# Matrix Transpose with Local Array¶

This section extends the discussion in Tiled Matrix Transpose, where only loop tiling is considered. Here, we combine loop tiling with RAJA::LocalArray objects which enable us to store data for each tile in CPU stack-allocated arrays or GPU thread local and shared memory to be used within kernels. For more information about RAJA::LocalArray, please see Local Array.

Key RAJA features shown in this example include:

• RAJA::kernel_param method with multiple lambda expressions
• RAJA::statement::Tile type
• RAJA::statement::ForICount type
• RAJA::LocalArray
• Specifying lambda arguments through statements

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. The algorithm 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 start with a non-RAJA C++ implementation to show the algorithm pattern. 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 A. 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.

  const int N_r = 267;
const int N_c = 251;

const int TILE_DIM = 16;

const int outer_Dimc = (N_c - 1) / TILE_DIM + 1;
const 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++ 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 Version of Tiled Loops with Local Array¶

RAJA provides mechanisms to tile loops and use local arrays in kernels so that algorithm patterns like we just described can be implemented with RAJA. A RAJA::LocalArray type specifies an object whose memory is created inside a kernel using a RAJA::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: array data type, 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 implementations of matrix transpose with RAJA.

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::RangeSegment(0, N_c),
RAJA::RangeSegment(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);

});


The RAJA::statement::Tile types in the execution policy define tiling of the outer ‘row’ (iteration space tuple index ‘1’) and ‘col’ (iteration space tuple index ‘0’) loops, including tile sizes (RAJA::tile_fixed types) and loop execution policies. Next, the RAJA::statement::InitLocalMem type initializes the local stack 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 which can be used in multiple lambdas in the kernel. Finally, we have two sets of nested inner loops for reading the input matrix entries into the local array and writing them out to the output matrix transpose. The inner bodies of each of these loop nests are identified by lambda expression arguments ‘0’ and ‘1’, respectively.

Note that the loops over tiles use RAJA::statement::ForICount types rather than RAJA::statement::For types that we have seen in other nested loop examples. The RAJA::statement::ForICount type generates local tile indices that are passed to lambda loop body expressions. 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; this is similar to RAJA::statement::For usage. 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. The loop execution policy template argument that follows works the same as in RAJA::statement::For usage. 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. The first argument is a tuple of iteration spaces that define the iteration ranges for the level in the loop nest. Again, the first integer parameters given to the RAJA::statement::Tile and RAJA::statement::ForICount types identify the tuple entry they apply to. The second argument 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 that each lambda expression takes five arguments. The first two are the matrix 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.

## RAJA::kernel Version of Tiled Loops with Local Array Specifying Lambda Arguments¶

The second RAJA variant works the same as the one above. The main differences between the two variants is due to the fact that in this second one, we use RAJA::statement::Lambda types to indicate which arguments each lambda 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::RangeSegment(0, N_c),
RAJA::RangeSegment(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 args; 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 are not needed.

The file RAJA/examples/tut_matrix-transpose-local-array.cpp` contains the complete working example code for the examples described in this section along with OpenMP, CUDA, and HIP variants.