Tiled Matrix Transpose with Local Array¶

This section extends the discussion in Tiled Matrix Transpose by adding local array objects which are used to store data for each tile in CPU stack-allocated arrays or GPU thread local and shared memory to be used within kernels.

There are exercise files RAJA/exercises/kernel-matrix-transpose-local-array.cpp and RAJA/exercises/launch-matrix-transpose-local-array.cpp for you to work through if you wish to get some practice with RAJA. The files RAJA/exercises/kernel-matrix-transpose-local-array._solutioncpp and RAJA/exercises/launch-matrix-transpose-local-array_solution.cpp contain complete working code for the examples. You can use the solution files to check your work and for guidance if you get stuck. To build the exercises execute make (kernel/launch)-matrix-transpose-local-array and make (kernel/launch)-matrix-transpose-local-array_solution from the build directory.

Key RAJA features shown in this example are:

RAJA::kernel_param method and execution policy usage with multiple lambda expressions

RAJA::statement::Tile type for loop tiling

RAJA::statement::ForICount type for generating local tile indices

RAJA::LocalArray type for thread-local tile memory arrays

RAJA::launch kernel execution interface

RAJA::expt::tile type for loop tiling

RAJA::expt::loop_icount method to generate local tile indices for Launch

RAJA_TEAM_SHARED macro for thread-local tile memory arrays

As in Tiled Matrix Transpose, this example computes the transpose of an input matrix \(A\) of size \(N_r \times N_c\) and stores the result in a second matrix \(At\) of size \(N_c \times N_r\). The operation uses a local memory tiling algorithm, which tiles the outer loops and iterates over tiles in inner loops. The algorithm first loads input matrix entries into a local two-dimensional array for a tile, and then reads from the tile swapping the row and column indices to generate the output matrix.

We choose tile dimensions smaller than the dimensions of the matrix and note that it is not necessary for the tile dimensions to divide evenly the number of rows and columns in the matrix. As in the Tiled Matrix Transpose example, we start by defining the number of rows and columns in the matrices, the tile dimensions, and the number of tiles.

  constexpr int N_r = 267;
  constexpr int N_c = 251;

  constexpr int TILE_DIM = 16;

  constexpr int outer_Dimc = (N_c - 1) / TILE_DIM + 1;
  constexpr int outer_Dimr = (N_r - 1) / TILE_DIM + 1;

We also use RAJA View objects to simplify the multi-dimensional indexing as in the Tiled Matrix Transpose example.

  RAJA::View<int, RAJA::Layout<DIM>> Aview(A, N_r, N_c);
  RAJA::View<int, RAJA::Layout<DIM>> Atview(At, N_c, N_r);

The complete sequential C-style implementation of the tiled transpose operation using a stack-allocated local array for the tiles is:

  //
  // (0) Outer loops to iterate over tiles
  //
  for (int by = 0; by < outer_Dimr; ++by) {
    for (int bx = 0; bx < outer_Dimc; ++bx) {

      // Stack-allocated local array for data on a tile
      int Tile[TILE_DIM][TILE_DIM];

      //
      // (1) Inner loops to read input matrix tile data into the array
      //
      //     Note: loops are ordered so that input matrix data access
      //           is stride-1.
      //
      for (int ty = 0; ty < TILE_DIM; ++ty) {
        for (int tx = 0; tx < TILE_DIM; ++tx) {

          int col = bx * TILE_DIM + tx;  // Matrix column index
          int row = by * TILE_DIM + ty;  // Matrix row index

          // Bounds check
          if (row < N_r && col < N_c) {
            Tile[ty][tx] = Aview(row, col);
          }
        }
      }

      //
      // (2) Inner loops to write array data into output array tile
      //
      //     Note: loop order is swapped from above so that output matrix
      //           data access is stride-1.
      //
      for (int tx = 0; tx < TILE_DIM; ++tx) {
        for (int ty = 0; ty < TILE_DIM; ++ty) {

          int col = bx * TILE_DIM + tx;  // Matrix column index
          int row = by * TILE_DIM + ty;  // Matrix row index

          // Bounds check
          if (row < N_r && col < N_c) {
            Atview(col, row) = Tile[ty][tx];
          }
        }
      }

    }
  }

Note

To prevent indexing out of bounds, when the tile dimensions do not divide evenly the matrix dimensions, we use a bounds check in the inner loops.
For efficiency, we order the inner loops so that reading from the input matrix and writing to the output matrix both use stride-1 data access.

`RAJA::kernel` Variants¶

The RAJA::kernel interface provides mechanisms to tile loops and use local arrays in kernels so that algorithm patterns like the C-style kernel above can be implemented with RAJA. When using RAJA::kernel, a RAJA::LocalArray type specifies an object whose memory is created inside a kernel using a statement type in a RAJA kernel execution policy. The local array data is only usable within the kernel. See Local Array for more information.

RAJA::kernel methods also support loop tiling statements which determine the number of tiles needed to perform an operation based on tile size and extent of the corresponding iteration space. Moreover, lambda expressions for the kernel will not be invoked for iterations outside the bounds of an iteration space when tile dimensions do not divide evenly the size of the iteration space; thus, no conditional checks on loop bounds are needed inside inner loops.

For the RAJA version of the matrix transpose kernel above, we define the type of the RAJA::LocalArray used for matrix entries in a tile and create an object to represent it:

  using TILE_MEM =
    RAJA::LocalArray<int, RAJA::Perm<0, 1>, RAJA::SizeList<TILE_DIM, TILE_DIM>>;
  TILE_MEM Tile_Array;

The template parameters that define the type are: the array data type, the data stride permutation for the array indices (here the identity permutation is given, so the default RAJA conventions apply; i.e., the rightmost array index will be stride-1), and the array dimensions. Next, we compare two RAJA::kernel implementations of the matrix transpose operation.

The complete RAJA sequential CPU variant with kernel execution policy and kernel is:

  using SEQ_EXEC_POL_I =
    RAJA::KernelPolicy<
      RAJA::statement::Tile<1, RAJA::tile_fixed<TILE_DIM>, RAJA::loop_exec,
        RAJA::statement::Tile<0, RAJA::tile_fixed<TILE_DIM>, RAJA::loop_exec,

          RAJA::statement::InitLocalMem<RAJA::cpu_tile_mem, RAJA::ParamList<2>,

          RAJA::statement::ForICount<1, RAJA::statement::Param<0>, RAJA::loop_exec,
            RAJA::statement::ForICount<0, RAJA::statement::Param<1>, RAJA::loop_exec,
              RAJA::statement::Lambda<0>
            >
          >,

          RAJA::statement::ForICount<0, RAJA::statement::Param<1>, RAJA::loop_exec,
            RAJA::statement::ForICount<1, RAJA::statement::Param<0>, RAJA::loop_exec,
              RAJA::statement::Lambda<1>
            >
          >

          >
        >
      >
    >;

  RAJA::kernel_param<SEQ_EXEC_POL_I>( 
    RAJA::make_tuple(RAJA::TypedRangeSegment<int>(0, N_c),
                     RAJA::TypedRangeSegment<int>(0, N_r)),

    RAJA::make_tuple((int)0, (int)0, Tile_Array),

    [=](int col, int row, int tx, int ty, TILE_MEM &Tile_Array) {
      Tile_Array(ty, tx) = Aview(row, col);
    },

    [=](int col, int row, int tx, int ty, TILE_MEM &Tile_Array) {
      Atview(col, row) = Tile_Array(ty, tx);
    }

  );

In the execution policy, the RAJA::statement::Tile types define tiling of the outer ‘row’ (iteration space tuple index ‘1’) and ‘col’ (iteration space tuple index ‘0’) loops, as well as tile sizes (RAJA::tile_fixed types) and loop execution policies. Next, the RAJA::statement::InitLocalMem type allocates the local tile array based on the memory policy type (here, we use RAJA::cpu_tile_mem for a CPU stack-allocated array). The RAJA::ParamList<2> parameter indicates that the local array object is associated with position ‘2’ in the parameter tuple argument passed to the RAJA::kernel_param method. The first two entries in the parameter tuple indicate storage for the local tile indices that are used in the two lambda expressions that comprise the kernel body. Finally, we have two sets of nested inner loops for reading the input matrix entries into the local tile array and writing them out to the output matrix transpose. The inner bodies of each of these loop nests are identified by lambda expression invocation statements RAJA::statement::Lambda<0> for the first lambda passed as an argument to the RAJA::kernel_param method and RAJA::statement::Lambda<1> for the second lambda argument.

Note that the loops within tiles use RAJA::statement::ForICount types rather than RAJA::statement::For types that we saw in the tiled matrix transpose example in Tiled Matrix Transpose. The RAJA::statement::ForICount type generates local tile indices that are passed to lambda loop body expressions to index into the local tile memory array. As the reader will observe, there is no local tile index computation needed in the lambdas for the RAJA version of the kernel as a result. The first integer template parameter for each RAJA::statement::ForICount type indicates the item in the iteration space tuple passed to the RAJA::kernel_param method to which it applies. The second template parameter for each RAJA::statement::ForICount type indicates the position in the parameter tuple passed to the RAJA::kernel_param method that will hold the associated local tile index. For more detailed discussion of RAJA loop tiling statement types, please see Loop Tiling.

Now that we have described the execution policy in some detail, let’s pull everything together by briefly walking though the call to the RAJA::kernel_param method, which is similar to RAJA::kernel but takes additional arguments needed to execute the operations involving local tile indices and the local memory array. The first argument is a tuple of iteration spaces that define the iteration ranges for the levels in the loop nest. Again, the first integer parameters given to the RAJA::statement::Tile and RAJA::statement::ForICount types identify the tuple entry to which they apply. The second argument:

RAJA::make_tuple((int)0, (int)0, Tile_Array)

is a tuple of data parameters that will hold the local tile indices and RAJA::LocalArray tile memory. The tuple entries are associated with various statements in the execution policy as we described earlier. Next, two lambda expression arguments are passed to the RAJA::kernel_param method for reading and writing the input and output matrix entries, respectively.

Note

RAJA::kernel_param accepts a parameter tuple argument after the iteration space tuple, which enables the parameters to be used in multiple lambda expressions in a kernel.

In the kernel, both lambda expressions take the same five arguments. The first two are the matrix global column and row indices associated with the iteration space tuple. The next three arguments correspond to the parameter tuple entries. The first two of these are the local tile indices used to access entries in the RAJA::LocalArray object memory. The last argument is a reference to the RAJA::LocalArray object itself.

The next RAJA::kernel_param variant we present works the same as the one above. It is different from the previous version since we include additional template parameters in the RAJA::statement::Lambda types to indicate which arguments each lambda expression takes and in which order. Here is the complete version including execution policy and kernel:

  using SEQ_EXEC_POL_II =
    RAJA::KernelPolicy<
      RAJA::statement::Tile<1, RAJA::tile_fixed<TILE_DIM>, RAJA::loop_exec,
        RAJA::statement::Tile<0, RAJA::tile_fixed<TILE_DIM>, RAJA::loop_exec,

          RAJA::statement::InitLocalMem<RAJA::cpu_tile_mem, RAJA::ParamList<0>,

          RAJA::statement::For<1, RAJA::loop_exec,
            RAJA::statement::For<0, RAJA::loop_exec,
              RAJA::statement::Lambda<0, Segs<0>, Segs<1>, Offsets<0>, Offsets<1>, Params<0> >
            >
          >,

          RAJA::statement::For<0, RAJA::loop_exec,
            RAJA::statement::For<1, RAJA::loop_exec,
              RAJA::statement::Lambda<1, Segs<0, 1>, Offsets<0, 1>, Params<0> >
            >
          >

          >
        >
      >
    >;

  RAJA::kernel_param<SEQ_EXEC_POL_II>( 
    RAJA::make_tuple(RAJA::TypedRangeSegment<int>(0, N_c),
                     RAJA::TypedRangeSegment<int>(0, N_r)),

    RAJA::make_tuple(Tile_Array),

    [=](int col, int row, int tx, int ty, TILE_MEM &Tile_Array) {
        Tile_Array(ty, tx) = Aview(row, col);
    },

    [=](int col, int row, int tx, int ty, TILE_MEM &Tile_Array) {
      Atview(col, row) = Tile_Array(ty, tx);
    }
  );

Here, the two RAJA::statement::Lambda types in the execution policy show two different ways to specify the segments (RAJA::Segs) associated with the matrix column and row indices. That is, we can use a Segs statement for each argument, or include multiple segment ids in one statement.

Note that we are using RAJA::statement::For types for the inner tile loops instead of RAJA::statement::ForICount` types used in the first variant. As a consequence of specifying lambda arguments, there are two main differences. The local tile indices are properly computed and passed to the lambda expressions as a result of the RAJA::Offsets types that appear in the lambda statement types. The RAJA::statement::Lambda type for each lambda shows the two ways to specify the local tile index arguments; we can use an Offsets statement for each argument, or include multiple segment ids in one statement. Lastly, there is only one entry in the parameter tuple in this case, the local tile array. The placeholders in the previous example are not needed.

Note

In this example, we need all five arguments in each lambda expression so the lambda expression argument lists are the same. Another use case for the template parameter argument specification described here is to be able to pass only the arguments used in a lambda expression. In particular when we use multiple lambda expressions to represent a kernel, each lambda can have a different argument lists from the others.

`RAJA::expt::launch` Variants¶

The RAJA::expt::launch interface provides mechanisms to tile loops and use local arrays in kernels to support algorithm patterns like the C-style kernel above. When, using RAJA::expt::launch, the RAJA_TEAM_SHARED macro is used to create a GPU shared memory array or a CPU stack memory array inside a kernel.

RAJA::expt::launch support methods for tiling over an iteration space using RAJA::expt::tile and RAJA::expt::loop_icount methods to tile loops and generate global iteration indices and local tile offsets. Moreover, lambda expressions for these methods will not be invoked for iterations outside the bounds of an iteration space when tile dimensions do not divide evenly the size of the iteration space; thus, no conditional checks on loop bounds are needed inside inner loops.

A complete RAJA sequential CPU variant with kernel execution policy and kernel is:

  using loop_pol_1 = RAJA::LoopPolicy<RAJA::loop_exec>;
  using launch_policy_1 = RAJA::LaunchPolicy<RAJA::seq_launch_t>;

  RAJA::launch<launch_policy_1>(
    RAJA::LaunchParams(), //LaunchParams may be empty when only running on the cpu
    [=] RAJA_HOST_DEVICE (RAJA::LaunchContext ctx) {

      RAJA::tile<loop_pol_1>(ctx, TILE_DIM, RAJA::TypedRangeSegment<int>(0, N_r), [&] (RAJA::TypedRangeSegment<int> const &row_tile) {

        RAJA::tile<loop_pol_1>(ctx, TILE_DIM, RAJA::TypedRangeSegment<int>(0, N_c), [&] (RAJA::TypedRangeSegment<int> const &col_tile) {

          RAJA_TEAM_SHARED double Tile_Array[TILE_DIM][TILE_DIM];

          RAJA::loop_icount<loop_pol_1>(ctx, row_tile, [&] (int row, int ty) {
            RAJA::loop_icount<loop_pol_1>(ctx, col_tile, [&] (int col, int tx) {

              Tile_Array[ty][tx] = Aview(row, col);

            });
          });

          RAJA::loop_icount<loop_pol_1>(ctx, col_tile, [&] (int col, int tx) {
            RAJA::loop_icount<loop_pol_1>(ctx, row_tile, [&] (int row, int ty) {

              Atview(col, row) = Tile_Array[ty][tx];

            });
          });

        });
      });

    });

Here, the RAJA::expt::tile method is used to create tilings of the outer ‘row’ and ‘col’ iteration spaces. The RAJA::expt::tile method takes an additional argument specifying the tile size for the corresponding loop. To traverse the tile, we use the RAJA::expt::loop_icount method, which is similar to the RAJA::ForICount statement used in a RAJA::kernel execution policy as shown above. A RAJA::expt::loop_icount method call will generate local tile index associated with the outer global index. The local tile index is necessary as we use it to read and write entries from/to global memory to RAJA_TEAM_SHARED memory array.

Tiled Matrix Transpose with Local Array¶

RAJA::kernel Variants¶

RAJA::expt::launch Variants¶

`RAJA::kernel` Variants¶

`RAJA::expt::launch` Variants¶