A Low-rank Interpolatory Projection Algorithm for Solving Large-scale T-Sylvester Equations

Abstract

This paper considers large-scale T-Sylvester equations of the form $AX - X^\top E^\top + B_1B_2^\top = 0$, which admit a low-rank solution. It is shown that when the unique solution of the T-Sylvester equation is low-rank, the problem naturally reduces to a tangential interpolation problem via oblique projection. The specific interpolation points and tangential directions needed to obtain the low-rank solution are not known a priori, thus requiring an iterative approach. An iterative interpolatory projection algorithm is proposed based on these interpolation conditions, which iteratively refines the interpolation data as the projection matrices expand in the number of columns. Numerical examples demonstrate that the proposed algorithm converges with projection matrices having significantly fewer columns compared to existing Krylov-subspace-based projection methods, confirming the superiority of the proposed algorithm over existing approaches.