Master ’ s Thesis Proposal , 20 credits : ScaLAPACK - style algorithms for Periodic Matrix Equations

1 Motivation This Master's Thesis project considers software for solving periodic Sylvester-type matrix equations. Recently, the ScaLAPACK-style library SCASY was completed. SCASY is a parallel HPC software library that solves for 42 sign and transpose variant of 8 common standard and generalized Sylvester-type matrix equations (see Table 1) which builds on the Table 1: The Sylvester-type matrix equations considered in the SCASY library. CT and DT denote the continuous-time and discrete-time variants, respectively. Name Matrix Equation Standard Sylvester (CT) AX − XB = C Standard Lyapunov (CT) AX + XA T = C Standard Sylvester (DT) AXB T − X = C Standard Lyapunov (DT) AXA T − X = C Generalized Coupled Sylvester (AX − Y B, DX − Y E) = (C, F) Generalized Sylvester AXB T − CXD T = E Generalized Lyapunov (CT) AXE T + EXA T = C Generalized Lyapunov (DT) AXA T − EXE T = C An example of a periodic Sylvester-type matrix equation is the periodic Sylvester equation (PSE): A k X k − X k⊕1 B k = C k , R m×n and a ⊕ b = a + b mod K. Serial algorithms for solving this equations was developed in, e.g., [7]. Periodic counterparts of many of the equations listed in Table 1 can be formulated, see, e.g., [13, 15, 16]. Generalized coupled matrix equations also arises in the context of computing periodic deflating subspaces with specified eigenvalues, see [8]. In the project, we would like to port the algorithms in SCASY to the periodic Sylvester-type matrix equations.