Direct methods for matrix Sylvester and Lyapunov equations

Abstract

We revisit the two standard dense methods for matrix Sylvester and Lyapunov equations: the Bartels-Stewart method for ${\mathbf{A}}_1\mathbf{X}+\mathbf{X}{\mathbf{A}}_2+\mathbf{D}=\mathbf{0}$ and Hammarling's method for $\mathbf{A}\mathbf{X}+\mathbf{X}{\mathbf{A}}^T+\mathbf{B}{\mathbf{B}}^T=\mathbf{0}$ 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 sllyap in the currently possibly the most efficient control library package SLICOT; numerical results show our scheme to be competitive.