A Schur Method for Solving Projected Continuous-Time Sylvester Equations

Authors: Yiqin Lin, Liang Bao, Qinghua Wu, Liping Zhou

Abstract:

In this paper, we propose a direct method based on the real Schur factorization for solving the projected Sylvester equation with relatively small size. The algebraic formula of the solution of the projected continuous-time Sylvester equation is presented. The computational cost of the direct method is estimated. Numerical experiments show that this direct method has high accuracy.

Keywords: Projected Sylvester equation, Schur factorization, Spectral projection, Direct method.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1056382

References:

[1] B. Anderson and J. Moore, Optimal Control-Linear Quadratic Methods, Prentice-Hall, Englewood Cliffs, NJ, 1990.
[2] L. Bao, Y. Lin and Y. Wei, A new projection method for solving large Sylvester equations, Appl. Numer. Math., 57(2007), 521-532.
[3] L. Bao, Y. Lin, L. Zhou and Y. Wei, A new iterative method for solving projected generalized continuous-time Lyapunov equations, preprint.
[4] R. H. Bartels and G. W. Stewart, Solution of the equation AX+XB = C, Comm. ACM, 15(1972), 820-826.
[5] U. Baur and P. Benner, Factorized solution of Lyapunov equations based on hierarchical matrix arithmetic, Computing, 78(2006), 211-234.
[6] P. Benner and E. S. Quintana-Ort'─▒, Solving stable generalized Lyapunov equations with the matrix sign function, Numer. Algorithms, 20(1999), 75-100.
[7] P. Benner, E. S. Quintana-Ort'─▒ and G. Quintana-Ort'─▒, Solving stable Sylvester equations via rational iterative schemes, J. Sci. Comput., 28(2006), 51-83.
[8] F. R. Gantmacher, Theory of Matrices, Chelsea, New York, 1959.
[9] G. H. Golub, S. Nash and C. Van Loan, A Hessenberg-Schur method for the problem AX+XB = C, IEEE Trans. Automat. Control, 24(1979), 909-913.
[10] G. H. Golub and C. F. Van Loan, Matrix Computations, 3rd edition, Johns Hoplins University Press, Baltimore, 1996.
[11] A. El Guennouni, K. Jbilou and J. Riquet, Krylov subspace methods for solving large Sylvester equations, Numer. Algorithms, 29(2002), 75-96.
[12] S. Gugercin, D. C. Sorensen, and A. C. Antoulas, A modified low-rank Smith method for large-scale Lyapunov equations, Numer. Algorithms, 32(2003), 27-55.
[13] S. J. Hammarling, Numerical solution of the stable non-negative definite Lyapunov equation, IMA J. Numer. Anal., 2(1982), 303-323.
[14] U. Helmke and J. Moore, Optimization and Dynamical Systems, Springer, London, 1994.
[15] D. Y. Hu and L. Reichel, Krylov-subspace methods for the Sylvester equation, Linear Algebra Appl., 172(1992), 283-313.
[16] I. M. Jaimoukha and E. M. Kasenally, Krylov subspace methods for solving large Lyapunov equations, SIAM J. Numer. Anal., 31(1994), 227-251.
[17] K. Jbilou, A. Messaoudi and H. Sadok, Global FOM and GMRES algorithms for matrix equations, Appl. Numer. Math., 31(1999), 49-63.
[18] K. Jbilou and A. J. Riquet, Projection methods for large Lyapunov matrix equations, Linear Algebra Appl., 415(2006), 344-358.
[19] D. Kressner, Block variants of Hammarling-s method for solving Lyapunov equations, ACM Trans. Math. Software, 34(2008), 1-15.
[20] J.-R. Li and J. White, Low rank solution of Lyapunov equations, SIAM J. Matrix Anal. Appl., 24(2002), 260-280.
[21] A. Lu and E. Wachspress, Solution of Lyapunov equations by alternating direction implicit iteration, Comput. Math. Appl., 21(1991), 43-58.
[22] V. Mehrmann, The Autonomous Linear Quadratic Control Problem, Theory and Numerical Solution, Lecture Notes in Control and Information Sciences, Vol. 163, Springer, Heidelberg, 1991.
[23] T. Penzl, A cyclic low-rank Smith method for large sparse Lyapunov equations, SIAM J. Sci. Comput., 21(2000), 1401-1418.
[24] P. Petkov, N. Christov and M. Konstantinov, Computational Methods for Linear Control Systems, Prentice-Hall, Hertfordshire, UK, 1991.
[25] M. Robbe and M. Sadkane, A convergence analysis of GMRES and FOM methods for Sylvester equations, Numer. Algorithms, 30(2002), 71-84.
[26] R. Sch¨upphaus, Regelungstechnische Analyse und Synthese von Mehrk ¨orpersystemen in Deskriptorform, Fortschritt- Berichte VDI, Reihe 8, Nr. 478. VDI Verlag, D¨usseldorf, 1995.
[27] R. Smith, Matrix equation XA + BX = C, SIAM J. Appl. Math., 16(1968), 198-201.
[28] C. Sorensen and Y. Zhou, Direct methods for matrix Sylvester and Lyapunov equations, J. Appl. Math., 6(2003), 277-303.
[29] T. Stykel, Gramian-based model reduction for descriptor systems, Math. Control Signals Systems, 16(2004), 297-319.
[30] T. Stykel, Balanced truncation model reduction for semidiscretized Stokes equation, Linear Algebra Appl., 415(2006), 262-289.
[31] T. Stykel, A modified matrix sign function method for projected Lyapunov equations, Systems Control Lett., 56(2007), 695-701.
[32] T. Stykel, Low-rank iterative methods for projected generalized Lyapunov equations, Elect. Trans. Numer. Anal., 30(2008), 187-202.
[33] E. Wachspress, Iterative solution of the Lyapunov matrix equation, Appl. Math. Lett., 107(1988), 87-90.