# 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.