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A rational Lanczos algorithm for model reduction

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Gallivan, K. Grimme, E Van Dooren, Paul [UCL]

This paper presents a model reduction method for large-scale linear systems that is based on a Lanczos-type approach. A variant of the nonsymmetric Lanczos method, rational Lanczos, is shown to yield a rational interpolant (multi-point Padi approximant) for the large-scale system. An exact expression for the error in the interpolant is derived. Examples are utilized to demonstrate that the rational Lanczos method provides opportunities for significant improvements in the rate of convergence over single-point Lanczos approaches.

Document type Article de périodique (Journal article) – Article de recherche
Access type Accès restreint
Publication date 1996
Language Anglais
Journal information "Numerical Algorithms" - Vol. 12, no. 1-2, p. 33-63 (1996)
Peer reviewed yes
Publisher Baltzer Sci Publ Bv (Amsterdam)
issn 1017-1398
e-issn 1572-9265
Publication status Publié
Affiliation UCL - FSA/INMA - Département d'ingénierie mathématique
Keywords state space systems ; nonsymmetric Lanczos algorithm ; Pade approximation ; rational interpolation ; model reduction
Links
  1. A.C. Antoulas and B.D.O. Anderson, Rational interpolation and state variable realizations, Lin. Alg. Appl. 137 (1990) 479–509.
  2. G.A. Baker Jr.,Essentials of Padé Approximants (Academic Press, New York, 1975).
  3. D.L. Boley and G. Golub, The nonsymmetric Lanczos algorithm and controllability, Syst. Contr. Lett. 16 (1991) 97–105.
  4. D.L. Boley, Krylov space methods on state-space control models, Report No. TR92-18, Department of Computer Science, University of Minnesota, MN (1992).
  5. R.R. Craig Jr., Recent literature on structural modeling, identification, and analysis, in:Mechanics and Control of Large Flexible Structures, ed. J.L. Junkins (AIAA, Washington, 1990).
  6. Chiprout Eli, Nakhla Michel S., Asymptotic Waveform Evaluation, ISBN:9781461363637, 10.1007/978-1-4615-3116-6
  7. P. Feldman and R.W. Freund, Efficient linear circuit analysis by Padé approximation via the Lanczos process, IEEE Trans. Comp.-Aided Design 14 (1995) 639–649.
  8. Fortuna L., Nunnari G., Gallo A., Model Order Reduction Techniques with Applications in Electrical Engineering, ISBN:9781447132004, 10.1007/978-1-4471-3198-4
  9. R.W. Freund, M.H. Gutknecht and N.M. Nachtigal, An implementation of the look-ahead Lanczos algorithm for non-Hermitian matrices, SIAM J. Sci. Comp. 14 (1993) 137–158.
  10. K. Gallivan, E. Grimme and P. Van Dooren, Asymptotic waveform evaluation via a Lanczos method, Appl. Math. Lett. 7 (1994) 75–80.
  11. K. Gallivan, E. Grimme and P. Van Dooren, Padé Approximation of large-scale dynamic systems with Lanczos methods,Proc. 33rd IEEE Conf. on Decision and Control, Lake Buena Vista, FL (1994).
  12. E. Gallopoulos and Y. Saad, Efficient solution of parabolic equations by Krylov approximation methods, SIAM J. Sci. Statist. Comp. 13 (1992) 1236–1264.
  13. G.H. Golub and C.F. Van Loan,Matrix Computations, 2nd ed. (Johns Hopkins University Press, Baltimore, MD, 1989).
  14. W.B. Gragg and A. Lindquist, On the partial realization problem, Lin. Alg. Appl. 50 (1983) 277–319.
  15. E. Grimme, D. Sorensen and P. Van Dooren, Model reduction of state space systems via an implicitly restarted Lanczos method, Numer. Algor. (1996), this issue.
  16. C. Hwang and M.-Y. Chen, A multi-point continued-fraction expansion for linear system reduction, IEEE Trans. Auto. Contr. AC-31 (1986) 648–651.
  17. I.M. Jaimoukha and E.M. Kasenally, Oblique projection methods for large scale model reduction, SIAM J. Matrix Anal. Appl. (1995), to appear.
  18. C. Kenney, A.J. Laub and S. Stubberud, Frequency response computation via rational interpolation, IEEE Trans. Auto. Contr. AC-38 (1993) 1203–1213.
  19. Lanczos C., An iteration method for the solution of the eigenvalue problem of linear differential and integral operators, 10.6028/jres.045.026
  20. B. Nour-Omid and R.W. Clough, Dynamic analysis of structures using Lanczos co-ordinates, Earthquake Eng. Struc. Dyn. 12 (1984) 565–577.
  21. B.N. Parlett, D.R. Taylor and Z.S. Liu, A look-ahead Lanczos algorithm for unsymmetric matrices, Math. Comp. 44 (1985) 105–124.
  22. B.N. Parlett, Reduction to tridiagonal form and minimal realizations, SIAM J. Matrix Anal. Appl. 13 (1992) 567–593.
  23. A. Ruhe, Rational Krylov sequence methods for eigenvalue computation, Lin. Alg. Appl. 58 (1984) 391–405.
  24. A. Ruhe, Rational Krylov algorithms for nonsymmetric eigenvalue problems II: Matrix pairs, Lin. Alg. Appl. 197 (1994) 283–295.
  25. A. Ruhe, The rational Krylov algorithm for nonsymmetric eigenvalue problems III: Complex shifts for real matrices, BIT 34 (1994) 165–176.
  26. Y. Saad, Analysis of some Krylov subspace approximations to the matrix exponential operator, SIAM J. Numer. Anal. 29 (1992) 209–228.
  27. T.-J. Su and R.R. Craig Jr., Krylov vector methods for model reduction and control of flexible structures, in:Control and Dynamic Systems: Integrated Technology Methods in Aerospace Systems Design, ed. C.T. Leondes (Academic Press, London, 1992).
  28. P. Van Dooren, Numerical linear algebra techniques for large scale matrix problems in systems and control,Proc. 31st IEEE Conf. on Decision and Control, Tucson, AZ (1992).
  29. C.D. Villemagne and R.E. Skelton, Model reduction using a projection formulation, Int. J. Control 46 (1987) 2141–2169.
  30. E.L. Wilson, M.-W. Yuan and J.M. Dickens, Dynamic analysis by direct superposition of Ritz vectors, Earthquake Eng. Struc. Dyn. 10 (1982) 813–821.
  31. P. Wortelboer, Frequency-weighted balanced reduction of closed-loop mechanical servosystems: theory and tools, PhD thesis, Technical University, Delft (1994).
  32. H. Xiheng, FF-Padé method of model reduction in frequency domain, IEEE Trans. Auto. Contr. AC-32 (1987) 243–246.
Bibliographic reference Gallivan, K. ; Grimme, E ; Van Dooren, Paul. A rational Lanczos algorithm for model reduction. In: Numerical Algorithms, Vol. 12, no. 1-2, p. 33-63 (1996)
Permanent URL http://hdl.handle.net/2078.1/47137