PARALLEL ALGORITHMS FOR THE QUASI-TRIANGULAR GENERALIZED SYLVESTER MATRIX EQUATION ON A SHARED MEMORY MULTIPROCESSOR

Abstract

In this paper we discuss the resolution of the generalized Sylvester matrix equation AXB +CXD = E, where A, C ∈Rmxm, B, D ∈Rmxm, E ∈Rmxn, 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 linear systems (Lewis [17]). Particular cases of this equation are the Sylvester equation AX +XD = E, and the Stein equation AXB +X = E. There are several sequential algorithms for solving both Sylvester and Stein equations. The most effective methods for the Sylvester matrix equation are the Bartels-Stewart algorithm [2] and the algorithm proposed by Golub, Nash and Van Loan [11]. Recently, some authors (Chu [3], Kågström-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 Kågström, Nyström and Poromaa [14] for solving the triangular Sylvester matrix equation and by Marqués and Hernández [18] for the quasi-triangular Stein matrix equation.