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.
Perturbation theory and backward error forAX−XB=C
Because of the special structure of the equationsAX−XB=C the usual relation for linear equations “backward error = relative residual” does not hold, and application of the standard perturbation result forAx=b yields a perturbation bound involving sep (A, B)−1 that is not always attainable. An expression is derived for the backward error of an approximate solutionY; it shows that the backward error can exceed the relative residual by an arbitrary factor. A sharp perturbation bound is derived and it is shown that the condition number it defines can be arbitrarily smaller than the sep(A, B)−1-based quantity that is usually used to measure sensitivity. For practical error estimation using the residual of a computed solution an “LAPACK-style” bound is shown to be efficiently computable and potentially much smaller than a sep-based bound. A Fortran 77 code has been written that solves the Sylvester equation and computes this bound, making use of LAPACK routines.
On series expansions and stochastic matrices
Let $P( 0 ) \in R^{n \times n} $ be a stochastic matrix representing transition probabilities in a Markov chain, which is completely decomposable into m independent chains plus a number of transient states. Also, suppose that for all $\varepsilon > 0$ small enough $P( \varepsilon ) \equiv P( 0 ) + \varepsilon C$ is a stochastic matrix representing a unichain Markov process. Let $\pi ( \varepsilon )$ be the stationary distribution of $P ( \varepsilon )$ and let $Y ( \varepsilon )$ be the deviation matrix of $P ( \varepsilon )$ for $\varepsilon > 0$. It was proved by Schweitzer that $\pi ( \varepsilon )$ has a series expansion around zero whose terms form a geometric sequence. He also showed that $Y ( \varepsilon )$ admits a Laurent expansion. In order to compute the series expansion of $\pi ( \varepsilon )$, a system of equations is defined resulting from equating coefficients of identical powers in the identity $\pi ( \varepsilon )( I - P( \varepsilon ) ) = \b{0} ^T $. The authors prove that the minimal n...