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Fast enclosure for solutions of generalized Sylvester equations

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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.

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Authors and Affiliations

  1. Faculty of Engineering, Gifu University, 1-1 Yanagido, Gifu-shi, Gifu, 501-1193, Japan

    Shinya Miyajima

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  1. Shinya Miyajima
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Correspondence to Shinya Miyajima.

Additional information

This research was partially supported by Grant-in-Aid for Scientific Research (C) (23560066, 2011–2015) from the Ministry of Education, Science, Sports and Culture of Japan.

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Miyajima, S. Fast enclosure for solutions of generalized Sylvester equations. Japan J. Indust. Appl. Math. 31, 293–304 (2014). https://doi.org/10.1007/s13160-014-0139-3

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  • DOI: https://doi.org/10.1007/s13160-014-0139-3

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