elpa2.F90 170 KB
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!    This file is part of ELPA.
!
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!    The ELPA library was originally created by the ELPA consortium,
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!    consisting of the following organizations:
!
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!    - Rechenzentrum Garching der Max-Planck-Gesellschaft (RZG),
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!    - Bergische Universität Wuppertal, Lehrstuhl für angewandte
!      Informatik,
!    - Technische Universität München, Lehrstuhl für Informatik mit
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!      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
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!    - 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
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!    it under the terms of the version 3 of the license of the
!    GNU Lesser General Public License as published by the Free
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!    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
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!
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! 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
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!
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! 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

  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
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#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
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   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)

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#ifdef WITH_OPENMP
   integer :: omp_get_max_threads
#endif
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   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
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#ifdef WITH_OPENMP
   allocate(global_id_tmp(0:np_rows-1,0:np_cols-1))
#endif
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#ifndef WITH_OPENMP
911
   call mpi_allreduce(mpi_in_place, global_id, np_rows*np_cols, mpi_integer, mpi_sum, mpi_comm, mpierr)
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#else
    global_id_tmp(:,:) = global_id(:,:)
    call mpi_allreduce(global_id_tmp, global_id, np_rows*np_cols, mpi_integer, mpi_sum, mpi_comm, mpierr)
    deallocate(global_id_tmp)
#endif
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   ! 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

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#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

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

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#ifdef WITH_OPENMP
   do istep=1,na-1-block_limits(my_pe)*nb
#else
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   do istep=1,na-1
1055
#endif
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      if(my_pe==0) then
         n = MIN(na-na_s,nb) ! number of rows to be reduced
         hv(:) = 0
         tau = 0
         ! The last step (istep=na-1) is only needed for sending the last HH vectors.
         ! We don't want the sign of the last element flipped (analogous to the other sweeps)
         if(istep < na-1) then
            ! Transform first column of remaining matrix
            vnorm2 = sum(ab(3:n+1,na_s-n_off)**2)
            call hh_transform_real(ab(2,na_s-n_off),vnorm2,hf,tau)
            hv(1) = 1
            hv(2:n) = ab(3:n+1,na_s-n_off)*hf
         endif
         d(istep) = ab(1,na_s-n_off)
         e(istep) = ab(2,na_s-n_off)
         if(istep == na-1) then
            d(na) = ab(1,na_s+1-n_off)
            e(na) = 0
         endif
      else
         if(na>na_s) then
            ! Receive Householder vector from previous task, from PE owning subdiagonal
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#ifdef WITH_OPENMP
            call mpi_recv(hv,nb,mpi_real8,my_pe-1,2,mpi_comm,MPI_STATUS,mpierr)
#else
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            call mpi_recv(hv,nb,mpi_real8,my_pe-1,2,mpi_comm,MPI_STATUS_IGNORE,mpierr)
1083
#endif
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            tau = hv(1)
            hv(1) = 1.
         endif
      endif

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


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#ifdef WITH_OPENMP
      if(max_threads > 1) then
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1100
        ! Codepath for OpenMP
1101

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        ! 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.
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         hv_t(:,1) = hv
         tau_t(1) = tau
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1112
         do iter = 1, 2
1113

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          ! 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
1256 1257 1258 1259 1260 1261 1262

         ! 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.

1263
#endif /* WITH_OPENMP */
1264

1265
         do iblk=1,nblocks
1266

1267 1268
            ns = na_s + (iblk-1)*nb - n_off ! first column in block
            ne = ns+nb-1                    ! last column in block
1269

1270
            if(ns+n_off>na) exit
1271

1272
            ! Store Householder vector for back transformation
1273

1274
            hh_cnt(iblk) = hh_cnt(iblk) + 1
1275

1276 1277
            hh_gath(1   ,hh_cnt(iblk),iblk) = tau
            hh_gath(2:nb,hh_cnt(iblk),iblk) = hv(2:nb)
1278

1279 1280 1281
#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
1282

1283
               call mpi_wait(ireq_hhs(iblk), MPI_STATUS_IGNORE, mpierr)
1284

1285 1286 1287 1288 1289 1290 1291 1292 1293 1294
               ! 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
1295

1296 1297 1298 1299 1300 1301 1302 1303 1304
            ! 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)!
1305

1306
            ! Multiply diagonal block and subdiagonal block with Householder vector
1307

1308
            if(iblk==nblocks .and. nc==nb) then
1309

1310 1311
            ! We need the last column from the next PE.
            ! First do the matrix multiplications without last column ...
1312

1313 1314 1315
            ! 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)
1316

1317 1318
            ! 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)
1319

1320 1321 1322 1323 1324
            ! ... 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)
1325

1326
#endif
1327

1328 1329 1330 1331 1332 1333 1334 1335 1336 1337
            ! ... 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)
               
1338 1339
            endif

1340 1341
            ! Calculate first column of subdiagonal block and calculate new
            ! Householder transformation for this column
1342

1343 1344
            hv_new(:) = 0 ! Needed, last rows must be 0 for nr < nb
            tau_new = 0
1345

1346
            if(nr>0) then
1347

1348 1349 1350
               ! 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
1351

1352 1353 1354 1355 1356 1357 1358 1359 1360 1361 1362 1363 1364 1365 1366 1367 1368 1369 1370 1371 1372 1373 1374
            ! 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
1375

1376 1377 1378
            ! 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)
1379

1380
            if(my_pe>0 .and. iblk==1) then
1381

1382 1383
               ! The first column of the diagonal block has to be send to the previous PE
               ! Calculate first column only ...
1384

1385
               ab(1:nc,ns) = ab(1:nc,ns) - hd(1:nc)*hv(1) - hv(1:nc)*hd(1)
1386

1387
               ! ... send it away ...
1388

1389 1390 1391 1392 1393 1394 1395 1396 1397 1398
#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)
1399
            else
1400 1401 1402 1403 1404 1405 1406 1407 1408 1409 1410 1411 1412 1413 1414 1415 1416 1417
               ! 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
1418 1419 1420 1421
                  ab(2+nb-i,i+ns-1) = ab(2+nb-i,i+ns-1) - hs(1)*hv(i)
               enddo
            endif
         endif
1422
         
1423 1424 1425 1426 1427 1428
         ! Use new HH vector for the next block
         hv(:) = hv_new(:)
         tau = tau_new

      enddo

1429 1430 1431 1432 1433 1434 1435 1436 1437 1438 1439 1440 1441 1442 1443 1444 1445 1446 1447 1448 1449 1450 1451 1452 1453
#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
      
1454
   enddo
1455 1456 1457 1458
#endif
enddo


1459 1460

   ! Finish the last outstanding requests
1461 1462 1463 1464 1465 1466 1467 1468 1469
#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
1470 1471 1472 1473 1474
   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)
1475
#endif
1476 1477 1478 1479 1480 1481 1482 1483 1484 1485 1486 1487 1488 1489 1490 1491 1492 1493 1494 1495 1496 1497 1498 1499 1500 1501 1502 1503 1504 1505 1506 1507 1508 1509 1510 1511 1512 1513 1514 1515 1516 1517 1518 1519 1520 1521 1522 1523 1524 1525 1526 1527 1528 1529 1530

   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
1531 1532 1533
#ifdef WITH_OPENMP
    integer thread_width, csw, b_off, b_len
#endif
1534 1535 1536
    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
1537 1538 1539
#ifdef WITH_OPENMP
    integer mpi_status(MPI_STATUS_SIZE)
#endif
1540 1541
    logical flag

1542 1543 1544
#ifdef WITH_OPENMP
    real*8, allocatable :: a(:,:,:,:), row(:)
#else
1545
    real*8, allocatable :: a(:,:,:), row(:)
1546 1547 1548 1549 1550 1551
#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
1552 1553
    real*8, allocatable :: top_border_send_buffer(:,:,:), top_border_recv_buffer(:,:,:)
    real*8, allocatable :: bottom_border_send_buffer(:,:,:), bottom_border_recv_buffer(:,:,:)
1554
#endif
1555 1556 1557 1558 1559 1560 1561
    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(:)
1562 1563 1564
#ifdef WITH_OPENMP
    integer, allocatable :: mpi_statuses(:,:)
#endif
1565 1566 1567 1568 1569 1570 1571 1572 1573 1574
    ! 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

1575 1576 1577 1578
#ifdef WITH_OPENMP
    integer max_threads, my_thread
    integer omp_get_max_threads
#endif
1579 1580 1581 1582

    kernel_time = 1.d-100
    kernel_flops = 0

1583 1584 1585 1586
#ifdef WITH_OPENMP
    max_threads = 1
    max_threads = omp_get_max_threads()
#endif
1587 1588 1589 1590 1591 1592 1593 1594 1595 1596 1597 1598 1599 1600 1601 1602 1603 1604 1605 1606 1607

    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
1608 1609 1610
#ifdef WITH_OPENMP
        thread_width = 0
#endif
1611 1612 1613 1614 1615 1616
        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
1617 1618 1619
#ifdef WITH_OPENMP
       thread_width = (l_nev-1)/max_threads + 1 ! number of eigenvectors per OMP thread
#endif
1620
        stripe_width = 48 ! Must be a multiple of 4
1621 1622 1623
#ifdef WITH_OPENMP
        stripe_count = (thread_width-1)/stripe_width + 1
#else
1624
        stripe_count = (l_nev-1)/stripe_width + 1
1625
#endif
1626
        ! Adapt stripe width so that last one doesn't get too small
1627 1628 1629
#ifdef WITH_OPENMP
        stripe_width = (thread_width-1)/stripe_count + 1
#else
1630
        stripe_width = (l_nev-1)/stripe_count + 1
1631
#endif