{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,28]],"date-time":"2026-02-28T16:17:50Z","timestamp":1772295470723,"version":"3.50.1"},"reference-count":20,"publisher":"MDPI AG","issue":"4","license":[{"start":{"date-parts":[[2013,11,25]],"date-time":"2013-11-25T00:00:00Z","timestamp":1385337600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/3.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Algorithms"],"abstract":"<jats:p>We address the numerical solution of Lyapunov, algebraic and differential Riccati equations, via the matrix sign function, on platforms equipped with general-purpose multicore processors and, optionally, one or more graphics processing units (GPUs). In particular, we review the solvers for these equations, as well as the underlying methods, analyze their concurrency and scalability and provide details on their parallel implementation. Our experimental results show that this class of hardware provides sufficient computational power to tackle large-scale problems, which only a few years ago would have required a cluster of computers.<\/jats:p>","DOI":"10.3390\/a6040857","type":"journal-article","created":{"date-parts":[[2013,11,25]],"date-time":"2013-11-25T12:20:55Z","timestamp":1385382055000},"page":"857-870","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":9,"title":["Solving Matrix Equations on Multi-Core and Many-Core Architectures"],"prefix":"10.3390","volume":"6","author":[{"given":"Peter","family":"Benner","sequence":"first","affiliation":[{"name":"Max Planck Institute for Dynamics of Complex Technical Systems, Sandtorstr 1, Magdeburg 39106, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Pablo","family":"Ezzatti","sequence":"additional","affiliation":[{"name":"Instituto de Computaci\u00f3n, Univ. de la Rep\u00fablica, Julio Herrera y Reissig 565, Montevideo 11300, Uruguay"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Hermann","family":"Mena","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Innsbruck, Technikerstr. 19a, Innsbruck 6020, Austria"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Enrique","family":"Quintana-Ort\u00ed","sequence":"additional","affiliation":[{"name":"Departamento de Ingenier\u00eda y Ciencia de Computadores, Universidad Jaime I, Av. de Vicent Sos Baynat s\/n, Castell\u00f3n 12071, Spain"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Alfredo","family":"Rem\u00f3n","sequence":"additional","affiliation":[{"name":"Max Planck Institute for Dynamics of Complex Technical Systems, Sandtorstr 1, Magdeburg 39106, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2013,11,25]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Abou-Kandil, H., Freiling, G., Ionescu, V., and Jank, G. (2003). Matrix Riccati Equations in Control and Systems Theory, Birkh\u00e4user.","DOI":"10.1007\/978-3-0348-8081-7"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"677","DOI":"10.1080\/00207178008922881","article-title":"Linear model reduction and solution of the algebraic Riccati equation by use of the sign function","volume":"32","author":"Roberts","year":"1980","journal-title":"Int. J. Control"},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"1701","DOI":"10.1016\/j.parco.2003.05.013","article-title":"State-space truncation methods for parallel model reduction of large-scale systems","volume":"29","author":"Benner","year":"2003","journal-title":"Parallel Comput."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"879","DOI":"10.1080\/10556780802058721","article-title":"Solving linear-quadratic optimal control problems on parallel computers","volume":"23","author":"Benner","year":"2008","journal-title":"Optim. Methods Softw."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"88","DOI":"10.1007\/978-3-642-28145-7_9","article-title":"Accelerating Model Reduction of Larger Linear Systems with Graphics Processors","volume":"Volume 7134","year":"2012","journal-title":"Applied Parallel and Scientific Computing"},{"key":"ref_6","unstructured":"Benner, P., Ezzatti, P., Mena, H., Quintana-Ort\u00ed, E.S., and Rem\u00f3n, A. (2011, January 26). Solving Differential Riccati Equations on Multi-GPU Platforms. Proceedings of the 10th International Conference on Computational and Mathematical Methods in Science and Engineering\u2013CMMSE 2011, Benidorm, Spain."},{"key":"ref_7","first-page":"132","article-title":"Using Hybrid CPU-GPU Platforms to Accelerate the Computation of the Matrix Sign Function","volume":"Volume 6043","author":"Lin","year":"2009","journal-title":"Proceedings of the 7th International Workshop on Algorithms, Models and Tools for Parallel Computing on Heterogeneous Networks"},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"549","DOI":"10.1007\/978-3-642-31464-3_56","article-title":"Accelerating BST Methods for Model Reduction with Graphics Processors","volume":"Volume 7203","author":"Wyrzykowski","year":"2012","journal-title":"Parallel Processing and Applied Mathematics"},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"1114","DOI":"10.1007\/s11227-013-0889-8","article-title":"Accelerating the Lyapack library using GPUs","volume":"65","author":"Dufrechu","year":"2013","journal-title":"J. Supercomput."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"439","DOI":"10.1016\/j.parco.2010.12.002","article-title":"A mixed-precision algorithm for the solution of Lyapunov equations on hybrid CPU-GPU platforms","volume":"37","author":"Benner","year":"2011","journal-title":"Parallel Comput."},{"key":"ref_11","unstructured":"Golub, G.H., and van Loan, C.F. (1996). Matrix Computations, Johns Hopkins University Press. 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Max Planck Institute Magdeburg Preprint MPIMD\/12-13, 2012. Available online: http:\/\/www.mpi-magdeburg.mpg.de\/preprints\/."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"422","DOI":"10.1145\/504210.504213","article-title":"FLAME: Formal linear algebra methods environment","volume":"27","author":"Gunnels","year":"2001","journal-title":"ACM Trans. Math. Softw."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"1762","DOI":"10.1137\/S1064827598345679","article-title":"A note on parallel matrix inversion","volume":"22","author":"Sun","year":"2001","journal-title":"SIAM J. Sci. Comput."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"429","DOI":"10.1007\/s11227-011-0606-4","article-title":"Using graphics processors to accelerate the computation of the matrix inverse","volume":"58","author":"Ezzatti","year":"2011","journal-title":"J. Supercomput."},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Ezzatti, P., Quintana-Ort\u00ed, E.S., and Rem\u00f3n, A. (,  2011). High Performance Matrix Inversion on a Multi-Core Platform with Several GPUs. Proceedings of the 19th International Euromicro Conference on Parallel, Distributed and Network-Based Processing, Berlin\/Heidelberg, Germany.","DOI":"10.1109\/PDP.2011.66"},{"key":"ref_20","unstructured":"Oberwolfach Model Reduction Benchmark Collection. Available online: http:\/\/simulation.uni-freiburg.de\/downloads\/benchmark."}],"container-title":["Algorithms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1999-4893\/6\/4\/857\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T21:50:52Z","timestamp":1760219452000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1999-4893\/6\/4\/857"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2013,11,25]]},"references-count":20,"journal-issue":{"issue":"4","published-online":{"date-parts":[[2013,12]]}},"alternative-id":["a6040857"],"URL":"https:\/\/doi.org\/10.3390\/a6040857","relation":{},"ISSN":["1999-4893"],"issn-type":[{"value":"1999-4893","type":"electronic"}],"subject":[],"published":{"date-parts":[[2013,11,25]]}}}