dbcsr_multiply_generic Subroutine

public subroutine dbcsr_multiply_generic(transa, transb, alpha, matrix_a, matrix_b, beta, matrix_c, first_row, last_row, first_column, last_column, first_k, last_k, retain_sparsity, filter_eps, flop)

Performs a multiplication of two dbcsr_type matrices, as C := alpha * op( A ) * op( B ) + beta * C.

Matrices m_a and m_b are multiplied into the m_c product matrix. If the dist2d parameter is not specified, then a new distribution_2d is determined for it.

Non-equal column dimensions of the right and product matrices The right and product matrix are allowed to have different (full) column dimensions. If they differ, there are certain peculiar behaviors, then the last_column is effectively set to the minimal of the two.

Beta scaling of the right product matrix If the effective last_column is less than the full column dimension of the product matrix, then the scaling of the product matrix with beta is limited to the submatrix specified by last_column.

Filtering The filter_eps parameter, if present, is used to filter the resulting matrix. The filtering criterion is whether the block-frobenius norm is less than the specified epsilon. One-the-fly filtering is done such that individual multiplications are skipped if the product of the frobenius norms of the left- and right-matrix blocks are less than the specified epsilon divided by the maximum number of possible multiplies in each row. In addition a final filtering is done as well with the same epsilon value.

Arguments

TypeIntentOptionalAttributesName
character(len=1), intent(in) :: transa

specifies the form of op( A ) to be used in the matrix multiplication transa = 'N' or 'n', op( A ) = A. transa = 'T' or 't', op( A ) = transpose(A). transa = 'C' or 'c', op( A ) = transpose(conjg(A)). specifies the form of op( B ) to be used in the matrix multiplication transb = 'N' or 'n', op( B ) = B. transb = 'T' or 't', op( B ) = transpose(B). transb = 'C' or 'c', op( B ) = transpose(conjg(B)).

character(len=1), intent(in) :: transb

specifies the form of op( A ) to be used in the matrix multiplication transa = 'N' or 'n', op( A ) = A. transa = 'T' or 't', op( A ) = transpose(A). transa = 'C' or 'c', op( A ) = transpose(conjg(A)). specifies the form of op( B ) to be used in the matrix multiplication transb = 'N' or 'n', op( B ) = B. transb = 'T' or 't', op( B ) = transpose(B). transb = 'C' or 'c', op( B ) = transpose(conjg(B)).

type(dbcsr_scalar_type), intent(in) :: alpha

scaling of product

type(dbcsr_type), intent(in) :: matrix_a

left BCSR matrix right BCSR matrix

type(dbcsr_type), intent(in) :: matrix_b

left BCSR matrix right BCSR matrix

type(dbcsr_scalar_type), intent(in) :: beta

scaling of existing data

type(dbcsr_type), intent(inout) :: matrix_c

resulting BCSR product matrix.

integer, intent(in), optional :: first_row

first full row of limiting submatrix last full row of limiting submatrix first full column of limiting submatrix last full column of limiting submatrix first full column of limiting inner product last full column of limiting inner product

integer, intent(in), optional :: last_row

first full row of limiting submatrix last full row of limiting submatrix first full column of limiting submatrix last full column of limiting submatrix first full column of limiting inner product last full column of limiting inner product

integer, intent(in), optional :: first_column

first full row of limiting submatrix last full row of limiting submatrix first full column of limiting submatrix last full column of limiting submatrix first full column of limiting inner product last full column of limiting inner product

integer, intent(in), optional :: last_column

first full row of limiting submatrix last full row of limiting submatrix first full column of limiting submatrix last full column of limiting submatrix first full column of limiting inner product last full column of limiting inner product

integer, intent(in), optional :: first_k

first full row of limiting submatrix last full row of limiting submatrix first full column of limiting submatrix last full column of limiting submatrix first full column of limiting inner product last full column of limiting inner product

integer, intent(in), optional :: last_k

first full row of limiting submatrix last full row of limiting submatrix first full column of limiting submatrix last full column of limiting submatrix first full column of limiting inner product last full column of limiting inner product

logical, intent(in), optional :: retain_sparsity

enforce the sparsity pattern of the existing product matrix; default is no

real(kind=real_8), intent(in), optional :: filter_eps

Filtering of the matrix

integer(kind=int_8), intent(out), optional :: flop

effective flop


Contents


Source Code

   SUBROUTINE dbcsr_multiply_generic(transa, transb, &
                                     alpha, matrix_a, matrix_b, beta, matrix_c, &
                                     first_row, last_row, first_column, last_column, first_k, last_k, &
                                     retain_sparsity, filter_eps, &
                                     flop)
      !! Performs a multiplication of two dbcsr_type matrices,
      !! as  C := alpha * op( A ) * op( B ) + beta * C.
      !!
      !! Matrices m_a and m_b are multiplied into the m_c product matrix. If the
      !! dist2d parameter is not specified, then a new distribution_2d is
      !! determined for it.
      !!
      !! Non-equal column dimensions of the right and product matrices
      !! The right and product matrix are allowed to have different
      !! (full) column dimensions. If they differ, there are certain
      !! peculiar behaviors, then the last_column is effectively set to
      !! the minimal of the two.
      !!
      !! Beta scaling of the right product matrix
      !! If the effective last_column is less than the full column
      !! dimension of the product matrix, then the scaling of the
      !! product matrix with beta is limited to the submatrix specified
      !! by last_column.
      !!
      !! Filtering
      !! The filter_eps parameter, if present, is used to filter the
      !! resulting matrix.  The filtering criterion is whether the
      !! block-frobenius norm is less than the specified epsilon.
      !! One-the-fly filtering is done such that individual
      !! multiplications are skipped if the product of the frobenius
      !! norms of the left- and right-matrix blocks are less than the
      !! specified epsilon divided by the maximum number of possible
      !! multiplies in each row.  In addition a final filtering is done
      !! as well with the same epsilon value.

      CHARACTER(LEN=1), INTENT(IN)                       :: transa, transb
         !! specifies the form of op( A ) to be used in the matrix multiplication transa = 'N' or 'n',  op( A ) = A. transa = 'T' or
         !! 't',  op( A ) = transpose(A). transa = 'C' or 'c',  op( A ) = transpose(conjg(A)).
         !! specifies the form of op( B ) to be used in the matrix multiplication transb = 'N' or 'n',  op( B ) = B. transb = 'T' or
         !! 't',  op( B ) = transpose(B). transb = 'C' or 'c',  op( B ) = transpose(conjg(B)).
      TYPE(dbcsr_scalar_type), INTENT(IN)                :: alpha
         !! scaling of product
      TYPE(dbcsr_type), INTENT(IN)                       :: matrix_a, matrix_b
         !! left BCSR matrix
         !! right BCSR matrix
      TYPE(dbcsr_scalar_type), INTENT(IN)                :: beta
         !! scaling of existing data
      TYPE(dbcsr_type), INTENT(INOUT)                    :: matrix_c
         !! resulting BCSR product matrix.
      INTEGER, INTENT(IN), OPTIONAL                      :: first_row, last_row, first_column, &
                                                            last_column, first_k, last_k
         !! first full row of limiting submatrix
         !! last full row of limiting submatrix
         !! first full column of limiting submatrix
         !! last full column of limiting submatrix
         !! first full column of limiting inner product
         !! last full column of limiting inner product
      LOGICAL, INTENT(IN), OPTIONAL                      :: retain_sparsity
         !! enforce the sparsity pattern of the existing product matrix; default is no
      REAL(KIND=real_8), INTENT(IN), OPTIONAL            :: filter_eps
         !! Filtering of the matrix
      INTEGER(KIND=int_8), INTENT(OUT), OPTIONAL         :: flop
         !! effective flop

      CHARACTER(len=*), PARAMETER :: routineN = 'dbcsr_multiply_generic'
      LOGICAL, PARAMETER                                 :: dbg = .FALSE.
      REAL(real_8), PARAMETER                            :: make_dense_occ_thresh = 1.0_dp

      CHARACTER                                          :: transa_l, transb_l
      INTEGER :: comm, f_col, f_k, f_row, handle, handle2, ithread, l_col, l_k, l_row, &
                 nimages_left_rows, nimages_match, nimages_right_cols, npcols, nprows, numnodes, &
                 output_unit
      INTEGER(KIND=int_8)                                :: my_flop
      LOGICAL :: ab_dense, keep_product_data, keep_sparsity, product_reindex, release_tdist, &
                 transpose_left, transpose_right, use_dense_mult, use_mempools, thread_dist_force
      REAL(KIND=dp)                                      :: cs
      TYPE(array_i1d_obj) :: dense_col_sizes, dense_k_sizes, dense_row_sizes, k_vmap, m_map, &
                             n_map, old_product_col_blk_offsets, old_product_col_blk_sizes, &
                             old_product_row_blk_offsets, old_product_row_blk_sizes, &
                             matrix_c_thread_dist
      TYPE(dbcsr_2d_array_type), POINTER                 :: m2s_left, m2s_right
      TYPE(dbcsr_distribution_obj)                       :: dense_product_distribution, &
                                                            old_product_distribution
      TYPE(dbcsr_imagedistribution_obj)                  :: dense_rdist_left, dense_rdist_right, &
                                                            rdist_left, rdist_right
      TYPE(dbcsr_mp_obj)                                 :: mp_obj
      TYPE(dbcsr_type)                                   :: matrix_left, matrix_right, product_matrix

      CALL timeset(routineN, handle)

      IF (dbcsr_get_occupation(matrix_a) .GT. 1) &
         DBCSR_ABORT("Matrix A occupation > 1")

      IF (dbcsr_get_occupation(matrix_b) .GT. 1) &
         DBCSR_ABORT("Matrix B occupation > 1")

      IF (dbcsr_get_occupation(matrix_c) .GT. 1) &
         DBCSR_ABORT("Matrix C occupation > 1")

      CALL array_nullify(dense_k_sizes)
      CALL array_nullify(dense_col_sizes)
      CALL array_nullify(dense_row_sizes)

      ! Reset GPU errors
      IF (has_acc) THEN
         CALL dbcsr_acc_clear_errors()
      END IF

      ! Check if RMA is used with OpenMPI, if so disabled it
      ! (OpenMPI has several bugs with RMA and it does not
      ! give any performance benefit)
      CALL check_openmpi_rma()

      use_mempools = dbcsr_cfg%use_mempools_cpu .OR. has_acc

      ! setup driver-dependent memory-types and their memory-pools ---------------

      ! the ab_buffers are shared by all threads
      IF (has_acc) THEN
         IF (.NOT. acc_stream_associated(stream_1)) THEN
            CALL acc_stream_create(stream_1, "MemCpy (odd ticks)")
            CALL acc_stream_create(stream_2, "MemCpy (even ticks)")
         END IF

         CALL dbcsr_memtype_setup(memtype_abpanel_1, has_pool=.TRUE., &
                                  acc_hostalloc=.TRUE., acc_devalloc=.TRUE., acc_stream=stream_1, &
                                  mpi=.TRUE., oversize_factor=default_resize_factor)

         CALL dbcsr_memtype_setup(memtype_abpanel_2, has_pool=.TRUE., &
                                  acc_hostalloc=.TRUE., acc_devalloc=.TRUE., acc_stream=stream_2, &
                                  mpi=.TRUE., oversize_factor=default_resize_factor)

         !TODO: ensure capacity 2/3?
         CALL dbcsr_memtype_setup(memtype_trsbuffer_1, has_pool=.TRUE., &
                                  acc_hostalloc=.TRUE., acc_devalloc=.TRUE., acc_stream=stream_1)
         CALL dbcsr_memtype_setup(memtype_trsbuffer_2, has_pool=.TRUE., &
                                  acc_hostalloc=.TRUE., acc_devalloc=.TRUE., acc_stream=stream_2)
         CALL dbcsr_mempool_limit_capacity(memtype_trsbuffer_1%pool, capacity=1)
         CALL dbcsr_mempool_limit_capacity(memtype_trsbuffer_2%pool, capacity=1)
      END IF

      CALL dbcsr_memtype_setup(memtype_mpi_buffer, mpi=.TRUE.)
      CALL dbcsr_memtype_setup(memtype_mpi_product, mpi=.TRUE., has_pool=use_mempools)

      ! check parameters ---------------------------------------------------------
      transa_l = transa
      transb_l = transb
      CALL uppercase(transa_l)
      CALL uppercase(transb_l)
      IF (transa_l .NE. dbcsr_no_transpose .AND. &
          transa_l .NE. dbcsr_transpose .AND. &
          transa_l .NE. dbcsr_conjugate_transpose) &
         DBCSR_ABORT("Invalid transa_l = "//transa_l)

      IF (transb_l .NE. dbcsr_no_transpose .AND. &
          transb_l .NE. dbcsr_transpose .AND. &
          transb_l .NE. dbcsr_conjugate_transpose) &
         DBCSR_ABORT("Invalid transb_l = "//transb_l)

      IF (dbg) THEN
         WRITE (*, *) '========== MULTIPLICATION ========================'
         CALL dbcsr_verify_matrix(matrix_a)
         CALL dbcsr_verify_matrix(matrix_b)
         CALL dbcsr_verify_matrix(matrix_c)
         WRITE (*, *) routineN//" ABC checksums", &
            dbcsr_checksum(matrix_a), &
            dbcsr_checksum(matrix_b), &
            dbcsr_checksum(matrix_c)
         IF (dbg) THEN
            CALL dbcsr_print(matrix_a, nodata=.TRUE.)
            CALL dbcsr_print(matrix_b, nodata=.TRUE.)
            CALL dbcsr_print(matrix_c, nodata=.TRUE.)
         END IF
      END IF

      ! transpose/conjg left and/or right matrices if needed
      transpose_left = .FALSE.
      SELECT CASE (transa_l)
      CASE (dbcsr_no_transpose)
         matrix_left = matrix_a
         transpose_left = .FALSE.
      CASE (dbcsr_transpose)
         matrix_left = dbcsr_type()
         IF (dbcsr_get_matrix_type(matrix_a) .EQ. dbcsr_type_antisymmetric) THEN
            !
            ! For antisymmetric matrix, we need to do a hard copy
            ! shallow_data_copy=.TRUE. does not handle properly antisymm matrices
            CALL dbcsr_new_transposed(matrix_left, matrix_a, &
                                      shallow_data_copy=.FALSE., redistribute=.FALSE., &
                                      transpose_distribution=.FALSE.)
         ELSE
            CALL dbcsr_new_transposed(matrix_left, matrix_a, &
                                      shallow_data_copy=.TRUE., redistribute=.FALSE., &
                                      transpose_distribution=.FALSE.)
         END IF
         transpose_left = .TRUE.
      CASE (dbcsr_conjugate_transpose)
         matrix_left = dbcsr_type()
         CALL dbcsr_new_transposed(matrix_left, matrix_a, &
                                   shallow_data_copy=.FALSE., redistribute=.FALSE., &
                                   transpose_distribution=.FALSE.)
         CALL dbcsr_conjg(matrix_left)
         transpose_left = .TRUE.
      CASE DEFAULT
         DBCSR_ABORT("wrong transa_l = "//transa_l)
      END SELECT

      transpose_right = .FALSE.
      SELECT CASE (transb_l)
      CASE (dbcsr_no_transpose)
         matrix_right = matrix_b
         transpose_right = .FALSE.
      CASE (dbcsr_transpose)
         matrix_right = dbcsr_type()
         IF (dbcsr_get_matrix_type(matrix_b) .EQ. dbcsr_type_antisymmetric) THEN
            !
            ! For antisymmetric matrix, we need to do a hard copy
            ! shallow_data_copy=.TRUE. does not handle properly antisymm matrices
            CALL dbcsr_new_transposed(matrix_right, matrix_b, &
                                      shallow_data_copy=.FALSE., redistribute=.FALSE., &
                                      transpose_distribution=.FALSE.)
         ELSE
            CALL dbcsr_new_transposed(matrix_right, matrix_b, &
                                      shallow_data_copy=.TRUE., redistribute=.FALSE., &
                                      transpose_distribution=.FALSE.)
         END IF
         transpose_right = .TRUE.
      CASE (dbcsr_conjugate_transpose)
         matrix_right = dbcsr_type()
         CALL dbcsr_new_transposed(matrix_right, matrix_b, &
                                   shallow_data_copy=.FALSE., redistribute=.FALSE., &
                                   transpose_distribution=.FALSE.)
         CALL dbcsr_conjg(matrix_right)
         transpose_right = .TRUE.
      CASE DEFAULT
         DBCSR_ABORT("wrong transb_l = "//transb_l)
      END SELECT
      !
      ! Ensure matrix compatibility.
      IF (.NOT. array_equality(matrix_c%row_blk_offset, matrix_left%row_blk_offset)) &
         DBCSR_ABORT("C/A rows not equal")
      IF (.NOT. array_equality(matrix_c%col_blk_offset, matrix_right%col_blk_offset)) &
         DBCSR_ABORT("C/B columns not equal")
      IF (.NOT. array_equality(matrix_left%col_blk_offset, matrix_right%row_blk_offset)) &
         DBCSR_ABORT("A cols/B rows not equal")
      !
      ! No dense multiplication when filtering is used.
      use_dense_mult = dbcsr_cfg%mm_densification .AND. (.NOT. PRESENT(filter_eps))
      !
      mp_obj = dbcsr_distribution_mp(matrix_c%dist)
      numnodes = dbcsr_mp_numnodes(mp_obj)
      nprows = dbcsr_mp_nprows(mp_obj)
      npcols = dbcsr_mp_npcols(mp_obj)
      !
      ! 3D layers
      CALL make_layers_3D_C_reduction(dbcsr_cfg%num_layers_3D, mp_obj)
      !
      ! No dense multiplication when RMA is used.
      IF (dbcsr_cfg%use_mpi_rma) THEN
         use_dense_mult = .FALSE.
      END IF
      ! we skip dense multiply for (anti)symmetric matrices (slowdown for S/H * C)
      IF (use_dense_mult) THEN
         IF (dbcsr_has_symmetry(matrix_left) .OR. &
             dbcsr_has_symmetry(matrix_right)) THEN
            use_dense_mult = .FALSE.
         ELSE
            use_dense_mult = dbcsr_may_be_dense(matrix_left, make_dense_occ_thresh) &
                             .AND. dbcsr_may_be_dense(matrix_right, make_dense_occ_thresh)
         END IF
      END IF
      ab_dense = use_dense_mult
      ! Use memory pools when no dense
      IF (.NOT. has_acc) THEN
         CALL dbcsr_memtype_setup(memtype_abpanel_1, has_pool=.NOT. ab_dense .AND. use_mempools, mpi=.TRUE.)
         CALL dbcsr_memtype_setup(memtype_abpanel_2, has_pool=.NOT. ab_dense .AND. use_mempools, mpi=.TRUE.)
      END IF
      !
      ! Submatrix selection
      f_row = 1
      l_row = dbcsr_nfullrows_total(matrix_c)
      f_col = 1
      l_col = dbcsr_nfullcols_total(matrix_c)
      f_k = 1
      l_k = dbcsr_nfullcols_total(matrix_left)
      IF (PRESENT(first_row)) THEN
         IF (first_row .LT. 1 .OR. first_row .GT. dbcsr_nfullrows_total(matrix_c)) &
            DBCSR_ABORT("Invalid first row specified")
         f_row = first_row
      END IF
      IF (PRESENT(last_row)) THEN
         IF (last_row .GT. dbcsr_nfullrows_total(matrix_c)) &
            DBCSR_ABORT("Invalid last row specified")
         l_row = last_row
      END IF
      IF (PRESENT(first_column)) THEN
         IF (first_column .LT. 1 .OR. first_column .GT. dbcsr_nfullcols_total(matrix_c)) &
            DBCSR_ABORT("Invalid first col specified")
         f_col = first_column
      END IF
      IF (PRESENT(last_column)) THEN
         IF (last_column .GT. dbcsr_nfullcols_total(matrix_c)) &
            DBCSR_ABORT("Invalid last column specified (C)")
         IF (last_column .GT. dbcsr_nfullcols_total(matrix_right)) &
            DBCSR_ABORT("Invalid last column specified (B)")
         l_col = last_column
      END IF
      IF (PRESENT(first_k)) THEN
         IF (first_k .LT. 1 .OR. first_k .GT. dbcsr_nfullcols_total(matrix_left)) &
            DBCSR_ABORT("Invalid first k specified (A)")
         f_k = first_k
      END IF
      IF (PRESENT(last_k)) THEN
         IF (last_k .GT. dbcsr_nfullcols_total(matrix_left)) &
            DBCSR_ABORT("Invalid last k specified (A)")
         l_k = last_k
      END IF
      !
      ! update statistics (we count marketing flops per MPI rank)
      dbcsr_mpi_statistics%last_mpi_ranks_used = numnodes
      marketing_flops = marketing_flops + &
                        (2.0*(l_row - f_row + 1.0)*(l_col - f_col + 1.0)/numnodes)*(l_k - f_k + 1.0)
      !
      ! Now optimize the default submatrix selection values away
      IF (f_row .EQ. 1) f_row = 0
      IF (l_row .EQ. dbcsr_nfullrows_total(matrix_left)) l_row = 0
      IF (f_col .EQ. 1) f_col = 0
      ! The last column must be set if the right and product matrices
      ! differ.
      l_col = MIN(l_col, dbcsr_nfullcols_total(matrix_right))
      l_col = MIN(l_col, dbcsr_nfullcols_total(matrix_c))
      IF (f_col .LE. 1 .AND. &
          l_col .EQ. dbcsr_nfullcols_total(matrix_right) .AND. &
          dbcsr_nfullcols_total(matrix_right) .EQ. &
          dbcsr_nfullcols_total(matrix_c)) l_col = 0
      IF (f_k .EQ. 1) f_k = 0
      IF (l_k .EQ. dbcsr_nfullcols_total(matrix_left)) l_k = 0
      IF (.NOT. PRESENT(last_column) .AND. &
          .NOT. array_equality(matrix_right%col_blk_size, &
                               matrix_c%col_blk_size)) THEN
         l_col = MIN(dbcsr_nfullcols_total(matrix_right), &
                     dbcsr_nfullcols_total(matrix_c))
      END IF
      IF (f_row .GT. l_row .AND. l_row .GT. 0) &
         DBCSR_ABORT("Last row smaller than first row")
      IF (f_col .GT. l_col .AND. l_col .GT. 0) &
         DBCSR_ABORT("Last col smaller than first col")
      !
      ! Product data needs to be retained when
      ! * beta != 0; or
      ! * there is column limiting (l_col > 0) and the limiting column
      !   is less than the number of full columns in the product matrix
      keep_sparsity = .FALSE.
      IF (PRESENT(retain_sparsity)) keep_sparsity = retain_sparsity
      !
      keep_product_data = keep_sparsity &
                          .OR. .NOT. dbcsr_scalar_are_equal(beta, dbcsr_scalar_zero(beta%data_type)) &
                          .OR. (l_col .GT. 0 .AND. l_col .LT. dbcsr_nfullcols_total(matrix_c)) &
                          .OR. (l_row .GT. 0 .AND. l_row .LT. dbcsr_nfullrows_total(matrix_c))
      !
      IF (.NOT. dbcsr_scalar_are_equal(beta, dbcsr_scalar_one(beta%data_type)) .AND. keep_product_data) THEN
         CALL dbcsr_scale(matrix_c, alpha_scalar=beta, &
                          limits=(/f_row, l_row, f_col, l_col/))
      END IF
      !
      ! The index of the product matrix is twiddled into canonical form
      ! if it is (anti)symmetric (i.e., rows and columns are where the
      ! row/column distributions say they are). Doing this in advance
      ! makes the local multiply more efficient.
      IF (dbcsr_has_symmetry(matrix_c)) THEN
         product_reindex = .TRUE.
      ELSE
         product_reindex = .FALSE.
      END IF
      ! Product can not be made dense; however, A & B may still be made
      ! dense unless previously determined otherwise.
      IF (product_reindex .OR. keep_sparsity) THEN
         use_dense_mult = .FALSE.
      END IF
      !
      ! The thread distribution must reflect the current (possibly
      ! dense) distribution
      thread_dist_force = .FALSE.
      IF (.NOT. dbcsr_distribution_has_threads(matrix_c%dist)) THEN
         release_tdist = .TRUE.
         CALL dbcsr_distribution_make_threads(matrix_c%dist)
      ELSE
         release_tdist = .FALSE.
         ! Make sure matrix_c thread dist == matrix_left thread dist
         ! This is currently a workaround
         IF (dbcsr_distribution_has_threads(matrix_left%dist)) THEN
            matrix_c_thread_dist = matrix_c%dist%d%thread_dist
            matrix_c%dist%d%thread_dist = matrix_left%dist%d%thread_dist
            CALL array_hold(matrix_left%dist%d%thread_dist)
            thread_dist_force = .TRUE.
         END IF
      END IF
      !
      ! Compute number of images (rows and columns)
      nimages_left_rows = dbcsr_mp_nprows(dbcsr_distribution_mp(matrix_left%dist))
      nimages_match = dbcsr_distribution_get_num_images_1d( &
                      dbcsr_nfullcols_total(matrix_left), &
                      dbcsr_nblkcols_total(matrix_left), &
                      dbcsr_mp_nprows(dbcsr_distribution_mp(matrix_left%dist)), &
                      dbcsr_mp_npcols(dbcsr_distribution_mp(matrix_left%dist)))
      nimages_right_cols = dbcsr_mp_npcols(dbcsr_distribution_mp(matrix_right%dist))
      !
      ! Create imaged distributions for the multiply.
      CALL dbcsr_create_image_dist(rdist_right, matrix_right%dist, &
                                   match_row_nbins=dbcsr_mp_npcols(dbcsr_distribution_mp(matrix_left%dist)), &
                                   match_col_nbins=npcols, &
                                   match_col_pdist=dbcsr_distribution_col_dist(matrix_c%dist), &
                                   nimages_rows=nimages_match, &
                                   nimages_cols=nimages_right_cols)
      !
      CALL dbcsr_create_image_dist(rdist_left, matrix_left%dist, &
                                   match_row_pdist=dbcsr_distribution_row_dist(matrix_c%dist), &
                                   match_row_nbins=nprows, &
                                   match_col_pdist=dbcsr_distribution_row_dist(rdist_right%i%main), &
                                   match_col_idist=array_data(rdist_right%i%row_image), &
                                   match_col_nbins=dbcsr_mp_nprows(dbcsr_distribution_mp(matrix_right%dist)), &
                                   nimages_rows=nimages_left_rows, &
                                   nimages_cols=nimages_match)
      !
      IF (ab_dense) THEN
         CALL dbcsr_make_dists_dense(dbcsr_distribution(matrix_c), &
                                     rdist_left, rdist_right, dense_product_distribution, &
                                     dense_rdist_left, dense_rdist_right,.NOT. use_dense_mult, &
                                     m_map, k_vmap, n_map, matrix_c%row_blk_size)
         CALL make_sizes_dense(matrix_c%row_blk_size, m_map, &
                               dbcsr_distribution_nrows(dense_product_distribution), &
                               dense_row_sizes)
         CALL make_sizes_dense(matrix_c%col_blk_size, n_map, &
                               dbcsr_distribution_ncols(dense_product_distribution), &
                               dense_col_sizes)
         CALL make_sizes_dense(matrix_right%row_blk_size, k_vmap, &
                               dbcsr_distribution_nrows(dense_rdist_right%i%main), &
                               dense_k_sizes)
      END IF
      !
      IF (use_dense_mult .AND. .NOT. ab_dense) &
         DBCSR_ABORT("Wrong logic when making dense matrices.")
      IF (use_dense_mult) THEN
         old_product_row_blk_offsets = matrix_c%row_blk_offset
         old_product_col_blk_offsets = matrix_c%col_blk_offset
         old_product_row_blk_sizes = matrix_c%row_blk_size
         old_product_col_blk_sizes = matrix_c%col_blk_size
         CALL array_hold(old_product_row_blk_offsets)
         CALL array_hold(old_product_col_blk_offsets)
         CALL array_hold(old_product_row_blk_sizes)
         CALL array_hold(old_product_col_blk_sizes)
         old_product_distribution = dbcsr_distribution(matrix_c)
         CALL dbcsr_distribution_hold(old_product_distribution)
         product_matrix = dbcsr_type()
         CALL dbcsr_make_dense(matrix_c, product_matrix, &
                               dense_product_distribution, &
                               dense_row_sizes, dense_col_sizes, &
                               m_map, n_map)
      ELSE
         product_matrix = dbcsr_type()
         CALL dbcsr_copy(product_matrix, matrix_c, shallow_data=.TRUE.)
      END IF
      IF (ab_dense) THEN
         CALL dbcsr_distribution_release(dense_product_distribution)
      END IF
      !
      ! This is needed to build the hash tables because they are
      ! locally indexed.
      CALL dbcsr_reset_locals(product_matrix)
      !
      IF (debug_mod) THEN
         WRITE (*, *) routineN//" Matrices ", dbcsr_get_matrix_type(matrix_a), &
            dbcsr_get_matrix_type(matrix_b), dbcsr_get_matrix_type(matrix_c)
         WRITE (*, *) routineN//" Matrices ", transa_l, transb_l, "keep", keep_product_data
      END IF
      IF (keep_product_data) THEN
         IF (product_reindex) THEN
            IF (debug_mod) WRITE (*, *) routineN//" Making canonical index"
            CALL dbcsr_make_index_canonical(product_matrix)
         END IF
         IF (ASSOCIATED(product_matrix%wms)) &
            DBCSR_ABORT("Product matrix should be finalized!")
         CALL dbcsr_make_untransposed_blocks(product_matrix)
!$OMP PARALLEL &
!$OMP DEFAULT (NONE) SHARED (product_matrix)
         ! For the multiply logic to work correctly, existing data must
         ! be added only after the index has been transformed into the
         ! canonical form.
         CALL dbcsr_add_wm_from_matrix(product_matrix)
!$OMP END PARALLEL
      ELSE
!$OMP PARALLEL DEFAULT(NONE) PRIVATE(ithread) &
!$OMP SHARED(product_matrix, memtype_product_wm)
         ithread = 0
!$       ithread = OMP_GET_THREAD_NUM()
         CALL dbcsr_work_create(product_matrix, work_mutable=.FALSE., &
                                memory_type=memtype_product_wm(ithread)%p)
!$OMP END PARALLEL
      END IF
      !
      IF (dbcsr_cfg%use_mpi_rma) THEN
         ! Check for previous multiplication completeness
         IF (request_sync_mult .NE. mp_request_null) THEN
            CALL timeset(routineN//"_sync_mult", handle2)
            CALL mp_wait(request_sync_mult)
            CALL timestop(handle2)
            request_sync_mult = mp_request_null
         END IF
         !
         ! Left buffer images
         CALL dbcsr_make_buffers(matrix_left, rdist_left, .TRUE., &
                                 f_row, l_row, f_k, l_k, &
                                 PRESENT(filter_eps))
         !
         ! Right buffer images
         CALL dbcsr_make_buffers(matrix_right, rdist_right, .FALSE., &
                                 f_k, l_k, f_col, l_col, &
                                 PRESENT(filter_eps), &
                                 alpha)
      ELSE
         product_matrix%nblks = 0
         product_matrix%nze = 0
         product_matrix%row_p(:) = 0
         CALL dbcsr_data_set_size_referenced(product_matrix%data_area, 0)
         product_matrix%valid = .FALSE.
         !
         ! Right images
         CALL make_m2s(matrix_right, m2s_right, rdist_right, dense_rdist_right, &
                       use_dense_mult, ab_dense, "R", &
                       f_k, l_k, f_row, l_row, f_col, l_col, &
                       dense_k_sizes, dense_col_sizes, &
                       k_vmap, m_map, n_map, &
                       alpha)
         !
         ! Left images
         CALL make_m2s(matrix_left, m2s_left, rdist_left, dense_rdist_left, &
                       use_dense_mult, ab_dense, "L", &
                       f_k, l_k, f_row, l_row, f_col, l_col, &
                       dense_row_sizes, dense_k_sizes, &
                       k_vmap, m_map, n_map)
      END IF
      !
      IF (ab_dense) THEN
         CALL array_release(k_vmap)
         CALL array_release(dense_row_sizes)
         CALL array_release(dense_col_sizes)
         CALL array_release(dense_k_sizes)
      END IF
      !
      ! The limits were already used.  Reset them.
      f_row = 0; l_row = 0
      f_col = 0; l_col = 0
      f_k = 0; l_k = 0
      !
      my_flop = 0
      IF (dbcsr_cfg%use_mpi_rma) THEN
         CALL multiply_3D(rdist_left, rdist_right, &
                          matrix_left, matrix_right, product_matrix, &
                          retain_sparsity=retain_sparsity, &
                          filter_eps=filter_eps, &
                          flop=my_flop, keep_product_data=keep_product_data)
      ELSE
         CALL multiply_cannon(m2s_left, m2s_right, product_matrix, &
                              retain_sparsity=retain_sparsity, &
                              filter_eps=filter_eps, &
                              flop=my_flop, keep_product_data=keep_product_data)
         CALL dbcsr_finalize(product_matrix, reshuffle=PRESENT(filter_eps) .AND. .NOT. keep_sparsity)
      END IF
      !
      ! RMA implementation algorithm has to synchronize at the end of each multiplication
      comm = dbcsr_mp_group(dbcsr_distribution_mp(dbcsr_distribution(matrix_c)))
      IF (PRESENT(flop)) THEN
         ! return the average number of flops per MPI rank.
         ! Variance (which is fairly large) could be computed as well.
         CALL timeset(routineN//"_mpsum_flop", handle2)
         numnodes = dbcsr_mp_numnodes(dbcsr_distribution_mp(dbcsr_distribution(matrix_c)))
         CALL mp_sum(my_flop, comm)
         IF (PRESENT(flop)) THEN
            flop = (my_flop + numnodes - 1)/numnodes
         END IF
         CALL timestop(handle2)
      ELSEIF (dbcsr_cfg%use_mpi_rma) THEN
         CALL mp_isync(comm, request_sync_mult)
      END IF
      !
      IF (release_tdist) THEN
         CALL dbcsr_distribution_no_threads(product_matrix%dist)
      ELSEIF (thread_dist_force) THEN
         ! Restore matrix_c thread-dist
         matrix_c%dist%d%thread_dist = matrix_c_thread_dist
         CALL array_release(matrix_left%dist%d%thread_dist)
      END IF
      IF (transpose_left) CALL dbcsr_release(matrix_left)
      IF (transpose_right) CALL dbcsr_release(matrix_right)
      !
      CALL dbcsr_release_locals(product_matrix)
      ! The index of the product matrix is reset to the CP2K form if it
      ! was previously set to the canonical form.
      IF (product_reindex) THEN
         IF (debug_mod) WRITE (*, *) routineN//" Making CP2K index"
         CALL dbcsr_make_index_canonical(product_matrix, cp2k=.TRUE.)
      END IF
      IF (use_dense_mult) THEN
         CALL dbcsr_release(matrix_c)
         matrix_c = dbcsr_type()
         CALL dbcsr_make_undense(product_matrix, matrix_c, &
                                 old_product_distribution, &
                                 old_product_row_blk_offsets, old_product_col_blk_offsets, &
                                 old_product_row_blk_sizes, old_product_col_blk_sizes, &
                                 m_map, n_map)
         CALL dbcsr_release(product_matrix)
         CALL array_release(old_product_row_blk_offsets)
         CALL array_release(old_product_col_blk_offsets)
         CALL array_release(old_product_row_blk_sizes)
         CALL array_release(old_product_col_blk_sizes)
         CALL dbcsr_distribution_release(old_product_distribution)
      ELSE
         CALL dbcsr_release(matrix_c)
         matrix_c = dbcsr_type()
         CALL dbcsr_copy(matrix_c, product_matrix, shallow_data=.TRUE.)
         CALL dbcsr_release(product_matrix)
      END IF
      !
      IF (.NOT. dbcsr_cfg%use_mpi_rma) THEN
         CALL dbcsr_destroy_array(m2s_left)
         DEALLOCATE (m2s_left)
         CALL dbcsr_destroy_array(m2s_right)
         DEALLOCATE (m2s_right)
      END IF
      !
      CALL dbcsr_image_dist_release(rdist_left)
      CALL dbcsr_image_dist_release(rdist_right)
      IF (ab_dense) THEN
         CALL array_release(m_map)
         CALL array_release(n_map)
      END IF
      !
      ! To support the canonical multiply (all non-transposed blocks),
      ! blocks may have to be transposed according to the CP2K
      ! triangular index.
      CALL dbcsr_make_untransposed_blocks(matrix_c)
      !
      IF (debug_mod .OR. careful_mod) THEN
         IF (debug_mod) &
            WRITE (*, *) routineN//" Use dense mult, symm", &
            use_dense_mult, ab_dense, dbcsr_has_symmetry(matrix_c)
         CALL dbcsr_verify_matrix(matrix_c)
         IF (debug_mod) THEN
            cs = dbcsr_checksum(matrix_c)
            WRITE (*, *) routineN//" Multiplication", &
               num_multiplications, " Product checksum", cs
         END IF
      END IF

      ! This tends to trigger only when all of these conditions are fulfilled:
      !  - transa=="T"
      !  - matrix_c contains already blocks and beta is not zero
      !  - GPU-acceleration is enabled
      !  - multiple OpenMP threads are used
      IF (INT(matrix_c%nblks, KIND=int_8) > &
          INT(SIZE(array_data(matrix_c%row_blk_size)), KIND=int_8)* &
          INT(SIZE(array_data(matrix_c%col_blk_size)), KIND=int_8)) &
         DBCSR_ABORT("Bug: Matrix contains too many blocks")
      output_unit = default_output_unit
      num_multiplications = num_multiplications + 1
      CALL timestop(handle)
   END SUBROUTINE dbcsr_multiply_generic