Direct methods for matrix Sylvester and Lyapunov equations (Q1811877)

Summary: We revisit the two standard dense methods for matrix Sylvester and Lyapunov equations: the \textit{R. H. Bartels} and \textit{G. W. Stewart} [Commun. ACM 15, No. 9, 820--826 (1972; Zbl 1372.65121)] method for \(\mathbf{A}_{1}\mathbf{X}+ \mathbf{XA}_{2}+\mathbf{D}=\mathbf{0}\) and \textit{S. J. Hammarling}'s method [IMA J. Numer. Anal. 2, 303-323 (1982; Zbl 0492.65017)] for \(\mathbf{AX}+\mathbf{XA}^{T}+\mathbf{BB}^{T}=\mathbf{0}\) with \(\mathbf{A}\) stable. We construct three schemes for solving the unitarily reduced quasitriangular systems. We also construct a new rank-1 updating scheme in Hammarling's method. This new scheme is able to accommodate a \(\mathbf{B}\) with more columns than rows as well as the usual case of a \(\mathbf{B}\) with more rows than columns, while Hammarling's original scheme needs to separate these two cases. We compared all of our schemes with the Matlab Sylvester and Lyapunov solver \(lyap.m\); the results show that our schemes are much more efficient. We also compare our schemes with the Lyapunov solver \textit{sllyap} in the currently possibly the most efficient control library package SLICOT; numerical results show our scheme to be competitive.