Report
Error analysis of matrix multiplication with narrow range floating-point arithmetic
| العنوان: | Error analysis of matrix multiplication with narrow range floating-point arithmetic |
|---|---|
| المؤلفون: | Mary, Théo, Mikaitis, Mantas |
| المساهمون: | Performance et Qualité des Algorithmes Numériques (PEQUAN), LIP6, Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), University of Leeds, ANR-22-EXNU-0002,Exa-MA,Methods and Algorithms for Exascale(2022), ANR-20-CE46-0009,INTERFLOP,Plateforme d'analyse pour l'arithmétique flottante(2020), ANR-23-CE46-0005,MixHPC,Algorithmes en précision mixte pour le calcul haute performance(2023) |
| المصدر: | https://hal.science/hal-04671474 ; 2024. |
| بيانات النشر: | HAL CCSD |
| سنة النشر: | 2024 |
| مصطلحات موضوعية: | matrix multiplication, rounding error analysis, floating-point arithmetic, underflow, overflow, scaling, multiword arithmetic, reduced precision, mixed precision, GPUs, [INFO]Computer Science [cs], [MATH]Mathematics [math] |
| الوصف: | High-performance computing hardware now supports many different floating-pointformats, from 64 bits to only 4 bits. While the effects of reducing precision in numerical linearalgebra computations have been extensively studied, some of these low precision formats also possessa very narrow range of representable values, meaning underflow and overflow are very likely. Thegoal of this article is to analyze the consequences of this narrow range on the accuracy of matrixmultiplication. 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 remaindominated by the rounding errors in most practical scenarios. We also show that this conclusionremains true when multiword arithmetic is used. We perform extensive numerical experiments thatconfirm that the narrow range of low precision arithmetics should not significantly affect the accuracyof matrix multiplication—provided a suitable scaling is used. |
| نوع الوثيقة: | report |
| اللغة: | English |
| Relation: | hal-04671474; https://hal.science/hal-04671474; https://hal.science/hal-04671474/document; https://hal.science/hal-04671474/file/manuscript.pdf |
| الاتاحة: | https://hal.science/hal-04671474 https://hal.science/hal-04671474/document https://hal.science/hal-04671474/file/manuscript.pdf |
| Rights: | info:eu-repo/semantics/OpenAccess |
| رقم الانضمام: | edsbas.C3FCF2E7 |
| قاعدة البيانات: | BASE |
| الوصف غير متاح. |