elpa2.F90

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



! ELPA2 -- 2-stage solver for ELPA
!
! Copyright of the original code rests with the authors inside the ELPA
! consortium. The copyright of any additional modifications shall rest
! with their original authors, but shall adhere to the licensing terms
! distributed along with the original code in the file "COPYING".


#include "config-f90.h"

module ELPA2

! Version 1.1.2, 2011-02-21

  use elpa_utilities
  USE ELPA1
  use elpa2_utilities
  use elpa_pdgeqrf

  implicit none

  PRIVATE ! By default, all routines contained are private

  ! The following routines are public:

  public :: solve_evp_real_2stage
  public :: solve_evp_complex_2stage

  public :: bandred_real
  public :: tridiag_band_real
  public :: trans_ev_tridi_to_band_real
  public :: trans_ev_band_to_full_real

  public :: bandred_complex
  public :: tridiag_band_complex
  public :: trans_ev_tridi_to_band_complex
  public :: trans_ev_band_to_full_complex

  public :: band_band_real
  public :: divide_band

  integer, public :: which_qr_decomposition = 1     ! defines, which QR-decomposition algorithm will be used
                                                    ! 0 for unblocked
                                                    ! 1 for blocked (maxrank: nblk)
!-------------------------------------------------------------------------------

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

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

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

  include 'mpif.h'


!******
contains

function solve_evp_real_2stage(na, nev, a, lda, ev, q, ldq, nblk,        &
                               matrixCols,                               &
                                 mpi_comm_rows, mpi_comm_cols,           &
                                 mpi_comm_all, THIS_REAL_ELPA_KERNEL_API,&
                                 useQR) result(success)

!-------------------------------------------------------------------------------
!  solve_evp_real_2stage: Solves the real eigenvalue problem with a 2 stage approach
!
!  Parameters
!
!  na          Order of matrix a
!
!  nev         Number of eigenvalues needed
!
!  a(lda,matrixCols)    Distributed matrix for which eigenvalues are to be computed.
!              Distribution is like in Scalapack.
!              The full matrix must be set (not only one half like in scalapack).
!              Destroyed on exit (upper and lower half).
!
!  lda         Leading dimension of a
!  matrixCols  local columns of matrix a and q
!
!  ev(na)      On output: eigenvalues of a, every processor gets the complete set
!
!  q(ldq,matrixCols)    On output: Eigenvectors of a
!              Distribution is like in Scalapack.
!              Must be always dimensioned to the full size (corresponding to (na,na))
!              even if only a part of the eigenvalues is needed.
!
!  ldq         Leading dimension of q
!
!  nblk        blocksize of cyclic distribution, must be the same in both directions!
!
!  mpi_comm_rows
!  mpi_comm_cols
!              MPI-Communicators for rows/columns
!  mpi_comm_all
!              MPI-Communicator for the total processor set
!
!-------------------------------------------------------------------------------
#ifdef HAVE_DETAILED_TIMINGS
 use timings
#endif
   implicit none
   logical, intent(in), optional :: useQR
   logical                       :: useQRActual, useQREnvironment
   integer, intent(in), optional :: THIS_REAL_ELPA_KERNEL_API
   integer                       :: THIS_REAL_ELPA_KERNEL

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

   integer                       :: my_pe, n_pes, my_prow, my_pcol, np_rows, np_cols, mpierr
   integer                       :: nbw, num_blocks
   real*8, allocatable           :: tmat(:,:,:), e(:)
   real*8                        :: ttt0, ttt1, ttts
   integer                       :: i
   logical                       :: success
   logical, save                 :: firstCall = .true.
   logical                       :: wantDebug

#ifdef HAVE_DETAILED_TIMINGS
   call timer%start("solve_evp_real_2stage")
#endif
   call mpi_comm_rank(mpi_comm_all,my_pe,mpierr)
   call mpi_comm_size(mpi_comm_all,n_pes,mpierr)

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


   wantDebug = .false.
   if (firstCall) then
     ! are debug messages desired?
     wantDebug = debug_messages_via_environment_variable()
     firstCall = .false.
   endif

   success = .true.

   useQRActual = .false.

   ! set usage of qr decomposition via API call
   if (present(useQR)) then
     if (useQR) useQRActual = .true.
     if (.not.(useQR)) useQRACtual = .false.
   endif

   ! overwrite this with environment variable settings
   if (qr_decomposition_via_environment_variable(useQREnvironment)) then
     useQRActual = useQREnvironment
   endif

   if (useQRActual) then
     if (mod(na,nblk) .ne. 0) then
       if (wantDebug) then
         write(error_unit,*) "solve_evp_real_2stage: QR-decomposition: blocksize does not fit with matrixsize"
       endif
     print *, "Do not use QR-decomposition for this matrix and blocksize."
     success = .false.
     return
     endif
   endif


   if (present(THIS_REAL_ELPA_KERNEL_API)) then
     ! user defined kernel via the optional argument in the API call
     THIS_REAL_ELPA_KERNEL = THIS_REAL_ELPA_KERNEL_API
   else

     ! if kernel is not choosen via api
     ! check whether set by environment variable
     THIS_REAL_ELPA_KERNEL = get_actual_real_kernel()
   endif

   ! check whether choosen kernel is allowed
   if (check_allowed_real_kernels(THIS_REAL_ELPA_KERNEL)) then

     if (my_pe == 0) then
       write(error_unit,*) " "
       write(error_unit,*) "The choosen kernel ",REAL_ELPA_KERNEL_NAMES(THIS_REAL_ELPA_KERNEL)
       write(error_unit,*) "is not in the list of the allowed kernels!"
       write(error_unit,*) " "
       write(error_unit,*) "Allowed kernels are:"
       do i=1,size(REAL_ELPA_KERNEL_NAMES(:))
         if (AVAILABLE_REAL_ELPA_KERNELS(i) .ne. 0) then
           write(error_unit,*) REAL_ELPA_KERNEL_NAMES(i)
         endif
       enddo

       write(error_unit,*) " "
       write(error_unit,*) "The defaul kernel REAL_ELPA_KERNEL_GENERIC will be used !"
     endif
     THIS_REAL_ELPA_KERNEL = REAL_ELPA_KERNEL_GENERIC

   endif

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

   nbw = (31/nblk+1)*nblk

   num_blocks = (na-1)/nbw + 1

   allocate(tmat(nbw,nbw,num_blocks))

   ! Reduction full -> band

   ttt0 = MPI_Wtime()
   ttts = ttt0
   call bandred_real(na, a, lda, nblk, nbw, matrixCols, num_blocks, mpi_comm_rows, mpi_comm_cols, &
                     tmat, wantDebug, success, useQRActual)
   if (.not.(success)) return
   ttt1 = MPI_Wtime()
   if (my_prow==0 .and. my_pcol==0 .and. elpa_print_times) &
      write(error_unit,*) 'Time bandred_real               :',ttt1-ttt0

   ! Reduction band -> tridiagonal

   allocate(e(na))

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

   call mpi_bcast(ev,na,MPI_REAL8,0,mpi_comm_all,mpierr)
   call mpi_bcast(e,na,MPI_REAL8,0,mpi_comm_all,mpierr)

   ttt1 = MPI_Wtime()
   time_evp_fwd = ttt1-ttts

   ! Solve tridiagonal system

   ttt0 = MPI_Wtime()
   call solve_tridi(na, nev, ev, e, q, ldq, nblk, matrixCols, mpi_comm_rows,  &
                    mpi_comm_cols, wantDebug, success)
   if (.not.(success)) return

   ttt1 = MPI_Wtime()
   if (my_prow==0 .and. my_pcol==0 .and. elpa_print_times) &
     write(error_unit,*) 'Time solve_tridi                :',ttt1-ttt0
   time_evp_solve = ttt1-ttt0
   ttts = ttt1

   deallocate(e)

   ! Backtransform stage 1

   ttt0 = MPI_Wtime()
   call trans_ev_tridi_to_band_real(na, nev, nblk, nbw, q, ldq, matrixCols, mpi_comm_rows, &
                                    mpi_comm_cols, wantDebug, success, THIS_REAL_ELPA_KERNEL)
   if (.not.(success)) return
   ttt1 = MPI_Wtime()
   if (my_prow==0 .and. my_pcol==0 .and. elpa_print_times) &
      write(error_unit,*) 'Time trans_ev_tridi_to_band_real:',ttt1-ttt0

   ! We can now deallocate the stored householder vectors
   deallocate(hh_trans_real)

   ! Backtransform stage 2

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

   deallocate(tmat)
#ifdef HAVE_DETAILED_TIMINGS
   call timer%stop("solve_evp_real_2stage")
#endif
1  format(a,f10.3)

end function solve_evp_real_2stage

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

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

function solve_evp_complex_2stage(na, nev, a, lda, ev, q, ldq, nblk, &
                                  matrixCols, mpi_comm_rows, mpi_comm_cols,      &
                                    mpi_comm_all, THIS_COMPLEX_ELPA_KERNEL_API) result(success)

!-------------------------------------------------------------------------------
!  solve_evp_complex_2stage: Solves the complex eigenvalue problem with a 2 stage approach
!
!  Parameters
!
!  na          Order of matrix a
!
!  nev         Number of eigenvalues needed
!
!  a(lda,matrixCols)    Distributed matrix for which eigenvalues are to be computed.
!              Distribution is like in Scalapack.
!              The full matrix must be set (not only one half like in scalapack).
!              Destroyed on exit (upper and lower half).
!
!  lda         Leading dimension of a
!  matrixCols  local columns of matrix a and q
!
!  ev(na)      On output: eigenvalues of a, every processor gets the complete set
!
!  q(ldq,matrixCols)    On output: Eigenvectors of a
!              Distribution is like in Scalapack.
!              Must be always dimensioned to the full size (corresponding to (na,na))
!              even if only a part of the eigenvalues is needed.
!
!  ldq         Leading dimension of q
!
!  nblk        blocksize of cyclic distribution, must be the same in both directions!
!
!  mpi_comm_rows
!  mpi_comm_cols
!              MPI-Communicators for rows/columns
!  mpi_comm_all
!              MPI-Communicator for the total processor set
!
!-------------------------------------------------------------------------------
#ifdef HAVE_DETAILED_TIMINGS
 use timings
#endif
   implicit none
   integer, intent(in), optional :: THIS_COMPLEX_ELPA_KERNEL_API
   integer                       :: THIS_COMPLEX_ELPA_KERNEL
   integer, intent(in)           :: na, nev, lda, ldq, nblk, matrixCols, mpi_comm_rows, mpi_comm_cols, mpi_comm_all
   complex*16, intent(inout)     :: a(lda,matrixCols), q(ldq,matrixCols)
   real*8, intent(inout)         :: ev(na)

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

   logical                       :: success, wantDebug
   logical, save                 :: firstCall = .true.

#ifdef HAVE_DETAILED_TIMINGS
   call timer%start("solve_evp_complex_2stage")
#endif
   call mpi_comm_rank(mpi_comm_all,my_pe,mpierr)
   call mpi_comm_size(mpi_comm_all,n_pes,mpierr)

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

   wantDebug = .false.
   if (firstCall) then
     ! are debug messages desired?
     wantDebug = debug_messages_via_environment_variable()
     firstCall = .false.
   endif


   success = .true.

   if (present(THIS_COMPLEX_ELPA_KERNEL_API)) then
     ! user defined kernel via the optional argument in the API call
     THIS_COMPLEX_ELPA_KERNEL = THIS_COMPLEX_ELPA_KERNEL_API
   else
     ! if kernel is not choosen via api
     ! check whether set by environment variable
     THIS_COMPLEX_ELPA_KERNEL = get_actual_complex_kernel()
   endif

   ! check whether choosen kernel is allowed
   if (check_allowed_complex_kernels(THIS_COMPLEX_ELPA_KERNEL)) then

     if (my_pe == 0) then
       write(error_unit,*) " "
       write(error_unit,*) "The choosen kernel ",COMPLEX_ELPA_KERNEL_NAMES(THIS_COMPLEX_ELPA_KERNEL)
       write(error_unit,*) "is not in the list of the allowed kernels!"
       write(error_unit,*) " "
       write(error_unit,*) "Allowed kernels are:"
       do i=1,size(COMPLEX_ELPA_KERNEL_NAMES(:))
         if (AVAILABLE_COMPLEX_ELPA_KERNELS(i) .ne. 0) then
           write(error_unit,*) COMPLEX_ELPA_KERNEL_NAMES(i)
         endif
       enddo

       write(error_unit,*) " "
       write(error_unit,*) "The defaul kernel COMPLEX_ELPA_KERNEL_GENERIC will be used !"
     endif
     THIS_COMPLEX_ELPA_KERNEL = COMPLEX_ELPA_KERNEL_GENERIC
   endif
   ! Choose bandwidth, must be a multiple of nblk, set to a value >= 32

   nbw = (31/nblk+1)*nblk

   num_blocks = (na-1)/nbw + 1

   allocate(tmat(nbw,nbw,num_blocks))

   ! Reduction full -> band

   ttt0 = MPI_Wtime()
   ttts = ttt0
   call bandred_complex(na, a, lda, nblk, nbw, matrixCols, num_blocks, mpi_comm_rows, mpi_comm_cols, &
                        tmat, wantDebug, success)
   if (.not.(success)) then
#ifdef HAVE_DETAILED_TIMINGS
     call timer%stop()
#endif
     return
   endif
   ttt1 = MPI_Wtime()
   if (my_prow==0 .and. my_pcol==0 .and. elpa_print_times) &
      write(error_unit,*) 'Time bandred_complex               :',ttt1-ttt0

   ! Reduction band -> tridiagonal

   allocate(e(na))

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

   call mpi_bcast(ev,na,MPI_REAL8,0,mpi_comm_all,mpierr)
   call mpi_bcast(e,na,MPI_REAL8,0,mpi_comm_all,mpierr)

   ttt1 = MPI_Wtime()
   time_evp_fwd = ttt1-ttts

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

   allocate(q_real(l_rows,l_cols))

   ! Solve tridiagonal system

   ttt0 = MPI_Wtime()
   call solve_tridi(na, nev, ev, e, q_real, ubound(q_real,dim=1), nblk, matrixCols, &
                    mpi_comm_rows, mpi_comm_cols, wantDebug, success)
   if (.not.(success)) return

   ttt1 = MPI_Wtime()
   if (my_prow==0 .and. my_pcol==0 .and. elpa_print_times)  &
      write(error_unit,*) 'Time solve_tridi                   :',ttt1-ttt0
   time_evp_solve = ttt1-ttt0
   ttts = ttt1

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

   deallocate(e, q_real)

   ! Backtransform stage 1

   ttt0 = MPI_Wtime()
   call trans_ev_tridi_to_band_complex(na, nev, nblk, nbw, q, ldq,  &
                                       matrixCols, mpi_comm_rows, mpi_comm_cols,&
                                       wantDebug, success,THIS_COMPLEX_ELPA_KERNEL)
   if (.not.(success)) return
   ttt1 = MPI_Wtime()
   if (my_prow==0 .and. my_pcol==0 .and. elpa_print_times) &
      write(error_unit,*) 'Time trans_ev_tridi_to_band_complex:',ttt1-ttt0

   ! We can now deallocate the stored householder vectors
   deallocate(hh_trans_complex)

   ! Backtransform stage 2

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

   deallocate(tmat)
#ifdef HAVE_DETAILED_TIMINGS
   call timer%stop("solve_evp_complex_2stage")
#endif

1  format(a,f10.3)

end function solve_evp_complex_2stage

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

subroutine bandred_real(na, a, lda, nblk, nbw, matrixCols, numBlocks, mpi_comm_rows, mpi_comm_cols, &
                        tmat, wantDebug, success, useQR)

!-------------------------------------------------------------------------------
!  bandred_real: Reduces a distributed symmetric matrix to band form
!
!  Parameters
!
!  na          Order of matrix
!
!  a(lda,matrixCols)    Distributed matrix which should be reduced.
!              Distribution is like in Scalapack.
!              Opposed to Scalapack, a(:,:) must be set completely (upper and lower half)
!              a(:,:) is overwritten on exit with the band and the Householder vectors
!              in the upper half.
!
!  lda         Leading dimension of a
!  matrixCols  local columns of matrix a
!
!  nblk        blocksize of cyclic distribution, must be the same in both directions!
!
!  nbw         semi bandwith of output matrix
!
!  mpi_comm_rows
!  mpi_comm_cols
!              MPI-Communicators for rows/columns
!
!  tmat(nbw,nbw,numBlocks)    where numBlocks = (na-1)/nbw + 1
!              Factors for the Householder vectors (returned), needed for back transformation
!
!-------------------------------------------------------------------------------
#ifdef HAVE_DETAILED_TIMINGS
 use timings
#endif
   implicit none

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

   integer             :: my_prow, my_pcol, np_rows, np_cols, mpierr
   integer             :: l_cols, l_rows
   integer             :: i, j, lcs, lce, lre, lc, lr, cur_pcol, n_cols, nrow
   integer             :: istep, ncol, lch, lcx, nlc
   integer             :: tile_size, l_rows_tile, l_cols_tile

   real*8              :: vnorm2, xf, aux1(nbw), aux2(nbw), vrl, tau, vav(nbw,nbw)

   real*8, allocatable :: tmp(:,:), vr(:), vmr(:,:), umc(:,:)

   ! needed for blocked QR decomposition
   integer             :: PQRPARAM(11), work_size
   real*8              :: dwork_size(1)
   real*8, allocatable :: work_blocked(:), tauvector(:), blockheuristic(:)

   logical, intent(in) :: wantDebug
   logical, intent(out):: success

   logical, intent(in) :: useQR

#ifdef HAVE_DETAILED_TIMINGS
   call timer%start("bandred_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)
   success = .true.


   ! Semibandwith nbw must be a multiple of blocksize nblk
   if (mod(nbw,nblk)/=0) then
     if (my_prow==0 .and. my_pcol==0) then
       if (wantDebug) then
         write(error_unit,*) 'ELPA2_bandred_real: ERROR: nbw=',nbw,', nblk=',nblk
         write(error_unit,*) 'ELPA2_bandred_real: ELPA2 works only for nbw==n*nblk'
       endif
       success = .false.
       return
     endif
   endif

   ! Matrix is split into tiles; work is done only for tiles on the diagonal or above

   tile_size = nblk*least_common_multiple(np_rows,np_cols) ! minimum global tile size
   tile_size = ((128*max(np_rows,np_cols)-1)/tile_size+1)*tile_size ! make local tiles at least 128 wide

   l_rows_tile = tile_size/np_rows ! local rows of a tile
   l_cols_tile = tile_size/np_cols ! local cols of a tile

   if (useQR) then
     if (which_qr_decomposition == 1) then
       call qr_pqrparam_init(pqrparam,    nblk,'M',0,   nblk,'M',0,   nblk,'M',1,'s')
       allocate(tauvector(na))
       allocate(blockheuristic(nblk))
       l_rows = local_index(na, my_prow, np_rows, nblk, -1)
       allocate(vmr(max(l_rows,1),na))

       call qr_pdgeqrf_2dcomm(a, lda, vmr, max(l_rows,1), tauvector(1), tmat(1,1,1), nbw, dwork_size(1), -1, na, &
                             nbw, nblk, nblk, na, na, 1, 0, PQRPARAM, mpi_comm_rows, mpi_comm_cols, blockheuristic)
       work_size = dwork_size(1)
       allocate(work_blocked(work_size))

       work_blocked = 0.0d0
       deallocate(vmr)
     endif
   endif

   do istep = (na-1)/nbw, 1, -1

     n_cols = MIN(na,(istep+1)*nbw) - istep*nbw ! Number of columns in current step

     ! Number of local columns/rows of remaining matrix
     l_cols = local_index(istep*nbw, my_pcol, np_cols, nblk, -1)
     l_rows = local_index(istep*nbw, my_prow, np_rows, nblk, -1)

     ! Allocate vmr and umc to their exact sizes so that they can be used in bcasts and reduces

     allocate(vmr(max(l_rows,1),2*n_cols))
     allocate(umc(max(l_cols,1),2*n_cols))

     allocate(vr(l_rows+1))

     vmr(1:l_rows,1:n_cols) = 0.
     vr(:) = 0
     tmat(:,:,istep) = 0

     ! Reduce current block to lower triangular form

     if (useQR) then
       if (which_qr_decomposition == 1) then
         call qr_pdgeqrf_2dcomm(a, lda, vmr, max(l_rows,1), tauvector(1), &
                                  tmat(1,1,istep), nbw, work_blocked,       &
                                  work_size, na, n_cols, nblk, nblk,        &
                                  istep*nbw+n_cols-nbw, istep*nbw+n_cols, 1,&
                                  0, PQRPARAM, mpi_comm_rows, mpi_comm_cols,&
                                  blockheuristic)
       endif
     else

       do lc = n_cols, 1, -1

         ncol = istep*nbw + lc ! absolute column number of householder vector
         nrow = ncol - nbw ! Absolute number of pivot row

         lr  = local_index(nrow, my_prow, np_rows, nblk, -1) ! current row length
         lch = local_index(ncol, my_pcol, np_cols, nblk, -1) ! HV local column number

         tau = 0

         if (nrow == 1) exit ! Nothing to do

         cur_pcol = pcol(ncol, nblk, np_cols) ! Processor column owning current block

         if (my_pcol==cur_pcol) then

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

           vr(1:lr) = a(1:lr,lch) ! vector to be transformed

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

           call mpi_allreduce(aux1,aux2,2,MPI_REAL8,MPI_SUM,mpi_comm_rows,mpierr)

           vnorm2 = aux2(1)
           vrl    = aux2(2)

           ! Householder transformation

           call hh_transform_real(vrl, vnorm2, xf, tau)

           ! Scale vr and store Householder vector for back transformation

           vr(1:lr) = vr(1:lr) * xf
           if (my_prow==prow(nrow, nblk, np_rows)) then
             a(1:lr-1,lch) = vr(1:lr-1)
             a(lr,lch) = vrl
             vr(lr) = 1.
           else
             a(1:lr,lch) = vr(1:lr)
           endif

         endif

         ! Broadcast Householder vector and tau along columns

         vr(lr+1) = tau
         call MPI_Bcast(vr,lr+1,MPI_REAL8,cur_pcol,mpi_comm_cols,mpierr)
         vmr(1:lr,lc) = vr(1:lr)
         tau = vr(lr+1)
         tmat(lc,lc,istep) = tau ! Store tau in diagonal of tmat

         ! Transform remaining columns in current block with Householder vector
         ! Local dot product

         aux1 = 0

         nlc = 0 ! number of local columns
         do j=1,lc-1
           lcx = local_index(istep*nbw+j, my_pcol, np_cols, nblk, 0)
           if (lcx>0) then
             nlc = nlc+1
             if (lr>0) aux1(nlc) = dot_product(vr(1:lr),a(1:lr,lcx))
           endif
         enddo

         ! Get global dot products
         if (nlc>0) call mpi_allreduce(aux1,aux2,nlc,MPI_REAL8,MPI_SUM,mpi_comm_rows,mpierr)

         ! Transform

         nlc = 0
         do j=1,lc-1
           lcx = local_index(istep*nbw+j, my_pcol, np_cols, nblk, 0)
           if (lcx>0) then
             nlc = nlc+1
             a(1:lr,lcx) = a(1:lr,lcx) - tau*aux2(nlc)*vr(1:lr)
           endif
         enddo

       enddo

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

       vav = 0
       if (l_rows>0) &
           call dsyrk('U','T',n_cols,l_rows,1.d0,vmr,ubound(vmr,dim=1),0.d0,vav,ubound(vav,dim=1))
       call symm_matrix_allreduce(n_cols,vav, nbw, nbw,mpi_comm_rows)

       ! Calculate triangular matrix T for block Householder Transformation

       do lc=n_cols,1,-1
         tau = tmat(lc,lc,istep)
         if (lc<n_cols) then
           call dtrmv('U','T','N',n_cols-lc,tmat(lc+1,lc+1,istep),ubound(tmat,dim=1),vav(lc+1,lc),1)
           tmat(lc,lc+1:n_cols,istep) = -tau * vav(lc+1:n_cols,lc)
         endif
       enddo
     endif

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

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

    ! Calculate umc = A**T * vmr
    ! Note that the distributed A has to be transposed
    ! Opposed to direct tridiagonalization there is no need to use the cache locality
    ! of the tiles, so we can use strips of the matrix

    umc(1:l_cols,1:n_cols) = 0.d0
    vmr(1:l_rows,n_cols+1:2*n_cols) = 0
    if (l_cols>0 .and. l_rows>0) then
      do i=0,(istep*nbw-1)/tile_size

        lcs = i*l_cols_tile+1
        lce = min(l_cols,(i+1)*l_cols_tile)
        if (lce<lcs) cycle

        lre = min(l_rows,(i+1)*l_rows_tile)
        call DGEMM('T','N',lce-lcs+1,n_cols,lre,1.d0,a(1,lcs),ubound(a,dim=1), &
                     vmr,ubound(vmr,dim=1),1.d0,umc(lcs,1),ubound(umc,dim=1))

        if (i==0) cycle
        lre = min(l_rows,i*l_rows_tile)
        call DGEMM('N','N',lre,n_cols,lce-lcs+1,1.d0,a(1,lcs),lda, &
                     umc(lcs,n_cols+1),ubound(umc,dim=1),1.d0,vmr(1,n_cols+1),ubound(vmr,dim=1))
      enddo
    endif

    ! Sum up all ur(:) parts along rows and add them to the uc(:) parts
    ! on the processors containing the diagonal
    ! This is only necessary if ur has been calculated, i.e. if the
    ! global tile size is smaller than the global remaining matrix

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

    if (l_cols>0) then
      allocate(tmp(l_cols,n_cols))
      call mpi_allreduce(umc,tmp,l_cols*n_cols,MPI_REAL8,MPI_SUM,mpi_comm_rows,mpierr)
      umc(1:l_cols,1:n_cols) = tmp(1:l_cols,1:n_cols)
      deallocate(tmp)
    endif

    ! U = U * Tmat**T

    call dtrmm('Right','Upper','Trans','Nonunit',l_cols,n_cols,1.d0,tmat(1,1,istep),ubound(tmat,dim=1),umc,ubound(umc,dim=1))

    ! VAV = Tmat * V**T * A * V * Tmat**T = (U*Tmat**T)**T * V * Tmat**T

    call dgemm('T','N',n_cols,n_cols,l_cols,1.d0,umc,ubound(umc,dim=1),umc(1,n_cols+1), &
               ubound(umc,dim=1),0.d0,vav,ubound(vav,dim=1))
    call dtrmm('Right','Upper','Trans','Nonunit',n_cols,n_cols,1.d0,tmat(1,1,istep),    &
               ubound(tmat,dim=1),vav,ubound(vav,dim=1))

    call symm_matrix_allreduce(n_cols,vav, nbw, nbw ,mpi_comm_cols)

    ! U = U - 0.5 * V * VAV
    call dgemm('N','N',l_cols,n_cols,n_cols,-0.5d0,umc(1,n_cols+1),ubound(umc,dim=1),vav, &
                ubound(vav,dim=1),1.d0,umc,ubound(umc,dim=1))

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

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

    ! A = A - V*U**T - U*V**T

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

    deallocate(vmr, umc, vr)

  enddo

  if (useQR) then
    if (which_qr_decomposition == 1) then
      deallocate(work_blocked)
      deallocate(tauvector)
    endif
  endif

#ifdef HAVE_DETAILED_TIMINGS
  call timer%stop("bandred_real")
#endif
end subroutine bandred_real

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

subroutine symm_matrix_allreduce(n,a,lda,ldb,comm)

!-------------------------------------------------------------------------------
!  symm_matrix_allreduce: Does an mpi_allreduce for a symmetric matrix A.
!  On entry, only the upper half of A needs to be set
!  On exit, the complete matrix is set
!-------------------------------------------------------------------------------
#ifdef HAVE_DETAILED_TIMINGS
 use timings
#endif
   implicit none
   integer  :: n, lda, ldb, comm
   real*8   :: a(lda,ldb)

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

#ifdef HAVE_DETAILED_TIMINGS
  call timer%start("symm_matrix_allreduce")
#endif

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

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

   nc = 0
   do i=1,n
     a(1:i,i) = h2(nc+1:nc+i)
     a(i,1:i-1) = a(1:i-1,i)
     nc = nc+i