Why is BLAS so much faster than my matrix multiplication implementation?

BLAS's Superior Performance in Matrix Multiplication

You've witnessed a striking difference in the performance of your own matrix multiplication function compared to that of BLAS. This brings up two questions:

1. How does BLAS achieve extreme performance?

BLAS is divided into three levels based on complexity and optimization techniques:

• Level 1: Operates on vectors and benefits from vectorization.
• Level 2: Handles matrix-vector operations and leverages multiprocessor architectures.
• Level 3: Performs matrix-matrix operations and optimizes for cache hierarchy. This optimization is crucial for the significant performance boost in Level 3 functions.

2. Why is your implementation slower?

Your implementation lacks the cache optimization employed by BLAS. The O(N^3) operations in matrix-matrix multiplication result in significant data movement between memory and cache. By implementing dedicated algorithms that minimize cache conflicts, BLAS significantly accelerates this process.

While modern compilers help optimize code, they cannot fully compensate for the specialized techniques used in BLAS implementations like ATLAS, GotoBLAS, and OpenBLAS.

Algorithms Used by BLAS

BLAS does not utilize complex algorithms like Coppersmith–Winograd or Strassen due to:

• Potential difficulty in providing cache-optimized implementations.
• Lack of numerical stability, which prohibits their use in LAPACK's computational kernel.
• High constant factor in their time complexity, making them only beneficial for exceptionally large matrices.

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