Numerical Solution of Generalized Lyapunov Equations

Two efficient methods for solving generalized Lyapunov equations and their implementations in FORTRAN 77 are presented. The first one is a generalization of the Bartels--Stewart method and the second is an extension of Hammarling's method to generalized Lyapunov equations. Our LAPACK based subroutines are implemented in a quite flexible way. They can handle the transposed equations and provide scaling to avoid overflow in the solution. Moreover, the Bartels--Stewart subroutine offers the optional estimation of the separation and the reciprocal condition number. A brief description of both algorithms is given. The performance of the software is demonstrated by numerical experiments. Key Words: Mathematical software, generalized Lyapunov equation, generalized Stein equation, condition estimation. AMS(MOS) subject classifications: 65F05, 65F15, 93B40, 93B51 1 Basic Properties of Generalized Lyapunov Equations Lyapunov equations play an essential role in control theory. In the past few y..