elpa1_compute_private.F90 107 KB
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!    This file is part of ELPA.
!
!    The ELPA library was originally created by the ELPA consortium,
!    consisting of the following organizations:
!
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!    - Max Planck Computing and Data Facility (MPCDF), formerly known as
!      Rechenzentrum Garching der Max-Planck-Gesellschaft (RZG),
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!    - Bergische Universität Wuppertal, Lehrstuhl für angewandte
!      Informatik,
!    - Technische Universität München, Lehrstuhl für Informatik mit
!      Schwerpunkt Wissenschaftliches Rechnen ,
!    - Fritz-Haber-Institut, Berlin, Abt. Theorie,
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!    - Max-Plack-Institut für Mathematik in den Naturwissenschaften,
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!      Leipzig, Abt. Komplexe Strukutren in Biologie und Kognition,
!      and
!    - IBM Deutschland GmbH
!
!    This particular source code file contains additions, changes and
!    enhancements authored by Intel Corporation which is not part of
!    the ELPA consortium.
!
!    More information can be found here:
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!    http://elpa.mpcdf.mpg.de/
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!
!    ELPA is free software: you can redistribute it and/or modify
!    it under the terms of the version 3 of the license of the
!    GNU Lesser General Public License as published by the Free
!    Software Foundation.
!
!    ELPA is distributed in the hope that it will be useful,
!    but WITHOUT ANY WARRANTY; without even the implied warranty of
!    MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the
!    GNU Lesser General Public License for more details.
!
!    You should have received a copy of the GNU Lesser General Public License
!    along with ELPA.  If not, see <http://www.gnu.org/licenses/>
!
!    ELPA reflects a substantial effort on the part of the original
!    ELPA consortium, and we ask you to respect the spirit of the
!    license that we chose: i.e., please contribute any changes you
!    may have back to the original ELPA library distribution, and keep
!    any derivatives of ELPA under the same license that we chose for
!    the original distribution, the GNU Lesser General Public License.
!
!
! ELPA1 -- Faster replacements for ScaLAPACK symmetric eigenvalue routines
!
! Copyright of the original code rests with the authors inside the ELPA
! consortium. The copyright of any additional modifications shall rest
! with their original authors, but shall adhere to the licensing terms
! distributed along with the original code in the file "COPYING".

#include "config-f90.h"

module ELPA1_compute
  use elpa_utilities
#ifdef HAVE_DETAILED_TIMINGS
  use timings
#endif
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  use elpa_mpi
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  implicit none

  PRIVATE ! set default to private

  public :: tridiag_real               ! Transform real symmetric matrix to tridiagonal form
  public :: trans_ev_real              ! Transform eigenvectors of a tridiagonal matrix back

  public :: tridiag_complex            ! Transform complex hermitian matrix to tridiagonal form
  public :: trans_ev_complex           ! Transform eigenvectors of a tridiagonal matrix back

  public :: solve_tridi                ! Solve tridiagonal eigensystem with divide and conquer method

  public :: local_index                ! Get local index of a block cyclic distributed matrix
  public :: least_common_multiple      ! Get least common multiple

  public :: hh_transform_real
  public :: hh_transform_complex

  public :: elpa_reduce_add_vectors_complex, elpa_reduce_add_vectors_real
  public :: elpa_transpose_vectors_complex, elpa_transpose_vectors_real

  contains

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#define DATATYPE REAL(kind=rk)
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#define BYTESIZE 8
#define REALCASE 1
#include "elpa_transpose_vectors.X90"
#include "elpa_reduce_add_vectors.X90"
#undef DATATYPE
#undef BYTESIZE
#undef REALCASE

    subroutine tridiag_real(na, a, lda, nblk, matrixCols, mpi_comm_rows, mpi_comm_cols, d, e, tau)

    !-------------------------------------------------------------------------------
    !  tridiag_real: Reduces a distributed symmetric matrix to tridiagonal form
    !                (like Scalapack Routine PDSYTRD)
    !
    !  Parameters
    !
    !  na          Order of matrix
    !
    !  a(lda,matrixCols)    Distributed matrix which should be reduced.
    !              Distribution is like in Scalapack.
    !              Opposed to PDSYTRD, a(:,:) must be set completely (upper and lower half)
    !              a(:,:) is overwritten on exit with the Householder vectors
    !
    !  lda         Leading dimension of a
    !  matrixCols  local columns of matrix
    !
    !  nblk        blocksize of cyclic distribution, must be the same in both directions!
    !
    !  mpi_comm_rows
    !  mpi_comm_cols
    !              MPI-Communicators for rows/columns
    !
    !  d(na)       Diagonal elements (returned), identical on all processors
    !
    !  e(na)       Off-Diagonal elements (returned), identical on all processors
    !
    !  tau(na)     Factors for the Householder vectors (returned), needed for back transformation
    !
    !-------------------------------------------------------------------------------
#ifdef HAVE_DETAILED_TIMINGS
      use timings
#endif
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      use precision
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      implicit none

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      integer(kind=ik)            :: na, lda, nblk, matrixCols, mpi_comm_rows, mpi_comm_cols
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      real(kind=rk)               :: d(na), e(na), tau(na)
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#ifdef USE_ASSUMED_SIZE
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      real(kind=rk)               :: a(lda,*)
#else
      real(kind=rk)               :: a(lda,matrixCols)
#endif
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      integer(kind=ik), parameter :: max_stored_rows = 32
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      integer(kind=ik)            :: my_prow, my_pcol, np_rows, np_cols, mpierr
      integer(kind=ik)            :: totalblocks, max_blocks_row, max_blocks_col, max_local_rows, max_local_cols
      integer(kind=ik)            :: l_cols, l_rows, nstor
      integer(kind=ik)            :: istep, i, j, lcs, lce, lrs, lre
      integer(kind=ik)            :: tile_size, l_rows_tile, l_cols_tile
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#ifdef WITH_OPENMP
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      integer(kind=ik)            :: my_thread, n_threads, max_threads, n_iter
      integer(kind=ik)            :: omp_get_thread_num, omp_get_num_threads, omp_get_max_threads
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#endif

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      real(kind=rk)               :: vav, vnorm2, x, aux(2*max_stored_rows), aux1(2), aux2(2), vrl, xf
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      real(kind=rk), allocatable  :: tmp(:), vr(:), vc(:), ur(:), uc(:), vur(:,:), uvc(:,:)
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#ifdef WITH_OPENMP
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      real(kind=rk), allocatable  :: ur_p(:,:), uc_p(:,:)
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#endif
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      integer(kind=ik)            :: istat
      character(200)              :: errorMessage
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#ifdef HAVE_DETAILED_TIMINGS
      call timer%start("tridiag_real")
#endif

      call mpi_comm_rank(mpi_comm_rows,my_prow,mpierr)
      call mpi_comm_size(mpi_comm_rows,np_rows,mpierr)
      call mpi_comm_rank(mpi_comm_cols,my_pcol,mpierr)
      call mpi_comm_size(mpi_comm_cols,np_cols,mpierr)
      ! 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


      totalblocks = (na-1)/nblk + 1
      max_blocks_row = (totalblocks-1)/np_rows + 1
      max_blocks_col = (totalblocks-1)/np_cols + 1

      max_local_rows = max_blocks_row*nblk
      max_local_cols = max_blocks_col*nblk

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      allocate(tmp(MAX(max_local_rows,max_local_cols)), stat=istat, errmsg=errorMessage)
      if (istat .ne. 0) then
        print *,"tridiag_real: error when allocating tmp "//errorMessage
        stop
      endif

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

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

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

      allocate(uc(max_local_cols), stat=istat, errmsg=errorMessage)
      if (istat .ne. 0) then
        print *,"tridiag_real: error when allocating uc "//errorMessage
        stop
      endif
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#ifdef WITH_OPENMP
      max_threads = omp_get_max_threads()

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      allocate(ur_p(max_local_rows,0:max_threads-1), stat=istat, errmsg=errorMessage)
      if (istat .ne. 0) then
        print *,"tridiag_real: error when allocating ur_p "//errorMessage
        stop
      endif

      allocate(uc_p(max_local_cols,0:max_threads-1), stat=istat, errmsg=errorMessage)
      if (istat .ne. 0) then
        print *,"tridiag_real: error when allocating uc_p "//errorMessage
        stop
      endif

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#endif

      tmp = 0
      vr = 0
      ur = 0
      vc = 0
      uc = 0

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      allocate(vur(max_local_rows,2*max_stored_rows), stat=istat, errmsg=errorMessage)
      if (istat .ne. 0) then
        print *,"tridiag_real: error when allocating vur "//errorMessage
        stop
      endif

      allocate(uvc(max_local_cols,2*max_stored_rows), stat=istat, errmsg=errorMessage)
      if (istat .ne. 0) then
        print *,"tridiag_real: error when allocating uvc "//errorMessage
        stop
      endif
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      d(:) = 0
      e(:) = 0
      tau(:) = 0

      nstor = 0

      l_rows = local_index(na, my_prow, np_rows, nblk, -1) ! Local rows of a
      l_cols = local_index(na, my_pcol, np_cols, nblk, -1) ! Local cols of a
      if(my_prow==prow(na, nblk, np_rows) .and. my_pcol==pcol(na, nblk, np_cols)) d(na) = a(l_rows,l_cols)

      do istep=na,3,-1

         ! Calculate number of local rows and columns of the still remaining matrix
         ! on the local processor

         l_rows = local_index(istep-1, my_prow, np_rows, nblk, -1)
         l_cols = local_index(istep-1, my_pcol, np_cols, nblk, -1)

         ! Calculate vector for Householder transformation on all procs
         ! owning column istep

         if(my_pcol==pcol(istep, nblk, np_cols)) then

            ! Get vector to be transformed; distribute last element and norm of
            ! remaining elements to all procs in current column

            vr(1:l_rows) = a(1:l_rows,l_cols+1)
            if(nstor>0 .and. l_rows>0) then
               call DGEMV('N',l_rows,2*nstor,1.d0,vur,ubound(vur,dim=1), &
                          uvc(l_cols+1,1),ubound(uvc,dim=1),1.d0,vr,1)
            endif

            if(my_prow==prow(istep-1, nblk, np_rows)) then
               aux1(1) = dot_product(vr(1:l_rows-1),vr(1:l_rows-1))
               aux1(2) = vr(l_rows)
            else
               aux1(1) = dot_product(vr(1:l_rows),vr(1:l_rows))
               aux1(2) = 0.
            endif

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#ifdef WITH_MPI
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            call mpi_allreduce(aux1,aux2,2,MPI_REAL8,MPI_SUM,mpi_comm_rows,mpierr)
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#else
            aux2 = aux1
#endif
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            vnorm2 = aux2(1)
            vrl    = aux2(2)

            ! Householder transformation

            call hh_transform_real(vrl, vnorm2, xf, tau(istep))

            ! Scale vr and store Householder vector for back transformation

            vr(1:l_rows) = vr(1:l_rows) * xf
            if(my_prow==prow(istep-1, nblk, np_rows)) then
               vr(l_rows) = 1.
               e(istep-1) = vrl
            endif
            a(1:l_rows,l_cols+1) = vr(1:l_rows) ! store Householder vector for back transformation

         endif

         ! Broadcast the Householder vector (and tau) along columns

         if(my_pcol==pcol(istep, nblk, np_cols)) vr(l_rows+1) = tau(istep)
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#ifdef WITH_MPI
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         call MPI_Bcast(vr,l_rows+1,MPI_REAL8,pcol(istep, nblk, np_cols),mpi_comm_cols,mpierr)
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#endif
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         tau(istep) =  vr(l_rows+1)

         ! Transpose Householder vector vr -> vc

         call elpa_transpose_vectors_real  (vr, ubound(vr,dim=1), mpi_comm_rows, &
                                            vc, ubound(vc,dim=1), mpi_comm_cols, &
                                            1, istep-1, 1, nblk)


         ! Calculate u = (A + VU**T + UV**T)*v

         ! For cache efficiency, we use only the upper half of the matrix tiles for this,
         ! thus the result is partly in uc(:) and partly in ur(:)

         uc(1:l_cols) = 0
         ur(1:l_rows) = 0
         if (l_rows>0 .and. l_cols>0) then

#ifdef WITH_OPENMP

#ifdef HAVE_DETAILED_TIMINGS
           call timer%start("OpenMP parallel")
#endif

!$OMP PARALLEL PRIVATE(my_thread,n_threads,n_iter,i,lcs,lce,j,lrs,lre)

           my_thread = omp_get_thread_num()
           n_threads = omp_get_num_threads()

           n_iter = 0

           uc_p(1:l_cols,my_thread) = 0.
           ur_p(1:l_rows,my_thread) = 0.
#endif
           do i=0,(istep-2)/tile_size
             lcs = i*l_cols_tile+1
             lce = min(l_cols,(i+1)*l_cols_tile)
             if (lce<lcs) cycle
             do j=0,i
               lrs = j*l_rows_tile+1
               lre = min(l_rows,(j+1)*l_rows_tile)
               if (lre<lrs) cycle
#ifdef WITH_OPENMP
               if (mod(n_iter,n_threads) == my_thread) then
                 call DGEMV('T',lre-lrs+1,lce-lcs+1,1.d0,a(lrs,lcs),lda,vr(lrs),1,1.d0,uc_p(lcs,my_thread),1)
                 if (i/=j) call DGEMV('N',lre-lrs+1,lce-lcs+1,1.d0,a(lrs,lcs),lda,vc(lcs),1,1.d0,ur_p(lrs,my_thread),1)
               endif
               n_iter = n_iter+1
#else
               call DGEMV('T',lre-lrs+1,lce-lcs+1,1.d0,a(lrs,lcs),lda,vr(lrs),1,1.d0,uc(lcs),1)
               if (i/=j) call DGEMV('N',lre-lrs+1,lce-lcs+1,1.d0,a(lrs,lcs),lda,vc(lcs),1,1.d0,ur(lrs),1)

#endif
             enddo
           enddo
#ifdef WITH_OPENMP
!$OMP END PARALLEL
#ifdef HAVE_DETAILED_TIMINGS
           call timer%stop("OpenMP parallel")
#endif

           do i=0,max_threads-1
             uc(1:l_cols) = uc(1:l_cols) + uc_p(1:l_cols,i)
             ur(1:l_rows) = ur(1:l_rows) + ur_p(1:l_rows,i)
           enddo
#endif
           if (nstor>0) then
             call DGEMV('T',l_rows,2*nstor,1.d0,vur,ubound(vur,dim=1),vr,1,0.d0,aux,1)
             call DGEMV('N',l_cols,2*nstor,1.d0,uvc,ubound(uvc,dim=1),aux,1,1.d0,uc,1)
           endif

         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-1) then
          call elpa_reduce_add_vectors_REAL  (ur, ubound(ur,dim=1), mpi_comm_rows, &
                                        uc, ubound(uc,dim=1), mpi_comm_cols, &
                                        istep-1, 1, nblk)
        endif

        ! Sum up all the uc(:) parts, transpose uc -> ur

        if (l_cols>0) then
          tmp(1:l_cols) = uc(1:l_cols)
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#ifdef WITH_MPI
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          call mpi_allreduce(tmp,uc,l_cols,MPI_REAL8,MPI_SUM,mpi_comm_rows,mpierr)
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#else
          uc = tmp
#endif
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        endif

        call elpa_transpose_vectors_real  (uc, ubound(uc,dim=1), mpi_comm_cols, &
                                         ur, ubound(ur,dim=1), mpi_comm_rows, &
                                         1, istep-1, 1, nblk)

        ! calculate u**T * v (same as v**T * (A + VU**T + UV**T) * v )

        x = 0
        if (l_cols>0) x = dot_product(vc(1:l_cols),uc(1:l_cols))
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#ifdef WITH_MPI
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        call mpi_allreduce(x,vav,1,MPI_REAL8,MPI_SUM,mpi_comm_cols,mpierr)
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#else
        vav = x
#endif
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        ! store u and v in the matrices U and V
        ! these matrices are stored combined in one here

        do j=1,l_rows
          vur(j,2*nstor+1) = tau(istep)*vr(j)
          vur(j,2*nstor+2) = 0.5*tau(istep)*vav*vr(j) - ur(j)
        enddo
        do j=1,l_cols
          uvc(j,2*nstor+1) = 0.5*tau(istep)*vav*vc(j) - uc(j)
          uvc(j,2*nstor+2) = tau(istep)*vc(j)
        enddo

        nstor = nstor+1

        ! If the limit of max_stored_rows is reached, calculate A + VU**T + UV**T

        if (nstor==max_stored_rows .or. istep==3) then

          do i=0,(istep-2)/tile_size
            lcs = i*l_cols_tile+1
            lce = min(l_cols,(i+1)*l_cols_tile)
           lrs = 1
            lre = min(l_rows,(i+1)*l_rows_tile)
            if (lce<lcs .or. lre<lrs) cycle
            call dgemm('N','T',lre-lrs+1,lce-lcs+1,2*nstor,1.d0, &
                       vur(lrs,1),ubound(vur,dim=1),uvc(lcs,1),ubound(uvc,dim=1), &
                       1.d0,a(lrs,lcs),lda)
          enddo

          nstor = 0

        endif

        if (my_prow==prow(istep-1, nblk, np_rows) .and. my_pcol==pcol(istep-1, nblk, np_cols)) then
          if (nstor>0) a(l_rows,l_cols) = a(l_rows,l_cols) &
                        + dot_product(vur(l_rows,1:2*nstor),uvc(l_cols,1:2*nstor))
          d(istep-1) = a(l_rows,l_cols)
        endif

      enddo

      ! Store e(1) and d(1)

      if (my_prow==prow(1, nblk, np_rows) .and. my_pcol==pcol(2, nblk, np_cols)) e(1) = a(1,l_cols) ! use last l_cols value of loop above
      if (my_prow==prow(1, nblk, np_rows) .and. my_pcol==pcol(1, nblk, np_cols)) d(1) = a(1,1)

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      deallocate(tmp, vr, ur, vc, uc, vur, uvc, stat=istat, errmsg=errorMessage)
      if (istat .ne. 0) then
        print *,"tridiag_real: error when deallocating uvc "//errorMessage
        stop
      endif

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      ! distribute the arrays d and e to all processors

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      allocate(tmp(na),  stat=istat, errmsg=errorMessage)
      if (istat .ne. 0) then
        print *,"tridiag_real: error when allocating tmp "//errorMessage
        stop
      endif
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#ifdef WITH_MPI
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      tmp = d
      call mpi_allreduce(tmp,d,na,MPI_REAL8,MPI_SUM,mpi_comm_rows,mpierr)
      tmp = d
      call mpi_allreduce(tmp,d,na,MPI_REAL8,MPI_SUM,mpi_comm_cols,mpierr)
      tmp = e
      call mpi_allreduce(tmp,e,na,MPI_REAL8,MPI_SUM,mpi_comm_rows,mpierr)
      tmp = e
      call mpi_allreduce(tmp,e,na,MPI_REAL8,MPI_SUM,mpi_comm_cols,mpierr)
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#endif
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      deallocate(tmp,  stat=istat, errmsg=errorMessage)
      if (istat .ne. 0) then
        print *,"tridiag_real: error when deallocating tmp "//errorMessage
        stop
      endif

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

    end subroutine tridiag_real

    subroutine trans_ev_real(na, nqc, a, lda, tau, q, ldq, nblk, matrixCols, mpi_comm_rows, mpi_comm_cols)

    !-------------------------------------------------------------------------------
    !  trans_ev_real: Transforms the eigenvectors of a tridiagonal matrix back
    !                 to the eigenvectors of the original matrix
    !                 (like Scalapack Routine PDORMTR)
    !
    !  Parameters
    !
    !  na          Order of matrix a, number of rows of matrix q
    !
    !  nqc         Number of columns of matrix q
    !
    !  a(lda,matrixCols)    Matrix containing the Householder vectors (i.e. matrix a after tridiag_real)
    !              Distribution is like in Scalapack.
    !
    !  lda         Leading dimension of a
    !  matrixCols  local columns of matrix a and q
    !
    !  tau(na)     Factors of the Householder vectors
    !
    !  q           On input: Eigenvectors of tridiagonal matrix
    !              On output: Transformed eigenvectors
    !              Distribution is like in Scalapack.
    !
    !  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
    !
    !-------------------------------------------------------------------------------
#ifdef HAVE_DETAILED_TIMINGS
      use timings
#endif
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      use precision
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      implicit none

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      integer(kind=ik)           :: na, nqc, lda, ldq, nblk, matrixCols, mpi_comm_rows, mpi_comm_cols
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      real(kind=rk)              :: tau(na)
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#ifdef USE_ASSUMED_SIZE
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      real(kind=rk)              :: a(lda,*), q(ldq,*)
#else
      real(kind=rk)              :: a(lda,matrixCols), q(ldq,matrixCols)
#endif
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      integer(kind=ik)           :: max_stored_rows
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      integer(kind=ik)           :: my_prow, my_pcol, np_rows, np_cols, mpierr
      integer(kind=ik)           :: totalblocks, max_blocks_row, max_blocks_col, max_local_rows, max_local_cols
      integer(kind=ik)           :: l_cols, l_rows, l_colh, nstor
      integer(kind=ik)           :: istep, i, n, nc, ic, ics, ice, nb, cur_pcol
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      real(kind=rk), allocatable :: tmp1(:), tmp2(:), hvb(:), hvm(:,:)
      real(kind=rk), allocatable :: tmat(:,:), h1(:), h2(:)
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      integer(kind=ik)           :: istat
      character(200)             :: errorMessage
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#ifdef HAVE_DETAILED_TIMINGS
      call timer%start("trans_ev_real")
#endif

      call mpi_comm_rank(mpi_comm_rows,my_prow,mpierr)
      call mpi_comm_size(mpi_comm_rows,np_rows,mpierr)
      call mpi_comm_rank(mpi_comm_cols,my_pcol,mpierr)
      call mpi_comm_size(mpi_comm_cols,np_cols,mpierr)

      totalblocks = (na-1)/nblk + 1
      max_blocks_row = (totalblocks-1)/np_rows + 1
      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

      max_stored_rows = (63/nblk+1)*nblk

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

      allocate(h1(max_stored_rows*max_stored_rows), stat=istat, errmsg=errorMessage)
      if (istat .ne. 0) then
        print *,"trans_ev_real: error when allocating h1 "//errorMessage
        stop
      endif

      allocate(h2(max_stored_rows*max_stored_rows), stat=istat, errmsg=errorMessage)
      if (istat .ne. 0) then
        print *,"trans_ev_real: error when allocating h2 "//errorMessage
        stop
      endif

      allocate(tmp1(max_local_cols*max_stored_rows), stat=istat, errmsg=errorMessage)
      if (istat .ne. 0) then
        print *,"trans_ev_real: error when allocating tmp1 "//errorMessage
        stop
      endif

      allocate(tmp2(max_local_cols*max_stored_rows), stat=istat, errmsg=errorMessage)
      if (istat .ne. 0) then
        print *,"trans_ev_real: error when allocating tmp2 "//errorMessage
        stop
      endif

      allocate(hvb(max_local_rows*nblk), stat=istat, errmsg=errorMessage)
      if (istat .ne. 0) then
        print *,"trans_ev_real: error when allocating hvn "//errorMessage
        stop
      endif

      allocate(hvm(max_local_rows,max_stored_rows), stat=istat, errmsg=errorMessage)
      if (istat .ne. 0) then
        print *,"trans_ev_real: error when allocating hvm "//errorMessage
        stop
      endif
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      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

      nstor = 0

      do istep=1,na,nblk

        ics = MAX(istep,3)
        ice = MIN(istep+nblk-1,na)
        if (ice<ics) cycle

        cur_pcol = pcol(istep, nblk, np_cols)

        nb = 0
        do ic=ics,ice

          l_colh = local_index(ic  , my_pcol, np_cols, nblk, -1) ! Column of Householder vector
          l_rows = local_index(ic-1, my_prow, np_rows, nblk, -1) ! # rows of Householder vector


          if (my_pcol==cur_pcol) then
            hvb(nb+1:nb+l_rows) = a(1:l_rows,l_colh)
            if (my_prow==prow(ic-1, nblk, np_rows)) then
              hvb(nb+l_rows) = 1.
            endif
          endif

          nb = nb+l_rows
        enddo

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#ifdef WITH_MPI
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        if (nb>0) &
            call MPI_Bcast(hvb,nb,MPI_REAL8,cur_pcol,mpi_comm_cols,mpierr)
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#endif
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        nb = 0
        do ic=ics,ice
          l_rows = local_index(ic-1, my_prow, np_rows, nblk, -1) ! # rows of Householder vector
          hvm(1:l_rows,nstor+1) = hvb(nb+1:nb+l_rows)
          nstor = nstor+1
          nb = nb+l_rows
        enddo

        ! Please note: for smaller matix sizes (na/np_rows<=256), a value of 32 for nstor is enough!
        if (nstor+nblk>max_stored_rows .or. istep+nblk>na .or. (na/np_rows<=256 .and. nstor>=32)) then

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

          tmat = 0
          if (l_rows>0) &
               call dsyrk('U','T',nstor,l_rows,1.d0,hvm,ubound(hvm,dim=1),0.d0,tmat,max_stored_rows)

          nc = 0
          do n=1,nstor-1
            h1(nc+1:nc+n) = tmat(1:n,n+1)
            nc = nc+n
          enddo
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#ifdef WITH_MPI
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          if (nc>0) call mpi_allreduce(h1,h2,nc,MPI_REAL8,MPI_SUM,mpi_comm_rows,mpierr)
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#else
          if (nc>0) h2 = h1
#endif
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          ! Calculate triangular matrix T

          nc = 0
          tmat(1,1) = tau(ice-nstor+1)
          do n=1,nstor-1
            call dtrmv('L','T','N',n,tmat,max_stored_rows,h2(nc+1),1)
            tmat(n+1,1:n) = -h2(nc+1:nc+n)*tau(ice-nstor+n+1)
            tmat(n+1,n+1) = tau(ice-nstor+n+1)
            nc = nc+n
          enddo

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

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

      enddo

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      deallocate(tmat, h1, h2, tmp1, tmp2, hvb, hvm, stat=istat, errmsg=errorMessage)
      if (istat .ne. 0) then
        print *,"trans_ev_real: error when deallocating hvm "//errorMessage
        stop
      endif
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#ifdef HAVE_DETAILED_TIMINGS
      call timer%stop("trans_ev_real")
#endif

    end subroutine trans_ev_real

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#define DATATYPE COMPLEX(kind=ck)
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#define BYTESIZE 16
#define COMPLEXCASE 1
#include "elpa_transpose_vectors.X90"
#include "elpa_reduce_add_vectors.X90"
#undef DATATYPE
#undef BYTESIZE
#undef COMPLEXCASE

    subroutine tridiag_complex(na, a, lda, nblk, matrixCols, mpi_comm_rows, mpi_comm_cols, d, e, tau)

    !-------------------------------------------------------------------------------
    !  tridiag_complex: Reduces a distributed hermitian matrix to tridiagonal form
    !                   (like Scalapack Routine PZHETRD)
    !
    !  Parameters
    !
    !  na          Order of matrix
    !
    !  a(lda,matrixCols)    Distributed matrix which should be reduced.
    !              Distribution is like in Scalapack.
    !              Opposed to PZHETRD, a(:,:) must be set completely (upper and lower half)
    !              a(:,:) is overwritten on exit with the Householder vectors
    !
    !  lda         Leading dimension of a
    !  matrixCols  local columns of matrix a
    !
    !  nblk        blocksize of cyclic distribution, must be the same in both directions!
    !
    !  mpi_comm_rows
    !  mpi_comm_cols
    !              MPI-Communicators for rows/columns
    !
    !  d(na)       Diagonal elements (returned), identical on all processors
    !
    !  e(na)       Off-Diagonal elements (returned), identical on all processors
    !
    !  tau(na)     Factors for the Householder vectors (returned), needed for back transformation
    !
    !-------------------------------------------------------------------------------
#ifdef HAVE_DETAILED_TIMINGS
      use timings
#endif
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      use precision
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      implicit none

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      integer(kind=ik)              :: na, lda, nblk, matrixCols, mpi_comm_rows, mpi_comm_cols
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      complex(kind=ck)              :: tau(na)
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#ifdef USE_ASSUMED_SIZE
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      complex(kind=ck)              :: a(lda,*)
#else
      complex(kind=ck)              :: a(lda,matrixCols)
#endif
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      real(kind=rk)                 :: d(na), e(na)
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      integer(kind=ik), parameter   :: max_stored_rows = 32
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      complex(kind=ck), parameter   :: CZERO = (0.d0,0.d0), CONE = (1.d0,0.d0)
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      integer(kind=ik)              :: my_prow, my_pcol, np_rows, np_cols, mpierr
      integer(kind=ik)              :: totalblocks, max_blocks_row, max_blocks_col, max_local_rows, max_local_cols
      integer(kind=ik)              :: l_cols, l_rows, nstor
      integer(kind=ik)              :: istep, i, j, lcs, lce, lrs, lre
      integer(kind=ik)              :: tile_size, l_rows_tile, l_cols_tile
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#ifdef WITH_OPENMP
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      integer(kind=ik)              :: my_thread, n_threads, max_threads, n_iter
      integer(kind=ik)              :: omp_get_thread_num, omp_get_num_threads, omp_get_max_threads
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#endif

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      real(kind=rk)                 :: vnorm2
      complex(kind=ck)              :: vav, xc, aux(2*max_stored_rows),  aux1(2), aux2(2), vrl, xf
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      complex(kind=ck), allocatable :: tmp(:), vr(:), vc(:), ur(:), uc(:), vur(:,:), uvc(:,:)
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#ifdef WITH_OPENMP
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      complex(kind=ck), allocatable :: ur_p(:,:), uc_p(:,:)
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#endif
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      real(kind=rk), allocatable    :: tmpr(:)
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      integer(kind=ik)              :: istat
      character(200)                :: errorMessage
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#ifdef HAVE_DETAILED_TIMINGS
      call timer%start("tridiag_complex")
#endif

      call mpi_comm_rank(mpi_comm_rows,my_prow,mpierr)
      call mpi_comm_size(mpi_comm_rows,np_rows,mpierr)
      call mpi_comm_rank(mpi_comm_cols,my_pcol,mpierr)
      call mpi_comm_size(mpi_comm_cols,np_cols,mpierr)

      ! 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


      totalblocks = (na-1)/nblk + 1
      max_blocks_row = (totalblocks-1)/np_rows + 1
      max_blocks_col = (totalblocks-1)/np_cols + 1

      max_local_rows = max_blocks_row*nblk
      max_local_cols = max_blocks_col*nblk

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      allocate(tmp(MAX(max_local_rows,max_local_cols)), stat=istat, errmsg=errorMessage)
      if (istat .ne. 0) then
       print *,"tridiag_complex: error when allocating tmp "//errorMessage
       stop
      endif

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

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

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

      allocate(uc(max_local_cols), stat=istat, errmsg=errorMessage)
      if (istat .ne. 0) then
       print *,"tridiag_complex: error when allocating uc "//errorMessage
       stop
      endif
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#ifdef WITH_OPENMP
      max_threads = omp_get_max_threads()

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      allocate(ur_p(max_local_rows,0:max_threads-1), stat=istat, errmsg=errorMessage)
      if (istat .ne. 0) then
       print *,"tridiag_complex: error when allocating ur_p "//errorMessage
       stop
      endif

      allocate(uc_p(max_local_cols,0:max_threads-1), stat=istat, errmsg=errorMessage)
      if (istat .ne. 0) then
       print *,"tridiag_complex: error when allocating uc_p "//errorMessage
       stop
      endif
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#endif

      tmp = 0
      vr = 0
      ur = 0
      vc = 0
      uc = 0

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      allocate(vur(max_local_rows,2*max_stored_rows), stat=istat, errmsg=errorMessage)
      if (istat .ne. 0) then
       print *,"tridiag_complex: error when allocating vur "//errorMessage
       stop
      endif

      allocate(uvc(max_local_cols,2*max_stored_rows), stat=istat, errmsg=errorMessage)
      if (istat .ne. 0) then
       print *,"tridiag_complex: error when allocating uvc "//errorMessage
       stop
      endif
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      d(:) = 0
      e(:) = 0
      tau(:) = 0

      nstor = 0

      l_rows = local_index(na, my_prow, np_rows, nblk, -1) ! Local rows of a
      l_cols = local_index(na, my_pcol, np_cols, nblk, -1) ! Local cols of a
      if (my_prow==prow(na, nblk, np_rows) .and. my_pcol==pcol(na, nblk, np_cols)) d(na) = a(l_rows,l_cols)

      do istep=na,3,-1

        ! Calculate number of local rows and columns of the still remaining matrix
        ! on the local processor

        l_rows = local_index(istep-1, my_prow, np_rows, nblk, -1)
        l_cols = local_index(istep-1, my_pcol, np_cols, nblk, -1)

        ! Calculate vector for Householder transformation on all procs
        ! owning column istep

        if (my_pcol==pcol(istep, nblk, np_cols)) then

          ! Get vector to be transformed; distribute last element and norm of
          ! remaining elements to all procs in current column

          vr(1:l_rows) = a(1:l_rows,l_cols+1)
          if (nstor>0 .and. l_rows>0) then
            aux(1:2*nstor) = conjg(uvc(l_cols+1,1:2*nstor))
            call ZGEMV('N',l_rows,2*nstor,CONE,vur,ubound(vur,dim=1), &
                        aux,1,CONE,vr,1)
          endif

          if (my_prow==prow(istep-1, nblk, np_rows)) then
            aux1(1) = dot_product(vr(1:l_rows-1),vr(1:l_rows-1))
            aux1(2) = vr(l_rows)
          else
            aux1(1) = dot_product(vr(1:l_rows),vr(1:l_rows))
            aux1(2) = 0.
          endif
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          call mpi_allreduce(aux1,aux2,2,MPI_DOUBLE_COMPLEX,MPI_SUM,mpi_comm_rows,mpierr)
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#else
          aux2 = aux1
#endif
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          vnorm2 = aux2(1)
          vrl    = aux2(2)

          ! Householder transformation

          call hh_transform_complex(vrl, vnorm2, xf, tau(istep))

          ! Scale vr and store Householder vector for back transformation

          vr(1:l_rows) = vr(1:l_rows) * xf
          if (my_prow==prow(istep-1, nblk, np_rows)) then
            vr(l_rows) = 1.
            e(istep-1) = vrl
          endif
          a(1:l_rows,l_cols+1) = vr(1:l_rows) ! store Householder vector for back transformation

        endif

        ! Broadcast the Householder vector (and tau) along columns

        if (my_pcol==pcol(istep, nblk, np_cols)) vr(l_rows+1) = tau(istep)
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        call MPI_Bcast(vr,l_rows+1,MPI_DOUBLE_COMPLEX,pcol(istep, nblk, np_cols),mpi_comm_cols,mpierr)
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#endif
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        tau(istep) =  vr(l_rows+1)

        ! Transpose Householder vector vr -> vc

!        call elpa_transpose_vectors  (vr, 2*ubound(vr,dim=1), mpi_comm_rows, &
!                                      vc, 2*ubound(vc,dim=1), mpi_comm_cols, &
!                                      1, 2*(istep-1), 1, 2*nblk)

        call elpa_transpose_vectors_complex  (vr, ubound(vr,dim=1), mpi_comm_rows, &
                                              vc, ubound(vc,dim=1), mpi_comm_cols, &
                                              1, (istep-1), 1, nblk)
        ! Calculate u = (A + VU**T + UV**T)*v

        ! For cache efficiency, we use only the upper half of the matrix tiles for this,
        ! thus the result is partly in uc(:) and partly in ur(:)

        uc(1:l_cols) = 0
        ur(1:l_rows) = 0
        if (l_rows>0 .and. l_cols>0) then

#ifdef WITH_OPENMP

#ifdef HAVE_DETAILED_TIMINGS
          call timer%start("OpenMP parallel")
#endif

!$OMP PARALLEL PRIVATE(my_thread,n_threads,n_iter,i,lcs,lce,j,lrs,lre)

          my_thread = omp_get_thread_num()
          n_threads = omp_get_num_threads()

          n_iter = 0

          uc_p(1:l_cols,my_thread) = 0.
          ur_p(1:l_rows,my_thread) = 0.
#endif

          do i=0,(istep-2)/tile_size
            lcs = i*l_cols_tile+1
            lce = min(l_cols,(i+1)*l_cols_tile)
            if (lce<lcs) cycle
            do j=0,i
              lrs = j*l_rows_tile+1
              lre = min(l_rows,(j+1)*l_rows_tile)
              if (lre<lrs) cycle
#ifdef WITH_OPENMP
              if (mod(n_iter,n_threads) == my_thread) then
                call ZGEMV('C',lre-lrs+1,lce-lcs+1,CONE,a(lrs,lcs),lda,vr(lrs),1,CONE,uc_p(lcs,my_thread),1)
                if (i/=j) call ZGEMV('N',lre-lrs+1,lce-lcs+1,CONE,a(lrs,lcs),lda,vc(lcs),1,CONE,ur_p(lrs,my_thread),1)
              endif
              n_iter = n_iter+1
#else
              call ZGEMV('C',lre-lrs+1,lce-lcs+1,CONE,a(lrs,lcs),lda,vr(lrs),1,CONE,uc(lcs),1)
              if (i/=j) call ZGEMV('N',lre-lrs+1,lce-lcs+1,CONE,a(lrs,lcs),lda,vc(lcs),1,CONE,ur(lrs),1)
#endif
            enddo
          enddo

#ifdef WITH_OPENMP
!$OMP END PARALLEL
#ifdef HAVE_DETAILED_TIMINGS
          call timer%stop("OpenMP parallel")
#endif

          do i=0,max_threads-1
            uc(1:l_cols) = uc(1:l_cols) + uc_p(1:l_cols,i)
            ur(1:l_rows) = ur(1:l_rows) + ur_p(1:l_rows,i)
          enddo
#endif

          if (nstor>0) then
            call ZGEMV('C',l_rows,2*nstor,CONE,vur,ubound(vur,dim=1),vr,1,CZERO,aux,1)
            call ZGEMV('N',l_cols,2*nstor,CONE,uvc,ubound(uvc,dim=1),aux,1,CONE,uc,1)
          endif

        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-1) then
          call elpa_reduce_add_vectors_COMPLEX  (ur, ubound(ur,dim=1), mpi_comm_rows, &
                                          uc, ubound(uc,dim=1), mpi_comm_cols, &
                                          (istep-1), 1, nblk)
        endif

        ! Sum up all the uc(:) parts, transpose uc -> ur

        if (l_cols>0) then
          tmp(1:l_cols) = uc(1:l_cols)
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          call mpi_allreduce(tmp,uc,l_cols,MPI_DOUBLE_COMPLEX,MPI_SUM,mpi_comm_rows,mpierr)
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#else
          uc = tmp
#endif
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        endif

!        call elpa_transpose_vectors  (uc, 2*ubound(uc,dim=1), mpi_comm_cols, &
!                                      ur, 2*ubound(ur,dim=1), mpi_comm_rows, &
!                                      1, 2*(istep-1), 1, 2*nblk)

        call elpa_transpose_vectors_complex  (uc, ubound(uc,dim=1), mpi_comm_cols, &
                                              ur, ubound(ur,dim=1), mpi_comm_rows, &
                                              1, (istep-1), 1, nblk)



        ! calculate u**T * v (same as v**T * (A + VU**T + UV**T) * v )

        xc = 0
        if (l_cols>0) xc = dot_product(vc(1:l_cols),uc(1:l_cols))
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        call mpi_allreduce(xc,vav,1,MPI_DOUBLE_COMPLEX,MPI_SUM,mpi_comm_cols,mpierr)
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#else
        vav = xc
#endif
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        ! store u and v in the matrices U and V
        ! these matrices are stored combined in one here

        do j=1,l_rows
          vur(j,2*nstor+1) = conjg(tau(istep))*vr(j)
          vur(j,2*nstor+2) = 0.5*conjg(tau(istep))*vav*vr(j) - ur(j)
        enddo
        do j=1,l_cols
          uvc(j,2*nstor+1) = 0.5*conjg(tau(istep))*vav*vc(j) - uc(j)
          uvc(j,2*nstor+2) = conjg(tau(istep))*vc(j)
        enddo

        nstor = nstor+1

        ! If the limit of max_stored_rows is reached, calculate A + VU**T + UV**T

        if (nstor==max_stored_rows .or. istep==3) then

          do i=0,(istep-2)/tile_size
            lcs = i*l_cols_tile+1
            lce = min(l_cols,(i+1)*l_cols_tile)
            lrs = 1
            lre = min(l_rows,(i+1)*l_rows_tile)
            if (lce<lcs .or. lre<lrs) cycle
            call ZGEMM('N','C',lre-lrs+1,lce-lcs+1,2*nstor,CONE, &
                         vur(lrs,1),ubound(vur,dim=1),uvc(lcs,1),ubound(uvc,dim=1), &
                         CONE,a(lrs,lcs),lda)
          enddo

          nstor = 0

        endif

        if (my_prow==prow(istep-1, nblk, np_rows) .and. my_pcol==pcol(istep-1, nblk, np_cols)) then
          if (nstor>0) a(l_rows,l_cols) = a(l_rows,l_cols) &
                          + dot_product(vur(l_rows,1:2*nstor),uvc(l_cols,1:2*nstor))
          d(istep-1) = a(l_rows,l_cols)
        endif

      enddo ! istep

      ! Store e(1) and d(1)

      if (my_pcol==pcol(2, nblk, np_cols)) then
        if (my_prow==prow(1, nblk, np_rows)) then
          ! We use last l_cols value of loop above
          vrl = a(1,l_cols)
          call hh_transform_complex(vrl, 0.d0, xf, tau(2))
          e(1) = vrl
          a(1,l_cols) = 1. ! for consistency only
        endif
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#ifdef WITH_MPI
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        call mpi_bcast(tau(2),1,MPI_DOUBLE_COMPLEX,prow(1, nblk, np_rows),mpi_comm_rows,mpierr)
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#endif
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      endif
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#ifdef WITH_MPI
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      call mpi_bcast(tau(2),1,MPI_DOUBLE_COMPLEX,pcol(2, nblk, np_cols),mpi_comm_cols,mpierr)
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#endif
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      if (my_prow==prow(1, nblk, np_rows) .and. my_pcol==pcol(1, nblk, np_cols)) d(1) = a(1,1)

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      deallocate(tmp, vr, ur, vc, uc, vur, uvc, stat=istat, errmsg=errorMessage)
      if (istat .ne. 0) then
       print *,"tridiag_complex: error when deallocating tmp "//errorMessage
       stop
      endif
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      ! distribute the arrays d and e to all processors

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      allocate(tmpr(na), stat=istat, errmsg=errorMessage)
      if (istat .ne. 0) then
       print *,"tridiag_complex: error when allocating tmpr "//errorMessage
       stop
      endif
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      tmpr = d
      call mpi_allreduce(tmpr,d,na,MPI_REAL8,MPI_SUM,mpi_comm_rows,mpierr)
      tmpr = d
      call mpi_allreduce(tmpr,d,na,MPI_REAL8,MPI_SUM,mpi_comm_cols,mpierr)
      tmpr = e
      call mpi_allreduce(tmpr,e,na,MPI_REAL8,MPI_SUM,mpi_comm_rows,mpierr)
      tmpr = e
      call mpi_allreduce(tmpr,e,na,MPI_REAL8,MPI_SUM,mpi_comm_cols,mpierr)
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#endif
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      deallocate(tmpr, stat=istat, errmsg=errorMessage)
      if (istat .ne. 0) then
       print *,"tridiag_complex: error when deallocating tmpr "//errorMessage
       stop
      endif

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

    end subroutine tridiag_complex

    subroutine trans_ev_complex(na, nqc, a, lda, tau, q, ldq, nblk, matrixCols, mpi_comm_rows, mpi_comm_cols)

    !-------------------------------------------------------------------------------
    !  trans_ev_complex: Transforms the eigenvectors of a tridiagonal matrix back
    !                    to the eigenvectors of the original matrix
    !                    (like Scalapack Routine PZUNMTR)
    !
    !  Parameters
    !
    !  na          Order of matrix a, number of rows of matrix q
    !
    !  nqc         Number of columns of matrix q
    !
    !  a(lda,matrixCols)    Matrix containing the Householder vectors (i.e. matrix a after tridiag_complex)
    !              Distribution is like in Scalapack.
    !
    !  lda         Leading dimension of a
    !
    !  tau(na)     Factors of the Householder vectors
    !
    !  q           On input: Eigenvectors of tridiagonal matrix
    !              On output: Transformed eigenvectors
    !              Distribution is like in Scalapack.
    !
    !  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
    !
    !-------------------------------------------------------------------------------
#ifdef HAVE_DETAILED_TIMINGS
      use timings
#endif
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      use precision
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      implicit none

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      integer(kind=ik)              ::  na, nqc, lda, ldq, nblk, matrixCols, mpi_comm_rows, mpi_comm_cols
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      complex(kind=ck)              ::  tau(na)
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#ifdef USE_ASSUMED_SIZE
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      complex(kind=ck)              :: a(lda,*), q(ldq,*)
#else
      complex(kind=ck)              ::  a(lda,matrixCols), q(ldq,matrixCols)
#endif
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      integer(kind=ik)              :: max_stored_rows
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      complex(kind=ck), parameter   :: CZERO = (0.d0,0.d0), CONE = (1.d0,0.d0)
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      integer(kind=ik)              :: my_prow, my_pcol, np_rows, np_cols, mpierr
      integer(kind=ik)              :: totalblocks, max_blocks_row, max_blocks_col, max_local_rows, max_local_cols
      integer(kind=ik)              :: l_cols, l_rows, l_colh, nstor
      integer(kind=ik)              :: istep, i, n, nc, ic, ics, ice, nb, cur_pcol
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      complex(kind=ck), allocatable :: tmp1(:), tmp2(:), hvb(:), hvm(:,:)
      complex(kind=ck), allocatable :: tmat(:,:), h1(:), h2(:)
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      integer(kind=ik)              :: istat
      character(200)                :: errorMessage
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#ifdef HAVE_DETAILED_TIMINGS
      call timer%start("trans_ev_complex")
#endif
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      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)

      totalblocks = (na-1)/nblk + 1
      max_blocks_row = (totalblocks-1)/np_rows + 1
      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

      max_stored_rows = (63/nblk+1)*nblk

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

      allocate(h1(max_stored_rows*max_stored_rows), stat=istat, errmsg=errorMessage)
      if (istat .ne. 0) then
       print *,"trans_ev_complex: error when allocating h1 "//errorMessage
       stop
      endif

      allocate(h2(max_stored_rows*max_stored_rows), stat=istat, errmsg=errorMessage)
      if (istat .ne. 0) then
       print *,"trans_ev_complex: error when allocating h2 "//errorMessage
       stop
      endif

      allocate(tmp1(max_local_cols*max_stored_rows), stat=istat, errmsg=errorMessage)
      if (istat .ne. 0) then
       print *,"trans_ev_complex: error when allocating tmp1 "//errorMessage
       stop
      endif

      allocate(tmp2(max_local_cols*max_stored_rows), stat=istat, errmsg=errorMessage)
      if (istat .ne. 0) then
       print *,"trans_ev_complex: error when allocating tmp2 "//errorMessage
       stop
      endif

      allocate(hvb(max_local_rows*nblk), stat=istat, errmsg=errorMessage)
      if (istat .ne. 0) then
       print *,"trans_ev_complex: error when allocating hvb "//errorMessage
       stop
      endif

      allocate(hvm(max_local_rows,max_stored_rows), stat=istat, errmsg=errorMessage)
      if (istat .ne. 0) then
       print *,"trans_ev_complex: error when allocating hvm "//errorMessage
       stop
      endif
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      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

      nstor = 0

      ! In the complex case tau(2) /= 0
      if (my_prow == prow(1, nblk, np_rows)) then
        q(1,1:l_cols) = q(1,1:l_cols)*((1.d0,0.d0)-tau(2))
      endif

      do istep=1,na,nblk

        ics = MAX(istep,3)
        ice = MIN(istep+nblk-1,na)
        if (ice<ics) cycle

        cur_pcol = pcol(istep, nblk, np_cols)

        nb = 0
        do ic=ics,ice

          l_colh = local_index(ic  , my_pcol, np_cols, nblk, -1) ! Column of Householder vector
          l_rows = local_index(ic-1, my_prow, np_rows, nblk, -1) ! # rows of Householder vector


          if (my_pcol==cur_pcol) then
            hvb(nb+1:nb+l_rows) = a(1:l_rows,l_colh)
            if (my_prow==prow(ic-1, nblk, np_rows)) then
              hvb(nb+l_rows) = 1.
            endif
          endif

          nb = nb+l_rows
        enddo

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#ifdef WITH_MPI
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        if (nb>0) &
           call MPI_Bcast(hvb,nb,MPI_DOUBLE_COMPLEX,cur_pcol,mpi_comm_cols,mpierr)
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#endif
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        nb = 0
        do ic=ics,ice
          l_rows = local_index(ic-1, my_prow, np_rows, nblk, -1) ! # rows of Householder vector
          hvm(1:l_rows,nstor+1) = hvb(nb+1:nb+l_rows)
          nstor = nstor+1
          nb = nb+l_rows
        enddo

        ! Please note: for smaller matix sizes (na/np_rows<=256), a value of 32 for nstor is enough!
        if (nstor+nblk>max_stored_rows .or. istep+nblk>na .or. (na/np_rows<=256 .and. nstor>=32)) then

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

          tmat = 0
          if (l_rows>0) &
             call zherk('U','C',nstor,l_rows,CONE,hvm,ubound(hvm,dim=1),CZERO,tmat,max_stored_rows)

          nc = 0
          do n=1,nstor-1
            h1(nc+1:nc+n) = tmat(1:n,n+1)
            nc = nc+n
          enddo
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          if (nc>0) call mpi_allreduce(h1,h2,nc,MPI_DOUBLE_COMPLEX,MPI_SUM,mpi_comm_rows,mpierr)
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#else
          if (nc>0) h2=h1
#endif
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          ! Calculate triangular matrix T

          nc = 0
          tmat(1,1) = tau(ice-nstor+1)
          do n=1,nstor-1
            call ztrmv('L','C','N',n,tmat,max_stored_rows,h2(nc+1),1)
            tmat(n+1,1:n) = -conjg(h2(nc+1:nc+n))*tau(ice-nstor+n+1)
            tmat(n+1,n+1) = tau(ice-nstor+n+1)
            nc = nc+n
          enddo

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

          if (l_rows>0) then
            call zgemm('C','N',nstor,l_cols,l_rows,CONE,hvm,ubound(hvm,dim=1), &
                        q,ldq,CZERO,tmp1,nstor)
          else
            tmp1(1:l_cols*nstor) = 0
          endif
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          call mpi_allreduce(tmp1,tmp2,nstor*l_cols,MPI_DOUBLE_COMPLEX,MPI_SUM,mpi_comm_rows,mpierr)
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#else
          tmp2 = tmp1
#endif
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          if (l_rows>0) then
            call ztrmm('L','L','N','N',nstor,l_cols,CONE,tmat,max_stored_rows,tmp2,nstor)
            call zgemm('N','N',l_rows,l_cols,nstor,-CONE,hvm,ubound(hvm,dim=1), &
                        tmp2,nstor,CONE,q,ldq)
          endif
          nstor = 0
        endif

      enddo

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      deallocate(tmat, h1, h2, tmp1, tmp2, hvb, hvm, stat=istat, errmsg=errorMessage)
      if (istat .ne. 0) then
       print *,"trans_ev_complex: error when deallocating hvb "//errorMessage
       stop
      endif
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#ifdef HAVE_DETAILED_TIMINGS
      call timer%stop("trans_ev_complex")
#endif

    end subroutine trans_ev_complex

    subroutine solve_tridi( na, nev, d, e, q, ldq, nblk, matrixCols, mpi_comm_rows, mpi_comm_cols, wantDebug, success )
#ifdef HAVE_DETAILED_TIMINGS
      use timings
#endif
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      use precision
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      implicit none

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      integer(kind=ik)              :: na, nev, ldq, nblk, matrixCols, mpi_comm_rows, mpi_comm_cols
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      real(kind=rk)                 :: d(na), e(na)
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#ifdef USE_ASSUMED_SIZE
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      real(kind=rk)                 :: q(ldq,*)
#else
      real(kind=rk)                 :: q(ldq,matrixCols)
#endif
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      integer(kind=ik)              :: i, j, n, np, nc, nev1, l_cols, l_rows
      integer(kind=ik)              :: my_prow, my_pcol, np_rows, np_cols, mpierr
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      integer(kind=ik), allocatable :: limits(:), l_col(:), p_col(:), l_col_bc(:), p_col_bc(:)
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      logical, intent(in)           :: wantDebug
      logical, intent(out)          :: success
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      integer(kind=ik)              :: istat
      character(200)                :: errorMessage

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#ifdef HAVE_DETAILED_TIMINGS
      call timer%start("solve_tridi")
#endif
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      call mpi_comm_rank(mpi_comm_rows,my_prow,mpierr)
      call mpi_comm_size(mpi_comm_rows,np_rows,mpierr)
      call mpi_comm_rank(mpi_comm_cols,my_pcol,mpierr)
      call mpi_comm_size(mpi_comm_cols,np_cols,mpierr)
      success = .true.

      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

      ! Set Q to 0

      q(1:l_rows, 1:l_cols) = 0.

      ! Get the limits of the subdivisons, each subdivison has as many cols
      ! as fit on the respective processor column

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      allocate(limits(0:np_cols), stat=istat, errmsg=errorMessage)
      if (istat .ne. 0) then
        print *,"solve_tridi: error when allocating limits "//errorMessage
        stop
      endif
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      limits(0) = 0
      do np=0,np_cols-1
        nc = local_index(na, np, np_cols, nblk, -1) ! number of columns on proc column np

        ! Check for the case that a column has have zero width.
        ! This is not supported!
        ! Scalapack supports it but delivers no results for these columns,
        ! which is rather annoying
        if (nc==0) then
#ifdef HAVE_DETAILED_TIMINGS
          call timer%stop("solve_tridi")
#endif
          if (wantDebug) write(error_unit,*) 'ELPA1_solve_tridi: ERROR: Problem contains processor column with zero width'
          success = .false.
          return
        endif
        limits(np+1) = limits(np) + nc
      enddo

      ! Subdivide matrix by subtracting rank 1 modifications

      do i=1,np_cols-1
        n = limits(i)
        d(n) = d(n)-abs(e(n))
        d(n+1) = d(n+1)-abs(e(n))
      enddo

      ! Solve sub problems on processsor columns

      nc = limits(my_pcol) ! column after which my problem starts

      if (np_cols>1) then
        nev1 = l_cols ! all eigenvectors are needed
      else
        nev1 = MIN(nev,l_cols)
      endif
      call solve_tridi_col(l_cols, nev1, nc, d(nc+1), e(nc+1), q, ldq, nblk,  &
                        matrixCols, mpi_comm_rows, wantDebug, success)
      if (.not.(success)) then
#ifdef HAVE_DETAILED_TIMINGS
        call timer%stop("solve_tridi")
#endif
        return
      endif
      ! If there is only 1 processor column, we are done

      if (np_cols==1) then
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        deallocate(limits, stat=istat, errmsg=errorMessage)
        if (istat .ne. 0) then
          print *,"solve_tridi: error when deallocating limits "//errorMessage
          stop
        endif

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#ifdef HAVE_DETAILED_TIMINGS
        call timer%stop("solve_tridi")
#endif
        return
      endif

      ! Set index arrays for Q columns

      ! Dense distribution scheme:

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

      allocate(p_col(na), stat=istat, errmsg=errorMessage)
      if (istat .ne. 0) then
        print *,"solve_tridi: error when allocating p_col "//errorMessage
        stop
      endif
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      n = 0
      do np=0,np_cols-1
        nc = local_index(na, np, np_cols, nblk, -1)
        do i=1,nc
          n = n+1
          l_col(n) = i
          p_col(n) = np
        enddo
      enddo

      ! Block cyclic distribution scheme, only nev columns are set:

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

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

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      p_col_bc(:) = -1
      l_col_bc(:) = -1

      do i = 0, na-1, nblk*np_cols
        do j = 0, np_cols-1
          do n = 1, nblk
            if (i+j*nblk+n <= MIN(nev,na)) then
              p_col_bc(i+j*nblk+n) = j
              l_col_bc(i+j*nblk+n) = i/np_cols + n
             endif
           enddo
         enddo
      enddo

      ! Recursively merge sub problems

      call merge_recursive(0, np_cols, wantDebug, success)
      if (.not.(success)) then
#ifdef HAVE_DETAILED_TIMINGS
        call timer%stop("solve_tridi")
#endif
        return
      endif

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      deallocate(limits,l_col,p_col,l_col_bc,p_col_bc, stat=istat, errmsg=errorMessage)
      if (istat .ne. 0) then
        print *,"solve_tridi: error when deallocating l_col "//errorMessage
        stop
      endif

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#ifdef HAVE_DETAILED_TIMINGS
      call timer%stop("solve_tridi")
#endif
      return

      contains
        recursive subroutine merge_recursive(np_off, nprocs, wantDebug, success)
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           use precision
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