A periodic Krylov-Schur algorithm for large matrix products

Numerische Mathematik

Volumn 103, Issue 3, 2006, Pages 461-483

  Kressner, Daniel ^a  

^a UNIVERSITY OF ZAGREB   (Croatia)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 33646488027     PISSN: 0029599X     EISSN: None     Source Type: Journal    
DOI: 10.1007/s00211-006-0682-1     Document Type: Article

References (42)

1. Antoulas, A.C.: Approximation of Large-Scale Dynamical Systems. SIAM, Philadelphia, PA, 2005
2. Antoulas, A.C., Sorensen, D.C., Gugercin, S.: A survey of model reduction methods for large-scale systems. In: Structured matrices in mathematics, computer science, and engineering. I (Boulder, CO, 1999). Vol. 280 of Contemp. Math. Amer. Math. Soc., Providence, RI, 2001, pp. 193-219
3. Beattie, C.A., Embree, M., Sorensen, D.C.: Convergence of polynomial restart Krylov methods for eigenvalue computations. SIAM Rev. 47(3), 492-515 (2005)
4. Bhatia, R.: Matrix Analysis. Springer-Verlag, New York, 1997
5. Bojanczyk, A., Golub, G.H., Van Dooren, P.: The periodic Schur decomposition; algorithm and applications. In: Proc. SPIE Conference, Vol. 1770, pp. 31-42, 1992
6. Bojanczyk, A., Van Dooren, P.: On propagating orthogonal transformations in a product of 2×2 triangular matrices. In: (Reichel, L. Ruttan, A. and R. S. Varga, eds.), Numerical Linear Algebra (Kent. OH, 1992), pp. 1-9. de Gruyter, 1993
7. Bomhol, W.: Iterative and parallel methods for linear systems, with applications in circuit simulation. PhD thesis. Universiteit Utrecht, Faculteit der Wiskunde en Informatica, Utrecht, Netherlands, 2001
8. Bomhof, W., van der Vorst, H.: A parallelizable GMRES-type method for p-cyclic matrices, with applications in circuit simulation. In: U. Van Rienen, M. Gunther, and D. Hecht, eds.). Scientific computing in electrical engineering: proceedings of the 3rd international workshop, August 20-23, 2000, Warnemünde, Germany, volume 18 of Lecture Notes in Computational Science and Engineering, Berlin, Heidelberg, London, 2001. Springer Verlag, pp. 293-300
9. Bonhoure, F., Dallery, Y., Stewart, W.J.: On the use of periodicity properties for the efficient numerical solution of certain Markov chains. Numer. Linear Algebra Appl. 1(3), 265-286 (1994)
10. Bru, R., Cantó, R., Ricarte, B.: Modelling nitrogen dynamics in citrus trees. Math and Comp. Mod 38, 975-987 (2003)
11. Bunch, J.R.: The weak and strong stability of algorithms in numerical linear algebra. Linear Algebra Appl. 88/89, 49-66 (1987)
12. Chahlaoui, Y., Van Dooren, P.: Benchmark examples for model reduction of linear time-invariant dynamical systems. In: (P. Benner, V. Mehrmann, and D. C. Sorensen eds.), Dimension Reduction of Large-Scale Systems, Vol. 45 of Lecture Notes in Computational Science and Engineering, Springer, Heidelberg, 2005, pp. 379-392
13. Daniel, J., Gragg, W.B., Kaufman, L., Stewart, G.W.: Reorthogonalization and stable algorithms for updating the Gram - Schmidt QR factorization. Math. Comput. 30, 772-795 (1976)
14. Ernst, O.G.: Equivalent iterative methods for p-cyclic matrices. Numer. Algorithms 25(1-4), 161-180 (2000)
15. Freund, R.W., Golub, G.H., Nachtigal, N.M.: Recent advances in Lanczos-based iterative methods for nonsymmetric linear systems. In: Algorithmic trends in computational fluid dynamics (1991). ICASE/NASA LaRC Ser., Springer, New York, 1993, pp. 137-162
16. Golub, G.H., Van Loan, C.F.: Matrix Computations. Johns Hopkins University Press, Baltimore, MD, third edition, 1996
17. Granat, R., Kågström, B.: Direct eigenvalue reordering in a product of matrices in extended periodic Schur form. Report UMINE-05.05, Department of Computing Science, Umeå University, Umeå, Sweden, 2005
18. Hammarling, S.J.: Numerical solution of the stable, non-negative definite Lyapunov equation. IMA J. Numer. Anal. 2, 303-323 (1982)
19. Hench, J.J., Laub, A.J.: Numerical solution of the discrete-time periodic Riccati equation. IEEE Trans. Automat. Control 39(6), 1197-1210 (1994)
20. Jaimoukha, I.M., Kasenally, E.M.: Krylov subspace methods for solving large Lyapunov equations. SIAM J. Numer. Anal. 31, 227-251 (1994)
21. Kressner, D.: The periodic QR algorithm is a disguised QR algorithm. 2003. To appear in Linear Algebra Appl.
22. Kressner, D.: Numerical Methods and Software for General and Structured Eigenvalue Problems. PhD thesis, TU Berlin, Institut für Mathematik, Berlin, Germany, 2004
23. Lehoucq, R.B.: Analysis and Implementation of an Implicitly Restarted Arnoldi Iteration. PhD thesis, Rice U., 1995
24. Lehoucq, R.B., Sorensen, D.C.: Deflation techniques for an implicitly restarted Arnoldi iteration. SIAM J. Matrix Anal. Appl. 17(4), 789-821 (1996)
25. Lchoucq, R.B., Sorensen, D.C., Yang, C.: ARPACK users' guide. SIAM, Philadelphia, PA, 1998. Solution of large-scale eigenvalue problems with implicitly restarted Arnoldi methods
26. Lin, W.-W., Van Dooren, P., Xu, Q.-F.: Periodic invariant subspaces in control. In: Proc. of IFAC Workshop on Periodic Control Systems, Como, Italy, 2001
27. Lust, K.: Numerical Bifurcation Analysis of Periodic Solutions of Partial Differential Equations. PhD thesis, Department of Computer Science. KU Leuven, Belgium, 1997
28. Lust, K.: Improved numerical Floquet multipliers. Internat. J. Bifur. Chaos Appl. Sci. Eng. 11(9), 2389-2410 (2001)
29. Parlett, B. N., Le, J.: Forward instability of tridiagonal QR. SIAM J. Matrix Anal. Appl. 14(1), 279-316 (1993)
30. Saad, Y.: Numerical Methods for Large Eigenvalue Problems: Theory and Algorithms. John Wiley, New York, 1992
31. Sorensen, D.C.: Implicit application of polynomial filters in a k-step Arnoldi method. SIAM J. Matrix Anal. Appl. 13, 357-385 (1992)
32. Sorensen, D.C.: Implicitly restarted Arnoldi/Lanczos methods for large scale eigenvalue calculations. In: Parallel numerical algorithms (Hampton, VA, 1994), Vol. 4 of ICASE/LaRC Interdiscip. Ser. Sci. Eng. Kluwer Acad. Publ., Dordrecht, 1997, pp. 119-165
33. Sreedhar, J., Van Dooren, P.: Pole placement via the periodic Schur decomposition. In: Proceedings Amer. Contr. Conf. pp. 1563-1567, 1993
34. Sreedhar, J., Van Dooren, P.: A Schur approach for solving some periodic matrix equations. In: (U. Helmke, R. Mennicken, and J. Saurer, eds.), Systems and Networks: Mathematical Theory and Applications, volume 77, Akademie Verlag, Berlin, 1994. pp. 339-362
35. Stewart, G.W.: Matrix Algorithms. Vol. II. SIAM, Philadelphia, PA, 2001. Eigensystems
36. Stewart, G.W.: A Krylov-Schur algorithm for large eigenproblems. SIAM J. Matrix Anal. Appl. 23(3), 601-614 (2001/02)
37. Stewart, W.J.: Introduction to the Numerical Solution of Markov Chains. Princeton University Press, Princeton, NJ, 1994
38. Van Loan, C.R: A general matrix eigenvalue algorithm. SIAM J. Numer. Anal. 12(6), 819-834 (1975)
39. Varga, A.: Balancing related methods for minimal realization of periodic systems. Systems Control Lett. 36(5), 339-349 (1999)
40. Varga, A., Van Dooren, P.: Computational methods for periodic systems - an overview. In: Proc. of IFAC Workshop on Periodic Control Systems, Como, Italy, 2001, pp. 171-176
41. Walker, H.R: Implementation of the GMRES method using Householder transformations. SIAM J. Sci. Stat. Comp. 9, 152-163 (1988)
42. Watkins, D.S.: Product eigenvalue problems. SIAM Rev. 47, 3-10 (2005)