elpa2.f90 116 KB
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! 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".

module ELPA2

! Version 1.1.2, 2011-02-21

  USE ELPA1

  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

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

  ! 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) &
      print 1,'Time bandred_real               :',ttt1-ttt0

   ! 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) &
      print 1,'Time tridiag_band_real          :',ttt1-ttt0

   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) &
      print 1,'Time solve_tridi                :',ttt1-ttt0
   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) &
      print 1,'Time trans_ev_tridi_to_band_real:',ttt1-ttt0

   ! 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) &
      print 1,'Time trans_ev_band_to_full_real :',ttt1-ttt0
   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) &
      print 1,'Time bandred_complex               :',ttt1-ttt0

   ! 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) &
      print 1,'Time tridiag_band_complex          :',ttt1-ttt0

   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)  &
      print 1,'Time solve_tridi                   :',ttt1-ttt0
   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) &
      print 1,'Time trans_ev_tridi_to_band_complex:',ttt1-ttt0

   ! 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) &
      print 1,'Time trans_ev_band_to_full_complex :',ttt1-ttt0
   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
         print *,'ERROR: nbw=',nbw,', nblk=',nblk
         print *,'ELPA2 works only for nbw==n*nblk'
         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
   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)


   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

   call mpi_allreduce(mpi_in_place, global_id, np_rows*np_cols, mpi_integer, mpi_sum, mpi_comm, mpierr)


   ! 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

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

   do istep=1,na-1

      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
            call mpi_recv(hv,nb,mpi_real8,my_pe-1,2,mpi_comm,MPI_STATUS_IGNORE,mpierr)
            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

      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)

         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.

         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 ...
            call mpi_recv(ab(1,ne),nb+1,mpi_real8,my_pe+1,1,mpi_comm,MPI_STATUS_IGNORE,mpierr)

            ! ... 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
               call mpi_wait(ireq_hv,MPI_STATUS_IGNORE,mpierr)
               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 ...

            call mpi_wait(ireq_ab,MPI_STATUS_IGNORE,mpierr)
            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 the is no BLAS routine like DGER2 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

   enddo

   ! Finish the last outstanding requests
   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)

   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

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

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
    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
    logical flag

    real*8, allocatable :: a(:,:,:), row(:)
    real*8, allocatable :: top_border_send_buffer(:,:,:), top_border_recv_buffer(:,:,:)
    real*8, allocatable :: bottom_border_send_buffer(:,:,:), bottom_border_recv_buffer(:,:,:)
    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(:)

    ! 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


    kernel_time = 1.d-100
    kernel_flops = 0


    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
         print *,'ERROR: nbw=',nbw,', nblk=',nblk
         print *,'band backtransform works only for nbw==n*nblk'
         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
        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
        stripe_width = 48 ! Must be a multiple of 4
        stripe_count = (l_nev-1)/stripe_width + 1
        ! Adapt stripe width so that last one doesn't get too small
        stripe_width = (l_nev-1)/stripe_count + 1
        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

    allocate(a(stripe_width,a_dim2,stripe_count))
    a(:,:,:) = 0

    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

    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
                    call MPI_Recv(row, l_nev, MPI_REAL8, src, 0, mpi_comm_rows, MPI_STATUS_IGNORE, mpierr)
                    call unpack_row(row,i-limits(ip))
                elseif(src==my_prow) then
                    src_offset = src_offset+1
                    row(:) = q(src_offset, 1:l_nev)
                    call unpack_row(row,i-limits(ip))
                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
                    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
            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

    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

    ! Initialize broadcast buffer

    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
                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)
            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

            !wait_b
            if(current_n_end < current_n) then
                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)
                if(next_n_end < next_n) then
                    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

            if(current_local_n <= bottom_msg_length + top_msg_length) then

                !wait_t
                if(top_msg_length>0) then
                    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

                !compute
                call compute_hh_trafo(0, current_local_n, 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

            else

                !compute
                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

                !compute
                call compute_hh_trafo(top_msg_length, current_local_n-top_msg_length-bottom_msg_length, i)

                !wait_t
                if(top_msg_length>0) then
                    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

                !compute
                call compute_hh_trafo(0, top_msg_length, i)
            endif

            if(next_top_msg_length > 0) then
                !request top_border data
                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

            !send_t
            if(my_prow > 0) then
                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

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

          enddo

          top_msg_length = next_top_msg_length

        else
            ! wait for last top_send_request
          do i = 1, stripe_count
            call MPI_Wait(top_send_request(i), MPI_STATUS_IGNORE, mpierr)
          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

                call MPI_Wait(result_send_request(nbuf), MPI_STATUS_IGNORE, mpierr)

                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
                    call MPI_Test(result_recv_request(nbuf), flag, MPI_STATUS_IGNORE, mpierr)
                    if(.not.flag) exit
                else
                    call MPI_Wait(result_recv_request(nbuf), MPI_STATUS_IGNORE, mpierr)
                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
            print *,'internal error, offset for shifting = ',offset
            call MPI_Abort(MPI_COMM_WORLD, 1, mpierr)
        endif
        a_off = a_off + offset
        if(a_off + next_local_n + nbw > a_dim2) then
            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)
                enddo
            enddo
            a_off = 0
        endif

    enddo

    ! Just for safety:
    if(ANY(top_send_request    /= MPI_REQUEST_NULL)) print *,'*** ERROR top_send_request ***',my_prow,my_pcol
    if(ANY(bottom_send_request /= MPI_REQUEST_NULL)) print *,'*** ERROR bottom_send_request ***',my_prow,my_pcol
    if(ANY(top_recv_request    /= MPI_REQUEST_NULL)) print *,'*** ERROR top_recv_request ***',my_prow,my_pcol
    if(ANY(bottom_recv_request /= MPI_REQUEST_NULL)) print *,'*** ERROR bottom_recv_request ***',my_prow,my_pcol

    if(my_prow == 0) then
        call MPI_Waitall(num_result_buffers, result_send_request, MPI_STATUSES_IGNORE, mpierr)
    endif

    if(ANY(result_send_request /= MPI_REQUEST_NULL)) print *,'*** ERROR result_send_request ***',my_prow,my_pcol
    if(ANY(result_recv_request /= MPI_REQUEST_NULL)) print *,'*** ERROR result_recv_request ***',my_prow,my_pcol

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

    ! deallocate all working space

    deallocate(a)
    deallocate(row)
    deallocate(limits)
    deallocate(result_send_request)
    deallocate(result_recv_request)
    deallocate(top_border_send_buffer)
    deallocate(top_border_recv_buffer)
    deallocate(bottom_border_send_buffer)
    deallocate(bottom_border_recv_buffer)
    deallocate(result_buffer)
    deallocate(bcast_buffer)
    deallocate(top_send_request)
    deallocate(top_recv_request)
    deallocate(bottom_send_request)
    deallocate(bottom_recv_request)

contains

    subroutine pack_row(row, n)
        real*8 row(:)
        integer