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While extended and rational block Krylov subspace methods provide a major reduction in iteration counts over polynomial block KSMs, they also require reliable solvers for the coefficient matrices, and these solvers are often iterative methods themselves. It is not hard to devise scenarios in which the available memory, and consequently the dimension of the Krylov subspace, is limited. In such scenarios for linear systems and eigenvalue problems, restarting is a well\u2010explored technique for mitigating memory constraints. In this work, such restarting techniques are applied to polynomial KSMs for matrix equations with a compression step to control the growing rank of the residual. An error analysis is also performed, leading to heuristics for dynamically adjusting the basis size in each restart cycle. A panel of numerical experiments demonstrates the effectiveness of the new method with respect to extended block KSMs.<\/jats:p>","DOI":"10.1002\/nla.2339","type":"journal-article","created":{"date-parts":[[2020,10,13]],"date-time":"2020-10-13T08:15:10Z","timestamp":1602576910000},"update-policy":"http:\/\/dx.doi.org\/10.1002\/crossmark_policy","source":"Crossref","is-referenced-by-count":3,"title":["Compress\u2010and\u2010restart block Krylov subspace methods for Sylvester matrix equations"],"prefix":"10.1002","volume":"28","author":[{"ORCID":"http:\/\/orcid.org\/0000-0003-3369-2958","authenticated-orcid":false,"given":"Daniel","family":"Kressner","sequence":"first","affiliation":[{"name":"Institute for Mathematics EPF Lausanne Lausanne Switzerland"}]},{"ORCID":"http:\/\/orcid.org\/0000-0001-9851-6061","authenticated-orcid":false,"given":"Kathryn","family":"Lund","sequence":"additional","affiliation":[{"name":"Department of Numerical Mathematics, Faculty of Mathematics and Physics Charles University Prague Czech Republic"}]},{"ORCID":"http:\/\/orcid.org\/0000-0003-1813-4181","authenticated-orcid":false,"given":"Stefano","family":"Massei","sequence":"additional","affiliation":[{"name":"Centre for Analysis, Scientific Computing and Applications (CASA) TU\/e Eindhoven The Netherlands"}]},{"ORCID":"http:\/\/orcid.org\/0000-0002-6987-4430","authenticated-orcid":false,"given":"Davide","family":"Palitta","sequence":"additional","affiliation":[{"name":"Research Group Computational Methods in Systems and Control Theory (CSC) Max Planck Institute for Dynamics of Complex Technical Systems Magdeburg Germany"}]}],"member":"311","published-online":{"date-parts":[[2020,10,13]]},"reference":[{"key":"e_1_2_8_2_1","doi-asserted-by":"publisher","DOI":"10.1137\/140993867"},{"key":"e_1_2_8_3_1","doi-asserted-by":"publisher","DOI":"10.1002\/nla.366"},{"key":"e_1_2_8_4_1","doi-asserted-by":"publisher","DOI":"10.1016\/S0167-6911(00)00010-4"},{"key":"e_1_2_8_5_1","doi-asserted-by":"publisher","DOI":"10.1137\/1.9780898718713"},{"key":"e_1_2_8_6_1","volume-title":"Robust and optimal control","author":"Zhou K","year":"1996"},{"key":"e_1_2_8_7_1","doi-asserted-by":"publisher","DOI":"10.1007\/s10543-015-0575-8"},{"key":"e_1_2_8_8_1","volume-title":"Numerical solution of algebraic Riccati equations, vol. 9 of Fundamentals of Algorithms","author":"Bini DA","year":"2012"},{"key":"e_1_2_8_9_1","volume-title":"Solving second and third\u2010order approximations to DSGE models: A recursive Sylvester equation solution, Norges bank working paper","author":"Binning A","year":"2013"},{"key":"e_1_2_8_10_1","doi-asserted-by":"publisher","DOI":"10.1002\/gamm.201310003"},{"key":"e_1_2_8_11_1","doi-asserted-by":"publisher","DOI":"10.1137\/130912839"},{"key":"e_1_2_8_12_1","doi-asserted-by":"publisher","DOI":"10.1017\/S0962492916000076"},{"key":"e_1_2_8_13_1","volume-title":"Boundary element methods, vol. 39 of Springer series in computational mathematics","author":"Sauter S. 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