Convergence analysis of projection methods for the numerical solution of large lyapunov equations

SIAM Journal on Numerical Analysis

Volumn 47, Issue 2, 2009, Pages 828-843

  Simoncini, V ^a   Druskin, V ^b  

^a UNIVERSITY OF BOLOGNA   (Italy)

^b SCHLUMBERGER DOLL RESEARCH   (United States)

Author keywords

Faber polynomials; Krylov subspace; Lyapunov equation; Matrix exponential

Indexed keywords

APPLICATION AREAS; ASYMPTOTIC ESTIMATES; COEFFICIENT MATRICES; CONTINUOUS-TIME; CONVERGENCE ANALYSIS; ERROR MATRICES; FABER POLYNOMIALS; GALERKIN; KRYLOV SUBSPACE; LINEAR CONVERGENCES; LYAPUNOV EQUATION; LYAPUNOV MATRIX EQUATIONS; MATRIX EXPONENTIAL; NUMERICAL EXPERIMENTS; NUMERICAL SOLUTIONS; POSITIVE DEFINITE; PROJECTION METHODS; SOLUTION MATRIXES; TYPE METHODS; UPPER BOUNDS;

GALERKIN METHODS; LYAPUNOV FUNCTIONS; LYAPUNOV METHODS; NUMERICAL METHODS; TELECOMMUNICATION NETWORKS;

CONVERGENCE OF NUMERICAL METHODS;

EID: 60049085652     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/070699378     Document Type: Article

References (39)

1. M. ABRAMOWITZ AND I. A. STEGUN, Handbook of Mathematical Functions, Dover, New York, 1965.
2. A. C. ANTOULAS, Approximation of Large-Scale Dynamical Systems, Adv. Des. Control 6, SIAM, Philadelphia, 2008.
3. B. BECKERMANN, Image numérigue, GMRES et polynômes de Faber, C, R. Acad. Sci. Paris Ser. I, 340 (2005), pp. 855-860.
4. B. BECKERMANN AND A. B, J. KUIJLAARS, Superlinear convergence of conjugate gradients, SIAM J. Numer. Anal., 39 (2001), pp. 300-329.
5. B. BECKERMANN AND A. B. J. KUIJLAARS, Superlinear CG convergence for special right-hand-sides. Electron. Trans. Numer. Anal., 14 (2002), pp. 1-19.
6. P. BENNER, Control theory, in Handbook of Linear Algebra, Chapman & Hall/CRC, Boca Raton, FL, 2006, Chapter 57.
7. J. P. COLEMAN AND N. J. MYERS, The Faber polynomials for annular sectors, Math. Comp., 64 (1995), pp. 181-203.
8. M. J. CORLESS AND A. E. FRAZHO, Linear Systems and Control-An Operator Perspective, Pure Appl. Math., Marcel Dekker, New York, Basel, 2003.
9. B. N. DATTA, Krylov subspace methods for large-scale matrix problems in control, Future Generation Computer Systems, 19 (2003), pp. 1253-1263.
10. T. A. DRISCOLL AND L. N. TREFETHEN, Schwarz-Christoffel Mapping, Cambridge Monogr. Appl. Comput. Math. 8, Cambridge University Press, Cambridge, UK, 2002.
11. T. DRISCOLL, Algorithm 756: A MATLAB Toolbox for Schwarz-Christoffel mapping, ACM Trans. Math. Software, 22 (1996), pp. 168-186.
12. V. DRUSKIN AND L. KNIZHNERMAN, Two polynomial methods of calculating functions of symmetric, matrices, U.S.S.R. Comput. Math. Math. Phys., 29 (1989), pp. 112-121.
13. V. DRUSKIN AND L. KNIZHNERMAN, Extended Krylov subspaces: Approximation of the matrix square root and related functions, SIAM J. Matrix Anal. Appl., 19 (1998), pp. 755-771.
14. M. ELERMANN, On semiiterative methods generated by Faber polynomials, Numer. Math., 56 (1989), pp. 139-156.
15. M. EIERMANN, Field of values and iterative methods, Linear Algebra Appl., 180 (1993), pp. 167-197.
16. S. W. ELLACOTT, Computation of Faber series with application to numerical polynomial approximation in the complex plane, Math. Comp., 40 (1983), pp. 575-587.
17. K. O. GEDDES, Near-minimax polynomial approximation in an elliptical region, SIAM J. Numer. Anal., 15 (1978), pp. 1225-1233.
18. G. GOLUB AND C. F. VAN LOAN, Matrix Computations, 3rd ed., The Johns Hopkins University Press, Baltimore, MD, 1996.
19. I. S. GRADSHTEYN AND I. M. RYZHIK, Table of Integrals, Series, and Products (corrected and enlarged edition), Academic Press, San Diego, CA, 1980.
20. N. J. HIGHAM, The Matrix Computation Toolbox, http://www.ma.man.ac.uk/~higham/mctoolbox.
21. M. HOCHBRUCK AND C. LUBICH, On Krylov subspace approximations to the matrix exponential operator, SIAM J. Numer. Anal., 34 (1997), pp. 1911-1925.
22. I. M. JAIMOUKHA AND E. M. KASENALLY', Krylov subspace methods for solving large Lyapunov equations, SIAM J. Numer. Anal., 31 (1994), pp. 227-251.
23. K. JBILOU AND A. J. RIQUET, Projection methods for large Lyapunov matrix equations, Linear Algebra Appl., 415 (2006), pp. 344-358.
24. L. KNIZHNERMAN, Calculus of functions of unsymmetric matrices using Arnoldi's method, Comput. Math. Math. Phys., 31 (1991), pp. 1-9.
25. T. KOCH AND J. LLIESEN, The conformal "bratwurst" maps and associated Faber polynomials, Numer. Math., 86 (2000), pp. 173-191.
26. J. LIESEN, Construction and Analysis of Polynomial Iterative Methods for Non-Hermitian Systems of Linear Equations, Ph.D. thesis, Fakultät für Mathematik, Universität Bielefeld, 1998.
27. T. A. MANTEUFFEL, The Tchebychev iteration for nonsymmetric linear systems, Numer. Math., 28 (1977), pp. 307-327.
28. MATRIX MARKET, A Visual Repository of Test Data for Use in Comparative Studies of Algorithms for Numerical Linear Algebra, Mathematical and Computational Sciences Division, National Institute of Standards and Technology; available online at http://math.nist.gov/MatrixMarket.
29. O. NEVANLINNA, Convergence of Iterations for Linear Equations, Birkhauser, Basel, 1993.
30. M. ROBBÉ AND M. SADKANE, A convergence analysis of GMRES and FOM for Sylvester equations, Numer. Algorithms, 30 (2002), pp. 71-89.
31. Y. SAAD, Numerical solution of large Lyapunov equations, in Signal Processing, Scattering, Operator Theory, and Numerical Methods, Proceedings of the International Symposium MTNS-89, Vol. III, M. A. Kaashoek, J. H. van Schuppen, and A. C. Ran, eds., Birkhäuser, Boston, 1990, pp. 503-511.
32. Y. SAAD, Iterative Methods for Sparse Linear Systems, PWS, Boston, 1996.
33. V. SIMONCINI, A new iterative method for solving large-scale Lyapunov matrix equations, SIAM J. Sci. Comput., 29 (2007), pp. 1268-1288.
34. V. SIMONCINI AND D. B. SZYLD, On the occurrence of superlinear convergence of exact and inexact Krylov subspace methods, SIAM Rev., 47 (2005), pp. 247-272.
35. V. I. SMIRNOV AND N. A. LEBEDEV, Functions of a Complex Variable, Constructive Theory, MIT Press, Cambridge, MA, 1968.
36. D. E. STEWART AND T. S. LEYK, Error estimates for Krylov subspace approximations of matrix exponentials, J. Comput. Appl. Math., 72 (1996), pp. 359-369.
37. P. K. SUETIN, Fundamental properties of Faber polynomials, Russian Math. Surv., 19 (1964), pp. 121-149.
38. P. K. SUTETTN, Series of Faber Polynomials (Analytical Methods and Special Functions), Gordon and Breach Science Publishers, Amsterdam, 1998 (translated by E. V. Pankratiev).
39. H. TAL-EZER, Spectral methods in time for parabolic problems, SIAM J. Numer. Anal., 26 (1989), pp. 1-11.