{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,4]],"date-time":"2026-06-04T22:43:14Z","timestamp":1780612994057,"version":"3.54.1"},"reference-count":98,"publisher":"SAGE Publications","issue":"4","license":[{"start":{"date-parts":[[2021,3,19]],"date-time":"2021-03-19T00:00:00Z","timestamp":1616112000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/journals.sagepub.com\/page\/policies\/text-and-data-mining-license"}],"funder":[{"DOI":"10.13039\/100000015","name":"US Department of Energy","doi-asserted-by":"crossref","award":["17-SC-20-SC"],"award-info":[{"award-number":["17-SC-20-SC"]}],"id":[{"id":"10.13039\/100000015","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["journals.sagepub.com"],"crossmark-restriction":true},"short-container-title":["The International Journal of High Performance Computing Applications"],"published-print":{"date-parts":[[2021,7]]},"abstract":"<jats:p>The efficient utilization of mixed-precision numerical linear algebra algorithms can offer attractive acceleration to scientific computing applications. Especially with the hardware integration of low-precision special-function units designed for machine learning applications, the traditional numerical algorithms community urgently needs to reconsider the floating point formats used in the distinct operations to efficiently leverage the available compute power. In this work, we provide a comprehensive survey of mixed-precision numerical linear algebra routines, including the underlying concepts, theoretical background, and experimental results for both dense and sparse linear algebra problems.<\/jats:p>","DOI":"10.1177\/10943420211003313","type":"journal-article","created":{"date-parts":[[2021,3,19]],"date-time":"2021-03-19T06:02:38Z","timestamp":1616133758000},"page":"344-369","update-policy":"https:\/\/doi.org\/10.1177\/sage-journals-update-policy","source":"Crossref","is-referenced-by-count":118,"title":["A survey of numerical linear algebra methods utilizing mixed-precision arithmetic"],"prefix":"10.1177","volume":"35","author":[{"given":"Ahmad","family":"Abdelfattah","sequence":"first","affiliation":[{"name":"University of Tennessee, Knoxville, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2177-952X","authenticated-orcid":false,"given":"Hartwig","family":"Anzt","sequence":"additional","affiliation":[{"name":"University of Tennessee, Knoxville, USA"},{"name":"Karlsruhe Institute of Technology, Karlsruhe, Germany"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Erik G","family":"Boman","sequence":"additional","affiliation":[{"name":"Sandia National Lab, Albuquerque, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Erin","family":"Carson","sequence":"additional","affiliation":[{"name":"Charles University, Prague, Czech Republic"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Terry","family":"Cojean","sequence":"additional","affiliation":[{"name":"Karlsruhe Institute of Technology, Karlsruhe, Germany"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Jack","family":"Dongarra","sequence":"additional","affiliation":[{"name":"University of Tennessee, Knoxville, USA"},{"name":"Oak Ridge National Lab, Oak Ridge, USA"},{"name":"University of Manchester, Manchester, UK"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Alyson","family":"Fox","sequence":"additional","affiliation":[{"name":"Lawrence Livermore National Lab, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Mark","family":"Gates","sequence":"additional","affiliation":[{"name":"University of Tennessee, Knoxville, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Nicholas J","family":"Higham","sequence":"additional","affiliation":[{"name":"University of Manchester, Manchester, UK"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Xiaoye S","family":"Li","sequence":"additional","affiliation":[{"name":"Lawrence Berkeley National Lab, Berkeley, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-3018-7190","authenticated-orcid":false,"given":"Jennifer","family":"Loe","sequence":"additional","affiliation":[{"name":"Sandia National Lab, Albuquerque, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Piotr","family":"Luszczek","sequence":"additional","affiliation":[{"name":"University of Tennessee, Knoxville, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Srikara","family":"Pranesh","sequence":"additional","affiliation":[{"name":"University of Manchester, Manchester, UK"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Siva","family":"Rajamanickam","sequence":"additional","affiliation":[{"name":"Sandia National Lab, Albuquerque, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3023-1849","authenticated-orcid":false,"given":"Tobias","family":"Ribizel","sequence":"additional","affiliation":[{"name":"Karlsruhe Institute of Technology, Karlsruhe, Germany"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Barry F","family":"Smith","sequence":"additional","affiliation":[{"name":"Argonne National Lab, Argonne, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Kasia","family":"Swirydowicz","sequence":"additional","affiliation":[{"name":"National Renewable Energy Lab, Boulder, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Stephen","family":"Thomas","sequence":"additional","affiliation":[{"name":"National Renewable Energy Lab, Boulder, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Stanimire","family":"Tomov","sequence":"additional","affiliation":[{"name":"University of Tennessee, Knoxville, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Yaohung M","family":"Tsai","sequence":"additional","affiliation":[{"name":"University of Tennessee, Knoxville, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"given":"Ulrike Meier","family":"Yang","sequence":"additional","affiliation":[{"name":"Lawrence Livermore National Lab, USA"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"179","published-online":{"date-parts":[[2021,3,19]]},"reference":[{"key":"bibr2-10943420211003313","doi-asserted-by":"publisher","DOI":"10.1109\/IPDPS.2019.00022"},{"key":"bibr3-10943420211003313","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-030-50417-5_18"},{"key":"bibr4-10943420211003313","doi-asserted-by":"publisher","DOI":"10.1088\/1742-6596\/180\/1\/012037"},{"key":"bibr5-10943420211003313","doi-asserted-by":"publisher","DOI":"10.1007\/s13160-019-00360-8"},{"key":"bibr6-10943420211003313","unstructured":"Anderson E, Bai Z, Bischof CH, et al. (1999) LAPACK Users\u2019 Guide. 3rd edn. Philadelphia, PA: Society for Industrial and Applied Mathematics. ISBN 0-89871-447-8. Available at: http:\/\/www.netlib.org\/lapack\/lug\/ (accessed 2020)."},{"key":"bibr7-10943420211003313","doi-asserted-by":"publisher","DOI":"10.21105\/joss.02260"},{"key":"bibr8-10943420211003313","doi-asserted-by":"publisher","DOI":"10.1002\/cpe.4460"},{"key":"bibr9-10943420211003313","doi-asserted-by":"publisher","DOI":"10.1177\/1094342019846547"},{"key":"bibr10-10943420211003313","doi-asserted-by":"publisher","DOI":"10.1016\/j.parco.2017.10.003"},{"key":"bibr11-10943420211003313","first-page":"31","volume":"33","author":"Arioli M","year":"2009","journal-title":"Electronic Transactions on Numerical Analysis"},{"key":"bibr12-10943420211003313","doi-asserted-by":"publisher","DOI":"10.1137\/060661545"},{"key":"bibr13-10943420211003313","doi-asserted-by":"publisher","DOI":"10.1137\/18M1197400"},{"key":"bibr14-10943420211003313","doi-asserted-by":"publisher","DOI":"10.1007\/BF01939321"},{"key":"bibr15-10943420211003313","doi-asserted-by":"publisher","DOI":"10.1007\/BF01934122"},{"key":"bibr16-10943420211003313","doi-asserted-by":"crossref","first-page":"249","DOI":"10.1093\/oso\/9780198535645.003.0016","volume-title":"Reliable Numerical Computation","author":"Bj\u00f6rck \u00c5","year":"1990"},{"key":"bibr17-10943420211003313","doi-asserted-by":"publisher","DOI":"10.1016\/0024-3795(94)90493-6"},{"key":"bibr18-10943420211003313","doi-asserted-by":"publisher","DOI":"10.1137\/0613015"},{"key":"bibr19-10943420211003313","doi-asserted-by":"publisher","DOI":"10.1137\/19M1289546"},{"key":"bibr20-10943420211003313","doi-asserted-by":"publisher","DOI":"10.1137\/17M1122918"},{"key":"bibr21-10943420211003313","doi-asserted-by":"publisher","DOI":"10.1137\/17M1140819"},{"key":"bibr22-10943420211003313","doi-asserted-by":"crossref","unstructured":"Carson E, Higham NJ, Pranesh S (2020) Three-precision GMRES-based iterative refinement for least squares problems. SIAM Journal on Scientific Computing 42(6): A4063\u2013A408.","DOI":"10.1137\/20M1316822"},{"key":"bibr23-10943420211003313","volume-title":"Communication-avoiding Krylov subspace methods in theory and practice","author":"Carson EC","year":"2015"},{"key":"bibr24-10943420211003313","volume-title":"Smoky Mountain Conference 2019","author":"Clark K","year":"2019"},{"key":"bibr25-10943420211003313","doi-asserted-by":"publisher","DOI":"10.1016\/j.cpc.2010.05.002"},{"key":"bibr26-10943420211003313","doi-asserted-by":"publisher","DOI":"10.1137\/S0895479800373498"},{"key":"bibr27-10943420211003313","doi-asserted-by":"publisher","DOI":"10.1145\/1141885.1141894"},{"key":"bibr28-10943420211003313","doi-asserted-by":"publisher","DOI":"10.1145\/356012.356016"},{"key":"bibr29-10943420211003313","doi-asserted-by":"publisher","DOI":"10.1137\/0720002"},{"key":"bibr30-10943420211003313","doi-asserted-by":"publisher","DOI":"10.1007\/s10589-011-9420-4"},{"key":"bibr31-10943420211003313","first-page":"175","volume-title":"Proceedings of International Conference on Computational Science, ICCS 2010","volume":"1","author":"Emans M","year":"2012"},{"key":"bibr32-10943420211003313","doi-asserted-by":"publisher","DOI":"10.1145\/3441850"},{"key":"bibr33-10943420211003313","unstructured":"Fox A, Kolasinski A (2019) Error analysis of inline ZFP compression for multigrid methods. In: 2019 Copper Mountain Conference for Multigrid Methods, Copper Mountain, CO, March 25, 2019."},{"key":"bibr34-10943420211003313","doi-asserted-by":"publisher","DOI":"10.1137\/18M1218212"},{"key":"bibr35-10943420211003313","doi-asserted-by":"publisher","DOI":"10.1137\/S0895479803424347"},{"key":"bibr36-10943420211003313","volume-title":"Proceedings of ENUMATH 2011","author":"Glimberg SL","year":"2011"},{"key":"bibr37-10943420211003313","author":"Gratton S","year":"2019","journal-title":"arXiv preprint"},{"key":"bibr38-10943420211003313","doi-asserted-by":"publisher","DOI":"10.1016\/0024-3795(89)90285-1"},{"key":"bibr39-10943420211003313","doi-asserted-by":"publisher","DOI":"10.1137\/S0895479895284944"},{"key":"bibr40-10943420211003313","doi-asserted-by":"crossref","first-page":"e5418","DOI":"10.1002\/cpe.5418","volume":"32","author":"Gr\u00fctzmacher T","year":"2019","journal-title":"Concurrency and Computation: Practice and Experience"},{"key":"bibr41-10943420211003313","first-page":"1737","volume-title":"Proceedings of the 32nd International Conference on International Conference on Machine Learning","volume":"37","author":"Gupta S","year":"2015"},{"key":"bibr42-10943420211003313","unstructured":"Gustafson J (2015) The End of Error: Unum Computing. Milton Park: Chapman & Hall\/CRC Computational Science. Taylor & Francis. ISBN 9781482239867. Available at: https:\/\/books.google.de\/books?id=W2ThoAEACAAJ (accessed 2020)."},{"key":"bibr43-10943420211003313","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-93698-7_45"},{"key":"bibr44-10943420211003313","doi-asserted-by":"crossref","unstructured":"Haidar A, Bayraktar H, Tomov S, et al. (2020) Mixed-precision solution of linear systems using accelerator-based computing. Mixed-precision iterative refinement using tensor cores on GPUs to accelerate solution of linear systems. Proceedings of the Royal Society A 476(2243): 20200110, 2020.","DOI":"10.1098\/rspa.2020.0110"},{"key":"bibr45-10943420211003313","first-page":"1","volume-title":"Proceedings of the International Conference for High Performance Computing, Networking, Storage, and Analysis","author":"Haidar A","year":"2018"},{"key":"bibr46-10943420211003313","doi-asserted-by":"publisher","DOI":"10.1145\/3148226.3148237"},{"key":"bibr47-10943420211003313","doi-asserted-by":"publisher","DOI":"10.1137\/0726070"},{"key":"bibr48-10943420211003313","doi-asserted-by":"publisher","DOI":"10.1093\/imanum\/17.4.495"},{"key":"bibr49-10943420211003313","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898718027"},{"key":"bibr50-10943420211003313","unstructured":"Higham NJ (2019) Error analysis for standard and GMRES-based iterative refinement in two and three-precisions. MIMS EPrint 2019.19, Manchester: Manchester Institute for Mathematical Sciences, The University of Manchester. Available at: http:\/\/eprints.maths.manchester.ac.uk\/2735\/ (accessed 2020)."},{"key":"bibr51-10943420211003313","doi-asserted-by":"publisher","DOI":"10.1137\/18M1226312"},{"key":"bibr52-10943420211003313","doi-asserted-by":"crossref","unstructured":"Higham NJ, Mary T (2020) Sharper probabilistic backward error analysis for basic linear algebra kernels with random data. SIAM Journal on Scientific Computing 42(5): A3427\u2013A3446.","DOI":"10.1137\/20M1314355"},{"key":"bibr53-10943420211003313","doi-asserted-by":"crossref","unstructured":"Higham NJ, Pranesh S (2021) Exploiting lower precision arithmetic in solving symmetric positive definite linear systems and least squares problems. SIAM Journal on Scientific Computing 43(1): A258\u2013A277.","DOI":"10.1137\/19M1298263"},{"key":"bibr54-10943420211003313","doi-asserted-by":"publisher","DOI":"10.1137\/19M1251308"},{"key":"bibr55-10943420211003313","doi-asserted-by":"publisher","DOI":"10.1137\/18M1229511"},{"key":"bibr56-10943420211003313","doi-asserted-by":"publisher","DOI":"10.1145\/1731022.1731027"},{"key":"bibr57-10943420211003313","volume-title":"IEEE Standard for Floating-Point Arithmetic, IEEE Std 754-2019 (Revision of IEEE 754-2008)","author":"IEEE","year":"2019"},{"key":"bibr58-10943420211003313","doi-asserted-by":"publisher","DOI":"10.1145\/1141885.1141886"},{"key":"bibr59-10943420211003313","doi-asserted-by":"publisher","DOI":"10.1137\/110825753"},{"key":"bibr60-10943420211003313","doi-asserted-by":"publisher","DOI":"10.1109\/SC.2006.30"},{"key":"bibr61-10943420211003313","doi-asserted-by":"publisher","DOI":"10.1002\/nla.1839"},{"key":"bibr62-10943420211003313","unstructured":"Lindstrom P (2018) Zfp version 0.5.3. Available at: https:\/\/zfp.readthedocs.io\/en\/release0.5.3\/index.htm (accessed 2020)."},{"key":"bibr63-10943420211003313","volume-title":"Multigrid for matrix-free finite element computations on graphics processors","author":"Ljungkvist K","year":"2017"},{"key":"bibr64-10943420211003313","first-page":"3322813","volume":"6","author":"Ljungkvist K","year":"2019","journal-title":"ACM Transactions on Parallel Processing"},{"key":"bibr65-10943420211003313","doi-asserted-by":"publisher","DOI":"10.1109\/SHPCC.1994.296698"},{"key":"bibr66-10943420211003313","doi-asserted-by":"publisher","DOI":"10.1017\/S096249290626001X"},{"key":"bibr67-10943420211003313","doi-asserted-by":"publisher","DOI":"10.1145\/321386.321394"},{"key":"bibr1-10943420211003313","unstructured":"NVIDIA (2017) Nvidia Tesla V100 GPU Architecture. Available at: https:\/\/images.nvidia.com\/content\/volta-architecture\/pdf\/volta-architecture-whitepaper.pdf (accessed 2020)."},{"key":"bibr68-10943420211003313","doi-asserted-by":"publisher","DOI":"10.1007\/s13160-018-0310-3"},{"key":"bibr69-10943420211003313","doi-asserted-by":"publisher","DOI":"10.1007\/s13160-019-00348-4"},{"key":"bibr70-10943420211003313","doi-asserted-by":"publisher","DOI":"10.1016\/0024-3795(80)90167-6"},{"key":"bibr71-10943420211003313","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-95168-3_29"},{"key":"bibr72-10943420211003313","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827500381239"},{"key":"bibr73-10943420211003313","doi-asserted-by":"publisher","DOI":"10.1137\/120897687"},{"key":"bibr74-10943420211003313","doi-asserted-by":"publisher","DOI":"10.1137\/050630416"},{"key":"bibr75-10943420211003313","doi-asserted-by":"publisher","DOI":"10.1137\/130911561"},{"key":"bibr76-10943420211003313","doi-asserted-by":"publisher","DOI":"10.1137\/0913042"},{"key":"bibr77-10943420211003313","first-page":"1","author":"Richter C","year":"2014","journal-title":"9th IET International Conference on Computation in Electromagnetics (CEM 2014)"},{"key":"bibr78-10943420211003313","doi-asserted-by":"publisher","DOI":"10.1137\/0907058"},{"key":"bibr79-10943420211003313","volume-title":"Introduction to Data Compression, Fourth Edition","author":"Sayood K","year":"2012","edition":"4"},{"key":"bibr80-10943420211003313","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827502406415"},{"key":"bibr81-10943420211003313","doi-asserted-by":"publisher","DOI":"10.1090\/S0025-5718-1980-0572859-4"},{"key":"bibr82-10943420211003313","doi-asserted-by":"publisher","DOI":"10.1007\/BF02309342"},{"key":"bibr83-10943420211003313","doi-asserted-by":"publisher","DOI":"10.1007\/s00211-006-0042-1"},{"key":"bibr84-10943420211003313","volume-title":"Introduction to Matrix Computations","author":"Stewart GW","year":"1973"},{"key":"bibr85-10943420211003313","doi-asserted-by":"publisher","DOI":"10.1145\/2642769.2642794"},{"key":"bibr86-10943420211003313","volume-title":"Aggregations of elementary transformations","author":"Sun X","year":"1996"},{"key":"bibr87-10943420211003313","doi-asserted-by":"crossref","first-page":"e2343","DOI":"10.1002\/nla.2343","volume":"28","author":"\u015awirydowicz K","year":"2020","journal-title":"Numerical Linear Algebra with Applications"},{"key":"bibr88-10943420211003313","author":"Tamstorf R","year":"2020","journal-title":"SIAM Journal on Scientific Computing"},{"key":"bibr89-10943420211003313","author":"Tamstorf R","year":"2020","journal-title":"SIAM Journal on Scientific Computing"},{"key":"bibr90-10943420211003313","unstructured":"The Trilinos Project Team (2020) The Trilinos Project Website. Available at: https:\/\/trilinos.github.io (accessed 2020)."},{"key":"bibr91-10943420211003313","doi-asserted-by":"publisher","DOI":"10.1137\/S0895479899359837"},{"key":"bibr92-10943420211003313","doi-asserted-by":"publisher","DOI":"10.1137\/S0895479802403459"},{"key":"bibr93-10943420211003313","doi-asserted-by":"publisher","DOI":"10.1137\/S1064827599353865"},{"key":"bibr94-10943420211003313","doi-asserted-by":"publisher","DOI":"10.1137\/0909010"},{"key":"bibr95-10943420211003313","volume-title":"Rounding Errors in Algebraic Processes","author":"Wilkinson JH","year":"1963"},{"key":"bibr96-10943420211003313","doi-asserted-by":"publisher","DOI":"10.1016\/j.procs.2016.05.502"},{"key":"bibr97-10943420211003313","doi-asserted-by":"publisher","DOI":"10.1137\/14M0973773"},{"key":"bibr98-10943420211003313","doi-asserted-by":"publisher","DOI":"10.1145\/2898347"}],"container-title":["The International Journal of High Performance Computing Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/journals.sagepub.com\/doi\/pdf\/10.1177\/10943420211003313","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/journals.sagepub.com\/doi\/full-xml\/10.1177\/10943420211003313","content-type":"application\/xml","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/journals.sagepub.com\/doi\/pdf\/10.1177\/10943420211003313","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,4,29]],"date-time":"2026-04-29T08:17:11Z","timestamp":1777450631000},"score":1,"resource":{"primary":{"URL":"https:\/\/journals.sagepub.com\/doi\/10.1177\/10943420211003313"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,3,19]]},"references-count":98,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2021,7]]}},"alternative-id":["10.1177\/10943420211003313"],"URL":"https:\/\/doi.org\/10.1177\/10943420211003313","relation":{},"ISSN":["1094-3420","1741-2846"],"issn-type":[{"value":"1094-3420","type":"print"},{"value":"1741-2846","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,3,19]]}}}