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The most straightforward approach computes <jats:inline-graphic xmlns:xlink=\"http:\/\/www.w3.org\/1999\/xlink\" xlink:href=\"graphic\/nla1973-math-0002.png\" xlink:title=\"urn:x-wiley:nla:media:nla1973:nla1973-math-0002\"\/> from the solution of an <jats:italic>m<\/jats:italic><jats:italic>n<\/jats:italic> \u00d7 <jats:italic>m<\/jats:italic><jats:italic>n<\/jats:italic> linear system, typically limiting the feasible values of <jats:italic>m<\/jats:italic>,<jats:italic>n<\/jats:italic> to a few hundreds at most. Our new approach exploits the fact that <jats:italic>X<\/jats:italic> can often be well approximated by a low\u2010rank matrix. It combines greedy low\u2010rank techniques with Galerkin projection and preconditioned gradients. In turn, only linear systems of size <jats:italic>m<\/jats:italic> \u00d7 <jats:italic>m<\/jats:italic> and <jats:italic>n<\/jats:italic> \u00d7 <jats:italic>n<\/jats:italic> need to be solved. Moreover, these linear systems inherit the sparsity of the coefficient matrices, which allows to address linear matrix equations as large as <jats:italic>m<\/jats:italic> = <jats:italic>n<\/jats:italic> = <jats:italic>O<\/jats:italic>(10<jats:sup>5<\/jats:sup>). Numerical experiments demonstrate that the proposed methods perform well for generalized Lyapunov equations. Even for the case of standard Lyapunov equations, our methods can be advantageous, as we do not need to assume that <jats:italic>C<\/jats:italic> has low rank. Copyright \u00a9 2015 John Wiley &amp; Sons, Ltd.<\/jats:p>","DOI":"10.1002\/nla.1973","type":"journal-article","created":{"date-parts":[[2015,3,4]],"date-time":"2015-03-04T09:21:16Z","timestamp":1425460876000},"page":"564-583","source":"Crossref","is-referenced-by-count":30,"title":["Truncated low\u2010rank methods for solving general linear matrix equations"],"prefix":"10.1002","volume":"22","author":[{"given":"Daniel","family":"Kressner","sequence":"first","affiliation":[{"name":"Chair of Numerical Algorithms and HPC MATHICSE, EPF Lausanne  CH\u20101015 Lausanne Switzerland"}]},{"given":"Petar","family":"Sirkovi\u0107","sequence":"additional","affiliation":[{"name":"Chair of Numerical Algorithms and HPC MATHICSE, EPF Lausanne  CH\u20101015 Lausanne Switzerland"}]}],"member":"311","published-online":{"date-parts":[[2015,3,4]]},"reference":[{"key":"e_1_2_8_2_1","doi-asserted-by":"publisher","DOI":"10.1002\/gamm.201310003"},{"key":"e_1_2_8_3_1","unstructured":"SimonciniV.Computational methods for linear matrix equations 2013. 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