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This paper considers the solution of large-scale Lyapunov matrix equations of the form AX +XA T = −bb T . The Arnoldi method is a simple but sometimes ineffective approach to deal with such equations. One of its major drawbacks is excessive memory consumption caused by slow convergence. To overcome this disadvantage, we propose two-pass Krylov subspace methods, which only compute the solution of the compressed equation in the first pass. The second pass computes the product of the Krylovdoi:10.1109/cacsd.2008.4627370 dblp:conf/cacsd/Kressner08 fatcat:eame625xjbetxeywn3arg2ieda