Specialized Parallel Algorithms forSolving Linear Matrix Equations in Control

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 diicult to parallelize. We investigate iterative solvers based on the matrix sign function and the Smith iteration which are highly eecient on parallel distributed computers. We also show that by coding using the Parallel Linear Algebra Package (PLAPACK) it is possible to exploit 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 eeort. 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 eeciency of these parallel solvers.

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