Direct methods for matrix Sylvester and Lyapunov equations

[摘要] We revisit the two standard dense methods for matrixSylvester and Lyapunov equations: the Bartels-Stewart method forA1X+XA2+D=0andHammarling's method forAX+XAT+BBT=0withAstable. We construct three schemes for solving the unitarily reducedquasitriangular systems. We also construct a new rank-1 updatingscheme in Hammarling's method. This new scheme is able toaccommodate aBwith more columns than rows as well as theusual case of aBwith more rows than columns, whileHammarling's original scheme needs to separate these two cases.We compared all of our schemes with the Matlab Sylvester andLyapunov solverlyap.m; the results show that ourschemes are much more efficient. We also compare our schemes withthe Lyapunov solversllyapin the currently possibly themost efficient control library package SLICOT; numerical resultsshow our scheme to be competitive.

[学科分类] 应用数学