In this work we introduce a low-rank algorithm designed to compute low-rank approximations of large-scale Lyapunov operator φ-functions. These computations are crucial for the implementation of matrix-valued exponential integrators tailored for large-scale stiff matrix differential equations, where the (approximate) solutions are of low rank. The method is developed using a scaling and recursive procedure, supplemented by a quasi-backward error analysis to determine the optimal parameters. The computational cost of the method primarily arises from the products of sparse matrix and block vectors.Numerical experiments confirm the effectiveness of the proposed method as a foundational tool for matrix-valued exponential integrators in addressing large-scale differential Lyapunov equations and Riccati equations.
MSC Classification: 65L05 , 65F10 , 65F30