------------------------------

!-------------------------------------------------------------------------------

subroutine solve_evp_complex_2stage(na, nev, a, lda, ev, q, ldq, nblk, mpi_comm_rows, mpi_comm_cols, mpi_comm_all)

!-------------------------------------------------------------------------------
!  solve_evp_complex_2stage: Solves the complex eigenvalue problem with a 2 stage approach
!
!  Parameters
!
!  na          Order of matrix a
!
!  nev         Number of eigenvalues needed
!
!  a(lda,*)    Distributed matrix for which eigenvalues are to be computed.
!              Distribution is like in Scalapack.
!              The full matrix must be set (not only one half like in scalapack).
!              Destroyed on exit (upper and lower half).
!
!  lda         Leading dimension of a
!
!  ev(na)      On output: eigenvalues of a, every processor gets the complete set
!
!  q(ldq,*)    On output: Eigenvectors of a
!              Distribution is like in Scalapack.
!              Must be always dimensioned to the full size (corresponding to (na,na))
!              even if only a part of the eigenvalues is needed.
!
!  ldq         Leading dimension of q
!
!  nblk        blocksize of cyclic distribution, must be the same in both directions!
!
!  mpi_comm_rows
!  mpi_comm_cols
!              MPI-Communicators for rows/columns
!  mpi_comm_all
!              MPI-Communicator for the total processor set
!
!-------------------------------------------------------------------------------

   implicit none

   integer, intent(in) :: na, nev, lda, ldq, nblk, mpi_comm_rows, mpi_comm_cols, mpi_comm_all
   complex*16, intent(inout) :: a(lda,*), q(ldq,*)
   real*8, intent(inout) :: ev(na)

   integer my_prow, my_pcol, np_rows, np_cols, mpierr
   integer l_cols, l_rows, l_cols_nev, nbw, num_blocks
   complex*16, allocatable :: tmat(:,:,:)
   real*8, allocatable :: q_real(:,:), e(:)
   real*8 ttt0, ttt1, ttts

   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)

   ! Choose bandwidth, must be a multiple of nblk, set to a value >= 32

   nbw = (31/nblk+1)*nblk

   num_blocks = (na-1)/nbw + 1

   allocate(tmat(nbw,nbw,num_blocks))

   ! Reduction full -> band

   ttt0 = MPI_Wtime()
   ttts = ttt0
   call bandred_complex(na, a, lda, nblk, nbw, mpi_comm_rows, mpi_comm_cols, tmat)
   ttt1 = MPI_Wtime()
   if(my_prow==0 .and. my_pcol==0 .and. elpa_print_times) &
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      write(error_unit,*) 'Time bandred_complex               :',ttt1-ttt0
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   ! Reduction band -> tridiagonal

   allocate(e(na))

   ttt0 = MPI_Wtime()
   call tridiag_band_complex(na, nbw, nblk, a, lda, ev, e, mpi_comm_rows, mpi_comm_cols, mpi_comm_all)
   ttt1 = MPI_Wtime()
   if(my_prow==0 .and. my_pcol==0 .and. elpa_print_times) &
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      write(error_unit,*) 'Time tridiag_band_complex          :',ttt1-ttt0
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   call mpi_bcast(ev,na,MPI_REAL8,0,mpi_comm_all,mpierr)
   call mpi_bcast(e,na,MPI_REAL8,0,mpi_comm_all,mpierr)

   ttt1 = MPI_Wtime()
   time_evp_fwd = ttt1-ttts

   l_rows = local_index(na, my_prow, np_rows, nblk, -1) ! Local rows of a and q
   l_cols = local_index(na, my_pcol, np_cols, nblk, -1) ! Local columns of q
   l_cols_nev = local_index(nev, my_pcol, np_cols, nblk, -1) ! Local columns corresponding to nev

   allocate(q_real(l_rows,l_cols))

   ! Solve tridiagonal system

   ttt0 = MPI_Wtime()
   call solve_tridi(na, nev, ev, e, q_real, ubound(q_real,1), nblk, mpi_comm_rows, mpi_comm_cols)
   ttt1 = MPI_Wtime()
   if(my_prow==0 .and. my_pcol==0 .and. elpa_print_times)  &
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      write(error_unit,*) 'Time solve_tridi                   :',ttt1-ttt0
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   time_evp_solve = ttt1-ttt0
   ttts = ttt1

   q(1:l_rows,1:l_cols_nev) = q_real(1:l_rows,1:l_cols_nev)

   deallocate(e, q_real)

   ! Backtransform stage 1

   ttt0 = MPI_Wtime()
   call trans_ev_tridi_to_band_complex(na, nev, nblk, nbw, q, ldq, mpi_comm_rows, mpi_comm_cols)
   ttt1 = MPI_Wtime()
   if(my_prow==0 .and. my_pcol==0 .and. elpa_print_times) &
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      write(error_unit,*) 'Time trans_ev_tridi_to_band_complex:',ttt1-ttt0
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   ! We can now deallocate the stored householder vectors
   deallocate(hh_trans_complex)

   ! Backtransform stage 2

   ttt0 = MPI_Wtime()
   call trans_ev_band_to_full_complex(na, nev, nblk, nbw, a, lda, tmat, q, ldq, mpi_comm_rows, mpi_comm_cols)
   ttt1 = MPI_Wtime()
   if(my_prow==0 .and. my_pcol==0 .and. elpa_print_times) &
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      write(error_unit,*) 'Time trans_ev_band_to_full_complex :',ttt1-ttt0
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   time_evp_back = ttt1-ttts

   deallocate(tmat)

1  format(a,f10.3)

end subroutine solve_evp_complex_2stage

!-------------------------------------------------------------------------------

subroutine bandred_real(na, a, lda, nblk, nbw, mpi_comm_rows, mpi_comm_cols, tmat)

!-------------------------------------------------------------------------------
!  bandred_real: Reduces a distributed symmetric matrix to band form
!
!  Parameters
!
!  na          Order of matrix
!
!  a(lda,*)    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
!
!  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,num_blocks)    where num_blocks = (na-1)/nbw + 1
!              Factors for the Householder vectors (returned), needed for back transformation
!
!-------------------------------------------------------------------------------

   implicit none

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

   integer my_prow, my_pcol, np_rows, np_cols, mpierr
   integer l_cols, l_rows
   integer i, j, lcs, lce, lre, lc, lr, cur_pcol, n_cols, nrow
   integer istep, ncol, lch, lcx, nlc
   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:: tmp(:,:), vr(:), vmr(:,:), umc(:,:)

   integer pcol, prow
   pcol(i) = MOD((i-1)/nblk,np_cols) !Processor col for global col number
   prow(i) = MOD((i-1)/nblk,np_rows) !Processor row for global row number


   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)

   ! Semibandwith nbw must be a multiple of blocksize nblk

   if(mod(nbw,nblk)/=0) then
      if(my_prow==0 .and. my_pcol==0) then
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         write(error_unit,*) 'ERROR: nbw=',nbw,', nblk=',nblk
         write(error_unit,*) 'ELPA2 works only for nbw==n*nblk'
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         call mpi_abort(mpi_comm_world,0,mpierr)
      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

   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

      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) ! 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)) 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)) 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)
         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

         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

      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,1),0.d0,vav,ubound(vav,1))
      call symm_matrix_allreduce(n_cols,vav,ubound(vav,1),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,1),vav(lc+1,lc),1)
            tmat(lc,lc+1:n_cols,istep) = -tau * vav(lc+1:n_cols,lc)
         endif
      enddo

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

      call elpa_transpose_vectors  (vmr, ubound(vmr,1), mpi_comm_rows, &
                                    umc(1,n_cols+1), ubound(umc,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

      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,1), &
                       vmr,ubound(vmr,1),1.d0,umc(lcs,1),ubound(umc,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,1),1.d0,vmr(1,n_cols+1),ubound(vmr,1))
         enddo
      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

      if(tile_size < istep*nbw) then
         call elpa_reduce_add_vectors  (vmr(1,n_cols+1),ubound(vmr,1),mpi_comm_rows, &
                                        umc, ubound(umc,1), mpi_comm_cols, &
                                        istep*nbw, n_cols, nblk)
      endif

      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

      ! U = U * Tmat**T

      call dtrmm('Right','Upper','Trans','Nonunit',l_cols,n_cols,1.d0,tmat(1,1,istep),ubound(tmat,1),umc,ubound(umc,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,1),umc(1,n_cols+1),ubound(umc,1),0.d0,vav,ubound(vav,1))
      call dtrmm('Right','Upper','Trans','Nonunit',n_cols,n_cols,1.d0,tmat(1,1,istep),ubound(tmat,1),vav,ubound(vav,1))

      call symm_matrix_allreduce(n_cols,vav,ubound(vav,1),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,1),vav,ubound(vav,1),1.d0,umc,ubound(umc,1))

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

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

      ! 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 dgemm('N','T',lre,lce-lcs+1,2*n_cols,-1.d0, &
                    vmr,ubound(vmr,1),umc(lcs,1),ubound(umc,1), &
                    1.d0,a(1,lcs),lda)
      enddo

      deallocate(vmr, umc, vr)

   enddo

end subroutine bandred_real

!-------------------------------------------------------------------------------

subroutine symm_matrix_allreduce(n,a,lda,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
!-------------------------------------------------------------------------------

   implicit none
   integer n, lda, comm
   real*8 a(lda,*)

   integer i, nc, mpierr
   real*8 h1(n*n), h2(n*n)

   nc = 0
   do i=1,n
      h1(nc+1:nc+i) = a(1:i,i)
      nc = nc+i
   enddo

   call mpi_allreduce(h1,h2,nc,MPI_REAL8,MPI_SUM,comm,mpierr)

   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

end subroutine symm_matrix_allreduce

!-------------------------------------------------------------------------------

subroutine trans_ev_band_to_full_real(na, nqc, nblk, nbw, a, lda, tmat, q, ldq, mpi_comm_rows, mpi_comm_cols)

!-------------------------------------------------------------------------------
!  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,*)    Matrix containing the Householder vectors (i.e. matrix a after bandred_real)
!              Distribution is like in Scalapack.
!
!  lda         Leading dimension of a
!
!  tmat(nbw,nbw,.) 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
!
!-------------------------------------------------------------------------------

   implicit none

   integer na, nqc, lda, ldq, nblk, nbw, mpi_comm_rows, mpi_comm_cols
   real*8 a(lda,*), q(ldq,*), tmat(nbw, nbw, *)

   integer my_prow, my_pcol, np_rows, np_cols, mpierr
   integer max_blocks_row, max_blocks_col, max_local_rows, max_local_cols
   integer l_cols, l_rows, l_colh, n_cols
   integer istep, lc, ncol, nrow, nb, ns

   real*8, allocatable:: tmp1(:), tmp2(:), hvb(:), hvm(:,:)

   integer pcol, prow, i
   pcol(i) = MOD((i-1)/nblk,np_cols) !Processor col for global col number
   prow(i) = MOD((i-1)/nblk,np_rows) !Processor row for global row number


   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

   allocate(tmp1(max_local_cols*nbw))
   allocate(tmp2(max_local_cols*nbw))
   allocate(hvb(max_local_rows*nbw))
   allocate(hvm(max_local_rows,nbw))

   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

   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)) 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),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)) 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,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,1),tmp2,n_cols)
         call dgemm('N','N',l_rows,l_cols,n_cols,-1.d0,hvm,ubound(hvm,1), &
                    tmp2,n_cols,1.d0,q,ldq)
      endif

   enddo

   deallocate(tmp1, tmp2, hvb, hvm)


end subroutine trans_ev_band_to_full_real

! --------------------------------------------------------------------------------------------------

subroutine tridiag_band_real(na, nb, nblk, a, lda, d, e, 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,*)    Distributed system matrix reduced to banded form in the upper diagonal
!
!  lda         Leading dimension of 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
!-------------------------------------------------------------------------------

   implicit none

   integer, intent(in) ::  na, nb, nblk, lda, mpi_comm_rows, mpi_comm_cols, mpi_comm
   real*8, intent(in)  :: a(lda,*)
   real*8, intent(out) :: d(na), e(na) ! set only on PE 0


   real*8 vnorm2, hv(nb), tau, x, h(nb), ab_s(1+nb), hv_s(nb), hv_new(nb), tau_new, hf
   real*8 hd(nb), hs(nb)

   integer i, j, n, nc, nr, ns, ne, istep, iblk, nblocks_total, nblocks, nt
   integer my_pe, n_pes, mpierr
   integer my_prow, np_rows, my_pcol, np_cols
   integer ireq_ab, ireq_hv
   integer na_s, nx, num_hh_vecs, num_chunks, local_size, max_blk_size, n_off
#ifdef WITH_OPENMP
   integer max_threads, my_thread, my_block_s, my_block_e, iter
   integer mpi_status(MPI_STATUS_SIZE)
   integer, allocatable :: mpi_statuses(:,:), global_id_tmp(:,:)
   integer, allocatable :: omp_block_limits(:)
   real*8, allocatable :: hv_t(:,:), tau_t(:)
#endif
   integer, allocatable :: ireq_hhr(:), ireq_hhs(:), global_id(:,:), hh_cnt(:), hh_dst(:)
   integer, allocatable :: limits(:), snd_limits(:,:)
   integer, allocatable :: block_limits(:)
   real*8, allocatable :: ab(:,:), hh_gath(:,:,:), hh_send(:,:,:)
   ! dummies for calling redist_band
   complex*16 :: c_a(1,1), c_ab(1,1)

#ifdef WITH_OPENMP
   integer :: omp_get_max_threads
#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

#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

   ! 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(.true., a, c_a, lda, na, nblk, nb, mpi_comm_rows, mpi_comm_cols, mpi_comm, ab, c_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
         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)
         num_hh_vecs = num_hh_vecs + local_size
      endif
      nx = nx - nb
      if(n == block_limits(nt+1)) then
         nt = nt + 1
      endif
   enddo

   ireq_hhs(:) = MPI_REQUEST_NULL

   ! 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

   ireq_ab = MPI_REQUEST_NULL
   ireq_hv = MPI_REQUEST_NULL

   ! 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)
      call mpi_isend(ab_s,nb+1,mpi_real8,my_pe-1,1,mpi_comm,ireq_ab,mpierr)
   endif

#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
#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
            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

!$omp parallel do private(my_thread, my_block_s, my_block_e, iblk, ns, ne, hv, tau, &
!$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

            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
                  call mpi_wait(ireq_ab,mpi_status,mpierr)
                  ab_s(1:nb+1) = ab(1:nb+1,na_s-n_off)
                  call mpi_isend(ab_s,nb+1,mpi_real8,my_pe-1,1,mpi_comm,ireq_ab,mpierr)
               endif

               ! Request last column from next PE
               ne = na_s + nblocks*nb - (max_threads-1) - 1
               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

            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
                  call mpi_wait(ireq_hv,mpi_status,mpierr)
                  hv_s(1) = tau_t(max_threads)
                  hv_s(2:) = hv_t(2:,max_threads)
                  call mpi_isend(hv_s,nb,mpi_real8,my_pe+1,2,mpi_comm,ireq_hv,mpierr)
               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

               call mpi_wait(ireq_hhs(iblk), MPI_STATUS_IGNORE, mpierr)

               ! Copy vectors into send buffer
               hh_send(:,1:hh_cnt(iblk),iblk) = hh_gath(:,1:hh_cnt(iblk),iblk)
               ! Send to destination
               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)
            ! 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.
#endif
            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 ...
#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

            ! ... 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
#ifdef WITH_OPENMP
                  call mpi_wait(ireq_hv,MPI_STATUS_IGNORE,mpierr)
#else
                  call mpi_wait(ireq_ab,MPI_STATUS_IGNORE,mpierr)
#endif
                  hv_s(1) = tau_new
                  hv_s(2:) = hv_new(2:)
                  call mpi_isend(hv_s,nb,mpi_real8,my_pe+1,2,mpi_comm,ireq_hv,mpierr)
               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 ...

#ifdef WITH_OPENMP               
               call mpi_wait(ireq_ab,MPI_STATUS,mpierr)
#else
               call mpi_wait(ireq_ab,MPI_STATUS_IGNORE,mpierr)
#endif
               ab_s(1:nb+1) = ab(1:nb+1,ns)
               call mpi_isend(ab_s,nb+1,mpi_real8,my_pe-1,1,mpi_comm,ireq_ab,mpierr)
               
               ! ... 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
            endif
         endif
         
         ! Use new HH vector for the next block
         hv(:) = hv_new(:)
         tau = tau_new

      enddo

#ifdef WITH_OPENMP
   endif


   do iblk = 1, nblocks



      if(hh_dst(iblk) >= np_rows) exit
      if(snd_limits(hh_dst(iblk)+1,iblk) == snd_limits(hh_dst(iblk),iblk)) exit
      
      if(hh_cnt(iblk) == snd_limits(hh_dst(iblk)+1,iblk)-snd_limits(hh_dst(iblk),iblk)) then
         ! Wait for last transfer to finish
         call mpi_wait(ireq_hhs(iblk), mpi_status, mpierr)
         ! Copy vectors into send buffer
         hh_send(:,1:hh_cnt(iblk),iblk) = hh_gath(:,1:hh_cnt(iblk),iblk)
         ! Send to destination
         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)
         ! Reset counter and increase destination row
         hh_cnt(iblk) = 0
         hh_dst(iblk) = hh_dst(iblk)+1
      endif
      
   enddo
#endif
enddo



   ! Finish the last outstanding requests
#ifdef WITH_OPENMP
   call mpi_wait(ireq_ab,MPI_STATUS,mpierr)
   call mpi_wait(ireq_hv,MPI_STATUS,mpierr)

   allocate(mpi_statuses(MPI_STATUS_SIZE,max(nblocks,num_chunks)))
   call mpi_waitall(nblocks, ireq_hhs, MPI_STATUSES, mpierr)
   call mpi_waitall(num_chunks, ireq_hhr, MPI_STATUSES, mpierr)
   deallocate(mpi_statuses)
#else
   call mpi_wait(ireq_ab,MPI_STATUS_IGNORE,mpierr)
   call mpi_wait(ireq_hv,MPI_STATUS_IGNORE,mpierr)

   call mpi_waitall(nblocks, ireq_hhs, MPI_STATUSES_IGNORE, mpierr)
   call mpi_waitall(num_chunks, ireq_hhr, MPI_STATUSES_IGNORE, mpierr)
#endif

   call mpi_barrier(mpi_comm,mpierr)

   deallocate(ab)
   deallocate(ireq_hhr, ireq_hhs)
   deallocate(hh_cnt, hh_dst)
   deallocate(hh_gath, hh_send)
   deallocate(limits, snd_limits)
   deallocate(block_limits)
   deallocate(global_id)

 end subroutine tridiag_band_real

! --------------------------------------------------------------------------------------------------

subroutine trans_ev_tridi_to_band_real(na, nev, nblk, nbw, q, ldq, mpi_comm_rows, mpi_comm_cols)

!-------------------------------------------------------------------------------
!  trans_ev_tridi_to_band_real:
!  Transforms the eigenvectors of a tridiagonal matrix back to the eigenvectors of the band matrix
!
!  Parameters
!
!  na          Order of matrix a, number of rows of matrix q
!
!  nev         Number eigenvectors to compute (= columns of matrix q)
!
!  nblk        blocksize of cyclic distribution, must be the same in both directions!
!
!  nb          semi bandwith
!
!  q           On input: Eigenvectors of tridiagonal 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/both
!
!-------------------------------------------------------------------------------

    implicit none

    integer, intent(in) :: na, nev, nblk, nbw, ldq, mpi_comm_rows, mpi_comm_cols
    real*8 q(ldq,*)

    integer np_rows, my_prow, np_cols, my_pcol

    integer i, j, ip, sweep, nbuf, l_nev, a_dim2
    integer current_n, current_local_n, current_n_start, current_n_end
    integer next_n, next_local_n, next_n_start, next_n_end
    integer bottom_msg_length, top_msg_length, next_top_msg_length
    integer stripe_width, last_stripe_width, stripe_count
#ifdef WITH_OPENMP
    integer thread_width, csw, b_off, b_len
#endif
    integer num_result_blocks, num_result_buffers, num_bufs_recvd
    integer a_off, current_tv_off, max_blk_size
    integer mpierr, src, src_offset, dst, offset, nfact, num_blk
#ifdef WITH_OPENMP
    integer mpi_status(MPI_STATUS_SIZE)
#endif
    logical flag

#ifdef WITH_OPENMP
    real*8, allocatable :: a(:,:,:,:), row(:)
#else
    real*8, allocatable :: a(:,:,:), row(:)
#endif

#ifdef WITH_OPENMP
    real*8, allocatable :: top_border_send_buffer(:,:), top_border_recv_buffer(:,:)
    real*8, allocatable :: bottom_border_send_buffer(:,:), bottom_border_recv_buffer(:,:)
#else
    real*8, allocatable :: top_border_send_buffer(:,:,:), top_border_recv_buffer(:,:,:)
    real*8, allocatable :: bottom_border_send_buffer(:,:,:), bottom_border_recv_buffer(:,:,:)
#endif
    real*8, allocatable :: result_buffer(:,:,:)
    real*8, allocatable :: bcast_buffer(:,:)

    integer n_off
    integer, allocatable :: result_send_request(:), result_recv_request(:), limits(:)
    integer, allocatable :: top_send_request(:), bottom_send_request(:)
    integer, allocatable :: top_recv_request(:), bottom_recv_request(:)
#ifdef WITH_OPENMP
    integer, allocatable :: mpi_statuses(:,:)
#endif
    ! MPI send/recv tags, arbitrary

    integer, parameter :: bottom_recv_tag = 111
    integer, parameter :: top_recv_tag    = 222
    integer, parameter :: result_recv_tag = 333

    ! Just for measuring the kernel performance
    real*8 kernel_time
    integer*8 kernel_flops

#ifdef WITH_OPENMP
    integer max_threads, my_thread
    integer omp_get_max_threads
#endif

    kernel_time = 1.d-100
    kernel_flops = 0

#ifdef WITH_OPENMP
    max_threads = 1
    max_threads = omp_get_max_threads()
#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)

    if(mod(nbw,nblk)/=0) then
      if(my_prow==0 .and. my_pcol==0) then
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         write(error_unit,*) 'ERROR: nbw=',nbw,', nblk=',nblk
         write(error_unit,*) 'band backtransform works only for nbw==n*nblk'
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         call mpi_abort(mpi_comm_world,0,mpierr)
      endif
    endif

    nfact = nbw / nblk


    ! local number of eigenvectors
    l_nev = local_index(nev, my_pcol, np_cols, nblk, -1)

    if(l_nev==0) then
#ifdef WITH_OPENMP
        thread_width = 0
#endif
        stripe_width = 0
        stripe_count = 0
        last_stripe_width = 0
    else
        ! Suggested stripe width is 48 since 48*64 real*8 numbers should fit into
        ! every primary cache
#ifdef WITH_OPENMP
       thread_width = (l_nev-1)/max_threads + 1 ! number of eigenvectors per OMP thread
#endif
        stripe_width = 48 ! Must be a multiple of 4
#ifdef WITH_OPENMP
        stripe_count = (thread_width-1)/stripe_width + 1
#else
        stripe_count = (l_nev-1)/stripe_width + 1
#endif
        ! Adapt stripe width so that last one doesn't get too small
#ifdef WITH_OPENMP
        stripe_width = (thread_width-1)/stripe_count + 1
#else
        stripe_width = (l_nev-1)/stripe_count + 1
#endif
        stripe_width = ((stripe_width+3)/4)*4 ! Must be a multiple of 4 !!!
        last_stripe_width = l_nev - (stripe_count-1)*stripe_width
    endif

    ! Determine the matrix distribution at the beginning

    allocate(limits(0:np_rows))

    call determine_workload(na, nbw, np_rows, limits)

    max_blk_size = maxval(limits(1:np_rows) - limits(0:np_rows-1))

    a_dim2 = max_blk_size + nbw

!DEC$ ATTRIBUTES ALIGN: 64:: a
#ifdef WITH_OPENMP
    allocate(a(stripe_width,a_dim2,stripe_count,max_threads))
    ! a(:,:,:,:) should be set to 0 in a parallel region, not here!
#else
    allocate(a(stripe_width,a_dim2,stripe_count))
    a(:,:,:) = 0
#endif

    allocate(row(l_nev))
    row(:) = 0

    ! Copy q from a block cyclic distribution into a distribution with contiguous rows,
    ! and transpose the matrix using stripes of given stripe_width for cache blocking.

    ! The peculiar way it is done below is due to the fact that the last row should be
    ! ready first since it is the first one to start below

#ifdef WITH_OPENMP
    ! Please note about the OMP usage below:
    ! This is not for speed, but because we want the matrix a in the memory and
    ! in the cache of the correct thread (if possible)

!$omp parallel do private(my_thread), schedule(static, 1)
    do my_thread = 1, max_threads
        a(:,:,:,my_thread) = 0 ! if possible, do first touch allocation!
    enddo
#endif

    do ip = np_rows-1, 0, -1
        if(my_prow == ip) then
            ! Receive my rows which have not yet been received
            src_offset = local_index(limits(ip), my_prow, np_rows, nblk, -1)
            do i=limits(ip)+1,limits(ip+1)
                src = mod((i-1)/nblk, np_rows)
                if(src < my_prow) then
#ifdef WITH_OPENMP
                   call MPI_Recv(row, l_nev, MPI_REAL8, src, 0, mpi_comm_rows, MPI_STATUS, mpierr)
!$omp parallel do private(my_thread), schedule(static, 1)
                    do my_thread = 1, max_threads
                        call unpack_row(row,i-limits(ip),my_thread)
                    enddo
#else
                    call MPI_Recv(row, l_nev, MPI_REAL8, src, 0, mpi_comm_rows, MPI_STATUS_IGNORE, mpierr)
                    call unpack_row(row,i-limits(ip))
#endif
                elseif(src==my_prow) then
                    src_offset = src_offset+1
                    row(:) = q(src_offset, 1:l_nev)
#ifdef WITH_OPENMP
!$omp parallel do private(my_thread), schedule(static, 1)
                    do my_thread = 1, max_threads
                        call unpack_row(row,i-limits(ip),my_thread)
                    enddo
#else
                    call unpack_row(row,i-limits(ip))
#endif
                endif
            enddo
            ! Send all rows which have not yet been send
            src_offset = 0
            do dst = 0, ip-1
              do i=limits(dst)+1,limits(dst+1)
                if(mod((i-1)/nblk, np_rows) == my_prow) then
                    src_offset = src_offset+1
                    row(:) = q(src_offset, 1:l_nev)
                    call MPI_Send(row, l_nev, MPI_REAL8, dst, 0, mpi_comm_rows, mpierr)
                endif
              enddo
            enddo
        else if(my_prow < ip) then
            ! Send all rows going to PE ip
            src_offset = local_index(limits(ip), my_prow, np_rows, nblk, -1)
            do i=limits(ip)+1,limits(ip+1)
                src = mod((i-1)/nblk, np_rows)
                if(src == my_prow) then
                    src_offset = src_offset+1
                    row(:) = q(src_offset, 1:l_nev)
                    call MPI_Send(row, l_nev, MPI_REAL8, ip, 0, mpi_comm_rows, mpierr)
                endif
            enddo
            ! Receive all rows from PE ip
            do i=limits(my_prow)+1,limits(my_prow+1)
                src = mod((i-1)/nblk, np_rows)
                if(src == ip) then
#ifdef WITH_OPENMP
                    call MPI_Recv(row, l_nev, MPI_REAL8, src, 0, mpi_comm_rows, MPI_STATUS, mpierr)
!$omp parallel do private(my_thread), schedule(static, 1)
                    do my_thread = 1, max_threads
                        call unpack_row(row,i-limits(my_prow),my_thread)
                     enddo
#else
                    call MPI_Recv(row, l_nev, MPI_REAL8, src, 0, mpi_comm_rows, MPI_STATUS_IGNORE, mpierr)
                    call unpack_row(row,i-limits(my_prow))
#endif
                endif
            enddo
        endif
    enddo


    ! Set up result buffer queue

    num_result_blocks = ((na-1)/nblk + np_rows - my_prow) / np_rows

    num_result_buffers = 4*nfact
    allocate(result_buffer(l_nev,nblk,num_result_buffers))

    allocate(result_send_request(num_result_buffers))
    allocate(result_recv_request(num_result_buffers))
    result_send_request(:) = MPI_REQUEST_NULL
    result_recv_request(:) = MPI_REQUEST_NULL

    ! Queue up buffers

    if(my_prow > 0 .and. l_nev>0) then ! note: row 0 always sends
        do j = 1, min(num_result_buffers, num_result_blocks)
            call MPI_Irecv(result_buffer(1,1,j), l_nev*nblk, MPI_REAL8, 0, result_recv_tag, &
                           mpi_comm_rows, result_recv_request(j), mpierr)
        enddo
    endif

    num_bufs_recvd = 0 ! No buffers received yet

    ! Initialize top/bottom requests

    allocate(top_send_request(stripe_count))
    allocate(top_recv_request(stripe_count))
    allocate(bottom_send_request(stripe_count))
    allocate(bottom_recv_request(stripe_count))

    top_send_request(:) = MPI_REQUEST_NULL
    top_recv_request(:) = MPI_REQUEST_NULL
    bottom_send_request(:) = MPI_REQUEST_NULL
    bottom_recv_request(:) = MPI_REQUEST_NULL

#ifdef WITH_OPENMP
    allocate(top_border_send_buffer(stripe_width*nbw*max_threads, stripe_count))
    allocate(top_border_recv_buffer(stripe_width*nbw*max_threads, stripe_count))
    allocate(bottom_border_send_buffer(stripe_width*nbw*max_threads, stripe_count))
    allocate(bottom_border_recv_buffer(stripe_width*nbw*max_threads, stripe_count))

    top_border_send_buffer(:,:) = 0
    top_border_recv_buffer(:,:) = 0
    bottom_border_send_buffer(:,:) = 0
    bottom_border_recv_buffer(:,:) = 0

    ! Initialize broadcast buffer
#else
    allocate(top_border_send_buffer(stripe_width, nbw, stripe_count))
    allocate(top_border_recv_buffer(stripe_width, nbw, stripe_count))
    allocate(bottom_border_send_buffer(stripe_width, nbw, stripe_count))
    allocate(bottom_border_recv_buffer(stripe_width, nbw, stripe_count))

    top_border_send_buffer(:,:,:) = 0
    top_border_recv_buffer(:,:,:) = 0
    bottom_border_send_buffer(:,:,:) = 0
    bottom_border_recv_buffer(:,:,:) = 0
#endif

    allocate(bcast_buffer(nbw, max_blk_size))
    bcast_buffer = 0

    current_tv_off = 0 ! Offset of next row to be broadcast


    ! ------------------- start of work loop -------------------

    a_off = 0 ! offset in A (to avoid unnecessary shifts)

    top_msg_length = 0
    bottom_msg_length = 0

    do sweep = 0, (na-1)/nbw

        current_n = na - sweep*nbw
        call determine_workload(current_n, nbw, np_rows, limits)
        current_n_start = limits(my_prow)
        current_n_end   = limits(my_prow+1)
        current_local_n = current_n_end - current_n_start

        next_n = max(current_n - nbw, 0)
        call determine_workload(next_n, nbw, np_rows, limits)
        next_n_start = limits(my_prow)
        next_n_end   = limits(my_prow+1)
        next_local_n = next_n_end - next_n_start

        if(next_n_end < next_n) then
            bottom_msg_length = current_n_end - next_n_end
        else
            bottom_msg_length = 0
        endif

        if(next_local_n > 0) then
            next_top_msg_length = current_n_start - next_n_start
        else
            next_top_msg_length = 0
        endif

        if(sweep==0 .and. current_n_end < current_n .and. l_nev > 0) then
            do i = 1, stripe_count
#ifdef WITH_OPENMP
                csw = min(stripe_width, thread_width-(i-1)*stripe_width) ! "current_stripe_width"
                b_len = csw*nbw*max_threads
                call MPI_Irecv(bottom_border_recv_buffer(1,i), b_len, MPI_REAL8, my_prow+1, bottom_recv_tag, &
                           mpi_comm_rows, bottom_recv_request(i), mpierr)
#else
                call MPI_Irecv(bottom_border_recv_buffer(1,1,i), nbw*stripe_width, MPI_REAL8, my_prow+1, bottom_recv_tag, &
                     mpi_comm_rows, bottom_recv_request(i), mpierr)
#endif
            enddo
        endif

        if(current_local_n > 1) then
            if(my_pcol == mod(sweep,np_cols)) then
                bcast_buffer(:,1:current_local_n) = hh_trans_real(:,current_tv_off+1:current_tv_off+current_local_n)
                current_tv_off = current_tv_off + current_local_n
            endif
            call mpi_bcast(bcast_buffer, nbw*current_local_n, MPI_REAL8, mod(sweep,np_cols), mpi_comm_cols, mpierr)
        else
            ! for current_local_n == 1 the one and only HH vector is 0 and not stored in hh_trans_real
            bcast_buffer(:,1) = 0
        endif

        if(l_nev == 0) cycle

        if(current_local_n > 0) then

          do i = 1, stripe_count
#ifdef WITH_OPENMP
            ! Get real stripe width for strip i;
            ! The last OpenMP tasks may have an even smaller stripe with,
            ! but we don't care about this, i.e. we send/recv a bit too much in this case.
            ! csw: current_stripe_width

            csw = min(stripe_width, thread_width-(i-1)*stripe_width)
#endif
            !wait_b
            if(current_n_end < current_n) then
#ifdef WITH_OPENMP
                call MPI_Wait(bottom_recv_request(i), MPI_STATUS, mpierr)
!$omp parallel do private(my_thread, n_off, b_len, b_off), schedule(static, 1)
                do my_thread = 1, max_threads
                    n_off = current_local_n+a_off
                    b_len = csw*nbw
                    b_off = (my_thread-1)*b_len
                    a(1:csw,n_off+1:n_off+nbw,i,my_thread) = &
                      reshape(bottom_border_recv_buffer(b_off+1:b_off+b_len,i), (/ csw, nbw /))
                enddo
#else
                call MPI_Wait(bottom_recv_request(i), MPI_STATUS_IGNORE, mpierr)
                n_off = current_local_n+a_off
                a(:,n_off+1:n_off+nbw,i) = bottom_border_recv_buffer(:,1:nbw,i)

#endif
                if(next_n_end < next_n) then
#ifdef WITH_OPENMP
                    call MPI_Irecv(bottom_border_recv_buffer(1,i), csw*nbw*max_threads, &
                                   MPI_REAL8, my_prow+1, bottom_recv_tag, &
                                   mpi_comm_rows, bottom_recv_request(i), mpierr)
#else
                    call MPI_Irecv(bottom_border_recv_buffer(1,1,i), nbw*stripe_width, MPI_REAL8, my_prow+1, bottom_recv_tag, &

                                   mpi_comm_rows, bottom_recv_request(i), mpierr)
#endif
                endif
            endif

            if(current_local_n <= bottom_msg_length + top_msg_length) then

                !wait_t
                if(top_msg_length>0) then
#ifdef WITH_OPENMP
                    call MPI_Wait(top_recv_request(i), MPI_STATUS, mpierr)
#else
                    call MPI_Wait(top_recv_request(i), MPI_STATUS_IGNORE, mpierr)
                    a(:,a_off+1:a_off+top_msg_length,i) = top_border_recv_buffer(:,1:top_msg_length,i)
#endif
                endif

                !compute
#ifdef WITH_OPENMP
!$omp parallel do private(my_thread, n_off, b_len, b_off), schedule(static, 1)
                do my_thread = 1, max_threads
                    if(top_msg_length>0) then
                        b_len = csw*top_msg_length
                        b_off = (my_thread-1)*b_len
                        a(1:csw,a_off+1:a_off+top_msg_length,i,my_thread) = &
                          reshape(top_border_recv_buffer(b_off+1:b_off+b_len,i), (/ csw, top_msg_length /))
                    endif
                    call compute_hh_trafo(0, current_local_n, i, my_thread)
                enddo
#else
                call compute_hh_trafo(0, current_local_n, i)
#endif
                !send_b
#ifdef WITH_OPENMP
                call MPI_Wait(bottom_send_request(i), mpi_status, mpierr)
                if(bottom_msg_length>0) then
                    n_off = current_local_n+nbw-bottom_msg_length+a_off
                    b_len = csw*bottom_msg_length*max_threads
                    bottom_border_send_buffer(1:b_len,i) = &
                        reshape(a(1:csw,n_off+1:n_off+bottom_msg_length,i,:), (/ b_len /))
                    call MPI_Isend(bottom_border_send_buffer(1,i), b_len, MPI_REAL8, my_prow+1, &
                                   top_recv_tag, mpi_comm_rows, bottom_send_request(i), mpierr)
                endif
#else
                call MPI_Wait(bottom_send_request(i), MPI_STATUS_IGNORE, mpierr)
                if(bottom_msg_length>0) then
                    n_off = current_local_n+nbw-bottom_msg_length+a_off
                    bottom_border_send_buffer(:,1:bottom_msg_length,i) = a(:,n_off+1:n_off+bottom_msg_length,i)
                    call MPI_Isend(bottom_border_send_buffer(1,1,i), bottom_msg_length*stripe_width, MPI_REAL8, my_prow+1, &


                                   top_recv_tag, mpi_comm_rows, bottom_send_request(i), mpierr)
                endif
#endif
            else

                !compute
#ifdef WITH_OPENMP
!$omp parallel do private(my_thread, b_len, b_off), schedule(static, 1)
                do my_thread = 1, max_threads
                    call compute_hh_trafo(current_local_n - bottom_msg_length, bottom_msg_length, i, my_thread)
                enddo

                !send_b
                call MPI_Wait(bottom_send_request(i), mpi_status, mpierr)
                if(bottom_msg_length > 0) then
                    n_off = current_local_n+nbw-bottom_msg_length+a_off
                    b_len = csw*bottom_msg_length*max_threads
                    bottom_border_send_buffer(1:b_len,i) = &
                      reshape(a(1:csw,n_off+1:n_off+bottom_msg_length,i,:), (/ b_len /))
                    call MPI_Isend(bottom_border_send_buffer(1,i), b_len, MPI_REAL8, my_prow+1, &
                                   top_recv_tag, mpi_comm_rows, bottom_send_request(i), mpierr)
                endif
#else
                call compute_hh_trafo(current_local_n - bottom_msg_length, bottom_msg_length, i)




                !send_b
                call MPI_Wait(bottom_send_request(i), MPI_STATUS_IGNORE, mpierr)
                if(bottom_msg_length > 0) then
                    n_off = current_local_n+nbw-bottom_msg_length+a_off
                    bottom_border_send_buffer(:,1:bottom_msg_length,i) = a(:,n_off+1:n_off+bottom_msg_length,i)
                    call MPI_Isend(bottom_border_send_buffer(1,1,i), bottom_msg_length*stripe_width, MPI_REAL8, my_prow+1, &


                                   top_recv_tag, mpi_comm_rows, bottom_send_request(i), mpierr)
                endif
#endif

                !compute
#ifdef WITH_OPENMP
!$omp parallel do private(my_thread), schedule(static, 1)
                do my_thread = 1, max_threads
                    call compute_hh_trafo(top_msg_length, current_local_n-top_msg_length-bottom_msg_length, i, my_thread)
                enddo
#else
                call compute_hh_trafo(top_msg_length, current_local_n-top_msg_length-bottom_msg_length, i)

#endif
                !wait_t
                if(top_msg_length>0) then
#ifdef WITH_OPENMP
                    call MPI_Wait(top_recv_request(i), mpi_status, mpierr)
#else
                    call MPI_Wait(top_recv_request(i), MPI_STATUS_IGNORE, mpierr)
                    a(:,a_off+1:a_off+top_msg_length,i) = top_border_recv_buffer(:,1:top_msg_length,i)
#endif
                endif

                !compute
#ifdef WITH_OPENMP
!$omp parallel do private(my_thread, b_len, b_off), schedule(static, 1)
                do my_thread = 1, max_threads
                    if(top_msg_length>0) then
                        b_len = csw*top_msg_length
                        b_off = (my_thread-1)*b_len
                        a(1:csw,a_off+1:a_off+top_msg_length,i,my_thread) = &
                          reshape(top_border_recv_buffer(b_off+1:b_off+b_len,i), (/ csw, top_msg_length /))
                    endif
                    call compute_hh_trafo(0, top_msg_length, i, my_thread)
                enddo
#else
                call compute_hh_trafo(0, top_msg_length, i)
#endif
            endif

            if(next_top_msg_length > 0) then
                !request top_border data
#ifdef WITH_OPENMP
                b_len = csw*next_top_msg_length*max_threads
                call MPI_Irecv(top_border_recv_buffer(1,i), b_len, MPI_REAL8, my_prow-1, &
                               top_recv_tag, mpi_comm_rows, top_recv_request(i), mpierr)
#else
                call MPI_Irecv(top_border_recv_buffer(1,1,i), next_top_msg_length*stripe_width, MPI_REAL8, my_prow-1, &

                               top_recv_tag, mpi_comm_rows, top_recv_request(i), mpierr)
#endif
            endif

            !send_t
            if(my_prow > 0) then
#ifdef WITH_OPENMP
                call MPI_Wait(top_send_request(i), mpi_status, mpierr)
                b_len = csw*nbw*max_threads
                top_border_send_buffer(1:b_len,i) = reshape(a(1:csw,a_off+1:a_off+nbw,i,:), (/ b_len /))
                call MPI_Isend(top_border_send_buffer(1,i), b_len, MPI_REAL8, &
                               my_prow-1, bottom_recv_tag, &
                               mpi_comm_rows, top_send_request(i), mpierr)
#else
                call MPI_Wait(top_send_request(i), MPI_STATUS_IGNORE, mpierr)
                top_border_send_buffer(:,1:nbw,i) = a(:,a_off+1:a_off+nbw,i)
                call MPI_Isend(top_border_send_buffer(1,1,i), nbw*stripe_width, MPI_REAL8, my_prow-1, bottom_recv_tag, &
                               mpi_comm_rows, top_send_request(i), mpierr)

#endif
            endif

            ! Care that there are not too many outstanding top_recv_request's
            if(stripe_count > 1) then
                if(i>1) then
#ifdef WITH_OPENMP
                     call MPI_Wait(top_recv_request(i-1), MPI_STATUS, mpierr)
#else
                     call MPI_Wait(top_recv_request(i-1), MPI_STATUS_IGNORE, mpierr)
#endif
                 else
#ifdef WITH_OPENMP
                     call MPI_Wait(top_recv_request(stripe_count), MPI_STATUS, mpierr)
#else
                     call MPI_Wait(top_recv_request(stripe_count), MPI_STATUS_IGNORE, mpierr)
#endif
                 endif
             endif

           enddo

           top_msg_length = next_top_msg_length

         else
             ! wait for last top_send_request
           do i = 1, stripe_count
#ifdef WITH_OPENMP
             call MPI_Wait(top_send_request(i), MPI_STATUS, mpierr)
#else
             call MPI_Wait(top_send_request(i), MPI_STATUS_IGNORE, mpierr)
#endif
           enddo
         endif

         ! Care about the result

         if(my_prow == 0) then

             ! topmost process sends nbw rows to destination processes

             do j=0,nfact-1

                 num_blk = sweep*nfact+j ! global number of destination block, 0 based
                 if(num_blk*nblk >= na) exit

                 nbuf = mod(num_blk, num_result_buffers) + 1 ! buffer number to get this block

#ifdef WITH_OPENMP
                 call MPI_Wait(result_send_request(nbuf), MPI_STATUS, mpierr)
#else
                 call MPI_Wait(result_send_request(nbuf), MPI_STATUS_IGNORE, mpierr)
#endif
                 dst = mod(num_blk, np_rows)

                 if(dst == 0) then
                     do i = 1, min(na - num_blk*nblk, nblk)
                         call pack_row(row, j*nblk+i+a_off)
                         q((num_blk/np_rows)*nblk+i,1:l_nev) = row(:)
                     enddo
                 else
                     do i = 1, nblk
                         call pack_row(result_buffer(:,i,nbuf),j*nblk+i+a_off)
                     enddo
                     call MPI_Isend(result_buffer(1,1,nbuf), l_nev*nblk, MPI_REAL8, dst, &
                                    result_recv_tag, mpi_comm_rows, result_send_request(nbuf), mpierr)
                 endif
             enddo

         else

            ! receive and store final result

             do j = num_bufs_recvd, num_result_blocks-1

                 nbuf = mod(j, num_result_buffers) + 1 ! buffer number to get this block

                 ! If there is still work to do, just test for the next result request
                 ! and leave the loop if it is not ready, otherwise wait for all
                 ! outstanding requests

                 if(next_local_n > 0) then
#ifdef WITH_OPENMP
                     call MPI_Test(result_recv_request(nbuf), flag, MPI_STATUS, mpierr)
#else
                     call MPI_Test(result_recv_request(nbuf), flag, MPI_STATUS_IGNORE, mpierr)

#endif
                     if(.not.flag) exit
                 else
#ifdef WITH_OPENMP
                     call MPI_Wait(result_recv_request(nbuf), MPI_STATUS, mpierr)
#else
                     call MPI_Wait(result_recv_request(nbuf), MPI_STATUS_IGNORE, mpierr)
#endif
                 endif

                 ! Fill result buffer into q
                 num_blk = j*np_rows + my_prow ! global number of current block, 0 based
                 do i = 1, min(na - num_blk*nblk, nblk)
                     q(j*nblk+i, 1:l_nev) = result_buffer(1:l_nev, i, nbuf)
                 enddo

                 ! Queue result buffer again if there are outstanding blocks left
                 if(j+num_result_buffers < num_result_blocks) &
                     call MPI_Irecv(result_buffer(1,1,nbuf), l_nev*nblk, MPI_REAL8, 0, result_recv_tag, &
                                    mpi_comm_rows, result_recv_request(nbuf), mpierr)

             enddo
             num_bufs_recvd = j

         endif

         ! Shift the remaining rows to the front of A (if necessary)

         offset = nbw - top_msg_length
         if(offset<0) then
2189
             write(error_unit,*) 'internal error, offset for shifting = ',offset
2190
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2210
2211
2212
2213
             call MPI_Abort(MPI_COMM_WORLD, 1, mpierr)
         endif
         a_off = a_off + offset
         if(a_off + next_local_n + nbw > a_dim2) then
#ifdef WITH_OPENMP
 !$omp parallel do private(my_thread, i, j), schedule(static, 1)
             do my_thread = 1, max_threads
                 do i = 1, stripe_count
                     do j = top_msg_length+1, top_msg_length+next_local_n
                        A(:,j,i,my_thread) = A(:,j+a_off,i,my_thread)
                     enddo
#else
             do i = 1, stripe_count
                 do j = top_msg_length+1, top_msg_length+next_local_n
                    A(:,j,i) = A(:,j+a_off,i)
#endif
                 enddo
             enddo
             a_off = 0
         endif

     enddo

     ! Just for safety:
2214
2215
2216
2217
     if(ANY(top_send_request    /= MPI_REQUEST_NULL)) write(error_unit,*) '*** ERROR top_send_request ***',my_prow,my_pcol
     if(ANY(bottom_send_request /= MPI_REQUEST_NULL)) write(error_unit,*) '*** ERROR bottom_send_request ***',my_prow,my_pcol
     if(ANY(top_recv_request    /= MPI_REQUEST_NULL)) write(error_unit,*) '*** ERROR top_recv_request ***',my_prow,my_pcol
     if(ANY(bottom_recv_request /= MPI_REQUEST_NULL)) write(error_unit,*) '*** ERROR bottom_recv_request ***',my_prow,my_pcol
2218
2219
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2221
2222
2223
2224
2225
2226
2227
2228

     if(my_prow == 0) then
#ifdef WITH_OPENMP
         allocate(mpi_statuses(MPI_STATUS_SIZE,num_result_buffers))
         call MPI_Waitall(num_result_buffers, result_send_request, mpi_statuses, mpierr)
         deallocate(mpi_statuses)
#else
         call MPI_Waitall(num_result_buffers, result_send_request, MPI_STATUSES_IGNORE, mpierr)
#endif
     endif

2229
2230
     if(ANY(result_send_request /= MPI_REQUEST_NULL)) write(error_unit,*) '*** ERROR result_send_request ***',my_prow,my_pcol
     if(ANY(result_recv_request /= MPI_REQUEST_NULL)) write(error_unit,*) '*** ERROR result_recv_request ***',my_prow,my_pcol
2231
2232

     if(my_prow==0 .and. my_pcol==0 .and. elpa_print_times) &
2233
         write(error_unit,'(" Kernel time:",f10.3," MFlops: ",f10.3)')  kernel_time, kernel_flops/kernel_time*1.d-6
2234
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