elpa1_tridiag_template.X90

#if 0
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!      Schwerpunkt Wissenschaftliches Rechnen ,
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!      Leipzig, Abt. Komplexe Strukutren in Biologie und Kognition,
!      and
!    - IBM Deutschland GmbH
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! ELPA1 -- Faster replacements for ScaLAPACK symmetric eigenvalue routines
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#endif


!> \brief Reduces a distributed symmetric matrix to tridiagonal form (like Scalapack Routine PDSYTRD)
!>
!  Parameters
!
!> \param na          Order of matrix
!>
!> \param a_mat(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
!>
!> \param lda         Leading dimension of a
!>
!> \param nblk        blocksize of cyclic distribution, must be the same in both directions!
!>
!> \param matrixCols  local columns of matrix
!>
!> \param mpi_comm_rows        MPI-Communicator for rows
!> \param mpi_comm_cols        MPI-Communicator for columns
!>
!> \param d_vec(na)       Diagonal elements (returned), identical on all processors
!>
!> \param e_vec(na)       Off-Diagonal elements (returned), identical on all processors
!>
!> \param tau(na)     Factors for the Householder vectors (returned), needed for back transformation
!>
!> \param useGPU      If true,  GPU version of the subroutine will be used
!>

    subroutine tridiag_&
    &MATH_DATATYPE&
    &_&
    &PRECISION &
    (na, a_mat, lda, nblk, matrixCols, mpi_comm_rows, mpi_comm_cols, d_vec, e_vec, tau, useGPU)
      use cuda_functions
      use iso_c_binding
#ifdef HAVE_DETAILED_TIMINGS
      use timings
#else
      use timings_dummy
#endif
      use precision
      implicit none

      logical, parameter      :: new_GPU_multiply = .true.
      
      integer(kind=ik), intent(in)                  :: na, lda, nblk, matrixCols, mpi_comm_rows, mpi_comm_cols
      logical, intent(in)                           :: useGPU

#if REALCASE == 1
      real(kind=REAL_DATATYPE), intent(out)         :: tau(na)
#endif
#if COMPLEXCASE == 1
      complex(kind=COMPLEX_DATATYPE), intent(out)   :: tau(na)
#endif
#if REALCASE == 1
#ifdef USE_ASSUMED_SIZE
      real(kind=REAL_DATATYPE), intent(inout)       :: a_mat(lda,*)
#else
      real(kind=REAL_DATATYPE), intent(inout)       :: a_mat(lda,matrixCols)
#endif
#endif
#if COMPLEXCASE == 1
#ifdef USE_ASSUMED_SIZE
      complex(kind=COMPLEX_DATATYPE), intent(inout) :: a_mat(lda,*)
#else
      complex(kind=COMPLEX_DATATYPE), intent(inout) :: a_mat(lda,matrixCols)
#endif
#endif
      real(kind=REAL_DATATYPE), intent(out)         :: d_vec(na), e_vec(na)

      integer(kind=ik), parameter                   :: max_stored_uv = 32

      ! id in processor row and column and total numbers of processor rows and columns
      integer(kind=ik)                              :: my_prow, my_pcol, np_rows, np_cols, my_rank
      integer(kind=ik)                              :: mpierr
#if REALCASE == 1
#ifdef DOUBLE_PRECISION_REAL
      real(kind=rk8), parameter                     :: ZERO=0.0_rk8, ONE = 1.0_rk8
#else
      real(kind=rk4), parameter                     :: ZERO=0.0_rk4, ONE = 1.0_rk4
#endif
#endif
#if COMPLEXCASE == 1
#ifdef DOUBLE_PRECISION_COMPLEX
      complex(kind=ck8), parameter                  :: ZERO = (0.0_rk8,0.0_rk8), ONE = (1.0_rk8,0.0_rk8)
#else
      complex(kind=ck4), parameter                  :: ZERO = (0.0_rk4,0.0_rk4), ONE = (1.0_rk4,0.0_rk4)
#endif
#endif
      integer(kind=ik)                              :: totalblocks, max_loc_block_rows, max_loc_block_cols, max_local_rows, &
                                                       max_local_cols
      ! updated after each istep (in the main cycle) to contain number of
      ! local columns and rows of the remaining part of the matrix
      !integer(kind=ik)                             :: l_cols, l_rows
      integer(kind=ik)                              :: l_cols, l_rows
      integer(kind=ik)                              :: n_stored_vecs

      integer(kind=C_intptr_T)                      :: a_dev, v_row_dev, v_col_dev, u_row_dev, u_col_dev, vu_stored_rows_dev, &
                                                       uv_stored_cols_dev
      logical                                       :: successCUDA

      integer(kind=ik)                              :: istep, i, j, l_col_beg, l_col_end, l_row_beg, l_row_end
      integer(kind=ik)                              :: tile_size, l_rows_per_tile, l_cols_per_tile
      integer(kind=c_size_t)                        :: a_offset

#ifdef WITH_OPENMP
      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
#endif

      real(kind=REAL_DATATYPE)                      :: vnorm2
#if REALCASE == 1
      real(kind=REAL_DATATYPE)                      :: vav, x, aux(2*max_stored_uv), aux1(2), aux2(2), vrl, xf
#endif
#if COMPLEXCASE == 1
      complex(kind=COMPLEX_DATATYPE)                :: vav, xc, aux(2*max_stored_uv),  aux1(2), aux2(2), aux3(1), vrl, xf
#endif

#if REALCASE == 1
      real(kind=REAL_DATATYPE), allocatable         :: tmp(:), &
                                                       v_row(:), &   ! used to store calculated Householder vector
                                                       v_col(:), &   ! the same vector, but transposed - differently distributed among MPI tasks
                                                       u_row(:), &
                                                       u_col(:)
#endif
#if COMPLEXCASE == 1
      complex(kind=COMPLEX_DATATYPE), allocatable   :: tmp(:), v_row(:), v_col(:), u_row(:), u_col(:)
#endif
      ! the following two matrices store pairs of vectors v and u calculated in each step
      ! at most max_stored_uv vector pairs are stored, than the matrix A_i is explicitli updated
      ! u and v are stored both in row and vector forms
      ! pattern: v1,u1,v2,u2,v3,u3,....
      ! todo: It is little bit confusing, I think, that variables _row actually store columns and vice versa
#if REALCASE == 1
      real(kind=REAL_DATATYPE), allocatable         :: vu_stored_rows(:,:)
      ! pattern: u1,v1,u2,v2,u3,v3,....
      real(kind=REAL_DATATYPE), allocatable         :: uv_stored_cols(:,:)
#endif
#if COMPLEXCASE == 1
      complex(kind=COMPLEX_DATATYPE), allocatable   :: vu_stored_rows(:,:), uv_stored_cols(:,:)
#endif

#ifdef WITH_OPENMP
#if REALCASE == 1
      real(kind=REAL_DATATYPE), allocatable         :: ur_p(:,:), uc_p(:,:)
#endif
#if COMPLEXCASE == 1
      complex(kind=COMPLEX_DATATYPE), allocatable   :: ur_p(:,:), uc_p(:,:)
#endif
#endif

#if COMPLEXCASE == 1
      real(kind=REAL_DATATYPE), allocatable         :: tmp_real(:)
#endif
      integer(kind=ik)                              :: istat
      character(200)                                :: errorMessage
      integer(kind=c_size_t), parameter             :: size_of_datatype = size_of_&
                                                                          &PRECISION&
                                                                          &_&
                                                                          &MATH_DATATYPE
      call timer%start("tridiag_&
      &MATH_DATATYPE&
      &" // &
      PRECISION_SUFFIX &
      )


      call timer%start("mpi_communication")
      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)
      call timer%stop("mpi_communication")

      if((my_prow == 0) .and. (my_pcol == 0)) then
        if( new_GPU_multiply) then
          write(*,*) "mat vec multiply version 1"
        else
          write(*,*) "mat vec multiply version 0"
        endif
      endif

      ! Matrix is split into tiles; work is done only for tiles on the diagonal or above
      ! seems that tile is a square submatrix, consisting by several blocks
      ! it is a smallest possible square submatrix, where blocks being distributed among
      ! processors are "aligned" in both rows and columns
      !  -----------------
      ! | 1 4 | 1 4 | 1 4 | ...
      ! | 2 5 | 2 5 | 2 5 | ...
      ! | 3 6 | 3 6 | 3 6 | ...
      !  ----------------- ...
      ! | 1 4 | 1 4 | 1 4 | ...
      ! | 2 5 | 2 5 | 2 5 | ...
      ! | 3 6 | 3 6 | 3 6 | ...
      !  ----------------- .
      !   : :   : :   : :    .
      !   : :   : :   : :      .
      !
      ! this is a tile, where each number represents block, assigned to a processor with the shown number
      ! size of this small block is nblk
      ! Image is for situation with 6 processors, 3 processor rows and 2 columns
      ! tile_size is thus nblk * 6
      !
      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_per_tile = tile_size/np_rows ! local rows of a tile
      l_cols_per_tile = tile_size/np_cols ! local cols of a tile

      totalblocks = (na-1)/nblk + 1
      max_loc_block_rows = (totalblocks-1)/np_rows + 1
      max_loc_block_cols = (totalblocks-1)/np_cols + 1

      ! localy owned submatrix has size at most max_local_rows x max_local_cols at each processor
      max_local_rows = max_loc_block_rows*nblk
      max_local_cols = max_loc_block_cols*nblk

      ! allocate memmory for vectors
      ! todo: It is little bit confusing, I think, that variables _row actually store columns and vice versa
      ! todo: if something has length max_local_rows, it is actually a column, no?
      ! todo: probably one should read it as v_row = vector v distributed among rows
      !
      allocate(tmp(MAX(max_local_rows,max_local_cols)), stat=istat, errmsg=errorMessage)
      call check_alloc("tridiag_&
      &MATH_DATATYPE ", "tmp", istat, errorMessage)

      ! allocate v_row 1 element longer to allow store and broadcast tau together with it
      allocate(v_row(max_local_rows+1), stat=istat, errmsg=errorMessage)
      call check_alloc("tridiag_&
      &MATH_DATATYPE ", "v_row", istat, errorMessage)

      allocate(u_row(max_local_rows), stat=istat, errmsg=errorMessage)
      call check_alloc("tridiag_&
      &MATH_DATATYPE ", "u_row", istat, errorMessage)

      allocate(v_col(max_local_cols), stat=istat, errmsg=errorMessage)
      call check_alloc("tridiag_&
      &MATH_DATATYPE ", "v_col", istat, errorMessage)

      allocate(u_col(max_local_cols), stat=istat, errmsg=errorMessage)
      call check_alloc("tridiag_&
      &MATH_DATATYPE ", "u_col", istat, errorMessage)

#ifdef WITH_OPENMP
      max_threads = omp_get_max_threads()

      allocate(ur_p(max_local_rows,0:max_threads-1), stat=istat, errmsg=errorMessage)
      call check_alloc("tridiag_&
      &MATH_DATATYPE ", "ur_p", istat, errorMessage)

      allocate(uc_p(max_local_cols,0:max_threads-1), stat=istat, errmsg=errorMessage)
      call check_alloc("tridiag_&
      &MATH_DATATYPE ", "uc_p", istat, errorMessage)
#endif /* WITH_OPENMP */

      tmp = 0
      v_row = 0
      u_row = 0
      v_col = 0
      u_col = 0

      allocate(vu_stored_rows(max_local_rows,2*max_stored_uv), stat=istat, errmsg=errorMessage)
      call check_alloc("tridiag_&
      &MATH_DATATYPE ", "vu_stored_rows", istat, errorMessage)

      allocate(uv_stored_cols(max_local_cols,2*max_stored_uv), stat=istat, errmsg=errorMessage)
      call check_alloc("tridiag_&
      &MATH_DATATYPE ", "uv_stored_cols", istat, errorMessage)

      if (useGPU) then
         successCUDA = cuda_malloc(v_row_dev, max_local_rows * size_of_datatype)
         check_alloc_cuda("tridiag: v_row_dev", successCUDA)

         successCUDA = cuda_malloc(u_row_dev, max_local_rows * size_of_datatype)

         check_alloc_cuda("tridiag: u_row_dev", successCUDA)

         successCUDA = cuda_malloc(v_col_dev, max_local_cols * size_of_datatype)
         check_alloc_cuda("tridiag: v_col_dev", successCUDA)

         successCUDA = cuda_malloc(u_col_dev, max_local_cols * size_of_datatype)
         check_alloc_cuda("tridiag: u_col_dev", successCUDA)

         successCUDA = cuda_malloc(vu_stored_rows_dev, max_local_rows * 2 * max_stored_uv * size_of_datatype)
         check_alloc_cuda("tridiag: vu_stored_rows_dev", successCUDA)

         successCUDA = cuda_malloc(uv_stored_cols_dev, max_local_cols * 2 * max_stored_uv * size_of_datatype)
         check_alloc_cuda("tridiag: vu_stored_rows_dev", successCUDA)
      endif !useGPU


      d_vec(:) = 0
      e_vec(:) = 0
      tau(:) = 0

      n_stored_vecs = 0

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

      if (useGPU) then
        ! allocate memmory for matrix A on the device and than copy the matrix

        successCUDA = cuda_malloc(a_dev, lda * matrixCols * size_of_datatype)
        check_alloc_cuda("tridiag: a_dev", successCUDA)

        successCUDA = cuda_memcpy(a_dev, loc(a_mat(1,1)), lda * matrixCols * size_of_datatype, cudaMemcpyHostToDevice)
        check_alloc_cuda("tridiag: a_dev", successCUDA)
      endif

      ! main cycle of tridiagonalization
      ! in each step, 1 Householder vector is calculated
      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

          ! copy l_cols + 1 column of A to v_row
          if (useGPU) then
	    a_offset = l_cols * lda * size_of_datatype
            ! we use v_row on the host at the moment! successCUDA = cuda_memcpy(v_row_dev, a_dev + a_offset, (l_rows)*size_of_PRECISION_real, cudaMemcpyDeviceToDevice)

            successCUDA = cuda_memcpy(loc(v_row(1)), a_dev + a_offset, (l_rows)* size_of_datatype, cudaMemcpyDeviceToHost)
            check_memcpy_cuda("tridiag a_dev 1", successCUDA)
          else
            v_row(1:l_rows) = a_mat(1:l_rows,l_cols+1)
          endif

            if (n_stored_vecs > 0 .and. l_rows > 0) then
	      call timer%start("blas")
#if COMPLEXCASE == 1
              aux(1:2*n_stored_vecs) = conjg(uv_stored_cols(l_cols+1,1:2*n_stored_vecs))
#endif
              call PRECISION_GEMV('N',   &
	                          l_rows, 2*n_stored_vecs,                                  &
                                  ONE, vu_stored_rows, ubound(vu_stored_rows,dim=1),        &
#if REALCASE == 1
                                  uv_stored_cols(l_cols+1,1), ubound(uv_stored_cols,dim=1), &
#endif
#if COMPLEXCASE == 1
                                   aux, 1,  &

#endif
                                  ONE, v_row, 1)
	       call timer%stop("blas")

            endif

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

#ifdef WITH_MPI
            call timer%start("mpi_communication")
            call mpi_allreduce(aux1, aux2, 2, MPI_MATH_DATATYPE_PRECISION, &
                               MPI_SUM, mpi_comm_rows, mpierr)
            call timer%stop("mpi_communication")
#else /* WITH_MPI */
          aux2 = aux1
#endif /* WITH_MPI */
          vnorm2 = aux2(1)
          vrl    = aux2(2)

          ! Householder transformation
#if REALCASE == 1
          call hh_transform_real_&
#endif
#if COMPLEXCASE == 1
          call hh_transform_complex_&
#endif
               &PRECISION &
                             (vrl, vnorm2, xf, tau(istep))
          ! Scale v_row and store Householder vector for back transformation

          v_row(1:l_rows) = v_row(1:l_rows) * xf
          if (my_prow == prow(istep-1, nblk, np_rows)) then
            v_row(l_rows) = 1.

            ! vrl is newly computed off-diagonal element of the final tridiagonal matrix
            e_vec(istep-1) = vrl
          endif

          ! store Householder vector for back transformation
          a_mat(1:l_rows,l_cols+1) = v_row(1:l_rows)

          ! add tau after the end of actuall v_row, to be broadcasted with it
          v_row(l_rows+1) = tau(istep)
         endif !(my_pcol == pcol(istep, nblk, np_cols))

#ifdef WITH_MPI
         ! Broadcast the Householder vector (and tau) along columns
         call MPI_Bcast(v_row, l_rows+1, MPI_MATH_DATATYPE_PRECISION,    &
                        pcol(istep, nblk, np_cols), mpi_comm_cols, mpierr)
#endif /* WITH_MPI */

        !recover tau, which has been broadcasted together with v_row
        tau(istep) =  v_row(l_rows+1)

        ! Transpose Householder vector v_row -> v_col
	call elpa_transpose_vectors_&
             &MATH_DATATYPE&
             &_&
             &PRECISION &
                                   (v_row, ubound(v_row,dim=1), mpi_comm_rows, v_col, ubound(v_col,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 u_col(:) and partly in u_row(:)

        u_col(1:l_cols) = 0
        u_row(1:l_rows) = 0
        if (l_rows > 0 .and. l_cols> 0 ) then
          if(useGPU) then
            successCUDA = cuda_memset(u_col_dev, 0, l_cols * size_of_datatype)
            check_memcpy_cuda("tridiag: u_col_dev", successCUDA)

            successCUDA = cuda_memset(u_row_dev, 0, l_rows * size_of_datatype)
            check_memcpy_cuda("tridiag: u_row_dev", successCUDA)

            successCUDA = cuda_memcpy(v_col_dev, loc(v_col(1)), l_cols * size_of_datatype, cudaMemcpyHostToDevice)

            check_memcpy_cuda("tridiag: v_col_dev", successCUDA)

            successCUDA = cuda_memcpy(v_row_dev, loc(v_row(1)), l_rows * size_of_datatype, cudaMemcpyHostToDevice)
            check_memcpy_cuda("tridiag: v_row_dev", successCUDA)
          endif ! useGU

#if REALCASE == 1
!            if (useGPU) then
             !u_col_dev(1:l_cols) = 0.
             !u_row_dev(1:l_rows) = 0.
             !v_col_dev(1:l_cols) = v_col(1:l_cols)
             !v_row_dev(1:l_rows) = v_row(1:l_rows)
!                do i=0,(istep-2)/tile_size
!                  l_col_beg = i*l_cols_per_tile+1
!                  l_col_end = min(l_cols,(i+1)*l_cols_per_tile)
!                  if(l_col_end<l_col_beg) cycle
!                  do j=0,i
!                     l_row_beg = j*l_rows_per_tile+1
!                     l_row_end = min(l_rows,(j+1)*l_rows_per_tile)
!                     if(l_row_end<l_row_beg) cycle
!                     if(mod(n_iter,n_threads) == my_thread) then
!                       call cublasDGEMV('T',l_row_end-l_row_beg+1,l_col_end-l_col_beg+1,1.d0,a_dev(l_row_beg,l_col_beg),lda,v_row_dev(l_row_beg),1,1.d0,u_col_dev(l_col_beg),1)
!                       if(i/=j) call cublasDGEMV('N',l_row_end-l_row_beg+1,l_col_end-l_col_beg+1,1.d0,a_dev(l_row_beg,l_col_beg),lda,v_col_dev(l_col_beg),1,1.d0,u_row_dev(l_row_beg),1)
!                     endif
!                     n_iter = n_iter+1
!                  enddo
!                enddo
!
             !--- for now, just use DSYMV!!!
             ! a_dev -> a_mat ?
             !write(*,*) "ubound ", ubound(a_mat,1), "lda", lda, "lcols", l_cols
!              call PRECISION_SYMV('U', l_cols,  &
!                         1.d0, a_mat, ubound(a_mat,1),  &
!                         v_row, 1,  &
!                         0.d0, u_col, 1)
             !u_col(1:l_cols) = u_col_dev(1:l_cols)
             !u_row(1:l_rows) = u_row_dev(1:l_rows)

!            else !do not use GPU
#endif

#ifdef WITH_OPENMP
          call timer%start("OpenMP parallel")
!$OMP PARALLEL PRIVATE(my_thread,n_threads,n_iter,i,l_col_beg,l_col_end,j,l_row_beg,l_row_end)

          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 /* WITH_OPENMP */
          do i= 0, (istep-2)/tile_size
            l_col_beg = i*l_cols_per_tile+1
            l_col_end = min(l_cols,(i+1)*l_cols_per_tile)
            if (l_col_end < l_col_beg) cycle
            do j = 0, i
              l_row_beg = j*l_rows_per_tile+1
              l_row_end = min(l_rows,(j+1)*l_rows_per_tile)
              if (l_row_end < l_row_beg) cycle
#ifdef WITH_OPENMP
              if (mod(n_iter,n_threads) == my_thread) then
	        call timer%start("blas")
                call PRECISION_GEMV(BLAS_TRANS_OR_CONJ, &
                                    l_row_end-l_row_beg+1, l_col_end-l_col_beg+1, &
                                    ONE, a_mat(l_row_beg,l_col_beg), lda,         &
                                    v_row(l_row_beg), 1, ONE, uc_p(l_col_beg,my_thread), 1)

                if (i/=j) then
                  call PRECISION_GEMV('N', l_row_end-l_row_beg+1, l_col_end-l_col_beg+1,          &
                                      ONE, a_mat(l_row_beg,l_col_beg), lda, v_col(l_col_beg), 1,  &

                                      ONE, ur_p(l_row_beg,my_thread), 1)
                endif
	        call timer%stop("blas")
              endif
              n_iter = n_iter+1
#else /* WITH_OPENMP */

              if (useGPU) then
                if(.not. new_GPU_multiply) then 
                  a_offset = ((l_row_beg-1) + (l_col_beg - 1) * lda) * size_of_datatype
                  call timer%start("cublas")
                  call cublas_PRECISION_GEMV(BLAS_TRANS_OR_CONJ,  &
                                           l_row_end-l_row_beg+1,l_col_end-l_col_beg+1,  &
                                           ONE, a_dev + a_offset, lda,                   &
                                           v_row_dev + (l_row_beg - 1) *                 &
                                           size_of_datatype, 1,                          &
                                           ONE, u_col_dev + (l_col_beg - 1) *            &
                                           size_of_datatype, 1)

                  if(i/=j) then
                    call cublas_PRECISION_GEMV('N', l_row_end-l_row_beg+1,l_col_end-l_col_beg+1,  &
                                             ONE, a_dev + a_offset, lda,                        &
                                             v_col_dev + (l_col_beg - 1) *                      &
                                             size_of_datatype, 1,                          &
                                             ONE, u_row_dev + (l_row_beg - 1) *                 &
                                             size_of_datatype, 1)
                  endif
                  call timer%stop("cublas")
                endif
              else ! useGPU
                call timer%start("blas")
                call PRECISION_GEMV(BLAS_TRANS_OR_CONJ,  &
                            l_row_end-l_row_beg+1, l_col_end-l_col_beg+1, &
                            ONE, a_mat(l_row_beg, l_col_beg), lda,         &
                            v_row(l_row_beg), 1,                           &
                            ONE, u_col(l_col_beg), 1)

                if (i/=j) then

                  call PRECISION_GEMV('N', l_row_end-l_row_beg+1, l_col_end-l_col_beg+1,  &
                                      ONE, a_mat(l_row_beg,l_col_beg), lda,               &
                                      v_col(l_col_beg), 1,                                &
                                      ONE, u_row(l_row_beg), 1)
                endif
	        call timer%stop("blas")
              endif ! useGPU

#endif /* WITH_OPENMP */
            enddo  ! j=0,i
          enddo  ! i=0,(istep-2)/tile_size

          if (useGPU) then
            if(new_GPU_multiply) then 
            
                do i=0,(istep-2)/tile_size
                    l_col_beg = i*l_cols_per_tile+1
                    l_col_end = min(l_cols,(i+1)*l_cols_per_tile)
                    if(l_col_end<l_col_beg) cycle
                                            
                    l_row_beg = 1
                    l_row_end = min(l_rows,(i+1)*l_rows_per_tile)
                    
                    a_offset = ((l_row_beg-1) + (l_col_beg - 1) * lda) * &
                           size_of_datatype
                                
                    call cublas_PRECISION_GEMV(BLAS_TRANS_OR_CONJ, &
                               l_row_end-l_row_beg+1, l_col_end-l_col_beg+1, &
                               ONE, a_dev + a_offset, lda,  &
                               v_row_dev + (l_row_beg - 1) * size_of_datatype, 1,  &
                               ONE, u_col_dev + (l_col_beg - 1) * size_of_datatype, 1)
                enddo
                
                do i=0,(istep-2)/tile_size
                    l_col_beg = i*l_cols_per_tile+1
                    l_col_end = min(l_cols,(i+1)*l_cols_per_tile)
                    if(l_col_end<l_col_beg) cycle
                    
                    l_row_beg = 1
                    l_row_end = min(l_rows,i*l_rows_per_tile)

                    a_offset = ((l_row_beg-1) + (l_col_beg - 1) * lda) * &
                           size_of_datatype
                           
                    call cublas_PRECISION_GEMV('N', l_row_end-l_row_beg+1, l_col_end-l_col_beg+1, & 
                               ONE, a_dev + a_offset, lda, &
                               v_col_dev + (l_col_beg - 1) * size_of_datatype,1, &
                               ONE, u_row_dev + (l_row_beg - 1) * size_of_datatype, 1)
                enddo

            
            endif          
          
            successCUDA = cuda_memcpy(loc(u_col(1)), u_col_dev, l_cols * size_of_datatype, cudaMemcpyDeviceToHost)
            check_memcpy_cuda("tridiag: u_col_dev 1", successCUDA)

            successCUDA = cuda_memcpy(loc(u_row(1)), u_row_dev, l_rows * size_of_datatype, cudaMemcpyDeviceToHost)
            check_memcpy_cuda("tridiag: u_row_dev 1", successCUDA)
          endif

!              call PRECISION_SYMV('U', l_cols,  &
!                         1.d0, a_mat, ubound(a_mat,1),  &
!                         v_row, 1,  &
!                         0.d0, u_col, 1)

!            endif ! useGPU

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

          do i=0,max_threads-1
            u_col(1:l_cols) = u_col(1:l_cols) + uc_p(1:l_cols,i)
            u_row(1:l_rows) = u_row(1:l_rows) + ur_p(1:l_rows,i)
          enddo
#endif /* WITH_OPENMP */

          if (n_stored_vecs > 0) then
	    call timer%start("blas")
#if REALCASE == 1
            call PRECISION_GEMV('T',     &
#endif
#if COMPLEXCASE == 1
            call PRECISION_GEMV('C',     &
#endif
	                        l_rows, 2*n_stored_vecs,   &
                                ONE, vu_stored_rows, ubound(vu_stored_rows,dim=1),   &
                                v_row,  1, ZERO, aux, 1)

            call PRECISION_GEMV('N', l_cols, 2*n_stored_vecs,   &
                                ONE, uv_stored_cols, ubound(uv_stored_cols,dim=1),   &
                                aux, 1, ONE, u_col,  1)
	    call timer%stop("blas")
          endif

        endif  ! (l_rows>0 .and. l_cols>0)

        ! Sum up all u_row(:) parts along rows and add them to the u_col(:) parts
        ! on the processors containing the diagonal
        ! This is only necessary if u_row 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_&
	  &MATH_DATATYPE&
	  &_&
	  &PRECISION &
          (u_row, ubound(u_row,dim=1), mpi_comm_rows, u_col, ubound(u_col,dim=1), &
	   mpi_comm_cols, istep-1, 1, nblk)

        endif

        ! Sum up all the u_col(:) parts, transpose u_col -> u_row

        if (l_cols>0) then
          tmp(1:l_cols) = u_col(1:l_cols)
#ifdef WITH_MPI
          call timer%start("mpi_communication")
          call mpi_allreduce(tmp, u_col, l_cols, MPI_MATH_DATATYPE_PRECISION,    &
                             MPI_SUM, mpi_comm_rows, mpierr)
          call timer%stop("mpi_communication")
#else /* WITH_MPI */
          u_col = tmp
#endif /* WITH_MPI */
        endif

        call elpa_transpose_vectors_&
	&MATH_DATATYPE&
	&_&
	&PRECISION &
        (u_col, ubound(u_col,dim=1), mpi_comm_cols, u_row, ubound(u_row,dim=1), &
	 mpi_comm_rows, 1, istep-1, 1, nblk)

        ! calculate u**T * v (same as v**T * (A + VU**T + UV**T) * v )
#if REALCASE == 1
        x = 0
        if (l_cols>0)  &
           x = dot_product(v_col(1:l_cols),u_col(1:l_cols))
#endif
#if COMPLEXCASE == 1
        xc = 0
        if (l_cols>0)  &
           xc = dot_product(v_col(1:l_cols),u_col(1:l_cols))
#endif

#ifdef WITH_MPI
        call timer%start("mpi_communication")
#if REALCASE == 1
        call mpi_allreduce(x, vav, 1, MPI_MATH_DATATYPE_PRECISION, MPI_SUM, mpi_comm_cols, mpierr)
#endif
#if COMPLEXCASE == 1
        call mpi_allreduce(xc, vav, 1 , MPI_MATH_DATATYPE_PRECISION, MPI_SUM, mpi_comm_cols, mpierr)
#endif
        call timer%stop("mpi_communication")
#else /* WITH_MPI */

#if REALCASE == 1
        vav = x
#endif
#if COMPLEXCASE == 1
        vav = xc
#endif

#endif /* WITH_MPI */

        ! store u and v in the matrices U and V
        ! these matrices are stored combined in one here

        do j=1,l_rows
#if REALCASE == 1
          vu_stored_rows(j,2*n_stored_vecs+1) = tau(istep)*v_row(j)
          vu_stored_rows(j,2*n_stored_vecs+2) = 0.5*tau(istep)*vav*v_row(j) - u_row(j)
#endif
#if COMPLEXCASE == 1
          vu_stored_rows(j,2*n_stored_vecs+1) = conjg(tau(istep))*v_row(j)
          vu_stored_rows(j,2*n_stored_vecs+2) = 0.5*conjg(tau(istep))*vav*v_row(j) - u_row(j)
#endif
        enddo
        do j=1,l_cols
#if REALCASE == 1
          uv_stored_cols(j,2*n_stored_vecs+1) = 0.5*tau(istep)*vav*v_col(j) - u_col(j)
          uv_stored_cols(j,2*n_stored_vecs+2) = tau(istep)*v_col(j)
#endif
#if COMPLEXCASE == 1
          uv_stored_cols(j,2*n_stored_vecs+1) = 0.5*conjg(tau(istep))*vav*v_col(j) - u_col(j)
          uv_stored_cols(j,2*n_stored_vecs+2) = conjg(tau(istep))*v_col(j)
#endif
        enddo

        ! We have calculated another Hauseholder vector, number of implicitly stored increased
        n_stored_vecs = n_stored_vecs+1

        ! If the limit of max_stored_uv is reached, calculate A + VU**T + UV**T
        if (n_stored_vecs == max_stored_uv .or. istep == 3) then

          if (useGPU) then
            successCUDA = cuda_memcpy(vu_stored_rows_dev, loc(vu_stored_rows(1,1)), &
                                      max_local_rows * 2 * max_stored_uv *          &
                                      size_of_datatype, cudaMemcpyHostToDevice)
            check_memcpy_cuda("tridiag: vu_stored_rows_dev", successCUDA)

            successCUDA = cuda_memcpy(uv_stored_cols_dev, loc(uv_stored_cols(1,1)), &
                                      max_local_cols * 2 * max_stored_uv *          &
                                      size_of_datatype, cudaMemcpyHostToDevice)
            check_memcpy_cuda("tridiag: uv_stored_cols_dev", successCUDA)
          endif

          do i = 0, (istep-2)/tile_size
            l_col_beg = i*l_cols_per_tile+1
            l_col_end = min(l_cols,(i+1)*l_cols_per_tile)
            l_row_beg = 1
            l_row_end = min(l_rows,(i+1)*l_rows_per_tile)
            if (l_col_end<l_col_beg .or. l_row_end<l_row_beg) &
              cycle

            if (useGPU) then
              call timer%start("cublas")
              call cublas_PRECISION_GEMM('N', BLAS_TRANS_OR_CONJ,     &
                                         l_row_end-l_row_beg+1, l_col_end-l_col_beg+1, 2*n_stored_vecs,                      &
                                         ONE, vu_stored_rows_dev + (l_row_beg - 1) *                                         &
                                         size_of_datatype,  &
                                         max_local_rows, uv_stored_cols_dev + (l_col_beg - 1) *                              &
                                         size_of_datatype,  &
                                         max_local_cols, ONE, a_dev + ((l_row_beg - 1) + (l_col_beg - 1) * lda) *            &
                                         size_of_datatype , lda)
              call timer%stop("cublas")
            else !useGPU
              call timer%start("blas")
              call PRECISION_GEMM('N', BLAS_TRANS_OR_CONJ,                &
                                   l_row_end-l_row_beg+1, l_col_end-l_col_beg+1, 2*n_stored_vecs,    &
                                   ONE, vu_stored_rows(l_row_beg,1), ubound(vu_stored_rows,dim=1),   &
                                   uv_stored_cols(l_col_beg,1), ubound(uv_stored_cols,dim=1),        &
                                   ONE, a_mat(l_row_beg,l_col_beg), lda)
              call timer%stop("blas")
            endif !useGPU
          enddo

          n_stored_vecs = 0

        endif

        if (my_prow == prow(istep-1, nblk, np_rows) .and. my_pcol == pcol(istep-1, nblk, np_cols)) then
          if (useGPU) then
            !a_mat(l_rows,l_cols) = a_dev(l_rows,l_cols)
             a_offset = ((l_rows - 1) + lda * (l_cols - 1)) * size_of_datatype

             successCUDA = cuda_memcpy(loc(a_mat(l_rows, l_cols)), a_dev + a_offset, &
                                       1 *  size_of_datatype, cudaMemcpyDeviceToHost)
             check_memcpy_cuda("tridiag: a_dev 3", successCUDA)

          endif
          if (n_stored_vecs > 0) then
            a_mat(l_rows,l_cols) = a_mat(l_rows,l_cols) &
                        + dot_product(vu_stored_rows(l_rows,1:2*n_stored_vecs),uv_stored_cols(l_cols,1:2*n_stored_vecs))
          end if
          d_vec(istep-1) = a_mat(l_rows,l_cols)

          if (useGPU) then
            !a_dev(l_rows,l_cols) = a_mat(l_rows,l_cols)
            !successCUDA = cuda_threadsynchronize()
            !check_memcpy_cuda("tridiag: a_dev 4a5a", successCUDA)

             successCUDA = cuda_memcpy(a_dev + a_offset, int(loc(a_mat(l_rows, l_cols)),kind=c_size_t), &
                                       int(1 * size_of_datatype, kind=c_size_t), cudaMemcpyHostToDevice)
             check_memcpy_cuda("tridiag: a_dev 4", successCUDA)
          endif
        endif

      enddo ! main cycle over istep=na,3,-1

#if COMPLEXCASE == 1
      ! Store e_vec(1) and d_vec(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
          if(useGPU) then
            successCUDA = cuda_memcpy(loc(aux3(1)), a_dev + (lda * (l_cols - 1)) * size_of_datatype, &
                                    1 * size_of_datatype, cudaMemcpyDeviceToHost)
            check_memcpy_cuda("tridiag: a_dev 5", successCUDA)
            vrl = aux3(1)
          else !useGPU
            vrl = a_mat(1,l_cols)
          endif !useGPU
          call hh_transform_complex_&
                                    &PRECISION &
                                    (vrl, CONST_REAL_0_0, xf, tau(2))
          e_vec(1) = vrl

          a_mat(1,l_cols) = 1. ! for consistency only
        endif
#ifdef WITH_MPI
        call timer%start("mpi_communication")
        call mpi_bcast(tau(2), 1, MPI_COMPLEX_PRECISION, prow(1, nblk, np_rows), mpi_comm_rows, mpierr)
        call timer%stop("mpi_communication")

#endif /* WITH_MPI */
      endif

#ifdef WITH_MPI
      call timer%start("mpi_communication")
      call mpi_bcast(tau(2), 1, MPI_COMPLEX_PRECISION, pcol(2, nblk, np_cols), mpi_comm_cols, mpierr)
      call timer%stop("mpi_communication")

#endif /* WITH_MPI */
      if (my_prow == prow(1, nblk, np_rows) .and. my_pcol == pcol(1, nblk, np_cols))  then
        if(useGPU) then
          successCUDA = cuda_memcpy(loc(aux3(1)), a_dev, &
                                    1 * size_of_datatype, cudaMemcpyDeviceToHost)
          check_memcpy_cuda("tridiag: a_dev 6", successCUDA)
          d_vec(1) = PRECISION_REAL(aux3(1))
        else !useGPU
          d_vec(1) = PRECISION_REAL(a_mat(1,1))
        endif !useGPU
      endif

#endif /* COMPLEXCASE == 1 */

#if REALCASE == 1
      ! Store e_vec(1)

      if (my_prow==prow(1, nblk, np_rows) .and. my_pcol==pcol(2, nblk, np_cols)) then
        if(useGPU) then
          successCUDA = cuda_memcpy(loc(e_vec(1)), a_dev + (lda * (l_cols - 1)) * size_of_datatype, &
                                    1 * size_of_datatype, cudaMemcpyDeviceToHost)
          check_memcpy_cuda("tridiag: a_dev 7", successCUDA)
        else !useGPU
          e_vec(1) = a_mat(1,l_cols) ! use last l_cols value of loop above
        endif !useGPU
      endif

     ! Store d_vec(1)
      if (my_prow==prow(1, nblk, np_rows) .and. my_pcol==pcol(1, nblk, np_cols)) then
        if(useGPU) then
          successCUDA = cuda_memcpy(loc(d_vec(1)), a_dev, 1 * size_of_datatype, cudaMemcpyDeviceToHost)
          check_memcpy_cuda("tridiag: a_dev 8", successCUDA)
        else !useGPU
          d_vec(1) = a_mat(1,1)
        endif !useGPU
      endif
#endif

      deallocate(tmp, v_row, u_row, v_col, u_col, vu_stored_rows, uv_stored_cols, stat=istat, errmsg=errorMessage)
      if (istat .ne. 0) then
#if REALCASE == 1
        print *,"tridiag_real: error when deallocating uv_stored_cols "//errorMessage
#endif
#if COMPLEXCASE == 1
        print *,"tridiag_complex: error when deallocating tmp "//errorMessage
#endif