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.

翻译：参数化李雅普诺夫方程序列广泛存在于众多应用场景中。此外，此类方程的解常作为整体计算流程的中间步骤，其主要目标是计算 $\text{trace}(EX)$，其中 $X$ 表示李雅普诺夫方程的解，$E$ 为给定矩阵。本文关注系数矩阵的参数依赖性可表示为固定“种子”矩阵低秩修正的一类问题。我们提出了两种新型数值方法，以充分利用这种共有结构。第一种方法基于 Sherman-Morrison-Woodbury (SMW) 公式与循环 Krylov 技术，适用于小规模问题，因其采用了稠密数值线性代数工具。第二种算法则能通过基于扩展 Krylov 子空间的前沿投影技术处理大规模问题。我们在阻尼振动系统研究与多智能体系统输出同步分析等若干问题上测试了新算法。结果表明，所提算法相比现有技术具有显著优势，能够大幅加速精确解的求解过程。