The Nine Circles of Matmul

The nine circles of matmul for row in A for col in B for pair in (row,col) triply nested for loop in memory managed language "lol how is this hard?" realizing it's O(N^3) triply nested for loop in non memory managed language "wow it's 10,000x faster this is the end right?" lowered time complexity matmuls like strassen's algorithm "wow multiplications are expensive" "wow it's O^2.something) now, this is great!" multiplication factoring trick optimized triply nested for loop in non memory managed language block partitioning loop strength realizing that lowered time complexity is garbage SIMD cache hierarchies cache locality single threaded happy times stop here. now everything is multiprocessed autotuning multithreaded matmuls deadlocks cache coherence killing children load balancing race conditions mutexes and semaphores massively parallel matmuls (GPUs) hyperdimensional cache/ thread hierarchies warp scheduling CUDA lol you're learning a new botched matmuls computing architecture now, sorry bucko MASSIVELY PARALLEL MATMULS (distributed) heterogeneous computing MapReduce sharding matrices benchmarks cost $$$ fault tolerance hardware starts failing network topology memory alignment Arcane optimizations hardware specific hand tuning kernel search leveraging algebras mixed precision arithmetic sparse matmuls square vs rectangular vs triangular optimizations fused kernels

A programming meme presents the “nine circles” of increasingly difficult matrix multiplication optimization, from a simple triply nested loop to distributed massively parallel systems. The joke is that improving performance keeps exposing deeper problems: complexity, memory, synchronization, hardware failures, and obscure optimization techniques, turning a simple algorithm into an infernal engineering ordeal. EN: A programming meme presents the “nine circles” of increasingly difficult matrix multiplication optimization, from a simple triply nested loop to distributed massively parallel systems. The joke is that improving performance keeps exposing deeper problems: complexity, memory, synchronization, hardware failures, and obscure optimization techniques, turning a simple algorithm into an infernal engineering ordeal.

• @ithueti / 13950 2024-09-05