Solving linear matrix equations in control problems on distributed memory multiprocessors

Abstract

Linear matrix equations such as Sylvester, Lyapunov and commutant matrix equations play an important role in many control problems, like the design of Luenberger's observers, pole assignment problems, system balancing and model reduction, inertia and stability problems, generic matrix function computation, etc. Two of the most efficient methods for solving linear matrix equations are the Schur algorithm and the Hessenberg-Schur algorithm. In this paper, we present parallel cyclic algorithms based on the Schur and Hessenberg-Schur methods for solving the Sylvester matrix equation. We also present parallel cyclic algorithms based on the Schur method for solving Lyapunov and commutant matrix equations. In the case of Lyapunov equations we also consider the problem of computing the Cholesky factor of the unknown matrix.<>