Contributions to Parallel Algorithms for Sylvester-type Matrix Equations and Periodic Eigenvalue Reordering in Cyclic Matrix Products

This Licentiate Thesis contains contributions in two different subfields of Computing Science: parallel ScaLAPACK-style algorithms for Sylvester-type matrix equations and periodic eigenvalue reordering in a cyclic product of matrices. Sylvester-type matrix equations, like the continuous-time Sylvester equation AX −XB = C, where A of size m×m, B of size n×n and C of size m×n are general matrices with real entries, have applications in many areas. For example, the continuous-time Sylvester equation shows up in eigenvalue problems, condition estimation of eigenvalue problems, e.g., sensitivity analysis of invariant subspaces corresponding to a specified spectrum, and in control and system theory. This thesis contributes to the area of parallel library ScaLAPACK-style software for solving Sylvester-type matrix equations. The algorithms and library software presented are based on the well-known Bartels–Stewart’s method and extend earlier work on triangular Sylvester-type matrix equations to general Sylvester matrix equations. The developed methods will serve as foundation for a future parallel software library for solving 42 sign and transpose variants of eight common Sylvester-type matrix equations. Many real world phenomena behave periodically, e.g., helicopter rotors and revolving satellites, and can be described in terms of periodic eigenvalue problems. Typically, eigenvalues and invariant subspaces (eigenvectors) to certain periodic matrix products are of interest and have direct physical interpretations. The eigenvalues of a cyclic matrix product can be computed via the periodic Schur decomposition. Our contribution in this area is a direct method for periodic eigenvalue reordering in the periodic real Schur form which extends earlier work on the standard and the generalized eigenvalue problems. Periodic eigenvalue reordering is vital in the computation of periodic eigenspaces corresponding to specified spectra and is utilized and requested, e.g., in recently proposed methods for solving periodic differential matrix equations arising in the analysis of the observability/controllability of linear continuous-time periodic systems and for solving discrete-time periodic Riccati equations arising in linear quadratic (LQ) optimal control problems. The proposed direct reordering method relies on orthogonal transformations only, i.e., is backward stable, and can be generalized to more general periodic matrix products arising in generalizations of the periodic Schur form.