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

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