Fast verified computation for the solution of the T-congruence Sylvester equation
An algorithm is proposed to compute an interval matrix that contains the symmetric matrix $X$ that solves $AX+X^TB=C$ (all matrices are in $\mathbb{C}^{n\times n}$). It was shown in [\textit{M. Oozawa} et al., J. Comput. Appl. Math. 329, 51--56 (2018; Zbl 1373.15023)] how, under certain conditions, the problem can be transformed into a Lyapunov equation. Based on this result, conditions on the numerically computed generalized eigendecomposition of the pencil $(B,A^T)$ or $(A,B^T)$ are formulated to guarantee the existence of a unique solution. The algorithm finding the required interval matrix has complexity $O(n^3)$ assuming that the employed enclosure routines for products and inverses of matrices have complexity at most $O(n^3)$. The MATLAB code using INTLAB (interval arithmetic toolbox) is available online.
scientific article; zbMATH DE number 6996761
scientific article