06028205

j

06028205 Xue, Jungong Xu, Shufang Li, Ren-Cang Accurate solutions of $M$-matrix Sylvester equations. Numer. Math. 120, No. 4, 639-670 (2012). 2012 Springer-Verlag, Berlin EN $M$-matrix Sylvester equation relative perturbation theory numerical examples

doi:10.1007/s00211-011-0420-1

The authors present a relative perturbation theory for an $M$-matrix Sylvester equation (MSE). Specifically, the MSE is meant by the matrix equation $AX + XB = C$ where $A$ and $B$ have positive diagonal entries and nonpositive off-diagonal entries; $P = I_m \otimes A + B^T \otimes I_n$ is a nonsingular $M$-matrix; and $C$ is entry-wise nonnegative. By the authors, it has been proved that the small relative perturbations to the entries of $A$, $B$, and $C$ introduce small relative errors to the entries of $X$. This is unlike the existing perturbation theory on a (general) Sylvester equation. The authors propose some important modifications to the existing numerical methods for the computation of $X$. Numerical examples are given to verify their claims as well. Reviewer's remark: A matrix can be considered as an linear ``operator", and the main concern of this paper is the relative perturbation analysis related to the matrix equations. Jong Hyuk Park (Ulsan)