Matrix Multiplication in Multiword Arithmetic: Error Analysis and Application to GPU Tensor Cores

Abstract

In multiword arithmetic, a matrix is represented as the unevaluated sum of two or more lower precision matrices, and a matrix product is formed by multiplying the constituents in low precision. We investigate the use of multiword arithmetic for improving the performance-accuracy tradeoff of matrix multiplication with mixed precision block fused multiply–add (FMA) hardware, focusing especially on the tensor cores available on NVIDIA GPUs. Building on a general block FMA framework, we develop a comprehensive error analysis of multiword matrix multiplication. After confirming the theoretical error bounds experimentally by simulating low precision in software, we use the cuBLAS and CUTLASS libraries to implement a number of matrix multiplication algorithms using double-fp16 (double-binary16) arithmetic. When running the algorithms on NVIDIA V100 and A100 GPUs, we find that double-fp16 is not as accurate as fp32 (binary32) arithmetic despite satisfying the same worst-case error bound. Using probabilistic error analysis, we explain why this issue is likely to be caused by the rounding mode used by the NVIDIA tensor cores, and we propose a parameterized blocked summation algorithm that alleviates the problem and significantly improves the performance-accuracy tradeoff.