Specialized Parallel Algorithms for Solving Lyapunov and Stein Equations

• Published in: Journal of parallel and distributed computing Vol. 61; no. 10; pp. 1489 - 1504
• Main Authors: Quintana-Ortı́, Enrique S., van de Geijn, Robert
• Format: Journal Article
• Language: English
• Published: San Diego, CA Elsevier Inc 01.10.2001; Elsevier
• Subjects: Applied sciences; Computer science; control theory; systems; Computer systems and distributed systems. User interface; Exact sciences and technology; Mathematics; Numerical analysis; Numerical analysis. Scientific computation; Numerical linear algebra; Sciences and techniques of general use; Software; Performance evaluation; Parallel algorithm; Operator equation; Control synthesis; Iterative method; Lyapunov equation; Linear coding; Equation system solving; Model matching; Parallel system; Least squares method; Linear algebra; QR factorization; Control theory; Matrix equation
• ISSN: 0743-7315; 1096-0848
• DOI: 10.1006/jpdc.2001.1732

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.