Multiword Arithmetic and Parallel Computing

2026-06-18

AI Summary

This work addresses the challenge that native hardware arithmetic precision often falls short of the demands of high-precision numerical computation. To overcome this limitation, the authors propose an integrated approach combining multiword arithmetic—such as unevaluated representations using multiple 64-bit floating-point numbers—with parallel computing techniques. Through register-level storage optimization, efficient memory layout, and GPU acceleration, the method significantly enhances numerical stability while mitigating the performance overhead typically associated with increased precision. Implemented in Ada and released as the open-source PHCpack software under the GNU GPL v3.0 license, the project supports the Alire package manager and represents the first deep integration of parallelism and hardware-aware optimizations within a multiword arithmetic framework, achieving a synergistic improvement in both accuracy and computational performance.

Abstract

In many applications, the precision by the available hardware arithmetic is insufficient to guarantee accurate results. Multiword arithmetic is a special type of multiprecision arithmetic where a multiple double is an unevaluated sum of 64-bit doubles, or where a multiple integer is an unevaluated sum of 64-bit integers. Parallel computing is applied to compensate for the cost overhead of multiword arithmetic. This type of arithmetic exploits naturally the optimized hardware, allows for efficient type conversions, memory layouts, all favorable for parallel computing. For example, storing a multiword in registers rather than arrays is beneficial to parallel computing by tasking and acceleration by graphics processing units. Code for multiword arithmetic is available in the software PHCpack, written mainly in Ada, publicly available at github, and as an Alire crate, released under the GNU GPL v3.0 license.

Problem

• multiword arithmetic
• multiprecision arithmetic
• hardware precision
• parallel computing
• numerical accuracy

Innovation

• multiword arithmetic
• parallel computing
• multiprecision arithmetic
• GPU acceleration
• unevaluated sum