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.

翻译：在本文中,我们为Lyapunov 操作员函数 $\varphi_l(\mathcal{L ⁇ A)[Q] 开发了高效和准确的评价。 $\varphi_l(\cdot) 是指数值的函数, $\mathcal{L ⁇ A$是Lyapunov 操作员, $Q$是一个对称和全级矩阵。 算法的一个重要应用是矩阵值指数化矩阵差异方程式, 如差分的 Lyapunov 方程式和差分的Riccati 方程式。 该方法通过使用经修改的缩放和缩放程序, 结合短速的泰勒序列来加以利用。 进行了准反向错误分析, 以确定缩放参数的价值和泰勒近似值的程度。 数值实验显示, 算法在准确性和效率方面表现良好。