Tiled Matrix Multiplication

fn matmul_tiled[
    layout: Layout, size: Int
](
    output: LayoutTensor[mut=False, dtype, layout],
    a: LayoutTensor[mut=False, dtype, layout],
    b: LayoutTensor[mut=False, dtype, layout],
):
    local_row = thread_idx.y
    local_col = thread_idx.x
    tiled_row = block_idx.y * TPB + local_row
    tiled_col = block_idx.x * TPB + local_col

    a_shared = tb[dtype]().row_major[TPB, TPB]().shared().alloc()
    b_shared = tb[dtype]().row_major[TPB, TPB]().shared().alloc()

    var acc: output.element_type = 0

    # Iterate over tiles to compute matrix product
    @parameter
    for tile in range((size + TPB - 1) // TPB):
        # Load A tile - global row stays the same, col determined by tile
        if tiled_row < size and (tile * TPB + local_col) < size:
            a_shared[local_row, local_col] = a[
                tiled_row, tile * TPB + local_col
            ]

        # Load B tile - row determined by tile, global col stays the same
        if (tile * TPB + local_row) < size and tiled_col < size:
            b_shared[local_row, local_col] = b[
                tile * TPB + local_row, tiled_col
            ]

        barrier()

        # Matrix multiplication within the tile
        if tiled_row < size and tiled_col < size:

            @parameter
            for k in range(min(TPB, size - tile * TPB)):
                acc += a_shared[local_row, k] * b_shared[k, local_col]

        barrier()

    # Write out final result
    if tiled_row < size and tiled_col < size:
        output[tiled_row, tiled_col] = acc

The tiled matrix multiplication implementation demonstrates efficient handling of matrices \((9 \times 9)\) using small tiles \((3 \times 3)\). Here’s how it works:

1. Shared memory allocation

   Input matrices (9×9) - Perfect fit for (3×3) tiling:
   A = [0  1  2  3  4  5  6  7  8 ]    B = [0  2  4  6  8  10 12 14 16]
       [9  10 11 12 13 14 15 16 17]        [18 20 22 24 26 28 30 32 34]
       [18 19 20 21 22 23 24 25 26]        [36 38 40 42 44 46 48 50 52]
       [27 28 29 30 31 32 33 34 35]        [54 56 58 60 62 64 66 68 70]
       [36 37 38 39 40 41 42 43 44]        [72 74 76 78 80 82 84 86 88]
       [45 46 47 48 49 50 51 52 53]        [90 92 94 96 98 100 102 104 106]
       [54 55 56 57 58 59 60 61 62]        [108 110 112 114 116 118 120 122 124]
       [63 64 65 66 67 68 69 70 71]        [126 128 130 132 134 136 138 140 142]
       [72 73 74 75 76 77 78 79 80]        [144 146 148 150 152 154 156 158 160]
   
   Shared memory per block (3×3):
   a_shared[TPB, TPB]  b_shared[TPB, TPB]
   

2. Tile processing loop

   Number of tiles = 9 // 3 = 3 tiles (perfect division!)
   
   For each tile:
   1. Load tile from A and B
   2. Compute partial products
   3. Accumulate in register
   

3. Memory loading pattern

   • With perfect \((9 \times 9)\) tiling, bounds check is technically unnecessary but included for defensive programming and consistency with other matrix sizes.

        # Load A tile - global row stays the same, col determined by tile
        if tiled_row < size and (tile * TPB + local_col) < size:
            a_shared[local_row, local_col] = a[
                tiled_row, tile * TPB + local_col
            ]
     
        # Load B tile - row determined by tile, global col stays the same
        if (tile * TPB + local_row) < size and tiled_col < size:
            b_shared[local_row, local_col] = b[
                tile * TPB + local_row, tiled_col
            ]
     

4. Computation within tile

   for k in range(min(TPB, size - tile * TPB)):
       acc += a_shared[local_row, k] * b_shared[k, local_col]
   

   • Avoids shared memory bank conflicts:

     Bank Conflict Free (Good):        Bank Conflicts (Bad):
     Thread0: a_shared[0,k] b_shared[k,0]  Thread0: a_shared[k,0] b_shared[0,k]
     Thread1: a_shared[0,k] b_shared[k,1]  Thread1: a_shared[k,0] b_shared[1,k]
     Thread2: a_shared[0,k] b_shared[k,2]  Thread2: a_shared[k,0] b_shared[2,k]
     ↓                                     ↓
     Parallel access to different banks    Serialized access to same bank of b_shared
     (or broadcast for a_shared)           if shared memory was column-major
     

     Shared memory bank conflicts explained:

     • Left (Good): b_shared[k,threadIdx.x] accesses different banks, a_shared[0,k] broadcasts to all threads
     • Right (Bad): If b_shared were column-major, threads would access same bank simultaneously
     • Key insight: This is about shared memory access patterns, not global memory coalescing
     • Bank structure: Shared memory has 32 banks; conflicts occur when multiple threads access different addresses in the same bank simultaneously

5. Synchronization points

   barrier() after:
   1. Tile loading
   2. Tile computation
   

Key performance features:

• Processes \((9 \times 9)\) matrix using \((3 \times 3)\) tiles (perfect fit!)
• Uses shared memory for fast tile access
• Minimizes global memory transactions with coalesced memory access
• Optimized shared memory layout and access pattern to avoid shared memory bank conflicts

6. Result writing:

   if tiled_row < size and tiled_col < size:
      output[tiled_row, tiled_col] = acc
   

   • Defensive bounds checking included for other matrix sizes and tiling strategies
   • Direct assignment to output matrix
   • All threads write valid results

Key optimizations

1. Layout optimization:

   • Row-major layout for all tensors
   • Efficient 2D indexing

2. Memory access:

   • Coalesced global memory loads
   • Efficient shared memory usage

3. Computation:

   • Register-based accumulation i.e. var acc: output.element_type = 0
   • Compile-time loop unrolling via @parameter

This implementation achieves high performance through:

• Efficient use of LayoutTensor for memory access
• Optimal tiling strategy
• Proper thread synchronization
• Careful boundary handling

from gpu.memory import async_copy_wait_all
from layout.layout_tensor import copy_dram_to_sram_async


fn matmul_idiomatic_tiled[
    layout: Layout, size: Int
](
    output: LayoutTensor[mut=False, dtype, layout],
    a: LayoutTensor[mut=False, dtype, layout],
    b: LayoutTensor[mut=False, dtype, layout],
):
    local_row = thread_idx.y
    local_col = thread_idx.x
    tiled_row = block_idx.y * TPB + local_row
    tiled_col = block_idx.x * TPB + local_col

    # Get the tile of the output matrix that this thread block is responsible for
    out_tile = output.tile[TPB, TPB](block_idx.y, block_idx.x)
    a_shared = tb[dtype]().row_major[TPB, TPB]().shared().alloc().fill(0)
    b_shared = tb[dtype]().row_major[TPB, TPB]().shared().alloc().fill(0)

    var acc: output.element_type = 0

    alias load_a_layout = Layout.row_major(1, TPB)  # Coalesced loading
    alias load_b_layout = Layout.row_major(1, TPB)  # Coalesced loading
    # Note: Both matrices stored in same orientation for correct matrix multiplication
    # Transposed loading would be useful if B were pre-transposed in global memory

    @parameter
    for idx in range(size // TPB):  # Perfect division: 9 // 3 = 3 tiles
        # Get tiles from A and B matrices
        a_tile = a.tile[TPB, TPB](block_idx.y, idx)
        b_tile = b.tile[TPB, TPB](idx, block_idx.x)

        # Asynchronously copy tiles to shared memory with consistent orientation
        copy_dram_to_sram_async[thread_layout=load_a_layout](a_shared, a_tile)
        copy_dram_to_sram_async[thread_layout=load_b_layout](b_shared, b_tile)

        # Wait for all async copies to complete
        async_copy_wait_all()
        barrier()

        # Compute partial matrix multiplication for this tile
        @parameter
        for k in range(TPB):
            acc += a_shared[local_row, k] * b_shared[k, local_col]

        barrier()

    # Write final result to output tile
    if tiled_row < size and tiled_col < size:
        out_tile[local_row, local_col] = acc

The idiomatic tiled matrix multiplication leverages Mojo’s LayoutTensor API and asynchronous memory operations for a beautifully clean implementation.

🔑 Key Point: This implementation performs standard matrix multiplication A × B using coalesced loading for both matrices.

What this implementation does:

• Matrix operation: Standard \(A \times B\) multiplication (not \(A \times B^T\))
• Loading pattern: Both matrices use Layout.row_major(1, TPB) for coalesced access
• Computation: acc += a_shared[local_row, k] * b_shared[k, local_col]
• Data layout: No transposition during loading - both matrices loaded in same orientation

What this implementation does NOT do:

• Does NOT perform \(A \times B^T\) multiplication
• Does NOT use transposed loading patterns
• Does NOT transpose data during copy operations

With the \((9 \times 9)\) matrix size, we get perfect tiling that eliminates all boundary checks:

1. LayoutTensor tile API

   out_tile = output.tile[TPB, TPB](block_idx.y, block_idx.x)
   a_tile = a.tile[TPB, TPB](block_idx.y, idx)
   b_tile = b.tile[TPB, TPB](idx, block_idx.x)
   

   This directly expresses “get the tile at position (block_idx.y, block_idx.x)” without manual coordinate calculation. See the documentation for more details.

2. Asynchronous memory operations

   copy_dram_to_sram_async[thread_layout=load_a_layout](a_shared, a_tile)
   copy_dram_to_sram_async[thread_layout=load_b_layout](b_shared, b_tile)
   async_copy_wait_all()
   

   These operations:

   • Use dedicated copy engines that bypass registers and enable compute-memory overlap via copy_dram_to_sram_async
   • Use specialized thread layouts for optimal memory access patterns
   • Eliminate the need for manual memory initialization
   • Important: Standard GPU loads are already asynchronous; these provide better resource utilization and register bypass

3. Optimized memory access layouts

   alias load_a_layout = Layout.row_major(1, TPB)    # Coalesced loading
   alias load_b_layout = Layout.row_major(1, TPB)    # Coalesced loading
   # Note: Both matrices use the same layout for standard A × B multiplication
   

   Memory Access Analysis for Current Implementation:

   Both matrices use Layout.row_major(1, TPB) for coalesced loading from global memory:

   • load_a_layout: Threads cooperate to load consecutive elements from matrix A rows
   • load_b_layout: Threads cooperate to load consecutive elements from matrix B rows
   • Key insight: Thread layout determines how threads cooperate during copy, not the final data layout

   Actual Computation Pattern (proves this is A × B):

   # This is the actual computation in the current implementation
   acc += a_shared[local_row, k] * b_shared[k, local_col]
   
   # This corresponds to: C[i,j] = Σ(A[i,k] * B[k,j])
   # Which is standard matrix multiplication A × B
   

   Why both matrices use the same coalesced loading pattern:

   Loading tiles from global memory:
   - Matrix A tile: threads load A[block_row, k], A[block_row, k+1], A[block_row, k+2]... (consecutive)
   - Matrix B tile: threads load B[k, block_col], B[k, block_col+1], B[k, block_col+2]... (consecutive)
   
   Both patterns are coalesced with Layout.row_major(1, TPB)
   

   Three separate memory concerns:

   1. Global-to-shared coalescing: Layout.row_major(1, TPB) ensures coalesced global memory access
   2. Shared memory computation: a_shared[local_row, k] * b_shared[k, local_col] avoids bank conflicts
   3. Matrix operation: The computation pattern determines this is A × B, not A × B^T

4. Perfect tiling eliminates boundary checks

   @parameter
   for idx in range(size // TPB):  # Perfect division: 9 // 3 = 3
   

   With \((9 \times 9)\) matrices and \((3 \times 3)\) tiles, every tile is exactly full-sized. No boundary checking needed!

5. Clean tile processing with defensive bounds checking

   # Defensive bounds checking included even with perfect tiling
   if tiled_row < size and tiled_col < size:
       out_tile[local_row, local_col] = acc
   

   With perfect \((9 \times 9)\) tiling, this bounds check is technically unnecessary but included for defensive programming and consistency with other matrix sizes.

Performance considerations

The idiomatic implementation maintains the performance benefits of tiling while providing cleaner abstractions:

1. Memory locality: Exploits spatial and temporal locality through tiling
2. Coalesced access: Specialized load layouts ensure coalesced memory access patterns
3. Compute-memory overlap: Potential overlap through asynchronous memory operations
4. Shared memory efficiency: No redundant initialization of shared memory
5. Register pressure: Uses accumulation registers for optimal compute throughput

This implementation shows how high-level abstractions can express complex GPU algorithms without sacrificing performance. It’s a prime example of Mojo’s philosophy: combining high-level expressiveness with low-level performance control.

Key differences from manual tiling

The idiomatic approach is not just cleaner but also potentially more performant due to the use of specialized memory layouts and asynchronous operations.

Educational: When would transposed loading be useful?

The current implementation does NOT use transposed loading. This section is purely educational to show what’s possible with the layout system.

Current implementation recap:

• Uses Layout.row_major(1, TPB) for both matrices
• Performs standard A × B multiplication
• No data transposition during copy

Educational scenarios where you WOULD use transposed loading:

While this puzzle uses standard coalesced loading for both matrices, the layout system’s flexibility enables powerful optimizations in other scenarios:

# Example: Loading pre-transposed matrix B^T to compute A × B
# (This is NOT what the current implementation does)
alias load_b_layout = Layout.row_major(TPB, 1)   # Load B^T with coalesced access
alias store_b_layout = Layout.row_major(1, TPB)  # Store as B in shared memory
copy_dram_to_sram_async[src_thread_layout=load_b_layout, dst_thread_layout=store_b_layout](b_shared, b_tile)

Use cases for transposed loading (not used in this puzzle):

1. Pre-transposed input matrices: When \(B\) is already stored transposed in global memory
2. Different algorithms: Computing \(A^T \times B\), \(A \times B^T\), or \(A^T \times B^T\)
3. Memory layout conversion: Converting between row-major and column-major layouts
4. Avoiding transpose operations: Loading data directly in the required orientation

Key distinction:

• Current implementation: Both matrices use Layout.row_major(1, TPB) for standard \(A \times B\) multiplication
• Transposed loading example: Would use different layouts to handle pre-transposed data or different matrix operations

This demonstrates Mojo’s philosophy: providing low-level control when needed while maintaining high-level abstractions for common cases.

---

Summary: Key takeaways

What the idiomatic tiled implementation actually does:

1. Matrix Operation: Standard A × B multiplication
2. Memory Loading: Both matrices use Layout.row_major(1, TPB) for coalesced access
3. Computation Pattern: acc += a_shared[local_row, k] * b_shared[k, local_col]
4. Data Layout: No transposition during loading

Why this is optimal:

• Coalesced global memory access: Layout.row_major(1, TPB) ensures efficient loading
• Bank conflict avoidance: Shared memory access pattern avoids conflicts
• Standard algorithm: Implements the most common matrix multiplication pattern