A low-rank algorithm for solving Lyapunov operator $\varphi$-functions within the matrix-valued exponential integrators

In this work we develop a low-rank algorithm for the computation of low-rank approximations to the large-scale Lyapunov operator $\varphi$-functions. Such computations constitute the important ingredient to the implementation of matrix-valued exponential integrators for large-scale stiff matrix differential equations where the (approximate) solutions are of low rank. We evaluate the approximate solutions of $LDL^T$-type based on a scaling and recursive procedure. The key parameters of the method are determined using a quasi-backward error bound combined with the computational cost. Numerical results demonstrate that our method can be used as a basis of matrix-valued exponential integrators for solving large-scale differential Lyapunov equations and differential Riccati equations.