Direct methods for matrix Sylvester and Lyapunov equations

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We revisit the two standard dense methods for matrix Sylvester and Lyapunov equations: the Bartels-Stewart method for A1X

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An improved numerical method for balanced truncation for symmetric second-order systems (2013, background): “Since direct methods [14, 15, 16] for solving (15) have a cubic computational complexity and quadratic memory demands for finding the solution X, their usage is limited to small or moderately large n.”
Computational Methods for Linear Matrix Equations (2016, methodology): “In [241], a variant of the Bartels-Stewart algorithm is proposed: the forward-backward substitution in step 2 is performed by a column-wise block scheme, which seems to be better suited for modern computer architectures than the original complex version; we also refer to [146],[147] for even more ef”
A New Iterative Method for Solving Large-Scale Lyapunov Matrix Equations (2007, methodology): “The function lyap computes the numerical solution of the small dimensional Lyapunov equation by means of a Schur decomposition type method; in our experiments, we used the algorithm proposed by Sorensen and Zhou [38], which implements the Bartels-Stewart method with only real arithmetic; other alter”