Parallel Matrix Multiplication (GEMM) is the operation C = α×A×B + β×C for dense matrices — the most important computational kernel in high-performance computing, deep learning, and scientific simulations, extensively optimized in BLAS libraries.

GEMM Complexity

• Naive triple-loop: O(N³) operations for N×N matrices.
• N=1024: 2 billion operations — serial takes ~2 seconds on single core.
• N=4096: 128 billion operations — needs parallelism.

BLAS (Basic Linear Algebra Subprograms)

• Level 3 BLAS: Matrix-matrix operations (GEMM, TRSM, SYRK).
• Vendor implementations: MKL (Intel), OpenBLAS, BLIS, cuBLAS (NVIDIA), rocBLAS (AMD).
• Drop-in replacement: Same API, dramatically different performance.
• MKL DGEMM: 90%+ of peak FLOPS on modern Intel CPUs.

CPU GEMM Optimization Techniques

Blocking (Tiling):

• Break matrices into blocks that fit in cache.
• Reduce cache miss rate from O(N³) → O(N³/B) for block size B.
• Three levels: Register tile → L1 tile → L2/L3 tile.

SIMD Vectorization:

• Use AVX-512 FMA to process 16 float/8 double FMAs per cycle.
• Register microkernel: 6×16 float accumulation in 48 registers (AVX-512).

Packing:

• Reorganize matrix panels into contiguous memory for stride-1 access.
• Eliminates TLB misses and streaming prefetch issues.

GPU GEMM

• cuBLAS GEMM: Uses Tensor Cores (16x16x16 matrix multiply per cycle).
• CUTLASS: Open-source CUDA GEMM with tunable tile sizes.
• Flash Attention: Fused attention is essentially batched GEMM with online softmax.

Distribution Across Nodes

• 2D block distribution: A and B partitioned in 2D across P processors.
• Cannon's algorithm: Communication-optimal 2D GEMM.
• SUMMA: Scalable Universal Matrix Multiplication Algorithm.
• ScaLAPACK, libflame: Distributed GEMM implementations.

GEMM optimization is the foundation of modern AI computing — 70–90% of transformer inference and training time is spent in matrix multiplications, and every 10% GEMM efficiency improvement translates directly to training cost and inference latency reduction.