Algorithms and Library Software for Periodic and Parallel Eigenvalue Reordering and Sylvester-Type Matrix Equations with Condition Estimation

Abstract: This Thesis contains contributions in two different but closely related subfields of Scientific and Parallel Computing which arise in the context of various eigenvalue problems: periodic and parallel eigenvalue reordering and parallel algorithms for Sylvestertype matrix equations with applications in condition estimation.Many real world phenomena behave periodically, e.g., helicopter rotors, revolving satellites and dynamic systems corresponding to natural processes, like the water flow in a system of connected lakes, and can be described in terms of periodic eigenvalue problems. Typically, eigenvalues and invariant subspaces (or, specifically, eigenvectors) to certain periodic matrix products are of interest and have direct physical interpretations. The eigenvalues of a matrix product can be computed without forming the product explicitly via variants of the periodic Schur decomposition. In the first part of the Thesis, we propose direct methods for eigenvalue reordering in the periodic standard and generalized real Schur forms which extend earlier work on the standard and generalized eigenvalue problems. The core step of the methods consists of solving periodic Sylvester-type equations to high accuracy. Periodic eigenvalue reordering is vital in the computation of periodic eigenspaces corresponding to specified spectra. The proposed direct reordering methods rely on orthogonal transformations and can be generalized to more general periodic matrix products where the factors have varying dimensions and ±1 exponents of arbitrary order.In the second part, we consider Sylvester-type matrix equations, like the continuoustime 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, which have applications in many areas. Examples include eigenvalue problems and condition estimation, and several problems in control system design and analysis. The parallel algorithms presented are based on the well-known Bartels–Stewart’s method and extend earlier work on triangular Sylvester-type matrix equations resulting in a novel software library SCASY. The parallel library provides robust and scalable software for solving 44 sign and transpose variants of eight common Sylvester-type matrix equations. SCASY also includes a parallel condition estimator associated with each matrix equation.In the last part of the Thesis, we propose parallel variants of the direct eigenvalue reordering method for the standard and generalized real Schur forms. Together with the existing and future parallel implementations of the non-symmetric QR/QZ algorithms and the parallel Sylvester solvers presented in the Thesis, the developed software can be used for parallel computation of invariant and deflating subspaces corresponding to specified spectra and associated reciprocal condition number estimates.