  In this paper, we develop efficient and accurate evaluation for the Lyapunov
operator function $\varphi_l(\mathcal{L}_A)[Q],$ where $\varphi_l(\cdot)$ 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.
