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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References
Chu K.E.: The solution of the matrix equations AXB − CXD = E AND (YA − DZ, YC − BZ) = (E, F). Linear Alg. Appl. 93, 93–105 (1987)
Datta B.N., Datta K.: Theoretical and computational aspects of some linear algebra problems in control theory. In: Byrnes, C.I., Lindquist, A. (eds) Computational and Combinatorial Methods in Systems Theory, pp. 201–212. Elsevier, Amsterdam (1986)
Frommer A., Hashemi B.: Verified error bounds for solutions of Sylvester matrix equations. Linear Alg. Appl. 436, 405–420 (2012)
Hashemi, B.: Verified computation of symmetric solutions to continuous-time algebraic Riccati matrix equations. Proc. SCAN conference, pp. 54–56. Novosibirsk (2012)
Higham N.: Accuracy and Stability of Numerical Algorithms. SIAM Publications, Philadelphia (2002)
Horn R.A., Johnson C.R.: Topics in Matrix Analysis. Cambridge University Press, Cambridge (1994)
Kressner, D., Mehrmann, V., Penzl, T.: CTLEX—a collection of benchmark examples for continuous-time Lyapunov equations. Tech. Rep. SLICOT Working Note 1999–2016 (1999) http://www.slicot.org/REPORTS/SLWN1999-6.ps.gz
Luther W., Otten W.: Verified calculation of the solution of algebraic Riccati equation. In: Csendes, T. (eds) Developments in Reliable Computing, pp. 105–118. Kluwer Academic Publishers, Dordrecht (1999)
Luther, W., Otten, W., Traczinski, H.: Verified calculation of the solution of continuous- and discrete time algebraic Riccati equation. Schriftenreihe des Fachbereichs Mathematik der Gerhard-Mercator-Universität Duisburg SM-DU-422 (1998)
Luther, W., Otten, W., Traczinski, H.: Verified calculation of the solution of discrete time algebraic Riccati equation. Proc. MISC workshop, pp. 411–421. Girona (1999)
Meyer C.D.: Matrix Analysis and Applied Linear Algebra. SIAM Publications, Philadelphia (2000)
Miyajima S.: Numerical enclosure for each eigenvalue in generalized eigenvalue problem. J. Comput. Appl. Math. 236, 2545–2552 (2012)
Miyajima S: Fast enclosure for solutions of Sylvester equations. Linear Alg. Appl. 439, 856–878 (2013)
Rohn, J.: VERSOFT: Verification software in MATLAB / INTLAB. http://uivtx.cs.cas.cz/~rohn/matlab/
Seif N., Hussein S., Deif A.: The interval Sylvester matrix equation. Computing 52, 233–244 (1994)
Shashikhin V.: Robust assignment of poles in large-scale interval systems. Autom. Remote Control 63, 200–208 (2002)
Shashikhin V: Robust stabilization of linear interval systems. J. Appl. Math. Mech. 66, 393–400 (2002)
Yamamoto T.: Error bounds for approximate solutions of systems of equations. Japan J. Indust. Appl. Math. 1, 157–171 (1984)
Yano, K., Koga, M.: Verified numerical computation in LQ control problem. Trans. SICE. 45:261–267 (2009) (in Japanese)
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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