A modified low-rank Smith method for large-scale Lyapunov equations

Numerical Algorithms

Volumn 32, Issue 1, 2003, Pages 27-55

  Gugercin, S ^a   Sorensen, D C ^a   Antoulas, A C ^a  

^a Rice University   (United States)

Author keywords

ADI iteration; Lyapunov equation; Model reduction; Smith method

Indexed keywords

EID: 0037276767     PISSN: 10171398     EISSN: None     Source Type: Journal    
DOI: 10.1023/A:1022205420182     Document Type: Article

References (26)

1. A.C. Antoulas, Lectures on the Approximation of Linear Dynamical Systems, Draft (SIAM, Philadelphia, PA, 2002) to appear.
2. A.C. Antoulas and D.C. Sorensen, Projection methods for balanced model reduction, Technical Report ECE-CAAM Departments, Rice University (March 2001).
3. A.C. Antoulas, D.C. Sorensen and S. Gugercin, A survey of model reduction methods for large-scale systems, in: Structured Matrices in Operator Theory, Numerical Analysis, Control, Signal and Image Processing, Contemporary Mathematics (Amer. Math. Soc., Providence, RI, 2001).
4. A.C. Antoulas, D.C. Sorensen and Y.K. Zhou, On the decay rate of Hankel singular values and related issues, Systems Control Lett. (2002) to appear.
5. R.H. Bartels and G.W. Stewart, Solution of the matrix equation AX + XA = C: Algorithm 432, Comm. ACM 15 (1972) 820-826.
6. N. Ellner and E. Wachspress, Alternating direction implicit iteration for systems with complex spectra, SIAM J. Numer. Anal. 28 (1991) 859-870.
7. ∞-error bounds, Internat. J. Control 39 (1984) 1115-1193.
8. G.H. Golub and C.F. Van Loan, Matrix Computations, 3rd ed. (Johns Hopkins Univ. Press, Baltimore, MD, 1996).
9. S. Gugercin and A.C. Antoulas, A comparative study of 7 algorithms for model reduction, in: Proc. of 39th CDC, Sydney, Australia, December 2000.
10. S. Gugercin and A.C. Antoulas, Approximation of the International Space Station 1R and 12A models, in: Proc. of the 40th CDC, December 2001.
11. S. Hammarling, Numerical solution of the stable, non-negative definite Lyapunov equation, IMA J. Numer. Anal. 2 (1982) 303-323.
12. A.S. Hodel, K.P. Poola and B. Tenison, Numerical solution of the Lyapunov equation by approximate power iteration, Linear Algebra Appl. 236 (1996) 205-230.
13. D.Y. Hu and L. Reichel, Krylov subspace methods for the Sylvester equation, Linear Algebra Appl. 172 (1992) 283-313.
14. I.M. Jaimoukha and E.M. Kasenally, Krylov subspace methods for solving large Lyapunov equations, SIAM J. Numer. Anal. 31 (1994) 227-251.
15. B.C. Moore, Principal component analysis in linear system: Controllability, observability and model reduction, IEEE Trans. Automat. Control 26 (1981) 17-32.
16. A. Lu and E.L. Wachspress, Solution of Lyapunov equations by alternating direction implicit iteration, Comput. Math. Appl. 21(9) (1991) 43-58.
17. D.W. Peaceman and H.H. Rachford, The numerical solutions of parabolic and elliptic differential equations, J. SIAM 3 (1955) 28-41.
18. T. Penzl, Algorithms for model reduction of large dynamical systems, Technical Report SFB393/99-40, Sonderforschungsbereich 393, Numerische Simulation auf massiv parallelen Rechern, TU Chemnitz, FRG (1999); available from http://www.tu-chemnitz.de/sfb393/sfb99pr.html.
19. T. Penzl, Eigenvalue decay bounds for solutions of Lyapunov equations: The symmetric case, Systems Control Lett. 40 (2000) 139-144.
20. T. Penzl, A cyclic low-rank Smith method for large sparse Lyapunov equations, SIAM J. Sci. Comput. 21(4) (2000) 1401-1418.
21. J.D. Roberts, Linear model reduction and solution of the algebraic Riccati equation by use of the sign function, Internat. J. Control 32 (1980) 677-687.
22. G. Starke, Optimal alternating direction implicit parameters for nonsymmetric systems of linear equations, SIAM J. Numer. Anal. 28 (1991) 129-141.
23. R.A. Smith, Matrix equation, XA + BX = C, SIAM J. Appl. Math. 16 (1968) 198-201.
24. E. Wachspress, Iterative solution of the Lyapunov matrix equation, Appl. Math. Let. 1 (1988) 87-90.
25. E. Wachspress, The ADI minimax problem for complex spectra, in: Iterative Methods for Large Linear Systems, eds. D. Kincaid and L. Hayes (Academic Press, San Diego, CA, 1990) pp. 251-271.
26. W. Xing, Q. Zhang and Q. Wang, A trace bound for a general square matrix product, IEEE Trans. Automat. Control 45(8) (2000) 1563-1565.