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 quantities of the form $f(X)$ where $X$ denotes the solution of a Lyapunov equation. 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 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.

翻译：参数化Lyapunov方程序列出现在众多应用场景中。此外，此类方程的解通常是整体计算流程中的中间步骤，其主要目标是计算形如$f(X)$的量，其中$X$表示Lyapunov方程的解。我们关注系数矩阵的参数依赖性可编码为对固定“种子”矩阵的低秩修正的问题。我们提出了两种充分利用这种共性结构的新型数值方法。第一种方法基于Krylov回收技术，适用于小规模问题，因为它利用了稠密数值线性代数工具。第二种算法则通过依赖基于扩展Krylov子空间的最新投影技术，能够解决大规模问题。我们在阻尼振动系统研究及多智能体系统输出同步问题分析中产生的多个问题上测试了新算法。结果表明，我们提出的算法优于现有技术，能够显著加速精确解的求解过程。