A Rational Krylov Method for Large‐Scale Linear Multidimensional Dynamical Systems

ABSTRACT In this paper, we investigate the use of multilinear algebra for reducing the order of multidimensional linear time‐invariant systems. Our main tools are tensor rational Krylov subspace methods, which enable us to approximate the system's solution within a low‐dimensional subspace. We introduce the tensor rational block Arnoldi and tensor rational block Lanczos algorithms. By utilizing these methods, we develop a model reduction approach based on projection techniques. Additionally, we demonstrate how these approaches can be applied to large‐scale Lyapunov tensor equations, which are critical for the balanced truncation method, a well‐known technique for order reduction. An adaptive method for choosing the interpolation points is also introduced. Finally, some numerical experiments are reported to show the effectiveness of the proposed adaptive approaches.