  Sequences of parametrized Lyapunov equations can be encountered in many
application settings. Moreover, solutions of such equations are often
intermediate steps of an overall procedure whose main goal is the computation
of $\text{trace}(EX)$ where $X$ denotes the solution of a Lyapunov equation and
$E$ is a given matrix. We are interested in addressing problems where the
parameter dependency of the coefficient matrix is encoded as a low-rank
modification to a \emph{seed}, fixed matrix. We propose two novel numerical
procedures that fully exploit such a common structure. The first one builds
upon the Sherman-Morrison-Woodbury (SMW) formula and recycling Krylov
techniques, and it is well-suited for small dimensional problems as it makes
use of dense numerical linear algebra tools. The second algorithm can instead
address large-scale problems by relying on state-of-the-art projection
techniques based on the extended Krylov subspace. We test the new algorithms on
several problems arising in the study of damped vibrational systems and the
analyses of output synchronization problems for multi-agent systems. Our
results show that the algorithms we propose are superior to state-of-the-art
techniques as they are able to remarkably speed up the computation of accurate
solutions.
