Analysis of Floating-Point Matrix Multiplication Computed via Integer Arithmetic
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arXiv
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| Autores principales: | , , , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866915990122528768 |
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| author | Abdelfattah, Ahmad Dongarra, Jack Fasi, Massimiliano Mikaitis, Mantas Tisseur, Françoise |
| author_facet | Abdelfattah, Ahmad Dongarra, Jack Fasi, Massimiliano Mikaitis, Mantas Tisseur, Françoise |
| contents | Ootomo, Ozaki, and Yokota [Int. J. High Perform. Comput. Appl., 38 (2024), p. 297-313] have proposed a strategy to recast a floating-point matrix multiplication in terms of integer matrix products. The factors A and B are split into integer slices, the product of these slices is computed exactly, and AB is approximated by accumulating these integer products in floating-point arithmetic. This technique is particularly well suited to mixed-precision matrix multiply-accumulate units with integer support, such as the NVIDIA tensor cores or the AMD matrix cores. The number of slices allows for performance-accuracy tradeoffs: more slices yield better accuracy but require more multiplications, which in turn reduce performance. We propose an inexpensive way to estimate the minimum number of multiplications needed to achieve a prescribed level of accuracy. Our error analysis shows that the algorithm may become inaccurate (or inefficient) if rows of A or columns of B are badly scaled. We perform a range of numerical experiments, both in simulation and on the latest NVIDIA GPUs, that confirm the analysis and illustrate strengths and weaknesses of the algorithm. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_11277 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Analysis of Floating-Point Matrix Multiplication Computed via Integer Arithmetic Abdelfattah, Ahmad Dongarra, Jack Fasi, Massimiliano Mikaitis, Mantas Tisseur, Françoise Numerical Analysis Mathematical Software 65F99, 65G50, 65Y10 Ootomo, Ozaki, and Yokota [Int. J. High Perform. Comput. Appl., 38 (2024), p. 297-313] have proposed a strategy to recast a floating-point matrix multiplication in terms of integer matrix products. The factors A and B are split into integer slices, the product of these slices is computed exactly, and AB is approximated by accumulating these integer products in floating-point arithmetic. This technique is particularly well suited to mixed-precision matrix multiply-accumulate units with integer support, such as the NVIDIA tensor cores or the AMD matrix cores. The number of slices allows for performance-accuracy tradeoffs: more slices yield better accuracy but require more multiplications, which in turn reduce performance. We propose an inexpensive way to estimate the minimum number of multiplications needed to achieve a prescribed level of accuracy. Our error analysis shows that the algorithm may become inaccurate (or inefficient) if rows of A or columns of B are badly scaled. We perform a range of numerical experiments, both in simulation and on the latest NVIDIA GPUs, that confirm the analysis and illustrate strengths and weaknesses of the algorithm. |
| title | Analysis of Floating-Point Matrix Multiplication Computed via Integer Arithmetic |
| topic | Numerical Analysis Mathematical Software 65F99, 65G50, 65Y10 |
| url | https://arxiv.org/abs/2506.11277 |