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Mixed precision algorithms in numerical linear algebra

Published online by Cambridge University Press:  09 June 2022

Nicholas J. Higham Open the ORCID record for Nicholas J. Higham [Opens in a new window]  and
Theo Mary
Nicholas J. Higham
Affiliation:
Department of Mathematics, University of Manchester, Manchester, M13 9PL, UK E-mail: [email protected]
Theo Mary
Affiliation:
Sorbonne Université, CNRS, LIP6, Paris, F-75005, France E-mail: [email protected]
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Abstract

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Today’s floating-point arithmetic landscape is broader than ever. While scientific computing has traditionally used single precision and double precision floating-point arithmetics, half precision is increasingly available in hardware and quadruple precision is supported in software. Lower precision arithmetic brings increased speed and reduced communication and energy costs, but it produces results of correspondingly low accuracy. Higher precisions are more expensive but can potentially provide great benefits, even if used sparingly. A variety of mixed precision algorithms have been developed that combine the superior performance of lower precisions with the better accuracy of higher precisions. Some of these algorithms aim to provide results of the same quality as algorithms running in a fixed precision but at a much lower cost; others use a little higher precision to improve the accuracy of an algorithm. This survey treats a broad range of mixed precision algorithms in numerical linear algebra, both direct and iterative, for problems including matrix multiplication, matrix factorization, linear systems, least squares, eigenvalue decomposition and singular value decomposition. We identify key algorithmic ideas, such as iterative refinement, adapting the precision to the data, and exploiting mixed precision block fused multiply–add operations. We also describe the possible performance benefits and explain what is known about the numerical stability of the algorithms. This survey should be useful to a wide community of researchers and practitioners who wish to develop or benefit from mixed precision numerical linear algebra algorithms.

Type
Research Article
Information
Acta Numerica , Volume 31 , May 2022 , pp. 347 - 414
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2022. Published by Cambridge University Press

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