On BLAS Level-3 Implementations of Common Solvers for (Quasi-) Triangular Generalized Lyapunov Equations

Abstract

The solutions of Lyapunov and generalized Lyapunov equations are a key player in many applications in systems and control theory. Their stable numerical computation, when the full solution is sought, is considered solved since the seminal work of Bartels and Stewart [1972] and its generalization by Penzl [1997]. Those variants do not go completely beyond BLAS level-2 style implementation. On modern computers, however, the formulation of level-3 BLAS type implementations is crucial to enable optimal usage of cache hierarchies and modern block scheduling methods based on directed acyclic graphs describing the interdependence of single block computations. Although there exists a recursive blocked solution scheme for (quasi-) triangular generalized Lyapunov equations [Jonsson and Kågström 2002b], we focus on standard blocking techniques. Using the standard blocking approach our contribution lifts the aforementioned level-2 algorithm by Penzl to BLAS level-3 for (quasi-) triangular equations. Especially, we consider the solution of the appearing Sylvester equations and provide a hybrid algorithm merging our strategy with the recursive blocking method above.