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Sequences of parametrized Lyapunov equations can be encountered in manyapplication settings. Moreover, solutions of such equations are oftenintermediate steps of an overall procedure whose main goal is the computationof quantities of the form $f(X)$ where $X$ denotes the solution of a Lyapunovequation. We are interested in addressing problems where the parameterdependency of the coefficient matrix is encoded as a low-rank modification to a\emph{seed}, fixed matrix. We propose two novel numerical procedures that fullyexploit such a common structure. The first one builds upon recycling Krylovtechniques, and it is well-suited for small dimensional problems as it makesuse of dense numerical linear algebra tools. The second algorithm can insteadaddress large-scale problems by relying on state-of-the-art projectiontechniques based on the extended Krylov subspace. We test the new algorithms onseveral problems arising in the study of damped vibrational systems and theanalyses of output synchronization problems for multi-agent systems. Ourresults show that the algorithms we propose are superior to state-of-the-arttechniques as they are able to remarkably speed up the computation of accuratesolutions.

Efficient solution of sequences of parametrized Lyapunov equations with applications