Parallel Algorithms for the Quasi-Triangular Generalized Sylvester Matrix Equation on a Shared Memory Multiprocessor

Abstract

Abstract In this paper we discuss the resolution of the generalized Sylvester matrix equation AXB + CXD = E, where A, C ∈ R n×n B, D ∈ R m×n E ∈ R m × n , and the unknown X is m × n. This equation is related to different topics in Control Theory and Linear Algebra; perturbation analysis of the generalized eigenvalue problem (Stewart-Sun [19]), solution of implicit linear differential equations (Epton [4)), analysis and design of control problems for descriptor linearsystems(Lewis [17))Particular cases of this equation are the Sylvester equation AX + X D =E, and the Stein equation AXB +X = E. There are several sequential algorithms for solving both Sylvester and Stein equations. The mosteffective methods for the Sylvester matrix equation are the Bartels-Stewart algorithm [2] and the algorithm proposedby Golub, Nash and Van Loan [11]. Recently, some authors (Chu [3], Kagstrom-Westin [16], Gardiner and colleagues[9, 10)) have improved the Bartels-Stewart algorithm for solving the generalized Sylvester matrix equation. Our objective in this paper is the study of the parallel solution of the quasi-triangular matrix equation AXB + CXD = E, using coarse-grain algorithms on a shared memory multiprocessor with available BLAS routines [8]. Our approach extends the results obtained by Kagstrom, Nystrom and Poromaa [14] for solving the triangular Sylvestermatrix equation and by Marques and Hernadez [18] for the quasi-triangular Stein matrix equation.