Analytic perturbation of Sylvester and Lyapunov matrix equations
We consider an analytic perturbation of the Sylvester matrix equation. Mainly we are interested in the singular case, that is, when the null space of the unperturbed Sylvester operator is not trivial, but the perturbed equation has a unique solution. In this case, the solution of the perturbed equation can be given in terms of a Laurent series. We provide a necessary and sufficient condition for the existence of a Laurent series with a first order pole. An efficient recursive procedure for the calculation of the Laurent series' coefficients is given. Finally, we show that in the particular, but practically important case of semisimple eigenvalues, the recursive procedure can be written in a compact matrix form.
An implicit deflation method for ill-conditioned Sylvester and Lyapunov equations
This paper describes an effective algorithm for solving ill-conditioned Sylvester or Lyapunov equations. These equations can be solved by conventional methods for ill-conditioned linear systems. However, such methods are not efficient since they require on the order of n 6 arithmetic operations on order n 2 data. For these ill-conditioned matrix equations, an implicit deflation algorithm is proposed to implement a certain SVD-based minimum norm least-squares approximate solution. The method is practical since it relies only on the ability to solve a Sylvester or Lyapunov equation. Certain practical details are also discussed.
Conserved Quantities of a Class of Nonlinear Difference-Difference Equations
Higher-order conserved quantities of a class of nonlinear difference-difference equations which are related to the discrete-time Toda equation are obtained using the Miura transformations.