Tiled Matrix Multiplication

fn matmul_tiled[
    layout: Layout, size: Int
](
    out: LayoutTensor[mut=False, dtype, layout],
    a: LayoutTensor[mut=False, dtype, layout],
    b: LayoutTensor[mut=False, dtype, layout],
):
    local_row = thread_idx.x
    local_col = thread_idx.y
    global_row = block_idx.x * TPB + local_row
    global_col = block_idx.y * 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: out.element_type = 0

    # Iterate over tiles to compute matrix product
    @parameter
    for tile in range((size + TPB - 1) // TPB):
        # Reset shared memory tiles
        if local_row < TPB and local_col < TPB:
            a_shared[local_row, local_col] = 0
            b_shared[local_row, local_col] = 0

        barrier()

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

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

        barrier()

        # Matrix multiplication within the tile
        if global_row < size and global_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 global_row < size and global_col < size:
        out[global_row, global_col] = acc

The tiled implementation with LayoutTensor handles large matrices efficiently by processing them in blocks. Here’s a comprehensive analysis:

Implementation Architecture

1. Layout Configuration:

   alias layout_tiled = Layout.row_major(SIZE_TILED, SIZE_TILED)
   

   • Defines row-major layout for all tensors
   • Ensures consistent memory access patterns
   • Enables efficient 2D indexing

2. Shared Memory Setup:

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

   • Uses TensorBuilder for structured allocation
   • Maintains row-major layout for consistency
   • Enables efficient tile processing

3. Thread and Block Organization:

   local_row = thread_idx.x
   local_col = thread_idx.y
   global_row = block_idx.x * TPB + local_row
   global_col = block_idx.y * TPB + local_col
   

   • Maps threads to matrix elements
   • Handles 2D indexing efficiently
   • Maintains proper boundary checks

Tile Processing Pipeline

1. Tile Iteration:

   @parameter
   for tile in range((size + TPB - 1) // TPB):
   

   • Compile-time unrolled loop
   • Handles matrix size not divisible by TPB
   • Processes matrix in TPB×TPB tiles

2. Shared Memory Reset:

   if local_row < TPB and local_col < TPB:
       a_shared[local_row, local_col] = 0
       b_shared[local_row, local_col] = 0
   

   • Clears previous tile data
   • Ensures clean state for new tile
   • Prevents data corruption

3. Tile Loading:

   # Load A tile
   if global_row < size and (tile * TPB + local_col) < size:
       a_shared[local_row, local_col] = a[global_row, tile * TPB + local_col]
   
   # Load B tile
   if (tile * TPB + local_row) < size and global_col < size:
       b_shared[local_row, local_col] = b[tile * TPB + local_row, global_col]
   

   • Handles boundary conditions
   • Uses LayoutTensor indexing
   • Maintains memory coalescing

4. Computation:

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

   • Processes current tile
   • Uses shared memory for efficiency
   • Handles partial tiles correctly

Memory Access Optimization

1. Global Memory Pattern:

   A[global_row, tile * TPB + local_col] → Coalesced reads
   B[tile * TPB + local_row, global_col] → Transposed access
   

   • Maximizes memory coalescing
   • Minimizes bank conflicts
   • Efficient for transposed access

2. Shared Memory Usage:

   a_shared[local_row, k] → Row-wise access
   b_shared[k, local_col] → Column-wise access
   

   • Optimized for matrix multiplication
   • Reduces bank conflicts
   • Enables data reuse

Synchronization and Safety

1. Barrier Points:

   barrier()  # After shared memory reset
   barrier()  # After tile loading
   barrier()  # After computation
   

   • Ensures shared memory consistency
   • Prevents race conditions
   • Maintains thread cooperation

2. Boundary Handling:

   if global_row < size and global_col < size:
       out[global_row, global_col] = acc
   

   • Prevents out-of-bounds access
   • Handles matrix edges
   • Safe result writing

Performance Characteristics

1. Memory Efficiency:

   • Reduced global memory traffic through tiling
   • Efficient shared memory reuse
   • Coalesced memory access patterns

2. Computational Throughput:

   • High data locality in shared memory
   • Efficient thread utilization
   • Minimal thread divergence

3. Scalability:

   • Handles arbitrary matrix sizes
   • Efficient for large matrices
   • Good thread occupancy

Key Optimizations

1. Layout Optimization:

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

2. Memory Access:

   • Coalesced global memory loads
   • Efficient shared memory usage
   • Minimal bank conflicts

3. Computation:

   • Register-based accumulation
   • Compile-time loop unrolling
   • Efficient tile processing

This implementation achieves high performance through:

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