Fast Multidimensional Matrix Multiplication on CPU from Scratch

The article discusses the performance of multidimensional matrix multiplication on a CPU, specifically using Numpy and C++. It highlights that Numpy can multiply two 1024x1024 matrices in approximately 8 milliseconds on a 4-core Intel CPU, achieving around 18 FLOPs per core per cycle. This performance is attributed to the use of optimized BLAS libraries, such as Intel's MKL, which implement functions like SGEMM for efficient matrix operations. The author explores the challenges of replicating this performance using plain C++, noting that a naive implementation takes significantly longer.

The article details various optimization techniques, including compiler flags, register accumulation, and cache-aware loop reordering, which can drastically reduce computation time. For instance, a cache-aware implementation improved performance to 89 milliseconds, demonstrating the importance of memory access patterns in matrix multiplication. The author emphasizes that while achieving performance close to Numpy's implementation is challenging, understanding these optimizations is crucial for developing efficient numerical algorithms.

Ultimately, the article serves as a guide for programmers interested in enhancing matrix multiplication performance in C++, illustrating the impact of hardware architecture and optimization strategies on computational efficiency. The author concludes that while their implementation achieves 9 FLOPs per core per cycle, it is not intended to compete with established BLAS libraries but rather to provide insights into performance optimization techniques.