Numerical solution of singular Sylvester equations

We consider the numerical solution of the large scale singular Sylvester equation of the form AX - XD inverted perpendicular = E (or AXB inverted perpendicular inverted perpendicular - CXD inverted perpendicular inverted perpendicular = E ), where the spectra Lambda(A) A ) and Lambda(D) D ) (or, Lambda(A A - lambda C) C ) and Lambda(D D - lambda B)) B )) have a nonempty intersection. Using appropriate invariant subspaces, the singular Sylvester equation is rewritten as four Sylvester equations, three of which are nonsingular and one singular. When the invariant subspaces are small, so are three of the equations (including the singular one) which can be solved efficiently. The fourth is large but nonsingular with structures and may be solved using the projection method with Krylov subspaces or techniques involving hierarchical matrices. Some numerical examples for the subspace method are provided. Crown Copyright (c) 2023 Published by Elsevier B.V. All rights reserved.