Error Analysis of Matrix Multiplication with Narrow Range Floating-Point Arithmetic

High-performance computing hardware now supports many different floating-point formats, from 64 bits to only 4 bits. While the effects of reducing precision in numerical linear algebra computations have been extensively studied, some of these low precision formats also possess a very narrow range of representable values, meaning underflow and overflow are very likely. The goal of this article is to analyze the consequences of this narrow range on the accuracy of matrix multiplication. We describe a simple scaling that can prevent overflow while minimizing underflow. We carry out an error analysis to bound the underflow errors and show that they should remain dominated by the rounding errors in most practical scenarios. We also show that this conclusion remains true when multiword arithmetic is used. We perform extensive numerical experiments that confirm that the narrow range of low precision arithmetics should not significantly affect the accuracy of matrix multiplication, provided a suitable scaling is used.