Computing the Lyapunov operator \varphi-functions, with an application to matrix-valued exponential integrators

In this paper, we develop efficient and accurate evaluation for the Lyapunov operator function $\varphil(\mathcal{L}A)[Q],$ where $\varphil(\cdot)$ is the function related to the exponential, $\mathcal{L}A$ is a Lyapunov operator and $Q$ is a symmetric and full-rank matrix. An important application of the algorithm is to the matrix-valued exponential integrators for matrix differential equations such as differential Lyapunov equations and differential Riccati equations. The method is exploited by using the modified scaling and squaring procedure combined with the truncated Taylor series. A quasi-backward error analysis is presented to determine the value of the scaling parameter and the degree of the Taylor approximation. Numerical experiments show that the algorithm performs well in both accuracy and efficiency.