Specialized Parallel Algorithms for Solving Lyapunov and Stein Equations

Abstract

Lyapunov and Stein matrix equations arise in many important analysis and synthesis applications in control theory. The traditional approach to solving these equations relies on the QR algorithm which is notoriously difficult to parallelize. We investigate iterative solvers based on the matrix sign function and the squared Smith iteration which are highly efficient on parallel distributed computers. We also show that by coding using the Parallel Linear Algebra Package (PLAPACK) it is possible to exploit the structure in the matrices and reduce the cost of these solvers. While the performance improvements due to the optimizations are modest, so is the coding effort. One of the optimizations, the updating of a QR factorization, has important applications elsewhere, e.g., in applications requiring the solution of a linear least-squares problem when the linear system is periodically updated. The experimental results on a Cray T3E attest to the high efficiency of these parallel solvers.