SIMD Vectorized Matrix Multiplication - Block Tiling Implementation

Setup & Configuration
VEC_WIDTH: 4  |  TILE_SIZE: 4  |  MATRIX_SIZE: 8
Block Grid: 2×2 blocks  |  Thread Grid: 4×4 threads per block
Memory Pattern: Each thread processes 4 consecutive columns using SIMD
Current Computation State
Current Row (A)
Current Columns (B)
SIMD Vector Group
Active Computation
Matrix A (8×8)
Matrix B (8×8)
Result C (8×8)
Current Thread Execution
Thread Coordinates
tx: 0, ty: 0
bx: 0, by: 0
Matrix Position
Row: 0
Base Col: 0
SIMD Vector
Processing 4 elements
Columns: 0-3
K-Loop Progress
k = 0 / 8
Accumulating...
Algorithm Implementation
Step 1: Thread Index Mapping
// Calculate global position from thread indices tx = thread_idx.x // 0 to TILE_SIZE-1 ty = thread_idx.y // 0 to TILE_SIZE-1 bx = block_idx.x // Block column by = block_idx.y // Block row // Map to global matrix coordinates row = by * TILE_SIZE + ty base_col = bx * TILE_SIZE + tx * VEC_WIDTH // Each thread handles VEC_WIDTH columns // Thread (0,0): processes columns 0-3 // Thread (1,0): processes columns 4-7
Step 2: SIMD Accumulator and Bounds Check
# SIMD accumulator var vec_sum = SIMD[dtype, VEC_WIDTH](0.0) # Bounds checking before computation if row < size and base_col < size: # Only proceed if thread is within matrix bounds # This prevents out-of-bounds memory access
Step 3: Main Computation Loop
for k in range(size): # Load single element from A matrix let a_val = a.load[1](row, k) # Create SIMD vector from B matrix elements let remaining = min(VEC_WIDTH, size - base_col) var b_vec = SIMD[dtype, VEC_WIDTH](0.0) for i in range(remaining): if base_col + i < size: b_vec[i] = b.load[1](k, base_col + i) # SIMD multiply-accumulate vec_sum += a_val * b_vec
Step 4: Vectorized Result Storage
# Store vectorized results let remaining = min(VEC_WIDTH, size - base_col) for i in range(remaining): if base_col + i < size: output.store[1](row, base_col + i, vec_sum[i]) # Each thread stores up to VEC_WIDTH results # Bounds checking ensures no buffer overflows
Performance Characteristics
SIMD Benefits
• Parallelism: 4 multiplications per instruction
• Memory Bandwidth: Vector loads improve throughput
• Register Efficiency: Fewer load/store operations
• Cache Locality: Sequential column access
Implementation Notes
• Bounds Checking: Prevents buffer overflows
• Vector Width: Must align with hardware
• Memory Alignment: Critical for performance
• Zero Padding: Handles partial vectors safely
Step Analysis
Current Operation: Initializing...
Memory Access: Starting computation...
SIMD Operation: Preparing vectors...
Performance Impact: 4x parallelization active
Thread Computation Examples
Block (0,0) - Thread (0,0)
Position: Row 0, Cols 0-3
SIMD Vector: A[0][k] × B[k][0:3]
Operation: 4 parallel multiplications
Block (0,0) - Thread (1,0)
Position: Row 0, Cols 4-7
SIMD Vector: A[0][k] × B[k][4:7]
Operation: 4 parallel multiplications
Block (1,0) - Thread (0,0)
Position: Row 0, Cols 0-3
SIMD Vector: A[0][k] × B[k][0:3]
Operation: Same data, different block
Block (0,1) - Thread (0,0)
Position: Row 4, Cols 0-3
SIMD Vector: A[4][k] × B[k][0:3]
Operation: Different row processing