A Comparison of Subspace methods for Sylvester Equations

Abstract

Sylvester equations AX_BX=C play an important roleinnumerical linear algebra. For example, they arise in the computation of invariant subspaces, in control problems, as linearizations of algebraic Riccati equations, and in the discretization of partial dierential equations. For small systems, direct methods are feasible. For large systems, iterative solution methods are available, like Krylov subspace methods. It can be observed that there are essentially two types of subspace methods for Sylvester equations: one in which block matrices are treated as rigid objects (functions on a grid), and one in which the blocks are seen as a basis of a subspace. In this short note we compare the two dierent types, and aim to identify which applications should make use of which solution methods.