Fast enclosure for solutions of generalized Sylvester equations

Abstract

Fast methods for enclosing solutions of generalized Sylvester equations $${{AXB + CXD = E, A, C \in \mathbb{C}^{m \times m}, B, D \in \mathbb{C}^{n \times n}, X, E \in \mathbb{C}^{m \times n}}}$$ are proposed. To develop these methods, theories which supply error bounds of numerical solutions are established. These methods require only $${\mathcal{O}(m^3 + n^3)}$$ operations, and give error bounds which are “verified” in the sense that all the possible rounding errors have been taken into account. At least one of these methods are applicable when B and C are nonsingular, and C −1 A and B −1 D are diagonalizable, or A and D are nonsingular, and A −1 C and D −1 B are diagonalizable. A technique for obtaining smaller error bounds is introduced. Numerical results show the properties of the proposed methods.